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\begin{document}
\begin{abstract} We prove the Liv\v{s}ic Theorem for H\"{o}lder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${\ensuremath{\mathbf{\sigma}}:X\to X}$ that satisfies the Anosov Closing Lemma, and a H\"{o}lder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a H\"{o}lder continuous transition function if and only if ${a(\ensuremath{\mathbf{\sigma}}^{n-1}p)\ldots a(\ensuremath{\mathbf{\sigma}} p)a(p)=e}$ for each periodic point $p=\ensuremath{\mathbf{\sigma}}^n p$.
\end{abstract}
\title{ Liv {s}
\section{Introduction}
We assume that $X$ is a compact metric space, $G$ a complete metric group, and $\ensuremath{\mathbf{\sigma}}:X\to X$ a homeomorphism.
We say that a map $a:\mathbb{Z}\times X\to G$ is {\it a cocycle} over \ensuremath{\mathbf{\sigma}}\ if
$$a(n,x)=a(n-k,\ensuremath{\mathbf{\sigma}}^kx)a(k,x)\quad\text{for any }n,k\in\ensuremath{\mathbb{Z}}$$
Every map $a:X\to G$ generates a cocycle $a(n,x)$ defined as $$a(n,x)=a(\ensuremath{\mathbf{\sigma}}^{n-1}x)a(\ensuremath{\mathbf{\sigma}}^{n-2}x)\ldots a(x) \quad n>0$$
$$a(0,x)=Id$$
$$a(n,x)= a^{-1}(\ensuremath{\mathbf{\sigma}}^{n}x)\ldots a^{-1}(\ensuremath{\mathbf{\sigma}}^{-2}x)a^{-1}(\ensuremath{\mathbf{\sigma}}^{-1}x)\quad n<0$$
We see that $a(1,x)=a(x)$. In this paper we consider only cocycles generated by H\"older continuous maps $a:X\to G$.
We say that a H\"older continuous map $a:X\to G$ is a {\it coboundary } (or more precisely generates a cocycle which is a coboundary) if there is a H\"older continuous function $t:X\to G$ such that
$$ a(x)=t(\ensuremath{\mathbf{\sigma}} x)t^{-1}(x)$$
The function $t(x)$ is a called a {transition map}.
If $a(x)$ is a coboundary then it is clear that
$$a(n,x)=t(\ensuremath{\mathbf{\sigma}}^n x)t^{-1}(x)$$
A question whether some cocycle is a coboundary or not appears naturally in many important problems in dynamical systems.
There is a simple necessary condition for a cocycle to be a coboundary. If $a(x)$ is a coboundary and $p\in X$ is a periodic point $\ensuremath{\mathbf{\sigma}}^n p=p$ then
$$a(\ensuremath{\mathbf{\sigma}}^{n-1}p)\ldots a(\ensuremath{\mathbf{\sigma}} p)a(p)=a(n,p)=t(\ensuremath{\mathbf{\sigma}}^n p)t^{-1}(p)=e$$
where $e$ is the identity element in the group $G$.
We say that for a cocycle $a(n,x)$ {\it periodic obstruction vanish} if
\begin{equation}\ensuremath{\lambda}bel{e0} a(\ensuremath{\mathbf{\sigma}}^{n-1}p)\ldots a(\ensuremath{\mathbf{\sigma}} p)a(p)=e\quad\forall p\in X\text{ with } \ensuremath{\mathbf{\sigma}}^np=p, n\in\mathbb{N} \end{equation}
A.Liv\v{s}ic (see ~\cite{L1,L2}) proved that when \ensuremath{\mathbf{\sigma}}\ is a transitive Anosov map and the group $G$ is $\mathbb{R}$ or $\mathbb{R}^n$ then a cocycle $a(x)$ is a coboundary if and only if the periodic obstruction vanish. This result is called Liv\v{s}ic theorem. The proof of the Liv\v{s}ic theorem for other groups turned out to be harder. Nevertheless, in the last twenty years in the series of papers (see \cite{BN},\cite{PW},\cite{P},\cite{KS},\cite{NT},\cite{LW}) it was shown that for some groups under an additional assumption on the growth rates of the cocycle $a(n,x)$ the condition (\ref{e0}) is also sufficient. For example, in \cite{BN} it was shown that if $G=B^\times$ the set of invertible elements of some Banach algebra then if periodic obstruction vanish and $a(x)$ is close to the identity element $e$ then it is a coboundary.
The question remained if this additional assumption will follow from the fact that the products along periodic points are equal to $e$. In 2011 B.Kalinin in \cite{Ka} made a breakthrough by proving the Liv\v{s}ic theorem for functions with values in $GL(n,\mathbb{R})$ and more generally in a connected Lie group assuming only that condition (\ref{e0}) is satisfied.
He used Lyapunov exponents for different invariant measures to estimate the rate of the cocycle growth and then approximated Lyapunov exponents for all invariant measures by Lyapunov exponents only at periodic points. To do the latter the Oseledets Theorem was used. In this paper, we are proving that a cocycle with values in invertible elements of Banach ring is a coboundary if and only if periodic obstructions vanish. There is no analogs of the Oseledets Theorem for Banach rings ( or even Banach algebras). Still we can define analogs of the highest and lowest Lyapunov exponents and using a different argument show that they could be approximated by the values of the cocycle at periodic points. Examples of Banach rings include, Banach algebras, and Banach algebras with $\ensuremath{\mathbb{F}}$ as a field of scalars, where $\ensuremath{\mathbb{F}}$ is a local field. For them it is a new result. Also several already known results follow: Liv\v{s}ic Theorem for cocycles with values in $GL(n,\ensuremath{\mathbb{R}})$ (see \cite{Ka}) and $GL(n,\ensuremath{\mathbb{F}})$ (see \cite{LZ}).
As in \cite{Ka} we require that the map \ensuremath{\mathbf{\sigma}}\ was transitive and had the following property
\begin{definition} We say that a homeomorphism $\ensuremath{\mathbf{\sigma}}:X\to X$ has a {\it closing property} if there exist positive numbers $\delta_0, \ensuremath{\lambda},C$ such that for any $x\in X$ and $n>0$ with $\text{dist}(x,\ensuremath{\mathbf{\sigma}}^n x)\le \delta_0$ we can find points $p,z\in X$ where
$$\ensuremath{\mathbf{\sigma}}^n p=p$$
and for every $i=0,1,\ldots, n$
$$\text{dist}(\ensuremath{\mathbf{\sigma}}^i p,\ensuremath{\mathbf{\sigma}}^i z)\le e^{-i\ensuremath{\lambda}}C\text{dist}(x,\ensuremath{\mathbf{\sigma}}^n x)\quad \text{dist}(\ensuremath{\mathbf{\sigma}}^i x,\ensuremath{\mathbf{\sigma}}^i z)\le e^{-(n-i)\ensuremath{\lambda}}C\text{dist}(x,\ensuremath{\mathbf{\sigma}}^n x) $$
We will call $\ensuremath{\lambda}mbda$ the expansion constant for the map \ensuremath{\mathbf{\sigma}}.
\end{definition}
Anosov maps and shifts of finite types are main examples of maps with closing property.
\begin{definition} An associative (non--commutative) ring $B$ with the unity element $e$ is called {\it Banach ring} if there is a function $\|\cdot\|:B\to\ensuremath{\mathbb{R}}$ such that
\begin{enumerate}
\item $\|a\|\ge 0$ and $\|a\|=0$ if and only if $a=0$.
\item $\|a+b\|\le \|a\|+\|b\|$.
\item $\|a\cdot b\|\le \|a\|\cdot \|b\|$.
\item The ring $B$ is a complete metric space with respect to the distance defined as $dist(a,b)=\|a-b\|$.
\end{enumerate}
\end{definition}
We denote as $B^\times$ the set of invertible elements of a Banach ring $B$. The main result of this paper is:
\begin{main}\ensuremath{\lambda}bel{t2} Let $X$ be a compact metric space, $\ensuremath{\mathbf{\sigma}}:X\to X$ a transitive homeomorphism with closing property. If $a:X\to B^\times $ is an $\alpha$-H\"older continuous function such that
$$ a(\ensuremath{\mathbf{\sigma}}^{n-1}p)\ldots a(\ensuremath{\mathbf{\sigma}} p)a(p)=e\quad \forall p\in X, n\in \mathbb{N} \text{ with } \ensuremath{\mathbf{\sigma}}^np=p$$
then there exists an $\alpha$-H\"older function $t:X\to B^\times$ such that
$$ a(x)=t(\ensuremath{\mathbf{\sigma}} x)t^{-1}(x)$$
\end{main}
\section{Subadditive Cocycles}
Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a continuous function. We will call a continuous function $s(n,x):\mathbb{Z}\times X\to \mathbb{R}$ \textit{ a subadditive cocycle} over the function $\ensuremath{\mathbf{\sigma}}$ if
$$s(n+m,x)\le s(n,\ensuremath{\mathbf{\sigma}}^m x)+s(m,\ensuremath{\mathbf{\sigma}} x)$$
From the Kingman's Theorem about subadditive cocycles \cite{Ki, Furstenberg} follows that for every $\ensuremath{\mathbf{\sigma}}$-invariant measure $\mu$ and for almost all $x$ there exists a number
\begin{equation}\ensuremath{\lambda}bel{e1} r(x)=\lim_{n\to\infty} \frac{s(n,x)}{n}\end{equation}
If $\mu$ is ergodic then this number is the same for a.a $x$ and equals $\displaystyle{\inf_{n\ge 1}\int_X \frac{s(n,x)}{n}d\mu}$. For an ergodic $\mu$ we will call this number $r_\mu$. The set of all \ensuremath{\mathbf{\sigma}}-invariant ergodic measures we denote as $\mathcal{M}$. The set of points $x\in X$ for which limit $(\ref{e1})$ exists we call regular and denote as $\mathcal{R}$
Of course, there could be points for which the limit in $(\ref{e1})$ does not exist.
We can also consider numbers $s_n=\displaystyle{\max_x s(n,x)}$. It is a subadditive sequence of numbers $s_{n+m}\le s_n+s_m$ and we denote as $r$ the following number:
\begin{equation}\ensuremath{\lambda}bel{e2}\displaystyle{r=\lim_{n\to\infty}\frac{s_n}{n}=\inf_{n\ge 1} \frac{s_n}{n}}
\end{equation}
It is known (see \cite{S}) that if $\ensuremath{\mathbf{\sigma}}$ is continuous and $X$ is compact then
\begin{equation}\ensuremath{\lambda}bel{e3} r=\sup_{x\in\mathcal{R}} r(x)=\sup_{\mu\in\mathcal{M}} r_\mu\end{equation}
For a periodic point $p=\ensuremath{\mathbf{\sigma}}^k p$ we denote as $r_{p}$ the following quantity
$$r_{p}=\frac{ s(k,p)}{k}$$
It is easy to see that $r(p)$ exists (but can be $-\infty$) and $r(p)\le r_p$.
We will show that if $\ensuremath{\mathbf{\sigma}}$ has a closing property we can prove that:
\begin{theorem}\ensuremath{\lambda}bel{t3} Let $X$ be a compact metric space, $\ensuremath{\mathbf{\sigma}}:X\to X$ a homeomorphism with closing property. We denote as $\mathcal{P}$ the set of all periodic points. If $a:X\to B^\times $ is an $\alpha$-H\"older continuous function, $a(n,x)$ is a cocycle generated by it and ${s(n,x)=\ln \|a(n,x)\| }$ then
\begin{equation}\ensuremath{\lambda}bel{e3} r=\sup_{x\in\mathcal{R}} r(x)=\sup_{\mu\in\mathcal{M}} r_\mu\le \sup_{p\in\mathcal{P}}r_p \end{equation}
\end{theorem}
An easy corollary of this theorem is the following important for us result.
\begin{corollary} \ensuremath{\lambda}bel{c3} Let $X$ be a compact metric space, $\ensuremath{\mathbf{\sigma}}:X\to X$ a homeomorphism with closing property. If $a(n,p)=e$ for every periodic point $p$ with period $n$ then for any $\ensuremath{\varepsilon}>0$ there exists $C$ such that for all integer positive $n$ and all $x\in X$
$$\|a(n,x)\|\le Ce^{\ensuremath{\varepsilon} n}$$
$$\|a(-n,x)\|\le Ce^{\ensuremath{\varepsilon} n}$$
$$\|[a(n,x)]^{-1}\|\le Ce^{\ensuremath{\varepsilon} n}$$
\end{corollary}
\begin{proof} The first inequality follows from the fact that if $s(n,x)=\ln\|a(n,x)\|$ then for this subadditive cocycle $r_p=0$ for every periodic point $p$ and from (\ref{e3}) follows that $r=0$. For the second inequality we can consider a cocycle $b(n,x)$ over $\ensuremath{\mathbf{\sigma}}^{-1}$ generated by $a^{-1}(x)$. Below, we will prove that if $a(x)$ is H\"{o}lder continuous then $a^{-1}(x)$ is also H\"{o}lder continuous, and therefore we can apply Theorem \ref{t3} to the cocycle $b(n,x)$ also. But $a(-n,x)=b(n,x)$ and if $a(n,p)=e$ for every periodic point then
$$b(n,p)=a(-n,p)=a(-n,\ensuremath{\mathbf{\sigma}}^np)=[a(n,p)]^{-1}=e$$
So the rate of growth $r$ for $b(n,x)$ is also 0.
The last inequality follows from the fact that
$$[a(n,x)]^{-1}=b(n,\ensuremath{\mathbf{\sigma}}^n x)$$
The only thing left to show is that if $a(x)$ is $\alpha$-H\"{o}lder continuous then $a^{-1}(x)$ is also $\alpha$-H\"{o}lder continuous. For normed rings the operation of taking the inverse element is continuous (see \cite{Na}). Therefore the function $a^{-1}(x)$ is bounded. But
$$\|a^{-1}-b^{-1}\|=\|b^{-1}(b-a)a^{-1}\|\le\|b^{-1}\|\cdot\|(b-a)\|\cdot\|a^{-1}\|$$
Therefore, if the function $a(x)$ is $\alpha$-H\"{o}lder continuous, then $a^{-1}(x)$ is also $\alpha$-H\"{o}lder continuous.
\end{proof}
\section{Proof of the Theorem \ref{t3}}
The following result proven in \cite[Proposition 4.2]{MK} will be used.
\begin{lemman}[A. Karlsson, G. A. Margulis]\ensuremath{\lambda}bel{MK} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a measurable map, $\mu$ an ergodic measure, $s(n,x)$ a subadditive cocycle. For any $\ensuremath{\epsilon}>0$, let $E_\ensuremath{\epsilon}$ be the set of $x$ in $X$ for which there exist an integer $K(x)$ and infinitely many $n$ such that
$$s(n,x)-s(n-k,\ensuremath{\mathbf{\sigma}}^kx)\ge (r_\mu-\ensuremath{\epsilon})k$$
for all $k, K(x)\le k\le n$. Let $E=\cap_{\ensuremath{\epsilon}>0} E_{\ensuremath{\epsilon}}$ then $\mu(E)=1$.
\end{lemman}
If $s(n,x)=\ln\|a(n,x)\|$ then the inequality in the lemma could be rewritten as
\begin{equation}\|a(n-k,\ensuremath{\mathbf{\sigma}}^k x)\|\le \|a(n,x)\|e^{-(r_\mu-\ensuremath{\epsilon})k}\ensuremath{\lambda}bel{fmk}\end{equation}
\begin{definition} Let $\gamma,\delta$ be some positive numbers and $n$ is a natural number. We say that a point $y$ is $(\gamma,\delta,n)-$close to $x$ if
$$dist(\ensuremath{\mathbf{\sigma}}^k x,\ensuremath{\mathbf{\sigma}}^k y)\le \delta e^{-\gamma k} \quad \text{for all}\quad 0\le k\le n$$
\end{definition}
\begin{prop} \ensuremath{\lambda}bel{l5} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a homeomorphism and $a:X\to B^\times$ be an $\alpha$-H\"older continuous function, and $s(n,x)=\ln\|a(n,x)\|$. For any $\gamma>0$ let $S_\gamma$ be the set of points $x$ in $X$ for which there exist a number $\delta(x)>0$ and infinitely many $n$ such that for any point $y$ which is $(\gamma,\delta,n)-$close to $x$
\begin{equation}\ensuremath{\lambda}bel{f11}
\|a(n,y)\|\ge \frac12\|a(n,x)\|
\end{equation}
Then $\mu(S_\gamma)=1$ for any ergodic invariant measure $\mu$ with $r-r_\mu<\alpha\gamma$.
\end{prop}
\begin{proof}
Let $\mu$ be an ergodic invariant measure with $r-r_\mu<\alpha\gamma$. We choose a number $0<\ensuremath{\varepsilon}<\frac13(\gamma\alpha-(r-r_\mu))$. Almost all points with respect to this measure and \ensuremath{\varepsilon}\ satisfy Karlsson-Margulis Lemma and for almost all points the number $r(x)=r_\mu$. Take a point $x$ from the intersection of those two sets. Using the identity
$$b_na_{n-1}\ldots b_1-a_na_{n-1}\ldots a_1=\sum_{k=1}^n b_n\ldots b_{k+1}(b_{k}-a_{k})a_{k-1}\ldots a_1$$
we can see that
$$\|a(n,x)-a(n,y)\|=\|\sum_{k=0}^{n-1} a(n-k-1,\ensuremath{\mathbf{\sigma}}^{k+1} x)[a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)]a(k,y)\|\le$$
\begin{equation}\ensuremath{\lambda}bel{f4}
\sum_{k=0}^{n-1} \|a(n-k-1,\ensuremath{\mathbf{\sigma}}^{k+1} x)\|\cdot\|a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)\|\cdot \|a(k,y)\|
\end{equation}
Our goal is to show that if we choose a sufficiently small $\delta$ then for infinitely many numbers $n$ and for every $(\gamma,\delta,n)-$close $y$ the sum $(\ref{f4})$ is smaller than $ \frac12\|a(n,x)\|$.
Let $K(x,\ensuremath{\varepsilon})$ and $n$ be as in the Karlsson-Margulis Lemma and a point $y$ is $(\gamma,\delta,n)-$close to $x$ for some $\delta$ that we specify later. By definition $\displaystyle{r=\lim_{k\to\infty} s_k/k}$, so we can find $K\ge K(x,\ensuremath{\varepsilon})$ such that $s_{k}<k(r+\ensuremath{\varepsilon})$ for all $k\ge K$, or $\|a(k,x)\|<e^{k(r+\ensuremath{\varepsilon})}$. For every $k> K$ factors in the product
\begin{equation}\ensuremath{\lambda}bel{f3} \|a(n-k-1,\ensuremath{\mathbf{\sigma}}^{k+1} x)\|\cdot\|a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)\|\cdot \|a(k,y)\|
\end{equation} could be bounded from above as \\
$\|a(n-k-1,\ensuremath{\mathbf{\sigma}}^k x)\|\le \|a(n,x)\|e^{- (r_\mu-\ensuremath{\epsilon}silon)(k+1)}\quad$ It follows from the Karlsson-Margulis Lemma. \\
$\|a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)\|\le H\delta^\alpha e^{-k\gamma\alpha}$ where $H$ is some positive constant. It follows from the fact that $a(x)$ is H\"older continuous and $y$ is $(\gamma,\delta,n)$-close to $x$.\\
$ \|a(k,y)\|\le e^{s_{k}}\le e^{k(r+\ensuremath{\varepsilon})}$ It follows from the definition of $K$.\\
If we combine those inequalities we can see that that the number in the product (\ref{f3}) is smaller than
$$ \|a(n,x)\|e^{- (r_\mu-\ensuremath{\epsilon}silon)(k+1)}\cdot H\delta^\alpha e^{-k\gamma\alpha}\cdot e^{k(r+\ensuremath{\varepsilon})}\le \|a(n,x)\|H\delta^\alpha e^{-k(\gamma\alpha-(r-r_\mu)-2\ensuremath{\varepsilon})}$$
After simplification we can write that
$$ \|a(n-k-1,\ensuremath{\mathbf{\sigma}}^{k+1} x)\|\cdot\|a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)\|\cdot \|a(k,y)\|\le \|a(n,x)\|H\delta^\alpha e^{-k\ensuremath{\varepsilon}}
$$
If we add those inequalities for $k\ge K$ we can see that
$$\|\sum_{k=K}^n a(n-k+1,\ensuremath{\mathbf{\sigma}}^{k+1} x)[a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)]a(k,y)\|\le$$
$$ \|a(n,x)\|H\delta^\alpha \sum_{k=0}^\infty e^{-k\ensuremath{\varepsilon}}= \|a(n,x)\|\frac{H\delta^\alpha}{1-e^{-\ensuremath{\varepsilon}}}$$
To estimate the number in the formula (\ref{f3}) for $k<K$ we denote as
$$M=1+\max_x ||a(x)||$$
$$m=1+\max_x ||a^{-1}(x)||$$
Then as before
$$\|a(\ensuremath{\mathbf{\sigma}}^{k}x)-a(\ensuremath{\mathbf{\sigma}}^{k}y)\|\le H \delta^\alpha e^{-k\gamma\alpha}$$
but
$$||a(n-k+1,\ensuremath{\mathbf{\sigma}}^{k+1} x)||=||a(-k+1,\ensuremath{\mathbf{\sigma}}^{n}x)a(n,x)||\le ||a(n,x)||\cdot m^k$$
and
$$||a(k,y)||\le M^{k}$$
So for $k<K$ the expression (\ref{f3}) is bounded by
$$||a(n,x)|| m^k\cdot H\delta^\alpha e^{-k\gamma\alpha}\cdot M^{k}\le ||a(n,x)||H\delta^\alpha (mM)^K$$
Finally,
$$||a(n,x)-a(n,y)||\le ||a(n,x)||\delta^\alpha \left(\frac{H}{1-e^{-\ensuremath{\varepsilon}}}+K(mM)^K\right)=||a(n,x)||\delta'$$
By choosing $\delta$ sufficiently small we can make $\delta'<1/2$. Then
$$||a(n,y)||= ||a(n,x)-(a(n,x)-a(n,y))||\ge ||a(n,x)||-||a(n,x)-a(n,y)||\ge$$
$$\ge \frac12|| a(n,x)||$$
\end{proof}
To finish the proof of the Theorem \ref{t3} we will need the following features of the maps with closing property.
\begin{lemma}\ensuremath{\lambda}bel{l6} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a homeomorphism with closing property and the expansion constant $\ensuremath{\lambda}$, then for any positive numbers $\ensuremath{\varepsilon}$ and $\delta$ there is a number $\delta'$ such that if $dist(x,\ensuremath{\mathbf{\sigma}}^kx)\le\delta'$ and $k\ge n(1+\ensuremath{\varepsilon})$ then there is a point $p$ such that $\ensuremath{\mathbf{\sigma}}^k p=p$ and $p$ is $(\gamma,\delta,n)-$close to $x$, where $\gamma=\ensuremath{\varepsilon}\ensuremath{\lambda}mbda$.
\end{lemma}
\begin{proof} It follows from the definition of the closing property that for $0\le i\le k$
$$dist(\ensuremath{\mathbf{\sigma}}^i x,\ensuremath{\mathbf{\sigma}}^i p)\le dist(\ensuremath{\mathbf{\sigma}}^i x,\ensuremath{\mathbf{\sigma}}^i z)+dist(\ensuremath{\mathbf{\sigma}}^i z,\ensuremath{\mathbf{\sigma}}^i p)\le 2C\delta' e^{-\ensuremath{\lambda}mbda \min(i,k-i) }$$
The function $-\ensuremath{\lambda}mbda\min(x,k-x)$ is convex downward so the segment connecting points $(0,0)$ and $(n,-\ensuremath{\lambda}mbda\min(n,k-n))$ on the graph of this function stays above the graph. The linear function that corresponds to this segment is $-\gamma x$ where
$$\gamma=\frac{k-n}{n}\ensuremath{\lambda}mbda>\ensuremath{\varepsilon}\ensuremath{\lambda}mbda$$
Therefore the point $p$ satisfies the following inequalities:
$$dist(\ensuremath{\mathbf{\sigma}}^i x,\ensuremath{\mathbf{\sigma}}^i p)\le 2 C\delta' e^{-\gamma i } \quad 0\le i\le n$$
If we take $\delta'=\frac{\delta}{2 C}$ we can see that $p$ is $(\gamma,\delta,n)-$close to $x$.
\end{proof}
\begin{lemma}\ensuremath{\lambda}bel{l7} Let $\ensuremath{\mathbf{\sigma}}:X\to X$ be a homeomorphism. For any $\ensuremath{\varepsilon},\delta>0$ let $P_{\ensuremath{\epsilon},\delta}$ be the set of points $x$ in $X$ for which there is an integer number $N=N(x,\ensuremath{\epsilon},\delta)$ such that if $n>N$ then there is an integer $ n(1+\ensuremath{\varepsilon})<k<n(1+2\ensuremath{\varepsilon})$ for which
$$dist(x,\ensuremath{\mathbf{\sigma}}^k x)<\delta$$
If $\displaystyle{P=\cap_{\ensuremath{\varepsilon}>0,\delta>0} P_{\ensuremath{\varepsilon},\delta}}$ then $\mu(P)=1$ for any invariant measure $\mu$.
\end{lemma}
\begin{proof} It is enough to prove it only for ergodic invariant measures. Let $\mu$ be an invariant ergodic measure. The support of a measure is the the set of all points in $X$ such that the measure of any open ball centered at $x$ is not 0. The support of a measure on a compact metric space has always full measure {(see \cite{Fe})}. $X$ is compact so there is a sequence of balls $B_i$ which is a base of the topology. If we define as $f_i(n,x)$ the number of such $k$ that $\ensuremath{\mathbf{\sigma}}^k x\in B_i$ and $1\le k\le n$ then by Birkhoff's Ergodic Theorem $\displaystyle{\lim_{n\to\infty} \frac{f_i(n,x)}{n}}$ exists and equals $\mu(B_i)$ for almost all $x$. It is easy to see that any $x$ that belongs to the support of the measure and satisfies Birkhoff's Ergodic Theorem for all $i$ will belong to the set $P$. Indeed, if we choose $\delta>0$ then we know that the ball $B_\delta$ centered at $x$ should have measure greater than 0. This ball is a countable union of some of the balls $B_i$, therefore there exists at least one ball $B_{i_0}$ such that $\mu(B_{i_0})>0$ and $B_{i_0}\subset B_\delta$. Now, using the numbers $\ensuremath{\varepsilon}$ and $\mu(B_{i_0})$ we choose a very small $\ensuremath{\epsilon}$. How small we specify later. For this $\ensuremath{\epsilon}>0$ we can find $N$ such that if $n>N$ then $|f_{i_0}(n,x)-\mu(B_{i_0}n)|<\ensuremath{\epsilon} n$. If $n>N$ and there is no $k$ such that ${n(1+\ensuremath{\varepsilon})<k<n(a+2\ensuremath{\varepsilon})}$ and ${\ensuremath{\mathbf{\sigma}}^k x\in B_{i_0}}$ then $f_{i_0}(n(1+\ensuremath{\varepsilon}),x)=f_{i_0}(n(1+2\ensuremath{\varepsilon}),x)$. It is impossible if we choose $\ensuremath{\epsilon}$ very small because in this case
$$ (\mu(B_{i_0})+\ensuremath{\epsilon})n(1+\ensuremath{\varepsilon})\ge f_{i_0}(n(1+\ensuremath{\varepsilon},x)=f_{i_0}(n(1+2\ensuremath{\varepsilon},x)\ge (\mu(B_{i_0})-\ensuremath{\epsilon})n(1+2\ensuremath{\varepsilon})$$
or
$$\frac{\mu(B_{i_0})+\ensuremath{\epsilon}}{\mu(B_{i_0})-\ensuremath{\epsilon}}\ge \frac{1+2\ensuremath{\varepsilon}}{1+\ensuremath{\varepsilon}}$$
When $\ensuremath{\epsilon}$ is small the left side is as close to 1 as we want, so we get a contradiction. It means, if $N$ is sufficiently big and $n>N$ then there is $k$ such that ${\ensuremath{\mathbf{\sigma}}^k\in B_{i_0}\subset B_\delta }$ and ${n(1+\ensuremath{\varepsilon})\le k\le n(1+2\ensuremath{\varepsilon})}$. Therefore the set $P$ includes the intersection of two sets of full measure and has the full measure.
\end{proof}
{\it Proof of the Theorem \ref{t3}:} Choose any $\ensuremath{\varepsilon}>0$. We can find an ergodic invariant measure $\mu$ such that ${r-r_\mu<min(\ensuremath{\varepsilon},\ensuremath{\varepsilon}\alpha\ensuremath{\lambda}mbda)}$. Choose a point $x$ such that $r(x)=r_\mu$ and $x$ belongs to the set $S_{\ensuremath{\varepsilon}\ensuremath{\lambda}mbda}\cap P$ where $S_{\ensuremath{\varepsilon}\ensuremath{\lambda}mbda}$ as in the Proposition \ref{l5} and $P$ as in the Lemma \ref{l7}. All those sets have full support, so their intersection is not empty. For the point $x$ we can find $\delta$ such that for infinitely many $n_i$ if a point $p$ is $(\ensuremath{\varepsilon}\ensuremath{\lambda}mbda,\delta,n_i)-$close to $x$ then
\begin{equation}\ensuremath{\lambda}bel{f10}\|a(n_i,p)\|\ge\frac12\|a(n_i,x)\|\end{equation}
For this $\delta$ we can find $\delta'$ from Lemma \ref{l6}. Using this $\delta'$ and $\ensuremath{\varepsilon}$ we can find $N=N(\ensuremath{\varepsilon},\delta')$ from the Lemma \ref{l7} such that if $n_i>N$ then there is $k$ such that $n_i(1+\ensuremath{\varepsilon})\le k\le n_i(1+2\ensuremath{\varepsilon})$ and $dist(\ensuremath{\mathbf{\sigma}}^kx,x)<\delta'$, then from Lemma \ref{l6} follows that there is a periodic point $p$ with the period $k$ such that it is $(\ensuremath{\varepsilon}\ensuremath{\lambda}mbda,\delta,n_i)-$close to $x$ and therefore satisfies the {inequality (\ref{f10})}.
Now, we estimate $\|a(k,p)\|$. Let $N'$ be a number such that if $n>N'$ then
$$\|a(n,x)\|\ge e^{n(r_\mu-\ensuremath{\varepsilon})}\ge e^{n(r-2\ensuremath{\varepsilon})}$$
We always can choose $n_i$ bigger not only than $N$ but also and $N'$. Denote as ${m=\ln\max_y\|a^{-1}(y)\|}$. Then
$$\|a(n_i,p)\|=\|a(-(k-n_i),p)a(k,p)\|\le \|a(-(k-n_i),p)\|\cdot\|a(k,p)\|$$
so
$$\|a(k,p)\|\ge \frac{\|a(n_i,p)\|}{e^{m(k-n_i)}}\ge\frac12\frac{\|a(n_i,x)\|}{e^{2m\ensuremath{\varepsilon} n_i}}\ge \frac12 e^{(r-2\ensuremath{\varepsilon}-2m\ensuremath{\varepsilon})n_i}$$
We see that
$$r_p=\frac{\ln\|a(k,p)\|}{k}\ge\frac{(r-2\ensuremath{\varepsilon}-2m\ensuremath{\varepsilon})n_i-\ln 2}{(1+2\ensuremath{\varepsilon})n_i}$$
Number $m$ does not depend on the choice of $x,n_i$ and $p$, so by choosing $\ensuremath{\varepsilon}$ very small and $n_i$ very big we can make $r_p$ as close to $r$ as we want.
\qed\\
\section{Proof of the Main Theorem }
After Theorem \ref{t3} is established we can use Corollary \ref{c3} to show that the growth of $\|a(n,x)\|$ is sub-exponential. It allows to use the idea of the original Liv\v{s}ic proof for cocycles with values in Banach rings.
H.Bercovici and V.Nitica in \cite{BN} (Theorem 3.2) showed that if $\ensuremath{\mathbf{\sigma}}$ is a transitive Anosov map, periodic obstructions vanish and
\begin{equation}\ensuremath{\lambda}bel{f11}\begin{split}\|a(x)\|&\le 1+\delta\\
\|a^{-1}(x)\|&\le 1+\delta
\end{split}\end{equation}
for some $\delta$ that depends on $\ensuremath{\mathbf{\sigma}}$, then $a(x)$ is a coboundary. From Corollary \ref{c3} we can get a little bit less. If periodic obstructions vanish then for any $\delta>0$ there exists $C>0$ such that for any positive integer $n$
\begin{equation*}\begin{split}\|a(n,x)\|&\le C(1+\delta)^n\\
\|[a(n,x)]^{-1}\|&\le C(1+\delta)^n
\end{split}\end{equation*}
Those inequalities are actually enough to repeat the arguments from \cite{BN} with some small changes, but we also refer to more general theorem proven in \cite{G} that considers cocycles over maps that satisfy closing property and with values in abstract groups satisfying some conditions. But we will need couple of more definitions.
\begin{definition} If $G$ is a group with metric denoted as $dist$ and $g\in G$ we define the distortion of the element $g$ as
$$|g|=\sup_{f\neq g}\left[ \frac{dist(gf,gh)}{dist{(f,h)}},\frac{dist(fg,hg)}{dist{(f,h)}},\frac{dist(g^{-1}f,g^{-1}h)}{dist{(f,h)}},\frac{dist(fg^{-1},hg^{-1})}{dist{(f,h)}}\right]$$
We say that a group is Lipschitz if $|g|<\infty$ for all $g\in G$.
\end{definition}
It is easy to see that for Banach rings if we define $$dist(f,h)=\max(\|f-h\|,\|f^{-1}-g^{-1}\|)$$ then $|g|\le \max(\|g\|,\|g^{-1}\|)$ and $B^\times$ is Lipschitz.
\begin{definition} We call the rate of distortion of a cocycle $a(x):X\to G$ the following number
$$r(a)=\lim_{n\to\infty} \frac{\sup_{x\in X}\ln |a(n,x)|}{n}$$
\end{definition}
\begin{theorem}\ensuremath{\lambda}bel{t9} Let $G$ be a Lipschitz group with the property that there are numbers $\ensuremath{\epsilon}$ and $D$ such that $dist(g,e)\le\ensuremath{\epsilon}$ implies $|g|\le D$, \ensuremath{\mathbf{\sigma}}\ be a transitive homeomorphism with $\ensuremath{\lambda}mbda$-closing property. If the rate of the distortion of an $\alpha$-H\"older continuous cocycle $a:X\to G$ is smaller than $\alpha\ensuremath{\lambda}/2$ and the periodic obstructions vanish then $a(x)$ is a coboundary with $\alpha$-H\"older continuous transition function $t(x)$.
\end{theorem}
\begin{proof} See \cite{G}\end{proof}
{\it Proof of the Main Theorem.} If $a(n,p)=e$ for every periodic point then it follows from the Corollary \ref{c3} that the distortion rate of the $a(n,x)$ is less or equal than 0. In the group $B^\times$ if $dist(e,g)<\frac12$ then $\|e-g\|<\frac12$ and $\|e-g^{-1}\|<\frac12$ so $\|g\|,\|g^{-1}\|\le\frac32$. We see that $|g|<\frac32$, therefore by Theorem \ref{t9} the cocycle $a(n,x)$ is a coboundary with
$\alpha$-H\"older continuous transition function.\qed
\end{document} |
\begin{document}
\pagenumbering{gobble}
\begin{titlepage}
\title{Sensitivity Oracles for All-Pairs Mincuts}
\author{
Surender Baswana\thanks{Department of Computer Science \& Engineering, IIT Kanpur, Kanpur -- 208016, India, sbaswana@cse.iitk.ac.in}
\and
Abhyuday Pandey\thanks{Department of Computer Science \& Engineering, IIT Kanpur, Kanpur -- 208016, India, pandey.abhyuday07@gmail.com}
}
\maketitle
\begin{abstract}
{
Let $G=(V,E)$ be an undirected unweighted graph on $n$ vertices and $m$ edges. We address the problem of sensitivity oracle for all-pairs mincuts in $G$ defined as follows.
Build a compact data structure that, on receiving any pair of vertices $s,t\in V$ and failure (or insertion) of any edge as query, can efficiently report the mincut between $s$ and $t$ after the failure (or the insertion).
To the best of our knowledge, there exists no data structure for this problem which takes $o(mn)$ space and a non-trivial query time.
We present the following results.
\begin{enumerate}
\item Our first data structure occupies ${\cal O}(n^2)$ space and guarantees ${\cal O}(1)$ query time to report the value of resulting $(s,t)$-mincut upon failure (or insertion) of any edge. Moreover, the set of vertices defining a resulting $(s,t)$-mincut after the update can be reported in ${\cal O}(n)$ time which is worst-case optimal.
\item
Our second data structure optimizes space at the expense of increased query time. It takes ${\cal O}(m)$ space -- which is also the space taken by $G$. The query time is ${\cal O}(\min(m,n c_{s,t}))$ where $c_{s,t}$ is the value of the mincut between $s$ and $t$ in $G$. This query time is faster by a factor of $\Omega(\min(m^{1/3},\sqrt{n}))$ compared to the best known deterministic algorithm \cite{DBLP:conf/focs/GoldbergR97a,DBLP:conf/stoc/KargerL98,DBLP:journals/corr/abs-2003-08929} to compute a $(s,t)$-mincut from scratch.
\item
If we are only interested in knowing if failure (or insertion) of an edge changes the value of $(s,t)$-mincut, we can distribute our ${\cal O}(n^2)$ space data structure evenly among $n$ vertices. For any failed (or inserted) edge we only require the data structures stored at its endpoints to determine if the value of $(s,t)$-mincut has changed for any $s,t \in V$.
Moreover, using these data structures we can also output efficiently a compact encoding of all pairs of vertices whose mincut value is changed after the failure (or insertion) of the edge.
\end{enumerate}
}
\end{abstract}
\end{titlepage}
\pagebreak
\pagenumbering{arabic}
\section{Introduction}
\subfile{src/introduction}
\section{Preliminaries} \label{sec:prelimiaries}
\subfile{src/preliminaries}
\section{Insights into \texorpdfstring{$3$}{3}-vertex mincuts} \label{sec:query-transformation}
\subfile{src/compact-graph-query-transf}
\section{A Compact Graph for Query Transformation}
\subfile{src/ft-steiner-connectivity}
\section{Compact Data Structure for Sensitivity Query} \label{sec:final-ds}
\subfile{src/graph-contractions}
\section{Distributed Sensitivity Oracle}
\label{sec:distributed-sensitivity-oracle}
\subfile{src/distributed-sensitivity-oracle}
\section{Conclusion}
\label{sec:conclusion}
\subfile{src/conclusion}
\pagebreak
\pagebreak
\appendix
\subfile{src/appendix}
\end{document} |
\begin{document}
\mainmatter
\title{Learning with a Drifting Target Concept}
\titlerunning{Learning with a Drifting Target Concept}
\author{Steve Hanneke \and Varun Kanade \and Liu Yang}
\authorrunning{Steve Hanneke, Varun Kanade, and Liu Yang}
\institute{Princeton, NJ USA.\\
\email{steve.hanneke@gmail.com}
\and
D\'{e}partement d'informatique, \'{E}cole normale sup\'{e}rieure, Paris, France.\\
\email{varun.kanade@ens.fr}
\and
IBM T.J. Watson Research Center, Yorktown Heights, NY USA.\\
\email{yangli@us.ibm.com}
}
\maketitle
\begin{abstract}
We study the problem of learning in the presence of a drifting target concept. Specifically,
we provide bounds on the error rate at a given time, given a learner with access to a history
of independent samples labeled according to a target concept that can change on each round.
One of our main contributions is a refinement of the best previous results for
polynomial-time algorithms for the space of linear separators under a uniform distribution.
We also provide general results for an algorithm capable of adapting to a variable rate of drift
of the target concept.
Some of the results also describe an active learning variant of this setting, and provide bounds on the
number of queries for the labels of points in the sequence sufficient to obtain the stated bounds
on the error rates.
\end{abstract}
\section{Introduction}
Much of the work on statistical learning has focused on
learning settings in which the concept to be learned is static
over time.
However, there are many application areas where this is not
the case. For instance, in the problem of face recognition,
the concept to be learned actually changes over time as
each individual's facial features evolve over time. In this
work, we study the problem of learning with a drifting
target concept. Specifically, we consider a statistical
learning setting, in which data arrive i.i.d. in a stream,
and for each data point, the learner is required to predict
a label for the data point at that time. We are then
interested in obtaining low error rates for these predictions.
The target labels are generated from a function known to reside
in a given concept space, and at each time $t$ the target function
is allowed to change by at most some distance $\mathcal Delta_{t}$: that is,
the probability the new target function disagrees with the previous
target function on a random sample is at most $\mathcal Delta_{t}$.
This framework has previously been studied in a number of articles.
The classic works of \cite{helmbold:91,helmbold:94,bartlett:96,long:99,bartlett:00} and \cite{barve:97}
together provide a general analysis of a
very-much related setting. Though the objectives in these works are
specified slightly differently, the results established there are
easily translated into our present framework,
and we summarize many of the relevant results from this literature
in Section~\ref{sec:background}.
While the results in these classic works are general, the best guarantees
on the error rates are only known for methods having no guarantees
of computational efficiency.
In a more recent effort, the work of \cite{min_concept} studies this problem
in the specific context of learning a homogeneous linear separator,
when all the $\mathcal Delta_{t}$ values are identical.
They propose a polynomial-time algorithm (based on the modified Perceptron
algorithm of \cite{stream_perceptron}),
and prove a bound on the number of mistakes it makes as a function of
the number of samples, when the data distribution satisfies a
certain condition called ``$\lambda$-good'' (which generalizes a useful
property of the uniform distribution on the origin-centered unit sphere).
However, their result is again worse than that obtainable by the known
computationally-inefficient methods.
Thus, the natural question is whether there exists a polynomial-time algorithm
achieving roughly the same guarantees on the error rates known for the inefficient methods.
In the present work, we resolve this question in the case of learning homogeneous
linear separators under the uniform distribution, by proposing a polynomial-time
algorithm that indeed achieves roughly the same bounds on the error rates
known for the inefficient methods in the literature.
This represents the main technical contribution of this work.
We also study the interesting problem of \emph{adaptivity} of an
algorithm to the sequence of $\mathcal Delta_{t}$ values, in the setting where
$\mathcal Delta_{t}$ may itself vary over time. Since the values $\mathcal Delta_{t}$
might typically not be accessible in practice, it seems important to
have learning methods having no explicit dependence on the sequence $\mathcal Delta_{t}$.
We propose such a method below, and prove that it achieves roughly the
same bounds on the error rates known for methods in the literature
which require direct access to the $\mathcal Delta_{t}$ values.
Also in the context of variable $\mathcal Delta_{t}$ sequences, we discuss
conditions on the sequence $\mathcal Delta_{t}$ necessary and sufficient
for there to exist a learning method guaranteeing a \emph{sublinear}
rate of growth of the number of mistakes.
We additionally study an \emph{active learning} extension to this
framework, in which, at each time, after making its prediction,
the algorithm may decide whether or not to request access to the
label assigned to the data point at that time. In addition to guarantees on the
error rates (for \emph{all} times, including those for which the label was not observed),
we are also interested in bounding the number of labels we expect the algorithm to
request, as a function of the number of samples encountered thus far.
\section{Definitions and Notation}
\label{sec:definitions}
Formally, in this setting, there is a fixed distribution $\mathcal{P}$ over the instance space $\mathcal X$,
and there is a sequence of independent $\mathcal{P}$-distributed unlabeled data $X_{1},X_{2},\ldots$.
There is also a concept space $\mathbb C$, and a sequence of target functions $h^{*}seq = \{h^{*}_{1},h^{*}_{2},\ldots\}$ in $\mathbb C$.
Each $t$ has an associated target label $Y_{t} = h^{*}_{t}(X_{t})$.
In this context, a (passive) learning algorithm is required, on each round $t$,
to produce a classifier $\hat{h}_{t}$ based on the observations $(X_{1},Y_{1}),\ldots,(X_{t-1},Y_{t-1})$,
and we denote by $\hat{Y}_{t} = \hat{h}_{t}(X_{t})$ the corresponding prediction by the algorithm
for the label of $X_{t}$. For any classifier $h$, we define ${\rm er}_{t}(h) = \mathcal{P}(x : h(x) \neq h^{*}_{t}(x))$.
We also say the algorithm makes a ``mistake'' on instance $X_{t}$ if $\hat{Y}_{t} \neq Y_{t}$;
thus, ${\rm er}_{t}(\hat{h}_{t}) = \mathbb P( \hat{Y}_{t} \neq Y_{t} | (X_{1},Y_{1}),\ldots,(X_{t-1},Y_{t-1}) )$.
For notational convenience, we will suppose the $h^{*}_{t}$ sequence is
chosen independently from the $X_{t}$ sequence (i.e., $h^{*}_{t}$ is chosen prior
to the ``draw'' of $X_{1},X_{2},\ldots \sim \mathcal{P}$), and is not random.
In each of our results, we will suppose $h^{*}seq$ is chosen from some set $S$ of
sequences in $\mathbb C$. In particular, we are interested in describing the sequence $h^{*}seq$
in terms of the magnitudes of \emph{changes} in $h^{*}_{t}$ from one time to the next.
Specifically, for any sequence $\mathcal Deltaseq = \{\mathcal Delta_{t}\}_{t=2}^{\infty}$ in $[0,1]$,
we denote by $S_{\mathcal Deltaseq}$ the set of all sequences $h^{*}seq$ in $\mathbb C$ such that,
$\forall t \in \mathbb{N}$, $\mathcal{P}(x : h_{t}(x) \neq h_{t+1}(x)) \leq \mathcal Delta_{t+1}$.
Throughout this article, we denote by $d$ the VC dimension of $\mathbb C$ \cite{vapnik:71},
and we suppose $\mathbb C$ is such that $1 \leq d < \infty$.
Also, for any $x \in \mathbb{R}$, define ${\rm Log}(x) = \ln(\max\{x,e\})$.
\section{Background: $(\epsilon,S)$-Tracking Algorithms}
\label{sec:background}
As mentioned, the classic literature on learning with a drifting target concept
is expressed in terms of a slightly different model. In order to relate those
results to our present setting, we first introduce the classic setting.
Specifically, we consider a model introduced by \cite{helmbold:94},
presented here in a more-general form inspired by \cite{bartlett:00}.
For a set $S$ of sequences $\{h_{t}\}_{t=1}^{\infty}$ in $\mathbb C$,
and a value $\epsilon > 0$, an algorithm $\mathcal A$ is said to be
\emph{$(\epsilon,S)$-tracking} if $\exists t_{\epsilon} \in \mathbb{N}$ such that,
for any choice of $h^{*}seq \in S$,
$\forall T \geq t_{\epsilon}$,
the prediction $\hat{Y}_{T}$ produced by $\mathcal A$ at time $T$ satisfies
\begin{equation*}
\mathbb P\left( \hat{Y}_{T} \neq Y_{T} \right) \leq \epsilon.
\end{equation*}
Note that the value of the probability in the above expression
may be influenced by $\{X_{t}\}_{t=1}^{T}$, $\{h^{*}_{t}\}_{t=1}^{T}$,
and any internal randomness of the algorithm $\mathcal A$.
The focus of the results expressed in this classical model is determining
sufficient conditions on the set $S$ for there to exist an $(\epsilon,S)$-tracking algorithm,
along with bounds on the sufficient size of $t_{\epsilon}$.
These conditions on $S$ typically take the form of an assumption on the
drift rate, expressed in terms of $\epsilon$. Below, we summarize
several of the strongest known results for this setting.
\subsection{Bounded Drift Rate}
\label{sec:classic-constant-drift}
The simplest, and perhaps most elegant, results for $(\epsilon,S)$-tracking algorithms
is for the set $S$ of sequences with a bounded drift rate. Specifically, for any $\mathcal Delta \in [0,1]$,
define $S_{\mathcal Delta} = S_{\mathcal Deltaseq}$, where $\mathcal Deltaseq$ is such that $\mathcal Delta_{t+1} = \mathcal Delta$ for every $t \in \mathbb{N}$.
The study of this problem was initiated in the original work of \cite{helmbold:94}.
The best known general results are due to \cite{long:99}: namely,
that for some $\mathcal Delta_{\epsilon} = \Theta( \epsilon^{2} / d )$,
for every $\epsilon \in (0,1]$, there exists an $(\epsilon,S_{\mathcal Delta})$-tracking algorithm for all values
of $\mathcal Delta \leq \mathcal Delta_{\epsilon}$.\footnote{In fact, \cite{long:99} also allowed the distribution
$\mathcal{P}$ to vary gradually over time. For simplicity, we will only discuss the case of fixed $\mathcal{P}$.}
This refined an earlier result of \cite{helmbold:94} by a logarithmic factor.
\cite{long:99} further argued that this result can be achieved with $t_{\epsilon} = \Theta(d/\epsilon)$.
The algorithm itself involves a beautiful modification of the one-inclusion graph prediction
strategy of \cite{haussler:94}; since its specification is somewhat involved,
we refer the interested reader to the original work of \cite{long:99} for the details.
\subsection{Varying Drift Rate: Nonadaptive Algorithm}
\label{sec:classic-varying-drift}
In addition to the concrete bounds for the case $h^{*}seq \in S_{\mathcal Delta}$,
\cite{helmbold:94} additionally present an elegant general result. Specifically,
they argue that, for any $\epsilon > 0$, and any $m = \Omega\left( \frac{d}{\epsilon}{\rm Log}\frac{1}{\epsilon} \right)$,
if $\sum_{i=1}^{m} \mathcal{P}(x : h^{*}_{i}(x) \neq h^{*}_{m+1}(x)) \leq m \epsilon / 24$, then
for $\hat{h} = \mathop{\rm argmin}_{h \in \mathbb C} \sum_{i=1}^{m} \mathbbm{1}[ h(X_{i}) \neq Y_{i} ]$,
$\mathbb P( \hat{h}(X_{m+1}) \neq h^{*}_{m+1}(X_{m+1}) ) \leq \epsilon$.\footnote{They in fact
prove a more general result, which also applies to methods approximately minimizing
the number of mistakes, but for simplicity we will only discuss this basic version of the result.}
This result immediately inspires an algorithm $\mathcal A$ which, at every time $t$,
chooses a value $m_{t} \leq t-1$, and predicts $\hat{Y}_{t} = \hat{h}_{t}(X_{t})$,
for $\hat{h}_{t} = \mathop{\rm argmin}_{h \in \mathbb C} \sum_{i=t-m_{t}}^{t-1} \mathbbm{1}[ h(X_{i}) \neq Y_{i} ]$.
We are then interested in choosing $m_{t}$ to minimize the value of $\epsilon$ obtainable
via the result of \cite{helmbold:94}. However, that method is based on the
values $\mathcal{P}( x : h^{*}_{i}(x) \neq h^{*}_{t}(x) )$, which would typically not
be accessible to the algorithm. However, suppose instead we have access to a
sequence $\mathcal Deltaseq$ such that $h^{*}seq \in S_{\mathcal Deltaseq}$.
In this case, we could approximate $\mathcal{P}( x : h^{*}_{i}(x) \neq h^{*}_{t}(x) )$
by its \emph{upper bound} $\sum_{j = i+1}^{t} \mathcal Delta_{j}$. In this case,
we are interested choosing $m_{t}$ to minimize the smallest value of $\epsilon$
such that $\sum_{i=t-m_{t}}^{t-1} \sum_{j=i+1}^{t} \mathcal Delta_{j} \leq m_{t} \epsilon / 24$
and $m_{t} = \Omega\left( \frac{d}{\epsilon} {\rm Log}\frac{1}{\epsilon} \right)$.
One can easily verify that this minimum is obtained at a value
\begin{equation*}
m_{t} = \Theta\left( \mathop{\rm argmin}_{m \leq t-1} \frac{1}{m} \sum_{i=t-m}^{t-1} \sum_{j=i+1}^{t} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d)}{m} \right),
\end{equation*}
and via the result of \cite{helmbold:94} (applied to the sequence $X_{t-m_{t}},\ldots,X_{t}$)
the resulting algorithm has
\begin{equation}
\label{eqn:hl94}
\mathbb P\left( \hat{Y}_{t} \neq Y_{t} \right) \leq O\left( \min_{1 \leq m \leq t-1} \frac{1}{m} \sum_{i=t-m}^{t-1} \sum_{j=i+1}^{t} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d)}{m} \right).
\end{equation}
As a special case, if every $t$ has $\mathcal Delta_{t} = \mathcal Delta$ for a fixed value $\mathcal Delta \in [0,1]$,
this result recovers the bound $\sqrt{ d \mathcal Delta {\rm Log}(1/\mathcal Delta) }$,
which is only slightly larger than that obtainable from the best bound of \cite{long:99}.
It also applies to far more general and more intersting sequences $\mathcal Deltaseq$,
including some that allow periodic large jumps (i.e., $\mathcal Delta_{t} = 1$ for some indices $t$),
others where the sequence $\mathcal Delta_{t}$ converges to $0$, and so on.
Note, however, that the algorithm obtaining this bound
directly depends on the sequence $\mathcal Deltaseq$.
One of the contributions of the present work is to remove this requirement, while
maintaining essentially the same bound, though in a slightly different form.
\subsection{Computational Efficiency}
\label{sec:classic-consistency}
\cite{helmbold:94} also proposed a reduction-based approach, which
sometimes yields computationally efficient methods, though the tolerable $\mathcal Delta$
value is smaller. Specifically, given any (randomized) polynomial-time algorithm $\mathcal A$
that produces a classifier $h \in \mathbb C$ with $\sum_{t=1}^{m} \mathbbm{1}[ h(x_{t}) \neq y_{t} ] = 0$
for any sequence $(x_1,y_1),\ldots,(x_m,y_m)$ for which such a classifier $h$ exists
(called the \emph{consistency problem}),
they propose a polynomial-time algorithm that is $(\epsilon,S_{\mathcal Delta})$-tracking
for all values of $\mathcal Delta \leq \mathcal Delta_{\epsilon}^{\prime}$,
where $\mathcal Delta_{\epsilon}^{\prime} = \Theta\left( \frac{\epsilon^{2}}{d^{2} {\rm Log}(1/\epsilon)} \right)$.
This is slightly worse (by a factor of $d {\rm Log}(1/\epsilon)$) than the drift rate tolerable by the
(typically inefficient) algorithm mentioned above.
However, it does sometimes yield computationally-efficient methods.
For instance, there are known polynomial-time algorithms for the consistency problem for the classes of
linear separators, conjunctions, and axis-aligned rectangles.
\subsection{Lower Bounds}
\label{sec:classic-lower-bound}
\cite{helmbold:94} additionally prove \emph{lower bounds} for specific concept spaces:
namely, linear separators and axis-aligned rectangles. They specifically argue that, for
$\mathbb C$ a concept space
\begin{equation*}
{\rm BASIC}_{n} = \{ \cup_{i=1}^{n} [i/n,(i+a_i)/n) : \mathbf{a} \in [0,1]^{n} \}
\end{equation*}
on $[0,1]$, under $\mathcal{P}$ the uniform distribution on $[0,1]$,
for any $\epsilon \in [0,1/e^{2}]$ and $\mathcal Delta_{\epsilon} \geq e^{4} \epsilon^{2} / n$,
for any algorithm $\mathcal A$, and any $T \in \mathbb{N}$, there exists a choice of $h^{*}seq \in S_{\mathcal Delta_{\epsilon}}$
such that the prediction $\hat{Y}_{T}$ produced by $\mathcal A$ at time $T$ satisfies
$\mathbb P\left( \hat{Y}_{T} \neq Y_{T} \right) > \epsilon$.
Based on this, they conclude that no $(\epsilon,S_{\mathcal Delta_{\epsilon}})$-tracking algorithm exists.
Furthermore, they observe that the space ${\rm BASIC}_{n}$ is embeddable in many
commonly-studied concept spaces, including halfspaces and axis-aligned
rectangles in $\mathbb{R}^{n}$, so that for $\mathbb C$ equal to either of these spaces,
there also is no $(\epsilon,S_{\mathcal Delta_{\epsilon}})$-tracking algorithm.
\section{Adapting to Arbitrarily Varying Drift Rates}
\label{sec:general}
This section presents a general bound on the error rate at each time,
expressed as a function of the rates of drift, which are allowed to be \emph{arbitrary}.
Most-importantly, in contrast to the methods from the literature discussed above,
the method achieving this general result is \emph{adaptive} to the drift rates,
so that it requires no information about the drift rates in advance. This is an
appealing property, as it essentially allows the algorithm to learn under an \emph{arbitrary}
sequence $h^{*}seq$ of target concepts; the difficulty of the task
is then simply reflected in the resulting bounds on the error rates:
that is, faster-changing sequences of target functions result in larger bounds on
the error rates, but do not require a change in the algorithm itself.
\subsection{Adapting to a Changing Drift Rate}
\label{sec:adaptive-varying-rate}
Recall that the method yielding \eqref{eqn:hl94} (based on the work of \cite{helmbold:94})
required access to the sequence $\mathcal Deltaseq$ of changes to achieve the stated guarantee
on the expected number of mistakes. That method is based on choosing a classifier to predict $\hat{Y}_{t}$
by minimizing the number of mistakes among the previous $m_{t}$ samples, where $m_{t}$ is a value
chosen based on the $\mathcal Deltaseq$ sequence. Thus, the key to modifying this algorithm to make it
adaptive to the $\mathcal Deltaseq$ sequence is to determine a suitable choice of $m_{t}$ without reference
to the $\mathcal Deltaseq$ sequence. The strategy we adopt here is to use the \emph{data} to determine
an appropriate value $\hat{m}_{t}$ to use. Roughly (ignoring logarithmic factors for now), the insight
that enables us to achieve this feat is that,
for the $m_{t}$ used in the above strategy, one can show that $\sum_{i=t-m_{t}}^{t-1} \mathbbm{1}[ h^{*}_{t}(X_{i}) \neq Y_{i} ]$
is roughly $\tilde{O}(d)$, and that
making the prediction $\hat{Y}_{t}$ with \emph{any} $h \in \mathbb C$ with roughly $\tilde{O}(d)$ mistakes
on these samples will suffice to obtain the stated bound on the error rate (up to logarithmic factors).
Thus, if we replace $m_{t}$ with the largest value $m$ for which $\min_{h \in \mathbb C} \sum_{i=t-m}^{t-1} \mathbbm{1}[ h(X_{i}) \neq Y_{i}]$
is roughly $\tilde{O}(d)$, then the above observation implies $m \geq m_{t}$. This then
implies that, for $\hat{h} = \mathop{\rm argmin}_{h \in \mathbb C} \sum_{i=t-m}^{t-1} \mathbbm{1}[ h(X_{i}) \neq Y_{i} ]$,
we have that $\sum_{i=t-m_{t}}^{t-1} \mathbbm{1}[ \hat{h}(X_{i}) \neq Y_{i} ]$ is also roughly $\tilde{O}(d)$,
so that the stated bound on the error rate will be achieved (aside from logarithmic factors)
by choosing $\hat{h}_{t}$ as this classifier $\hat{h}$.
There are a few technical modifications to this argument needed to get the logarithmic factors to work out properly,
and for this reason the actual algorithm and proof below are somewhat more involved.
Specifically, consider the following algorithm (the value of the universal constant $K \geq 1$ will be specified below).
\begin{bigboxit}
0. For $T = 1,2,\ldots$\\
1. \quad Let $\hat{m}_{T} \!=\! \max\!\left\{ m \!\in\! \{1,\ldots,T\!-\!1\} : \min\limits_{h \in \mathbb C} \max\limits_{m^{\prime} \leq m} \frac{\sum_{t=T-m^{\prime}}^{T-1} \mathbbm{1}[h(X_{t}) \neq Y_{t}]}{d {\rm Log}(m^{\prime}/d) + {\rm Log}(1/\delta)} < K \right\}$\\
2. \quad Let $\hat{h}_{T} = \mathop{\rm argmin}\limits_{h \in \mathbb C} \max\limits_{m^{\prime} \leq \hat{m}_{T}} \frac{\sum_{t=T-m^{\prime}}^{T-1} \mathbbm{1}[h(X_{t}) \neq Y_{t}]}{d {\rm Log}(m^{\prime}/d) + {\rm Log}(1/\delta)}$
\end{bigboxit}
Note that the classifiers $\hat{h}_{t}$ chosen by this algorithm have no dependence on $\mathcal Deltaseq$,
or indeed anything other than the data $\{(X_{i},Y_{i}) : i < t\}$, and the concept space $\mathbb C$.
\begin{theorem}
\label{thm:epst-adaptive}
Fix any $\delta \in (0,1)$, and let $\mathcal A$ be the above algorithm.
For any sequence $\mathcal Deltaseq$ in $[0,1]$, for any $\mathcal{P}$ and any choice of $h^{*}seq \in S_{\mathcal Deltaseq}$,
for every $T \in \mathbb{N} \setminus \{1\}$, with probability at least $1-\delta$,
\begin{equation*}
{\rm er}_{T}\left( \hat{h}_{T} \right)
\leq O\left( \min_{1 \leq m \leq T-1} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m} \right).
\end{equation*}
\end{theorem}
Before presenting the proof of this result, we first state a crucial lemma, which follows immediately
from a classic result of \cite{vapnik:82,vapnik:98}, combined with the fact (from \cite{vidyasagar:03}, Theorem 4.5)
that the VC dimension of the collection of sets $\{ \{x : h(x) \neq g(x)\} : h,g \in \mathbb C \}$ is at most $10 d$.
\begin{lemma}
\label{lem:vc-ratio}
There exists a universal constant $c \in [1,\infty)$ such that,
for any class $\mathbb C$ of VC dimension $d$, $\forall m \in \mathbb{N}$, $\forall \delta \in (0,1)$,
with probability at least $1-\delta$,
every $h,g \in \mathbb C$ have
\begin{multline*}
\left| \mathcal{P}(x : h(x) \neq g(x)) - \frac{1}{m}\sum_{t=1}^{m} \mathbbm{1}[h(X_{t}) \neq g(X_{t})] \right|
\\ \leq c \sqrt{ \left(\frac{1}{m}\sum_{t=1}^{m} \mathbbm{1}[h(X_{t}) \neq g(X_{t})] \right) \frac{d {\rm Log}(m/d)+{\rm Log}(1/\delta)}{m}}
\\ + c \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m}.
\end{multline*}
\end{lemma}
We are now ready for the proof of Theorem~\ref{thm:epst-adaptive}.
For the constant $K$ in the algorithm, we will choose $K = 145 c^{2}$,
for $c$ as in Lemma~\ref{lem:vc-ratio}.
\begin{proof}[Proof of Theorem~\ref{thm:epst-adaptive}]
Fix any $T \in \mathbb{N}$ with $T \geq 2$, and define
\begin{multline*}
m_{T}^{*} = \max\left\{ m \in \{1,\ldots,T-1\} : \forall m^{\prime} \leq m, \phantom{\sum_{t=T-m^{\prime}}^{T-1}} \right.
\\ \left. \sum_{t=T-m^{\prime}}^{T-1} \mathbbm{1}[h^{*}_{T}(X_{t}) \neq Y_{t}] < K ( d {\rm Log}(m^{\prime}/d) + {\rm Log}(1/\delta) )\right\}.
\end{multline*}
Note that
\begin{equation}
\label{eqn:adaptive-target-mistakes}
\sum_{t=T-m_{T}^{*}}^{T-1} \mathbbm{1}[h^{*}_{T}(X_{t}) \neq Y_{t}] \leq K (d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)),
\end{equation}
and also note that (since $h^{*}_{T} \in \mathbb C$) $\hat{m}_{T} \geq m_{T}^{*}$, so that (by definition of $\hat{m}_{T}$ and $\hat{h}_{T}$)
\begin{equation*}
\sum_{t=T-m_{T}^{*}}^{T-1} \mathbbm{1}[\hat{h}_{T}(X_{t}) \neq Y_{t}] \leq K ( d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta) )
\end{equation*}
as well.
Therefore,
\begin{align*}
\sum_{t=T-m_{T}^{*}}^{T-1} \!\!\mathbbm{1}[h^{*}_{T}(X_{t}) \neq \hat{h}_{T}(X_{t})]
& \leq
\sum_{t=T-m_{T}^{*}}^{T-1} \!\!\mathbbm{1}[h^{*}_{T}(X_{t}) \neq Y_{t}]
+
\sum_{t=T-m_{T}^{*}}^{T-1} \!\!\mathbbm{1}[Y_{t} \neq \hat{h}_{T}(X_{t})]
\\ & \leq
2 K ( d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta) ).
\end{align*}
Thus, by Lemma~\ref{lem:vc-ratio}, for each $m \in \mathbb{N}$,
with probability at least $1-\delta / (6 m^{2})$, if $m_{T}^{*} = m$, then
\begin{equation*}
\mathcal{P}(x : \hat{h}_{T}(x) \neq h^{*}_{T}(x))
\leq
(2K+c \sqrt{2K} + c) \frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(6(m_{T}^{*})^{2}/\delta)}{m_{T}^{*}}.
\end{equation*}
Furthermore, since
${\rm Log}(6(m_{T}^{*})^{2}) \leq \sqrt{2K} d {\rm Log}(m_{T}^{*} / d)$,
this is at most
\begin{equation*}
2(K+c \sqrt{2K}) \frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)}{m_{T}^{*}}.
\end{equation*}
By a union bound (over values $m \in \mathbb{N}$), we have that with probability at least $1-\sum_{m=1}^{\infty} \delta/(6 m^{2}) \geq 1 - \delta/3$,
\begin{equation*}
\mathcal{P}(x : \hat{h}_{T}(x) \neq h^{*}_{T}(x))
\leq 2(K+c \sqrt{2K}) \frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)}{m_{T}^{*}}.
\end{equation*}
Let us denote
\begin{equation*}
\tilde{m}_{T} = \mathop{\rm argmin}_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m}.
\end{equation*}
Note that, for any $m^{\prime} \in \{1,\ldots,T-1\}$ and $\delta \in (0,1)$,
if $\tilde{m}_{T} \geq m^{\prime}$, then
\begin{align*}
& \min_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m}
\\ & \geq \min_{m \in \{m^{\prime},\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j}
= \frac{1}{m^{\prime}} \sum_{i=T-m^{\prime}}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j},
\end{align*}
while if $\tilde{m}_{T} \leq m^{\prime}$, then
\begin{align*}
& \min_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m}
\\ & \geq \min_{m \in \{1,\ldots,m^{\prime}\}} \frac{d {\rm Log}(m/d)+{\rm Log}(1/\delta)}{m}
= \frac{d {\rm Log}(m^{\prime}/d) + {\rm Log}(1/\delta)}{m^{\prime}}.
\end{align*}
Either way, we have that
\begin{align}
& \min_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d) + {\rm Log}(1/\delta)}{m} \notag
\\ & \geq \min\left\{ \frac{d {\rm Log}(m^{\prime}/d) + {\rm Log}(1/\delta)}{m^{\prime}}, \frac{1}{m^{\prime}} \sum_{i=T-m^{\prime}}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} \right\}. \label{eqn:adaptive-min-lb}
\end{align}
For any $m \in \{1,\ldots,T-1\}$,
applying Bernstein's inequality (see \cite{boucheron:13}, equation 2.10) to the random variables $\mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ]/d$, $i \in \{T-m,\ldots,T-1\}$,
and again to the random variables $-\mathbbm{1}[h^{*}_{T}(X_{i}) \neq Y_{i}]/d$, $i \in \{T-m,\ldots,T-1\}$, together with a union bound,
we obtain that, for any $\delta \in (0,1)$, with probability at least $1 - \delta / (3m^{2})$,
\begin{align}
& \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) ) \notag
\\ & {\hskip 1cm}- \sqrt{ \left( \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) ) \right) \frac{2\ln(3m^{2}/\delta)}{m} } \notag
\\ & < \frac{1}{m} \sum_{i=T-m}^{T-1} \mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ] \notag
\\ & < \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) ) \notag
\\ & {\hskip 1cm}+ \max\begin{cases}
\sqrt{ \left( \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) ) \right) \frac{4\ln(3m^{2}/\delta)}{m} }
\\\frac{(4/3)\ln(3m^{2}/\delta)}{m} \end{cases}.\label{eqn:adaptive-empirical-ub}
\end{align}
The left inequality implies that
\begin{equation*}
\frac{1}{m} \!\sum_{i=T-m}^{T-1}\!\!\! \mathcal{P}( x \!:\! h^{*}_{T}(x) \neq h^{*}_{i}(x) )
\leq \max\left\{ \frac{2}{m} \!\sum_{i=T-m}^{T-1} \!\!\!\mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ], \frac{8\ln(3m^{2}/\delta)}{m} \right\}\!.
\end{equation*}
Plugging this into the right inequality in \eqref{eqn:adaptive-empirical-ub}, we obtain that
\begin{multline*}
\frac{1}{m} \sum_{i=T-m}^{T-1} \mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ]
< \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) )
\\ + \max\left\{ \sqrt{ \left(\frac{1}{m} \sum_{i=T-m}^{T-1} \mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ] \right) \frac{8\ln(3m^{2}/\delta)}{m} }, \frac{\sqrt{32}\ln(3m^{2}/\delta)}{m} \right\}.
\end{multline*}
By a union bound, this holds simultaneously for all $m \in \{1,\ldots,T-1\}$ with probability at least $1-\sum_{m = 1}^{T-1} \delta / (3m^{2}) > 1 - (2/3)\delta$.
Note that, on this event,
we obtain
\begin{multline*}
\frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) )
>
\frac{1}{m} \sum_{i=T-m}^{T-1} \mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ]
\\ - \max\left\{ \sqrt{ \left(\frac{1}{m} \sum_{i=T-m}^{T-1} \mathbbm{1}[ h^{*}_{T}(X_{i}) \neq Y_{i} ] \right) \frac{8\ln(3m^{2}/\delta)}{m} }, \frac{\sqrt{32}\ln(3m^{2}/\delta)}{m} \right\}.
\end{multline*}
In particular, taking $m = m_{T}^{*}$, and invoking maximality of $m_{T}^{*}$, if $m_{T}^{*} < T-1$, the right hand side is at least
\begin{equation*}
(K - 6c\sqrt{K}) \frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)}{m_{T}^{*}}.
\end{equation*}
Since $\frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} \geq \frac{1}{m} \sum_{i=T-m}^{T-1} \mathcal{P}( x : h^{*}_{T}(x) \neq h^{*}_{i}(x) )$,
taking $K = 145 c^{2}$,
we have that with probability at least $1-\delta$, if $m_{T}^{*} < T-1$, then
\begin{align*}
& 10(K+c \sqrt{2K})\min_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d)+{\rm Log}(1/\delta)}{m}
\\ & \geq
10(K+c \sqrt{2K})\min\left\{ \frac{d {\rm Log}(m_{T}^{*}/d)+{\rm Log}(1/\delta)}{m_{T}^{*}}, \frac{1}{m_{T}^{*}} \sum_{i=T-m_{T}^{*}}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} \right\}
\\ & \geq
10(K+c \sqrt{2K})\frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)}{m_{T}^{*}}
\\ & \geq \mathcal{P}(x : \hat{h}_{T}(x) \neq h^{*}_{T}(x)).
\end{align*}
Furthermore, if $m_{T}^{*} = T-1$, then we trivially have (on the same $1-\delta$ probability event as above)
\begin{align*}
& 10(K+c \sqrt{2K})\min_{m \in \{1,\ldots,T-1\}} \frac{1}{m} \sum_{i=T-m}^{T-1} \sum_{j=i+1}^{T} \mathcal Delta_{j} + \frac{d {\rm Log}(m/d)+{\rm Log}(1/\delta)}{m}
\\ & \geq 10(K+c \sqrt{2K}) \min_{m \in \{1,\ldots,T-1\}} \frac{d {\rm Log}(m/d)+{\rm Log}(1/\delta)}{m}
\\ & = 10(K+c \sqrt{2K}) \frac{d {\rm Log}((T-1)/d)+{\rm Log}(1/\delta)}{T-1}
\\ & = 10(K+c \sqrt{2K})\frac{d {\rm Log}(m_{T}^{*}/d) + {\rm Log}(1/\delta)}{m_{T}^{*}}
\geq \mathcal{P}(x : \hat{h}_{T}(x) \neq h^{*}_{T}(x)).
\end{align*}
\qed
\end{proof}
\subsection{Conditions Guaranteeing a Sublinear Number of Mistakes}
\label{sec:sublinear}
\input{tex-files/sublinear.tex}
\section{Polynomial-Time Algorithms for Linear Separators}
\label{sec:halfspaces}
In this section, we suppose $\mathcal Delta_{t} = \mathcal Delta$ for every $t \in \mathbb{N}$, for a fixed constant $\mathcal Delta > 0$,
and we consider the special case of learning homogeneous linear separators in $\mathbb{R}^{k}$ under a uniform distribution
on the origin-centered unit sphere.
In this case, the analysis of \cite{helmbold:94} mentioned in Section~\ref{sec:classic-consistency} implies that it is possible to achieve a
bound on the error rate that is $\tilde{O}(d \sqrt{\mathcal Delta})$,
using an algorithm that runs in time ${\rm poly}(d,1/\mathcal Delta,\log(1/\delta))$ (and independent of $t$) for each prediction.
This also implies that it is possible to achieve expected number of mistakes among $T$ predictions that is $\tilde{O}(d \sqrt{\mathcal Delta}) \times T$.
\cite{min_concept}\footnote{This
work in fact studies a much broader model of drift, which in fact allows the distribution $\mathcal{P}$ to vary with time as well. However, this $\tilde{O}((d \mathcal Delta)^{1/4}) \times T$ result can be obtained
from their more-general theorem by calculating the various parameters for this particular setting.}
have since proven that a variant of the Perceptron algorithm is capable of achieving an expected number of mistakes $\tilde{O}( (d \mathcal Delta)^{1/4} ) \times T$.
Below, we improve on this result by showing that there exists an efficient algorithm that achieves a
bound on the error rate that is $\tilde{O}(\sqrt{d \mathcal Delta})$,
as was possible with the inefficient algorithm of \cite{helmbold:94,long:99} mentioned in Section~\ref{sec:classic-constant-drift}.
This leads to a bound on the expected number of mistakes that is $\tilde{O}(\sqrt{d \mathcal Delta}) \times T$.
Furthermore, our approach also allows us to present the method as an \emph{active learning}
algorithm, and to bound the expected number of queries, as a function of the
number of samples $T$, by $\tilde{O}(\sqrt{d \mathcal Delta}) \times T$.
The technique is based on a modification of the algorithm of \cite{helmbold:94},
replacing an empirical risk minimization step with (a modification of) the computationally-efficient algorithm of \cite{awasthi:13}.
Formally, define the class of homogeneous linear separators as the set of classifiers
$h_{w} : \mathbb{R}^{d} \to \{-1,+1\}$, for $w \in \mathbb{R}^{d}$ with $\|w\|=1$,
such that $h_{w}(x) = {\rm sign}( w \cdot x )$ for every $x \in \mathbb{R}^{d}$.
\subsection{An Improved Guarantee for a Polynomial-Time Algorithm}
\label{sec:efficient-linsep}
We have the following result.
\begin{theorem}
\label{thm:linsep-uniform}
When $\mathbb C$ is the space of homogeneous linear separators (with $d \geq 4$)
and $\mathcal{P}$ is the uniform distribution on the surface of
the origin-centered unit sphere in $\mathbb{R}^{d}$,
for any fixed $\mathcal Delta > 0$,
for any $\delta \in (0,1/e)$,
there is an algorithm that runs in time ${\rm poly}(d,1/\mathcal Delta,\log(1/\delta))$ for each time $t$,
such that for any $h^{*}seq \in S_{\mathcal Delta}$,
for every sufficiently large $t \in \mathbb{N}$, with probability at least $1-\delta$,
\begin{equation*}
{\rm er}_{t}(\hat{h}_{t}) = O\left( \sqrt{\mathcal Delta d \log\left(\frac{1}{\delta}\right) } \right).
\end{equation*}
Also, running this algorithm with $\delta = \sqrt{\mathcal Delta d} \land 1/e$,
the expected number of mistakes among the first $T$ instances is
$O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\mathcal Delta d}\right) } T \right)$.
Furthermore, the algorithm can be run as an \emph{active learning} algorithm,
in which case, for this choice of $\delta$, the expected number of labels
requested by the algorithm among the first $T$ instances is
$O\left( \sqrt{\mathcal Delta d} \log^{3/2}\left(\frac{1}{\mathcal Delta d}\right) T \right)$.
\end{theorem}
We first state the algorithm used to obtain this result. It is primarily based on a
margin-based learning strategy of \cite{awasthi:13}, combined with an initialization
step based on a modified Perceptron rule from \cite{stream_perceptron,min_concept}.
For $\tau > 0$ and $x \in \mathbb{R}$, define $\ell_{\tau}(x) = \max\left\{0, 1 - \frac{x}{\tau}\right\}$.
Consider the following algorithm and subroutine;
parameters $\delta_{k}$, $m_{k}$, $\tau_{k}$, $r_{k}$, $b_{k}$, $\alpha$, and $\kappa$
will all be specified in the context of the proof; we suppose $M = \sum_{k=0}^{\lceil \log_{2}(1/\alpha) \rceil} m_{k}$.
\begin{bigboxit}
Algorithm: DriftingHalfspaces\\
0. Let $\tilde{h}_{0}$ be an arbitrary classifier in $\mathbb C$\\
1. For $i = 1,2,\ldots$\\
2. \quad $\tilde{h}_{i} \gets {\rm ABL}(M (i-1), \tilde{h}_{i-1})$\\
\end{bigboxit}
\begin{bigboxit}
Subroutine: ${\rm ModPerceptron}(t,\tilde{h})$\\
0. Let $w_{t}$ be any element of $\mathbb{R}^{d}$ with $\|w_{t}\| = 1$\\
1. For $m = t+1,t+2,\ldots,t+m_{0}$\\
2. \quad Choose $\hat{h}_{m} = \tilde{h}$ (i.e., predict $\hat{Y}_{m} = \tilde{h}(X_{m})$ as the prediction for $Y_{m}$)\\
3. \quad Request the label $Y_{m}$\\
4. \quad If $h_{w_{m-1}}(X_{m}) \neq Y_{m}$\\
5. \qquad $w_{m} \gets w_{m-1} - 2(w_{m-1} \cdot X_{m}) X_{m}$\\
6. \quad Else $w_{m} \gets w_{m-1}$\\
7. Return $w_{t+m_{0}}$
\end{bigboxit}
\begin{bigboxit}
Subroutine: ${\rm ABL}(t,\tilde{h})$\\
0. Let $w_{0}$ be the return value of ${\rm ModPerceptron}(t,\tilde{h})$\\
1. For $k = 1,2,\ldots,\lceil \log_{2}(1/\alpha) \rceil$\\
2. \quad $W_{k} \gets \{\}$\\
3. \quad For $s = t + \sum_{j=0}^{k-1} m_{j} + 1, \ldots, t + \sum_{j=0}^{k} m_{j}$\\
4. \qquad Choose $\hat{h}_{s} = \tilde{h}$ (i.e., predict $\hat{Y}_{s} = \tilde{h}(X_{s})$ as the prediction for $Y_{s}$)\\
5. \qquad If $|w_{k-1} \cdot X_{s}| \leq b_{k-1}$, Request label $Y_{s}$ and let $W_{k} \gets W_{k} \cup \{(X_{s},Y_{s})\}$\\
6. \quad Find $v_{k} \in \mathbb{R}^{d}$ with $\|v_{k} - w_{k-1}\| \leq r_{k}$, $0 < \|v_{k}\| \leq 1$,
and\\ {\hskip 7mm}$\sum\limits_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (v_{k} \cdot x)) \leq \inf\limits_{v : \|v-w_{k-1}\| \leq r_{k}} \sum\limits_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (v \cdot x)) + \kappa |W_{k}|$\\
7. \quad Let $w_{k} = \frac{1}{\|v_{k}\|} v_{k}$\\
8. Return $h_{w_{\lceil \log_{2}(1/\alpha) \rceil-1}}$
\end{bigboxit}
Before stating the proof, we have a few additional lemmas that will be needed.
The following result for ${\rm ModPerceptron}$ was proven by \cite{min_concept}.
\begin{lemma}
\label{lem:perceptron}
Suppose $\mathcal Delta < \frac{1}{512}$.
Consider the values $w_{m}$ obtained during the execution of ${\rm ModPerceptron}(t,\tilde{h})$.
$\forall m \in \{t+1,\ldots, t+ m_{0}\}$, $\mathcal{P}(x : h_{w_{m}}(x) \neq h_{m}^{*}(x)) \leq \mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x))$.
Furthermore, letting $c_{1} = \frac{\pi^{2}}{d \cdot 400 \cdot 2^{15}}$, if
$\mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) \geq 1/32$,
then with probability at least $1/64$,
$\mathcal{P}(x : h_{w_{m}}(x) \neq h_{m}^{*}(x)) \leq (1 - c_{1}) \mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x))$.
\end{lemma}
This implies the following.
\begin{lemma}
\label{lem:perceptron-init}
Suppose $\mathcal Delta \leq \frac{\pi^{2}}{400 \cdot 2^{27} (d+\ln(4/\delta))}$.
For $m_{0} = \max\{\lceil 128 (1/c_{1}) \ln(32) \rceil,$ $\lceil 512 \ln(\frac{4}{\delta}) \rceil \}$,
with probability at least $1-\delta/4$,
${\rm ModPerceptron}(t,\tilde{h})$ returns a vector $w$ with
$\mathcal{P}(x : h_{w}(x) \neq h_{t+m_{0}+1}^{*}(x)) \leq 1/16$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:perceptron} and a union bound, in general we have
\begin{equation}
\label{eqn:perceptron-weak-update}
\mathcal{P}(x : h_{w_{m}}(x) \neq h_{m+1}^{*}(x)) \leq \mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) + \mathcal Delta.
\end{equation}
Furthermore, if $\mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) \geq 1/32$,
then wth probability at least $1/64$,
\begin{equation}
\label{eqn:perceptron-strong-update}
\mathcal{P}(x : h_{w_{m}}(x) \neq h_{m+1}^{*}(x)) \leq (1-c_{1}) \mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) + \mathcal Delta.
\end{equation}
In particular, this implies that the number $N$ of values $m \in \{t+1,\ldots,t+m_{0}\}$ with either
$\mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) < 1/32$ or $\mathcal{P}(x : h_{w_{m}}(x) \neq h_{m+1}^{*}(x)) \leq (1-c_{1}) \mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) + \mathcal Delta$
is lower-bounded by a ${\rm Binomial}(m,1/64)$ random variable.
Thus, a Chernoff bound implies that with probability at least $1 - \exp\{ - m_{0} / 512 \} \geq 1 - \delta/4$,
we have $N \geq m_{0} / 128$. Suppose this happens.
Since $\mathcal Delta m_{0} \leq 1/32$, if any $m \in \{t+1,\ldots,t+m_{0}\}$ has $\mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) < 1/32$,
then inductively applying \eqref{eqn:perceptron-weak-update} implies that
$\mathcal{P}(x : h_{w_{t+m_{0}}}(x) \neq h_{t+m_{0}+1}^{*}(x)) \leq 1/32 + \mathcal Delta m_{0} \leq 1/16$.
On the other hand, if all $m \in \{t+1,\ldots,t+m_{0}\}$ have $\mathcal{P}(x : h_{w_{m-1}}(x) \neq h_{m}^{*}(x)) \geq 1/32$,
then in particular we have $N$ values of $m \in \{t+1,\ldots,t+m_{0}\}$ satisfying \eqref{eqn:perceptron-strong-update}.
Combining this fact with \eqref{eqn:perceptron-weak-update} inductively, we have that
\begin{multline*}
\mathcal{P}(x : h_{w_{t+m_{0}}}(x) \neq h_{t+m_{0}+1}^{*}(x))
\leq (1-c_{1})^{N} \mathcal{P}(x : h_{w_{t}}(x) \neq h_{t+1}^{*}(x)) + \mathcal Delta m_{0}
\\ \leq (1-c_{1})^{(1/c_{1}) \ln(32) } \mathcal{P}(x : h_{w_{t}}(x) \neq h_{t+1}^{*}(x)) + \mathcal Delta m_{0}
\leq \frac{1}{32} + \mathcal Delta m_{0}
\leq \frac{1}{16}.
\end{multline*}
\qed
\end{proof}
Next, we consider the execution of ${\rm ABL}(t,\tilde{h})$, and let the sets $W_{k}$ be as in that execution.
We will denote by $w^{*}$ the weight vector with $\|w^{*}\|=1$ such that $h_{t+m_{0}+1}^{*} = h_{w^{*}}$.
Also denote by $M_{1} = M-m_{0}$.
The proof relies on a few results proven in the work of \cite{awasthi:13}, which we summarize in the following lemmas.
Although the results were proven in a slightly different setting in that work (namely, agnostic learning under a fixed joint distribution),
one can easily verify that their proofs remain valid in our present context as well.
\begin{lemma}
\label{lem:denoised-risk}
\cite{awasthi:13}
Fix any $k \in \{1,\ldots,\lceil \log_{2}(1/\alpha) \rceil\}$.
For a universal constant $c_{7} > 0$, suppose $b_{k-1} = c_{7} 2^{1-k} / \sqrt{d}$,
and let $z_{k} = \sqrt{r_{k}^{2}/(d-1) + b_{k-1}^{2}}$.
For a universal constant $c_{1} > 0$, if $\|w^{*} - w_{k-1}\| \leq r_{k}$,
\begin{multline*}
{\hskip -3mm}\left| \mathbb E\!\left[ \sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}(|w^{*} \cdot x|) \Big| w_{k-1}, |W_{k}| \right]
- \mathbb E\!\left[ \sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (w^{*} \cdot x)) \Big| w_{k-1}, |W_{k}| \right] \right|
\\ \leq c_{1} |W_{k}| \sqrt{2^{k} \mathcal Delta M_{1}} \frac{z_{k}}{\tau_{k}}.
\end{multline*}
\end{lemma}
\begin{lemma}
\label{lem:margin-error-concentration}
\cite{balcan:13}
For any $c > 0$, there is a constant $c^{\prime} > 0$ depending only on $c$ (i.e., not depending on $d$)
such that, for any $u,v \in \mathbb{R}^{d}$ with $\|u\|=\|v\|=1$, letting $\sigma = \mathcal{P}(x : h_{u}(x) \neq h_{v}(x))$,
if $\sigma < 1/2$, then
\begin{equation*}
\mathcal{P}\left( x : h_{u}(x) \neq h_{v}(x) \text{ and } |v \cdot x| \geq c^{\prime} \frac{\sigma}{\sqrt{d}} \right) \leq c \sigma.
\end{equation*}
\end{lemma}
The following is a well-known lemma concerning concentration around the equator for the uniform distribution (see e.g., \cite{stream_perceptron,balcan:07,awasthi:13});
for instance, it easily follows from the formulas for the area in a spherical cap derived by \cite{li:11}.
\begin{lemma}
\label{lem:uniform-P-concentration}
For any constant $C > 0$, there are constants $c_{2},c_{3} > 0$ depending only on $C$ (i.e., independent of $d$) such that,
for any $w \in \mathbb{R}^{d}$ with $\|w\|=1$, $\forall \gamma \in [0, C/\sqrt{d}]$,
\begin{equation*}
c_{2} \gamma \sqrt{d} \leq \mathcal{P}\left( x : |w \cdot x| \leq \gamma \right) \leq c_{3} \gamma \sqrt{d}.
\end{equation*}
\end{lemma}
Based on this lemma, \cite{awasthi:13} prove the following.
\begin{lemma}
\label{lem:opt-margin-loss}
\cite{awasthi:13}
For $X \sim \mathcal{P}$, for any $w \in \mathbb{R}^{d}$ with $\|w\|=1$, for any $C > 0$ and $\tau, b \in [0,C/\sqrt{d}]$,
for $c_{2},c_{3}$ as in Lemma~\ref{lem:uniform-P-concentration},
\begin{equation*}
\mathbb E\left[ \ell_{\tau}( |w^{*} \cdot X| ) \Big| |w \cdot X| \leq b \right] \leq \frac{c_{3} \tau}{c_{2} b}.
\end{equation*}
\end{lemma}
The following is a slightly stronger version of a result of \cite{awasthi:13} (specifically,
the size of $m_{k}$, and consequently the bound on $|W_{k}|$, are both improved by a factor of $d$
compared to the original result).
\begin{lemma}
\label{lem:margin-error-bound}
Fix any $\delta \in (0,1/e)$.
For universal constants $c_{4},c_{5},c_{6},c_{7},c_{8},c_{9},c_{10} \in (0,\infty)$,
for an appropriate choice of $\kappa \in (0,1)$ (a universal constant),
if $\alpha = c_{9} \sqrt{\mathcal Delta d \log\left(\frac{1}{\kappa\delta}\right)}$,
for every $k \in \{1,\ldots,\lceil \log_{2}(1/\alpha) \rceil\}$,
if $b_{k-1} = c_{7} 2^{1-k} / \sqrt{d}$, $\tau_{k} = c_{8} 2^{-k} / \sqrt{d}$, $r_{k} = c_{10} 2^{-k}$, $\delta_{k} = \delta / (\lceil \log_{2}(4/\alpha) \rceil - k)^{2}$,
and $m_{k} = \left\lceil c_{5} \frac{2^{k}}{\kappa^{2}} d \log\left(\frac{1}{\kappa\delta_{k}} \right)\right\rceil$,
and if $\mathcal{P}(x : h_{w_{k-1}}(x) \neq h_{w^{*}}(x)) \leq 2^{-k-3}$,
then with probability at least $1-(4/3)\delta_{k}$,
$|W_{k}| \leq c_{6} \frac{1}{\kappa^{2}} d \log\left(\frac{1}{\kappa\delta_{k}}\right)$
and
$\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w^{*}}(x)) \leq 2^{-k-4}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:uniform-P-concentration}, and a Chernoff and union bound,
for an appropriately large choice of $c_{5}$ and any $c_{7} > 0$,
letting $c_{2},c_{3}$ be as in Lemma~\ref{lem:uniform-P-concentration} (with $C=c_{7} \lor (c_{8}/2)$),
with probability at least $1-\delta_{k}/3$,
\begin{equation}
\label{eqn:Wk-bounds}
c_{2} c_{7} 2^{-k} m_{k}
\leq |W_{k}| \leq
4 c_{3} c_{7} 2^{-k} m_{k}.
\end{equation}
The claimed upper bound on $|W_{k}|$ follows from this second inequality.
Next note that, if $\mathcal{P}(x : h_{w_{k-1}}(x) \neq h_{w^{*}}(x)) \leq 2^{-k-3}$,
then
\begin{equation*}
\max\{ \ell_{\tau_{k}}(y (w^{*} \cdot x)) : x \in \mathbb{R}^{d}, |w_{k-1} \cdot x| \leq b_{k-1}, y \in \{-1,+1\} \} \leq c_{11} \sqrt{d}
\end{equation*}
for some universal constant $c_{11} > 0$.
Furthermore, since $\mathcal{P}(x : h_{w_{k-1}}(x) \neq h_{w^{*}}(x)) \leq 2^{-k-3}$,
we know that the angle between $w_{k-1}$ and $w^{*}$ is at most $2^{-k-3} \pi$,
so that
\begin{multline*}
\|w_{k-1} - w^{*}\|
= \sqrt{ 2 - 2 w_{k-1} \cdot w^{*} }
\leq \sqrt{ 2 - 2 \cos(2^{-k-3} \pi) }
\\ \leq \sqrt{ 2 - 2 \cos^{2}(2^{-k-3} \pi) }
= \sqrt{2} \sin(2^{-k-3} \pi) \leq 2^{-k-3} \pi \sqrt{2}.
\end{multline*}
For $c_{10} = \pi\sqrt{2} 2^{-3}$, this is $r_{k}$.
By Hoeffding's inequality (under the conditional distribution given $|W_{k}|$), the law of total probability,
Lemma~\ref{lem:denoised-risk}, and linearity of conditional expectations,
with probability at least $1-\delta_{k}/3$, for $X \sim \mathcal{P}$,
\begin{multline}
\label{eqn:opt-loss-bound}
\sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}( y ( w^{*} \cdot x) )
\leq |W_{k}| \mathbb E\left[ \ell_{\tau_{k}}(|w^{*} \cdot X|) \Big| w_{k-1}, |w_{k-1} \cdot X| \leq b_{k-1} \right]
\\ + c_{1} |W_{k}| \sqrt{2^{k} \mathcal Delta M_{1}} \frac{z_{k}}{\tau_{k}}
+ \sqrt{ |W_{k}| (1/2) c_{11}^{2} d \ln(3/\delta_{k}) }.
\end{multline}
We bound each term on the right hand side separately.
By Lemma~\ref{lem:opt-margin-loss}, the first term is at most $|W_{k}|\frac{c_{3} \tau_{k}}{c_{2} b_{k-1}} = |W_{k}|\frac{c_{3} c_{8}}{2 c_{2} c_{7}}$.
Next,
\begin{equation*}
\frac{z_{k}}{\tau_{k}}
= \frac{\sqrt{c_{10}^{2} 2^{-2k}/(d-1) + 4 c_{7}^{2} 2^{-2k}/d}}{c_{8} 2^{-k} / \sqrt{d}}
\leq \frac{\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}}{c_{8}},
\end{equation*}
while $2^{k} \leq 2/\alpha$
so that the second term is at most
\begin{equation*}
\sqrt{2} c_{1} \frac{\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}}{c_{8}} |W_{k}| \sqrt{ \frac{\mathcal Delta m}{\alpha} }.
\end{equation*}
Noting that
\begin{equation}
\label{eqn:m-bound}
M_{1} = \sum_{k^{\prime}=1}^{\lceil \log_{2}(1/\alpha) \rceil} m_{k^{\prime}}
\leq \frac{32 c_{5}}{\kappa^{2}} \frac{1}{\alpha} d \log\left(\frac{1}{\kappa\delta}\right),
\end{equation}
we find that the second term on the right hand side of \eqref{eqn:opt-loss-bound} is at most
\begin{equation*}
\sqrt{\frac{c_{5}}{c_{9}}} \frac{8 c_{1}}{\kappa} \frac{\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}}{c_{8}} |W_{k}| \sqrt{ \frac{\mathcal Delta d \log\left(\frac{1}{\kappa\delta}\right)}{\alpha^{2}} }
= \frac{8 c_{1} \sqrt{c_{5}}}{\kappa} \frac{\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}}{c_{8}c_{9}} |W_{k}|.
\end{equation*}
Finally, since $d \ln(3/\delta_{k}) \leq 2 d \ln(1/\delta_{k}) \leq \frac{2 \kappa^{2}}{c_{5}} 2^{-k} m_{k}$,
and \eqref{eqn:Wk-bounds} implies $2^{-k} m_{k} \leq \frac{1}{c_{2} c_{7}} |W_{k}|$,
the third term on the right hand side of \eqref{eqn:opt-loss-bound} is at most
\begin{equation*}
|W_{k}| \frac{c_{11} \kappa}{ \sqrt{c_{2} c_{5} c_{7}} }.
\end{equation*}
Altogether, we have
\begin{equation*}
\sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}( y ( w^{*} \cdot x) )
\leq |W_{k}| \left(
\frac{c_{3} c_{8}}{2 c_{2} c_{7}}
+ \frac{8 c_{1} \sqrt{c_{5}}}{\kappa} \frac{\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}}{c_{8}c_{9}}
+ \frac{c_{11} \kappa}{ \sqrt{c_{2} c_{5} c_{7}} }\right).
\end{equation*}
Taking $c_{9} = 1/\kappa^{3}$ and $c_{8} = \kappa$, this is at most
\begin{equation*}
\kappa |W_{k}| \left(
\frac{c_{3}}{2 c_{2} c_{7}}
+ 8 c_{1} \sqrt{c_{5}}\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}
+ \frac{c_{11}}{ \sqrt{c_{2} c_{5} c_{7}} }\right).
\end{equation*}
Next, note that because $h_{w_{k}}(x) \neq y \Rightarrow \ell_{\tau_{k}}(y (v_{k} \cdot x)) \geq 1$,
and because (as proven above) $\|w^{*} - w_{k-1}\| \leq r_{k}$,
\begin{equation*}
|W_{k}| {\rm er}_{W_{k}}( h_{w_{k}} )
\leq \sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (v_{k} \cdot x))
\leq \sum_{(x,y) \in W_{k}} \ell_{\tau_{k}}(y (w^{*} \cdot x)) + \kappa |W_{k}|.
\end{equation*}
Combined with the above, we have
\begin{equation*}
|W_{k}| {\rm er}_{W_{k}}( h_{w_{k}} )
\leq \kappa |W_{k}| \left(
1 + \frac{c_{3}}{2 c_{2} c_{7}}
+ 8 c_{1} \sqrt{c_{5}}\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}}
+ \frac{c_{11}}{ \sqrt{c_{2} c_{5} c_{7}} }\right).
\end{equation*}
Let $c_{12} = 1 + \frac{c_{3}}{2 c_{2} c_{7}} + 8 c_{1} \sqrt{c_{5}}\sqrt{ 2c_{10}^{2} + 4 c_{7}^{2}} + \frac{c_{11}}{ \sqrt{c_{2} c_{5} c_{7}} }$.
Furthermore,
\begin{multline*}
|W_{k}|{\rm er}_{W_{k}}( h_{w_{k}} )
= \sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq y ]
\\ \geq \sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq h_{w^{*}}(x) ] - \sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w^{*}}(x) \neq y ].
\end{multline*}
For an appropriately large value of $c_{5}$,
by a Chernoff bound, with probability at least $1-\delta_{k}/3$,
\begin{equation*}
\sum_{s=t+\sum_{j=0}^{k-1}m_{j} + 1}^{t+\sum_{j=0}^{k} m_{j}} \mathbbm{1}[ h_{w^{*}}(X_{s}) \neq Y_{s} ]
\leq 2 e \mathcal Delta M_{1} m_{k} + \log_{2}(3/\delta_{k}).
\end{equation*}
In particular, this implies
\begin{equation*}
\sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w^{*}}(x) \neq y ]
\leq 2 e \mathcal Delta M_{1} m_{k} + \log_{2}(3/\delta_{k}),
\end{equation*}
so that
\begin{equation*}
\sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq h_{w^{*}}(x) ]
\leq |W_{k}|{\rm er}_{W_{k}}( h_{w_{k}} ) + 2 e \mathcal Delta M_{1} m_{k} + \log_{2}(3/\delta_{k}).
\end{equation*}
Noting that \eqref{eqn:m-bound} and \eqref{eqn:Wk-bounds} imply
\begin{align*}
\mathcal Delta M_{1} m_{k} & \leq \mathcal Delta \frac{32 c_{5}}{\kappa^{2}} \frac{ d \log\left(\frac{1}{\kappa\delta}\right) }{c_{9} \sqrt{ \mathcal Delta d \log\left(\frac{1}{\kappa\delta}\right)}} \frac{2^{k}}{c_{2} c_{7}} |W_{k}|
\leq \frac{32 c_{5}}{c_{2} c_{7} c_{9} \kappa^{2}} \sqrt{ \mathcal Delta d \log\left(\frac{1}{\kappa\delta}\right) } 2^{k} |W_{k}|
\\ & = \frac{32 c_{5}}{c_{2} c_{7} c_{9}^{2} \kappa^{2}} \alpha 2^{k} |W_{k}|
= \frac{32 c_{5} \kappa^{4}}{c_{2} c_{7}} \alpha 2^{k} |W_{k}|
\leq \frac{32 c_{5} \kappa^{4}}{c_{2} c_{7}} |W_{k}|,
\end{align*}
and \eqref{eqn:Wk-bounds} implies $\log_{2}(3/\delta_{k}) \leq \frac{2\kappa^{2}}{c_{2}c_{5}c_{7}}|W_{k}|$,
altogether we have
\begin{align*}
\sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq h_{w^{*}}(x) ]
& \leq |W_{k}|{\rm er}_{W_{k}}( h_{w_{k}} ) + \frac{64 e c_{5} \kappa^{4}}{c_{2} c_{7}} |W_{k}| + \frac{2\kappa^{2}}{c_{2}c_{5}c_{7}}|W_{k}|
\\ & \leq \kappa |W_{k}| \left( c_{12} + \frac{64 e c_{5} \kappa^{3}}{c_{2} c_{7}} + \frac{2\kappa}{c_{2}c_{5}c_{7}} \right).
\end{align*}
Letting $c_{13} = c_{12} + \frac{64 e c_{5}}{c_{2} c_{7}} + \frac{2}{c_{2}c_{5}c_{7}}$, and noting $\kappa \leq 1$,
we have
$\sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq h_{w^{*}}(x) ] \leq c_{13} \kappa |W_{k}|$.
Lemma~\ref{lem:vc-ratio} (applied under the conditional distribution given $|W_{k}|$)
and the law of total probability imply that with probability at least $1-\delta_{k}/3$,
\begin{align*}
|W_{k}| &\mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \Big| |w_{k-1} \cdot x| \leq b_{k-1}\right)
\\ & \leq \sum_{(x,y) \in W_{k}} \mathbbm{1}[ h_{w_{k}}(x) \neq h_{w^{*}}(x)]
+ c_{14} \sqrt{ |W_{k}| (d \log(|W_{k}|/d) + \log(1/\delta_{k})) },
\end{align*}
for a universal constant $c_{14} > 0$.
Combined with the above, and the fact that \eqref{eqn:Wk-bounds} implies
$\log(1/\delta_{k}) \leq \frac{\kappa^{2}}{c_{2}c_{5}c_{7}}|W_{k}|$
and
\begin{align*}
d \log(|W_{k}|/d) & \leq d \log\left(\frac{8c_{3}c_{5}c_{7} \log\left(\frac{1}{\kappa\delta_{k}}\right)}{\kappa^{2}}\right)
\\ & \leq d \log\left(\frac{8 c_{3} c_{5} c_{7}}{\kappa^{3} \delta_{k}}\right)
\leq 3\log(8 \max\{c_{3},1\} c_{5} ) c_{5} d \log\left(\frac{1}{\kappa \delta_{k}}\right)
\\ & \leq 3 \log(8 \max\{c_{3},1\}) \kappa^{2} 2^{-k} m_{k}
\leq \frac{3 \log(8 \max\{c_{3},1\})}{c_{2} c_{7}} \kappa^{2} |W_{k}|,
\end{align*}
we have
\begin{align*}
|W_{k}| & \mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \Big| |w_{k-1} \cdot x| \leq b_{k-1}\right)
\\ & \leq c_{13} \kappa |W_{k}|
+ c_{14} \sqrt{ |W_{k}| \left( \frac{3 \log(8 \max\{c_{3},1\})}{c_{2} c_{7}} \kappa^{2} |W_{k}| + \frac{\kappa^{2}}{c_{2}c_{5}c_{7}}|W_{k}| \right)}
\\ & = \kappa |W_{k}| \left( c_{13} + c_{14} \sqrt{ \frac{3 \log(8 \max\{c_{3},1\})}{c_{2} c_{7}} + \frac{1}{c_{2}c_{5}c_{7}}}\right).
\end{align*}
Thus, letting $c_{15} = \left( c_{13} + c_{14} \sqrt{ \frac{3 \log(8 \max\{c_{3},1\})}{c_{2} c_{7}} + \frac{1}{c_{2}c_{5}c_{7}}}\right)$,
we have
\begin{equation}
\label{eqn:conditional-error-bound}
\mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \Big| |w_{k-1} \cdot x| \leq b_{k-1}\right)
\leq c_{15} \kappa.
\end{equation}
Next, note that $\|v_{k} - w_{k-1}\|^{2} = \|v_{k}\|^{2} + 1 - 2 \|v_{k}\| \cos( \pi \mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) )$.
Thus, one implication of the fact that $\|v_{k} - w_{k-1}\| \leq r_{k}$ is that
$\frac{\|v_{k}\|}{2} + \frac{1-r_{k}^{2}}{2\|v_{k}\|} \leq \cos( \pi \mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) )$;
since the left hand side is positive, we have $\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) < 1/2$.
Additionally, by differentiating, one can easily verify that for $\phi \in [0,\pi]$,
$x \mapsto \sqrt{ x^{2} + 1 - 2 x \cos(\phi) }$ is minimized at $x=\cos(\phi)$,
in which case $\sqrt{x^{2} + 1 - 2 x \cos(\phi) } = \sin(\phi)$.
Thus, $\|v_{k} - w_{k-1}\| \geq \sin( \pi \mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x) ) )$.
Since $\|v_{k} - w_{k-1}\| \leq r_{k}$,
we have $\sin(\pi \mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x))) \leq r_{k}$.
Since $\sin(\pi x) \geq x$ for all $x \in [0,1/2]$,
combining this with the fact (proven above) that $\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) < 1/2$
implies $\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) \leq r_{k}$.
In particular, we have that both $\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x)) \leq r_{k}$ and $\mathcal{P}(x : h_{w^{*}}(x) \neq h_{w_{k-1}}(x)) \leq 2^{-k-3} \leq r_{k}$.
Now Lemma~\ref{lem:margin-error-concentration} implies that, for any universal constant $c > 0$,
there exists a corresponding universal constant $c^{\prime} > 0$ such that
\begin{equation*}
\mathcal{P}\left(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x) \text{ and } |w_{k-1} \cdot x| \geq c^{\prime} \frac{r_{k}}{\sqrt{d}} \right) \leq c r_{k}
\end{equation*}
and
\begin{equation*}
\mathcal{P}\left(x : h_{w^{*}}(x) \neq h_{w_{k-1}}(x) \text{ and } |w_{k-1} \cdot x| \geq c^{\prime} \frac{r_{k}}{\sqrt{d}} \right) \leq c r_{k},
\end{equation*}
so that (by a union bound)
\begin{align*}
& \mathcal{P}\left(x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \text{ and } |w_{k-1} \cdot x| \geq c^{\prime} \frac{r_{k}}{\sqrt{d}} \right)
\\ & \leq
\mathcal{P}\left(x : h_{w_{k}}(x) \neq h_{w_{k-1}}(x) \text{ and } |w_{k-1} \cdot x| \geq c^{\prime} \frac{r_{k}}{\sqrt{d}} \right)
\\ & +
\mathcal{P}\left(x : h_{w^{*}}(x) \neq h_{w_{k-1}}(x) \text{ and } |w_{k-1} \cdot x| \geq c^{\prime} \frac{r_{k}}{\sqrt{d}} \right)
\leq 2 c r_{k}.
\end{align*}
In particular, letting $c_{7} = c^{\prime} c_{10} / 2$, we have $c^{\prime} \frac{r_{k}}{\sqrt{d}} = b_{k-1}$.
Combining this with \eqref{eqn:conditional-error-bound}, Lemma~\ref{lem:uniform-P-concentration}, and a union bound, we have that
\begin{align*}
& \mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x)\right)
\\ & \leq \mathcal{P}\left(x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \text{ and } |w_{k-1} \cdot x| \geq b_{k-1} \right)
\\ & {\hskip 3mm}+ \mathcal{P}\left(x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \text{ and } |w_{k-1} \cdot x| \leq b_{k-1} \right)
\\ & \leq 2 c r_{k} + \mathcal{P}\left( x : h_{w_{k}}(x) \neq h_{w^{*}}(x) \Big| |w_{k-1} \cdot x| \leq b_{k-1} \right) \mathcal{P}\left(x : |w_{k-1} \cdot x| \leq b_{k-1}\right)
\\ & \leq 2 c r_{k} + c_{15} \kappa c_{3} b_{k-1} \sqrt{d}
= \left( 2^{5} c c_{10} + c_{15} \kappa c_{3} c_{7} 2^{5} \right) 2^{-k-4}.
\end{align*}
Taking $c = \frac{1}{2^{6} c_{10}}$ and $\kappa = \frac{1}{2^{6} c_{3} c_{7} c_{15}}$,
we have $\mathcal{P}(x : h_{w_{k}}(x) \neq h_{w^{*}}(x)) \leq 2^{-k-4}$, as required.
By a union bound, this occurs with probability at least $1 - (4/3)\delta_{k}$.
\qed
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:linsep-uniform}]
We begin with the bound on the error rate.
If $\mathcal Delta > \frac{\pi^{2}}{400 \cdot 2^{27} (d+\ln(4/\delta))}$, the result trivially holds, since then $1 \leq \frac{400 \cdot 2^{27}}{\pi^{2}} \sqrt{\mathcal Delta (d+\ln(4/\delta))}$.
Otherwise, suppose $\mathcal Delta \leq \frac{\pi^{2}}{400 \cdot 2^{27} (d+\ln(4/\delta))}$.
Fix any $i \in \mathbb{N}$.
Lemma~\ref{lem:perceptron-init} implies that, with probability at least $1-\delta/4$,
the $w_{0}$ returned in Step 0 of ${\rm ABL}(M(i-1),\tilde{h}_{i-1})$ satisfies
$\mathcal{P}(x : h_{w_{0}}(x) \neq h_{M(i-1) + m_{0}+1}^{*}(x)) \leq 1/16$.
Taking this as a base case, Lemma~\ref{lem:margin-error-bound} then inductively implies that,
with probability at least
\begin{multline*}
1 - \frac{\delta}{4} - \sum_{k=1}^{\lceil \log_{2}(1/\alpha) \rceil} (4/3) \frac{\delta}{2(\lceil \log_{2}(4/\alpha) \rceil - k)^{2}}
\geq 1 - \frac{\delta}{2} \left(1 + (4/3) \sum_{\ell=2}^{\infty} \frac{1}{\ell^{2}} \right)
\geq 1 - \delta,
\end{multline*}
every $k \in \{ 0, 1, \ldots, \lceil \log_{2}(1/\alpha) \rceil \}$ has
\begin{equation}
\label{eqn:abl-mistake-prob-raw}
\mathcal{P}(x : h_{w_{k}}(x) \neq h_{M(i-1)+m_{0}+1}^{*}(x)) \leq 2^{-k-4},
\end{equation}
and furthermore the number of labels requested during ${\rm ABL}(M(i-1),\tilde{h}_{i-1})$ total to at most (for appropriate universal constants $\hat{c}_{1},\hat{c}_{2}$)
\begin{align*}
m_{0} + \!\!\!\!\sum_{k=1}^{\lceil \log_{2}(1/\alpha) \rceil} |W_{k}|
& \leq \hat{c}_{1} \left(d + \ln\left(\frac{1}{\delta}\right) + \sum_{k=1}^{\lceil \log_{2}(1/\alpha) \rceil} d \log\left(\frac{( \lceil \log_{2}(4/\alpha) \rceil - k )^{2}}{\delta}\right) \right)
\\ & \leq \hat{c}_{2} d \log\left(\frac{1}{\mathcal Delta d}\right)\log\left(\frac{1}{\delta}\right).
\end{align*}
In particular, by a union bound, \eqref{eqn:abl-mistake-prob-raw} implies that for every $k \in \{1,\ldots,\lceil \log_{2}(1/\alpha) \rceil\}$,
every
\begin{equation*}
m \in \left\{ M(i-1) + \sum_{j=0}^{k-1} m_{j} + 1, \ldots, M(i-1) + \sum_{j=0}^{k} m_{j} \right\}
\end{equation*}
has
\begin{align*}
& \mathcal{P}(x : h_{w_{k-1}}(x) \neq h_{m}^{*}(x))
\\ & \leq \mathcal{P}(x : h_{w_{k-1}}(x) \neq h_{M(i-1)+m_{0}+1}^{*}(x)) + \mathcal{P}(x : h_{M(i-1)+m_{0}+1}^{*}(x) \neq h_{m}^{*}(x))
\\ & \leq 2^{-k-3} + \mathcal Delta M.
\end{align*}
Thus, noting that
\begin{align*}
M & = \sum_{k=0}^{\lceil \log_{2}(1/\alpha) \rceil} m_{k}
= \Theta\left( d + \log\left(\frac{1}{\delta}\right) + \sum_{k=1}^{\lceil \log_{2}(1/\alpha) \rceil} 2^{k} d \log\left(\frac{\lceil \log_{2}(1/\alpha) \rceil - k}{\delta}\right) \right)
\\ & = \Theta\left( \frac{1}{\alpha} d \log\left(\frac{1}{\delta}\right) \right)
= \Theta\left(\sqrt{\frac{d}{\mathcal Delta} \log\left(\frac{1}{\delta}\right)} \right),
\end{align*}
with probability at least $1-\delta$,
\begin{equation*}
\mathcal{P}(x : h_{w_{\lceil \log_{2}(1/\alpha) \rceil-1}}(x) \neq h^{*}_{M i}(x) ) \leq O\left( \alpha + \mathcal Delta M \right) = O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\delta}\right) } \right).
\end{equation*}
In particular, this implies that, with probability at least $1-\delta$, every $t \in \{M i + 1, \ldots, M (i+1)-1\}$ has
\begin{align*}
{\rm er}_{t}(\hat{h}_{t}) & \leq \mathcal{P}(x : h_{w_{\lceil \log_{2}(1/\alpha) \rceil-1}}(x) \neq h^{*}_{M i}(x) ) + \mathcal{P}( x : h^{*}_{M i}(x) \neq h^{*}_{t}(x) )
\\ & \leq O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\delta}\right) } \right) + \mathcal Delta M
= O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\delta}\right) } \right),
\end{align*}
which completes the proof of the bound on the error rate.
Setting $\delta = \sqrt{\mathcal Delta d}$, and noting that $\mathbbm{1}[ \hat{Y}_{t} \neq Y_{t} ] \leq 1$, we have that for any $t > M$,
\begin{equation*}
\mathbb P\left( \hat{Y}_{t} \neq Y_{t} \right)
= \mathbb E\left[ {\rm er}_{t}(\hat{h}_{t}) \right]
\leq O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\delta}\right) } \right) + \delta
= O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\mathcal Delta d}\right) } \right).
\end{equation*}
Thus, by linearity of the expectation,
\begin{equation*}
\mathbb E\left[ \sum_{t=1}^{T} \mathbbm{1}\left[ \hat{Y}_{t} \neq Y_{t} \right] \right]
\leq M + O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\mathcal Delta d}\right) } T \right)
= O\left( \sqrt{ \mathcal Delta d \log\left(\frac{1}{\mathcal Delta d}\right) } T \right).
\end{equation*}
Furthermore, as mentioned, with probability at least $1-\delta$,
the number of labels requested during the execution of ${\rm ABL}(M(i-1),\tilde{h}_{i-1})$ is at most
\begin{equation*}
O\left( d \log\left(\frac{1}{\mathcal Delta d}\right)\log\left(\frac{1}{\delta}\right) \right).
\end{equation*}
Thus, since the number of labels requested during the execution of ${\rm ABL}(M(i-1),\tilde{h}_{i-1})$ cannot exceed $M$,
letting $\delta = \sqrt{\mathcal Delta d}$, the expected number of requested labels during this execution is at most
\begin{align*}
O\left( d \log^{2}\left(\frac{1}{\mathcal Delta d}\right) \right) + \sqrt{\mathcal Delta d} M
& \leq O\left( d \log^{2}\left(\frac{1}{\mathcal Delta d}\right) \right) + O\left( d \sqrt{\log\left(\frac{1}{\mathcal Delta d}\right) } \right)
\\ & = O\left( d \log^{2}\left(\frac{1}{\mathcal Delta d}\right) \right).
\end{align*}
Thus, by linearity of the expectation, the expected number of labels requested among the first $T$ samples is at most
\begin{equation*}
O\left( d \log^{2}\left(\frac{1}{\mathcal Delta d}\right) \left\lceil \frac{T}{M} \right\rceil \right)
= O\left( \sqrt{\mathcal Delta d} \log^{3/2}\left(\frac{1}{\mathcal Delta d}\right) T \right),
\end{equation*}
which completes the proof.
\qed
\end{proof}
\paragraph{Remark:} The original work of \cite{min_concept} additionally allowed for some number $K$ of ``jumps'':
times $t$ at which $\mathcal Delta_{t} = 1$. Note that, in the above algorithm, since the influence of each sample is localized to the predictors trained
within that ``batch'' of $M$ instances, the effect of allowing such jumps would only change the bound on the number of
mistakes to $\tilde{O}\left(\sqrt{d \mathcal Delta} T + \sqrt{\frac{d}{\mathcal Delta}} K \right)$. This compares favorably to the
result of \cite{min_concept}, which is roughly $O\left( (d \mathcal Delta)^{1/4} T + \frac{d^{1/4}}{\mathcal Delta^{3/4}} K \right)$.
However, the result of \cite{min_concept} was proven for a more general setting, allowing distributions $\mathcal{P}$
that are not uniform (though they do require a relation between the angle between any two separators and the
probability mass they disagree on, similar to that holding for the uniform distribution, which seems to require that the
distributions approximately retain some properties of the uniform distribution). It is not clear whether Theorem~\ref{thm:linsep-uniform} can be
generalized to this larger family of distributions.
\section{General Results for Active Learning}
\label{sec:general-active}
As mentioned, the above results on linear separators also provide results
for the number of queries in \emph{active learning}. One can also state
quite general results on the expected number of queries and mistakes
achievable by an active learning algorithm.
This section provides such results, for an algorithm based on the
the well-known strategy of \emph{disagreement-based} active learning.
Throughout this section, we suppose $h^{*}seq \in S_{\mathcal Delta}$,
for a given $\mathcal Delta \in (0,1]$: that is, $\mathcal{P}( x : h^{*}_{t+1}(x) \neq h^{*}_{t}(x)) \leq \mathcal Delta$
for all $t \in \mathbb{N}$.
First, we introduce a few definitions.
For any set $\mathcal H \subseteq \mathbb C$, define the \emph{region of disagreement}
\begin{equation*}
\mathcal DIS(\mathcal H) = \{x \in \mathcal X : \exists h,g \in \mathcal H \text{ s.t. } h(x) \neq g(x) \}.
\end{equation*}
The analysis in this section is centered around the following algorithm.
The ${\rm Active}$ subroutine is from the work of \cite{hanneke:activized} (slightly modified here),
and is a variant of the $A^2$ (Agnostic Acive) algorithm of \cite{A2};
the appropriate values of $M$ and $\hat{T}_{k}(\cdot)$ will be discussed below.
\begin{bigboxit}
Algorithm: ${\rm DriftingActive}$\\
0. For $i = 1,2,\ldots$\\
1. \quad ${\rm Active}(M (i-1))$\\
\end{bigboxit}
\begin{bigboxit}
Subroutine: ${\rm Active}(t)$\\
0. Let $\hat{h}_{0}$ be an arbitrary element of $\mathbb C$, and let $V_{0} \gets \mathbb C$\\
1. Predict $\hat{Y}_{t+1} = \hat{h}_{0}(X_{t+1})$ as the prediction for the value of $Y_{t+1}$\\
2. For $k = 0,1,\ldots,\log_{2}(M/2)$\\
3. \quad $Q_{k} \gets \{\}$\\
4. \quad For $s = 2^{k}+1,\ldots,2^{k+1}$\\
5. \qquad Predict $\hat{Y}_{s} = \hat{h}_{k}(X_{s})$ as the prediction for the value of $Y_{s}$\\
6. \qquad If $X_{s} \in \mathcal DIS(V_{k})$\\
7. \quad\qquad Request the label $Y_{s}$ and let $Q_{k} \gets Q_{k} \cup \{(X_{s},Y_{s})\}$\\
8. \quad Let $\hat{h}_{k+1} = \mathop{\rm argmin}_{h \in V_{k}} \sum_{(x,y) \in Q_{k}} \mathbbm{1}[h(x) \neq y]$\\
9. \quad Let $V_{k+1} \gets \{h \in V_{k} : \sum_{(x,y) \in Q_{k}} \mathbbm{1}[h(x) \neq y] - \mathbbm{1}[\hat{h}_{k+1}(x) \neq y] \leq \hat{T}_{k}\}$
\end{bigboxit}
As in the ${\rm DriftingHalfspaces}$ algorithm above, this ${\rm DriftingActive}$
algorithm proceeds in batches, and in each batch runs an active learning algorithm
designed to be robust to classification noise. This robustness to classification noise
translates into our setting as tolerance for the fact that there is no classifier in $\mathbb C$
that perfectly classifies all of the data. The specific algorithm employed here maintains
a set $V_{k} \subseteq \mathbb C$ of candidate classifiers, and requests the labels of samples $X_{s}$
for which there is some disagreement on the classification among classifiers in $V_{k}$.
We maintain the invariant that there is a low-error classifier contained in $V_{k}$ at all
times, and thus the points we query provide some information to help us determine
which among these remaining candidates has low error rate. Based on these queries,
we periodically (in Step 9) remove from $V_{k}$ those classifiers making a relatively excessive
number of mistakes on the queried samples, relative to the minimum among classifiers in $V_{k}$.
All predictions are made with an element of $V_{k}$.\footnote{One could alternatively proceed
as in ${\rm DriftingHalfspaces}$, using the final classifier from the previous batch, which
would also add a guarantee on the error rate achieved at all sufficiently large $t$.}
We prove an abstract bound on the number of labels requested by this algorithm,
expressed in terms of the \emph{disagreement coefficient} \cite{hanneke:07b},
defined as follows. For any $r \geq 0$ and any classifier $h$, define ${\rm B}(h,r) = \{g \in \mathbb C : \mathcal{P}(x : g(x) \neq h(x)) \leq r\}$.
Then for $r_{0} \geq 0$ and any classifier $h$, define the disagreement coefficient of $h$ with respect to $\mathbb C$ under $\mathcal{P}$:
\begin{equation*}
\theta_{h}(r_{0}) = \sup_{r > r_{0}} \frac{ \mathcal{P}( \mathcal DIS( {\rm B}( h, r ) ) ) }{r}.
\end{equation*}
Usually, the disagreement coefficient would be used with $h$ equal the target concept;
however, since the target concept is not fixed in our setting,
we will make use of the worst-case value of the disagreement coefficient:
$\theta_{\mathbb C}(r_{0}) = \sup_{h \in \mathbb C} \theta_{h}(r_{0})$.
This quantity has been bounded for a variety of spaces $\mathbb C$ and distributions $\mathcal{P}$
(see e.g., \cite{hanneke:07b,el-yaniv:12,balcan:13}).
It is useful in bounding how quickly the region $\mathcal DIS(V_{k})$ collapses in the
algorithm. Thus, since the probability the algorithm requests the label of the next instance
is $\mathcal{P}(\mathcal DIS(V_{k}))$, the quantity $\theta_{\mathbb C}(r_{0})$ naturally arises in characterizing the
number of labels we expect this algorithm to request.
Specifically, we have the following result.\footnote{Here,
we define $\lceil x \rceil_{2} = 2^{\lceil \log_{2}(x) \rceil}$, for $x \geq 1$.}
\begin{theorem}
\label{thm:general-active}
For an appropriate universal constant $c_{1} \in [1,\infty)$,
if $h^{*}seq \in S_{\mathcal Delta}$ for some $\mathcal Delta \in (0,1]$,
then taking $M = \left\lceil c_{1} \sqrt{\frac{d}{\mathcal Delta}} \right\rceil_{2}$,
and $\hat{T}_{k} = \log_{2}(1/\sqrt{d \mathcal Delta}) + 2^{2k+2} e \mathcal Delta$,
and defining $\epsilon_{\mathcal Delta} = \sqrt{d\mathcal Delta} {\rm Log}(1/(d\mathcal Delta))$,
the above ${\rm DriftingActive}$ algorithm makes an expected number of mistakes among the
first $T$ instances that is
\begin{equation*}
O\left(\epsilon_{\mathcal Delta} {\rm Log}(d/\mathcal Delta) T \right) = \tilde{O}\left( \sqrt{d\mathcal Delta} \right) T
\end{equation*}
and requests an expected number of labels among the first $T$ instances that is
\begin{equation*}
O\left( \theta_{\mathbb C}( \epsilon_{\mathcal Delta} ) \epsilon_{\mathcal Delta} {\rm Log}(d/\mathcal Delta) T \right) = \tilde{O}\left( \theta_{\mathbb C}(\sqrt{d \mathcal Delta}) \sqrt{d \mathcal Delta} \right) T.
\end{equation*}
\end{theorem}
The proof of Theorem~\ref{thm:general-active} relies on an analysis of the behavior of the ${\rm Active}$ subroutine,
characterized in the following lemma.
\begin{lemma}
\label{lem:active-subroutine}
Fix any $t \in \mathbb{N}$, and consider the values obtained in the execution of ${\rm Active}(t)$.
Under the conditions of Theorem~\ref{thm:general-active},
there is a universal constant $c_{2} \in [1,\infty)$ such that,
for any $k \in \{0,1,\ldots,\log_{2}(M/2)\}$,
with probability at least $1-2\sqrt{d \mathcal Delta}$, if
$h^{*}_{t+1} \in V_{k}$,
then $h^{*}_{t+1} \in V_{k+1}$ and
$\sup_{h \in V_{k+1}} \mathcal{P}(x : h(x) \neq h^{*}_{t+1}(x)) \leq c_{2}
2^{-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta})$.
\end{lemma}
\begin{proof}
By a Chernoff bound, with probability at least $1-\sqrt{d \mathcal Delta}$,
\begin{equation*}
\sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[h^{*}_{t+1}(X_{s}) \neq Y_{s}]
\leq \log_{2}(1/\sqrt{d \mathcal Delta}) + 2^{2k+2} e \mathcal Delta
= \hat{T}_{k}.
\end{equation*}
Therefore, if $h^{*}_{t+1} \in V_{k}$, then since every $g \in V_{k}$
agrees with $h^{*}_{t+1}$ on those points $X_{s} \notin \mathcal DIS(V_{k})$,
in the update in Step 9 defining $V_{k+1}$,
we have
\begin{align*}
& \sum_{(x,y) \in Q_{k}} \mathbbm{1}[h^{*}_{t+1}(x) \neq y] - \mathbbm{1}[\hat{h}_{k+1}(x) \neq y]
\\ & = \sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[h^{*}_{t+1}(X_{s}) \neq Y_{s}]
- \min_{g \in V_{k}} \sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[g(X_{s}) \neq Y_{s}]
\\ & \leq \sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[h^{*}_{t+1}(X_{s}) \neq Y_{s}] \leq \hat{T}_{k},
\end{align*}
so that $h^{*}_{t+1} \in V_{k+1}$ as well.
Furthermore, if $h^{*}_{t+1} \in V_{k}$,
then by the definition of $V_{k+1}$,
we know every $h \in V_{k+1}$ has
\begin{equation*}
\sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[ h(X_{s}) \neq Y_{s} ]
\leq \hat{T}_{k} + \sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[ h^{*}_{t+1}(X_{s}) \neq Y_{s} ],
\end{equation*}
so that a triangle inequality implies
\begin{align*}
\sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[ h(X_{s}) \neq h^{*}_{t+1}(X_{s}) ]
& \leq
\sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[ h(X_{s}) \neq Y_{s} ]
+ \mathbbm{1}[ h^{*}_{t+1}(X_{s}) \neq Y_{s} ]
\\ & \leq
\hat{T}_{k} + 2 \sum_{s=2^{k}+1}^{2^{k+1}} \mathbbm{1}[ h^{*}_{t+1}(X_{s}) \neq Y_{s} ]
\leq 3 \hat{T}_{k}.
\end{align*}
Lemma~\ref{lem:vc-ratio} then implies that, on an additional event of
probability at least $1-\sqrt{d \mathcal Delta}$,
every $h \in V_{k+1}$ has
\begin{align*}
& \mathcal{P}(x : h(x) \neq h^{*}_{t+1}(x))
\\ & \leq 2^{-k} 3\hat{T}_{k} + c 2^{-k} \sqrt{3\hat{T}_{k} (d {\rm Log}(2^{k}/d)+{\rm Log}(1/\sqrt{d\mathcal Delta}))}
\\ & \phantom{\leq } + c 2^{-k} (d {\rm Log}(2^{k}/d) + {\rm Log}(1/\sqrt{d\mathcal Delta}))
\\ & \leq
2^{-k} 3 \log_{2}(1/\sqrt{d\mathcal Delta})
+ 2^{k} 12 e \mathcal Delta
+ c 2^{-k} \sqrt{ 6 \log_{2}(1/\sqrt{d\mathcal Delta}) d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta})}
\\ & \phantom{\leq } + c 2^{-k} \sqrt{ 2^{2k} 24 e \mathcal Delta d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}) }
+ 2 c 2^{-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta})
\\ &
\leq
2^{-k} 3 \log_{2}(1/\sqrt{d\mathcal Delta})
+ 12 e c_{1} \sqrt{d\mathcal Delta}
+ 3 c 2^{-k} \sqrt{ d } {\rm Log}(c_{1} / \sqrt{d\mathcal Delta})
\\ & \phantom{\leq } + \sqrt{24 e} c \sqrt{d \mathcal Delta {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}) }
+ 2 c 2^{-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}),
\end{align*}
where $c$ is as in Lemma~\ref{lem:vc-ratio}.
Since $\sqrt{d \mathcal Delta} \leq 2 c_{1} d / M \leq c_{1} d 2^{-k}$,
this is at most
\begin{equation*}
\left(5 + 12 e c_{1}^{2} + 3 c + \sqrt{24 e} c c_{1} + 2 c\right)
2^{-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}).
\end{equation*}
Letting $c_{2} = 5 + 12 e c_{1}^{2} + 3 c + \sqrt{24 e} c c_{1} + 2 c$,
we have the result by a union bound.
\qed
\end{proof}
We are now ready for the proof of Theorem~\ref{thm:general-active}.
\begin{proof}[Proof of Theorem~\ref{thm:general-active}]
Fix any $i \in \mathbb{N}$, and consider running ${\rm Active}(M(i-1))$.
Since $h^{*}_{M(i-1)+1} \in \mathbb C$,
by Lemma~\ref{lem:active-subroutine}, a union bound, and induction,
with probability at least $1-2\sqrt{d\mathcal Delta} \log_{2}(M/2)
\geq 1 - 2 \sqrt{d\mathcal Delta} \log_{2}(c_{1}\sqrt{d/\mathcal Delta})$,
every $k \in \{0,1,\ldots,\log_{2}(M/2)\}$ has
\begin{equation}
\label{eqn:general-active-radius}
\sup_{h \in V_{k}} \mathcal{P}(x : h(x) \neq h^{*}_{M(i-1)+1}(x)) \leq
c_{2} 2^{1-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}).
\end{equation}
Thus, since $\hat{h}_{k} \in V_{k}$ for each $k$,
the expected number of mistakes among the predictions
$\hat{Y}_{M(i-1)+1},\ldots,\hat{Y}_{M i}$
is
\begin{align*}
& 1 + \sum_{k=0}^{\log_{2}(M/2)} \sum_{s=2^{k}+1}^{2^{k+1}} \mathbb P(\hat{h}_{k}(X_{M(i-1)+s}) \neq Y_{M(i-1)+s})
\\ & \leq 1 + \sum_{k=0}^{\log_{2}(M/2)} \sum_{s=2^{k}+1}^{2^{k+1}}
\mathbb P(h^{*}_{M(i-1)+1}(X_{M(i-1)+s}) \neq Y_{M(i-1)+s})
\\ & \phantom{\leq } + \sum_{k=0}^{\log_{2}(M/2)} \sum_{s=2^{k}+1}^{2^{k+1}} \mathbb P(\hat{h}_{k}(X_{M(i-1)+s}) \neq h^{*}_{M(i-1)+1}(X_{M(i-1)+s}))
\\ & \leq
1 + \mathcal Delta M^{2} +
\sum_{k=0}^{\log_{2}(M/2)} 2^{k} \left( c_{2} 2^{1-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}) + 2\sqrt{d\mathcal Delta}\log_{2}(M/2)\right)
\\ & \leq
1 + 4 c_{1}^{2} d + 2 c_{2} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta}) \log_{2}(2 c_{1} \sqrt{d/\mathcal Delta})
+ 4c_{1} d \log_{2}(c_{1} \sqrt{d/\mathcal Delta})
\\ & =
O\left( d {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) \right).
\end{align*}
Furthermore, \eqref{eqn:general-active-radius} implies the algorithm only
requests the label $Y_{M(i-1)+s}$ for $s \in \{2^{k}+1,\ldots,2^{k+1}\}$
if $X_{M(i-1)+s} \in \mathcal DIS({\rm B}(h^{*}_{M(i-1)+1}, c_{2} 2^{1-k} d {\rm Log}(c_{1} / \sqrt{d\mathcal Delta})))$,
so that the expected number of labels requested among $Y_{M(i-1)+1},\ldots,Y_{M i}$ is at most
\begin{align*}
& 1 + \sum_{k=0}^{\log_{2}(M/2)} 2^{k} \left(\mathbb E[ \mathcal{P}(\mathcal DIS({\rm B}(h^{*}_{M(i-1)+1}, c_{2} 2^{1-k} d {\rm Log}(c_{1}/\sqrt{d\mathcal Delta}))))] \right.
\\ & {\hskip 6cm}\left.+ 2 \sqrt{d\mathcal Delta} \log_{2}(c_{1}\sqrt{d/\mathcal Delta})\right)
\\ & \leq
1 + \theta_{\mathbb C}\left(4 c_{2} d {\rm Log}(c_{1}/\sqrt{d\mathcal Delta}) / M\right) 2 c_{2} d {\rm Log}(c_{2}/\sqrt{d\mathcal Delta}) \log_{2}(2 c_{1} \sqrt{d/\mathcal Delta})
\\ & {\hskip 6cm}+ 4 c_{1} d \log_{2}(c_{1}\sqrt{d/\mathcal Delta})
\\ & =
O\left( \theta_{\mathbb C}\left( \sqrt{d\mathcal Delta} {\rm Log}(1/(d\mathcal Delta)) \right) d {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) \right).
\end{align*}
Thus, the expected number of mistakes among indices $1,\ldots,T$ is at most
\begin{equation*}
O\left( d {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) \left\lceil \frac{T}{M} \right\rceil \right)
= O\left( \sqrt{d\mathcal Delta} {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) T \right),
\end{equation*}
and the expected number of labels requested among indices $1,\ldots,T$ is at most
\begin{multline*}
O\left( \theta_{\mathbb C}\left( \sqrt{d\mathcal Delta} {\rm Log}(1/(d\mathcal Delta)) \right) d {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) \left\lceil \frac{T}{M} \right\rceil \right)
\\ = O\left( \theta_{\mathbb C}\left( \sqrt{d\mathcal Delta} {\rm Log}(1/(d\mathcal Delta)) \right) \sqrt{d\mathcal Delta} {\rm Log}(d/\mathcal Delta) {\rm Log}(1/(d\mathcal Delta)) T \right).
\end{multline*}
\qed
\end{proof}
\end{document} |
\begin{equation}gin{document}
\date{}
\title{ON THE UNIVERSALITY OF SOME SMARANDACHE LOOPS OF BOL-MOUFANG TYPE
\footnote{2000 Mathematics Subject Classification. Primary 20NO5 ;
Secondary 08A05.}
\thanks{{\bf Keywords and Phrases :} Smarandache quasigroups, Smarandache loops, universality, $f,g$-principal isotopes}}
\author{T\`em\'it\'op\'e Gb\'ol\'ah\`an Ja\'iy\'e\d ol\'a\thanks{On Doctorate Programme at
the University of Agriculture Abeokuta, Nigeria.}
\thanks{All correspondence to be addressed to this author}\\
Department of Mathematics,\\
Obafemi Awolowo University, Ile Ife, Nigeria.\\
jaiyeolatemitope@yahoo.com, tjayeola@oauife.edu.ng} \maketitle
\begin{equation}gin{abstract}
A Smarandache quasigroup(loop) is shown to be universal if all its
$f,g$-principal isotopes are Smarandache $f,g$-principal isotopes.
Also, weak Smarandache loops of Bol-Moufang type such as
Smarandache: left(right) Bol, Moufang and extra loops are shown to
be universal if all their $f,g$-principal isotopes are Smarandache
$f,g$-principal isotopes. Conversely, it is shown that if these weak
Smarandache loops of Bol-Moufang type are universal, then some
autotopisms are true in the weak Smarandache sub-loops of the weak
Smarandache loops of Bol-Moufang type relative to some Smarandache
elements. Futhermore, a Smarandache left(right) inverse property
loop in which all its $f,g$-principal isotopes are Smarandache
$f,g$-principal isotopes is shown to be universal if and only if it
is a Smarandache left(right) Bol loop in which all its
$f,g$-principal isotopes are Smarandache $f,g$-principal isotopes.
Also, it is established that a Smarandache inverse property loop in
which all its $f,g$-principal isotopes are Smarandache
$f,g$-principal isotopes is universal if and only if it is a
Smarandache Moufang loop in which all its $f,g$-principal isotopes
are Smarandache $f,g$-principal isotopes. Hence, some of the
autotopisms earlier mentioned are found to be true in the
Smarandache sub-loops of universal Smarandache: left(right) inverse
property loops and inverse property loops.
\end{abstract}
\section{Introduction}
W. B. Vasantha Kandasamy initiated the study of Smarandache loops
(S-loop) in 2002. In her book \cite{phd75}, she defined a
Smarandache loop (S-loop) as a loop with at least a subloop which
forms a subgroup under the binary operation of the loop called a
Smarandache subloop (S-subloop). In \cite{sma2}, the present author
defined a Smarandache quasigroup (S-quasigroup) to be a quasigroup
with at least a non-trivial associative subquasigroup called a
Smarandache subquasigroup (S-subquasigroup). Examples of Smarandache
quasigroups are given in Muktibodh \cite{muk}. For more on
quasigroups, loops and their properties, readers should check
\cite{phd3}, \cite{phd41},\cite{phd39}, \cite{phd49}, \cite{phd42}
and \cite{phd75}. In her (W.B. Vasantha Kandasamy) first paper
\cite{phd83}, she introduced Smarandache : left(right) alternative
loops, Bol loops, Moufang loops, and Bruck loops. But in
\cite{sma1}, the present author introduced Smarandache : inverse
property loops (IPL), weak inverse property loops (WIPL), G-loops,
conjugacy closed loops (CC-loop), central loops, extra loops,
A-loops, K-loops, Bruck loops, Kikkawa loops, Burn loops and
homogeneous loops. The isotopic invariance of types and varieties of
quasigroups and loops described by one or more equivalent
identities, especially those that fall in the class of Bol-Moufang
type loops as first named by Fenyves \cite{phd56} and \cite{phd50}
in the 1960s and later on in this $21^{st}$ century by Phillips and
Vojt\v echovsk\'y \cite{phd9}, \cite{phd61} and \cite{phd124} have
been of interest to researchers in loop theory in the recent past.
For example, loops such as Bol loops, Moufang loops, central loops
and extra loops are the most popular loops of Bol-Moufang type whose
isotopic invariance have been considered. Their identities relative
to quasigroups and loops have also been investigated by Kunen
\cite{ken1} and \cite{ken2}. A loop is said to be universal relative
to a property ${\cal P}$ if it is isotopic invariant relative to
${\cal P}$, hence such a loop is called a universal ${\cal P}$ loop.
This language is well used in \cite{phd88}. The universality of most
loops of Bol-Moufang types have been studied as summarised in
\cite{phd3}. Left(Right) Bol loops, Moufang loops, and extra loops
have all been found to be isotopic invariant. But some types of
central loops were shown to be universal in Ja\'iy\'e\d ol\'a
\cite{tope} and \cite{phdtope} under some conditions. Some other
types of loops such as A-loops, weak inverse property loops and
cross inverse property loops (CIPL) have been found be universal
under some neccessary and sufficient conditions in \cite{phd40},
\cite{phd43} and \cite{phd30} respectively. Recently, Michael Kinyon
et. al. \cite{phd95}, \cite{phd118}, \cite{phd119} solved the
Belousov problem concerning the universality of F-quasigroups which
has been open since 1967 by showing that all the isotopes of
F-quasigroups are Moufang loops.
In this work, the universality of the Smarandache concept in loops
is investigated. That is, will all isotopes of an S-loop be an
S-loop? The answer to this could be 'yes' since every isotope of a
group is a group (groups are G-loops). Also, the universality of
weak Smarandache loops, such as Smarandache Bol loops (SBL),
Smarandache Moufang loops (SML) and Smarandache extra loops (SEL)
will also be investigated despite the fact that it could be expected
to be true since Bol loops, Moufang loops and extra loops are
universal. The universality of a Smarandache inverse property loop
(SIPL) will also be considered.
\section{Preliminaries}
\begin{equation}gin{mydef}
A loop is called a Smarandache left inverse property loop (SLIPL) if
it has at least a non-trivial subloop with the LIP.
A loop is called a Smarandache right inverse property loop (SRIPL)
if it has at least a non-trivial subloop with the RIP.
A loop is called a Smarandache inverse property loop (SIPL) if it
has at least a non-trivial subloop with the IP.
A loop is called a Smarandache right Bol-loop (SRBL) if it has at
least a non-trivial subloop that is a right Bol(RB)-loop.
A loop is called a Smarandache left Bol-loop (SLBL) if it has at
least a non-trivial subloop that is a left Bol(LB)-loop.
A loop is called a Smarandache central-loop (SCL) if it has at least
a non-trivial subloop that is a central-loop.
A loop is called a Smarandache extra-loop (SEL) if it has at least a
non-trivial subloop that is a extra-loop.
A loop is called a Smarandache A-loop (SAL) if it has at least a
non-trivial subloop that is a A-loop.
A loop is called a Smarandache Moufang-loop (SML) if it has at least
a non-trivial subloop that is a Moufang-loop.
\end{mydef}
\begin{equation}gin{mydef}
Let $(G,\oplus)$ and $(H,\otimes)$ be two distinct quasigroups. The
triple $(A,B,C)$ such that $A,B,C~:~(G,\oplus)\rightarrow
(H,\otimes)$ are bijections is said to be an isotopism if and only
if
\begin{equation}gin{displaymath}
xA\otimes yB=(x\oplus y)C~\forall~x,y\in G.
\end{displaymath}
Thus, $H$ is called an isotope of $G$ and they are said to be
isotopic. If $C=I$, then the triple is called a principal isotopism
and $(H,\otimes)=(G,\otimes )$ is called a principal isotope of
$(G,\oplus )$. If in addition, $A=R_g$, $B=L_f$, then the triple is
called an $f,g$-principal isotopism, thus $(G,\otimes )$ is reffered
to as the $f,g$-principal isotope of $(G,\oplus )$.
A subloop(subquasigroup) $(S,\otimes )$ of a loop(quasigroup)
$(G,\otimes )$ is called a Smarandache $f,g$-principal isotope of
the subloop(subquasigroup) $(S,\oplus )$ of a loop(quasigroup)
$(G,\oplus )$ if for some $f,g\in S$,
\begin{equation}gin{displaymath}
xR_g\otimes yL_f=(x\oplus y)~\forall~x,y\in S.
\end{displaymath}
On the other hand $(G,\otimes )$ is called a Smarandache
$f,g$-principal isotope of $(G,\oplus )$ if for some $f,g\in S$,
\begin{equation}gin{displaymath}
xR_g\otimes yL_f=(x\oplus y)~\forall~x,y\in G
\end{displaymath}
where $(S,\oplus )$ is a S-subquasigroup(S-subloop) of $(G,\oplus
)$. In these cases, $f$ and $g$ are called Smarandache
elements(S-elements).
\end{mydef}
\begin{equation}gin{myth}\leftarrowbel{1:1}(\cite{phd41})
Let $(G,\oplus)$ and $(H,\otimes)$ be two distinct isotopic
loops(quasigroups). There exists an $f,g$-principal isotope
$(G,\circ )$ of $(G,\oplus)$ such that $(H,\otimes)\cong (G,\circ
)$.
\end{myth}
\begin{equation}gin{mycor}\leftarrowbel{1:2}
Let ${\cal P}$ be an isotopic invariant property in
loops(quasigroups). If $(G,\oplus)$ is a loop(quasigroup) with the
property ${\cal P}$, then $(G,\oplus)$ is a universal
loop(quasigroup) relative to the property ${\cal P}$ if and only if
every $f,g$-principal isotope $(G,\circ )$ of $(G,\oplus)$ has the
property ${\cal P}$.
\end{mycor}
{\bf Proof}\\
If $(G,\oplus)$ is a universal loop relative to the property ${\cal
P}$ then every distinct loop isotope $(H,\otimes)$ of $(G,\oplus)$
has the property ${\cal P}$. By Theorem~\ref{1:1}, there exists an
$f,g$-principal isotope $(G,\circ )$ of $(G,\oplus)$ such that
$(H,\otimes)\cong (G,\circ )$. Hence, since ${\cal P}$ is an
isomorphic invariant property, every $(G,\circ )$ has it.\\
Conversely, if every $f,g$-principal isotope $(G,\circ )$ of
$(G,\oplus)$ has the property ${\cal P}$ and since by
Theorem~\ref{1:1} for each distinct isotope $(H,\otimes)$ there
exists an $f,g$-principal isotope $(G,\circ )$ of $(G,\oplus)$ such
that $(H,\otimes)\cong (G,\circ )$, then all $(H,\otimes)$ has the
property, Thus, $(G,\oplus)$ is a universal loop relative to the
property ${\cal P}$.
\begin{equation}gin{mylem}\leftarrowbel{1:3}
Let $(G,\oplus)$ be a loop(quasigroup) with a subloop(subquasigroup)
$(S,\oplus )$. If $(G,\circ )$ is an arbitrary $f,g$-principal
isotope of $(G,\oplus)$, then $(S,\circ )$ is a
subloop(subquasigroup) of $(G,\circ)$ if $(S,\circ )$ is a
Smarandache $f,g$-principal isotope of $(S,\oplus )$.
\end{mylem}
{\bf Proof}\\
If $(S,\circ )$ is a Smarandache $f,g$-principal isotope of
$(S,\oplus )$, then for some $f,g\in S$,
\begin{equation}gin{displaymath}
xR_g\circ yL_f=(x\oplus y)~\forall~x,y\in SI\!\!Rightarrow x\circ
y=xR_g^{-1}\oplus yL_f^{-1}\in S~\forall~x,y\in S
\end{displaymath}
since $f,g\in S$. So, $(S,\circ )$ is a subgroupoid of $(G,\circ )$.
$(S,\circ )$ is a subquasigroup follows from the fact that
$(S,\oplus )$ is a subquasigroup. $f\oplus g$ is a two sided
identity element in $(S,\circ )$. Thus, $(S,\circ )$ is a subloop of
$(G,\circ )$.
\section{Main Results}
\subsection*{Universality of Smarandache Loops}
\begin{equation}gin{myth}\leftarrowbel{1:4}
A Smarandache quasigroup is universal if all its $f,g$-principal
isotopes are Smarandache $f,g$-principal isotopes.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a Smarandache quasigroup with a S-subquasigroup
$(S,\oplus )$. If $(G,\circ )$ is an arbitrary $f,g$-principal
isotope of $(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a
subquasigroup of $(G,\circ)$ if $(S,\circ )$ is a Smarandache
$f,g$-principal isotope of $(S,\oplus )$. Let us choose all
$(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath} It shall now be shown that
\begin{equation}gin{displaymath}(x\circ y)\circ z=x\circ (y\circ z)~\forall~x,y,z\in
S.
\end{displaymath}
But in the quasigroup $(G,\oplus )$, $xy$ will have preference over
$x\oplus y~\forall~x,y\in G$.
\begin{equation}gin{displaymath}
(x\circ y)\circ z=(xR_g^{-1}\oplus yL_f^{-1})\circ z=(xg^{-1}\oplus
f^{-1}y)\circ z=(xg^{-1}\oplus f^{-1}y)R_g^{-1}\oplus
zL_f^{-1}
\end{displaymath}
\begin{equation}gin{displaymath}
=(xg^{-1}\oplus f^{-1}y)g^{-1}\oplus f^{-1}z=xg^{-1}\oplus
f^{-1}yg^{-1}\oplus f^{-1}z.
\end{displaymath}
\begin{equation}gin{displaymath}
x\circ (y\circ z)=x\circ (yR_g^{-1}\oplus zL_f^{-1})=x\circ
(yg^{-1}\oplus f^{-1}z)=xR_g^{-1}\oplus (yg^{-1}\oplus
f^{-1}z)L_f^{-1}
\end{displaymath}
\begin{equation}gin{displaymath}
=xg^{-1}\oplus f^{-1}(yg^{-1}\oplus
f^{-1}z)=xg^{-1}\oplus f^{-1}yg^{-1}\oplus f^{-1}z.
\end{displaymath}
Thus, $(S,\circ )$ is an S-subquasigroup of $(G,\circ )$ hence,
$(G,\circ )$ is a S-quasigroup. By Theorem~\ref{1:1}, for any
isotope $(H,\otimes )$ of $(G,\oplus)$, there exists a $(G,\circ )$
such that $(H,\otimes )\cong (G,\circ )$. So we can now choose the
isomorphic image of $(S,\circ)$ which will now be an S-subquasigroup
in $(H,\otimes )$. So, $(H,\otimes )$ is an S-quasigroup. This
conclusion can also be drawn straight from Corollary~\ref{1:2}.
\begin{equation}gin{myth}\leftarrowbel{1:5}
A Smarandache loop is universal if all its $f,g$-principal isotopes
are Smarandache $f,g$-principal isotopes. Conversely, if a
Smarandache loop is universal then
\begin{equation}gin{displaymath}
(I,L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})
\end{displaymath} is an autotopism of an S-subloop of the S-loop such that $f$ and $g$ are S-elements.
\end{myth}
{\bf Proof}\\
Every loop is a quasigroup. Hence, the first claim follows from
Theorem~\ref{1:4}. The proof of the converse is as follows. If a
Smarandache loop $(G,\oplus )$ is universal then every isotope
$(H,\otimes)$ is an S-loop i.e there exists an S-subloop $(S,\otimes
)$ in $(H,\otimes )$. Let $(G,\circ )$ be the $f,g$-principal
isotope of $(G,\oplus)$, then by Corollary~\ref{1:2}, $(G,\circ)$ is
an S-loop with say an S-subloop $(S,\circ)$. So,
\begin{equation}gin{displaymath}
(x\circ y)\circ z=x\circ (y\circ z)~\forall~x,y,z\in S
\end{displaymath}
where \begin{equation}gin{displaymath} x\circ y=xR_g^{-1}\oplus
yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
\begin{equation}gin{displaymath}
(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus zL_f^{-1}=xR_g^{-1}\oplus
(yR_g^{-1}\oplus zL_f^{-1})L_f^{-1}. \end{displaymath} Replacing
$xR_g^{-1}$ by $x'$, $yL_f^{-1}$ by $y'$ and taking $z=e$ in
$(S,\oplus)$ we have; \begin{equation}gin{displaymath} (x'\oplus
y')R_g^{-1}R_{f^\rho}=x'\oplus
y'L_fR_g^{-1}R_{f^\rho}L_f^{-1}I\!\!Rightarrow (I,L_fR_
g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho}) \end{displaymath} is an
autotopism of an S-subloop $(S,\oplus )$ of the S-loop $(G,\oplus )$
such that $f$ and $g$ are S-elements.
\subsection*{Universality of Smarandache Bol, Moufang and Extra Loops}
\begin{equation}gin{myth}\leftarrowbel{1:6}
A Smarandache right(left)Bol loop is universal if all its
$f,g$-principal isotopes are Smarandache $f,g$-principal isotopes.
Conversely, if a Smarandache right(left)Bol loop is universal then
\begin{equation}gin{displaymath}
{\cal
T}_1=(R_gR_{f^\rho}^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})\bigg({\cal
T}_2=(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda})\bigg)
\end{displaymath}
is an autotopism of an SRB(SLB)-subloop of the SRBL(SLBL) such that
$f$ and $g$ are S-elements.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a SRBL(SLBL) with a S-RB(LB)-subloop $(S,\oplus
)$. If $(G,\circ )$ is an arbitrary $f,g$-principal isotope of
$(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of
$(G,\circ)$ if $(S,\circ )$ is a Smarandache $f,g$-principal isotope
of $(S,\oplus )$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
It is already known from \cite{phd3} that RB(LB) loops are
universal, hence $(S,\circ )$ is a RB(LB) loop thus an
S-RB(LB)-subloop of $(G,\circ)$. By Theorem~\ref{1:1}, for any
isotope $(H,\otimes )$ of $(G,\oplus)$, there exists a $(G,\circ )$
such that $(H,\otimes )\cong (G,\circ )$. So we can now choose the
isomorphic image of $(S,\circ)$ which will now be an
S-RB(LB)-subloop in $(H,\otimes )$. So, $(H,\otimes )$ is an
SRBL(SLBL). This conclusion can also be drawn straight from
Corollary~\ref{1:2}.
The proof of the converse is as follows. If a SRBL(SLBL) $(G,\oplus
)$ is universal then every isotope $(H,\otimes)$ is an SRBL(SLBL)
i.e there exists an S-RB(LB)-subloop $(S,\otimes )$ in $(H,\otimes
)$. Let $(G,\circ )$ be the $f,g$-principal isotope of $(G,\oplus)$,
then by Corollary~\ref{1:2}, $(G,\circ)$ is an SRBL(SLBL) with say
an SRB(SLB)-subloop $(S,\circ)$. So for an SRB-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
[(y\circ x)\circ z]\circ x=y\circ [(x\circ z)\circ
x]~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
[(yR_g^{-1}\oplus xL_f^{-1})R_g^{-1}\oplus zL_f^{-1}]R_g^{-1}\oplus
xL_f^{-1}=yR_g^{-1}\oplus [(xR_g^{-1}\oplus zL_f^{-1})R_g^{-1}\oplus
xL_f^{-1}]L_f^{-1}. \end{displaymath} Replacing $yR_g^{-1}$ by $y'$,
$zL_f^{-1}$ by $z'$ and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
(y'R_{f^\rho}R_g^{-1}\oplus z')R_g^{-1}R_{f^\rho}=y'\oplus
z'L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_f^{-1}. \end{displaymath} Again,
replace $y'R_{f^\rho}R_g^{-1}$ by $y''$ so that
\begin{equation}gin{displaymath}
(y''\oplus z')R_g^{-1}R_{f^\rho}=y''R_gR_{f^\rho}^{-1}\oplus
z'L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_f^{-1}I\!\!Rightarrow
(R_gR_{f^\rho}^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})
\end{displaymath}
is an autotopism of an SRB-subloop $(S,\oplus )$ of the S-loop $(G,\oplus )$ such that $f$ and $g$ are S-elements.\\
On the other hand, for a SLB-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
[x\circ (y\circ x)]\circ z=x\circ [y\circ (x\circ
z)]~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
[xR_g^{-1}\oplus (yR_g^{-1}\oplus xL_f^{-1})L_f^{-1}]R_g^{-1}\oplus
zL_f^{-1}=xR_g^{-1}\oplus [yR_g^{-1}\oplus (xR_g^{-1}\oplus
zL_f^{-1})L_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus z'=(y'\oplus
z'L_{g^\leftarrowmbda}L_f^{-1})L_f^{-1}L_{g^\leftarrowmbda}.
\end{displaymath} Again, replace $z'L_{g^\leftarrowmbda}L_f^{-1}$ by $z''$
so that
\begin{equation}gin{displaymath}
y'R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z''L_fL_{g^\leftarrowmbda}^{-1}=(y'\oplus
z'')L_f^{-1}L_{g^\leftarrowmbda}I\!\!Rightarrow
(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda})
\end{displaymath}
is an autotopism of an SLB-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
\begin{equation}gin{myth}\leftarrowbel{1:7}
A Smarandache Moufang loop is universal if all its $f,g$-principal
isotopes are Smarandache $f,g$-principal isotopes. Conversely, if a
Smarandache Moufang loop is universal then
\begin{equation}gin{displaymath}
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}),
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda}),
(R_gR_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1},R_g^{-1}R_{f^\rho}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_{g^\leftarrowmbda}^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho})
\end{displaymath}
are autotopisms of an SM-subloop of the SML such that $f$ and $g$
are S-elements.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a SML with a SM-subloop $(S,\oplus )$. If
$(G,\circ )$ is an arbitrary $f,g$-principal isotope of
$(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of
$(G,\circ)$ if $(S,\circ )$ is a Smarandache $f,g$-principal isotope
of $(S,\oplus )$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
It is already known from \cite{phd3} that Moufang loops are
universal, hence $(S,\circ )$ is a Moufang loop thus an SM-subloop
of $(G,\circ)$. By Theorem~\ref{1:1}, for any isotope $(H,\otimes )$
of $(G,\oplus)$, there exists a $(G,\circ )$ such that $(H,\otimes
)\cong (G,\circ )$. So we can now choose the isomorphic image of
$(S,\circ)$ which will now be an SM-subloop in $(H,\otimes )$. So,
$(H,\otimes )$ is an SML. This conclusion can also be drawn straight
from Corollary~\ref{1:2}.
The proof of the converse is as follows. If a SML $(G,\oplus )$ is
universal then every isotope $(H,\otimes)$ is an SML i.e there
exists an SM-subloop $(S,\otimes )$ in $(H,\otimes )$. Let $(G,\circ
)$ be the $f,g$-principal isotope of $(G,\oplus)$, then by
Corollary~\ref{1:2}, $(G,\circ)$ is an SML with say an SM-subloop
$(S,\circ)$. For an SM-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
(x\circ y)\circ (z\circ x)=[x\circ (y\circ z)]\circ
x~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus (zR_g^{-1}\oplus
xL_f^{-1})L_f^{-1}=[xR_g^{-1}\oplus (yR_g^{-1}\oplus
zL_f^{-1})L_f^{-1}]R_g^{-1}\oplus xL_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}=(y'\oplus
z')L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}I\!\!Rightarrow
\end{displaymath}
\begin{equation}gin{displaymath}
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho})
\end{displaymath}
is an autotopism of an SM-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Again, for an SM-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
(x\circ y)\circ (z\circ x)=x\circ [(y\circ z)\circ x]
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus (zR_g^{-1}\oplus
xL_f^{-1})L_f^{-1}=xR_g^{-1}\oplus [(yR_g^{-1}\oplus
zL_f^{-1})R_g^{-1}\oplus xL_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}=(y'\oplus
z')R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}I\!\!Rightarrow
\end{displaymath}
\begin{equation}gin{displaymath}
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda})
\end{displaymath}
is an autotopism of an SM-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Also, if $(S,\circ)$ is an SM-subloop then,
\begin{equation}gin{displaymath}
[(x\circ y)\circ x]\circ z=x\circ [y\circ (x\circ z)]
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
[(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus xL_f^{-1}]R_g^{-1}\oplus
zL_f^{-1}=xR_g^{-1}\oplus [yR_g^{-1}\oplus (xR_g^{-1}\oplus
zL_f^{-1})L_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1}\oplus
z'=(y'\oplus z'L_{g^\leftarrowmbda}L_f^{-1})L_f^{-1}L_{g^\leftarrowmbda}.
\end{displaymath}
Again, replace $z'L_{g^\leftarrowmbda}L_f^{-1}$ by $z''$ so that
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1}\oplus
z''L_fL_{g^\leftarrowmbda}^{-1}=(y'\oplus
z'')L_f^{-1}L_{g^\leftarrowmbda}I\!\!Rightarrow
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda})
\end{displaymath}
is an autotopism of an SM-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Furthermore, if $(S,\circ)$ is an SM-subloop then,
\begin{equation}gin{displaymath}
[(y\circ x)\circ z]\circ x=y\circ [x\circ (z\circ x)]
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
[(yR_g^{-1}\oplus xL_f^{-1})R_g^{-1}\oplus zL_f^{-1}]R_g^{-1}\oplus
xL_f^{-1}=yR_g^{-1}\oplus [xR_g^{-1}\oplus (zR_g^{-1}\oplus
xL_f^{-1})L_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
(y'R_{f^\rho}R_g^{-1}\oplus z')R_g^{-1}R_{f^\rho}=y'\oplus
z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1}.
\end{displaymath}
Again, replace $y'R_{f^\rho}R_g^{-1}$ by $y''$ so that
\begin{equation}gin{displaymath}
(y''\oplus z')R_g^{-1}R_{f^\rho}=y''R_gR_{f^\rho}^{-1}\oplus
z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1}I\!\!Rightarrow
(R_gR_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1},R_g^{-1}R_{f^\rho})
\end{displaymath}
is an autotopism of an SM-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Lastly, $(S,\oplus)$ is an SM-subloop if and only if $(S,\circ)$ is
an SRB-subloop and an SLB-subloop. So by Theorem~\ref{1:6}, ${\cal
T}_1$ and ${\cal T}_2$ are autotopisms in $(S,\oplus)$, hence ${\cal
T}_1{\cal T}_2$ and ${\cal T}_2{\cal T}_1$ are autotopisms in
$(S,\oplus)$.
\begin{equation}gin{myth}\leftarrowbel{1:8}
A Smarandache extra loop is universal if all its $f,g$-principal
isotopes are Smarandache $f,g$-principal isotopes. Conversely, if a
Smarandache extra loop is universal then
$(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_{f^\rho}^{-1}R_gL_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1}R_g)$,
\begin{equation}gin{displaymath}
(R_gR_{f^\rho}^{-1}R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_{g^\leftarrowmbda}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}),
(R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1}L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})
\end{displaymath}
\begin{equation}gin{displaymath}
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}),
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda}),
(R_gR_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1},R_g^{-1}R_{f^\rho}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_{g^\leftarrowmbda}^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}),
\end{displaymath}
are autotopisms of an SE-subloop of the SEL such that $f$ and $g$
are S-elements.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a SEL with a SE-subloop $(S,\oplus )$. If
$(G,\circ )$ is an arbitrary $f,g$-principal isotope of
$(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of
$(G,\circ)$ if $(S,\circ )$ is a Smarandache $f,g$-principal isotope
of $(S,\oplus )$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
In \cite{phd34} and \cite{phd36} respectively, it was shown and
stated that a loop is an extra loop if and only if it is a Moufang
loop and a CC-loop. But since CC-loops are G-loops(they are
isomorphic to all loop isotopes) then extra loops are universal,
hence $(S,\circ )$ is an extra loop thus an SE-subloop of
$(G,\circ)$. By Theorem~\ref{1:1}, for any isotope $(H,\otimes )$ of
$(G,\oplus)$, there exists a $(G,\circ )$ such that $(H,\otimes
)\cong (G,\circ )$. So we can now choose the isomorphic image of
$(S,\circ)$ which will now be an SE-subloop in $(H,\otimes )$. So,
$(H,\otimes )$ is an SEL. This conclusion can also be drawn straight
from Corollary~\ref{1:2}.
The proof of the converse is as follows. If a SEL $(G,\oplus )$ is
universal then every isotope $(H,\otimes)$ is an SEL i.e there
exists an SE-subloop $(S,\otimes )$ in $(H,\otimes )$. Let $(G,\circ
)$ be the $f,g$-principal isotope of $(G,\oplus)$, then by
Corollary~\ref{1:2}, $(G,\circ)$ is an SEL with say an SE-subloop
$(S,\circ)$. For an SE-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
[(x\circ y)\circ z]\circ x=x\circ [y\circ (z\circ
x)]~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
[(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus zL_f^{-1}]R_g^{-1}\oplus
xL_f^{-1}=xR_g^{-1}\oplus [yR_g^{-1}\oplus (zR_g^{-1}\oplus
xL_f^{-1})L_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
(y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z')R_g^{-1}R_{f^\rho}=(y'\oplus
z'L_fR_g^{-1}R_{f^\rho}L_f^{-1})L_f^{-1}L_{g^\leftarrowmbda}.
\end{displaymath}
Again, replace $z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}$ by $z''$ so that
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z''L_fR_{f^\rho}^{-1}R_gL_f^{-1}=(y'\oplus
z'')L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1}R_gI\!\!Rightarrow
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_{f^\rho}^{-1}R_gL_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1}R_g)
\end{displaymath}
is an autotopism of an SE-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Again, for an SE-subloop $(S,\circ)$,
\begin{equation}gin{displaymath}
(x\circ y)\circ (x\circ z)=x\circ [(y\circ x)\circ z]
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
(xR_g^{-1}\oplus yL_f^{-1})R_g^{-1}\oplus (xR_g^{-1}\oplus
zL_f^{-1})L_f^{-1}=xR_g^{-1}\oplus [(yR_g^{-1}\oplus
xL_f^{-1})R_g^{-1}\oplus zL_f^{-1}]L_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z'L_{g^\leftarrowmbda}L_f^{-1}=(y'R_{f^\rho}R_g^{-1}\oplus
z')L_f^{-1}L_{g^\leftarrowmbda}.
\end{displaymath}
Again, replace $y'R_{f^\rho}R_g^{-1}$ by $y''$ so that
\begin{equation}gin{displaymath}
y''R_gR_{f^\rho}^{-1}R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}\oplus
z'L_{g^\leftarrowmbda}L_f^{-1}=(y''\oplus
z')L_f^{-1}L_{g^\leftarrowmbda}I\!\!Rightarrow
(R_gR_{f^\rho}^{-1}R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_{g^\leftarrowmbda}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda})
\end{displaymath}
is an autotopism of an SE-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Also, if $(S,\circ)$ is an SE-subloop then,
\begin{equation}gin{displaymath}
(y\circ x)\circ (z\circ x)=[y\circ (x\circ z)]\circ x
~\forall~x,y,z\in S\end{displaymath} where
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
Thus,
\begin{equation}gin{displaymath}
(yR_g^{-1}\oplus xL_f^{-1})R_g^{-1}\oplus (zR_g^{-1}\oplus
xL_f^{-1})L_f^{-1}= [(yR_g^{-1}\oplus (xR_g^{-1}\oplus
zL_f^{-1})L_f^{-1}]R_g^{-1}\oplus xL_f^{-1}.
\end{displaymath} Replacing $yR_g^{-1}$ by $y'$, $zL_f^{-1}$ by $z'$
and taking $x=e$ in $(S,\oplus)$ we have
\begin{equation}gin{displaymath}
y'R_{f^\rho}R_g^{-1}\oplus z'L_fR_g^{-1}R_{f^\rho}L_f^{-1}=(y'\oplus
z'L_{g^\leftarrowmbda}L_f^{-1})R_g^{-1}R_{f^\rho}.
\end{displaymath}
Again, replace $z'L_{g^\leftarrowmbda}L_f^{-1}$ by $z''$ so that
\begin{equation}gin{displaymath}
y'R_{f^\rho}R_g^{-1}\oplus
z''L_fL_{g^\leftarrowmbda}^{-1}L_fR_g^{-1}R_{f^\rho}L_f^{-1}=(y'\oplus
z')R_g^{-1}R_{f^\rho}I\!\!Rightarrow
(R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1}L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})
\end{displaymath}
is an autotopism of an SE-subloop $(S,\oplus )$ of the S-loop
$(G,\oplus )$ such that $f$ and $g$ are S-elements.
Lastly, $(S,\oplus)$ is an SE-subloop if and only if $(S,\circ)$ is
an SM-subloop and an SCC-subloop. So by Theorem~\ref{1:7}, the six
remaining triples are autotopisms in $(S,\oplus)$.
\subsection*{Universality of Smarandache Inverse Property Loops}
\begin{equation}gin{myth}\leftarrowbel{1:9}
A Smarandache left(right) inverse property loop in which all its
$f,g$-principal isotopes are Smarandache $f,g$-principal isotopes is
universal if and only if it is a Smarandache left(right) Bol loop in
which all its $f,g$-principal isotopes are Smarandache
$f,g$-principal isotopes.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a SLIPL with a SLIP-subloop $(S,\oplus )$. If
$(G,\circ )$ is an arbitrary $f,g$-principal isotope of
$(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of
$(G,\circ)$ if $(S,\circ )$ is a Smarandache $f,g$-principal isotope
of $(S,\oplus )$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
$(G,\oplus)$ is a universal SLIPL if and only if every isotope
$(H,\otimes )$ is a SLIPL. $(H,\otimes )$ is a SLIPL if and only if
it has at least a SLIP-subloop $(S,\otimes )$. By Theorem~\ref{1:1},
for any isotope $(H,\otimes )$ of $(G,\oplus)$, there exists a
$(G,\circ )$ such that $(H,\otimes )\cong (G,\circ )$. So we can now
choose the isomorphic image of $(S,\circ)$ to be $(S,\otimes )$
which is already a SLIP-subloop in $(H,\otimes )$. So, $(S,\circ)$
is also a SLIP-subloop in $(G,\circ )$. As shown in \cite{phd3},
$(S,\oplus )$ and its $f,g$-isotope(Smarandache $f,g$-isotope)
$(S,\circ)$ are SLIP-subloops if and only if $(S,\oplus )$ is a left
Bol subloop(i.e a SLB-subloop). So, $(G,\oplus)$ is SLBL.
Conversely, if $(G,\oplus)$ is SLBL, then there exists a
SLB-subloop $(S,\oplus )$ in $(G,\oplus)$. If $(G,\circ )$ is an
arbitrary $f,g$-principal isotope of $(G,\oplus)$, then by
Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of $(G,\circ)$ if
$(S,\circ )$ is a Smarandache $f,g$-principal isotope of $(S,\oplus
)$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
By Theorem~\ref{1:1}, for any isotope $(H,\otimes )$ of
$(G,\oplus)$, there exists a $(G,\circ )$ such that $(H,\otimes
)\cong (G,\circ )$. So we can now choose the isomorphic image of
$(S,\circ)$ to be $(S,\otimes )$ which is a SLB-subloop in
$(H,\otimes )$ using the same reasoning in Theorem~\ref{1:6}. So,
$(S,\circ)$ is a SLB-subloop in $(G,\circ )$. Left Bol loops have
the left inverse property(LIP), hence, $(S,\oplus )$ and $(S,\circ)$
are SLIP-subloops in $(G,\oplus)$ and $(G,\circ )$ respectively.
Thence, $(G,\oplus)$ and $(G,\circ )$ are SLBLs. Therefore,
$(G,\oplus)$ is
a universal SLIPL.\\
The proof for a Smarandache right inverse property loop is similar
and is as follows. Let $(G,\oplus)$ be a SRIPL with a SRIP-subloop
$(S,\oplus )$. If $(G,\circ )$ is an arbitrary $f,g$-principal
isotope of $(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a
subloop of $(G,\circ)$ if $(S,\circ )$ is a Smarandache
$f,g$-principal isotope of $(S,\oplus )$. Let us choose all
$(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
$(G,\oplus)$ is a universal SRIPL if and only if every isotope
$(H,\otimes )$ is a SRIPL. $(H,\otimes )$ is a SRIPL if and only if
it has at least a SRIP-subloop $(S,\otimes )$. By Theorem~\ref{1:1},
for any isotope $(H,\otimes )$ of $(G,\oplus)$, there exists a
$(G,\circ )$ such that $(H,\otimes )\cong (G,\circ )$. So we can now
choose the isomorphic image of $(S,\circ)$ to be $(S,\otimes )$
which is already a SRIP-subloop in $(H,\otimes )$. So, $(S,\circ)$
is also a SRIP-subloop in $(G,\circ )$. As shown in \cite{phd3},
$(S,\oplus )$ and its $f,g$-isotope(Smarandache $f,g$-isotope)
$(S,\circ)$ are SRIP-subloops if and only if $(S,\oplus )$ is a
right Bol subloop(i.e a SRB-subloop). So, $(G,\oplus)$ is SRBL.
Conversely, if $(G,\oplus)$ is SRBL, then there exists a
SRB-subloop $(S,\oplus )$ in $(G,\oplus)$. If $(G,\circ )$ is an
arbitrary $f,g$-principal isotope of $(G,\oplus)$, then by
Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of $(G,\circ)$ if
$(S,\circ )$ is a Smarandache $f,g$-principal isotope of $(S,\oplus
)$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
By Theorem~\ref{1:1}, for any isotope $(H,\otimes )$ of
$(G,\oplus)$, there exists a $(G,\circ )$ such that $(H,\otimes
)\cong (G,\circ )$. So we can now choose the isomorphic image of
$(S,\circ)$ to be $(S,\otimes )$ which is a SRB-subloop in
$(H,\otimes )$ using the same reasoning in Theorem~\ref{1:6}. So,
$(S,\circ)$ is a SRB-subloop in $(G,\circ )$. Right Bol loops have
the right inverse property(RIP), hence, $(S,\oplus )$ and
$(S,\circ)$ are SRIP-subloops in $(G,\oplus)$ and $(G,\circ )$
respectively. Thence, $(G,\oplus)$ and $(G,\circ )$ are SRBLs.
Therefore, $(G,\oplus)$ is a universal SRIPL.
\begin{equation}gin{myth}\leftarrowbel{1:10}
A Smarandache inverse property loop in which all its $f,g$-principal
isotopes are Smarandache $f,g$-principal isotopes is universal if
and only if it is a Smarandache Moufang loop in which all its
$f,g$-principal isotopes are Smarandache $f,g$-principal isotopes.
\end{myth}
{\bf Proof}\\
Let $(G,\oplus)$ be a SIPL with a SIP-subloop $(S,\oplus )$. If
$(G,\circ )$ is an arbitrary $f,g$-principal isotope of
$(G,\oplus)$, then by Lemma~\ref{1:3}, $(S,\circ )$ is a subloop of
$(G,\circ)$ if $(S,\circ )$ is a Smarandache $f,g$-principal isotope
of $(S,\oplus )$. Let us choose all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
$(G,\oplus)$ is a universal SIPL if and only if every isotope
$(H,\otimes )$ is a SIPL. $(H,\otimes )$ is a SIPL if and only if it
has at least a SIP-subloop $(S,\otimes )$. By Theorem~\ref{1:1}, for
any isotope $(H,\otimes )$ of $(G,\oplus)$, there exists a $(G,\circ
)$ such that $(H,\otimes )\cong (G,\circ )$. So we can now choose
the isomorphic image of $(S,\circ)$ to be $(S,\otimes )$ which is
already a SIP-subloop in $(H,\otimes )$. So, $(S,\circ)$ is also a
SIP-subloop in $(G,\circ )$. As shown in \cite{phd3}, $(S,\oplus )$
and its $f,g$-isotope(Smarandache $f,g$-isotope) $(S,\circ)$ are
SIP-subloops if and only if $(S,\oplus )$ is a Moufang subloop(i.e a
SM-subloop). So, $(G,\oplus)$ is SML.
Conversely, if $(G,\oplus)$ is SML, then there exists a SM-subloop
$(S,\oplus )$ in $(G,\oplus)$. If $(G,\circ )$ is an arbitrary
$f,g$-principal isotope of $(G,\oplus)$, then by Lemma~\ref{1:3},
$(S,\circ )$ is a subloop of $(G,\circ)$ if $(S,\circ )$ is a
Smarandache $f,g$-principal isotope of $(S,\oplus )$. Let us choose
all $(S,\circ )$ in this manner. So,
\begin{equation}gin{displaymath}
x\circ y=xR_g^{-1}\oplus yL_f^{-1}~\forall~x,y\in S.
\end{displaymath}
By Theorem~\ref{1:1}, for any isotope $(H,\otimes )$ of
$(G,\oplus)$, there exists a $(G,\circ )$ such that $(H,\otimes
)\cong (G,\circ )$. So we can now choose the isomorphic image of
$(S,\circ)$ to be $(S,\otimes )$ which is a SM-subloop in
$(H,\otimes )$ using the same reasoning in Theorem~\ref{1:6}. So,
$(S,\circ)$ is a SM-subloop in $(G,\circ )$. Moufang loops have the
inverse property(IP), hence, $(S,\oplus )$ and $(S,\circ)$ are
SIP-subloops in $(G,\oplus)$ and $(G,\circ )$ respectively. Thence,
$(G,\oplus)$ and $(G,\circ )$ are SMLs. Therefore, $(G,\oplus)$ is a
universal SIPL.
\begin{equation}gin{mycor}\leftarrowbel{1:11}
If a Smarandache left(right) inverse property loop is universal then
\begin{equation}gin{displaymath}
(R_gR_{f^\rho}^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho})\bigg(
(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda})\bigg)
\end{displaymath}
is an autotopism of an SLIP(SRIP)-subloop of the SLIPL(SRIPL) such
that $f$ and $g$ are S-elements.
\end{mycor}
{\bf Proof}\\
This follows by Theorem~\ref{1:9} and Theorem~\ref{1:11}.
\begin{equation}gin{mycor}\leftarrowbel{1:12}
If a Smarandache inverse property loop is universal then
\begin{equation}gin{displaymath}
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}),
(R_{g}L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}R_g^{-1},L_fL_{g^\leftarrowmbda}^{-1},L_f^{-1}L_{g^\leftarrowmbda}),
(R_gR_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}L_f^{-1},R_g^{-1}R_{f^\rho}),
\end{displaymath}
\begin{equation}gin{displaymath}
(R_gL_f^{-1}L_{g^\leftarrowmbda}R_g^{-1},L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho}L_{g^\leftarrowmbda}^{-1},R_g^{-1}R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}),
(R_{f^\rho}L_f^{-1}L_{g^\leftarrowmbda}R_{f^\rho}^{-1},L_fR_g^{-1}R_{f^\rho}L_f^{-1},L_f^{-1}L_{g^\leftarrowmbda}R_g^{-1}R_{f^\rho})
\end{displaymath}
are autotopisms of an SIP-subloop of the SIPL such that $f$ and $g$
are S-elements.
\end{mycor}
{\bf Proof}\\
This follows from Theorem~\ref{1:10} and Theorem~\ref{1:7}.
\begin{equation}gin{thebibliography}{99}
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\end{document} |
\begin{document}
\newcounter{algnum}
\newcounter{step}
\newtheorem{alg}{Algorithm}
\newenvironment{algorithm}{\begin{alg}\mathcal End{alg}}
\mathcal Title[Joint spectral radius, Sturmian measures, finiteness conjecture]
{Joint spectral radius, Sturmian measures, and the finiteness conjecture}
\author{O.~Jenkinson \& M. Pollicott}
\address{Oliver Jenkinson;
School of Mathematical Sciences, Queen
Mary, University of London, Mile End Road, London, E1 4NS, UK.
\newline {\mathcal Tt omj@maths.qmul.ac.uk}}
\address{Mark Pollicott;
Mathematics Institute,
University of Warwick,
Coventry, CV4 7AL, UK.
\newline {\mathcal Tt mpollic@maths.warwick.ac.uk}}
\subjclass[2010]{Primary 15A18, 15A60; Secondary 37A99, 37B10, 68R15}
\begin{abstract}
The joint spectral radius of a pair of $2 \mathcal Times 2$ real matrices $(A_0,A_1)\in M_2(\mathcal Mathbb{R})^2$
is defined to be
$r(A_0,A_1)= \mathcal Limsup_{n\mathcal To\infty} \mathcal Max \{\|A_{i_1}\cdots A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$,
the optimal growth rate of the norm of products of these matrices.
The Lagarias-Wang finiteness conjecture \cite{lagariaswang}, asserting that $r(A_0,A_1)$ is
always the $n$th root of the spectral radius of some length-$n$ product $A_{i_1}\cdots A_{i_n}$, has been refuted by
Bousch \& Mairesse \cite{bouschmairesse}, with subsequent counterexamples presented by
Blondel, Theys \& Vladimirov \cite{btv}, Kozyakin \cite{kozyakin}, Hare, Morris, Sidorov \& Theys
\cite{hmst}.
In this article we introduce a new approach to generating finiteness counterexamples,
and use this to exhibit an open subset of $M_2(\mathcal Mathbb{R})^2$ with the property that each member $(A_0,A_1)$ of the subset generates
uncountably many counterexamples of the form $(A_0,tA_1)$.
Our methods employ ergodic theory, in particular the analysis of Sturmian invariant measures;
this approach allows a short proof
that the relation between the parameter $t$ and the Sturmian parameter $\mathcal Mathcal{P}(t)$ is a devil's staircase.
\mathcal End{abstract}
\mathcal Maketitle
\section{Introduction}\mathcal Label{generalsection}
\subsection{Problem and setting}\mathcal Label{problemsetting}
For a square matrix $A$ with real entries, its \mathcal Emph{spectral radius} $r(A)$, defined as the maximum modulus of its eigenvalues, satisfies \mathcal Emph{Gelfand's formula}
$$
r(A)= \mathcal Lim_{n\mathcal To\infty} \|A^n\|^{1/n}\,,
$$
where $\|\cdot\|$ is a matrix norm.
More generally, for a finite collection $\mathcal Mathcal A=\{A_0,\mathcal Ldots, A_l\}$ of real square matrices, all of the same size,
the \mathcal Emph{joint spectral radius} $r(\mathcal Mathcal A)$ is defined by
\begin{equation}\mathcal Label{jsrlimsup}
r(\mathcal Mathcal A) =
\mathcal Limsup_{n\mathcal To\infty} \mathcal Max \{\|A_{i_1}\cdots A_{i_n}\|^{1/n}: i_j\in\{0,\mathcal Ldots, l\}\} \,,
\mathcal End{equation}
or equivalently (see e.g.~\cite{jungers}) by
\begin{equation}\mathcal Label{rjsr}
r(\mathcal Mathcal A) =
\mathcal Lim_{n \mathcal To +\infty} \mathcal Max
\{ r(A_{i_1}\cdots A_{i_n})^{1/n}: i_j\in\{0, \dots,l\}\} \,.
\mathcal End{equation}
The notion of joint spectral radius was introduced by Rota \& Strang \cite{rotastrang},
and notably popularised by Daubechies \& Lagarias \cite{daubechieslagarias} in their work on wavelets.
Since the 1990s it has become an area of very active research interest, from both a pure and an applied perspective
(see e.g.~\cite{blondel, jungers, kozyakinbiblio, strang}).
The set $\mathcal Mathcal A$ is said to have the \mathcal Emph{finiteness property} if $r(\mathcal Mathcal A)=r(A_{i_1}\cdots A_{i_n})^{1/n}$
for some $i_1,\mathcal Ldots, i_n\in\{0,\mathcal Ldots, l\}$.
It was conjectured by Lagarias \& Wang \cite{lagariaswang}
(see also Gurvits \cite{gurvits}) that every such $\mathcal Mathcal A$ enjoys the finiteness property.
This so-called finiteness conjecture was, however, refuted by Bousch \& Mairesse \cite{bouschmairesse}, and a number of authors (see \cite{btv,hmst, kozyakin, morrissidorov}) have subsequently given examples of sets $\mathcal Mathcal A$ for which the finiteness property fails.
A common feature of these finiteness counterexamples has been a judicious choice of a pair of $2 \mathcal Times 2$ matrices $A_0,A_1$, followed by an argument that for certain $t>0$, the finiteness property fails for the set $\mathcal Mathcal A(t)= \{ A_0^{(t)}, A_1^{(t)}\} = \{A_0, tA_1\}$.
In fact for many of these examples it has been observed that the family $(\mathcal Mathcal A(t))_{t>0}$ can be associated with the class of \mathcal Emph{Sturmian} sequences of Morse \& Hedlund \cite{morsehedlund}:
for a given $t>0$ an appropriate Sturmian sequence $(i_n)_{n=1}^\infty \in\{0,1\}^\mathbb N$ turns out to give the optimal matrix product, in the sense that the joint spectral radius $r(\mathcal Mathcal A(t))$ equals
$\mathcal Lim_{n\mathcal To\infty} r(A_{i_1}^{(t)}\cdots A_{i_n}^{(t)})^{1/n}$
(see \cite{btv,bouschmairesse,hmst,kozyakin,morrissidorov} for further details).
A Sturmian sequence $(i_n)_{n=1}^\infty$ has a well-defined \mathcal Emph{1-frequency}
$\mathcal Mathcal{P} = \mathcal Lim_{N\mathcal To\infty} \frac{1}{N} \sum_{n=1}^N i_n$, and it is those sets $\mathcal Mathcal A(t)$ whose associated Sturmian sequences\footnote{We follow the definition of Sturmian sequence given in \cite{bullettsentenac}, though note that
some authors refer to these as \mathcal Emph{balanced} sequences,
reserving the nomenclature \mathcal Emph{Sturmian} precisely for those balanced sequences with irrational 1-frequency.}
have \mathcal Emph{irrational} 1-frequency which yield counterexamples to the finiteness conjecture
(see Proposition \mathcal Ref{irrationalcounterexampleconnection} below for a more precise description of the connection between finiteness counterexamples and Sturmian sequences with irrational 1-frequency).
For certain such families $(\mathcal Mathcal A(t))_{t>0}$ (which henceforth we refer to as \mathcal Emph{Sturmian families}), it has been proved by Morris \& Sidorov \cite{morrissidorov}
(see also \cite[p.~109]{bouschmairesse})
that
if $\mathcal Mathcal{P}(t)$ denotes the 1-frequency associated
to $\mathcal Mathcal A(t)$, then the parameter
mapping $t\mathcal Mapsto \mathcal Mathcal{P}(t)$ is continuous and monotone, but \mathcal Emph{singular}
in the sense that
$\{t>0:\mathcal Mathcal{P}(t)\notin \mathcal Mathbb{Q}\}$ is nowhere dense; in other words,
the uncountably many parameters
$t$ for which finiteness counterexamples occur only constitute a thin subset\footnote{The belief that finiteness counterexamples are rare appears to be widespread; for example Maesumi \cite{maesumi} conjectures
that they consitute a set of (Lebesgue) measure zero in the space of matrices.}
of $\mathcal Mathbb{R}^+$.
Examples of Sturmian families $(\mathcal Mathcal A(t))_{t>0}$
have been given by
Bousch \& Mairesse \cite{bouschmairesse}, who considered the family
generated by matrix pairs of the form
\begin{equation}\mathcal Label{bmfamily}
\mathcal Mathcal A= \mathcal Left( \begin{pmatrix} e^{\kappa h_0}+1 & 0 \cr e^\kappa & 1 \cr \mathcal End{pmatrix}\, , \
\begin{pmatrix} 1 & e^\kappa \cr 0 & e^{\kappa h_1} +1 \cr \mathcal End{pmatrix} \mathcal Right)\,,\,
\kappa>0\,, h_0,h_1>0\,, h_0+h_1 < 2\,,
\mathcal End{equation}
by Kozyakin \cite{kozyakin}, who studied the family
generated by pairs of the form
\begin{equation}\mathcal Label{kozyakinpairs}
\mathcal Mathcal A = \mathcal Left( \begin{pmatrix} 1 & 0 \cr c & d \cr \mathcal End{pmatrix}\, , \
\begin{pmatrix} a & b \cr 0 & 1 \cr \mathcal End{pmatrix} \mathcal Right) \,,\,
0<a,d<1\mathcal Le bc\,,
\mathcal End{equation}
and by various authors \cite{btv,hmst,morrissidorov} focusing on the family generated by
the particular pair
\begin{equation}\mathcal Label{standardpair}
\mathcal Mathcal A = \mathcal Left( \begin{pmatrix} 1 & 0 \cr 1 & 1 \cr \mathcal End{pmatrix}\, , \
\begin{pmatrix} 1 & 1 \cr 0 & 1 \cr \mathcal End{pmatrix} \mathcal Right) \,.
\mathcal End{equation}
For an invertible matrix $P$,
the simultaneous similarity $(A_0,A_1)\mathcal Mapsto (P^{-1}A_0P, P^{-1}A_1P)$
leaves invariant the joint spectral radius,
and does not change the sequences $(i_n)_{n=1}^\infty$ attaining the optimal matrix product,
while if $u,v>0$ then $(uA_0,vA_1)$
has the same optimizing sequences as $(A_0, (v/u)A_1)$.
Therefore, declaring $\mathcal Mathcal A=(A_0,A_1)$ and $\mathcal Mathcal A'=(A_0',A_1')$ to be equivalent if
$A_0'=uP^{-1}A_0P$ and $A_1'=vP^{-1}A_1P$ for some invertible $P$ and $u,v>0$, we see that the
equivalence of $\mathcal Mathcal A$ and $\mathcal Mathcal A'$ implies that $(\mathcal Mathcal A(t))_{t>0}$ is a Sturmian family if and only if
$(\mathcal Mathcal A'(t))_{t>0}$ is.
In particular, $(\mathcal Mathcal A'(t))_{t>0}$ is a Sturmian family whenever $\mathcal Mathcal A'$ is equivalent to a matrix pair $\mathcal Mathcal A$ of the form
(\mathcal Ref{bmfamily}), (\mathcal Ref{kozyakinpairs}), or (\mathcal Ref{standardpair}).
The purpose of this article is to introduce an approach to studying the joint spectral radius and
generating finiteness counterexamples, which in particular yields new examples of Sturmian
families $(\mathcal Mathcal A(t))_{t>0}$, i.e.~where $\mathcal Mathcal A$ is not equivalent to a matrix pair of the form
(\mathcal Ref{bmfamily}), (\mathcal Ref{kozyakinpairs}), or (\mathcal Ref{standardpair}).
Our method is conceptually different to previous authors, employing notions from
dynamical systems, ergodic theory, and in particular ergodic optimization (see e.g.~\cite{jeo}).
Specifically, we identify a dynamical system $T_\mathcal Mathcal A$ with the matrix pair $\mathcal Mathcal A=(A_0,A_1)$,
and cast the problem of determining the joint spectral radius $r(\mathcal Mathcal A)$ in terms of ergodic optimization
(see Theorem \mathcal Ref{maxtheoremintro} below): it suffices to determine the
$T_\mathcal Mathcal A$-invariant probability measure which maximizes the integral of a certain auxiliary real-valued function $f_\mathcal Mathcal A$.
Working with the family of \mathcal Emph{$\mathcal Mathcal A$-Sturmian measures} (certain probability measures
invariant under $T_\mathcal Mathcal A$) instead of Sturmian sequences, we exploit a characterisation of these measures in terms of the smallness of their support to show that they give precisely the family
of $f_{\mathcal Mathcal A(t)}$-maximizing measures, $t>0$.
In particular, whenever the $f_{\mathcal Mathcal A(t)}$-maximizing measure is Sturmian of \mathcal Emph{irrational} parameter
$\mathcal Mathcal{P}(t)$ then $\mathcal Mathcal A(t)$ is a finiteness counterexample (cf.~Proposition \mathcal Ref{irrationalcounterexampleconnection}).
The $\mathcal Mathcal A$-Sturmian measures are naturally identified with Sturmian measures on $\Omega = \{0,1\}^\mathbb N$, the full shift on two symbols (see Notation \mathcal Ref{sturmianfullshift}). A notable feature of our approach is that the singularity of the parameter mapping
$t\mathcal Mapsto \mathcal Mathcal{P}(t)$ (and in particular the fact that $\{t>0:\mathcal Mathcal{P}(t)\notin\mathcal Mathbb{Q}\}$ is nowhere dense in $\mathcal Mathbb{R}^+$)
is then readily deduced (see Theorem \mathcal Ref{deviltheorem} in \S \mathcal Ref{devilsection}) as a consequence of classical facts about
parameter dependence of Sturmian measures on $\Omega$
(i.e.~rather than requiring the {\it ab initio} approach of \cite{morrissidorov}).
\subsection{Statement of results}\mathcal Label{statementsubsection}
We use $M_2(\mathcal Mathbb{R})$ to denote the set of real $2 \mathcal Times 2$ matrices,
and focus attention on certain of its open subsets:
\begin{notation}
$M_2(\mathcal Mathbb{R}^+)$ will denote the set of \mathcal Emph{positive matrices}, i.e.~matrices in $M_2(\mathcal Mathbb{R})$
with entries in $\mathcal Mathbb{R}^+ =\{x\in\mathcal Mathbb{R}:x>0\}$,
and
$M_2^+(\mathcal Mathbb{R}^+)=\{A\in M_2(\mathcal Mathbb{R}^+):\det A>0\}$
will denote the set of \mathcal Emph{positive orientation-preserving matrices}.
\mathcal End{notation}
Turning to \mathcal Emph{pairs} of matrices, we shall consider the following open subset
of $M_2^+(\mathcal Mathbb{R}^+)^2$:
\begin{defn}\mathcal Label{frakmdefn}
Let $\mathcal Mm\subset M_2^+(\mathcal Mathbb{R}^+)^2$ denote the set of matrix pairs
\begin{gather*}
\mathcal Left( A_0, A_1\mathcal Right)
= \mathcal Left( \begin{pmatrix} a_0&b_0\\c_0&d_0\mathcal End{pmatrix} ,
\begin{pmatrix} a_1&b_1\\ c_1& d_1\mathcal End{pmatrix} \mathcal Right) \in M_2^+(\mathcal Mathbb{R}^+)^2
\mathcal End{gather*}
satisfying
\begin{equation}\mathcal Label{definequal1}
\frac{a_0}{c_0} < \frac{b_1}{d_1}
\mathcal End{equation}
and
\begin{equation}\mathcal Label{definequal2}
a_1+c_1-b_1-d_1 < 0 < a_0+c_0-b_0-d_0\,.
\mathcal End{equation}
For reasons which will become apparent later (see Proposition \mathcal Ref{concavconvexprop}),
$\mathcal Mm$ will be referred to as the set of \mathcal Emph{concave-convex matrix pairs}.
\mathcal End{defn}
Finally, our counterexamples to the Lagarias-Wang finiteness conjecture will be drawn from
a certain open subset $\mathfrak D$
(given by Definition \mathcal Ref{defnofnn} below) of $\mathcal Mm$
which is conveniently described in terms of quantities $\mathcal Rho_A$ and $\sigma_A$ defined as follows:
\begin{defn}
For $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$,
define
$$
\mathcal Rho_A = \frac{2b}{a-d-2b+\sqrt{(a-d)^2+4bc}}\,,
$$
and if $a+c\neq b+d$ then define
$$
\sigma_A = \frac{b-a}{a+c-b-d}\,.
$$
\mathcal End{defn}
It turns out (see Corollary \mathcal Ref{rhoposlessminusone}) that if
$(A_0,A_1)\in\mathcal Mm$
then
$\sigma_{A_0}<0<\mathcal Rho_{A_0}$ and $\mathcal Rho_{A_1}<-1$.
The set $\mathfrak D$ is defined by imposing two inequalities:
\begin{defn}\mathcal Label{defnofnn}
Define
\begin{equation*}
\mathfrak D
=
\mathcal Left\{\, (A_0,A_1)\in\mathcal Mm : \
\mathcal Rho_{A_1} < \sigma_{A_0}
\ \mathcal Text{and}\
\sigma_{A_1} < \mathcal Rho_{A_0} \mathcal Right\}\,.
\mathcal End{equation*}
\mathcal End{defn}
Clearly $\mathfrak D$ is an open subset of $\mathcal Mm$, hence also of $M_2(\mathcal Mathbb{R})^2$.
It is also non-empty: for example
it is readily verified that the two-parameter family
\begin{equation}\mathcal Label{2dimreg}
\mathfrak D_2 = \mathcal Left\{
\mathcal Left( \begin{pmatrix} 1&b\\c&1\mathcal End{pmatrix} ,
\begin{pmatrix} 1& c\\ b & 1\mathcal End{pmatrix} \mathcal Right)
: \
bc < 1 < c \ \ , \ \ (b,c)\in (\mathcal Mathbb{R}^+)^2
\mathcal Right\}
\mathcal End{equation}
is a subset of $\mathfrak D$.
Note that the pair (\mathcal Ref{standardpair})
studied in \cite{btv, hmst,morrissidorov}, and
corresponding to $(b,c)=(0,1)$ in (\mathcal Ref{2dimreg}), lies on the boundary of both $\mathfrak D$ and $\mathfrak D_2$.
\mathcal Medskip
A version of our main result is the following:
\begin{theorem}\mathcal Label{nontechnictheorem}
The open subset $\mathfrak D\subset M_2(\mathcal Mathbb{R})^2$ is such that if $\mathcal Mathcal A=(A_0,A_1)\in\mathfrak D$ then
for uncountably many $t\in\mathcal Mathbb{R}^+$, the matrix
pair $(A_0,tA_1)$
is a
finiteness counterexample.
\mathcal End{theorem}
If we define $\mathcal Ee \subset M_2(\mathbb R)^2$
to be the set of matrix pairs which are equivalent to some pair in $\mathfrak D$
(recall that
$\mathcal Mathcal A=(A_0,A_1)$ and $\mathcal Mathcal A'=(A_0',A_1')$ are equivalent if
$A_0'=uP^{-1}A_0P$ and $A_1'=vP^{-1}A_1P$ for some invertible $P$ and $u,v>0$) then clearly:
\begin{cor}\mathcal Label{nontechnictheoremcor}
The open subset $\mathcal Ee\subset M_2(\mathcal Mathbb{R})^2$ is such that if $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Ee$ then
for uncountably many $t\in\mathcal Mathbb{R}^+$, the matrix
pair $(A_0,tA_1)$
is a
finiteness counterexample.
\mathcal End{cor}
\begin{remark}
Theorem \mathcal Ref{nontechnictheorem} yields new finiteness counterexamples, in the sense that $\mathfrak D$ contains matrix pairs
which are not equivalent to pairs satisfying (\mathcal Ref{bmfamily}), (\mathcal Ref{kozyakinpairs}), or (\mathcal Ref{standardpair}).
To see this, note for example that
\begin{equation}\mathcal Label{exampleofa}
\mathcal Mathcal A = (A_0,A_1) =
\mathcal Left( \begin{pmatrix} 5/8 & 3/112 \cr 7/8 & 15/16 \cr \mathcal End{pmatrix}\, , \
\begin{pmatrix} 15/16 & 1 \cr 1/128 & 7/8 \cr \mathcal End{pmatrix} \mathcal Right)
\mathcal End{equation}
belongs to $\mathfrak D$.
Both $A_0$ and $A_1$ have their larger eigenvalue equal to 1, and smaller eigenvalues given by
$\mathcal Lambda_0=9/16$ and $\mathcal Lambda_1=13/16$, respectively.
Now both matrices in (\mathcal Ref{standardpair}) have the single eigenvalue 1, so (\mathcal Ref{exampleofa}) cannot be equivalent to (\mathcal Ref{standardpair}); moreover (\mathcal Ref{exampleofa}) is not equivalent to any matrix pair satisfying (\mathcal Ref{bmfamily}), since both matrices in (\mathcal Ref{bmfamily}) have the property that the larger eigenvalue is more than double the smaller eigenvalue.
Lastly, we show that $\mathcal Mathcal A$ in (\mathcal Ref{exampleofa}) is not equivalent to any pair $\mathcal Mathcal A'$ satisfying (\mathcal Ref{kozyakinpairs}),
i.e.~$\mathcal Mathcal A'=(A_0',A_1')$ with
$A_0' = \begin{pmatrix} 1 & 0 \cr c & d \cr \mathcal End{pmatrix}$,
$A_1'= \begin{pmatrix} a & b \cr 0 & 1 \cr \mathcal End{pmatrix}$,
where $0<a,d<1\mathcal Le bc$.
Note that both $A_0'$ and $A_1'$ have their larger eigenvalue equal to 1 (as is the case for $A_0$ and $A_1$),
and smaller eigenvalues equal to $d$ and $a$, respectively.
Thus if $\mathcal Mathcal A$ and $\mathcal Mathcal A'$ were equivalent then there would exist an invertible $P$ such that
$A_0'=P^{-1}A_0P$ and $A_1'=P^{-1}A_1P$ (i.e.~the positive reals $u,v$ in the above definition of equivalence must both equal 1), so that $d=\mathcal Lambda_0=9/16$, $a=\mathcal Lambda_1=13/16$, and $\mathcal Text{trace}(A_0A_1)=\mathcal Text{trace}(A_0'A_1')$.
In particular,
$1 \mathcal Le bc = \mathcal Text{trace}(A_0'A_1')-d-a = \mathcal Text{trace}(A_0A_1)-\mathcal Lambda_0-\mathcal Lambda_1 = 12995/14336 <1$, a contradiction.
It follows that $\mathcal Mathcal A\in\mathfrak D$ given by (\mathcal Ref{exampleofa}) is not equivalent to any matrix pair satisfying
(\mathcal Ref{kozyakinpairs}).
\mathcal End{remark}
A key tool in proving Theorem \mathcal Ref{nontechnictheorem} is the following Theorem \mathcal Ref{maxtheoremintro}
(proved in \S \mathcal Ref{induceddynsyssection} as Theorem \mathcal Ref{maxtheorem})
characterising the joint spectral radius of $\mathcal Mathcal A\in\mathcal Mm$ in terms of maximizing the integral of a certain
function $f_\mathcal Mathcal A$ over the set $\mathcal M_\mathcal Mathcal A$ of probability measures invariant under an associated mapping $T_\mathcal Mathcal A$.
More precisely, the action of any positive matrix $A$ on $(\mathcal Mathbb{R}^+)^2$ induces a projective map $T_A$ (see \S \mathcal Ref{inducproj}),
and if $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$
then the inverses $T_{A_0}^{-1}$, $T_{A_1}^{-1}$
together define a two-branch
dynamical system $T_\mathcal Mathcal A$ (see \S \mathcal Ref{induceddynsyssection}) on a subset of the unit interval $X$.
Defining the real-valued function $f_\mathcal Mathcal A$,
in terms of the derivative $T_\mathcal Mathcal A'$ and characteristic functions of the images $T_{A_0}(X)$ and $T_{A_1}(X)$,
by
\begin{equation*}
f _\mathcal Mathcal A =
\frac{1}{2}
\mathcal Left(
\mathcal Log T_\mathcal Mathcal A' +(\mathcal Log \det A_0)\mathcal Mathbbm{1}_{T_{A_0}(X)} +(\mathcal Log \det A_1)\mathcal Mathbbm{1}_{T_{A_1}(X)}
\mathcal Right)
\mathcal End{equation*}
then gives:
\begin{theorem}\mathcal Label{maxtheoremintro}
If $\mathcal Mathcal A\in\mathcal Mm$ then
\begin{equation}\mathcal Label{maxtheoremintroeq}
\mathcal Log r(\mathcal Mathcal A) =
\mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu \,.
\mathcal End{equation}
\mathcal End{theorem}
\mathcal Medskip
In order to state a more precise version of Theorem \mathcal Ref{nontechnictheorem},
we first need
some
basic facts concerning ergodic theory, symbolic dynamics, and Sturmian measures:
\begin{notation}\mathcal Label{sturmianfullshift}
Let $\Omega = \{0,1\}^\mathbb N$ denote the set of one-sided sequences $\omega=(\omega_n)_{n=1}^\infty$,
where $\omega_n\in\{0,1\}$ for all $n\ge1$.
When equipped with the product topology, $\Omega$ becomes a compact space,
and the shift map $\sigma:\Omega\mathcal To\Omega$ defined by
$(\sigma\omega)_n = \omega_{n+1}$ for all $n\ge1$ is then continuous.
Let $\mathcal M$ denote the set of shift-invariant Borel probability measures
on $\Omega$; when equipped with the weak-$*$ topology $\mathcal M$ is compact (see \cite[Thm.~6.10]{walters}).
We equip $\Omega$ with the lexicographic order $<$,
and write $[\omega^-,\omega^+]=\{\omega\in\Omega: \omega^-\mathcal Le \omega \mathcal Le \omega^+\}$.
A \mathcal Emph{Sturmian interval} is one of the form $[0\omega,1\omega]$, for some $\omega\in\Omega$.
A measure $\mathcal Mu\in\mathcal M$ is called \mathcal Emph{Sturmian} (see e.g.~\cite[Prop~1.5]{bouschmairesse}, \cite{bullettsentenac})
if its support is contained in a Sturmian interval.
Let $\mathcal Mathcal{S}\subset \mathcal M$ denote the class of Sturmian measures on $\Omega$.
For a Sturmian measure $\mathcal Mu\in\mathcal Mathcal{S}$, the value
$\mathcal Mu([1])$, denoted $\mathcal Mathcal{P}(\mathcal Mu)$, is called its \mathcal Emph{(Sturmian) parameter}\footnote{This
corresponds to the \mathcal Emph{1-frequency} mentioned in \S \mathcal Ref{problemsetting}, sometimes called
the \mathcal Emph{1-ratio} (see e.g.~\cite{hmst, morrissidorov}),
or the \mathcal Emph{rotation number} (see e.g.~\cite{bullettsentenac}).}, where $[1]$ denotes the (cylinder) set
$\{\omega\in\Omega : \omega_1=1\}$.
A \mathcal Emph{Sturmian sequence} of parameter $\mathcal Mathcal{P}$ is any point in the support of the Sturmian measure of parameter $\mathcal Mathcal{P}$.
\mathcal End{notation}
The following are classical facts about Sturmian measures (see e.g.~\cite[\S 1.1]{bouschmairesse} or \cite{bullettsentenac}):
\begin{prop}\mathcal Label{sturmianclassical}
\item[\, (a)]
For each Sturmian interval $[0\omega,1\omega]\subset\Omega$ there exists a unique Sturmian measure
whose support is contained in this interval.
\item[\, (b)]
The mapping $\mathcal Mathcal{P}:\mathcal Mathcal{S}\mathcal To[0,1]$ is a homeomorphism.
If $\mathcal Mu\in\mathcal Mathcal{S}$ has $\mathcal Mathcal{P}(\mathcal Mu)\in\mathcal Mathbb{Q}$ then its support is a single $\sigma$-periodic orbit, while if $\mathcal Mathcal{P}(\mathcal Mu)\notin\mathcal Mathbb{Q}$ then its support
is a Cantor subset of $\Omega$ which supports no other $\sigma$-invariant measure (and in particular contains no periodic orbit).
\item[\, (c)]
If $d(\omega)$ denotes the Sturmian parameter of the Sturmian
measure supported by the Sturmian interval $[0\omega,1\omega]\subset \Omega$,
then the map $d:\Omega\mathcal To [0,1]$ is
continuous, non-decreasing, and surjective.
The preimage $d^{-1}(\mathcal Mathcal{P})$ is a singleton if $\mathcal Mathcal{P}$ is irrational,
and a positive-length closed interval if $\mathcal Mathcal{P}$ is rational.
\mathcal End{prop}
For example the Sturmian measures of parameter
$1/2$, $1/3$, $2/5$, $3/8$ and $5/13$ are, respectively,
supported by the $\sigma$-periodic orbits generated by the finite words
$$
01
\, ,\
001
\, ,\
00101
\, ,\
00100101
\, ,\
0010010100101
\,,
$$
whereas the Sturmian measure of parameter
$(3-\sqrt{5})/2$ is
supported by
the smallest Cantor set
containing the $\sigma$-orbit of
$$
0010010100100101001010010010100101
\mathcal Ldots
$$
In view of Theorem \mathcal Ref{maxtheoremintro}, for a matrix pair $\mathcal Mathcal A\in\mathcal Mm$ we are interested in
measures $\nu\in\mathcal M_\mathcal Mathcal A$ attaining the maximum in (\mathcal Ref{maxtheoremintroeq}),
i.e.~satisfying $\int f_\mathcal Mathcal A\, d\nu = \mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu$;
such $\nu$ will be called \mathcal Emph{$f_\mathcal Mathcal A$-maximizing}.
There is a topological conjugacy between $T_\mathcal Mathcal A$ and the shift map
$\sigma:\Omega\mathcal To\Omega$,
and this induces a natural homeomorphism between $\mathcal M_\mathcal Mathcal A$ and $\mathcal M$;
the image of any $f_\mathcal Mathcal A$-maximizing measure under this homeomorphism will be
called a \mathcal Emph{maximizing measure for $\mathcal Mathcal A$}.
We then say that $\mathcal Mathcal A=(A_0,A_1)\in \mathcal Mm$ \mathcal Emph{generates a full Sturmian family}
if the set of maximizing measures for the family $\mathcal Mathcal A(t) = (A_0,tA_1)$, $t\in\mathcal Mathbb{R}^+$,
is precisely the set $\mathcal Mathcal{S}$ of all Sturmian measures on $\Omega$.
A more precise version of our main result Theorem \mathcal Ref{nontechnictheorem} is then the following:
\begin{theorem}\mathcal Label{maintheorem}
Every matrix pair in the open subset $\mathfrak D\subset M_2(\mathcal Mathbb{R})^2$
(and hence the open subset $\mathcal Ee\subset M_2(\mathcal Mathbb{R})^2$)
generates a full Sturmian family.
\mathcal End{theorem}
Note that Theorem \mathcal Ref{maintheorem} will follow from a more detailed version,
Theorem \mathcal Ref{deviltheorem}, which in particular incorporates the
statement that the parameter map $t\mathcal Mapsto \mathcal Mathcal{P}(t)$ is a devil's staircase.
\subsection{Relation with previous results}\mathcal Label{relatsubsection}
The methods of this paper can also be used to give an alternative proof of some of the results mentioned above,
namely establishing the analogue of Theorem \mathcal Ref{maintheorem} in certain cases treated by
Bousch \& Mairesse \cite{bouschmairesse} and Kozyakin \cite{kozyakin},
and the case considered by
Blondel, Theys \& Vladimirov \cite{btv}, Hare, Morris, Sidorov \& Theys \cite{hmst}, and Morris \& Sidorov \cite{morrissidorov}.
As already noted, the matrix pair (\mathcal Ref{standardpair}) lies on the boundary of our open set $\mathfrak D$,
and clearly it also lies on the boundary of the set $\mathcal Mathfrak{K}\subset M_2^+(\mathcal Mathbb{R}^+)^2$ defined by
Kozyakin's conditions (\mathcal Ref{kozyakinpairs}).
It can be checked that $\mathcal Mathfrak{K}$ itself lies in the boundary of our set $\mathcal Mm$, but not in the boundary of $\mathfrak D$.
However, the subset $\mathcal Mathfrak{K}' \subset \mathcal Mathfrak{K}$ defined by
\begin{equation}\mathcal Label{kprime}
\mathcal Mathfrak{K}'
=
\mathcal Left\{
\mathcal Left( \begin{pmatrix} 1 & 0 \cr c & d \cr \mathcal End{pmatrix}\, , \
\begin{pmatrix} a & b \cr 0 & 1 \cr \mathcal End{pmatrix} \mathcal Right) \in\mathcal Mathfrak{K}
:
a\mathcal Le b\ \mathcal Text{and}\ d\mathcal Le c\mathcal Right\} \,,
\mathcal End{equation}
can be readily checked to lie in the boundary of $\mathfrak D$.
Matrices in the Bousch-Mairesse family (\mathcal Ref{bmfamily}) do not all satisfy our condition
(\mathcal Ref{definequal1}), or indeed the corresponding weak inequality,
so do not automatically belong to the boundary of $\mathcal Mm$.
However, imposing the additional condition
\begin{equation}\mathcal Label{bmadditional}
e^{2\kappa} \ge (e^{\kappa h_0}+1)(e^{\kappa h_1}+1)
\mathcal End{equation}
ensures that a matrix pair satisfying (\mathcal Ref{bmfamily}) belongs to the boundary
of $\mathcal Mm$, and indeed also belongs to the boundary of $\mathfrak D$.
In \S \mathcal Ref{adaptations} we will indicate the minor modifications to our approach needed to
handle the case of (\mathcal Ref{standardpair}), and the sub-cases of (\mathcal Ref{bmfamily}) and (\mathcal Ref{kozyakinpairs})
defined by (\mathcal Ref{bmadditional}) and (\mathcal Ref{kprime}) respectively.
\subsection{Organisation of article}
The article is organised as follows. Section \mathcal Ref{prelimsection} consists of preliminaries:
maps induced by matrices acting on projective space, Perron-Frobenius theory, and some useful notation
and identities. Section \mathcal Ref{projconvprojconc} develops the notions of projective convexity and projective concavity.
Section \mathcal Ref{induceddynsyssection} introduces the induced dynamical system $T_\mathcal Mathcal A$ for concave-convex matrix pairs $\mathcal Mathcal A$,
the formulation of joint spectral radius in terms of ergodic optimization (Theorem \mathcal Ref{maxtheorem}),
and the connection between the finiteness property and $T_\mathcal Mathcal A$-periodic orbits.
Section \mathcal Ref{sturmasection} introduces Sturmian measures and Sturmian intervals for the dynamical system $T_\mathcal Mathcal A$,
and makes the connection between finiteness counterexamples and unique maximizing measures which are Sturmian of irrational parameter.
Section \mathcal Ref{transfersection} establishes the existence of an important technical tool, the Sturmian transfer function.
After deriving some explicit formulae for extremal Sturmian intervals in Section \mathcal Ref{extsturmsection},
the key Section \mathcal Ref{particularsection} establishes the link between
Sturmian intervals and the parameter $t$ of the pair $\mathcal Mathcal A(t)$.
Section \mathcal Ref{dominatessec} treats the case of those parameters $t$ such that one matrix in the pair $\mathcal Mathcal A(t)$
dominates the other, so that the joint spectral radius $r(\mathcal Mathcal A(t))$ is simply the spectral radius of the dominating matrix.
All other parameters are considered in Section \mathcal Ref{technicalsection}, establishing that the joint spectral radius
is always attained by a unique Sturmian measure.
Finally, in Section \mathcal Ref{devilsection} we show that the map taking parameter values $t$ to the associated Sturmian
parameter
$\mathcal Mathcal{P}(t)$ is a devil's staircase.
\section{Preliminaries}\mathcal Label{prelimsection}
\subsection{The induced map for a positive matrix}\mathcal Label{inducproj}
\begin{notation}
Throughout we use the notation $X=[0,1]$.
\mathcal End{notation}
A positive matrix $A\in M_2(\mathcal Mathbb{R}^+)$ gives a self-map
$v \mathcal Mapsto Av$
of $(\mathcal Mathbb{R}^+)^2$.
This lifts to a self-map $\widetilde A: [v]\mathcal Mapsto [Av]$
of projective space $(\mathcal Mathbb{R}^+)^2/\sim$, the equivalence relation $\sim$ being defined by
$v\sim v'$ if $v=sv'$ for some $s>0$, and $[v]$ denoting the equivalence class containing $v\in(\mathcal Mathbb{R}^+)^2$.
It is convenient to identify projective space with
\begin{equation*}
\Sigma=\mathcal Left\{ \begin{pmatrix} x\\ 1-x\mathcal End{pmatrix} : x\in (0,1)\mathcal Right\} \,,
\mathcal End{equation*}
so that the projection $\pi:(\mathcal Mathbb{R}^+)^2 \mathcal To \Sigma$ takes the form
$$
\pi: \begin{pmatrix} x\\ y\mathcal End{pmatrix} \mathcal Mapsto \begin{pmatrix} \frac{x}{x+y} \\ \frac{y}{x+y} \mathcal End{pmatrix}\,,
$$
and the projective map is represented as
$\pi\circ A:\Sigma \mathcal To\Sigma$,
taking the explicit form
\begin{equation*}
\pi\circ A:
\begin{pmatrix} x\\ 1-x \mathcal End{pmatrix} \mathcal Mapsto
\begin{pmatrix} \frac{(a-b)x+b}{(a+c-b-d)x + b+d} \\ \frac{(c-d)x+d}{(a+c-b-d)x+b+d}\mathcal End{pmatrix}\,.
\mathcal End{equation*}
This projective mapping is completely determined by its first coordinate, thereby
motivating the following definition of
the self-map $T_A$ of the unit interval $X=[0,1]$:
\begin{defn}\mathcal Label{taxasa}
For $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$,
the \mathcal Emph{induced map} $T_A:X\mathcal To X$ is defined by
$$
T_A(x)=\frac{(a-b)x+b}{(a+c-b-d)x+b+d}\,,
$$
the \mathcal Emph{induced image} $X_A$ is defined by
$$X_A=T_A(X)=\mathcal Left[\frac{b}{b+d},\frac{a}{a+c}\mathcal Right]\,,$$
and the \mathcal Emph{induced inverse map} $S_A:X_A\mathcal To X$ is given by
$$
S_A(x)= T_A^{-1}(x) = \frac{ (b+d)x - b}{-(a+c-b-d)x+a-b}
\,.
$$
\mathcal End{defn}
\begin{remark}
Defining $P=\begin{pmatrix} 1 & 0 \\ 1 & 1 \mathcal End{pmatrix}$,
the M\"obius maps $T_A$ and $S_A$ are represented, respectively, by the matrices
$PAP^{-1}$ and $PA^{-1}P^{-1}$.
\mathcal End{remark}
\begin{remark}\mathcal Label{invariantposmultiple}
The objects defined in Definition \mathcal Ref{taxasa} do not change if the matrix $A$
is multiplied by a positive real number; that is,
if $t>0$, $A \in M_2^+(\mathcal Mathbb{R}^+)$,
then $T_{tA}=T_A$ (hence $S_{tA}=S_A$), and $X_{tA}=X_A$.
\mathcal End{remark}
In view of (\mathcal Ref{definequal2}) in the definition of $\mathcal Mm$,
it suffices to restrict attention to matrices
of the following form:
\begin{notation}
Let $\mathbb M$ denote the set of matrices $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$
such that $a+c\neq b+d$.
\mathcal End{notation}
\begin{lemma}
For $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$, the map $T_A$ has a single fixed point $p_A=T_A(p_A)$ in $X$.
If $A \in \mathbb M$ then
\begin{equation}\mathcal Label{paformula}
p_A
= \frac{a-d-2b+\sqrt{(a-d)^2+4bc}}{2(a+c-b-d)}\,,
\mathcal End{equation}
and if $A\notin\mathbb M$ then
\begin{equation}\mathcal Label{paformula2}
p_A = \frac{b}{2b+d-a}\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Uniqueness follows from the fact that $A$ has all entries strictly positive, and the formulae (\mathcal Ref{paformula})
and (\mathcal Ref{paformula2})
are
straightforward computations.
\mathcal End{proof}
\subsection{Notation and matrix preliminaries}
For a matrix $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$,
it will be useful to write
\begin{equation}\mathcal Label{alphadef}
\alpha_A = a+c-b-d \,,
\mathcal End{equation}
\begin{equation}\mathcal Label{betadef}
\beta_A = a-d-2b \,,
\mathcal End{equation}
\begin{equation}\mathcal Label{gammadef}
\gamma_A = \sqrt{(a-d)^2+4bc}\,,
\mathcal End{equation}
noting that these quantities are related by the following identity:
\begin{lemma}\mathcal Label{alphabetagammalemma}
For $A \in M_2^+(\mathcal Mathbb{R}^+)$,
\begin{equation}\mathcal Label{alphabetagamma}
\gamma_A^2 - \beta_A^2
=
4b \alpha_A \,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Straightforward computation.
\mathcal End{proof}
For ease of reference it will be convenient to collect together
various previously defined objects expressed in terms of the above notation.
\begin{prop}
For $A \in M_2^+(\mathcal Mathbb{R}^+)$,
\begin{equation}\mathcal Label{rhoaredone}
\mathcal Rho_A = \frac{2b}{\beta_A + \gamma_A}\,,
\mathcal End{equation}
\begin{equation*}
T_A(x) = \frac{(a-b)x+b}{\alpha_Ax+b+d}\,,
\mathcal End{equation*}
\begin{equation*}
S_A(x)
= \frac{(b+d)x-b}{-\alpha_{A}(x+\sigma_{A})} \,,
\mathcal End{equation*}
and if moreover $A\in\mathbb M$ then
\begin{equation}\mathcal Label{sigmaaredone}
\sigma_A = \frac{b-a}{\alpha_A}\,,
\mathcal End{equation}
\begin{equation}\mathcal Label{paformularedone}
p_A = \frac{\beta_A+\gamma_A}{2\alpha_A}
= \frac{b\, \sigma_A}{(b-a)\mathcal Rho_A}\,.
\mathcal End{equation}
The set $\mathcal Mm$ can be written as
\begin{equation*}
\mathcal Mm
=
\mathcal Left\{ (A_0,A_1) \in \mathbb M^2 : \frac{a_0}{c_0}<\frac{b_1}{d_1}
\ \mathcal Text{and}\
\alpha_{A_1} < 0 < \alpha_{A_0} \mathcal Right\}\,.
\mathcal End{equation*}
\mathcal End{prop}
\subsection{Perron-Frobenius theory and the joint spectral radius}
\begin{lemma}\mathcal Label{pflemma}
The dominant (Perron-Frobenius) eigenvalue $\mathcal Lambda_A>0$ of the matrix
$A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in M_2^+(\mathcal Mathbb{R}^+)$
is given by
\begin{equation*}
\mathcal Label{lambdaadef}
\mathcal Lambda_A = \frac{1}{2}\mathcal Left( a+d +\gamma_A\mathcal Right)
=\frac{b}{p_A}+a-b
\,,
\mathcal End{equation*}
with corresponding left eigenvector
\begin{equation*}
w_A = ( a-d +\gamma_A, 2b)\,,
\mathcal End{equation*}
and right eigenvector
\begin{equation*}
v_A= \begin{pmatrix} p_A \\ 1-p_A \mathcal End{pmatrix} \,.
\mathcal End{equation*}
\mathcal End{lemma}
For $A \in M_2^+(\mathcal Mathbb{R}^+)$,
the derivative of $T_A$ at its fixed point $p_A$ is related to the determinant
and Perron-Frobenius eigenvalue of $A$ as follows:
\begin{lemma}\mathcal Label{derivfixedpt}
If $A \in M_2^+(\mathcal Mathbb{R}^+)$ then
\begin{equation*}
T_A'(p_A) = \frac{\det A}{\mathcal Lambda_A^2}\,.
\mathcal End{equation*}
\mathcal End{lemma}
\begin{proof}
This is a straightforward computation. If $A\in\mathbb M$ we can use the
expression
$p_A=\frac{\beta_A+\gamma_A}{2\alpha_A}$
(see (\mathcal Ref{paformularedone})), the
derivative formula
$T_A'(x)
=
\det A\mathcal Left( \alpha_A x+b+d\mathcal Right)^{-2}
$, and the fact that
$\mathcal Lambda_A = \frac{1}{2}\mathcal Left( a+d +\gamma_A\mathcal Right)
=\frac{1}{2}(\beta_A+\gamma_A)+b+d$
(see (\mathcal Ref{lambdaadef})).
If $A\notin\mathbb M$ then $T_A'\mathcal Equiv \frac{a-b}{b+d}$,
$\mathcal Lambda_A=b+d$, and the relation $a+c=b+d$ means that $\det A = (a-b)(b+d)$, so the result follows.
\mathcal End{proof}
Since the Perron-Frobenius eigenvalue $\mathcal Lambda_A$ is also the spectral radius $r(A)$,
we obtain the following corollary:
\begin{cor}\mathcal Label{specradiusta}
If $A \in M_2^+(\mathcal Mathbb{R}^+)$ then its spectral radius $r(A)$ satisfies
\begin{equation}\mathcal Label{specradta}
r(A) = \mathcal Left( \frac{\det A}{T_A'(p_A)}\mathcal Right)^{1/2}\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{proof}
Immediate from Lemma \mathcal Ref{derivfixedpt}.
\mathcal End{proof}
\begin{notation}
Let us write finite words using the alphabet $\{0,1\}$ as
$\underline{i}=(i_1,\mathcal Ldots,i_n)$, and their length as $|\underline{i}|=n$.
Let $\Omega^*$ denote the set of all such finite words; that is, $\Omega^*=\cup_{n\ge1} \{0,1\}^n$.
Given $\mathcal Mathcal A=(A_0,A_1)\in M_2(\mathcal Mathbb{R})^2$, and $\underline{i}\in\{0,1\}^n$, let $A(\underline{i})$
denote the product
\begin{equation}\mathcal Label{matrixproduct}
A(\underline{i}) = A_{i_1}\cdots A_{i_n}\,.
\mathcal End{equation}
\mathcal End{notation}
Corollary \mathcal Ref{specradiusta} then allows us to express
the joint spectral radius of a matrix pair
$\mathcal Mathcal A=(A_0,A_1)\in M_2^+(\mathcal Mathbb{R}^+)^2$
in terms of induced maps of the products $A(\underline{i})$ as follows:
\begin{prop}\mathcal Label{jsrderiv}
If $\mathcal Mathcal A=(A_0,A_1)\in M_2^+(\mathcal Mathbb{R}^+)^2$, then its joint spectral radius $r(\mathcal Mathcal A)$ satisfies
\begin{equation}\mathcal Label{jsrderiveq}
r(\mathcal Mathcal A)=
\sup_{\underline{i}\in \Omega^*}
\mathcal Left( \frac{\det A(\underline{i}) }{T_{A(\underline{i}) }' (p_{A(\underline{i} )})} \mathcal Right)^{1/2|\underline{i}|}\,.
\mathcal End{equation}
\mathcal End{prop}
\begin{proof}
The
expression (\mathcal Ref{rjsr}) for the joint spectral radius can be written as
$$r(\mathcal Mathcal A)
=\sup\mathcal Left\{ r(A_{i_1}\cdots A_{i_n})^{1/n}:n\ge 1, i_j\in\{0,1\}\mathcal Right\}
=
\sup_{\underline{i}\in \Omega^*} r(A(\underline{i}))^{1/|\underline{i}|}
\,,
$$
so applying Corollary \mathcal Ref{specradiusta} with $A$ replaced by $A(\underline{i})$ yields the result.
\mathcal End{proof}
\subsection{Some useful formulae}
The purpose of this short subsection is to collect together various formulae
which will prove useful in the sequel.
Firstly, we have the following two expressions for the determinant of $A$ involving $\alpha_A$ and
$\sigma_A$:
\begin{lemma}\mathcal Label{detalphasigma}
For $A\in \mathbb M$, its determinant can be expressed as
\begin{equation}\mathcal Label{detac}
\det A = -\alpha_A(a+c)\mathcal Left( \frac{a}{a+c} + \sigma_A\mathcal Right)
\mathcal End{equation}
and
\begin{equation}\mathcal Label{detbd}
\det A = -\alpha_A(b+d)\mathcal Left( \frac{b}{b+d} + \sigma_A\mathcal Right) \,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Straightforward computation.
\mathcal End{proof}
There is a useful alternative way of expressing the quantity $\mathcal Rho_A$:
\begin{lemma}\mathcal Label{rhoquadratic}
For $A\in \mathbb M$,
\begin{equation}\mathcal Label{altrho}
\mathcal Rho_A = \frac{\gamma_A -\beta_A}{2\alpha_A}\,,
\mathcal End{equation}
and $\mathcal Rho_A$
is the larger
root of the quadratic polynomial $q_A$ defined by
\begin{equation}\mathcal Label{qdef}
q_A(z) = \alpha_A z^2 +\beta_A z-b
\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
The expression (\mathcal Ref{altrho}) follows from
(\mathcal Ref{rhoaredone}) and the identity
(\mathcal Ref{alphabetagamma}).
The larger root of $q_A$ is computed to be
$$
\frac{-\beta_A+\sqrt{\beta_A^2+4\alpha_A b}}{2\alpha_A}
= \frac{-\beta_A +\gamma_A}{2\alpha_A}\,,
$$
again using (\mathcal Ref{alphabetagamma}).
\mathcal End{proof}
Clearly
\begin{equation}\mathcal Label{qdet}
q_A(z)
=\det \begin{pmatrix} 1 & z \\ -\alpha_A z & \beta_A z - b \mathcal End{pmatrix} \,,
\mathcal End{equation}
though the following expression will prove to be more useful:
\begin{lemma}\mathcal Label{qpolylemma}
For $A\in \mathbb M$,
\begin{equation*}
q_A(z)
=\det \begin{pmatrix} 1 & z \\ b +d -\alpha_A z & (a-b) z -b \mathcal End{pmatrix} \,.
\mathcal End{equation*}
\mathcal End{lemma}
\begin{proof}
Straightforward computation.
\mathcal End{proof}
\section{Projective convexity and projective concavity}\mathcal Label{projconvprojconc}
\begin{remark}\mathcal Label{derivativesremark}
\item[\, (a)]
For $A\in\mathbb M$ and $x\in X$, the derivative formula
\begin{equation}\mathcal Label{taderiv}
T_A'(x)
=
\det A\mathcal Left( \alpha_A x+b+d\mathcal Right)^{-2}
\mathcal End{equation}
implies that if $\mathcal Mathcal A\in \mathbb M^2$ then
$T_{A_0}$ and $T_{A_1}$ are orientation preserving.
\item[\, (b)]
For $A\in\mathbb M$ and $x\in X$,
the second derivative formula
\begin{equation}\mathcal Label{tasecondderiv}
T_A''(x)
=
-2\alpha_A \det A \mathcal Left( \alpha_A x+b+d\mathcal Right)^{-3}
\mathcal End{equation}
implies that if $\mathcal Mathcal A\in\mathcal Mm$ then
$T_{A_0}'' < 0$ and $T_{A_1}'' > 0$,
i.e.~$T_{A_0}$ is strictly concave and $T_{A_1}$ is strictly convex.
\mathcal End{remark}
Part (b) of Remark \mathcal Ref{derivativesremark} motivates the following definition, partitioning $\mathbb M$ into two subsets:
\begin{defn}
A matrix $A\in \mathbb M$ will be called \mathcal Emph{projectively convex} if the induced map $T_A$ is strictly convex, and
\mathcal Emph{projectively concave} if the induced map $T_A$ is strictly concave.
\mathcal End{defn}
\begin{remark}
The set $\mathbb M$ is the disjoint union of the subset of
projectively convex matrices and
the subset of projectively concave matrices.
\mathcal End{remark}
Recall that
\begin{equation}\mathcal Label{weig}
w_A=(w^{(1)}_A,w^{(2)}_A)
= ( a-d +\gamma_A, 2b)
\mathcal End{equation}
denotes the Perron-Frobenius left eigenvector of $A\in \mathbb M$,
and that (consequently) the right eigenvector for the other eigenvalue of $A$
is $\begin{pmatrix} w^{(2)}_A \\ -w^{(1)}_A\mathcal End{pmatrix}$.
It is useful to record the following identity:
\begin{lemma}\mathcal Label{wrholem}
For $A = \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in \mathbb M$,
\begin{equation}\mathcal Label{wrho}
\mathcal Rho_A = \frac{w_A^{(2)}}{w_A^{(1)}-w_A^{(2)}}\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Immediate from
(\mathcal Ref{rhoaredone}) and
(\mathcal Ref{weig}).
\mathcal End{proof}
\begin{cor}\mathcal Label{rhosiminvt}
For $A\in \mathbb M$, if $Q\in M_2(\mathcal Mathbb{R})$ is non-singular then
$\mathcal Rho_{Q^{-1}AQ}=\mathcal Rho_A$; that is, $\mathcal Rho_A$ is invariant under similarities.
\mathcal End{cor}
\begin{proof}
Immediate from Lemma \mathcal Ref{wrholem}, and the fact that the eigenvector $w_A$ is invariant under similarities.
\mathcal End{proof}
There are various useful characterisations of projective convexity and projective concavity:
\begin{lemma}\mathcal Label{concaveequivconditions}
For $A \in \mathbb M$, the following are equivalent
\item[\, (i)]
$A$ is projectively concave,
\item[\, (ii)]
$\alpha_A>0$,
\item[\, (iii)]
$\mathcal Rho_A>0$,
\item[\, (iv)]
$w^{(1)}_A > w^{(2)}_A$.
\mathcal End{lemma}
\begin{proof}
As noted in Remark \mathcal Ref{derivativesremark} (b), the second derivative formula
(\mathcal Ref{tasecondderiv}) yields the equivalence of (i) and (ii), since $\det A >0$, and a function
is strictly concave if and only if its second derivative is strictly negative.
To prove the equivalence of (ii) and (iii), we consider separately the cases where $\beta_A\ge0$ and
$\beta_A<0$.
If $\beta_A\ge0$ then $\alpha_A=\beta_A+b+c>0$, so we must simply
show that $\mathcal Rho_A>0$.
But $\gamma_A>0$ by definition, hence $\beta_A+\gamma_A>0$, and therefore
(\mathcal Ref{rhoaredone}) implies that $\mathcal Rho_A = \frac{2b}{\beta_A + \gamma_A}>0$, as required.
If on the other hand $\beta_A<0$ then $\gamma_A - \beta_A>0$ is automatically true,
again since $\gamma_A>0$ by definition.
Using (\mathcal Ref{alphabetagamma}) and (\mathcal Ref{rhoaredone}) we see that
\begin{equation*}
2\alpha_A \mathcal Rho_A = \gamma_A-\beta_A >0\,,
\mathcal End{equation*}
so indeed $\alpha_A>0$ if and only if $\mathcal Rho_A>0$, as required.
Lastly, the equivalence of (iii) and (iv) is immediate from (\mathcal Ref{wrho}), since $w_A^{(2)}>0$.
\mathcal End{proof}
\begin{lemma}\mathcal Label{convexequivconditions}
For $A \in \mathbb M$, the following are equivalent
\item[\, (i)]
$A$ is projectively convex,
\item[\, (ii)]
$\alpha_A<0$,
\item[\, (iii)]
$\mathcal Rho_A < -1$,
\item[\, (iv)]
$w^{(1)}_A < w^{(2)}_A$.
\mathcal End{lemma}
\begin{proof}
A function
is strictly convex if and only if its second derivative is strictly positive, so
the equivalence of (i) and (ii)
follows from
(\mathcal Ref{tasecondderiv}), since $\det A >0$
and $\alpha_Ax+b+d = a+c+(b+d)(1-x)>0$ for all $x\in X$.
To prove that (iii) is equivalent to (iv), note that (\mathcal Ref{wrho})
gives $w^{(1)}_A=w^{(2)}_A(1+\mathcal Rho_A^{-1})$; therefore $\mathcal Rho_A<-1$ if and only if
$1+\mathcal Rho_A^{-1}\in (0,1)$, if and only if $w^{(1)}_A\in(0,w^{(2)}_A)$.
Lastly, to prove the equivalence of (ii) and (iii), it follows from
Lemma \mathcal Ref{concaveequivconditions} that $\alpha_A<0$ if and only if $\mathcal Rho_A<0$,
but this latter inequality in fact implies $w^{(2)}_A-w^{(1)}_A>0$ by (\mathcal Ref{wrho}), so
$$\mathcal Rho_A = -1 - \frac{w_A^{(1)}}{w_A^{(2)}-w_A^{(1)}} <-1\,,$$
as required.
\mathcal End{proof}
Note that in Lemma \mathcal Ref{convexequivconditions} the assertion is not merely that $\mathcal Rho_A<0$, but that $\mathcal Rho_A<-1$; this should be contrasted with the inequality $\mathcal Rho_A>0$ in Lemma
\mathcal Ref{concaveequivconditions}.
It is now clear why $\mathcal Mm$ is described as the set of
\mathcal Emph{concave-convex pairs}\footnote{Note, however, the restriction that the induced images be
\mathcal Emph{disjoint}, with the concave image to the \mathcal Emph{left} of the convex one.}:
\begin{prop}\mathcal Label{concavconvexprop}
The set $\mathcal Mm$ consists of those matrix pairs $(A_0,A_1)\in \mathbb M^2$ such that
$A_0$ is projectively concave, $A_1$ is projectively convex, and the induced
image for $A_0$ is strictly to the left of the induced image of $A_1$.
\mathcal End{prop}
\begin{proof}
Lemmas \mathcal Ref{concaveequivconditions} and \mathcal Ref{convexequivconditions}
imply that the inequality $\alpha_{A_1}<0<\alpha_{A_0}$
in Definition \mathcal Ref{frakmdefn}
is equivalent to
$A_0$ being projectively concave and $A_1$ being projectively convex.
The inequality $\frac{a_0}{c_0} < \frac{b_1}{d_1}$ in Definition \mathcal Ref{frakmdefn}
is equivalent to
$T_{A_0}(1) = \frac{a_0}{a_0+c_0} < \frac{b_1}{b_1+d_1}=T_{A_1}(0)$, which asserts that the right
endpoint of the induced image $X_{A_0}$ is strictly to the left of the left endpoint
of the induced image $X_{A_1}$.
\mathcal End{proof}
\begin{lemma}\mathcal Label{lemma1}
If $A\in \mathbb M$ is projectively concave then
$\sigma_{A}<0$.
\mathcal End{lemma}
\begin{proof}
Projective concavity of $A$ means that $\alpha_A>0$,
so
by (\mathcal Ref{sigmaaredone})
it suffices to show that $b<a$.
Since $\det A = ad-bc>0$
and $\alpha_A=a+c-b-d>0$
we derive
$$
a-b>d-c > \frac{bc}{a}-c = -\frac{c}{a}(a-b)\,,
$$
or in other words
$$
(a-b)\mathcal Left(1+\frac{c}{a}\mathcal Right)>0\,,
$$
and hence $a-b>0$,
as required.
\mathcal End{proof}
We can now prove the following result mentioned in \S \mathcal Ref{statementsubsection}
(note, however, that there is no constraint on the sign of $\sigma_{A_1}$ when $(A_0,A_1)\in\mathcal Mm$):
\begin{cor}\mathcal Label{rhoposlessminusone}
If $(A_0,A_1)\in \mathcal Mm$ then
$\sigma_{A_0}<0<\mathcal Rho_{A_0}$ and $\mathcal Rho_{A_1}<-1$.
\mathcal End{cor}
\begin{proof}
Immediate from Lemmas \mathcal Ref{concaveequivconditions}, \mathcal Ref{convexequivconditions},
and \mathcal Ref{lemma1}
\mathcal End{proof}
An important result is the following:
\begin{lemma}\mathcal Label{posneglinfactors}
If $A\in \mathbb M$ then
\begin{equation}\mathcal Label{alwayspos}
-\alpha_{A}(x+\sigma_{A}) > 0 \quad\mathcal Text{for all } x\in X_{A}\,.
\mathcal End{equation}
In particular, if $A\in \mathbb M$ is projectively concave then
\begin{equation}\mathcal Label{posneglinfactorseq0}
x+\sigma_{A}<0 \quad\mathcal Text{for all }x\in X_{A}\,,
\mathcal End{equation}
and if $A\in \mathbb M$ is projectively convex then
\begin{equation}\mathcal Label{posneglinfactorseq1}
x+\sigma_{A}>0 \quad\mathcal Text{for all }x\in X_{A}\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Clearly (\mathcal Ref{alwayspos}) follows from (\mathcal Ref{posneglinfactorseq0}) and
(\mathcal Ref{posneglinfactorseq1}),
since $\alpha_A$ is positive if $A$ is projectively concave, and negative if
$A$ is projectively convex,
by Lemmas
\mathcal Ref{concaveequivconditions} and \mathcal Ref{convexequivconditions}.
To prove (\mathcal Ref{posneglinfactorseq0}), note that
$ -\alpha_A(a+c)\mathcal Left( \frac{a}{a+c} + \sigma_A\mathcal Right) = \det A >0$
by (\mathcal Ref{detac}), and if $A$ is projectively concave then $\alpha_A>0$, so
$\frac{a}{a+c} + \sigma_A < 0$. But $\frac{a}{a+c}$ is the righthand endpoint of $X_A$, so if $x\in X_A$ then $x\mathcal Le \frac{a}{a+c}$, therefore
$
x+\sigma_A \mathcal Le \frac{a}{a+c} + \sigma_A < 0
$, as required.
To prove (\mathcal Ref{posneglinfactorseq1}), note that
$-\alpha_A(b+d)\mathcal Left( \frac{b}{b+d} + \sigma_A\mathcal Right) = \det A >0$
by (\mathcal Ref{detbd}), and if $A$ is projectively convex then $\alpha_A<0$, so
$\frac{b}{b+d} + \sigma_A >0$.
But $\frac{b}{b+d}$ is the lefthand endpoint of $X_A$, so if $x\in X_A$ then $x\ge \frac{b}{b+d}$, therefore
$
x+\sigma_A \ge \frac{b}{b+d} + \sigma_A > 0
$, as required.
\mathcal End{proof}
\begin{cor}\mathcal Label{alwaysposcor}
If $(A_0,A_1)\in\mathcal Mm$ then
$
x+\sigma_{A_0} <0
$
for $x\in X_{A_0}$,
and
$x+\sigma_{A_1} >0
$
for $x\in X_{A_1}$, and
$-\alpha_{A_i}(x+\sigma_{A_i}) > 0$ for all $x\in X_{A_i}$, $i\in\{0,1\}$.
\mathcal End{cor}
\begin{proof}
Immediate from Lemma \mathcal Ref{posneglinfactors}.
\mathcal End{proof}
\begin{lemma}\mathcal Label{qrhoineq}
If $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$ then
\begin{equation*}
q_{A_1}(\mathcal Rho_{A_0}) < 0
<
q_{A_0}(\mathcal Rho_{A_1})\,.
\mathcal End{equation*}
\mathcal End{lemma}
\begin{proof}
The larger root of $q_{A_1}$ is $\mathcal Rho_{A_1}$, by Lemma \mathcal Ref{rhoquadratic}.
It follows that $q_{A_1}(z)= \alpha_{A_1} z^2 +\beta_{A_1} z-b_1 <0$ for all $z>\mathcal Rho_{A_1}$, since
the leading coefficient $\alpha_{A_1}<0$, since $A_1$ is projectively convex.
But by Lemmas \mathcal Ref{concaveequivconditions} and \mathcal Ref{convexequivconditions}
we know that $\mathcal Rho_{A_0} > 0 > -1 > \mathcal Rho_{A_1}$, so indeed
$q_{A_1}(\mathcal Rho_{A_0}) < 0$, as required.
The smaller root of $q_{A_0}$, which we shall denote by $r_{A_0}$, is given by
$$
r_{A_0} = \frac{-(\gamma_{A_0}+\beta_{A_0})}{2\alpha_{A_0}}\,.
$$
It follows that
\begin{equation}\mathcal Label{qlarger}
q_{A_0}(z)= \alpha_{A_0} z^2 +\beta_{A_0} z-b_0 >0
\quad\mathcal Text{for all }
z<r_{A_0}\,,
\mathcal End{equation}
since
the leading coefficient $\alpha_{A_0}>0$, since $A_0$ is projectively concave.
Now $\mathcal Rho_{A_1}<-1$ by Lemma \mathcal Ref{convexequivconditions}, and if we can show that $r_{A_0}> -1$
then it follows that $\mathcal Rho_{A_1}<r_{A_0}$, and hence $q_{A_0}(\mathcal Rho_{A_1})>0$ by (\mathcal Ref{qlarger}).
To show that indeed $r_{A_0}> -1$, note that this inequality is equivalent to $2\alpha_A-\beta_A>\gamma_A$.
Both sides are positive, so this is equivalent to
$(2\alpha_A-\beta_A)^2>\gamma_A^2$,
which
using (\mathcal Ref{alphabetagamma})
becomes $4\alpha_A(\alpha_A-\beta_A) > 4b\alpha_A$.
This latter inequality is equivalent to $\alpha_A-\beta_A>b$, which is true because in fact
$\alpha_A-\beta_A=b+c>b$.
\mathcal End{proof}
We deduce the following technical lemma, which will be used in \S \mathcal Ref{dominatessec}:
\begin{lemma}\mathcal Label{techderivlemma}
If $\mathcal Mathcal A\in\mathcal Mm$ then the M\"obius function
\begin{equation*}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_0}}{(b_1+d_1-\alpha_{A_1}\mathcal Rho_{A_0})x +(a_1-b_1)\mathcal Rho_{A_0}-b_1}
\mathcal End{equation*}
has strictly negative derivative,
while
the M\"obius function
\begin{equation*}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_1}}{(b_0+d_0-\alpha_{A_0}\mathcal Rho_{A_1})x +(a_0-b_0)\mathcal Rho_{A_1}-b_0}
\mathcal End{equation*}
has strictly positive derivative.
\mathcal End{lemma}
\begin{proof}
A general M\"obius map $x\mathcal Mapsto \frac{Px+Q}{Rx+S}$ has derivative
$D(Rx+S)^{-2}$, where
$D=PS-QR$,
so the derivative is strictly negative if $D<0$, and strictly positive if $D>0$.
For our first M\"obius map we have
$$
D=\det \begin{pmatrix} 1 & \mathcal Rho_{A_0} \\ b_1 +d_1 -\alpha_{A_1} \mathcal Rho_{A_0} & (a_1-b_1) \mathcal Rho_{A_0} -b_1 \mathcal End{pmatrix}
= q_{A_1}(\mathcal Rho_{A_0})
$$
by Lemma \mathcal Ref{qpolylemma},
and $q_{A_1}(\mathcal Rho_{A_0})$ is strictly negative by Lemma \mathcal Ref{qrhoineq}, so the derivative of the map is strictly negative, as required.
For our second M\"obius map we have
$$
D=\det \begin{pmatrix} 1 & \mathcal Rho_{A_1} \\ b_0 +d_0 -\alpha_{A_0} \mathcal Rho_{A_1} & (a_0-b_0) \mathcal Rho_{A_1} -b_0 \mathcal End{pmatrix}
= q_{A_0}(\mathcal Rho_{A_1})
$$
by Lemma \mathcal Ref{qpolylemma},
and $q_{A_0}(\mathcal Rho_{A_1})$ is strictly positive by Lemma \mathcal Ref{qrhoineq}, so the derivative of the map is strictly positive, as required.
\mathcal End{proof}
\section{The induced dynamical system for a concave-convex matrix pair}\mathcal Label{induceddynsyssection}
\subsection{The induced dynamical system and joint spectral radius}
\begin{defn}\mathcal Label{induceddefn}
For a matrix pair $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$,
define the
\mathcal Emph{induced space} $X_\mathcal Mathcal A$ to be
$$
X_\mathcal Mathcal A = X_{A_0}\cup X_{A_1}\,,
$$
and define the \mathcal Emph{induced dynamical system}
$T_\mathcal Mathcal A:X_\mathcal Mathcal A\mathcal To X$ by
\begin{equation*}
T_\mathcal Mathcal A(x) =
\begin{cases}
S_{A_0}(x)&\mathcal Text{if } x\in X_{A_0}\\
S_{A_1}(x)&\mathcal Text{if }x\in X_{A_1}\,.
\mathcal End{cases}
\mathcal End{equation*}
\mathcal End{defn}
\begin{remark}\mathcal Label{taremarks}
\item[\, (a)]
The map $T_\mathcal Mathcal A:X_\mathcal Mathcal A\mathcal To X$ is Lipschitz continuous since $X_\mathcal Mathcal A=X_{A_0}\cup X_{A_1}$ is the union of disjoint intervals $X_{A_0}$ and $X_{A_1}$, and the restriction of $T_\mathcal Mathcal A$ to $X_{A_i}$ is the M\"obius mapping $S_{A_i}$, which is certainly Lipschitz continuous.
\item[\, (b)]
Note that the (surjective) induced dynamical system $T_\mathcal Mathcal A$ is naturally defined as a mapping
from $X_\mathcal Mathcal A$ to $X=[0,1]$.
To view it as a surjective \mathcal Emph{self}-mapping of some set (the natural setting for a dynamical system) we consider its restriction to
the \mathcal Emph{induced Cantor set}
$Y_\mathcal Mathcal A:=\cap_{n\ge0}T_\mathcal Mathcal A^{-n}(X_\mathcal Mathcal A)$,
and note that $T_\mathcal Mathcal A:Y_\mathcal Mathcal A\mathcal To Y_\mathcal Mathcal A$ is topologically conjugate to the shift map
$\sigma$ on $\Omega = \{0,1\}^\mathbb N$.
\mathcal End{remark}
\begin{prop}\mathcal Label{jsrderivta}
If $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$ then its joint spectral radius $r(\mathcal Mathcal A)$ satisfies
\begin{equation}\mathcal Label{jsrderivtaeq}
r(\mathcal Mathcal A)=
\sup_{\underline{i}\in \Omega^*}
\mathcal Left( \det A(\underline{i}) \,\, (T_{\mathcal Mathcal A}^{|\underline{i}|})' (p_{A(\underline{i} )}) \mathcal Right)^{1/2|\underline{i}|}\,.
\mathcal End{equation}
\mathcal End{prop}
\begin{proof}
From Definition \mathcal Ref{induceddefn} we see that $T_\mathcal Mathcal A\circ T_{A_i}$ is the identity map on $X$,
for $i\in\{0,1\}$, since $T_\mathcal Mathcal A$ is defined in terms of the inverses $S_{A_i}=T_{A_i}^{-1}$.
Similarly, for any $\underline{i}\in\{0,1\}^n$ we see that $T_\mathcal Mathcal A^n\circ T_{A(\underline{i})}$ is also the
identity map on $X$, so
$(T_\mathcal Mathcal A^n)'(T_{A(\underline{i})}(x)) \, T_{A(\underline{i})}'(x) =1$
for all $x\in X$, by the chain rule. Setting $x=p_{A(\underline{i})} = T_{A(\underline{i})}(p_{A(\underline{i})})$ we obtain
\begin{equation}\mathcal Label{inversederivformu}
(T_\mathcal Mathcal A^n)'(p_{A(\underline{i})}) = \frac{1}{ T_{A(\underline{i})}'( p_{A(\underline{i})} )} \,,
\mathcal End{equation}
and combining this with (\mathcal Ref{jsrderiveq}) gives the required formula (\mathcal Ref{jsrderivtaeq}).
\mathcal End{proof}
\subsection{Invariant measures for the induced dynamical system}
\begin{defn}
For $\mathcal Mathcal A\in\mathcal Mm$,
let $\mathcal M_\mathcal Mathcal A$ denote the set
of $T_\mathcal Mathcal A$-invariant Borel probability measures on $X=[0,1]$;
the support of any such measure is contained in $Y_\mathcal Mathcal A=\cap_{n\ge0} T_\mathcal Mathcal A^{-n}(X_\mathcal Mathcal A)$.
\mathcal End{defn}
The following is a well known consequence of the compactness of $Y_\mathcal Mathcal A$ and continuity of $T_\mathcal Mathcal A$
(see e.g.~\cite[Thm.~6.10]{walters}):
\begin{lemma}\mathcal Label{macompactlemma}
The set $\mathcal M_\mathcal Mathcal A$ is compact with respect to the weak$^*$ topology.
\mathcal End{lemma}
\begin{defn}\mathcal Label{periodicorbitmeasuredefn}
For any $p\in X_\mathcal Mathcal A$ that is a periodic point for $T_\mathcal Mathcal A$, with $T_\mathcal Mathcal A^n(p)=p$, we say that the
probability measure $\mathcal Mu\in\mathcal M_\mathcal Mathcal A$ defined by
\begin{equation}
\mathcal Mu
=\frac{1}{n}\sum_{j=0}^{n-1} \delta_{T_\mathcal Mathcal A^j (p)}
\mathcal End{equation}
is the corresponding \mathcal Emph{periodic orbit measure} (or \mathcal Emph{$T_\mathcal Mathcal A$-periodic orbit measure}).
\mathcal End{defn}
\begin{remark}\mathcal Label{conjugacymeasures}
The topological conjugacy $h_\mathcal Mathcal A:\Omega\mathcal To Y_\mathcal Mathcal A$ between the shift map $\sigma:\Omega\mathcal To\Omega$
and $T_\mathcal Mathcal A:Y_\mathcal Mathcal A\mathcal To Y_\mathcal Mathcal A$ (cf.~Remark \mathcal Ref{taremarks} (b))
induces a one-to-one correspondence
$h_\mathcal Mathcal A^*:\mathcal M\mathcal To\mathcal M_\mathcal Mathcal A$ between invariant measures.
\mathcal End{remark}
\begin{defn}\mathcal Label{eomax}
For a bounded Borel function $f:X_\mathcal Mathcal A\mathcal To\mathcal Mathbb{R}$,
a measure $m\in\mathcal M_\mathcal Mathcal A$ is called \mathcal Emph{$f$-maximizing} if
$$
\int f\, dm = \sup_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f\, d\mathcal Mu
\,.
$$
\mathcal End{defn}
In the generality of Definition \mathcal Ref{eomax}, the notion of an $f$-maximizing invariant measure
is part of the wider field of
so-called ergodic optimization, see e.g.~\cite{jeo}.
\begin{defn}
For $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$, define the \mathcal Emph{induced function} $f_\mathcal Mathcal A:X_\mathcal Mathcal A\mathcal To\mathcal Mathbb{R}$ by
\begin{equation}\mathcal Label{function}
f _\mathcal Mathcal A = \frac{1}{2}\mathcal Left( \mathcal Log T_\mathcal Mathcal A' +(\mathcal Log \det A_0)\mathcal Mathbbm{1}_{X_{A_0}} +(\mathcal Log \det A_1)\mathcal Mathbbm{1}_{X_{A_1}} \mathcal Right) \,. \
\mathcal End{equation}
That is,
\begin{equation}\mathcal Label{fthatis}
f_\mathcal Mathcal A(x)=
\begin{cases}
\frac{1}{2}\mathcal Left( \mathcal Log S_{A_0}'(x) + \mathcal Log \det A_0 \mathcal Right) &\mathcal Text{if }x\in X_{A_0} \\
\frac{1}{2} \mathcal Left(\mathcal Log S_{A_1}'(x) + \mathcal Log \det A_1 \mathcal Right) &\mathcal Text{if }x\in X_{A_1} \,,
\mathcal End{cases}
\mathcal End{equation}
so writing $A_i= \begin{pmatrix} a_i&b_i\\c_i&d_i\mathcal End{pmatrix} $ gives
\begin{equation}\mathcal Label{explicitfalt}
f_\mathcal Mathcal A(x) = \mathcal Log \mathcal Left( \frac{\det A_i}{-\alpha_{A_i}(x+ \sigma_{A_i})} \mathcal Right)
\quad\mathcal Text{for }x\in X_{A_i}\,,
\mathcal End{equation}
where we recall from
Corollary \mathcal Ref{alwaysposcor} that $\frac{\det A_i}{-\alpha_{A_i}(x+ \sigma_{A_i})}>0$
for all $x\in X_{A_i}$.
\mathcal End{defn}
\begin{remark}\mathcal Label{faremark}
The function $f_\mathcal Mathcal A$ is clearly Lipschitz continuous on each $X_{A_i}$, hence Lipschitz continuous
on $X_\mathcal Mathcal A = X_{A_0}\cup X_{A_1}$, since the intervals $X_{A_0}$ and $X_{A_1}$ are disjoint.
\mathcal End{remark}
The reason for introducing the function $f_\mathcal Mathcal A$ is provided by the following characterisation of the joint spectral radius in terms of ergodic optimization:
\begin{theorem}\mathcal Label{maxtheorem}
If $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$ then its joint spectral radius $r(\mathcal Mathcal A)$ satisfies
\begin{equation}\mathcal Label{logra}
\mathcal Log r(\mathcal Mathcal A) =
\mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu \,.
\mathcal End{equation}
\mathcal End{theorem}
\begin{proof}
From Proposition \mathcal Ref{jsrderivta} we have
\begin{equation}\mathcal Label{jsrderivtalog}
\mathcal Log r(\mathcal Mathcal A)=
\sup_{\underline{i}\in \Omega^*}
\mathcal Log \mathcal Left( \det A(\underline{i}) \,\, (T_{\mathcal Mathcal A}^{|\underline{i}|})' (p_{A(\underline{i} )}) \mathcal Right)^{1/2|\underline{i}|}\,.
\mathcal End{equation}
If $\underline{i}\in\{0,1\}^n$ then
\begin{align}\mathcal Label{falogdet}
\mathcal Log \mathcal Left( \det A(\underline{i}) \,\, (T_{\mathcal Mathcal A}^n)' (p_{A(\underline{i} )}) \mathcal Right)^{1/2|\underline{i}|}
&=
\frac{1}{2n} \mathcal Left( \mathcal Log \det \mathcal Left( A_{i_1}\cdots A_{i_n} \mathcal Right) + \mathcal Log (T_{\mathcal Mathcal A}^n)' (p_{A(\underline{i} )}) \mathcal Right) \cr
&=
\frac{1}{2n} \mathcal Left( \mathcal Log \prod_{j=1}^n \det A_{i_j} + \mathcal Log \prod_{j=0}^{n-1} T_{\mathcal Mathcal A}' (T_\mathcal Mathcal A^j (p_{A(\underline{i} )})) \mathcal Right) \cr
&=
\frac{1}{n} \sum_{j=0}^{n-1}\ \frac{1}{2}\mathcal Left( \mathcal Log \det A_{i_{j+1}} + \mathcal Log T_{\mathcal Mathcal A}' (T_\mathcal Mathcal A^j (p_{A(\underline{i} )})) \mathcal Right) \cr
&=
\frac{1}{n} \sum_{j=0}^{n-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j (p_{A(\underline{i} )})) \,,
\mathcal End{align}
where the last step uses (\mathcal Ref{function}) together with the fact that
$$
\mathcal Log \det A_{i_{j+1}}
=
\mathcal Left( \mathcal Left( \mathcal Log \det A_0 \mathcal Right) \mathcal Mathbbm{1}_{X_{A_0}} +(\mathcal Log \det A_1)\mathcal Mathbbm{1}_{X_{A_1}} \mathcal Right)(T_\mathcal Mathcal A^j (p_{A(\underline{i} )})) \,,
$$
because
\begin{equation*}
\mathcal Mathbbm{1}_{X_{A_k}}(T_\mathcal Mathcal A^j (p_{A(\underline{i} )}))
=
\begin{cases}
1& \mathcal Text{if } i_{j+1}=k\cr
0& \mathcal Text{if } i_{j+1}\neq k \,.
\mathcal End{cases}
\mathcal End{equation*}
Combining (\mathcal Ref{jsrderivtalog}) and (\mathcal Ref{falogdet}) gives
\begin{equation}\mathcal Label{perptmeasures}
\mathcal Log r(\mathcal Mathcal A)=
\sup_{\underline{i}\in \Omega^*}
\frac{1}{|\underline{i}|} \sum_{j=0}^{|\underline{i}|-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j (p_{A(\underline{i} )}))
= \sup_{\underline{i}\in \Omega^*}
\int f_\mathcal Mathcal A\, d\mathcal Mu_{\underline{i}}
\,,
\mathcal End{equation}
where
$$\mathcal Mu_{\underline{i}}=\frac{1}{|\underline{i}|}\sum_{j=0}^{|\underline{i}|-1} \delta_{T_\mathcal Mathcal A^j (p_{A(\underline{i} )})}$$
is the
periodic orbit measure (see Definition \mathcal Ref{periodicorbitmeasuredefn})
for the period-$|\underline{i}|$ point
$p_{A(\underline{i} )}=T_\mathcal Mathcal A^{|\underline{i}|}(p_{A(\underline{i} )})$, i.e.~the
unique measure in $\mathcal M_\mathcal Mathcal A$ whose support equals the
periodic orbit $\{ T_\mathcal Mathcal A^j (p_{A(\underline{i} )})\}_{j= 0}^{ |\underline{i}|-1}$.
By a result of Parthasarathy \cite{parthasarathy} (see also Sigmund \cite{sigmund}),
the collection of periodic orbit measures $\{\mathcal Mu_{\underline{i}}:\underline{i}\in\Omega^*\}$
is weak$^*$ dense in the weak$^*$ compact space $\mathcal M_\mathcal Mathcal A$, so
\begin{equation}\mathcal Label{parthsigmund}
\sup_{\underline{i}\in \Omega^*}
\int f_\mathcal Mathcal A\, d\mathcal Mu_{\underline{i}}
= \mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu \,,
\mathcal End{equation}
and combining with (\mathcal Ref{perptmeasures}) gives the required equality
(\mathcal Ref{logra}).
\mathcal End{proof}
\subsection{The finiteness property and periodic orbits}
In view of Theorem \mathcal Ref{maxtheorem}, we shall be interested in those measures $m\in\mathcal M_\mathcal Mathcal A$ which
are $f_\mathcal Mathcal A$-maximizing, in the sense of Definition \mathcal Ref{eomax},
i.e.~$m$ attains the maximum in (\mathcal Ref{logra}): $\mathcal Log r(\mathcal Mathcal A) =
\mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu =\int f_\mathcal Mathcal A\, dm$.
The finiteness property for $\mathcal Mathcal A$
(which we recall means that $r(\mathcal Mathcal A)=r(A_{i_1}\cdots A_{i_n})^{1/n}$
for some $i_1,\mathcal Ldots, i_n\in\{0,1\}$)
corresponds to existence of a periodic orbit measure which
is $f_\mathcal Mathcal A$-maximizing:
\begin{prop}\mathcal Label{finitenessperiodicequivalenceprop}
$\mathcal Mathcal A\in\mathcal Mm$ has the finiteness property if and only if some $T_\mathcal Mathcal A$-periodic orbit measure is $f_\mathcal Mathcal A$-maximizing.
\mathcal End{prop}
\begin{proof}
If $\mathcal Mathcal A\in\mathcal Mm$ has the finiteness property,
and $\underline{i}\in\{0,1\}^n$ satisfies $r(\mathcal Mathcal A)= r(A(\underline{i}))^{1/n}$,
then we claim that the corresponding periodic orbit measure
$\mathcal Mu_{\underline{i}}=\frac{1}{n}\sum_{j=0}^{n-1} \delta_{T_\mathcal Mathcal A^j (p_{A(\underline{i} )})}$
is $f_\mathcal Mathcal A$-maximizing.
To see this, first note that (\mathcal Ref{logra}) gives
\begin{equation}\mathcal Label{firstnote}
\mathcal Log r(A(\underline{i}))^{1/n} = \mathcal Log r(\mathcal Mathcal A) = \mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu\,,
\mathcal End{equation}
and the lefthand side of (\mathcal Ref{firstnote}) can be written as
\begin{equation}\mathcal Label{intermediate1}
\mathcal Log r(A(\underline{i}))^{1/n} = \mathcal Log \mathcal Left( \frac{\det A(\underline{i})}{T_{A(\underline{i})}'(p_{A(\underline{i})})}\mathcal Right)^{1/2n}
= \mathcal Log \mathcal Left( \det A(\underline{i}) \,\, (T_{\mathcal Mathcal A}^n)' (p_{A(\underline{i} )}) \mathcal Right)^{1/2n}
\mathcal End{equation}
using Corollary \mathcal Ref{specradiusta} and (\mathcal Ref{inversederivformu}),
and therefore (\mathcal Ref{falogdet}) gives
\begin{equation}\mathcal Label{intermediate2}
\mathcal Log r(A(\underline{i}))^{1/n} = \frac{1}{n} \sum_{j=0}^{n-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j (p_{A(\underline{i} )}))
=\int f_\mathcal Mathcal A\, d\mathcal Mu_{\underline{i}}\,,
\mathcal End{equation}
so (\mathcal Ref{firstnote}) implies that
$\mathcal Mu_{\underline{i}}$
is indeed $f_\mathcal Mathcal A$-maximizing.
Conversely, if some $T_\mathcal Mathcal A$-periodic orbit measure is $f_\mathcal Mathcal A$-maximizing,
then this measure is necessarily of the form $\mathcal Mu_{\underline{i}}=\frac{1}{n}\sum_{j=0}^{n-1} \delta_{T_\mathcal Mathcal A^j (p_{A(\underline{i} )})}$
for some $n\in\mathbb N$ and $\underline{i}\in\{0,1\}^n$,
and satisfies
$\int f_\mathcal Mathcal A\, d\mathcal Mu_{\underline{i}} = \mathcal Max_{\mathcal Mu\in\mathcal M_\mathcal Mathcal A} \int f_\mathcal Mathcal A\, d\mathcal Mu$.
Combining
(\mathcal Ref{logra}) with
(\mathcal Ref{intermediate1}) and (\mathcal Ref{intermediate2}) gives
$\mathcal Log r(A(\underline{i}))^{1/n} = \mathcal Log r(\mathcal Mathcal A)$, so $\mathcal Mathcal A$ has the finiteness property, as required.
\mathcal End{proof}
Note that the above proof has also established:
\begin{prop}
Suppose that $\mathcal Mathcal A\in\mathcal Mm$,
and $\underline{i}\in\{0,1\}^n$ for some $n\in\mathbb N$.
Then $r(\mathcal Mathcal A)= r(A(\underline{i}))^{1/n}$
if and only if the periodic orbit measure
$\mathcal Mu_{\underline{i}}=\frac{1}{n}\sum_{j=0}^{n-1} \delta_{T_\mathcal Mathcal A^j (p_{A(\underline{i} )})}$
is $f_\mathcal Mathcal A$-maximizing.
\mathcal End{prop}
Recall that we say $\mathcal Mathcal A\in\mathcal Mm$ is a \mathcal Emph{finiteness counterexample} if
$r(\mathcal Mathcal A) > r(A_{i_1}\cdots A_{i_n})^{1/n}$
for all $n\in\mathbb N$ and all choices $i_1,\mathcal Ldots, i_n\in\{0,1\}$. We have:
\begin{prop}\mathcal Label{counterexampleequivalenceprop}
$\mathcal Mathcal A\in\mathcal Mm$ is a finiteness counterexample if and only if no $T_\mathcal Mathcal A$-periodic orbit measure is $f_\mathcal Mathcal A$-maximizing.
In this case there exists at least one measure $\mathcal Mu\in\mathcal M_\mathcal Mathcal A$ that is $f_\mathcal Mathcal A$-maximizing, and there exist uncountably many sequences $\omega\in\Omega$ such that
\begin{equation}
\mathcal Label{inthelimit}
r(\mathcal Mathcal A) = \mathcal Lim_{n\mathcal To\infty} r(A_{\omega_1}\cdots A_{\omega_n})^{1/n} \,.
\mathcal End{equation}
\mathcal End{prop}
\begin{proof}
The first statement is equivalent to that of Proposition \mathcal Ref{finitenessperiodicequivalenceprop}, while the existence of
an $f_\mathcal Mathcal A$-maximizing measure $\mathcal Mu$ is a consequence
(see e.g.~\cite[Prop.~2.4 (i)]{jeo})
of the continuity of $f_\mathcal Mathcal A$ and the weak$^*$ compactness of
$\mathcal M_\mathcal Mathcal A$ (see Lemma \mathcal Ref{macompactlemma}). In fact
$\mathcal Mu$ may be chosen to be an ergodic measure,
since it is readily shown that the set of $f_\mathcal Mathcal A$-maximizing measures is convex, and any of its extremal points
is ergodic (see e.g.~\cite[Prop.~2.4]{jeo}).
The ergodic
theorem (see e.g.~\cite[Thm.~1.14]{walters}) then implies that
\begin{equation}\mathcal Label{inthelimit2}
\mathcal Log r(\mathcal Mathcal A) = \int f_\mathcal Mathcal A\, d\mathcal Mu = \mathcal Lim_{n\mathcal To\infty} \frac{1}{n}\sum_{j=0}^{n-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j(p))
\mathcal End{equation}
for $\mathcal Mu$-almost every $p\in Y_\mathcal Mathcal A$.
Since no periodic orbit measure is $f_\mathcal Mathcal A$-maximizing, the measure $\mathcal Mu$
must have uncountable support, and therefore
(\mathcal Ref{inthelimit2}) holds for an uncountable set of points $p\in Y_\mathcal Mathcal A$.
We may use the topological conjugacy $h_\mathcal Mathcal A:\Omega\mathcal To Y_\mathcal Mathcal A$
to define the image measure $m=(h_\mathcal Mathcal A^{-1})^*(\mathcal Mu)$, which also has uncountable support, and
if we write $h_\mathcal Mathcal A(\omega)=p$ then
\begin{equation}\mathcal Label{sumproductexp}
r(A_{\omega_1}\cdots A_{\omega_n})^{1/n} = \mathcal Exp\mathcal Left( \frac{1}{n}\sum_{j=0}^{n-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j(p)) \mathcal Right) \,,
\mathcal End{equation} so
(\mathcal Ref{inthelimit2}) implies $
r(\mathcal Mathcal A) = \mathcal Lim_{n\mathcal To\infty} r(A_{\omega_1}\cdots A_{\omega_n})^{1/n}
$
for $m$-almost every $\omega\in\Omega$, hence for uncountably many $\omega\in\Omega$.
\mathcal End{proof}
\subsection{Monotonicity properties and formulae}
The following simple lemma records that for $\mathcal Mathcal A\in\mathcal Mm$,
the induced dynamical system $T_{\mathcal Mathcal A(t)}$
is independent of $t$, and that the induced function $f_{\mathcal Mathcal A(t)}$ differs from $f_\mathcal Mathcal A$
only by the addition of a scalar multiple of the characteristic function
for the image $X_{A_1}$.
\begin{lemma}\mathcal Label{simple}
For $\mathcal Mathcal A=(A_0,A_1)\in \mathcal Mm$,
and all $t>0$,
\begin{itemize}
\item[(i)] $T_{\mathcal Mathcal A(t)} = T_\mathcal Mathcal A$ \,,
\item[(ii)] $f_{\mathcal Mathcal A(t)} = f_\mathcal Mathcal A + (\mathcal Log t)\mathcal Mathbbm{1}_{X_{A_1}}$,
\item[(iii)]
$
f_{\mathcal Mathcal A(t)}(T_{A_0}(1)) - f_{\mathcal Mathcal A(t)}(T_{A_1}(0))
=
f_{\mathcal Mathcal A}(T_{A_0}(1)) - f_{\mathcal Mathcal A}(T_{A_1}(0)) - \mathcal Log t\,,
$
\item[(iv)] $f_{\mathcal Mathcal A(t)}'=f_\mathcal Mathcal A'$, with
\begin{equation}\mathcal Label{fderivative}
f_\mathcal Mathcal A' (x)=
-( x+\sigma_{A_i})^{-1}
\quad\mathcal Text{for }x\in X_{A_i}, \ i\in\{0,1\}\,.
\mathcal End{equation}
\mathcal End{itemize}
\mathcal End{lemma}
\begin{proof}
(i) From Remark \mathcal Ref{invariantposmultiple} we see that if $t>0$ then
$T_{tA_1}=T_{A_1}$, hence
$T_{\mathcal Mathcal A(t)} = T_\mathcal Mathcal A$.
(ii)
Formula (\mathcal Ref{fthatis}) gives $f_\mathcal Mathcal A = f_{\mathcal Mathcal A(t)}$ on $X_{A_0}$,
while for $x\in X_{A_1}$ we have
$$f_{\mathcal Mathcal A(t)}(x) = \frac{1}{2}\mathcal Left( \mathcal Log S_{A_1}'(x) + \mathcal Log \det tA_1 \mathcal Right)
= \mathcal Log t +f_\mathcal Mathcal A(x)$$
since $\mathcal Log \det tA_1 = \mathcal Log \mathcal Left( t^2 \det A_1 \mathcal Right) = 2\mathcal Log t +\mathcal Log \det A_1$,
thus
$f_{\mathcal Mathcal A(t)} = f_\mathcal Mathcal A + (\mathcal Log t)\mathcal Mathbbm{1}_{X_{A_1}}$.
(iii) This is immediate from part (ii).
(iv) The formula for $f_\mathcal Mathcal A'$ follows readily from the explicit formula (\mathcal Ref{explicitfalt}) for $f_\mathcal Mathcal A$,
and is equal to $f_{\mathcal Mathcal A(t)}'$ by (ii) above.
\mathcal End{proof}
\begin{lemma}\mathcal Label{posnegf}
If $\mathcal Mathcal A=(A_0,A_1)\in\mathcal Mm$ then
\begin{enumerate}
\item[(i)]
$f_\mathcal Mathcal A'$ is strictly positive on $X_{A_0}$ and strictly negative on $X_{A_1}$,
\item[(ii)]
$f_\mathcal Mathcal A$ is strictly increasing on $X_{A_0}$ and strictly decreasing on $X_{A_1}$,
\item[(iii)]
$(f_\mathcal Mathcal A\circ T_{A_0}^i)'(x)>0$ and $(f_\mathcal Mathcal A\circ T_{A_1}^i)'(x)<0$ for all $i\ge1$, $x\in X$.
\mathcal End{enumerate}
\mathcal End{lemma}
\begin{proof}
(i)
In view of formula (\mathcal Ref{fderivative}),
it suffices to note that by Corollary \mathcal Ref{alwaysposcor},
$
x+\sigma_{A_0} <0
$
for $x\in X_{A_0}$,
and
$x+\sigma_{A_1} >0
$
for $x\in X_{A_1}$.
(ii) This is an immediate consequence of (i).
(iii)
By the chain rule,
\begin{equation}\mathcal Label{chain}
(f_\mathcal Mathcal A\circ T_{A_j}^i)'(x) = f_\mathcal Mathcal A'(T_{A_j}^i (x)) (T_{A_j}^i)'(x)\,.
\mathcal End{equation}
The second factor $(T_{A_j}^i)'(x)$ on the righthand side of (\mathcal Ref{chain}) is strictly positive
for all $x\in X$, $i\ge1$, $j\in\{0,1\}$, since $T_{A_j}$ is orientation-preserving, as noted in Remark \mathcal Ref{derivativesremark}.
Regarding the sign of the first factor $f_\mathcal Mathcal A'(T_{A_j}^i (x))$ on the righthand side of (\mathcal Ref{chain}),
note that since $i\ge1$ then $T_{A_j}^i(x)\in X_{A_j}=T_{A_j}(X)$ for all $x\in X$.
Part (i) above then implies that $f_\mathcal Mathcal A'(T_{A_j}^i (x))$
is strictly positive when $j=0$ and strictly negative when $j=1$.
It follows that
$(f_\mathcal Mathcal A\circ T_{A_j}^i)'(x)$
is strictly positive when $j=0$ and strictly negative when $j=1$, as required.
\mathcal End{proof}
For the purposes of the following Lemma \mathcal Ref{usefullem}, it will be convenient to introduce the following notation:
\begin{notation}
For a matrix $A= \begin{pmatrix} a&b\\c&d\mathcal End{pmatrix} \in \mathbb M$,
define
$$
\delta_A = \frac{b+d}{\alpha_A} = \frac{b+d}{a+c-b-d}\,.
$$
\mathcal End{notation}
We can now give another characterisation of $\mathcal Rho_A$:
\begin{lemma}\mathcal Label{usefullem}
For $A\in \mathbb M$,
\begin{equation*}
\mathcal Rho_A = \mathcal Lim_{k\mathcal To \infty} \delta_{A^k}\,.
\mathcal End{equation*}
\mathcal End{lemma}
\begin{proof}
Perron-Frobenius theory (see e.g.~\cite[Thm.~0.17]{walters}) gives
$$
\mathcal Lim_{k\mathcal To\infty} \mathcal Lambda_A^{-k} A^k = v \, w_A
=
\begin{pmatrix} v^{(1)} w_A^{(1)} & v^{(1)} w_A^{(2)} \\ v^{(2)} w_A^{(1)} & v^{(2)} w_A^{(2)} \mathcal End{pmatrix}\,,
$$
where the positive dominant eigenvalue $\mathcal Lambda_A>0$ and corresponding left eigenvector
$w_A$ are as in Lemma \mathcal Ref{pflemma}, and $v$ is a corresponding right eigenvector
(a suitable multiple of $v_A$ from Lemma \mathcal Ref{pflemma}), normalised so that
$w_A v=1$.
It follows that
\begin{equation}\mathcal Label{rklimit}
\mathcal Lim_{k\mathcal To\infty} \delta_{A^k}
= \frac{v^{(1)} w^{(2)}_A +v^{(2)} w_A^{(2)} }{v^{(1)} w_A^{(1)}+v^{(2)} w^{(1)}_A - v^{(1)} w^{(2)}_A -v^{(2)} w^{(2)}_A }
=\frac{w^{(2)}_A}{w^{(1)}_A -w^{(2)}_A}\,,
\mathcal End{equation}
so the formula
$\mathcal Rho_A = \frac{w_A^{(2)}}{w_A^{(1)}-w_A^{(2)}}$ from Lemma \mathcal Ref{wrholem} concludes the proof.
\mathcal End{proof}
\begin{cor}\mathcal Label{useful}
For $A\in \mathbb M$, $x\in X$,
\begin{equation}\mathcal Label{rhooccurence}
\sum_{n=1}^\infty (\mathcal Log S_A'\circ T_A^n)'(x) = \frac{2}{x+\mathcal Rho_A}\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{proof}
A simple calculation using the chain rule
yields
\begin{equation}\mathcal Label{negativeformula}
\sum_{n=1}^k ( \mathcal Log S_A'\circ T_A^n)'(x) =
- (\mathcal Log T_{A^k}')'(x)
= \frac{2}{x+\delta_{A^k}}
\mathcal End{equation}
for all $k\ge1$, so
letting $k\mathcal To\infty$
we see that the result follows from Lemma \mathcal Ref{usefullem}.
\mathcal End{proof}
Recalling from (\mathcal Ref{fthatis}) that $f_\mathcal Mathcal A= \frac{1}{2}\mathcal Left( \mathcal Log S_{A_i}' +\mathcal Log \det A_i\mathcal Right)$ on $X_{A_i}$,
the following result is an immediate consequence of Corollary \mathcal Ref{useful}:
\begin{cor}\mathcal Label{usefulcor}
If $\mathcal Mathcal A=(A_0,A_1)\in \mathbb M^2$ then for $i\in \{0,1\}$,
\begin{equation}\mathcal Label{sumformula}
\sum_{n=1}^\infty (f_\mathcal Mathcal A\circ T_{A_i}^n)'(x) = \frac{1}{x+\varrho_{A_i}}
\quad\mathcal Text{for }x\in X_{A_i}\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{cor}\mathcal Label{posneg}
If $\mathcal Mathcal A=(A_0,A_1)\in \mathcal Mm$
then for all $x\in X$,
\begin{equation}\mathcal Label{posnegsums}
\sum_{n=1}^\infty (f_\mathcal Mathcal A\circ T_{A_0}^n)'(x) >0
\quad\mathcal Text{and}\quad
\sum_{n=1}^\infty (f_\mathcal Mathcal A\circ T_{A_1}^n)'(x) <0\,,
\mathcal End{equation}
and
\begin{equation}\mathcal Label{posnegxrhos}
x+\varrho_{A_0}>0
\quad\mathcal Text{and}\quad
x+\varrho_{A_1}<0\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{proof}
The inequalities in (\mathcal Ref{posnegsums}) follow from Lemma \mathcal Ref{posnegf} (iii),
while (\mathcal Ref{posnegxrhos}) is an immediate consequence of
(\mathcal Ref{sumformula}) and
(\mathcal Ref{posnegsums}).
\mathcal End{proof}
\begin{remark}
The inequality $x+\varrho_{A_0}>0$ in (\mathcal Ref{posnegxrhos}) can also be deduced from the fact that
$\mathcal Rho_{A_0}>0$ (by Corollary \mathcal Ref{rhoposlessminusone}) and
$x\ge0$.
\mathcal End{remark}
\section{Sturmian measures associated to a concave-convex matrix pair}\mathcal Label{sturmasection}
For $\mathcal Mathcal A\in\mathcal Mm$, the induced space $X_\mathcal Mathcal A$ becomes an ordered set when equipped
with the usual order on $X=[0,1]$. In particular,
by a \mathcal Emph{sub-interval} of $X_\mathcal Mathcal A$ we mean any subset of $X_\mathcal Mathcal A$ of the form $I\cap X_\mathcal Mathcal A$
where $I$ is some sub-interval of $X$. Note that a sub-interval of $X_\mathcal Mathcal A$ is
a sub-interval of $X$ if it is contained in either $X_{A_0}$ or $X_{A_1}$; otherwise it is
a union of two disjoint intervals in $X$.
\begin{defn}
Given a matrix pair $\mathcal Mathcal A \in \mathcal Mm$,
a closed interval $\Gamma\subset X_\mathcal Mathcal A$ is called
\mathcal Emph{$\mathcal Mathcal A$-Sturmian} (or simply \mathcal Emph{Sturmian})
if $T_\mathcal Mathcal A(\mathcal Min\Gamma)=T_\mathcal Mathcal A(\mathcal Max\Gamma)$,
i.e.~its two endpoints $\mathcal Min \Gamma$ and $\mathcal Max \Gamma$ have the same image under
the induced dynamical system
$T_\mathcal Mathcal A$.
\mathcal End{defn}
\begin{remark}\mathcal Label{sturmianasturmian}
\item[\, (a)]
The topological conjugacy $h_\mathcal Mathcal A: \Omega\mathcal To Y_\mathcal Mathcal A$ (cf.~Remark \mathcal Ref{conjugacymeasures}) is
order preserving, so
if $\Gamma\subset X_\mathcal Mathcal A$ is an $\mathcal Mathcal A$-Sturmian interval,
then $h_\mathcal Mathcal A^{-1}(\Gamma\cap Y_\mathcal Mathcal A)$ is
a Sturmian interval
as defined in Notation \mathcal Ref{sturmianfullshift}
(i.e. of the form $[0\omega,1\omega]$ for some $\omega\in\Omega$).
\item[\, (b)]
For all $t>0$, an interval is $\mathcal Mathcal A$-Sturmian if and only if it is $\mathcal Mathcal A(t)$-Sturmian.
\mathcal End{remark}
\begin{defn}\mathcal Label{henceforthca}
Let $\mathcal I_\mathcal Mathcal A$ denote the collection of all $\mathcal Mathcal A$-Sturmian intervals.
Note that $\mathcal I_\mathcal Mathcal A$ is naturally parametrized by $X=[0,1]$:
for each $c\in X$ there is a unique $\Gamma\in\mathcal I_\mathcal Mathcal A$ such that
$T_\mathcal Mathcal A(\mathcal Min \Gamma)= T_\mathcal Mathcal A(\mathcal Max\Gamma) =c$.
Henceforth we shall write $c_\mathcal Mathcal A(\Gamma)$ to denote the common value
$T_\mathcal Mathcal A(\mathcal Min \Gamma)= T_\mathcal Mathcal A(\mathcal Max\Gamma)$ for an $\mathcal Mathcal A$-Sturmian interval $\Gamma\in\mathcal I_\mathcal Mathcal A$, noting that
\begin{equation*}
c_\mathcal Mathcal A:\mathcal I_\mathcal Mathcal A\mathcal To X
\mathcal End{equation*}
is a bijection.
As a subset of $X$, we can express $\Gamma\in\mathcal I_\mathcal Mathcal A$ as
\begin{equation}\mathcal Label{disjunionsturm}
\Gamma = [T_{A_0}(c_\mathcal Mathcal A(\Gamma)), T_{A_0}(1)] \cup [T_{A_1}(0),T_{A_1}(c_\mathcal Mathcal A(\Gamma))]\,.
\mathcal End{equation}
\mathcal End{defn}
\begin{remark}\mathcal Label{neglectsingleton}
It is apparent from (\mathcal Ref{disjunionsturm}) that, viewed as a subset of $X=[0,1]$, an $\mathcal Mathcal A$-Sturmian interval $\Gamma$ is
\mathcal Emph{always} a disjoint union of two closed intervals. Note, however, that for the two extremal cases where $c_\mathcal Mathcal A(\Gamma)=0$ or $1$, one of the intervals in the disjoint union is a singleton set (and the other interval is, respectively, either $X_{A_0}$ or $X_{A_1}$).
These extremal cases are particularly significant, and in the calculations of \S \mathcal Ref{extsturmsection}
onwards it is convenient to neglect the singleton set, thereby identifying the extremal $\mathcal Mathcal A$-Sturmian interval
with either $X_{A_0}$ or $X_{A_1}$.
\mathcal End{remark}
\begin{defn}\mathcal Label{sturmianmeasdef}
We say that a $T_\mathcal Mathcal A$-invariant Borel probability measure on $X_\mathcal Mathcal A$ is
\mathcal Emph{$\mathcal Mathcal A$-Sturmian}
if its support is contained in
some $\mathcal Mathcal A$-Sturmian interval.
Let
$\mathcal S_\mathcal Mathcal A$ denote the collection of $\mathcal Mathcal A$-Sturmian measures.
\mathcal End{defn}
\begin{remark}
\item[\, (a)]
In view of Remarks \mathcal Ref{conjugacymeasures} and \mathcal Ref{sturmianasturmian},
the class of $\mathcal Mathcal A$-Sturmian measures on $X_\mathcal Mathcal A$ is just the $h_\mathcal Mathcal A^*$-image of the class of Sturmian measures
on the shift space $\Omega$,
i.e.~$\mathcal S_\mathcal Mathcal A = h_\mathcal Mathcal A^*(\mathcal Mathcal{S})$.
In particular (cf.~Proposition \mathcal Ref{sturmianclassical}\,(b)), $\mathcal S_\mathcal Mathcal A$ is also naturally parametrized by $X=[0,1]$: the map $\mathcal Mathcal{P}\circ (h_\mathcal Mathcal A^*)^{-1}:\mathcal S_\mathcal Mathcal A\mathcal To[0,1]$ is a homeomorphism,
and for $\mathcal Mu\in\mathcal S_\mathcal Mathcal A$ we refer to $\mathcal Mathcal{P}\circ (h_\mathcal Mathcal A^*)^{-1}(\mathcal Mu) = \mathcal Mu(X_{A_1})$
as its \mathcal Emph{(Sturmian) parameter}.
\item[\, (b)]
For all $t>0$, a measure is $\mathcal Mathcal A$-Sturmian if and only if it is $\mathcal Mathcal A(t)$-Sturmian.
\mathcal End{remark}
In \S \mathcal Ref{technicalsection} we shall identify cases where $\mathcal Mathcal A$-Sturmian measures arise as unique maximizing measures
for $f_{\mathcal Mathcal A(t)}$, $t>0$.
In particular, for certain $t$ the unique $f_{\mathcal Mathcal A(t)}$-maximizing measure is a Sturmian measure of \mathcal Emph{irrational} parameter, and
such $\mathcal Mathcal A(t)$ turn out to be \mathcal Emph{finiteness counterexamples}:
\begin{prop}\mathcal Label{irrationalcounterexampleconnection}
If $\mathcal Mathcal A\in\mathcal Mm$ is such that there is a unique $f_\mathcal Mathcal A$-maximizing measure, and this measure is an $\mathcal Mathcal A$-Sturmian measure with irrational parameter $\mathcal Mathcal{P}$, then $\mathcal Mathcal A$ is a finiteness counterexample
(i.e.~$r(\mathcal Mathcal A) > r(A_{i_1}\cdots A_{i_n})^{1/n}$
for all $n\in\mathbb N$ and all choices $i_1,\mathcal Ldots, i_n\in\{0,1\}$).
In this case
\begin{equation}
\mathcal Label{inthelimit3}
r(\mathcal Mathcal A) = \mathcal Lim_{n\mathcal To\infty} r(A_{\omega_1}\cdots A_{\omega_n})^{1/n} \,,
\mathcal End{equation}
holds for the uncountably many Sturmian sequences $\omega=(\omega_n)_{n=1}^\infty$ of parameter $\mathcal Mathcal{P}$.
\mathcal End{prop}
\begin{proof}
By assumption there is a unique $f_\mathcal Mathcal A$-maximizing measure $\mathcal Mu$,
and this measure is an $\mathcal Mathcal A$-Sturmian measure with irrational parameter $\mathcal Mathcal{P}$, which in particular is not a periodic orbit measure. It follows that no $T_\mathcal Mathcal A$-periodic orbit measure is $f_\mathcal Mathcal A$-maximizing, so Proposition
\mathcal Ref{counterexampleequivalenceprop} implies that $\mathcal Mathcal A$ is a finiteness counterexample,
and that there exist uncountably many sequences $\omega\in\Omega$ such that (\mathcal Ref{inthelimit3}) holds.
In fact the support of any $\mathcal Mathcal A$-Sturmian measure $\mathcal Mu$ is uniquely ergodic (see e.g.~\cite[Cor.~1.6]{bouschmairesse}),
so the ergodic theorem holds for the uncountably many points in the support of $\mathcal Mu$ (see e.g.~\cite[Thm.~6.19]{walters}),
and therefore $\mathcal Lim_{n\mathcal To\infty} \frac{1}{n}\sum_{j=0}^{n-1} f_\mathcal Mathcal A(T_\mathcal Mathcal A^j(p)) = \int f_\mathcal Mathcal A\, d\mathcal Mu = \mathcal Log r(\mathcal Mathcal A)$
for all $p$ in the support of $\mathcal Mu$.
Writing $p=h_\mathcal Mathcal A(\omega)$ as in the proof of Proposition \mathcal Ref{counterexampleequivalenceprop},
the relation (\mathcal Ref{sumproductexp}) then implies that
(\mathcal Ref{inthelimit3}) holds for all
points in the support of the Sturmian measure $m$,
i.e.~for all Sturmian sequences of parameter $\mathcal Mathcal{P}$.
\mathcal End{proof}
\section{The Sturmian transfer function}\mathcal Label{transfersection}
In order to
show that the maximizing measure for $f_{\mathcal Mathcal A(t)}$ is supported in some $\mathcal Mathcal A$-Sturmian interval
$\Gamma\in\mathcal I_\mathcal Mathcal A$, our strategy will be to add a coboundary $\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$,
where the corresponding \mathcal Emph{Sturmian transfer function} $\varphi_\Gamma$ is introduced below,
so that the new function $ f_\mathcal Mathcal A + \varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$
takes a constant value on all of $\Gamma$, and
is strictly smaller than this constant value on the complement of $\Gamma$.
This approach is patterned on ideas of Bousch \cite{bousch}
in the setting of the angle-doubling map
and degree-one trigonometric polynomials.
To proceed, it is convenient to introduce the following:
\begin{defn}
For $\mathcal Mathcal A\in\mathcal Mm$, to each $\mathcal Mathcal A$-Sturmian interval $\Gamma$ we associate the \mathcal Emph{hybrid contraction}
$\mathcal Tau_\Gamma: X\mathcal To X_\mathcal Mathcal A$, defined by
\begin{equation}\mathcal Label{hybrid}
\mathcal Tau_\Gamma(x)=
\begin{cases}
T_{A_1}(x)&\mathcal Text{if }x\in[0,c(\Gamma))\\
T_{A_0}(x)&\mathcal Text{if }x\in[c(\Gamma),1]\,.
\mathcal End{cases}
\mathcal End{equation}
\mathcal End{defn}
\begin{remark}\mathcal Label{taugammaremark}
The hybrid contraction $\mathcal Tau_\Gamma$ satisfies $\mathcal Tau_\Gamma(X)=\Gamma$,
and is piecewise Lipschitz continuous.
More precisely, its restriction to $[0,c_\mathcal Mathcal A(\Gamma))$ is Lipschitz,
as is its restriction to $[c_\mathcal Mathcal A(\Gamma),1]$.
\mathcal End{remark}
\begin{lemma}\mathcal Label{phiexists}
Given $\mathcal Mathcal A\in \mathcal Mm$, and
an $\mathcal Mathcal A$-Sturmian interval $\Gamma\in \mathcal I_\mathcal Mathcal A$,
there exists a unique Lipschitz continuous function $\varphi_{\mathcal Mathcal A,\Gamma}:X\mathcal To\mathcal Mathbb{R}$
which simultaneously satisfies\footnote{The substantial condition is (\mathcal Ref{defphiformula}),
which determines $\varphi_{\mathcal Mathcal A,\Gamma}$ up to an additive constant.
The extra condition
(\mathcal Ref{zeroconv}) is useful in that it removes any ambiguity when discussing $\varphi_{\mathcal Mathcal A,\Gamma}$.}
\begin{equation}\mathcal Label{defphiformula}
\varphi_{\mathcal Mathcal A, \Gamma}' = \sum_{n=1}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_{\Gamma}^n)'
\quad\mathcal Text{Lebesgue a.e.,}
\mathcal End{equation}
and
\begin{equation}\mathcal Label{zeroconv}
\varphi_{\mathcal Mathcal A, \Gamma}(0)=0\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
The function $f_\mathcal Mathcal A$
is Lipschitz,
and $\mathcal Tau_\Gamma$ is piecewise Lipschitz (cf.~Remark \mathcal Ref{taugammaremark}),
so each $\mathcal Tau_\Gamma^n$ is piecewise Lipschitz, so by Rademacher's Theorem is differentiable Lebesgue
almost everywhere, with $L^\infty$ derivative.
Now
$\|(\mathcal Tau_\Gamma^n)'\|_\infty = O(\mathcal Theta^n)$ as $n\mathcal To\infty$ for some $\mathcal Theta \in(0,1)$,
so the sum
$$\sum_{n=1}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'
= \sum_{n=1}^\infty f_\mathcal Mathcal A'\circ \mathcal Tau_\Gamma^n .( \mathcal Tau_\Gamma^n)'
$$
is Lebesgue almost everywhere convergent (as its $n$th term is
$O(\mathcal Theta^n)$),
and defines an $L^\infty$
function with respect to Lebesgue measure on $X$.
In particular, it has a Lipschitz antiderivative $\varphi_\Gamma$,
which is the unique Lipschitz antiderivative up to an additive constant,
hence uniquely defined if it satisfies the additional condition
$\varphi_{\mathcal Mathcal A, \Gamma}(0)=0$.
\mathcal End{proof}
\begin{notation}
For $\mathcal Mathcal A\in\mathcal Mm$, $\Gamma\in\mathcal I_\mathcal Mathcal A$,
the function $\varphi_\Gamma=\varphi_{\mathcal Mathcal A,\Gamma}$ whose existence and uniqueness is guaranteed
by Lemma \mathcal Ref{phiexists}
will be referred to as the corresponding \mathcal Emph{Sturmian transfer function}.
\mathcal End{notation}
\begin{remark}
Note that although the induced function $f_\mathcal Mathcal A$ is only defined on $X_\mathcal Mathcal A$, the Sturmian transfer function
$\varphi_\Gamma$ is actually defined on all of $X=[0,1]$. For the most part, however, we shall only
be interested in the restriction of $\varphi_\Gamma$ to $X_\mathcal Mathcal A$.
More precisely, we shall be interested in certain properties of $f_\mathcal Mathcal A+\varphi_\Gamma$,
or of $f_\mathcal Mathcal A+ \varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$,
considered as functions defined on $X_\mathcal Mathcal A$, beginning with the following Corollary \mathcal Ref{lipcont}.
\mathcal End{remark}
\begin{cor}\mathcal Label{lipcont}
If $\mathcal Mathcal A\in\mathcal Mm$, and $\Gamma$ is any $\mathcal Mathcal A$-Sturmian interval,
then both $f_\mathcal Mathcal A+\varphi_\Gamma$
and $f_\mathcal Mathcal A+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$ are Lipschitz continuous functions on $X_\mathcal Mathcal A$.
\mathcal End{cor}
\begin{proof}
Both $T_\mathcal Mathcal A$ and $f_\mathcal Mathcal A$ are Lipschitz continuous on $X_\mathcal Mathcal A$, as noted in Remarks \mathcal Ref{taremarks} and \mathcal Ref{faremark}, and $\varphi_\Gamma$ is Lipschitz continuous on $X$ as noted in Lemma \mathcal Ref{phiexists}, hence Lipschitz continuous
on $X_\mathcal Mathcal A$. It follows that
both $f_\mathcal Mathcal A+\varphi_\Gamma$
and $f_\mathcal Mathcal A+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$ are Lipschitz continuous on $X_\mathcal Mathcal A$.
\mathcal End{proof}
\begin{lemma}\mathcal Label{flatxai}
Suppose $\mathcal Mathcal A\in\mathcal Mm$, $t>0$, and $\Gamma$ is any $\mathcal Mathcal A$-Sturmian interval.
The Lipschitz continuous function
$f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A:X_\mathcal Mathcal A\mathcal To\mathcal Mathbb{R}$ has the property that its restriction to
$\Gamma\cap X_{A_0}$ is a constant function, and
its restriction to
$\Gamma\cap X_{A_1}$ is a constant function.
\mathcal End{lemma}
\begin{proof}
By Corollary \mathcal Ref{lipcont}, the function $f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$ is Lipschitz continuous on $X_\mathcal Mathcal A$, because $\mathcal Mathcal A(t)\in\mathcal Mm$.
So by the fundamental theorem of calculus for Lipschitz functions (see e.g.~\cite[Thm.~7.1.15]{kk}),
the required result will follow if it can be shown that
\begin{equation}\mathcal Label{derivativezero2}
(f_{\mathcal Mathcal A(t)}+\varphi_\Gamma -\varphi_\Gamma \circ T_\mathcal Mathcal A)'=0\quad \mathcal Text{Lebesgue a.e. on $\Gamma$}\,.
\mathcal End{equation}
But $f_{\mathcal Mathcal A(t)}'=f_\mathcal Mathcal A'$, so
(\mathcal Ref{derivativezero2}) is equivalent to proving that
\begin{equation}\mathcal Label{derivativezero}
(f_\mathcal Mathcal A +\varphi_\Gamma-\varphi_\Gamma\circ T_\mathcal Mathcal A)'=0\quad \mathcal Text{Lebesgue a.e. on $\Gamma$}\,.
\mathcal End{equation}
To establish this almost everywhere equality,
note that
$$
f_\mathcal Mathcal A' + \varphi_\Gamma' =
f_\mathcal Mathcal A' + \sum_{n=1}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'
=
\sum_{n=0}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'
=
\sum_{n=0}^\infty f_\mathcal Mathcal A' \circ \mathcal Tau_\Gamma^n. (\mathcal Tau_\Gamma^n)'
$$
and
$$
(\varphi_\Gamma\circ T_\mathcal Mathcal A)' =
\sum_{n=1}^\infty f_\mathcal Mathcal A' \circ \mathcal Tau_\Gamma^n \circ T_\mathcal Mathcal A . (\mathcal Tau_\Gamma^n)'\circ T_\mathcal Mathcal A . T_\mathcal Mathcal A'
= \sum_{n=1}^\infty f_\mathcal Mathcal A' \circ \mathcal Tau_\Gamma^{n-1}. (\mathcal Tau_\Gamma^{n-1})',
$$
since $(\mathcal Tau_\Gamma\circ T_\mathcal Mathcal A)' = \mathcal Tau_\Gamma^{n-1}$, so
indeed (\mathcal Ref{derivativezero}) holds.
\mathcal End{proof}
\begin{remark}
In the generality of Lemma \mathcal Ref{flatxai}, the constant values assumed by
$f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$ on
$\Gamma\cap X_{A_0}$ and
$\Gamma\cap X_{A_1}$
do not coincide.
However, we shall shortly give (see Lemma \mathcal Ref{flattenlemma}) an extra condition
which does ensure
that $f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$ takes the \mathcal Emph{same} constant value
on the whole of $\Gamma$. Indeed this possibility is a key tool in our strategy.
\mathcal End{remark}
\section{The extremal Sturmian intervals}\mathcal Label{extsturmsection}
\subsection{Formulae involving extremal intervals}
As noted in Remark \mathcal Ref{neglectsingleton}, an $\mathcal Mathcal A$-Sturmian interval is the disjoint union of two closed intervals when viewed as a subset of $X=[0,1]$. However, the two \mathcal Emph{extremal} cases yield a leftmost
$\mathcal Mathcal A$-Sturmian interval equal to $X_{A_0}\cup\{T_{A_1}(0)\}$,
and a rightmost $\mathcal Mathcal A$-Sturmian interval equal to $\{T_{A_0}(1)\}\cup X_{A_1}$.
The presence of singleton sets in these expressions is notationally inconvenient, and unnecessary
for our purposes, so henceforth we neglect them.
More precisely, henceforth
the leftmost $\mathcal Mathcal A$-Sturmian interval is taken to be $X_{A_0}=T_{A_0}(X)$, and denoted
by $\Gamma_0$, so that $\mathcal Tau_{\Gamma_0}=T_{A_0}$;
the rightmost $\mathcal Mathcal A$-Sturmian interval is taken to be $X_{A_1}=T_{A_1}(X)$, and denoted
by $\Gamma_1$, so that $\mathcal Tau_{\Gamma_1}=T_{A_1}$.
When the $\mathcal Mathcal A$-Sturmian interval $\Gamma$ is either $\Gamma_0$ or $\Gamma_1$, there is an explicit formula for the Sturmian transfer function $\varphi_\Gamma$:
\begin{lemma}\mathcal Label{easyphiformulae}
Suppose $\mathcal Mathcal A\in\mathcal Mm$.
For $i\in\{0,1\}$, and all $x\in X$,
\begin{equation}\mathcal Label{phiiformulae}
\varphi_{\Gamma_i}(x) = \mathcal Log \mathcal Left( \frac{x+\mathcal Rho_{A_i}}{\mathcal Rho_{A_i}} \mathcal Right) \,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Now $\mathcal Tau_{\Gamma_i} = T_{A_i}$, so the defining formula (\mathcal Ref{defphiformula}) becomes
\begin{equation}\mathcal Label{defphiiformula}
\varphi_{\Gamma_i}'(x) = \sum_{n=1}^\infty (f_\mathcal Mathcal A\circ T_{A_i}^n)'(x)\,,
\mathcal End{equation}
and then (\mathcal Ref{sumformula}) implies that
\begin{equation*}
\varphi_{\Gamma_i}'(x) = \frac{1}{x+\varrho_{A_i}}\,.
\mathcal End{equation*}
Noting that the sign of $x+\varrho_{A_i}$ is positive when $i=0$ and negative when $i=1$
(see Corollary \mathcal Ref{posneg}), as well as the convention that $\varphi_{\Gamma_i}(0)=0$ (see Definition \mathcal Ref{phiexists}),
we deduce the required expression (\mathcal Ref{phiiformulae}).
\mathcal End{proof}
\begin{defn}\mathcal Label{Deltadefndefn}
Given $\mathcal Mathcal A=(A_0,A_1)\in \mathcal Mm$ and
$\Gamma\in\mathcal I_\mathcal Mathcal A$,
define $\Delta_\mathcal Mathcal A(\Gamma) \in \mathcal Mathbb{R}$ by
\begin{equation}\mathcal Label{Deltadefn}
\Delta_\mathcal Mathcal A(\Gamma)
=
\mathcal Left(\varphi_\Gamma(1)-\varphi_\Gamma(0) \mathcal Right)
-
\mathcal Left( \varphi_\Gamma(T_{A_0}(1))-\varphi_\Gamma(T_{A_1}(0)) \mathcal Right) \,,
\mathcal End{equation}
noting the equivalent expression
\begin{equation}\mathcal Label{Deltadefnalt}
\Delta_\mathcal Mathcal A(\Gamma)
=
\varphi_\Gamma(1)
-
\mathcal Left( \varphi_\Gamma(T_{A_0}(1))-\varphi_\Gamma(T_{A_1}(0)) \mathcal Right)
\mathcal End{equation}
as a consequence of the convention that $\varphi_\Gamma(0)=0$
(see Lemma \mathcal Ref{phiexists}).
\mathcal End{defn}
The values $\Delta_\mathcal Mathcal A(\Gamma_i)$ play an important role, so it will be useful to record
the following explicit formulae:
\begin{lemma}\mathcal Label{deltagammailemma}
Suppose $\mathcal Mathcal A\in\mathcal Mm$.
For $i\in\{0,1\}$,
\begin{equation}\mathcal Label{deltagammai}
\Delta_\mathcal Mathcal A(\Gamma_i) =
\mathcal Log \mathcal Left(
\frac{(1+\mathcal Rho_{A_i}) \mathcal Left(\frac{b_1}{b_1+d_1}+\mathcal Rho_{A_i}\mathcal Right)}{ \mathcal Rho_{A_i} \mathcal Left( \frac{a_0}{a_0+c_0}+\mathcal Rho_{A_i}\mathcal Right)}
\mathcal Right) \,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
This is immediate from the defining formula (\mathcal Ref{Deltadefn})
(or (\mathcal Ref{Deltadefnalt})) for $\Delta_\mathcal Mathcal A(\Gamma_i)$, together with formula
(\mathcal Ref{phiiformulae}) for $\varphi_{\Gamma_i}$, and the fact that $T_{A_0}(1)=\frac{a_0}{a_0+c_0}$
and $T_{A_1}(0) = \frac{b_1}{b_1+d_1}$.
\mathcal End{proof}
\begin{cor}\mathcal Label{deltasigns}
If $\mathcal Mathcal A\in\mathcal Mm$ then
\begin{equation*}
\mathcal Label{starrr}
\Delta_\mathcal Mathcal A(\Gamma_1) < 0 < \Delta_\mathcal Mathcal A(\Gamma_0)\,.
\mathcal End{equation*}
\mathcal End{cor}
\begin{proof}
The four terms
$
\mathcal Rho_{A_i}
$,
$
1+\mathcal Rho_{A_i}
$,
$
\frac{a_0}{a_0+c_0}+\mathcal Rho_{A_i}
$,
$
\frac{b_1}{b_1+d_1}+\mathcal Rho_{A_i}
$
in (\mathcal Ref{deltagammai}) are all positive if $i=0$, and all negative if $i=1$, by
the inequalities (\mathcal Ref{posnegxrhos})
in Corollary \mathcal Ref{posneg}.
Now $\mathcal Mathcal A\in\mathcal Mm$ implies that (\mathcal Ref{definequal1}) holds, so
$
\frac{b_1}{b_1+d_1}+\mathcal Rho_{A_i}
>
\frac{a_0}{a_0+c_0}+\mathcal Rho_{A_i}
$,
and clearly $1+\mathcal Rho_{A_i}>\mathcal Rho_{A_i}$
Consequently
$$
\frac{(1+\mathcal Rho_{A_i}) \mathcal Left(\frac{b_1}{b_1+d_1}+\mathcal Rho_{A_i}\mathcal Right)}{ \mathcal Rho_{A_i} \mathcal Left( \frac{a_0}{a_0+c_0}+\mathcal Rho_{A_i}\mathcal Right)}
$$
is strictly greater than $1$ if $i=0$, and strictly smaller than $1$ if $i=1$.
The result then follows from Lemma \mathcal Ref{deltagammailemma}.
\mathcal End{proof}
\subsection{Adaptations for other matrix pairs}\mathcal Label{adaptations}
As mentioned in \S \mathcal Ref{relatsubsection}, the methods of this paper can be adapted
so as to give alternative proofs
of certain results
(analogues of Theorem \mathcal Ref{maintheorem})
mentioned in \S \mathcal Ref{generalsection},
namely establishing that a full Sturmian family is generated
by the matrix pair (\mathcal Ref{standardpair}), and for matrix pairs corresponding to sub-cases of (\mathcal Ref{bmfamily}) and (\mathcal Ref{kozyakinpairs})
which lie in the boundary of $\mathfrak D$.\footnote{Note that all
of the matrix pairs in (\mathcal Ref{bmfamily}), (\mathcal Ref{kozyakinpairs}), (\mathcal Ref{standardpair}) have the property that
$A_0$ is projectively concave and $A_1$ is projectively convex.}
In this subsection we indicate the modifications necessary to handle these cases.
Firstly, the induced space $X_\mathcal Mathcal A$ may be the whole of $X=[0,1]$ rather than a disjoint union of two closed intervals:
this occurs if $a_0/c_0=b_1/d_1$ (i.e.~when (\mathcal Ref{definequal1}) becomes an equality), which is the case for the pair
(\mathcal Ref{standardpair}), and for (\mathcal Ref{kozyakinpairs}) if $bc=1$.
Secondly, in each of the cases (\mathcal Ref{bmfamily}), (\mathcal Ref{kozyakinpairs}) and (\mathcal Ref{standardpair}), the
induced maps $T_{A_0}$ and $T_{A_1}$ have fixed points
at
0 and 1 respectively, so that the dynamical system $T_\mathcal Mathcal A$ also fixes these points.
For (\mathcal Ref{standardpair}), both 0 and 1 are indifferent fixed points, i.e.~$T_{A_0}'(0)=1=T_{A_1}'(1)$.
For (\mathcal Ref{bmfamily}) and (\mathcal Ref{kozyakinpairs}) these fixed points are unstable for the induced maps $T_{A_0}$ and $T_{A_1}$, i.e.~$T_{A_0}'(0)>1$ and $T_{A_1}'(1)>1$, but both of these maps also have stable fixed points in the interior of $X=[0,1]$.
Consequently for (\mathcal Ref{standardpair}) the dynamical system $T_\mathcal Mathcal A:X\mathcal To X$ has indifferent fixed points at 0 and 1,
and no other fixed points, while for (\mathcal Ref{bmfamily}) and (\mathcal Ref{kozyakinpairs}) the dynamical system $T_\mathcal Mathcal A$ has stable fixed points at 0 and 1, and two further unstable fixed points in the interior of $X$.
The potentially problematic stable fixed points for $T_\mathcal Mathcal A$ can in fact be avoided by omitting to consider
the two extremal $\mathcal Mathcal A$-Sturmian intervals: this ensures the
asymptotic $\|(\mathcal Tau_\Gamma^n)'\|_\infty = O(\mathcal Theta^n)$ as $n\mathcal To\infty$,
$\mathcal Theta \in(0,1)$, and the existence of Sturmian transfer functions is proved as in Lemma \mathcal Ref{phiexists}.
In the case where $T_\mathcal Mathcal A$ has indifferent fixed points, it is even possible to consider
extremal $\mathcal Mathcal A$-Sturmian intervals, as the series defining the Sturmian transfer function is nonetheless convergent.
The existence of Sturmian transfer functions then allows the
remainder of the method of proof to proceed essentially as for matrix pairs in $\mathfrak D$,
ultimately establishing analogues of the main result Theorem \mathcal Ref{maintheorem}.
\section{Associating $\mathcal Mathcal A$-Sturmian intervals to parameter values}\mathcal Label{particularsection}
\begin{notation}
For a Sturmian interval $\Gamma\in\mathcal I_\mathcal Mathcal A$,
let $s_\Gamma\in\mathcal S_\mathcal Mathcal A$
denote the $\mathcal Mathcal A$-Sturmian measure supported by $\Gamma$, i.e.~$s_\Gamma$ is
the unique $T_\mathcal Mathcal A$-invariant probability measure whose support is contained in $\Gamma$.
\mathcal End{notation}
\begin{lemma}\mathcal Label{flattenlemma}
Suppose $\mathcal Mathcal A\in \mathcal Mm$.
If $t\in\mathcal Mathbb{R}^+$
and
$\Gamma\in \mathcal I_\mathcal Mathcal A$ are such that
\begin{equation}\mathcal Label{keyflatteningequation}
f_{\mathcal Mathcal A(t)}(T_{A_0}(1)) - f_{\mathcal Mathcal A(t)}(T_{A_1}(0))
= \Delta_\mathcal Mathcal A( \Gamma)\,,
\mathcal End{equation}
then the Lipschitz continuous function
$f_{\mathcal Mathcal A(t)}+\varphi_{\Gamma} - \varphi_{\Gamma}\circ T_\mathcal Mathcal A$ is
equal to the constant value
$\int f_{\mathcal Mathcal A(t)}\, ds_\Gamma$ when restricted to $\Gamma$.
\mathcal End{lemma}
\begin{proof}
By Lemma \mathcal Ref{flatxai} we know that
$f_{\mathcal Mathcal A(t)}+\varphi_\Gamma-\varphi_\Gamma\circ T_\mathcal Mathcal A$
is constant when restricted to
$\Gamma\cap X_{A_0}$, and also constant
when restricted to $\Gamma\cap X_{A_1}$.
To prove that these constant values are the \mathcal Emph{same},
it suffices to show that $f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A$
takes the same value at the point $T_{A_0}(1)\in X_{A_0}$ as
it does at the point $T_{A_1}(0)\in X_{A_1}$.
But the equality
$$
\mathcal Left(f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A\mathcal Right)(T_{A_0}(1))
=
\mathcal Left(f_{\mathcal Mathcal A(t)}+\varphi_\Gamma - \varphi_\Gamma\circ T_\mathcal Mathcal A\mathcal Right)(T_{A_1}(0))
$$
holds if and only if
$$
f_{\mathcal Mathcal A(t)}(T_{A_0}(1)) - f_{\mathcal Mathcal A(t)}(T_{A_1}(0)) = \varphi_\Gamma(1)-\varphi_\Gamma(0) - \mathcal Left(\varphi_\Gamma(T_{A_0}(1)) - \varphi_\Gamma(T_{A_1}(0))\mathcal Right)\,,
$$
in other words
$f_{\mathcal Mathcal A(t)}(T_{A_0}(1)) - f_{\mathcal Mathcal A(t)}(T_{A_1}(0)) = \Delta_\mathcal Mathcal A( \Gamma)$, which is precisely
the hypothesis
(\mathcal Ref{keyflatteningequation}).
\mathcal End{proof}
\begin{cor}\mathcal Label{flattencor}
Given $\mathcal Mathcal A\in \mathcal Mm$, if $t\in\mathcal Mathbb{R}^+$
and
$\Gamma\in \mathcal I_\mathcal Mathcal A$ are such that
\begin{equation}\mathcal Label{keyflatteningequation1}
\mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t^{-1} \mathcal Right)
= \Delta_\mathcal Mathcal A(\Gamma) \,,
\mathcal End{equation}
then the Lipschitz continuous function
$f_{\mathcal Mathcal A(t)}+\varphi_{\Gamma} - \varphi_{\Gamma}\circ T_\mathcal Mathcal A$ is
equal to the constant value
$\int f_{\mathcal Mathcal A(t)}\, ds_\Gamma$ on $\Gamma$.
\mathcal End{cor}
\begin{proof}
By Lemma \mathcal Ref{flattenlemma} it suffices to show that
$$f_{\mathcal Mathcal A(t)}(T_{A_0}(1)) - f_{\mathcal Mathcal A(t)}(T_{A_1}(0))
=
\mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t^{-1} \mathcal Right)\,,$$
and by
Lemma \mathcal Ref{simple}(iii) this is equivalent to showing that
$$f_{\mathcal Mathcal A}(T_{A_0}(1)) - f_{\mathcal Mathcal A}(T_{A_1}(0))
=
\mathcal Log \mathcal Left( \frac{a_0+c_0}{b_1+d_1} \mathcal Right)\,.$$
Substituting $T_{A_0}(1)=\frac{a_0}{a_0+c_0}$ and $T_{A_1}(0)=\frac{b_1}{b_1+d_1}$
into, respectively, the formulae (\mathcal Ref{explicitfalt}) for $f_\mathcal Mathcal A$ on $X_{A_0}$ and $X_{A_1}$ yields
\begin{equation}\mathcal Label{fata01}
f_\mathcal Mathcal A(T_{A_0}(1)) = \mathcal Log(a_0+c_0)
\mathcal End{equation}
and
\begin{equation}\mathcal Label{fata10}
f_\mathcal Mathcal A(T_{A_1}(0)) = \mathcal Log(b_1+d_1)\,,
\mathcal End{equation}
so the result follows.
\mathcal End{proof}
In view of equation
(\mathcal Ref{keyflatteningequation1})
we make the following definition:
\begin{defn}
For $\mathcal Mathcal A\in\mathcal Mm$ and $i\in\{0,1\}$, define $t_i=t_i(\mathcal Mathcal A)$ by
\begin{equation}\mathcal Label{tiadef}
t_i = t_i(\mathcal Mathcal A) = \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) e^{-\Delta_\mathcal Mathcal A(\Gamma_i)}\,,
\mathcal End{equation}
so that
\begin{equation*}
\mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t_i^{-1} \mathcal Right)
= \Delta_\mathcal Mathcal A(\Gamma_i)\,.
\mathcal End{equation*}
\mathcal End{defn}
\begin{remark}
Since $e^{-\Delta_\mathcal Mathcal A(\Gamma_0)} < 1 < e^{-\Delta_\mathcal Mathcal A(\Gamma_1)}$
by (\mathcal Ref{starrr}), it follows that
\begin{equation*}
t_0(\mathcal Mathcal A) <t_1(\mathcal Mathcal A)\,.
\mathcal End{equation*}
\mathcal End{remark}
\begin{lemma}
For $\mathcal Mathcal A\in\mathcal Mm$ and
$i\in\{0,1\}$,
\begin{equation}\mathcal Label{initialtiaexpressions}
t_i(\mathcal Mathcal A)
=
\frac{\mathcal Rho_{A_i} \mathcal Left( a_0 + \mathcal Rho_{A_i}(a_0+c_0) \mathcal Right)}{(1+\mathcal Rho_{A_i})\mathcal Left(b_1+\mathcal Rho_{A_i}(b_1+d_1)\mathcal Right)} \,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
From (\mathcal Ref{deltagammai}) we see that for $i\in\{0,1\}$,
\begin{equation*}
e^{-\Delta_\mathcal Mathcal A(\Gamma_i)} =
\frac{ \mathcal Rho_{A_i} \mathcal Left( \frac{a_0}{a_0+c_0}+\mathcal Rho_{A_i}\mathcal Right)}{(1+\mathcal Rho_{A_i}) \mathcal Left(\frac{b_1}{b_1+d_1}+\mathcal Rho_{A_i}\mathcal Right)}\,,
\mathcal End{equation*}
so that (\mathcal Ref{tiadef}) gives
\begin{equation}\mathcal Label{firsttexpression}
t_i(\mathcal Mathcal A) =
\mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) e^{-\Delta_\mathcal Mathcal A(\Gamma_i)}
=
\frac{\mathcal Rho_{A_i} \mathcal Left( a_0 + \mathcal Rho_{A_i}(a_0+c_0) \mathcal Right)}{(1+\mathcal Rho_{A_i})\mathcal Left(b_1+\mathcal Rho_{A_i}(b_1+d_1)\mathcal Right)} \,,
\mathcal End{equation}
which is the required expression (\mathcal Ref{initialtiaexpressions}).
\mathcal End{proof}
A consequence is the following property:
\begin{cor}
For $\mathcal Mathcal A\in \mathcal Mm$, $t\in\mathcal Mathbb{R}^+$, and $i\in\{0,1\}$,
\begin{equation}\mathcal Label{tti}
t_i( \mathcal Mathcal A(t)) = \frac{t_i(\mathcal Mathcal A)}{t}\,.
\mathcal End{equation}
\mathcal End{cor}
\begin{proof}
This follows easily from
(\mathcal Ref{initialtiaexpressions}),
and the easily verified fact (used only in the proof of the $i=1$ case) that
$\mathcal Rho_{tA_1}=\mathcal Rho_{A_1}$.
Specifically, for $i\in\{0,1\}$,
\begin{equation*}
t_i(\mathcal Mathcal A(t))
=
\frac{\mathcal Rho_{A_i} \mathcal Left( a_0 + \mathcal Rho_{A_i}(a_0+c_0) \mathcal Right)}{(1+\mathcal Rho_{A_i})\mathcal Left(tb_1+\mathcal Rho_{A_i}(tb_1+td_1)\mathcal Right)}
=\frac{1}{t}
\frac{\mathcal Rho_{A_i} \mathcal Left( a_0 + \mathcal Rho_{A_i}(a_0+c_0) \mathcal Right)}{(1+\mathcal Rho_{A_i})\mathcal Left(b_1+\mathcal Rho_{A_i}(b_1+d_1)\mathcal Right)}
= \frac{t_i(\mathcal Mathcal A)}{t}\,.
\mathcal End{equation*}
\mathcal End{proof}
\begin{lemma}
For $\mathcal Mathcal A\in\mathcal Mm$, the quantities $t_0(\mathcal Mathcal A)$ and $t_1(\mathcal Mathcal A)$ admit the following alternative expressions:
\begin{equation}\mathcal Label{tiaexpressions}
t_0(\mathcal Mathcal A)
=
\frac{\det A_0}{ \mathcal Left( a_0 - b_0 ( 1+\mathcal Rho_{A_0}^{-1})\mathcal Right) \mathcal Left( d_1 +b_1(1+\mathcal Rho_{A_0}^{-1})\mathcal Right)}
\mathcal End{equation}
and
\begin{equation}\mathcal Label{tiaexpressionsi1}
t_1(\mathcal Mathcal A)
=
\frac{ \mathcal Left(a_0+c_0(1+\mathcal Rho_{A_1}^{-1})^{-1}\mathcal Right) \mathcal Left(a_1-b_1(1+\mathcal Rho_{A_1}^{-1})\mathcal Right)}{\det A_1}\,.
\mathcal End{equation}
\mathcal End{lemma}
\begin{proof}
Since (\mathcal Ref{firsttexpression}) implies
\begin{equation}\mathcal Label{intermediatetexpression}
t_0(\mathcal Mathcal A) =
\frac{a_0 + \mathcal Rho_{A_0}(a_0+c_0) }{(1+\mathcal Rho_{A_0})\mathcal Left(d_1+b_1(1+\mathcal Rho_{A_0}^{-1})\mathcal Right)} \,,
\mathcal End{equation}
we see that $t_0(\mathcal Mathcal A)$ is equal to (\mathcal Ref{tiaexpressions}) if and only if
\begin{equation}\mathcal Label{ifftexpression}
\frac{a_0 + \mathcal Rho_{A_0}(a_0+c_0) }{1+\mathcal Rho_{A_0}}
=
\frac{a_0d_0-b_0c_0}{a_0-b_0(1+\mathcal Rho_{A_0}^{-1})}\,.
\mathcal End{equation}
Clearing fractions in (\mathcal Ref{ifftexpression}) reveals it to be equivalent to the equation
$$
q_{A_0}(\mathcal Rho_{A_0}) = \alpha_{A_0} \mathcal Rho_{A_0}^2+\beta_{A_0}\mathcal Rho_{A_0}-b_0 =0\,,
$$
which is true by Lemma \mathcal Ref{rhoquadratic}.
Since (\mathcal Ref{firsttexpression}) implies
\begin{equation}\mathcal Label{intermediatetexpression1}
t_1(\mathcal Mathcal A) =
\frac{\mathcal Rho_{A_1}\mathcal Left( a_0+c_0(1+\mathcal Rho_{A_1}^{-1})^{-1}\mathcal Right)}{b_1+\mathcal Rho_{A_1}(b_1+d_1)} \,,
\mathcal End{equation}
we see that $t_1(\mathcal Mathcal A)$ is equal to (\mathcal Ref{tiaexpressionsi1}) if and only if
\begin{equation}\mathcal Label{ifftexpression1}
\frac{\mathcal Rho_{A_1}}{b_1+\mathcal Rho_{A_1}(b_1+d_1)}
=
\frac{a_1-b_1(1+\mathcal Rho_{A_1}^{-1})}{\det A_1}\,.
\mathcal End{equation}
Clearing fractions in (\mathcal Ref{ifftexpression1}) reveals it to be equivalent to the equation
$$
q_{A_1}(\mathcal Rho_{A_1}) = \alpha_{A_1} \mathcal Rho_{A_1}^2+\beta_{A_1}\mathcal Rho_{A_1}-b_1 =0\,,
$$
which is true by Lemma \mathcal Ref{rhoquadratic}.
\mathcal End{proof}
\begin{notation}
For $\mathcal Mathcal A\in\mathcal Mm$, let $\mathcal T_\mathcal Mathcal A$ denote the open interval $\mathcal Left(t_0(\mathcal Mathcal A), t_1(\mathcal Mathcal A)\mathcal Right)$.
\mathcal End{notation}
\begin{prop}\mathcal Label{analogue}
Let $\mathcal Mathcal A\in\mathcal Mm$.
For each $t \in \mathcal T_\mathcal Mathcal A$ there exists
an $\mathcal Mathcal A$-Sturmian interval $\Gamma_\mathcal Mathcal A(t)\in \mathcal I_\mathcal Mathcal A$ such that
$f_{\mathcal Mathcal A(t)}+\varphi_{\Gamma_t} - \varphi_{\Gamma_t}\circ T_\mathcal Mathcal A$ is
equal to the constant value
$\int f_{\mathcal Mathcal A(t)}\, ds_{\Gamma_\mathcal Mathcal A(t)}$ on $\Gamma_\mathcal Mathcal A(t)$.
\mathcal End{prop}
\begin{proof}
First we show that $\Delta_\mathcal Mathcal A: \Gamma \mathcal Mapsto \Delta_\mathcal Mathcal A(\Gamma)$ is continuous.
The formula (\mathcal Ref{Deltadefnalt}) defines
\begin{align*}
\Delta_\mathcal Mathcal A(\Gamma)
& =
\varphi_\Gamma(1)
-
\varphi_\Gamma(T_{A_0}(1)) + \varphi_\Gamma(T_{A_1}(0)) \cr
& =
\varphi_\Gamma(1)
-
\varphi_\Gamma\mathcal Left(\frac{a_0}{a_0+c_0}\mathcal Right) + \varphi_\Gamma\mathcal Left(\frac{b_1}{b_1+d_1}\mathcal Right) \,,
\mathcal End{align*}
so the continuity of $ \Delta_\mathcal Mathcal A$ will follow from the fact
that $\Gamma\mathcal Mapsto \varphi_\Gamma(z)$ is continuous for each $z\in X$.
To see this, first note that Definition \mathcal Ref{phiexists} gives
\begin{equation*}
\varphi_\Gamma(z)
= \varphi_\Gamma(z)-\varphi_\Gamma(0)
=\int_0^z \varphi_\Gamma'
= \int_0^z \sum_{n=1}^\infty (f_\mathcal Mathcal A \circ \mathcal Tau_\Gamma^n)' \,,
\mathcal End{equation*}
and re-writing this integral as
\begin{equation*}
\sum_{n=1}^\infty \int_0^z (f_\mathcal Mathcal A \circ \mathcal Tau_\Gamma^n)'
= \sum_{n=1}^\infty \int_{\mathcal Tau^n_\Gamma[0,z]} f_\mathcal Mathcal A'
= \sum_{n=1}^\infty \int \mathcal Mathbbm{1}_{\mathcal Tau^n_\Gamma[0,z]} f_\mathcal Mathcal A'
=\int f_\mathcal Mathcal A' \sum_{n=1}^\infty \mathcal Mathbbm{1}_{\mathcal Tau^n_\Gamma[0,z]}
\mathcal End{equation*}
gives
\begin{equation}\mathcal Label{phigammaexpression}
\varphi_\Gamma(z) = \int f_\mathcal Mathcal A' H_z(\Gamma) \,,
\mathcal End{equation}
where
\begin{equation*}
H_z(\Gamma) = \sum_{n=1}^\infty \mathcal Mathbbm{1}_{\mathcal Tau^n_\Gamma[0,z]} \,.
\mathcal End{equation*}
Now each map $H_{z,n}:\Gamma\mathcal Mapsto \mathcal Mathbbm{1}_{\mathcal Tau^n_\Gamma[0,z]}$
clearly belongs to $C([\Gamma_0,\Gamma_1],L^1)$, the space of
continuous functions from $[\Gamma_0,\Gamma_1]$ to $L^1=L^1(dx)$,
and $\sum_{n=1}^\infty H_{z,n}$
is convergent in
$C([\Gamma_0,\Gamma_1],L^1)$, so $H_z(\cdot) \in C([\Gamma_0,\Gamma_1],L^1)$.
It then follows from (\mathcal Ref{phigammaexpression}) that
$\Gamma\mathcal Mapsto \varphi_\Gamma(z)$ is continuous, as required.
Now
note that
the function $G_\mathcal Mathcal A$ defined by
\begin{equation}\mathcal Label{gadef}
G_\mathcal Mathcal A(t) = \mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t^{-1} \mathcal Right)
\mathcal End{equation}
is strictly decreasing, since $a_0,c_0,b_1,d_1>0$,
so if $t \in \mathcal T_\mathcal Mathcal A = (t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A))$ then
\begin{equation}\mathcal Label{ginside}
G_\mathcal Mathcal A(t)\in \mathcal Left(G_\mathcal Mathcal A(t_1(\mathcal Mathcal A)),G_\mathcal Mathcal A(t_0(\mathcal Mathcal A))\mathcal Right) = (\Delta_\mathcal Mathcal A(\Gamma_1),\Delta_\mathcal Mathcal A(\Gamma_0))\,.
\mathcal End{equation}
Now $\Delta_\mathcal Mathcal A$ is continuous, so applying
the intermediate value theorem to this function (defined on the interval $[\Gamma_0,\Gamma_1]$) we see that in view of (\mathcal Ref{ginside}), there exists an $\mathcal Mathcal A$-Sturmian interval, which we denote by $\Gamma_\mathcal Mathcal A(t)$, such that
$\Gamma_\mathcal Mathcal A(t)\in (\Gamma_0,\Gamma_1)$
and
\begin{equation}\mathcal Label{deltatdefeq}
\Delta_\mathcal Mathcal A(\Gamma_\mathcal Mathcal A(t))=G_\mathcal Mathcal A(t)\,.
\mathcal End{equation}
In other words,
\begin{equation*}
\mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t^{-1} \mathcal Right)
= \Delta_\mathcal Mathcal A(\Gamma_\mathcal Mathcal A(t)) \,,
\mathcal End{equation*}
so that Corollary \mathcal Ref{flattencor} implies that
$f_{\mathcal Mathcal A(t)}+\varphi_{\Gamma_t} - \varphi_{\Gamma_t}\circ T_\mathcal Mathcal A
= \int f_{\mathcal Mathcal A(t)}\, ds_{\Gamma_\mathcal Mathcal A(t)}$ on $\Gamma_\mathcal Mathcal A(t)$, as required.
\mathcal End{proof}
\section{The case when one matrix dominates}\mathcal Label{dominatessec}
It will be useful to record the value of the induced function
$f_\mathcal Mathcal A$ at the two fixed points of $T_\mathcal Mathcal A$:
\begin{lemma}\mathcal Label{fapai}
For $\mathcal Mathcal A\in\mathcal Mm$ and $i\in\{0,1\}$,
\begin{equation*}
f_\mathcal Mathcal A(p_{A_i})
=
\mathcal Log\mathcal Left( \frac{\det A_i}{a_i - b_i ( 1+\mathcal Rho_{A_i}^{-1})}
\mathcal Right)
=
\mathcal Log\mathcal Left( \frac{\det A_i}{a_i-b_i -\frac{1}{2}(\beta_{A_i}+\gamma_{A_i})}\mathcal Right)
\,.
\mathcal End{equation*}
\mathcal End{lemma}
\begin{proof}
Straightforward computation using (\mathcal Ref{rhoaredone}),
(\mathcal Ref{paformularedone}),
and (\mathcal Ref{explicitfalt}).
\mathcal End{proof}
We first consider a sufficient condition for
the projectively concave matrix
$A_0$ to be the dominant matrix of the pair $\mathcal Mathcal A=(A_0,A_1)$:
\begin{theorem}\mathcal Label{t0larger1theorem}
If $\mathcal Mathcal A\in\mathcal Mm$ is such that
\begin{equation}\mathcal Label{t0larger1}
t_0(\mathcal Mathcal A) \ge 1\,,
\mathcal End{equation}
then the Dirac measure at the fixed point $p_{A_0}$ is the unique $f_\mathcal Mathcal A$-maximizing measure;
in particular, the joint spectral radius of $\mathcal Mathcal A$ is equal to the spectral radius of $A_0$.
\mathcal End{theorem}
\begin{proof}
Choosing
$\varphi(x) = \varphi_{\Gamma_0}(x) = \mathcal Log \mathcal Left(\frac{x+\mathcal Rho_{A_0}}{\mathcal Rho_{A_0}}\mathcal Right)$
ensures,
by Lemma \mathcal Ref{flatxai},
that
$f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is constant when restricted to $X_{A_0} = \Gamma_0$,
and the constant value assumed by this function is clearly $f_\mathcal Mathcal A(p_{A_0})$.
The result will follow if we can show that $f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is strictly decreasing
on $X_{A_1}$,
and that the value $(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(\frac{b_1}{b_1+d_1})$
at the left endpoint of $X_{A_1}$ is no greater than the constant value $f_\mathcal Mathcal A(p_{A_0})$.
This is because
the Dirac measure $\delta_{p_{A_0}}$ will then clearly be the unique maximizing measure for
$f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$, and hence the unique maximizing measure for $f_\mathcal Mathcal A$.
To compute the value $(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(\frac{b_1}{b_1+d_1})$ we recall
from (\mathcal Ref{fata10}) that
$$
f_\mathcal Mathcal A\mathcal Left( \frac{b_1}{b_1+d_1}\mathcal Right)
= f_\mathcal Mathcal A(T_{A_1}(0)) = \mathcal Log(b_1+d_1)\,,
$$
and note that
$$
\varphi\mathcal Left(T_\mathcal Mathcal A\mathcal Left(\frac{b_1}{b_1+d_1}\mathcal Right)\mathcal Right)=\varphi(0) = 0
\,,
$$
and
$$
\varphi\mathcal Left( \frac{b_1}{b_1+d_1}\mathcal Right)=\mathcal Log\mathcal Left( \frac{ \frac{b_1}{b_1+ d_1} +\mathcal Rho_{A_0}}{\mathcal Rho_{A_0}}\mathcal Right)
=\mathcal Log\mathcal Left(\frac{\mathcal Left( d_1 +b_1(1+\mathcal Rho_{A_0}^{-1})\mathcal Right)}{b_1+d_1}\mathcal Right)
\,.
$$
Therefore
\begin{equation}\mathcal Label{endpointformula}
(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)\mathcal Left(\frac{b_1}{b_1+d_1}\mathcal Right)
=
\mathcal Log \mathcal Left( d_1 +b_1(1+\mathcal Rho_{A_0}^{-1})\mathcal Right)
\,.
\mathcal End{equation}
By Lemma \mathcal Ref{fapai},
\begin{equation}\mathcal Label{fpt0}
f_\mathcal Mathcal A(p_{A_0})
=
\mathcal Log\mathcal Left( \frac{\det A_0}{a_0 - b_0 ( 1+\mathcal Rho_{A_0}^{-1})}
\mathcal Right)\,,
\mathcal End{equation}
so (\mathcal Ref{endpointformula}) and (\mathcal Ref{fpt0}) imply that the desired inequality
$$(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)\mathcal Left(\frac{b_1}{b_1+d_1}\mathcal Right)\mathcal Le f_\mathcal Mathcal A(p_{A_0})$$
is precisely the hypothesis (\mathcal Ref{t0larger1}),
since
\begin{equation*}
t_0(\mathcal Mathcal A)
=
\frac{\det A_0}{ \mathcal Left( a_0 - b_0 ( 1+\mathcal Rho_{A_0}^{-1})\mathcal Right) \mathcal Left( d_1 +b_1(1+\mathcal Rho_{A_0}^{-1})\mathcal Right)}
\mathcal End{equation*}
by (\mathcal Ref{tiaexpressions}).
It remains to show that $f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is strictly decreasing on $X_{A_1}$.
Suppose $x\in X_{A_1}$. We know by (\mathcal Ref{explicitfalt}) that
$$
f_\mathcal Mathcal A(x) = \mathcal Log \mathcal Left( \frac{\det A_1}{-\alpha_{A_1}(x+ \sigma_{A_1})} \mathcal Right)\,.
$$
Now
$$
\varphi(x) = \mathcal Log \mathcal Left( \frac{x+\mathcal Rho_{A_0}}{\mathcal Rho_{A_0}}\mathcal Right)\,,
$$
so
$$
\varphi(T_\mathcal Mathcal A(x))
=
\mathcal Log \mathcal Left( \frac{S_{A_1}(x)+\mathcal Rho_{A_0}}{\mathcal Rho_{A_0}}\mathcal Right)\,,
$$
and therefore
$$
(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(x)
=
\mathcal Log\mathcal Left( \frac{\det A_1 (x+\mathcal Rho_{A_0})}{-\alpha_{A_1}(x+\sigma_{A_1})(S_{A_1}(x)+\mathcal Rho_{A_0})}\mathcal Right)\,.
$$
It therefore suffices to show that
\begin{equation}\mathcal Label{suffdec}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_0}}{-\alpha_{A_1}(x+\sigma_{A_1})(S_{A_1}(x)+\mathcal Rho_{A_0})}
\mathcal End{equation}
is strictly decreasing.
For this note that
$$
S_{A_1}(x)+\mathcal Rho_{A_0}
= \frac{(b_1+d_1)x-b_1}{-\alpha_{A_1}(x+\sigma_{A_1})}+\mathcal Rho_{A_0}
=
\frac{ (b_1+d_1-\alpha_{A_1}\mathcal Rho_{A_0})x +(a_1-b_1)\mathcal Rho_{A_0}-b_1}{-\alpha_{A_1}(x+\sigma_{A_1})}
$$
so (\mathcal Ref{suffdec}) is seen to be the M\"obius function
\begin{equation*}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_0}}{ (b_1+d_1-\alpha_{A_1}\mathcal Rho_{A_0})x +(a_1-b_1)\mathcal Rho_{A_0}-b_1} \,,
\mathcal End{equation*}
which is known to be strictly decreasing by Lemma \mathcal Ref{techderivlemma}.
\mathcal End{proof}
As a consequence of Theorem \mathcal Ref{t0larger1theorem} we obtain:
\begin{cor}\mathcal Label{t0largerttheorem}
If $\mathcal Mathcal A\in\mathcal Mm$ and $t\in \mathcal Mathbb{R}^+$ are such that
\begin{equation}\mathcal Label{t0largert}
t \mathcal Le t_0(\mathcal Mathcal A) \,,
\mathcal End{equation}
then the Dirac measure at the fixed point $p_{A_0}$ is the unique $f_{\mathcal Mathcal A(t)}$-maximizing measure;
in particular, the joint spectral radius of $\mathcal Mathcal A(t)$ is equal to the spectral radius of $A_0$.
\mathcal End{cor}
\begin{proof}
The assumption (\mathcal Ref{t0largert}) means, using (\mathcal Ref{tti}),
that $t_0(\mathcal Mathcal A(t))\ge 1$, so the result follows by applying
Theorem \mathcal Ref{t0larger1theorem}
with $\mathcal Mathcal A$ replaced by $\mathcal Mathcal A(t)$.
\mathcal End{proof}
We now turn to an analogous sufficient condition for the projectively convex matrix $A_1$
to be dominant:
\begin{theorem}\mathcal Label{t1smaller1theorem}
If $\mathcal Mathcal A\in\mathcal Mm$ is such that
\begin{equation}\mathcal Label{t1smaller1}
t_1(\mathcal Mathcal A) \mathcal Le 1\,,
\mathcal End{equation}
then the Dirac measure at the fixed point $p_{A_1}$ is the unique $f_\mathcal Mathcal A$-maximizing measure;
in particular, the joint spectral radius of $\mathcal Mathcal A$ is equal to the spectral radius of $A_1$.
\mathcal End{theorem}
\begin{proof}
Choosing
$\varphi(x) = \varphi_{\Gamma_1}(x) = \mathcal Log \mathcal Left(\frac{x+\mathcal Rho_{A_1}}{\mathcal Rho_{A_1}}\mathcal Right)$
ensures,
by Lemma \mathcal Ref{flatxai},
that
$f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is constant when restricted to $X_{A_1} = \Gamma_1$,
and the constant value assumed by this function is clearly $f_\mathcal Mathcal A(p_{A_1})$.
The result will follow if we can show that $f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is strictly increasing
on $X_{A_0}$,
and that the value $(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(\frac{a_0}{a_0+c_0})$
at the right endpoint of $X_{A_0}$ is no greater than the constant value $f_\mathcal Mathcal A(p_{A_1})$.
This is because
the Dirac measure $\delta_{p_{A_1}}$ will then clearly be the unique maximizing measure for
$f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$, and hence the unique maximizing measure for $f_\mathcal Mathcal A$.
To compute the value $(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(\frac{a_0}{a_0+c_0})$ we recall
from (\mathcal Ref{fata01}) that
$$
f_\mathcal Mathcal A\mathcal Left( \frac{a_0}{a_0+c_0}\mathcal Right)
= f_\mathcal Mathcal A(T_{A_0}(1)) = \mathcal Log(a_0+c_0)\,,
$$
and note that
$$
\varphi\mathcal Left(T\mathcal Left(\frac{a_0}{a_0+c_0}\mathcal Right)\mathcal Right)=\varphi(1) = \mathcal Log\mathcal Left( \frac{1+\mathcal Rho_{A_1}}{\mathcal Rho_{A_1}}\mathcal Right)
= \mathcal Log(1+\mathcal Rho_{A_1}^{-1})
\,,
$$
and
$$
\varphi\mathcal Left( \frac{a_0}{a_0+c_0}\mathcal Right)=\mathcal Log\mathcal Left( \frac{ \frac{a_0}{a_0+c_0} +\mathcal Rho_{A_1}}{\mathcal Rho_{A_1}}\mathcal Right)
=\mathcal Log\mathcal Left(\frac{\mathcal Left( c_0 +a_0(1+\mathcal Rho_{A_1}^{-1})\mathcal Right)}{a_0+c_0}\mathcal Right)
\,.
$$
Therefore
\begin{equation}\mathcal Label{endpointformula2}
(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)\mathcal Left(\frac{a_0}{a_0+c_0}\mathcal Right)
=
\mathcal Log \mathcal Left( a_0 +c_0(1+\mathcal Rho_{A_1}^{-1})^{-1} \mathcal Right)
\,.
\mathcal End{equation}
By Lemma \mathcal Ref{fapai},
\begin{equation}\mathcal Label{fpt02}
f_\mathcal Mathcal A(p_{A_1})
=
\mathcal Log\mathcal Left( \frac{\det A_1}{a_1 - b_1 ( 1+\mathcal Rho_{A_1}^{-1})}
\mathcal Right)\,,
\mathcal End{equation}
so (\mathcal Ref{endpointformula2}) and (\mathcal Ref{fpt02}) imply that the desired inequality
$$(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)\mathcal Left(\frac{a_0}{a_0+c_0}\mathcal Right)\mathcal Le f_\mathcal Mathcal A(p_{A_1})$$
is precisely the hypothesis (\mathcal Ref{t1smaller1}),
since
\begin{equation*}
t_1(\mathcal Mathcal A)
=
\frac{ \mathcal Left(a_0+c_0(1+\mathcal Rho_{A_1}^{-1})^{-1}\mathcal Right) \mathcal Left(a_1-b_1(1+\mathcal Rho_{A_1}^{-1})\mathcal Right)}{\det A_1}
\mathcal End{equation*}
by (\mathcal Ref{tiaexpressions}).
It remains to show that $f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A$ is strictly increasing on $X_{A_0}$.
Suppose $x\in X_{A_0}$. We know by (\mathcal Ref{explicitfalt}) that
$$
f_\mathcal Mathcal A(x) = \mathcal Log \mathcal Left( \frac{\det A_0}{-\alpha_{A_0}(x+ \sigma_{A_0})} \mathcal Right)\,.
$$
Now
$$
\varphi(x) = \mathcal Log \mathcal Left( \frac{x+\mathcal Rho_{A_1}}{\mathcal Rho_{A_1}}\mathcal Right)\,,
$$
so
$$
\varphi(T_\mathcal Mathcal A(x))
=
\mathcal Log \mathcal Left( \frac{S_{A_0}(x)+\mathcal Rho_{A_1}}{\mathcal Rho_{A_1}}\mathcal Right)\,,
$$
and therefore
$$
(f_\mathcal Mathcal A+\varphi-\varphi\circ T_\mathcal Mathcal A)(x)
=
\mathcal Log\mathcal Left( \frac{\det A_0 (x+\mathcal Rho_{A_1})}{-\alpha_{A_0}(x+\sigma_{A_0})(S_{A_0}(x)+\mathcal Rho_{A_1})}\mathcal Right)\,.
$$
It therefore suffices to show that
\begin{equation}\mathcal Label{suffinc}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_1}}{-\alpha_{A_0}(x+\sigma_{A_0})(S_{A_0}(x)+\mathcal Rho_{A_1})}
\mathcal End{equation}
is strictly increasing.
For this note that
$$
S_{A_0}(x)+\mathcal Rho_{A_1}
= \frac{(b_0+d_0)x-b_0}{-\alpha_{A_0}(x+\sigma_{A_0})}+\mathcal Rho_{A_1}
=
\frac{ (b_0+d_0-\alpha_{A_0}\mathcal Rho_{A_1})x +(a_0-b_0)\mathcal Rho_{A_1}-b_0}{-\alpha_{A_0}(x+\sigma_{A_0})}
$$
so (\mathcal Ref{suffinc}) is seen to be the M\"obius function
\begin{equation*}
x\mathcal Mapsto \frac{x+\mathcal Rho_{A_1}}{ (b_0+d_0-\alpha_{A_0}\mathcal Rho_{A_1})x +(a_0-b_0)\mathcal Rho_{A_1}-b_0} \,,
\mathcal End{equation*}
which is known to be strictly increasing by Lemma \mathcal Ref{techderivlemma}.
\mathcal End{proof}
As a consequence of Theorem \mathcal Ref{t1smaller1theorem}
we obtain:
\begin{cor}\mathcal Label{t1smallerttheorem}
If $\mathcal Mathcal A\in\mathcal Mm$ and $t\in \mathcal Mathbb{R}^+$ are such that
\begin{equation}\mathcal Label{t1smallert}
t \ge t_1(\mathcal Mathcal A) \,,
\mathcal End{equation}
then the Dirac measure at the fixed point $p_{A_1}$ is the unique $f_{\mathcal Mathcal A(t)}$-maximizing measure;
in particular, the joint spectral radius of $\mathcal Mathcal A(t)$ is equal to the spectral radius of $tA_1$.
\mathcal End{cor}
\begin{proof}
The assumption (\mathcal Ref{t1smallert}) means, using (\mathcal Ref{tti}),
that $t_1(\mathcal Mathcal A(t))\mathcal Le 1$, so the result follows by applying
Theorem \mathcal Ref{t1smaller1theorem}
with $\mathcal Mathcal A$ replaced by $\mathcal Mathcal A(t)$.
\mathcal End{proof}
\section{Sturmian maximizing measures}\mathcal Label{technicalsection}
It is at this point that we make the extra hypothesis
that the matrix pair $\mathcal Mathcal A$ lies in the class $\mathfrak D \subset \mathcal Mm$.
By Lemma \mathcal Ref{posnegf}(ii) we know that if $\mathcal Mathcal A\in\mathcal Mm$ then $f_\mathcal Mathcal A$ is strictly increasing on $X_{A_0}$ and strictly decreasing on $X_{A_1}$; the following result asserts that
if we make the stronger hypothesis that $\mathcal Mathcal A\in\mathfrak D$ then
these monotonicity properties
are inherited by all functions formed by adding a Sturmian transfer function $\varphi_\Gamma$ to $f_\mathcal Mathcal A$.
\begin{prop}\mathcal Label{increasingdecreasingprop}
Let $\mathcal Mathcal A\in\mathfrak D$.
For each $\mathcal Mathcal A$-Sturmian interval $\Gamma\in \mathcal I_\mathcal Mathcal A$,
the function $f_\mathcal Mathcal A+\varphi_\Gamma:X_\mathcal Mathcal A\mathcal To\mathcal Mathbb{R}$ is strictly increasing on $X_{A_0}$,
and strictly decreasing on $X_{A_1}$.
\mathcal End{prop}
\begin{proof}
First suppose $x\in X_{A_0}$.
Let $0=i_0<i_1<i_2<\mathcal Ldots$ be the sequence of all
integers such that $\mathcal Tau_\Gamma^{i_k}(x) \in X_{A_0}$.
For $k\ge0$, writing $z=\mathcal Tau_\Gamma^{i_k}(x)$ we see that
if $1\mathcal Le i<i_{k+1}-i_k$ then
$\mathcal Tau_\Gamma^i(z)\in X_{A_1}$,
and thus
$\mathcal Tau_\Gamma^i(z)=T_{A_1}^i(z)$, so that
\begin{equation}\mathcal Label{tauta1pos}
\sum_{i=0}^{i_{k+1}-i_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z)
=
f_\mathcal Mathcal A'(z) + \sum_{i=1}^{i_{k+1}-i_k-1} (f_\mathcal Mathcal A\circ T_{A_1}^i)'(z)
>
f_\mathcal Mathcal A'(z) + \sum_{i=1}^{\infty} (f_\mathcal Mathcal A\circ T_{A_1}^i)'(z)\,,
\mathcal End{equation}
where the inequality is because
$(f_\mathcal Mathcal A\circ T_{A_1}^i)'(z) <0$ for all $i\ge1$, by Lemma \mathcal Ref{posnegf}.
Now
$z\in X_{A_0}$, so
(\mathcal Ref{fderivative}) in Lemma \mathcal Ref{simple} (iii)
gives
$f_\mathcal Mathcal A' (z)= -(z+\sigma_{A_0})^{-1}$ (which is positive),
and formula
(\mathcal Ref{sumformula}) from Corollary \mathcal Ref{usefulcor} gives
$\sum_{i=1}^\infty (f_\mathcal Mathcal A\circ T_{A_1}^i)'(z) = (z+\varrho_{A_1})^{-1}$ (which is negative), so
(\mathcal Ref{tauta1pos}) implies that
\begin{equation}\mathcal Label{sigrho}
\sum_{i=0}^{i_{k+1}-i_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z)
>
\frac{-1}{z+\sigma_{A_0}} + \frac{1}{z+\varrho_{A_1}} \,.
\mathcal End{equation}
However $\mathcal Mathcal A\in\mathfrak D$, so $\mathcal Rho_{A_1}<\sigma_{A_0}$, and therefore the righthand side of
(\mathcal Ref{sigrho}) is positive, so we have shown that
\begin{equation*}
\sum_{i=0}^{i_{k+1}-i_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z)
>
0\,.
\mathcal End{equation*}
It follows that for all $k\ge0$,
$$
\sum_{n=i_k}^{i_{k+1}-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x)
=
(\mathcal Tau_\Gamma^{i_k})'(x) \sum_{i=0}^{i_{k+1}-i_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z) > 0\,,
$$
and hence
$$
(f_\mathcal Mathcal A+\varphi_\Gamma)'(x)
=\sum_{n=0}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x)
=\sum_{k=0}^\infty \sum_{n=j_k}^{j_{k+1}-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x) >0\,,
$$
so $f_\mathcal Mathcal A+\varphi_\Gamma$ is strictly increasing on $X_{A_0}$.
Now suppose $x\in X_{A_1}$. The proof proceeds analogously to the above.
Let $0=j_0<j_1<j_2<\mathcal Ldots$ be the sequence of all
integers such that $\mathcal Tau_\Gamma^{j_k}(x) \in X_{A_1}$.
For $k\ge0$, writing $z=\mathcal Tau_\Gamma^{j_k}(x)$ we see that
if $1\mathcal Le i<j_{k+1}-j_k$ then
$\mathcal Tau_\Gamma^i(z)\in X_{A_0}$,
and thus
$\mathcal Tau_\Gamma^i(z)=T_{A_0}^i(z)$, so that
\begin{equation}\mathcal Label{sigrho2}
\sum_{i=0}^{j_{k+1}-j_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z)
=
f_\mathcal Mathcal A'(z) + \sum_{i=1}^{j_{k+1}-j_k-1} (f_\mathcal Mathcal A\circ T_{A_0}^i)'(z)
<
f_\mathcal Mathcal A'(z) + \sum_{i=1}^{\infty} (f_\mathcal Mathcal A\circ T_{A_0}^i)'(z) \,,
\mathcal End{equation}
using the fact that
$(f_\mathcal Mathcal A\circ T_{A_0}^i)'(z) >0$ for all $i\ge1$, by Lemma \mathcal Ref{posnegf}.
The righthand side of (\mathcal Ref{sigrho2}) can be written as
$-(z+\sigma_{A_1})^{-1} + (z+\varrho_{A_0})^{-1}$
using Lemma \mathcal Ref{simple} (iii)
and
Corollary \mathcal Ref{usefulcor},
and this is strictly negative since $\sigma_{A_1}<\varrho_{A_0}$
because $\mathcal Mathcal A\in\mathfrak D$,
so we have shown that
\begin{equation*}
\sum_{i=0}^{j_{k+1}-j_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z)
<
\frac{-1}{z+\sigma_{A_1}} + \frac{1}{z+\varrho_{A_0}} < 0\,.
\mathcal End{equation*}
It follows that for all $k\ge0$,
$$
\sum_{n=j_k}^{j_{k+1}-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x)
=
(\mathcal Tau_\Gamma^{j_k})'(x) \sum_{i=0}^{j_{k+1}-j_k-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^i)'(z) < 0\,,
$$
and hence
$$
(f_\mathcal Mathcal A+\varphi_\Gamma)'(x)
=\sum_{n=0}^\infty (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x)
=\sum_{k=0}^\infty \sum_{n=j_k}^{j_{k+1}-1} (f_\mathcal Mathcal A\circ \mathcal Tau_\Gamma^n)'(x) <0\,,
$$
so $f_\mathcal Mathcal A+\varphi_\Gamma$ is strictly decreasing on $X_{A_1}$.
\mathcal End{proof}
\begin{theorem}\mathcal Label{thm6}
Let $\mathcal Mathcal A\in\mathfrak D$ and $t\in \mathcal T_\mathcal Mathcal A= (t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A))$.
The $\mathcal Mathcal A$-Sturmian measure
supported by
the $\mathcal Mathcal A$-Sturmian interval $\Gamma_\mathcal Mathcal A(t)$ is the unique maximizing measure
for $f_{\mathcal Mathcal A(t)}$; thus the corresponding Sturmian measure on $\Omega = \{0,1\}^\mathbb N$ is the unique $\mathcal Mathcal A(t)$-maximizing measure.
\mathcal End{theorem}
\begin{proof}
Let us write $\varphi = \varphi_{\Gamma_\mathcal Mathcal A(t)}$ and $T=T_\mathcal Mathcal A=T_{\mathcal Mathcal A(t)}$.
We know that
$f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T$
is a constant function when restricted to $\Gamma_\mathcal Mathcal A(t)=: [\gamma_t^-,\gamma_t^+] \cap X_\mathcal Mathcal A$,
by Proposition \mathcal Ref{analogue}.
In particular,
$$(f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T)\mathcal Left(\gamma_t^-\mathcal Right)
=
(f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T)\mathcal Left(\gamma_t^+\mathcal Right)\,,
$$
and because
$T(\gamma_t^-)=T(\gamma_t^+)$,
we deduce that
\begin{equation}\mathcal Label{equality}
(f_{\mathcal Mathcal A(t)}+\varphi)\mathcal Left(\gamma_t^-\mathcal Right)
=
(f_{\mathcal Mathcal A(t)}+\varphi)\mathcal Left(\gamma_t^+\mathcal Right)
\,.
\mathcal End{equation}
But Proposition \mathcal Ref{increasingdecreasingprop} implies that
$f_{\mathcal Mathcal A(t)}+\varphi$ is strictly increasing on $X_{A_0}$,
and strictly decreasing on $X_{A_1}$, so together with (\mathcal Ref{equality}) we deduce that
\begin{equation}\mathcal Label{strongsturmian}
(f_{\mathcal Mathcal A(t)}+\varphi)(x) > (f_{\mathcal Mathcal A(t)}+\varphi)(y)\quad\mathcal Text{for all }x\in \Gamma_\mathcal Mathcal A(t),\ y\in X_\mathcal Mathcal A \setminus \Gamma_\mathcal Mathcal A(t)\,.
\mathcal End{equation}
Consequently, if $z,z'$ are such that $T(z)=T(z')$, with $z\in \Gamma_\mathcal Mathcal A(t)$ and $z'\notin \Gamma_\mathcal Mathcal A(t)$, then
$$
(f_{\mathcal Mathcal A(t)}+\varphi)(z) > (f_{\mathcal Mathcal A(t)}+\varphi)(z')\,,
$$
and hence
$$
(f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T)(z) > (f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T)(z')\,.
$$
In other words, the constant value of
$f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T$ on $\Gamma_\mathcal Mathcal A(t)$ is its global maximum,
and this value is not attained at any point in $X_\mathcal Mathcal A \setminus \Gamma_\mathcal Mathcal A(t)$.
It follows that the Sturmian measure supported by $\Gamma_\mathcal Mathcal A(t)$ is the unique maximizing measure
for $f_{\mathcal Mathcal A(t)}+\varphi-\varphi\circ T$,
and hence the unique maximizing measure for $f_{\mathcal Mathcal A(t)}$.
Thus the corresponding Sturmian measure on $\Omega=\{0,1\}^\mathbb N$ is the unique $\mathcal Mathcal A(t)$-maximizing measure.
\mathcal End{proof}
Recall from \S \mathcal Ref{generalsection} that
$\mathcal Ee \subset M_2(\mathbb R)^2$
denotes the set of matrix pairs which are equivalent to some pair in $\mathfrak D$, where
equivalence of
$\mathcal Mathcal A=(A_0,A_1)$ and $\mathcal Mathcal A'=(A_0',A_1')$ means that
$A_0'=uP^{-1}A_0P$ and $A_1'=vP^{-1}A_1P$ for some invertible $P$ and $u,v>0$.
We deduce the following theorem:
\begin{theorem}\mathcal Label{deducedthm}
If $\mathcal Mathcal A\in\mathcal Ee$ and $t\in\mathcal Mathbb{R}^+$, then $\mathcal Mathcal A(t)$ has a unique maximizing measure, and this maximizing measure is Sturmian.
\mathcal End{theorem}
\begin{proof}
It suffices to prove the result for $\mathcal Mathcal A\in\mathfrak D$, and this is
immediate from Corollaries \mathcal Ref{t0largerttheorem} and \mathcal Ref{t1smallerttheorem},
Theorem \mathcal Ref{thm6}, and
the fact that $\mathfrak D \subset \mathcal Mm$.
\mathcal End{proof}
\section{The parameter map is a devil's staircase}\mathcal Label{devilsection}
As noted in Remark \mathcal Ref{conjugacymeasures}, if $\mathcal Mathcal A\in\mathcal Mm$ then there is a topological conjugacy
$h_\mathcal Mathcal A:\Omega\mathcal To Y_\mathcal Mathcal A$ between the the shift map $\sigma:\Omega\mathcal To\Omega$
and the restriction of $T_\mathcal Mathcal A$ to the Cantor set $Y_\mathcal Mathcal A\subset X_\mathcal Mathcal A$; the map $h_\mathcal Mathcal A$ is strictly increasing with respect to the orders on $\Omega$ and $Y_\mathcal Mathcal A$ (cf.~Remark \mathcal Ref{sturmianasturmian}).
If $d:\Omega\mathcal To [0,1]$ is as in Proposition \mathcal Ref{sturmianclassical}\,(c), associating to
$\omega\in\Omega$ the Sturmian parameter of the measure supported by $[0\omega,1\omega]$,
then the map $d_\mathcal Mathcal A:Y_\mathcal Mathcal A\mathcal To[0,1]$ given by
$d_\mathcal Mathcal A=d\circ h_\mathcal Mathcal A^{-1}$
enjoys the same properties as $d$:
\begin{lemma}
The map $d_\mathcal Mathcal A: Y_\mathcal Mathcal A\mathcal To [0,1]$
is continuous, non-decreasing, and surjective.
The preimage $d_\mathcal Mathcal A^{-1}(\mathcal Mathcal{P})$ is a singleton if $\mathcal Mathcal{P}$ is irrational,
and a positive-length closed interval if $\mathcal Mathcal{P}$ is rational.
\mathcal End{lemma}
\begin{proof}
Immediate from Proposition \mathcal Ref{sturmianclassical}\,(c), and the fact that $h_\mathcal Mathcal A$ is strictly increasing.
\mathcal End{proof}
Note that $d_\mathcal Mathcal A$ associates to $y\in Y_\mathcal Mathcal A$
the parameter of the $\mathcal Mathcal A$-Sturmian measure supported by the $\mathcal Mathcal A$-Sturmian
interval
$c_\mathcal Mathcal A^{-1}(y)$,
where
we recall from Definition \mathcal Ref{henceforthca} that the identification map $c_\mathcal Mathcal A:\mathcal I_\mathcal Mathcal A\mathcal To [0,1]$ is defined by
$c_\mathcal Mathcal A(\Gamma)=T_\mathcal Mathcal A(\mathcal Min \Gamma) = T_\mathcal Mathcal A(\mathcal Max \Gamma)$.
Of the extensions of the function $d_\mathcal Mathcal A$ from the Cantor set $Y_\mathcal Mathcal A$ to the interval $X=[0,1]$,
there is a unique one giving a non-decreasing self-map $d_\mathcal Mathcal A:X\mathcal To [0,1]$.
This extension, which we shall also denote by $d_\mathcal Mathcal A$, is continuous, and $d_\mathcal Mathcal A(c)$ is just the parameter of the $\mathcal Mathcal A$-Sturmian measure
$s_{c_\mathcal Mathcal A^{-1}(c)}$ (i.e.~of the $\mathcal Mathcal A$-Sturmian measure supported by
the $\mathcal Mathcal A$-Sturmian interval $c_\mathcal Mathcal A^{-1}(c)$) for each $c\in X$.
We therefore have the following:
\begin{cor}\mathcal Label{dadev}
The map $d_\mathcal Mathcal A: X\mathcal To [0,1]$
is continuous, non-decreasing, and surjective.
The preimage $d_\mathcal Mathcal A^{-1}(\mathcal Mathcal{P})$ is a singleton if $\mathcal Mathcal{P}$ is irrational,
and a positive-length closed interval if $\mathcal Mathcal{P}$ is rational.
\mathcal End{cor}
\begin{defn}
For $\mathcal Mathcal A\in\mathfrak D$, let $\mathcal Mathcal{P}_\mathcal Mathcal A(t)$ denote the parameter of the Sturmian maximizing measure for $\mathcal Mathcal A(t)$,
or equivalently of the $\mathcal Mathcal A$-Sturmian $f_{\mathcal Mathcal A(t)}$-maximizing measure.
This defines the \mathcal Emph{parameter map} $\mathcal Mathcal{P}_\mathcal Mathcal A:\mathcal Mathbb{R}^+\mathcal To[0,1]$.
\mathcal End{defn}
Recalling
(see Proposition \mathcal Ref{analogue})
the map $t\mathcal Mapsto \Gamma_\mathcal Mathcal A(t)$ associating $\mathcal Mathcal A$-Sturmian interval to parameter
$t\in\mathcal T_\mathcal Mathcal A = (t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A))$,
we see that in fact the map $\mathcal Mathcal{P}_\mathcal Mathcal A:\mathcal T_\mathcal Mathcal A\mathcal To X$ can be written as
\begin{equation}\mathcal Label{rfactor}
\mathcal Mathcal{P}_\mathcal Mathcal A = d_\mathcal Mathcal A\circ c_\mathcal Mathcal A\circ \Gamma_\mathcal Mathcal A\,.
\mathcal End{equation}
This means that $\mathcal Mathcal{P}_\mathcal Mathcal A$ will enjoy the same properties as established for $d_\mathcal Mathcal A$ in Corollary
\mathcal Ref{dadev}, provided $c_\mathcal Mathcal A\circ \Gamma_\mathcal Mathcal A$ is strictly increasing:
\begin{lemma}\mathcal Label{strictinccomp}
For $\mathcal Mathcal A\in\mathcal Mm$, the map $c_\mathcal Mathcal A\circ \Gamma_\mathcal Mathcal A: \mathcal T_\mathcal Mathcal A\mathcal To X$ is strictly increasing and surjective.
\mathcal End{lemma}
\begin{proof}
Recall from
(\mathcal Ref{gadef})
the function $G_\mathcal Mathcal A$ given by
$$G_\mathcal Mathcal A(t) = \mathcal Log \mathcal Left( \mathcal Left(\frac{a_0+c_0}{b_1+d_1}\mathcal Right) t^{-1} \mathcal Right)\,,$$
and that $\Gamma_\mathcal Mathcal A(t)\in \mathcal I_A$ is defined
(see (\mathcal Ref{deltatdefeq})) by the identity
\begin{equation*}
\Delta_\mathcal Mathcal A \circ \Gamma_\mathcal Mathcal A
= G_\mathcal Mathcal A
\,.
\mathcal End{equation*}
Now $G_\mathcal Mathcal A$ is strictly decreasing, so in particular injective,
therefore the map $\Gamma_\mathcal Mathcal A$ is necessarily injective.
Note that $\Gamma_\mathcal Mathcal A$ clearly extends to a continuous injection on
$\overline{\mathcal T_\mathcal Mathcal A}= [t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A)]$, with $\Gamma_\mathcal Mathcal A(t_i(\mathcal Mathcal A))=\Gamma_i$ for $i\in\{0,1\}$.
Now $c_\mathcal Mathcal A:\mathcal I_\mathcal Mathcal A\mathcal To X$ is a bijection,
so $c_\mathcal Mathcal A\circ \Gamma_\mathcal Mathcal A:\overline{\mathcal T_\mathcal Mathcal A}\mathcal To X$ is injective, and its continuity means it is strictly monotone.
But
$c_\mathcal Mathcal A(\Gamma_\mathcal Mathcal A(t_0(\mathcal Mathcal A)))=0$
and $c_\mathcal Mathcal A(\Gamma_\mathcal Mathcal A(t_1(\mathcal Mathcal A)))=1$, so the map
$c_\mathcal Mathcal A\circ\Gamma_\mathcal Mathcal A$ must be strictly increasing and surjective, as required.
\mathcal End{proof}
We can now prove that the parameter map $\mathcal Mathcal{P}_\mathcal Mathcal A:\mathcal Mathbb{R}^+\mathcal To[0,1]$ is
singular. More specifically, its properties described by the following Theorem \mathcal Ref{deviltheorem}
mean it is a \mathcal Emph{devil's staircase}. These properties of the parameter map had been noted
by Bousch \& Mairesse \cite{bouschmairesse} in the context of the family (\mathcal Ref{bmfamily}),
and proved in detail by Morris \& Sidorov \cite{morrissidorov} for the family (\mathcal Ref{standardpair}).
The following result can be viewed as a
more detailed version of Theorem
\mathcal Ref{maintheorem} from \S \mathcal Ref{generalsection}:
\begin{theorem}\mathcal Label{deviltheorem}
If $\mathcal Mathcal A\in\mathcal Ee$ and $t\in\mathcal Mathbb{R}^+$, then $\mathcal Mathcal A(t)$ has a unique maximizing measure, and this maximizing measure is Sturmian.
Let $\mathcal Mathcal{P}_\mathcal Mathcal A(t)$ denote the parameter of the Sturmian maximizing measure for $\mathcal Mathcal A(t)$.
The parameter map $\mathcal Mathcal{P}_\mathcal Mathcal A:\mathcal Mathbb{R}^+\mathcal To [0,1]$
is continuous, non-decreasing, and surjective.
The preimage $\mathcal Mathcal{P}_\mathcal Mathcal A^{-1}(\mathcal Mathcal{P})$ is a singleton if $\mathcal Mathcal{P}$ is irrational,
and a positive-length closed interval if $\mathcal Mathcal{P}$ is rational.
\mathcal End{theorem}
\begin{proof}
The set $\mathcal Ee$ consists of matrix pairs which are equivalent to a matrix pair in $\mathfrak D$, so it suffices to prove the result
for $\mathcal Mathcal A\in\mathfrak D$.
Theorem \mathcal Ref{deducedthm} gives that
$\mathcal Mathcal A(t)$ has a unique maximizing measure, and that this maximizing measure is Sturmian.
For $t\in \mathcal Mathbb{R}^+\setminus \mathcal T_\mathcal Mathcal A$ we know that
\begin{equation}\mathcal Label{rot0}
\mathcal Mathcal{P}_\mathcal Mathcal A(t)=0 \quad\mathcal Text{for }t\in (0,t_0(\mathcal Mathcal A))
\mathcal End{equation}
by Theorem \mathcal Ref{t0largerttheorem},
and
\begin{equation}\mathcal Label{rot1}
\mathcal Mathcal{P}_\mathcal Mathcal A(t)=1 \quad\mathcal Text{for }t\in (t_1(\mathcal Mathcal A),\infty)
\mathcal End{equation}
by Theorem \mathcal Ref{t1smallerttheorem}, since the Dirac measures at the fixed points
$p_{A_0}$ and $p_{A_1}$ are $\mathcal Mathcal A$-Sturmian measures of parameters 0 and 1 respectively.
In view of (\mathcal Ref{rot0}) and (\mathcal Ref{rot1}), it suffices to
establish the required properties of $\mathcal Mathcal{P}_\mathcal Mathcal A$ on the sub-interval $\mathcal T_\mathcal Mathcal A=(t_0(\mathcal Mathcal A),t_1(\mathcal Mathcal A))$.
Using
the factorisation
(\mathcal Ref{rfactor}),
we see that this follows
from Corollary \mathcal Ref{dadev}
and Lemma \mathcal Ref{strictinccomp}.
\mathcal End{proof}
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\mathcal End{document} |
\begin{document}
\begin{center}
{A heuristic for the non-unicost set covering problem using local branching} ~\\
~\\
{J.E. Beasley} ~\\
~\\
~\\
Mathematics, Brunel University, Uxbridge UB8 3PH, UK
~\\
~\\
john.beasley@brunel.ac.uk
~\\
{http://people.brunel.ac.uk/$\sim$mastjjb/jeb/jeb.html}
~\\
~\\
April 2023\\
\end{center}
\begin{abstract}
In this paper we present a heuristic for the non-unicost set covering problem using local branching.
Local branching eliminates the need to define a problem specific search neighbourhood for any particular (zero-one) optimisation problem. It does this by incorporating a generalised Hamming distance neighbourhood into the problem, and this leads naturally to an appropriate neighbourhood search procedure.
We apply our approach to the non-unicost set covering problem. Computational results are presented for 65 test problems that have been widely considered in the literature. Our results indicate that our approach is very competitive in terms of solution quality with other approaches from the literature.
We believe that the work described here illustrates that the potential for using local branching, operating as a stand-alone matheuristic,
has not been fully exploited in the literature.
\end{abstract}
\sloppy Keywords: Hamming distance; heuristics; local branching; integer programming; neighbourhood search; set covering
\section{Introduction}
As the reader may be aware a common approach to the heuristic solution of many zero-one integer programming problems is to apply neighbourhood search. By this we mean that given a (typically feasible) solution to the problem at hand we examine \enquote{small} changes to this solution. So we examine solutions in the \enquote{neighbourhood} of this feasible solution. If we find a better feasible solution then this typically becomes the new solution and the process repeats until some termination condition is satisfied
(e.g.~computational time limit, or failure to improve on the solution).
There are a number of general neighbourhood search approaches in the literature such as simulated annealing~\cite{kirkpatrick83},
tabu search~\cite{glover90} and
variable neighbourhood search~\cite{mladenovic97,hansen01} that can be applied. Such approaches set out a general search procedure, but need particularisation for the problem at hand, e.g.~in defining the neighbourhood of a solution. Typically the neighbourhood of a solution is defined by specifying the possible moves away from a solution. These neighbourhood search approaches have been extensively used in the literature. For example a recent search using Web of Science (http://www.webofscience.com) listed approximately 28,000 papers referring to simulated annealing, 11,000 papers referring to tabu search and 7,000 papers referring to variable neighbourhood search
In this paper we present an optimisation based approach to neighbourhood search based upon local branching.
Local branching eliminates the need to define a problem specific search neighbourhood for any particular (zero-one) optimisation problem.
It does this by incorporating a generalised Hamming distance neighbourhood into the problem, and this leads naturally to an appropriate
neighbourhood search procedure.
The structure of this paper is as follows. In Section~\ref{opt} we define what we mean by the neighbourhood of a feasible solution to a general (zero-one) optimisation problem. We then go on to outline a search procedure that we can adopt to successively search for improved solutions.
In Section~\ref{app} we consider the example optimisation problem, the non-unicost set covering problem, to which we are going to apply
our
approach. We define the problem and consider relevant literature on the problem with especial reference to papers in the literature which report good computational results.
In Section~\ref{results} we present computational results based on applying our
approach to 65 non-unicost set covering problems that have been extensively considered by others in the literature.
Finally in Section~\ref{conc} we present our conclusions.
\section{Optimisation based neighbourhood search}
\label{opt}
In this section we first define
what we mean by the neighbourhood of a feasible solution to a general (zero-one) optimisation problem.
We then go on to outline a search procedure that we can adopt to successively search for improved solutions.
\subsection{Neighbourhood}
To illustrate our approach suppose that we have a general zero-one integer programming problem
involving $n$ zero-one variables $[x_i,~i=1,\ldots,n]$ and $m$ constraints where the optimisation problem is:
\begin{equation}
\mbox{minimise}~~~~\sum_{i=1}^n c_i x_i
\label{eq1}
\end{equation}
subject to:
\begin{equation}
\sum_{j=1}^n a_{ij} x_j \geq b_i ~~~~ i = 1, \ldots, m
\label{eq2}
\end{equation}
\begin{equation}
x_i \in \{0,1\} ~~~~i=1,\ldots,n
\label{eq3}
\end{equation}
Equation~(\ref{eq1}) is a minimisation objective. Without significant loss of generality we shall henceforth assume that all the objective function coefficients $[c_i]$ are integer. Equation~(\ref{eq2}) represents the constraints of the problem and Equation~(\ref{eq3}) the integrality condition.
Let $[X_i]$ be some feasible solution to the problem. Then in this paper we define the neighbourhood of $[X_i]$ to be any set of zero-one variable values $[x_i]$ satisfying $1 \leq \sum_{i=1}^n |x_i - X_i| \leq K$, where $K$ is a known positive constant of our choice. In other words the neighbourhood of $[X_i]$ is any set of zero-one values $[x_i]$ such that the Hamming distance between $[x_i]$ and $[X_i]$ lies between one and $K$.
Then consider the optimisation problem:
\begin{equation}
\mbox{minimise}~~~~\sum_{i=1}^n c_i x_i
\label{eq1a}
\end{equation}
subject to Equations~(\ref{eq2}),(\ref{eq3}) and:
\begin{equation}
1 \leq \sum_{i=1~X_i=0}^n x_i + \sum_{i=1~X_i=1}^n (1- x_i ) \leq K
\label{eqs1}
\end{equation}
\begin{equation}
\sum_{i=1}^n c_i x_i \leq \sum_{i=1}^n c_i X_i - 1
\label{eqs2}
\end{equation}
Here we have added two constraints to our original optimisation problem. In Equation~(\ref{eqs1}) the expression seen is a linearisation of the nonlinear Hamming distance $\sum_{i=1}^n |x_i - X_i|$.
Equation~(\ref{eqs1}) ensures that the Hamming distance between $[x_i]$ and $[X_i]$ is at least one (so we have a solution different from $[X_i]$) and is also less than or equal to $K$.
Equation~(\ref{eqs2}) implies that we are only interested in improved feasible solutions in the neighbourhood of $[X_i]$, i.e.~those that strictly improve on the solution value $\sum_{i=1}^n c_i X_i $ associated with the current solution. Improving on the current feasible solution cannot be guaranteed by Equation~(\ref{eqs1}) since it only constrains the structural (Hamming distance) difference between two solutions, it does not address their objective function values.
With regard to a minor technical issue here we have that in integer programming terms use of Equation~(\ref{eqs2}) automatically implies that the Hamming distance between $[x_i]$ and $[X_i]$ is at least one.
This is because any improved feasible solution must be different from $[X_i]$. However it could be that including an explicit lower limit on the Hamming distance of one (as in Equation~(\ref{eqs1})) improves computational performance, e.g.~by improving the linear programming relaxation solution, so we include it here.
\sloppy Our amended optimisation problem is now
optimise Equation~(\ref{eq1a}) subject to Equations~(\ref{eq2}),(\ref{eq3}),(\ref{eqs1}),(\ref{eqs2}). In essence
here we amend the original optimisation problem to restrict attention to distinctly different (and improved) solutions within the $K$ neighbourhood of $[X_i]$.
Note here that because of the extra constraints added to the original optimisation problem
any feasible solution
must be an improved solution as compared to $[X_i]$.
Use of a constraint based upon Hamming distance has previously been given in the literature by Fischetti and Lodi~\cite{fischetti03}. In their approach, which they call \enquote{local branching}, once a feasible solution $[X_i]$ is found within an enumerative scheme, for example linear programming based tree search, two tree branches are created. One of these branches, which they call the left branch, has
$ \sum_{i=1}^n |x_i - X_i| \leq K$.
The other branch, which they call the right branch, has
$ \sum_{i=1}^n |x_i - X_i| \geq K +1$.
They suggested tactical exploration of the left branch, using standard branching procedures, in the hope of finding an improved feasible solution within the Hamming distance $K$ neighbourhood of the current feasible solution before proceeding with exploration of the right branch.
They proposed varying the value of $K$ depending upon search progress: for example reducing $K$ if the left branch has not resulted in an improved solution within a specified time limit; increasing $K$ to diversify the search.
Our approach differs from local branching as described in~\cite{fischetti03} in one significant respect, namely that
we only focus on the left branch, no exploration is attempted with regard to the right branch. This is because we are focusing on generating good quality heuristic solutions, abandoning any attempt to achieve a provably optimal solution for the original problem (Equations~(\ref{eq1})-(\ref{eq3})) under investigation.
In general terms it is clear that if we solve our amended optimisation problem to proven global optimality (e.g.~using a package such as Cplex~\cite{cplex1210}) then we will either:
\begin{compactitem}
\item find an improved feasible solution within the $K$ neighbourhood of $[X_i]$, technically the minimum feasible solution within the neighbourhood; or
\item prove that there is no improved feasible solution within the neighbourhood.
\end{compactitem}
Obviously computational considerations may mean that we do not solve the amended optimisation problem to proven global optimality, but within computational limits we may still find an improved feasible solution.
As noted above any feasible solution to the amended optimisation problem must by definition be an improved solution as compared to $[X_i]$.
Obviously, as with standard neighbourhood search procedures, an improved feasible solution can be used to replace $[X_i]$ and the process repeated in a natural way. The search procedure we adopted based upon the amended optimisation problem is detailed below.
\subsection{Search procedure}
Our search procedure requires an initial feasible solution $[X_i]$ as well as three parameter values. These are:
an initial value for $K$; a value $\delta$ for incrementing $K$ so as to increase the size of the neighbourhood and a value $L$ for the number of successive iterations we allow without improving the solution before terminating the search.
\newline
\newline
\noindent Our search procedure is:
\begin{enumerate}[label=(\alph*), noitemsep]
\item Initialise $[X_i]$, $K$, $\delta$ and $L$. Set $t \leftarrow 0$, where $t$ is the iteration counter.
\item Set $t \leftarrow t+1$ Solve the amended optimisation problem. If we find an improved feasible solution replace $[X_i]$ with this solution.
\item If $L$ successive iterations have been performed without improving the current feasible solution then stop, else set $K \leftarrow K + \delta$ and go to step (b).
\end{enumerate}
\noindent In this procedure we increase the size of the neighbourhood (increment $K$ by $\delta$) at each iteration, irrespective as to whether an improved solution has been found or not. This ensures that we continually expand the search space around the (current) feasible solution. The procedure only terminates once $L$ successive iterations have been performed without finding an improved feasible solution.
\subsection{Comment}
We would make a number of comments as to our
approach:
\begin{itemize}
\item Our approach draws directly on the mathematical formulation of the problem and hence can be classed as a matheuristic~\cite{boschetti2009, boschetti23, maniezzo21}.
\begin{comment}
\item It is general in nature, potentially making little use of any problem specific knowledge.
\end{comment}
\item It eliminates the need to design problem specific search neighbourhoods, since the neighbourhood is automatically incorporated into
the amended optimisation problem using the Hamming distance (Equation~(\ref{eqs1})) as discussed above.
\item If the amended optimisation problem associated with the final feasible solution found has been solved to proven global optimality then we have an \emph{\textbf{absolute guarantee}} that there is no improved feasible solution within the Hamming distance $K$ neighbourhood associated with that final feasible solution.
\item Clearly local branching is not a new concept. Indeed various authors in the literature have mentioned its use as a heuristic, e.g.~most recently~\cite{boschetti23}. \textbf{\emph{However, we believe that work described here illustrates that the potential for using local branching, operating as a stand-alone matheuristic as in the approach described in this paper,
has not been fully exploited in the literature.}}
\end{itemize}
\section{The set covering problem}
\label{app}
\begin{comment}It is clear that to investigate the worth of the
approach given above we need to apply it to an example optimisation problem.
For this purpose we choose to use a classical zero-one optimisation problem, the set covering problem.
\end{comment}
The set covering problem is the problem of choosing a minimum cost set of columns $[x_i]$ that collectively cover each of the $m$ rows in the problem. Referring back to Equation~(\ref{eq2}) above we have that $[a_{ij}]$ is a known matrix with $a_{ij}=1$ if column $j$ covers row $i$, $a_{ij}=0$ otherwise. The values $[b_i]$ are all one.
There are two variants of the problem, one where the column costs $[c_i]$ are all one (known as the unicost set covering problem), one where the column costs $[c_i]$ are general
non-negative values (referred to as the non-unicost set covering problem, or more commonly as just the set covering problem). In the results given below we apply our approach to the
non-unicost problem. Of the two variants of the problem the non-unicost variant has attracted greater attention in the literature.
The (non-unicost) set covering problem has been considered by a number of authors in the literature as discussed below.
It is not our intention here to give a comprehensive and detailed review of the literature for the set covering problem. Indeed that would be a mammoth task, since a recent search using Web of Science (http://www.webofscience.com) listed nearly 500 papers that included the phrase \enquote{set covering} in their title.
Rather our intention is to highlight significant papers in the literature which report good computational results on set covering instances. This is because the focus of this paper is whether, for the specific example problem (set covering) considered, our
approach can yield good quality results as compared with those already reported in the literature.
\subsection{Relevant literature}
\sloppy Caprara and Toth~\cite{caprara00} give a survey of algorithms for the set covering problem prior to 2000. As is clear from their paper most of the authors in the literature since 1990 have made use of the test problems publicly available from
OR-Library~\cite{beasley1990}, see
http://people.brunel.ac.uk/$\sim$mastjjb/jeb/info.html.
In this paper we also make use of these test problems.
Lan et al~\cite{lan07} presented a heuristic approach which they called Meta-RPS (Meta-heuristic for Randomized Priority Search). They stressed the use of randomness to avoid local optima. Their approach is a repeated application of: firstly a constructive heuristic to find a feasible solution (but including randomisation); secondly a local improvement heuristic based on neighbourhood search. Their approach also included preprocessing, both to exclude columns from consideration and to include them if a column is the only one that covers a row. To reduce the computation time associated with their neighbourhood search procedure they defined a core problem consisting of a small subset of columns.
Lan et al~\cite{lan07} reported that their heuristic is one of only two to find all optimal/best-known solutions for non-unicost instances.
They gave a table illustrating the effectiveness of different heuristics on 65 non-unicost set covering problems. Of note there is
the indirect genetic algorithm of Aickelin~\cite{aickelin02};
the genetic algorithm of Beasley and Chu~\cite{beasley96}; and
the lagrangian heuristic of Caprara et al~\cite{caprara99}.
We consider each of these three approaches below.
\begin{comment}
In terms of unicost problems they compared their approach against that of six different heuristics for 15 unicost instances with the result given showing that their approach outperforms (on average) these other heuristics.
\end{comment}
\begin{comment}
In brief, of the competitive approaches for non-unicost problem cited by Lan et al~\cite{lan07}, Caprara et al~\cite{caprara99} is a lagrangian heuristic approach using dynamic pricing for the variables and systematic use of column fixing to improve the solution. Beasley and Chu~\cite{beasley96} is a genetic algorithm approach including a new fitness-based crossover operator (fusion), a variable mutation
rate and a heuristic feasibility operator tailored specifically for the set covering problem. Aickelin~\cite{aickelin02} is a genetic algorithm approach with a decoder which works on a permuted list of the rows to be covered, with hill-climbing to improve the solution applied after the decoder has provided a suitable solution.
\end{comment}
Aickelin~\cite{aickelin02} presented a genetic algorithm approach with a decoder which works on a permuted list of the rows to be covered, with hill-climbing to improve the solution applied after the decoder has provided a suitable solution. For 65 non-unicost set covering problems they compare their approach with other approaches,~\cite{beasley96,caprara99}, taken from the literature.
Beasley and Chu~\cite{beasley96} presented a genetic algorithm approach including a new fitness-based crossover operator (fusion), a variable mutation
rate and a heuristic feasibility operator tailored specifically for the non-unicost set covering problem. They reported computational experience for
their approach on 65 non-unicost set covering problems.
Caprara et al~\cite{caprara99} presented a lagrangian heuristic approach using dynamic pricing for the variables and systematic use of column fixing to improve the solution. They made use of a number of improvements in the subgradient optimisation procedure as well as a refining procedure to improve upon any given solution. They gave a table illustrating the effectiveness of different heuristics on 65 non-unicost set covering problems showing that their approach performs well.
Naji-Azimi et al~\cite{naji10} presented an electromagnetic metaheuristic approach drawing on the work of Birbil and Fang~\cite{birbil03}. Their approach involves an initial preprocessing step and then repetitively adjusting a pool of solutions to which local search is first applied and where the solutions are then changed based on the force generated by the \enquote{charge} associated with each solution. Mutation was also applied to perturb solutions. They considered 65 non-unicost problems and compared their results with those of Lan et al~\cite{lan07}. Their results indicated that their approach was competitive with that of Lan et al~\cite{lan07}.
\begin{comment}
For 15 unicost problems they compared their results with those of Lan et al~\cite{lan07} and the GRASP algorithm of Bautista and Pereira\cite{bautista07} which indicated that (on average) they outperformed Bautista and Pereira\cite{bautista07}, but were in turn out-performed by Lan et al~\cite{lan07}. We would note here that GRASP, Greedy Randomised Adaptive Search Procedures, was originally proposed by Feo and Resende~\cite{feo95} and is an approach that first constructs an initial
solution via an adaptive randomised greedy function and then applies a local search procedure
to the constructed solution.
Naji-Azimi et al~\cite{naji10} also modified the genetic algorithm
of Beasley and Chu~\cite{beasley96} to more effectively deal with unicost problems and compared their approach with that modified genetic algorithm on 55 test problems. Their results indicated that (on average) their electromagnetic approach out-performed the modified genetic algorithm. Although their algorithm was originally designed for unicost problems they modified their approach for non-unicost problems and presented results for 65 non-unicost problems and compared their results with those of Lan et al~\cite{lan07}. Their results indicated that their approach was competitive with that of Lan et al~\cite{lan07}.
\end{comment}
\begin{comment}
Gao et al~\cite{gao15} presented a local search approach for the unicost set covering problem based upon row weighting. They used tabu search strategies to prevent cycling between solutions. They presented computational results for 70 unicost problems and compared their results with those of Bautista and Pereira\cite{bautista07}, Musliu~\cite{musliu06} and
Naji-Azimi et al~\cite{naji10}.
They reported that their approach improved the best-known solutions in 14 of the instances. Muslio~\cite{musliu06} had good overall performance on their test problems and that approach is a local search approach based on a new method for neighbourhood generation from the current solution at
each iteration as well as making use of search history in conjunction with a tabu search mechanism to avoid cycles.
\end{comment}
Reyes and Araya~\cite{reyes21} presented a greedy randomised adaptive search procedure (GRASP~\cite{feo95, festa02}) based strategy for the non-unicost set covering problem. They proposed iterated local search and reward/penalty procedures in order to accelerate convergence and improve upon the GRASP solutions. Their approach also included preprocessing both to exclude columns from consideration and to include them. They presented results, based upon 30 trials, for 65 non-unicost set covering problems.
In recent years a number of papers in the literature have applied algorithms based upon paradigms drawn from the natural world (sometimes referred to as bio-inspired metaheuristics). One example of work of this kind is Soto et al~\cite{soto17} who presented approaches based on cuckoo search and black hole optimisation.
Cuckoo search (see Yang and Deb~\cite{yang09}) is a population based approach where each \enquote{nest} in the population contains a number of \enquote{eggs} (solutions) and \enquote{cuckoos} lay eggs in randomly chosen nests. The best nests carry over to the next generation.
Black hole optimisation (see Kumar et al~\cite{kumar15}) is also a population based approach where a \enquote{black hole} attracts \enquote{stars} (solutions). Stars change locations as they are attracted by the black hole. Some stars are absorbed by the black hole and replaced by newly generated stars (solutions).
Soto et al~\cite{soto17} applied these two approaches to the non-unicost set covering problem. They applied preprocessing and presented results for 65 non-unicost set covering problems (based on 30 trials for each instance).
\section{Computational results}
\label{results}
In this section we first discuss the non-unicost set covering test problems which we used. We then give computational results for our
approach when applied to these test problems. We also give a comparison between the results from our approach and eight other approaches
presented previously in the literature.
\subsection{Test problems}
We used the standard set of 65 non-unicost set covering test problems that are available from
OR-Library~\cite{beasley1990}, see http://people.brunel.ac.uk/$\sim$mastjjb/jeb/info.html.
We used a Windows pc with 8GB of memory and an Intel Core i5-1137G7 2.4Ghz processor, a multi-core pc with four cores. The initial value of $K$ was set to 5, the neighbourhood increment $\delta$ was set to 5 and the number successive iterations $L$ allowed without improving the current feasible solution before termination was set to 5. These values were set based on limited computational experimentation.
The initial feasible solution required to start the search was produced using a simple greedy heuristic for the set covering problem: repetitively choosing a column with the minimum value of the ratio (column cost/number of uncovered rows covered by the column). Once a solution covering all rows had been found we removed any redundant columns (those columns for which all rows which they cover are also covered by other columns).
Note here that, unlike a number of other papers in the literature
(e.g.~\cite{lan07,naji10,reyes21,soto17}), we made no use of problem-specific preprocessing
to eliminate columns.
Table~\ref{table1} shows the characteristics of the 65 non-unicost problems considered. In that table we show the problem set name, the number of instances in that set, the number of rows ($m$) and columns ($n$) and the problem density ($[\sum_{i=1}^m \sum_{j=1}^n a_{ij}/mn]$ expressed as a percentage). We set a time limit of 15 seconds for the solution of each optimisation
for all the problems with $m \leq 500$, where Cplex~\cite{cplex1210} with default parameter settings was used as the solver.
For the larger problems with $m=1000$ we increased this time limit to 45 seconds.
\begin{table}[hbt!]
\centering
\renewcommand{1mm}{1mm}
\renewcommand{1.0}{1.0}
\begin{tabular}{|c|c|c|c|c|}
\hline
Problem set name & Number of instances & Number of rows ($m$) & Number of columns ($n$)
& Density (\%) \\
\hline
4 & 10 & 200 & 1000 & 2 \\
5 & 10 & 200 & 2000 & 2 \\
6 & 5 & 200 & 1000 & 5 \\
A & 5 & 300 & 3000 & 2 \\
B & 5 & 300 & 3000 & 5 \\
C & 5 & 400 & 4000 & 2 \\
D & 5 & 400 & 4000 & 5 \\
NRE & 5 & 500 & 5000 & 10 \\
NRF & 5 & 500 & 5000 & 20 \\
NRG & 5 & 1000 & 10000 & 2 \\
NRH & 5 & 1000 & 10000 & 5 \\
\hline
\end{tabular}
\caption{Test problem characteristics}
\label{table1}
\end{table}
\subsection{Results}
Table~\ref{table2} shows the results obtained by our optimisation approach.
In that table we show the optimal/best-known solution value (OBK) for each problem, as taken from Lan et al~\cite{lan07}. We also show the solution value as obtained by our approach,
the final value of $K$ at termination,
the total computation time (in seconds), and whether the amended optimisation problem for the final value of $K$ was solved to proven optimality or not. We have not given in Table~\ref{table2} the number of iterations made in our procedure as this can easily deduced by dividing the final value of $K$ by the neighbourhood increment $\delta$, where we used $\delta=5$.
So for example consider problem 4.10 in Table~\ref{table2}. The optimal/best-known solution for this instance is 514 and the \enquote{o} signifies that this was the solution found by our approach. The value of $K$, the size of the neighbourhood at the final iteration, was 45 and the total solution time was 0.4 seconds. The \enquote{yes} in the solution guarantee column signifies that the amended optimisation problem associated with this final value of $K$ was solved to proven optimality, indicating that we have an absolute guarantee that there is no improved solution within a Hamming distance of $K=45$ from the solution associated with the value of 514 as found by our approach.
Considering Table~\ref{table2} it is clear that for all but one of the 65 test problems we found the optimal/best-known solution. From Table~\ref{table1} these problems are of increasing size and for all 45 problems up to and including problem set D we have the guarantee on solution quality. However for the 20 larger problems we only have one instance in which this is the case (recall here that we impose a time limit for the solution of each and every amended optimisation problem encountered during the process).
\begin{table}[hbt!]
\footnotesize
\centering
\renewcommand{1mm}{1mm}
\renewcommand{1.0}{0.85}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Instance & Optimal/best-known (OBK) & Solution & Final $K$ & Time (secs)
& Solution guarantee \\
\hline
4.1 & 429 & o & 45 & 0.4 & yes \\
4.2 & 512 & o & 50 & 0.6 & yes \\
4.3 & 516 & o & 55 & 0.5 & yes \\
4.4 & 494 & o & 50 & 0.5 & yes \\
4.5 & 512 & o & 45 & 0.4 & yes \\
4.6 & 560 & o & 55 & 0.7 & yes \\
4.7 & 430 & o & 50 & 0.4 & yes \\
4.8 & 492 & o & 45 & 0.8 & yes \\
4.9 & 641 & o & 55 & 1.0 & yes \\
4.10 & 514 & o & 45 & 0.4 & yes \\
\hline
5.1 & 253 & o & 50 & 1.1 & yes \\
5.2 & 302 & o & 50 & 1.3 & yes \\
5.3 & 226 & o & 50 & 0.7 & yes \\
5.4 & 242 & o & 45 & 1.1 & yes \\
5.5 & 211 & o & 45 & 0.7 & yes \\
5.6 & 213 & o & 45 & 0.8 & yes \\
5.7 & 293 & o & 50 & 1.1 & yes \\
5.8 & 288 & o & 45 & 1.0 & yes \\
5.9 & 279 & o & 50 & 0.8 & yes \\
5.10 & 265 & o & 50 & 0.7 & yes \\
\hline
6.1 & 138 & o & 60 & 2.0 & yes \\
6.2 & 146 & o & 40 & 1.1 & yes \\
6.3 & 145 & o & 45 & 1.5 & yes \\
6.4 & 131 & o & 45 & 1.4 & yes \\
6.5 & 161 & o & 50 & 1.8 & yes \\
\hline
A1 & 253 & o & 50 & 3.7 & yes \\
A2 & 252 & o & 55 & 4.1 & yes \\
A3 & 232 & o & 50 & 2.9 & yes \\
A4 & 234 & o & 50 & 2.4 & yes \\
A5 & 236 & o & 65 & 3.3 & yes \\
\hline
B1 & 69 & o & 40 & 3.5 & yes \\
B2 & 76 & o & 45 & 9.4 & yes \\
B3 & 80 & o & 45 & 4.7 & yes \\
B4 & 79 & o & 45 & 10.2 & yes \\
B5 & 72 & o & 40 & 3.4 & yes \\
\hline
C1 & 227 & o & 55 & 4.6 & yes \\
C2 & 219 & o & 60 & 5.8 & yes \\
C3 & 243 & o & 80 & 19.6 & yes \\
C4 & 219 & o & 50 & 4.0 & yes \\
C5 & 215 & o & 55 & 5.0 & yes \\
\hline
D1 & 60 & o & 50 & 7.6 & yes \\
D2 & 66 & o & 50 & 23.0 & yes \\
D3 & 72 & o & 55 & 23.5 & yes \\
D4 & 62 & o & 45 & 10.6 & yes \\
D5 & 61 & o & 45 & 5.1 & yes \\
\hline
NRE1 & 29 & o & 40 & 78.5 & no \\
NRE2 & 30 & o & 65 & 160.2 & no \\
NRE3 & 27 & o & 60 & 139.8 & no \\
NRE4 & 28 & o & 50 & 105.3 & no \\
NRE5 & 28 & o & 40 & 76.8 & no \\
\hline
NRF1 & 14 & o & 45 & 114.2 & no \\
NRF2 & 15 & o & 35 & 78.5 & no \\
NRF3 & 14 & o & 45 & 44.4 & yes \\
NRF4 & 14 & o & 40 & 93.4 & no \\
NRF5 & 13 & o & 45 & 109.4 & no \\
\hline
NRG1 & 176 & o & 65 & 308.3 & no \\
NRG2 & 154 & o & 60 & 219.7 & no \\
NRG3 & 166 & o & 85 & 565.3 & no \\
NRG4 & 168 & o & 115 & 821.2 & no \\
NRG5 & 168 & o & 75 & 475.4 & no \\
\hline
NRH1 & 63 & 64 & 55 & 344.9 & no \\
NRH2 & 63 & o & 70 & 501.0 & no \\
NRH3 & 59 & o & 100 & 773.0 & no \\
NRH4 & 58 & o & 85 & 618.6 & no \\
NRH5 & 55 & o & 55 & 342.0 & no \\
\hline
\end{tabular}
\caption{Computational results}
\label{table2}
\end{table}
\normalsize
In order to compare our results with previous results in the literature we have taken the detailed results given by various authors and computed percentage deviation from the \\
optimal/best-known value, OBK, as shown in Table~\ref{table2}. In other words using the solution values given by authors in their papers we computed 100(solution value - OBK)/OBK for each individual problem and then averaged the percentage deviations. Note here that whilst previous authors may have used this percentage deviation approach their OBK values may be different from those shown in Table~\ref{table2} (since obviously best-known values may be updated over time).
Table~\ref{table3} shows this comparison, where in each case we give the average percentage deviation; the number of solutions equal
to the optimal/best-known solution (OBK) and the average computation time (in seconds), as calculated
using solution details in the papers cited.
All of these averages relate to the same 65 instances which we considered, as detailed in Table~\ref{table1}.
So for example, for the work presented in this paper the average percentage deviation is 0.02\%,
for 64 of the 65 test problems the solution found was equal to the OBK solution (as detailed in Table~\ref{table2}) and the average computation time was 94.6 seconds.
As we would expect different authors have used different hardware and so a direct comparison between computation times is difficult, but the times given are an indication of how quickly problems are solved on average. In order to help set the times shown in context, as the papers cited range from 1996-2023, so over 25 years during which hardware has improved immensely, we also show in Table~\ref{table3} the year each paper was published.
With regard to Table~\ref{table3}:
\begin{compactitem}
\item Aickelin~\cite{aickelin02} and Beasley and Chu~\cite{beasley96} used ten trials of their algorithms. So for both these papers
in Table~\ref{table3} we have used the best solution found over the ten trials in calculating percentage deviation, and the time seen is the total time for these ten trials.
\item For Caprara et al~\cite{caprara99} the detailed times given in their paper are the times as to when the final best solution was first found. As such we have no information as to the total time taken, which is the value given for the other papers cited.
\item For Lan et al~\cite{lan07} the detailed times given in their paper are the times as to when the final best solution was first found. For this reason the time given in
Table~\ref{table3} is taken from the best performing of three different variants of their approach (the variant Meta-RaPs w/randomized priority rules, see Table 4 in\cite{lan07}) and the time given is as reported by them in their paper as the total time taken for that variant.
\item For Reyes and Araya~\cite{reyes21} the times given in their paper appear to be the average time for each trial, where they used 30 trials. So in Table~\ref{table3} we give the total time for 30 trials and in calculating percentage deviation we used the best solution found over these 30 trials.
\item For Soto et al~\cite{soto17} the times given in their paper are the average time for each trial~\cite{private22}, where they used 30 trials. So in Table~\ref{table3} we give the total time for 30 trials and in calculating percentage deviation we used the best solution found over these 30 trials.
\end{compactitem}
\noindent \emph{\textbf{We should stress here that the papers considered in Table~\ref{table3}
represent, based upon our literature review, the most effective heuristics for the set covering problem given previously in the literature.}}
Considering Table~\ref{table3}
then it seems reasonable to conclude that our
approach is very competitive in terms of solution quality for the non-unicost set covering problem as compared with other approaches given in the literature.
In order to compare our results with those obtained using Cplex alone we solved all of the 65 test problems,
using Cplex~\cite{cplex1210} with default parameter settings, but imposed a time limit for each problem equal to the corresponding solution time as shown in Table~\ref{table2}.
Over the 65 test problems Cplex gave an average percentage deviation of 0.12\%, where for 60 of the 65 test problems the solution found was equal to the OBK solution. Considering the comparative values for the approach given in this paper as in Table~\ref{table3} (average percentage deviation of 0.02\%, 64 of the 65 test problems equal to the OBK solution) it is clear that our approach is adding value as compared with using Cplex alone.
\begin{table}[hbtp!]
\centering
\renewcommand{1mm}{1mm}
\renewcommand{1.0}{1.0}
\begin{tabular}{|l|c|c|c|c|}
\hline
Approach & Average & Number of & Average & Year\\
& \% deviation & OBK solutions & time (secs) & published\\
\hline
This paper & 0.02 & 64 & 94.6 & 2023\\
Aickelin~\cite{aickelin02} & 0.13 & 61 & 1179.5 & 2002 \\
Beasley and Chu~\cite{beasley96} & 0.07 & 61 & 14694.3 &1996 \\
Caprara et al~\cite{caprara99} & 0 & 65 & not known &1999 \\
Lan et al~\cite{lan07} & 0 & 65 & 878.4 & 2007 \\
Naji-Azimi et al~\cite{naji10} & 0.18 & 53& 118.4 & 2010\\
Reyes and Araya~\cite{reyes21} & 0.25 & 50& 244.7 & 2021 \\
Soto et al~\cite{soto17} black hole & 1.55 & 35 & 184.5 & 2017\\
Soto et al~\cite{soto17} cuckoo search & 1.01 &35 & 151.0 & 2017 \\
\hline
\end{tabular}
\caption{Comparison of results}
\label{table3}
\end{table}
\section{Conclusions}
\label{conc}
In this paper we have presented
a heuristic for the non-unicost set covering problem using local branching.
Local branching eliminates the need to define a problem specific search neighbourhood for any particular (zero-one) optimisation problem. It does this by incorporating a generalised Hamming distance neighbourhood into the problem, and this leads naturally to an appropriate neighbourhood search procedure.
We applied our approach to the non-unicost set covering problem and presented computational results for 65 test problems that have been widely considered in the literature. Our results indicated that our heuristic for the set covering problem using local branching
is very competitive in terms of solution quality with other approaches from the literature.
We would also stress here the relative simplicity involved in creating a local branching based matheuristic
using the approach given in this paper. Obviously one needs a mathematical formulation of the problem at hand, but aside
from that the most that is needed is some (relatively simple) problem-specific procedure to generate an initial feasible solution.
We believe that the work described here illustrates that the potential for using local branching, operating as a stand-alone matheuristic as in the approach described in this paper,
has not been fully exploited in the literature.
We hope that the work presented here will encourage others to explore using it for other problems.
\FloatBarrier
\pagestyle{empty}
\linespread{1}
\small \normalsize
\end{document} |
\betaegin{document}
\title{Differential systems with reflection\ and matrix invariants}
\footnotetext{\footnotemark Département de Physique Théorique et Section de Mathématiques. Université de Genève, Genève, CH-1211 Switzerland. \href{mailto:santiago.codesido@unige.ch}{santiago.codesido@unige.ch}}
\footnotetext{Corresponding author. Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Facultade de Matemáticas, Santiago, Spain. \href{mailto:fernandoadrian.fernandez@usc.es}{fernandoadrian.fernandez@usc.es}. Partially supported by project MTM2016-75140-P (AEI/FEDER, UE) and Xunta de Galicia (Spain), project EM2014/032.}
\medbreak
\betaegin{abstract}
In this work we derive important properties regarding matrix invariants which occur in the theory of differential equations with reflection.
\varepsilonnd{abstract}
\textbf{Keywords:} differential equations with reflection, matrix invariants.
\section{Introduction}
In recent works regarding the solution and Green's functions of Differential Equations with Reflection (see for instance \gammaite{Toj3, TojMI, CTMal,CT17}) the strong relation between linear analysis and linear algebra is highlighted. In particular, in the most recent of the aforementioned works, the authors obtain an explicit fundamental matrix for the system of differential equations with reflection
\betaegin{equation}\lambdaabel{hlsystem}Hu(t):=Fu'(t)+Gu'(-t)+A u(t)+Bu(-t)=0, t\in{\mathbb R},
\varepsilonnd{equation}
where $n\in{\mathbb N}$, $A,B,F,G\in{\mathcal M}_n({\mathbb R})$ and $u:{\mathbb R}\to{\mathbb R}^n$. To be precise, they prove the following result.
\betaegin{thm}[\gammaite{CT17}]\lambdaabel{thmexpfm}
Assume $F-G$ and $F+G$ are invertible. Then
\betaegin{equation*}\lambdaabel{Xseries}X(t): = \sum_{k=0}^\infty\frac{E^k t^{2k}}{(2k)!} -(F+G)^{-1}(A+B)\sum_{k=0}^\infty\frac{E^k t^{2k+1}}{(2k+1)!},\varepsilonnd{equation*}
where $E=(F-G)^{-1}(A-B)(F+G)^{-1}(A+B)$, is a fundamental matrix of problem \varepsilonqref{hlsystem}. If we further assume $A-B$ and $A+B$ are invertible, then $E$ is invertible and we can consider a square root $\Omega$ of $E$. Then,
\betaegin{equation*}\lambdaabel{fme}X(t)=\gammaosh \Omega t -(F+G)^{-1}(A+B)\Omega^{-1}\sinh\Omega t.\varepsilonnd{equation*}
\varepsilonnd{thm}
What is more, in another recent work the authors proved an analog of the Liouville's formula for the case with reflections in systems of order two.
\betaegin{thm}[Abel-Jacobi-Liouville Identity \gammaite{CoTo17}] \lambdaabel{AJLid}Let $n=2$ in equation \varepsilonqref{hlsystem}. Then $(|X|,|X'|)$ is the unique solution of the system of differential equations
\betaegin{equation*}\lambdaabel{n2e}\betaegin{aligned}x''= &\tr(E)x-2y, \\y''= &-2|E|x+\tr(E)y,\varepsilonnd{aligned}
\varepsilonnd{equation*}
subject to the one point conditions \[x(0)=1,\quad y(0)=|M_+|,\quad x'(0)=-\tr(M_+),\quad y'(0)=\tr(\operatorname{Adj}(M_+)E).\]
\varepsilonnd{thm}
The authors also presented in that work the following conjecture:
\betaegin{con} For any $n\ge 1$, if $X(t)$ is a fundamental matrix of problem \varepsilonqref{hlsystem}, then $|X(t)|$ can be obtained as a component of the solution of a linear system of differential equations with constant coefficients, those coefficients depending only on the different matrix invariants of $E$, which is defined as in Theorem \ref{thmexpfm}.
\varepsilonnd{con}
In order to attempt proving this conjecture, and taking into account the proof of Theorem \ref{AJLid}, we need to study the different matrix invariants of the matrices appearing in the theory.
\section{The $\mathbf Y$ matrix}
For $X(t)$ the fundamental matrix of the problem, define
\betaegin{equation}
Y(t) := X(t)^{-1} X'(t).
\lambdaabel{Ydef}
\varepsilonnd{equation}
We have that $X = S_1-M_+ S_2$ where $S_1$ and $S_2$ s are power series in $E$ which we can formally give, by using $\Omega = \sqrt{E}$, as $S_1=\gammaosh(\Omega t),\ S_2=\Omega^{-1} \sinh(\Omega t).$
Notice both are indeed power series in $\Omega^2 = E$. Since $X'' = X E$ and $ (Y^{-1})'= - Y^{-1} Y'Y^{-1}$ we have that
\betaegin{equation}
Y'= E - Y^2.
\lambdaabel{ricattiEq}
\varepsilonnd{equation}
Using the construction of \gammaite{levin} we build an associated ODE system
\betaegin{equation*}
z' = \lambdaeft(\betaegin{array}{cc}
0 & E \\ I & 0
\varepsilonnd{array}\right)z.
\varepsilonnd{equation*}
The system has as fundamental matrix
\betaegin{equation*}
\lambdaeft(\betaegin{array}{cc}
\gammaosh(\Omega t) & \Omega^{-1}\sinh(\Omega t) \\ \Omega \sinh(\Omega t) & \gammaosh(\Omega t)
\varepsilonnd{array}\right).
\varepsilonnd{equation*}
The solution of equation (\ref{ricattiEq}) is then given by
\betaegin{equation*}
Y(t) = \lambdaeft[\gammaosh(\Omega t) Y(0)+\Omega \sinh(\Omega t) \right] \lambdaeft[\Omega^{-1}\sinh(\Omega t) Y(0)+\gammaosh(\Omega t) \right]^{-1},
\varepsilonnd{equation*}
which in terms of the $S$ functions is
$Y(t) = \lambdaeft[-S_1 M_+ +E S_2\right] \lambdaeft[-S_2 M_+ + S_1\right]^{-1}$,
where we fix the initial condition with $Y(0)=X'(0)=-M_+$.
This seems like a commuted version of expression (\ref{Ydef}), but it is nothing more than the hypergeometric identity.
Consider the Liouville equation for $Y$ itself, that is,
\[
\lambdaeft(\lambdaog |Y|\right) '= \mathrm{Tr}\lambdaeft(Y^{-1} Y'\right) = \mathrm{Tr}\lambdaeft( Y - Y^{-1}E \right).\] Then we have
$\mathrm{Tr}(Y^{-1}E)=\mathrm{Tr}(Y)-\lambdaeft(\lambdaog |Y|\right)'$, which can be calculated in terms of invariants of $|X|$.
\section{Complex systems}
The main involution occurring in the theory of complex variable is the complex conjugation ${\mathcal C}:{\mathbb C}\to {\mathbb C}$, ${\mathcal C}(z)=\overline z$. It is, in fact, a reflection with respect to the second variable if we write $z=(x,y)\in{\mathbb R}^2$: ${\mathcal C}(x,y)=(x,-y)$.
We consider now an operator $L$ acting on $z(t)$ as
\betaegin{equation}
\lambdaabel{mainComplexEquation}
A_0 z(t) + A_1 \overline{z(t)} + B_0 z(t)+ B_1 \overline{z'(t)},
\varepsilonnd{equation}
where $z: \mathbb{R}\to \mathbb{C}^n$, and $A_i,B_i \in \mathcal{M}_{n\times n}\lambdaeft(\mathbb{C}\right):=\mathcal{M}$.
We can consider a extended algebra $\mathcal{M}^*$ by with the linear operation of complex conjugation which acts as
$
\mathcal{C} z = \overline{z}, z\in \mathbb{C}^n.
$
It is easy to see that the following properties hold\footnote{In fact, one could use here any involution for which matrix conjugation verifies $\mathcal{C}A\mathcal{C}\in \mathcal{M}$
by defining $\overline{A}$ suitably.},
\betaegin{equation}
\lambdaabel{conjugationProperties}
\mathcal{C}^2 = I, \mathcal{C} A = \overline{A} \mathcal{C},
\varepsilonnd{equation}
where $\overline{A}$ is the complex conjugate of $A$.
The we consider the free product quotiented by these relations,
\betaegin{equation*}
\mathcal{M}^* = \mathcal{M} \subsetar \lambdaeft\{\mathcal{C}\right\} \lambdaeft/ \lambdaeft( \mathcal{C}^2 = I, \mathcal{C} A = \overline{A} \mathcal{C}\right).\right.
\varepsilonnd{equation*}
Now, we can see that this is in fact a $\mathbb{Z}_2$-graded algebra. Due to the conditions (\ref{conjugationProperties}), we can move any $\mathcal{C}$'s to the
right, and any power of it is reduced modulo $2$. Therefore, any element of $A\in \mathcal{M}^*$ can be written as
$
\mathbf{A} = A_0 + A_1 \mathcal{C}$ with $A_0,A_1 \in \mathcal{M}
$.
As a vector space, $
\mathcal{M}^* = \mathcal{M} \oplus \mathcal{M}$.
The grading is clear by looking at the product of two generic elements,
\betaegin{equation}
\lambdaabel{conjugationAlgebraProduct}
\mathbf{A} \mathbf{B}=\lambdaeft(A_0 + A_1\mathcal{C}\right)\lambdaeft(B_0 + B_1\mathcal{C}\right) =\lambdaeft(A_0 B_0 + A_1 \overline{B_1}\right) + \lambdaeft( A_0 B_1 + A_1 \overline{B_0} \right) \mathcal{C}.
\varepsilonnd{equation}
We can also calculate a explicit inverse, using $
\lambdaeft(I-A \mathcal{C}\right)\lambdaeft(I+A \mathcal{C}\right) = I - A \overline{A}
$,
from where
$\lambdaeft(I+A \mathcal{C}\right)^{-1} = \lambdaeft(I-A \overline{A}\right) \lambdaeft(I - A \mathcal{C}\right)$.
Since $\lambdaeft(A B\right)^{-1}=B^{-1} A^{-1}$, we can generalize it to
\betaegin{equation*}
\lambdaeft(A_0 + A_1 \mathcal{C}\right)^{-1} = \lambdaeft(A_1^{-1}A_0 - \overline{A_0^{-1}A_1}\right)^{-1} \lambdaeft(A_1^{-1}-\overline{A_0^{-1}}\mathcal{C}\right).
\varepsilonnd{equation*}
As it is, the expression is unclear when either $A_i=0$. We rewrite
\betaegin{equation}
\lambdaabel{conjugationAlgebraInverse}
\mathbf{A}^{-1} = \Delta\lambdaeft(A_0,A_1\right) + \Delta\lambdaeft(\overline{A_1},\overline{A_0}\right) \mathcal{C},
\varepsilonnd{equation}
with
\betaegin{equation*}
\Delta\lambdaeft(A_0,A_1\right)=\betaegin{cases}
0, & A_0 = 0, A_1 {n\in\betaN}eq 0, \\
\lambdaeft(A_0 - A_1 \overline{A_0^{-1}A_1}\right)^{-1}, & A_0 {n\in\betaN}eq 0, A_1 {n\in\betaN}eq 0.
\varepsilonnd{cases}
\varepsilonnd{equation*}
As a last note, this can of course be realized as a matrix algebra over $\mathbb{R}^{2n}$, although it does not lead to anything new (other than clutter). For the record,
one can take a representation
\[
\rho\lambdaeft(z\right) = \lambdaeft(\betaegin{array}{c}
\Re z \\ \hline \Im z
\varepsilonnd{array}\right), \quad
\rho\lambdaeft(A\right) = \lambdaeft(\betaegin{array}{c|c}
\Re A & -\Im A \\ \hline
\Im A & \Re A
\varepsilonnd{array}\right), \quad
\rho\lambdaeft(\mathcal{C}\right) = \lambdaeft(\betaegin{array}{c|c}
I & 0 \\ \hline 0 & -I
\varepsilonnd{array}\right),
\]
for which it is easy to see that properties such as (\ref{conjugationProperties}) or (\ref{conjugationAlgebraProduct}) hold.
\subsection{Equation reduction}
With these tools, one can rewrite equation (\ref{mainComplexEquation}) as
$
\mathbf{B} z'(t) + \mathbf{A} z(t) = 0,
$
which can be reduced to
$
z'(t) + \lambdaeft(\mathbf{B}^{-1} \mathbf{A}\right) z(t) =0
$.
This means we can just focus on the study of
\betaegin{equation*}
z'(t) + A_0 z(t) + A_1 \overline{z(t)} =0.
\varepsilonnd{equation*}
Of course, one could now look for a fundamental operator inside the $\mathcal{M}^*$ algebra, such that solutions fulfill
\betaegin{equation*}
z(t) = \mathbf{X}(t) z(0),
\varepsilonnd{equation*}
which is
\betaegin{equation*}
\mathbf{X} = \gammaosh \lambdaeft(\mathbf{A} t\right) - \sinh \lambdaeft(\mathbf{A} t\right).
\varepsilonnd{equation*}
Unfortunately, it does not seem easy to calculate terms like $\mathbf{A}^n$ at the moment. We could in principle directly get
an explicit fundamental matrix if we could write a manageable expression. However, we do learn something important. Since $\mathbf{A}^n \in \mathcal{M}^*$, then
$\gammaosh\lambdaeft( \mathbf{A} t\right) \in \mathcal{M}^*$ too. Hence, if we want to find fundamental matrices for the problem, the ansatz must be of the form
\betaegin{equation*}
z(t) = \lambdaeft( X_0(t) + X_1(t) \mathcal{C} \right)z(0)= X_0(t) z(0) + X_1(t) \overline{z(0)},
\varepsilonnd{equation*}
where $X(0)=I$ and $Y(0)=0$ as to agree with $\mathbf{X}(0)=I$.
\subsection{$\mathbf{A^n}$ generating function}
The components of $\mathbf{A}^n$ can still be algorithmically computed by using the expression
\betaegin{equation*}
\lambdaeft(I-t \mathbf{A}\right)^{-1} = \sum_{n=0}^\infty \lambdaeft(t \mathbf{A}\right).
\varepsilonnd{equation*}
By writing the explicit inverse of $I-t\mathbf{A}=\lambdaeft(I-t A_0\right)-t A_1 \mathcal{C}$, we get
\betaegin{small}
\betaegin{equation*}
\mathbf{A}^n = \frac{1}{n!} \frac{\mathrm{d}}{\mathrm{d}t}\lambdaeft[ \lambdaeft(A_1^{-1}\lambdaeft( I - t A_0\right) - \overline{\lambdaeft(I -t A_0\right)^{-1}A_1}\right)^{-1}
\lambdaeft(A_1^{-1} - \overline{ \lambdaeft(t^{-1}I-A_0\right)^{-1} } \mathcal{C}\right)\right]_{t=0}.
\varepsilonnd{equation*}
\varepsilonnd{small}
\subsection{Ansatz}
We take the system
\betaegin{equation*}
z'+A z+\overline{B z} = 0, z: \mathbb{R}\to \mathbb{C}^n, A,B \in \mathcal{M}
\varepsilonnd{equation*}
and introduce the ansatz
\betaegin{equation*}
z = X z_0 + \overline{Y z_0}, z_0=z(0), X,Y: \mathbb{R} \to \mathcal{M},
\varepsilonnd{equation*}
\betaegin{equation*}
\lambdaeft(X' + A X + \overline{B} Y \right) z_0 + \lambdaeft(Y'+ \overline{A} Y + B X\right) \overline{z_0} = 0,
\varepsilonnd{equation*}
\betaegin{equation}
\lambdaabel{fundamentalSystem}
\lambdaeft\{ \betaegin{array}{r}
X' + A X + \overline{B} Y = 0, \\
Y'+ \overline{A} Y + B X = 0.
\varepsilonnd{array} \right.
\varepsilonnd{equation}
This a ordinary system. Take $X''$ and substitute $Y'$ and $Y$ through equation (\ref{fundamentalSystem}),
\betaegin{align*}
\betaegin{split}
& X'' + A X' + \overline{B} \lambdaeft(-B X - \overline{A} \overline{B^{-1}}\lambdaeft(-X'-A X\right)\right) = 0, \\
& X'' + \lambdaeft(A + \overline{B A B^{-1}}\right) X' + \lambdaeft(\overline{B A B^{-1}}A - \overline{B}B\right) X = 0.
\varepsilonnd{split}
\varepsilonnd{align*}
Unsurprisingly, we get a very similar structure to the inverses in expression (\ref{conjugationAlgebraInverse}).For the sake of notation, we will rename the coefficients as
\betaegin{equation*}
\lambdaabel{singleFundamentalMatrixEquation}
X'' + F X' + G X= 0.
\varepsilonnd{equation*}
Repeating the process, we get
\betaegin{equation*}
Y'' + \overline{F} Y' + \overline{G} Y = 0.
\varepsilonnd{equation*}
The initial conditions for this second order problem are given by $\mathbf{X}(0)=I$ and equation (\ref{fundamentalSystem}). That is,
\[X(0) = I, \quad Y(0)=0,\quad
X'(0) = -A, \quad Y'(0) = -\overline{A}.
\]
In principle, we could now take as an ansatz
\betaegin{equation*}
X = \alphalpha e^{(\Gamma + \Omega) t} + \betaeta e^{(\Gamma-\Omega) t},
\varepsilonnd{equation*}
subject to the conditions
\betaegin{equation*}
X(0) = \alphalpha + \betaeta, X'(0) = \alphalpha (\Gamma+\Omega) + \betaeta (\Gamma-\Omega),
\varepsilonnd{equation*}
which can be inverted into
\betaegin{align*}
\betaegin{split}
\alphalpha = &\frac{1}{2} \lambdaeft[ X'(0) - X(0) \lambdaeft(\Gamma-\Omega\right) \right] \Omega^{-1},\\
\betaeta = &-\frac{1}{2} \lambdaeft[ X'(0) - X(0) \lambdaeft(\Gamma+\Omega\right) \right] \Omega^{-1} .
\varepsilonnd{split}
\varepsilonnd{align*}
\section{Generalized matrix invariants}
In the following section we use the concept of crossed or generalized matrix invariants which can be found in \gammaite{simon} and \gammaite{cgm} among others.
\subsection{Definition and basic properties}
Let $X_1,\deltaots,X_N \in GL\lambdaeft(n\right)$. Define
\betaegin{equation}\lambdaabel{z} Z\lambdaeft(X_1,\deltaots,X_N\right) := \deltaet\lambdaeft( I + \sum_{i=1}^N \alphalpha_i X_i \right) := \sum_{m_i} \alphalpha_1^{m_1}\deltaots\alphalpha_N^{m_N} Z_{m_1,\deltaots,m_N}\lambdaeft(X_1,\deltaots,X_N\right) .\varepsilonnd{equation}
Since $\deltaet$ is an algebraic combination of matrix entries, the expansion is a polynomial in the $\alphalpha_i$ variables. We can however take the sum to be over
all integer values of $m_i$ by suitably defining its $\alphalpha$-coefficients, $Z_{m_1,\deltaots,m_N}$, as zero when not corresponding to any power that appears in the $\deltaet$ expansion.
In particular,
\betaegin{equation}
Z_{m_1,\deltaots,m_N}\lambdaeft(X_1,\deltaots,X_n\right) = 0 \text{ if } \min\lambdaeft\{m_i\right\}<0.
\lambdaabel{noNegativeMultiTraces}
\varepsilonnd{equation}
These $Z$ coefficients then give us the \textit{generalized matrix invariants}, which reduce to the usual ones when we only consider one matrix (or set the other indices to $0$).
We can get explicit expressions in terms of traces via
\betaegin{equation}
\deltaet(I+\alphalpha X) = e^{\mathrm{Tr}\lambdaog (I+\alphalpha X)}
\lambdaabel{detExp}
\varepsilonnd{equation}
by expanding the Taylor series around $\alphalpha=0$ and using the linearity of the trace. This already gives, looking at the leading order of the exponential expansion,
\betaegin{equation}
Z_{0,\deltaots,0} (X_1,\deltaots,X_n) = 1.
\lambdaabel{zeroMultiTrace}
\varepsilonnd{equation}
In the same way, we can reduce any expression with a $0$ index,
\betaegin{equation*}
Z_{n_1,\deltaots,n_{N-1},0} \lambdaeft(X_1,\deltaots,X_N\right) = Z_{n_1,\deltaots,n_{N-1}} \lambdaeft(X_1,\deltaots,X_{N-1}\right).
\varepsilonnd{equation*}
Looking at higher coefficients upon expanding the exponential returns higher invariants. For instance,
\betaegin{align*}
\mathrm{Tr}\lambdaog\lambdaeft(I+\alphalpha X\right) =& \alphalpha \mathrm{Tr}\lambdaeft(X\right) -\frac{\alphalpha^2}{2} \mathrm{Tr}\lambdaeft(X^2\right) + \frac{\alphalpha^3}{3} \mathrm{Tr}\lambdaeft(X^3\right)
+ O\lambdaeft(\alphalpha^4\right), \\
Z_{1}(X) =& \mathrm{Tr}(X),\\
Z_{2}(X) =& \frac{1}{2} \lambdaeft( \mathrm{Tr}(X)^2-\mathrm{Tr}(X^2) \right),\\
Z_{3}(X) =& \frac{1}{6} \lambdaeft( \mathrm{Tr}(X)^3-3\mathrm{Tr}(X^2)\mathrm{Tr}(X)+\mathrm{Tr}(X^3) \right),
\varepsilonnd{align*}
etc, but also
\betaegin{equation*}
Z_{1,1}(X,Y) = \mathrm{Tr}(X) \mathrm{Tr}(Y)-\mathrm{Tr}(X Y).
\varepsilonnd{equation*}
Of course, equation (\ref{detExp}) is usually proven using Liouville's formula. We will make contact with it again later, when looking at the derivatives of the $Z$ invariants themselves.
\subsection{Factorization}
Consider
\betaegin{equation*}
\deltaet\lambdaeft(1+\alphalpha A + \sum_{i} \betaeta_i B\right),
\varepsilonnd{equation*}
and the fact that
\betaegin{equation*}
\deltaet\lambdaeft( \alphalpha A \right) =\alphalpha^n \deltaet A.
\varepsilonnd{equation*}
Extract this determinant from the original expansion,
\betaegin{align*}
&\deltaet\lambdaeft(I+\alphalpha A + \sum_{i} \betaeta_i B\right) = \sum_{l,m_i} \alphalpha^{l} \lambdaeft(\prod \betaeta^{m_i}\right) Z_{l, m_1,\deltaots,m_N}\lambdaeft(A,B_1,\deltaots,B_N\right) \\ = &\deltaet\lambdaeft(A\right) \sum_{l,m_i} \alphalpha^{l} \lambdaeft(\prod \betaeta_i^{m_i}\right) Z_{n-l-\sum m_i,m_1,\deltaots,m_N} \lambdaeft(A^{-1},A^{-1}B_1,\deltaots,A^{-1}B_N\right).
\varepsilonnd{align*}
To equate the two polynomials, we equate every coefficient and get a \textbf{duality} relationship
\betaegin{equation}\lambdaabel{multiFactorization}
Z_{l, m_1,\deltaots,m_N}\lambdaeft(A,B_1,\deltaots,B_N\right) = \deltaet\lambdaeft(A\right) Z_{n-l-\sum m_i,m_1,\deltaots,m_N} \lambdaeft(A^{-1},A^{-1}B_1,\deltaots,A^{-1}B_N\right).\varepsilonnd{equation}
Already an interesting property comes from the fact that any $Z$ with negative indices must be $0$, by equation (\ref{noNegativeMultiTraces}). The dual of this statement is then
\betaegin{equation*}
Z_{m_1,\deltaots,m_N}\lambdaeft(X_1,\deltaots,X_N\right) = 0 \text{ if } \sum_{i}m_i > n.
\varepsilonnd{equation*}
We will call the sum of all indices $\sum m_i$ the \textbf{order} of the trace $Z_{m_1,\deltaots,m_N}$.
That an invariant of order higher than the size of the matrix is zero
reduces, as expected, to the usual property of matrix invariants when we have a single matrix, and together with expression ($\ref{noNegativeMultiTraces}$) ensures that only
a finite number of $Z$ invariants for any given set of $X_i$ is non-zero.
We can also take a dual of equation $(\ref{zeroMultiTrace})$, which is the well known
\betaegin{equation*}
Z_{n}\lambdaeft(X\right) = \deltaet\lambdaeft(X\right)
\varepsilonnd{equation*}
or
\betaegin{equation*}
1 = \deltaet\lambdaeft(X\right) Z_{n}\lambdaeft(X^{-1}\right).
\varepsilonnd{equation*}
Now, this statement gets interesting when we introduce more matrices. Consider the two matrix case,
\betaegin{equation*}
Z_{l,m} \lambdaeft(A,B\right) = \deltaet\lambdaeft(A\right) Z_{n-l-m,m}\lambdaeft(A^{-1},A^{-1}B\right)
\varepsilonnd{equation*}
and set $A=X$, $B=X Y$, and $l+m=n$,
\betaegin{equation*}
Z_{n-m,m}\lambdaeft(X, X Y\right) = \deltaet\lambdaeft(X\right) Z_{0,m}\lambdaeft(X^{-1},Y\right) = \deltaet\lambdaeft(X\right) Z_m\lambdaeft(Y\right).
\varepsilonnd{equation*}
This, which is the generalization of
\betaegin{equation*}
\deltaet\lambdaeft(X Y\right) = \deltaet\lambdaeft(X\right) \deltaet\lambdaeft(Y\right),
\varepsilonnd{equation*}
allows us to decompose order $n$ invariants of a product into products of invariants. In particular, we get the $n=2$ expression with which
we built the ODE system.
More generally,
\betaegin{equation}
Z_{n-\sum m_i,m_1,\deltaots,m_N} \lambdaeft(X,X Y_1,\deltaots,X Y_N\right) = \deltaet\lambdaeft(X\right) Z_{m_1,\deltaots,m_N}\lambdaeft(Y_1,\deltaots,Y_N\right).
\lambdaabel{determinantFactorization}
\varepsilonnd{equation}
\subsection{Small-$\mathbf\varepsilonpsilon$ expansion}
By using expression \varepsilonqref{z} we can easily derive distributivity properties, which can be applied to calculate
\betaegin{align*}
& Z_{l,m_1,\deltaots,m_N}\lambdaeft(A_1+\varepsilonpsilon A_2+O\lambdaeft(\varepsilonpsilon^2\right),B_1,\deltaots,B_N\right) \\
=&\sum_{i=0}^{l} \varepsilonpsilon^i Z_{l-i,i,m_1,\deltaots,m_N}\lambdaeft(A_1,A_2+O\lambdaeft(\varepsilonpsilon\right),B_1,\deltaots,B_N\right) \\
=&\sum_{i=0}^{l} \varepsilonpsilon^i \lambdaeft[Z_{l-i,i,m_1,\deltaots,m_N}\lambdaeft(A_1,A_2,B_1,\deltaots,B_N\right) +O\lambdaeft(\varepsilonpsilon\right) \right] \\
=&\sum_{i=0}^{1} \varepsilonpsilon^i Z_{l-i,i,m_1,\deltaots,m_N}\lambdaeft(A_1,A_2,B_1,\deltaots,B_N\right) +O\lambdaeft(\varepsilonpsilon^2\right) \\
=&Z_{l,m_1,\deltaots,m_N}\lambdaeft(A_1,B_1,\deltaots,B_N\right) + \varepsilonpsilon Z_{l-1,1,m_1,\deltaots,m_N}\lambdaeft(A_1,A_2,B_1,\deltaots,B_N\right) +O\lambdaeft(\varepsilonpsilon^2\right).
\varepsilonnd{align*}
\subsection{Derivatives}
As we have seen before, the derivatives of the invariants play an essential role in the theory. We would now like to have a formula for derivatives of the form
\betaegin{equation*}
\frac{\mathrm{d}}{\mathrm{d}t} Z_m\lambdaeft(X\lambdaeft(t\right)\right).
\varepsilonnd{equation*}
Consider
\betaegin{equation*}
Z^{(m_0,m_1,m_2,\deltaots)}\lambdaeft(X\right) := Z_{m_0,m_1,m_2,\deltaots}\lambdaeft(X,X',X'',\deltaots\right),
\varepsilonnd{equation*}
such that for some $N$ we have $m_i=0$ for every $i>N$.
We can retrieve its first derivative from its Taylor series, which we can in turn get from its small $\varepsilonpsilon$ expansion.
\betaegin{align*}
& Z^{(m_0,m_1,\deltaots)}\lambdaeft( X+\varepsilonpsilon X'+O\lambdaeft(\varepsilonpsilon^2\right) \right) \\
=&Z^{(m_0,m_1,m_2,\deltaots)}\lambdaeft(X\right)+\varepsilonpsilon \lambdaeft(m_1+1\right) Z_{m_0-1,m_1+1,m_2,\deltaots}\lambdaeft(X,X',X'',\deltaots\right) \\ & +\varepsilonpsilon \lambdaeft(m_2+1\right) Z_{m_0,m_1-1,m_2+1,\deltaots}\lambdaeft(X,X',X'',\deltaots\right)+\gammadots
\varepsilonnd{align*}
Taking the $\varepsilonpsilon$ term we get the first coefficient of the Taylor series, i.\,e., the first derivative,
\[
\lambdaeft(Z^{(m_0,m_1,m_2,\deltaots)}\lambdaeft(X\right)\right)' =\sum_{i=1}^\infty \lambdaeft(m_i+1\right) Z^{ (m_0,\deltaots,m_{i-1}-1,m_i+1,\deltaots ) }\lambdaeft(X\right).
\]
Notice that the infinite sum is merely formal, since by equation (\ref{noNegativeMultiTraces}) it is guaranteed to terminate as soon as all the remaining $m_i$ are $0$, due
to the $m_{i-1}-1$ index at every term.
For the first few derivatives, we find via recursion the general expressions
\betaegin{align*}
Z_{m} \lambdaeft(X\right)' = \lambdaeft(Z^{(m)}\right)'=&Z^{(m-1,1)},
\\
Z_{m} \lambdaeft(X\right)''= \lambdaeft(Z^{(m)}\right)''=& 2 Z^{(m-2,2)} + Z^{(m-1,0,1)},
\\
Z_{m} \lambdaeft(X\right)''' = \lambdaeft(Z^{(m)}\right)''' =& 6 Z^{(m-3,3)} + 3 Z^{(m-2,1,1)} + Z^{(m-1,0,0,1)}.
\varepsilonnd{align*}
Something very important (albeit somehow obvious, following Leibniz's rule for matrices), is that the order of the invariants involved in the expressions is preserved.
This allows us to use the factorization formula (\ref{determinantFactorization}) over the derivatives of the determinant, which corresponds to $Z_n$.
As a small note, if we take in the first derivative $m=n$ together with expression (\ref{multiFactorization}), we get
\betaegin{equation*}
\deltaet\lambdaeft(X\right)'=Z_{n-1,1}\lambdaeft(X,X'\right) = \deltaet\lambdaeft(X\right) Z_1\lambdaeft(X^{-1} X'\right) = \deltaet\lambdaeft(X\right) \mathrm{Tr}\lambdaeft(X^{-1} X'\right),
\varepsilonnd{equation*}
the usual Liouville's Formula.
\subsection{Application to the differential system of invariants for $\mathbf{n>2}$}
In the matrix dimension $m=2$ case, taking derivatives of the determinant eventually closes, since $X''=XE$. This follows from
\[\deltaet(X)'' = Z_m(X)'' = 2 Z^{(m-2,2)}(X) + Z^{(m-1,0,1)}(X).\]
The $Z^{(m-1,0,1)}(X)$ can be immediately rewritten as a determinant by using the duality formula,
\[Z_{m-1,0,1} (X, X', X'') = \deltaet(X) Z_1 (X^{-1} X'') = \deltaet(X) \tr(E),\]
and, when $m=2$,
\[Z^{(m-2,2)}(X) = Z^{(0,2)}(X) = \deltaet(X').\]
Of course, now we can do the same for $X'$,
\[\deltaet(X')'' = 2 Z^{(m-2,2)}(X) + Z^{(m-1,0,1)}(X)\]
and, for $m=2$,
\[\deltaet(X')'' = 2 \deltaet(X'') + \deltaet(X') \tr(E) = 2 \deltaet(E) \deltaet(X) + \deltaet(X') \tr(E),\]
closing the system as we had found in Theorem \ref{AJLid}. The problem is now obvious, since for $m>2$, $Z^{(m-2,2)}(X)$ will involve a non trivial product between $X$ and $X'$. One could consider this as a new variable for the system, but its derivatives will now concern objects of the form $Z^{(m-2,1,1)}(X)$ which, if understood as yet another variable of the system, would yield upon derivation
\[Z^{(m-2,1,0,1)}(X),\ Z^{(m-2,1,0,0,1)}(X),\ Z^{(m-2,1,0,...,0,1)}, ...\]
Notice that this will always involve a term in $X'$, and a term in $X$, so that we cannot perform the same trick as we did for $Z^{(m-1,0,1)}(X)$ --namely, using $X''=XE$ to factor the determinant out. Hence, the system of second derivatives of invariants for $m>2$ does not close.
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\providecommand{\url}[1]{{#1}}
\providecommand{\urlprefix}{URL }
\varepsilonxpandafter\ifx\gammasname urlstyle\varepsilonndcsname\relax
\providecommand{\deltaoi}[1]{DOI~\deltaiscretionary{}{}{}#1}\varepsilonlse
\providecommand{\deltaoi}{DOI~\deltaiscretionary{}{}{}\betaegingroup
\urlstyle{rm}\Url}\fi
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Cabada, A., Tojo, F.A.F.: \varepsilonmph{Solutions and {Green's} function of the first
order linear equation with reflection and initial conditions}.
{n\in\betaN}ewblock Bound. Value Probl. \textbf{2014}(1), 99 (2014)
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Cabada, A., Tojo, F.A.F.: \varepsilonmph{{Green's} functions for reducible functional
differential equations}.
{n\in\betaN}ewblock Bull. Malays. Math. Sci. Soc. pp. 1--22 (2016)
\betaibitem{CT17}
Cabada, A., Tojo, F.A.F.: \varepsilonmph{On linear differential equations and systems
with reflection}.
{n\in\betaN}ewblock Applied Mathematics and Computation \textbf{305}, 84--102 (2017)
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Codesido, S., Grassi, A., Marino, M.: \varepsilonmph{Spectral theory and mirror curves
of higher genus}.
{n\in\betaN}ewblock In: Annales Henri Poincar{\'e}, pp. 1--64. Springer (2015)
\betaibitem{CoTo17}
Codesido, S., Tojo, F.A.F.: \varepsilonmph{A Liouville's Formula for systems with
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Levin, J.: \varepsilonmph{On the matrix Riccati equation}.
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Simon, B.: \varepsilonmph{Notes on infinite determinants of Hilbert space operators}.
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\varepsilonnd{thebibliography}
\varepsilonnd{document} |
\begin{document}
\title{Discriminating between L\"uders and von Neumann measuring devices: \\ An NMR investigation }
\author{C. S. Sudheer Kumar, Abhishek Shukla, and T. S. Mahesh
}
\email{mahesh.ts@iiserpune.ac.in}
\affiliation{Department of Physics and NMR Research Center,\\
Indian Institute of Science Education and Research, Pune 411008, India}
\begin{abstract}
Measurement of an observable on a quantum system involves a probabilistic collapse of the quantum state and a corresponding measurement outcome. L\"uders and von Neumann state update rules attempt to describe the above phenomenological observations. These rules are identical for a nondegenerate observable, but differ for a degenerate observable. While L\"uders rule preserves superpositions within a degenerate subspace under a measurement of the corresponding degenerate observable, the von Neumann rule does not. Recently Hegerfeldt and Mayato
[Phys. Rev. A, 85, 032116 (2012)]
had formulated a protocol to discriminate between the two types of measuring devices. Here we have reformulated this protocol for quantum registers comprising of system and ancilla qubits. We then experimentally investigated this protocol using nulear spin systems with the help of NMR techniques, and found that L\"uders rule is favoured.
\end{abstract}
\keywords{Degeneracy, State update Rules }
\pacs{ 03.65.Ta, 03.67.-a, 03.67.Ac}
\maketitle
\section{Introduction}
Quantum measurement paradox lies at the heart of foundations of quantum mechanics\cite{D_Home_book}. It's an experimental fact that, upon measurement, a quantum state collapses into an eigenstate of the observable being measured. However there is no collapse in the unitary evolution described by Schr\"odinger equation, and therefore, the collapse has to be imposed from outside the formalism.
Let us assume an observable $A_N$ with discrete and nondegenerate eigenspectrum. In that case, the measurement leads to a collapse of the state to one of the eigenstates of $A_N$ (see Fig. \ref{luder}).
On the other hand, if we consider an observable $A$ with a degenerate eigenspectrum, there are two extreme rules to update the state after the measurement. The most commonly used rule was postulated by Gerhart L\"uders in 1951 \cite{luders1951zustandsanderung,Luders_originl_paper}. According to it, a system existing in a superposition of degenerate eigenstates is unaffected by the measurement such that the superposition is preserved.
However, an earlier postulate by von Neumann, proposed in 1932 \cite{von_math_foundof_QM}, does not preserve such a superposition. In the latter postulate,
the measuring device refines the observable $A$ into another commuting observable $A'$ (actual system observable) having a nondegenerate spectrum. The resulting measurement collapses the state to an eigenstate of $A'$, and the original superposition is not preserved under the measurement as if the degeneracy has been lifted \cite{Discriminate_von_Luder_protocol}.
Although, one generally assumes L\"uders state update rule implicitly in quantum physics, occassionally one encounters applications of the von Neumann state update rule.
One example is in the context of Leggett-Garg inequality in multilevel quantum systems \cite{multi_level_LGI}.
In principle, measurements which are intermediate between L\"uders and von Neumann can also be conceived \cite{Discriminate_von_Luder_protocol,multi_level_LGI}.
\begin{figure}
\caption{Comparison between L\"uders and von Neumann measurement postulates. }
\label{luder}
\end{figure}
Recently, Hegerfeldt and Mayato have proposed a general protocol (HM protocol) to discriminate between L\"uders and von Neumann kind of measuring devices \cite{Discriminate_von_Luder_protocol}.
To explain this protocol we consider an observable $A$, having two-fold degenerate eigenvalues, say $+1$ and $-1$ (see Fig. \ref{HMprotocolfig}). The HM protocol involves the following steps: (i) prepare an eigenstate $\ket{\xi_\mathrm{in}}$ of $A$, (ii) let the device measure $A$, and (iii) characterize the output state. In step (ii) a L\"uders measurement will preserve the state, while a von Neumann measurement may not. The last step is simply to determine if the step (ii) has changed the state or not. If the state has changed, we conclude that the device is von Neumann. Else, either the device is of L\"uders type, or the chosen initial state $\ket{\xi_\mathrm{in}}$ happens to be a nondegenerate eigenstate of the actual system observable $A'$. To rule out the latter possibility, one may change the initial state and repeat the above steps (Fig. \ref{HMprotocolfig}). This way one can attempt to discriminate between the L\"uders and von Neumann measurement devices.
\begin{figure}
\caption{HM protocol for discriminating between L\"uders and von Neumann measurements. }
\label{HMprotocolfig}
\end{figure}
In this work, we reformulate the HM protocol for a quantum register and try to investigate it using experiments. Nuclear spin ensembles in liquid, liquid-crystalline, or solid-state systems have often been chosen as convenient testbeds for studying foundations of quantum physics \cite{suter1988study,Moussa,LGI_Soumya,Context_cssk}. Their main advantages are long coherence times and excellent control over quantum dynamics via highly developed nuclear magnetic resonance (NMR) techniques.
In section II, we briefly explain the HM protocol as adapted to an NMR setup.
The experimental details to discriminate between the L\"uders and von Neumann measuring devices is described in section III. Finally we conclude in section IV.
\section{Theory}
For the sake of clarity, and also to match the experimental details described in the next section, we consider a system of two qubits.
Since the system is to be measured projectively, dimension of the pointer basis should be greater than or equal to that of the system, and hence we need at least two ancillary qubits.
We refer to the ancillary qubits as (1,2) and system qubits as (3,4).
We use Zeeman product basis as our computational basis and denote eigenkets of $\sigma_z$, the Pauli $z$-operator,
by $\ket{0}$ and $\ket{1}$. We denote the basis vectors of system qubits as
\begin{eqnarray}
\ket{\phi_{0}} = \ket{00}, \ket{\phi_{1}} = \ket{01}, \ket{\phi_{2}} = \ket{10},
\ket{\phi_{3}} = \ket{11}.
\label{phis}
\end{eqnarray}
Let us assume a two-fold degenerate system-observable with spectral decomposition
\begin{eqnarray}
A = (\Pi_0+\Pi_1)-(\Pi_2+\Pi_3), ~~~~
\label{A}
\end{eqnarray}
where the projectors are defined as $\Pi_j=\proj{\chi_j},\ket{\chi_0}=\alpha_0\ket{\phi_0}+\beta_0\ket{\phi_1}$, $\ket{\chi_1}=\alpha_1\ket{\phi_0}+\beta_1\ket{\phi_1}$, $\ket{\chi_2}=\alpha_2\ket{\phi_2}+\beta_2\ket{\phi_3}$, $\ket{\chi_3}=\alpha_3\ket{\phi_2}+\beta_3\ket{\phi_3}$ are eigenvectors of $A$.
The projectors have the property $\Pi_k\Pi_l=\delta_{kl}\Pi_k$. We note that $A$ has no unique spectral decomposition due to the degeneracy.
We consider a measurement model, wherein a quantum system being measured undergoes a joint evolution with the measuring device, ultimately forming an entangled state. When the measuring device collapses to a particular pointer state, the system also collapses to the corresponding eigenstate. Let $Q$ be the observable corresponding to the ancilla (measuring device) and $g$ be the system-ancilla interaction strength. The joint evolution is then of the form
\begin{eqnarray}
U_\mathrm{int} = \exp(-i{\cal H}_\mathrm{int}\tau),
\end{eqnarray}
where ${\cal H}_\mathrm{int} = g ~Q \otimes A$ is the interaction Hamiltonian in units of angular frequency.
To fix the basis inside a degenerate subspace, we should choose a nondegenerate observable $A'$ which commutes with $A$, so that they are simultaneously diagonalizable and hence we can find a common eigenbasis. For simplicity we choose the computional basis $\{\ket{\phi_j}\}$ as the common eigenbasis. Then the observable $A'$ must have the following spectral decomposition
\begin{eqnarray}
A'= \sum_{j=0}^3 a'_j P_j,
\label{A'_defined}
\end{eqnarray}
where $P_j = \proj{\phi_j}$ and the nondegenerate eigenvalues $a'_j$ are yet to be determined.
Let us assume the device to be von Neumann which refines the degenerate observable $A$ that is being measured, into a nondegenerate observable $A'$, via a mapping $f(A')=A$.
As the refined observable $A'$ has nondegenerate eigenvalues and commutes with $A$, it fixes the basis inside the degenerate subspace. However, the choice of $A'$ is not unique, i.e., any orthonormal basis inside the degenerate subspace can be nondegenerate eigenkets of $A'$, and the von Neumann device has the freedom to choose among them \cite{von_math_foundof_QM}.
The measurement outcome is passed via the refining function $f$, such that $f(a'_0)=f(a'_1)=+1$ and $f(a'_2)=f(a'_3)=-1$. Hence the outcome is same as if $A$ is being measured. To projectively measure the observable $A'$, the measuring device has to jointly evolve with the system under the interaction Hamiltonian,
\begin{eqnarray}
{\cal H}'_\mathrm{int}= g ~Q\otimes A'.
\end{eqnarray}
For instance, we choose $Q=q_1\sigma_{1z}+q_2\sigma_{2z}$, where
$\sigma_{1z} = \sigma_z \otimes\mathbbm{1}_2$,
$\sigma_{2z} = \mathbbm{1}_2\otimes\sigma_z$ and $\mathbbm{1}_2$ is $2\times 2$ identity operator.
The joint evolution between the measuring device (ancillary qubits) and the system is described by the unitary operator
\begin{eqnarray}
U'_\mathrm{int} = \exp(-i\cal{H}_\mathrm{int}' \tau),
\end{eqnarray}
where $\tau$ is duration of the evolution.
\begin{figure}
\caption{An interpolating function $a = f(a') = (-a'^3+7a')/6$ mapping the nondegenerate eigenvalues $a'$ of $A'$ onto degenerate eigenvalues $a$ of $A$.}
\label{fofx}
\end{figure}
If each of the quantum register is initially prepared in $\ket{\Phi_0} = \ket{++++}$, with $\ket{+} = (\ket{0}+\ket{1})/\sqrt{2}$, the state after the joint evolution is given by
\begin{eqnarray}
U'_\mathrm{int}\ket{\Phi_0} &=& \frac{1}{2} \bigg( e^{-iga'_0Q \tau}\ket{++}\ket{\phi_0}+
e^{-iga'_1Q \tau}\ket{++}\ket{\phi_1}+ \nonumber \\
&& e^{-iga'_2Q \tau}\ket{++}\ket{\phi_2}+
e^{-iga'_3Q \tau}\ket{++}\ket{\phi_3} \bigg) \nonumber \\
&=& \frac{1}{2} \sum_{j=0}^3 \ket{\psi_j}\ket{\phi_j},
\label{von_evolutn}
\end{eqnarray}
where $\ket{\phi_j}$ are as defined in Eqs. \ref{phis} and $\ket{\psi_j} = \exp(-iga'_jQ \tau)\ket{++}$ represent states of the ancillary qubits. To realize the projective measurement, the pointer basis $\{\ket{\psi_j}\}$ must be orthonormal. Imposing the mutual orthogonality condition results in trigonometric constraint equations leading to a set of possible solutions. One such possible solution is
\begin{eqnarray}
\begin{array}{l|l}
a'_0 = -a'_2 = -3 & q_1 = \pi/(4g\tau) \\
a'_1 = -a'_3 = 1 & q_2 = -q_1/2.\\
\end{array}
\end{eqnarray}
Again, the von Neumann measuring device has the freedom to choose a particular pointer basis among several possible ones. Substituting the above values in Eq. \ref{A'_defined}, we obtain,
\begin{eqnarray}
A'=-3 P_0+ P_1 +3 P_2 -P_3,
\end{eqnarray}
which is obviously nondegenerate in the computational basis. The refining function $f$ can now be setup by interpolating the eigenvalue distribution (see Fig. \ref{fofx}). For the above example, we find a possible map to be $f(A') = (-A'^3+7A')/6 = A$.
The quantum circuit for discriminating L\"uders and von Neumann devices, illustrated in Fig. \ref{fig_discrimnt_von_lud}, involves four qubits each of which is initialized in state $\ket{+}$. If the device is L\"uders, the system undergoes a joint evolution $U_\mathrm{int}$ with the ancilla resulting in the state
\begin{eqnarray}
U_\mathrm{int} \ket{\Phi_0} &=& \frac{1}{\sqrt{2}}
\bigg( e^{-igQ \tau}\ket{++} \frac{d_0\ket{\chi_0}+d_1\ket{\chi_1}}{\sqrt{2}} + \nonumber \\
&& e^{igQ \tau}\ket{++} \frac{d_2\ket{\chi_2}+d_3\ket{\chi_3}}{\sqrt{2}} \bigg) \nonumber \\
& = & \frac{1}{\sqrt{2}} \bigg(
\ket{\psi_1} \frac{\ket{\phi_0}+\ket{\phi_1}}{\sqrt{2}} +
\ket{\psi_3} \frac{\ket{\phi_2}+\ket{\phi_3}}{\sqrt{2}}
\bigg),~~~~
\label{Luder_evolution}
\end{eqnarray}
where the coefficients $d_j$ depend on the choice of $\ket{\chi_j}$ (defined after Eq. \ref{A}) \cite{ludernote1}.
\begin{figure}
\caption{(a) Quantum circuit to discriminate L\"uders and von Neumann devices.
(b) The NMR pulse-scheme to implement the circuit in (a).}
\label{fig_discrimnt_von_lud}
\end{figure}
Note that if the measuring device is of von Neumann type, it will instead measure $A'$, and pass the measurement outcome via the function $f$, as explained before.
After the joint evolution of system and ancilla, a selective measurement of ancilla qubits is carried out.
Generally in a quantum measurement the measuring device collapses to its pointer basis. In our scheme, we perform the projective measurement in the computational basis after transforming the ancilla qubits onto the computational basis using a similarity transformation $U_{a}^\dagger$, such that
\begin{eqnarray}
U_{a}\ket{00}=\ket{\psi_0},U_{a}\ket{01}=\ket{\psi_1},\nonumber\\
U_{a}\ket{10}=\ket{\psi_2},U_{a}\ket{11}=\ket{\psi_3}.
\label{U_ra_constraints}
\end{eqnarray}
By substituting the explicit forms of $\ket{\psi_j}$, we obtain
\begin{eqnarray}
U_{a}=\frac{1}{2}
\left[
\begin{array}{cccc}
z^3 & z^{-1} & z^{-3} & z \\
z^{9} & z^{-3} & z^{-9} & z^{3} \\
z^{-9} & z^{3} & z^{9} & z^{-3} \\
z^{-3} & z & z^{3} & z^{-1}
\end{array}
\right],
\label{U_ra}
\end{eqnarray}
where $z=\exp(i\pi/8)$.
Finally, the ancilla is traced-out and the state of system qubits is characterized with the help of quantum state tomography.
According to the L\"uders state update rule, if a degenerate observable $A$ (as in Eq. \ref{A}) is measured on a system in state $\rho_0$, then the postmeasurement state of the ensemble is described by
\begin{eqnarray}
\rho_{L}=\sum_{l=\pm 1}{\mathbb P}_l\rho_{0}{\mathbb P}_l,
\label{rhoLgen}
\end{eqnarray}
where ${\mathbb P}_{+1} = \Pi_{0}+\Pi_{1}$,
${\mathbb P}_{+1} = \Pi_{2}+\Pi_{3}$.
For the initial state $\rho_0 = \proj{\Phi_0}$,
we obtain
\begin{eqnarray}
\rho_L =
(\mathbbm{1}_4 + \mathbbm{1}_2 \otimes \sigma_x)/4.
\label{rhoL}
\end{eqnarray}
However according to von Neumann's degeneracy breaking state update rule, the postmeasurement state of the ensemble is given by
\begin{eqnarray}
\rho_N = \sum_{j=0}^3 \Pi_j \rho_0 \Pi_j,
\label{rhoNgen}
\end{eqnarray}
where, $\Pi_j$'s are fixed by the refining observable $A'$.
Therefore, for the initial state $\rho_0 = \proj{\Phi_0}$ and the observable $A'$ (Eq. \ref{A'_defined}), the postmeasurement state collapses to a maximally mixed state, i.e.,
\begin{eqnarray}
\rho_N = \mathbbm{1}_4/4.
\label{rhoN}
\end{eqnarray}
In both the cases, the probabilities of obtaining the eigenvalues $\pm 1$ are identical, i.e.,
\begin{eqnarray}
p_{+1} &=& \mathrm{Tr}({\mathbb P}_{+1}\rho_{0}{\mathbb P}_{+1})=\sum_{j=0,1}\mathrm{Tr}(\Pi_j \rho_0 \Pi_j) ~~\mathrm{and,}\nonumber \\
p_{-1} &=& \mathrm{Tr}({\mathbb P}_{-1}\rho_{0}{\mathbb P}_{-1})=\sum_{j=2,3}\mathrm{Tr}(\Pi_j \rho_0\Pi_j).
\end{eqnarray}
Thus although, the measurement outcomes (eigenvalues) and their probabilities are identical, the postmeasurement states $\rho_L$ and $\rho_N$ are different \cite{Luders_originl_paper,von_math_foundof_QM,Discriminate_von_Luder_protocol,multi_level_LGI}.
In fact, the Uhlmann fidelity between $\rho_L$ and $\rho_N$ turns out to be $F(\rho_L,\rho_N)=\mbox{Tr}\sqrt{\sqrt{\rho_N}\rho_L\sqrt{\rho_N}}=1/\sqrt{2}$ \cite{quant_info_neilson_chuang}.
Therefore, it is possible to discriminate between the Luders and von Neumann devices by simply characterizing the final state of the system as shown by the circuit in Fig. \ref{fig_discrimnt_von_lud}.
\begin{figure}
\caption{Molecular structure of 1,2-Dibromo-3,5-difluorobenzene, Hamiltonian parameters, and the relaxation parameters. In the table, the diagonal values indicate resonance offsets ($\omega_j/2\pi$); off-diagonal values $(J_{ij}
\label{mol_H}
\end{figure}
\section{Experiment}
We utilize the four spin-1/2 nuclei of 1,2-dibromo-3,5-difluorobenzene (DBDF) as our quantum register.
About 12 mg of DBDF was partially oriented in 600 $\upmu$l of liquid crystal MBBA. The molecular structure of DBDF and its NMR Hamiltonian parameters are shown in Fig. \ref{mol_H}.
The experiments were performed at 300 K on a 500 MHz Bruker UltraShield NMR spectrometer.
The secular part of the spin-Hamiltonian is of the form \cite{cavanagh},
\begin{eqnarray}
{\cal H}_0 &=& -\sum_{j=1}^4 \omega_j I_{jz}
+ 2\pi\sum_{j,k>j} (J_{jk}+2D_{jk}) I_{jz}I_{kz} \nonumber \\
&&+ 2\pi (J_{12}-D_{12})( I_{1x}I_{2x}+ I_{1y}I_{2y}),
\end{eqnarray}
where $\omega_j$, $J_{ij}$, and $D_{ij}$ are the resonance off-sets, indirect scalar coupling constants, and direct dipole-dipole coupling constants (Fig. \ref{mol_H}).
The strong-coupling term (i.e., the last term) is relevant only for (H$_1$, H$_2$) spins since $\vert\omega_1-\omega_2\vert < 2\pi \vert D_{12} \vert$.
We choose H$_1$, H$_2$ as ancilla (qubits 1, 2) and F$_3$, F$_4$ as the system (qubits 3, 4).
The NMR pulse diagram to implement the quantum circuit in Fig. \ref{fig_discrimnt_von_lud}(a) is shown
in Fig. \ref{fig_discrimnt_von_lud}(b). It begins with the initial state preparation. The thermal equilibrium state of the NMR system in the Zeeman eigenbasis under high-field, high-temperature, and secular approximation is given by \cite{Levitt_Spindynabook,cavanagh},
\begin{equation}
\rho_{\mathrm{eq}} =
\mathbbm{1}_{16}/16 + \sum_{j=1}^4 \epsilon_j I_{zj},
\label{rho_eq}
\end{equation}
where $\epsilon_j \sim 10^{-5}$ are the purity factors and the second term in the right hand side corresponds to the traceless deviation density matrix. The identity part is invariant under the unitary transformations and does not give rise to observable signal. Therefore only the deviation part is generally considered for both state preparation and characterization \cite{cory}.
The initial state of the quantum register assumed in the theory section, i.e., $\ket{\Phi_0}$ can be prepared by applying an Hadamard operator on each of the four qubits in a pure $\ket{0}$ state. However, in NMR, the preparation of such pure states is difficult and instead a pseudopure state is used \cite{cory}.
In our work, we utilize a technique based on preparing a pair of pseudopure states (POPS) \cite{POPS_Fung}. It involves inverting a single transition and subtracting the resulting spectrum from that of the thermal equilibrium. By inverting the transition $\ket{0000}$ to $\ket{0001}$ transition using a transition selective $\pi$ pulse, followed by Hadamard gates ({\textsf H}) on all the spins we obtain the POPS deviation density matrix:
\begin{eqnarray}
\rho_\mathrm{POPS} = \big(\proj{++++}-\proj{+++-}\big) \nonumber.
\end{eqnarray}
We then implemented the quantum circuit shown in Fig. \ref{fig_discrimnt_von_lud} (a) using the pulse sequence in Fig. \ref{fig_discrimnt_von_lud} (b).
As evident from circuit in Fig. \ref{fig_discrimnt_von_lud}, controls are designed to implement $U_{\mathrm{int}}$ (Eq. \ref{Luder_evolution}) since we intend to measure $A$. Whether to map it to $A'$ or not is left to the device.
The unitary operators $U_\mathrm{int}$ and $U_{a}^\dagger$ were realized by
bang-bang optimal control \cite{Bangbang_TSM}. Hadamard and tomography operations were only few hundred micro seconds long and had a simulated fidelity of about 0.99, when averaged over $\pm 10\%$ inhomogeneous RF fields.
The combined operation of $U_\mathrm{int}$ and $U_a^\dagger$ was about 17 ms in duration and had an average fidelity over $0.933$.
The intermediate measurement on ancilla was realized by applying strong pulse-field-gradients (PFG). By applying a $\pi_x$ pulse on the system spins in between two symmetrically spaced PFG pulses, we realize the selective dephasing of the ancilla spins (Fig. \ref{fig_discrimnt_von_lud} (b)). The central $\pi_x$ also refocuses all the system-ancilla coherent evolutions during the ancilla measurement. When averaged over the sample volume this process retains only the diagonal terms in the density matrix of the ancialla spins and thus simulates a projective measurement of ancilla. Setting the total duration of this process to $1/(J_{34}+2D_{34})$ also ensures refocusing of (F$_3$, F$_4$) interactions.
Finally, the density matrix of the system qubits was characterized using quantum state tomography. It involved nine independent measurements with different tomography pulses ({\textsf T}) (Fig. \ref{fig_discrimnt_von_lud} (b)) \cite{Tomography_Chuang447,Tomo_Soumya}.
The results of the quantum circuit (Fig. \ref{fig_discrimnt_von_lud}) on $\proj{++++}$ state by L\"uders and von Neumann devices are described in Eqs. \ref{rhoL} and \ref{rhoN} respectively.
For L\"uders measurement with the POPS input state
$\proj{++++}-\proj{+++-}$, the final deviation density matrix (in circuit \ref{fig_discrimnt_von_lud}) is expected to be
\begin{eqnarray}
\rho_L' = \mathbbm{1}_2 \otimes \sigma_x/2.
\label{Lud_finalstate}
\end{eqnarray}
On the other hand, for von Neumann measurement, the POPS input state leads to a maximally mixed final state with a null deviation density matrix $(\rho_N')$.
Fig. \ref{fig_tomo} compares the experimental results with the theoretically expected deviation density matrices. The correlation \cite{Correltn_fidlty_cory}
\begin{eqnarray}
C = \frac{\mathrm{Tr}[\rho'_{L}\rho'_\mathrm{exp}]}{\sqrt{\mathrm{Tr}[\rho_{L}^{\prime 2}]\mathrm{Tr}[\rho_\mathrm{exp}^{\prime 2}]}}
\label{C_corelatn}
\end{eqnarray}
between the theoretical ($\rho'_{L}$, Eq. \ref{Lud_finalstate}) and
the experimental ($\rho'_\mathrm{exp}$) deviation density matrices was 0.923. The reduction in the correlation is mainly due to coherent errors caused by imperfect unitary operators, fluctuations in the dipolar coupling constants due to temperature gradients over the sample volume, inhomogeneous RF fields, as well as due to decoherence.
The correlation expression in Eq. \ref{C_corelatn} is not directly applicable for the null-matrix $\rho'_N$. Therefore, we replace $\rho'_N$ with random traceless diagonal matrices, and obtained 0.28 as the upper bound for the correlation of $\rho'_\mathrm{exp}$ with $\rho'_N$.
Therefore we conclude that the experimental deviation density matrix is much closer to $\rho'_L$ (Eq. \ref{Lud_finalstate}), and strongly favors the L\"uders update rule.
\begin{figure}
\caption{Real (a) and imaginary (b) parts of the theoretically expected deviation density matrix for a L\"uders device ($\rho'_{L}
\label{fig_tomo}
\end{figure}
\section{Conclusions}
Quantum measurements, involving probabilistic state collapse and corresponding measurement outcomes, has always been mysterious. There have been attempts to deduce rules based on phenomenological observations. According to one of the earliest reduction rules, given by von Neumann, superposition in a degenerate subspace is destroyed by the measurement of the respective degenerate observable. This rule was later substantially modified by Gerhart L\"uders. The modified rule, which is most commonly used, implies that superpositions within the degenerate subspaces are preserved under such a measurement.
A protocol to determine whether a given measuring device is L\"uders or von Neumann was recently formulated by Hegerfeldt and Mayato \cite{Discriminate_von_Luder_protocol}. In this work, we have adapted this protocol for quantum information systems, and utilize ancilla qubits for performing a desired measurement on system qubits. Moreover, we describe an NMR experiment, with two system qubits and two ancilla qubits, to discriminate between L\"uders and von Neumann devices. Within the limitations of experimental NMR techniques, we found that the measurements are of L\"uders type.
There is a possibility that the above measurement is still of von Neumann type, if the chosen initial state happens to be a nondegenerate eigenstate of the actual system observable ($A'$). One way to rule out this possibility is by changing the initial state (Fig. \ref{HMprotocolfig}). However, it is also possible that the actual system observable is dynamic, in which case it is even more difficult to discriminate between L\"uders and von Neumann measurements. In this work we have not excluded these possibilities.
Nevertheless, the present work opens many interesting questions. For example, how can we build a von Neumann measuring device, or even an intermediate measuring device that partly breaks the degeneracy? More importantly, further research in this direction may throw some light on fundamental aspects of quantum measurement itself.
\end{document} |
\begin{document}
\begin{center} {Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment}
\end{center}
\begin{center}
{\small \textsc{Erika Rivero-Esquivel}\footnote{email: erika.rivero@correo.uady.mx}, \textsc{Eric \'Avila-Vales}\footnote{email: avila@uady.mx}, \textsc{Gerardo E. Garc\'ia-Almeida}\footnote{Corresponding author. email: galmeida@uady.mx Tel. (999) 942 31 40 Ext. 1108.}
}
\end{center}
\begin{center} {\small \sl $^{1,2,3}$ Facultad de Matem\'aticas, Universidad Aut\'onoma de Yucat\'an, \\ Anillo Perif\'erico Norte, Tablaje 13615, C.P. 97119, M\'erida, Yucat\'an, Mexico}
\end{center}
{\small \centerline{\bf Abstract}
\begin{quote}
We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.
\end{quote}}
\noindent {\bf Keywords}: Local Stability, Hopf Bifurcation, Global Stability, Backward Bifurcation
\section{Introduction}
Mathematical models that describe the dynamics of infectious diseases in
communities, regions and countries can contribute to have better approaches in
the disease control in epidemiology. Researchers always look for thresholds,
equilibria, periodic solutions, persistence and eradication of the disease.
For classical disease transmission models, it is common to have one endemic
equilibrium and that the basic reproduction number tells us that a disease is
persistent if is greater than 1,and dies out if is less than 1. This kind of
behaviour associates to forward bifurcation. However, there are epidemic
models with multiple endemic equilibria
\cite{hadeler1997,dushoff1998,driessche2000,brauer2004}, within these models
it can happen that a stable endemic equilibrium coexist with a disease free
equilibrium, this phenomenon is called backward bifurcation \cite{hadeler1995}.
In order to prevent and control the spread of infectious diseases like,
measles, tuberculosis and influenza, treatment is an important and effective
method. In classical epidemic models, the treatment rate of the infectious is
assumed to be proportional to the number of the infective individuals
\cite{anderson1991}. Therefore we need to investigate how the application of
treatment affects the dynamical behaviour of these diseases. In that direction
in \cite{wang2004}, Wang and Ruan, considered the removal rate
\[
T(I)=\left\{
\begin{array}
[c]{ll}
k, & \text{if }I>0\\
0, & \text{if }I=0.
\end{array}
\right.
\]
In the following model
\[
\begin{aligned}
\frac{dS}{dt}&= A-dS-\lambda SI, \\
\frac{dI}{dt}&=\lambda SI-(d+\gamma)I-T(I), \\
\frac{dR}{dt}&=\gamma I+T(I)-dR,
\end{aligned}
\]
where S, I , and R denote the numbers of the susceptible, infective and
recovered individuals at time t , respectively. The authors study the
stability of equilibria and prove the model exhibits Bogdanov-Takens
bifurcation, Hopf bifurcation and Homoclinic bifurcation. In \cite{zhangliu},
the authors introduce a saturated treatment
\[
T(I)=\frac{\beta I}{1+\alpha I}.
\]
A related work is \cite{zhoufan}, \cite{wancui}. \par
Hu, Ma and Ruan \cite{humaruan} studied the model
\begin{equation}
\begin{aligned} \dfrac{dS}{dt} =& bm(S+R) - \frac{\beta SI}{1+\alpha I} -bS+p\delta I\\ \dfrac{dI}{dt} =& \frac{\beta SI}{1+\alpha I}+(q\delta -\delta -\gamma)I - T(I)\\ \dfrac{dR}{dt} =& \gamma I-bR+bm'(S+R)+T(I) \label{hmr1}
\end{aligned}
\end{equation}
the basic assumptions for the model \eqref{hmr1} are, the total population
size at time $t$ is denoted by $N=S+I+R$. The newborns of $S$ and $R$ are
susceptible individuals, and the newborns of $I$ who are not vertically
infected are also susceptible individuals, $b$ denotes the death rate and
birth rate of susceptible and recovered individuals, $\delta $ denotes the death
rate and birth rate of infective individuals, $\gamma $ is the natural recovery rate
of infective individuals. $q$ ($q\leq1$) is the vertical transmission rate,
and note $p=1-q$, then $0\leq p\leq1$. Fraction $m^{\prime}$ of all newborns
with mothers in the susceptible and recovered classes are vaccinated and
appeared in the recovered class, while the remaining fraction, $m=1-m^{\prime
}$, appears in the susceptible class, the incidence rate is described by a
nonlinear function $\beta SI/(1+\alpha I)$, where $\beta$ is a positive
constant describing the infection rate and $\alpha$ is a nonnegative constant.
The treatment rate of the disease is
\[
T(I)=\left\{
\begin{array}
[c]{ll}
kI, & \text{if }0\leq I\leq I_{0},\\
u=kI_{0}, & \text{if }I>I_{0}
\end{array}
\right.
\]
where $I_{0}$ is the infective level at which the healthcare systems reaches
capacity.\newline
In this work we will extend model \eqref{hmr1} introducing the treatment rate
$\frac{\beta_{2} I}{1+\alpha_{2} I}$, where $\alpha_{2}$, $\beta_{2}>0$,
obtaining the following model
\begin{equation}
\label{hmr3}\begin{aligned} \dfrac{dS}{dt} =& bm(S+R) - \frac{\beta SI}{1+\alpha I} -bS+p\delta I\\
\dfrac{dI}{dt} =& \frac{\beta SI}{1+\alpha I}+(q\delta -\delta -\gamma)I - \frac{\beta_2 I}{1+\alpha_2 I}\\
\dfrac{dR}{dt} =& \gamma I-bR+bm'(S+R)+\frac{\beta_2 I}{1+\alpha_2 I}. \end{aligned}
\end{equation}
Because $\frac{dN}{dt}=0$, the total number of population $N$ is constant. For
convenience, it is assumed that $N=S+I+R=1$. By using $S+R=1-I$, the first two
equations of \eqref{hmr3} do not contain the variable $R$. Therefore, system
\eqref{hmr3} is equivalent to the following 2-dimensional system:
\begin{equation}
\label{ruanmod}\begin{aligned} \dfrac{dS}{dt} &=-\dfrac{\beta SI}{1+\alpha I}-bS+bm(1-I)+p\delta I \\ \dfrac{dI}{dt} &=\dfrac{\beta SI}{1+\alpha I}-p\delta I -\gamma I-\dfrac{\beta _2 I}{1+\alpha _2 I} . \end{aligned}
\end{equation}
The parameters in the model are described below:
\begin{itemize}
\item $S,I,R$ are the normalized susceptible, infected, and recovered
population, respectively, therefore it follows that $S,I,R\leq1.$
\item $b$ is a positive number representing the birth and death rate of
susceptible and recovered population.
\item $\delta$ is a positive number representing the birth and death rate of
infected population.
\item $\gamma$ is a positive number giving the natural recovery rate of
infected population.
\item $q$ is positive ($q\leq1$) representing the vertical transmission rate
(disease transmission from mother to son before or during birth). It is
assumed that descendents of the susceptible and recovered classes belong to
the susceptible class, in the same way to the fraction of the newborns of the
infected class not affected by vertical transmission.
\item $p=1-q$ therefore $0\leq p\leq1$.
\item $m^{\prime}$ is positive and it is the fraction of vaccinated newborns
from susceptible and recovered mothers and therefore belong to the recovered
class. $m=1-m^{\prime}\geq0$ is the rest of newborns, which belong to the
susceptible class.
\item $\beta$ is positive, representing the infection rate, $\alpha$ is a
positive saturation constant (In the model the incidence rate is given by the
nonlinear function $\frac{\beta SI}{1+\alpha I}$ ).
\item $\frac{\beta_{2}I}{1+\alpha_{2}I}$ is the treatment function, satisfying
$\lim_{I\rightarrow\infty}\frac{\beta_{2}I}{1+\alpha_{2}I}=\frac{\beta_{2}
}{\alpha_{2}},$ where $\alpha_{2},\beta_{2}>0.$
\end{itemize}
We note that if $\alpha_{2}=0$ the treatment becomes bilinear, case considered
in \cite{humaruan}, whereas if $\beta_{2}=0$ treatment is null, not being of
interest here. Therefore we will assume $\beta_{2},\alpha_{2}>0.$ \par
The paper in distributed as follows: in section 2 we compute the equilibria points and determine the conditions of its existence (as real values) and positivity, in section 3 we analyze the stability of the disease free equilibrium and endemic equilibria points in terms of value of $\mathcal{R}_{0} $ and the parameters of treatment function. Section 4 is dedicated to study Hopf bifurcation of the endemic equilibria points and section 5 shows discussion of all our results and we give some control measures that could be effective to eradicate the disease in each case. \par
Following \cite{humaruan} we define
\begin{equation}
\mathcal{R}_{0}:=\dfrac{\beta m}{\beta_{2}+p\delta+\gamma}.
\end{equation}
When $ \beta_2 =0 $, $\mathcal{R}_{0} $ reduces to
\begin{equation}
\mathcal{R}_{0} ^{*} = \dfrac{ \beta m}{ p \delta + \gamma },
\end{equation}
which is the basic reproduction number of model \eqref{ruanmod} without treatment.
\begin{lemma}
Given the initial conditions $S(0)=S_{0}>0,I(0)=I_{0}>0$, then the solution of
(\ref{ruanmod}) satisfies $S(t),I(t)>0\quad\forall t>0$ and $S(t)+I(t)\leq1$.
\label{teo1}
\end{lemma}
\begin{proof}
Take the solution $S(t),I(t)$ satisfying the initial conditions $S(0)=S_{0}
>0,I(0)=I_{0}>0$. Assume that the solution is not always positive, i.e., there
exists a $t_{0}$ such that $S(t_{0})\leq0$ or $I(t_{0})\leq0$. By Bolzano's
theorem there exists a $t_{1}\in(0,t_{0}]$ such that $S(t_{1})=0$ or
$I(t_{1})=0$, which can be written as $S(t_{1})I(t_{1})=0$ for some $t_{1}
\in(0,t_{0}]$. Let
\begin{equation}
t_{2}=\min\{t_{i},S(t_{i})I(t_{i})=0\}.
\end{equation}
Assume first that $S(t_{2})=0$, then $\frac{dS\left( t_{2}\right) }{dt}>0$
implying that $S$ is increasing at $t=t_{2}.$ Hence $S(t)$ is negative for
values of $t<t_{2}$ near $t_{2},$ a contradiction. Therefore $S(t)>0$ $\forall
t>0$ and we must have $I(t_{2})=0,$ implying $\frac{dI(t_{2})}{dt}=0.$ Note
that if for some $t\geq0$ $I(t)=0,$ then $\frac{dI(t)}{dt}=0.$ Then any
solution with $I(0)=I_{0}=0$ will satisfy $I(t)=0$ $\forall t>0.$ By
uniqueness of solutions this fact implies that if $I(0)=I_{0}>0,$ then $I(t)$
will remain positive for all $t>0.$ Therefore $I(t_{2})=0$ leads to a
contradiction. Hence both $S$ and $I$ are nonnegative for all $t>0.$ Finally,
adding both derivatives of $S(t)$ and $I(t)$ we get:
\begin{equation}
\dfrac{d(S+I)}{dt}=-bS+bm-bmI-\gamma I-\dfrac{\beta_{2}I}{1+\alpha_{2}I}
\end{equation}
Being $S,I\geq0$, if $S+I=1$ then $0\leq S\leq1$, $0\leq I\leq1$. Analyzing
the expression $-bS+bm-bmI$,
\[
-bS+bm-bmI=b(m-mI-S)=b(m-mI-1+I)=b(m-1+I(1-m)).
\]
Note that by the definition of the model parameters, $1-m=m^{\prime}\geq0$.
Knowing that $I\leq1,$ then
\begin{equation}
I(1-m)\leq1-m\Rightarrow I(1-m)+m-1\leq0.
\end{equation}
Therefore $-bS+bm-bmI\leq0$. Hence $\frac{d(S+I)}{dt}\leq0$ and $S+I$ is non
increasing along the line $S+I=1$, implying that $S+I\leq1$. Note also that
$S+I$ cannot be grater than 1, otherwise from $R=1-(S+I),$ $R$ would be
negative, a nonsense.
\end{proof}
\section{Existence and positivity of equilibria}
Assume that system (\ref{ruanmod}) has a constant solution $(S_{0},I_{0})$, then:
\begin{align}
-\dfrac{\beta S_{0}I_{0}}{1+\alpha I_{0}}-bS_{0}+bm(1-I_{0})+p\delta I_{0} &
=0\label{eqec1}\\
\dfrac{\beta S_{0}I_{0}}{1+\alpha I_{0}}-p\delta I_{0}-\gamma I_{0}
-\dfrac{\beta_{2}I_{0}}{1+\alpha_{2}I_{0}} & =0 \label{eqec2}
\end{align}
From (\ref{eqec1}) we obtain
\begin{equation}
S_{0}=\dfrac{(1+\alpha I_{0})(bm(1-I_{0})+p\delta I_{0})}{\beta
I_{0}+b(1+\alpha I_{0})}. \label{eqec3}
\end{equation}
And we get from (\ref{eqec2}):
\begin{align}
& I_{0}\left( \dfrac{\beta S_{0}}{1+\alpha I_{0}}-p\delta-\gamma-\dfrac
{\beta_{2}}{1+\alpha_{2}I_{0}}\right) =0\nonumber\\
& \Rightarrow I_{0}=0\quad \text{or}\quad\dfrac{\beta S_{0}}{1+\alpha I_{0}
}-p\delta-\gamma-\dfrac{\beta_{2}}{1+\alpha_{2}I_{0}}=0. \label{eqec4}
\end{align}
If $I_{0}=0$ then $S_{0}=m$, obtaining in that way the disease-free
equilibrium $E=(m,0)$.
\begin{theorem}
System (\ref{ruanmod}) has a positive disease-free equilibrium $E=(m,0)$.
\end{theorem}
In order to obtain positive solutions of system \ref{ruanmod} if $I_{0}\neq0$ then:
\begin{align}
& \dfrac{\beta S_{0}}{1+\alpha I_{0}}-p\delta-\gamma-\dfrac{\beta_{2}}
{1+\alpha_{2}I_{0}} =0\nonumber\\
& \Rightarrow S_{0} =\dfrac{1+\alpha I_{0}}{\beta}\left( p\delta
+\gamma+\dfrac{\beta_{2}}{1+\alpha_{2}I_{0}}\right) \label{eqec5}
\end{align}
We obtain the following quadratic equation:
\begin{equation}
AI_{0}^{2}+BI_{0}+C=0. \label{eqec7}
\end{equation}
Or
\begin{equation}
I_{0}^{2}+(B/A)I_{0}+C/A=0, \label{eqec8}
\end{equation}
where the coefficients are given by:
\begin{align}
A & =\alpha_{2}(\beta(\gamma+bm)+\alpha b(p\delta+\gamma))>0\nonumber\\
B & =\beta(\gamma+\beta_{2}+bm(1-\alpha_{2}))+b\alpha(p\delta+\gamma
+\beta_{2})+b\alpha_{2}(p\delta+\gamma)\nonumber\\
& =\beta(\gamma+\beta_{2}+bm-bm\alpha_{2})+b\alpha(1-\mathcal{R}_{0}
)(p\delta+\gamma+\beta_{2})+\beta mb\alpha+b\alpha_{2}(p\delta+\gamma
)\nonumber\\
C & =b(p\delta+\gamma+\beta_{2}-\beta m)=b(p\delta+\gamma+\beta
_{2})(1-\mathcal{R}_{0}).\label{abc}
\end{align}
Its roots are:
\begin{align}
I_{1} & =\dfrac{-B-\sqrt{B^{2}-4AC}}{2A}\nonumber\\
I_{2} & =\dfrac{-B+\sqrt{B^{2}-4AC}}{2A}. \label{I0}
\end{align}
And using these values in (\ref{eqec5}) we obtain its respective values
\begin{align}
S_{1} & =\dfrac{1+\alpha I_{1}}{\beta}\left( p\delta+\gamma+\dfrac
{\beta_{2}}{1+\alpha_{2}I_{1}}\right) \nonumber\\
S_{2} & =\dfrac{1+\alpha I_{2}}{\beta}\left( p\delta+\gamma+\dfrac
{\beta_{2}}{1+\alpha_{2}I_{2}}\right) . \label{eqec14}
\end{align}
Then our candidate for endemic equilibria are $E_{1}=(S_{1},I_{1})$, $E_{2}
=(S_{2},I_{2})$.
Note that $C=0$ if and only if $\mathcal{R}_{0}=1$, $C>0$ if and only if
$\mathcal{R}_{0}<1,$ and $C<0$ if and only if $\mathcal{R}_{0}>1$ .
\begin{figure}
\caption{Location of the sets
$A_{1}
\label{fig1}
\end{figure}
For $ \mathcal{R}_{0} ^{*}>1 $ we define the following sets:
\begin{align}
A_{1} & =\{(\beta_{2},\alpha_{2}): \beta_{2}>0,0<\alpha_{2}\leq \alpha_2^{0} ,\nonumber\\
A_{2} & =\{(\beta_{2},\alpha_{2}): \beta_{2}\geq g(\alpha_{2}),\alpha
_{2}> \alpha_2^{0}
>0\},\nonumber\\
A_{3} & =\{(\beta_{2},\alpha_{2}): 0<\beta_{2}<g(\alpha_{2}),\alpha_{2}
> \alpha_2^{0} >0\}.
\end{align}
Where
$$ \alpha_2^{0} = \frac
{-\beta(mb\alpha+\gamma+bm)}{b(p\delta+\gamma-\beta m)}$$
\[
g(\alpha_{2})=-\frac{1}{\beta}(b\alpha_{2}(p\delta+\gamma-\beta m)+\beta
(\gamma+bm+mb\alpha)).
\]
Define :
\begin{align}
P_{1} & = 1 + \dfrac{1}{b \alpha( p \delta+ \gamma+ \beta_{2})} [ \beta(
\gamma+ \beta_{2} + bm - bm \alpha_{2} ) + \beta m b \alpha+ b \alpha_{2} ( p
\delta+ \gamma) ]\nonumber\\
R_{0}^{+} & = 1 - \dfrac{1}{b \alpha^{2} ( p \delta+ \gamma+ \beta_{2}
)}\nonumber\\
& \left[ \sqrt{- \beta\alpha( bm \alpha+ \beta_{2} + \gamma+ bm - \alpha
_{2}bm ) + \beta\alpha_{2} ( \gamma+ bm ) } - \sqrt{ \alpha_{2} ( \beta\gamma+
\beta b m + \alpha b p \delta+ \alpha b \gamma) } \right] ^{2} .
\end{align}
Figure \ref{fig1} shows the location of these sets.
\begin{theorem}
If $\mathcal{R}_{0}>1$ the system (\ref{ruanmod}) has a unique (positive)
endemic equilibrium $E_{2}$. \label{teo10}
\end{theorem}
\begin{proof}
If $\mathcal{R}_{0}>1$ then $C<0$, then using Routh Hurwitz criterion for $n=2$, the
quadratic equation has two real roots with different sign, $I_{1}$ and $I_{2}
$, where $I_{1}<I_{2}$. Hence there exists a unique
positive endemic equilibrium $E_{2}=(S_{2},I_{2})$.
\end{proof}
\begin{theorem}
Let $0<\mathcal{R}_{0}\leq1$. For system (\ref{ruanmod}), if $ \mathcal{R}_{0} ^{*} \leq 1 $ then there are no positive endemic equilibria. Otherwise, if
$ \mathcal{R}_{0} ^{*}>1 $ the following propositions hold:
\begin{enumerate}
\item If $\mathcal{R}_{0}=1$ and $(\beta_{2},\alpha_{2})\in A_{3}$ the system
(\ref{ruanmod}) has a unique positive endemic equilibrium $E_2=(S_2
,I_2)$, where
\[
I_2=-B/A,\quad S_2=\dfrac{1+\alpha I_2}{\beta}\left( p\delta
+\gamma+\dfrac{\beta_{2}}{1+\alpha_{2}I_2}\right) .
\]
.
\item If $\max\{P_{1},R_{0}^{+}\}<\mathcal{R}_{0}<1$ and $(\beta_{2}
,\alpha_{2})\in A_{3},$ the system (\ref{ruanmod}) has a pair of positive
endemic equilibria $E_{1},E_{2}$.
\item If $1>\mathcal{R}_{0}=R_{0}^{+}>P_{1}$ and $(\beta_{2},\alpha_{2})\in
A_{3},$ the system (\ref{ruanmod}) has a unique positive endemic equilibrium
$E_{1}=E_{2}$.
\item If $1>\mathcal{R}_{0}=P_{1}$ and $(\beta_{2},\alpha_{2})\in A_{3},$ the
system (\ref{ruanmod}) has no positive endemic equilibria.
\item If $0<\mathcal{R}_{0}\leq1$ and $(\beta_{2},\alpha_{2})\in A_{1}\cup
A_{2},$ the system (\ref{ruanmod}) has no positive endemic equilibria.
\item If $(\beta_{2},\alpha_{2})\in A_{3}$ and $0<\mathcal{R}_{0}<\max
(R_{0}^{+},P_{1})<1,$ then there are no positive endemic equilibria.
\end{enumerate}
\label{teo4}
\end{theorem}
\begin{proof}
If $0<\mathcal{R}_{0}\leq1,$ then $C\geq0$, so the roots of the
equation $AI^{2}+BI+C=0$ are not real with different sign, but real with equal
signs, complex conjugate or some of them are zero. If endemic equilibria exist
and are positive, it is necessary that $B<0$. After some calculations we can see that:
\begin{align}
& B<0 \Leftrightarrow\mathcal{R}_{0}>1+\dfrac{\beta(\gamma+\beta_{2}
+bm-bm\alpha_{2})+\beta mb\alpha+b\alpha_{2}(p\delta+\gamma)}{b\alpha
(p\delta+\gamma+\beta_{2})}:=P_{1}.
\end{align}
From the assumption that $\mathcal{R}_{0}\leq1$ then $P_{1}<1$, hence the
expression $\beta(\gamma+\beta_{2}+bm-bm\alpha_{2})+\beta mb\alpha+b\alpha
_{2}(p\delta+\gamma)$ must be negative, this happens if and only if
\begin{gather}
\beta_{2}<-\frac{1}{\beta}(b\alpha_{2}(p\delta+\gamma-\beta
m)+\beta(\gamma+bm+mb\alpha))=g(\alpha_{2}).
\end{gather}
If $ \mathcal{R}_{0} ^{*} \leq 1 $ then $-\frac{1}{\beta}(b\alpha_{2}
(p\delta+\gamma-\beta m)+\beta(\gamma+bm+mb\alpha))<0$ and it is not possible
to find a value of $\beta_{2}$ fulfilling the previous inequality, therefore
there are no positive endemic equilibria. \par
Now, if $ \mathcal{R}_{0} ^{*}>1 $ we have that.
\begin{enumerate}
\item If $\mathcal{R}_{0}=1$ then $C=0$ and the equation (\ref{eqec7}) is
transformed into
\begin{equation}
AI_{0}^{2}+BI_{0}=0,
\end{equation}
with $A>0$. Its roots are $I_{1}=0$ and $I_2=-B/A,$ and there
exists a unique endemic equilibrium that is positive if and only if $B<0$, that is given by $E_2=(S_2,I_2)$, where
\begin{align}
I_2 & =-B/A\nonumber\\
S_2 & =\dfrac{1+\alpha I_2}{\beta}\left( p\delta+\gamma+\dfrac{\beta
_{2}}{1+\alpha_{2}I_2}\right) . \label{eqec13}
\end{align}
Note that if $\alpha_{2}> \alpha_2^{0} \quad
\text{and} \quad \mathcal{R}_{0} ^{*}>1 $ then $g(\alpha_{2})>0$.
Hence $A_{3}$ is nonempty and its elements satisfy $B<0$, therefore if
$(\beta_{2},\alpha_{2})\in A_{3}$ there exists a unique positive endemic
equilibrium $E_2$.
\item If $\mathcal{R}_{0}<1$ then $C>0$ and the roots of the quadratic equation
for $I_{0}$ must be real of
equal sign or complex conjugate. By the previous part we know that if
$(\beta_{2},\alpha_{2})\in A_{3}$ then $P_{1}<1$, moreover if $\mathcal{R}
_{0}>P_{1}$ then $B<0$ and therefore both roots must have positive real part.
Finally, to assure that equilibria are both real, we demand that $\Delta\geq0$
. Computing $\Delta$:
\begin{align}
\Delta & =B^{2}-4AC\nonumber\\
& =A_{2}\mathcal{R}_{0}^{2}+B_{2}\mathcal{R}_{0}+C_{2}=\Delta(\mathcal{R}
_{0}),
\end{align}
where:
\begin{align}
A_2 &= {\alpha}^{2}{b}^{2}\left( p\delta+\gamma+\mathit{\beta_{2}}\right)
^{2}\\
B_2 &= -2\,[\beta\,\left( \gamma+\mathit{\beta_{2}}+bm\left( 1-\mathit{\alpha
_{2}}\right) \right) +\alpha\,b\left( p\delta+\gamma+\mathit{\beta_{2}
}\right) +\beta\,mb\alpha\nonumber\\
& +b\mathit{\alpha_{2}}\,\left( p\delta+\gamma\right) ]\alpha\,b\left(
p\delta+\gamma+\mathit{\beta_{2}}\right) +4\,\mathit{\alpha_{2}}\,\left(
\beta\,\left( \gamma+bm\right) \alpha\,b\left( p\delta+\gamma\right)
\right) \nonumber \\
& b\left( p\delta+\gamma+\mathit{\beta_{2}}\right) \\
C_2 &= \left( \beta\,\left( \gamma+\mathit{\beta_{2}}+bm\left(
1-\mathit{\alpha_{2}}\right) \right) +\alpha\,b\left( p\delta
+\gamma+\mathit{\beta_{2}}\right) +\beta\,mb\alpha+b\mathit{\alpha_{2}
}\,\left( p\delta+\gamma\right) \right) ^{2}\nonumber\\
& -4\,\mathit{\alpha_{2}}\,\left( \beta\,\left( \gamma+bm\right)
+\alpha\,b\left( p\delta+\gamma\right) \right) b\left( p\delta
+\gamma+\mathit{\beta_{2}}\right).
\end{align}
The previous expression is a quadratic function of $\mathcal{R}_{0}$. To
establish the region where $\Delta\geq0$, it is necessary to know how the roots
of $\Delta(\mathcal{R}_{0})$ behave. The discriminant of the quadratic
function $\Delta(\mathcal{R}_{0})$ is
\begin{align}
\Delta_{2} & =-16\,\mathit{\alpha_{2}}\,{b}^{2}\beta\,\left( p\delta
+\gamma+\mathit{\beta_{2}}\right) ^{2}\left( \beta\,\gamma+\beta
\,bm+\alpha\,bp\delta+\alpha\,b\gamma\right) \nonumber\\
& \left( \alpha(\alpha bm+\beta_{2}+\gamma+bm)-\alpha_{2}(\gamma+bm+\alpha
bm)\right) . \label{ec15}
\end{align}
If we assume that $\Delta_{2}<0,$ then $\alpha_{2}<\frac{\alpha(bm\alpha
-\beta_{2}+\gamma+bm)}{\gamma+bm+\alpha bm}$ and in this case we have that:
\begin{align}
\gamma+\beta_{2}+bm-bm\alpha_{2}+bm\alpha>\frac{2\beta_{2}\alpha
bm+(\gamma+bm)(\gamma+\beta_{2}+bm+bm\alpha)}{\gamma+bm+\alpha bm}>0.
\end{align}
So we get that $P_{1}>1>\mathcal{R}_{0}$, which is a contradiction with
the assumption in this part, therefore $\Delta_{2}\geq0$ and in consequence
$\Delta(\mathcal{R}_{0})$ has two real roots,
\begin{align}
R_{0}^{-} & =\dfrac{-B_{2}-\sqrt{\Delta_{2}}}{2A_{2}}\nonumber\\
& =1-\dfrac{1}{b\alpha^{2}(p\delta+\gamma+\beta_{2})}[\sqrt{-\beta
(\alpha(bm\alpha+\beta_{2}+\gamma+bm-bm\alpha_{2})-\alpha_{2}(\gamma
+bm))}\nonumber\\
& +\sqrt{\alpha_{2}(\beta(\gamma+bm)+\alpha b(p\delta+\gamma))}
]^{2},\nonumber\\
R_{0}^{+} & =\dfrac{-B_{2}+\sqrt{\Delta_{2}}}{2A_{2}}\nonumber\\
& 1-\dfrac{1}{b\alpha^{2}(p\delta+\gamma+\beta_{2})}[\sqrt{-\beta
(\alpha(bm\alpha+\beta_{2}+\gamma+bm-bm\alpha_{2})-\alpha_{2}(\gamma
+bm))}\nonumber\\
& -\sqrt{\alpha_{2}(\beta(\gamma+bm)+\alpha b(p\delta+\gamma))}]^{2}.
\end{align}
Note that due to the positivity of $\Delta_{2}$ and (\ref{ec15}),
we have that
$$-\beta(\alpha(bm\alpha+\beta_{2}+\gamma+bm-bm\alpha
_{2})-\alpha_{2}(\gamma+bm))$$
is positive, allowing its roots to be well
defined. Analyzing the derivative of
$\Delta(\mathcal{R}_{0})$ we have that $$\Delta^{\prime}(R_{0}^{+}
)=\sqrt{\Delta_{2}}>0 \quad \text{and} \quad \Delta^{\prime}(R_{0}^{-})=-\sqrt{\Delta_{2}}<0,$$
moreover $R_{0}^{-}<R_{0}^{+}$ making $\Delta$ positive for $\mathcal{R}
_{0}>R_{0}^{+}$ or $\mathcal{R}_{0}<R_{0}^{-}$. Nevertheless $$R_{0}
^{-}=1+\frac{1}{b\alpha(p\delta+\gamma+\beta_{2})}(\beta(\gamma+\beta
_{2}+bm-bm\alpha_{2}+bm \alpha))-\epsilon,$$ while $$P_{1}=1+\frac
{1}{b\alpha(p\delta+\gamma+\beta_{2})}(\beta(\gamma+\beta_{2}+bm-bm\alpha
_{2}+bm \alpha))+\epsilon_{2},$$
with $ \epsilon, \epsilon_{2}>0$, making $R_{0}^{-}<P_{1}<R_{0}$.
Therefore for $\mathcal{R}_{0}>\max(P_{1},R_{0}^{+})$, we have that there exists
two positive endemic equilibria
$E_{1},E_{2}$, proving this part.
\item If $(\beta_{2},\alpha_{2})\in A_{3}$ then $P_{1}<1$. If $1>\mathcal{R}
_{0}>P_{1},$ then we have that $B<0$ and $C>0$, therefore we have a pair of
roots of the quadratic for $I$ with positive real part. In the previous part
it was proven that for $P_{1}<1$ the discriminant $\Delta_{2}\geq0$ and both
roots $R_{0}^{+},R_{0}^{-}$ are real and less than one. If $\mathcal{R}
_{0}=R_{0}^{+}$ then $\Delta=0$ and both roots are fused in one $I_{1}
=-B/2A=I_{2}$. Therefore we have a unique positive endemic equilibrium
$E_{1}=E_{2}$.
\item If $(\beta_{2},\alpha_{2})\in A_{3}$ then $P_{1}<1$. If $\mathcal{R}
_{0}=P_{1}<1$ then $C>0,$ implying that the roots are complex conjugate or
real of the same sign. Being $\mathcal{R}_{0}=P_{1}$ then $B=0,$ implying that
both roots have real part equal to zero, therefore there are no positive
endemic equilibria.
\item If $0<\mathcal{R}_{0}\leq1$ and $(\beta_{2},\alpha_{2})\in A_{1}\cup
A_{2}$ then $P_{1}\geq1$, therefore $\mathcal{R}_{0}\leq P_{1}$ , $B\geq0,$
and $C\geq0$. Hence there are two roots with real part zero or negative, which
are not positive equilibria.
\item If $(\beta_{2},\alpha_{2})\in A_{3}$ we have that $P_{1}<1$ and the
roots of the discriminant $R_{0}^{+},R_{0}^{-}$ are real, in addition that
$R_{0}^{-}<P_{1}$ and $R_{0}^{+}<1$ by definition of this case. If
$0<\mathcal{R}_{0}<\max\{R_{0}^{+},P_{1}\}<1,$ then $C>0$ and the roots
$I_{2},I_{3}$ are complex conjugate or real with the same sign. If
$\mathcal{R}_{0}<P_{1}$ then $B>0,$ and the roots have negative real part, so
there are not positive endemic equilibria. If $0<\mathcal{R}_{0}<R_{0}^{+}$
and $\mathcal{R}_{0}>R_{0}^{-}$, then $\Delta<0$ and the roots are complex
conjugate, therefore there is not real endemic equilibria. If $0<\mathcal{R}
_{0}<R_{0}^{+}$ and $\mathcal{R}_{0}\leq R_{0}^{-}<P_{1},$ then it reduces to
the first case in which there are not positive endemic equilibria.
\end{enumerate}
\begin{figure}
\caption{Graph of
$\mathcal{R}
\label{fig12}
\end{figure}
\begin{figure}
\caption{Graph of
$\mathcal{R}
\label{fig2}
\end{figure}
Theorem \ref{teo4} gives us a complete scenario of the existence of endemic equilibria. When $\mathcal{R}_{0} ^{*} \leq 1 $ we have that $\mathcal{R}_{0} <1 $, it follows from the fact that $\mathcal{R}_{0} < \mathcal{R}_{0} ^{*}$ whenever $ \beta_2>0 $; then system \ref{ruanmod} has only a disease free equilibrium and no endemic equilibria. \par
Otherwise, when $\mathcal{R}_{0} ^{*}>1$ . If $ ( \beta_2, \alpha_2 ) \in A_1 \cup A_2 $ then we have no endemic equilibria for $0<\mathcal{R}_{0}<1 $ and a unique endemic equilibria $E_2$ when $\mathcal{R}_{0} >1$, so there exists a forward bifurcation in $\mathcal{R}_{0} =1$ from the disease free equilibrium to $E_2$ (see figure \ref{fig12} ). If $ ( \beta_2, \alpha_2 ) \in A_3$ there exist two positive endemic equilibria whenever $\max\{P_{1},R_{0}^{+}\}<\mathcal{R}_{0}<1$ ( $P_1$ and $\mathcal{R}_{0} ^{+}$ depend on $ \beta_2 $), we can observe the backward bifurcation of the equilibrium $E$ to two endemic equilibria (see figure \ref{fig2} ).
As an immediate consequence of the previous theorem we have that if
$\mathcal{R}_{0}>1$ there exists a unique positive endemic equilibrium, while
if $\mathcal{R}_{0}<1$ and the conditions of the second part are fulfilled,
there exist two positive endemic equilibria. Hence we have the following corollary:
\end{proof}
\begin{corollary}
If $\mathcal{R}_{0}=1$, $ \mathcal{R}_{0} ^{*}>1 $ and $(\beta,\alpha_{2})\in A_{3}$, system
(\ref{ruanmod}) has a backward bifurcation of the disease-free equilibrium $E$.
\end{corollary}
\begin{proof}
First we note that if $(\beta_{2},\alpha_{2})\in A_{3}$ then $R_{0}^{+}$ is
real less than one and $P_{1}<1$, therefore we can find a neighborhood of
points in the interval $(\max\{R_{0}^{+},P_{1}\},1)$. By case 2 of
theorem, if $\mathcal{R}_{0}$ lies in this neighborhood there exist two
positive endemic equilibria $E_{1},E_{2}$; for $\mathcal{R}_{0}=1$ there
exists a unique positive endemic equilibrium $E_2$, while the other endemic
equilibrium becomes zero. Finally for $\mathcal{R}_{0}>1$ there exists a
unique positive endemic equilibrium as the zero "endemic" equilibrium becomes
negative.
\end{proof}
\section{Characteristic Equation and Stability}
The characteristic equation of the linearization of system (\ref{ruanmod}) in
the equilibrium $(S_{0},I_{0})$ is given by:
\begin{equation}
\det(DF-\lambda I),
\end{equation}
where
\begin{equation}
DF=\left(
\begin{matrix}
\frac{\partial f_{1}}{\partial S} & \frac{\partial f_{1}}{\partial I}\\
\frac{\partial f_{2}}{\partial S} & \frac{\partial f_{2}}{\partial I}
\end{matrix}
\right) .
\end{equation}
Matrix is evaluated in the equilibrium $(S_{0},I_{0})$. Functions
$f_{1},f_{2}$ are the following:
\begin{align}
f_{1} & =-\dfrac{\beta SI}{1+\alpha I}-bS+bm(1-I)+p\delta I\\
f_{2} & =\dfrac{\beta SI}{1+\alpha I}-p\delta I-\gamma I-\dfrac{\beta_{2}
I}{1+\alpha_{2}I}.
\end{align}
Computing the matrix $DF$ we obtain:
\begin{equation}
DF (S,I) = \left(
\begin{matrix}
\dfrac{-\beta I}{1 + \alpha I } - b & \dfrac{-\beta S }{ (1+ \alpha I)^{2} }
-bm + p \delta\\
\dfrac{\beta I}{1 + \alpha I } & \dfrac{\beta S}{ (1+ \alpha I)^{2} } - p
\delta- \gamma- \dfrac{ \beta_{2} }{(1+ \alpha_{2} I)^{2}}
\end{matrix}
\right).
\end{equation}
\subsection{Stability of \textit{disease free }equilibrium}
For the disease free equilibrium $E=(m,0)$ the Jacobian matrix is:
\[
DF(m,0)=\left(
\begin{matrix}
-b & -\beta m-bm+p\delta\\
0 & \beta m-p\delta-\gamma-\beta_{2}
\end{matrix}
\right).
\]
\begin{theorem}
If $\mathcal{R}_{0}<1$ then the equilibrium $E=(m,0)$ of model (\ref{ruanmod})
is locally asymptotically stable, while if $\mathcal{R}_{0}>1$ then it is
unstable. \label{teo5}
\end{theorem}
\begin{proof}
The characteristic equation for the equilibrium $E$ is given by
\begin{align}
P(\lambda) & =\det(DF(m,0)-\lambda I_{2x2})\nonumber\\
& =\det\left(
\begin{matrix}
-b-\lambda & -\beta m-bm+p\delta\\
0 & \beta m-p\delta-\gamma-\beta_{2}-\lambda
\end{matrix}
\right) \nonumber\\
& =(-b-\lambda)(\beta m-p\delta-\gamma-\beta_{2}-\lambda). \label{carec1}
\end{align}
The equation (\ref{carec1}) has two real roots $\lambda_{1}=-b$ and
$\lambda_{2}=\beta m-p\delta-\gamma-\beta_{2}$. By Hartman-Grobman's theorem,
if the roots of (\ref{carec1}) have non-zero real part then the solutions of
system (\ref{ruanmod}) and its linearization are qualitatively equivalent. If
both roots have negative real part then the equilibrium $E$ is locally
asymptotically stable, whilst if any of the roots has positive real part the
equilibrium is unstable. Clearly $\lambda_{1}<0,$ but $\lambda_{2}<0$ if and
only if
\[
\beta m-p\delta-\gamma<\beta_{2},
\]
if and only if $\mathcal{R}_{0}<1$.
\end{proof}
According to the previous theorem and theorem \ref{teo4} we obtain the
following result for the global stability of equilibrium $E$\ :
\begin{theorem}
If $0<\mathcal{R}_{0}<1$ and one of the following conditions holds:
\begin{itemize}
\item $ \mathcal{R}_{0} ^{*} \leq 1 $.
\item $\mathcal{R}_{0}=P_{1}$ and $(\beta_{2},\alpha_{2})\in A_{3}$.
\item $(\beta_{2}, \alpha_{2}) \in A_{1} \cup A_{2} $.
\item $(\beta_{2},\alpha_{2})\in A_{3}$ and $0<\mathcal{R}_{0}<\max\{R_{0}
^{+},P_{1}\}$.
\end{itemize}
Then equilibrium $E$ of system (\ref{ruanmod}) is globally asymptoticaly stable.
\label{teo6}
\end{theorem}
\begin{proof}
If $0<\mathcal{R}_{0}<1$ then by theorem \ref{teo5} the equilibrium $E$ is
locally asymptotically stable. If any of the given conditions holds then by
theorem \ref{teo4} there are no endemic equilibria in the region
$D=\{S(t),I(t)\geq0\quad\forall t>0,\quad S(t)+I(t)\leq1\}$, which it was
proven to be positively invariant in theorem \ref{teo1}. By \cite{perko} (page
245) any solution of (\ref{ruanmod}) starting in $D$ must approach either an
equilibrium or a closed orbit in $D$. By \cite{kelley} (theorem 3.41) if the
solution path approaches a closed orbit, then this closed orbit must enclose
an equilibrium. Nevertheless, the only equilibrium existing in $D$ is $E$ and
it is located in the boundary of $D$, therefore there is no closed orbit
enclosing it, totally contained in $D$. Hence any solution of system
(\ref{ruanmod}) with initial conditions in $D$ must approach the point $E$ as
$t$ tends to infinity. \begin{figure}
\caption{Global stability of
equilibrium $E$.}
\label{fig3}
\end{figure}
\end{proof}
\begin{example}
Take the following values for the parameters: $\alpha=0.4,\alpha_{2}
=10,\beta=0.2,b=0.2,\gamma=0.01,\delta=0.01,p=0.02,m=0.3,\beta_{2}=0.1$.
Equilibrium $E=(0.3,0)$, $\mathcal{R}_{0}=0.5445<1$. By theorem \ref{teo5}, $E$
is locally asymptotically stable, $ \alpha_2^{0} =7.42<\alpha_{2}$ and $g(\alpha_{2})=-0.1864<\beta_{2}$,
therefore $(\beta_{2},\alpha_{2})\in A_{2}$. By theorem \ref{teo4} there are
no positive endemic equilibria. Finally by theorem \ref{teo6} we have that $E$
is globally stable. See figure \ref{fig3}.
\end{example}
\begin{theorem}
If $\mathcal{R}_{0}=1$ and $\beta_{2}\neq g(\alpha_{2})$ then equilibrium $E$
is a saddle point. Moreover, if $(\beta_{2},\alpha_{2})\in A_{1}\cup A_{2}$
the region $D$ is contained in the stable manifold of $E$. \label{teo7}
\end{theorem}
\begin{proof}
If $\mathcal{R}_{0}=1$ one of the eigenvalues of the Jacobian matrix of the
system is zero, hence we cannot apply Hartman-Grobman's theorem. In order to
establish the stability of equilibrium $E$ we apply central manifold theory.
Making the change of variables, $\hat{S}=S-m$, $ \hat{I}=I $, we obtain the equivalent system
\begin{align}
\dfrac{d\hat{S}}{dt} & =-\dfrac{\beta(\hat{S}+m) \hat{I} }{1+\alpha \hat{I} }-b\hat
{S}-bm \hat{I} +p\delta \hat{I} \nonumber\\
\dfrac{d \hat{I} }{dt} & =\dfrac{\beta(\hat{S}+m) \hat{I} }{1+\alpha \hat{I} }-p\delta \hat{I} -\gamma
\hat{I} -\dfrac{\beta_{2} \hat{I} }{1+\alpha_{2} \hat{I} }. \label{carac6}
\end{align}
Because $ \hat{I}=I $ we ignore the hat and use only $I$. This new system has an equilibrium in $\hat{E}=(0,0)$ and its Jacobian matrix
in that point is
\begin{equation}
DF(m,0)=\left(
\begin{matrix}
-b & - \beta m - bm + p \delta\\
0 & 0
\end{matrix}
\right) .\label{eqst1}
\end{equation}
Using change of variables $S=u-{\frac{\left( \gamma+\mathit{beta2}+bm\right) v}{b}},I=v$
and $\beta m=p\delta+\gamma+\beta_{2}$ we obtain the equivalent system (see appendix A):
\begin{align}
\frac{dv}{dt} & =0u+f(v,u)\nonumber\\
\frac{du}{dt} & =-bu+g(v,u), \label{eqst2}
\end{align}
where $f$ and $g$ are defined in Appendix A. \par
By \cite{carr}, system \eqref{ruanmod} has a center manifold of the form
$u=h(v)$ and the flow in the
center manifold ( and therefore in the system ) is given by the equation
\[
v^{\prime}=f(v,h(v))\sim f(v,\phi(v)),
\]
where $h(v) = a_0 v^{2} + a_1 v^{3} + O(v^{4}) $, and $a_i$'s are given in Appendix A. Expanding the Taylor series of $f$ we obtain the flow equation
\begin{align}
v^{\prime} & =-{\frac{{b}^{3}\beta\,m+{b}^{2}{\beta}^{2}m+{b}^{3}
\gamma\,\mathit{\alpha_{2}}-{b}^{2}\beta\,p\delta+{b}^{3}\alpha\,\beta
\,m+{b}^{3}p\delta\,\mathit{\alpha_{2}}-{b}^{3}\beta\,m\mathit{\alpha_{2}}
}{{b}^{3}}}v^{2}+O(v^{3})\nonumber\\
& =Hv^{2}+O(v^{3}).
\end{align}
Therefore the dynamics of solutions near the equilibrium $\hat{E}=(0,0)$
is given by the quadratic term, whenever this term is not zero. We note that
$H=0$ if and only if
\begin{equation}
\alpha_{2}=\frac{-\beta(bm+\beta m-p\delta+b\alpha m)}{b(p\delta+\gamma-\beta
m)}.
\end{equation}
Substituting again $\mathcal{R}_{0}=1$, expressed as $\beta m=p\delta
+\gamma+\beta_{2}$, we obtain $H=0$ if and only if $\beta_{2}=g(\alpha_{2})$. \par
If $(\beta_{2},\alpha_{2})\in A_{3}$ then $H>0$. $v^{\prime}>0$ for $v\neq0$.
If $(\beta_{2},\alpha_{2})\in A_{1}\cup A_{2}$ then $H<0$, $v^{\prime}<0$ for
$v\neq0$. In both cases $\hat{E}$ is a saddle point. Moreover, if $(\beta
_{2},\alpha_{2})\in A_{1}\cup A_{2}$ then $H<0$ and $v^{\prime}<0$ for $v>0$.
Recalling $v(t)=I(t)$ we have under this assumption that $I^{\prime}(t)<0$ for
$I>0$ therefore $I(t)\rightarrow0^{+}$, while as $v_{1}=(1,0)$ is the stable
direction of the point $E$ then $S(t)\rightarrow0$, therefore the solutions in
the region $D$ approach the equilibrium $E$ as $t\rightarrow\infty$.
\begin{figure}
\caption{Phase plane of the system
for $\mathcal{R}
\label{fig9}
\end{figure}
\end{proof}
\begin{example}
Take the following values for the parameters: $\beta=0.2,\alpha=0.4,\delta
=0.01,\gamma=0.01,\alpha_{2}=10,m=0.3,p=0.02,b=0.2, \beta_2 = 0.0498$. In this case
$\mathcal{R}_{0}=1$, $ \alpha_2^{0} =1.8876$
and $g(\alpha_{2})=0.4040$, hence $(\beta_{2},\alpha_{2})\in A_{3}$.
By the first case of theorem \ref{teo4} the system has a unique endemic
equilibrium in $S_2=0.11210,I_2=0.4781$. By theorem \ref{teo7} the
equilibrium $E$ is a saddle point, see figure \ref{fig9}.
\end{example}
\begin{figure}
\caption{Phase plane for
$\mathcal{R}
\label{fig10}
\end{figure}
\begin{example}
If we take the same values as in the previous example except $\alpha
_{2}=2$, then
$g(\alpha_{2})=0.0056<\beta_{2}$, hence $(\beta_{2},\alpha_{2})\in A_{2}$. By
theorem \ref{teo4} the system has no endemic equilibria, and by theorem
\ref{teo7} the point $E$ is a saddle point. Moreover, the region $D$ is
totally contained in the stable manifold, see figure \ref{fig10}.
\end{example}
\subsection{Stability of endemic equilibria}
The general form of the Jacobian matrix is
\begin{equation}
DF=\left(
\begin{matrix}
-\dfrac{\beta I}{1+\alpha I}-b & & -\dfrac{\beta S}{(1+\alpha I)^{2}
}-bm+p\delta\\
\dfrac{\beta I}{1+\alpha I} & & \dfrac{\beta S}{(1+\alpha I)^{2}}
-p\delta-\gamma-\dfrac{\beta_{2}}{(1+\alpha_{2}I)^{2}}
\end{matrix}
\right) .
\end{equation}
Therefore the characteristic equation for an endemic equilibrium is
\begin{align}
P(\lambda) & =\left( -\dfrac{\beta I}{1+\alpha I}-b-\lambda\right) \left(
\dfrac{\beta S}{(1+\alpha I)^{2}}-p\delta-\gamma-\dfrac{\beta_{2}}
{(1+\alpha_{2}I)^{2}}-\lambda\right) \nonumber\\
- & \left( \dfrac{\beta I}{1+\alpha I}\right) \left( -\dfrac{\beta
S}{(1+\alpha I)^{2}}-bm+p\delta\right).
\end{align}
If we denote by
\begin{align}
C_{I} & :=\frac{\beta I}{1+\alpha I}\\
C_{S} & :=\frac{\beta S}{(1+\alpha I)^{2}}\\
D_{I} & :=\frac{\beta_{2}}{(1+\alpha_{2}I)^{2}}.
\end{align}
Then the characteristic polynomial is rewritten as
\begin{align}
P(\lambda) & =\lambda^{2}+W\lambda+U \label{carec2}.
\end{align}
Where:
\begin{align}
W &= C_I+b-C_S + p \delta + \gamma + D_I \\
U &= C_I \gamma + C_I D_I - b C_S + b p \delta + b \gamma + b D_I + C_I b m .
\end{align}
By proposition Routh Hurwitz criteria for $n=2$ if the coefficient $W$ and the independent
term $U$ are positive then the roots of the characteristic equation have negative
real part and therefore the endemic equilibrium is locally asymptotically stable.
Note that whenever the equilibriums are positive, $C_{I},C_{S},D_{I}$ will be
positive as well. Let us analyze the stability according to the value of
$\mathcal{R}_{0}$.
\begin{theorem}
Whenever the equilibrium $E_1$ exists it is a saddle and therefore unstable.
\label{teo8}
\end{theorem}
\begin{proof}
Consider $E_{1}=(S_{1},I_{1})$ and its characteristic polynomial
(\ref{carec2}). By Routh-Hurwitz criterion for quadratic polynomials, its
roots have negative real part if and only if $U>0$ and $W>0$ , where $U,W$
depend on $E_{1}$. Moreover, when $U<0$ its roots are both real with different sign and when $U>0$ and $W<0$ the roots have positive real part. Computing the value of $U$ and expressing $S_{1}$ in terms
of $I_{1}$ we obtain
\begin{equation}
U=\dfrac{I_1(a_{1}I_{1}^{2}+b_{1}I_{1}+c_{1})}{(1+\alpha I_{1})(1+\alpha
_{2}I_{1})^{2}}=\dfrac{I_1 F(I_{1})}{(1+\alpha I_{1})(1+\alpha_{2}I_{1})^{2}}.
\end{equation}
Where:
\begin{align}
a_{1} & =\alpha_{2}^{2}(\beta\gamma+bp\alpha\delta+b\alpha\gamma
+bm\beta) = \alpha_ 2 A >0, \nonumber\\
b_{1} & =2\alpha_{2}(\beta\gamma+bp\alpha\delta+b\alpha\gamma+bm\beta
) = 2A>0,\nonumber\\
c_{1} & =\beta\beta_{2}+bm\beta+bp\alpha\delta+b\alpha\beta_{2}+\beta
\gamma-b\alpha_{2}\beta_{2}+b\alpha\gamma = B- \alpha_2 C.
\end{align}
We are assuming that equilibrium $E_1$ exists and it is positive, and these happens (by previous section) when $B<0$ and $C>0$, so $c_1<0$. The sign of $U$ is equal to $\text{sgn}(F(I_{1}))$. $F(I_{1})$ has two roots of the
form:
\begin{align}
I\ast &=\dfrac{-b_{1}+\sqrt{b_{1}^{2}-4a_{1}c_{1}}}{2a_{1}} \\
I\ast\ast &=\dfrac{-b_{1}-\sqrt{b_{1}^{2}-4a_{1}c_{1}}}{2a_{1}}.
\end{align}
Where $b_1^{2}-4a_1 c_1>0$ and therefore $I \ast $ and $I \ast \ast$ are both real values with
$I\ast\ast<0$. $F(I_{1})>0$ for $I_{1}>I\ast$ and $I_{1}<I\ast\ast$, but second condition never holds because $I_{1}>0$, so $F(I_1)<0$ for $0 < I_1 < I\ast $. \par
Computing $I*$ in terms of $A,B,C$:
\begin{align}
I \ast &= - \dfrac{1}{\alpha_2} + \dfrac{1}{\alpha_2 A} \sqrt{( A^{2}- \alpha_2 AB + \alpha_2^{2} AC )} .
\end{align}
Substituting $\Delta = B^{^2}-4AC>0$
\begin{align}
I \ast &= - \dfrac{1}{\alpha_2} + \dfrac{1}{\alpha_2 A} \sqrt{\left( A^{2}- \alpha_2 AB + \frac{\alpha_2 ^{2}}{4} (B^{2}- \Delta ) \right)} \nonumber \\
&= - \dfrac{1}{\alpha_2} + \dfrac{1}{2 \alpha_2 A} \sqrt{(2A- \alpha_2 B )^{2}- \alpha_2^{2} \Delta } \nonumber \\
&> - \dfrac{1}{\alpha_2} + \dfrac{1}{2 \alpha_2 A} \left( \sqrt{(2A- \alpha_2 B )^{2}} - \sqrt{\alpha_2^{2} \Delta} \right) \nonumber \\
&= \dfrac{-B- \sqrt{\Delta} }{2A} = I_1.
\end{align}
Therefore $U<0$ and the equilibrium $E_1$ is a saddle.
\end{proof}
\begin{theorem}
Assume the conditions of theorem \ref{teo4} for existence and positivity of the endemic equilibrium $E_2$.
If $I_2< I*$ the equilibrium $E_2$ is unstable, else if $I_2> I*$ then $E_2$ is locally asymptotically stable for $s>0$ and unstable for $s<0$. \par
Where $s=m_{1}(-B+\sqrt{B^{2}-4AC})+2Am_{2}$,
\begin{align}
m_{1} & =(r+\beta_{2}\alpha-\beta_{2}\alpha_{2}+2B\alpha_{2})A^{2}-\alpha
_{2}^{2}rAC-AB\alpha_{2}(b\alpha_{2}+2r+B^{2}\alpha_{2}^{2}r),\nonumber\\
m_{2} & =bA^{2}-AC\alpha_{2}(b\alpha_{2}+2r)+\alpha_{2}^{2}rBC,\nonumber\\
r & =\alpha(p\delta+b+\gamma)+\beta.
\end{align}
\label{teo9}
\end{theorem}
\begin{proof}
Consider $E_{2}=(S_{2},I_{2})$ be real and positive, and its characteristic polynomial (\ref{carec2}). We will have that the equilibrium is unstable when $U<0$ and locally asymptotically stable when $U>0, W>0$.
Following the previous proof
\begin{equation}
U=\dfrac{I_2 (a_{1}I_{2}^{2}+b_{1}I_{2}+c_{1})}{(1+\alpha I_{2})(1+\alpha
_{2}I_{2})^{2}}=\dfrac{I_2 F(I_{2})}{(1+\alpha I_{2})(1+\alpha_{2}I_{2})^{2}}.
\end{equation}
Where $a_1,b_1,c_1$ are the same as in previous theorem. Therefore $\text{sgn}(U)=\text{sgn}(F(I_{2}))$. We have seen that $F(I_{2})$ has two real roots $I \ast$ and $ I \ast \ast$. Again $F(I_{2})>0$ for $I_{2}>I\ast$ and $I_{2}<I\ast\ast$ (which does not holds because $I \ast \ast<0$), and $F(I_2)<0$ for $0 < I_2 < I\ast $ . So if $I_2< I*$ the equilibrium $E_2$ is unstable. \par
When $I_2> I*$ then $U>0$ and
\begin{align}
& W=\dfrac{1}{(1+\alpha I_{2})(1+\alpha_{2}I_{2})^{2}}[{\alpha_{{2}}}
^{2}\left( \alpha\,\gamma+b\alpha+\beta+\alpha\,p\delta\right) {I_{{2}}}
^{3}\nonumber\\
& +\alpha_{{2}}\left( b\alpha_{{2}}+2\,\alpha\,p\delta+2\,b\alpha
+2\,\alpha\,\gamma+2\,\beta\right) {I_{{2}}}^{2}\nonumber\\
& +\left( \alpha\,p\delta+b\alpha+\beta+\alpha\,\mathit{\beta_{2}
}-\mathit{\beta_{2}}\,\alpha_{{2}}+\alpha\,\gamma+2\,b\alpha_{{2}}\right)
I_{{2}}+b]\nonumber\\
& =\dfrac{G(I_{2})}{(1+\alpha I_{2})(1+\alpha_{2}I_{2})^{2}}.
\end{align}
By using the division algorithm,
\begin{align}
G(I_{2}) & =(AI_{2}^{2}+BI_{2}+C)P(I_{2})\nonumber\\
& +\frac{1}{A^{2}}[(r+\beta_{2}\alpha-\beta_{2}\alpha_{2}+2B\alpha_{2}
)A^{2}-\alpha_{2}^{2}rAC-AB\alpha_{2}(b\alpha_{2}+2r+B^{2}\alpha_{2}
^{2}r)I_{2}\nonumber\\
& +bA^{2}-AC\alpha_{2}(b\alpha_{2}+2r)+\alpha_{2}^{2}rBC],\nonumber\\
& =(AI_{2}^{2}+BI_{2}+C)P(I_{2})+\dfrac{m_{1}I_{2}+m_{2}}{A^{2}}.
\end{align}
Where $P(I_{2})$ is a polynomial in $I_{2}$ of degree one. Being $I_{2}$ a
coordinate of an equilibrium then $AI_{2}^{2}+BI_{2}+C=0$ and
\[
G(I_{2})=\dfrac{m_{1}I_{2}+m_{2}}{A^{2}}.
\]
Hence $\text{sgn}(W)=\text{sgn}(G(I_{2}))=\text{sgn}(\frac{m_{1}I_{2}+m_{2}}{A^{2}})=\text{sgn}(m_{1}
I_{2}+m_{2}).$ Substituting the value of $I_{2},$
\[
m_{1}I_{2}+m_{2}=\frac{m_{1}}{2A}(-B+\sqrt{B^{2}-4AC})+m_{2}.
\]
It follows that $\text{sgn}(m_{1}I_{2}+m_{2})=\text{sgn}(m_{1}(-B+\sqrt{B^{2}-4AC}
)+2Am_{2})=\text{sgn}(s).$ Therefore $E_2$ is unstable if $s<0$ and locally asymptotically stable if $s>0$.
\end{proof}
\section{Hopf bifurcation}
By previous section we know that the system \eqref{ruanmod} has two positive endemic equilibria under the conditions of theorem \eqref{teo4} . Equilibrium $E_1$ is always a saddle, so its stability does not change and there is no possibility of a Hopf bifurcation in it. So let us analyse the existence of a Hopf bifurcation of equilibrium $E_2=(S_2,I_2)$. Analysing the characteristic equation for $E_2$, it has a pair of pure imaginary roots if and only if $U>0$ and $W=0$ . \par
\begin{theorem}
System \eqref{ruanmod} undergoes a Hopf bifurcation of the endemic equilibrium $E_2$ (whenever it exists) if $I_2>I^{*}$ and $s= 0$. Moreover, if $ \bar{a}_{2}<0 $, there is a family of stable periodic orbits of
\eqref{ruanmod} as $s$ decreases from 0; if $ \bar{a}_2>0 $, there is a family of unstable periodic orbits of \eqref{ruanmod} as $s$ increases from 0. \label{biftheo1}
\end{theorem}
The characteristical polinomial for $E_2$ has a pair of pure imaginary roots iff $U>0$ and $W=0$. From the proof of theorem \ref{teo8} we have that $ U>0$ if and only if one of the conditions (i),(ii) is satisfied .
Although, sgn$(W)=$sgn$(s)$, so $W=0$ if and only if $s=0$. By first part of theorem 3.4.2 of \cite{guckenheimer} the roots $ \lambda $ and $ \bar{ \lambda } $ of \eqref{carec2} for $E_2$ vary smoothly, so we can affirm that near $ s=0 $ these roots are still complex conjugate and
\begin{align}
\frac{d Re( \lambda ( s))}{d s} \mid_{s = 0} &= \frac{d}{d s } \left( \frac{1}{2} W(s) \right) \nonumber \\
&= \frac{1}{2} \frac{d}{d s } \left( \frac{1}{2 A ^{3} (1+ \alpha_1 I_1)( 1+ \alpha_2 I_1)^{2} } s \right) \nonumber \\
&= \frac{1}{ 4 A^{3} (1+ \alpha_1 I_1)( 1+ \alpha_2 I_1)^{2} } \neq 0.
\end{align}
Therefore $s=0$ is the Hopf bifurcation point for \eqref{ruanmod} .\par
To analyze the behaviour of the solutions of \eqref{ruanmod} when $ s=0 $ we make a change of coordinates to obtain a new equivalent system to \eqref{ruanmod} with an equilibrium in $(0,0)$ in the $x-y$ plane ( see appendix B ). Under this change the system becomes:
\begin{align}
\frac{dx}{dt} &= {\frac {a_{{11}}x+a_{{12}}y+c_{{1}}xy+c_{{2}}{y}^{2} }{1+\alpha_{{1}}y+
\alpha_{{1}}I_{{2}}}}, \nonumber \\
\frac{dy}{dt} &= {\frac {a_{{21}}x+a_{{22}}y+c_{{3}}xy+c_{{4}}x{y}^{2}+c_{{5}}{y}^{2}+c
_{{6}}{y}^{3} }{ \left( 1+\alpha_{{1}}y+\alpha_{{1}}I_{{2}} \right)
\left( 1+\alpha_{{2}}y+\alpha_{{2}}I_{{2}} \right) }}. \label{hmrhb1}
\end{align}
Where the $a_{ij}$'s and $c_i$'s are defined in appendix B.
System \eqref{hmrhb1} and \eqref{ruanmod} are equivalent ( appendix B ), so we can work with \eqref{hmrhb1}. This system has a pair of pure imaginary eigenvalues if and only if \eqref{ruanmod} has them too. As we said before it happens if and only if any of conditions (i),(ii) is satisfied and $s=0$. Computing jacobian matrix $DF(0,0)$ of \eqref{hmrhb1}
\begin{equation}
DF(0,0)= \left[ \begin {array}{cc} {\dfrac {a_{{11}}}{1+\alpha_{{1}}I_{{2}}}}&
{\dfrac {a_{{12}}}{ \left( 1+\alpha_{{1}}I
_{{2}} \right) }}\\ \noalign{
}{\dfrac {a_{{21}}}{ \left( 1+
\alpha_{{2}}I_{{2}} \right) \left( 1+\alpha_{{1}}I_{{2}} \right) }}&{
\dfrac {a_{{22}}}{ \left( 1+\alpha_{{2}}I_{{2}} \right) \left( 1+
\alpha_{{1}}I_{{2}} \right) }}\end {array} \right].
\end{equation}
$$ Tr (DF(0,0)) = Tr(Df(S_2,I_2)) , \quad \det (DF(0,0)) = \det (Df(S_2,I_2)).$$
So condition $s=0$ is equivalent to $ a_{11}(1+\alpha_2 I_2)+ a_{22}=0 $ and (i),(ii) are equivalent to $ a_{22} a_{11}- a_{12} a_{21} > 0 $. \par
System \eqref{hmrhb1} can be rewritten as
\begin{align}
\dfrac{dx}{dt} &= {\dfrac {a_{{11}}x}{1+\alpha_{{1}}I_{{2}}}}+{\frac {a_{{12}}y}{1+\alpha
_{{1}}I_{{2}}}} + G_1 (x,y) \\
\dfrac{dy}{dt} &= {\frac {a_{{21}}x}{ \left( 1+\alpha_{{1}}I_{{2}} \right) \left( 1+
\alpha_{{2}}I_{{2}} \right) }}+{\frac {a_{{22}}y}{ \left( 1+\alpha_{{1
}}I_{{2}} \right) \left( 1+\alpha_{{2}}I_{{2}} \right) }} + G_2(x,y).
\end{align}
Where $G_1,G_2$ are defined in appendix B.
Let $ \Lambda = \sqrt{ \det (DF(0,0)) } $. We use the change of variable $u=x, v= \frac{a_{11}}{ \Lambda( 1+ \alpha_1 I_2)} + \frac{a_{12} y}{ \Lambda(1 + \alpha_{1} I_2 ) } $, to obtain the following equivalent system:
\begin{equation}
\left( \begin{matrix}
u \\ v
\end{matrix}\right)= \left( \begin{matrix}
0 & \Lambda \\ - \Lambda & 0
\end{matrix}\right) \left( \begin{matrix}
u \\ v
\end{matrix}\right) + \left( \begin{matrix}
H_1 (u,v) \\ H_2(u,v)
\end{matrix}\right).
\end{equation}
Where
\begin{align}
H_1(u,v) &= -\dfrac{\left( \left( -a_{{12}}c_{{1}}+a_{{11}}c_{{2}} \right) u+ \left( -
\Lambda c_{{2}}\alpha_{{1}}I_{{2}}+ \Lambda a_{{12}
}\alpha_{{1}}- \Lambda c_{{2}} \right) v \right) \left(
\left( \Lambda+ \Lambda \alpha_{{1}}I_{{2}} \right) v-a_{{11}}u
\right)
}{a_{{12}} \left( \left( \alpha_{{1}} \Lambda + \Lambda {\alpha_{{1}}}^{2}I_{{2}} \right) v+a_{{12}}-\alpha_{{1}}a_{{
11}}u+a_{{12}}\alpha_{{1}}I_{{2}} \right)
} \\
H_2(u,v) &= - \dfrac{1}{h(u,v)} \left[ (\Lambda(1+ \alpha_1 I_2)v-a_{11}u) \left( A_1 v^{2}+A_2 uv + A_3 v + A_4 u^{2} + A_5 u \right) \right]
\end{align}
And $A_1,...,A_5, h(u,v)$ are defined in appendix B.
Let
\begin{align}
\bar{a}_2 &= \dfrac{1}{16} [ (H_1)_{uuu} + (H_1)_{uvv} + (H_{2})_{uuv} + (H_2)_{vvv} ] + \dfrac{1}{16( - \Lambda)} [(H_1)_{uv} ((H_1)_{uu} + (H_1)_{vv}) \nonumber \\
& - (H_{2})_{uv} ( (H_{2})_{uu} + (H_2)_{vv} ) - (H_1)_{uu}(H_2)_{uu} + (H_1)_{vv} (H_2)_{vv}].
\end{align}
Where $$(H_1)_{uuu}= \dfrac{ \partial }{ \partial u } \left( \dfrac{ \partial }{ \partial u } \left( \dfrac{ \partial H_1}{ \partial u } \right) \right)(0,0) ,$$
and so on ($ \bar{a}_2 $ is explicitly expressed in appendix B) . \par
Then by theorem 3.4.2 of \cite{guckenheimer} if $ \bar{a }_{2} \neq 0 $ then there exist a surface of periodic solutions, if $ \bar{a}_{2}<0 $ then these cycles are stable, but if $ \bar{a}_{2}>0 $ then cycles are repelling. \par
\section{Discussion}
As we said in the introduction, traditional epidemic models have always stability results in terms of $\mathcal{R}_{0} $, such that we need only reduce $\mathcal{R}_{0} <1$ to eradicate the disease. However, including the treatment function brings new epidemic equilibria that make the dynamics of the model more complicated. Now, let's discuss some control strategies for the infectious disease, analysing the parameters of the treatment function ($ \alpha_{2}, \beta_{2} $) and looking for conditions that allow us to eliminate the disease. We make this study by cases.\par
A first approach is focus on the definition of $\mathcal{R}_{0} $, we can see that $\mathcal{R}_{0} $ decreases when $ \beta_2 $ increases, so the first measure suggesting control is a big value for $ \beta_2 $. But this is not always a good way to proceed. Let us divide our analysis in the following cases: \par
\textit{Case 1: There is no positive endemic equilibrium for $\mathcal{R}_{0} \leq 1$. } This happens when $\mathcal{R}_{0} ^{*} \leq 1 $ ( by theorem \ref{teo4} ) or when $ \mathcal{R}_{0} ^{*}>1$ and $ ( \alpha_{2}, \beta_{2} ) \in A_1 \cup A_2 $ ( theorem \ref{teo4} , number 5). In this case if $\mathcal{R}_{0} >1$ there is a unique positive endemic equilibrium, therefore there exists a bifurcation at $\mathcal{R}_{0} =1$ : from the disease free equilibrium, which is globally asymptotically stable for $0<\mathcal{R}_{0} <1$ (by theorem \ref{teo5}) and a saddle for $\mathcal{R}_{0} =1$ and $ \beta_2 \neq g( \alpha_2 ) $ (theorem \ref{teo7} ), to the positive endemic equilibrium $E_2$ as $\mathcal{R}_{0} $ increase. $E_2$ will be locally asymptotic stable or unstable depending on theorem \ref{teo9} or surrounded by a limit cycle (theorem \ref{biftheo1} ) . If conditions for Hopf bifurcation hold then the stability of the limit cycle is determined by $ \bar{a}_{2} $; when $ \bar{a}_2<0 $ the periodic orbit is stable and therefore $E_2$ is unstable, while if $ \bar{a}_2>0 $ then the periodic orbit is unstable and $E_2$ is stable . In this case the best way to eradicate the disease is finding parameters that allow $\mathcal{R}_{0} <1$, because then all the infectious states tend to $I=0$. \par
\begin{figure}
\caption{Bifurcation diagram in terms of $ \beta_2 $ and $ \alpha_2 $. The values of the parameters taken are $ \alpha = 0.4, \beta = 0.3, b= 0.2, \gamma = 0.03, \delta = 0.05, p=0.3, m=0.3 $. Here $ \mathcal{R}
\label{biffig1}
\end{figure}
\textit{Case 2: There exist endemic equilibria for $\mathcal{R}_{0} \leq 1$}. This happens when $ ( \alpha_{2}, \beta_{2} ) \in A_3 $. The existence of endemic equilibria is determined by the relationship between $\mathcal{R}_{0} $ and $ \max \{ P_1, \mathcal{R}_{0} ^{+} \} $. Let $F( \alpha_2, \beta_2 )= \mathcal{R}_{0} -\mathcal{R}_{0} ^{+}$, $G( \alpha_2, \beta_2 ) = \mathcal{R}_{0} - P_1$, and focus on the implicit curves defined by $F=0$ and $G=0$. These curves divide the domain $A_3$ in another ones (see figure \ref{biffig1} ):
\begin{align}
A_{3}^{1} & = \{( \alpha_2, \beta_2 ) \in A_3, 0< \mathcal{R}_{0} <\mathcal{R}_{0} ^{+} \} \nonumber \\
A_{3}^{2} & = \{( \alpha_2, \beta_2 ) \in A_3, \mathcal{R}_{0} >\mathcal{R}_{0} ^{+} \} \nonumber \\
A_{3}^{3} & = \{( \alpha_2, \beta_2 ) \in A_3, 0< \mathcal{R}_{0} <P_1 \} \nonumber \\
A_{3}^{4} & = \{( \alpha_2, \beta_2 ) \in A_3, \mathcal{R}_{0} > P_1 \}.
\end{align}
If $ ( \alpha_{2}, \beta_{2} ) \in A_{3}^{2} \cap A_{3}^{4} $ then there exist two endemic equilibria $E_1$( a saddle ) and $E_2$ ( stable or unstable depending on conditions of theorems \ref{teo9} and possibly with a periodic orbit around (theorems \ref{biftheo1} )), but when $\mathcal{R}_{0} =1$ one of them becomes negative, leaving us with $E_2$. In this case $\mathcal{R}_{0} <1$ is not a sufficient condition to control the disease, because even with $\mathcal{R}_{0} <1$ we have endemic positive equilibria that could be stable and then the disease will tend to a non zero value; also we have the possibility of a periodic solution, or biologically, an outbreak that will apparently `` disappear '' but will re-emerge after some time. \par
The best way in this case is ensuring $ ( \alpha_{2}, \beta_{2} ) \in ( A_{3}^{2} \cap A_{3}^{4})^{c} $ because then we don't have endemic equilibria for $\mathcal{R}_{0} <1$ and the disease free will be globally asymptotically stable.
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\appendix
\section{Computing center manifold}
The Jacobian matrix of system \eqref{carac6} is
\begin{equation}
DF(m,0)=\left(
\begin{matrix}
-b & - \beta m - bm + p \delta\\
0 & 0
\end{matrix}
\right) . \label{eqst1}
\end{equation}
With eigenvalues $\lambda_{1}=-b$ and $\lambda_{2}=0$ and
respective eigenvectors $v_{1}=(1,0)$ and $v_{2}=(-\frac{\gamma+\beta
_{2}+bm}{b},1)$. Using the eigenvectors to establish a new coordinate system
we define:
\begin{equation}
\left(
\begin{matrix}
\hat{S}\\
I
\end{matrix}
\right) =\left(
\begin{array}
[c]{cc}
1 & -{\frac{\gamma+\beta_{2}+bm}{b}}\\
\noalign{
}0 & 1
\end{array}
\right) \left(
\begin{matrix}
u\\
v
\end{matrix}
\right) \quad\text{, or}\quad\left(
\begin{matrix}
u\\
v
\end{matrix}
\right) =\left(
\begin{array}
[c]{cc}
1 & {\frac{\gamma+\beta_{2}+bm}{b}}\\
\noalign{
}0 & 1
\end{array}
\right) \left(
\begin{matrix}
S\\
I
\end{matrix}
\right).
\end{equation}
Under this transformation the system becomes
\begin{align}
\frac{du}{dt} & ={\frac{d}{dt}}\hat{S}\left( t\right) +{\frac{\left(
\gamma+\mathit{\beta_{2}}+bm\right) {\frac{d}{dt}}I\left( t\right) }{b}
}\nonumber\\
& =-{\frac{\beta\,\left( \hat{S}+m\right) I}{1+\alpha\,I}}-b\hat
{S}-bmI+p\delta\,I+\left( \gamma+\mathit{\beta_{2}}+bm\right) \nonumber\\
& \left( {\frac{\beta\,\left( \hat{S}+m\right) I}{1+\alpha\,I}}-\left(
p\delta+\gamma\right) I-{\frac{\mathit{\beta_{2}}\,I}{1+\mathit{\alpha_{2}
}\,I}}\right) \frac{1}{b},\nonumber\\
\frac{dv}{dt} & =\frac{dI}{dt}\\
& ={\frac{\beta\,\left( \hat{S}+m\right) I}{1+\alpha\,I}}-\left(
p\delta+\gamma\right) I-{\frac{\mathit{\beta_{2}}\,I}{1+\mathit{\alpha_{2}
}\,I}}.
\end{align}
Substituting $S=u-{\frac{\left( \gamma+\beta_2 +bm\right) v}{b}},I=v$
and $\beta m=p\delta+\gamma+\beta_{2}$ we obtain:
\begin{align}
\frac{dv}{dt} & =0u+f(v,u)\nonumber\\
\frac{du}{dt} & =-bu+g(v,u), \label{eqst2}
\end{align}
where
\begin{align}
& f(u,v)=-{\frac{v\left( -\beta\,b-\beta\,b\mathit{\alpha_{2}}\,v\right)
u}{\left( 1+\alpha\,v\right) \left( 1+\mathit{\alpha_{2}}\,v\right) b}
}\nonumber\\
& -\frac{v}{\left( 1+\alpha\,v\right) \left( 1+\mathit{\alpha_{2}
}\,v\right) b}(\left( \beta\,bm\mathit{\alpha_{2}}+b\gamma\,\alpha
\,\mathit{\alpha_{2}}-\beta\,\mathit{\alpha_{2}}\,p\delta+bp\delta
\,\alpha\,\mathit{\alpha_{2}}+{\beta}^{2}\mathit{\alpha_{2}}\,m\right)
{v}^{2}\nonumber\\
& +\left( bp\delta\,\mathit{\alpha_{2}}+\beta\,bm-\beta\,bm\mathit{alpha2}
+b\gamma\,\mathit{\alpha_{2}}+{\beta}^{2}m-\beta\,p\delta+b\alpha
\,\beta\,m\right) v),\nonumber\\
& g(u,v)=-\frac{1}{\left( 1+\alpha\,v\right) \left( 1+\mathit{\alpha_{2}
}\,v\right) b^{2}}[v((m{b}^{2}\gamma\,\alpha\,\mathit{\alpha_{2}}+2\,{\beta
}^{2}b{m}^{2}\mathit{\alpha_{2}}+\beta\,\mathit{\alpha_{2}}\,{p}^{2}{\delta
}^{2}\nonumber \\
&+{\beta}^{3}\mathit{\alpha_{2}}\,{m}^{2}-b\gamma\,p\delta\,\alpha
\,\mathit{\alpha_{2}}+b\gamma\,\alpha\,\mathit{\alpha_{2}}\,\beta\,m-2\,{\beta}^{2}
\mathit{\alpha_{2}}\,mp\delta+bp\delta\,\alpha\,\mathit{\alpha_{2}}
\,\beta\,m\nonumber \\
&-{b}^{2}m\alpha\,\mathit{\alpha_{2}}\,\beta-{\beta}^{2}
bm\mathit{\alpha_{2}}+{b}^{2}{m}^{2}\beta\,\mathit{\alpha_{2}}+{b}^{2}
mp\delta\,\alpha\,\mathit{\alpha_{2}}-b{p}^{2}{\delta}^{2}\alpha\,\mathit{\alpha_{2}}\nonumber \\
&-2\,\beta\,bm\mathit{\alpha_{2}}\,p\delta-{b}^{2}m\beta\,\mathit{\alpha_{2}}+b\beta\,\mathit{\alpha_{2}
}\,p\delta)v^{2}+({b}^{2}{m}^{2}\beta-2\,\beta\,bmp\delta-\beta\,{b}
^{2}m+2\,{\beta}^{2}b{m}^{2}\nonumber\\
& -{\beta}^{2}bm+\beta\,{p}^{2}{\delta}^{2}+2\,\beta\,bm\mathit{\alpha_{2}
}\,p\delta-bp\delta\,\alpha\,\beta\,m+{\beta}^{3}{m}^{2}+\beta\,bp\delta
-{b}^{2}\alpha\,\beta\,m-2\,{\beta}^{2}mp\delta\nonumber \\
&+{b}^{2}{m}^{2}\alpha\,\beta+b\alpha\,{\beta}^{2}{m}^{2}-u\beta\,{b}^{2}m\alpha_{{2}}-b{\beta}
^{2}u\alpha_{{2}}m+b\beta\,u\alpha_{{2}}p\delta-\gamma\,bp\delta\,\alpha_{{2}
}+\gamma\,\beta\,bm\alpha_{{2}}\nonumber \\
&+{b}^{2}mp\delta\,\alpha_{{2}}-{b}^{2}{m}
^{2}\beta\,\alpha_{{2}}+u\beta\,{b}^{2}\alpha_{{2}}-b{p}^{2}{\delta}^{2}\alpha_{{2}}-{\beta}
^{2}b{m}^{2}\alpha_{{2}}+{b}^{2}m\gamma\,\alpha_{{2}})v-{b}^{2}m\beta
\,u\nonumber \\
&+u\beta\,{b}^{2}-b{\beta}^{2}um+b\beta\,up\delta)].
\end{align}
By \cite{carr} the system (\ref{eqst2}) has a center manifold of the form
$u=h(v)$. Let $\phi:\mathbb{R}\rightarrow\mathbb{R}$ and define the
annihilator:
\begin{align}
N\phi & =\phi^{\prime}(v)(f(v,\phi(v)))+b\phi-g(v,\phi(v))\nonumber\\
& =\frac{1}{b^{2}(1+\alpha v)(1+\alpha_{2}v)}[bp\delta\,\alpha\,{v}^{3}
\alpha_{{2}}\beta\,m+{b}^{2}{m}^{2}\beta\,{v}^{2}-\beta\,{v}^{2}{b}^{2}
m+{b}^{3}\phi+{b}^{3}\phi\,\alpha\,v\nonumber \\
&+{b}^{3}\phi\,\alpha_{{2}}v+{b}^{2}m\gamma\,\alpha_{2}\,{v}^{2}+\phi\,\beta\,v{b}^{2}+vb\phi
\,\beta\,p\delta+{b}^{2}mp\delta\,\alpha_{{2}}{v}^{2}-\phi\,\beta\,{v}^{2}{b}^{2}m\alpha_{{2}}\nonumber \\
&-\gamma\,bp\delta\,\alpha_{{2}}{v}^{2}+{b}^{2}m\gamma\,\alpha\,{v}^{3}\alpha_{{2}}+\gamma\,\beta\,{v}^{2}
bm\alpha_{{2}}+{b}^{2}{m}^{2}\alpha\,{v}^{2}\beta-2\,{\beta}^{2}{v}
^{2}mp\delta-{\beta}^{2}{v}^{2}b{m}^{2}\alpha_{2}\nonumber \\
&+\beta\,{v}^{2}bp\delta\-{b}^{2}{v}^{2}\alpha\,\beta\,m+b\alpha\,{v}^{2}{\beta}^{2}{m}^{2}
-bp\delta\,\alpha\,{v}^{2}\beta\,m+{\beta}^{3}{v}^{2}{m}^{2}-2\,\beta\,{v}
^{2}bmp\delta+{\beta}^{3}{v}^{3}\alpha_{{2}}{m}^{2}\nonumber \\
&+2\,{\beta}^{2}{v}^{2}b{m}^{2}+\beta\,{v}^{2}{p}^{2}{\delta}^{2}-{\beta}^{2}{v}^{2}bm-\phi\,\beta
\,v{b}^{2}m-vb\phi\,{\beta}^{2}m+\beta\,{v}^{3}b\alpha_{{2}}p\delta-{b}^{2}
{v}^{3}\alpha\,\alpha_{{2}}\beta\,m\nonumber \\
&-b{p}^{2}{\delta}^{2}\alpha\,{v}^{3}\alpha_{{2}}-2\,{\beta}^{2}{v}^{3}\alpha_{{2}}mp\delta+\phi\,\beta\,{v}^{2}{b}
^{2}\alpha_{{2}}-{b}^{2}{m}^{2}\beta\,{v}^{2}\alpha_{{2}}+{b}^{2}{m}^{2}
\beta\,{v}^{3}\alpha_{{2}}-\beta\,{v}^{3}{b}^{2}m\alpha_{{2}}\nonumber \\
&-{\beta}^{2}{v}^{3}b\alpha_{{2}}m+2\,{\beta}^{2}{v}^{3}b{m}^{2}\alpha_{{2}}-b{p}^{2}{\delta}^{2}{v}
^{2}\alpha_{{2}}+\beta\,{v}^{3}\alpha_{{2}}{p}^{2}{\delta}^{2}+{b}^{3}
\phi\,\alpha\,{v}^{2}\alpha_{{2}}-{v}^{2}b\phi\,{\beta}^{2}m\alpha_{{2}}\nonumber \\
&+{v}^{2}b\phi\,\beta\,p\delta\,\alpha_{{2}}+b\gamma\,\alpha\,{v}^{3}\alpha_{{2}}\beta\,m+{b}^{2}mp\delta\,\alpha
\,{v}^{3}\alpha_{{2}}-\gamma\,bp\delta\,\alpha\,{v}^{3}\alpha_{{2}}-2\,\beta\,{v}^{3}bm\alpha_{{2}}p\delta\nonumber \\
&+2\,\beta\,{v}^{2}bm\alpha_{{2}}p\delta].
\end{align}
Assume that $\phi=a_{0}v^{2}+a_{1}v^{3}+O(v^{4})$, then by substituting $\phi$
and $\frac{d\phi}{dv}$ in the annihilator $N\phi$ and expanding its Taylor
series we get:
\begin{align}
N\phi & =\frac{1}{b^{2}}((\gamma\,\beta\,bm\alpha_{2}+{b}^{2}mp\delta
\,\alpha_{2}-{b}^{2}{m}^{2}\beta\,\mathit{\alpha_{2}}+2\,\beta
\,bm\mathit{\alpha_{2}}\,p\delta+2\,{\beta}^{2}b{m}^{2}+{b}^{2}{m}^{2}
\beta\nonumber\\
& -\beta\,{b}^{2}m+{b}^{3}\mathit{a_{0}}-{\beta}^{2}b{m}^{2}\mathit{\alpha
_{2}}-2\,\beta\,bmp\delta+{b}^{2}m\gamma\,\mathit{\alpha_{2}}-\gamma
\,bp\delta\,\mathit{\alpha_{2}}-{b}^{2}\alpha\,\beta\,m-{\beta}^{2}
bm\nonumber\\
& +b\alpha\,{\beta}^{2}{m}^{2}+{b}^{2}{m}^{2}\alpha\,\beta-2\,{\beta}
^{2}mp\delta+\beta\,{p}^{2}{\delta}^{2}-bp\delta\,\alpha\,\beta\,m-b{p}
^{2}{\delta}^{2}\mathit{\alpha_{2}}+\beta\,bp\delta\nonumber \\
&+{\beta}^{3}{m}^{2})v^{2}-\frac{1}{b^{2}}[\alpha\,{\beta}^{3}{m}^{2}-\mathit{a_{0}}\,\beta\,{b}
^{2}-{b}^{3}\mathit{a_{1}}-2\,bp\delta\,\alpha\,\beta\,m-{\beta}^{2}b{m}
^{2}{\mathit{\alpha_{2}}}^{2}-b{p}^{2}{\delta}^{2}{\mathit{\alpha_{2}}}^{2}\nonumber \\
&+m{b}^{2}\gamma\,{\mathit{\alpha_{2}}}^{2}-{b}^{2}{m}^{2}\beta
\,{\mathit{\alpha_{2}}}^{2}-{b}^{2}\alpha\,\beta\,m-{b}^{2}{\alpha}^{2}\beta\,m-\alpha\,{\beta}
^{2}bm+b{\alpha}^{2}{\beta}^{2}{m}^{2}+{b}^{2}{m}^{2}{\alpha}^{2}\beta\nonumber \\
&+\alpha\,\beta\,{p}^{2}{\delta}^{2}+3\,\mathit{a_{0}}\,\beta\,{b}^{2}m+3\,\mathit{a_{0}}\,b{\beta}^{2}m+2\,\mathit{a_{0}}\,{b}^{2}\gamma
\,\mathit{\alpha_{2}}+2\,\mathit{a_{0}}\,{b}^{2}p\delta\,\mathit{\alpha_{2}
}-2\,\mathit{a_{0}}\,\beta\,{b}^{2}m\mathit{\alpha_{2}}\nonumber \\
&-3\,\mathit{a_{0}}\,b\beta\,p\delta+2\,\mathit{a_{0}}\,{b}^{2}\alpha\,\beta\,m-2\,\alpha\,{\beta}^{2}mp\delta+\alpha\,\beta\,bp\delta+{b}^{2}
mp\delta\,{\mathit{\alpha_{2}}}^{2}+b\gamma\,{\mathit{\alpha_{2}}}^{2}\beta\,m\nonumber \\
&-b\gamma\,p\delta\,{\mathit{\alpha_{2}}}^{2}+2\,\beta\,bm{\mathit{\alpha_{2}}}^{2}p\delta-bp\delta\,{\alpha}^{2}\beta\,m+{b}^{2}{m}^{2}\alpha\,
\beta+2\,b\alpha\,{\beta}^{2}{m}^{2}]v^{3}+O(v^{4})).
\end{align}
By choosing the coefficients of $v^{2}$ and $v^{3}$ in order to have
$N\phi=O(v^{4})$ we obtain that $a_{0}$ and $a_{1}$ must be the following:
\begin{align}
a_{0} & =-\frac{1}{b^{3}}[{b}^{2}{m}^{2}\beta+\beta\,bp\delta-{b}^{2}
\alpha\,\beta\,m+{b}^{2}mp\delta\,\alpha_{2}-\gamma\,bp\delta\,\alpha
_{2}+\gamma\,\beta\,bm\alpha_{2}-2\,\beta\,bmp\delta\nonumber\\
& -\beta\,{b}^{2}m+2\,{\beta}^{2}b{m}^{2}-{\beta}^{2}bm+\beta\,{p}^{2}
{\delta}^{2}+2\,\beta\,bm\alpha_{2}\,p\delta-bp\delta\,\alpha\,\beta
\,m+{\beta}^{3}{m}^{2}-b{p}^{2}{\delta}^{2}\alpha_{2}\nonumber\\
& -{b}^{2}{m}^{2}\beta\,\alpha_{2}-2\,{\beta}^{2}mp\delta+{b}^{2}
m\gamma\,\alpha_{2}+b\alpha\,{\beta}^{2}{m}^{2}+{b}^{2}{m}^{2}\alpha
\,\beta-{\beta}^{2}b{m}^{2}\alpha_{2}],\\
a_{1} & =\frac{1}{b^{3}}[\alpha\,{\beta}^{3}{m}^{2}-ao\,\beta\,{b}
^{2}-2\,bp\delta\,\alpha\,\beta\,m-{\beta}^{2}b{m}^{2}\alpha_{2}-b{p}
^{2}{\delta}^{2}{\mathit{\alpha_{2}}}^{2}+m{b}^{2}\gamma\,\alpha_{2}^{2}
-{b}^{2}{m}^{2}\beta\alpha_{2}^{2}\nonumber\\
& -{b}^{2}\alpha\,\beta\,m-{b}^{2}{\alpha}^{2}\beta\,m-\alpha\,{\beta}
^{2}bm+b{\alpha}^{2}{\beta}^{2}{m}^{2}+{b}^{2}{m}^{2}{\alpha}^{2}\beta
+\alpha\,\beta\,{p}^{2}{\delta}^{2}+3\,\mathit{a_{0}}\,\beta\,{b}
^{2}m\nonumber\\
& +3\,a_{0}\,b{\beta}^{2}m+2\,a_{0}\,{b}^{2}\gamma\,\alpha_{2}
+2\,\mathit{a_{0}}\,{b}^{2}p\delta\,\alpha_{2}-2\,\mathit{a_{0}}\,\beta
\,{b}^{2}m\mathit{\alpha_{2}}-3\,\mathit{a_{0}}\,b\beta\,p\delta
+2\,\mathit{a_{0}}\,{b}^{2}\alpha\,\beta\,m\nonumber\\
& -2\,\alpha\,{\beta}^{2}mp\delta+\alpha\,\beta\,bp\delta+{b}^{2}
mp\delta\,{\mathit{\alpha_{2}}}^{2}+b\gamma\,{\mathit{\alpha_{2}}}^{2}
\beta\,m-b\gamma\,p\delta\,{\mathit{\alpha_{2}}}^{2}+2\,\beta
\,bm{\mathit{\alpha_{2}}}^{2}p\delta\nonumber\\
&-bp\delta\,{\alpha}^{2}\beta\,m+{b}^{2}{m}^{2}\alpha\,\beta+2\,b\alpha\,{\beta}^{2}{m}^{2}].
\end{align}
Hence $h(v)=a_{0}v^{2}+a_{1}v^{3}+O(v^{4})$.
\section{Hopf bifurcation}
To analyze the behaviour of the solutions of \eqref{ruanmod} when $ s=0 $ we make a change of coordinates $x=S-S_2$, $y=I-I_2$, to obtain a new equivalent system to \eqref{ruanmod} with an equilibrium in $(0,0)$ in the $x-y$ plane. Under this change the system becomes in:
\begin{align}
\frac{dx}{dt} &= {\frac {a_{{11}}x+a_{{12}}y+c_{{1}}xy+c_{{2}}{y}^{2} + c_7}{1+\alpha_{{1}}y+
\alpha_{{1}}I_{{2}}}}, \nonumber \\
\frac{dy}{dt} &= {\frac {a_{{21}}x+a_{{22}}y+c_{{3}}xy+c_{{4}}x{y}^{2}+c_{{5}}{y}^{2}+c
_{{6}}{y}^{3} + c_8}{ \left( 1+\alpha_{{1}}y+\alpha_{{1}}I_{{2}} \right)
\left( 1+\alpha_{{2}}y+\alpha_{{2}}I_{{2}} \right) }}. \label{hmrhb2}
\end{align}
Where:
\begin{align}
a_{11} &= -b-\beta_{{1}}I_{{2}}-b\alpha_{{1}}I_{{2}} \\
a_{12} &= -2\,bm\alpha_{{1}}I_{{2}}+bm\alpha_{{1}}-b\alpha_{{1}}S_{{2}}+2\,p
\delta\,\alpha_{{1}}I_{{2}}+p\delta-bm-\beta_{{1}}S_{{2}} \\
c_{1} &=-b\alpha_{{1}}-\beta_{{1}} \\
c_{2} &= -bm\alpha_{{1}}+p\delta\,\alpha_{{1}} \\
a_{21} &=-I_{{2}} \left( -\beta_{{1}}-\beta_{{1}}\alpha_{{2}}I_{{2}} \right) \\
a_{22} &= -2\,p\delta\,\alpha_{{1}}I_{{2}}+2\,\beta_{{1}}\alpha_{{2}}S_{{2}}I_{{
2}}-3\,p\delta\,\alpha_{{1}}\alpha_{{2}}{I_{{2}}}^{2}-2\,\gamma\,
\alpha_{{1}}I_{{2}} \nonumber \\
&-2\,\gamma\,\alpha_{{2}}I_{{2}}-2\,p\delta\,\alpha_
{{2}}I_{{2}}-2\,\beta_{{2}}\alpha_{{1}}I_{{2}}-3\,\gamma\,\alpha_{{1}}
\alpha_{{2}}{I_{{2}}}^{2}-\gamma-p\delta-\beta_{{2}}+\beta_{{1}}S_{{2}
} \\
c_{3} &= 2\,\beta_{{1}}\alpha_{{2}}I_{{2}}+\beta_{{1}} \\
c_{4} &= \beta_{{1}}\alpha_{{2}}{y}^{2} \\
c_{5} &= -3\,p\delta\,\alpha_{{1}}\alpha_{{2}}I_{{2}}-3\,\gamma\,\alpha_{{1}}
\alpha_{{2}}I_{{2}}-p\delta\,\alpha_{{1}}+\beta_{{1}}\alpha_{{2}}S_{{2
}}-\gamma\,\alpha_{{1}}-\gamma\,\alpha_{{2}}-p\delta\,\alpha_{{2}}\nonumber \\
&-\beta_{{2}}\alpha_{{1}} \\
c_{6} &= -p\delta\,\alpha_{{1}}\alpha_{{2}}-\gamma\,\alpha_{{1}}\alpha_{{2}} \\
c_7 &= -(\beta_{{1}}S_{{2}}I_{{2}}-bm\alpha_{{1}}I_{{2}}+bS_{{2}}-p\delta\,I_{{
2}}-p\delta\,\alpha_{{1}}{I_{{2}}}^{2}+b\alpha_{{1}}S_{{2}}I_{{2}}+bmI
_{{2}} \nonumber \\
&-bm+bm\alpha_{{1}}{I_{{2}}}^{2})\\
c_8 &=- I_2 [p\delta\,\alpha_{{1}}I_{{2}}+p\delta+p\delta\,\alpha_{{2}}I_{{2}}+
\gamma\,\alpha_{{2}}I_{{2}}-\beta_{{1}}\alpha_{{2}}S_{{2}}I_{{2}}+
\gamma\,\alpha_{{1}}I_{{2}}+\beta_{{2}}\alpha_{{1}}I_{{2}} \nonumber \\
&+\gamma+ \gamma\,\alpha_{{1}}\alpha_{{2}}{I_{{2}}}^{2}-\beta_{{1}}S_{{2}}+\beta
_{{2}}+p\delta\,\alpha_{{1}}\alpha_{{2}}{I_{{2}}}^{2}].
\end{align}
But from the equations for the equilibrium point we can prove that $c_7=c_8=0$, so the system we will work on is
\begin{align}
\frac{dx}{dt} &= {\frac {a_{{11}}x+a_{{12}}y+c_{{1}}xy+c_{{2}}{y}^{2} }{1+\alpha_{{1}}y+
\alpha_{{1}}I_{{2}}}}, \nonumber \\
\frac{dy}{dt} &= {\frac {a_{{21}}x+a_{{22}}y+c_{{3}}xy+c_{{4}}x{y}^{2}+c_{{5}}{y}^{2}+c
_{{6}}{y}^{3} }{ \left( 1+\alpha_{{1}}y+\alpha_{{1}}I_{{2}} \right)
\left( 1+\alpha_{{2}}y+\alpha_{{2}}I_{{2}} \right) }}.
\end{align}
If we denote system \eqref{ruanmod} as $ (S,I)' = f(S,I) $ and system \eqref{hmrhb2} as $(x,y)' = F(x,y)$, $ f=(f_1,f_2) $, $F=(F_1,F_2)$ then
$$ F(x,y) = f(x+S_2,y+I_2), $$
and
$$ \frac{ \partial F_i }{ \partial x } (x,y) = \frac{ \partial f_i }{ \partial S } (x+S_2,y+I_2)\frac{ \partial S }{ \partial x} (x,y) + \frac{ \partial f_i }{ \partial I } (x+S_2,y+I_2)\frac{ \partial I }{ \partial x} (x,y)= \frac{ \partial f_i }{ \partial S } (x+S_2,y+I_2)$$
$$ \frac{ \partial F_i }{ \partial y } (x,y) = \frac{ \partial f_i }{ \partial S } (x+S_2,y+I_2)\frac{ \partial S }{ \partial y} (x,y) + \frac{ \partial f_i }{ \partial I } (x+S_2,y+I_2)\frac{ \partial I }{ \partial y} (x,y)= \frac{ \partial f_i }{ \partial S } (x+S_2,y+I_2) .$$
So, the jacobian matrix of \eqref{hmrhb1} $ DF(0,0) $ in the equilibrium is equal to the jacobian matrix of system \eqref{ruanmod} $Df(S_1,I_1)$. We can also compute the partial derivatives of system \eqref{hmrhb2} and \eqref{hmrhb1} to prove that they are equal,ie,
\begin{equation}
Df(S_2,I_2) = DF(0,0).
\end{equation}
Therefore the system \eqref{hmrhb1} and \eqref{ruanmod} are equivalent and we can work with system \eqref{hmrhb1}. The jacobian matrix $DF(0,0)$ of \eqref{hmrhb1} is:
\begin{equation}
DF(0,0)= \left[ \begin {array}{cc} {\dfrac {a_{{11}}}{1+\alpha_{{1}}I_{{2}}}}&
{\dfrac {a_{{12}}}{ \left( 1+\alpha_{{1}}I
_{{2}} \right) }}\\ \noalign{
}{\dfrac {a_{{21}}}{ \left( 1+
\alpha_{{2}}I_{{2}} \right) \left( 1+\alpha_{{1}}I_{{2}} \right) }}&{
\dfrac {a_{{22}}}{ \left( 1+\alpha_{{2}}I_{{2}} \right) \left( 1+
\alpha_{{1}}I_{{2}} \right) }}\end {array} \right].
\end{equation}
So system \eqref{hmrhb1} can be rewritten as
\begin{align}
\dfrac{dx}{dt} &= {\dfrac {a_{{11}}x}{1+\alpha_{{1}}I_{{2}}}}+{\frac {a_{{12}}y}{1+\alpha
_{{1}}I_{{2}}}} + G_1 (x,y) \\
\dfrac{dy}{dt} &= {\frac {a_{{21}}x}{ \left( 1+\alpha_{{1}}I_{{2}} \right) \left( 1+
\alpha_{{2}}I_{{2}} \right) }}+{\frac {a_{{22}}y}{ \left( 1+\alpha_{{1
}}I_{{2}} \right) \left( 1+\alpha_{{2}}I_{{2}} \right) }} + G_2(x,y).
\end{align}
Where
\begin{align}
G_1 &= \dfrac{1}{(1 + \alpha_1 y+ \alpha_1 I_2)(1 + \alpha_1 I_2)} \{[(1 + \alpha_1 I_2)c_1-a_{11} \alpha_1] xy + [ c_2 ( 1 + \alpha_1 I_2 ) - \alpha_1 a_{12} ] y^{2} \} \\
G_2 &= \dfrac{1}{(1 + \alpha_1 y+ \alpha_1 I_2)(1 + \alpha_2 y+ \alpha_2 I_2)(1 + \alpha_1 I_2)(1 + \alpha_2 I_2)} \{ [c_3 (1 + \alpha_1 I_2)(1 + \alpha_2 I_2)\nonumber \\ &- a_{21} ( \alpha_2 + \alpha_1 + 2 \alpha_1 \alpha_2 I_2 ) ] x y + [ c_4 ( 1 + \alpha_1 I_2 )( 1 + \alpha_2 I_2 ) - a_{21} \alpha_1 \alpha_2 ] xy^{2}\nonumber \\ &+ [c_{5}( 1 + \alpha_1 I_2 )( 1 + \alpha_2 I_2) - a_{22} ( \alpha_2 + \alpha_1 + 2 \alpha_1 \alpha_2 I_1 ) ] y^{2} + [ c_{6} ( 1 + \alpha_1 I_2 )( 1 + \alpha_2 I_2 )\nonumber \\
&- a_{22} \alpha_1 \alpha_2 ] y^{3} \}.
\end{align}
We need the normal form of the system \eqref{hmrhb1}. The eigenvalues of $ DF(0,0) $ when $s_2=0$ and (i),(ii) are satisfied are:
$$ \Lambda i , - \Lambda i .$$
With complex eigenvector $$ v= \left( \begin{matrix}
- 1 \\ \dfrac{ - \Lambda i(1 + \alpha_1 I_2)+ a_{11} }{a_{12}}
\end{matrix}\right), \quad \bar{v} = \left( \begin{matrix}
- 1 \\ \dfrac{ \Lambda i(1 + \alpha_1 I_2)+ a_{11} }{a_{12}}
\end{matrix}\right) .$$
Using the Jordan Canonical form of matrix $DF(0,0)$ and the procedure in \cite{perko} (p. 107, 108) we use the change of variable $u=x, v= \frac{a_{11}}{ \Lambda( 1+ \alpha_1 I_2)} + \frac{a_{12} y}{ \Lambda(1 + \alpha_{1} I_2 ) } $, to obtain the following equivalent system:
\begin{equation}
\left( \begin{matrix}
u \\ v
\end{matrix}\right)= \left( \begin{matrix}
0 & \Lambda \\ - \Lambda & 0
\end{matrix}\right) \left( \begin{matrix}
u \\ v
\end{matrix}\right) + \left( \begin{matrix}
H_1 (u,v) \\ H_2(u,v)
\end{matrix}\right).
\end{equation}
Where
\begin{align}
H_1(u,v) &= -\dfrac{\left( \left( -a_{{12}}c_{{1}}+a_{{11}}c_{{2}} \right) u+ \left( -
\Lambda c_{{2}}\alpha_{{1}}I_{{2}}+ \Lambda a_{{12}
}\alpha_{{1}}- \Lambda c_{{2}} \right) v \right) \left(
\left( \Lambda+ \Lambda \alpha_{{1}}I_{{2}} \right) v-a_{{11}}u
\right)
}{a_{{12}} \left( \left( \alpha_{{1}} \Lambda + \Lambda {\alpha_{{1}}}^{2}I_{{2}} \right) v+a_{{12}}-\alpha_{{1}}a_{{
11}}u+a_{{12}}\alpha_{{1}}I_{{2}} \right)
} \\
H_2(u,v) &= - \dfrac{1}{h(u,v)} \left[ (\Lambda(1+ \alpha_1 I_2)v-a_{11}u) \left( A_1 v^{2}+A_2 uv + A_3 v + A_4 u^{2} + A_5 u \right) \right].
\end{align}
And:
\begin{align*}
A_1 &= {\Lambda}^{2} \left( 1+\alpha_{{1}}I_{{2}} \right) ^{2} [ -a_{{12
}}c_{{6}}\alpha_{{2}}{I_{{2}}}^{2}\alpha_{{1}}-a_{{11}}c_{{2}}\alpha_{
{1}}{I_{{2}}}^{2}{\alpha_{{2}}}^{2}-a_{{11}}c_{{2}}\alpha_{{1}}I_{{2}}
\alpha_{{2}}\nonumber \\
&-a_{{12}}c_{{6}}\alpha_{{1}}I_{{2}}+a_{{11}}a_{{12}}\alpha
_{{1}}{\alpha_{{2}}}^{2}I_{{2}}+a_{{11}}a_{{12}}\alpha_{{1}}\alpha_{{2
}} +a_{{12}}a_{{22}}\alpha_{{1}}\alpha_{{2}}\nonumber \\
&-a_{{11}}c_{{2}}{\alpha_{{2
}}}^{2}I_{{2}}-a_{{12}}c_{{6}}\alpha_{{2}}I_{{2}}-a_{{11}}c_{{2}}
\alpha_{{2}}-a_{{12}}c_{{6}} ]
\\
A_2 &= -\Lambda\, \left( 1+\alpha_{{1}}I_{{2}} \right) [ a_{{11}}a_{{12
}}c_{{1}}{\alpha_{{2}}}^{2}\alpha_{{1}}{I_{{2}}}^{2}+{a_{{12}}}^{2}c_{
{4}}\alpha_{{2}}{I_{{2}}}^{2}\alpha_{{1}}-2\,a_{{12}}a_{{11}}c_{{6}}
\alpha_{{2}}{I_{{2}}}^{2}\alpha_{{1}}\nonumber \\
&-2\,c_{{2}}\alpha_{{1}}{I_{{2}}}^{2}{\alpha_{{2}}}^{2}{a_{{11}}}^{2} +a_{{12}}\alpha_{{1}}{\alpha_{{2}}}^{2}{a_{{11}}}^{2}I_{{2}}-2\,a_{{12}}a_{{11}}c_{{6}}\alpha_{{1}}I_{{2}
}-2\,c_{{2}}\alpha_{{1}}I_{{2}}\alpha_{{2}}{a_{{11}}}^{2}\nonumber \\
&+{a_{{12}}}^{2}c_{{4}}\alpha_{{1}}I_{{2}}+a_{{11}}a_{{12}}c_{{1}}\alpha_{{2}}\alpha
_{{1}}I_{{2}}+a_{{11}}a_{{12}}c_{{1}}{\alpha_{{2}}}^{2}I_{{2}}+{a_{{12
}}}^{2}c_{{4}}\alpha_{{2}}I_{{2}}-2\,a_{{12}}a_{{11}}c_{{6}}\alpha_{{2
}}I_{{2}}\nonumber \\
&-2\,{a_{{11}}}^{2}c_{{2}}{\alpha_{{2}}}^{2}I_{{2}}+{a_{{12}}}
^{2}c_{{4}}-2\,{a_{{11}}}^{2}c_{{2}}\alpha_{{2}}-2\,a_{{12}}a_{{11}}c_{{6}}+a_{{11}}a_{{12}}c_{{1}}\alpha_{{2}}+a_{{12}}\alpha_{{1}}\alpha_{
{2}}{a_{{11}}}^{2}\nonumber \\
&-{a_{{12}}}^{2}\alpha_{{1}}a_{{21}}\alpha_{{2}}+2\,a_{{12}}a_{{11}}a_{{22}}\alpha_{{1}}\alpha_{{2}} ] \\
A_3 &= \Lambda\, \left( 1+\alpha_{{1}}I_{{2}} \right) a_{{12}} [ -a_{{12
}}c_{{5}}\alpha_{{1}}{I_{{2}}}^{2}\alpha_{{2}}+a_{{12}}a_{{11}}\alpha_
{{1}}{\alpha_{{2}}}^{2}{I_{{2}}}^{2}+2\,a_{{12}}a_{{22}}\alpha_{{1}}
\alpha_{{2}}I_{{2}}\nonumber \\
&+2\,a_{{12}}a_{{11}}\alpha_{{1}}\alpha_{{2}}I_{{2}}
-a_{{12}}c_{{5}}\alpha_{{1}}I_{{2}}+a_{{12}}a_{{22}}\alpha_{{1}}+a_{{
11}}a_{{12}}\alpha_{{1}}-a_{{12}}c_{{5}}\alpha_{{2}}I_{{2}}+a_{{12}}a_
{{22}}\alpha_{{2}}\nonumber \\
&-2\,a_{{11}}c_{{2}}\alpha_{{1}}{I_{{2}}}^{2}\alpha_{{2}}-a_{{11}}c_{{2}}\alpha_{{1}}I_{{2}}-a_{{11}}c_{{2}}
-a_{{11}}c_{{2}}{\alpha_{{2}}}^{2}{I_{{2}}}^{2}-2\,a_{{11}}c_{{2}}\alpha_{{2}}I_{{2}} ] \\
A_4 &= -a_{{11}} [-{a_{{12}}}^{2}c_{{4}}\alpha_{{2}}I_{{2}}-a_{{11}}a_{
{12}}c_{{1}}\alpha_{{2}}\alpha_{{1}}I_{{2}}+c_{{2}}\alpha_{{1}}I_{{2}}
\alpha_{{2}}{a_{{11}}}^{2}-{a_{{12}}}^{2}c_{{4}}\alpha_{{2}}{I_{{2}}}^
{2}\alpha_{{1}}\nonumber \\
&-{a_{{12}}}^{2}c_{{4}}\alpha_{{1}}I_{{2}}-{a_{{12}}}^{2
}c_{{4}}+{a_{{12}}}^{2}\alpha_{{1}}a_{{21}}\alpha_{{2}}-a_{{12}}a_{{11
}}a_{{22}}\alpha_{{1}}\alpha_{{2}}-a_{{11}}a_{{12}}c_{{1}}{\alpha_{{2}
}}^{2}\alpha_{{1}}{I_{{2}}}^{2}\nonumber \\
&+a_{{12}}a_{{11}}c_{{6}}+{a_{{11}}}^{2}
c_{{2}}{\alpha_{{2}}}^{2}I_{{2}}+a_{{12}}a_{{11}}c_{{6}}\alpha_{{2}}{I
_{{2}}}^{2}\alpha_{{1}}+{a_{{11}}}^{2}c_{{2}}\alpha_{{2}}-a_{{11}}a_{{
12}}c_{{1}}\alpha_{{2}}\nonumber \\
&+a_{{12}}a_{{11}}c_{{6}}\alpha_{{2}}I_{{2}}+a_{{12}}a_{{11}}c_{{6}}\alpha_{{1}}I_{{2}}+c_{{2}}\alpha_{{1}}{I_{{2}}}^{
2}{\alpha_{{2}}}^{2}{a_{{11}}}^{2}-a_{{11}}a_{{12}}c_{{1}}{\alpha_{{2}}}^{2}I_{{2}} ] \\
A_5 &= a_{{12}} [ 2\,{a_{{11}}}^{2}c_{{2}}\alpha_{{2}}I_{{2}}+{a_{{11}}}
^{2}c_{{2}}\alpha_{{1}}I_{{2}}+{a_{{12}}}^{2}\alpha_{{1}}a_{{21}}+{a_{
{11}}}^{2}c_{{2}}{\alpha_{{2}}}^{2}{I_{{2}}}^{2}-a_{{12}}a_{{11}}a_{{
22}}\alpha_{{2}}\nonumber \\
&-a_{{12}}a_{{11}}a_{{22}}\alpha_{{1}}-{a_{{12}}}^{2}c_{{3}}\alpha_{{2}}I_{{2}}-{a_{{12}}}^{2}c_{{3}}\alpha_{{1}}I_{{2}}+a_{{
11}}a_{{12}}c_{{5}}+{a_{{12}}}^{2}\alpha_{{2}}a_{{21}}-a_{{12}}a_{{11}
}c_{{1}}\nonumber \\
&+2\,{a_{{12}}}^{2}\alpha_{{1}}a_{{21}}\alpha_{{2}}I_{{2}}-a_{{
12}}a_{{11}}c_{{1}}{\alpha_{{2}}}^{2}{I_{{2}}}^{2}-a_{{12}}a_{{11}}c_{
{1}}\alpha_{{1}}I_{{2}}-2\,a_{{12}}a_{{11}}c_{{1}}\alpha_{{2}}I_{{2}}\nonumber \\
&+a_{{11}}a_{{12}}c_{{5}}\alpha_{{2}}I_{{2}}+a_{{11}}a_{{12}}c_{{5}}
\alpha_{{1}}I_{{2}}-{a_{{12}}}^{2}c_{{3}}\alpha_{{1}}{I_{{2}}}^{2}
\alpha_{{2}}-{a_{{12}}}^{2}c_{{3}}+a_{{11}}a_{{12}}c_{{5}}\alpha_{{1}}
{I_{{2}}}^{2}\alpha_{{2}}\nonumber \\
&-2\,a_{{12}}a_{{11}}a_{{22}}\alpha_{{1}}
\alpha_{{2}}I_{{2}}-2\,a_{{12}}a_{{11}}c_{{1}}\alpha_{{1}}{I_{{2}}}^{2
}\alpha_{{2}}-a_{{12}}a_{{11}}c_{{1}}\alpha_{{1}}{I_{{2}}}^{3}{\alpha_
{{2}}}^{2}+2\,{a_{{11}}}^{2}c_{{2}}\alpha_{{1}}{I_{{2}}}^{2}\alpha_{{2
}}\nonumber \\
&+{a_{{11}}}^{2}c_{{2}}\alpha_{{1}}{I_{{2}}}^{3}{\alpha_{{2}}}^{2}+{a
_{{11}}}^{2}c_{{2}} ]
\\
h(u,v) &=\Lambda\, \left( 1+\alpha_{{1}}I_{{2}} \right) ^{2}a_{{12}} [
\left( \alpha_{{1}}\Lambda+\Lambda\,{\alpha_{{1}}}^{2}I_{{2}}
\right) v+a_{{12}}-\alpha_{{1}}a_{{11}}u+a_{{12}}\alpha_{{1}}I_{{2}}
] \nonumber \\
& [ \left( \alpha_{{2}}\Lambda+\alpha_{{2}}\Lambda\,
\alpha_{{1}}I_{{2}} \right) v+a_{{12}}-\alpha_{{2}}a_{{11}}u+\alpha_{{
2}}I_{{2}}a_{{12}} ] \left( 1+\alpha_{{2}}I_{{2}} \right) .
\end{align*}
Let
\begin{align}
\bar{a}_2 &= \dfrac{1}{16} [ (H_1)_{uuu} + (H_1)_{uvv} + (H_{2})_{uuv} + (H_2)_{vvv} ] + \dfrac{1}{16( - \Lambda)} [(H_1)_{uv} ((H_1)_{uu} + (H_1)_{vv}) \nonumber \\
& - (H_{2})_{uv} ( (H_{2})_{uu} + (H_2)_{vv} ) - (H_1)_{uu}(H_2)_{uu} + (H_1)_{vv} (H_2)_{vv}].
\end{align}
Then
\begin{align}
\bar{a}_{2} &= \dfrac{3\left( \left( -c_{{1}}\Lambda\,v{\alpha_{{1}}}^{2}I_{{2}}+\Lambda\,va_{{11}}
{\alpha_{{1}}}^{2}-a_{{12}}c_{{1}}\alpha_{{1}}I_{{2}}+a_{{11}}c_{{2}}
\alpha_{{1}}I_{{2}}-c_{{1}}\Lambda\,v\alpha_{{1}}-a_{{12}}c_{{1}}+a_{{
11}}c_{{2}} \right) a_{{12}}{a_{{11}}}^{2}\alpha_{{1}}
\right)}{8 \left( a_{{12}}+\alpha_{{1}}\Lambda\,v \right) ^{4} \left( 1+\alpha_{
{1}}I_{{2}} \right) ^{3}
} \nonumber \\
& - \dfrac{\left( -3\,a_{{11}}c_{{2}}-3\,a_{{11}}c_{{2}}\alpha_{{1}}I_{{2}}+2\,a
_{{12}}a_{{11}}\alpha_{{1}}+a_{{12}}c_{{1}}+a_{{12}}c_{{1}}\alpha_{{1}
}I_{{2}} \right) \alpha_{{1}}{\Lambda}^{2}
}{8 \left( 1+\alpha_{{1}}I_{{2}} \right) {a_{{12}}}^{3}} \nonumber \\
& - \dfrac{1}{8 \Lambda\, \left( 1+\alpha_{{1}}I_{{2}} \right) ^{4}{a_{{12}}}^{4}
\left( 1+\alpha_{{2}}I_{{2}} \right) ^{3}
}[ 2\,a_{{11}}A_{{5}}\alpha_{{1}}\Lambda+6\,a_{{11}}A_{{5}}\alpha_{{1}}
\Lambda\,\alpha_{{2}}I_{{2}}+2\,a_{{11}}A_{{5}}{\alpha_{{1}}}^{2}
\Lambda\,I_{{2}} \nonumber \\
&+4\,a_{{11}}A_{{5}}{\alpha_{{1}}}^{2}\Lambda\,{I_{{2}}
}^{2}\alpha_{{2}}+2\,a_{{11}}A_{{5}}\alpha_{{2}}\Lambda-{a_{{11}}}^{2}
A_{{3}}\alpha_{{1}}-2\,{a_{{11}}}^{2}A_{{3}}\alpha_{{1}}\alpha_{{2}}I_
{{2}}-{a_{{11}}}^{2}A_{{3}}\alpha_{{2}}-a_{{11}}A_{{2}}a_{{12}} \nonumber \\
&-a_{{11
}}A_{{2}}a_{{12}}\alpha_{{2}}I_{{2}}-a_{{11}}A_{{2}}a_{{12}}\alpha_{{1
}}I_{{2}}-a_{{11}}A_{{2}}a_{{12}}\alpha_{{1}}{I_{{2}}}^{2}\alpha_{{2}}
+A_{{4}}\Lambda\,a_{{12}}+A_{{4}}\Lambda\,a_{{12}}\alpha_{{2}}I_{{2}}\nonumber \\
&+2\,A_{{4}}\Lambda\,a_{{12}}\alpha_{{1}}I_{{2}}+2\,A_{{4}}\Lambda\,a_{{
12}}\alpha_{{1}}{I_{{2}}}^{2}\alpha_{{2}}+A_{{4}}\Lambda\,a_{{12}}{
\alpha_{{1}}}^{2}{I_{{2}}}^{2}+A_{{4}}\Lambda\,a_{{12}}{\alpha_{{1}}}^
{2}{I_{{2}}}^{3}\alpha_{{2}}]\nonumber \\
&+ \dfrac{3}{8} {\frac {(-A_{{1}}a_{{12}}-A_{{1}}a_{{12}}\alpha_{{2}}I_{{2}}+A_{{3}}
\alpha_{{1}}\Lambda+2\,A_{{3}}\alpha_{{1}}\Lambda\,\alpha_{{2}}I_{{2}}
+A_{{3}}\alpha_{{2}}\Lambda)}{ \left( 1+\alpha_{{1}}I_{{2}} \right) ^{2
}{a_{{12}}}^{4} \left( 1+\alpha_{{2}}I_{{2}} \right) ^{3}}} \nonumber \\
&- \dfrac{1}{16 \Lambda }[ -2\,{\frac {\Lambda\, \left( -2\,a_{{11}}c_{{2}}-2\,a_{{11}}c_{{2}}
\alpha_{{1}}I_{{2}}+a_{{12}}a_{{11}}\alpha_{{1}}+a_{{12}}c_{{1}}+a_{{
12}}c_{{1}}\alpha_{{1}}I_{{2}} \right) }{{a_{{12}}}^{4} \left( 1+\alpha_{{1}}I_{{2}}
\right) ^{2}}} \nonumber \\
& -2\,{\frac { \left( A_{{5}}\Lambda+A_{{5}}\Lambda\,\alpha_{{1}}I_{{2}}
-a_{{11}}A_{{3}} \right) \left( -a_{{11}}A_{{5}}+A_{{3}}\Lambda+A_{{3
}}\Lambda\,\alpha_{{1}}I_{{2}} \right) }{{\Lambda}^{2} \left( 1+\alpha
_{{1}}I_{{2}} \right) ^{6}{a_{{12}}}^{6} \left( 1+\alpha_{{2}}I_{{2}}
\right) ^{4}}} \nonumber \\
& -4\,{\frac { \left( -a_{{12}}c_{{1}}+a_{{11}}c_{{2}} \right) {a_{{11}}
}^{2}A_{{5}}}{{a_{{12}}}^{5} \left( 1+\alpha_{{1}}I_{{2}} \right) ^{4}
\Lambda\, \left( 1+\alpha_{{2}}I_{{2}} \right) ^{2}}}
4\,{\frac { \left( -c_{{2}}\alpha_{{1}}I_{{2}}+a_{{12}}\alpha_{{1}}-c_
{{2}} \right) {\Lambda}^{2}A_{{3}}}{{a_{{12}}}^{5} \left( 1+\alpha_{{1
}}I_{{2}} \right) ^{2} \left( 1+\alpha_{{2}}I_{{2}} \right) ^{2}}}
].
\end{align}
\end{document} |
\begin{document}
\title{Temporal profile of biphotons generated from a hot atomic vapor and spectrum of electromagnetically induced transparency}
\author{
Shih-Si Hsiao,$^{1,}$\footnote{Electronic address: {\tt shihsi.hsiao@gmail.com}}
Wei-Kai Huang,$^1$
Yi-Min Lin,$^1$
Jia-Mou Chen,$^1$
Chia-Yu Hsu,$^1$ and
Ite A. Yu$^{1,2,}$}\email{yu@phys.nthu.edu.tw}
\address{$^1$Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan \\
$^2$Center for Quantum Technology, Hsinchu 30013, Taiwan
}
\begin{abstract}
We systematically studied the temporal profile of biphotons, i.e., pairs of time-correlated single photons, generated from a hot atomic vapor via the spontaneous four-wave mixing process. The measured temporal width of biphoton wave packet or two-photon correlation function against the coupling power was varied from about 70 to 580 ns. We derived an analytical expression of the biphoton's spectral profile in the Doppler-broadened medium. The analytical expression reveals that the spectral profile is mainly determined by the effect of electromagnetically induced transparency (EIT), and behaves like a Lorentzian function with a linewidth approximately equal to the EIT linewidth. Consequently, the biphoton's temporal profile influenced by the Doppler broadening is an exponential-decay function, which was consistent with the experimental data. Employing a weak input probe field of classical light, we further measured the EIT spectra under the same experimental conditions as those in the biphoton measurements. The theoretical predictions of the biphoton wave packets calculated with the parameters determined by the classical-light EIT spectra are consistent with the experimental data. The consistency demonstrates that in the Doppler-broadened medium, the classical-light EIT spectrum is a good indicator for the biphoton's temporal profile. Besides, the measured biphoton's temporal widths well approximated to the predictions of the analytical formula based on the biphoton's EIT effect. This study provides an analytical way to quantitatively understand the biphoton's spectral and temporal profiles in the Doppler-broadened medium.
\end{abstract}
\maketitle
\newcommand{\FigOne}{
\begin{figure}
\caption{
Transition diagram of the SFWM process. Under the presence of the pump and coupling fields, the vacuum fluctuation induces the generation of a pair of the signal and probe single photons. The pump transition has a large magnitude of the one-photon detuning, $\Delta_p$, such that the excitation to state $|4\rangle$ is negligible. In the experiment, a hot vapor of $^{87}
\label{fig:transition_diagram}
\end{figure}
}
\newcommand{\FigTwo}{
\begin{figure}
\caption{
(a) Theoretical predictions of the biphoton's EIT spectrum of a Doppler-broadened medium. Solid lines are the numerical results calculated from Eq.~(\ref{eq:T_exact}
\label{fig:eit_simulation}
\end{figure}
}
\newcommand{\FigThree}{
\begin{figure}
\caption{
(a) Theoretical predictions of the biphoton's FWM spectrum of a Doppler-broadened medium. Solid lines are the numerical results calculated from Eq.~(\ref{eq:FWM_part}
\label{fig:fwm_simulation}
\end{figure}
}
\newcommand{\FigFour}{
\begin{figure}
\caption{
Comparisons between the linewidths of biphoton's EIT, FWM, and overall spectra, i.e., $\Gamma_{\rm EIT}
\label{fig:three_linewidth_comparison}
\end{figure}
}
\newcommand{\FigFive}{
\begin{figure}
\caption{
Theoretical predictions of the biphoton's overall spectrum of a Doppler-broadened medium. Solid lines are the numerical results given by Eq.~(\ref{eq:biphoton_frequency}
\label{fig:biphoton_simulation}
\end{figure}
}
\newcommand{\FigSix}{
\begin{figure}
\caption{
(a)-(e) From top to bottom, blue lines are the measured EIT spectra at the coupling powers of 0.5, 1, 2, 4, and 8 mW, and red lines are the theoretical predictions calculated from Eq.~(\ref{eq:T_exact}
\label{fig:experiment_data}
\end{figure}
}
\newcommand{\FigSeven}{
\begin{figure}
\caption{
Temporal width of the biphoton wave packet as a function of the coupling power. Red circles are obtained from the best fits of the biphoton wave packets as the examples shown in Figs.~\ref{fig:experiment_data}
\label{fig:eit_biphoton_linewidth}
\end{figure}
}
\section{Introduction} \label{sec:introduction}
The biphoton is a pair of time-correlated single photons. When the first photon is detected to trigger a quantum operation, the second photon in the same pair will be employed in the quantum operation. The first and second photons can be regarded as the heralding and heralded single photons. Biphotons can be used in quantum information processing~\cite{QI1,QI2,QI3,QI4,QI5,QI6}, such as quantum communication~\cite{Q_communication_1,Q_communication_2,Q_communication_3}, quantum memory~\cite{Q_memory_1,Q_memory_2,Q_memory_3,Q_memory_4}, and quantum interference~\cite{Q_interference_1,Q_interference_2,Q_interference_3}.
One of the mechanisms to produce biphotons is the spontaneous four-wave mixing (SFWM) process~\cite{SWFM1,SWFM2,SWFM3,SWFM4,SWFM5,SWFM6,SWFM7,SWFM8,SWFM9,SWFM10,SWFM11,SWFM12,SWFM13,SWFM14,SWFM15,SWFM16,SWFM17,SWFM18,SWFM19, SWFM20,SWFM21,SWFM22,SWFM23,SWFM24}, which has been commonly employed with media of cold or hot atomic vapors. There are several theoretical models to study the temporal profile of the biphoton. In Ref.~\cite{SWFM14}, S.~Du {\it et al}. developed the theory of the time-correlation function of biphotons in a cold atomic vapor. The authors systematically studied the two-photon correlation function, i.e., the biphoton wave packet, discussed the properties of each term in wave packet, and predicted the effects of the dephasing rate and propagation delay time on the temporal profile of the biphoton wave packet. Based on the theoretical model in Ref.~\cite{SWFM14}, several groups conducted experiments to study the temporal profiles of biphotons with cold atoms~\cite{SWFM7,SWFM8, SWFM9,SWFM10,ColdAtoms1,ColdAtoms2}. Moreover, in Ref.~\cite{SWFM8} and \cite{SWFM14} the authors derive the analytical expressions of time correlated function, providing the convenient method to expected the temporal profile of biphoton in cold atom system.
The SFWM-generated biphoton source of a room-temperature or hot atomic vapor has the merits of adjustable linewidth, tunable frequency, and high generation rate. In Ref.~\cite{SWFM18}, C.~Shu {\it et al}. reported the generation of biphotons from a hot atomic vapor, and provided the numerical simulation for the time-correlation function of biphotons. After Ref.~\cite{SWFM18}, several groups improved the efficiency of biphoton generation with hot-atom sources, and compared data with results of numerical simulations~\cite{SWFM23, SWFM24, HotAtom1}.
However, there is no analytical formula available for the temporal profile of biphoton's time-correlation function or wave packet in the Doppler-broadened medium so far, and the systematic study on the temporal profile has not been reported before.
In this article, we report the systematic study on the temporal profile of the SFWM biphotons generated from a Doppler-broadened medium. Although the spectral profile of the biphoton is influenced by the electromagnetically induced (EIT) effect, the four-wave mixing (FWM) process, the phase matching condition, and the Doppler effect as well as the Boltzmann distribution of the velocity, we derived an analytical expression to show that the biphoton's overall spectrum is approximately Lorentzian. Thus, the temporal profile of the biphoton wave packet behaves like an exponential-decay function with the time constant, which is mainly determined by the EIT effect. Furthermore, we experimentally demonstrated that the measured biphoton's temporal profile is predominately characterized by the measured EIT spectrum. The predictions of the biphoton's temporal width calculated from the analytical formula are in good agreement with the experimental data.
We organize the article as follows. In Sec.~II, we start with the time-correlation function or the wave packet of the SFWM-generated biphoton in a Doppler-broadened medium, and derive the analytical expressions or formulas of the biphoton's spectral and temporal profiles. In Sec.~III, we describe our experimental setup for the measurement of the EIT spectrum with classical light and that of the biphoton wave packet. In Sec.~IV, we presented the data of the EIT spectra and the biphoton wave packets, and compare them with the predictions from the theoretical model. Finally, we give a conclusion in Sec.~V.
\section{Theoretical Predictions}
Biphotons or a pair of single photons are produced from a hot atomic vapor with the spontaneous four-wave mixing process (SFWM). Figure~\ref{fig:transition_diagram} shows the relevant energy levels and transitions of SFWM process for the generation of biphotons. All population is placed in the ground state of $|1\rangle$. The pump field is far detuned from the transition of $|1\rangle$ $\rightarrow$ $|4\rangle$, and the coupling field drive the transition of $|2\rangle$ $\rightarrow$ $|3\rangle$ resonantly. Due to the vacuum fluctuation, a pair of anti-Stokes and Stokes photons can be spontaneously emitted. The frequencies of the pump field, anti-Stokes photon, coupling field, and Stokes photon form the resonant four-photon transition. Note that, we employ the hyperfine optical pumping (HOP) field to empty the population in $|2\rangle$, which may contribute some residual light of the HOP field in its hollow region. The anti-Stokes photon propagates with the speed of light in vacuum since the pump field and anti-Stokes form the Raman process with the large one-photon detuning. The Stokes photon is slow light due to the effect of electromagnetically induced transparency (EIT) in two-photon on resonance of the coupling field and Stokes photon.
\FigOne
The temporal profile of biphoton wave packet is determined by the theory of the time correlation function between the anti-Stokes and Stokes photons. Considering phase-match condition, the time correlation function is shown below \cite{SWFM14}.
\begin{eqnarray}
G^{(2)}(\tau) =
\left| \int_{-\infty}^{\infty} d\delta \frac{1}{2\pi} e^{-i\delta\tau} F(\delta) \right|^2,
\label{eq:biphoton}
\end{eqnarray}
\begin{eqnarray}
F(\delta) =
\frac{\sqrt{k_{as} k_s}L}{2} \chi(\delta)
{\rm sinc} \left[ \frac{k_s L}{4} \xi(\delta) \right]
e^{i(k_s L/4) \xi(\delta)} ,
\label{eq:biphoton_frequency}
\end{eqnarray}
where $\delta$ is the two-photon detuning between the anti-Stokes photon and pump field (or $-\delta$ is that between the Stokes photon and coupling field), $\tau$ is the delay time of the Stokes photon, $k_{as}$ and $k_s$ are the wave vectors of the two photons, $L$ is the medium length, $\chi(\delta)$ is the cross-susceptibility of the Stokes photon induced by the anti-Stokes photon, and $\xi(\delta)$ is the self-susceptibility of the Stokes photon. Considering the Doppler-broadened medium and the all-copropagating scheme, the self-susceptibility and cross-susceptibility are given by
\begin{eqnarray}
\label{eq:self_chi}
\frac{k_s L}{4} \xi(\delta) \!\!\! &=& \!\!\! \frac{\alpha_s\Gamma_3}{2} \int_{-\infty}^{\infty} d\omega_D
\frac{e^{-\omega_D^2/\Gamma_D^2}}{\sqrt{\pi}\Gamma_D}
\\ &\times&
\frac{\delta+i\gamma}{\Omega_c^2-4(\delta+i\gamma)(\delta+\omega_D+i\Gamma_3/2)},
\nonumber
\\
\label{eq:cross_chi}
\frac{\sqrt{k_{as} k_s}L}{2} \chi(\delta) \!\!\! &=& \!\!\!
\frac{\sqrt{\alpha_{as}\alpha_s}\sqrt{\Gamma_3\Gamma_4}}{4} \int_{-\infty}^{\infty} d\omega_D
\frac{e^{-\omega_D^2/\Gamma_D^2}}{\sqrt{\pi}\Gamma_D}
\\ &\times&
\frac{\Omega_p}{\Delta_p-\omega_D + i\Gamma_4/2}
\nonumber \\ &\times&
\frac{\Omega_c}{\Omega_c^2-4(\delta+i\gamma)(\delta+\omega_D+i\Gamma_3/2)},~~
\nonumber
\end{eqnarray}
where $\omega_D$ is the Doppler shift, $\Gamma_D$ is the Doppler width, $\alpha_s = n \sigma_s L$ ($n$ is the atomic density and $\sigma_s$ is the resonant absorption cross section of the Stokes transition) is the optical depth of the entire atoms interacting with the Stokes photon on resonance, $\alpha_{as}$ means the similar optical depth of the anti-Stokes transition, $\Omega_p$ and $\Omega_c$ are the Rabi frequencies of the pump and coupling fields, $\Gamma_3$ and $\Gamma_4$ are the spontaneous decay rates of the excited states (i.e., $|3\rangle$ and $|4\rangle$ in Fig.~\ref{fig:transition_diagram}) in the Stokes and anti-Stokes transitions, respectively, $\Delta_p$ is the detuning of the pump field, and $\gamma$ is the dephasing rate of the ground-state coherence, i.e., the decoherence rate. Since the difference between $\Gamma_3$ and $\Gamma_4$ is merely about 5\% in our case, we neglect the difference and set $\Gamma_3 = \Gamma_4 \equiv \Gamma = 2\pi\times6~{\rm MHz}$ in this work. Due to the temperature of the vapor cell being 57 $^{\circ}$C, $\Gamma_D =$ 54$\Gamma$ in the calculation~\cite{OurSR2018}.
In the biphoton's spectral profile shown in Eq.~(\ref{eq:biphoton_frequency}), we first derive the analytical expression corresponding to the EIT effect, i.e.,
\begin{eqnarray}
T_{\rm EIT} \!\!&=&\!\!
\left| \exp\!\! \left[ i\frac{k_s L}{4} \xi(\delta) \right] \right|^2
\nonumber \\
\!\!&=&\!\!
\exp\!\! \left[ -\alpha_s' \!\! \int_{-\infty}^{\infty} \!\! d\omega_D
\:e^{-\omega_D^2/\Gamma_D^2}
\frac{\beta}{(\omega_D-\omega_0)^2+\beta^2} \right],~~
\label{eq:T_exact}
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:alpha_s_p}
\alpha_s'&=&\alpha_s \frac{\sqrt{\pi}\Gamma}{4\Gamma_D} \\
\label{eq:x_0}
\omega_0 &=&
\frac{\delta \Omega_c^2}{4(\delta^2+\gamma^2)} -\delta, \\
\label{eq:beta}
\beta &=&
\frac{2 \delta^2 \Gamma+\gamma(2\gamma\Gamma+\Omega_c^2)}
{4(\delta^2+\gamma^2)}
\approx \frac{2 \delta^2 \Gamma+\gamma\Omega_c^2}{4(\delta^2+\gamma^2)}.
\end{eqnarray}
The above approximation is valid under the assumption of $2\gamma\Gamma \ll \Omega_c^2$, which is very reasonable in typical EIT experiments. We name $T_{\rm EIT}$ as the biphoton's EIT spectrum.
In Eq.~(\ref{eq:T_exact}), the integral results in the Voigt function, i.e., the convolution between a Gaussian and a Lorentzian functions. The Voigt function can be expressed as the real part of the Faddeeva function, $w(z)$. Thus,
\begin{equation}
T_{\rm EIT} = \exp \left\{ - \alpha_s' {\rm Re}[w(z)] \right\},
\end{equation}
where $w(z)\equiv {\rm exp}(-z^2) {\rm erfc}(-i z)$ and $z=(\omega_0+i \beta)/\Gamma_D$. Re[$w(z)$] can be approximated as a Lorentzian function of the two-photon detuning ($\delta$) given by
\begin{equation}
{\rm Re}[w(z)] \approx R\left(1-\frac{A}{1+4\delta^2/\Gamma_L^2}\right),
\label{eq:F_approximation}
\end{equation}
where
\begin{eqnarray}
\label{eq:R}
R &=& \:\exp\! \left( \frac{1}{4 \Gamma_D^2} \right)
\:{\rm erfc}\! \left( \frac{1}{2 \Gamma_D} \right), \\
\label{eq:A}
A &=& 1-\frac{1}{R} \:\exp\! \left( \frac{\Omega_c^4}{16 \gamma^2 \Gamma_D^2} \right)
\:{\rm erfc}\! \left( \frac{\Omega_c^2}{4 \gamma \Gamma_D} \right), \\
\label{eq:linewidth}
\Gamma_L &=& 2\gamma
\left( 1 +\frac{\Omega_c^2}{4\gamma \Gamma_D} \right).
\end{eqnarray}
In Eq.~(\ref{eq:F_approximation}), $A/(1+4\delta^2/\Gamma_L^2)$ is the Lorentzian function and $\Gamma_L$ is its full width at half maximum (FWHM). Denote the Lorentzian function as $f(\delta)$ and $1-{\rm Re}[w(z)]/R$ as $g(\delta)$. The percentage difference between them, i.e., $\int (f- g)^2 d\delta / \int (f \cdot g) d\delta$, is always less than 5\% as long as $\gamma >$ 0.01$\Gamma$. When $\Omega_c^2/(4\gamma \Gamma_D) \gg 1$, Eq.~(\ref{eq:F_approximation}) becomes not a good approximation. Using Eq.~(\ref{eq:F_approximation}), we obtain the biphoton's EIT spectrum in the Doppler-broadened medium as the following:
\begin{eqnarray}
\label{eq:T_approximation}
T_{\rm EIT} &=& \:\exp\! \left[ -\alpha_s' R
\left( 1-\frac{A}{1+ 4\delta^2/\Gamma_L^2} \right) \right].
\end{eqnarray}
The FWHM, peak height, and baseline in the above equation are exactly the same as those in the equation below.
\begin{eqnarray}
\label{eq:T_approximation2}
T_{\rm EIT} &\approx& e^{-\alpha_s' R}
\left[1+ \frac{ e^{\alpha_s' RA}-1}{1+4\delta^2/\Gamma_{\rm EIT}^2} \right],\\
\label{eq:Gamma_EIT}
\Gamma_{\rm EIT} &=&
\Gamma_L \sqrt{\frac{ \alpha_s' R A }{ \ln[(1+e^{\alpha_s' R A})/2] } -1}.
\end{eqnarray}
The percentage difference between the right-hand sides of Eqs.~(\ref{eq:T_approximation}) and (\ref{eq:T_approximation2}) is always less than 5\%, and a smaller value of $\alpha_s' R A$ makes the difference less. According to Eq.~(\ref{eq:T_approximation2}), the biphoton's EIT spectrum is approximated as a Lorentzian peak on top of a baseline \cite{OurSR2018}.
\FigTwo
We compare the biphoton's EIT spectrum calculated from the numerical integral of Eq.~(\ref{eq:T_exact}) with that calculated from the analytical formula of Eq.~(\ref{eq:T_approximation}). Figure~\ref{fig:eit_simulation}(a) shows the comparisons at different values of the coupling Rabi frequency, $\Omega_c$. As $\Omega_c$ becomes larger, i.e., a larger value of $\Omega_c^2/(4\gamma\Gamma_D)$, the deviation between numerical and analytical spectra becomes larger. Figure~\ref{fig:eit_simulation}(b) shows the percentage difference between the numerical and analytical values of the FWHM of biphoton's EIT spectrum. One can observe the percentage difference increases with the value of $\Omega_c^2/(4 \gamma \Gamma_D)$.
We next derive the four-wave mixing (FWM) effect in the biphoton spectral profile shown in Eq.~(\ref{eq:biphoton_frequency}), named the biphoton's FWM spectrum thereafter. The spectrum of FWM transmission is the square of absolute value of Eq.~(\ref{eq:cross_chi}),
\begin{eqnarray}
\label{eq:FWM_part}
T_{\rm FWM}= \left| \frac{\sqrt{k_{as} k_s}L}{2} \chi(\delta) \right|^2.
\end{eqnarray}
At a large value of the pump detuning $\Delta_p \gg \Gamma$, which is the typical experimental condition in the biphoton generation, the term $\Omega_p/(\Delta_p-\omega_D + i\Gamma/2 )$ in Eq.~(\ref{eq:cross_chi}) is treated as a constant of $\Omega_p/\Delta_p$. The integration can be approximated as the following complex Lorentz function:
\begin{eqnarray}
\label{eq:FWM_part_th}
T_{\rm FWM} &=&
\left| \frac{\Gamma\sqrt{\alpha_{as}\alpha_s}}{4}
\frac{\Omega_p}{\Delta_p} \right. \left. \frac{B}{1-i (2 \delta/\Gamma_{\rm FWM})} \right|^2, \\
\label{eq:Gamma_FWM}
\Gamma_{\rm FWM} &=& 2\gamma
\left( 1 +\frac{\Omega_c^2}{4\gamma \Gamma_D} \right)
= \Gamma_L, \\
\label{eq:B}
B &=& \frac{\sqrt{\pi}\Omega_c}{4 \gamma \Gamma_D}
\:{\rm exp}\! \left( \frac{\Omega_c^4}{16 \gamma^2 \Gamma_D^2} \right)
\:{\rm erfc}\! \left( \frac{\Omega_c^2}{4 \gamma \Gamma_D} \right).
\end{eqnarray}
The above approximation is valid under the condition of $\Gamma_{\rm FWM} \gg \Omega_c^2/(5.4 \Gamma_D)$, where $\Omega_c^2/(5.4 \Gamma_D)$, obtained numerically, is the separation between the two transmission peaks in the biphoton's FWM spectrum. Thus, the spectral profile of cross-susceptibility is a Lorentzian function with a FWHM of $\Gamma_{\rm FWM}$, which is equal to $\Gamma_L$.
\FigThree
We compare the biphoton's EIT spectrum calculated from the numerical integral of Eq.~(\ref{eq:FWM_part}) with that calculated from the analytical formula of Eq.~(\ref{eq:FWM_part_th}). Figure~\ref{fig:fwm_simulation}(a) shows the numerical and analytical spectra of FWM transmission at the coupling Rabi frequencies $\Omega_c$ of 2.7$\Gamma$, 3.8$\Gamma$ and 5.4$\Gamma$, respectively. The deviation between the numerical and analytical spectra is more prominent as $\Omega_c$ increases, because the separation between the two transmission peaks in the FWM spectrum gets larger due to a larger value of $\Omega_c$. Figure~\ref{fig:fwm_simulation}(b) shows the percentage difference between the numerical and analytical values of the FWHM of biphoton's FWM spectrum. One can observe the difference increases with the value of $\Omega_c^2/(4 \gamma \Gamma_D)$.
\FigFour
The biphoton's overall spectrum shown by Eq.~(\ref{eq:biphoton_frequency}) is approximately equal to the product of the biphoton's EIT spectrum, $T_{\rm EIT}$, and the biphoton's FWM spectrum. $T_{\rm FWM}$. Since $|{\rm sinc} \! \left[ k_s L \xi(\delta) /4 \right]|$ is slowly-varying within the biphoton's spectral width, Eq.~(\ref{eq:biphoton_frequency}) without ${\rm sinc} \! \left[ k_s L \xi(\delta) /4 \right]$ changes the result little. As shown in Eq.~(\ref{eq:T_approximation2}), the analytical expression of biphoton's EIT spectrum is proportional to
\begin{equation}
\label{eq:eit_bi}
1+ \frac{\sigma-1}{1+ 4\delta^2/\Gamma_{\rm EIT}^2},
\end{equation}
where $\sigma = e^{\alpha'_s R A}$ is a constant independent of $\delta$. As shown in Eq.~(\ref{eq:FWM_part_th}), the analytical expression of the biphoton's FWM spectrum is proportional to
\begin{equation}
\label{eq:fmw_bi}
\frac{1}{1+4\delta^2/(\eta \Gamma_{\rm EIT})^2},
\end{equation}
where $\eta^{-1} = \sqrt{\alpha'_s R A / \ln[(1+e^{\alpha'_s R A})/2] -1}$ based on Eqs.~(\ref{eq:Gamma_EIT}) and (\ref{eq:Gamma_FWM}), which is independent of $\delta$. As the value of $\alpha'_s R A$ is not large, we have $\sigma \approx \eta^2$, and the product of the biphoton's EIT and FWM spectral profiles gives
\begin{eqnarray}
&& \left( 1+ \frac{\sigma-1}{1+4\delta^2/\Gamma_{\rm EIT}^2} \right) \times
\frac{1}{1+4\delta^2/(\eta \Gamma_{\rm EIT})^2} \nonumber \\
&& ~\approx
\frac{\eta^2}{1+4\delta^2/\Gamma_{\rm EIT}^2}.
\end{eqnarray}
As the value of $\alpha'_s R A$ is large, $\eta$ and $\sigma$ become large. A large $\eta$ makes the biphoton's FWM linewidth far greater than the biphoton's EIT linewidth, indicating that the product of the biphoton's EIT and FWM spectral profiles is dominated by the EIT profile. We obtain
\begin{eqnarray}
&& \left( 1+ \frac{\sigma-1}{1+4\delta^2/\Gamma_{\rm EIT}^2} \right) \times
\frac{1}{1+4\delta^2/(\eta \Gamma_{\rm EIT})^2} \nonumber \\
&& ~\approx
1+ \frac{\sigma-1}{1+4\delta^2/\Gamma_{\rm EIT}^2}
\approx \frac{\sigma}{1+4\delta^2/\Gamma_{\rm EIT}^2}.
\end{eqnarray}
The second approximation in the above is due to $\sigma \gg 1$. Thus, the linewidth, $\Gamma_{\rm BI}$, of biphoton's overall spectrum is close to that, $\Gamma_{\rm EIT}$, of the biphoton's EIT spectrum.
In Fig.~\ref{fig:three_linewidth_comparison}, we compare the ratio of $\Gamma_{\rm EIT}$ to $\Gamma_{\rm BI}$ and that of $\Gamma_{\rm FWM}$ to $\Gamma_{\rm BI}$, where $\Gamma_{\rm EIT}$, $\Gamma_{\rm BI}$, and $\Gamma_{\rm FWM}$ are numerically calculated from Eqs.~(\ref{eq:biphoton_frequency}), (\ref{eq:T_exact}), and (\ref{eq:FWM_part}), respectively. The ratios are plotted against $\Omega_c^2$ at various values of the OD ($\alpha_s$) and decoherence rate ($\gamma$).
In all the cases, it is apparent that the linewidth of the biphoton's overall spectrum is nearly the same that of the biphoton's EIT spectrum, i.e.,
\begin{equation}
\label{eq:linewidth_bi}
\Gamma_{\rm BI} \approx \Gamma_{\rm EIT}.
\end{equation}
The outcome of the comparison between the numerical results shown in Fig.~\ref{fig:three_linewidth_comparison} is consistent with the expectation of the discussion utilizing the analytical expressions in Eqs.~(\ref{eq:eit_bi}) and (\ref{eq:fmw_bi}).
The numerical and analytical results of the biphoton's overall spectrum are compared in Fig.~\ref{fig:biphoton_simulation}(a). The solid lines are numerically calculated from the square of Eq.~(\ref{eq:biphoton_frequency}), and the dashed lines are the results of the analytical formula in Eq.~(\ref{eq:T_approximation2}).
Figure~\ref{fig:biphoton_simulation}(b) shows the percentage difference between the numerical and analytical values of the FWHM of biphoton's overall spectrum. The numerical linewidth is determined by the Lorentzian best fit of the biphoton spectrum calculated from Eq.~(\ref{eq:biphoton_frequency}). The analytical linewidth is given by the formula in Eq.~(\ref{eq:linewidth_bi}) or equivalently Eq.~(\ref{eq:Gamma_EIT}). In Fig.~\ref{fig:biphoton_simulation}(b), the percentage difference increases with the value of $\Omega_c^2/(4 \gamma \Gamma_D)$.
\FigFive
We have derived the analytical formula of the biphoton's self-susceptibility [$(k_s L/4) \xi(\delta)$] or EIT spectrum as shown by Eq.~(\ref{eq:T_approximation2}), and that of the biphoton's cross-susceptibility [$(\sqrt{k_{as} k_s}L/2) \chi(\delta)$] or FWM spectrum as shown by Eq.~(\ref{eq:FWM_part_th}). In addition, ${\rm sinc} \! \left[ k_s L \xi(\delta) /4 \right]$ in Eq.~(\ref{eq:biphoton_frequency}) plays little role in the spectral profile. Therefore, the analytical expression of Eq.~(\ref{eq:biphoton_frequency}) is obtained, and the time-correlation function or the wave packet of biphotons in Eq.~(\ref{eq:biphoton}) becomes
\begin{eqnarray}
\label{eq:g2_approx}
G^{(2)}(\tau) &=&
C e^{-\Gamma_{\rm BI} t}, \\
\label{eq:C_value}
C & \propto & (\alpha_{as}\alpha_s \Gamma^2) \frac{\Omega_p^2}{\Delta_p^2} B^2 e^{-\alpha_s'R(1-A)}.
\end{eqnarray}
The biphoton's wave packet is an exponential-decay function with the decay time constant of $1/\Gamma_{\rm BI}$. As long as the value of $\Omega_c^2/(4\gamma \Gamma_D)$ is not far greater than 1, Eq.~(\ref{eq:g2_approx}) is a reasonable approximation.
\section{Experimental Setup}
We experimentally studied the temporal width and spectral linewidth of SFWM biphotons generated from a hot vapor of $^{87}$Rb atoms. The SFWM transition scheme is shown in Fig.~\ref{fig:transition_diagram}, and the actual energy levels in the experiment are specified in the caption. The time correlation function (i.e., the wave packet) of biphotons and the corresponding EIT spectrum were measured as functions of the coupling power or the square of the coupling Rabi frequency $\Omega_c^2$. In the measurement of the EIT spectrum, a weak probe laser field was employed in additional to the coupling field, and its Rabi frequency is far less than $\Omega_c$ to satisfy the perturbation limit and to affect the ground-state population distribution very little. The two laser fields form the $\Lambda$-type transition scheme with zero one-photon detuning, and their transitions are depicted in Fig.~1(a). Both laser fields have the wavelengths of about 795 nm. The probe laser was injection-locked by the coupling laser with an offset frequency, which was provided by an fiber-based electro-optic modulator (EOM) with an input RF frequency of around 6835 MHz. The injection-lock scheme ensures the two-photon detuning, $\delta$, between the probe and coupling fields to be stable and free of the laser frequency fluctuations. We swept the RF frequency of the EOM, i.e., the probe frequency which is equal to the two-photon detuning, and measured the probe transmission in the spectroscopic measurement.
The coupling and input probe fields completely overlapped and propagated through the same direction in the measurement of the EIT spectrum, minimizing the decoherence rate induced by the Doppler effect. The two fields had the $s$ and $p$ polarizations, and the $e^{-2}$ full widths of 1.5 and 1.4 mm. The output probe beam was collected by a polarization-maintained fiber (PMF), and its power after the PMF was detected by a Thorlabs PDA36A photo detector. The PMF's collection efficiency for the probe field is 70\%. We produced the coupling (or pump) field from an external-cavity diode laser of Toptica DL DLC pro 795 (or 780). A homemade 795 nm bare-diode laser under the injection lock generated the probe field. The coupling frequency was stabilized by the saturated absorption spectroscopy, and the pump frequency was stabilized by a wave meter.
In the measurement of the biphoton wave packet or two-photon correlation function as a function of the delay time between the photon pair, we employed only the coupling and pump fields. The coupling field is the same as that used in the measurements of the EIT spectrum. The pump field had the $p$ polarization. We set the pump detuning to $2$ GHz. The $e^{-2}$ full width of the pump field is 1.2 mm. Pairs of anti-Stokes (or signal) and Stokes (or probe) photons were generated by the spontaneous four-wave mixing (SFWM) process. The coupling and pump laser fields and the signal and Stokes probe photons all propagated in the same direction. This all-copropagation scheme can satisfy the phase match condition. The biphotons or the pairs of anti-Stokes and Stokes photons were collected by two PMFs, and their counts after the PMFs detected by two single-photon counting modules (SPCMs). To block the leakages of the strong coupling and pump fields to the SPCMs, we utilized polarization filters and etalon filters which totally provide the extinction ratios of about 130 dB.
We applied a hyperfine optical pumping (HOP) field to optically pump the population out of the state of $|5S_{1/2}, F=1\rangle$ during the measurements of the biphoton wave packet and the EIT spectrum. The HOP field has a hollow-shaped beam profile, while the interaction region of the SFWM process and the EIT spectrum is in its hollow region. We set the power of the HOP field to 9 mW and stabilized its frequency to 80 MHz below the transition frequency of $|5S_{1/2},F=1\rangle$ $\rightarrow$ $|5P_{3/2},F=2\rangle$. The HOP field increased decoherence rate a little in the experimental system. Other details of the experimental setup and measurement method of the biphoton wave packet can be found in our previous works \cite{Ourhotbiphoton1, Ourhotbiphoton2}.
\section{Results and Discussion} \label{sec:results}
We systematically measured the EIT spectra and the biphoton wave packets. The temperature of the vapor cell, which consists of nearly only $^{87}$Rb atoms, was maintained at 57 $^{\circ}$C throughout the experiment. At this temperature, $\Gamma_D =$ 54$\Gamma$. We utilized the absorption spectrum of a weak probe field in the presence of the coupling and HOP fields to determine the value of $\alpha_s$ (OD). In the measurement of the absorption spectrum, the probe frequency was swept across the entire Doppler-broadened spectral lines of $|5S_{1/2},F=2\rangle$ $\rightarrow$ $|5P_{1/2},F=1\rangle$ and $|5S_{1/2},F=2\rangle$ $\rightarrow$ $|5P_{1/2},F=2\rangle$. We fitted the absorption spectrum and determined $\alpha_s =$ 360$\pm$10.
In the measurement of biphoton wave packets, we used five different coupling powers of 0.5, 1, 2, 4, and 8 mW. The coupling Rabi frequency was determined by the EIT spectrum, and the determination method will be explained in the next paragraph. The pump power or Rabi frequency was fixed to 0.5 mW or approximately 2.0$\Gamma$, which is estimated according to the peak intensity of the pump beam. In the measurement of EIT spectra, we set the probe power to 0.5 $\mu$W. The probe field was weak enough that it can be treated as the perturbation. The coupling power in each EIT spectrum was the same as that in the corresponding biphoton wave packet.
\FigSix
Figures~\ref{fig:experiment_data}(a)-(e) show the EIT spectra at various coupling powers. The blue lines represent the experimental data. The red lines are the theoretical predictions calculated from with Eq.~(\ref{eq:T_exact}). In the theoretical calculation, we set the OD ($\alpha_s$) to 360 as determined by the absorption spectrum, and kept the ratio between the five coupling Rabi frequencies ($\Omega_c$) as the square root of the ratio between the five coupling powers, i.e., the Rabi frequencies correspond to $\Omega_{c0}$, $\sqrt{2}\Omega_{c0}$, $2\Omega_{c0}$, $2\sqrt{2}\Omega_{c0}$, and $4\Omega_{c0}$. We tuned a single value of $\Omega_{c0}$ for all EIT spectra, and varied the value of the decoherence rate ($\gamma$) for each EIT spectrum to get a good match between the data and the predictions. The value of $\Omega_{c0}$ used in the calculation was 1.75$\Gamma$, while the Rabi frequency estimated from the peak intensity of the 0.5-mW coupling power was 2.05$\Gamma$. When we boosted the coupling power by 16 folds, the value of $\gamma$ increased by a factor of 4.3. The change of the biphoton's temporal width was mainly influenced by the coupling power. The consistency between the experimental data and the theoretical predictions is satisfactory.
The green dashed lines in Figs.~\ref{fig:experiment_data}(a)-(e) are the best fits of Lorentzian functions, since the EIT spectral profile is approximately a Lorentzian function as shown by Eq.~(\ref{eq:T_approximation2}). The measured EIT spectral profile of a Doppler-broadened medium is very close to a Lorentzian function. Please note that the biphoton's EIT spectrum is given by $|\exp[i(k_s L/4)\xi]|^2$ as shown in Eq.~(\ref{eq:T_exact}), and the EIT spectrum with an input probe field is given by $|\exp[i(k_s L/2) \xi]|^2$. The difference of a factor of 2 between the two expressions is as a matter of fact that the probe photons resulted from the SFWM process are generated everywhere in a medium, and they propagate through a half of the medium length on average. To apply Eq.~(\ref{eq:T_approximation}) or (\ref{eq:T_approximation2}) to the EIT spectrum with an input probe field, $\alpha_s'$ in the equation needs to be redefined as $\alpha_s \sqrt{\pi}\Gamma/(2\Gamma_D)$, which differs from Eq.~(\ref{eq:alpha_s_p}) by a factor of 2.
\FigSeven
The blue lines in Figs.~\ref{fig:experiment_data}(f)-(j) are the experimental data of biphoton wave packets taken at the same coupling powers as those in Figs.~\ref{fig:experiment_data}(a)-(e), respectively. The red lines are the theoretical predictions calculated from Eq.~(\ref{eq:biphoton}). We set the calculation parameters of $\Omega_c$, $\alpha_s$, and $\gamma$ to the values determined by the absorption and EIT spectra. A Lorentzian function with a linewidth of 35 MHz, corresponding to the spectral profile of the etalon filter used in the experiment, was added to Eq.~(\ref{eq:biphoton_frequency}) in the calculation. Since the biphoton wave packets have the linewidths between 270 kHz and 2.2 MHz, the etalon filter's linewidth affects the calculation results little. The green dashed lines are the best fits of exponential-decay functions, i.e., $y(t) = y_0+A \times{\rm exp} [-(t-t0)/\tau]$, since the biphoton wave packet behaves approximately like an exponential-decay function as shown by Eq.~(\ref{eq:g2_approx}). We first determined the value of $y_0$ from the baseline of data, then fixed the value of $t_0$, and finally fitted the values of $A$ and $\tau$. The theoretical predictions and the exponential-decay best fits are in good agreement with the experimental data. In Fig.~\ref{fig:experiment_data}, we have demonstrated that the EIT spectrum measured with classical light can be a good indicator of the biphoton wave packet generated by the SFWM process.
To verify whether the temporal width of the measured biphoton wave packet is consistent with the analytical formula of $1/\Gamma_{\rm BI}$, we plot the biphoton's temporal width as a function of the coupling power in Fig.~\ref{fig:eit_biphoton_linewidth}. The red circles are the biphoton's temporal widths obtained from the $e^{-1}$ time constants of the best fits as the examples shown in Figs.~\ref{fig:experiment_data}(f)-(j). Each data point is the average of repeated measurements. As for the analytical formulas of $1/\Gamma_{\rm BI}$, the blue squares are the results calculated from Eq.~(\ref{eq:Gamma_EIT}) or (\ref{eq:linewidth_bi}). The calculation parameters of $\Omega_c$, $\alpha_s$, and $\gamma$ are the values determined in Figs.~\ref{fig:experiment_data}(a)-(e). Since the values of $\Omega_c^2/ (4\gamma\Gamma_D)$ of the five coupling powers ranged between 0.71 and 2.6, Eq.~(\ref{eq:g2_approx}) as well as the formula of $1/\Gamma_{\rm BI}$ is a good approximation. As expected, the agreement between the data of the measured temporal width and the predictions of the analytical formula is satisfactory.
\section{Conclusion}
We have systematically studied the temporal profile of biphotons generated from a hot atomic vapor via the SFWM. Taking into account all velocity groups in a Doppler-broadened medium, we were able to derive the analytical formula of the biphoton wave packet or the two-photon correlation function. The derived formula shows that the biphoton's spectral profile is a Lorentzian function with the linewidth mainly determined by the EIT linewidth, $\Gamma_{\rm EIT}$, where the formula of $\Gamma_{\rm EIT}$ is given by Eq.~(\ref{eq:Gamma_EIT}). Thus, the biphoton wave packet is close to an exponential-decay function with the $e^{-1}$ decay time constant of $1/\Gamma_{\rm EIT}$.
Since the coupling power can significantly affect the EIT effect and the biphoton's temporal width, we measured the EIT spectra with classical light and the biphoton wave packets generated from the SFWM process at various coupling powers from 0.5 to 8 mW. The measured EIT spectrum exhibited a Lorentzian profile, which is consistent with the prediction of the analytical formula shown in Eq.~(\ref{eq:T_approximation2}). Under the same experimental condition as the EIT spectrum, the measured biphoton wave packet behaved mainly like an exponential-decay function, in agreement with the analytical formula shown in Eq.~(\ref{eq:g2_approx}). We utilized the experimental parameters determined by the EIT spectra to calculate the theoretical predictions of the biphoton wave packets. The agreement between the theoretical predictions and experimental data is satisfactory. Furthermore, we compared the biphoton temporal width obtained from the measurement with that calculated from the analytical formula given by Eq.~(\ref{eq:Gamma_EIT}) or (\ref{eq:linewidth_bi}). The comparison verifies the analytical formula is a reasonable approximation as long as the value of $\Omega_c^2/(4\gamma \Gamma_D)$ is not far greater than 1.
In conclusion, the EIT spectrum measured with classical light is a good indicator for the biphoton wave packet generated by the SFWM process. The analytical formulas derived in this work provide a simple way to quantitatively understand the temporal and spectral profiles of the biphotons .
\section*{Acknowledgments}
This work was supported by Grant Nos.~109-2639-M-007-001-ASP and 110-2639-M-007-001-ASP of the Ministry of Science and Technology, Taiwan.
\end{document} |
\begin{document}
\title{Galloping in fast-growth natural merge sorts}
\begin{abstract}
We study the impact of sub-array merging routines
on merge-based sorting algorithms.
More precisely, we focus on the
\emph{galloping} sub-routine that
TimSort\xspace uses to merge monotonic (non-decreasing)
sub-arrays, hereafter called \emph{runs},
and on the impact on the
number of element comparisons performed
if one uses this sub-routine instead of a naïve
merging routine.
The efficiency of
TimSort\xspace and of similar sorting algorithms has often been
explained by using the notion of \emph{runs} and the
associated \emph{run-length entropy}. Here, we focus
on the related notion of \emph{dual runs},
which was introduced in the 1990s, and the
associated \emph{dual run-length entropy}.
We prove, for this complexity measure,
results that are similar to those already known
when considering standard run-induced measures:
in particular, TimSort\xspace requires only~$\mathcal{O}(n + n \log(\sigma))$ element comparisons
to sort arrays of length~$n$ with~$\sigma$
distinct values.
In order to do so, we introduce new notions
of \emph{fast-} and \emph{middle-growth} for
natural merge sorts (i.e., algorithms based on
merging runs). By using these notions, we prove
that several merge sorting algorithms,
provided that they use TimSort\xspace's galloping
sub-routine for merging runs,
are as efficient as TimSort\xspace at sorting arrays
with low run-induced or dual-run-induced
complexities.
\end{abstract}
\section{Introduction}\label{sec:intro}
In 2002, Tim Peters, a software engineer, created a new sorting algorithm,
which was called TimSort\xspace~\cite{Peters2015} and was built
on ideas from McIlroy~\cite{McIlroy1993}. This algorithm immediately
demonstrated its efficiency for sorting actual data, and was
adopted as the standard sorting algorithm in core libraries of
widespread programming languages such as Python and Java.
Hence, the prominence of such a custom-made algorithm over previously
preferred \emph{optimal} algorithms contributed to the regain of interest
in the study of sorting algorithms.
Among the best-identified reasons
behind the success of TimSort\xspace are the fact that
this algorithm is well adapted to the architecture of
computers (e.g., for dealing with cache issues) and to realistic
distributions of data. In particular, the very conception
of TimSort\xspace makes it particularly well-suited
to sorting data whose \emph{run decompositions}~\cite{BaNa13,EsCaWo92} (see Figure~\ref{fig:runs})
are simple.
Such decompositions were already used in 1973 by
Knuth's NaturalMergeSort\xspace~\cite[Section 5.2.4]{Knuth98},
which adapted the traditional
MergeSort\xspace algorithm as follows: NaturalMergeSort\xspace is based on splitting arrays into monotonic subsequences, also called \emph{runs}, and on merging these runs together.
Thus, all algorithms sharing this feature of NaturalMergeSort\xspace are also called
\emph{natural} merge sorts.
\begin{figure}
\caption{A sequence and its \emph{run decomposition}
\label{fig:runs}
\end{figure}
In addition to being a natural merge sort, TimSort\xspace includes many
optimisations, which were carefully engineered, through extensive
testing, to offer the best complexity performances.
As a result,
the general structure of TimSort\xspace can be split into three main components:
(i) a variant of an insertion sort,
which is used to deal with \emph{small} runs,
e.g., runs of length less than 32,
(ii) a simple policy for choosing which \emph{large} runs to merge,
(iii) a sub-routine for merging these runs, based on a so-called
\emph{galloping} strategy.
The second component has been subject to
an intense scrutiny these last few years,
thereby giving birth to a great variety of {TimSort\xspace}-like
algorithms, such as
\textalpha-StackSort\xspace~\cite{AuNiPi15},
\mbox{\textalpha-MergeSort\xspace}~\cite{BuKno18},
ShiversSort\xspace~\cite{shivers02} (which \emph{predated} TimSort\xspace), adaptive ShiversSort\xspace~\cite{Ju20}, PeekSort\xspace and
PowerSort\xspace~\cite{munro2018nearly}.\label{citesorts}
On the contrary,
the first and third components,
which seem more complicated and whose effect might be
harder to quantify, have often been used as black boxes
when studying TimSort\xspace or designing variants thereof.
In what follows, we focus on the third component and prove
that it is very efficient:
whereas TimSort\xspace requires~$\mathcal{O}(n+n \log(\rho))$
comparisons to sort arrays of length~$n$
that can be decomposed as a concatenation of~$\rho$ non-decreasing arrays,
this component makes TimSort\xspace require
only~$\mathcal{O}(n+n \log(\sigma))$
comparisons to sort arrays of length~$n$
with~$\sigma$ distinct values.
\paragraph*{Context and related work}
The success of TimSort\xspace has nurtured the interest in the quest for
sorting algorithms that would be both excellent all-around and
adapted to arrays with few runs.
However, its \emph{ad hoc} conception made its complexity analysis
harder than what one might have hoped, and it is only in 2015,
a decade after TimSort\xspace had been largely deployed, that Auger et
al.~\cite{AuNiPi15} proved that
TimSort\xspace required~$\mathcal{O}(n \log(n))$ comparisons
for sorting arrays of length~$n$.
This is optimal in the model of sorting by comparisons,
if the input array can be
an arbitrary array of length~$n$.
However, taking into account the run decompositions
of the input array allows using finer-grained
complexity classes, as follows.
First, one may consider only arrays
whose run decomposition consists of~$\rho$
monotonic runs. On such arrays, the best worst-case
time complexity one may hope for is~$\mathcal{O}(n + n \log(\rho))$~\cite{Mannila1985}.
Second, we may consider even more restricted classes of
input arrays, and focus only on
those arrays that consist of~$\rho$ runs of lengths~$r_1,\ldots,r_\rho$. In that case,
every comparison-based sorting algorithm
requires at least~$n \mathcal{H} + \mathcal{O}(n)$ element comparisons,
where~$\mathcal{H}$ is defined as~$\mathcal{H} = H(r_1/n,\ldots,r_\rho/n)$ and~$H(x_1,\ldots,x_\rho) = -\sum_{i=1}^\rho x_i \log_2(x_i)$
is the general entropy function~\cite{BaNa13,Ju20,McIlroy1993}.
The number~$\mathcal{H}$ is called the
\emph{run-length entropy} of the array.
Since the early 2000s,
several natural merge sorts were proposed,
all of which were meant to offer easy-to-prove
complexity guarantees:
ShiversSort\xspace, which runs in
time~$\mathcal{O}(n \log(n))$;
\textalpha-StackSort\xspace, which, like NaturalMergeSort\xspace,
runs in
time~$\mathcal{O}(n + n \log(\rho))$;
\mbox{\textalpha-MergeSort\xspace}, which, like TimSort\xspace, runs in
time~$\mathcal{O}(n + n \mathcal{H})$;
adaptive ShiversSort\xspace, PeekSort\xspace and PowerSort\xspace, which run in time~$n \mathcal{H} + \mathcal{O}(n)$.
Except TimSort\xspace, these algorithms are, in fact, described only
as policies for deciding which runs to merge, the actual sub-routine used for merging runs
being left implicit:
since choosing a naïve merging sub-routine
does not harm the worst-case time complexities
considered above,
all authors identified
the cost of merging two runs of lengths~$m$ and~$n$
with the sum~$m+n$, and the complexity of the algorithm
with the sum of the costs of the merges performed.
One notable exception is that
of Munro and Wild~\cite{munro2018nearly}.
They compared the
running times of TimSort\xspace and of TimSort\xspace's variant obtained by using
a naïve merging routine
instead of TimSort\xspace's galloping sub-routine.
However, and although they mentioned the
challenge of finding distributions on arrays
that might benefit from galloping,
they did not address this challenge, and focused
only on arrays with a low entropy~$\mathcal{H}$.
As a result,
they unsurprisingly observed that the
galloping sub-routine looked \emph{slower}
than the naïve one.
Galloping turns out to be very efficient when sorting
arrays with few distinct values, a class of arrays that had
also been intensively studied.
As soon as 1976, Munro and Spira~\cite{MuSpi76} proposed a complexity measure~$\mathcal{H}^\ast$
related to the run-length entropy,
with the property that~$\mathcal{H}^\ast \leqslant
\log_2(\sigma)$ for arrays with~$\sigma$ values.
They also proposed an algorithm for sorting arrays of length~$n$ with~$\sigma$
values by using~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons.
McIlroy~\cite{McIlroy1993} then extended their work to arrays representing a permutation~$\pi$,
identifying~$\mathcal{H}^\ast$ with the run-length entropy
of~$\pi^{-1}$ and proposing a variant of
Munro and Spira's algorithm that would use~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons in this generalised
setting.
Similarly, Barbay et al.~\cite{BaOchoaSatti16} invented the
algorithm QuickSynergySort\xspace, which aimed at minimising the number of
comparisons, achieving a~$\mathcal{O}(n + n \mathcal{H}^\ast)$ upper bound and further refining the parameters it used, by taking into account
the interleaving between runs and dual runs.
Yet, all of these algorithms
require~$\omega(n + n \mathcal{H})$ element moves in the worst case.
Furthermore, as a side effect of being rather complicated and
lacking a proper analysis, except that
of~\cite{munro2018nearly} that hinted at its inefficiency,
the galloping sub-routine has been omitted
in various mainstream implementations of natural merge sorts,
in which it was replaced by its naïve variant.
This is the case, for instance, in library TimSort\xspace implementations
of the programming languages Swift~\cite{Swift} and
Rust~\cite{Rust}.
On the contrary, TimSort\xspace's implementation in
other languages, such as Java~\cite{Java},
Octave~\cite{Octave} or the V8 JavaScript
engine~\cite{V8}, and PowerSort\xspace's implementation in Python~\cite{Python}
include the galloping sub-routine.
\paragraph*{Contributions}
We study the time complexity of various natural
merge sort algorithms in a context where arrays are not just
parametrised by their lengths.
More precisely, we focus on a
decomposition of input arrays that is
dual to the decomposition of arrays into
monotonic runs,
and that was proposed by McIlroy~\cite{McIlroy1993}.
Consider an array~$A$ that we want to sort in a
\emph{stable} manner, i.e., in which two elements can always
considered to be distinct, if only because their positions in~$A$ are distinct. Without loss of generality,
we identify the values~$A[1],A[2],\ldots,A[n]$ with the integers from~$1$ to~$n$,
thereby making~$A$ a permutation of the set~$\{1,2,\ldots,n\}$.
A common measure of presortedness
consists in subdividing~$A$ into
distinct monotonic \emph{runs},
i.e., partitioning the set~$\{1,2,\ldots,n\}$
into intervals~$R_1,R_2,\ldots,R_\rho$
on which the function~$x \mapsto A[x]$ is monotonic.
\begin{figure}
\caption{The arrays~$A$ and~$B$ are lexicographically
equivalent to the permutation~$\tau$. Their dual runs,
represented with gray and white horizontal stripes,
have respective lengths~$4$,~$3$ and~$3$.
The mappings~$A \mapsto \tau$ and~$B \mapsto \tau$ identify them with the dual runs of~$\tau$,
i.e., with the runs of the permutation~$\tau^{-1}
\label{fig:dual-runs}
\end{figure}
Here, we adopt a dual approach,
which consists in partitioning the set~$\{1,2,\ldots,n\}$ into the \emph{increasing} runs~$S_1,S_2,\ldots,S_\sigma$ of the inverse permutation~$A^{-1}$.
These intervals~$S_i$ are already known under the name of
\emph{shuffled up-sequences}~\cite{BaNa13,LevPet90} or
\emph{riffle shuffles}~\cite{McIlroy1993}. In order to
underline their connection with runs, we say that these intervals are
the \emph{dual runs} of~$A$, and we denote their lengths by~$s_i$.
The process of transforming an array into a permutation and then
extracting its dual runs is illustrated in
Figure~\ref{fig:dual-runs}.
When~$A$ is \emph{not} a permutation of~$\{1,2,\ldots,n\}$,
the dual runs of~$A$ are simply the maximal
intervals~$S_i$ such that~$A$ is non-decreasing on the
set of positions~$\{j \colon A[j] \in S_i\} \subseteq \{1,2,\ldots,n\}$.
The length of a dual run is then defined as the cardinality of
that set of positions.
Thus, two lexicographically equivalent arrays have dual runs
of the same lengths:
this is the case, for instance, of the arrays~$A$,~$B$ and~$\tau$ in Figure~\ref{fig:dual-runs}.
In particular, we may see~$\tau$ and~$B$ as
canonical representatives of the array~$A$:
these are the unique permutation
of~$\{1,2,\ldots,n\}$ and the unique
array with values in~$\{1,2,\ldots,\sigma\}$
that are lexicographically equivalent with~$A$.
More generally,
an array that contains~$\sigma$ distinct values cannot have
more than~$\sigma$ dual runs.
Note that, in general, there is no non-trivial connection between
the runs of a permutation and its dual runs. For instance, a
permutation with a given number of runs may have arbitrarily many
(or few) dual runs, and conversely.
In this article, we prove that,
by using TimSort\xspace's galloping sub-routine,
several natural merge sorts
require only~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons,
or even~$n \mathcal{H}^\ast + \mathcal{O}(n)$ comparisons,
where~$\mathcal{H}^\ast = H(s_1/n,\ldots,s_\sigma/n) \leqslant \log_2(\sigma)$
is called the \emph{dual run-length entropy} of the array,~$s_i$ is the length of the dual run~$S_i$, and~$H$ is the general entropy function already mentioned above.
This legitimates using TimSort\xspace's arguably complicated galloping
sub-routine rather than its naïve alternative,
in particular when sorting arrays that are
constrained to have relatively few distinct values.
This also subsumes results that have been known
since the 1970s. For instance, adapting the optimal constructions
for alphabetic Huffman codes by Hu and Tucker~\cite{HuTucker71}
or Garsia and Wachs~\cite{Garsia77} to \emph{merge trees}
(described in Section~\ref{sec:fast-growth})
already provided sorting algorithms working in time~$n \mathcal{H} + \mathcal{O}(n)$.
Our new results rely on notions that we call
\emph{fast-} and \emph{middle-growth} properties,
and which are found in natural merge sorts like \textalpha-MergeSort\xspace,
\textalpha-StackSort\xspace, adaptive ShiversSort\xspace, ShiversSort\xspace, PeekSort\xspace, PowerSort\xspace or TimSort\xspace.
More precisely, we prove that merge sorts
require~$\mathcal{O}(n + n \mathcal{H})$ comparisons
\emph{and} element moves when they possess the
fast-growth property, thereby encompassing complexity
results that were proved separately for each of these algorithms~\cite{auger2018worst,
BuKno18,Ju20,munro2018nearly},
and~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons when they possess the
fast- or middle-growth property, which is a completely
new result.
Finally, we prove finer complexity bounds on the number
of comparisons used by
adaptive ShiversSort\xspace, ShiversSort\xspace, NaturalMergeSort\xspace, PeekSort\xspace and PowerSort\xspace, which require only~$n \mathcal{H}^\ast + \mathcal{O}(n\log(\mathcal{H}^\ast+1)+n)$ comparisons,
nearly matching the~$n \mathcal{H} + \mathcal{O}(n)$ (or~$n \log_2(n) + \mathcal{O}(n)$ and~$n \log_2(\rho) + \mathcal{O}(n)$, in the cases of ShiversSort\xspace and NaturalMergeSort\xspace) complexity upper bound they already enjoy in terms of comparisons
and element moves.
\section{The galloping sub-routine for merging runs}
\label{sec:description}
Here, we describe the galloping sub-routine that
the algorithm TimSort\xspace uses to merge adjacent non-decreasing runs.
This sub-routine is a blend between a naïve
merging algorithm, which requires~$a+b-1$
comparisons to merge runs~$A$ and~$B$ of lengths~$a$ and~$b$,
and a dichotomy-based algorithm,
which requires~$\mathcal{O}(\log(a+b))$ comparisons in the best case,
and~$\mathcal{O}(a+b)$ comparisons in the worst case.
It depends on a parameter~$\mathbf{t}$, and
works as follows.
When merging runs~$A$ and~$B$ into one large run~$C$,
we first need to find the least integers~$k$ and~$\ell$ such that~$B[0] < A[k] \leqslant B[\ell]$: the~$k+\ell$ first elements of~$C$
are~$A[0],A[1],\ldots,A[k-1]$,~$B[0],B[1],\ldots,B[\ell-1]$,
and the remaining elements of~$C$ are obtained by merging
the sub-array of~$A$ that spans positions~$k$ to~$a$ and the
sub-array of~$B$ that spans positions~$\ell$ to~$b$.
Computing~$k$ and~$\ell$ efficiently is therefore
a crucial step towards reducing the number of comparisons
required by the merging sub-routine (and, thus, by the sorting
algorithm).
This computation is a special case of the following problem:
if one wishes to find a secret integer~$m \geqslant 1$
by choosing integers~$x \geqslant 1$
and testing whether~$x \geqslant m$,
what is, as a function of~$m$, the least number of tests that one
must perform?
Bentley and Yao~\cite{BeYa76} answer this question
by providing simple strategies, which they number~$\mathsf{B}_0, \mathsf{B}_1,\ldots$:
\begin{itemize}
\item[$\mathsf{B}_0$:] choose~$x = 1$, then~$x = 2$, and so on, until
one chooses~$x = m$, thereby finding~$m$ in~$m$ queries;
\item[$\mathsf{B}_1$:] first use~$\mathsf{B}_0$ to find~$\lceil \log_2(m) \rceil + 1$ in~$\lceil \log_2(m) \rceil + 1$
queries, i.e., choose~$x = 2^k$ until~$x \geqslant m$, then compute the bits of~$m$ (from the most significant bit
of~$m$ to the least significant one) in~$\lceil \log_2(m)
\rceil - 1$
additional queries;
Bentley and Yao call this strategy a \emph{galloping} (or \emph{exponential
search}) technique;
\item[$\mathsf{B}_{k+1}$:] like~$\mathsf{B}_1$, except that one
finds~$\lceil \log_2(m) \rceil + 1$ by using~$\mathsf{B}_k$
instead of~$\mathsf{B}_0$.
\end{itemize}
Strategy~$\mathsf{B}_0$ uses~$m$ queries,~$\mathsf{B}_1$ uses~$2 \lceil \log_2(m) \rceil$ queries
(except for~$m = 1$, where it uses one query), and each strategy~$\mathsf{B}_k$ with~$k \geqslant 2$
uses~$\log_2(m) + o(\log(m))$ queries.
Thus, if~$m$ is known to be arbitrarily large,
one should favour some strategy~$\mathsf{B}_k$
(with~$k \geqslant 1$) over the naïve strategy~$\mathsf{B}_0$.
However, when
merging runs taken from a permutation chosen uniformly at
random over the~$n!$ permutations of~$\{1,2,\ldots,n\}$, the integer~$m$ is frequently small,
which makes~$\mathsf{B}_0$ suddenly more
attractive.
In particular, the overhead of using~$\mathsf{B}_1$ instead of~$\mathsf{B}_0$ is a prohibitive~$+20\%$ or~$+33\%$ when~$m = 5$ or~$m = 3$, as illustrated
in the black cells of Table~\ref{table:1}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|C{3.75mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|C{3.35mm}|}
\hline
\clap{$m$} & \clap{$1$} & \clap{$2$} & \clap{$3$} & \clap{$4$} & \clap{$5$} & \clap{$6$} & \clap{$7$} & \clap{$8$} & \clap{$9$} & \clap{$10$} & \clap{$11$} & \clap{$12$} & \clap{$13$} & \clap{$14$} & \clap{$15$} & \clap{$16$} & \clap{$17$} \\
\hline
\clap{$\mathsf{B}_0$} & \clap{$1$} & \clap{$2$} & \clap{$3$} & \clap{$4$} & \clap{$5$} & \clap{$6$} & \clap{$7$} & \clap{$8$} & \clap{$9$} & \clap{$10$} & \clap{$11$} & \clap{$12$} & \clap{$13$} & \clap{$14$} & \clap{$15$} & \clap{$16$} & \clap{$17$} \\
\hline
\clap{$\mathsf{B}_1$} & \clap{$1$} & \clap{$2$} & \clap{\cellcolor{black}\color{white}$\mathbf{4}$} & \clap{$4$} & \clap{\cellcolor{black}\color{white}$\mathbf{6}$} & \clap{$6$} & \clap{$6$} & \clap{$6$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$8$} & \clap{$10$} \\
\hline
\end{tabular}
\end{center}
\caption{Comparison requests needed by
strategies~$\mathsf{B}_0$ and~$\mathsf{B}_1$
to find a secret integer~$m \geqslant 1$.
\label{table:1}}
\end{table}
McIlroy~\cite{McIlroy1993} addresses this issue by choosing a
parameter~$\mathbf{t}$ and using a blend between the strategies~$\mathsf{B}_0$ and~$\mathsf{B}_1$,
which consists in two successive steps~$\mathsf{C}_1$ and~$\mathsf{C}_2$:
\begin{enumerate}
\item[$\mathsf{C}_1$:] one first follows~$\mathsf{B}_0$ for up to~$\mathbf{t}$ steps, thereby choosing~$x = 1$,~$x = 2$, \ldots,~$x = \mathbf{t}$
(if~$m \leqslant \mathbf{t}-1$, one stops after choosing~$x = m$);
\item[$\mathsf{C}_2$:] if~$m \geqslant \mathbf{t}+1$, one switches to~$\mathsf{B}_1$
(or, more precisely, to a version of~$\mathsf{B}_1$ translated by~$\mathbf{t}$, since the precondition~$m \geqslant 1$ is now~$m \geqslant \mathbf{t}+1$).
\end{enumerate}
Once such a parameter~$\mathbf{t}$ is fixed,
McIlroy's mixed strategy allows retrieving~$m$
in~$\mathsf{cost}_{\mathbf{t}}(m)$ queries, where~$\mathsf{cost}_{\mathbf{t}}(m) = m$ if~$m \leqslant \mathbf{t}+2$, and~$\mathsf{cost}_{\mathbf{t}}(m) =
\mathbf{t} + 2 \lceil \log_2(m - \mathbf{t}) \rceil$
if~$m \geqslant \mathbf{t}+3$.
In practice, however, we will replace this cost function by
the following simpler upper bound.
\begin{lemma}\label{lem:cost-bound}
For all~$\mathbf{t} \geqslant 0$ and~$m \geqslant 1$, we have~$\mathsf{cost}_{\mathbf{t}}(m) \leqslant \mathsf{cost}_{\mathbf{t}}^\ast(m)$,
where
\[\mathsf{cost}_{\mathbf{t}}^\ast(m) =
\min\{(1 + 1 / (\mathbf{t}+3)) m, \mathbf{t} + 2 +
2 \log_2(m + 1)\}.\]
\end{lemma}
\begin{proof}
Since the desired inequality is immediate when~$m \leqslant \mathbf{t}+2$, we assume
that~$m \geqslant \mathbf{t}+3$.
In that case, we already have~$\mathsf{cost}_{\mathbf{t}}(m) \leqslant
\mathbf{t} + 2 (\log_2(m-\mathbf{t})+1) \leqslant
\mathbf{t}+2+\log_2(m+1)$,
and we prove now that~$\mathsf{cost}_\mathbf{t}(m) \leqslant m+1$.
Indeed, let~$u = m - \mathbf{t}$ and let~$f \colon x \mapsto x-1-2\log_2(x)$.
The function~$f$ is positive and increasing
on the interval~$[7,+\infty)$.
Thus, it suffices to check by hand that~$(m+1) - \mathsf{cost}_\mathbf{t}(m) = 0,1,0,1$
when~$u = 3,4,5,6$,
and that~$(m+1) - \mathsf{cost}_\mathbf{t}(m) \geqslant f(u) > 0$ when~$u \geqslant 7$. It follows, as expected, that~$\mathsf{cost}_{\mathbf{t}}(m) \leqslant m+1 \leqslant
(1 + 1/(\mathbf{t}+3))m$.
\end{proof}
The above discussion immediately provides us with
a cost model for
the number of comparisons performed when merging two runs.
\begin{proposition}\label{pro:2}
Let~$A$ and~$B$ be two non-decreasing runs of
lengths~$a$ and~$b$, with values in~$\{1,2,\ldots,\sigma\}$.
For each integer~$i \leqslant \sigma$, let~$a_{\rightarrow i}$ (respectively,~$b_{\rightarrow i}$) be the number of elements
in~$A$ (respectively, in~$B$) with value~$i$.
Using a merging sub-routine based on McIlroy's mixed strategy
for a fixed parameter~$\mathbf{t}$,
we need at most
\[1 + \sum_{i=1}^\sigma
\mathsf{cost}_{\mathbf{t}}^\ast(a_{\rightarrow i}) +
\mathsf{cost}_{\mathbf{t}}^\ast(b_{\rightarrow i})\]
element comparisons to merge the runs~$A$ and~$B$.
\end{proposition}
\begin{proof}
First, assume that~$a_{\rightarrow i} = 0$ for some~$i \geqslant 2$.
Replacing every value~$j \geqslant i+1$ with the value~$j-1$
in both arrays~$A$ and~$B$ does not change the behaviour
of the sub-routine and decreases the value of~$\sigma$.
Moreover, the function~$\mathsf{cost}_{\mathbf{t}}^\ast$ is
sub-additive, i.e., we have~$\mathsf{cost}_{\mathbf{t}}^\ast(m) +
\mathsf{cost}_{\mathbf{t}}^\ast(m') \geqslant \mathsf{cost}_{\mathbf{t}}^\ast(m+m')$ for all~$m \geqslant 0$ and~$m' \geqslant 0$.
Hence, without loss of generality, we assume
that~$a_{\rightarrow i} \geqslant 1$ for all~$i \geqslant 2$.
Similarly, we assume without loss of generality that~$b_{\rightarrow i} \geqslant 1$ for all~$i \leqslant \sigma-1$.
Under these assumptions, the array~$C$ that results from merging~$A$ and~$B$ consists of~$a_{\rightarrow 1}$ elements from~$A$, then~$b_{\rightarrow 1}$ elements
from~$B$,~$a_{\rightarrow 2}$ elements from~$A$,~$b_{\rightarrow 2}$ elements from~$B$,
\ldots,~$a_{\rightarrow \sigma}$ elements
from~$A$ and~$b_{\rightarrow \sigma}$ elements from~$B$.
Thus, the galloping
sub-routine consists in discovering successively
the integers~$a_{\rightarrow 1},b_{\rightarrow 1},a_{\rightarrow 2},b_{\rightarrow 2},\ldots,a_{\rightarrow \sigma}$,
each time using McIlroy's strategy based on the two steps~$\mathsf{C}_1$ and~$\mathsf{C}_2$;
checking whether~$a_{\rightarrow 1} = 0$ requires one more comparison
than prescribed by McIlroy's strategy, and the integer~$b_{\rightarrow \sigma}$
does not need to be discovered once the entire
run~$A$ has been scanned be the merging sub-routine.
\end{proof}
We simply call~$\mathbf{t}$-\emph{galloping sub-routine} the
merging sub-routine based on McIlroy's mixed strategy
for a fixed parameter~$\mathbf{t}$; when the value of~$\mathbf{t}$ is irrelevant, we simply omit mentioning it.
Then, the quantity
\[1 + \sum_{i=1}^\sigma \mathsf{cost}_{\mathbf{t}}^\ast(a_i) +
\mathsf{cost}_{\mathbf{t}}^\ast(b_i)\]
is called the ($\mathbf{t}$-)\emph{galloping cost} of merging~$A$ and~$B$.
By construction, this cost never exceeds~$1 + 1/(\mathbf{t}+3)$ times the \emph{naïve
cost} of merging~$A$ and~$B$, which is simply defined as~$a + b$.
Below, we study the impact of using the galloping sub-routine
instead of the naïve one, which amounts to replacing
naïve merge costs by their galloping variants.
Note that using this new galloping
cost measure is relevant
only if the cost of element comparisons is significantly
larger than the cost of element (or pointer) moves.
For example, even if we were lucky enough to observe that
each element in~$B$ is smaller than
each element in~$A$, we would perform
only~$\mathcal{O}(\log(a+b))$ element comparisons, but as many
as~$\mathcal{T}heta(a+b)$ element moves.
\paragraph*{Updating the parameter~$\mathbf{t}$}
We assumed above that the parameter~$\mathbf{t}$ did not
vary while the runs~$A$ and~$B$ were being merged with each other.
This is not how~$\mathbf{t}$ behaves in TimSort\xspace's
implementation of the galloping sub-routine.
Instead, the parameter~$\mathbf{t}$ is
initially set to a
constant ($\mathbf{t} = 7$ in Java), and
may change during the algorithm as follows.
In step~$\mathsf{C}_2$,
after using the strategy~$\mathsf{B}_1$,
and depending on the value of~$m$ that we found,
one may realise that using~$\mathsf{B}_0$ might have been
less expensive than using~$\mathsf{B}_1$.
In that case, the value of~$\mathbf{t}$ increases by~$1$,
and otherwise (i.e., if using~$\mathsf{B}_1$ was indeed a smart
move), it decreases by~$1$ (with a minimum of~$0$).
When sorting a random permutation,
changing the value of~$\mathbf{t}$
in that way decreases the average overhead of
sometimes using~$\mathsf{B}_1$ instead of~$\mathsf{B}_0$ to a constant.
More generally, the worst-case overhead is limited
to a sub-linear~$\mathcal{O}(\sqrt{n \log(n)})$,
as proved in Proposition~\ref{pro:TS-update}.
Deciding whether our results remain valid
when~$\mathbf{t}$ is updated
like in TimSort\xspace remains an open question.
However, in Section~\ref{subsec:variable-t},
we propose and study the following alternative update policy:
when merging runs of lengths~$a$ and~$b$,
we set~$\mathbf{t} = \lceil \log_2(a+b) \rceil$.
\begin{proposition}\label{pro:TS-update}
Let~$\mathcal{A}$ be a stable merge sort algorithm with the
middle-growth property, and let~$A$ be an array
of length~$n$.
Let~$\mathsf{c}_1$ be the number of comparisons
that~$\mathcal{A}$ requires to sort~$A$ when it uses
the naïve sub-routine, and let~$\mathsf{c}_2$
be the number of comparisons that~$\mathcal{A}$ requires
to sort~$A$ when it uses the galloping sub-routine
with TimSort\xspace's update policy for the parameter~$\mathbf{t}$.
We have~$\mathsf{c}_2 \leqslant \mathsf{c}_1 +
\mathcal{O}(\sqrt{n \log(n)})$.
\end{proposition}
\begin{proof}
Below, we group the comparisons
that~$\mathcal{A}$ performs while sorting~$A$ into
\emph{steps}, which we will consider as individual
units. Steps are formed as follows.
Let~$R$ and~$R'$ be consecutive runs
that~$\mathcal{A}$ is about to merge,
and let us subdivide their concatenation~$R \cdot R'$ into~$\sigma$ dual runs~$S_1,S_2,\ldots,S_\sigma$ (note that these are
the dual runs of~$R \cdot R'$ and not the dual
runs of~$A$, i.e., some elements of~$R$ may belong
to a given dual run~$S_i$ of~$R \cdot R'$ while
belonging to distinct dual runs or~$A$).
Each step consists of those comparisons used to
discover the
elements of~$R$ (resp.,~$R'$) that belong to
a given dual run~$S_i$.
Thus, the comparisons used to merge~$R$ and~$R'$
are partitioned into~$2\sigma-1$ steps:
in the first step, we discover those elements of~$R$ that belong to~$S_1$; in the second step, those elements of~$R'$ that belong to~$S_1$; then, those elements of~$R$ that belong to~$S_2$ ; \ldots; and, finally,
those elements of~$R$ that belong to~$S_\sigma$.
Let~$s_1,s_2,\ldots,s_\ell$ be the steps into
which the comparisons performed by~$\mathcal{A}$ are
grouped.
By construction, each step~$s_i$ consists in
finding an integer~$m_i \geqslant 1$ (or, possibly,~$m_i = 0$ if~$s_i$ is the first step of a merge
between two consecutive runs).
If~$\mathcal{A}$ uses TimSort\xspace's update policy, the step~$s_i$
consists in using
McIlroy's strategy for a given parameter~$\mathbf{t}(s_i)$ that depends on the run~$s_i$.
We also denote by~$\mathbf{t}(s_{\ell+1})$ the parameter
value obtained after~$\mathcal{A}$ has finished sorting
the array~$A$.
Let~$\delta_i = 0$ if~$m_i \leqslant \mathbf{t}(s_i)$,
$\delta_i = 1$ if~$\mathbf{t}(s_i)+1 \leqslant m_i
\leqslant \mathbf{t}(s_i)+6$
and~$\delta_i = -1$ if~$\mathbf{t}(s_i)+7 \leqslant m_i$.
The naïve strategy~$\mathsf{B}_0$ requires~$m_i$ comparisons
to find that integer~$m_i$, and McIlroy's strategy
requires up to~$m_i + \delta_i$ comparisons.
Since~$\mathbf{t}(s_{i+1}) = \max\{0,\mathbf{t}(s_i) + \delta_i\}$, McIlroy's strategy never uses
more than~$m_i + \mathbf{t}(s_{i+1}) - \mathbf{t}(s_i)$
comparisons.
Consequently, the overhead of using TimSort\xspace's update
policy instead of a naïve merging sub-routine
is at most~$\mathbf{t}(s_{\ell+1}) - \mathbf{t}(s_1)$.
Finally, let~$\mu_i = m_1+\ldots+m_i$ for all~$i \leqslant \ell$.
We show that~$3\mu_i \geqslant \tau^2$ whenever~$\mathbf{t}(s_i) = \mathbf{t}(s_1) + \tau$.
Indeed, if~$\tau \geqslant 1$ and if~$s_i$ is
the first step for which~$\mathbf{t}(s_i) = \mathbf{t}(s_1) + \tau$,
we have~$\mathbf{t}(s_{i-1}) = \mathbf{t}(s_1) + \tau-1$.
It follows that~$3 \mu_{i-1} \geqslant (\tau-1)^2$ and~$m_i \geqslant \mathbf{t}(s_1) + \tau \geqslant \tau$,
which proves that~$3 \mu_i \geqslant (\tau-1)^2 + 3 \tau \geqslant
\tau^2$. Thus, we conclude that~$\mathbf{t}(s_\ell+1) - \mathbf{t}(s_1) \leqslant \sqrt{3 \mu_\ell}$. Since
Theorem~\ref{thm:middle-few-naïve} below proves that~$\mu_\ell = \mathcal{O}(n \log(n))$, the desired result
follows.
\end{proof}
Note that, in general, if $\mathcal{A}$ is any stable
natural merge sort algorithm, it will require at
most $\mathcal{O}(n^2)$ comparisons in the worst case,
and therefore using the galloping sub-routine
with TimSort\xspace's update may not increase the number of
comparisons performed by more than $\mathcal{O}(n)$.
\section{Fast-growth and (tight) middle-growth properties}
\label{sec:fast-growth}
In this section, we focus on two novel properties of
stable natural merge sorts, which we call \emph{fast-growth}
and \emph{middle-growth}, and on a variant of
the latter property, which we call
\emph{tight middle-growth}.
These properties capture
all TimSort\xspace-like natural merge sorts invented in the last
decade, and explain why these sorting algorithms require
only~$\mathcal{O}(n + n \mathcal{H})$ element moves and~$\mathcal{O}(n + n \min\{\mathcal{H},\mathcal{H}^\ast\})$ element comparisons.
We will prove in Section~\ref{sec:pos-fast-growth}
that many algorithms have these properties.
When applying a stable natural merge sort on an array~$A$, the elements of~$A$ are clustered into
monotonic sub-arrays called \emph{runs},
and the algorithm consists in repeatedly merging
consecutive runs into one larger run until the array itself
contains only one run.
Consequently, each element may undergo
several successive merge operations.
\emph{Merge trees}~\cite{BaNa13,Ju20,munro2018nearly} are
a convenient way to represent the succession of runs
that ever occur while~$A$ is being sorted.
\begin{definition}\label{def:merge-tree}
The \emph{merge tree} induced by a stable natural merge
sort algorithm on an array~$A$ is
the binary rooted tree~$\mathcal{T}$ defined as follows.
The nodes of~$\mathcal{T}$ are all the runs
that were present in the initial array~$A$
or that resulted from merging two runs.
The runs of the initial array are the leaves of~$\mathcal{T}$,
and when two consecutive runs~$R_1$ and~$R_2$ are merged with
each other into a new run~$\overline{R}$,
the run~$R_1$ spanning positions immediately to the left
of those of~$R_2$, they form the
left and the right children of the node~$\overline{R}$,
respectively.
\end{definition}
Such trees ease the task of referring to
several runs that might not have occurred
simultaneously. In particular, we will often refer
to the~$i$\textsuperscript{th} \emph{ancestor}
or a run~$R$, which is just~$R$ itself if~$i = 0$,
or the parent, in the tree~$\mathcal{T}$,
of the~$(i-1)$\textsuperscript{th}
ancestor of~$R$ if~$i \geqslant 1$.
That ancestor will be denoted by~$\aR{i}$.
Before further manipulating these runs,
let us first present some notation about runs and
their lengths, which we will frequently use.
We will commonly denote runs with capital letters,
possibly with some index or adornment,
and we will then denote the length of such a run
with the same small-case letter and the same
index or adornment. For instance,
runs named~$R$,~$R_i$,~$\aR{j}$,~$Q'$ and~$\overline{S}$
will have respective lengths~$r$,~$r_i$,~$\ar{j}$,~$q'$ and~$\overline{s}$.
Finally, we say that a run~$R$ is a \emph{left} run if it
is the left child of its parent, and that it is a
\emph{right} run if it is a right child. The root
of a merge tree is neither left nor right.
\begin{definition}\label{def:fast-growth}
We say that a stable natural merge sort algorithm~$\mathcal{A}$ has the \emph{fast-growth
property} if it satisfies the following statement:
\begin{quote}
There exist an integer~$\ell \geqslant 1$ and a real number~$\alpha > 1$
such that, for every merge tree~$\mathcal{T}$
induced by~$\mathcal{A}$ and every run~$R$ at depth~$\ell$
or more in~$\mathcal{T}$, we have~$\ar{\ell} \geqslant
\alpha r$.
\end{quote}
We also say that~$\mathcal{A}$ has the
\emph{middle-growth property} if it satisfies
the following statement:
\begin{quote}
There exists a real number~$\beta > 1$
such that, for every merge tree~$\mathcal{T}$
induced by~$\mathcal{A}$, every integer~$h \geqslant 0$
and every run~$R$ of height~$h$ in~$\mathcal{T}$, we have~$r \geqslant \beta^h$.
\end{quote}
Finally, we say that~$\mathcal{A}$ has the
\emph{tight middle-growth property} if
it satisfies the following statement:
\begin{quote}
There exists an integer~$\gamma \geqslant 0$
such that, for every merge tree~$\mathcal{T}$
induced by~$\mathcal{A}$, every integer~$h \geqslant 0$
and every run~$R$ of height~$h$ in~$\mathcal{T}$, we have~$r \geqslant 2^{h-\gamma}$.
\end{quote}
\end{definition}
Since every node of height~$h \geqslant 1$
in a merge tree
is a run of length at least~$2$,
each algorithm with the fast-growth property
or with the tight middle-growth property
also has the middle-growth property:
indeed, it suffices to
choose~$\beta = \min\{2,\alpha\}^{1/\ell}$
in the first case,
and~$\beta = 2^{1/(\gamma+1)}$ in the second one.
As a result, the first and third properties
are stronger than
the latter one, and indeed they have stronger consequences.
\begin{theorem}\label{thm:fast-few-naïve}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the fast-growth property.
If~$\mathcal{A}$ uses either
the galloping or the naïve
sub-routine for merging runs, it
requires~$\mathcal{O}(n + n\mathcal{H})$ element comparisons and
moves to sort arrays of length~$n$ and run-length entropy~$\mathcal{H}$.
\end{theorem}
\begin{proof}
Let~$\ell \geqslant 1$ and~$\alpha > 1$ be the integer
and the real number
mentioned in the definition of the statement
``$\mathcal{A}$ has the fast-growth property''.
Let~$A$ be an array of length~$n$ with~$\rho$ runs of lengths~$r_1,r_2,\ldots,r_\rho$,
let~$\mathcal{T}$ be the merge tree induced
by~$\mathcal{A}$ on~$A$, and let~$d_i$ be the depth
of the run~$R_i$ in the tree~$\mathcal{T}$.
The algorithm~$\mathcal{A}$ uses~$\mathcal{O}(n)$ element comparisons and
element moves to
delimit the runs it will then merge and to make
them non-decreasing.
Then, both the galloping and the naïve merging sub-routine
require~$\mathcal{O}(a+b)$ element comparisons and moves to merge two
runs~$A$ and~$B$ of lengths~$a$ and~$b$.
Therefore, it suffices to prove that~$\sum_{R \in \mathcal{T}} r = \mathcal{O}(n + n \mathcal{H})$.
Consider some leaf~$R_i$ of the tree~$\mathcal{T}$, and let~$k = \lfloor d_i / \ell \rfloor$. The run
$\aR{k\ell}_i$ is a run of size~$\ar{k\ell}_i \geqslant \alpha^k r_i$, and thus~$n \geqslant
\ar{k\ell}_i \geqslant \alpha^k r_i$. Hence,~$d_i+1 \leqslant \ell(k+1) \leqslant \ell \left(\log_\alpha(n/r_i) + 1\right)$, and we conclude that
\[
\sum_{R \in \mathcal{T}} r = \sum_{i=1}^\rho (d_i+1) r_i \leqslant
\ell \sum_{i=1}^\rho \left(r_i \log_\alpha(n/r_i) + r_i\right)
= \ell (n \mathcal{H} / \log_2(\alpha) + n) = \mathcal{O}(n + n\mathcal{H}).\qedhere\]
\end{proof}
Similar, weaker results also holds for
algorithms with the (tight)
middle-growth property.
\begin{theorem}
\label{thm:middle-few-naïve}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the middle-growth property.
If~$\mathcal{A}$ uses either
the galloping or the naïve
sub-routine for merging runs, it
requires~$\mathcal{O}(n \log(n))$ element comparisons and
moves to sort arrays of length~$n$.
If, furthermore,~$\mathcal{A}$ has the tight middle-growth
property, it requires at most~$n \log_2(n) + \mathcal{O}(n)$ element comparisons and moves to sort arrays
of length~$n$.
\end{theorem}
\begin{proof}
Let us borrow the notations from the previous
proof, and
let~$\beta > 1$ be the real number mentioned in
the definition of the statement ``$\mathcal{A}$ has the
middle-growth property''.
Like in the proof of
Theorem~\ref{thm:fast-few-naïve}, it suffices
to show that~$\sum_{R \in \mathcal{T}} r = \mathcal{O}(n \log(n))$.
The~$d_i$\textsuperscript{th} ancestor of a run~$R_i$ is the root of~$\mathcal{T}$, and thus~$n \geqslant
\beta^{d_i}$. Hence,~$d_i \leqslant
\log_\beta(n)$, which proves that
\[\sum_{R \in \mathcal{T}} r = \sum_{i=1}^\rho (d_i+1) r_i \leqslant
\sum_{i=1}^\rho (\log_\beta(n)+1) r_i =
(\log_\beta(n)+1) n = \mathcal{O}(n \log(n)).\]
Similarly, if~$\mathcal{A}$ has the tight middle-growth
property, let~$\gamma$ be the integer mentioned in
the definition of the statement ``$\mathcal{A}$ has the
tight middle-growth property''.
This
time,~$n \geqslant 2^{d_i-\gamma}$, which proves
that~$d_i \leqslant \log_2(n) + \gamma$ and that
\[\sum_{R \in \mathcal{T}} r = \sum_{i=1}^\rho (d_i+1) r_i \leqslant
\sum_{i=1}^\rho (\log_2(n)+\gamma+1) r_i =
(\log_2(n)+\gamma+1) n = n \log_2(n) + \mathcal{O}(n). \qedhere\]
\end{proof}
Theorems~\ref{thm:fast-few-naïve}
and~\ref{thm:middle-few-naïve} provide us with
a simple framework for recovering well-known
results on the complexity of many algorithms.
By contrast, Theorem~\ref{thm:middle-few} consists
in new complexity guarantees
on the number of element comparisons
performed by algorithms with the middle-growth property, provided that they use the galloping
sub-routine.
\begin{theorem}\label{thm:middle-few}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the middle-growth property. If~$\mathcal{A}$ uses the galloping
sub-routine for merging runs,
it requires~$\mathcal{O}(n + n\mathcal{H}^\ast)$ element comparisons
to sort arrays of length~$n$ and dual run-length entropy~$\mathcal{H}^\ast$.
\end{theorem}
\begin{proof}
All comparisons performed by the galloping sub-routine
are of the form~$A[i] \leqslant^?\! A[j]$,
where~$i$ and~$j$ are positions such that~$i < j$.
Thus, the behaviour of~$\mathcal{A}$, i.e., the element comparisons
and element moves it performs, is invariant
under lexicographic equivalence, as illustrated in
Figure~\ref{fig:dual-runs}.
Consequently, starting from an array~$A$ of length~$n$
with~$\sigma$ dual runs
setting~$B[j] \eqdef i$ whenever~$A[j]$ belongs to the
dual run~$S_i$,
and we may now assume that~$A$ coincides with~$B$.
This assumption allows us to directly use
Proposition~\ref{pro:2}, whose presentation would have been
more complicated if we had referred to dual runs of an
underlying array instead of referring directly to distinct
values.
Let~$\beta > 1$ be the real number
mentioned in the definition of the statement
``$\mathcal{A}$ has the middle-growth property''.
Let~$A$ be an array of length~$n$ and whose values are integers
from~$1$ to~$\sigma$,
let~$s_1,s_2,\ldots,s_\sigma$ be the lengths of its
dual runs, and let~$\mathcal{T}$ be the merge tree induced
by~$\mathcal{A}$ on~$A$.
The algorithm~$\mathcal{A}$ uses~$\mathcal{O}(n)$ element comparisons to
delimit the runs it will then merge and to make
them non-decreasing. We prove now
that merging these runs requires
only~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons.
For every run~$R$ in~$\mathcal{T}$ and every integer~$i \leqslant
\sigma$, let~$r_{\rightarrow i}$
be the number of elements of~$R$
with value~$i$.
In the galloping cost model,
merging two runs~$R$ and~$R'$ requires at most
$1 + \sum_{i=1}^\sigma \mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}) + \mathsf{cost}_{\mathbf{t}}^\ast(r'_{\rightarrow i})$
element comparisons.
Since less than~$n$ such merge operations are performed,
and since~$n = \sum_{i=1}^\sigma s_i$ and~$n \mathcal{H}^\ast = \sum_{i=1}^\sigma s_i \log(n / s_i)$,
it remains to show that
\[\sum_{R \in \mathcal{T}} \mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}) = \mathcal{O}(s_i + s_i \log(n / s_i))\]
for all~$i \leqslant \sigma$.
Then, since~$\mathsf{cost}_{\mathbf{t}}^\ast(m) \leqslant (\mathbf{t}+1) \mathsf{cost}_0^\ast(m)$
for all parameter values~$\mathbf{t} \geqslant 0$ and
all~$m \geqslant 0$, we assume without loss of
generality that~$\mathbf{t} = 0$.
Consider now some integer~$h \geqslant 0$,
and let~$\mathsf{C}_0(h) = \sum_{R \in \mathcal{R}_h} \mathsf{cost}_0^\ast(r_{\rightarrow i})$,
where~$\mathcal{R}_h$ denotes the set of runs at height~$h$
in~$\mathcal{T}$.
Since no run in~$\mathcal{R}_h$ descends
from another one, we already have~$\mathsf{C}_0(h) \leqslant
2 \sum_{R \in \mathcal{R}_h} r_{\rightarrow i} \leqslant 2 s_i$ and~$\sum_{R \in \mathcal{R}_h} r \leqslant n$.
Moreover, by definition of~$\beta$, each run~$R \in \mathcal{R}_h$
is of length~$r \geqslant \beta^h$, and thus~$|\mathcal{R}_h| \leqslant n / \beta^h$.
Let also~$f \colon x \mapsto \mathbf{t}+2 + 2 \log_2(x+1)$,~$g \colon x \mapsto x \, f(s_i / x)$
and~$\lambda = \lceil\log_\beta(n / s_i) \rceil$.
Both~$f$ and~$g$ are positive and concave on the interval~$(0,+\infty)$, thereby also being increasing.
It follows that, for all~$h \geqslant 0$,
\begin{align*}
\mathsf{C}_0(\lambda + h) & \leqslant
\sum_{R \in \mathcal{R}_{\lambda + h}} f(r_{\rightarrow i})
\leqslant
|\mathcal{R}_{\lambda + h}| \, f\big({\textstyle\sum_{R \in \mathcal{R}_{\lambda + h}} r_{\rightarrow_i} / |\mathcal{R}_{\lambda + h}|}\big) \leqslant
g\big(|\mathcal{R}_{\lambda + h}|\big) \\
& \leqslant
g\big(n / \beta^{\lambda + h}\big) \leqslant
g\big(s_i \beta^{-h}\big) =
\big(2+2 \log_2(\beta^h+1)\big) s_i \beta^{-h} \\
& \leqslant \big(2 + 2 \log_2(2 \beta^h)\big) s_i \beta^{-h} =
\big(4 + 2 h \log_2(\beta)\big) s_i \beta^{-h}.
\end{align*}
Inequalities on the first line hold
by definition of~$\mathsf{cost}_1^\ast$,
because~$f$ is concave,
and because~$f$ is increasing;
inequalities on the second line hold
because~$g$ is increasing and
because~$|\mathcal{R}_h| \leqslant n / \beta^h$.
We conclude that
\begin{align*}
\sum_{R \in \mathcal{T}} \mathsf{cost}_0^\ast(r_{\rightarrow i})
& = \sum_{h \geqslant 0} \mathsf{C}_0(h)
= \sum_{h=0}^{\lambda-1} \mathsf{C}_0(h) + \sum_{h \geqslant 0}
\mathsf{C}_0(\lambda + h) \\
& \leqslant 2 \lambda s_i +
4 s_i \sum_{h \geqslant 0} \beta^{-h} +
2 \log_2(\beta) s_i \sum_{h \geqslant 0} h \beta^{-h} \\
& \leqslant
\mathcal{O}\big(s_i (1 + \log(n / s_i)\big) +
\mathcal{O}\big(s_i\big) +
\mathcal{O}\big(s_i\big) = \mathcal{O}\big(s_i+ s_i \log(n / s_i)\big). &&\qedhere
\end{align*}
\end{proof}
\section{A few algorithms with the fast- and (tight) middle-growth properties}
\label{sec:pos-fast-growth}
In this section, we briefly present the algorithms
mentioned in Section~\ref{sec:intro} and prove
that each of them enjoys the fast-growth property
and/or the (tight) middle-growth property.
Before treating these algorithms one by one,
we first sum up our results.
\begin{theorem}\label{thm:fast}
The algorithms TimSort\xspace, \textalpha-MergeSort\xspace, PowerSort\xspace, PeekSort\xspace and adaptive ShiversSort\xspace
have the fast-growth property.
\end{theorem}
An immediate consequence of Theorems~\ref{thm:fast-few-naïve}
and~\ref{thm:middle-few} is that
these algorithms sort arrays of length~$n$ and
run-length entropy~$\mathcal{H}$ in time~$\mathcal{O}(n + n\mathcal{H})$,
which was already well-known,
and that, if used with the galloping merging sub-routine,
they only need~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons
to sort arrays of length~$n$
and dual run-length entropy~$\mathcal{H}^\ast$, which is a new result.
\begin{theorem}\label{thm:middle}
The algorithms NaturalMergeSort\xspace, ShiversSort\xspace and \textalpha-StackSort\xspace
have the middle-growth property.
\end{theorem}
Theorem~\ref{thm:middle-few} proves that,
if these three algorithms are used with the galloping
merging sub-routine,
they only need~$\mathcal{O}(n + n \mathcal{H}^\ast)$ comparisons
to sort arrays of length~$n$
and dual run-length entropy~$\mathcal{H}^\ast$.
By contrast, observe that they can be implemented
by using a stack, following TimSort\xspace's own implementation,
but where only the two top runs of the stack could be merged.
It is proved in~\cite{Ju20} that such algorithms may require~$\omega(n + n \mathcal{H})$ comparisons to sort
arrays of length~$n$ and run-length entropy~$\mathcal{H}$.
Hence, Theorem~\ref{thm:fast-few-naïve} shows that
these three algorithms do \emph{not} have the
fast-growth property.
\begin{theorem}\label{thm:tight-middle}
The algorithms NaturalMergeSort\xspace, ShiversSort\xspace and PowerSort\xspace
have the tight middle-growth property.
\end{theorem}
Theorem~\ref{thm:middle-few-naïve} proves that
these algorithms sort arrays of length~$n$
in time~$n \log_2(n) + \mathcal{O}(n)$, which was already
well-known.
In Section~\ref{sec:precise-bounds-PoS}, we will
further improve our upper bounds on the number of
comparisons that these algorithms require when
sorting arrays of length~$n$
and dual run-length entropy~$\mathcal{H}^\ast$.
Note that the algorithms \textalpha-StackSort\xspace, \textalpha-MergeSort\xspace and TimSort\xspace
may require more than~$n \log_2(n) + \mathcal{O}(n)$
comparisons to sort arrays of length~$n$,
which proves that they do not have the tight
middle-growth property.
In Section~\ref{sec:precise-bounds-PoS},
we will prove that adaptive ShiversSort\xspace and PeekSort\xspace also fail to have
the this property, although
they enjoy still complexity upper bounds similar to
those of algorithms with the tight middle-growth
property.
\subsection{Algorithms with the fast-growth property}
\label{subsec:other-fast}
\subsubsection{PowerSort\xspace}
\label{subsubsec:PoS}
The algorithm PowerSort\xspace is best defined by introducing
the notion of \emph{power}
of a run endpoint or of a run, and then
characterising the merge trees that PowerSort\xspace induces.
\begin{definition}\label{def:PoS:power}
Let~$A$ be an array of length~$n$, whose run decomposition
consists of runs~$R_1,R_2,\ldots,R_\rho$,
ordered from left to right.
For all integers~$i \leqslant \rho$, let~$e_i = r_1+\ldots+r_i$. We also abusively set~$e_{-1} = -\infty$ and~$e_{\rho+1} = n$.
When~$0 \leqslant i \leqslant \rho$, we denote by~$\mathbf{I}(i)$ the half-open interval~$(e_{i-1}+e_i,e_i+e_{i+1}]$.
The \emph{power} of~$e_i$, which we denote by~$p_i$,
is then defined as the least integer~$p$ such that~$\mathbf{I}(i)$ contains an element of the set~$\{k n / 2^{p-1} \colon k \in \mathbb{Z}\}$.
Thus, we (abusively) have~$p_0 = -\infty$ and~$p_\rho = 0$.
Finally, let~$R_{i \ldots j}$
be a run obtained by merging consecutive runs~$R_i,R_{i+1},\ldots,R_j$.
The \emph{power} of the run~$R$
is defined as~$\max\{p_{i-1},p_j\}$.
\end{definition}
\begin{lemma}\label{lem:unique-min-power}
Each non-empty sub-interval~$I$ of the set~$\{0,\ldots,\rho\}$ contains exactly one integer~$i$
such that~$p_i \leqslant p_j$ for all~$j \in I$.
\end{lemma}
\begin{proof}
Assume that the integer~$i$ is not unique.
Since~$e_0$ is the only endpoint with power~$-\infty$,
we know that~$0 \notin I$.
Then, let~$a$ and~$b$ be elements of~$I$ such that~$a < b$ and~$p_a = p_b \leqslant p_j$ for all~$j \in I$,
and let~$p = p_a = p_b$.
By definition of~$p_a$ and~$p_b$, there exist odd integers~$k$ and~$\ell$ such that~$k n / 2^{p - 1} \in \mathbf{I}(a)$ and~$\ell n / 2^{p - 1} \in \mathbf{I}(b)$.
Since~$j \geqslant k+1$,
the fraction~$(k + 1) n / 2^{p-1}$ belongs to
some interval~$\mathbf{I}(j)$ such that~$a \leqslant j \leqslant b$.
But since~$k+1$ is even, we know that~$p_j < p$,
which is absurd.
Thus, our initial assumption is invalid,
which completes the proof.
\end{proof}
\begin{corollary}
\label{cor:construction}
Let~$R_1,\ldots,R_\rho$ be the run decomposition of an array~$A$. There is exactly one tree~$\mathcal{T}$ that is induced on~$A$ and
in which every inner node has a smaller power than its
children. Furthermore, for every
run~$R_{i \ldots j}$ in~$\mathcal{T}$, we have~$\max\{p_{i-1},p_j\} < \min\{p_i,p_{i+1},\ldots,p_{j-1}\}$.
\end{corollary}
\begin{proof}
Given a merge tree~$\mathcal{T}$,
let us prove that the following statements are equivalent:
\begin{itemize}
\item[$\mathsf{S}_1$:]\label{item:unique:1}
each inner node of~$\mathcal{T}$ has a smaller power than its children;
\item[$\mathsf{S}_2$:]\label{item:unique:2}
each run~$R_{i \ldots j}$ that belongs to~$\mathcal{T}$ has a power that
is smaller than all of~$p_i,\ldots,p_{j-1}$;
\item[$\mathsf{S}_3$:]\label{item:unique:3}
if a run~$R_{i \ldots j}$ is an inner node of~$\mathcal{T}$,
its children are the two runs~$R_{i \ldots k}$ and~$R_{k+1 \ldots j}$ such that~$p_k = \min\{p_i,\ldots,p_{j-1}\}$.
\end{itemize}
First, if~$\mathsf{S}_1$ holds,
we prove~$\mathsf{S}_3$ by induction on
the height~$h$ of the run~$R_{i \ldots j}$.
Indeed, if the restriction of~$\mathsf{S}_3$ to runs of
height less than~$h$ holds,
let~$R_{i \ldots k}$ and~$R_{k+1 \ldots j}$ be the
children of a run~$R_{i \ldots j}$ of height~$h$.
If~$i < k$, the run~$R_{i \ldots k}$ has two children~$R_{i \ldots \ell}$ and~$R_{\ell+1 \ldots k}$ such that~$p_\ell = \min\{p_i,\ldots,p_{k-1}\}$,
and the powers of these runs, i.e.,~$\max\{p_{i-1},p_\ell\}$
and~$\max\{p_\ell,p_k\}$, are greater than the power of~$R_{i \ldots k}$, i.e.,~$\max\{p_{i-1},p_k\}$, which proves
that~$p_\ell > p_k$.
It follows that~$p_k = \min\{p_i,\ldots,p_k\}$, and one proves
similarly that~$p_k = \min\{p_k,\ldots,p_{j-1}\}$,
thereby showing that~$\mathsf{S}_3$ also holds for runs
of height~$h$.
Then, if~$\mathsf{S}_3$ holds,
we prove~$\mathsf{S}_2$ by
induction on the depth~$d$ of the run~$R_{i \ldots j}$.
Indeed, if the restriction of~$\mathsf{S}_2$ to runs of
depth less than~$d$ holds,
let~$R_{i \ldots k}$ and~$R_{k+1 \ldots j}$ be the
children of a run~$R_{i \ldots j}$ of depth~$d$.
Lemma~\ref{lem:unique-min-power} and~$\mathsf{S}_3$
prove that~$p_k$ is the unique smallest element of~$\{p_i,\ldots,p_{j-1}\}$, and the induction hypothesis proves
that~$\max\{p_{i-1},p_j\} < p_k$.
It follows that both powers~$\max\{p_{i-1},p_k\}$ and~$\max\{p_k,p_j\}$ are smaller than all of~$p_i,\ldots,p_{k-1},p_{k+1},\ldots,p_{j-1}$,
thereby showing that~$\mathsf{S}_2$ also holds for runs
of depth~$d$.
Finally, if~$\mathsf{S}_2$ holds, let~$R_{i \ldots j}$ be
an inner node of~$\mathcal{T}$, with children~$R_{i \ldots k}$ and~$R_{k+1 \ldots j}$.
Property~$\mathsf{S}_2$ ensures that~$\max\{p_{i-1},p_j\} < p_k$, and thus that~$\max\{p_{i-1},p_j\}$ is smaller than both~$\max\{p_{i-1},p_k\}$ and~$\max\{p_k,p_j\}$,
i.e., that~$R_{i \ldots j}$ has a smaller power that its
children, thereby proving~$\mathsf{S}_1$.
In particular, once the array~$A$ and its run decomposition~$R_1,\ldots,R_\rho$ are fixed,~$\mathsf{S}_3$ provides us with a
deterministic top-down construction of the unique merge tree~$\mathcal{T}$ induced on~$A$ and that satisfies~$\mathsf{S}_1$:
the root of~$\mathcal{T}$ must be the run~$R_{1 \ldots \rho}$ and,
provided that some run~$R_{i \ldots j}$ belongs to~$\mathcal{T}$, where~$i < j$, Lemma~\ref{lem:unique-min-power}
proves that the integer~$k$ mentioned in~$\mathsf{S}_3$
is unique, which means that~$\mathsf{S}_3$ unambiguously
describes the children of~$R_{i \ldots j}$ in the tree~$\mathcal{T}$.
This proves the first claim of
Corollary~\ref{cor:construction}, and the second claim of
Corollary~\ref{cor:construction}
follows from the equivalence between the
statements~$\mathsf{S}_1$ and~$\mathsf{S}_2$.
\end{proof}
This leads to the following characterisation of
the algorithm PowerSort\xspace, which is proved in~\cite[Lemma 4]{munro2018nearly}, and which
Corollary~\ref{cor:construction} allows us to
consider as an alternative
definition of PowerSort\xspace.
\begin{definition}\label{def:PoS}
In every merge tree that
PowerSort\xspace induces, inner nodes have a smaller power than their children.
\end{definition}
\begin{lemma}\label{lem:PoS:power}
Let~$\mathcal{T}$ be a merge tree induced by PowerSort\xspace,
let~$R$ be a run of~$\mathcal{T}$ with power~$p$,
and let~$\aR{2}$ be its grandparent.
We have~$2^{p-2} r < n < 2^p \ar{2}$.
\end{lemma}
\begin{proof}
Let~$R_{i \ldots j}$ be the run~$R$.
Without loss of generality, we assume that~$p = p_j$, the case~$p = p_{i-1}$ being entirely similar.
Corollary~\ref{cor:construction} states that
all of~$p_i,\ldots,p_{j-1}$ are larger than~$p$,
and therefore that~$p \leqslant \min\{p_i,\ldots,p_j\}$.
Thus, the union of intervals~$\mathbf{I}(i) \cup \ldots \cup \mathbf{I}(j) = (e_{i-1}+e_i,e_j+e_{j+1}]$ does not contain any element of the set~$\mathcal{S} = \{k n / 2^{p-2} \colon k \in \mathbb{Z}\}$.
Consequently, the bounds~$e_{i-1}+e_i$ and~$e_j+e_{j+1}$ are contained
between two consecutive elements of~$\mathcal{S}$, i.e.,
there exists an integer~$\ell$ such that
\[\ell n / 2^{p-2} \leqslant e_{i-1}+e_i \leqslant
e_j+e_{j+1} < (\ell+1) n / 2^{p-2},\]
and we conclude that
\[
r = e_j - e_{i-1} \leqslant (e_j+e_{j+1}) - (e_{i-1}+e_i)
< n / 2^{p-2}.\]
We prove now that~$n \leqslant 2^p \ar{2}$.
To that end, we assume that both~$R$ and~$\aR{1}$ left children, the other possible cases being
entirely similar.
There exist integers~$u$ and~$v$ such that~$\aR{1} = R_{i \ldots u}$ and~$\aR{2} = R_{i \ldots v}$.
Hence,~$\max\{p_{i-1},p_u\} < \max\{p_{i-1},p_j\} = p$,
which shows that~$p_u < p_j = p$.
Thus, both intervals~$\mathbf{I}(j)$ and~$\mathbf{I}(u)$, which are subintervals
of~$(2e_{i-1},2e_v]$,
contain elements of the set~$\mathcal{S}' = \{k n / 2^{p-1} \colon k \in \mathbb{Z}\}$.
This means that there exist two integers~$k$ and~$\ell$ such
that~$2e_{i-1} < k n / 2^{p-1} < \ell n / 2^{p-1}
\leqslant 2 e_v$, from which we conclude that
\[
\ar{2} = e_v - e_{i-1} > (\ell-k) n / 2^p \geqslant
n / 2^p. \qedhere\]
\end{proof}
\begin{theorem}\label{thm:fast-growth-PoS}
The algorithm PowerSort\xspace has the fast-growth property.
\end{theorem}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by PowerSort\xspace.
Then, let~$R$ be a run in~$\mathcal{T}$, and let~$p$ and~$\ap{3}$ be the respective powers of the runs~$R$ and~$\aR{3}$.
Definition~\ref{def:PoS} ensures
that~$p \geqslant \ap{3}+3$, and therefore
Lemma~\ref{lem:PoS:power} proves that
\[2^{\ap{3}+1} r \leqslant 2^{p-2} r < n < 2^{\ap{3}} \ar{5}.\]
This means that~$\ar{5} \geqslant 2 r$, and therefore
that PowerSort\xspace has the fast-growth property.
\end{proof}
\subsubsection{PeekSort\xspace}
\label{subsubsec:PeS}
Like its sibling PowerSort\xspace, the algorithm PeekSort\xspace is best defined
by characterizing the merge trees it induces.
\begin{definition}\label{def:tree:PeS}
Let~$\mathcal{T}$ be the merge tree induced by PeekSort\xspace on an array~$A$.
The children of each internal node~$R_{i \ldots j}$ of~$\mathcal{T}$
are the runs~$R_{i \ldots k}$ and~$R_{k+1 \ldots j}$ for which the quantity
\[ |2 e_k - e_j - e_{i-1}|\]
is minimal. In case of equality, the integer~$k$ is chosen to
be as small as possible.
\end{definition}
\begin{proposition}\label{pro:fast-growth-PeS}
The algorithm PeekSort\xspace has the fast-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by PeekSort\xspace,
and let~$R$ be a run in~$\mathcal{T}$.
We prove that~$\ar{3} \geqslant 2 r$.
Indeed, let us assume that a
majority of the runs~$R = \aR{0}$,~$\aR{1}$ and~$\aR{2}$ are left runs.
The situation is entirely similar if a majority of these runs
are right runs.
Let~$i < j$ be the two smallest integers such that~$\aR{i}$ and~$\aR{j}$ are left runs, and let~$S$ and~$T$ be their right siblings,
as illustrated in Figure~\ref{fig:pro:fast-growth-PeS}.
We can write these runs as~$\aR{i} = R_{w+1 \ldots x}$,~$S = R_{x+1 \ldots y}$,~$\aR{j} = R_{v+1 \ldots y}$ and~$T = R_{y+1 \ldots z}$
for some integers~$v \leqslant w < x < y < z$.
Definition~\ref{def:tree:PeS} states that
\[|\ar{j} - t| = |2 e_y - e_z - e_v| \leqslant
|2 e_{y-1} - e_z - e_v| = |\ar{j} - t - 2 r_y|.\]
Thus,~$\ar{j} - t - 2 r_y$ is negative, i.e.,~$t \geqslant \ar{j} - 2 r_y$, and
\[\ar{3} \geqslant \ar{j+1} = \ar{j} + t \geqslant
2 \ar{j} - 2 r_y = 2 (e_{y-1} - e_v) \geqslant 2 (e_x - e_w) = 2 \ar{i} \geqslant 2 r. \qedhere\]
\end{proof}
\begin{figure}
\caption{Runs~$\aR{i}
\label{fig:pro:fast-growth-PeS}
\end{figure}
\subsubsection{Adaptive ShiversSort\xspace}
\label{subsubsec:cASS}
The algorithm adaptive ShiversSort\xspace is presented in
Algorithm~\ref{alg:cASS}.
It is based on an \emph{ad hoc} tool,
which we call \emph{level} of a run :
the level of a run~$R$ is defined as the
number~$\ell = \lfloor \log_2(r) \rfloor$.
In practice, and following our naming conventions,
we denote by~$\al{i}$
and~$\ell_i$ the respective levels
of the runs~$\aR{i}$ and~$R_i$.
\begin{algorithm}[h]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on the
stack.}
\Note{We denote the height of the stack~$\mathcal{S}$ by~$h$, and its~$i$\textsuperscript{th} deepest run by~$R_i$. The length of~$R_i$ is denoted by~$r_i$, and we set~$\ell_i = \lfloor \log_2(r_i) \rfloor$. When two consecutive runs of~$\mathcal{S}$ are merged, they are replaced, in~$\mathcal{S}$, by the run resulting from the merge. \justifying}
\BlankLine~$\mathcal{S} \ensuremath{\leftarrow}~$ an empty stack\;
\While(\label{main-loop:cASS:start}){\textbf{true}\xspace}{
\If(\label{test:cASS:case-1})
{\textrm{$h \geqslant 3$ and~$\ell_{h-2}
\leqslant \max\{\ell_{h-1},\ell_h\}$}}
{merge the runs~$R_{h-2}$ and~$R_{h-1}$\label{alg:cASS:merge}}
\mathsf{E}lseIf{\textrm{the end of the array has not yet been reached}}
{find a new monotonic run~$R$, make it non-decreasing,
and push it onto~$\mathcal{S}$\label{cASS:case-push}}
\mathsf{E}lse{break\label{algline:cASS:end_inner_loop}}
}
\While{$h \geqslant 2$\label{alg:cASS:trigger:collapse}}{
merge the runs~$R_{h-1}$ and~$R_h$
\label{alg:cASS:collapse}
}
\end{small}
\caption{adaptive ShiversSort\xspace\label{alg:cASS}}
\end{algorithm}
Observe that appending a fictitious run
of length~$2n$ to the array~$A$ and
stopping our sequence of merges just before
merging that fictitious run does not modify
the sequence of merges performed by the algorithm,
but allows us to assume that every merge was
performed in line~\ref{alg:cASS:merge}.
Therefore, we work below under that assumption.
Our proof is based on the following result,
which was stated and proved in~\cite[Lemma 7]{Ju20}. The inequality (ii) is to be considered
only when~$h \geqslant 4$.
\begin{lemma}\label{lemma:cASS:invariant}
Let~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ be a stack
obtained while executing adaptive ShiversSort\xspace.
We have (i)~$\ell_i \geqslant \ell_{i+1}+1$
whenever~$1 \leqslant i \leqslant h-3$ and
(ii)~$\ell_{h-3} \geqslant \ell_{h-1}+1$.
\end{lemma}
Then, Lemmas~\ref{lem:cASS::right} and~\ref{lem:cASS::left}
focus on inequalities involving the levels
of a given run~$R$ belonging to a merge tree
induced by adaptive ShiversSort\xspace, and of its ancestors.
In each case, we denote by~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$
the stack at the moment where the run~$R$ is merged.
\begin{lemma}\label{lem:cASS::right}
If~$\aR{1}$ is a right run,~$\al{2} \geqslant \ell+1$.
\end{lemma}
\begin{proof}
The run~$R$ coincides with either~$R_{h-2}$ or~$R_{h-1}$, and~$\aR{1}$ is the parent of the runs~$R_{h-2}$ and~$R_{h-1}$. Hence,
the run~$R_{h-3}$ descends from the left sibling of~$\aR{1}$
and from~$\aR{2}$.
Thus, inequalities (i) and (ii) prove that~$\al{2} \geqslant \ell_{h-3}
\geqslant \max\{\ell_{h-2},\ell_{h-1}\}+1
\geqslant \ell+1$.
\end{proof}
\begin{lemma}\label{lem:cASS::left}
If~$R$ is a left run,~$\al{2} \geqslant \ell+1$.
\end{lemma}
\begin{proof}
Since~$R$ is a left run, it coincides with~$R_{h-2}$,
and~$\ell_{h-2} \leqslant \max\{\ell_{h-1},\ell_h\}$.
Then, if~$\aR{1}$ is a right run,
Lemma~\ref{lem:cASS::right} already proves that~$\al{2} \geqslant \ell+1$.
Otherwise,~$\aR{1}$ is a left run, and the run~$R_h$ descends from~$\aR{2}$, thereby proving that
\[\ar{2} \geqslant r + \max\{r_{h-1},r_h\}
\geqslant {2^\ell + 2^{\max\{\ell_{h-1},\ell_h\}}}
\geqslant 2^{\ell+1},\]
and thus that~$\al{2} \geqslant \ell+1$ too.
\end{proof}
\begin{proposition}\label{pro:fast-growth-cASS}
The algorithm adaptive ShiversSort\xspace has the fast-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by adaptive ShiversSort\xspace,
and let~$R$ be a run in~$\mathcal{T}$.
We will prove that~$\ar{6} \geqslant 2r$.
In order to do so,
we first prove that~$\al{3} \geqslant \ell+1$.
Indeed, Lemma~\ref{lem:cASS::right} shows that~$\al{3} \geqslant \al{2} \geqslant \ell+1$ if
if~$\aR{1}$ is a right run, and
Lemma~\ref{lem:cASS::left} shows that~$\al{3} \geqslant \al{1}+1 \geqslant \ell+1$
if~$\aR{1}$ is a left run.
We show similarly that~$\al{6} \geqslant \al{3}+1$,
and we conclude that~$\ar{6} \geqslant 2^{\al{6}} \geqslant 2^{\ell+2} \geqslant 2r$.
\end{proof}
\subsubsection{TimSort\xspace}
\label{subsubsec:TS}
The algorithm TimSort\xspace is presented in
Algorithm~\ref{alg:TS}.
\begin{algorithm}[h]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on the
stack.}
\Note{We denote the height of the stack~$\mathcal{S}$ by~$h$, and its~$i$\textsuperscript{th} deepest run by~$R_i$. The length of~$R_i$ is denoted by~$r_i$. When two consecutive runs of~$\mathcal{S}$ are merged, they are replaced, in~$\mathcal{S}$, by the run resulting from the merge. \justifying}
\BlankLine~$\mathcal{S} \ensuremath{\leftarrow}~$ an empty stack\;
\While(\label{main-loop:start}){\textbf{true}\xspace}{
\If(\label{test:case-1})
{\textrm{$h \geqslant 3$ and~$r_{h-2} < r_h$}}
{merge the runs~$R_{h-2}$ and~$R_{h-1}$
\tcp*[f]{case \#1}\label{case-1}}
\mathsf{E}lseIf(\label{test:case-2})
{\textrm{$h \geqslant 2$ and~$r_{h-1} \leqslant r_h$}}
{merge the runs~$R_{h-1}$ and~$R_h$
\tcp*[f]{case \#2}\label{case-2}}
\mathsf{E}lseIf(\label{test:case-3})
{\textrm{$h \geqslant 3$ and~$r_{h-2} \leqslant r_{h-1} + r_h$}}
{merge the runs~$R_{h-1}$ and~$R_h$
\tcp*[f]{case \#3}\label{case-3}}
\mathsf{E}lseIf(\label{test:case-4})
{\textrm{$h \geqslant 4$ and~$r_{h-3} \leqslant r_{h-2} + r_{h-1}$}}
{merge the runs~$R_{h-1}$ and~$R_h$
\tcp*[f]{case \#4}\label{case-4}}
\mathsf{E}lseIf{\textrm{the end of the array has not yet been reached}}
{find a new monotonic run~$R$, make it non-decreasing,
and push it onto~$\mathcal{S}$\label{case-push}}
\mathsf{E}lse{break\label{algline:end_inner_loop}}
}
\While{$h \geqslant 2$}{
merge the runs~$R_{h-1}$ and~$R_h$
\label{alg:collapse}
}
\end{small}
\caption{TimSort\xspace\label{alg:TS}}
\end{algorithm}
We say that a run~$R$ is
a \#1-, a \#2-, a \#3 or a \#4-run
if is merged in line~\ref{case-1},~\ref{case-2},
~\ref{case-3} or~\ref{case-4},
respectively. We also say that~$R$ is a \#\rlap{\textsuperscript{\phantom{2}1}}\textsubscript{2\phantom{1}4}\xspace-run if it is a \#1-, a \#2- or a
\#4-run, and that~$R$ is a \#\rlap{\textsuperscript{\phantom{3}2}}\textsubscript{3\phantom{2}4}\xspace-run if it is a
\#2-, a \#3- or a \#4-run.
Like in Section~\ref{subsubsec:cASS},
appending a fictitious run of length~$2n$ that we
will avoid merging allows us to assume that
every run is merged in line~\ref{case-1},~\ref{case-2},
~\ref{case-3} or~\ref{case-4}, i.e.,
is a \#1-, a \#2-, a \#3 or a \#4-run.
Our proof is then based on the following result,
which extends~\cite[Lemma 5]{auger2018worst}
by adding the inequality (v).
The inequality (ii) is to be considered
only when~$h \geqslant 3$,
and the inequalities (iii) to (v) are to be
considered only when~$h \geqslant 4$.
\begin{lemma}\label{lemma:invariant}
Let~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ be a stack
obtained while executing TimSort\xspace.
We have (i)~$r_i > r_{i+1} + r_{i+2}$
whenever~$1 \leqslant i \leqslant h-4$,
(ii)~$3 r_{h-2} > r_{h-1}$,
(iii)~$r_{h-3} > r_{h-2}$,
(iv)~$r_{h-3} + r_{h-2} > r_{h-1}$ and
(v)~$\max\{r_{h-3}/r,4 r_h\} > r_{h-1}$.
\end{lemma}
\begin{proof}
Lemma 5 from~\cite{auger2018worst} already
proves the inequalities (i) to (iv).
Therefore, we prove, by a direct induction on the
number of (push or merge) operations
performed before obtaining the stack~$\mathcal{S}$,
that~$\mathcal{S}$ also satisfies (v).
When the algorithm starts, we have~$h \leqslant 4$,
and therefore there is nothing to prove in that
case.
Then, when a stack~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ obeying the inequalities (i) to (v)
is transformed into a stack~$\overline{\mathcal{S}} = (\overline{R}_1,\overline{R}_2,
\ldots,\overline{R}_{\overline{h}})$
\begin{itemize}
\item by inserting a run,~$\overline{h}
= h+1$ and~$\overline{r}_{\overline{h}-3}
= r_{h-2} > r_{h-1} + r_h > 2 r_h = 2 \overline{r}_{\overline{h}-1}$;
\item by merging the runs~$R_{h-1}$ and~$R_h$,
we have~$\overline{h} = h-1$ and
\[\overline{r}_{\overline{h}-3} = r_{h-4} >
r_{h-3} + r_{h-2} > 2 r_{h-2} = 2 \overline{r}_{\overline{h}-1};\]
\item by merging the runs~$R_{h-2}$ and~$R_{h-1}$,
case \#1 just occurred and~$\overline{h} = h-1$,
so that
\[4 \overline{r}_{\overline{h}} = 4 r_h > 4 r_{h-2}
> r_{h-2} + r_{h-1} = \overline{r}_{\overline{h}-1}.\]
\end{itemize}
In each case,~$\overline{\mathcal{S}}$ satisfies (v),
which completes the induction and the proof.
\end{proof}
Then, Lemmas~\ref{lem::right} and~\ref{lem::left}
focus on inequalities involving the lengths
of a given run~$R$ belonging to a merge tree
induced by TimSort\xspace, and of its ancestors.
In each case, we denote by~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$
the stack at the moment where the run~$R$ is merged.
\begin{lemma}\label{lem::right}
If~$\aR{1}$ is a right run,~$\ar{2} \geqslant 4r/3$.
\end{lemma}
\begin{proof}
Let~$S$ be the left sibling of the run~$\aR{1}$,
and let~$i$ be the integer such that~$R = R_i$.
If~$i = h-2$, (iii) shows that~$r_{i-1} > r_i$.
If~$i = h-1$, (ii) shows that~$r_{i-1} > r_i / 3$.
In both cases, the run~$R_{i-1}$
descends from~$\aR{2}$, and
thus~$\ar{2} \geqslant r + r_{i-1}
\geqslant 4 r / 3$.
Finally, if~$i = h$, the run~$R$ is a \#\rlap{\textsuperscript{\phantom{3}2}}\textsubscript{3\phantom{2}4}\xspace-right run,
which means both that~$r_{h-2} \geqslant r$ and that~$R_{h-2}$ descends from~$S$.
It follows that~$\ar{2} \geqslant r + r_{h-2} \geqslant 2 r$.
\end{proof}
\begin{lemma}\label{lem::left}
If~$R$ is a left run,~$\ar{2} \geqslant 5 r / 4$.
\end{lemma}
\begin{proof}
We treat four cases independently, depending on
whether~$R$ is a \#1-, a \#2-, a \#3- or a \#4-left run.
In each case, we assume that the run~$\aR{1}$ is a left run,
since Lemma~\ref{lem::right} already proves
that~$\ar{2} \geqslant 4r/3$
when~$\aR{1}$ is a right run.
\begin{itemize}
\item If~$R$ is a \#1-left run,
the run~$R = R_{h-2}$ is merged with~$R_{h-1}$ and~$r_{h-2} < r_h$. Since~$\aR{1}$ is a left run,~$R_h$ descends from~$\aR{2}$, and thus~$\ar{2} \geqslant r + r_h \geqslant 2r$.
\item If~$R$ is a \#2-left run, the run~$R = R_{h-1}$ is merged with~$R_h$ and~$r_{h-1} \leqslant r_h$. It follows, in that case,
that~$\ar{2} \geqslant \ar{1} = r + r_h \geqslant 2 r$.
\item If~$R$ is a \#3-left run, the run~$R = R_{h-1}$ is merged with~$R_h$, and~$r_{h-2} \leqslant r_{h-1} + r_h =
\ar{1}$.
Due to this inequality,
our next journey through the
loop of lines~\ref{main-loop:start}
to~\ref{algline:end_inner_loop} must trigger
another merge.
Since~$\aR{1}$ is a left run, that merge
must be a \#1-merge, which means that~$r_{h-3} < \ar{1}$.
Consequently, (v) proves that
\[\ar{1} \geqslant \max\{r_{h-3},r_{h-1}+r_h\} \geqslant
5 r_{h-1}/4 = 5 r/4.\]
\item We prove that~$R$ cannot be a \#4-left runs.
Indeed, if~$R$ is a \#4-left run, the run~$R = R_{h-1}$ is merged with~$R_h$,
and we both have~$r_{h-2} > \ar{1}$ and~$r_{h-3} \leqslant r_{h-2} + r_{h-1} \leqslant
r_{h-2} + \ar{1}$.
Due to the latter inequality,
our next journey through the
loop of lines~\ref{main-loop:start}
to~\ref{algline:end_inner_loop} must trigger
another merge.
Since~$\ar{1} < r_{h-2} < r_{h-3}$, that new merge
cannot be a \#1-merge, and thus~$\aR{1}$ is a right run,
contradicting our initial assumption. \qedhere
\end{itemize}
\end{proof}
\begin{proposition}\label{pro:fast-growth-TS}
The algorithm TimSort\xspace has the fast-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by TimSort\xspace,
and let~$R$ be a run in~$\mathcal{T}$.
We will prove that~$\ar{3} \geqslant
5 r/4$.
Indeed, if~$\aR{1}$ is a right run,
Lemma~\ref{lem::right} proves that~$\ar{3} \geqslant \ar{2} \geqslant 4r/3$.
Otherwise,~$\aR{1}$ is a left run, and
Lemma~\ref{lem::left}
proves that~$\ar{3} \geqslant 5 \ar{1}/4 \geqslant 5r/4$.
\end{proof}
\subsubsection{\textalpha-MergeSort\xspace}
\label{subsubsec:aMS}
The algorithm \textalpha-MergeSort\xspace is
parametrised by a real number~$\alpha > 1$ and
is presented in Algorithm~\ref{alg:aMS}.
\begin{algorithm}[h]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort, parameter~$\alpha > 1$}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on the
stack.}
\Note{We denote the height of the stack~$\mathcal{S}$ by~$h$, and its~$i$\textsuperscript{th} deepest run by~$R_i$. The length of~$R_i$ is denoted by~$r_i$. When two consecutive runs of~$\mathcal{S}$ are merged, they are replaced, in~$\mathcal{S}$, by the run resulting from the merge. \justifying}
\BlankLine~$\mathcal{S} \ensuremath{\leftarrow}~$ an empty stack\;
\While(\label{aMS:main-loop:start}){\textbf{true}\xspace}{
\If(\label{test:aMS:case-1})
{\textrm{$h \geqslant 3$ and~$r_{h-2} < r_h$}}
{merge the runs~$R_{h-2}$ and~$R_{h-1}$
\tcp*[f]{case \#1}\label{aMS:case-1}}
\mathsf{E}lseIf(\label{test:aMS:case-2})
{\textrm{$h \geqslant 2$ and~$r_{h-1} < \alpha r_h$}}
{merge the runs~$R_{h-1}$ and~$R_h$
\tcp*[f]{case \#2}\label{aMS:case-2}}
\mathsf{E}lseIf(\label{test:aMS:case-3})
{\textrm{$h \geqslant 3$ and~$r_{h-2} < \alpha r_{h-1}$}}
{merge the runs~$R_{h-1}$ and~$R_h$
\tcp*[f]{case \#3}\label{aMS:case-3}}
\mathsf{E}lseIf{\textrm{the end of the array has not yet been reached}}
{find a new monotonic run~$R$, make it non-decreasing,
and push it onto~$\mathcal{S}$\label{aMS:case-push}}
\mathsf{E}lse{break\label{algline:aMS:end_inner_loop}}
}
\While{$h \geqslant 2$}{
merge the runs~$R_{h-1}$ and~$R_h$
\label{alg:aMS:collapse}
}
\end{small}
\caption{\textalpha-MergeSort\xspace\label{alg:aMS}}
\end{algorithm}
Like in Section~\ref{subsubsec:TS},
we say that a run~$R$ is
a \#1-, a \#2- or a \#3-run
if is merged in line~\ref{aMS:case-1},~\ref{aMS:case-2}
or~\ref{aMS:case-3}.
We also say that~$R$ is
a \#\rlap{\textsuperscript{1}}\textsubscript{2}\xspace-run if it is a \#1- or a \#2-run, and that~$R$ is a \#\rlap{\textsuperscript{2}}\textsubscript{3}\xspace-run if it is a \#2- or a \#3-run.
In addition, still like in Sections~\ref{subsubsec:cASS}
and~\ref{subsubsec:TS}, we safely assume that
each run is a \#1-, a \#2- or a \#3-run.
Our proof is then based on the following result,
which extends~\cite[Theorem 14]{BuKno18}
by adding the inequalities (ii) and (iii),
which are to be considered only when~$h \geqslant 3$.
\begin{lemma}\label{lemma:aMS:invariant}
Let~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ be a stack
obtained while executing \mbox{\textalpha-MergeSort\xspace}.
We have (i)~$r_i \geqslant \alpha r_{i+1}$
whenever~$1 \leqslant i \leqslant h-3$,
(ii)~$r_{h-2} \geqslant (\alpha-1) r_{h-1}$ and
(iii)~$\max\{r_{h-2}/\alpha,\alpha r_h/(\alpha-1)\} \geqslant r_{h-1}$.
\end{lemma}
\begin{proof}
Theorem 14 from~\cite{BuKno18} already proves the
inequality (i). Therefore, we prove,
by a direct induction on the number
of (push or merge) operations
performed before obtaining the stack~$\mathcal{S}$,
that~$\mathcal{S}$ satisfies (ii) and (iii).
When the algorithm starts, we have~$h \leqslant 2$,
and therefore there is nothing to prove in that
case.
Then, when a stack~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ obeying (i), (ii)
and (iii) is transformed into a stack~$\overline{\mathcal{S}} = (\overline{R}_1,\overline{R}_2,
\ldots,\overline{R}_{\overline{h}})$
\begin{itemize}
\item by inserting a run,~$\overline{h}
= h+1$ and~$\overline{r}_{\overline{h}-2}
= r_{h-1} \geqslant \alpha r_h = \overline{r}_{\overline{h}-1}$;
\item by merging the runs~$R_{h-1}$ and~$R_h$,
we have~$\overline{h} = h-1$ and~$\overline{r}_{\overline{h}-2}
= r_{h-3} \geqslant \alpha r_{h-2} = \overline{r}_{\overline{h}-1}$;
\item by merging the runs~$R_{h-2}$ and~$R_{h-1}$,
case \#1 just occurred and~$\overline{h} = h-1$,
so that
\[\min\{\overline{r}_{\overline{h}-2},\alpha\overline{r}_{\overline{h}}\} =
\min\{r_{h-3},\alpha r_h\} \geqslant \alpha r_{h-2}
\geqslant
(\alpha-1)(r_{h-2} + r_{h-1}) = (\alpha-1) \overline{r}_{\overline{h}-1}.\]
\end{itemize}
In each case,~$\overline{\mathcal{S}}$ satisfies (ii) and (iii),
which completes the induction and the proof.
\end{proof}
Lemmas~\ref{lem:aMS::right} and~\ref{lem:aMS::left}
focus on inequalities involving the lengths
of a given run~$R$ belonging to a merge tree
induced by \textalpha-MergeSort\xspace, and of its ancestors.
In each case, we denote by~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$
the stack at the moment where the run~$R$ is merged.
In what follows, we also set~$\alpha^\star = \min\{\alpha,1+1/\alpha,1+(\alpha-1)/\alpha\}$.
\begin{lemma}\label{lem:aMS::right}
If~$\aR{1}$ is a right run,~$\ar{2} \geqslant \alpha^\star r$.
\end{lemma}
\begin{proof}
Let~$i$ be the integer such that~$R = R_i$.
If~$i = h-2$, (i) shows that~$r_{i-1} \geqslant \alpha r_i$.
If~$i = h-1$, (ii) shows that~$r_{i-1} \geqslant (\alpha-1) r_i$.
In both cases,~$R_{i-1}$
descends from~$\aR{2}$, and
thus~$\ar{2} \geqslant r + r_{i-1}
\geqslant \alpha r$.
Finally, if~$i = h$, the run~$R$ is a \#\rlap{\textsuperscript{2}}\textsubscript{3}\xspace-right run,
which means that~$r_{h-2} \geqslant r$ and that~$R_{h-2}$
descends from the left sibling of~$\ar{1}$.
It follows that~$\ar{2} \geqslant r + r_{h-2} \geqslant 2 r \geqslant
(1 + 1/\alpha) r$.
\end{proof}
\begin{lemma}\label{lem:aMS::left}
If~$R$ is a left run,~$\ar{2} \geqslant \alpha^\star r$.
\end{lemma}
\begin{proof}
We treat three cases independently, depending on
whether~$R$ is a \#1-, a \#2 or a \#3-left run.
In each case, we assume that~$\aR{1}$ is a left run,
since Lemma~\ref{lem:aMS::right} already proves that~$\ar{2} \geqslant \alpha^\star r$ when~$\aR{1}$ is a right
run.
\begin{itemize}
\item If~$R$ is a \#1-left run,
the run~$R = R_{h-2}$ is merged with~$R_{h-1}$ and~$r_{h-2} < r_h$. Since~$\aR{1}$ is a left run,~$R_h$ descends from~$\aR{2}$, and thus~$\ar{2} \geqslant r + r_h \geqslant 2r \geqslant
(1 + 1/\alpha) r$.
\item If~$R$ is a \#2-left run, the run~$R = R_{h-1}$ is merged with~$R_h$ and~$r < \alpha r_h$. It follows, in that case, that~$\ar{2} \geqslant \ar{1} =
r + r_h \geqslant (1 + 1/\alpha) r$.
\item If~$R$ is a \#3-left run, the run~$R = R_{h-1}$ is merged with~$R_h$ and~$r_{h-2} < \alpha r_{h-1}$.
Hence, (iii) proves that~$(\alpha-1) r \leqslant \alpha r_h$, so that~$\ar{2} \geqslant \ar{1} =
r + r_h \geqslant (1 + (\alpha-1)/\alpha) r$. \qedhere
\end{itemize}
\end{proof}
\begin{proposition}\label{pro:fast-growth-aMS}
The algorithm \textalpha-MergeSort\xspace has the fast-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by \textalpha-MergeSort\xspace,
and let~$R$ be a run in~$\mathcal{T}$.
We will prove that~$\ar{3} \geqslant
\alpha^\star r$.
Indeed, if~$\aR{1}$ is a right run,
Lemma~\ref{lem:aMS::right} proves that~$\ar{3} \geqslant \ar{2} \geqslant
\alpha^\ast r$.
Otherwise,~$\aR{1}$ is a left run, and
Lemma~\ref{lem:aMS::left} proves that~$\ar{3} \geqslant \alpha^\ast \ar{1}
\geqslant \alpha^\ast r$.
\end{proof}
\subsection{Algorithms with the tight middle-growth property}
\label{subsec:other-tight}
\subsubsection{PowerSort\xspace}
\label{subsubsec:PoS:2}
\begin{proposition}
\label{pro:POS:tight}
The algorithm PowerSort\xspace has the tight middle-growth
property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by PowerSort\xspace and
let~$R$ be a run in~$\mathcal{T}$ at depth at least~$h$.
We will prove that~$\ar{h} \geqslant 2^{h-4}$.
If~$h \leqslant 4$, the desired inequality
is immediate.
Then, if~$h \geqslant 5$,
let~$n$ be the length of the array on which~$\mathcal{T}$ is induced.
Let also~$p$ and~$\ap{h-2}$ be the
respective powers of the runs~$R$ and~$\aR{h-2}$.
Definition~\ref{def:PoS} and
Lemma~\ref{lem:PoS:power} prove that~$2^{\ap{h-2}+h-4} \leqslant
2^{p-2} \leqslant 2^{p-2} r < n <
2^{\ap{h-2}} \ar{h}$.
\end{proof}
\subsubsection{NaturalMergeSort\xspace}
\label{subsubsec:NMS}
The algorithm NaturalMergeSort\xspace consists in a plain binary merge
sort, whose unit pieces of data to be merged
are runs instead of being single elements.
Thus, we identify NaturalMergeSort\xspace with the fundamental
property that describes those merge trees it induces.
\begin{definition}\label{def:tree:NMS}
Let~$\mathcal{T}$ be a merge tree induced by NaturalMergeSort\xspace,
and let~$R$ and~$\overline{R}$ be two runs that are
siblings of each other in~$\mathcal{T}$.
Denoting by~$n$ and~$\overline{n}$
the respective numbers of leaves of~$\mathcal{T}$ that
descend from~$R$ and from~$\overline{R}$,
we have~$|n - \overline{n}| \leqslant 1$.
\end{definition}
\begin{proposition}\label{pro:fast-growth-NMS}
The algorithm NaturalMergeSort\xspace has the tight
middle-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by NaturalMergeSort\xspace,
let~$R$ be a run in~$\mathcal{T}$, and let~$h$
be its height.
We will prove by induction on~$h$
that, if~$h \geqslant 1$, the run~$R$ is an ancestor
of at least~$2^{h-1}+1$ leaves of~$\mathcal{T}$,
thereby showing that~$r \geqslant 2^{h-1}$.
First, this is the case if~$h = 1$.
Then, if~$h \geqslant 2$, let~$R_1$ and~$R_2$ be the
two children of~$R$. One of them, say~$R_1$,
has height~$h-1$. Let us denote by~$n$,~$n_1$ and~$n_2$ the number of
leaves that descend from~$R$,~$R_1$ and~$R_2$, respectively.
The induction hypothesis shows that
\[n = n_1 + n_2 \geqslant 2 n_1-1 \geqslant
2 \times (2^{h-2}+1) - 1 = 2^{h-1}+1,\]
which completes the proof.
\end{proof}
\subsubsection{ShiversSort\xspace}
\label{subsubsec:ShS}
The algorithm ShiversSort\xspace is presented
in Algorithm~\ref{alg:ShS}.
Like adaptive ShiversSort\xspace, it relies on the notion of
\emph{level} of a run.
\begin{algorithm}[ht]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on the
stack.}
\Note{We denote the height of the stack~$\mathcal{S}$ by~$h$, and its~$i$\textsuperscript{th} deepest run by~$R_i$. The length of~$R_i$ is denoted by~$r_i$, and we set~$\ell_i = \lfloor \log_2(r_i) \rfloor$. When two consecutive runs of~$\mathcal{S}$ are merged, they are replaced, in~$\mathcal{S}$, by the run resulting from the merge. \justifying}
\BlankLine~$\mathcal{S} \ensuremath{\leftarrow}~$ an empty stack\;
\While(\label{main-loop:ShS:start}){\textbf{true}\xspace}{
\If(\label{test:ShS:case-1})
{\textrm{$h \geqslant 1$ and~$\ell_{h-1}
\leqslant \ell_h$}}
{merge the runs~$R_{h-1}$ and~$R_h$\label{ShS:std:merge}}
\mathsf{E}lseIf{\textrm{the end of the array has not yet been reached}}
{find a new monotonic run~$R$, make it non-decreasing,
and push it onto~$\mathcal{S}$\label{ShS:case-push}}
\mathsf{E}lse{break\label{algline:ShS:end_inner_loop}}
}
\While{$h \geqslant 2$}{
merge the runs~$R_{h-1}$ and~$R_h$
\label{alg:ShS:collapse}
}
\end{small}
\caption{ShiversSort\xspace\label{alg:ShS}}
\end{algorithm}
Our proof is based on the following result,
which appears in the proof
of~\cite[Theorem 11]{BuKno18}.
\begin{lemma}\label{lemma:ShS:invariant}
Let~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ be a stack
obtained while executing ShiversSort\xspace.
We have~$\ell_i \geqslant \ell_{i+1}+1$
whenever~$1 \leqslant i \leqslant h-2$.
\end{lemma}
Lemmas~\ref{lem:ShS:right}
and~\ref{lem:ShS:left-right}
focus on inequalities involving the lengths
of a given run~$R$ belonging to a merge tree
induced by adaptive ShiversSort\xspace, and of its ancestors.
In each case, we denote by~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$
the stack at the moment where the run~$R$ is merged.
However, and unlike
Sections~\ref{subsubsec:cASS}
to~\ref{subsubsec:aMS}, we cannot simulate
the merge operations that occur in
line~\ref{alg:ShS:collapse} as if they had
occurred in line~\ref{ShS:std:merge}.
Instead, we say that a run~$R$ is \emph{rightful}
if~$R$ and its ancestors are all right runs
(i.e., if~$R$ belongs to the rightmost branch
of the merge tree), and that~$R$ is
\emph{standard} otherwise.
\begin{lemma}\label{lem:ShS:right}
Let~$R$ be a run in~$\mathcal{T}$, and let~$k \geqslant 1$
be an integer. If each of the~$k$ runs~$\aR{0},\ldots,\aR{k-1}$ is a right
run,~$\al{k} \geqslant k-1$.
\end{lemma}
\begin{proof}
For all~$i \leqslant k$, let~$u(i)$ be the least
integer such that~$R_{u(i)}$
descends from~$\aR{i}$.
Since~$\aR{0},\ldots,\aR{k-1}$ are right runs,
we know that~$u(k) < u(k-1) < \ldots < u(0) = h$.
Thus, Lemma~\ref{lemma:ShS:invariant} proves that
\[\al{k} \geqslant \ell_{u(k)} \geqslant
\ell_{u(1)} + (k-1) \geqslant k-1. \qedhere\]
\end{proof}
\begin{lemma}\label{lem:ShS:left-right}
Let~$R$ be a run in~$\mathcal{T}$, and let~$k \geqslant 1$
be an integer. If~$R$ is a left run and
the~$k-1$ runs~$\aR{1},\ldots,\aR{k-1}$
are right runs, we have~$\al{k} \geqslant \ell+k-1$ if~$\aR{1}$ is
a rightmost run, and~$\al{k} \geqslant \ell+k$
if~$\aR{1}$ is a standard run.
\end{lemma}
\begin{proof}
First, assume that~$k = 1$.
If~$\aR{1}$ is rightful, the desired inequality
is immediate.
If~$\aR{1}$ is standard, however,
the left run~$R = R_{h-1}$ was merged with
the run~$R_h$ because~$\ell \leqslant \ell_h$.
In that case, it follows that~$\ar{1} = r + r_h \geqslant 2^\ell + 2^{\ell_h}
\geqslant 2^\ell + 2^\ell = 2^{\ell+1}$,
i.e., that~$\al{1} \geqslant \ell+1$.
Assume now that~$k \geqslant 2$.
Note that~$\aR{k}$ is rightful if and only if~$\aR{1}$ is also rightful.
Then, for all~$i \leqslant k$, let~$u(i)$ be the least
integer such that the run~$R_{u(i)}$
descends from~$\aR{i}$.
Since~$\aR{1},\ldots,\aR{k-1}$ are right runs,
we know that
\[u(k) < u(k-1) < \ldots < u(1) = h-1.\]
In particular, let~$R'$ be the left sibling of~$\aR{k-1}$: this is an ancestor of~$R_{u(k)}$,
and the left child of~$\aR{k}$.
Consequently, Lemma~\ref{lemma:ShS:invariant} and
our study of the case~$k = 1$ conjointly prove that
\begin{itemize}
\item
$\al{k} \geqslant \ell' \geqslant \ell_{u(k)}
\geqslant \ell_{u(1)}+k-1 = \ell+k-1$ if~$\aR{1}$ and~$\aR{k}$ are rightful;
\item $\al{k} \geqslant \ell'+1 \geqslant \ell_{u(k)}+1
\geqslant \ell_{u(1)}+k = \ell+k$
if~$\aR{1}$ and~$\aR{k}$ are standard. \qedhere
\end{itemize}
\end{proof}
\begin{proposition}\label{pro:middle-growth-ShS}
The algorithm ShiversSort\xspace has the tight
middle-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by ShiversSort\xspace
and let~$R$ be a run in~$\mathcal{T}$ at depth at least~$h$. We will prove that~$\ar{h} \geqslant 2^{h-2}$.
Let~$a_1 < a_2 < \ldots < a_k$ be the
non-negative integers smaller than~$h$ and
for which~$\aR{a_i}$ is a left run. We also set~$a_{k+1} = h$.
Lemma~\ref{lem:ShS:right} proves that~$\al{a_1} \geqslant a_1-1$.
Then, for all~$i < k$, the run~$\aR{a_i+1}$
is standard, since it descends from the left run~$\aR{a_k}$, and thus
Lemma~\ref{lem:ShS:left-right} proves that~$\al{a_{i+1}} \geqslant \al{a_i} + a_{i+1} - a_i$.
Lemma~\ref{lem:ShS:left-right} also proves that~$\al{a_{k+1}} \geqslant \al{a_k} + a_{k+1} - a_k-1$.
It follows that~$\al{h} = \al{a_{k+1}} \geqslant
h-2$, and therefore that~$\ar{h} \geqslant 2^{\al{h}} \geqslant 2^{h-2}$.
\end{proof}
\subsection{Algorithms with the middle-growth property}
\label{subsec:other-middle}
\subsubsection{\textalpha-StackSort\xspace}
\label{subsubsec:aSS}
The algorithm \textalpha-StackSort\xspace, which
predated and inspired its variant
\textalpha-MergeSort\xspace, is presented in
Algorithm~\ref{alg:aSS}.
\begin{algorithm}[ht]
\begin{small}
\mathcal{S}etArgSty{texttt}
\mathsf{D}ontPrintSemicolon
\Input{Array~$A$ to sort, parameter~$\alpha > 1$}
\mathcal{R}esult{The array~$A$ is sorted into a single run.
That run remains on the
stack.}
\Note{We
denote
the
height
of
the
stack
~$\mathcal{S}$
by
~$h$,
and
its
~$i$\textsuperscript{th}
deepest
run
by
~$R_i$.
The
length
of
~$R_i$
is
denoted
by
~$r_i$.
When
two
consecutive
runs
of
~$\mathcal{S}$
are
merged,
they
are replaced, in~$\mathcal{S}$,
by the run resulting from the merge.}
\BlankLine~$\mathcal{S} \ensuremath{\leftarrow}~$ an empty stack\;
\While(\label{aSS:main-loop:start}){\textbf{true}\xspace}{
\If{\textrm{$h \geqslant 2$ and
~$r_{h-1} \leqslant \alpha r_h$}}
{merge the runs~$R_{h-1}$ and~$R_h$}
\mathsf{E}lseIf{\textrm{the end of the array has not yet been reached}}
{find a new monotonic run~$R$, make it non-decreasing,
and push it onto~$\mathcal{S}$\label{aSS:case-push}}
\mathsf{E}lse{break\label{algline:aSS:end_inner_loop}}
}
\While{$h \geqslant 2$}{
merge the runs~$R_{h-1}$ and~$R_h$
\label{alg:aSS:collapse}
}
\end{small}
\caption{\textalpha-StackSort\xspace\label{alg:aSS}}
\end{algorithm}
Our proof is based on the following result,
which appears in~\cite[Lemma 2]{AuNiPi15}.
\begin{lemma}\label{lemma:aSS:invariant}
Let~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$ be a stack
obtained while executing \mbox{\textalpha-StackSort\xspace}.
We have~$r_i > \alpha r_{i+1}$
whenever~$1 \leqslant i \leqslant h-2$.
\end{lemma}
Lemmas~\ref{lem:aSS::right} and~\ref{lem:aSS::left}
focus on inequalities involving the lengths
of a given run~$R$ belonging to a merge tree
induced by \textalpha-StackSort\xspace, and of its ancestors.
In each case, we denote by~$\mathcal{S} = (R_1,R_2,\ldots,R_h)$
the stack at the moment where the run~$R$ is merged.
Furthermore, like in
Section~\ref{subsubsec:ShS},
we say that a run~$R$ is \emph{rightful}
if~$R$ and its ancestors are all right runs,
and that~$R$ is
\emph{standard} otherwise.
\begin{lemma}\label{lem:aSS::right}
Let~$R$ be a run in~$\mathcal{T}$, and let~$k$ and~$m$
be two integers. If~$k$ of the~$m+1$ runs~$\aR{0}, \aR{1}, \ldots, \aR{m}$
are right runs,~$\ar{m+1} \geqslant
\alpha^{k-1}$.
\end{lemma}
\begin{proof}
Let~$a_1 < a_2 < \ldots < a_k$ be the smallest
integers such that~$\aR{a_1},\aR{a_2},
\ldots,\aR{a_k}$ are right runs.
For all~$i \leqslant k-1$,
let~$\aS{i}$ be the left sibling of~$\aR{a_i}$,
and let~$R_{u(i)}$ be a leaf of~$\aS{i}$.
Since~$R_{u(i)}$ descends from~$\aR{a_{i+1}}$, we
know that~$u(i+1) < u(i)$, so
that~$u(k) \leqslant u(1) - (k-1)$.
Thus, Lemma~\ref{lemma:aSS:invariant} proves that~$\ar{m+1} \geqslant r_{u(k)} \geqslant
\alpha^{k-1} r_{u(1)} \geqslant \alpha^{k-1}$.
\end{proof}
\begin{lemma}\label{lem:aSS::left}
Let~$R$ be a run in~$\mathcal{T}$.
If~$R$ is a left run and if its
parent is a standard
run, we have~$\ar{1} \geqslant (1+1/\alpha)r$.
\end{lemma}
\begin{proof}
Since~$R$ is a standard left run, it coincides with~$R_{h-1}$ and
~$r_{h-1} \leqslant \alpha r_h$. It follows
immediately that~$\ar{1} = r_{h-1} + r_h \geqslant (1+1/\alpha) r$.
\end{proof}
\begin{proposition}\label{pro:middle-growth-aSS}
The algorithm \textalpha-StackSort\xspace has the middle-growth property.
\end{proposition}
\begin{proof}
Let~$\mathcal{T}$ be a merge tree induced by ShiversSort\xspace, let~$R$ be a run in~$\mathcal{T}$ at depth at least~$h$,
and let~$\beta = \min\{\alpha,1+1/\alpha\}^{1/4}$. We will prove that~$\ar{h} \geqslant \beta^h$.
Indeed, let~$k = \lceil h/2 \rceil$.
We distinguish two cases, which are not
mutually exclusive if $h$ is even.
\begin{itemize}
\item If at least~$k$ of the~$h$ runs~$R, \aR{1},\aR{2}, \ldots, \aR{h-1}$
are right runs,
Lemma~\ref{lem:aSS::right} shows that
\[\ar{h} \geqslant \alpha^{k-1}
\geqslant \beta^{4(k-1)}.\]
\item If at least~$k$ of the~$h$ runs~$R, \aR{1},\aR{2}, \ldots, \aR{h-1}$
are left runs,
let~$a_1 < a_2 < \ldots < a_k$ be the smallest
integers such that~$\aR{a_1},\aR{a_2}, \ldots,
\aR{a_k}$ are left runs.
When~$j < k$, the run~$\aR{a_j+1}$ is standard,
since it descends from the left run~$\aR{a_k}$.
Therefore, due to Lemma~\ref{lem:aSS::left},
an immediate induction on~$j$ shows that~$\ar{a_j+1} \geqslant (1+1/\alpha)^j$
for all~$j \leqslant k-1$.
It follows that that
\[\ar{h} \geqslant \ar{a_{k-1}+1} \geqslant
(1+1/\alpha)^{k-1} \geqslant \beta^{4(k-1)}.\]
\end{itemize}
Consequently,~$\ar{h} \geqslant \beta^{4(k-1)} \geqslant \beta^{2h-4} \geqslant \beta^h$ whenever~$h \geqslant 4$,
whereas~$\ar{h} \geqslant 1 = \beta^h$
whenever~$h = 0$
and~$\ar{h} \geqslant 2 \geqslant 1+1/\alpha
\geqslant \beta^h$ whenever~$1 \leqslant h \leqslant 3$.
\end{proof}
\section{Refined complexity bounds}
\label{sec:precise-bounds-PoS}
One weakness of Theorem~\ref{thm:middle-few} is that it cannot
help us to distinguish the complexity upper bounds of those
algorithms that have the middle-growth property, although
the constants hidden in the~$\mathcal{O}$ symbol could be dramatically
different. Below, we study these constants,
and focus on upper bounds of the type~$\mathsf{c} n \mathcal{H}^\ast + \mathcal{O}(n)$ or~$\mathsf{c} n (1 + o(1)) \mathcal{H}^\ast + \mathcal{O}(n)$.
Since sorting arrays of length~$n$, in general, requires
at least~$\log_2(n!) = n \log_2(n) + \mathcal{O}(n)$ comparisons,
and since~$\mathcal{H}^\ast \leqslant \log_2(n)$ for all arrays,
we already know that~$\mathsf{c} \geqslant 1$ for any such
constant~$\mathsf{c}$.
Below, we focus on finding matching upper bounds in two
regimes: first using a fixed parameter~$\mathbf{t}$,
thereby obtaining a constant~$\mathsf{c} > 1$,
and then letting~$\mathbf{t}$ depend on the lengths
of those runs that are being merged, in which case we
reach the constant~$\mathsf{c} = 1$.
Inspired by the success of
Theorem~\ref{thm:middle-few-naïve}, which states
that algorithms with
the tight middle-growth property sort array
of length~$n$ by using
only~$\mathsf{c} n \log_2(n) + \mathcal{O}(n)$ element
comparisons with~$\mathsf{c} = 1$,
we focus primarily on that property,
while not forgetting other algorithms
that also enjoyed~$n \mathcal{H} + \mathcal{O}(n)$
or~$n \log_2(n) + \mathcal{O}(n)$ complexity upper
bounds despite not having the tight
middle-growth property.
\subsection{Fixed parameter t and middle-growth property}
\label{subsec:fixed-t}
\begin{theorem}\label{thm:PoS-constant}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the tight middle-growth property.
For each parameter~$\mathbf{t} \geqslant 0$,
if~$\mathcal{A}$ uses the~$\mathbf{t}$-galloping
sub-routine for merging runs,
it requires at most
\[(1 + 1 / (\mathbf{t}+3)) n \mathcal{H}^\ast + \log_2(\mathbf{t}+1) n +
\mathcal{O}(n)\] element comparisons
to sort arrays of length~$n$ and dual run-length entropy~$\mathcal{H}^\ast$.
\end{theorem}
\begin{proof}
Let us follow a variant of the proof
of Theorem~\ref{thm:middle-few}.
Let~$\gamma$ be the integer mentioned
in the definition of the statement
``$\mathcal{A}$ has the tight middle-growth property'',
let~$\mathcal{T}$ be the merge tree induced by~$\mathcal{A}$
on an array~$A$ of length~$n$,
and let~$s_1,s_2,\ldots,s_\sigma$ be the lengths
of the dual runs of~$A$.
Like in the proof of Theorem~\ref{thm:middle-few},
we just need to prove that
\[ \sum_{R \in \mathcal{T}} \mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}) \leqslant
(1+1/(\mathbf{t}+3)) s_i \log_2(n / s_i) +
s_i \log_2(\mathbf{t}+1) +
\mathcal{O}(s_i)\]
for all~$i \leqslant \sigma$.
Let~$\mathcal{R}_h$ be the set of runs
at height~$h$ in~$\mathcal{T}$. By construction, no run
in~$\mathcal{R}_h$ descends from another one, which
proves that~$\sum_{R \in \mathcal{R}_h} r_{\rightarrow i} \leqslant s_i$ and that~$\sum_{R \in \mathcal{R}_h} r \leqslant n$.
Since each run~$R \in \mathcal{R}_h$
is of length~$r \geqslant 2^{h-\gamma}$,
it follows that~$|\mathcal{R}_h| \leqslant n / 2^{h-\gamma}$.
Then, consider the function
\[\mathsf{C}_{\mathbf{t}}(h) = \sum_{R \in\mathcal{R}_h}
\mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}).\]
We noted above that
\[ \mathsf{C}_{\mathbf{t}}(h) \leqslant
(1+1/(\mathbf{t}+3)) \sum_{R \in \mathcal{R}_h} r_{\rightarrow i} \leqslant (1+1/(\mathbf{t}+3))
s_i\]
for all~$h \geqslant 0$.
Let also~$f \colon x \mapsto \mathbf{t} + 2 + 2 \log_2(x+1)$,~$g \colon x \mapsto x \, f(s_i / x)$ and~$\mu = \lceil\log_2((\mathbf{t}+1) n / s_i)\rceil$.
Both functions~$f$ and~$g$ are positive, concave
and increasing on~$(0,+\infty)$, which shows that
\begin{align*}
\mathsf{C}_{\mathbf{t}}(\mu + \gamma + h) & \leqslant
\sum_{R \in \mathcal{R}_{\mu + \gamma+h}} f(r_{\rightarrow i})
\leqslant
|\mathcal{R}_{\mu + \gamma+h}| \, f\big({\textstyle\sum_{R \in \mathcal{R}_{\mu + \gamma+h}} r_{\rightarrow_i} / |\mathcal{R}_{\mu + \gamma+h}|}\big) \leqslant
g\big(|\mathcal{R}_{\mu + \gamma+h}|\big) \\
& \leqslant
g\big(n / 2^{\mu + h}\big) \leqslant
g\big(2^{-h} s_i / (\mathbf{t}+1)\big) \\
& \leqslant
\big(\mathbf{t}+2+
2 \log_2\!\big( 2^h (\mathbf{t}+1) + 1 \big)\big)
2^{-h} s_i / (\mathbf{t}+1)
\\
& \leqslant \big(\mathbf{t} + 2 + 2(h(\mathbf{t}+1)+1)\big)2^{-h} s_i /(\mathbf{t}+1) \leqslant (4+2h) 2^{-h} s_i.
\end{align*}
We conclude that
\begin{align*}
\sum_{R \in \mathcal{T}} \mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i})
& = \sum_{h \geqslant 0} \mathsf{C}_{\mathbf{t}}(h)
= \sum_{h=0}^{\mu+\gamma-1} \mathsf{C}_{\mathbf{t}}(h) + \sum_{h \geqslant 0}
\mathsf{C}_{\mathbf{t}}(\mu + \gamma+h) \\
& \leqslant (1+1/(\mathbf{t}+3)) (\mu+\gamma) s_i +
4 s_i \sum_{h \geqslant 0} 2^{-h} +
2 s_i \sum_{h \geqslant 0} h 2^{-h} \\
& \leqslant
(1+1/(\mathbf{t}+3))(\log_2(n / s_i) + \log_2(\mathbf{t}+1)) s_i + \mathcal{O}(s_i) \\
& \leqslant
(1+1/(\mathbf{t}+3)) \log_2(n / s_i) s_i +
\log_2(\mathbf{t}+1) s_i + \mathcal{O}(s_i). &&\qedhere
\end{align*}
\end{proof}
\subsection{Parameter t with logarithmic growth and tight middle-growth property}
\label{subsec:variable-t}
Letting the parameter~$\mathbf{t}$ vary,
we minimise the upper bound provided by Theorem~\ref{thm:PoS-constant}
by choosing~$\mathbf{t} = \mathcal{T}heta(\mathcal{H}^\ast+1)$, in which case
this upper bound simply becomes~$n \mathcal{H}^\ast + \log_2(\mathcal{H}^\ast+1)n + \mathcal{O}(n)$.
However, computing~$\mathcal{H}^\ast$ before
starting the actual sorting process is not reasonable.
Instead, we update the parameter~$\mathbf{t}$
as follows, which will provide us with a
slightly larger upper bound.
\begin{definition}
\label{def:log-galloping}
We call \emph{logarithmic} galloping sub-routine
the merging sub-routine that, when merging adjacent runs
of lengths~$a$ and~$b$, performs the same comparisons and
element moves as the~$\mathbf{t}$-galloping sub-routine
for~$\mathbf{t} = \lceil \log_2(a+b) \rceil$.
\end{definition}
\begin{theorem}\label{thm:PoS-log}
Let~$\mathcal{A}$ be a stable natural merge sort algorithm
with the tight middle-growth property.
If~$\mathcal{A}$ uses the logarithmic galloping
sub-routine for merging runs,
it requires at most
\[n \mathcal{H}^\ast +
2 \log_2(\mathcal{H}^\ast+1) n + \mathcal{O}(n)\]
element comparisons
to sort arrays of length~$n$ and dual run-length entropy~$\mathcal{H}^\ast$.
\end{theorem}
\begin{proof}
Let us refine and adapt the proofs
of Theorems~\ref{thm:middle-few}
and~\ref{thm:PoS-constant}.
Let~$\gamma$ be the integer mentioned
in the definition of the statement
``$\mathcal{A}$ has the tight middle-growth property'',
let~$\mathcal{T}$ be the merge tree induced by~$\mathcal{A}$
on an array~$A$ of length~$n$,
and let~$s_1,s_2,\ldots,s_\sigma$ be the lengths
of the dual runs of~$A$.
Using a parameter~$\mathbf{t} = \lceil \log_2(r) \rceil$
to merge runs~$R'$ and~$R''$ into one run~$R$
requires at most
\[1 + \sum_{i=1}^\sigma
\mathsf{cost}_{\lceil \log_2(r) \rceil}^\ast(r'_{\rightarrow i}) +
\mathsf{cost}_{\lceil \log_2(r) \rceil}^\ast(r''_{\rightarrow i})\]
element comparisons.
Given that
\begin{align*}
\mathsf{cost}_{\lceil \log_2(r) \rceil}^\ast(r'_{\rightarrow i}) & \leqslant
\min\{(1+1/\log_2(r)) r'_{\rightarrow i}, \log_2(r)+3+2 \log_2(r'_{\rightarrow i}+1)\} \\
& \leqslant \min\{(1+1/\log_2(r)) r'_{\rightarrow i}, 3\log_2(r+1)+3\},
\end{align*}
and that~$r'_{\rightarrow i} + r''_{\rightarrow i}=r_{\rightarrow i}$,
this makes a total of at most
\[1 + \sum_{i=1}^\sigma \mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i})\]
element comparisons, where~$\mathsf{cost}_{\log}^\ast(r,m) = \min\{(1+1/\log_2(r)) m,6\log_2(r+1)+6\}$.
Then, let~$\mathcal{T}^\ast$ denote the tree obtained
after removing the leaves of~$\mathcal{T}$.
We focus on proving that
\[\sum_{R \in \mathcal{T}^\ast}
\mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i}) \leqslant
s_i \log_2(n/s_i) + 2 s_i \log_2(\log_2(2n/s_i)) +
\mathcal{O}(s_i)\]
for all~$i \leqslant \sigma$.
Indeed, finding the run decomposition~$R_1,R_2,\ldots,R_\rho$ of~$A$
requires~$n-1$ comparisons,
and~$\rho-1 \leqslant n-1$ merges
are then performed, which will make a total of
up to
\begin{align*}
2n+\sum_{R \in \mathcal{T}^\ast}\sum_{i=1}^\sigma
\mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i})
& \leqslant
2 n + \sum_{i=1}^\sigma
s_i \log_2(n/s_i) + 2 s_i \log_2(\log_2(n/s_i)+1) +
\mathcal{O}(s_i) \\
& \leqslant n \mathcal{H}^\ast + 2 \log_2(\mathcal{H}^\ast+1) n + \mathcal{O}(n)
\end{align*}
comparisons, the latter inequality being due to
the concavity of the function~$x \mapsto \log_2(x+1)$.
Then, let~$\mathcal{R}_h$ be the set of runs
at height~$h$ in~$\mathcal{T}$, and let
\[\mathsf{C}_{\log}(h) = \sum_{R \in\mathcal{R}_h}
\mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i}).\]
No run in~$\mathcal{R}_h$ descends
from another one, and each run~$R \in \mathcal{R}_h$ has length~$r \geqslant 2^{\max\{1,h-\gamma\}}$, which proves that~$|\mathcal{R}_h| \leqslant n / 2^{h-\gamma}$ and that
\[\mathsf{C}_{\log}(h) \leqslant
\sum_{R \in \mathcal{R}_h}
(1 + 1 / \log_2(r)) r_{\rightarrow i} \leqslant
\sum_{R \in \mathcal{R}_h}
(1 + 1 / \!\max\{1,h-\gamma\}) r_{\rightarrow i}
\leqslant
(1 + 1 / \!\max\{1,h-\gamma\}) s_i.\]
Let also~$f \colon x \mapsto 1 + \log_2(x+1)$,~$g \colon x \mapsto x \, f(n / x)$,~$z = n/s_i \geqslant 1$ and~$\nu = \lceil\log_2(z \log_2(2 z))\rceil + \gamma$.
Both functions~$f$ and~$g$ are concave,
positive and increasing on~$(0,+\infty)$,
which proves that
\begin{align*}
\mathsf{C}_{\log}(\nu + h) / 6 & \leqslant
\sum_{R \in \mathcal{R}_{\nu +h}}
f(r)
\leqslant
|\mathcal{R}_{\nu+h}| \, f\big({\textstyle\sum_{R \in \mathcal{R}_{\nu+h}} r / |\mathcal{R}_{\nu + h}|}\big) \leqslant
g(|\mathcal{R}_{\nu+h}|) \leqslant
g(n / 2^{\nu-\gamma+h}) \\
& \leqslant
g\big(2^{-h} s_i/\log_2(2z)\big) =
2^{-h} s_i f(2^h z \log_2(2z)) / \log_2(2z) \\
& \leqslant 2^{-h} s_i \big(1 + \log_2(2^h
(2z)^2)\big) / \log_2(2z) =
2^{-h} (h+1) s_i / \log_2(2z) +
2^{1-h} s_i \\
& \leqslant (h+3) 2^{-h} s_i,
\end{align*}
where the inequality between the second and
third line simply comes from the fact that
\[1 + 2^h z \log_2(2z) \leqslant
1 + 2^{h+1} z^2 \leqslant 2^h (1+2z^2) \leqslant
2^h \times (2z)^2\]
whenever~$h \geqslant 0$ and~$z \geqslant 1$.
It follows that
\[\sum_{h=1}^\gamma \mathsf{C}_{\log}(h) +
\sum_{h \geqslant 0} \mathsf{C}_{\log}(\nu + h)
\leqslant 2 \gamma s_i + 6 \sum_{h \geqslant 0}(h+3)2^{-h}s_i =
\mathcal{O}(s_i),\]
whereas
\[\sum_{h=1}^{\nu-\gamma-1} \mathsf{C}_{\log}(\gamma+h) \leqslant
\sum_{h=1}^{\nu-1} (1+1/h) s_i \leqslant ((\nu-1)+1+\ln(\nu-1)) s_i = (\nu+\ln(\nu)) s_i.\]
Thus, we conclude that
\begin{align*}
\sum_{R \in \mathcal{T}^\ast} \mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i})
& = \sum_{h \geqslant 1} \mathsf{C}_{\log}(h) \leqslant
s_i(\nu + \ln(\nu)) + \mathcal{O}(s_i) \\
& \leqslant
s_i \log_2(z) + s_i \log_2(\log_2(2z)) + s_i \log_2(\log_2(2z^2)) + \mathcal{O}(s_i) \\
& \leqslant
s_i \log_2(z) + 2 s_i \log_2(\log_2(2z)) + \mathcal{O}(s_i). &&\qedhere
\end{align*}
\end{proof}
It is of course possible to approach
the~$n \mathcal{H}^\ast + \log_2(\mathcal{H}^\ast+1)n + \mathcal{O}(n)$
upper bound by improving our
update policy. For instance,
choosing~$\mathbf{t} =
\tau \lceil \log_2(a+b) \rceil$
for a given constant~$\tau \geqslant 1$ provides us with an
\[n \mathcal{H}^\ast + (1+1/\tau) \log_2(\mathcal{H}^\ast+1)n +
\log_2(\tau) + \mathcal{O}(n)\]
upper bound, and choosing~$\mathbf{t} =
\lceil \log_2(a+b) \rceil \times
\lceil \log_2(\log_2(a+b))\rceil$ further improves that upper bound.
However, such improvements soon become negligible
in comparison with the overhead of having to
compute the value of~$\mathbf{t}$.
\subsection{Refined upper bounds for adaptive ShiversSort\xspace}
\label{subsec:cASS:precise}
\begin{proposition}
\label{pro:cASS:not-tight}
The algorithm adaptive ShiversSort\xspace does not have the
tight middle-growth property.
\end{proposition}
\begin{proof}
Let~$A_k$ be an array whose run decomposition
consists in runs of lengths~$1 \cdot 2 \cdot 1 \cdot 4 \cdot 1 \cdot 8 \cdot 1 \cdot 16 \cdot 1 \cdots 1 \cdot 2^k \cdot 1$
for some integer~$k \geqslant 0$.
When sorting the array~$A_k$, the algorithm
adaptive ShiversSort\xspace keeps merging the two leftmost
runs at its disposal.
The resulting tree, represented in
Figure~\ref{fig:22-cASS-tree},
has height~$h = 2k$ and its root
is a run of length~$r = 2^{k+1}+k-1 = o(2^h)$.
\end{proof}
\begin{figure}
\caption{Merge tree induced by adaptive ShiversSort\xspace on~$A_4$.
Each run is labelled by its length.\label{fig:22-cASS-tree}
\label{fig:22-cASS-tree}
\end{figure}
\begin{theorem}
\label{thm:cASS-log}
Theorems~\ref{thm:PoS-constant}
and~\ref{thm:PoS-log} remain valid if
we consider the algorithm
adaptive ShiversSort\xspace instead of
an algorithm with the tight middle-growth property.
\end{theorem}
\begin{proof}
Both proofs of Theorems~\ref{thm:PoS-constant}
and~\ref{thm:PoS-log} are based on the following
construction.
Given a constant~$\gamma$ and a merge tree~$\mathcal{T}$,
we partition the runs of~$\mathcal{T}$
into sets~$\mathcal{R}_h$ of pairwise incomparable runs,
such that each run~$R$ in each set~$\mathcal{R}_h$
has a length~$r \geqslant 2^{h-\gamma}$.
In practice, in both proofs, we define~$\mathcal{R}_h$ as
be the set of runs with height~$h$.
Below, we adapt this approach.
Let~$\mathcal{T}$ be the merge tree induced by adaptive ShiversSort\xspace on an
array~$A$ of length~$n$
and dual runs~$S_1,S_2,\ldots,S_\sigma$.
Following the terminology of~\cite{Ju20},
we say that a run~$R$ in~$\mathcal{T}$ is
\emph{non-expanding} if it has the same level
as its parent~$\aR{1}$, i.e., if~$\ell = \al{1}$.
It is shown, in the proof
of~\cite[Theorem 3.2]{Ju20},
that the lengths of the non-expanding runs
sum up to an integer smaller than~$3n$.
Hence, we partition~$\mathcal{T}$ as follows.
First, we place all the non-expanding runs
into one set~$\mathcal{R}_{-1}$.
Then, for each~$\ell \geqslant 0$,
we define~$\mathcal{R}_\ell$ as the set of expanding runs
with level~$\ell$:
by construction, each run~$R$ in~$\mathcal{R}_\ell$
has a length~$r \geqslant 2^\ell$,
and the elements of~$\mathcal{R}_\ell$ are pairwise
incomparable.
Thus, if we use a constant parameter~$\mathbf{t}$,
we perform a total of at most~$2n + \sum_{h \geqslant -1} \mathsf{C}_{\mathbf{t}}^\ast(h)$
element comparisons, where
\[\mathsf{C}_{\mathbf{t}}^\ast(h) = \sum_{R \in \mathcal{R}_h}
\sum_{i=1}^\sigma
\mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}).\]
The proof of
Theorem~\ref{thm:PoS-constant} then shows that
\[\sum_{h \geqslant 0}
\mathsf{C}_{\mathbf{t}}^\ast(h) \leqslant (1+1/(\mathbf{t}+3)) n \mathcal{H}^\ast
+ \log_2(\mathbf{t}+1)n + \mathcal{O}(n),\]
whereas
\[\mathsf{C}_{\mathbf{t}}^\ast(-1) =
\sum_{R \in \mathcal{R}_{-1}}
\sum_{i=1}^\sigma
\mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i})
\leqslant
\sum_{R \in \mathcal{R}_{-1}}
\sum_{i=1}^\sigma 2 r_{\rightarrow i} =
\sum_{R \in \mathcal{R}_{-1}} 2 r
\leqslant 6 n,\]
thereby proving that adaptive ShiversSort\xspace has the desired
complexity upper bound when using the~$\mathbf{t}$-galloping
sub-routine.
Similarly, we perform at most~$2n + \mathsf{C}_{\log}^\ast(-1) + \sum_{h \geqslant 1}
\mathsf{C}_{\log}^\ast(h)$ element comparisons
if we use the logarithmic
galloping sub-routine, where we set
\[\mathsf{C}_{\log}^\ast(h) = \sum_{R \in \mathcal{R}^\ast_h}
\sum_{i=1}^\sigma
\mathsf{cost}_{\log}^\ast(r,r_{\rightarrow i}).\]
and~$\mathcal{R}^\ast_h$ denotes the set of runs in~$\mathcal{R}_h$
that are not leaves of~$\mathcal{T}$.
By following the proof of
Theorem~\ref{thm:PoS-log},
one concludes similarly that
\[\sum_{h \geqslant 1}
\mathsf{C}_{\log}^\ast(h) \leqslant n \mathcal{H}^\ast
+ 2 \log_2(\mathcal{H}^\ast+1)n + \mathcal{O}(n),\]
and observing that
\[\mathsf{C}_{\log}^\ast(-1) \leqslant \sum_{R \in \mathcal{R}^\ast_{-1}} (1+1/\log_2(r)) r \leqslant
\sum_{R \in \mathcal{R}^\ast_{-1}} 2 r \leqslant 6 n\]
completes the proof.
\end{proof}
\subsection{Refined upper bounds for PeekSort\xspace}
\label{subsec:PeS:precise}
\begin{proposition}
\label{pro:PeS:not-tight}
The algorithm PeekSort\xspace does not have the
tight middle-growth property.
\end{proposition}
\begin{proof}
For all~$k \geqslant 0$,
let~$\mathcal{S}_k$ be the integer-valued
sequence defined inductively by~$\mathcal{S}_0 = 1$ and~$\mathcal{S}_{k+1} = \mathcal{S}_k \cdot
3^k \cdot \mathcal{S}_k$, where~$u \cdot v$ denotes the concatenation of two
sequences~$u$ and~$v$.
Then, let~$B_k$ be an array whose run decomposition
consists in runs whose lengths form the sequence~$\mathcal{S}_k$.
When sorting the array~$B_k$, the algorithm
PeekSort\xspace performs two merges per
sub-sequence~$S_1 = 1 \cdot 1 \cdot 1$,
thereby obtaining a sequence
of run lengths that is~$3 \mathcal{S}_{k-1}$, i.e.,
the sequence~$\mathcal{S}_{k-1}$ whose elements were
tripled. It then works on that new
sequence, obtaining the sequences~$9 \mathcal{S}_{k-2},
\ldots,3^k \mathcal{S}_0$.
The resulting tree, represented in
Figure~\ref{fig:32-PeS-tree},
has height~$h = 2k$ and its root
is a run of length~$r = 3^k = o(2^h)$.
\end{proof}
\begin{figure}
\caption{Merge tree induced by PeekSort\xspace on~$B_3$.
Each run is labelled by its length.\label{fig:32-PeS-tree}
\label{fig:32-PeS-tree}
\end{figure}
The method of casting aside runs with a limited
total length, which we employed while adapting
the proofs Theorems~\ref{thm:PoS-constant}
and~\ref{thm:PoS-log} to adaptive ShiversSort\xspace, does not work with
the algorithm PeekSort\xspace.
Instead, we rely on the approach employed by
Bayer~\cite{Ba75} to prove the~$n\mathcal{H}+2n$ upper
bound on the number of element comparisons and
moves when PeekSort\xspace uses the naïve merging
sub-routine. This approach is based on the
following result, which relies on the notions of
\emph{split runs} and \emph{growth rate}
of a run.
In what follows, let us recall some notations:
given an array~$A$ of length~$n$ whose run decomposition
consists of runs~$R_1,R_2,\ldots,R_\rho$, we set~$e_i = r_1+ \ldots + r_i$ for all integers~$i \leqslant \rho$,
and we denote by~$R_{i \ldots j}$ a run that results from
merging consecutive runs~$R_i,R_{i+1},\ldots,R_j$.
\begin{definition}
\label{def:split:runs}
Let~$\mathcal{T}$ be a merge tree, and let~$R$ and~$R'$
be the children of a run~$\overline{R}$.
The \emph{split run} of~$\overline{R}$ is
defined as the rightmost leaf of~$R$ if~$r \geqslant r'$, and as the leftmost leaf
of~$R'$ otherwise.
The length of that split run is then called the
\emph{split length} of~$\overline{R}$, and it is
denoted by~$\mathsf{sl}(\overline{R})$.
Finally, the quantity~$\log_2(\overline{r}) -
\max\{\log_2(r),\log_2(r')\}$ is called \emph{growth rate}
of the run~$\overline{R}$, and denoted by~$\mathsf{gr}(\overline{R})$.
\end{definition}
\begin{lemma}
\label{lem:bayer}
Let~$\mathcal{T}$ be a merge tree induced by PeekSort\xspace.
We have~$\mathsf{gr}(\overline{R}) \overline{r} + 2 \mathsf{sl}(\overline{R})
\geqslant \overline{r}$ for each internal node~$\overline{R}$ of~$\mathcal{T}$.
\end{lemma}
\begin{proof}
Let~$R = R_{i \ldots k}$ and~$R' = R_{k+1 \ldots j}$
be the children of the run~$\overline{R}$.
We assume, without loss of generality, that~$r \geqslant r'$. The case~$r < r'$
is entirely symmetric.
Definition~\ref{def:tree:PeS} then states that
\[|r - r'| = |2 e_k - e_j - e_{i-1}| \leqslant
|2 e_{k-1} - e_j - e_{i-1}| = |r - r' - 2 r_k|,\]
which means that~$r- r' - 2 r_k$ is negative
and that~$r - r' \leqslant 2 r_k + r' - r$.
Then, consider the function~$f \colon t \mapsto
4t-3-\log_2(t)$. This function is non-negative
and increasing on the interval~$[1/2,+\infty)$.
Finally, let~$z = r / \overline{r}$, so that~$\mathsf{gr}(\overline{R}) =
-\log_2(z)$ and~$r' = (1-z) \overline{r}$. Since~$r \geqslant r'$, we have~$z \geqslant 1/2$,
and therefore
\[\mathsf{gr}(\overline{R}) \overline{r} + 2 \mathsf{sl}(\overline{R})
= 2 r_k - \log_2(z) \overline{r} \geqslant
2 (r - r') - \log_2(z) \overline{r} =
(f(z)+1) \overline{r} \geqslant \overline{r}. \qedhere\]
\end{proof}
\begin{theorem}
\label{thm:PeS-constant}
Theorem~\ref{thm:PoS-constant} remains valid if
we consider the algorithm
PeekSort\xspace instead of
an algorithm with the tight middle-growth property.
\end{theorem}
\begin{proof}
Let~$\mathcal{T}$ be the merge tree induced by PeekSort\xspace on an
array~$A$ of length~$n$,
runs~$R_1,R_2,\ldots,R_\rho$
and dual runs~$S_1,S_2,\ldots,S_\sigma$,
and let~$\mathcal{T}^\ast$ be the tree obtained by deleting
the~$\rho$ leaves~$R_i$ from~$\mathcal{T}$.
Using~$\mathbf{t}$-galloping
to merge runs~$R$ and~$R'$ into one run~$\overline{R}$
requires at most
\begin{align*}
\mathcal{C}_{\overline{R}} & = 1 + \sum_{i=1}^\sigma
\mathsf{cost}_{\mathbf{t}}^\ast(r_{\rightarrow i}) +
\mathsf{cost}_{\mathbf{t}}^\ast(r'_{\rightarrow i}) \\
& \leqslant 1 + \sum_{i=1}^\sigma
\min\{(1+1/(\mathbf{t}+3)) \overline{r}_{\rightarrow i},
2\mathbf{t} + 4 + 4 \log_2(\overline{r}_{\rightarrow i}+1)\}
\end{align*}
element comparisons. Since~$1 + 1/(\mathbf{t}+3) \leqslant 2$ and
\[2 \, \mathsf{sl}(\overline{R}) \geqslant (1 - \mathsf{gr}(\overline{R})) \overline{r}
= \sum_{i=1}^\sigma (1 - \mathsf{gr}(\overline{R})) \overline{r}_{\rightarrow i},\]
we even have
\[\mathcal{C}_{\overline{R}} \leqslant 1 + 4 \mathsf{sl}(\overline{R}) + \sum_{i=1}^\sigma
\mathsf{cost}_{\mathbf{t},\mathsf{gr}(\overline{R})}^\heartsuit(\overline{r}_{\rightarrow i}),\]
where we set~$\mathsf{cost}_{\mathbf{t},\gamma}^\heartsuit(m) =
\min\{(1+1/(\mathbf{t}+3)) \gamma m,2\mathbf{t}+4+4\log_2(m+1)\}$.
Then, note that each leaf run~$R_i$ is a split run
of at most two internal nodes of~$\mathcal{T}$: these are
the parents of the least ancestor of~$R_i$ that
is a left run
(this least ancestor can be~$R_i$ itself) and of the least ancestor of~$R_i$
that is a right run.
It follows that
\[\sum_{R \in \mathcal{T}^\ast} \mathsf{sl}(R) \leqslant 2 \sum_{i=1}^\rho r_i = 2 n.\]
Consequently, overall, and including the~$n-1$
comparisons needed to find the run decomposition
of~$A$, the algorithm PeekSort\xspace performs a total of
at most
\[n + \sum_{R \in \mathcal{T}^\ast} \mathcal{C}_{R}
\leqslant 8 n + \sum_{i=1}^\sigma
\sum_{R \in \mathcal{T}^\ast} \mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i})\]
elements comparisons.
Now, given an integer~$i \leqslant \sigma$,
we say that a run~$R$ is \emph{small}
if it has a length~$r < (\mathbf{t}+1) n / s_i$, and
that~$R$ is \emph{large} otherwise.
Then, we denote by~$\mathcal{T}^{\mathsf{small}}$ the set of
small runs of~$\mathcal{T}$, and by~$\mathcal{T}^{\mathsf{large}}$
the set of large runs of~$\mathcal{T}$.
We will prove both inequalities
\begin{align}
\sum_{R \in \mathcal{T}^{\mathsf{small}}} \!\!\mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) \leqslant
(1+1/(\mathbf{t}+3)) \log_2((\mathbf{t}+1) n / s_i) s_i \\
\sum_{R \in \mathcal{T}^{\mathsf{large}}} \!\!\mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) = \mathcal{O}(s_i),
\end{align}
from which Theorem~\ref{thm:PeS-constant} will follow.
We first focus on proving the inequality (1).
For all~$j \leqslant \rho$, let~$R_j^\uparrow$
be the set of those strict ancestors of
the run~$R_j$ that are small, i.e., of the runs~$\aR{k}_j$ such
that~$1 \leqslant k \leqslant |R_j^\uparrow|$.
With these notations, it already follows that
\begin{align*}
\sum_{R \in \mathcal{T}^{\mathsf{small}}}
\mathsf{gr}(R) r_{\rightarrow i} & =
\sum_{R \in \mathcal{T}^{\mathsf{small}}}
\sum_{j=1}^\rho \mathbf{1}_{R \in R_j^\uparrow} \mathsf{gr}(R) (r_j)_{\rightarrow i} =
\sum_{j=1}^\rho \sum_{k=1}^{|R_j^\uparrow|} \mathsf{gr}(\aR{k}_j) (r_j)_{\rightarrow i} \\
& \leqslant
\sum_{j=1}^\rho \sum_{k=1}^{|R_j^\uparrow|}
\log_2(\ar{k}_j / \ar{k-1}_j) (r_j)_{\rightarrow i}
=
\sum_{j=1}^\rho \log_2(\ar{|R_j^\uparrow|}_j / r_j) (r_j)_{\rightarrow i} \\
& \leqslant \sum_{j=1}^\rho \log_2((\mathbf{t}+1) n/s_i) (r_j)_{\rightarrow i} = \log_2((\mathbf{t}+1) n/s_i) s_i,
\end{align*}
which proves that
\[\sum_{R \in \mathcal{T}^{\mathsf{small}}} \mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) \leqslant
(1+1/(\mathbf{t}+3)) \sum_{R \in \mathcal{T}^{\mathsf{small}}} \mathsf{gr}(R)
r_{\rightarrow i} \leqslant
(1+1/(\mathbf{t}+3)) \log_2((\mathbf{t}+1) n / s_i) s_i.\]
Finally, let
\[\mathsf{C}_{\mathbf{t}}^{\mathsf{large}}(h) = \sum_{R \in \mathcal{R}^{\mathsf{large}}_h}
\mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}),\]
where~$\mathcal{R}^{\mathsf{large}}_h$ denotes the set
of runs with height~$h \geqslant 0$
in the sub-tree~$\mathcal{T}^{\mathsf{large}}$.
Since PeekSort\xspace has the fast-growth property, there is an integer~$\kappa$ such that~$\ar\kappa \geqslant 2 r$ for all runs~$R$ in~$\mathcal{T}$.
Thus, each run~$R$ in~$\mathcal{R}^{\mathsf{large}}_h$ is of length~$r \geqslant 2^{h/\kappa-1} (\mathbf{t}+1) n / s_i$, and~$|\mathcal{R}^{\mathsf{large}}_h| \leqslant 2^{1-h/\kappa} s_i / (\mathbf{t}+1)$.
Then, since the functions~$f \colon x \mapsto 2\mathbf{t}+4+4\log_2(x+1)$
and~$g \colon x \mapsto x f(s_i/x)$
are positive, increasing and
concave on~$(0,+\infty)$, we have
\begin{align*}
\mathsf{C}_{\mathbf{t}}^{\mathsf{large}}(h)
& \leqslant
\sum_{R \in \mathcal{R}^{\mathsf{large}}_h}f(r_{\rightarrow i})
\leqslant
|\mathcal{R}^{\mathsf{large}}_h| \, f\big({\textstyle\sum_{R \in \mathcal{R}^{\mathsf{large}}_h} r / |\mathcal{R}^{\mathsf{large}}_h|}\big) \leqslant
g(|\mathcal{R}^{\mathsf{large}}_h|) \leqslant
g\big(2^{1-h/\kappa}s_i / (\mathbf{t}+1)\big) \\
& \leqslant
2^{1-h/\kappa}s_i \big(2\mathbf{t}+4+4
\log_2(2^{h/\kappa-1} (\mathbf{t}+1) + 1)\big)/ (\mathbf{t}+1) \\
& \leqslant
2^{1-h/\kappa}s_i \big(2\mathbf{t}+4+4
(h + \mathbf{t} + 2)\big)/ (\mathbf{t}+1)
\leqslant
(12 + 4 h) 2^{1-h/\kappa}s_i
\end{align*}
for all~$h \geqslant 0$. This allows us to
conclude that
\[\sum_{R \in \mathcal{T}^{\mathsf{large}}} \mathsf{cost}_{\mathbf{t},\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) =
\sum_{h \geqslant 0} \mathsf{C}_{\mathbf{t}}^{\mathsf{large}}(h)
\leqslant \sum_{h \geqslant 0}(12+4h)2^{1-h/\kappa} s_i
= \mathcal{O}(s_i). \qedhere\]
\end{proof}
\begin{theorem}
\label{thm:PeS-log}
Theorem~\ref{thm:PoS-log} remains valid if
we consider the algorithm
PeekSort\xspace instead of
an algorithm with the tight middle-growth property.
\end{theorem}
\begin{proof}
Let us reuse the notations we introduced while
proving Theorem~\ref{thm:PeS-constant}.
With these notations, the algorithm PeekSort\xspace
performs at total of at most
\[\mathcal{C} = 8n + \sum_{i=1}^\sigma \sum_{R \in \mathcal{T}^\ast}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^{\heartsuit}(r_{\rightarrow i})\]
element comparisons.
Let~$\theta = 2^{\ln(2)}$, and let~$\mathcal{T}^\ast$ be the tree obtained after removing
the leaves of~$\mathcal{T}$.
We say that an inner node~$R$ or~$\mathcal{T}$
is \emph{biased} if it has a child~$R'$ such that~$r \leqslant \theta r'$, i.e.,
if~$\mathsf{gr}(R) \leqslant \ln(2)$.
Let~$\mathcal{T}^{\mathsf{biased}}$ be the set of such runs.
In~$R$ is biased,
Lemma~\ref{lem:bayer} proves that
the split run of~$R$ is a run~$\overline{R}$ of length
\[\overline{r} = \mathsf{sl}(R) \geqslant (1 - \mathsf{gr}(R)) r / 2
\geqslant \ln(e/2) r / 2.\]
Since each run~$R_j$ is the split run of at most
two nodes~$R \in \mathcal{T}^\ast$, the lengths of the biased runs
sum up to at most~$4 n / \ln(e/2)$, and therefore
\[\sum_{i=1}^\sigma \sum_{R \in \mathcal{T}^{\mathsf{biased}}}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^{\heartsuit}(r_{\rightarrow i}) \leqslant
\sum_{i=1}^\sigma \sum_{R \in \mathcal{T}^{\mathsf{biased}}}
2 r_{\rightarrow i} =
2 \sum_{R \in \mathcal{T}^{\mathsf{biased}}} r = \mathcal{O}(n).\]
Then, given an integer~$i \leqslant \sigma$,
we set~$z = n / s_i \geqslant 1$ and~$\mu = \lceil \log_\theta(z \log_2(2 z)) \rceil$.
We also denote by~$\mathcal{R}_h$ the set of unbiased runs~$R$ such that~$\lfloor \log_\theta(r) \rfloor = h$.
Since these runs are unbiased, each set~$\mathcal{R}_h$
consists of pairwise incomparable runs.
Then, let
\[\mathsf{C}(h) = \sum_{R \in \mathcal{R}_h} \mathsf{gr}(R) r_{\rightarrow i}
\text{, }
\mathsf{D}(h) = \sum_{R \in \mathcal{R}_h}
r_{\rightarrow i} / \log_2(r)
\text{ and }
\mathsf{E}(h) = \sum_{R \in \mathcal{R}_h} (1+\log_2(r+1)).\]
By following the proof of
Theorem~\ref{thm:PeS-constant}, we first observe
that
\[\sum_{h=1}^{\mu-1} \mathsf{C}(h) \leqslant \log_2(\theta^{\mu}) s_i =
s_i \log_2(z) + s_i \log_2(\log_2(2z)) +
\mathcal{O}(s_i).\]
We also observe that
\[\sum_{h=1}^{\mu-1} \mathsf{D}(h) \leqslant \sum_{h=1}^{\mu-1} \sum_{R \in \mathcal{R}_h} \frac{r_{\rightarrow i}}{h \log_2(\theta)} \leqslant \sum_{h=1}^{\mu-1}
\frac{s_i}{h \ln(2)} \leqslant s_i (1 + \log_2(\mu)) =
s_i \log_2(\log_2(2z)) + \mathcal{O}(s_i).\]
Finally, consider the functions~$f \colon x \mapsto 1 + \log_2(x+1)$ and~$g \colon x \mapsto x \, f(n / x)$.
Both functions~$f$ and~$g$ are concave,
positive and increasing on~$(0,+\infty)$, which proves that
\begin{align*}
\mathsf{E}(\mu + h) & \leqslant
\sum_{R \in \mathcal{R}_{\mu +h}}
f(r)
\leqslant
|\mathcal{R}_{\mu+h}| \, f\big({\textstyle\sum_{R \in \mathcal{R}_{\mu+h}} r / |\mathcal{R}_{\mu + h}|}\big) \leqslant
g(|\mathcal{R}_{\mu+h}|) \leqslant
g(n / \theta^{\mu+h}) \\
& \leqslant
g\big(\theta^{-h} s_i/\log_2(2z)\big) =
\theta^{-h} s_i f(\theta^h z \log_2(2z)) / \log_2(2z) \\
& \leqslant \theta^{-h} s_i \big(1 + \log_2(\theta^h
(2z)^2)\big) / \log_2(2z) =
\theta^{-h} (h \ln(2)+1) s_i / \log_2(2z) +
2 \theta^{-h} s_i \\
& \leqslant (h+3) \theta^{-h} s_i,
\end{align*}
and therefore that
\[\sum_{h \geqslant \mu} \mathsf{E}(h) \leqslant
\sum_{h \geqslant 0} (h+3) \theta^{-h} s_i =
\mathcal{O}(s_i).\]
These three summations allow us to conclude that
\begin{align*}
\sum_{R \notin \mathcal{T}^{\mathsf{biased}}}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) & =
\sum_{h \geqslant 1} \sum_{R \in \mathcal{R}_h}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) \\
& \leqslant \sum_{h \geqslant 1} \sum_{R \in \mathcal{R}_h}
\min\{\mathsf{gr}(R) r_{\rightarrow i} + r_{\rightarrow i}/\log_2(r),6 (1+\log_2(r+1))\} \\
& \leqslant \sum_{h \geqslant 1} \min\{\mathsf{C}(h) + \mathsf{D}(h), 6 \mathsf{E}(h)\} \\
& \leqslant \sum_{h=1}^{\mu-1} (\mathsf{C}(h) + \mathsf{D}(h)) + 6 \sum_{h \geqslant \mu} \mathsf{E}(h) \\
& \leqslant s_i \log_2(n/s_i) +
2 s_i \log_2(\log_2(2n/s_i)) + \mathcal{O}(s_i),
\end{align*}
from which it follows that
\begin{align*}
\mathsf{C} & = \sum_{i=1}^\sigma \sum_{R \in \mathcal{T}^\ast}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) + \mathcal{O}(n) \\
& = \sum_{i=1}^\sigma \sum_{R \in \mathcal{T}^{\mathsf{biased}}}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) +
\sum_{i=1}^\sigma \sum_{R \notin \mathcal{T}^{\mathsf{biased}}}
\mathsf{cost}_{\lceil \log_2(r) \rceil,\mathsf{gr}(R)}^\heartsuit(r_{\rightarrow i}) + \mathcal{O}(n) \\
& \leqslant \sum_{i=1}^\sigma
\big(s_i \log_2(n/s_i) +
2 s_i \log_2(\log_2(2n/s_i)) + \mathcal{O}(s_i)\big) +
\mathcal{O}(n) \\
& \leqslant n \mathcal{H}^\ast + 2 n \ln(\mathcal{H}^\ast+1) + \mathcal{O}(n). &&\qedhere
\end{align*}
\end{proof}
\section{Conclusion: An idealistic galloping cost model}
In the above sections, we observed the impact of
using a galloping sub-routine for a fixed or a
variable parameter~$\mathbf{t}$.
Although choosing a constant value of~$\mathbf{t}$ (e.g.,~$\mathbf{t} = 7$, as advocated in~\cite{McIlroy1993}) already
leads to very good results, letting~$\mathbf{t}$ vary, for instance
by using the logarithmic variant of the sub-routine,
provides us with even better complexity guarantees,
with an often negligible overhead of~$\mathcal{O}(n \log(\mathcal{H}^\ast+1) + n)$ element comparisons:
up to a small error, this provides us with the following
\emph{idealistic} cost model for run merges, allowing us to
simultaneously identify the parameter~$\mathbf{t}$ with~$+\infty$ and
with a constant.
\begin{definition}
Let~$A$ and~$B$ be two non-decreasing runs
with~$a_{\rightarrow i}$ (respectively,~$b_{\rightarrow i}$) elements of value~$i$ for all~$i \in {\{1,2,\ldots,\sigma\}}$.
The \emph{idealistic galloping cost} of merging~$A$ and~$B$
is defined as the quantity
\[\sum_{i=1}^\sigma \mathsf{cost}_{\mathsf{ideal}}^\ast(a_{\rightarrow i}) + \mathsf{cost}_{\mathsf{ideal}}^\ast(b_{\rightarrow i}),\]
where~$\mathsf{cost}_{\mathsf{ideal}}^\ast(m) = \min\{m, \log_2(m+1) + \mathcal{O}(1)\}$.
\end{definition}
We think that
this idealistic cost model
is both simple and precise enough to
allow studying the complexity of natural merge
sorts in general, provided that they use
the galloping sub-routine.
Thus, it would be interesting to use that
cost model in order to study, for
instance, the least constant~$\mathsf{c}$
for which various algorithms such as
TimSort\xspace or \textalpha-MergeSort\xspace require up to~$\mathsf{c} n (1 + o(1)) \mathcal{H}^\ast + \mathcal{O}(n)$
element comparisons.
We also hope that this simpler framework
will foster the interest for the galloping merging
sub-routine of TimSort\xspace, and possibly lead to
amending Swift and Rust
implementations of TimSort\xspace to
include that sub-routine, which we believe is
too efficient in relevant cases to be omitted.
\end{document} |
\betaegin{equation}gin{document}
\title{Accelerated Particle Swarm Optimization and Support Vector Machine for Business Optimization and Applications }
\alphauthor{Xin-She Yang$^1$, Suash Deb$^2$ and Simon Fong$^3$ \\ \\
1) Department of Engineering,
University of Cambridge, \\
Trumpinton Street,
Cambridge CB2 1PZ, UK. \\
\alphand
2) Department of Computer Science \& Engineering, \\
C. V. Raman College of Engineering, \\
Bidyanagar, Mahura, Janla,
Bhubaneswar 752054, INDIA. \\
\alphand
3) Department of Computer and Information Science, \\
Faculty of Science and Technology, \\
University of Macau, Taipa, Macau. \\
}
\deltaate{}
\maketitle
\betaegin{equation}gin{abstract}
Business optimization is becoming increasingly important because all business activities
aim to maximize the profit and performance of products and services, under
limited resources and appropriate constraints. Recent developments in support vector machine
and metaheuristics show many advantages of these techniques.
In particular, particle swarm optimization is now widely used in solving tough optimization
problems. In this paper, we use a
combination of a recently developed Accelerated PSO and a nonlinear support vector machine to
form a framework for solving business optimization problems.
We first apply the proposed APSO-SVM to production optimization, and then use it for
income prediction and project scheduling. We also carry out some parametric studies and discuss the
advantages of the proposed metaheuristic SVM. \\ \\
{\betaf Keywords:} Accelerated PSO, business optimization, metaheuristics, PSO, support vector machine,
project scheduling. \\
\noindent Reference to this paper should be made as follows: \\ \\
Yang, X. S., Deb, S., and Fong, S., (2011), Accelerated Particle Swarm Optimization
and Support Vector Machine for Business Optimization and Applications, in: Networked Digital Technologies (NDT2011),
Communications in Computer and Information Science, Vol. 136, Springer,
pp. 53-66 (2011).
\epsilonnd{abstract}
\qquadection{Introduction}
Many business activities often have to deal with large, complex databases.
This is partly driven by information technology, especially the Internet,
and partly driven by the need to extract meaningful knowledge by data mining.
To extract useful information among a huge amount of data requires
efficient tools for processing vast data sets. This is equivalent to
trying to find an optimal solution to a highly nonlinear problem with multiple, complex
constraints, which is a challenging task. Various techniques for such data mining and optimization have been
developed over the past few decades. Among these techniques,
support vector machine is one of the best techniques for regression,
classification and data mining \cite{Howley,Kim,Pai,Shi,Shi2,Vapnik}.
On the other hand, metaheuristic algorithms also become powerful for solving
tough nonlinear optimization problems \cite{Blum,Kennedy,Kennedy2,Yang,Yang2}.
Modern metaheuristic algorithms have been developed with an aim to carry out
global search, typical examples are genetic algorithms \cite{Gold},
particle swarm optimisation (PSO) \cite{Kennedy}, and Cuckoo Search \cite{YangDeb,YangDeb2}.
The efficiency of metaheuristic algorithms can be attributed to the
fact that they imitate the best features in nature, especially the selection of the fittest
in biological systems which have evolved by natural selection over millions of years.
Since most data have noise or associated randomness, most these algorithms
cannot be used directly. In this case, some form of averaging or reformulation of the problem
often helps. Even so, most algorithms become difficult to implement for such type of optimization.
In addition to the above challenges, business optimization often concerns
with a large amount but often incomplete data, evolving dynamically over
time. Certain tasks cannot start before other required tasks are completed,
such complex scheduling is often NP-hard and no universally efficient tool exists.
Recent trends indicate that metaheuristics can be very promising, in combination with
other tools such as neural networks and support vector machines \cite{Howley,Kim,Tabu,Smola}.
In this paper, we intend to present a simple framework of business optimization using
a combination of support vector machine with accelerated PSO. The paper is
organized as follows: We first will briefly review particle swarm optimization and accelerated PSO,
and then introduce the basics of support vector machines (SVM). We then
use three case studies to test the proposed framework. Finally, we discussion its implications
and possible extension for further research.
\qquadection{Accelerated Particle Swarm Optimization}
\qquadubsection{PSO}
Particle swarm optimization (PSO) was developed by Kennedy and
Eberhart in 1995 \cite{Kennedy,Kennedy2}, based on the swarm behaviour such
as fish and bird schooling in nature. Since then, PSO has
generated much wider interests, and forms an exciting, ever-expanding
research subject, called swarm intelligence. PSO has been applied
to almost every area in optimization, computational intelligence,
and design/scheduling applications. There are at least two dozens of
PSO variants, and hybrid algorithms by combining PSO
with other existing algorithms are also increasingly popular.
PSO searches the space of an objective function
by adjusting the trajectories of individual agents,
called particles, as the piecewise paths formed by positional
vectors in a quasi-stochastic manner. The movement of a swarming particle
consists of two major components: a stochastic component and a deterministic component.
Each particle is attracted toward the position of the current global best
$\ff{g}^*$ and its own best location $\ff{x}_i^*$ in history,
while at the same time it has a tendency to move randomly.
Let $\ff{x}_i$ and $\ff{v}_i$ be the position vector and velocity for
particle $i$, respectively. The new velocity vector is determined by the
following formula
\betaegin{equation} \ff{v}_i^{t+1}= \ff{v}_i^t + \alpha \ff{\epsilon}_1
[\ff{g}^*-\ff{x}_i^t] + \beta \ff{\epsilon}_2 [\ff{x}_i^*-\ff{x}_i^t].
\label{pso-speed-100}
\epsilonnd{equation}
where $\ff{\epsilon}_1$ and $\ff{\epsilon}_2$ are two random vectors, and each
entry taking the values between 0 and 1.
The parameters $\alpha$ and $\beta$ are the learning parameters or
acceleration constants, which can typically be taken as, say, $\alpha\approx\beta \approx2$.
There are many variants which extend the standard PSO
algorithm, and the most noticeable improvement is probably to use an inertia function $\theta
(t)$ so that $\ff{v}_i^t$ is replaced by $\theta(t) \ff{v}_i^t$
\betaegin{equation} \ff{v}_i^{t+1}=\theta \ff{v}_i^t + \alpha \ff{\epsilon}_1
[\ff{g}^*-\ff{x}_i^t] + \beta \ff{\epsilon}_2 [\ff{x}_i^*-\ff{x}_i^t],
\label{pso-speed-150}
\epsilonnd{equation}
where $\theta \in (0,1)$ \cite{Chat,Clerc}. In the simplest case,
the inertia function can be taken as a constant, typically $\theta \approx 0.5 \qquadim 0.9$.
This is equivalent to introducing a virtual mass to stabilize the motion
of the particles, and thus the algorithm is expected to converge more quickly.
\qquadubsection{Accelerated PSO}
The standard particle swarm optimization uses both the current global best
$\ff{g}^*$ and the individual best $\ff{x}^*_i$. The reason of using the individual
best is primarily to increase the diversity in the quality solutions, however,
this diversity can be simulated using some randomness. Subsequently, there is
no compelling reason for using the individual best, unless the optimization
problem of interest is highly nonlinear and multimodal.
A simplified version
which could accelerate the convergence of the algorithm is to use the global
best only. Thus, in the accelerated particle swarm optimization (APSO) \cite{Yang,Yang2}, the
velocity vector is generated by a simpler formula
\betaegin{equation} \ff{v}_i^{t+1}=\ff{v}_i^t + \alpha \ff{\epsilon}_n + \beta (\ff{g}^*-\ff{x}_i^t), \label{pso-sys-10} \epsilonnd{equation}
where $\ff{\epsilon}_n$ is drawn from $N(0,1)$
to replace the second term.
The update of the position
is simply \betaegin{equation} \ff{x}_i^{t+1}=\ff{x}_i^t + \ff{v}_i^{t+1}. \label{pso-sys-20} \epsilonnd{equation} In order to
increase the convergence even further, we can also write the
update of the location in a single step
\betaegin{equation} \ff{x}_i^{t+1}=(1-\beta) \ff{x}_i^t+\beta \ff{g}^* +\alpha \ff{\epsilon}_n. \label{APSO-500} \epsilonnd{equation}
This simpler version will give the same order of convergence.
Typically, $\alphalpha = 0.1 L \qquadim 0.5 L$ where $L$ is the scale of each variable, while $\betaegin{equation}ta= 0.1 \qquadim 0.7$
is sufficient for most applications. It is worth pointing out that
velocity does not appear in equation (\ref{APSO-500}), and there is no need to deal with
initialization of velocity vectors.
Therefore, APSO is much simpler. Comparing with many PSO variants, APSO uses only two parameters,
and the mechanism is simple to understand.
A further improvement to the accelerated PSO is to reduce the randomness
as iterations proceed.
This means that we can use a monotonically decreasing function such as
\betaegin{equation} \alpha =\alpha_0 e^{-\qquadigmaamma t}, \epsilonnd{equation}
or \betaegin{equation} \alpha=\alpha_0 \qquadigmaamma^t, \qquad (0<\qquadigmaamma<1), \epsilonnd{equation}
where $\alpha_0 \approx0.5 \qquadim 1$ is the initial value of the randomness parameter.
Here $t$ is the number of iterations or time steps.
$0<\qquadigmaamma<1$ is a control parameter \cite{Yang2}. For example, in our implementation, we will use
\betaegin{equation} \alpha=0.7^t, \epsilonnd{equation}
where $t \in [0,t_{\max}]$ and $t_{\max}$ is the maximum of iterations.
\qquadection{Support Vector Machine}
Support vector machine (SVM) is an efficient tool for data mining and classification
\cite{Vapnik2,Vapnik3}. Due to the vast volumes of
data in business, especially e-commerce, efficient
use of data mining techniques becomes a necessity.
In fact, SVM can also be considered as an optimization tool, as its objective is
to maximize the separation margins between data sets. The proper combination of SVM
with metaheuristics could be advantageous.
\qquadubsection{Support Vector Machine}
A support vector machine essentially transforms a set of data
into a significantly higher-dimensional space by nonlinear transformations
so that regression and data fitting can be carried out in this high-dimensional space.
This methodology can be used for data classification, pattern recognition,
and regression, and its theory was based on statistical machine learning theory
\cite{Smola,Vapnik,Vapnik2}.
For classifications with the learning examples
or data $(\ff{x}_i, y_i)$ where $i=1,2,...,n$ and $y_i \in \{-1,+1\}$,
the aim of the learning is to find a function
$\partialhi_{\alpha} (\ff{x})$ from allowable functions $\{\partialhi_{\alpha}: \alpha \in \Omega \}$
such that $\partialhi_{\alpha}(\ff{x}_i) \mapsto y_i$ for $(i=1,2,...,n)$
and that the expected risk $E(\alpha)$ is minimal. That is the minimization
of the risk \betaegin{equation} E(\alpha)=\kk{1}{2} \int |\partialhi_{\alpha}(x) -y| dQ(\ff{x},y), \epsilonnd{equation}
where $Q(\ff{x},y)$ is an unknown probability distribution, which makes
it impossible to calculate $E(\alpha)$ directly. A simple approach is
to use the so-called empirical risk
\betaegin{equation} E_p(\alpha) \approx \kk{1}{2 n} \qquadum_{i=1}^n \betaig|\partialhi_{\alpha}(\ff{x}_i)-y_i \betaig|. \epsilonnd{equation}
However, the main flaw of this approach is that a small risk or error on
the training set does not necessarily guarantee a small
error on prediction if the number $n$ of training data is small \cite{Vapnik3}.
For a given probability of at least $1-p$, the Vapnik bound for the
errors can be written as
\betaegin{equation} E(\alpha) \le R_p(\alpha) + \Psi \Big(\kk{h}{n}, \kk{\log (p)}{n} \Big), \epsilonnd{equation}
where
\betaegin{equation} \Psi \betaig(\kk{h}{n}, \kk{\log(p)}{n} \betaig) =\qquadqrt{\kk{1}{n} \betaig[h (\log \kk{2n}{h}+1)
-\log(\kk{p}{4})\betaig]}. \epsilonnd{equation}
Here $h$ is a parameter, often referred to as the Vapnik-Chervonenskis
dimension or simply VC-dimension \cite{Vapnik}, which describes the capacity
for prediction of the function set $\partialhi_{\alpha}$.
In essence, a linear support vector machine is to
construct two hyperplanes as far away as possible
and no samples should be between these two planes.
Mathematically, this is equivalent to two equations
\betaegin{equation} \ff{w} \cdot \ff{x} + \ff{b} = \partialm 1, \epsilonnd{equation}
and a main objective of
constructing these two hyperplanes is to maximize the distance (between the two planes)
\betaegin{equation} d=\kk{2}{||\ff{w}||}. \epsilonnd{equation}
Such maximization of $d$ is equivalent to the minimization of $||w||$ or more conveniently $||w||^2$.
From the optimization point of view, the maximization of margins can be written as
\betaegin{equation} \textrm{minimize } \kk{1}{2} ||\ff{w}||^2 = \kk{1}{2} (\ff{w} \cdot \ff{w}). \epsilonnd{equation}
This essentially becomes an optimization problem
\betaegin{equation} \textrm{minimize } \Psi= \kk{1}{2} || \ff{w} ||^2 +\lambda \qquadum_{i=1}^n \epsilonta_i, \epsilonnd{equation}
\betaegin{equation} \textrm{subject to } y_i (\ff{w} \cdot \ff{x}_i + \ff{b}) \qquadigmae 1-\epsilonta_i, \label{svm-ineq-50} \epsilonnd{equation}
\betaegin{equation} \qquad \qquad \qquad \epsilonta_i \qquadigmae 0, \qquad (i=1,2,..., n), \epsilonnd{equation}
where $\lambda>0$ is a parameter to be chosen appropriately.
Here, the term $\qquadum_{i=1}^n \epsilonta_i$ is essentially a measure of
the upper bound of the number of misclassifications on the training data.
\qquadubsection{Nonlinear SVM and Kernel Tricks}
The so-called kernel trick is an important technique, transforming data dimensions
while simplifying computation.
By using Lagrange multipliers $\alpha_i \qquadigmae 0$, we can rewrite the above constrained optimization
into an unconstrained version, and we have
\betaegin{equation} L=\kk{1}{2} ||\ff{w}||^2 +\lambda \qquadum_{i=1}^n \epsilonta_i - \qquadum_{i=1}^n \alpha_i [y_i (\ff{w} \cdot \ff{x}_i + \ff{b}) -(1-\epsilonta_i)]. \epsilonnd{equation}
From this, we can write the Karush-Kuhn-Tucker conditions
\betaegin{equation} \partialab{L}{\ff{w}}=\ff{w} - \qquadum_{i=1}^n \alpha_i y_i \ff{x}_i =0, \epsilonnd{equation}
\betaegin{equation} \partialab{L}{\ff{b}} = -\qquadum_{i=1}^n \alpha_i y_i =0, \epsilonnd{equation}
\betaegin{equation} y_i (\ff{w} \cdot \ff{x}_i+\ff{b})-(1-\epsilonta_i) \qquadigmae 0, \epsilonnd{equation}
\betaegin{equation} \alpha_i [y_i (\ff{w} \cdot \ff{x}_i + \ff{b}) -(1-\epsilonta_i)]=0, \qquad (i=1,2,...,n), \label{svm-KKT-150} \epsilonnd{equation}
\betaegin{equation} \alpha_i \qquadigmae 0, \qquad \epsilonta_i \qquadigmae 0, \qquad (i=1,2,...,n). \epsilonnd{equation}
From the first KKT condition, we get
\betaegin{equation} \ff{w}=\qquadum_{i=1}^n y_i \alpha_i \ff{x}_i. \epsilonnd{equation}
It is worth pointing out here that
only the nonzero $\alpha_i$ contribute to overall solution. This comes from the
KKT condition (\ref{svm-KKT-150}),
which implies that when $\alpha_i \ne 0$, the inequality (\ref{svm-ineq-50}) must be
satisfied exactly, while $\alpha_0=0$ means the inequality is automatically
met. In this latter case, $\epsilonta_i=0$. Therefore, only the corresponding training
data $(\ff{x}_i, y_i)$ with $\alpha_i>0$ can contribute to the solution, and thus such
$\ff{x}_i$ form the support vectors (hence, the name support vector machine). \index{support vectors}
All the other data with $\alpha_i=0$ become irrelevant.
It has been shown that the solution for $\alpha_i$ can be found
by solving the following quadratic programming \cite{Vapnik,Vapnik3}
\betaegin{equation} \textrm{maximize } \qquadum_{i=1}^n \alpha_i -\kk{1}{2} \qquadum_{i,j=1}^n \alpha_i \alpha_j y_i y_j (\ff{x}_i \cdot \ff{x}_j), \epsilonnd{equation}
subject to
\betaegin{equation} \qquadum_{i=1}^n \alpha_i y_i=0, \qquad 0 \le \alpha_i \le \lambda, \qquad (i=1,2,...,n). \epsilonnd{equation}
From the coefficients $\alpha_i$, we can write the final classification or decision
function as \betaegin{equation} f(\ff{x}) =\textrm{sgn} \betaig [ \qquadum_{i=1}^n \alpha_i y_i (\ff{x} \cdot \ff{x}_i) + \ff{b} \betaig ], \epsilonnd{equation}
where sgn is the classic sign function.
As most problems are nonlinear in business applications, and the above linear SVM cannot be
used. Ideally, we should find some nonlinear transformation $\partialhi$ so that the
data can be mapped onto a high-dimensional space where the classification
becomes linear. The transformation should be
chosen in a certain way so that their dot product leads to a
kernel-style function $K(\ff{x},\ff{x}_i)=\partialhi(\ff{x}) \cdot \partialhi(\ff{x}_i)$.
In fact, we do not need to know such transformations,
we can directly use the kernel functions $K(\ff{x},\ff{x}_i)$ to complete this task.
This is the so-called kernel function trick. Now the main task is to chose
a suitable kernel function for a given, specific problem.
For most problems in nonlinear support vector machines, we can
use $K(\ff{x},\ff{x}_i)=(\ff{x} \cdot x_i)^d$ for polynomial classifiers,
$K(\ff{x},\ff{x}_i)=\tanh[k (\ff{x} \cdot \ff{x}_i) +\Theta)]$ for neural networks,
and by far the most widely used kernel is the Gaussian radial basis function (RBF)
\betaegin{equation} K(\ff{x},\ff{x}_i)=\epsilonxp \Big[-\kk{||\ff{x}-\ff{x}_i||^2}{(2 \qquadigma^2)} \Big]
=\epsilonxp \Big[-\qquadigmaamma ||\ff{x}-\ff{x}_i||^2 \Big],\epsilonnd{equation}
for the nonlinear classifiers. This kernel can easily be extended to
any high dimensions. Here $\qquadigma^2$ is the variance and $\qquadigmaamma=1/2\qquadigma^2$ is
a constant. In general, a simple bound of $0 < \qquadigmaamma \le C$ is used, and
here $C$ is a constant.
Following the similar procedure as discussed earlier for linear SVM,
we can obtain the coefficients $\alpha_i$ by solving the following optimization
problem \betaegin{equation} \textrm{maximize } \qquadum_{i=1}^n \alpha_i -\kk{1}{2} \alpha_i \alpha_j y_i y_j K(\ff{x}_i,\ff{x}_j). \epsilonnd{equation}
It is worth pointing out under Mercer's conditions for the kernel function,
the matrix $\ff{A}=y_i y_j K(\ff{x}_i, \ff{x}_j)$ is a symmetric positive definite matrix \cite{Vapnik3}, which
implies that the above maximization is a quadratic programming problem, and can thus
be solved efficiently by standard QP techniques \cite{Smola}.
\qquadection{Metaheuristic Support Vector Machine with APSO}
\qquadubsection{Metaheuristics}
There are many metaheuristic algorithms for optimization and most these algorithms
are inspired by nature \cite{Yang}. Metaheuristic algorithms such as genetic
algorithms and simulated annealing are widely used, almost routinely, in many
applications, while relatively new algorithms such as particle swarm optimization \cite{Kennedy},
firefly algorithm and cuckoo search are becoming more and more popular \cite{Yang,Yang2}.
Hybridization of these algorithms with existing algorithms are also emerging.
The advantage of such a combination is to use a balanced tradeoff between
global search which is often slow and a fast local search. Such a balance
is important, as highlighted by the analysis by Blum and Roli \cite{Blum}.
Another advantage of this method is that we can use any algorithms we like
at different stages of the search or even at different stage of iterations.
This makes it easy to combine the advantages of various algorithms so as
to produce better results.
Others have attempted to carry out parameter optimization associated with neural
networks and SVM. For example, Liu et al. have used SVM optimized by PSO for
tax forecasting \cite{Liu}. Lu et al. proposed a model for finding optimal parameters in SVM by PSO
optimization \cite{Lu}. However, here we intend to propose a generic framework
for combining efficient APSO with SVM, which can be extended to
other algorithms such as firefly algorithm \cite{YangFA,Yang2010}.
\qquadubsection{APSO-SVM}
Support vector machine has a major advantage, that is, it is less likely to
overfit, compared with other methods such as regression and neural networks.
In addition, efficient quadratic programming can be used for training support vector machines.
However, when there is noise in the data, such algorithms are not quite suitable.
In this case, the learning or training to estimate the parameters in the
SVM becomes difficult or inefficient.
Another issue is that the choice of the values of
kernel parameters $C$ and $\qquadigma^2$ in the kernel functions;
however, there is no agreed guideline on how
to choose them, though the choice of their values should make
the SVM as efficiently as possible. This itself
is essentially an optimization problem.
Taking this idea further, we first use an educated guess set of
values and use the metaheuristic algorithms such as accelerated PSO
or cuckoo search to find the best kernel parameters
such as $C$ and $\qquadigma^2$ \cite{Yang,YangDeb}.
Then, we used these parameters to construct the support vector machines
which are then used for solving the problem of interest. During the iterations
and optimization, we can also modify kernel parameters and
evolve the SVM accordingly. This framework can be called a metaheuristic
support vector machine. Schematically, this Accelerated PSO-SVM can be represented as
shown in Fig. 1.
\vcode{0.7}{
Define the objective; \\
Choose kernel functions; \\
Initialize various parameters; \\
{\betaf while} (criterion) \\
\indent $\quad$ Find optimal kernel parameters by APSO; \\
\indent $\quad$ Construct the support vector machine; \\
\indent $\quad$ Search for the optimal solution by APSO-SVM; \\
\indent $\quad$ Increase the iteration counter; \\
{\betaf end} \\
Post-processing the results;
}{Metaheuristic APSO-SVM. }
For the optimization of parameters and business applications discussed below, APSO
is used for both local and global search \cite{Yang,Yang2}.
\qquadection{Business Optimization Benchmarks}
Using the framework discussed earlier, we can easily implement it in
any programming language, though we have implemented using Matlab.
We have validated our implementation using the standard test
functions, which confirms the correctness of the implementation.
Now we apply it to carry out case studies with known analytical solution
or the known optimal solutions. The Cobb-Douglas production
optimization has an analytical solution which can be used for
comparison, while the second case is a standard benchmark in
resource-constrained project scheduling \cite{Kol}.
\qquadubsection{Production Optimization}
Let us first use the proposed approach to study
the classical Cobb-Douglas production optimization.
For a production of a series of products and the labour costs,
the utility function can be written
\betaegin{equation} q=\partialrod_{j=1}^n u_j^{\alpha_j} =u_1^{\alpha_1} u_2^{\alpha_2} \cdots u_n^{\alpha_n}, \epsilonnd{equation}
where all exponents $\alpha_j$ are non-negative, satisfying
\betaegin{equation} \qquadum_{j=1}^n \alpha_j =1. \epsilonnd{equation}
The optimization is the minimization of the utility
\betaegin{equation} \textrm{minimize } q \epsilonnd{equation}
\betaegin{equation} \textrm{subject to } \qquadum_{j=1}^n w_j u_j =K, \epsilonnd{equation}
where $w_j (j=1,2,...,n)$ are known weights.
This problem can be solved using the Lagrange multiplier method as
an unconstrained problem
\betaegin{equation} \partialsi=\partialrod_{j=1}^n u_j^{\alpha_j} + \lambda (\qquadum_{j=1}^n w_j u_j -K), \epsilonnd{equation}
whose optimality conditions are
\betaegin{equation} \partialab{\partialsi}{u_j} = \alpha_j u_j^{-1} \partialrod_{j=1}^n u_j^{\alpha_j} + \lambda w_j =0, \quad (j=1,2,...,n), \epsilonnd{equation}
\betaegin{equation} \partialab{\partialsi}{\lambda} = \qquadum_{j=1}^n w_j u_j -K =0. \epsilonnd{equation}
The solutions are
\betaegin{equation} u_1=\kk{K}{w_1 [1+\kk{1}{\alpha_1} \qquadum_{j=2}^n \alpha_j ]}, \;
u_j=\kk{w_1 \alpha_j}{w_j \alpha_1} u_1, \epsilonnd{equation}
where $(j=2,3, ..., n)$.
For example, in a special case of $n=2$, $\alpha_1=2/3$, $\alpha_2=1/3$, $w_1=5$,
$w_2=2$ and $K=300$, we have
\[ u_1=\kk{Q}{w_1 (1+\alpha_2/\alpha_1)} =40, \;
u_2=\kk{K \alpha_2}{w_2 \alpha_1 (1+\alpha_2/\alpha_1)}=50. \]
As most real-world problem has some uncertainty, we can now add
some noise to the above problem. For simplicity, we just modify
the constraint as
\betaegin{equation} \qquadum_{j=1}^n w_j u_j = K (1+ \betaegin{equation}ta \epsilonpsilon), \epsilonnd{equation}
where $\epsilonpsilon$ is a random number drawn from a Gaussian distribution with a zero mean
and a unity variance,
and $0 \le \betaegin{equation}ta \ll 1$ is a small positive number.
We now solve this problem as an optimization problem by the proposed APSO-SVM.
In the case of $\betaegin{equation}ta=0.01$,
the results have been summarized in Table 1
where the values are provided with different problem size $n$ with
different numbers of iterations. We can see that the results converge
at the optimal solution very quickly.
\betaegin{equation}gin{table}[ht]
\caption{Mean deviations from the optimal solutions.}
\centering
\betaegin{equation}gin{tabular}{lllll}
\hline \hline
size $n$ & Iterations & deviations \\
\hline
10 & 1000 & 0.014 \\
20 & 5000 & 0.037 \\
50 & 5000 & 0.040 \\
50 & 15000 & 0.009 \\
\hline
\epsilonnd{tabular}
\epsilonnd{table}
\qquadection{Income Prediction}
Studies to improve the accuracy of classifications are extensive. For example, Kohavi proposed a
decision-tree hybrid in 1996 \cite{UCI}. Furthermore, an efficient training algorithm for support vector machines was proposed by Platt in 1998 \cite{Platt,Platt2},
and it has some significant impact on machine learning, regression and data mining.
A well-known benchmark for classification and regression is the income prediction using the
data sets from a selected 14 attributes of a household from a sensus form \cite{UCI,Platt}.
We use the same data sets at ftp://ftp.ics.uci.edu/pub/machine-learning-databases/adult
for this case study. There are 32561 samples in the training set with 16281 for testing.
The aim is to predict if an individual's income is above or below 50K ?
Among the 14 attributes, a subset can be selected, and a subset such as age, education level,
occupation, gender and working hours are commonly used.
Using the proposed APSO-SVM and choosing the limit value of $C$ as $1.25$,
the best error of $17.23\%$ is obtained (see Table \ref{table-3}), which is comparable with most accurate predictions
reported in \cite{UCI,Platt}.
\betaegin{equation}gin{table}[ht]
\caption{Income prediction using APSO-SVM. \label{table-3}}
\centering
\betaegin{equation}gin{tabular}{l|l|l}
\hline \hline
Train set (size) & Prediction set & Errors (\%) \\
\hline
512 & 256 & $24.9$ \\
1024 & 256 & $20.4$ \\
16400 & 8200 & $17.23$ \\
\hline
\epsilonnd{tabular}
\epsilonnd{table}
\qquadubsection{Project Scheduling}
Scheduling is an important class of discrete optimization with a wider range
of applications in business intelligence. For resource-constrained project scheduling problems,
there exists a standard benchmark
library by Kolisch and Sprecher \cite{Kol,Kol2}. The basic model consists
of $J$ activities/tasks, and some activities cannot start before all its predecessors $h$
are completed. In addition, each activity $j=1,2,...,J$
can be carried out, without interruption,
in one of the $M_j$ modes, and performing any activity $j$ in any chosen
mode $m$ takes $d_{jm}$ periods, which is supported by a set of renewable
resource $R$ and non-renewable resources $N$. The project's makespan or upper
bound is T, and the overall capacity of non-renewable resources is
$K_r^{\nu}$ where $r \in N$. For an activity $j$ scheduled in mode $m$,
it uses $k^{\rho}_{jmr}$ units of renewable resources
and $k^{\nu}_{jmr}$ units of non-renewable resources
in period $t=1,2,..., T$.
For activity $j$, the shortest duration is fit into the time
windows $[EF_j, LF_j]$ where $EF_j$ is the earliest finish times,
and $LF_j$ is the latest finish times. Mathematically, this
model can be written as \cite{Kol}
\betaegin{equation} \textrm{Minimize }\; \Psi (\ff{x}) \qquadum_{m=1}^{M_j} \qquadum_{t=EF_j}^{LF_j} t \cdot x_{jmt}, \epsilonnd{equation}
subject to
\[ \qquadum_{m=1}^{M_h} \qquadum_{t=EF_j}^{LF_j} t \;\; x_{hmt} \le \qquadum_{m=1}^{M_j} \qquadum_{t=EF_j}^{LF_j} (t-d_{jm}) x_{jmt},
(j=2,..., J), \]
\[ \qquadum_{j=1}^J \qquadum_{m=1}^{M_j} k^{\rho}_{jmr} \qquadum_{q=\max\{t,EF_j\}}^{\min\{t+d_{jm}-1,LF_j\}} x_{jmq} \le K_r^{\rho},
(r \in R), \]
\betaegin{equation} \qquadum_{j=1}^J \qquadum_{m=1}^{M_j} k_{jmr}^{\nu} \qquadum_{t=EF_j}^{LF_j} x_{jmt} \le K^{\nu}_r, (r \in N), \epsilonnd{equation}
and
\betaegin{equation} \qquadum_{j=1}^{M_j} \qquadum{t=EF_j}^{LF_j} =1, \qquad j=1,2,...,J, \epsilonnd{equation}
where $x_{jmt} \in \{0,1\}$ and $t=1,...,T$.
As $x_{jmt}$ only takes two values $0$ or $1$, this problem
can be considered as a classification problem, and metaheuristic
support vector machine can be applied naturally.
\betaegin{equation}gin{table}[ht]
\caption{Kernel parameters used in SVM.}
\centering
\betaegin{equation}gin{tabular}{l|l}
\hline \hline
Number of iterations & SVM kernel parameters \\
\hline
1000 & $C=149.2$, $\qquadigma^2=67.9$ \\
5000 & $C=127.9$, $\qquadigma^2=64.0$ \\
\hline
\epsilonnd{tabular}
\epsilonnd{table}
Using the online benchmark library \cite{Kol2}, we have solved this type
of problem with $J=30$ activities (the standard test set j30). The run time
on a modern desktop computer is about 2.2 seconds for $N=1000$ iterations
to 15.4 seconds for $N=5000$ iterations. We have run
the simulations for 50 times so as to obtain meaningful statistics.
The optimal kernel parameters found for the support vector machines
are listed in Table 3, while the deviations from the known best solution
are given in Table 4 where the results by other methods are also compared.
\betaegin{equation}gin{table}[ht]
\caption{Mean deviations from the optimal solution (J=30).}
\centering
\betaegin{equation}gin{tabular}{lllll}
\hline \hline
Algorithm & Authors & $N=1000$ & $5000$ \\
\hline
PSO \cite{Tcho} & Kemmoe et al. (2007) & 0.26 & 0.21 \\
hybribd GA \cite{Valls} & Valls eta al. (2007) & 0.27 & 0.06 \\
Tabu search \cite{Tabu} & Nonobe \& Ibaraki (2002) & 0.46 & 0.16 \\
Adapting GA \cite{Hart} & Hartmann (2002) & 0.38 & 0.22 \\
{\betaf Meta APSO-SVM } & this paper & {\betaf 0.19 } & {\betaf 0.025} \\
\hline
\epsilonnd{tabular}
\epsilonnd{table}
From these tables, we can see that the proposed metaheuristic support vector machine
starts very well, and results are comparable with those by other methods such as hybrid
genetic algorithm. In addition, it converges more quickly, as the number of
iterations increases. With the same amount of function evaluations involved,
much better results are obtained, which implies that APSO is very efficient,
and subsequently the APSO-SVM is also efficient in this context. In addition, this
also suggests that this proposed framework is appropriate for automatically choosing
the right parameters for SVM and solving nonlinear optimization problems.
\qquadection{Conclusions}
Both PSO and support vector machines are now widely used as optimization techniques
in business intelligence. They can also be used for data mining to extract useful information
efficiently. SVM can also be considered as an optimization
technique in many applications including business optimization. When there is noise in data,
some averaging or reformulation may lead to better performance. In addition, metaheuristic
algorithms can be used to find the optimal kernel parameters for a support vector machine
and also to search for the optimal solutions. We have used three very different case studies to demonstrate
such a metaheuristic SVM framework works.
Automatic parameter tuning and efficiency improvement will be an important topic for
further research. It can be expected that this framework can be used for other applications.
Furthermore, APSO can also be used to combine with other algorithms such as neutral networks
to produce more efficient algorithms \cite{Liu,Lu}. More studies in this area are highly needed.
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\begin{document}
\title{Graph2Graph Learning\ with Conditional Autoregressive Models}
\begin{abstract}
We present a graph neural network model for solving graph-to-graph learning problems. Most deep learning on graphs considers ``simple'' problems such as graph classification or regressing real-valued graph properties. For such tasks, the main requirement for intermediate representations of the data is to maintain the structure needed for output, i.e.~keeping classes separated or maintaining the order indicated by the regressor. However, a number of learning tasks, such as regressing graph-valued output, generative models, or graph autoencoders, aim to predict a graph-structured output. In order to successfully do this, the learned representations need to preserve far more structure. We present a conditional auto-regressive model for graph-to-graph learning and illustrate its representational capabilities via experiments on challenging subgraph predictions from graph algorithmics; as a graph autoencoder for reconstruction and visualization; and on pretraining representations that allow graph classification with limited labeled data.
\end{abstract}
\section{Introduction}
Graphs are everywhere! While machine learning and deep learning on graphs have for long caught wide interest, most research continues to focus on relatively simple tasks, such as graph classification~\cite{ying2018hierarchical,xu2018powerful}, or regressing a single continuous value from graph-valued input~\cite{chen2019alchemy,yang2019analyzing,dwivedi2020benchmarkgnns,ok20similarity,bianchi21arma,zhang20dlgsurvey,wu21comprehensivesurvey}. While such tasks are relevant and challenging, they ask relatively little from the learned intermediate representations: For graph classification, performance relies on keeping classes separate, and for regressing a single real variable, performance relies on ordering graphs according to the output variable, but within those constraints, intermediate graph embeddings can shuffle the graphs considerably without affecting performance.
In this paper, we consider Graph2Graph learning, {\it{i.e., }} problems whose input features and output predictions are both graphs. Such problems include both Graph2Graph regression and graph autoencoder models. For such problems, the model has to learn intermediate representations that carry rich structural information about the encoded graphs.
\paragraph{Related work.} While most existing work on graph neural networks (GNNs) centers around graph classification, some work moves in the direction of more complex output. Within chemoinformatics, several works utilize domain specific knowledge to predict graph-valued output, e.g.~ chemical reaction outcomes predicted via difference networks~\cite{jin2017predicting}. Similarly utilizing domain knowledge, junction trees are used to define graph variational autoencoders~\cite{junctionVAE} and Graph2Graph translation models~\cite{jin2018learning} designed specifically for predicting molecular structures, and interpretable substructures and structural motifs are used to design generative models of molecular Graph2Graphs in~\cite{jin2020multi} and~\cite{jin2020hierarchical}, respectively. Some work on encoder-decoder networks involving graph pooling and unpooling exist, but are only applied to node- and graph classification~\cite{gao2019graph,ying2018hierarchical}. More general generative models have also appeared, starting with the Variational Graph Autoencoder~\cite{kipf2016variational} which is primarily designed for link prediction in a single large graph. In~\cite{li2020dirichlet}, clustering is leveraged as part of a variational graph autoencoder to ensure common embedding of clusters. This, however, leads to a strong dependence on the similarity metric used to define clusters, which may be unrelated to the task at hand. In~\cite{you2018graphrnn}, a graph RNN, essentially structured as a residual decoder architecture, is developed by using the adjacency matrix to give the graph an artificial sequential structure. This generative model samples from a distribution fitted to a population of data, and does not generalize to graph-in, graph-out prediction problems.
\textbf{We contribute} a model for Graph2Graph prediction with flexible modelling capabilities. Utilizing the graph representation from~\cite{you2018graphrnn}, we build a full encoder-decoder network analogous to previous work in sequence-to-sequence learning~\cite{sutskever2014sequence}. Drawing on tools from image segmentation, we obtain edge-wise probabilities for the underlying discrete graphs, as well as flexible loss functions for handling the class imbalance implied by sparse graphs. Our graph RNN creates rich graph representations for complex graph prediction tasks. We illustrate its performance both for graph regression, as a graph autoencoder including visualization, and for unsupervised pretraining of graph representations to allow graph classification with limited labeled data.
\section{Method}
\label{sec:Method}
Our framework aims to generate new output graphs as a function of input graphs. Below, we achieve this by treating Graph2Graph regression as a sequence-to-sequence learning task by representing the graph as a sequence structure.
\subsection{Graph Representation}
Our framework takes as input a collection of undirected graphs with node- and edge attributes, denoted as $G = \{V, E\}$, where $V$ is the set of nodes and $E$ is the set of edges between nodes. Given a fixed node ordering $\pi$, a graph $G$ is uniquely represented by its attributed adjacency matrix
$A \in \{ \mathbb{R}^\gamma\}^{n\times n}$, where $\gamma$ is the attribute dimension for edges in $G$ and $n$ is the number of nodes. Moreover, $A_{i,j}$ is not $null$ iff $(v_{i}, v_{j}) \in E$.
Consistent with the representation in~\cite{you2018graphrnn}~\cite{popova2019molecularrnn}, a graph $G$ will be represented as sequences of adjacency vectors $\{X_1, X_2,\cdots, X_n\}$ obtained by breaking $A$ up by rows. $X_i = (A_{i, i-1},A_{i, i-2},\cdots,A_{i,1})$ encodes the sequence of attributes of edges connecting the node $v_i$ to its previous nodes $\{v_{i-1}, v_{i-2}, ..., v_1\}$. The graph $G$ is thus transformed into a sequence of sequences, spanning across graph nodes $V$ and edges $E$. A toy example (one graph with 4 nodes) is shown in Fig.~\ref{fig:graph_representation}. This representation allows us to build on graph representation learning for sequences.
\begin{figure}
\caption{Graph representation: a toy example.}
\label{fig:graph_representation}
\caption{Rolled representation of the proposed Graph2Graph network}
\label{fig:rolling_network_architecture}
\end{figure}
\subsection{Graph2Graph Network Architecture}
Using the above graph representation, we propose a novel encoder-decoder architecture for Graph2Graph predictions building on previous models for sequence-to-sequence prediction~\cite{sutskever2014sequence}. At the encoding phase, graph information will be captured at edge level as well as node level, and analogously, the decoder will infer graph information at edge and node level in turn. A graphical illustration of proposed network is shown in Fig.~\ref{fig:rolling_network_architecture}, with its unrolled version in Fig.~\ref{fig:unrolling_network_architecture}.
\begin{figure}
\caption{\textbf{Left:}
\label{fig:unrolling_network_architecture}
\end{figure}
\subsubsection{Encoder}
The encoder aims to extract useful information from the input graph to feed into the decoding inference network. Utilizing the sequential representation of the input graph $G$, we use expressive recurrent neural networks (RNN) to encode $G$. The structure and attributes in $G$ are summarized across two levels of RNNs, each based on~\cite{sutskever2014sequence}: the node level RNN, denoted as \textit{encNodeRNN}, and the edge level RNN, denoted as \textit{encEdgeRNN}. For the \textit{encNodeRNN}, we apply bidirectional RNNs~\cite{schuster1997bidirectional} to encode information from both previous and following nodes, see Fig.~\ref{fig:unrolling_network_architecture} right subfigure.
The encoder network \textit{encEdgeRNN} reads elements in $X_i$ as they are ordered, {\it{i.e., }} from $X_{i,1}$ to $X_{i,i-1}$, using the forward RNN $\overrightarrow{g}_{edge}$. Here, $\overrightarrow{g}_{edge}$ is a state-transition function, more precisely a Gated Recurrent Unit (GRU). The GRU $\overrightarrow{g}_{edge}$ produce a sequence of forward hidden states $(\overrightarrow{h}^{edge}_{i,1},\overrightarrow{h}^{edge}_{i,2}, \cdots, \overrightarrow{h}^{edge}_{i,i-1})$. Then we pick $\overrightarrow{h}^{edge}_{i,i-1}$ to be the context vector of $X_i$, and input it into \textit{encNodeRNN}
The encoder network \textit{encNodeRNN} is a bidirectional RNN, which receives input from \textit{encEdgeRNN} and transmits hidden states over time, as shown in Figs.~\ref{fig:rolling_network_architecture} and~\ref{fig:unrolling_network_architecture}. This results in concatenated hidden states $h^{node}_{i} = \overrightarrow{h}^{node}_i\|\overleftarrow{h}^{node}_{n-i}$. The final hidden state $\overrightarrow{h}^{node}_{n}$ and $\overleftarrow{h}^{node}_{1}$ are concatenated and used as the initial hidden state for the decoder.
\subsubsection{Decoder}
Once the input graph is encoded, the decoder seeks to infer a corresponding output graph for a given problem. To help understand how the decoder works, we consider a particular graph prediction problem motivated by an NP hard problem from graph algorithms, namely the identification of a \emph{maximum clique}. In this application, we take a graph as input and aim to predict which nodes and edges belong to its maximum clique. Thus, the decoder predicts whether to retain or discard the next edge given all previous nodes, predicted edges and the context vector of the input graph.
The decoder defines a probability distribution $p(Ys)$, where $Ys = (Y_0,Y_1,\dots,Y_{m-1}) $ is the sequence of adjacency vectors that forms the output as predicted by the decoder. Here, $p(Ys)$ is a joint probability distribution on the adjacency vector predictions and can be decomposed as the product of ordered conditional distributions:
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
p(Ys) = \prod_{i=0}^{m-1}p(Y_i|Y_0,Y_1,\dots,Y_{i-1}, C),
\label{eq:dec_nodeProbDistribution}
\end{equation}
\noindentndent where $m$ is the node number of the predicted graph, and $C$ is the context vector used to facilitate predicting $Y_i$, as explained in Sec.~\ref{sec:attention} below. Denote $p(Y_i|Y_0,Y_1,\dots,Y_{i-1}, C)$ as $p(Y_i|Y_{<i}, C)$ for simplication.
In order to maintain edge dependencies in the prediction phase, $p(Y_i|Y_{<i}, C)$ is factorized as composed as a product of conditional probabilities.
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
p(Y_i|Y_{<i}, C) = \prod_{k=0}^{i-1}p(Y_{i,k}|Y_{i,<k},c; Y_{<i}, C)
\label{eq:dec_edgeProbDistribution}
\end{equation}
\noindentndent where $c$ is the edge context vector for predicting $Y_{i,k}$.
The cascaded relation between Equation~\eqref{eq:dec_nodeProbDistribution} and Equation~\eqref{eq:dec_edgeProbDistribution}, is reflected in practice by their approximation by the two cascaded RNNs called \textit{decNodeRNN} and \textit{decEdgeRNN}. Here, \textit{decNodeRNN} transits graph information from node $i-1$ to node $i$ hence generating a node (see Eq.~\eqref{eq:decNodeRNN}), and \textit{decEdgeRNN} generates edge predictions for the generated node (Eq.~\eqref{eq:decEdgeRNN}). The output of each \textit{decNodeRNN} cell will serve as an initital hidden state for \textit{decEdgeRNN}. Moreover, at each step, the output of the \textit{decEdgeRNN} will be fed into an MLP head with sigmoid activation function, generating a probability of keeping this edge in the output graph (Eq.~\eqref{eq:decEdgeMLP}). This use of sigmoid activation is similar to its use in image segmentation, where a pixel-wise foreground probability is obtained from it.
\begin{align}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\label{eq:decNodeRNN}
&s^{node}_i = f_{node}(s^{node}_{i-1}, f_{edge,down}(Y_{i-1}), C_i ) \\
\label{eq:decEdgeRNN}
&s^{edge}_{i,j}=f_{edge}(s^{edge}_{i,j-1}, emb(Y_{i,j-1}), c_{i,j}) \\
\label{eq:decEdgeMLP}
&o_{i,j} = \text{MLP}(s^{edge}_{i,j})
\end{align}
Our decoder has some features in common with the graph RNN~\cite{you2018graphrnn}. We extend this by utilizing attention based on the decoder, as well as by adding in extra GRUs in the $f_{edge,down}$, encoding the predicted adjacency vector from the previous step in order to improve model expressive capability.
\begin{algorithm}[t]
\caption{\label{alg:overall algorithm}Graph to Graph learning algorithm.}
\textit{Input:} graph $\textbf{X} = (X_1,\dots, X_n)$.
\textit{Output:} graph $\textbf{Y} = (Y_1,\dots, Y_m)$.
\begin{algorithmic}[1]
\STATE For each $X_i, i=1,\dots,n$, use edge RNN $g_{edge}$ to calculate their encodings$\{XEmb_1,\dots,XEmb_n\}$ and hidden states $h^{edge}_{i, j}$.
\STATE Forward node RNN $\overrightarrow{g}_{node}$ reads $\{XEmb_1,\dots,XEmb_n\}$ and calculates forward node hidden states $\{\overrightarrow{h}^{node}_1,\dots,\overrightarrow{h}^{node}_n\}$
\STATE Reverse node RNN $\overleftarrow{g}_{node}$ reads $\{XEmb_n,\dots,XEmb_1\}$ and calculates reverse node hidden states $\{\overleftarrow{h}^{node}_1,\dots,\overleftarrow{h}^{node}_n\}$
\STATE Final hidden state $h^{node}_i = \textsc{Concat}(\overrightarrow{h}^{node}_i, \overleftarrow{h}^{node}_{n-i+1})$
\STATE Set initial hidden state of decoder node RNN as $s^{node}_1 = h^{node}_n$, initial edge sequence encoding $YEmb_0 = \mathrm{SOS}, i=1$
\STATE While $i<=m$ do
\begin{itemize}
\item $f_{edge,down}$ encodes $Y_{i-1}$ to be $YEmb_{i-1}$
\item $s^{node}_i = f_{node}(s^{node}_{i-1},YEmb_{i-1}, \textup{NodeAttn}(h^{node}_{1:n},s^{node}_{i-1} ))$
\item $s^{edge}_{i,0} = s^{node}_i, j=1, Y_{i,0} = \mathrm{SOS_{decEdge1}}$
\item While $j<=i$ do
\begin{itemize}
\item $s^{edge}_{i,j} = f_{edge}(s^{edge}_{i,j-1}, Y_{i,j-1}, \text{EdgeAttn}(s^{edge}_{i,k}, h^{edge}_{:,:}))$
\item $Y_{i,j} = \text{MLP}(s^{edge}_{i,j})$
\item $j\leftarrow j+1$
\end{itemize}
\item $i\leftarrow i+1$
\end{itemize}
\STATE Return predicted graph sequence $Y_{1:m}$
\end{algorithmic}
\label{alg:overall overall algorithm}
\end{algorithm}
\subsubsection{Attention mechanism} \label{sec:attention}
Seen from Eq.~\eqref{eq:dec_nodeProbDistribution}~\eqref{eq:dec_edgeProbDistribution}, the final edge probabilities are conditioned on a node context vector $C$, as well as an edge context vector $c$. We derive these following~\cite{bahdanau2014neural}:
The node context vector $C_i$, is defined from the hidden vectors $(h^{node}_1, h^{node}_2, \dots, h^{node}_n)$ generated by the encoder. Each hidden vector $h^{node}_i$ has captured global information from the graph $G$, with a main focus on the $i$-th node. Now, the $i^{th}$ node context vector $C_i$ is computed as the weighted sum across all these hidden vectors:
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
C_i = \sum_{j=1}^{n}vHiddenWeight_{i,j}h^{node}_j.
\label{eq:nodeAttnComputation}
\end{equation}
The weight assigned to $h^{node}_j$ is computed as the normalized compatibility of $h^{node}_j$ with $s^{node}_{i-1}$:
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
vHiddenWeight_{i,j} = \frac{\exp(\phi(s^{node}_{i-1}, h^{node}_j))}
{\sum_{k=1}^{n}\exp(\phi(s^{node}_{i-1}, h^{node}_k))}
\label{eq:nodeAttnWeightComputation}
\end{equation}
\noindentndent where $\phi$ is a compatibility function, parameterized as a feedforward neural network and trained jointly along with other modules in the whole model.
The computation of the edge level context $c_{i,j}$ follows the same scheme. The overall framwork is summarized in Algorithm~\ref{alg:overall algorithm}.
\section{Experiments and Evaluation}
\label{sec:ExperimentsAndEvaluation}
Next, we show how our model can be applied in a wide range of tasks, including Graph2Graph regression applied as a heuristic to solve challenging graph algorithmic problems; representation learning as a graph autoencoder; and utilizing the graph autoencoder to learn semantic representations for graph classification with limited labeled data.
\paragraph{General experimental setup.} All models were trained on a single NVIDIA Titan GPU with 12GB memory, using the Adam~\cite{Kingma2015} optimizer with learning rate $\in \{0.01, 0.003\}$ and batch size $\in \{64,128\}$.
\paragraph{Loss Function.} For all Graph2Graph learning tasks, we used the Focal loss~\cite{lin2017focal} function known from segmenting images with unbalanced classes, analogous to the relatively sparse graphs. Note that image segmentations and binary graphs are similar as target objects, in the sense that both consist of multiple binary classifications.
For an input graph $\textbf{X}$, a ground truth output graph $\mathbb{Y}$ and a Graph2Graph network $\mathcal{M}$, the loss $\ell_{i,j}$ for the edge between node $i$ and its $j$-th previous node is :
\begin{equation}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt}
\ell_{i,j} = -(1-p^t_{i,j})^\gamma \log(p^t_{i,j}),
\label{eq:lossFuncSingle}
\end{equation}
\noindentndent where $p^t_{i,j}$ denotes likelihood of \emph{correct prediction}, and $\gamma>0$ is a hyperparameter used to reduce relative loss for well classified edges. In our experiment $\gamma=2$.
The final loss is a sum of edge-wise losses: $\mathcal{L}(\mathcal{M}(\textbf{X}), \mathbb{Y}) = \sum_{(i, j)\in\mathcal{I}}\ell_{i,j}$, where $\mathcal{I}$ is the set of node index pairs. This varies depending on application: For the maximum clique prediction (Sec.~\ref{subsec:maxCliquePrediction}), we have $\forall a \in \mathcal{I}, \textbf{X}_a = 1$, restricting to predicting subgraphs of the input. For the autoencoder (Sec.~\ref{sec:ae} and~\ref{sec:graphClassification}), on the other hand, $\mathcal{I}$ contains all index pairs in \textbf{X}.
\begin{table}[t]
\center
{
\caption{\label{tab:datasets_3tasks}Summary of datasets used in our experiments on three applications.}
\resizebox{\textwidth}{!}
{
\setlength{\tabcolsep}{4pt}{
\begin{tabular}{@{}llcccccc@{}}
\toprule
~ &\textbf{Dataset} &\# Graphs &\# Node (avg) &\# Node (max) &\# Training &\# Validation &\# Test \\
\midrule
\multirowcell{3}{Sec.~\ref{subsec:maxCliquePrediction}\\ \textbf{Maximum Clique}} &DBLP\_v1 &{$14488$} &{$11.7$} &{$39$} &{$8692$} &{$2898$} &{$2898$} \\
~ &IMDB-MULTI &{$1500$} &{$13.0$} &{$89$} &{$900$} &{$300$} &{$300$} \\
~ &deezer\_ego\_nets &{$2558$} &{$16.7$} &{$30$} &{$1552$} &{$518$} &{$518$} \\
\midrule
\multirowcell{3}{Sec.~\ref{sec:ae}\\ \textbf{AutoEncoder}} &DBLP\_v1 &{$19455$} &{$10.5$} &{$39$} &{$11673 $} &{$3891 $} &{$3891$} \\
~ &IMDB-MULTI &{$1500$} &{$13.0$} &{$89$} &{$900$} &{$300$} &{$300$} \\
~ &MUTAG &{$188$} &{$17.9$} &{$28$} &{$112$} &{$38$} &{$38$} \\
\midrule
\multirowcell{3}{Sec.~\ref{sec:graphClassification}\\ \textbf{Classification}} &DBLP\_v1 &{$19455$} &{$10.5$} &{$39$} &{$11673 $} &{$3891 $} &{$3891$} \\
~ &IMDB-MULTI &{$1500$} &{$13.0$} &{$89$} &{$1200$} &{$150$} &{$150$} \\
~ &AIDS &{$2000$} &{$15.7$} &{$\emph{50}$} &{$1600$} &{$200$} &{$200$} \\
~ &NCI1 &{$4100$} &{$29.9$} &{$\emph{50}$} &{$3288$} &{$411$} &{$411$} \\
~ &IMDB-BINARY &{$1000$} &{$19.8$} &{$\emph{100}$} &{$800$} &{$100$} &{$100$} \\
\bottomrule
\end{tabular}
}
}
}
\end{table}
\begin{table}[t]
\center
\setlength\tabcolsep{10pt}
{
\caption{\label{tab:maxClique_acc_iou}Results of maximal clique prediction in term of accuracy(\%) and edge IOUs(\%) on DBLP\_v1, IMDB-MULTI and deezer\_ego\_nets dataset. OOM means out of memory even if batch size is 1, Graph2Graph denotes our proposed model. Results are based on a random split.}
\resizebox{\textwidth}{!}
{
\begin{tabular}{@{}lcccccc@{}}
\toprule
~ & \multicolumn{2}{c}{DBLP\_v1}& \multicolumn{2}{c}{IMDB-MULTI}& \multicolumn{2}{c}{deezer\_ego\_nets} \\
\midrule
\textbf{Models} & Accuracy &edge IoU & Accuracy &edge IoU & Accuracy & edge IoU \\
\cmidrule(r){1-1} \cmidrule(r){2-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
MLP &$85.0$ & $93.8$ & $ 61.0 $ & $85.7 $ & $33.2$ & $66.7$ \\
GRU w/o Attn &$85.96$ & $95.47$ &$54.33$ &$79.82$ &$42.86$ & $69.76 $\\
GRU with Attn &\multicolumn{2}{c}{>3days} &\multicolumn{2}{c}{OOM}& $46.53$& $76.75$ \\
Graph2Graph &$\textbf{95.51}$ & $\textbf{97.43}$ & $\textbf{82.3}$ & $\textbf{92.5}$ & $\textbf{58.5}$ & $\textbf{81.8}$ \\
\bottomrule
\end{tabular}
}
}
\end{table}
\begin{table}[!t]
\center
{
\caption{\label{tab:ablation_maxClique_acc_iou}Ablation study on maximal clique prediction in term of accuracies and edge IOUs on DBLP\_v1, IMDB-MULTI and deezer\_ego\_nets dataset. Graph2Graph denotes our proposed model. Results are computed on a random split}
\resizebox{\textwidth}{!}
{
\begin{tabular}{@{}lcccccc@{}}
\toprule
~ & \multicolumn{2}{c}{DBLP\_v1}& \multicolumn{2}{c}{IMDB-MULTI}& \multicolumn{2}{c}{deezer\_ego\_nets} \\
\cmidrule(r){2-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
\textbf{Models} & Accuracy &edge IoU & Accuracy &edge IoU& Accuracy & edge IoU \\
\midrule
Graph2Graph w/o NodeAttn &$93.48$ & $96.15$ &$63.67$ &$83.21$ &$50.39$ &$78.90$ \\
Graph2Graph w/o EdgeAttn &$94.58$ & $96.58$ &$80.00$ &$90.51$ &$56.76$ &$\textbf{82.90}$\\
Graph2Graph full &$\textbf{95.51}$ &$\textbf{97.43}$ & $\textbf{82.3}$ & $\textbf{92.5}$ & $\textbf{58.5}$ & $81.8$ \\
\bottomrule
\end{tabular}
}
}
\end{table}
\begin{figure}
\caption{Examples of maximal clique predictions; edge probability shown in colorbar.}
\label{fig:maxClique_example_1}
\label{fig:maxClique_example_2}
\label{fig:maxClique_example_colorBar}
\label{fig:maxClique_visualization}
\end{figure}
\subsection{Solving graph algorithmic problems via Graph2Graph regression}\label{subsec:maxCliquePrediction}
Many important graph problems are NP complete~\cite{bomze1999maximum, alekseev2007np}, enhancing the need for efficient heuristics. While most existing heuristics are deterministic in their nature, and hence also in the mistakes that they make, a neural network would be trained to perform the task at hand for a particular data distribution. This gives possibilities for improving both quality, tailoring the predictions to the data, and computational complexity, which is low at test time.
Here, we train the Graph2Graph architecture to predict maximum cliques, an NP complete problem, and illustrate its performance on several benchmark datasets.
\paragraph{Problem Definition and settings} The Maximal Clique (MC) is the \textit{complete} subgraph $\graph^{*}$ of a given graph $G$, which contains the maximal number of nodes.
To reduce computational time, we employed a fixed attention mechanism for both node level and edge level by setting $vHiddenWeight_{i,j} =1, \text{if } i=j$.
\paragraph{Data.}
Our experiments are carried out using the datasets DBLP\_v1~\cite{pan2013graph}, IMDB-MULTI~\cite{yanardag2015deep}, and deezer\_ego\_nets~\cite{rozemberczki2020api} from the TUD graph benchmark database~\cite{tud}. Graphs whose maximum clique contained less than 3 nodes were excluded, and for the deezer\_ego\_nets, we excluded those graphs that had more than 30 nodes, giving the dataset statistics shown in Table~\ref{tab:datasets_3tasks}. In each graph dataset, 60\% were used for training, 20\% for validation and 20\% for test.
\paragraph{Results.}
Performance was quantified both in terms of accuracy and Intersection over Union (IoU) on correctly predicted edges. While the former counts completely correctly predicted subgraphs, the latter quantifies near-successes, analogous to its use in image segmentation. We compare our performance with other similar autoencoder architectures with backbones as MLP, GRU with~\cite{bahdanau2014neural} and without Attention~\cite{cho2014learning}, by flattening a graph as one-level sequence. The results found in Table~\ref{tab:maxClique_acc_iou} clearly show that the Graph2Graph architecture outperforms the alternative models. This high performance is also illustrated in Fig.~\ref{fig:maxClique_visualization}, which shows visual examples of predicted maximal cliques. These are illustrated prior to thresholding, with edge probabilities indicated by edge color. More examples are found in the Supplementary Material.
\paragraph{Ablation Study.} By removing node level attention (\textit{Graph2Graph w/o NodeAttn}) and edge level attention (\textit{Graph2Graph w/o EdgeAttn}) from the original model, we investigate the components' contributions to the MC prediction; see Table~\ref{tab:ablation_maxClique_acc_iou} for results. We see that Graph2Graph Models have better performance over Graph2Graph without NodeAttn and that without EdgeAttn on all three datasets under both metrics, except the edgeIoU on deezer\_ego\_nets. These results demonstrate the contributions of attention mechanism at node level and edge level to performance improvement.
\begin{figure}
\caption{Latent representation for DBLP\_v1 test set.}
\label{fig:dblp_tsne}
\end{figure}
\begin{figure}
\caption{Latent representation for test sets of IMDB-MULTI (left) and MUTAG (right).}
\label{fig:imdb_mutag_tsne}
\end{figure}
\subsection{Graph autoencoder via Graph2Graph prediction} \label{sec:ae}
It is well known from image- and language processing~\cite{cho2014learning}~\cite{sutskever2014sequence} that encoder-decoder networks often learn semantically meaningful embeddings when trained to reproduce their input as an autoencoder. We utilize the encoder-decoder structure of our proposed Graph2Graph model to train the network as a graph autoencoder, mapping graphs to continuous context vectors, and back to an approximation of the original graph.
\paragraph{Problem definition and settings.} Given input graphs $G$, train the Graph2Graph network $\mathcal{M} \colon G \mapsto H$ for the prediction $H = \hat{G}$ to reconstruct $G$ as well as possible. The encoder uses single directional RNNs for both edgeRNN and nodeRNN, {\it{i.e., }} $h^{edge}_i = \overrightarrow{h}^{edge}_i$, $h^{node}_i = \overrightarrow{h}^{node}_i$. We use no edge context $c$, and constrain all node contexts $C$ as $h^{node}_n$ to obtain more compressed encodings. The resulting $h^{node}_n$ serves as a latent representation of $G$ in a learned latent space, which in our experiments has dimension 128.
\paragraph{Data.} We use the full TU~\cite{tud} datasets DBLP\_v1, IMDB-MULTI and MUTAG~\cite{debnath1991structure}, using 60/20/20\% for training/validation/test, respectively; see Table~\ref{tab:datasets_3tasks} for details.
\paragraph{Results.} A visual comparison of original and reconstructed graphs is found in Fig.~\ref{fig:graphReconstruction}, and Figs.~\ref{fig:dblp_tsne} and~\ref{fig:imdb_mutag_tsne} visually demonstrate the ability of the learned representation to preserve graph structure by visualizing the the test-set latent features in a 2D t-SNE plot~\cite{van2008visualizing}. Note, in particular, how graphs that are visually similar are found nearby each other. It is evident from the t-SNE plots that the Graph2Graph model has captured semantic information of the graphs, whereas adjacency matrix embeddings (see supplementary material) fail to capture such patterns. Note also that even on a the very small trainingset of MUTAG, the embedding still preserves semantic structure. The expressiveness of the latent space embeddings is further validated on the proxy task of graph classification with limited labeled data below. More visualization results can be found in the supplementary material, including a comparison with t-SNE on the original adjacency matrices.
\begin{figure}
\caption{Autoencoder reconstructions from DBLP\_v1. Edge probability in grayscale (see colorbar).}
\label{fig:graphReconstruction_1}
\label{fig:graphReconstruction_2}
\label{fig:graphReconstruction_colorBar}
\label{fig:graphReconstruction}
\end{figure}
\subsection{Pretraining semantic representations for graph classification with limited labeled data}
\label{sec:graphClassification}
In this section, we utilize the graph autoencoder from Sec.~\ref{sec:ae} to learn semantic, low-dimensional embeddings of the dataset graphs, and apply an MLP on the latent variables for classification. In particular, we investigate the ``limited labels'' setting, which appears frequently in real life settings where data is abundant, but labeled data is scarce and expensive.
\paragraph{Problem formulation and settings.} The graph autoencoder is trained on a large training set (not using graph labels). An MLP is subsequently trained for classification on labeled training set subsamples of the latent representations, to study how the resulting model's performance depends on the size of the labeled training set. We compare our own subset model with the state-of-the-art Graph Isomorphism Network~\cite{xu2018powerful} (GIN) using similar hyperparameters (2 layers of MLP, each with hidden dimension 64; batch size 32) and selecting, for each model, the best performing epoch out of 100. Both models are given purely structural data, supplying GIN with node degrees as node labels. Both models are trained on randomly sampled subsets consisting of 0.1\%, 0.25\%, 0.5\%, 1\%, 5\%, 10\% and 100\% of the labeled training data, respectively; keeping the validation and training sets fixed. Training is repeated on new random subsets 10 times.
\paragraph{Data.} We use the DBLP\_v1, NCI1~\cite{wale2008comparison,kim2016pubchem,shervashidze2011weisfeiler}, AIDS~\cite{aidsdata,riesen2008iam}, IMDB-BINARY~\cite{yanardag2015deep} and IMDB-MULTI datasets for classification. The rather large DBLP\_v1 is divided into 60/20/20 \% splits for training/validation/testing, whereas the remaining datasets are divided into 80/10/10 \% splits to ensure that the best models are robustly trained. These splits were kept fixed across models.
\paragraph{Results.}
As shown in Fig.~\ref{fig:small_labels}, the model pretrained as a Graph2Graph autoencoder outperforms GIN on the small training sets, and approaches similar performance on the full training set.
\begin{figure}
\caption{Results from classification with features pre-trained as a Graph2Graph autoencoder.}
\label{fig:small_labels}
\end{figure}
\section{Discussion and Conclusion}
\label{sec:DiscussionConclusion}
In this paper, we have turned our attention to graph-to-graph prediction; a problem which has so far seen very limited development within graph deep learning. We have utilized this both for graph-in, graph-out regression; a graph autoencoder; and for unsupervised pretraining of semantic representations that allow learning discriminative classification models with very little labeled data.
Our paper proposes a new, general family of problems for deep learning on graphs; namely predictions whose input and output are all graphs. We propose several tasks for learning with such models, and establish methods for validating them using publicly available benchmark data.
Our experimental validation differs from state-of-the-art in graph classification, in that we work with fixed training/validation/test splits as is commonly done in deep learning. To make this feasible, we have chosen to validate on datasets that are larger than the most commonly used graph benchmark datasets. Nevertheless, we have worked with publicly available benchmark data, making it easy for others to improve on and compare to our models. Further encouraging this, our code will be made publicly available upon publication.
While our model opens up for general solutions to new problems, it also has weaknesses. First, our current implementation assumes that all graphs have the same size, obtaining this by zero-padding all graphs to the maximal size. While this assumption is also found in other graph deep learning work, it is an expensive one, and in future work we will seek to remove it.
Our model depends on the order of the nodes used to create the adjacency matrix, and thus per se depends on node permutation. However, in a similar fashion as~\cite{you2018graphrnn}, all graphs are represented using a depth first order before feeding them into the model, which ensures that different permutations to the input graph still gives consistent output.
The performance of the state-of-the-art benchmark GIN is lower than that reported in the literature~\cite{xu2018powerful}, for two main reasons. First, as has previously been pointed out~\cite{xu2018powerful}, the most common way to report performance for graph neural networks is by reporting the largest encountered validation performance; an approach that is explained by the widespread use of small datasets. As we have chosen to perform validation with larger datasets, we do not do this. Second, this tendency is emphasized by our use of structural information alone in order to assess differences in models rather than differences in information supplied.
This also emphasizes the potential for even stronger representations. The Graph2Graph network currently only uses structural information, and the extension to graphs with node labels or node- and edge weights, as well as graphs whose structures or attributes are stochastic, forms important directions for future work.
In conclusion, we present an autoregressive model for graph-to-graph predictions and show its utility in several different tasks, ranging from graph-valued regression, via autoencoders and their use for visualization, to graph classification with limited labeled data based on latent representations pretrained as an autoencoder.
Most work in deep learning for graphs addresses problems whose output is ``simple'', such as classification with discrete output, or regression with real-valued output. This paper demonstrates, quantitatively and visually, that graph neural networks can be used to learn far richer outputs, with corresponding rich internal representations.
\small
\end{document} |
\begin{document}
\title{Steiner trees and higher geodecity}
\begin{abstract}
Let~$G$ be a connected graph and $\ell : E(G) \to \mathbb{R}^+$ a length-function on the edges of~$G$. The \emph{Steiner distance} $\rm sd_G(A)$ of $A \subseteq V(G)$ within~$G$ is the minimum length of a connected subgraph of~$G$ containing~$A$, where the length of a subgraph is the sum of the lengths of its edges.
It is clear that every subgraph $H \subseteq G$, with the induced length-function $\ell|_{E(H)}$, satisfies $\rm sd_H(A) \geq \rm sd_G(A)$ for every $A \subseteq V(H)$. We call $H \subseteq G$ \emph{$k$-geodesic in~$G$} if equality is attained for every $A \subseteq V(H)$ with $|A| \leq k$. A subgraph is \emph{fully geodesic} if it is $k$-geodesic for every $k \in \mathbb{N}$. It is easy to construct examples of graphs $H \subseteq G$ such that~$H$ is $k$-geodesic, but not $(k+1)$-geodesic, so this defines a strict hierarchy of properties. We are interested in situations in which this hierarchy collapses in the sense that if $H \subseteq G$ is $k$-geodesic, then~$H$ is already fully geodesic in~$G$.
Our first result of this kind asserts that if~$T$ is a tree and $T \subseteq G$ is 2-geodesic with respect to some length-function~$\ell$, then it is fully geodesic. This fails for graphs containing a cycle. We then prove that if~$C$ is a cycle and $C \subseteq G$ is 6-geodesic, then~$C$ is fully geodesic. We present an example showing that the number~6 is indeed optimal.
We then develop a structural approach towards a more general theory and present several open questions concerning the big picture underlying this phenomenon.
\end{abstract}
\begin{section}{Introduction}
\label{intro}
Let~$G$ be a graph and $\ell : E(G) \to \mathbb{R}^+$ a function that assigns to every edge $e \in E(G)$ a positive \emph{length} $\ell(e)$. This naturally extends to subgraphs $H \subseteq G$ as $\ell(H) := \sum_{e \in E(H)} \ell(e)$. The \emph{Steiner distance} $\rm sd_G(A)$ of a set $A \subseteq V(G)$ is defined as the minimum length of a connected subgraph of~$G$ containing~$A$, where $\rm sd_G(A) := \infty$ if no such subgraph exists. Every such minimizer is necessarily a tree and we say it is a \emph{Steiner tree for~$A$ in~$G$}. In the case where $A = \{ x, y\}$, the Steiner distance of~$A$ is the ordinary distance~$\rm d_G(x,y)$ between~$x$ and~$y$. Hence this definition yields a natural extension of the notion of ``distance'' for sets of more than two vertices. Corresponding notions of radius, diameter and convexity have been studied in the literature \cite{chartrand, steinerdiamdeg, steinerdiamgirth, steinerdiamplanar, steinerconvexnhood, steinerconvexgeom}. Here, we initiate the study of \emph{Steiner geodecity}, with a focus on structural assumptions that cause a collapse in the naturally arising hierarchy.
Let $H \subseteq G$ be a subgraph of~$G$, equipped with the length-function $\ell|_{E(H)}$. It is clear that for every $A \subseteq V(H)$ we have $\rm sd_H(A) \geq \rm sd_G(A)$. For a natural number~$k$, we say that~$H$ is \emph{$k$-geodesic in~$G$} if $\rm sd_H(A) = \rm sd_G(A)$ for every $A \subseteq V(H)$ with $|A| \leq k$. We call~$H$ \emph{fully geodesic in~$G$} if it is $k$-geodesic for every $k \in \mathbb{N}$.
By definition, a $k$-geodesic subgraph is $m$-geodesic for every $m \leq k$. In general, this hierarchy is strict: In Section~\ref{general theory} we provide, for every $k \in \mathbb{N}$, examples of graphs $H \subseteq G$ and a length-function $\ell : E(G) \to \mathbb{R}^+$ such that~$H$ is $k$-geodesic, but not $(k+1)$-geodesic. On the other hand, it is easy to see that if $H \subseteq G$ is a 2-geodesic \emph{path}, then it is necessarily fully geodesic, because the Steiner distance of any $A \subseteq V(H)$ in~$H$ is equal to the maximum distance between two $a, b \in A$. Our first result extends this to all trees.
\begin{theorem} \label{tree 2-geo}
Let~$G$ be a graph with length-function~$\ell$ and $T \subseteq G$ a tree. If~$T$ is 2-geodesic in~$G$, then it is fully geodesic.
\end{theorem}
Here, it really is necessary for the subgraph to be acyclic (see Corollary~\ref{H_2 forests}). Hence the natural follow-up question is what happens in the case where the subgraph is a cycle.
\begin{theorem} \label{cycle 6-geo}
Let~$G$ be a graph with length-function~$\ell$ and $C \subseteq G$ a cycle. If~$C$ is 6-geodesic in~$G$, then it is fully geodesic.
\end{theorem}
Note that the number~6 cannot be replaced by any smaller integer.
In Section~\ref{preliminaries} we introduce notation and terminology needed in the rest of the paper. Section~\ref{toolbox} contains observations and lemmas that will be used later. We then prove Theorem~\ref{tree 2-geo} in Section~\ref{sct on trees}. In Section~\ref{sct on cycles} we prove Theorem~\ref{cycle 6-geo} and provide an example showing that the number~6 is optimal. Section~\ref{general theory} contains an approach towards a general theory, aiming at a deeper understanding of the phenomenon displayed in Theorem~\ref{tree 2-geo} and Theorem~\ref{cycle 6-geo}. Finally, we take the opportunity to present the short and easy proof that in any graph~$G$ with length-function~$\ell$, the cycle space of~$G$ is generated by the set of fully geodesic cycles.
\end{section}
\begin{section}{Preliminaries}
\label{preliminaries}
All graphs considered here are finite and undirected. It is convenient for us to allow parallel edges. In particular, a cycle may consist of just two vertices joined by two parallel edges. Loops are redundant for our purposes and we exclude them to avoid trivialities. Most of our notation and terminology follows that of~\cite{diestelbook}, unless stated otherwise.
A set~$A$ of vertices in a graph~$G$ is called \emph{connected} if and only if $G[A]$ is.
Let $G, H$ be two graphs. A \emph{model of~$G$ in~$H$} is a family of disjoint connected \emph{branch-sets} $B_v \subseteq V(H)$, $v \in V(G)$, together with an injective map $\beta : E(G) \to E(H)$, where we require that for any $e \in E(G)$ with endpoints $u, v \in V(G)$, the edge $\beta(e) \in E(H)$ joins vertices from~$B_u$ and~$B_v$. We say that~$G$ \emph{is a minor of~$H$} if~$H$ contains a model of~$G$.
We use additive notation for adding or deleting vertices and edges. Specifically, let~$G$ be a graph, $H$ a subgraph of~$G$, $v \in V(G)$ and $e =xy \in E(G)$. Then $H + v$ is the graph with vertex-set $V(H) \cup \{ v \}$ and edge-set $E(H) \cup \{ vw \in E(G) \colon w \in V(H) \}$. Similarly, $H + e$ is the graph with vertex-set $V(H) \cup \{ x, y\}$ and edge-set $E(H) \cup \{ e \}$.
Let~$G$ be a graph with length-function~$\ell$. A \emph{walk} in~$G$ is an alternating sequence $W = v_1e_1v_2 \ldots e_kv_{k+1}$ of vertices~$v_i$ and edges~$e_i$ such that $e_i = v_i v_{i+1}$ for every $1 \leq i \leq k$. The walk~$W$ is \emph{closed} if $v_1 = v_{k+1}$. Stretching our terminology slightly, we define the \emph{length} of the walk as $\rm len_G(W) := \sum_{1 \leq i \leq k} \ell(e_i)$. The \emph{multiplicity}~$\rm m_W(e)$ of an edge $e \in E(G)$ is the number of times it is traversed by~$W$, that is, the number of indices $1 \leq j \leq k$ with $e = e_j$. It is clear that
\begin{equation} \label{length walk}
\rm len_G(W) = \sum_{e \in E(G)} \rm m_W(e) \ell(e) .
\end{equation}
Let~$G$ be a graph and~$C$ a cycle with $V(C) \subseteq V(G)$. We say that a walk~$W$ in~$G$ is \emph{traced by~$C$} in~$G$ if it can be obtained from~$C$ by choosing a starting vertex $x \in V(C)$ and an orientation~$\overrightarrow{C}$ of~$C$ and replacing every $\overrightarrow{ab} \in E(\overrightarrow{C})$ by a shortest path from~$a$ to~$b$ in~$G$. A cycle may trace several walks, but they all have the same length: Every walk~$W$ traced by~$C$ satisfies
\begin{equation} \label{length traced walk}
\rm len_G(W) = \sum_{ab \in E(C)} \rm d_G(a, b) .
\end{equation}
Even more can be said if the graph~$G$ is a tree. Then all the shortest $a$-$b$-paths for $ab \in E(C)$ are unique and all walks traced by~$C$ differ only in their starting vertex and/or orientation. In particular, every walk~$W$ traced by~$C$ in a tree~$T$ satisfies
\begin{equation} \label{multiplicities traced walk tree}
\forall e \in E(T): \, \, \rm m_W(e) = | \{ ab \in E(C) \colon e \in aTb \} | ,
\end{equation}
where $aTb$ denotes the unique $a$-$b$-path in~$T$.
Let~$T$ be a tree and $X \subseteq V(T)$. Let $e \in E(T)$ and let $T_1^e, T_2^e$ be the two components of $T -e$. In this manner, $e$ induces a bipartition $X = X_1^e \cup X_2^e$ of~$X$, given by $X_i^e = V(T_i^e) \cap X$ for $i \in \{ 1, 2 \}$. We say that the bipartition is \emph{non-trivial} if neither of $X_1^e, X_2^e$ is empty. The set of leaves of~$T$ is denoted by~$L(T)$. If $L(T) \subseteq X$, then every bipartition of~$X$ induced by an edge of~$T$ is non-trivial.
Let~$G$ be a graph with length-function~$\ell$, $A \subseteq V(G)$ and~$T$ a Steiner tree for~$A$ in~$G$. Since $\ell(e) >0 $ for every $e \in E(G)$, every leaf~$x$ of~$T$ must lie in~$A$, for otherwise $T - x$ would be a tree of smaller length containing~$A$.
In general, Steiner trees need not be unique. If~$G$ is a tree, however, then every $A \subseteq V(G)$ has a unique Steiner tree given by $\bigcup_{a, b \in A} aTb$.
\end{section}
\begin{section}{The toolbox}
\label{toolbox}
The first step in all our proofs is a simple lemma that guarantees the existence of a particularly well-behaved substructure that witnesses the failure of a subgraph to be $k$-geodesic.
Let~$H$ be a graph, $T$ a tree and~$\ell$ a length-function on~$T \cup H$. We call~$T$ a \emph{shortcut tree for~$H$} if the following hold:
\begin{enumerate}[ itemindent=0.8cm, label=(SCT\,\arabic*)]
\item $V(T) \cap V(H) = L(T)$, \label{sct vert}
\item $E(T) \cap E(H) = \emptyset$, \label{sct edge}
\item $\ell(T) < \rm sd_H( L(T) )$, \label{sct shorter}
\item For every $B \subseteqsetneq L(T)$ we have $\rm sd_H(B) \leq \rm sd_T(B)$. \label{sct minim}
\end{enumerate}
Note that, by definition, $H$ is not $|L(T)|$-geodesic in~$T \cup H$.
\begin{lemma} \label{shortcut tree}
Let~$G$ be a graph with length-function~$\ell$, $k$ a natural number and $H \subseteq G$. If~$H$ is not $k$-geodesic in~$G$, then~$G$ contains a shortcut tree for~$H$ with at most~$k$ leaves.
\end{lemma}
\begin{proof}
Among all $A \subseteq V(H)$ with $|A| \leq k$ and $\rm sd_G(A) < \rm sd_H(A)$, choose~$A$ such that $\rm sd_G(A)$ is minimum. Let $T \subseteq G$ be a Steiner tree for~$A$ in~$G$. We claim that~$T$ is a shortcut tree for~$H$.
\textit{Claim 1:} $L(T) = A = V(T) \cap V(H)$.
The inclusions $L(T) \subseteq A \subseteq V(T) \cap V(H)$ are clear. We show $V(T) \cap V(H) \subseteq L(T)$. Assume for a contradiction that $x \in V(T) \cap V(H)$ had degree $d \geq 2$ in~$T$. Let $T_1, \ldots, T_d$ be the components of $T - x$ and for $j \in [d]$ let $A_j := A \cap V(T_j) \cup \{ x \}$. Since $L(T) \subseteq A$, every tree~$T_i$ contains some $a \in A$ and so $A \not \subseteq A_j$. In particular $|A_j| \leq k$. Moreover $\rm sd_G(A_j) \leq \ell( T_j + x) < \ell(T)$, so by our choice of~$A$ and~$T$ it follows that $\rm sd_G(A_j) = \rm sd_H(A_j)$. Therefore, for every $j \in [d]$ there exists a connected $S_j \subseteq H$ with $A_j \subseteq V(S_j)$ and $\ell(S_j) \leq \ell(T_j + x)$. But then $S := \bigcup_j S_j \subseteq H$ is connected, contains~$A$ and satisfies
\[
\ell(S) \leq \sum_{j = 1}^d \ell(S_j) \leq \sum_{j=1}^d \ell(T_j + x) = \ell(T) ,
\]
which contradicts the fact that $\rm sd_H(A) > \ell(T)$ by choice of~$A$ and~$T$.
\textit{Claim 2:} $E(T) \cap E(H) = \emptyset$.
Assume for a contradiction that $xy \in E(T) \cap E(H)$. By Claim~1, $x, y \in L(T)$ and so~$T$ consists only of the edge~$xy$. But then $T \subseteq H$ and $\rm sd_H(A) \leq \ell(T)$, contrary to our choice of~$A$ and~$T$.
\textit{Claim 3:} $\ell(T) < \rm sd_H(L(T))$.
We have $\ell(T) = \rm sd_G(A) < \rm sd_H(A)$. By Claim~1, $A = L(T)$.
\textit{Claim 4:} For every $B \subseteqsetneq L(T)$ we have $\rm sd_H(B) \leq \rm sd_T(B)$.
Let $B \subseteqsetneq L(T)$ and let $T' := T - (A \setminus B)$. By Claim~1, $T'$ is the tree obtained from~$T$ by chopping off all leaves not in~$B$ and so
\[
\rm sd_G(B) \leq \ell(T') < \ell(T) = \rm sd_G(A) .
\]
By minimality of~$A$, it follows that $\rm sd_H(B) = \rm sd_G(B) \leq \rm sd_T(B)$.
\end{proof}
Our proofs of Theorem~\ref{tree 2-geo} and Theorem~\ref{cycle 6-geo} proceed by contradiction and follow a similar outline. Let $H \subseteq G$ be a subgraph satisfying a certain set of assumptions. The aim is to show that~$H$ is fully geodesic. Assume for a contradiction that it was not and apply Lemma~\ref{shortcut tree} to find a shortcut tree~$T$ for~$H$. Let~$C$ be a cycle with $V(C) \subseteq L(T)$ and let $W_H, W_T$ be walks traced by~$C$ in~$H$ and~$T$, respectively. If $|L(T)| \geq 3$, then it follows from~(\ref{length traced walk}) and~\ref{sct minim} that $\rm len(W_H) \leq \rm len(W_T)$.
Ensure that $\rm m_{W_T}(e) \leq 2$ for every $e \in E(T)$ and that $\rm m_{W_H}(e) \geq 2$ for all $e \in E(S)$, where $S \subseteq H$ is connected with $L(T) \subseteq V(S)$. Then
\[
2\, \rm sd_H(L(T)) \leq 2\, \ell(S) \leq \rm len(W_H) \leq \rm len(W_T) \leq 2 \, \ell(T) ,
\]
which contradicts~\ref{sct shorter}.
The first task is thus to determine, given a tree~$T$, for which cycles~$C$ with $V(C) \subseteq V(T)$ we have $m_W(e) \leq 2$ for all $e \in E(T)$, where~$W$ is a walk traced by~$C$ in~$T$. Let $S \subseteq T$ be the Steiner tree for~$V(C)$ in~$T$. It is clear that~$W$ does not traverse any edges $e \in E(T) \setminus E(S)$ and $L(S) \subseteq V(C) \subseteq V(S)$. Hence we can always reduce to this case and may for now assume that $S = T$ and $L(T) \subseteq V(C)$.
\begin{lemma} \label{pos even}
Let~$T$ be a tree, $C$ a cycle with $L(T) \subseteq V(C) \subseteq V(T)$ and~$W$ a walk traced by~$C$ in~$T$. Then $m_W(e)$ is positive and even for every $e \in E(T)$.
\end{lemma}
\begin{proof}
Let $e \in E(T)$ and let $V(C) = V(C)_1 \cup V(C)_2$ be the induced bipartition. Since $L(T) \subseteq V(C)$, this bipartition is non-trivial. By~(\ref{multiplicities traced walk tree}), $m_W(e)$ is the number of $ab \in E(C)$ such that $e \in aTb$. By definition, $e \in aTb$ if and only if~$a$ and~$b$ lie in different sides of the bipartition. Every cycle has a positive even number of edges across any non-trivial bipartition of its vertex-set.
\end{proof}
\begin{lemma} \label{equality achieved}
Let~$T$ be a tree, $C$ a cycle with $L(T) \subseteq V(C) \subseteq V(T)$. Then
\[
2 \ell(T) \leq \sum_{ab \in E(C)} \rm d_T(a, b) .
\]
Moreover, there is a cycle~$C$ with $V(C) = L(T)$ for which equality holds.
\end{lemma}
\begin{proof}
Let~$W$ be a walk traced by~$C$ in~$T$. By Lemma~\ref{pos even}, (\ref{length walk}) and~(\ref{length traced walk})
\[
2 \ell(T) \leq \sum_{e \in E(T)} \rm m_W(e) \ell(e) = \rm len(W) = \sum_{ab \in E(C)} \rm d_T(a, b) .
\]
To see that equality can be attained, let~$2T$ be the multigraph obtained from~$T$ by doubling all edges. Since all degrees in~$2T$ are even, it has a Eulerian trail~$W$, which may be considered as a walk in~$T$ with $\rm m_W(e) = 2$ for all $e \in E(T)$. This walk traverses the leaves of~$T$ in some cyclic order, which yields a cycle~$C$ with $V(C) = L(T)$. It is easily verified that~$W$ is traced by~$C$ in~$T$ and so
\[
2 \ell(T) = \sum_{e \in E(T)} \rm m_W(e) \ell(e) = \rm len(W) = \sum_{ab \in E(C)} \rm d_T(a, b) .
\]
\end{proof}
We have now covered everything needed in the proof of Theorem~\ref{tree 2-geo}, so the curious reader may skip ahead to Section~\ref{sct on trees}.
\begin{figure}
\caption{A tree with four leaves}
\label{4leaftree}
\end{figure}
In general, not every cycle~$C$ with $V(C) = L(T)$ achieves equality in Lemma~\ref{equality achieved}. Consider the tree~$T$ from Figure~\ref{4leaftree} and the following three cycles on $L(T)$
\[
C_1 = abcda, \, \, C_2 = acdba, \, \, C_3 = acbda .
\]
\begin{figure}
\caption{The three cycles on~$T$}
\label{4leaftree3cyc}
\end{figure}
For the first two, equality holds, but not for the third one. But how does~$C_3$ differ from the other two? It is easy to see that we can add~$C_1$ to the planar drawing of~$T$ depicted in Figure~\ref{4leaftree}: There exists a planar drawing of $T \cup C_1$ extending this particular drawing. This is not true for~$C_2$, but it can be salvaged by exchanging the positions of~$a$ and~$b$ in Figure~\ref{4leaftree}. Of course, this is merely tantamount to saying that $T \cup C_i$ is planar for $i \in \{ 1, 2\}$.
On the other hand, it is easy to see that $T \cup C_3$ is isomorphic to~$K_{3,3}$ and therefore non-planar.
\begin{lemma} \label{euler planar}
Let~$T$ be a tree and~$C$ a cycle with $V(C) = L(T)$. Let~$W$ be a walk traced by~$C$ in~$T$. The following are equivalent:
\begin{enumerate}[label=(a)]
\item $T \cup C$ is planar.
\item For every $e \in E(T)$, both $V(C)_1^e, V(C)_2^e$ are connected in~$C$.
\item $W$ traverses every edge of~$T$ precisely twice.
\end{enumerate}
\end{lemma}
\begin{proof}
(a) $\Rightarrow$ (b): Fix a planar drawing of $T \cup C$. The closed curve representing~$C$ divides the plane into two regions and the drawing of~$T$ lies in the closure of one of them. By symmetry, we may assume that it lies within the closed disk inscribed by~$C$. Let $A \subseteq V(C)$ disconnected and choose $a, b \in A$ from distinct components of~$C[A]$. $C$ is the disjoint union of two edge-disjoint $a$-$b$-paths $S_1, S_2$ and both of them must meet $C \setminus A$, say $c \in V(S_1) \setminus A$ and $d \in V(S_2) \setminus A$.
The curves representing $aTb$ and $cTd$ lie entirely within the disk and so they must cross. Since the drawing is planar, $aTb$ and~$cTd$ have a common vertex. In particular, $A$ cannot be the set of leaves within a component of $T - e$ for any edge $e \in E(T)$.
(b) $\Rightarrow$ (c): Let $e \in E(T)$. By assumption, there are precisely two edges $f_1, f_2 \in E(C)$ between $V(C)_1^e$ and $V(C)_2^e$. These edges are, by definition, the ones whose endpoints are separated in~$T$ by~$e$. By~(\ref{multiplicities traced walk tree}), $m_W(e) =2$.
(c) $\Rightarrow$ (a): For $ab \in E(C)$, let $D_{ab} := aTb + ab \subseteq T \cup C$. The set $\mathcal{D} := \{ D_{ab} \colon ab \in E(C) \}$ of all these cycles is the fundamental cycle basis of $T \cup C$ with respect to the spanning tree~$T$. Every edge of~$C$ occurs in only one cycle of~$\mathcal{D}$. By assumption and~(\ref{multiplicities traced walk tree}), every edge of~$T$ lies on precisely two cycles in~$\mathcal{D}$. Covering every edge of the graph at most twice, the set~$\mathcal{D}$ is a \emph{sparse basis} of the cycle space of $T \cup C$. By MacLane's Theorem, $T \cup C$ is planar.
\end{proof}
\end{section}
\begin{section}{Shortcut trees for trees}
\label{sct on trees}
\begin{proof}[Proof of Theorem~\ref{tree 2-geo}]
Assume for a contradiction that $T \subseteq G$ was not fully geodesic and let $R \subseteq T$ be a shortcut tree for~$T$. Let $T' \subseteq T$ be the Steiner tree for $L(R)$ in~$T$. By Lemma~\ref{equality achieved}, there is a cycle~$C$ with $V(C) = L(R)$ such that
\[
2 \ell(R) = \sum_{ab \in E(C)} \rm d_R(a, b) .
\]
Note that~$T'$ is 2-geodesic in~$T$ and therefore in~$G$, so that $\rm d_{T'}(a,b) \leq \rm d_R(a,b)$ for all $ab \in E(C)$. Since every leaf of~$T'$ lies in $L(R) = V(C)$, we can apply Lemma~\ref{equality achieved} to~$T'$ and~$C$ and conclude
\[
2 \ell(T') \leq \sum_{ab \in E(C)} \rm d_{T'}(a, b) \leq \sum_{ab \in E(C)} \rm d_R(a, b) = 2\ell(R) ,
\]
which contradicts~\ref{sct shorter}.
\end{proof}
\end{section}
\begin{section}{Shortcut trees for cycles}
\label{sct on cycles}
By Lemma~\ref{shortcut tree}, it suffices to prove the following.
\begin{theorem} \label{shortcut tree cycle strong}
Let~$T$ be a shortcut tree for a cycle~$C$. Then $T \cup C$ is a subdivision of one of the five (multi-)graphs in Figure~\ref{five shortcut trees}. In particular, $C$ is not 6-geodesic in $T \cup C$.
\end{theorem}
\begin{figure}
\caption{The five possible shortcut trees for a cycle}
\label{five shortcut trees}
\end{figure}
Theorem~\ref{shortcut tree cycle strong} is best possible in the sense that for each of the graphs in Figure~\ref{five shortcut trees} there exists a length-function which makes the tree inside a shortcut tree for the outer cycle, see Figure~\ref{cycleshortcutlength}. These length-functions were constructed in a joint effort with Pascal Gollin and Karl Heuer in an ill-fated attempt to prove that a statement like Theorem~\ref{cycle 6-geo} could not possibly be true.
\begin{figure}
\caption{Shortcut trees for cycles}
\label{cycleshortcutlength}
\end{figure}
This section is devoted entirely to the proof of Theorem~\ref{shortcut tree cycle strong}. Let~$T$ be a shortcut tree for a cycle~$C$ with length-function $\ell : E(T \cup C) \to \mathbb{R}^+$ and let $L := L(T)$.
The case where $|L| = 2$ is trivial, so we henceforth assume that $|L| \geq 3$. By suppressing any degree-2 vertices, we may assume without loss of generality that $V(C) = L(T)$ and that~$T$ contains no vertices of degree~2.
\begin{lemma} \label{cover disjoint trees}
Let $T_1, T_2 \subseteq T$ be edge-disjoint trees. For $i \in \{ 1, 2\}$, let $L_i := L \cap V(T_i)$. If $L = L_1 \cup L_2$ is a non-trivial bipartition of~$L$, then both $C[L_1], C[L_2]$ are connected.
\end{lemma}
\begin{proof}
By~\ref{sct minim} there are connected $S_1, S_2 \subseteq C$ with $\ell(S_i) \leq \rm sd_T(L_i) \leq \ell(T_i)$ for $i \in \{ 1, 2 \}$. Assume for a contradiction that~$C[L_1]$ was not connected. Then $V(S_1) \cap L_2$ is non-empty and $S_1 \cup S_2 $ is connected, contains~$L$ and satisfies
\[
\ell( S_1 \cup S_2) \leq \ell(S_1) + \ell(S_2) \leq \ell(T_1) + \ell(T_2) \leq \ell(T) ,
\]
which contradicts~\ref{sct shorter}.
\end{proof}
\begin{lemma} \label{planar 3reg}
$T \cup C$ is planar and 3-regular.
\end{lemma}
\begin{proof}
Let $e \in E(T)$, let $T_1, T_2$ be the two components of $T - e$ and let $L = L_1 \cup L_2$ be the induced (non-trivial) bipartition of~$L$. By Lemma~\ref{cover disjoint trees}, both $C[L_1], C[L_2]$ are connected. Therefore $T \cup C$ is planar by Lemma~\ref{euler planar}.
To see that $T \cup C$ is 3-regular, it suffices to show that no $t \in T$ has degree greater than~3 in~$T$. We just showed that $T \cup C$ is planar, so fix some planar drawing of it. Suppose for a contradiction that $t \in T$ had $d \geq 4$ neighbors in~$T$. In the drawing, these are arranged in some cyclic order as $t_1, t_2, \ldots, t_d$. For $j \in [d]$, let $R_j := T_j + t$, where~$T_j$ is the component of $T - t$ containing~$t_j$. Let~$T_{\rm odd}$ be the union of all~$R_j$ for odd $j \in [d]$ and~$T_{\rm even}$ the union of all~$R_j$ for even $j \in [d]$. Then $T_{\rm odd}, T_{\rm even} \subseteq T$ are edge-disjoint and yield a nontrivial bipartition $L = L_{\rm odd} \cup L_{\rm even}$ of the leaves. But neither of $C[L_{\rm odd}], C[L_{\rm even}]$ is connected, contrary to Lemma~\ref{cover disjoint trees}.
\end{proof}
\begin{lemma} \label{consec cycle long}
Let $e_0 \in E(C)$ arbitrary. Then for any two consecutive edges $e_1, e_2$ of~$C$ we have $\ell(e_1) + \ell(e_2) > \ell(e_0)$. In particular $\ell(e_0) < \ell(C)/2$.
\end{lemma}
\begin{proof}
Suppose that $e_1, e_2 \in E(C)$ are both incident with $x \in L$. Let $S \subseteq C$ be a Steiner tree for $B := L \setminus \{ x \}$ in~$C$. By~\ref{sct minim} and~\ref{sct shorter} we have
\[
\ell(S) \leq \rm sd_T(B) \leq \ell(T) < \rm sd_C(L) .
\]
Thus $x \notin S$ and $E(S) = E(C) \setminus \{ e_1, e_2 \}$. Thus $P := C - e_0$ is not a Steiner tree for~$B$ and we must have $\ell(P) > \ell(S)$.
\end{proof}
Let $t \in T$ and~$N$ its set of neighbors in~$T$. For every $s \in N$ the set~$L_s$ of leaves~$x$ with $s \in tTx$ is connected in~$C$. Each $C[L_s]$ has two edges $f_s^1, f_s^2 \in E(C)$ incident to it.
\begin{lemma} \label{good root}
There is a $t \in T$ such that for every $s \in N$ and any $f \in \{ f_s^1, f_s^2 \}$ we have $\ell(C[L_s] + f) < \ell(C)/2$.
\end{lemma}
\begin{proof}
We construct a directed graph~$D$ with $V(D) = V(T)$ as follows. For every $t \in T$, draw an arc to any $s \in N$ for which $\ell(C[L_s] + f_s^i) \geq \ell(C)/2$ for some $i \in \{ 1, 2 \}$.
\textit{Claim:} If $\overrightarrow{ts} \in E(D)$, then $\overrightarrow{st} \notin E(D)$.
Assume that there was an edge $st \in E(T)$ for which both $\overrightarrow{st}, \overrightarrow{ts} \in E(D)$. Let $T_s, T_t$ be the two components of $T - st$, where $s \in T_s$, and let $L = L_s \cup L_t$ be the induced bipartition of~$L$. By Lemma~\ref{cover disjoint trees}, both $C[L_s]$ and $C[L_t]$ are connected paths, say with endpoints $a_s, b_s$ and $a_t, b_t$ (possibly $a_s = b_s$ or $a_t = b_t$) so that $a_sa_t \in E(C)$ and $b_sb_t \in E(C)$ (see Figure~\ref{badneighbors}). Without loss of generality $\ell(a_sa_t) \leq \ell(b_sb_t)$. Since $\overrightarrow{ts} \in E(D)$ we have $\ell(C[L_t] + b_sb_t) \geq \ell(C)/2$ and therefore $C[L_s] + a_sa_t$ is a shortest $a_t$-$b_s$-path in~$C$.
Similarly, it follows from $\overrightarrow{st} \in E(D)$ that $\rm d_C( a_s, b_t) = \ell(C[L_t] + a_sa_t)$.
\begin{figure}
\caption{The setup in the proof of Lemma~\ref{good root}
\label{badneighbors}
\end{figure}
Consider the cycle $Q := a_tb_sa_sb_ta_t$ and let $W_T, W_C$ be walks traced by~$Q$ in~$T$ and in~$C$, respectively. Then $\rm len(W_T) \leq 2 \, \ell(T)$, whereas
\[
\rm len(W_C) = 2 \, \ell (C - b_sb_t) \geq 2 \, \rm sd_C(L) .
\]
By~\ref{sct minim} we have $\rm d_C(x,y) \leq \rm d_T(x,y)$ for all $x, y \in L$ and so $\rm len(W_C) \leq \rm len(W_T)$. But then $\rm sd_C(L) \leq \ell(T)$, contrary to~\ref{sct shorter}. This finishes the proof of the claim.
Since every edge of~$D$ is an orientation of an edge of~$T$ and no edge of~$T$ is oriented both ways, it follows that~$D$ has at most $|V(T)| - 1$ edges. Since~$D$ has $|V(T)|$ vertices, there is a $t \in V(T)$ with no outgoing edges.
\end{proof}
Fix a node $t \in T$ as guaranteed by the previous lemma. If~$t$ was a leaf with neighbor~$s$, say, then $\ell(f_s^1) = \ell(C) - \ell(C[L_s] + f_s^2) > \ell(C)/2$ and, symmetrically, $\ell(f_s^2) > \ell(C)/2$, which is impossible. Hence by Lemma~\ref{planar 3reg}, $t$ has three neighbors $s_1, s_2, s_3 \in T$ and we let $L_i := C[L_{s_i}]$ and $\ell_i := \ell(L_i)$. There are three edges $f_1, f_2, f_3 \in E(C) \setminus \bigcup E(L_i)$, where $f_1$ joins~$L_1$ and~$L_2$, $f_2$ joins~$L_2$ and~$L_3$ and~$f_3$ joins~$L_3$ and~$L_1$. Each~$L_i$ is a (possibly trivial) path whose endpoints we label $a_i, b_i$ so that, in some orientation, the cycle is given by
\[
C = a_1L_1b_1 + f_1 + a_2L_2b_2 + f_2 + a_3L_3b_3 + f_3 .
\]
Hence $f_1 = b_1a_2$, $f_2 = b_2a_3$ and $f_3 = b_3a_1$ (see Figure~\ref{trail Q}).
The fact that $\ell_1 + \ell(f_1) \leq \ell(C)/2$ means that $L_1 + f_1$ is a shortest $a_1$-$a_2$-path in~$C$ and so $\rm d_C(a_1, a_2) = \ell_1 + \ell(f_1)$. Similarly, we thus know the distance between all other pairs of vertices with just one segment~$L_i$ and one edge~$f_j$ between them.
\begin{figure}
\caption{The cycle~$Q$}
\label{trail Q}
\end{figure}
If $|L_i| \leq 2$ for every $i \in [3]$, then $T \cup C$ is a subdivision of one the graphs depicted in Figure~\ref{five shortcut trees} and we are done. Hence from now on we assume that at least one~$L_i$ contains at least~3 vertices.
\begin{lemma} \label{jumps}
Suppose that $\max \{ |L_s| \colon s \in N \} \geq 3$. Then there is an $s \in N$ with $\ell( f_s^1 + C[L_s] + f_s^2) \leq \ell(C)/2$.
\end{lemma}
\begin{proof}
For $j \in [3]$, let $r_j := \ell( f_{s_j}^1 + L_j + f_{s_j}^2)$. Assume wlog that $|L_1| \geq 3$. Then~$L_1$ contains at least two consecutive edges, so by Lemma~\ref{consec cycle long} we must have $\ell_1 > \ell(f_2)$. Therefore
\[
r_2 + r_3 = \ell(C) + \ell(f_2) - \ell_1 < \ell(C) ,
\]
so the minimum of $r_2, r_3$ is less than $\ell(C)/2$.
\end{proof}
By the previous lemma, we may wlog assume that
\begin{equation} \label{jump edge}
\ell(f_2) + \ell_3 + \ell(f_3) \leq \ell(C)/2 ,
\end{equation}
so that $f_2 + L_3 + f_3$ is a shortest $a_1$-$b_2$-path in~$C$. Together with the inequalities from Lemma~\ref{good root}, this will lead to the final contradiction.
Consider the cycle $Q = a_1b_2a_2a_3b_3b_1a_1$ (see Figure~\ref{trail Q}). Let~$W_T$ be a walk traced by~$Q$ in~$T$. Every edge of~$T$ is traversed at most twice, hence
\begin{equation}
\sum_{ab \in E(Q)} \rm d_T(a, b) = \ell(W_T) \leq 2\ell(T) . \label{sum through tree}
\end{equation}
Let~$W_C$ be a walk traced by~$Q$ in~$C$. Using~(\ref{jump edge}) and the inequalities from Lemma~\ref{good root}, we see that
\begin{align*}
\ell(W_C) &= \sum_{ab \in E(Q)} \rm d_C(a, b) = 2 \ell_1 + 2 \ell_2 + 2 \ell_3 + 2 \ell(f_2) + 2 \ell(f_3) \\
&= 2 \ell(C) - 2 \ell(f_1) .
\end{align*}
But by~\ref{sct minim} we have $\rm d_C(a,b) \leq \rm d_T(a,b)$ for all $a, b \in L(T)$ and therefore $\ell(W_C) \leq \ell(W_T)$. Then by~(\ref{sum through tree})
\[
2 \ell(C) - 2 \ell(f_1) = \ell(W_C) \leq \ell(W_T) \leq 2 \ell(T).
\]
But then $S := C - f_1$ is a connected subgraph of~$C$ with $L(T) \subseteq V(S)$ satisfying $\ell(S) \leq \ell(T)$. This contradicts~\ref{sct shorter} and finishes the proof of Theorem~\ref{shortcut tree cycle strong}.
\qed
\end{section}
\begin{section}{Towards a general theory}
\label{general theory}
We have introduced a notion of higher geodecity based on the concept of the Steiner distance of a set of vertices. This introduces a hierarchy of properties: Every $k$-geodesic subgraph is, by definition, also $m$-geodesic for any $m < k$. This hierarchy is strict in the sense that for every~$k$ there are graphs~$G$ and $H \subseteq G$ and a length-function~$\ell$ on~$G$ such that~$H$ is $k$-geodesic in~$G$, but not $(k+1)$-geodesic. To see this, let~$G$ be a complete graph with $V(G) = [k+1] \cup \{ 0 \}$ and let~$H$ be the subgraph induced by $[k+1]$. Define $\ell(0j) := k-1$ and $\ell(ij) := k$ for all $i, j \in [k+1]$. If~$H$ was not $k$-geodesic, then~$G$ would contain a shortcut tree~$T$ for~$H$ with $|L(T)| \leq k$. Then~$T$ must be a star with center~$0$ and
\[
\ell(T) = (k-1)|L(T)| \geq k(|L(T)|-1) .
\]
But any spanning tree of $H[L(T)]$ has length $k(|L(T)|-1) $ and so $\rm sd_H(L(T)) \leq \ell(T)$, contrary to~\ref{sct shorter}. Hence~$H$ is a $k$-geodesic subgraph of~$G$. However, the star~$S$ with center~$0$ and $L(S) = [k+1]$ shows that
\[
\rm sd_G(V(H)) \leq (k+1)(k-1) < k^2 = \rm sd_H(V(H)) = k^2 .
\]
Theorem~\ref{tree 2-geo} and Theorem~\ref{cycle 6-geo} demonstrate a rather strange phenomenon by providing situations in which this hierarchy collapses.
For a given natural number $k \geq 2$, let us denote by~$\mathcal{H}_k$ the class of all graphs~$H$ with the property that whenever~$G$ is a graph with $H \subseteq G$ and~$\ell$ is a length-function on~$G$ such that~$H$ is $k$-geodesic, then~$H$ is also fully geodesic.
By definition, this yields an ascending sequence $\mathcal{H}_2 \subseteq \mathcal{H}_3 \subseteq \ldots $ of classes of graphs. By Theorem~\ref{tree 2-geo} all trees lie in~$\mathcal{H}_2$. By Theorem~\ref{cycle 6-geo} all cycles are contained in~$\mathcal{H}_6$. The example above shows that $K_{k+1} \notin \mathcal{H}_k$.
We now describe some general properties of the class~$\mathcal{H}_k$.
\begin{theorem} \label{H_k minor closed}
For every natural number $k \geq 2$, the class~$\mathcal{H}_k$ is closed under taking minors.
\end{theorem}
To prove this, we first provide an easier characterization of the class~$\mathcal{H}_k$.
\begin{proposition} \label{H_k sct}
Let $k \geq 2$ be a natural number and~$H$ a graph. Then $H \in \mathcal{H}_k$ if and only if every shortcut tree for~$H$ has at most~$k$ leaves.
\end{proposition}
\begin{proof}
Suppose first that $H \in \mathcal{H}_k$ and let~$T$ be a shortcut tree for~$H$. By~\ref{sct shorter}, $H$ is not $|L(T)|$-geodesic in $T \cup H$. Let~$m$ be the minimum integer such that~$H$ is not $m$-geodesic in $T \cup H$. By Lemma~\ref{shortcut tree}, $T \cup H$ contains a shortcut tree~$S$ with at most~$m$ leaves for~$H$. But then by~\ref{sct vert} and~\ref{sct edge}, $S$ is the Steiner tree in~$T$ of $B := L(S) \subseteq L(T)$. If $B \subseteqsetneq L(T)$, then $\ell(S) = \rm sd_T(B) \geq \rm sd_H(B)$ by~\ref{sct minim}, so we must have $B = L(T)$ and $m \geq |L(T)|$. Thus~$H$ is $(|L(T)| - 1)$-geodesic in $T \cup H$, but not $|L(T)|$-geodesic. As $H \in \mathcal{H}_k$, it must be that $|L(T)| - 1 < k$.
Suppose now that every shortcut tree for~$H$ has at most~$k$ leaves and let $H \subseteq G$ $k$-geodesic with respect to some length-function $\ell : E(G) \to \mathbb{R}^+$. If~$H$ was not fully geodesic, then~$G$ contained a shortcut tree~$T$ for~$H$. By assumption, $T$ has at most~$k$ leaves. But then $\rm sd_G( L(T)) \leq \ell(T) < \rm sd_H(L(T))$, so~$H$ is not $k$-geodesic in~$G$.
\end{proof}
\begin{lemma} \label{wlog connected}
Let $k \geq 2$ be a natural number and~$G$ a graph. Then $G \in \mathcal{H}_k$ if and only if every component of~$G$ is in~$\mathcal{H}_k$.
\end{lemma}
\begin{proof}
Every shortcut tree for a component~$K$ of~$G$ becomes a shortcut tree for~$G$ by taking $\ell(e) := 1$ for all $e \in E(G) \setminus E(K)$. Hence if $G \in \mathcal{H}_k$, then every component of~$G$ is in~$\mathcal{H}_k$ as well.
Suppose now that every component of~$G$ is in~$\mathcal{H}_k$ and that~$T$ is a shortcut tree for~$G$. If there is a component~$K$ of~$G$ with $L(T) \subseteq V(K)$, then~$T$ is a shortcut tree for~$K$ and so $|L(T)| \leq k$ by assumption. Otherwise, let $t_1 \in L(T) \cap V(K_1)$ and $t_2 \in L(T) \cap V(K_2)$ for distinct components $K_1, K_2$ of~$G$. By~\ref{sct minim}, it must be that $L(T) = \{ t_1, t_2 \}$ and so $|L(T)| = 2 \leq k$.
\end{proof}
\begin{lemma} \label{sct minor closed}
Let~$G, H$ be two graphs and let~$T$ be a shortcut tree for~$G$. If~$G$ is a minor of~$H$, then there is a shortcut tree~$T'$ for~$H$ which is isomorphic to~$T$.
\end{lemma}
\begin{proof}
Since~$G$ is a minor of~$H$, there is a family of disjoint connected sets $B_v \subseteq V(H)$, $v \in V(G)$, and an injective map $\beta : E(G) \to E(H)$ such that for $uv \in E(G)$, the end vertices of $\beta(uv) \in E(H)$ lie in~$B_u$ and~$B_v$.
Let~$T$ be a shortcut tree for~$G$ with $\ell : E(T \cup G) \to \mathbb{R}^+$. By adding a small positive real number to every $\ell(e)$, $e \in E(T)$, we may assume that the inequalities in~\ref{sct minim} are strict, that is
\[
\rm sd_G(B) \leq \rm sd_T(B) - \epsilon
\]
for every $B \subseteq L(T)$ with $2 \leq |B| < |L(T)|$, where $\epsilon > 0$ is some constant.
Obtain the tree~$T'$ from~$T$ by replacing every $t \in L(T)$ by an arbitrary $x_t \in B_t$ and every $t \in V(T) \setminus L(T)$ by a new vertex~$x_t$ not contained in $V(H)$, maintaining the adjacencies. It is clear by definition that $V(T') \cap V(H) = L(T')$ and $E(T') \cap E(H) = \emptyset$. We now define a length-function $\ell' : E(T' \cup H) \to \mathbb{R}^+$ as follows.
For every edge $st \in E(T)$, the corresponding edge $x_sx_t \in E(T')$ receives the same length $\ell'(x_sx_t) := \ell(st)$. Every $e \in E(H)$ that is contained in one of the branchsets~$B_v$ is assigned the length $\ell '(e) := \delta$, where $\delta := \epsilon / |E(H)| $. For every $e \in E(G)$ we let $\ell '( \beta(e)) := \ell(e)$. To all other edges of~$H$ we assign the length~$\ell(T) + 1$.
We now show that~$T'$ is a shortcut tree for~$H$ with the given length-function~$\ell'$. Suppose that $S' \subseteq H$ was a connected subgraph with $L(T') \subseteq V(S')$ and $\ell'(S') \leq \ell'(T')$. By our choice of~$\ell'$, every edge of~$S'$ must either lie in a branchset~$B_v$ or be the image under~$\beta$ of some edge of~$G$, since otherwise $\ell'(S') > \ell(T) = \ell'(T')$. Let $S \subseteq V(G)$ be the subgraph where $v \in V(S)$ if and only if $V(S') \cap B_v$ is non-empty and $e \in E(S)$ if and only if $\beta(e) \in E(S')$. Since~$S'$ is connected, so is~$S$: For any non-trivial bipartition $V(S) = U \cup W$ the graph~$S'$ contains an edge between $\bigcup_{u \in U} B_u$ and $\bigcup_{w \in W} B_w$, which in turn yields an edge of~$S$ between~$U$ and~$W$. Moreover $L(T) \subseteq V(S)$, since~$V(S')$ contains~$x_t$ and thus meets~$B_t$ for every $t \in L(T)$. Finally, $\ell(S) \leq \ell(S')$, which contradicts our assumption that~$T$ is a shortcut tree for~$G$.
For $B' \subseteq L(T')$ with $2 \leq |B'| < |L(T')|$, let $B := \{ t \in T \colon x_t \in B' \}$. By assumption, there is a connected $S \subseteq G$ with $B \subseteq V(S)$ and $\ell(S) \leq \rm sd_T(B) - \epsilon$. Let
\[
S' := \bigcup_{v \in V(S)} H[B_v] + \{ \beta(e) \colon e \in E(S) \}.
\]
For every $x_t \in B'$ we have $t \in B \subseteq V(S)$ and so $x_t \in B_t \subseteq V(S')$. Since~$S$ is connected and every $H[B_v]$ is connected, $S'$ is connected as well. Moreover
\[
\ell'(S') \leq \delta |E(H)| + \ell(S) \leq \rm sd_T(B) = \rm sd_{T'}(B') .
\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{H_k minor closed}]
Let~$H$ be a graph in~$\mathcal{H}_k$ and~$G$ a minor of~$H$. Let~$T$ be a shortcut tree for~$G$. By Lemma~\ref{sct minor closed}, $H$ has a shortcut tree~$T'$ which is isomorphic to~$T$. By Proposition~\ref{H_k sct} and assumption on~$H$, $T$ has $|L(T')| \leq k$ leaves. Since~$T$ was arbitrary, it follows from Proposition~\ref{H_k sct} that $G \in \mathcal{H}_k$.
\end{proof}
\begin{corollary} \label{H_2 forests}
$\mathcal{H}_2$ is the class of forests.
\end{corollary}
\begin{proof}
By Theorem~\ref{tree 2-geo} and Lemma~\ref{wlog connected}, every forest is in~$\mathcal{H}_2$. On the other hand, if~$G$ contains a cycle, then it contains the triangle~$C_3$ as a minor. We saw in Section~\ref{sct on cycles} that~$C_3$ has a shortcut tree with~3 leaves. By Lemma~\ref{sct minor closed}, so does~$G$ and hence $G \notin \mathcal{H}_2$ by Proposition~\ref{H_k sct}.
\end{proof}
\begin{corollary} \label{num leaf bounded}
For every natural number $k \geq 2$ there exists an integer $m = m(k)$ such that every graph that is not in~$\mathcal{H}_k$ has a shortcut tree with more than~$k$, but not more than~$m$ leaves.
\end{corollary}
\begin{proof}
Let $k \geq 2$ be a natural number. By Theorem~\ref{H_k minor closed} and the Graph Minor Theorem of Robertson and Seymour~\cite{graphminorthm} there is a finite set~$R$ of graphs such that for every graph~$H$ we have $H \in \mathcal{H}_k$ if and only if~$H$ does not contain any graph in~$R$ as a minor. Let $m(k) := \max_{G \in R} |G|$.
Let~$H$ be a graph and suppose $H \notin \mathcal{H}_k$. Then~$H$ contains some $G \in R$ as a minor. By Proposition~\ref{H_k sct}, this graph~$G$ has a shortcut tree~$T$ with more than~$k$, but certainly at most~$|G|$ leaves. By Lemma~\ref{sct minor closed}, $H$ has a shortcut tree isomorphic to~$T$.
\end{proof}
We remark that we do not need the full strength of the Graph Minor Theorem here: We will see in a moment that the tree-width of graphs in~$\mathcal{H}_k$ is bounded for every $k \geq 2$, so a simpler version of the Graph Minor Theorem can be applied, see~\cite{excludeplanar}. Still, it seems that Corollary~\ref{num leaf bounded} ought to have a more elementary proof.
\begin{question}
Give a direct proof of Corollary~\ref{num leaf bounded} that yields an explicit bound on~$m(k)$. What is the smallest possible value for~$m(k)$?
\end{question}
In fact, we are not even aware of any example that shows one cannot simply take $m(k) = k+1$.
Given that~$\mathcal{H}_2$ is the class of forests, it seems tempting to think of each class~$\mathcal{H}_k$ as a class of ``tree-like'' graphs. In fact, containment in~$\mathcal{H}_k$ is related to the tree-width of the graph, but the relation is only one-way.
\begin{proposition} \label{low tw example}
For any integer $k \geq 1$, the graph $K_{2, 2k}$ is not in~$\mathcal{H}_{2k-1}$.
\end{proposition}
\begin{proof}
Let~$H$ be a complete bipartite graph $V(H) = A \cup B \cup \{ x, y \}$ with $|A| = |B| = k$, where $uv \in E(H)$ if and only if $u \in A \cup B$ and $v \in \{ x, y \}$ (or vice versa). We construct a shortcut tree for $H \cong K_{2,2k}$ with~$2k$ leaves.
For $x', y' \notin V(H)$, let~$T$ be the tree with $V(T) = A \cup B \cup \{ x', y' \}$, where~$x'$ is adjacent to every $a \in A$, $y'$ is adjacent to every $b \in B$ and $x'y' \in E(T)$. It is clear that $V(T) \cap V(H) = L(T)$ and~$T$ and~$H$ are edge-disjoint. We now define a length-function $\ell : E(T \cup H) \to \mathbb{R}^+$ that turns~$T$ into a shortcut tree for~$H$.
For all $a \in A$ and all $b \in B$, let
\begin{gather*}
\ell ( a x) = \ell (a x' ) = \ell ( b y) = \ell (b y') = k-1, \\
\ell ( a y ) = \ell ( a y') = \ell ( b x ) = \ell ( b x' ) = k, \\
\ell ( x' y' ) = k - 1 .
\end{gather*}
Let $A' \subseteq A, B' \subseteq B$. We determine $\rm sd_H(A' \cup B')$. By symmetry, it suffices to consider the case where $|A'| \geq |B'|$. We claim that
\[
\rm sd_H( A' \cup B') = (k-1)|A' \cup B'| + |B'| .
\]
It is easy to see that $\rm sd_H( A' \cup B') \leq (k-1)|A'| + k|B'|$, since $S^* := H[A' \cup B' \cup \{ x \}]$ is connected and achieves this length. Let now $S \subseteq H$ be a tree with $A' \cup B' \subseteq V(S)$.
If every vertex in $A' \cup B'$ is a leaf of~$S$, then~$S$ can only contain one of~$x$ and~$y$, since it does not contain a path from~$x$ to~$y$. But then~$S$ must contain one of $H[A' \cup B' \cup \{x \}]$ and $H[ A' \cup B' \cup \{ y \} ]$, so $\ell(S) \geq \ell(S^*)$.
Suppose now that some $x \in A' \cup B'$ is not a leaf of~$S$. Then~$x$ has two incident edges, one of length~$k$ and one of length~$k-1$. For $s \in S$, let $r(s)$ be the sum of the lengths of all edges of~$S$ incident with~$s$. Then $\ell(s) \geq k-1$ for all $s \in A' \cup B'$ and $\ell(x) \geq 2k-1$. Since $A' \cup B'$ is independent in~$H$ (and thus in~$S$), it follows that
\begin{align*}
\ell(S) &\geq \sum_{s \in A' \cup B'} r(s) \geq |( A' \cup B') \setminus \{ x \}| (k-1) + (2k-1) \\
&= |A' \cup B'|(k-1) + k \geq (k-1)|A' \cup B'| + |B'| .
\end{align*}
Thus our claim is proven. For $A', B'$ as before, it is easy to see that
\[
\rm sd_T( A' \cup B') = \begin{cases}
(k-1)|A' \cup B'| , &\text{ if } B' = \emptyset \\
(k-1)|A' \cup B'| + k- 1, &\text{ otherwise.}
\end{cases}
\]
We thus have $\rm sd_T( A' \cup B' ) < \rm sd_H(A' \cup B')$ if and only if $|A'| = |B'| = k$. Hence~\ref{sct shorter} and~\ref{sct minim} are satisfied and~$T$ is a shortcut tree for~$H$ with~$2k$ leaves.
\end{proof}
Note that the graph $K_{2,k}$ is planar and has tree-width~2. Hence there is no integer~$m$ such that all graphs of tree-width at most~2 are in~$\mathcal{H}_m$. Using Theorem~\ref{H_k minor closed}, we can turn Proposition~\ref{low tw example} into a positive result, however.
\begin{corollary} \label{H_k exclude K_2,k+2}
For any $k \geq 2$, no $G \in \mathcal{H}_k$ contains $K_{2, k+2}$ as a minor.
\qed
\end{corollary}
In particular, it follows from the Grid-Minor Theorem~\cite{excludeplanar} and planarity of~$K_{2,k}$ that the tree-width of graphs in~$\mathcal{H}_k$ is bounded. Bodlaender et al~\cite{excludeK2k} gave a more precise bound for this special case, showing that graphs excluding $K_{2,k}$ as a minor have tree-width at most~$2(k-1)$.
It seems plausible that a qualitative converse to Corollary~\ref{H_k exclude K_2,k+2} might hold.
\begin{question} \label{K_2,k as culprit}
Is there a function $q : \mathbb{N} \to \mathbb{N}$ such that every graph that does not contain $K_{2,k}$ as a minor is contained in~$\mathcal{H}_{q(k)}$?
\end{question}
Since no subdivision of a graph~$G$ contains $K_{2, |G| + e(G) + 1}$ as a minor, a positive answer would prove the following.
\begin{conjecture}
For every graph~$G$ there exists an integer~$m$ such that every subdivision of~$G$ lies in~$\mathcal{H}_m$.
\end{conjecture}
\end{section}
\begin{section}{Generating the cycle space}
Let~$G$ be a graph with length-function~$\ell$. It is a well-known fact (see e.g.~\cite[Chapter~1, exercise~37]{diestelbook}) that the set of 2-geodesic cycles generates the cycle space of~$G$. This extends as follows, showing that fully geodesic cycles abound.
\begin{proposition} \label{generate cycle space}
Let~$G$ be a graph with length-function~$\ell$. The set of fully geodesic cycles generates the cycle space of~$G$.
\end{proposition}
We remark, first of all, that the proof is elementary and does not rely on Theorem~\ref{cycle 6-geo}, but only requires Lemma~\ref{shortcut tree} and Lemma~\ref{pos even}.
Let~$\mathcal{D}$ be the set of all cycles of~$G$ which cannot be written as a 2-sum of cycles of smaller length. The following is well-known.
\begin{lemma}
The cycle space of~$G$ is generated by~$\mathcal{D}$.
\end{lemma}
\begin{proof}
It suffices to show that every cycle is a 2-sum of cycles in~$\mathcal{D}$. Assume this was not the case and let $C \subseteq G$ be a cycle of minimum length that is not a 2-sum of cycles in~$\mathcal{D}$. In particular, $C \notin \mathcal{D}$ and so there are cycles $C_1, \ldots, C_k$ with $C = C_1 \oplus \ldots \oplus C_k$ and $\ell(C_i) < \ell(C)$ for every $i \in [k]$. By our choice of~$C$, every~$C_i$ can be written as a 2-sum of cycles in~$\mathcal{D}$. But then the same is true for~$C$, which is a contradiction.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{generate cycle space}]
We show that every $C \in \mathcal{D}$ is fully geodesic. Indeed, let $C \subseteq G$ be a cycle which is not fully geodesic and let $T\subseteq G$ be a shortcut tree for~$C$. There is a cycle~$D$ with $V(D) = L(T)$ such that~$C$ is a union of edge-disjoint $L(T)$-paths $P_{ab}$ joining~$a$ and~$b$ for $ab \in E(D)$.
For $ab \in E(D)$ let $C_{ab} := aTb + P_{ab}$. Every edge of~$C$ lies in precisely one of these cycles. An edge $e \in E(T)$ lies in $C_{ab}$ if and only if $e \in aTb$. By Lemma~\ref{pos even} and~(\ref{multiplicities traced walk tree}), every $e \in E(T)$ lies in an even number of cycles~$C_{ab}$. Therefore $C = \bigoplus_{ab \in E(D)} C_{ab}$.
For every $ab \in E(D)$, $C$ contains a path~$S$ with $E(S) = E(C) \setminus E(P_{ab})$ with $L(T) \subseteq V(S)$. Since~$T$ is a shortcut tree for~$C$, it follows from~\ref{sct shorter} that
\[
\ell(C_{ab}) \leq \ell(T) + \ell(P_{ab}) < \ell(S) + \ell(P_{ab}) = \ell(C) .
\]
In particular, $C \notin \mathcal{D}$.
\end{proof}
The fact that 2-geodesic cycles generate the cycle space has been extended to the topological cycle space of locally finite graphs graphs by Georgakopoulos and Spr\"{u}ssel~\cite{agelos}. Does Proposition~\ref{generate cycle space} have a similar extension?
\end{section}
\end{document} |
\begin{document}
\title{Proving UNSAT in SMT: \ The Case of Quantifier Free Non-Linear Real Arithmetic}
\begin{abstract}
We discuss the topic of unsatisfiability proofs in SMT, particularly with reference to quantifier free non-linear real arithmetic. We outline how the methods here do not admit trivial proofs and how past formalisation attempts are not sufficient. We note that the new breed of local search based algorithms for this domain may offer an easier path forward.
\end{abstract}
\section{Introduction}
Since 2013, SAT Competitions have required certificates for unsatisfiability which are verified offline \cite{HJS18}. As the SAT problems tackled have grown larger, and the solvers have grown more complicated, such proofs have become more important for building trust in these solvers. The SAT community has agreed on DRAT as a common format for presenting such proofs (although within this there are some flavours \cite{RB19}).
The SMT community has long recognized the value of proof certificates, but alas producing them turned our to be much more difficult than for the SAT case. The current version of the SMT-LIB Language (v2.6) \cite{SMTLIB} specifies API commands for requesting and inspecting proofs from solvers but sets no requirements on the form those proofs take. In fact on page 66 it writes explicitly: ``\emph{The format of the proof is solver-specific}''. We assume that this is a place holder for future work on an SMT-LIB proof format, rather than a deliberate design. The paper \cite{BdMF15} summarises some of the requirements, challenges and various approaches taken to proofs in SMT. Key projects that have been working on this issue include LFSC \cite{Stump2012} and veriT \cite{Barbosa2020}, but there has not been a general agreement in the community yet.
Our long-term vision is that an SMT solver would be able to emit a ``proof'' that covers both the Boolean reasoning and the theory reasoning (possibly from multiple theories) such that a theorem prover (or a combination of multiple theorem provers) could verify its correctness, where the inverted commas indicate that some programming linkage between the theorem provers might be necessary. We would still be some way from having a fully verified one-stop checker as in GRAT \cite{Lammich2020}, but would be a lot closer to it than we are now.
In \cite{BdMF15} the authors explain that since in SMT the propositional and theory reasoning are not strongly mixed, an SMT proof can be an interleaving of SAT proofs and theory reasoning proofs in the shape of a Boolean resolution tree whose leaves are clauses. They identify the main challenge of proof production as keeping enough information to produce proofs, without hurting efficiency too much. This may very well be true for many cases, but for the area of interest for the authors, \verb+QF_NRA+ (Quantifier-Free Nonlinear Real Arithmetic), there is the additional challenge of providing the proofs of the theory lemmas themselves.
\section{Quantifier Free Non-Linear Real Arithmetic}
\verb+QF_NRA+ typically considers a logical formula $\Phi$ where the literals are statements about the signs of polynomials with rational coefficients, i.e. $f_i(x_1,\ldots,x_n)\sigma_i0$ with $\sigma_i\in\{=,\ne,>,\ge,<,\le\}$.
Any SMT solver which claims to tackle this logic completely relies in some way on the theory of Cylindrical Algebraic Decomposition (CAD). This was initiated by Collins \cite{Collins1975} in the 1970s with many subsequent developments since: see for example the collection \cite{CJ98} or the introduction of the recent paper \cite{EBD20}. The key idea is to decompose infinite space $\mathbb{R}^n$ into a finite number of disjoint regions upon each of which the truth of the constraints is constant. This may be achieved by decomposing to ensure the signs of the polynomials involved are invariant, although optimisations can produce a coarser, and thus cheaper, decomposition.
In the case of unsatisfiability an entire CAD truth invariant for the constraints may be produced, and the solver can check that the formula is unsatisfiable for a sample of each cell. How may this be verified? The cylindrical condition\footnote{Formally, the condition is that projection of any two cells onto a lower dimensional space with respect to the variable ordering are either equal or disjoint. Informally, this means the cells are stacked in cylinders over a decomposition in lower dimensional space.} means that checking our cells decompose the space is trivial, but the fact that the constraints have invariant truth-value is a deep implication of the algorithm, not necessarily apparent from the output.
\subsection*{Past QF\_NRA Formalisation Attempts}
There was a project in Coq to formalise Quantifier Elimination in Real Closed Fields. This may also be tackled by CAD, and of course has SMT in \verb+QF_NRA+ as a sub-problem. Work began on an implementation of CAD in Coq with some of the underlying infrastructure formalised \cite{Mahboubi2007}, but the project proceeded to instead formalise QE via alternative methods \cite{CM10}, \cite{CM12b} which are far less efficient\footnote{Although CAD is doubly exponential in the number of variables, the methods verified do not even have worst case complexity bound by a finite tower of exponentials!}. We learn that the CAD approach was not proven correct in the end \cite[bottom of p. 38]{CM12b}. Thus while it is formalised that Real QE (and thus satisfiability) is decidable, this does not offer a route to verifying current solver results.
The only other related work in the literature we found is \cite{NMD15} which essentially formalises something like CAD but only for problems in one variable.
\section{Potential from Coverings Instead of Decompositions?}
There has been recent interaction between the SMT community and the computer algebra community \cite{SC2} from which many of these methods originate. Computer algebra implementations are being adapted for SMT compliance \cite{SC2}, as CAD was in \cite{KA20}, and there has also be success when they are used directly \cite{FOSKT18}. Most excitingly, there have been some entirely new algorithmic approaches developed.
Perhaps most notable is the NLSAT algorithm of Jovanovi\'{c} and de Moura \cite{JdM12}, introduced in 2012 and since generalised into the model constructing satisfiability calculus (mcSAT) framework \cite{dMJ13}. In mcSAT the search for a Boolean model and a theory model are mutually guided by each other away from unsatisfiable regions. Partial solution candidates for the Boolean structure and for the corresponding theory constraints are constructed incrementally in parallel. Boolean conflicts are generalised using propositional resolution as normal. At the theory level, when an assignment (sample point) is determined not to satisfy all constraints then this is generalised from the point to a region containing the point on which the same constraints fail for the same reason.
In NLSAT, which only considers \verb+QF_NRA+, the samples are generalised to CAD cells\footnote{But not necessarily one that would be produced within any entire CAD for the problem.} being excluded by adding a new clause with the negation of the algebraic description of the cell. In UNSAT cases these additional clauses become mutually exclusive, in effect the cells generated cover all possible space in $\mathbb{R}^n$. However, as these are not arranged cylindrically, this may not be trivial to check from the output. We note also the more efficient algorithm to compute these single CAD cells in \cite{BK15}, and the new type of decomposition they inspired in \cite{Brown2015}.
Another new approach was presented recently in \cite{ADEK21}: conflict driven cylindrical algebraic covering (CDCAC). Like NLSAT this produces a covering of $\mathbb{R}^n$ to show unsatisfiability. Essentially, a depth first search is performed according to the theory variables. Conflicts over particular assignments are generalised to cells until a covering of a dimension is obtained, and then this covering is generalised to a cell in the dimension below. In this procedure the covering itself is explicit and easy to verify. Further, CDCAD computes the covering relative to a set of constraints to check their consistency independent of the Boolean search, meaning it can be more easily integrated into an CDCL(T)-style SMT solver and combined with its other theory solving modules than NLSAT, which is a solving framework on its own.
Both NLSAT and CDCAC rely on CAD theory to conclude that the generalisations of conflicts from models to cells are valid, and so the verification of such theory is still a barrier to verifiable proofs. But unlike CAD itself, the conflicts that are being generalized are local for both NLSAT and CDCAC. This may allow a simpler path for verification of individual cases based on the particular relationships of the polynomials involved. It was observed in \cite{ADEKT20} that a trace of the computation from CDCAC appears far closer to a human derived proof than any of the other algorithms discussed here. Whether this means it will be more susceptible to machine verification remains to be seen.
\section{Other approaches for QF\_NRA}
We wrote earlier that all solvers tackling \verb+QF_NRA+ in a complete manner rely on CAD based approaches, as these are the only complete methods that have been implemented.
However, we should note that most solvers also employ a variety of incomplete methods for \verb+QF_NRA+ which tend to be far more efficient than CAD based ones and so are attempted first, and may also be used to solve sub-problems or simplify the input to CAD. These include incremental linearisation \cite{CGIRS18c}, interval constraint propagation \cite{TVO17}, virtual substitution \cite{Weispfenning1997a}, subtropical satisfiability \cite{FOSV17} and Gr\"obner bases \cite{HEDP16}.
So, although we think there is potential for verifying output of a cylindrical covering based algorithm, we caution that to obtain fully verified proofs for \verb+QF_NRA+ problems we must take on a greater body of work: to generate proofs for all these methods and furthermore integrate them into or combine them with the CAD proofs.
\label{sect:bib}
\end{document} |
\begin{document}
\title{PriSTI: A Conditional Diffusion Framework for Spatiotemporal Imputation \\
}
\author{\IEEEauthorblockN{Mingzhe Liu$^{*1}$, Han Huang$^{*1}$, Hao Feng$^1$, Leilei Sun\textsuperscript{\Letter}$^{1}$, Bowen Du$^{1}$, Yanjie Fu$^{2}$}
\IEEEauthorblockA{$^1$State Key Laboratory of Software Development Environment, Beihang University, Beijing 100191, China\\
$^2$Department of Computer Science, University of Central Florida, FL 32816, USA\\
\{mzliu1997, h-huang, pinghao, leileisun, dubowen\}@buaa.edu.cn, yanjie.fu@ucf.edu
}\thanks{$^*\,$Equal contribution. \Letter $\,$ Corresponding author.}
}
\maketitle
\begin{abstract}
Spatiotemporal data mining plays an important role in air quality monitoring, crowd flow modeling, and climate forecasting. However, the originally collected spatiotemporal data in real-world scenarios is usually incomplete due to sensor failures or transmission loss. Spatiotemporal imputation aims to fill the missing values according to the observed values and the underlying spatiotemporal dependence of them.
The previous dominant models impute missing values autoregressively and suffer from the problem of error accumulation.
As emerging powerful generative models, the diffusion probabilistic models can be adopted to impute missing values conditioned by observations and avoid inferring missing values from inaccurate historical imputation.Â
However, the construction and utilization of conditional information are inevitable challenges when applying diffusion models to spatiotemporal imputation.
To address above issues, we propose a conditional diffusion framework for spatiotemporal imputation with enhanced prior modeling, named PriSTI.
Our proposed framework provides a conditional feature extraction module first to extract the coarse yet effective spatiotemporal dependencies from conditional information as the global context prior. Then, a noise estimation module transforms random noise to realistic values, with the spatiotemporal attention weights calculated by the conditional feature, as well as the consideration of geographic relationships.
PriSTI outperforms existing imputation methods in various missing patterns of different real-world spatiotemporal data, and effectively handles scenarios such as high missing rates and sensor failure.
The implementation code is available at \url{https://github.com/LMZZML/PriSTI}.
\end{abstract}
\begin{IEEEkeywords}
Spatiotemporal Imputation, Diffusion Model, Spatiotemporal Dependency Learning
\end{IEEEkeywords}
\section{Introduction}
Spatiotemporal data is a type of data with intrinsic spatial and temporal patterns, which is widely applied in the real world for tasks such as air quality monitoring \cite{cao2018brits, yi2016st}, traffic status forecasting \cite{li2017diffusion, wu2019graph}, weather prediction \cite{bauer2015quiet} and so on.
However, due to the sensor failures and transmission loss \cite{yi2016st}, the incompleteness in spatiotemporal data is a common problem, characterized by the randomness of missing value's positions and the diversity of missing patterns, which results in incorrect analysis of spatiotemporal patterns and further interference on downstream tasks.
In recent years, extensive research \cite{cao2018brits, liu2019naomi, cini2021filling} has dived into spatiotemporal imputation, with the goal of exploiting spatiotemporal dependencies from available observed data to impute missing values.
\begin{figure*}
\caption{The motivation of our proposed methods. We summarize the existing methods that can be applied to spatiotemporal imputation, and compare our proposed methods with the recent existing methods. The grey shadow represents the missing part, while the rest with blue solid line represents observed values $X$.}
\label{fig:motivation}
\end{figure*}
The early studies applied for spatiotemporal imputation usually impute along the temporal or spatial dimension with statistic and classic machine learning methods, including but not limited to autoregressive moving average (ARMA) \cite{ansley1984estimation, harvey1990forecasting}, expectation-maximization algorithm (EM) \cite{shumway1982approach, nelwamondo2007missing}, k-nearest neighbors (KNN) \cite{trevor2009elements, beretta2016nearest}, etc.
But these methods impute missing values based on strong assumptions such as the temporal smoothness and the similarity between time series, and ignore the complexity of spatiotemporal correlations.
With the development of deep learning, most effective spatiotemporal imputation methods \cite{cao2018brits, yoon2018gain, cini2021filling} use the recurrent neural network (RNN) as the core to impute missing values by recursively updating their hidden state, capturing the temporal correlation with existing observations.
Some of them also simply consider the feature correlation \cite{cao2018brits} by the multilayer perceptron (MLP) or spatial similarity between different time series \cite{cini2021filling} by graph neural networks.
However, these approaches inevitably suffer from error accumulation \cite{liu2019naomi}, i.e., inference missing values from inaccurate historical imputation, and only output the deterministic values without reflecting the uncertainty of imputation.
More recently, diffusion probabilistic models (DPM) \cite{sohl2015deep, ho2020denoising, song2020score}, as emerging powerful generative models with impressive performance on various tasks, have been adopted to impute multivariate time series. These methods impute missing values starting from randomly sampled Gaussian noise, and convert the noise to the estimation of missing values \cite{tashiro2021csdi}.
Since the diffusion models are flexible in terms of neural network architecture, they can circumvent the error accumulation problem from RNN-based methods through utilizing architectures such as attention mechanisms when imputation,
which also have a more stable training process than generative adversarial networks (GAN).
However, when applying diffusion models to imputation problem, the modeling and introducing of the conditional information in diffusion models are the inevitable challenges. For spatiotemporal imputation, the challenges can be specific to the construction and utilization of conditional information with spatiotemporal dependencies.
Tashiro et al. \cite{tashiro2021csdi} only model temporal and feature dependencies by attention mechanism when imputing, without considering spatial similarity such as geographic proximity and time series correlation.
Moreover, they combine the conditional information (i.e., observed values) and perturbed values directly as the input for models during training, which may lead to inconsistency inside the input spatiotemporal data, increasing the difficulty for the model to learn spatiotemporal dependencies.
To address the above issues, we propose a conditional diffusion framework for SpatioTemporal Imputation with enhanced Prior modeling (PriSTI).
We summarize the existing methods that can be applied to spatiotemporal imputation, and compare the differences between our proposed method and the recent existing methods, as shown in Figure \ref{fig:motivation}.
Since the main challenge of applying diffusion models on spatiotemporal imputation is how to model and utilize the spatiotemporal dependencies in conditional information for the generation of missing values, our proposed method reduce the difficulty of spatiotemporal dependencies learning by extracting conditional feature from observation as a global context prior.
The imputation process of spatiotemporal data with our proposed method is shown in the right of Figure \ref{fig:motivation}, which gradually transform the random noise to imputed missing values by the trained PriSTI.
PriSTI takes observed spatiotemporal data and geographic information as input. During training, the observed values are randomly erased as imputation target through a specific mask strategy.
The incomplete observed data is first interpolated to obtain the enhanced conditional information for diffusion model.
For the construction of conditional information, a conditional feature extraction module is provided to extract the feature with spatiotemporal dependencies from the interpolated information.
Considering the imputation of missing values not only depends on the values of nearby time and similar time series, but also is affected by geographically surrounding sensors, we design the specialized spatiotemporal dependencies learning methods. The proposed method comprehensively aggregates spatiotemporal global features and geographic information to fully exploit the explicit and implicit spatiotemporal relationships in different application scenarios.
For the utilization of conditional information, we design a noise estimation module to mitigate the impact of the added noise on the spatiotemporal dependencies learning. The noise estimation module utilizes the extracted conditional feature, as the global context prior, to calculate the spatiotemporal attention weights, and predict the added Gaussian noise by spatiotemporal dependencies.
PriSTI performs well in spatiotemporal data scenarios with spatial similarity and feature correlation.
For three real-world datasets in the fields of air quality and traffic, our proposed method outperforms existing methods in various missing patterns. Moreover, PriSTI can support downstream tasks through imputation, and effectively handles the case of high missing rates and sensor failure.
Our contributions are summarized as follows:
\begin{itemize}
\item We propose PriSTI, a conditional diffusion framework for spatiotemporal imputation, which constructs and utilizes conditional information with spatiotemporal global correlations and geographic relationships.
\item To reduce the difficulty of learning spatiotemporal dependencies, we design a specialized noise prediction model that extracts conditional features from enhanced observations, calculating the spatiotemporal attention weights using the extracted global context prior.
\item Our proposed method achieves the best performance on spatiotemporal data in various fields, and effectively handles application scenarios such as high missing rates and sensor failure.
\end{itemize}
The rest of this paper is organized as follows. We first state the definition of the spatiotemporal imputation problem and briefly introduce the background of diffusion models in Section \ref{sec:problem_def}. Then we introduce how the diffusion models are applied to spatiotemporal imputation, as well as the details of our proposed framework in Section \ref{sec:method}. Next, we evaluate the performance of our proposed method in various missing patterns in Section \ref{sec:exp}. Finally, we review the related work for spatiotemporal imputation in Section \ref{sec:related_work} and conclude our work in Section \ref{sec:conclusion}.
\section{Preliminaries}\label{sec:problem_def}
In this section, we introduce some key definitions in spatiotemporal imputation, state the problem definition and briefly introduce the diffusion probabilistic models.
\textbf{Spatiotemporal data. }
We formalize spatiotemporal data as a sequence $X_{1:L}=\{X_1, X_2,\cdots, X_L\}\in \mathbb{R}^{N\times L}$ over consecutive time, where $X_l\in\mathbb{R}^N$ is the values observed at time $l$ by $N$ observation nodes, such as air monitoring stations and traffic sensors. Not all observation nodes have observed values at time $l$. We use a binary mask $M_l\in\{0,1\}^N$ to represent the observed mask at time $l$, where $m_l^{i,j}=1$ represents the value is observed while $m_l^{i,j}=0$ represents the value is missing.
Since there is no ground truth for real missing data in practice, we manually select the imputation target $\widetilde{X}\in \mathbb{R}^{N\times L}$ from available observed data for training and evaluation, and identify them with the binary mask $\widetilde{M}\in \mathbb{R}^{N\times L}$.
\textbf{Adjacency matrix. }
The observation nodes can be formalized as a graph $G=\langle V,E\rangle$, where $V$ is the node set and $E$ is the edge set measuring the pre-defined spatial relationship between nodes, such as geographical distance.
We denote $A\in \mathbb{R}^{N\times N}$ to represent the geographic information as the adjacency matrix of the graph $G$. In this work, we only consider the setting of static graph, i.e., the geographic information $A$ does not change over time.
\textbf{Problem statement. }
Given the incomplete observed spatiotemporal data $X$ and geographical information $A$, our task of spatiotemporal imputation is to estimate the missing values or corresponding distributions in spatiotemporal data $X_{1:L}$.
\textbf{Diffusion probabilistic models. }
Diffusion probabilistic models \cite{dickstein15, ho2020denoising} are deep generative models that have achieved cutting-edge results in the field of image synthesis \cite{rombach2022high}, audio generation \cite{kong2020diffwave}, etc., which generate samples consistent with the original data distribution by adding noise to the samples and learning the reverse denoising process.
The diffusion probabilistic model can be formalized as two Markov chain processes of length $T$, named the \textit{diffusion process} and the \textit{reverse process}.
Let $\widetilde{X}^0\sim p_{data}$ where $p_{data}$ is the clean data distribution, and $\widetilde{X}^t$ is the sampled latent variable sequence, where $t=1,\cdots,T$ is the diffusion step. $\widetilde{X}^T\sim \mathcal{N}(0, \bm{I})$ where $\mathcal{N}$ is Gaussian distribution. The diffusion process adds Gaussian noise gradually into $\widetilde{X}^0$ until $\widetilde{X}^0$ is close to $\widetilde{X}^T$, while the reverse process denoises $\widetilde{X}^t$ to recover $\widetilde{X}^0$.
More details about applying the diffusion models on spatiotemporal imputation are introduced in Section \ref{sec:method}.
\begin{table}[t]
\centering
\caption{Important notations and corresponding descriptions.}
\label{tab:notation}
\setlength{\tabcolsep}{1mm}
\resizebox{0.95\columnwidth}{!}{
\begin{tabular}{c|l}
\toprule
Notations & Descriptions\cr
\midrule
$\bm{X}$ & Spatiotemporal data \cr
$\bm{\widetilde{X}}$ & Manually selected imputation target \cr
$N$ & The number of the observation nodes \cr
$L, l$ & Length of the observed time and observed time step \cr
$T, t$ & Length of the diffusion steps and diffusion step \cr
$\bm{A}$ & Adjacency matrix of geographic information \cr
$\mathcal{X}$ & Interpolated conditional information \cr
$\beta_t, \alpha_t, \hat{\alpha}_t$ & Constant hyperparameters of diffusion model \cr
$\epsilon_{\theta}$ & Noise prediction model \cr
\bottomrule
\end{tabular}}
\end{table}
\section{Methodology}\label{sec:method}
The pipeline of our proposed spatiotemporal imputation framework, PriSTI, is shown in Figure \ref{fig:framework}. PriSTI adopts a conditional diffusion framework to exploit spatiotemporal global correlation and geographic relationship for imputation. To address the challenge about the construction and utilization of conditional information when impute by diffusion model, we design a specialized noise prediction model to enhance and extract the conditional feature.
In this section, we first introduce how the diffusion models are applied to spatiotemporal imputation, and then introduce the detail architecture of the noise prediction model, which is the key to success of diffusion models.
\subsection{Diffusion Model for Spatiotemporal Imputation}\label{sec:ddpm4imp}
To apply the diffusion models on spatiotemporal imputation, we regard the spatiotemporal imputation problem as a conditional generation task.
The previous studies \cite{tashiro2021csdi} have shown the ability of conditional diffusion probabilistic model for multivariate time series imputation. The spatiotemporal imputation task can be regarded as calculating the conditional probability distribution $q(\widetilde{X}_{1:L}^0|X_{1:L})$, where the imputation of $\widetilde{X}_{1:L}^0$ is conditioned by the observed values $X_{1:L}$.
However, the previous studies impute without considering the spatial relationships, and simply utilize the observed values as conditional information. In this section, we explain how our proposed framework impute spatiotemporal data by the diffusion models.
In the following discussion, we use the superscript $t\in\{0, 1, \cdots, T\}$ to represent the diffusion step, and omit the subscription $1:L$ for conciseness.
As mentioned in Section \ref{sec:problem_def}, the diffusion probabilistic model includes the \textit{diffusion process} and \textit{reverse process}.
The \textit{diffusion process} for spatiotemporal imputation is irrelavant with conditional information, adding Gaussian noise into original data of the imputation part, which is formalized as:
\begin{equation}
\begin{aligned}
& q(\widetilde{X}^{1:T}|\widetilde{X}^{0})=\prod_{t=1}^T q(\widetilde{X}^{t}|\widetilde{X}^{t-1}), \\
& q(\widetilde{X}^{t}|\widetilde{X}^{t-1})=\mathcal{N}(\widetilde{X}^t; \sqrt{1-\beta_t}\widetilde{X}^{t-1}, \beta_t \bm{I}),
\end{aligned}
\end{equation}
where $\beta_t$ is a small constant hyperparameter that controls the variance of the added noise.
The $\widetilde{X}^t$ is sampled by $\widetilde{X}^t=\sqrt{\bar{\alpha}_t}\widetilde{X}^0+\sqrt{1-\bar{\alpha}_t}\epsilon$, where $\alpha_t=1-\beta_t$, $\bar{\alpha}_t=\prod_{i=1}^t\alpha_i$, and $\epsilon$ is the sampled standard Gaussian noise. When $T$ is large enough, $q(\widetilde{X}^T|\widetilde{X}^0)$ is close to standard normal distribution .
The \textit{reverse process} for spatiotemporal imputation gradually convert random noise to missing values with spatiotemporal consistency based on conditional information. In this work, the reverse process is conditioned on the interpolated conditional information $\mathcal{X}$ that enhances the observed values, as well as the geographical information $A$. The reverse process can be formalized as:
\begin{equation}\label{eq:reverse_process}
\begin{aligned}
& p_{\theta}(\widetilde{X}^{0:T-1}|\widetilde{X}^{T}, \mathcal{X}, A)=\prod_{t=1}^T p_{\theta}(\widetilde{X}^{t-1}|\widetilde{X}^{t}, \mathcal{X}, A), \\
& p_{\theta}(\widetilde{X}^{t-1}|\widetilde{X}^{t}, \mathcal{X}, A)=\mathcal{N}(\widetilde{X}^{t-1}; \mu_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t), \sigma_t^2 \bm{I}).
\end{aligned}
\end{equation}
Ho et al. \cite{ho2020denoising} introduce an effective parameterization of $\mu_{\theta}$ and $\sigma_t^2$. In this work, they can be defined as:
\begin{equation}\label{eq:mu_sigma}
\begin{aligned}
& \mu_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t)=\frac{1}{\sqrt{\bar{\alpha}_t}}\left(\widetilde{X}^{t}-\frac{\beta_t}{\sqrt{1-\bar{\alpha}_t}}\epsilon_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t)\right), \\
& \sigma_t^2=\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_t}\beta_t,
\end{aligned}
\end{equation}
where $\epsilon_{\theta}$ is a neural network parameterized by $\theta$, which takes the noisy sample $\widetilde{X}^{t}$ and conditional information $\mathcal{X}$ and adjacency matrix $A$ as input, predicting the added noise $\epsilon$ on imputation target to restore the original information of the noisy sample.
Therefore, $\epsilon_{\theta}$ is often named \textit{noise prediction model}. The noise prediction model does not limit the network architecture, whose flexibility is benificial for us to design the model suitable for spatiotemporal imputation.
\begin{figure}
\caption{The pipeline of PriSTI. PriSTI takes observed values and geographic information as input. It first interpolates observations and models the global context prior by the conditional feature extraction module, and then utilizes the noise estimation module to predict noise with help of the conditional information.}
\label{fig:framework}
\end{figure}
\textbf{Training Process.}
During training, we mask the input observed value $X$ through a random mask strategy to obtain the imputation target $\widetilde{X}^t$, and the remaining observations are used to serve as the conditional information for imputation. Similar to CSDI \cite{tashiro2021csdi}, we provide the mask strategies including point strategy, block strategy and hybrid strategy (see more details in Section \ref{sec:exp_set}). Mask strategies produce different masks for each training sample.
After obtaining the training imputation target $\widetilde{X}^0$ and the interpolated conditional information $\mathcal{X}$, the training objective of spatiotemporal imputation is:
\begin{equation}\label{eq:loss}
\mathcal{L}(\theta)=\mathbb{E}_{\widetilde{X}^0\sim q(\widetilde{X}^0), \epsilon\sim\mathcal{N}(0,I)}\left\Vert\epsilon-\epsilon_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t)\right\Vert^2.
\end{equation}
Therefore, in each iteration of the training process, we sample the Gaussian noise $\epsilon$, the imputation target $\widetilde{X}^t$ and the diffusion step $t$, and obtain the interpolated conditional information $\mathcal{X}$ based on the remaining observations.
More details on the training process of our proposed framework is shown in Algorithm \ref{alg:train}.
\begin{algorithm}[t]
\caption{Training process of PriSTI.}
\label{alg:train}
\hspace*{0.02in} {\bf Input:}Incomplete observed data $X$, the adjacency matrix $A$, the number of iteration $N_{it}$, the number of diffusion steps $T$, noise levels sequence $\bar{\alpha}_t$. \\
\hspace*{0.02in} {\bf Output:}{Optimized noise prediction model $\epsilon_{\theta}$.}
\begin{algorithmic}[1]
\For {$i=1$ \text{to} $N_{it}$}
\State $\widetilde{X}^0 \gets \text{Mask}(X)$;
\State $\mathcal{X} \gets \text{Interpolate}(\widetilde{X}^0)$;
\State Sample $t \sim \text{Uniform}(\{1,\cdots,T\})$, $\epsilon\sim\mathcal{N}(0,\textbf{\text{I}})$;
\State $\widetilde{X}^t \gets \sqrt{\bar{\alpha}_t}\widetilde{X}^0+\sqrt{1-\bar{\alpha}_t}\epsilon$;
\State Updating the gradient $\nabla_{\theta}\left\Vert\epsilon-\epsilon_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t)\right\Vert^2$.
\EndFor
\end{algorithmic}
\end{algorithm}
\textbf{Imputation Process.}
When using the trained noise prediction model $\epsilon_{\theta}$ for imputation, the observed mask $\widetilde{M}$ of the data is available, so the imputation target $\widetilde{X}$ is the all missing values in the spatiotemporal data, and the interpolated conditional information $\mathcal{X}$ is constructed based on all observed values.
The model receives $\widetilde{X}^T$ and $\mathcal{X}$ as inputs and generates samples of the imputation results through the process in Equation (\ref{eq:reverse_process}).
The more details on the imputation process of our proposed framework is shown in Algorithm \ref{alg:impute}.
\begin{algorithm}[t]
\caption{Imputation process with PriSTI.}
\label{alg:impute}
\hspace*{0.02in} {\bf Input:}A sample of incomplete observed data $X$, the adjacency matrix $A$, the number of diffusion steps $T$, the optimized noise prediction model $\epsilon_{\theta}$.\\
\hspace*{0.02in} {\bf Output:}{Missing values of the imputation target $\widetilde{X}^0$.}
\begin{algorithmic}[1]
\State $\mathcal{X} \gets \text{Interpolate}(X)$;
\State Set $\widetilde{X}^T\sim\mathcal{N}(0, \textbf{\text{I}})$;
\For {$t=T$ \text{to} $1$}
\State $\mu_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t) \gets \frac{1}{\sqrt{\bar{\alpha}_t}}\left(\widetilde{X}^{t}-\frac{\beta_t}{\sqrt{1-\bar{\alpha}_t}}\epsilon_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t)\right)$
\State $\widetilde{X}^{t-1} \gets \mathcal{N}(\mu_{\theta}(\widetilde{X}^{t}, \mathcal{X}, A, t), \sigma_t^2 \bm{I})$
\EndFor
\end{algorithmic}
\end{algorithm}
Through the above framework, the diffusion model can be applied to spatiotemporal imputation with the conditional information. However, the construction and utilization of conditional information with spatiotemporal dependencies are still challenging. It is necessary to design a specialized noise prediction model $\epsilon_{\theta}$ to reduce the difficulty of learning spatiotemporal dependencies with noisy information, which will be introduced in next section.
\subsection{Design of Noise Prediction Model}
In this section, we illustrate how to design the noise prediction model $\epsilon_{\theta}$ for spatiotemporal imputation.
Specifically, we first interpolate the observed value to obtain the enhanced coarse conditional information. Then, a \textit{conditional feature extraction module} is designed to model the spatiotemporal correlation from the coarse interpolation information. The output of the conditional feature extraction module is utilized in the designed \textit{noise estimation module} to calculate the attention weights, which provides a better global context prior for spatiotemporal dependencies learning.
\subsubsection{Conditional Feature Extraction Module}
\begin{figure}
\caption{The architecture of the conditional feature extraction module (left) and noise estimation module (right). Both modules utilize the same components, including a temporal attention $\text{Attn}
\label{fig:STDL}
\end{figure}
The conditional feature extraction module is dedicated to model conditional information when the diffusion model is applied to spatiotemporal imputation.
According to the diffusion model for imputation described above, $\epsilon_{\theta}$ takes the conditional information $\mathcal{X}$ and the noisy information $\widetilde{X}^t$ as input. The previous studies, such as CSDI \cite{tashiro2021csdi}, regards the observed values as conditional information, and takes the concatenation of conditional information and perturbed values as the input, which are distinguished only by a binary mask.
However, the trend of the time series in imputation target are unstable due to the randomness of the perturbed values, which may cause the noisy sample to have an inconsistent trend with the original time series (such as the $\widetilde{X}_{1:L}^t$ in Figure \ref{fig:framework}), especially when the diffusion step $t$ is close to $T$.
Although CSDI utilizes two different Transformer layers to capture the temporal and feature dependencies, the mixture of conditional and noisy information increases the learning difficulty of the noise prediction model, which can not be solved by a simple binary mask identifier.
To address the above problem, we first enhance the obvserved values for the conditional feature extraction, expecting the designed model to learn the spatiotemporal dependencies based on this enhanced information.
In particular, inspired by some spatiotemporal forecasting works based on temporal continuity \cite{choi2022graph}, we apply linear interpolation to the time series of each node to initially construct a coarse yet effective interpolated conditional information $\mathcal{X}$ for denoising.
Intuitively, this interpolation does not introduce randomness to the time series, while also retaining a certain spatiotemporal consistency.
From the test results of linear interpolation on the air quality and traffic speed datasets (see Table \ref{tab:overallmae}), the spatiotemporal information completed by the linear interpolation method is available enough for a coarse conditional information.
Moreover, the fast computation of linear interpolation satisfies the training requirements of real-time construction under random mask strategies in our framework.
Although linear interpolation solves the completeness of observed information simply and efficiently, it only simply describes the linear uniform change in time, without modeling temporal nonlinear relationship and spatial correlations.
Therefore, we design a learnable module $\gamma(\cdot)$ to model a conditional feature $H^{pri}$ with spatiotemporal information as a global context prior, named Conditional Feature Extraction Module.
The module $\gamma(\cdot)$ takes the interpolated conditional information $\mathcal{X}$ and adjacency matrix $A$ as input, extract the spatiotemporal dependencies from $\mathcal{X}$ and output $H^{pri}$ as the global context for the calculation of spatiotemporal attention weights in noise prediction.
In particular, the conditional feature $H^{pri}\in\mathbb{R}^{N\times L\times d}$ is obtained by $H^{pri}=\gamma(\mathcal{H}, A)$, where $\mathcal{H}=\text{Conv}(\mathcal{X})$ and $\text{Conv}(\cdot)$ is $1\times1$ convolution, $\mathcal{H}\in \mathbb{R}^{N\times L\times d}$, and $d$ is the channel size.
The conditional feature extraction module $\gamma(\cdot)$ comprehensively combines the spatiotemporal global correlations and geographic dependency, as shown in the left of Figure \ref{fig:STDL}, which is formalized as:
\begin{equation}\label{eq:gps}
\begin{aligned}
& H^{pri} = \gamma(\mathcal{H}, A)=\text{MLP}(\varphi_{\text{SA}}(\mathcal{H})+\varphi_{\text{TA}}(\mathcal{H})+\varphi_{\text{MP}}(\mathcal{H}, A)),\\
& \varphi_{\text{SA}}(\mathcal{H})=\text{Norm}(\text{Attn}_{spa}(\mathcal{H})+\mathcal{H}),\\
& \varphi_{\text{TA}}(\mathcal{H})=\text{Norm}(\text{Attn}_{tem}(\mathcal{H})+\mathcal{H}),\\
& \varphi_{\text{MP}}(\mathcal{H}, A)=\text{Norm}(\text{MPNN}(\mathcal{H}, A)+\mathcal{H}),\\
\end{aligned}
\end{equation}
where $\text{Attn}(\cdot)$ represents the global attention, and the subscripts $spa$ and $tem$ represent spatial attention and temporal attention respectively. We use the dot-product multi-head self-attention in Transformer \cite{vaswani2017attention} to implement $\text{Attn}(\cdot)$. And $\text{MPNN}(\cdot)$ represents the spatial message passing neural network, which can be implemented by any graph neural network. We adopt the graph convolution module from Graph Wavenet \cite{wu2019graph}, whose adjacency matrix includes a bidirectional distance-based matrix and an adaptively learnable matrix.
The extracted conditional feature $H^{pri}$ solves the problem of constructing conditional information. It does not contain the added Gaussian noise, and includes temporal dependencies, spatial global correlations and geographic dependencies compared with observed values. To address the remaining challenge of utilizing conditional information, $H^{pri}$ serves as a coarse prior to guide the learning of spatiotemporal dependencies, which is introduced in the next section.
\subsubsection{Noise Estimation Module}
The noise estimation module is dedicated to the utilization of conditional information when the diffusion model is applied to spatiotemporal imputation.
Since the information in noisy sample may have a wide deviation from the real spatiotemporal distribution because of the randomness of the Gaussian noise, it is difficult to learn spatiotemporal dependencies directly from the mixture of conditional and noisy information.
Our proposed noise estimation module captures spatiotemporal global correlations and geographical relationships by a specialized attention mechanism, which reduce the difficulty of spatiotemporal dependencies learning caused by the sampled noise.
Specifically, the inputs of the noise estimation module include two parts: the noisy information $H^{in}=\text{Conv}(\mathcal{X} || \widetilde{X}^t)$ that consists of interpolation information $\mathcal{X}$ and noise sample $\widetilde{X}^t$, and the prior information including the conditional feature $H^{pri}$ and adjacency matrix $A$.
To comprehensively consider the spatiotemporal global correlation and geographic relationship of the missing data, the temporal features $H^{tem}$ are first learned through a temporal dependency learning module $\gamma_{\mathcal{T}}(\cdot)$, and then the temporal features are aggregated through a spatial dependency learning module $\gamma_{\mathcal{S}}(\cdot)$.
The architecture of the noise estimation module is shown as the right of Figure \ref{fig:STDL}, which is formalized as follows:
\begin{equation}\label{eq:gps_deep}
\begin{aligned}
& H^{tem}=\gamma_{\mathcal{T}}(H^{in})=\text{Attn}_{tem}(H^{in}), \\
& H^{spa}=\gamma_{\mathcal{S}}(H^{tem}, A)=\text{MLP}(\varphi_{\text{SA}}(H^{tem})+\varphi_{\text{MP}}(H^{tem}, A)),
\end{aligned}
\end{equation}
where $\text{Attn}_{tem}(\cdot)$, $\varphi_{\text{SA}}(\cdot)$ and $\varphi_{\text{MP}}(\cdot)$ are same as the components in Equation (\ref{eq:gps}), which are used to capture spatiotemporal global attention and geographic similarity, and $H^{tem}, H^{spa}\in \mathbb{R}^{N \times L \times d}$ are the outputs of temporal and spatial dependencies learning modules.
However, in Eq. (\ref{eq:gps_deep}), spatiotemporal dependencies learning is performed on the mixture of conditional noisy information, i.e. $H^{in}$. When the diffusion step $t$ approaches $T$, noise sample $\widetilde{X}^t$ would increase the difficulty of spatiotemporal dependencies learning. To reduce the impact of $\widetilde{X}^t$ while convert it into Gaussian noise,
we change the input of the attention components $\text{Attn}_{tem}(\cdot)$ and $\text{Attn}_{spa}(\cdot)$, which calculate the attention weights by using the conditional feature $H^{pri}$.
In particular, take temporal attention $\text{Attn}_{tem}(\cdot)$ as an example, we rewrite the dot-product attention $\text{Attn}_{tem}(Q_{\mathcal{T}},K_{\mathcal{T}},V_{\mathcal{T}})=\text{softmax}(\frac{Q_{\mathcal{T}}K_{\mathcal{T}}^T}{\sqrt{d}})\cdot V_{\mathcal{T}}$ as $\text{Attn}_{tem}(\mathcal{A}_{\mathcal{T}}, V_{\mathcal{T}}) = \mathcal{A}_{\mathcal{T}} \cdot V_{\mathcal{T}}$, where $\mathcal{A}_{\mathcal{T}}=\text{softmax}(\frac{Q_{\mathcal{T}}K_{\mathcal{T}}^T}{\sqrt{d}})$ is the attention weight.
We calculate the attention weight $\mathcal{A}_{\mathcal{T}}$ by the conditional feature $H^{pri}$, i.e., we set the input $Q_{\mathcal{T}}$, $K_{\mathcal{T}}$ and $V_{\mathcal{T}}$ as:
\begin{equation}\label{eq:cross_att}
Q_{\mathcal{T}}=H^{pri}\cdot W^Q_{\mathcal{T}}, K_{\mathcal{T}}=H^{pri}\cdot W^K_{\mathcal{T}}, V_{\mathcal{T}}=H^{in}\cdot W^V_{\mathcal{T}},
\end{equation}
where $W^Q_{\mathcal{T}}, W^K_{\mathcal{T}}, W^V_{\mathcal{T}}\in\mathbb{R}^{d\times d}$ are learnable projection parameters.
The spatial attention $\text{Attn}_{spa}(\mathcal{A}_{\mathcal{S}}, V_{\mathcal{S}})$ calculates the attention weight in the same way:
\begin{equation}\label{eq:cross_att_spa}
Q_{\mathcal{S}}=H^{pri}\cdot W^Q_{\mathcal{S}}, K_{\mathcal{S}}=H^{pri}\cdot W^K_{\mathcal{S}}, V_{\mathcal{S}}=H^{tem}\cdot W^V_{\mathcal{S}}.
\end{equation}
The noise estimation module consists of Equation (\ref{eq:gps_deep}) - (\ref{eq:cross_att_spa}), which has the same attention and MPNN components as the conditional feature extraction module with different input and architecture.
The conditional feature extraction module models spatiotemporal dependencies only from the interpolated conditional information $\mathcal{X}$ in a single layer, so it extracts information through a wide network architecture, i.e., directly aggregates the spatiotemporal global correlation and geographic dependency.
Since the noise estimation module needs to convert the noisy sample to standard Gaussian distribution in multiple layers, it learns spatiotemporal dependencies from the noisy samples with help of the conditional feature through a deep network architecture, i.e., extracts the temporal correlation first and aggregates the temporal feature through the spatial global correlation and geographic information.
In addition, when the number of nodes in the spatiotemporal data is large, the computational cost of spatial global attention is high, and the time complexity of its similarity calculation and weighted summation are both $O(N^2d)$.
Therefore, we map $N$ nodes to $k$ virtual nodes, where $k<N$.
We rewrite the $K_{\mathcal{S}}$ and $V_{\mathcal{S}}$ in Equation (\ref{eq:cross_att_spa}) when attention is used for spatial dependencies learning as:
\begin{equation}\label{eq:node_samp}
K_{\mathcal{S}}=H^{pri}\cdot P^K_{\mathcal{S}} W^K_{\mathcal{S}} , V_{\mathcal{S}}= H^{tem}\cdot P^V_{\mathcal{S}} W^V_{\mathcal{S}},
\end{equation}
where $P^K_{\mathcal{S}}, P^V_{\mathcal{S}}\in\mathbb{R}^{N\times d}$ is the downsampling parameters. And the time complexity of the modified spatial attention is reduced to $O(Nkd)$.
\subsubsection{Auxiliary Information and Output}
We add auxiliary information $U=\text{MLP}(U_{tem}, U_{spa})$ to both the conditional feature extraction module and the noise estimation module to help the imputation, where $U_{tem}$ is the sine-cosine temporal encoding \cite{vaswani2017attention}, and $U_{spa}$ is learnable node embedding.
We expand and concatenate $U_{tem}\in\mathbb{R}^{L\times 128}$ and $U_{spa}\in\mathbb{R}^{N\times 16}$, and obtain auxiliary information $U\in\mathbb{R}^{N\times L\times d}$ that can be input to the model through an MLP layer.
The noise estimation module stacks multiple layers, and the output $H^{spa}$ of each layer is divided into residual connection and skip connection after a gated activation unit. The residual connection is used as the input of the next layer, and the skip connections of each layer are added and through two layers of $1\times 1$ convolution to obtain the output of the noise prediction model $\epsilon_{\theta}$. The output only retains the value of imputation target, and the loss is calculated by Equation (\ref{eq:loss}).
\section{Experiments}\label{sec:exp}
In this section, we first introduce the dataset, baselines, evaluation metrics and settings of our experiment. Then, we evaluate our proposed framework PriSTI with a large amount of experiments for spatiotemporal imputation to answer the following research questions:
\begin{itemize}
\item \textbf{RQ1}: Can PriSTI provide superior imputation performance in various missing patterns compared to several state-of-the-art baselines?
\item \textbf{RQ2}: How is the imputation performance for PriSTI for different missing rate of spatiotemporal data?
\item \textbf{RQ3}: Does PriSTI benefit from the construction and utilization of the conditional information?
\item \textbf{RQ4}: Does PriSTI extract the temporal and spatial dependencies from the observed spatiotemporal data?
\item \textbf{RQ5}: Can PriSTI impute the time series for the unobserved sensors only based on the geographic location?
\end{itemize}
\subsection{Dataset}
We conduct experiments on three real-world datasets: an air quality dataset AQI-36, and two traffic speed datasets METR-LA and PEMS-BAY.
AQI-36 \cite{yi2016st} contains hourly sampled PM2.5 observations from 36 stations in Beijing, covering a total of 12 months. METR-LA \cite{li2017diffusion} contains traffic speed collected by 207 sensors in the highway of Los Angeles County \cite{jagadish2014big} in 4 months, and PEMS-BAY \cite{li2017diffusion} contains traffic speed collected by 325 sensors on highways in the San Francisco Bay Area in 6 months. Both traffic datasets are sampled every 5 minutes.
For the geographic information, the adjacency matrix is obtained based on the geographic distances between monitoring stations or sensors followed the previous works \cite{li2017diffusion}. We build the adjacency matrix for the three datasets using thresholded Gaussian kernel \cite{shuman2013emerging}.
\subsection{Baselines}
To evaluate the performance of our proposed method, we compare with classic models and state-of-the-art methods for spatiotemporal imputation. The baselines include statistic methods (MEAN, DA, KNN, Lin-ITP), classic machine learning methods (MICE, VAR, Kalman), low-rank matrix factorization methods (TRMF, BATF), deep autoregressive methods (BRITS, GRIN) and deep generative methods (V-RIN, GP-VAE, rGAIN, CSDI).
We briefly introduce the baseline methods as follows:
(1)\textbf{MEAN}: directly use the historical average value of each node to impute.
(2)\textbf{DA}: impute missing values with the daily average of corresponding time steps.
(3)\textbf{KNN}: calculate the average value of nearby nodes based on geographic distance to impute.
(4)\textbf{Lin-ITP}: linear interpolation of the time series for each node, as implemented by torchcde\footnote{https://github.com/patrick-kidger/torchcde}.
(5)\textbf{KF}: use Kalman Filter to impute the time series for each node, as implemented by filterpy\footnote{https://github.com/rlabbe/filterpy}.
(6)\textbf{MICE} \cite{white2011multiple}: multiple imputation method by chain equations;
(7)\textbf{VAR}: vector autoregressive single-step predictor.
(8)\textbf{TRMF} \cite{yu2016temporal}: a temporal regularized matrix factorization method.
(9)\textbf{BATF} \cite{chen2019missing}: a Bayesian augmented tensor factorization model, which incorporates the generic forms of spatiotemporal domain knowledge. We implement TRMF and BATF using the code in the Transdim\footnote{https://github.com/xinychen/transdim} repository. The rank is set to 10, 40 and 50 on AQI-36, METR-LA and PEMS-BAY, respectively.
(10)\textbf{V-RIN} \cite{mulyadi2021uncertainty}: a method to improve deterministic imputation using the quantified uncertainty of VAE, whose probability imputation result is provided by the quantified uncertainty.
(11)\textbf{GP-VAE} \cite{fortuin2020gp}: a method for time series probabilistic imputation by combining VAE with Gaussian process.
(12)\textbf{rGAIN}: GAIN \cite{yoon2018gain} with a bidirectional recurrent encoder-decoder, which is a GAN-based method.
(13)\textbf{BRITS} \cite{cao2018brits}: a multivariate time series imputation method based on bidirectional RNN.
(14)\textbf{GRIN} \cite{cini2021filling}: a bidirectional GRU based method with graph neural network for multivariate time series imputation.
(15)\textbf{CSDI} \cite{tashiro2021csdi}: a probability imputation method based on conditional diffusion probability model, which treats different nodes as multiple features of the time series, and using Transformer to capture feature dependencies.
In the experiment, the baselines MEAN, KNN, MICE, VAR, rGAIN, BRITS and GRIN are implemented by the code\footnote{https://github.com/Graph-Machine-Learning-Group/grin} provided by the authors of GRIN \cite{cini2021filling}.
We reproduced these baselines, and the results are consistent with their claims, so we retained the results claimed in GRIN for the above baselines.
The implementation details of the remaining baselines have been introduced as above.
\subsection{Evaluation metrics}
We apply three evaluation metrics to measure the performance of spatiotemporal imputation: Mean Absolute Error (MAE), Mean Squared Error (MSE) and Continuous Ranked Probability Score (CRPS) \cite{matheson1976scoring}.
MAE and MSE reflect the absolute error between the imputation values and the ground truth,
and CRPS evaluates the compatibility of the estimated probability distribution with the observed value.
We introduce the calculation details of CRPS as follows.
For a missing value $x$ whose estimated probability distribution is $D$, CRPS measures the compatibility of $D$ and $x$, which can be defined as the integral of the quantile loss $\Lambda_{\alpha}$:
\begin{equation}
\begin{aligned}
\text{CRPS}(D^{-1},x) & =\int^1_0 2\Lambda_{\alpha}(D^{-1}(\alpha), x)d\alpha,\\
\Lambda_{\alpha}(D^{-1}(\alpha), x) & =(\alpha-\mathbb{I}_{x<D^{-1}(\alpha)})(x-D^{-1}(\alpha)),
\end{aligned}
\end{equation}
where $\alpha\in[0,1]$ is the quantile levels, $D^{-1}(\alpha)$ is the $\alpha$-quantile of distribution $D$, $\mathbb{I}$ is the indicator function.
Since our distribution of missing values is approximated by generating 100 samples, we compute quantile losses for discretized quantile levels with 0.05 ticks following \cite{tashiro2021csdi} as:
\begin{equation}
\text{CRPS}(D^{-1},x) \simeq \sum_{i=1}^{19}2\Lambda_{i\times 0.05}(D^{-1}(i\times 0.05), x)/19.
\end{equation}
We compute CRPS for each estimated missing value and use the average as the evaluation metric, which is formalized as:
\begin{equation}
\text{CRPS}(D, \widetilde{X})=\frac{\sum_{\tilde{x}\in\widetilde{X}}\text{CRPS}(D^{-1},\tilde{x})}{|\widetilde{X}|}.
\end{equation}
\subsection{Experimental settings}\label{sec:exp_set}
\textbf{Dataset.}
We divide training/validation/test set following the settings of previous work \cite{yi2016st, cini2021filling}.
For AQI-36, we select Mar., Jun., Sep., and Dec. as the test set, the last 10\% of the data in Feb., May, Aug., and Nov. as the validation set, and the remaining data as the training set.
For METR-LA and PEMS-BAY, we split the training/validation/test set by $70\%/10\%/20\%$.
\textbf{Imputation target.}
For air quality dataset AQI-36, we adapt the same evaluation strategy as the previous work provided by \cite{yi2016st}, which simulates the distribution of real missing data.
For the traffic datasets METR-LA and PEMS-BAY, we use the artificially injected missing strategy provided by \cite{cini2021filling} for evaluation, as shown in Figure \ref{fig:bp-missing}, which includes two missing patterns:
(1) \textbf{Block missing}: based on randomly masking 5\% of the observed data, mask observations ranging from 1 to 4 hours for each sensor with 0.15\% probability;
(2) \textbf{Point missing}: randomly mask 25\% of observations.
The missing rate of each dataset under different missing patterns has been marked in Table \ref{tab:overallmae}. It is worth noting that in addition to the manually injected faults, each dataset also has original missing data (13.24\% in AQI-36, 8.10\% in METR-LA and 0.02\% in PEMS-BAY). All evaluations are performed only on the manually masked parts of the test set.
\begin{figure}
\caption{The illustration of some missing patterns.}
\label{fig:bp-missing}
\end{figure}
\textbf{Training strategies.}
As mentioned in Section \ref{sec:ddpm4imp}, on the premise of known missing patterns in test data, we provide three mask strategies.
The details of these mask strategies are described as follows:
\begin{itemize}
\item Point strategy: take a random value $m$ between [0, 100], randomly select $m$\% of the data from $X$ as the imputation target $\widetilde{X}$, and the remaining unselected data is regarded as observed values in training process.
\item Block strategy: For each node, a sequence with a length in the range $[L/2, L]$ is selected as the imputation target with a probability from 0 to 15\%. In addition, 5\% of the observed values are randomly selected and added to the imputation target.
\item Hybrid strategy: For each training sample $X$, it has a 50\% probability to be masked by the point strategy, and the other 50\% probability to be masked by the block strategy or a historical missing pattern i.e., the missing patterns of other samples in the training set.
\end{itemize}
We utilize different mask strategies for various missing patterns and datasets to make the model simulate the corresponding missing patterns as much as possible during training. Since AQI-36 has much original missing in the training set, which is fewer in traffic datasets, during the training process of PriSTI, we adopt hybrid strategy with historical missing pattern on AQI-36, hybrid strategy with block strategy on block-missing of traffic datasets, and point strategy on point-missing of traffic datasets.
\textbf{Hyperparameters of PriSTI.} For the hyperparameters of PriSTI, the batch size is 16. The learning rate is decayed to 0.0001 at 75\% of the total epochs, and decayed to 0.00001 at 90\% of the total epochs. The hyperparameter for diffusion model include a minimum noise level $\beta_1$ and a maximum noise level $\beta_T$. We adopted the quadratic schedule for other noise levels following \cite{tashiro2021csdi}, which is formalized as:
\begin{equation}
\beta_t=\left(\frac{T-t}{T-1}\sqrt{\beta_1}+\frac{t-1}{T-1}\sqrt{\beta_T}\right)^2.
\end{equation}
The diffusion time embedding and temporal encoding are implemented by the sine and cosine embeddings followed previous works \cite{kong2020diffwave,tashiro2021csdi}.
We summary the hyperparameters of PriSTI in Table \ref{tab:exp_setting}. All the experiments are run for 5 times.
\begin{table}[t]
\centering
\caption{The hyperparameters of PriSTI for all datasets.}
\label{tab:exp_setting}
\setlength{\tabcolsep}{1mm}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{cccc}
\toprule
Description & AQI-36 & METR-LA& PEMS-BAY\cr
\midrule
Batch size & 16 & 16 & 16 \cr
Time length $L$ & 36 & 24 & 24 \cr
Epochs & 200 & 300 & 300 \cr
Learning rate & 0.001 & 0.001 & 0.001 \cr
Layers of noise estimation & 4 & 4 & 4 \cr
Channel size $d$ & 64 & 64 & 64 \cr
Number of attention heads & 8 & 8 & 8 \cr
Minimum noise level $\beta_1$ & 0.0001 & 0.0001 & 0.0001 \cr
Maximum noise level $\beta_T$ & 0.2 & 0.2 & 0.2 \cr
Diffusion steps $T$ & 100 & 50 & 50 \cr
Number of virtual nodes $k$ & 16 & 64 & 64 \cr
\bottomrule
\end{tabular}}
\end{table}
\begin{table*}[ht]
\centering
\caption{The results of MAE and MSE for spatiotemporal imputation.}
\label{tab:overallmae}
\resizebox{0.95\textwidth}{!}{
\setlength{\tabcolsep}{1mm}{
\renewcommand{1}{1}
\begin{tabular}{ccccccccccc}
\toprule
\multirow{3}{*}{Method}&
\multicolumn{2}{c}{AQI-36}& \multicolumn{4}{c}{METR-LA}& \multicolumn{4}{c}{PEMS-BAY}\cr
\cmidrule(lr){2-3} \cmidrule(lr){4-7} \cmidrule(lr){8-11}
& \multicolumn{2}{c}{Simulated failure (24.6\%)}& \multicolumn{2}{c}{Block-missing (16.6\%)}& \multicolumn{2}{c}{Point-missing (31.1\%)}& \multicolumn{2}{c}{Block-missing (9.2\%)}& \multicolumn{2}{c}{Point-missing (25.0\%)}\cr
\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(lr){6-7} \cmidrule(lr){8-9} \cmidrule(lr){10-11}
& MAE & MSE & MAE & MSE & MAE & MSE & MAE & MSE & MAE & MSE \cr
\midrule
Mean & 53.48$\pm$0.00 & 4578.08$\pm$0.00 & 7.48$\pm$0.00 & 139.54$\pm$0.00 & 7.56$\pm$0.00 & 142.22$\pm$0.00 & 5.46$\pm$0.00 & 87.56$\pm$0.00 & 5.42$\pm$0.00 & 86.59$\pm$0.00 \cr
DA & 50.51$\pm$0.00 & 4416.10$\pm$0.00 & 14.53$\pm$0.00 & 445.08$\pm$0.00 & 14.57$\pm$0.00 & 448.66$\pm$0.00 & 3.30$\pm$0.00 & 43.76$\pm$0.00 & 3.35$\pm$0.00 & 44.50$\pm$0.00 \cr
KNN & 30.21$\pm$0.00 & 2892.31$\pm$0.00 & 7.79$\pm$0.00 & 124.61$\pm$0.00 & 7.88$\pm$0.00 & 129.29$\pm$0.00 & 4.30$\pm$0.00 & 49.90$\pm$0.00 & 4.30$\pm$0.00 & 49.80$\pm$0.00 \cr
Lin-ITP & 14.46$\pm$0.00 & 673.92$\pm$0.00 & 3.26$\pm$0.00 & 33.76$\pm$0.00 & 2.43$\pm$0.00 & 14.75$\pm$0.00 & 1.54$\pm$0.00 & 14.14$\pm$0.00 & 0.76$\pm$0.00 & 1.74$\pm$0.00 \cr
\midrule
KF & 54.09$\pm$0.00 & 4942.26$\pm$0.00 & 16.75$\pm$0.00 & 534.69$\pm$0.00 & 16.66$\pm$0.00 & 529.96$\pm$0.00 & 5.64$\pm$0.00 & 93.19$\pm$0.00 & 5.68$\pm$0.00 & 93.32$\pm$0.00 \cr
MICE & 30.37$\pm$0.09 & 2594.06$\pm$7.17 & 4.22$\pm$0.05 & 51.07$\pm$1.25 & 4.42$\pm$0.07 & 55.07$\pm$1.46 & 2.94$\pm$0.02 & 28.28$\pm$0.37 & 3.09$\pm$0.02 & 31.43$\pm$0.41 \cr
VAR & 15.64$\pm$0.08 & 833.46$\pm$13.85 & 3.11$\pm$0.08 & 28.00$\pm$0.76 & 2.69$\pm$0.00 & 21.10$\pm$0.02 & 2.09$\pm$0.10 & 16.06$\pm$0.73 & 1.30$\pm$0.00 & 6.52$\pm$0.01 \cr
TRMF & 15.46$\pm$0.06 & 1379.05$\pm$34.83 & 2.96$\pm$0.00 & 22.65$\pm$0.13 & 2.86$\pm$0.00 & 20.39$\pm$0.02 & 1.95$\pm$0.01 & 11.21$\pm$0.06 & 1.85$\pm$0.00 & 10.03$\pm$0.00 \cr
BATF & 15.21$\pm$0.27 & 662.87$\pm$29.55 & 3.56$\pm$0.01 & 35.39$\pm$0.03 & 3.58$\pm$0.01 & 36.05$\pm$0.02 & 2.05$\pm$0.00 & 14.48$\pm$0.01 & 2.05$\pm$0.00 & 14.90$\pm$0.06 \cr
\midrule
V-RIN & 10.00$\pm$0.10 & 838.05$\pm$24.74 & 6.84$\pm$0.17 & 150.08$\pm$6.13 & 3.96$\pm$0.08 & 49.98$\pm$1.30 & 2.49$\pm$0.04 & 36.12$\pm$0.66 & 1.21$\pm$0.03 & 6.08$\pm$0.29 \cr
GP-VAE & 25.71$\pm$0.30 & 2589.53$\pm$59.14 & 6.55$\pm$0.09 & 122.33$\pm$2.05 & 6.57$\pm$0.10 & 127.26$\pm$3.97 & 2.86$\pm$0.15 & 26.80$\pm$2.10 & 3.41$\pm$0.23 & 38.95$\pm$4.16 \cr
rGAIN & 15.37$\pm$0.26 & 641.92$\pm$33.89 & 2.90$\pm$0.01 & 21.67$\pm$0.15 & 2.83$\pm$0.01 & 20.03$\pm$0.09 & 2.18$\pm$0.01 & 13.96$\pm$0.20 & 1.88$\pm$0.02 & 10.37$\pm$0.20 \cr
BRITS & 14.50$\pm$0.35 & 622.36$\pm$65.16 & 2.34$\pm$0.01 & 17.00$\pm$0.14 & 2.34$\pm$0.00 & 16.46$\pm$0.05 & 1.70$\pm$0.01 & 10.50$\pm$0.07 & 1.47$\pm$0.00 & 7.94$\pm$0.03 \cr
GRIN & 12.08$\pm$0.47 & 523.14$\pm$57.17 & 2.03$\pm$0.00 & 13.26$\pm$0.05 & 1.91$\pm$0.00 & 10.41$\pm$0.03 & 1.14$\pm$0.01 & 6.60$\pm$0.10 & 0.67$\pm$0.00 & 1.55$\pm$0.01 \cr
CSDI & 9.51$\pm$0.10 & 352.46$\pm$7.50 & 1.98$\pm$0.00 & 12.62$\pm$0.60 & 1.79$\pm$0.00 & 8.96$\pm$0.08 & 0.86$\pm$0.00 & 4.39$\pm$0.02 & 0.57$\pm$0.00 & 1.12$\pm$0.03 \cr
\midrule
PriSTI & \textbf{9.03$\pm$0.07} & \textbf{310.39$\pm$7.03} & \textbf{1.86$\pm$0.00} & \textbf{10.70$\pm$0.02} & \textbf{1.72$\pm$0.00} & \textbf{8.24$\pm$0.05} & \textbf{0.78$\pm$0.00} & \textbf{3.31$\pm$0.01} & \textbf{0.55$\pm$0.00} & \textbf{1.03$\pm$0.00} \cr
\bottomrule
\end{tabular}}}
\end{table*}
\begin{table}[t]
\centering
\caption{The results of CRPS for spatiotemporal imputation.}
\label{tab:pro_est}
\renewcommand{1}{1}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{cccccc}
\toprule
\multirow{2}{*}{Method}&
AQI-36& \multicolumn{2}{c}{METR-LA}& \multicolumn{2}{c}{PEMS-BAY}\cr
\cmidrule(lr){2-2} \cmidrule(lr){3-4} \cmidrule(lr){5-6}
& {SF}& {Block}& {Point}& {Block}& {Point}\cr
\midrule
V-RIN & 0.3154 & 0.1283 & 0.0781 & 0.0394 & 0.0191 \cr
GP-VAE & 0.3377 & 0.1118 & 0.0977 & 0.0436 & 0.0568 \cr
CSDI & 0.1056 & 0.0260 & 0.0235 & 0.0127 & 0.0067 \cr
\midrule
PriSTI & \textbf{0.0997} & \textbf{0.0244} & \textbf{0.0227} & \textbf{0.0093} & \textbf{0.0064} \cr
\bottomrule
\end{tabular}}
\end{table}
\begin{table}[t]
\centering
\caption{The prediction on AQI-36 after imputation.}
\label{tab:prediction}
\renewcommand{1}{1}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Metric & Ori. & BRITS & GRIN & CSDI & PriSTI \cr
\midrule
MAE & 36.97 & 34.61 & 33.77 & 30.20 & \textbf{29.34} \cr
RMSE &60.37 & 56.66 & 54.06 & 46.98 & \textbf{45.08} \cr
\bottomrule
\end{tabular}}
\end{table}
\subsection{Results}
\subsubsection{Overall Performance (RQ1)}
We first evaluate the spatiotemporal imputation performance of PriSTI compared with other baselines.
Since not all methods can provide the probability distribution of missing values, i.e. evaluated by the CRPS metric, we show the deterministic imputation result evaluated by MAE and MSE in Table \ref{tab:overallmae}, and select V-RIN, GP-VAE, CSDI and PriSTI to be evaluated by CRPS, which is shown in Table \ref{tab:pro_est}.
The probability distribution of missing values of these four methods are simulated by generating 100 samples, while their deterministic imputation result is the median of all generated samples.
Since CRPS fluctuates less across 5 times experiments (the standard error is less than 0.001 for CSDI and PriSTI), we only show the mean of 5 times experiments in Table \ref{tab:pro_est}.
It can be seen from Table \ref{tab:overallmae} and Table \ref{tab:pro_est} that our proposed method outperforms other baselines on various missing patterns in different datasets. We summarize our findings as follows:
(1) The statistic methods and classic machine learning methods performs poor on all the datasets. These methods impute missing values based on assumptions such as stability or seasonality of time series, which can not cover the complex temporal and spatial correlations in real-world datasets.
The matrix factorization methods also perform not well due to the low-rank assumption of data.
(2) Among deep learning methods, GRIN, as an autoregressive state-of-the-art multivariate time series imputation method, performs better than other RNN-based methods (rGAIN and BRITS) due to the extraction of spatial correlations. However, the performance of GRIN still has a gap compared to the diffusion model-based methods (CSDI), which may be caused by the inherent defect of error accumulation in autoregressive models.
(3) For deep generative models, the VAE-based methods (V-RIN and GP-VAE) can not outperform CSDI and PriSTI. Our proposed method PriSTI outperforms CSDI in various missing patterns of every datasets, which indicates that our design of conditional information construction and spatiotemporal correlation can improve the performance of diffusion model for imputation task.
In addition, for the traffic datasets, we find that our method has a more obvious improvement than CSDI in the block-missing pattern compared with point-missing, which indicates that the interpolated information may provide more effective conditional information than observed values especially when the missing is continuous at time.
In addition, we select the methods of the top 4 performance rankings (i.e., the average ranking of MAE and MSE) in Table \ref{tab:overallmae} (PriSTI, CSDI, GRIN and BRITS) to impute all the data in AQI-36, and then use the classic spatiotemporal forecasting method Graph Wavenet \cite{wu2019graph} to make predictions on the imputed dataset. We divide the imputed dataset into training, valid and test sets as 70\%/10\%/20\%, and use the data of the past 12 time steps to predict the next 12 time steps. We use the MAE and RMSE (the square root of MSE) for evaluation. The prediction results is shown in Table \ref{tab:prediction}. Ori. represents the raw data without imputation.
The results in Table \ref{tab:prediction} indicate that the prediction performance is affected by the data integrity, and the prediction performance on the data imputed by different methods also conforms to the imputation performance of these methods in Table \ref{tab:overallmae}. This demonstrates that our method can also help the downstream tasks after imputation.
\subsubsection{Sensitivity analysis (RQ2)}
It is obvious that the performance of model imputation is greatly affected by the distribution and quantity of observed data. For spatiotemporal imputation, sparse and continuously missing data is not conducive to the model learning the spatiotemporal correlation.
To test the imputation performance of PriSTI when the data is extremely sparse, we evaluate the imputation ability of PriSTI in the case of 10\%-90\% missing rate compared with the three baselines with the best imputation performance (BRITS, GRIN and CSDI).
We evaluate in the block-missing and point-missing patterns of METR-LA, respectively. To simulate the sparser data in different missing patterns, for the block missing pattern, we increase the probability of consecutive missing whose lengths in the range $[12, 48]$; for the point missing pattern, we randomly drop the observed values according to the missing rate.
We train one model for each method, and use the trained model to test with different missing rates for different missing patterns. For BRITS and GRIN, their models are trained on data that is randomly masked by 50\% with the corresponding missing pattern. For CSDI and PriSTI, their models are trained with the mask strategies consistent with the original experimental settings.
The MAE of each method under different missing rates are shown in Figure \ref{fig:sensitivity_analysis}. When the missing rate of METR-LA reaches 90\%, PriSTI improves the MAE performance of other methods by 4.67\%-34.11\% in block-missing pattern and 3.89\%-43.99\% in point-missing pattern.
The result indicates that our method still has better imputation performance than other baselines at high missing rate, and has a greater improvement than other methods when data is sparser.
We believe that this is due to the interpolated conditional information we construct retains the spatiotemporal dependencies that are more in line with the real distribution than the added Gaussian noise as much as possible when the data is highly sparse.
\begin{figure}
\caption{The imputation results of different missing rates.}
\label{fig:sens_a}
\label{fig:sens_b}
\label{fig:sensitivity_analysis}
\end{figure}
\subsubsection{Ablation study (RQ3 and RQ4)}
We design the ablation study to evaluate the effectiveness of the conditional feature extraction module and noise estimation module. We compare our method with the following variants:
\begin{itemize}
\item \textit{mix-STI}: the input of noise estimation module is the concatenation of the observed values $X$ and sampled noise $\widetilde{X}^T$, and the interpolated conditional information $\mathcal{X}$ and conditional feature extraction module are not used.
\item \textit{w/o CF}: remove conditional features in Equation (\ref{eq:cross_att}) and (\ref{eq:cross_att_spa}), i.e., the conditional feature extraction module is not used, and all the $Q$, $K$, and $V$ are the concatenation of interpolated conditional information $\mathcal{X}$ and sampled noise $\widetilde{X}^T$ when calculating the attention weights.
\item \textit{w/o spa}: remove the spatial dependencies learning module $\gamma_\mathcal{S}$ in Equation (\ref{eq:gps_deep}).
\item \textit{w/o tem}: remove the temporal dependencies learning module $\gamma_\mathcal{T}$ in Equation (\ref{eq:gps_deep}).
\item \textit{w/o MPNN}: remove the component of message passing neural network $\varphi_{\text{MP}}$ in the spatial dependencies learning module $\gamma_\mathcal{S}$.
\item \textit{w/o Attn}: remove the component of spatial global attention $\varphi_{\text{SA}}$ in the spatial dependencies learning $\gamma_\mathcal{S}$.
\end{itemize}
\begin{table}[t]
\centering
\caption{Ablation studies.}
\label{tab:abl}
\renewcommand{1}{1}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{cccc}
\toprule
\multirow{2}{*}{Method}&
AQI-36 & \multicolumn{2}{c}{METR-LA}\cr
\cmidrule(lr){2-2} \cmidrule(lr){3-4}
& {Simulated failure}& {Block-missing}& {Point-missing}\cr
\midrule
mix-STI & 9.83$\pm$0.04 & 1.93$\pm$0.00 & 1.74$\pm$0.00 \cr
w/o CF & 9.28$\pm$0.05 & 1.95$\pm$0.00 & 1.75$\pm$0.00 \cr
\midrule
w/o spa & 10.07$\pm$0.04 & 3.51$\pm$0.01 & 2.23$\pm$0.00 \cr
w/o tem & 10.95$\pm$0.08 & 2.43$\pm$0.00 & 1.92$\pm$0.00 \cr
w/o MPNN & 9.10$\pm$0.10 & 1.92$\pm$0.00 & 1.75$\pm$0.00 \cr
w/o Attn & 9.15$\pm$0.08 & 1.91$\pm$0.00 & 1.74$\pm$0.00 \cr
\midrule
PriSTI & \textbf{9.03$\pm$0.07} & \textbf{1.86$\pm$0.00} & \textbf{1.72$\pm$0.00}\cr
\bottomrule
\end{tabular}}
\end{table}
The variants \textit{mix-STI} and \textit{w/o CF} are used to evaluate the effectiveness of the construction and utilization of the conditional information, where \textit{w/o CF} utilizes the interpolated information $\mathcal{X}$ while \textit{mix-STI} does not.
The remaining variants are used to evaluate the spatiotemporal dependencies learning of PriSTI. \textit{w/o spa} and \textit{w/o tem} are used to prove the necessity of learning temporal and spatial dependencies in spatiotemporal imputation, and \textit{w/o MPNN} and \textit{w/o Attn} are used to evaluate the effectiveness of spatial global correlation and geographic dependency.
Since the spatiotemporal dependencies and missing patterns of the two traffic datasets are similar, we perform ablation study on datasets AQI-36 and METR-LA, and their MAE results are shown in Table \ref{tab:abl}, from which we have the following observations:
(1) According to the results of \textit{mix-STI}, the enhanced conditional information and the extraction of conditional feature is effective for spatiotemporal imputation. We believe that the interpolated conditional information is effective for the continuous missing, such as the simulated failure in AQI-36 and the block-missing in traffic datasets.
And the results of \textit{w/o CF} indicate that the construction and utilization of conditional feature improve the imputation performance for diffusion model, which can demonstrate that the conditional feature extraction module and the attention weight calculation in the noise estimation module is beneficial for the spatiotemporal imputation of PriSTI, since they model the spatiotemporal global correlation with the less noisy information.
2) The results of \textit{w/o spa} and \textit{w/o tem} indicate that both temporal and spatial dependencies are necessary for the imputation. This demonstrate that our proposed noise estimation module captures the spatiotemporal dependencies based on the conditional feature, which will also be validated qualitatively in Section \ref{sec:case_study}.
3) From the results of \textit{w/o MPNN} and \textit{w/o Attn}, the components of the spatial global attention and message passing neural network have similar effects on imputation results, but using only one of them is not as effective as using both, which indicates that spatial global correlation and geographic information are both necessary for the spatial dependencies.
Whether the lack of geographic information as input or the lack of captured implicit spatial correlations, the imputation performance of the model would be affected.
We believe that the combination of explicit spatial relationships and implicit spatial correlations can extract useful spatial dependencies in real-world datasets.
\begin{figure}
\caption{The visualization of the probabilistic imputation in AQI-36 and block-missing pattern of METR-LA. Each subfigure represents a sensor, and the time windows of all sensors are aligned. The black crosses represent observations, and dots of various colors represent the ground truth of missing values. The solid green line is the deterministic imputation result, and the green shadow represents the quantile between 0.05 to 0.95.}
\label{fig:visual_aqi}
\label{fig:visual_la_block}
\label{fig:visual_la_block}
\end{figure}
\subsubsection{Case study (RQ4)}\label{sec:case_study}
We plot the imputation result at the same time of some nodes in AQI-36 and the block-missing pattern of METR-LA to qualitatively analyze the spatiotemporal imputation performance of our method, as shown in Figure \ref{fig:visual_la_block}.
Each of the subfigure represents a sensor, the black cross represents the observed value, and the dots of other colors represent the ground truth of the part to be imputed. The green area is the part between the 0.05 and 0.95 quantiles of the estimated probability distribution, and the green solid line is the median, that is, the deterministic imputation result.
We select 5 sensors in AQI-36 and METR-LA respectively, and display their geographic location on the map. Taking METR-LA as the example, it can be observed from Figure \ref{fig:visual_la_block} that sensor 188 and 194 have almost no missing values in the current time window, while their surrounding sensors have continuous missing values, and sensor 192 even has no observed values, which means its temporal information is totally unavailable.
However, the distribution of our generated samples still covers most observations, and the imputation results conform to the time trend of different nodes.
This indicates that on the one hand, our method can capture the temporal dependencies by the given observations for imputation, and on the other hand, when the given observations are limited, our method can utilize spatial dependencies to impute according to the geographical proximity or nodes with similar temporal pattern. For example, for the traffic system, the time series corresponding to sensors with close geographical distance are more likely to have similar trends.
\begin{figure}
\caption{The imputation for unobserved sensors in AQI-36. The orange dotted line represents the ground truth, the green solid line represents the deterministic imputation result, and the green shadow represents the quantile between 0.05 to 0.95.}
\label{fig:mask_sensor}
\end{figure}
\subsubsection{Imputation for sensor failure (RQ5)}
We impute the spatiotemporal data in a more extreme case: some sensors fail completely, i.e., they cannot provide any observations. The available information is only their location, so we can only impute by the observations from other sensors.
This task is often studied in the research related to Kriging \cite{stein1999interpolation}, which requires the model to reconstruct a time series for a given location based on geographic location and observations from other sensors.
We perform the experiment of sensor failure on AQI-36. According to \cite{cini2021filling}, we select the air quality monitoring station with the highest (station 14) and lowest (station 31) connectivity. Based on the original experimental settings unchanged, all observations of these two nodes are masked during the training process. The results of the imputation for unobserved sensors are shown in Figure \ref{fig:mask_sensor}, where the orange dotted line is the ground truth, the green solid line is the median of the generated samples, and the green shadow is the quantiles between 0.05 and 0.95.
We use MAE to quantitatively evaluate the results, the MAE of station 14 is 10.23, and the MAE of station 31 is 15.20.
Since among all baselines only GRIN can impute by geographic information, the MAE compared to the same experiments in GRIN is also shown in Figure \ref{fig:mask_sensor}, and PriSTI has a better imputation performance on the unobserved nodes.
This demonstrates the effectiveness of PriSTI in exploiting spatial relationship for imputation.
Assuming that the detected spatiotemporal data are sufficiently dependent on spatial geographic location, our proposed method may be capable of reconstructing the time series for a given location within the study area, even if no sensors are deployed at that location.
\subsubsection{Hyperparameter analysis and time costs}
We conduct analysis and experiments on some key hyperparameters in PriSTI to illustrate the setting of hyperparameters and the sensitivity of the model to these parameters. Taking METR-LA as example, we analyze the three key hyperparameters: the channel size of hidden state $d$, maximum noise level $\beta_T$ and number of virtual nodes $k$, as shown in Figure \ref{fig:sensitivity_parameter}.
Among them, $d$ and $k$ affect the amount of information learned by the model. $\beta_T$ affects the level of sampled noise, and too large or too too small value is not conducive to the learning of noise prediction model.
According to the results in Figure \ref{fig:sensitivity_parameter}, we set 0.2 to $\beta_T$ which is the optimal value. For $d$ and $k$, although the performance is better with larger value, we set both $d$ and $k$ to 64 considering the efficiency.
In addition, we select several deep learning methods with the higher imputation performance ranking to compare their efficiency with PriSTI on dataset AQI-36 and METR-LA. The total training time and inference time of different methods are shown in Figure \ref{fig:efficiency}.
The experiments are conducted on AMD EPYC 7371 CPU, NVIDIA RTX 3090.
It can be seen that the efficiency gap between methods on datasets with less nodes (AQI-36) is not large, but the training time cost of generative methods CSDI and PriSTI on datasets with more nodes (METR-LA) is higher. Due to the construction of conditional information, PriSTI has 25.7\% more training time and 17.9\% more inference time on METR-LA than CSDI.
\begin{figure}
\caption{The sensitivity study of key parameters.}
\label{fig:hp_a}
\label{fig:hp_b}
\label{fig:hp_c}
\label{fig:sensitivity_parameter}
\end{figure}
\begin{figure}
\caption{The time costs of PriSTI and other baselines.}
\label{fig:eff_a}
\label{fig:eff_b}
\label{fig:efficiency}
\end{figure}
\section{Related Work}\label{sec:related_work}
Since the spatiotemporal data can be imputed along temporal or spatial dimension, there are a large amount of literature for missing value imputation in spatiotemporal data.
For the time series imputation, the early studies imputed missing values by statistical methods such as local interpolation \cite{kreindler2006effects, acuna2004treatment}, which reconstructs the missing value by fitting a smooth curve to the observations.
Some methods also impute missing values based on the historical time series through EM algorithm \cite{shumway1982approach, nelwamondo2007missing} or the combination of ARIMA and Kalman Filter \cite{harvey1990forecasting, ansley1984estimation}.
There are also some early studies filling in missing values through spatial relationship or neighboring sequence, such as KNN \cite{trevor2009elements, beretta2016nearest} and Kriging \cite{stein1999interpolation}.
In addition, the low-rank matrix factorization \cite{salakhutdinov2008bayesian, yu2016temporal, chen2019missing, chen2021bayesian} is also a feasible approach for spatiotemporal imputation, which exploits the intrinsic spatial and temporal patterns based on the prior knowledge.
For instance, TRMF \cite{yu2016temporal} incorporates the structure of temporal dependencies into a temporal regularized matrix factorization framework.
BATF \cite{chen2019missing} incorporates domain knowledge from transportation systems into an augmented tensor factorization model for traffic data modeling.
In recent years, there have been many studies on spatiotemporal data imputation through deep learning methods \cite{liu2019naomi, ma2019cdsa}.
Most deep learning imputation methods focus on the multivariate time series and use RNN as the core to model temporal relationships \cite{che2018recurrent, yoon2018estimating, cao2018brits, cini2021filling}.
The RNN-based approach for imputation is first proposed by GRU-D \cite{che2018recurrent} and is widely used in deep autoregressive imputation methods. Among the RNN-based methods, BRITS \cite{cao2018brits} imputes the missing values on the hidden state through a bidirectional RNN and considers the correlation between features.
GRIN \cite{cini2021filling} introduces graph neural networks based on BRITS to exploit the inductive bias of historical spatial patterns for imputation.
In addition to directly using RNN to estimate the hidden state of missing parts, there are also a number of methods using GAN to generate missing data \cite{luo2018multivariate, yoon2018gain, miao2021generative}.
For instance, GAIN \cite{yoon2018gain} imputes data conditioned on observation values by the generator, and utilizes the discriminator to distinguish the observed and imputed part.
SSGAN \cite{miao2021generative} proposes a semi-supervised GAN to drive the generator to estimate missing values using observed information and data labels.
However, these methods are still RNN-based autoregressive methods, which are inevitably affected by the problem of error accumulation, i.e., the current missing value is imputed by the inaccurate historical estimated values in a sequential manner.
To address this problem, Liu et al. \cite{liu2019naomi} proposes NAOMI, developing a non-autoregressive decoder that recursively updates the hidden state, and using generative adversarial training to impute.
Fortuin et al. \cite{fortuin2020gp} propose a multivariate time series imputation method utilizing the VAE architecture with a Gaussian process prior in the latent space to capture temporal dynamics.
Some other works capture spatiotemporal dependencies through the attention mechanism \cite{ma2019cdsa, shukla2021multi, du2022saits}, which not only consider the temporal dependencies but also exploit the geographic locations \cite{ma2019cdsa} and correlations between different time series \cite{du2022saits}.
Recently, a diffusion model-based generative imputation framework CSDI \cite{tashiro2021csdi} shows the performance advantages of deep generative models in multivariate time series imputation tasks.
The Diffusion Probabilistic Models (DPM) \cite{sohl2015deep, ho2020denoising, song2020score}, as deep generative models, have achieved great performance than other generative methods in several fields such as image synthesis \cite{rombach2022high, ho2020denoising}, audio generation \cite{kong2020diffwave, goel2022s}, and graph generation \cite{huang2022graphgdp, huang2023conditional}.
In terms of imputation tasks, there are existing methods for 3D point cloud completion \cite{lyu2021conditional} and multivariate time series imputation \cite{tashiro2021csdi} through conditional DPM.
CSDI imputes the missing data through score-based diffusion models conditioned on observed data, exploiting temporal and feature correlations by a two dimensional attention mechanism.
However, CSDI takes the concatenation of observed values and noisy information as the input when training, increasing the difficulty of the attention mechanism's learning.
Different from existing diffusion model-based imputation methods, our proposed method construct the prior and imputes spatiotemporal data based on the extracted conditional feature and geographic information.
\section{Conclusion}\label{sec:conclusion}
We propose PriSTI, a conditional diffusion framework for spatiotemporal imputation, which imputes missing values with help of the extracted conditional feature to calculate temporal and spatial global correlations.
Our proposed framework captures spatiotemporal dependencies by comprehensively considering spatiotemporal global correlation and geographic dependency.
PriSTI achieves more accurate imputation results than state-of-the-art baselines in various missing patterns of spatiotemporal data in different fields, and also handles the case of high missing rates and sensor failure.
In future work, we will consider improving the scalability and computation efficiency of existing frameworks on larger scale spatiotemporal datasets, and how to impute by longer temporal dependencies with refined conditional information.
\section*{Acknowledgment}
We thank anonymous reviewers for their helpful comments.
This research is supported by the National Natural Science Foundation of China (62272023).
\end{document} |
\begin{document}
\title{ Quantum measurement bounds beyond the uncertainty relations}
\author{Vittorio Giovannetti$^1$, Seth Lloyd$^2$, Lorenzo Maccone$^3$}
\affiliation{ $^1$ NEST, Scuola Normale Superiore and Istituto
Nanoscienze-CNR,
piazza dei Cavalieri 7, I-56126 Pisa, Italy \\
$^2$Dept.~of Mechanical Engineering, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA \\
$^3$Dip.~Fisica ``A.~Volta'', INFN Sez.~Pavia, Universit\`a di
Pavia, via Bassi 6, I-27100 Pavia, Italy}
\begin{abstract}
We give a bound to the precision in the estimation of a parameter in
terms of the expectation value of an observable. It is an extension
of the Cram\'er-Rao inequality and of the Heisenberg uncertainty
relation, where the estimation precision is typically bounded in
terms of the variance of an observable.
\end{abstract}
\maketitle
Quantum measurements are limited by bounds such as the Heisenberg
uncertainty relations \cite{heisenberg,robertson} or the quantum
Cram\'er-Rao inequality \cite{holevo,helstrom,BRAU96,BRAU94}, which
typically constrain the ability in recovering a target quantity
(e.g.~a relative phase) through the {\em standard deviation} of a
conjugate one (e.g.~the energy) evaluated on the state of the probing
system. Here we give a new bound related to the {\em expectation
value}: we show that the precision in the quantity cannot scale
better than the inverse of the expectation value (above a ``ground
state'') of its conjugate counterpart. It is especially relevant in
the expanding field of quantum metrology \cite{review}: it settles in
the positive the longstanding conjecture of quantum optics
\cite{caves,yurke,barry,ou,bollinger,smerzi}, recently challenged
\cite{dowling,rivasluis,zhang}, that the ultimate phase-precision
limit in interferometry is lower bounded by the inverse of the total
number of photons employed in the estimation process.
The aim of Quantum Parameter
Estimation~\cite{holevo,helstrom,BRAU96,BRAU94} is to recover the {\em
unknown} value $x$ of a parameter that is written into the state
$\rho_x$ of a probe system through some {\em known} encoding mechanism
$U_x$. For example, we can recover the relative optical delay $x$
among the two arms of a Mach-Zehnder interferometer described by its
unitary evolution $U_x$ using as probe a light beam fed into the
interferometer. The statistical nature of quantum mechanics induces
fluctuations that limit the ultimate precision which can be achieved
(although we can exploit quantum ``tricks'' such as entanglement and
squeezing in optimizing the state preparation of the probe and/or the
detection stage \cite{GIOV06}).
In particular, if the encoding stage is repeated
several times using $\nu$ identical copies of the same probe input
state $\rho_x$, the root mean square error (RMSE) $\Delta X$ of the
resulting estimation process is limited by the quantum Cram\'er-Rao
bound~\cite{holevo,helstrom,BRAU96,BRAU94} $\Delta X\geqslant
1/\sqrt{\nu Q(x)}$, where ${Q}(x)$ is the quantum Fisher information.
For pure probe states and unitary encoding mechanism $U_x$, ${Q}(x)$
is equal to the variance $(\Delta H)^2$ (calculated on the probe
state) of the generator $H$ of the transformation $U_x=e^{-ixH}$.
In this case, the Cram\'er-Rao bound takes the form
\begin{eqnarray}
\Delta
X\geqslant 1/(\sqrt{\nu} \Delta H)\label{QC}\;
\end{eqnarray}
of an uncertainty relation \cite{BRAU94,BRAU96}. In
fact, if the parameter $x$ can be connected to an observable,
Eq.~\eqref{QC} corresponds to the Heisenberg uncertainty relation for
conjugate variables~\cite{heisenberg,robertson}. This bound is
asymptotically achievable in the limit of $\nu \rightarrow \infty$
\cite{holevo,helstrom}.
\begin{figure}
\caption{ Lower bounds to the precision estimation $\Delta X$ as a
function of the experimental repetitions $\nu$. The green area in
the graph represents the forbidden values due to our bound
\eqref{ris}
\end{figure}
Here we will derive a bound in terms of the expectation value of $H$,
which (in the simple case of constant $\Delta X$) takes the form (see
Fig.~\ref{f:compar})
\begin{eqnarray}
\Delta X\geqslant \kappa/[\nu(\langle H\rangle-E_0)]
\labell{ris}\;,
\end{eqnarray}
where $E_0$ is the value of a ``ground state'', the minimum eigenvalue
of $H$ whose eigenvector is populated in the probe state (e.g.~the
ground state energy when $H$ is the probe's Hamiltonian), and $\kappa\simeq
0.091$ is a constant of order one. Our bound holds both for biased and
unbiased measurement procedures, and for pure and mixed probe states.
When $\Delta X$ is dependent on $x$, a constraint of the
form~(\ref{ris}) can be placed on the average value of $\Delta X(x)$
evaluated on any two values $x$ and $x'$ of the parameter which are
sufficiently separated, namely
\begin{eqnarray} \label{bd1} \frac{\Delta X(x) + \Delta X(x')}{2} &\geqslant
& \frac{\kappa}{\nu(\langleH\rangle - E_0)} \;.
\end{eqnarray}
Hence, we cannot exclude that strategies whose error $\Delta X$ depend
on $x$ may have a ``sweet spot'' where the bound \eqref{ris} may be
beaten \cite{rivasluis}, but inequality \eqref{bd1} shows that the
average value of $\Delta X$ is subject to the bound. Thus, these
strategies are of no practical use, since the sweet spot depends on
the unknown parameter $x$ to be estimated and the extremely good
precision in the sweet spot must be counterbalanced by a
correspondingly bad precision nearby.
Proving the bound~\eqref{ris} in full generality is clearly not a
trivial task since no definite relation can be established between
$\nu(\langle H\rangle-E_0)$ and the term $\sqrt{\nu} \Delta H$ on
which the Cram\'er-Rao bound is based. In particular, scaling
arguments on $\nu$ cannot be used since, on one hand, the value of
$\nu$ for which Eq.~(\ref{QC}) saturates is not known (except in the
case in which the estimation strategy is fixed \cite{caves}, which has
little fundamental relevance) and, on the other hand, input probe
states $\rho$ whose expectation values $\langle H \rangle$ depend
explicitly on $\nu$ may be employed, e.g.~see Ref.~\cite{rivasluis}.
To circumvent these problems our proof is based on the quantum speed
limit~\cite{qspeed}, a generalization of the Margolus-Levitin
~\cite{margolus} and Bhattacharyya bounds~\cite{bhatta,man} which
links the fidelity $F$ between the two joint states
$\rho_x^{\otimes\nu}$ and $\rho_{x'}^{\otimes\nu}$ to the difference
$x'-x$ of the parameters $x$ and $x'$ imprinted on the states through
the mapping $U_x=e^{-ixH}$ [The fidelity between two states $\rho$ and
$\sigma$ is defined as $F=\{\mbox{Tr}[\sqrt{\sqrt{\rho}
\sigma\sqrt{\rho}}]\}^2$. A connection between quantum metrology and
the Margolus-Levitin theorem was proposed in \cite{kok}, but this
claim was subsequently retracted in \cite{erratum}.] In the case of
interest here, the quantum speed limit \cite{qspeed} implies
\begin{eqnarray}
|x'-x|\geqslant\frac\pi2
\max \left[\frac{\alpha(F)}{\nu(\langleH\rangle-E_0)}\;,\
\frac{\beta(F)}{\sqrt{\nu} \Delta H}\right] \;\labell{newqsl}\;,
\end{eqnarray}
where the $\nu$ and $\sqrt{\nu}$ factors at the denominators arise
from the fact that here we are considering $\nu$ copies of the probe
states $\rho_x$ and $\rho_{x'}$, and where
$\alpha(F)\simeq\beta^2(F)=4\arccos^2(\sqrt{F})/\pi^2$ are the
functions plotted in Fig.~\ref{f:qsl} of the supplementary material.
The inequality~\eqref{newqsl} tells us that the parameter difference
$|x'-x|$ induced by a transformation $e^{-i(x'-x)H}$ which employs
resources $\langleH\rangle-E_0$ and $\Delta H$ cannot be arbitrarily small (when
the parameter $x$ coincides with the evolution time, this sets a limit
to the ``speed'' of the evolution, the quantum speed limit).
We now give the main ideas of the proof of \eqref{ris} by focusing on
a simplified scenario, assuming pure probe states $|\psi_x\rangle=U_x
|\psi\rangle$, and unbiased estimation strategies constructed in terms
of projective measurements with RSME $\Delta X$ that do not depend on
$x$ (all these assumptions are dropped in the supplementary material).
For unbiased estimation, $x=\sum_j P_j(x) x_j$ and the RMSE coincides
with the variance of the distribution $P_j(x)$, i.e.~$\Delta
X=\sqrt{\sum_j P_j(x) [ x_j-x]^2}$, where $P_j(x) = |\langle x_j |
\psi_x \rangle^{\otimes\nu}|^2$ is the probability of obtaining the
result $x_j$ while measuring the joint state
$|\psi_x\rangle^{\otimes\nu}$ with a projective measurement on the
joint basis $|x_j\rangle$. Let us consider two values $x$ and $x'$ of
the parameter that are further apart than the measurement's RMSE,
i.e.~$x'-x=2 \lambda \Delta X$ with $\lambda>1$. If no such $x$ and
$x'$ exist, the estimation is extremely poor: basically the whole
domain of the parameter is smaller than the RMSE. Hence, for
estimation strategies that are sufficiently accurate to be of
interest, we can always assume that such a choice is possible (see
below). The Tchebychev inequality states that for an arbitrary
probability distribution $p$, the probability that a result $x$ lies
more than $\lambda\Delta X$ away from the average $\mu$ is upper
bounded by $1/\lambda^2$, namely $p(|x-\mu|\geqslant \lambda\Delta
X)\leqslant 1/\lambda^2$. It implies that the probability that
measuring $|\Psi_{x'}\rangle :=|\psi_{x'}\rangle^{\otimes\nu}$ the
outcome $x_j$ lies within $\lambda\Delta X$ of the mean value
associated with $|\Psi_x\rangle:=|\psi_x\rangle^{\otimes\nu}$ cannot
be larger $1/\lambda^2$. By the same reasoning, the probability that
measuring $|\Psi_{x}\rangle$ the outcome $x_j$ will lie within
$\lambda\Delta X$ of the mean value associated with
$|\Psi_{x'}\rangle$ cannot be larger $1/\lambda^2$. This implies that
the overlap between the states $|\Psi_{x}\rangle$ and
$|\Psi_{x'}\rangle$ cannot be too large: more precisely, $F=|
\langle\Psi_x|\Psi_{x'} \rangle|^2\leqslant 4/\lambda^2$. Replacing
this expression into \eqref{newqsl} (exploiting the fact that $\alpha$
and $\beta$ are decreasing functions) we obtain
\begin{eqnarray}
2\lambda \Delta X\geqslant
\frac\pi{2}
\max \left[\frac{\alpha(4/\lambda^2)}{\nu(\langleH\rangle-E_0)}\;,\
\frac{\beta(4/\lambda^2)}{\sqrt{\nu} \Delta H}\right]
\labell{ineq}\;,
\end{eqnarray}
whence we obtain \eqref{ris} by optimizing over $\lambda$ the first
term of the $\max$, i.e.~choosing
$\kappa=\sup_\lambda\pi\:\alpha(4/\lambda^2)/(4\lambda)\simeq 0.091$. The
second term of the $\max$ gives rise to a quantum Cram\'er-Rao type
uncertainty relation (or a Heisenberg uncertainty relation) which,
consistently with the optimality of Eq.~(\ref{QC}) for $\nu\gg1$, has
a pre-factor $\pi \beta(4/\lambda^2)/ (4 \lambda)$ which is smaller
than $1$ for all $\lambda$. This means that for large $\nu$ the bound
\eqref{ris} will be asymptotically superseded by the Cram\'er-Rao
part, which scales as $\propto 1/\sqrt{\nu}$ and is achievable in this
regime.
Analogous results can be obtained (see supplementary material) when
considering more general scenarios where the input states of the
probes are not pure, the estimation process is biased, and it is
performed with arbitrary POVM measurements. (In the case of biased
measurements, the constant $\kappa$ in \eqref{ris} and \eqref{bd1}
must be replaced by $\kappa= \sup_{\lambda} \pi
\alpha(4/\lambda^2)/[4(\lambda+1)]\simeq 0.074$, where a $+1$ term
appears in the denominator.) In this generalized context, whenever the
RMSE depends explicitly on the value $x$ of the parameter, the
result~\eqref{ris} derived above is replaced by the weaker
relation~\eqref{bd1}. Such inequality clearly does not necessarily
exclude the possibility that at a ``sweet spot'' the estimation might
violate the scaling~(\ref{ris}). However, Eq.~(\ref{bd1}) is still
sufficient strong to exclude accuracies of the form $\Delta X(x)
=1/R(x,\nu\langle H\rangle)$ where, as in Refs.~\cite{ssw,rivasluis},
$R(x,z)$ is a function of $z$ which, for all $x$, increases more than
linearly, i.e.~$\lim_{z\rightarrow \infty} z/R(x,z)=0$.
The bound~\eqref{ris} has been derived under the explicit assumption
that $x$ and $x'$ exists such that $x'-x\geqslant 2 \lambda \Delta X$
for some $\lambda >1$, which requires one to have $x'-x\geqslant 2
\Delta X$. This means that the estimation strategy must be good
enough: the probe is sufficiently sensitive to the transformation
$U_x$ that it is shifted by more than $\Delta X$ during the
interaction. The existence of pathological estimation strategies which
violate such condition cannot be excluded {\em a priori}. Indeed
trivial examples of this sort can be easily constructed, a fact which
may explain the complicated history of the Heisenberg bound with
claims \cite{caves,yurke,barry,ou,bollinger,smerzi} and counterclaims
\cite{dowling,rivasluis,zhang,ssw}. It should be stressed however,
that the assumption $x'-x\geqslant 2 \Delta X$ is always satisfied
except for extremely poor estimation strategies with such large errors
as to be practically useless. One may think of repeating such a poor
estimation strategy $\nu>1$ times and of performing a statistical
average to decrease its error. However, for sufficiently large $\nu$
the error will decrease to the point in which the $\nu$ repetitions of
the poor strategy are, collectively, a good strategy, and hence again
subject to our bounds \eqref{ris} and \eqref{bd1}.
Our findings are particularly relevant in the field of quantum optics,
where a controversial and longly debated problem
\cite{caves,yurke,barry,ou,bollinger,smerzi,ssw,dowling,rivasluis,zhang}
is to determine the scaling of the ultimate limit in the
interferometric precision of estimating a phase as a function of the
energy $\langle H\rangle$ devoted to preparing the $\nu$ copies of the
probes: it has been conjectured
\cite{caves,yurke,barry,ou,bollinger,smerzi} that the phase RMSE is
lower bounded by the inverse of the total number of photons employed
in the experiment, the ``Heisenberg bound'' for
interferometry\footnote{This ``Heisenberg''
bound~\cite{caves,yurke,barry,ou,bollinger,smerzi} should not be
confused with the Heisenberg scaling defined for general quantum
estimation problem~\cite{review} in which the $\sqrt{\nu}$ at the
denominator of Eq.~(\ref{QC}) is replaced by $\nu$ by feeding the
$\nu$ inputs with entangled input states -- e.g. see
Ref.~\cite{review,GIOV06}.}. Its achievability has been recently
proved \cite{HAYA10-1}, and, in the context of quantum parameter
estimation, it corresponds to an equation of the form of
Eq.~\eqref{ris}, choosing $x=\phi$ (the relative phase between the
modes in the interferometer) and $H=a^\dag a$ (the number operator).
The validity of this bound has been questioned several times
\cite{ssw,dowling,rivasluis,zhang}. In particular schemes have been
proposed~\cite{ssw,rivasluis} that apparently permit better scalings
in the achievable RMSE (for instance $\Delta X \approx (\nu\langle
H\rangle)^{-\gamma}$ with $\gamma>1$). None of these protocols have
conclusively proved such scalings for arbitrary values of the
parameter $x$, but a sound, clear argument against the possibility of
breaking the $\gamma=1$ scaling of Eq.~(\ref{ris}) was missing up to
now. Our results validate the Heisenberg bound by showing that it
applies to all those estimation strategies whose RMSE $\Delta X$ do
not depend on the value of the parameter $x$, and that the remaining
strategies can only have good precision for isolated (hence
practically useless) values of the unknown parameter $x$.\newline
V.G. acknowledges support by MIUR through FIRB-IDEAS Project No.
RBID08B3FM. S.L. acknowledges Intel, Lockheed Martin, DARPA, ENI under
the MIT Energy Initiative, NSF, Keck under xQIT, the MURI QUISM
program, and Jeffrey Epstein. L.M. acknowledges useful discussions
with Prof. Alfredo Luis.
\subsection*{Supplementary material}
Our bound refers to the estimation of the value $x$ of a real
parameter $X$ that identifies a unitary transformation $U_x=e^{-iHx}$,
generated by an Hermitian operator $H$. The usual setting in quantum
channel parameter estimation~(see \cite{review} for a recent review)
is to prepare $\nu$ copies of a probe system in a fiducial state
$\rho$, apply the mapping $U_x$ to each of them as
$\rho\to\rho_x=U_x\rho {U_x}^\dag$, and then perform a (possibly
joint) measurement on the joint output state $\rho_x^{\otimes \nu}$,
the measurement being described by a generic Positive Operator-Valued
Measure (POVM) of elements $\{ E_j\}$. [The possibility of applying a
joint transformation on the $\nu$ probes before the interaction $U_x$
(e.g.~to entangle them as studied in \cite{GIOV06}) can also be
considered, but it is useless in this context, since it will not
increase the linear scaling in $\nu$ of the term $\nu(\langle
H\rangle-E_0)$ that governs our bounds.] The result $j$ of the measurement is finally
used to recover the quantity $x$ through some data processing which
assigns to each outcome $j$ of the POVM a value $x_j$ which represents
the estimation of $x$. The accuracy of the process can be gauged by
the RMSE of the problem, i.e.~by the quantity
\begin{eqnarray}\label{defdelta}
\Delta X := \sqrt{\sum_{j} P_j(x) [ x_j -x ]^2 } = \sqrt{ \delta^2X +
(\bar{x}-x )^2 },
\end{eqnarray}
where $P_j(x) = \mbox{Tr}[ E_j \rho_x^{\otimes \nu}]$ is the
probability of getting the outcome $j$ when measuring $\rho_x^{\otimes
\nu}$, $\bar{x} := \sum_{j} P_j(x) x_j$ is the average of the
estimator function, and where
\begin{eqnarray}
\delta^2 X := \sum_{j} P_j(x) [ x_j - \bar{x}]^2\;,
\end{eqnarray}
is the variance of the random variable $x_j$. The estimation is said
to be unbiased if $\bar{x}$ coincides with the real value $x$,
i.e.~$\bar{x}=x$, so that, in this case, $\Delta X$ coincides with
$\delta X$. General estimators however may be biased with $\bar{x}\neq
x$, so that $\Delta X > \delta X$ (in this case, they are called
asymptotically unbiased if $\bar{x}$ converges to $x$ in the limit
$\nu\rightarrow \infty$).
In the main text we restricted our analysis to pure states of the
probe $\rho=|\psi\rangle\langle \psi|$ and focused on projective
measurements associated to unbiased estimation procedures whose RMSE
$\Delta X$ is independent on $x$.
Here we extend the proof to drop the above simplifying assumptions,
considering a generic (non necessarily unbiased) estimation process
which allows one to determine the value of the real parameter $X$
associated with the non necessarily pure input state $\rho$.
Take two values $x$ and $x'$ of $X$ such that their associated RMSE
verifies the following constraints
\begin{eqnarray}
&&\Delta X(x) \neq 0\;,\label{pos}\\
&&|x-x'| =
( \lambda+1) [\Delta X(x)+\Delta X(x') ] \;,
\label{dist}
\end{eqnarray}
for some fixed value $\lambda$ greater than 1 (the right hand side of
Eq.~(\ref{dist}) can be replaced by $\lambda [\Delta X(x) +\Delta
X(x')]$ if the estimation is unbiased). In these expressions $\Delta
X(x)$ and $\Delta X(x')$ are the RMSE of the estimation evaluated
through Eq.~(\ref{defdelta}) on the output states $\rho_x^{\otimes
\nu}$ and $\rho_{x'}^{\otimes \nu}$ respectively (to include the
most general scenario we do allow them to depend explicitly on the
values taken by the parameter $X$). In the case in which the
estimation is asymptotically unbiased and the quantum Fisher
information $Q(x)$ of the problem takes finite values, the condition
(\ref{pos}) is always guaranteed by the quantum Cram\'{e}r-Rao
bound~\cite{holevo,helstrom,BRAU96,BRAU94} (but notice that our proof
holds also if the quantum Cram\'{e}r-Rao bound does not apply -- in
particular, we do not require the estimation to be asymptotically
unbiased). The condition~(\ref{dist}) on the other hand is verified
by any estimation procedure which achieves a reasonable level of
accuracy: indeed, if it is not verified, then this implies that the
interval over which $X$ can span is not larger than twice the average
RMSE achievable in the estimation.
Since the fidelity between two quantum states is the minimum of the
classical fidelity of the probability distributions from arbitrary
POVMs~\cite{nc}, we can bound the fidelity between $\rho_x^{\otimes
\nu}$ and $\rho_{x'}^{\otimes \nu}$ as follows
\begin{eqnarray}
F :=\Big[ \mbox{Tr} \sqrt{\sqrt{\rho_x^{\otimes \nu}}
\rho_{x'}^{\otimes \nu} \sqrt{\rho_x^{\otimes \nu}} }\Big]^2
\leqslant \Big[
\sum_j\sqrt{ P_j(x) P_j(x') } \Big]^2\;,\nonumber\\\label{fid}
\end{eqnarray}
with $P_j(x) = \mbox{Tr} [ E_j \rho_x^{\otimes \nu}]$ and $P_j(x') = \mbox{Tr} [ E_j \rho_{x'}^{\otimes \nu}]$.
The right-hand-side of this expression can be bound as
\begin{widetext}
\begin{eqnarray}
\sum_j\sqrt{ P_j(x) P_j(x') }
&=& \sum_{j\in I} \sqrt{ P_j(x) P_j(x') }+ \sum_{j\notin I} \sqrt{
P_j(x) P_j(x') } \nonumber \\
&\leqslant& \sqrt{\sum_{j\in I}{ P_j(x) } \sum_{j'\in I}{
P_{j'}(x') }} +\sqrt{ \sum_{j\notin I} { P_j(x) }
\sum_{j'\notin I}{P_{j'}(x') } }
\nonumber \\ &\leqslant&
\sqrt{\sum_{j\in I}{ P_j(x) } } +\sqrt{
\sum_{j'\notin I}{P_{j'}(x') } } \;,\label{boun1}
\end{eqnarray}
where $I$ is a subset of the domain of possible outcomes $j$ that we
will specify later, and where we used the Cauchy-Schwarz inequality
and the fact that $ \sum_{j'\in I}{ P_{j'}(x') }\leqslant 1$ and $
\sum_{j\notin I} { P_j(x) }\leqslant 1 $ independently from $I$. Now,
take $I$ to be the domain of the outcomes $j$ such that
\begin{eqnarray}
|x_j - \bar{x}'| \leqslant \lambda \delta X',
\labell{lam}\;
\end{eqnarray}
where $\lambda$ is a positive parameter (here $\bar{x}'$ and $(\delta X')^2$ are the average and the variance value of $x_j$ computed with the probability distribution $P_{j}(x')$).
From the Tchebychev inequality it then follows that
\begin{eqnarray}
\sum_{j'\notin I}{P_{j'}(x') } \leqslant 1 /\lambda^2\;,\label{asa1}
\end{eqnarray}
which gives a significant bound only when $\lambda>1$.
To bound the other term on the rhs of Eq.~(\ref{boun1}) we
notice that $|x-x'| \leqslant \big|x- \bar{x} \big| + \big| x'- \bar{x'}\big| + \big|\bar{x}-\bar{x}'\big|$ and use
Eq.~(\ref{dist}) and (\ref{defdelta}) to write
\begin{eqnarray}
\big|\bar{x}-\bar{x}' \big| &\geqslant& (\lambda +1) (\Delta X + \Delta X') - \big|x- \bar{x} \big| - \big| x'- \bar{x}'\big| \nonumber \\
&=& (\lambda +1) (\Delta X + \Delta X') - \sqrt{\Delta^2X - \delta^2 X} - \sqrt{\Delta^2X' - \delta^2 X'} \geqslant \lambda (\Delta X + \Delta X')\;.
\end{eqnarray}
From Eq.~(\ref{lam}) we also notice that for $j\in I$ we have
\begin{eqnarray}
\big|\bar{x}-\bar{x}'\big|
\leqslant \big|\bar{x}- x_j\big| + \big| x_j - \bar{x}'\big|
\leqslant \big|\bar{x}- x_j\big| + \lambda \delta X'\;,
\end{eqnarray}
which with the previous expression gives us
\begin{eqnarray}
\big|\bar{x}- x_j\big| \geqslant \lambda (\Delta X + \Delta X')
- \lambda \delta X'\geqslant \lambda \Delta X \geqslant \lambda \delta X\;,
\end{eqnarray}
and hence (using again the Tchebychev inequality)
\begin{eqnarray}
\sum_{j\in I}{P_{j}(x) } \leqslant 1 /\lambda^2\;.\label{17}
\end{eqnarray}
Replacing \eqref{asa1} and \eqref{17} into (\ref{fid}) and
(\ref{boun1}) we obtain
\begin{eqnarray}\label{fidnew}
F \leqslant 4 /\lambda^2 \;.
\end{eqnarray}
We can now employ the quantum speed limit inequality~(\ref{newqsl})
from \cite{qspeed} and merge it with the condition (\ref{dist}) to
obtain
\begin{eqnarray}
(\lambda +1 )(\Delta X + \Delta X')= |x'-x| &\geqslant&
\frac{\pi}{2} \max \left\{ \frac{\alpha(F)}{\nu(\langleH\rangle- E_0)},
\frac{\beta(F)}{\sqrt{\nu} \Delta H }\right\}
\geqslant \frac{\pi}{2} \max \left\{
\frac{\alpha(4/\lambda^2)}{\nu(\langleH\rangle - E_0)},
\frac{\beta(4/\lambda^2)}{\sqrt{\nu} \Delta H}\right\},\label{ila}
\end{eqnarray}
\end{widetext}
where, as in the main text, we used the fact that $\alpha$ and $\beta$
are decreasing functions of their arguments, and the fact that the
expectation and variances of $H$ over the family $\rho_x$ is
independent of $x$ (since $H$ is independent of $x$). The first term
of Eq.~\eqref{ila} together with the first part of the $\max$ implies
Eq.~(\ref{bd1}), choosing $\kappa= \sup_{\lambda} \pi
\alpha(4/\lambda^2)/[4(\lambda+1)]\simeq 0.074$, which for unbiased
estimation can be replaced by $\kappa=\sup_{\lambda} \pi
\alpha(4/\lambda^2)/[4\lambda] \simeq 0.091$. In the case in which
$\Delta X(x) =\Delta X(x')=\Delta X$ we then immediately obtain the
bound~(\ref{ris}).
\begin{figure}
\caption{ Plot of the functions $\alpha(F)$ and $\beta(F)$ appearing
in Eq.~(\ref{newqsl}
\end{figure}
\begin{figure}
\caption{Plot of the function $\pi\: \alpha(4/\lambda^2)/(4\lambda)$
as a function of $\lambda$ (blue continuous line). The function
$\alpha$ is evaluated numerically according to the prescription of
\cite{qspeed}
\end{figure}
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\end{document} |
\begin{document}
\title{Paraconsistent Machines and their Relation to Quantum Computing}
\begin{abstract}
We describe a method to axiomatize computations in deterministic
Turing machines. When applied to computations in
non-deterministic Turing machines, this method may produce
contradictory (and therefore trivial) theories, considering
classical logic as the underlying logic. By substituting in such
theories the underlying logic by a paraconsistent logic
we define a new computation model, the
\emph{paraconsistent Turing machine}. This
model allows a partial simulation of superposed states of
quantum computing. Such a feature allows the definition of
paraconsistent algorithms which solve (with some restrictions)
the well-known Deutsch's and Deutsch-Jozsa
problems. This first model of computation, however,
does not adequately represent the notions of \emph{entangled
states} and \emph{relative phase}, which are key features in quantum computing.
In this way, a more sharpened model of paraconsistent Turing machines is defined,
which better approaches quantum computing features.
Finally, we define complexity classes for such models, and establish some
relationships with classical complexity classes.
\end{abstract}
\section{Introduction}
The undecidability of first-order logic was first proved by
Alonzo Church in \cite{Church-1936b} and an alternative proof with
the same result was presented by Alan Turing in
\cite{Turing-1936}. In his paper, Turing defined an abstract
model of automatic machines, now known as \emph{Turing machines}
(TMs), and demonstrated that there are unsolvable problems for that
class of machines. By axiomatizing machine
computations in first-order theories, he could then prove that
the decidability of first-order logic would imply the solution
of established unsolvable problems. Consequently, by
\emph{reductio ad absurdum}, first-order logic is shown to be
undecidable. Turing's proof was simplified by Richard B\"uchi in
\cite{Buchi-1962}, and a more recent and clear version of this
proof is presented by George Boolos and Richard Jeffrey in
\cite[Chap. 10]{Boolos-Jeffrey-1989}.
By following \cite{Boolos-Jeffrey-1989} and adding new axioms, we
define a method to obtain adequate theories for computations in
\emph{deterministic} TMs (DTMs) (Sec. \ref{axio-TM-comp}), which
verify a formal notion of representation of TM computation (here introduced),
therefore enhancing the standard way in which TMs are expressed by means of
classical first-order logic.
Next, we will show that by using the same axiomatization method for
\emph{non-deterministic} TMs (NDTMs), we obtain (in some cases)
contradictory theories, and therefore trivial theories in view of the
underlying logic. At this point, we have two options in sight to
avoid triviality: (a) The first option, consisting in the classical
move of restricting theories in a way that contradictions could
not be derived (just representing computations in NDTMs); (b) the
second option, consisting in substituting the underlying logic by
a paraconsistent logic, supporting contradictions and providing a way to define new models of
computation through the interpretation of the theories. The first
option is sterile: incapable of producing a new model of
computation. In this paper, we follow the second option,
initially defining in Sec. \ref{ptms} a model of
\emph{paraconsistent} TMs (ParTMs), using the paraconsistent
logic $LFI1^*$ (see \cite{Carnielli-Coniglio-Marcos-2007}). We
then show that ParTMs allow a partial simulation of `superposed states', an important
feature of quantum computing (Sec. \ref{sim-qc-ptms}). By using this property,
and taking advantage of `conditions of inconsistency' added to the instructions,
we show that the quantum solution of Deutsch's and
Deutsch-Jozsa problems can be simulated in ParTMs (Sec.
\ref{sim-D-DJ-prob}). However, as explained in Sec.
\ref{sim-ent-states-rel-phases}, ParTMs are not adequate to simulate
\emph{entangled states} and \emph{relative phases}, which are key features in quantum computing.
Thus, still in Sec. \ref{sim-ent-states-rel-phases}, we define a
paraconsistent logic with a \emph{non-separable} conjunction and,
with this logic, a new model of paraconsistent TMs is defined,
which we call \emph{entangled paraconsistent} TMs (EParTMs).
In EParTMs, uniform entangled states can be successfully
simulated. Moreover, we describe how the notion of relative phase (see \cite[p.
193]{Nielsen-Chuang-2000}) can be introduced
in this model of computation.\footnote{ParTMs were first presented in \cite{Agudelo-Sicard-2004}
and relations with quantum computing were presented in \cite{Agudelo-Carnielli-2005},
but here we obtain some improvements and introduce the model of EParTMs,
which represents a better approach to quantum computing.}
The \emph{paraconsistent computability theory} has already been
mentioned in \cite[p. 196]{Sylvan-Copeland-2000} as an `emerging
field of research'. In that paper, \emph{dialethic machines} are
summarily described as Turing machines acting under dialethic
logic (a kind of paraconsistent logic) in the presence of a
contradiction, but no definition of any computation model is
presented.\footnote{The authors say that ``it is not difficult to
describe how a machine might encounter a contradiction: For some
statement $A$, both $A$ and $\neg A$ appear in its output or
among its inputs'' (cf. \cite[p. 196]{Sylvan-Copeland-2000});
but how can $\neg A$ `appear'? They also claim that ``By contrast
[with a classical machine], a machine programmed with a dialethic
logic can proceed with its computation satisfactorily [when a
contradiction appears]''; but how would they proceed?} Also,
an obscure argument is presented in an attempt to show how dialethic machines
could be used to compute classically uncomputable functions.
Contrarily, our models of ParTMs and EParTMs do not intend to
break the Church-Turing thesis, i.e., all problems computed by
ParTMs and EParTMs can also be computed by TMs (Sec.
\ref{comp-power-partms-EParTMs}). However, such models are
advantageous for the understanding of quantum computing and
parallel computation in general. Definitions of computational
complexity classes for ParTMs and EParTMs, in addition to interesting
relations with existing classical and quantum computational
complexity classes, are presented in Sec. \ref{comp-power-partms-EParTMs}.
The paraconsistent approach to quantum computing presented here
is just one way to describe the role of quantum features in the
process of computation by means of non-classical logics; in
\cite{Agudelo-Carnielli-2007} we presented another way to define
a model of computation based on another paraconsistent logic, also
related with quantum computing. The relationship between these
two different definitions of paraconsistent computation is a task
that needs to be addressed in future work.
\section{Axiomatization of TM Computations}\label{axio-TM-comp}
As mentioned above, a method to define first-order theories
for Turing machine computations has already been introduced in \cite{Turing-1936}.
Although this is a well-known construction, in view of the important role of
this method in our definition of paraconsistent Turing machines we will
describe it in detail, following \cite[Chap. 10]{Boolos-Jeffrey-1989}.
This method will be extended to deal with a formal notion of representation (here introduced) of TM computation.
Considering $Q = \{q_1, \ldots, q_n \}$ as a finite set of states
and $\Sigma = \{s_1, \ldots, s_m \}$ as a finite set of
read/write symbols, we will suppose that TM instructions are defined by
quadruples of one of the following types (with the usual interpretations, where
$R$ means a movement to the right, and $L$ means a movement to the left):
\begin{align}
&q_i s_j s_k q_l,\tag{I}\label{inst-i}\\
&q_i s_j R q_l,\tag{II}\label{inst-ii}\\
&q_i s_j L q_l.\tag{III}\label{inst-iii}
\end{align}
By convention, we will enumerate the instants of time and
the cells of the tape by integer numbers, and we will consider that
machine computations begin at time $0$, with a symbol sequence on
the tape (the \emph{input} of the computation), and with the
machine in state $q_1$ scanning the symbol on cell $0$. $s_1$ will be
assumed to be an \emph{empty} symbol.
In order to represent the computation of a TM $\mc{M}$ with input
$\alpha$ (hereafter $\mc{M}(\alpha)$), we initially define the
first-order theory $\Delta_{FOL}(\mc{M}(\alpha))$ over the first-order language $\mc{L}
= \{Q_1, \ldots, Q_n, S_1, \ldots, S_{m}, <, ', 0\}$,\footnote{The
subscript $FOL$ on $\Delta$ aims to emphasize the fact that we
are considering the \emph{classical first-order logic} (FOL) as
the underlying logic of the theory, i.e. $\Delta_{FOL} \vdash A$
means $\Delta \vdash_{FOL} A$. A different subscript will
indicate that another (non-classical) first-order logic is being
taken into consideration.} where
symbols $Q_i$, $S_j$ and $<$ are binary predicate symbols, $'$ is
a unary function symbol and $0$ is a constant symbol. In the
intended interpretation $\mc{I}$ of the sentences in
$\Delta_{FOL}(\mc{M}(\alpha))$, variables are interpreted as
integer numbers, and symbols in $\mc{L}$ are interpreted in the
following way:
\begin{itemize}
\item $Q_i(t, x)$ indicates that $\mc{M}(\alpha)$ is in state $q_i$, at time $t$,
scanning the cell $x$;
\item $S_j(t, x)$ indicates that $\mc{M}(\alpha)$ contains the symbol $s_j$, at
time $t$, on cell $x$;
\item $<(x, y)$ indicates that $x$ is less than $y$, in the standard order of
integer numbers;
\item $'(x)$ indicates the successor of $x$;
\item $0$ indicates the number $0$.
\end{itemize}
To simplify notation, we will use $x < y$ instead of $<(x, y)$ and
$x'$ instead of $'(x)$. The theory $\Delta_{FOL}(\mc{M}(\alpha))$
consists of the following axioms:
\begin{itemize}
\item Axioms establishing the properties of $'$ and $<$:
\begin{align}
&\forall z \exists x (z = x'), \tag{A1} \label{existe-suc}\\
&\forall z \forall x \forall y (((z=x') \wedge (z=y')) \to (x=y)), \tag{A2} \label{unicidad-suc}\\
&\forall x \forall y \forall z (((x<y) \wedge (y<z)) \to (x<z)), \tag{A3} \label{trans-menorque}\\
&\forall x (x < x'), \tag{A4} \label{relac-suc-menorque}\\
&\forall x \forall y ((x<y) \to (x \neq y)). \tag{A5} \label{antireflex-menorque}
\end{align}
\item An axiom for each instruction $i_j$ of $\mc{M}$. The axiom is defined depending
respectively on the instruction type \eqref{inst-i}, \eqref{inst-ii} or \eqref{inst-iii}
as:
\begin{multline}
\forall t \forall x \Biggl(\biggl(Q_i(t, x) \wedge S_j(t, x)\biggr) \to \biggl(Q_l(t', x) \wedge S_k(t', x) \wedge \\ \forall y \Bigl((y \neq x) \to
\Bigl(\bigwedge_{i=1}^{m}\bigr(S_i(t, y) \to S_i(t', y)\bigr)\Bigr)\Bigr)\biggr)\Biggr), \tag{A$i_{\msf{j}}$ \eqref{inst-i}} \label{ax-inst-i}
\end{multline}
\begin{equation}
\forall t \forall x \Biggl(\biggl(Q_i(t, x) \wedge S_j(t, x)\biggr) \to \biggl(Q_l(t', x') \wedge \\ \forall y \Bigl(\bigwedge_{i=1}^{m}\bigl(S_i(t, y) \to
S_i(t', y)\bigr)\Bigr)\biggr)\Biggr), \tag{A$i_{\msf{j}}$ \eqref{inst-ii}} \label{ax-inst-ii}
\end{equation}
\begin{equation}
\forall t \forall x \Biggl(\biggl(Q_i(t, x') \wedge S_j(t, x')\biggr) \to \biggl(Q_l(t', x) \wedge \\ \forall y \Bigl(\bigwedge_{i=1}^{m}\bigl(S_i(t, y) \to
S_i(t', y)\bigr)\Bigr)\biggr)\Biggr). \tag{A$i_{\msf{j}}$ \eqref{inst-iii}} \label{ax-inst-iii}
\end{equation}
\item An axiom to specify the initial configuration of the machine. Considering the input
$\alpha = s_{i_0} s_{i_1} \ldots s_{i_{p-1}}$, where $p$ represents the length of
$\alpha$, this axiom is defined by:
\begin{multline}
Q_1(0, 0) \wedge \left(\bigwedge_{j=0}^{p-1} S_{i_{j}}(0, 0^{j})\right) \wedge \forall y \left(\left(\bigwedge_{j=0}^{p-1} y \neq 0^{j}\right) \to S_1(0, y)\right), \tag{A$\alpha$} \label{init-conf}
\end{multline}
where $0^{j}$ means $j$ iterations of the successor ($'$) function to constant $0$.
\end{itemize}
In \cite{Boolos-Jeffrey-1989}, a sentence $H$ is defined to
represent the halting of the computation, and it is thus proved
that $\Delta_{FOL}(\mc{M}(\alpha)) \vdash H$ iff the machine
$\mc{M}$ with input $\alpha$ halts. In this way, the decidability
of first-order logic implies the solution for the \emph{halting
problem}, a well-known unsolvable problem; this proves (by
\emph{reductio ad absurdum}) the undecidability of first-order
logic. For Boolos and Jeffrey's aims,
$\Delta_{FOL}(\mc{M}(\alpha))$ theories are strong enough, but
our purpose here is to attain a precise logical representation of TM
computations. Therefore, we will formally define the notion of
representability of a TM computation and show that new axioms must
be added to $\Delta_{FOL}(\mc{M}(\alpha))$ theories. Our
definition of the representation of a TM computation (Definition
\ref{def-rep-comp}) is founded upon the definitions of the
representation of functions and relations (Definition
\ref{def-rep-func} and \ref{def-rep-rel}) in theories introduced
by Alfred Tarski in collaboration with Andrzej Mostowski and
Raphael M. Robinson in \cite{Tarski-Mostowski-Robinson-1953}.
\begin{definition}\label{def-rep-func}
Let $f$ be a function of arity $k$, $\Delta$ an arbitrary theory and $\varphi(x_1, \ldots, x_k, x)$ a wff (with $k + 1$ free variables) in $\Delta$. The function $f$ is \emph{represented} by $\varphi$ in $\Delta$ if $f(m_1, \ldots, m_k) = n$ implies (bars are used to denote numerals):
\begin{enumerate}
\item $\Delta \vdash \varphi(\bar{m_1}, \ldots, \bar{m_k}, \bar{n})$,\label{def-rep-func-cond-i}
\item if $n \neq p$ then $\Delta \vdash \neg \varphi(\bar{m_1}, \ldots, \bar{m_k}, \bar{p})$, and\label{def-rep-func-cond-ii}
\item $\Delta \vdash \varphi(\bar{m_1}, \ldots, \bar{m_k}, \bar{q}) \rightarrow \bar{q} = \bar{n}$.\label{def-rep-func-cond-iii}
\end{enumerate}
\end{definition}
\begin{definition}\label{def-rep-rel}
Let $R$ be a relation of arity $k$, $\Delta$ an arbitrary theory and $\varphi(x_1, \ldots, x_k)$ a wff (with $k$ free variables) in $\Delta$. The relation $R$ is \emph{represented} by $\varphi$ in $\Delta$ if:
\begin{enumerate}
\item $(m_1, \ldots, m_k) \in R$ implies $\Delta \vdash \varphi(\bar{m_1}, \ldots, \bar{m_k})$, and\label{def-rep-rel-cond-i}
\item $(m_1, \ldots, m_k) \notin R$ implies $\Delta \vdash \neg \varphi(\bar{m_1}, \ldots, \bar{m_k})$.\label{def-rep-rel-cond-ii}
\end{enumerate}
\end{definition}
\begin{definition}\label{def-rep-comp}
Let $\mc{M}$ be a TM, $\alpha$ the input for $\mc{M}$, and
$\mu(\mc{M}(\alpha)) = \langle \mbb{Z}, Q_1^{\mu}, Q_2^{\mu}, \dots, Q_n^{\mu}, S_1^{\mu}, S_1^{\mu}, \dots, S_{m}^{\mu}, <^{\mu}, '^{\mu}, 0^{\mu} \rangle$
the structure determined by the intended interpretation $\mc{I}$.\footnote{
$\mbb{Z}$ represents the integers, the
relations $Q_i^{\mu}$ express couples of instants of time and positions for
states $q_i$ in the computation of $\mc{M}(\alpha)$, relations $S_j^{\mu}$
express couples of instants of time and positions for symbols $s_j$ in the computation
of $\mc{M}(\alpha)$, $<^{\mu}$ is the standard strict order on $\mbb{Z}$, $'^{\mu}$
is the successor function on $\mbb{Z}$ and $0^{\mu}$ is the integer $0$.}.
A theory $\Delta$, in the language $\mc{L} = \{Q_1, Q_2, \ldots, Q_n, S_0, S_1, \ldots, S_{m-1}, <, ', 0\}$,
\emph{represents the computation of $\mc{M}(\alpha)$} if:
\begin{enumerate}
\item $<^{\mu}$ is represented by $\varphi(x, y) := x < y$ in $\Delta$,
\item $'^{\mu}$ is represented by $\varphi(x, y) := x' = y$ in $\Delta$,
\item $Q_i^{\mu}$ ($i = 1, \ldots, n$) are represented by $\varphi(x, y) := Q_i(x, y)$ in $\Delta$, and
\item $S_j^{\mu}$ ($j = 1, \ldots, m$) are represented by $\varphi(x, y) := S_j(x, y)$ in $\Delta$.
\end{enumerate}
\end{definition}
\begin{theorem}\label{theo-delta-not-rep}
Let $\mc{M}$ be a TM and $\alpha$ the input for $\mc{M}$. The
theory $\Delta_{FOL}(\mc{M}(\alpha))$ cannot represent the
computation of $\mc{M}(\alpha)$.
\begin{proof}
We show that condition \ref{def-rep-rel-cond-ii} of Definition \ref{def-rep-rel}
cannot be satisfied for relations $Q_i$ and $S_j$: Indeed, when $\mc{M}(\alpha)$ is in
state $q_i$, at time $t$ and position $x$, it is not in any other state
$q_j$ ($i \neq j$); in this case, we have
that $\Delta_{FOL}(\mc{M}(\alpha)) \vdash Q_i(\bar{t}, \bar{x})$
(by the proof in \cite[Chap. 10]{Boolos-Jeffrey-1989}),
but on the other hand we have
that $\Delta_{FOL}(\mc{M}(\alpha)) \nvdash \neg Q_j(\bar{t}, \bar{x})$,
because a non-standard TM with the same instructions of $\mc{M}$ (but
allowing multiple simultaneous states: starting the computation in
two-different simultaneous states, for example) also validates all
axioms in $\Delta_{FOL}(\mc{M}(\alpha))$. A similar situation occurs
with relations $S_j$. We can also define other non-standard TMs which
allow different symbols and states, on different positions of the tape,
at times before the beginning or after the end of the computation,
in such a way that the machine validates all axioms in $\Delta_{FOL}(\mc{M}(\alpha))$.
\end{proof}
\end{theorem}
Theorem \ref{theo-delta-not-rep} shows that it is necessary to
expand the theories $\Delta_{FOL}(\mc{M}(\alpha))$ in order to disallow
non-standard interpretations and to grant representation of
computations in accordance with Definition \ref{def-rep-comp}. We thus
define the notion of an \emph{intrinsic theory of the
computation of} $\mc{M}(\alpha)$ as the theory
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ by specifying which new
axioms have to be added to $\Delta_{FOL}(\mc{M}(\alpha))$
theories, so that these extended theories are able to represent
their respective TM computations (Theorem
\ref{theo-delta-exp-rep}). For the specification of such axioms,
we will suppose that before the beginning of any computation and
after the end of any computation (if the computation halts), the
machine is in none of its states and no symbols (not even the
empty symbol) occurs anywhere in its tape. New axioms are defined
as follows:
\begin{itemize}
\item An axiom to define the situation of $\mc{M}(\alpha)$ before the beginning of
the computation:
\begin{equation}
\forall t \forall x \left((t < 0) \to \left(\left(\bigwedge_{i=1}^{n} \neg Q_i(t, x)\right) \wedge \left(\bigwedge_{j=1}^{m} \neg S_j(t, x)\right)\right)\right).\tag{A$t0$} \label{ax-t-0}
\end{equation}
\item An axiom to define the situation of $\mc{M}(\alpha)$ after the
end of the computation (if the computation halts):
\begin{multline}
\forall t \forall x \Biggr(\neg \Biggl(\bigvee_{q_i s_j \in I} \biggl(Q_i(t, x) \wedge S_j(t, x)\biggr)\Biggr) \to \\
\forall u \forall y \Biggl( t < u \to \biggl(\biggl(\bigwedge_{i=1}^{n}\neg Q_i(u, y) \biggr) \wedge \biggl(\bigwedge_{j=1}^{m}\neg S_j(u, y)\biggr)\biggr)\Biggr)\Biggr), \tag{A$th$} \label{ax-t-halt}
\end{multline}
where subscript $q_i s_j \in I$ means that, in the disjunction, only
combinations of $q_i s_j$ coincident with the first two symbols of some instruction
of $\mc{M}$ are taken into account.
\item An axiom for any state symbol $q_i$ of $\mc{M}$
establishing the uniqueness of any state and any position in a given instant of time:
\begin{equation}
\forall t \forall x \left(Q_i(t, x) \to \left(\left(\bigwedge_{j \neq i} \neg Q_j(t, x)\right) \wedge \forall y \left(y \neq x \to \bigwedge_{i=1}^{n}\neg Q_i(t, y)\right)\right)\right). \tag{A$q_i$} \label{ax-unity-state}
\end{equation}
\item An axiom for any read/write symbol $s_i$ of $\mc{M}$
establishing the uniqueness of any symbol in a given instant of time and position:
\begin{equation}
\forall t \forall x \left(S_i(t, x) \to \bigwedge_{i \neq j} \neg S_j(t, x)\right).\tag{A$s_j$}\label{ax-unity-symbol}
\end{equation}
\end{itemize}
\begin{theorem}\label{theo-delta-exp-rep}
Let $\mc{M}$ be a TM and $\alpha$ the input for $\mc{M}$. Then, the
intrinsic theory $\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ represents
the computation of $\mc{M}(\alpha)$.
\begin{proof}
Representation for the relation $<^{\mu}$ and for the function
$'^{\mu}$ is easy to proof. Representation for relations
$Q_i^{\mu}$ and $S_j^{\mu}$ follows from the proof in \cite[Chap.
10]{Boolos-Jeffrey-1989} and direct applications of the new
axioms in the intrinsic theory
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$.
\end{proof}
\end{theorem}
The definitions and theorems above consider only
DTMs (i.e, TMs with no pairs of instructions with the same two initial symbols);
the next theorem establishes that the method of axiomatization
defined above, when used to NDTMs, produces
contradictory theories (in some cases).
\begin{theorem}\label{theo-NDTM-cont}
Let $\mc{M}$ be a NDTM and $\alpha$ an input for $\mc{M}$. If
$\mc{M}(\alpha)$ reaches an ambiguous configuration (i.e. a configuration where multiple instructions can be executed), then its
intrinsic theory $\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ is
contradictory.
\begin{proof}
By the proof in \cite[Chap. 10]{Boolos-Jeffrey-1989}, it is deduced a formula that expresses the ambiguous configuration. Then, by using theorems corresponding to the possible instructions that can be executed in the ambiguous configuration, there are deduced formulas expressing multiplicity of states, positions or symbols in some cell of the tape. Thus, by using axiom \eqref{ax-unity-state} or \eqref{ax-unity-symbol}, a contradiction is deduced.
\end{proof}
\end{theorem}
In \cite[p. 48]{Odifreddi-1989}, Odifreddi, in his definition of a
TM, establishes a condition of ``consistency'' for the machine
disallowing the existence of ``contradictory'' instructions (i.e.,
instructions with the same two initial symbols), which
corresponds to the notion of DTM. Thus, NDTMs are those that do not accomplish
the condition of consistency. Theorem
\ref{theo-NDTM-cont} shows that Odifreddi's idea of consistency in
TMs coincides with the consistency of the intrinsic theories
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$.
Note that contradictions in intrinsic theories
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ arise by the multiple use
of axioms \eqref{ax-inst-i}, \eqref{ax-inst-ii} and
\eqref{ax-inst-iii} for the same instance of $t$ in combination
with the use of axioms \eqref{ax-unity-state} or
\eqref{ax-unity-symbol}. The multiple use of axioms
\eqref{ax-inst-i}, \eqref{ax-inst-ii} and \eqref{ax-inst-iii} for
the same instance of $t$ indicates the simultaneous execution of
multiple instructions, which can derive (at time $t + 1$)
multiplicity of symbols in the cell of the tape where the
instruction is executed, or multiplicity of states and positions,
while axioms \eqref{ax-unity-state} and \eqref{ax-unity-symbol}
establish the uniqueness of such elements. Intrinsic theories
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ can be easily adapted to
deal with the idea that only one instruction is chosen to be
executed when the machine reaches an ambiguous configuration,
obtaining adequate theories for NDTMs computations. However, this
is not our focus in this paper. We are interested in
generalizing the notion of TM by using a paraconsistent logic, as
this is a fecund way of approaching quantum computing from a
logical viewpoint.
\section{Paraconsistent TMs}\label{ptms}
There are many paraconsistent logics. They are proposed from different
philosophical perspectives but share the common feature of being
logics which support contradictions without falling into deductive
trivialization. Although in the definition of ParTMs we could, in
principle, depart from any first-order paraconsistent logic, we
will use the logic $LFI1^*$ (see
\cite{Carnielli-Marcos-deAmo-2000}), because it possesses an already
established proof-theory and first-order semantics, has properties
that allows natural interpretations of consequences of
$\Delta^{\star}_{LFI1^*}(\mc{M}(\alpha))$
theories\footnote{Intrinsic theories
$\Delta^{\star}_{LFI1^*}(\mc{M}(\alpha))$ are obtained
by substituting the underlying logic of
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ theories by $LFI1^*$.} as
`paraconsistent computations', and also allows the addition of
conditions to control the execution of instructions involving
multiplicity of symbols and states.\footnote{It is worth to
remark that the choice of another paraconsistent logic, with
other features, can lead to different notions of ParTMs, as is
the case in Sec. \ref{sim-ent-states-rel-phases}.} $LFI1^*$ is the
first-order extension of $LFI1$, which is an LFI that extends
positive classical logic, defines connectives of
consistency $\circ$ and inconsistency $\bullet$,
and identifies inconsistency with contradiction by means of the
equivalence $\bullet A \leftrightarrow (A \wedge \neg A)$.
For intrinsic theories $\Delta^{\star}_{LFI1^*}(\mc{M}(\alpha))$
the proof in \cite[Chap. 10]{Boolos-Jeffrey-1989} continues to
hold, because $LFI1^*$ is an extension of positive classical
logic. Thus, as described above, the use of multiple axioms
describing instructions for the same instance of $t$ indicates
simultaneous execution of the instructions, which gives place to
multiplicity of symbols in the cell and multiplicity of states
and positions. Such a multiplicity, in conjunction with axioms
\eqref{ax-unity-state} and \eqref{ax-unity-symbol}, entails
contradictions which are identified in $LFI1^*$ with
inconsistencies. Thus, inconsistency in $\Delta^{\star}_{LFI1^*}(\mc{M}(\alpha))$ theories
characterize multiplicity.
By taking advantage of the robustness of $LFI1^*$ in the presence
of inconsistencies and their interpretation as multiplicity, we
can supply the ParTMs with conditions of inconsistency in the
two initial symbols of instructions in order to control the
process of computation. $q_i^{\bullet}$ will indicate that the
instruction will only be executed in configurations where the
machine is in multiple states or multiple positions, and
$s_j^{\bullet}$ will indicate that the instruction will only be
executed in cells with multiple symbols. These conditions
correspond to put the connective $\bullet$, respectively, in
front of the predicate $Q_i$ or $S_j$ in the antecedent of the axioms
related to the instructions. These apparently innocuous conditions are
essential for taking advantage of the parallelism provided by ParTMs and EParTMs.
As will be argued below, inconsistency conditions on the instructions seem to be
a more powerful mechanism than quantum interference, which is the instrument provided
by quantum computation taking advantage of quantum parallelism.
Note that axioms \eqref{ax-inst-i}, \eqref{ax-inst-ii} and
\eqref{ax-inst-iii} not only express the action of instructions
but also specify the preservation of symbols unmodified by the
instructions. Thus, in ParTMs, we have to take into account that any
instruction is executed in a specific position on the tape,
carrying symbols from cells not modified by the instruction to the next
instant of time; this is completed independently of the execution of other
instructions.
A ParTM is then defined as:
\begin{definition}
A \emph{ParTM} is a NDTM such that:
\begin{itemize}
\item When the machine reaches an ambiguous configuration it \emph{simultaneously} executes
all possible instructions, which can produce multiplicity on states, positions and symbols
in some cells of the tape;
\item Each instruction is executed in the position corresponding to the respective
state; symbols in cells unmodified by the instructions are carried to the next instant
of time;
\item \emph{Inconsistency} (or \emph{multiplicity}) conditions are allowed on the first
two symbols of the instructions (as described above);
\item The machine stops when there are no instructions to be executed; at this stage some
cells of the tape can contain multiple symbols, any choice of them represents a result of
the computation.
\end{itemize}
\end{definition}
The next example illustrates how a ParTM performs computations:
\begin{example}\label{exam-ParTM}
Let $\mc{M}$ be a ParTM with instructions: $i_1: q_1 0 0 q_2$,
$i_2: q_1 0 1 q_2$, $i_3: q_2 0 R q_3$, $i_4: q_2 1 R q_3$, $i_5:
q_3 \emptyset 1 q_4$, $i_6: q_4 0 0 q_5$, $i_7: q_4 1 0 q_5$,
$i_8: q_4 1^{\bullet} * q_5$, $i_9: q_5 * 1 q_5$. Figure
\ref{fig-comp-ParTM} schematizes the computation of $\mc{M}$,
beginning in position $0$, state $q_1$ and reading the symbol $0$
(with symbol $\emptyset$ in all other cells of the tape).
Instructions to be executed in each instant of time $t$ are
written within parentheses (note that instruction $i_8$ is not
executed at time $t = 3$ because of the inconsistency condition
on the scanned symbol). $\mc{M}$ will
be useful in the paraconsistent solution of Deutsch's and
Deutsch-Jozsa problems (Sec. \ref{sim-D-DJ-prob}).
\begin{figure}
\caption{\scriptsize Example of computation in a ParTM}
\label{fig-comp-ParTM}
\end{figure}
\end{example}
\subsection{Simulating Quantum Computation through Paraconsistent TMs}\label{sim-qc-ptms}
In the Neumann-Dirac formulation, quantum mechanics is
synthesized in four postulates (cf. \cite[Sec.
2.2]{Nielsen-Chuang-2000}): The first postulate establishes that
states of isolated physical systems are represented by unit
vectors in a Hilbert space (known as the \emph{space state} of
the system); the second postulate claims that the evolution of closed
quantum systems is described by unitary transformations in the
Hilbert space; the third postulate deals with observations of
physical properties of the system by relating physical properties
with Hermitian operators (called \emph{observables}) and
establishing that, when a measurement is performed, an eigenvalue
of the observable is obtained (with a certain probability
depending upon the state of the system) and the system collapses to
its respective eigenstate; finally, the fourth postulate
establishes the tensor product of the state spaces of the component
systems as being the state space of the compound system,
allowing us to represent the state of a compound system as the tensor
product of the state of its subsystems (when the states of the
subsystems are known).
The best known models of quantum computation, namely \emph{quantum
Turing machines} (QTMs) and \emph{quantum circuits} (QCs), are
direct generalizations of TMs and boolean
circuits, respectively, using the laws of quantum mechanics.
By taking into account the postulates of quantum mechanics briefly
described above, a QTM (introduced in \cite{Deutsch-1985}) is
defined by considering elements of a TM (state, position and
symbols on the tape) as being observables of a quantum
system. The configuration of a QTM is thus represented by a unit vector in a Hilbert
space and the evolution is described by a unitary operator (with
some restrictions in order to satisfy the requirement that the machine
operates by `finite means', see \cite[p. 7]{Deutsch-1985} and
\cite{Ozawa-Nishimura-2000}). Since the configuration of a QTM
is described by a unit vector, it is generally a linear
superposition of basis states (called a \emph{superposed
state}), where the basis states represent classical TM
configurations. In quantum mechanics, superposed states can be
interpreted as the coexistence of multiple states, thus a QTM
configuration can be interpreted as the simultaneous existence of
multiple classical TM configurations. The linearity of the
operator describing the evolution of a QMT allows us to think in
the parallel execution of the instructions over the different states
(possibly an exponential number) present in the superposition.
Unfortunately, to know the result of the computation, we have to
perform a measurement of the system and, by the third postulate
of quantum mechanics, we can obtain only one classical TM
configuration in a probabilistic way, in effect, irredeemably losing all
other configurations. The art of `quantum programming' consists
in taking advantage of the intrinsic parallelism of the model,
by using quantum interference\footnote{Quantum interference is
expressed by the addition of amplitudes corresponding to equal basis states in a superposed state.
When signs of amplitudes are the same, their sum obtains a greater amplitude; in this case, we say that the interference is \emph{constructive}.
Otherwise, amplitudes subtract and we say that the interference is \emph{destructive}. Quantum interference occurs in the evolution of one quantum state to another.} to increase amplitudes of desired states
before the measurement of the system, with the aim to solve problems more
efficiently than in the classical case.
As shown in \cite{Ozawa-Nishimura-2000}, the evolution of a QTM
can be equivalently specified by a \emph{local transition
function}\footnote{Some changes were made in the definition of
$\delta$ to deal with the quadruple notation for instructions we
are using here.} $\fp{\delta} Q \times \Sigma \times \{\Sigma \cup
\{R, L\}\} \times Q \to \mbb{C}$, which validates some conditions related
to the unitary of state vectors and the reversibility of operators. In
this definition, the transition $\delta(q_i, s_j, Op, q_l) = c$
can be interpreted as the following action of the QTM: If the
machine is in state $q_i$ reading the symbol $s_j$, it follows
with the probability amplitude $c$ that the machine will perform the
operation $Op$ (which can be either to write a symbol or to move
on the tape) and reaches the state $q_l$. The amplitude $c$ cannot
be interpreted as the probability of performing the respective
transition, as with probabilistic TMs. Indeed, QTMs do not choose
only one transition to be executed; they can perform multiple
transitions simultaneously in a single instant of time in
a superposed configuration. Moreover, configurations resulting from different
transitions can interfere constructively or destructively (if they represent the same classical configuration), respectively increasing
or decreasing the amplitude of the configuration in the superposition.
By taking into account that each choice function on the elements of
a ParTM, in an instant of time $t$, gives a classical TM
configuration; a configuration of a ParTM can be viewed as a
\emph{uniform}\footnote{A superposed state is said to be \emph{uniform} if
all states in the superposition, with amplitude different of $0$,
have amplitudes with the same magnitude.}
superposition of classical TM configurations. This way, ParTMs
seem to be similar to QTMs: We could see ParTMs as QTMs without
amplitudes (which allows us only to represent uniform superpositions).
However, in ParTMs, actions performed by different instructions
mix indiscriminately, and thus all combinations of the singular
elements in a ParTM configuration are taken into account, which
makes it impossible to represent entangled states by only considering the
multiplicity of elements as superposed states (this point is
discussed in Sec. \ref{sim-ent-states-rel-phases}). Another difference
between the ParTMs and QTMs models is that `superposed states'
in the former model do not supply a notion of relative phase (corresponding to signs of basis states in uniform superpositions),
an important feature of quantum superpositions required for quantum interference.
Such a feature, as mentioned before, is the key mechanism for taking advantage of quantum parallelism.
However, inconsistency conditions on the instructions of ParTMs allows us to take
advantage of `paraconsistent parallelism', and this seems to be a
more powerful property than quantum interference (this point is fully discussed in
Sec. \ref{sim-ent-states-rel-phases}). In spite of the differences
between ParTMs and QTMs, ParTMs are able to simulate important
features of quantum computing; in particular, they can simulate uniform non-entangled superposed quantum states and solve the Deutsch and Deutsch-Jozsa problems preserving the efficiency of the quantum algorithms, but with certain restrictions (see Sec. \ref{sim-D-DJ-prob}). In Sec.
\ref{sim-ent-states-rel-phases}, we define another model of ParTMs, based on
a paraconsistent logic endowed with a `non-separable' conjunction, which
enables the simulation of uniform entangled states and represents a
better approach for the model of QTMs. We also show that a notion
of `relative phase' can be introduced in this new model of computation.
\subsubsection{Paraconsistent Solutions for Deutsch and Deutsch-Jozsa Problems}\label{sim-D-DJ-prob}
Given an arbitrary function $\fp{f} \{0, 1\} \to \{0, 1\}$ and an
`oracle' (or black box) that computes $f$, Deutsch's problem
consists in defining a procedure to determine if $f$ is
\emph{constant} ($f(0) = f(1)$) or \emph{balanced} ($f(0) \neq
f(1)$) allowing only one query to the oracle. Classically, the procedure
seems to require two queries to the oracle in order to compute $f(0)$
and $f(1)$, plus a further step for the comparison; but by taking
advantage of the quantum laws the problem can be solved in a more efficient way,
by executing just a single query.
A probabilistic quantum solution to Deutsch's problem was first
proposed in \cite{Deutsch-1985} and a deterministic quantum
algorithm was given in
\cite{Cleve-Ekert-Macchiavello-Mosca-1998}. The deterministic
solution is usually formulated in the QCs formalism, so we
briefly describe this model of computation before presenting the
quantum algorithm.
The model of QCs (introduced in \cite{Deutsch-1989}) is defined
by generalizing the boolean circuit model in accordance with
the postulates of quantum mechanics: The classical unit of
information, the \emph{bit}, is generalized as the \emph{quantum
bit} (or \emph{qubit}), which is mathematically represented by a
unit vector in a two-dimensional Hilbert space; classical
logic gates are replaced by unitary operators; registers of
qubits are represented by tensor products and measurements
(following conditions of the third postulate above) are accomplished
at the end of the circuits in order to obtain the output of the
computation.\footnote{For a detailed introduction to QCs see
\cite{Nielsen-Chuang-2000}.} Under this definition, the QC
depicted in Figure \ref{fig-qc-Deutsch-problem} represents a
deterministic solution to Deutsch's problem.
\begin{figure}
\caption{\scriptsize QC to solve Deutsch's problem}
\label{fig-qc-Deutsch-problem}
\end{figure}
In the figure, squares labeled by $H$ represent \emph{Hadamard}
gates. A Hadamard gate is a quantum gate which performs the
following transformations ($\ket{\cdot}$ representing a vector in
Dirac's notation):
\begin{align}
\fp{H} &\ket{0} \mapsto \frac{1}{\sqrt{2}} \left(\ket{0} + \ket{1}\right) \nonumber \\
&\ket{1} \mapsto \frac{1}{\sqrt{2}} \left(\ket{0} - \ket{1}\right).
\end{align}
The rectangle labeled by $U_f$ represents the quantum oracle that
performs the operation $U_f(\ket{x, y}) = \ket{x, y \oplus
f(x)}$, where $\ket{x, y}$ represents the tensor product
$\ket{x} \otimes \ket{y}$ and $\oplus$ represents the addition module
2. Vectors $\ket{\psi_i}$ are depicted to explain, step by step, the process
of computation:
\begin{enumerate}
\item At the beginning of the computation, the input register takes the value $\ket{\psi_0} = \ket{0, 1}$;
\item After performing the two first Hadamard gates, the following superposition
is obtained:
\begin{equation}
\ket{\psi_1} = H \ket{0} \otimes H \ket{1} = \frac{1}{2}\left((\ket{0} + \ket{1}) \otimes (\ket{0} - \ket{1})\right);\label{eq-state1-qc-dp}
\end{equation}
\item By applying the operation $U_f$, one obtains:
\begin{align}
\ket{\psi_2} &= U_f \left(\frac{1}{2}\left(\ket{0, 0} - \ket{0, 1} + \ket{1, 0} - \ket{1, 1}\right)\right) \label{eq-state2-qc-dp} \\
&= \frac{1}{2}\left(\ket{0, 0 \oplus f(0)} - \ket{0, 1 \oplus f(0)} + \ket{1, 0 \oplus f(1)} - \ket{1, 1 \oplus f(1)}\right) \nonumber \\
&= \frac{1}{2}\left((-1)^{f(0)} (\ket{0} \otimes (\ket{0} - \ket{1})) + (-1)^{f(1)} (\ket{1} \otimes (\ket{0} - \ket{1}))\right) \nonumber \\
&=
\begin{cases}
\pm \left(\frac{1}{\sqrt{2}}(\ket{0} + \ket{1})\right) \otimes \left(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\right) \mbox{ if } f(0) = f(1),\\
\pm \left(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\right) \otimes \left(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\right) \mbox{ if } f(0) \neq f(1).
\end{cases} \nonumber
\end{align}
\item By applying the last Hadamard gate, one finally reaches:
\begin{equation}
\ket{\psi_3} =
\begin{cases}
\pm \ket{0} \otimes \left(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\right) \mbox{ if } f(0) = f(1),\\
\pm \ket{1} \otimes \left(\frac{1}{\sqrt{2}}(\ket{0} - \ket{1})\right) \mbox{ if } f(0) \neq f(1).
\end{cases}\label{eq-state3-qc-dp}
\end{equation}
\end{enumerate}
After a measurement of the first qubit of the state $\ket{\psi_3}$
is accomplished (on the standard basis, cf. \cite{Nielsen-Chuang-2000}),
one obtains $0$ (with probability $1$) if $f$ is constant or $1$
(with probability $1$) if $f$ is balanced.
The first step of the above QC generates a superposed state
(Eq. \eqref{eq-state1-qc-dp}), which is taken into the next step
to compute the function $f$ in parallel (Eq.
\eqref{eq-state2-qc-dp}), generating a superposition in such a way that
the relative phase of $\ket{1}$ in the first qubit differs depending on if $f$ is constant or balanced.
By applying again a Hadamard gate on the first qubit, quantum interference acts leaving the first qubit in the
basis state $\ket{0}$ if $f$ is constant or in the basis state $\ket{1}$ if $f$ is balanced (Eq. \eqref{eq-state3-qc-dp}). Thus,
by performing a measurement of the first qubit, we determine
with certainty if $f$ is constant or balanced. Note that
$U_f$ is used only once in the computation.
The ParTM in Example \ref{exam-ParTM} gives a `paraconsistent' simulation of
the quantum algorithm that solves Deutsch's problem, for the
particular case where $f$ is the constant function $1$.
Instructions $i_1$ and $i_2$, executed simultaneously at time $t
= 0$, simulate the generation of the superposed state.
Instructions $i_3$ to $i_5$ compute the constant function $1$
over the superposed state, performing in parallel the computation of
$f(0)$ and $f(1)$, and writing the results on position $1$ of the
tape. Instructions $i_6$ to $i_9$ check whether $f(0) = f(1)$, writing
$0$ on position $1$ of the tape if there is no multiplicity of
symbols on the cell (meaning that $f$ is constant) or writing $1$
in another case (meaning that $f$ is balanced). In the present case
$f(0) = f(1) = 1$, thus the execution of the instructions from $i_6$ to $i_9$
gives as result the writing of $0$ on position $1$ of the tape.
Consider a TM $\mc{M}'$ a black box that computes a function
$\fp{f} \{0, 1\} \to \{0, 1\}$. We could substitute instructions
$i_3$ to $i_5$ in Example \ref{exam-ParTM} (adequately renumbering
instructions and states from $i_6$ to $i_9$ if necessary) with the
instructions from $\mc{M}'$ in order to determine if $f$ is constant or
balanced. In this way, we define a paraconsistent simulation of the
quantum algorithm that solves Deutsch's problem. In the simulation,
$\mc{M}'$ is the analog of $U_f$ and quantum parallelism is
mimicked by the parallelism provided by the multiplicity
allowed in the ParTMs.
Notwithstanding the parallelism provided
by ParTMs, this first paraconsistent model of computation has some peculiar
properties which could give rise to `anomalies' in the process of
computation. For instance, consider a TM $\mc{M}'$ with
instructions: $i_1 = q_1 0 0 q_2$, $i_2 = q_1 1 1 q_3$, $i_3 =
q_2 0 R q_4$, $i_4 = q_3 1 R q_4$, $i_5 = q_2 1 R q_5$, $i_6 =
q_4 \emptyset 1 q_4$, $i_7 = q_5 \emptyset 0 q_5$. If $\mc{M}'$
starts the computation on position $0$ and state $q_1$, with a
single symbol ($0$ or $1$) on position $0$ of the tape, and with
symbol $\emptyset$ on all other cells of the tape, then $\mc{M}'$
computes the constant function $1$. Nevertheless, if $\mc{M}'$
begins reading symbols $0$ and $1$ (both simultaneously on cell
$0$), then it produces $0$ and $1$ as its outputs, as if $\mc{M}'$
had computed a balanced function. This example shows well how different
paths of a computation can mix indiscriminately, producing paths of computation
not possible in the TM considered as an oracle (when viewed as a classical TM) and generating undesirable results.
Therefore, only TMs that exclusively perform their possible paths of computation
when executed on a superposed state can be considered as oracles.
TMs with this property will be called \emph{parallelizable}.
Note that this restriction is not a serious limitation to our paraconsistent
solution of Deutsch's problem, because parallelizable TMs that compute any
function $\fp{f} \{0, 1\} \to \{0, 1\}$ are easy to define.
The Deutsch-Jozsa problem, first presented in
\cite{Deutsch-Jozsa-1992}, is the generalization of Deutsch's
problem to functions $\fp{f} \{0, 1\}^n \to \{0, 1\}$, where $f$
is assumed to be either constant or balanced.\footnote{$f$ is balanced
if $\card{f^{-1}(0)} = \card{f^{-1}(1)}$, where $\card{A}$
represents the cardinal of the set $A$.} A quantum solution to the
Deutsch-Jozsa problem is a direct generalization of the quantum
solution to Deutsch's problem presented above: The input register
is now constituted of $n + 1$ qubits and takes the value
$\ket{0}^{\otimes n} \otimes \ket{1}$ (where
$\ket{\cdot}^{\otimes n}$ represents the tensor product of $n$
qubits $\ket{\cdot}$); new Hadamard gates are added to act on the
new qubits in the input register and also on the first $n$
outputs of $U_f$, and $U_f$ is now a black box acting on $n + 1$
qubits, performing the operation $U_f(\ket{x_1, \ldots, x_n, y})
= \ket{x_1, \ldots, x_n, y \oplus f(x_1, \ldots, x_n)}$. In this
case, when a measurement of the first $n$ qubits is accomplished
at the end of the computation, if all the values obtained are $0$,
then $f$ is constant (with probability $1$); in another case $f$ is
balanced (with probability $1$).\footnote{Calculations are not
presented here, for details see \cite{Nielsen-Chuang-2000}.}
The paraconsistent solution to Deutsch's problem can be easily
generalized to solve the Deutsch-Jozsa problem as well: The input to
$\mc{M}$ must be a sequence of $n$ symbols $0$; instructions
$i_1$ and $i_2$ must be substituted by instructions $i_1 = q_1 0
0 q_2$, $i_2 = q_1 0 1 q_2$, $i_3 = q_1 0 R q_1$, $i_4 = q_1
\emptyset L q_3$, $i_5 = q_3 0 L q_3$, $i_6 = q_3 \emptyset R
q_4$, and the machine $\mc{M}'$ is now considered to be a parallelizable TM computing a
constant or balanced function $\fp{f} \{0, 1\}^n \to \{0, 1\}$.
Note that the paraconsistent solution to the Deutsch-Jozsa problem does not depend on
the assumption of the function to be constant or balanced; in fact, the solution can be
applied in order to distinguish between constant and non-constant functions.
This can lead to the erroneous conclusion that PartTMs are too powerful machines, in which all NP-Problems could be solved in polynomial time: Someone could mistakenly
think that, considering an oracle
evaluating propositional formulas in polynomial time, we could immediately define a ParTM solving SATISFIABILIY in polynomial time
(by defining, for instance, instructions to set values $0$ and $1$ to propositional variables, invoking the oracle to simultaneously evaluate
all possible values of the formula, and then using instructions with inconsistent conditions to establish whether any value $1$ was obtained).
However, this is not the case, because only parallelizable TMs can be taken as oracles.
As proven in Theorem \eqref{eq-comp-temp-partm-dtm}, ParTMs can be efficiently simulated
by DTMs. Then, if we had a ParTM solving SATISFIABILITY in polynomial time, this would lead to the surprising result that $P = NP$.
To avoid such a mistake, we have to take into account the restriction
of parallelizability imposed to oracles in our model: If we had a parallelizable TM to
evaluate propositional formulas, it would be easy to define a ParTM solving SATISFIABILIY in polynomial time and,
by Theorem \eqref{eq-comp-temp-partm-dtm}, we would conclude $P = NP$. This only
shows the difficulty (or impossibility) in defining a parallelizable TM able to evaluate propositional formulas.
On the other hand, Grover's quantum search algorithm and its proven optimality (see \cite[Chap. 6]{Nielsen-Chuang-2000}) implies
the non-existence of a `naive' search-based method to determine whether a function is constant or not in a time
less than $O(\sqrt{2^n})$. This shows that, in order to take advantage of
parallelism, inconsistency conditions on instructions featured by ParTMs is a
more powerful property than quantum interference. However, in the case of ParTMs, this feature does not allow us to
define more efficient algorithms than otherwise defined by classical means. The reason is that the different paths of computations may mix in this model, and consequently
we have to impose the parallelizability restriction on oracles.
In the EParTMs model defined in the next section, different paths of computation do not mix indiscriminately as in ParTMs. Thus, no restrictions on the oracles are necessary, and, as it is shown in Theorem \eqref{EParTM-csat},
this new model of computation solves all NP-problems in polynomial time. This result shows that conditions of inconsistency on the instructions
are a really efficient method to take advantage of parallelism, and that this mechanism is more powerful than quantum interference.
\subsubsection{Simulating Entangled States and Relative Phases}\label{sim-ent-states-rel-phases}
In quantum theory, if we have $n$ physical systems with state
spaces $H_1, \ldots, H_n$ respectively, the system composed by
the $n$ systems has the state space $H_1 \otimes
\ldots \otimes H_n$ (in this case, $\otimes$ represent the
tensor product between state spaces) associated to it. Moreover, if we have
that the states of the $n$ component systems are respectively
$\ket{\psi_1}, \ldots, \ket{\psi_n}$, then the state of the
composed system is $\ket{\psi_1} \otimes \ldots \otimes
\ket{\psi_n}$. However, there are states in composed systems that
cannot be described as tensor products of the states of the
component systems; these states are known as \emph{entangled
states}. An example of a two qubit entangled state is $\ket{\psi}
= \frac{1}{\sqrt{2}} (\ket{00} + \ket{11})$.
Entangled states enjoy the property that a measurement of one state in the
component system affects the state of other component systems,
even when systems are spatially separated. In this way, singular (one particle)
systems lose identity, because their states are only describable in
conjunction with other systems. Entanglement is one of the
more (if not the most) puzzling characteristics of quantum
mechanics, with no analogue in classical physics. Many quantum
computing researchers think that entangled states play a
definite role in the definition of efficient quantum algorithms,
but this is not a completely established fact; any elucidation
about this would be of great relevance. In this direction, we are going
to show how the concept of entanglement can be expressed in
logical terms, and we will define a new model of paraconsistent
TMs (EParTMs) in which uniform entangled states are well
represented.
As mentioned before, choice functions over the different
elements (state, position and symbol on the cells of the tape) of
a ParTM, in a given instant of time $t$, determine a classical TM
configuration. Then, a configuration of a ParTM can be viewed as
a uniform superposition of classical TM configurations where all
combinations of singular elements are taken into account.
Ignoring amplitudes, the tensor product of composed systems coincides with
all combinations of the basis states present (with amplitude greater than $0$) in the component systems.
For instance, if a system $S_1$ is in state $\ket{\psi_1} =
\ket{a_{i_1}} + \ldots + \ket{a_{i_n}}$ and a system $S_2$ is in
state $\ket{\psi_2} = \ket{b_{j_1}} + \ldots + \ket{b_{j_m}}$,
then the composed system of $S_1$ and $S_2$ is in state
$\ket{\psi_{1,2}} = \ket{a_{i_1} b_{j_1}} + \ldots +
\ket{a_{i_1} b_{j_m}} + \ldots + \ket{a_{i_n} b_{j_1}} + \ldots +
\ket{a_{i_n} b_{j_m}}$. This rule can be applied $n - 1$ times to
obtain the state of a system composed by $n$ subsystems.
Consequently, just by interpreting multiplicity of elements as superposed
states, ParTMs cannot represent entangled states, because all of
their configurations can be expressed as tensor products of
their singular elements. This is why we define the new model
of EParTMs, or ``entangled paraconsistent TMs'' (cf.
Definition~\ref{EParTM}).
In a ParTM configuration all combinations of its singular elements
are taken into account in the execution of its instructions (and also in the reading of the results).
This is because the logic $LFI1^*$, used in
the definition of the model, validates the \emph{rule of
separation} (i.e. $\vdash_{\text{LFI1}^*} A \wedge B$ implies
$\vdash_{\text{LFI1}^*} A$ and $\vdash_{\text{LFI1}^*} B$) and the
\emph{rule of adjunction} (i.e. $\vdash_{\text{LFI1}^*} A$ and
$\vdash_{\text{LFI1}^*} B$ implies $\vdash_{\text{LFI1}^*} A
\wedge B$). Then, for instance, if
$\Delta^{\star}_{\text{LFI1}^*}(\mc{M}(n))\vdash Q_1(\overline{t},
\overline{x}) \wedge S_1(\overline{t}, \overline{x})$ and
$\Delta^{\star}_{\text{LFI1}^*}(\mc{M}(n))\vdash Q_2(\overline{t},
\overline{x}) \wedge S_2(\overline{t}, \overline{x})$, it is also
possible to deduce $\Delta^{\star}_{\text{LFI1}^*}(\mc{M}(n))\vdash
Q_1(\overline{t}, \overline{x}) \wedge S_2(\overline{t},
\overline{x})$ and $\Delta^{\star}_{\text{LFI1}^*}(\mc{M}(n))\vdash
Q_2(\overline{t}, \overline{x}) \wedge S_1(\overline{t},
\overline{x})$.
By the previous explanation, if we want to define a model of
paraconsistent TMs where configurations are not totally mixed,
we have to consider a paraconsistent logic where the rule of
separation or the rule of adjunction are not both valid.
There exist non-adjunctive paraconsistent logics,\footnote{The
most famous of them is the \emph{discussive} (or
\emph{discursive}) logic $D2$, introduced by Stanis\l aw
Ja\'{s}kowski in \cite{Jaskowski-1948} and \cite{Jaskowski-1949},
with extensions to first order logic and with possible
applications in the axiomatization of quantum theory, cf.
\cite{daCosta-Doria-1995}.} but paraconsistent systems where the rule
of separation fails have never been proposed.
Moreover, despite the fact that non-adjunctive paraconsistent logics appear
to be an acceptable solution to avoid the phenomenon of complete mixing in ParTMs, the
notion of entanglement seems to be more related with the failure of
the rule of separation: Indeed, an entangled state describes the
`conjunctive' state of a composed system, but not the state of
each single subsystem. Thus, in order to define a model of
paraconsistent TMs better approaching the behavior of QTMs, we
first define a paraconsistent logic with a \emph{non-separable}
conjuction.
By following the ideas in \cite{Beziau-2002} (see also
\cite{Beziau-2005}), a paraconsistent negation $\neg_{\diamond}$ is defined into the
well-known modal system $S5$ (departing from classical negation $\neg$)
by $\neg_{\diamond} A \eqdef \diamond \neg A$
(some properties of this negation are presented in the referred
papers). We now define a \emph{non-separable} conjunction $\wedge_{\diamond}$
into $S5$ by $A \wedge_{\diamond} B \eqdef \diamond (A \wedge B)$, where $\wedge$
is the classical conjunction. Some properties of this conjunction are
the following:
\begin{align}
&\vdash_{S5} A \wedge_{\diamond} B \mbox{ does not imply neither } \vdash_{S5} A \mbox{ nor } \vdash_{S5} B, \tag{$\wedge_{\diamond}1$}\label{ns-conj-prop-ns}\\
&\vdash_{S5} A \mbox{ and } \vdash_{S5} B \mbox{ implies } \vdash_{S5} A \wedge_{\diamond}B, \tag{$\wedge_{\diamond}2$}\label{ns-conj-prop-ad}\\
&\nvdash_{S5} \left(A \wedge_{\diamond}(B \wedge_{\diamond}C)\right) \leftrightarrow \left((A \wedge_{\diamond}B) \wedge_{\diamond}C\right), \tag{$\wedge_{\diamond}3$}\label{ns-conj-prop-nass}\\
&\vdash_{S5} (A \wedge_{\diamond}B) \leftrightarrow (B \wedge_{\diamond}A), \tag{$\wedge_{\diamond}4$}\label{ns-conj-prop-conm}\\
&\nvdash_{S5} \left((A \wedge_{\diamond}B) \wedge (C \wedge_{\diamond}D)\right) \rightarrow \left((A \wedge_{\diamond}D) \vee (C \wedge_{\diamond} B))\right)\tag{$\wedge_{\diamond}5$}\label{ns-conj-prop-comb}\\
&\vdash_{S5} (A_1 \wedge_{\diamond}(A_2 \wedge \ldots \wedge A_n)) \leftrightarrow \diamond (A_1 \wedge \ldots \wedge A_n). \tag{$\wedge_{\diamond}6$}\label{ns-conj-prop-mult-conj}
\end{align}
Property \eqref{ns-conj-prop-ns} reflects the non-separable
character of $\wedge_{\diamond}$, while \eqref{ns-conj-prop-ad} shows that
$\wedge_{\diamond}$ validates the rule of adjunction and
\eqref{ns-conj-prop-nass} grants the non-associativity
of $\wedge_{\diamond}$. \eqref{ns-conj-prop-conm} shows that $\wedge_{\diamond}$ is
commutative, \eqref{ns-conj-prop-comb} is a consequence of
\eqref{ns-conj-prop-ns} related with the expression of entangled
states, and \eqref{ns-conj-prop-mult-conj} is a simple
application of the definition of $\wedge_{\diamond}$ which will be useful
below.
A paraconsistent non-separable logic, which we will call $PNS5$, can be `extracted'
from the modal logic $S5$ (as much as done for negation in \cite{Beziau-2002})
by inductively defining a translation $\fp{*} ForPNS5 \to ForS5$
as:\footnote{Where $ForPNS5$ is the set of propositional formulas
generated over the signature $\sigma = \{\neg_{\diamond}, \wedge_{\diamond}, \vee,
\to\}$ (defined in the usual way) and $ForS5$ is the set of
formulas of $S5$.}
\begin{align*}
&A^* = A \mbox{ if $A$ is atomic},\\
&(\neg_{\diamond} A)^* = \diamond \neg (A)^*, \\
&(A \wedge_{\diamond}B)^* = \diamond (A^* \wedge B^*),\\
&(A \# B)^* = A^* \# B^* \mbox{ for $\# \in \{\vee, \to\}$};
\end{align*}
and by defining a consequence relation in the wffs of $PNS5$ as:
\begin{equation*}
\Gamma \vdash_{PNS5} A \mbox{ iff } \Gamma^* \vdash_{S5} A^*,
\end{equation*}
where $\Gamma$ represents a subset of $ForPNS5$ and $\Gamma^* =
\{B^* | B \in \Gamma \}$. This translation completely specifies
$PNS5$ as a sublogic of $S5$ with the desired properties.
In the spirit of the LFIs (see
\cite{Carnielli-Coniglio-Marcos-2007}), we can define a
connective $\bullet$ of `inconsistency' in $PNS5$ by $\bullet A \eqdef A
\wedge_{\diamond}\neg_{\diamond} A$ (which is equivalent to $\diamond A
\wedge \diamond \neg A$ in $S5$), a connective $\circ$ of `consistency'
by $\circ A \eqdef \neg_{\diamond} \bullet A$ (which is equivalent
to $\square \neg A \vee \square A$ in $S5$), a classical negation $\neg$
by $\neg A \eqdef \neg_{\diamond} A \wedge_{\diamond}\circ A$ (which is
equivalent to $\diamond \neg A \wedge (\square \neg A \vee
\square A)$ in $S5$, entailing $\neg A$) and a
classical conjunction by $A \wedge B \eqdef (A \wedge_{\diamond} B) \wedge_{\diamond} (\circ A \wedge_{\diamond} \circ B)$
(which is equivalent to $\diamond (A \wedge B) \wedge (\square (A \wedge B) \vee
\square (A \wedge \neg B) \vee \square (\neg A \wedge B) \vee
\square (\neg A \wedge \neg B))$ in $S5$, entailing $A \wedge B$). Consequently, the ``explosion principles''
$(A \wedge \neg_{\diamond} A \wedge \circ A) \rightarrow B$, $(A \wedge_{\diamond} (\neg_{\diamond} A \wedge_{\diamond} \circ A)) \rightarrow B$,
$((A \wedge_{\diamond} \neg_{\diamond} A) \wedge_{\diamond} \circ A) \rightarrow B$ and
$((A \wedge_{\diamond} \circ_{\diamond} A) \wedge_{\diamond} \neg A) \rightarrow B$
are theorems of $PNS5$; in this way, $PNS5$ is a legitimate logic of formal inconsistency (cf. \cite{Carnielli-Coniglio-Marcos-2007}).
These definitions also allow us to fully embed classical
propositional logic into $PNS5$.
With the aim to use the logic $PNS5$ (instead of $LFI1^*$) in the definition of
EParTMs, we first need to extend $PNS5$ to first-order logic
with equality. This can be obtain by considering $S5Q^{=}$ (the first-order version of $S5$, with equality)
instead of $S5$ in the definition of the logic, and adjusting the translation function $*$ to deal with quantifiers and equality.
However, for the shake of simplicity, we will consider just $S5Q^{=}$ in the definition of the model,
and we will regard the connectives $\neg_{\diamond}, \wedge_{\diamond}, \bullet$
and $\circ$ as definitions into this logic. Then, we will substitute the underlying logic of intrinsic theories
$\Delta^{\star}_{FOL}(\mc{M}(\alpha))$ by $S5Q^{=}$, and through the
Kripkean interpretation of $\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha))$
theories, we will define what is a EParTM. Before that, we need
to identify which kind of negation ($\neg$ or $\neg_{\diamond}$) and
conjunction ($\wedge$ or $\wedge_{\diamond}$) are adequate in each axiom
of $\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha))$ (we will consider right-associative
conjunction, i.e., $A \wedge B \wedge C$ always mean $A \wedge (B
\wedge C)$; this is necessary a proviso because of the non-associativity of
$\wedge_{\diamond}$, cf. property \eqref{ns-conj-prop-nass}):
\begin{enumerate}
\item In axioms \eqref{existe-suc}-\eqref{antireflex-menorque}, negations and conjunctions are the classical ones;
\item in axioms \eqref{ax-inst-i}-\eqref{ax-inst-iii}, the conjunction in the antecedent is $\wedge_{\diamond}$, (considering \eqref{ns-conj-prop-mult-conj}) only the first conjunction in the consequent is $\wedge_{\diamond}$ (other conjunctions are classical), and negation in \eqref{ax-inst-i} is classical;
\item in axioms \eqref{init-conf} and \eqref{ax-t-0}, negations and conjunctions are the classical ones;
\item in axiom \eqref{ax-t-halt}, only the conjunction in the antecedent is $\wedge_{\diamond}$, all other connectives are classical;
\item in axioms \eqref{ax-unity-state} and \eqref{ax-unity-symbol}, all conjunctions are classical, but negations are $\neg_{\diamond}$ (except in $y \neq x$), and it is also necessary to add the connective $\diamond$ before the predicates $Q_i$ and $S_i$ into the antecedent of the axioms.
\end{enumerate}
We also need to define a notion of \emph{representation} for the
configurations of the TMs by worlds in a (possible-worlds) kripkean structure:
\begin{definition}
Let $w$ be a world in a kripkean structure. If $Q_i(\overline{t}, \overline{x}), \ldots, S_{j_{-1}}(\overline{t}, -1), S_{j_{0}}(\overline{t}, 0), S_{j_{1}}(\overline{t}, 1), \ldots$ are valid predicates on $w$, we say that $w$ \emph{represents} a configuration for a TM $\mc{M}$ at time $\overline{t}$, and the configuration is given by the intended interpretation $I$ presented above.
\end{definition}
By considering the choices of connectives and the definition above,
worlds in the kripkean interpretation of
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha))$ represent the parallel
computation of all possible computational paths of a NDTM
$\mc{M}$ for the input $\alpha$:
\begin{enumerate}
\item By axiom \eqref{init-conf}, there will be a world $w_{0}$ representing the initial configuration of $\mc{M}(\alpha)$;
\item by axioms \eqref{ax-inst-i}-\eqref{ax-inst-iii}, if $w_{t}$ represents a non-final configuration of $\mc{M}(\alpha)$ at time $t$, by any instruction $i_j$ to be executed at time $t$ (on such configuration), there will be a world $w_{t+1,j}$ representing a configuration of $\mc{M}(\alpha)$ at time $t+1$.
\end{enumerate}
Configurations represented by worlds for the same instant of time
$t$ can be considered \emph{superposed configurations}. In a
superposed configuration, a state on position $x$ and a symbol on
position $y$ are said to be \emph{entangled} if there exist $i, j, k, l$
($i \neq k$ and $j \neq l$) such that
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \vdash Q_i(\overline{t},
\overline{x}) \wedge_{\diamond}S_j(\overline{t}, \overline{y})$,
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \vdash Q_k(\overline{t},
\overline{x}) \wedge_{\diamond}S_l(\overline{t}, \overline{y})$ and
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \nvdash Q_i(\overline{t},
\overline{x}) \wedge_{\diamond}S_l(\overline{t}, \overline{y})$ or
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \nvdash Q_k(\overline{t},
\overline{x}) \wedge_{\diamond}S_j(\overline{t}, \overline{y})$. In a
similar way, the notion of entangled symbols on positions $x$ and $y$ can
also be defined.
In order to exemplify the definition above, consider the two qubits entangled state
$\ket{\psi} = \frac{1}{\sqrt{2}} (\ket{00} + \ket{11})$. Suppose the first qubit of $\ket{\psi}$
represents the state on position $x$ of an EParTM $\mc{M}$ (value $\ket{0}$ representing state $q_1$ and value $\ket{1}$ representing state $q_2$),
and the second qubit of $\ket{\psi}$ represents the symbol on position $y$ of $\mc{M}$
(value $\ket{0}$ representing symbol $s_1$ and value $\ket{1}$ representing symbol $s_2$).
Regarding only the state in position $x$ and the symbol in position $y$ of $\mc{M}$, state $\ket{\psi}$ represents a configuration of $\mc{M}$,
at time instant $t$, in which only the combinations $q_1 s_1$ and $q_2 s_2$ are possible, all other combinations being impossible.
This is expressed in the theory $\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha))$ by $\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \vdash Q_1(\overline{t},
\overline{x}) \wedge_{\diamond}S_1(\overline{t}, \overline{y})$,
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \vdash Q_2(\overline{t},
\overline{x}) \wedge_{\diamond}S_2(\overline{t}, \overline{y})$,
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \nvdash Q_1(\overline{t},
\overline{x}) \wedge_{\diamond}S_2(\overline{t}, \overline{y})$ and
$\Delta^{\star}_{S5Q^{=}}(\mc{M}(\alpha)) \nvdash Q_2(\overline{t},
\overline{x}) \wedge_{\diamond}S_1(\overline{t}, \overline{y})$.
Taking into account the definition of the inconsistency
connective in $S5Q^{=}$, as in the model of ParTMs, we can
define conditions of inconsistency in the execution of
instructions in the EParTMs. In this case, by the definition of
the inconsistency connective in $S5Q^{=}$ and its kripkean
interpretation, condition $q_i^\bullet$ will indicate that the
instruction will be executed only when at least two
configurations in the superposition differ in the current state
or position, while condition $s_j^\bullet$ will indicate that the
instruction will be executed only when at least two
configurations in the superposition differ in the symbol on the
position where the instruction can be executed.
A EParTM is then defined as:
\begin{definition}\label{EParTM}
A \emph{EParTM} is a NDTM such that:
\begin{itemize}
\item When the machine reaches an ambiguous configuration with $n$ possible instructions to be executed, the machine configuration \emph{splits} into $n$ copies, executing a different instruction in each copy; the set of the distinct configurations for an instant of time $t$ is called a \emph{superposed configuration};
\item \emph{Inconsistency} conditions are allowed on the first two symbols of instructions (as indicated above);
\item When there are no instructions to be executed (in any current configuration), the machine stops; at this stage the machine can be in a superposed configuration, each configuration in the superposition represents a result of the computation.
\end{itemize}
\end{definition}
Note that a EParTM parallelly performs all possible paths of
computation of a NDTM, and only such paths. This differs from the
previous model of ParTMs, where the combination of actions of
different instructions had led to computational paths not possible
in the corresponding NDTM.
Following \cite{Bennett-1973}, it is possible to define a
reversible EParTM for any EParTM without inconsistency
conditions in instructions.\footnote{In the case of EParTMs, it
is only necessary to avoid overlapping in the ranges of
instructions; the parallel execution of all possible instructions
in an ambiguous configuration does not imply irreversibility.}
This way, EParTMs almost coincide with QTMs without amplitudes;
EParTMs represent uniform superpositions with no
direct representation of the notion of relative phase,
but does allow conditions of inconsistency on the instructions.
As mentioned before, the notion of relative phase is a key
ingredient in allowing interference between different paths
of computation in QTMs, which is essential in order to take advantage of quantum
parallelism in the efficient solution of problems; however, this method has theoretical restrictions
which disable any definition of an efficient (polynomial time) quantum algorithm solving an NP-complete problem
by a naive search-based method (see \cite[Chap. 6]{Nielsen-Chuang-2000}). On the other
hand, conditions of inconsistency on instructions provided by EParTMs
are an efficient mechanism to accomplish actions
depending on different paths of computation. In fact, Theorem \ref{EParTM-csat} proves that
all NP-problems can be efficiently solved by EParTMs.
EParTMs represent an abstract model of computation, independent of any physical implementation.
However, if we think from the physical construction of EParTMs, quantum mechanics provides a way to implement the
simultaneous execution of different paths of computation, but does not provide
what it seems to be a simple operation over the superpositions obtained by quantum parallelism: The execution
of instructions depending on differences in elements on the superposed states (which correspond to
conditions of inconsistency on instructions of EParTMs). In this way, quantum mechanics does not supply a direct theoretical frame for the implementation
of EParTMs, but this definitely does not forbid the possibility of a physical implementation of EParTMs (perhaps conditions of inconsistency could be
implemented by a sophisticated quantum physical procedure, or by a new physical theory).
We could also modify the definition of EParTMs to capture more properties of QTMs. In this way, conditions of inconsistency in instructions could be avoided, and a notion of `relative phase' could be introduced in EParTMs. This could be achieved by extending $S5Q^=$ with a new connective of possibility. Thus, the possibility connective of $S5$ (now denoted by $\diamond_1$) would represent `positive configurations' and the other possibility connective ($\diamond_2$) would represent `negative configurations' (axioms establishing the behavior of $\diamond_2$ and their interrelation with other connectives would need to be added; in particular, combinations of connectives $\diamond_1$ and $\diamond_2$ would have to behave in an analogous way to combinations of symbols $+$ and $-$). The connective $\diamond_2$ could be used to define a new paraconsistent negation as well as a new non-separable conjunction. Thus, by specifying which connectives would be used in each axiom, we could obtain a different definition of EParTMs. In this new definition, a concept of `interference' can be specified; equal configurations with the same possibility connective interfere constructively, while equal configurations with different possibility connectives interfere destructively. Although details are not given here, this construction shows once more how we can define computation models with distinct computational power by just substituting the logic underlying theories $\Delta^{\star}_{FOL}(\mc{M}(\alpha))$. In this sense, computability can be seen as relative to logic. Alternatively, we can add a new element on the EParTMs: a sign indicating the `relative phase' of the configuration, and a new kind of instructions to change the relative phase.
\subsection{About the Power of ParTMs and EParTMs}\label{comp-power-partms-EParTMs}
In order to estimate the computational power of ParTMs and EParTMs,
we first define what the `deciding' of a language
(i.e. a set of strings of symbols $L \subset \Sigma^*$, where
$\Sigma$ is a set of symbols and $^*$ represents the Kleene
closure) means in these models of computation.
In the definition, we will consider multiple results in a computation as being possible
responses from which we have to randomly select only one.
We will also suppose that ParTMs and EParTMs have two distinguished
states: $q_y$ (the \emph{accepting state}) and $q_n$ (the
\emph{rejecting state}), and that all final states of the machine
(if it halts) are $q_y$ or $q_n$.
\begin{definition}
Let $\mc{M}$ be a ParTM (EParTM) and $x$ be a string of symbols in the input/output alphabet
of $\mc{M}$. We say that $\mc{M}$ \emph{accepts} $x$
with probability $\frac{m}{n}$ if $\mc{M}(x)$ halts in a
superposition of $n$ configurations and $m$ of them are in
state $q_y$; conversely, we say that $\mc{M}$ \emph{rejects} $x$
with probability $\frac{m}{n}$ if $\mc{M}(x)$ halts in a `superposition'
of $n$ configurations and $m$ of them are in state $q_n$.
Consequently, we say that $\mc{M}$ \emph{decides} a language $L$,
with error probability at most $1 - \frac{m}{n}$,
if for any string $x \in L$, $\mc{M}$ accepts $x$ with probability at least $\frac{m}{n}$,
and for any string $x \notin L$, $\mc{M}$ rejects $x$ with probability at least $\frac{m}{n}$.
\end{definition}
Bounded-error probabilistic time complexity classes are
defined for ParTMs and EParTMs as:
\begin{definition}
BParTM-PTIME (BEParTM-PTIME) is the class of
languages decided in \emph{polynomial time} by some ParTM (EParTM), with error probability at most $\frac{1}{3}$.
BParTM-EXPTIME (BEParTM-EXPTIME) is the class of
languages decided in \emph{exponential time} by some ParTM (EParTM), with error probability at most $\frac{1}{3}$.
\end{definition}
Space complexity classes can be defined in an analogous way, considering only the
largest space used for the different superposed configurations.
Now, we will prove that ParTMs are computationally equivalent to
DTMs, showing how to simulate the computation of ParTMs by DTMs
(Theorem \ref{eq-comp-partm-dtm}). As a consequence, we have
that the class of languages decided by both models of computation
is the same. It is obvious that computations performed by DTMs
can be computed also by ParTMs, because DTMs are particular cases
of ParTMs. What is surprising is that the simulation of ParTMs by
DTMs is performed with \emph{only} a polynomial slowdown in time
(Theorem \ref{eq-comp-temp-partm-dtm}) and a constant factor
overhead in space (direct consequence of the proof of Theorem
\ref{eq-comp-partm-dtm}). Theorems \ref{eq-comp-partm-dtm} and
\ref{eq-comp-temp-partm-dtm} are inspired in the simulation of
multi-tape TMs by one-tape TMs as presented in
\cite{Hopcroft-Motwani-Ullman-2001}, and show once more how
powerful the classical model of TMs is.
\begin{theorem}\label{eq-comp-partm-dtm}
Any ParTM can be simulated by a DTM.
\begin{proof}
Let $\mc{M}$ be a ParTM with $n$ states and $m$ input/output symbols. Define a DTM $\mc{M}'$ and suppose its tape is divided into $2n + m$ tracks. Symbols $1$ and $0$ on track $i$ ($1 \leq i \leq n$) and position $p$ of $\mc{M}'$ represent respectively that $q_i$ is or is not one of the states of $\mc{M}$ in position $p$. In a similar way, symbols $1$ and $0$ on track $j$ ($n + 1 \leq j \leq n + m$) and position $p$ of $\mc{M}'$ respectively represent the occurrence or non-occurrence of symbol $s_j$ on position $p$ of $\mc{M}$. Tracks $n + m + 1$ to $2n + m$ are used to calculate states resulting from the parallel execution of instructions in $\mc{M}$, and values on these tracks represent states of $\mc{M}$ in the same way as tracks $1$ to $n$. The symbol $\$$ is used on track $1$ of $\mc{M}'$ to delimitate the area where $\mc{M}$ is in any state (i.e., where any symbol $1$ appears on some track $i$ associated to states of $\mc{M}$). To simulate a step of the computation of $\mc{M}$, $\mc{M}'$ scans the tape between delimiters $\$$ in four times. In the first scan (from left to right), $\mc{M}'$ simulates the parallel execution of instructions where the action is a movement to right: In each position, $\mc{M}'$ writes values in tracks $n + m + 1$ to $2n + m$ in accordance with states `remembered' from the previous step and collects (in the state of the machine, depending on the content of tracks $1$ to $n + m$ and the instructions of movement to right of $\mc{M}$) the states to be written in the next position of the tape; $\mc{M}'$ also moves delimiters $\$$ if necessary. The second scan is similar to the first one, but in the opposite direction and simulating instructions of movement to the left, taking care in the writing of values so as not to delete values $1$ written in the previous scan. In the third scan (from left to right), $\mc{M}'$ simulates the parallel execution of instructions where the action is the modification of symbols on the tape: In each position, depending on the content of tracks $1$ to $n + m$ and in accordance with the writing instructions of $\mc{M}$, $\mc{M}'$ writes values on tracks $n + 1$ to $n + m$ (corresponding to symbols written by instructions of $\mc{M}$) and also on tracks $n + m + 1$ to $2n + m$ (corresponding to changes of states from the writing instructions of $\mc{M}$, taking care in the writing of values so as not to delete values $1$ written in the previous scans). Finally, $\mc{M}'$ performs a fourth scan (from right to left) copying values from tracks $n + m + 1$ to $2n + m$ on tracks $1$ to $n$ and writing $0$ on tracks $n + m + 1$ to $2n + m$.
\end{proof}
\end{theorem}
\begin{theorem}\label{eq-comp-temp-partm-dtm}
The DTM of Theorem \ref{eq-comp-partm-dtm} simulates $n$ steps of the corresponding ParTM in time $O(n^2)$.
\begin{proof}
Let $\mc{M}$ be a ParTM and $\mc{M}'$ be the DTM described in the proof of Theorem \ref{eq-comp-partm-dtm} such that $\mc{M}'$ simulates the behavior of $\mc{M}$. After $n$ steps of computation, the leftmost state and the rightmost state of $\mc{M}$ cannot be separated by more than $2n$ cells, consequently this is the separation of $\$$ delimiters in the first track of $\mc{M}'$. In any scan of $\mc{M}'$, in the simulation of a step of computation of $\mc{M}$, $\mc{M}'$ has to move between $\$$ delimiters, and a writing operation can be performed in any position, thus any scan takes at most $4 n$ steps within the computation (ignoring steps due to scanning of delimiters $\$$ and their possible relocation). Therefore, the simulation of the $n$ step in the computation of $\mc{M}$ takes at most $16 n$ steps, i.e., time $O(n)$. Consequently, for the simulation of $n$ steps in $\mc{M}$, $\mc{M}'$ requires no more than $n$ times this amount, i.e., time $O(n^2)$.
\end{proof}
\end{theorem}
\begin{corollary}
The class of languages decided by ParTMs and DTMs are the same, and languages are decided with the same temporal and spatial complexity in both models.
\begin{proof}
Direct consequence of theorems \ref{eq-comp-partm-dtm} and \ref{eq-comp-temp-partm-dtm}; it is only necessary to add another scan between delimiters $\$$ at the end of the simulation to search for an accepting state, finalizing $\mc{M}'$ in its accepting state if symbol $1$ is found in the track corresponding to the accepting state of $\mc{M}$, or finalizing $\mc{M}'$ in its rejecting state if no symbol $1$ is found in the track corresponding to the accepting state of $\mc{M}$. Clearly, this additional scan takes at most a polynomial number of steps (thus preserving the temporal complexity) and does not use new space (thus preserving the spatial complexity).
\end{proof}
\end{corollary}
For EParTMs the situation is different: The class
of languages decided in both models continues to be the same
(DTMs can simulate all paths of computation of a EParTM, writing
different configurations in separate portions of the tape and
considering the different configurations in the simulation of
instructions with inconsistency conditions), but all $NP$-problems
can be \emph{deterministically} (with error probability $0$) computed
in polynomial time by EParTMs (a direct consequence of Theorem
\ref{EParTM-csat}, since the satisfiability of propositional formulas
in conjunctive normal form (CSAT) is $NP$-complete). Thus, time complexity
of EParTMs and DTMs are equal only if $P = NP$, which is broadly
believed to be false.
\begin{theorem}\label{EParTM-csat}
CSAT is in BEParTMs-PTIME.
\begin{proof}
It is not difficult to define a NDTM $\mc{M}$ deciding CSAT in polynomial time in which all computational paths have the same depth and finish in the same position of the tape. By considering $\mc{M}$ as a EParTM, all computational paths are performed in parallel, obtaining a superposition of configurations in which at least one of them is in state $q_y$ if the codified conjunctive normal form formula is satisfiable, or with all configurations in $q_n$ otherwise. Thus, by adding the instructions $i_{n+j}: q_y^\bullet s_j s_j q_y$ and $i_{n+m+j}: q_n^\bullet s_j s_j q_y$ to $\mc{M}$ (where $m$ is the number of input/output symbols of $\mc{M}$ and $1 \leq j \leq m$) we have the acceptance or rejection with probability 1.
\end{proof}
\end{theorem}
\section{Final Remarks}
In this paper, we generalize a method for axiomatize Turing
machine computations not only with foundational aims, but
also envisaging new models of computation by logical handling (basically through the
substitution of the underlying logic of the intrinsic theories in the computation),
showing a way in which logical representations can be used in the
construction of new concepts.
The new models of computation defined here use a sophisticated
logical language which permits us to express some important features of
quantum computing. The first model allows the simulation of superposed
states by means of multiplicity of elements in TMs, enabling
the simulation of some quantum algorithms but unable to
speed up classical computation. In order to overcome
this weakness, we define a second model which is able to represent entangled
states, in this way, reaching an exponential speed-up of an
$NP$-complete problem. Both models are grounded on paraconsistent
logic (LFIs). In particular, the only element in the language
that cannot be directly simulated in quantum computing is the
``inconsistency operator'' of the second model. As this is a key component
in the efficiency of the whole model, an important
problem is to decide whether it can or cannot be characterized by
quantum means.
In spite of \emph{paraconsistent computational theory} being only an
emerging field of research, we believe that this logic
relativization of the notion of computation is really promising
in the search of efficient solutions to problems, particularly
helping in the understanding of the role of quantum
features and indeterminism in computation processes.
\end{document} |
\begin{document}
\title{A Unified and Strengthened Framework for the Uncertainty Relation}
\author{Xiao Zheng}
\affiliation{
Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), School of Physics and Nuclear Energy Engineering, Beihang University, Xueyuan Road No. 37, Beijing 100191, China
}
\author{Shao-Qiang Ma}
\affiliation{
Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), School of Physics and Nuclear Energy Engineering, Beihang University, Xueyuan Road No. 37, Beijing 100191, China
}
\author{Guo-Feng Zhang}
\email{gf1978zhang@buaa.edu.cn}
\affiliation{
Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), School of Physics and Nuclear Energy Engineering, Beihang University, Xueyuan Road No. 37, Beijing 100191, China
}
\author{Heng Fan}
\affiliation{
Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China
}
\author{Wu-Ming Liu}
\affiliation{
Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China
}
\date{\today}
\begin{abstract}
We provide a unified and strengthened framework for the product form and the sum form variance-based uncertainty relations by constructing a unified uncertainty relation. In the unified framework, we deduce that the uncertainties of the incompatible observables are bounded by not only the commutator of themselves, but also the quantities related with the other operator. This operator can provide information so that we can capture the uncertainty of the measurement result more accurately, and thus is named as the information operator. The introduction of the information operator can fix the deficiencies in both the product form and the sum form uncertainty relations, and provides a more accurate description of the quantum uncertainty relation. The unified framework also proposes a new interpretation of the uncertainty relation for non-Hermitian operators; i.e., the ``observable" second-order origin moments of the non-Hermitian operators cannot be arbitrarily small at the same time when they are generalized-incompatible on the new definition of the generalized commutator.
\end{abstract}
\maketitle
Quantum uncertainty relations \cite{1,2,3}, expressing the impossibility of the joint sharp preparation of the incompatible observables \cite{4w,4}, are the most fundamental differences between quantum and classical mechanics \cite{5,6,7,28S}. The uncertainty relation has been widely used in the quantum information science, such as quantum non-cloning theorem \cite{7P,7H}, quantum cryptography \cite{CW,C7,8,9}, entanglement detection \cite{10,10C,24,11,12}, quantum spins squeezing \cite{13,14,15,16}, quantum metrology \cite{7V,17,18}, quantum synchronization \cite{18A,18F} and mixedness detection \cite{19,20}. In general, the improvement in uncertainty relations will greatly promote the development of quantum information science \cite{7V,10,20B,21P,21X}.
The variance-based uncertainty relations for two incompatible observables $A$ and $B$ can be divided into two forms: the product form ${\Delta A}^2{\Delta B}^2\geq LB_{p}$ \cite{2,3,4,28L} and the sum form ${\Delta A}^2+{\Delta B}^2\geq LB_{s}$ \cite{21,22,23,25}, where $LB_{p}$ and $LB_s$ represent the lower bounds of the two forms uncertainty relations, and ${\Delta Q}^2$ is the variance of $Q$ \cite{25F}. The product form uncertainty relation cannot fully capture the concept of incompatible observables, because it can be trivial; i.e., the lower bound $LB_{p}$ can be null even for incompatible observables \cite{21,22,4JL,28}. This deficiency is referred to as the triviality problem of the product form uncertainty relation. In order to fix the triviality problem, Maccone and Pati deduced a sum form uncertainty relation with a nonzero lower bound for incompatible observables \cite{28}, showing that the triviality problem can be addressed by the sum form uncertainty relation. Since then, lots of effort has been made to investigate the uncertainty relation in the sum form \cite{10,21,26,27,H2,42}. However, most of the sum form uncertainty relations depend on the orthogonal state to the state of the system, and thus are difficult to apply to the high dimension Hilbert space \cite{21}. There also exist the uncertainty relations based on the entropy \cite{5,6,7P,34} and skew information \cite{34S}, which may not suffer the triviality problem, but they cannot capture the incompatibility in terms of the experimentally measured error bars, namely variances \cite{28,28S}.
Here we only focus on the uncertainty relation based on the variance. Despite the significant progress on the variance-based uncertainty relation, previous work mainly studies the product form and the sum form uncertainty relations, separately. A natural question is raised : can the uncertainty relations in the two forms be integrated into a unified framework? If so, can the unified framework fix the deficiencies in the traditional uncertainty relations and provide a more accurate description of the quantum uncertainty relation? In other words, can the unified framework provide a stronger theoretical system for the quantum uncertainty relation?
In this Letter, we provide a unified framework for the product form and the sum form variance-based uncertainty relations by constructing a unified uncertainty relation. The unified framework shows that the uncertainties of the incompatible observables $A$ and $B$ are bounded by not only their commutator, but also the quantities related with the other operator, named as the information operator. Actually, the deficiencies in both the product form and the sum form uncertainty relations can be considered as having not taken the information operator into consideration and can be completely fixed by the introduction of the information operator. Furthermore, the uncertainty inequality will become an uncertainty equality when a specific number of information operators are introduced, which means the uncertainty relation can be expressed exactly with the help of the information operators. Thus the unified framework provides a strengthened theoretical system for the uncertainty relation. Meanwhile, our uncertainty relation provides a new interpretation of the uncertainty relation for non-Hermitian operators; i.e., the ``observable" second-order origin moments of the non-Hermitian operators cannot be arbitrarily small at the same time when they are generalized-incompatible on the new definition of the generalized commutator. The new interpretation reveals some novel quantum properties that the traditional uncertainty relation cannot do.
\emph{Unified Uncertainty Relation.--- }The Schr\"{o}dinger uncertainty relation (SUR) is the initial as well as the most widely used product form uncertainty relation \cite{3}:
\begin{align}
{\Delta A}^2{\Delta B}^2\geq\frac{1}{4}|\langle[A,B]\rangle|^2+\frac{1}{4}|\langle\{\check{A},\check{B}\}\rangle|^2\tag{1},
\end{align}
where $\langle Q\rangle$ represents the expected value of $Q$, $\check{Q}=Q-\langle Q\rangle$, $[A,B]=AB-BA$ and $\{\check{A},\check{B}\}=\check{A}\check{B}+\check{B}\check{A}$ represent the commutator and anti-commutator, respectively. One of the most famous sum form uncertainty relations, which have fixed the triviality problem in the product form uncertainty relation, takes the form \cite{28}:
\begin{align}
{\Delta A}^2+{\Delta B}^2\geq|\langle\psi|A\pm iB|\psi^\perp\rangle|^2\pm i\langle[A,B]\rangle \tag{2},
\end{align}
where $|\psi^\bot\rangle$ is the state orthogonal to the state of the system $|\psi\rangle$.
Before constructing the unified uncertainty relation, we first consider the non-Hermitian extension of the commutator and anti-commutator. There exist two kinds of operators in quantum mechanics: Hermitian and non-Hermitian operators, but it should be paid particular attention that lots of uncertainty relations are invalid for non-Hermitian operators \cite{37,38,39}. For instance, $|[\sigma_+,\sigma_-]|^2/4+|\{\check{\sigma}_+,\check{\sigma}_-\}|^2/4\geq{\Delta\sigma_+}^2{\Delta\sigma_-}^2$ , where the non-Hermitian operator $\sigma_+(\sigma_-)$ is the raising (lowering) operator of the single qubit system. That is to say, different from the Hermitian operators, the uncertainties of the non-Hermitian operators are not lower-bounded by quantities related with the commutator. The essential reason for this phenomenon is that $i[\mathcal{A},\mathcal{B}]$ and $\{\mathcal{A},\mathcal{B}\}$ cannot be guaranteed to be Hermitian by the existing definition of commutator and anti-commutator when the operator $\mathcal{A}$ or $\mathcal{B}$ is non-Hermitian. To fix this problem, we define the generalized commutator and anti-commutator as:
\begin{align}
[\mathcal{A},\mathcal{B}]_{\mathcal{G}}=\mathcal{A}^\dag\mathcal{B}-\mathcal{B}^\dag\mathcal{A}, \quad \{\mathcal{A},\mathcal{B}\}_{\mathcal{G}}=\mathcal{A}^\dag\mathcal{B}+\mathcal{B}^\dag\mathcal{A}\tag{3}.
\end{align}
The generalized commutator and anti-commutator will reduce to the normal ones when $\mathcal{A}$ and $\mathcal{B}$ are both Hermitian. We say that $\mathcal{A}$ and $\mathcal{B}$ are generalized-incompatible (generalized-anti-incompatible) with each other hereafter when $\langle[\mathcal{A},\mathcal{B}]_{\mathcal{G}}\rangle\neq0$ $(\langle\{\mathcal{A},\mathcal{B}\}_{\mathcal{G}}\rangle\neq0)$. Then, one can obtain a new uncertainty relation for both Hermitian and non-Hermitian operators (for more detail, please see the Unified Uncertainty Relation in the Supplemental Material \cite{35}):
\begin{align}
\langle\mathcal{A}^\dag\mathcal{A}\rangle\langle\mathcal{B}^\dag\mathcal{B}\rangle=\frac{|\langle [{\mathcal{A}},{\mathcal{B}}]_{\mathcal{G}}\rangle|^2}{4}+\frac{|\langle\{{\mathcal{A}},{\mathcal{B}}\}_{\mathcal{G}}\rangle|^2}{4}+\langle\mathcal{C}^\dag\mathcal{C}\rangle\langle\mathcal{B}^\dag\mathcal{B}\rangle \tag{4},
\end{align}
where the remainder $\langle\mathcal{C}^\dag\mathcal{C}\rangle\langle\mathcal{B}^\dag\mathcal{B}\rangle\geq0$ with $\mathcal{C}=\mathcal{A}-\langle\mathcal{B}^\dag\mathcal{A}\rangle\mathcal{B}/\langle\mathcal{B}^\dag\mathcal{B}\rangle$, and $\langle\mathcal{Q}^\dag\mathcal{Q}\rangle$ is the second-order origin moment of the operator $\mathcal{Q}$.
In fact, the traditional interpretation of the uncertainty relation is invalid for non-Hermitian operators, because, as mentioned above, most of the uncertainty relations will be violated when applied to non-Hermitian operators. The uncertainty relation (4) provides a new interpretation of the uncertainty relation for non-Hermitian operators; i.e., the second-order origin moments $\langle\mathcal{A}^\dag\mathcal{A}\rangle$ and $\langle\mathcal{B}^\dag\mathcal{B}\rangle$ cannot be arbitrarily small at the same time when $\mathcal{A}$ and $\mathcal{B}$ are generalized-incompatible or generalized-anti-incompatible with each other. Remarkably, the operators $\mathcal{A}^\dag\mathcal{A}$, $\mathcal{B}^\dag\mathcal{B}$, $i[\mathcal{A},\mathcal{B}]_{\mathcal{G}}$, and $\{\mathcal{A},\mathcal{B}\}_{\mathcal{G}}$ are Hermitian even when $\mathcal{A}$ and $\mathcal{B}$ are non-Hermitian. That is to say, different from the variance, the second-order origin moment is observable for both the Hermitian and non-Hermitian operators. The new interpretation reveals some novel quantum properties that the traditional uncertainty relations cannot do. Such as, applying the new uncertainty relation (4) to the annihilation operators $a_1$ and $a_2$ of two continuous variable subsystems, one can deduce that the product of the expected energy of two subsystems $\langle a_1^\dag a_1\rangle\langle a_2^\dag a_2\rangle$ is lower-bounded by $|\langle[a_1,a_2]_{\mathcal{G}}\rangle|^2/4+|\langle\{a_1,a_2\}_{\mathcal{G}}\rangle|^2/4$. Especially, the energy of two subsystems cannot be arbitrarily small at the same time, when the annihilation operators of the two systems are generalized-incompatible or generalized-anti-incompatible on the state of the system, which means $\langle[a_1,a_2]_{\mathcal{G}}\rangle$ or $\langle\{a_1,a_2\}_{\mathcal{G}}\rangle$ does not equal or tend to zero.
The new uncertainty relation (4) expresses the quantum uncertainty relation in terms of the second-order origin moment, instead of the variance, but can unify the uncertainty relations based on the variance. Then, we demonstrate that some well-known uncertainty relations in either the sum form or the product form can be unified by the new uncertainty relation. Firstly, the new uncertainty relation turns into the product form uncertainty relation SUR, if we replace the operators $\mathcal{A}$ and $\mathcal{B}$ with the Hermitian operators $\check{A}=A-\langle A\rangle$ and $\check{B}=B-\langle B\rangle$. Secondly, assuming the system is in the pure state $|\psi\rangle$ and substituting the non-Hermitian operators $\mathcal{A}=\check{A}\pm i\check{B}$ and $\mathcal{B}=|\psi^\bot\rangle\langle\psi|$ into the uncertainty relation (4), one can obtain the sum form uncertainty relation (2). Here, the product form $\langle\mathcal{A}^\dag\mathcal{A}\rangle\langle\mathcal{B}^\dag\mathcal{B}\rangle=\Delta(A\pm iB)^{2}\Delta(|\psi^\bot\rangle\langle\psi|)^{2}=\Delta A^{2}+\Delta B^{2}\pm i\langle[A,B]\rangle$ turns into the sum form. That is to say, the product form uncertainty relation is the new uncertainty relation for Hermitian operators and the sum form uncertainty relation is actually the new uncertainty relation for non-Hermitian operators. The other uncertainty relations in the two forms \cite{26,42,27,41,37,38,23,28} can also be recovered by the uncertainty relation (4) in the similar way, and thus the equality (4) provides a unified uncertainty relation.
The uncertainty relations in the two forms can be divided into several categories with respect to their purposes and applications, such as the uncertainty relations focused on the effect of the incompatibility of observables on the uncertainty \cite{26,42}, the uncertainty relations used to investigate the relation between the variance of the sum of the observables and the sum variances of the observables \cite{23,28}, and even the uncertainty relations for three and more observables \cite{27,41}. The unified uncertainty indicates that they can all be integrated into a unified framework. Besides, by the introduction of the information operator, the unified framework provides a strengthened theoretical system for the quantum uncertainty relation. That is to say, the unified framework can fix the deficiencies in the traditional uncertainty relations, and provides a more accurate description of the uncertainty relation. The corresponding discussion will be presented in the next section.
\begin{figure}
\caption{The spin-1 system is chosen as the platform to demonstrate the new uncertainty inequality (8). We take $A=J_x$, $B=J_z$, $\hbar=1$ , and the state is parameterized by $\alpha$ as $\rho=\cos^2(\alpha)|1\rangle\langle1|+\sin^2(\alpha)|-1\rangle\langle-1|$, with $|\pm1\rangle$ and $|0\rangle$ being the eigenstates of $J_z$. The green dash-dotted line represents the lower bound of the SUR (denoted by $LB_{SUR}
\end{figure}
\begin{figure}
\caption{Illustration to demonstrate the function of the information operator is presented. We take $\hbar=1$, and assume that the state of the spin-1 system is in the pure state $|\psi\rangle=\cos(\beta)|1\rangle+\sin(\beta)|-1\rangle$ with $|\pm1\rangle$ being the eigenstates of $J_z$. The operator set $\Theta=\{\mathcal{O}
\end{figure}
\emph{Information operator.---}Based on the initial spirit of Schr\"{o}dinger, the SUR can be derived as follows \cite{25V}. Assume $\mathcal{F}=\sum^N_{m=1}x_m\check{A}_m$, where $A_m$ stands for an arbitrary operator, $N$ is the number of the operators and $x_m\in C$ represents a random complex number. Using the non-negativity of the second-order origin moment of $\mathcal{F}$ \cite{25V}, namely $\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq0$, one can obtain:
\begin{align}
&\mathbb{D}:\geq0\tag{5},
\end{align}
where $\mathbb{D}$ is the $N\times N$ dimension matrix with the elements $\mathbb{D}(m,n)=\langle\check{A}_m^\dag\check{A}_n\rangle$ and $\mathbb{D}:\geq0$ means that $\mathbb{D}$ is a positive semidefinite matrix. As for the positive semidefinite matrix $\mathbb{D}$, we have ${\rm Det}(\mathbb{D})\geq0$ with ${\rm Det}(\mathbb{D})$ being the determinant value of ${\mathbb{D}}$, and $X^\dag.\mathbb{D}.X\geq0$ with $X\in C^N$ being a random column vector. In fact, ${\rm Det}(\mathbb{D})\geq0$ turns into the product form uncertainty relation and $X^\dag.\mathbb{D}.X\geq0$ becomes the sum form uncertainty relation. For instance, taking $N=2$ and $X=\{1,\mp i\}^T$, one can obtain that ${\rm Det}(\mathbb{D})\geq0$ is the SUR and $X^\dag.\mathbb{D}.X\geq0$ is the sum form uncertainty relation $\Delta A^{2}+\Delta B^{2}=|\langle [A,B]\rangle|$. Thus, the SUR can be interpreted as the fundamental inequality $\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq0$ or $\mathbb{D}:\geq0$, and so do the other uncertainty relations deduced in Refs. \cite{28LS,52,53,54,55}.
However, the quantum properties of the operator $\mathcal{F}$, in most cases, cannot be fully expressed by $\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq0$, because the non-negativity of the second-order origin moment $\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq0$ cannot provide any information of $\mathcal{F}$ in the quantum level. Considering an arbitrary operator $\mathcal{O}$ , based on the unified uncertainty relation (4), one has:
\begin{align}
\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq\dfrac{|\langle i[{\mathcal{F}},{\mathcal{O}}]_{\mathcal{G}}\rangle|^2+|\langle\{{\mathcal{F}},{\mathcal{O}}\}_{\mathcal{G}}\rangle|^2} {4\langle\mathcal{O}^\dag\mathcal{O}\rangle} \tag{6}.
\end{align}
Especially, we have $|\langle i[{\mathcal{F}},{\mathcal{O}}]_{\mathcal{G}}\rangle|^2+|\langle\{{\mathcal{F}},{\mathcal{O}}\}_{\mathcal{G}}\rangle|^2/4\langle\mathcal{O}^\dag\mathcal{O}\rangle>0$ when the operator $\mathcal{O}$ is generalized-incompatible or generalized-anti-incompatible with $\mathcal{F}$. Obviously, the introduction of $\mathcal{O}$ provides a more accurate description for the second-order origin moment $\langle \mathcal{F}^\dag \mathcal{F}\rangle$. That is to say, the operator $\mathcal{O}$ can provide information for the second-order origin moment of $\mathcal{F}$ that $\langle \mathcal{F}^\dag \mathcal{F}\rangle\geq0$ cannot do, and thus we name $\mathcal{O}$ as the information operator. In order to investigate the quantum uncertainty relation more accurately, the information operator should be introduced. Using (6), we have:
\begin{align}
&\mathbb{D}:\geq \mathbb{V}\tag{7},
\end{align}
where $\mathbb{V}$ is the $N\times N$ dimension positive semidefinite matrix with the elements $\mathbb{V}(m,n)=\langle\check{A}_m^\dag\mathcal{O}\rangle\langle\mathcal{O}^\dag\check{A}_n\rangle/\langle\mathcal{O}^\dag\mathcal{O}\rangle$ and $\mathbb{D}:\geq \mathbb{V}$ means $\mathbb{D}-\mathbb{V}$ is a positive semidefinite matrix. Based on the properties of the positive semidefinite matrix, we can obtain a series of uncertainty relations for $N$ observables in both the product form and the sum form.
To demonstrate the importance of the information operator, we will investigate its function on fixing the deficiencies appearing in the traditional uncertainty relations. The triviality problem of the SUR occurs when the state of the system happens to be the eigenstate of $A$ or $B$. For instance, one has $|\langle[A,B]\rangle/2i|^2+|\langle\{\check{A},\check{B}\}\rangle/2|^2\equiv{\Delta A}^2{\Delta B}^2\equiv0$ in the finite-dimension Hilbert space when ${\Delta A}^2=0$ or ${\Delta B}^2=0$. Different from ${\Delta A}^2{\Delta B}^2$, the sum of the variances ${\Delta A}^2+{\Delta B}^2$ will never equal zero for incompatible observables even when the state of the system is the eigenstate of $A$ or $B$. Thus the sum form has the advantage in expressing the uncertainty relation. However, the lower bounds of the most sum form uncertainty relations depend on the state $|\psi^\perp\rangle$, making them difficult to apply to the high dimension Hilbert space \cite{21}. Based on the analysis in the previous section, the sum form uncertainty relation (2) can be written as
$\Delta\mathcal{A}^{2}\Delta\mathcal{B}^{2}\geq|\langle\psi|[\mathcal{\check{A}},\mathcal{\check{B}}]_{\mathcal{G}}|\psi\rangle|^{2}/4+|\langle\psi|\{\mathcal{\check{A}},\mathcal{\check{B}}\}_{\mathcal{G}}|\psi\rangle|^{2}/4$, where $\mathcal{A}=A\pm iB$ and $\mathcal{B}=|\psi^\bot\rangle\langle\psi|$, which means the uncertainty relation (2) is still a type of the SUR. Obviously, the state $|\psi\rangle$ will never be the eigenstate of $\mathcal{B}$ when we take $\mathcal{B}=|\psi^\bot\rangle\langle\psi|$, and therefore the triviality problem of the SUR can be remedied by (2) \cite{40}. However, it is due to the existence of $|\psi^\bot\rangle\langle\psi|$ that the uncertainty relation (2) cannot be applied to the high dimension system. Thus, the triviality problem of the product form uncertainty can be considered as the essential reason for the phenomenon that lots of sum form uncertainty relations are difficult to apply to the high dimension system.
In fact, the physical essence of the triviality problem can be described as that we cannot obtain any information of the uncertainty of $A(B)$ by the product form uncertainty relation, when the state of the system happens to be the eigenstate of $B(A)$. Thus, the information operator, which can provide the information for the uncertainty relation, can be used to fix this triviality problem. Here, two generalized-incompatible operators $\mathcal{R}$ and $\mathcal{S}$ will be introduced as the information operators. According to (4) and (6), the information operator $\mathcal{O}$ will not contain any effective information of $\mathcal{F}$ when $\langle\mathcal{O}^\dag\mathcal{O}\rangle=0$, and thus the information operator introduced to fix the triviality problem should satisfy $\langle\mathcal{O}^\dag\mathcal{O}\rangle\neq0$. Based on the unified uncertainty relation (4), the second-order origin moments of the generalized-incompatible operators $\mathcal{R}$ and $\mathcal{S}$ will never be zero at the same time, hence at least one of the two information operators can provide effective information to fix the triviality problem. The corresponding uncertainty relation is obtained as (please see the Information operator in the Supplemental Material \cite{35}):
\begin{align}
{\Delta A}^2+{\Delta B}^2\geq&\max_{\mathcal{O}\in\{\mathcal{R},\mathcal{S}\}}\{\frac{|\langle\mathcal{O}^\dag(\check{A}+e^{i\theta}\check{B})\rangle|^2}{\langle\mathcal{O}^\dag\mathcal{O}\rangle}\}-\langle\{\check{A},e^{i\theta}\check{B}\}_{\mathcal{G}}\rangle \tag{8},
\end{align}
where $\theta\in[0,2\pi]$ should be chosen to maximize the lower bound. The triviality problem can be completely fixed by the uncertainty relation (8) for almost any choice of the generalized-incompatible operators $\mathcal{R}$ and $\mathcal{S}$ : choose $\mathcal{R}$ and $\mathcal{S}$ that can avoid $\langle\check{A}\check{B}\rangle\equiv\langle\mathcal{R}^\dag\check{A}\rangle\equiv\langle\mathcal{R}^\dag\check{B}\rangle\equiv\langle\mathcal{S}^\dag\check{A}\rangle\equiv\langle\mathcal{S}^\dag\check{B}\rangle\equiv0$. Such a choice is always possible, as shown in Fig.1.
Due to the absence of $|\psi^\bot\rangle$, the uncertainty relation (8) can be well applied to the high dimension system. Meanwhile, the uncertainty relation (8) has a tighter lower bound than the uncertainty realtion depending on $|\psi^\bot\rangle$ by limiting the choice of the information operator, as shown in Fig.1. Furthermore, the inequality (8) will become an equality on the condition that $\mathcal{R}$ or $\mathcal{S}=\lambda_1 \check{A}+\lambda_2 \check{B} $ with ${|\lambda_1|}^2={|\lambda_2|}^2\neq0$ and $\lambda_1,\lambda_2\in C$. The condition is independent on the state $|\psi^\bot\rangle$, and thus can be easily satisfied even for the high dimension Hilbert space. Besides, the uncertainty relation (8) will reduce to the uncertainty relation (2) when taking $\mathcal{R}=|\psi^\bot\rangle\langle\psi|$ and ignoring the influence of the other information operator $\mathcal{S}$, which means that the uncertainty relation (2) can also be considered as taking $|\psi^\bot\rangle\langle\psi|$ as the information operator.
The introduction of the information operator makes us express the uncertainty relation more accurately. Based on the unified uncertainty relation (4), we can obtain the following uncertainty equality (please see the Information operator in the Supplemental Material \cite{35}):
\begin{align}
&\mathbb{D}=\sum^r_{k=1}\mathbb{V}_k \tag{9},
\end{align}
where $\mathbb{V}_k$ is the $N\times N$ dimension positive semidefinite matrix with the elements $\mathbb{V}_k(m,n)=\langle\check{A}_m^\dag\mathcal{O}_k\rangle\langle\mathcal{O}_k^\dag\check{A}_n\rangle/\langle\mathcal{O}_k^\dag\mathcal{O}_k\rangle$,
$\mathcal{O}_k$ is the element of the operator set $\Theta=\{\mathcal{O}_1,\mathcal{O}_2,\cdots,\mathcal{O}_r\}$ in which the elements satisfy $\langle\mathcal{O}^\dag_i\mathcal{O}_j\rangle=\langle\mathcal{O}^\dag_i\mathcal{O}_j\rangle\delta_{ij}$ and $\langle\mathcal{O}^\dag_k\mathcal{O}_k\rangle\neq 0$ with $k,i,j\in\{1,2,\cdots,r\}$, and $r$ is the maximum number of the elements that the set $\Theta$ can hold. The set can be obtained by the Schmidt transformation (please see the Schmidt transformation process in the Supplemental Material \cite{35}) \cite{LM,NI}. The value of $r$ is equal to the rank of the Metric matrix corresponding to the bilinear operator function$\langle\mathcal{A}^\dagger\mathcal{B}\rangle$, and only depends on the state of the system. It is worth mentioning that $r$ is less than $d$ for the pure state and less than $d^2$ for the mixed state in the $d$-dimension system, and $r$ will tend to the infinity when considering the infinite-dimension system. The uncertainty equality indicates that the information of the uncertainties for incompatible observables can be captured accurately when $r$ information operators are introduced, as shown in Fig.2.
\emph{Discussion.--- }The variance-based uncertainty relations can be divided into the product form and the sum form. The product form uncertainty relation cannot fully capture the concept of the incompatible observables, and the problem is referred to as the triviality problem of the product form uncertainty relation. The triviality problem can be fixed by the sum form uncertainty relation, and thus lots of effort has been made to investigate the sum form uncertainty relation. However, most of the sum form uncertainty relations depend on the orthogonal state to the state of the system, and are difficult to apply to the high dimension Hilbert space.
We provide a unified uncertainty relation for the two forms uncertainty relations, and deduce that the essences of the product form and the sum form uncertainty relations are actually the unified uncertainty relation for Hermitian operators and non-Hermitian operators, respectively.
Thus, the unified uncertainty relation provides a unified framework for the two forms uncertainty relations.
In the unified framework, we deduce that the uncertainty relation for incompatible observables is bounded by not only the commutator of themselves, but also the quantities related with the other operator, which can provide information for the uncertainty and thus is named as the information operator. The deficiencies in the product form and the sum form uncertainty relations are actually identical in essence, and can be completely fixed by the introduction of the information operators. Furthermore, the uncertainty inequality will become an uncertainty equality when a specific number of information operators are introduced, which means the uncertainty relation can be expressed exactly with the help of the information operators. Thus, the unified framework provides a strengthened theoretical system for the uncertainty relation.
The unified framework also provides a new interpretation of the quantum uncertainty relation for the non-Hermitian operators, i.e., the ``observable" second-order origin moments of the non-Hermitian operators cannot be arbitrarily small at the same time when they are generalized-incompatible or generalized-anti-incompatible with each other. The new interpretation reveals some novel quantum properties that the traditional uncertainty relation cannot do
This work is supported by the National Natural Science Foundation of China (Grant Nos.11574022, 61227902, 11774406, 11434015, 61835013), MOST of China (Nos. 2016YFA0302104, 2016YFA0300600), the Chinese Academy of Sciences Central of Excellence in Topological Quantum Computation (XDPB-0803), the National Key R\&D Program of China under grants Nos. 2016YFA0301500, SPRPCAS under grants No. XDB01020300, XDB21030300.
X. Z. and S. Q. M. contributed equally to this work.
\end{document} |
\begin{document}
\title{Tight bounds from multiple-observable entropic uncertainty relations}
\author{Alberto Riccardi}
\affiliation{INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100 Pavia, Italy}
\author{Giovanni Chesi}
\affiliation{INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100 Pavia, Italy}
\author{Chiara Macchiavello}
\affiliation{Dipartimento di Fisica, Universit\`{a} degli Studi di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy \\
INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy}
\author{Lorenzo Maccone}
\affiliation{Dipartimento di Fisica, Universit\`{a} degli Studi di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy \\
INFN Sezione di Pavia, Via Agostino Bassi 6, I-27100, Pavia, Italy}
\begin{abstract}
We investigate the additivity properties for both bipartite and multipartite
systems by using entropic uncertainty relations (EUR) defined in terms
of the joint Shannon entropy of probabilities of local measurement outcomes and we apply them to entanglement detection.
In particular, we introduce state-independent and state-dependent
entropic inequalities whose violation certifies the presence of quantum
correlations. We show that the additivity of EUR holds only for EUR
that involve two observables, while inequalities that consider more
than two observables or the addition of the Von Neumann entropy of
a subsystem enable to detect quantum correlations. Furthermore, we
study their detection power for bipartite systems and for several
classes of states of a three-qubit system.
\end{abstract}
\maketitle
Entropic uncertainty relations (EUR) are inequalities that express
preparation uncertainty relations (UR) as sums of Shannon entropies
of probability distributions of measurement outcomes. First introduced
for continuous variables systems \cite{EUR1,EUR2,EUR3,EUR4}, they
were then generalized for pair of observables with discrete spectra
\cite{EUR5,EUR6,EUR7,EUR9,EUR10} (see \cite{EUR8} for a review of
the topic). Conversely to the most known UR defined for product of
variances \cite{Heis1,Robertson1}, which are usually state-dependent,
EUR provide lower bounds, which quantify the knowledge trade-off between
the different observables, that are state-independent.
Variance-based UR for the sum of variances \cite{SVar4} in some cases
also provide state-independent bounds \cite{SVar1,SVar2,SVar3,SVar5,SVar6}.
EUR, due to their simple structure, allow to consider UR for more
than two observables in a natural way by simply adding more entropies,
a task that is not straightforward for UR based on the product of variances. However,
tight bounds for multiple-observable EUR are known only for small
dimensions and for restricted sets of observables, typically for complementary
observables \cite{MultiAzarchs,MultiBallesterWehner,MultiIvanovic,MultiSanchez,TightAR,MultiCHina,MultiMolner},
namely the ones that have mutually unbiased bases as eigenbases, and
for angular momentum observables \cite{URAngular-1,TightAR}. \\
Besides their importance from a fundamental point of view as preparation
uncertainty relations, EUR have recently been used to investigate
the nature of correlations in composite quantum systems, providing
criteria that enable to detect the presence of different types of
quantum correlations, both for bipartite and multipartite systems.
Entanglement criteria based on EUR were defined in \cite{EntGuh,Ent1,Ent2,Ent3,Huang},
while steering inequalities in \cite{Steering1,Steering2,Steering3,Steering4,Steering5,SteeringAR}.
\\
Almost all of these criteria are based on EUR for conditional Shannon
entropies, where one tries to exploit, in the presence of correlations,
side information about some subsystems to reduce global uncertainties,
while only partial results for joint Shannon entropies are known \cite{EntGuh,Ent4}.
Moreover, it has been recently proven in \cite{Additivity} that if
one considers EUR defined for the joint Shannon entropy and only pairs
of observables, then it is not possible to distinguish between separable and
entangled states since in this case additivity holds. \\
In this paper we show that if we consider EUR for more than two observables
the additivity of EUR does no longer hold. This result implies that it is possible
to define criteria that certify the presence of entanglement by using
the joint Shannon entropy for both the bipartite and the multipartite
case. We investigate which criteria can be derived from EUR based on the joint Shannon entropy and their performance. We then provide some examples of entangled states that violate our criteria.
This paper is organized as follows: in Section I we briefly review
some concepts of single system EUR, in particular we discuss the case
of multiple observables. In Section II we establish the entanglement
criteria for bipartite systems and in Section III we address the problem
in the multipartite scenario. Finally, in Section IV we consider some
examples of entangled states that are detected by these criteria, in
particular we focus on the multi-qubit case.
\section{Entropic uncertainty relations Review}
The paradygmatic example of EUR for observables with a discrete
non-degenerate spectrum is due to Maassen and Uffink \cite{EUR7},
and it states that for any two observables $A_{1}$ and $A_{2}$,
defined on a $d$-dimensional system, the following inequality holds:
\begin{equation}
H(A_{1})+H(A_{2})\geqslant-2\log_{2}c=q_{MU},\label{MUF}
\end{equation}
where $H(A_{1})$ and $H(A_{2})$ are the Shannon entropies of the
measurement outcomes of two observables $A_{1}=\sum_{j}a_{j}^{1}\ket{a_{j}^{1}}\bra{a_{j}^{1}}$
and $A_{2}=\sum_{j}a_{j}^{2}\ket{a_{j}^{2}}\bra{a_{j}^{2}}$, namely
$H(A_{I})=-\sum_{j}p(a_{j}^{I})\log p(a_{j}^{I})$ being $p(a_{j}^{I})$
the probability of obtaining the outcome $a_{J}^{I}$ of $A_{I}$,
and $c=\max_{j,k}\left|\braket{a_{j}^{1}|a_{k}^{2}}\right|$ is the
maximum overlap between their eigenstates. The bound (\ref{MUF})
is known to be tight if $A_{1}$ and $A_{2}$ are complementary observables.
We remind that two observables $A_{1}$ and $A_{2}$ are said to be
complementary iff their eigenbases are mutually unbiased, namely iff
$\left|\braket{a_{j}^{1}|a_{k}^{2}}\right|=\frac{1}{\sqrt{d}}$ for
all eigenstates, where $d$ is the dimension of the system (see \cite{MUBs}
for a review on MUBs). In this case $q_{MU}=\log_{2}d$, hence we
have:
\begin{equation}
H(A_{1})+H(A_{2})\geqslant\log_{2}d.
\end{equation}
The above relation has a clear interpretation as UR:
let us suppose that $H(A_{1})=0$, which means that the state of the
system is an eigenstate of $A_{1}$, then the other entropy $H(A_{2})$
must be maximal, hence if we have a perfect knowledge of one observable
the other must be completely undetermined. For arbitrary observables
stronger bounds, that involve the second largest term in $\left|\braket{a_{j}|b_{k}}\right|$,
were derived in \cite{EUR10,EUR9}.
An interesting feature of EUR is that they can be generalized
to an arbitrary number of observables in a straightforward way from
Maassen and Uffink's EUR. Indeed, let us consider for simplicity the
case of three observables $A_{1}$, $A_{2}$ and $A_{3}$, which
mutually satisfy the following EURs:
\begin{equation}
H(A_{i})+H(A_{j})\geqslant q_{MU}^{ij},\label{MUF2}
\end{equation}
where $i,j=1,2,3$ labels the three observables. Then, we have:
\begin{align}
\sum_{k=1}^{3}H(A_{k}) & =\frac{1}{2}\sum_{k=1}^{3}\sum_{j\neq k}H(A_{k})+H(A_{j})\nonumber \\
& \geq\frac{1}{2}\left(q_{MU}^{12}+q_{MU}^{13}+q_{MU}^{23}\right)
\end{align}
where we have applied (\ref{MUF2}) to each pair. If we have $L$
observables, the above inequality becomes:
\begin{equation}
\sum_{k=1}^{L}H(A_{k})\geq\frac{1}{\left(L-1\right)}\sum_{t\in T_{2}}q_{MU}^{t},\label{MultiObsEUR}
\end{equation}
where $t$ takes values in the set $T_{2}$ of labels of all the
possible $L(L-1)/2$ pairs of observables. For example if
$L=4$, then $T_{2}=\{12,13,14,23,24,34\}$. However EUR in the form
(\ref{MultiObsEUR}) are usually not tight, i.e. in most cases the
lower bounds can be improved. Tight bounds are known only for small
dimensions and for complementary or angular momentum observables.
For the sake of simplicity, henceforth all explicit examples will
be discussed only for complementary observables. The maximal number
of complementary observables for any given dimension is an open problem
\cite{MUBs}, which finds its roots in the classification of all complex
Hadamard matrices. However, if $d$ is a power of a prime then $d+1$
complementary observables always exist. For any $d$, even
if it is not a power of a prime, it is possible to find at least three
complementary observables \cite{MUBs}. The method that we will define
in the next Section can be therefore used in any dimension. The qubit
case, where at most three complementary observables exist, which are
in correspondence with the three Pauli matrices, was studied in \cite{MultiSanchez},
while for systems with dimension three to five tight bounds for an
arbitrary number of complementary observables were derived in \cite{TightAR}.
For example in the qubit case, where the three observables $A_{1},A_{2}$
and $A_{3}$ correspond to the three Pauli matrices $\sigma_{x},\sigma_{y}$
and $\sigma_{z},$ we have:
\begin{equation}
H(A_{1})+H(A_{2})+H\left(A_{3}\right)\geqslant2,\label{MultiQubit}
\end{equation}
and the minimum is achieved by the eigenstates of one of the $A_{i}$.
In the case of a qutrit, where four complementary observables exist,
we instead have:
\begin{align}
& H(A_{1})+H(A_{2})+H(A_{3})\geqslant3,\label{3d 3Mubs}\\
& H(A_{1})+H(A_{2})+H(A_{3})+H(A_{4})\geqslant4.\label{3d 4mubs}
\end{align}
The minimum values are achieved by:
\begin{align}
& \frac{e^{i\varphi}\ket0+\ket1}{\sqrt{2}},\ \frac{e^{i\varphi}\ket0+\ket2}{\sqrt{2}},\ \frac{e^{i\varphi}\ket1+\ket2}{\sqrt{2}},
\end{align}
where $\varphi=\frac{\pi}{3},\pi,\frac{5\pi}{3}$. Another result,
for $L<d+1$, can be found in \cite{MultiBallesterWehner}, where
it has been shown that if the Hilbert space dimension is a square,
that is $d=r^{2},$ then for $L<r+1$ the inequality (\ref{MultiObsEUR})
is tight, namely:
\begin{equation}
\sum_{i=1}^{L}H(A_{i})\geqslant\frac{L}{2}\log_{2}d=q_{BW}.\label{Ballester}
\end{equation}
In order to have a compact expression to use, we express the EUR for
$L$ observables in the following way:
\begin{equation}
\sum_{i=1}^{L}H(A_{i})\geq f\left(\mathcal{A},L\right),\label{L-ObsEur}
\end{equation}
where $f\left(\mathcal{A},L\right)$ indicates the lower bound, which
can be tight or not, and it depends on the set $\mathcal{A}=\left\{ A_{1},...,A_{L}\right\} $
of $L$ observables considered. Here we also point out in the lower
bound how many observables are involved. When we refer explicitly to
tight bounds we will use the additional label $T$, namely $f^{T}\left(\mathcal{A},L\right)$
expresses a lower bound that we know is achievable via some states.
\section{Bipartite entanglement criteria}
In this Section we discuss bipartite entanglement criteria based on
EUR, defined in terms of joint Shannon entropies. The framework consists
in two parties, say Alice and Bob, who share a quantum state $\rho_{AB}$,
and they want to establish if their state is entangled. Alice
and Bob can perform $L$ measurements each, that we indicate respectively
as $A_{1},..,A_{L}$ and $B_{1},..,B_{L}$. Alice and Bob measure the
observables $A_{i}\otimes B_{j}$ and they want to have a criterion
defined in terms of the joint Shannon entropies $H\left(A_{i},B_{j}\right)$
which certifies the presence of entanglement. As a reminder, in a
bipartite scenario we say that the state $\rho_{AB}$ is entangled
iff it cannot be expressed as a convex combination of product states,
which are represented by separable states, namely iff:
\begin{equation}
\rho_{AB}\neq\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{B}^{i},
\end{equation}
where $p_{i}\geq0$, $\sum_{i}p_{i}=1$, and $\rho_{A}^{i}$, $\rho_{B}^{i}$
are Alice and Bob's states respectively.
\begin{prop}
If the state $\rho_{AB}$ is separable, then the following EUR must
hold:
\begin{equation}
\sum_{i=1}^{L}H(A_{i},B_{i})\geq f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right),\label{Primo criterio}
\end{equation}
where $f\left(\mathcal{A},L\right)$ and $f\left(\mathcal{B},L\right)$
are the lower bounds of the single system EUR, namely
\begin{equation}
\sum_{i=1}^{L}H(A_{i})\geq f\left(\mathcal{A},L\right),
\end{equation}
\begin{equation}
\sum_{i=1}^{L}H(B_{i})\geq f\left(\mathcal{B},L\right).
\end{equation}
\end{prop}
\begin{proof}
Let us focus first on $H(A_{i},B_{i})$ which, for the properties
of the Shannon entropy, can be expressed as:
\begin{equation}
H(A_{i},B_{i})=H(A_{i})+H(B_{i}|A_{i}).
\end{equation}
We want to bound $H\left(B_{i}|A_{i}\right)$ which is computed over
the state $\rho_{AB}=\sum_{j}p_{j}\rho_{A}^{j}\otimes\rho_{B}^{j}$
. Through the convexity of the relative entropy, one can prove that
the conditional entropy H(B|A) is concave in $\rho_{AB}$. Then we
have:
\begin{equation}
H(B_{i}|A_{i})_{\sum_{j}p_{j}\rho_{A}^{j}\otimes\rho_{B}^{j}}\geq\sum_{j}p_{j}H(B_{i}|A_{i})_{\rho_{A}^{j}\otimes\rho_{B}^{j}},
\end{equation}
thus, since the right-hand side of the above Equation is evaluated on a product
state, we have:
\begin{equation}
H(B_{i}|A_{i})_{\sum_{j}p_{j}\rho_{A}^{j}\otimes\rho_{B}^{j}}\geq\sum_{j}p_{j}H(B_{i})_{\rho_{B}^{j}}.
\end{equation}
Therefore, considering $\sum_{i=1}^{L}H(A_{i},B_{i})$, we derive
the following:
\begin{equation}
\sum_{i=1}^{L}H(A_{i},B_{i})\geq\sum_{i}H(A_{i})+\sum_{j}p_{j}\sum_{i}H(B_{i})_{\rho_{B}^{j}}.\label{Proof}
\end{equation}
Then we can observe that $\sum_{i=1}^{L}H(A_{i})\geq f\left(\mathcal{A},L\right)$
and $\sum_{i}H(B_{i})_{\rho_{B}^{j}}\geq f\left(\mathcal{B},L\right)$,
the latter holding due to EUR being state-independent
bounds. Therefore we have:
\begin{align}
\sum_{i=1}^{L}H(A_{i},B_{i}) & \geq f\left(\mathcal{A},L\right)+\sum_{j}p_{j}f\left(\mathcal{B},L\right)\nonumber \\
& =f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right),
\end{align}
since $\sum_{j}p_{j}=1.$
\end{proof}
Any state that violates the inequality $\sum_{i=1}^{L}H(A_{i},B_{i})\geq f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right)$
must be therefore entangled. However this is not sufficient to have
a proper entanglement criterion. Indeed, if we consider the observables
$A_{i}\otimes B_{i}$ as ones of the bipartite system then they must
satisfy an EUR for all states, even the entangled ones, which can
be expressed as:
\begin{equation}
\sum_{i=1}^{L}H(A_{i},B_{i})\geq f(\mathcal{AB},L),
\end{equation}
where the lower bound now depends on the observables $A_{i}\otimes B_{i}$,
while $f\left(\mathcal{A},L\right)$ and $f\left(\mathcal{B},L\right)$
depend respectively on $A_{i}$ and $B_{i}$ individually. In order
to have a proper entanglement criterion then we should have that
\begin{equation}
f(\mathcal{AB},L)<f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right),
\end{equation}
which means that the set of entangled states that violate the inequality
is not empty. As it was shown in \cite{Additivity}, for $L=2$, we
have $f(\mathcal{AB},2)=f\left(\mathcal{A},2\right)+f\left(\mathcal{B},2\right)$
for any observables, which expresses the additivity of EUR for pairs
of observables. A counterexample of this additivity property for $L>3$
is provided by the complete set of complementary observables for two
qubits, indeed we have:
\begin{equation}
H(A_{1},B_{1})+H(A_{2},B_{2})+H(A_{3},B_{3})\geq3,
\end{equation}
and the minimum is attained by the Bell states while $f\left(\mathcal{A},3\right)+f\left(\mathcal{B},3\right)=4,$
which provides the threshold that enables entanglement detection
in the case of two qubits. \\
Let us now clarify the difference of this result with respect to those
defined in terms of EUR based on conditional entropies, in particular
to entropic steering inequalities. Indeed, if one looks at the proof
of Proposition 1, it could be claimed that there is no difference at all
since we used the fact that $\sum_{i}H(B_{i}|A_{i})\geq f(\mathcal{B},L)$,
which is a steering inequality, namely violation of it witnesses the
presence of quantum steering from Alice to Bob. However the difference
is due to the symmetric behavior of the joint entropy, which contrasts
with the asymmetry of quantum steering. To be more formal, the joint
Shannon entropy $H(A_{i},B_{i})$ can be rewritten in two forms:
\begin{align}
H(A_{i},B_{i})= & H(A_{i})+H(B_{i}|A_{i})\\
= & H(B_{i})+H(A_{i}|B_{i}),\nonumber
\end{align}
then:
\begin{equation}
\sum_{i}H(A_{i},B_{i})=\sum_{i}\left(H(A_{i})+H(B_{i}|A_{i})\right),\label{SI 1}
\end{equation}
and
\begin{equation}
\sum_{i}H(A_{i},B_{i})=\sum_{i}\left(H(B_{i})+H(A_{i}|B_{i})\right).\label{SI2}
\end{equation}
If now the state is not steerable from Alice to Bob, we have $\sum_{i}H(B_{i}|A_{i})\geq f(\mathcal{B},L)$,
which implies $\sum_{i=1}^{L}H(A_{i},B_{i})\geq f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right)$.
Note that in this case if we look at $\sum_{i}H(A_{i}|B_{i})$ no
bound can be derived, apart from the trivial bound $\sum_{i}H(A_{i}|B_{i})\geq0$,
since there are no assumptions on the conditioning from Bob to Alice.
Conversely, if the state is not steerable from Bob to Alice, i.e.
we exchange the roles, we have $\sum_{i}H(B_{i}|A_{i})\geq0$ and
$\sum_{i}H(A_{i}|B_{i})\geq f(\mathcal{A},L)$, which implies again
$\sum_{i=1}^{L}H(A_{i},B_{i})\geq f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right)$.
Therefore if we just look at the inequality (\ref{Primo criterio}),
we cannot distinguish between entanglement, or the two possible forms
of quantum steering and since the presence of steering, for bipartite
systems, implies entanglement it is more natural to think about Eq.~(\ref{Primo criterio})
as an entanglement criterion, while if we want to investigate steering
properties of the state we should look at the violation of the criteria
$\sum_{i}H(B_{i}|A_{i})\geq f(\mathcal{B},L)$ and $\sum_{i}H(A_{i}|B_{i})\geq f(\mathcal{A},L).$
\subsubsection*{State-dependent bounds}
A stronger entanglement criteria can be derived by considering the
state-dependent EUR:
\begin{equation}
\sum_{i=1}^{L}H(A_{i})\geq f\left(\mathcal{A},L\right)+S\left(\rho_{A}\right),\label{State dependent}
\end{equation}
or the corresponding version for Bob's system $\sum_{i=1}^{L}H(B_{i})\geq f\left(\mathcal{B},L\right)+S\left(\rho_{B}\right)$,
where $S\left(\rho_{A}\right)$ and $S\left(\rho_{B}\right)$ are
the Von Neumann entropies of the marginal states of $\rho_{AB}.$
\begin{prop}
If the state $\rho_{AB}$ is separable, then the following EUR must
hold:
\begin{equation}
\sum_{i=1}^{L}H(A_{i},B_{i})\geq f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right)+\max\left(S\left(\rho_{A}\right),S\left(\rho_{B}\right)\right).\label{Primo criterio-1}
\end{equation}
\end{prop}
\begin{proof}
The proof is the same of Proposition 1 where we use (\ref{State dependent})
in (\ref{Proof}), instead of the state-dependent bound (\ref{L-ObsEur}).
The same holds if we use the analogous version for Bob. Then, aiming
at the strongest criterion, we can take the maximum between the two
Von Neumann entropies.
\end{proof}
The edge in using these criteria, instead of the one defined in Proposition
1, is such that even for $L=2$ the bound is meaningful. Indeed
a necessary condition to the definition of a proper criterion is that:
\begin{equation}
f^{T}(\mathcal{AB},2)<f\left(\mathcal{A},2\right)+f\left(\mathcal{B},2\right)+S\left(\rho_{X}\right),\label{VN criteria}
\end{equation}
where $X=A,B$ with the additional requirement that the bound on the
left is tight, i.e. there exist states the violate the criterion.
As an example we can consider a two-qubit system, the observables
$X_{AB}=\sigma_{X}^{A}\otimes\sigma_{X}^{B}$ and $Z_{AB}=\sigma_{Z}^{A}\otimes\sigma_{Z}^{B}$,
which for all states of the whole system satisfy $H(X_{AB})+H(Z_{AB})\geq2$,
and the state $\rho_{AB}=\ket{\phi^{+}}\bra{\phi^{+}}$, indeed for
this scenario the entanglement criterion reads:
\begin{equation}
H(X_{AB})+H(Z_{AB})\geq3,
\end{equation}
which is actually violated since the left-hand side is equal to 2.
Note that in general the condition $f^{T}(\mathcal{AB},L)<f\left(\mathcal{A},L\right)+f\left(\mathcal{B},L\right)+S\left(\rho_{X}\right)$
is necessary to the usefulness of the corresponding entanglement criteria.
\section{Multipartite entanglement criteria}
We now extend the results of Propositions 1 and 2 for multipartite
systems, where the notion of entanglement has to be briefly discussed
since it has a much richer structure than the bipartite case.
Indeed, we can distinguish among different levels of separability.
First, we say that a state $\rho_{V_{1},..,V_{n}}$ of $n$ systems
$V_{1},..,V_{n}$ is fully separable iff it can be written
in the form:
\begin{equation}
\rho_{V_{1},..,V_{n}}^{FS}=\sum_{i}p_{i}\rho_{V_{1}}^{i}\otimes...\otimes\rho_{V_{n}}^{i},\label{Fully sep}
\end{equation}
with $\sum_{i}p_{i}=1$, namely it is a convex combination of product
states of the single subsystems. As a case of study we will always
refer to tripartite systems, where there are three parties, say Alice,
Bob and Charlie. In this case a fully separable state can be written
as:
\begin{equation}
\rho_{ABC}^{FS}=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{B}^{i}\otimes\rho_{C}^{i}.
\end{equation}
Any state that does not admit such a decomposition contains entanglement among some subsystems. However, we can
define different levels of separability. Hence, we say that the state $\rho_{V1,..,V_{n}}$
of $n$ systems is separable with respect to a given partition $\{I_{1},..,I_{k}\}$,
where $I_{i}$ are disjoint subsets of the indices $I=\{1,..,n\}$,
such that $\cup_{j=1}^{k}I_{j}=I$, iff it can be expressed as:
\begin{equation}
\rho_{V_{1},..,V_{n}}^{1,..,k}=\sum_{i}p_{i}\rho_{1}^{i}\otimes..\otimes\rho_{k}^{i},
\end{equation}
namely some systems share entangled states, while the state is separable
with respect to the partition considered. For tripartite system we
have three different possible bipartitions: $1|23$, $2|13$ and $3|12$.
As an example, if the state $\rho_{ABC}$ can be expressed as:
\begin{equation}
\rho_{ABC}^{1|23}=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{BC}^{i},
\end{equation}
then there is no entanglement between Alice and Bob+Charlie, while
these last two share entanglement. If a state does not admit such
a decomposition, it is entangled with respect to this partition.
Finally, we say that $\rho_{V_{1},..,V_{n}}$ of $n$ systems can
have at most $m$-system entanglement iff it is a mixture of all states
such that each of them is separable with respect to some partition
$\{I_{1},..,I_{k}\}$, where all sets of indices $I_{k}$ have cardinality
$N\leq m$. For tripartite systems this corresponds to the notion of
biseparability, namely the state can have at most 2-system entanglement.
A biseparable state can be written as:
\begin{equation}
\rho_{ABC}=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{BC}^{i}+\sum_{j}q_{j}\rho_{B}^{j}\otimes\rho_{AC}^{j}+\sum_{k}m_{k}\rho_{C}^{k}\otimes\rho_{AB}^{k},
\end{equation}
with $\sum_{i}p_{i}+\sum_{j}q_{j}+\sum_{k}m_{k}=1.$ For $n=3$ a state is then said
to be genuine tripartite entangled if it is $3$-system entangled,
namely if it does not admit such a decomposition.
\subsubsection*{Full separability}
Let us clarify the scenario: in each system $V_{i}$ we consider a
set of $L$ observables $V_{i}^{1},..,V_{i}^{L}$ that we indicate
as $\mathcal{V}_{i}.$ The single system EUR is expressed as:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{i}^{j}\right)\geq f\left(\mathcal{V}_{i},L\right).\label{EUR Vi}
\end{equation}
We are interested in defining criteria in terms of $\sum_{j=1}^{L}H\left(V_{1}^{j},..,V_{n}^{j}\right)$.
A first result regards the notion of full separability.
\begin{prop}
If the state $\rho_{V_{1},..,V_{n}}$ is fully separable, then the
following EUR must hold:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{1}^{j},..,V_{n}^{j}\right)\geq\sum_{i=1}^{n}f\left(\mathcal{V}_{i},L\right).
\end{equation}
\end{prop}
\begin{proof}
Let us consider the case $n=3$. For a given $j$ we have:
\begin{equation}
H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)=H\left(V_{1}^{j}\right)+H\left(V_{2}^{j}V_{3}^{J}|V_{1}^{j}\right).
\end{equation}
Since the state is separable with respect to the partition $23|1$, due
to concavity of the Shannon entropy, we have:
\begin{equation}
H\left(V_{2}^{j}V_{3}^{J}|V_{1}^{j}\right)\geq\sum_{i}p_{i}H(V_{2}^{j}V_{3}^{j})_{\rho_{2}^{i}\otimes\rho_{3}^{i}}.
\end{equation}
By using the chain rule of the Shannon entropy, the above right-hand side
can be rewritten as:
\begin{align}
\sum_{i}p_{i}H(V_{2}^{j}V_{3}^{j})_{\rho_{2}^{i}\otimes\rho_{3}^{i}}= & \sum_{i}p_{i}H(V_{2}^{j})_{\rho_{2}^{i}}\nonumber \\
& +\sum_{i}p_{i}H(V_{3}^{j}|V_{2}^{j})_{\rho_{2}^{i}\otimes\rho_{3}^{i}},
\end{align}
where the last term can be lower bounded by exploiting the separability
of the state and the concavity of the Shannon entropy, namely:
\begin{equation}
\sum_{i}p_{i}H(V_{3}^{j}|V_{2}^{j})_{\rho_{2}^{i}\otimes\rho_{3}^{i}}\geq\sum_{i}p_{i}H(V_{3}^{j})_{\rho_{3}^{i}}.
\end{equation}
By summing over $j$ we arrive at the thesis:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)\geq\sum_{i=1}^{3}f\left(\mathcal{V}_{i},L\right),
\end{equation}
since $\sum_{j}H\left(V_{1}^{j}\right)\geq f\left(\mathcal{V}_{1},L\right)$,
$\sum_{i}p_{i}\sum_{j}H(V_{2}^{j})_{\rho_{2}^{i}}\geq f\left(\mathcal{V}_{2},L\right)$
and $\sum_{i}p_{i}\sum_{j}H(V_{3}^{j})_{\rho_{3}^{i}}\geq f\left(\mathcal{V}_{3},L\right)$
because of the state-independent EUR. The extension of the proof to
$n$ systems is straightforward.
\end{proof}
The following proposition follows directly by considering the state-dependent
bound:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{i}^{j}\right)\geq f\left(\mathcal{V}_{i},L\right)+S\left(\rho_{i}\right).\label{dh}
\end{equation}
\begin{prop}
If the state $\rho_{V_{1},..,V_{n}}$ is fully separable, then the
following EUR must hold:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{1}^{j},..,V_{n}^{j}\right)\geq\sum_{i=1}^{n}f\left(\mathcal{V}_{i},L\right)+\max\left(S\left(\rho_{1}\right),...,S\left(\rho_{n}\right)\right).
\end{equation}
\end{prop}
Note that only the Von Neumann entropy of one system is present in
the above inequality. This is due to the fact that we use only (\ref{dh})
in the first step of the proof, otherwise we would end with criteria
that require the knowledge of the decomposition (\ref{Fully sep}).
\subsubsection*{Genuine multipartite entanglement}
We now analyze the strongest form of multipartite entanglement in
the case of three systems, say Alice, Bob and Charlie. We make the
further assumptions that the three systems have the same dimension
and in each system the parties perform the same set of measurements,
which implies that there is only one bound of the single system EUR
that we indicate as $\mathcal{F}_{1}\left(L\right).$ We indicate
the bound on a pair of systems as $F_{2}\left(L\right)$, namely $\sum_{i=1}^{L}H\left(A_{i},B_{i}\right)\geq F_{2}\left(L\right)$
and the same by permuting the three systems. This will contribute
to the readability of the paper. With this notation the criterion
defined in Proposition 3 for three systems reads as $\sum_{j=1}^{L}H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)\geq3F_{1}\left(L\right)$,
and must be satisfied by all fully separable states.
\begin{prop}
If $\rho_{ABC}$ is not genuine multipartite entangled, namely
it is biseparable, then the following EUR must hold:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)\geq\frac{5}{3}\mathcal{F}_{1}\left(L\right)+\frac{1}{3}F_{2}(L).\label{Prop 5}
\end{equation}
\end{prop}
\begin{proof}
Let us assume that $\rho_{ABC}$ is biseparable, that is:
\begin{equation}
\rho_{ABC}=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{BC}^{i}+\sum_{l}q_{l}\rho_{B}^{l}\otimes\rho_{AC}^{l}+\sum_{k}m_{k}\rho_{C}^{k}\otimes\rho_{AB}^{k}.
\end{equation}
The joint Shannon entropy $H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)$
can be expressed as:
\begin{align}
H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)= & \frac{1}{3}\left[H(V_{1}^{j})+H\left(V_{2}^{j},V_{3}^{j}|V_{1}^{j}\right)\right]\label{chai rules}\\
& +\frac{1}{3}\left[H(V_{2}^{j})+H\left(V_{1}^{j},V_{3}^{j}|V_{2}^{j}\right)\right]\nonumber \\
& +\frac{1}{3}\left[H(V_{3}^{j})+H\left(V_{1}^{j},V_{2}^{j}|V_{3}^{j}\right)\right].\nonumber
\end{align}
By using the concavity of Shannon entropy and the fact that the state
is biseparable we find these relations:
\begin{align}
H\left(V_{2}^{j},V_{3}^{j}|V_{1}^{j}\right) & \geq\sum_{i}p_{i}H\left(V_{2}^{j},V_{3}^{j}\right)_{\rho_{BC}^{i}}\\
& +\sum_{l}q_{l}H(V_{2}^{j})_{\rho_{B}^{l}}+\sum_{l}m_{k}H(V_{3}^{j})_{\rho_{C}^{k}};\nonumber
\end{align}
\begin{align}
H\left(V_{1}^{j},V_{2}^{j}|V_{3}^{j}\right) & \geq\sum_{i}p_{i}H\left(V_{1}^{j}\right)_{\rho_{A}^{i}}\\
& +\sum_{l}q_{l}H(V_{2}^{j})_{\rho_{B}^{l}}+\sum_{l}m_{k}H(V_{1}^{j},V_{2}^{j})_{\rho_{AB}^{k}};\nonumber
\end{align}
\begin{align}
H\left(V_{1}^{j},V_{3}^{j}|V_{2}^{j}\right) & \geq\sum_{i}p_{i}H\left(V_{1}^{j}\right)_{\rho_{A}^{i}}\\
& +\sum_{l}q_{l}H(V_{1}^{j},V_{3}^{j})_{\rho_{AC}^{l}}+\sum_{l}m_{k}H(V_{3}^{j})_{\rho_{C}^{k}}.\nonumber
\end{align}
Then, by considering the sum over $j$ of the sum of the above entropies,
and using EUR, we find:
\begin{equation}
\begin{array}{c}
\sum_{j}H\left(V_{2}^{j},V_{3}^{j}|V_{1}^{j}\right)+H\left(V_{1}^{j},V_{2}^{j}|V_{3}^{j}\right)+H\left(V_{1}^{j},V_{3}^{j}|V_{2}^{j}\right)\\
\geq2\mathcal{F}_{1}\left(L\right)+F_{2}(L).
\end{array}
\end{equation}
The thesis (\ref{Prop 5}) is now implied by combining the expression above, Eq.
(\ref{chai rules}) and the following EUR:
\begin{equation}
\sum_{j}H(V_{1}^{j})+H(V_{2}^{j})+H(V_{3}^{j})\geq3\mathcal{F}_{1}\left(L\right).\label{EURs}
\end{equation}
\end{proof}
\begin{prop}
If $\rho_{ABC}$ is not genuine multipartite entangled, namely
it is biseparable, then the following EUR must hold:
\begin{equation}
\sum_{j=1}^{L}H\left(V_{1}^{j},V_{2}^{j},V_{3}^{j}\right)\geq\frac{5}{3}\mathcal{F}_{1}\left(L\right)+\frac{1}{3}F_{2}(L)+\frac{1}{3}\sum_{x=A,B,C}S\left(\rho_{X}\right).\label{Prop 5-1}
\end{equation}
\end{prop}
The above proposition follows from the proof of Prop. 5, where we consider
the single system state-dependent EUR.
\section{Entanglement Detection}
Here we discuss our criteria for bipartite
and multipartite systems. We will mainly focus on pure states and
multi-qubit systems. We inspect in detail how many entangled states and which levels of separability can be detected with the different criteria derived from the EUR. We point out that, if one focuses just on the entanglement-detection efficiency, bounds retrieved from EUR based on joint Shannon entropy are not as good as some existing criteria. On the other hand, note that the experimental verification of our EUR-based criteria may require less measurements. For instance, if we want to detect the entanglement of a multipartite state through the PPT method, we need to perform a tomography of the state, which involves the measurement of $d^4$ observables. On the contrary, the evaluation of the entropies just needs the measurements of the observables involved in the EUR and its number can be fixed at $3$ independently of the dimension $d$.
\subsection{Bipartite systems}
Let us start with the easiest case of two qubits. In this scenario
we will consider complementary observables and the tight EUR \cite{MultiSanchez,TightAR},
hence the considered criteria read as:
\begin{equation}
H(A_{1},B_{1})+H(A_{2},B_{2})<2+\max(S(\rho_{A}),S(\rho_{B})),\label{criterio1}
\end{equation}
\begin{equation}
\sum_{i=1}^{3}H(A_{i},B_{i})<4,\label{criterio2}
\end{equation}
\begin{align}
\sum_{i=1}^{3}H(A_{i},B_{i}) & <4+\max(S(\rho_{A}),S(\rho_{B})).\label{criterio3}
\end{align}
where $A_{1}=Z_{1},A_{2}=X_{1}$ and $A_{3}=Y_{1}$ being $Z_{1},X_{1}$
and $Y_{1}$ the usual Pauli matrices for the first qubit; the same
holds for the second qubit. In the case of two qubits we have already
shown in Section II that maximally entangled states are detected by
the above criteria.\\
We can then consider the family of entangled two-qubit states given
by:
\begin{equation}
\ket{\psi_{\epsilon}}=\epsilon\ket{00}+\sqrt{1-\epsilon^{2}}\ket{11},\label{entangled states1}
\end{equation}
where $\epsilon\in(0,1).$ We first note that for this family we have
$S(\rho_{A})=S(\rho_{B})=-\epsilon^{2}\log_{2}\epsilon^{2}-(1-\epsilon^{2})\log_{2}(1-\epsilon^{2}),$
which is equal to $H(A_{1},B_{1}).$ Conversely, we have instead
$H(A_{2},B_{2})=-\frac{1}{2}(1-\bar{\epsilon})\log_{2}(\frac{1}{4}(1-\bar{\epsilon}))-\frac{1}{2}(1+\bar{\epsilon})\log_{2}(\frac{1}{4}(1+\bar{\epsilon})),$
with $\bar{\epsilon}=2\epsilon\sqrt{1-\epsilon^{2}}$ and $H(A_{2},B_{2})=H(A_{3},B_{3})$.
The family of states (\ref{entangled states1}) is then completely
detected by (\ref{criterio1}) and (\ref{criterio3}), since $H(A_{2},B_{2})<2$, while
(\ref{criterio2}) fails to detect all states (see Fig.~\ref{fig1}).
Let us consider now the entangled two-qudit states given by:
\begin{equation}
\ket{\psi_{\lambda}}=\sum_{i=0}^{d-1}\lambda_{i}\ket{ii},\label{entangled states1-1}
\end{equation}
where $\sum_{i}\lambda_{i}^{2}=1$ and $0<\lambda_{i}<1$. As an entanglement
criterion we consider:
\begin{equation}
H(A_{1},B_{1})+H(A_{2},B_{2})\geq2\log_{2}d+\max(S(\rho_{A}),S(\rho_{B})),\label{qudit criterion}
\end{equation}
where the first observable $A_{1}$ is the computational basis and
$A_{2}$ is its Fourier transform, which is well-defined in any dimension,
and the same for $B_{1}$ and $B_{2}$. First, we can observe that
for these states we have $S(\rho_{A})=S(\rho_{B})=-\sum_{i}\lambda_{i}^{2}\log_{2}\lambda_{i}^{2}$. Moreover, since $A_{1}$ and $B_{1}$ are respectively represented
by the computational bases, we have $H(A_{1},B_{1})=-\sum_{i}\lambda_{i}^{2}\log_{2}\lambda_{i}^{2}$.
Hence, the entanglement condition becomes:
\begin{equation}
H(A_{2},B_{2})<2\log_{2}d.
\end{equation}
\begin{figure}
\caption{Pure Bipartite States: the continuous line represents $\sum_{i=1}
\label{fig1}
\end{figure}
However, for any two-qudit states we have $H(A_{2},B_{2})\leq2\log_{2}d$
and the maximum is achieved by states that give uniform probability
distributions for $A_{2}\otimes B_{2}$. Since $A_{2}$ and $B_{2}$
are the Fourier transformed bases of the computational ones, the family
(\ref{entangled states1-1}) cannot give a uniform probability distribution
due the definition of the Fourier transform. The maximum value could
be attained only by states of the form $\ket{ii}$, hence by separable
states which are excluded in (\ref{entangled states1-1}). Thus, our
criterion (\ref{qudit criterion}) detects all two-qudit entangled
states of the form (\ref{entangled states1-1}).\\
\begin{figure}
\caption{GHZ States: on the left, the continuous line represents $\sum_{i=1}
\label{fig2}
\end{figure}
\subsection{Multipartite systems}
As an example of multipartite systems we focus on the case of a three
qubit system. In this case a straightforward generalization of the
Schmidt decomposition is not available. However, the pure states can be
parameterized and classified in terms of five real parameters:
\begin{equation}
\ket{\psi}=\lambda_{0}\ket{000}+\lambda_{1}e^{i\phi}\ket{100}+\lambda_{2}\ket{101}+\lambda_{3}\ket{110}+\lambda_{4}\ket{111},
\end{equation}
where $\sum_{i}\lambda_{i}^{2}=1$. In particular we are interested
in two classes of entangled states, the GHZ states, given by
\begin{equation}
\ket{GHZ}=\lambda_{0}\ket{000}+\lambda_{4}\ket{111},
\end{equation}
and the $W$-states, which are
\begin{equation}
\ket{W}=\lambda_{0}\ket{000}+\lambda_{2}\ket{101}+\lambda_{3}\ket{110}.
\end{equation}
\begin{figure}
\caption{Tripartite W-States: The above shows in which areas in the plane $\lambda_{0}
\label{fig3}
\end{figure}
\begin{figure}
\caption{W-states Non-Separability: the plot shows the effectiveness of the state-dependent criterion
(\ref{multi_ent2}
\label{fig4}
\end{figure}
\begin{figure}
\caption{W-State Genuine Multipartite Entanglement: the plot shows the performance of the state-dependent criterion (\ref{gen_ent2}
\label{fig5}
\end{figure}
The three observables considered in each system
are the Pauli matrices, hence $A_{1}=Z_{1},A_{2}=X_{1}$ and
$A_{3}=Y_{1}$ and the same for the other subsystems. The criteria
for detecting the presence of entanglement, namely states that are not fully separable,
in this case read as:
\begin{equation}
\sum_{i=1}^{3}H(A_{i},B_{i},C_{i})<6,\label{multi_ent1}
\end{equation}
\begin{equation}
\sum_{i=1}^{3}H(A_{i},B_{i},C_{i})<6+\max(S(\rho_{A}),S(\rho_{B}),S(\rho_{C})),\label{multi_ent2}
\end{equation}
while the criteria for genuine multipartite entanglement are:
\begin{equation}
\sum_{i=1}^{3}H(A_{i},B_{i},C_{i})<\text{\ensuremath{\frac{13}{3}}},\label{gen_ent1}
\end{equation}
\begin{align}
\sum_{i=1}^{3}H(A_{i},B_{i},C_{i}) & <\ensuremath{\frac{13}{3}}+\frac{1}{3}\sum_{x=A,B,C}S\left(\rho_{X}\right).\label{gen_ent2}
\end{align}
For the class of GHZ states the sum of the three entropies $\sum_{i=1}^{3}H(A_{i},B_{i},C_{i})$
is plotted as a function of $\lambda_{0}$ in Fig.~\ref{fig2} with respect
to the state-independent and dependent bounds. We can therefore see
that the state-independent bounds fail to detect even the weakest
form of entanglement. Conversely, the state-dependent bounds identify
all states as non-separable but none as genuine multipartite entangled.
\\
For the class of the W states the effectiveness of our criteria is
shown in Figs.~\ref{fig3},~\ref{fig4} and~\ref{fig5}. Since the W-states depend on two parameters
we decided to use contour plots in the plane $\lambda_{0}\times\lambda_{2}$
showing which subsets of W states are detected as non-fully separable or
genuine multipartite entangled. As we can see, the state-independent
bounds (Fig.~\ref{fig3}) detect the non-separable character of the W-states
for a large subset of them. Conversely, no state is identified as
genuine multipartite entangled. By using the state-dependent bounds
(Figs.~\ref{fig4} and~\ref{fig5}) we are able to detect almost all non separable W-states
and, above all, we can also identify a small subset of W states as
genuine multipartite entangled.
\section{Conclusions}
In conclusion, we derived and characterized a number of entropic
uncertainty inequalities, defined in terms of the joint Shannon entropy,
whose violation guarantees the presence of entanglement. On
a theoretical level, which was the main aim of this work, we clarified
that EUR entanglement criteria for the joint Shannon entropy require
at least three different observables or, if one considers only two
measurements, the addition of the Von Neumann entropy of a subsystem,
showing thus that the additivity character of the EUR holds only for
two measurements \cite{Additivity}. We also extended our
criteria to the case of multipartite systems, which enable us to discriminate
between different types of multipartite entanglement. We then showed
how these criteria perform for both bipartite and multipartite systems,
providing several examples of states that are detected by the proposed
criteria.
This material is based upon work supported by the U.S. Department
of Energy, Office of Science, National Quantum Information Science
Research Centers, Superconducting Quantum Materials and Systems Center
(SQMS) under contract number DEAC02-07CH11359 and by the EU H2020
QuantERA ERA-NET Cofund in Quantum Technologies project QuICHE.
\end{document} |
\begin{document}
\title{Chern classes in equivariant bordism}
\date{\longrightarrowday; 2020 AMS Math.\ Subj.\ Class.: 55N22, 55N91, 55P91, 57R85}
\author{Stefan Schwede}
\address{Mathematisches Institut, Universit\"at Bonn, Germany}
\email{schwede@math.uni-bonn.de}
\begin{abstract} We introduce Chern classes in $U(m)$-equivariant homotopical bordism
that refine the Conner-Floyd-Chern classes in the $\mathbf{MU}$-cohomology of $B U(m)$.
For products of unitary groups, our Chern classes form regular sequences
that generate the augmentation ideal of the equivariant bordism rings.
Consequently, the Greenlees-May local homology spectral sequence collapses for products of unitary groups.
We use the Chern classes to reprove the $\mathbf{MU}$-completion theorem of Greenlees-May and La Vecchia.
\end{abstract}
\maketitle
\section*{Introduction}
Complex cobordism $\mathbf{MU}$ is arguably the most important cohomology theory in algebraic topology.
It represents the bordism theory of stably almost complex manifolds,
and it is the universal complex oriented cohomology theory;
via Quillen's celebrated theorem \cite{quillen:formal_group},
$\mathbf{MU}$ is the entry gate for the theory of formal group laws into
stable homotopy theory, and thus the cornerstone of chromatic stable homotopy theory.
Tom Dieck's homotopical equivariant bordism $\mathbf{MU}_G$ \cite{tomDieck:bordism_integrality},
defined with the help of equivariant Thom spaces,
strives to be the legitimate equivariant refinement of complex cobordism,
for compact Lie groups $G$.
The theory $\mathbf{MU}_G$ is the universal equivariantly complex oriented theory;
and for abelian compact Lie groups, the coefficient ring $\mathbf{MU}_G^*$ carries the universal
$G$-equivariant formal group law \cite{hausmann:group_law}.
Homotopical equivariant bordism receives a homomorphism
from the geometrically defined equivariant bordism theory; due to the lack
of equivariant transversality, this homomorphism is {\em not} an isomorphism for non-trivial groups.
In general, the equivariant bordism ring $\mathbf{MU}^*_G$
is still largely mysterious; the purpose of this paper is to elucidate its structure
for unitary groups, and for products of unitary groups.
Chern classes are important characteristic classes for complex vector bundles
that were originally introduced in singular cohomology.
Conner and Floyd \cite[Corollary 8.3]{conner-floyd:relation_cobordism}
constructed Chern classes for complex vector bundles in complex cobordism;
in the universal cases, these yield classes $c_k\in \mathbf{MU}^{2 k}(B U(m))$
that are nowadays referred to as Conner-Floyd-Chern classes.
Conner and Floyd's construction works in much the same way for any complex oriented cohomology theory,
see \cite[Part II, Lemma 4.3]{adams:stable_homotopy};
in singular cohomology, it reduces to the classical Chern classes.
The purpose of this note is to define and study Chern classes
in $U(m)$-equivariant homotopical bordism $\mathbf{MU}^*_{U(m)}$
that map to the Conner-Floyd-Chern classes
under tom Dieck's bundling homomorphism \cite[Proposition 1.2]{tomDieck:bordism_integrality}.
Our classes satisfy the analogous formal properties as their classical counterparts,
including the equivariant refinement of the Whitney sum formula, see Theorem \ref{thm:CFC main}.
Despite the many formal similarities, there are crucial qualitative differences
compared to Chern classes in complex oriented cohomology theories: our
Chern classes are {\em not} characterized by their restriction to the maximal torus, and some
of our Chern classes are zero-divisors, see Remark \ref{rk:torus_restriction}.
We will use our Chern classes and the splitting of \cite{schwede:split BU}
to prove new structure results about the equivariant bordism rings $\mathbf{MU}^*_{U(m)}$
for unitary groups, or more generally for products of unitary groups.
To put this into context, we recall that in the special case when $G$ is an {\em abelian} compact Lie group,
the graded ring $\mathbf{MU}^*_G$ is concentrated in even degrees and free as a module
over the non-equivariant cobordism ring $\mathbf{MU}^*$ \cite[Theorem 5.3]{comezana}, \cite{loeffler:equivariant},
and the bundling homomorphism $\mathbf{MU}^*_G\longrightarrow \mathbf{MU}^*(B G)$ is completion
at the augmentation ideal of $\mathbf{MU}^*_G$ \cite[Theorem 1.1]{comezana-may}, \cite{loeffler:bordismengruppen}.
For non-abelian compact Lie groups $G$, however, the equivariant bordism rings $\mathbf{MU}^*_G$
are still largely mysterious.
\pagebreak
The main result of this note is the following:\wedgeallskip
{\bf Theorem.} {\em Let $m\geq 1$ be a natural number.
\begin{enumerate}[\em (i)]
\item
The sequence of Chern classes $c_m^{(m)},c_{m-1}^{(m)},\dots,c_1^{(m)}$
is a regular sequence that generates the augmentation ideal of the graded-commutative ring $\mathbf{MU}^*_{U(m)}$.
\item
The completion of $\mathbf{MU}_{U(m)}^*$ at the augmentation ideal
is a graded $\mathbf{MU}^*$-power series algebra in the above Chern classes.
\item
The bundling homomorphism $\mathbf{MU}_{U(m)}^*\longrightarrow \mathbf{MU}^*(B U(m))$ extends to an isomorphism
\[ ( \mathbf{MU}_{U(m)}^*)^\wedge_I \ \longrightarrow \ \mathbf{MU}^*(BU(m)) \]
from the completion at the augmentation ideal.
\end{enumerate}
}
We prove this result as a special case of Theorem \ref{thm:completions} below;
the more general version applies to products of unitary groups.
As we explain in Remark \ref{rk:degenerate}, the regularity of the Chern classes
also implies that the Greenlees-May local homology spectral sequence
converging to $\mathbf{MU}^*(BU(m))$ degenerates
because the relevant local homology groups vanish in positive degrees.
As another application we use the Chern classes in equivariant bordism
to give a reformulation and self-contained proof of work of Greenlees-May \cite{greenlees-may:completion}
and La Vecchia \cite{lavecchia} on the completion theorem for $\mathbf{MU}_G$,
see Theorem \ref{thm:completion}.
\section{Equivariant \texorpdfstring{$\mathbf{MU}$}{MU}-Chern classes}
In this section we introduce the Chern classes in $U(m)$-equivariant homotopical bordism,
see Definition \ref{def:CFC}. We establish their basic properties
in Theorem \ref{thm:CFC main}, including a Whitney sum formula and the fact that the bundling homomorphism
takes our Chern classes to the Conner-Floyd-Chern classes in $\mathbf{MU}$-cohomology.
We begin by fixing our notation.
For a compact Lie group $G$, we write $\mathbf{MU}_G$ for the $G$-equivariant homotopical bordism spectrum
introduced by tom Dieck \cite{tomDieck:bordism_integrality}.
For our purposes, it is highly relevant that the theories $\mathbf{MU}_G$ for varying compact Lie groups $G$ assemble
into a global stable homotopy type, see \cite[Example 6.1.53]{schwede:global}.
For an integer $n$, we write $\mathbf{MU}_G^n=\pi_{-n}^G(\mathbf{MU})$ for the $G$-equivariant coefficient group
in cohomological degree $n$.
Since $\mathbf{MU}$ comes with the structure of a global ring spectrum, it supports
graded-commutative multiplications on $\mathbf{MU}_G^*$, as well as external multiplication pairings
\[ \times \ : \ \mathbf{MU}_G^k\times \mathbf{MU}_K^l \ \longrightarrow \ \mathbf{MU}_{G\times K}^{k+l} \]
for all pairs of compact Lie groups $G$ and $K$.
We write $\nu_k$ for the tautological representation
of the unitary group $U(k)$ on $\mathbb C^k$; we denote its Euler class by
\[ e_k \ = \ e(\nu_k) \ \in \ \mathbf{MU}^{2 k}_{U(k)}\ ,\]
compare \cite[page 347]{tomDieck:bordism_integrality}.
We write $U(k,m-k)$ for the block subgroup of $U(m)$ consisting of matrices of the form
$(\begin{smallmatrix}A & 0 \\ 0 & B \end{smallmatrix})$
for $(A,B)\in U(k)\times U(m-k)$.
We write $\tr_{U(k,m-k)}^{U(m)}:\mathbf{MU}_{U(k,m-k)}^*\longrightarrow\mathbf{MU}_{U(m)}^*$
for the transfer associated to the inclusion $U(k,m-k)\longrightarrow U(m)$,
see for example \cite[Construction 3.2.11]{schwede:global}.
\begin{defn}\label{def:CFC}
For $0\leq k\leq m$, the {\em $k$-th Chern class}
in equivariant complex bordism is the class
\[ c_k^{(m)} \ = \ \tr_{U(k,m-k)}^{U(m)}(e_k\times 1_{m-k})\ \in \ \mathbf{MU}^{2 k}_{U(m)}\ , \]
where $1_{m-k}\in\mathbf{MU}_{U(m-k)}^0$ is the multiplicative unit. We also set $c_k^{(m)} =0$ for $k>m$.
\end{defn}
In the extreme cases $k=0$ and $k=m$, we recover familiar classes:
since $e_0$ is the multiplicative unit in the non-equivariant cobordism ring $\mathbf{MU}^*$,
the class $c_0^{(m)}=1_m$ is the multiplicative unit in $\mathbf{MU}_{U(m)}^0$.
In the other extreme, $c_m^{(m)}=e_m=e(\nu_m)$ is the Euler class of
the tautological $U(m)$-representation.
As we will show in Theorem \ref{thm:CFC main} (ii), the classes $c_k^{(m)}$
are compatible in $m$ under restriction to smaller unitary groups.
\begin{rk}\label{rk:torus_restriction}
We alert the reader that the restriction homomorphism $\res^{U(m)}_{T^m}:\mathbf{MU}^*_{U(m)}\longrightarrow \mathbf{MU}^*_{T^m}$
is not injective for $m\geq 2$, where $T^m$ denotes a maximal torus in $U(m)$.
So the Chern classes in $\mathbf{MU}^*_{U(m)}$ are not characterized by their restrictions
to the maximal torus -- in contrast to the non-equivariant situation for complex oriented cohomology theories.
To show this we let $N$ denote the maximal torus normalizer inside $U(m)$. The class
\[ 1- \tr_N^{U(m)}(1) \ \in \ \mathbf{MU}^0_{U(m)} \]
has infinite order because the $U(m)$-geometric fixed point map
takes it to the multiplicative unit; in particular, this class is nonzero.
The double coset formula \cite[IV Corollary 6.7 (i)]{lms}
\[ \res^{U(m)}_{T^m}(\tr_N^{U(m)}(1))\ = \ \res^N_{T^m}(1)\ = \ 1 \]
implies that the class $ 1- \tr_N^{U(m)}(1)$ lies in the kernel of the restriction homomorphism
$\res^{U(m)}_{T^m}:\mathbf{MU}^0_{U(m)}\longrightarrow \mathbf{MU}^0_{T^m}$.
Moreover, the Chern class $c_1^{(2)}$ is a zero-divisor in the ring $\mathbf{MU}^*_{U(2)}$,
also in stark contrast to Chern classes in complex oriented cohomology theories.
Indeed, reciprocity for restriction and transfers \cite[Corollary 3.5.17 (v)]{schwede:global}
yields the relation
\begin{align*}
c_1^{(2)}\cdot (1-\tr_N^{U(2)}(1)) \
&= \ \tr_{U(1,1)}^{U(2)}(e_1\times 1)\cdot (1-\tr_N^{U(2)}(1)) \\
&= \
\tr_{U(1,1)}^{U(2)}((e_1\times 1)\cdot \res^{U(2)}_{U(1,1)}(1-\tr_N^{U(2)}(1))) \ = \ 0 \ .
\end{align*}
One can also show that the class $1-\tr_N^{U(2)}(1)$ is infinitely divisible by the Euler class $e_2=c_2^{(2)}$;
so it is also in the kernel of the completion map at the ideal $(e_2)$.
\end{rk}
The Chern class $c_k^{(m)}$ is defined as a transfer; so identifying its restriction
to a subgroup of $U(m)$ involves a double coset formula.
The following double coset formula will take care of all cases we need in this paper;
it ought to be well-known to experts, but I do not know a reference.
The case $l=1$ is established in \cite[Lemma 4.2]{symonds-splitting},
see also \cite[Example 3.4.13]{schwede:global}.
The double coset space $U(i,j)\backslash U(m)/ U(k,l)$ is discussed at various places in the
literature, for example \cite[Example 3]{matsuki:double_coset},
but I have not seen the resulting double coset formula spelled out.
\begin{prop}[Double coset formula]\label{prop:double coset}
Let $i,j,k,l$ be positive natural numbers such that $i+j=k+l$. Then
\[ \res^{U(i+j)}_{U(i,j)}\circ\tr_{U(k,l)}^{U(k+l)} \ = \
\sum_{0,k-j\leq d\leq i,k}\, \tr_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}\circ\gamma_d^*\circ \res^{U(k,l)}_{U(d,k-d,i-d,l-i+d)}\ ,\]
where $\gamma_d\in U(i+j)$ is the permutation matrix of the shuffle permutation $\chi_d\in\Sigma_{i+j}$
given by
\[ \chi_d(a) \ = \
\begin{cases}
a & \text{ for $1\leq a\leq d$,}\\
a-d+i& \text{ for $d+1\leq a\leq k$,}\\
a+d-k& \text{ for $k+1\leq a\leq k+i-d$, and}\\
a & \text{ for $a > k+i-d$.}
\end{cases}
\]
\end{prop}
\begin{proof}
We refer to \cite[IV 6]{lms} or \cite[Theorem 3.4.9]{schwede:global} for the general
double coset formula for $\res^G_K\circ\tr_H^G$ for two closed subgroups $H$ and $K$
of a compact Lie group $G$; we need to specialize it to the situation at hand.
We first consider a matrix $A\in U(m)$ such that the center $Z$ of $U(i,j)$
is {\em not} contained in the $U(i,j)$-stabilizer
\[ S_A\ = \ U(i,j)\cap {^A U(k,l)} \]
of the coset $A\cdot U(k,l)$.
Then $S_A\cap Z$ is a proper subgroup of the center $Z$ of $U(i,j)$, which is isomorphic
to $U(1)\times U(1)$. So $S_A\cap Z$ has strictly smaller dimension than $Z$.
Since the center of $U(i,j)$ is contained in the normalizer of $S_A$,
we conclude that the group $S_A$ has an infinite Weyl group inside $U(i,j)$.
All summands in the double coset formula indexed by such points then involve transfers with infinite Weyl groups,
and hence they vanish.
So all non-trivial contributions to the double coset formula
stem from double cosets $U(i,j)\cdot A\cdot U(k,l)$ such that $S_A$ contains the center of $U(i,j)$.
In particular the matrix
$ \left( \begin{smallmatrix} - E_i & 0 \\ 0 & E_j \end{smallmatrix} \right)$ then belongs to $S_A$.
We write $L=A\cdot (\mathbb C^k\oplus 0^l)$, a complex $k$-plane in $\mathbb C^{k+l}$;
we consider $x\in\mathbb C^i$ and $y\in\mathbb C^j$ such that $(x,y)\in L$.
Because $ \left( \begin{smallmatrix} - E_i & 0 \\ 0 & E_j \end{smallmatrix} \right)\cdot L=L$,
we deduce that $(-x,y)\in L$. Since $(x,y)$ and $(-x,y)$ belong to $L$, so do the vectors $(x,0)$ and $(y,0)$.
We have thus shown that the $k$-plane $L=A\cdot(\mathbb C^k\oplus 0^l)$ is spanned by the intersections
\[ L\cap (\mathbb C^i\oplus 0^j) \text{\qquad and\qquad} L\cap (0^i\oplus\mathbb C^j)\ . \]
We organize the cosets with this property by the dimension of the first intersection:
we define $M_d$ as the closed subspace of $U(m)/U(k,l)$
consisting of those cosets $A\cdot U(k,l)$ such that
\[ \dim_\mathbb C( L\cap (\mathbb C^i\oplus 0^j))\ = \ d
\text{\qquad and\qquad}
\dim_\mathbb C( L\cap (0^i\oplus\mathbb C^j))\ = \ k-d\ . \]
If $M_d$ is non-empty, we must have $0, k-j\leq d\leq i,k$.
The group $U(i,j)$ acts transitively on $M_d$, and the coset $\gamma_d\cdot U(k,l)$ belongs to $M_d$;
so $M_d$ is the $U(i,j)$-orbit type manifold of $U(m)/U(k,l)$ for the conjugacy class of
\[ S_{\gamma_d}\ = \ U(i,j)\cap {^{\gamma_d} U(k,l)} \ = \ U(d,i-d,k-d,j-k+d)\ . \]
The corresponding orbit space $U(i,j)\backslash M_d=U(i,j)\cdot\gamma_d\cdot U(k,l)$
is a single point inside the double coset space, so its internal Euler characteristic is 1.
This orbit type thus contributes the summand
\[ \tr_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}\circ\gamma_d^*\circ \res^{U(k,l)}_{U(d,k-d,i-d,l-i+d)} \]
to the double coset formula.
\end{proof}
In \cite[Corollary 8.3]{conner-floyd:relation_cobordism},
Conner and Floyd define Chern classes for complex vector bundles
in the non-equivariant $\mathbf{MU}$-cohomology rings.
In the universal cases, these yield classes $c_k\in \mathbf{MU}^{2 k}(B U(m))$
that are nowadays referred to as Conner-Floyd-Chern classes.
The next theorem spells out the key properties of our Chern classes $c_k^{(m)}$;
parts (i), (ii) and (iii) roughly say that all the familiar structural properties
of the Conner-Floyd-Chern classes in $\mathbf{MU}^*(B U(m))$
already hold for our Chern classes in $U(m)$-equivariant $\mathbf{MU}$-theory.
Part (iv) of the theorem refers to the bundling maps $\mathbf{MU}_G^*\longrightarrow \mathbf{MU}^*(B G)$
defined by tom Dieck in \cite[Proposition 1.2]{tomDieck:bordism_integrality}.
\begin{thm}\label{thm:CFC main} The Chern classes in homotopical equivariant bordism enjoy the following properties.
\begin{enumerate}[\em (i)]
\item For all $0\leq k\leq m=i+j$, the relation
\[ \res^{U(m)}_{U(i,j)}(c_k^{(m)})\ = \ \sum_{d=0,\dots,k} c_d^{(i)}\times c_{k-d}^{(j)}\]
holds in the group $\mathbf{MU}_{U(i,j)}^{2 k}$.
\item The relation
\[ \res^{U(m)}_{U(m-1)}(c_k^{(m)})\ = \
\begin{cases}
c_k^{(m-1)} & \text{ for $0\leq k\leq m-1$, and}\\
\ 0 & \text{ for $k=m$}
\end{cases}\]
holds in the group $\mathbf{MU}_{U(m-1)}^{2 k}$.
\item Let $T^m$ denote the diagonal maximal torus of $U(m)$. Then the restriction homomorphism
\[ \res^{U(m)}_{T^m} \ : \ \mathbf{MU}_{U(m)}^{2 k} \ \longrightarrow \ \mathbf{MU}^{2 k}_{T^m} \]
takes the class $c_k^{(m)}$ to the $k$-th elementary symmetric polynomial
in the classes $p_1^*(e_1),\dots,p_m^*(e_1)$,
where $p_i:T^m\longrightarrow T=U(1)$ is the projection to the $i$-th factor.
\item The bundling map
\[ \mathbf{MU}_{U(m)}^* \ \longrightarrow \ \mathbf{MU}^*(BU(m)) \]
takes the class $c_k^{(m)}$ to the $k$-th Conner-Floyd-Chern class.
\end{enumerate}
\end{thm}
\begin{proof}
(i) This property exploits the double coset formula for
$\res^{U(m)}_{U(i,j)}\circ\tr_{U(k,m-k)}^{U(m)}$ recorded in Proposition \ref{prop:double coset},
which is the second equation in the following list:
\begin{align*}
\res^{U(m)}_{U(i,j)}(c_k^{(m)})\
&= \ \res^{U(m)}_{U(i,j)}(\tr_{U(k,m-k)}^{U(m)}(e_k\times 1_{m-k})) \\
&= \
\sum_{d=0,\dots,k} \tr_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(\gamma_d^*(\res^{U(k,m-k)}_{U(d,k-d,i-d,j-k+d)}(e_k\times 1_{m-k})))\\
&= \
\sum_{d=0,\dots,k} \tr_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(\gamma_d^*(e_d\times e_{k-d}\times 1_{i-d}\times 1_{j-k+d}))\\
&= \
\sum_{d=0,\dots,k} \tr_{U(d,i-d,k-d,j-k+d)}^{U(i,j)}(e_d\times 1_{i-d}\times e_{k-d}\times 1_{j-k+d})\\
&= \
\sum_{d=0,\dots,k} \tr_{U(d,i-d)}^{U(i)}(e_d\times 1_{i-d})\times
\tr_{U(k-d,j-k+d)}^{U(j)}(e_{k-d}\times 1_{j-k+d})\\
&= \ \sum_{d=0,\dots,k} c_d^{(i)}\times c_{k-d}^{(j)}
\end{align*}
Part (ii) for $k<m$ follows from part (i) by restriction from $U(m-1,1)$ to $U(m-1)$:
\begin{align*}
\res^{U(m)}_{U(m-1)}(c_k^{(m)})\
&= \ \res^{U(m-1,1)}_{U(m-1)}(\res^{U(m)}_{U(m-1,1)}(c_k^{(m)}))\\
&= \ \res^{U(m-1,1)}_{U(m-1)}(c_{k-1}^{(m-1)}\times c_1^{(1)}\ + \ c_k^{(m-1)}\times c_0^{(1)})\\
&= \ c_{k-1}^{(m-1)}\times \res^{U(1)}_1(c_1^{(1)})\ +\ c_k^{(m-1)}\times \res^{U(1)}_1(c_0^{(1)})\ = \ c_k^{(m-1)}\ .
\end{align*}
We have used that the class $c_1^{(1)}=e_1$ is in the kernel of the augmentation
$\res^{U(1)}_1:\mathbf{MU}_{U(1)}^*\longrightarrow \mathbf{MU}^*$. The Euler class $c_m^{(m)}=e(\nu_m)$
restricts to 0 in $\mathbf{MU}^*_{U(m-1)}$ because the restriction
of the tautological $U(m)$-representation to $U(m-1)$ splits off a trivial 1-dimensional summand.
(iii) An inductive argument based on property (i) shows the desired relation:
\begin{align*}
\res^{U(m)}_{T^m}(c_k^{(m)}) \ &= \
\res^{U(m)}_{U(1,\dots,1)}(c_k^{(m)}) \\
&= \ \sum_{A\subset\{1,\dots,m\}, |A|=k}\quad \prod_{a\in A} p_a^*(c_1^{(1)})\cdot\prod_{b\not\in A}p_b^*(c_0^{(1)}) \\
&= \ \sum_{A\subset\{1,\dots,m\}, |A|=k} \quad \prod_{a\in A} p_a^*(e_1)\ .
\end{align*}
(iv)
As before we let $T^m$ denote the diagonal maximal torus in $U(m)$.
The splitting principle holds for non-equivariant complex oriented cohomology theories,
see for example \cite[Proposition 8.10]{dold:Chern_classes}.
In other words, the right vertical map in the commutative square of graded rings is injective:
\[ \xymatrix{ \mathbf{MU}^*_{U(m)}\ar[r]\ar[d]_{\res^{U(m)}_{T^m}} & \mathbf{MU}^*(B U(m))\ar[d]^-{ (B i)^*} \\
\mathbf{MU}^*_{T^m}\ar[r] & \mathbf{MU}^*(B T^m) \ar@{=}[r] & \mathbf{MU}^*[[p_1^*(e_1),\dots,p_m^*(e_1)]]
} \]
The $k$-th Conner-Floyd-Chern class is characterized as the unique element
of $\mathbf{MU}^{2 k}(B U(m))$ that maps to
the $k$-th elementary symmetric polynomial in the classes $p_1^*(e_1),\dots,p_m^*(e_1)$.
Together with part (iii), this proves the claim.
\end{proof}
\section{Regularity results}
In this section we use the Chern classes to formulate new structural properties of the equivariant
bordism ring $\mathbf{MU}_{U(m)}^*$. In particular, we can say what $\mathbf{MU}_{U(m)}^*$
looks like after dividing out some of the Chern classes, and after completing at the Chern classes.
The following theorem states these facts more generally for $U(m)\times G$
instead of $U(m)$; by induction on the number of factors, we can then deduce corresponding results
for products of unitary groups, see Theorem \ref{thm:completions}.
The results in this section make crucial use of the splitting theorem for
global functors established in \cite{schwede:split BU}.
\begin{thm}\label{thm:structure}
For every compact Lie group $G$ and all $0\leq k\leq m$, the sequence of Chern classes
\[ (c_m^{(m)}\times 1_G,\ c_{m-1}^{(m)}\times 1_G,\dots,\ c_{k+1}^{(m)}\times 1_G) \]
is a regular sequence in the graded-commutative ring $\mathbf{MU}^*_{U(m)\times G}$
that generates the kernel of the surjective restriction homomorphism
\[ \res^{ U(m)\times G}_{U(k)\times G}\ :\ \mathbf{MU}_{U(m)\times G}^*\ \longrightarrow \mathbf{MU}_{U(k)\times G}^*\ . \]
In particular, the sequence of Chern classes $(c_m^{(m)},c_{m-1}^{(m)},\dots,c_1^{(m)})$
is a regular sequence that generates the augmentation ideal of the graded-commutative ring $\mathbf{MU}^*_{U(m)}$.
\end{thm}
\begin{proof}
We argue by downward induction on $k$. The induction starts with $k=m$, where there is nothing to show.
Now we assume the claim for some $k\leq m$, and we deduce it for $k-1$.
The inductive hypothesis shows that
$c_m^{(m)}\times 1_G,\dots,c_{k+1}^{(m)}\times 1_G$
is a regular sequence in the graded-commutative ring $\mathbf{MU}^*_{U(m)\times G}$,
and that the restriction homomorphism $\res^{U(m)\times G}_{U(k)\times G}$
factors through an isomorphism
\[ \mathbf{MU}_{U(m)\times G}^*/(c_m^{(m)}\times 1_G,\dots,c_{k+1}^{(m)}\times 1_G)
\ \cong \mathbf{MU}_{G\times U(k)}^*\ . \]
We exploit that the various equivariant bordism spectra $\mathbf{MU}_G$ underlie a global spectrum,
see \cite[Example 6.1.53]{schwede:global};
thus the restriction homomorphism $\res^{U(k)\times G}_{U(k-1)\times G}$ is surjective by
Theorem 1.4 and Proposition 2.2 of \cite{schwede:split BU}.
Hence the standard long exact sequence unsplices into
a short exact sequence of graded $\mathbf{MU}^*$-modules:
\[ 0\ \longrightarrow\ \mathbf{MU}_{U(k)\times G}^{*-2 k}\ \xrightarrow{(e_k\times 1_G)\cdot -\ }\
\mathbf{MU}_{U(k)\times G}^* \xrightarrow{\res^{U(k)\times G}_{U(k-1)\times G}}\ \mathbf{MU}_{U(k-1)\times G}^*\ \longrightarrow\ 0
\]
Because
\[ \res^{U(m)\times G}_{U(k)\times G}(c_k^{(m)}\times 1_G)\ = \ c_k^{(k)}\times 1_G\ = \ e_k\times 1_G\ , \]
we conclude that $c_k^{(m)}\times 1_G$ is a non zero-divisor in
$\mathbf{MU}_{U(m)\times G}^*/(c_m^{(m)}\times 1_G,c_{m-1}^{(m)}\times 1_G,\dots,c_{k+1}^{(m)}\times 1_G)$,
and that additionally dividing out $c_k^{(m)}\times 1_G$ yields $\mathbf{MU}_{U(k-1)\times G}^*$.
This completes the inductive step.
\end{proof}
We can now identify the completion of $\mathbf{MU}^*_{U(m)}$ at the augmentation ideal
as an $\mathbf{MU}^*$-power series algebra on the Chern classes.
We state this somewhat more generally for products of unitary groups, which we write as
\[ U(m_1,\dots,m_l)\ = \ U(m_1)\times\dots\times U(m_l)\ , \]
for natural numbers $m_1,\dots,m_l\geq 1$.
For $1\leq i\leq l$, we write $p_i:U(m_1,\dots,m_l)\longrightarrow U(m_i)$ for the projection to the $i$-th factor,
and we set
\[ c^{[i]}_k \ = \ p_i^*(c_k^{(m_i)})\ = \ 1_{U(m_1,\dots,m_{i-1})}\times c_k^{(m_i)}\times 1_{U(m_{i+1},\dots,m_l)}
\ \in \ \mathbf{MU}_{U(m_1,\dots,m_l)}^{2 k}\ .\]
The following theorem was previously known for tori, i.e., for $m_1=\dots=m_l=1$.
\begin{thm}\label{thm:completions}
Let $m_1,\dots,m_l\geq 1$ be positive integers.
\begin{enumerate}[\em (i)]
\item
The sequence of Chern classes
\begin{equation}\label{eq:Chern_for_products}
c_{m_1}^{[1]},\dots,c_1^{[1]},c_{m_2}^{[2]},\dots,c_1^{[2]},\dots, c_{m_l}^{[l]},\dots,c_1^{[l]}
\end{equation}
is a regular sequence that generates the augmentation ideal of the graded-commutative ring $\mathbf{MU}^*_{U(m_1,\dots,m_l)}$.
\item
The completion of $\mathbf{MU}_{U(m_1,\dots,m_l)}^*$ at the augmentation ideal
is a graded $\mathbf{MU}^*$-power series algebra in the Chern classes \eqref{eq:Chern_for_products}.
\item
The bundling map $\mathbf{MU}_{U(m_1,\dots,m_l)}^*\longrightarrow \mathbf{MU}^*(B U(m_1,\dots,m_l))$ extends to an isomorphism
\[ ( \mathbf{MU}_{U(m_1,\dots,m_l)}^*)^\wedge_I \ \longrightarrow \ \mathbf{MU}^*(BU(m_1,\dots,m_l)) \]
from the completion at the augmentation ideal.
\end{enumerate}
\end{thm}
\begin{proof}
Part (i) follows from Theorem \ref{thm:structure} by induction on the number $l$ of factors.
We prove parts (ii) and (iii) together.
We must show that for every $n\geq 1$, $\mathbf{MU}_{U(m_1,\dots,m_l)}^*/I^n$ is free as an $\mathbf{MU}^*$-module on the monomials
of degree less than $n$ in the Chern classes \eqref{eq:Chern_for_products}.
There is nothing to show for $n=1$.
The short exact sequence
\[ 0\ \longrightarrow\ I^n/I^{n+1}\ \longrightarrow\ \mathbf{MU}_{U(m_1,\dots,m_l)}^*/I^{n+1}\ \longrightarrow\ \mathbf{MU}_{U(m_1,\dots,m_l)}^*/I^n\ \longrightarrow\ 0\]
and the inductive hypothesis reduce the claim to showing that
$I^n/I^{n+1}$ is free as an $\mathbf{MU}^*$-module on the monomials of degree exactly $n$ in the Chern classes
\eqref{eq:Chern_for_products}.
Since the augmentation ideal $I$ is generated by these Chern classes,
the $n$-th power $I^n$ is generated, as a module over $\mathbf{MU}_{U(m_1,\dots,m_l)}^*$, by the monomials
of degree $n$.
So $I^n/I^{n+1}$ is generated by these monomials as a module over $\mathbf{MU}^*$.
The bundling map $\mathbf{MU}_{U(m_1,\dots,m_l)}^*\longrightarrow \mathbf{MU}^*(B U(m_1,\dots,m_l))$
is a homomorphism of augmented $\mathbf{MU}^*$-algebras,
and it takes the Chern class $c_k^{[i]}$ to the inflation of the $k$-th Conner-Floyd-Chern class
along the projection to the $i$-th factor.
By the theory of complex orientations, the collection of these
Conner-Floyd-Chern classes are $\mathbf{MU}^*$-power series generators of $\mathbf{MU}^*(B U(m_1,\dots,m_l))$;
in particular,
the images of the Chern class monomials are $\mathbf{MU}^*$-linearly independent in $\mathbf{MU}^*(B U(m_1,\dots,m_l))$.
Hence these classes are themselves linearly independent in $I^n/I^{n+1}$.
\end{proof}
\begin{rk}\label{rk:degenerate}
Greenlees and May \cite[Corollary 1.6]{greenlees-may:completion} construct
a local homology spectral sequence
\[ E_2^{p,q}\ = \ H^I_{-p,-p}(\mathbf{MU}_G^*)\ \Longrightarrow \ \mathbf{MU}^{p+q}( B G )\ .\]
The regularity results about Chern classes from Theorem \ref{thm:completions} imply that
whenever $G=U(m_1,\dots,m_l)$ is a product of unitary groups, the $E_2^{p,q}$-term vanishes for all $p\ne 0$,
and the spectral sequence degenerates into the isomorphism
\[ E_2^{0,*}\ \cong \ (\mathbf{MU}_{U(m_1,\dots,m_l)}^*)^\wedge_I \ \cong \ \mathbf{MU}^*( B U(m_1,\dots,m_l)) \]
of Theorem \ref{thm:completions} (iii).
\end{rk}
\begin{rk}
The previous regularity theorems are special cases of the following more general results
that hold for every global $\mathbf{MU}$-module $E$:
\begin{itemize}
\item For every compact Lie group $G$, the sequence of Chern classes
$c_m^{(m)}\times 1_G,\dots,c_1^{(m)}\times 1_G$
acts regularly on the graded $\mathbf{MU}^*_{U(m)\times G}$-module $E^*_{U(m)\times G}$.
\item The restriction homomorphism
\[ \res^{ U(m)\times G}_ G\ :\ E_{U(m)\times G}^*\ \longrightarrow \ E_G^*\]
factors through an isomorphism
\[ E_{U(m)\times G}^*/(c_m^{(m)}\times 1_G,\dots, c_1^{(m)}\times 1_G)\ \cong \ E_G^* \ .\]
\item
For all $m_1,\dots,m_l\geq 1$, the sequence of Chern classes \eqref{eq:Chern_for_products}
acts regularly on the graded $\mathbf{MU}^*_{U(m_1,\dots,m_l)}$-module $E^*_{U(m_1,\dots,m_l)}$.
\end{itemize}
As in Remark \ref{rk:degenerate}, the regularity properties also imply the degeneracy
of the Greenlees-May local homology spectral sequence converging to $E^*(B U(m_1,\dots,m_l))$.
\end{rk}
\section{The \texorpdfstring{$\mathbf{MU}$}{MU}-completion theorem via Chern classes}
In this section we use the Chern classes to reformulate the $\mathbf{MU}_G$-completion theorem
of Greenlees-May \cite{greenlees-may:completion} and La Vecchia \cite{lavecchia},
for any compact Lie group $G$, and we give a short and self-contained proof.
We emphasize that the essential arguments of this section are all contained in
\cite{greenlees-may:completion} and \cite{lavecchia};
the Chern classes let us arrange them in a more conceptual and concise way.
The references \cite{greenlees-may:completion, lavecchia}
ask for a finitely generated ideal of $\mathbf{MU}_G^*$ that is `sufficiently large' in the
sense of \cite[Definition 2.4]{greenlees-may:completion};
while we have no need to explicitly mention sufficiently large ideals,
the new insight is that the ideal generated by the Chern classes
of any faithful $G$-representation is `sufficiently large'.
\begin{con}[Chern classes of representations]
We let $V$ be a complex representation of a compact Lie group $G$. We let
$\rho:G\longrightarrow U(m)$ be a continuous homomorphism that classifies $V$, i.e., such that $\rho^*(\nu_m)$
is isomorphic to $V$; here $m=\dim_\mathbb C(V)$.
The {\em $k$-th Chern class} of $V$ is
\[ c_k(V)\ = \ \rho^*(c_k^{(m)})\ \in \ \mathbf{MU}_G^{2 k}\ .\]
In particular, $c_0(V)=1$, $c_m(V)=e(V)$ is the Euler class, and $c_k(V)=0$ for $k>m$.
\end{con}
\begin{eg} As an example, we consider the tautological representation $\nu_2$ of $S U(2)$ on $\mathbb C^2$.
By the general properties of Chern classes we have
$c_0(\nu_2)=1$, $c_2(\nu_2)=e(\nu_2)$ is the Euler class,
and $c_k(\nu_2)=0$ for $k\geq 3$. The first Chern class of $\nu_2$ can be rewritten
by using a double coset formula as follows:
\begin{align*}
c_1(\nu_2)\
&= \ \res^{U(2)}_{S U(2)}(c_1^{(2)}) \ = \ \res^{U(2)}_{S U(2)}(\tr_{U(1,1)}^{U(2)}(e_1\times 1)) \\
&= \ \tr_T^{S U(2)}(\res^{U(1,1)}_T(e_1\times 1)) \ = \ \tr_T^{S U(2)}(e(\chi)) \ .
\end{align*}
Here $T=\{
(\begin{smallmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{smallmatrix}) \ : \ \lambda\in U(1)
\}$
is the diagonal maximal torus of $S U(2)$, $\chi:T\cong U(1)$ is the character that projects
onto the upper left diagonal entry, and $e(\chi)\in\mathbf{MU}^2_T$ is its Euler class.
\end{eg}
\begin{con}
We construct a $G$-equivariant $\mathbf{MU}_G$-module $K(G,V)$ associated
to a complex representation $V$ of a compact Lie group $G$.
The construction is a special case of one used by Greenlees and May
\cite[Section 1]{greenlees-may:completion},
based on the sequence of Chern classes $c_1(V),\dots,c_m(V)$, where $m=\dim_\mathbb C(V)$.
For any equivariant homotopy class $x\in \mathbf{MU}_G^l$,
we write $\mathbf{MU}_G[1/x]$ for the $\mathbf{MU}_G$-module localization of $\mathbf{MU}_G$ with $x$ inverted;
in other words, $\mathbf{MU}_G[1/x]$ is a homotopy colimit (mapping telescope)
in the triangulated category of the sequence
\[ \mathbf{MU}_G\ \xrightarrow{-\cdot x} \ \Sigma^l \mathbf{MU}_G\ \xrightarrow{-\cdot x} \Sigma^{2 l}\mathbf{MU}_G\ \xrightarrow{-\cdot x} \ \Sigma^{3 l}\mathbf{MU}_G\ \xrightarrow{-\cdot x} \ \dots \ \ .\]
We write $K(x)$ for the fiber of the morphism $\mathbf{MU}_G\longrightarrow\mathbf{MU}_G[1/x]$.
Then we define
\[ K(G,V)\ = \ K(c_1(V))\wedge_{\mathbf{MU}_G}\dots \wedge_{\mathbf{MU}_G}K(c_m(V))\ . \]
The smash product of the morphisms $K(c_i(V))\longrightarrow\mathbf{MU}_G$ provides a morphism of $G$-equivariant $\mathbf{MU}_G$-module spectra
\[ \epsilon_V\ : \ K(G,V)\ \longrightarrow \ \mathbf{MU}_G .\]
By general principles, the module $K(G,V)$
only depends on the radical of the ideal generated by the classes $c_1(V),\dots,c_m(V)$.
But more is true: as a consequence of Theorem \ref{thm:completion} below,
$K(G,V)$ is entirely independent, as a $G$-equivariant $\mathbf{MU}_G$-module, of the faithful representation $V$.
\end{con}
\begin{prop}\label{prop:characterize K(G,V)}
Let $V$ be a faithful complex representation of a compact Lie group $G$.
\begin{enumerate}[\em (i)]
\item The morphism
$\epsilon_V:K(G,V)\longrightarrow \mathbf{MU}_G$ is an equivalence of underlying non-equivariant spectra.
\item For every non-trivial closed subgroup $H$ of $G$, the $H$-geometric fixed point spectrum
$\Phi^H(K(G,V))$ is trivial.
\end{enumerate}
\end{prop}
\begin{proof}
(i) We set $m=\dim_\mathbb C(V)$.
The Chern classes $c_1(V),\dots,c_m(V)$ belong to the augmentation ideal
of $\mathbf{MU}_G^*$, so they restrict to 0 in $\mathbf{MU}_{\{1\}}^*$, and
hence the underlying non-equivariant spectrum of $\mathbf{MU}_G[1/c_i(V)]$ is trivial
for each $i=1,\dots,m$.
Hence the morphisms $K(c_i(V))\longrightarrow \mathbf{MU}_G$ are underlying non-equivariant
equivalences for $i=1,\dots,m$.
So also the morphism $\epsilon_V$ is an underlying non-equivariant equivalence.
(ii) We let $H$ be a non-trivial closed subgroup of $G$.
We set $W=V-V^H$, the orthogonal complement of the $H$-fixed points.
This is a complex $H$-representation with $W^H=0$; moreover, $W$ is nonzero because
$H$ acts faithfully on $V$ and $H\ne\{1\}$.
For $k=\dim_\mathbb C(W)$ we then have
\[ e(W) \ = \ c_k(W)\ = \ c_k(W\oplus V^H)\ = \ c_k(\res^G_H(V))\ = \ \res^G_H( c_k(V)) \ ;\]
the second equation uses the fact that adding a trivial representation
leaves Chern classes unchanged, by part (ii) of Theorem \ref{thm:CFC main}.
Since $W^H=0$, the geometric fixed point homomorphism $\Phi^H:\mathbf{MU}_H^*\longrightarrow \Phi_H^*(\mathbf{MU})$
sends the Euler class $e(W) = \res^G_H( c_k(V))$ to an invertible element.
The functor $\Phi^H\circ \res^G_H$ commutes with inverting elements.
Since the class $\Phi^H(\res^G_H(c_k(V)))$ is already invertible,
the localization morphism $\mathbf{MU}_G\longrightarrow \mathbf{MU}_G[1/c_k(V)]$ induces an equivalence on $H$-geometric fixed points.
Since the functor $\Phi^H\circ\res^G_H$ is exact, it annihilates the fiber $K(c_k(V))$
of the localization $\mathbf{MU}_G\longrightarrow \mathbf{MU}_G[1/c_k(V)]$.
The functor $\Phi^H\circ\res^G_H$ is also strong monoidal, in the sense of a natural
equivalence of non-equivariant spectra
\[ \Phi^H(X\wedge_{\mathbf{MU}_G}Y) \ \simeq \ \Phi^H(X)\wedge_{\Phi^H(\mathbf{MU}_G)}\Phi^H(Y) \ , \]
for all $G$-equivariant $\mathbf{MU}_G$-modules $X$ and $Y$.
Since $K(G,V)$ contains $K(c_k(V))$ as a factor (with respect to $\wedge_{\mathbf{MU}_G}$),
we conclude that the spectrum $\Phi^H(K(G,V))$ is trivial.
\end{proof}
The following 'completion theorem' is a reformulation of the combined
work of Greenlees-May \cite[Theorem 1.3]{greenlees-may:completion}
and La Vecchia \cite{lavecchia}. It is somewhat more precise in that an unspecified
`sufficiently large' finitely generated ideal of $\mathbf{MU}_G^*$ is replaced by the
ideal generated by the Chern classes of a faithful $G$-representation.
The proof is immediate from the properties of $K(G,V)$ listed in
Proposition \ref{prop:characterize K(G,V)}.
We emphasize, however, that our proof is just a different way of arranging some arguments from
\cite{greenlees-may:completion} and \cite{lavecchia} while taking advantage of the Chern class formalism.
Since the morphism $\epsilon_V:K(G,V)\longrightarrow \mathbf{MU}_G$ is a non-equivariant equivalence
of underlying spectra, the morphism $E G_+\wedge \mathbf{MU}_G\longrightarrow \mathbf{MU}_G$ that collapses
the universal space $E G$ to a point admits a unique lift to a morphism
of $G$-equivariant $\mathbf{MU}_G$-modules $\psi: E G_+\wedge \mathbf{MU}_G\longrightarrow K(G,V)$ across $\epsilon_V$.
\begin{thm}\label{thm:completion}
Let $V$ be a faithful complex representation of a compact Lie group $G$.
Then the morphism
\[ \psi\ : \ E G_+\wedge \mathbf{MU}_G\ \longrightarrow\ K(G,V) \]
is an equivalence of $G$-equivariant $\mathbf{MU}_G$-module spectra.
\end{thm}
\begin{proof}
Because the underlying space of $E G$ is contractible, the composite
\[ E G_+\wedge \mathbf{MU}_G\ \xrightarrow{\ \psi\ } \ K(G,V)\ \xrightarrow{\ \epsilon_V\ }\ \mathbf{MU}_G \]
is an equivariant equivalence of underlying non-equivariant spectra.
Since $\epsilon_V$ is an equivariant equivalence of underlying non-equivariant spectra
by Proposition \ref{prop:characterize K(G,V)}, so is $\psi$.
For all non-trivial closed subgroups $H$ of $G$, source and target of $\psi$
have trivial $H$-geometric fixed points spectra,
again by Proposition \ref{prop:characterize K(G,V)}. So the morphism $\psi$ induces
an equivalence on geometric fixed point spectra for all closed subgroup of $G$,
and it is thus an equivariant equivalence.
\end{proof}
\end{document} |
\begin{document}
\ifreport
\title{Improved Static Analysis for\\ Parameterised Boolean Equation Systems\\
using Control Flow Reconstruction}
\else
\title{Liveness Analysis for\\ Parameterised Boolean Equation Systems}
\fi
\author{Jeroen J. A. Keiren\inst{1} \and Wieger Wesselink\inst{2} \and Tim A.
C. Willemse\inst{2}}
\institute{VU University Amsterdam, The Netherlands\\
\email{j.j.a.keiren@vu.nl}
\and
Eindhoven University of Technology, The Netherlands\\
\email{\{j.w.wesselink, t.a.c.willemse\}@tue.nl}
}
\maketitle
\begin{abstract}
We present a sound static analysis technique for fighting the
combinatorial explosion
of parameterised Boolean equation systems (PBESs). These essentially
are systems of mutually recursive fixed point equations ranging over
first-order logic formulae.
Our method detects parameters that are not live by analysing
a control
flow graph of a PBES, and it subsequently eliminates such parameters. We show
that a naive approach to constructing a control flow graph, needed for the analysis,
may suffer from an
exponential blow-up,
and we define
an approximate analysis that avoids this problem.
The effectiveness of our techniques is
evaluated using a number of case studies.
\end{abstract}
\section{Introduction}
\emph{Parameterised Boolean equation systems (PBESs)}~\cite{GW:05tcs} are
systems of fixpoint equations that range over first-order formulae;
they are essentially an equational variation of \emph{Least Fixpoint
Logic (LFP)}. Fixpoint logics such as PBESs have applications in
database theory and computer aided verification. For instance, the
CADP~\cite{Gar+:11} and mCRL2~\cite{Cra+:13} toolsets use PBESs
for model checking and equivalence checking and in \cite{AFJV:11} PBESs are
used to solve Datalog queries.
In practice, the predominant problem for PBESs is evaluating (henceforth
referred to as \emph{solving}) them so as to
answer the decision problem encoded
in them. There are a variety of techniques for solving
PBESs, see~\cite{GW:05tcs}, but the most straightforward method is
by instantiation to a \emph{Boolean equation system (BES)}
\cite{Mad:97}, and then solving this BES.
This process is similar to the explicit generation
of a behavioural state space from its symbolic description, and it
suffers from a combinatorial explosion that is akin to the state
space explosion problem. Combatting this combinatorial
explosion is therefore instrumental in speeding up the process of
solving the problems encoded by PBESs.
While several static analysis techniques have been described using fixpoint
logics, see \emph{e.g.}\xspace \cite{CC:77}, with the exception of the static analysis
techniques for PBESs, described in~\cite{OWW:09}, no such techniques seem to
have been applied to fixpoint
logics themselves.
Our main contribution in this paper is a static analysis method for PBESs that
significantly improves over the aforementioned
techniques for simplifying
PBESs.
In our method, we construct a \emph{control flow graph}
(CFG) for
a given PBES and subsequently apply state space reduction
techniques~\cite{FBG:03,YG:04}, combined with liveness analysis
techniques
from compiler technology~\cite{ASU:86}.
These typically scrutinise syntactic
descriptions of behaviour to detect and eliminate
variables that at some point become irrelevant (dead, not live) to the
behaviour,
thereby decreasing the complexity.
The notion of control flow of a PBES is not self-evident: formulae
in fixpoint logics (such as PBESs) do not have a notion of a program
counter. Our notion of control flow is based on the concept of
\emph{control flow parameters} (CFPs), which induce a CFG.
Similar notions exist in the context of state space
exploration, see~\emph{e.g.}\xspace~\cite{PT:09atva}, but so far, no such concept exists
for fixpoint logics.
The size of the CFGs is potentially exponential
in the number of CFPs. We therefore also describe a modification of
our analysis---in which reductive power is traded against a lower
complexity---that does not suffer from this problem.
Our static analysis technique allows for solving PBESs using
instantiation that hitherto could not be solved this way, either
because the underlying BESs would be infinite or they would be extremely large.
We show that our methods are
sound; \emph{i.e.}\xspace, simplifying PBESs using our analyses lead to PBESs with the
same solution.
Our static analysis techniques have been implemented in the
mCRL2 toolset~\cite{Cra+:13} and applied to a set of model
checking and equivalence checking problems. Our experiments show that the
implementations
outperform existing static analysis techniques for PBESs~\cite{OWW:09} in
terms of reductive power, and that reductions of almost 100\% of the size of the
underlying BESs can be achieved. Our experiments confirm that the optimised
version sometimes achieves slightly less reduction
than our
non-optimised version, but is faster.
Furthermore, in cases where no additional reduction is achieved compared
to existing techniques, the overhead is mostly
neglible.
\paragraph{Structure of the paper.}
In Section~\ref{sec:preliminaries} we give a cursory overview of basic
PBES theory and in Section~\ref{sec:example}, we present an example to
illustrate the difficulty of using instantiation to
solve a PBES and to sketch our solution. In
Section~\ref{sec:CFP_CFG} we describe our construction of control flow graphs
for PBESs and in Section~\ref{sec:dataflow} we describe our live parameter
analysis. We present an optimisation of
the analysis in Section~\ref{sec:local}. The approach is evaluated
in Section~\ref{sec:experiments}, and
Section~\ref{sec:conclusions} concludes.
\paperonly{\textbf{We refer to~\cite{KWW:13report} for
proofs and additional results.}}
\section{Preliminaries}\label{sec:preliminaries}
Throughout this paper, we work in a setting of \emph{abstract data
types} with non-empty data sorts $\sort{D_1}, \sort{D_2}, \ldots$,
and operations on these sorts, and a set $\varset{D}$ of sorted
data variables. We write vectors in boldface, \emph{e.g.}\xspace $\var{d}$ is
used to denote a vector of data variables. We write $\var[i]{d}$
to denote the $i$-th element of a vector $\var{d}$.
A semantic set $\semset{D}$ is associated to every sort $\sort{D}$,
such that each term of sort $\sort{D}$, and all operations on
$\sort{D}$ are mapped to the elements and operations of $\semset{D}$
they represent. \emph{Ground terms} are terms that do not contain
data variables. For terms that contain data variables, we use an
environment $\ensuremath{\delta}$ that maps each variable from $\varset{D}$
to a value of the associated type. We assume an interpretation
function $\sem{\_}{}{}$ that maps every term $t$ of sort $\sort{D}$
to the data element $\sem{t}{}{\ensuremath{\delta}}$ it represents, where the
extensions of $\ensuremath{\delta}$ to open terms and vectors are standard. Environment
updates are denoted $\ensuremath{\delta}[\subst{d}{v}]$, where
$\ensuremath{\delta}[\subst{d}{v}](d') = v$ if $d' = d$, and $\ensuremath{\delta}(d')$
otherwise.
We specifically assume the existence of a sort $\sort{B}$ with
elements $\ensuremath{\mathit{true}}$ and $\ensuremath{\mathit{false}}$ representing the Booleans $\semset{B}$
and a sort $\sort{N} = \{0, 1, 2, \ldots \}$ representing the natural
numbers $\semset{N}$. For these sorts, we assume that the usual operators
are available and, for readability, these are written the same
as their semantic counterparts.
\emph{Parameterised Boolean equation systems}~\cite{Mat:98} are
sequences of
fixed-point equations ranging over \emph{predicate formulae}. The
latter are first-order formulae extended with predicate
variables, in which the non-logical symbols are taken from the data
language.
\begin{definition}
\label{def:formula}
\label{def:semFormula}
\emph{Predicate formulae} are defined through the following grammar:
$$
\varphi, \psi ::= b \mid X(\val{e}) \mid \varphi \land \psi \mid \varphi \lor
\psi \mid \forall d \colon D. \varphi \mid \exists d \colon D. \varphi$$
in which $b$ is a data term of sort $\sort{B}$, $X(\val{e})$ is a \emph{predicate
variable instance} (PVI) in which $X$ is a predicate variable of
sort $\vec{\sort{D}} \to \sort{B}$, taken from some sufficiently large set
$\varset{P}$ of predicate variables, and $\val{e}$ is a vector of data terms of
sort $\vec{\sort{D}}$.
The interpretation of a predicate formula $\varphi$ in the
context of a predicate
environment $\ensuremath{\eta} \colon \varset{P} \to \semset{D} \to \semset{B}$
and data environment $\ensuremath{\delta}$ is denoted as
$\sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}}$, where:
\[
\begin{array}{ll}
\sem{b}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=}
\left \{
\begin{array}{ll} \text{true} & \text{if $\ensuremath{\delta}(b)$ holds} \\
\text{false} & \text{otherwise}
\end{array}
\right . \\[5pt]
\sem{X(\val{e})}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=}
\left \{
\begin{array}{ll} \text{true}& \text{if $\ensuremath{\eta}(X)(\ensuremath{\delta}(\val{e}))$
holds} \\
\text{false} & \text{otherwise}
\end{array}
\right . \\[5pt]
\sem{\phi \land \psi}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=} \sem{\phi}{\ensuremath{\eta}\ensuremath{\delta}} \text{ and } \sem{\psi}{\ensuremath{\eta}\ensuremath{\delta}} \text{ hold} \\[5pt]
\sem{\phi \lor \psi}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=} \sem{\phi}{\ensuremath{\eta}\ensuremath{\delta}} \text{ or } \sem{\psi}{\ensuremath{\eta}\ensuremath{\delta}} \text{ hold} \\[5pt]
\sem{\forall{d \colon \sort{D}}.~ \phi}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=} \text{for all ${v \in \semset{D}}$, }~\sem{\phi}{\ensuremath{\eta}\ensuremath{\delta}[v/d]} \text{ holds} \\[5pt]
\sem{\exists{d \colon \sort{D}}.~ \phi}{\ensuremath{\eta}\ensuremath{\delta}}
&\ensuremath{=} \text{for some ${v \in \semset{D}}$, }~\sem{\phi}{\ensuremath{\eta}\ensuremath{\delta}[v/d]} \text{ holds}
\end{array}
\]
\end{definition}
We assume the usual precedence rules for the logical operators.
\emph{Logical equivalence} between two predicate formulae $\varphi,
\psi$, denoted $\varphi \equiv \psi$, is defined as
$\sem{\varphi}{\ensuremath{\eta}\ensuremath{\delta}}
= \sem{\psi}{\ensuremath{\eta}\ensuremath{\delta}}$ for all $\ensuremath{\eta}, \ensuremath{\delta}$.
Freely
occurring data variables
in $\varphi$ are denoted by $\free{\varphi}$. We refer to $X(\val{e})$ occuring
in a predicate formula as a \emph{predicate variable instance} (PVI).
For simplicity, we assume that if a data variable is bound by a quantifier
in a formula $\varphi$, it does not also occur free within $\varphi$.
\begin{definition}
\label{def:PBES}
PBESs are defined by the following grammar:
$$
\ensuremath{\mathcal{E}} ::= \ensuremath{\emptyset} \mid (\nu X(\var{d} \colon \vec{D}) = \varphi) \ensuremath{\mathcal{E}}
\mid (\mu X(\var{d} \colon \vec{D}) = \varphi) \ensuremath{\mathcal{E}}
$$
in which $\ensuremath{\emptyset}$ denotes the empty equation system; $\mu$ and $\nu$ are the
least and greatest fixed point signs, respectively; $X$ is a sorted predicate
variable of sort $\vec{\sort{D}} \to \sort{B}$, $\var{d}$ is a vector of formal
parameters,
and $\varphi$ is a predicate formula. We henceforth omit a trailing $\ensuremath{\emptyset}$.
\end{definition}
By convention $\rhs{X}$ denotes the right-hand side of the
defining equation for $X$ in a PBES $\ensuremath{\mathcal{E}}$;
$\param{X}$ denotes the set of \emph{formal parameters} of $X$ and
we assume that $\free{\rhs{X}} \subseteq
\param{X}$. By superscripting a formal parameter with the predicate
variable to which it belongs, we distinguish between formal parameters for
different predicate variables, \emph{i.e.}\xspace, we
write $d^X$ when $d \in \param{X}$. We write $\sigma$ to stand for
either $\mu$ or $\nu$.
The set of \emph{bound predicate variables} of some PBES $\ensuremath{\mathcal{E}}$, denoted
$\bnd{\ensuremath{\mathcal{E}}}$, is the set of predicate variables occurring
at the left-hand sides of the equations in $\ensuremath{\mathcal{E}}$. Throughout this
paper, we deal with PBESs that are both \emph{well-formed}, \emph{i.e.}\xspace for every
$X \in \bnd{\ensuremath{\mathcal{E}}}$ there is exactly one equation in $\ensuremath{\mathcal{E}}$, and
\emph{closed}, \emph{i.e.}\xspace for every $X \in \bnd{\ensuremath{\mathcal{E}}}$, only predicate variables taken
from $\bnd{\ensuremath{\mathcal{E}}}$ occur in $\rhs{X}$.
To each PBES $\ensuremath{\mathcal{E}}$ we associate a \emph{top assertion}, denoted
$\mcrlKw{init}~X(\val{v})$, where we require $X \in \bnd{\ensuremath{\mathcal{E}}}$. For
a parameter $\var[m]{d} \in \param{X}$ for the top assertion
$\mcrlKw{init}~X(\val{v})$ we define the value $\init{\var[m]{d}}$
as $\val[m]{v}$.\\
We next define a PBES's semantics. Let $\semset{B}^{\vec{\semset{D}}}$
denote the set of functions $f \colon \vec{\semset{D}} \to \semset{B}$,
and define the ordering $\sqsubseteq$ as $f \sqsubseteq g$ iff for
all $\vec{v} \in \vec{\semset{D}}$, $f(\vec{v})$ implies $g(\vec{v})$.
For a given pair of environments $\ensuremath{\delta}, \ensuremath{\eta}$, a predicate
formula $\varphi$ gives rise to a predicate transformer $T$
on the complete lattice
$(\semset{B}^{\vec{\semset{D}}}, \sqsubseteq)$ as follows:
$
T(f) = \lambda \vec{v} \in \vec{\semset{D}}.
\sem{\varphi}{\ensuremath{\eta}[\subst{X}{f}]}{\ensuremath{\delta}[\subst{\vec{d}}{\vec{v}}]}
$.
Since the predicate transformers defined this way are monotone,
their extremal fixed points exist. We denote the least fixed point of
a given predicate transformer $T$ by $\mu T$, and the greatest fixed point
of $T$ is denoted $\nu T$.
\begin{definition}
The \emph{solution} of an equation system in the context of a predicate
environment $\ensuremath{\eta}$ and data environment $\ensuremath{\delta}$ is defined inductively
as follows:
\begin{align*}
\sem{\ensuremath{\emptyset}}{\ensuremath{\eta}}{\ensuremath{\delta}} & \ensuremath{=} \ensuremath{\eta} \\
\sem{(\mu X(\var{d} \colon \vec{D}) = \rhs{X}) \ensuremath{\mathcal{E}}}{\ensuremath{\eta}}{\ensuremath{\delta}}
& \ensuremath{=} \sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}[\subst{X}{\mu T}]}{\ensuremath{\delta}}\\
\sem{(\nu X(\var{d} \colon \vec{D}) = \rhs{X}) \ensuremath{\mathcal{E}}}{\ensuremath{\eta}}{\ensuremath{\delta}}
& \ensuremath{=} \sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}[\subst{X}{\nu T}]}{\ensuremath{\delta}}
\end{align*}
with $T(f) = \lambda \val{v} \in \val{\semset{D}}.
\sem{\varphi}{(\sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}[\subst{X}{f}]}{\ensuremath{\delta}})}
{\ensuremath{\delta}[\subst{\var{d}}{\val{v}}]}$
\end{definition}
The solution prioritises the fixed point signs of left-most equations
over the fixed point signs of equations that follow, while respecting
the equations. Bound predicate variables of closed PBESs have a
solution that is independent of the predicate and data environments
in which it is evaluated. We therefore omit these environments and
write $\sem{\ensuremath{\mathcal{E}}}(X)$ instead of $\sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}}{\ensuremath{\delta}}(X)$.
\reportonly{
\newcommand{\sigma}{\sigma}
\newcommand{\rankE}[2]{\ensuremath{\mathsf{rank}_{#1}(#2)}}
\newcommand{\rank}[1]{\ensuremath{\mathsf{rank}(#1)}}
The \emph{signature} \cite{Wil:10} of a predicate variable $X$ of sort
$\vec{\sort{D}} \to \sort{B}$, $\signature{X}$, is the product $\{X\} \times
\semset{D}$.
The notion of signature is lifted to sets of predicate variables $P \subseteq
\varset{P}$ in the natural way, \emph{i.e.}\xspace
$\signature{P} = \bigcup_{X \in P} \signature{X}$.\footnote{Note that in
\cite{Wil:10} the notation $\mathsf{sig}$ is used to denote the signature. Here
we deviate from this notation due to the naming conflict with the
\emph{significant parameters} of a formula, which also is standard notation
introduced in \cite{OWW:09}, and which we introduce in
Section~\ref{sec:dataflow}.}
\begin{definition}[{\cite[Definition~6]{Wil:10}}]\label{def:r-correlation}
Let ${\rel{R}} \subseteq \signature{\varset{P}} \times \signature{\varset{P}}$
be
an arbitrary
relation. A predicate environment $\ensuremath{\eta}$ is an $\rel{R}$-correlation iff
$(X, \val{v}) {\rel{R}} (X', \val{v'})$ implies $\ensuremath{\eta}(X)(\val{v}) =
\ensuremath{\eta}(X')(\val{v'})$.
\end{definition}
A \emph{block} is a non-empty equation system of like-signed fixed point
equations. Given an equation system $\ensuremath{\mathcal{E}}$, a block $\mathcal{B}$ is maximal
if its neighbouring equations in $\ensuremath{\mathcal{E}}$ are of a different sign than the
equations in $\mathcal{B}$. The $i^\mathit{th}$ maximal block in $\ensuremath{\mathcal{E}}$ is
denoted by
$\block{i}{\ensuremath{\mathcal{E}}}$.
For relations $\rel{R}$ we write $\correnv{\rel{R}}$ for the set of
$\rel{R}$-correlations.
\begin{definition}[{\cite[Definition~7]{Wil:10}}]
\label{def:consistent-correlation} Let $\ensuremath{\mathcal{E}}$ be an equation system.
Relation ${\rel{R}} \subseteq \signature{\varset{P}} \times
\signature{\varset{P}}$ is
a \emph{consistent correlation} on $\ensuremath{\mathcal{E}}$, if for $X, X' \in \bnd{\ensuremath{\mathcal{E}}}$,
$(X, \val{v}) \rel{R} (X', \val{v'})$ implies:
\begin{compactenum}
\item for all $i$, $X \in \bnd{\block{i}{\ensuremath{\mathcal{E}}}}$ iff $X' \in
\bnd{\block{i}{\ensuremath{\mathcal{E}}}}$
\item for all $\ensuremath{\eta} \in \correnv{\rel{R}}$, $\ensuremath{\delta}$, we have
$\sem{\rhs{X}}{\ensuremath{\eta} \ensuremath{\delta}[\subst{\var{d}}{\val{v}}]} =
\sem{\rhs{X'}}{\ensuremath{\eta} \ensuremath{\delta}[\subst{\var{d'}}{\val{v'}}]}$
\end{compactenum}
For $X, X' \in \bnd{\ensuremath{\mathcal{E}}}$, we say $(X, \val{v})$ and $(X', \val{v'})$
consistently
correlate, denoted as $(X, \val{v}) \ensuremath{\doteqdot} (X', \val{v'})$ iff there
exists a correlation $\rel{R} \subseteq
\signature{\bnd{\ensuremath{\mathcal{E}}}} \times \signature{\bnd{\ensuremath{\mathcal{E}}}}$ such that $(X, \val{v})
\rel{R} (X', \val{v'})$ .
\end{definition}
Consistent
correlations can be lifted to variables in different equation systems in $\ensuremath{\mathcal{E}}$
and $\ensuremath{\mathcal{E}}'$, assuming that the variables in the equation systems do not
overlap.
We call such equation systems \emph{compatible}.
Lifting consistent correlations to different equation systems can, \emph{e.g.}\xspace, be
achieved by merging the equation systems to an
equation system $\mathcal{F}$, in which, if $X \in \bnd{\ensuremath{\mathcal{E}}}$, then
$X \in \bnd{\block{i}{\ensuremath{\mathcal{E}}}}$ iff $X \in \bnd{\block{i}{\mathcal{F}}}$,
and likewise for $\ensuremath{\mathcal{E}}'$.
The consistent correlation can then be defined on $\mathcal{F}$.
The following theorem \cite{Wil:10} shows the relation between consistent
correlations and the solution of a PBES.
\begin{theorem}[{\cite[Theorem~2]{Wil:10}}]\label{thm:willemse}\label{thm:cc}
Let $\ensuremath{\mathcal{E}}$, $\ensuremath{\mathcal{E}}'$ be compatible equation systems, and $\ensuremath{\doteqdot}$ a
consistent correlation. Then for all $X \in \bnd{\ensuremath{\mathcal{E}}}$,
$X' \in \bnd{\ensuremath{\mathcal{E}}'}$ and all $\ensuremath{\eta} \in \correnv{\ensuremath{\doteqdot}}$, we have
$(X, \val{v}) \ensuremath{\doteqdot} (X', \val{v'}) \implies \sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}
\ensuremath{\delta}}(X)(\val{v}) =
\sem{\ensuremath{\mathcal{E}}'}{\ensuremath{\eta} \ensuremath{\delta}}(X')(\val{v'})$
\end{theorem}
We use this theorem in proving the correctness of our static analysis
technique.
}
\section{A Motivating Example}\label{sec:example}
In practice, solving PBESs proceeds via \emph{instantiating}~\cite{PWW:11}
into \emph{Boolean equation systems (BESs)}, for which solving is
decidable. The latter is the fragment of PBESs with equations
that range over propositions only, \emph{i.e.}\xspace, formulae without data and
quantification. Instantiating a PBES to a BES is akin to state space
exploration and suffers from a similar combinatorial
explosion. Reducing the time spent on it is thus instrumental in speeding
up, or even enabling the solving process.
We illustrate this using the following (academic) example, which we also
use as our running example:
\[
\begin{array}{lll}
\nu X(i,j,k,l\colon\sort{N}) & = &
( i \not= 1 \vee j \not= 1 \vee X(2,j,k,l+1)) \wedge \forall m\colon\sort{N}. Z(i,2,m+k,k) \\
\mu Y(i,j,k,l\colon\sort{N}) & = &
k = 1 \vee (i = 2 \wedge X(1,j,k,l) ) \\
\nu Z(i,j,k,l\colon\sort{N}) & = &
(k < 10 \vee j = 2) \wedge (j \not= 2 \vee Y(1,1,l,1) ) \wedge
Y(2,2,1,l)
\end{array}
\]
The presence of PVIs $X(2,j,k,l+1)$ and $Z(i,2,m+k,k)$ in $X$'s
equation means the solution to $X(1,1,1,1)$ depends on the
solutions to $X(2,1,1,2)$ and $Z(1,2,v+1,1)$,
for all values $v$, see Fig.~\ref{fig:instantiate}. Instantiation
finds these dependencies by simplifying the right-hand
side of $X$ when its parameters have been assigned value $1$:
\[
( 1 \not= 1 \vee 1 \not= 1 \vee X(2,1,1,1+1))
\wedge \forall m\colon\sort{N}. Z(1,2,m+1,1)
\]
Since for an infinite number of different arguments the solution
to $Z$ must be computed, instantiation does not terminate. The problem is with
the third parameter ($k$) of $Z$. We cannot simply assume
that values assigned to the third parameter of $Z$ do not matter;
in fact, only when $j =2$, $Z$'s right-hand side predicate formula
does not depend on $k$'s value. This is where our developed method will
come into play: it automatically
determines that it is sound to replace PVI
$Z(i,2,m+k,k)$ by, \emph{e.g.}\xspace, $Z(i,2,1,k)$ and to remove the universal
quantifier, enabling us to solve $X(1,1,1,1)$ using
instantiation.
Our technique uses a \emph{Control Flow Graph} (CFG) underlying the
PBES for analysing which parameters of a PBES are \emph{live}.
The CFG is a finite abstraction of the dependency graph
that would result from instantiating a PBES. For instance, when
ignoring the third and fourth parameters in our example PBES,
we find that the solution to $X(1,1,*,*)$ depends
on the first PVI, leading to $X(2,1,*,*)$ and the second PVI in $X$'s
equation, leading to $Z(1,2,*,*)$. In the same way we can determine
the dependencies for $Z(1,2,*,*)$, resulting in the finite structure
depicted in Fig.~\ref{fig:CFG}. The subsequent liveness analysis annotates
each vertex with a label indicating which parameters
cannot (cheaply) be excluded from
having an impact on the solution to the equation system; these are
assumed to be live. Using these labels, we modify the PBES automatically.
\begin{figure}
\caption{Dependency graph}
\caption{Control flow graph for the running example}
\label{fig:instantiate}
\label{fig:CFG}
\end{figure}
Constructing a good CFG is a major difficulty, which we address in
Section~\ref{sec:CFP_CFG}. The liveness analysis and the subsequent
modification of the analysed PBES is described in
Section~\ref{sec:dataflow}. Since the CFG constructed in
Section~\ref{sec:CFP_CFG} can still suffer from a combinatorial
explosion, we present an optimisation of our analysis in
Section~\ref{sec:local}.
\section{Constructing Control Flow Graphs for PBESs}
\label{sec:CFP_CFG}
The vertices in the control flow graph we constructed in the previous section
represent the values assigned to a subset of the equations' formal
parameters whereas an edge between two vertices captures the
dependencies among (partially instantiated) equations. The better
the control flow graph approximates the dependency graph resulting
from an instantiation, the more precise the resulting liveness
analysis.
Since computing a precise control flow graph is expensive,
the problem is to
compute the graph effectively and balance
precision and cost.
To this end, we first identify a set of
\emph{control flow parameters}; the values to
these parameters will make up the vertices in the control flow
graph. While there is some choice for control flow parameters,
we require that these are parameters for which we can
\emph{statically} determine:
\begin{compactenum}
\item the (finite set of) values these parameters can assume,
\item the set of PVIs on which the truth of a right-hand
side predicate formula may depend, given a concrete value for each control flow
parameter, and
\item the values assigned to the control flow parameters by all
PVIs on which the truth of a right-hand side predicate formula
may depend.
\end{compactenum}
In addition to these requirements, we impose one other restriction:
control flow parameters of one equation must be \emph{mutually
independent}; \emph{i.e.}\xspace, we have to be able to determine their values
independently of each other. Apart from being a natural requirement
for a control flow parameter, it enables us to devise optimisations of
our liveness analysis.
We now formalise these ideas. First, we characterise three partial
functions that together allow to relate values of formal parameters
to the dependency of a formula on a given PVI. Our formalisation
of these partial functions is based on the following observation:
if in a formula $\varphi$, we can replace a particular PVI $X(\val{e})$
with the subformula $\psi \wedge X(\val{e})$ without this affecting
the truth value of $\varphi$, we know that $\varphi$'s truth value
only depends on $X(\val{e})$'s whenever $\psi$ holds. We will
choose $\psi$ such that it allows us to pinpoint exactly what value
a formal parameter of an equation has (or will be assigned through
a PVI). Using these functions, we then identify our
control flow parameters by eliminating variables that do
not meet all of the aforementioned requirements.
In order to reason about individual PVIs occurring in predicate
formulae we introduce the notation necessary to do so. Let
$\npred{\varphi}$ denote the number of PVIs occurring in a predicate
formula $\varphi$. The function $\predinstphi{\varphi}{i}$ is the
formula representing the $i^\text{th}$ PVI in $\varphi$, of which
$\predphi{\varphi}{i}$ is the name and $\dataphi{\varphi}{i}$
represents the term that appears as the argument of the instance. In general
$\dataphi{\varphi}{i}$ is a vector, of which we denote the $j^{\text{th}}$
argument by $\dataphi[j]{\varphi}{i}$.
Given predicate formula $\psi$ we write $\varphi[i \mapsto \psi]$
to indicate that the PVI at position $i$ is replaced syntactically
by $\psi$ in $\varphi$.
\reportonly{
Formally we define $\varphi[i \mapsto
\psi]$,
as follows.
\begin{definition}
Let $\psi$ be a predicate formula, and let $i \leq \npred{\varphi}$,
$\varphi[i \mapsto \psi]$ is defined inductively as follows.
\begin{align*}
b[i \mapsto \psi] & \ensuremath{=} b \\
Y(e)[i \mapsto \psi] & \ensuremath{=} \begin{cases} \psi & \text{if $i = 1$}\\ Y(e) &
\text{otherwise} \end{cases} \\
(\forall d \colon D . \varphi)[i \mapsto \psi] & \ensuremath{=} \forall d \colon D .
\varphi[i \mapsto \psi]\\
(\exists d \colon D . \varphi)[i \mapsto \psi] & \ensuremath{=} \exists d \colon D .
\varphi[i \mapsto \psi]\\
(\varphi_1 \land \varphi_2)[i \mapsto \psi] & \ensuremath{=} \begin{cases}
\varphi_1 \land \varphi_2[(i - \npred{\varphi_1}) \mapsto \psi] & \text{if } i
> \npred{\varphi_1} \\
\varphi_1[i \mapsto \psi] \land \varphi_2 & \text{if } i \leq \npred{\varphi_1}
\end{cases}\\
(\varphi_1 \lor \varphi_2)[i \mapsto \psi] & \ensuremath{=} \begin{cases}
\varphi_1 \lor \varphi_2[(i - \npred{\varphi_1}) \mapsto \psi] & \text{if } i >
\npred{\varphi_1} \\
\varphi_1[i \mapsto \psi] \lor \varphi_2 & \text{if } i \leq \npred{\varphi_1}
\end{cases}
\end{align*}
\end{definition}
}
\begin{definition} Let $s \colon \varset{P} \times \mathbb{N} \times \mathbb{N}
\to D$, $t \colon \varset{P} \times \mathbb{N} \times \mathbb{N} \to D$, and
$c \colon \varset{P} \times \mathbb{N} \times \mathbb{N} \to \mathbb{N}$
be partial
functions, where $D$ is the union of all ground terms.
The triple $(s,t,c)$ is a \emph{unicity constraint} for PBES $\ensuremath{\mathcal{E}}$ if for all
$X \in \bnd{\ensuremath{\mathcal{E}}}$, $i,j,k \in \mathbb{N}$ and
ground terms $e$:
\begin{compactitem}
\item (source) if $s(X,i,j) {=} e$ then
$\rhs{X} \equiv
\rhs{X}[i \mapsto (\var[j]{d} = e \wedge
\predinstphi{\rhs{X}}{i})]$,
\item (target) if $t(X,i,j) {=} e$ then
$\rhs{X}
\equiv \rhs{X}[i \mapsto (\dataphi[j]{\rhs{X}}{i} = e \wedge
\predinstphi{\rhs{X}}{i})]$,
\item (copy) if $c(X,i,j) {=} k$ then $\rhs{X} \equiv \rhs{X}[i \mapsto
(\dataphi[k]{\rhs{X}}{i} = \var[j]{d} \wedge
\predinstphi{\rhs{X}}{i} )]$.
\end{compactitem}
\end{definition}
Observe that indeed, function $s$ states that, when defined, formal
parameter $\var[j]{d}$ must have value $s(X,i,j)$ for $\rhs{X}$'s
truth value to depend on that of $\predinstphi{\rhs{X}}{i}$. In
the same vein $t(X,i,j)$, if defined, gives the fixed value of the
$j^\text{th}$ formal parameter of $\predphi{\rhs{X}}{i}$.
Whenever $c(X,i,j) = k$ the value of variable $\var[j]{d}$ is
transparently copied to position $k$ in the $i^\text{th}$ predicate
variable instance of $\rhs{X}$. Since $s,t$ and $c$ are partial
functions, we do not require them to be defined; we use $\bot$ to
indicate this.
\begin{example}\label{exa:unicity_constraint}
A unicity constraint $(s,t,c)$ for our running example
could be one that assigns $s(X,1,2) = 1$, since parameter $j^X$
must be $1$ to make $X$'s right-hand side formula depend on PVI
$X(2,j,k,l+1)$. We can set $t(X,1,2) = 1$, as one can deduce that
parameter $j^X$ is set to $1$ by the PVI $X(2,j,k,l+1)$;
furthermore, we can set $c(Z,1,4) = 3$, as parameter $k^Y$ is
set to $l^Z$'s value by PVI $Y(1,1,l,1)$.
\end{example}
\reportonly{
The requirements allow unicity constraints to be underspecified. In practice,
it is desirable to choose the constraints as complete as possible. If, in a
unicity constraint $(s,t,c)$, $s$ and $c$ are defined for a predicate variable
instance, it can immediately be established that we can define $t$ as well.
This is formalised by the following property.
\begin{property}\label{prop:sourceCopyDest}
Let $X$ be a predicate variable, $i \leq \npred{\rhs{X}}$, let $(s,t,c)$ be
a unicity constraint, and let $e$ be a value, then
$$
(s(X, i, n) = e \land c(X, i, n) = m) \implies t(X, i, m) = e.
$$
\end{property}
Henceforth we assume that all unicity constraints satisfy this property.
The overlap between $t$ and $c$ is now straightforwardly formalised in the
following lemma.
\begin{lemma}\label{lem:copyDest}
Let $X$ be a predicate variable, $i \leq \npred{\rhs{X}}$, and let
$(s,t,c)$ be a unicity constraint, then if
$(s(X, i, n)$ and $t(X, i, m)$ are both defined,
$$
c(X, i, n) = m \implies s(X, i, n) = t(X, i, m).
$$
\end{lemma}
\begin{proof}
Immediately from the definitions and Property~\ref{prop:sourceCopyDest}.
\end{proof}
}
From hereon, we assume that $\ensuremath{\mathcal{E}}$ is an arbitrary PBES
with $(\ensuremath{\mathsf{source}}\xspace,\ensuremath{\mathsf{target}}\xspace,\ensuremath{\mathsf{copy}}\xspace)$
a unicity constraint we can deduce for it.
Notice that for each formal parameter for which either \ensuremath{\mathsf{source}}\xspace
or \ensuremath{\mathsf{target}}\xspace is defined for some PVI, we have a finite set of values
that this parameter can assume. However, at this point we do not
yet know whether this set of values is exhaustive: it may be that
some PVIs may cause the parameter to take on arbitrary values.
Below, we will narrow down for which parameters we \emph{can}
ensure that the set of values is exhaustive. First, we eliminate
formal parameters that do not meet conditions 1--3 for PVIs that
induce self-dependencies for an equation.
\begin{definition}\label{def:LCFP}
A parameter $\var[n]{d} \in \param{X}$ is a \emph{local
control flow parameter} (LCFP) if for all $i$ such that $\predphi{\rhs{X}}{i}
= X$, either $\source{X}{i}{n}$ and $\dest{X}{i}{n}$ are defined, or
$\copied{X}{i}{n} = n$.
\end{definition}
\begin{example} Formal parameter $l^X$ in our running example
does not meet the conditions of Def.~\ref{def:LCFP} and is therefore not
an LCFP. All other parameters in all other equations are still LCFPs since
$X$ is the only equation with a self-dependency.
\end{example}
From the formal parameters that are LCFPs, we
next eliminate those parameters that do not meet conditions 1--3
for PVIs that induce dependencies among \emph{different} equations.
\begin{definition}\label{def:GCFP}
A parameter $\var[n]{d} \in \param{X}$
is a \emph{global control flow parameter} (GCFP)
if it is an LCFP, and for all $Y \in \bnd{\mathcal{E}}\setminus \{X\}$ and
all $i$ such that $\predphi{\rhs{Y}}{i} = X$, either
$\dest{Y}{i}{n}$ is defined, or $\copied{Y}{i}{m} = n$
for some GCFP $\var[m]{d} \in \param{Y}$.
\end{definition}
The above definition is recursive in nature: if a parameter does
not meet the GCFP conditions then this may result in another parameter
also not meeting the GCFP conditions. Any set of parameters that
meets the GCFP conditions is a good set, but larger sets possibly lead to
better information about the control flow in a PBES.
\begin{example}
Formal parameter $k^Z$ in our running example
is not a GCFP since in PVI $Z(i,2,m+k,1)$ from $X$'s equation,
the value assigned to $k^Z$ cannot be determined.
\end{example}
The parameters that meet the GCFP conditions satisfy the conditions
1--3 that we imposed on control flow parameters: they assume a
finite set of values, we can deduce which PVIs may affect the
truth of a right-hand side predicate formula, and we can deduce how
these parameters evolve as a result of all PVIs in a PBES. However,
we may still have parameters of a given equation that are mutually
dependent. Note that this dependency can only arise as a result of
copying parameters: in all other cases, the functions \ensuremath{\mathsf{source}}\xspace
and \ensuremath{\mathsf{target}}\xspace provide the information to deduce concrete values.
\begin{example}
GCFP $k^Y$ affects GCFP $k^X$'s value through PVI $X(1,j,k,l)$; likewise,
$k^X$ affects
$l^Z$'s value through PVI $Z(i,2,m+k,k)$.
Through the PVI $Y(2,2,1,l)$ in $Z$'s equation,
GCFP $l^Z$ affects GCFPs $l^Y$ value. Thus, $k^Y$ affects $l^Y$'s value
transitively.
\end{example}
We identify parameters that, through copying, may become
mutually dependent. To this end, we use a relation $\sim$, to
indicate that GCFPs are \emph{related}. Let $\var[\!\!\!\!n]{d^X}$ and
$\var[\!\!\!\!m]{d^Y}$ be GCFPs; these are \emph{related}, denoted
$\var[\!\!\!\!n]{d^X}
\sim \var[\!\!\!\!m]{d^Y}$, if $n = \copied{Y}{i}{m}$ for some $i$. Next,
we characterise when a set of GCFPs does not introduce mutual
dependencies.
\begin{definition}
\label{def:control_structure}
Let $\mathcal{C}$ be a set of GCFPs, and let $\sim^*$ denote the reflexive,
symmetric and transitive closure of $\sim$ on $\mathcal{C}$.
Assume ${\approx} \subseteq \mathcal{C} \times \mathcal{C}$ is an equivalence
relation that subsumes $\sim^*$; \emph{i.e.}\xspace,
that satisfies $\sim^* \subseteq \approx$. Then the pair
$\langle \mathcal{C}, \approx \rangle$ defines a \emph{control structure}
if for all $X \in \bnd{\mathcal{E}}$ and all $d,d' \in \mathcal{C} \cap
\param{X}$, if $d \approx d'$, then $d = d'$.
\end{definition}
We say that a unicity constraint is a \emph{witness} to a control
structure $\langle \varset{C},\approx\rangle$ if the latter can be
deduced from the unicity constraint through
Definitions~\ref{def:LCFP}--\ref{def:control_structure}.
The equivalence $\approx$ in a control structure
also serves to identify GCFPs that take on the same role in
\emph{different} equations: we say that two parameters $c,c' \in
\varset{C}$ are \emph{identical} if $c \approx c'$.
As a last step, we formally define our notion of a
control flow parameter.
\begin{definition}
A formal parameter $c$ is a \emph{control flow parameter (CFP)} if there
is a control structure $\langle \varset{C},\approx\rangle$ such
that $c \in \varset{C}$.
\end{definition}
\begin{example}\label{exa:CFP}
Observe that there is a unicity constraint that
identifies that parameter
$i^X$ is copied to $i^Z$ in our running example. Then necessarily $i^Z \sim
i^X$ and thus
$i^X \approx i^Z$ for a control structure $\langle \mathcal{C},\approx \rangle$
with $i^X,i^Z \in \mathcal{C}$.
However, $i^X$ and $i^Y$
do not have to be related, but we have the option to define $\approx$
so that they are. In fact, the structure $\langle
\{i^X,j^X,i^Y,j^Y,i^Z,j^Z\}, \approx \rangle$ for which $\approx$
relates all (and only) identically named parameters is a control
structure.
\end{example}
Using a control structure $\langle \varset{C},\approx\rangle$,
we can ensure that all equations have the same set of CFPs. This can
be done by assigning unique names to
identical CFPs and by adding CFPs that
do not appear in an equation as formal parameters for this equation.
Without loss of generality
we therefore continue to work under the following assumption.
\begin{assumption} \label{ass:names}
The set of CFPs is the same for every equation in
a PBES; that is, for all $X, Y \in \bnd{\ensuremath{\mathcal{E}}}$ in a PBES $\ensuremath{\mathcal{E}}$ we have
$d^X \in \param{X}$ is a CFP iff $d^Y \in \param{Y}$ is a CFP, and $d^X \approx
d^Y$.
\end{assumption}
From hereon, we call any formal parameter that is not a control flow parameter
a \emph{data parameter}. We make this
distinction explicit by partitioning $\varset{D}$ into CFPs $\varset{C}$
and data parameters $\varset{D}^{\mathit{DP}}$. As a consequence of Assumption~\ref{ass:names},
we may assume that every PBES we consider has equations with the same sequence of CFPs;
\emph{i.e.}\xspace, all equations are of the form
$\sigma X(\var{c} \colon \vec{C}, \var{d^X} \colon \vec{D^X})
= \rhs{X}(\var{c}, \var{d^X})$, where $\var{c}$ is the (vector of) CFPs, and
$\var{d^X}$ is the (vector of) data parameters of the equation for $X$.
Using the CFPs, we next construct a control flow graph. Vertices in this
graph represent valuations for the vector of CFPs and
the edges capture dependencies on PVIs.
The set of potential valuations for the CFPs is bounded by $\values{\var[k]{c}}$,
defined as:
\[
\{ \init{\var[k]{c}} \} \cup \bigcup\limits_{i \in \mathbb{N}, X \in \bnd{\ensuremath{\mathcal{E}}}}
\{ v \in D \mid
\source{X}{i}{k} = v \lor \dest{X}{i}{k} = v \}.
\]
We generalise $\ensuremath{\mathsf{values}}$ to the vector $\var{c}$ in the obvious way.
\begin{definition}
\label{def:globalCFGHeuristic}
The control flow graph (CFG) of $\ensuremath{\mathcal{E}}$ is a directed graph $(V^{\semantic}syn,
{\smash{\xrightarrow{\semantic}syn}})$
with:
\begin{compactitem}
\item $V^{\semantic}syn \subseteq \bnd{\ensuremath{\mathcal{E}}} \times \values{\var{c}}$.
\item ${\smash{\xrightarrow{\semantic}syn}} \subseteq V \times \mathbb{N} \times
V$ is the least relation for which, whenever $(X,\val{v}) \xrightarrow{\semantic}syn[i]
(\predphi{\rhs{X}}{i},\val{w})$ then for every $k$ either:
\begin{compactitem}
\item $\source{X}{i}{k} = \val[k]{v}$ and $\dest{X}{i}{k} = \val[k]{w}$, or
\item $\source{X}{i}{k} = \bot$, $\copied{X}{i}{k} = k$ and $\val[k]{v} =
\val[k]{w}$, or
\item $\source{X}{i}{k} = \bot$, and $\dest{X}{i}{k} = \val[k]{w}$.
\end{compactitem}
\end{compactitem}
\end{definition}
We refer to the vertices in the CFG as \emph{locations}. Note that
a CFG is finite since the set $\values{\var{c}}$ is finite.
Furthermore, CFGs are complete in the sense that all PVIs on which
the truth of some $\rhs{X}$ may depend when $\val{c} = \val{v}$ are neighbours
of
location $(X, \val{v})$.
\reportonly{
\begin{restatable}{lemma}{resetrelevantpvineighbours}
\label{lem:relevant_pvi_neighbours}
Let $(V^{\semantic}syn, {\smash{\xrightarrow{\semantic}syn}})$ be $\ensuremath{\mathcal{E}}$'s control flow graph. Then
for all $(X, \val{v}) \in V^{\semantic}syn$ and all predicate environments
$\ensuremath{\eta}, \ensuremath{\eta}'$ and data environments $\ensuremath{\delta}$:
$$\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}}]}
= \sem{\rhs{X}}{\ensuremath{\eta}'}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}}]}$$
provided that
$\ensuremath{\eta}(Y)(\val{w}) = \ensuremath{\eta}'(Y)(\val{w})$ for all
$(Y, \val{w})$ satisfying
$(X, \val{v}) \xrightarrow{\semantic}syn[i] (Y, \val{w})$.
\end{restatable}
\begin{proof}
Let $\ensuremath{\eta}, \ensuremath{\eta}'$ be predicate environments,$\ensuremath{\delta}$ a data
environment, and let $(X, \val{v}) \in V$.
Suppose that for all $(Y, \val{w})$ for which
$(X, \val{v}) \xrightarrow{\semantic}syn[i] (Y, \val{w})$, we know that $\ensuremath{\eta}(Y)(\val{w})
=
\ensuremath{\eta}'(Y)(\val{w})$.
Towards a contradiction, let
$\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]}
\neq
\sem{\rhs{X}}{\ensuremath{\eta}'}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]}$.
Then there must be a predicate variable instance $\predinstphi{\rhs{X}, i}$
such that
\begin{equation}\label{eqn:ass_pvi}
\begin{array}{cl}
&
\ensuremath{\eta}(\predphi{\rhs{X}}{i})(\sem{\dataphi{\rhs{X}}{i}}{}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]})\\
\neq &
\ensuremath{\eta}'(\predphi{\rhs{X}}{i})(\sem{\dataphi{\rhs{X}}{i}}{}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]}).
\end{array}
\end{equation}
Let
$\dataphi{\rhs{X}}{i} = (\val{e},\val{e'})$, where $\val{e}$ are the values
of the control flow parameters, and $\val{e'}$ are the values of the data
parameters.
Consider an arbitrary control flow parameter $\var[\ell]{c}$. We
distinguish two cases:
\begin{compactitem}
\item $\source{X}{i}{\ell} \neq \bot$. Then we know
$\dest{X}{i}{\ell} \neq \bot$, and the
requirement for the edge $(X, \val{v}) \xrightarrow{\semantic}syn[i]
(\predphi{\rhs{X}}{i}, \val{e})$
is satisfied for $\ell$.
\item $\source{X}{i}{\ell} = \bot$. Since $\var[\ell]{c}$
is a control flow parameter, we can distinguish two cases based on
Definitions~\ref{def:LCFP} and \ref{def:GCFP}:
\begin{compactitem}
\item $\dest{X}{i}{\ell} \neq \bot$. Then parameter $\ell$
immediately satisfies the requirements that show the existence
of the edge $(X, \val{v}) \xrightarrow{\semantic}syn[i]
(\predphi{\rhs{X}}{i}, \val{e})$ in the third clause in the
definition of CFG.
\item $\copied{X}{i}{\ell} = \ell$.
According to the definition of $\ensuremath{\mathsf{copy}}\xspace$, we now know that
$\val[\ell]{v} = \val[\ell]{e}$, hence the edge
$(X, \val{v}) \xrightarrow{\semantic}syn[i]
(\predphi{\rhs{X}}{i}, \val{e})$ exists according to the
second requirement in the definition of CFG.
\end{compactitem}
\end{compactitem}
Since we have considered an arbitrary $\ell$, we know that for all $\ell$
the requirements are satisfied, hence
$(X, \val{v}) \xrightarrow{\semantic}syn[i] (\predphi{\rhs{X}}{i}, \val{e})$. Then
according to the definition of $\ensuremath{\eta}$ and $\ensuremath{\eta}'$,
$\ensuremath{\eta}(\predphi{\rhs{X}}{i})(\sem{\val{e}}{}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]})
=
\ensuremath{\eta}'(\predphi{\rhs{X}}{i})(\sem{\val{e}}{}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\val{v}}{}{}}]})$.
This contradicts \eqref{eqn:ass_pvi}, hence we find that
$\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\var{v}}{}{}}]}
=
\sem{\rhs{X}}{\ensuremath{\eta}'}{\ensuremath{\delta}[\subst{\var{c}}{\sem{\var{v}}{}{}}]}$.\qed
\end{proof}
}
\begin{example}
Using the CFPs identified earlier and an appropriate unicity constraint, we can
obtain the CFG depicted in Fig.~\ref{fig:CFG} for our running example.
\end{example}
\paragraph{Implementation.} CFGs are defined in terms of
CFPs, which in turn are obtained from a unicity constraint.
Our definition of a unicity constraint is not constructive.
However, a unicity constraint can be derived from \emph{guards} for
a PVI. While computing the exact guard, \emph{i.e.}\xspace the strongest formula $\psi$ satisfying
$\varphi \equiv
\varphi[i \mapsto (\psi \wedge \predinstphi{\varphi}{i})]$, is
computationally
hard, we can efficiently approximate it as follows:
\begin{definition}
Let $\varphi$ be a predicate formula. We define the \emph{guard}
of the $i$-th PVI in $\varphi$,
denoted $\guard{i}{\varphi}$, inductively as follows:
\begin{align*}
\guard{i}{b} & = \ensuremath{\mathit{false}} &
\guard{i}{Y} & = \ensuremath{\mathit{true}} \\
\guard{i}{\forall d \colon D . \varphi} & = \guard{i}{\varphi} &
\guard{i}{\exists d \colon D . \varphi} & = \guard{i}{\varphi} \\
\guard{i}{\varphi \land \psi} & = \begin{cases}
s(\varphi) \land \guard{i - \npred{\varphi}}{\psi} & \text{if } i > \npred{\varphi} \\
s(\psi) \land \guard{i}{\varphi} & \text{if } i \leq \npred{\varphi}
\end{cases} \span\omit\span\omit\\
\guard{i}{\varphi \lor \psi} & = \begin{cases}
s(\lnot \varphi) \land \guard{i - \npred{\varphi}}{\psi} & \text{if } i >
\npred{\varphi} \\
s(\lnot \psi) \land \guard{i}{\varphi} & \text{if } i \leq \npred{\varphi}
\end{cases}\span\omit\span\omit
\end{align*}
where
$s(\varphi) = \varphi$ if $\npred{\varphi} = 0$, and $\ensuremath{\mathit{true}}$ otherwise.
\end{definition}
We have $\varphi \equiv \varphi[i \mapsto (\guard{i}{\varphi} \wedge \predinstphi{\varphi}{i})]$;
\emph{i.e.}\xspace,
$\predinstphi{\varphi}{i}$ is relevant to $\varphi$'s truth value only if
$\guard{i}{\varphi}$ is satisfiable.
\reportonly{
This is formalised int he following lemma.
\begin{lemma}\label{lem:addGuard}
Let $\varphi$ be a predicate formula, and let $i \leq \npred{\varphi}$, then
for every predicate environment $\ensuremath{\eta}$ and data environment $\ensuremath{\delta}$,
$$
\sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\varphi[i \mapsto (\guard{i}{\varphi} \land
\predinstphi{\varphi}{i})]}{\ensuremath{\eta}}{\ensuremath{\delta}}.
$$
\end{lemma}
\begin{proof}
Let $\ensuremath{\eta}$ and $\ensuremath{\delta}$ be arbitrary. We proceed by induction on
$\varphi$.
The base cases where $\varphi = b$ and $\varphi = Y(\val{e})$ are trivial, and
$\forall d \colon D . \psi$ and $\exists d \colon D . \psi$ follow immediately
from the induction hypothesis. We
describe the case where $\varphi = \varphi_1 \land \varphi_2$ in detail,
the $\varphi = \varphi_1 \lor \varphi_2$ is completely analogous.
Assume that $\varphi = \varphi_1 \land \varphi_2$.
Let $i \leq \npred{\varphi_1 \land \varphi_2}$.
Without loss of generality assume that $i \leq \npred{\varphi_1}$, the other
case is analogous. According to the induction hypothesis,
\begin{equation}
\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\varphi_1[i \mapsto (\guard{i}{\varphi_1} \land
\predinstphi{\varphi_1}{i})]}{\ensuremath{\eta}}{\ensuremath{\delta}} \label{eq:ih}
\end{equation}
We distinguish two cases.
\begin{compactitem}
\item $\npred{\varphi_2} \neq 0$. Then
$\sem{\guard{i}{\varphi_1}}{\ensuremath{\delta}}{\ensuremath{\eta}}
= \sem{\guard{i}{\varphi_1 \land \varphi_2}}{\ensuremath{\delta}}{\ensuremath{\eta}}$
according to the definition of $\mathsf{guard}$. Since $i \leq \npred{\varphi_1}$,
we find that
$
\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{(\varphi_1 \land \varphi_2)[i \mapsto (\guard{i}{\varphi_1 \land
\varphi_2} \land \predinstphi{\varphi_1 \land
\varphi_2}{i})]}{\ensuremath{\eta}}{\ensuremath{\delta}}.
$
\item $\npred{\varphi_2} = 0$.
We have to show that
$$\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\varphi_1[i \mapsto (\guard{i}{\varphi_1 \land \varphi_2} \land
\predinstphi{\varphi_1}{i})] \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}$$
From the semantics, it follows that
$\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}} \land
\sem{\varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}.
$
Combined with \eqref{eq:ih}, and an application of the semantics, this yields
$$
\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\varphi_1[i \mapsto (\guard{i}{\varphi_1} \land
\predinstphi{\varphi_1}{i})] \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}.
$$
According to the definition
of $\mathsf{guard}$, $\guard{i}{\varphi_1 \land \varphi_2} = \varphi_2 \land
\guard{i}{\varphi_1}$.
Since $\varphi_2$ is present in the context, the desired result
follows.\qed
\end{compactitem}
\end{proof}
We can generalise the above, and guard every predicate variable
instance in a formula with its guard, which preserves the solution of the
formula. To this end we introduce the function $\mathsf{guarded}$.
\begin{definition}\label{def:guarded}
Let $\varphi$ be a predicate formula, then
$$\guarded{\varphi} \ensuremath{=} \varphi[i \mapsto (\guard{i}{\varphi} \land
\predinstphi{\varphi}{i})]_{i \leq \npred{\varphi}}$$
where $[i \mapsto \psi_i]_{i \leq \npred{\varphi}}$ is the simultaneous
syntactic substitution of all $\predinstphi{\varphi}{i}$ with $\psi_i$.
\end{definition}
The following corollary follows immediately from Lemma~\ref{lem:addGuard}.
\begin{corollary}\label{cor:guardedPreservesSol}
For all formulae $\varphi$, and for all predicate environments $\ensuremath{\eta}$,
and data environments $\ensuremath{\delta}$,
$
\sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}} = \sem{\guarded{\varphi}}{\ensuremath{\eta}}{\ensuremath{\delta}}
$
\end{corollary}
This corollary confirms our intuition that indeed the guards we compute
effectively guard the recursions in a formula.
}
A good heuristic for defining the unicity constraints is looking
for positive occurrences of constraints of the form $d
= e$ in the guards and using this information to see if the arguments
of PVIs reduce to constants.
\section{Data Flow Analysis}\label{sec:dataflow}
Our liveness analysis is built on top of CFGs constructed using
Def.~\ref{def:globalCFGHeuristic}. The analysis proceeds as follows:
for each location in the CFG, we first identify the data parameters
that may directly affect the truth value of the corresponding predicate
formula. Then we inductively identify data parameters that can
affect such parameters through PVIs as live as well. Upon termination,
each location is labelled by the \emph{live} parameters at that
location.
The set $\significant{\varphi}$ of parameters that affect the truth value of
a predicate formula $\varphi$, \emph{i.e.}\xspace, those parameters that occur in
Boolean data terms, are approximated as follows:
\begin{align*}
\significant{b} & = \free{b} &
\significant{Y(e)} & = \emptyset \\
\significant{\varphi \land \psi} & = \significant{\varphi} \cup \significant{\psi} &
\significant{\varphi \lor \psi} & = \significant{\varphi} \cup \significant{\psi} \\
\significant{\exists d \colon D . \varphi} & = \significant{\varphi} \setminus \{ d \} &
\significant{\forall d \colon D . \varphi} & = \significant{\varphi} \setminus \{ d \}
\end{align*}
Observe that $\significant{\varphi}$ is not invariant under logical
equivalence. We use this fact to our advantage: we assume the
existence of a function $\mathsf{simplify}$ for which we require
$\simplify{\varphi} \equiv \varphi$, and
$\significant{\simplify{\varphi}}\subseteq \significant{\varphi}$.
An appropriately chosen function $\mathsf{simplify}$ may help to narrow
down the parameters that affect the truth value of predicate
formulae in our base case. Labelling the CFG with live variables
is achieved as follows:
\begin{definition}
\label{def:markingHeuristic}
Let $\ensuremath{\mathcal{E}}$ be a PBES and let
$(V^{\semantic}syn, \xrightarrow{\semantic}syn)$ be its CFG.
The labelling $\markingsn{\syntactic}{} \colon V^{\semantic}syn \to
\mathbb{P}({\varset{D}^{\mathit{DP}}})$ is defined as
$\marking{\markingsn{\syntactic}{}}{X, \val{v}} = \bigcup_{n \in \ensuremath{\mathbb{N}}}
\marking{\markingsn{\syntactic}{n}}{X, \val{v}}$, with
$\markingsn{\syntactic}{n}$ inductively defined as:
\[
\begin{array}{ll}
\marking{\markingsn{\syntactic}{0}}{X, \val{v}} & =
\significant{\simplify{\rhs{X}[\var{c} := \val{v}]}}\\
\marking{\markingsn{\syntactic}{n+1}}{X, \val{v}} & =
\marking{\markingsn{\syntactic}{n}}{X, \val{v}} \\
& \cup \{ d \in \param{X} \cap \varset{D}^{\mathit{DP}} \mid \exists {i \in \mathbb{N},
(Y,\val{w}) \in V}:
(X, \val{v}) \xrightarrow{\semantic}syn[i] (Y, \val{w}) \\
& \qquad \land \exists {\var[\ell]{d} \in
\marking{\markingsn{\syntactic}{n}}{Y, \val{w}}}:~
\affects{d}{\dataphi[\ell]{\rhs{X}}{i}}
\}
\end{array}
\]
\end{definition}
The set $\marking{\markingsn{\syntactic}{}}{X, \val{v}}$ approximates the set of
parameters potentially live at location $(X, \val{v})$; all other data
parameters are guaranteed to be ``dead'', \emph{i.e.}\xspace, irrelevant.
\begin{example}\label{exa:globalCFGLabelled} The
labelling computed for our running example is depicted in Fig.~\ref{fig:CFG}.
One can cheaply establish that
$k^Z \notin \marking{\markingsn{\syntactic}{0}}{Z,1,2}$ since assigning
value $2$ to $j^Z$ in $Z$'s right-hand side effectively allows to
reduce subformula $(k < 10 \vee j =2)$ to $\ensuremath{\mathit{true}}$. We have
$l \in \marking{\markingsn{\syntactic}{1}}{Z,1,2}$ since
we have $k^Y \in \marking{\markingsn{\syntactic}{0}}{Y,1,1}$.
\end{example}
\reportonly{
The labelling from Definition~\ref{def:markingHeuristic} induces
a relation $\markrel{\markingsn{\syntactic}{}}$ on signatures as follows.
\begin{definition}\label{def:markrelSyntactic}
Let $\markingsn{\syntactic}{} \colon V \to \mathbb{P}(\varset{D}^{\mathit{DP}})$ be a
labelling. $\markingsn{\syntactic}{}$ induces a relation
$\markrel{\markingsn{\syntactic}{}}$ such that $(X, \sem{\val{v}}{}{},
\sem{\val{w}}{}{})
\markrel{\markingsn{\syntactic}{}} (Y, \sem{\val{v'}}{}{}, \sem{\val{w'}}{}{})$
if and only if
$X = Y$, $\sem{\val{v}}{}{} = \sem{\val{v'}}{}{}$, and
$\forall \var[k]{d} \in \marking{\markingsn{\syntactic}{}}{X,
\val{v}}: \semval[k]{w} = \semval[k]{w'}$.
\end{definition}
Observe that the relation $\markrel{\markingsn{\syntactic}{}}$
allows for relating \emph{all} instances of the non-labelled data parameters at
a given control flow location.
We prove that, if locations are related using the relation
$\markrel{\markingsn{\syntactic}{}}$, then the corresponding instances in the
PBES have the same solution by showing that $\markrel{\markingsn{\syntactic}{}}$
is a consistent correlation.
In order to prove this, we first show that given a predicate environment and two
data environments, if the solution of a formula differs between those
environments, and all predicate variable instances in the formula have the same
solution, then there must be a significant parameter $d$ in
the formula that gets a different value in the two data environments.
\begin{lemma}
\label{lem:non_recursive_free}
For all formulae $\varphi$, predicate environments $\ensuremath{\eta}$,
and data environments $\ensuremath{\delta}, \ensuremath{\delta}'$, if
$\sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}} \neq \sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}'}$
and for all $i \leq \npred{\varphi}$,
$\sem{\predinstphi{\varphi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\predinstphi{\varphi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}'}$,
then $\exists d \in \significant{\varphi}: \ensuremath{\delta}(d) \neq \ensuremath{\delta}'(d)$.
\end{lemma}
\begin{proof}
We proceed by induction on $\varphi$.
\begin{compactitem}
\item $\varphi = b$. Trivial.
\item $\varphi = Y(e)$. In this case the two preconditions
contradict, and the result trivially follows.
\item $\varphi = \forall e \colon D . \psi$. Assume that
$\sem{\forall e \colon D . \psi}{\ensuremath{\eta}}{\ensuremath{\delta}}
\neq \sem{\forall e \colon D . \psi}{\ensuremath{\eta}}{\ensuremath{\delta}'}$, and furthermore,
$\forall i \leq \npred{\forall e \colon D . \psi}:
\sem{\predinstphi{\forall e \colon D . \psi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\predinstphi{\forall e \colon D . \psi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}'}$.
According to the semantics, we have
$\forall u \in \semset{D} . \sem{\psi}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{e}{u}]}
\neq \forall u' \in \semset{D} .
\sem{\psi}{\ensuremath{\eta}}{\ensuremath{\delta}'[\subst{e}{u'}]}$,
so $\exists u \in \semset{D}$ such that
$\sem{\psi}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{e}{u}]}
\neq \sem{\psi}{\ensuremath{\eta}}{\ensuremath{\delta}'[\subst{e}{u}]}$.
Choose an arbitrary such $u$. Observe that also
for all $i \leq \npred{\psi}$, we know that
$$\sem{\predinstphi{\psi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{e}{u}]}
= \sem{\predinstphi{\psi}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}'[\subst{e}{u}]}.$$
According to the induction hypothesis, there exists some $d \in
\significant{\psi}$
such that $\ensuremath{\delta}[\subst{e}{u}](d) \neq \ensuremath{\delta}'[\subst{e}{u}](d)$.
Choose such a $d$, and observe that $d \neq e$ since otherwise $u \neq u$,
hence $d \in \significant{\forall e \colon D . \psi}$,
which is the desired result.
\item $\varphi = \exists e \colon D . \psi$. Analogous to the previous case.
\item $\varphi = \varphi_1 \land \varphi_2$. Assume that
$\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}}
\neq \sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}'}$, and suppose
that that for all $i \leq \npred{\varphi_1 \land \varphi_2}$, we know that
$\sem{\predinstphi{\varphi_1 \land \varphi_2}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\predinstphi{\varphi_1 \land \varphi_2}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}'}$.
According to the first assumption, either
$\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}} \neq
\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}'}$,
or $\sem{\varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}} \neq
\sem{\varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}'}$.
Without loss of generality, assume that
$\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}} \neq
\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}'}$,
the other case is completely analogous.
Observe that from our second assumption it follows that
$\forall i \leq \npred{\varphi_1}:
\sem{\predinstphi{\varphi_1}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}}
= \sem{\predinstphi{\varphi_1}{i}}{\ensuremath{\eta}}{\ensuremath{\delta}'}$.
According to the induction hypothesis, we now find some
$d \in \significant{\varphi_1}$ such that $\ensuremath{\delta}(d) \neq \ensuremath{\delta}'(d)$.
Since $\significant{\varphi_1} \subseteq \significant{\varphi_1 \land
\varphi_2}$,
our result follows.
\item $\varphi = \varphi_1 \lor \varphi_2$. Analogous to the previous case.
\qed
\end{compactitem}
\end{proof}
This is now used in proving the following proposition, that shows that related
signatures have the same solution. This result follows from the fact that
$\markrel{\markingsn{\syntactic}{}}$ is a consistent correlation.
\begin{proposition}
\label{prop:ccSyn}
Let $\ensuremath{\mathcal{E}}$ be a PBES, with global control flow graph $(V^{\semantic}syn, \xrightarrow{\semantic}syn)$,
and labelling $\markingsn{\syntactic}{}$. For all predicate environments
$\ensuremath{\eta}$ and data environments $\ensuremath{\delta}$,
$$(X, \semval{v}, \semval{w}) \markrel{\markingsn{\syntactic}{}}
(Y, \semval{v'}, \semval{w'})
\implies \sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}}{\ensuremath{\delta}}(X(\val{v},\val{w})) =
\sem{\ensuremath{\mathcal{E}}}{\ensuremath{\eta}}{\ensuremath{\delta}}(Y(\val{v'},\val{w'})).$$
\end{proposition}
\begin{proof}
We show that $\markrel{\markingsn{\syntactic}{}}$ is a consistent correlation.
The result then follows immediately from Theorem~\ref{thm:cc}.
Let $n$ be the smallest number such that for all $X, \val{v}$,
$\markingsn{\syntactic}{n+1}(X, \val{v})
= \markingsn{\syntactic}{n}(X, \val{v})$, and hence
$\markingsn{\syntactic}{n}(X, \val{v}) = \markingsn{\syntactic}{}(X, \val{v})$.
Towards a contradiction, suppose that $\markrel{\markingsn{\syntactic}{}}$
is not a consistent correlation. Since $\markrel{\markingsn{\syntactic}{}}$
is not a consistent correlation, there exist
$X, X', \val{v}, \val{v'}, \val{w}, \val{w'}$ such that
$(X, \semval{v}, \semval{w}) \markrel{\markingsn{\syntactic}{n}} (X',
\semval{v'}, \semval{w'})$,
and
\begin{equation*}
\exists \ensuremath{\eta} \in \correnv{\markrel{\markingsn{\syntactic}{n}}}, \ensuremath{\delta}:
\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},
\subst{\var{d}}{\semval{w}}]}
\neq \sem{\rhs{X'}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v'}},
\subst{\var{d}}{\semval{w'}}]}.
\end{equation*}
According to Definition~\ref{def:markrelSyntactic}, $X = X'$, and $\semval{v} =
\semval{v'}$,
hence this is equivalent to
\begin{equation}\label{eq:equal_phi}
\exists \ensuremath{\eta} \in \correnv{\markrel{\markingsn{\syntactic}{n}}}, \ensuremath{\delta}:
\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},
\subst{\var{d}}{\semval{w}}]}
\neq \sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},
\subst{\var{d}}{\semval{w'}}]}.
\end{equation}
Let $\ensuremath{\eta}$ and $\ensuremath{\delta}$ be such, and let
$\ensuremath{\delta}_1 = \ensuremath{\delta}[\subst{\var{c}}{\semval{v}},
\subst{\var{d}}{\semval{w}}]$
and $\ensuremath{\delta}_2 = \ensuremath{\delta}[\subst{\var{c}}{\semval{v}},
\subst{\var{d}}{\semval{w'}}]$.
Define $ \varphi'_X \ensuremath{=} \simplify{\rhs{X}[\var{c} := \val{v}]}.$
Since the values in $\val{v}$ are closed, and from the definition of
$\mathsf{simplify}$,
we find that $\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}_1} =
\sem{\varphi'_X}{\ensuremath{\eta}}{\ensuremath{\delta}_1}$,
and likewise for $\ensuremath{\delta}_2$. Therefore, we know that
\begin{equation}
\label{eq:equal_phi'}
\sem{\varphi'_X}{\ensuremath{\eta}}{\ensuremath{\delta}_1} \neq
\sem{\varphi'_X}{\ensuremath{\eta}}{\ensuremath{\delta}_2}.
\end{equation}
Observe that for all $\var[k]{d} \in \marking{\markingsn{\syntactic}{}}{X,
\val{v}}$,
$\semval[k]{w} = \semval[k]{w'}$ by definition of
$\markrel{\markingsn{\syntactic}{}}$.
Every predicate variable instance that might change the solution of $\varphi'_X$
is a neighbour of $(X, \val{v})$ in the control flow graph, according to Lemma
\ref{lem:relevant_pvi_neighbours}.
Take an arbitrary predicate variable instance
$\predinstphi{\rhs{X}}{i} = Y(\val{e}, \val{e'})$ in $\varphi'_X$.
We first show that $\sem{\val[\ell]{e'}}{}{\ensuremath{\delta}_1}
= \sem{\val[\ell]{e'}}{}{\ensuremath{\delta}_2}$ for all $\ell$.
Observe that $\sem{\val{e}}{}{\ensuremath{\delta}_1} = \sem{\val{e}}{}{\ensuremath{\delta}_2}$ since
$\val{e}$ are expressions substituted for control flow parameters, and hence
are either constants, or the result of copying.
Furthermore, there is no unlabelled parameter $\var[k]{d}$ that can influence a
labelled parameter $\var[\ell]{d}$ at location $(Y, \val{u})$. If there is a
$\var[\ell]{d} \in \markingsn{\syntactic}{n}(Y, \val{u})$ such that
$\var[k]{d} \in \free{\val[\ell]{e'}}$, and
$\var[k]{d} \not \in \markingsn{\syntactic}{n}(X, \val{v})$, then by
definition of labelling $\var[k]{d} \in \markingsn{\syntactic}{n+1}(X,
\val{v})$,
which contradicts the assumption that the labelling is stable, so it follows
that
\begin{equation} \label{eq:equivalent_arguments_Xi}
\sem{\val[\ell]{e'}}{}{\ensuremath{\delta}_1}
= \sem{\val[\ell]{e'}}{}{\ensuremath{\delta}_2}\text{ for all }\ell.
\end{equation}
From \eqref{eq:equivalent_arguments_Xi}, and since we have chosen the predicate
variable instance arbitrarily, it follows that for all $1 \leq i \leq
\npred{\varphi'_X}$,
$\sem{X(\val{e},\val{e'})}{\ensuremath{\eta}}{\ensuremath{\delta}_1}
= \sem{X(\val{e},\val{e'})}{\ensuremath{\eta}}{\ensuremath{\delta}_2}$.
Together with \eqref{eq:equal_phi'}, according to
Lemma~\ref{lem:non_recursive_free},
this implies that there is some $d \in \significant{\varphi'_X}$
such that $\ensuremath{\delta}_1(d) \neq \ensuremath{\delta}_2(d)$. From the definition of
$\markingsn{\syntactic}{0}$, however, it follows that $d$ must be labelled
in $\marking{\markingsn{\syntactic}{0}}{X, \val{v}}$, and hence also in
$\marking{\markingsn{\syntactic}{n}}{X, \val{v}}$.
According to the definition of $\markrel{\markingsn{\syntactic}{n}}$ it then is
the case that $\ensuremath{\delta}_1(d) = \ensuremath{\delta}_2(d)$, which is a contradiction.
Since also in this case we derive a contradiction, the original assumption that
$\markrel{\markingsn{\syntactic}{}}$ is not a consistent correlation does not
hold, and we conclude that $\markrel{\markingsn{\syntactic}{}}$ is a consistent
correlation. \qed
\end{proof}
}
\label{sec:reset}
A parameter $d$ that is not live at a location
can be assigned a fixed default value. To this end
the corresponding data argument of the PVIs that lead to that location
are replaced by a default value $\init{d}$. This is achieved by
function $\ensuremath{\mathsf{Reset}}$, defined below:
\reportonly{
\begin{definition}
\label{def:reset}
Let $\ensuremath{\mathcal{E}}$ be a PBES, let $(V, \to)$ be its CFG, with labelling
$\markingsn{\syntactic}{}$\!. Resetting a PBES is inductively defined on the
structure of
$\ensuremath{\mathcal{E}}$.
$$
\begin{array}{lcl}
\reset{\markingsn{\syntactic}{}}{\ensuremath{\emptyset}} & \ensuremath{=} & \ensuremath{\emptyset} \\
\reset{\markingsn{\syntactic}{}}{\sigma X(\var{c} \colon \vec{C}, \var{d}
\colon \vec{D}) = \varphi) \ensuremath{\mathcal{E}}'} & \ensuremath{=}
& (\sigma \changed{X}(\var{c} \colon \vec{C}, \var{d} \colon\vec{D}) =
\reset{\markingsn{\syntactic}{}}{\varphi})
\reset{\markingsn{\syntactic}{}}{\ensuremath{\mathcal{E}}'} \\
\end{array}
$$
Resetting for formulae is defined inductively as follows:
$$
\begin{array}{lcl}
\reset{\markingsn{\syntactic}{}}{b} & \ensuremath{=} & b\\
\reset{\markingsn{\syntactic}{}}{\varphi \land \psi} & \ensuremath{=} &
\reset{\markingsn{\syntactic}{}}{\varphi} \land
\reset{\markingsn{\syntactic}{}}{\psi}\\
\reset{\markingsn{\syntactic}{}}{\varphi \lor \psi} & \ensuremath{=} &
\reset{\markingsn{\syntactic}{}}{\varphi} \lor
\reset{\markingsn{\syntactic}{}}{\psi}\\
\reset{\markingsn{\syntactic}{}}{\forall d \colon D. \varphi} & \ensuremath{=} &
\forall d \colon D. \reset{\markingsn{\syntactic}{}}{\varphi} \\
\reset{\markingsn{\syntactic}{}}{\exists d \colon D. \varphi} & \ensuremath{=} &
\exists d \colon D. \reset{\markingsn{\syntactic}{}}{\varphi} \\
\reset{\markingsn{\syntactic}{}}{X(\val{e}, \val{e'})} & \ensuremath{=} &
\bigwedge_{\val{v} \in \values{\var{c}}} (\val{e} = \val{v} \implies
\changed{X}(\val{v}, \resetvars{(X,
\val{v})}{\markingsn{\syntactic}{}}{\val{e'}}))
\end{array}
$$
With $\val{e} = \val{v}$ we denote that for all $i$,
$\val[i]{e} = \var[i]{v}$.
The function $\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{e'}}$ is
defined
positionally as follows:
$$
\ind{\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{e'}}}{i} =
\begin{cases}
\val[i]{e'} & \text{ if } \var[i]{d} \in
\marking{\markingsn{}{}}{X, \val{v}}
\\
\init{\var[i]{d}} & \text{otherwise}.
\end{cases}
$$
\end{definition}
\begin{remark}
We can reduce the number of equivalences we introduce in resetting a recurrence.
This effectively reduces the guard as follows.
Let $X \in \bnd{\ensuremath{\mathcal{E}}}$, such that $Y(\val{e}, \val{e'}) =
\predinstphi{\rhs{X}}{i}$,
and let $I = \{ j \mid \dest{X}{i}{j} = \bot \} $ denote the indices of the
control flow parameters for which the destination is undefined.
Define $\var{c'} = \var[i_1]{c}, \ldots, \var[i_n]{c}$ for $i_n \in I$,
and $\val{f} = \val[i_1]{e}, \ldots, \val[i_n]{e}$ to be the vectors
of control flow parameters for which the destination is undefined, and the
values that are assigned to them in predicate variable instance $i$. Observe
that these are the only control flow parameters that we need to constrain in
the guard while resetting.
We can redefine $\reset{\markingsn{\syntactic}{}}{X(\val{e}, \val{e'})}$ as
follows.
$$\reset{\markingsn{\syntactic}{}}{X(\val{e}, \val{e'})} \ensuremath{=}
\bigwedge_{\val{v'} \in \values{\var{c'}}}
( \val{f} = \val{v'} \implies
\changed{X}(\val{v}, \resetvars{\markingsn{\syntactic}{}}{(X,
\val{v})}{\val{e'}}) ).$$
In this definition
$\val{v}$ is defined positionally as $$\val[j]{v} =
\begin{cases}
\val[j]{v'} & \text{if } j \in I \\
\dest{X}{i}{j} & \text{otherwise}
\end{cases}$$
\end{remark}
Resetting dead parameters preserves the solution of the PBES. We formalise
this in Theorem~\ref{thm:resetSound} below. Our proof is based on consistent
correlations. We first define the relation $R^{\ensuremath{\mathsf{Reset}}}$, and we show that
this is indeed a consistent correlation. Soundness then follows from
Theorem~\ref{thm:cc}. Note that $R^{\ensuremath{\mathsf{Reset}}}$ uses the relation
$\markrel{\markingsn{\syntactic}{}}$ from Definition~\ref{def:markrelSyntactic}
to relate predicate variable instances of the original equation system. The
latter
is used in the proof of Lemma~\ref{lem:resetRecursion}.
\begin{definition}\label{def:resetRelSyn}
Let $R^{\ensuremath{\mathsf{Reset}}}$ be the relation defined as follows.
$$
\begin{cases}
X(\semval{v}, \semval{w})) R^{\ensuremath{\mathsf{Reset}}} \changed{X}(\semval{v},
\sem{\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{w})}}{}{}) \\
X(\semval{v}, \semval{w}) R^{\ensuremath{\mathsf{Reset}}} X(\semval{v}, \semval{w'}) & \text{if }
X(\semval{v}, \sem{\val{w}}{}{}) \markrel{\markingsn{\syntactic}{}}
X(\semval{v}, \semval{w'})
\end{cases}
$$
\end{definition}
We first show
that we can unfold the values of the control flow parameters in every predicate
variable instance, by duplicating the predicate variable instance, and
substituting the values of the CFPs.
\begin{lemma}\label{lem:unfoldCfl}
Let $\ensuremath{\eta}$ and $\ensuremath{\delta}$ be environments, and let $X \in \bnd{\ensuremath{\mathcal{E}}}$,
then for all $i \leq \npred{\rhs{X}}$, such that $\predinstphi{\rhs{X}}{i}
= Y(\val{e},\val{e'})$,
$$\sem{Y(\val{e}, \val{e'}))}{\ensuremath{\eta}}{\ensuremath{\delta}} =
\sem{\bigwedge_{\val{v}\in \values{\var{c}}} (\val{e} = \val{v} \implies
Y(\val{v}, \val{e'})}{\ensuremath{\eta}}{\ensuremath{\delta}}$$
\end{lemma}
\begin{proof}
Straightforward; observe that $\val{e} = \val{v}$ for
exactly one $\val{v} \in \values{\var{c}}$, using that $\val{v}$ is
closed.\qed
\end{proof}
Next we establish that resetting dead parameters is sound, \emph{i.e.}\xspace it
preserves the solution of the PBES. We first show that resetting
a predicate variable instance in an $R^{\ensuremath{\mathsf{Reset}}}$-correlating environment
and a given data environment is sound.
\begin{lemma}\label{lem:resetRecursion}
Let $\ensuremath{\mathcal{E}}$ be a PBES, let $(V, \to)$ be its CFG, with labelling
$\markingsn{\syntactic}{}$ such that $\markrel{\markingsn{\syntactic}{}}$ is a
consistent correlation, then
$$\forall \ensuremath{\eta} \in
\correnv{R^{\ensuremath{\mathsf{Reset}}}}, \ensuremath{\delta}: \sem{Y(\val{e},
\val{e'})}{\ensuremath{\eta}}{\ensuremath{\delta}} =
\sem{\reset{\markingsn{\syntactic}{}}{Y(\val{e},
\val{e'})}}{\ensuremath{\eta}}{\ensuremath{\delta}}$$
\end{lemma}
\begin{proof}
Let $\ensuremath{\eta} \in \correnv{R^{\ensuremath{\mathsf{Reset}}}}$, and $\ensuremath{\delta}$ be arbitrary. We
derive this as follows.
$$
\begin{array}{ll}
& \sem{\reset{\markingsn{\syntactic}{}}{Y(\val{e},
\val{e'}))}}{\ensuremath{\eta}}{\ensuremath{\delta}} \\
= & \{ \text{Definition~\ref{def:reset}} \} \\
& \sem{\bigwedge_{\val{v} \in \cfl{Y}} (\val{e} = \val{v}
\implies \changed{Y}(\val{v},\resetvars{(Y,
\val{v})}{\markingsn{\syntactic}{}}{\val{e'}}))}{\ensuremath{\eta}}{\ensuremath{\delta}}
\\
=^{\dagger} & \bigwedge_{\val{v} \in \cfl{Y}} (\sem{\val{e}}{}{\ensuremath{\delta}}
= \sem{\val{v}}{}{} \implies
\sem{\changed{Y}(\val{v},\resetvars{(Y,
\val{v})}{\markingsn{\syntactic}{}}{\val{e'}}}{\ensuremath{\eta}}{\ensuremath{\delta}}))
\\
=^{\dagger} & \bigwedge_{\val{v} \in \cfl{Y}} (\sem{\val{e}}{}{\ensuremath{\delta}}
= \semval{v} \implies
\ensuremath{\eta}(\changed{Y})(\sem{\val{v}}{}{\ensuremath{\delta}},\sem{\resetvars{(Y,
\val{v})}{\markingsn{\syntactic}{}}{\val{e'}}}{}{\ensuremath{\delta}}))\\
= & \{ \ensuremath{\eta} \in \correnv{R^{\ensuremath{\mathsf{Reset}}}} \} \\
& \bigwedge_{\val{v} \in \cfl{Y}} (\sem{\val{e}}{}{\ensuremath{\delta}} =
\semval{v} \implies
\ensuremath{\eta}(Y)(\sem{\val{v}}{}{\ensuremath{\delta}},\sem{\val{e'}}{}{\ensuremath{\delta}})))\\
=^{\dagger} & \bigwedge_{\val{v} \in \cfl{Y}} (\sem{\val{e}}{}{\ensuremath{\delta}}
= \semval{v} \implies
\sem{Y(\val{v}, \val{e'})}{\ensuremath{\eta}}{\ensuremath{\delta}})) \\
=^{\dagger} & \sem{\bigwedge_{\val{v} \in \cfl{Y}} (\val{e} = \val{v}
\implies Y(\val{v}, \val{e'}))}{\ensuremath{\eta}}{\ensuremath{\delta}} \\
= & \{ \text{Lemma~\ref{lem:unfoldCfl}} \}\\
& \sem{Y(\val{e}, \val{e'}))}{\ensuremath{\eta}}{\ensuremath{\delta}}
\end{array}
$$
Here at $^{\dagger}$ we have used the semantics.\qed
\end{proof}
By extending this result to the right-hand sides of equations, we can prove that
$R^{\ensuremath{\mathsf{Reset}}}$ is a consistent correlation.
\begin{proposition}
\label{prop:resetCc}
Let $\ensuremath{\mathcal{E}}$ be a PBES, and let $(V, \to)$ be a CFG, with labelling
$\markingsn{\syntactic}{}$ such that $\markrel{\markingsn{\syntactic}{}}$ is a
consistent correlation. Let $X \in \bnd{\ensuremath{\mathcal{E}}}$, with $\val{v} \in \ensuremath{\mathit{CFL}}(X)$,
then for all $\val{w}$, and for all predicate environments $\ensuremath{\eta} \in
\correnv{R^{\ensuremath{\mathsf{Reset}}}}$ and data environments $\ensuremath{\delta}$
$$
\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},\subst{\var{d}}{\semval{w}}]}
=
\sem{\reset{\markingsn{\syntactic}{}}{\rhs{X}}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},\subst{\var{d}}{\sem{\resetvars{(X,
\var{v})}{\markingsn{\syntactic}{}}{\val{w}}}{}{}}]} $$
\end{proposition}
\begin{proof}
Let $\ensuremath{\eta}$ and $\ensuremath{\delta}$ be arbitrary, and define $\ensuremath{\delta}_r \ensuremath{=}
\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},\subst{\var{d}}{\sem{\resetvars{(X,
\val{v})}{\markingsn{\syntactic}{}}{\val{w}}}{}{}}]$.
We first prove that
\begin{equation}
\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}
=
\sem{\reset{\markingsn{\syntactic}{}}{\rhs{X}}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}
\end{equation}
We proceed by induction on $\rhs{X}$.
\begin{compactitem}
\item $\rhs{X} = b$. Since $\reset{\markingsn{\syntactic}{}}{b} = b$ this
follows immediately.
\item $\rhs{X} = Y(\val{e})$. This follows immediately from
Lemma~\ref{lem:resetRecursion}.
\item $\rhs{X} = \forall y \colon D . \varphi$. We derive that
$\sem{\forall y \colon D . \varphi}{\ensuremath{\eta}}{\ensuremath{\delta}_r} = \forall v
\in \semset{D} . \sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}_r[\subst{y}{v}]}$.
According to the induction hypothesis, and since we applied only a
dummy transformation on $y$, we find that
$\sem{\varphi}{\ensuremath{\eta}}{\ensuremath{\delta}_r[\subst{y}{v}]}
=
\sem{\reset{\markingsn{\syntactic}{}}{\varphi}}{\ensuremath{\eta}}{\ensuremath{\delta}_r[\subst{y}{v}]}$,
hence $\sem{\forall y \colon D . \varphi}{\ensuremath{\eta}}{\ensuremath{\delta}_r} =
\sem{\reset{\markingsn{\syntactic}{}}{\forall y \colon D .
\varphi}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}$.
\item $\rhs{X} = \exists y \colon D . \varphi$. Analogous to the previous
case.
\item $\rhs{X} = \varphi_1 \land \varphi_2$. We derive that
$\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}_r} =
\sem{\varphi_1}{\ensuremath{\eta}}{\ensuremath{\delta}_r} \land
\sem{\varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}_r}$.
If we apply the induction hypothesis on both sides we get
$\sem{\varphi_1 \land \varphi_2}{\ensuremath{\eta}}{\ensuremath{\delta}_r} =
\sem{\reset{\markingsn{\syntactic}{}}{\varphi_1}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}
\land
\sem{\reset{\markingsn{\syntactic}{}}{\varphi_2}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}$.
Applying the semantics, and the definition of $\ensuremath{\mathsf{Reset}}$ we find this
is equal to
$\sem{\reset{\markingsn{\syntactic}{}}{\varphi_1 \land
\varphi_2}}{\ensuremath{\eta}}{\ensuremath{\delta}_r}$.
\item $\rhs{X} = \varphi_1 \lor \varphi_2$. Analogous to the previous case.
\end{compactitem}
Hence we find that
$\sem{\reset{\markingsn{\syntactic}{}}{\rhs{X}}}{\ensuremath{\eta}}{\ensuremath{\delta}_{r}}
= \sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}_{r}}$.
It now follows immediately from the observation that
$\markrel{\markingsn{\syntactic}{}}$ is a consistent correlation, and
Definition~\ref{def:reset}, that
$\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}_{r}} =
\sem{\rhs{X}}{\ensuremath{\eta}}{\ensuremath{\delta}[\subst{\var{c}}{\semval{v}},\subst{\var{d}}{\semval{w}}]}$.
Our result follows by transitivity of $=$. \qed
\end{proof}
The theory of consistent correlations now gives an immediate proof of soundness
of resetting dead parameters, which is formalised by the following
theorem.
\begin{theorem}
\label{thm:resetSound}
Let $\ensuremath{\mathcal{E}}$ be a PBES, with control flow graph $(V, \to)$ and labelling
$\markingsn{}{}$\!. For all $X$, $\val{v}$ and $\val{w}$:
$$\sem{\ensuremath{\mathcal{E}}}{}{}(X(\sem{\val{v}},\sem{\val{w}})) =
\sem{\reset{\markingsn{\syntactic}{}}{\ensuremath{\mathcal{E}}}}{}{}(\changed{X}(\sem{\val{v}},\sem{\val{w}})).$$
\end{theorem}
\begin{proof}
Relation $R^{\ensuremath{\mathsf{Reset}}}$ is a consistent correlation, as witnessed by
Proposition~\ref{prop:resetCc}. From Theorem~\ref{thm:cc} the result
now follows immediately.\qed
\end{proof}
}
\paperonly{
\begin{definition}
\label{def:reset}
Let $\ensuremath{\mathcal{E}}$ be a PBES, let $(V, \to)$ be its CFG, with labelling
$\markingsn{\syntactic}{}$\!. The PBES
$\reset{\markingsn{\syntactic}{}}{\ensuremath{\mathcal{E}}}$ is obtained
from $\ensuremath{\mathcal{E}}$ by replacing every PVI
$X(\val{e},\val{e'})$ in every $\rhs{X}$ of $\ensuremath{\mathcal{E}}$ by the formula
$\bigwedge_{\val{v} \in \values{\var{c}}} (\val{v} \not= \val{e} \vee
X(\val{e}, \resetvars{(X,\val{v})}{\markingsn{\syntactic}{}}{\val{e'}}))$.
The function $\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{e'}}$ is
defined
positionally as follows:
$$
\text{if $\var[i]{d} \in \marking{\markingsn{}{}}{X,\val{v}}$ we set }
\ind{\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{e'}}}{i} =
\val[\!\!i]{e'},
\text{ else }
\ind{\resetvars{(X, \val{v})}{\markingsn{\syntactic}{}}{\val{e'}}}{i} =
\init{\var[i]{d}}.
$$
\end{definition}
Resetting dead parameters preserves the solution of the PBES, as we claim below.
\begin{restatable}{theorem}{resetSound}
\label{thm:resetSound}
Let $\ensuremath{\mathcal{E}}$ be a PBES, and
$\markingsn{}{}$ a labelling. For all predicate variables $X$, and ground terms
$\val{v}$ and $\val{w}$:
$\sem{\ensuremath{\mathcal{E}}}{}{}(X(\sem{\val{v}},\sem{\val{w}})) =
\sem{\reset{\markingsn{\syntactic}{}}{\ensuremath{\mathcal{E}}}}{}{}(X(\sem{\val{v}},\sem{\val{w}}))$.
\end{restatable}
}
As a consequence of the above theorem, instantiation of a PBES may become
feasible where this was not the case for the original PBES.
This is nicely illustrated by our running example, which now indeed
can be instantiated to a BES.
\begin{example}\label{exa:reset} Observe that parameter
$k^Z$ is not labelled in any of the $Z$ locations. This means that
$X$'s right-hand side essentially changes to:
$$
\begin{array}{c}
( i \not= 1 \vee j \not= 1 \vee X(2,j,k,l+1)) \wedge \\
\forall m\colon\sort{N}. (i \not= 1 \vee Z(i,2,1,k)) \wedge
\forall m\colon\sort{N}. (i \not= 2 \vee Z(i,2,1,k))
\end{array}
$$
Since variable $m$ no longer occurs in the above formula, the
quantifier can be eliminated. Applying the reset function on the
entire PBES leads to a PBES that we \emph{can} instantiate to a BES
(in contrast to the original PBES),
allowing us to compute that the
solution to $X(1,1,1,1)$ is $\ensuremath{\mathit{true}}$.
This BES has only 7 equations.
\end{example}
\section{Optimisation}\label{sec:local}
Constructing a CFG can suffer from a combinatorial explosion; \emph{e.g.}\xspace,
the size of the CFG underlying the following PBES
is exponential in the number of detected CFPs.
\[
\begin{array}{lcl}
\nu X(i_1,\dots,i_n \colon \sort{B}) & =&
(i_1 \wedge X(\ensuremath{\mathit{false}}, \dots, i_n)) \vee
(\neg i_1 \wedge X(\ensuremath{\mathit{true}}, \dots, i_n)) \vee \\
&\dots \vee
& (i_n \wedge X(i_1, \dots, \ensuremath{\mathit{false}})) \vee
(\neg i_n \wedge X(i_1, \dots, \ensuremath{\mathit{true}}))
\end{array}
\]
In this section we develop an alternative to the analysis of the
previous section which mitigates the combinatorial explosion
but still yields sound results. The
correctness of our alternative is based on the following proposition,
which states that resetting using any labelling that approximates that of
Def.~\ref{def:markingHeuristic} is sound.
\begin{proposition}\label{prop:approx}
Let, for given PBES $\ensuremath{\mathcal{E}}$, $(V^{\semantic}syn, {\smash{\xrightarrow{\semantic}syn}})$ be a
CFG with labelling $\markingsn{\syntactic}{}$, and let $L'$ be
a labelling such that $\marking{\markingsn{\syntactic}{}}{X,\val{v}} \subseteq
L'(X,\val{v})$ for all $(X, \val{v})$. Then for all $X, \val{v}$ and
$\val{w}$:
$\sem{\ensuremath{\mathcal{E}}}{}{}(X(\sem{\val{v}},\sem{\val{w}})) =
\sem{\reset{L'\!}{\ensuremath{\mathcal{E}}}}{}{}(X(\sem{\val{v}},\sem{\val{w}}))$
\end{proposition}
\reportonly{
\begin{proof}
Let $(V^{\semantic}syn, {\smash{\xrightarrow{\semantic}syn}})$ be a
CFG with labelling $\markingsn{\syntactic}{}$, and let $L'$ be
a labelling such that $\marking{\markingsn{\syntactic}{}}{X,\val{v}}
\subseteq L'(X,\val{v})$ for all $(X, \val{v})$.
Define relation $R^{\ensuremath{\mathsf{Reset}}}_{\markingsn{\syntactic}{},L'}$ as follows.
$$
\begin{cases}
X(\semval{v}, \semval{w})) R^{\ensuremath{\mathsf{Reset}}}_{L,L'} \changed{X}(\semval{v},
\sem{\resetvars{(X, \val{v})}{L'}{\val{w})}}{}{}) \\
X(\semval{v}, \semval{w}) R^{\ensuremath{\mathsf{Reset}}}_{L,L'} X(\semval{v}, \semval{w'}) &
\text{if }
X(\semval{v}, \sem{\val{w}}{}{}) \markrel{\markingsn{\syntactic}{}}
X(\semval{v}, \semval{w'})
\end{cases}
$$
The proof using $R^{\ensuremath{\mathsf{Reset}}}_{L,L'}$ now follows the exact same line of
reasoning as the proof of Theorem~\ref{thm:resetSound}.\qed
\end{proof}
}
The idea is to analyse a CFG consisting of disjoint subgraphs for
each individual CFP, where each
subgraph captures which PVIs are under the control of a CFP: only if the
CFP can confirm whether a predicate formula potentially depends on
a PVI, there will be an edge in the graph.
As before, let
$\ensuremath{\mathcal{E}}$ be an arbitrary but fixed PBES, $(\ensuremath{\mathsf{source}}\xspace, \ensuremath{\mathsf{target}}\xspace,
\ensuremath{\mathsf{copy}}\xspace)$ a unicity constraint derived from $\ensuremath{\mathcal{E}}$, and
$\var{c}$ a vector of CFPs.
\begin{definition}
\label{def:localCFGHeuristic}
The \emph{local} control flow graph (LCFG) is a graph $(V^{\semantic}loc, \xrightarrow{\semantic}loc)$ with:
\begin{compactitem}
\item $V^{\semantic}loc = \{ (X, n, v) \mid X \in \bnd{\ensuremath{\mathcal{E}}} \land
n \le |\var{c}| \land v \in \values{\var[n]{c}} \}$, and
\item $\xrightarrow{\semantic}loc \subseteq V^{\semantic}loc \times \mathbb{N} \times V^{\semantic}loc$ is the
least relation
satisfying $(X,n,v) \xrightarrow{\semantic}loc[i] (\predphi{\rhs{X}}{i},n,w)$ if:
\begin{compactitem}
\item $\source{X}{i}{n} = v$ and $\dest{X}{i}{n} = w$, or
\item $\source{X}{i}{n} = \bot$, $\predphi{\rhs{X}}{i} \neq X$ and
$\dest{X}{i}{n} = w$, or
\item $\source{X}{i}{n} = \bot$, $\predphi{\rhs{X}}{i} \neq X$ and
$\copied{X}{i}{n} = n$ and $v = w$.
\end{compactitem}
\end{compactitem}
\end{definition}
We write $(X, n, v) \xrightarrow{\semantic}loc[i]$ if there exists some $(Y, m, w)$ such that
$(X, n, v) \xrightarrow{\semantic}loc[i] (Y, m, w)$.
Note that the size of an LCFG is $\mathcal{O}(|\bnd{\ensuremath{\mathcal{E}}}| \times |\var{c}|
\times \max\{ |\values{\var[k]{c}}| ~\mid~ 0 \leq k \leq |\var{c}| \})$.
We next describe how to label the LCFG in such a way that the
labelling meets the condition of Proposition~\ref{prop:approx},
ensuring soundness of our liveness analysis. The idea of using LCFGs
is that in practice, the use and alteration of a data parameter is entirely
determined by a single CFP, and that only on ``synchronisation points''
of two CFPs (when the values of the two CFPs are such that they
both confirm that a formula may depend on the same PVI) there is
exchange of information in the data parameters.
We first formalise when a
data parameter is involved in a recursion (\emph{i.e.}\xspace, when the parameter may
affect whether a formula depends on a PVI, or when a PVI may modify
the data parameter through a self-dependency or uses it to change another parameter).
Let $X \in \bnd{\ensuremath{\mathcal{E}}}$ be an arbitrary bound predicate variable in the
PBES $\ensuremath{\mathcal{E}}$.
\begin{definition}
\label{def:used}
\label{def:changed} Denote $\predinstphi{\rhs{X}}{i}$ by $Y(\var{e})$.
Parameter $\var[j]{d} \in \param{X}$ is:
\begin{compactitem}
\item \emph{used for}
$Y(\var{e})$
if $\var[j]{d} \in \free{\guard{i}{\rhs{X}}}$;
\item \emph{used in}
$Y(\var{e})$ if for some $k$, we have $\var[j]{d} \in \free{\var[k]{e}}$,
($k \not=j$ if
$X =Y$)
;
\item
\emph{changed} by
$Y(\var{e})$ if both $X = Y$ and
$\var[j]{d} \neq \var[j]{e}$.
\end{compactitem}
\end{definition}
We say that a data parameter \emph{belongs to} a CFP if it controls
its complete dataflow.
\begin{definition}
\label{def:belongsTo}
\label{def:rules}
CFP $\var[j]{c}$ \emph{rules}
$\predinstphi{\rhs{X}}{i}$ if $(X, j, v) \xrightarrow{\semantic}loc[i]$ for some $v$.
Let $d \in \param{X} \cap \varset{D}^{\mathit{DP}}$ be a data parameter;
$d$ \emph{belongs to} $\var[j]{c}$ if and only if:
\begin{compactitem}
\item whenever $d$ is
used for \emph{or} in $\predinstphi{\rhs{X}}{i}$, $\var[j]{c}$ rules
$\predinstphi{\rhs{X}}{i}$, and
\item whenever $d$ is changed by $\predinstphi{\rhs{X}}{i}$,
$\var[j]{c}$ rules $\predinstphi{\rhs{X}}{i}$.
\end{compactitem}
The set of data parameters that belong to $\var[j]{c}$ is denoted
by $\belongsto{\var[j]{c}}$.
\end{definition}
By adding dummy CFPs that can only take on one value, we can ensure that
every data parameter belongs to at least one CFP.
For simplicity and without loss of generality, we can therefore
continue to work under the following assumption.
\begin{assumption}\label{ass:belongs}
Each data parameter in an equation
belongs to at least one CFP.
\end{assumption}
We next describe how to conduct the liveness analysis using the
LCFG. Every live data parameter is only labelled in those subgraphs
corresponding to the CFPs to which it belongs. The labelling
itself is constructed in much the same way as was done in the previous
section.
Our base case labels a vertex $(X, n, v)$ with those parameters
that belong to the CFP and that are significant in $\rhs{X}$ when
$\var[n]{c}$ has value $v$.
The backwards reachability now dinstinguishes two cases,
based on whether the influence on live variables is internal to the CFP
or via an external CFP.
\begin{definition}
\label{def:relevanceLocalHeuristic}
Let $(V^{\semantic}loc\!, \xrightarrow{\semantic}loc)$ be a LCFG for PBES
$\ensuremath{\mathcal{E}}$. The labelling $\markingsn{\mathit{l}}{} \colon V^{\semantic}loc
\to \mathbb{P}(\varset{D}^{\mathit{DP}})$
is defined as
$\marking{\markingsn{\mathit{l}}{}}{X, n, v} = \bigcup_{k \in \ensuremath{\mathbb{N}}} \marking{\markingsn{\mathit{l}}{k}}{X, n, v}$,
with
$\markingsn{\mathit{l}}{k}$ inductively defined as:
\[
\begin{array}{ll}
\marking{\markingsn{\mathit{l}}{0}}{X, n, v} & =
\{ d \in \belongsto{\var[n]{c}} \mid d \in
\significant{\simplify{\rhs{X}[\var[n]{c} := v]}} \} \\
\marking{\markingsn{\mathit{l}}{k+1}}{X, n, v} & =
\marking{\markingsn{\mathit{l}}{k}}{X, n, v} \\
& \quad \cup \{ d \in \belongsto{\var[n]{c}} \mid
\exists i,w \text{ such that }~ \exists \var[\!\!\!\!\ell]{{d}^{Y}} \in
\marking{\markingsn{\mathit{l}}{k}}{Y,n,w}: \\
& \qquad (X, n, v) \xrightarrow{\semantic}loc[i] (Y, n, w) \land
\affects{d}{\dataphi[\ell]{\rhs{X}}{i}} \} \\
& \quad \cup \{ d \in \belongsto{\var[n]{c}} \mid
\exists i,m,v',w' \text{ such that } (X, n, v) \xrightarrow{\semantic}loc[i] \\
& \qquad \land\ \exists \var[\!\!\!\!\ell]{d^Y} \in
\marking{\markingsn{\mathit{l}}{k}}{Y, m, w'}: \var[\!\!\!\!\ell]{d^Y} \not \in
\belongsto{\var[n]{c}} \\
& \qquad \land\ (X,m,v') \xrightarrow{\semantic}loc[i] (Y,m,w') \land
\affects{d}{\dataphi[\ell]{\rhs{X}}{i}} \}
\end{array}
\]
\end{definition}
On top of this labelling we define the induced labelling
$\marking{\markingsn{\mathit{l}}{}}{X, \val{v}}$, defined as $d \in
\marking{\markingsn{\mathit{l}}{}}{X, \val{v}}$ iff for all $k$ for
which $d \in \belongsto{\var[k]{c}}$ we have $d \in
\marking{\markingsn{\mathit{l}}{}}{X, k, \val[k]{v}}$. This labelling
over-approximates the labelling of Def.~\ref{def:markingHeuristic}; \emph{i.e.}\xspace, we
have $\marking{\markingsn{\syntactic}{}}{X,\val{v}} \subseteq
\marking{\markingsn{\mathit{l}}{}}{X,\val{v}}$ for all $(X,\val{v})$.
\reportonly{
We formalise this in the following lemma.
\begin{lemma}
Let, for given PBES $\ensuremath{\mathcal{E}}$, $(V^{\semantic}syn, {\smash{\xrightarrow{\semantic}syn}})$ be a global
control flow graph with
labelling $\markingsn{\syntactic}{}$, and let $(V^{\semantic}loc, \xrightarrow{\semantic}loc)$ be
a local control flow graph with
labelling $\markingsn{\mathit{l}}{}$, that has been lifted to the global CFG. Then
$\marking{\markingsn{\syntactic}{}}{X,\val{v}} \subseteq
\marking{\markingsn{\mathit{l}}{}}{X,\val{v}}$ for all $(X,\val{v})$.
\end{lemma}
\begin{proof}
We prove the more general statement that for all natural numbers $n$ it
holds
that
$\forall (X, \val{v}) \in V^{\semantic}syn, \forall d \in
\marking{\markingsn{\syntactic}{n}}{X,
\val{v}}: (\forall j: d \in \belongsto{\var[j]{c}} \implies
d \in \marking{\markingsn{\mathit{l}}{n}}{X, j, \val[j]{v}})$. The lemma
then is an immediate consequence.
We proceed by induction on $n$.
\begin{compactitem}
\item $n = 0$. Let $(X, \val{v})$ and $d \in
\marking{\markingsn{\syntactic}{0}}{X, \val{v}}$
be arbitrary. We need to show that $\forall j: d \in
\belongsto{\var[j]{c}}
\implies d \in \marking{\markingsn{\mathit{l}}{0}}{X, j, \val[j]{v}}$.
Let $j$ be arbitrary such that $d \in \belongsto{\var[j]{c}}$.
Since $d \in \marking{\markingsn{\syntactic}{0}}{X, \val{v}}$, by
definition
$d \in \significant{\simplify{\rhs{X}[\var{c} := \val{v}]}}$, hence also
$d \in \significant{\simplify{\rhs{X}[\var[j]{c} :=
\val[j]{v}}]}$.
Combined with the assumption that $d \in \belongsto{\var[j]{c}}$,
this gives us $d \in \marking{\markingsn{\mathit{l}}{0}}{X, j, \val[j]{v}}$
according to Definition~\ref{def:localCFGHeuristic}.
\item $n = m + 1$. As induction hypothesis assume for all $(X, \val{v})
\in V$:
\begin{equation}\label{eq:IHlocalapprox}
\forall d: d \in
\marking{\markingsn{\syntactic}{m}}{X, \val{v}}
\implies (\forall j: d \in \belongsto{\var[j]{c}}
\implies d \in \marking{\markingsn{\mathit{l}}{m}}{X,j,\val[j]{v}}).
\end{equation}
Let $(X, \val{v})$ be arbitrary with
$d \in \marking{\markingsn{\syntactic}{m+1}}{X, \val{v}}$. Also let $j$
be arbitrary, and assume that $d \in \belongsto{\var[j]{c}}$.
We show that $d \in
\marking{\markingsn{\mathit{l}}{m+1}}{X,j,\val[j]{v}}$ by distinguishing the
cases of Definition~\ref{def:markingHeuristic}.
If $d \in \marking{\markingsn{\syntactic}{m}}{X, \val{v}}$ the
result follows immediately from the induction hypothesis. For the second
case, suppose there are $i \in \mathbb{N}$ and $(Y, \val{w}) \in V$ such
that $(X, \val{v}) \smash{\xrightarrow{\semantic}syn[i]} (Y, \val{w})$,
also assume there is some $\var[\ell]{d} \in
\marking{\markingsn{\syntactic}{m}}{Y, \val{w}}$
with $d \in \free{\dataphi[\ell]{\rhs{X}}{i}}$. Let
$i$ and $\var[\ell]{d}$ be such, and observe that
$Y = \predphi{\rhs{X}}{i}$ and $i \leq \npred{\rhs{X}}$.
According to the induction hypothesis,
$\forall k: \var[\ell]{d} \in \belongsto{\var[k]{c}}
\implies \var[\ell]{d} \in
\marking{\markingsn{\mathit{l}}{m}}{Y, k,
\val[k]{w}}$.
We distinguish two cases.
\begin{compactitem}
\item $\var[\ell]{d}$ belongs to $\var[j]{c}$. According to
\eqref{eq:IHlocalapprox}, we know
$\var[\ell]{d} \in
\marking{\markingsn{\mathit{l}}{m}}{Y, j,
\val[j]{w}}$.
Since $d \in \free{\dataphi[\ell]{\rhs{X}}{i}}$,
we only need to show that $(X, j, \val[j]{v}) \xrightarrow{\semantic}loc[i]
(Y, j, \val[j]{w})$.
We distinguish the cases for $j$ from
Definition~\ref{def:globalCFGHeuristic}.
\begin{compactitem}
\item $\source{X}{i}{j} = \val[j]{v}$ and $\dest{X}{i}{j} =
\val[j]{w}$,
then according to Definition~\ref{def:localCFGHeuristic} $(X, j,
\val[j]{v}) \xrightarrow{\semantic}loc[i]
(Y, j, \val[k]{w})$ .
\item $\source{X}{i}{j} = \bot$, $\copied{X}{i}{j} = j$ and
$\val[j]{v} = \val[j]{w}$.
In case $Y \neq X$ the edge exists locally, and
we are done.
Now suppose that $Y = X$. Then
$\predinstphi{\rhs{X}}{i}$
is not ruled by $\var[j]{c}$. Furthermore, $\var[\ell]{d}$
is changed in $\predinstphi{\rhs{X}}{i}$, hence
$\var[\ell]{d}$
cannot belong to $\var[j]{c}$, which is a contradiction.
\item $\source{X}{i}{j} = \bot$, $\copied{X}{i}{j} = \bot$ and
$\dest{X}{i}{j} = \val[j]{w}$. This is completely analogous to
the previous case.
\end{compactitem}
\item $\var[\ell]{d}$ does not belong to $\var[j]{c}$.
Recall that there must be some $\var[k]{c}$ such that
$\var[\ell]{d}$ belongs to $\var[k]{c}$, and by assumption now
$\var[\ell]{d}$ does not belong to $\var[j]{c}$. Then
according to Definition~\ref{def:relevanceLocalHeuristic}, $d$ is
marked in $\marking{\markingsn{\mathit{l}}{m+1}}{X, j, \val[j]{v}}$,
provided that $(X, j, \val[j]{v}) \xrightarrow{\semantic}loc[i]$ and $(X, k, v')
\xrightarrow{\semantic}loc[i] (Y, k, \val[k]{w})$ for some
$v'$. Let $v' = \val[k]{v}$ and $w' = \val[j]{w}$, according to the
exact same reasoning as
before, the existence of the edges $(X,j,\val[j]{v}) \xrightarrow{\semantic}loc[i] (Y, j,
\val[j]{w})$ and $(X, k, \val[k]{v}) \xrightarrow{\semantic}loc[i] (Y,
k, \val[k]{w})$ can be shown, completing the proof.\qed
\end{compactitem}
\end{compactitem}
\end{proof}
}
Combined with Prop.~\ref{prop:approx}, this leads to the following
theorem.
\begin{theorem}
We have
$\sem{\ensuremath{\mathcal{E}}}(X(\semval{v}, \semval{w})) =
\sem{\reset{\markingsn{\mathit{l}}{}}{\ensuremath{\mathcal{E}}}}{}{}(\changed{X}(\sem{\val{v}},\sem{\val{w}}))$
for all
predicate variables $X$ and ground terms $\val{v}$ and $\val{w}$.
\end{theorem}
The induced labelling $\markingsn{\mathit{l}}{}$ can remain
implicit; in an implementation, the labelling constructed
by Def.~\ref{def:relevanceLocalHeuristic} can be used directly, sidestepping a
combinatorial explosion.
\section{Case Studies}\label{sec:experiments}
We implemented our techniques
in the tool \texttt{pbesstategraph} of the mCRL2
toolset~\cite{Cra+:13}. Here, we report on the tool's effectiveness
in simplifying the PBESs originating from model checking problems and
behavioural equivalence checking problems: we compare sizes of the BESs
underlying the original PBESs to those for the PBESs obtained after
running the tool \texttt{pbesparelm} (implementing the techniques
from~\cite{OWW:09}) and those for the PBESs obtained after running
our tool. Furthermore, we compare the total times needed for reducing the PBES,
instantiating it into a BES, and solving this BES.
\begin{table}[!ht]
\small
\caption{Sizes of the BESs underlying (1) the original PBESs, and the
reduced PBESs using (2)
\texttt{pbesparelm}, (3) \texttt{pbesstategraph} (global)
and (4) \texttt{pbesstategraph} (local).
For the original PBES, we report the number of generated BES equations,
and the time required for generating and
solving the resulting BES. For the other PBESs, we state the total
reduction in percentages (\emph{i.e.}\xspace, $100*(|original|-|reduced|)/|original|$),
and the reduction of the times (in percentages, computed in the same way),
where for times we additionally include the \texttt{pbesstategraph/parelm}
running times.
Verdict $\surd$ indicates the problem
has solution $\ensuremath{\mathit{true}}$; $\times$ indicates it is $\ensuremath{\mathit{false}}$.
}
\label{tab:results}
\centering
\scriptsize
\begin{tabular}{lc@{\hspace{5pt}}|@{\hspace{5pt}}rrrr@{\hspace{5pt}}|@{\hspace{5pt}}rrrr@{\hspace{5pt}}|@{\hspace{5pt}}c@{}}
& \multicolumn{1}{c@{\hspace{10pt}}}{} & \multicolumn{4}{c}{Sizes} &
\multicolumn{4}{c}{Times}&Verdict\\
\cmidrule(r){3-6}
\cmidrule(r){7-10}
\cmidrule{11-11}\\[-1.5ex]
& \multicolumn{1}{c}{} & \multicolumn{1}{@{\hspace{5pt}}c}{Original} &
\multicolumn{1}{c}{\texttt{parelm}} &
\multicolumn{1}{c}{\texttt{st.graph}} &
\multicolumn{1}{c@{\hspace{5pt}}}{\texttt{st.graph}} &
\multicolumn{1}{c}{Original}
& \multicolumn{1}{c}{\texttt{parelm}} & \multicolumn{1}{c}{\texttt{st.graph}} &
\multicolumn{1}{c@{\hspace{5pt}}}{\texttt{st.graph}} & \\
& \multicolumn{1}{c@{\hspace{10pt}}}{$|D|$}
&&& \multicolumn{1}{c}{\texttt{(global)}} &
\multicolumn{1}{c}{\texttt{(local)}} &&& \multicolumn{1}{c}{\texttt{(global)}}
&
\multicolumn{1}{c}{\texttt{(local)}}& \\
\\[-1ex]
\toprule
\\[-1ex]
\multicolumn{4}{c}{Model Checking Problems} \\
\cmidrule{1-4} \\[-1ex]
\multicolumn{11}{l}{\textbf{No deadlock}} \\[.5ex]
\emph{Onebit} & $2$ & 81,921 & 86\% & 89\% & 89\% & 15.7 & 90\% & 85\% & 90\% & $\surd$ \\
& $4$ & 742,401 & 98\% & 99\% & 99\% & 188.5 & 99\% & 99\% & 99\% & $\surd$ \\
\emph{Hesselink} & $2$ & 540,737 & 100\% & 100\% & 100\% & 64.9 & 99\% & 95\% & 99\% & $\surd$ \\
& $3$ & 13,834,801 & 100\% & 100\% & 100\% & 2776.3 & 100\% & 100\% & 100\% & $\surd$ \\
\\[-1ex]
\multicolumn{11}{l}{\textbf{No spontaneous generation of messages}} \\[.5ex]
\emph{Onebit} & $2$ & 185,089 & 83\% & 88\% & 88\% & 36.4 & 87\% & 85\% & 88\% & $\surd$ \\
& $4$ & 5,588,481 & 98\% & 99\% & 99\% & 1178.4 & 99\% & 99\% & 99\% & $\surd$ \\
\\[-1ex]
\multicolumn{11}{l}{\textbf{Messages that are read are inevitably sent}}
\\[.5ex]
\emph{Onebit} & $2$ & 153,985 & 63\% & 73\% & 73\% & 30.8 & 70\% & 62\% & 73\% & $\times$ \\
& $4$ & 1,549,057 & 88\% & 92\% & 92\% & 369.6 & 89\% & 90\% & 92\% & $\times$ \\
\\[-1ex]
\multicolumn{11}{l}{\textbf{Messages can overtake one another}} \\[.5ex]
\emph{Onebit} & $2$ & 164,353 & 63\% & 73\% & 70\% & 36.4 & 70\% & 67\% & 79\% & $\times$ \\
& $4$ & 1,735,681 & 88\% & 92\% & 90\% & 332.0 & 88\% & 88\% & 90\% & $\times$ \\
\\[-1ex]
\multicolumn{11}{l}{\textbf{Values written to the register can be read}}
\\[.5ex]
\emph{Hesselink} & $2$ & 1,093,761 & 1\% & 92\% & 92\% & 132.8 & -3\% & 90\% & 91\% & $\surd$ \\
& $3$ & 27,876,961 & 1\% & 98\% & 98\% & 5362.9 & 25\% & 98\% & 99\% & $\surd$ \\
\\[-1ex]
\multicolumn{4}{c}{Equivalence Checking Problems} \\
\cmidrule{1-4} \\[-1ex]
\multicolumn{11}{l}{\textbf{Branching bisimulation equivalence}} \\[.5ex]
\emph{ABP-CABP} & $2$ & 31,265 & 0\% & 3\% & 0\% & 3.9 & -4\% & -1880\% & -167\% & $\surd$ \\
& $4$ & 73,665 & 0\% & 5\% & 0\% & 8.7 & -7\% & -1410\% & -72\% & $\surd$ \\
\emph{Buf-Onebit} & $2$ & 844,033 & 16\% & 23\% & 23\% & 112.1 & 30\% & 28\% & 31\% & $\surd$ \\
& $4$ & 8,754,689 & 32\% & 44\% & 44\% & 1344.6 & 35\% & 44\% & 37\% & $\surd$ \\
\emph{Hesselink I-S} & $2$ & 21,062,529 & 0\% & 93\% & 93\% & 4133.6 & 0\% & 74\% & 91\% & $\times$ \\
\\[-1ex]
\multicolumn{11}{l}{\textbf{Weak bisimulation equivalence}} \\[.5ex]
\emph{ABP-CABP} & $2$ & 50,713 & 2\% & 6\% & 2\% & 5.3 & 2\% & -1338\% & -136\% & $\surd$ \\
& $4$ & 117,337 & 3\% & 10\% & 3\% & 13.0 & 4\% & -862\% & -75\% & $\surd$ \\
\emph{Buf-Onebit} & $2$ & 966,897 & 27\% & 33\% & 33\% & 111.6 & 20\% & 29\% & 28\% & $\surd$ \\
& $4$ & 9,868,225 & 41\% & 51\% & 51\% & 1531.1 & 34\% & 49\% & 52\% & $\surd$ \\
\emph{Hesselink I-S} & $2$ & 29,868,273 & 4\% & 93\% & 93\% & 5171.7 & 7\% & 79\% & 94\% & $\times$ \\
\\[-1ex]
\bottomrule
\end{tabular}
\end{table}
Our cases are taken from the literature. We here present a selection of the
results. For the model checking problems,
we considered the \emph{Onebit} protocol, which is a complex sliding window
protocol, and Hesselink's handshake register~\cite{Hes:98}.
Both protocols are parametric in the set of values that can be read
and written. A selection of properties of varying complexity and
varying nesting degree, expressed in the data-enhanced modal
$\mu$-calculus are checked.\footnote{\reportonly{The formulae are contained in
the appendix;}
\paperonly{The formulae are contained in \cite{KWW:13report};} here we
use textual characterisations instead.}
For the behavioural equivalence checking problems, we considered a
number of communication protocols such as the \emph{Alternating Bit
Protocol} (ABP), the \emph{Concurrent Alternating Bit Protocol} (CABP),
a two-place buffer (Buf) and the aforementioned Onebit protocol. Moreover,
we compare an implementation of Hesselink's register to a specification
of the protocol that is correct with respect to trace equivalence (but
for which currently no PBES encoding exists) but not with respect to the
two types of behavioural equivalence checking problems we consider here:
branching bisimilarity and weak bisimilarity.
The experiments were performed on a 64-bit Linux machine with kernel
version 2.6.27, consisting of 28 Intel\textregistered\ Xeon\copyright\ E5520
Processors running
at 2.27GHz, and 1TB of shared main memory. None of our experiments use
multi-core features. We used revision 12637 of the mCRL2 toolset,
and the complete scripts for our test setup are available at
\url{https://github.com/jkeiren/pbesstategraph-experiments}.
The results are reported in Table~\ref{tab:results};
higher percentages mean better reductions/\-smaller
runtimes.\reportonly{\footnote{The absolute sizes and times are included in the
appendix.}}
The experiments confirm our technique can achieve as much as an
additional reduction of about 97\% over \texttt{pbesparelm}, see the
model checking and equivalence problems
for Hesselink's
register. Compared to the sizes of the BESs underlying the original PBESs,
the reductions can be immense. Furthermore,
reducing the PBES using the local stategraph algorithm, instantiating, and
subsequently solving it is typically faster than using the global stategraph
algorithm,
even when the reduction achieved by the first is less.
For the equivalence checking
cases, when no reduction is achieved the local version of stategraph sometimes
results in substantially larger running times than parelm, which in turn already
adds an overhead compared to the original; however, for the cases in which this
happens the original running time is around or below 10 seconds, so the
observed increase may be due to inaccuracies in measuring.
\section{Conclusions and Future Work}\label{sec:conclusions}
We described a static analysis technique for PBESs that uses
a notion of control flow to determine when data parameters become
irrelevant. Using this information, the PBES can be simplified, leading
to smaller underlying BESs. Our static analysis technique
enables the solving of PBESs using instantiation that so far could not be solved
this way as shown by our running example.
Compared to existing techniques, our new static analysis technique can lead to
additional reductions of up-to 97\% in practical cases, as illustrated by our
experiments. Furthermore, if a reduction can be achieved the technique can
significantly speed up instantiation and solving, and in case no reduction is
possible, it typically does not negatively impact the total running time.
Several techniques described in this paper can be used
to enhance existing reduction techniques for PBESs. For instance,
our notion of a \emph{guard} of a predicate variable instance
in a PBES can be put to use to cheaply improve on the heuristics for
constant elimination~\cite{OWW:09}. Moreover, we believe that our
(re)construction of control flow graphs from PBESs can be used to
automatically generate invariants for PBESs. The theory on invariants
for PBESs is well-established, but still lacks proper
tool support.
\ifreport
\appendix
\section{$\mu$-calculus formulae}\label{app:experiments}
Below, we list the formulae that were verified in
Section~\ref{sec:experiments}. All formulae are denoted in
the the first order modal $\mu$-calculus, an mCRL2-native data
extension of the modal $\mu$-calculus. The formulae assume that
there is a data specification defining a non-empty sort $D$ of
messages, and a set of parameterised actions that are present
in the protocols. The scripts we used to generate our results,
and the complete data of the experiments are available from
\url{https://github.com/jkeiren/pbesstategraph}-\url{experiments}
\subsection{Onebit protocol verification}
\begin{itemize}
\item No deadlock:
\[
\nu X. [\ensuremath{\mathit{true}}]X \wedge \langle \ensuremath{\mathit{true}} \rangle \ensuremath{\mathit{true}}
\]
Invariantly, over all reachable states at least one action is enabled.
\item Messages that are read are inevitably sent:
\[
\nu X. [\ensuremath{\mathit{true}}]X \wedge \forall d\colon D.[ra(d)]\mu Y.([\overline{sb(d}]Y \wedge \langle \ensuremath{\mathit{true}} \rangle \ensuremath{\mathit{true}}))
\]
The protocol receives messages via action $ra$ and tries to send these
to the other party. The other party can receive these via action $sb$.
\item Messages can be overtaken by other messages:
\[
\begin{array}{ll}
\mu X. \langle \ensuremath{\mathit{true}} \rangle X \vee
\exists d\colon D. \langle ra(d) \rangle
\mu Y. \\
\qquad \ensuremath{\mathbb{B}}ig ( \langle \overline{sb(d)}Y \vee \exists d'\colon D. d \neq d' \wedge
\langle ra(d') \rangle \mu Z. \\
\qquad \qquad ( \langle \overline{sb(d)} \rangle Z \vee
\langle sb(d') \rangle \ensuremath{\mathit{true}}) \\
\qquad \ensuremath{\mathbb{B}}ig )
\end{array}
\]
That is, there is a trace in which message $d$ is read, and is still in the
protocol when another message $d'$ is read, which then is sent to the receiving
party before message $d$.
\item No spontaneous messages are generated:
\[
\begin{array}{ll}
\nu X.
[\overline{\exists d\colon D. ra(d)}]X \wedge\\
\qquad \forall d':D. [ra(d')]\nu Y(m_1\colon D = d'). \\
\qquad\qquad \ensuremath{\mathbb{B}}ig ( [\overline{\exists d:D. ra(d) \vee sb(d) }]Y(m_1) \wedge \\
\qquad\qquad\quad \forall e\colon D.[sb(e)]((m_1 = e) \wedge X) \wedge \\
\qquad\qquad\quad \forall e':D. [ra(e')]\nu Z(m_2\colon D = e'). \\
\qquad\qquad\qquad \ensuremath{\mathbb{B}}ig ([\overline{\exists d \colon D. ra(d) \vee sb(d)}]Z(m_2) \wedge \\
\qquad\qquad\qquad\quad \forall f:D. [sb(f)]((f = m_1) \wedge Y(m_2))\\
\qquad\qquad\qquad\quad \ensuremath{\mathbb{B}}ig )\\
\qquad\qquad \ensuremath{\mathbb{B}}ig )
\end{array}
\]
Since the onebit protocol can contain two messages at a time, the
formula states that only messages that are received can be subsequently
sent again. This requires storing messages that are currently in the buffer
using parameters $m_1$ and $m_2$.
\end{itemize}
\subsection{Hesselink's register}
\begin{itemize}
\item No deadlock:
\[\nu X. [\ensuremath{\mathit{true}}]X \wedge \langle \ensuremath{\mathit{true}} \rangle \ensuremath{\mathit{true}} \]
\item Values that are written to the register can be read from the
register if no other value is written to the register in the meantime.
\[
\begin{array}{l}
\nu X. [\ensuremath{\mathit{true}}]X \wedge
\forall w \colon D . [begin\_write(w)]\nu Y.\\
\qquad\qquad \ensuremath{\mathbb{B}}ig ( [\overline{end\_write}]Y \wedge
[end\_write]\nu Z. \\
\qquad\qquad\qquad \ensuremath{\mathbb{B}}ig( [\overline{\exists d:D.begin\_write(d)}]Z \wedge
[begin\_read]\nu W.\\
\qquad\qquad\qquad\qquad ([\overline{\exists d:D.begin\_write(d)}]W \wedge \\
\qquad\qquad\qquad\qquad\qquad \forall w': D . [end\_read(w')](w = w') ) \\
\qquad\qquad\qquad \ensuremath{\mathbb{B}}ig) \\
\qquad\qquad \ensuremath{\mathbb{B}}ig) \\
\end{array}
\]
\end{itemize}
\section{Absolute sizes and times for the experiments}
\begin{table}[!ht]
\small
\caption{Sizes of the BESs underlying (1) the original PBESs, and the
reduced PBESs using (2)
\texttt{pbesparelm}, (3) \texttt{pbesstategraph} (global)
and (4) \texttt{pbesstategraph} (local).
For each PBES, we report the number of generated BES equations,
and the time required for generating and
solving the resulting BES. For the other PBESs, we additionally include the
\texttt{pbesstategraph/parelm}
running times.
Verdict $\surd$ indicates the problem
has solution $\ensuremath{\mathit{true}}$; $\times$ indicates it is $\ensuremath{\mathit{false}}$.
}
\label{tab:results_absolute}
\input{table_r12637_absolute}
\end{table}
\fi
\end{document} |
\begin{document}
\title{The growth of fine Selmer groups}
\author{Meng Fai Lim}
\email{limmf@mail.ccnu.edu.cn}
\address{School of Mathematics and Statistics, Central China Normal University, No.152, Luoyu Road, Wuhan, Hubei 430079, CHINA}
\author{V. Kumar Murty}
\email{murty@math.toronto.edu}
\address{Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, CANADA}
\date{\today}
\twoheadrightarrownks{Research of VKM partially supported by an NSERC Discovery grant}
\keywords{Fine Selmer groups, abelian variety,
class groups, $p$-rank.}
\mathfrak{m}aketitle
\begin{abstract}
Let $A$ be an abelian variety defined over a number field
$F$. In this paper, we will investigate the growth of the $p$-rank
of the fine Selmer group in three situations. In particular, in
each of these situations, we show that there is a strong analogy
between the growth of the $p$-rank of the fine Selmer group and the
growth of the $p$-rank of the class groups.
\end{abstract}
\section{Introduction}
In the study of rational points on Abelian varieties, the Selmer group plays an important role. In Mazur's fundamental work \cite{Mazur}, the
Iwasawa theory of Selmer groups was introduced. Using this theory, Mazur was able to describe the growth of the
size of the $p$-primary part of the Selmer group in $\mathfrak{m}athds{Z}_p$-towers. Recently, several authors have initiated the study of a certain subgroup, called the fine Selmer group. This subgroup, as well as the `fine' analogues of the Mordell-Weil group and Shafarevich-Tate group, seem to have stronger finiteness properties than the classical Selmer group (respectively,
Mordell-Weil or Shafarevich-Tate groups). The fundamental paper of Coates and Sujatha \cite{CS} explains some of these properties.
\mathfrak{m}edskip
Let $F$ be a number field and $p$ an odd prime. Let $A$ be an Abelian variety defined over $F$ and let $S$ be a finite set
of primes of $F$ including the infinite primes, the primes where $A$ has bad reduction, and the primes of $F$ over $p$.
Fix an algebraic closure $\overline{F}$ of $F$ and denote by $F_S$ the maximal subfield of $\overline{F}$ containing
$F$ which is unramified outside $S$. We set $G = \Gammal(\overline{F}/F)$ and $G_S = \Gammal(F_S/F)$.
\mathfrak{m}edskip
The usual $p^{\infty}$-Selmer group of $A$ is defined by
$$
\Sel_{p^{\infty}}(A/F) = \ker\Big(H^1(G,A[p^{\infty}])\longrightarrow
\bigoplus_{v}H^1(F_v, A)[p^{\infty}]\Big).
$$
Here, $v$ runs through all the primes of $F$ and as usual, for a
$G$-module $M$, we write $H^*(F_v,M)$ for the Galois cohomology of
the decomposition group at $v$. Following \cite{CS}, the
$p^{\infty}$-fine Selmer group of $A$ is defined by
\[
R_{p^\infty}(A/F) = \ker\Big(H^1(G_S(F),A[p^{\infty}])\longrightarrow \bigoplus_{v\in S}H^1(F_v,
A[p^{\infty}])\Big).
\]
This definition is in fact independent of the choice of $S$ as can be seen from the exact sequence (Lemma \ref{indep of S})
$$
0 \longrightarrow R_{p^\infty}(A/F) \longrightarrow \Sel_{p^{\infty}}(A/F)
\longrightarrow \bigoplus_{v|p}A(F_v)\otimesimes\mathfrak{m}athds{Q}p/\mathfrak{m}athds{Z}p.
$$
\mathfrak{m}edskip
Coates and Sujatha study this group over a field $F_\infty$ contained in $F_S$ and for which $\Gammal(F_\infty/F)$
is a $p$-adic Lie group. They set
$$
R_{p^\infty}(A/F_\infty)\ =\ \displaystyle \mathop{\varinjlim}\limits R_{p^\infty}(A/L)
$$
where the inductive limit ranges over finite extensions $L$ of $F$
contained in $F_\infty$. When $F_\infty = F^{cyc}$ is the cyclotomic
$\mathfrak{m}athds{Z}_p$-extension of $F$, they conjecture that the Pontryagin dual
$Y_{p^\infty}(A/F_\infty)$ is a finitely generated $\mathfrak{m}athds{Z}_p$-module.
This is known {\em not} to be true for the dual of the classical
Selmer group. A concrete example of such is the elliptic curve
$E/\mathfrak{m}athds{Q}$ of conductor 11 which is given by
\[ y^2 +y=x^3 -x^2 -10 x- 20. \]
For the prime $p = 5$, it is known that the Pontryagin dual of the
$5^{\infty}$-Selmer group over $\mathfrak{m}athds{Q}^{\mathrm{cyc}}$ is not finitely generated
over $\mathfrak{m}athds{Z}_5$ (see \cite[\S 10, Example 2]{Mazur}). On the other hand
it is expected to be true if the Selmer group is replaced by a
module made out of class groups. Thus, in some sense, the fine
Selmer group seems to approximate the class group. One of the themes
of our paper is to give evidence of that by making the relationship
precise in three instances.
\mathfrak{m}edskip
Coates and Sujatha also study extensions for which $G_\infty = \Gammal(F_\infty/F)$ is a $p$-adic Lie group of
dimension larger than $1$ containing $F^{cyc}$. They make a striking conjecture that the dual
$Y_{p^\infty}(A/F_\infty)$ of the fine Selmer group is pseudo-null as a module over the Iwasawa algebra
$\Lambdambda(G_\infty)$. While we have nothing to say about this conjecture, we investigate the growth of $p$-ranks of
fine Selmer groups in some pro-$p$ towers that are {\em not} $p$-adic analytic.
\mathfrak{m}edskip
\section{Outline of the Paper}
Throughout this paper, $p$ will always denote an odd prime. In the
first situation, we study the growth of the fine Selmer groups over
certain $\mathfrak{m}athds{Z}p$-extensions. It was first observed in \cite{CS} that
over a cyclotomic $\mathfrak{m}athds{Z}p$-extension, the growth of the fine Selmer
group of an abelian variety in a cyclotomic $\mathfrak{m}athds{Z}p$-extension exhibits
phenomena parallel to the growth of the $p$-part of the class groups
over a cyclotomic $\mathfrak{m}athds{Z}p$-extension. Subsequent papers \cite{A, JhS,
LimFine} in this direction has further confirmed this observation.
(Actually, in \cite{JhS, LimFine}, they have also considered the
variation of the fine Selmer group of a more general $p$-adic
representation. In this article, we will only be concerned with the
fine Selmer groups of abelian varieties.) In this paper, we will
show that the growth of the $p$-rank of fine Selmer group of an
abelian variety in a certain class of $\mathfrak{m}athds{Z}p$-extension is determined
by the growth of the $p$-rank of ideal class groups in the
$\mathfrak{m}athds{Z}p$-extension in question (see Theorem \ref{asymptotic compare})
and vice versa. We will also specialize our theorem to the
cyclotomic $\mathfrak{m}athds{Z}p$-extension to recover a theorem of Coates-Sujatha
\cite[Theorem 3.4]{CS}.
\mathfrak{m}edskip
In the second situation, we investigate the growth of the fine
Selmer groups over $\mathfrak{m}athds{Z}/p$-extensions of a fixed number field. We
note that it follows from an application of the Grunwald-Wang
theorem that the $p$-rank of the ideal class groups grows
unboundedly in $\mathfrak{m}athds{Z}/p$-extensions of a fixed number field. Recently,
many authors have made analogous studies in this direction replacing
the ideal class group by the classical Selmer group of an abelian
variety (see \cite{Ba, Br, Ce, Mat09}). In this article, we
investigate the analogous situation for the fine Selmer group of an
abelian variety, and we show that the $p$-rank of the fine Selmer
group of the abelian variety grows unboundedly in $\mathfrak{m}athds{Z}/p$-extensions
of a fixed number field (see Theorem \ref{class Z/p}). Note that the
fine Selmer group is a subgroup of the classical Selmer group, and
therefore, our results will also recover some of their results.
\mathfrak{m}edskip
In the last situation, we consider the growth of the fine Selmer
group in an infinite unramified pro-$p$ extensions. It is known that
the $p$-rank of the class groups is unbounded in such tower under
suitable assumptions. Our result will again show that we have the
same phenomenon for the $p$-rank of fine Selmer groups (see Theorem
\ref{Fine Sel in class tower}). As above, our result will also imply
some of the main results in \cite{LM, Ma, MO}, where analogous
studies in this direction have been made for the classical Selmer
group of an abelian variety.
\section{$p$-rank} \lambdabel{some cohomology lemmas}
In this section, we record some basic results on Galois cohomology
that will be used later. For an abelian group $N$, we define its
$p$-rank to be the $\mathfrak{m}athds{Z}/p$-dimension of $N[p]$ which we denote by
$r_p(N)$. If $G$ is a pro-$p$ group, we write $h_i(G) =
r_p\big(H^i(G,\mathfrak{m}athds{Z}/p)\big)$. We now state the following lemma which
gives an estimate of the $p$-rank of the first cohomology group.
\begin{lem} \lambdabel{cohomology rank inequalities} Let $G$ be a pro-$p$ group,
and let $M$ be a discrete $G$-module which is cofinitely generated
over $\mathfrak{m}athds{Z}p$.
If $h_1(G)$ is finite, then $r_p\big(H^1(G,M)\big)$ is finite, and
we have the following estimates for
$r_p\big(H^1(G,M)\big)$
\[
h_1(G)r_p(M^G) -r_p \big( (M/M^G)^G\big)
\leq r_p\big(H^1(G,M)\big) \leq h_1(G)\big(\mathfrak{m}athrm{corank}_{\mathfrak{m}athds{Z}p}(M)
+ \log_p(\big| M/M_{\mathfrak{m}athrm{div}}\big|)\big).
\]
\end{lem}
\begin{prop}f
See \cite[Lemma
3.2]{LM}. \end{prop}f
We record another useful estimate.
\begin{lem} \lambdabel{estimate lemma}
Let
\[ W \longrightarrow X \longrightarrow Y \longrightarrow Z\]
be an exact sequence of cofinitely generated abelian groups. Then we have
\[ \Big| r_p(X) - r_p(Y) \Big| \leq 2r_p(W) + r_p(Z).\]
\end{lem}
\begin{prop}f It suffices to show the lemma for the exact sequence
\[ 0\longrightarrow W \longrightarrow X
\longrightarrow Y \longrightarrow Z \longrightarrow 0.\]
We break up the exact sequence into two short exact sequences
\[ 0\longrightarrow W \longrightarrow X
\longrightarrow C \longrightarrow 0, \]
\[ 0\longrightarrow C \longrightarrow Y
\longrightarrow Z \longrightarrow 0.\]
From these short exact sequences, we obtain two exact sequences of
finite dimensional $\mathfrak{m}athds{Z}/p$-vector spaces (since $W$, $X$, $Y$ and $Z$
are cofinitely generated abelian groups)
\[ 0\longrightarrow W[p] \longrightarrow X[p]
\longrightarrow C[p] \longrightarrow P \longrightarrow 0 , \]
\[ 0\longrightarrow C[p] \longrightarrow Y[p]\longrightarrow Q
\longrightarrow 0, \]
where $P\subseteq W/p$ and $Q\subseteq Z[p]$. It follows from these
two exact sequences and a straightforward calculation that we have
\[ r_p(X) - r_p(Y) = r_p(W) - r_p(P) - r_p(Q). \]
The inequality of the lemma is immediate from this.
\end{prop}f
\mathfrak{m}edskip
\section{Fine Selmer groups} \lambdabel{Fine Selmer group section}
As before, $p$ will denote an odd prime. Let $A$ be an abelian
variety over a number field $F$. Let $S$ be a finite set of primes
of $F$ which contains the primes above $p$, the primes of bad
reduction of $A$ and the archimedean primes. Denote by $F_S$ the
maximal algebraic extension of $F$ unramified outside $S$. We will
write $G_S(F) = \Gammal(F_S/F)$.
\mathfrak{m}edskip
As stated in the introduction and following
\cite{CS}, the fine Selmer group of $A$ is defined by
\[ R(A/F) = \ker\Big(H^1(G_S(F),A[p^{\infty}])\longrightarrow \bigoplus_{v\in S}H^1(F_v,
A[p^{\infty}])\Big). \]
(Note that we have dropped the subscript $p^\infty$ on $R(A/F)$ as $p$ is fixed.)
\mathfrak{m}edskip
To facilitate further discussion, we also recall the definition of
the classical Selmer group of $A$ which is given by
\[ \Sel_{p^{\infty}}(A/F) = \ker\Big(H^1(F,A[p^{\infty}])\longrightarrow
\bigoplus_{v}H^1(F_v, A)[p^{\infty}]\Big), \] where $v$ runs through
all the primes of $F$. (Note the difference of the position of the
``$[p^{\infty}]$'' in the local cohomology groups in the
definitions.)
\mathfrak{m}edskip
At first viewing, it will seem that the definition of the fine
Selmer group depends on the choice of the set $S$. We shall show
that this is not the case.
\begin{lem} \lambdabel{indep of S}
We have an exact sequence
\[ 0 \longrightarrow R(A/F) \longrightarrow \Sel_{p^{\infty}}(A/F)
\longrightarrow \bigoplus_{v|p}A(F_v)\otimesimes\mathfrak{m}athds{Q}p/\mathfrak{m}athds{Z}p. \]
In particular, the definition of the fine
Selmer group does not depend on the choice of the set $S$. \end{lem}
\begin{prop}f Let $S$ be a finite set of primes of $F$ which contains the
primes above $p$, the primes of bad reduction of $A$ and the
archimedean primes. Then by \cite[Chap. I, Corollary 6.6]{Mi}, we
have the following description of the Selmer group
\[ \Sel_{p^{\infty}}(A/F) = \ker\Big(H^1(G_S(F),A[p^{\infty}])\longrightarrow
\bigoplus_{v\in S}H^1(F_v, A)[p^{\infty}]\Big). \]
Combining this description with the definition of the fine Selmer
group and an easy diagram-chasing argument, we obtain the required
exact sequence (noting that $A(F_v)\otimesimes\mathfrak{m}athds{Q}p/\mathfrak{m}athds{Z}p =0$ for $v\nmid
p$). \end{prop}f
\begin{remark} In \cite{Wu}, Wuthrich used the exact sequence in the lemma for
the definition of the fine Selmer group.
\end{remark}
We end the section with the following simple lemma which gives a
lower bound for the $p$-rank of the fine Selmer group in terms of the
$p$-rank of the $S$-class group. This will be used in Sections
\ref{unboundness} and \ref{unramified pro-p}.
\begin{lem} \lambdabel{lower bound}
Let $A$ be an abelian variety defined over a
number field $F$. Suppose that $A(F)[p]\neq 0$. Then we have
\[r_p\big(R(A/F)\big) \geq r_p(\Cl_S(F))r_p(A(F)[p])-2d, \]
where $d$ denotes the dimension of the abelian variety $A$.
\end{lem}
\begin{prop}f
Let $H_S$ be the $p$-Hilbert $S$-class field of $F$ which, by definition, is the maximal abelian unramified $p$-extension
of $F$ in which all primes in $S$ split completely. Consider the
following diagram
\[ \entrymodifiers={!! <0pt, .8ex>+} \SelectTips{eu}{}
\xymatrix{
0 \ar[r] & R(A/F) \ar[d]^{\alpha} \ar[r] & H^1(G_S(F),A[p^{\infty}]) \ar[d]^{\beta}
\ar[r]^{} & \displaystyle\bigoplus_{v\in S}H^1(F_v,
A[p^{\infty}]) \ar[d]^{\gammamma} \\
0 \ar[r] & R(A/H_S) \ar[r] & H^1(G_S(H_S),A[p^{\infty}])
\ar[r] & \displaystyle\bigoplus_{v\in S}\bigoplus_{w|v}H^1(H_{S,w},
A[p^{\infty}]) }
\]
with exact rows. Here the vertical maps are given by the restriction maps.
Write $\gammamma = \oplus_v \gammamma_v$, where
\[ \gammamma_v : H^1(F_v,
A[p^{\infty}]) \longrightarrow \bigoplus_{w|v}H^1(H_{S,w},
A[p^{\infty}]). \]
It follows from the inflation-restriction sequence that $\ker
\gammamma_v = H^1(G_v,
A(H_{S,v})[p^{\infty}])$, where $G_v$ is the decomposition group of
$\Gammal(H_S/F)$ at $v$. On the other hand, by the definition of $H_S$,
all the primes of $F$ in $S$ split completely in $H_S$, and
therefore, we have $G_v=1$ which in turn implies that $\ker \gammamma
=0$. Similarly, the inflation-restriction sequence gives the
equality $\ker \beta = H^1(\Gammal(H_S/F), A(H_S)[p^{\infty}])$.
Therefore, we obtain an injection
\[
H^1(\Gammal(H_S/F), A(H_S)[p^{\infty}])\hookrightarrow R(A/F) \]
It follows from this injection that we have
\[ r_p(R(A/F)) \geq r_p\big(H^1(\Gammal(H_S/F), A(H_S)[p^{\infty}])\big). \]
By Lemma \ref{cohomology rank inequalities}, the latter quantity is
greater or equal to
\[ h_1(\Gammal(H_S/F))r_p(A(F)[p^{\infty}]) - 2d. \]
By class field theory, we have $\Gammal(H_S/F)\cong \Cl_S(F)$, and
therefore,
\[ h_1(\Gammal(H_S/F)) = r_p(\Cl_S(F)/p) = r_p(\Cl_S(F)),\]
where the last equality follows from the fact that $\Cl_S(F)$ is
finite. The required estimate is now established (and noting that
$r_p(A(F)[p]) = r_p(A(F)[p^{\infty}]))$.
\end{prop}f
\begin{remark}
Since the fine Selmer group is contained in the classical Selmer group
(cf. Lemma \ref{indep of S}), the above estimate also gives a lower
bound for the classical Selmer group. \end{remark}
\section{Growth of fine Selmer groups in a $\mathfrak{m}athds{Z}p$-extension}
\lambdabel{cyclotomic Zp-extension}
As before, $p$ denotes an odd prime. In this section, $F_{\infty}$
will always denote a fixed $\mathfrak{m}athds{Z}p$-extension of $F$. We will denote
$F_n$ to be the subfield of $F_{\infty}$ such that $[F_n : F] =
p^n$. If $S$ is a finite set of primes of $F$, we denote by $S_f$
the set of finite primes in $S$.
\mathfrak{m}edskip
We now state the main theorem of this section which compares the
growth of the fine Selmer groups and the growth of the class groups
in the $\mathfrak{m}athds{Z}p$-extension of $F$. To simplify our discussion, we will
assume that $A[p]\subseteq A(F)$.
\mathfrak{m}edskip
\begin{thm} \lambdabel{asymptotic compare}
Let $A$ be a $d$-dimensional abelian variety defined over a number field
$F$. Let $F_{\infty}$ be a fixed $\mathfrak{m}athds{Z}p$-extension of $F$ such that
the primes of $F$ above $p$ and the bad reduction primes of $A$
decompose finitely in $F_{\infty}/F$. Furthermore, we assume that
$A[p]\subseteq A(F)$. Then we have
\[ \Big|r_p(R(A/F_n)) - 2dr_p(\Cl(F_n))\Big|=
O(1).\]
\end{thm}
In preparation for the proof of the theorem, we require a few
lemmas.
\begin{lem}
Let $F_{\infty}$ be a $\mathfrak{m}athds{Z}p$-extension of $F$ and
let $F_n$ be the subfield of $F_{\infty}$ such that $[F_n : F] =
p^n$. Let $S$ be a given finite set of primes of $F$ which contains
all the primes above $p$ and the archimedean primes. Suppose that
all the primes in $S_f$ decompose finitely in $F_{\infty}/F$. Then
we have
\[ \Big|r_p(\Cl(F_n)) - r_p(\Cl_S(F_n))\Big|=
O(1). \]
\end{lem}
\begin{prop}f
For each $F_n$, we write $S_f(F_n)$ for the set of finite primes of
$F_n$ above $S_f$. For each $n$, we have the following exact
sequence (cf. \cite[Lemma 10.3.12]{NSW})
\[ \mathfrak{m}athds{Z}^{|S_f(F_n)|} \longrightarrow \Cl(F_n)\longrightarrow \Cl_S(F_n)
\longrightarrow 0. \]
Denote by $C_n$ the kernel of $\Cl(F_n)\longrightarrow
\Cl_S(F_n)$. Note that $C_n$ is finite, since it is contained in
$\Cl(F_n)$. Also, it is clear from the above exact sequence that
$r_p(C_n) \leq |S_f(F_n)|$ and $r_p(C_n/p) \leq |S_f(F_n)|$. By
Lemma \ref{estimate lemma}, we have
\[ \Big| r_p(\Cl(F_n)) - r_p(\Cl_S(F_n)) \Big| \leq 3|S_f(F_n)| =
O(1), \]
where the last equality follows from the assumption that
all the primes in $S_f$ decompose finitely in $F_{\infty}/F$. \end{prop}f
\mathfrak{m}edskip
Before stating the next lemma, we introduce the $p$-fine Selmer
group of an abelian variety $A$. Let $S$ be a finite set of primes
of $F$ which contains the primes above $p$, the primes of bad
reduction of $A$ and the archimedean primes. Then the $p$-fine
Selmer group (with respect to $S$) is defined to be
\[ R_S(A[p]/F) = \ker\Big(H^1(G_S(F),A[p])\longrightarrow \bigoplus_{v\in S}H^1(F_v,
A[p])\Big). \] Note that the $p$-fine Selmer group may be dependent
on $S$. In fact, as we will see in the proof of Theorem
\ref{asymptotic compare} below, when $F = F(A[p])$, we have
$R_S(A[p]/F) = \Cl_S(F)[p]^{2d}$, where the latter group is clearly
dependent on $S$.
We can now state the following lemma which compare the growth of
$r_p(R_S(A[p]/F_n))$ and $r_p(R(A/F_n))$.
\begin{lem}
Let $F_{\infty}$ be a $\mathfrak{m}athds{Z}p$-extension of $F$ and
let $F_n$ be the subfield of $F_{\infty}$ such that $[F_n : F] =
p^n$. Let $A$ be an abelian variety defined over $F$. Let $S$ be a
finite set of primes of $F$ which contains the primes above $p$, the
primes of bad reduction of $A$ and the archimedean primes. Suppose
that all the primes in $S_f$ decompose finitely in $F_{\infty}/F$.
Then we have
\[ \Big|r_p(R_S(A[p]/F_n)) -r_p(R(A/F_n))\Big| =
O(1).\]
\end{lem}
\begin{prop}f
We have a commutative diagram
\[ \entrymodifiers={!! <0pt, .8ex>+} \SelectTips{eu}{}
\xymatrix{
0 \ar[r] & R_S(A[p]/F_n) \ar[d]^{s_n} \ar[r] & H^1(G_S(F_n),A[p]) \ar[d]^{h_n}
\ar[r]^{} & \displaystyle\bigoplus_{v_n\in S(F_n)}H^1(F_{n,v_n},
A[p]) \ar[d]^{g_n} \\
0 \ar[r] & R(A/F_n)[p] \ar[r] & H^1(G_S(F_n),A[p^{\infty}])[p]
\ar[r] & \displaystyle\bigoplus_{v_n\in S(F_n)} H^1(F_{n,v_n},
A[p^{\infty}])[p] }
\]
with exact rows. It is an easy exercise to show that the maps
$h_n$ and $g_n$ are surjective, that $\ker h_n =
A(F_n)[p^{\infty}]/p$ and that
\[ \ker g_n = \displaystyle\bigoplus_{v_n\in S(F_n)}A(F_{n,v_n})[p^{\infty}]/p.
\]
Since we are assuming $p$ is odd, we have $r_p(\ker g_n)\leq 2d|S_f(F_n)|$.
By an application of Lemma \ref{estimate lemma}, we have
\[ \begin{array}{rl} \Big| r_p(R_S(A[p]/F_n))) - r_p(R(A/F_n)) \Big|\!
&\leq ~2r_p(\ker s_n) + r_p(\mathrm{coker}\, s_n) \\
& \leq ~ 2r_p(\ker h_n) + r_p(\ker g_n) \\
& \leq ~ 4d + 2d|S_f(F_n)| = O(1),\\
\end{array} \] where the last equality
follows from the assumption that all the primes in $S_f$ decompose
finitely in $F_{\infty}/F$. \end{prop}f
We are in the position to prove our theorem.
\begin{prop}f[Proof of Theorem \ref{asymptotic compare}]
Let $S$ be the
finite set of primes of $F$ consisting precisely of the primes above
$p$, the primes of bad reduction of $A$ and the archimedean primes.
By the hypothesis $A[p]\subseteq A(F)$ ($\subseteq A(F_n)$) of the
theorem, we have $A[p] \cong (\mathfrak{m}athds{Z}/p)^{2d}$ as $G_S(F_n)$-modules.
Therefore, we have $H^1(G_S(F_n),A[p]) = \Hom(G_S(F_n),A[p])$. We
have similar identification for the local cohomology groups, and it
follows that
$$
R_S(A[p]/F_n) = \Hom(\Cl_S(F_n),A[p])\cong \Cl_S(F_n)[p]^{2d}
$$
as abelian groups. Hence we have $r_p(R_S(A[p]/F_n)) = 2d
r_p(\Cl_S(F_n))$. The conclusion of the theorem is now immediate
from this equality and the above two lemmas. \end{prop}f
\begin{cor} \lambdabel{asymptotic compare corollary}
Retain the notations and assumptions of Theorem \ref{asymptotic compare}.
Then we have
\[ r_p(R(A/F_n)) = O(1)\]
if and only if
\[ r_p(\Cl(F_n)) = O(1).\]
\end{cor}
For the remainder of the section, $F_{\infty}$ will be taken to be
the cyclotomic $\mathfrak{m}athds{Z}p$-extension of $F$. As before, we denote by $F_n$
the subfield of $F_{\infty}$ such that $[F_n : F] = p^n$. Denote by
$X_{\infty}$ the Galois group of the maximal abelian unramified
pro-$p$ extension of $F_{\infty}$ over $F_{\infty}$. A well-known
conjecture of Iwasawa asserts that $X_{\infty}$ is finitely
generated over $\mathfrak{m}athds{Z}p$ (see \cite{Iw, Iw2}). We will call this
conjecture the \textit{Iwasawa $\mathfrak{m}u$-invariant conjecture} for
$F_{\infty}$. By \cite[Proposition 13.23]{Wa}, this is also
equivalent to saying that $r_p(\Cl(F_n)/p)$ is bounded independently
of $n$. Now, by the finiteness of class groups, we have
$r_p(\Cl(F_n))= r_p(\Cl(F_n)/p)$. Hence the Iwasawa $\mathfrak{m}u$-invariant
conjecture is equivalent to saying that $r_p(\Cl(F_n))$ is bounded
independently of $n$.
\mathfrak{m}edskip
We consider the analogous situation for the fine Selmer group. Define
$R(A/F_{\infty}) = \displaystyle \mathop{\varinjlim}\limits_nR(A/F_n)$ and denote by $Y(A/F_{\infty})$
the Pontryagin dual of $R(A/F_{\infty})$. We may now recall the
following conjecture which was first introduced in \cite{CS}.
\noindent \textbf{Conjecture A.} For any number field $F$,
$Y(A/F_{\infty})$ is a finitely generated $\mathfrak{m}athds{Z}p$-module, where
$F_{\infty}$ is the cyclotomic $\mathfrak{m}athds{Z}p$-extension of $F$.
\mathfrak{m}edskip
We can now give the proof of \cite[Theorem 3.4]{CS}. For another
alternative approach, see \cite{JhS, LimFine}.
\begin{thm} \lambdabel{Coates-Sujatha}
Let $A$ be a $d$-dimensional abelian variety defined over a number field
$F$ and let $F_{\infty}$ be the cyclotomic $\mathfrak{m}athds{Z}p$-extension of $F$.
Suppose that $F(A[p])$ is a finite $p$-extension of $F$.
Then Conjecture A holds for $A$ over $F_{\infty}$ if and only if
the Iwasawa $\mathfrak{m}u$-invariant conjecture holds for $F_{\infty}$.
\end{thm}
\begin{prop}f
Now if $L'/L$ is a finite $p$-extension, it follows from \cite[Theorem 3]{Iw}
that the Iwasawa $\mathfrak{m}u$-invariant conjecture holds for $L_{\infty}$
if and only if the Iwasawa $\mathfrak{m}u$-invariant conjecture holds for
$L'_{\infty}$. On the other hand, it is not difficult to show that
the map
\[ Y(A/L'_{\infty})_{G}\longrightarrow Y(A/L_{\infty})\]
has finite kernel and cokernel, where $G=\Gammal(L'/L)$. It follows
from this observation that Conjecture A holds for $A$ over
$L_{\infty}$ if and only if $Y(A/L'_{\infty})_{G}$ is finitely
generated over $\mathfrak{m}athds{Z}p$. Since $G$ is a $p$-group, $\mathfrak{m}athds{Z}p[G]$ is local
with a unique maximal (two-sided) ideal $p\mathfrak{m}athds{Z}p[G]+I_G$, where $I_G$
is the augmentation ideal (see \cite[Proposition 5.2.16(iii)]{NSW}).
It is easy to see from this that
\[ Y(A/L'_{\infty})/\mathfrak{m} \cong Y(A/L'_{\infty})_{G}/pY(A/L'_{\infty})_{G}.
\] Therefore, Nakayama's lemma
for $\mathfrak{m}athds{Z}p$-modules tells us that $Y(A/L'_{\infty})_{G}$ is finitely
generated over $\mathfrak{m}athds{Z}p$ if and only if $Y(A/L'_{\infty})/\mathfrak{m}$ is finite.
On the other hand, Nakayama's lemma for $\mathfrak{m}athds{Z}p[G]$-modules tells us
that $Y(A/L'_{\infty})/\mathfrak{m}$ is finite if and only if
$Y(A/L'_{\infty})$ is finitely generated over $\mathfrak{m}athds{Z}p[G]$. But since
$G$ is finite, the latter is equivalent to $Y(A/L'_{\infty})$ being
finitely generated over $\mathfrak{m}athds{Z}p$. Hence we have shown that Conjecture A
holds for $A$ over $L_{\infty}$ if and only if Conjecture A holds
for $A$ over $L_{\infty}'$.
\mathfrak{m}edskip
Therefore, replacing $F$ by $F(A[p])$, we may assume that
$A[p]\subseteq A(F)$. Write $\Gamma_n = \Gammal(F_{\infty}/F_n)$. Consider
the following commutative diagram
\[ \entrymodifiers={!! <0pt, .8ex>+} \SelectTips{eu}{}
\xymatrix{
0 \ar[r] & R(A/F_n) \ar[d]^{r_n} \ar[r] &
H^1(G_S(F_n),A[p^{\infty}]) \ar[d]^{f_n}
\ar[r]^{} & \displaystyle\bigoplus_{v_n\in S(F_n)}H^1(F_{n,v_n},
A[p^{\infty}]) \ar[d]^{\gammamma_n} \\
0 \ar[r] & R(A/F_{\infty})^{\Gamma_n} \ar[r] &
H^1(G_S(F_{\infty}),A[p^{\infty}])^{\Gamma_n}
\ar[r] & \Big(\displaystyle \displaystyle \mathop{\varinjlim}\limits_n\bigoplus_{v_n\in S(F_n)}
H^1(F_{n,v_n}, A[p^{\infty}])\Big)^{\Gamma_n} }
\]
with exact rows, and the vertical maps given by the restriction maps.
It is an easy exercise to show that $r_p(\ker f_n) \leq 2d$,
$r_p(\ker \gammamma_n) \leq 2d|S_f(F_n)|$, and that $f_n$ and
$\gammamma_n$ are surjective. It then follows from these estimates and
Lemma \ref{estimate lemma} that we have
\[ \Big| r_p\big(R(A/F_n))\big) -
r_p\big(R(A/F_{\infty})^{\Gamma_n}\big) \Big| = O(1).
\] Combining this observation with \cite[Lemma 13.20]{Wa}, we have
that Conjecture A holds for $A$ over $F_{\infty}$ if and only if
$r_p(R(A/F_n))=O(1)$. The conclusion of the theorem is now immediate
from Corollary \ref{asymptotic compare corollary}. \end{prop}f
\section{Unboundedness of fine Selmer groups in $\mathfrak{m}athds{Z}/p$-extensions} \lambdabel{unboundness}
In this section, we will study the question of unboundedness of fine
Selmer groups in $\mathfrak{m}athds{Z}/p$-extensions. We first recall the case of
class groups. Since for a number field $L$, the $S$-class
group $\Cl_{S}(L)$ is finite, we have
$r_p(\Cl_{S}(L)) = \dim_{\mathfrak{m}athds{Z}/p}\big(\Cl_S(L)/p\big)$.
\begin{prop} \lambdabel{class Z/p}
Let $S$ be a finite set of primes of $F$ which contains all the
the archimedean primes. Then there exists a sequence $\{L_n\}$ of
distinct number fields such that each $L_n$ is a $\mathfrak{m}athds{Z}/p$-extension of
$F$ and such that
\[ r_p(\Cl_{S}(L_n)) \geq n \] for every $n \geq 1$.
\end{prop}
\begin{prop}f
Denote $r_1$ (resp. $r_2$) be the number of real places of $F$
(resp. the number of the pairs of complex places of $F$). Let $S_1$
be a set of primes of $F$ which contains $S$ and such that
\[ |S_1| \geq |S| + r_1 + r_2 + \delta+1. \]
Here $\delta = 1$ if $F$ contains a primitive $p$-root of unity, and
0 otherwise. By the theorem of Grunwald-Wang (cf. \cite[Theorem
9.2.8]{NSW}), there exists a $\mathfrak{m}athds{Z}/p$-extension $L_1$ of $F$ such that
$L_1/F$ is ramified at all the finite primes of $S_1$ and unramified
outside $S_1$. By \cite[Proposition 10.10.3]{NSW}, we have
\[ r_p(\Cl_{S}(L_1)) \geq |S_1| - |S| -r_1 - r_2 -\delta
\geq 1.\]
Choose $S_2$ to be a set of primes of $F$ which contains
$S_1$ (and hence $S_0$) and which has the property that
\[ |S_2| \geq |S_1| + 1 \geq |S| + r_1 + r_2 + \delta+2. \]
By the theorem of Grunwald-Wang, there exists a $\mathfrak{m}athds{Z}/p$-extension
$L_2$ of $F$ such that $L_2/F$ is ramified at all the finite primes
of $S_2$ and unramified outside $S_2$. In particular, the fields
$L_1$ and $L_2$ are distinct. By an application of \cite[Proposition
10.10.3]{NSW} again, we have
\[ r_p(\Cl_{S}(L_2)) \geq |S_2| - |S| -r_1 - r_2 -\delta
\geq 2.\]
Note that since there are infinitely many primes in $F$, we can always continue
the above process iteratively. Also, it is clear from our choice of
$L_n$, they are mutually distinct. Therefore, we have the required
conclusion. \end{prop}f
For completeness and for ease of later comparison, we record the
following folklore result.
\begin{thm} Let $F$ be a number field. Then we have
\[ \sup\{ r_p\big(\Cl(L)\big)~ |~
\mathfrak{m}box{L/F is a cyclic extension of degree p}\} = \infty \]\end{thm}
\begin{prop}f
Since $\Cl(L)$ surjects onto $\Cl_S(L)$, the theorem follows from
the preceding proposition.
\end{prop}f
We now record the analogous statement for the fine Selmer groups.
\begin{thm} \lambdabel{theorem Z/p} Let $A$ be an abelian variety defined over a
number field $F$. Suppose that $A(F)[p]\neq 0$. Then we have
\[ \sup\{ r_p\big(R(A/L)\big)~ |~
\mathfrak{m}box{L/F is a cyclic extension of degree p}\} = \infty \]\end{thm}
\begin{prop}f
This follows immediately from combining Lemma \ref{lower bound} and
Proposition \ref{class Z/p}. \end{prop}f
In the case, when $A(F)[p]=0$, we have the following weaker
statement.
\begin{cor} \lambdabel{theorem Z/p corollary} Let $A$ be a $d$-dimensional
abelian variety defined over a number field $F$. Suppose that
$A(F)[p]=0$. Define
\[m = \mathfrak{m}in\{[K:F]~|~ A(K)[p]\neq 0\}.\]
Then we have
\[ \sup\{ r_p\big(R(A/L)\big)~ |~
\mathfrak{m}box{L/F is an extension of degree pm}\} = \infty \]\end{cor}
\begin{prop}f
This follows from an application of the previous theorem to
the field $K$. \end{prop}f
\begin{remark} Clearly $1< m\leq |\mathfrak{m}athrm{GL}_{2d}(\mathfrak{m}athds{Z}/p)| = (p^{2d}-1)(p^{2d}
-p)\cdots (p^{2d}-p^{2d-1})$. In fact, we can even do
better\footnote{We thank Christian Wuthrich for pointing this out to
us.}. Write $G=\Gammal(F(A[p])/F)$. Note that this is a subgroup of
$\mathfrak{m}athrm{GL}_{2d}(\mathfrak{m}athds{Z}/p)$. Let $P$ be a nontrivial point in $A[p]$
and denote by $H$ the subgroup of $G$ which fixes $P$. Set $K =
F(A[p])^H$. It is easy to see that $[K:F] = [G:H] = |O_G(P)|$, where
$O_G(P)$ is the orbit of $P$ under the action of $G$. Since $O_G(P)$
is contained in $A[p]\setminus\{0\}$, we have $m \leq [K:F] =
|O_G(P)| \leq p^{2d}-1$. \end{remark}
As mentioned in the introductory section, analogous result to the
above theorem for the classical Selmer groups have been studied (see
\cite{Ba, Br, Ce, K, KS, Mat06, Mat09}). Since the fine Selmer group
is contained in the classical Selmer group (cf. Lemma \ref{indep of
S}), our result recovers the above mentioned results (under our
hypothesis). We note that the work of \cite{Ce} also considered the
cases of a global field of positive characteristic. We should also
mention that in \cite{ClS, Cr}, they have even established the
unboundness of $\Sha(A/L)$ over $\mathfrak{m}athds{Z}/p$-extensions $L$ of $F$ in
certain cases (see also \cite{K, Mat09} for some other related
results in this direction). In view of these results on $\Sha(A/L)$,
one may ask for analogous results for a `fine' Shafarevich-Tate
group.
\mathfrak{m}edskip
Wuthrich \cite{Wu} introduces such a group as follows. One first
defines a `fine' Mordell-Weil group $M_{p^{\infty}}(A/L)$ by the
exact sequence
$$
0 \longrightarrow\ M_{p^\infty}(A/L) \longrightarrow \ A(L) \otimesimes \mathfrak{m}athds{Q}_p/\mathfrak{m}athds{Z}_p
\longrightarrow\ \displaystyle\bigoplus_{v|p} A(L_v) \otimesimes \mathfrak{m}athds{Q}_p/\mathfrak{m}athds{Z}_p.
$$
Then, the `fine' Shafarevich-Tate group is defined by the exact sequence
$$
0 \ \longrightarrow\ M_{p^\infty}(A/L)\ \longrightarrow\ R_{p^\infty}(A/L)\ \longrightarrow\ \mathfrak{m}athds{Z}he_{p^\infty}(A/L)\ \longrightarrow
\ 0.
$$
In fact, it is not difficult to show that $\mathfrak{m}athds{Z}he(A/L)$ is contained
in the ($p$-primary) classical Shafarevich-Tate group (see loc.
cit.).
One may therefore think of $\mathfrak{m}athds{Z}he(A/L)$ as
the `Shafarevich-Tate part' of the fine Selmer group.
\mathfrak{m}edskip
With this definition in hand, one is naturally led to the following question for which we do not have
an answer at present.
\mathfrak{m}edskip \noindent \textbf{Question.} Retaining the assumptions of
Theorem \ref{theorem Z/p}, do we also have
\[ \sup\{ r_p\big(\mathfrak{m}athds{Z}he_{p^\infty}(A/L)\big)~ |~
\mathfrak{m}box{L/F is a cyclic extension of degree p}\} = \infty ?\]
\mathfrak{m}edskip
\section{Growth of fine Selmer groups in
infinite unramified pro-$p$ extensions} \lambdabel{unramified pro-p}
We introduce an interesting class of infinite unramified extensions
of $F$. Let $S$ be a finite set (possibly empty) of primes in $F$.
As before, we denote the $S$-ideal class group of $F$ by $\Cl_S(F)$.
For the remainder of the section, $F_{\infty}$ will denote the
maximal unramified $p$-extension of $F$ in which all primes in $S$
split completely. Write $\Sigma = \Sigma_F= \Gammal(F_{\infty}/F)$, and
let $\{ \Sigma_n\}$ be the derived series of $\Sigma$. For each $n$,
the fixed field $F_{n+1}$ corresponding to $\Sigma_{n+1}$ is the
$p$-Hilbert $S$-class field of $F_n$.
\mathfrak{m}edskip
Denote by $S_{\infty}$ the collection of infinite primes of $F$, and
define $\delta$ to be 0 if $\mathfrak{m}u_p\subseteq F$ and 1 otherwise. Let
$r_1(F)$ and $r_2(F)$ denote the number of real and complex places
of $F$ respectively. It is known that if the inequality
\[ r_p(\Cl_S(F)) \geq 2+ 2\sqrt{r_1(F)+ r_2(F) + \delta +
|S\setminus S_{\infty}|}\]
holds, then $\Sigma$ is infinite (see
\cite{GS}, and also \cite[Chap.\ X, Theorem 10.10.5]{NSW}). Stark
posed the question on whether $r_p(\Cl_S(F_n))$ tends to infinity in
an infinite $p$-class field tower as $n$ tends to infinity. By class
field theory, we have $r_p(\Cl_S(F_n)) = h_1(\Sigma_n)$. It then
follows from the theorem of Lubotzsky and Mann \cite{LuM} that
Stark's question is equivalent to whether the group $\Sigma$ is
$p$-adic analytic. By the following conjecture of Fontaine-Mazur
\cite{FM}, one does not expect $\Sigma$ to be an analytic group if
it is infinite.
\mathfrak{m}edskip\par
\mathfrak{m}edskip \noindent \textbf{Conjecture} (Fontaine-Mazur) \textit{For
any number field $F$,\, the group $\Sigma_F$ has no infinite
$p$-adic analytic quotient.}
\mathfrak{m}edskip
Without assuming the Fontaine-Mazur Conjecture, we have the
following unconditional (weaker) result, proven by various authors.
\begin{thm} \lambdabel{p-adic class tower} Let $F$ be a number field. If the
following inequality
\[ r_p(\Cl_S(F)) \geq 2+ 2\sqrt{r_1(F)+ r_2(F) +
\delta + |S\setminus S_{\infty}|}\] holds, then the group $\Sigma_F$
is not $p$-adic analytic. \end{thm}
\begin{prop}f When $S$ is the empty set, this theorem has been proved
independently by Boston \cite{B} and Hajir \cite{Ha}. For a general
nonempty $S$, this is proved in \cite[Lemma 2.3]{Ma}. \end{prop}f
\mathfrak{m}edskip\par
Collecting all the information we have, we obtain the following
result which answers an analogue of Stark's question, namely the
growth of the $p$-rank of the fine Selmer groups.
\begin{thm} \lambdabel{Fine Sel in class tower}
Let $A$ be an Abelian
variety of dimension $d$ defined over $F$ and let $S$ be a finite
set of primes which contains the primes above $p$, the primes of bad
reduction of $A$ and the archimedean primes. Let $F_{\infty}$ be the
maximal unramified $p$-extension of $F$ in which all primes of the
given set $S$ split completely, and let $F_n$ be defined as above.
Suppose that
\[ r_p(\Cl_S(F)) \geq 2+ 2\sqrt{r_1(F)+ r_2(F) + \delta +
|S\setminus S_{\infty}|}\] holds, and suppose that $A(F)[p]\neq 0$.
Then the $p$-rank of $R(A/F_n)$ is unbounded as $n$ tends to
infinity. \end{thm}
\begin{prop}f
By Lemma \ref{lower bound}, we have
\[ r_p(R(A/F_n)) \geq r_p(\Cl_S(F_n))r_p(A(F)[p])-2d. \]
Now by the hypothesis of the theorem, it follows from Theorem \ref{p-adic class
tower} that $\Sigma_F$ is not $p$-adic analytic. By the theorem of Lubotzsky and Mann
\cite{LuM}, this in turn implies that $r_p(\Cl_S(F_n))$ is
unbounded as $n$ tends to infinity. Hence we also have that $r_p(R(A/F_n))$ is unbounded as $n$ tends to
infinity (note here we also make use of the fact that $r_p(A(F)[p])\neq 0$ which comes from the hypothesis that $A(F)[p]\neq 0$). \end{prop}f
\mathfrak{m}edskip
\begin{remark}
(1) The analogue of the above result for the classical Selmer group has been
established in \cite{LM, Ma}. In particular, our result here refines
(and implies) those proved there.
(2) Let $A$ be an abelian variety defined over $F$ with complex
multiplication by $K$, and suppose that $K\subseteq F$. Let
$\mathfrak{m}athfrak{p}$ be a prime ideal of $K$ above $p$. Then one can
define a $\mathfrak{m}athfrak{p}$-version of the fine Selmer group replacing
$A[p^{\infty}]$ by $A[\mathfrak{m}athfrak{p}^{\infty}]$ in the definition of
the fine Selmer group. The above arguments carry over to establish
the fine version of the results in \cite{MO}. \end{remark}
\mathfrak{m}edskip
\end{ack}
\mathfrak{m}edskip
\end{document} |
\begin{document}
\title{Bayesian testing of linear versus nonlinear effects using Gaussian process priors}
\author{Joris Mulder}
\date{}
\thispagestyle{empty} \maketitle
\begin{abstract}
A Bayes factor is proposed for testing whether the effect of a key predictor variable on the dependent variable is linear or nonlinear, possibly while controlling for certain covariates. The test can be used (i) when one is interested in quantifying the relative evidence in the data of a linear versus a nonlinear relationship and (ii) to quantify the evidence in the data in favor of a linear relationship (useful when building linear models based on transformed variables). Under the nonlinear model, a Gaussian process prior is employed using a parameterization similar to Zellner's $g$ prior resulting in a scale-invariant test. Moreover a Bayes factor is proposed for one-sided testing of whether the nonlinear effect is consistently positive, consistently negative, or neither. Applications are provides from various fields including social network research and education.
\end{abstract}
\section{Introduction}
Linearity between explanatory and dependent variables is a key assumption in most statistical models. In linear regression models, the explanatory variables are assumed to affect the dependent variables in a linear manner, in logistic regression models it is assumed that the explanatory variables have a linear effect on the logit of the probability of a success on the outcome variable, in survival or event history analysis a log linear effect is generally assumed between the explanatory variables and the event rate, etc. Sometimes nonlinear functions (e.g., polynomials) are included of certain explanatory variables (e.g., for modeling curvilinear effects), or interaction effects are included between explanatory variables, which, in turn, are assumed to affect the dependent variable(s) in a linear manner.
Despite the central role of linear effects in statistical models, statistical tests of linearity versus nonlinearity are only limitedly available. In practice researchers tend to eyeball the relationship between the variables based on a scatter plot. When a possible nonlinear relationship is observed, various linear transformations (e.g., polynomial, logarithmic, Box-Cox) are applied and significance tests are executed to see if the coefficients of the transformed variables are significant or not. Eventually, when the nonlinear trend results in a reasonable fit, standard statistical inferential methods are applied (such as testing whether certain effects are zero and/or evaluating interval estimates).
This procedure is problematic for several reasons. First, executing many different significance tests on different transformed variables may result in $p$-hacking and inflated type I errors. Second, regardless of the outcome of a significance test, e.g., when testing whether the coefficient of the square of the predictor variable, $X^2$, equals zero, $H_0:\beta_{X^2}=0$ versus $H_1:\beta_{X^2}\not=0$, we would not learn whether $X^2$ has a linear effect on $Y$ or not; only whether an increase of $X^2$ results \textit{on average} in an increase/decrease of $Y$ or not. Third, nonlinear transformations (e.g., polynomials, logarithmic, Box-Cox) are only able to create approximate linearity for a limited set of nonlinear relationships. Fourth, eyeballing the relationship can be subjective, and instead a principle approach is needed.
To address these shortcomings this paper proposes a Bayes factor for the following hypothesis test
\begin{eqnarray}
\nonumber\text{$M_0:$ ``$X$ has a linear effect on $Y$''}~~\\
\label{Htest}\text{versus}~~~~~~~~~~~~~~~~~~~\\
\nonumber \text{$M_1:$ ``$X$ has a nonlinear effect on $Y$'',}
\end{eqnarray}
possibly while controlling for covariates. Unlike $p$ value significance tests, a Bayes factor can be used for quantifying the relative evidence in favor of linearity \citep{Wagenmakers:2007}. Furthermore, Bayes factors are preferred for large samples as significance tests may indicate that the null model needs to be rejected even though inspection may not show striking discrepancies from linearity. This behavior is avoided when using Bayesian model selection \citep{Raftery:1995}.
Under the alternative model, a Gaussian process prior is used to model the nonlinear effect. A Gaussian process is employed due to its flexibility to model nonlinear relationships \citep{Rasmussen:2007}. Because nonlinear relationships are generally fairly smooth, the Gaussian process is modeled using a squared exponential kernel. Furthermore, under both models a $g$ prior approach is considered \cite{Zellner:1986} so that the test is scale-invariant of the dependent variable. To our knowledge a $g$ prior was not used before for parameterizing a Gaussian process. As a result of the common parameterization under both models, the test comes down to testing whether a specific scale parameter equals zero or not, where a zero value implies linearity. Under the alternative the scale parameter is modeled using a half-Cauchy prior with a scale hyperparameter that can be chosen depending on the expected deviation from linearity under the alternative model.
Furthermore, in the case of a nonlinear effect, a Bayes factor is proposed for testing whether the effect is consistently increasing, consistently decreasing or neither. This test can be seen as a novel nonlinear extension to one-sided testing.
Finally note that the literature on Gaussian processes has mainly focused on estimating nonlinear effects \citep[e.g.,][]{Rasmussen:2007,Duvenaud:2011,Cheng:2019}, and not testing nonlinear effects, with an exception of \cite{Liu:2017} who proposed a significance (score) test, which has certain drawbacks as mentioned above. Further note that spline regression analysis is also typically used for estimating nonlinear effects, and not for testing (non)linearity.
The paper is organized as follows. Section 2 describes the linear and nonlinear Bayesian models and the corresponding Bayes factor. Its behavior is also explored in a numerical simulation. Section 3 describes the nonlinear one-sided Bayesian test. Subsequently, Section 4 presents 4 applications of the proposed methodology in different research fields. We end the paper with a short discussion in Section 5.
\section{A Bayes factor for testing (non)linearity}
\subsection{Model specification}
Under the standard linear regression model, denoted by $M_0$, we assume that the mean of the dependent variable $Y$ depends proportionally on the key predictor variable $X$, possibly while correcting for certain covariates. Mathematically, this implies that the predictor variable is multiplied with the same coefficient, denoted by $\beta$, to compute the (corrected) mean of the dependent variable for all values of $X$. The linear model can then be written as
\begin{equation}
M_0:\textbf{y}\sim\mathcal{N}(\beta\textbf{x} + \bm\gamma\textbf{Z},\sigma^2 \textbf{I}_n),
\end{equation}
where $\textbf{y}$ is a vector containing the $n$ observations of the dependent variable, $\textbf{x}$ contains the $n$ observations of the predictor variable, $\textbf{Z}$ is a $n\times k$ matrix of covariates (which are assumed to be orthogonal to the key predictor variable) with corresponding coefficients $\bm\gamma$, and $\sigma^2$ denotes the error variance which is multiplied with the identity matrix of size $n$, denoted by $\textbf{I}_n$. To complete the Bayesian model, we adopt the standard $g$ prior approach \citep{Zellner:1986} by setting a Gaussian prior on $\beta$ where the variance is scaled based on the error variance, the scale of the predictor variable, and the sample size, with a flat prior for the nuisance regression coefficients, and the independence Jeffreys prior for the error variance, i.e.,
\begin{eqnarray*}
\beta |\sigma^2 &\sim & N(0,\sigma^2g(\textbf{x}'\textbf{x})^{-1})\\
p(\bm\gamma) & \propto & 1\\
p(\sigma^2) & \propto & \sigma^{-2}.
\end{eqnarray*}
The prior mean is set to the default value of 0 so that, a priori, small effects in absolute value are more likely than large effects (as is common in applied research) and positive effects are equally likely as negative effects (an objective choice in Bayesian one-sided testing \citep{Jeffreys,Mulder:2010}). By setting $g=n$ we obtain a unit-information prior \citep{KassWasserman:1995,Liang:2008} which will be adopted throughout this paper\footnote{Note that we don't place a prior on $g$, as is becoming increasingly common \citep{Liang:2008,Rouder:2009,Bayarri:2007}, because we are not specifically testing whether $\beta$ equals 0 and to keep the model as simple as possible.}.
Under the alternative nonlinear model, denoted by $M_1$, we assume that the mean of the dependent variable does not depend proportionally on the predictor variable. This implies that the observations of the predictor variable can be multiplied with different values for different values of the predictor variable $X$. This can be written as follows
\begin{equation}
M_1:\textbf{y}\sim\mathcal{N}(\bm\beta(\textbf{x})\circ\textbf{x} + \bm\gamma\textbf{Z},\sigma^2 \textbf{I}_n),
\end{equation}
where $\bm\beta(\textbf{x})$ denotes a vector of length $n$ containing the coefficients of the corresponding $n$ observations of the predictor variable $\textbf{x}$, and $\circ$ denotes the Hadamard product. The vector $\bm\beta(\textbf{x})$ can be viewed as the $n$ realizations when plugging the different values of $\textbf{x}$ in a unknown theoretical function $\beta(x)$. Thus, in the special case where $\beta(x)$ is a constant function, say, $\beta(x)=\beta$, model $M_1$ would be equivalent to the linear model $M_0$.
Next we specify a prior probability distribution for the function of the coefficients. Because we are testing for linearity, it may be more likely to expect relatively smooth changes between different values, say, $\beta_i(x_i)$ and $\beta_j(x_j)$ than large changes when the values $x_i$ and $x_j$ are close to each other. A Gaussian process prior for the function $\bm\beta(\textbf{x})$ has this property which is defined by
\begin{equation}
\bm\beta(\textbf{x}) | \tau^2,\xi \sim \mathcal{GP}(\textbf{0},\tau^2k(\textbf{x},\textbf{x}'|\xi)),
\end{equation}
which has a zero mean function and a kernel function $k(\cdot,\cdot)$ which defines the covariance of the coefficients as a function of the distance between values of the predictor variable. A squared exponential kernel will be used which is given by
\begin{equation}
\label{kernel}
k(x_i,x_j|\xi) = \exp\left\{ -\tfrac{1}{2}\xi^2(x_i-x_j)^2 \right\},
\end{equation}
for $i,j=1,\ldots,n$. As can be seen, predictor variables $x_i$ and $x_j$ that are close to (far away from) each other have a larger (smaller) covariance, and thus, are on average closer to (further away from) each other. The hyperparameter $\xi$ controls the smoothness of the function where values close to 0 imply very smooth function shapes and large values imply highly irregular shapes (as will be illustrated later). Note that typically the smoothness is parameterized via the reciprocal of $\xi$. Here we use the current parameterization so that the special value $\xi=0$ would come down to a constant function, say $\beta(x)=\beta$, which would correspond to a linear relationship between the predictor and the outcome variable.
The hyperparameter $\tau^2$ controls the prior magnitude of the coefficients, i.e., the overall prior variance for the coefficients. We extend the $g$ prior formulation to the alternative model by setting $\tau^2=\sigma^2g(\textbf{x}'\textbf{x})^{-1}$ and specify the same priors for $\bm\gamma$ and $\sigma^2$ as under $M_0$. Furthermore, by taking into account that the Gaussian process prior implies that the coefficients for the observed predictor variables follow a multivariate normal distribution, the priors under $M_1$ given the predictor variables can be formulated as
\begin{eqnarray*}
\bm\beta(\textbf{x})|\sigma^2,\xi,\textbf{x} & \sim & \mathcal{N}(\textbf{0}, \sigma^2g(\textbf{x}'\textbf{x})^{-1}k(\textbf{x},\textbf{x}'|\xi))\\
p(\bm\gamma) & \propto & 1\\
p(\sigma^2) & \propto & \sigma^{-2}.
\end{eqnarray*}
To complete the model a half-Cauchy prior is specified for the key $\xi$ having prior scale $s_{\xi}$, i.e.,
\[
\xi \sim \text{half-}\mathcal{C}(s_{\xi}).
\]
The motivation for this prior is based on one of \cite{Jeffreys} desiderata which states that small deviations from the null value are generally more likely a priori than large deviations otherwise there would be no point in testing the null value. In the current setting this would imply that small deviations from linearity are more likely to be expected than large deviations. This would imply that values of $\xi$ close to 0 are more likely a priori than large values, and thus that the prior distribution for $\xi$ should be a decreasing function. The half-Cauchy distribution satisfies this property. Further note that the half-Cauchy prior is becoming increasingly popular for scale parameters in Bayesian analyses \citep{Gelman:2006,Polson:2012,MulderPericchi:2018}.
The prior scale for key parameter $\xi$ under $M_1$ should be carefully specified as it defines which deviations from linearity are most plausible. To give the reader more insight about how $\xi$ affects the distribution of the slopes of $\textbf{y}$ as function of $\textbf{x}$, Figure \ref{fig1} displays 10 random draws of the function of slopes when setting $\xi_1=0$ (Figure \ref{fig1}a), $\xi=\exp(-2)$ (Figure \ref{fig1}b), $\xi=\exp(-1)=1$ (Figure \ref{fig1}c), $\xi=\exp(0)$ (Figure \ref{fig1}d) while fixing $\tau^2=\sigma^2 g (\textbf{x}'\textbf{x})^{-1}=1$, where the slope function is defined by
\begin{equation}
\bm\eta (\textbf{x}) = \frac{d}{d\textbf{x}}[\bm\beta(\textbf{x})\circ\textbf{x}] = \bm\beta(\textbf{x}) +
\frac{d}{d\textbf{x}}[\bm\beta(\textbf{x})]\circ\textbf{x}.
\end{equation}
The figure shows that by increasing $\xi$ we get larger deviations from a constant slope. Based on these plots we qualify the choices $\xi=\exp(-2)$, $\exp(-1)$, and 1 as small deviations, medium deviations, and large deviations from linearity, respectively.
\begin{figure}
\caption{Ten random slope functions $\bm\eta(\textbf{x}
\label{fig1}
\end{figure}
Because the median of a half-Cauchy distribution is equal to the scale parameter $s_{\xi}$, the scale parameter could be set based on the expected deviation from linearity. It is important to note here that the expected deviation depends on the range of the predictor variable: In a very small range it may be expected that the effect is close to linear but in a wide range of the predictor variable, large deviations from linearity may be expected. Given the plots in Figure \ref{fig1}, one could set the prior scale equal to $s_{\xi} = 6e/\text{range}(\textbf{x})$, where $e$ can be interpreted as a standardized measure for the deviation from linearity such that setting $e = \exp(-2), \exp(-1)$, or $\exp(0)$ would imply small, medium, or large deviations from linearity, respectively. Thus, if the range of $\textbf{x}$ would be equal to 6 (as in the plots in Figure \ref{fig1}), the median of $\xi$ would be equal to $\exp(-2), \exp(-1)$, and $\exp(0)$, as plotted in Figure \ref{fig1}.
\subsection{Bayes factor computation}
The Bayes factor is defined as the ratio of the marginal (or integrated) likelihoods under the respective models. For this reason it is useful to integrate out the coefficient $\beta$ under $M_0$ and the coefficients $\bm\beta(\textbf{x})$ under $M_1$, which are in fact nuisance parameters in the test. This yields the following integrated models
\begin{align}
M_0 : &
\begin{cases}
\textbf{y} | \textbf{x},\bm\gamma,\sigma^2 \sim \mathcal{N}(\textbf{Z}\bm\gamma,\sigma^2 g (\textbf{x}'\textbf{x})^{-1}\textbf{x}\textbf{x}'+\sigma^2\textbf{I}_n) \\
p(\bm\gamma) \propto 1\\
p(\sigma^2) \propto \sigma^{-2}
\end{cases}\\
M_1 : &
\begin{cases}
\textbf{y} | \textbf{x},\bm\gamma,\sigma^2,\xi \sim \mathcal{N}(\textbf{Z}\bm\gamma,\sigma^2 g (\textbf{x}'\textbf{x})^{-1} k(\textbf{x},\textbf{x}'|\xi) \circ \textbf{x}\textbf{x}'+\sigma^2\textbf{I}_n) \\
p(\bm\gamma) \propto 1\\
p(\sigma^2) \propto \sigma^{-2}\\
\xi \sim \text{half-}\mathcal{C}(s_{\xi}),
\end{cases}
\end{align}
As can be seen $\sigma^2$ is a common factor in all (co)variances of $\textbf{y}$ under both models. This makes inferences about $\xi$ invariant to the scale of the outcome variable. Finally note that the integrated models clearly show that the model selection problem can concisely be written as
\begin{eqnarray*}
M_0&:&\xi = 0\\
M_1&:&\xi > 0.
\end{eqnarray*}
because $k(\textbf{x},\textbf{x}'|\xi)=1$ when setting $\xi=0$.
Using the above integrated models, the Bayes factor can be written as
\begin{equation*}
B_{01} = \frac{
\iint p(\textbf{y}|\textbf{x},\bm\gamma,\sigma^2,\xi=0)\pi(\bm\gamma)\pi(\sigma^2)
d\bm\gamma d\sigma^2
}{
\iiint p(\textbf{y}|\textbf{x},\bm\gamma,\sigma^2,\xi)\pi(\bm\gamma)\pi(\sigma^2)\pi(\xi)d\bm\gamma d\sigma^2 d\xi
},
\end{equation*}
which quantifies the relative evidence in the data between the linear model $M_0$ and the nonlinear model $M_1$.
Different methods can be used for computing marginal likelihoods. Throughout this paper we use an importance sample estimate. The R code for the computation of the marginal likelihoods and the sampler from the posterior predictive distribution can be found in the supplementary material.
\subsection{Numerical behavior}
Numerical simulation were performed to evaluate the performance of the proposed Bayes factor. The nonlinear function was set equal to $\beta(x)=3h\phi(x)$, for $h=0,\ldots,.5$, where $\phi$ is the standard normal probability density function (Figure \ref{simulfig}; upper left panel). In the case $h=0$, the effect is linear, and as $h$ increase, the effect becomes increasingly nonlinear. The dependent variable was computed as $\bm\beta(\textbf{x})\circ\textbf{x}+\bm\epsilon$, where $\bm\epsilon$ was sampled from a normal distribution with mean 0 and $\sigma=.1$.
The logarithm of the Bayes factor, denoted by $\log(B_{01})$, was computed between the linear model $M_0$ and the nonlinear model $M_1$ (Figure \ref{simulfig}; lower left panel) while setting the prior scale equal to $s_{\xi}=\exp(-2)$ (small prior scale; solid line), $\exp(-1)$ (medium prior scale; dashed line), and $\exp(0)$ (large prior scale; dotted line) for sample size $n=20$ (black lines), $50$ (red lines), and 200 (green lines) for equally distant predictor values in the interval $(-3,3)$. Overall we see the expected trend where we obtain evidence in favor of $M_0$ in the case $h$ is close to zero and evidence in favor of $M_1$ for larger values of $h$. Moreover the evidence for $M_0$ ($M_1$) is larger for larger sample sizes and larger prior scale when $h=0$ ($h\gg 0$) as anticipated given the consistent behavior of the Bayes factor.
\begin{figure}
\caption{Left panels. (Upper) Example functions of $\beta(x)=3h\phi(x)$, for $h=0,\ldots,.5$, where $\phi$ is the standard normal probability density function. Left lower panel and (lower) corresponding logarithm of the Bayes factor as function of $h$ for $n=20$ (black lines), $50$ (red lines), and 200 (green lines) for a small prior scale $s_{\xi}
\label{simulfig}
\end{figure}
Next we investigated the robustness of the test to nonlinear relationships that are not smooth as in the Gaussian processes having a squared exponential kernel. A similar analysis was performed when using the nonsmooth, discontinuous step function $\beta(x)=h~1(x>0)$, where $1(\cdot)$ is the indicator function, for $h=0,\ldots,.5$ (Figure \ref{simulfig}; upper right panel). Again the dependent variable was computed as $\bm\beta(\textbf{x})\circ\textbf{x}+\bm\epsilon$ and the logarithm of the Bayes factor was computed (Figure \ref{simulfig}; lower right panel). The Bayes factor shows a similar behavior as the above example where the data came from a smooth nonlinear alternative. The similarity of the results can be explained by the fact that even though the step function cannot be generated using a Gaussian process with a squared exponential kernel, the closest approximation of the step function is still nonlinear, and thus evidence is found against the linear model $M_0$ in the case $h>0$. This illustrates that the proposed Bayes factor is robust to nonsmooth nonlinear alternative models.
\section{Extension to one-sided testing}
When testing linear effects, the interest is often on whether the effect is either positive or negative if the null does not hold. Equivalently in the case of nonlinear effects the interest would be whether the effect is consistently increasing or consistently decreasing over the range of $X$. To model this we divide the parameter space under the nonlinear model $M_1$ in three subspaces:
\begin{eqnarray}
\nonumber M_{1,\text{positive}} &:& \text{``the nonlinear effect of $X$ on $Y$ is consistently positive''}\\
\nonumber M_{1,\text{negitive}} &:& \text{``the nonlinear effect of $X$ on $Y$ is consistently negative''}\\
\nonumber M_{1,\text{complement}}&:& \text{``the nonlinear effect of $X$ on $Y$ is neither consistently}\\
\label{onesided}&&\text{positive, nor consistently negative''.}
\end{eqnarray}
Note that the first model implies that the slope function is consistently positive, i.e., $\eta(x)>0$, the second implies that the slope is consistently negative, i.e., $\eta(x)<0$, while the third complement model assumes that the slope function is neither consistently positive nor negative.
Following standard Bayesian methodology using truncated priors for one-sided testing problems \citep{Klugkist:2005,Mulder:2020}, we set truncated Gaussian process priors on each of these three models, e.g., for model $M_{1,\text{positive}}$, this comes down to
\[
\pi_{1,\text{pos}}(\bm\beta(\textbf{x})|\xi_1,\tau_1) = \pi_1(\bm\beta(\textbf{x})|\xi,\tau)
\text{Pr}(\bm\eta(\textbf{x})>\textbf{0}|M_1,\xi,\tau)^{-1} 1_{\bm\eta(\textbf{x})>\textbf{0}}(\bm\beta(\textbf{x})),
\]
where $1_{\{\cdot\}}(\cdot)$ denotes the indicator function, and the prior probability, which serves as normalizing constant, equals
\[
\text{Pr}(\bm\eta(\textbf{x})>\textbf{0}|M_1,\xi,\tau) = \int_{\bm\eta(\textbf{x})>\textbf{0}} \pi_1(\bm\beta(\textbf{x})|\xi,\tau) d\bm\beta(\textbf{x}).
\]
Note that the prior probability for a consistently positive effect is equal because the prior mean of $\bm\beta(\textbf{x})$ equals $\textbf{0}$. Given this prior, the Bayes factor of each constrained model against the unconstrained model $M_1$ is then given by the ratio of the posterior and prior probabilities that the constraints hold under $M_1$, e.g.,
\[
B_{(1,\text{pos})u} = \frac{\text{Pr}(\bm\eta(\textbf{x})>\textbf{0}|M_1,\textbf{y})}{\text{Pr}(\bm\eta(\textbf{x})>\textbf{0}|M_1)}.
\]
Bayes factors between the above three models can then be computed using the transitive property of the Bayes factor, e.g., $B_{(1,\text{pos})(1,\text{comp})}=B_{(1,\text{pos})u}/B_{(1,\text{comp})u}$.
The choice of the prior of $\xi$ (which reflects the expected deviation from linearity before observing the data) implicitly determines the prior probability that the nonlinear effect is consistently positive or negative effect. This makes intuitive sense as large (small) deviations from linearity make it less (more) likely that the effect is either consistently positive or negative. This can also be observed from a careful inspection of the random draws in Figure \ref{fig1}. When $\xi=\exp(-2)$, we see that 4 out of 10 random functions in Figure \ref{fig1}b are consistently positive and 2 functions are consistently negative; when $\xi=\exp(-1)$ we see 1 random function that is consistently positive and 1 function that is consistently negative; and when $\xi=\exp(0)$ none of the 10 draws are either consistently positive or negative. The probabilities for a consistently positive (or negative) effect can simply be computed as the proportion of draws of random functions that is consistently positive (or negative). The choices $s_{\xi}=\exp(-2),~\exp(-1),$ and $\exp(0)$ result in prior probabilities for a consistently positive effect are approximately 0.25, 0.14, and 0.06.
\section{Empirical applications}
\subsection{Neuroscience: Facebook friends vs grey matter}
\cite{Kanai:2012} studied the relationship between the number of facebook friends and the grey matter density in regions of the brain that are related to social perception and associative memory to better understand the reason reasons for people to participate in online social networking. Here we analyze the data from the right entorhinal cortex ($n=41$). Due to the nature of the variables a positive relationship was expected. Based on a significance test \citep{Kanai:2012} and a Bayes factor \citep{Wetzels:2012} on a sample of size 41, it was concluded that there is evidence for a nonzero correlation between the square root of the number of Facebook friends and the grey matter density. In order for a correlation to be meaningful however it is important that the relationship is (approximately) linear. Here we test whether the relationship is linear or nonlinear. Furthermore, in the case of a nonlinear relationship, we test whether the relationships are consistently positive, consistently negative, or neither. Besides the predictor variable, the employed model has an intercept. The predictor variable is shifted to have a mean of 0 so that it is independent of the vector of ones for the intercept.
The Bayes factor between the linear model against the nonlinear model when using a prior scale of $\exp(-1)$ (medium effect) was equal to $B_{01}=2.50$ (with $\log(B_{01})=0.917$). This implies very mild evidence for a linear relationship between the square root of the number of Facebook friends and grey matter density in this region of the predictor variable. When assuming equal prior model probabilities, this would result in posterior model probabilities of .714 and .286 for $M_0$ and $M_0$, respectively. Thus if we would conclude that the relation is linear there would be a conditional error probability of drawing the wrong conclusion. Table \ref{appresults} presents the Bayes factors also for the other prior scales which tell a similar tale. Figure \ref{appfigure} (upper left panel) displays the data (circles; replotted from Kanai et al., 2012) and 50 draws of the posterior distribution density for the mean function under the nonlinear model at the observed values of the predictor variable. As can be seen most draws are approximately linear, and because the Bayes factor functions as an Occam's razor, the (linear) null model receives most evidence.
Even though we found evidence for a linear effect, there is still posterior model uncertainty and therefore we computed the Bayes factors between the one-sided models \eqref{onesided} under the nonlinear model $M_1$. This resulted in Bayes factors for the consistently positive, consistently negative, and the complement model against the unconstrained model of $B_{(1,\text{pos})u}=\frac{.825}{.140}=5.894$, $B_{(2,\text{pos})u}=\frac{.000}{.140}=0.000$, and $B_{(1,\text{comp})u}=\frac{.175}{.720}=0.242$, and thus most evidence for a consistently positive effect $B_{(1,\text{pos})(1,\text{neg})}\approx\infty$ and $B_{(1,\text{pos})(1,\text{neg})}\approx24.28$. These results are confirmed when checking the slopes of the posterior draws of the nonlinear mean function in Figure \ref{appfigure} (upper left panel).
\begin{table}[t]
\begin{center}
\caption{Log Bayes factors for the linear model versus the nonlinear model using different prior scales $s_{\xi}$.}
\begin{tabular}{lccccccccccc}
\hline
& sample size & $s_{\xi}=\mbox{e}^{-2}$ & $s_{\xi}=\mbox{e}^{-1}$ & $s_{\xi}=1$\\
\hline
Fb friends \& grey matter & 41 & 0.508 & 0.917 & 1.45\\
Age \& knowing gay & 63 & -37.7 & -38.3 & -38.1\\
Past activity \& waiting time & 500 & -0.776 & -0.361 & 0.394\\
Mother's IQ \& child test scores & 434 & -2.46 & -2.07 & -1.38\\
\hline
\end{tabular}\label{appresults}
\end{center}
\end{table}
\begin{figure}
\caption{Observations (circles) and 50 draws of the mean function (lines) under the nonlinear model $M_1$ for the four different applications. In the lower left panel draws are given in the case the mother finished her high school (light blue lines) or not (dark blue lines).}
\label{appfigure}
\end{figure}
\subsection{Sociology: Age and attitude towards gay}
We consider data presented in \cite{Gelman:2014} from the 2004 National Annenberg Election Survey containing respondents' age, sex, race, and attitude on three gay-related questions from the 2004 National Annenberg Election Survey. Here we are interested in the relationship between age and the proportion of people who know someone who's gay ($n=63$). It may be expected that older people may know less people who are gay and thus a negative relationship may be expected. Here we test whether the relationship between these variables is linear or not. In the case of a nonlinear relationship we also perform the one-sided test whether the relationship is consistently positive, negative, or neither. Again the employed model also has an intercept.
When setting the prior scale to a medium deviation from linearity, the logarithm of the Bayes factor between the linear model against the nonlinear model was approximately equal to $-38.3$, which corresponds to a Bayes factor of 0. This implies convincing evidence for a nonlinear effect. When using a small or large prior scale, the Bayes factors result in the same conclusion (Table \ref{appresults}). Figure \ref{appfigure} (upper right panel) displays the data (black circles) and 50 posterior draws of the mean function, which have clear nonlinear curves which fit the observed data.
Next we computed the Bayes factors for the one-sided test which results in decisive evidence for the complement model that the relationship is neither consistently positive nor consistently negative, with $B_{(1,\text{comp})(1,\text{pos})}=\infty$ and $B_{(1,\text{comp})(1,\text{neg})}=\infty$. This is confirmed when checking the posterior draws of the mean function in Figure \ref{appfigure} (upper right panel). We see a slight increase of the proportion of respondents who know someone who's gay towards the age of 45, and a decrease afterwards.
\subsection{Social networks: inertia and dyadic waiting times}
In dynamic social network data it is often assumed that actors in a network have a tendency to continue initiate social interactions with each other as a function of the volume of past interactions. This is also called inertia. In the application of the relational event model \cite{Butts:2008}, it is often assumed that the expected value of the logarithm of the waiting time between social interactions depends linearly on the number of past social interactions between actors. Here we consider relational (email) data from the Enron e-mail corpus \citep{Cohen:2009}. We consider a subset of the last $n=500$ emails (excluding 4 outliers) in a network of 156 employees in the Enron data \citep{Cohen:2009}. We use a model with an intercept.
Based on a medium prior scale under the nonlinear model, the logarithm of the Bayes factor between the linear model against the nonlinear model equals $\log(B_{01})=-0.361$, which corresponds to $B_{10}=1.43$, implying approximately equal evidence for both models. The posterior probabilities for the two models would be 0.412 and 0.588 for model $M_0$ and $M_1$, respectively. When using the small and large prior scale the Bayes factors are similar (Table \ref{appresults}), where the direction of the evidence flips towards the null when using a large prior scale. This could be interpreted as that a large deviation from linearity is least likely. Based on the posterior draws of the mean function in Figure \ref{appfigure} (lower left panel) we also see an approximate linear relationship. The nonlinearity seems to be mainly caused by the larger observations of the predictor variable. As there are relatively few large observations, the evidence is inconclusive about the nature of the relationship (linear or nonlinear). This suggests that more data would be needed in the larger region of the predictor variable.
The Bayes factors for the one-sided tests yield most evidence for a consistent decrease but the evidence is not conclusive in comparison to the complement model with $B_{(1,\text{neg})(1,\text{pos})}=\infty$ and $B_{(1,\text{neg})(1,\text{comp})}=5.14$. This suggests that there is most evidence that dyads (i.e., pairs of actors) that have been more active in the past will communicate more frequently.
\subsection{Education: Mother's IQ and child test scores}
In \cite{GelmanHill} the relationship between the mother's IQ and the test scores of her child is explored while controlling for whether the mother finished her high school. The expectation is that there is a positive relationship between the two key variables, and additionally there may be a positive effect of whether the mother went to high school. Here we explore whether the relationship between the mother's IQ and child test scores is linear. An ANCOVA model is considered with an intercept and a covariate that is either 1 or 0 depending on whether the mother finished high school or not\footnote{As discussed by \cite{GelmanHill} an interaction effect could also be reasonable to consider. Here we did not add the interaction effect for illustrative purposes. We come back to this in the Discussion.}.
Based on a medium prior scale, we obtain a logarithm of the Bayes factor for $M_0$ against $M_1$ of $-2.07$. This corresponds to a Bayes factor of $B_{10}=7.92$ which implies positive evidence for the nonlinear model. Table \ref{appresults} shows that the evidence for $M_1$ is slightly higher (lower) when using a smaller (larger) prior scale. This suggests that a small deviation from linearity is more likely than a large deviation a posteriori.
Next we computed the Bayes factors for testing whether the relationships are consistently increasing, consistently decreasing, or neither. We found clear evidence for a consistently increasing effect with Bayes factors equal to $B_{(1,\text{pos})(1,\text{neg})}=\infty$ and $B_{(1,\text{pos})(1,\text{comp})}=13.3$. This implies that, within this range of the predictor variable, a higher IQ of the mother always results in a higher expected test score of the child. This is also confirmed from the random posterior curves in Figure \ref{appfigure} (lower right panel) where we correct for whether the mother finished high
school (blue lines) or not (green lines).
\section{Discussion}
In order to make inferences about the nature of the relationship between two variables principled statistical tests are needed. In this paper a Bayes factor was proposed that allows one to quantify the relative evidence in the data between a linear relationship and a nonlinear relationship, possibly while controlling for certain covariates. The test is useful (i) when one is interested in assessing whether the relationship between variables is more likely to be linear or more likely to be nonlinear, and (ii) to determine whether a certain relationship is linear after transformation.
A Gaussian process prior with a square exponential kernel was used to model the nonlinear relationship under the alternative (nonlinear) model. The model was parameterized similar as Zellner's $g$ prior to make inferences that are invariant of the scale of the dependent variable and predictor variable. Moreover the Gaussian process was parameterized using the reciprocal of the scale length parameter which controls the smoothness of the nonlinear trend so that the linear model would be obtained when setting this parameter equal to 0. Moreover a standardized scale for this parameter was proposed to quantify the deviation from linearity under the alternative model.
In the case of a nonlinear effect a Bayes factor was proposed for testing whether the effect was consistently positive, consistently negative, or neither. This test can be seen as a nonlinear extension of Bayesian one-sided testing. Unlike the linear one-sided test, the Bayes factor depends on the prior scale for the nonlinear one-sided test. Thus, the prior scale also needs to be carefully chosen for the one-sided test depending on the expected deviation from linearity.
As a next step it would be useful to extend the methodology to correct covariates that have a nonlinear effect on the outcome variable \citep[e.g., using additive Gaussian processes;][]{Cheng:2019,Duvenaud:2011}, to test nonlinear interaction effects, or to allow other kernels to model other nonlinear forms. We leave this for future work.
\end{document} |
\begin{document}
\title{Beyond the $Q$-process: various ways of conditioning the multitype Galton-Watson process}
\author{Sophie P\'{e}nisson
\thanks{\texttt{sophie.penisson@u-pec.fr}}}
\affil{Universit\'e Paris-Est, LAMA (UMR 8050), UPEMLV, UPEC, CNRS, 94010 Cr\'{e}teil, France}
\date{ }
\mathbf{m}aketitle
\begin{abstract}
Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated $Q$-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the $Q$-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained $Q$-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a $Q$-process. It does not however coincide with the original associated $Q$-process, except in the critical regime.
\end{abstract}
{\bf Keywords:} multitype branching process, conditioned limit theorem, quasi-stationary distribution, $Q$-process, size-biased distribution, total progeny
{\bf 2010 MSC:} 60J80, 60F05
\mathbf{m}athbf{s}ection{Introduction}
The benchmark of our study is the $Q$-process associated with a multitype Galton-Watson (GW) process, obtained by conditioning the branching process $\mathbf{X}_k$ on not being extinct in the distant future ($\{\mathbf{X}_{k+n}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0}\}$, with $n\to +\infty$) and on the event that extinction takes place ($\{\lim_l \mathbf{X}_l=\mathbf{m}athbf{0}\}$) (see \cite{Naka78}). Our goal is to investigate some seemingly comparable conditioning results and to relate them to the $Q$-process.
After a description of the basic assumptions on the multitype GW process, we start in Subsection \ref{sec:asso} by describing the "associated" branching process, which will be a key tool when conditioning on the event that extinction takes place, or when conditioning on an infinite total progeny.
We shall first prove in Section \ref{sec:threshold} that by replacing in what precedes the conditioning event $\{\mathbf{X}_{k+n}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0}\}$ by $\{\mathbf{X}_{k+n}\in S\}$, where $S$ is a subset which does not contain $\mathbf{m}athbf{0}$, the obtained limit process remains the $Q$-process. This means in particular that conditioning in the distant future on reaching a non-zero state or a positive threshold, instead of conditioning on non-extinction, does not alter the result.
In a second instance, we focus in the noncritical case on the stationary measure of the positive recurrent $Q$-process. Formulated in a loose manner, this measure is obtained by considering $\{\mathbf{X}_k\mathbf{m}id \mathbf{X}_{k+n}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0}\}$, by delaying the extinction time ($n\to\infty$), and by studying the long-time behavior of the limit process ($k\to\infty$). It is already known (\cite{Naka78}) that inverting the limits leads to the same result. We prove in Section \ref{sec:Yaglom double} that the convergence to the stationary measure still holds even if $n$ and $k$ simultaneously grow to infinity. This requires an additional second-order moment assumption if the process is subcritical.
Finally, we investigate in Section \ref{sec:totalprog} the distribution of the multitype GW process conditioned on having an infinite total progeny. This is motivated by Kennedy's result, who studies in \cite{Ken75} the behavior of a monotype GW process $ X_k$ conditioned on the event $\{N = n\}$ as $n\to+\infty$, where $N=\mathbf{m}athbf{s}um_{k=0}^{+\infty}X_k$ denotes the total progeny. Note that the latter conditioning seems comparable to the device of conditioning on the event that extinction occurs but has not done so by generation $n$. It is indeed proven in the aforementioned paper that in the critical case, conditioning on the total progeny or on non-extinction indifferently results in the $Q$-process. This result has since then been extended for instance to monotype GW trees and to other conditionings: in the critical case, conditioning a GW tree by its height, by its total progeny or by its number of leaves leads to the same limiting tree (see e.g. \cite{AbrDel14,Jan12}). However, in the noncritical case, the two methods provide different limiting results: the limit process is always the $Q$-process of some critical process, no matter the class of criticality of the original process. Under a moment assumption (depending on the number of types of the process), we generalize this result to the multitype case. For this purpose we assume that the total progeny increases to infinity according to the "typical" limiting type proportions of the associated critical GW process, by conditioning on the event $\{\mathbf{N} = \left\lfloor n\mathbf{m}athbf{w}\right\rfloor\}$ as $n\to\infty$, where $\mathbf{m}athbf{w}$ is a left eigenvector related to the maximal eigenvalue 1 of the mean matrix of the critical process.
\mathbf{m}athbf{s}ubsection{Notation}
\label{sec:notation}
Let $d\geqslant 1$. In this paper, a generic point in $ \mathbb{R}^{d}$ is denoted by $\mathbf{m}athbf{x}=(x_1,\ldots,x_d)$, and its transpose is written $\mathbf{m}athbf{x}^T$. By $\mathbf{m}athbf{e}_i=(\delta_{i,j}) _{1\leqslant j\leqslant d}$ we denote the $i$-th unit vector in $ \mathbb{R}^{d}$, where $\delta_{i,j}$ stands for the Kronecker
delta. We write $\mathbf{m}athbf{0}=\left( 0,\ldots,0\right) $ and $\mathbf{1}=\left( 1,\ldots,1\right) $. The notation $\mathbf{m}athbf{x}\mathbf{m}athbf{y}$ (resp. $\left\lfloor \mathbf{m}athbf{x} \right\rfloor$) stands for the vector with coordinates $x_iy_i$ (resp. $\left\lfloor x_i \right\rfloor$, the integer part of $x_i$). We denote by $\mathbf{m}athbf{x}^{\mathbf{m}athbf{y}}$ the product $\prod_{i=1}^dx_i^{y_i}$. The obvious partial order on $ \mathbb{R}^{d}$ is $\mathbf{m}athbf{x}\leqslant \mathbf{m}athbf{y}$, when $x_i\leqslant y_i$ for each $i$, and $\mathbf{m}athbf{x}<\mathbf{m}athbf{y}$ when $x_i< y_i$ for each $i$. Finally, $\mathbf{m}athbf{x} \cdot \mathbf{m}athbf{y}$ denotes the scalar product in $ \mathbb{R}^{d}$, $\|\mathbf{m}athbf{x}\|_1$ the $L^1$-norm and $\|\mathbf{m}athbf{x}\|_2$ the $L^2$-norm.
\mathbf{m}athbf{s}ubsection{Multitype GW processes}
Let $( \mathbf{X}_k)_{k\geqslant 0}$ denote a $d$-type GW process, with $n$-th transition probabilities $P_n\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)= \mathbf{m}athbb{P}( \mathbf{X}_{k+n}=\mathbf{m}athbf{y}\mathbf{m}id\mathbf{X}_{k}=\mathbf{m}athbf{x})$, $ k$, $n\in\mathbf{m}athbb{N}$, $\mathbf{m}athbf{x}$, $\mathbf{m}athbf{y}\in\mathbf{N}N$. Let $\mathbf{m}athbf{f}=\left( f_1,\ldots,f_d\right) $ be its offspring generating function, where for each $i=1\ldots d$ and $\mathbf{m}athbf{r}\in[0,1]^d$, $f_i\left( \mathbf{m}athbf{r}\right) =\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_i}(\mathbf{m}athbf{r}^{\mathbf{X}_1})=\mathbf{m}athbf{s}um_{\mathbf{k}\in\mathbf{N}N}p_i\left( \mathbf{k}\right) \mathbf{m}athbf{r}^{\mathbf{k}}$, the subscript $\mathbf{m}athbf{e}_i$ denoting the initial condition, and $p_i$ the offspring probability distribution of type $i$. For each $i$, we denote by $\mathbf{m}^i=\left( m_{i1},\ldots,m_{id}\right) $ (resp. $\mathbf{m}athbf{\Sigma}^{i}$) the mean vector (resp. covariance matrix) of the offspring probability distribution $p_i$. The mean matrix is then given by $\mathbf{m}athbf{M}=(m_{ij})_{1\leqslant i,j \leqslant d}$. If it exists, we denote by $\rho$ its Perron's root, and by $\mathbf{u}$ and $\mathbf{v}$ the associated right and left eigenvectors (i.e. such that $\mathbf{m}athbf{M}\mathbf{u}^T=\rho\mathbf{u}^T$, $\mathbf{v}\mathbf{m}athbf{M}=\rho\mathbf{v}$), with the normalization convention $\mathbf{u}\cdot \mathbf{1}=\mathbf{u}\cdot\mathbf{v}=1$. The process is then called critical (resp. subcritical, supercritical) if $\rho=1$ (resp. $\rho<1$, $\rho>1$). In what follows we shall denote by $\mathbf{m}athbf{f}_n$ the $n$-th iterate of the function $\mathbf{m}athbf{f}$, and by $\mathbf{m}athbf{M}^n=(m_{ij}^{(n)})_{1\leqslant i,j \leqslant d}$ the $n$-th power of the matrix $\mathbf{M}$, which correspond respectively to the generating function and mean matrix of the process at time $n$. By the branching property, for each $\mathbf{m}athbf{x}\in\mathbf{N}N$, the function $\mathbf{m}athbf{f}_n^\mathbf{m}athbf{x}$ then corresponds to the generating function of the process at time $n$ with initial state $\mathbf{m}athbf{x}$, namely $\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}(\mathbf{m}athbf{r}^{\mathbf{X}_n})=\mathbf{m}athbf{f}_n\left( \mathbf{m}athbf{r}\right)^\mathbf{m}athbf{x}$. Finally, we define the extinction time $T=\inf\{k\in\mathbf{m}athbb{N},\,\mathbf{X}_k=\mathbf{m}athbf{0}\}$, and the extinction probability vector $\mathbf{q}=(q_1,\ldots,q_d)$, given by $q_i=\mathbf{m}athbb{P}_{\mathbf{m}athbf{e}_i}\left( T<+\infty\right)$, $i=1\ldots d$.
\mathbf{m}athbf{s}ubsection{Basic assumptions}
\label{sec:basic assumptions}
\begin{enumerate}
\item[$(A_1)$] The mean matrix $\mathbf{m}athbf{M}$ is finite. The process is nonsingular ($\mathbf{m}athbf{f}(\mathbf{m}athbf{r})\mathbf{m}athbf{n}eq \mathbf{m}athbf{M}\mathbf{m}athbf{r}$), is positive regular (there exists some $n\in\mathbf{N}et$ such that each entry of $\mathbf{m}athbf{M}^n$ is positive), and is such that $\mathbf{m}athbf{q}>\mathbf{m}athbf{0}$.
\end{enumerate}
The latter statement will always be assumed. It ensures in particular the existence of the Perron's root $\rho$ and that (\cite{Karl66}),
\begin{equation}\label{mean}
\lim_{n\to+\infty}\rho^{-n}m^{(n)}_{ij}=u_iv_j.
\end{equation} When necessary, the following additional assumptions will be made.
\begin{enumerate}
\item[$(A_2)$] For each $i,j=1\ldots d$, $\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_i}( X_{1,j}\ln X_{1,j})<+\infty$.
\item[$(A_3)$] The covariance matrices $\mathbf{m}athbf{\Sigma}^i$, $i=1\ldots d$, are finite.
\end{enumerate}
\mathbf{m}athbf{s}ubsection{The associated process}
\label{sec:asso}
For any vector $\mathbf{a}>\mathbf{m}athbf{0}$ such that for each $i=1\ldots d$, $f_i(\mathbf{a})<+\infty$, we define the generating function $\overline{\mathbf{m}athbf{f}}=\left( \overline{f}_1,\ldots,\overline{f}_d\right) $ on $[0,1]^d$ as follows: \[\overline{f}_i\left( \mathbf{m}athbf{r}\right)=\frac{f_i\left( \mathbf{a} \mathbf{m}athbf{r}\right)}{f_i\left( \mathbf{a}\right)},\ \ i=1\ldots d.\]
We then denote by $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ the GW process with offspring generating function $\overline{\mathbf{m}athbf{f}}$, which will be referred to as the \textit{associated process} with respect to $\mathbf{a}$. We shall denote by $\overline{P}_n$, $\overline{p}_i$ etc. its transition probabilities, offspring probability distributions etc. We easily compute that for each $n\geqslant 1$, $i=1\ldots d$, $\mathbf{k}\in\mathbf{N}N$ and $\mathbf{m}athbf{r}\in[0,1]^d$, denoting by $*$ the convolution product,
\begin{align}\label{off}
\overline{p}_i^{*n}\left(\mathbf{k}\right)=\frac{\mathbf{a}^{\mathbf{k}}}{f_i\left( \mathbf{a}\right)^{n} }p_i^{*n}\left(\mathbf{k}\right),\ \ \
\overline{f}_{n,i}\left(\mathbf{m}athbf{r}\right)=\frac{f_{n,i}\left(\mathbf{a}\mathbf{m}athbf{r}\right)}{f_i\left( \mathbf{a}\right)^{n} }.
\end{align}
\begin{remark}\label{rem: subcritic} It is known (\cite{JagLag08}) that a supercritical GW process conditioned on the event $\{T<+\infty\}$ is subcritical. By construction, its offspring generating function is given by $\mathbf{m}athbf{r}\mathbf{m}apsto f_i(\mathbf{m}athbf{q}\mathbf{m}athbf{r})/q_i$. Since the extinction probability vector satisfies $\mathbf{m}athbf{f}(\mathbf{m}athbf{q})=\mathbf{m}athbf{q}$ (\cite{Har63}), this means that the associated process $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ with respect to $\mathbf{m}athbf{q}$ is subcritical.
\end{remark}
\mathbf{m}athbf{s}ection{Classical results: conditioning on non-extinction}
\label{sec:nonext}
\mathbf{m}athbf{s}ubsection{The Yaglom distribution (\cite{JofSpit67}, Theorem 3)}
\label{sec:Yaglom}
Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a subcritical multitype GW process satisfying $(A_1)$. Then for all $\mathbf{m}athbf{x}_0,\mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$,
\begin{equation}\label{Yaglom}
\lim_{k\to+\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k}=\mathbf{z} \mathbf{m}id \mathbf{X}_k\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}\right)=\mathbf{m}athbf{n}u(\mathbf{z}),
\end{equation}
where $\mathbf{m}athbf{n}u$ is a probability distribution on $\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ independent of the initial state $\mathbf{m}athbf{x}_0$. This quasi-stationary distribution is often referred to as the Yaglom distribution associated with $\left(\mathbf{X}_k\right)_{k\geqslant 0}$. We shall denote by $g$ its generating function $g(\mathbf{m}athbf{r})=\mathbf{m}athbf{s}um_{\mathbf{z}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}}\mathbf{m}athbf{n}u(\mathbf{z})\mathbf{m}athbf{r}^{\mathbf{z}}$. Under $(A_2)$, $\mathbf{m}athbf{n}u$ admits finite and positive first moments
\begin{equation}\label{first moment g}
\frac{\partial g \left( \mathbf{m}athbf{1}\right) }{\partial r_i}=v_i\gamma^{-1},\ \ i=1\ldots d,
\end{equation}
where $\gamma>0$ is a limiting quantity satisfying for each $\mathbf{m}athbf{x}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$,
\begin{equation}\label{againbasic}
\lim_{k\to +\infty}\rho^{-k}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}}\left( \mathbf{X}_k\mathbf{m}athbf{n}eq\mathbf{m}athbf{0}\right) = \gamma\,\mathbf{m}athbf{x}\cdot\mathbf{u}.
\end{equation}
\mathbf{m}athbf{s}ubsection{The $Q$-process (\cite{Naka78}, Theorem 2)}
\label{sec:Q process}
Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a multitype GW process satisfying $(A_1)$. Then for all $\mathbf{m}athbf{x}_0\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, $k_1\leqslant\ldots\leqslant k_j\in\mathbf{m}athbb{N}$, and $\mathbf{m}athbf{x}_1,\ldots,\mathbf{m}athbf{x}_j\in\mathbf{N}N$,
\begin{multline}\label{limext}
\lim_{n\to+\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id \mathbf{X}_{k_j+n}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0},\,T<+\infty\right)\\=\frac{1}{\overline{\rho}^{k_j}}\frac{\mathbf{m}athbf{x}_j\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}_0\cdot\overline{\mathbf{u}}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j\right),
\end{multline}
where $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ is the associated process with respect to $\mathbf{q}$. As told in the introduction, this limiting process is the $Q$-process associated with $\left(\mathbf{X}_k\right)_{k\geqslant 0}$. It is Markovian with transition probabilities
\begin{equation*}
Q_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)=\frac{1}{\overline{\rho}}\frac{\mathbf{m}athbf{y}\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}\cdot\overline{\mathbf{u}}}\overline{P}_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)=\frac{1}{\overline{\rho}}\mathbf{q}^{\mathbf{m}athbf{y}-\mathbf{m}athbf{x}}\frac{\mathbf{m}athbf{y}\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}\cdot\overline{\mathbf{u}}}P_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right),\ \ \ \ \ \ \mathbf{m}athbf{x},\mathbf{m}athbf{y}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}.
\end{equation*}
If $\rho>1$, the $Q$-process is positive recurrent. If $\rho=1$, it is transient. If $\rho<1$, the $Q$-process is positive recurrent if and only if $(A_2)$ is satisfied. In the positive recurrent case, the stationary measure for the $Q$-process is given by the size-biased Yaglom distribution
\begin{equation}\label{size biased}
\overline{\mathbf{m}u}\left( \mathbf{z}\right)=\frac{\mathbf{z}\cdot\mathbf{u}\,\overline{\mathbf{m}athbf{n}u}\left( \mathbf{z}\right)}{\mathbf{m}athbf{s}um_{\mathbf{m}athbf{y}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}}\mathbf{m}athbf{y}\cdot\mathbf{u}\,\overline{\mathbf{m}athbf{n}u}\left( \mathbf{m}athbf{y}\right)},\ \ \mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\},
\end{equation}
where $\overline{\mathbf{m}athbf{n}u}$ is the Yaglom distribution associated with the subcritical process $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$.
\mathbf{m}athbf{s}ubsection{A Yaglom-type distribution (\cite{Naka78}, Theorem 3)}
\label{sec:Yaglom type}
Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a noncritical multitype GW process satisfying $(A_1)$. Then for all $\mathbf{m}athbf{x}_0,\mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ and $n\in\mathbf{N}$,
$
\lim_{k}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k}=\mathbf{z} \mathbf{m}id \mathbf{X}_{k+n}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0},\,T<+\infty\right)=\overline{\mathbf{m}athbf{n}u}^{(n)}(\mathbf{z})$,
where $\overline{\mathbf{m}athbf{n}u}^{(n)}$ is a probability distribution on $\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ independent of the initial state $\mathbf{m}athbf{x}_0$. In particular, $\overline{\mathbf{m}athbf{n}u}^{(0)}=\overline{\mathbf{m}athbf{n}u}$ is the Yaglom distribution associated with $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$, the associated subcritical process with respect to $\mathbf{q}$. Moreover, assuming in addition $(A_2)$ if $\rho<1$, then for each $\mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, $
\lim_{n}\overline{\mathbf{m}athbf{n}u}^{(n)}(\mathbf{z})=\overline{\mathbf{m}u}\left( \mathbf{z}\right).$
\mathbf{m}athbf{s}ection{Conditioning on reaching a certain state or threshold}
\label{sec:threshold}
In this section we shall generalize \eqref{limext} by proving that by replacing the conditioning event $\{\mathbf{X}_{k_j+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}\}$ by $\{\mathbf{X}_{k_j+n}\in S\}$, where $S$ is a subset of $\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, the obtained limit process remains the $Q$-process. In particular, conditioning the process on reaching a certain non-zero state or positive threshold in a distant future, i.e. with \begin{equation*}S=\{\mathbf{m}athbf{y}\},\ S=\{\mathbf{m}athbf{x}\in\mathbf{N}N,\ \|\mathbf{m}athbf{x}\|_1= m\}\ \mathbf{m}box{or}\ S=\{\mathbf{m}athbf{x}\in\mathbf{N}N,\ \|\mathbf{m}athbf{x}\|_1\geqslant m\},\end{equation*} ($\mathbf{m}athbf{y}\mathbf{m}athbf{n}eq\mathbf{m}athbf{0}, m>0$), leads to the same result as conditioning the process on non-extinction.
In what follows we call a subset $S$ accessible if for any $\mathbf{m}athbf{x}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, there exists some $n\in\mathbf{m}athbb{N}$ such that $\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}}\left( \mathbf{X}_n\in S\right) >0$. For any subset $S$ we shall denote $S^c=\mathbf{N}N\mathbf{m}athbf{s}etminus\left(\{\mathbf{m}athbf{0}\}\cup S\right)$.
\begin{theorem}Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a multitype GW process satisfying $(A_1)$, and let $S$ be a subset of $\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$. If $\rho\leqslant 1$ we assume in addition one of the following assumptions:
\begin{itemize}
\item[$(a_1)$] $S$ is finite and accessible,
\item[$(a_2)$] $S^c$ is finite,
\item[$(a_3)$] $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ is subcritical and satisfies $(A_2)$.
\end{itemize}
Then for all $\mathbf{m}athbf{x}_0\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, $k_1\leqslant\ldots\leqslant k_j\in\mathbf{m}athbb{N}^*$ and $\mathbf{m}athbf{x}_1,\ldots,\mathbf{m}athbf{x}_j\in\mathbf{N}N$,
\begin{multline}\label{result}
\lim_{n\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{X}_{k_j+n}\in S,\ T<+\infty\right)
\\=\frac{1}{\overline{\rho}^{k_j}}\frac{\mathbf{m}athbf{x}_j\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}_0\cdot\overline{\mathbf{u}}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots,
\overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j\right),
\end{multline}
where $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ is the associated process with respect to $\mathbf{q}$.
\end{theorem}
\begin{proof}Note that if $\rho>1$, then $\mathbf{q}<\mathbf{1}$ (\cite{AthNey}) which implies that $\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_i}( \overline{X}_{1,j}\ln \overline{X}_{1,j}) <+\infty$, meaning that $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0} $ automatically satisfies $(A_2)$. Thanks to Remark \ref{rem: subcritic}, we can thus assume without loss of generality that $\rho \leqslant 1$ and simply consider the limit \begin{multline}\label{reaching}\lim_{n\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{X}_{k_j+n}\in S\right)\\=\lim_{n\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\right) \frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left( \mathbf{X}_n\in S\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_j+n}\in S\mathbf{i}g)}.\end{multline}
Let us recall here some of the technical results established for $\rho \leqslant 1$ in \cite{Naka78}, essential to our proof. First, for each $\mathbf{m}athbf{0}\leqslant \mathbf{b}<\mathbf{c}\leqslant \mathbf{m}athbf{1}$ and $\mathbf{m}athbf{x}\in\mathbf{N}N$,
\begin{equation}\label{Nak1}
\lim_{n\to +\infty}\frac{\mathbf{v}\cdot\left( \mathbf{m}athbf{f}_{n+2}\left( \mathbf{m}athbf{0}\right)-\mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) \right) }{\mathbf{v}\cdot\left( \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right)-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) \right) }=\rho,
\end{equation}
\begin{equation}\label{Nak2}
\lim_{n\to +\infty}\frac{ \mathbf{m}athbf{f}_{n}\left( \mathbf{c}\right)^{\mathbf{m}athbf{x}}-\mathbf{m}athbf{f}_{n}\left( \mathbf{b}\right)^{\mathbf{m}athbf{x}} }{\mathbf{v}\cdot\left( \mathbf{m}athbf{f}_{n}\left( \mathbf{c}\right)-\mathbf{m}athbf{f}_{n}\left( \mathbf{b}\right) \right) }=\mathbf{m}athbf{x}\cdot\mathbf{u},
\end{equation}
Moreover, for each $\mathbf{m}athbf{x},\mathbf{m}athbf{y}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$,
\begin{equation}\label{Nak3}
\lim_{n\to +\infty}\frac{1-\mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}}{1-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}}=\rho,
\end{equation}
\begin{equation}\label{Nak4}
P_n\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right) =\left( \pi\left( \mathbf{m}athbf{y}\right) +\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)\right) \left( \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}\right),
\end{equation}where $\lim_{n}\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)=0$ and $\pi$ is the unique measure (up to multiplicative constants) on $\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ not identically zero satisfying $\mathbf{m}athbf{s}um_{\mathbf{m}athbf{y}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}}\pi( \mathbf{m}athbf{y}) P( \mathbf{m}athbf{y},\mathbf{z})=\rho\pi( \mathbf{z})$ for each $\mathbf{z}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}$. In particular, if $\rho<1$, $\pi=\left( 1-\rho\right)^{-1}\mathbf{m}athbf{n}u$, where $\mathbf{m}athbf{n}u$ is the probability distribution defined by \eqref{Yaglom}.
Let us first assume $(a_1)$. By \eqref{Nak4}
\begin{align}\label{toworkon1}
\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left( \mathbf{X}_n\in S\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_j+n}\in S\mathbf{i}g)}&=\frac{\mathbf{m}athbf{s}um_{\mathbf{z} \in S}P_n\left( \mathbf{m}athbf{x}_j,\mathbf{z}\right) }{\mathbf{m}athbf{s}um_{\mathbf{z} \in S}P_{n+k_j}\left( \mathbf{m}athbf{x}_0,\mathbf{z}\right)}\mathbf{m}athbf{n}onumber\\&=\frac{\pi\left( S\right) +\varepsilon_n\left( \mathbf{m}athbf{x}_j\right) }{\pi\left( S\right) +\varepsilon_{n+k_j}\left( \mathbf{m}athbf{x}_0\right)}\frac{ \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}}{\mathbf{m}athbf{f}_{n+k_j+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}-\mathbf{m}athbf{f}_{n+k_j}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}},
\end{align}
where $\lim_{n}\varepsilon_n\left( \mathbf{m}athbf{x}\right)=\lim_{n}\mathbf{m}athbf{s}um_{\mathbf{z}\in S}\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)=0$ since $S$ is finite. On the one hand, we can deduce from \eqref{Nak1} and \eqref{Nak2} that
\begin{equation*}
\lim_{n\to +\infty}\frac{ \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}}{\mathbf{m}athbf{f}_{n+k_j+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}-\mathbf{m}athbf{f}_{n+k_j}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}}
=\frac{1}{\rho^{k_j}}\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}}.
\end{equation*}
On the other hand, $\pi$ being not identically zero, there exists some $\mathbf{m}athbf{y}_0\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ such that $\pi\left( \mathbf{m}athbf{y}_0\right) >0$. Since $S$ is accessible, there exists some $\mathbf{z}_0\in S$ and $k\in\mathbf{m}athbb{N}^*$ such that $P_k\left( \mathbf{m}athbf{y}_0,\mathbf{z}_0\right)>0$, and thus
\[+\infty>\pi\left( S\right) \geqslant\pi\left( \mathbf{z}_0\right) =\rho^{-k}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{y}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}}\pi\left( \mathbf{m}athbf{y}\right) P_k\left( \mathbf{m}athbf{y},\mathbf{z}_0\right)\geqslant \rho^{-k}\pi\left( \mathbf{m}athbf{y}_0\right) P_k\left( \mathbf{m}athbf{y}_0,\mathbf{z}_0\right)>0.\]
From \eqref{toworkon1} we thus deduce that \eqref{reaching} leads to \eqref{result}.
Let us now assume $(a_2)$. We can similarly deduce from \eqref{Nak4} that
\begin{multline}\label{toworkon}
\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left( \mathbf{X}_n\in S\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}( \mathbf{X}_{k_j+n}\in S)}\\=\frac{1-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}- \left( \pi\left( S^c\right) +\varepsilon_n\left( \mathbf{m}athbf{x}_j\right)\right) \left( \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}\right) }{1-\mathbf{m}athbf{f}_{n+k_j}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}-( \pi\left( S^c\right) +\varepsilon_{n+k_j}\left( \mathbf{m}athbf{x}_0\right)) (\mathbf{m}athbf{f}_{n+k_j+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}-\mathbf{m}athbf{f}_{n+k_j}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}) },
\end{multline}
with $0\leqslant \pi\left( S^c\right) <+\infty$ and $\lim_{n}\varepsilon_n\left( \mathbf{m}athbf{x}\right)=\lim_{n}\mathbf{m}athbf{s}um_{\mathbf{z}\in S^c}\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)=0$ since $S^c$ is finite. Note that \eqref{limext} implies that
\begin{equation*}
\ \lim_{n\to +\infty}\frac{1-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_j}}{1-\mathbf{m}athbf{f}_{n+k_j}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}_0}}=\frac{1}{\rho^{k_j}}\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}},
\end{equation*} which together with \eqref{Nak3} enables to show that \eqref{toworkon} tends to $\rho^{-k_j}\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}}$ as $n$ tends to infinity, leading again to \eqref{result}.
Let us finally assume $(a_3)$. Then we know from \cite{Naka78} (Remark 2) that $\pi(\mathbf{z})>0$ for each $\mathbf{z}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}$, hence automatically $0<\pi\left( S\right) =\left( 1-\rho\right)^{-1}\mathbf{m}athbf{n}u\left( S\right)<+\infty$. Moreover, $\mathbf{m}athbf{n}u$ admits finite first-order moments (see \eqref{first moment g}). Hence for any $a>0$, by Markov's inequality,
\begin{align*}
\left| \mathbf{m}athbf{s}um_{\mathbf{z}\in S}\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)\right|&\leqslant \mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{z}\in S\\\|\mathbf{z}\|_1<a}}\left|\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)\right|+\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{z}\in S\\\|\mathbf{z}\|_1 \geqslant a}}\left|\frac{P_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)}{\mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}}- \pi\left( \mathbf{z}\right)\right|\\&\leqslant \mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{z}\in S\\\|\mathbf{z}\|_1<a}}\left|\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)\right|+\frac{1}{a}\frac{\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}\left(\|\mathbf{X}_n\|_1\right)}{\mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}}+\frac{1}{1-\rho}\frac{1}{a}
\mathbf{m}athbf{s}um_{i=1}^d\frac{\partial g \left( \mathbf{m}athbf{1}\right) }{\partial r_i}.
\end{align*}
We recall that by \eqref{againbasic}, $\lim_n \rho^{-n}\left( \mathbf{m}athbf{f}_{n+1}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) ^{\mathbf{m}athbf{x}}\right) =\left( 1-\rho\right) \gamma\,\mathbf{m}athbf{x}\cdot\mathbf{u}$, while by \eqref{mean}, $\lim_n \rho^{-n}\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}\left(\|\mathbf{X}_n\|_1\right)=\mathbf{m}athbf{s}um_{i,j=1}^dx_iu_iv_j$. Hence the previous inequality ensures that $\lim_n \mathbf{m}athbf{s}um_{\mathbf{z}\in S}\varepsilon_n\left( \mathbf{m}athbf{x},\mathbf{z}\right)=0$. We can thus write \eqref{toworkon1} even without the finiteness assumption of $S$, and prove \eqref{result} as previously.
\end{proof}
\mathbf{m}athbf{s}ection{The size-biased Yaglom distribution as a double limit}
\label{sec:Yaglom double}
From Subsection \ref{sec:Q process} and Subsection \ref{sec:Yaglom type} we know that in the noncritical case, assuming $(A_2)$ if $\rho<1$,
\begin{align*}
\lim_{k\to +\infty}\lim_{n\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_k=\mathbf{z}\mathbf{m}id\mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0},\,T<+\infty \right) &=\lim_{k\to +\infty}Q_k\left(\mathbf{m}athbf{x}_0, \mathbf{z}\right) =\overline{\mathbf{m}u}\left( \mathbf{z}\right),\\
\lim_{n\to +\infty}\lim_{k\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_k=\mathbf{z}\mathbf{m}id\mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0},\,T<+\infty \right) &=\lim_{n\to +\infty}\overline{\mathbf{m}athbf{n}u}^{(n)}\left(\mathbf{z}\right) =\overline{\mathbf{m}u}\left( \mathbf{z}\right).
\end{align*} We prove here that, under the stronger assumption $(A_3)$ if $\rho<1$, this limiting result also holds when $k$ and $n$ simultaneously tend to infinity.
\begin{theorem}Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a noncritical multitype GW process satisfying $(A_1)$. If $\rho<1$, we assume in addition $(A_3)$. Then for all $\mathbf{m}athbf{x}_0\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$ and $\mathbf{z}\in\mathbf{N}N$, \[\lim_{\mathbf{m}athbf{s}ubstack{n\to +\infty\\k\to+\infty}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_k=\mathbf{z}\mathbf{m}id\mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0},\,T<+\infty \right) =\overline{\mathbf{m}u}\left( \mathbf{z}\right),\] where $\overline{\mathbf{m}u}$ is the size-biased Yaglom distribution of $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$, the associated process with respect to $\mathbf{q}$. \end{theorem}
\begin{remark}
This implies in particular that for any $0<t<1$,
\[\lim_{k\to +\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{\left\lfloor kt\right\rfloor}=\mathbf{z}\mathbf{m}id\mathbf{X}_{k}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0},\,T<+\infty \right) =\overline{\mathbf{m}u}\left( \mathbf{z}\right).\]
\end{remark}
\begin{remark}In the critical case, the $Q$-process is transient and the obtained limit is degenerate. A suitable normalization in order to obtain a non-degenerate probability distribution is of the form $\mathbf{X}_k/k$. However, even with this normalization, the previous result does not hold in the critical case. Indeed, we know for instance that in the monotype case, a critical process with finite variance $\mathbf{m}athbf{s}igma^2>0$ satisfies for each $z\geqslant 0$ (\cite{LaNey68}),
\begin{align*}\lim_{k\to +\infty}\lim_{n\to +\infty}\mathbf{m}athbb{P}_{1}\left( \frac{X_k}{k}\leqslant z\mathbf{m}id X_{k+n}\mathbf{m}athbf{n}eq 0\right) &=1-e^{-\frac{2z}{\mathbf{m}athbf{s}igma^2}} ,\\
\lim_{n\to +\infty}\lim_{k\to +\infty}\mathbf{m}athbb{P}_{1}\left( \frac{X_k}{k}\leqslant z\mathbf{m}id X_{k+n}\mathbf{m}athbf{n}eq 0 \right)&=1-e^{-\frac{2z}{\mathbf{m}athbf{s}igma^2}}-\frac{2z}{\mathbf{m}athbf{s}igma^2}e^{-\frac{2z}{\mathbf{m}athbf{s}igma^2}}.
\end{align*}
\end{remark}
\begin{proof}Thanks to Remark \ref{rem: subcritic} and to the fact that if $\rho>1$, $\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_i}( \overline{X}_{1,j} \overline{X}_{1,l})<+\infty$, we can assume without loss of generality that $\rho < 1$. For each $n$, $k\in\mathbf{m}athbb{N}$ and $\mathbf{m}athbf{r}\in[0,1]^d$, \[\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left(\mathbf{m}athbf{r}^{\mathbf{X}_k}\mathbf{m}athbf{1}_{\mathbf{X}_{k+n}=\mathbf{m}athbf{0}} \right)=\mathbf{m}athbf{s}um_{\mathbf{m}athbf{y}\in\mathbf{N}N}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left(\mathbf{X}_k=\mathbf{m}athbf{y}\right) \mathbf{m}athbf{r}^{\mathbf{m}athbf{y}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{y}}\left(\mathbf{X}_{n}=\mathbf{m}athbf{0} \right)=\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) \right)^{\mathbf{m}athbf{x}_0},\] which leads to
\begin{align}\label{begin}
\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left[\mathbf{m}athbf{r}^{\mathbf{X}_k}\mathbf{m}id \mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0} \right]&=\frac{\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left(\mathbf{m}athbf{r}^{\mathbf{X}_k} \right) -\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left(\mathbf{m}athbf{r}^{\mathbf{X}_k}\mathbf{m}athbf{1}_{\mathbf{X}_{k+n}=\mathbf{m}athbf{0}} \right)}{1-\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left(\mathbf{X}_{k+n}= \mathbf{m}athbf{0} \right)}\mathbf{m}athbf{n}onumber\\&=\frac{\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\right)^{\mathbf{m}athbf{x}_0} -\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) \right)^{\mathbf{m}athbf{x}_0} }{1-\mathbf{m}athbf{f}_{k+n}\left( \mathbf{m}athbf{0}\right)^{\mathbf{m}athbf{x}_0}}.
\end{align}
By Taylor's theorem,
\begin{multline}\label{Taylor}\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\right)^{\mathbf{m}athbf{x}_0} -\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right) \right)^{\mathbf{m}athbf{x}_0}=\mathbf{m}athbf{s}um_{i=1}^d\frac{\partial \mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}_0}\left( \mathbf{m}athbf{r}\right)}{\partial r_i}r_i\left( 1-f_{n,i}\left( \mathbf{m}athbf{0}\right) \right) \\-\mathbf{m}athbf{s}um_{i,j=1\ldots d} \frac{r_ir_j\left( 1-f_{n,i}\left( \mathbf{m}athbf{0}\right) \right)( 1-f_{n,j}\left( \mathbf{m}athbf{0}\right) )}{2}\int_0^1\left( 1-t\right) \frac{\partial ^2\mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}_0}\left( \mathbf{m}athbf{r}-t \mathbf{m}athbf{r}\left( \mathbf{m}athbf{1}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right)\right) \right)}{\partial r_i\partial r_j}dt,
\end{multline}
with
\begin{equation}\label{calculus}\frac{\partial \mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}_0}\left( \mathbf{m}athbf{r}\right)}{\partial r_i}=\mathbf{m}athbf{s}um_{j=1}^dx_{0,j}\frac{\partial f_{k,j}\left( \mathbf{m}athbf{r}\right) }{\partial r_i}\mathbf{m}athbf{f}_k\left( \mathbf{m}athbf{r}\right)^{\mathbf{m}athbf{x}_0-\mathbf{m}athbf{e}_j}.
\end{equation}
Let us first prove the existence of $\lim_k\rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}\right)}{\partial r_i}$ for each $i,j$ and $\mathbf{m}athbf{r}\in[0,1]^d$. For each $k$, $p\in\mathbf{m}athbb{N}$ and $a>0$,
\begin{multline}\label{en fait si}
\Big| \rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}\right)}{\partial r_i}- \rho^{-(k+p)}\frac{\partial f_{k+p,j}\left(\mathbf{m}athbf{r}\right)}{\partial r_i} \Big|\\\leqslant \mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{z}\in\mathbf{N}N\\\|\mathbf{z}\|_2< a}}z_i\mathbf{m}athbf{r}^{\mathbf{z}-\mathbf{m}athbf{e}_i}\Big|\rho^{-k}P_k(\mathbf{m}athbf{e}_j, \mathbf{z}) - \rho^{-(k+p)}P_{k+p}(\mathbf{m}athbf{e}_j, \mathbf{z})\Big| \\+ \rho^{-k}\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_j}\left( X_{k,i}\mathbf{m}athbf{1}_{\|\mathbf{X}_k\|_2\geqslant a}\right) +\rho^{-(k+p)}\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_j}\left( X_{k+p,i}\mathbf{m}athbf{1}_{\|\mathbf{X}_{k+p}\|_2\geqslant a}\right).
\end{multline}
By Cauchy-Schwarz and Markov's inequalities, $ \mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_j}( X_{k,i}\mathbf{m}athbf{1}_{\|\mathbf{X}_k\_2|\geqslant a})\leqslant \frac{1}{a}\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_j}( \|\mathbf{X}_{k}\|_2^2).$ For each $\mathbf{m}athbf{x}\in\mathbf{N}N$, let $\mathbf{m}athbf{C}_{\mathbf{m}athbf{x},k}$ be the matrix $( \mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}( X_{k,i}X_{k,j}))_{1\leqslant i,j\leqslant d}$. According to \cite{Har63},
\begin{equation}\label{har}
\mathbf{m}athbf{C}_{\mathbf{m}athbf{x},k}=( \mathbf{m}athbf{M}^T) ^k\mathbf{m}athbf{C}_{\mathbf{m}athbf{x},0}\mathbf{m}athbf{M}^k+\mathbf{m}athbf{s}um_{n=1}^k( \mathbf{m}athbf{M}^T) ^{k-n}\left( \mathbf{m}athbf{s}um_{i=1}^d\mathbf{m}athbf{\Sigma}^{i}\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}\left(X_{n-1,i} \right) \right) \mathbf{m}athbf{M}^{k-n}.
\end{equation}
Thanks to \eqref{mean} this implies the existence of some $C>0$ such that for all $k\in\mathbf{m}athbb{N}$,
$\rho^{-k}\mathbf{m}athbb{E}_{\mathbf{m}athbf{e}_j}( \|\mathbf{X}_{k}\|_2^2)=\rho^{-k}\mathbf{m}athbf{s}um_{i=1}^d[ \mathbf{m}athbf{C}_{\mathbf{m}athbf{e}_j,k}]_{ii} \leqslant C$, and the two last right terms in \eqref{en fait si} can be bounded by $2Ca^{-1}$. As for the first right term in \eqref{en fait si}, it is thanks to \eqref{againbasic} and \eqref{Nak4} as small as desired for $k$ large enough. This proves that $( \rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}\right)}{\partial r_i})_k$ is a Cauchy sequence.
Its limit is then necessarily, for each $\mathbf{m}athbf{r}\in[0,1]^d$,
\begin{equation}\label{limit to prove}
\lim_{k\to+\infty} \rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}\right)}{\partial r_i}=\gamma u_j\frac{\partial g\left(\mathbf{m}athbf{r}\right)}{\partial r_i},
\end{equation}
where $g$ is defined in Subsection \ref{sec:nonext}. Indeed, since assumption $(A_3)$ ensures that $(A_2)$ is satisfied, we can deduce from \eqref{Yaglom} and \eqref{againbasic}
that $\lim_{k}\rho^{-k}(f_{k,j}( \mathbf{m}athbf{r})-f_{k,j}( \mathbf{m}athbf{0}))=\gamma u_j g(\mathbf{m}athbf{r})$. Hence, using the fact that $0\leqslant \rho^{-k}\frac{\partial f_{k,j}\left( \mathbf{m}athbf{r}\right)}{\partial r_i} \leqslant \rho^{-k}m^{(k)}_{ji}$, which thanks to \eqref{mean} is bounded, we obtain by Lebesgue's dominated convergence theorem that for each $h\in\mathbf{m}athbb{R}$ such that $\mathbf{m}athbf{r}+h\mathbf{m}athbf{e}_i\in[0,1]^d$, \begin{equation*}\gamma u_j g(\mathbf{m}athbf{r}+h\mathbf{m}athbf{e}_i)-\gamma u_j g(\mathbf{m}athbf{r})
=\lim_{k\to+\infty} \int_0^h\rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}+t\mathbf{m}athbf{e}_{i}\right)}{\partial r_i}dt= \int_0^h\lim_{k\to+\infty}\rho^{-k}\frac{\partial f_{k,j}\left(\mathbf{m}athbf{r}+t\mathbf{m}athbf{e}_{i}\right)}{\partial r_i}dt,\end{equation*}
proving \eqref{limit to prove}.
In view of \eqref{Taylor}, let us note that for each $\mathbf{m}athbf{r}\in[0,1]^ d$, there exists thanks to \eqref{mean} and \eqref{har} some $C>0$ such that for each $k\in\mathbf{m}athbb{N}$,
$ 0\leqslant \rho^{-k}\frac{\partial ^2\mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}}\left( \mathbf{m}athbf{r} \right)}{\partial r_i\partial r_j}\leqslant\rho^{-k}\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}}[ X_{k,j}( X_{k,i}-\delta_{ij})]\leqslant C$, hence for each $k$, $n\in\mathbf{m}athbb{N}$,\[\rho^{-k}\int_0^1\left( 1-t\right) \frac{\partial ^2\mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}}\left( \mathbf{m}athbf{r}-t \mathbf{m}athbf{r}\left( \mathbf{m}athbf{1}-\mathbf{m}athbf{f}_{n}\left( \mathbf{m}athbf{0}\right)\right) \right)}{\partial r_i\partial r_j}dt\leqslant \frac{C}{2}.\]
Together with \eqref{againbasic} this entails that the last right term in \eqref{Taylor} satisfies
\[\lim_{\mathbf{m}athbf{s}ubstack{n\to +\infty\\k\to+\infty}}\rho^{-(k+n)}\mathbf{m}athbf{s}um_{i,j=1\ldots d}\frac{r_ir_j\left( 1-f_{n,i}\left( \mathbf{m}athbf{0}\right) \right)( 1-f_{n,j}\left( \mathbf{m}athbf{0}\right) )}{2} \int_0^1\ldots\ dt=0.\]
Moreover, we deduce from \eqref{calculus}, \eqref{limit to prove} and $\lim_n\mathbf{m}athbf{f}_n(\mathbf{m}athbf{r})=\mathbf{m}athbf{1}$ that the first right term in \eqref{Taylor} satisfies
\[\lim_{\mathbf{m}athbf{s}ubstack{n\to +\infty\\k\to+\infty}}\rho^{-(k+n)}\mathbf{m}athbf{s}um_{i=1}^d\frac{\partial \mathbf{m}athbf{f}_k^{\mathbf{m}athbf{x}_0}\left( \mathbf{m}athbf{r}\right)}{\partial r_i}r_i\left( 1-f_{n,i}\left( \mathbf{m}athbf{0}\right) \right)=\gamma^2\,\mathbf{m}athbf{x}_0\cdot\mathbf{u} \mathbf{m}athbf{s}um_{i=1}^dr_iu_i\frac{\partial g\left(\mathbf{m}athbf{r}\right)}{\partial r_i}.\]
Recalling \eqref{begin} and \eqref{againbasic}, we have thus proven that for each $\mathbf{m}athbf{r}\in[0,1]^d$,
\[\lim_{\mathbf{m}athbf{s}ubstack{n\to +\infty\\k\to+\infty}}\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left[\mathbf{m}athbf{r}^{\mathbf{X}_k}\mathbf{m}id \mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0} \right] =\gamma \mathbf{m}athbf{s}um_{i=1}^dr_iu_i\frac{\partial g\left(\mathbf{m}athbf{r}\right)}{\partial r_i}=\gamma \mathbf{m}athbf{s}um_{\mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}}\mathbf{z}\cdot\mathbf{u}\,\mathbf{m}athbf{n}u\left( \mathbf{z}\right)\mathbf{m}athbf{r}^{\mathbf{z}}.\]
Finally, \eqref{first moment g} leads to
$\gamma \mathbf{m}athbf{s}um_{i=1}^du_i\frac{\partial g\left(\mathbf{m}athbf{1}\right)}{\partial r_i}=1$, and thus
\[\lim_{\mathbf{m}athbf{s}ubstack{n\to +\infty\\k\to+\infty}}\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_0}\left[\mathbf{m}athbf{r}^{\mathbf{X}_k}\mathbf{m}id \mathbf{X}_{k+n}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0} \right] =\frac{\mathbf{m}athbf{s}um_{\mathbf{z}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}}\mathbf{z}\cdot\mathbf{u}\,\mathbf{m}athbf{n}u\left( \mathbf{z}\right)\mathbf{m}athbf{r}^{\mathbf{z}}}{\mathbf{m}athbf{s}um_{\mathbf{m}athbf{y}\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}}\mathbf{m}athbf{y}\cdot\mathbf{u}\, \mathbf{m}athbf{n}u\left( \mathbf{m}athbf{y}\right)},\]
which by \eqref{size biased} is a probability generating function.
\end{proof}
\mathbf{m}athbf{s}ection{Conditioning on the total progeny}
\label{sec:totalprog}
Let $\mathbf{m}athbf{N}=\left( N_1,\ldots,N_d\right)$ denote the total progeny of the process $\left(\mathbf{X}_k\right)_{k\geqslant 0} $, where for each $i=1\ldots d$,
$N_i=\mathbf{m}athbf{s}um_{k=0}^{+\infty}X_{k,i}$,
and $N_i=+\infty$ if the sum diverges. Our aim is to study the behavior of $\left(\mathbf{X}_k \right)_{k\geqslant 0} $ conditioned on the event $\{\mathbf{N}=\left\lfloor n\mathbf{m}athbf{w} \right\rfloor\}$, as $n$ tends to infinity, for some specific positive vector $\mathbf{m}athbf{w}$. We recall that in the critical case, the GW process suitably normalized and conditioned on non-extinction in the same fashion as in \eqref{Yaglom}, converges to a limit law supported by the ray $\{\lambda\mathbf{v}: \lambda\geqslant 0\}\mathbf{m}athbf{s}ubset \mathbf{m}athbb{R}_+^d$. In this sense,
its left eigenvector $\mathbf{v}$ describes "typical limiting type proportions", as pointed out in \cite{FleiVat06}. As we will see in Lemma \ref{lem1}, conditioning a GW process on a given total progeny size comes down to conditioning an associated critical process on the same total progeny size. For this reason, the vector $\mathbf{m}athbf{w}$ will be chosen to be the left eigenvector of the associated critical process. It then appears that, similarly as in the monotype case (\cite{Ken75}), the process conditioned on an infinite total progeny $\{\mathbf{N}=\left\lfloor n\mathbf{m}athbf{w} \right\rfloor\}$, $n\to\infty$, has the structure of the $Q$-process of a critical process, and is consequently transient. This is the main result, stated in Theorem \ref{thm2}.
\begin{theorem}\label{thm2} Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a multitype GW process satisfying $(A_1)$. We assume in addition that
\begin{enumerate}
\item[$(A_4)$] there exists $\mathbf{a}>\mathbf{m}athbf{0}$ such that the associated process with respect to $\mathbf{a}$ is critical,
\item[$(A_5)$] for each $j=1\ldots d$, there exist $i=1\ldots d$ and $\mathbf{k}\in\mathbf{N}N$ such that $p_i\left(\mathbf{k}\right) >0$ and $p_i\left(\mathbf{k}+\mathbf{m}athbf{e}_j\right) >0$,
\item[$(A_6)$] the associated process with respect to $\mathbf{a}$ admits moments of order $d+1$, and its covariance matrices are positive-definite.
\end{enumerate}
Then for all $\mathbf{m}athbf{x}_0\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\}$, $k_1\leqslant\ldots\leqslant k_j\in\mathbf{m}athbb{N}$, and $\mathbf{m}athbf{x}_1,\ldots,\mathbf{m}athbf{x}_j\in\mathbf{N}N$,
\begin{equation}\label{lim}
\lim_{n\to+\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{N}=\left\lfloor n\overline{\mathbf{v}} \right\rfloor\mathbf{i}g)\\=\frac{\mathbf{m}athbf{x}_j\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}_0\cdot\overline{\mathbf{u}}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{i}g),
\end{equation}
where $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ is the associated process with respect to $\mathbf{a}$.
\end{theorem}
The limiting process defined by \eqref{lim} is thus Markovian with transition probabilities
\begin{equation*}
\overline{Q}_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right) =\frac{\mathbf{m}athbf{y}\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}\cdot\overline{\mathbf{u}}}\overline{P}_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right)=\frac{\mathbf{a}^{\mathbf{m}athbf{y}}}{\mathbf{m}athbf{f}\left( \mathbf{a}\right) ^{\mathbf{m}athbf{x}}}\frac{\mathbf{m}athbf{y}\cdot\overline{\mathbf{u}}}{\mathbf{m}athbf{x}\cdot\overline{\mathbf{u}}}P_1\left( \mathbf{m}athbf{x},\mathbf{m}athbf{y}\right),\ \ \ \ \ \ \mathbf{m}athbf{x},\mathbf{m}athbf{y}\in\mathbf{N}N\mathbf{m}athbf{s}etminus\{\mathbf{m}athbf{0}\},
\end{equation*}and corresponds to the $Q$-process associated with the critical process $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$.
\begin{remark}$ $
\begin{itemize}
\item If $d=1$, the conditional event $\{\mathbf{N}=\left\lfloor n\overline{\mathbf{v}} \right\rfloor\}$ reduces to $\{N=n\}$, as studied in \cite{Ken75}, in which assumptions $(A_4)$--$(A_6)$ are also required\footnote{Since the author's work, it has been proved in \cite{AbrDelGuo15} that in the critical case and under $(A_1)$, Theorem \ref{thm2} holds true under the minimal assumptions of aperiodicity of the offspring distribution (implied by $(A_5)$) and the finiteness of its first order moment.}.
\item If $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ is critical, assumption $(A_4)$ is satisfied with $\mathbf{a}=\mathbf{1}$. This assumption is also automatically satisfied if $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ is supercritical. Indeed, as mentioned in Remark \ref{rem: subcritic}, the associated process with respect to $\mathbf{m}athbf{0}<\mathbf{q}<\mathbf{1}$ is subcritical and thus satisfies $\overline{\rho}< 1$. The fact that $\rho>1$ and the continuity of the Perron's root as a function of the mean matrix coefficients then ensures the existence of some $\mathbf{q}\leqslant \mathbf{a} \leqslant \mathbf{1}$ satisfying $(A_4)$. Note however that such an $\mathbf{a}$ is not unique.
\item For any $\mathbf{a}>\mathbf{m}athbf{0}$, $p_i$ and $\overline{p}_i$ share by construction the same support. As a consequence, $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ satisfies $(A_5)$ if and only if $\left(\overline{\mathbf{X}}_k\right)_{k\geqslant 0} $ does. Moreover, a finite covariance matrix $\mathbf{m}athbf{\Sigma}^i$ is positive-definite if and only if there does not exist any $c\in\mathbf{m}athbb{R}$ and $\mathbf{m}athbf{x}\mathbf{m}athbf{n}eq \mathbf{m}athbf{0}$ such that $\mathbf{m}athbf{x}\cdot\mathbf{X}=c$ $\mathbf{m}athbb{P}_{\mathbf{m}athbf{e}_i}$-almost-surely, hence if and only if $\mathbf{m}athbf{x}\cdot\overline{\mathbf{X}}=c$ $\mathbf{m}athbb{P}_{\mathbf{m}athbf{e}_i}$-almost-surely. Consequently, provided it exists, $\mathbf{m}athbf{\Sigma}^i$ is positive-definite if and only if $\overline{\mathbf{m}athbf{\Sigma}}^i$ is positive-definite as well.
\end{itemize}
\end{remark}
We shall first show in Lemma \ref{lem1} that for any $\mathbf{a}$, the associated process $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$ with respect to $\mathbf{a}$, conditioned on $\{\overline{\mathbf{N}}=\mathbf{m}athbf{n}\}$, has the same probability distribution as the original process conditioned on $\{\mathbf{N}=\mathbf{m}athbf{n}\}$, for any $\mathbf{m}athbf{n}\in\mathbf{N}N$. It is thus enough to prove Theorem \ref{thm2} in the critical case, which is done at the end of the article.
It follows from Proposition 1 in \cite{Good75} or directly from Theorem 1.2 in \cite{ChauLiu11} that the probability distribution of the total progeny in the multitype case is given for each $\mathbf{m}athbf{x}_0$, $\mathbf{m}athbf{n}\in\mathbf{N}N$ with $\mathbf{m}athbf{n}>\mathbf{m}athbf{0}$, $\mathbf{m}athbf{n}\geqslant \mathbf{m}athbf{x}_0 $ by
\begin{equation}\label{totprogeny}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0 }\left( \mathbf{N}=\mathbf{m}athbf{n}\right)=\frac{1}{n_1\ldots n_d}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{k}^{1},\ldots,\mathbf{k}^{d}\in\mathbf{N}N\\\mathbf{k}^{1}+\ldots+\mathbf{k}^{d}=\mathbf{m}athbf{n}-\mathbf{m}athbf{x}_0}}\det \begin{pmatrix}
n_1\mathbf{m}athbf{e}_1-\mathbf{k}^1\\\cdots \\mathbf{m}athbf{n}_{d}\mathbf{m}athbf{e}_{d}-\mathbf{k}^{d}
\end{pmatrix} \prod_{i=1}^d p_i^{*n_i}\left( \mathbf{k}^i\right).\end{equation}
\begin{lemma}\label{lem1}Let $\left(\mathbf{X}_k \right)_{k\geqslant 0} $ be a multitype GW process. Then, for any $\mathbf{a}>\mathbf{m}athbf{0}$, the associated process $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g) _{k\geqslant 0}$ with respect to $\mathbf{a}$ satisfies for any $\mathbf{m}athbf{x}_0\in\mathbf{N}N$, $k_1\leqslant\ldots\leqslant k_j\in\mathbf{m}athbb{N}$, $\mathbf{m}athbf{x}_1,\ldots,\mathbf{m}athbf{x}_j\in\mathbf{N}N$ and $\mathbf{m}athbf{n}\in\mathbf{N}N$,
\begin{equation*}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{N}=\mathbf{m}athbf{n}\right)=\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\overline{\mathbf{N}}=\mathbf{m}athbf{n}\right).
\end{equation*}
\end{lemma}
\begin{proof}From \eqref{off} and \eqref{totprogeny} ,
$\mathbf{m}athbb{P}_{\mathbf{k}}\left( \overline{\mathbf{N}}=\mathbf{m}athbf{n}\right)
=\frac{\mathbf{a}^{\mathbf{m}athbf{n}-\mathbf{k}}}{\mathbf{m}athbf{f}\left( \mathbf{a}\right)^{\mathbf{m}athbf{n}} } \mathbf{m}athbb{P}_{\mathbf{k}}\left( \mathbf{N}=\mathbf{m}athbf{n}\right)$.
For all $n\in\mathbf{m}athbb{N}$, we denote by $\mathbf{N}_n=\mathbf{m}athbf{s}um_{k=0}^n\mathbf{X}_k$ (resp. $\overline{\mathbf{N}}_n=\mathbf{m}athbf{s}um_{k=0}^n\overline{\mathbf{X}}_k$) the total progeny up to generation $n$ of $\left(\mathbf{X}_k \right)_{k\geqslant 0} $ (resp. $\mathbf{i}g(\overline{\mathbf{X}}_k\mathbf{i}g)_{k\geqslant 0}$). Then
\begin{align*}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left(\overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j,\overline{\mathbf{N}}_{k_j}=\mathbf{m}athbf{l}\right)&=\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{i}_1,\ldots,\mathbf{i}_{k_j-1}\in\mathbf{N}N\\\mathbf{i}_1+\ldots+\mathbf{i}_{k_j-1}=\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_0-\mathbf{m}athbf{x}_j}}\overline{P}_1\left( \mathbf{m}athbf{x}_0,\mathbf{i}_1\right) \ldots\overline{P}_1( \mathbf{i}_{k_j-1},\mathbf{m}athbf{x}_j) \\
&=\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{i}_1,\ldots,\mathbf{i}_{k_j-1}\in\mathbf{N}N\\\mathbf{i}_1+\ldots+\mathbf{i}_{k_j-1}=\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_0-\mathbf{m}athbf{x}_j}}\frac{\mathbf{a}^{\mathbf{i}_1}P_1\left( \mathbf{m}athbf{x}_0,\mathbf{i}_1\right) }{\mathbf{m}athbf{f}\left( \mathbf{a}\right) ^{\mathbf{m}athbf{x}_0}}\ldots\frac{\mathbf{a}^{\mathbf{m}athbf{x}_j}P_1( \mathbf{i}_{k_j-1},\mathbf{m}athbf{x}_j) }{\mathbf{m}athbf{f}\left( \mathbf{a}\right) ^{\mathbf{i}_{k_j-1}}}\\&=\frac{\mathbf{a}^{\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_0}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left(\mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j,\mathbf{N}_{k_j}=\mathbf{m}athbf{l}\right)}{\mathbf{m}athbf{f}\left( \mathbf{a}\right) ^{\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_j}} ,
\end{align*}
and similarly
\begin{equation*}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\Big( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j,\overline{\mathbf{N}}_{k_j}=\mathbf{m}athbf{l}\Big)\\=\frac{\mathbf{a}^{\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_0}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j,\mathbf{N}_{k_j}=\mathbf{m}athbf{l}\right)}{\mathbf{m}athbf{f}\left( \mathbf{a}\right) ^{\mathbf{m}athbf{l}-\mathbf{m}athbf{x}_j}} .
\end{equation*}
Consequently, thanks to the Markov property,
\begin{align*}
&\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\overline{\mathbf{N}}=\mathbf{m}athbf{n}\right)
\\
&\ \ =\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{m}athbf{l}\in\mathbf{N}N\\\mathbf{m}athbf{l}\leqslant \mathbf{m}athbf{n}}}\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \overline{\mathbf{X}}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \overline{\mathbf{X}}_{k_j}=\mathbf{m}athbf{x}_j,\overline{\mathbf{N}}_{k_j}=\mathbf{m}athbf{l}\right)\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left(\overline{\mathbf{N}}=\mathbf{m}athbf{n}-\mathbf{m}athbf{l}+\mathbf{m}athbf{x}_j\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left(\overline{\mathbf{N}}=\mathbf{m}athbf{n}\right)} \\
&\ \ =\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{N}=\mathbf{m}athbf{n}\right).
\end{align*}
\end{proof}
Thanks to Lemma \ref{lem1}, it suffices to prove Theorem \ref{thm2} in the critical case. For this purpose, we prove the following convergence result for the total progeny of a critical GW process.
\begin{proposition}\label{prop: convergence}
Let $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ be a critical multitype GW process satisfying $(A_1)$, $(A_5)$ and $(A_6)$. Then there exists $C> 0$ such that for all $\mathbf{m}athbf{x}_0\in\mathbf{N}N$,
\begin{equation}
\lim_{n\to+\infty}n^{\frac{d}{2}+1}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0 }\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)=C\mathbf{m}athbf{x}_0\cdot \mathbf{u}.
\end{equation}
\end{proposition}
\begin{proof}
From \eqref{totprogeny}, for each $n\geqslant \mathbf{m}ax_{i}v_i^{-1}$, $n\geqslant \mathbf{m}ax_{i} x_{0,i}v_i^{-1}$,
\begin{align*}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0 }\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)&=\frac{1}{\prod_{i=1}^{d} \left\lfloor nv_i\right\rfloor }\mathbf{m}athbb{E}\left[ \det \hspace{-1mm}\begin{pmatrix}
\left\lfloor nv_1\right\rfloor \mathbf{m}athbf{e}_1-\mathbf{S}^1_{\left\lfloor nv_1\right\rfloor }\\\cdots\\\left\lfloor nv_d\right\rfloor \mathbf{m}athbf{e}_{d}-\mathbf{S}^{d}_{\left\lfloor nv_d\right\rfloor }
\end{pmatrix}\hspace{-1mm}\mathbf{1}_{\mathbf{m}athbf{s}um_{i=1}^d\mathbf{S}^i_{\left\lfloor nv_i\right\rfloor }=\left\lfloor n\mathbf{v}\right\rfloor -\mathbf{m}athbf{x}_0}\right]\mathbf{m}athbf{n}onumber\\
&=\frac{1}{ \left\lfloor nv_d\right\rfloor }\mathbf{m}athbb{E}\left[\det \hspace{-1mm}\begin{pmatrix}\mathbf{m}athbf{e}_1-\mathbf{S}^1_{\left\lfloor nv_1\right\rfloor }/\left\lfloor nv_1\right\rfloor\\\cdots\\ \mathbf{m}athbf{e}_{d-1}-\mathbf{S}^{d-1}_{\left\lfloor nv_{d-1}\right\rfloor}/\left\lfloor nv_{d-1}\right\rfloor \\\mathbf{m}athbf{x}_0 \end{pmatrix}\hspace{-1mm}\mathbf{1}_{\mathbf{m}athbf{s}um_{i=1}^d\mathbf{S}^i_{\left\lfloor nv_i\right\rfloor }=\left\lfloor n\mathbf{v}\right\rfloor -\mathbf{m}athbf{x}_0}\right]\hspace{-1mm},
\end{align*}
where the family $(\mathbf{m}athbf{S}_{\left\lfloor nv_i\right\rfloor}^{i})_{i=1\ldots d}$ is independent and is such that for each $i$, $\mathbf{m}athbf{S}_{\left\lfloor nv_i\right\rfloor}^{i}$ denotes the sum of $\left\lfloor nv_i\right\rfloor$ independent and identically distributed random variables with probability distribution $p_i$.
Let us consider the event $A_n=\mathbf{i}g\{ \mathbf{m}athbf{s}um_{i=1}^d\mathbf{S}^i_{\left\lfloorloor nv_i\right\rfloorloor }=\left\lfloor n\mathbf{v}\right\rfloor -\mathbf{m}athbf{x}_0\mathbf{i}g\}$. We define the covariance matrix $\mathbf{m}athbf{\Sigma}=\mathbf{m}athbf{s}um_{i=1}^dv_i\mathbf{m}athbf{\Sigma}^{i}$, which since $\mathbf{v}>\mathbf{m}athbf{0}$ is positive-definite under $(A_6)$.
Theorem 1.1 in \cite{Bent05} for nonidentically distributed independent variables ensures that $\mathbf{m}athbf{s}um_{i=1}^d ( \mathbf{S}^i_{\left\lfloor nv_i\right\rfloor }-\left\lfloor nv_i\right\rfloor \mathbf{m}^i)n^{-\frac{1}{2}}$ converges in distribution as $n\to+\infty$ to the multivariate normal distribution $\mathbf{m}athcal{N}_d\left(\mathbf{m}athbf{0},\mathbf{m}athbf{\Sigma} \right)$ with density $\phi$. Under $(A_5)$ we have
\[\limsup_n \frac{n}{\mathbf{m}in_{j=1\ldots d}\mathbf{m}athbf{s}um_{i=1}^d \frac{n_i}{d}\mathbf{m}athbf{s}um_{\mathbf{k}\in\mathbf{N}N}\mathbf{m}in\left( p_i\left( \mathbf{k}\right) , p_i\left( \mathbf{k}+\mathbf{m}athbf{e}_j\right) \right) }<+\infty,\]
which by Theorem 2.1 in \cite{DavMcDon} ensures the following local limit theorem for nonidentically distributed independent variables:
\begin{equation}\label{local}
\lim_{n\to\infty}\mathbf{m}athbf{s}up_{\mathbf{k}\in\mathbf{N}N}\left|n^{\frac{d}{2}}\mathbf{m}athbb{P}\left(\mathbf{m}athbf{s}um_{i=1}^d \mathbf{S}^i_{\left\lfloor nv_i\right\rfloor }=\mathbf{k}\right)-\phi\left(\frac{\mathbf{k}-\mathbf{m}athbf{s}um_{i=1}^d \left\lfloor nv_i\right\rfloor \mathbf{m}^i}{\mathbf{m}athbf{s}qrt{n}} \right) \right|=0.
\end{equation}
In the critical case, the left eigenvector $\mathbf{v}$ satisfies for each $j$, $v_j=\mathbf{m}athbf{s}um_{i=1}^d v_i m_{ij}$, hence $ 0\leqslant |\left\lfloorloor n v_j\right\rfloorloor -\mathbf{m}athbf{s}um_{i=1}^d\left\lfloorloor n v_i\right\rfloorloor m_{ij}|<\mathbf{m}ax(1,\mathbf{m}athbf{s}um_{i=1}^d m_{ij})$ and \eqref{local} implies in particular that
\begin{equation}\label{local2}
\lim_{n\to+\infty} n^{\frac{d}{2}}\mathbf{m}athbb{P}\left(A_n\right)=\phi\left( \mathbf{m}athbf{0}\right) =\frac{1 }{\left( 2\pi\right) ^{\frac{d}{2}}\left(\det \mathbf{m}athbf{\Sigma}\right) ^{\frac{1}{2}}}.
\end{equation}
Now, denoting by $\mathbf{m}athfrak{S}_d$ the symmetric group of order $d$ and by $\epsilon(\mathbf{m}athbf{s}igma)$ the signature of a permutation $\mathbf{m}athbf{s}igma\in\mathbf{m}athfrak{S}_d$, we obtain by Leibniz formula that
\begin{multline}\label{Leib} \left\lfloor nv_d\right\rfloor\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0 }\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)= \mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}igma\in \mathbf{m}athfrak{S}_d}\varepsilon\left( \mathbf{m}athbf{s}igma\right)x_{0,\mathbf{m}athbf{s}igma(d)} \mathbf{m}athbb{E}\Big[\prod_{i=1}^{d-1}\Big( \delta_{i,\mathbf{m}athbf{s}igma(i)}-\frac{S^i_{\left\lfloor nv_i\right\rfloor,\mathbf{m}athbf{s}igma(i) }}{\left\lfloor nv_i\right\rfloor}\Big)\Big]\\=\hspace{-4mm}\mathbf{m}athbf{s}um_{I\mathbf{m}athbf{s}ubset\{1,\ldots, d-1\}}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}igma\in \mathbf{m}athfrak{S}_d}\hspace{-2mm}\varepsilon\left( \mathbf{m}athbf{s}igma\right)x_{0,\mathbf{m}athbf{s}igma(d)} \mathbf{m}athbb{E}\Big[\prod_{i\in I}\Big(\hspace{-1mm}-\frac{S^i_{\left\lfloor nv_i\right\rfloor,\mathbf{m}athbf{s}igma(i) }}{\left\lfloor nv_i\right\rfloor}+ m_{i,\mathbf{m}athbf{s}igma(i)}\Big)\mathbf{1}_{A_n}\Big]\hspace{-1mm}\prod_{i\mathbf{m}athbf{n}otin I}( \delta_{i,\mathbf{m}athbf{s}igma(i)}-m_{i,\mathbf{m}athbf{s}igma(i) }).
\end{multline}
Let $\varepsilon>0$. Since on the event $A_n$ each $S_{\left\lfloor nv_i\right\rfloor,j}^{i}/\left\lfloor nv_i\right\rfloor$ is bounded, there exists some constant $A>0$ such that for each $i,j=1\ldots d$,
\begin{align*} \mathbf{m}athbb{E}\left(\left|\frac{S^i_{\left\lfloor nv_i\right\rfloor,j }}{\left\lfloor nv_i\right\rfloor}- m_{i,j}\right|\mathbf{1}_{A_n}\right)&\leqslant \varepsilon\mathbf{m}athbb{P}\left( A_n\right) +\frac{A}{\varepsilon^{d+1}}\mathbf{m}athbb{E}\left(\left|\frac{S^i_{\left\lfloor nv_i\right\rfloor,j}}{\left\lfloor nv_i\right\rfloor}- m_{i,j}\right|^{d+1}\right)\\&\leqslant \varepsilon\mathbf{m}athbb{P}\left( A_n\right) +\frac{AB}{\varepsilon^{d+1}\left\lfloor nv_i\right\rfloor^{\frac{d+1}{2}}}\mathbf{m}athbb{E}\left(\left|S^i_{1,j}- m_{i,j}\right|^{d+1}\right),
\end{align*}
for some constant $B>0$. The second inequality on the $d+1$-th central moment can be found for instance in \cite{DharJog69}, Theorem 2. From \eqref{local2} it thus appears that for each non-empty subset $I\mathbf{m}athbf{s}ubset\{1,\ldots, d-1\}$,
\[\lim_{n\to+\infty}n^{\frac{d}{2}}\hspace{-2mm}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}igma\in \mathbf{m}athfrak{S}_d}\hspace{-2mm}\varepsilon\left( \mathbf{m}athbf{s}igma\right)x_{0,\mathbf{m}athbf{s}igma(d)} \mathbf{m}athbb{E}\Big[\prod_{i\in I}\Big(-\frac{S^i_{\left\lfloor nv_i\right\rfloor,\mathbf{m}athbf{s}igma(i) }}{\left\lfloor nv_i\right\rfloor}+ m_{i,\mathbf{m}athbf{s}igma(i)}\Big)\mathbf{1}_{A_n}\Big]\hspace{-1mm}\prod_{i\mathbf{m}athbf{n}otin I} \left( \delta_{i,\mathbf{m}athbf{s}igma(i)}-m_{i,\mathbf{m}athbf{s}igma(i) }\right)\hspace{-1mm}=0.\] Consequently, considering the remaining term in \eqref{Leib} corresponding to $I=\emptyset$, we obtain that
\begin{align*}&\lim_{n\to+\infty}n^{\frac{d}{2}+1}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0 }\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)\\&\ \ \ \ =\lim_{n\to+\infty}n^{\frac{d}{2}} \mathbf{m}athbb{P}\left(A_n\right)\frac{1}{v_d}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}igma\in \mathbf{m}athfrak{S}_d}\varepsilon\left( \mathbf{m}athbf{s}igma\right)x_{0,\mathbf{m}athbf{s}igma(d)}\prod_{i=1}^{d-1} \left( \delta_{i,\mathbf{m}athbf{s}igma(i)}-m_{i,\mathbf{m}athbf{s}igma(i)}\right) \\&\ \ \ \ = \frac{1 }{v_d\left( 2\pi\right) ^{\frac{d}{2}}\left(\det \mathbf{m}athbf{\Sigma}\right) ^{\frac{1}{2}}}\det\begin{pmatrix}\mathbf{m}athbf{e}_1-\mathbf{m}^1\\\cdots\\ \mathbf{m}athbf{e}_{d-1}-\mathbf{m}^{d-1} \\\mathbf{m}athbf{x}_0 \end{pmatrix}=\frac{\mathbf{m}athbf{x}_0\cdot \mathbf{m}athbf{D}}{v_d\left( 2\pi\right) ^{\frac{d}{2}}\left(\det \mathbf{m}athbf{\Sigma}\right) ^{\frac{1}{2}}},
\end{align*}
where $\mathbf{m}athbf{D}=(D_1,\ldots,D_d)$ is such that $D_i$ is the $(d,i)$-th cofactor of the matrix $\mathbf{m}athbf{I}-\mathbf{M}$.
The criticality of $\left(\mathbf{X}_k\right)_{k\geqslant 0} $ implies that $\det\left( \mathbf{m}athbf{I}-\mathbf{M}\right)=( \mathbf{m}athbf{e}_d-\mathbf{m}^d )\cdot \mathbf{m}athbf{D}=0$. Moreover, for each $j=1\ldots d-1$,
$( \mathbf{m}athbf{e}_j-\mathbf{m}^j )\cdot \mathbf{m}athbf{D}$ corresponds to the determinant of $ \mathbf{m}athbf{I}-\mathbf{M}$ in which the $d$-th row has been replaced by the $j$-th row, and is consequently null. We have thus proven that for each $j=1\ldots d$, $( \mathbf{m}athbf{e}_j-\mathbf{m}^j )\cdot \mathbf{m}athbf{D}=0$, or equivalently that $\mathbf{m}athbf{s}um_{i=1}^d m_{ji} D_i=D_j.$ Hence $\mathbf{m}athbf{D}$ is a right eigenvector of $\mathbf{M}$ for the Perron's root 1, which implies the existence of some nonnull constant $c$ such that $\mathbf{m}athbf{D}=c\mathbf{u}$, leading to the desired result.
\end{proof}
\textit{Proof of Theorem \ref{thm2}}
Let us assume that $\left(\mathbf{X}_k\right)_{k\geqslant 0}$ is critical and satisfies $(A_1)$, $(A_5)$ and $(A_6)$. Let $\mathbf{m}athbf{x}_0\in\mathbf{N}N$, $k_1\leqslant\ldots\leqslant k_j\in\mathbf{m}athbb{N}$, and $\mathbf{m}athbf{x}_1,\ldots,\mathbf{m}athbf{x}_j\in\mathbf{N}N$ and let us show that
\begin{equation}\label{toprove}
\lim_{n\to+\infty}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\mathbf{i}g)\\=\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{i}g).
\end{equation}
Let $\frac{3}{4}<\varepsilon<1$. The Markov property entails that
\begin{multline}\label{second term}
\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{m}id\mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)
\\=\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{m}athbf{l}\in\mathbf{N}N\\\mathbf{m}athbf{l}< \left\lfloorloor n^{\varepsilon}\mathbf{v} \right\rfloorloor}} \mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j,\mathbf{N}_{k_j}=\mathbf{m}athbf{l}\right)\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left(\mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor-\mathbf{m}athbf{l}+\mathbf{m}athbf{x}_j\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)}\\\ \ +\mathbf{m}athbf{s}um_{\mathbf{m}athbf{s}ubstack{\mathbf{m}athbf{l}\in\mathbf{N}N\\ \left\lfloorloor n^{\varepsilon}\mathbf{v} \right\rfloorloor\leqslant\mathbf{m}athbf{l}\leqslant \left\lfloor n\mathbf{v} \right\rfloor}} \mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j,\mathbf{N}_{k_j}=\mathbf{m}athbf{l}\right)\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left(\mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor-\mathbf{m}athbf{l}+\mathbf{m}athbf{x}_j\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)}.
\end{multline}
Note that \eqref{local} ensures that
\[\lim_{n} n^{\frac{d}{2}}\mathbf{m}athbb{P}\left(\mathbf{m}athbf{s}um_{i=1}^d \mathbf{S}^i_{\left\lfloor nv_i\right\rfloor -l_i+x_{j,i}}=\left\lfloor n\mathbf{v}\right\rfloor -\mathbf{m}athbf{l}\right)=\frac{1 }{\left( 2\pi\right) ^{\frac{d}{2}}\left(\det \mathbf{m}athbf{\Sigma}\right) ^{\frac{1}{2}}}, \]
uniformly in $\mathbf{m}athbf{l}< \left\lfloorloor n^{\varepsilon}\mathbf{v} \right\rfloorloor$, and that the proof of Proposition \ref{prop: convergence} can be used to show that
\[
\lim_{n\to+\infty}n^{\frac{d}{2}+1}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j }\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor-\mathbf{m}athbf{l}+\mathbf{m}athbf{x}_j\right)=\frac{C\mathbf{m}athbf{x}_j\cdot \mathbf{u} }{v_d\left( 2\pi\right) ^{\frac{d}{2}}\left(\det \mathbf{m}athbf{\Sigma}\right) ^{\frac{1}{2}}},
\]uniformly in $\mathbf{m}athbf{l}< \left\lfloorloor n^{\varepsilon}\mathbf{v} \right\rfloorloor$. Together with Proposition \ref{prop: convergence}, this shows that the first sum in \eqref{second term} converges to
\begin{equation}\label{hop}
\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}}\mathbf{m}athbf{s}um_{\mathbf{m}athbf{l}\in\mathbf{N}N} \mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j,\mathbf{N}_{k_j}=\mathbf{m}athbf{l}\mathbf{i}g)\\=\frac{\mathbf{m}athbf{x}_j\cdot\mathbf{u}}{\mathbf{m}athbf{x}_0\cdot\mathbf{u}}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\mathbf{i}g( \mathbf{X}_{k_1}=\mathbf{m}athbf{x}_1,\ldots, \mathbf{X}_{k_j}=\mathbf{m}athbf{x}_j\mathbf{i}g)
\end{equation}
as $n\to+\infty$. The second sum in \eqref{second term} can be bounded by
\begin{align*}
\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left(\mathbf{N}_{k_j}\geqslant \left\lfloor n^{\varepsilon}\mathbf{v}\right\rfloor\right) }{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)}&\leqslant
\frac{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_j}\left(\|\mathbf{N}_{k_j}\|_1^{d+1}\geqslant n^{(d+1)\varepsilon}\|\mathbf{v}\|_1^{d+1}\right)}{\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)}\\
&\leqslant \frac{\mathbf{m}athbb{E}_{\mathbf{m}athbf{x}_j}\left(\|\mathbf{N}_{k_j}\|_1^{d+1}\right)}{\|\mathbf{v}\|_1^{d+1}n^{(d+1)\varepsilon}\mathbf{m}athbb{P}_{\mathbf{m}athbf{x}_0}\left( \mathbf{N}=\left\lfloor n\mathbf{v} \right\rfloor\right)}.
\end{align*}
Thanks to $(A_5)$, the moments of order $d+1$ of the finite sum $\mathbf{N}_{k_j}$ are finite, and since $(d+1)\varepsilon>\frac{d}{2}+1$, the right term of the last inequality converges to 0 as $n\to +\infty$ thanks to Proposition \ref{prop: convergence}. This together with \eqref{hop} in \eqref{second term} finally proves \eqref{toprove}.
\end{document} |
\begin{document}
\title{A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds}
\begin{abstract}
In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds
of dimension \(m=\dim M\geq 3\) with infinite volume that admit an Euclidean type Sobolev inequality
into general Riemannian manifolds by assuming finiteness of $\|\tau(\phi)\|_{L^p(M)}, p>1$ and smallness of $\|d\phi\|_{L^m(M)}$.
This is an improvement of a recent result of the first named author, where he assumed $2<p<m$.
As applications we also get several nonexistence results for proper biharmonic submersions from complete non-compact manifolds into general Riemannian manifolds.
\end{abstract}
\,\,\,\,ection{Introduction}
Let $(M,g)$ be a Riemannian manifold and $(N,h)$ a Riemannian manifold without boundary. For a $W^{1,2}(M,N)$ map $\phi$, the energy density of $\phi$ is defined by
$$ e(\phi)=|d\phi|^2=\rm{Tr_g}(\phi^\ast h),$$
where $\phi^\ast h$ is the pullback of the metric tensor $h$. The energy functional of the map $\phi$ is defined as $$E(\phi)=\frac{1}{2}\int_Me(\phi)dv_g.$$
The Euler-Lagrange equation of $E(\phi)$ is $\tau(\phi)=\rm Tr_g\bar{\nabla} d\phi=0$ and $\tau(\phi)$ is called the \textbf{tension field} of $\phi$. A map is called a \textbf{harmonic map} if $\tau(\phi)=0$.
The theory of harmonic maps has many important applications in various fields of differential geometry, including minimal surface theory, complex geometry, see \cite{SY} for a survey.
Much effort has been paid in the last several decades to generalize the notion of harmonic maps. In 1983, Eells and Lemaire (\cite{EL}, see also \cite{ES}) proposed to consider the bienergy functional
$$E_2(\phi)=\frac{1}{2}\int_M|\tau(\phi)|^2dv_g$$ for smooth maps between Riemannian manifolds. Stationary points of the bienergy functional are called \textbf{biharmonic maps}.
We see that harmonic maps are biharmonic maps and even more, minimizers of the bienergy functional. In 1986, Jiang \cite{Ji} derived the first and second variational formulas of the bienergy functional and initiated the study of biharmonic maps. The Euler-Lagrange equation of $E_2(\phi)$ is given by
$$\tau_2(\phi):=-\Delta^\phi\tau(\phi)-\,\,\,\,um_{i=1}^mR^N(\tau(\phi), d\phi(e_i))d\phi(e_i)=0,$$
where $\Delta^\phi:=\,\,\,\,um_{i=1}^m(\bar{\nabla}_{e_i}\bar{\nabla}_{e_i}-\bar{\nabla}_{\nabla_{e_i}e_i})$. Here, $\nabla$ is the Levi-Civita connection on $(M,g)$, $\bar{\nabla}$ is the induced connection on the pullback bundle $\phi^{\ast}TN$, and $R^N$ is the Riemannian curvature tensor on $N$.
The first nonexistence result for biharmonic maps was obtained by Jiang \cite{Ji}. He proved that biharmonic maps from a compact, orientable Riemannian manifold into a Riemannian manifold of nonpositive curvature are harmonic.
Jiang's theorem is a direct application of the Weitzenb\"ock formula. If $\phi$ is biharmonic, then
\begin{eqnarray*}
-\frac{1}{2}\Delta|\tau(\phi)|^2&=&\langlegle-\Delta^\phi\tau(\phi), \tau(\phi)\ranglegle-|\bar{\nabla}\tau(\phi)|^2
\\&=&Tr_g\langlegle R^N(\tau(\phi), d\phi)d\phi, \tau(\phi)\ranglegle-|\bar{\nabla}\tau(\phi)|^2
\\&\leq&0.
\end{eqnarray*}
The maximum principle implies that $|\tau(\phi)|^2$ is constant. Therefore $\bar{\nabla}\tau(\phi)=0$ and so by
$$div\langlegle d\phi, \tau(\phi)\ranglegle=|\tau(\phi)|^2+\langlegle d\phi, \bar{\nabla}\tau(\phi)\ranglegle,$$
we deduce that $div\langlegle d\phi, \tau(\phi)\ranglegle=|\tau(\phi)|^2$. Then, by the divergence theorem, we have $\tau(\phi)=0$. Generalizations of this result by making use of similar ideas are given in \cite{On}.
If $M$ is non-compact, the maximum principle is no longer applicable. In this case, Baird et al. \cite{BFO} proved that biharmonic maps
from a complete non-compact Riemannian manifold with nonnegative Ricci curvature into a nonpositively curved manifold with finite bienergy are harmonic.
It is natural to ask whether we can abandon the curvature restriction on the domain manifold and weaken the integrability condition on the bienergy.
In this direction, Nakauchi et al. \cite{NUG} proved that biharmonic maps from a complete manifold to a nonpositively curved manifold are harmonic if ($p=2$)
\\(i) $\int_M|d\phi|^2dv_g<\infty$ and $\int_M|\tau(\phi)|^pdv_g<\infty$, or
\\(ii) $Vol(M, g)=\infty$ and $\int_M|\tau(\phi)|^pdv_g<\infty.$
Later Maeta \cite{Ma} generalized this result by assuming that $p\geq2$ and further generalizations are given by the second named author in \cite{Luo1}, \cite{Luo2}.
Recently, the first named author proved a nonexistence result for proper biharmonic maps from complete non-compact manifolds into general target manifolds \cite{Ba}, by only assuming that the sectional curvatures of the target manifold have an upper bound.
Explicitly, he proved the following theorem.
\begin{thm}[Branding]\label{Bra}
Suppose that $(M,g)$ is a complete non-compact Riemannian manifold of dimension \(m=\dim M\geq 3\)
whose Ricci curvature is bounded from below and with positive injectivity radius.
Let $\phi: (M,g)\to (N,h)$ be a smooth biharmonic map, where \(N\) is another Riemannian manifold.
Assume that the sectional curvatures of $N$ satisfy $K^N\leq A,$ where $A$ is a positive constant.
If $$\int_M|\tau(\phi)|^pdv_g<\infty$$ and $$\int_M|d\phi|^mdv_g<\epsilon$$ for $2<p<m$ and $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)) sufficiently small, then $\phi$ must be harmonic.
\end{thm}
The central idea in the proof of Theorem \ref{Bra}
is the use of an \emph{Euclidean type Sobolev inequality} that allows to control the curvature term
in the biharmonic map equation. However, in order for this inequality to hold one has to make stronger
assumptions on the domain manifold \(M\) as in Theorem \ref{Bra}, which we will correct below.
We say that a complete non-compact Riemannian manifold of infinite volume admits
an \emph{Euclidean type Sobolev inequality} if the following inequality holds (assuming \(m=\dim M\geq 3\))
\begin{align}
\label{sobolev-inequality}
(\int_M|u|^{2m/(m-2)}dv_g)^\frac{m-2}{m}\leq C_{sob}^M\int_M|\nabla u|^2dv_g
\end{align}
for all \(u\in W^{1,2}(M)\) with compact support,
where \(C_{sob}^M\) is a positive constant that depends on the geometry of \(M\).
Such an inequality holds in \(\mathbb{R}^m\) and is well-known as \emph{Gagliardo-Nirenberg inequality} in this case.
One way of ensuring that \eqref{sobolev-inequality} holds is the following:
If \((M,g)\) is a complete, non-compact Riemannian manifold of dimension \(m\)
with nonnegative Ricci curvature, and if for some point \(x\in M\)
\begin{align*}
\lim_{R\to\infty}\frac{vol_g(B_R(x))}{\omega_mR^m}>0
\end{align*}
holds, then \eqref{sobolev-inequality} holds true, see \cite{Sh}.
Here, \(\omega_m\) denotes the volume of the unit ball in \(\mathbb{R}^m\).
For further geometric conditions ensuring that \eqref{sobolev-inequality} holds
we refer to \cite[Section 3.7]{He}.
In this article we will correct the assumptions that are needed for Theorem \ref{Bra} to hold
and extend it to the case of $p=2$, which is a more natural integrability condition.
Motivated by these aspects, we actually can prove the following result:
\begin{thm}\label{main1}
Suppose that $(M,g)$ is a complete, connected non-compact Riemannian manifold of dimension \(m=\dim M\geq 3\) with infinite volume that admits
an Euclidean type Sobolev inequality of the form \eqref{sobolev-inequality}.
Moreover, suppose that \((N,h)\) is another Riemannian manifold
whose sectional curvatures satisfy $K^N\leq A,$ where $A$ is a positive constant.
Let $\phi: (M,g)\to (N,h)$ be a smooth biharmonic map.
If $$\int_M|\tau(\phi)|^pdv_g<\infty$$ and $$\int_M|d\phi|^mdv_g<\epsilon$$
for $p>1$ and $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)) sufficiently small, then $\phi$ must be harmonic.
\end{thm}
Similar ideas have been used to derive Liouville type results for \(p\)-harmonic maps in \cite{NT}, see also \cite{zc} for a more general result.
In the proof we choose a test function of the form $(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\tau(\phi), (p>1, \delta>0)$ to avoid problems that may be caused by the zero points of $\tau(\phi)$.
When we take the limit $\delta\to 0$, we also need to be careful about the set of zero points of $\tau(\phi)$, and a delicate analysis is given.
For details please see the proof in section 2.
Moreover, we can get the following Liouville type result.
\begin{thm}\label{main2}
Suppose that $(M,g)$ is a complete, connected non-compact Riemannian manifold of \(m=\dim M\geq 3\) with nonnegative Ricci curvature that admits
an Euclidean type Sobolev inequality of the form \eqref{sobolev-inequality}.
Moreover, suppose that \((N,h)\) is another Riemannian manifold
whose sectional curvatures satisfy $K^N\leq A,$ where $A$ is a positive constant.
Let $\phi: (M,g)\to (N,h)$ be a smooth biharmonic map.
If $$\int_M|\tau(\phi)|^pdv_g<\infty$$ and $$\int_M|d\phi|^mdv_g<\epsilon$$
for $p>1$ and $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)) sufficiently small, then $\phi$ is a constant map.
\end{thm}
Note that due to a classical result of Calabi and Yau \cite[Theorem 7]{Yau} a complete non-compact Riemannian manifold
with nonnegative Ricci curvature has infinite volume.
\begin{rem}
Due to Theorem \ref{main1} we only need to prove that harmonic maps satisfying the assumption of Theorem \ref{main2} are constant maps.
Such a result was proven in \cite{NT} and thus Theorem \ref{main2} is a corollary of Theorem 1.5 in \cite{NT}. Conversely, Theorem \ref{main2} generalizes related Liouville type results for harmonic maps in \cite{NT}.
\end{rem}
\quad \\
\textbf{Organization:} Theorem \ref{main1} is proved in section 2.
In section 3 we apply Theorem \ref{main1} to get several nonexistence results for proper biharmonic submersions.
\,\,\,\,ection{Proof of the main result}
In this section we will prove Theorem \ref{main1}.
Assume that $x_0\in M$. We choose a cutoff function $0\leq\eta\leq1$ on $M$ that satisfies
\begin{equation}\label{flow1}
\left\{\begin{array}{rcl}
\eta(x)&=&1, \quad \forall \ x\in B_R(x_0), \\
\eta(x)&=&0,\quad \forall \ x\in M\,\,\,\,etminus B_{2R}(x_0),\\
|\nabla\eta(x)|&\leq& \frac{C}{R}, \quad \forall \ x \in M.
\end{array}\right.
\end{equation}
\begin{lem}\label{lem1}
Let $\phi:(M,g)\to (N,h)$ be a smooth biharmonic map and assume that the sectional curvatures of $N$ satisfy $K^N\leq A.$
Let $\delta$ be a positive constant. Then the following inequalities hold.
\\(1) If $1<p<2$, we have
\begin{eqnarray}\label{ine1}
&&(1-\frac{p-1}{2})\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g \nonumber
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R}(x_0)}(|\tau(\phi)|^2+\delta)^\frac{p}{2}dv_g\nonumber
\\&-&(p-2)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g;
\end{eqnarray}
\\(2) If $p\geq2$, we have
\begin{eqnarray}\label{ine1'}
&&\frac{1}{2}\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g \nonumber
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R}(x_0)}(|\tau(\phi)|^2+\delta)^\frac{p}{2}dv_g.
\end{eqnarray}
\end{lem}
\proof Multiplying the biharmonic map equation by $\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\tau(\phi)$ we get
$$\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\Delta^\phi\tau(\phi), \tau(\phi)\ranglegle=-\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\,\,\,\,um_{i=1}^mR^N(\tau(\phi),d\phi(e_i),\tau(\phi),d\phi(e_i)).$$
Integrating over $M$ and using integration by parts we get
\begin{eqnarray}\label{inem1}
&&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\Delta^\phi\tau(\phi), \tau(\phi)\ranglegle dv_g\nonumber
\\&=&-2\int_M(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle\eta\nabla\eta dv_g\nonumber
\\&-&(p-2)\int_M\eta^2|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}dv_g\nonumber
\\&-&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}|\bar{\nabla}\tau(\phi)|^2dv_g\nonumber
\\&\leq&-2\int_M(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle\eta\nabla\eta dv_g
\\&-&(p-2)\int_M\eta^2|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}dv_g\nonumber
\\&-&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g.\nonumber
\end{eqnarray}
Therefore when $1<p<2$ we have
\begin{eqnarray*}
&&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\Delta^\phi\tau(\phi), \tau(\phi)\ranglegle dv_g
\\&\leq&(\frac{p-1}{2}-1)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&+&\frac{2}{p-1}\int_M(|\tau(\phi)|^2+\delta)
^\frac{p}{2}|\nabla\eta|^2dv_g
\\&-&(p-2)\int_M\eta^2|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}dv_g
\\&\leq& (\frac{p-1}{2}-1)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&+&\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g
\\&-&(p-2)\int_M\eta^2|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}dv_g
\\&\leq& (\frac{p-1}{2}-1)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&+&\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g
\\&-&(p-2)\int_M\eta^2|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}dv_g,
\end{eqnarray*}
where in the last inequality we used $1<p<2$ and
$$|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2\leq|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2.$$
Therefore we find
\begin{eqnarray*}
&&(1-\frac{p-1}{2})\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g \nonumber
\\&\leq& \int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\,\,\,\,um_{i=1}^mR^N(\tau(\phi),d\phi(e_i),\tau(\phi),d\phi(e_i))dv_g
\\&+&\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g\nonumber
\\&-&(p-2)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}|\tau(\phi)|^2|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g\nonumber
\\&-&(p-2)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g\nonumber
\\&-&(p-2)\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g,
\end{eqnarray*}
which proves the first claim.\\
When $p\geq2$ equation \eqref{inem1} gives
\begin{eqnarray*}
&&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\Delta^\phi\tau(\phi), \tau(\phi)\ranglegle dv_g
\\&\leq&-2\int_M(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle\eta\nabla\eta dv_g
\\&-&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&\leq&-\frac{1}{2}\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g+2\int_M(|\tau(\phi)|^2+\delta)
^\frac{p}{2}|\nabla\eta|^2dv_g.
\end{eqnarray*}
Therefore we have
\begin{eqnarray*}
&&\frac{1}{2}\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\\&\leq&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}\,\,\,\,um_{i=1}^mR^N(\tau(\phi),d\phi(e_i),\tau(\phi),d\phi(e_i))dv_g+\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-2}{2}|\tau(\phi)|^2|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g
\\&\leq& A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g+\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)
^\frac{p}{2}dv_g.
\end{eqnarray*}
This completes the proof of Lemma \ref{lem1}.
$
\Box$\\
In the following we will estimate the term $$A\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g.$$
\begin{lem}\label{lem2}
Assume that $(M,g)$ satisfies the assumptions of Theorem \ref{main1}. Then the following inequality holds
\begin{eqnarray}\label{ine2}
&&\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g\nonumber
\\&\leq & C(\int_M|d\phi|^mdv_g)^{\frac{2}{m}}\times
\\&&(\frac{1}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)^\frac{p}{2}dv_g\nonumber
+\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g),
\end{eqnarray}
where $C$ is a constant depending on $p,A$ and the geometry of $M$.
\end{lem}
\proof Set $f=(|\tau(\phi)|^2+\delta)^\frac{p}{4}$, then we have
$$\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p}{2}|d\phi|^2dv_g=\int_M\eta^2f^2|d\phi|^2dv_g.$$ Then by H\"older's inequality we get
$$\int_M\eta^2f^2|d\phi|^2dv_g\leq (\int_M(\eta f)^\frac{2m}{m-2}dv_g)^\frac{m-2}{m}(\int_M|d\phi|^mdv_g)^\frac{2}{m}.$$
Applying \eqref{sobolev-inequality} to $u=\eta f$ we get
$$(\int_M(\eta f)^\frac{2m}{m-2}dv_g)^\frac{m-2}{m}\leq C^M_{sob}\int_M|d(\eta f)|^2dv_g,$$
which leads to
\begin{eqnarray}\label{ine7}
\int_M\eta^2f^2|d\phi|^2dv_g\leq 2C^M_{sob}(\int_M|d\phi|^mdv_g)^{\frac{2}{m}}(\int_M|d\eta|^2f^2dv_g+\int_M\eta^2|df|^2dv_g).
\end{eqnarray}
Note that $f=(|\tau(\phi)|^2+\delta)^\frac{p}{4}$ and $$|df|^2=\frac{p^2}{4}(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\langlegle\bar{\nabla}\tau(\phi), \tau(\phi)\ranglegle|^2\leq \frac{p^2}{4}(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2.$$
This completes the proof of Lemma \ref{lem2}.
$
\Box$\\
When $1<p<2$, due to Lemmas \ref{lem1}, \ref{lem2}, we see that by choosing $\epsilon$ sufficiently small such that $AC\epsilon^\frac{2}{m}\leq\frac{p-1}{4},$ we have
\begin{eqnarray}\label{ine3}
\frac{p-1}{4}\int_M\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\leq\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)^\frac{p}{2}dv_g,
\end{eqnarray}
where $C$ is a constant depending on $p,A$ and the geometry of $M$.
Now, set $M_1:=\{x\in M| \tau(\phi)(x)=0\}$, and $M_2=M\,\,\,\,etminus M_1.$
If $M_2$ is an empty set, then we are done. Hence we assume that $M_2$ is nonempty and we will get a contradiction below.
Note that since $\phi$ is smooth, $M_2$ is an open set.
From \eqref{ine3} we have
\begin{eqnarray}
\frac{p-1}{4}\int_{M_2}\eta^2(|\tau(\phi)|^2+\delta)^\frac{p-4}{2}|\bar{\nabla}\tau(\phi)|^2|\tau(\phi)|^2dv_g
\leq\frac{C}{R^2}\int_{B_{2R(x_0)}}(|\tau(\phi)|^2+\delta)^\frac{p}{2}dv_g.
\end{eqnarray}
Letting $\delta\to 0$ we get
$$\frac{p-1}{4}\int_{M_2}\eta^2|\tau(\phi)|^{p-2}|\bar{\nabla}\tau(\phi)|^2dv_g\leq\frac{C}{R^2}\int_{B_{2R(x_0)}}|\tau(\phi)|^pdv_g\leq
\frac{C}{R^2}\int_{M}|\tau(\phi)|^pdv_g.$$
Letting $R\to \infty$ we get
$$\frac{p-1}{4}\int_{M_2}|\tau(\phi)|^{p-2}|\bar{\nabla}\tau(\phi)|^2dv_g=0.$$
When $p\geq2$ by a similar discussion we can prove that
$$\frac{1}{4}\int_{M_2}|\tau(\phi)|^{p-2}|\bar{\nabla}\tau(\phi)|^2dv_g=0.$$
Therefore we have that $\bar{\nabla}\tau(\phi)=0$ everywhere in $M_2$ and hence $M_2$ is an \textbf{open} and \textbf{closed} nonempty set, thus $M_2=M$ (as we assume that $M$ is a connected manifold)
and $|\tau(\phi)|\equiv c$ for some constant $c\neq 0$. Thus $Vol(M)<\infty$ by $\int_Mc^pdv_g<\infty$.
In the following we will need Gaffney's theorem \cite{Ga}, stated below:
\begin{thm}[Gaffney]
Let $(M, g)$ be a complete Riemannian manifold. If a $C^1$ 1-form $\omega$ satisfies
that $\int_M|\omega|dv_g<\infty \ and \ \int_M|\delta\omega| dv_g<\infty,$ or equivalently, a $C^1$ vector field $X$ defined by
$\omega(Y) = \langlegle X, Y \ranglegle, (\forall Y \in TM)$ satisfies that $\int_M|X|dv_g<\infty \ and \ \int_M|div X|dv_g<\infty,$ then $$\int_M\delta\omega dv_g=\int_Mdiv Xdv_g=0.$$
\end{thm}
Define a l-form on $M$ by
$$\omega(X):=\langlegle d\phi(X),\tau(\phi)\ranglegle,~(X\in TM).$$
Then
\begin{eqnarray*}
\int_M|\omega|dv_g&=&\int_M(\,\,\,\,um_{i=1}^m|\omega(e_i)|^2)^\frac{1}{2}dv_g
\\&\leq&\int_M|\tau(\phi)||d\phi|dv_g
\\&\leq&c Vol(M)^{1-\frac{1}{m}}(\int_M|d\phi|^mdv_g)^\frac{1}{m}
\\&<&\infty.
\end{eqnarray*}
In addition, we calculate $-\delta\omega=\,\,\,\,um_{i=1}^m(\nabla_{e_i}\omega)(e_i)$:
\begin{eqnarray*}
-\delta\omega&=&\,\,\,\,um_{i=1}^m\nabla_{e_i}(\omega(e_i))-\omega(\nabla_{e_i}e_i)
\\&=&\,\,\,\,um_{i=1}^m\{\langlegle\bar{\nabla}_{e_i}d\phi(e_i),\tau(\phi)\ranglegle
-\langlegle d\phi(\nabla_{e_i}e_i),\tau(\phi)\ranglegle\}
\\&=&\,\,\,\,um_{i=1}^m\langlegle \bar{\nabla}_{e_i}d\phi(e_i)-d\phi(\nabla_{e_i}e_i),\tau(\phi)\ranglegle
\\&=&|\tau(\phi)|^2,
\end{eqnarray*}
where in the second equality we used $\bar{\nabla}\tau(\phi)=0$. Therefore $$\int_M|\delta\omega|dv_g=c^2Vol(M)<\infty.$$
Now by Gaffney's theorem and the above equality we have that
$$0=\int_M(-\delta\omega)dv_g=\int_M|\tau(\phi)|^2dv_g=c^2Vol(M),$$
which implies that $c=0$, a contradiction. Therefore we must have $M_1=M$, i.e. $\phi$ is a harmonic map. This completes the proof of Theorem \ref{main1}.
$
\Box$\\
\,\,\,\,ection{Applications to biharmonic submersions}
In this section we give some applications of our result to biharmonic submersions.
First we recall some definitions \cite{BW}.
Assume that $\phi: (M, g)\to (N, h)$ is a smooth map between Riemannian manifolds and $x\in M$. Then $\phi$ is called {\bf horizontally weakly conformal} if either
(i) $d\phi_x=0$, or
(ii) $d\phi_x$ maps the horizontal space $\rm \mathcal{H}_x=\{Ker~d\phi_x\}^\bot$ conformally \textbf{onto} $T_{\phi(x)}N$, i.e.
$$h(d\phi_x(X), d\phi_x(Y))=\lambda^2 g(X, Y), (X, Y\in \mathcal{H}_x),$$
for some $\lambda=\lambda(x)>0,$ called the {\bf dilation} of $\phi$ at $x$.
A map $\phi$ is called {\bf horizontally weakly conformal} or {\bf semiconformal} on $M$ if it is horizontally weakly conformal at every point of $M$. Furthermore, if $\phi$ has no critical points, then we call it a {\bf horizontally conformal submersion}: In this case the dilation $\lambda:M \to (0,\infty)$ is a smooth function. Note that if $\phi: (M, g)\to (N, h)$ is a horizontally weakly conformal map and $\dim M<\dim N$, then $\phi$ is a constant map.
If for every harmonic function $f: V\to \mathbb{R}$ defined on an open subset $V$ of $N$ with $\phi^{-1}(V)$ nonempty, the composition $f\circ\phi$ is harmonic on $\phi^{-1}(V)$, then $\phi$ is called a {\bf harmonic morphism}. Harmonic morphisms are characterized as follows (cf. \cite{Fu, Is}).
\begin{thm}[\cite{Fu, Is}]\label{thm4}
A smooth map $\phi: (M, g)\to (N, h)$ between Riemannian manifolds is a harmonic morphism if and only if $\phi$ is both harmonic and horizontally weakly conformal.
\end{thm}
When $\phi:(M^m, g)\to (N^n, h),(m>n\geq2)$ is a horizontally conformal submersion, the tension field is given by
\begin{eqnarray}\label{eq5}
\tau(\phi)=\frac{n-2}{2}\lambda^2d\phi(grad_\mathcal{H}(\frac{1}{\lambda^2}))
-(m-n)d\phi(\hat{H}),
\end{eqnarray}
where $grad_\mathcal{H}(\frac{1}{\lambda^2})$ is the horizontal component of $\rm grad(\frac{1}{\lambda^2})$, and $\hat{H}$ is the {\bf mean curvature} of the fibres given by the trace
$$\hat{H}=\frac{1}{m-n}\,\,\,\,um_{i=n+1}^m\mathcal{H}(\nabla_{e_i}e_i).$$
Here, $\{e_i, i=1,...,m\}$ is a local orthonormal frame field on $M$ such that $\{e_{i}, i=1,...,n\}$ belongs to $\mathcal{H}_x$ and $\{e_{j}, j=n+1,...,m \}$ belongs to $\mathcal{V}_x$ at each point $x\in M$, where $T_xM=\mathcal{H}_x\oplus \mathcal{V}_x$.
Nakauchi et al. \cite{NUG}, Maeta \cite{Ma} and Luo \cite{Luo2} applied their nonexistence result for biharmonic maps to get conditions for which biharmonic submersions are harmonic morphisms.
Here, we give another such result by using Theorem \ref{main1}. We have
\begin{pro}
Let $\phi:(M^m, g)\to (N^n, h), (m>n\geq2)$ be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold $(M,g)$ with infinite volume,
that admits an Euclidean Sobolev type inequality of the form \eqref{sobolev-inequality},
into a Riemannian manifold $(N, h)$ with sectional curvatures $K^N\leq A$ and $p$ a real constant satisfying $1<p<\infty$. If
$$ \int_M\lambda^p|\frac{n-2}{2}\lambda^2grad_\mathcal{H}(\frac{1}{\lambda^2})
-(m-n)\hat{H}|_g^pdv_g<\infty,$$
and $$\int_M\lambda^mdv_g<\epsilon$$ for sufficiently small $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)), then $\phi$ is a harmonic morphism.
\end{pro}
\proof By (\ref{eq5}) we have,
$$\int_M|\tau(\phi)|_h^pdv_g=\int_M\lambda^p|\frac{n-2}{2}\lambda^2grad_\mathcal{H}(\frac{1}
{\lambda^2})
-(m-n)\hat{H}|_g^pdv_g<\infty,$$
and since $|d\phi(x)|^2=n\lambda^2(x)$, we get that $\phi$ is harmonic by Theorem \ref{main1}. Since $\phi$ is also a horizontally conformal submersion, $\phi$ is a harmonic morphism by Theorem \ref{thm4}. $
\Box$\\
In particular, if $\dim N=2$, we have
\begin{cor}
Let $\phi:(M^m, g)\to (N^2, h)$ be a biharmonic horizontally conformal submersion from a complete, connected non-compact Riemannian manifold $(M, g)$ with infinite volume,
that admits an Euclidean Sobolev type inequality of the form \eqref{sobolev-inequality},
into a Riemannian surface $(N, h)$ with Gauss curvature bounded from above and $p$ a real constant satisfying $1<p<\infty$. If
$$\int_M\lambda^p|\hat{H}|_g^pdv_g<\infty,$$
and $$\int_M\lambda^mdv_g<\epsilon$$ for sufficiently small $\epsilon>0$ (depending on $p,A$ and the geometry of \(M\)), then $\phi$ is a harmonic morphism.
\end{cor}
\quad\\
\textbf{Acknowledgements:}
The first named author gratefully acknowledges the support of the Austrian Science Fund (FWF)
through the project P30749-N35 ``Geometric variational problems from string theory''.
The second named author is supported by the NSF of China(No.11501421, No.11771339). Part of the work was finished when the second named author
was a visiting scholar at Tsinghua University. He would like to express his gratitude to
Professor Yuxiang Li and Professor Hui Ma for their invitation and to Tsinghua University for their hospitality. The second named author also would like to thank Professor Ye Lin Ou for his interest in this work and discussion.
{
}
\,\,\,\,c
Volker Branding
Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
{\tt volker.branding@univie.ac.at}
\,\,\,\,c
Yong Luo
School of mathematics and statistics,
Wuhan university, Wuhan 430072, China
{\tt yongluo@whu.edu.cn}
\,\,\,\,c
\end{document} |
\begin{document}
\title{Regular Variation in a \rev{Fixed-Point} Problem for Single- and Multiclass Branching Processes and Queues }
\authorone[Aarhus University]{S\O ren Asmussen}
\addressone{Department of Mathematics, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark}
\authortwo[Heriot-Watt University and\\
. \hspace{3.1cm} Novosibirsk State University]{Sergey Foss}
\addresstwo{School of Mathematical and Computer Sciences, Heriot-Watt University, EH14 4AS, Edinburgh, United Kingdom. Research supported by RSF grant No. 17-11-01173.}
\begin{abstract}
Tail asymptotics of the solution $R$ to a \rev{fixed-point} problem of type $R \ \mathrm{e}qdistr\ Q + \sum_1^N R_m$
is derived under heavy-tailed conditions allowing both dependence between $Q$ and $N$ and the
tails to be of the same order of magnitude. Similar results are derived for a $K$-class version with applications
to multitype branching processes and busy periods in multiclass queues.\mathrm{e}nd{abstract}
\keywords{Busy period; Galton-Watson process; intermediate regular variation; multivariate regular variation;
random recursion; random sums}
\ams{60H25}{60J80; 60K25; 60F10}
\section{Introduction}\label{S:Intr}
\setcounter{equation}{0}
This paper is concerned with the tail asymptotics of the solution $R$
to the \rev{fixed-point} problem
\begin{equation}\label{eq1}
R \ \mathrm{e}qdistr\ Q + \sum_{m=1}^N R_m
\mathrm{e}nd{equation}
under suitable regular variation (RV) conditions
and the similar problem in a multidimensional setting stated below as \mathrm{e}qref{30.6a}. Here
in \mathrm{e}qref{eq1} $Q,N$ are (possibly dependent) non-negative non-degenerate r.v.'s where $N$ is integer-valued, $R_1,R_2,\ldots$
are i.i.d.\ and distributed as $R$, and ${\overline n}=\mathbb{E} N<1$ (similar notation for expected values is used in the following).
A classical example is $R$ being the M/G/1 busy period, cf.\ \cite{deMeyer}, \cite{Bert}, where $Q$ is the service
time of the first customer in the busy period and $N$ the number of arrivals during his service. Here
$Q$ and $N$ are indeed heavily dependent, with tails of the same order of magnitude when $Q$ has a regularly varying (RV) distribution; more precisely,
$N$ is Poisson$(\lambda q)$ given $Q=q$. Another example is the total progeny of a subcritical branching
process, where $Q\mathrm{e}quiv 1$ and $N$ is the number of children of the ancestor, More generally, $R$ could be
the total life span of the individuals in a Crump-Mode-Jagers process (\cite{Jagers}), corresponding to
$Q$ being the lifetime of the ancestor and $N$ the number of her children. Related examples are weighted
branching processes, see~\cite{Mariana} for references. Note that connections between branching processes and RV
have a long history, going back at least to \cite{Seneta69}, \cite{Seneta74}.
Recall some definitions of classes of heavy-tailed distributions.
A distribution $F$ on the real line is {\it long-tailed}, $F\in {\cal L}$,
if, for some $y>0$
\begin{equation}\label{long}
\frac{\overline{F}(x+y)}{\overline{F}(x)} \rightarrow 1\quad\text{as}\ x\to\infty ;
\mathrm{e}nd{equation}
$F$ is {\it regularly varying}, $F\in {\cal RV}$,
if, for some
$\beta >0$,
$
\overline{F}(x) = x^{-\beta} L(x),
$
where $L(x)$ is a {\it slowly varying} (at infinity) function;\\
$F$ is {\it intermediate regularly varying}, $F\in {\cal IRV}$,
if
\begin{equation}\label{IRV1}
\lim_{\alpha \uparrow 1} \limsup_{x\to\infty}
\frac{\overline{F}(\alpha x)}{
\overline{F}(x)} = 1.
\mathrm{e}nd{equation}
It is known that ${\cal L} \supset {\cal IRV} \supset {\cal RV}$
and if $F$ has a finite mean, then ${\cal L} \supset {\cal S}^*
\supset {\cal IRV}$
where ${\cal S}^*$ is the class of so-called {\it strong subexponential
distributions}, see e.g. \cite{EKM1997} or \cite{FKZ2013} for further
definitions and properties of heavy-tailed distributions.
Tail asymptotics of quantities related to $R$ have earlier been studied in
\cite{Mariana}, \cite{Litvak} under RV conditions (see also~\cite{BDM}).
Our main result is the following:
\begin{theorem}\label{Th:3.8a}
Assume ${\overline n}<1$ and $ {\overline q}<\infty$. Then:\\[1mm]
{\rm (i)}
There is only one \rev{non-negative} solution $R$ to equation \mathrm{e}qref{eq1} with finite mean. For this solution,
${\overline r}={\overline q}/(1-{\overline n})$.\\[1mm]
{\rm (ii)} If further\\
{\bf (C)}
the distribution of $Q+cN$ is intermediate regularly varying for
all $c>0$ in the interval $({\overline r} -\mathrm{e}psilon, {\overline r}+\mathrm{e}psilon )$ where ${\overline r}$ is as in {\rm (i)} and
$\mathrm{e}psilon >0$ is any small number,\\
then
\begin{equation}\label{eq10}
\mathbb{P} (R>x) \sim \frac{1}{1-{\overline n}} \mathbb{P}(Q+{\overline r}
N >x)\quad \text{ as }x\to\infty\,.
\mathrm{e}nd{equation}
{\rm (iii)} In particular, condition {\bf (C)} holds in the following three cases:\\
{\rm (a)} $(Q,N)$ has a 2-dimensional regularly varying distribution;\\
{\rm (b)} $Q$ has an intermediate regularly varying distribution and $\mathbb{P} (N>x) = {\mathrm{o}}(\mathbb{P} (Q>x))$;\\
{\rm (c)} $N$ has an intermediate regularly varying distribution and $\mathbb{P} (Q>x) = {\mathrm{o}}(\mathbb{P} (N>x))$.
\mathrm{e}nd{theorem}
Part (i) is well known from several sources and not deep (see
the proof of the more general
Proposition~\ref{Prop:6.8a} below and the references at the end of
the section for more general versions).
Part (ii) generalizes and unifies results of \cite{Mariana}, \cite{Litvak} in several ways.
Motivated from the Google page rank
algorithm, both of these papers consider a more general recursion
\begin{equation}\label{eq1A}
R \ \mathrm{e}qdistr\ Q + \sum_{m=1}^N A_mR_m\,.
\mathrm{e}nd{equation}
However, \cite{Mariana} does not allow dependence and/or the tails of $Q$ and $N$ to be equally heavy.
These features are incorporated in \cite{Litvak}, but on the other hand that paper require strong conditions
on the $A_i$ which do not allow to take $A_i\mathrm{e}quiv 1$ when dealing with sharp asymptotics. To remove all of these restrictions is essential for
the applications to queues and branching processes we have in mind.
Also, our proofs are considerably simpler and shorter
than those of \cite{Mariana}, \cite{Litvak}. The key tool is a general result of \cite{FossZ}
giving the tail asymptotics of the maximum of a random walk up to a (generalised) stopping time.
\begin{remark}
\rev{In Theorem \ref{Th:3.8a}, we considered the case $A_i\mathrm{e}quiv 1$ only. However, our approach may work in the more
general setting of~\mathrm{e}qref{eq1A} with i.i.d. positive $\{A_m\}$ that do not depend on $Q,N$ and $\{R_m\}$. For example, if we assume, in addition to
${\overline n} <1$,
that $\mathbb{P} (0<A_1\le 1) =1$,
then the exact tail asymptotics for $\mathbb{P} (R>x)$ may be easily found using the upper bound
\mathrm{e}qref{eq10} and the principle of a single big jump. However, the formula
for the tail asymptotics in this case is much more complicated that \mathrm{e}qref{eq10}.}
\mathrm{e}nd{remark}
The multivariate version involves a \rev{set} $\bigl(R(1),\ldots,(R(K)\bigr)$ of r.v.'s satisfying
\begin{equation}\label{30.6a}R(i) \ \mathrm{e}qdistr\ Q(i)+ \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R_{m}(k)\mathrm{e}nd{equation}
In the branching setting, this relates to $K$-type processes by thinking of $N^{(k)}(i)$ as the number of type $k$
children of a type $i$ ancestor.
One example is the total progeny where $Q(i)\mathrm{e}quiv 1$, others relate as above to the total life span and
weighted branching processes.
A queueing example is the busy periods $R(i)$ in the multiclass queue in \cite{AEH}, with $i$ the class of the first
customer in the busy period and $Q(i)$ the service time of a class $i$ customer; the model states that during
service of a class $i$ customer, class $k$ customers arrive at rate $\lambda_{ik}$. One
should note for this example \cite{AEH} \rev{gives} only lower asymptotic bounds,
whereas we here provide sharp asymptotics.
The treatment of \mathrm{e}qref{30.6a} is considerably more involved than for \mathrm{e}qref{eq1}, and we defer the details
of assumptions and results to Section~\ref{S:MRV}. We remark here only that the concept of multivariate
regular variation (MRV) will play a key role; that
the analogue of the crucial assumption ${\overline n}<1$ above is subcriticality, $\rho=$spr$(\mbox{\boldmath$ M$})<1$ where spr means spectral radius and
$\mbox{\boldmath$ M$}$ is the offspring mean matrix with elements
$m_{ik}=\mathbb{E} N^{(k)}(i)$; and that the argument will involve a recursive procedure from \cite{Foss1, Foss2},
reducing $K$ to $K-1$ so that in the end we are back to the case $K=1$ of \mathrm{e}qref{eq1} and Theorem~\ref{Th:3.8a}.
\rev{\subsubsection*{Bibliographical remarks}
An $R$, or its distribution, satisfying \mathrm{e}qref{eq1A} is often called a fixed point of the smoothing transform (going back to~\cite{Liggett}). There is an extensive literature
on this topic, but rather than on tail asymptotics, the emphasis is most often on existence and uniqueness questions
(these are easy in our context with all r.v.'s non-negative with finite mean and we give
short self-contained proofs). Also the assumption $A_i\neq 1$ is crucial for most of this
literature. See further \cite{Aldous}, \cite{Gerold1}, \cite{Gerold2}
and references there.}
\rev{It should be noted that the term ``multivariate smoothing transform"
(e.g.~\cite{Mentemeier}) means to a recursion
of vectors, that is, a version of \mathrm{e}qref{eq1} with $R,Q\in \mathbb{R}^K$. This is different from our set-up because in~\mathrm{e}qref{30.6a} we are only interested in the one-dimensional
distributions of the $R(i)$. In fact, for our applications there is no interpretation of a vector with $i$th marginal having the distribution of $R(i)$.
}
In \cite{Vatutin}, tail asymptotics for the
total progeny of a multitype branching process is studied by
different techniques€‹ in the critical case $\rho =1$.
\section{One-dimensional case: equation \mathrm{e}qref{eq1}}\label{S:1D}
\setcounter{equation}{0}
The heuristics behind \mathrm{e}qref{eq10} is the principle of a single large jump: for $R$ to exceed $x$, either one or both elements of $(Q,N)$
must be large, or the independent
event occurs that $R_m>x$ for some $m\le N$, in which case $N$ is small or moderate.
If $N$ is large, $\sum_1^NR_m$ is approximately ${\overline r} N$, so roughly the probability of the first possibility is
$\mathbb{P}(Q+{\overline r} N>x)$. On the other hand, results for compound heavy-tailed sums suggest that the approximate probability of the second
possibility is ${\overline n}\mathbb{P}(R>x)$. We thus arrive at
\[\mathbb{P}(R>x) \approx\ \mathbb{P}(Q+{\overline r} N>x)\,+\,{\overline n}\mathbb{P}(R>x)\]
and \mathrm{e}qref{eq10}.
In the proof of Theorem~\ref{Th:3.8a}, let $(Q_1,N_1),(Q_2,N_2),\ldots$ be an i.i.d.\ sequence of pairs distributed as the (possibly dependent)
pair $(Q,N)$ in \mathrm{e}qref{eq1}. Then $S_n=\sum_{i=1}^n \xi_i$, $i=0,1,\ldots$ where $\xi_i=N_i-1$
is a random walk. Clearly,
$\mathbb{E} \xi_i <0$. Let
\begin{equation}\label{eq2}
\tau = \min \{n\ge 1 \ : \ S_n<0\}= \min \{n\ge 1 \ : \ S_n=-1\}\,.
\mathrm{e}nd{equation}
Note that by Wald's identity $\mathbb{E} S_\tau=\mathbb{E}\tau\cdot\mathbb{E} (N-1)$ and $S_\tau=-1$ we have
\begin{equation}\label{3.8b}
\mathbb{E}\tau = \frac{1}{1-\mathbb{E} N}
\mathrm{e}nd{equation}
Now either $N_1=0$, in which case $\tau=1$, or $N_1>0$ so that $S_1=N_1-1$ and to proceed to level -1,
the random walk must go down one level $N_1$ times. This shows that (in obvious notation)
\begin{equation}\label{3.8d}
\tau \ \mathrm{e}qdistr\ 1 + \sum_{i=1}^N \tau_i
\mathrm{e}nd{equation}
That is, $\tau$ is a solution to \mathrm{e}qref{eq1} with
$Q\mathrm{e}quiv 1$. On the other hand, the total progeny in a Galton-Watson process with the number of offsprings
of an individual distributed as $N$ obviously also satisfies \mathrm{e}qref{3.8d}, and hence by uniqueness
must have the same distribution as $\tau$. This result first occurs as equation (4) in \cite{Dwass}, but note that
an alternative representation (1) in that paper appears to have been the one receiving the most attention
in the literature.
Now define $\varphi_i=k_0+k_1Q_i$,
\begin{equation}\label{eq3}
V=\sum_{i=1}^{\tau} \varphi_i
\mathrm{e}nd{equation}
Here the $k_0,k_1$ are non-negative constants, $k_0+k_1>0$.
In particular, if $k_0=1,k_1=0$, then $V=\tau$, and
further
\begin{equation}\label{3.8c}
k_0=0, k_1=1\qquad\Rightarrow\qquad V\mathrm{e}qdistr R.
\mathrm{e}nd{equation}
\rev{Indeed, arguing as before, we conclude that equation $V\mathrm{e}qdistr \varphi +\sum_1^N V_i$ has only one integrable positive solution, and, clearly,
$$
V \mathrm{e}qdistr \varphi + \sum_1^N V_i \mathrm{e}qdistr \varphi + \sum_1^N \varphi_i + \sum_1^N \sum_1^{N_i} \varphi_{i,j} + \sum_1^N \sum_1^{N_i} \sum_1^{N_{i,j}} \varphi_{i,j,k} +
\ldots \mathrm{e}qdistr \sum_1^{\tau} \varphi_i
$$
where, like before, $(\varphi, N)$, $(\varphi_i,N_i)$, $(\varphi_{i,j},N_{i,j})$, etc. are
i.i.d. vectors. In particular, $V$ becomes $R$ when replacing $\varphi$ by $Q$.}
\begin{proof}[Proof of Theorem~\ref{Th:3.8a}]
It remains to find the asymptotics of $\mathbb{P} (V>x)$ as
$x\to\infty$. Throughout the proof, we assume $k_1>0$.
Let $r_0$ be the solution to the
equation
$$
\mathbb{E} \varphi_1+r_0 \mathbb{E} \xi_1=0.
$$
Note that in the particular case where $k_0=0$ and $k_1=1$,
\begin{equation}\label{eq5}
r_0= \frac{\mathbb{E} Q}{1-\mathbb{E} N} = \overline{r}.
\mathrm{e}nd{equation}
Choose $r>r_0$ as close to $r_0$ as needed and let
$$
\psi_i=\varphi_i+ r\xi_i.
$$
We will find upper and lower bounds for the asymptotics
of
$\mathbb{P}(V>x)$ and show that they are asymptotically equivalent.
Since $k_1>0$ and $Q+Nr/k_1$
has an IRV distribution, the distribution of $k_1Q+rN$ is IRV, too.
\\[2mm]
{\bf Upper bound.} The key is to apply the main result of~\cite{FossZ}
to obtain the
following upper bound.
\begin{align*}\MoveEqLeft
\mathbb{P} (V>x) \ = \
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \varphi_i > x\Bigr) \ = \
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \psi_i > x + r
{S}_{\tau}\Bigr)\\
&=\
\mathbb{P} \Bigl(\sum_{i=1}^{\tau} \psi_i > x-r\Bigr)\ \le\
\mathbb{P} \Bigl(\max_{1\le k \le \tau}
\sum_{i=1}^{k} \psi_i > x-r\Bigr)\\
&\sim \
\mathbb{E} \tau
\mathbb{P} (\psi_1 > x-r) \ \sim \
\mathbb{E} \tau
\mathbb{P} (\psi_1 > x-r+k_0) \ = \
\mathbb{E} \tau
\mathbb{P} (k_1Q+rN > x)
\mathrm{e}nd{align*}
Here the first equivalence follows from~\cite{FossZ}, noting that the distribution of $\psi_1$
belongs to the class $S^*$ and that~\cite{FossZ} only requires $\varphi_1, \varphi_2,\ldots$ to be i.i.d.\
w.r.t.\ some filtration w.r.t.\ which $\tau$ is a stopping time. For the second, we used
the long-tail property \mathrm{e}qref{long} of the distribution of $\psi_1$.
Let $F$ be the distribution function of $k_1Q+r_0N$.
Then, as $x\to\infty$,
$$
\overline{F}(x) \le \mathbb{P} (k_1Q+rN>x) \le
\mathbb{P} (rk_1Q/r_0+rN>x) \le \overline{F}(\alpha x) \le
(1+o(1)) c(\alpha )\overline{F}(x)
$$
where $\alpha = r_0/r<1$ and $c(\alpha ) =\limsup_{y\to\infty} \overline{F}(\alpha y)/\overline{F}(y)$.
Now we assume the IRV condition to hold, let $r\downarrow r_0$ and apply
\mathrm{e}qref{IRV1} to obtain the upper bound
\begin{equation}\label{eq4}
\mathbb{P} (R>x) \le (1+o(1))
\mathbb{E} \tau \mathbb{P} (k_1Q+r_0N>x)
\mathrm{e}nd{equation}
In particular, if $k_0=0$ and $k_1=1$, then
$r_0=\overline{r}$ is as in~\mathrm{e}qref{eq5}. \\[2mm]
{\bf Lower bound.}
Here we put $\psi_n = \varphi_n + r\xi_n$ where
$r$ is any positive number strictly smaller than $r_0$.
Then the $\psi_n$ are i.i.d.\ random variables with common mean $\mathbb{E} \psi_1 >0$.
We have, for any fixed $C>0$, $L>0$, $n=1,2,\ldots$ and $x\ge 0$ that
\begin{equation}\label{double}
\mathbb{P} (V>x) \ \ge \
\mathbb{P}\Bigl(\sum_{i=1}^{\tau } \psi_i >x\Bigr)\ \ge\ \sum_{i=1}^n \mathbb{P} (D_i \cap A_i)
\mathrm{e}nd{equation}
where
\[
D_i \ =\ \Bigl\{ \sum_{j=1}^{i-1} |\psi_j|\le C, \tau \ge i,
\psi_i>x+C+L \Bigr\}
\quad\text{and}\quad
A_i \ =\ \bigcap_{\mathrm{e}ll\ge 1}\Bigl\{\sum_{j=1}^\mathrm{e}ll \psi_{i+j}\ge -L \Bigr\}.
\]
Indeed, the first inequality in \mathrm{e}qref{double} holds since $S_{\tau}$ is non-positive. Next, the events $D_i$
are disjoint and, given $D_i$, we have $\sum_1^i \psi_j >x+L$. Then, given $D_i\cap A_i$,
we have $\sum_1^k \psi_j \ge x$ for all $k\ge i$ and, in particular, $\sum_{j=1}^{\tau}\psi_j >x$. Thus, \mathrm{e}qref{double} holds.
The events $\{A_i\}$ form a stationary sequence. Due to the SLLN, for any $\varepsilon >0$,
one can choose
$L=L_0$ so large that $\mathbb{P} (A_i) \ge 1-\varepsilon$.
For this $\varepsilon$, choose $n_0$ and $C_0$ such that
$$
\sum_{i=1}^{n_0} \mathbb{P} \Bigl(\sum_{j=1}^{i-1} |\psi_j|\le C_0, \tau \ge i\Bigr)\ \ge\ (1-\varepsilon ) \mathbb{E} \tau.
$$
Since the random variables $(\{\psi_j\}_{j<i},{\mathbf I}(\tau\le i))$ are independent of $\{\psi_j\}_{j\ge i}$, we obtain further that, for any
$\varepsilon \in (0,1)$ and for any $n\ge n_0$, $C\ge C_0$ and $L\ge L_0$,
\begin{eqnarray*}
\mathbb{P} (V>x) &\ge &
\sum_{i=1}^n \mathbb{P} \Bigl(\sum_{j=1}^{i-1} |\psi_j|\le C, \tau \ge i\Bigr)
\mathbb{P} (\psi_i>x+C+L) \mathbb{P} (A_i) \\
&\ge &
(1-\varepsilon )^2 \mathbb{P} (\psi_1 >x+C+L) \sum_{i=1}^n \mathbb{P} (\tau \ge i)\\
&\sim &
(1-\varepsilon )^2 \mathbb{P} (\psi_1 >x) \sum_{i=1}^n \mathbb{P} (\tau \ge i),
\mathrm{e}nd{eqnarray*}
as $x\to\infty$. Here the final equivalence follows from the long-tailedness
of $\psi_1$. Letting first $n$ \rev{go} to infinity and then $\varepsilon$ to zero, we get $\liminf_{x\to\infty} \mathbb{P} (V>x)/\mathbb{E} \tau \mathbb{P} (\psi_1>x) \ge 1.$ Then we let $r\uparrow r_0$ and use the IRV property \mathrm{e}qref{IRV1}. In the particular case
$k_0=0,k_1=1$ we obtain an asymptotic lower bound that is equivalent
to the upper bound derived above
\mathrm{e}nd{proof}
\begin{remark}
A slightly more intuitive approach to the lower bound is to bound $\mathbb{P}(R>x)$ below by the sum of the contributions from the
disjoint events $B_1,B_2,B_3$ where \[
B_1=B\cap\{{\overline r} N\ >\mathrm{e}psilon x\},\quad B_2=B\cap\{A< {\overline r} N\le\mathrm{e}psilon x\},\quad
B_3= \{{\overline r} N\le A\}\]
with $B=\{Q+{\overline r} N>(1+\mathrm{e}psilon) x\}$.
Here for large $x,A$ and small $\mathrm{e}psilon$,
\begin{align*}\mathbb{P}(R>x;\,B_1)\ &\sim\ \mathbb{P}(Q+{\overline r} N>x,\,{\overline r} N>\mathrm{e}psilon x)\\
\mathbb{P}(R>x;\,B_2)\ &\ge \mathbb{P}(Q>x,{\overline r} N\le\mathrm{e}psilon x)\ \sim \ \mathbb{P}(Q+{\overline r} N>x,\,{\overline r} N\le\mathrm{e}psilon x)\\
\mathbb{P}(R>x;B_3)\ &\ge\ \sum_{n=0}^{A/{\overline r}}\mathbb{P}(R_1+\cdots+R_n>x)\mathbb{P}(N=n)\\ & \ \ge\
\sum_{n=0}^{A/{\overline r}}\mathbb{P}\bigl(\max(R_1,\ldots,R_n)>x\bigr)\mathbb{P}(N=n)
\\ & \ \sim\ \sum_{n=0}^{A/{\overline r}}n\mathbb{P}(R>x)\mathbb{P}(N=n)\ \sim\ \mathbb{E}[N\wedge A/{\overline r}]\mathbb{P}(R>x) \ \sim\ {\overline n} \mathbb{P}(R>x)
\mathrm{e}nd{align*}
We omit the full details since they are close to arguments given in Section~\ref{S:FayeMRV} for the
multivariate case.\mathrm{e}nd{remark}
\section{Multivariate version}\label{S:MRV}
\setcounter{equation}{0}
\rev{The assumptions for \mathrm{e}qref{30.6a} are that all $R_{m}(k)$ are independent of
the vector
\begin{equation}\label{30.6ex}\mbox{\boldmath$ V$}(i)\ =\ \bigl(Q(i),N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)\,,
\mathrm{e}nd{equation}
that they are mutually independent and that $R_{m}(k)\mathrm{e}qdistr R(k)$.
Recall that we are only interested in the one-dimensional
distributions of the $R(i)$. Accordingly, for a solution to \mathrm{e}qref{30.6a}
we only require the validity for each fixed $i$. }
Recall that the offspring mean matrix is denoted $\mbox{\boldmath$ M$}$ where $m_{ik}=\mathbb{E} N^{(k)}(i)$,
and that $\rho=$spr($\mbox{\boldmath$ M$})$; $\rho$ is the Perron-Frobenius root if $\mbox{\boldmath$ M$}$ is irreducible which it is
not necessary to assume. \rev{No restrictions on the dependence structure of the vectors in \mathrm{e}qref{30.6ex} }need to be imposed for the following result to hold (but later we need MRV!):
\begin{proposition}\label{Prop:6.8a} Assume $\rho<1$. Then:\\[1mm]
{\rm (i)} the \rev{fixed-point} problem \mathrm{e}qref{30.6a} has a unique non-negative solution
with ${\overline r}_i=\mathbb{E} R(i)<\infty$ for all $i$;\\[1mm] {\rm (ii)} the ${\overline r}_i=\mathbb{E} R(i)<\infty$ are given as the unique solution to the set
\begin{equation}\label{6.8dd} {\overline r}_i\ =\ {\overline q}_i+\sum_{k=1}^K m_{ik}{\overline r}_k\,,\qquad i=1,\ldots,K,
\mathrm{e}nd{equation}
of linear equations.
\mathrm{e}nd{proposition}
\begin{proof} (i) Assume first $Q(i)\mathrm{e}quiv 1$, $i=1,\ldots,K$. \rev{The existence of a
solution to \mathrm{e}qref{30.6a} is then clear since we may take $R(i)$ as the total progeny
of a type $i$ ancestor in a $K$-type Galton-Watson process where the vector of children of a type $j$ individual is distributed as $\bigl(N^{(1)}(j),\ldots, N^{(K)}(j)\bigr)$.
For uniqueness, let $\bigl(R(1),\ldots, R(K)\bigr)$ be any solution and}
consider the $K$-type Galton-Watson trees ${\mathcal G}(i)$, $ i=1,\ldots,K$, where ${\mathcal G}(i)$ corresponds to an
ancestor of type $i$.
If we define $R^{(0)}(i)=1$,
\[R^{(n)}(i) \ \mathrm{e}qdistr\ 1+ \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R^{(n-1)}_{m}(k)\,,\]
\rev{with similar conventions as for \mathrm{e}qref{30.6a},}
then $R^{(n)}(i)$ is the total progeny of a type $i$ ancestor under the restriction
that the depth of the tree is at most $n$. Induction easily gives that
\rev{$R^{(n)}(i)\preceq_{{\rm st}} R(i)$ ($\preceq_{{\rm st}} \ =\ $ stochastic order) for
each $i$. Since also $R^{(n)}(i) \preceq R^{(n+1)}(i)$, limits
$R^{(\infty)}(i)$ exist, $R^{(\infty)}(i)$ must simply be the unrestricted vector of total progeny of different types,
and $ R^{(\infty)}(i)\preceq_{{\rm st}} R(i)$.
Assuming the $R(i)$} to have finite mean, \mathrm{e}qref{6.8dd} clearly holds with
${\overline q}_i=1$, and so the $\Delta_i={\overline r}_i-\mathbb{E} R^{(\infty)}(i)$ satisfy $\Delta_i=\sum_1^K m_{ik}\Delta_k$. But
$\rho<1$ implies that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible so the only solution is $\Delta_i=0$ which in view of
\rev{$R^{(\infty)}(i)\preceq_{{\rm st}} R(i)$ implies $R^{(\infty)}(i) \mathrm{e}qdistr R(i)$} and the stated uniqueness when $Q(i)\mathrm{e}quiv 1$.
For more general $Q(i)$, we equip each individual of type $j$ in ${\mathcal G}(i)$ with a weight distributed as $Q(j)$,
such that the dependence between her $Q(j)$ and her offspring vector has the given structure. The argument
is then a straightforward generalization and application of what was done above for $Q(i)\mathrm{e}quiv 1$.
(ii) Just take expectations in \mathrm{e}qref{30.6a} and note as before that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible.
\mathrm{e}nd{proof}
For tail asymptotics, we need an MRV assumption. The definition of MRV exists in some equivalent variants,
cf.\ \cite{Sid87}, \cite{Meerschaert}, \cite{Basrak}, \cite{Sid07}, but we shall use the one in polar $L_1$-coordinates
adapted to deal with several random vectors at a time as in \mathrm{e}qref{30.6ex}. Fix here and in the following
a reference RV tail ${\overline F}(x)=L(x)/x^\alpha$ on $(0,\infty)$,
for $\mbox{\boldmath$ v$}=(v_1,\ldots, v_p)$ define $\|\mbox{\boldmath$ v$}\|=\|\mbox{\boldmath$ v$}\|_1=$ $|v_1|+\cdots+|v_p|$ and let ${\cal B}={\cal B}_p=$ $\{\mbox{\boldmath$ v$}:\, \|\mbox{\boldmath$ v$}\|=1\}$.
We then say that a random vector $\mbox{\boldmath$ V$}=(V_1,\ldots, V_p)$
satisfies {\mbox{{\rm MRV}$(F)$}}\ or has property {\mbox{{\rm MRV}$(F)$}}\ if
$\mathbb{P}\bigl(\|\mbox{\boldmath$ V$}\|>x\bigr)$ $\sim b{\overline F}(x)$ where either (1) $b=0$ or (2) $b>0$
and the angular part $\mbox{\boldmath$ T$}heta_{\mbox{\scriptsize\boldmath$ V$}}=\mbox{\boldmath$ V$}/\|\mbox{\boldmath$ V$}\|$ satisfies
\[\mathbb{P}\bigl(\mbox{\boldmath$ T$}heta_{\mbox{\scriptsize\boldmath$ V$}}\in\cdot\,\big|\,\|\mbox{\boldmath$ V$}\|>x\bigr)\ \convdistr \ \mu \text{ as }x\to\infty\]
for some measure $\mu$ on ${\cal B}$ (the angular measure).
Our basic condition is then that for the given reference RV tail ${\overline F}(x)$ we have that\\[1mm]
(MRV) For any $i=1,\ldots,K$ the vector $\mbox{\boldmath$ V$}(i)$ in \mathrm{e}qref{30.6ex} satisfies {\mbox{{\rm MRV}$(F)$}}, where
$b=b(i)>0$ for at least one $i$.\\[1mm]
Note that $F$ is the same for all $i$ but the angular measures $ \mu_i$ not necessarily so.
We also assume that the mean ${\overline z}$ of $F$ is finite, which will ensure that all expected values coming up in the following
are finite.
\rev{Assumption {\mbox{{\rm MRV}$(F)$}}\ implies RV of linear combinations, in particular
marginals. More precisely (see the Appendix),}
\begin{equation}\label{30.6ax}\mathbb{P}\bigl(a_0Q(i)+a_1N^{(1)}(i)+\cdots+a_KN^{(K)}(i) >x\bigr)\ \sim\ c_i(a_0,\ldots,a_K){\overline F}(x)
\mathrm{e}nd{equation} where $\displaystyle
c_i(a_0,\ldots,a_K)\ =\ b(i)\int_{{\mathcal B}}(a_0\theta_0+\cdots+a_K\theta_K)^\alpha\mu_i(\mathrm{d}\theta_0,\ldots,\mathrm{d}\theta_K)\,.$
\begin{theorem}\label{Th:6.8a} Assume that $\rho<1$, ${\overline z}<\infty$ and that \mathrm{e}mph{(MRV)} holds.
Then there are constants $d_1,\ldots,d_K$ such that
\begin{equation}\label{30.6d}
\mathbb{P}(R(i)>x)\,\sim\,d_i{\overline F}(x)\ \ \text{as}\ x\to\infty.
\mathrm{e}nd{equation}
Here the $d_i$ are given as the unique solution to the set
\begin{equation}\label{6.8d} d_i\ =\ c_i(1,{\overline r}_1,\ldots,{\overline r}_K)+\sum_{k=1}^K m_{ik}d_k\,,\qquad i=1,\ldots,K,
\mathrm{e}nd{equation}
of linear equation\rev{s} where the ${\overline r}_i$ are as in Proposition~\ref{Prop:6.8a} and the $c_i$ as in
\mathrm{e}qref{30.6ax}.
\mathrm{e}nd{theorem}
\noindent The proof follows in Sections~\ref{S:Outline}--\ref{S:ProofCompl}.
\section{Outline of proof}\label{S:Outline}
\setcounter{equation}{0}
When $K>1$, we did not manage to find a random walk argument extending Section~\ref{S:1D}.
Instead, we shall use a recursive procedure, going back to \cite{Foss1, Foss2} in a queueing setting, for eventually
being able to infer \mathrm{e}qref{30.6d}. The identification \mathrm{e}qref{6.8d} of the $d_i$ then follows immediately
from the following result to be proved in Section~\ref{S:FayeMRV} (the case $p=1$ is Lemma 4.7 of \cite{Fay}):
\begin{proposition}\label{Prop:2.8b} Let $\mbox{\boldmath$ N$}=(N_1,\ldots,N_p)$ be MRV with
$\mathbb{P}\bigl(\|\mbox{\boldmath$ N$}\|>x\bigr)\sim c_{\mbox{\scriptsize\boldmath$ N$}}{\overline F}(x)$ and let the r.v.'s $Z_m^{(i)}$, $i=1,\ldots,p$, $m=1,2,\ldots$, be
independent with distribution $F_j$ for $Z_i^{(j)}$, independent of $\mbox{\boldmath$ N$}$
and having finite mean $\overline z_j=\mathbb{E} Z_m^{(j)}$. Define
$S_{m}^{(j)}=Z_1^{(j)}+\cdots+Z_{m}^{(j)}$. If ${\overline F}_j(x)\sim c_j{\overline F}(x)$, then \[
\mathbb{P}\bigl(S_{N_1}^{(1)}+\cdots+S_{N_p}^{(p)}>x\bigr)\ \sim\ \mathbb{P}({\overline z}_1N_1+\cdots+{\overline z}_1N_p\,>\,x)+c_0{\overline F}(x)\]
where $c_0\,=\,c_1\mathbb{E} N_1+\cdots+c_p\mathbb{E} N_p$ . \mathrm{e}nd{proposition}
The recursion idea in \cite{Foss1, Foss2} amounts in a queueing context to let
all class $K$ customers be served first. We implement it here in the branching setting.
Consider the multitype Galton-Watson tree
${\mathcal G}$.
For an ancestors of type $i<K$ and any of her daughters $m=1,\ldots, N^{(K)}(i)$
of type $K$, consider the family tree ${\mathcal G}_m(i)$ formed by $m$ and all her type $K$ descendant in \mathrm{e}mph{direct}
line. For \rev{a vertex} $g\in{\mathcal G}_m(i)$ and $k<K$, let $N^{(k)}_g(K)$ denote the number of type $k$ daughters
of $g$.
Note that ${\mathcal G}_m(i)$ is simply a one-type Galton-Watson tree with the number
of daughters distributed as $N^{(K)}(K)$ and starting from a single ancestor.
In particular, the expected size of ${\mathcal G}_m(i)$ is $1/(1-m_{KK})$. We further have
\begin{align}\label{30.6aa}R(i) \ &\mathrm{e}qdistr\ \widetilde Q(i)+ \sum_{k=1}^{K-1}\sum_{m=1}^{\widetilde N^{(k)}(i)}R_{m}(k)\,,\quad i=1,\ldots,K-1,
\\ \intertext{where} \label{30.6aaa}
\widetilde Q(i)\ &=\ Q(i)+\sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)} Q_g(K)\,, \\ \label{30.6aab}
\widetilde N^{(k)}(i) \ &=\ N^{(k)}(i)+\sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)}N^{(k)}_g(K) \,.
\mathrm{e}nd{align}
that is, a \rev{fixed-point} problem with one type less.
\begin{example}\label{Ex:5.8a}\rm Let $K=2$ and consider the 2-type family tree in Fig.~\ref{treefig},
where type $i=1$ has
green color, the type $2$ descendants of the ancestor in direct line red, and the remaining type 2 individuals blue.
The green type 1 individuals marked with a triangle are the ones that are counted as extra type 1 children
in the reduced recursion \mathrm{e}qref{30.6aa}.
We have $N^{(2)}(1)=2$ and if $m$ is the upper red individual
of type 2, then ${\mathcal G}_m(2)$ has size 4. Further $\sum_{g\in{\mathcal G}_m(1)}N^{(1)}_g=2$, with $m$ herself and her upper
daughter each contributing with one individual.
{\mathcal G}fig
The offspring mean in the reduced 1-type family tree is $\widetilde m=m_{11}+m_{12}m_{21}/(1-m_{22})$.
Indeed, the first term is the expected number of original type 1 offspring of the ancestor and in the second term,
$m_{12}$ is the expected number of type 2 offspring of the ancestor, $1/(1-m_{22})$ the size of the direct line
type 2 family tree produced by each of them, and $m_{21}$ the expected number of type 1 offspring
of each individual in this tree.
Since the original 2-type tree is finite, the reduced 1-type tree must necessarily also
be so, so that $\widetilde m\le 1$. A direct verification of this is instructive. First note that
\[ \widetilde m\le 1\ \iff\ m_{11}-m_{11}m_{22}+m_{12}m_{21}\le 1-m_{22}\ \iff\ \text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\le1\]
But the characteristic polynomial of the
2-type offspring mean matrix $\mbox{\boldmath$ M$}$ is $\lambda^2-\lambda\,\text{tr}(\mbox{\boldmath$ M$})+\text{det}(\mbox{\boldmath$ M$})$.
Further the dominant eigenvalue $\rho$ of $\mbox{\boldmath$ M$}$ satisfies $\rho< 1$ so that
\[ \text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\ \le\ \rho\,\text{tr}(\mbox{\boldmath$ M$})-\text{det}(\mbox{\boldmath$ M$})\ =\ \rho^2\ <\ 1.\]
\mathrm{e}nd{example}
\section{Proof of Proposition~\ref{Prop:2.8b}}\label{S:FayeMRV}
\setcounter{equation}{0}
We shall need the following result of Nagaev et al.\ (see the discussion in \cite{Fay}
around equation (4.2) there for references):
\begin{lemma}\label{Lemma:11.7aa} Let $Z_1,Z_2,\ldots$ be i.i.d.\ and RV with finite mean $\overline z$ and
define $S_k=Z_1+\cdots+Z_k$. Then for any $\delta>0$
\[ \sup_{y\ge \delta k}\Bigl|\frac{\mathbb{P}(S_k>k{\overline z}+y)}{k{\overline F}(y)}-1\Bigr|\ \to\ 0,\ k\to\infty.\]
\mathrm{e}nd{lemma}
\begin{corollary}\label{Cor:2.8a} Under the assumptions of Lemma~\ref{Lemma:11.7aa},
it holds for $0<\mathrm{e}psilon<1/{\overline z}$ that
\[d(F,\mathrm{e}psilon)\ =\ \limsup_{x\to\infty}\sup_{k<\mathrm{e}psilon x}\frac{\mathbb{P}(S_{k}>x)}{k{\overline F}(x)}\ < \ \infty\]
\mathrm{e}nd{corollary}
\begin{proof} Define $\delta=(1-\mathrm{e}psilon{\overline z})/\mathrm{e}psilon$. We can write $x=k{\overline z}+y$ where
\[y\ =\ y(x,k)\ =\ x-k{\overline z}\ \ge\ x(1-\mathrm{e}psilon {\overline z})\ =\ x\mathrm{e}psilon\delta\ \ge\ \delta k\,.\]
Lemma~\ref{Lemma:11.7aa} therefore gives that for all large $x$ we can bound $\mathbb{P}(S_{k}>x)$
by $Ck{\overline F}(y)$ where $C$ does not depend on $x$. Now just note that by RV
\[{\overline F}(y) \ \le\ {\overline F}(x\mathrm{e}psilon\delta)\ \sim (\mathrm{e}psilon\delta)^{-\alpha}{\overline F}(x)\,.\]
\mathrm{e}nd{proof}
\begin{proof}[Proof of Proposition~\ref{Prop:2.8b}]
For ease of exposition, we start by the case $p=2$. We split the probability in question into four parts
\begin{align*}p_1(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1\le \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{21}(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{22}(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1\le \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x\bigr)\\
p_3(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x\bigr)
\\ \intertext{Here}
p_1(x)\ &=\ \sum_{k_1,k_2=0}^{\mathrm{e}psilon x}\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)
\mathbb{P}(N_1=k_1,N_2=k_2)
\mathrm{e}nd{align*}
Since $S_{k_1}^{(1)},\,S_{k_2}^{(2)}$ are independent, we have by standard RV theory that
\[\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)\ \sim\ (k_1c_1+k_2c_2){\overline F}(x)\]
as $x\to\infty$. Further Corollary~\ref{Cor:2.8a} gives that for $k_1,k_2\le \mathrm{e}psilon x$ and all large $x$ we have
\begin{align*}\mathbb{P}\bigl(S_{k_1}^{(1)}+S_{k_2}^{(2)}>x\bigr)\ &\le\
\mathbb{P}\bigl(S_{k_1}^{(1)}>x/2\bigr)+\mathbb{P}\bigl(S_{k_2}^{(2)}>x/2\bigr)\\ &\le\
2\bigl(d(F_1,2\mathrm{e}psilon)k_1+d(F_2,2\mathrm{e}psilon)k_2\bigr){\overline F}(x)\,.
\mathrm{e}nd{align*}
Hence by dominated convergence
\[\frac{p_1(x)}{{\overline F}(x)}\ \to \ \sum_{k_1,k_2=0}^{\infty}(k_1c_1+k_2c_2)
\mathbb{P}(N_1=k_1,N_2=k_2)\ =\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2\,.\]
For $p_3(x)$, denote by $A_j(m)$ the event that $S_{k_j}^{(j)}/k_j\le{\overline z}_j/(1-\mathrm{e}psilon)$
for all $k_j>m$. Then by the LNN there are constants $r(m)$ converging to 0 as $m\to\infty$
such that $\mathbb{P} \rev{\bigl(}A_j(m)^{\mbox{\normalsize c}}\rev{\bigr)} \le r(m)$ for $j=1,2$. It follows that
\begin{align*}
p_3(x)\ &\le \ \bigl(\mathbb{P} \rev{\bigl(}A_1(\mathrm{e}psilon x)^{\mbox{\normalsize c}}\rev{\bigl)}+
\mathbb{P}\rev{\bigl(} A_2(\mathrm{e}psilon x)^{\mbox{\normalsize c}}\rev{\bigl)}\bigr)
\mathbb{P}(N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)\\ &
+\mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x,A_1(\mathrm{e}psilon x),A_2(\mathrm{e}psilon x)\bigr)
\\ &\le r(\mathrm{e}psilon x){\mathrm{O}}\bigl({\overline F}(x)\bigr)+\mathbb{P}\bigl(({\overline z}_1N_1+{\overline z}_2N_2)/(1-\mathrm{e}psilon)>x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)\\ &\le\ {\mathrm{o}}\bigl({\overline F}(x)\bigr) \mathbb{P}\bigl(({\overline z}_1N_1+{\overline z}_2N_2)>\mathrm{e}ta x,\, N_1> \mathrm{e}psilon x,\, N_2> \mathrm{e}psilon x)
\mathrm{e}nd{align*}
\rev{as $x\to\infty$},
where $\mathrm{e}ta<1-\mathrm{e}psilon$ will be specified later.
For $p_{21}(x)$, we write $p_{21}(x)=p_{21}'(x)+p_{21}''(x)$ where
\begin{align*}
p_{21}'(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,S_{N_2}^{(2)}\le \gamma x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\
p_{21}''(x)\ &=\ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x,S_{N_2}^{(2)}> \gamma x,\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\mathrm{e}nd{align*}
with $\gamma=2\mathrm{e}psilon{\overline z}_2$. Here
\begin{align*}
p_{21}''(x)\ &\le \ \mathbb{P}\bigl(S_{N_1}^{(1)}+S_{\mathrm{e}psilon x}^{(2)}>x,S_{\mathrm{e}psilon x}^{(2)}> \gamma x,\,
N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\ &\le\
\mathbb{P}\bigl(S_{ \mathrm{e}psilon x}^{(2)}> \gamma x,\,
N_1> \mathrm{e}psilon x\bigr)\ =\ \mathbb{P}\bigl(S_{ \mathrm{e}psilon x}^{(2)}> \gamma x\Bigr)\,
\mathbb{P}(N_1> \mathrm{e}psilon x)\\ &=\ {\mathrm{o}}(1){\mathrm{O}}\bigl({\overline F}(x)\bigr)\ =\ {\mathrm{o}}\bigl({\overline F}(x)\bigr)\,,
\mathrm{e}nd{align*}
using the LLN in the fourth step. Further as in the estimates above
\begin{align*}
p_{21}'(x)\ &\le\ \mathbb{P}\bigl(S_{N_1}^{(1)}>x(1-\gamma),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\\ &\le \
{\mathrm{o}}\bigl({\overline F}(x)\bigr)\ +\ \mathbb{P}\bigl({\overline z}_1N_1>x(1-\gamma)(1-\mathrm{e}psilon),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\\ &\le \ \mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>x(1-\gamma)(1-\mathrm{e}psilon),\, N_1> \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)
\mathrm{e}nd{align*}
We can now finally put the above estimates together. For ease of notation, write $\mathrm{e}ta=\mathrm{e}ta(\mathrm{e}psilon)=
(1-\gamma)(1-\mathrm{e}psilon)$ and note that $\mathrm{e}ta\uparrow 1$ as $\mathrm{e}psilon\downarrow 0$. Using a similar
estimate for $p_{12}(x)$ as for $p_{21}(x)$ and noting that
\[\mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>\mathrm{e}ta x,\, N_1\le \mathrm{e}psilon x,\, N_2\le \mathrm{e}psilon x\bigr)\ =\ 0\]
for $\mathrm{e}psilon$ small enough, we get
\begin{align*} \MoveEqLeft \limsup_{x\to\infty}\frac{1}{{\overline F}(x)}\mathbb{P}\bigl(S_{N_1}^{(1)}+S_{N_2}^{(2)}>x\bigr)\\ &
=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2+ \limsup_{x\to\infty}\frac{1}{{\overline F}(x)}\mathbb{P}\bigl({\overline z}_1N_1+{\overline z}_2N_2>\mathrm{e}ta x\bigr)\\
&=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2 +c({\overline z}_1,{\overline z}_2) \limsup_{x\to\infty}\frac{{\overline F}(\mathrm{e}ta x)}{{\overline F}(x)}\\
&=\ c_1\mathbb{E} N_1+c_2\mathbb{E} N_2 +c({\overline z}_1,{\overline z}_2)\frac{1}{\mathrm{e}ta^\alpha}
\mathrm{e}nd{align*}
Letting $\mathrm{e}psilon\downarrow 0$ gives that the $\limsup$ is bounded by $c_0\rev{+}c({\overline z}_1,{\overline z}_2)$. Similar estimates
for the $\liminf$ complete the proof for $p=2$.
If $p>2$, the only essential difference is that $p_{21}(x),p_{22}(x)$ need to be replaced by the $2^p-2$ terms
corresponding to all combinations of some $N_i$ being $\le \mathrm{e}psilon x$ and the others $>\mathrm{e}psilon x$,
with the two exceptions being the ones where either all are $\le \mathrm{e}psilon x$ or all are $>\mathrm{e}psilon x$.
However, to each of these similar estimates as the above ones for $p_{21}(x)$ apply.
\mathrm{e}nd{proof}
\section{Preservation of MRV under sum operations}\label{S:PresMRV}
\setcounter{equation}{0}
Before giving our main auxiliary result, Proposition~\ref{Prop:5.8c}, it is instructive to recall two extremely simple example of MRV. The first is two i.i.d.\ RV$(F)$ r.v.'s $X_1,X_2$, where a big value of the $X_1+X_2$
can only occur if one variable is big and the other small, which gives MRV with the angular measure
concentrated on the points $(1,0),\,(0,1)\in {\cal B}_2$ with mass 1/2 for each. Slightly more complicated:
\begin{proposition}\label{Prop:5.8b}
Let $N,Z,Z_1,Z_2,\ldots$ be non-negative r.v.'s such that $N\in{\mathbb N}$, $Z,Z_1,Z_2,\ldots$ are i.i.d., non-negative
and independent of $N$. Assume that $\mathbb{P}(N>x)\sim c_N{\overline F}(x)$, $\mathbb{P}(Z>x)\sim c_Z{\overline F}(x)$
for some RV \rev{tail ${\overline F}(x)=L(x)/x^\alpha$} and write $S=\sum_1^NZ_i$, ${\overline n}=\mathbb{E} N$, ${\overline z}=\mathbb{E} Z$,
where $c_N+c_Z>0$. Then:\\[1mm]
\mathrm{e}mph{(i)} $\mathbb{P}(S>x)\sim (c_N {\overline z}^{\alpha}+c_Z{\overline n}){\overline F}(x)$;\\[1mm]
\mathrm{e}mph{(ii)} The random vector $(N,S)$ is \mathrm{e}mph{MRV} with \[\mathbb{P}\bigl(\|(N,S)\|>x\bigr)\sim c_{N,S}{\overline F}(x)\quad
\text{where}\quad c_{N,S}=c_N (1+{\overline z}^{\alpha})+c_Z{\overline n}\]
and angular measure $\mu_{N,S}$ concentrated on the points $\mbox{\boldmath$ b$}_1=\bigl(1/(1+{\overline z}),{\overline z}/(1+{\overline z})\bigr)$ and $\mbox{\boldmath$ b$}_2=(0,1)$
with \[\mu_{N,S}(\mbox{\boldmath$ b$}_1)=\frac{c_N}{c_N+c_Z{\overline n}}\,,\qquad \mu_{N,S}(\mbox{\boldmath$ b$}_2)=\frac{c_Z{\overline n}}{c_N+c_Z{\overline n}}\,.\]
\mathrm{e}nd{proposition}
\begin{proof} Part (i) is Lemma 4.7 of~\cite{Fay} (see also~\cite{Denisov}). The proof in~\cite{Fay} also shows that is $S>x$, then
either approximately $N{\overline z}>x$, occuring w.p.\ $c_N{\overline F}(x/{\overline z})\sim c_N {\overline z}^{\alpha}{\overline F}(x)$, or $N\le \mathrm{e}psilon x$ and $Z_i>x$,
occuring w.p.\ $c_Z\mathbb{E}[N\wedge \mathrm{e}psilon x]{\overline F}(x)$. The first possibility is what gives the atom of $\mu_{N,S}$ at $b_1$
and the second gives the atom at $b_2$ since $\mathbb{E}[N\wedge \mathrm{e}psilon x]\uparrow{\overline n}$.
\mathrm{e}nd{proof}
\begin{proposition}\label{Prop:5.8c}
Let $\mbox{\boldmath$ V$}=(\mbox{\boldmath$ T$},N)\in[0,\infty)^{p}\times{\mathbb N}$ satisfy {\mbox{{\rm MRV}$(F)$}},
let $\mbox{\boldmath$ Z$},\mbox{\boldmath$ Z$}_1,\mbox{\boldmath$ Z$}_2,\ldots$\,$\in[0,\infty)^q$ be i.i.d.\ and independent of $(\mbox{\boldmath$ T$},N)$ and
satisfying {\mbox{{\rm MRV}$(F)$}}, and
define $\mbox{\boldmath$ S$}=\sum_1^N\mbox{\boldmath$ Z$}_i$. Then $\mbox{\boldmath$ V$}^*=(\mbox{\boldmath$ T$},N,\mbox{\boldmath$ S$})$ satisfies {\mbox{{\rm MRV}$(F)$}}.
\mathrm{e}nd{proposition}
\begin{proof}
Let $\overline \mbox{\boldmath$ z$}\in[0,\infty)^q$ be the mean of $\mbox{\boldmath$ Z$}$.
Similar arguments as in Section~\ref{S:FayeMRV} show that $\|V^*\|>x$ basically occurs when either
$\|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|>x$ or when $ \|\mbox{\boldmath$ V$}\|\le\mathrm{e}psilon x$ and some $\|\mbox{\boldmath$ Z$}_i\|>x$.
The probabilities of these events are approximately of the form $c'{\overline F}(x)$ and $c''{\overline F}(x)$, so the radial part
of $\mbox{\boldmath$ V$}^*$ is RV with asymptotic tail $c_{\mbox{\scriptsize\boldmath$ V$}^*}{\overline F}(x)$ where $c_{\mbox{\scriptsize\boldmath$ V$}^*}=c'+c''$.
Now
\[\mathbb{P}\Bigl(\frac{(\mbox{\boldmath$ T$},N)}{\|(\mbox{\boldmath$ T$},N)\|}\in \cdot\,\Big|\, \|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|>x\Big)\ \to\ \mu'\]
for some probability measure $\mu'$ on the $(p+1)$-dimensional unit sphere ${\cal B}_{p+1}$; this follows since
$\|\mbox{\boldmath$ T$}\|+N+N\|\overline\mbox{\boldmath$ z$}\|$ is a norm and the MRV property of a vector is independent of the choice of norm.
Letting $\delta'_0$ be Dirac measure at $(0,\ldots,0)\in\mathbb{R}^q$, $\delta''_0$ be Dirac measure at $(0,\ldots,0)\in\mathbb{R}^{p+1} $
and $\mu''=\mu_{\mbox{\boldmath$ Z$}}$ \rev{the angular measure of $\mbox{\boldmath$ Z$}$}, we obtain the desired conclusion with $c_{\mbox{\scriptsize\boldmath$ V$}^*}=c'+c''$ and \rev{the angular measure of $\mbox{\boldmath$ V$}^*$
given by}
\[\mu_{\mbox{\scriptsize\boldmath$ V$}^*}\ =\ \frac{c'}{c'+c''}\,\mu'\otimes\delta''_0+\frac{c''}{c'+c''}\,\delta'_0\otimes\mu''\]
\mathrm{e}nd{proof}
In calculations to follow (Lemma~\ref{Lemma:7.8a}), extending some $\mbox{\boldmath$ V$}$ to some $\mbox{\boldmath$ V$}^*$ in a number of steps, expressions for $c_{\mbox{\scriptsize\boldmath$ V$}^*},\mu_{\mbox{\scriptsize\boldmath$ V$}^*}$ can be deduced along the lines of the proof of
Propositions~\ref{Prop:5.8b}--\ref{Prop:5.8c} but the expression and details become extremely tedious. Fortunately, they won't be needed and are therefore omitted --- all that matters is existence.
If $\alpha$ is not an even integer,
the MRV alone of $\mbox{\boldmath$ V$}^*$ can alternatively (and slightly easier) be obtained from
Theorem 1.1(iv) of~\cite{Basrak}, stating that by non-negativity it suffices to verify MRV of any linear combination.
\section{Proof of Theorem~\ref{Th:6.8a} completed}\label{S:ProofCompl}
\setcounter{equation}{0}
\begin{lemma}\label{Lemma:7.8a} In the setting of \mathrm{e}qref{30.6aa},
the random vector \[\mbox{\boldmath$ V$}^*(i)\ =\ \bigl(\widetilde Q(i), \widetilde N^{(1)}(i),\ldots,\widetilde N^{(K-1)}(i)\bigr)\] satisfies {\mbox{{\rm MRV}$(F)$}}\
for all $i$.
\mathrm{e}nd{lemma}
\begin{proof}
Let $\bigl|{\mathcal G}_m(i)\bigr|$ be the
number of elements of ${\mathcal G}_m(i)$ and
\begin{align*} M_1(i)\ &=\ \sum_{m=1}^{N^{(K)}(i)}\bigl|{\mathcal G}_m(i)\bigr|\,,\\
M_2(i)\ &=\ \sum_{m=1}^{N^{(K)}(i)}\sum_{g\in{\mathcal G}_m(i)}\bigl(Q_g(K),N_g^{(1)}(K),\ldots,N_g^{(1)}(K-1)\bigr)\mathrm{e}nd{align*}
Recall that our basic assumption is that the
\begin{equation}\label{7.8d} \mbox{\boldmath$ V$}^*(i)\ =\ \bigl(Q(i), N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)\mathrm{e}nd{equation}
satisfy {\mbox{{\rm MRV}$(F)$}}. The connection to a Galton-Watson tree
and Theorem~\ref{Th:3.8a} with $Q\mathrm{e}quiv 1$, $N=N^{(K)}(i)$ therefore imply that so does any $\bigl|{\mathcal G}_m(i)\bigr|$,
and since these r.v.'s are i.i.d.\ and independent of $N^{(K)}(i)$, Proposition~\ref{Prop:5.8c} gives that
$\mbox{\boldmath$ V$}_1(i)=\bigl(\mbox{\boldmath$ V$}(i), M_1(i)\bigr)$ satisfies {\mbox{{\rm MRV}$(F)$}}. Now the {\mbox{{\rm MRV}$(F)$}}\ property of \mathrm{e}qref{7.8d} with $i=K$
implies that the vectors $\bigl(Q_g(K),N_g^{(1)}(K),\ldots,N_g^{(K-1)}(K)\bigr)$, being distributed as
$\bigl(Q(K),N^{(1)}(K),\ldots,N^{(K-1)}(K)\bigr)$
again satisfy {\mbox{{\rm MRV}$(F)$}}. But $M_2(i)$ is a sum of $M_1(i)$ such vectors that are i.i.d.\ given $M_1(i)$.
Using Proposition~\ref{Prop:5.8c} once more gives that $\mbox{\boldmath$ V$}_2(i)=\bigl(\mbox{\boldmath$ V$}(i), M_1(i),M_2(i)\bigr)$ satisfies {\mbox{{\rm MRV}$(F)$}}.
But $\mbox{\boldmath$ V$}^*(i)$ is a function of $\mbox{\boldmath$ V$}_2(i)$. Since this function is linear, property {\mbox{{\rm MRV}$(F)$}}\ of
$\mbox{\boldmath$ V$}_2(i)$ carries over to $\mbox{\boldmath$ V$}^*(i)$.
\mathrm{e}nd{proof}
\begin{proof}[Proof of Theorem~\ref{Th:6.8a}]
We use induction in $K$. The case $K=1$ is just Theorem~\ref{Th:3.8a}, so assume
Theorem~\ref{Th:6.8a} shown for $K-1$.
The induction hypothesis and Lemma~\ref{Lemma:7.8a} implies that $\mathbb{P}(R(i)>x)\sim d_i{\overline F}(x)$ for $i=1,...,K-1$.
Rewriting \mathrm{e}qref{30.6a} for $i=K$ as
\begin{equation*} R(K) \ \mathrm{e}qdistr\ Q^*(K)+\sum_{m=1}^{N^{(K)}(K)}R_{m}(K)
\ \ \text{where}\ \ Q^*(K)\,=\,\sum_{k=1}^{K-1}\sum_{m=1}^{N^{(k)}(K)}R_{m}(k)\,,\mathrm{e}nd{equation*}
we have a \rev{fixed-point} problem of type \mathrm{e}qref{eq1} and can then use Theorem~\ref{Th:6.8a} to conclude that
also $\mathbb{P}(R(K)>x)\sim d_K{\overline F}(x)$, noting that the needed MRV condition on $\bigl(Q^*(K),N^{(k)}(K)\bigr)$
follows by another application of Proposition~\ref{Prop:5.8c}.
Finally, to identify the $d_i$ via \mathrm{e}qref{6.8d}, appeal to Proposition~\ref{Prop:2.8b} with
$\mbox{\boldmath$ N$}=\bigl(Q(i), N^{(1)}(i),\ldots,N^{(K)}(i)\bigr)$, writing the r.h.s.\ of \mathrm{e}qref{30.6a} as
\[{\mathrm{O}}(1)\,+\,\sum_{m=1}^{\lfloor Q(i)\rfloor} 1\,+\, \sum_{k=1}^K\sum_{m=1}^{N^{(k)}(i)}R_{m}(k)\,.\]
Existence and uniqueness of a solution to \mathrm{e}qref{6.8d} follows by once more noticing that $\rho<1$
implies that $\mbox{\boldmath$ I$}-\mbox{\boldmath$ M$}$ is invertible.
\mathrm{e}nd{proof}
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\appendix
\section{\rev{Proof of \mathrm{e}qref{30.6ax}}}
\rev{The RV of linear combinations subject to MRV assumptions has received considerable attention, see e.g.~\cite{Basrak}, but we could not find explicit
formulas like \mathrm{e}qref{30.6ax} for the relevant constants so we give a
self-contained proof. The formula is a special case of the following: if $\mbox{\boldmath$X$}=(X_1\,\ldots\,X_n)\in\mathbb{R}^n$
is a random vector such that $\mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>t\big)\sim L(t)/t^\alpha$
and $\mbox{\boldmath$ T$}heta=\mbox{\boldmath$X$}/\|\mbox{\boldmath$X$}\|$ has conditional limit distribution
$\mu$ in ${\mathcal B}_1$ given $\|\mbox{\boldmath$X$}\|>t$ as $t\to\infty$, then
\[\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>x)\ =\ \mathbb{P}(a_1X_1+\cdots+a_nX_n>t)\ \sim\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha\mu(\mathrm{d}\mbox{\boldmath$ \theta$})\]}
\rev{To see this, note that given $\mbox{\boldmath$ T$}heta=\mbox{\boldmath$ \theta$}\in{\mathcal B}_1$,
$\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$} =$ $ \|\mbox{\boldmath$X$}\|(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})$
will exceed $t>0$ precisely when $\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0$ and $\|\mbox{\boldmath$X$}\|>t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}$.
Thus one expects that
\begin{align*}\MoveEqLeft
\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)\ \sim\
\int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)\mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\bigr)\,\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\\ & \sim\
\int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)\frac{L(t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})}{(t/\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha}\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\ \sim\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}\mathbb{I}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}>0)(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^\alpha\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\mathrm{e}nd{align*}
which is the same as asserted.}
\rev{For the rigorous proof, assume $\|\mbox{\boldmath$a$}\|=1$. Then ${\mathcal B}_1$ is the disjoint union of the sets
$B_{1,n},\ldots,B_{n,n}$ where $B_{i,n}=$
$\{\mbox{\boldmath$ \theta$}\in{\mathcal B}_1:\,(i-1)/n<\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\le i/n\}$
for $i=2,\ldots,n$ and $B_{1,n}=$ $\{\mbox{\boldmath$ \theta$}\in{\mathcal B}_1:\,\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$}\le 1/n\}$. Assuming $\mathbb{P}(\mbox{\boldmath$ T$}heta=i/n)=0$ for all integers $i,n$, we get
\begin{align*}
\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)\ &=\ \sum_{i=1}^n \mathbb{P}\bigl(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t,\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)\
\le\ \sum_{i=1}^n \mathbb{P}\bigl(\|\mbox{\boldmath$X$}\|>ti/n,\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)
\\& \sim\ \sum_{i=1}^n \frac{L(ti/n)}{(ti/n)^\alpha}
\mathbb{P}\bigl(\mbox{\boldmath$ T$}heta\in B_{i,n}\,\big|\,|\mbox{\boldmath$X$}\|>t\bigr)\
\sim\ \frac{L(t)}{t^\alpha} \sum_{i=1}^n (i/n)^\alpha \mathbb{P}\bigl(\mbox{\boldmath$ T$}heta\in B_{i,n}\bigr)\\ &=\
\frac{L(t)}{t^\alpha} \int_{{\mathcal B}_1}f_{+,n}(\mbox{\boldmath$ \theta$})\,\mu(\mathrm{d}\mbox{\boldmath$ \theta$})
\mathrm{e}nd{align*}
where $ f_{+,n}$ is the step function taking value $(i/n)^\alpha$ on $B_{i,n}$.
A similar argument gives the asymptotic lower bound $\int \!f_{-,n}\,\mathrm{d}\mu\,L(t)/t^\alpha$ for
$\mathbb{P}(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$X$}>t)$ where $ f_{-,n}$ equals $\bigl((i-1)/n)\bigr)^\alpha$ on $B_{i,n}$
for $i>1$ and $0$ on $B_{1,n}$.
But $f_{\pm,n}(\mbox{\boldmath$ \theta$})$ both have limits $\bigl[(\mbox{\boldmath$a$}\cdot\mbox{\boldmath$ \theta$})^+\bigr]^\alpha$
as $n\to\infty$ and are
bounded by 1. Letting $n\to\infty$ and using dominated convergence completes the proof.}
\rev{The case $\mathbb{P}(\mbox{\boldmath$ T$}heta=i/n)>0$ for some $i,n$ is handled by a trivial
redefinition of the $B_{i,n}$.}
\mathrm{e}nd{document}
The referee has asked us for a reference for the the value of the
constant $c(a_0,\ldots,a_K)$ in the following result:
if the non-negative vector $(X_0,\ldots,X_K)$ is MRV with index $\alpha$ and angular measure $\mu$, then
\begin{equation}\label{30.6ax}\mathbb{P}\bigl(a_0X_0+a_1X_1+\cdots+a_KX_K >x\bigr)\ \sim\ c(a_0,\ldots,a_K){\overline F}(x)
\mathrm{e}nd{equation} where $\displaystyle
c(a_0,\ldots,a_K)\ =\ (a_0\theta_0+\cdots+a_K\theta_K)^\alpha\mu(\mathrm{d}\theta_0,\ldots,\mathrm{d}\theta_K)\,.$
I have it in my notes used for my course \mathrm{e}mph{Risk}, but when I wrote them,
I didn't see the formula anywhere in the papers on MRV and linear combinations.
It must of course be known.
\mathrm{e}nd{document} |
\begin{document}
\title[Carath\'eodory functions]{Carath\'eodory
functions in the Banach space setting}
\author[D. Alpay]{Daniel Alpay}
\address{Department of Mathematics\\
Ben--Gurion University of the Negev\\
Beer-Sheva 84105\\ Israel} \email{dany@math.bgu.ac.il}
\thanks{D. Alpay thanks the Earl Katz family for endowing the chair which
supports his research}
\author[O. Timoshenko]{Olga Timoshenko}
\address{Department of Mathematics\\
Ben--Gurion University of the Negev\\
Beer-Sheva 84105\\ Israel} \email{olgat@math.bgu.ac.il}
\author[D. Volok]{Dan Volok}
\address{Department of Mathematics\\
Kansas State University\\
Manhattan, KS 66506-2602\\USA}\email{danvolok@hotmail.com}
\subjclass[2000]{30A86; 47A56.}
\date{}
\begin{abstract}
We prove representation theorems for Carath\'eodory functions in the setting of Banach spaces.
\end{abstract}
\maketitle
\section{Introduction}
L. de Branges and J. Rovnyak introduced in \cite{dbr1}, \cite{dbr2}
various families of reproducing kernel Hilbert spaces of functions which take
values in a Hilbert space and are analytic in the open unit disk
or in the open upper half-plane. These spaces play an important role
in operator theory, interpolation theory,
inverse scattering, the theory of wide sense stationary stochastic
processes and related topics; see for instance \cite{Dmk},
\cite{ad1}, \cite{ad2}. In the case of the open unit disk
$\mathbb{D}$, of particular importance are the following two
kinds of reproducing kernels:
\begin{align}
\label{avron} k_\phi(z,w)&=\frac{\phi(z)+\phi(w)^*}{2(1-zw^*)},\\
\nonumber
k_s(z,w)&=\frac{I-s(z)s(w)^*}{1-zw^*}.
\end{align}
In these expressions, $s(z)$ and $\phi(z)$ are operator-valued functions analytic
in $\mathbb{D}$, $*$ denotes
the Hilbert space adjoint, and $I$ denotes the identity operator.
The functions for which the kernels $k_\phi(z,w)$ and $k_s(z,w)$ are positive
are called respectively Carath\'eodory and Schur functions.
We remark that one can use the Cayley
transform
$$s(z)=(I-\phi(z))(I+\phi(z))^{-1}$$
to reduce the study of the kernels $k_\phi(z,w)$ to the study of
the kernels $k_s(z,w)$. For these latter it is
well known that the positivity of the kernel $k_s(z,w)$ implies analyticity of $s(z)$.\\
Every Carath\'eodory function
admits two equivalent representations. The first, called the
Riesz -- Herglotz representation, reads as follows:
\begin{equation}
\label{anaelle}
\phi(z)=ia+\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}d\mu(t)
\end{equation}
where $a$ is a real number and where $\mu(t)$ is an increasing
function such that $\mu(2\pi)<\infty$. The integral is a
Stieltjes integral and the proof relies on Helly's theorem; see
\cite[pp. 19--27]{MR48:904}.\\
The second representation reads:
\begin{equation}
\label{anaelle2}
\phi(z)=ia+\Gamma(U+zI)(U-zI)^{-1}\Gamma^*
\end{equation}
where $a\in{\mathbb R}$ and where $U$ is a unitary operator in an
auxiliary Hilbert space ${\mathcal H}$ and $\Gamma$ is a bounded
operator from ${\mathcal H}$ into ${\mathbb C}$.\\
The expression \eqref{anaelle2} still makes sense in a more
general setting when the kernel $k_\phi(z,w)$ has a finite
number of negative squares. The space ${\mathcal H}$ is then a
Pontryagin space. This is the setting in which Kre\u\i n and
Langer proved this result; see \cite[Satz 2.2 p. 361]{kl1}. They
allowed the values of the function $\phi(z)$ to be operators
between Pontryagin spaces and required weak continuity at the
origin. Without this hypothesis one can find functions for which
the kernel $k_\phi(z,w)$ has a finite number of negative squares
but which are not meromorphic in ${\mathbb D}$ and in particular
cannot admit representations of the form \eqref{anaelle2}; for
instance the function
$$\phi(z)=\begin{cases} 0\,\,{\rm if}\,\, z\not=0\\
1\,\,{\rm if}\,\, z=0
\end{cases}
$$
defines a kernel $k_\phi(z,w)$ which has one negative square; see
\cite[p. 82]{adrs} for an analogue for $k_s(z,w)$
kernels.\\
Operator--valued Carath\'eodory functions were extensively studied in the
Hilbert space case; see e.g. \cite{MR1821917}, \cite{MR1759548},
\cite{MR899274}, \cite{MR1036844}. We would also like to mention the
non-stationary setting, where an analogue of the representation
\eqref{anaelle2} was obtained for upper-triangular operators; see
\cite{MR2003f:47021} and \cite{MR1704663}.\\
The notion of reproducing kernel space (with positive or
indefinite metric) can also be introduced for functions which
take values in Banach spaces and even topological vector spaces.
The positive case was studied already by Pedrick for functions
with values in certain topological vector spaces in an
unpublished report \cite{pedrick} and studied further by P. Masani in his 1978 paper
\cite{masani78}. Motivations originate from the theory of partial
differential equations (see e.g. \cite{MR1973084}) and the theory
of stochastic processes (see e.g. \cite[\S 4]{MR647140} and \cite{MR626346}).\\
The present paper is devoted to the study of Carath\'eodory functions
whose values are bounded operators between appropriate Banach
spaces. It seems that there are no natural analogs of
Schur functions or of the Cayley transform in this setting.\\
Let ${\mathcal B}$ be a Banach space. We denote by ${\mathcal B}^*$ the
space of anti-linear bounded
functionals (that is, its conjugate dual space). The duality between ${\mathcal B}$
and ${\mathcal B}^*$ is
denoted by
\[
\langle b_*, b\rangle_{\mathcal B}\defi b_*(b),\quad \text{where
} b\in {\mathcal B}\text{ and } b_*\in{\mathcal B}^*.\] An
${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function $\phi(z)$ defined in
some open neighborhood $\Omega$ of the origin
and weakly continuous at the origin will be called a Carath\'eodory
function if the ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
kernel
\begin{equation}
\label{toto}
k_\phi(z,w)=\frac{\phi(z)+\phi(w)^*\big|_{\mathcal B}}{2(1-zw^*)}
\end{equation}
is positive in $\Omega$. The notion of positivity for bounded
operators and kernels from ${\mathcal B}$
into ${\mathcal
B}^*$ is reviewed in the next section. We shall prove (see Theorem \ref{Ivry-sur-Seine})
that every Carath\'eodory function admits a representation of the form
\eqref{anaelle2} and, in particular, admits an analytic extension to
$\mathbb{D}$; see e.g. \cite[pp. 189--190]{MR58:12429a} for
information on vector-valued analytic functions. We note that the proof of
this theorem can be adapted to the case when the kernel
$k_\phi(z,w)$ has a finite number of negative squares;
see Remark \ref{indef}.\\
Furthermore, if $\mathcal{B}$ is a separable Banach space
then ${\mathbf L}({\mathcal B},{\mathcal
B}^*)$-valued Carath\'eodory functions can be characterized as
functions analytic in $\mathbb{D}$ and such that
\[
\phi(z)+\phi(z)^*\bigm|_{\mathcal B}\ge 0,\quad z\in{\mathbb D}.
\]
Moreover, in this case we have an analogue of the Riesz -- Herglotz
representation \eqref{anaelle}; see Theorem
\ref{Antony}.\\
We conclude with the outline of the paper; the next three sections
are of preliminary nature, and deal with positive operators, Stieltjes
integrals and Helly's theorem respectively. Representation theorems for
Carath\'eodory functions are proved in Section 5. Two cases are to be
distinguished, as whether ${\mathcal B}$ is separable or not.
The case of ${\mathbf L}({\mathcal B}^*,{\mathcal B})$-valued
Carath\'eodory functions will be treated in the last section of this paper.
This case is of special importance. Indeed, if $\phi(z)$ is a ${\mathbf L}
({\mathcal B}^*, {\mathcal B})$-valued Carath\'eodory function which takes
invertible values, its inverse is a
${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued Carath\'eodory
function. In the Hilbert space case, this fact has important connections
with operator models for pairs of unitary
operators (see \cite{dbs} and, for the
analogue for self--adjoint operators, \cite{dbr1}, \cite{MR1960423}).
We will explore the Banach space generalizations of these results
in a future publication.
\section{Positive operators and positive kernels}
In this section we review for the
convenience of the reader various facts on bounded positive operators from
${\mathcal B}$ into ${\mathcal B}^*$.
First some notations and a definition.
\begin{definition} Let ${\mathcal B}$ be a complex Banach space and let
$A\in{\mathcal L}({\mathcal B},{\mathcal B}^*).$ The operator $A$ is said to
be {\em positive} if
\[
\langle Ab,b\rangle_{\mathcal B}\geq 0,\quad \forall b\in {\mathcal B}.\]
\end{definition}
Note that a positive operator is in particular self-adjoint in the sense
that $A=A^*\bigm|_{\mathcal B}$, that
is,
\begin{equation}
\label{gabriel} \langle Ab,c\rangle_{{\mathcal B}}=
\overline{\langle Ac,b\rangle_{{\mathcal B}}}.
\end{equation}
Indeed, \eqref{gabriel} holds for $b=c$ in view of the positivity. It then
holds for all choices of $b,c\in{\mathcal B}$ by polarization:
\[
\begin{split}
\langle Ab,c\rangle_{{\mathcal B}}&=\frac{1}{4} \sum_{k=0}^3i^k\langle A(b+i^kc),(b+i^kc)
\rangle_{{\mathcal B}}\\
&=\frac{1}{4}
\sum_{k=0}^3i^k\langle A(c+i^{-k}b),(c+i^{-k}b)
\rangle_{{\mathcal B}}\\
&=\frac{1}{4}\overline{
\sum_{k=0}^3i^{-k}\langle A(c+i^{-k}b),(c+i^{-k}b)
\rangle_{{\mathcal B}}}\\
&=\overline{\langle Ac,b\rangle_{{\mathcal B}}}.
\end{split}
\]
Now, let $\tau$ be the natural injection
from ${\mathcal B}$ into
${\mathcal B}^{**}$:
\begin{equation}
\label{bastille}
\langle \tau(b), b_*\rangle_{{\mathcal B}^*} =
\overline{\langle b_*, b\rangle_{\mathcal B}}.
\end{equation}
We have for $b,c\in{\mathcal B}$:
\[
\begin{split}
\langle A^*\tau c,b\rangle_{\mathcal B}&=\langle \tau c, Ab
\rangle_{{\mathcal B}^*}\\
&=\overline{\langle Ab,c\rangle_{\mathcal B}}\\
&=\langle Ac,b\rangle_{\mathcal B}
\end{split}
\]
in view of \eqref{gabriel}, and hence $A=A^*\big|_{\mathcal B}$.\\
The following factorization result is well known and originates
with the works of Pedrick \cite{pedrick} (in the case of
topological vector spaces with appropriate properties) and
Vakhania \cite[\S 4.3.2 p.101]{MR626346} (for positive elements
in ${\mathbf L}({\mathcal B}^*,{\mathcal B})$); see the
discussion in \cite[p. 416]{masani78}. We refer also to
\cite{MR647140} for the case of barreled spaces and to
\cite{MR1973084} for the case of unbounded operators.
\begin{theorem}
The operator $A\in{\mathcal L}({\mathcal B},{\mathcal B}^*)$ is
positive if and only if there exist a Hilbert space ${\mathcal H}$
and a bounded operator $T\in{\mathbf L} ({\mathcal B},{\mathcal
H})$ such that $A=T^*T$. Moreover,
\begin{equation}
\label{samantha}
\langle Ab,c\rangle_{{\mathcal B}}=\langle Tb,
Tc \rangle_{\mathcal H},\quad b,c\in {\mathcal B}
\end{equation}
and
\begin{equation}
\label{bagdad} \sup_{\|b\|=1}\langle Ab,b\rangle_{{\mathcal B}} =\|A\|=\|T\|^2.
\end{equation}
Finally we have
\begin{equation}
\label{sof-sof}
|\langle Ab,c\rangle_{\mathcal B}|\le\langle Ab,b\rangle^{1/2}
\langle Ac,c\rangle^{1/2}.
\end{equation}
\label{rosh-hashana}
\end{theorem}
\begin{proof} For $Ab$ and $Ac$ in the range of $A$ the expression
\begin{equation}
\label{mind your step} \langle Ab,Ac\rangle_A= \langle Ab,c\rangle_{{\mathcal B}}=
\overline{\langle
Ac,b\rangle_{{\mathcal B}}}.
\end{equation}
is well defined in the sense that $Ab=0$ (resp. $Ac=0$) implies that
\eqref{mind your step} is equal to $0$.
Thus formula \eqref{mind your step} defines a
sesquilinear form on $\ran A$. It is
positive since the operator $A$ is positive. Moreover, it is non-degenerate because
if $\langle Ab,b\rangle_{\mathcal{B}}=0$ then $Ab=0.$ \\
Indeed, if $c$ is such that $\langle Ac,c\rangle_{\mathcal{B}}=0$ then
\eqref{gabriel} implies that
the real and the imaginary parts of $\langle Ab,c\rangle_{\mathcal{B}}$ are equal, respectively, to
\[\dfrac{1}{2}\langle A(b+c),b+c\rangle_{\mathcal{B}}\quad\text{and}\quad\dfrac{1}{2}\langle A(b+ic),b+ic\rangle_{\mathcal{B}}\] and, therefore, are non-negative. Then the same can be said about
$\langle Ab,-c\rangle_{\mathcal{B}},$ hence
\begin{equation}\label{referee}
\langle Ab,c\rangle_{\mathcal{B}}=0.
\end{equation}
Furthermore, if $c$ is such that $\langle Ac,c\rangle_{\mathcal{B}}>0$ then
we have \[0\leq \left\langle A\left(b-\dfrac{\langle Ab,c\rangle_{\mathcal{B}}}{\langle Ac,c\rangle_{\mathcal{B}}}c\right),b-\dfrac{\langle Ab,c\rangle_{\mathcal{B}}}{\langle Ac,c\rangle_{\mathcal{B}}}c\right\rangle_{\mathcal{B}}=-\dfrac{|\langle Ab,c\rangle_{\mathcal{B}}|^2}{\langle Ac,c\rangle_{\mathcal{B}}},\]
hence \eqref{referee} holds in this case, as well.\\
Thus, $(\ran A,\langle\cdot,\cdot\rangle_A)$ is a
pre-Hilbert
space. We will denote by ${\mathcal H}_A$ its completion and define
$$T\,:\,{\mathcal B}\longrightarrow {\mathcal H}_A,\quad Tb\defi Ab.$$
We have for $b,c\in{\mathcal B}$
\[
\langle T^*(Ac),b\rangle_{{\mathcal B}}= \langle Ac,Tb\rangle_{{\mathcal H}_A} =
\langle Ac,b\rangle_{{\mathcal
B}}.
\]
Hence, $T^*$ extends continuously to the injection map from ${\mathcal H}_A$
into ${\mathcal B}^*$. We note that
\[
\langle Tb,Tc\rangle_{{\mathcal H}_A}=\langle T^*Tb,c\rangle_{{\mathcal B}}
=\langle Ab,c\rangle_{{\mathcal B}}\]
(that is, \eqref{samantha} holds). The claim on the norms is proved as follows: we have
$$\|A\|=\|T^*T\|\le \|T\|^2$$
on the one hand and
\[
\|A\|\ge\sup_{\|b\|=1}\langle Ab,b\rangle_{{\mathcal B}} =\sup_{\|b\|=1} \langle T^*Tb,b\rangle_{{\mathcal B}}
=\sup_{\|b\|=1}\langle Tb,Tb\rangle_{{\mathcal H}_A} =\|T\|^2,
\]
that is, $\|A\|\ge\|T\|^2$ on the other hand.
Combining the two inequalities we obtain \eqref{bagdad}. We now prove
\eqref{sof-sof}. We have:
\[
\begin{split}
|\langle Ab,c\rangle_{\mathcal B}|&=\langle Tb,Tc\rangle_{\mathcal H}\\
&\le \langle Tb,Tb\rangle_{\mathcal H}^{1/2}
\langle Tc,Tc\rangle_{\mathcal H}^{1/2}\\
&=\langle Ab,b\rangle_{\mathcal B}\langle Ac,c\rangle_{\mathcal B}.
\end{split}
\]
\end{proof}
We will say that $A\leq B$ if $B-A\geq 0.$ Note that
\begin{equation}
\label{etoile}
A\leq B \implies \|A\|\leq\|B\|.
\end{equation}
Indeed, from \eqref{bagdad} we have:
$$\|A\|=\sup_{\|b\|=1}\langle Ab,b
\rangle_{{\mathcal B}}\le \sup_{\|b\|=1}\langle Bb,b \rangle_{{\mathcal B}}=\|B\|.$$
\begin{definition}\label{defrk}
Let ${\mathcal H}$ be a Hilbert space of ${\mathcal B}^*$-valued functions
defined on a set $\Omega$ and let $K(z,w)$ be an ${\mathbf L}({\mathcal B},
{\mathcal B}^*)$-valued kernel
defined on $\Omega\times\Omega.$ The kernel $K(z,w)$ is called the reproducing
kernel of the Hilbert space
${\mathcal H}$ if for every $w\in\Omega$ and $b\in{\mathcal B}$
$K(\cdot,w)b\in\mathcal{H}$ and
\[
\langle f, K(\cdot,w)b\rangle_{\mathcal H}= \langle f(w),
b\rangle_{{\mathcal B}},\quad\forall f\in\mathcal{H}.
\]
\end{definition}
\begin{definition}
Let $K(z,w)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
kernel defined on $\Omega\times\Omega.$ The
kernel $K(z,w)$ is said to be positive if for any choice of $z_1,\dots, z_n\in\Omega$ and
$b_1,\dots,b_n\in\mathcal{B}$ it holds that
\[\sum_{j=1}^n\langle K(z_i,z_j)b_j,b_i\rangle_{\mathcal{B}}\geq 0.\]
\end{definition}
\begin{proposition}
\label{p1}
The reproducing kernel $K(z,w)$ of a Hilbert space of ${\mathcal B}^*$-valued functions,
when it exists, is unique and positive.
\end{proposition}
\begin{proposition}
\label{p2}
Let
$K(z,w)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
positive kernel defined on
$\Omega\times\Omega.$ Then there exists a unique Hilbert space of
${\mathcal B}^*$-valued functions defined on
$\Omega$ with the reproducing kernel $K(z,w)$.
\end{proposition}
The proofs of these propositions are the same as in the Hilbert space case
and are therefore omitted.
\begin{remark}\label{brem}
One can derive the notion of a reproducing kernel Hilbert space of
${\mathcal B}$-valued functions from
Definition \ref{defrk} above, using the natural injection
$\tau$ from ${\mathcal B}$ into
${\mathcal B}^{**}$ defined by \eqref{bastille}.
\end{remark}
\section{Stieltjes integral}
In this section we define the Stieltjes
integral of a scalar function with respect to a ${\mathbf
L}({\mathcal B},{\mathcal B}^*)$--valued positive measure. We
here follows the analysis presented in \cite[\S 4 p. 19]{MR48:904} for
the case of operators in Hilbert spaces. We consider a separable
Banach space ${\mathcal B}$ and an increasing positive function
$$M\,:\, [a,b]\longrightarrow {\mathbf
L}({\mathcal B},{\mathcal B}^*).$$
Thus, $M(t)\geq 0$ for all $t\in[a,b]$ and moreover
$$
a\le t_1\le t_2\le b\Longrightarrow M(t_2)-M(t_1)\ge 0.$$
Let $f(t)$ be a scalar continuous function on $[a,b]$ and let
$$a=t_0\le \xi_1\le t_1\le \xi_2\le t_2\le\cdots\le \xi_{m}\le
t_m=b$$ be a subdivision of $[a,b]$.
The Stieltjes integral $\int_a^b f(t)dM(t)$
is defined to be the limit (in the ${\bf L}({\mathcal B},{\mathcal B}^*)$ topology) of the
sums of the form
$$\sum_{j=1}^m f(\xi_j)(M(t_j)-M(t_{j-1}))$$
as $\sup_{j}|t_j-t_{j-1}|$ goes to $0$.
\begin{theorem}
\label{Sevres-Babylone}
The integral $\int_a^bf(t)dM(t)$ exists.
\end{theorem}
The proof of Theorem \ref{Sevres-Babylone}
is done along the lines of
\cite{MR48:904}. First we need the following lemma.
\begin{lemma}\label{brod}
Let $\alpha_j$ and $\beta_j$ be complex numbers such that
$|\alpha_j|\le |\beta_j|$ ($j=1,2,\ldots m$). Let $H_1,\ldots,
H_m$ be positive operators from ${\mathcal B}$ into ${\mathcal
B}^*$. Then it holds that
\begin{equation*}
\|\sum_{j=1}^m\alpha_j H_j\|\le \|\sum_{j=1}^m |\beta_j|H_j\|.
\end{equation*}
\end{lemma}
\begin{proof} For each $j$ we write $H_j=T_j^*T_j$ where $T_j$ is a
bounded operator from ${\mathcal B}$ into some
Hilbert space ${\mathcal H}_j$ as in Theorem \ref{rosh-hashana}. Then
for $b,c\in{\mathcal B}$ of modulus $1$ we
have:
\[
\begin{split}
|\langle \sum_{j=1}^m\alpha_j H_j b,c\rangle_{{\mathcal B}}| &=
|\sum_{j=1}^m\langle \alpha_j H_j b,c\rangle_{{\mathcal B}}|\\
&\le
\sum_{j=1}^m|\alpha_j| \cdot|\langle T_jb,T_jc\rangle_{{\mathcal H}_j}|\\
&\le \sum_{j=1}^m\sqrt{|\beta_j|} \|T_jb\|_{{\mathcal H}_j}
\sqrt{|\beta_j|}\|T_jc\|_{{\mathcal H}_j}\\
&\le \left(\sum_{j=1}^m|\beta_j|\|T_jb\|^2_{{\mathcal
H}_j}\right)^{1/2}
\left(\sum_{j=1}^m|\beta_j|\|T_jc\|^2_{{\mathcal H}_j}\right)^{1/2}\\
& \le \left(\sum_{j=1}^m|\beta_j|\langle H_jb,
b\rangle_{{\mathcal B}}\right)^{1/2}
\left(\sum_{j=1}^m|\beta_j|\langle H_jc,
c\rangle_{{\mathcal B}}\right)^{1/2}\\
&\le \|\sum_{j=1}^m|\beta_j| H_j\|
\end{split}
\]
where we have used \eqref{bagdad} to get the last inequality.
Thus, taking the supremum on $c$ (of unit norm) we have
$\|\sum_{j=1}^m\alpha_jH_jb\|\le\|\sum_{j=1}^m|\beta_j| H_j\|$,
and hence the required inequality.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Sevres-Babylone}]
It suffices to show that for every $\epsilon>0$ there exists $\delta>0$ such that if
\begin{equation}\label{subdiv}a=t_0\leq t_1\leq\dots\leq t_m=b\end{equation} is a
subdivision of $[a,b]$ such that
$\max_{j}|t_j-t_{j-1}|\leq \delta$ and
$$a=t_0=t_1^0\leq \dots\leq t_1^{k_1}=t_1=t_2^0\leq\dots\leq t_{m}^{k_{m}} =t_m=b$$
is a continuation of the subdivision \eqref{subdiv} then
for every choice of $\xi_j\in[t_{j-1},t_j]$ and
$\xi_j^\ell\in[t_j^{\ell-1},t_j^\ell]$ it holds that
$$\left\|\sum_{j=1}^m f(\xi_j)(M(t_j)-M(t_{j-1}))-\sum_{j=1}^m
\sum_{\ell=1}^{k_j}
f(\xi_j^\ell)(M(t_j^\ell)-M(t_j^{\ell-1}))\right\|\leq\epsilon.$$
Let us take $\delta$ such that
$$|t^\prime-t^{\prime\prime}|\leq\delta\implies
|f(t^\prime)-f(t^{\prime\prime})|\leq\dfrac{\epsilon}{\|M(b)-M(a)\|}.$$
Then the desired conclusion follows from Lemma \ref{brod}.
\end{proof}
\section{Helly's theorem}
The following theorem is proved in the case of {\sl separable} Hilbert space
in \cite{MR48:904} (see Theorem 4.4 p. 22 there). The proof goes in the same
way in the case of separable Banach spaces. We quote it in a version
adapted to the present setting.
\begin{theorem}
\label{helly}
Let $F_n(t)$ ($t\in[0,2\pi]$) be a sequence of positive increasing
${\mathbf L}({\mathcal B},{\mathcal B}^*)$--valued functions such that
$$F_n(t)\le F_0,\quad n=0,1,\ldots\quad\text{and}\quad t\in[0,2\pi],$$
where $F_0$ is some bounded positive
operator. Then, there exists a subsequence of $F_n$
(which we still denote by $F_n$)
which converges weakly for every $t\in[0,2\pi]$. Moreover, for $f(t)$ a
continuous scalar function we have (in the weak sense, and via the
subsequence):
$$
\int_0^{2\pi}f(t)dF(t)=\lim_{n\rightarrow \infty}\int_0^{2\pi}f(t)dF_n(t).
$$
\end{theorem}
The proof given in \cite{MR48:904} relies on the hypothesis of separability
and on the inequalities
\begin{equation}
\label{rastignac}
\begin{split}
|\langle F_n(t)x,y\rangle_{\mathcal B}|&\le\|F_0\|\cdot\|x\|\cdot\|y\|,\quad
x,y\in{\mathcal B}\\
\sum_{\ell=1}^m |
\langle \Delta_{\ell,n}Fx,y\rangle_{\mathcal B}|&\le 2\|F_0\|\cdot
\|x\|\cdot\|y\|
\end{split}
\end{equation}
where $0=t_0\le t_1<\cdots <t_m=2\pi$ and
$\Delta_{\ell,n}F=F_n(t_\ell)-F_n(t_{\ell-1})$.
The first inequality follows from \eqref{etoile}. We prove the second
one using the factorization given in
Theorem \ref{rosh-hashana}. Using this theorem we write:
$$\Delta_{\ell,n}=T_{\ell,n}^*T_{\ell,n}$$
where $T_{\ell,n}$ is a bounded operator from some Hilbert space
${\mathcal H}_{\ell,n}$ into ${\mathcal B}.$
Then, using \eqref{sof-sof} we have:
\[
\begin{split}
\sum_{\ell =1}^m
|\langle\Delta F_{\ell,n}x,y\rangle_{\mathcal B}|&\le
\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}x,x\rangle_{\mathcal B}^{1/2}
\langle\Delta F_{\ell,n}y,y\rangle_{\mathcal B}^{1/2}\\
&\le
\left(\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}x,x\rangle_{\mathcal B}\right)^{1/2}
\left(\sum_{\ell=1}^m
\langle\Delta F_{\ell,n}y,y\rangle_{\mathcal B}\right)^{1/2}\\
&=\langle (F(2\pi)-F(0))x,x\rangle^{1/2}\langle (F(2\pi)-F(0))y,y
\rangle^{1/2}\\
&\le \langle 2F_0x,x\rangle_{\mathcal B}^{1/2}
\langle 2F_0y,y\rangle_{\mathcal B}^{1/2}\\
&\le 2\|F_0\|\cdot \|x\|\cdot\|y\|.
\end{split}
\]
The proof then proceeds as follows (see
\cite[p. 22]{MR48:904}). One applies the first inequality in \eqref{rastignac}
for $x,y$ in a dense countable set $E$ of ${\mathcal B}$. The functions
$t\mapsto \langle F_n(t)x,y\rangle_{\mathcal B}$ are of bounded variation.
An application of the scalar case of Helly's theorem and
the diagonal process allows to find a subsequence of $F_n$ such that
for all $x,y\in E$ and every $t\in[0,2\pi]$ the limit
$$\lim_{n\rightarrow\infty}\langle F_n(t)x,y\rangle_{\mathcal B}$$
exists. We refer the reader to \cite{MR48:904} for more details.
\section{${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued
Carath\'eodory functions}
\begin{definition}
Let $\phi(z)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function,
weakly continuous at the origin
in the sense that
\begin{equation}\label{weak} \langle\phi(z)b,b\rangle_{\mathcal{B}}\rightarrow
\langle\phi(0)b,b\rangle_{\mathcal{B}}\text{ as }z\rightarrow 0,\quad\forall b\in\mathcal{B}.
\end{equation}
For a Carath\'eodory function $\phi(z)$ we shall denote by
$\mathcal{L}(\phi)$ the Hilbert space of $\mathcal{B}^*$-valued
functions with the reproducing kernel $k_\phi(z,w)$.
\end{definition}
We give two representation theorems for Carath\'eodory
functions. In the first we make no assumption on the space ${\mathcal B}$.
Following arguments of Krein
and Langer (see \cite{kl1}), we prove the existence of a realization of the
form \eqref{anaelle2}. The second theorem assumes that
the space ${\mathcal B}$ is separable. We prove that in this case
the Carath\'eodory functions can be characterized as functions
analytic in the open unit disk with positive real part. Then we
derive a Herglotz-type representation formula.
\begin{theorem} Let $\Omega$ be a neighborhood of the origin and
let $\phi(z)$ be an ${\mathbf L}({\mathcal B},{\mathcal B}^*)$-valued function
defined in $\Omega$ and weakly continuous at
the origin (in the sense \eqref{weak}).
Then $\phi(z)$ is a Carath\'eodory function if and only if it admits
the representation
\begin{equation*}
\phi(z)^*\big|_{\mathcal B}=D+C^*(I-z^*V)^{-1}(I+z^*V)C,\quad z\in\Omega,
\end{equation*}
or equivalently,
\begin{equation}
\label{sarah}
\phi(z)=D^*\big|_{\mathcal B}+C^*(I+zV^*)(I-zV^*)^{-1}C,\quad z\in\Omega,
\end{equation}
where $V$ is an isometric operator in some Hilbert space
${\mathcal H}$, $C$ is a bounded operator from ${\mathcal
B}$ into ${\mathcal H}$ and $D$ is a purely imaginary
operator from ${\mathcal B}$ into ${\mathcal B}^{*}$ in the sense that
\begin{equation}
\label{D1}
D+D^*\big|_{{\mathcal B}}=0.
\end{equation}
In particular, every Carath\'eodory function has an analytic extension to
the whole open unit disk.
\label{Ivry-sur-Seine}
\end{theorem}
\begin{proof}
Let $\phi(z)$ be a Carath\'eodory function. First we observe that elements of
$\mathcal{L}(\phi)$ are weakly continuous at the origin:
\[\langle f(w),b\rangle_{\mathcal{B}}\rightarrow
\langle f(0),b\rangle_{\mathcal{B}}\text{ as }w\rightarrow 0,\quad
\forall\,f\in\mathcal{L}(\phi),b\in\mathcal{B}.\]
Indeed, this is a consequence of the Cauchy -- Schwarz inequality as
\[\langle f(w),b\rangle_{\mathcal{B}}-
\langle f(0),b\rangle_{\mathcal{B}}=
\langle f,(k_\phi(\cdot,w)-k_\phi(\cdot,0))b\rangle_{\mathcal{L}(\phi)}\]
and
\[
\|(k_\phi(\cdot, w)-k_\phi(\cdot,0))b\|^2_{\mathcal{L}(\phi)}
=\frac{|w|^2}{1-|w|^2}\Re \langle \phi(w)b,b\rangle_{{\mathcal B}}.
\]
We consider in ${\mathcal L}(\phi)\times{\mathcal L}(\phi)$ the linear
relation $R$ spanned by the pairs
\[
R=
\left(\begin{array}{cc}
\sum k_\phi(z,w_i)w_i^*b_i,\sum k_\phi(z,w_i)b_i-k_\phi(z,0)(\sum b_i)
\end{array}\right)
\]
where all the $b_i\in{\mathcal B}$, the $w_i\in\Omega$ and where all the
sums are finite. This
relation is densely defined because of the weak continuity of the elements of
${\mathcal L}(\phi)$ at the origin.
Indeed, let $f\in{\mathcal L}(\phi)$ be orthogonal to the domain of $R$. Then,
\[
\langle f, k_\phi(\cdot,w)b\rangle_{{\mathcal L}(\phi)}=\langle f(w),b
\rangle_{{\mathcal B}}=0\]
for all $b\in{\mathcal B}$ and all
points $w\not =0$ in the domain of $f$. Thus $f(w)=0$ at all these points $w$
and the continuity hypothesis implies that also $f(0)=0$.
The relation $R$ is readily seen to be isometric. Its closure is thus the
graph of an isometry, which we call $V$. We have:
\[
V(k_\phi(z,w)w^*b)=k_\phi(z,w)b-k_\phi(z,0)b,\]
and in particular
\begin{equation}
\label{Alfort-Ecole-veterinaire}
(I-w^*V)^{-1}k_\phi(\cdot, 0)b=k_\phi(\cdot, w)b.
\end{equation}
Denote by $C$ the map
$$C\,:\,{\mathcal B}\longrightarrow {\mathcal L}(\phi),\quad(Cb)(z)
\defi k_\phi(z,0)b.$$
Then, for $f\in{\mathcal L}(\phi)$, $C^*f=f(0)$.
Applying $C$ on the left on both sides of
\eqref{Alfort-Ecole-veterinaire} we obtain
\[\frac{\phi(0)+\phi(w)^*\big|_{\mathcal B}}{2}b=C^*(I-w^*V)^{-1}Cb.\]
Since $$C^*Cb=\frac{\phi(0)+\phi(0)^*\big|_{\mathcal B}}{2}b$$
we obtain
\[
\begin{split}
\phi(0)+\phi(w)^*\big|_{\mathcal B}&=2C^*(I-w^*V)^{-1}C-C^*C+C^*C\\
&=C^*(I-w^*V)^{-1}(I+w^*V)C+C^*C
\end{split}
\]
so that
\[
\phi(w)^*\big|_{{\mathcal B}}+\frac{\phi(0)-\phi(0)^*\big|_{\mathcal B}}{2}
=
C^*(I-w^*V)^{-1}(I+w^*V)C,
\]
which gives the required formula with
\begin{equation}
\label{D} D=\frac{\phi(0)-\phi(0)^*\big|_{\mathcal B}}{2}.
\end{equation}
We now prove the converse statement and first compute
$$\langle k_\phi(z,w)x,y\rangle_{{\mathcal B}}
\quad{\rm for}\quad x,y\in{\mathcal B}.$$
We have
\[
\begin{split}
\langle\phi(w)^*\big|_{\mathcal B}x,y\rangle_{{\mathcal B}}&=
\langle Dx,y\rangle_{{\mathcal B}}+\langle
(I-w^*V)^{-1}(I+w^*V)Cx,Cy\rangle_{{\mathcal L}(\phi)}.
\end{split}
\]
We have, with $\tau$ the natural injection from ${\mathcal B}$ into
${\mathcal B}^{**}$ (see \eqref{bastille}):
\begin{equation}
\begin{split}
\langle \phi(z)x,y\rangle_{{\mathcal B}} &= \overline{\langle \tau
y ,\phi(z)x\rangle_{{\mathcal B}^*}}\\
&=\overline{\langle \phi(z)^*\tau y,x\rangle_{{\mathcal B}^*}}\\
&= \overline{\langle \phi(z)^*\big|_{\mathcal B}y,x
\rangle_{{\mathcal B}}}\\
&=\overline{\langle D y,x\rangle_{{\mathcal B}}}+
\langle(I-zV^*)^{-1}(I+zV^*)Cx,Cy\rangle_{{\mathcal L}(\phi)}\\
&=\langle D^*\big|_{\mathcal B}x ,y\rangle_{{\mathcal B}}+
\langle(I-zV^*)^{-1}(I+zV^*)Cx,Cy\rangle_{{\mathcal L}(\phi)},
\end{split}
\label{elinor}
\end{equation}
and so
$$\langle k_\phi(z,w)x,y\rangle_{{\mathcal B}}
=\langle (I-zV^*)^{-1}Cx,(I-wV^*)^{-1}Cy\rangle_{{\mathcal L}(\phi)},$$
from which follows the positivity of $k_\phi(z,w)$. Finally,
\eqref{elinor} also implies \eqref{sarah} and this concludes the proof.
\end{proof}
\begin{remark}
Although the above argument is very close to the one in
\cite[p. 365--366]{kl1} we note the following: we use a concrete
space (the space ${\mathcal L}(\phi)$) to build the relation rather than
abstract elements and the relation $R$ is defined slightly differently.
\end{remark}
\begin{remark}\label{indef}
As already mentioned, the above argument still goes through when the
kernel has a finite number of negative squares. In this case the space
$\mathcal{L}(\phi)$ is a Pontryagin space. For $b\in
{\mathcal B}$ and sufficiently small $h\in{\mathbb C}$ we consider
the functions $f_h(z)=(K(z,w+h)-K(z,w))b$, which have the following
properties:
\begin{align}
\nonumber
\lim_{h\rightarrow 0}\langle f_h,f_h\rangle_{{\mathcal L}(\phi)}&=0,\\
\lim_{h\rightarrow 0}\langle f,f_h\rangle_{{\mathcal L}(\phi)}&=0,
\quad\forall f\in\spa\{k_\phi(\cdot,w)b\}.
\nonumber
\end{align}
It follows from the convergence criteria in Pontryagin spaces (see
\cite{ikl}, \cite[p. 356]{kl1}) that
$$\lim_{h\rightarrow 0}\langle f,f_h\rangle_{{\mathcal L}(\phi)}=0,\quad
\forall f\in\mathcal{L}(\phi).$$ The fact that the relation is the graph of
an isometric operator is
proved in \cite[Theorem 1.4.2 p. 29]{adrs}. This follows from a theorem of
Shmulyan which states that a
contractive relation between Pontryagin spaces of same index is the graph
of a contractive operator; see
\cite{s} and \cite[Theorem 1.4.1 p. 27]{adrs}.
\end{remark}
\begin{theorem}
Let ${\mathcal B}$ be a separable Banach space and let $\phi(z)$ be a
${\mathbf L}({\mathcal B},{\mathcal
B}^*)$--valued function analytic in the open unit disk, such that
\[\phi(z)+{\phi(z)^*}\bigm|_{ \mathcal{B}}\,
\geq 0.\] Then there exists an increasing ${\mathbf L}({\mathcal B},{\mathcal B}^*)$--valued
function $M(t)$
($t\in[0,2\pi]$) and a purely imaginary operator $D$ (that is, satisfying \eqref{D1})
such that
\[
\phi(z)=D+\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}dM(t),
\]
where the integral is defined in the weak sense. Furthermore the
kernel $k_\phi(z,w)$ is positive in ${\mathbb D}$.
\label{Antony}
\end{theorem}
\begin{proof} We follow the arguments in \cite{MR48:904}, and will apply
Theorem \ref{helly}. The separability hypothesis of ${\mathcal B}$ is used at
this point.\\
We first assume that $\phi(z)$
is analytic in $|z|<1+\epsilon$ with $\epsilon>0$. We have
(the existence of the integrals follows from Theorem
\ref{Sevres-Babylone}):
\[
\begin{split}
\frac{1}{4\pi}
\int_0^{2\pi}\phi(e^{it})\frac{e^{it}+z}{e^{it}-z}dt&=\phi(z)-\frac{\phi(0)}{2}\\
\frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{it})^*\big|_{\mathcal B}
\frac{e^{it}+z}{e^{it}-z}dt&= \frac{1}{2\pi
}\int_0^{2\pi}(\phi(e^{-it})^*\big|_{\mathcal B}
\frac{e^{-it}+z}{e^{-it}-z}\\
&=\frac{(\phi(0))^*\big|_{\mathcal B}}{2}.
\end{split}
\]
Thus,
\[
\phi(z)=D+\int_0^{2\pi}\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{e^{it}+z}{e^{it}-z}dt,
\] with $D$ as in \eqref{D} and the formula for general
$\phi(z)$ follows from Helly's theorem applied to the measures
$$
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}dt,\quad r<1$$
(or more precisely to a sequence $r_n\rightarrow 1$).\\
We now prove the positivity of the kernel $k_\phi(z,w)$ and first
assume that $\phi(z)$ is analytic in $|z|<1+\epsilon$ as above.
We have:
\[
\frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{-it}))^*\big|_{\mathcal B}
\frac{e^{it}+z}{e^{it}-z}dt=(\phi(z^*))^*\big|_{\mathcal B}
-\frac{(\phi(0))^*\big|_{\mathcal B}}{2}.
\]
Thus
\[
(\phi(z^*))^*\big|_{\mathcal B} -\frac{(\phi(0))^*\big|_{\mathcal
B}}{2}= \frac{1}{4\pi}\int_0^{2\pi}(\phi(e^{it}))^*\big|_{\mathcal
B} \frac{e^{-it}+z}{e^{-it}-z}dt
\]
and so
$$
(\phi(z^*))^*\big|_{\mathcal B}=
-D+\int_0^{2\pi}\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{1+ze^{it}}{1-ze^{it}}dt
$$
since
$$
\frac{1}{4\pi}\int_0^{2\pi}\phi(e^{it})\frac{1+ze^{it}}{1-ze^{it}}dt
=
\frac{1}{4\pi}\int_0^{2\pi}\phi(e^{-it})\frac{e^{it}+z}{e^{it}-z}dt=\frac{\phi(0)}{2}.$$
Thus,
$$k_\phi(z,w)=\int_0^{2\pi}
\frac{\left(\phi(e^{it})+(\phi(e^{it}))^*\big|_{\mathcal B}\right)}{4\pi}\frac{1}{(e^{it}-z)(e^{it}-w)^*}dt.$$
The positivity follows. The case of general $\phi(z)$ is done by approximation using Helly's theorem; indeed
using Theorem \ref{helly} we have for general $\phi(z)$:
\[
\begin{split}
\langle k_\phi(w_\ell,w_j)b_j,b_\ell\rangle_{\mathcal B}&=
\\
&\hspace{-2cm}=
\lim_{r\rightarrow 1}\int_0^{2\pi}
\left\langle
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{1}{(e^{it}-w_\ell)(e^{it}-w_j)^*}b_j,b_\ell
\right\rangle_{\mathcal B}dt\\
&\hspace{-2cm}=
\lim_{r\rightarrow 1}\int_0^{2\pi}
\left\langle
\frac{\left(\phi(re^{it})+(\phi(re^{it}))^*\big|_{\mathcal
B}\right)}{4\pi}\frac{b_j}{(e^{it}-w_j)^*},\frac{b_\ell}{(e^{it}-w_\ell)^*}
\right\rangle_{\mathcal B}dt.
\end{split}
\]
\end{proof}
\section{The case of ${\mathbf L}({\mathcal B}^{*},{\mathcal B})$--valued
functions} We turn to the case of ${\mathbf L}({\mathcal
B}^{*},{\mathcal B})$--valued functions.
Using the natural
injection $\tau$
$$\mathcal{B}\stackrel{\tau}{\mapsto}\mathcal{B}^{**}$$
defined by \eqref{bastille} we shall say that an ${\mathbf
L}({\mathcal B}^{*},{\mathcal B})$--valued function $\phi(z)$ is
a Carath\'eodory function if the ${\mathbf L}({\mathcal
B}^{*},{\mathcal B}^{**})$-valued function $\tau\phi(z)$ is a
Carath\'eodory function.
\begin{theorem} An ${\mathbf L}({\mathcal B}^*,{\mathcal B})$--valued
function $\phi(z)$ defined in a neighborhood of the origin
and weakly continuous at the origin is a Carath\'eodory function
if and only if it admits the representation
\begin{equation*}
\phi(z)^*=D+C^*(I-z^*V)^{-1}(I+z^*V)C,
\end{equation*}
or, equivalently,
\[
\tau\phi(z)=D^*\big|_{{\mathcal B}_*}+C^*(I+zV^*)(I-zV^*)^{-1}C,\]
where $V$ is an isometric operator in some Hilbert space
${\mathcal H}$, where $C$ is a bounded operator from ${\mathcal
B}^*$ into ${\mathcal H}$ and where $D$ is a purely imaginary
operator from ${\mathcal B}^*$ into ${\mathcal B}^{**}$.
In particular $\phi(z)$ has an analytic extension to the whole open unit disk.
\end{theorem}
\begin{proof} By Theorem \ref{Ivry-sur-Seine} (with $\mathcal{B}$ replaced by
$\mathcal{B}^*$),
$\phi(z)$ is a Carath\'eodory function if and only if
\[(\tau\phi(z))^*\bigm|_{\mathcal{B}^*}=D+C^*(I-z^*V)^{-1}(I+z^*V)C,\]
where $C, V, D$ have the stated properties.
But $(\tau\phi(z))^*\bigm|_{\mathcal{B}^*}=\phi(z)^*.$
\end{proof}
\end{document} |
\begin{document}
\title{Most incompatible measurements for robust steering tests}
\author{Jessica Bavaresco}
\email{jessica.bavaresco@oeaw.ac.at}
\affiliation{Departamento de F\'isica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\affiliation{Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria}
\author{Marco T\'ulio Quintino}
\affiliation{Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan}
\author{Leonardo Guerini}
\affiliation{Departamento de Matem\'atica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\affiliation{ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain}
\author{Thiago O. Maciel}
\affiliation{Departamento de F\'isica, Universidade Federal de Minas Gerais, Caixa Postal 702, 31270-901, Belo Horizonte, MG, Brazil}
\author{Daniel Cavalcanti}
\affiliation{ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain}
\author{Marcelo Terra Cunha}
\affiliation{Departamento de Matem\'atica Aplicada, IMECC-Unicamp, 13084-970, Campinas, S\~ao Paulo, Brazil}
\date{\today}
\begin{abstract}
We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to $N$ measurements of $k$ outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a $d$-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements.
We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering isotropic states and provide candidates to the most incompatible sets of measurements in each case.
Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
Correlations arising from local measurements on entangled states can lead to statistics that cannot be explained by any local causal theory \cite{RevNL}. This nonlocal aspect of quantum mechanics can be analyzed from two nonequivalent perspectives, Bell nonlocality \cite{Bell} and Einstein-Podolsky-Rosen (EPR) steering \cite{EPR,Wiseman}. On a bipartite scenario, while Bell nonlocality deals with a full device-independent approach where a correlation experiment is analyzed only considering the probability relations between inputs and outcomes, EPR steering plays an intermediate role between entanglement and Bell nonlocality by doing a device-independent analysis only on one side of the experiment while treating the other side in a device-dependent manner (e.g., performing full state tomography).
Although the first notions of EPR steering date back to 1935 \cite{EPR}, its modern mathematical formulation only appeared in 2007 \cite{Wiseman}, and many of its fundamental properties \cite{MTGeneralMeas,MTOneWay,MultipartiteDani,aolita14,quintino17} and applications to semi-device independent protocols \cite{tomamichel11,tomamichel12,OneSidedDIQC,law14,passaro15,kogias17} are only being understood recently. In order to get a better understanding of EPR steering and make use of its practical applications, an important task is to determine which states can lead to these nonlocal correlations.
EPR steering can be certified with the use of steering witnesses \cite{SteIneq}, but finding suitable inequalities and choosing appropriate measurements to reveal this sort of nonlocality of a given quantum state remains a nontrivial task. On the other hand, proving that a quantum state cannot lead to EPR steering can be done by constructing a local hidden state (LHS) model that is able to simulate the statistics of the quantum state \cite{WernerModel,Wiseman,sania14,JoeCriterion}. Recently, a general algorithm to construct LHS models for quantum states was presented \cite{LHScavalcanti,LHShirsch}, regarding scenarios where all (i.e., infinitely many) measurements are considered. However, from a practical perspective, it is important to characterize what one can do with a limited number of measurements and outcomes, or yet, when even the structure of the allowed measurements is restricted.
Since the ability to demonstrate steering is intimately related to the ability to perform incompatible measurements \cite{krausbook,buschbook}, by addressing the problem of characterizing the steerability of quantum states under restrictive scenarios one can simultaneously address the problem of characterizing the ability to jointly perform a set of unknown measurements subjected to the same restrictions \cite{MTJM,UolaJM1}. Although the question of whether a set of fixed (known) measurements is jointly measurable can be decided by semidefinite programming (SDP) \cite{wolf09} -- which is also the case for deciding whether a given quantum state is steerable when subjected to a set of fixed (known) measurements \cite{MattPusey,SteeringWeight} -- if the complete description of the measurements is not known, there do not exist general methods to characterize steerability or joint measurability.
In this paper we consider steering and joint measurability in scenarios where the number of measurements and outcomes is finite. By systematically applying adaptations of the parametric search, the seesaw algorithm \cite{RevSDP}, and the outer polytope approximation \cite{SuperPOVM}, we derive upper and lower bounds to the maximal amount of white noise that a quantum state can endure before it is no longer able to demonstrate steering when subjected to a set of $N$ general local measurements with $k$ outcomes. Using the same methods, we calculate upper and lower bounds for the minimum amount of white noise that must be applied to \textit{any} set of general $N$ qudit $k$-outcome measurements so that it is assured that they can be jointly performed. By imposing further restrictions on the measurement scenarios, we also study prominent classes of measurements that are known to be useful in many quantum information tasks, such as projective measurements, symmetric informationally complete (SIC) measurements \cite{renes04}, and measurements from mutually unbiased bases (MUBs) \cite{durt10}.
We present our calculations for qubit states in scenarios ranging from $2$ to $18$ projective, planar projective, symmetric, and general measurements and provide strong evidence that, in the considered scenarios, general positive-operator valued measures (POVMs) do not outperform projective measurements for exhibiting steering. We also show that our optimal sets of qubit measurements are not distributed in the Bloch sphere according to any of the most intuitive patterns, such as the vertices of Platonic solids, the distribution of electrons on a sphere in the Thomson problem \cite{Thomson}, and the Fibonacci spiral \cite{Fibonacci}. We present an alternative candidate for this distribution for the cases of $N\in\{2,\ldots,6\}$ measurements, supported by our numerical findings. For higher dimensions, we present evidence that increasing the number of outcomes beyond the value of the local dimension of the state does not improve white-noise robustness, again implying that projective measurements are optimal for steering. We also prove that, in many cases, incomplete sets of MUB measurements are not optimal, while providing numerical evidence that complete sets may be optimal for steering isotropic states.
\section{Preliminaries}
\subsection{Einstein-Podolsky-Rosen steering}
Bipartite steerability is usually defined in terms of an assemblage. Let $\rho_{AB}$ be a bipartite quantum state shared by Alice and Bob and let $\{M_{a|x}\}$ be a set of measurements on Alice's subsystems. Then, an assemblage $\{\sigma_{a|x}\}$ is defined as
\begin{equation}
\sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}\,\rho_{AB}),
\end{equation}
for all $a,x$, where $x\in\{1,\ldots,N\}$ and $a\in\{1,\ldots,k\}$ label Alice's measurements and outcomes, respectively, and $\text{\normalfont Tr}_A$ denotes the partial trace over the Hilbert space of Alice. An assemblage does not demonstrate steering when it admits an LHS model, namely, when there exists $\Lambda$ such that
\begin{equation}\label{eqlhs}
\sigma_{a|x} = \sum_\lambda \pi(\lambda)p_A(a|x,\lambda)\rho_\lambda,
\end{equation}
for all $a,x$, where $\lambda\in\Lambda$ are the possible values that can be assumed by a local hidden variable with probability $\pi(\lambda)$, $p_A(a|x,\lambda)$ is the probability of Alice's obtaining outcome $a$ conditioned on her choice of measurement $x$ and $\lambda$, and, finally, $\rho_\lambda$ is a local hidden state held by Bob that is conditioned by the value $\lambda$ and independent of Alice's measurements and outcomes. An assemblage demonstrates steering when it does not admit such decomposition \cite{Wiseman} or, equivalently, when it violates a steering inequality \cite{SteIneq}. A quantum state $\rho_{AB}$ is unsteerable if all assemblages that can be generated by performing local measurements on it admit an LHS model. On the other hand, a quantum state is steerable if there exists a set of measurements that, when locally performed on it, generates an assemblage that violates a steering inequality.
\subsection{Measurement incompatibility}
A set of measurements $\{M_{a|x}\}$, where $x\in\{1,\ldots,N\}$ labels the measurements in the set and $a\in\{1,\ldots,k\}$ labels the outcomes of each measurement, is jointly measurable, or compatible, if there exists a \textit{joint measurement}, $\{M_\lambda\}$, such that
\begin{equation}
M_{a|x} = \sum_\lambda \pi(\lambda) p(a|x,\lambda) M_\lambda,
\end{equation}
for all $a,x$, where $\pi(\lambda)$ and $p(a|x,\lambda)$ are elements of probability distributions. Hence, all POVM elements $M_{a|x}$ can be recovered by coarse-graining over the joint measurement $\{M_\lambda\}$.
Although for projective measurements joint measurability is equivalent to commutation, general POVMs from a jointly measurable set may not commute \cite{kru,teiko08}. In this sense, joint measurability is a more general definition of incompatibility.
\subsection{Main problem} \label{Quantify}
Consider the depolarizing map $\Lambda_\eta$ acting on the Hermitian operator $A$ of a $d$-dimensional Hilbert space $\set{H}$, defined as
\begin{equation}\label{eqdepolmap}
A\mapsto\Lambda_{\eta}(A) = \eta A+(1-\eta)\text{\normalfont Tr}(A)\frac{\mathbb{1}}{d}.
\end{equation}
The depolarizing map can be physically interpreted as the effect of the presence of white noise in the implementation of $A$.
When applied to elements of an assemblage it defines a steering quantifier, the white-noise robustness of an assemblage,
\begin{equation}
\eta(\{\sigma_{a|x}\})=\max\left\{\eta \ | \ \{\Lambda_{\eta}(\sigma_{a|x})\}_{a,x} \in LHS\right\},
\end{equation}
where $LHS$ is the set of assemblages that admit an LHS model and, hence, do not demonstrate steering. Therefore, $\eta(\{\sigma_{a|x}\})$ is the exact value of $\eta$, called the critical visibility of the assemblage, above which $\{\sigma_{a|x}\}$ no longer admits an LHS model. Since $\{\Lambda_1(\sigma_{a|x})\}$ is the assemblage itself and $\{\Lambda_0(\sigma_{a|x})\}$ is such that each of its elements corresponds to a multiple of the identity (and therefore it always admits an LHS model), by convexity it is guaranteed that the critical visibility of the assemblage $\eta(\{\sigma_{a|x}\})$ lies in $[0,1]$.
Given an assemblage, its critical visibility can be calculated by an SDP (see Sec. \ref{upper} and see Ref. \cite{RevSDP} for a review of SDP characterization of steering). Similarly, by applying the depolarizing map to a set of measurements $\{M_{a|x}\}$ instead of an assemblage, one can define the critical visibility for a set of measurements to be incompatible, i.e., a value of $\eta$ above which a set of measurements can no longer be described by a joint POVM.
Here we are interested in calculating the minimum of the quantity $\eta(\{\sigma_{a|x}\})$ among all the possible choices of $N$ measurements with $k$ outputs for a fixed quantum state $\rho_{AB}$. Formally this quantity can be defined as
\begin{equation}
\begin{split}
\eta^*(\rho_{AB},N,k) = \min_{\{M_{a|x}\}} \Big\{ &\eta(\{\sigma_{a|x}\}) \ | \\
&\sigma_{a|x}=\text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}\,\rho_{AB}) \Big\},
\end{split}
\end{equation}
where the minimization runs over sets $\{M_{a|x}\}$ of $N$ $k$-outcome measurements. The value $\eta^*(\rho_{AB}, N, k)$ is the critical visibility of the quantum state $\rho_{AB}$ when subjected to $N$ measurements of $k$ outcomes. Note that for $\eta\leq\eta^*(\rho_{AB}, N, k)$, the state $\rho_{AB}$ is unsteerable for {\it all} sets of $N$ $k$-outcome measurements, and for $\eta>\eta^*(\rho_{AB}, N, k)$, $\rho_{AB}$ is steerable for {\it at least one} set of $N$ $k$-outcome measurements.
Unlike the critical visibility of an assemblage $\eta(\{\sigma_{a|x}\})$, the critical visibility of a quantum state $\eta^*(\rho_{AB}, N, k)$ is the solution of a min-max optimization problem and cannot be calculated by an SDP. In this work we provide methods to obtain upper and lower bounds for $\eta^*(\rho_{AB},N,k)$.
\subsection{Connection to the most incompatible measurements}\label{JM=STE}
In Refs. \cite{MTJM,UolaJM1} the authors have proved that a set of measurements $\{M_{a|x}\}$ is not jointly measurable \textit{if and only if} Alice can steer Bob by performing the same measurements on her share of a maximally entangled state $\ket{\phi_d^+}:=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}$, where $d$ stands for the local dimension of the quantum system. Hence, the critical visibility $\eta^*(\ket{\phi_d^+}, N,k)$ coincides with the critical visibility for which \textit{any} set of $N$ measurements with $k$ outcomes is jointly measurable.
Following from the definition of the depolarizing map (Eq. (\ref{eqdepolmap})) and the maximally entangled states, it is easy to show that the noisy assemblage $\{\Lambda_\eta(\sigma_{a|x})\}$, resulting from applying the depolarizing map to an assemblage generated by performing local measurements $\{M_{a|x}\}$ on a maximally entangled state $\ket{\phi_d^+}$, is equivalent to the assemblage resulting from locally performing measurements $\{M_{a|x}\}$ on the noisy state $(\mathbb{1}\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})$. Namely,
\begin{equation}
\Lambda_\eta(\sigma_{a|x})=\text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1}_d\,(\mathbb{1}_d\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})),
\end{equation}
where
\begin{equation}
(\mathbb{1}_d\otimes\Lambda_\eta)(\ketbra{\phi_d^+}{\phi_d^+})=\eta\ketbra{\phi_d^+}{\phi_d^+}+(1-\eta)\frac{\mathbb{1}_{d^2}}{d^2}
\end{equation}
is the isotropic state of local dimension $d$. Therefore, the critical visibility of the maximally entangled states $\eta^*(\ket{\phi_d^+},N,k)$ is equal to the critical value of the parameter $\eta$ of the isotropic states for which they can demonstrate steering, which, in turn, is equal to the critical visibility for any set of $N$ unknown qudit measurements with $k$ outcomes to be compatible. For this reason we speak equivalently of the critical visibility of the maximally entangled states, isotropic states, and joint measurability. To simplify notation, we define this quantity as $\eta^*(d, N,k)\coloneqq\eta^*(\ket{\phi_d^+}, N,k)$.
For general states, one can lower-bound the noise robustness of joint measurability by that of steerability \cite{QuantifiersDani}.
\section{Methods}\label{methods}
In the following we describe three methods we used to characterize the steerability of quantum states subjected to restricted measurement scenarios. The first method provides upper bounds for $\eta^*(\rho_{AB}, N,k)$ in scenarios where not only the number of measurements and outcomes is fixed but possibly also the structure of the POVMs. The second one provides upper bounds for $\eta^*(\rho_{AB},N,k)$ when only the number of measurements and outcomes is fixed (considering general measurements). Both methods provide candidates for the optimal set of measurements in a given scenario. The third method provides lower bounds for $\eta^*(\rho_{AB},N,k)$ and constructs LHS models for quantum states when the number of measurements and outcomes is fixed.
All code used in this work is available in Ref. \cite{Code}.
\subsection{Upper bounds for $\eta^*(\rho_{AB},N,k)$}\label{upper}
\textit{Search algorithm}. For a given quantum state $\rho_{AB}$ and a fixed set of measurements $\{M_{a|x}\}$, the critical visibility $\eta(\{\sigma_{a|x}\})$ of the assemblage $\{\sigma_{a|x}\}$, which is generated by locally performing these measurements on the given state, can be calculated by SDP
\begin{align}
\begin{split}\label{wnrsdp}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{M_{a|x}\} \\
\max &\hspace{0.2cm} \eta \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}),\ \forall\, a,x \\
& \hspace{0.2cm} \eta\sigma_{a|x}+(1-\eta)\text{\normalfont Tr}(\sigma_{a|x})\frac{\mathbb{1}}{d}=\sum_\lambda D(a|x,\lambda)\sigma_\lambda,\ \forall\, a,x \\
& \hspace{0.2cm} \sigma_\lambda \geq 0, \ \forall\,\lambda,
\end{split}
\end{align}
where $D(a|x,\lambda)$ are elements of deterministic probability distributions and $\lambda\in\{1,\ldots,k^N\}$.
For a fixed quantum state $\rho_{AB}$, different sets of measurements can be tested, each set requiring one SDP to calculate the value of $\eta(\{\sigma_{a|x}\})$. The first method we propose is to parametrize the sets of measurements allowed in a given scenario and, by varying these parameters, explore the solution of multiple SDPs to calculate a bound for $\eta^*(\rho_{AB},N,k)$.
Two important facts can be explored to facilitate this task. The first one is that it is only necessary to optimize over extremal measurements. This is due to the fact that the critical value of $\eta$ depends linearly on the choice of measurements, hence, by convexity, the optimal value will be obtained over extremal measurements. The second fact is that for a system of dimension $d$, extremal measurements have at most $d^2$ outcomes \cite{ExtremalPOVM}.
Aside from the restriction on the number of measurements and outcomes, it is possible to impose restrictions on the parametrization that specify a certain kind of measurement that can be more relevant to the problem one wishes to approach. For instance, it is possible to perform an optimization over only projective measurements or other POVMs with some specific structure (e.g., SIC-POVMs).
The optimization tools chosen for this work are the MATLAB functions \code{fminsearch} \cite{fminsearch}, an unconstrained nonlinear multivariable optimization tool, and \code{fmincon} \cite{fmincon}, a constrained nonlinear multivariable optimization tool. These methods are heuristic and, as such, are not guaranteed to find a global minimum. In order to improve the bound they provide for $\eta^*(\rho_{AB},N,k)$, multiple different initial points can be tested. They also provide a candidate for the optimal set of $N$ $k$-outcome measurements in the given scenario, the one that generates the most robust assemblage when locally performed on $\rho_{AB}$.
\\
\textit{See-saw algorithm}. The seesaw algorithm is an iterative method for solving some nonlinear optimization problems that has found many applications in quantum information theory. In Refs. \cite{CounterPeres,DisprovePeres,RevSDP} seesaw algorithms are used as methods of measurement optimization that are here adapted to approach our problem.
Our seesaw iterates two SDPs. The first one is the dual formulation of SDP (\ref{wnrsdp}):
\begin{subequations}
\begin{align}
\begin{split}\label{seesaw1}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{M_{a|x}\} \\
\min_{\{F_{a|x}\}} &\hspace{0.2cm} 1 - \sum_{a,x} \text{\normalfont Tr}(F_{a|x}\sigma_{a|x}) \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}) ,\ \forall\, a,x \\
& \hspace{0.2cm} 1 - \sum_{a,x} \text{\normalfont Tr} (F_{a|x}\sigma_{a|x}) + \frac{1}{d} \sum_{a,x} \text{\normalfont Tr} (F_{a|x})\text{\normalfont Tr}(\sigma_{a|x}) = 0 \\
& \hspace{0.2cm} \sum_{a,x} D_\lambda(a|x)F_{a|x} \leq 0, \ \forall\,\lambda.
\end{split}
\end{align}
This SDP returns the coefficients $\{F_{a|x}\}$ of a steering inequality of the form $\sum_{a,x}\text{\normalfont Tr}(F_{a|x}\sigma_{a|x})\geq0$. The value obtained by the assemblage $\{\sigma_{a|x}\}$, which is generated by the input state $\rho_{AB}$ and set of measurements $\{M_{a|x}\}$, for the left hand side of this inequality is precisely $1-\eta(\{\sigma_{a|x}\})$. This is due to the fact that primal and dual problems satisfy strong duality. As part of the seesaw, this SDP starts by taking a randomly chosen set of $N$ $k$-outcome measurements and the quantum state whose steerability one wishes to characterize.
The second SDP of the seesaw is
\begin{align}
\begin{split}\label{seesaw2}
\text{given} &\hspace{0.2cm} \rho_{AB}, \{F_{a|x}\} \\
\max_{\{M_{a|x}\}} &\hspace{0.2cm} \sum_{a,x} \text{\normalfont Tr}(F_{a|x}\sigma_{a|x}) \\
\text{s.t.} &\hspace{0.2cm} \sigma_{a|x} = \text{\normalfont Tr}_A(M_{a|x}\otimes\mathbb{1} \, \rho_{AB}) ,\ \forall\, a,x \\
&\hspace{0.2cm} M_{a|x}\geq 0, \ \forall\, a,x \\
& \hspace{0.2cm} \sum_a M_{a|x} = \mathbb{1}, \ \forall\,x.
\end{split}
\end{align}
\end{subequations}
This SDP takes the coefficients $\{F_{a|x}\}$ of the steering inequality that were outputted by the first SDP, (\ref{seesaw1}), as input and, for the same quantum state $\rho_{AB}$, finds the set of POVMs $\{M_{a|x}\}$ that generates the assemblage that maximally violates this inequality. The measurement set that is the output of this SDP, (\ref{seesaw2}), will be the input of the first SDP, (\ref{seesaw1}), in the next round of the iteration. When performed locally on the fixed quantum state, it will necessarily generate an assemblage that has the same or a lower critical visibility than the measurement set from the previous round. When some convergence condition is satisfied (e.g., the diference between the solutions of SDP (\ref{seesaw1}), in two subsequent rounds is less than a certain value) the iteration is halted. The final value for $\eta$ found by the seesaw is an upper bound for $\eta^*(\rho_{AB},N,k)$ of the input state and the set of measurements found by the seesaw is a candidate for the optimal set of $N$ general measurements with $k$ outcomes for steering the state $\rho_{AB}$.
This is also a heuristic method, hence, by itself, it does not prove that the obtained bound is tight. However, it is possible to improve the result by testing multiple different initial points. Contrary to the search algorithm, which allows for constraints on the structure of the POVMs, this method optimizes over all possible sets of $N$ $k$-outcome general POVMs. Our calculations have shown that even though the seesaw algorithm does not have this extra feature, when the interest is in optimizing over general POVMs, it is more effective in doing so than the search algorithm (in the sense that the seesaw demands computational times that are orders of magnitude smaller than the search algorithm for the same number of measurements and outcomes). In all cases tested, for the same state and scenario the solutions of both methods coincide.
\subsection{Lower bounds for $\eta^*(\rho_{AB},N,k)$}\label{lower}
\textit{Outer polytope approximation.} Consider the set $\set{A}$ of all assemblages that can be generated by performing $N$ local measurements of $k$ outcomes on a fixed quantum state $\rho_{AB}$. This set is convex but not a polytope. In order to guarantee that all assemblages in $\set{A}$ admit an LHS model it is sufficient to guarantee that this holds for all of the extremal assemblages of the set. However, since there is an \textit{infinite number} of extremal assemblages in this set (each one corresponding to an extremal set of $N$ $k$-outcome measurements) it is not viable to test each and every one of them.
The method we propose to overcome this problem is based on the techniques presented in Ref. \cite{SuperPOVM}, where the authors approximate the set of quantum measurements by outer polytopes. The idea is to construct an external polytope $\Delta$ that contains $\set{A}$, such that every assemblage in $\set{A}$ can be expressed as a convex combination of the \textit{finitely many} extremal points of $\Delta$. We call the extremal points of $\Delta$ quasiassemblages (they are nonpositive operators that sum to a reduced state $\rho_B$). The way we generate these quasi-assemblages is by applying well-chosen quasi-POVMs \cite{SuperPOVM} (nonpositive operators that sum to the identity) to a fixed quantum state. One can calculate the white-noise robustness of each quasi-assemblage in $\Delta$ using SDP (\ref{wnrsdp}), and the lowest value among them will be a lower bound for $\eta^*(\rho_{AB}, N,k)$. The SDP will return LHS models for the quasi-assemblages which can be used to construct, by simple convex combination, LHS models for all assemblages in the depolarized set.
We now detail the construction of these polytopes for the case where the dimension of Alice's system is $d=2$. A generalization to higher dimensions follows analogously, and we refer to Ref. \cite{SuperPOVM} for more details. In this case, any measurement operator $M$ can be written as $M =\alpha\mathbb{1}+\vec{v}\cdot\vec{\sigma}$, where $\vec{v}$ is a three-dimensional real vector and $\vec{\sigma}$ is the vector of Pauli matrices. By checking the eigenvalues we see that $M\geq0$ if and only if $||\vec{v}||\leq\alpha$, where $||\cdot||$ is the Euclidian norm, which is equivalent to saying that $\vec{v}$ is contained in a real sphere of radius $\alpha\geq 0$. This allows one to represent each measurement operator as a vector in a rescaled Bloch sphere of radius $\alpha$. In order to approximate the set of all POVMs in $d=2$ it is sufficient to approximate the Bloch sphere by an outer polyhedron, which is a simple task in $\mathbb{R}^3$ (see Fig. \ref{blochpolytope}).
\begin{figure}
\caption{\small Example of an approximation of the Bloch sphere by an outer polytope.}
\label{blochpolytope}
\end{figure}
Since the extremal points of the polytope are outside the Bloch sphere, i.e., $||\vec{v}||>\alpha$, they violate the positivity condition for operators in a two-dimensional Hilbert space. Hence, the extremal points of the polytope do not correspond to positive semidefinite operators. They are represented by a vector $v$ such that $v\cdot w_i\leq\alpha$, for some finite set $\{w_i\}$ of vectors defining facets of a polytope that contains the sphere of radius $\alpha$. Accordingly, a quasi-POVM is a set of these nonpositive operators that sum up to the identity. All sets of $N$ quasi-POVMs that can be constructed from the extremal points of the polytope that approximates the Bloch sphere are then locally performed on $\rho_{AB}$ to obtain the quasi-assemblages that define the polytope $\Delta$ that approximates the set $\set{A}$.
The lower bound provided by this method can be improved by increasing the number of tangency points of the outer polytope on the sphere. Contrary to the search and seesaw algorithms, the outer polytope method converges to the exact value of $\eta^*(\rho_{AB},N,k)$ with probability $p=1$ in the limit of an infinite number of generic extremal points. Hence, the bound can be improved as much as necessary, up to available computational resources.
\subsection{Brief discussion of the methods}\label{methoddiscussion}
\textit{Different quantifiers of steering and joint measurability.} We start our discussion by remarking that although the presented methods were based on the white-noise robustness of steering, they can be easily adapted to estimate other quantifiers of steering that can be calculated by an SDP for fixed state and measurements. Some examples are the steering weight \cite{SteeringWeight} and the generalized robustness of steering \cite{Piani}. Also, given the strong connection between joint measurability and steering discussed in Sec. \ref{JM=STE}, analogues of all these steering quantifiers also exist for joint measurability \cite{QuantifiersDani} and our methods can be used to obtain upper and lower bounds for these quantities as well.
\textit{Convergence.} As discussed in Sec. \ref{lower}, the method to calculate lower bounds for $\eta^*(\rho_{AB},N,k)$ is constituted by a sequence of algorithms that converges to the precise value in the limit of infinite extremal points. The cumbersome feature is that the precise value cannot be attained within a finite number of steps. On the other hand, the upper-bound methods consist of heuristic optimization algorithms that may return the exact critical visibility, but there is no guarantee of that. We note that although we did not present a sequence of converging algorithms for calculating upper bounds for $\eta^*(\rho_{AB},N,k)$, one can be constructed by simply testing every possible combination of measurements, possibly with the assistance of polytopes that approximate the set of assemblages from the inside. Since the set of measurements is convex, it can be approximated by a converging sequence of polytopes, guaranteeing the existence of this sequence of algorithms \cite{siberian}. The drawback of this ``brute force'' converging method is that it may take an impractical amount of time to find useful bounds, which is not the case for the heuristic upper-bounds methods discussed in Sec. \ref{upper}.
\textit{Lower bounds for a finite vs. an infinite number of measurements.} In Refs. \cite{LHScavalcanti,LHShirsch}, the authors have presented a method for constructing LHS models for quantum states when all possible measurements (hence an infinite number) are considered. Here, we address a similar question, but in cases where a finite number of measurements is considered. Perhaps surprisingly, our algorithm suggests that constructing local hidden state models for only a finite number of measurements is considerably harder than for an infinite number of measurements. For instance, calculating (good) lower bounds for the critical visibility of two-qubit Werner states subjected to five dichotomic measurements was a very computationally challenging task. Nonetheless, when all possible measurements are considered, the numerical methods of Refs. \cite{LHScavalcanti,LHShirsch} can find good lower bounds in a reasonably small time.
\textit{Numerical stability of the seesaw method.} When implementing the seesaw method with the visibility parametrization described in Sec. \ref{Quantify}, we faced some numerical instability. To overcome this problem, the parametrization
\begin{equation}
\frac{\sigma_{a|x} + t \text{\normalfont Tr}(\sigma_{a|x})\frac{1}{d}}{1+t} \in LHS
\end{equation}
was used instead. The SDPs were then rewritten as a minimization over the parameter $t$ with the correspondence $\eta^*=\frac{1}{1+t^*}$, where the superscript $^*$ denotes the optimal value. Although the interpretation of the visibility parametrization is more straightforward due to its relation with the depolarazing map, the formulation of the problem with the $t$ parameter is equivalent. The numerical stability was also improved by avoiding redundant constraints on normalization and nonsignaling conditions.
We also used the seesaw method to calculate upper bounds for the generalized robustness of steering \cite{Piani} of a quantum state. The seesaw for this quantifier was shown to be more numerically stable than the one for white-noise robustness. As a consequence, the generalized robustness seesaw was used to approach the scenarios with the largest number of parameters in this work.
\section{Results}
We now present the results we obtained by applying the machinery developed in the last section to some specific quantum states. In order to tackle the steering and the joint measurability problem simultaneously, we concentrate on isotropic states in our examples. Also, two-qubit isotropic states can be mapped into two-qubit Werner states \cite{WernerModel} via a local unitary transformation, which always preserves the steerability \cite{MTGeneralMeas}. For this reason, we present our two-qubit results in terms of Werner states, which in this case are given by
\begin{equation}
(\mathbb{1}\otimes\Lambda_\eta)(\ketbra{\psi^-}{\psi^-})=\eta\ketbra{\psi^-}{\psi^-}+(1-\eta)\frac{\mathbb{1}}{4},
\end{equation}
where $\ket{\psi^-}=\frac{1}{\sqrt{2}}(\ket{01}-\ket{10})$ is the singlet state.
To simplify notation we refer to the critical visibility of these two-qubit states as simply $\eta^*(N,k)\coloneqq\eta^*(2,N,k)$.
\subsection{Planar qubit projective measurements}\label{planar}
We start with a simple family of qubit measurements, the planar projective measurements. These are qubit projective measurements whose Bloch vectors are confined to the same plane. The reasons for studying this kind of measurement include its simple experimental implementation \cite{WisemanPlanar} and the low computational cost required to optimize over these measurements, compared to more general ones.
Initially, we use the search algorithm with the constraint that all measurement vectors are coplanar to calculate upper bounds for the critical visibility $\eta^*(N,2)$ of two-qubit Werner states. Calculations were performed for sets of $N\in\{2,...,15\}$ planar projective measurements. The results are presented in Fig. \ref{graphprojective} and Table \ref{tablenum}.
For all trials performed with multiple different initial points, the result for both the objective function--the parameter $\eta$--and the optimization variables--the angles between the Bloch vectors of the measurements--were the same for all values of $N$ tested. In all cases, the optimal set of measurements found by the algorithm is the one in which the Bloch vectors of all measurements are equally spaced on a plane, i.e., each Bloch vector is separated from its next neighbors by an angle of $\frac{\pi}{N}$, as represented for the cases of $N\in\{2,...,5\}$ in Fig. \ref{figoptsetplan}.
\begin{figure}
\caption{\small Plot of the upper bounds of the critical visibility of two-qubit Werner states subjected to $N$ regular tetrahedron and regular trine measurements and the upper and lower bounds for $N$ planar projective and general projective measurements. Dotted black lines correspond to the values $\eta=\frac{1}
\label{graphprojective}
\end{figure}
Next, we calculated lower bounds for $\eta^*(N,2)$ in the restricted scenario of planar projective measurements using the method of outer polytope approximation. Results are reported for the cases of $N\in\{2,\ldots,5\}$ planar projective measurements also in Fig. \ref{graphprojective} and Table \ref{tablenum}. The lower bound for $\eta^*(N,2)$ found by the outer polytope approximation method matches the upper bound found by the search algorithm up to three or four decimal places for all cases tested. We consider this to be enough evidence to claim that for the cases of $N\in\{2,\ldots,5\}$ planar projective measurements, the optimal set of measurements for steering two-qubit Werner states is the set of equally spaced measurements. This is equivalent to stating that the most incompatible set of $N\in\{2,\ldots,5\}$ planar projective qubit measurements is the set of equally spaced measurements. We also conjecture this result to be valid for any number of planar projective measurements. The values we calculated match the analytical results for the incompatibility of equally spaced planar projective qubit measurements presented in Refs. \cite{WisemanPlanar,Siegen}.
\begin{figure}
\caption{\small Optimal set of $N\in\{2,\ldots,5\}
\label{figoptsetplan}
\end{figure}
\subsection{General qubit projective measurements}\label{spherical}
Since the optimal sets of measurements for our problem in the case of planar projective measurements appear to be the sets of equally spaced measurements on a plane, we hypothesize that the optimal sets of general qubit projective measurements correspond to some notion of equally spaced points on a sphere. Unfortunately, contrarily to the equivalent problem on a circumference, the problem of equally distributing points on a sphere is not trivial and many different sets of points can be defined using different notions of distance. This problem is particularly difficult in the regime of few points. For this work we chose the equally spaced notion of the Thomson problem \cite{Thomson} and the Fibonacci problem \cite{Fibonacci}. The former, for the particular cases of $N\in\{2,3,4,6,10\}$ projective measurements corresponding to $4,6,8,12,$ and $20$ vertices, is equivalent to the Platonic solids. The results for the critical visibility of two-qubit Werner states subjected to sets of $N\in\{2,\ldots,18\}$ local measurements constructed from these two notions of equal spacing are listed in Table \ref{tablenum}.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c | c c |}
\multicolumn{7}{c}{Projective qubit measurements} \\
\hline
\multicolumn{1}{|c}{} & \multicolumn{2}{c }{Gen. opt.} & \multicolumn{2}{c |}{Planar opt.} & \multicolumn{2}{c |}{Fixed sets} \\
\hline \hline
$N$ & Upper & Lower & Upper & Lower & Thomson & Fibonnaci \\
\hline
$\quad$ $2$ $\quad$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7071$ & $0.7102$\\
$3$ & $0.5774$ & $0.5755$ & $0.6667$ & $0.6667$ & $0.5774$ & $0.6981$ \\
$4$ & $0.5547$ & $0.5437$ & $0.6533$ & $0.6532$ & $0.5774$ & $0.6114$ \\
$5$ & $0.5422$ & $0.5283$ & $0.6472$ & $0.6470$ & $0.5513$ & $0.5653$ \\
$6$ & $0.5270$ & $$ & $0.6440$ & $$ & $0.5393$ & $0.5561$ \\
$7$ & $0.5234$ & $$ & $0.6420$ & $$ & $0.5234$ & $0.5533$ \\
$8$ & $0.5202$ & $$ & $0.6407$ & $$ & $0.5250$ & $0.5508$ \\
$9$ & $0.5149$ & $$ & $0.6399$ & $$ & $0.5209$ & $0.5359$ \\
$10$ & $0.5144$ & $$ & $0.6392$ & $$ & $0.5191$ & $ 0.5302$ \\
$11$ & $0.5132$ & $$ & $0.6388$ & $$ & $0.5148$ & $0.5274$ \\
$12$ & $0.5117$ & $$ & $0.6384$ & $$ & $0.5152$ & $0.5261$ \\
$13$ & $0.5105$ & $$ & $0.6382$ & $$ & $0.5126$ & $0.5220$ \\
$14$ & $$ & $$ & $ 0.6380$ & $$ & $0.5114$ & $ 0.5180$ \\
$15$ & $$ & $$ & $0.6378$ & $$ & $ 0.5107$ & $0.5158$ \\
$16$ & $$ & $$ & $$ & $$ & $0.5106$ & $0.5158$ \\
$17$ & $$ & $$ & $$ & $$ & $0.5086$ & $0.5150$ \\
$18$ & $$ & $$ & $$ & $$ & $0.5079$ & $0.5136$ \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for the critical visibility of two-qubit Werner states subjected to $N$ projective measurements.}
\label{tablenum}
\end{table}
To test whether these sets of measurements are indeed optimal, we once again use the search algorithm, now with the only restriction that the measurement operators correspond to projectors. We report upper bounds for the value of $\eta^*(N,2)$ in scenarios of $N\in\{2,\ldots,13\}$ general projective measurements. In all cases the search algorithm was able to improve the bound provided by both the Thomson and the Fibonacci measurements (see Table \ref{tablenum}), proving that they are actually not optimal. The best upper bounds are plotted in Fig. \ref{graphprojective} and the Bloch vectors of the measurement elements that form the best candidate for the optimal set of measurements in the cases of $N\in\{2,\ldots,6\}$ projective measurements are plotted in Fig. \ref{figoptsetsph}.
\begin{figure}
\caption{\small Candidates for the optimal set of $N\in\{2,\ldots,6\}
\label{figoptsetsph}
\end{figure}
For these measurements, the vectors are distributed in a particular way: for $2$ and $3$ measurements, we have sets of orthogonal vectors; for $4$ measurements we have $3$ coplanar and equally distributed vectors and $1$ vector orthogonal to the other $3$; for $5$ and $6$ measurements, the structure of $3$ coplanar equally spaced vectors is maintained and the other vectors are agglomerated around the poles of the sphere with the same $z$-projection. For the cases of $7$ or more measurements, this apparent symmetry is no longer necessarily respected.
Using the outer polytope approximation we calculated the lower bounds for the cases of $N\in\{2,\ldots,5\}$ projective measurements that can be seen on Fig. \ref{graphprojective} and Table \ref{tablenum}. In this case the gap between upper and lower bounds for the general projective case is larger than for the planar projective case. This is due to the increase in the number of parameters in the former case as compared to the latter. However, due to the convergence properties of our outer polytope method, as discussed on Secs. \ref{lower} and \ref{methoddiscussion}, these lower bounds can be improved beyond what is the scope of this work.
\subsection{General POVM relevance for qubits}\label{povm}
An old standing question in nonlocality is to understand when general POVMs are useful to reveal this property in a given quantum state \cite{WernerModel,Barrett,WernerPOVM,nguyen16}. It is well known that constructing local hidden variable (LHV) and LHS models for general POVMs is considerably harder than constructing these models for projective measurements \cite{WernerModel,Barrett,WernerPOVM,nguyen16}. Moreover, it is not known whether nonprojective measurements are more useful than projective ones to demonstrate EPR steering or Bell nonlocality. For some particular fixed (nontight) Bell inequalities, it is known that general POVMs can lead to a larger Bell violation than projective measurements \cite{RelevantPOVM}, but the existence of a quantum state that has a LHV/LHS model for projective measurements but displays Bell nonlocality/EPR steering when general POVMs are considered is still an open question.
We have applied our seesaw method to two-qubit Werner states where the noncharacterized party has access to $N\in\{2,\ldots,7\}$ general POVMs with $2,3,$ and $4$ outcomes. We recall that for the case of $2$ outputs, nonprojective POVMs can never be useful for nonlocality, since they can always be written as convex combinations of projective measurements \cite{ExtremalPOVM}. Also, qubit POVMs with more than $4$ outcomes are never extremal \cite{ExtremalPOVM}, hence these measurements could never lead to better bounds for the critical visibility. For this reason we now define the quantity $\eta^*(N)\coloneqq\eta^*(N,d^2=4)$, the critical visibility of two-qubit Werner states when subjected to $N$ POVMs of an arbitrary number of outcomes.
In addition to using the seesaw method to explore general POVMs, we have applied the search algorithm to the specific case where Alice is required to perform symmetric $3$- and $4$-outcome POVMs on her side of a maximally entangled two-qubit state. In the $4$-outcome case, we have fixed all measurements to be SIC-POVMs \cite{renes04}, which are extremal measurements whose Bloch vectors correspond to the vertices of a regular tetrahedron. In the $3$ outcomes case, the chosen POVM was the symmetric extremal measurement whose Bloch vectors correspond to the ``Mercedes-Benz star'', also called the regular trine \cite{Trine} (see Fig. \ref{optpovm}). These particular symmetric nonprojective measurements are known to be useful in tasks such as tomography \cite{rehacek04,SICPOVMTomography} and cryptography \cite{fuchs03}, hence they are interesting examples of extremal nonprojective qubit POVMs \cite{ExtremalPOVM}.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c |}
\multicolumn{4}{c}{Symmetric qubit POVMs} \\
\hline
$N$ & Proj. ($k=2$) & Trine ($k=3$) & Tetra. ($k=4$) \\
\hline
$\quad$ $2$ $\quad$ & $\quad$ $0.7071$ $\quad$ & $\quad$ $0.7739$ $\quad$ & $\quad$ $0.8165$ $\quad$\\
$3$ & $0.5774$ & $0.7202$ & $0.7829$ \\
$4$ & $0.5547$ & $0.6917$ & $0.7716$ \\
$5$ & $0.5422$ & $0.6791$ & $0.7653$ \\
$6$ & $0.5270$ & $0.6690$ & $0.7617$ \\
$7$ & $0.5234$ & $0.6656$ & $0.7605$ \\
$8$ & $0.5202$ & $0.6647$ & -- \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for upper bounds of the critical visibility of two-qubit Werner states subjected to $N$ extremal symmetric POVMs.}
\label{tablesic}
\end{table}
Our results for symmetric qubit POVMs are plotted in Fig. \ref{graphprojective} and listed in Table \ref{tablesic} for the cases of $N\in\{2,\ldots,8\}$, including the results for projective measurements, which are symmetric $2$-outcome POVMs, for the sake of comparison. In the case of $N=2$, the optimal set of regular trine and regular tetrahedron POVMs is plotted in Fig. \ref{optpovm}. It is easy to see that under none of the analyzed scenarios were the symmetric nonprojective POVMs able to show more steering than the projective measurements. In fact, the bounds for symmetric $3$- and $4$-outcome qubit POVMs are considerably worse than for projective qubit measurements.
\begin{figure}
\caption{\small Candidates for the optimal set of $N=2$ regular trine (left; $k=3$) and regular tetrahedron (right; $k=4$) symmetric qubit POVMs for steering two-qubit Werner states.}
\label{optpovm}
\end{figure}
As for the optimization over general $3$- and $4$-outcome qubit POVMs, we could \textit{not} find any set of $N$ general POVMs that are able to overperform projective ones. For $N=2$ and $3$ and $k=4$, the seesaw algorithm ran $10^5$ times, each time with a different initial point; for $N=4$ and $k=3$, the seesaw ran $4\times10^4$ times, and for $k=4$, $3\times10^4$ times; for $N=5$, $6$, and $7$, and $k=3$, it ran $2\times10^4$, $2\times10^3$, and $200$ times, respectively. Without exception, our algorithm recovered the bound for $\eta^*(N)$ obtained by the optimization over projective measurements using general POVMs, usually by nulling two measurement outcomes and ``simulating'' a projective measurement. However, it was never able to surpass it.
Strictly speaking, the results presented in this section are only upper bounds for the critical visibility $\eta^*(N)$. Nevertheless, given the small number of parameters in the two-qubit scenario and the number of times we have ran our heuristic method, we believe that these results are strong evidence that general POVMs are not useful to reveal EPR-steering in two-qubit Werner states.
\subsection{Higher dimension states and measurements}\label{higherd}
We now explore the generality of our seesaw method on quantum systems of dimension $d>2$ by calculating bounds for the critical visibility $\eta^*(d,N,k)$ of higher dimension maximally entangled states. Let us start with the simple case where these states are subjected to only $2$ local general measurements. These calculations are reported for states of dimensions $d\in\{2,\ldots,6\}$ in Table \ref{tablen2}. We note that by increasing the number of outcomes in the measurements from $k=2$ up to $k=d$ the bounds for $\eta^*(d,2,k)$ are significantly improved and the optimal sets of measurements are always composed by projective measurements--even though most of these scenarios allow extremal nonprojective POVMs. However, once the number of outcomes achieves $k=d+1$ the bound for $\eta^*(d,2,k)$ provided by the seesaw method ceases to decrease and it seems that increasing the number of outputs beyond this point does not improve the results. Since there only exist projective measurements with up to $k=d$ outcomes, this result is evidence that allowing POVMs more general than projective measurements does not increase the robustness of the steerability of isotropic states. Following the connection between the steerability of these states and joint measurability, this is also evidence that sets of $2$ general qudit POVMs cannot be more incompatible than sets of $2$ projective qudit measurements.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c c |}
\multicolumn{6}{c}{$N=2$} \\
\hline
$\quad$ $k$ $\quad$& $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.7000$ & $0.6901$ & $0.6812$ & $0.6736$ \\
$3$ & $0.7071$ & $0.6794$ & $0.6722$ & $0.6621$ & $0.6527$ \\
$4$ & $$ & $0.6794$ & $0.6665$ & $0.6544$ & $0.6448$ \\
$5$ & $$ & $$ & $0.6665$ & $0.6483$ & $0.6429$ \\
$6$ & $$ & $$ & $$ & $0.6483$ & $0.6390$ \\
$7$ & $$ & $$ & $$ & $$ & 0.6390 \\
\hline
\end{tabular}
}
\end{center}
\caption{Summary of numerical results for upper bounds of the critical visibility of $d$-dimensional isotropic states subjected to $2$ general POVMs of $k\in\{2,\ldots,d+1\}$ outcomes.}
\label{tablen2}
\end{table}
Since the scenario where the uncharacterized party is allowed to perform only $2$ measurements is very particular, we performed the same calculations reported above for $d$-dimensional isotropic states allowing scenarios with $3$ and $4$ general measurements with outcomes up to $d+1$ as well. In these broader scenarios, we also calculated upper bounds for the critical visibility $\eta^*(d,N,k)$ of the isotropic states. However, since the number of parameters increases too rapidly (exponentially on the number of measurements), the seesaw method presented some numerical instability, and for this reason we are not able to reach any conclusions about the relevance of general POVMs in these scenarios.
\subsection{Mutually unbiased bases}\label{MUB}
A set of MUBs consists of 2 or more orthonormal bases $\{\ket{a_x}\}_a$ in a $d$-dimensional Hilbert space that satisfy
\begin{equation}
|\braket{a_x}{b_y}|^2=\frac{1}{d}, \quad \forall\, a,b \in\{1,\ldots,d\}, \ x\neq y,
\end{equation}
for all bases $x,y$ \cite{durt10}. A set of MUBs is called complete if for a Hilbert space of dimension $d$ there exists $d+1$ MUBs. These bases can be used to construct sets of mutually unbiased projective measurements with a high level of symmetry, and for this reason one might think they would be good candidates for the optimal set of measurements for measurement incompatibility and for EPR steering with a maximally entangled state.
We have calculated the critical visibility of isotropic states of dimension $d\in\{2,\ldots,6\}$ when subjected to local MUB measurements using SDP (\ref{wnrsdp}) and listed the results in Table \ref{tablemub}. These exact values calculated by our SDP (\ref{wnrsdp}), show significant improvement over the analytical bounds obtained in Refs. \cite{marciniak15,hgsieh16} for steering with MUBs and maximally entangled states. Next, we used the seesaw method to calculate upper bounds for $\eta^*(d,N,d)$ of the isotropic states when locally subjected to sets of general POVMs with $d$ outcomes for some number of measurements $N$ where MUB measurements are known to exist. Perhaps surprisingly, in many cases we found sets of measurements with greater or equal robustness, showing that MUBs are not necessarily the best choice of measurements to reveal quantum steering nor are they the most incompatible ones. The results are listed in Table \ref{tablemub}. The optimal measurements found by the seesaw method are all projective measurements in these cases as well. We remark that in Refs. \cite{carmeli11,heinosaari13}, the authors have computed (analytically) the required visibility $\eta^*(d,2,d)$ for any pair of $d$-dimensional MUB measurements to be jointly performed; here we have shown that there exist pairs of measurements that are even more incompatible than mutually unbiased ones. However, in scenarios where there exist complete sets of MUB measurements, for dimensions $2$, $3$, and $4$, we were not able to find measurements more resistant to white noise and better for steering isotropic states than the MUB ones, which is evidence that they may be optimal for this task.
\begin{table}[h!]
\begin{center}
{\renewcommand{0.75}{0.75}
\begin{tabular}{| c | c c c c c |}
\multicolumn{6}{c}{MUBs} \\
\hline
$\quad$ $N$ $\quad$ & $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.6830$ & $0.6667$ & $0.6545$ & $0.6449$ \\
$3$ & $0.5774$ & $0.5686$ & $0.5469$ & $0.5393$ & $0.5204$ \\
$4$ & $$ & $0.4818$ & $0.5000$ & $0.4615$ & $$ \\
$5$ & $$ & $$ & $0.4309$ & $0.4179$ & $$ \\
$6$ & $$ & $$ & $$ & $0.3863$ & $$ \\
\hline
\multicolumn{6}{c}{} \\
\multicolumn{6}{c}{General $d$-outcome POVMs} \\
\hline
$N$& $d=2$ & $3$ & $4$ & $5$ & $6$ \\
\hline
$2$ & $0.7071$ & $0.6794$ & $0.6665$ & $0.6483$ & $0.6395$ \\
$3$ & $0.5774$ & $0.5572$ & $0.5412$ & $0.5266$ & $0.5139$ \\
$4$ & $$ & $0.4818$ & $0.4797$ & $0.4615$ & $$ \\
$5$ & $$ & $$ & $0.4309$ & -- & $$ \\
$6$ & $$ & $$ & $$ & -- & $$ \\
\hline
\end{tabular}
}
\end{center}
\caption{Comparison between the exact critical visibility of isotropic states in dimension $d$ subject to local mutually unbiased measurements and the upper bound of the same states when optimizing over general POVMs with $k=d$.}
\label{tablemub}
\end{table}
\section{Discussion}
We have used three methods for investigating EPR steering and joint measurability under restrictive measurement scenarios and discussed the applicability of each one. Using white-noise robustness as a quantifier, we have presented two heuristic methods for calculating the critical visibility of quantum states subjected to a finite number of measurements and one converging method for lower-bounding the same quantity. Our methods can be easily adapted to other steering and joint measurability quantifiers.
For two-qubit Werner states, we showed that the best sets of $N\in\{2,\ldots,5\}$ planar projective measurements are equally spaced measurements and conjecture this result to be valid for all $N\in\mathbb{N}$. Our upper bounds for the critical visibility of two-qubit Werner states subjected to planar projective measurements match the analytical expressions derived in Refs. \cite{WisemanPlanar,Siegen} for equally spaced measurements. We proved that intuitive notions of equally spaced measurements in the Bloch sphere, like the vertices of Platonic solids, do not correspond to the best measurements to show steering with two-qubit Werner states, nor are they the most incompatible sets of measurements. We showed that symmetric $3$- and $4$-outcome qubit POVMs are not optimal for steering two-qubit Werner states as well. Upper bounds for the critical visibility of two-qubit Werner states subjected to $N\in\{2,\ldots,18\}$ general measurements were calculated. We provided strong numerical evidence that general POVMs are not more suitable for steering two-qubit Werner states than projective measurements, and suggested candidates for the optimal sets of $N\in\{2,\ldots,6\}$ qubit measurements that are projective and follow a nonintuitive pattern.
Our results for higher dimension isotropic states indicate that increasing the number of outcomes until $k=d$ improves the bound for the critical visibility of the state. However, increasing the number of outcomes beyond the value of the local dimension of the state does not seem to improve the bounds, which strengthens the idea that nonprojective POVMs are not relevant for steering. The candidates for optimal measurements in all higher dimension scenarios are projective measurements. Finally, we proved that many incomplete sets of MUB measurements are not optimal for steering and provided numerical evidence that complete sets of MUB measurements could be optimal for steering isotropic states.
Although we presented numerical evidence against the relevance of nonprojective POVMs for EPR steering, deciding if projective measurements are indeed optimal for steering in all scenarios and for all quantum states still remains an open question.
One future direction is to apply similar techniques for the study of Bell nonlocality. Although some simple adaptation of our methods can be used to tackle the analogous problem for Bell nonlocality, the number of parameters in the problem could make our algorithms impracticable even in simple scenarios.
All code written for this work is available in a repository \cite{Code}.
{\small{
}}
\end{document} |
\betaegin{document}
\deltaate{\tauoday}
\tauitle{Jet schemes of quasi-ordinary surface singularities}
\alphauthor{Helena Cobo}
\varepsilonmail{helenacobo@gmail.com}
\alphauthor{Hussein Mourtada}
\alphaddress{Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7, B\^atiment Sophie Germain, 75013 Paris, France.}
\varepsilonmail{hussein.mourtada@imj-prg.fr}
\betaegin{abstract}
In this paper we give a complete description of the irreducible components of the jet schemes (with origin in the singular locus) of a two-dimensional quasi-ordinary hypersurface singularity.
We associate with these components and with their codimensions and embedding dimensions, a weighted graph. We prove that the data of this weighted graph is equivalent to the data of the
topological type of the singularity. We also determine a component of the jet schemes (or equivalently, a divisor on $\mathbb A^3$), that computes the log canonical threshold
of the singularity embedded in $\mathbb A^3$. This provides us with pairs $X\sigmaubset\mathbb A^3$ whose log canonical thresholds are not contributed by monomial divisorial valuations.
Note that for a pair $C\sigmaubset\mathbb A^2$, where $C$ is a plane curve, the log canonical threshold is always contributed by a monomial divisorial valuation (in suitable coordinates of $\mathbb A^2$).
\varepsilonnd{abstract}
\sigmaubjclass[2010]{14E18,14J17.}
\keywords{Singularities, Jet schemes, Quasi-ordinary singularities, log canonical threshold.}
\title{Jet schemes of quasi-ordinary surface singularities}
\sigmaection{Introduction}
A quasi-ordinary singularity $(X,0)$ of dimension $d,$ comes with a finite projection $p:X\lambdaongrightarrow \mathbb{A}^d,$ whose discriminant is a normal crossing
divisor. These singularities appear in the Jungian approach to resolution of singularities (see \cite{PP-HJ}). We are interested in irreducible quasi-ordinary
hypersurfaces. Thanks to Abhyankar-Jung theorem, we know that a hypersurface of this type can be parametrized by a Puiseux series (i.e an element in
$\mathbb C[[x_1^{\pihirac{1}{n}},\lambdadots,x_d^{\pihirac{1}{n}}]]).$ Moreover, some special exponents (called the characteristic exponents) which belong to the support of
this series, are complete invariants of the topological type of the singularity (see \cite{Gau}). In particular, they determine invariants which come from
resolution of singularities, like the log canonical threshold or the Motivic zeta functions (\cite{BGG}, \cite{ACLM}, \cite{CoGPqo}). They also give
insights about the construction of a resolution of singularities (\cite{BMc0}, \cite{BMc}, \cite{GP4}, \cite{Vill}).
Our aim is to construct some comparable complete invariants for all type of singularities. We search for such invariants in the jet schemes. For $m\in \mathbb{N},$ the
$m$-jet scheme, denoted by $X_m$, is a scheme that parametrizes morphisms $\mbox{Spec }\mathbb C[t]/(t^{m+1})\lambdaongrightarrow X.$ Intuitively we can think of it as
parametrizing arcs in an ambient space, which have a large contact, depending on $m$, with $X.$ We know already that some invariants which come from resolution of singularities
are encoded in jet schemes (\cite{Mus}, \cite{EinMustata}).
We want to extract from the jet schemes information about the singularity, which can be expressed in terms of invariants of resolutions of singularities.
With this, our next goal is to construct a resolution of singularities by using invariants of jet schemes. For specific types of singularities, the knowledge of the irreducible components of the jet schemes $X_m$
of a singular variety X, together with some invariants of them, like dimension or embedding dimension, permits to determine deep invariants of the singularity of $X$: the topological type in the case of curves (see \cite{Hcur}),
and the analytical type in the case of normal toric surfaces (see \cite{HCRAS} and \cite{Htor}). Moreover, in the case of irreducible plane curves, the minimal embedded resolution can be constructed from the jet schemes (\cite{LMR}), and the same for rational double point singularities (\cite{MP}).
Notice that, understanding the structure of jet schemes for particular singularities, remains a difficult problem. These structures have been studied in \cite{Yuen}
and \cite{DoC} for determinantal varieties, in \cite{Hcur}
for plane curve singularities, \cite{HCRAS} and \cite{Htor} for normal toric surfaces, in \cite{Hrat} for rational double point surface singularities, and in
\cite{SS} for commuting matrix pairs schemes. In the toric case, no result is known for dimension bigger than two.
More general jet schemes can be defined, and there are relations with invariants of singularities (see \cite{Mus14}).
In this paper, we study jet schemes of a two-dimensional quasi-ordinary hypersurface singularity $X$. We will give a combinatorial description of the irreducible components of the set of $m$-jets with center
in the singular locus of $X$, in terms of the following invariants of the singularity: the lattices $N_0,N_1,\lambdadots,N_g,$ the minimal system of generators $\gammaamma_1,\lambdadots,\gammaamma_g$ of the semigroup $\Gamma$ of $X,$
and the numerical data attached to them, $n_1,\lambdadots,n_g$ and $e_1,\lambdadots,e_g$ (see Section \rhoef{secQO} for definitions). Given $h\in\mathbb C[x_1,\lambdadots,x_n]$, an algebraic variety $X$, and $p,m\in\mathbb Z_{>0}$ with $p\lambdaeq m$,
let $\mbox{Cont}_X^p(h)_m$ be the locally closed set defined as $\mbox{Cont}_X^p(h)_m=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(h\circ\gammaamma)=p\}$. Then we associate with any lattice point $\nu\in N_0$ the constructible set
\[D_m^\nu=(\mbox{Cont}_X^{\nu_1}(x_1)_m\cap\mbox{Cont}_X^{\nu_2}(x_2)_m)_{red},\]
and its closure $C_m^\nu=\mathbb overline{D_m^\nu}$. The sets $C_m^\nu$ are the candidates to be the irreducible components, but there are many inclusions among them. We study these inclusions by defining, on the lattices
associated with $X$, subtle relations which depend on the singular loci of the quasi-ordinary surfaces defined by the approximated roots.
For some components the relation is very easy, is given by the product ordering $\lambdaeq_p$, and we can prove that
\[\mbox{ if }\nu\lambdaeq_p\nu'\mbox{ then }C_m^{\nu'}\sigmaubset C_m^\nu.\]
For other components the relation is more complicated, we have that
\[\mbox{ if }\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}\mbox{ then }C_m^{\nu'}\sigmaubset C_m^\nu\]
where $j'(m,\nu)$ is the integer $j\in\{0,\lambdadots,g\}$ defined by
\[n_je_j\lambdaangle\nu,\gammaamma_j\rhoangle+e_j\lambdaeq m<n_{j+1}e_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+e_{j+1},\]
and $\sigma_{Reg,j'(m,\nu)}$ is defined to keep track of the singular locus of the $j'(m,\nu)$-th approximated root as follows.
Since the normalization of any quasi-ordinary singularity is a toric variety, let $\nu_j: Z_{\sigma,N_j}\lambdaongrightarrow V(f_j)$ be the normalization of $V(f_j)$ ($f_j$ being the $j$-th approximated root). Then $Z_{\sigma,N_j}\sigmaetminus\nu_j^{-1}(Sing(V(f_j)))$ is an open set of the toric variety $Z_{\sigma,N_j}$, it is a union of orbits, and we denote by $\sigma_{Reg,j}$ the fan formed by the faces of $\sigma$ corresponding to the orbits.
Then, with the minimal elements with respect to these relations we define a set $F_m\sigmaubset\mathbb Z^2$, and for any $\nu\in F_m,$ we have a component $C_m^\nu\sigmaubset X_m$. We prove that these are the irreducible components.
\betaegin{The}
For any $m\in\mathbb Z_{>0}$ we have that the space of $m$-jets of $X$ with center in the singular locus has the following decomposition into irreducible components
\[\pii_m^{-1}(X_{Sing})=\cup_{\nu\in F_m}C_m^\nu.\]
\varepsilonnd{The}
Hence, for
every $m \in \mathbb Z_{>0}$, we determine the irreducible components of $X_m$ with center in the singular locus of $X$. Moreover, we give a formula for the codimension of $C_m^\nu$ in Proposition \rhoef{Prop1}.
To prove the theorem we need to understand the geometry of the sets $D_m^\nu$. This is done by comparing their geometries with
the geometries of the corresponding subsets for the approximate roots. The following proposition describes this comparison.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in\sigma\cap N_j$ such that $n_1\cdots n_g\lambdaangle\nu,\gammaamma_1\rhoangle\lambdaeq m$, we have that
\[D_m^\nu=(\pii_{m,[\pihirac{m}{e_j}]}^a)^{-1}(D_{j,[\pihirac{m}{e_j}]}^\nu)\]
where the integer $j\in\{1,\lambdadots,g\}$ satisfies the relation
\[n_j\cdots n_g\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq m<n_{j+1}\cdots n_g\lambdaangle\nu,\gammaamma_{j+1}\rhoangle,\]
by $D_{j,m}^\nu$ we denote the corresponding set $D_m^\nu$ for the quasi-ordinary surface defined by the $j$-th approximated root, and for $q>p$, $\pii_{q,p}^a:\mathbb A^3_q\lambdaongrightarrow\mathbb A^3_p$ is the projection of the jet schemes of the affine ambient space.
\varepsilonnd{Pro}
This explains in part how the singular locus of the approximated roots play a role in our problem.
The irreducible components of the jet schemes fit in natural projective systems, to which we associate a weighted graph. The vertices of the graph correspond to irreducible components, and to every vertex we attach the corresponding codimension and embedded dimension. We will prove the following result.
\betaegin{The}
The weighted graph determines and it is determined by the topological type of the singularity.
\varepsilonnd{The}
This theorem achieves one of our goals for this type of singularities: constructing a complete invariant of the singularity from its jet schemes. Note that other invariants involving
arcs and jets, like motivic zeta functions, do not determine the topological type in the case of quasi ordinary singularities, see \cite{CoGPqo} and \cite{Nuelo}.
In another direction, using Mustata's formula (\cite{Mus}), we will determine an irreducible component of an $m$-jet scheme, or equivalently a divisor on the ambient space $\mathbb A^3$, which contributes the log canonical threshold
of the pair $X\sigmaubset\mathbb A^3$ (note that the log canonical threshold for such a pair has been computed in \cite{BGG}, by looking at the poles of the motivic zeta function). This provides us
with pairs $X\sigmaubseteq\mathbb A^3$ whose log canonical threshold is not contributed by a monomial divisorial valuation. For instance, for the quasi-ordinary surface defined by $f=(z^2-x_1x_2)^2-x_1^3x_2z$,
the log canonical thresholds satisfy this property. Note that for a pair $C\sigmaubseteq\mathbb A^2$, where $C$ is a plane curve, the log canonical threshold is always contributed by a monomial valuation. See \cite{AN} and \cite{ACLM2} for the computation of the log canonical threshold for plane curves.
Along with the same ideas of \cite{Gen}, we are working to construct an embedded resolution of singularities of $X$ from the data of the graph
constructed in this paper. We think that such a resolution puts light on the resolution of singularities obtained by González Pérez in \cite{GP4}, and give in
some sense an answer to the question of Lipman (\cite{Lipman-Eq}) on the construction of a canonical resolution of singularity of a quasi-ordinary hypersurface from the
characteristic exponents.
Moreover, understanding the surface case is an important step in the understanding of the general case.
The structure of the paper is as follows. In Section \rhoef{Sec2} we introduce jet schemes. A brief exposition on quasi-ordinary singularities is given in Section
\rhoef{secQO}, together with some useful definitions at the end of the section. Section \rhoef{Sec4} is the heart of the paper; it is devoted to the study of the irreducible components of the
jet schemes of quasi-ordinary surface singularities. In Section \rhoef{Proofs} we state and proof some results which are useful but technical, and moreover we leave the proofs of some previous results, to make Section \rhoef{Sec4} more readable.
{\betaf Acknowledgments.}
We thank Pedro González Pérez and the referees, for comments and suggestions which improved substantially the presentation of this paper.
The beginning of this work was done during a stay of HC supported by the
European Research Council under the European Community’s Seventh Framework
Programme (FP7/2007-2013) / ERC Grant Agreement nr. 246903 NMNAG. She thanks Institut Mathématique de
Jussieu for hospitality.
HM is partially supported by ANR-12-JS01-0002-01 SUSI.
\sigmaection{Jet schemes}
\lambdaabel{Sec2}
In this section we define jet schemes of an affine scheme $X,$ see \cite{Ishii-07} for details.
Let $X=\mbox{Spec }\mathbb C[x_1,\lambdadots,x_n]/I$ be an affine scheme of finite type. For $m\in\mathbb Z_{>0}$ the functor
$F_m:\ \mathbb C\mbox{-Schemes}\lambdaongrightarrow \mbox{Sets}$ which, with an affine scheme defined by a $\mathbb C$-algebra $A$, associates
\[F_m(\mbox{Spec}(A))=\mbox{Hom}_\mathbb C(\mbox{Spec}(A[t]/(t^{m+1})),X),\]
is representable by a $\mathbb C$-scheme, denoted by $X_m$. This is the scheme of $m$-jets. Its closed points are morphisms of the form
\[\gammaamma:\mbox{Spec}(\mathbb C[t]/(t^{m+1}))\lambdaongrightarrow X.\]
Such a morphism $\gammaamma$ is equivalent to a $\mathbb C$-algebra homomorphism
\[\gammaamma^\alphast:\mathbb C[x_1,\lambdadots,x_n]/I\lambdaongrightarrow \mathbb C[t]/(t^{m+1}).\]
If we fix a set of generators $f_1,\lambdadots,f_r$ for the ideal $I$, the map $\gammaamma^\alphast$ is determined by the image of the $x_i^{'s}$
\[x_i\mapsto x_i^{(0)}+x_i^{(1)}t+\cdots+x_i^{(m)}t^m,\ 1\lambdaeq i\lambdaeq n,\]
where the relations
\betaegin{equation}
f_i(x_1^{(0)}+\cdots+x_1^{(m)}t^m,\lambdadots,x_n^{(0)}+\cdots+x_n^{(m)}t^m)\varepsilonquiv 0\mbox{ mod }t^{m+1}
\lambdaabel{relation}
\varepsilonnd{equation}
must hold for each $f_i$, with $1\lambdaeq i\lambdaeq r$.
If we write equations in (\rhoef{relation}) as
\betaegin{equation}
\betaegin{array}{cc}
f_i(x_1^{(0)}+x_1^{(1)}t+\cdots+x_1^{(m)}t^m,\lambdadots,x_n^{(0)}+x_n^{(1)}t+\cdots+x_n^{(m)}t^m)=\\
\\
=\sigmaum_{j=0}^mF_i^{(j)}(x_1^{(0)},\lambdadots,x_1^{(j)},\lambdadots,
x_n^{(0)},\lambdadots,x_n^{(j)})\ t^j\mbox{ mod }t^{m+1},\\
\varepsilonnd{array}
\lambdaabel{expf}
\varepsilonnd{equation}
we have that giving a closed point of $X_m$ is equivalent to giving a point in $V(F_l^{(j)})_{0\lambdaeq j\lambdaeq m,1\lambdaeq l\lambdaeq r}\sigmaubset\mathbb A_m^n$, where
$\mathbb A_m^n=\mbox{Spec}(\mathbb C[x_i^{(0)},\lambdadots,x_i^{(m)}]_{i=1,\lambdadots,n})$. Hence we can make the
following identification
\betaegin{equation}
X_m=\mbox{Spec }\lambdaeft(\pihirac{\mathbb C[x_i^{(0)},\lambdadots,x_i^{(m)}]_{i=1,\lambdadots,n}}{(F_l^{(j)})_{0\lambdaeq j\lambdaeq m,\ 1\lambdaeq l\lambdaeq r}}\rhoight).
\lambdaabel{Desc1}
\varepsilonnd{equation}
\betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The equations defining the $3$-jets are
\[\betaegin{array}{ll}
F^{(0)} & = {z^{(0)}}^3-{x_1^{(0)}}^3{x_2^{(0)}}^2\\
\\
F^{(1)} & =3{z^{(0)}}^2z^{(1)}-3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(0)}}^2-2{x_1^{(0)}}^3{x_2^{(0)}}x_2^{(1)}\\
\\
F^{(2)} & =3{z^{(0)}}^2z^{(2)}+3z^{(0)}{z^{(1)}}^2-6{x_1^{(0)}}^2{x_1^{(1)}}x_2^{(0)}x_2^{(1)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(2)}-3{x_1^{(0)}}^2x_1^{(2)}{x_2^{(0)}}^2\\
& \ \ -{x_1^{(0)}}^3{x_2^{(1)}}^2-3x_1^{(0)}{x_1^{(1)}}^2{x_2^{(0)}}^2\\
\\
F^{(3)} & = {z^{(1)}}^3+6z^{(0)}z^{(1)}z^{(2)}+3{z^{(0)}}^2z^{(3)}-2{x_1^{(0)}}^3x_2^{(0)}x_2^{(3)}-2{x_1^{(0)}}^3x_2^{(1)}x_2^{(2)}-6{x_1^{(0)}}^2x_1^{(1)}x_2^{(0)}x_2^{(2)}\\
& \ \ -3{x_1^{(0)}}^2x_1^{(1)}{x_2^{(1)}}^2-6{x_1^{(0)}}^2x_1^{(2)}x_2^{(0)}x_2^{(1)}-6x_1^{(0)}{x_1^{(1)}}^2x_2^{(0)}x_2^{(1)}-3{x_1^{(0)}}^2x_1^{(3)}{x_2^{(0)}}^2\\
& \ \ -6x_1^{(0)}x_1^{(1)}x_1^{(2)}{x_2^{(0)}}^2-{x_1^{(1)}}^3{x_2^{(0)}}^2\\
\varepsilonnd{array}\]
\lambdaabel{Ex0}
\varepsilonnd{Exam}
\betaegin{Rem}
Every polynomial $F^{(l)}$ is non-zero and quasi-homogeneous of degree $l$ in the variables $x_i^{(0)},\lambdadots,x_i^{(l)}$, $z^{(0)},\lambdadots,z^{(l)}$ for $i=1,2$. In $F^{(0)},\lambdadots,F^{(l)}$ the variables $x_1^{(l)},\ x_2^{(l)}$ and $z^{(l)}$ appear only in $F^{(l)}$.
\lambdaabel{RemTonto}
\varepsilonnd{Rem}
\sigmaection{Quasi-ordinary surface singularities}
\lambdaabel{secQO}
In this section we collect some well known facts about quasi-ordinary hypersurface singularities of dimension two, and we prove some lemmas which will be used in the next section. We state everything for the case of dimension two, though the definitions and results hold in any dimension.
An equidimensional germ $(X, 0)$, of dimension $2$, is {\varepsilonm quasi-ordinary} (q.o. for short) if there exists a finite projection $p: (X, 0) \rhoightarrow (\mathbb C^2,0)$ which is a local isomorphism outside a normal crossing divisor. If $(X, 0)$ is a hypersurface there is an embedding $(X, 0) \sigmaubset (\mathbb C^{3},0) $, where $X$ is
defined by an equation $f= 0$, and $f \in \mathbb C \{ x_1,x_2 \} [z]$ is a {\it quasi-ordinary ~polynomial}; that is, a Weierstrass polynomial with discriminant
$\Delta_z f$ of the form $\Delta_z f = x_1^{\delta_1}\cdot x_2^{\delta_2} \varepsilonpsilon$ for a unit $\varepsilonpsilon$ in the ring $ \mathbb C \{ x_1,x_2 \}$ of convergent power series and
$(\delta_1,\delta_2) \in \mathbb Z^2_{\gammaeqslant 0}$. In these coordinates the projection $p$ is the restriction of the projection
\[\mathbb C^{3}\rhoightarrow\mathbb C^2,\ \ \ \ (x_1,x_2,z)\mapsto(x_1,x_2).\]
From now on we assume that $(X,0)$ is analytically irreducible, that is $f \in \mathbb C \{ x_1, x_2 \} [z]$ is irreducible (see \cite{Ass} and \cite{Ev} for criteria of irreducibility of q.o. polynomial). The Jung-Abhyankar theorem guarantees
that the roots of a q.o.~ polynomial $f$, called {\it q.o.~branches}, are fractional power series in $\mathbb C \{ x_1^{1/n},x_2^{1/n}\}$, for $n =\deltaeg f$ (see
\cite{Abhyankar}). The difference $\zetaeta^{(i)}-\zetaeta^{(j)}$ of two different roots of $f$ divides the discriminant of $f$ in the ring $\mathbb C\{x_1^{1/n},
x_2^{1/n}\}$. Therefore $\zetaeta^{(i)}-\zetaeta^{(j)}=x_1^{\lambda_{ij}^{(1)}}x_2^{\lambda_{ij}^{(2)}}u_{ij}$ where $u_{ij}$ is a unit in $\mathbb C\{x_1^{1/n},x_2^{1/n}\}$. The exponents $\lambda_{ij}$
are characterized in the following Lemma:
\betaegin{Lem} \lambdaabel{expo} {\rhom (see \cite{Gau}, Prop. 1.3)}
Let $f \in \mathbb C \{ x_1, x_2 \} [z] $ be an irreducible
q.o.~polynomial. Let $\zeta$ be a root of $f$ with expansion:
\betaegin{equation} \lambdaabel{expan}
\zeta = \sigmaum \betaeta_{\lambda} \betaold{x}^\lambda.
\varepsilonnd{equation}
There exists $0 \ne \lambda_1, \deltaots, \lambda_g \in \mathbb Q^2_{\gammaeqslant 0}$
such that $\lambda_1\lambdaeq\lambda_2\lambdaeq\cdots\lambdaeq\lambda_g$, and if $M_0 :=\mathbb Z^2 $ and $M_j := M_{j-1} + \mathbb Z \lambda_j$ for
$j=1, \deltaots, g$, then:
\betaegin{enumerate}
\item [(i)] $\betaeta_{\lambda_i} \ne 0$ and if $\betaeta_{\lambda} \ne 0$ then $\lambda \in M_j$ where
$j$ is the unique integer such that $\lambda_j \lambdaeqslant \lambda$ and
$\lambda_{j+1} \nleq \lambda$ (where $\lambdaeqslant$ means coordinate-wise and we convey that $\lambda_{g +1} = \infty$).
\item [(ii)] For $j=1, \deltaots, g$, we have $\lambda_j \notin M_{j-1}$, hence the index
$n_j = [M_{j-1} : M_j]$ is $> 1$.
\varepsilonnd{enumerate}
Moreover if $\zetaeta\in\mathbb C\{x_1^{1/n},x_2^{1/n}\}$ is a fractional power series satisfying the conditions above, then $\zetaeta$ is a quasi-ordinary branch.
\varepsilonnd{Lem}
\betaegin{Defi}
The exponents $\lambda_1 , \deltaots, \lambda_g $ in Lemma \rhoef{expo} are called {\varepsilonm characteristic exponents} of the q.o.~branch $\zeta$. We denote by $M$ the lattice
$M_g$ and we call it the lattice associated to the q.o.~branch $\zeta$. We denote by $N$ (resp. $N_i$) the dual lattice of $M$ (resp. $M_i$ for $i=1,\lambdadots,g$). For
convenience we denote $\lambda_0:=(0,0)$ and $n_0:=1$.
\lambdaabel{defExp}
\varepsilonnd{Defi}
In \cite{Gau} Gau proved that the characteristic exponents determine and are determined by the embedded topological type of $(X,0)$.
As a consequence of Lemma \rhoef{expo} we have the following result:
\betaegin{Lem}
If $\zetaeta$ is a quasi-ordinary branch of the form (\rhoef{expan}) then the series $\zetaeta_{j-1}:=\sigmaum_{\lambdaambda\not\gammaeq\lambdaambda_j}\betaeta_\lambdaambda\betaold{x}^\lambdaambda$ is a
quasi-ordinary branch with characteristic exponents $\lambda_1,\lambdadots,\lambda_{j-1}$, for $j=1,\lambdadots,g$.
\lambdaabel{LemConseq}
\varepsilonnd{Lem}
\betaegin{Defi}
For $0\lambdaeq j\lambdaeq g-1$ we have the germ of quasi-ordinary hypersurface $(X^{(j)},0)$, where $X^{(j)}$ is parametrized by the branch $\zetaeta_j$. For convenience we also denote $\zetaeta$ by $\zetaeta_g$ and $X$ by $X^{(g)}$.
\lambdaabel{SemiX}
\varepsilonnd{Defi}
Without loss of generality we relabel the variables $x_1, x_2$ in such a way that if $\lambda_j = ( \lambda_j^{(1)}, \lambda_j^{(2)}) \in
\mathbb Q^2$ for $j=1, \deltaots, g$, then we have:
\betaegin{equation}
(\lambda_1^{(1)}, \deltaots, \lambda_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}} (\lambda_1^{(2)}, \deltaots, \lambda_g^{(2)}),
\lambdaabel{lex}
\varepsilonnd{equation}
where $\gammaeqslant_{\mbox{\rhom lex}}$ is lexicographic order. The
q.o.~branch $\zeta$
is said to be normalized if
$\lambda_1$ is not of the form $(\lambda_1^{(1)}, 0)$ with $\lambda_1^{(1)} <
1$. Lipman proved that the germ $(X,0)$ can be parametrized by a
normalized q.o.~branch (see \cite{Gau}, Appendix).
We assume from now on that the q.o.~branch $\zeta$ is normalized.
The semigroup $\mathbb Z^2_{\gammaeqslant 0}$ has a minimal set of generators
$v_1,v_2$, which is a basis of the lattice $M_0$. The dual
basis, $\{w_1,w_2\}$, is a basis of the dual lattice $N_0$, and spans a regular cone $\sigma$ in
$N_{0,\mathbb R}=N_0\mathbb otimes_\mathbb Z\mathbb R$. It follows that $\mathbb Z^2_{\gammaeqslant 0} = \sigma^\vee \cap
M_0$, where $\sigma^\vee = \mathbb R^2_{\gammaeqslant 0}$ is the dual cone of
$\sigma$. The $\mathbb C$-algebra $\mathbb C \{ x_1,x_2 \} $ is isomorphic
to $\mathbb C \{ \sigma^\vee \cap M_0 \} $. The
local algebra ${\mathcal O}_X = \mathbb C\{x_1, x_2\}[z]/(f)$ of the singularity $(X,0)$ is isomorphic to
$\mathbb C\{\sigma^\vee \cap M_0\}[\zeta]$. By Lemma \rhoef{expo} the series $\zeta
$ can be viewed as an element $\sigmaum \betaeta_\lambda \betaold x^\lambda$ of the algebra
$ \mathbb C\{\sigma^\vee \cap M\}$.
\betaegin{Lem} (see \cite{GP4})
The homomorphism $\mathcal O_X\lambdaongrightarrow\mathbb C\{\sigma^\vee\cap M\}$ is the inclusion of $\mathcal O_X$ in its integral closure in its field of fractions.
\lambdaabel{NormTor}
\varepsilonnd{Lem}
This Lemma shows that the normalization of a quasi-ordinary hypersurface $(X,0)$ is the germ of the toric variety $X(\sigma,N)=Z^{\sigma^\vee\cap M}$ at the distinguished point.
The elements of $M$ defined by:
\betaegin{equation}
{\gamma}_1 = \lambda_1\mbox{ and } {\gamma}_{j+1}- n_j {\gamma}_{j} = \lambda_{j+1} - \lambda_{j}\mbox{ for } j= 1, \deltaots, g-1,
\lambdaabel{defSem}
\varepsilonnd{equation}
span the semigroup $\Gamma := \mathbb Z^2_{\gammaeqslant 0} + \gamma_1 \mathbb Z_{\gammaeqslant
0} + \cdots + \gamma_g \mathbb Z_{\gammaeqslant 0} \sigmaubset \sigma^\vee \cap M$. For convenience we denote $\gammaamma_0:=0$. The
semigroup $\Gamma$ defines an analytic invariant of the germ
$(X,0)$ (see \cite{Jussieu},\cite{PPP04},\cite{KM}).
\betaegin{Defi}
The monomial variety associated to $(X,0)$ is the toric variety
\[X^\Gamma:=\mbox{Spec }\mathbb C[\Gamma].\]
\lambdaabel{defMV}
\varepsilonnd{Defi}
Moreover we associate with the characteristic exponents the following sequence of semigroups:
\[\Gamma_j=\sigma^\vee\cap M+\gammaamma_1\mathbb Z_{\gammaeq 0}+\cdots+\gammaamma_j\mathbb Z_{\gammaeq 0},\mbox{ for }j=0,\lambdadots,g.\]
And we have the corresponding monomial varieties associated to $\Gamma_j$. We denote by $e_{i-1}:=n_i\cdots n_g$ for $1<i\lambdaeq g$ and set $e_g:=1$.
Notice that, by (\rhoef{lex}) and the definition of $\gammaamma_1,\lambdadots,\gammaamma_g$, we deduce that
\betaegin{equation}
(\gammaamma_1^{(1)}, \deltaots, \gammaamma_g^{(1)}) \gammaeqslant_{\mbox{\rhom lex}}(\gammaamma_1^{(2)}, \deltaots, \gammaamma_g^{(2)}).
\lambdaabel{lexGamma}
\varepsilonnd{equation}
The following Lemma gathers some important facts about the generators $\gammaamma_j$ and the semigroups $\Gamma_j$.
\betaegin{Lem}(see Lemma 3.3 in \cite{Jussieu})
\betaegin{enumerate}
\item[(i)] We have that $\gammaamma_j>n_{j-1}\gammaamma_{j-1}$ for $j=2,\lambdadots,g$, where $<$ means $\neq$ and $\lambdaeq$ coordinate-wise.
\\
\item[(ii)] If a vector $u_j\in\sigma^\vee\cap M_j$, then we have $u_j+n_j\gammaamma_j\in\Gamma_j$.
\\
\item[(iii)] The vector $n_j\gammaamma_j$ belongs to the semigroup $\Gamma_{j-1}$ for $j=1,\lambdadots,g$. Moreover, we have a unique relation
\betaegin{equation}
n_j\gammaamma_j=\alphalpha^{(j)}+r_1^{(j)}\gammaamma_1+\cdots+r_{j-1}^{(j)}\gammaamma_{j-1}
\lambdaabel{relgam}
\varepsilonnd{equation}
such that $0\lambdaeq r_i^{(j)}\lambdaeq n_i-1$ and $\alphalpha^{(j)}\in M_0$ for $j=1,\lambdadots,g$.
\varepsilonnd{enumerate}
\lambdaabel{LemaPedro}
\varepsilonnd{Lem}
\betaegin{Defi}
Given two irreducible quasi-ordinary polynomials $f$ and $g$ in $\mathbb C\{x_1,x_2\}[z]$ such that $fg$ is a quasi-ordinary polynomial, we say that $f$ and $g$
have order of coincidence $\alphalpha\in \mathbb Q^2$ if $\alphalpha$ is the largest exponent on the set
\[\{\lambda_{ij}\ |\ f(\zetaeta^{(i)})=g(\zetaeta^{(j)})=0\},\]
where $\zetaeta^{(i)}$ and $\zetaeta^{(j)}$ are roots of $fg$.
\lambdaabel{OrdCon}
\varepsilonnd{Defi}
\betaegin{Defi}
We associate to $f$ a set of semi-roots
\[z=f_0,f_1\lambdadots,f_g=f\in \mathbb C\{x_1,x_2\}[z].\]
Every $f_j$ is an irreducible quasi-ordinary polynomial of degree $n_0\cdots n_j$ with order of coincidence with $f$ equal to $\lambda_{j+1}$ for $j=0,\lambdadots,g$.
\lambdaabel{defAroots}
\varepsilonnd{Defi}
They are parametrized by truncations of a root $\zetaeta(x_1^{1/n},x_2^{1/n})$ of $f$ in the following sense:
\betaegin{Pro}
(see \cite{Jussieu}) Let $q\in \mathbb C\{x_1,x_2\}[z]$ be a monic polynomial of degree $n_0\cdots n_j$. Then $q$ is a $j$-th semi-root of $f$ if and only if
$q(\zetaeta)=\betaold{x}^{\gammaamma_j}\varepsilonpsilon_j$ for a unit $\varepsilonpsilon_j$ in $\mathbb C\{x_1,x_2\}[z]$.
\varepsilonnd{Pro}
\betaegin{Cor}
The quasi-ordinary polynomials $f_j\in\mathbb C\{x_1,x_2\}[z]$ defining $X^{(j)}$ (see Definition \rhoef{SemiX}) for $j=0,\lambdadots,g$ form a system of semiroots of $f$.
\lambdaabel{CorSR}
\varepsilonnd{Cor}
In what follows we state some results about quasi-ordinary polynomials and approximated roots. Moreover we give some definitions and notations that will be
used in the next section.
Approximated roots play an important role in the understanding of quasi-ordinary singularities. We have the following expansions of the semiroots in terms of the previous ones:
\betaegin{Lem} (See Lemma 35 in \cite{GP4})
The expansion of the approximated roots is of the following form:
\betaegin{equation}
c_j^*f_j=f_{j-1}^{n_j}-c_jx_1^{\alphalpha_1^{(j)}}x_2^{\alphalpha_2^{(j)}}f_0^{r_1^{(j)}}\cdots f_{j-2}^{r_{j-1}^{(j)}}+\sigmaum c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}f_0^{r_1}\cdots f_{j-1}^{r_j},
\lambdaabel{expSR}
\varepsilonnd{equation}
where $c_j^*,c_j\in\mathbb C^*$, $0\lambdaeq r_i^{(j)},r_i<n_i$ for $i=1,\lambdadots,j$, and \[n_j\gammaamma_j=(\alphalpha_1^{(j)},\alphalpha_2^{(j)})+r_1^{(j)}\gammaamma_1+\cdots+r_{j-1}^{(j)}\gammaamma_{j-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j.\]
\lambdaabel{Lema35}
\varepsilonnd{Lem}
Let $(X,0)\sigmaubset(\mathbb C^3,0)$ be a germ of quasi-ordinary surface with characteristic exponents $\lambda_1,\lambdadots,\lambda_g$. We denote by $\lambda_1=(\pihirac{a_1}{n_1},\pihirac{b_1}{n_1})$ the first
characteristic exponent.
Notice that, by (\rhoef{lex}), we have that $a_1\gammaeq b_1\gammaeq 0$.
\betaegin{Lem}
We have that
\betaegin{equation}
f_1=z^{n_1}-c_1x_1^{a_1}x_2^{b_1}+\sigmaum c_{ijk}x_1^ix_2^jz^k,
\lambdaabel{eqSR1}
\varepsilonnd{equation}
with $(i,j)+k\gammaamma_1>n_1\gammaamma_1$ and $k<n_1$ whenever $c_{ijk}\neq 0$.
And for $1\lambdaeq l\lambdaeq g-1$ we have
\betaegin{equation}
f=f_l^{e_l}+\sigmaum_{(i,j)+k\gammaamma_1>n_le_l\gammaamma_l}c_{ijk}^{(l)}x_1^ix_2^jz^k.
\lambdaabel{eqSRpqprima}
\varepsilonnd{equation}
Moreover, the following expansions will be useful. For $0\lambdaeq j<g-1$
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}
\lambdaabel{grgr1}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}>n_{j+1}e_{j+1}\gammaamma_{j+1}$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\betaegin{equation}
f =f_j^{e_j}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j-1}^{r_j}
\lambdaabel{grgr2}
\varepsilonnd{equation}
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j>n_je_j\gammaamma_j$ whenever $c_{\underline{\alphalpha},\underline{r}}\neq 0$.
\lambdaabel{LemExpSR}
\varepsilonnd{Lem}
{\varepsilonm Proof.} By applying recursively Lemma \rhoef{Lema35} and using Lemma \rhoef{LemaPedro} (i).\pisifill $\Box$
\sigmaubsection{Some toric geometry}
\lambdaabel{toric}
See \cite{F} for a reference on toric geometry.
Given a lattice $N$ we denote by $N_\mathbb R$ the vector space spanned by $N$ over the field $\mathbb R$. We denote by $M$ the dual lattice, $M=\mbox{Hom}(N,\mathbb Z)$, and by $\lambdaangle,\rhoangle : N\tauimes M\lambdaongrightarrow\mathbb Z$ the duality pairing between the lattices $N$ and $M$.
A rational convex polyhedral cone (or simply a cone) is the set of non-negative linear combinations of vectors $v_1,\lambdadots,v_r\in N$. A cone is strictly convex if it contains no lines. The dual cone of $\sigma$, denoted by $\sigma^\vee$, is the set
\[\sigma^\vee=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle\gammaeq 0\ \pihiorall u\in\sigma\},\]
and the orthogonal of $\sigma$, denoted by $\sigma^\betaot$, is
\[\sigma^\betaot=\{\mathbb omega\in M_\mathbb R\ |\ \lambdaangle\mathbb omega,u\rhoangle=0\ \pihiorall u\in\sigma\}.\]
We denote by $\sigmatackrel{\circ}{\sigma}$ the relative interior of the cone $\sigma$. A fan $\Sigma$ is a family of strictly convex cones in $N_\mathbb R$ such that for any $\sigma\in\Sigma$ any face of $\sigma$ belongs to $\Sigma$, and for any $\sigma,\tau\in\Sigma$, the intersection $\sigma\cap\tau$ is a face of both.
The relation $\tau\lambdaeq\sigma$ denotes that $\tau$ is a face of $\sigma$. The support of the fan $\Sigma$ is the set $|\Sigma|:=\cup_{\tau\in\Sigma}\tau\sigmaubset N_\mathbb R$.
Let $\tau$ be a strictly convex cone, rational for the lattice $N$. By Gordan's Lemma the semigroup $\sigma^\vee\cap M$ is finitely generated. We denote by $\mathbb C[\sigma^\vee\cap M]$ the semigroup algebra of $\sigma^\vee\cap M$ with complex coefficients. The toric variety $Z(\tau,N):=\mbox{Spec }\mathbb C[\tau^\vee\cap M]$ is normal. The torus $T_N:=Z(N)$ is an open dense subset of $Z(\tau,N)$ which acts on $Z(\tau,N)$ and the action extends the action of the torus on itself by multiplication. There is a one to one correspondence between the faces $\tauheta$ of $\tau$ and the orbits orb$(\tauheta)$ of the torus action on $Z(\tau,N)$, which reverses the inclusions of their closures. The closure of orb$(\tauheta)$ is the toric variety $Z((\sigma^\vee\cap\tauheta^\betaot)^\vee,N)$ for $\tau\lambdaeq\sigma$.
\sigmaubsection{Definitions and Notations}
\lambdaabel{defsnots}
We introduce now some definitions and notations which will be used throughout the paper.
The singular locus of a quasi-ordinary singularity is determined, after Lipman, by its characteristic exponents (see \cite{Lipman2} and \cite{PPP04}).
\betaegin{Defi} We define
\[\betaegin{array}{ll}
Z_i=X\cap\{x_i=0\}, & \mbox{for }i=1,2\\
\\
Z_{12}=X\cap\{x_1=x_2=0\}. & \\
\varepsilonnd{array}\]
Moreover, the smallest number $c\in\{1,2\}$ with the property that
\[\lambda_i^{(j)}=0,\mbox{ for all }1\lambdaeq i\lambdaeq g\mbox{ and }c+1\lambdaeq j\lambdaeq 2\]
is called the equisingular dimension of the quasi-ordinary projection $p$.
\lambdaabel{equi}
\varepsilonnd{Defi}
In \cite{Lipman2}
Lipman proved that the spaces $Z_1,Z_2$ and $Z_{12}$ are irreducible.
By condition (\rhoef{lex}) we have that $c$ gives the number of variables appearing in the monomials $\betaold x^{\lambda_1},\lambdadots,\betaold x^{\lambda_g}$.
\betaegin{Defi}
Let $X$ be a quasi-ordinary surface singularity with $g\gammaeq 1$ characteristic exponents. We define the integers $g_1\gammaeq 0$ and $g_2\in\{g_1,g_1+1\}$ as follows
\[\mbox{ if }c=1\mbox{ we set }g_1=g_2=g+1,\]
otherwise
\[\betaegin{array}{l}
\gammaamma_{g_1}^{(2)}=0\mbox{ and }\gammaamma_{g_1+1}^{(2)}\neq 0,\\
\\
g_2=\lambdaeft\{\betaegin{array}{cl}
g_1+1 & \mbox{ if }\gammaamma_{g_1+1}^{(2)}=\pihirac{1}{n_{g_1+1}}\\
\\
g_1 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\\
\varepsilonnd{array}\]
\lambdaabel{defg1}
\varepsilonnd{Defi}
Note that these integers can be defined with the same property for the characteristic exponents. Now we use them to describe the singular locus of $X$.
Lipman's theorem describes the singular locus $X_{Sing}$ of a quasi-ordinary hypersurface $X$. We state it here in the particular case of surfaces.
\betaegin{The} (See Theorem 7.3 in \cite{Lipman2})
Let $X$ be a quasi-ordinary surface singularity with characteristic exponents $\lambda_1,\lambdadots,\lambda_g$. Then we have:
\betaegin{enumerate}
\item[(i)] $X_{Sing}=Z_{12}$ if and only if $g=1$ and $\lambda_1=(\pihirac{1}{n},\pihirac{1}{n})$.
\item[(ii)] If $c=1$ then $X_{Sing}=Z_1$.
\item[(iii)] Otherwise $c=2$, and since $\lambda_1^{(1)}\neq 0$, $Z_1\sigmaubset X$ is a component of $X_{Sing}$. Moreover, if we do not have
simultaneously $\lambda_k^{(2)}=0$ for all $1\lambdaeq k\lambdaeq g-1$ and $\lambda_g^{(2)}=\pihirac{1}{n_g}$ then the singular locus is reducible of the form $X_{Sing}=Z_1\cup Z_2$.
\varepsilonnd{enumerate}
\lambdaabel{CorSing}
\varepsilonnd{The}
\betaegin{Rem}
Notice that
\[\betaegin{array}{l}
Z_1=\{x_1=z=0\}\\
\\
Z_2=\{x_2=f_{g_1}=0\}\\
\\
Z_{12}=\{(0,0,0)\}\\
\varepsilonnd{array}\]
and hence the singular locus of a quasi-ordinary surface singularity is either a point, or a line, or two lines, or a line and a singular curve.
\varepsilonnd{Rem}
Then, geometrically, the meaning of the integer $g_2$ is to measure the irreducibility of the singular locus of the approximated roots. Indeed,
\[\betaegin{array}{l}
X_{Sing}^{(j)} \mbox{ is irreducible, for }1\lambdaeq j\lambdaeq g_2\\
\\
X_{Sing}^{(j)} \mbox{ is reducible, for }g_2<j\lambdaeq g\\
\varepsilonnd{array}\]
Now we define a sequence of semi-open cones keeping track of the singular locus of the quasi-ordinary hypersurfaces $X^{(j)}$ for $j=1,\lambdadots,g$ (see Definition
\rhoef{SemiX}). Let
\[\nu_j:X(\sigma,N_j)\lambdaongrightarrow X^{(j)}\]
be the normalization of $X^{(j)}$ (see Lemma \rhoef{NormTor}). Consider $\nu_j^{-1}(X_{Sing}^{(j)})\sigmaubseteq X(\sigma,N_j)$, it is a disjoint union of orbits
\[\nu_j^{-1}(X_{Sing}^{(j)})=\betaigsqcup_\tau\mbox{orb}(\tau),\ \mbox{ for some }\tau\mbox{ faces of }\sigma.\]
We also have that the complement of $\nu_j^{-1}(X_{Sing}^{(j)})$ in the toric variety $X(\sigma,N_j)$ is a union of orbits.
\betaegin{Defi}
We define, for $j=1,\lambdadots,g$
\[\betaegin{array}{ll}
\sigma_{Sing,j} & \mbox{ the fan associated to }\nu_j^{-1}(X_{Sing}^{(j)})\\
\\
\sigma_{Reg,j} & \mbox{ the fan associated to }X(\sigma,N_j)\sigmaetminus\nu_j^{-1}(X_{Sing}^{(j)})\\
\varepsilonnd{array}\]
For $j=g$ we just denote them by $\sigma_{Sing}$ and $\sigma_{Reg}$ respectively.
\lambdaabel{DefTSigmas}
\varepsilonnd{Defi}
\betaegin{Rem}
Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$, let $\rho_1$ and $\rho_2$ be its one-dimensional faces. For $1\lambdaeq j\lambdaeq g$ we have
\[\sigma_{Sing,j}=\lambdaeft\{\betaegin{array}{cll}
\sigma\sigmaetminus(\rho_1\cup\rho_2) & \mbox{ if } & X_{Sing}^{(j)}=Z_{12}\\
\\
\sigma\sigmaetminus\rho_2 & \mbox{ if } & X_{Sing}^{(j)}=Z_1\\
\\
\sigma\sigmaetminus\{(0,0)\} & \mbox{ if } & X_{Sing}^{(j)}=Z_1\cup Z_2\\
\varepsilonnd{array}\rhoight.\]
and by definition $\sigma_{Reg,j}=\sigma\sigmaetminus\sigma_{Sing,j}$.
\lambdaabel{defSigmas}
\varepsilonnd{Rem}
The fan $\sigma_{Sing}$ will turn out to be necessary in our description of $\pii_m^{-1}(X_{Sing})$ (see Lemma \rhoef{LemGamma}), while the fans $\sigma_{Reg,j}$ will be important in the description of the irreducible components (see Proposition \rhoef{PropC1}).
The sequence
$\{\sigma_{Reg,1},\lambdadots,\sigma_{Reg,g}\}$ is not very complicated, in the sense that most of the elements are the same.
Since by definition $\gammaamma_{g_1+1}^{(2)}=\lambdaambda_{g_1+1}^{(2)}$ then, by Theorem \rhoef{CorSing}, we deduce
\betaegin{equation}
\betaegin{array}{ll}
\mbox{for }1\lambdaeq j\lambdaeq g_2 & \sigmaigma_{Reg,j}=\lambdaeft\{\betaegin{array}{cl}
\rho_1\cup\rho_2 & \mbox{ if }\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }j=1\\
\rho_2 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\\
\\
\mbox{for }g_2+1\lambdaeq j\lambdaeq g & \sigmaigma_{Reg,j}=\{(0,0)\}\\
\varepsilonnd{array}
\lambdaabel{SigRegj}
\varepsilonnd{equation}
Moreover notice that, by definition, we have $\sigma_{Sing,j}\sigmaubseteq\sigma_{Sing,j+1}$.
\betaegin{Defi}
Given $\nu\in\sigma\cap N_0$, we define the following sequence of real numbers
\[l_i(\nu):=n_ie_i\lambdaangle\nu,\gammaamma_i\rhoangle,\mbox{ for }1\lambdaeq i\lambdaeq g.\]
Set $l_0(\nu)=0$ and $l_{g+1}(\nu)=\infty$ for any $\nu\in\mathbb Z^2$.
Moreover, we define
\[i(\nu)=\lambdaeft\{\betaegin{array}{cl}
g+1 & \mbox{ if }\nu\in N_g\\
\\
\mbox{min }\{i\in\{1,\lambdadots,g\}\ |\ \nu\notin N_i\} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
and
\[\betaegin{array}{l}
c(\nu)=\mbox{max }\{0\lambdaeq i\lambdaeq g\}\ |\ \lambdaangle\nu,\gammaamma_i\rhoangle=0\}\\
\\
m(\nu)=\mbox{min }\{1\lambdaeq i\lambdaeq g\ |\ n_i\lambdaangle\nu,\gammaamma_i\rhoangle<\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\}\\
\varepsilonnd{array}\]
\lambdaabel{defs}
\varepsilonnd{Defi}
Notice that, by definition, $l_1(\nu)=l_2(\nu)=\cdots=l_{m(\nu)}(\nu)<l_{m(\nu)+1}(\nu)$. Moreover, for $\nu\in\sigma_{Sing}\cap N_0$, we have that
\[\betaegin{array}{ccc}
c(\nu)=\lambdaeft\{\betaegin{array}{cc}
g_1 & \mbox{ if }g_1>0\mbox{ and }\nu\in\rho_2\\
\\
0 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
&
\mbox{ and }
&
m(\nu)=\lambdaeft\{\betaegin{array}{cc}
\in\{1,\lambdadots,g\} & \mbox{ if }\nu\in\rho_1\\
\\
g_1 & \mbox{ if } \nu\in\rho_2\mbox{ and }g_1>0\\
\\
1 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
\varepsilonnd{array}\]
It is straightforward to check that $c(\nu)\lambdaeq m(\nu)\lambdaeq i(\nu)$.
\betaegin{Lem}
For any $\nu\in\sigma_{Sing}\cap N_0$, we have that
\[l_i(\nu)\in\mathbb Z \mbox{ for }1\lambdaeq i\lambdaeq\mbox{min }\{i(\nu),g\}.\]
Moreover the integers $l_i(\nu)$ are ordered as
\betaegin{equation}
0= l_{c(\nu)}(\nu)< l_{c(\nu)+1}(\nu)\lambdaeq\cdots \lambdaeq l_g(\nu)<l_{g+1}(\nu)=\infty,
\lambdaabel{ineq}
\varepsilonnd{equation}
and we have the equality $l_i(\nu)=l_{i+1}(\nu)$ if and only if $\gammaamma_{i+1}^{(j)}=n_i\gammaamma_i^{(j)}$ for $j$ either $1$ or $2$, and $\nu\in\rho_j$.
For $i>c(\nu)$ we have that the following statements are equivalent:
\betaegin{enumerate}
\item[(i)] $\nu\in N_i$
\item[(ii)] for $1\lambdaeq j\lambdaeq i-1,\ \nu\in N_j$ and the number $\pihirac{l_{j+1}(\nu)-l_j(\nu)}{e_j}$ is a positive integer.
\varepsilonnd{enumerate}
\lambdaabel{l-ord}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
For $1\lambdaeq i<i(\nu)$ we have that $\nu\in N_i$, therefore $\lambdaangle\nu,\gammaamma_i\rhoangle\in\mathbb Z$ and $l_i(\nu)$ is an integer. If $i(\nu)<g+1$, then
$\nu\in N_{i(\nu)-1}$ and by (\rhoef{relgam}) we deduce that $l_{i(\nu)}(\nu)$ is an integer.
By definition of $\gammaamma_i$ it follows that $e_i\gammaamma_{i+1}=e_{i-1}\gammaamma_i+e_i(\lambda_{i+1}-\lambda_i)$. Then
\[l_{i+1}(\nu)=l_i(\nu)+e_i\lambdaangle\nu,\lambda_{i+1}-\lambda_i\rhoangle,\]
and the second claim of the Lemma follows because the exponents $\lambda_i$ are ordered lexicographically as (\rhoef{lex}).
The equivalence follows by the fact that
\[l_j(\nu)-l_{j-1}(\nu)=e_{j-1}(\lambdaangle\nu,\gammaamma_j\rhoangle-n_{j-1}\lambdaangle\nu,\gammaamma_{j-1}\rhoangle).\]
\pisifill $\Box$
\sigmaection{Jet schemes of quasi-ordinary surface singularities}
\lambdaabel{Sec4}
In this section we describe the irreducible components of $\pii_m^{-1}(X_{Sing})\sigmaubset X_m$. We begin with an overview of the section.
We will associate, to any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$, a family of $m$-jets that we call
$C_m^\nu$. Roughly speaking, it is the Zariski closure of the set of $m$-jets whose order of contact with the hyperplane coordinate $x_i$ is bigger or equal to $\nu_i$, for $i=1,2$.
We divide the sets $C_m^\nu$ into two types. Sets defined by the annihilation of hyperplane coordinates in $\mathbb A_m^3=
\mbox{Spec }\mathbb C[x_1^{(i)},x_2^{(i)},z^{(i)}]_{i=0,\lambdadots,m}$, which are associated with $\nu$ in a certain set $H_m\sigmaubset\sigmaigma_{Sing}\cap N_0$ (where $H$ stands for hyperplane). The $C_m^\nu$ of the second type have a more complicated geometry and are associated with $\nu$ in a certain set $L_m$ (where $L$ stands for
lattice, because for such components, $\nu$ belongs to one of the lattices $N_i$ for $1\lambdaeq i\lambdaeq g$).
The geometry of $C_m^\nu$ for $\nu\in H_m$ is evident and the possible inclusions among different $C_m^\nu$ are determined only by looking at the
product ordering of the associated vectors $\nu$. For $\nu\in L_m$, to understand the geometry of $C_m^\nu$, we consider the order of contact of the generic point
of $C_m^\nu$ with the different approximated roots. This allows us to detect certain dense subset of $C_m^\nu$, which is isomorphic to the cartesian product of an open set of the spectrum of the graded algebra (associated with the last approximated root appearing in the equations defining $C_m^{\nu}$) and an affine
space. This permits to prove that each $C_m^\nu$ is irreducible and to compute its codimension (see Proposition \rhoef{Prop1}).
The inclusions among the $C_m^\nu$ for $\nu\in H_m\cup L_m$ are more delicate. We introduce in Definition \rhoef{orden2} a new relation to detect such
inclusions. In this definition, the singular locus of the last approximated root affecting the geometry of $C_m^\nu$, plays a crucial role.
Finally, with the collection of sets $C_m^\nu$ left, we prove in Theorem \rhoef{TheCaso1} that they are the irreducible components of the $m$-jets
through the singular locus.
\betaegin{Defi}
Let $h\in \mathbb C[x_1,\lambdadots,x_n]$ and let $X$ be an algebraic variety. For $p,m\in\mathbb Z_{>0}$ with $p\lambdaeq m$ we set
\[\mbox{Cont}_X^p(h)_m:=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(h\circ\gammaamma)=p\}.\]
And, for $m\in\mathbb Z_{>0}$ and any $\nu=(\nu_1,\nu_2)\in\sigma_{Sing}\cap N_0$ with $\nu_i\lambdaeq m$ we define the constructible set
\[D_m^\nu=(\mbox{Cont}_X^{\nu_1}(x_1)_m\cap\mbox{Cont}_X^{\nu_2}(x_2)_m)_{red},\]
where we consider the reduced structure, and
\[C_m^\nu=\mathbb overline{D_m^\nu},\]
its Zariski closure. We denote by $D(f)$ the open set
\[D(f)=\mbox{Spec }R_f\]
where $R$ is the ring $R=\mathbb C[x_1^{(j)},x_2^{(j)},z^{(j)}]_{j\gammaeq 0}$.
\lambdaabel{DefDC}
\varepsilonnd{Defi}
Given a jet $\gammaamma\in X_m$, if $x_i\circ\gammaamma\neq0$ for $i=1,2$, the vector $\nu=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))$ belongs to $\sigma\cap N_0$ and $0\lambdaeq\nu_i\lambdaeq m$. Moreover it is trivial that $\gammaamma\in C_m^\nu$. Now we look at $m$-jets with origin at the singular locus.
\betaegin{Lem}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, there exists $\nu\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$, such that $\gammaamma\in C_m^\nu$.
Moreover
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu.\]
\lambdaabel{LemGamma}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
Given $\gammaamma\in\pii_m^{-1}(X_{Sing})$, suppose first that $x_i\circ\gammaamma\neq 0$ for $i=1,2$. Then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and obviously $\gammaamma\in D_m^\nu\sigmaubseteq C_m^\nu$. We have to prove that $\nu\in\sigma_{Sing}\cap N_0$, and this follows by Remark \rhoef{defSigmas}, since:
\betaegin{enumerate}
\item[(i)] If $X_{Sing}=\{(0,0,0)\}$, then $\gammaamma(0)=(0,0,0)$ and ord$_t(x_i\circ\gammaamma)>0$ for $i=1,2$.
\item[(ii)] If $X_{Sing}=Z_1$, then $\gammaamma(0)=(0,x_2(0),0)$, and $\mbox{ord}_t(x_1\circ\gammaamma)>0,\ \mbox{ord}_t(x_2\circ\gammaamma)\gammaeq 0$.
\item[(iii)] If $X_{Sing}=Z_1\cup Z_2$, then
\[\mbox{if }\gammaamma(0)\in Z_1, \mbox{ we have ord}_t(x_1\circ\gammaamma)>0\mbox{ and ord}_t(x_2\circ\gammaamma)\gammaeq 0\]
\[\mbox{if }\gammaamma(0)\in Z_2, \mbox{ we have ord}_t(x_1\circ\gammaamma)\gammaeq 0\mbox{ and ord}_t(x_2\circ\gammaamma)>0\]
\varepsilonnd{enumerate}
To deal with the other cases, notice that
\[C_m^\nu=\{\gammaamma\in X_m\ |\ \mbox{ord}_t(x_i\circ\gammaamma)\gammaeq\nu_i,\ i=1,2\}.\]
If $x_i\circ\gammaamma=0$ for $i=1,2$, then $\gammaamma\in C_m^\nu$ for any $\nu\in\sigma_{Sing}\cap N_0$ with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$.
If $x_1\circ\gammaamma=0$ and $x_2\circ\gammaamma\neq 0$, then we denote $\alphalpha:=\mbox{ord}_t(x_2\circ\gammaamma)$. We have $0\lambdaeq\alphalpha\lambdaeq m$, and $\gammaamma\in C_m^\nu$
for any $\nu\in\sigma_{Sing}\cap N_0$, with $0\lambdaeq\nu_i\lambdaeq m$ for $i=1,2$, and $\nu_2\lambdaeq\alphalpha$.
The left case $x_1\circ\gammaamma\neq 0$ and $x_2\circ\gammaamma=0$ is analogous to the last one.
We prove the other inclusion. If $\gammaamma\in X_m\sigmaetminus\pii_m^{-1}(X_{Sing})$, then $\gammaamma(0)\notin X_{Sing}$. Again distinguishing cases depending on the singular locus, we can prove that $\nu=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\notin\sigma_{Sing}$.
\pisifill$\Box$
\sigmaubsection{Description of the sets $C_m^\nu$}
The sets $C_m^\nu$ are the candidates to be irreducible components of $\pii_m^{-1}(X_{Sing})$. We proceed to study these sets. Notice that, by definition, it follows that
\[C_m^\nu\sigmaubset V(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)},i=1,2).\]
\betaegin{Defi}
For $\nu\in\sigma\cap N_0$ and $m\in\mathbb Z_{>0}$ we define the ideals
\[I^\nu=\lambdaeft(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)}\rhoight)_{i=1,2},\]
\[J_m^\nu=Rad\lambdaeft((F^{(i)}\mbox{ mod }I^\nu)_{0\lambdaeq i\lambdaeq m}\rhoight).\]
Moreover we define the integers $j(m,\nu)\in\{0,\lambdadots,i(\nu)-1\}$ and $j'(m,\nu)\in\{0,\lambdadots,j(m,\nu)\}$ by the
inequalities
\[\betaegin{array}{c}
l_j(\nu)\lambdaeq m< l_{j+1}(\nu),\\
\\
l_j(\nu)+e_j\lambdaeq m<l_{j+1}(\nu)+e_{j+1},\\
\varepsilonnd{array}\]
respectively.
\lambdaabel{IJ}
\varepsilonnd{Defi}
Then we have that
\betaegin{equation}
D_m^\nu=V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)}),
\lambdaabel{eqast}
\varepsilonnd{equation}
where the fact of taking the radical in the definition of $J_m^\nu$ corresponds to taking the reduced structure in the definition of $D_m^\nu$.
We have to study the polynomials $F^{(j)}\mbox{ mod }I^\nu$ for $0\lambdaeq j\lambdaeq m$. Thanks to the identification in (\rhoef{Desc1}), these polynomials have to be seen as the defining equations of $D_m^\nu$.
\betaegin{Exam}
Let $X$ be the quasi-ordinary surface defined by the polynomial $f=z^3-x_1^3x_2^2$. The singular locus of $X$ is reducible
\[X_{Sing}=\{x_1=z=0\}\cup\{x_2=z=0\}.\]
We described in Example \rhoef{Ex0} the equations of the $3$-jets, and if we look at $3$-jets with origin in the singular locus, then we have to add the condition either $x_1^{(0)}=z^{(0)}=0$ or $x_2^{(0)}=z^{(0)}=0$. This is equivalent to consider the equations modulo the ideal $I^\nu$.
Then
\[\betaegin{array}{lcl}
(\pii_1^{-1}(X_{Sing}))_{red} & = & \pii_1^{-1}(\{x_1=z=0\})\cup\pii_1^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},z^{(0)})=C_1^{(1,0)}\cup C_1^{(0,1)}\sigmaubset\mathbb A_1^3.\\
\\
(\pii_2^{-1}(X_{Sing}))_{red} & = & \pii_2^{-1}(\{x_1=z=0\})\cup\pii_2^{-1}(\{x_2=z=0\})\\
& = & V(x_1^{(0)},z^{(0)})\cup (V(x_2^{(0)},z^{(0)},{x_1^{(0)}}^3{x_2^{(1)}}^2))_{red}=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_1^{(0)},x_2^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=\\
& = & V(x_1^{(0)},z^{(0)})\cup V(x_2^{(0)},x_2^{(1)},z^{(0)})=C_2^{(1,0)}\cup C_2^{(0,2)}\sigmaubset\mathbb A_2^3,\\
\varepsilonnd{array}\]
since $V(x_1^{(0)},x_2^{(0)},z^{(0)})\sigmaubset V(x_1^{(0)},z^{(0)})$. And
\[(\pii_3^{-1}(X_{Sing}))_{red}=V(x_1^{(0)},z^{(0)},{z^{(1)}}^3-{x_1^{(1)}}^3{x_2^{(0)}}^2)\cup V(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)})=C_3^{(1,0)}\cup C_3^{(0,2)}
\sigmaubset\mathbb A_3^3.\]
\lambdaabel{Ex1}
\varepsilonnd{Exam}
In this example we see how the components are defined by hyperplane coordinates for $m<3$, and at level $m=3$ the equation $f$ starts playing a role. When there are more than one approximated root, the approximated roots affect the geometry of $C_m^\nu$ one after the other as $m$ grows. This will be explained in Proposition \rhoef{Cgeom}. We illustrate this with another example.
\betaegin{Exam}
Consider the quasi-ordinary surface $f=(z^2-x_1^3)^3-x_1^{10}x_2^4$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},0)$ and $\gammaamma_2=(\pihirac{10}{3},
\pihirac{4}{3})$, and the singular locus is $X_{Sing}=\{x_1=z=0\}\cup\{x_2=z^2-x_1^3=0\}$.
If we lift the component of the singular locus $Z_2=\{x_2=f_1=0\}$ at level $3$, we have that
$(\pii_3^{-1}(Z_2))_{red}=V(x_2^{(0)},F_1^{(0)},F_2^{(1)},F_2^{(2)},F_2^{(3)})$, where $F_1^{(0)}={z^{(0)}}^2-{x_1^{(0)}}^3$, and
\[\betaegin{array}{ll}
F_2^{(1)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(1)}\mbox{ mod }(x_2^{(0)},z^{(0)})\\
& \varepsilonquiv 0\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)})\\
\\
F_2^{(2)} & \varepsilonquiv 3{F_1^{(0)}}^2F_1^{(2)}+6F_1^{(0)}{F_1^{(1)}}^2\mbox{ mod }(x_2^{(0)},z^{(0)})\\
& \varepsilonquiv 0\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)})\\
\\
F_2^{(3)} & \varepsilonquiv {F_1^{(1)}}^3\mbox{ mod }(x_2^{(0)},z^{(0)},F_1^{(0)}),\\
\varepsilonnd{array}\]
and then $(\pii_3^{-1}(Z_2))_{red}=V(x_2^{(0)},F_1^{(0)},F_1^{(1)})$. Notice that it is not a component of $(\pii_3^{-1}(X_{Sing}))_{red},$ since it is not irreducible. Indeed, it decomposes as
\[(\pii_3^{-1}(Z_2))_{red}=V(x_1^{(0)},x_2^{(0)},z^{(0)})\cup\mathbb overline{V(x_2^{(0)},F_1^{(0)},F_1^{(1)})\cap D(x_1^{(0)})}.\]
\lambdaabel{Ex2}
\varepsilonnd{Exam}
We have seen in this example how, to give a minimal set of generators of $J_m^\nu$, we need to study the polynomials $F^{(l)}\mbox{ mod }J_{l-1}^\nu$.
Therefore we introduce the following definition.
\betaegin{Defi}
For $\nu\in \sigma_{Sing}\cap N_0$ and $0\lambdaeq l\lambdaeq m$, we denote by
\[F_{\nu}^{(l)}:=F^{(l)}\mbox{ mod }(I^\nu,J_{l-1}^\nu),\]
and for the approximated roots the notation is $F_{j,\nu}^{(l)}$.
\lambdaabel{Fnu}
\varepsilonnd{Defi}
And now we obviously have that
\[J_m^\nu=Rad(F_\nu^{(0)},\lambdadots,F_\nu^{(m)}).\]
Regarding the claim in Remark \rhoef{RemTonto}, once we consider $F_\nu^{(l)}$, it is not true anymore that the polynomials are non-zero. But, whenever $F_\nu^{(l)}$ is non-zero, then it is quasi-homogeneous of degree $l$.
In general, the first approximated root which appears is not necessarily the first one, and the process does not finish with the last one.
To control, for a given $\nu$, all this behaviour, we defined the integers $i(\nu),c(\nu)$ and $m(\nu)$ in Definition \rhoef{defs}.
Indeed, given $m\in\mathbb Z_{>0}$ and $\nu\in\sigma_{Sing}\cap N_0$ such that $l_{c(\nu)}(\nu)\lambdaeq m$, the approximated roots which will influence the defining ideal of $D_m^\nu$, are
\[f_{c(\nu)},\lambdadots,f_{j(m,\nu)},\]
where remember the convention $f_0=z$. Moreover, the moment when $f_i$ begins to influence the defining equations of $C_m^\nu$ (or in other words, the generators of $J_m^\nu$) for the first time is exactly at $m=l_i(\nu)$. This is the content of Corollary \rhoef{Corolario3}.
The meaning of the integer $i(\nu)$ is that, at $m=l_{i(\nu)}(\nu)$, $\nu$ does no longer give rise to an irreducible component (see Lemma \rhoef{LemD}).
The integer $j(m,\nu)$ will be useful to describe the component $C_m^\nu$ (see Proposition \rhoef{Cgeom}), while $j'(m,\nu)$ will be crucial when studying the inclusion $C_m^{\nu'}\sigmaubseteq C_m^\nu$ (see Proposition \rhoef{PropC1}).
\betaegin{Exam}
We revisit Example \rhoef{Ex1}. If we lift the component $C_3^{(0,2)}$ to level $4$, we have
\[\pii_{4,3}^{-1}(C_3^{(0,2)})=V(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)},F^{(4)})\]
where $F^{(4)}\varepsilonquiv {x_1^{(0)}}^3{x_2^{(2)}}^2\mbox{ mod }(x_2^{(0)},x_2^{(1)},z^{(0)},z^{(1)})$. Therefore
\[\pii_{4,3}^{-1}(C_3^{(0,2)})=C_4^{(1,2)}\cup C_4^{(0,3)}.\]
Then at level $m=4$, the vector $(0,2)$ does not give rise to an irreducible component any longer. The reason is that $(0,2)\notin N_1$ and $4=l_1(0,2)$.
\lambdaabel{Ex1re}
\varepsilonnd{Exam}
This is the case in general as we claim in the next Lemma, whose proof is left to Section \rhoef{Proofs}.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$, we have
\[D_m^\nu=\varepsilonmptyset\mbox{ if and only if }m\gammaeq l_{i(\nu)}(\nu).\]
\lambdaabel{LemD}
\varepsilonnd{Lem}
As a consequence of this Lemma, we are going to prove an improvement of Lemma \rhoef{LemGamma}, namely, for $m\in\mathbb Z_{>0}$, to cover $\pii_m^{-1}(X_{Sing})$ it is enough to consider $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$ with $m<l_{i(\nu)}(\nu)$.
Notice that, if $l_{m(\nu)}(\nu)\lambdaeq m$, then $l_1(\nu)\lambdaeq l_{m(\nu)}(\nu)\lambdaeq m$, and by the previous Lemma, we have to ask $\nu\in N_1$ whenever $l_{m(\nu)}\lambdaeq m$.
\betaegin{Defi}
Given $m\in\mathbb Z_{> 0}$ we define the sets:
\[\betaegin{array}{l}
H_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0\ |\ l_{m(\nu)}(\nu)\gammaeq m+1\},\\
\\
L_m=\{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_1\ |\ l_{m(\nu)}(\nu)\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
It will be necessary later to subdivide the set $L_m$ as
\[\betaegin{array}{l}
L_m^==\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)\lambdaeq m<\mbox{min}\{l_{m(\nu)}(\nu)+e_{m(\nu)},l_{i(\nu)}(\nu)\}\},\\
\\
L_m^<=\{\nu\in L_m\ |\ l_{m(\nu)}(\nu)+e_{m(\nu)}\lambdaeq m<l_{i(\nu)}(\nu)\}.\\
\varepsilonnd{array}\]
\lambdaabel{DefsSets}
\varepsilonnd{Defi}
If we come back to Example \rhoef{Ex1}, we have that
\[\betaegin{array}{ll}
(\pii_1^{-1}(X_{Sing}))_{red}=C_1^{(1,0)}\cup C_1^{(0,1)} & \mbox{ with }(1,0),(0,1)\in H_1,\\
\\
(\pii_2^{-1}(X_{Sing}))_{red}=C_2^{(1,0)}\cup C_2^{(0,2)} & \mbox{ with }(1,0),(0,2)\in H_2,\\
\\
(\pii_3^{-1}(X_{Sing}))_{red}=C_3^{(1,0)}\cup C_3^{(0,2)} & \mbox{ with }(1,0)\in L_3^=\mbox{ and }(0,2)\in H_3.\\
\varepsilonnd{array}\]
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$, we have that $H_m\cup L_m\neq \varepsilonmptyset$, and
\[\pii_m^{-1}(X_{Sing})=\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu.\]
\lambdaabel{LemTh}
\varepsilonnd{Lem}
{\varepsilonm Proof.} The first claim follows because $(m,m)\in H_m$ for any $m\in\mathbb Z_{>0}$. Indeed, since $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=e_1m(a_1+b_1)>m$, where the last inequality holds because $a_1+b_1>1$, since the branch is normalized.
By Lemma \rhoef{LemGamma}
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu\sigmaubseteq\betaigcup_{\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0}C_m^\nu=\pii_m^{-1}(X_{Sing}).\]
We prove the other inclusion. Notice that $\nu\notin H_m\cup L_m$ is equivalent to $l_{i(\nu)}(\nu)\lambdaeq m$, because $m(\nu)\lambdaeq i(\nu)$. For any $\gammaamma\in\pii_m^{-1}(X_{Sing})$,
$\betaullet$ if $x_i\circ\gammaamma\neq 0$ for $i=1,2$, then $\nu:=(\mbox{ord}_t(x_1\circ\gammaamma),\mbox{ord}_t(x_2\circ\gammaamma))\in[0,m]^2$ and $\gammaamma\in D_m^\nu$. Hence, by Lemma \rhoef{LemD} we have that $m<l_{i(\nu)}(\nu)$ and therefore $\nu\in H_m\cup L_m$.
Otherwise,
$\betaullet$ if $x_i\circ\gammaamma=0$ for $i=1,2$ we saw in the proof of Lemma \rhoef{LemGamma} that $\gammaamma\in C_m^\nu$ for any $\nu\in[0,m]^2\cap\sigma_{Sing}\cap N_0$. Therefore, since $H_m\cap L_m\neq\varepsilonmptyset$, there exists $\nu$ with $\gammaamma\in C_m^\nu$.
$\betaullet$ if $x_1\circ\gammaamma=0$ and $x_2\circ\gammaamma\neq 0$, then by the proof of Lemma \rhoef{LemGamma}, we have that $\gammaamma\in C_m^{(m,\alphalpha)}$, where $\alphalpha=\mbox{ord}_t(x_2\circ\gammaamma)$. We have to prove that $\nu:=(m,\alphalpha)\in H_m$, and this follows since $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=e_1(a_1m+b_1\alphalpha)>m$, again using that the branch is normalized.
$\betaullet$ if $x_1\circ\gammaamma\neq 0$ and $x_2\circ\gammaamma=0$, then by the proof of Lemma \rhoef{LemGamma}, $\gammaamma\in C_m^\nu$ for any $\nu$ with $\nu_1\lambdaeq\alphalpha$ and $\nu_2\lambdaeq m$, where $\alphalpha=\mbox{ord}_t(x_1\circ\gammaamma)$. Hence we only have to prove that $([0,\alphalpha]\tauimes[0,m])\cap(H_m\cup L_m)\neq\varepsilonmptyset$. If $b_1\gammaeq 1$ then $\nu:=(\alphalpha,m)\in H_m$. Indeed, if $b_1>1$ clearly $l_{m(\nu)}(\nu)\gammaeq l_1(\nu)=e_1(a_1\alphalpha+b_1m)>m$. The same works if $b_1=1$ and $g>1$, because then $e_1>1$. If $b_1=1$ and $g=1$, then by Theorem \rhoef{CorSing} we deduce that $\rho_2\nsubseteq\sigma_{Sing}$. Then $\alphalpha>0$ and it follows that $\nu\in H_m$.
The case left is $b_1=0$. In this case, if we set $\nu:=(\alphalpha,m)$, we have $m(\nu)=g_1$ and $i(\nu)\gammaeq g_1+1$. Then $l_{i(\nu)}(\nu)\gammaeq l_{g_1+1}(\nu)=e_{g_1+1}(\alphalpha n_{g_1+1}\gammaamma_{g_1+1}^{(1)}+mn_{g_1+1}\gammaamma_{g_1+1}^{(2)})>m$, where we are using that if $\alphalpha=0$ then $\rho_2\sigmaubseteq\sigma_{Sing}$ and $n_{g_1+1}\gammaamma_{g_1+1}^{(2)}>1$. If $\alphalpha\gammaeq n_1$, the same argument shows that $(n_1,m)\in L_m$. Otherwise $(\alphalpha,m)\in L_m$, since $\gammaamma\in X_m$ and $\gammaamma_1=(\pihirac{a_1}{n_1},0)$, therefore $a_1\mbox{ord}_t(x_1\circ\gammaamma)=n_1\mbox{ord}_t(z\circ\gammaamma)$, which implies that $\alphalpha\pihirac{a_1}{n_1}\in\mathbb Z$ or in other words, $(\alphalpha,m)\in N_1$.
\pisifill $\Box$
Given $\nu\in\sigma_{Sing}\cap N_0$, it gives rise to a candidate of irreducible component at level $m$, $C_m^\nu$, for
\[0=l_{c(\nu)}(\nu)\lambdaeq m<l_{i(\nu)}(\nu).\]
\betaegin{Rem}
\betaegin{enumerate}
\item[(i)] If $\nu\in H_m$, then $c(\nu)=0$, since otherwise $m(\nu)=c(\nu)$ and $l_{m(\nu)}(\nu)=0<m+1$, or in other words, if $g_1>0$ then $H_m\cap\rho_2=\varepsilonmptyset$.
\item[(ii)] It is clear that
$j(m,\nu)=0 \mbox{ if and only if }\nu\in H_m$ and $j(m,\nu)\gammaeq 1 \mbox{ if and only if }\nu\in L_m$.
\varepsilonnd{enumerate}
\lambdaabel{RemDefHL}
\varepsilonnd{Rem}
For $\nu\in H_m$ the sets $C_m^\nu$ are very easy to describe, as we see in the next Proposition.
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m$ we have that
\[J_m^\nu=(z^{(0)},\lambdadots,z^{([m/n])}),\]
and hence the set $C_m^\nu$ is defined by hyperplane coordinates in $\mathbb A_m^3$ as
\[C_m^\nu=V(x_i^{(0)},\lambdadots,x_i^{(\nu_i-1)},i=1,2;z^{(0)},\lambdadots,z^{([\pihirac{m}{n}])}).\]
\lambdaabel{CHm}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
The proof is by induction on $m$. For $m=1$ we have
\[H_1=\lambdaeft\{\betaegin{array}{ll}
\{(1,1)\} & \mbox{ if }\gammaamma_1=(\pihirac{1}{n},\pihirac{1}{n})\mbox{ and }g=1\\
\\
\{(1,0),(1,1)\} & \mbox{ if }g_1>0\mbox{ (recall that the branch is normalized)}\\
\\
\{(1,0),(1,1),(0,1)\} &\mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
and the claim follows, since
\[C_1^{(1,0)}=V(x_1^{(0)},z^{(0)}),\ C_1^{(1,1)}=V(x_1^{(0)},x_2^{(0)},z^{(0)})\mbox{ and }\ C_1^{(0,1)}=V(x_2^{(0)},z^{(0)})\mbox{ if }g_1=0.\]
Suppose the claim is true for $m$ and we will prove it for $m+1$. Given $\nu\in H_{m+1}$, since $\nu\in H_m$, by induction hypothesis we have that
\[J_{m+1}^\nu=(z^{(0)},\lambdadots,z^{([m/n])},F_\nu^{(m+1)}).\]
We claim that
\[F_\nu^{(m+1)}=\lambdaeft\{\betaegin{array}{cl}
0 & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
{z^{(\pihirac{m+1}{n})}}^n & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
which proves the result, since
\[\lambdaeft[\pihirac{m+1}{n}\rhoight]=\lambdaeft\{\betaegin{array}{cl}
[\pihirac{m}{n}] & \mbox{ if }m+1\not\varepsilonquiv 0\mbox{ mod }n\\
\\
\pihirac{m+1}{n} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
If $F_\nu^{(m+1)}\neq 0$ then it is a quasi-homogeneous polynomial of degree $m+1$. By the expansion (\rhoef{grgr1}) given in Lemma \rhoef{LemExpSR}
\[f=z^n+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1>n_1e_1\gammaamma_1=n\gammaamma_1$. Then any monomial in $f-z^n$ verifies for any $\gammaamma\in D_m^\nu$
\[\betaegin{array}{ll}
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\mbox{ord}_t(z\circ\gammaamma)\\
& \gammaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)\rhoangle+r_1\pihirac{m+1}{n}\\
& \gammaeq n\lambdaangle\nu,\gammaamma_1\rhoangle-r_1\lambdaangle\nu,\gammaamma_1\rhoangle+r_1\pihirac{m+1}{n}\\
& =(n-r_1)\pihirac{l_1(\nu)}{n}+r_1\pihirac{m+1}{n}>(n-r_1)\pihirac{m+1}{n}+r_1\pihirac{m+1}{n}=m+1\\
\varepsilonnd{array}\]
because $\nu\in H_{m+1}$ and by induction hypothesis ord$_t(z\circ\gammaamma)>[\pihirac{m}{n}]$ (and therefore $\gammaeq\pihirac{m+1}{n}$). Hence these monomials do not contribute to $F_\nu^{(m+1)}$ and the result follows by the quasi-homogeneity of $F_\nu^{(m+1)}$.\pisifill$\Box$
For $\nu\in L_m$ the geometry of $C_m^\nu$ is much more complicated, the ideal $J_m^\nu$ is described in Corollary \rhoef{Corolario3}.
In the next Proposition we compare jet schemes of a quasi-ordinary singularity with jet schemes of its approximated roots.
For $1\lambdaeq i\lambdaeq g$, $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we denote by $D_{i,m}^\nu$ the set
\[D_{i,m}^\nu=\{\gammaamma\in X_m^{(i)}\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k, k=1,2\}_{red},\]
and we denote $D_{g,m}^\nu$ simply by $D_m^\nu$ (see
Definition \rhoef{SemiX} for the definition of $X^{(i)}$).
\betaegin{Pro}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we have that
\[D_m^\nu=(\pii_{m,[\pihirac{m}{e_j}]}^a)^{-1}(D_{j,[\pihirac{m}{e_j}]}^\nu)\]
where $j=j(m,\nu)$, and for $q>p$, $\pii_{q,p}^a:\mathbb A^3_q\lambdaongrightarrow\mathbb A^3_p$ is the projection on the jet schemes of the affine ambient space.
\lambdaabel{Cgeom}
\varepsilonnd{Pro}
Hence, for $m\in\mathbb Z_{>0}$ and $\nu\in L_m$ with $j(m,\nu)=j$, the geometry of $C_m^\nu$ is determined
by the geometry of the $j$-th approximated root.
Before proving the Proposition we need the following technical result, whose proof is moved to Section \rhoef{Proofs}.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, we have that for all $\gammaamma\in D_m^\nu$,
\[\betaegin{array}{l}
\mbox{ord}_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle,\ \mbox{ for }0\lambdaeq k\lambdaeq j(m,\nu)-1\\
\\
\mbox{ord}_t(f_k\circ\gammaamma)>\pihirac{m}{e_k},\ \mbox{ for }j(m,\nu)\lambdaeq k\lambdaeq g\\
\varepsilonnd{array}\]
\lambdaabel{Lemfk}
\varepsilonnd{Lem}
{\betaf\varepsilonm Proof of Proposition \rhoef{Cgeom}.} For $\nu\in H_m$ we have $j(m,\nu)=0$, and the claim follows by Proposition \rhoef{CHm}. For $\nu\in L_m$ it is enough to prove that, if $j(m,\nu)=j$ we have
\betaegin{equation}
D_m^\nu=\lambdaeft\{\gammaamma\in\mathbb A_m^3\ |\ \mbox{ord}_t(x_i\circ\gammaamma)=\nu_i, i=1,2\mbox{ and }\mbox{ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}\rhoight\}.
\lambdaabel{Claim2}
\varepsilonnd{equation}
By Lemma \rhoef{Lemfk} it follows that
\[D_m^\nu\sigmaubseteq\lambdaeft\{\gammaamma\in\mathbb A_m^3\ |\ \mbox{ord}_t(x_i\circ\gammaamma)=\nu_i,i=1,2\mbox{ and ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}\rhoight\}\]
We prove the other inclusion. Let $\gammaamma$ be a jet with ord$_t(x_i\circ\gammaamma)=\nu_i$ for $i=1,2$ and ord$_t(f_j\circ\gammaamma)>\pihirac{m}{e_j}$. We want to prove that it is indeed an $m$-jet in $X$, or in other words, that ord$_t(f\circ\gammaamma)\gammaeq m+1$. Notice that if $j=g$ there is nothing to prove.
Then $j<g$, and first we will prove that
\[\mbox{ord}_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}.\]
Indeed, consider $f_j$ quasi-ordinary surface with $j$ characteristic exponents. If $\betaar m:=[\pihirac{m}{e_j}]$ and $\betaar\gammaamma=\pii_{m,\betaar m}(\gammaamma)$, then we have that $\betaar\gammaamma\in D_{j,\betaar m}^\nu$. Moreover $n_j\lambdaangle\nu,\gammaamma_j\rhoangle\lambdaeq \betaar m$, and then, by Lemma \rhoef{Lemfk} applied to $f_j$, we have
\[\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle,\ \mbox{ for }0\lambdaeq k\lambdaeq j-1.\]
Since $\lambdaangle\nu,\gammaamma_{k+1}\rhoangle\lambdaeq\lambdaangle\nu,\gammaamma_j\rhoangle<\betaar m<m$, we deduce that ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)$.
Now we consider $f_{j+1}$. By Lemma \rhoef{Lema35} we have
\[f_{j+1}=f_j^{n_{j+1}}-c_{j+1}x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}}\]
and using that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k\lambdaeq j-1$ we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_j^{n_{j+1}}\circ\gammaamma) & =n_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)>\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((x_1^{\alphalpha_1^{(j+1)}}x_2^{\alphalpha_2^{(j+1)}}z^{r_1^{(j+1)}}\cdots f_{j-1}^{r_j^{(j+1)}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1^{(j+1)},\alphalpha_2^{(j+1)})+r_1^{(j+1)}\gammaamma_1+\cdots+r_j^{(j+1)}\gammaamma_j\rhoangle\\
& =n_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle >\pihirac{m}{e_{j+1}}\\
\\
\mbox{ord}_t((c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_j^{r_{j+1}})\circ\gammaamma) & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\varepsilonnd{array}\]
If ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$, then there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
\[\mbox{ord}_t(f_{j+1}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}\]
and we get the following inequalities
\[\betaegin{array}{rl}
(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j} & \lambdaeq\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\pihirac{m}{e_j}\\
\\
& <\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_j\gammaamma_j\rhoangle+r_{j+1}\mbox{ord}_t(f_j\circ\gammaamma)\\
\\
& \lambdaeq\pihirac{m}{e_{j+1}}\\
\varepsilonnd{array}\]
Hence $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle+r_{j+1}\pihirac{m}{e_j}<\pihirac{m}{e_{j+1}}=n_{j+1}\pihirac{m}{e_j}$, and then $(n_{j+1}-r_{j+1})\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<(n_{j+1}-r_{j+1})\pihirac{m}{e_j}$. Since $r_{j+1}<n_{j+1}$, we have
\[\lambdaangle\nu,\gammaamma_{j+1}\rhoangle<\pihirac{m}{e_j}\]
which is in contradiction with $j(m,\nu)=j$. Therefore we have just proved that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$.
To finish, by Lemma \rhoef{LemExpSR} we have
\[f=f_{j+1}^{e_{j+1}}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_{j+1}e_{j+1}\gammaamma_{j+1}} c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k,\]
and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{j+1}^{e_{j+1}}\circ\gammaamma) & =e_{j+1}\mbox{ord}_t(f_{j+1}\circ\gammaamma)>m\\
\\
\mbox{ord}_t((c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k)\circ\gammaamma) & \gammaeq n_{j+1}e_{j+1}\lambdaangle\nu,\gammaamma_{j+1}\rhoangle=l_{j+1}(\nu)>m\\
\varepsilonnd{array}\]
Hence ord$_t(f\circ\gammaamma)>m$ as we wanted to prove. \pisifill$\Box$
As a consequence of Proposition \rhoef{Cgeom}, we have the following algebraic counterpart, where we explain how the equations of the approximated roots appear as generators of $J_m^\nu$, and therefore a minimal presentation of the ideal $J_m^\nu$ is given.
\betaegin{Cor} Given $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$, for $l_i(\nu)\lambdaeq l<l_{i+1}(\nu)$ (resp. $l_i(\nu)\lambdaeq l\lambdaeq m$) if $c(\nu)\lambdaeq i< j(m,\nu)$ (resp. $i=j(m,\nu)$), we have that
\[F_\nu^{(l)}=\lambdaeft\{\betaegin{array}{cl}
{F_{i,\nu}^{(\pihirac{l}{e_i})}}^{e_i} & \mbox{ if }l\varepsilonquiv 0\mbox{ mod }e_i\\
\\
0 & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
Hence the ideal $J_m^\nu$ is generated by the polynomials
\[J_m^\nu=\lambdaeft(F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+k_i)}\rhoight)_i\]
for $c(\nu)\lambdaeq i\lambdaeq j(m,\nu)$ such that $l_i(\nu)<l_{i+1}(\nu)$, and $0\lambdaeq k_i<\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}\mbox{ if }i<j(m,\nu)$ and $0\lambdaeq k_{j(m,\nu)}\lambdaeq
[\pihirac{m-l_{j(m,\nu)}(\nu)}{e_{j(m,\nu)}}]$. Moreover, we have
\betaegin{equation}
F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+G_{i,\nu},
\lambdaabel{EqP}
\varepsilonnd{equation}
where $G_{i,\nu}$ is the polynomial
\[G_{i,\nu}=\sigmaum c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}{F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{r_i}\]
with $c_{\underline{\alphalpha},\underline{r}}$ are the coefficients appearing in the expansion given in Lemma \rhoef{Lema35}, and such that
\betaegin{enumerate}
\item[(i)] $c_{\underline{\alphalpha},\underline{r}}\neq 0$
\item[(ii)] $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$
\varepsilonnd{enumerate}
Notice that condition (ii) does not hold when $\nu\notin\rho_1\cup\rho_2$, and hence $G_{i,\nu}=0$ in these cases.
\lambdaabel{Corolario3}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It is a consequence of Proposition \rhoef{Cgeom}, applied to any $l_i(\nu)\lambdaeq l<l_{i+1}(\nu)$, and using the trivial observation that $J_{m'}^\nu\sigmaubseteq J_m^\nu$ for any $m'<m$. Indeed, for any $l_i(\nu)\lambdaeq l< l_{i+1}(\nu)$ we study the polynomials $F_\nu^{(l_i(\nu))},\lambdadots,F_\nu^{(l)}$ (note that we need $l_i(\nu)<l_{i+1}(\nu)$). We have that $j(l,\nu)=i$, and, by Proposition \rhoef{Cgeom}, $D_l^\nu=(\pii_{l,[\pihirac{l}{e_i}]}^a)^{-1}(D_{i,[\pihirac{l}{e_i}]}^\nu)$, or in other words
\[D_l^\nu=\lambdaeft\{\gammaamma\in\mathbb A_l^3\ |\ \mbox{ord}_t(x_k\circ\gammaamma)=\nu_k,\ k=1,2,\mbox{ ord}_t(f_i\circ\gammaamma)>\lambdaeft[\pihirac{l}{e_i}\rhoight]\rhoight\}\]
Then the ideal $J_l^\nu$ only depends on $f_i$ (and hence on its approximated roots). Moreover, by Lemma \rhoef{Lemfk}, we deduce that for $0\lambdaeq k<i$
\[F_{k,\nu}^{(r_k)}\in J_l^\nu,\ \mbox{ for }0\lambdaeq r_k<\lambdaangle\nu,\gammaamma_{k+1}\rhoangle.\]
By the expansion (\rhoef{grgr1}) in Lemma \rhoef{LemExpSR}
\[f=f_i^{e_i}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$.
The part $f_i^{e_i}$ contributes to $F^{(l)}$ with ${F_{i,\nu}^{(\pihirac{l}{e_i})}}^{e_i}$ and only when $l$ is divisible by $e_i$. While for the {\varepsilonm monomials}
$x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}$ the contribution is given by
\[{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots {F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{r_i}{F_{i,\nu}^{(\pihirac{a}{r_{i+1}})}}^{r_{i+1}}\]
with $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+a=l$, and only when $a$ is divisible by $r_{i+1}$. Set $a=br_{i+1}$, we claim that $F_{i,\nu}^{(b)}$ belongs to $J_l^\nu$ and hence the {\varepsilonm monomial} does not contribute to $F^{(l)}$. Indeed, at level $l'=be_i$ it appears as ${F_{i,\nu}^{(b)}}^{e_i}$, and we only need to prove that $l'<l$. Since
\[\betaegin{array}{ll}
l & =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}b\\
& =\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle+r_{i+1}\pihirac{l'}{e_i}\\
\varepsilonnd{array}\]
we have
\[\betaegin{array}{ll}
l' & =\pihirac{e_i}{r_{i+1}}(l-\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle)\\
& \lambdaeq\pihirac{e_i}{r_{i+1}}(l-l_{i+1}(\nu)+r_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle)\\
& =\pihirac{e_i}{r_{i+1}}l-(\pihirac{e_i}{r_{i+1}}-1)l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Therefore $r_{i+1}l'\lambdaeq e_il-(e_i-r_{i+1})l_{i+1}(\nu)$. Suppose that $l'\gammaeq l$, then
\[(e_i-r_{i+1})l_{i+1}(\nu)\lambdaeq e_il-r_{i+1}l'\lambdaeq (e_i-r_{i+1})l\]
and since $e_i>r_{i+1}$ it contradicts the fact that $j(l,\nu)=i$.
Now equation (\rhoef{EqP}) follows by Lemma \rhoef{Lema35}.
\pisifill$\Box$
From Corollary \rhoef{Corolario3} we deduce the following:
\betaegin{equation}
D_m^\nu\sigmaubset D(F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}\cdots F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}),\ \mbox{ for any }\nu\in L_m\mbox{ with }j(m,\nu)=j.
\lambdaabel{eqO}
\varepsilonnd{equation}
To illustrate the description of $J_m^\nu$ given in Corollary \rhoef{Corolario3}, we consider some particular cases.
$\betaullet$ First, the simplest case, when $\nu\notin\rho_1\cup\rho_2$ we have $G_{j,\nu}=0$ for any $j$, and $m(\nu)=1$. Hence (note that we also have $c(\nu)=0$):
\betaegin{equation}
\betaegin{array}{lcl}
F_{0,\nu}^{(r)} & = & z^{(r)}, \mbox{ for }0\lambdaeq r<\pihirac{l_{1}(\nu)}{e_0}\\
\\
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})} & = & {z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}\\
\\
F_1^{(\pihirac{l_1(\nu)}{e_1}+r)} & &\mbox{ for }1\lambdaeq r<\pihirac{l_2(\nu)-l_1(\nu)}{e_1}\\
\ \ \ \ \ \vdots & \vdots & \\
F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g-1}})} & = & {F_{g_1-2,\nu}}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-2}})}-{x_1^{(\nu_1)}}^{n_{g_1-1}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1-1)}}\cdots
{F_{g_1-3,\nu}^{(\pihirac{l_{g_1-2}(\nu)}{e_{g_1-3}})}}^{r_{g_1-2}^{(g_1-1)}}\\
\\
F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g-1}}+r)} & & \mbox{ for }1\lambdaeq r<\pihirac{l_{g_1}(\nu)-l_{g_1-1}(\nu)}{e_{g_1-1}}\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}})} & = & {F_{g_1-1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{n_{g_1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1)}}{z^{(\lambdaangle\nu,
\gammaamma_1\rhoangle)}}^{r_1^{(g_1)}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}^{r_2^{(g_1)}} \cdots {F_{g_1-2,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-2}})}}^{r_{g_1-1}^{(g_1)}}\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+r)} & &\mbox{ for }1\lambdaeq r<\pihirac{l_{g_1+1}(\nu)-l_{g_1}(\nu)}{e_{g_1}}\\
\\
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & = & {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}
{x_2^{(\nu_2)}}^{\alphalpha_2^{(g_1+1)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}\\
\ \ \ \ \ \vdots & \vdots &\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j})} & = & {F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}}^{n_j}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(j)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(j)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(j)}}\cdots {F_{j-2,\nu}^{(\pihirac{l_{j-1}(\nu)}{e_{j-2}})}}^{r_{j-1}^{(j)}}\\
\\
F_{j,\nu}^{(\pihirac{l_j(\nu)}{e_j}+r)} & & \mbox{ for }1\lambdaeq r\lambdaeq[\pihirac{m-l_j(\nu)}{e_j}]\\
\varepsilonnd{array}
\lambdaabel{ecsFs}
\varepsilonnd{equation}
Notice that the variable $x_2^{(\nu_2)}$ appears for the first time in the equation $F_{g_1+1}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}$, and raised to the power one or bigger depending on whether $g_2=g_1+1$ or $g_2=g_1$ respectively.
$\betaullet$ Another example is when $j(m,\nu)=c(\nu)$, then
\[J_m^\nu=\lambdaeft(F_{c(\nu),\nu}^{(0)},\lambdadots,F_{c(\nu),\nu}^{([\pihirac{m}{e_{c(\nu)}}])}\rhoight)\]
and if moreover $c(\nu)=0$ this is just the content of Proposition \rhoef{CHm}. Notice that the description of $J_m^\nu$ for $\nu\in H_m$ given in Proposition \rhoef{CHm} is contained in Corollary \rhoef{Corolario3}, but we wanted to stress the fact that for $\nu\in H_m$ the description is particularly simple.
Now we can prove the irreducibility of the sets $C_m^\nu$.
\betaegin{Pro}
For any $m\in\mathbb Z_{> 0}$ and $\nu\in H_m\cup L_m$, the set $C_m^\nu$ is irreducible and has codimension
\[\nu_1+\nu_2+\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}+\lambdaeft[\pihirac{m-l_{j(m,\nu)}(\nu)}{e_{j(m,\nu)}}\rhoight]+1\]
\lambdaabel{Prop1}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
To simplify notation we will denote along this proof $j(m,\nu)$ just by $j$ and by $k_i(\nu)$, or simply by $k_i$ when $\nu$ is clear from the context, we denote
the quotient $\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}$.
First notice that, by definition of $c(\nu)$, $\sigmaum_{k=0}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}=\sigmaum_{k=c(\nu)}^{j-1}\pihirac{l_{k+1}(\nu)-l_k(\nu)}{e_k}$.
If $\nu\in H_m$ we have $j(m,\nu)=0$ and the claim about the codimension follows by Proposition \rhoef{CHm}. The closed set $C_m^\nu$ is
irreducible since it is defined by hyperplane coordinates.
If $\nu\in L_m^=$, then by Lemma \rhoef{LemCm}, we have that $C_m^\nu=V(I^\nu,J_m^\nu)$, where the ideal $J_m^\nu$ is described in equation (\rhoef{eqJ}). Then
Codim$(C_m^\nu)=\nu_1+\nu_2+\pihirac{l_1(\nu)}{n}+1$. Notice that if $c(\nu)>0$ then $l_1(\nu)=0$. The
irreducibility of $C_m^\nu$ follows from the irreducibility of $F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}$.
Now let $\nu$ be an element in $L_m^<$. We have to study carefully the generators of $J_m^\nu$ given in Corollary \rhoef{Corolario3}. Any $F_{i,\nu}^{(l)}$ is quasi-homogeneous of degree $l$, but the second property described in Remark \rhoef{RemTonto} is not true anymore once we consider the equations modulo $I^\nu$. We need to know when a certain variable $x_k^{(l)}$ or $z^{(l)}$ appear for the first time in the generators of $J_m^\nu$. Notice that for any $\gammaamma\in D_m^\nu$ we have ord$_t(x_i\circ\gammaamma)=\nu_i$ for $i=1,2$, and ord$_t(z\circ\gammaamma)=\lambdaangle\nu,\gammaamma_1\rhoangle$. It is clear that the variables $x_k^{(\nu_k)}$ for $k=1,2$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$ appear for the first time in the first non-monomial equation $F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}$. In the next equation appear $x_k^{(\nu_k+1)}$, $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}$, and so on. Then looking at the generators of $J_m^\nu$ described in Corollary \rhoef{Corolario3}, we deduce that, for $m(\nu)\lambdaeq i<j$ and $0\lambdaeq r<k_i$, or $0\lambdaeq r<[\pihirac{m-l_j(\nu)}{e_j}]$ when $i=j$,
\[(\alphast)\ \ \ \ \ \ \ \mbox{ the variables }x_k^{(\nu_k+k_{m(\nu)}+\cdots+k_{i-1}+r)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k_{m(\nu)}+\cdots+k_{i-1}+r)}\mbox{ appear for the first time in }F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\]
Notice that for $1\lambdaeq l<m(\nu)$ we have $k_l=0$.
We divide the set of generators of $J_m^\nu$, given in Corollary \rhoef{Corolario3}, in two sets:
\[\betaegin{array}{l}
\mathcal C_1=\lambdaeft\{F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}\rhoight\}_{c(\nu)\lambdaeq i\lambdaeq j,\ l_i(\nu)<l_{i+1}(\nu)}\\
\\
\mathcal C_2=\lambdaeft\{F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\rhoight\}_{(i,r)\in A_2}\\
\varepsilonnd{array}\]
where $A_2=\{(i,r)\ |\ c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu),\ 0<r<k_i\}\cup\{(j,r)\ |\ 0<r<[\pihirac{m-l_j(\nu)}{e_j}]\}$.
We claim:
\betaegin{enumerate}
\item[(i)] $V(\mathcal C_1)\sigmaimeq Z^{\Gamma_m^\nu}$, the toric variety defined by the semigroup $\Gamma_m^\nu$ generated by
\[\{\gammaamma_i\}_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\]
If $\nu\notin\rho_1\cup\rho_2$ then $\Gamma_m^\nu=\Gamma_{j(m,\nu)}$ and $V(\mathcal C_1)$ is isomorphic to the monomial variety associated to $X^{(j(m,\nu))}$ (see Definition \rhoef{defMV}).
\
\item[(ii)] any $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}\in\mathcal C_2$ is linear over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$ with respect to one of the variables described in ($\alphast$), which appears for the first time on this equation.
\varepsilonnd{enumerate}
Since any of these equations in $\mathcal C_2$ is linear in a different variable, and, by ($\alphast$) we have that it appears for the first time in $\mathcal C_2$, we deduce
\[V(I^\nu,\mathcal C_2)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaimeq\mathbb A^{\alphalpha(m,\nu)}\]
where $\alphalpha(m,\nu)=3(m+1)-\nu_1-\nu_2-\sigmaum_{i=c(\nu),...,j-1,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)-[\pihirac{m-l_j(\nu)}{e_j}]$, because $V(I^\nu,\mathcal C_2)\sigmaubseteq\mathbb A_m^3\sigmaimeq\mathbb A^{3(m+1)}$. Hence
\[D_m^\nu\sigmaimeq\lambdaeft(Z^{\Gamma_m^\nu}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\rhoight)\tauimes\mathbb A^{\alphalpha(m,\nu)}\]
The toric variety $Z^{\Gamma_m^\nu}$ is irreducible and hence the irreducibility of $C_m^\nu$ follows by the previous isomorphism. Moreover $Z^{\Gamma_m^\nu}$ is complete intersection, hence the codimension equals the number of defining equations, which is the cardinal of $\mathcal C_1$.
Therefore
\[\betaegin{array}{ll}
\mbox{Codim}(C_m^\nu) & =\sigmaharp\mathcal C_1+\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}(k_i-1)+[\pihirac{m-l_j(\nu)}{e_j}]\\
\\
& =\nu_1+\nu_2+\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i+[\pihirac{m-l_j(\nu)}{e_j}]+\sigmaharp\mathcal C_1-\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}\\
\varepsilonnd{array}\]
Finally the statement about the codimension follows now by these two remarks:
$\betaullet$ $\sigmaum_{c(\nu)\lambdaeq i<j,\ l_i(\nu)<l_{i+1}(\nu)}k_i=\sigmaum_{i=c(\nu)}^{j-1}k_i$, since $k_i=0$ whenever $l_i(\nu)=l_{i+1}(\nu)$.
$\betaullet$ $l_j(\nu)<l_{j+1}(\nu)$ by definition of $j(m,\nu)$, and therefore
\[\sigmaharp\{c(\nu)\lambdaeq i<j\ |\ l_i(\nu)<l_{i+1}(\nu)\}=\sigmaharp\mathcal C_1-1.\]
Now we prove the claim. To prove (i), notice that we can write equation (\rhoef{EqP}) as
\[F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}={F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i}-c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\cdot U\]
where, arguing as in the proof of Lemma \rhoef{TechLem} (i), we have $U\neq 0$. Then we have the isomorphism
\[V(F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\sigmaimeq V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}\]
where $h_i=w_i^{n_i}-x_1^{\alphalpha_1^{(i)}}
x_2^{\alphalpha_2^{(i)}}z^{r_1^{(i)}}w_1^{r_2^{(i)}}\cdots w_{i-2}^{r_{i-1}^{(i)}}$, with the relation $n_i\gammaamma_i=(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1^{(i)}\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}$. And $V(h_i)_{c(\nu)\lambdaeq i\lambdaeq j(m,\nu),\ l_i(\nu)<l_{i+1}(\nu)}$ is isomorphic to the toric variety $Z^{\Gamma_m^\nu}$.
To prove the claim (ii), we distinguish three cases, depending on $m$.
(a) For $m<l_{g_1+1}(\nu)+e_{g_1+1}$. In this case we have that $g_1>0$ and therefore $m(\nu)$ is either $g_1$ or $1$.
Suppose first that $m(\nu)=1$. Then
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})}= {z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}+G_{1,\nu},\]
where $G_{1,\nu}$ is the polynomial $\sigmaum c_{i_1i_2k}{x_1^{(\nu_1)}}^{i_1}{x_2^{(\nu_2)}}^{i_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k$ with $\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle=n_1\lambdaangle\nu,\gammaamma_1\rhoangle=a_1\nu_1$. Moreover, for $0<r<k_1$, $F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}$ is a quasi-homogeneous polynomial of degree $\pihirac{l_1(\nu)}{e_1}+r=a_1\nu_1+r$, and using ($\alphast$),
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}={x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+r)}\cdot U_1^{(r)}+H_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}\]
where $H_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}$ is a polynomial in which $x_1^{(\nu_1+r)}$ do not appear, and $U_1^{(r)}\neq 0$. Analogously, for $1<i<j(m,\nu)$ and $0<r<k_i(\nu)$ we have
\[F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}= n_i{F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}}^{n_i-1}F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}}+r)}+H_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)},\]
where $H_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}$ is a polynomial in which the variable $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ does not appear. Moreover
\[F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}}+r)}= a_1{x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}+H_{i-1,\nu}^{(r)},\]
where the variable $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ does not appear in the polynomial $H_{i-1,\nu}^{(r)}$. Then,
by Lemma \rhoef{TechLem} (ii) it follows that for $1=m(\nu)\lambdaeq i<j(m,\nu)$ and $0<r<k_i(\nu)$ the equation
$F_i^{(\pihirac{l_i(\nu)}{e_i}+r)}$ is linear on $x_1^{(\nu_1+k_1+\cdots+k_{i-1}+r)}$ over $D(x_1^{(\nu_1)})$.
We still have to deal with the equations $F_j^{(\pihirac{l_j(\nu)}{e_j}+r)}$ with $1\lambdaeq r\lambdaeq [\pihirac{m-l_j(\nu)}{e_j}]$. Notice that $j(m,\nu)\lambdaeq g_1+1$. If $j(m,
\nu)<g_1+1$ then the argument is exactly as before, and if $j(m,\nu)=g_1+1$ then $\lambdaeft[\pihirac{m-l_j(\nu)}{e_j}\rhoight]=0$.
Suppose now that $m(\nu)=g_1>1$. Then $\nu=(0,\nu_2)$ and the generators of $J_m^\nu$ are
\[\betaegin{array}{ll}
F_{g_1}^{(r)} \mbox{ for }0\lambdae r\lambdaeq[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}] & \mbox{ if }j(m,\nu)=g_1\\
\\
F_{g_1}^{(r)}\mbox{ for }0\lambdaeq r<k_{g_1}(\nu)\mbox{ and }F_{g_1+1}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & \mbox{ if }j(m,\nu)=g_1+1\\
\varepsilonnd{array}\]
For $r>0$ we have that $F_{g_1}^{(r)}\varepsilonquiv \pihirac{\piartial F_{g_1}}{\piartial x_1}(x_1^{(0)},z^{(0)})x_1^{(r)}+\pihirac{\piartial F_{g_1}}{\piartial z}(x_1^{(0)},
z^{(0)})z^{(0)}+H_r\mbox{ mod }I^\nu$, with $H_r$ a polynomial where the variables $x_1^{(r)}$ and $z^{(r)}$ do not appear. Since we are looking at $\{x_1^{(0)}
\neq 0\}$, we are outside the singular locus, and we deduce that these equations are linear either in $x_1^{(r)}$ or in $z^{(r)}$. The rest of the proof follows as in the previous part of this case.
(b) For $l_{g_1+1}(\nu)+e_{g_1+1}\lambdaeq m<l_{g_2+1}(\nu)+e_{g_2+1}$. This is only possible when $g_2=g_1+1$.
Notice that if $j(m,\nu)=g_2+1$ then $\lambdaeft[\pihirac{m-l_j(\nu)}{e_j}\rhoight]=0$. We just have to study the generators $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r)}$ for $i>g_1$
(which are all the generators of $J_m^\nu$ when $g_1=0$) since the others were studied in the previous case. That is, we have the equations $F_{g_2,\nu}^{(\pihirac{l_{g_2}
(\nu)}{e_{g_2}}+r)}$ for $0\lambdaeq r\lambdaeq k_{g_2}(\nu)-1$ if $j(m,\nu)=g_2+1$ and $0\lambdaeq r\lambdaeq\lambdaeft[\pihirac{m-l_{g_2}(\nu)}{e_{g_2}}\rhoight]$ otherwise. If $j(m,\nu)=g_2+1$
we also have the generator $F_{g_2+1,\nu}^{(\pihirac{l_{g_2+1}(\nu)}{e_{g_2+1}})}$.
\[\betaegin{array}{lcl}
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}})} & = & {F_{g_1}^{(\pihirac{l_{g_2}(\nu)}{e_{g_1}})}}^{n_{g_2}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2)}{z^{(\lambdaangle
\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots {F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}+G_{g_2,\nu}\\
\\
F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)} & = & {x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2+r)}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_2)}}\cdots
{F_{g_1-1}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_2)}}\cdot U_{g_2}^{(r)}+H_r\\
\varepsilonnd{array}\]
where $U_{g_2}^{(r)}$ is $1$ if $\nu\notin\rho_1\cup\rho_2$ and $\neq 0$ otherwise, and $H_r$ is a polynomial in which the variable $x_2^{(\nu_2+r)}$ does not appear. Then by Lemma \rhoef{TechLem} (ii) we deduce that every $F_{g_2,\nu}^
{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)}$ is linear on $x_2^{(\nu_2+r)}$ over $D(x_1^{(\nu_1)})$. And the rest of the argument goes as in the previous case.
(c) For $l_{g_2+1}(\nu)+e_{g_2+1}\lambdaeq m$, with the same arguments it is easy to see that for $i>g_2$ and $1\lambdaeq r<k_i(\nu)$ each $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+
r)}$ is linear on $x_1^{(\nu_1+k_1+\cdots+k_i+r)}$ over $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$.
\pisifill$\Box$
In particular we have the following variation of the codimension of $C_m^\nu$ as $m$ grows.
\betaegin{Cor}
For $\nu\in H_m\cup L_m$ such that $\nu\in H_{m-1}\cup L_{m-1}$ we have that
\[\mbox{Codim}(C_m^\nu)=\lambdaeft\{\betaegin{array}{ll}
\mbox{Codim}(C_{m-1}^\nu)+1 & \mbox{ if }m\varepsilonquiv 0\mbox{ mod }e_{j(m-1,\nu)}\\
\\
\mbox{Codim}(C_{m-1}^\nu) & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
\lambdaabel{CorCodim}
\varepsilonnd{Cor}
\sigmaubsection{Inclusions among the $C_m^\nu$}
We have a collection of irreducible sets $\{C_m^\nu\ |\ \nu\in H_m\cup L_m\}$ covering $\pii_m^{-1}(X_{Sing})$, but in general it is not its decomposition in irreducible components. We have to study the inclusions
\betaegin{equation}
C_m^{\nu'}\sigmaubseteq C_m^\nu\mbox{ for different }\nu,\nu'\in H_m\cup L_m.
\lambdaabel{ContGen}
\varepsilonnd{equation}
We need to see $C_m^\nu$ as the closure of a set, which is slightly different from $D_m^\nu$, though described by the ideals $I^\nu$ and $J_m^\nu$. For instance, when $j(m,\nu)=c(\nu)=g_1>0$, by Corollary \rhoef{Corolario3} $J_m^\nu=(F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])})$, and
\[C_m^\nu=\mathbb overline{V(I^\nu,F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])})\cap D(x_1^{(\nu_1)})},\]
because the polynomials $F_{g_1,\nu}^{(0)},\lambdadots,F_{g_1,\nu}^{([\pihirac{m}{e_{g_1}}])}$ do not depend on $x_2^{(\nu_2)}$, and hence, when taking the Zariski closure, we can drop the condition $D(x_2^{(\nu_2)})$ in the description of $D_m^\nu$ given in (\rhoef{eqast}). This is the description we are looking for, and it is the content of the next Lemma.
\betaegin{Lem}
For $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ we have that $C_m^\nu=\mathbb overline{O_m^\nu}$, where
\[O_m^\nu:=\lambdaeft\{\betaegin{array}{cl}
V(I^\nu,J_m^\nu) & \mbox{ if }j'(m,\nu)<m(\nu)\\
\\
V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)}) & \mbox{ if }m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_2\\
\\
V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)}) & \mbox{ if }j'(m,\nu)\gammaeq g_2+1\\
\varepsilonnd{array}\rhoight.\]
Notice that when $j'(m,\nu)<m(\nu)$ then $O_m^\nu=C_m^\nu$, and if $j'(m,\nu)\gammaeq g_2+1$ then $O_m^\nu=D_m^\nu$.
\lambdaabel{LemCm}
\varepsilonnd{Lem}
{\varepsilonm Proof.} If $j'(m,\nu)<m(\nu)$ we have two possibilities regarding $j(m,\nu)$, either $j(m,\nu)<m(\nu)$ or $j(m,\nu)=m(\nu)$. Suppose that $j(m,\nu)<m(\nu)$, then $j(m,\nu)=0$, since $l_{j(m,\nu)}(\nu)<l_{j(m,\nu)+1}(\nu)$ and $l_1(\nu)=\cdots=l_{m(\nu)}(\nu)<l_{m(\nu)+1}(\nu)$. Then $\nu\in H_m$, and by Proposition \rhoef{CHm} we have that $C_m^\nu=V(I^\nu,J_m^\nu)$ where $J_m^\nu=(z^{(0)},\lambdadots,z^{([\pihirac{m}{n}])})$. If $j(m,\nu)=m(\nu)$ we have that $J_m^\nu$
is the ideal
\betaegin{equation}
J_m^\nu=\lambdaeft\{\betaegin{array}{cl}
\lambdaeft(F_{g_1,\nu}^{(0)}\rhoight) & \mbox{ if }c(\nu)=g_1>0\\
\\
\lambdaeft(z^{(0)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle-1)},F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})}\rhoight) & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.
\lambdaabel{eqJ}
\varepsilonnd{equation}
Therefore the conditions $D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})$ disappear when taking the Zariski closure.
Suppose now that $m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_2$. We prove that
\betaegin{equation}
\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})}=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})},
\lambdaabel{igD}
\varepsilonnd{equation}
or in other words, the open condition $x_2^{(\nu_2)}\neq 0$ is superfluous when taking the Zariski closure. This claim is obvious for $m(\nu)\lambdaeq j'(m,\nu)\lambdaeq g_1$, since $x_2^{(\nu_2)}$ appears in the generators of $J_m^\nu$ at most once (in $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}$ if
$l_{g_1+1}(\nu)\lambdaeq m<l_{g_1+1}(\nu)+e_{g_1+1}$). For $g_1<j'(m,\nu)\lambdaeq g_2$, we are necessarily in the case $g_2=g_1+1$ and we prove the
equality (\rhoef{igD}) by induction on $m$. For $m=l_{g_2}(\nu)+e_{g_2}$, if the equality (\rhoef{igD}) does not hold, then
\[C:=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}}\nsubseteq C_m^\nu.\]
Notice that by Corollary \rhoef{Corolario3}
\[F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}})}={F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}}^{n_{g_2}}-c_{g_2}{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2)}
\cdots {F_{g_2-2,\nu}^{(\pihirac{l_{g_2-1}(\nu)}{e_{g_2-2}})}}^{r_{g_2-1}^{(g_2)}}+G_{g_2,\nu}\]
and by Lemma \rhoef{TechLem} (i) we deduce that, if $x_2^{(\nu_2)}=0$ then $F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}$. Moreover,
since the polynomials $F_\nu^{(l)}$ are quasi-homogeneous
\[F_{g_2,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+1)}\varepsilonquiv\betaar c_{g_2}{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_2)}}x_2^{(\nu_2+1)}\cdots {F_{g_2-2,\nu}^{(\pihirac{l_{g_2-1}(\nu)}{e_{g_2-2}})}}^{r_{g_2-1}^{(g_2)}}\mbox{ mod }x_2^{(\nu_2)}\]
and, by (\rhoef{eqO}), we deduce that $x_2^{(\nu_2+1)}=0$. Hence \[C=V(I^\nu,J_{l_{g_2}(\nu)-1}^\nu,x_2^{(\nu_2)},x_2^{(\nu_2+1)},F_{g_2-1,\nu}^{(\pihirac{l_{g_2}(\nu)}{e_{g_2-1}})}),\]
and now consider the closed
set $C':=\pii_{m,m-1}(C_m^\nu)=\mathbb overline{V(I^\nu,J_{m-1}^\nu)\cap D(x_1^{(\nu_1)})}$. We have that $\pii_{m,m-1}^{-1}(C')=\mathbb overline{V(I^\nu,
J_m^\nu)\cap D(x_1^{(\nu_1)})}=C_m^\nu\cup C$ with Codim$(C_m^\nu)=\mbox{Codim}(C')+1$ and Codim$(C)=\mbox{Codim}(C')+2$, which is a contradiction. Suppose it true for $m$ and we prove it for $m+1$. Consider $C':=\pii_{m+1,m}(C_m^\nu)=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})}$. By
induction hypothesis $C'=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})}$, and then $\pii_{m+1,m}^{-1}(C')=\mathbb overline{V(I^\nu,J_m^\nu,
F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})}$. If $F_\nu^{(m+1)}= 0$ then we are done. Otherwise, by Corollary \rhoef{Corolario3}, $F_\nu^{(m+1)}={F_{g_2,\nu}^
{(\pihirac{l_{g_2}(\nu)}{e_{g_2}}+r)}}^{e_{g_2}}$ where $r=\pihirac{m+1-l_{g_2}(\nu)}{e_{g_2}}$, and, as in the first step of
induction, if it subdivides as
\[C_{m+1}^\nu\cup \mathbb overline{V(I^\nu,J_m^\nu,F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}},\]
then Codim$(\mathbb overline{V(I^\nu,J_m^\nu,F_\nu^{(m+1)})\cap D(x_1^{(\nu_1)})\cap\{x_2^{(\nu_2)}=0\}})=\mbox{Codim}(C')+2$ which is a contradiction.
Finally, if $j'(m,\nu)\gammaeq g_2+1$ there is nothing to prove.
\pisifill $\Box$
We will describe a set $F_m\sigmaubset H_m\cup L_m$ such that $\{C_m^\nu\ |\ \nu\in F_m\}$ is the set of irreducible components. The process of
defining $F_m$ as a subset of $H_m\cup L_m$ is done in two steps. The first reduction is easy. We consider the product ordering $\lambdaeq_p$ in $\mathbb Z^2$ given by:
\betaegin{equation}
\nu\lambdaeq_p\nu'\mbox{ if and only if }\nu_i\lambdaeq\nu'_i\mbox{ for }i=1,2.
\lambdaabel{ordenGen}
\varepsilonnd{equation}
\betaegin{Pro}
For $\nu,\nu'\in H_m\cup L_m$ we have that
\betaegin{enumerate}
\item [(i)] If $C_m^{\nu'}\sigmaubseteq C_m^\nu$ then $\nu\lambdaeq_p\nu'$.
\\
\item [(ii)] Moreover if $\nu,\nu'\in H_m\cup L_m^=$ then we have
\[C_m^{\nu'}\sigmaubseteq C_m^\nu\mathbb Longleftrightarrow \nu\lambdaeq_p\nu'.\]
\varepsilonnd{enumerate}
\lambdaabel{Lema2Gen}
\varepsilonnd{Pro}
{\varepsilonm Proof.}
\betaegin{enumerate}
\item [(i)] Suppose that $\nu$ and $\nu'$ are not comparable. Then we can assume that $\nu_1<\nu_1'$ and $\nu_2>\nu_2'$. Then, since $C_m^\nu\sigmaubseteq V(I^\nu)$, and $C_m^{\nu'}\sigmaubset V(I^{\nu'})$, it follows that
\[C_m^\nu\nsubseteq C_m^{\nu'}\mbox{ and }C_m^{\nu'}\nsubseteq C_m^\nu.\]
\item[(ii)] The claim follows by (i) and the definition of $C_m^\nu$ for $\nu\in
H_m\cup L_m^=$.
\varepsilonnd{enumerate}
\pisifill$\Box$
\betaegin{Defi}
According to Proposition \rhoef{Lema2Gen} we define the set:
\[P_m=min_{\lambdaeq_p}\{H_m\cup L_m^=\}.\]
\lambdaabel{defPm}
\varepsilonnd{Defi}
The second reduction, which defines the set $F_m\sigmaubseteq P_m\cup L_m^<$, is much more involved, and the singular locus of the approximated roots play a role now when studying the inclusions $C_m^{\nu'}\sigmaubseteq C_m^\nu$ for different elements $\nu$ and $\nu'$ in $P_m\cup L_m^<$. By Proposition \rhoef{Lema2Gen} (i)
we have to consider $\nu\lambdaneq_p\nu'$, where, by definition of $P_m$, $\nu\in L_m^<$ and $\nu'\in P_m\cup L_m^<$.
\betaegin{Pro}
Given $m\in\mathbb Z_{>0}$, $\nu\in L_m^<$ and $\nu'\in P_m\cup L_m^<$ with $\nu\lambdaneq_p\nu'$, such that $\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}$ then $C_m^{\nu'}\sigmaubseteq C_m^\nu$.
\lambdaabel{PropC1}
\varepsilonnd{Pro}
{\varepsilonm Proof.} We simplify notation by setting $k_i(\nu)=\pihirac{l_{i+1}(\nu)-l_i(\nu)}{e_i}$, for $1\lambdaeq i\lambdaeq g$. By the
description of $\sigma_{Reg,j}$ given in (\rhoef{SigRegj}) we have to prove the inclusion $C_m^{\nu'}\sigmaubseteq C_m^\nu$ when $\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}$ and
$1\lambdaeq j'(m,\nu)\lambdaeq g_2$. Then, by Lemma \rhoef{LemCm}, we have that
\[C_m^\nu=\mathbb overline{V(I^\nu,J_m^\nu)\cap D(x_1^{(\nu_1)})}.\]
Suppose first that $\sigma_{Reg,j'(m,\nu)}=\rho_2$, then $\nu'=\nu+(0,\betaeta)$ with $\betaeta>0$. We distinguish two cases:
(i) If $\nu'\in H_m$, then by Proposition \rhoef{CHm},
\[C_m^{\nu'}=V(I^\nu,x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+\betaeta-1)},z^{(0)},\lambdadots,z^{([m/n])}).\]
Note that $g_1=0$, because otherwise $l_1(\nu')=l_1(\nu)$ and $\nu'\notin H_m$. Then, since $1\lambdaeq j'(m,\nu)\lambdaeq g_2$, we deduce that $g_2=1$.
There exists $1\lambdaeq r<k_1(\nu)$ such that
\[l_1(\nu)+re_1\lambdaeq m<l_1(\nu)+(r+1)e_1\]
since $\nu\in L_m^<$. Then
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+r)}\rhoight).\]
Notice that $[\pihirac{m}{n}]=[\pihirac{l_1(\nu)+re_1}{n}]=\lambdaangle\nu,\gammaamma_1\rhoangle+\alphalpha$, where $\alphalpha=[\pihirac{r}{n_1}]$. Now, since $\nu'=\nu+(0,\betaeta)
\in H_m$ and $g_2=1$, we have that $l_1(\nu')=l_1(\nu)+e_1\betaeta\gammaeq m+1$ and it follows that $\betaeta>r$.
Hence we have to prove
\betaegin{equation}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+l)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+\betaeta-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+l-1)})
\lambdaabel{eqqq}
\varepsilonnd{equation}
for $0\lambdaeq l\lambdaeq r$. By Corollary \rhoef{Corolario3} we have $F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})}={z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-{x_1^{(\nu_1)}}^{a_1}x_2^{(\nu_2)}+G_{1,\nu}$.
And by Lemma \rhoef{TechLem} (i), if $x_2^{(\nu_2)}=0$ then $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}=0$, and hence $G_{1,\nu}=0$.
By quasi-homogeneity we can write
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}=c_1{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1-1}z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}+c_2{x_1^{(\nu_1)}}^{a_1-1}x_1^{(\nu_1+1)}
x_2^{(\nu_2)}+c_3{x_1^{(\nu_1)}}^{a_1}x_2^{(\nu_2+1)}+G_{1,\nu}^{(1)}\]
where $c_1,c_2,c_3$ are certain coefficients and $G_{1,\nu}^{(1)}$ is a quasi-homogeneous polynomial of degree $\pihirac{l_1(\nu)}{e_1}+1$. We have that $G_{1,\nu}^{(1)}=0$ when $\nu\notin\rho_1\cup\rho_2$, and otherwise, we can apply the same arguments as in the proof of Lemma \rhoef{TechLem} we deduce that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},x_2^{(\nu_2+1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}).\]
Again by quasi-homogeneity, if $n_1<r$,
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)} & =z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+n_1)})+
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}}^{n_1}+x_2^{(\nu_2)}h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\\
\\
& \ x_2^{(\nu_2+1)}h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+\cdots+x_2^{(\nu_2+n_1)}h_{2,n_1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+n_1)})+G_{1,\nu}^{(n_1)}.\\
\varepsilonnd{array}\]
And analogously we prove that
\[F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+n_1)}\varepsilonquiv 0\mbox{ mod }(x_2^{(\nu_2)},\lambdadots,x_2^{(\nu_2+n_1)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}).\]
And in general, for $1\lambdaeq k\lambdaeq r$
\[\betaegin{array}{ll}
F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1}+k)} & = z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}h_{1,0}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})+
z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)}
h_{1,1}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+1)},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+k)})\\
\\
& \ +\ \cdots\ +z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])}h_{1,[k/n_1]}(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle+[\pihirac{k}{n_1}])},\lambdadots,z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle+k)})+\\
\\
& +h_{2,0}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2)}+h_{2,1}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+1)}+\cdots+\\
\\
& +h_{2,k}(x_1^{(\nu_1)},\lambdadots,x_1^{(\nu_1+k)})x_2^{(\nu_2+k)}\\
\varepsilonnd{array}\]
where $h_{1,0},h_{1,1},\lambdadots,h_{1,[k/n_1]},h_{2,0},\lambdadots,h_{2,k}$ are polynomials. And since $k\lambdaeq r<\betaeta$ and $[\pihirac{m}{n}]=\lambdaangle\nu,\gammaamma_1\rhoangle+\alphalpha$
with $\alphalpha=[\pihirac{r}{n_1}]\gammaeq[\pihirac{k}{n_1}]$, we have proved (\rhoef{eqqq}) as we wanted.
(ii) If $\nu'\in L_m$, then by Lemma \rhoef{LemCm}
\[C_m^{\nu'}=\mathbb overline{V(I^{\nu'},J_m^{\nu'})\cap D(x_1^{(\nu_1')})}.\]
Since $\nu\lambdaneq_p\nu'$ we have that $I^\nu\sigmaubseteq I^{\nu'}$. We are going to prove that the generators of
$J_m^\nu$ modulo $(I^{\nu'},z^{(l)})_{0\lambdaeq l<\pihirac{l_1(\nu')}{n}}$ belong to $J_m^{\nu'}$. Since $l_i(\nu)=l_i(\nu')$ for $1\lambdaeq i\lambdaeq g_1$, we have that
\[\betaegin{array}{ll}
F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r_i)}=F_{i,\nu'}^{(\pihirac{l_i(\nu')}{e_i}+r_i)} & \mbox{ for }1\lambdaeq i\lambdaeq g_1-1
\mbox{ and }0\lambdaeq r_i<k_i(\nu)=k_i(\nu'),\\
\\
F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+r)}=F_{g_1,\nu'}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+r)} & \mbox{ for }
0\lambdaeq r<k_{g_1}(\nu)<k_{g_1}(\nu').\\
\varepsilonnd{array}\]
When $J_m^\nu$ has more generators, that is, when $j(m,\nu)>g_1$, then $j'(m,\nu)\gammaeq g_1$. We distinguish two cases.
\betaegin{enumerate}
\item[(ii.a)] If $j'(m,\nu)=g_1$, then, by Corollary \rhoef{Corolario3},
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)};F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i}+r_i)},1\lambdaeq i\lambdaeq g_1,0\lambdaeq r_i<k_i(\nu);F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}\rhoight)_{l_i(\nu)<l_{i+1}(\nu)},\]
and $l_{g_1+1}(\nu)\lambdaeq m<l_{g_1+1}(\nu)+e_{g_1+1}$. By (\rhoef{ecsFs}) we have that
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}= {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}},\]
since $\nu_2'>\nu_2$. Notice that $l_{g_1+1}(\nu)=l_{g_1}(\nu)+l_{g_1+1}(\nu)-l_{g_1}(\nu)=l_{g_1}(\nu')+\alphalpha\lambdaeq m$, with $\alphalpha>0$. Then $l_{g_1}(\nu')<m$ and
therefore $j(m,\nu')\gammaeq g_1$.
If $j(m,\nu')>g_1$ then
\[F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}\in J_m^{\nu'},\]
since $\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}=
\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu)$ and $k_{g_1}(\nu)<k_{g_1}(\nu')$. If $j(m,\nu')=g_1$ then
\[F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}\in J_m^{\nu'},\]
since $\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}=\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu)$, and $k_{g_1}(\nu)\lambdaeq[\pihirac{m-l_{g_1}
(\nu')}{e_{g_1}}]$, because $l_{g_1}(\nu)=l_{g_1}(\nu')$ and $m\gammaeq l_{g_1+1}(\nu)$.
\item[(ii.b)] If $j'(m,\nu)>g_1$, then we are in the case $g_2=g_1+1$ and $j'(m,\nu)=g_1+1$. There exists an integer $1\lambdaeq r<k_{g_1+1}(\nu)$ such that
\betaegin{equation}
l_{g_1+1}(\nu)+re_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu)+(r+1)e_{g_1+1}.
\lambdaabel{toro}
\varepsilonnd{equation}
Then $J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}}{e_{g_1}}+k_{g_1}(\nu)-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+r)}\rhoight)$, where
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})}= {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}x_2^{(\nu_2)}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}.\]
And, analogously to case (i), we can write the polynomials $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ for $1\lambdaeq k\lambdaeq r$ as
\[\betaegin{array}{ll}
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)} & =F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}h_{1,0}+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+1)}h_{1,1}+
\cdots+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+
[\pihirac{k}{n_{g_1+1}}])}h_{1,[k/n_{g_1+1}]}+\\
\\
& h_{2,0}x_2^{(\nu_2)}+\cdots+h_{2,k}x_2^{(\nu_2+k)},\\
\varepsilonnd{array}\]
for certain polynomials $h_{1,0},\lambdadots,h_{1,[k/n_{g_1+1}]},h_{2,0},\lambdadots,h_{2,k}$.
If $j(m,\nu')=g_1$ then $l_{g_1+1}(\nu')>m$ and by (\rhoef{toro}) it follows that $\betaeta>r$. Moreover
\[J_m^{\nu'}=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu')}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu')}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}})},\lambdadots,F_{g_1,\nu}^
{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+
\alphalpha)}\rhoight)\]
with $\alphalpha=[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]$.
We have that $[\pihirac{m-l_{g_1}(\nu')}{e_{g_1}}]=[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}]=[\pihirac{m}{e_{g_1}}]-\pihirac{l_{g_1}(\nu)}{e_{g_1}}$, since $\pihirac{l_{g_1}(\nu)}
{e_{g_1}}$ is an integer, and by (\rhoef{toro}) we have that $\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+\pihirac{r}{n_{g_1+1}}\lambdaeq\pihirac{m}{e_{g_1}}<\lambdaangle\nu,\gammaamma_{g_1+1}
\rhoangle+\pihirac{r+1}{n_{g_1+1}}$. Then $[\pihirac{m}{e_{g_1}}]=\lambdaangle\nu,\gammaamma_{g_1+1}\rhoangle+[\pihirac{r}{n_{g_1+1}}]$, and we have that
\[\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{k}{n_{g_1+1}}\rhoight]\lambdaeq\pihirac{l_{g_1}(\nu)}{e_{g_1}}+\lambdaeft[\pihirac{m-l_{g_1}(\nu)}{e_{g_1}}\rhoight].\]
It follows that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+k)}$ belongs to $J_m^{\nu'}$.
If $j(m,\nu')=g_1+1$ then $\nu'\in N_{g_+1}$ and therefore $\betaeta=qn_{g_1+1}$ with $q\in\mathbb Z_{>0}$. Moreover there exists an integer $0<r'<k_{g_1+1}(\nu')$ such that
\[l_{g_1+1}(\nu')+r'e_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu')+(r'+1)e_{g_1+1}.\]
Or equivalently $l_{g_1+1}(\nu)+(r'+qn_{g_1+1})e_{g_1+1}\lambdaeq m<l_{g_1+1}(\nu)+(r'+qn_{g_1+1}+1)e_{g_1+1}$. Then by (\rhoef{toro}) it follows that
\[r=r'+qn_{g_1+1},\]
and therefore $r>\betaeta$. Moreover $k_{g_1}(\nu')=k_{g_1}(\nu)+qn_{g_1+1}$. Then
\[J_m^{\nu'}=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu')}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu')}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu')-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}}+r')}\rhoight),\]
where notice that $\pihirac{l_{g_1}(\nu')}{e_{g_1}}+k_{g_1}(\nu')-1=\pihirac{l_{g_1}(\nu)}{e_{g_1}}+k_{g_1}(\nu)+qn_{g_1+1}-1$, and
\[J_m^\nu=\lambdaeft(z^{(0)},\lambdadots,z^{(\pihirac{l_1(\nu)}{n}-1)},F_{1,\nu}^{(\pihirac{l_1(\nu)}{e_1})},\lambdadots,F_{g_1,\nu}^{(\pihirac{l_{g_1}(\nu)}{e_{g_1}}+k_{g_1}(\nu)-1)},
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})},\lambdadots,F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+r)}\rhoight).\]
Since $I^{\nu'}=(x_1^{(0)},\lambdadots,x_1^{(\nu_1-1)},x_2^{(0)},\lambdadots,x_2^{(\nu_2+qn_{g_1+1}-1)})$ it follows
that for $0\lambdaeq k\lambdaeq r'$ and $s=qn_{g_1+1}+k$
\[F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}= F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu')}{e_{g_1+1}}+k)}.\]
Then finally we have to prove that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}\in J_m^{\nu'}$ for
$0\lambdaeq s<qn_{g_1+1}$. This follows as in the previous cases, since
\[\betaegin{array}{ll}
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}})} & = {F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}}^{n_{g_1+1}}-{x_1^{(\nu_1)}}^{\alphalpha_1^{(g_1+1)}}x_2^{(\nu_2)}
{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(g_1+1)}}\cdots {F_{g_1-1,\nu}^{(\pihirac{l_{g_1-1}(\nu)}{e_{g_1-1}})}}^{r_{g_1}^{(g_1+1)}}\\
\\
& \mbox{ and for }0\lambdaeq s<qn_{g_1+1}\\
\\
F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)} & = F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}})}h_{1,0}+\cdots+F_{g_1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1}}+
[\pihirac{s}{n_{g_1+1}}])}h_{1,[s/n_{g_1+1}]}\\
\\
&\ -h_{2,0}x_2^{(\nu_2)} -\cdots-h_{2,s}x_2^{(\nu_2+s)}\\
\varepsilonnd{array}\]
for polynomials $h_{1,0},\lambdadots,h_{1,[s/n_{g_1+1}]},h_{2,0},\lambdadots,h_{2,s}$. And it is clear that $F_{g_1+1,\nu}^{(\pihirac{l_{g_1+1}(\nu)}{e_{g_1+1}}+s)}\in J_m^{\nu'}$.
\varepsilonnd{enumerate}
The key point in all the cases is that $\alphalpha_2^{(g_1+1)}=1$.
If $\nu'-\nu\in\rho_1$ then we are in the case $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Now $\nu_1'>\nu_1$ and $\nu_2'=\nu_2$, and similar arguments apply to this case to get the inclusion we want to prove.
\pisifill$\Box$
The previous Proposition motivates the following definition.
\betaegin{Defi}
We consider the relation in $N_0$, depending on $m$ and denoted by $<_{R,m}$, given by
\[\nu\lambdaeq_{R,m}\nu'\mbox{ if and only if }\nu\lambdaeq_p\nu'\mbox{ and }\nu'-\nu\in\sigma_{Reg,j'(m,\nu)}.\]
We define the set $F_m=\mbox{min}_{\lambdaeq_{R,m}}\{P_m\cup L_m^<\}$.
\lambdaabel{orden2}
\varepsilonnd{Defi}
Notice that, by (\rhoef{SigRegj}), for $m$ and $\nu$ such that $j'(m,\nu)>g_2$, this order is just equality.
It is worth pointing out that the inclusions which are described by this last relation in Proposition \rhoef{PropC1}, can be explained by the fact that even though a curve may be in the singular locus of a quasi-ordinary surface, it may not be part of the singular locus of its first approximated quasi-ordinary surfaces. And as Proposition \rhoef{Cgeom} explains, the geometry of $C_m^\nu$ is only determined by the geometry of one of its approximated roots, for $m$ small enough. Hence, the jets which project to the singular locus of the surface but not to the singular locus of the approximated surfaces will not give rise to irreducible components of the jet schemes for $m$ small enough, and they will be included in other components.
Now we prove that all possible inclusions are controlled by the relation defined in Definition \rhoef{orden2} and the product ordering, that is, in the set $F_m$.
\betaegin{Pro}
Given $m\in\mathbb Z_{>0}$ and $\nu,\nu'\in F_m$ with $\nu\lambdaeq_p\nu'$ then $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$.
\lambdaabel{PropC2}
\varepsilonnd{Pro}
{\varepsilonm Proof.} We will prove that $C_m^{\nu'}\not\sigmaubseteq C_m^\nu$ by showing that
\betaegin{equation}
\mbox{Codim}(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^\nu).
\lambdaabel{eqCodim}
\varepsilonnd{equation}
First notice that $\nu\in L_m^<$, since otherwise there would not exist $\nu'\neq \nu$ such that $\nu'\in F_m$ and $\nu\lambdaeq_p\nu'$. Recall that $\sigma=\mathbb R_{\gammaeq 0}^2$,
we define the set
\[\betaegin{array}{cl}
E(\nu)_m & =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and it is minimal with respect to }\lambdaeq_{R,m}\mbox{ in }\nu+\sigma\}\\
\\
& =\{\nu'\in(\nu+\sigma)\cap(P_m\cup L_m^<)\ |\ \nu'\neq\nu\mbox{ and }\nexists\ \omegaidetilde\nu\in(\nu+\sigma)\cap(P_m\cup L_m^<)\mbox{ such that }\omegaidetilde\nu\lambdaeq_{R,m}\nu'\}.\\
\varepsilonnd{array}\]
We claim that for any $\nu'\in E(\nu)_m$ we have that
\betaegin{equation}
\mbox{Codim}(C_{m_0}^{\nu'})\lambdaeq\mbox{Codim}(C_{m_0}^\nu)\ \mbox{ for }l_1(\nu)+e_1\lambdaeq m_0<l_{i(\nu)}(\nu).
\lambdaabel{eqCodim0}
\varepsilonnd{equation}
We prove this claim by induction on $m$. For $m=l_1(\nu)+e_1$ we have that
(i) if $a_1=1$ then we are in the case $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $E(\nu)_m=\varepsilonmptyset$, because $\nu+(1,0),\nu+(0,1)\notin N_1$ and $\nu+
(2,0),\nu+(0,2)\in H_m$ but $\nu\lambdaeq_{R,m}\nu+(2,0)$ and $\nu\lambdaeq_{R,m}\nu+(0,2)$.
(ii) If $a_1>1$ then $\nu+(1,0)\in H_m$ and it follows that in fact $E(\nu)_m=\{\nu+(1,0)\}$, because the only other possible $\nu'$ is
\[\betaegin{array}{ll}
\nu'=\nu+(0,1)\in P_m\cup L_m & \mbox{ if }b_1\varepsilonquiv 0\mbox{ mod }n_1\\
\nu'=\nu+(0,2)\in P_m\cup L_m & \mbox{ otherwise}\\
\varepsilonnd{array}\]
and in both cases we have that $\nu\lambdaeq_{R,m}\nu'$. Now, by Lemma \rhoef{Prop1} we have that for $\nu'=\nu+(1,0)$,
\[\mbox{Codim}(C_m^{\nu'})=\nu_1+\nu_2+\pihirac{l_1(\nu)}{n}+2=\mbox{Codim}(C_m^\nu).\]
Suppose that the claim is true for $m-1$ and we prove it for $m$. Let $\nu'$ be an element in $E(\nu)_m$.
(i) If $\nu'\in E(\nu)_{m-1}$, by induction hypothesis, we have that $\mbox{Codim}(C_{m-1}^{\nu'})\lambdaeq\mbox{Codim}(C_{m-1}^\nu)$. By Corollary \rhoef{CorCodim}
we know that, passing from $m-1$ to $m$, the codimension of $C_m^\nu$ grows if and only if $m$ is divisible by $e_{j(m-1,\nu)}$, and it grows by one. But since
$\nu\lambdaeq_p\nu'$ we have that $j(m-1,\nu')\lambdaeq j(m-1,\nu)$ and therefore if $e_{j(m-1,\nu')}$ divides $m$ then $e_{j(m-1,\nu)}$ divides $m$, and it follows that
$\mbox{Codim}(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^\nu)$.
(ii) If $\nu'\notin E(\nu)_{m-1}$, there exists $\omegaidetilde\nu\in E(\nu)_{m-1}$ such that $\omegaidetilde\nu\lambdaeq_{R,m-1}\nu'$ and $\omegaidetilde\nu\not\lambdaeq_{R,m}
\nu'$. By induction hypothesis we have that $\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})\lambdaeq\mbox{Codim}(C_{m-1}^\nu)$, and again, since $\nu\lambdaneq\omegaidetilde\nu$ then
$j(m,\nu)\gammaeq j(m,\omegaidetilde\nu)$ and therefore $\mbox{Codim}(C_m^{\omegaidetilde\nu})\lambdaeq\mbox{Codim}(C_m^\nu)$. Now we are going to prove that $\mbox{Codim}
(C_m^{\nu'})\lambdaeq\mbox{Codim}(C_m^{\omegaidetilde\nu})$. We have two possibilities, either $\omegaidetilde\nu\in L_m^<$ or $\omegaidetilde\nu\notin L_m^<$.
If $\omegaidetilde\nu\in L_m^<$, then $m=l_{g_2+1}(\omegaidetilde\nu)+e_{g_2+1}$ and
\[\pii_{m,m-1}^{-1}(O_{m-1}^{\omegaidetilde\nu})=V(I^{\omegaidetilde\nu},z^{(0)},\lambdadots,z^{(\lambdaangle\omegaidetilde\nu,\gammaamma_1\rhoangle-1)},F_{1,\omegaidetilde\nu}^
{(\pihirac{l_1(\omegaidetilde\nu)}{e_1})},\lambdadots,F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})},
F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)})\cap D(x_1^{(\omegaidetilde\nu_1)}),\]
where $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}})}= {F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}}-{x_1^{(\omegaidetilde\nu_1)}}^
{\alphalpha_1^{(g_2+1)}}{x_2^{(\omegaidetilde\nu_2)}}^{\alphalpha_2^{(g_2+1)}}\cdots F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})}+G_{g_2+1,\omegaidetilde\nu}$, with $\alphalpha_2^{(g_2+1)}>1$, and $F_{g_2+1,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2+1}}+1)}= n_{g_2+1}
{F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}})}}^{n_{g_2+1}-1}F_{g_2,\omegaidetilde\nu}^{(\pihirac{l_{g_2+1}(\omegaidetilde\nu)}{e_{g_2}}+1)}-x_2^{(\omegaidetilde\nu_2)}H$,
where $H$ is a polynomial in the variables
$H(x_1^{(\omegaidetilde\nu_1)},x_1^{(\omegaidetilde\nu_1+1)},x_2^{(\omegaidetilde\nu_2)},x_2^{(\omegaidetilde\nu_2+1)},\lambdadots,F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}})},
F_{g_2-1,\omegaidetilde\nu}^{(\pihirac{l_{g_2-1}(\omegaidetilde\nu)}{e_{g_2-1}}+1)})$. Then
\[(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap
V(x_2^{(\omegaidetilde\nu_2)})}\cup\mathbb overline{V(I^{\omegaidetilde\nu},J_m^{\omegaidetilde\nu})\cap D(x_1^{(\omegaidetilde\nu_1)})\cap D(x_2^{(\omegaidetilde\nu_2)})},\]
and it is not difficult to see that $(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=C_m^{\nu'}\cup C_m^{\omegaidetilde\nu}$, where $\nu'=\omegaidetilde\nu+(0,\alphalpha)$, with
\[\alphalpha=\lambdaeft\{\betaegin{array}{cl}
1 & \mbox{ if }g_2=g_1\\
\mbox{min}\{n_{g_1+1},k_{g_1+1}(\omegaidetilde\nu)\} & \mbox{ otherwise}\\
\varepsilonnd{array}\rhoight.\]
where remember that $k_i(\omegaidetilde\nu)$ denotes $\pihirac{l_{i+1}(\omegaidetilde\nu)-l_i(\omegaidetilde\nu)}{e_i}$. In both cases we have, by Proposition \rhoef{Prop1}, that $\mbox{Codim}(C_m^{\nu'})=\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})+1=\mbox{Codim}(C_m^{\omegaidetilde\nu})$.
If $\omegaidetilde\nu\notin L_m^<$ then $m=l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)$ with $i(\omegaidetilde\nu)\lambdaeq g_2+1$, since $j'(m-1,\omegaidetilde\nu)\lambdaeq g_2$. We have that
$(\pii_{m,m-1}^{-1}(O_{m-1}^{\omegaidetilde\nu}))_{red}=V(I^{\omegaidetilde\nu},J_{m-1}^{\omegaidetilde\nu},F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac
{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})})\cap D(x_1^{(\omegaidetilde\nu_1)})$, where
\[F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac
{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})}={x_1^{(\omegaidetilde\nu_1)}}^{\alphalpha_1^{(i(\omegaidetilde\nu))}}{x_2^{(\omegaidetilde\nu_2)}}^{\alphalpha_2^
{(i(\omegaidetilde\nu))}} {z^{(\lambdaangle\omegaidetilde\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i(\omegaidetilde\nu))}}\cdots {F_{i(\omegaidetilde\nu)-2,\omegaidetilde\nu}^{(\pihirac{l_{i(\omegaidetilde\nu)-2}
(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)-2}})}}^{r_{i(\omegaidetilde\nu)-1}^{(i(\omegaidetilde\nu))}}+G_{i(\omegaidetilde\nu),\omegaidetilde\nu}.\]
Therefore, by Lemma \rhoef{TechLem}, $F_{i(\omegaidetilde\nu),\omegaidetilde\nu}^{(\pihirac{l_{i(\omegaidetilde\nu)}(\omegaidetilde\nu)}{e_{i(\omegaidetilde\nu)}})}=0$ implies that $ x_2^{(\omegaidetilde
\nu_2)}=0$ because $i(\omegaidetilde\nu)-2<g_2$. And, as before, if $g_2=g_1+1$ and $i(\omegaidetilde\nu)=g_2+1$ then we have that $\nu'=\omegaidetilde\nu+(0,\alphalpha)$ with
$\alphalpha=\mbox{min}\{n_{g_1+1},k_{g_1+1}(\omegaidetilde\nu)\}$. Otherwise $\nu'=\omegaidetilde\nu+(0,1)$, and in both cases we have
\[(\pii_{m,m-1}^{-1}(C_{m-1}^{\omegaidetilde\nu}))_{red}=C_m^{\nu'}\]
with $\mbox{Codim}(C_m^{\nu'})=\mbox{Codim}(C_{m-1}^{\omegaidetilde\nu})+1$. Since $\omegaidetilde\nu\in E(\nu)_{m-1}$, it follows that $j(m-1,\nu)>j(m-1,\omegaidetilde\nu)=
i(\omegaidetilde\nu)-1$ and by Corollary \rhoef{CorCodim} we have that $\mbox{Codim}(C_m^\nu)=\mbox{Codim}(C_{m-1}^\nu)+1$, which finishes the proof.
\pisifill $\Box$
Now we can prove the main theorem of this section.
\betaegin{The}
For $m\in\mathbb Z_{>0}$ the decomposition of $\pii_m^{-1}(X_{Sing})$ in irreducible components is given by
\[(\pii_m^{-1}(X_{Sing}))_{red}=\betaigcup_{\nu\in F_m}C_m^\nu.\]
\lambdaabel{TheCaso1}
\varepsilonnd{The}
{\varepsilonm Proof.} The irreducibility of the sets $C_m^\nu$ was proven in Proposition \rhoef{Prop1}. And by Proposition \rhoef{Lema2Gen}, Proposition \rhoef{PropC1} and Proposition \rhoef{PropC2} we have that
\[\betaigcup_{\nu\in H_m\cup L_m}C_m^\nu=\betaigcup_{\nu\in F_m}C_m^\nu.\]
Hence the result follows by Lemma \rhoef{LemTh}.
\pisifill$\Box$
\betaegin{Rem}
For $\nu\in N_g$, if $\nu\in F_{l_g(\nu)}$, then $\nu\in F_m$ for every $m\gammaeq l_g(\nu)$, or in other words, $\nu$ gives rise to an irreducible component for any $m\gammaeq l_g(\nu)$.
\lambdaabel{RemarkFinal}
\varepsilonnd{Rem}
\betaegin{Rem}
When the equisingular dimension is $c=1$ (see Definition \rhoef{equi}), then $g_1=g_2=g$. Moreover we have the following properties for $1\lambdaeq i\lambdaeq g$
\[\betaegin{array}{l}
l_i(\nu)=l_i(\nu+(0,r)),\ \mbox{ for all }r\in\mathbb Z\\
\\
\mbox{if }\nu\in N_i\mbox{ then }\nu+(0,r)\in N_i,\ \mbox{ for all }r\in\mathbb Z\\
\varepsilonnd{array}\]
Hence we deduce that for any $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ we have $\sigma_{Reg,j'(m,\nu)}=\rho_2$, and therefore $F_m=(P_m\cup L_m^<)\cap\rho_1$.
The behaviour of the jet schemes is exactly as the plane curve defined by the Puiseux pairs $\lambdaambda_1^{(1)},\lambdadots,\lambdaambda_g^{(1)}$.
In \cite{Hcur} the second author describes the irreducible components of jets through the origin in the case of plane curves .
\varepsilonnd{Rem}
The previous remark is the simplest evidence of the fact that the irreducible components are only affected by the topological type. This is proved in Theorem \rhoef{topTree}
To any quasi-ordinary surface singularity we can associate a weighted graph, containing information about the irreducible components of jet schemes and how they behave under truncation maps.
\betaegin{Defi}
The weighted graph of the jet schemes of $X$ is the leveled weighted graph $\Gamma$ defined as follows:
\betaegin{itemize}
\item for $m\gammaeq 1$ we represent every irreducible components of $\pii_m^{-1}(X_{Sing})$ by a vertex $V_m$, the sub-index $m$ being the level of the vertex;
\item we join the vertices $V_{m+1}$ and $V_m$ if the canonical morphism $\pii_{m+1,m}$ induces a morphism between the corresponding irreducible components;
\item we weight each vertex by the dimension of the corresponding irreducible component.
\varepsilonnd{itemize}
We define $E\Gamma$ to be the weighted graph that we obtain from $\Gamma$
by weighting any vertex of $\Gamma$ by the embedding dimensions of the corresponding irreducible components (note that by the definition of $\Gamma,$ these vertices are also weighted by their dimensions).
\lambdaabel{defGrafo0}
\varepsilonnd{Defi}
Notice that the data of the codimension together with the embedded dimension permits to distinguish when the vertex corresponds to a hyperplane or a lattice component. Indeed, given a vertex of the graph, let $e$ be the embedded dimension and $c$ the codimension, then the vertex corresponds to a hyperplane component if and only if $e+c=3(m+1)$. Therefore we can extract from $E\Gamma$ a subgraph $\Gamma'$ as follows.
\betaegin{Defi}
We define a weighted subgraph $\Gamma'$ of $E\Gamma$ by adding the condition that we join the vertices $V_m$ (corresponding to a certain component, say $C_m^{\nu'}$) and $V_{m-1}$ (corresponding to $C_m^\nu$) only if
\betaegin{itemize}
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)\lambdaeq g_2$ then $\nu'=\nu+(0,\alphalpha)$ with $\alphalpha$ minimal among the elements in $F_{m}$.
\item if $\nu\in L_{m-1}^<$ with $j(m-1,\nu)>g_2$ then $\nu'=\nu$.
\varepsilonnd{itemize}
\lambdaabel{defGrafo}
\varepsilonnd{Defi}
The important thing about this new graph $\Gamma'$ is that, with the weights, we are able to detect when we pass from a hyperplane component at level $m$ to a lattice component at level $m+1$, as we also do in the graph $E\Gamma$, but now we can follow this component in a unique path in the graph as $m$ grows. This will be useful to prove the following result.
\betaegin{The}
The graph $\Gamma'$ determines and it is determined by the topological type of the singularity.
\lambdaabel{topTree}
\varepsilonnd{The}
{\varepsilonm Proof.} Obviously the graph is determined by the semigroup, and therefore, by \cite{Gau}, by the topological type.
To prove the converse we consider two different sets of generators of the semigroup $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1',\lambdadots,\gammaamma_{g'}'\}$ and we will prove that the corresponding weighted graphs are different too.
Given a weighted graph we can recover the number of characteristic exponents in the following way. Any vertex $V_m$ on the graph comes with the codimension $c(V_m)$ and the embedded dimension $e(V_m)$. Take an infinite branch, and consider the finite part that starts at
\[m_0=\mbox{max }\{m\ |\ V_{m-1}\mbox{ is a hyperplane component and }V_m\mbox{ is a lattice component}\},\]
and ends at
\[m_1=\mbox{min }\{m\ |\ c(V_m)=c(V_{m-1})+1\mbox{ for all }m>m_1\}.\]
We can read $e_0,\lambdadots,e_{g-1}$ making use of Corollary \rhoef{CorCodim}. Indeed, along the piece of branch, the vertex $V_m$ corresponds to a component $C_m^\nu$ with $\nu\in N_g$, $m_0=l_1(\nu)$ and $m_1=l_g(\nu)$. To read this data we consider only branches that projects into the component $Z_1$ of the singular locus, since otherwise we can only assure that $m_0=l_{m(\nu)}(\nu)$, and we do not have all the information whenever $m(\nu)>1$. Notice that $Z_1$ is always a component of the singular locus unless we are in the case $g=1$ and $\gammaamma=(\pihirac{1}{n},\pihirac{1}{n})$, which is very easy to recognize. Indeed, it is the only case when at level $m=1$ we have only one component, with codimension 3 and embedded dimension 0. Moreover the multiplicity $n$ equals the first time $m$ when we have a lattice component. Therefore this simple case is very easily understood in the graph. For the rest of the cases, since we know that $\nu\in N_g$, going backwards we look for the biggest $m'$ such that $c(V_{m'})=c(V_{m_0})-1$. Then $n=m_0-m'$. Now, going from level $m_0$ to $m_1$, we know that the codimension grows by one exactly every $e_1$ steps at first, after every $e_2$ steps, and so on. Since $e_1>e_2>\cdots>e_g=1$ we can read these numbers on the graph. Notice that equivalently we get $n_1,\lambdadots,n_g$, and in particular we have $g$, the number of characteristic exponents.
Suppose now that the number of generators of the semigroups is the same, say $g$. We will prove by induction on $g$ that the graphs corresponding to different sets of generators, are different. We denote the vertices at level $m$ by $V_m(c(V_m),e(V_m))$. For $g=1$, the multiplicity is read from the graph as was explained before, and the situation for $m=1$ is:
\[\betaegin{array}{cl}
\betaullet\ V_1(3,0) & \mbox{ if }\gammaamma=(\pihirac{1}{n},\pihirac{1}{n})\\
\\
\betaullet\ V_1(2,1) & \mbox{ if }\gammaamma=(\pihirac{a}{n},\pihirac{1}{n}),\mbox{ with }a>1\\
\\
\betaullet\ V_1(2,1)\ \ \betaullet\ V_1(2,1) & \mbox{ if }\gammaamma=(\pihirac{a}{n},\pihirac{b}{n}),\mbox{ with }b>1\\
\varepsilonnd{array}\]
If we want to compare the graph associated to $\gammaamma$ and the graph associated to $\gammaamma'$, we just have to consider the cases $\gammaamma=(\pihirac{a}{n},\pihirac{1}{n})$, $\gammaamma'=(\pihirac{a'}{n},\pihirac{1}{n})$ with $a\neq a'$, and $\gammaamma=(\pihirac{a}{n},\pihirac{b}{n})$, $\gammaamma'=(\pihirac{a'}{n},\pihirac{b'}{n})$ with $b,b'\neq 1$ and $\gammaamma\neq\gammaamma'$. The first case is very easy to distinguish, since the first moment a component splits in two is at $m=a$ for one graph, and at $m=a'$ for the other. For the other case, first note that the graph of any quasi-ordinary with only one characteristic exponent $\gammaamma=(\pihirac{a}{n},\pihirac{b}{n})$ is the graph associated to $z^n-x_1^ax_2^b$. The key point is that, when $b>1$ we have in the graph a branch which generically corresponds to $x_2^{(0)}\neq 0$ (resp. $x_1^{(0)}\neq 0$), that is, it behaves like the graph of the curve $z^n-x_1^a$ (resp. $z^n-x_2^b$). Therefore comparing graphs associated to $\gammaamma$ and $\gammaamma'$ with $\gammaamma\neq\gammaamma'$, we deduce from Theorem 3.3 in \cite{Hcur}, that the graphs must be different.
Now, suppose it is true for $g-1$ characteristic exponents, and we will prove it for $g$. From Proposition \rhoef{Cgeom} we deduce that is sufficient to prove that the graphs associated to the sets $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$ are different, since otherwise it holds by induction hypothesis. Moreover, since we read the integers $n_1,\lambdadots,n_g$ in the graph, we assume that $n_g'=n_g$. As in the case $g=1$, by looking at the singular locus (which is seen at $m=1$) we just have to consider the case
$\gammaamma_g^{(2)}=\gammaamma_g'^{(2)}=\pihirac{1}{n_g}$ and the case $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$. In the first case $\gammaamma_g^{(1)}\neq\gammaamma_g'^{(1)}$ and $\gammaamma_i^{(2)}=\gammaamma_i'^{(2)}=0$ for $1\lambdaeq i\lambdaeq g-1$. Therefore the graphs are the same till we get to level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$, where $\nu=(\nu_1,0)\in\sigma_{Sing}\cap N_{g-1}$ with $\nu_1$ smallest with this property. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ the graphs must differ at some moment.
Finally, when $\gammaamma\neq\gammaamma'$ with $\gammaamma_g^{(2)},\gammaamma_g'^{(2)}>\pihirac{1}{n_g}$, again by Proposition \rhoef{Cgeom}, the graphs must be the same for $\{\gammaamma_1,\lambdadots,\gammaamma_g\}$ and $\{\gammaamma_1,\lambdadots,\gammaamma_{g-1},\gammaamma_g'\}$, till the last approximated root, that is, $f$, starts playing a role in the definition of a component, say $C^\nu$. Since $\lambdaangle\nu,\gammaamma_g\rhoangle\neq\lambdaangle\nu,\gammaamma_g'\rhoangle$ we will see the difference on the graphs at level $m=\mbox{min }\{n_g\lambdaangle\nu,\gammaamma_g\rhoangle,n_g\lambdaangle\nu,\gammaamma_g'\rhoangle\}$.
\pisifill$\Box$
\sigmaubsection{Log-canonical threshold}
In \cite{Mus}, Musta\c ta gave a formula of the log-canonical threshold in terms of the codimension of jet schemes, which in our setting can be stated as
\betaegin{equation}
lct(f)=\mbox{min}_{m\gammaeq 0}\pihirac{\mbox{Codim}(X_m)}{m+1}.
\lambdaabel{lct}
\varepsilonnd{equation}
Then, as an application to Theorem \rhoef{TheCaso1}, we can recover, for the case of surfaces, the result in \cite{BGG}.
\betaegin{Cor}
The log-canonical threshold of a quasi-ordinary surface singularity is given by:
\[lct_0(X,\mathbb A^3)=\lambdaeft\{\betaegin{array}{cl}
\pihirac{1+\lambda_1^{(1)}}{e_0\lambda_1^{(1)}} & \ \mbox{ if }\lambda_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\\
\\
1 & \mbox{ if }\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g=1\\
\\
\pihirac{n_1(1+\lambda_2^{(1)})}{e_1(n_1(1+\lambda_2^{(1)})-1)} & \ \mbox{ if }\lambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ and }g>1\\
\varepsilonnd{array}\rhoight.\]
Moreover, the components that contribute to the log canonical threshold are
\[\betaegin{array}{ll}
C_{l_1(\nu)-1}^\nu & \mbox{ if }\gammaamma_1\neq(\pihirac{1}{n_1},\pihirac{1}{n_1})\mbox{ or }g=1\\
\\
C_{l_2(\nu)-1}^\nu & \mbox{ otherwise}\\
\varepsilonnd{array}\]
where $\nu=(l,0)\in N_1$ if $\gammaamma_1\neq (\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $\nu=(l,0)\in N_2$ otherwise.
\lambdaabel{Thlct}
\varepsilonnd{Cor}
{\varepsilonm Proof.}
The case $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g=1$ behaves as an $A_n$-singularity, and then $lct(f)=1$. For the rest of the cases, by Corollary \rhoef{CorCodim}, the codimension of a component grows faster as $m$ grows, for
bigger $j(m,\nu)$. Therefore, the smaller codimension will be attached for $\nu\in P_m\cap F_m$, and more concretely for
$\nu\in H_m\cap F_m$ whenever $H_m\cap F_m\neq\varepsilonmptyset$. If $g_1=0$, since $a_1\gammaeq b_1$, we deduce that the minimal codimension among the elements in $P_m\cap F_m$
is attached for $\nu$ of the form $\nu=(l,0)$, while if $g_1>0$ then $P_m\cap F_m$ consists of just a point of the form $\nu=(l,0)$.
We want to minimize not just the codimension, but the quotient $\pihirac{\mbox{Codim}(X_m)}{m+1}$. That is, to find the biggest $m$ such that $\nu$ still
belongs to $P_m\cap F_m$. Then, when the first characteristic exponent is different from $(\pihirac{1}{n_1},\pihirac{1}{n_1})$, this is
attached for $m=l_1(\nu)-1$ such that $\nu\in L_{m+1}^=$. Then $m=l_1(l,0)-1$ and Codim$(C_m^\nu)=l+[\pihirac{m}{n}]+1$, and since $\nu\in L_{m+1}^=$, $(l,0)\in N_1$
and therefore Codim$(C_m^\nu)=l+l\pihirac{a_1}{n_1}$, which implies that $\pihirac{\mbox{Codim}(C_m^\nu)}{m+1}=\pihirac{a_1+n_1}{na_1}$.
If $\gammaamma_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g>1$, what happens is that when $m=l_1(\nu)$ there is no subdivision of the component and $\sigma_{Reg,1}=\rho_1\cup\rho_2$.
If we denote the second exponent by $\gammaamma_2=(\pihirac{\alphalpha_2}{n_1n_2},\pihirac{\betaeta_2}{n_1n_2})$, we look for $\nu$ of the form $(l,0)$ such that $m+1=l_2(\nu)$
with $\nu\in N_2$. Then $\mbox{Codim}(C_m^\nu)=l+\pihirac{l_1(\nu)}{n}+[\pihirac{m-l_1(\nu)}{e_1}]+1=l+\pihirac{l_1(\nu)}{n}+\pihirac{l_2(\nu)-l_1(\nu)}{e_1}$, and therefore
$\pihirac{\mbox{Codim}(C_m^\nu)}{m+1}=\pihirac{l+l\pihirac{1}{n_1}+\pihirac{1}{e_1}(e_2n_2l\pihirac{\alphalpha_2}{n_1n_2}-e_1n_1l\pihirac{1}{n_1})}{e_2n_2l\pihirac{\alphalpha_2}{n_1n_2}}=
\pihirac{1+\pihirac{1}{n_1}+\pihirac{\alphalpha_2}{n_1n_2}-1}{e_2\pihirac{\alphalpha_2}{n_1}}=\pihirac{1+\pihirac{\alphalpha_2}{n_2}}{e_1\pihirac{\alphalpha_2}{n_2}}$.
This coincides with the statement since $\lambda_2=(\pihirac{\alphalpha_2}{n_1n_2}-\pihirac{n_1-1}{n_1},\pihirac{\betaeta_2}{n_1n_2}-\pihirac{n_1-1}{n_1})$.
\pisifill $\Box$
\betaegin{Rem}
Notice that $\pihirac{1+\lambdaambda_1^{(1)}}{e_0\lambdaambda_1^{(1)}}\lambdaeq 1$ except in the case $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ and $g=1$. Moreover in the case of
surfaces the condition $\lambda_1^{(1)}=\pihirac{1}{n_1}$ is equivalent to $\lambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$ since the branch is normalized. Then, see notations in
\cite{BGG}, $\varepsilonll_1=\varepsilonll_2$ and in Theorem \rhoef{Thlct} we recover, for the case of surfaces, the formula given in Theorem 3.1 in \cite{BGG}.
\varepsilonnd{Rem}
We now deduce a family of examples whose log canonical threshold can not be computed by a monomial valuation.
\betaegin{Cor}
Let $X$ be a quasi-ordinary surface singularity with $g>1$ characteristic exponents, and such that $\lambdaambda_1=(\pihirac{1}{n_1},\pihirac{1}{n_1})$. Then $lct(X,\mathbb A^3)$ can not be contributed by monomial valuations in
any variables.
\lambdaabel{CorMV}
\varepsilonnd{Cor}
{\varepsilonm Proof.} It follows from Corollary \rhoef{Thlct} that $lct(X,\mathbb A^3)$ is contributed by $C_{l_1(\nu)}^\nu$, for $\nu$ as is made precise in the above statement.
This is equivalent to say that the valuation
\[\betaegin{array}{ll}
\mathcal V_{C_{l_2(\nu)-1}^\nu} :& \mathbb C[[x_1,x_2,z]]\lambdaongrightarrow\mathbb N\\
\\
& \ \ \ \ \ \ h\ \lambdaongmapsto\ \mbox{ord}_t(h\circ\varepsilonta)\\
\varepsilonnd{array}\]
where $\varepsilonta$ is the generic point of $({\mathbb P}si_{l_2(\nu)-1}^{\mathbb A^3})^{-1}(C_{l_2(\nu)-1}^\nu)$. Note that $\nu$ can take all the values described in Corollary \rhoef{Thlct} but since ${z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{n_1}-x_1^{(\nu_1)}x_2^{(\nu_2)}=0$ is one of the defining equations of $C_{l_2(\nu)-1}^\nu$, then $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>n_1\mathcal V_{C_{l_2(\nu)-1}^\nu}(z)$ and $\mathcal V_{C_{l_2(\nu)-1}^\nu}(z^{n_1}-x_1x_2)>\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_1)+\mathcal V_{C_{l_2(\nu)-1}^\nu}(x_2)$. Therefore $\mathcal V_{C_{l_2(\nu)-1}^\nu}$ is not a monomial valuation.
\pisifill$\Box$
\sigmaubsection{Example}
\lambdaabel{exs}
Consider the quasi-ordinary surface $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$. The semigroup is generated by the vectors
\[\gammaamma_1=\lambdaeft(\pihirac{3}{2},0\rhoight),\ \gammaamma_2=\lambdaeft(\pihirac{7}{2},\pihirac{3}{2}\rhoight)\mbox{ and }\gammaamma_3=\lambdaeft(\pihirac{29}{4},\pihirac{13}{4}\rhoight).\]
We have that $g_1=g_2=1$.
The singular locus is reducible, of the form
\[X_{Sing}=\{z=x_1=0\}\cup\{x_2=z^2-x_1^3=0\}=Z_1\cup Z_2.\]
Then $\sigma_{Sing}=\mathbb R_{\gammaeq 0}^2\sigmaetminus\{0\}$ and $\sigma_{Reg,1}=\rho_2,\ \sigma_{Reg,2}=\sigma_{Reg,3}=\{(0,0)\}$.
The set $F_m$ describing the irreducible components is the following, for some $m$:
\[\betaegin{array}{l}
F_m=\{(1,0),(0,1)\}, \ \mbox{ for }1\lambdaeq m<6\\
\\
F_m=\{(1,0),(0,2)\}, \ \mbox{ for }6\lambdaeq m<12\\
\\
F_{12}=\{(2,0),(0,2)\}\\
\\
F_{13}=\{(2,0),(0,3)\} \\
\\
F_{18}=\{(2,0),(0,4)\} \\
\\
F_{26}=\{(2,0),(0,4),(0,5)\} \\
\\
F_{28}=\{(3,0),(2,0),(0,4),(0,5)\} \\
\varepsilonnd{array}\]
and the result can be checked by lifting the components $Z_1$ and $Z_2$ of the singular locus to level $m$ as the following graph shows (we did not draw the weights of the vertices for clearness).
\betaegin{figure}[h]
\unitlength=1mm
\betaegin{center}
\betaegin{picture}(110,100)(0,3)
\lambdainethickness{0.15mm}
\piut(24,0){$Z_1$}\piut(80,0){$Z_2$}
\piut(6,4){$m=1$}\piut(25,4){$\betaullet$}\piut(80,4){$\betaullet$}
\piut(26,5){\lambdaine(0,1){5}}\piut(81,5){\lambdaine(0,1){5}}
\piut(6,9){$m=2$}\piut(25,9){$\betaullet$}\piut(80,9){$\betaullet$}
\piut(11,13){$\vdots$}\piut(25,13){$\vdots$}\piut(80,13){$\vdots$}
\piut(6,20){$m=25$}\piut(25,20){$\betaullet$}\piut(80,20){$\betaullet$}
\piut(26,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(0,1){5}}\piut(81,21){\lambdaine(1,1){5}}
\piut(6,25){$m=26$}\piut(25,25){$\betaullet$}\piut(80,25){$\betaullet$}\piut(85,25){$\betaullet$}
\piut(26,26){\lambdaine(0,1){5}}\piut(81,26){\lambdaine(0,1){5}}
\piut(6,30){$m=27$}\piut(25,30){$\betaullet$}\piut(80,30){$\betaullet$}\piut(86,26){\lambdaine(1,1){5}}\piut(90,31){$\nearrow$}
\piut(26,31){\lambdaine(0,1){5}}\piut(26,31){\lambdaine(4,1){20}}\piut(81,31){\lambdaine(0,1){5}}
\piut(6,35){$m=28$}\piut(25,35){$\betaullet$}\piut(45,35){$\betaullet$}\piut(80,35){$\betaullet$}
\piut(11,39){$\vdots$}\piut(25,39){$\vdots$}\piut(45,39){$\vdots$}\piut(80,39){$\vdots$}
\piut(6,45){$m=37$}\piut(25,45){$\betaullet$}\piut(45,45){$\betaullet$}\piut(80,45){$\betaullet$}
\piut(26,46){\lambdaine(0,1){5}}\piut(46,46){\lambdaine(0,1){5}}\piut(81,46){\lambdaine(0,1){5}}\piut(81,46){\lambdaine(1,1){5}}
\piut(6,50){$m=38$}\piut(25,50){$\betaullet$}\piut(45,50){$\betaullet$}\piut(80,50){$\betaullet$}\piut(85,50){$\betaullet$}
\piut(26,51){\lambdaine(0,1){5}}\piut(46,51){\lambdaine(0,1){5}}\piut(81,51){\lambdaine(0,1){5}}
\piut(6,55){$m=39$}\piut(25,55){$\betaullet$}\piut(45,55){$\betaullet$}\piut(80,55){$\betaullet$}
\piut(26,56){\lambdaine(0,1){5}}\piut(46,56){\lambdaine(0,1){5}}\piut(81,56){\lambdaine(0,1){5}}
\piut(6,60){$m=40$}\piut(25,60){$\betaullet$}\piut(45,60){$\betaullet$}\piut(80,60){$\betaullet$}
\piut(26,61){\lambdaine(0,1){5}}\piut(46,61){\lambdaine(0,1){5}}\piut(81,61){\lambdaine(0,1){5}}
\piut(6,65){$m=41$}\piut(25,65){$\betaullet$}\piut(45,65){$\betaullet$}\piut(80,65){$\betaullet$}
\piut(26,66){\lambdaine(0,1){5}}\piut(46,66){\lambdaine(0,1){5}}\piut(46,67){$\nearrow$}\piut(81,66){\lambdaine(0,1){5}}
\piut(6,70){$m=42$}\piut(25,70){$\betaullet$}\piut(45,70){$\betaullet$}\piut(80,70){$\betaullet$}
\piut(11,74){$\vdots$}\piut(25,74){$\vdots$}\piut(45,74){$\vdots$}\piut(80,74){$\vdots$}
\piut(6,80){$m=49$}\piut(25,80){$\betaullet$}\piut(45,80){$\betaullet$}\piut(80,80){$\betaullet$}
\piut(26,81){\lambdaine(0,1){5}}\piut(46,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(0,1){5}}\piut(81,81){\lambdaine(1,1){5}}
\piut(6,85){$m=50$}\piut(25,85){$\betaullet$}\piut(45,85){$\betaullet$}\piut(80,85){$\betaullet$}\piut(85,85){$\betaullet$}
\piut(26,86){\lambdaine(0,1){5}}\piut(46,86){\lambdaine(0,1){5}}\piut(81,86){\lambdaine(0,1){5}}\piut(86,86){\lambdaine(0,1){5}}
\piut(6,90){$m=51$}\piut(25,90){$\betaullet$}\piut(45,90){$\betaullet$}\piut(80,90){$\betaullet$}\piut(85,90){$\betaullet$}
\piut(26,91){\lambdaine(0,1){5}}\piut(26,91){\lambdaine(1,1){5}}\piut(46,91){\lambdaine(0,1){5}}\piut(81,91){\lambdaine(0,1){5}}\piut(86,91){\lambdaine(0,1){5}}
\piut(6,95){$m=52$}\piut(25,95){$\betaullet$}\piut(30,95){$\betaullet$}\piut(45,95){$\betaullet$}\piut(80,95){$\betaullet$}\piut(85,95){$\betaullet$}
\piut(26,96){\lambdaine(0,1){5}}\piut(31,96){\lambdaine(0,1){5}}\piut(46,96){\lambdaine(0,1){5}}\piut(81,96){\lambdaine(0,1){5}}\piut(86,97){$\nearrow$}
\piut(6,100){$m=53$}\piut(25,100){$\betaullet$}\piut(30,100){$\betaullet$}\piut(45,100){$\betaullet$}\piut(80,100){$\betaullet$}
\varepsilonnd{picture}
\varepsilonnd{center}
\caption{The graph of the surface defined by $f=((z^2-x_1^3)^2-x_1^7x_2^3)^2-x_1^{11}x_2^5(z^2-x_1^3)$.}
\lambdaabel{figEx1}
\varepsilonnd{figure}
The arrows in the figure represent the behaviour explained in Remark \rhoef{RemarkFinal}.
Now we give some explanations to illustrate how Proposition \rhoef{Lema2Gen} and Proposition \rhoef{PropC1} work.
For $m=1$, we have $H_1=\{(1,0),(0,1)\}$, $L_1^=\{(0,1)\}$ and $L_1^<=\varepsilonmptyset$. The claim on $F_1$ in this case follows easily by Proposition \rhoef{Lema2Gen}.
At level $m=6$ we have $H_6=\{\nu\in[0,m]^2\cap N_0\ |\ \nu_1\neq 0\}$, $L_6^==\varepsilonmptyset$ and $L_6^<=\{(0,\nu_2)\ |\ 2\lambdaeq\nu_2\lambdaeq 6\}$ with $j'(6,(0,2))=1$, then, by Proposition \rhoef{PropC1} $C_6^{\nu'}\sigmaubseteq C_6^{(0,2)}$ for $\nu'\in\{(0,3),(0,4),(0,5),(0,6)\}$, because $\sigma_{Reg,1}=\rho_2=\{0\}\tauimes\mathbb R_{\gammaeq 0}$. By Proposition \rhoef{Lema2Gen}, only $\nu=(1,0)$ contributes to $F_6$ from the vectors in $H_6$, and the claim on $F_6$ follows. Note how at this level $\nu=(0,1)$ does no longer give rise to an irreducible component, since $l_2(0,1)=6$ and $(0,1)\notin N_2$. Then we have that $(0,2)\in F_6$ and the vertex associated with $C_5^{(0,1)}$ and the one associated with $C_6^{(0,2)}$ are joined in the graph $\Gamma'$.
\sigmaection{Technical results and proofs.}
\lambdaabel{Proofs}
In this section we state and prove some results which are used along the paper but only in the proofs of other results, and can be skipped to read Section \rhoef{Sec4}. Moreover there are some proofs which we leave to this section.
Recall that we denote the first characteristic exponent by $\lambdaambda_1=\gammaamma_1=(\pihirac{a_1}{n_1},\pihirac{b_1}{n_1})$ with $a_1\gammaeq b_1,\ a_1>0\mbox{ and }b_1\gammaeq 0$,
and if $b_1=0$ then $a_1>n_1$ (we always consider normalized branches). If $g_1>0$ then, by Lemma \rhoef{Lema35}, we have that $n_{g_1+1}\gammaamma_{g_1+1}=
(\alphalpha_1^{(g_1+1)},\alphalpha_2^{(g_1+1)})+(r_1^{(g_1+1)}\gammaamma_1^{(1)}+\cdots+r_{g_1}^{(g_1+1)}\gammaamma_{g_1}^{(1)},0)$, therefore $n_{g_1+1}\gammaamma_{g_1+1}^{(2)}\in\mathbb Z$, or in other words, $\gammaamma_{g_1+1}=(\gammaamma_{g_1+1}^{(1)},\pihirac{b_{g_1+1}}{n_{g_1+1}})$
with $b_{g_1+1}\gammaeq 1$.
\betaegin{Lem}
In the relation $n_{g_2+1}\gammaamma_{g_2+1}=(\alphalpha_1^{(g_2+1)},\alphalpha_2^{(g_2+1)})+r_1^{(g_2+1)}\gammaamma_1+\cdots+r_{g_2}^{(g_2+1)}\gammaamma_{g_2}$ given in Lemma
\rhoef{LemaPedro}, we have that $\alphalpha_2^{(g_2+1)}>1$.
\lambdaabel{alphaM1}
\varepsilonnd{Lem}
{\varepsilonm Proof.} If $g_2=g_1$ the claim is trivial since $\alphalpha_2^{(g_2+1)}=b_{g_2+1}>1$. Otherwise $n_{g_2+1}\gammaamma_{g_2+1}^{(2)}=\alphalpha_2^{(g_2+1)}+
r_{g_2}^{(g_2+1)}\pihirac{1}{n_{g_2}}$, and since, by Lemma \rhoef{LemaPedro}, $\gammaamma_{g_2+1}^{(2)}\gammaeq n_{g_2}\gammaamma_{g_2}^{(2)}=1$ and $0\lambdaeq r_{g_2}^{(g_2+1)}<
n_{g_2}$, then $\alphalpha_2^{(g_2+1)}\gammaeq n_{g_2+1}-\pihirac{r_{g_2}^{(g_2+1)}}{n_{g_2}}>1$, because $n_{g_2+1}\gammaeq 2$. \pisifill $\Box$
In Corollary \rhoef{Corolario3} we describe the generators of $J_m^\nu$ for $\nu\in H_m\cup L_m$. But we also need to describe the polynomial $F_{\nu}^{(l_{i(\nu)}(\nu))}$ (recall that by definition $\nu\notin H_{l_{i(\nu)}(\nu)}\cup L_{l_{i(\nu)}(\nu)}$).
We do this in the next Lemma, but before we look at an example.
\betaegin{Exam}
Let $X$ be a quasi-ordinary surface defined by $f=((z^2-x_1^3x_2^2)^2-x_1^6x_2^4z)^3-x_1^{23}x_2^{14}z$. The generators of the semigroup are $\gammaamma_1=(\pihirac{3}{2},1),\ \gammaamma_2=(\pihirac{15}{4},\pihirac{5}{2}) \mbox{ and }\gammaamma_3=(\pihirac{49}{6},5)$. Notice that $\nu=(0,3)\notin N_2$, and $l_2(\nu)=l_3(\nu)$. At level $m=45$ we have the set
\[D_{45}^{(0,3)}=V(x_2^{(0)},x_2^{(1)},x_2^{(2)},z^{(0)},z^{(1)},z^{(2)},F_{1,\nu}^{(6)},F_{1,\nu}^{(7)},F_{3,\nu}^{(45)})\cap D(x_1^{(0)})\cap D(x_2^{(3)}),\]
where
\[\betaegin{array}{rl}
F_{3,\nu}^{(45)} & ={F_{2,\nu}^{(15)}}^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
& =({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3-{x_1^{(0)}}^{23}{x_2^{(3)}}^{14}z^{(3)}\\
= & ({x_1^{(0)}}^6{x_2^{(3)}}^4z^{(3)})^3(1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}),\\
\varepsilonnd{array}\]
since $D_{45}^{(0,3)}\sigmaubset D(z^{(3)})$. Since $\gammaamma(0)=(x_1^{(0)},0,0)\in X$, and we are considering germs of quasi-ordinary singularities, we have that $|x_1^{(0)}|<<1$ and we deduce that $1-\pihirac{{x_1^{(0)}}^8{x_2^{(3)}}^3}{{z^{(3)}}^2}\neq 0$.
\lambdaabel{Ex4}
\varepsilonnd{Exam}
This example illustrates the fact that we are looking at jet schemes of a germ of quasi-ordinary singularity, instead of jet schemes of the whole affine surface. If we looked at the whole surface there would be other irreducible components that we do not consider here. This is expectable because the components we consider are determined by the invariants of the topological type at the origin, so they describe only what happens in a small neighbourhood of zero. Actually the other components that may appear when looking at the whole affine surface, will project on closed points, different from the origin, of the singular locus.
\betaegin{Lem}
Given $m\in\mathbb Z_{>0}$ and $\nu\in H_m\cup L_m$ with $m+1=l_{i(\nu)}(\nu)$, then
\[F_\nu^{(l_{i(\nu)}(\nu))}= \lambdaeft({x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\rhoight)^{e_{i(\nu)}}\cdot U,\]
where $U$ is a unit in $\mathbb C[[{x_1^{(\nu_1)}}^{\pim 1},{x_2^{(\nu_2)}}^{\pim 1},{F_0^{(\pihirac{l_1(\nu)}{e_0})}}^{\pim 1},\lambdadots,{F_{i(\nu)-2}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{\pim 1}]]$.
When $\nu\notin\rho_1\cup\rho_2$, then $U=1$.
\lambdaabel{LemF'}
\varepsilonnd{Lem}
{\varepsilonm Proof.} We have that $j(m,\nu)=i(\nu)-1$, and, by Lemma \rhoef{Lemfk}, for any $\gammaamma\in D_m^\nu$
\[\betaegin{array}{ll}
\mbox{ord}_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle & \mbox{ for }0\lambdaeq k\lambdaeq i(\nu)-2\\
\\
\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)>\pihirac{m}{e_{i(\nu)-1}}\\
\varepsilonnd{array}\]
Then ord$_t(f_{i(\nu)-1}\circ\gammaamma)\gammaeq\pihirac{m+1}{e_{i(\nu)-1}}=\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)-1}}=\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, and since $\nu\notin N_{i(\nu)}$, $\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$ is not an integer. Hence
\[\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)>\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle.\]
We have by Lemma \rhoef{Lema35}
\[f_{i(\nu)}=f_{i(\nu)-1}^{n_{i(\nu)}}-c_{i(\nu)}x_1^{\alphalpha_1^{(i(\nu))}}x_2^{\alphalpha_2^{(i(\nu))}}f_0^{r_1^{(i(\nu))}}\cdots f_{i(\nu)-2}^{r_{i(\nu)-1}^{(i(\nu))}}+\sigmaum c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}f_0^{r_1}\cdots f_{i(\nu)-1}^{r_{i(\nu)}},\]
and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{i(\nu)-1}^{n_{i(\nu)}}\circ\gammaamma) & >n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle\\
\\
\mbox{ord}_t((c_{i(\nu)}x_1^{\alphalpha_1^{(i(\nu))}}x_2^{\alphalpha_2^{(i(\nu))}}\cdots f_{i(\nu)-2}^{r_{i(\nu)-1}^{(i(\nu))}})\circ\gammaamma) & =n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle\\
\\
\mbox{ord}_t((c_{\underline\alphalpha,\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}\cdots f_{i(\nu)-1}^{r_{i(\nu)}})\circ\gammaamma) &
=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i(\nu)-1}\gammaamma_{i(\nu)-1}\rhoangle+r_{i(\nu)}\mbox{ord}_t(f_{i(\nu)-1}\circ\gammaamma)\\
& >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i(\nu)}\gammaamma_{i(\nu)}\rhoangle\\
& \gammaeq n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle.\\
\varepsilonnd{array}\]
Then ord$_t(f_{i(\nu)}\circ\gammaamma)=n_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle=\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}}$, and
\[F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}= {x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\]
By Lemma \rhoef{LemExpSR}
\[f=f_{i(\nu)}^{e_{i(\nu)}}+\sigmaum c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k\]
where $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$. Then
\[\betaegin{array}{ll}
\mbox{ord}_t(f_{i(\nu)}^{e_{i(\nu)}}\circ\gammaamma)=l_{i(\nu)}(\nu)\\
\\
\mbox{ord}_t((c_{ijk}^{(i(\nu))}x_1^ix_2^jz^k)\circ\gammaamma)\gammaeq l_{i(\nu)}(\nu)\\
\varepsilonnd{array}\]
and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}+G_{i(\nu),\nu},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
and the sum runs over $i,j,k$ such that
\betaegin{enumerate}
\item[(i)] $c_{ijk}^{(i(\nu))}\neq 0$
\item[(ii)] $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=l_{i(\nu)}(\nu)$
\varepsilonnd{enumerate}
Notice that if $\nu\notin\rho_1\cup\rho_2$, then condition (ii) never holds and $G_{\nu,i(\nu)}=0$. In this case we are done. Otherwise, from (\rhoef{eqO}) we deduce that $\gammaamma\in D_m^\nu\sigmaubset D(F_{\nu,i(\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})})$, and hence
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\lambdaeft(1+G_{i(\nu),\nu}/{F_{i(\nu,\nu)}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}
^{e_{i(\nu)}}\rhoight).\]
\pisifill$\Box$
{\betaf\varepsilonm Proof of Lemma \rhoef{LemD}.}
If $m<l_{i(\nu)}(\nu)$, then we deduce from Proposition \rhoef{Prop1} that $D_m^\nu$ is non-empty.
Otherwise $m\gammaeq l_{i(\nu)}(\nu)$, and by definition $F^{(l_{i(\nu)}(\nu))}\in J_m^\nu$, and by Lemma \rhoef{LemF'} (and its proof)
\[F_\nu^{(l_{i(\nu)}(\nu))}={F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}\cdot U,\]
where
\[F_{i_(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}={x_1^{(\nu_1)}}^{\alphalpha_1^{(i(\nu))}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i(\nu))}}{F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}}^{r_1^{(i(\nu))}}{F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})}}
^{r_2^{(i(\nu))}}\cdots{F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})}}^{r_{i(\nu)-1}^{(i(\nu))}}\]
with $(\alphalpha_1^{(i(\nu))},\alphalpha_2^{(i(\nu))})+r_1^{(i(\nu))}\gammaamma_1+\cdots+r_{i(\nu)-1}^{(i(\nu))}\gammaamma_{i(\nu)-1}=n_{i(\nu)}\gammaamma_{i(\nu)}$. And
\[U=1+G_{i(\nu),\nu}/{F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}},\]
where
\[G_{i(\nu),\nu}=\sigmaum c_{ijk}^{(i(\nu))}{x_1^{(\nu_1)}}^i{x_2^{(\nu_2)}}^j{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^k\]
with $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=n_{i(\nu)}e_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, though $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$.
Notice that at level $l_{i(\nu)}(\nu)-1$ we can apply Corollary \rhoef{Corolario3}, and deduce, as in (\rhoef{eqO}), that
\[D_{l_{i(\nu)-1}(\nu)-1}^\nu\sigmaubseteq D(F_{0,\nu}^{(\pihirac{l_1(\nu)}{e_0})}\cdots F_{i(\nu)-2,\nu}^{(\pihirac{l_{i(\nu)-1}(\nu)}{e_{i(\nu)-2}})})\]
Hence $D_{l_{i(\nu)}(\nu)}^\nu$ satisfies the same property, and since, by definition, $D_{l_{i(\nu)}(\nu)}^\nu\sigmaubseteq D(x_1^{(\nu_1)}\cdot x_2^{(\nu_2)})$, we just have to argue that $U\neq 0$.
In Lemma \rhoef{LemF'} we prove that $U=1$ when $\nu\notin\rho_1\cup\rho_2$. Suppose the contrary, then either $\nu_1=0$ or $\nu_2=0$. We work it all out and write $F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}$ in terms of $x_1^{(\nu_1)}$, $x_2^{(\nu_2)}$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$. The key point is that when $G_{i(\nu),\nu}\neq 0$ is because $(i,j)+k\gammaamma_1>n_{i(\nu)}e_{i(\nu)}\gammaamma_{i(\nu)}$ but $\lambdaangle\nu,(i,j)+k\gammaamma_1\rhoangle=n_{i(\nu)}e_{i(\nu)}\lambdaangle\nu,\gammaamma_{i(\nu)}\rhoangle$, because $\nu\in\rho_1\cup\rho_2$. Therefore either $x_1^{(0)}$ or $x_2^{(0)}$ do appear in U (it do not cancel in the quotient $G_{i(\nu),\nu}/{F_{i(\nu),\nu}^{(\pihirac{l_{i(\nu)}(\nu)}{e_{i(\nu)}})}}^{e_{i(\nu)}}$), depending on whether $\nu_1=0$ or $\nu_2=0$. Then we have that $U$ depends on the origin of the jet $\gammaamma(0)\in X$, and since we are dealing with germs of quasi-ordinary singularities, we have that $|x_i^{(0)}|<<1$ for $i=1,2$, and we deduce that $U\neq 0$.
\pisifill$\Box$
{\betaf\varepsilonm Proof of Lemma \rhoef{Lemfk}.} We distinguish the cases $\nu\in H_m$ and $\nu\in L_m$. For $\nu\in H_m$, $j(m,\nu)=0$ and we have to prove that $\mbox{ord}_t(f_k\circ\gammaamma)>\pihirac{m}{e_k}$ for $0\lambdaeq k\lambdaeq g$. By Proposition \rhoef{CHm} it is true for $k=0$. For $k=g$ the claim is obvious, and for $1\lambdaeq k\lambdaeq g-1$ we use the expansion (\rhoef{eqSRpqprima}) in Lemma \rhoef{LemExpSR},
\[f=f_k^{e_k}+\sigmaum_{(i,j)+r\gammaamma_1>n_ke_k\gammaamma_k}c_{ijr}^{(k)}x_1^ix_2^jz^r.\]
Suppose that there exist $c_{ijr}^{(k)}\neq 0$ such that $\lambdaangle\nu,(i,j)\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq m$. Then, using that ord$_t(z\circ\gammaamma)>\pihirac{m}{n}$ and $k\gammaeq 1$, we have the following inequalities
\[l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\pihirac{m}{n}<l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq l_k(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle+r\mbox{ ord}_t(z\circ\gammaamma)\lambdaeq m.\]
Then $l_1(\nu)-r\lambdaangle\nu,\gammaamma_1\rhoangle<m(1-\pihirac{r}{n})$, and since $r<n$ this is equivalent to $\lambdaangle\nu,\gammaamma_1\rhoangle<\pihirac{m}{n}$, which contradicts the fact that $j(m,\nu)=0$. Hence ord$_t(c_{ijr}^{(k)}x_1^ix_2^jz^r\circ\gammaamma)>m$ and therefore ord$_t(f_k^{e_k}\circ\gammaamma)>m$ as we wanted to prove.
For $\nu\in L_m$ the proof is by induction on $j(m,\nu)$. And we will make use repeatedly of the following observation. In general, for any function $f\in\mathbb C[x_1,x_2,z]$, and any $m$-jet $\gammaamma$, there is no relation among ord$_t(f\circ\gammaamma)$ and ord$_t(f\circ\pii_{m,m'}(\gammaamma))$ with $m'<m$. But if ord$_t(x_i\circ\gammaamma)\neq 0$ for $i=1,2$, ord$_t(z\circ\gammaamma)\neq 0$, and for $m'<m$, the $m'$-jet $\gammaamma':=\pii_{m,m'}(\gammaamma)$ verifies that ord$_t(x_i\circ\gammaamma')\neq 0$ and ord$_t(z\circ\gammaamma')\neq 0$, then
\[\mbox{ord}_t(f\circ\gammaamma)=\mbox{ord}_t(f\circ\gammaamma').\]
By Remark \rhoef{RemDefHL} (ii) the first case of induction is $j(m,\nu)=1$. Then in particular $l_1(\nu)<l_2(\nu)$. For $\gammaamma\in D_m^\nu$, set $\betaar m:=l_1(\nu)-1<m$ and $\betaar\gammaamma:=\pii_{m,\betaar m}(\gammaamma)$. Then $j(\betaar m,\nu)=0$ and $\betaar\gammaamma\in D_{\betaar m}^\nu$. By Proposition \rhoef{CHm} we have
\[\mbox{ord}_t(z\circ\betaar\gammaamma)>\pihirac{\betaar m}{n}=\pihirac{l_1(\nu)-1}{n}=\lambdaangle\nu,\gammaamma_1\rhoangle-\pihirac{1}{n}\]
Hence ord$_t(z\circ\betaar\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$ and therefore ord$_t(z\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_1\rhoangle$. Suppose that the inequality is strict, ord$_t(z\circ\gammaamma)>\lambdaangle\nu,\gammaamma_1\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_1=z^{n_1}-x_1^{a_1}x_2^{b_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1\gammaamma_1}x_1^{i_1}x_2^{i_2}z^k\]
Then
\[\betaegin{array}{rl}
\mbox{ord}_t(z^{n_1}\circ\gammaamma) & >n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{a_1}x_2^{b_1}\circ\gammaamma) & =n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1\lambdaangle\nu,\gammaamma_1\rhoangle\\
\varepsilonnd{array}\]
and hence ord$_t(f_1\circ\gammaamma)=n_1\lambdaangle\nu,\gammaamma_1\rhoangle$. Again by Lemma \rhoef{LemExpSR} we have
\[f=f_1^{e_1}+\sigmaum_{(i_1,i_2)+k\gammaamma_1>n_1e_1\gammaamma_1}c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\]
and since
\[\betaegin{array}{rl}
\mbox{ord}_t(f_1^{e_1}\circ\gammaamma) & =e_1n_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\\
\mbox{ord}_t(x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma) & >\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq n_1e_1\lambdaangle\nu,\gammaamma_1\rhoangle=l_1(\nu)\\
\varepsilonnd{array}\]
we deduce ord$_t(f\circ\gammaamma)=l_1(\nu)\lambdaeq m$, which is a contradiction. Then ord$_t(z\circ\gammaamma)=\lambdaangle\nu,\gammaamma_1\rhoangle$.
Now we prove that ord$_t(f_1\circ\gammaamma)>\pihirac{m}{e_1}$. Suppose the contrary, ord$_t(f_1\circ\gammaamma)\lambdaeq\pihirac{m}{e_1}$. If $g=1$ there is nothing to prove.
If $g=2$, then we consider the expansion given in Lemma \rhoef{Lema35}
\[f=f_2=f_1^{n_2}-c_2x_1^{\alphalpha_1^{(2)}}x_2^{\alphalpha_2^{(2)}}z^{r_1^{(2)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{i_1}x_2^{i_2}z^{r_1}f_1^{r_2}\]
where $(i_1,i_2)+r_1\gammaamma_1+r_2\gammaamma_2>n_2\gammaamma_2$, and
\[\betaegin{array}{rl}
\mbox{ord}_t(f_1^{n_2}\circ\gammaamma) & \lambdaeq m,\\
\\
\mbox{ord}_t(x_1^{\alphalpha_1^{(2)}}x_2^{\alphalpha_2^{(2)}}z^{r_1^{(2)}}\circ\gammaamma) & =n_2\lambdaangle\nu,\gammaamma_2\rhoangle=l_2(\nu)>m\\
\varepsilonnd{array}\]
Then there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
\[n_2\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r_1\gammaamma_1\rhoangle+r_2\mbox{ord}_t(f_1\circ\gammaamma)\]
or equivalently
\[(n_2-r_2)\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r_1\gammaamma_1\rhoangle\gammaeq (n_2-r_2)\lambdaangle\nu,\gammaamma_2\rhoangle\]
And since $r_2<n_2$ we conclude ord$_t(f_1\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_2\rhoangle>\pihirac{m}{e_1}$, which is a contradiction.
If $g>2$, by Lemma \rhoef{LemExpSR}
\[f=f_1^{e_1}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}f_1^{r_2}\]
and since we are supposing that ord$_t(f_1^{e_1}\circ\gammaamma)\lambdaeq m$, there must exists $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
$\mbox{ord}_t(f_1^{e_1}\circ\gammaamma)=\mbox{ord}_t(x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}f_1^{r_2}\circ\gammaamma)$,
hence
\[e_1\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1\rhoangle+r_2\mbox{ord}_t(f_1\circ\gammaamma)\]
or equivalently
\[(e_1-r_2)\mbox{ord}_t(f_1\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1\rhoangle\gammaeq n_2e_2\lambdaangle\nu,\gammaamma_2\rhoangle-r_2\lambdaangle\nu,\gammaamma_2\rhoangle\]
and since $r_2<e_1$ we conclude that ord$_t(f_1\circ\gammaamma)\gammaeq \lambdaangle\nu,\gammaamma_2\rhoangle$, which is a contradiction.
The rest is simple, by Lemma \rhoef{LemExpSR}, for $k>1=j(m,\nu)$, we have
\[f=f_k^{e_k}+\sigmaum c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\]
and since ord$_t(c_{i_1i_2k}x_1^{i_1}x_2^{i_2}z^k\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+k\gammaamma_1\rhoangle\gammaeq l_k(\nu)>m$, we deduce ord$_t(f_k^{e_k}\circ\gammaamma)>m$.
Suppose now that the claim is true for $j(m,\nu)=j$ and we will prove it for $j(m,\nu)=j+1$. Let $\gammaamma\in D_m^\nu$, with
$l_{j+1}(\nu)\lambdaeq m<l_{j+1}(\nu)$. We set $\betaar m=l_{j+1}(\nu)-1$ and $\betaar\gammaamma=\pii_{m,\betaar m}(\gammaamma)$. Then $\gammaamma\in D_{\betaar m}^\nu$ and $j(\betaar m,\nu)=i\lambdaeq j$, where
\[l_i(\nu)\lambdaeq\betaar m<l_{i+1}(\nu)=\cdots=l_{j+1}(\nu).\]
Then, by Lemma \rhoef{l-ord}, this is equivalent to
\[\betaegin{array}{c}
n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle=\lambdaangle\nu,\gammaamma_{i+2}\rhoangle\\
\\
n_{i+2}\lambdaangle\nu,\gammaamma_{i+2}\rhoangle=\lambdaangle\nu,\gammaamma_{i+3}\rhoangle\\
\vdots\\
n_j\lambdaangle\nu,\gammaamma_j\rhoangle=\lambdaangle\nu,\gammaamma_{j+1}\rhoangle\\
\varepsilonnd{array}\]
By induction hypothesis we deduce ord$_t(f_k\circ\gammaamma)=\mbox{ord}_t(f_k\circ\betaar\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $0\lambdaeq k<i$. We are going to prove that ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By induction we have
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\mbox{ord}_t(f_i\circ\betaar\gammaamma)>\pihirac{\betaar m}{e_i}=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle-\pihirac{1}{e_i}\]
Therefore
\[\mbox{ord}_t(f_i\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\]
Let us suppose that ord$_t(f_i\circ\gammaamma)>\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f_{i+1}=f_i^{n_{i+1}}-c_{i+1}x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\cdots f_{i-1}^{r_i^{(i+1)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
and we have
\[\betaegin{array}{rl}
\mbox{ord}_t(f_i^{n_{i+1}}\circ\gammaamma) & > n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(x_1^{\alphalpha_1^{(i+1)}}x_2^{\alphalpha_2^{(i+1)}}z^{r_1^{(i+1)}}\circ\gammaamma) & =n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle\\
\varepsilonnd{array}\]
Therefore ord$_t(f_{i+1}\circ\gammaamma)=n_{i+1}\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$. By Lemma \rhoef{LemExpSR}
\[f=f_{i+1}^{e_{i+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\]
where $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}>n_{i+1}e_{i+1}\gammaamma_{i+1}$. Then
\[\betaegin{array}{rl}
\mbox{ord}_t(f_{i+1}^{e_{i+1}}\circ\gammaamma) & =l_{i+1}(\nu)\\
\\
\mbox{ord}_t(c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_i^{r_{i+1}}\circ\gammaamma) & >\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i+1}\gammaamma_{i+1}\rhoangle\gammaeq l_{i+1}(\nu)\\
\varepsilonnd{array}\]
Then ord$_t(f\circ\gammaamma)=l_{i+1}(\nu)=l_{j+1}(\nu)\lambdaeq m$, which is a contradiction. Therefore ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We can prove that ord$_t(f_k\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{k+1}\rhoangle$ for $i<k\lambdaeq j$ one after the other exactly as the proof of ord$_t(f_i\circ\gammaamma)=\lambdaangle\nu,\gammaamma_{i+1}\rhoangle$.
We prove now that ord$_t(f_{j+1}\circ\gammaamma)>\pihirac{m}{e_{j+1}}$. Suppose the contrary, ord$_t(f_{j+1}\circ\gammaamma)\lambdaeq\pihirac{m}{e_{j+1}}$.
If $j+2=g$ then by Lemma \rhoef{LemExpSR}
\[f=f_{j+2}=f_{j+1}^{n_{j+2}}-c_{j+2}x_1^{\alphalpha_1^{(g)}}x_2^{(g)}z^{r_1^{(g)}}\cdots f_j^{r_{j+1}^{(g)}}+\sigmaum c_{\underline{\alphalpha},\underline{r}} x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
while, if $j+2<g$ we have the expansion
\[f=f_{j+1}^{e_{j+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}} x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
with $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+2}\gammaamma_{j+2}>n_{j+2}e_{j+2}\gammaamma_{j+2}$. In both cases we have
\[f=f_{j+1}^{e_{j+1}}+\sigmaum c_{\underline{\alphalpha},\underline{r}}x_1^{\alphalpha_1}x_2^{\alphalpha_2}z^{r_1}\cdots f_{j+1}^{r_{j+2}}\]
with $(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+2}\gammaamma_{j+2}\gammaeq n_{j+2}e_{j+2}\gammaamma_{j+2}$. Looking at the expansion, since ord$_t(f_{j+1}^{e_{j+1}}\circ\gammaamma)\lambdaeq m$, there must exist $c_{\underline{\alphalpha},\underline{r}}\neq 0$ such that
ord$_t(f_{j+1}^{e_{j+1}}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}\rhoangle+r_{j+2}\mbox{ord}_t(f_{j+1}\circ\gammaamma)$, or equivalently
\[(e_{j+1}-r_{j+2})\mbox{ord}_t(f_{j+1}\circ\gammaamma)=\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{j+1}\gammaamma_{j+1}\rhoangle\gammaeq n_{j+2}e_{j+2}\lambdaangle\nu,\gammaamma_{j+2}\rhoangle-r_{j+2}\lambdaangle\nu,\gammaamma_{j+2}\rhoangle\]
And since $r_{j+2}<e_{j+1}$ we deduce that ord$_t(f_{j+1}\circ\gammaamma)\gammaeq\lambdaangle\nu,\gammaamma_{j+2}\rhoangle>\pihirac{m}{e_{j+1}}$, which is a contradiction.
Finally we prove that ord$_t(f_k^{e_k}\circ\gammaamma)>m$. By Lemma \rhoef{LemExpSR}, $f=f_k^{e_k}+\sigmaum c_{i_1i_2r}x_1^{i_1}x_2^{i_2}z^r$
with $(i_1,i_2)+r\gammaamma_1>n_ke_k\gammaamma_k$. Then ord$_t(x_1^{i_1}x_2^{i_2}z^r\circ\gammaamma)=\lambdaangle\nu,(i_1,i_2)+r\gammaamma_1\rhoangle\gammaeq l_k(\nu)>m$, and the result follows.
\pisifill$\Box$
\betaegin{Lem} For $m\in\mathbb Z_{>0}$ and $\nu\in L_m$, we have the following.
\betaegin{enumerate}
\item[(i)] If $i\lambdaeq j(m,\nu)$, then
\[F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}=0\mbox{ if and only if }{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}=0\]
Roughly speaking the part $G_{i,\nu}$ in equation (\rhoef{EqP}) is not meaningful.
\
\item[(ii)] For $m(\nu)\lambdaeq j\lambdaeq g_1$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})\cap
D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}),\]
and for $g_1<j\lambdaeq g$ we have
\[V(I^\nu,F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})})_{m(\nu)\lambdaeq i\lambdaeq j}\cap D(x_1^{(\nu_1)})\cap D(x_2^{(\nu_2)})\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1
\rhoangle)})\cap D(F_{1,\nu}^{(\pihirac{l_2(\nu)}{e_1})})\cap\cdots\cap D(F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}).\]
\varepsilonnd{enumerate}
\lambdaabel{TechLem}
\varepsilonnd{Lem}
{\varepsilonm Proof.}
(i) First observe that if $j(m,\nu)\gammaeq i$, then $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}\in J_m^\nu$ and it has the form given in equation (\rhoef{EqP}) in Corollary \rhoef{Corolario3}. The proof is obvious if $G_{i,\nu}=0$. Suppose the contrary. Then $\nu\in\rho_1\cup\rho_2$, and the claim is not obvious if $\betaar G_{i,\nu}\neq 0$, where
\[\betaar G_{i,\nu}=\sigmaum c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}\]
with the conditions $c_{\underline{\alphalpha},\underline{r}}\neq 0$, $\lambdaangle\nu,(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_i\gammaamma_i\rhoangle=n_i\lambdaangle\nu,\gammaamma_i\rhoangle$ and $r_i=0$. Then, $F_{i-1,\nu}^{(\pihirac{l_i(\nu)}{e_{i-1}})}=0$ if and only if
\[c_i{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}+\betaar G_{i,\nu}=0\]
where, remember that any term $c_{\underline{\alphalpha},\underline{r}}{x_1^{(\nu_1)}}^{\alphalpha_1}{x_2^{(\nu_2)}}^{\alphalpha_2}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1}\cdots
{F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}}$ in particular appears in $F_{i,\nu}^{(\pihirac{l_i(\nu)}{e_i})}$, and hence it satisfies that $(\alphalpha_1^{(i)},\alphalpha_2^{(i)})+r_1\gammaamma_1+\cdots+r_{i-1}^{(i)}\gammaamma_{i-1}<(\alphalpha_1,\alphalpha_2)+r_1\gammaamma_1+\cdots+r_{i-1}\gammaamma_{i-1}$. Then we can write last equation as
\[{x_1^{(\nu_1)}}^{\alphalpha_1^{(i)}}{x_2^{(\nu_2)}}^{\alphalpha_2^{(i)}}{z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}}^{r_1^{(i)}}\cdots {F_{i-2,\nu}^{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})}}^{r_{i-1}^{(i)}}\lambdaeft(c_i+P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)},\lambdadots,F_{i-2,\nu}^
{(\pihirac{l_{i-1}(\nu)}{e_{i-2}})})\rhoight)\]
where $P$ is a polynomial non-unit. If we work it all out, then we can write $P$ as a polynomial in $x_1^{(\nu_1)}$, $x_2^{(\nu_2)}$ and $z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}$. Now we use that $\nu\in\rho_1\cup\rho_2$, and hence either $\nu_1=0$ or $\nu_2=0$. Then $P(x_1^{(\nu_1)},x_2^{(\nu_2)},z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)})$ depends on the origin $\gammaamma(0)$ of the jets, and, since we are dealing with germs of quasi-ordinary singularities we can always suppose that $|x_i^{(\nu_i)}|<<1$ for $i=1$ or $2$ depending on whether $\nu_1=0$ or $\nu_2=0$. As a consequence we can always suppose that $P<<c_i$.
(ii) The inclusions follow directly from Corollary \rhoef{Corolario3} when $G_{j,\nu}=0$. If $G_{j,\nu}\neq 0$, the proof is by induction on $j$. For $j=m(\nu)$ the claim says that
\[V(I^\nu,F_{m(\nu),\nu}^{(\pihirac{l_{m(\nu)}(\nu)}{e_{m(\nu)}})})\cap D(M)\sigmaubset D(z^{(\lambdaangle\nu,\gammaamma_1\rhoangle)}),\]
where
\[M=\lambdaeft\{\betaegin{array}{cl}
x_1^{(\nu_1)} & \mbox{ if }g_1\gammaeq 1\\
\\
x_1^{(\nu_1)}x_2^{(\nu_2)} & \mbox{ if }g_1=0\\
\varepsilonnd{array}\rhoight.\]
and it follows by Lemma \rhoef{Lemfk}. Suppose it is true for $j-1$ and we prove it for $j$. We only have to prove that $F_{j-1,\nu}^{(\pihirac{l_j(\nu)}{e_{j-1}})}\neq 0$. But this follows by (i).
\pisifill$\Box$
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\begin{document}
\title[Nakamaye's theorem on log canonical pairs]{Nakamaye's theorem on log canonical pairs}
\dedicatory{\normalsize \dag \ Dedicated to the memory of Fassi}
\author[S. Cacciola and A.F. Lopez]{Salvatore Cacciola* and Angelo Felice Lopez*}
\thanks{* Research partially supported by the MIUR national project ``Geometria delle variet\`a
algebriche" PRIN 2010-2011.}
\address{\hspace{.1in}kip -.43cm Dipartimento di Matematica e Fisica, Universit\`a di Roma
Tre, Largo San Leonardo Murialdo 1, 00146, Roma, Italy. e-mail {\tt cacciola@mat.uniroma3.it, lopez@mat.uniroma3.it}}
\thanks{{^{-1}t 2010 Mathematics Subject Classification} : Primary 14C20, 14F18. Secondary 14E15, 14B05}
\begin{abstract}
We generalize Nakamaye's description, via intersection theory, of the augmented base locus of a big and nef divisor on a normal pair with log-canonical singularities or, more generally, on a normal variety with non-lc locus of dimension $\leq 1$. We also generalize Ein-Lazarsfeld-Musta{\c{t}}{\u{a}}-Nakamaye-Popa's description, in terms of valuations, of the subvarieties of the restricted base locus of a big divisor on a normal pair with klt singularities.
\end{abstract}
\maketitle
\section{Introduction}
\label{intro}
Let $X$ be a normal complex projective variety and let $D$ be a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. The stable base locus
\[ \mathbf{B}(D)= \bigcap_{E \geq 0 : E \sim _{{\mathbb Q}} D} {\rm Supp}(E) \]
is an important closed subset associated to $D$, but it is often difficult to handle. On the other hand, there are two, perhaps even more important, base loci associated to $D$.
One of them is the augmented base locus (\cite{n}, \cite[Def. 1.2]{elmnp1})
\[ \mathbf{B}_+(D)= \bigcap_{E \geq 0 : D - E \mbox{\ ample}} {\rm Supp}(E) \]
where $E$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor. Since this locus measures the failure of $D$ to be ample, it has proved to be a key tool in several recent important results
in birational geometry, such as Takayama \cite{t}, Hacon and McKernan's \cite{hm} effective birationality of pluricanonical maps or Birkar, Cascini, Hacon and McKernan's \cite{bchm} finite generation of the canonical ring, just to mention a few.
One way to compute $\mathbf{B}_+(D)$ is to pick a sufficiently small ample ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $A$ on $X$, because then one knows that $\mathbf{B}_+(D) = \mathbf{B}(D-A)$ by \cite[Prop. 1.5]{elmnp1}.
In the case when $D$ is also nef, for every subvariety $V \subseteqset X$ of dimension $d \geq 1$ such that $D^d \cdot V = 0$, we have that $D_{|V}$ is not big, whence $s(D-A)_{|V}$ cannot be effective for any $s ^{-1}n {\mathbb N}$ and therefore $V \subseteqseteq \mathbf{B}(D-A) = \mathbf{B}_+(D)$. Now define
\[ {\rm Null}(D) = \bigcup_{V \subseteqset X: D^d \cdot V = 0} V \]
so that, by what we just said,
\begin{equation}
\label{nak}
{\rm Null}(D) \subseteqseteq \mathbf{B}_+(D).
\end{equation}
A somewhat surprising result of Nakamaye \cite[Thm. 0.3]{n} (see also \cite[\S 10.3]{laz}) asserts that, if $X$ is {^{-1}t smooth} and $D$ is big and nef, then in fact equality holds in \eqref{nak}.
As is well-known, in birational geometry, one must work with normal varieties with some kind of (controlled) singularities. In the light of this, it becomes apparent that it would be nice to have a generalization of Nakamaye's Theorem to normal varieties. While in positive characteristic the latter has been recently proved to hold, on any projective scheme, by Cascini, McKernan and Musta{\c{t}}{\u{a}} \cite[Thm. 1.1]{cmm}, we will show in this article a generalization to normal complex varieties with log canonical singularities. This partially answers a question in \cite{cmm}.
More precisely let us define
\begin{defn}
{\rm Let $X$ be a normal projective variety. The {\bf non-lc locus} of $X$ is
\[ X_{\mathrm{nlc}} =\bigcap_{{\Delta}} \mathrm{Nlc}(X,{\Delta}) \]
where ${\Delta}$ runs among all effective Weil ${\mathbb Q}$-divisors such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})$ is the locus of points $x ^{-1}n X$ such that $(X, {\Delta})$ is not log canonical at $x$.}
\end{defn}
Using Ambro's and Fujino's theory of non-lc ideal sheaves \cite{a}, \cite{f} and a modification of some results of de Fernex and Hacon \cite{dh}, we prove
^{\vee}skip .3cm
\begin{bthm}
\label{main}
Let $X$ be a normal projective variety such that $\dim X_{\mathrm{nlc}} \leq 1$.
Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bthm}
^{\vee}skip .2cm
This easily gives the following
\begin{bcor}
\label{dim}
Let $X$ be a normal projective variety such that $\dim \operatorname{Sing}(X) \leq 1$ or $\dim X \leq 3$ or there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ such that $(X, {\Delta})$ is log canonical.
Let $D$ be a big and nef ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$. Then
\[ \mathbf{B}_+(D)=\operatorname{Null}(D). \]
\end{bcor}
^{\vee}skip .2cm
Moreover, using a striking result of Gibney, Keel and Morrison \cite[Thm. 0.9]{gkm}, we can give a very quick application to the moduli space of stable pointed curves.
\begin{bcor}
\label{mod}
Let $g \geq 1$ and let $D$ be a big and nef ${\mathbb Q}$-divisor on $\overline{M}_{g,n}$. Then
\[ \mathbf{B}_+(D) \subseteqseteq \partial \overline{M}_{g,n}. \]
\end{bcor}
Thus, for example, one gets new compactifications of $M_{g,n}$ by taking rational maps associated to such divisors.
The second base locus associated to any pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor $D$, measuring how far $D$ is from being nef, is the restricted base locus \cite[Def. 1.12]{elmnp1}.
\begin{defn}
{\rm Let $X$ be a normal projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. The {\bf restricted base locus} of $D$ is
\[ \mathbf{B}_-(D)= \bigcup_{A {\ \rm ample}} \mathbf{B}(D+A) \]
where $A$ runs among all ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors such that $D+A$ is a ${\mathbb Q}$-divisor.}
\end{defn}
Restricted base loci are countable unions of subvarieties by \cite[Prop. 1.19]{elmnp1}, but not always closed \cite[Thm. 1.1]{les}.
For a big ${\mathbb Q}$-divisor $D$ on a {^{-1}t smooth} variety $X$, the subvarieties of $\mathbf{B}_-(D)$ are precisely described in \cite[Prop. 2.8]{elmnp1} (also in positive characteristic in \cite[Thm. 6.2]{mus}) in terms of asymptotic valuations.
\begin{defn}
\label{as1}
{\rm (\cite[Def. III.2.1]{nak}, \cite[Lemma 3.3]{elmnp1}, \cite[\S 1.3]{bbp}, \cite[\S 2]{dh})
Let $X$ be a normal projective variety, let $D$ be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$, that is $v$ is a positive integer multiple of the valuation associated to a prime divisor $\Gamma$ lying on a birational model $f : Y \to X$. The center of $v$ on $X$ is $c_X(v)=f(\Gamma)$.
If $D$ is big, we set
\[ v(\|D\|) = ^{-1}nf \{v(E), E \ \mbox{effective} \ {\mathbb R}\mbox{-Cartier} \ {\mathbb R}\mbox{-divisor on X such that} \ E \equiv D\}; \]
if $D$ is pseudoeffective, we pick an ample divisor $A$ and set
\[ v(\|D\|) = \lim_{^{\vee}arepsilon \to 0^+} v(\|D + ^{\vee}arepsilon A\|). \]
If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and $b ^{-1}n {\mathbb N}$ is such that $bD$ is Cartier and $|bD| \neq \emptyset$, we set
(see \cite[Def. 2.14]{cd} or \cite[Def. 2.2]{elmnp1} for the case $D$ big)
\[ v(\langle D \rangle) = \lim_{m \to + ^{-1}nfty}\frac{v(|mbD|)}{mb} \]
where, if $g$ is an equation, at the generic point of $c_X(v)$, of a general element in $|mbD|$, then $v(|mbD|) = v(g)$.}
\end{defn}
Now the main content of \cite[Prop. 2.8]{elmnp1} is that, given a discrete valuation $v$ on a {^{-1}t smooth} $X$ with center $c_X(v)$ and a big divisor $D$, then $c_X(v) \subseteqseteq \mathbf{B}_-(D)$ if and only if $v(\|D\|) > 0$. Using the main result of \cite{cd} we give a generalization to normal pairs with klt singularities.
\begin{bthm}
\label{main2}
Let $X$ be a normal projective variety such that there exists an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ with $(X,{\Delta})$ a klt pair. Let $v$ be a divisorial valuation on $X$.
Then
\begin{itemize}
^{-1}tem [(i)] If $D$ is a big Cartier divisor on $X$ we have
\[ v(\langle D \rangle) > 0 \ \mbox{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D) \ \mbox{if and only if} \ \limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty. \]
^{-1}tem [(ii)] If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
\[ v(\|D\|) > 0 \ \textrm{if and only if} \ c_X(v)\subseteqseteq \mathbf{B}_-(D). \]
\end{itemize}
\end{bthm}
^{\vee}skip .2cm
\noindent {^{-1}t Acknowledgments}. We wish to thank Lorenzo Di Biagio for some helpful discussions.
\section{Non-lc ideal sheaves}
\label{basic}
\begin{notn}
{\rm Throughout the article we work over the complex numbers. Given a variety $X$ and a coherent sheaf of ideals $\mathcal{J} \subseteqset \mathcal{O}_X$, we denote by $\mathcal{Z}(\mathcal{J})$ the closed subscheme of $X$ defined by $\mathcal{J}$. If $X$ is a normal projective variety and ${\Delta}$ is a Weil ${\mathbb Q}$-divisor on $X$, we call $(X, {\Delta})$ a {\bf pair} if $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier. We refer to \cite[Def. 2.34]{km} for the various notions of singularities of pairs.}
\end{notn}
\begin{defn}
{\rm Let $X$ be a normal projective variety and let ${\Delta} = \sum\limits_{i=1}^s d_i D_i$ be a Weil ${\mathbb Q}$-divisor on $X$, where the ${D_i}'s$ are distinct prime divisors.
Given $a ^{-1}n {\mathbb R}$ we set ${\Delta}^{>a}=\sum\limits_{1 \leq i \leq s : d_i>a} d_i D_i$, ${\Delta}^+={\Delta}^{>0}$, ${\Delta}^- =(-{\Delta})^+$ and ${\Delta}^{<a}=-((-{\Delta})^{>-a})$. The {\bf round up} of ${\Delta}$ is $\lceil {\Delta} \rceil = \sum\limits_{i=1}^s \lceil d_i \rceil D_i$ and the {\bf round down} is $\lfloor {\Delta} \rfloor = \sum\limits_{i=1}^s \lfloor d_i \rfloor D_i$. We also set ${\Delta}^{\#} = {\Delta}^{<-1} + {\Delta}^{>-1}$.}
\end{defn}
The following is easily proved.
\begin{rem}
\label{sha}
{\rm Let $X$ be a normal projective variety and let ${\Delta}, {\Delta}'$ be Weil ${\mathbb Q}$-divisors on $X$. Then
\begin{itemize}
^{-1}tem[(i)] $\lceil (-{\Delta})^{\#} \rceil = \lceil -({\Delta}^{<1}) \rceil - \lfloor {\Delta}^{>1} \rfloor$;
^{-1}tem[(ii)] If ${\Delta} \leq {\Delta}'$, then $\lceil {\Delta}^{\#} \rceil \leq \lceil ({\Delta}')^{\#} \rceil$.
\end{itemize}}
\end{rem}
We recall the definition of non-lc ideal sheaves \cite[Def. 4.1]{a}, \cite[Def. 2.1]{f}.
\begin{defn}
\label{inlc}
{\rm Let $(X, {\Delta})$ be a pair and let $f: Y \to X$ be a resolution of $X$ such that ${\Delta}_Y :=f^{\ast}(K_X+{\Delta})-K_Y$ has simple normal crossing support.
The {\bf non-lc ideal sheaf associated to} $(X,{\Delta})$ is
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor). \]}
\end{defn}
\begin{rem}
\label{nlc}
{\rm Non-lc ideal sheaves are well-defined by \cite[Prop. 2.6]{f}, \cite[Rmk. 4.2(iv)]{a}. Moreover, when ${\Delta}$ is effective and $f : Y \to X$ is a log-resolution of $(X, {\Delta})$, we have that the non-lc locus of $(X, {\Delta})$ is, set-theoretically, $\operatorname{Nlc}(X,{\Delta})= f(\operatorname{Supp}({\Delta}_Y^{>1})) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))$ \cite[Lemma 2.2]{f}.}
\end{rem}
\begin{rem}
\label{ic}
{\rm The non-lc ideal sheaf of a pair $(X,{\Delta})$ with ${\Delta}$ effective is an integrally closed ideal.}
\begin{proof}
With notation as in Definition \ref{inlc}, set $G = \lceil - ({\Delta}_Y^{<1}) \rceil$ and $N= \lfloor {\Delta}_Y^{>1} \rfloor$, so that $G$ and $N$ are effective divisors without common components, $G$ is $f$-exceptional and $\mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(G-N) = f_{\ast}\mathcal{O}_Y(-N)$ by Fujita's lemma \cite[Lemma 2.2]{fta}, \cite[Lemma 1-3-2]{kmm}, \cite[Lemma 4.5]{dh}. Therefore $\mathcal{J}_{NLC}(X,{\Delta})$ is an ideal sheaf and it is integrally closed by \cite[Prop. 9.6.11]{laz}.
\end{proof}
\end{rem}
Our next goal is to prove, using techniques and results in de Fernex-Hacon \cite{dh}, that non-lc ideal sheaves have a unique maximal element. To this end we will use some results of Fujino \cite{f} and de Fernex-Hacon \cite{dh} that we wish to recall for the reader's convenience.
\begin{lemma} \cite[Lemma 2.7]{f}
\label{fuj}
Let $g : Y' \to Y$ be a proper birational morphism between smooth varieties and let $B_Y$ be an ${\mathbb R}$-divisor on $Y$ having simple normal crossing support. Assume that $B_{Y'} := g^{\ast}(K_Y+B_Y)-K_{Y'}$ also has simple normal crossing support. Then
\[ g_{\ast} {\mathcal O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) \cong {\mathcal O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ). \]
\end{lemma}
\begin{defn}
\label{dfh1}
{\rm (\cite[Def. 3.1]{dh})
Let $f: Y \to X$ be a proper birational morphism between normal varieties and let $K_Y$ be a canonical divisor on $Y$ and $K_X = f_{\ast} K_Y$. For every $m \geq 1$ define $K_{m, Y/X} = K_Y - \frac{1}{m}f^{\natural}(mK_X)$, where $f^{\natural}(mK_X)$ is the divisor on $Y$ such that ${\mathcal O}_Y(-f^{\natural}(mK_X)) = ({\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y)^{^{\vee}ee ^{\vee}ee}$.}
\end{defn}
\begin{lemma}
\label{dh2}
Let $m \geq 1$. In (i)-(iv) below let $f: Y \to X$ be a proper birational morphism between normal varieties. Then
\begin{itemize}
^{-1}tem [(i)] If $X$ is Gorenstein then $K_{m, Y/X} = K_{Y/X} := K_Y + f^{\ast}(-K_X)$;
^{-1}tem [(ii)] \cite[Rmk 3.3]{dh} For all $q \geq 1$ we have $K_{m, Y/X} \leq K_{mq, Y/X}$;
^{-1}tem [(iii)] \cite[Lemma 3.5]{dh} Assume that $mK_Y$ is Cartier and ${\mathcal O}_X(-mK_X) \cdot {\mathcal O}_Y$ is invertible. Let $Y'$ be a normal variety and let $g : Y' \to Y$ be a proper birational morphism. Then $K_{m, Y'/X} = K_{m, Y'/Y} + g^{\ast} K_{m, Y/X}$;
^{-1}tem [(iv)] \cite[Rmk 3.9]{dh} Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and assume that $m(K_X + {\Delta}elta)$ is Cartier. Then $K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{m, Y/X}$;
^{-1}tem [(v)] \cite[Thm. 5.4 and its proof]{dh} For every $m \geq 2$ there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ such that $m{\Delta}_m$ is integral, $\lfloor {\Delta}_m \rfloor = 0$, $({\Delta}_m)_Y$ has simple normal crossing support, $f$ is a log-resolution for the log-pair $((X, {\Delta}_m), \mathcal{O}_X(-mK_X))$, $K_X + {\Delta}_m$ is ${\mathbb Q}$-Cartier and $K_{m, Y/X} = K_Y + f_{\ast}^{-1}({\Delta}_m) - f^{\ast}(K_X+{\Delta}_m)$.
\end{itemize}
\end{lemma}
In (iv) and (v) above $f_{\ast}^{-1}({\Delta})$ is the proper transform of ${\Delta}$. Note that our ${\Delta}_Y$ (Definition \ref{inlc}) is different from the one in \cite[Def. 3.8]{dh}.
Now we have
\begin{prop}
\label{max}
Let $X$ be a normal projective variety. Then there exists a Weil ${\mathbb Q}$-divisor ${\Delta}_0$ on $X$ such that $\lfloor {\Delta}_0 \rfloor = 0$,
$K_X + {\Delta}_0$ is ${\mathbb Q}$-Cartier and
\[ \mathcal{J}_{NLC}(X,{\Delta}) \subseteqseteq \mathcal{J}_{NLC}(X,{\Delta}_0) \]
for every pair $(X,{\Delta})$ with ${\Delta}$ effective.
\end{prop}
\begin{proof}
Fix a canonical divisor $K_X$ on $X$ and an integer $m \geq 2$. By Lemma \ref{dh2}(v) there exist a log-resolution $f: Y \to X$ of $(X, \mathcal{O}_X(-mK_X))$ and a Weil ${\mathbb Q}$-divisor ${\Delta}_m$ on $X$ with the properties in (v). In particular $K_{m, Y/X}$ is $f$-exceptional. Now set
\[ \mathfrak{a}_m(X) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil).\]
As in the proof of Remark \ref{ic} we get that $\mathfrak{a}_m(X)$ is a coherent ideal sheaf. Let us check that its definition is independent of the choice of $f$. Let $f': Y' \to X$ be another log-resolution of $(X, \mathcal{O}_X(-mK_X))$ and assume, as we may, that $f'$ factors through $f$ and a morphism $g : Y' \to Y$. By Lemma \ref{dh2}(iii) and (i) we have
$K_{m, Y'/X} = K_{Y'/Y} + g^{\ast} K_{m, Y/X} = K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X})$, whence
\begin{equation}
\label{ben}
(fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast}(g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil)).
\end{equation}
Now set $B_Y = - K_{m, Y/X}$ and $B_{Y'} = g^{\ast}(K_Y+B_Y) - K_{Y'}$ so that, using Remark \ref{sha}(i) and Lemma \ref{fuj}, we have
\[ g_{\ast} \mathcal{O}_{Y'} (\lceil (K_{Y'} - g^{\ast} (K_Y - K_{m, Y/X}))^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil (-B_{Y'})^{\#} \rceil) = g_{\ast} \mathcal{O}_{Y'} (\lceil -(B_{Y'}^{<1}) \rceil - \lfloor B_{Y'}^{>1} \rfloor ) = \]
\[ = \mathcal{O}_Y (\lceil -(B_Y^{<1}) \rceil - \lfloor B_Y^{>1} \rfloor ) = \mathcal{O}_Y (\lceil (-B_Y)^{\#} \rceil) = \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]
and by \eqref{ben} we get
\[ (fg)_{\ast}\mathcal{O}_{Y'}(\lceil (K_{m, Y'/X})^{\#} \rceil) = f_{\ast} \mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) \]
that is $\mathfrak{a}_m(X)$ is well defined.
We now claim that the set $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element. In fact, given $m, q \geq 2$, let $f: Y \to X$ be a log-resolution of $(X, \mathcal{O}_X(-mK_X)) + \mathcal{O}_X(-mqK_X))$. By Lemma \ref{dh2}(ii) and Remark \ref{sha}(ii) we have $\lceil (K_{m, Y/X})^{\#} \rceil \leq \lceil (K_{mq, Y/X})^{\#} \rceil$ and therefore $\mathfrak{a}_m(X) \subseteqseteq \mathfrak{a}_{mq}(X)$. Using the ascending chain condition on ideals we conclude that $\{\mathfrak{a}_m(X), m \geq 2 \}$ has a unique maximal element, which we will denote by $\mathfrak{a}_{\rm max}(X)$.
Next let us show that all the ideal sheaves $\mathfrak{a}_m(X)$, for $m \geq 2$ (whence in particular also $\mathfrak{a}_{\rm max}(X)$), are in fact non-lc ideal sheaves of a suitable pair.
Let ${\Delta}_m$ be as above, so that, by Remark \ref{sha}(i) and using $\lceil (-f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = 0$, we have
\[ \lceil - (({\Delta}_m)_Y^{<1}) \rceil - \lfloor ({\Delta}_m)_Y^{>1} \rfloor = \lceil (-({\Delta}_m)_Y)^{\#} \rceil = \lceil (K_{m, Y/X} - f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \]
\[ = \lceil (K_{m, Y/X})^{\#} \rceil + \lceil (- f_{\ast}^{-1}({\Delta}_m))^{\#} \rceil = \lceil (K_{m, Y/X})^{\#} \rceil \]
whence
\[ \mathcal{J}_{NLC}(X,{\Delta}_m)=f_{\ast}\mathcal{O}_Y(\lceil - (({\Delta}_m)_Y^{<1}) \rceil - \lfloor ({\Delta}_m)_Y^{>1} \rfloor) = f_{\ast}\mathcal{O}_Y( \lceil (K_{m, Y/X})^{\#} \rceil) = \mathfrak{a}_m(X). \]
To finish the proof, let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $q ^{-1}n {\mathbb N}$ be such that $q(K_X + {\Delta})$ is Cartier. Let $m_0 \geq 2$ be such that $\mathfrak{a}_{\rm max}(X) = \mathfrak{a}_{m_0}(X) = \mathfrak{a}_{qm_0}(X)$. By what we proved above, there exists ${\Delta}_0 := {\Delta}_{qm_0}$ such that $\mathcal{J}_{NLC}(X,{\Delta}_0) = \mathfrak{a}_{\rm max}(X)$. By Lemma \ref{dh2}(iv) we have that $- {\Delta}_Y \leq K_Y + f_{\ast}^{-1}({\Delta}) - f^{\ast}(K_X+{\Delta}) \leq K_{qm_0, Y/X}$, whence also, by Remark \ref{sha} (i) and (ii),
\[ \lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor = \lceil (- {\Delta}_Y)^{\#} \rceil \leq \lceil (K_{qm_0, Y/X})^{\#} \rceil \]
and therefore
\[ \mathcal{J}_{NLC}(X,{\Delta})=f_{\ast}\mathcal{O}_Y(\lceil - ({\Delta}_Y^{<1}) \rceil - \lfloor {\Delta}_Y^{>1} \rfloor) \subseteqseteq f_{\ast}\mathcal{O}_Y(\lceil (K_{qm_0, Y/X})^{\#} \rceil) = \mathfrak{a}_{\rm max}(X) = \mathcal{J}_{NLC}(X,{\Delta}_0). \]
\end{proof}
\section{Proof of Theorem \ref{main}}
\label{b+}
We record the following lemma, which is also of independent interest.
\begin{lemma}
\label{bs}
Let $(X,{\Delta})$ be a pair with ${\Delta}$ effective and let $D$ be an effective Cartier divisor on $X$. Then there exists $c = c(X,{\Delta}, D) ^{-1}n {\mathbb N}$ such that the set-theoretic equality
\[ \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E_1+\dots+E_c)) \]
holds for some $E_1,\dots, E_c ^{-1}n |D|$.
\end{lemma}
\begin{proof}
Let $f:Y \to X$ be a log-resolution of $(X,{\Delta})$ and of the linear series $|D|$ such that $f_{\ast}^{-1} {\Delta} + \mathbf{B}s|f^{\ast}D| + \operatorname{Exc}(f)$ has simple normal crossing support.
Write ${\Delta}_Y = {\Delta}_Y^+ - {\Delta}_Y^-$, where ${\Delta}_Y^+$ and ${\Delta}_Y^-$ are effective simple normal crossing support ${\mathbb Q}$-divisors without common components. Then ${\Delta}_Y^-= \sum_{i=1}^s \delta_i D_i$, for some non-negative $\delta_i ^{-1}n {\mathbb Q}$ and distinct prime divisors $D_i$'s and define
\[ c = \lceil \operatorname{max}\{\delta_i, 1 \leq i \leq s \} \rceil +2. \]
Moreover we have that $|f^{\ast}D|=|M|+F$, where $|M|$ is base-point free and $\operatorname{Supp}(F)= \mathbf{B}s|f^{\ast}D|$. By Bertini's Theorem and \cite[Lemma 9.1.9]{laz}, we can choose $M_1,\dots, M_c ^{-1}n |M|$ general divisors such that, for all $j=1,\dots,c$, $M_j$ is smooth, every component of $M_j$ is not a component of ${\Delta}_Y, M_1,\dots, M_{j-1}$ and ${\Delta}_Y+M_1+\dots+M_c+F$ has simple normal crossing support. Now, for all $j = 1, \ldots, c$, $M_j+F ^{-1}n |f^{\ast}D|$, so that there exists $E_j ^{-1}n |D|$ such that $M_j+F =f^{\ast}E_j$. Set $E=E_1+\dots +E_c$ and notice that $f$ is also a log-resolution of $(X,{\Delta}+E)$.
By Remark \ref{nlc} we have $\operatorname{Nlc}(X,{\Delta})=\mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}))\subseteqseteq \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$, the latter inclusion following by Remark \ref{sha}(i) and (ii), because $E$ is effective. Also, for every prime divisor $\Gamma$ in the support of $F$ we get for the discrepancies
\[ a(\Gamma,X,{\Delta}+E)=a(\Gamma,X,{\Delta})-\operatorname{ord}_{\Gamma}(f^{\ast} E) =-\operatorname{ord}_{\Gamma} ({\Delta}_Y) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq \]
\[ \leq \operatorname{ord}_{\Gamma} ({\Delta}_Y^-) - \operatorname{ord}_{\Gamma}(f^{\ast} E) \leq \operatorname{max}\{\delta_i, 1 \leq i \leq s \} -\operatorname{ord}_{\Gamma}(M_1+\dots+M_c+cF)\leq -2 \]
whence $f(\Gamma) \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E)$. As $\mathbf{B}s|D|$ is the union of such $f(\Gamma)$'s, using Remark \ref{nlc}, we get the inclusion $\mathbf{B}s |D| \subseteqseteq \operatorname{Nlc}(X, {\Delta}+E) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$.
On the other hand notice that $({\Delta}+E)_Y =f^{\ast}(K_X+{\Delta}+E)-K_Y= {\Delta}_Y +f^{\ast}E$. Also ${\Delta}_Y + f^{\ast} E = {\Delta}_Y + M_1+\dots+M_c+cF$, so that
\[ \operatorname{Supp}(({\Delta}+E)_Y^{>1}) = \operatorname{Supp}(({\Delta}_Y + f^{\ast} E)^{>1}) \subseteqseteq \operatorname{Supp}(F) \cup \operatorname{Supp} ({\Delta}_Y^{>1}) \]
whence
\[ f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq f(\operatorname{Supp}(F)) \cup f(\operatorname{Supp} ({\Delta}_Y^{>1})) = \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
Therefore, by Remark \ref{nlc},
\[ \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) = \operatorname{Nlc}(X, {\Delta}+E) = f(\operatorname{Supp}(({\Delta}+E)_Y^{>1})) \subseteqseteq \mathbf{B}s|D| \cup \operatorname{Nlc}(X,{\Delta}). \]
\end{proof}
Now we essentially follow the proof of Nakamaye's Theorem as in \cite[\S 10.3]{laz} and \cite[Thm. 0.3]{n}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main}}}
\begin{proof}
We can assume that $D$ is a Cartier divisor. The issue is of course to prove that $\mathbf{B}_+(D) \subseteqseteq \operatorname{Null}(D)$, since the opposite inclusion holds on any normal projective variety, as explained in the introduction.
By Proposition \ref{max} and Remark \ref{nlc} there is an effective Weil ${\mathbb Q}$-divisor ${\Delta}$ on $X$ such that $K_X + {\Delta}$ is ${\mathbb Q}$-Cartier and $\mathrm{Nlc}(X,{\Delta})= X_{\mathrm{nlc}}$, so that $\dim \mathrm{Nlc}(X,{\Delta}) \leq 1$.
Let $A$ be an ample Cartier divisor such that $A-(K_X+{\Delta})$ is ample. As in \cite[Proof of Thm. 10.3.5]{laz}) we can choose $a, p ^{-1}n {\mathbb N}$ sufficiently large such that
\[ \mathbf{B}_+(D)= \mathbf{B}(aD-2A)= \mathbf{B}s|paD-2pA|. \]
By Lemma \ref{bs} there exist $c ^{-1}n {\mathbb N}$ and a Cartier divisor $E$ on $X$ such that
\[ \mathbf{B}_+(D)\cup \operatorname{Nlc}(X,{\Delta}) = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E)) \]
and $E \equiv c(paD-2pA)=qaD-2qA$, where $q:= cp ^{-1}n {\mathbb N}$.
Set $Z = \mathcal{Z}(\mathcal{J}_{NLC}(X,{\Delta}+E))$. For $m \geq qa$, we get that
\[ mD-qA-(K_X+{\Delta}+E)\equiv (m-qa)D+qA-(K_X+{\Delta}) \]
is ample, whence $H^1(X, \mathcal{J}_{NLC}(X,{\Delta}+E) \otimes \mathcal{O}_X(mD-qA))=0$, for $m\geq qa$
by \cite[Thm. 3.2]{f}, \cite[Thm. 4.4]{a}, so that the restriction map
\begin{equation}
\label{uno}
H^0(X,\mathcal{O}_X(mD-qA))\to H^0(Z, \mathcal{O}_Z(mD-qA)) \mbox{\ is surjective for} \ m \geq qa.
\end{equation}
By contradiction let us assume that there exists an irreducible component $V$ of $\mathbf{B}_+(D)$, such that $V \not \subseteqseteq \operatorname{Null}(D)$.
Now $V \subseteqseteq \mathbf{B}_+(D) \subseteqseteq \mathbf{B}(D-\frac{q}{m}A)\subseteqseteq \mathbf{B}s |mD-qA|$ for $m ^{-1}n {\mathbb N}$, whence the restriction map
\[ H^0(X,\mathcal{O}_X(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m ^{-1}n {\mathbb N} \]
and therefore, by \eqref{uno}, also
\begin{equation}
\label{due}
H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA)) \mbox{\ is zero for} \ m \geq qa.
\end{equation}
On the other hand $\dim V\geq 1$, as $\mathbf{B}_+(D)$ does not contain isolated points by \cite[Proposition 1.1]{elmnp2}(which holds on $X$ normal).
As $\dim \operatorname{Nlc}(X,{\Delta}) \leq 1$, this implies that $V$ is an irreducible component of $Z$. Moreover, as $V\not \subseteqseteq \operatorname{Null}(D)$, we have that $D_{|_V}$ is big.
Now, by Remark \ref{ic}, $\mathcal{J}_{NLC}(X,{\Delta}+E)$ is integrally closed, and exactly as in \cite[Proof of Thm. 10.3.5]{laz} (the proof of this part holds on any normal projective variety) it follows that, for $m \gg 0$, $H^0(Z,\mathcal{O}_Z(mD-qA))\to H^0(V,\mathcal{O}_V(mD-qA))$ is not zero, thus contradicting \eqref{due}.
This concludes the proof.
\end{proof}
\renewcommand{Proof}{Proof}
\renewcommand{Proof}{Proof of Corollary {\rm \ref{dim}}}
\begin{proof}
Note that, on any normal projective variety $X$, we have $X_{\mathrm{nlc}} \subseteqseteq \operatorname{Sing}(X)$ (see for example \cite[Rmk 4.8]{cd}) and if $\dim X\leq 3$, then $\dim \operatorname{Sing}(X) \leq 1$. Then just apply Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof}
\renewcommand{Proof}{Proof of Corollary {\rm \ref{mod}}}
\begin{proof}
By \cite[Thm. 0.9]{gkm} we know that $\operatorname{Null}(D) \subseteqseteq \partial \overline{M}_{g,n}$. On the other hand it is well-known
(see for example \cite[Lemma 10.1]{bchm}) that $(\overline{M}_{g,n}, 0)$ is klt, whence the conclusion follows by Theorem \ref{main}.
\end{proof}
\renewcommand{Proof}{Proof}
\section{Restricted base loci on klt pairs}
\label{rests}
We first recall that, associated to a pseudoeffective divisor $D$, there are two more loci, one that also measures how far $D$ is from being nef and another one that measures how far $D$ is from being nef and abundant.
\begin{defn}
\label{as2}
{\rm Let $X$ be a normal projective variety and let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$. As in \cite[Def. 1.7]{bbp}, we define the {\bf non-nef locus}
\[ \operatorname{Nnef}(D) = \bigcup_{v : v(\|D\|) > 0} c_X(v) \]
where $v$ runs among all divisorial valuations on $X$, $c_X(v)$ is its center and $v(\|D\|)$ is as in Definition \ref{as1}.
Let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$. As in \cite[Def. 2.18]{cd}, we define the {\bf non nef-abundant locus}
\[ \operatorname{Nna}(D) = \bigcup_{v : v(\langle D \rangle) > 0} c_X(v) \]
where again $v$ runs among all divisorial valuations on $X$ and $v(\langle D \rangle)$ is as in Definition \ref{as1}.}
\end{defn}
In the sequel we will use the fact that, for $D$ big (\cite[Lemma 3.3]{elmnp1}) or even abundant (\cite[Prop. 6.4]{leh}), we have $v(\|D\|) = v(\langle D \rangle)$, while in general they are different when $D$ is only pseudoeffective (\cite[Rmk 2.16]{cd}).
We will also use (see \cite[page 2]{bfj} and references therein)
\noindent {\bf Izumi's Theorem}
\label{iz}
{^{-1}t Let $X$ be a normal variety over an algebraically closed field $k$ and let $0 ^{-1}n X$ be a closed point. Let $m_0$ be the maximal ideal of the local ring ${\mathcal O}_{X,0}$ and set, for any $f ^{-1}n {\mathcal O}_{X,0}$, $\operatorname{ord}_0 (f) = \operatorname{max}\{j \geq 0 : f ^{-1}n m_0^j \}$.
For any divisorial valuation $v$ of $k(X)$ centered at $0$, there exists a constant $C = C(v) > 0$ such that
\[ C^{-1} \operatorname{ord}_0(f) \leq v(f) \leq C \operatorname{ord}_0(f).\]}
We start by proving an analogue of \cite[Prop. 2.8]{elmnp1} for $\operatorname{Nna}(D)$.
\begin{thm}
\label{nna}
Let $X$ be a normal projective variety, let $D$ be a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor such that $\kappa(D) \geq 0$ and let $v$ be a divisorial valuation on $X$.
Then
\[ c_X(v)\subseteqseteq \operatorname{Nna}(D) \ \textrm{if and only if} \ v(\langle D \rangle)>0. \]
\end{thm}
\begin{proof}
We can assume that $D$ is Cartier and effective. By definition of $\operatorname{Nna}(D)$, we just need to prove that if $c_X(v) \subseteqseteq \operatorname{Nna}(D)$, then $v(\langle D \rangle)>0$.
We first prove the theorem when $X$ is smooth. For any $p ^{-1}n {\mathbb N}$ let $b(|pD|)$ be the base ideal of $|pD|$, $\mathcal{J}(X,\|pD\|))$ the asymptotic multiplier ideal and denote by $b_p$ and $j_p$ the corresponding images in $R_v$, the DVR associated to $v$. As in \cite[\S 2]{elmnp1}, we get
\begin{equation}
\label{j}
v(\langle D \rangle) = \lim_{p \to +^{-1}nfty} \frac{v(b_p)}{p} \geq \lim_{p \to +^{-1}nfty} \frac{v(j_p)}{p} = \sup_{p ^{-1}n {\mathbb N}} \{\frac{v(j_p)}{p}\}.
\end{equation}
By \cite[Cor. 5.2]{cd} we have the set-theoretic equality
\[ \operatorname{Nna}(D)=\bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}(X,\|pD\|)) \]
whence there exists $p_0 ^{-1}n {\mathbb N}$ such that $c_X(v) \subseteqseteq \mathcal{Z}(\mathcal{J}(X,\|p_0D\|))$, so that $v(j_{p_0})>0$ and \eqref{j} gives that $v(\langle D \rangle)>0$.
We now prove the theorem for a divisorial valuation $\nu$ on $X$ such that $c_X(\nu) = \{x\}$ is a closed point.
As $c_X(\nu) \subseteqseteq \operatorname{Nna}(D)$, there exists a divisorial valuation $v_0$ on $X$ such that $v_0(\langle D \rangle)>0$ and $x ^{-1}n c_X(v_0)$.
Let $E_0$ be a prime divisor over $X$ such that $v_0 = k \operatorname{ord}_{E_0}$ for some $k ^{-1}n {\mathbb N}$. We can assume that there is a birational morphism $\mu:Y\to X$ from a smooth
variety $Y$ such that $E_0 \subseteqset Y$. As $\mu(E_0)=c_X(\operatorname{ord}_{E_0}) = c_X(v_0 )$, there is a point $y ^{-1}n E_0$ such that $\mu(y) = x$. Let $\pi : Y' \to Y$ be the blow-up of $Y$ on $y$ with exceptional divisor $E_y$. For any $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$$\operatorname{ord}_{E_y}(G) =\operatorname{ord}_{E_y}(\pi^*(\mu^*G)) = \operatorname{ord}_{y}(\mu^*G) \geq \operatorname{ord}_{E_0}(\mu^*G) = \operatorname{ord}_{E_0}(G)$$
therefore $\operatorname{ord}_{E_y}(\langle D \rangle) \geq \operatorname{ord}_{E_0}(\langle D \rangle) = \frac{1}{k} v_0(\langle D \rangle)>0$.
Since $c_X(\operatorname{ord}_{E_y})=\{x\}$, by Izumi's Theorem applied twice, there exist $C > 0, C' > 0$ such that for all $m ^{-1}n {\mathbb N}$ and $G ^{-1}n |mD|$ we have
$\operatorname{ord}_{E_y}(G) \leq C' \operatorname{ord}_{x}(G) \leq C \nu(G)$. Hence $\nu(\langle D \rangle) \geq \frac{1}{C} \operatorname{ord}_{E_y}(\langle D \rangle)>0$.
Finally let $v$ be any divisorial valuation on $X$ with $c_X(v)\subseteqseteq \operatorname{Nna}(D)$. As above there is a birational morphism $f : Z \to X$ from a smooth
variety $Z$ and a prime divisor $E \subseteqset Z$ such that $v = h \operatorname{ord}_E$ for some $h ^{-1}n {\mathbb N}$. For every closed point $z ^{-1}n E$ we have that $\nu:= \operatorname{ord}_z$ is a divisorial valuation with $c_X(\nu) \subseteqseteq c_X(\operatorname{ord}_E) \subseteqseteq \operatorname{Nna}(D)$ and $c_X(\nu)$ is a closed point. Thus, by what we proved above, we have that $\operatorname{ord}_z(\langle f^*(D) \rangle) = \operatorname{ord}_z(\langle D \rangle)>0$ for all $z ^{-1}n E$, so that $E \subseteqseteq \operatorname{Nna}(f^*(D))$. As $Z$ is smooth, we get $v(\langle D \rangle) = h\operatorname{ord}_E(\langle D \rangle) = h \operatorname{ord}_E(\langle f^*(D) \rangle)>0$.
\end{proof}
We next prove an analogous result for $\operatorname{Nnef}(D)$.
\begin{thm}
\label{nnef}
Let $X$ be a normal projective variety, let $D$ be a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$ and let $v$ be a divisorial valuation on $X$.
Then
$$c_X(v) \subseteqseteq \operatorname{Nnef}(D) \ \textrm{if and only if} \ v(\|D\|)>0.$$
\end{thm}
\begin{proof} Again we need to prove that $v(\|D\|) > 0$ if $c_X(v) \subseteqseteq \operatorname{Nnef}(D)$.
By \cite[Lemmas 2.13 and 2.12]{cd}, there exists a sequence of ample ${\mathbb R}$-Cartier ${\mathbb R}$-divisors $\{A_m\}_{m ^{-1}n {\mathbb N}}$
such that $\|A_m\| \to 0$, $D+A_m$ is a big ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor for all $m ^{-1}n {\mathbb N}$ and
$$\operatorname{Nnef}(D)=\bigcup_{m^{-1}n {\mathbb N}} \operatorname{Nnef}(D+A_m).$$
Then there is $m_0 ^{-1}n {\mathbb N}$ such that $c_X(v)\subseteqseteq \operatorname{Nnef}(D+A_{m_0})$. As $D+A_{m_0}$ is big, we have $\operatorname{Nnef}(D+A_{m_0}) = \operatorname{Nna}(D+A_{m_0})$, whence $v(\|D+A_{m_0}\|) = v(\langle D+A_{m_0}\rangle) >0$ by Theorem \ref{nna}.
Therefore $0 < v(\|D+A_{m_0}\|) \leq v(\|D\|) + v(\|A_{m_0}\|) = v(\|D\|)$.
\end{proof}
\begin{rem}
{\rm Note that, given a normal projective variety $X$, Theorems {\rm \ref{nna}} and {\rm \ref{nnef}} can be rewritten as follows (where $x$ is a closed point).
If $D$ is a ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor on $X$ such that $\kappa(D)\geq 0$, then
$$\operatorname{Nna}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\langle D\rangle)>0\}.$$
If $D$ is a pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, then
$$\operatorname{Nnef}(D)=\bigcup_{x ^{-1}n X} \{x\;|\;\{x\}=c_X(v) \mbox{ for some divisorial valuation } v \mbox{ with } v(\|D\|)>0\}.$$}
\end{rem}
Next we will prove Theorem {\rm \ref{main2}. We will use a singular version (see for example \cite[Def. 2.2]{cd}) of standard asymptotic multiplier ideal sheaves \cite[Ch. 11]{laz}.
\renewcommand{Proof}{Proof of Theorem {\rm \ref{main2}}}
\begin{proof}
In both cases we have that $\operatorname{Nnef}(D) = \mathbf{B}_-(D)$ by \cite[Thm. 1.2]{cd}, whence also $\operatorname{Nna}(D) = \mathbf{B}_-(D)$ in case (i). Then (ii) follows by Theorem \ref{nnef} and the first equivalence in (i) by Theorem \ref{nna}. To complete the proof of (i) we need to show that if $\limsup_{m \to + ^{-1}nfty} v(|mD|) = + ^{-1}nfty$ then $v(\langle D\rangle) > 0$, the reverse implication being obvious. We will proceed similarly to \cite[Proof of Prop. 2.8]{elmnp1} and \cite[Proof of Lemma 4.1]{cd}. If $v(\langle D\rangle) = 0$, by what we just proved, we have that $c_X(v) \not\subseteqseteq \mathbf{B}_-(D)$ and, by \cite[Cor. 5.2]{cd}, we have the set-theoretic equality
\[ \mathbf{B}_-(D) = \bigcup_{p ^{-1}n {\mathbb N}} \mathcal{Z}(\mathcal{J}((X,{\Delta});\|pD\|)) \]
where $\mathcal{J}((X,{\Delta});\|pD\|)$ is as in \cite[Def. 2.2]{cd}. Therefore $c_X(v) \not\subseteqseteq \mathcal{Z}(\mathcal{J}((X,{\Delta}); \|pD\|))$ for any $p ^{-1}n {\mathbb N}$. Let $H$ be a very ample Cartier divisor such that $H - (K_X + {\Delta})$ is ample and let $n = \dim X$. By Nadel's vanishing theorem \cite[Thm. 9.4.17]{laz}, we deduce that $\mathcal{J}((X,{\Delta});\|pD\|) \otimes \mathcal{O}_X((n + 1)H + pD)$ is $0$-regular in the sense of Castelnuovo-Mumford, whence globally generated, for every $p ^{-1}n {\mathbb N}$, and therefore $c_X(v) \not\subseteqseteq \mathbf{B}s|(n+1)H + pD|$. On the other hand, as $D$ is big, there is $m_0 ^{-1}n {\mathbb N}$ such that $m_0 D \sim (n + 1)H + E$ for some effective Cartier divisor $E$. Hence, for any $m \geq m_0$, we get $v(|mD|) = v(|(m-m_0)D + (n + 1)H + E|) \leq v(|(m-m_0)D + (n + 1)H|) + v(|E|) = v(|E|)$ and the theorem follows.
\end{proof}
\renewcommand{Proof}{Proof}
We end the section with an observation on the behavior of these loci under birational maps.
\begin{cor}
Let $f:Y \to X$ be a projective birational morphism between normal projective varieties. Then:
\begin{itemize}
^{-1}tem [(i)] For every ${\mathbb Q}$-Cartier ${\mathbb Q}$-divisor $D$ on $X$ such that $\kappa(D)\geq 0$, we have
$$\operatorname{Nna}(f^*(D))=f^{-1}(\operatorname{Nna}(D));$$
^{-1}tem [(ii)] For every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
$$\operatorname{Nnef}(f^*(D))=f^{-1}(\operatorname{Nnef}(D));$$
^{-1}tem [(iii)] If there exist effective Weil ${\mathbb Q}$-divisors ${\Delta}_X$ on $X$ and ${\Delta}_Y$ on $Y$ such that $(X,{\Delta}_X)$ and $(Y,{\Delta}_Y)$ are klt pairs,
then, for every pseudoeffective ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on $X$, we have
$$\mathbf{B}_-(f^*(D))=f^{-1}(\mathbf{B}_-(D))$$
\end{itemize}
\end{cor}
\begin{proof}
To see (i), for every closed point $y ^{-1}n Y$, let $v_y$ be a divisorial valuation such that $c_Y(v_y)=\{y\}$.
Then, by Theorem \ref{nna}, we have,
\[ y^{-1}n f^{-1}(\operatorname{Nna}(D)){\mathcal L}eftrightarrow \{f(y)\}=c_X(v_y)\subseteqseteq \operatorname{Nna}(D){\mathcal L}eftrightarrow \]
\[ {\mathcal L}eftrightarrow v_y(\langle f^*(D) \rangle) = v_y(\langle D\rangle)>0{\mathcal L}eftrightarrow \{y\}=c_Y(v_y)\subseteqseteq \operatorname{Nna}(f^*(D)). \]
Now (ii) can be proved exactly in the same way by using Theorem \ref{nnef}, while (iii) follows from (ii) and \cite[Thm. 1.2]{cd}.
\end{proof}
\end{document} |
\begin{document}
\title{\Large \textbf{HALF-ISOMORPHISMS WHOSE INVERSES ARE ALSO HALF-ISOMORPHISMS}
\begin{abstract}
\noindent{}Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a \emph{half-isomorphism} if $f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$, for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a half-automorphism. A half-isomorphism $f$ is called \emph{special} if $f^{-1}$ is also a half-isomorphism. In this paper, necessary and sufficient conditions for the existence of special half-isomorphisms on groupoids and quasigroups are obtained. Furthermore, some examples of non-special half-automorphisms for loops of infinite order are provided.
\end{abstract}
\noindent{}{\it Keywords}: half-isomorphism, half-automorphism, special half-isomorphism, groupoid, quasigroup, loop.
\section{Introduction}
A \emph{groupoid} consists of a nonempty set with a binary operation. A groupoid $( Q,*) $ is called a \emph{quasigroup} if for each $ a, b \in Q $ the equations $ a * x = b $ and $ y * a = b$ have unique solutions for $x,y \in Q$. A \textit{quasigroup} $(L,*)$ is a \emph{loop} if there exists an identity element $1\in L$ such that $ 1 * x = x = x * 1$, for any $ x \in L $. The fundamental definitions and facts from groupoids, quasigroups, and loops can be found in \cite{B71,P90}.
Let $(G,*)$ and $(G',\cdot)$ be groupoids. A bijection $f: G \rightarrow G'$ is called a \emph{half-isomorphism} if \mbox{$f(x*y)\in\{f(x)\cdot f(y),f(y)\cdot f(x)\}$,} for any $ x, y \in G$. A half-isomorphism of a groupoid onto itself is a \emph{half-automorphism}. We say that a half-isomorphism is \emph{trivial} when it is either an isomorphism or an anti-isomorphism.
In 1957, Scott \cite{Sco57} showed that every half-isomorphism on groups is trivial. In the same paper, the author provided an example of a loop of order $8$ that has a nontrivial half-automorphism, then the result for groups can not be generalized to all loops. Recently, a similar version of Scott's result was proved for some subclasses of Moufang loops \cite{GG12,GGRS16,KSV16} and automorphic loops \cite{GA192}. A \emph{Moufang loop} is a loop that satisfies the identity $ x(y(xz))=((xy)x)z$, and an \emph{automorphic loop} is a loop in which every inner mapping is an automorphism \cite{BP56}. We note that there are Moufang loops and automorphic loops that have nontrivial half-automorphisms \cite{GG13,GA19,GPS17}.
In \cite{GA192}, the authors defined the concept of \emph{special half-isomorphism}. A half-isomorphism $f:G \rightarrow G'$ is called \emph{special} if the inverse mapping $f^{-1}:G' \rightarrow G$ is also a half-isomorphism. It is easy to construct an example of a
half-isomorphism that is not special, as we can see below.
\begin{exem}
\label{ex1} Let $G = \{1,2,...,6\}$ and consider the following Cayley tables of $(G,*)$ and $(G, \cdot)$:
\begin{center}
\begin{minipage}{.35\textwidth}
\centering
\begin{tabular}{c|cccccc}
$*$ & 1& 2& 3&4&5&6 \\
\hline
1 & 1&2 &3&4&5&6 \\
2 & 2&3 &4&5&6&1 \\
3 & 3&4 &5&6&1&2\\
4 & 4&5 &6&1&2&3 \\
5 & 5&6 &1&2&3&4 \\
6 & 6&1 &2&3&4&5 \\
\end{tabular}
\end{minipage}
\begin{minipage}{.35\textwidth}
\centering
\begin{tabular}{c|cccccc}
$\cdot $ & 1& 2& 3&4&5&6 \\
\hline
1 & 1&2 &3&4&5&6 \\
2 & 2&3 &4&5&6&1 \\
3 & 3&1 &5&6&4&2\\
4 & 4&5 &6&1&2&3 \\
5 & 5&6 &1&2&3&4 \\
6 & 6&4 &2&3&1&5 \\
\end{tabular}
\end{minipage}
\end{center}
Note that $(G,*)$ is isomorphic to $C_6$, the cyclic group of order $6$, and $(G,\cdot)= L$ is a nonassociative loop. Consider the mapping $f: C_6\rightarrow L$ defined by $f(x) = x$, for all $x\in G$. For $x,y\in G$ such that $x\leq y$ and $(x,y)\not = (3,5)$, we have $y*x = x*y = x\cdot y$. Furthermore, $3*5 = 5*3 = 5\cdot 3$. Thus, $f$ is a half-isomorphism. From $3\cdot 5 = 4$ and $3*5 = 5*3 = 1$, it follows that $f^{-1}(3\cdot 5)\not \in \{f^{-1}(3)*f^{-1}(5),f^{-1}(5)*f^{-1}(3)\}$, and hence $f^{-1}$ is not a half-isomorphism.
\qed
\end{exem}
We note that providing some examples for the case of non-special half-automorphisms can be very complicated. For finite loops, every half-automorphism is special \cite[Corollary 2.7]{GA192}, and in section~\ref{sec3} we show that the same is valid for finite groupoids.
As we can see in the example~\ref{ex1}, in general, a half-isomorphism does not preserve the structure of the loop. For instance, $C_6$ is associative and commutative and has a subgroup $H = \{1,3,5\}$, while $L$ is nonassociative and noncommutative, and $f(H)$ is not a subloop of $L$. However, the inverse mapping of a half-isomorphism can preserve some structure, like the commutative property and subloops \cite[Proposition 2.2]{GA192}. The same naturally holds for special half-isomorphisms.
This paper is organized as follows: Section~\ref{sec2} presents the definitions and basic results about half-isomorphisms. In section~\ref{sec3}, some presented results in \cite{GA192} on half-isomorphisms in loops are generalized to groupoids. In section~\ref{sec4}, the concept of \emph{principal h-groupoid} of a groupoid is defined, and then a necessary and sufficient condition for the existence of special half-isomorphisms between groupoids is obtained. Furthermore, equations related to the number of special half-automorphisms, automorphisms and anti-automorphisms of a groupoid are obtained. In section~\ref{sec5}, the concept of \emph{principal h-quasigroup} of a quasigroup is defined, and then the set of these quasigroups is described. Some examples of non-special half-automorphisms in loops are provided in section~\ref{sl}.
\section{Preliminaries}
\label{sec2}
Here, the required definitions and basic results on half-isomorphisms are stated.
\begin{defi} Let $G$ and $G'$ be groupoids. We will say that $G$ is \emph{half-isomorphic} to $G'$, denoted by $G\stackrel{H}{\cong} G'$, if there exists a special half-isomorphism between $G$ and $G'$. Note that $\stackrel{H}{\cong}$ is an equivalence relation. If $G$ is isomorphic to $G'$, we write $G \cong G'$.
\end{defi}
The next proposition assures that quasigroups half-isomorphic to loops are also loops.
\begin{prop}
\label{prop21}
Let $(G,*)$ and $(G',\cdot)$ be groupoids and $f:G \rightarrow G'$ be a half-isomorphism. If $G'$ has an identity element $1$, then $f^{-1}(1)$ is the identity element of $G$.
\end{prop}
\begin{proof}
Let $x = f^{-1}(1) \in G$. For $y\in G$, we have that $\{f(x*y),f(y*x)\}\subset \{1\cdot f(y),f(y)\cdot1\} = \{f(y)\}$. Since $f$ is a bijection, we have $x*y = y*x = y$. Therefore, $x$ is an identity element of $G$.
\end{proof}
Now, let $(G,*),(G',\cdot),(G'',\bullet)$ be groupoids, and $f:G \rightarrow G'$ and $g:G' \rightarrow G''$ be half-isomorphisms. For $x,y\in G$, we have
\begin{center}
$gf(x*y) \in \{g(f(x)\cdot f(y)),g(f(y)\cdot f(x))\} = \{gf(x)\bullet gf(y),gf(y)\bullet gf(x))\}.$
\end{center}
Thus, $gf$ is a half-isomorphism. If $f$ and $g$ are special half-isomorphisms, then $(gf)^{-1} = f^{-1}g^{-1}$ is also a special half-isomorphism.
We denote the sets of the half-automorphisms, special half-automorphisms, and trivial half-automorphisms of a groupoid $G$ by $\mathit{Half}(G)$, $\mathit{Half}_S(G)$, and $\mathit{Half_T}(G)$, respectively. Note that automorphisms and anti-automorphisms are always special half-automorphisms, and consequently $\mathit{Half_T}(G) \subset \mathit{Half}_S(G) \subset \mathit{Half}(G)$.
For $f,g\in \mathit{Half}(G)$, we already see that $fg\in \mathit{Half}(G)$. The identity mapping $I_d$ of $G$ is the identity element of $\mathit{Half}(G)$. Thus, $\mathit{Half}(G)$ is a group if and only if it is closed under inverses, which is equivalent to $\mathit{Half}(G) = \mathit{Half}_S(G)$. In particular, $\mathit{Half}_S(G)$ is always a group.
A composition of two automorphisms or two anti-automorphisms is an automorphism, and if $f$ is an automorphism and $g$ is an anti-automorphism, then $fg$ and $gf$ are anti-automorphisms and $g^{-1}fg$ is an automorphism. Thus, $\mathit{Half_T}(G)$ is a group and the automorphism group of $G$, denoted by $Aut(G)$, is a normal subgroup of $\mathit{Half_T}(G)$.
The following result summarizes the discussion above.
\begin{prop}
\label{prop22}
Let $G$ be a groupoid. Then:
\noindent{}(a) $\mathit{Half}_S(G)$ is a group and $\mathit{Half}_T(G)$ is a subgroup of $\mathit{Half}_S(G)$.
\\
(b) $\mathit{Half}(G)$ is a group if and only if $\mathit{Half}(G)=\mathit{Half}_S(G)$.
\\
(c) $Aut(G) \triangleleft \mathit{Half}_T(G)$.
\end{prop}
\begin{obs}
It is shown in section~\ref{sl} that in general $\mathit{Half}(G)$ is not a group.
\end{obs}
\section{Special half-isomorphisms on groupoids}
\label{sec3}
Considering $(G,*)$ and $(G',\cdot)$ as groupoids, define the following set:
\begin{equation*}
K(G) = \{(x,y)\in G\times G \mid xy = yx\}
\end{equation*}
The next two results are respectively extensions of Proposition 2.3 and Theorem 2.5 of \cite{GA192} to groupoids. We note that the proofs are similar to the ones for corresponding results given in \cite{GA192}.
\begin{lema}
\label{lema32}
Let $f:G \rightarrow G'$ be a half-isomorphism. Then
\begin{equation*}
\begin{aligned}
\psi_{(G,G')}:{} & K(G') &\rightarrow {}&K(G)\\
&(x,y)&\mapsto {}&(f^{-1}(x),f^{-1}(y))
\end{aligned}
\end{equation*}
\noindent{}is injective.
\end{lema}
\begin{proof}
For $(x,y)\in K(G')$, we have
\begin{equation*}
\{f(f^{-1}(x)*f^{-1}(y)),f(f^{-1}(y)*f^{-1}(x))\} \subseteq \{x\cdot y, y\cdot x\} = \{x\cdot y\}.
\end{equation*}
Then, $f(f^{-1}(x)*f^{-1}(y))=f(f^{-1}(y)*f^{-1}(x))$, and so $f^{-1}(x)*f^{-1}(y) = f^{-1}(y)*f^{-1}(x)$. Thus, $(f^{-1}(x),f^{-1}(y))\in K(Q)$ and the mapping $\psi_{(G,G')}$ is well-defined.
Now, let $(x,y),(x',y')\in K(G')$ such that $\psi_{(G,G')}((x,y)) = \psi_{(G,G')}((x',y'))$. Then, $f^{-1}(x) = f^{-1}(x')$ and $f^{-1}(y) = f^{-1}(y')$. Since $f$ is a bijection, the mapping $\psi_{(G,G')}$ is injective.
\end{proof}
\begin{teo}
\label{teo31}
Let $f:G \rightarrow G'$ be a half-isomorphism. Then, the following statements are equivalent:
\noindent{}(a) $f$ is special.\\
(b) $\{f(x*y),f(y*x)\} = \{f(x)\cdot f(y), f(y)\cdot f(x)\}$ for any $x,y\in G$.\\
(c) For all $x,y\in G$ such that $x*y = y*x$, we have $f(x)\cdot f(y) = f(y)\cdot f(x)$.\\
(d) $\psi_{(G,G')}$ is a bijection.
\end{teo}
\begin{proof}(a) $\Rightarrow$ (b) Let $x,y \in G$. Since $f$ is a half-isomorphism, we have \mbox{$\{f(x*y),f(y*x)\} \subseteq$} \mbox{$ \{f(x)\cdot f(y), f(y)\cdot f(x)\}$.} Since $f^{-1}$ is a half-isomorphism, we have $\{f^{-1}(f(x)\cdot f(y)), f^{-1}(f(y)\cdot f(x))\} \subseteq \{x*y,y*x\}$, and hence $\{f(x)\cdot f(y), f(y)\cdot f(x)\} \subseteq \{f(x*y),f(y*x)\}$.\\
\noindent{}(b) $\Rightarrow$ (c) Let $x,y \in G$ such that $x*y = y*x$. Then, $f(x*y) = f(y*x)$. Using the hypothesis, we get $\{f(x)\cdot f(y), f(y)\cdot f(x)\} = \{f(x*y),f(y*x)\} = \{f(x*y)\}$, and therefore $f(x)\cdot f(y) = f(y)\cdot f(x)$.\\
\noindent{}(c) $\Rightarrow$ (d) From Lemma~\ref{lema32}, we know that $\psi_{(G,G')}$ is injective. Let $(x,y) \in K(G)$. By hypothesis, we have $f(x)\cdot f(y) = f(y)\cdot f(x)$, and then $(f(x),f(y)) \in K(G')$. It is clear that $\psi_{(G,G')}((f(x),f(y))) = (x,y)$, and hence $\psi_{(G,G')}$ is a bijection.\\
\noindent{}(d) $\Rightarrow$ (a) Let $x,y\in G'$. If $(x,y)\in K(G')$, then $(f^{-1}(x),f^{-1}(y)) \in K(G)$ since $\psi_{(G,G')}$ is a bijection. Thus, $f(f^{-1}(x)* f^{-1}(y)) = x\cdot y$, and therefore $f^{-1}(x\cdot y) = f^{-1}(x)* f^{-1}(y)$. If $(x,y)\not \in K(G')$, then $(f^{-1}(x),f^{-1}(y)) \not \in K(G)$ since $\psi_{(G,G')}$ is a bijection. Consequently, we have
\begin{center}
$\{f(f^{-1}(x)* f^{-1}(y)), f(f^{-1}(y)* f^{-1}(x))\} = \{x\cdot y,y\cdot x\}$,
\end{center}
\noindent{}and hence $f^{-1}(x\cdot y) \in \{f^{-1}(x)* f^{-1}(y), f^{-1}(y)* f^{-1}(x)\}$.
\end{proof}
As direct consequences of Lemma~\ref{lema32} and Theorem~\ref{teo31}, we have the following corollaries.
\begin{cor}
\label{cor31}
Let $f:G \rightarrow G'$ be a half-isomorphism. If $|K(G)| = |K(G')| < \infty$, then $f$ is special.
\end{cor}
\begin{cor}
\label{cor32}
Let $G$ be a groupoid such that $|K(G)| < \infty$. Then, $\mathit{Half}(G)$ is a group.
\end{cor}
\begin{cor}
\label{cor33}
Let $G$ be a finite groupoid. Then, $\mathit{Half}(G)$ is a group.
\end{cor}
A loop is \emph{diassociative} if any two of its elements generate an associative subloop. Moufang loops and groups are examples of diassociative loops. In \cite[Lemma 2.1]{KSV16}, the authors showed that the item (c) of Theorem~\ref{teo31} holds for any half-isomorphism on diassociative loops. Therefore, we have the next result.
\begin{cor}
\label{cor34}
Let $(L,*)$ and $(L',\cdot)$ be diassociative loops. Then, every half-isomorphism between $L$ and $L'$ is special.
\end{cor}
\begin{obs} The Corollary~\ref{cor34} cannot be extended for some important classes of loops. In example~\ref{ex61}, a non-special half-isomorphism between a right Bol loop and a group is introduced. A loop is called \emph{right Bol loop} if it satisfies the identity $((xy)z)y = x((yz)y)$.
\end{obs}
This section is finished with a property of half-isomorphic groupoids.
\begin{prop}
\label{prop33}
If $G\stackrel{H}{\cong} G'$, then:
\noindent{}(a) $\mathit{Half}(G) \cong \mathit{Half}(G')$
\\
(b) $\mathit{Half}_S(G) \cong \mathit{Half}_S(G')$
\end{prop}
\begin{proof}
Let $\phi: G \rightarrow G'$ be a special half-isomorphism. Define $\varphi: \mathit{Half}(G) \rightarrow \mathit{Half}(G')$ by $\varphi(f) = \phi f \phi^{-1}$. It is clear that $\varphi$ is a bijection. For $f,g\in \mathit{Half}(G)$, we have $\varphi(fg) = \phi fg \phi^{-1} = \phi f\phi^{-1} \phi g \phi^{-1} = \varphi(f)\varphi(g)$. Thus, $\mathit{Half}(G) \cong \mathit{Half}(G')$. The rest of the claim is concluded from the fact that $\varphi(\mathit{Half}_S(G)) = \mathit{Half}_S(G')$.
\end{proof}
\begin{obs}
\label{ob31} If $G\stackrel{H}{\cong} G'$, then $Aut(G)$ is not isomorphic to $Aut(G')$ in general (see example~\ref{ex41}).
\end{obs}
\section{Principal h-groupoids of G}
\label{sec4}
In this section, $G_0 = (G,*)$ is considered as a noncommutative groupoid.
Let $(G',\bullet)$ be a groupoid such that $G_0\stackrel{H}{\cong} (G',\bullet)$. Then, there exists a special half-isomorphism $f$ of $G_0$ into $(G',\bullet)$. Define an operation $\cdot$ on $G$ by $x\cdot y = f^{-1}(f(x)\bullet f(y))$. Thus, $f$ is an isomorphism of $(G,\cdot)$ into $(G',\bullet)$, and hence $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism, where $I_d$ is the identity mapping of $G$.
A groupoid $(G,\cdot)$ for which $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism is called a \emph{principal h-groupoid of $G_0$}. Therefore, the following result is at hand.
\begin{prop}
\label{prop41}
Let $G'$ be a groupoid. Then, $G_0\stackrel{H}{\cong} G'$ if and only if $G'$ is isomorphic to a principal h-groupoid of $G_0$.
\end{prop}
Denote by $\mathcal{M}(G_0)$ the set of the principal h-groupoids of $G_0$. Note that for \mbox{$(G,\cdot),(G,\bullet)\in \mathcal{M}(G_0)$,} we have $(G,\cdot)=(G,\bullet)$ if $x\cdot y = x \bullet y$, for all $x,y\in G$, which is equivalent to $I_d$ being an isomorphism between $(G,\cdot)$ and $(G,\bullet)$.
Let $(G,\cdot)\in \mathcal{M}(G_0)$. Since $I_d: G_0 \rightarrow (G,\cdot)$ is a special half-isomorphism, we have
\begin{equation}
\label{eq41}
\{x*y,y*x\} = \{x\cdot y,y\cdot x\}, \textrm{ for all } x,y \in G.
\end{equation}
If $(x,y)\in K(G_0)$, then $x\cdot y = y\cdot x = x*y$. For each pair $(x,y),(y,x)\in G\times G\setminus K(G_0)$, there are two possible values for $x\cdot y$ and $y\cdot x$ by \eqref{eq41}. Thus, if $G$ is finite, we have $2^{| G\times G\setminus K(G_0)|/2}$ possibilities for a principal h-groupoid of $G_0$. Hence, the following result is at hand.
\begin{prop}
\label{prop42} If $G$ is finite, then $|\mathcal{M}(G_0)| =2^{(|G|^2-|K(G_0)|)/2}$.
\end{prop}
Define $\mathcal{M}_I(G_0) = \{G'\in \mathcal{M}(G_0)\,|\, G'\cong G_0\}$ and let $S(G)$ be the set of permutations of $G$. For $G' = (G,\cdot) \in \mathcal{M}_I(G)$, define $Iso(G',G_0) = \{f\in S(G)\,|\, f \textrm{ is an isomorphism of } G' \textrm{ into } G_0\}$. Note that $Iso(G_0,G_0) = Aut(G_0)$. In the next result, we determine a relationship between $\mathit{Half}_S(G_0)$, $Aut(G_0)$ and $\mathcal{M}_I(G_0)$.
\begin{prop}
\label{prop43} We have:
\noindent{}(a) $Iso(G',G_0) \subset \mathit{Half}_S(G_0)$, for every $G'\in \mathcal{M}_I(G_0)$.
\\
(b) For each $G'\in \mathcal{M}_I(G_0)$, $Iso(G',G_0)$ is a right coset of $Aut(G_0)$ in $\mathit{Half}_S(G_0)$, that is, there exists $f\in \mathit{Half}_S(G_0)$ such that $Iso(G',G_0) = Aut(G_0)f$.
\\
(c) For $G_1,G_2 \in \mathcal{M}_I(G_0)$, if $Iso(G_1,G_0) \cap Iso(G_2,G_0) \not = \emptyset$, then $G_1 = G_2$.
\\
(d) $\mathit{Half}_S(G_0) = \bigcup_{G'\in \mathcal{M}_I(G_0)} Iso(G',G_0)$.
\\
(e) $|\mathcal{M}_I(G_0)| = [\mathit{Half}_S(G_0):Aut(G_0)]$, which is the index of $Aut(G_0)$ in $\mathit{Half}_S(G_0)$.
\end{prop}
\begin{proof} (a) For $G' = (G,\cdot)\in \mathcal{M}_I(G_0)$, let $f\in Iso(G',G_0)$. Then $f(x\cdot y) = f(x)*f(y)$, for all $x,y\in G$. By \eqref{eq41}, $\{f(x\cdot y),f(y\cdot x)\} = \{f(x)*f(y),f(y)*f(x)\}$, for all $x,y \in G$. By Theorem~\ref{teo31}, $f\in \mathit{Half}_S(G_0)$.\\
\noindent{}(b) Fix $f\in Iso(G',G_0)$. It is clear that $gf^{-1}\in Aut(G_0)$, for every $g\in Iso(G',G_0)$, and $\alpha f \in Iso(G',G_0)$, for every $\alpha \in Aut(G_0)$. Hence, we have the desired result.\\
\noindent{}(c) Let $f\in Iso(G_1,G_0) \cap Iso(G_2,G_0)$. Note that $I_d= f^{-1}f:G_1 \rightarrow G_2$ is an isomorphism. From the definition of $\mathcal{M}(G_0)$, it follows that $G_1 = G_2$.\\
\noindent{}(d) Let $f \in \mathit{Half}_S(G_0)$. Define the operation $\cdot$ on $G$ by $x\cdot y = f^{-1}(f(x)*f(y))$, for all $x,y\in G$. Note that $f: (G,\cdot)\rightarrow (G,*)$ is an isomorphism. Furthermore, since $f \in \mathit{Half}_S(G_0)$, and $f(x\cdot y) = f(x)*f(y)$ and $f(y\cdot x) = f(y)*f(x)$, for all $x,y \in G$, we have $\{x\cdot y,y\cdot x\} = \{x*y,y*x\}$, for all $x,y \in G$. Thus, $G' = (G,\cdot)\in \mathcal{M}_I(G_0)$, and hence $f\in Iso(G',G_0)$.\\
\noindent{}(e) It is a consequence of the previous items.
\end{proof}
As a consequence of the Proposition~\ref{prop33} and the item (e) of Proposition~\ref{prop43}, we have the following result.
\begin{cor}
\label{cor41} Let $G',G''$ be groupoids such that $G'\stackrel{H}{\cong} G''$ and $\mathit{Half}_S(G')$ is finite. Then,
\begin{center}
$|\mathcal{M}_I(G')|.|Aut(G')| = |\mathcal{M}_I(G'')|.|Aut(G'')|$
\end{center}
\end{cor}
Define $G_0^T = (G,\cdot)$, where $x\cdot y = y*x$, for all $x,y\in G$, and denote the set of anti-automorphisms of $G_0$ by $Ant(G_0)$. Since $G_0$ is noncommutative, we have $Aut(G_0)\cap Ant(G_0) = \emptyset$.
\begin{prop}
\label{prop44} $G_0$ has an anti-automorphism if and only if $G_0^T\in \mathcal{M}_I(G_0)$. In this case, $|Ant(G)| = |Aut(G)|$.
\end{prop}
\begin{proof} Note that a bijection $f$ of $G$ is an anti-automorphism of $G_0$ if and only if $f$ is an isomorphism of $G_0$ into $G_0^T$. The rest of the claim is concluded from the item (b) of Proposition~\ref{prop43}.
\end{proof}
\begin{exem}
\label{ex41}
Let $Q = \{1,2,...,8\}$ and consider the following Cayley tables of $(Q,*)$ and $(Q,\cdot)$:
\begin{center}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$*$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&6&5&7&8\\
2&2&1&4&3&5&6&8&7\\
3&4&3&1&2&7&8&5&6\\
4&3&4&2&1&8&7&6&5\\
5&5&6&8&7&1&2&4&3\\
6&6&5&7&8&2&1&3&4\\
7&8&7&6&5&3&4&1&2\\
8&7&8&5&6&4&3&2&1\\
\end{tabular}
\end{minipage}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$\cdot $&1&2&3&4&5&6&7&8\\
\hline
1&1&2&4&3&6&5&7&8\\
2&2&1&3&4&5&6&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&1&8&7&6&5\\
5&5&6&8&7&1&2&4&3\\
6&6&5&7&8&2&1&3&4\\
7&8&7&6&5&3&4&1&2\\
8&7&8&5&6&4&3&2&1\\
\end{tabular}
\end{minipage}
\end{center}
We have $(Q,*)$ and $(Q,\cdot)$ being quasigroups. Note that, for $x,y\in Q$:
\begin{center}
$x*y = \left\{\begin{array}{l}
y\cdot x, \textrm{ if } (x,y) \in \{(1,3),(1,4),(2,3),(2,4),(3,1),(3,2),(4,1),(4,2)\},\\
x\cdot y, \textrm{ otherwise.}
\end{array}\right.$
\end{center}
Thus, $(Q,\cdot) \in \mathcal{M}((Q,*))$. Using the LOOPS package \cite{NV1} for GAP \cite{gap} we get $|Aut((Q,*))| = 4$ and $|Aut((Q,\cdot))| = 8$. This illustrates Remark~\ref{ob31}.
Note that $|K((Q,*))| = 16$, and hence $|\mathcal{M}((Q,*))| = 2^{24} = 16777216$. Using a GAP computation with the LOOPS package, we get that there are $64$ quasigroups in $\mathcal{M}((Q,*))$ and $|\mathcal{M}_I((Q,*))| = 12$. By Proposition~\ref{prop43}, we have $|\mathit{Half}((Q,*))|= 48$ and $|\mathcal{M}_I((Q,\cdot))| = 6$.
It is observed that the number of quasigroups in $\mathcal{M}((Q,*))$ is much smaller than $|\mathcal{M}((Q,*))|$. In the next section, we will see that the same occurs for any finite noncommutative quasigroup. \qed
\end{exem}
\section{Principal h-quasigroups of Q}
\label{sec5}
Here, $Q_0= (Q,*)$ is considered as a noncommutative quasigroup. A quasigroup $(Q,\cdot)$ is a \emph{principal h-quasigroup of $Q_0$} if $(Q,\cdot)\in \mathcal{M}(Q_0)$. Denote by $\mathcal{N}(Q_0)$ the set of the principal h-quasigroups of $Q_0$. It is clear that $\mathcal{M}_I(Q_0) \subset \mathcal{N}(Q_0) \subset \mathcal{M}(Q_0)$. The next result is concluded from Proposition~\ref{prop41}.
\begin{prop}
\label{prop51} Let $Q'$ be a quasigroup. Then $Q_0\stackrel{H}{\cong} Q'$ if and only if $Q'$ is isomorphic to a principal h-quasigroup of $Q_0$.
\end{prop}
Now, we describe $\mathcal{N}(Q_0)$. For $(x,y),(x',y')\in Q\times Q \setminus K(Q_0)$, we say that $(x,y)\sim (x',y')$ if one of the following holds:
\noindent{}(i) $(x',y') = (y,x)$,
\\
(ii) $x = x'$ and $\{x*y,y*x\}\cap \{x*y',y'*x\}\not = \emptyset$,
\\
(iii) $y = y'$ and $\{x*y,y*x\}\cap \{x'*y,y*x'\}\not = \emptyset$.
We say that $(x,y)\equiv (x',y')$ if there are $z_1,z_2,...,z_l \in Q\times Q \setminus K(Q_0)$ such that $(x,y)\sim z_1 \sim z_2\sim...\sim z_l \sim (x',y')$.
The relation $\sim$ is reflexive and symmetric, and hence $\equiv$ is an equivalence relation. Denote by $r(Q_0)$ the number of equivalence classes of $\equiv$ on $Q\times Q \setminus K(Q_0)$.
Suppose that $Q$ is finite and let $\tau = \{(x_1,y_1),(x_2,y_2),...,(x_{r(Q_0)},y_{r(Q_0)})\}$ be a set of representatives of the equivalence classes of $\equiv$ on $Q\times Q \setminus K(Q_0)$. Consider $\mathbb{Z}_2 = \{0,1\}$, and for $\sigma = \{\sigma_1,\sigma_2,...,\sigma_{r(Q_0)}\} \in \mathbb{Z}_2^{r(Q_0)}$, define the operation $\stackrel{\sigma}{\bullet}$ on $Q$ by:
\begin{center}
$x \stackrel{\sigma}{\bullet} y = \left\{\begin{array}{l}
x * y, \textrm{ if } (x,y)\in K(Q_0) \textrm{ or } (x,y)\equiv (x_i,y_i), \textrm{ where } \sigma_i = 0,\\
y * x, \textrm{ if } (x,y)\equiv (x_i,y_i), \textrm{ where } \sigma_i = 1.
\end{array}\right.$
\end{center}
Denote $(Q,\stackrel{\sigma}{\bullet})$ by $Q_\sigma$ and let $\mathcal{N}_\tau(Q_0) = \{Q_\sigma \,|\, \sigma \in \mathbb{Z}_2^{r(Q_0)}\}$. Note that $\mathcal{N}_\tau(Q_0)\subset \mathcal{M}(Q_0)$ and $|\mathcal{N}_\tau(Q_0)| = 2^{r(Q_0)}$.
\begin{teo}
\label{teo51} If $Q$ is finite, then $\mathcal{N}(Q_0) = \mathcal{N}_\tau(Q_0)$. In particular, $|\mathcal{N}(Q_0)| = 2^{r(Q_0)}$.
\end{teo}
\begin{proof}
Let $Q_\sigma \in \mathcal{N}_\tau(Q_0)$. Since $Q$ is finite, in order to prove that $Q_\sigma$ is a quasigroup, we only need to show that the cancellation laws are satisfied, that is, $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y' \Rightarrow y = y'$ and $x \stackrel{\sigma}{\bullet} y = x' \stackrel{\sigma}{\bullet} y \Rightarrow x = x'$.
Let $x,y,y'\in Q$ be such that $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y'$. If $(x,y)\in K(Q_0)$, then \mbox{$x*y = y*x \in $} $ \{x*y',y'*x\}$, and hence $y = y'$. Now suppose that $(x,y)\not \in K(Q_0)$. We have four possibilities:
\noindent{}(i) $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$,
\\
(ii) $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$,
\\
(iii) $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$,
\\
(iv) $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$.
In (i) and (ii), it is immediately seen that $y = y'$.
For (iii) and (iv), we have $(x,y)\sim (x,y')$. Hence, there exists $(x_i,y_i)\in \tau$ such that $(x,y) \equiv (x_i,y_i)$ and $(x,y') \equiv (x_i,y_i)$. By definition of $\stackrel{\sigma}{\bullet}$, we have either $x \stackrel{\sigma}{\bullet} y = x*y$ and $x \stackrel{\sigma}{\bullet} y' = x*y'$, or $x \stackrel{\sigma}{\bullet} y = y*x$ and $x \stackrel{\sigma}{\bullet} y' = y'*x$. Since $(x,y)\not \in K(Q_0)$, it follows that $(x,y')\in K(Q_0)$. Similarly to the case $(x,y)\in K(Q_0)$, one can conclude that $y = y'$.
Thus, the cancellation law $x \stackrel{\sigma}{\bullet} y = x \stackrel{\sigma}{\bullet} y' \Rightarrow y = y'$ holds in $Q_\sigma$. The second cancellation law can be proven similarly. Therefore, $Q_\sigma \in \mathcal{N}(Q_0)$.
Conversely, let $Q' = (Q,\cdot) \in \mathcal{N}(Q_0)$. Then, there exists $\sigma \in \mathbb{Z}_2^{r(Q_0)}$ such that $x_i\cdot y_i = x_i \stackrel{\sigma}{\bullet} y_i$, for any $(x_i,y_i)\in \tau$. For $(x,y)\in K(Q_0)$, it is vividly deduced that $x\cdot y = x \stackrel{\sigma}{\bullet} y$.
Consider $(x_i,y_i)\in \tau$. Then, $y_i\cdot x_i = y_i \stackrel{\sigma}{\bullet} x_i$. Let $(x,y) \in Q\times Q\setminus \{(x_i,y_i),(y_i,x_i)\}$ such that $(x,y)\sim (x_i,y_i)$. By \eqref{eq41} and the definition of $\stackrel{\sigma}{\bullet}$, we have $x\cdot y \not = x_i\cdot y_i = x_i\stackrel{\sigma}{\bullet} y_i$ and $x \stackrel{\sigma}{\bullet} y \not = x_i\stackrel{\sigma}{\bullet} y_i$, and therefore the only possibility is $x\cdot y = x \stackrel{\sigma}{\bullet} y$. For every $(x,y)\sim (x_i,y_i)$, one can use the previous arguments and result in $x'\cdot y' = x' \stackrel{\sigma}{\bullet} y'$, for all $(x',y')\sim (x,y)$. Since $Q$ is finite, this procedure must end at some point, and hence $x\cdot y = x \stackrel{\sigma}{\bullet} y$, for all $(x,y)\equiv (x_i,y_i)$. As a result, we have $Q' = Q_\sigma$.
\end{proof}
By Proposition~\ref{prop42}, if $Q$ is finite, then $r(Q_0)\leq (|Q|^2 - |K(Q_0)|)/2$. The next proposition provides a better estimate for $r(Q_0)$. According to this result, it is seen that $|\mathcal{N}(Q_0)|$ is much smaller that $|\mathcal{M}(Q_0)|$.
\begin{prop}
\label{prop52a} If $Q$ is finite, then $r(Q_0)\leq (|Q|^2 - |K(Q_0)|)/6$ and $|\mathcal{N}(Q_0)|\leq \sqrt[3]{|\mathcal{M}(Q_0)|}$. In particular, $|\mathcal{M}(Q_0)| \geq 8$.
\end{prop}
\begin{proof}
Let $(x,y)\in Q\times Q \setminus K(Q_0)$ and $[(x,y)]$ be the equivalence class of $(x,y)$ with respect to $\equiv$. Since $Q_0$ is a quasigroup, there are $x',y' \in Q$ such that $x'\not =x$, $y'\not = y$, $(x',y)\sim (x,y)$, and $(x,y')\sim (x,y)$. We have $x\not = y$, $x'\not = y$ and $x\not = y'$ since $(x,y),(x',y),(x,y')\not \in K(Q_0)$. Thus, $|[(x,y)]|\geq |\{(x,y),(x',y),(x,y'),(y,x),(y,x'),(y',x)\}| = 6$. Hence, $|Q\times Q \setminus K(Q_0)|\geq 6 \,r(Q_0)$. The rest of the claim follows from Proposition~\ref{prop42}, Theorem~\ref{teo51} and the fact that $r(Q_0)\geq 1$.
\end{proof}
If $Q$ is finite and $r(Q_0)$ is small, one can generate all quasigroups of $\mathcal{N}(Q_0)$ computationally. Then, by using Propositions~\ref{prop51} and \ref{prop44} it can be verified if a quasigroup $Q'$ is half-isomorphic to $Q_0$ and generated all elements of $\mathit{Half}(Q_0)$. However, $r(Q_0)$ can be a large number even for groups of small order, and therefore generating all the quasigroups of $\mathcal{N}(Q_0)$ becomes computationally unviable. The next example illustrates both situations. In this example, $r(Q_0)$ and $|\mathcal{M}(Q_0)|$ are obtained by using GAP computing with the LOOPS package \cite{gap,NV1}.
\begin{exem} (a) Let $A_5$ be the alternating group of order $60$. We have that $r(A_5) = 91$, and hence $|\mathcal{N}(A_5)| = 2^{91}$. Furthermore, $|\mathcal{M}(A_5)| = 2^{1650}$.
\noindent{}(b) The LOOPS package for GAP contains all nonassociative right Bol loops of order $141$ (there are $23$ such loops). The right Bol loops of this order were classified in \cite{KNV}. If $L$ is one of these loops, then $3\leq r(L) \leq 8$, and hence $|\mathcal{N}(L)| \leq 256$. Furthermore, $|\mathcal{M}(L)| \geq 2^{5405}$.\qed
\end{exem}
By Proposition~\ref{prop21}, every quasigroup half-isomorphic to a loop is also a loop. Consequently, the same results as those presented for quasigroups in this section can be proven for loops. For more structured classes of loops, as it is seen in the following result, one can provide more information about the loops of $\mathcal{N}(L)$.
\begin{prop}
\label{prop52}
Let $G$ be a finite noncommutative group. Then, $|\mathcal{M}_I(G)| = 2$.
\end{prop}
\begin{proof}
From Scott's result \cite{Sco57}, we have $\mathit{Half}(G) = \mathit{Half}_T(G)$. Since $G$ is noncommutative, the mapping $J: G \to G$, defined by $J(x) = x^{-1}$, is an anti-automorphism of $G$. By Proposition~\ref{prop44}, we have $|\mathit{Half}(G)| = 2|Aut(G)|$. Thus, the claim follows from Proposition~\ref{prop43}.
\end{proof}
In fact, the previous proposition can be extended to any noncommutative loop that has an anti-automorphism and where every half-automorphism is trivial, such as the noncommutative loops of the subclass of Moufang loops in \cite[Thereom 1.4]{KSV16}, which include the noncommutative Moufang loops of odd order \cite{GG12}. Notice that this result cannot be extended even to all Moufang loops. In \cite[Example 4.6]{G20}, a noncommutative Moufang loop $L$ of order $16$ is given for which $|\mathcal{M}_I(L)| = [\mathit{Half}(L):Aut(L)] = 16.$
\section{A construction of a non-special half-automorphism}
\label{sl}
Let $G$ be a nonempty set with binary operations $*$ and $\cdot$ such that there exists a non-special half-isomorphism $f:(G,*) \rightarrow (G,\cdot)$. Define $G_\infty = \prod_{i=1}^\infty G$. The elements of $G_\infty$ will be denoted by $(x_i) = (x_i)_{i=1}^\infty$, where $x_i \in G$, for all $i$. For $(x_i),(y_i)\in G_\infty$, define the operation $(x_i)\bullet (y_i) = (z_i)$, where
\begin{center}
$z_j = \left\{\begin{array}{l}
x_j*y_j, \textrm{ if } j \textrm{ is odd}, \\
x_j\cdot y_j, \textrm{ if } j \textrm{ is even.} \\
\end{array}\right.$
\end{center}
Then, $(G_\infty,\bullet)$ is a groupoid. It is easy to see that if $(G,*)$ and $(G,\cdot)$ are quasigroups (loops), then $(G_\infty,\bullet)$ is also a quasigroup (loop). Define the mapping $\phi: G_\infty\rightarrow G_\infty$ by $\phi(x_i) = (y_i)$, where
\begin{center}
$y_j = \left\{\begin{array}{l}
f(x_1), \textrm{ if } j = 2, \\
x_{j+2}, \textrm{ if } j \textrm{ is odd}, \\
x_{j-2}, \textrm{ if } j>2 \textrm{ and } j \textrm{ is even.} \\
\end{array}\right.$
\end{center}
Thus, $\phi$ is a bijection and in each entry of $(x_i)$ it behaves like an isomorphism or a half-isomorphism. Hence, $\phi$ is a half-automorphism of $G_\infty$. Since $f$ is a non-special half-isomorphism, there are $x,y \in G$ such that $f^{-1}(x\cdot y) \not \in \{f^{-1}(x)*f^{-1}(y),f^{-1}(y)*f^{-1}(x)\}$. Then,
\begin{center}
$\phi^{-1}((x)_{i=1}^\infty\bullet(y)_{i=1}^\infty) \not \in \{\phi^{-1}((x)_{i=1}^\infty)\bullet \phi^{-1}((y)_{i=1}^\infty),\phi^{-1}((y)_{i=1}^\infty)\bullet \phi^{-1}((x)_{i=1}^\infty)\}$.
\end{center}
Therefore, $\phi$ is a non-special half-automorphism of $G_\infty$.
In example~\ref{ex1}, we have loops $C_6 = (G,*)$ and $L=(G,\cdot)$ for the conditions above, hence the loop $G_\infty$ has a non-special half-automorphism. Note that $\mathit{Half}(G_\infty)$ is not a group.
In the following example, a non-special half-isomorphism between a right Bol loop and a group is provided. This example is obtained by using MACE4 \cite{mace}.
\begin{exem}
\label{ex61}
Let $G = \{1,2,...,8\}$ and consider the following Cayley tables of $(G,*)$ and $(G,\cdot)$:
\begin{center}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$*$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&6&3&5&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&8&1&7&6&5\\
5&5&6&7&1&8&2&3&4\\
6&6&5&8&7&2&1&4&3\\
7&7&8&5&3&6&4&1&2\\
8&8&7&6&5&4&3&2&1\\
\end{tabular}
\end{minipage}
\begin{minipage}{.4\textwidth}
\centering
\begin{tabular}{c|cccccccc}
$\cdot$&1&2&3&4&5&6&7&8\\
\hline
1&1&2&3&4&5&6&7&8\\
2&2&1&4&3&6&5&8&7\\
3&3&4&1&2&7&8&5&6\\
4&4&3&2&1&8&7&6&5\\
5&5&7&6&8&1&3&2&4\\
6&6&8&5&7&2&4&1&3\\
7&7&5&8&6&3&1&4&2\\
8&8&6&7&5&4&2&3&1\\
\end{tabular}
\end{minipage}
\end{center}
We have $(G,*) = L$ as a right Bol loop and $(G,\cdot)$ being isomorphic to $D_8$, which is the dihedral group of order $8$. The permutation $f = (3 \, 5 \, 7)(4\, 6\, 8)$ of $G$ is a half-isomorphism of $L$ into $D_8$. Since $|K(L)| = 56$ and $|K(D_8)| = 40$, $f$ is a non-special half-isomorphism by Theorem~\ref{teo31}. Since $L$ and $D_8$ are right Bol loops, $G_\infty$ is also a right Bol loop, and from the previous construction we have a non-special half-automorphism in a right Bol loop of infinite order.\qed
\end{exem}
\section*{Acknowledgments}
Some calculations in this work have been made by using the finite model builder MACE4, developed by McCune \cite{mace}, and the LOOPS package \cite{NV1} for GAP \cite{gap}.
\addcontentsline{toc}{section}{Acknowledgments}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Bayesian estimation for a parametric Markov Renewal model applied to seismic data}
\runtitle{Bayesian Markov Renewal model for seismic data}
\author{\fnms{Ilenia} \snm{Epifani}\corref{}\ead[label=e1]{ilenia.epifani@polimi.it}}
\address{Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy\\ \printead{e1}}
\and
\author{\fnms{Lucia} \snm{Ladelli}\ead[label=e2]{lucia.ladelli@polimi.it}}
\address{Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy\\ \printead{e2}}
\and
\author{\fnms{Antonio} \snm{Pievatolo}\ead[label=e3]{antonio.pievatolo@cnr.it}}
\address{IMATI-CNR, Via Bassini 15, 20133 Milano, Italy\\ \printead{e3}}
\runauthor{Epifani, Ladelli, Pievatolo}
\begin{abstract}
This paper presents a complete methodology for Bayesian inference on a semi-Markov process, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation. The holding times (conditional on the transition between two given states) are assumed to be Weibull-distributed. We examine the elicitation of the joint prior density of the shape and scale parameters of the Weibull distributions, deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning data'' and moment existence conditions. This framework is applied to data of earthquakes of three types of severity (low, medium and high size) that occurred in the central Northern Apennines in Italy and collected by the \cite{CPTI04} catalogue. Assumptions on two types of energy accumulation and release mechanisms are evaluated.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60K20}
\kwd{62F15}
\kwd{62M05}
\kwd{86A15}
\kwd[; secondary ]{65C05}
\end{keyword}
\begin{keyword}
\kwd{Bayesian inference}
\kwd{Earthquakes}
\kwd{Gibbs sampling}
\kwd{Markov Renewal process}
\kwd{Predictive distribution}
\kwd{semi-Markov process}
\kwd{Weibull distribution}
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:introduction}
Markov Renewal processes or their semi-Markov representation have been considered in the seismological literature as models which allow the distribution of the inter-occurrence times between earthquakes to depend on the last and the next earthquake and to be not necessarily exponential.
The time predictable and the slip predictable models studied in \cite{ShimazakiNakata}, \cite{GuagMol}, \cite{GuagMolMul} and \cite{BetroGaravGuagRotonTagl} are special cases of Markov Renewal processes. These models are capable of interpreting the predictable behavior of strong earthquakes in some seismogenic areas. In these processes the magnitude is a deterministic function of the inter-occurrence time.
A stationary Markov Renewal process with Weibull inter-occurrence times has been studied from a classical statistical point of view in \cite{Alvarez}. The Weibull model allows for the consideration of monotonic hazard rates; it contains the exponential model as a special case which gives a Markov Poisson point process. In \cite{Alvarez} the model parameters were fitted to the large earthquakes in the North Anatolian Fault Zone through maximum likelihood and the Markov Poisson point process assumption was tested.
In order to capture a non monotonic behavior in the hazard, in \cite{GaravagliaPavani} the model of Alvarez was modified and a Markov Renewal process with inter-occurrence times that are mixtures of an exponential and a Weibull distribution was fitted to the same Turkish data.
In \cite{Masala} a parametric semi-Markov model with a generalized Weibull distribution for the inter-occurrence times was adapted to Italian earthquakes. Actually the semi-Markov model with generalized Weibull distributed times was first used in \cite{Foucher} to study the evolution of HIV infected patients. \cite{Votsi} considered a semi-Markov model for the seismic hazard assessment in the Northern Aegean sea and estimated the quantities of interest (semi-Markov kernel, Markov Renewal functions, etc.) through a nonparametric method.
While a wide literature concerning classical inference for Markov Renewal models for earthquake forecasting exists, to our knowledge a Bayesian approach is limited in this context. \cite{PatwardhanEtAlii} considered a semi-Markov model with log-normal distributed discrete inter-occurrence times and applied it to the large earthquakes in the circum-Pacific belt.
They stressed the fact that it is relevant to use Bayesian techniques when prior knowledge is available and it is fruitful even if the sample size is small.
\cite{MarinPlaRiosInsua} also employed semi-Markov models in the Bayesian framework, applied to a completely different area: sow farm management. They used WinBugs to perform computations (but without giving details) and they elicited their prior distributions on parameters from knowledge on farming practices.
From a probabilistic viewpoint, a Bayesian statistical treatment of a semi-Markov process amounts to model the data as a mixture of semi-Markov processes, where the mixing measure is supported on the parameters, by means of their prior laws. A complete characterization of such a mixture has been given in
\cite{EpifaniFortiniLadelli}.
In this paper we develop a parametric Bayesian analysis for a Markov Renewal process modelling earthquakes in an Italian seismic region. The magnitudes are classified into three categories according to their severity: low, medium and high size, and these categories represent the states visited by the process. As in \cite{Alvarez}, the inter-occurrence times are assumed to be Weibull random variables. The ``current sample'' is formed by the sequences of earthquakes in a homogeneous seismic region and by the corresponding inter-occurrence times collected up to a time $T$. When $T$ does not coincide with an earthquake, the last observed inter-occurrence time is censored.
The prior distribution of the parameters of the model is elicited using a ``learning dataset'', i.e. data coming from a seismic region similar to that under analysis.
The posterior distribution of the parameters is obtained through Gibbs sampling and the following summaries are estimated: transition probabilities, shape and scale parameters of the Weibull holding times for each transition and the so-called cross state-probabilities (CSPs). The transition probabilities indicate whether the strength of the next earthquake is in some way dependent on the strength of the last one; the shape parameters of the holding times indicate whether the hazard rate between two earthquakes of given magnitude classes is decreasing or increasing; the CSPs give the probability that the next earthquake occurs at or before a given time and is of a given magnitude, conditionally on the time elapsed since the last earthquake and on its magnitude.
The paper is organized as follows. In Section~\ref{sec:1} we illustrate the dataset and we discuss the choice of the Weibull model in detail. Section~\ref{sec:2} introduces the parametric Markov Renewal model.
Section~\ref{sec:3} deals with the elicitation of the prior.
Section~\ref{sec:apennines} contains the Bayesian data analysis with the estimation of the above-mentioned summaries. We also test a time predictable and a slip predictable model against the data. Section~\ref{sec:6} is devoted to some concluding remarks. Appendix \ref{sec:gibbsampling} contains the detailed derivation of the full conditional distributions and the JAGS (Just Another Gibbs Sampler) implementation of the Gibbs sampler (\cite{Plummer}).
\section{A test dataset}
\label{sec:1}
We tested our method on a sequence of seismic events chosen among those examined in \cite{Rotondi}, which was given us by the author. The sequence collects events that occurred in a tectonically homogeneous macroregion, identified as $\text{MR}_3$ by Rotondi and corresponding to the central Northern Apennines in Italy. The subdivision of Italy into eight (tectonically homogeneous) seismic macroregions can be found in the \cite{DISS} and the data are collected in the \cite{CPTI04} catalogue. If one considers earthquakes with magnitude\footnote{We refer to the moment magnitude which is related to the seismic moment $M_0$ by the following relationship: $M_w = \frac{2}{3}(\log_{10}M_0-16.05$); see \cite{Hanks}, where it is denoted by \textbf{M}.} $M_w\ge 4.5$, the sequence is complete from year 1838: a lower magnitude would make the completeness of the series questionable, especially in its earlier part. The map of these earthquakes marked by dots appears in Figure~\ref{fig:mappa}.
\begin{figure}
\caption{Map of Italy with dots indicating earthquakes with magnitude $M_w\ge 4.5$ belonging to macroregion $\text{MR}
\label{fig:mappa}
\end{figure}
As a lower threshold for the class of strong earhquakes, we choose $M_w\ge 5.3$, as suggested by \cite{Rotondi}. Then a magnitude state space with three states is obtained by indexing an earthquake by 1, 2 or 3 if its magnitude belongs to intervals $[4.5,4.9)$, $[4.9,5.3)$, $[5.3,+\infty)$, respectively. Magnitude 4.9 is just the midpoint between 4.5 and 5.3 and the released energy increases geometrically as one moves through the endpoints, with a common ratio of 4: if $M_0(M_w)$ denotes the seismic moment $M_0$ associated with $M_w$, then $M_0(5.3)/M_0(4.9) = M_0(4.9)/M_0(4.5) = 10^{\frac{3}{2} 0.4} \simeq 4$.
The energy released from an earthquake with $M_w=4.9$ does not match the midpoint between seismic moments associated with magnitudes $4.5$ and $5.3$ (in fact, this correspondence holds if $M_w=5.1$). However, there seem to be no general rule in the literature for splitting magnitude intervals. For example, \cite{Votsi} used cut-points 5.5, 5.7 and 6.1, so that $M_0(6.1)/M_0(5.7)\simeq 4$ and $M_0(5.7)/M_0(5.5)\simeq 2$, while the energy midpoint is at $M_w=5.9$; following \cite{Altinok}, \cite{Alvarez} uses cut-points 5.5, 6.0 and 6.5; \cite{Masala} employed the magnitude classes $M_w<4.7$, $M_w\in [4.7, 5)$, $M_w\ge 5$. All these authors do not give any special reason for their choices.
A more structured approach is attempted by \cite{Sadeghian}, who applied a statistical clustering algorithm to magnitudes, and again by \cite{Votsi} when they propose a different classification of states that combines both magnitude and fault orientation information. From a modelling viewpoint, this latter approach is certainly preferable, because it is likely to produce more homogeneous classes, however we do not have enough additional information to attempt this type of classification of our data in a meaningful way. An entirely different approach is that based on risk, in which cut-points would change with the built environment.
We now examine inter-occurrence times. \cite{Rotondi} considers a nonparametric Bayesian model for the inter-occurrence times between strong earthquakes (i.e. $M_w\ge 5.3$), after a preliminary data analysis which rules out Weibull, Gamma, log-normal distributions among others frequently used.
On the other hand, with a Markov Renewal model, the sequence of all the inter-occurrence times is subdivided into shorter ones according to the magnitudes, so that we think that a parametric distribution is a viable option. In particular,
we focussed on the macroregion $\text{MR}_3$ because the Weibull distribution seems to fit the inter-occurrence times better than in other macroregions. This fact is based on qq-plots. The qq-plots for $\text{MR}_3$ are shown in Figure~\ref{fig:qqplots}. The plot for transitions from 1 to 3 shows a sample quantile that is considerably larger than expected. The outlying point corresponds to a long inter-occurrence time of about 9 years, between 1987 and 1996, while 99~percent of the inter-occurence times are below 5 years. Obviously, the classification into macroregions influences the way the earthquake sequence is subdivided.
Given the Markov Renewal model framework, holding time distributions other than the Weibull could be used, such as the inverse Gaussian, the log-normal and the Gamma. However, the inverse Gaussian qq-plots clearly indicate that this distribution does not fit the data. As for the log-normal, the outlying point in the qq-plot of the $(1,3)$ transition becomes only a little less isolated, but at the expense of introducing an evident curvature in the qq-plot of the $(1,1)$ transition, whereas the remaining qq-plots are unchanged. The Gamma qq-plots are indistinguishable from the Weibull qq-plots, but we prefer working with the Weibull in view of the existing literature on seismic data analysis where the Weibull is employed. In this respect, we could follow \cite{Masala} and choose the generalized Weibull, which includes the Weibull, but the qq-plots are unchanged even with the extra parameter.
From a Bayesian computational point of view, there is no special reason for preferring the (possibly generalized) Weibull to the Gamma, as neither of them possesses a conjugate prior distribution and numerical methods are needed in both cases for making inference.
In the existing literature, the Weibull distribution has been widely used to model holding times between earthquakes from different areas and with different motivations. In Section \ref{sec:introduction} we mentioned \cite{Alvarez}, \cite{GaravagliaPavani} and \cite{Masala}, but there are also other authors. \cite{AbaimovEtAlii} argued that the increase in stress caused by the motion of tectonic plates at plate boundary faults is adequately described by an increasing hazard function, such as the Weibull can have. Instead, other distributions have an inappropriate tail behaviour: the log-normal hazard tends to zero with time and the inverse Gaussian hazard tends to a constant. Goodness-of-fit checks for the recurrence times of slip events in the creeping section of the San Andreas fault in central California confirmed that the Weibull is preferable to the mentioned alternatives. \cite{HrostMousl} considered a Weibull model, for single faults (or fault systems with homogeneous strength statistics) and power law stress accumulation. They derived the Weibull model from a theoretical framework based on the statistical mechanics of brittle fracture and they applied it to microearthquake sequences (small magnitudes) from the island of Crete and from a seismic area of Southern California, finding agreement with the data except for some deviations in the upper tail. Regarding tail behaviour, we can make a connection with
\cite{HasumiEtAlii}, who analyzed a catalogue of the Japan Meteorological Agency. These data support the hypothesis that the holding times can be described by a mixture of a Weibull distribution and a log-Weibull distribution (which possesses a heavier tail); if only earthquakes with a magnitude exceeding a threshold are considered, the weight of the log-Weibull component becomes negligible as the threshold increases.
\begin{figure}
\caption{Weibull qq-plots of earthquake inter-occurrence times (central Northern Appennines) classified by transition between magnitude classes.}
\label{fig:qqplots}
\end{figure}
\section{Markov Renewal model}
\label{sec:2}
Let us observe, over a period of time $[0,T]$, a process in which different events occur, with random inter-occurence times.
Let us suppose that the possible states of the process are the points of a finite set
$E=\{1,\dots , s \}$ and that the process starts from state $j_0$.
Let us denote by $\tau$ the number of times the process changes states in the time interval $[0,T]$ and by
$t_i$ the time of the $i$-th change of state. Hence,
$0< t_1< \dots < t_{\tau}\leq T$. Let $j_0, j_1,\dots, j_{\tau}$ be the sequence of states visited by
the process and $x_i$ the holding time in the state $j_{i-1}$, for $i=1,\dots, \tau$. Then
\begin{equation*}
x_i= t_i-t_{i-1} \qquad \mbox{for} \; i=1,\dots, \tau
\end{equation*}
with $t_0:=0$. Furthermore, let $u_T$ be the time spent in $j_\tau$
\begin{equation*}
u_T= T-t_{\tau} ,
\end{equation*}
so the time $u_T$ is a right-censored time.
Finally, our data are collected in the vector $(\bj, \bx, u_T)$, where $ (\bj, \bx) =(j_{n}, x_{n})_{n=1,\ldots, \tau }$.
In what follows, we assume that the data $(\bj, \bx, u_T)$ are the result of the observation
of a homogeneous Markov Renewal process $(J_n, X_n)_{n\geq 0}$ starting from $j_0$.
This means that the sequence $(J_n, X_n)_{n\geq 0}$ satisfies
\begin{equation}
P(J_0=j_0) = 1, \quad P(X_0=0) =1
\label{eq:MRP_J_0}
\end{equation}
and for every $n\geq 0$, $j\in E$ and $t\geq 0$
\begin{equation}
\label{eq:MRP_n}
P(J_{n+1}=j, X_{n+1}\leq t | (J_{k}, X_{k})_{k\leq n})=
P(J_{n+1}=j, X_{n+1}\leq t | (J_{n}, X_{n}))= p_{J_{n} j} F_{J_{n} j}(t) \ .
\end{equation}
The transitions probabilities $p_{ij}$'s are collected in a transition matrix $\bp = (p_{ij})_{i,j \in E}$ and $(F_{ij})_{i,j \in E}$ is an array of
distribution functions on $\mathbb{R}_{+}=(0,+\infty)$.
For more details on Markov Renewal processes see, for example, \cite{LimniosOprisan}.
We just recall that, under Assumptions~\eqref{eq:MRP_J_0} and \eqref{eq:MRP_n}:
\begin{itemize}
\item[--] the process $(J_n)_{n\geq 0}$ is a Markov chain, starting from $j_0$, with transition matrix $\bp$,
\item[--] the holding times $(X_n)_{n\geq 0}$, conditionally on $(J_n)_{n\geq 0}$, form a sequence of independent positive random variables, with distribution function $F_{J_{n-1}\ J_n}$.
\end{itemize}
We assume that the functions $F_{ij}$ are absolutely continuous with respect to the Lebesgue measure
with density $f_{ij}$. Hence, the likelihood function of the data $(\bj, \bx, u_T)$ is
\begin{equation}
\label{eq:likelihood_general}
L(\bj, \bx, u_T)=
\left(\prod_{i=0}^{\t-1}p_{j_i j_{i+1}}f_{j_i j_{i+1}}(x_{i+1})\right)^{\mathds{1}(\tau>0)}\times
\sum_{k\in E}p_{j_{\t}k} \bar{F}_{j_{\t}k}(u_T),
\end{equation}
where, for every $x$, $\bar{F}_{ij}$ is the survival function
\begin{displaymath}
\bar{F}_{ij}(x)=1-F_{ij}(x)=P(X_{n+1}>x|J_n=i, J_{n+1}=j) \ .
\end{displaymath}
Furthermore, we assume that each inter-occurrence time has a Weibull density $f_{ij}$ with
shape parameter $\a_{ij}$ and scale parameter $\theta_{ij}$, i.e.
\begin{equation}
\label{eq:weib}
f_{ij}(x)=\frac{\a_{ij}}{\theta_{ij}}\left(\frac{x}{\theta_{ij}}\right)^{\a_{ij}-1}
\exp\left\{-\left(\frac{x}{\theta_{ij}}\right)^{\a_{ij}}\right \} \ , \quad x>0 , \; \a_{ij}>0, \;\theta_{ij}>0 \ .
\end{equation}
For conciseness, let
${\boldsymbol{\alpha}}=(\a_{ij})_{i,j\in E}$ and ${\boldsymbol{\theta}}=(\th_{ij})_{i,j\in E}$.
In order to write the likelihood in a more convenient way, let us introduce the following natural statistics. We will say that the process visits the string $(i,j)$ if a visit to $i$ is followed by a visit to $j$ and we denote by
\begin{itemize}
\item[--] $x^{\r}_{ij}$ the time spent in state $i$ at the $\r$-th visit to the string $(i,j)$,
\item[--] $N_{ij}$ the number of visits to the string $(i,j)$.
\end{itemize}
Then, assuming $\tau\ge 1$, Equations~\eqref{eq:likelihood_general} and \eqref{eq:weib} yield the following
representation of the likelihood function
\begin{multline}
\label{eq:likelihood-param}
L(\bj, \bx, u_T \, |\, \bp, \boldsymbol{\a},\boldsymbol{ \theta}) =\prod_{i,k\in E}p_{ik}^{N_{ik}}\times \\
\times \prod_{i,k\in E}\left[\a_{ik}^{N_{ik}}\frac{1}{\theta_{ik}^{\a_{ik}N_{ik}}}\left(\prod_{\rho=1}^{N_{ik}}x^{\r}_{ik}\right)^{\a_{ik}-1} \times
\exp\left\{-\frac{1}{\theta_{ik}^{\a_{ik}}}\sum_{\rho =1}^{N_{ik}}(x^{\r}_{ik})^{\a_{ik}}\right\}\right]\times \\
\times \left(\sum_{k\in E}p_{j_{\t}k}
\exp\left\{-\left(\frac{u_T}{\theta_{j_{\t} k}}\right)^{\a_{j_{\t} k}}\right\}\right)\ .
\end{multline}
Our purpose is now to perform a Bayesian analysis for $\bp, \boldsymbol{\alpha}$ and $ \boldsymbol{\theta}$ which allows us to introduce prior knowledge on the parameters. As shown in Appendix \ref{sec:gibbsampling}, this analysis is possible via a Gibbs sampling approach.
\section{Bayesian analysis}
\label{sec:3}
\subsection{The prior distribution}
\label{subsec:3.1}
Let us assume that a priori $\bp$ is independent of $\ba$ and $\bth$. In particular,
the rows of $\bp$ are $s$ independent vectors with Dirichlet distribution
with parameters $\bg_1, \cdots ,\bg_s$ and total mass $c_1, \cdots,c_s$, respectively. This means
that, for $i=1,\ldots ,s$, the prior density of the $i$-th row is
\begin{equation}
\label{eq:dirich}
\pi_{1,i}(p_{i1},\ldots , p_{is}) = \frac{\Gamma(c_i)}{\prod_{j=1}^{s}\Gamma(\g_{ij})} {\prod_{j=1}^{s}p_{ij}^{\g_{ij}-1}}
\end{equation}
on $T=\{ (p_{i1},\ldots , p_{is})| \ p_{ij} \geq 0,\ \sum_j p_{ij} =1\}$ where $\bg_i =(\g_{i1}, \cdots , \g_{is})$, with $\g_{ij}>0$ and $c_i=\sum_{j=1}^s \g_{ij}$.
As far as $ \ba$ and $\bth$ are concerned, the $\th_{ij}$'s, given the $\a_{ij}$'s, are independent with generalized inverse Gamma
densities
\begin{equation}
\label{eq:pi2}
\pi_{2,ij}(\th_{ij}| \ba) = \pi_{2,ij}(\th_{ij}| \a_{ij}) =
\frac{\a_{ij}b_{ij}(\a_{ij})^{m_{ij}}}{\Gamma(m_{ij})}\th_{ij}^{-(1+m_{ij}\a_{ij})}
\times \exp\left\{-\frac{b_{ij}(\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\},
\;\; \th_{ij}>0,
\end{equation}
where $m_{ij}>0$ and
\begin{equation}
\label{eq:b-q}
b_{ij}(\a_{ij}) = \left(t^{ij}_{q_{ij}}\right)^{\a_{ij}}[(1-q_{ij})^{-1/m_{ij}}-1]^{-1}
\end{equation}
with
$t^{ij}_{q_{ij}}>0$ and $q_{ij}\in (0,1)$.
In other terms, $\th_{ij}^{- \a_{ij}}$, given $ \a_{ij}$, has a prior Gamma density with shape $m_{ij}$ and scale $1/b_{ij}(\a_{ij})$. In symbols $\th_{ij}|
\a_{ij} \sim m_{\alpha}thcal{ GIG}(m_{ij}, b_{ij}(\a_{ij}), \a_{ij})$.
We borrow the expression of the $b_{ij}(\a_{ij})$'s in \eqref{eq:b-q} from \cite{Bousquet2} and, as a consequence of this choice, $t^{ij}_{q_{ij}}$ turns out to be the marginal quantile of order $q_{ij}$ of an inter-occurrence time between states $i$ and $j$. Indeed, if $\pi_{3,ij}$ denotes the density of $\a_{ij}$ and $X$ is such a random time, then
\begin{align*}
P(X> t) & = \int_{0}^{+\infty}\int_{0}^{+\infty}
P(X> t|\a_{ij},\theta_{ij})\pi_{2,ij}(\th_{ij}|\a_{ij})\pi_{3,ij}(\a_{ij})d\th_{ij} d\a_{ij}\\
& = \int_{0}^{+\infty}\Big[\frac{b_{ij}(\a_{ij})}{b_{ij}(\a_{ij})+t^{\a_{ij}}}\Big]^{m_{ij}}\pi_{3,ij}(\a_{ij})d\a_{ij} \ ,
\qquad \qquad \qquad \qquad\;\forall t>0.
\end{align*}
Hence, in view of \eqref{eq:b-q}, if $t=t^{ij}_{q_{ij}}$, we obtain $P(X>t^{ij}_{q_{ij}})=1-q_{ij}$, for every proper prior density $\pi_{3,ij}$.
Finally, a priori, the
components of $ \ba$ are independent and have
densities $\pi_{3,ij}$
such that
\begin{multline}
\label{eq:pi3}
\pi_{3,ij}(\a_{ij})\propto
\a_{ij}^{m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}\exp\{-m_{ij}d_{ij}\a_{ij}\}\mathds{1}(\a_{ij}\ge\a_{0,ij}), \\
\a_{0,ij}\geq 0,\; c_{ij}> 0,\; m_{ij}>0, \; d_{ij}\ge 0\ .
\end{multline}
As far as the prior $\pi_{3,ij}$ is concerned, it is easy to see that:
\begin{enumerate}[$a)$]
\item if $d_{ij} >0$, then $\pi_{3,ij}$ is a proper prior;
\item if $\a_{0,ij}=0$ and $d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density;
\item if $c_{ij}=1 , \ \a_{0,ij}>0 \mbox{ and } d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density
truncated from below at $\a_{0,ij}$;
\item if $c_{ij}=m_{ij}, \ \a_{0,ij}>0 \mbox{ and } d_{ij} >0$,
then $\pi_{3,ij}$ is a Gamma density shifted by
$\a_{0,ij}$;
\item if $c_{ij}=1$ and $m_{ij}\to 0$, then $\pi_{2,ij}(\theta_{ij}|\a_{ij})\pi_{3,ij}(\a_{ij})$ approaches the Jeffreys prior for the Weibull model: $1/\theta_{ij}\mathds{1}{(\theta_{ij}>0)}\mathds{1}{(\alpha_{ij} \geq \a_{0,ij})}$;
\item if $c_{ij} \ge 1$ \mbox{ and } $m_{ij}\ge 1$, then $\pi_{3,ij}$ is a log-concave function.
\end{enumerate}
The prior corresponding to the choices in $c)$ was first introduced in \cite{Bousquet} and \cite{Bousquet2}.
As discussed in \cite{GilksWild}, the log-concavity of $\pi_{3,ij}$ is necessary
in the implementation of the Gibbs sampler (see also \cite{BergerSun}), although adjustments exist for the non-log-concave case (see \cite{GilksBestTan}).
Furthermore, we will show later that a support suitably bounded away from zero ensures the existence of the posterior moments of the $\th_{ij}$'s.
\subsection{Elicitation of the hyperparameters}
In this section we focus our attention on the prior of $ ( \a_{ij},\th_{ij})$, for fixed $i,j$. Adapting the approach developed by Bousquet to our situation, we give a statistical justification of the prior introduced in Subsection~\ref{subsec:3.1}. An interpretation of the hyperparameters is also provided.
For the sake of semplicity, let us drop the indices $i,j$ in all the notations and quantities.
Suppose that a ``learning dataset'' $\by_{m} = (y_{1},\ldots , y_{m})$ of $m$ holding times in the state $i$ followed by a visit to the state $j$ is available from another seismic region similar to the one under analysis.
Therefore the prior scheme defined by Equations \eqref{eq:pi2}--\eqref{eq:pi3} can be interpreted as a suitable modification of a posterior distribution of $( \a, \th)$, given the learning dataset $\by_{m}$.
This approach allows us to elicit the hyperparameters.
More precisely, consider
for $(\a, \theta)$ the posterior density, conditionally on $\by_{m}$, when we start from the following improper prior:
\begin{equation}
\label{eq:pi_alpha1}
\tilde{\pi}(\a, \theta) \propto \theta^{-1}\left(1-\frac{\alpha_0}{\alpha}\right)^{c-1} \mathds{1}{(\theta \geq 0)} \mathds{1}{(\alpha \geq \alpha_0)} \ ,
\end{equation}
for some suitable $c\geq 1$ and $\alpha_0\geq 0$ (The condition $c\geq 1$ guarantees that $\tilde{\pi}(\a, \theta)$ is a log-concave function with respect to $\alpha$).
Consequently, the posterior density of $\theta$, given $\alpha$ and $\by_{m}$, is
\begin{equation}
\label{eq:prior-theta}
\tilde\pi_2(\theta| \by_{m}, \alpha ) = m_{\alpha}thcal{ GIG}(m, \tilde b(\by_{m} ,\alpha), \alpha)
\end{equation}
and the posterior density of $\alpha$ is
\begin{equation}
\label{eq:prior-alpha}
\tilde\pi_3(\alpha |\by_{m}) \propto \frac{\alpha^{m-c}(\alpha -\alpha_0)^{c-1}}{\tilde b^m(\by_{m}, \alpha)}
\exp\{{-m}{\beta(\by_{m})\alpha}\}\mathds{1}{(\alpha \geq \alpha_0)} \ ,
\end{equation}
with $\tilde b( \by_{m},\alpha)=\sum_{i=1}^{m}y^{\alpha}_{i}$ and $\beta(\by_{m})= {\sum_{i=1}^{m}\ln y_{i}}/{m}$.
Notice that the posterior we obtain
has a simple hierarchical structure: $\tilde\pi_2(\theta|\by_{m},\alpha)$ is a generalized inverse Gamma density and this provides both a justification of the form of the $\pi_2(\theta|\alpha)$ in \eqref{eq:pi2} and an interpretation of the first parameter $m$. Indeed $m$ is equal to the size of the learning dataset $\by_m$ and so it is a measure of prior uncertainty.
Now, if we replace the function $\tilde b(\by_{m},\alpha )$ in \eqref{eq:prior-theta} and \eqref{eq:prior-alpha} by the easier convex function of $\alpha$ introduced in \eqref{eq:b-q}, i.e.
$b(\alpha)= t^{\alpha}_q[(1-q)^{-1/m}-1]^{-1}$, with $t_q>0$ and $q\in (0,1)$,
then $\tilde\pi_3(\alpha |\by_{m})$ takes the same form as
in \eqref{eq:pi3} with
\begin{equation}
\label{eq:d_ij}
d=\ln {t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m} \ .
\end{equation}
In this way, we obtain a justification of the form of the prior densities $\pi_{3,ij}$'s in \eqref{eq:pi3} and an easy way to elicit its parameter $d_{ij}$ when the learning dataset is available.
Furthermore, $b(\alpha)$ can be also elicited once the predictive quantile $t_q$ is specified. Its specification can be accomplished, for example, in the two following different ways:
\begin{enumerate}
\item we estimate an empirical quantile $\hat t_q$ from the learning dataset;
\item an expert is asked about the chance, quantified by $q$, of an earthquake before $t_q$.
\end{enumerate}
\mbox{}\indent In the following, if a learning dataset of size $m\ge 2$ is available, we consider an empirical quantile $\hat{t}_q$ of order $q$ such that
\begin{displaymath}
\ln \hat{t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m}> 0\ .
\end{displaymath}
Therefore, letting $\hat{b}(\alpha)$ denote the value of $b(\a)$ corresponding to $\hat{t}_q$, we propose a Bayesian analysis based on the prior
\begin{equation}\label{pi-2}
\pi_2(\theta|\alpha )=m_{\alpha}thcal{ GIG}(m, \hat{b}(\alpha), \a)
\end{equation}
and
\begin{equation}\label{pi-3}
\pi_3(\alpha)\propto \a^{m-c}(\alpha-\a_0)^{c-1} \exp\left\{-m\left(\ln \hat{t}_q- \frac{\sum _{i=1}^{m}\ln y_{i}}{m}\right)\a\right\}\mathds{1}{(\alpha\geq \a_0)} \ ,
\end{equation}
where $m$ is the size of the learning dataset. In addition, we choose $c=m$ so that $\pi_3(\alpha)$ is a shifted Gamma prior and consequently it is proper and log-concave.
The remaining hyperparameter $\a_0$ is chosen so that the posterior second moment of $\theta$ is finite. If $\a$ is bounded away from zero, then
\begin{displaymath}
\E(\theta^2) = \E\left(\frac{\Gamma(m-2/\a)}{\Gamma(m)}\left[\hat{b}(\alpha)\right]^{2 / \a}\right)\leq \tilde{K} \E(\Gamma(m-2/\a))
\end{displaymath}
for a suitable constant $\tilde{K}$. As a consequence if $\a_0=2/m$, then $\E(\theta^2)<+\infty$ and hence also the posterior second moment of $\theta$ is finite.
The choice $\a_0=2/m$ is suitable only if $m>2$.
If $m=2$, then
$\a_0=2/m=1$ and decreasing hazard rates are ruled out.
In the absence of additional specific prior information, this is an arbitrary restriction, so a value for $\a_0$ smaller than 1 must be chosen. Then, the prior second moment of $\theta$ is not finite anymore. On the other hand, for the posterior second moment to be finite, we need $\alpha>2/(2+N)$, where $N$ is the number of transitions between the two concerned states in the (current) sample. Thus the second moment of $\theta$ can stay non-finite, even a posteriori, if $2/(2+N)>\alpha_0$. This would show that the data add little information for that specific transition. To avoid this, we may let $\a_0$ be the minimum between the value $2/3$, corresponding to the smallest learning sample size such that $\a_0<1$, and the value $2/(2+N)$, necessary for the finiteness of the posterior second moment. Therefore, $\displaystyle \alpha_0 = \min\{2/3, 2/(2+N)\}$.
Finally, if $\gamma$ denotes the hyperparameter corresponding to the indexes $i$ and $j$ in the Dirichlet prior \eqref{eq:dirich}, then we select $\gamma=m+1$, i.e. $\gamma$ is equal to the number of transitions from state $i$ to state $j$ in the learning dataset, plus one.
\subsection{Scarce prior information}\label{sec:scarce}
The construction of the prior distribution of $(\alpha,\theta)$ must be modified for those pairs of states between which no more than one transition was observed in the learning dataset.
If $m=1$, the single learning observation $y_{1}$ determines $\hat{b}(\alpha)$. As $\hat{t}_q=y_{1}$ for any $q$, it seems reasonable to use $q=0.5$, so $y_{1}$ would represent the prior opinion on the median holding time. Since $d=0$ when $m=1$, then $\pi_3(\alpha)$ is improper for any $c>0$. We make it proper by restricting its support to an interval $(\alpha_0,\alpha_1)$. The value $\alpha_1=10$ is suitable for all practical purposes. As before, the choice $\alpha_0 = 2/m=2$ would be too much restrictive, so we select again $\displaystyle \alpha_0 = \min\{2/3, 2/(2+N)\}$. With regard to $c$, we put $c=2$.
Furthermore, the elicitation of the hyperparameter of the Dirichlet prior is again $\gamma = m+1=2$, i.e. the number of transitions observed in the learning dataset (just one) plus one.
If $m=0$, the prior information on the number of transitions is that there have been no transitions, but there is no information on the holding times.
In this case we can represent in the model the absence of information, choosing
\[
q=0.5\quad\mbox{and}\quad \tilde t_{0.5}\sim U(t_1,t_2),
\]
that is $\tilde t_{0.5}$ is uniformly distributed over a big time interval $(t_1,t_2)$, independently from everything else.
Hence, we use $q=0.5$ and $\tilde{t}_{0.5}$ to obtain $\hat{b}(\alpha)$ and we fall in the previous case by substituting $m=1$ to $m=0$.
For clearness, in Table~\ref{table:hyperparameters}, we summarize the hyperparameter selection for priors \eqref{eq:dirich}, \eqref{pi-2} and \eqref{pi-3}.
\begin{table}[h]
\renewcommand*{\arraystretch}{1.4}
\begin{tabular}{r|cccc}
& $m>2$ & $m=2$ & $m=1$ & $m=0$\\
\hline
$t_q$ & $\hat{t}_q$ & $\hat{t}_q$ & $y_{1}$ & $\tilde{t}_q\sim U(t_1,t_2)$\\
$c$ & $m$ & $m$ & 2 & 2\\
$\a_0$ & $\frac{2}{m}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ & $\min\left\{\frac23, \frac2{2+N}\right\}$ \\
$\gamma$ & $m+1$ & $m+1$ & $m+1$ & $2$ \\
\end{tabular}
\caption{Hyperparameter selection as the learning sample size $m$ varies.}
\label{table:hyperparameters}
\end{table}
\section{Analysis of the central Northern Apennines sequence}\label{sec:apennines}
In this section we analyze the macroregion $\text{MR}_3$ sequence, using the semi-Markov model.
We coded the Gibbs sampling algorithm in the JAGS software package, which is designed to work
closely with the \cite{Rpackage} package, in which all statistical computations and
graphics were performed. Details of the Gibbs sampler are in Appendix \ref{sec:gibbsampling}. On the whole, 750,000 iterations for one chain were run for estimating the unknown
parameters in the model, and the first 250,000 were discarded as burn-in.
After the burn-in, one out of every 100 simulated values was kept for
posterior analysis, for a total sample size of 5,000. The
convergence diagnostics, such as those available in the R package CODA
(Geweke, Heidelberger and Welch stationarity test, interval halfwidth
test), were computed for all parameters, indicating that convergence
has been achieved.
Model fitting, model validation and an attempt at forecasting involve the following steps:
\begin{enumerate}
\item the learning dataset for the elicitation of the prior distribution is chosen;
\item model fit is assessed by comparing observed inter-occurrence times (grouped by transition) to posterior predictive intervals;
\item cross state-probabilities are estimated, as an indication to the most likely magnitude and time to the next event, given information up to the present time;
\item an interpretation in terms of slip predictable or time predictable model is provided.
\end{enumerate}
For the elicitation of the prior distribution, the learning data are taken from $\text{MR}_4$, another macroregion among those considered by \cite{Rotondi}, who examines statistical summaries of the holding times and suggests that $\text{MR}_4$ could be used as a learning set for the hyperparameters of $\text{MR}_3$. \cite{PeruggiaSantner}, in their analysis of the magnitudes and of the inter-occurence times of eartquakes from another Italian area, chose a subset of the incomplete older part of their series to elicit prior distributions. This procedure is justifiable in their case because the old and the new part of the series can be regarded as two different processes and the cut-point between them appears to be clearly identified. If we did the same with our series, we would alter the Bayesian learning process, because we would obtain different posterior distributions on changing the cut-point position.
Transition frequencies and median inter-occurrence times appear in Table \ref{table:empirical.P} for both the $\text{MR}_3$ and the $\text{MR}_4$ datasets. The Dirichlet hyperparameters $\bg_1, \cdots ,\bg_s$ are equalled to the rows of Table \ref{table:empirical.P}\subref{table:0.empirical.P} plus one. The medians are reported because we have selected $q=0.5$ in Table \ref{table:hyperparameters}: the medians in Table \ref{table:empirical.P}\subref{table:0.median} are smaller than the medians in Table \ref{table:empirical.P}\subref{table:1.median} in six entries out of nine, in some cases considerably.
\begin{table}[h]
\begin{center}
\subtable[\label{table:1.empirical.P}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 65 & 30 & 17 \\
from 2 & 32 & 15 & 7 \\
from 3 & 15 & 9 & 4 \\
\hline
\end{tabular}
}}\quad
\subtable[\label{table:0.empirical.P}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 114 & 51 & 13 \\
from 2 & 56 & 25 & 4 \\
from 3 & 8 & 8 & 3 \\
\hline
\end{tabular}
}}\quad\\
\subtable[\label{table:1.median}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 204 & 257 & 141 \\
from 2 & 150 & 122 & 219 \\
from 3 & 142 & 82 & 309 \\
\hline
\end{tabular}
}}\quad
\subtable[\label{table:0.median}]{
\scalebox{ 1}{ \begin{tabular}{rrrr}
\hline
& to 1 & to 2 & to 3 \\
\hline
from 1 & 105 & 61 & 193 \\
from 2 & 104 & 99 & 76 \\
from 3 & 209 & 117 & 78 \\
\hline
\end{tabular}
}}
\end{center}
\caption{Summaries of datasets $\text{MR}_3$ (tables on the left) and $\text{MR}_4$ (tables on the right); $\text{MR}_4$ is the learning dataset used for hyperparameter elicitation. (a) and (b): number of observed transitions; (c) and (d): median inter-occurrence times (in days).}
\label{table:empirical.P}
\end{table}
Let us consider the predictive check mentioned above. Figure~\ref{figure:caterpillar_times} shows posterior predictive 95~percent probability intervals of the inter-occurrence times for every transition, with the observed inter-occurrence times superimposed. These are empirical intervals computed by generating stochastic inter-occurrence times from their relevant distributions at every iteration of the Gibbs sampler. Possible outliers, represented as triangles, are those times with Bayesian $p$-value (that is the predictive tail probability) less than 2.5~percent.
\begin{figure}
\caption{Posterior predictive $95$~percent credible intervals of the inter-occurrence times in days with actual times denoted by (blue) solid dots. Suspect outliers are denoted by (red)-pointing triangles. The (green) dotted line shows the posterior median and the (violet) dashed line the posterior mean. The prior distribution was elicited from the $\text{MR}
\label{figure:caterpillar_times}
\end{figure}
In Table~\ref{table:posterior.time} we report the expected value (and the standard deviation) of the inter-occurrence times. In Table \ref{table:b.outliers} the numbers of upper and lower extreme points and their overall percentage are collected. While deviations from the nominal 95\% coverage are acceptable for transitions with low absolute frequency, such as $(2,3)$, $(3,3)$, $(3,2)$, the remaining transitions require attention. We see that the percentage of outliers higher than the nominal value is mostly due to the upper outliers, which occur as an effect of the difference between the prior opinion on the marginal median of the inter-occurrence times and the median of the observed sequence (compare Table \ref{table:empirical.P}\subref{table:0.median} to Table \ref{table:empirical.P}\subref{table:1.median}). A
few really extreme inter-occurrence times, such as the small values observed at transitions $(1,1)$, $(2,1)$ and the large one at transition $(1,3)$, match unsurprisingly the outlying points in the corresponding qq-plots in Figure~\ref{fig:qqplots}. This fact could be regarded as a lack of fit of the Weibull model, but it could also be due to an imperfect assignment of some events to the macroregion $\text{MR}_3$ or to an insufficient filtering of secondary events (i.e. aftershocks and foreshocks): earthquakes incorrectly assigned to $\text{MR}_3$ and aftershocks or foreshocks can give rise to very short inter-occurrence times; on the other hand, earthquakes which should be in $\text{MR}_3$ but which were attributed to other macroregions can produce very long inter-occurrence times.
\begin{table}[ht]
\begin{center}
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1 &191 (12) & 172 (18) & 331 (70)\\
2 &214 (22) & 238 (43) & 354 (145)\\
3 & 263 (58) & 203 (55)& 314 (134)\\
\hline
\end{tabular}
\end{center}
\caption{Predictive means (and standard deviations) of inter-occurrence times for each transition (in days); prior elicited from $\text{MR}_4$.}
\label{table:posterior.time}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{cccc|ccc|ccc}
& \multicolumn{3}{c}{Upper outliers} & \multicolumn{3}{c}{Lower outliers} & \multicolumn{3}{c}{\% of outliers}\\
\hline
& 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3\\
\hline
1 & 8 & 3 & 1 & 2 & 0 & 0 & 15.4\% & 10.0\% & 5.9\%\\
2 & 4& 3& 1& 1& 0 & 0 & 15.6\% &20.0\% & 14.3\%\\
3 & 1 & 0 & 0 & 0 & 1 & 0 & 6.7\% & 11.1\%& 0.0\%\\
\hline
\end{tabular}
\end{center}
\caption{Number of points having lower or upper posterior $p$-value less than $2.5$~percent and their percentage; prior elicited from $\text{MR}_4$.}
\label{table:b.outliers}
\end{table}
The shape parameters $\alpha_{ij}$ are particularly important as they reflect an increasing hazard if larger than 1, a decreasing hazard if smaller than 1 and a constant hazard if equal to 1. Table~\ref{table:posterior.alpha-theta}~\subref{table:posterior.alpha} displays the posterior means of these parameters (along with their posterior standard deviations). Finally Table~\ref{table:posterior.P} shows the posterior means of the transition probabilities. Notice that the last row departs from the other two; we will return to this in the following.
\begin{table}[ht]
\centering
\subtable[Shape parameter $\alpha$ \label{table:posterior.alpha}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1 & 1.18 (0.06) & 1.07 (0.08) & 0.94 (0.10) \\
2 & 1.07 (0.07) & 0.95 (0.10) & 0.89 (0.14) \\
3 & 1.04 (0.16) & 1.03 (0.16) & 1.11 (0.21)\\
\hline
\end{tabular}
}}\quad
\subtable[Scale parameter $\theta$ \label{table:posterior.theta}]
{\scalebox{1}{
\begin{tabular}{llll}
\hline
& 1 & 2 & 3 \\
\hline
1& 201.7 (13.2)& 175.6 (19.1)& 317.8 (67.0)\\
2 & 219.2 (22.5)& 231.0 (40.5)& 327.3 (132.4)\\
3 & 262.7 (57.2) & 201.9 (52.2) & 320.1 (133.5)\\
\hline
\end{tabular}
}}
\caption{Posterior means (with standard deviations) of the shape parameter $\alpha$ in \subref{table:posterior.alpha} and of the scale parameter $\theta$ in \subref{table:posterior.theta}; prior elicited from $\text{MR}_4$.}
\label{table:posterior.alpha-theta}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrr}
\hline
& 1 & 2 & 3 \\
\hline
1 & 0.614 (0.028) & 0.280 (0.026) & 0.106 (0.018)\\
2 & 0.626 (0.041) & 0.290 (0.038) & 0.085 (0.023)\\
3 & 0.479 (0.070) & 0.361 (0.067) & 0.160 (0.051)\\
\hline
\end{tabular}
\end{center}
\caption{Summaries of the posterior distributions of the transition matrix $\bp$. Posterior means (with standard deviations); prior elicited from $\text{MR}_4$.}
\label{table:posterior.P}
\end{table}
Cross state-probability plots are an attempt at predicting what type of event and when it is most likely to occur.
A cross state-probability (CSP) $P_{t_0| \Delta x}^{ij}$ represents the probability that the next event will be in state $j$ within a time interval $\Delta x$ under the assumption that the previous event was in state $i$ and $t_0$ time units have passed since its occurrence:
\begin{multline}
\label{eq:CSP}
P_{t_0| \Delta x}^{ij} = P\left(J_{n+1} =j, \ X_{n+1}\le t_0+\Delta x| \ J_n = i, \ X_{n+1} > t_0\right) = \\
= \frac{ p_{ij}\left(\bar{F}_{ij}(t_0) - \bar{F}_{ij}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{ik} \bar{F}_{ik}(t_0)} \ .
\end{multline}
Figure~\ref{figure:csp} displays the CSPs with time origin on 31 December 2002, the closing date of the \cite{CPTI04} catalogue. At this time, the last recorded event had been in class 2 and had occurred 965 days earlier (so $t_0$ is about 32 months). From these plots we can read out the probability that an event of any given type will occur before a certain number of months. For example, after 24 months, the sum of the mean CSPs in the three graphs indicates that the probability that an event will have occurred is around 88\%, with a larger probability assigned to an event of type 2, followed by type 1 and type 3. The posterior means of the CSPs are also reported in Table~\ref{table:csp}.
\begin{figure}
\caption{Posterior mean and median of CSPs with time origin on 31 December 2002 up to 48 months ahead, along with 90~percent posterior credible intervals. Transitions are from state 2 to state 1, 2 or 3 (first to third panel, respectively). Months since 31 December 2002 are along the x-axis. The learning set is $MR_4$.}
\label{figure:csp}
\end{figure}
\begin{table}[ht]
\begin{center}
\scalebox{ 0.9}{ \begin{tabular}{rrrrrrrrrrr}
\hline
&1 Month & 2 Months & 3 Months & 4 Months &5 Months & 6 Months & 1 Year& 2 Years & 3 Years & 4 Years \\
\hline
to 1 & 0.045 & 0.080 & 0.113 & 0.140 & 0.164 & 0.184& 0.256 & 0.296 & 0.303& 0.304\\
to 2 & 0.038 & 0.069 & 0.099 & 0.125 & 0.149 & 0.169& 0.257 & 0.327 & 0.348& 0.356\\
to 3 & 0.023 & 0.041 & 0.061 & 0.078 & 0.095 & 0.109& 0.180 & 0.256 & 0.291& 0.309\\
\hline
\end{tabular}}
\end{center}
\caption{CSPs with time origin on 31 December 2002, as represented in Figure~\ref{figure:csp}; prior elicited from $\text{MR}_4$.}
\label{table:csp}
\end{table}
The predictive capability of our model can be assessed by marking the time of the next event on the relevant CSP plot. In our specific case, the first event in 2003, which can be assigned to the macroregion $\text{MR}_3$ happened in the Forl\`{\i} area on 26 January and was of type 1, with a CSP of 4.5\%. This is a low probability, but a single case is not enough to judge our model, which would be a bad one if repeated comparisons did not reflect the pattern represented by the CSPs. Therefore we repeated the same comparison by re-estimating the model using only the data up to 31 December 2001, 31 December 2000, and so on backwards down to 1992. The results are shown in Table~\ref{table:31.12.92-01}. The boxed numbers correspond to the observed events and it is a good sign that they do not always correspond to very high or very low CSPs, as this would indicate that events occur too late or too early compared to the estimated model. If we were to plot the conditional densities obtained by differentiating the CSPs with respect to $\Delta x$, marking the observed inter-occurrence times on the x-axis, we would observe that very few of them appear in the tails.
\begin{table}[h!]
\centering
\scalebox{ 0.8}{\begin{tabular}{rrrrrrrrrrrr}
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2001;\quad previous event type:2;\quad holding time: 600 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 392 days & 2 Years & 3 Years & 4 Years \\
to 1 & 0.069 & 0.122 & 0.173 & 0.215 & 0.251 & 0.282 & 0.392 & \boxed{{\color{blue}0.401}} & 0.451 & 0.461 & 0.462 \\
to 2 & 0.038 & 0.068 & 0.097 & 0.123 & 0.146 & 0.166 & 0.248 & 0.256 & 0.310 & 0.328 & 0.333 \\
to 3 & 0.015 & 0.027 & 0.039 & 0.050 & 0.061 & 0.070 & 0.113 & 0.118 & 0.158 & 0.178 & 0.187 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/2000;\quad previous event type: 2;\quad holding time: 235 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1Year & 2 Years & 757 days & 3 Years & 4 Years \\
to 1 & 0.085 & 0.152 & 0.217 & 0.270 & 0.318 & 0.358 & 0.505 & 0.584 & \boxed{{\color{blue}0.586}} & 0.597 & 0.599 \\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & 0.134 & 0.152 & 0.223 & 0.273 & 0.275 & 0.287 & 0.290 \\
to 3 & 0.009 & 0.017 & 0.024 & 0.031 & 0.037 & 0.042 & 0.066 & 0.090 & 0.091 & 0.099 & 0.103\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1999;\quad previous event type: 1;\quad holding time: 177 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 130 days & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.086 & 0.157 & 0.222 & 0.277 & 0.292 & 0.325 & 0.366 & 0.518 & 0.600 & 0.613 & 0.615\\
to 2 & 0.035 & 0.063 & 0.090 & 0.113 & \boxed{{\color{blue}0.119}} & 0.133 & 0.151 & 0.220 & 0.269 & 0.281 & 0.284 \\
to 3 & 0.009 & 0.016 & 0.023 & 0.029 & 0.031 & 0.035 & 0.040 & 0.062 & 0.082 & 0.091 & 0.094 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1998;\quad previous event type: 3;\quad holding time: 280 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 188 days & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 & 0.085 & 0.151 & 0.214 & 0.267 & 0.314 & 0.353 & \boxed{{\color{blue}0.361}} & 0.496 & 0.573 & 0.585 & 0.587 \\
to 2 & 0.035 & 0.063 & 0.091 & 0.114 & 0.135 & 0.153 & 0.157 & 0.225 & 0.277 & 0.290 & 0.294 \\
to 3 & 0.010 & 0.018 & 0.025 & 0.032 & 0.039 & 0.045 & 0.046 & 0.071 & 0.096 & 0.106 & 0.111 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1997;\quad previous event type: 3;\quad holding time: 442 days}}\\
\hline
& 1 Month & 2 Months & 85 days & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1& 0.078 & 0.138 & 0.187 & 0.196 & 0.244 & 0.286 & 0.320 & 0.448 & 0.516 & 0.526 & 0.528 \\
to 2 & 0.036 & 0.065 & 0.090 & 0.094 & 0.118 & 0.140 & 0.159 & 0.237 & 0.294 & 0.310 & 0.315\\
to 3 & 0.012 & 0.022 & \boxed{{\color{blue}0.030}} & 0.031 & 0.040 & 0.048 & 0.056 & 0.090 & 0.123 & 0.138 & 0.145\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1996;\quad previous event type: 3;\quad holding time: 77 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 450 days & 2 Years & 3 Years & 4 Years \\
to 1 & 0.084 & 0.152 & 0.218 & 0.273 & 0.323 & 0.366 & 0.525 & 0.562 & 0.614 & 0.628 & 0.630 \\
to 2 & 0.036 & 0.063 & 0.090 & 0.113 & 0.133 & 0.150 & 0.219 & 0.236 & 0.266 & 0.277 & 0.281 \\
to 3 & 0.008 & 0.015 & 0.022 & 0.027 & 0.032 & 0.037 & 0.056 &\boxed{{\color{blue}0.062}} & 0.074 & 0.081 & 0.084 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1995;\quad previous event type: 1;\quad holding time: 3100 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 288 days & 1 Year & 2 Years & 3 Years & 4 Years \\
to 1 &0.004 & 0.008 & 0.011 & 0.014 & 0.017 & 0.019 & 0.024 & 0.027 & 0.033 & 0.035 & 0.035 \\
to 2 & 0.022 & 0.041 & 0.059 & 0.075 & 0.091 & 0.104 & 0.143 & 0.165 & 0.222 & 0.244 & 0.253\\
to 3 & 0.042 & 0.079 & 0.115 & 0.148 & 0.180 & 0.208 & \boxed{{\color{blue}0.296}} & 0.348 & 0.509 & 0.590 & 0.633 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1994;\quad previous event type: 1;\quad holding time: 2735 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 653 days & 2 Years & 3 Years & 4 Years \\
to 1& 0.005 & 0.010 & 0.014 & 0.018 & 0.021 & 0.023 & 0.034 & 0.041 & 0.041 & 0.043 & 0.043\\
to 2 & 0.024 & 0.043 & 0.062 & 0.080 & 0.096 & 0.110 & 0.174 & 0.225 & 0.233 & 0.256 & 0.265\\
to 3 & 0.041 & 0.076 & 0.112 & 0.144 & 0.175 & 0.204 & 0.341 & \boxed{{\color{blue}0.473}} & 0.497 & 0.575 & 0.617\\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1993;\quad previous event type: 1;\quad holding time: 2370 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 1018 days & 3 Years & 4 Years \\
to 1 & 0.007 & 0.013 & 0.019 & 0.024 & 0.028 & 0.031 & 0.045 & 0.055 & 0.056 & 0.057 & 0.057 \\
to 2 & 0.025 & 0.046 & 0.067 & 0.086 & 0.103 & 0.118 & 0.186 & 0.248 & 0.268 & 0.271 & 0.280\\
to 3 & 0.040 & 0.074 & 0.108 & 0.140 & 0.170 & 0.197 & 0.329 & 0.479 & \boxed{{\color{blue}0.542}} & 0.554 & 0.593 \\
\hline
& \multicolumn{11}{c}{\textbf{end of catalogue: 31/12/1992;\quad previous event type: 1;\quad holding time: 2005 days}}\\
\hline
& 1 Month & 2 Months & 3 Months & 4 Months & 5 Months & 6 Months & 1 Year & 2 Years & 3 Years & 1383 days & 4 Years \\
to 1 & 0.011 & 0.019 & 0.027 & 0.034 & 0.040 & 0.045 & 0.065 & 0.078 & 0.080 & 0.081 & 0.081\\
to 2 & 0.028 & 0.051 & 0.074 & 0.094 & 0.113 & 0.130 & 0.202 & 0.267 & 0.290 & 0.298 & 0.299\\
to 3 & 0.038 & 0.070 & 0.103 & 0.132 & 0.161 & 0.186 & 0.311 & 0.451 & 0.520 & \boxed{{\color{blue}0.551}} & 0.557 \\
\hline
\end{tabular}}
\caption{CSPs as the end of the catalogue shifts back by one-year steps. The numbers in boxes are the probability that the next observed event has occurred at or before the time when it occurred and is of the type that has been observed. The prior was elicited from $\text{MR}_4$.}
\label{table:31.12.92-01}
\end{table}
The examination of the posterior distributions of transition probabilities and of the predictive distributions of the inter-occurrence times can give some insight into the type of energy release and accumulation mechanism. We consider two mechanisms, the time predictable model (TPM) and the slip predictable model (SPM).
In the TPM, it is assumed that when a maximal energy threshold is reached, some fraction of it (not always the same) is released and an earthquake occurs. The consequence is that the time until the next earthquake increases with the amplitude of the last earthquake. So, the holding time distribution depends on the current event type, but not on the next event type, that is, we expect $F_{ij}(t)=F_{i\cdot}(t)$, $j=1,2,3$. The strength of an event does not depend on the strength of the previous one, because every time the same energy level has to be reached for the event to occur. So we expect $p_{ij}=p_{\cdot j}$, $j=1,2,3$, that is, a transition matrix with equal rows. If this is the case, the CSPs~\eqref{eq:CSP} would simplify as follows,
\begin{equation}\label{eq:CSPTPM}
P_{t_0|\Delta x}^{ij} = \frac{ p_{ij}\left(\bar{F}_{ij}(t_0) - \bar{F}_{ij}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{ik} \bar{F}_{ik}(t_0)}
= \frac{p_{\cdot j} \left(\bar{F}_{i\cdot}(t_0) - \bar{F}_{i\cdot}(t_0+ \Delta x )\right)}{\bar{F}_{i\cdot}(t_0)} \ ,
\end{equation}
so that, under the TPM assumption, given $i$, they are proportional to each other as $j=1,2,3$ for any $\Delta x$, and the ratio
$P_{t_0|\Delta x}^{ij}/P_{t_0|\Delta x}^{ik}$ equals $p_{\cdot j}/p_{\cdot k}$ for any pair $(j,k)$.
In the SPM, after an event, energy falls to a minimal threshold and increases until the next event, where it starts to increase again from the same threshold. The consequence is that the energy of the next earthquake increases with time since the last earthquake. So, the magnitude of an event depends on the length of the holding time, but not on the magnitude of the previous one, because energy always accumulates from the same threshold. In this case again $p_{ij} = p_{\cdot j}$, but $F_{ij}(t) = F_{\cdot j}(t)$, so
\begin{equation}
\label{eq:CSPSPM}
P_{t_0|\Delta x}^{ij} =
\frac{ p_{\cdot j}\left(\bar{F}_{\cdot j}(t_0) - \bar{F}_{\cdot j}(t_0+ \Delta x )\right)}{ \sum_{k \in E} p_{\cdot k} \bar{F}_{\cdot k}(t_0)} \ .
\end{equation}
Then, under the SPM assumption, CSPs are equal to each other as $i=1,2,3$ for any $\Delta x$, given $j$.
An additional feature that can help discriminate between the TPM and the SPM is the tail of the holding time distribution: for a TPM, the tail of the holding time distribution is thinner \textit{after} a weak earthquake than \textit{after} a strong one; for an SPM, the tail of the holding time is thinner \textit{before} a weak earthquake than \textit{before} a strong one.
In the present case the posterior mean of the third row of $\textbf{p}$, see Table \ref{table:posterior.P}, is clearly different from the other two rows, unlike the empirical transition matrix derived from Table \ref{table:empirical.P}\subref{table:1.empirical.P}, because of the prior information from $\text{MR}_4$. So, with this prior, both the TPM and the SPM are excluded.
On the other hand, things change with the noninformative prior elicited without a learning set.
In this case, we let all the Dirichlet hyperparameters $\g_{ij} $'s be equal to 2. Following Section~ \ref{sec:scarce}, the missing learning set for each string $(i,j)$ is substituted by a unique fictitious observation $\tilde{t}^{ij}_{0.5}$ uniformly distributed over $(1, 5000)$ days, and $m_{ij}$ is set to one; this establishes the prior for $\theta_{ij}$.
The prior of $\alpha_{ij}$ derived from Equation \eqref{pi-3} with $m_{ij}=1$ and $c_{ij}=2$ (taken from Table~\ref{table:hyperparameters}) is
$$
\pi_3(\alpha_{ij}) \propto \left(1-\frac{\alpha_{ij}}{\alpha_{0,ij}}\right) \mathds{1}(\alpha_{0,ij}\le\alpha_{ij}\le\alpha_{1,ij})
$$
with $\alpha_{0,ij}=2/(2+N_{ij})$ (see Table \ref{table:empirical.P}\subref{table:1.empirical.P} for the $N_{ij}$'s) and $\alpha_{1,ij}=10$. Note that on our current sample, the lower limit $\alpha_{0,ij}$ is always smaller than $2/3$.
With this prior specification, the posterior distributions of the rows of the transition matrix do not differ significantly, as seen from Table~\ref{table:lessinformative.posterior.P}, so we can assume $p_{ij}=p_{\cdot j}$ for all indexes $i$ and examine the ratios of CSPs to verify the TPM and the SPM hypotheses.
\begin{table}[ht]
\begin{center}
\begin{tabular}{rrrr}
\hline
& 1 & 2 & 3 \\
\hline
1 & 0.569 (0.046) & 0.271 (0.041)& 0.160 (0.033)\\
2 & 0.568 (0.064)& 0.283 (0.059)& 0.149 (0.046)\\
3 & 0.500 (0.085)& 0.323 (0.078)& 0.177 (0.065)\\
\hline
\end{tabular}
\end{center}
\caption{Posterior means (with standard deviations) of the transition matrix $\bp$ with the noninformative prior.}
\label{table:lessinformative.posterior.P}
\end{table}
Figures~\ref{figure:improper_ratio_TPM} and \ref{figure:improper_ratio_SPM} display the posterior means of the ratios of the CSPs as a function of $\Delta x$ for $t_0=0$, with the noninformative prior. For the TPM the generic ratio of two CSPs indexed by $(i,j)$ and $(i,k)$ should be approximately constant and close to $p_{\cdot j}/p_{\cdot k}$, where the $p_{\cdot j}$ represents the common values of the entries in the $j$-th column of $\bp$, under the TPM. The horizontal lines in Figure~\ref{figure:improper_ratio_TPM} are the posterior expectations of $p_{ij}/p_{ik}$, which would estimate $p_{\cdot j}/p_{\cdot k}$ if the TPM assumption were true. For the SPM, the ratio of CPSs, now indexed by $(i,j)$ and $(k,j)$, should be close to one. The plots indicate that it is not so, therefore neither the TPM nor the SPM are supported by the data.
As for the TPM, this finding is confirmed by the examination of the posterior probabilities that $\a_{ij}<\a_{ik}$ and $\theta_{ij}<\theta_{ik}$, for any given $i$ and $j\ne k$: $\Pr(\a_{ij}<\a_{ik}|\bj, \bx, u_T)$ is 0.55 for string $(2,1)$ versus string $(2,3)$ and 0.51 for $(3,1)$ versus $(3,2)$, but it is either larger than 0.75 or smaller than 0.35 for all the other strings; $\Pr(\theta_{ij}<\theta_{ik}|\bj, \bx, u_T)$ is 0.61 for $(1,2)$ versus $(1,3)$ and is 0.53 for $(3,1)$ versus $(3,2)$, but it is lower than 0.39 for all the other strings. As for the SPM, we have examined $\Pr(\a_{ij}<\a_{kj}|\bj, \bx, u_T)$ and $\Pr(\theta_{ij}<\theta_{kj}|\bj, \bx, u_T)$ for any $j$ and $i\ne k$: $\Pr(\a_{ij}<\a_{kj}|\bj, \bx, u_T)$ is 0.51 for $(2,2)$ versus $(3,2)$, but is either larger than 0.75 or smaller than 0.35 for all the other strings; $\Pr(\theta_{ij}<\theta_{kj}|\bj, \bx, u_T)$ is between 0.44 and 0.63 for three comparisons but is either larger than 0.73 or smaller than 0.30 for the remaining ones.
\begin{figure}
\caption{Checking the TPM: posterior means of ratios of CSPs $P_{0|\Delta x}
\label{figure:improper_ratio_TPM}
\end{figure}
\begin{figure}
\caption{Checking the SPM: posterior means of ratios of CSPs $P_{0|\Delta x}
\label{figure:improper_ratio_SPM}
\end{figure}
\section{Concluding remarks}
\label{sec:6}
We have presented a complete Bayesian methodology for the inference on semi-Markov processes, from the elicitation of the prior distribution, to the computation of posterior summaries, including a guidance for its JAGS implementation.
In particular, we have examined in detail the elicitation of the joint prior density of the shape and scale parameters of the Weibull-distributed holding times (conditional on the transition between two given states), deriving a specific class of priors in a natural way, along with a method for the determination of hyperparameters based on ``learning data'' and moment existence conditions. This framework has been applied to the analysis of seismic data, but it can be adopted for inference on any system for which a Markov Renewal process is plausible. A possible and not-yet explored application is the modelling of voltage sags (or voltage dips) in power engineering: the state space would be formed by different classes of voltage, starting from voltage around its nominal value, down to progressively deeper sags. In the engineering literature, the dynamic aspect of this problem is in fact disregarded, while it could help bring additional insight into this phenomenon.
With regard to the seismic data analysis, other uses of our model can be envisaged. The model can be applied to areas with a less complex tectonics, such as Turkey, by replicating for example Alvarez's analysis. Outliers, such as those appearing in Figure \ref{figure:caterpillar_times}, could point at events whose assignment to a specific seismogenic source should be re-discussed. The analysis of earthquake occurrence can support decision making related to the risk of future events. We have not examined this issue here, but a methodology is outlined by \cite{Cano}.
A final note concerns the more recent Italian seismic catalogue \cite{CPTI11}, including events up to the end of 2006. Every new release of the catalogue involves numerous changes in the parameterization of earthquakes; as the DISS event classification by macroregion is not yet available for events in this catalogue we cannot use this more recent source of data.
\appendix
\section{Gibbs sampling}
\label{sec:gibbsampling}
Here we derive the full conditional distributions involved in the Gibbs sampling and give indications on its JAGS implementation.
\subsection{Full conditional distributions}
Let the last holding time be censored i.e. $u_T>0$. Hence,
in order to obtain some simple full conditional distributions and then an efficient Gibbs sampling, we introduce the auxiliary variable $j_{\tau+1}$
which represents the unobserved state following the last visited state $j_\t$. Moreover, let
$\bt_{\bq} = (t^{ij}_{q_{ij}}, \;\; i,j\in E)$. Each hyperparameter $t^{ij}_{q_{ij}}$ may be either a known constant or $t^{ij}_{q_{ij}}$ is uniformly distributed over an interval $(t_1, t_2)$. Moreover, all of them are independent of each other.
Thus the state space of the Gibbs sampler is $(\bp, \ba, \bth,j_{\t+1}, \bt_{\bq})$ and the following full likelihood derived from \eqref{eq:likelihood-param}:
\begin{multline}
\label{eq:full-likelihood-param}
L(\bj, \bx, u_T , j_{\t+1}|\bp, \ba, \bth) =\prod_{i,k\in E}p_{ik}^{N_{ik}}\times \\
\times \prod_{i,k\in E}\left[\a_{ik}^{N_{ik}}\frac{1}{\theta_{ik}^{\a_{ik}N_{ik}}}\left(\prod_{\rho=1}^{N_{ik}}x^{\r}_{ik}\right)^{\a_{ik}-1} \times
\exp\left\{-\frac{1}{\theta_{ik}^{\a_{ik}}}\sum_{\rho =1}^{N_{ik}}(x^{\r}_{ik})^{\a_{ik}}\right\}\right]\times \\
\times \left(p_{j_{\t} j_{\t+1}}
\exp\left\{-\left(\frac{u_T}{\theta_{j_{\t} j_{\t+1}}}\right)^{\a_{j_{\t} j_{\t+1}}}\right\}\right)\
\end{multline}
is multiplied by the prior and used to determine the full conditionals.
For every $i$ and $j$ let
\begin{align*}
\bp_{(-i)} & = \mbox{the transition matrix $\bp$ without the $i$-th row} , \\
\ba_{(-ij)} & = (\a_{hk}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) \ , \\
\bth_{(-ij)} & = (\th_{hk}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) \ , \\
\bt_{{\bq}(-ij)} & = (t^{hk}_{q_{hk}}, \;\; h,k\in E,\quad (h,k) \neq (i,j) ) , \\
\tilde{N}_{ij} & = {N}_{ij} + \mathds{1}{\left((j_{\t},j_{\t+1})=(i,j)\right)}, \quad
\tilde{m_{\alpha}thbf N}_{i} =\left(\tilde{N}_{ij}, \;\; j=1,\ldots, s \right),\\
\tilde{M}_{ij}(\a_{ij}) & = \sum_{\rho=1}^{{N}_{ij}}(x_{ij}^{\rho})^{\a_{ij}} + \ u_{T}^{\a_{ij}}\mathds{1}( (j_{\t},j_{\t+1})=(i,j)), \\
{\C}_{ij} & = \prod_{\rho=1}^{{N}_{ij}}x_{ij}^{\rho} \ .
\end{align*}
The following result on the full conditional distributions of the Gibbs sampling holds.
\begin{proposition}
\label{prop:A.1}
Let the prior on $(\bp,\ \ba, \ \bth, \ \bt_{\bq})$ be the following
\begin{enumerate}[i)]
\item $\bp$ is independent of $\ba$ and $\bth$ and the rows of $\bp$ are $s$ independent vectors with Dirichlet distribution with parameters $\bg_1, \cdots ,\bg_s$ and total mass $c_1, \cdots c_s$, respectively,
\item the $\th_{ij}$'s, given the $\a_{ij}$'s and the $t^{ij}_{q_{ij}}$'s, are independent with $\th_{ij}|\a_{ij} \sim {m_{\alpha}thcal GIG}(m_{ij}, b_{ij}(t^{ij}_{q_{ij}},\a_{ij}), \a_{ij})$, where
\begin{equation*}
b_{ij}(t^{ij}_{q_{ij}},\a_{ij}) = \left(t^{ij}_{q_{ij}}\right)^{\a_{ij}}[(1-q_{ij})^{-1/m_{ij}}-1]^{-1} \ ,
\end{equation*}
and $t^{ij}_{q_{ij}}$ is either a known constant or $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$,
\item $\pi_{3,ij}(\a_{ij})\propto \a_{ij}^{m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}\exp\{-m_{ij}d_{ij}\a_{ij}\}\mathds{1}(\a_{ij}\in {I_{ij}})$,
$m_{ij}>0, c_{ij}>0$, $\a_{0,ij}>0 \mbox{ and } d_{ij}\ge0$ where
$$
I_{ij} =
\begin{cases}
(\a_{0,ij}, \a_{1,ij}) & \mbox{ if } d_{ij} = 0\\
(\a_{0,ij}, \infty) & \mbox{ if } d_{ij} > 0 \ .
\end{cases}
$$
\end{enumerate}
Then
\begin{enumerate}[$a)$]
\item the conditional distribution of $\bp_i$, given $\bj, \ \bx, \ u_T, \ j_{\t+1},\ \bp_{(-i)}, \ \ba, \ \bth$ and $\bt_{\bq}$ is a Dirichlet distribution with parameter $\tilde{m_{\alpha}thbf N}_{i}+\bg_i$;
\item the conditional distribution of $\th_{ij}^{\a_{ij}}$, given $\bj, \ \bx , \ u_T, \ j_{\t+1}, \ \bp , \ \ba, \ \bth_{(-ij)}$ and $\bt_{\bq}$ is an inverse Gamma distribution with shape $m_{ij}+{N}_{ij}$ and rate $b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij})$;
\item the conditional density of $\a_{ij}$, given $\bj , \ \bx, \ u_T, \ j_{\t+1}, \ \bp, \ \ba_{(-ij)}, \ \bth$ and $\bt_\bq$ is proportional to
\begin{multline}
\label{eq:pi3_posterior}
\a_{ij}^{N_{ij}+1+m_{ij}-c_{ij}}\left(\a_{ij}-\a_{0,ij}\right)^{c_{ij}-1}
\times
\exp\left\{ - \left(m_{ij}d_{ij} -\log \frac{\C_{ij} (t^{ij}_{q_{ij}})^{m_{ij}}}{\th_{ij}^{N_{ij} + m_{ij}}}\right)\a_{ij}\right\}
\times \\
\times \exp\left\{- \frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij}) }{\th_{ij}^{\a_{ij}}}\right\} \mathds{1}(\a_{ij}\in {I_{ij}})\ ,
\end{multline}
and it is log-concave if $c_{ij}\ge 1$;
\item the conditional density of the unseen state $J_{\t+1}$, given $\bj, \ \bx, \ u_T, \ \bp, \ \ba,\bth$ and $\bt_{\bq}$, is
\begin{equation*}
\frac{p_{j_{\t}j}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}j}}\right)^{\a_{j_{\t}j}}\right\}}{\sum_{k\in E}
p_{j_{\t}k}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}k}}\right)^{\a_{j_{\t}k}}\right\}} \, ;
\end{equation*}
\item if $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$, then its conditional distribution given $\bj, \ \bx, \ u_T, \ \bp, \ \ba, \ \bth$ and $\bt_{{\bq}(-ij)}$ is a doubly-truncated at $(t_1, t_2)$ generalized Gamma with parameters
$a= \th_{ij}[(1-q_{ij})^{-1/m_{ij}}-1]^{1 /\a_{ij}}$, $d=\a_{ij}m_{ij}+1$ and $p=\alpha_{ij}$, i.e.
$$
\pi(t_{q_{ij}}^{ij} | \bj, \bx, u_T, \bp, \ba, \bth, \bt_{{\bq}(-ij)}) =
\frac{p/a^d}{\Gamma(d/p)} \left(t_{q_{ij}}^{ij}\right)^{d-1} \exp\left\{-\left(t_{q_{ij}}^{ij} / a \right)^p \right\} \mathds{1}(t_1 <t^{ij}_{q_{ij}}< t_2) .
$$
\end{enumerate}
\end{proposition}
\begin{proof}
As the row $\bp_{i}$ is independent of $(\bp_{(-i)},\ba,\bth, \bt_{\bq})$, conditionally on the data and $j_{\t+1}$, then
\begin{equation*}
\pi(\bp_i|\bj, \bx , u_T, \ j_{\t+1} ,\bp_{(-i)}, \ba,\bth,\bt_{\bq}) \propto L(\bj, \bx, u_T, \ j_{\t+1}| \bp, \ba,\bth)
\times \pi_{1,i}(\bp_i) \propto \prod_{j\in E} p_{ij}^{\tilde{N}_{i,j}} \times \prod_{j\in E} p_{ij}^{\gamma_{ij}-1} ,
\end{equation*}
where $\pi_{1,i}$ denotes the Dirichlet prior of $\bp_i$. Hence point $a)$ of Proposition \ref{prop:A.1} follows.
As regards the full conditional distribution of $\theta_{ij}$,
we have
\begin{align*}
\pi(\th_{ij} & \ |\ \bj, \bx , u_T, \ j_{\t+1}, \bp, \ba,\bth_{(-ij)},\bt_{\bq})
\propto L(\bj, \bx, u_T, \ j_{\t+1} \ |\bp, \ba,\bth) \times \pi_{2,ij}(\th_{ij}|\alpha_{ij}, t^{ij}_{q_{ij}}) \\
& \propto
\prod_{i,k\in E} \left[\a_{ik}^{N_{ik}}\frac{\C_{ik}^{\a_{ik}-1}}{\theta_{ik}^{\a_{ik}N_{ik}}}
\times\exp\left\{-\frac{\tilde{M}_{ik}(\a_{ik})}{\theta_{ik}^{\a_{ik}}}\right\} \right]
\times \exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}\th_{ij}^{-[1+\a_{ij}m_{ij}]}\\
& \propto \; \; \th_{ij}^{-[1+\a_{ij}(m_{ij}+N_{ij})]}
\exp\left\{-\frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}.
\end{align*}
As one can see, the last function is the kernel of an inverse Gamma distribution with parameters $m_{ij}+N_{ij}$ and $b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{i,j}(\a_{ij})$ and point $b)$ follows. \newline
A similar reasoning yields a full conditional distribution for $\a_{ij}$
proportional to \eqref{eq:pi3_posterior}.
Furthermore, concerning its log-concavity, notice that the function in \eqref{eq:pi3_posterior} can be written as the product of the following four log-concave functions:
\begin{align*}
& \a_{ij}^{N_{ij}+m_{ij}} , \ \Bigl(1-\frac{\a_{ij}}{\a_{0,ij}}\Bigr)^{c_{ij}-1} , \
\exp\Bigl\{ - \bigl(m_{ij}d_{ij} -\log \frac{\C_{ij} (t^{ij}_{q_{ij}})^{m_{ij}}}{\th_{ij}^{N_{ij} + m_{ij}}}\bigr)\a_{ij}\Bigr\} , \
\exp\Bigl\{- \frac{b_{ij}(t^{ij}_{q_{ij}},\a_{ij})+\tilde{M}_{ij}(\a_{ij}) }{\th_{ij}^{\a_{ij}}}\Bigr\}
\end{align*}
In particular,
the second function is log-concave for $c_{ij} \ge 1$ and
the last term is a product of log-concave functions of the kind $\a_{ij} m_{\alpha}psto \exp\{-z^{\a_{ij}}\}$.
Hence the log-concavity follows from the property that the product of log-concave functions is log-concave too.
Regarding point $d)$, it is enough to observe that Equations~\eqref{eq:MRP_n} and \eqref{eq:full-likelihood-param} imply the following:
\begin{align*}
P(J_{\t+1}=j |\ \bj, \bx, u_T, \bp , \ba, \bth ) & =
\frac{P(J_{\t+1}=j, \ X_{\tau +1} >u_T | \ \bj, \bx,\bp , \ba, \bth)}{\sum_{k\in E}P(J_{\t+1}=k, \ X_{\tau +1} >u_T | \ \bj, \bx,\bp , \ba, \bth)} \\
& =\frac{p_{j_{\t}j}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}j}}\right)^{\a_{j_{\t}j}}\right\}}{\sum_{k\in E}
p_{j_{\t}k}\exp\left\{-\left(\frac{u_T}{\th_{j_{\t}k}}\right)^{\a_{j_{\t}k}}\right\}} \ .
\end{align*}
Finally, if $t^{ij}_{q_{ij}}$ is uniformly distributed over $(t_1, t_2)$, then
\begin{align*}
\pi(t_{q_{ij}}^{ij} &| \ \bj, \bx, u_T, \bp, \ba, \bth, \bt_{{\bq}(-ij)}) \propto
\pi_2(\th_{ij}|\a_{ij},t^{ij}_{q_{ij}})\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2) \\
&\propto b^{m_{ij}}(t^{ij}_{q_{ij}},\a_{ij})\exp\left\{-\frac{b(t^{ij}_{q_{ij}},\a_{ij})}{\th_{ij}^{\a_{ij}}}\right\}\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2) \\
&\propto \left(t_{q_{ij}}^{ij}\right)^{\a_{ij}m_{ij}} \exp\left\{-\left(\frac{t_{q_{ij}}^{ij}}{\th_{ij}[(1-q_{ij})^{-1/m_{ij}}-1]^{1 /\a_{ij}}}\right)^{\a_{ij}} \right\}
\mathds{1}(t_1 \le t_{q_{ij}}^{ij} \le t_2)
\end{align*}
The last equation is the kernel of a generalized Gamma, doubly-truncated at $(t_1, t_2)$, as introduced in \cite{Stacy}, so point $e)$ follows.
\end{proof}
\subsection{JAGS implementation} Proposition~\ref{prop:A.1} implies that JAGS should be able to run an exact Gibbs sampler.
The model description we have adopted in the JAGS language is based on the full likelihood \eqref{eq:full-likelihood-param}.
It is not important that the model description matches the actual model which generated the data as long as the full conditional distributions, which are determined by the joint distribution of the data and the parameters, remain unchanged. In detail, we consider the following joint distribution:
$$
L_1(\bj,\bx,u_T|j_{\tau+1},\bp,\ba,\bth) \pi(j_{\tau+1}|\bp_{j_\tau}) \pi(\bp,\ba,\bth)
$$
where $\pi(\bp,\ba,\bth)$ is the joint prior as derived in Section~\ref{sec:3} using Equations \eqref{eq:dirich}, \eqref{pi-2} and \eqref{pi-3} and
\begin{align*}
\pi(j_{\tau+1}|\bp_{j_\tau}) & = p_{j_\tau j_{\tau+1}}\\
L_1(\bj,\bx,u_T|j_{\tau+1},\bp,\ba,\bth) & = L(\bj,\bx,u_T,j_{\tau+1}|\bp,\ba,\bth)/p_{j_\tau j_{\tau+1}}
\end{align*}
The factors of the likelihood $L_1$, to be extracted from Equation \eqref{eq:full-likelihood-param}, are modelled in JAGS as follows.
For any value of $i$, the factor
$$
\prod_{k\in E} p_{ik}^{N_{ik}}
$$
is contributed by a multinomial likelihood with probability vector $\bp_i$ and $\sum_k N_{ik}$ trials.
The factors in square brackets in \eqref{eq:full-likelihood-param} are contributed by the uncensored Weibull holding times for every string $(i,k)$ and are obtained in JAGS as Weibull densities with parameters $\alpha_{ik}$ and $\theta_{ik}$, using a ``for'' loop sweeping the strings. The last factor, which accounts for the censored holding time $u_T$, is handled by a special instruction, by which $u_T$ is first declared to be a right censored time with upper censoring point $T-t_\tau$, and then is assigned a Weibull distribution with parameters $\alpha_{j_\tau j_{\tau+1}}$ and $\theta_{j_\tau j_{\tau+1}}$.
The factor $\pi(j_{\tau+1}|\bp_{j_\tau}) \pi(\bp,\ba,\bth)$, representing the prior associated with $L_1$, is handled as follows. The additional prior on $j_{\tau+1}$ is a discrete distribution on the integers 1, 2, 3, with probabilities taken from the row of $\bp$ indexed by $j_\tau$. Every row $\bp_i$ of $\bp$ is assigned a Dirichled distribution directly. Whenever $m_{ik}\ge 2$, as $c_{ik}=m_{ik}$, the shape parameters $\alpha_{ik}$ have a shifted Gamma prior, see Equation \eqref{pi-3}, obtainable by defining in JAGS a new non-shifted Gamma variable with shape $c_{ik}$ and rate $m_{ik}d_{ik}$, which, after summing the shift, is assigned to $\alpha_{ik}$; for the value of $d_{ik}$ see Equation \eqref{eq:d_ij}. The generalized Gamma for $\theta_{ik}$ is defined conditionally on $\alpha_{ik}$: first a Gamma prior with shape $m_{ij}$ and rate $\hat{b}_{ik}(\alpha_{ik})$ is assigned to a new random variable $a_{ik}$ and then $a_{ik}^{-1/\a_{ik}}$ is assigned to $\theta_{ik}$.
In case there is either just one observation or no learning dataset $\by_m$, some special instructions in the JAGS code are needed.
In particular, if there is no learning dataset, then the missing learning dataset is substituted, for every string $(i,k)$, by the fictitious observation $\tilde{t}_{ik}$ drawn from a uniform distribution over $(1,5000)$ days (so $m_{ik}=1$ for all strings). Then, the priors of the $\theta_{ik}$ retain the same form, whereas the priors of the $\alpha_{ik}$'s, derived from Equation \eqref{pi-3} with $m_{ik}=1$ and $c=2$ (a value taken from Table~\ref{table:hyperparameters}), are
$$
\pi_3(\alpha_{ik}) \propto \left(1-\frac{\alpha_{0,ik}}{\alpha_{ik}}\right) \mathds{1}(\alpha_{0,ik}\le\alpha_{ik}\le\alpha_{1,ik})
$$
with assigned $\alpha_{0,ik}$ and $\alpha_{1,ik}$.
This latter distribution is coded using the so-called zeros trick: a fictitious zero observation from a Poisson distribution with mean $\phi_{ik}=-\ln \pi_3(\alpha_{ik})$ is introduced; then a uniform prior over $[\a_{0,ik},\a_{1,ik}]$ is assigned to $\alpha_{ik}$. The effect on the formula of the joint distribution is that the likelihood $L_1$ gets multiplied by the factor $\exp(-\phi_{ik})$, contributed by the zero observation; the multiplication by the uniform density gives back the correct factor accounting for the prior of $\alpha_{ik}$.
\section*{\large{Acknowledgments}}
We are grateful to Renata Rotondi for providing us with data and the map of Italy along with her very helpful comments. We are solely responsible for any remaining inaccuracy.
\end{document} |
\betagin{document}
\title[Identities for a parametric Weyl algebra]{Identities for a parametric Weyl algebra over a ring}
\thanks{The first author was supported by CNPq 313358/2017-6, FAEPEX 2054/19 and FAEPEX 2655/19}
\author{Artem Lopatin}
\address{Artem Lopatin\\
State University of Campinas, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil}
\email{dr.artem.lopatin@gmail.com (Artem Lopatin)}
\author{Carlos Arturo Rodriguez Palma}
\address{Carlos Arturo Rodriguez Palma\\
State University of Campinas, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil; Industrial University of Santander, Cl.~9 \#Cra 27, Ciudad Universit\'aria,
Bucaramanga, Santander, Colombia}
\email{carpal1878@gmail.com (Carlos Arturo Rodriguez Palma)}
\betagin{abstract}
In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra $\mathsf{A}_h$ generated by elements $x,y,$ which satisfy the relation $yx-xy=h$ for some $0\neq h\in \mathbb{F}[x]$. We generalize this construction to $\mathsf{A}_h(\mathsf{B})$ by working over the fixed $\mathbb{F}$-algebra $\mathsf{B}$ instead of $\mathbb{F}$. We describe the polynomial identities for $\mathsf{A}_h(\mathsf{B})$ over the infinite field $\mathbb{F}$ in case $h\in\mathsf{B}[x]$ satisfies certain restrictions.
\noindent{\bf Keywords: } polynomial identities, matrix identities, Weyl algebra, Ore extensions, positive characteristic.
\noindent{\bf 2020 MSC: } 16R10; 16S32.
\end{abstract}
\maketitle
\section{Introduction}
Assume that $\mathbb{F}$ is a field of arbitrary characteristic $p=\mathop{\rm char}\mathbb{F}\geq0$. All vector spaces and algebras are over $\mathbb{F}$ and all algebras are associative, unless other\-wise is stated. For the fixed $\mathbb{F}$-algebra $\mathsf{B}$ with unity we write $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ for the $\mathbb{F}$-algebra of non-commutative $\mathsf{B}$-polynomials in variables $x_1,\ldots,x_m$, i.e., $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ is a free left (and a free right) $\mathsf{B}$-module with the basis given by the set of all non-commutative monomials in $x_1,\ldots,x_m$, where we assume that $\beta x_i = x_i \beta$ for all $\beta\in \mathsf{B}$ and $1\leq i\leq m$. The unity $1$ of $\mathsf{B}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ corresponds to the empty monomial. In case the variables are $x_1,x_2,\dotsc$ the algebra of non-commutative $\mathsf{B}$-polynomials is denoted by $\mathsf{B}\lambdangle X\mathsf{R}A$. Similarly, we define the algebra of commutative $\mathsf{B}$-polynomials $\mathsf{B}[x_1,\ldots,x_m]$ as a free left (and a free right) $\mathsf{B}$-module with the basis given by the set of all monomials in $x_1^{i_1}\cdots x_m^{i_m}$ with $i_1,\ldots,i_m\geq0$, where we assume that $\beta x_i = x_i \beta$ and $x_i x_j=x_j x_i$ for all $\beta\in \mathsf{B}$ and $1\leq i,j\leq m$. Note that $\mathsf{B}\lambdangle x\mathsf{R}A = \mathsf{B}[x]$.
\subsection{Parametric Weyl algebra $\mathsf{A}_h(\mathsf{B})$ as the Ore extension}
We study the polynomial identities for the following family of infinite-dimensional unital algebras $\mathsf{A}_h(\mathsf{B})$, which are parametrized by a polynomial $h$ from the center of $\mathsf{B}[x]$:
\betagin{defin}\lambdabel{Def Ah}
For $h\in Z(\mathsf{B})[x]$, the {\it parametric Weyl algebra} $\mathsf{A}_h(\mathsf{B})$ {\it over the ring} $\mathsf{B}$ is the unital associative algebra over $\mathbb{F}$ generated by $\mathsf{B}$ and letters $x$, $y$ commuting with $\mathsf{B}$ subject to the defining relation $yx=xy+h$ (equivalently, $[y,x]=h$, where $[y,x]=yx-xy$), i.e., $$\mathsf{A}_h(\mathsf{B})=\mathsf{B}\lambdangle x,y\rangle/\id{yx-xy-h}.$$
\end{defin}
For short, we denote $\mathsf{A}_h=\mathsf{A}_h(\mathbb{F})$. The partial cases of the given construction are the Weyl algebra $\mathsf{A}_1$, the polynomial algebra $\mathsf{A}_0=\mathbb{F}[x,y]$, and the universal enveloping algebra $\mathsf{A}_x$ of the two-dimensional nonabelian Lie algebra. For $h\in \mathbb{F}[x]$, the following isomorphism of $\mathbb{F}$-algebras holds:
\betagin{eq}\lambdabel{eq_tensor}
\mathsf{A}_h(\mathsf{B}) \sigmameq \mathsf{B} \otimes_{\mathbb{F}} \mathsf{A}_h.
\end{eq}
Note that in general the isomorphism~\mathsf{R}ef{eq_tensor} does not hold because $A_h$ is not well-defined in case $h\not\in\mathbb{F}[x]$. Given a polynomial $f= \eta_d x^d + \eta_{d-1} x^{d-1} + \cdots + \eta_0$ of $\mathsf{B}[x]$ with $d\geq0$, we say that $\eta_d$ is the {\it leading coefficient} of $f$ and the product $\eta_d x^d$ is the {\it leading term} of $f$.
The algebra $\mathsf{A}_h$ was introduced and studied by Benkart, Lopes, Ondrus~\cite{Benkart_Lopes_Ondrus_I, Benkart_Lopes_Ondrus_II, Benkart_Lopes_Ondrus_III} as a natural object in the theory of Ore extensions. In particular, they determined automorphisms of $\mathsf{A}_h$ over an arbitrary field $\mathbb{F}$ and the invariants of $\mathsf{A}_h$ under the automorphisms, completely described the simple modules and derivations of $\mathsf{A}_h$ over any field. Then Lopes and Solotar~\cite{Lopes_Solotar_2019} described the Hochschild cohomology ${\rm HH}^{\bullet}(\mathsf{A}_h)$ over a field of arbitrary characteristic. Over an algebraically closed field of zero characteristic simple $\mathsf{A}_h$-modules were independently classified by Bavula~\cite{Bavula_2020}. In recent preprints~\cite{Bavula_2021_zero},~\cite{Bavula_2021_prime} Bavula continued the study of the automorphism group of $\mathsf{A}_h$.
Let us recall that an Ore extension of $R$ (or, equivalently, a skew polynomial ring over $R$) $A=\mathsf{R}[y,\sigmagma,\deltalta]$ is given by a unital associative (not necessarily commutative) algebra $\mathsf{R}$ over a field $\mathbb{F}$, an $\mathbb{F}$-algebra endomorphism $\sigmagma:\mathsf{R}\rightarrow \mathsf{R}$, and a $\sigmagma$-derivation $\deltalta:\mathsf{R}\rightarrow \mathsf{R}$, i.e., $\deltalta$ is $\mathbb{F}$-linear map and $\deltalta(ab)=\deltalta(a)b+\sigmagma(a)\deltalta(b)$ for all $a,b\in \mathsf{R}$. Then $A=\mathsf{R}[y,\sigmagma,\deltalta]$ is the unital algebra generated by $y$ over $\mathsf{R}$ subject to the relation $$ya=\sigmagma(a)y+\deltalta(a) \quad \text{ for all } a\in \mathsf{R}.$$
Assume that $\mathsf{R}=\mathsf{B}[x]$, $\sigmagma={\rm id}_{\mathsf{R}}$ is the identity automorphism on $\mathsf{R}$, and $\deltalta:\mathsf{R}\rightarrow \mathsf{R}$ is given by $\deltalta(a)=a'h$ for all $a\in \mathsf{R}$, where $a'$ stands for the usual derivative of a $\mathsf{B}$-polynomial $a$ with respect to the variable $x$. Since $h\in Z(\mathsf{R})$, $\delta$ is a derivation of $\mathsf{R}$. Using the linearity of derivative and induction on the degree of $a\in\mathsf{B}[x]$ it is easy to see that
\betagin{eq}
[y,a]=a'h \text{ holds in }\mathsf{A}_h(\mathsf{B}) \text{ for all }a\in \mathsf{B}[x].
\end{eq}
\noindent{}Hence $\mathsf{A}_h(\mathsf{B})=\mathsf{R}[y,\sigmagma,\deltalta]$ is an Ore extension. The following lemma is a corollary of Observation 2.1 from~\cite{AVV} proven by Awami, Van den Bergh and Van Oystaeyen (see also Proposition 3.2 of \cite{AD2} and Lemma 2.2 of \cite{Benkart_Lopes_Ondrus_I}).
\betagin{lemma} Assume that $A=\mathsf{R}[y,\sigma,\delta]$ is an Ore extension of $\mathsf{R}=\mathbb{F}[x]$, where $\sigma$ is an automorphism of $\mathsf{R}$. Then $A$ is isomorphic to one of the following algebras:
\betagin{enumerate}
\item[$\bullet$] a quantum plane, i.e., $A\sigmameq\FF\lambdangle x,y\rangle/\id{yx-q xy}$ for some $q\in\mathbb{F}^{*}=\mathbb{F}\backslash \{0\}$;
\item[$\bullet$] a quantized Weyl algebra, i.e., $A\sigmameq\FF\lambdangle x,y\rangle/\id{yx-q xy-1}$ for some $q\in\mathbb{F}^{*}$;
\item[$\bullet$] an algebra $\mathsf{A}_h$ for some $h\in\mathbb{F}[x]$.
\end{enumerate}
\end{lemma}
\noindent{}Note that by Theorem 9.3 of~\cite{Benkart_Lopes_Ondrus_I} the algebra $\mathsf{A}_h$ is not a generalized Weyl algebra over $\mathbb{F}[x]$ in the sense of Bavula~\cite{Bavula_1992} in case $h\not\in\mathbb{F}$.
Since the algebra of $\mathsf{B}$-polynomials $\mathsf{B}[x,y]$ is well studied, in what follows we assume that $h$ is non-zero. Moreover, we assume that the following restriction holds:
\betagin{conv}\lambdabel{conv1} The leading coefficient of $h\in Z(\mathsf{B})[x]$ is not a zero divisor.
\end{conv}
\subsection{Polynomial identities}
A polynomial identity for a unital $\mathbb{F}$-algebra $\mathcal{A}$ is an element $f(x_1,\ldots,x_m)$ of $\mathbb{F}\lambdangle X\mathsf{R}A$ such that $f(a_1,\ldots,a_m)=0$ in $\mathcal{A}$ for all $a_1,\ldots,a_m\in \mathcal{A}$. The set $\Id{\mathcal{A}}$ of all polynomial identities for $\mathcal{A}$ is a T-ideal, i.e., $\Id{\mathcal{A}}$ is an ideal of $\FF\lambdangle X\rangle$ such that $\phi(\Id{\mathcal{A}})\subset \Id{\mathcal{A}}$ for every endomorphism $\phi$ of $\FF\lambdangle X\rangle$. An algebra that satisfies a nontrivial polynomial identity is called a PI-algebra. A T-ideal $I$ of $\mathbb{F}\lambdangle X\mathsf{R}A$ generated by $f_1,\ldots,f_k\in \mathbb{F}\lambdangle X\mathsf{R}A$ is the minimal T-ideal of $\mathbb{F}\lambdangle X\mathsf{R}A$ that contains $f_1,\ldots,f_k$. We say that $f\in \mathbb{F}\lambdangle X\mathsf{R}A$ follows from $f_1,\ldots,f_k$ if $f\in I$. Given a monomial $w\in \mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$, we write $\deltag_{x_i}(w)$ for the number of letters $x_i$ in $w$ and $\mathop{\rm mdeg}(w)$ for the multidegree $(\deltag_{x_1}(w),\ldots,\deltag_{x_m}(w))$ of $w$. An element $f\in\FF\lambdangle X\rangle$ is called multihomogeneous if it is a linear combination of monomials of the same multidegree. We say that algebras $\mathcal{A}$, $\mathcal{B}$ are called PI-equivalent and write $\mathcal{A} \sigmam_{\rm PI} \mathcal{B}$ if $\Id{\mathcal{A}} =\Id{\mathcal{B}}$.
Denote the $n^{\rm th}$ Weyl algebra by
$$\mathsf{A}W_n=\mathbb{F}\lambdangle x_1,\ldots,x_n,y_1,\ldots,y_n\mathsf{R}A/I,$$
where the ideal $I$ is generated by $[y_i,x_j]=\delta_{ij}$, $[x_i,x_j]=0$, $[y_i,y_j]=0$ for all $1\leq i,j\leq n$. Note that $\mathsf{A}_1=\mathsf{A}W_1$.
Assume that $p=0$. It is well-known that the algebra $\mathsf{A}W_n$ does not have nontrivial polynomial identities. Nevertheless, some subspaces of $\mathsf{A}W_n$ satisfy certain polynomial identities. Namely, denote by $\mathsf{A}W_n^{(1,1)}$ the $\mathbb{F}$-span of $x_i y_j$ in $\mathsf{A}W_n$ for all $1\leq i,j \leq n$ and by $\mathsf{A}W_n^{(-,r)}$ the $\mathbb{F}$-span of $a y_{j_1}\cdots y_{j_r}$ in $\mathsf{A}W_n$ for all $1\leq j_1,\ldots, j_r\leq n$ and $a\in \mathbb{F}[x_1,\ldots,x_n]$. Dzhumadil'daev~\cite{Askar_2004, Askar_2014} studied the standard polynomial
$${\rm St}_N(t_{1},\ldots,t_{N})=\sum_{\sigmagma\in S_{N}}(-1)^{\sigmagma}t_{\sigmagma(1)}\cdots t_{\sigmagma(N)}$$
over some subspaces of $\mathsf{A}W_n$. Namely, he showed that
\betagin{enumerate}
\item[$\bullet$] ${\rm St}_N$ is a polynomial identity for $\mathsf{A}W_n^{(-,1)}$ in case $N\geq n^2 + 2n$;
\item[$\bullet$] ${\rm St}_N$ is not a polynomial identity for $\mathsf{A}W_n^{(-,1)}$ in case $N< n^2 + 2n - 1$;
\item[$\bullet$] ${\rm St}_N$ is a polynomial identity for $\mathsf{A}W_1^{(-,r)}$ if and only if $N>2r$;
\item[$\bullet$] the minimal degree of nontrivial identity in $\mathsf{A}W_1^{(-,r)}$ is $2r+1$.
\end{enumerate}
Using graph--theoretic combinatorial approach Dzhumadil'daev and Yeliussizov~\cite{Askar_Yeliussizov_2015} established that
\betagin{enumerate}
\item[$\bullet$] ${\rm St}_{2n}$ is a polynomial identity for $\mathsf{A}W_n^{(1,1)}$ if and only if $n=1,2,3$.
\end{enumerate}
Note that the space $\mathsf{A}W_n^{(-,1)}$ together the multiplication given by the Lie bracket is the $n^{\rm th}$ Witt algebra $W_n$, which is a simple infinitely dimensional Lie algebra. The polynomial identities for the Lie algebra $W_n$ were studied by Mishchenko~\cite{Mishchenko_1989}, Razmyslov~\cite{Razmyslov_book} and others. The well-known open conjecture claims that all polynomial identities for $W_1$ follow from the standard Lie identity
$$\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}[[[[t_0,t_{\sigma(1)}]t_{\sigma(2)}]t_{\sigma(3)}]t_{\sigma(4)}].$$
\noindent{}$\mathbb{Z}$-graded identities for $W_1$ were described by Freitas, Koshlukov and Krasilnikov~\cite{W1_2015}.
\subsection{Results}
In Theorem~\ref{Teorema Principal} we prove that over an infinite field $\mathbb{F}$ of positive characteristic $p$ the algebra $\mathsf{A}_h(\mathsf{B})$ is PI-equivalent to the algebra of $p\times p$ matrices over $\mathsf{B}$ in case $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$. On the other hand, over a finite field the similar result does not hold in case $\mathsf{B}=\mathbb{F}$ (see Theorem~\ref{theo_finite}). As about the case of zero characteristic, in Theorem~\ref{theo0} we prove that similarly to $\mathsf{A}_1$, the algebra $\mathsf{A}_h(\mathsf{B})$ does not have nontrivial polynomial identities.
In Section~\ref{section_example} we consider the algebra $\mathsf{A}_{h}(\mathsf{B})=\mathsf{A}_{\zeta}(\mathbb{F}^2)$ such that $h=\zeta$ does not satisfy Convention~\ref{conv1} and the statements of Theorems \ref{theo0} and ~\ref{Teorema Principal} do not hold for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. We describe polynomial identities for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$ and compare them with the polynomial identities for the Grassmann unital algebra of finite rank.
\section{Properties of $\mathsf{A}_h(\mathsf{B})$}
Many properties of an Ore extension $A=\mathsf{R}[y,\sigmagma,\deltalta]$ are inherited from an underlying algebra $\mathsf{R}$. Namely, it is well-known that when $\sigmagma$ is an automorphism, then:
\betagin{itemize}
\item $A$ is a free right and a free left $\mathsf{R}$-module with the basis $\{y^i \ | \ i\geq 0\}$ (see Proposition 2.3 of~\cite{K. R. Goodearl});
\item in case $R$ is left (right, respectively) noetherian we have that $A$ is left (right, respectively) noetherian (see Theorem 2.6 of~\cite{K. R. Goodearl});
\item in case $\mathsf{R}$ is a domain we have that $A$ is a domain (see Exercise 2O~of \cite{K. R. Goodearl}).
\end{itemize}
\noindent{}In case $\mathsf{B}=\mathbb{F}$ the algebra $\mathsf{A}_h=\mathsf{A}_h(\mathsf{B})$ is a noetherian domain, but in general case $\mathsf{A}_h(\mathsf{B})$ lacks these properties, since $\mathsf{B}\subset \mathsf{A}_h(\mathsf{B})$ (see also Example~\ref{ex2} below).
In order to distinguish the generators for the algebras $\mathsf{A}_h(\mathsf{B})$ and $\mathsf{A}_1(\mathsf{B})$, we will use the following
\betagin{conv}\lambdabel{conv2} The generators of $\mathsf{A}_h(\mathsf{B})$ are denoted by $x,\widehat{y}, 1$ and the generators of $\mathsf{A}_1(\mathsf{B})$ are denoted by $x,y,1$.
\end{conv}
\betagin{lemma}\lambdabel{lemma_basis}
The sets $\{x^{i}\widehat{y}^{j} \ | \ i,j\geq 0\}$ and $\{\widehat{y}^{j}x^{i} \ | \ i,j\geq 0\}$ are $\mathsf{B}$-bases for $\mathsf{A}_h(B)$.
\end{lemma}
\betagin{proof}
Obviously, $\mathsf{A}_h(\mathsf{B})$ is the $\mathsf{B}$-span of each of the sets from the lemma. On the other hand, $\mathsf{B}$-linear independence of these sets follows from the fact that $\mathsf{A}_h(\mathsf{B})$ is a free right and a free left $\mathsf{B}[x]$-module with the basis $\{\widehat{y}^i \ | \ i\geq 0\}$.
\end{proof}
Introduce the following lexicographical order on $\mathbb{Z}^2$:
$(i,j)<(r,s)$ in case $j<s$ or $j=s,\;i<r$. Denote the multidegree of a monomial $w=x^i\widehat{y}^j$ of $\mathsf{A}_h(\mathsf{B})$ by $\mathop{\rm mdeg}(w)=(i,j)$. Given an arbitrary element
$a=\sum_{i,j\geq0}\beta_{ij} x^i \widehat{y}^j$ of $\mathsf{A}_h(\mathsf{B})$, where only finitely many $\beta_{ij}\in\mathsf{B}$ are non-zero, define its {\it multidegree} $\mathop{\rm mdeg}(a)=(d_x,d_y)$ as the maximal multidegree of its momomials, i.e. as the maximal element of the set $\{(i,j) \ | \ \beta_{ij}\neq0\}$. By Lemma~\ref{lemma_basis} the multidegree is well-defined. As above, the coefficient $\beta_{d_x,d_y}$ is called the {\it leading coefficient} of $a$ and the product $\beta_{d_x,d_y}\,x^{d_x} \widehat{y}^{d_y}$ is called the {\it leading term} of $a$. In case $a\in\mathsf{B}$ we set $\mathop{\rm mdeg}(a)=(0,0)$ and the leading coefficient as well as the leading term of $a$ is $a$.
\betagin{lemma}\lambdabel{lemma_mult}
Assume $i,j,r,s\geq 0$. Then
\betagin{enumerate}
\item[(a)] the leading term of $x^i \widehat{y}^j\cdot x^r \widehat{y}^s$ is $x^{i+r}\, \widehat{y}^{j+s}$;
\item[(b)] in case $h\in\mathsf{B}$ we have
$$x^i \widehat{y}^j\cdot x^r \widehat{y}^s = \sum_{k=0}^{\min\{j,r\}} k! \binom{j}{k} \binom{r}{k} x^{i+r-k} h^k\, \widehat{y}^{j+s-k}.$$
\end{enumerate}
\end{lemma}
\betagin{proof}
Recall that $\delta(a)=a'h$ for each $a\in\mathsf{B}[x]$. Since $[\widehat{y},a]=\delta(a)$, the induction on $j$ implies that
\betagin{eq}\lambdabel{eq1_lemma_mult}
\widehat{y}^j x^r = \sum_{k=0}^{j} \binom{j}{k} \delta^k(x^r) \widehat{y}^{j-k}
\end{eq}
(cf.~Lemma 5.2 of~\cite{Benkart_Lopes_Ondrus_I}). Taking $k=0$ in equality~\mathsf{R}ef{eq1_lemma_mult}, we obtain that the leading term of $\widehat{y}^j x^r$ is $x^r\widehat{y}^j$. Similarly we conclude the proof of part (a). Part (b) follows from~\mathsf{R}ef{eq1_lemma_mult} and
$$\delta^k(x^r) =
\left\{
\betagin{array}{cl}
\frac{r!}{(r-k)!} x^{r-k} h^k, & \text{if } k\leq r \\
0, & \text{if } k> r \\
\end{array}
\right..
$$
\end{proof}
\betagin{lemma}\lambdabel{lemma_deg}
If the leading coefficient of one of non-zero elements $a,b\in \mathsf{A}_h(\mathsf{B})$ is not a zero divisor, then $\mathop{\rm mdeg}(ab) = \mathop{\rm mdeg}(a) + \mathop{\rm mdeg}(b)$. In particular, $ab$ is not zero.
\end{lemma}
\betagin{proof} Consider $a=\sum_{i=1}^m \beta_{i} x^{r_i} \widehat{y}^{s_i}$ and $b=\sum_{j=1}^n \gamma_{j} x^{k_j} \widehat{y}^{l_j}$ for $m,n\geq1$ and non-zero $\beta_i,\gamma_j\in\mathsf{B}$, where we assume that elements of each of the sets $\{(r_i,s_i) \ | \ 1\leq i\leq m\}$ and $\{(k_j,l_j) \ | \ 1\leq j\leq n\}$ are pairwise different. Assume that $\mathop{\rm mdeg}(a)=(r_1,s_1)$ and $\mathop{\rm mdeg}(b)=(k_1,l_1)$. Part (a) of Lemma~\ref{lemma_mult} implies that
$$\mathop{\rm mdeg}(x^{r_1} \widehat{y}^{s_1} x^{k_1} \widehat{y}^{l_1})=(r_1\!+\!k_1, s_1\!+\!l_1) \;\text{ and }\; \mathop{\rm mdeg}(x^{r_i} \widehat{y}^{s_i} x^{k_j} \widehat{y}^{l_j})<(r_1\!+\!k_1, s_1\!+\!l_1)$$ if $(i,j)\neq (1,1)$. Since $\beta_1 \gamma_1\neq 0$, we obtain $\mathop{\rm mdeg}(ab) = (r_1+k_1, s_1 + l_1)$ and the proof is concluded.
\end{proof}
\betagin{lemma}\lambdabel{lemma_embedding}
\betagin{enumerate}
\item[(a)] The $\mathsf{B}$-linear homomorphism of $\mathbb{F}$-algebras $\psi: \mathsf{A}_h(\mathsf{B})\to \mathsf{A}_1(\mathsf{B})$, defined by
$$1\to 1,\quad x\to x,\quad \widehat{y}\to y h,$$
is an embedding $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$.
\item[(b)] $\{x^{i}h^{j}y^{j} \ | \ i,j\geq 0\}$ and $\{y^{j}h^{j}x^{i} \ | \ i,j\geq 0\}$ are $\mathsf{B}$-bases for $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$.
\end{enumerate}
\end{lemma}
\betagin{proof}
\noindent{\bf (a)} Since $\psi([\widehat{y},x]-h)=([y,x]-1)h=0$ in $\mathsf{A}_1(\mathsf{B})$, the homomorphism $\psi$ is well-defined. Assume that $\psi$ is not an embedding, i.e., there exists non-zero finite sum $a=\sum_{i,j\geq0}\beta_{ij} x^i \widehat{y}^j$ with $\beta_{ij}\in \mathsf{B}$ such that
$$
\psi(a)=\sum_{i,j\geq0}\beta_{ij} x^i (y h)^j =0 \quad\text{ in }\quad\mathsf{A}_1(\mathsf{B}).
$$
Denote $\mathop{\rm mdeg}(a)=(r,s)$. If $(r,s)=(0,0)$, then $a\in \mathsf{B}$ and $\psi(a)=a$ is not zero; a contradiction. Therefore, $(r,s)\neq (0,0)$.
Since $\mathop{\rm mdeg}(x^i (y h)^j)= (i + j \deltag(h), j)$ by Lemma~\ref{lemma_deg} and Convention~\ref{conv1}, we obtain that $\mathop{\rm mdeg}(\psi(a))= (r + s \deltag(h), s)$ is not zero; a contradiction.
\noindent{\bf (b)} Since $h$ lies in the center of $\mathsf{B}[x]$, repeating the proof of Lemma 3.4 from~\cite{Benkart_Lopes_Ondrus_I} for $\mathsf{A}_h(\mathsf{B})$ we can see that
$$\mathsf{A}_h(\mathsf{B})=\bigoplus_{j\geq 0}\mathsf{B}[x]h^{j}y^{j}=\bigoplus_{j\geq 0}y^{j}h^{j}\mathsf{B}[x].$$
Similarly to part (a), we conclude the proof by the reasoning with multidegree.
\end{proof}
\betagin{example}\lambdabel{ex2} Assume $\mathsf{B}$ is the $\mathbb{F}$-algebra of double numbers, i.e., $\mathsf{B}$ has an $\mathbb{F}$-basis $\{1,\zeta\}$ with $\zeta^2=0$. Then the ideal $I=\mathbb{F}\text{-span}\{\zeta x^i \widehat{y}^j \ | \ i,j\geq 0\}$ is a proper nilpotent ideal of $\mathsf{A}_h(\mathsf{B})$. In particular, the algebra $\mathsf{A}_h(\mathsf{B})$ is not semi-prime.
\end{example}
\section{$\mathsf{A}_h(\mathsf{B})$ as the algebra of differential operators}\lambdabel{section_diff}
Denote by ${\rm Map}(\mathsf{B}[z])$ the algebra of all $\mathbb{F}$-linear maps over $\mathsf{B}[z]$ with respect to composition. Assume that $\mathfrak{D}_h(\mathsf{B}[z])$ is the subalgebra of ${\rm Map}(\mathsf{B}[z])$ generated by the following maps: the multiplication $\beta\,{\rm Id}$ by an element $\beta$ of $\mathsf{B}$, i.e., $(\beta\,{\rm Id})(f)=\beta f$, the multiplication $\chi$ by $z$, i.e., $\chi(f)=zf$, and the derivation $\delta$ given by $\delta(f)=f'h(z)$ for all $f\in\mathsf{B}[z]$. Note that $\delta= h(\chi)\, \partial$, where $\partial$ stands for the operator of the usual derivative. Obviously, maps $\chi$, $h(\chi)$ and $\partial$ are $\mathsf{B}$-linear. For short, we write $\chi^0$ for ${\rm Id}$.
\betagin{prop}\lambdabel{prop_diff}
\betagin{enumerate}
\item[(a)] $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ i,j\geq 0\}$ is an $\mathsf{B}$-basis for $\mathfrak{D}_h(\mathsf{B}[z])$ in case $p=0$.
\item[(b)] $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ 0\leq i,\; 0\leq j<p \}$ is an $\mathsf{B}$-basis for $\mathfrak{D}_h(\mathsf{B}[z])$ in case $p>0$.
\item[(c)]$\mathsf{A}_h(\mathsf{B}) /\id{h^p y^p} \sigmameq \mathfrak{D}_h(\mathsf{B}[z])$ for each $p\geq 0$.
\end{enumerate}
\end{prop}
\betagin{proof}
Consider the $\mathsf{B}$-linear homomorphism of $\mathbb{F}$-algebras $\Phi: \mathsf{A}_1(\mathsf{B}) \to {\rm Map}(\mathsf{B}[z])$ given by
$1\to {\rm Id}$, $x\to \chi$, $y\to \partial$. Since $\Phi([y,x]-1)(f)=(\partial \chi - \chi \partial - {\rm Id})(f) = f + z f' - z f' - f = 0$ for all $f\in\mathsf{B}[z]$, the map $\Phi$ is a well-defined. Applying $\Phi$ to parts (a) and (b) of Lemma~\ref{lemma_embedding} we obtain that $\mathfrak{D}_h(\mathsf{B}[z]) = \Phi(\mathsf{A}_h(\mathsf{B}))$ is an $\mathsf{B}$-span of $\{\chi^{i} h(\chi)^j \partial^{j} \ | \ i,j\geq 0\}$.
Let $p=0$. Assume that some non-zero finite sum $\pi=\sum_{i,j\geq 0} \beta_{ij} \chi^{i} h(\chi)^j \partial^{j}$ with $\beta_{ij}\in\mathsf{B}$ belongs to the kernel of $\Phi$. Denote by $j_0$ the minimal $j\geq0$ with $\beta_{ij}\neq 0$ for some $i$ and denote by $i_0$ the maximal $i$ with $\beta_{ij_0}\neq 0$. Then $\pi(z^{j_0}) = j_0!\, h(z)^{j_0} \sum_{0\leq i\leq i_0} \beta_{i,j_0}\, z^i =0$ in $\mathsf{B}[z]$. Thus Convention~\ref{conv1} together with $j_0!\,\beta_{i_0,j_0} \neq 0$ implies a contradiction. Part~(a) is proven.
Assume that $p>0$ and some non-trivial finite sum
$$\pi=\sum\limits_{0\leq i,\; 0\leq j<p } \beta_{ij} \chi^{i} h(\chi)^j \partial^{j}$$
with $\beta_{ij}\in\mathsf{B}$ belongs to the kernel of $\Phi$. As above, we obtain a contradiction. Since $\partial^p=0$, we conclude the proof of part (b).
Parts (a) and (b) together with part (b) of Lemma~\ref{lemma_embedding} conclude the proof of part (c).
\end{proof}
\betagin{theo}\lambdabel{theo0}
In case $p=0$ the algebra $\mathsf{A}_h(\mathsf{B})$ does not have nontrivial polynomial identities.
\end{theo}
\betagin{proof}
Assume that $\mathbb{F}$-algebra $\mathsf{A}_h(\mathsf{B})$ has a nontrivial polynomial identity. Since $p=0$, there exists $N>0$ such that $\mathsf{A}_h(\mathsf{B})$ satisfies a nontrivial multilinear identity $f(x_1,\ldots,x_N)=\sum_{\sigma\in S_N}\alpha_{\sigma} x_{\sigma(1)}\cdots x_{\sigma(N)}$ with $\alpha_{\sigma}\in \mathbb{F}$. Moreover, we can assume that $\alpha_{\rm Id}\neq0$ for the identity permutation ${\rm Id}$. Given $j>0$, we write $F_j$ for a $\mathsf{B}$-linear map $\chi^{2j} h(\chi)^j \partial^{j}$ from $\mathfrak{D}_h(\mathsf{B}[z])$. Denote by $d\geq0$ the degree of $h$ and we write $\eta$ for the leading coefficient of $h$. Recall that $\eta$ is not a zero divisor by Convention~\ref{conv1}. Note that
\betagin{eq}\lambdabel{eq_theo0}
F_{j}(z^m)=\left\{
\betagin{array}{cc}
0, & \text{ in case }m<j\\
\frac{m!}{(m-j)!} \, z^{m+j} h(z)^j, & \text{ in case }m\geq j.\\
\end{array}
\right.
\end{eq}
In particular, $\deltag(F_{j}(z^j))=j(d+2)$ and the leading coefficient of $F_{j}(z^j)$ is $j!\,\eta^j$, which is not a zero divisor.
By parts (a) and (c) of Proposition~\ref{prop_diff}, the equality $f(F_{j_{1}},\ldots,F_{j_N})=0$ holds in $\mathfrak{D}_h(\mathsf{B}[z])$ for all $j_1,\ldots,j_{N}>0$. Consider $j_k=(d+2)^{N-k}$ for all $0\leq k\leq N$.
Note that $1=j_N<j_{N-1}<\cdots < j_0$. We claim that for any $\sigma\in S_N$ we have
\betagin{eq}\lambdabel{claim_theo0}
F_{j_{\sigma(1)}}\circ \cdots \circ F_{j_{\sigma(N)}}(z) \neq 0 \text{ if and only if } \sigma= {\rm Id}.
\end{eq}
\betagin{eq}\lambdabel{claim2_theo0}
\text{The leading term of } F_{j_1}\circ \cdots \circ F_{j_N}(z) \text{ is } j_{1}!\cdots j_N!\,\eta^{j_{1}+\cdots +j_N} z^{j_{0}}.
\end{eq}
Let us prove these claims. Assume that $F_{j_{\sigma(1)}}\circ \cdots \circ F_{j_{\sigma(N)}}(z) \neq 0$ for some $\sigma\in S_N$.
Since $F_{j_{\sigma(N)}}(z)\neq 0$, then equality~\mathsf{R}ef{eq_theo0} implies that $\sigma(N)=N$, $j_{\sigma(N)}=1$, $\deltag(F_{j_{\sigma(N)}}(z))=d+2=j_{N-1}$ and the leading coefficient of $F_{j_{\sigma(N)}}(z)$ is $\eta$, which is not a zero divisor.
Similarly, assume that for $1\leq l< N$ with $\sigma(l)\leq l$ the inequality $F_{j_{\sigma(l)}}(g)\neq 0$ holds for some $g\in \mathsf{B}[z]$ with the leading term $j_{l+1}!\cdots j_N!\,\eta^{j_{l+1}+\cdots + j_N} z^{j_l}$. Then equality~\mathsf{R}ef{eq_theo0} implies that $\sigma(l)=l$ and $\deltag(F_{j_{\sigma(l)}}(g))=j_{l-1}$. Moreover, the leading term of $F_{j_{\sigma(l)}}(g)$ is $j_{l}!\cdots j_N!\,\eta^{j_{l}+\cdots +j_N} z^{j_{l-1}}$. Consequently applying this reasoning to $l=N-1,\, l=N-2,\ldots, l=1$, we conclude the proof of claims~\mathsf{R}ef{claim_theo0} and~\mathsf{R}ef{claim2_theo0}.
Claims~\mathsf{R}ef{claim_theo0} and~\mathsf{R}ef{claim2_theo0} imply that $0=f(F_1,\ldots,F_N)(z)=\alpha_{\rm Id}\, F_{j_1}\circ \cdots \circ F_{j_N}(z)\neq 0$ by Convention~\ref{conv1}; a contradiction.
\end{proof}
\section{Polynomial Identities for $\mathsf{A}_h(\mathsf{B})$ in positive characteristic}\lambdabel{section_positive}
We write $M_n=M_n(\mathbb{F})$ for the algebra of all $n\times n$ matrices over $\mathbb{F}$ and denote by $\widetilde{M}_n$ the algebra of all $n\times n$ matrices over $\mathsf{B}[x,y]$. Denote by $I_n$ the identity $n\times n$ matrix and by $E_{ij}\in M_n$ the matrix such that the $(i,j)^{\rm th}$ entry is equal to one and the rest of entries are zeros. Consider the properties of the next two matrices of $M_p$:
$$A_{0}=\sum_{i=1}^{p-1}E_{i+1,i}\;\;\text{ and }\;\;B_{0}=\sum_{i=1}^{p-1}i\cdot E_{i,i+1}.$$
\betagin{lemma}\lambdabel{lemma_A0_B0}
\betagin{enumerate}
\item[(a)] For all $0\leq k<p$ we have that
$$A_{0}^{k}=\sum_{i=1}^{p-k}E_{k+i,i}\;\;\text{ and }\;\; B_{0}^{k}=\sum_{i=1}^{p-k}\frac{(k+i-1)!}{(i-1)!}E_{i,k+i},$$
\noindent{}where $A_0^0$ and $B_0^0$ stand for $I_p$.
\item[(b)] $B_{0}A_{0}-A_{0}B_{0}=I_{p}$.
\end{enumerate}
\end{lemma}
\betagin{proof} The formula for $A_{0}^{k}$ is trivial. We prove the formula for $B_{0}^{k}$ by induction on $k$. For $k=1$ the claim holds. Assume that the claim is valid for some $k<p-1$. Then
\betagin{align*}
B_{0}^{k+1}&=\left(\sum_{i=1}^{p-k}\frac{(k+i-1)!}{(i-1)!}E_{i,k+i}\right)\left(\sum_{r=1}^{p-1}r\cdot E_{r,r+1} \right)=\sum_{i=1}^{p-(k+1)}\frac{(i+k)!}{(i-1)!}E_{i,k+1+i}
\end{align*}
and the required is proven. Part (b) is straightforward.
\end{proof}
Define the $\mathsf{B}$-linear homomorphism $\varphi:\mathsf{B}\lambdangle x,y \mathsf{R}A \rightarrow \widetilde{M}_p$ of algebras by
$$x\mapsto A,\;\; y\mapsto B,\;\; 1\mapsto I_p,$$
where $A=xI_{p}+A_{0}$ and $B=yI_{p}+B_{0}$. Since $\beta A=A \beta$ and $\beta B = B\beta$ for each $\beta\in B$, the homomorphism $\varphi$ is well-defined.
\betagin{lemma}\lambdabel{A1 subset Map}
The homomorphism $\varphi$ induces the injective $\mathsf{B}$-linear homomorphism $\ov{\varphi}:\mathsf{A}_1(\mathsf{B})\rightarrow \widetilde{M}_p$. In particular, the restriction of $\ov{\varphi}$ to $\mathsf{A}_h(\mathsf{B})\subset \mathsf{A}_1(\mathsf{B})$ is the injective $\mathsf{B}$-linear homomorphism $\mathsf{A}_h(\mathsf{B})\rightarrow \widetilde{M}_p$.
\end{lemma}
\betagin{proof}
By part (b) of Lemma \ref{lemma_A0_B0} we have that $\varphi(yx-xy-1)=BA-AB-I_{p}=0$. Therefore, $\varphi$ induces a $\mathsf{B}$-linear homomorphism $\ov{\varphi}:\mathsf{A}_1(\mathsf{B})\rightarrow \widetilde{M}_p$ of algebras.
Assume that there exists a nonzero $a\in \mathsf{A}_1(\mathsf{B})$ such that $\ov{\varphi}(a)=0$. Since $\{x^{i}y^{j} \ | \ i,j\geq 0\}$ is an $\mathsf{B}$-basis for $\mathsf{A}_1(\mathsf{B})$ by Lemma~\ref{lemma_basis}, we have $a=\sum_{i,j\geq 0}\beta_{ij}x^{i}y^{j}$ for a finite sum with $\beta_{ij}\in \mathsf{B}$. Thus
$0= \ov{\varphi}(a) = \sum_{i,j\geq 0} \beta_{ij} A^i B^j$. The equalities
$$A^{i}=\betagin{pmatrix}
x^{i} & 0 &\cdots & 0 \\
* & * &\cdots & * \\
\vdots &\vdots&&\vdots\\
* & * &\cdots & * \\
\end{pmatrix}
\;\; \text{and} \;\;
B^{j}=\betagin{pmatrix}
y^{j} & * &\cdots & * \\
0 & * &\cdots & * \\
\vdots &\vdots&&\vdots\\
0 & * &\cdots & * \\
\end{pmatrix}$$
imply that the $(1,1)^{\rm th}$ entry of $A^{i}B^{j}$ is $(A^{i}B^{j})_{1,1}=x^{i}y^{j}$. Hence
$0 = (\varphi(a))_{1,1} = \sum_{i,j\geq 0} \beta_{ij} x^{i}y^{j}$ in $\mathsf{B}[x,y]$.
Hence $\beta_{ij}=0$ for all $i,j\geq 0$ and $a=0$; a contradiction. Therefore $\ov{\varphi}$ is injective.
\end{proof}
Given $1\leq i,j\leq p$ and $k\geq 1$, we write $z_{ij}(k)$ for $x_{i+p(j-1)+p^2(k-1)}\in \mathbb{F}\lambdangle X\mathsf{R}A$. The {\it generic $p\times p$ matrix $X_k$ with non-commutative elements} is the matrix $X_k=(z_{ij}(k))_{1\leq i,j\leq p}$.
\betagin{cor}\lambdabel{Id(Mp) subset Id(Ah)}
$\Id{M_p(\mathsf{B})}\subset \Id{\mathsf{A}_h(\mathsf{B})}$, if the field $\mathbb{F}$ is infinite.
\end{cor}
\betagin{proof}
Lemma~\ref{A1 subset Map} implies that $\Id{\widetilde{M}_p}\subset \Id{\mathsf{A}_h(\mathsf{B})}$.
Since $B\subset B[x,y]$, we have $\Id{\widetilde{M}_p}\subset \Id{M_p(\mathsf{B})}$. On the other hand, assume that $f=f(x_1,\ldots,x_n)$ is a polynomial identity for $M_p(\mathsf{B})$. Then $f(X_1,\ldots,X_n)= (f_{ij})_{1\leq i,j\leq n}$ for some $f_{ij}\in \mathbb{F}\lambdangle X\mathsf{R}A$ with $f_{ij}\in \Id{\mathsf{B}}$. It is well-known that for an infinite field $\mathbb{F}$ and a commutative unital $\mathbb{F}$-algebra $\mathcal{C}$ the polynomial identities for a unital $\mathbb{F}$-algebra $\mathcal{B}$ and $\mathcal{C}\otimes_{\mathbb{F}} \mathcal{B}$ are the same (for example, see Lemma 1.4.2 of~\cite{A.Gia}). Since $\mathsf{B}[x,y]=\mathbb{F}[x,y]\otimes_{\mathbb{F}} \mathsf{B}$, we obtain $f_{ij}\in \Id{\mathsf{B}[x,y]}$ and $f\in \Id{\widetilde{M}_p}$. The required is proven.
\end{proof}
For each $\alpha\in Z(\mathsf{B})$ consider the evaluation $\mathsf{B}$-linear homomorphism $\epsilon_{\alpha}:\mathsf{B}[x,y]\rightarrow \mathsf{B}$ of unital $\mathbb{F}$-algebras defined by
$$x\mapsto \alpha, \;\; \ y\mapsto 0$$
and extend it to the evaluation homomorphism $\varepsilon_{\alpha}:\widetilde{M}_p\rightarrow M_p(\mathsf{B})$. Since $\mathsf{A}_h(\mathsf{B})$ is a subalgebra of $\widetilde{M}_p$ by means of embedding $\ov{\varphi}$ (see Lemma \ref{A1 subset Map}), we can consider the images of $x,\widehat{y}\in \mathsf{A}_h(\mathsf{B})$ in $M_p(\mathsf{B})$, which we denote by $C_{\alpha}$ and $D_{\alpha}$, respectively:
$$\betagin{array}{cll}
C_{\alpha}=\varepsilon_{\alpha}(\ov{\varphi}(x))& =\varepsilon_{\alpha}(A) & = \alpha I_p + A_0,\\
D_{\alpha}=\varepsilon_{\alpha}(\ov{\varphi}(\widehat{y}))& =\varepsilon_{\alpha}(Bh(A)) & = B_{0}\varepsilon_{\alpha}(h(A)).
\end{array}
$$
\noindent{}Obviously, $\beta C_{\alpha} = C_{\alpha} \beta$ and $\beta D_{\alpha} = D_{\alpha} \beta$ for each $\beta\in B$. To obtain the explicit description of the matrix $D_{\alpha}$ we calculate $h(A)$. For $r\geq1$ denote the $r^{\rm th}$ derivative of $h\in\mathsf{B}[x]$ by $h^{(r)}=\frac{d^{r}h}{dx^{r}}$ and write $h^{(0)}$ for $h$. Note that $h^{(r)}(\alpha)$ lies in the center of $\mathsf{B}$.
\betagin{lemma}\lambdabel{lemma_hA}
$$h(A) =\sum_{i=1}^{p}\sum_{j=1}^{i} \frac{1}{(i-j)!} h^{(i-j)} E_{ij}.$$
\end{lemma}
\betagin{proof}
We start with the case of $h=x^{k}\in\mathsf{B}[x]$ for some $k\geq0$. Obviously, the claim of the lemma holds for $h=1$. Therefore, we assume that $k\geq1$. Since $A_{0}^{r}=0$ for all $r\geq p$, we have
$$h(A) = A^{k} = (xI_{p}+A_{0})^{k} = \sum_{r=0}^{\min\{k,p-1\}}\binom{k}{r}x^{k-r}A_{0}^{r}.$$
\noindent{}Part (a) of Lemma \ref{lemma_A0_B0} implies
$$h(A) = \sum_{r=0}^{\min\{k,p-1\}}\binom{k}{r}x^{k-r} \left(\sum_{i=1}^{p-r}E_{r+i,i}\right).$$
\noindent{}Regrouping the terms we obtain
\betagin{eq}\lambdabel{eq_lemma_hA}
h(A)=\sum_{i=1}^{p}\; \sum_{j=\max\{1,i-k\}}^{i} \binom{k}{i-j}x^{k-(i-j)}E_{ij}.
\end{eq}
\noindent{}Note that for $0\leq r< p$ we can rewrite
$$\frac{1}{r!} h^{(r)} = \left\{
\betagin{array}{rl}
\binom{k}{r}x^{k-r}, & \text{ if } r\leq k \\
0, & \text{ otherwise } \\
\end{array}
\right.,
$$
where $r!$ is not zero in $\mathbb{F}$. Hence equality~\mathsf{R}ef{eq_lemma_hA} implies that the claim holds for $h=x^{k}$.
The general case follows from the proven partial case and the $\mathsf{B}$-linearity of derivatives.
\end{proof}
Lemma~\ref{lemma_hA} together with the definition of $D_{\alpha}$ immediately implies the next corollary.
\betagin{cor}\lambdabel{cor_D0}
$$D_{\alpha} = \sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \frac{i}{(i-j+1)!} h^{(i-j+1)}(\alpha) E_{ij}.$$
\end{cor}
For short, denote the $(i,j)^{\rm th}$ entry of $D_{\alpha}$ by $\D{i}{j}\in Z(\mathsf{B})$ and for all $1\leq k < p$ define
\betagin{eq}\lambdabel{eq_Dak}
D_{\alpha,k} = D_{\alpha} - \sum_{r=0}^{k-1}\D{k}{k-r}A_0^{r} = D_{\alpha} - \sum_{r=0}^{k-1} \sum_{i=1}^{p-r} \D{k}{k-r}E_{r+i,i}.
\end{eq}
\noindent{}We apply the following technical lemma in the proof of key Proposition~\ref{Id(Ah) subset Id(Mp)} (see below).
\betagin{lemma}\lambdabel{Epk.Dk}
For all $1\leq r\leq p$ and $1\leq k < p$ we have
$$E_{rk}\, D_{\alpha,k}=k\, h(\alpha) E_{r,k+1}.$$
\end{lemma}
\betagin{proof}
We have
\betagin{align*}
E_{rk}D_{\alpha,k}&=\sum_{i=1}^{p-1} \sum_{j=1}^{i+1} \D{i}{j}E_{rk}E_{ij} - \sum_{j=0}^{k-1}\sum_{i=1}^{p-j} \D{k}{k-j} E_{rk} E_{i+j,i}\\
&= \sum_{j=1}^{k+1} \D{k}{j}E_{rj} - \sum_{j=1}^{k} \D{k}{j} E_{rj}\\
&= \D{k}{k+1}E_{r,k+1}.\\
\end{align*}
Equality $\D{k}{k+1} = k\, h(\alpha)$ concludes the proof.
\end{proof}
\betagin{prop}\lambdabel{Id(Ah) subset Id(Mp)}
\betagin{enumerate}
\item[(a)] Assume $\alpha\in Z(\mathsf{B})$. Then $\varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B}))$ contains $h(\alpha)^{2(p-1)} M_p(\mathsf{B})$.
\item[(b)] Assume $h(\alpha)$ is invertible in $\mathsf{B}$ for some $\alpha\in Z(\mathsf{B})$. Then $\Id{\mathsf{A}_h(\mathsf{B})}\subset \Id{M_p(\mathsf{B})}$.
\item[(c)] Assume $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$ and $\mathbb{F}$ is infinite. Then $\Id{\mathsf{A}_h(\mathsf{B})}\subset \Id{M_p(\mathsf{B})}$.
\end{enumerate}
\end{prop}
\betagin{proof} For short, we write $\beta$ for $h(\alpha)\in Z(\mathsf{B})$.
\noindent{\bf (a)}
Denote by $\mathcal{L} = \varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B})) = \alphag_{\mathsf{B}}\{I_p,C_{\alpha}, D_{\alpha}\}$ the $\mathbb{F}$-algebra generated by $\mathsf{B} I_p$, $C_{\alpha}$, $D_{\alpha}$. Since $A_0=C_{\alpha} - \alpha I_p$, we obtain that
$$A_0^{k}=\sum_{i=1}^{p-k}E_{k+i,i}\in \mathcal{L}$$
for all $0\leq k<p$. In particular, $E_{p1}=A_0^{p-1}\in\mathcal{L}$. Equality~\mathsf{R}ef{eq_Dak} implies that $D_{\alpha,k}\in\mathcal{L}$ for all $1\leq k < p$.
The statement of part (a) follows from the following claim:
\betagin{eq}\lambdabel{claim1}
\{ \beta^{p+k-r-1} E_{rk} \ | \ 1\leq r,k \leq p\}\subset \mathcal{L}.
\end{eq}
\noindent{}To prove the claim we use descending induction on $r$.
Assume $r=p$. We have $E_{p1}\in\mathcal{L}$. Lemma~\ref{Epk.Dk} implies that $E_{p1} D_{\alpha,1} = \beta E_{p2}$. Since $E_{p1}$, $D_{\alpha,1}$ belong to $\mathcal{L}$, we can see that $\beta E_{p2}\in\mathcal{L}$. Similarly, the equality $\beta E_{p2}D_{\alpha,2} = 2 \beta^2 E_{p3}$ implies $\beta^2 E_{p3}\in\mathcal{L}$. Repeating this reasoning we obtain that $\beta^{k-1} E_{pk}\in\mathcal{L}$ for all $1\leq k\leq p$.
Assume that for some $1\leq r< p$ claim~\mathsf{R}ef{claim1} holds for all $r'>r$, i.e., for every $1\leq k\leq p$ we have $\beta^{p+k-r'-1} E_{r'k}\in\mathcal{L}$. Since
$$\beta^{p-r}\left(A_0^{r-1} - \sum_{k=2}^{p-r+1} E_{(r-1)+k,k}\right) = \beta^{p-r} E_{r1},$$
we obtain $\beta^{p-r} E_{r1}\in\mathcal{L}$. Lemma~\ref{Epk.Dk} implies that $\beta^{p-r} E_{r1} D_{\alpha,1} = \beta^{p+1-r} E_{r2}$. Hence $\beta^{p+1-r} E_{r2}\in\mathcal{L}$. Repeating this reasoning we obtain that $\beta^{p+k-r-1} E_{rk} \in\mathcal{L}$ for all $1< k\leq p$, since $\beta^{p+k-r-2} E_{r,k-1} D_{\alpha,k-1} = (k-1)\, \beta^{p+k-r-1} E_{r,k}$. Claim~\mathsf{R}ef{claim1} is proven.
\noindent{\bf (b)} Since $h(\alpha)$ is invertible in $\mathsf{B}$, part (a) implies that $\varepsilon_{\alpha}(\mathsf{A}_h(\mathsf{B}))=M_p(\mathsf{B})$. Since $\varepsilon_{\alpha}$ is a homomorphism of $\mathbb{F}$-algebras, the required is proven.
\noindent{\bf (c)} Consider a polynomial identity $f\in \mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ for $\mathsf{A}_h(\mathsf{B})$. Since $\mathbb{F}$ is infinite, without loss of generality we can assume that $f$ is homogeneous with respect to the natural grading of $\mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ by degrees, i.e., each monomial of $f$ has one and the same degree $t>0$.
Part (a) implies that for every $A_1,\ldots,A_m$ from $M_p(\mathsf{B})$ there exist $a_1,\ldots,a_m$ from $A_h(\mathsf{B})$ such that
$$\beta^{2(p-1)t} f(A_1,\ldots,A_m) = f(\beta^{2(p-1)} A_1,\ldots,\beta^{2(p-1)} A_m) = f(\varepsilon_{\alpha}(a_1),\ldots,\varepsilon_{\alpha}(a_m)).$$
Since $\varepsilon_{\alpha}$ is a homomorphism of $\mathbb{F}$-algebras, we have $f(\varepsilon_{\alpha}(a_1),\ldots,\varepsilon_{\alpha}(a_m))=0$. Therefore $f$ is a polynomial identity for $\mathsf{A}_h(\mathsf{B})$ because $\beta$ is not a zero divisor.
\end{proof}
To illustrate the proof of part (a) of Proposition~\ref{Id(Ah) subset Id(Mp)}, we repeat it in the partial case of $p=3$ in the following example.
\betagin{example}\lambdabel{ex1}
Assume $p = 3$ and $h(\alpha)\neq0$ for some $\alpha\in Z(\mathsf{B})$. For short, denote $\beta=h(\alpha)$, $\beta'=h'(\alpha)$ and $\beta''=h''(\alpha)$. Then $A_0=E_{21} + E_{32}$,
$$ C_{\alpha}=\betagin{pmatrix}
\alpha& 0 & 0\\
1 & \alpha & 0\\
0 & 1 & \alpha \\
\end{pmatrix},\quad\text{ and }\quad
D_{\alpha}=\betagin{pmatrix}
\beta'&\beta&0\\
\beta''& 2\beta' & 2\beta\\
0&0&0
\end{pmatrix}.
$$
To show that $\mathcal{L} = \alphag_{\mathsf{B}}\{I_p,C_{\alpha}, D_{\alpha}\}$ contains $\beta^4 M_3(\mathsf{B})$, we consider the following elements of $\mathcal{L}$:
$$ D_{\alpha,1}=D_{\alpha} - \beta' I_3=
\betagin{pmatrix}
0 & \beta & 0\\
\beta'' & \beta' & 2 \beta\\
0 & 0 & -\beta' \\
\end{pmatrix},
$$
$$ D_{\alpha,2}=D_{\alpha} - 2 \beta' I_3 - \beta'' A_0 =
\betagin{pmatrix}
-\beta' & \beta & 0\\
0 & 0 & 2 \beta\\
0 & -\beta'' & -2 \beta' \\
\end{pmatrix}.
$$
\noindent{}Note that $A_0=C_{\alpha} - \alpha I_3$ and $A_0^2=E_{31}$ belong to $\mathcal{L}$. Since
$$E_{31} D_{\alpha,1} = \beta E_{32}\;\;\text{ and }\;\; \beta E_{32} D_{\alpha,2} = 2 \beta^2 E_{33},$$
we obtain that $ \beta E_{32},\, \beta^2E_{33}\in\mathcal{L}$. Thus $ \beta(A_0-E_{32})= \beta E_{21}$ lies in $\mathcal{L}$. Since
$$\beta E_{21} D_{\alpha,1} = \beta^2 E_{22}\;\;\text{ and }\;\; \beta^2 E_{22} D_{\alpha,2} = 2 \beta^3 E_{23},$$
we obtain
that $ \beta^2 E_{22},\, \beta^3 E_{23}\in\mathcal{L}$. Hence $ \beta^2(I_3 - E_{22} - E_{33}) = \beta^2 E_{11}$ lies in $\mathcal{L}$. Since
$$\beta^2 E_{11} D_{\alpha,1} = \beta^3 E_{12}\;\;\text{ and }\;\; \beta^3 E_{12} D_{\alpha,2} = 2 \beta^4 E_{13},$$ we obtain
that $\beta^3 E_{12},\, \beta^4 E_{13}\in\mathcal{L}$. Therefore, $\mathcal{L}$ contains $\beta^4 M_3(\mathsf{B})$.
\end{example}
\betagin{theo}\lambdabel{Teorema Principal}
Assume that $\mathbb{F}$ is an infinite field of characteristic $p>0$.
\betagin{enumerate}
\item[(a)] If $h(\alpha)$ is not a zero divisor for some $\alpha\in Z(\mathsf{B})$, then $\mathsf{A}_h(\mathsf{B})\sigmam_{\rm PI} M_p(\mathsf{B})$.
\item[(b)] If $\mathsf{B}=\mathbb{F}$, then $\mathsf{A}_h\sigmam_{\rm PI} M_p$.
\end{enumerate}
\end{theo}
\betagin{proof} Part (a) follows from Corollary \ref{Id(Mp) subset Id(Ah)} and part (c) of Proposition \ref{Id(Ah) subset Id(Mp)}.
Assume that $\mathsf{B}=\mathbb{F}$. Then there exists $\alpha\in\mathbb{F}$ with $h(\alpha)\neq0$, because $h\in\mathbb{F}[x]$ is not zero and $\mathbb{F}$ is infinite. Part (a) concludes the proof of part (b).
\end{proof}
\betagin{cor}\lambdabel{cor1}
Assume that $\mathbb{F}$ is an infinite field of characteristic $p>0$ and for $h= \eta_d x^d + \eta_{d-1} x^{d-1} + \cdots + \eta_0$ from $Z(\mathsf{B})[x]$ we have that $\eta_d$ and $\eta_0$ are not zero divisors. Then $\mathsf{A}_h(\mathsf{B})\sigmam_{\rm PI} M_p(\mathsf{B})$.
\end{cor}
\section{$\mathsf{A}_h$ over finite fields}\lambdabel{section_finite}
In this section we assume that $\mathsf{B}=\mathbb{F}$ is the field of finite of order $q=p^k$ and $\mathbb{F}\subset \mathbb{K}$ for an infinite field $\mathbb{K}$. Since $h\in\mathbb{F}[x]$, Convention~\ref{conv1} is equivalent to the inequality $h\neq0$. As above, we write $M_p$ for $M_p(\mathbb{F})$ and $\mathsf{A}_h$ for $\mathsf{A}_h(\mathbb{F})$. Given a $\mathbb{K}$-algebra $\mathcal{A}$, we write $\IdK{\mathcal{A}}$ for the ideal of $\mathbb{K}\lambdangle X\mathsf{R}A$ of polynomial identities for $\mathcal{A}$ over $\mathbb{K}$. In this section we proof the next result.
\betagin{theo}\lambdabel{theo_finite}
\betagin{enumerate}
\item[(a)] $\IdK{M_p(\mathbb{K})} \bigcap \mathbb{F}\lambdangle X\mathsf{R}A \subset \Id{\mathsf{A}_h}$.
\item[(b)] $\Id{\mathsf{A}_h} \subset \Id{M_p}$, if $h(\alpha)\neq0$ for some $\alpha\in\mathbb{F}$.
\item[(c)] $\mathsf{A}_h \not\sigmam_{\rm PI} M_p$.
\end{enumerate}
\end{theo}
\betagin{proof}
Since $\mathsf{A}_h\subset \mathsf{A}_h(\mathbb{K})= \mathsf{A}_h \otimes_{\mathbb{F}}\mathbb{K}$ as $\mathbb{F}$-algebras, we can see that
$$\Id{\mathsf{A}_h \otimes_{\mathbb{F}}\mathbb{K}}\; = \; \IdK{\mathsf{A}_h(\mathbb{K})} \cap \mathbb{F}\lambdangle X\mathsf{R}A \; \subset \; \Id{\mathsf{A}_h}.$$
Part (b) of Theorem~\ref{Teorema Principal} concludes the proof of part (a). Part (b) follows from part (b) of Proposition \ref{Id(Ah) subset Id(Mp)}.
Consider $F_{p,q}(x,y)=G_{p,q}(x)\, R_{p,q}(x,y)\, (y^{q}-y)$ of $\mathbb{F}\lambdangle x,y\mathsf{R}A$, where
\betagin{align*}
G_{p,q}(x)&=(x^{q^{2}}-x)(x^{q^{3}}-x)\cdots(x^{q^{p}}-x),\\
R_{p,q}(x,y)&= \left(1-(y\,({\rm ad}\, x)^{p-1})^{q-1}\right)
\left(1-(y\,({\rm ad}\, x)^{p-2})^{q-1}\right) \cdots
\left(1-(y\,{\rm ad}\, x)^{q-1}\right)
\end{align*}
for $y\,{\rm ad}\,x=[y,x]$.
Genov~\cite{G. Genov} proved that $F_{p,q}(x,y)$ is a polynomial identity for $M_{p}$.
Since $x\,{\rm ad}\,x = [x,x] = 0$, for $x\in\mathsf{A}_h$ we have $R_{p,q}(x,x)=1$ and $$F_{p,q}(x,x)=(x^{q}-x)(x^{q^{2}}-x)(x^{q^{3}}-x)\cdots(x^{q^{p}}-x).$$
By part (b) of Lemma~\ref{lemma_embedding} elements $x,x^2,x^3,\ldots $ are linearly independent in $\mathsf{A}_h$. Therefore, $F_{p,q}(x,x)\neq0$ in $\mathsf{A}_h$; part (c) is proven.
\end{proof}
\betagin{conj}\lambdabel{conj}
$\Id{M_p(\mathbb{K})} \bigcap \mathbb{F}\lambdangle X\mathsf{R}A = \Id{\mathsf{A}_h}$.
\end{conj}
\section{Counterexample}\lambdabel{section_example}
In this section we consider a counterexample to show that without Convention~\ref{conv1} Theorems~\ref{theo0} and~\ref{Teorema Principal} do not hold. Namely, we consider the commutative algebra $\mathsf{B}\sigmameq\mathbb{F}^2$ of double numbers from Example~\ref{ex2}, i.e., $\mathsf{B}$ has an $\mathbb{F}$-basis $\{1,\zeta\}$ with $\zeta^2=0$, and set $h=\zeta$. Note that Convention~\ref{conv1} does not hold for $h$. Then the statements of Theorems~\ref{theo0} and \ref{Teorema Principal} are not valid for $\mathsf{A}_{\zeta}(\mathbb{F}^2)=\mathsf{A}_h(\mathsf{B})$ (see Proposition~\ref{prop_ex} below).
\betagin{remark}\lambdabel{remark_ex}
If Convention~\ref{conv1} does not hold for $h$, then Lemmas~\ref{lemma_basis} and \ref{lemma_mult} are still valid for $\mathsf{A}_h(\mathsf{B})$.
\end{remark}
Part (b) of Lemma~\ref{lemma_mult} together with Remark~\ref{remark_ex} implies that for all $i,j,r,s\geq 0$ we have
\betagin{eq}\lambdabel{eq1_ex}
x^i \widehat{y}^j\cdot x^r \widehat{y}^s = x^{i+r}\, \widehat{y}^{s+j} + \zeta\, jr \, x^{i+r-1}\, \widehat{y}^{s+j-1},
\end{eq}
where we use conventions that $x^{-1}=0$ and $y^{-1}=0$. Then
\betagin{eq}\lambdabel{eq2_ex}
[x^i \widehat{y}^j, x^r \widehat{y}^s] = \zeta\, (jr-is) \, x^{i+r-1}\, \widehat{y}^{s+j-1} \;\text{ in }\; \mathsf{A}_{\zeta}(\mathbb{F}^2).
\end{eq}
The unital finite dimensional Grassmann algebra $\mathsf{G}_k$ of rank $k$ has an $\mathbb{F}$-basis
$$\{1,e_{i_1}\cdots e_{i_m} \ | \ 1\leq i_1<\cdots < i_m\leq k\}$$
and satisfies the defining relations $e_i^2=0$ and $e_i e_j = - e_j e_i$ for all $1\leq i,j\leq k$. The polynomial identities for $\mathsf{G}_k$ were described by Di Vincenzo~\cite{DiVincenzo_1991} for $p=0$ and by Giambruno, Koshlukov~\cite{GiambrunoKoshlukov_2001} for any infinite field.
\betagin{prop}\lambdabel{prop_ex} Assume that $\mathbb{F}$ is an infinite field.
\betagin{enumerate}
\item[(a)] The T-ideal of identities $\Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$ is generated by $$ f_1=[[x_1,x_2],x_3],\quad f_2=[x_1,x_2]\, [x_3,x_4].$$
\item[(b)] $\mathsf{A}_{\zeta}(\mathbb{F}^2)\not\sigmam_{\rm PI} M_t(\mathsf{C})$ for every $t\geq 2$ and every $\mathbb{F}$-algebra $\mathsf{C}$ with unity.
\item[(c)] $\mathsf{A}_{\zeta}(\mathbb{F}^2)\sigmam_{\rm PI} \mathsf{G}_k$ if and only if $k\in\{2,3\}$.
\end{enumerate}
\end{prop}
\betagin{proof}
\noindent{\bf (a)} By $\mathbb{F}$-linearity formula~\mathsf{R}ef{eq2_ex} implies that $[a,b]$ belongs to $\zeta\, \mathsf{A}_{\zeta}(\mathbb{F}^2)$ for all $a,b\in \mathsf{A}_{\zeta}(\mathbb{F}^2)$. Then $f_1,f_2\in \mathbb{F}\lambdangle X\mathsf{R}A$ are nontrivial polynomial identities for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$, since $\zeta^2=0$. Note that
$$f_3 = [x_1,x_2]\,x_3\, [x_4,x_5] = [[x_1,x_2],x_3] [x_4,x_5] + x_3 [x_1,x_2] [x_4,x_5]\in \Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$$
follows from $f_1,f_2$. Denote by $I$ the T-ideal generated by $f_1,f_2$.
Assume that $f=\sum_k \alpha_k w_k$ is a nontrivial identity for $\mathsf{A}_{\zeta}(\mathbb{F}^2)$, where $\alpha_k\in\mathbb{F}$ and $w_k\in\mathbb{F}\lambdangle x_1,\ldots,x_m\mathsf{R}A$ is a monomial. Since $\mathbb{F}$ is infinite, we can assume that $f$ is multihomogeneous. i.e., there exists $\un{d}\in\mathbb{N}^m$ with $\mathop{\rm mdeg}(w_k)=\un{d}$ for each $k$. We apply equalities
$$u x_j x_i v = u x_i x_j v - u [x_i,x_j] v,$$
$$u [x_i,x_j] v = [x_i,x_j] u v - [[x_i,x_j], u] v,$$
where monomials $u,v$ can be empty and $i<j$, to monomials $\{w_k\}$ and then repeat this procedure. Since $f_1,f_3\in \Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$, we finally obtain that there exist $g\in I$, $\alpha_0, \alpha_{ij}\in\mathbb{F}$ such that
$$f = g + \alpha_0 x_1^{d_1}\cdots x_m^{d_m} + \sum_{1\leq i<j\leq m} \alpha_{ij} [x_i,x_j] x_1^{d_1}\cdots x_i^{d_i-1} \cdots x_j^{d_j-1} \cdots x_m^{d_m} \;\text{ in }\;\mathbb{F}\lambdangle X\mathsf{R}A.$$
Since $f(1,\ldots,1) = g(1,\ldots,1) + \alpha_0$, we obtain that $\alpha_0=0$. Consider $i<j$ with $d_i,d_j\geq1$. Making substitutions $x_i\to x$, $x_j\to \widehat{y}$, $x_l\to 1$ for each $l$ different from $i$ and $j$, we can see that $0=0 + \alpha_{ij} [x,\widehat{y}] x^{d_i-1}\widehat{y}^{d_j-1}$ in $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. Thus $-\alpha_{ij} \zeta x^{d_i-1}\widehat{y}^{d_j-1}=0$ in $\mathsf{A}_{\zeta}(\mathbb{F}^2)$. Lemma~\ref{lemma_basis} together with Remark~\ref{remark_ex} implies that $\alpha_{ij}=0$. Therefore, $f=g$ lies in $I$.
\noindent{\bf (b)} Since $\mathbb{F}\subset \mathsf{C}$, every polynomial identity for $M_t(\mathsf{C})$ lies in $\Id{M_t(\mathbb{F})}$. By Amitsur--Levitzki Theorem~\cite{Amitsur_Levitzki} the minimal degree of a polynomial identity for $M_t(\mathbb{F})$ is $2t$. In particular, $f_1$ is not an identity for $M_t(\mathsf{C})$.
\noindent{\bf (c)} Since $\mathsf{G}_k$ is commutative in case $p=2$ or $k=1$, we can assume that $p\neq 2$ and $k\geq2$. Note that
\betagin{eq}\lambdabel{eq3_ex}
f_2(e_1,e_2,e_3,e_4)=4e_1 e_2 e_3 e_4 \neq 0 \;\text{ in }\; \mathsf{G}_k \;\text{ for }\; k\geq4.
\end{eq}
Thus we can assume that $k\in\{2,3\}$. The T-ideal $\Id{\mathsf{G}_k}$ is generated by
\betagin{enumerate}
\item[$\bullet$] $f_1,f_2$ in case $p=0$ or $p=k=3$.
\item[$\bullet$] $f_1,{\rm St}_4$ in case $p>k$, where $k\in\{2,3\}$.
\end{enumerate}
Since
$${\rm St}_4(x_1,x_2,x_3,x_4) = [x_1,x_2]\circ [x_3,x_4] - [x_1,x_3]\circ [x_2,x_4] + [x_1,x_4] \circ [x_2,x_3],$$
where $u\circ v$ stands for $uv+vu$, part (a) implies that ${\rm St}_4$ lies in $\Id{\mathsf{A}_{\zeta}(\mathbb{F}^2)}$. On the other hand, we can see that $f_2$ is a polynomial identity for $\mathsf{G}_k$ when $k\in\{2,3\}$. Part (c) is proven.
\end{proof}
\betagin{thebibliography}{99}
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\end{document}
\appendix
\section{Some remarks and proofs}
\betagin{remark}\lambdabel{remark0} Assume $p=0$ and $h\in \mathbb{F}^{*}$.
The classical proof of the fact that $\mathsf{A}_1$ does not have nontrivial polynomial identities is also valid for $\mathsf{A}_h$. Namely, Kaplansky's Theorem from
\betagin{enumerate}
\item[$\bullet$] I. Kaplansky, {\it Rings with a polynomial identity}, Bull. Amer. Math. Soc. {\bf 54} (1948), 575--580.
\end{enumerate}
states that any primitive $\mathbb{F}$-algebra $A$ satisfying a polynomial identity of degree $d>0$ is a simple central algebra of finite dimension and $\dim_{\mathrm{Z}(A)}A\leq[d/2]^{2}$.
Since the algebra $\mathsf{A}_h$ is simple by part (d) of Proposition \ref{Obs Prop Ah} and unitary, then it is also primitive. Moreover, ${\rm Z}(\mathsf{A}_h)=\mathbb{F}$ by part~(b) of Proposition \ref{Obs Prop Ah}. Assume that $\mathsf{A}_h$ has a nontrivial polynomial identity. Then Kaplansky's theorem implies that the dimension of $\mathsf{A}_h$ over $\mathbb{F}={\rm Z}(\mathsf{A}_h)$ is finite; a contradiction.
\end{remark}
\betagin{remark}
The matrices $A_0$ and $B_0$ of $M_p$ defined in Section~\ref{section_positive} are the following matrices:
$$A_{0}=\betagin{pmatrix}
0&0&\cdots&0&0\\
1&0&\cdots&0&0\\
0&1&\cdots&0&0\\
\vdots&\vdots&&\vdots&\vdots\\
0&0&\cdots&1&0
\end{pmatrix}\;\;\text{and}\;\; B_{0}=\betagin{pmatrix}
0&1&0&\cdots&0\\
0&0&2&\cdots&0\\
\vdots&\vdots&\vdots&&\vdots\\
0&0&0&\cdots&p-1\\
0&0&0&\cdots&0
\end{pmatrix}.$$
\end{remark}
\betagin{proof_of}{of Lemma~\ref{lemma_A0_B0}.}
\noindent{\bf (a)} [formula for $A_0^k$]
For $k = 1$ the statement holds by the definition of $A_{0}$. Suppose the statement holds for $k<p-1$. Then
\betagin{align*}
A_{0}^{k+1}&=\left(\sum_{j=1}^{p-k}E_{(j+k)j}\right)\left(\sum_{r=1}^{p-1}E_{(r+1)r} \right)=\sum_{j=1}^{p-k}E_{(j+k)(j-1)}=\sum_{i=1}^{p-(k+1)}E_{(i+k+1)i}.
\end{align*}
The claim is proven.
\noindent{\bf (b)}
\betagin{align*}
B_{0}A_{0}-A_{0}B_{0}&=\left(\sum_{i=1}^{p-1} iE_{i(i+1)} \right) \left(\sum_{j=1}^{p-1}E_{(j+1)j} \right)-\left(\sum_{j=1}^{p-1} E_{(j+1)j} \right) \left(\sum_{i=1}^{p-1} i E_{i(i+1)}\right)\\
&=\sum_{i=1}^{p-1} i E_{ii}-\sum_{i=1}^{p-1} i E_{(i+1)(i+1)}\\
&=I_{p}
\end{align*}
\end{proof_of}
\betagin{proof_of}{of Corollary~\ref{cor_D0}.}
By Lemma~\ref{lemma_hA} and the definition of $D_0$ we have
$$D_{0} = B_{0}\varepsilon_{\alpha}(h(A)) = \left(\sum_{r=1}^{p-1}r\, E_{r(r+1)} \right)
\left( \sum_{i=1}^{p}\sum_{j=1}^{i} \frac{1}{(i-j)!} h^{(i-j)}(\alpha) E_{ij} \right). $$
The required follows.
\end{proof_of}
\betagin{lemma}\lambdabel{base para Mp}
The set $\betata=\{A_{0}^{r}B_{0}^{s} \ | \ 0\leq r,s < p\}$ is a base for $M_{p}(\mathbb{F})$.
\end{lemma}
\betagin{proof}
Since $|\betata|=p^{2}=\dim_{\mathbb{F}}{M_{p}(\mathbb{F})}$ then it is sufficient to prove that $\betata $ is a linearly independent set.
Suppose that there are scalars, not all zero, $a_{rs}\in \mathbb{F}$ with $0\leq r,s<p$ such that: $$\sum_{0\leq r,s<p}a_{rs}A_{0}^{r}B_{0}^{s}=0.$$
Let $s_{0}=\min\{s \ | \ a_{rs}\neq 0 \ \ \text{for some} \ \ 0\leq r<p\}$ , then we have
\betagin{align*}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{s_{0}}+\sum_{\substack{0\leq r<p\\ s_{0}<s<p}}a_{rs}A_{0}^{r}B_{0}^{s}.\\
\intertext{Since $0\leq s_{0}<s<p$ then $0\leq p-s_{0}-1<p$ and $s+p-s_{0}-1>p-1$ hence, by parts (b) and (c) of Lemma~\ref{lemma_A0_B0}, $B_{0}^{s+p-s_{0}-1}=0$ e $B^{p-s_{0}-1}\neq 0$. thus}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}+\sum_{\substack{0\leq r<p\\ s_{0}<s<p}}a_{rs}A_{0}^{r}B_{0}^{s+p-s_{0}-1},\\
\intertext{then}
0&=\sum_{0\leq r<p}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}.
\end{align*}
Let $r_{0}=\max\{r \ | \ a_{rs_{0}\neq 0}\}$, then is immediate $a_{r_{0}s_{0}}\neq 0$. Hence
\betagin{align*}
0&=a_{r_{0}s_{0}}A_{0}^{r_{0}}B_{0}^{p-1}+\sum_{0\leq r<r_{0}}a_{rs_{0}}A_{0}^{r}B_{0}^{p-1}.\\
\intertext{As $r_{0}\geq 0$ then $p-r_{0}-1<p$, hence $A_{0}^{p-r_{0}-1}\neq 0.$ then}
0&=a_{r_{0}s_{0}}A_{0}^{p-1}B_{0}^{p-1}+\sum_{0\leq r<r_{0}}a_{rs_{0}}A_{0}^{p-r_{0}-1+r}B_{0}^{p-1}.\\
\intertext{Since $r<r_{0}$ then $p-r_{0}-1+r<p-1$, this implies that $A_{0}^{p-r_{0}-1+r}\neq A_{0}^{p-1}$. Hence}
0&= a_{r_{0}s_{0}}A_{0}^{p-1}B_{0}^{p-1}=a_{r_{0}s_{0}}(p-1)!E_{pp.}
\end{align*}
From where $a_{r_{0}s_{0}}(p-1)!=0$. Since $(p-1)!\neq 0$ module $p$, we have $a_{r_{0}s_{0}}=0$. this is a contradiction. Therefore, $\betata$ is a base for $M_{p}(\mathbb{F})$.
\end{proof}
\betagin{prop}\lambdabel{Obs Prop Ah}
\betagin{enumerate}
\item[(a)] (\cite{Benkart_Lopes_Ondrus_I}, Theorem 5.3) If $p=0$, then the center ${\rm Z}(\mathsf{A}_h)$ of $\mathsf{A}_h$ is $\mathbb{F}$; if $p>0$, then ${\rm Z}(\mathsf{A}_h)=\mathbb{F}[x^{p},h^{p}y^{p}]$.
\item[(b)] (\cite{Benkart_Lopes_Ondrus_I}, Proposition 5.5) If $p>0$, then $\mathsf{A}_h$ is a free module over ${\rm Z}(\mathsf{A}_h)$ and the set $\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$ is a basis.
\item[(c)] (\cite{Benkart_Lopes_Ondrus_I}, Corollary 7.4) The algebra $\mathsf{A}_h$ is simple if and only if $p=0$ and $h\in \mathbb{F}^{*}$.
\end{enumerate}
\end{prop}
\section{Some references}
\betagin{enumerate}
\item[(1)]
Kaplansky's Theorem (see Remark~\ref{remark0}) can be found in
\betagin{enumerate}
\item[$\diamond$] I.N. Herstein, {\it Noncommutative rings}, The Carus Math. Monographs 15, 1973 (see Theorem 6.3.2).
\item[$\diamond$] \cite{A.Gia} (see Theorem 1.11.7, page 31).
\end{enumerate}
\item[(2)] [Remark for the proof of Corollary~\ref{Id(Mp) subset Id(Ah)}].
If $R$ is an algebra over a commutative ring $C$, then $M_{n}(R)\cong R\otimes_{C}M_{n}(C)$. This remark can be found in
\betagin{enumerate}
\item[$\diamond$]L.H. Rowen, {\it Polynomial Identities in Rings Theory}, Academic Press, New York, 1980 (see Proposition 1.8.15)).
\end{enumerate}
\item[(3)] Assume that $\mathbb{F}$ is a finite field of order $q=p^{k}$. Then in
\betagin{enumerate}
\item[$\diamond$] Yu. N. Maltsev, E.N. Kuzmin, {\it A basis for identities of the algebra of second order matrices over a finite field}, Algebra Logic 17 (1978), 17-21.
\end{enumerate}
it was proved that $\Id{M_{2}(\mathbb{F})}$ is generated by
\betagin{align*}
f_{1}(x,y)&=(x-x^q)(y-y^{q^{2}})(1-[x,y]^{q-1}),\\
f_{2}(x,y)&=(x-x^q)\circ(y-y^{q^{2}})[(x-x^q)\circ(y-y^{q})]^{q},
\end{align*}
where $[x,y]=xy-yx$, $x\circ y=xy+yx$.
\end{enumerate}
\section{We do not need it!}
A noncommutative polynomial $f(x_{1},\ldots,x_{n})=\sum\alphapha_{i}x_{i_{1}}\cdots x_{i_{d}}$ of $\mathbb{F}\lambdangle x_1,\ldots,x_n\mathsf{R}A$ is called homogeneous of degree $d$, where $\alpha_i\in\mathbb{F}$. $f(x_{1},\ldots,x_{n})$ is homogeneous of multidegree $(d_{1},\ldots,d_{n})$ if each variable $x_{i}$ appears the same number of times $d_{i}$ in all monomials.
The polynomial $f(x_{1},\ldots,x_{n})$ is multilinear of degree $n$ if it homogeneous of multidegree $(1,\ldots,1$) ($n$ times), i.e.,
$$f(x_{1},\ldots,x_{n})=\sum_{\sigmagma\in S_{n}}\alphapha_{\sigmagma}x_{\sigmagma(1)}\cdots x_{\sigmagma(n)}$$ for some $\alphapha_{\sigmagma}\in\mathbb{F}$.
\betagin{remark}\lambdabel{Obs Prop Basicas PI} The following statements are well known.
\betagin{enumerate}
\item[a)] Let $A$ be an algebra over a field $\mathbb{F}$ spanned by a set $\betata$. It is easy to check that if a multilinear polynomial $f(x_{1},\cdots,x_{n})$ vanished on $\betata$, then $f(x_{1},\cdots,x_{n})$ is a polynomial identity for $A$.
\end{enumerate}
\end{remark}
Part~(c) of Proposition~\ref{Obs Prop Ah} together with Corollary 1.19 from
\betagin{enumerate}
\item[$\bullet$] V. Drensky, E. Formanek, {\it Polynomial identity rings}, CRM Barcelona, Springer Basel AG, 2004.
\end{enumerate}
implies that in case $p>0$ the algebra $\mathsf{A}_h$ have nontrivial polynomial identities.
The following result provides an important tool for study multilinear polynomial identities for the algebra $\mathsf{A}_h$.
\betagin{lemma}\lambdabel{Lema pol multilineal}
Assume $p>0$ and $f\in\FF\lambdangle X\rangle$ is a multilinear polynomial. Then $f$ is a polynomial identity for $\mathsf{A}_h$ if and only if $ f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}$ from $S=\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$.
\end{lemma}
\betagin{proof}
If $f$ is polynomial identity for $\mathsf{A}_h$, then $f(b_{1},\ldots,b_{n})=0$ for all $b_{1},\ldots,b_{n}\in \mathsf{A}_h$, in particularly $ f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in S=\{x^{i}h^{j}y^{j} \ | \ 0\leq i,j<p\}$, because $S\subset \mathsf{A}_h$. Reciprocally, from part (a) of Proposition \ref{Obs Prop Ah} it is clear that $\betata=\{x^{i}h^{j}y^{j} \ | \ i,j\geq 0\}$ is a generator set of $\mathsf{A}_h$, and since $f$ is multilinear polynomial it is sufficient to prove that $f$ vanish on $\betata$ (see Remark \ref{Obs Prop Basicas PI}, item $a)$). Let $b_{1},\ldots,b_{n}\in \betata$, then as the center of $\mathsf{A}_h$ is $\mathbb{F}[x^{p},h^{p}y^{p}]$ we have that $b_{k}=c_{k}w_{k}$, where $c_{k}\in {\rm Z}(\mathsf{A}_h)$ and $w_{k}\in S$ for all $1\leq k\leq n$. Hence, $$f(b_{1},\ldots,b_{n})=f(c_{1}w_{1},\ldots,c_{n}w_{n})=c_{1}\cdots c_{n}f(w_{1},\ldots,w_{n})=0,$$ because $f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in S$. Therefore, $f$ is polynomial identity for $\mathsf{A}_h$.
\end{proof}
The well-known Amitsur theorem (see \cite{A.Gia}, Theorem 1.7.7) states that the standard polynomial ${\rm St}_{2n}$ is a polynomial identity for $M_{n}$, the $n\times n$-matrix algebra over $\mathbb{F}$. The following example proves that the standard polynomial of degree $4$, ${\rm St}_{4}$, is a polynomial identity for $\mathsf{A}_h$ if the characteristic of $\mathbb{F}$ is $2$.
\betagin{example}
If the characteristic of $\mathbb{F}$ is 2, then $\mathsf{A}_h$ satisfies the standard identity ${\rm St}_{4}=0$.
\end{example}
\betagin{proof}
As ${\rm St}_{4}$ is a multilinear polynomial, by Lemma \ref{Lema pol multilineal}, it is sufficient to prove that ${\rm St}_{4}$ vanish on $S=\{1,x,hy,xhy\}$. Let $a_{1},a_{2},a_{3},a_{4}\in S$. Suppose $a_{i}=a_{j}$ for some $i\neq j$ then for each even permutation $\sigmagma\in S_{4}$ there is an odd permutation $\tau=\sigmagma\cdot(i \ j)$ such that $a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=a_{\tau(1)}a_{\tau(2)}a_{\tau(3)}a_{\tau(4)}$, hence \[{\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=2\sum_{\sigmagma\ par}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}.\] Then, ${\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=0$, because the characteristic of $\mathbb{F}$ is $2$. On the other hand, if $a_{i}\neq a_{j}$ for all $i\neq j$, then for each permutation $\sigmagma\in S_{4}$ there is $1\leq i\leq 4$ such that $a_{\sigmagma(i)}=1$, hence \[{\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=\sum_{\sigmagma\in S_{4}}(-1)^{\sigmagma}a_{\sigmagma(1)}a_{\sigmagma(2)}a_{\sigmagma(3)}a_{\sigmagma(4)}=4\sum_{\pi\in S_{3}} (-1)^{\pi}b_{\pi(1)}b_{\pi(2)}b_{\pi(3)},\] where $b_{1},b_{2},b_{3}\in \{x,hy,xhy\}$. As $\mathbb{F}$ has characteristic 2, ${\rm St}_{4}(a_{1},a_{2},a_{3},a_{4})=0$. Therefore, ${\rm St}_{4}$ is a polynomial identity for $\mathsf{A}_h$.
\end{proof}
\section{Open question: Is a polynomial $h(x)$ over an algebra with non divisor-zero coefficient of the highest term satisfies $h(\alpha)$ is not a divisor-zero for some $\alpha\in \mathbb{F}$? }
\noindent{\bf Question. }{\it
Assume that $\mathsf{B}$ is a unital algebra (maybe non-commutative) over an infinite field $\mathbb{F}$ and $h$ is a polynomial from $\mathsf{B}[x]$ with all coefficients from the center $Z(\mathsf{B})$.
Moreover, the coefficient of the highest term of $h$ is not a zero-divisor.
Is it true that $h(\alpha)$ is not a zero-divisor for some $\alpha\in Z(\mathsf{B})$?
}
\end{document} |
\begin{document}
\title{A Near Optimal Approximation Algorithm for Vertex-Cover Problem}
\author{Deepak Puthal\footnote{deepakpnitr@gmail.com} \\
Department of Computer Science \& Engineering\\
National Institute of Technology Silchar\\
Silchar, Assam, India}
\maketitle
\begin{abstract}
Recently, there has been increasing interest and progress in improvising the approximation algorithm for well-known NP-Complete problems, particularly the approximation algorithm for the Vertex-Cover problem. Here we have proposed a polynomial time efficient algorithm for vertex-cover problem for more approximate to the optimal solution, which lead to the worst time complexity $\Theta(V^2)$ and space complexity $\Theta(V+E)$. We show that our proposed method is more approximate with example and theorem proof. Our algorithm also induces improvement on previous algorithms for the independent set problem on graphs of small and high degree.
\end{abstract}
{\bf Keywords}:Approximation algorithm \and Vertex-Cover Problem \and Complexity \and Adjacency list.
\section{Introduction}
\label{intro}
A graph G represents as G = (V, E): V is number of vertices in graph and E is number of edges in graph and impliment as an adjacency lists or as an adjacency matrix for both directed and undirected graphs. There are two types of graph i.e. (I) Sparse graphs-those for which $|E|$ is much less than $|V|^2$ $(E<<V^2)$. (II) Dense graphs-those for which $|E|$ is close to $|V|^2 (E \simeq V^2)$. Here we presented the graph as the adjacency-list for the evaluation of our algorithm. \\
The vertex-cover problem is to find a minimum number of vertex to cover a given undirected graph. We call such a vertex cover an optimal vertex cover. This problem is the optimization version of an NP-complete decision problem. Proposed algorithm is polynomial time algorithm in order to find the set of vertex to cover the graph. Which shows the better performance than the traditional algorithm for vertex cover \cite{RefB1}.
\section{The vertex-cover problem}
As it is NP-Hard problem so it is hard to find an optimal solution of a graph G, but not difficult to find a near optimal solution. Our propose method gives very near optimal solution for Vertex-cover problem. The following approximation algorithm takes an undirected graph G as input \cite{RefB1} and returns a set of vertex to cover the graph and whose size is less than the previous method. \\
All graphs mentioned here are simple undirected graph. We follow \cite{RefB2} for definitions. Our proposed method (See algorithm and Fig.\ref{fig:2}) is on undirected graph. Here we used the adjacency list to represent graph G. We introduce a new field \textit{weight} in the \textit{list} to store the degree of each individual vertex. \textit{i.e.}
struct list
\{
\hspace{5 mm} char vertex;
\hspace{5 mm} int weight;
\hspace{5 mm} struct node *next;
\hspace{5 mm} struct node *ref;
\};
\begin{algorithm}
\caption{Approximate Vertex-Cover Algorithm}
\begin{algorithmic} [1]
\REQUIRE {{In the \textit{List} we introduce another field \textit{weight} \\ The value of \textit{weight} is number of node in reference \textit{(ref)}}}
\STATE $C^+ \leftarrow \emptyset $
\STATE L = \textit{List}
\STATE L[w] = Reference weight
\STATE (h, v) = highest weight of the list and respective vertex \label{marker}
\IF {$h \neq 0$}
\STATE $C^+ \leftarrow C^+ \cup {v}$
\STATE v[w] $\leftarrow 0$
\FOR{all vertex of \textit{List} L[ref] $\in \{v\}$}
\STATE L[w] $\leftarrow$ L[w]-1
\ENDFOR
\STATE \textbf{go to} \ref{marker}
\ELSE
\RETURN $C^+$
\ENDIF
\end{algorithmic}
\end{algorithm}
It's space complexity is $\Theta(V+E)$ \cite{RefB1}. For step 4 search in the list is O(V). In the for loop \textit{i.e. step 8 to 10} for each individual vertex need to search its reference vertices. So worst time complexity is $O(V*(V-1))$= $O(V^2)$. So the worst time complexity of the graph is $\Theta(V^2)$.
\newtheorem{mydef}{Theorem}
\begin{mydef}
\emph{(Thomas H. Cormen et.al. \cite{RefB1})}
\label{vcp1}
APPROX-VERTEX-COVER is a polynomial-time 2-approximation algorithm.
\end{mydef}
\begin{mydef}
\label{vcp2}
Proposed Approximate Vertex-Cover is a polynomial-time $(2-\varepsilon)$-approximation algorithm.
\end{mydef}
\begin{proof}
In Theorem ~\ref{vcp1} $C$ is the set of vertex for APPROX-VERTEX-COVER and $C^*$ in the optimal vertex cover $i.e. |C| \leq 2|C^*|$. In our approach we pick one vertex and remove the edges connected to that vertex. So most of the times we don't consider both end point of one edges, which followed in Theorem ~\ref{vcp1}. For our proposed method we consider the resultant set of vertex is $C^+$, then $|C^+| = |C| - \varepsilon$.\\
$ \Rightarrow |C^+| = (2- \varepsilon) |C^*|$, $0 \leq \varepsilon \leq 1$.
\qed
\end{proof}
In some cases proposed method$(C^+)$ approaches to optimal solution when $\varepsilon$ value is 1. Our method is shown in Fig. \ref{fig:1}, algorithm (Approximate Vertex-Cover), and proved in Theorem ~\ref{vcp2}. The comparison of the optimal vertex-cover, previous vertex-cover and proposed vertex-cover shown in Fig. \ref{fig:2}.
\begin{figure}
\caption{The operation of Approximate Vertex-Cover. \textit{(a)}
\label{fig:1}
\end{figure}
\begin{figure*}
\caption{Comparison of optimal, Traditional and proposed vertex cover result. \textit{(a)}
\label{fig:2}
\end{figure*}
\section{Conclusion}
Here in our proposed technique we produce the set of vertex for vertex-cover problem. Which is more near optimal solution and better than the previous technique.
\end{document} |
\begin{document}
\author{Henrik Schlichtkrull}
\address[Henrik Schlichtkrull]{Department of Mathematics, University of
Copenhagen\\ Universitetsparken 5, DK-2100 Copenhagen \O}
\email{schlicht@math.ku.dk}
\author{Peter Trapa}\thanks{The second author was supported in part by
NSF grant DMS-1302237.}
\address[Peter Trapa]{Department of Mathematics, University of Utah\\
Salt Lake City, UT 84112}
\email{ptrapa@math.utah.edu}
\author{David A. Vogan, Jr.}
\address[David Vogan]{Department of Mathematics, MIT \\ Cambridge, MA 02139}
\email{dav@math.mit.edu}
\title{Laplacians on spheres}
\date{\today}
\maketitle
\begin{abstract}
Spheres can be written as homogeneous spaces $G/H$ for compact Lie
groups in a small number of ways. In each case, the decomposition of
$L^2(G/H)$ into irreducible representations of $G$ contains
interesting information. We recall these decompositions, and see what
they can reveal about the analogous problem for noncompact real forms
of $G$ and $H$.
\end{abstract}
\noindent This paper is dedicated to Joe Wolf, in honor of all that we
have learned from him about the connections among geometry,
representation theory, and harmonic analysis; and in gratitude for
wonderful years of friendship.
\section{Introduction}\label{sec:intro}
\setcounter{equation}{0}
The sphere has a Riemannian metric, unique up to a positive scale,
that is preserved by the action of the orthogonal group. Computing the
spectrum of the Laplace operator is a standard and beautiful
application of representation theory. These notes will look at some
variants of this computation, related to interesting subgroups of the
orthogonal group.
The four variants presented in Sections \ref{sec:R}, \ref{sec:C},
\ref{sec:H}, and \ref{sec:O} correspond to the following very general
fact, due to \'Elie Cartan: if $G/K$ is an irreducible compact
Riemannian symmetric space of real rank $1$, then $K$ is
transitive on the unit sphere in $T_{eK}(G/K)$. (The only caveat is in
the case of the one-dimensional symmetric space ${S^1}$. In this case
one needs to use the full isometry group $O(2)$ rather than its
identity component to get the transitivity.) The isotropy group of
a point on the sphere is often called $M$ in the theory; so the
conclusion is that
\begin{equation}\label{e:rank1}
\text{sphere of dimension ($\dim G/K -1$)} \simeq K/M
\end{equation}
The calculations we do correspond to the rank one symmetric spaces
\begin{alignat*}{5}
&O(n+1)/O(n), && S^{n-1} \simeq O(n)/O(n-1) \qquad &&\text{(Section
\ref{sec:R}),}\\
&SU(n+1)/U(n),&& S^{2n-1} \simeq U(n)/U(n-1) &&\text{(Section \ref{sec:C}),}\\[.5ex]
&Sp(n+1)/[Sp(n)\times Sp(1)],\quad &&\begin{matrix} S^{4n-1} \simeq
[Sp(n)\times Sp(1)]/ \\[.2ex] \qquad [Sp(n-1)\times
Sp(1)_\Delta] \end{matrix} &&\text{(Section
\ref{sec:H}), and}\\[.5ex]
&F_4/\Spin(9) && S^{15} \simeq \Spin(9)/\Spin(7)' &&\text{(Section
\ref{sec:O}).}
\end{alignat*}
The representations of $O(n)$, $U(n)$, $Sp(n)\times Sp(1)$, and
$\Spin(9)$ that we are computing are exactly the $K$-types of the
spherical principal series representations for the noncompact forms of
the symmetric spaces.
Rank one symmetric spaces provide three infinite families (and one
exceptional example) of realizations of spheres as homogeneous spaces
(for compact Lie groups). A theorem due to Montgomery-Samelson and
Borel (\cite{MS} and \cite{BOct}; there is a nice account in
\cite{Wolf}*{(11.3.17)}) classifies {\em all} such realizations. In
addition to some minor variants on those above, like
$$S^{2n-1} \simeq SU(n)/SU(n-1),\qquad S^{4n-1} \simeq
Sp(n)/Sp(n-1),$$
the only remaining possibilities are
$$\begin{aligned}
S^6 \simeq G_{2,c}/SU(3) \qquad &\text{(Section \ref{sec:G2}), and}\\
S^7 \simeq \Spin(7)/G_{2,c} \qquad &\text{(Section \ref{sec:bigG2}).}\\
\end{aligned}$$
After recalling in Sections \ref{sec:R}--\ref{sec:bigG2} the classical
harmonic analysis related to these various realizations of spheres, we
will examine in Sections \ref{sec:invt}--\ref{sec:size} what these
classical results say about invariant differential operators.
In Sections \ref{sec:Opq}--\ref{sec:bigncG2} we examine what
this information about harmonic analysis on spheres can tell us about
harmonic analysis on hyperboloids. With $n=p+q$ the symmetric spaces
$$H_{p,q}=O(p,q)/O(p-1,q),\quad (0\le q\le n)$$
are said to be {\it real forms} of each other (and thus in
particular of
$$S^{n-1}=O(n)/O(n-1) = H_{n,0}).$$
Similarly, each of the realizations listed above of $S^{n-1}$ as a
non-symmetric homogeneous space for a subgroup of $O(n)$ corresponds
to one or more noncompact real forms, realizing some of the $H_{p,q}$
as non-symmetric homogeneous spaces for subgroups of $O(p,q)$. These
realizations exhibit the hyperbolic spaces as examples of real
spherical spaces of rank one, and as such our interest is primarily
with their discrete series. These and related spaces have previously been
studied by T.~Kobayashi (see \cite{Kobayashi-Stiefel,Kob,toshi:zuckerman,toshi:howe}). In Sections
\ref{sec:Upq}--\ref{sec:bigncG2} we give an essentially self-contained treatment,
in some cases giving slightly more refined information. In particular, we obtain some
interesting discrete series representations for small parameter values
for the real forms of
$S^6 \simeq G_{2,c}/SU(3)$.
\begin{subequations}\label{se:orbitmethod}
For information about real spherical spaces and their discrete series
in general we refer to \cite{KKOS}; this paper was intended in part to
examine some interesting examples of those results. In particular, we
are interested in formulating the parametrization of discrete series
in a way that may generalize as much as possible. We are very grateful
to Job Kuit for extensive discussions of this parametrization problem.
One such formulation involves the ``method of coadjoint orbits:''
representations of $G$ are parametrized by certain orbits $G\cdot
\lambda$ of $G$ on the real dual vector space
\begin{equation}\label{eq:coadjoint}
{\mathfrak g}_0^* =_{\text{def}} \Hom_{\mathbb R}(\Lie(G),{\mathbb R})
\end{equation}
(often together with additional data). The orbits corresponding to
representations appearing in $G/H$
typically have representatives
\begin{equation}\label{eq:GmodH}
\lambda \in [{\mathfrak g}_0/{\mathfrak h}_0]^*.
\end{equation}
We mention this at the beginning of the paper because this coadjoint
orbit parametrization is often {\em not} a familiar one (like that of
representations of compact groups by highest weights). We will write
something like
\begin{equation}\label{eq:orbitparam}
\pi(\text{orbit\ }\lambda,\Lambda)
\end{equation}
for the representation of $G$ parametrized by $G\cdot\lambda$ (and
sometimes additional data $\Lambda$). If $G$ is an equal-rank reductive group
and $\lambda \in {\mathfrak g}_0^*$ is a regular elliptic element
(never mind exactly what these terms mean), then
\begin{equation}\label{eq:dsparam}
\pi(\text{orbit\ }\lambda) = \text{discrete series with
Harish-Chandra parameter $i\lambda$;}
\end{equation}
so this looks like a moderately familiar parametrization. (Here
``discrete series representation'' has the classical meaning of an
irreducible summand of $L^2(G)$. Soon we will use the term more
generally to refer to summands of $L^2(G/H)$.) But notice
that \eqref{eq:dsparam} includes the case of $G$ compact. In that case
$\lambda$ is not the highest weight, but rather an exponent in the
Weyl character formula.
Here is how most of our discrete series will arise. Still for $G$
reductive, if $\lambda$ is elliptic but possibly singular, define
\begin{equation}\label{eq:cohindA}
G^\lambda = L, \qquad {\mathfrak q} = {\mathfrak l} + {\mathfrak u}
\end{equation}
to be the $\theta$-stable parabolic subalgebra defined by the
requirement that
\begin{equation}\label{eq:cohindB}
i\lambda(\alpha^\vee) > 0, \qquad (\alpha \in \Delta({\mathfrak
u},{\mathfrak h})).
\end{equation}
The ``additional data'' that we sometimes need is a one-dimensional character
\begin{equation}\label{eq:cohindC}
\Lambda \colon L \rightarrow {\mathbb C}^\times, \quad d\Lambda =
i\lambda + \rho({\mathfrak u}).
\end{equation}
(If $G^\lambda$ is connected, which is automatic if $G$ is connected
and $\lambda$ is elliptic, then $\Lambda$ is uniquely determined by
$\lambda$; the {\em existence} of $\Lambda$ is an {\em integrality}
constraint on $\lambda$.) Attached to $(\lambda,\Lambda)$ is a
cohomologically induced unitary
representation $\pi(\text{orbit\ }\lambda,\Lambda)$ satisfying
\begin{equation}\label{eq:cohindD}\begin{aligned}
\text{infinitesimal character} &= i\lambda - \rho_L = d\Lambda
-\rho.\\[1ex]
\text{lowest $K$-type} &= \Lambda - 2\rho({\mathfrak u}\cap
{\mathfrak k}) \\
&= i\lambda - \rho({\mathfrak u}\cap
{\mathfrak p}) + \rho({\mathfrak u}\cap {\mathfrak k}).
\end{aligned}
\end{equation}
If $\lambda$ is small, the formula for the lowest $K$-type can fail:
one thing that is true is that this representation of $K$ appears if
the weight is dominant for $K$.
In \cite{VZ}, the representation $\pi(\text{orbit\ }\lambda,\Lambda)$
was called $A_{\mathfrak q}(\Lambda-2\rho({\mathfrak u}))$.
If $G = K$ is compact, then
\begin{equation}\label{eq:cohindcpt}
\pi(\text{orbit\ }\lambda) = \text{repn of highest
weight\ } i\lambda + \rho({\mathfrak
u}).
\end{equation}
If this weight fails to be dominant, then (still in the
compact case) $\pi(\text{orbit\ }\lambda,\Lambda) = 0$. A confusing
but important aspect of this construction is that the same
representation of $G$ may be attached to several different coadjoint
orbits. Still for $G=K$ compact, the trivial representation is attached
to the orbit of $i\rho({\mathfrak u})$ for each of the
($2^{\text{semisimple rank($K$)}}$) different $K$ conjugacy classes of parabolic
subalgebras ${\mathfrak q}$. If we are looking at the trivial
representation inside functions on a homogeneous space $G/H$, then the
requirement \eqref{eq:GmodH} will ``prefer'' only some of these orbits:
different orbits for different $H$.
\end{subequations}
\begin{subequations}\label{se:invts}
{\bf Notational convention.} If $(\pi,V_\pi)$ is a representation of a
group $G$, and $H\subset G$ is a subgroup, we write
\begin{equation}\label{eq:invts}
(\pi^H,V_\pi^H),
\end{equation}
or often just $\pi^H$ for the subspace of $H$-fixed vectors in
$V_\pi$. If $T\in \End(V_\pi)$ preserves $V_\pi^H$, then we will write
\begin{equation}\label{eq:invtops}
\pi^H(T) =_{\text{def}} T|_{V_\pi^H}
\end{equation}
for the restriction of $T$ to the invariant vectors. This notation
may be confusing because we often write a family of representations
of $G$ as something like
\begin{equation}
\{\pi^G_s \mid s\in S\};
\end{equation}
then in the notation $[\pi^G_s]^H$, the superscripts $G$ and $H$ have
entirely different meanings. We hope that no essential ambiguity
arises in this way.
\end{subequations}
\section{The classical calculation}
\label{sec:R}
\setcounter{equation}{0}
\begin{subequations}\label{se:Rsphere}
Suppose $n\ge 1$ is an integer. Write $O(n)$ for the orthogonal group
of the standard inner product on ${\mathbb R}^n$, and
\begin{equation}
S^{n-1} = \{v\in {\mathbb R}^n \mid \langle v,v\rangle = 1\}
\end{equation}
for the $(n-1)$-dimensional sphere. We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $O(n)$ acts transitively on $S^{n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
O(n)^{e_1} \simeq O(n-1);
\end{equation}
we embed $O(n-1)$ in $O(n)$ by acting on the last $n-1$
coordinates. This shows
\begin{equation}
S^{n-1} \simeq O(n)/O(n-1).
\end{equation}
\end{subequations}
Now Frobenius reciprocity guarantees that if $H\subset G$ are compact
groups, then
\begin{equation}
L^2(G/H) \simeq \sum_{(\pi,V_\pi)\in \widehat G} V_\pi \otimes
(V_\pi^*)^H.
\end{equation}
In words, the multiplicity of an irreducible representation $\pi$ of
$G$ in $L^2(G/H)$ is equal to the dimension of the space of $H$-fixed
vectors in $\pi^*$. So understanding functions on $G/H$ amounts to
understanding representations of $G$ admitting an $H$-fixed
vector. All of the compact homogeneous spaces $G/H$ that we will
consider are {\em Gelfand pairs}, meaning that $\dim (V_{\pi^*})^H \le
1$ for every $\pi \in \widehat G$.
\begin{subequations}\label{se:Rreps}
Here's how that looks for our example. We omit the cases $n=1$ and $n=2$,
which are degenerate versions of the same thing; so assume $n\ge 3$. A
maximal torus in
$O(n)$ is
\begin{equation}
T = SO(2)^{[n/2]},
\end{equation}
so a weight is an $[n/2]$-tuple of integers. For every integer $a
\ge 0$ there is an irreducible representation $\pi^{O(n)}_a$ of highest
weight
\begin{equation}\label{eq:Rdim}
(a,0,\ldots,0), \qquad \dim \pi^{O(n)}_a = \frac{(a+n/2
-1)\prod_{j=1}^{n-3} (a+j)}{(n/2 -1)\cdot (n-3)!}.
\end{equation}
Notice that the polynomial function of $a$ giving the dimension has
degree $n-2$. One natural description of $\pi^{O(n)}_a$ is
\begin{equation}\label{eq:harmonic}
\pi^{O(n)}_a = S^a({\mathbb C}^n)/r^2 S^{a-2}({\mathbb C}^n);
\end{equation}
what we divide by is zero if $a< 2$. We will be interested in the
{\em infinitesimal characters} of the representations $\pi^{O(n)}_a$; that
is, the scalars by which elements of
\begin{equation}
{\mathfrak Z}({\mathfrak o}(n)) =_{\text{def}} U({\mathfrak
o}(n)_{\mathbb C})^{O(n)}
\end{equation}
act on $\pi^{O(n)}_a$. According to Harish-Chandra's theorem,
infinitesimal characters may be identified with Weyl group orbits of
complexified weights. The infinitesimal character of a
finite-dimensional representation of highest weight $\lambda$ is given
by $\lambda+\rho$, with $\rho$ half the sum of the positive
roots. Using the calculation of $\rho$ given in \eqref{eq:Orho}, we get
\begin{equation}\label{eq:Oinfchar}
\text{infinitesimal character}(\pi^{O(n)}_a) = (a+(n-2)/2, (n-4)/2,
(n-6)/2,\cdots).
\end{equation}
The key fact (in the notation explained in \eqref{se:invts}) is that
\begin{equation}\label{eq:Rkey}
\dim [\pi^{O(n)}_a]^{O(n-1)} = 1 \quad (a\ge 0), \qquad \dim\pi^{O(n-1)} = 0
\quad (\pi \not\simeq \pi^{O(n)}_a).
\end{equation}
Therefore
\begin{equation}\label{eq:Osphere}
L^2(S^{n-1}) \simeq \sum_{a=0}^\infty \pi^{O(n)}_a
\end{equation}
as representations of $O(n)$.
If $n=1$, the definition \eqref{eq:harmonic} of $\pi_a^{O(1)}$ is
still reasonable. Then $\pi_a^{O(1)}$ is one-dimensional if $a=0$ or
$1$, and zero for $a\ge 2$. The formula \eqref{eq:Osphere} is still
valid.
If $n=2$, the definition \eqref{eq:harmonic} of $\pi_a^{O(2)}$ is
still reasonable, and \eqref{eq:Osphere} is still valid. Then
$\pi_a^{O(2)}$ is one-dimensional if $a=0$, and two-dimensional for
$a\ge 1$.
\end{subequations}
\begin{subequations}\label{se:Rorbit}
Here is the orbit method perspective. The Lie algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-symmetric matrices; ${\mathfrak
h}_0$ is the subalgebra in which the first row and column are
zero. We can identify ${\mathfrak g}_0^*$ with ${\mathfrak g}_0$ using
the invariant bilinear form
$$B(X,Y) = \tr(XY).$$
Doing that, define
\begin{equation}
a_{\text{orbit}} = a+(n-2)/2
\end{equation}
{\small
\begin{equation}
\lambda(a_{\text{orbit}}) =\begin{pmatrix} 0& a_{\text{orbit}}/2 &
\quad 0&\dots & 0\\ -a_{\text{orbit}}/2 & 0& \quad 0 &\dots &0\\[1ex]
0&0 &\\
\vdots&\vdots & & \text{\Large $0_{(n-2)\times (n-2)}$}\\[-.5ex]
\\ 0& 0 &&&\end{pmatrix} \in ({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
The isotropy group for $\lambda(a_{\text{orbit}})$ is
\begin{equation}
O(n)^{\lambda(a_{\text{orbit}})} = SO(2)\times O(n-2)
=_{\text{def}} L.
\end{equation}
With this notation,
\begin{equation}
\pi_a^{O(n)} = \pi(\text{orbit\ } \lambda(a_{\text{orbit}})).
\end{equation}
The reason this is true is that the infinitesimal character of the
orbit method representation on the right is (by \eqref{eq:cohindD})
\begin{equation}\begin{aligned}
\lambda(a_{\text{orbit}}) -\rho_L &=
(a+(n-2)/2,-(n-4)/2,-(n-6)/2,\cdots)\\
&= \text{infinitesimal character of\ } \pi_a^{O(n)}.\end{aligned}
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a\ge 0$ but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -(n-2)/2.
\end{equation}
For the compact group $O(n)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0, \qquad 0 > a >
-(n-2)/2,
\end{equation}
(for example because the infinitesimal characters of these
representations are singular) so the difference is not important. But
matters will be more interesting in the noncompact case (Section
\ref{sec:Opq}).
\end{subequations}
Back in the general world of a homogeneous space $G/H$ for compact
groups, fix a (positive) $G$-invariant metric on ${\mathfrak g}_0 =
\Lie(G)$, and write
\begin{equation} \Omega_G = -\text{(sum of squares of an orthonormal
basis)}.
\end{equation}
for the corresponding Casimir operator. (We use a minus sign because
natural choices for the metric are negative definite rather than
positive definite.) The $G$-invariant metric on ${\mathfrak g}_0$
defines an $H$-invariant metric on
${\mathfrak g}_0/{\mathfrak h}_0 \simeq T_e(G/H)$, and therefore a
$G$-invariant Riemannian structure on $G/H$. Write
\begin{equation}
L = \text{negative of Laplace-Beltrami operator on $G/H$,}
\end{equation}
a $G$-invariant differential operator. According to
\cite{GGA}*{Exercise II.A4}, the action of $\Omega_G$ on functions on
$G/H$ is equal to the action of $L$. (The Exercise is stated for symmetric
spaces, but the proof on page 568 works in the present setting.) Consequently
\centerline{on an irreducible $G$-representation $\pi \subset
C^\infty(G/H)$,}
\centerline{$L$ acts by the scalar $\pi(\Omega_G)$.}
So we need to be able to calculate these scalars. If $T$ is a maximal
torus in $G$, and $\pi$ has highest weight $\lambda\in {\mathfrak
t}^*$, then
\begin{equation}
\pi(\Omega_G) = \langle \lambda + 2\rho,\lambda \rangle = \langle
\lambda+\rho, \lambda+ \rho\rangle - \langle \rho,\rho\rangle.
\end{equation}
Here $2\rho\in {\mathfrak t}^*$ is the sum of the positive roots. (The
second formula relates this scalar to the infinitesimal character
written in \eqref{eq:Oinfchar} above.)
\begin{subequations}\label{se:Rspec}
Now we're ready to calculate the spectrum of the spherical Laplace
operator $L$. We need to calculate
$\pi^{O(n)}_a(\Omega_{O(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Orho}
2\rho(O(n)) = (n-2, n-4,\cdots,n-2[n/2]).
\end{equation}
(Recall that we have identified weights of $T=SO(2)^{[n/2]}$ with
$[n/2]$-tuples of integers.) Because our highest weight is
\begin{equation}
\lambda = (a,0,\ldots,0),
\end{equation}
we find
\begin{equation}\label{eq:LR}
\pi^{O(n)}_a(\Omega_{O(n)}) = a^2 + (n-2)a = a_{\text{orbit}}^2 - (n-2)^2/4.
\end{equation}
\end{subequations}
\begin{theorem} \label{thm:Rspec} Suppose $n\ge 3$. The eigenvalues of
the (negative)
Laplace-Beltrami operator $L$ on
$S^{n-1}$ are $a^2 + (n-2)a$, for all non-negative integers $a$. The
multiplicity of this eigenvalue is
$$ \frac{(a+n/2 -1)\prod_{j=1}^{n-3} (a+j)}{(n/2 -1)\cdot (n-3)!},$$
a polynomial in $a$ of degree $n-2$.
\end{theorem}
In Sections \ref{sec:C}--\ref{sec:O} we'll repeat this calculation using other
groups.
\section{The complex calculation}
\label{sec:C}
\setcounter{equation}{0}
\begin{subequations}\label{se:Usphere}
Suppose $n\ge 1$ is an integer. Write $U(n)$ for the unitary group
of the standard Hermitian inner product on ${\mathbb C}^n$, and
\begin{equation}
S^{2n-1} = \{v\in {\mathbb C}^n \mid \langle v,v\rangle = 1\}
\end{equation}
for the $(2n-1)$-dimensional sphere. We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{2n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $U(n)$ acts transitively on $S^{2n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
U(n)^{e_1} \simeq U(n-1);
\end{equation}
we embed $U(n-1)$ in $U(n)$ by acting on the last $n-1$
coordinates. This shows
\begin{equation}
S^{2n-1} \simeq U(n)/U(n-1).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Ureps}
Here is the representation theory. We omit the case $n=1$,
which is a degenerate version of the same thing; so assume $n\ge 2$. A
maximal torus in
$U(n)$ is
\begin{equation}
T = U(1)^n
\end{equation}
so a weight is an $n$-tuple of integers. For all integers $b\ge 0$ and
$c\ge 0$ there is an irreducible representation $\pi^{U(n)}_{b,c}(n)$ of highest
weight
\begin{equation}
(b,0,\ldots,0,-c), \qquad \dim \pi^{U(n)}_{b,c}= \frac{(b+c+n-1)\prod_{j=1}^{n-2}
(b+j)
(c+j)}{(n -1)\cdot [(n-2)!]^2}.
\end{equation}
Notice that the polynomial giving the dimension has degree $2n-3$ in
the variables $b$ and $c$. A
natural description of the representation is
\begin{equation}
\pi^{U(n)}_{b,c} \simeq S^b({\mathbb C}^n) \otimes S^c(\overline{\mathbb
C}^n)/r^2 S^{b-1}({\mathbb C}^n) \otimes S^{c-1}(\overline{\mathbb
C}^n);
\end{equation}
what we divide by is zero if $b$ or $c$ is zero. The space is (a
quotient of) polynomial functions on ${\mathbb C}^n$, homogeneous of
degree $b$ in the holomorphic coordinates and homogeneous of degree
$c$ in the antiholomorphic coordinates.
Using the calculation of $\rho$ given in \eqref{eq:Urho} below, we
find
\begin{small}\begin{equation}\label{eq:Uinflchar}
\hbox{infl.~char.}\left(\pi^{U(n)}_{b,c}\right) = (b+(n-1)/2,(n-3)/2,\cdots,
-(n-3)/2,-(c+(n-1)/2).
\end{equation}\end{small}
The key fact (again in the notation of \eqref{se:invts}) is that
\begin{equation}\label{e:Ckey}
\dim [\pi^{U(n)}_{b,c}]^{U(n-1)} = 1 \quad (b\ge 0, \ c\ge 0),
\qquad \dim\pi^{U(n-1)} = 0 \quad (\pi \not\simeq \pi^{U(n)}_{b,c}).
\end{equation}
Therefore
\begin{equation}
L^2(S^{2n-1}) \simeq \sum_{b\ge 0, \ c\ge 0} \pi^{U(n)}_{b,c}
\end{equation}
as representations of $U(n)$.
We add one more piece of representation-theoretic information,
without explaining yet why it is useful. If we write $U(1)$ for the
multiplication by unit scalars in the first coordinate, then $U(1)$ commutes
with $U(n-1)$. In any representation of $U(n)$, $U(1)$ therefore
preserves the $U(n-1)$-fixed vectors. The last fact is
\begin{equation}\label{e:extraCreps}
\text{$U(1)$ acts on $[\pi^{U(n)}_{b,c}]^{U(n-1)}$ by the weight $b-c$.}
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Corbit}
Here is the orbit method perspective. The Lie algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-hermitian matrices; ${\mathfrak
h}_0$ is the subalgebra in which the last row and column are zero.
We can identify ${\mathfrak
g}_0^*$ with ${\mathfrak g}_0$ using the invariant bilinear form
$$B(X,Y) = \tr(XY).$$
Doing that, define
\begin{equation}
b_{\text{orbit}} = b+(n-1)/2, \qquad c_{\text{orbit}} = c+ (n-1)/2.
\end{equation}
We need also an auxiliary parameter
\begin{equation}
r_{\text{orbit}} = (b_{\text{orbit}}c_{\text{orbit}})^{1/2}.
\end{equation}
Now define a linear functional
{\small
\begin{equation}
\lambda(b_{\text{orbit}},c_{\text{orbit}}) =\begin{pmatrix} i(b_{\text{orbit}} -
c_{\text{orbit}})& r_{\text{orbit}} & \quad 0&\dots & 0\\ -r_{\text{orbit}}
& 0 & \quad 0 &\dots &0\\[1ex] 0&0 &\\ \vdots&\vdots & & \text{\Large
$0_{(n-2)\times (n-2)}$}\\[-.5ex] \\ 0& 0 &&&\end{pmatrix} \in
({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
This skew-hermitian matrix has been constructed to be orthogonal to
${\mathfrak h}_0$, and to have eigenvalues $ib_{\text{orbit}}$,
$-ic_{\text{orbit}}$, and $n-2$ zeros. Its isotropy group is (as long
as $r_{\text{orbit}} \ne 0$)
\begin{equation}
U(n)^{\lambda(b_{\text{orbit}},c_{\text{orbit}})} = U(1)
\times U(n-2) \times U(1) =_{\text{def}} L;
\end{equation}
the first and last $U(1)$ factors are not the usual ``coordinate''
$U(1)$ factors, but rather correspond to the $ib_{\text{orbit}}$ and
$-ic_{\text{orbit}}$ eigenspaces respectively.
With this notation,
\begin{equation}
\pi_{b,c}^{U(n)} = \pi(\text{orbit\ } \lambda(b_{\text{orbit}},c_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $b,c\ge 0$ but rather
\begin{equation}
b_{\text{orbit}} > 0 \iff b > -(n-1)/2, \qquad c_{\text{orbit}} > 0
\iff c > -(n-1)/2.
\end{equation}
For the compact group $U(n)$ we have
\begin{equation}\begin{aligned}
\pi(\text{orbit\ } \lambda(b_{\text{orbit}},c_{\text{orbit}}))=0
\quad &\text{if}\quad 0 > b > -(n-1)/2 \\
&\text{or}\quad 0 > c > -(n-1)/2,\end{aligned}
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:Upq}).
\end{subequations}
\begin{subequations}\label{se:Cspec}
Now we're ready for spectral theory. We need to calculate
$\pi^{U(n)}_{b,c}(\Omega_{U(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Urho}
2\rho(U(n)) = (n-1, n-3,\cdots,-(n-1)).
\end{equation}
(Recall that we have identified weights of $T=U(1)^n$ with
$n$-tuples of integers.) Because our highest weight is
\begin{equation}
\lambda = (b,0,\ldots,-c),
\end{equation}
we find
\begin{equation}\label{eq:LC}\begin{aligned}
\pi^{U(n)}_{b,c}(\Omega_{U(n)}) &= b^2 + c^2 + (n-1)(b+c)\\
&= b_{\text{orbit}}^2 + c_{\text{orbit}}^2 - (n-1)^2/2.
\end{aligned}
\end{equation}
Just as for the representation theory above, we'll add one more piece
of information without explaining why it will be useful:
\begin{equation}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(\Omega_{U(1)}) = (b-c)^2 = b^2 + c^2 - 2bc.
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraCspec}
[\pi^{U(n)}_{b,c}]^{U(n-1)}(2\Omega_{U(n)} - \Omega_{U(1)}) = (b+c)^2 +
(2n-2)(b+c).
\end{equation}
\end{subequations}
\begin{theorem} Suppose $n\ge 2$. The eigenvalues of the (negative)
Laplace-Beltrami operator $L_U$ on
$S^{2n-1}$ are $b^2 + c^2 + (n-1)(b+c)$, for all non-negative integers
$b$ and $c$. The
multiplicity of this eigenvalue is
$$ \frac{(b+c+n-1)\prod_{j=1}^{n-2}
(b+j) \prod_{k=1}^{n-2} (c+k)}{(n -1)\cdot (n-2)! \cdot (n-2)!}$$
a polynomial in $b$ and $c$ of total degree $2n-3$.
A little more precisely, the multiplicity of an eigenvalue $\lambda$
is the sum over all expressions
$$\lambda = b^2 + c^2 + (n-1)(b+c)$$
(with $b$ and $c$ nonnegative integers) of the indicated polynomial in
$b$ and $c$.
\end{theorem}
Let us compute the first few eigenvalues when $n=2$, so that we are
looking at $S^3$. Some numbers are in Table \ref{table:CLaplacian}. We
have also included eigenvalues and multiplicities from the
calculation with $O(4)$ acting on $S^3$, and the peculiar added
calculations from \eqref{e:extraCreps} and \eqref{e:extraCspec}.
\begin{table}\label{table:CLaplacian}
\caption{\bf Casimir eigenvalues and multiplicities on
$S^3$}
\begin{tabular}{|c|c|c|c|c || c | c | c|}
\hline
$b$ & $c$ & \Tstrut\Bstrut\small{$\pi^{U(n)}_{b,c}(\Omega_{U(2)})$}
&{\scriptsize $[\pi^{U(n)}_{b,c}]^{U(1)}(2\Omega_{U(2)} - \Omega_{U(1)})$}
& dim & $a$
& \small{$\pi^{O(4)}_a(\Omega_{O(4)})$} & dim \\[1ex]
\hline
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\
\hline
0 & 1 & 2 & 3 & 2 & 1 & 3 & 4\\
1 & 0 & 2 & 3 & 2 &&&\\
\hline
0 & 2 & 6 & 8 & 3 & 2 & 8 & 9\\
1 & 1 & 4 & 8 & 3&&&\\
2 & 0 & 6 & 8 & 3&&&\\
\hline
0 & 3 & 12 & 15 & 4 & 3 & 15 & 16\\
1 & 2 & 8 & 15 & 4 &&&\\
2 & 1 & 8 & 15 & 4 &&&\\
3 & 0 & 12 & 15 & 4 &&&\\
\hline
\end{tabular}
\end{table}
Since each half (left and the right) of the table concerns $S^3$,
there should be some relationship between them. There are indeed
relationships, but they are not nearly as close as one might
expect. What is being calculated in each case is the spectrum of a
Laplace-Beltrami operator. It is rather clear that the spectra are
quite different: the multiplicities calculated with $U(2)$ are
smaller than the multiplicities calculated with $O(4)$, and the actual
eigenvalues are smaller for $U(2)$ as well.
The reason for this is that metric $g_{O}$ that we
used in the $O(2n)$ calculation is not the same as the metric $g_{U}$
that we used in the $U(n)$ calculation. There are two aspects to the
difference. Recall that
\begin{equation}\label{eq:Rtan}
T_{e_1}(S^{n-1}) = \{(0,v_2,\cdots,v_{n}) \mid v_j \in {\mathbb
R}\} \simeq {\mathbb R}^{n-1}.
\end{equation}
In this picture, we will see that $g_{O}$ is the usual inner product
on ${\mathbb R}^{n-1}$. In the $U(n)$ picture,
\begin{equation}\label{eq:T2n-1}
T_{e_1}(S^{2n-1}) = \{(it_1,z_2,\cdots,z_{n}) \mid t\in {\mathbb
R},\ z_j \in {\mathbb C}\} \simeq {\mathbb R} + {\mathbb C}^{n-1}.
\end{equation}
In this picture, $g_U$ is actually {\em twice} the usual inner product
on ${\mathbb C}^{n-1}$:
\begin{equation}\label{eq:gOU}
|(0,x_2+iy_2,\cdots,x_n+iy_n)|^2_{g_U} =
2|(0,0,x_2,y_2,\cdots,x_n,y_n)|^2_{g_O}.
\end{equation}
Here is how to see this factor of two. The
Riemannian structure $g_O$ for $O(n)$ is
related to the invariant bilinear form on ${\mathfrak o}(n)$
\begin{equation}\label{eq:Oform}
\langle X,Y\rangle_{O(n)} = (1/2)\tr(XY).
\end{equation}
The reason for the
factor of $1/2$ is so that the form restricts to (minus) the ``standard''
inner product on the Cartan subalgebra ${\mathfrak s}{\mathfrak
o}(2)^{[n/2]} \simeq {\mathbb R}^{[n/2]}$. Now suppose that
$$v\in {\mathbb R}^{n-1} \simeq T_{e_1}(S^{n-1}).$$
The tangent vector $v$ is given by the $n\times n$ skew-symmetric
matrix $A(v)$ with first row $(0,v)$, first column $(0,-v)^t$, and all
other entries zero. Then
\begin{equation}\label{eq:gO}
|v|^2_{g_O} = -\langle A(v),A(v)\rangle_{O(n)} = -(1/2)(\tr(A(v)A(v)))
= |v|^2,
\end{equation}
proving the statement after \eqref{eq:Rtan} about $g_0$.
For similar reasons, $g_U$ is related
to the invariant form on ${\mathfrak u}(n)$
\begin{equation}\label{eq:Uform1}
\langle Z,W\rangle_{U(n)} = \RE\tr(ZW) = (1/2)(\tr(ZW) +
\overline{\tr(ZW)}).
\end{equation}
If $z\in {\mathbb C}^{n-1} \subset T_{e_1}(S^{2n-1})$, then the
tangent vector $z$ is given by the $n\times n$ skew-Hermitian matrix $B(z)$
with first row $(0,z)$, first column $(0,-\overline{z})^t$, and all
other entries zero. Therefore
\begin{equation}\label{eq:gU}
|z|^2_{g_U} = -\langle B(z),B(z)\rangle_{U(n)} = -\RE(\tr(B(z)B(z)) =
2|z|^2.
\end{equation}
Now equations \eqref{eq:gO} and \eqref{eq:gU} prove \eqref{eq:gOU}
Doubling the Riemannian metric has the effect of
dividing the Laplace operator by two, and so dividing the eigenvalues
by two. For this reason, the eigenvalues computed using $U(n)$ ought
to be half of those computed using $O(2n)$.
But that is still not what the table says. The reason is that in the
$U(n)$ picture, there is a ``preferred'' line in each tangent space,
corresponding to the fibration
$$S^1 \rightarrow S^{2n-1} \rightarrow {\mathbb C}{\mathbb P}^{n-1}.$$
In our coordinates in \eqref{eq:T2n-1}, it is the coordinate
$t_1$. The skew-Hermitian matrix $C(it_1)$ involved has $it_1$ in the first
diagonal entry, and all other entries zero.
\begin{equation}
|(it_1,0,\cdots,0)|^2_{g_U} = -\langle C(it_1),C(it_1)\rangle_{U(n)}
= t_1^2 = |(0,t_1,0,0,\cdots,0,0)|^2_{g_O}:
\end{equation}
no factor of two. So the metric attached to the $U(n)$ action is
fundamentally different from the metric attached to the $O(2n)$
action. In the $U(n)$ case, there is a new (non-elliptic) Laplacian $L_{U(1)}$
acting in the direction of the $S^1$ fibration only. The remarks about
metrics above say that
\begin{equation}\label{eq:LOU}
L_O = 2L_U - L_{U(1)}.
\end{equation}
(The reason is that the sum of squares of derivatives in $L_O$ is
almost exactly twice the sum of squares $L_U$; except that this factor
of two is not needed in the direction of the $U(1)$ fibration.)
The ``extra'' calculations \eqref{e:extraCreps} and
\eqref{e:extraCspec} are calculating the spectrum of
$L_{U(1)}$ representation-theoretically; so the column
$$[\pi^{U(n)}_{b,c}]^{U(1)}(2\Omega_{U(2)} - \Omega_{U(1)})$$
in the table above is calculating the spectrum of the classical Laplacian
$L_O$.
Here is a final representation-theoretic statement, explaining how the
$U(n)$ and $O(2n)$ calculations fit together.
\begin{theorem} \label{thm:OUcptbranch} Suppose $n\ge 2$, and $a$ is a
non-negative integer. Using the inclusion $U(n)\subset O(2n)$, we have
$$\pi^{O(2n)}_a|_{U(n)} = \sum_{\substack{0\le b,c \\[.1ex] b+c = a}} \pi^{U(n)}_{b,c}.$$
The contribution of these representations to the spectrum of the
$O(2n)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(2n)}_a(\Omega_{O(2n)}) &= a^2 + (2n-2)a \\
& = (b+c)^2 + 2(n-1)(b+c) \\
&= [\pi^{U(n)}_{b,c}]^{U(n-1)}(2\Omega_{U(n)} - \Omega_{U(1)}).\end{aligned}$$
\end{theorem}
\section{The quaternionic calculation}
\label{sec:H}
\setcounter{equation}{0}
\begin{subequations}\label{se:Hsphere}
Suppose $n\ge 1$ is an integer. Write $Sp(n)$ for the unitary group
of the standard Hermitian inner product on ${\mathbb H}^n$. This is a
group of ${\mathbb H}$-linear transformations; that is, ${\mathbb
R}$-linear transformations commuting with scalar multiplication by
${\mathbb H}$. Because ${\mathbb H}$ is noncommutative, these scalar
multiplications do {\em not} commute with each other, and so are {\em
not} linear. It is therefore possible and convenient to enlarge
$Sp(n)$ to
\begin{equation}\label{eq:Spbig}
Sp(n) \times Sp(1) = Sp(n)_{\text{linear}} \times
Sp(1)_{\text{scalar}};
\end{equation}
the second factor is scalar multiplication by unit quaternions. This
enlarged group acts on ${\mathbb H}^n$, by the
formula
\begin{equation}
(g_{\text{linear}},z_{\text{scalar}})\cdot v = gvz^{-1};
\end{equation}
we need the inverse to make the right action of scalar multiplication
into a left action. The action preserves length, and so can be
restricted to the $(4n-1)$-dimensional sphere
\begin{equation}
S^{4n-1} = \{v\in {\mathbb H}^n \mid \langle v,v\rangle = 1\}
\end{equation}
We choose as a base point
\begin{equation}
e_1 = (1,0,\ldots,0) \in S^{4n-1},
\end{equation}
which makes sense by our assumption that $n\ge 1$.
Then $Sp(n)\times Sp(1)$ acts transitively on $S^{2n-1}$, and the isotropy group at
$e_1$ is
\begin{equation}
[Sp(n)\times Sp(1)]^{e_1} \simeq Sp(n-1) \times Sp(1)_\Delta.
\end{equation}
Here we embed $Sp(n-1)$ in $Sp(n)$ by acting on the last $n-1$
coordinates, and the last factor is the diagonal subgroup in
$Sp(1)_{\text{linear}}$ (acting on the first coordinate) and
$Sp(1)_{\text{scalar}}$. This shows
\begin{equation}
S^{4n-1} \simeq [Sp(n)\times Sp(1)]/[Sp(n-1) \times Sp(1)_\Delta].
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Hreps}
Here is the representation theory. We omit the case $n=1$,
which is a degenerate version of the same thing; so assume $n\ge 2$. A
maximal torus in
$Sp(n)$ is
\begin{equation}
T = U(1)^n,
\end{equation}
$n$ copies of the unit complex numbers acting diagonally on ${\mathbb
H}^n$. A weight is therefore an $n$-tuple of integers. For all
integers $d\ge e\ge 0$ there is an irreducible representation
\begin{equation}\begin{aligned}
\pi^{Sp(n)}_{d,e} &\text{\ of highest weight\ } (d,e,0,\ldots,0,0), \\[.3ex]
\dim \pi^{Sp(n)}_{d,e}&= \frac{(d+e+2n-1)(d-e+1)\prod_{j=1}^{2n-3}
(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}.
\end{aligned}\end{equation}
A maximal torus in $Sp(1)$ is $U(1)$, and a weight is an integer. For
each integer $f\ge 0$ there is an irreducible representation
\begin{equation}
\pi^{Sp(1)}_f \text{\ of highest weight\ } f,\quad \dim \pi^{Sp(1)}_f =
f+1.
\end{equation}
We are interested in the representations (for $d\ge e \ge 0$)
\begin{small}
\begin{equation}\begin{aligned}
\pi^{Sp(n)\times Sp(1)}_{d,e} &= \pi^{Sp(n)}_{d,e}\otimes
\pi^{Sp(1)}_{d-e} \\
\dim \pi^{Sp(n)\times Sp(1)}_{d,e}&= \frac{(d+e+2n-1)(d-e+1)^2\prod_{j=1}^{2n-3}
(d+j+1)(e+j)}{(2n -1)(2n-2)\cdot [(2n-3)!]^2}.
\end{aligned}\end{equation}\end{small}
Notice that the polynomial giving the dimension has degree $4n-3$.
Using the calculation of $\rho$ given in \eqref{eq:Hrho} below, we
find
\begin{small}\begin{equation}\label{eq:Hinflchar}
\hbox{infl.~char.}(\pi^{Sp(n)\times Sp(1)}_{d,e}) =
(d+n,e+(n-1),n-2,\cdots,1)(d-e+1).
\end{equation}\end{small}
The key fact is that
\begin{equation}\label{e:Hkey}\begin{aligned}
\dim [\pi^{Sp(n)\times Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta} &= 1 \qquad (d\ge e \ge 0),\\
\dim\pi^{Sp(n-1)\times Sp(1)_\Delta} &= 0 \quad (\pi \not\simeq
\pi^{Sp(n)\times Sp(1)}_{d,e}).
\end{aligned}
\end{equation}
Therefore
\begin{equation}
L^2(S^{4n-1}) \simeq \sum_{d\ge e \ge 0} \pi^{Sp(n)\times Sp(1)}_{d,e}
\end{equation}
as representations of $Sp(n)\times Sp(1)$.
Here is one more piece of representation-theoretic information. We saw that
$Sp(n-1)\times Sp(1)_\Delta \subset Sp(n-1)\times Sp(1)\times Sp(1)
\subset Sp(n) \times Sp(1)$; so inside any
representation of $Sp(n)\times Sp(1)$ we get a natural representation of
$Sp(1)\times Sp(1)$ generated by the $Sp(n-1)\times Sp(1)_\Delta$
fixed vectors. The last fact is
\begin{equation}\label{e:extraHreps}\begin{aligned}
\ [Sp(1)\times Sp(1)] &\cdot [\pi^{Sp(n)\times
Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta} \\ &= \text{irr of
highest weight $(d-e,d-e)$.}
\end{aligned}\end{equation}
This representation has infinitesimal character
\begin{equation}\label{eq:Hsubinflchar}\begin{aligned}
\hbox{infl.~char.}\big([Sp(1)\times Sp(1)]&\cdot[\pi^{Sp(n)\times
Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta}\big) \\ &= (d-e+1,d-e+1).
\end{aligned}\end{equation}
\end{subequations}
\begin{subequations}\label{se:Horbit}
Here is the orbit method perspective. (To simplify the notation, we
will discuss only $G=Sp(n)$ rather than $Sp(n) \times Sp(1)$.) The Lie
algebra ${\mathfrak
g}_0$ consists of $n\times n$ skew-hermitian quaternionic matrices;
${\mathfrak h}_0$ is the subalgebra in which the last row and column are zero.
Define
\begin{equation}
d_{\text{orbit}} = d+(n-1), \qquad e_{\text{orbit}} = e+ (n-2).
\end{equation}
We need also an auxiliary parameter
\begin{equation}
r_{\text{orbit}} = (d_{\text{orbit}}e_{\text{orbit}})^{1/2}.
\end{equation}
Now define a linear functional
{\small
\begin{equation}
\lambda(d_{\text{orbit}},e_{\text{orbit}}) =\begin{pmatrix} i(d_{\text{orbit}} +
e_{\text{orbit}})& r_{\text{orbit}} & \quad 0&\dots & 0\\ -r_{\text{orbit}}
& 0 & \quad 0 &\dots &0\\[1ex] 0&0 &\\ \vdots&\vdots & & \text{\Large
$0_{(n-2)\times (n-2)}$}\\[-1ex] \\ 0& 0 &&&\end{pmatrix} \in
({\mathfrak g}_0/{\mathfrak h}_0)^*.
\end{equation}}
This skew-hermitian matrix has been constructed to be orthogonal to
${\mathfrak h}_0$, and to be conjugate by $G$ to
\begin{equation}
\begin{pmatrix} id_{\text{orbit}}& 0 &0& \dots & 0\\
0& ie_{\text{orbit}}& 0& \dots & 0\\[1ex] 0& 0 &&\\ \vdots& \vdots&
& \text{\Large $0_{(n-2)\times(n-2)}$}\\
0&0 \end{pmatrix}
\end{equation}
With this notation,
\begin{equation}
\pi_{d,e}^{Sp(n)} = \pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $d\ge e\ge 0$ but rather
\begin{equation}
d_{\text{orbit}} > e_{\text{orbit}} > 0 \iff d + 1 > e > -(n-1).
\end{equation}
For the compact group $Sp(n)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(d_{\text{orbit}},e_{\text{orbit}}))=0
\quad \text{if}\quad 0 > e > -(n-1)
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:Sppq}).
\end{subequations}
\begin{subequations}\label{se:Hspec}
Now we're ready for spectral theory. Because the group is a
product, it is natural to calculate the eigenvalues of the Casimir
operators from the two factors separately. We calculate first
$\pi^{Sp(n)\times Sp(1))}_{d,e}(\Omega_{Sp(n)})$. The sum of the positive roots is
\begin{equation}\label{eq:Hrho}
2\rho(Sp(n)) = (2n, 2n-2,\cdots,2).
\end{equation}
Because our highest weight for $Sp(n)$ is
\begin{equation}
\lambda = (d,e,0,\ldots,0),
\end{equation}
we find
\begin{equation}\label{eq:LH}\begin{aligned}
\pi^{Sp(n)}_{d,e}(\Omega_{Sp(n)}) &= d^2 + e^2 + 2nd + 2(n-1)e\\
&=d_{\text{orbit}}^2 + e_{\text{orbit}}^2 - n^2 - (n-1)^2. \end{aligned}
\end{equation}
Similarly
\begin{equation}
\pi^{Sp(n)\times Sp(1)}_{d,e}(\Omega_{Sp(1)}) = (d-e)^2 + 2(d-e) = d^2 + e^2 -
2de + 2(d-e).
\end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraHspec}\begin{aligned}
\pi^{Sp(n)\times Sp(1)}_{d,e}(2\Omega_{Sp(n)} - \Omega_{Sp(1)}) &= (d+e)^2 +
(4n-2)(d+e)\\ &=\pi^{O(4n)}_{d+e}(\Omega_{O(4n)}).
\end{aligned}\end{equation}
This formula is first of all just an algebraic identity, obtained by
plugging in $a=d+e$ and $4n$ in the formula \eqref{eq:LR}. But it has
a more serious meaning. Let us directly compare the metrics $g_O$ and
$g_{Sp}$ on $S^{4n-1}$, as we did for $g_{U}$ in Section
\ref{sec:C}. We find that on a $(4n-4)$-dimensional subspace of the
tangent space, $g_O$ is some multiple $x\cdot g_{Sp}$; and on the
orthogonal $3$-dimensional subspace (corresponding to the $Sp(1)\simeq
S^3$ fibers of the bundle $S^{4n-1} \rightarrow {\mathbb
P}^{n-1}({\mathbb H})$) there is a different relationship $g_O =
y\cdot g_{Sp}$. (It is not difficult to check by more careful
calculation that $x=2$ and $z=1$, but we are looking here for what is
obvious.) It follows that
$$ L_{O} = xL_{Sp} - zL_{Sp(1)},$$
exactly as in \eqref{eq:LOU}. If now
$$\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_a^{O(4n)},$$
then we conclude (by computing the Laplacian separately in these two
representations) that there is (for all integers $d\ge e \ge 0$) an
algebraic identity
$$ x(d^2+e^2+2nd +2(n-1)e) - z((d-e)^2 + 2(d-e)) = a^2 + (4n-2)a;$$
here $a\ge 0$ is some integer depending on $d$ and $e$. Since every
integer $a \ge 0$ must appear in such an identity, it follows easily
that $x=2$ and $z=1$, and that $a=d+e$. In particular,
\begin{equation}\label{eq:LOSp}
L_{O} = 2L_{Sp} - L_{Sp(1)}.
\end{equation}
This means that the equation \eqref{e:extraHspec} is describing two
calculations of $L_O$, in the subrepresentation
\begin{equation}
\pi_{d,e}^{Sp(n)\times Sp(1)} \subset \pi_{d+e}^{O(4n)}.
\end{equation}
\end{subequations}
Here is what we have proven about how the $Sp(n)$ and $O(4n)$
calculations fit together.
\begin{theorem} \label{thm:OSpcptbranch} Suppose $n\ge 2$, and $a$ is
a non-negative integer. Using the map $Sp(n)\times
Sp(1)\rightarrow O(4n)$, we have
$$\pi^{O(4n)}_a|_{Sp(n)\times Sp(1)} = \sum_{\substack{d\ge e \ge 0 \\ d+e= a}}
\pi^{Sp(n)\times Sp(1)}_{d,e}.$$
The contribution of these representations to the spectrum of the
$O(4n)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(4n)}_a(\Omega_{O(4n)}) &= a^2 + (4n-2)a \\
& = (d+e)^2 + (4n-2)(d+e) \\
&= \pi^{Sp(n)\times Sp(1)}_{d,e}(2\Omega_{Sp(n)} - \Omega_{Sp(1)}).\end{aligned}$$
\end{theorem}
\section{The octonionic calculation}
\label{sec:O}
\setcounter{equation}{0}
We will make no explicit discussion of octonions, except to say that
$F_4$ is related; and that the non-associativity of octonions makes it
impossible to define a ``projective space'' except in octonionic
dimension one. That is why this example is not part of an infinite
family like the real, complex, and quaternionic ones.
\begin{subequations}\label{se:Osphere}
Write $\Spin(9)$ for the compact spin double cover of $SO(9)$. This
group can be defined using a spin representation $\sigma$, which has dimension
$2^{(9-1)/2} = 16$. The representation is real, so we fix a
realization $(\sigma_{\mathbb R},V_{\mathbb R})$ on a
sixteen-dimensional real vector space. Of course the compact group
$\Spin(9)$ preserves a positive definite inner product on $V_{\mathbb
R}$, and
\begin{equation}
S^{15} = \{v\in V_{\mathbb R} \mid \langle v,v\rangle = 1\}
\end{equation}
We choose as a base point
\begin{equation}
v_1 \in S^{15},
\end{equation}
Then $\Spin(9)$ acts transitively on $S^{15}$. (Once one knows that
$F_{4,c}/\Spin(9)$ is a (sixteen-dimensional) rank one Riemannian
symmetric space, and that the action of $\Spin(9)$ on the tangent
space at the base point is the spin representation, then this is
Cartan's result \eqref{e:rank1}.) The isotropy group at $v_1$ is
\begin{equation}
\Spin(9)^{v_1} \simeq \Spin(7)'.
\end{equation}
The embedding of $\Spin(9)^{v_1}$ in $\Spin(9)$ can be described as follows.
First, we write
\begin{equation}
\Spin(8)\subset \Spin(9)
\end{equation}
for the double cover of $SO(8) \subset SO(9)$. Next, we embed
\begin{equation}
\Spin(7)' \ {\buildrel{\text{spin}}\over \longrightarrow}\ \Spin(8).
\end{equation}
(We use the prime to distinguish this subgroup from the double cover
of $SO(7) \subset SO(8)$, which we will call $\Spin(7) \subset
\Spin(8)$.) The way this works is that the spin representation of
$\Spin'(7)$ has dimension $2^{(7-1)/2}= 8$, is real, and preserves a
quadratic form, so $\Spin'(7) \subset SO(8)$. (Another explanation
appears in \eqref{se:bigG2sphere} below.) Now take the double cover
of this inclusion. This shows
\begin{equation}
S^{15} \simeq \Spin(9)/\Spin(7)'.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Oreps}
Here is the representation theory. A maximal torus in $\Spin(9)$ is a
double cover of $SO(2)^4 \subset SO(9)$. A weight is {\em either} a
$4$-tuple of integers (the weights factoring to $SO(2)^4$) {\em or} a
$4$-tuple from ${\mathbb Z} + 1/2$.
For all integers $x\ge 0$ and $y \ge 0$ there is an irreducible representation
\small
\begin{equation}\begin{aligned}
\pi^{\Spin(9)}_{x,y} &\text{\ of highest weight\ } (y/2 +
x,y/2,y/2,y/2),\\[.3ex]
\dim \pi^{\Spin(9)}_{x,y} &=
\frac{(2x+y+7)\prod_{j=1}^3
(x+j)(y+j+1)(y+2j-1)(x+y+j+3)}{7!\cdot 6! \cdot(1/2)}
\end{aligned}\end{equation}
\normalsize
Notice that the polynomial giving the dimension has degree $13$.
Using the calculation of $\rho$ given in \eqref{eq:octrho} below, we
find
\begin{small}
\begin{equation}\label{eq:octinflchar}
\hbox{infl.~char.}(\pi^{\Spin(9)}_{x,y}) =
((2x+y+7)/2,(y+5)/2,(y+3)/2,(y+1)/2).
\end{equation}
\end{small}
The key fact is that
\begin{equation}\label{e:Okey}\begin{aligned}
\dim [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'} &= 1 \qquad (x\ge 0,\ y \ge
0),\\
\dim\pi^{\Spin(7)'} &= 0 \quad (\pi \not\simeq \pi^{\Spin(9)}_{x,y}).
\end{aligned}\end{equation}
Therefore
\begin{equation}
L^2(S^{15}) \simeq \sum_{x\ge 0,\ y \ge 0} \pi^{\Spin(9)}_{x,y}
\end{equation}
as representations of $\Spin(9)$.
Here is one more piece of representation-theoretic information. We saw that
$\Spin(7)'\subset\Spin(8) \subset \Spin(9)$; so inside any
representation of $\Spin(9)$ we get a natural representation of
$\Spin(8)$ generated by the $\Spin(7)'$ fixed vectors. The last fact is
\begin{small}\begin{equation}\label{e:extraOreps}
\Spin(8)\cdot [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'} = \text{irr of
highest weight $(y/2,y/2,y/2,y/2)$.}
\end{equation}\end{small}
This representation has infinitesimal character
\begin{small}\begin{equation}\label{eq:octsubinflchar}
\hbox{infl.~char.}\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}\right) =
((y+6)/2,(y+4)/2,(y+2)/2,y/2).
\end{equation}\end{small}
Here is why this is true. Helgason's theorem about symmetric spaces says that
the representations of $\Spin(8)$ of highest weights
\begin{equation}\label{eq:87'}
(y/2,y/2,y/2,y/2)
\end{equation}
are precisely the ones having a $\Spin(7)'$-fixed
vector, and furthermore this fixed vector is unique. The
corresponding statement for $\Spin(8)/\Spin(7)$ is the case $n=8$ of
Theorem \ref{thm:Rspec}. In that case the highest weights for
$\Spin(8)$ appearing are the multiples of the fundamental weight
$(1,0,0,0)$ (corresponding to the simple root at the end of the
``long'' leg of the Dynkin diagram of $D_4$. For
$\Spin(8)/\Spin(7)'$, the weights appearing must therefore be
multiples of the fundamental weight $(1/2,1/2,1/2,1/2)$ for a simple
root on one of the ``short'' legs of the Dynkin diagram, proving
\eqref{eq:87'}.
To complete the proof of \eqref{e:Okey} using \eqref{eq:87'} we need only the
classical branching theorem for $\Spin(8) \subset \Spin(9)$ (see for example
\cite{KBeyond}*{Theorem 9.16}).
\end{subequations}
\begin{subequations}\label{se:Oorbit}
Here is the orbit method perspective. Define
\begin{equation}
x_{\text{orbit}} = x+ 2, \qquad y_{\text{orbit}} = y+3.
\end{equation}
Then it turns out that there is a $9\times 9$ real skew-symmetric
matrix $\lambda(x_{\text{orbit}}, y_{\text{orbit}})$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) &\in ({\mathfrak
g}_0/{\mathfrak h}_0)^*\\
\lambda(x_{\text{orbit}},y_{\text{orbit}}) &\ \text{has eigenvalues}\\
&\ \text{$\pm i(x_{\text{orbit}}/2+y_{\text{orbit}}/4)$ and
$\pm i(y_{\text{orbit}}/4)$ (three times).} \end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_{x,y}^{\Spin(9)} = \pi(\text{orbit\ }
\lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $x,y \ge 0$ and but rather
\begin{equation}
x_{\text{orbit}},\ y_{\text{orbit}} > 0 \iff x > -2, y > -3.
\end{equation}
For the compact group $\Spin(9)$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(x_{\text{orbit}},y_{\text{orbit}}))=0
\quad \text{if}\quad 0 > x > -2 \ \text{or} \ 0 > y > -3;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:spinp9-p}).
\end{subequations}
\begin{subequations}\label{se:Ospec}
Now we're ready for spectral theory. We need to calculate
$\pi^{\Spin(9)}_{x,y}(\Omega_{\Spin(9)})$. The sum of the positive roots is
\begin{equation}\label{eq:octrho}
2\rho(\Spin(9)) = (7,5,3,1).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (y/2 + x,y/2,y/2,y/2),
\end{equation}
we find
\begin{equation}\label{eq:LO}
\pi^{\Spin(9)}_{x,y}(\Omega_{\Spin(9)}) = x^2 + y^2 + xy +8y + 7x.
\end{equation}
Just as for the representation theory above, we'll add one more piece
of information without explaining why it will be useful:
\begin{equation}
\left(\Spin(8) \cdot [\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}
\right)(\Omega_{\Spin(8)}) = y^2 + 6y \end{equation}
Combining the last two equations gives
\begin{equation}\label{e:extraOspec}\begin{aligned}
\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}\right)(4\Omega_{\Spin(9)} -
3\Omega_{\Spin(8)}) &= (2x+y)^2 + 14(2x+y)\\
&=\pi^{O(16)}_{2x+y}(\Omega_{O(16)}).
\end{aligned}\end{equation}
The last equality can be established exactly as in
\eqref{e:extraHspec}.
\end{subequations}
Here is how the $\Spin(9)$ and $O(16)$ calculations fit together.
\begin{theorem}\label{thm:Ospin9cptbranch} Using the inclusion
$\Spin(9) \subset O(16)$ given by the spin representation, we have
$$\pi_a^{O(16)}|_{\Spin(9)} = \sum_{\substack{x\ge 0,\ y\ge 0
\\[.2ex] 2x+y=a}} \pi_{x,y}^{\Spin(9)}.$$
The contribution of these representations to the spectrum of the
$O(16)$-invariant Laplacian $L_O$ is
$$\begin{aligned}
\pi^{O(16)}_a(\Omega_{O(16)}) &= a^2 + (16-2)a\\ &= (2x+y)^2 +
14(2x+y) \\
&=
\left(\Spin(8)\cdot[\pi^{\Spin(9)}_{x,y}]^{\Spin(7)'}]\right)(4\Omega_{\Spin(9)}
- 3\Omega_{\Spin(8)}).\end{aligned}$$
\end{theorem}
\section{The small $G_2$ calculation}
\label{sec:G2}
\setcounter{equation}{0}
\begin{subequations}\label{se:G2sphere}
Write $G_{2,c}$ for the $14$-dimensional compact connected Lie group
of type $G_2$. There is a $7$-dimensional real representation
$(\tau_{\mathbb R},W_{\mathbb R})$ of
$G_{2,c}$, whose (complexified) weights are zero and the six short
roots. The representation $\tau_{\mathbb R}$ preserves a positive
definite inner product, and so defines inclusions
\begin{equation}
G_{2,c} \hookrightarrow SO(W_{\mathbb R}), \qquad G_{2,c} \hookrightarrow
\Spin(W_{\mathbb R}).
\end{equation}
The corresponding action of $G_{2,c}$ on $S^6$ is transitive. An
isotropy group is isomorphic to $SU(3)$; this is a subgroup generated
by a maximal torus and the long root $SU(2)$s. Therefore
\begin{equation}
S^{6} = \{w\in W_{\mathbb R} \mid \langle v,v\rangle = 1\} \simeq G_{2,c}/SU(3).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:G2reps}
Here is the representation theory. Having identified a subgroup of
$G_{2,c}$ with $SU(3)$, we may as well take for our maximal torus in
$G_{2,c}$ the diagonal torus
\begin{equation}
T = S(U(1)^3) \subset SU(3).
\end{equation}
The weights of $T$ are therefore
\begin{equation}
X^*(T) = \{\lambda=(\lambda_1,\lambda_2,\lambda_3) \mid \lambda_i -
\lambda_j \in {\mathbb Z},\quad \lambda_1+\lambda_2 + \lambda_3 = 0\}.
\end{equation}
For each integer $a\ge 0$ there is an irreducible representation
\begin{equation}\label{eq:G2dim}
\pi_{a} \text{\ highest wt\ }
(2a/3,-a/3,-a/3), \quad
\dim \pi_{a} = \frac{(2a+5)\prod_{j=1}^4 (a+j)}{5!}
\end{equation}
Notice that the polynomial giving the dimension has degree $5$. In
fact it is exactly the polynomial of \eqref{eq:Rdim} giving the dimension of
$\pi_a^{O(7)}$.
Using the calculation of $\rho$ given in \eqref{eq:G2rho} below, we
find
\begin{equation}\label{eq:G2inflchar}
\hbox{infinitesimal character of\ }\pi_a =
((2a+5)/3,-(a+1)/3,-(a+4)/3).
\end{equation}
The key fact is that
\begin{equation}\label{e:G2key}
\dim \pi_{a}^{SU(3)} = 1 \quad (a\ge 0), \qquad
\dim\pi^{SU(3)} = 0 \quad (\pi \not\simeq \pi_{a}).
\end{equation}
Therefore
\begin{equation}
L^2(S^{6}) \simeq \sum_{a\ge 0} \pi_{a}
\end{equation}
as representations of $G_{2,c}$.
\end{subequations}
\begin{subequations}\label{se:G2orbit}
Here is the orbit method perspective. Define
\begin{equation}
a_{\text{orbit}} = a+5/2.
\end{equation}
Then it turns out that there is an element $\lambda(a_{\text{orbit}})
\in {\mathfrak g}_0^*$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(a_{\text{orbit}}) &\in ({\mathfrak g}_0/{\mathfrak h}_0)^*\\
\lambda(a_{\text{orbit}}) &\ \text{is conjugate to}\\
&\quad a_{\text{orbit}}\cdot(2/3,-1/3,-1/3).
\end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_a = \pi(\text{orbit\ }
\lambda(a_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a \ge 0$ and but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -5/2.
\end{equation}
For the compact group $G_{2,c}$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0
\quad \text{if}\quad 0 > a > -5/2;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:G2s}).
\end{subequations}
\begin{subequations}\label{se:G2spec}
Now we're ready for spectral theory. We need to calculate
$\pi_{a}(\Omega_{G_{2,c}})$. The sum of the positive roots is
\begin{equation}\label{eq:G2rho}
2\rho(G_{2,c}) = (10/3,-2/3,-8/3).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (2a/3,-a/3,-a/3)
\end{equation}
we find
\begin{equation}\label{eq:LG2}\begin{aligned}
\pi_a(\Omega_{G_{2,c}}) &= 2a^2/3 + 10a/3 =
2(a^2 +5a)/3\\
&= (2/3)(a_{\text{orbit}}^2 - 25/4) \end{aligned}
\end{equation}
\end{subequations}
Here is how the $G_{2,c}$ and $O(7)$ calculations fit together.
\begin{theorem}
Using the inclusion $G_{2,c}\subset O(7)$, we have
$$\pi^{O(7)}_a|_{G_{2,c}} = \pi^{G_{2,c}}_a.$$
The contribution of these representations to the spectrum of the
$O(7)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(7)}_a(\Omega_{O(7)}) &= a^2 + 5a \\
&= \pi^{G_{2,c}}_a(3\Omega_{G_{2,c}}/2).\end{aligned}$$
\end{theorem}
This is a consequence of the equality of dimensions observed at
\eqref{eq:G2dim}, together with the fact that the inclusion of
$G_{2,c}$ in $O(7)$ carries (some) short roots to (some) short roots.
\section{The big $G_2$ calculation}
\label{sec:bigG2}
\setcounter{equation}{0}
\begin{subequations} \label{se:bigG2sphere}
Suppose $n$ is an integer at least two. The group $\Spin(2n)$, or
equivalently the Lie algebra ${\mathfrak s}{\mathfrak p}{\mathfrak
i}{\mathfrak n}(2n)$, has an interesting
outer automorphism of order two: conjugation by the orthogonal matrix
\begin{equation}
\sigma = \Ad \begin{pmatrix} 1 & 0 & \cdots & 0 & 0\\ 0 & 1 &
\cdots & 0 & 0 \\ & &
\ddots & & \\ 0 & 0 & \cdots & 1 & 0 \\ 0 & 0 & \cdots & 0 & -1
\end{pmatrix}.
\end{equation}
The group of fixed points of $\sigma$ is the ``first $2n-1$ coordinates''
\begin{equation}
\Spin(2n-1) = \Spin(2n)^{\sigma}.
\end{equation}
The automorphism $\sigma$ implements the automorphism of the Dynkin diagram
\setlength{\unitlength}{1cm}
\begin{picture}(2.8,1)(-2.7,.2)
\multiput(-.1,.5)(.7,0){3}{\circle*{.15}}
\multiput(0,.5)(.7,0){2}{\line(1,0){.5}}
\put(1.45,.48){\small{\dots}}
\put(2.0,.5){\circle*{.15}}
\multiput(2.4,-.01)(0,1.03){2}{\circle*{.15}}
\put(2.07,.57){\line(2,3){.25}}
\put(2.07,.43){\line(2,-3){.25}}
\put(2.7,.22){\begin{rotate}{90}$\longleftrightarrow$\end{rotate}}
\end{picture}
\noindent exchanging the two short legs. If $n=4$, the Dynkin diagram
\setlength{\unitlength}{1cm}
\begin{picture}(2.8,1)(-2,.2)
\multiput(1.3,.5)(.5,0){1}{\circle*{.15}}
\put(1.4,.5){\line(1,0){.5}}
\put(2.0,.5){\circle*{.15}}
\multiput(2.4,-.01)(0,1.03){2}{\circle*{.15}}
\put(2.07,.57){\line(2,3){.25}}
\put(2.07,.43){\line(2,-3){.25}}
\end{picture}
\noindent has two additional involutive automorphisms, exchanging the other two
pairs of legs. This gives rise to two additional (nonconjugate)
automorphisms $\sigma'$ and $\sigma''$ of $\Spin(8)$. Their fixed
point groups are isomorphic to $\Spin(7)$, but not conjugate to the
standard one (or to each other). We call them
\begin{equation}
\Spin(8)^{\sigma'} = \Spin(7)', \qquad \Spin(8)^{\sigma''} = \Spin(7)''.
\end{equation}
The full automorphism group of the Dynkin diagram is the symmetric
group $S_3$; $\sigma_0$ and $\sigma_\pm$ are the three
transpositions, any two of which generate $S_3$. The fixed point group
of the full $S_3$ is
\begin{equation}
\Spin(8)^{S_3} = G_{2,c} = \Spin(7) \cap \Spin(7)';
\end{equation}
this is a classical way to construct $G_{2,c}$. It follows that
\begin{equation}
S^7 = \Spin(8)/\Spin(7) \supset \Spin(7)'/G_{2,c}.
\end{equation}
Because the last homogeneous space is also seven-dimensional, the
inclusion is an equality
\begin{equation}
S^7 = \Spin(7)'/G_{2.c}.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:bigG2reps}
Here is the representation theory. We take for our maximal torus in
$\Spin(7)'$ the double cover $T_+$ of
\begin{equation}
SO(2)^3 \subset SO(7).
\end{equation}
The weights of $T_+$ are
\begin{equation}
X^*(T) = \{\lambda=(\lambda_1,\lambda_2,\lambda_3) \mid \lambda_i \in
{\mathbb Z} \text{\ (all $i$) or \ } \lambda_i \in
{\mathbb Z}+1/2 \text{\ (all $i$)}\}.
\end{equation}
For each integer $a\ge 0$ there is an irreducible representation
\begin{equation}\begin{aligned}
\pi^{\Spin(7)'}_{a} &\text{\ highest wt\ }
(a/2,a/2,a/2), \\
\dim \pi^{\Spin(7)'}_{a} &= \frac{(a+3)\prod_{j=1}^5(a+j)}{3\cdot 5!}
\end{aligned}\end{equation}
Notice that the polynomial giving the dimension has degree $6$; in
fact it is exactly the polynomial \eqref{eq:Rdim} giving the dimension of
$\pi_a^{O(8)}$.
Using the calculation of $\rho$ given in \eqref{eq:spinrho} below (or
in \eqref{eq:Orho}) we find
\begin{equation}\label{eq:bigG2inflchar}
\hbox{infl.~char.}(\pi^{\Spin(7)'}_a) =
((a+5)/2,(a+3)/2,(a+1)/2).
\end{equation}
The key fact is that
\begin{equation}\label{e:bigG2key}
\pi_{a}^{O(8)}|_{\Spin(7)'} = \pi_a^{\Spin(7)'}.
\end{equation}
Therefore
\begin{equation}
L^2(S^{7}) \simeq \sum_{a\ge 0} \pi^{\Spin(7)'}_{a}
\end{equation}
as representations of $\Spin(7)'$.
\end{subequations}
\begin{subequations}\label{se:bigG2orbit}
Here is the orbit method perspective. Define
\begin{equation}
a_{\text{orbit}} = a+3.
\end{equation}
Then it turns out that there is a $7\times 7$ skew-symmetric real
matrix $\lambda(a_{\text{orbit}})$ (which we will
not attempt to write down) with the properties
\begin{equation} \begin{aligned}
\lambda(a_{\text{orbit}}) &\in ({\mathfrak g}_0/{\mathfrak
h}_0)^*\\ \lambda(a_{\text{orbit}}) &\quad\text{has eigenvalues}\ \pm
a_{\text{orbit}}/4 \ \text{(three times)}.
\end{aligned}
\end{equation}
Consequently
\begin{equation}
\pi_{a}^{\Spin(7)'} = \pi(\text{orbit\ }
\lambda(a_{\text{orbit}})).
\end{equation}
An aspect of the orbit method perspective is that the ``natural''
dominance condition is no longer $a \ge 0$ and but rather
\begin{equation}
a_{\text{orbit}} > 0 \iff a > -3.
\end{equation}
For the compact group $G_{2,c}$ we have
\begin{equation}
\pi(\text{orbit\ } \lambda(a_{\text{orbit}}))=0
\quad \text{if}\quad 0 > a > -3;
\end{equation}
so the difference is not important. But matters will be more
interesting in the noncompact case (Section \ref{sec:bigncG2}).
\end{subequations}
\begin{subequations}\label{se:bigG2spec}
Now we're ready for spectral theory. We need to calculate
$\pi_{a}(\Omega_{\Spin(7)'})$. The sum of the positive roots is
\begin{equation}\label{eq:spinrho}
2\rho(\Spin(7)') = (5,3,1).
\end{equation}
Because our highest weight is
\begin{equation}
\lambda = (a/2,a/2,a/2)
\end{equation}
we find
\begin{equation}\label{eq:LbigG2}\begin{aligned}
\pi^{\Spin(7)'}_a(\Omega_{\Spin(7)'}) &= 3a^2/4 + 9a/2 =
3(a^2 +6a)/4\\ &= (3/4)(a_{\text{orbit}}^2 - 9)\\ &=
(3/4)\pi^{O(8)}_a(\Omega_{O(8)}).
\end{aligned}\end{equation}
\end{subequations}
Here is a summary.
\begin{theorem}
Using the inclusion $\Spin(7)'\subset O(8)$, we have
$$\pi^{O(8)}_a|_{\Spin(7)'} = \pi^{\Spin(7)'}_a.$$
The contribution of these representations to the spectrum of the
$O(8)$-invariant Laplacian $L_O$ is
$$\begin{aligned}\pi^{O(8)}_a(\Omega_{O(8)}) &= a^2 + 6a \\
&= \pi^{\Spin(7)'}_a(4\Omega_{\Spin(7)'}/3).\end{aligned}$$
\end{theorem}
\section{Invariant differential operators}
\label{sec:invt}
\setcounter{equation}{0}
\begin{subequations}\label{se:invt}
Suppose $H\subset G$ is a closed subgroup of a Lie group $G$. Write
\begin{equation}
{\mathbb D}(G/H) = \text{$G$-invariant differential operators on
$G/H$},
\end{equation}
an algebra. Following for example Helgason \cite{GGA}*{pages
274--275}, we wish to understand this algebra and its spectral
theory as a way to understand functions on $G/H$. A first step is to
describe the algebra in terms of the Lie algebras of $G$ and $H$. This
is done in \cite{Hinvt} when $H$ is reductive in $G$ (precisely, when
the Lie algebra ${\mathfrak h}_0$ has an $\Ad(H)$-stable complement in
${\mathfrak g}_0$). Ways to remove this hypothesis have been understood
for a long time; we follow the nice account in \cite{invt}.
Write
\begin{equation}\begin{aligned}
{\mathfrak g}_0 &= \Lie(G) = \text{real left-invariant vector
fields on $G$}\\
{\mathfrak g} &= {\mathfrak g}_0\otimes_{\mathbb R}{\mathbb C} =
\text{complex left-invariant vector fields on $G$}
\end{aligned}\end{equation}
These vector fields act on functions by differentiating ``on the
right:''
\begin{equation}
(Xf)(g) = \frac{d}{dt}\left( f(g\exp(tX))\right)|_{t=0} \qquad (X\in
{\mathfrak g}_0).
\end{equation}
As usual we can therefore identify the enveloping algebra
\begin{equation}
U({\mathfrak g}) = \text{left-invariant complex differential operators
on $G$}.
\end{equation}
We can identify
\begin{equation}
C^\infty(G/H) = \{f\in C^\infty(G) \mid f(xh) = f(x)\quad (x\in G,
h\in H)\}.
\end{equation}
Now consider the space
\begin{equation}
I(G/H) =_{\text{def}} \left[ U({\mathfrak g})\otimes_{U({\mathfrak
h})}{\mathbb C} \right]^{\Ad(H)\otimes 1}
\end{equation}
Before we pass to $\Ad(H)$-invariants, we have only a left
$U({\mathfrak g})$ module: no algebra structure. But
$\Ad(H)$-invariants inherit the algebra structure from $U({\mathfrak
g})\otimes_{\mathbb C} {\mathbb C}$; so $I(G/H)$ is an algebra. The
natural action
\begin{equation}
U({\mathfrak g})\otimes C^\infty(G) \rightarrow C^\infty(G)
\end{equation}
(which is a left algebra action, but comes by differentiating on the
right) restricts to a left algebra action
\begin{equation}
I(G/H) \otimes C^\infty(G/H) \rightarrow C^\infty(G/H)
\end{equation}
on the subspace $C^\infty(G/H) \subset C^\infty(G)$.
Suppose more generally that $(\tau,V_\tau)$ is a finite-dimensional (and
therefore smooth) representation of $H$. Then
\begin{equation}
{\mathcal V}_\tau = G \times_H V_\tau
\end{equation}
is a $G$-equivariant vector bundle on $G/H$. The space of smooth sections is
\begin{equation}
C^\infty({\mathcal V}_\tau) = \{f\in C^\infty(G,V_\tau) \mid f(xh) =
\tau(h)^{-1} f(x) \quad (x\in G, h\in H)\}.
\end{equation}
Now consider the space
\begin{equation}
I^\tau(G/H) = \left[ U({\mathfrak g})\otimes_{U({\mathfrak h})}\End(V_\tau)
\right]^{(\Ad\otimes \Ad)(H)}
\end{equation}
(The group $H$ acts by automorphisms on both the algebra $U({\mathfrak
g})$ and the algebra $\End(V_\tau)$, in the latter case by
conjugation by the operators $\tau(h)$. The $H$-invariants are taken
for the tensor product of these two actions.) Before we pass to
$\Ad(H)$-invariants, we have only a left
$U({\mathfrak g})$ module: no algebra structure. But
$\Ad(H)$-invariants inherit the algebra structure from $U({\mathfrak
g})\otimes_{\mathbb C} \End(V_\tau)$; so $I^\tau(G/H)$ is an algebra. The
natural action
\begin{equation}
[U({\mathfrak g})\otimes_{\mathbb C} \End(V_\tau)] \otimes
C^\infty(G,V_\tau) \rightarrow C^\infty(G,V_\tau)
\end{equation}
(which is a left algebra action, but comes by differentiating on the
right) restricts to a left algebra action
\begin{equation}\label{eq:bundleinvt}
I^\tau(G/H)\otimes C^\infty({\mathcal V}_\tau) \rightarrow
C^\infty({\mathcal V}_\tau).
\end{equation}
\end{subequations}
\begin{proposition}[Helgason \cite{Hinvt}*{Theorem
10}; \cite{HinvtBull}*{pages 758--759}; Koornwinder
\cite{invt}*{Theorem 2.10}]\label{prop:invt} Suppose $H$ is a
closed subgroup of the Lie group $G$. The action \eqref{eq:bundleinvt}
identifies the algebra $I^\tau(G/H)$ with
$${\mathbb D}^\tau(G/H) = \text{$G$-invariant differential operators
on the vector bundle ${\mathcal V}_\tau$}.$$
The action of $I^\tau(G/H)$ on formal power series sections of
${\mathcal V}_\tau$ at the identity is a faithful action.
\end{proposition}
\begin{subequations}\label{se:Helgharm}
Helgason's idea for invariant harmonic analysis (see for example
\cite{GGA}*{Introduction}) is to understand the
spectral theory of the algebra $I(G/H)={\mathbb D}(G/H)$ on
$C^\infty(G/H)$; or, more generally, of ${\mathbb D}^\tau(G/H)$ on
smooth sections of ${\mathcal V}_\tau$. Suppose for example that
${\mathbb D}(G/H)$ is {\em abelian}, and fix an algebra homomorphism
\begin{equation}
\lambda \colon {\mathbb D}(G/H) \rightarrow {\mathbb C}, \qquad
\lambda \in \Max\Spec({\mathbb D}(G/H)).
\end{equation}
Then the collection of simultaneous eigenfunctions
\begin{equation}
C^\infty(G/H)_\lambda =_{\text{def}} \{f\in C^\infty(G/H) \mid Df =
\lambda(D)f \mid D\in {\mathbb D}(G/H)\}
\end{equation}
is naturally a representation of $G$ (by left translation). The
question is for which $\lambda$ the space $C^\infty(G/H)_\lambda$ is
nonzero; and more precisely, what representation of $G$ it carries.
We can define
\begin{equation}
\Spec(G/H) = \{\lambda \in \Max\Spec({\mathbb D}(G/H)) \mid
C^\infty(G/H)_\lambda \ne 0\}.
\end{equation}
All of these remarks apply equally well to vector bundles.
\end{subequations}
\begin{subequations}\label{se:constructinvts1}
How can we identify interesting or computable invariant differential
operators? The easiest way is using the center of the enveloping
algebra
\begin{equation}\label{eq:Zg}
{\mathfrak Z}({\mathfrak g}) =_{\text{def}} U({\mathfrak g})^G.
\end{equation}
(If $G$ is disconnected, this may be a proper subalgebra of the
center.) The obvious map
\begin{equation}\label{eq:Zgi}
i_G\colon {\mathfrak Z}({\mathfrak g}) \rightarrow I^\tau(G/H),
\qquad z \mapsto z\otimes I_{V_\tau}
\end{equation}
is an algebra homomorphism. Here is how the spectral theory of the
differential operators $i_G( {\mathfrak Z}({\mathfrak g}))$ is related
to representation theory. Suppose that $(\pi,E_\pi)$ is a smooth
irreducible representation of $G$. Under a variety of mild assumptions
(for example, if $G$ is reductive and $\pi$ is quasisimple) there is a
homomorphism
\begin{equation}
\chi_\pi\colon {\mathfrak Z}({\mathfrak g}) \rightarrow {\mathbb C}
\end{equation}
called the {\em infinitesimal character of $\pi$} so that
\begin{equation}
d\pi(z) = \chi_\pi(z)\cdot I_{E_\pi}.
\end{equation}
Suppose now that there is a $G$-equivariant inclusion
\begin{equation}\label{eq:harmanalysis}
j_G\colon E_\pi \rightarrow C^\infty(G/H,{\mathcal V}_\tau).
\end{equation}
Finding inclusions like \eqref{eq:harmanalysis} is one of the things
harmonic analysis is about. One reason we care about it is the
consequences for spectral theory:
\begin{equation}\label{eq:spectral1}
i_G(z) \text{\ acts on $j_G(E_\pi) \subset C^\infty({\mathcal V}_\tau)$ by
the scalar $\chi_\pi(z)$} \qquad (z \in {\mathfrak Z}({\mathfrak
g})).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:constructinvts2}
Here is a generalization. Suppose $G_1$ is a subgroup of $G$
normalized by $H$:
\begin{equation}
G_1 \subset G, \qquad \Ad(H)(G_1) \subset G_1.
\end{equation}
(The easiest way for this to happen is for $G_1$ to contain $H$.) Then
$H$ acts on ${\mathfrak Z}({\mathfrak g}_1)$, so we get
\begin{equation}\label{eq:Zg1i}
i_{G_1}\colon {\mathfrak Z}({\mathfrak g}_1)^H \rightarrow I^\tau(G/H),
\qquad z_1 \mapsto z_1\otimes I_{V_\tau}.
\end{equation}
These invariant differential operators are acting along the
submanifolds
\begin{equation}
xG_1/(G_1\cap H) \subset G/H \qquad (x\in G)
\end{equation}
of $G/H$. An example is the first coordinate $G_1=U(1)$ introduced in
\eqref{se:Ureps}, for $H=U(n-1)$. The operator $\Omega_{U(1)}$ on
$S^{2n-1}$ (acting along the fibers of the map $S^{2n-1} \rightarrow
{\mathbb C}{\mathbb P}^{n-1}$) is one of these new invariant
operators. A more interesting example is $G_1=Sp(1)\times Sp(1)$ studied
in \eqref{e:extraHreps}.
Here is how the spectral theory of these new operators is related to
representation theory. The map \eqref{eq:harmanalysis} is (by
Frobenius reciprocity) the same thing as an $H$-equivariant map
\begin{equation}
j_H\colon E_\pi \rightarrow V_\tau
\end{equation}
or equivalently
\begin{equation}
j_H^*\colon V_\tau^* \rightarrow E_\pi^*.
\end{equation}
It makes sense to define
\begin{equation}
(E_\pi^*)^{G_1,j_H} = \text{$G_1$ representation generated by
$j_H^*(V_\tau^*)$} \subset \pi^*.
\end{equation}
If the $G_1$ representation $(\pi^*)^{G_1,j_H}$ has infinitesimal character
$\chi_1^*$ (the contragredient of the infinitesimal character $\chi_1$),
then
\begin{equation} \label{eq:spectral2}
i_{G_1}(z_1) \text{\ acts on $j_G(E_\pi) \subset C^\infty({\mathcal V}_\tau)$ by
the scalar $\chi_1(z_1)$} \qquad (z_1 \in {\mathfrak Z}({\mathfrak
g}_1)^H).
\end{equation}
The homomorphisms $i_G$ of \eqref{eq:Zgi} and \eqref{eq:Zg1i} define an
algebra homomorphism from the abstract (commutative) tensor product
algebra
\begin{equation}\label{eq:Zgprodi}
i_G\otimes i_{G_1} \colon {\mathfrak Z}({\mathfrak g}) \otimes_{\mathbb
C}{\mathfrak Z}({\mathfrak g}_1) \rightarrow I^\tau(G/H).
\end{equation}
The reason for this is that ${\mathfrak Z}({\mathfrak g})$ commutes
with all of $U({\mathfrak g})$.
\end{subequations}
Now that we understand the relationship between representations in
$C^\infty({\mathcal V}_\tau)$ and the spectrum of invariant
differential operators, let us see what the results of Sections
\ref{sec:R}--\ref{sec:bigG2} can tell us: in particular, about the
kernel of the homomorphism $i_G\otimes i_{G_1}$ of \eqref{eq:Zgprodi}.
\begin{subequations}\label{se:Rdiff}
We begin with $G=O(n)$, $H=O(n-1)$ as in Section
\ref{sec:R}. Write $n=2m+\epsilon$, with $\epsilon=0$ or
$1$. A maximal torus in $G$ is
\begin{equation}
T=SO(2)^m,\qquad {\mathfrak t}_0 = {\mathbb
R}^m, \qquad {\mathfrak t} = {\mathbb C}^m.
\end{equation}
The Weyl group $W(O(n))$ acts by permutation and sign changes on
these $m$ coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCO}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(O(n))} =
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^m/W(O(n)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}$ by \eqref{eq:HCO}. According to
\eqref{eq:spectral1} and \eqref{eq:Oinfchar}, the invariant
differential operator $i_G(z)$ will act on $\pi_a\subset
C^\infty(G/H)$ by the scalar
$$ p(a+(n-2)/2, (n-4)/2,\cdots,(n-2m)/2).$$
Recalling that $n-2m=\epsilon=0$ or $1$, we write this as
\begin{equation}\label{eq:Rscalar}
p(a+(n-2)/2, (n-4)/2,\cdots,\epsilon/2).
\end{equation}
\end{subequations}
Here is the consequence we want.
\begin{proposition}\label{prop:Rdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_m]^{W(O(n))}$$
vanishes on the (affine) line
$$\{(\alpha,(n-4)/2,\cdots,\epsilon/2) \mid \alpha\in {\mathbb C}\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{proof} The statement ``if'' is a consequence of
\eqref{eq:Rscalar}: if the differential operator is zero, then $p$
must vanish at all the points $(a+(n-2)/2, (n-4)/2,\cdots)$ with $a$
a non-negative integer. These points are Zariski dense in the
line. For ``only if,'' the vanishing of the polynomial makes the
differential operator act by zero on all the subspaces $\pi_a\subset
C^\infty(G/H)$. The sum of these subspaces is dense (for example as
a consequence of \eqref{eq:Osphere}); so the differential operator
acts by zero. The faithfulness statement in Proposition
\ref{prop:invt} then implies that $i_G(z)=0$. \end{proof}
\begin{corollary}\label{cor:Rdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The $O(n)$ infinitesimal characters
factoring to $i_G({\mathfrak Z}({\mathfrak g}))$ are indexed by
weights
\begin{equation}\label{eq:Rinfchar}
(\alpha,(n-4)/2,\cdots,\epsilon/2) \qquad (\alpha \in {\mathbb C}).
\end{equation}
Suppose $(\pi,E_\pi)$ is a representation of ${\mathfrak
o}(n,{\mathbb C})$ having an infinitesimal character, and that
$(E_\pi^*)^{{\mathfrak o}(n-1,{\mathbb C})} \ne 0$. Then $\pi$ has
infinitesimal character of the form \eqref{eq:Rinfchar}.
\end{subequations}
\end{corollary}
Exactly the same arguments apply to the other examples treated in
Sections \ref{sec:R}--\ref{sec:bigG2}. We will just state the
conclusions.
\begin{subequations}\label{se:Cdiff}
Suppose $G=U(n)$, $H=U(n-1)$ as in Section
\ref{sec:C}. A maximal torus in $G$ is
\begin{equation}
T=U(1)^n,\qquad {\mathfrak t}_0 = {\mathbb
R}^n, \qquad {\mathfrak t} = {\mathbb C}^n.
\end{equation}
The Weyl group $W(U(n))$ acts by permutation on
these $n$ coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCU}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(U(n))} =
{\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^n/W(U(n)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}$ by \eqref{eq:HCU}. According to
\eqref{eq:spectral1} and \eqref{eq:Uinflchar}, the invariant
differential operator $i_G(z)$ will act on $\pi_{b,c}\subset
C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:Cscalar}
p((b+(n-1))/2, (n-3)/2,\cdots,-(n-3)/2,-(c+(n-1))/2).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:Cdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_n]^{W(U(n))}$$
vanishes on the (affine) plane
$$\{(\xi,(n-3)/2,\cdots,-(n-3)/2,-\tau) \mid (\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:Cdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The $U(n)$ infinitesimal characters
factoring to $i_G({\mathfrak Z}({\mathfrak g}))$ are indexed by
weights
\begin{equation}\label{eq:Cinfchar}
(\xi,(n-3)/2,\cdots,-(n-3)/2,-\tau) \qquad ((\xi,\tau) \in
{\mathbb C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
u}(n,{\mathbb C})$ having an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak u}(n-1,{\mathbb C})} \ne 0$. Then
$F_\gamma$ has infinitesimal character of the form \eqref{eq:Cinfchar}. The
parameters $\xi$ and $\tau$ may be determined as follows. The
central character of $\gamma$ (scalars by which the one-dimensional center of
the Lie algebra acts) is given by $\xi-\tau$. If in addition
$F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(2n,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi+\tau =
\alpha$. (Replacing $\alpha$ by the equivalent infinitesimal
character parameter $-\alpha$
has the effect of interchanging $\xi$ and $-\tau$, which defines an
equivalent infinitesimal character parameter.)
\end{subequations}
\end{corollary}
\begin{subequations}\label{se:Hdiff}
Suppose next that $G=Sp(n)\times Sp(1)$, $H=Sp(n-1)\times
Sp(1)_\Delta$ as in Section \ref{sec:H}. A maximal torus in $G$ is
\begin{equation}
T=U(1)^n\times U(1),\qquad {\mathfrak t}_0 = {\mathbb
R}^n\times {\mathbb R}, \qquad {\mathfrak t} = {\mathbb C}^n
\times {\mathbb C}.
\end{equation}
The Weyl group $W(Sp(n)\times Sp(1))$ acts by sign changes on all
$n+1$ coordinates, and permutation of the first $n$
coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCSp}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(Sp(n)\times Sp(1))} =
{\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\leftrightarrow {\mathbb C}^{n+1}/W(Sp(n)\times Sp(1)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}$ by
\eqref{eq:HCSp}. According to \eqref{eq:spectral1} and
\eqref{eq:Hinflchar}, the invariant differential operator $i_G(z)$
will act on $\pi_{d,e}\subset C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:Hscalar}
p((d+n, e+(n-1),n-2,\cdots,1),(d-e+1)).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:Hdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_n,y]^{W(Sp(n)\times Sp(1))}$$
vanishes on the (affine) plane
$$\{(\xi,\tau,n-2,\cdots,1)(\xi-\tau) \mid (\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:Hdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The infinitesimal characters for
$Sp(n)\times Sp(1)$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:Hinfchar}
(\xi,\tau,n-2,\cdots,1)(\xi-\tau) \qquad ((\xi,\tau) \in {\mathbb
C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})$ having an infinitesimal
character, and that $(F_\gamma^*)^{{\mathfrak s}{\mathfrak
p}(n-1,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})_\Delta} \ne 0$. Then $F_\gamma$
has infinitesimal character of the form \eqref{eq:Hinfchar};
$\xi-\tau$ is the infinitesimal character of the ${\mathfrak
s}{\mathfrak p}(1,{\mathbb C})$ factor. If in
addition $F_\gamma \subset
E_\pi$ for some representation $(\pi,E_\pi)$ of ${\mathfrak
o}(4n,{\mathbb C})$ as in Corollary \ref{cor:Rdiff}, then we may take
$\xi+\tau = \alpha$.
\end{subequations}
\end{corollary}
This is a good setting in which to consider the more general invariant
differential operators from \eqref{se:constructinvts2}.
\begin{subequations}\label{se:HG1diff}
Suppose in that general setting that $G_1$ is reductive, and choose a Cartan
subalgebra ${\mathfrak t}_1\subset {\mathfrak g}_1$, with (finite) Weyl group
\begin{equation}
W(G_1) =_{\text{def}} N_{G_1({\mathbb C})}({\mathfrak
t}_1)/Z_{G_1({\mathbb C})}({\mathfrak t_1}) \subset
\Aut({\mathfrak t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)
\simeq S({\mathfrak t})^{W_1}.
\end{equation}
The adjoint action of $H$ on $G_1$ defines another Weyl group,
which normalizes $W(G_1)$:
\begin{equation}
W(G_1) \triangleleft W_H(G_1) =_{\text{def}} N_{H({\mathbb C})}({\mathfrak
t}_1)/Z_{H({\mathbb C})}({\mathfrak t_1}) \subset \Aut({\mathfrak
t}_1), \qquad {\mathfrak Z}({\mathfrak g}_1)^H
\simeq S({\mathfrak t}_1)^{W_H(G_1)}.
\end{equation}
Under mild hypotheses (for example $G_1$ is reductive algebraic and the
adjoint action of $H$ is algebraic) then $W_H(G_1)$ is finite, so the
algebra ${\mathfrak Z}({\mathfrak g}_1)$ is finite over ${\mathfrak
Z}({\mathfrak g}_1)^H$, and the maximal ideals in this smaller algebra
are given by evaluation at
\begin{equation}
\mu\in {\mathfrak t}_1^*/W_H(G_1).
\end{equation}
In the case $G_1=Sp(1)\times Sp(1)$, the adjoint action of $H$ on $G_1$ is
contained in that of $G_1$, so $W(G_1)=W_H(G_1)$, and ${\mathfrak Z}({\mathfrak
g}_1)^H = {\mathfrak Z}({\mathfrak g}_1)$. We have
\begin{equation}
T_1=U(1)^2, \qquad {\mathfrak t}_{1,0} = {\mathbb R}^2, \qquad {\mathfrak
t}_1 = {\mathbb C}^2.
\end{equation}
The Weyl group $W(G_1)=W_H(G_1)$ acts by sign changes on each
coordinate, so the Harish-Chandra isomorphism is
\begin{equation}\label{eq:HCH}
{\mathfrak Z}({\mathfrak g}_1)^H = {\mathfrak Z}({\mathfrak g}_1) \simeq
S({\mathfrak t}_1)^{W(G_1)} = {\mathbb C}[u_1,u_2]^{W(G_1)}
\end{equation}
Suppose therefore that $z_1\in {\mathfrak Z}({\mathfrak g}_1)$
corresponds to $p_1\in {\mathbb C}[x_1,x_2]^{W(G_1)}$. According to
\eqref{eq:spectral2} and
\eqref{eq:Hsubinflchar}, the invariant differential operator $i_{G_1}(z_1)$
acts on $\pi^{Sp(n)\times Sp(1)}_{d,e} \subset C^\infty(G/H)$ by the
scalar
\begin{equation}\label{eq:HG1scalar}
p_1(d-e+1,d-e+1).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:HG1diff} With notation as above,
suppose that
$$P\in {\mathbb C}[x_1,\cdots,x_n,y,u_1,u_2]^{W(G)\times W(G_1)},$$
and write $Z\in {\mathfrak Z}({\mathfrak g})\otimes {\mathfrak
Z}({\mathfrak g}_1)^H$ for the corresponding central element. Then $P$
vanishes on the affine plane
$$\{(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau)\mid
(\xi,\tau)\in {\mathbb C}^2\}.$$
if and only if $(i_G\otimes i_{G_1})(Z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:HG1diff}
\addtocounter{equation}{-1}
\begin{subequations}
In the setting $G/H =
(Sp(n)\times Sp(1))/(Sp(n-1)\times Sp(1)_\Delta)$, $G_1=Sp(1)\times Sp(1)$,
the characters of the tensor product algebra \eqref{eq:Zgprodi}
which factor to the image in $I(G/H)$ are indexed by weights
\begin{equation}\label{eq:HG1infchar}
(\xi,\tau,n-2,\cdots,1)(\xi-\tau)(\xi-\tau,\xi-\tau).
\end{equation}
Here the first $n$ coordinates are giving the infinitesimal
character for $Sp(n)$; the next is the infinitesimal character for
the $Sp(1)$ factor of $G$; and the last two are the infinitesimal
character for $G_1$.
Suppose $(\gamma,F_\gamma)$ is an ${\mathfrak
s}{\mathfrak p}(n,{\mathbb C})$ representation as in Corollary
\ref{cor:Hdiff}. Then the ${\mathfrak g}_1$ representation
generated by $(F_\gamma^*)^{{\mathfrak s}{\mathfrak p}(n-1,{\mathbb
C})}$ has infinitesimal character $(\xi-\tau,\xi-\tau)$.
\end{subequations}
\end{corollary}
\begin{subequations}\label{se:octdiff}
Suppose next that $G=\Spin(9)$, $H=\Spin(7)'$ as in Section
\ref{sec:O}. A maximal torus in $G$ is
\begin{equation}
T= \hbox{double cover of\ }SO(2)^4,\qquad {\mathfrak t}_0 = {\mathbb
R}^4, \qquad {\mathfrak t} = {\mathbb C}^4.
\end{equation}
The Weyl group $W(\Spin(9))$ acts by permutation and sign changes on these
four coordinates. Harish-Chandra's theorem identifies
\begin{equation}\label{eq:HCoct}
{\mathfrak Z}({\mathfrak g}) \simeq S({\mathfrak t})^{W(\Spin(9))} =
{\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}.
\end{equation}
Therefore
\begin{equation}
\hbox{(maximal ideals in ${\mathfrak Z}({\mathfrak g})$})
\longleftrightarrow {\mathbb C}^{4}/W(\Spin(9)).
\end{equation}
Suppose $z\in {\mathfrak Z}({\mathfrak g})$ corresponds to $p\in
{\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}$ by
\eqref{eq:HCoct}. According to \eqref{eq:spectral1} and
\eqref{eq:octinflchar}, the invariant differential operator $i_G(z)$
will act on $\pi^{\Spin(9)}_{x,y}\subset C^\infty(G/H)$ by the scalar
\begin{equation}\label{eq:octscalar}
p((2x+y+7)/2,(y+5)/2,(y+3)/2,(y+1)/2).
\end{equation}
\end{subequations}
\begin{proposition}\label{prop:octdiff} With notation as above,
the polynomial
$$p\in {\mathbb C}[x_1,\cdots,x_4]^{W(\Spin(9))}$$
vanishes on the (affine) plane
$$\{(\xi,\tau+5/2,\tau+3/2,\tau+1/2) \mid (\xi,\tau)\in
{\mathbb C}^2\}.$$
if and only if $i_G(z)\in I(G/H)$ is equal to zero.
\end{proposition}
\begin{corollary}\label{cor:octdiff}
\addtocounter{equation}{-1}
\begin{subequations}
The infinitesimal characters for
$\Spin(9)$ factoring to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:octinfchar}
(\xi,\tau+5/2,\tau+3/2,\tau+1/2) \qquad ((\xi,\tau) \in {\mathbb C}^2).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}{\mathfrak i}{\mathfrak n}(9,{\mathbb C})$ having
an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak h}({\mathbb C})} \ne 0$. Then $F_\gamma$ has
infinitesimal character of the form \eqref{eq:Hinfchar}. If the
${\mathfrak s} {\mathfrak p} {\mathfrak i} {\mathfrak n} (8,{\mathbb
C})$-module generated by $(F_\gamma^*)^{{\mathfrak h}({\mathbb
C})}$ has a submodule with an infinitesimal character, then we
may choose $\tau$ so that this infinitesimal character is
\begin{equation}
(\tau+3,\tau+2,\tau+1,\tau).
\end{equation}
If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(16,{\mathbb C})$
as in Corollary \ref{cor:Rdiff} (with infinitesimal character
parameter $\alpha$) then we may choose $\xi=\alpha/2$.
\end{subequations}
\end{corollary}
For the last two cases we write even less.
\begin{corollary}\label{cor:G2diff}
\addtocounter{equation}{-1}
\begin{subequations}
When $G/H = G_{2,c}/SU(3)$, the
infinitesimal characters for
$G_{2,c}$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:G2infchar}
(2\xi,(1/2)-\xi,-(1/2)-\xi) \qquad (\xi \in {\mathbb C}).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
g}_2({\mathbb C})$ having an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak u}(3,{\mathbb C})} \ne 0$. Then the
infinitesimal character of $F_\gamma$ is of the form in
\eqref{eq:G2infchar}. If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(7,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi = \alpha/3$.
\end{subequations}
\end{corollary}
\begin{corollary}\label{cor:bigG2diff}
\addtocounter{equation}{-1}
\begin{subequations}
When $G/H = \Spin(7)'/G_{2,c}$, the
infinitesimal characters for
$\Spin(7)'$ which factor to $i_G({\mathfrak Z}({\mathfrak
g}))$ are indexed by weights
\begin{equation}\label{eq:bigG2infchar}
(\xi+1,\xi,\xi-1) \qquad (\xi \in {\mathbb C}).
\end{equation}
Suppose $(\gamma,F_\gamma)$ is a representation of ${\mathfrak
s}{\mathfrak p}{\mathfrak i}{\mathfrak n}(7,{\mathbb C})'$ having
an infinitesimal character, and that
$(F_\gamma^*)^{{\mathfrak g}_2({\mathbb C})} \ne 0$. Then the
infinitesimal character of $F_\gamma$ is of the form in
\eqref{eq:bigG2infchar}. If in addition $F_\gamma \subset E_\pi$ for
some representation $(\pi,E_\pi)$ of ${\mathfrak o}(8,{\mathbb C})$
as in Corollary \ref{cor:Rdiff}, then we may take $\xi= \alpha/2$.
\end{subequations}
\end{corollary}
\section{Changing real forms}
\label{sec:changereal}
\setcounter{equation}{0}
Results like \eqref{eq:spectral1} and its generalization
\eqref{eq:spectral2} explain why it is interesting to study the
representations of $G$ appearing in $C^\infty({\mathcal V}_\tau)$ and
the invariant differential operators on this space. In this section we
state our first method for doing that.
\begin{definition}\label{def:realform}
\addtocounter{equation}{-1}
\begin{subequations}\label{se:realform}
Suppose $G_1$ and $G_2$ are Lie
groups with closed subgroups $H_1$ and $H_2$. Assume that there is
an isomorphism of complexified Lie algebras
\begin{equation}i\colon {\mathfrak g}_1 \buildrel{\sim}\over{\longrightarrow}
{\mathfrak g}_2, \qquad i({\mathfrak h}_1) = {\mathfrak h}_2.
\end{equation}
Finally, assume that $i$ identifies the Zariski closure of $\Ad(H_1)$
in $\Aut({\mathfrak g}_1)$ with the Zariski closure of $\Ad(H_2)$ in
$\Aut({\mathfrak g}_2)$. (This is automatic if $H_1$ and $H_2$ are
connected.) Then we say that the homogeneous space $G_2/H_2$ is {\em
another real form} of the homogeneous space $G_1/H_1$.
Given representations $(\tau_i,V_{\tau_i})$ of $H_i$, we say that
${\mathcal V}_{\tau_2}$ is {\em another real form} of ${\mathcal
V}_{\tau_1}$ if there is an isomorphism
\begin{equation}i\colon V_{\tau_1} \buildrel{\sim}\over{\longrightarrow}
V_{\tau_2}\end{equation}
respecting the actions of ${\mathfrak h}$, and identifying the
Zariski closure of $\Ad(H_1)$ in $\End(V_{\tau_1})$ with the Zariski
closure of $\Ad(H_2)$ in $\End(V_{\tau_2})$.
Whenever ${\mathcal V}_{\tau_2}$ is another real form of ${\mathcal
V}_{\tau_1}$, we get an algebra isomorphism
\begin{equation}\label{eq:realformisom}
i\colon {\mathbb D}^{\tau_1}(G_1/H_1)
\buildrel{\sim}\over{\longrightarrow} {\mathbb D}^{\tau_2}(G_2/H_2).
\end{equation}
\end{subequations}
\end{definition}
We will use these isomorphisms together with results like Corollaries
\ref{cor:Cdiff}--\ref{cor:bigG2diff} (proven using compact homogeneous
spaces $G_1/H_1$) to control the possible representations appearing in
some noncompact homogeneous spaces $G_2/H_2$.
\section{Changing the size of the group}
\label{sec:size}
\setcounter{equation}{0}
Our second way to study representations and invariant differential
operators is this.
\begin{subequations}\label{se:size}
In the setting \eqref{se:invt}, suppose that $S\subset G$ is a closed
subgroup, and that
\begin{equation}\label{eq:bigsub1}
\dim G/H = \dim S/(S\cap H).
\end{equation}
Equivalent requirements are
\begin{equation}\label{eq:bigsub2}
{\mathfrak s}/({\mathfrak s}\cap {\mathfrak h}) = {\mathfrak
g}/{\mathfrak h}
\end{equation}
or
\begin{equation}\label{eq:bigsub3}
{\mathfrak s} + {\mathfrak h} = {\mathfrak g}
\end{equation}
or
\begin{equation}\label{eq:bigsub4}
\hbox{$S/(S\cap H)$ is open in $G/H$}.
\end{equation}
Because of this open embedding, differential operators on $S/(S\cap
H)$ are more or less the same thing as differential operators on
$G/H$. The condition of $S$-invariance is weaker than the condition of
$G$-invariance, so we get natural inclusions
\begin{equation}\label{eq:opincl}
{\mathbb D}(G/H) \hookrightarrow {\mathbb D}(S/(S\cap H)), \qquad {\mathbb
D}^\tau(G/H) \hookrightarrow {\mathbb D}^\tau(S/(S\cap H)).
\end{equation}
(notation as in \eqref{se:invt}). In terms of the
algebraic description of these operators given in Proposition
\ref{prop:invt}, notice first that the condition in \eqref{eq:bigsub2}
shows that the inclusion ${\mathfrak s} \hookrightarrow {\mathfrak g}$
defines an isomorphism
\begin{equation}
U({\mathfrak s})\otimes_{{\mathfrak s}\cap{\mathfrak h}}\End(V_\tau)
\simeq U({\mathfrak g})\otimes_{{\mathfrak h}}\End(V_\tau)
\end{equation}
Therefore
\begin{equation}\begin{aligned}
\ [U({\mathfrak g})\otimes_{{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes \Ad)(H)} &\hookrightarrow
[U({\mathfrak g})\otimes_{{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes\Ad)(S\cap H)} \\
&\simeq [U({\mathfrak s})\otimes_{{\mathfrak s}\cap{\mathfrak
h}}\End(V_\tau)]^{(\Ad\otimes\Ad)(S\cap H)}.
\end{aligned} \end{equation}
That is,
\begin{equation}\label{eq:algincl}
I^\tau(G/H) \hookrightarrow I^\tau(S/(S\cap H)).
\end{equation}
This algebra inclusion corresponds to the differential operator
inclusion \eqref{eq:opincl} under the identification of Proposition
\ref{prop:invt}.
\end{subequations}
Here is a useful fact.
\begin{proposition} Let $G$ be a connected reductive Lie group,
and let $H$ and $S$ be closed connected reductive subgroups.
Assume the equivalent conditions (\ref{eq:bigsub1})-(\ref{eq:bigsub4}). Then
\begin{enumerate}
\item $G=SH$, and
\item there is a Cartan involution for $G$ preserving both $S$ and
$H$.
\end{enumerate}
\end{proposition}
\begin{proof} Part (1) is due to Onishchik \cite{Oni69}*{Theorem 3.1}.
For (2), since $H$ is reductive in $G$, there is a Cartan
involution $\theta_H$ for $G$ preserving $H$, and likewise there is
one $\theta_S$ preserving $S$. By the uniqueness of Cartan involutions
for $G$, $\theta_S$ is the conjugate of $\theta_H$ by some element $g\in G$,
which by (1) can be decomposed as $g=sh$. The $h$-conjugate of $\theta_H$,
which is also the $s^{-1}$-conjugate of $\theta_S$, has the required property.
\end{proof}
It follows from (1) that if $(G_c,S_c,H_c)$ is a triple of a compact Lie group
and two closed subgroups such that $G_c=S_cH_c$, and if $(G,S,H)$ is a triple
of real forms (that is, $G/S$ is a real form of $G_c/S_c$ and
$G/H$ a real form of $G_c/H_c$), then $S$ acts transitively on $G/H$.
Conversely, by (2) every transitive action on a reductive homogeneous space
$G/H$ by a reductive subgroup $S\subset G$ is obtained in this way.
In the following sections we shall apply this principle to the real hyperboloid
(\ref{eq:Hpq}), which is a real form of $S^{p+q-1}=O(p+q)/O(p+q-1)$.
The hypothesis that both $S$ and $H$ be reductive is certainly
necessary. Suppose for example that $S$ is a noncompact
real form of the complex reductive group $G$, and that
$H$ is a parabolic subgroup of $G$ (so that $S$ and $G$ are reductive,
but $H$ is not). Then $S$ has finitely many orbits on $G/H$
(\cite{Wolfflag}), and in particular has open orbits (so that the
conditions \eqref{eq:bigsub1}--\eqref{eq:bigsub4} are satisfied); but
the number of orbits is almost always greater than one (so $G \ne SH$).
\section{Classical hyperboloids}
\label{sec:Opq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Opq}
In this section we recall the classical representation-theoretic
decomposition of functions on real hyperboloids: that is, on other
real forms of spheres. The spaces are
\begin{equation}\label{eq:Hpq}
H_{p,q} = \{v\in {\mathbb R}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\} = O(p,q)/O(p-1,q).
\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard quadratic form of
signature $(p,q)$ on ${\mathbb R}^{p+q}$. The inclusion of the right
side of the equality in the middle is just given by the action of the
orthogonal group on the basis vector $e_1$; surjectivity is Witt's
theorem. This realization of the hyperboloid is a symmetric space, so
the Plancherel decomposition is completely known. In particular, the
discrete series may be described as follows. To avoid degenerate
cases, we assume that
\begin{equation}\label{eq:Obigp}
p \ge 2.
\end{equation}
There is a ``compact Cartan subspace'' with Lie algebra
\begin{equation}\label{eq:cptCartanOpq}
{\mathfrak a}_c = \langle e_{12} - e_{21} \rangle.
\end{equation}
The first requirement is that
\begin{equation}
{\mathfrak a}_c \subset {\mathfrak k} = {\mathfrak o}(p) \times
{\mathfrak o}(q).
\end{equation}
That this is satisfied is a consequence of \eqref{eq:Obigp}. The
second requirement is that ${\mathfrak a}_c$ belongs to the $-1$
eigenspace of the involutive automorphism
\begin{equation}
\sigma = \Ad\left(\diag(-1,1,1,\dots,1) \right)
\end{equation}
with fixed points the isotropy subgroup $O(p-1,q)$. (More precisely,
the group of fixed points of $\sigma$ is $O(1) \times O(p-1,q)$; so
our hyperboloid is a $2$-to-$1$ cover of the algebraic symmetric space
$O(p,q)/[O(1)\times O(p-1,q)]$. But the references also treat analysis on
this cover.)
For completeness, we mention that whenever
\begin{equation}\label{eq:Obigq}
q \ge 1.
\end{equation}
there is another conjugacy class of Cartan subspace, represented by
\begin{equation}\label{eq:splCartanOpq}
{\mathfrak a}_s = \langle e_{1,p+1} + e_{p+1,1} \rangle.
\end{equation}
This one is split, and corresponds to the continuous part of the
Plancherel formula.
The discrete series for the symmetric space $H_{p,q}$ is constructed
as follows. Using the compact Cartan subspace ${\mathfrak a}_c$,
construct a $\theta$-stable parabolic
\begin{equation}\label{eq:qOpq}
{\mathfrak q}^{O(p,q)} = {\mathfrak l}^{O(p,q)} + {\mathfrak u}^{O(p,q)} \subset
{\mathfrak o}(p+q,{\mathbb C});
\end{equation}
the corresponding Levi subgroup is
\begin{equation}
L^{O(p,q)} = [O(p,q)]^{{\mathfrak a}_c} = SO(2) \times O(p-2,q)
\end{equation}
We will need notation for the characters of $SO(2)$:
\begin{equation}
\widehat{SO(2)} = \{\chi_\ell \mid \ell \in {\mathbb Z}\}.
\end{equation}
The discrete series consists of certain irreducible representations
\begin{equation}
A_{{\mathfrak q}^{O(p,q)}}(\lambda), \qquad \lambda\colon L^{O(p,q)} \rightarrow
{\mathbb C}^\times.
\end{equation}
The allowed $\lambda$ are (first) those trivial on
\begin{equation}
L^{O(p,q)} \cap O(p-1,q) = O(p-2,q).
\end{equation}
These are precisely the characters of $SO(2)$, and so are indexed by
integers $\ell \in {\mathbb Z}$. Second, there is a positivity
requirement
\begin{equation}
\ell + (p+q-2)/2 > 0.
\end{equation}
We write
\begin{equation}\label{eq:OAq}\begin{aligned}
\lambda(\ell) &=_{\text{def}} \chi_\ell \otimes 1 \colon L^{O(p,q)}\rightarrow
{\mathbb C}^\times, \\ \pi^{O(p,q)}_\ell &= A_{{\mathfrak
q}^{O(p,q)}}(\lambda_\ell) \qquad
(\ell > (2-p-q)/2).
\end{aligned}\end{equation}
The infinitesimal character of this representation is
\begin{equation}
\text{infl char}(\pi_\ell^{O(p,q)}) = (\ell +
(p+q-2)/2, (p+q-4)/2, (p+q-6)/2,\cdots).
\end{equation}
The discrete part of the Plancherel decomposition is
\begin{equation}\label{eq:Opqdisc}
L^2(H_{p,q})_{\text{disc}} = \sum_{\ell > -(p+q-2)/2}
\pi_\ell^{O(p,q)}.
\end{equation}
This decomposition appears in \cite{StrH}*{page 360}, and
\cite{RossH}*{page 449, Theorem 10, and page 471}. What Strichartz
calls $N$ and $n$ are for us $p$ and $q$; his $d$ is our $\ell$. What
Rossmann calls $q$ and $p$ are for us $p$ and $q$; his $\nu -\rho$ is
our $\ell$; and his $\rho$ is $(p+q-2)/2$. The
identification of the representations as cohomologically induced
may be found in \cite{Virr}*{Theorem 2.9}.
\end{subequations}
\begin{subequations} \label{se:Opqorbit}
Here is the orbit method perspective. Just as for $O(n)$, we use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. We find
\begin{equation}
({\mathfrak g}_0/{\mathfrak h}_0)^* \simeq {\mathbb R}^{p-1,q},
\end{equation}
respecting the action of $H=O(p-1,q)$. The orbits of $H$ of largest
dimension are given by the value of the quadratic form: positive for
the orbits represented by nonzero elements $x(e_{12}-e_{21})$ of the
compact Cartan subspace of \eqref{eq:cptCartanOpq}; negative for
nonzero elements of the split Cartan subspace
$y(e_{1,p+1}+ e_{p+1,1})$; and zero for the nilpotent element
$(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$.
Define
\begin{equation}\begin{aligned}
\ell_{\text{orbit}} &= \ell + (n-2)/2,\\
\lambda(\ell_{\text{orbit}}) &=
\ell_{\text{orbit}}\cdot((e_{12}-e_{21})/2).
\end{aligned}
\end{equation}
Then the coadjoint orbits for discrete series have representatives
in the compact Cartan subspace
\begin{equation}
\pi_\ell^{O(p,q)} = \pi(\text{orbit}\ \lambda(\ell_{\text{orbit}})).
\end{equation}
Now this representation is an irreducible unitary cohomologically
induced representation whenever
\begin{equation}
\ell_{\text{orbit}} > 0 \iff \ell > -(n-2)/2.
\end{equation}
One of the advantages of the orbit method picture is that the
condition $\ell_{\text{orbit}} >0$ is simpler than the one
$\ell>(-(n-2)/2)$ arising from more straightforward representation
theory as in \eqref{eq:OAq}. Of course we always need also the
integrality condition
\begin{equation}
\ell_{\text{orbit}} \equiv (n-2)/2 \pmod{\mathbb Z} \iff \ell \equiv 0
\pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations} \label{se:Opqcont}
For completeness we mention also the continuous part of the Plancherel
decomposition. The split Cartan subspace ${\mathfrak a}_s$ (defined
above as long as $p$ and $q$ are each at least $1$) gives rise to a
real parabolic subgroup
\begin{equation}\label{eq:POpq}
P^{O(p,q)} = M^{O(p,q)} A_s N^{O(p,q)}, \qquad M^{O(p,q)} = \{\pm
1\} \times O(p-1,q-1).
\end{equation}
Here $A_s = \exp({\mathfrak a}_s) \simeq {\mathbb R}$, and $\{\pm 1\}$
is
$$O(1)_\Delta \subset O(1)\times O(1) \subset O(1,1);$$
we have
$$\{\pm 1\} \times A_s = SO(1,1) \simeq {\mathbb R}^\times,$$
an algebraic split torus. Therefore
\begin{equation}\label{eq:POpqB}
P^{O(p,q)} = SO(1,1)\times O(p-1,q-1)\times N^{O(p,q)}.
\end{equation}
The characters of $SO(1,1)$ are
\begin{equation}
\widehat{SO(1,1)} = \{\chi_{\epsilon,\nu} \mid \epsilon \in {\mathbb
Z}/2{\mathbb Z}, \nu \in {\mathbb C}\}, \qquad
\chi_{\epsilon,\nu}(r) = |r|^\nu \cdot \sgn(r)^\epsilon.
\end{equation}
We define
\begin{equation}
\pi_{\epsilon,\nu}^{O(p,q)} =
\Ind_{P^{O(p,q)}}^{O(p,q)}\left(\chi_{\epsilon,\nu}\otimes
1_{O(p-1,q-1)}\otimes 1_{N^{O(p,q)}}\right).
\end{equation}
Here (in contrast to the definition of discrete series
$\pi^{O(p,q)}_\ell$) we use normalized induction, with a $\rho$
shift. As a consequence, the infinitesimal character of this
representation is
\begin{equation}
\text{infl char}(\pi_{\epsilon,\nu}^{O(p,q)}) = (\nu, (p+q-4)/2,
(p+q-6)/2,\cdots);
\end{equation}
The continuous part of the Plancherel decomposition is
\begin{equation}\label{eq:Opqcont}
L^2(H_{p,q})_{\text{cont}} = \sum_{\epsilon \in {\mathbb
Z}/2{\mathbb Z}} \int_{\nu\in i{\mathbb R_{\ge 0}}}
\pi_{\epsilon,\nu}^{O(p,q)}.
\end{equation}
Just as for the discrete part of the decomposition, {\em all} (not
just almost all) of the representations $\pi_{\epsilon,\nu}^{O(p,q)}$
are irreducible (always for $\nu \in i{\mathbb R}$).
There is an orbit-theoretic formulation of these parameters as well,
corresponding to elements $-i\nu\cdot(e_{1,p+1} + e_{p+1,1})/2$ of the split
Cartan subspace. We omit the details.
\end{subequations}
\begin{subequations}\label{se:OKbranch}
We will need to understand the restriction of $\pi^{O(p,q)}_\ell$ to
the maximal compact subgroup
\begin{equation}
K = O(p)\times O(q) \subset O(p,q).
\end{equation}
This computation requires knowing
\begin{equation}
L^{O(p,q)} \cap K = SO(2)\times O(p-2)\times O(q), \qquad {\mathfrak
u\cap s} = \chi_{1}\otimes 1 \otimes {\mathbb C}^{q};
\end{equation}
here ${\mathfrak g} = {\mathfrak k}\oplus {\mathfrak s}$ is the
complexified Cartan decomposition. Consequently
\begin{equation}
S^m({\mathfrak u\cap s}) = \chi_m \otimes 1 \otimes S^m({\mathbb
C}^{q}) = \sum_{0\le k \le m/2} \chi_m \otimes 1 \otimes \pi^{O(q)}_{m-2k}.
\end{equation}
Now an analysis of the Blattner formula for restricting
cohomologically induced representations to $K$ gives
\begin{equation}\label{eq:Obranch}
\pi_\ell^{O(p,q)}|_{O(p)\times O(q)} =
\sum_{m=0}^\infty \quad \sum_{0\le k \le m/2}\pi^{O(p)}_{m+\ell + q}
\otimes \pi^{O(q)}_{m-2k}.
\end{equation}
If $p$ is much larger than $q$, then
some of the parameters for representations of $O(p)$ are negative.
Those representations should be understood to be zero.
A description of the restriction to $O(p)\times O(q)$ is in
\cite{RossH}*{Lemma 11}. In Rossmann's coordinates, what is written is
$$ \begin{aligned}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} &\mid -(m+(p-2)/2) +
(n+(q-2)/2) \ge \nu, \\ m+n &\equiv \nu -\rho - p \pmod{2}\}.
\end{aligned}$$
Converting to our coordinates as explained after \eqref{eq:Opqdisc}
gives
\begin{equation}\begin{aligned}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} &\mid (m+(p-2)/2) -
(n+(q-2)/2) \ge \nu, \\
m-n &\equiv \nu -\rho + q \pmod{2}\},
\end{aligned}\end{equation}
or equivalently
\begin{equation}
\{ \pi_m^{O(p)} \otimes \pi_n^{O(q)} \mid m - n \ge \ell + q-1, \quad
m-n \equiv \ell + q \pmod{2}\}.
\end{equation}
The congruence condition makes the inequality into
$$m-n \ge \ell + q,$$
which matches the description in \eqref{eq:Obranch}
Finally, we record the easier formulas
\begin{equation}\label{eq:Ocontbranch}
\pi_{\epsilon,\nu}^{O(p,q)}|_{O(p)\times O(q)} =
\sum_{\substack{m,m' \ge 0\\ m-m' \equiv \epsilon \pmod{2}}} \pi^{O(p)}_m
\otimes \pi^{O(q)}_{m'}.
\end{equation}
\end{subequations}
\section{Hermitian hyperboloids}
\label{sec:Upq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Upq}
In this section we see what the ideas from Sections \ref{sec:invt} and
\ref{sec:Opq} say about the discrete series of the non-symmetric
spherical spaces
\begin{equation}
H_{2p,2q} = \{v\in {\mathbb C}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\} = U(p,q)/U(p-1,q).
\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard Hermitian form of
signature $(p,q)$ on ${\mathbb C}^{p+q}$. The inclusion of the right side
in the middle is just given by the action of the
unitary group on the basis vector $e_1$; surjectivity is Witt's
theorem for Hermitian forms. These discrete series were
completely described by Kobayashi in \cite{Kob}*{Theorem 6.1}.
To simplify many formulas, we write in this section
\begin{equation}\label{eq:Unpq}
n = p+q.
\end{equation}
Our approach (like Kobayashi's) is to restrict the discrete series
representations $\pi_\ell^{O(2p,2q)}$ of \eqref{eq:Opqdisc} to
$U(p,q)$.
We should mention at this point that the homogeneous space
$U(n)/U(n-1)$ has another noncompact real form $GL(n,{\mathbb
R})/GL(n-1,{\mathbb R})$, arising from the inclusion
\begin{equation}
GL(n,{\mathbb R}) \hookrightarrow O(n,n)
\end{equation}
as a real Levi subgroup. For this real form (as Kobayashi observes)
the discrete series representations $\pi_\ell^{O(n,n)}$ decompose
continuously on restriction to $GL(n,{\mathbb R})$, and consequently
this homogeneous space has no discrete series. (More precisely, the
character $x-y$ of the center of $U(1)$ of $U(p,q)$ (an integer)
appearing in the analysis below must be replaced by a character of the
center ${\mathbb R}^\times$ of $GL(n,{\mathbb R})$ (a real number and
a sign).)
We begin by computing the restriction to $U(p)\times U(q)$.
What is good about this is that the representations of $O(2p)$ and
$O(2q)$ appearing in
\eqref{eq:Obranch} are representations appearing in the action of $O$
on spheres. We already computed (in Theorem \ref{thm:OUcptbranch}) how
those branch to unitary groups. The conclusion is
\begin{small}\begin{equation}\label{e:OUbranch}
\pi^{O(2p,2q)}_{_\ell}|_{U(p)\times U(q)} =
\sum_{\substack{0\le b,c \\[.2ex] b+c \ge \ell+2q}} \quad
\sum_{\substack{0\le b',c' \\[.2ex] b'+c' \le b+c -\ell-2q
\\[.2ex] b'+c'
\equiv b+c-\ell \pmod{2}}}
\pi^{U(p)}_{b,c}\otimes \pi^{U(q)}_{b',c'}.
\end{equation}\end{small}
\end{subequations}
This calculation, together with Corollary \ref{cor:Cdiff}, proves most of
\begin{proposition}\label{prop:OUbranch} Suppose $p$ and $q$ are
nonnegative integers, each at least two; and suppose $\ell >
-(n-1)$. Then the restriction of the discrete series representation
$\pi^{O(2p,2q)}_\ell$ to $U(p,q)$ is the direct sum of
the one-parameter family of representations
$$\pi^{U(p,q)}_{x,y},\quad
x,y \in {\mathbb Z}, \quad x+y = \ell.$$
The infinitesimal character of $\pi^{U(p,q)}_{x,y}$ corresponds to
the weight
$$(x+(n-1)/2,(n-3)/2,\ldots,-(n-3)/2, -y-(n-1)/2).$$
Restriction to the maximal compact subgroup is
$$\pi^{U(p,q)}_{x,y}|_{U(p)\times U(q)} = \sum_{r,s\ge
0}\sum_{k=0}^{\min(r,s)} \pi^{U(p)}_{x+q+r,y+q+s}\otimes
\pi^{U(q)}_{s-k,r-k}.$$
If one of the two subscripts in a $U(p)$ representation is negative,
that term is to be interpreted as zero.
Each of the representations $\pi^{U(p,q)}_{x,y}$ is irreducible.
\end{proposition}
The ``one parameter'' referred to in the proposition is $x-y$; the
pair $(x,y)$ can be thought of as a single parameter because of the
constraint $x+y=\ell$. What we have done is sorted the representations
of $U(p)\times U(q)$ appearing in \eqref{e:OUbranch} according to the
character of the center $U(1)$ of $U(p,q)$; this character is
$(b-c)+(b'-c')$, and we call it $x-y$ in the rearrangement in
Proposition \ref{prop:OUbranch}. The corresponding representation of
$U(p,q)$ (the part of $\pi^{O(2p,2q)}_\ell$ where $U(1)$ acts by
$x-y$) is what we call $\pi^{U(p,q)}_{x,y}$. In order to prove most of
the proposition, we just need to check that the same representations
of $U(p)\times U(q)$ appear in \eqref{e:OUbranch} and in Proposition
\ref{prop:OUbranch}, and this is easy. We will prove the
irreducibility assertion (using \cite{Kob}) after \eqref{e:Aq-smallx}
below.
\begin{subequations}\label{se:Upqrep}
Having identified the restriction to $U(p)\times U(q)$, we record
for completeness Kobayashi's identification of the actual representations of
$U(p,q)$. These come in three families, according to the values of
the integers $x$ and $y$. The families are cohomologically induced
from three $\theta$-stable parabolic subalgebras:
\begin{equation}
{\mathfrak q}^{U(p,q)}_+ = {\mathfrak l}^{U(p,q)}_+ + {\mathfrak
u}^{U(p,q)}_+ \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_+ = U(1)_p\times U(1)_q \times U(p-1,q-1);
\end{equation}
\begin{equation}
{\mathfrak q}^{U(p,q)}_0 = {\mathfrak l}^{U(p,q)}_0 + {\mathfrak
u}^{U(p,q)}_0 \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_0 = U(1)_p \times U(p-2,q)\times U(1)_p;
\end{equation}
and
\begin{equation}
{\mathfrak q}^{U(p,q)}_- = {\mathfrak l}^{U(p,q)}_- + {\mathfrak
u}^{U(p,q)}_- \subset {\mathfrak u}(n,{\mathbb C});
\end{equation}
with Levi subgroup
\begin{equation}
L^{U(p,q)}_- = U(p-1,q-1)\times U(1)_q \times U(1)_p.
\end{equation}
(We write $U(1)_p$ for a coordinate $U(1) \subset U(p)$, and $U(1)_q
\subset U(q)$ similarly. More complete descriptions of these
parabolics are in \cite{Kob}.)
Suppose first that
\begin{equation}\label{eq:bigx}
x> \ell + (n-1)/2, \qquad y < -(n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
$\xi_x$ for the character of $U(1)$ corresponding to $x\in {\mathbb Z}$.
Consider the one-dimensional character
\begin{equation}
\lambda^+_{x,y} = \xi_x\otimes \xi_{-(y + n-2)} \otimes {\det}^1
\end{equation}
of $L^{U(p,q)}_+$. What Kobayashi proves in \cite{Kob}*{Theorem 6.1}
is
\begin{equation}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_+}(\lambda^+_{x,y})
\qquad (x > \ell + (n-1)/2).
\end{equation}
Suppose next that
\begin{equation}\label{eq:middlex}
\ell + (n-1)/2 \ge x \ge -(n-1)/2, \qquad -(n-1)/2\le y \le
\ell + (n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two pairs of inequalities are equivalent.)
Consider the one-dimensional character
\begin{equation}
\lambda^0_{x,y} = \xi_x\otimes 1\otimes \xi_{-y}
\end{equation}
of $L^{U(p,q)}_0$. Kobayashi's result in \cite{Kob}*{Theorem 6.1}
is now
\begin{equation}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_0}(\lambda^0_{x,y})
\qquad (-(n-1)/2 \le x \le \ell + (n-1)/2).
\end{equation}
The remaining case is
\begin{equation}\label{eq:smallx}
x < -(n-1)/2, \qquad y > \ell + (n-1)/2.
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
\begin{equation}
\lambda^-_{x,y} = {\det}^{-1}\otimes \xi_{x+n-2}\otimes \xi_{-y}
\end{equation}
of $L^{U(p,q)}_-$. In this case Kobayashi proves
\begin{equation}\label{e:Aq-smallx}
\pi^{U(p,q)}_{x,y} = A_{{\mathfrak q}^{U(p,q)}_-}(\lambda^-_{x,y})
\qquad (x < -(n-1)/2).
\end{equation}
\end{subequations}
\begin{subequations} \label{se:Upqorbit}
Here is the orbit method perspective. Just as for $U(n)$, we use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. The linear functionals vanishing on ${\mathfrak h}_0^*$ are
\begin{equation}
\lambda(t,u,v) = \begin{pmatrix}it & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix} \simeq
{\mathbb R} + {\mathbb C}^{p-1,q}
\end{equation}
with $t\in {\mathbb R}$, $u \in {\mathbb C}^{p-1}$, $v\in {\mathbb C}^q$.
The orbits of $H=U(p-1,q)$ of largest dimension are given by the real
number $t$, and the value of the Hermitian form on the vector $(u,v)$:
positive for the orbits represented by nonzero elements
$r(e_{12}-e_{21})$ (nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 +
4r^2)^{1/2}$); negative for nonzero elements $s(e_{1,p+1}+ e_{p+1,1})$
(nonzero eigenvalues $i(t\pm a)/2$, with $a=(t^2 - 4s^2)^{1/2}$); and
zero for the nilpotent element $(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$
(two nonzero eigenvalues $it/2$).
Define
\begin{equation}
\ell_{\text{orbit}} = \ell + (n-1), \quad x_{\text{orbit}} = x +
(n-1)/2, \quad y_{\text{orbit}} = y + (n-1)/2.
\end{equation}
The coadjoint orbits for discrete series have representatives
\begin{equation}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = \begin{cases} ix_{\text{orbit}}e_1
- iy_{\text{orbit}}e_{p+1} & x_{\text{orbit}} > 0 >
y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 + (e_{2,p}-e_{p,2}) & \\
\quad+ (e_{2,p+1}+e_{p+1,2}) & x_{\text{orbit}} > 0 = y_{\text{orbit}}\\
ix_{\text{orbit}}e_1 - iy_{\text{orbit}}e_{p} & x_{\text{orbit}} >
y_{\text{orbit}} > 0\\
iy_{\text{orbit}}e_p + (e_{1,2}-e_{2,1})&\\
\quad +e_{1,p+1}+e_{p+1,1}) & x_{\text{orbit}} = 0 > y_{\text{orbit}}\\
ix_{\text{orbit}}e_p - iy_{\text{orbit}}e_{p+q} & 0 > x_{\text{orbit}} >
y_{\text{orbit}}.
\end{cases}
\end{equation}
Then
\begin{equation}
\pi_{x,y}^{U(p,q)} = \pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
(We have not discussed attaching representations to partly nilpotent
coadjoint orbits like $\lambda(x_{\text{orbit}},0)$ (with
$x_{\text{orbit}} >0$); suffice it to say that the definitions given
above using ${\mathfrak q}_0$ are reasonable ones. It would be equally
reasonable to use instead ${\mathfrak q}_+$. We will see in
\eqref{eq:Aq-big-middle-coincide} that this leads to the same
representation.)
In the orbit method picture the condition \eqref{eq:bigx} simplifies to
\begin{equation}\label{eq:bigxorbit}
x_{\text{orbit}} > y_{\text{orbit}} > 0.
\end{equation}
Similarly, \eqref{eq:smallx} becomes
\begin{equation}\label{eq:smallxorbit}
x_{\text{orbit}} < y_{\text{orbit}}<0.
\end{equation}
Finally, \eqref{eq:middlex} is
\begin{equation}\label{eq:middlexorbit}
x_{\text{orbit}} \ge 0 \ge y_{\text{orbit}};
\end{equation}
equality in either of these inequalities is the case of partially
nilpotent coadjoint orbits.
In all cases we need also the genericity condition
\begin{equation}
\ell_{\text{orbit}} >0 \iff \ell > -(n-1),
\end{equation}
and the integrality conditions
\begin{equation}
x_{\text{orbit}} \equiv (n-1)/2 \pmod{\mathbb Z}, y_{\text{orbit}}
\equiv (n-1)/2 \pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations}
Here now is a sketch of a proof of the irreducibility assertion from
Proposition \ref{prop:OUbranch}. Each of the cohomologically induced
representations above is in the
weakly fair range. The general results for the weakly fair range of
\cite{Vunit} together with \cite{VGLn}*{Section 16} apply to show that
they are irreducible or zero. The key point is that the moment map
for the cotangent bundle to a relevant partial flag variety is
birational onto its image. This is automatic in type $A$, which is why
the arguments in \cite{VGLn} for $GL(n,\mathbb{R})$ also apply to
$U(p,q)$.
We close with a comment about how the three series of derived functor
modules fit together. If we relax the strict inequalities on $x$ (and
$y$) in \eqref{eq:bigx}, then we are at one edge of the weak
inequalities in \eqref{eq:middlex}. For these values of $x$ and $y$
(which occur only when $n$ is odd), namely
\[
(x,y) = \left ( \ell + (n-1)/2, -(n-1)/2 \right),
\]
or equivalently
\[
(x_{\text{orbit}},y_{\text{orbit}}) = \left(
\ell_{\text{orbit}},0\right),
\]
we claim
\begin{equation}\label{eq:Aq-big-middle-coincide}
A_{{\mathfrak q}^{U(p,q)}_+}(\lambda^+_{x,y}) \simeq A_{{\mathfrak
q}^{U(p,q)}_0}(\lambda^0_{x,y}).
\end{equation}
To see this, one can begin by checking they have the same associated
variety: the $U(p,{\mathbb C}) \times U(q,{\mathbb C})$ saturations of
${\mathfrak u}^{U(p,q)}_+ \cap \mathfrak{s}$ and ${\mathfrak
u}^{U(p,q)}_0 \cap \mathfrak{s}$ coincide. (The dense orbit of
$U(p,{\mathbb C}) \times U(q,{\mathbb C})$ is one of the two
possibilities with one Jordan block of size $3$ and the others of size
$1$.) A little further checking shows that they also have the same
annihilator: for $\ell \leq (n-2)/2$, given the associated variety
calculation, there is a unique possibility for the annihilator; a
slightly more refined analysis handles larger $\ell$. Given that
their annihilators and associated varieties are the same, the main
result of \cite{BVUpq} implies \eqref{eq:Aq-big-middle-coincide}.
Similarly, for the other edge of the inequalities in
\eqref{eq:middlex}, namely
\[
(x,y) = \left (-(n-1)/2, \ell+(n-1)/2 \right),
\]
we have
\begin{equation}\label{eq:Aq-small-middle-coincide}
A_{{\mathfrak q}^{U(p,q)}_0}(\lambda^0_{x,y}) \simeq A_{{\mathfrak
q}^{U(p,q)}_-}(\lambda^-_{x,y})
\end{equation}
by a similar argument.
\end{subequations}
\section{Quaternionic hyperboloids}
\label{sec:Sppq}
\setcounter{equation}{0}
\begin{subequations}\label{se:Sppq}
In this section we use the ideas from Section \ref{sec:invt} to investigate the
discrete series of the non-symmetric spherical spaces
\begin{equation}\begin{aligned}
H_{4p,4q} &= \{v\in {\mathbb H}^{p,q} \mid \langle
v,v\rangle_{p,q} = 1\}\\ &= [Sp(p,q)\times Sp(1)]/[Sp(p-1,q)\times
Sp(1)_\Delta].
\end{aligned}\end{equation}
Here $\langle,\rangle_{p,q}$ is the standard Hermitian form of
signature $(p,q)$ on ${\mathbb H}^{p+q}$. We are using the action of
a real form of the enlarged group from \eqref{eq:Spbig}, namely
\begin{equation}
Sp(p,q) \times Sp(1) = Sp(p,q)_{\text{linear}} \times
Sp(1)_{\text{scalar}};
\end{equation}
The inclusion of the last
side of the equality (for $H_{4p,4q}$) in the middle is just given by
the action of this enlarged quaternionic unitary group on the basis
vector $e_1$; surjectivity is Witt's theorem for quaternionic
Hermitian forms. To avoid talking about degenerate cases, we will
assume
\begin{equation}
p, q \ge 2.
\end{equation}
Just as in Section \ref{sec:Upq}, we will simplify many formulas by
writing
\begin{equation}\label{eq:Spnpq}
n = p+q.
\end{equation}
The homogeneous space
$Sp(n)/Sp(n-1)$ has another noncompact real form $[Sp(2n,{\mathbb
R})\times Sp(2,{\mathbb R})]/[Sp(2(n-1),{\mathbb R})\times
Sp(2,{\mathbb R})_\Delta]$, arising from an inclusion
\begin{equation}
Sp(2n,{\mathbb R})\times Sp(2,{\mathbb R})\ \hookrightarrow O(2n,2n).
\end{equation}
This real form certainly has discrete series: we expect that the
discrete summands of the restriction of $\pi_\ell^{O(2n,2n)}$ are
indexed by discrete series representations of $Sp(2,{\mathbb R})$,
just as we find below (for $Sp(p,q)$) that they are indexed by
irreducible representations of the compact group $Sp(1)$. But we have
not carried out this analysis.
Our goal is to restrict the discrete series representations
$\pi^{O(4p,4q)}_\ell$ of \eqref{eq:Opqdisc} to $Sp(p,q)$, and so to
understand some representations in the discrete series of $(Sp(p,q)\times
Sp(1))/(Sp(p-1,q)\times Sp(1))$.
\end{subequations}
\begin{subequations}\label{se:OSpbranch}
We have calculated in Theorem \ref{thm:OSpcptbranch} how the $O(4p)$
and $O(4q)$ representations appearing in \eqref{eq:Obranch} restrict
to $Sp$. The result is
\begin{small}\begin{equation}\label{e:OSpBigBranch}
\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(1)]\times [Sp(q)\times Sp(1)]} =
\sum_{\substack {m=0\\[.1ex] 0 \le k \le m/2}}^\infty \
\sum_{\substack{0\le e \le d\\[.1ex] 0\le e'\le d'\\[.2ex] d+e =
m+\ell+4q\\[.1ex] d'+e' = m-2k}}
\pi^{Sp(p)\times Sp(1)}_{d,e}\otimes \pi^{Sp(q)\times Sp(1)}_{d',e'}.
\end{equation}\end{small}
The group to which we are restricting here is actually a little larger
than the maximal compact subgroup of $Sp(p,q)\times Sp(1)$, which is
\begin{equation}
Sp(p) \times Sp(q) \times Sp(1)_\Delta;
\end{equation}
the subscript $\Delta$ indicates that this $Sp(1)$ factor
(corresponding to scalar multiplication on ${\mathbb H}^{p,q})$) is
diagonal in the $Sp(1)\times Sp(1)$ of \eqref{e:OSpBigBranch}
(corresponding to separate scalar multiplications on ${\mathbb H}^p$
and ${\mathbb H}^q$).
The branching $(G\times G)|_{G_\Delta}$ is tensor product
decomposition, which is very simple for $Sp(1)$. We find
\begin{small}\begin{equation}\label{e:OSpcptBranch}\begin{aligned}
&\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(q)\times Sp(1)]} =\\
&\sum_{\substack {m=0\\[.2ex] 0 \le k \le m/2}}^\infty \
\sum_{\substack{0\le e \le d\\[.1ex] 0\le e'\le d'\\[.2ex] d+e =
m+\ell+4q\\[.1ex] d'+e' = m-2k}}
\sum_{j=0}^{\min(d-e,d'-e')}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_{d+d'-e-e'-2j}.
\end{aligned}\end{equation}\end{small}
It will be useful to rewrite this formula. The indices $m$ and $k$
serve only to bound some of the other indices, so we can eliminate
them by rewriting the bounds. We find
\begin{small}\begin{equation}\label{e:OSpcptBranch2}\begin{aligned}
&\pi_\ell^{O(4p,4q)}|_{[Sp(p)\times Sp(q)\times Sp(1)]} =\\[.5ex]
&\sum_{\substack{0\le e \le d\quad 0\le e'\le d'\\[.4ex] d'+e' \le d+e-\ell-4q
\\[.4ex] d'+e' \equiv d+e - \ell \pmod{2}}}
\sum_{\substack{|(d-e)-(d'-e')|\le f \\[.2ex] \quad \le
(d-e)+(d'-e')\\[.4ex] \quad f\equiv
(d-e)+(d'-e')\pmod{2}}}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_f.
\end{aligned}\end{equation}\end{small}
For each of these representations of $K$, define integers $x$ and $y$
by solving the equations
\begin{equation}
x+y = \ell, \qquad x-y = f.
\end{equation}
The congruence condition on $f$ guarantees that $x$ and $y$ are
indeed integers. Conversely, given any integers $x$ and $y$ satisfying
\begin{equation}
x+y=\ell, \qquad x\ge y
\end{equation}
we can define
\begin{equation}
\begin{split}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = \text{subrepresentation of
$\pi^{O(4p,4q)}_\ell|_{Sp(p,q)\times Sp(1)}$}\\
\; \text{where $Sp(1)$ acts
with infl. char. $x-y+1$.}
\end{split}
\end{equation}
Equivalently, we are asking that $Sp(1)$ act by a multiple of
$\pi^{Sp(1)}_{x-y}$.
\end{subequations}
This calculation, together with Corollary \ref{cor:Hdiff}, proves most of
\begin{proposition}\label{prop:OSpbranch} Suppose $p$ and $q$ are
nonnegative integers, each at least two; and suppose $\ell > -2n
+1$. Then the restriction of the discrete series representation
$\pi^{O(4p,4q)}_\ell$ to $Sp(p,q)\times Sp(1)$ is the direct sum of
the one-parameter family of representations
$$\pi^{Sp(p,q)\times Sp(1)}_{x,y},\quad
x\ge y \in {\mathbb Z}, \quad x+y =
\ell.$$
The infinitesimal character of $\pi^{Sp(p,q)\times Sp(1)}_{x,y}$ corresponds to
the weight
$$(x+n, y+n-1,n-2,\ldots,1)(x-y+1).$$
Restriction to the maximal compact subgroup is
\begin{small}\begin{equation*}\begin{aligned}
& \pi^{Sp(p,q)\times Sp(1)}_{x,y}|_{Sp(p)\times Sp(q) \times Sp(1)} =\\[.5ex]
&\sum_{\substack{0\le e \le d\quad 0\le e'\le d'\\[.4ex] d'+e' \le d+e-(x+y)-4q
\\[.4ex] d'+e' \equiv d+e - (x+y) \pmod{2}}}
\sum_{\substack{|(d-e)-(d'-e')|\le x-y \\[.2ex] x-y \le
(d-e)+(d'-e')\\[.4ex] \quad
(d-e)+(d'-e') \equiv x-y \pmod{2}}}
\pi^{Sp(p)}_{d,e}\otimes \pi^{Sp(q)}_{d',e'}\otimes \pi^{Sp(1)}_{x-y}.
\end{aligned}\end{equation*}\end{small}
Each of the representations $\pi^{Sp(p,q)}_{x,y}$ is irreducible.
\end{proposition}
We will prove the irreducibility assertions (using \cite{Kob}) after
\eqref{eq:Aq-Spsmallx} below.
\begin{subequations}\label{se:Sppqrep}
Having identified the restriction to $Sp(p)\times Sp(q)\times Sp(1)$,
we want to record Kobayashi's identification of the actual
representations of $Sp(p,q)\times Sp(1)$. These come in two
families, according to
the values of the integers $x$ and $y$. The families are
cohomologically induced from two $\theta$-stable parabolic
subalgebras. The first is
\begin{equation}
{\mathfrak q}_+^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_+^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_+^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_+^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_q\times Sp(p-1,q-1)] \times U(1).
\end{equation}
(The first three factors are in $Sp(p,q)$. We write $U(1)_p$ for a
coordinate $U(1) \subset U(p)$, and $U(1)_q \subset U(q)$ similarly.)
The second parabolic is
\begin{equation}
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)} = {\mathfrak l}_0^{Sp(p,q)\times
Sp(1)} + {\mathfrak u}_0^{Sp(p,q)\times Sp(1)} \subset
{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C}),
\end{equation}
with Levi subgroup
\begin{equation}
L_0^{Sp(p,q)\times Sp(1)} = [U(1)_p\times U(1)_p\times Sp(p-2,q)] \times U(1).
\end{equation}
More complete descriptions of these parabolics are in \cite{Kob}.)
Suppose first that
\begin{equation}\label{eq:Spbigx}
x> \ell + (n-1), \qquad y < -(n-1).
\end{equation}
(Since $x+y=\ell$, these two inequalities are equivalent.) Write
$\xi_x$ for the character of $U(1)$ corresponding to $x\in {\mathbb Z}$.
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spbigx}
\lambda^+_{x,y} = \left[\xi_x\otimes \xi_{-(y+2n-2)} \otimes
1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_+$. What Kobayashi proves in \cite{Kob}*{Theorem 6.1}
is
\begin{equation}\label{eq:Aq-Spbigx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_+}(\lambda^+_{x,y}) \qquad x > \ell + (n-1).
\end{equation}
Suppose next that
\begin{equation}\label{eq:Spsmallx}
\ell + (n-1) \ge x > \ell/2, \qquad -(n-1)\le y < \ell/2.
\end{equation}
(Since $x+y=\ell$, these two pairs of inequalities are equivalent.)
Consider the one-dimensional character
\begin{equation}\label{eq:lam-Spsmallx}
\lambda^0_{x,y} = \left[\xi_x\otimes \xi_y\otimes 1\right]\otimes \xi_{x-y}
\end{equation}
of $L^{Sp(p,q)\times Sp(1)}_0$. Kobayashi's result in \cite{Kob}*{Theorem 6.1}
is now
\begin{equation}\label{eq:Aq-Spsmallx}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_0}(\lambda^0_{x,y})
\qquad \ell/2 < x \le \ell + (n-1)).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Sppqorbit}
Here is the orbit method perspective. Use a
trace form to identify ${\mathfrak g}_0^*$ with ${\mathfrak
g}_0$. Linear functionals vanishing on ${\mathfrak h}_0^*$ are
quaternionic matrices
\begin{equation}
\lambda(z,u,v) = \left[\begin{pmatrix}z & \hskip -2ex u& \hskip -2ex
v\\[.5ex] -\overline u &
\\ \overline v & \text{\Large \hskip 3ex $0_{(n-1)\times (n-1)}$ \hskip
-2ex}\\ \end{pmatrix},-z\right] \simeq
{\mathfrak s}{\mathfrak p}(1) + {\mathbb H}^{p-1,q}
\end{equation}
with $z\in {\mathfrak s}{\mathfrak p}(1)$ (the purely imaginary
quaternions), $u \in {\mathbb H}^{p-1}$, $v\in {\mathbb H}^q$.
The orbits of $H=Sp(p-1,q)\times Sp(1)_\Delta$ of largest dimension are given
by $|z|$, and the value of the Hermitian form on the vector $(u,v)$:
positive for the orbits represented by nonzero elements
$r(e_{12}-e_{21})$ (nonzero eigenvalues $i(|z|\pm a)/2$, with $a=(|z|^2 +
4r^2)^{1/2}$); negative for nonzero elements $s(e_{1,p+1}+ e_{p+1,1})$
(nonzero eigenvalues $i(|z|\pm a)/2$, with $a=(|z|^2 - 4s^2)^{1/2}$); and
zero for the nilpotent element $(e_{12}-e_{21}+e_{1,p+1}+e_{p+1,1})$
(nonzero eigenvalues $i|z|/2$).
Define
\begin{equation}
\ell_{\text{orbit}} = \ell + (2n-1), \quad x_{\text{orbit}} = x +
n, \quad y_{\text{orbit}} = y + n-1.
\end{equation}
The coadjoint orbits for discrete series have representatives
\begin{equation}
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = \begin{cases} [ix_{\text{orbit}}e_1
-iy_{\text{orbit}}e_{p+1},i(x_{\text{orbit}} - y_{\text{orbit}})]&\\
\qquad x_{\text{orbit}} > 0 > y_{\text{orbit}}&\\[.5ex]
[ix_{\text{orbit}}e_1 +
(e_{2,p}-e_{p,2}+e_{2,p+1}+e_{p+1,2}),ix_{\text{orbit}}] &\\
\qquad x_{\text{orbit}} > 0 = y_{\text{orbit}}&\\[.5ex]
[ix_{\text{orbit}}e_1 + iy_{\text{orbit}}e_2,i(x_{\text{orbit}} -
y_{\text{orbit}})] & \\
\qquad x_{\text{orbit}} > y_{\text{orbit}} > 0 &
\end{cases}
\end{equation}
Then
\begin{equation}
\pi_{x,y}^{Sp(p,q)} = \pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})).
\end{equation}
(The partly nilpotent
coadjoint orbits $\lambda(x_{\text{orbit}},0)$ (with
$x_{\text{orbit}} >0$) can be treated as for $U(p,q)$.)
In the orbit method picture the condition \eqref{eq:Spbigx} simplifies to
\begin{equation}\label{eq:Spbigxorbit}
x_{\text{orbit}} > 0 > y_{\text{orbit}}.
\end{equation}
Similarly, \eqref{eq:Spsmallx} becomes
\begin{equation}\label{eq:Spsmallxorbit}
x_{\text{orbit}} > y_{\text{orbit}} \ge 0;
\end{equation}
equality in the inequality is the case of partially nilpotent coadjoint orbits.
In all cases we need also the genericity condition
\begin{equation}
\ell_{\text{orbit}} >0 \iff \ell > -(2n-1), \qquad
x_{\text{orbit}}-y_{\text{orbit}}>0 \iff x-y+1 > 0
\end{equation}
and the integrality conditions
\begin{equation}
x_{\text{orbit}} \equiv n \pmod{\mathbb Z},\qquad y_{\text{orbit}}
\equiv n-1 \pmod{\mathbb Z}.
\end{equation}
\end{subequations}
\begin{subequations}\label{se:Sppqirr}
Here is a sketch of proof of the irreducibility assertion from
Proposition \ref{prop:OSpbranch}. Each of the cohomologically induced
representations above is in the weakly fair range, so the general
theory of \cite{Vunit} applies. One conclusion of this theory is that
the cohomologically induced representations are irreducible modules
for a certain twisted differential operator algebra ${\mathcal
D}_{x,y}$; but in contrast to the $U(p,q)$ case, the natural map
$$U({\mathfrak s}{\mathfrak p}(p+q,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})) \rightarrow {\mathcal D}_{x,y}$$
{\em need not} be surjective:
some of the cohomologically induced modules corresponding to discrete
series for $[Sp(2n,R)/Sp(2n-4,R)\times Sp(2,R)_\Delta$ are
{\em reducible}.
Here is an irreducibility proof for the case
\eqref{eq:Aq-Spsmallx}. We begin by defining
\begin{equation}
{\mathfrak q}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} = {\mathfrak
l}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} + {\mathfrak
u}_{0,\text{big}}^{Sp(p,q)\times Sp(1)} \supset
{\mathfrak q}_0^{Sp(p,q)\times Sp(1)}
\end{equation}
with Levi subgroup
\begin{equation}
L_{0,big}^{Sp(p,q)\times Sp(1)} = [U(2)_p\times Sp(p-2,q)] \times U(1).
\end{equation}
Define
\begin{equation}
\lambda^{0,\text{big}}_{x,y} = \left[\pi^{U(2)}_{x,y}\otimes
1\right]\otimes \xi_{x-y}.
\end{equation}
Induction by stages proves that
\begin{equation}
\pi^{Sp(p,q)\times Sp(1)}_{x,y} = A_{{\mathfrak q}^{Sp(p,q)\times
Sp(1)}_{0,\text{big}}}(\lambda^{0,\text{big}}_{x,y})
\qquad \ell/2 < x \le \ell + (n-1)).
\end{equation}
In this realization, the irreducibility argument from the $U(p,q)$
case goes through. The moment map from the cotangent bundle of the
(smaller) partial flag variety {\em is} birational onto its (normal)
image; so the map
\[U({\mathfrak s}{\mathfrak p}(p+q,{\mathbb C}) \times {\mathfrak
s}{\mathfrak p}(1,{\mathbb C})) \rightarrow {\mathcal
D}^{\text{small}}_{x,y}\]
is surjective, proving irreducibility. (The big parabolic subalgebra
defines a small partial flag variety, which is why we label the
twisted differential operator algebra ``small.'')
This argument does not apply to the case \eqref{eq:Aq-Spbigx},
since the corresponding larger Levi subgroup has a factor $U(1,1)$,
and the corresponding representation there is a discrete series. In
that case we have found only an unenlightening computational argument
for the irreducibility, which we omit.
Finally, the two series of derived functor modules fit together as
follows. If we consider the edge of the inequalities in
\eqref{eq:Spbigx} and \eqref{eq:Spsmallx}, namely
\[
(x,y) = \left ( \ell + (n-1), -(n-1) \right ),
\]
then we have
\begin{equation}\label{eq:Aq-Sp-coincide}
A_{{\mathfrak q}^{Sp(p,q)\times Sp(1)}_+}(\lambda^+_{x,y}) =
A_{{\mathfrak q}^{Sp(p,q)\times Sp(1)}_0}(\lambda^0_{x,y}).
\end{equation}
For this equality, as for the irreducibility of $A_{{\mathfrak
q}^{Sp(p,q)\times Sp(1)}_+}(\lambda^+_{x,y})$, we have found only
an unenlightening computational argument, which we omit.
\end{subequations}
\section{Octonionic hyperboloids}
\label{sec:spinp9-p}
\setcounter{equation}{0}
\begin{subequations}\label{se:spinp9-p}
We look for noncompact forms of the non-symmetric spherical space
$$S^{15} = \Spin(9)/\Spin(7)'$$
studied in Section \ref{sec:O}. The map from $\Spin(p,q)$ (with
$p+q=9$) to a form of
$O(16)$ will be given by the spin representation, which is therefore
required to be real. The spin representation is real if and only if
$p+q$ and $p-q$ are each congruent to $0$, $1$, or $7$ modulo $8$. The
candidates are
\begin{equation}
G=\Spin(5,4) \quad\text{or}\quad G=\Spin(8,1),
\end{equation}
with maximal compact subgroups
\begin{equation}
K = \Spin(5)\times_{\{\pm 1\}} \Spin(4) \quad\text{or}\quad \Spin(8);
\end{equation}
in the first case this means that the natural central subgroups $\{\pm 1\}$ in
$\Spin(5)$ and $\Spin(4)$ are identified with each other (and with the
natural central $\{\pm 1\}$ in $\Spin(5,4)$). In each case the
sixteen-dimensional spin representation of $G$ is real and preserves a
quadratic form of signature $(8,8)$. One way to see this is to notice
that the restriction of the spin representation to $K$ is a sum
of two irreducible representations
\begin{equation}
\spin(5) \otimes \spin(4)_{\pm} \quad \text{or} \quad \spin(8)_\pm
\end{equation}
Here $\spin(2m)_\pm$ denotes
the two half-spin representations, each of dimension $2^{m-1}$, of
$\Spin(2m)$. We are therefore looking at the hyperboloid
\begin{equation}\begin{aligned}
H_{8,8} &= \{v\in {\mathbb R}^{8,8} \mid \langle v,v\rangle_{8,8} = 1\}\\
&= \Spin(5,4)/\Spin(3,4)'\quad \text{or}\\
&=\Spin(8,1)/\Spin(7)'.
\end{aligned}\end{equation}
\end{subequations}
\begin{subequations}\label{se:spin81}
The discrete series for the second case was described
by Kobayashi in connection with branching from $SO(8,8)$ to $\Spin(8,1)$ in
\cite[Section 5.2]{toshi:howe}. Here we carry out an
approach
using the development above.
The harmonic analysis problem is
\begin{equation}
L^2(H_{8,8}) \simeq L^2(\Spin(8,1))^{\Spin(7)'};
\end{equation}
the $\Spin(7)'$ action is on the right. This problem is resolved by
Harish-Chandra's Plancherel formula for $\Spin(8,1)$: the discrete
series are exactly those of Harish-Chandra's discrete series that
contain a $\Spin(7)'$-fixed vector, and the multiplicity is the
dimension of that fixed space. Because of Helgason's branching law
from $\Spin(7)'$ to $\Spin(8)$ \eqref{eq:87'}, the number in question is
the sum of the multiplicities of the $\Spin(8)$ representations of
highest weights
\begin{equation}
\mu_y = (y/2,y/2,y/2,y/2) \qquad (y \in {\mathbb N}).
\end{equation}
Corollary \ref{cor:octdiff} constrains the possible infinitesimal
characters, and therefore the Harish-Chandra parameters, of
representations appearing on this hyperboloid. Here are the discrete
series having these infinitesimal characters. Suppose $x$ is an integer
satisfying $2x+y+7 >0$. Define
\begin{equation}
\pi^{\Spin(8,1)}_{x,y,\pm} = \begin{cases} \substack{\text{discrete
series with
parameter}\\((2x+y+7)/2,(y+5)/2,(y+3)/2,\pm(y+1)/2)}& x \ge 0\\[.5ex]
\qquad\qquad 0 & 0> x > -4\\[.5ex]
\substack{\text{discrete series with
parameter}\\ ((y+5)/2,(y+3)/2,(y+1)/2,\pm(2x+y+7)/2)} &
-4 \ge x > -(y+7)/2.
\end{cases} \end{equation}
We can now use Blattner's formula to determine which of these discrete
series contain $\Spin(8)$ representations of highest weight
$\mu_y$. The representations with a subscript $-$ are immediately
ruled out (since the last coordinate of the highest weight of any
$K$-type of such a discrete series must be negative). Similarly, in
the first case with $+$ the lowest $K$-type has highest weight
$(2x+1,1,1,1)+\mu_y$, and all other highest weights of $K$-types arise
by adding positive integers to these coordinates; so $\mu_y$ cannot
arise.
In the third case with $+$ the lowest $K$-type has highest weight
$(0,0,0,x+4)+\mu_y$; we get to $\mu_y$ by adding the nonnegative
multiple $-x-4$ of the noncompact positive root $e_4$. A more careful
examination of Blattner's formula shows that in fact $\mu_y$ has
multiplicity one. This proves
\begin{equation}\label{eq:spin81ds}
L^2(H_{8,8})_{\text{disc}} = \sum_{
y\ge 1,\ -4 \ge x > -(y+7)/2}
\pi^{\Spin(8,1)}_{x,y,+}.
\end{equation}
Furthermore (by Corollary \ref{cor:octdiff})
\begin{equation}
\pi^{O(8,8)}_\ell|_{\Spin(9,1)} = \sum_{\substack{
y\ge 1,\ -4 \ge x \ge -(y+7)/2\\[.4ex] 2x+y=\ell}}
\pi^{\Spin(8,1)}_{x,y,+}.
\end{equation}
These discrete series are cohomologically induced from
one-dimensional characters of the spin double cover of the compact
Levi subgroup
\begin{equation}
SO(2)\times U(3) \subset SO(2) \times SO(6) \subset SO(8) \subset
SO(8,1).
\end{equation}
\end{subequations}
\begin{subequations}\label{se:spin81orbit}
Here is the orbit method perspective. We have
\[\label{eq:spin81orbits}
({\mathfrak g}_0/{\mathfrak h}_0)^* \simeq \Spin^8 + {\mathbb R}^7
\]
as a representation of $H=\Spin(7)'$; the first summand is the
$8$-dimensional spin representation. What distinguishes this from the
compact case analyzed in \eqref{se:Oorbit} is that the restriction of
the natural $G$-invariant form has opposite signs on the two
summands; we take it to be negative on the first and positive on the
second. Because of \eqref{eq:GmodH},
the orbits we want are represented by $H$ orbits of maximal dimension
on this space. A generic orbit on ${\mathbb R}^7$ is given by the
value of the quadratic form
length $a_7 > 0$, and the corresponding isotropy group is
$\Spin(6)'\simeq SU(4)$. As a representation of $SU(4)$,
\[
\Spin^8 \simeq {\mathbb C}^4
\]
regarded as a real vector space. Here again the nonzero orbits are
indexed by the value of the Hermitian form $b_{\text{spin}} < 0$. The
conclusion is
that the regular $H$ orbits on $({\mathfrak g}_0/{\mathfrak h}_0)^*$
are
\[
\lambda(a_7,b_{\text{spin}}) \qquad (a_7> 0, \quad b_{\text{spin}} < 0).
\]
It turns out that the eigenvalues of such a matrix are $\pm i(a_7/4)^{1/2}$
(repeated three times), $\pm i(a_7/4 +b_{\text{spin}})^{1/2}$, and
one more eigenvalue zero. Accordingly the element is elliptic if and
only if $a_7/4 + b_{\text{spin}} \ge 0$. In this case we write
\[
x_{\text{orbit}} = (a_7/4 + b_{\text{spin}})^{1/2} -a_7^{1/2}/2, \quad
y_{\text{orbit}} = a^{1/2}_7 \qquad (a_7/4 + b_{\text{spin}} \ge 0)
\]
The elliptic elements we want are
\[
\lambda(x_{\text{orbit}},y_{\text{orbit}}) = (y_{\text{orbit}}/2,
y_{\text{orbit}}/2, y_{\text{orbit}}/2, y_{\text{orbit}}/2 +x), \qquad
(y_{\text{orbit}}/2 > -x_{\text{orbit}} > 0);
\]
we have represented the element (in fairly standard coordinates) by
something in the dual of a compact Cartan subalgebra $[{\mathfrak
s}{\mathfrak o}(2)]^4$ to which it is conjugate.
If now we define
\[
y = y_{\text{orbit}} -3, \quad x = x_{\text{orbit}} -2,
\]
then
\begin{equation}
\pi_{x,y,+}^{\Spin(8,1)} =
\pi(\text{orbit}\ \lambda(x_{\text{orbit}},y_{\text{orbit}})) \qquad
(0 > x_{\text{orbit}} > -y_{\text{orbit}}).
\end{equation}
When $y_{\text{orbit}} = 1$ or $2$ or $3$, or $x_{\text{orbit}}=-1$, these
representations are zero; that is the source of the conditions
$$y_{\text{orbit}} \ge 4, \quad -2 \ge x_{\text{orbit}} -
y_{\text{orbit}}/2$$
in \eqref{eq:spin81ds}.
\end{subequations}
\begin{subequations}\label{se:spin54}
In the first case of \eqref{se:spinp9-p}, we are looking at
\begin{equation}
H_{8,8} \simeq \Spin(5,4))/{\Spin(4,3)'};
\end{equation}
this is the ${\mathbb R}$-split version of Section \ref{sec:O}, and so
arises from
\begin{equation}
\Spin(4,3)' \ {\buildrel{\text{spin}}\over
\longrightarrow}\ \Spin(4,4) \subset \Spin(5,4).
\end{equation}
We have not determined the discrete series for this homogeneous space;
of course we expect two-parameter families of representations
cohomologically induced from one-dimensional characters of spin double
covers of real forms of $SO(2)\times U(3)$.
\end{subequations}
\section{The split $G_2$ calculation}
\label{sec:G2s}
\setcounter{equation}{0}
\begin{subequations}\label{se:G2hyp}
Write $G_{2,s}$ for the $14$-dimensional split Lie group
of type $G_2$. There is a $7$-dimensional real representation
$(\tau_{{\mathbb R},s},W_{{\mathbb R},s})$ of
$G_{2,s}$, whose weights are zero and the six short
roots. This preserves an inner product of signature $(4,3)$, and so
defines an inclusion
\begin{equation}\label{eq:G2SO7nc}
G_{2,s} \hookrightarrow SO(4,3).
\end{equation}
The corresponding actions of $G_{2,s}$ on the hyperboloids
\begin{equation}
H_{4,3} = O(4,3)/O(3,3), \qquad H_{3,4} = O(3,4)/O(2,4)
\end{equation}
are transitive. The isotropy groups are real forms of $SU(3)$:
\begin{equation}
H_{4,3} \simeq G_{2,s}/SL(3,{\mathbb R}), \qquad H_{3,4} \simeq
G_{2,s}/SU(2,1).
\end{equation}
\end{subequations}
The discrete series for these cases are given by Kobayashi (up to
two questions of reducibility) in
\cite[Thm 6.4]{Kob} ; see also \cite[Theorem 3.5]{toshi:zuckerman}.
We now give a self-contained treatment of the
classification, and resolve the reducibility.
\begin{subequations}\label{se:G2sreps}
The (real forms of) $O(7)$ representations appearing on these
hyperboloids are all related to the flag variety
\begin{equation}\begin{aligned}
O(7,{\mathbb C})/P &= \text{isotropic lines in\ }{\mathbb C}^7, \\
P = MN,\qquad M&=GL(1,{\mathbb C}) \times O(5,{\mathbb C}).
\end{aligned}\end{equation}
What makes everything simple is that $G_2({\mathbb C})$ is transitive
on this flag variety:
\begin{equation}\begin{aligned}
\text{isotropic lines in\ }{\mathbb C}^7 &= G_2({\mathbb C})/Q, \\
Q=LU, \qquad L &= GL(2,{\mathbb C}).
\end{aligned}\end{equation}
Precisely, the discrete series for $H_{4,3}$ are cohomologically
induced from the $\theta$-stable parabolic
\begin{equation}
{\mathfrak p}_1 = {\mathfrak m}_1 + {\mathfrak n}_1, \qquad
M_1=SO(2)\times O(2,3).
\end{equation}
The discrete series representations are
\begin{equation}
\pi^{O(4,3)}_{1,\ell} = A_{{\mathfrak p}_1}(\lambda_1(\ell)), \qquad \ell+5/2> 0.
\end{equation}
(cf. \eqref{se:Upq}). The inducing representation is the $SO(2)$
character indexed by $\ell$, and trivial on $O(2,3)$. Similarly, the
discrete series for $H_{3,4}$ are cohomologically
induced from the $\theta$-stable parabolic
\begin{equation}
{\mathfrak p}_2 = {\mathfrak m}_2 + {\mathfrak n}_2, \qquad
M_2=SO(2)\times O(1,4).
\end{equation}
The discrete series are
\begin{equation}
\pi^{O(3,4)}_{2,\ell} = A_{{\mathfrak p}_2}(\lambda_2(\ell)),\qquad \ell+5/2 > 0.
\end{equation}
The intersections of these parabolics with $G_2$ are
\begin{equation}
{\mathfrak q}_1 = {\mathfrak l}_1 + {\mathfrak u}_1, \qquad
L_1 = \text{long root $U(1,1)$}.
\end{equation}
and
\begin{equation}
{\mathfrak q}_2 = {\mathfrak l}_2 + {\mathfrak u}_2, \qquad
L_2= \text{long root $U(2)$}.
\end{equation}
(The Levi subgroups are just {\em locally} of this form.)
Because the $G_2$ actions on the $O(4,3)$ partial flag varieties are
transitive, we get discrete series representations for $H_{4,3}$
\begin{equation}
\pi^{G_{2,s}}_{1,\ell} = A_{{\mathfrak q}_1}(\lambda_1(\ell)),\qquad \ell+5/2 > 0.
\end{equation}
The character is $\ell$ times the action of $L_1$ on the highest short
root defining ${\mathfrak q}_1$.
Similarly, for the action on $H_{3,4}$
\begin{equation}
\pi^{G_{2,s}}_{2,\ell} = A_{{\mathfrak q}_2}(\lambda_2(\ell)),\qquad \ell +5/2 > 0.
\end{equation}
The {\tt atlas} software \cite{atlas} tells us that all of these
discrete series representations of $G_2$ are irreducible, with the
single exception of $\pi^{G_{2,s}}_{1,-2} = A_{{\mathfrak
q}_1}(\lambda_1(-2)).$ That representation is a sum of two
irreducible constituents. One constituent is the unique non-generic
limit of discrete series of infinitesimal character a short root. In
\cite{G2}*{Theorem 18.5}, (describing some of Arthur's unipotent
representations) this is the representation described in (b). The
other constituent is described in part (c) of that same theorem. The
irreducible representation $\pi^{G_{2,s}}_{2,-2} = A_{{\mathfrak
q}_2}(\lambda_2(-2))$ appears in part (a) of the
theorem. All of these identifications (including the reducibility of
$\pi^{G_{2,s}}_{1,-2}$) follow from knowledge of the
$K$-types of these representations (given in \eqref{se:G2sK} below)
and the last assertion of \cite{G2}*{Theorem 18.5}.
Summarizing, in the notation of \cite{G2},
\begin{equation}\label{eq:G2unip}
\pi^{G_{2,s}}_{1,-2} \simeq J_{-}(H_2; (2, 0)) \oplus J(H_2; (1,1)),
\qquad \pi^{G_{2,s}}_{2,-2} \simeq J(H_1; (1,1)).
\end{equation}
That is, the first discrete series for these non-symmetric spherical
spaces include three of the five unipotent representations for the
split $G_2$ attached to the principal nilpotent in $SL(3) \subset
G_2$.
\end{subequations}
\begin{subequations}\label{G2sorbit}
Here is the orbit method perspective. For the case of $H_{4,3}$, the
representation of $H=SL(3,{\mathbb R})$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb R}^3 + ({\mathbb
R}^3)^*$. The generic orbits of $H$ are indexed by non-zero real
numbers $A$, the value of a linear functional on a vector. We can
arrange the normalizations so that the elliptic elements are exactly
those with $A>0$; if we define
\[ \ell_{\text{orbit}} = A^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 5/2, \]
and write $\lambda_1(\ell_{\text{orbit}})$ for a representative of this
orbit, then
\[
\pi_{1,\ell}^{G_{2,s}} =
\pi({\text{orbit}},\lambda_1(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
For the case of $H_{3,4}$, the representation of $H=SU(2,1)$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb C}^{2,1}$; generic orbits
are parametrized by the nonzero values $B$ of the Hermitian form of signature
$(2,1)$. The elliptic orbits are those with $B>0$; if we define
\[ \ell_{\text{orbit}} = B^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 5/2 \]
then
\[
\pi_{2,\ell}^{G_{2,s}} =
\pi({\text{orbit}},\lambda_2(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
\end{subequations}
\begin{subequations} \label{se:G2sK}
We conclude this section by calculating the restrictions to
\begin{equation}
K = SU(2)_{\text{long}} \times_{\{\pm 1\}} SU(2)_{\text{short}}
\subset G_{2,s}.
\end{equation}
We define
\begin{equation}\begin{aligned}
\gamma_d^{\text{long}} &= \text{$(d+1)$-diml irr of
$SU(2)_{\text{long}}$}\\
\gamma_d^{\text{short}} &= \text{$(d+1)$-diml irr of
$SU(2)_{\text{short}}$}
\end{aligned}\end{equation}
The maximal compact of $O(4,3)$ is $O(4)\times O(3)$. The embedding of
$G_{2,s}$ sends $SU(2)_{\text{long}}$ to one of the factors in
$$O(4) \supset SO(4) \simeq SU(2)\times_{\{\pm 1\}} SU(2),$$
and sends $SU(2)_{\text{short}}$ diagonally into the product of the
other $SU(2)$ factor and $SO(3)\subset O(3)$ (by the two-fold cover
$SU(2) \rightarrow SO(3)$). According to \eqref{eq:Obranch},
\begin{equation}\begin{aligned}
\pi^{O(4,3)}_{1,\ell}|_{O(4)\times O(3)} = \sum_{\substack{d-\ell-3
\ge e \ge 0 \\[.2ex] e\equiv d-\ell-3 \pmod{2}}} \pi^{O(4)}_d \otimes
\pi^{O(3)}_e\\
\pi^{O(3,4)}_{2,\ell}|_{O(3)\times O(4)} = \sum_{\substack{d' -\ell -4
\ge e' \ge 0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}} \pi^{O(3)}_{d'} \otimes
\pi^{O(3)}_{e'}.
\end{aligned}\end{equation}
By an easy calculation, we deduce
\begin{equation}\begin{aligned}
\pi^{G_{2,s}}_{1,\ell}|_K = \sum_{\substack{d-\ell-3 \ge e\ge 0\\[.2ex]
e\equiv d-\ell-3 \pmod{2}}} \gamma_d^{\text{long}} \otimes
\left[\gamma_d^{\text{short}} \otimes
\gamma_{2e}^{\text{short}}\right].\\
\pi^{G_{2,s}}_{2,\ell}|_K = \sum_{\substack{d'-\ell-4 \ge e'\ge
0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}}
\gamma^{\text{long}}_{e'} \otimes \left[ \gamma_{e'}^{\text{short}} \otimes
\gamma_{2d'}^{\text{short}}\right]
\end{aligned}\end{equation}
The internal tensor products in the short $SU(2)$ factors are of
course easy to compute:
\begin{equation}
\pi^{G_{2,s}}_{1,\ell}|_K = \sum_{\substack{d-\ell-3 \ge e\ge
0\\[.2ex] e\equiv d-\ell-3\pmod{2}}} \sum_{k=0}^{\min(d,2e)}
\gamma^{\text{long}}_{e'} \otimes \gamma_{d+2e-2k}^{\text{short}},
\end{equation}
\begin{equation}
\pi^{G_{2,s}}_{2,\ell}|_K = \sum_{\substack{d'-\ell-4 \ge e'\ge
0\\[.2ex] e'\equiv d'-\ell-4\pmod{2}}} \sum_{k'=0}^{e'}
\gamma^{\text{long}}_{e'} \otimes \gamma_{2d'+e'-2k'}^{\text{short}}
\end{equation}
\end{subequations}
\section{The noncompact big $G_2$ calculation}
\label{sec:bigncG2}
\setcounter{equation}{0}
\begin{subequations}\label{se:bigncG2hyp}
In this section we look at noncompact forms of $S^7 \simeq
\Spin(7)'/G_{2,c}$ from Section \ref{sec:bigG2}. The noncompact forms
of $\Spin(7)$ are $\Spin(p,q)$ with $p+q=7$, having maximal compact
subgroups $\Spin(p)\times_{\{\pm 1\}} \Spin(q)$. None of these compact
subgroups can contain $G_{2,c}$ (unless $pq=0$), so the isotropy subgroup we are
looking for is the split form $G_{2,s}$. The seven-dimensional
representation of $G_{2,s}$ is real, and its invariant bilinear form
is of signature $(3,4)$; so we are looking at
\begin{equation}
G_{2,s} \hookrightarrow \Spin(3,4),
\end{equation}
the double cover of the inclusion \eqref{eq:G2SO7nc}. This homogeneous
space is discussed briefly in \cite{Kob}*{Corollary 5.6(e)}, which is proven
in part (ii) of the proof on page 197. We will argue along similar
lines, but get more complete conclusions (parallel to
Kobayashi's results described in Sections
\ref{sec:Upq}--\ref{sec:Sppq}).
The
eight-dimensional spin representation of $\Spin(3,4)$ is real and of
signature $(4,4)$, so we get
\begin{equation}
\Spin(3,4)' \hookrightarrow \Spin(4,4), \qquad \Spin(3,4)'\cap
\Spin(3,4) = G_{2,s}.
\end{equation}
The $\Spin(3,4)'$ action on
\begin{equation}
H_{4,4} = \Spin(4,4)/\Spin(3,4)
\end{equation}
is transitive, so
\begin{equation}
H_{4,4}\simeq \Spin(3,4)'/G_{2,s}.
\end{equation}
In a similar fashion, we find an identification of six-dimensional
complex manifolds
\begin{equation}\label{eq:qmatchbigG2}
\Spin(4,4)/[\Spin(2)\times_{\{\pm 1\}}] \Spin(2,4)] \simeq
\Spin(3,4)'/\widetilde{U(1,2)}.
\end{equation}
The manifold on the left corresponds to the $\theta$-stable parabolic
${\mathfrak q}^{O(4,4)}$ described in \eqref{eq:qOpq}; the discrete
series $\pi^{O(4,4)}_\ell$ for $H_{4,4}$ are obtained from it by
cohomological induction.
The manifold on the right corresponds to the $\theta$-stable parabolic
\begin{equation}\label{eq:qSpin34}
{\mathfrak q}^{\Spin(3,4)'} = {\mathfrak l}^{\Spin(3,4)'} +
{\mathfrak u}^{\Spin(3,4)'} \subset
{\mathfrak o}(7,{\mathbb C});
\end{equation}
the corresponding Levi subgroup is
\begin{equation}
L^{\Spin(3,4)'} = \widetilde{U(1,2)}
\end{equation}
The covering here is the ``square root of determinant'' cover; the
one-dimensional characters are half integer powers of the
determinant. We are interested in
\begin{equation}\begin{aligned}
\lambda_\ell &= {\det}^{\ell/2} \in [L^{\Spin(3,4)'}]\,\widehat{\ }
\qquad (\ell +3 > 0).\\
\pi^{\Spin(3,4)'}_\ell &= A_{{\mathfrak q}^{\Spin(3,4)}}(\lambda_\ell) \qquad
(\ell > -3).
\end{aligned}\end{equation}
The infinitesimal character of this representation is
\begin{equation}
\text{infl char}(\pi_\ell^{\Spin(4,3)'}) = ((\ell+5)/2,(\ell+3)/2,(\ell+1)/2).
\end{equation}
As a consequence of \eqref{eq:qmatchbigG2},
\begin{equation}
\pi^{O(4,4)}_\ell|_{\Spin(3,4)'} \simeq \pi^{\Spin(3,4)'}_\ell.
\end{equation}
The discrete part of the Plancherel decomposition is therefore
\begin{equation}\label{eq:bigncG2disc}
L^2(H_{4,4})_{\text{disc}} = \sum_{\ell > -3}
\pi_\ell^{\Spin(3,4)'}.
\end{equation}
The ``weakly fair'' range for $\pi^{\Spin(3,4)'}_\ell$ is $\ell \ge
-3$, so all the representations $\pi^{\Spin(3,4)'}_\ell$ are
contained in the weakly fair range. In particular, \cite{Vunit}
establishes {\em a priori} the unitarity of what turn out to be the
discrete series representations.
But the results in \cite{Vunit} prove only
\begin{equation}\label{eq:bigG2unit}
\text{$\pi^{\Spin(3,4)'}_\ell$ is irreducible for $\ell \ge 0$.}
\end{equation}
The {\tt atlas} software \cite{atlas} proves the irreducibility of the
first two discrete series (those not covered by \eqref{eq:bigG2unit}).
\end{subequations}
\begin{subequations}\label{se:bigG2sorbit}
Here is the orbit method perspective. The
representation of $H=G_{2,s}$ on $[{\mathfrak
g}_0/{\mathfrak h}_0]^*$ is ${\mathbb R}^{3,4}$, the real
representation whose highest weight is a short
root. We have already said that this representation carries an
invariant quadratic form of signature $(3,4)$. The generic orbits of
$H$ are indexed by non-zero real
numbers $A$, the values of the quadratic form. We can
arrange the normalizations so that the elliptic elements are exactly
those with $A>0$; if we define
\[ \ell_{\text{orbit}} = A^{1/2}, \qquad \ell = \ell_{\text{orbit}} - 3, \]
and write $\lambda(\ell_{\text{orbit}})$ for a representative of this
orbit, then
\[
\pi_\ell^{\Spin(3,4)} =
\pi({\text{orbit}},\lambda(\ell_{\text{orbit}})) \qquad
(\ell_{\text{orbit}} > 0).
\]
\end{subequations}
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\begin{document}
\title{On crossed products of the Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups}
\author{Takeshi KATSURA\\
Department of Mathematical Sciences\\
University of Tokyo, Komaba, Tokyo, 153-8914, JAPAN\\
e-mail: {\tt katsu@ms.u-tokyo.ac.jp}}
\date{}
\maketitle
\begin{abstract}
{\footnotesize We investigate the structures of crossed products of the
Cuntz algebra ${\mathcal O}_\infty$ by quasi-free actions of abelian groups.
We completely determine their ideal structures and compute the strong Connes
spectra and K-groups.}
\varepsilonnd{abstract}
\section{Introduction}
The crossed products of $C^*$-algebras give us plenty of interesting examples,
and the structures of them have been examined by several authors.
In \cite{Ki}, A. Kishimoto gave a necessary and sufficient condition
that the crossed products by abelian groups become simple
in terms of the strong Connes spectrum.
For the case of the crossed products of Cuntz algebras
by so-called quasi-free actions of abelian groups,
he gave a condition for simplicity, which is easy to check.
In \cite{KK1} and \cite{KK2}, A. Kishimoto and A. Kumjian dealt with,
among others, the crossed products of Cuntz algebras by quasi-free actions of
the real group $\mathbb{R}$.
In our previous papers \cite{Ka1}, \cite{Ka2},
we examined the structures of crossed products of Cuntz algebras ${\mathcal O}_n$
by quasi-free actions of
arbitrary locally compact, second countable, abelian groups.
The class of our algebras has many examples of simple stably projectionless
$C^*$-algebras as well as AF-algebras and purely infinite $C^*$-algebras.
In \cite{Ka1}, we completely determined the ideal structures of our algebras,
and gave another proof of A. Kishimoto's result on the simplicity of them.
We also gave a necessary and sufficient condition that our algebras become
primitive, and computed the Connes spectra and K-groups of our algebras.
In \cite{Ka2}, we proved that our algebras become AF-embeddable
when actions satisfy certain conditions.
To the best of the author's knowledge, this is the first case
to have succeeded in embedding crossed products of purely infinite
$C^*$-algebras into AF-algebras except trivial cases.
We also gave a necessary and sufficient condition that our algebras become
simple and purely infinite,
and consequently our algebras are either purely infinite or AF-embeddable
when they are simple.
In this paper, we deal with crossed products of the Cuntz algebra ${\mathcal O}_\infty$
by quasi-free actions of arbitrary locally compact, second countable,
abelian groups.
From section 3 to section 6, we completely determine the ideal structures
of such algebras by using the technique developed in \cite{Ka1}.
We omit detailed computations if similar computations have been already done
in \cite{Ka1}.
Readers are referred to \cite{Ka1}.
In the last section, we gather some results on crossed products
of the Cuntz algebra ${\mathcal O}_\infty$.
Among others, we give another proof of the determination of the simplicity
of the crossed products done by A. Kishimoto,
and we succeed in computing the strong Connes spectra of quasi-free actions
on the Cuntz algebra ${\mathcal O}_\infty$.
The crossed products examined in this paper or in \cite{Ka1}, \cite{Ka2},
can be considered as continuous counterparts of
Cuntz-Krieger algebras or graph algebras (cf. \cite{D}).
From this point of view, the crossed products of ${\mathcal O}_n$ can be considered
as graph algebras of locally finite graphs,
and the ones of ${\mathcal O}_\infty$ can be considered as graph algebras of graphs
whose vertices emit and receive infinitely many edges.
Recently the ideal structures of graph algebras, which is not necessarily
locally finite, were deeply examined in \cite{BHRS} and \cite{HS}.
Compared with row finite case,
it is rather difficult to describe ideal structures of graph algebras
which have vertices emitting infinitely many edges.
This seems to be related to the difficulty of examination of the
ideal structures of the crossed products of ${\mathcal O}_\infty$
compared with the ones of ${\mathcal O}_n$ done in \cite{Ka1}.
{\bf Acknowledgment.}
The author would like to thank
to his advisor Yasuyuki Kawahigashi for his support and encouragement,
to Masaki Izumi for various comments and many suggestions.
He is also grateful to Iain Raeburn and Wojciech Szyma\'nski
for stimulating discussions.
This work was partially supported by Research Fellowship
for Young Scientists of the Japan Society for the Promotion of Science.
\section{Preliminaries}\label{PRE}
The Cuntz algebra ${\mathcal O}_\infty$ is the universal $C^*$-algebra generated
by infinitely many isometries
$S_1,S_2,\ldots$ satisfying $S_i^*S_j=\delta_{i,j}$.
For $n\in\mathbb{Z}_+:=\{1,2,\ldots\}$ and $k\in\mathbb{N}:=\{0,1,\ldots\}$,
we define the set ${\mathcal W}_n^{(k)}$ of words in $\{1,2,\ldots,n\}$
with length $k$ by ${\mathcal W}_n^{(0)}=\{\varepsilonmptyset\}$ and
$${\mathcal W}_n^{(k)}
=\big\{ (i_1,i_2,\ldots,i_k)\ \big|\ i_j\in\{1,2,\ldots,n\}\big\}$$
for $k\geq 1$.
Set ${\mathcal W}_n=\bigcup_{k=0}^\infty {\mathcal W}_n^{(k)}$ and
${\mathcal W}_\infty=\bigcup_{n=1}^\infty {\mathcal W}_n$.
For $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$,
we denote its length $k$ by $|\mu|$,
and set $S_\mu=S_{i_1}S_{i_2}\cdots S_{i_k}\in{\mathcal O}_\infty$.
Let $G$ be a locally compact abelian group which satisfies
the second axiom of countability and ${\mathcal G}amma$ be the dual group of $G$.
We use $+$ for multiplicative operations of abelian groups except for $\mathbb{T}$,
which is the group of the unit circle in the complex plane $\mathbb{C}$.
The pairing of $t\in G$ and $\gamma\in{\mathcal G}amma$ is denoted by
$\ip{t}{\gamma}\in\mathbb{T}$.
For $\omega=(\omega_1,\omega_2,\ldots)\in{\mathcal G}amma^\infty$,
we define an action $\alpha^\omega$ of abelian group $G$ on ${\mathcal O}_\infty$ by
$\alpha^\omega_t(S_i)=\ip{t}{\omega_i}S_i$ for $i\in\mathbb{Z}_+$ and $t\in G$.
The action $\alpha^\omega:G\curvearrowright{\mathcal O}_\infty$ becomes quasi-free
(for a definition of quasi-free actions on Cuntz algebras, see \cite{E}).
However, there exist quasi-free actions of abelian group $G$ on ${\mathcal O}_\infty$,
which are not conjugate to $\alpha^\omega$ for any $\omega\in{\mathcal G}amma^\infty$
though we do not deal with such actions.
The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ has a $C^*$-subalgebra
$\mathbb{C} 1{\rtimes_{\alpha^\omega}}G$ which is isomorphic to $C_0({\mathcal G}amma)$.
We consider $C_0({\mathcal G}amma)$ as a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
The Cuntz algebra ${\mathcal O}_\infty$ is naturally embedded into
the multiplier algebra $M({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For each $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$,
we define an element $\omega_\mu$ of ${\mathcal G}amma$
by $\omega_\mu=\sum_{j=1}^{k}\omega_{i_j}$.
For $\gamma_0\in{\mathcal G}amma$, we define a (reverse) shift automorphism
$\sigma_{\gamma_0}:C_0({\mathcal G}amma)\to C_0({\mathcal G}amma)$ by
$(\sigma_{\gamma_0} f)(\gamma)=f(\gamma+\gamma_0)$ for $f\in C_0({\mathcal G}amma)$.
Once noting that $\alpha^\omega_t(S_\mu)=\ip{t}{\omega_\mu}S_\mu$
for $\mu\in{\mathcal W}_\infty$, one can easily verify that
$fS_\mu =S_\mu\sigma_{\omega_\mu}f$ for any $f\in C_0({\mathcal G}amma)\subset {\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For a subset $X$ of a $C^*$-algebra,
we denote by $\spa X$ the linear span of $X$,
and by $\cspa X$ its closure.
We have
${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G=\cspa\{ S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma)\}$.
We denote by $\mathbb{M}_k$ the $C^*$-algebra of $k \times k$ matrices
for $k=1,2,\ldots$,
and by $\mathbb{K}$ the $C^*$-algebra of compact operators
of the infinite dimensional separable Hilbert space.
\section{Gauge invariant ideals}
In this section, we determine all the ideals
which are globally invariant under the gauge action.
Here an ideal means a closed two-sided ideal,
and the gauge action $\beta:\mathbb{T}\curvearrowright{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is defined by
$\beta_t(S_\mu fS_\nu^*)=t^{|\mu|-|\nu|}S_\mu fS_\nu^*$
for $\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma)$ and $t\in\mathbb{T}$.
For a positive integer $n$, we define a projection $p_n$ by
$p_n=1-\sum_{i=1}^nS_iS_i^*$.
We set $p_0=1$.
Since $p_n$ commutes with $C_0({\mathcal G}amma)$,
$p_n C_0({\mathcal G}amma)$ is a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
which is isomorphic to $C_0({\mathcal G}amma)$.
\begin{definition}\rm\label{omega}
Let $I$ be an ideal of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For each $n\in\mathbb{N}$,
we define the closed subset $X_I^{(n)}$ of ${\mathcal G}amma$ by
$$X_I^{(n)}=\{\gamma\in{\mathcal G}amma\mid
f(\gamma)=0\mbox{ for all }f\in C_0({\mathcal G}amma)\mbox{ with }p_nf\in I\}.$$
Set $X_I=X_I^{(0)}$, $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}$,
and denote by $\widetilde{X}_I$
the pair $(X_I,X_I^{(\infty)})$ of subsets of ${\mathcal G}amma$.
\varepsilonnd{definition}
In other words, $X_I^{(n)}$ is determined by
$p_n C_0({\mathcal G}amma\setminus X_I^{(n)})=I\cap p_n C_0({\mathcal G}amma)$.
One can easily see that
$X_{I_1\cap I_2}^{(n)}=X_{I_1}^{(n)}\cup X_{I_2}^{(n)}$ for any $n\in\mathbb{N}$,
hence $X_{I_1\cap I_2}=X_{I_1}\cup X_{I_2},\
X_{I_1\cap I_2}^{(\infty)}=X_{I_1}^{(\infty)}\cup X_{I_2}^{(\infty)}$
and that $I_1\subset I_2$ implies
$X_{I_1}^{(n)}\supset X_{I_2}^{(n)}$ for any $n\in\mathbb{N}$, hence implies
$X_{I_1}\supset X_{I_2},\ X_{I_1}^{(\infty)}\supset X_{I_2}^{(\infty)}$.
For $n\in\mathbb{N}$, the set $X_I^{(n)}$ can be described
only in terms of $X_I$ and $X_I^{(\infty)}$.
\begin{lemma}\label{X_I^{(n)}}
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have
\begin{align*}
X_I^{(n)}&=X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i),
\varepsilonnd{align*}
for any $n\in\mathbb{N}$.
\varepsilonnd{lemma}
\begin{proposition}f
Let $\gamma$ be an element of $X_I$
and $i$ be a positive integer grater than $n$.
Take $f\in C_0({\mathcal G}amma)$ with $p_nf\in I$.
Since
$$S_{i}^*p_nfS_{i}=S_{i}^*fS_{i}=S_{i}^*S_{i}\sigma_{\omega_{i}}f
=\sigma_{\omega_{i}}f,$$
we have $\sigma_{\omega_{i}}f\in I\cap C_0({\mathcal G}amma)$.
Since $\gamma\in X_I$, we have $\sigma_{\omega_{i}}f(\gamma)=0$.
Hence $f(\gamma+\omega_{i})=0$ for any $f\in C_0({\mathcal G}amma)$ with $p_nf\in I$.
It implies $\gamma+\omega_{i}\in X_I^{(n)}$.
Thus $X_I^{(n)}\supset X_I+\omega_i$ for any $i>n$.
For $n\leq m$, we have $X_I^{(n)}\supset X_I^{(m)}$ because $p_np_m=p_m$.
Therefore $X_I^{(n)}\supset X_I^{(\infty)}$.
Thus $X_I^{(n)}\supset
X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
Conversely, take
$\gamma\notin X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
Since $\gamma\notin X_I^{(\infty)}$, we can find a positive integer $m$ so that
$\gamma\notin X_I^{(m)}$.
When $m\leq n$, we see that $\gamma\notin X_I^{(n)}$.
We will show $\gamma\notin X_I^{(n)}$ in the case $m>n$.
Since $\gamma\notin X_I^{(m)}$,
there exists $f\in C_0({\mathcal G}amma)$ such that $p_mf\in I$ and $f(\gamma)\neq 0$.
For each $i=n+1,n+2,\ldots,m$, there exists $f_i\in C_0({\mathcal G}amma)\cap I$
such that $f_i(\gamma-\omega_i)\neq 0$ because $\gamma\notin X_I+\omega_i$.
Set $g=f\prod_{i=n+1}^m\sigma_{-\omega_i}f_i$.
We have $g(\gamma)\neq 0$ and
$$p_n g=p_m g+\sum_{i=n+1}^mS_iS_i^*g=p_m g+\sum_{i=n+1}^mS_i(\sigma_{\omega_i}g)S_i^*\in I.$$
Therefore $\gamma\notin X_I^{(n)}$.
Thus we have
$X_I^{(n)}=X_I^{(\infty)}\cup\bigcup_{i=n+1}^\infty (X_I+\omega_i)$.
\varepsilonnd{proposition}f
\begin{definition}\rm
A subset $X$ of ${\mathcal G}amma$ is called {\varepsilonm $\omega$-invariant}
if $X$ is a closed set with $X+\omega_i\subset X$ for any $i\in\mathbb{Z}_+$.
For an $\omega$-invariant set $X$, we define a closed set $H_X$ by
$$H_X=\overline{X\setminus\bigcup_{i=1}^\infty(X+\omega_i)}\ \cup\
\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}.$$
\varepsilonnd{definition}
Note that $H_X$ is a closed subset of $X$.
\begin{definition}\rm
A pair $\widetilde{X}=(X,X^\infty)$ of subsets of ${\mathcal G}amma$ is called
{\varepsilonm $\omega$-invariant} if $X$ is an $\omega$-invariant set, and
$X^\infty$ is a closed set satisfying $H_X\subset X^\infty\subset X$.
\varepsilonnd{definition}
\begin{proposition}
For any ideal $I$ of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the pair $\widetilde{X}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{X_I^{(n)}},
we have $X_I=X_I^{(\infty)}\cup \bigcup_{i=1}^\infty(X_I+\omega_i)$.
From this, we see that $X_I$ is $\omega$-invariant and that
$X_I\setminus\bigcup_{i=1}^\infty(X_I+\omega_i)\subset
X_I^{(\infty)}\subset X_I$.
By Lemma \ref{X_I^{(n)}},
we have $\overline{\bigcup_{i=n}^\infty(X+\omega_i)}\subset\overline{X_I^{(n)}}
=X_I^{(n)}$.
Hence $\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}\subset
\bigcap_{n=1}^\infty X_I^{(n)}=X_I^{(\infty)}$.
Therefore we get $H_X\subset X_I^{(\infty)}\subset X_I$.
\varepsilonnd{proposition}f
We will show that for an $\omega$-invariant pair $\widetilde{X}$,
there exists a gauge invariant ideal $I$ such that
$\widetilde{X}_I=\widetilde{X}$ (Proposition \ref{exist}).
\begin{lemma}\label{X^{(n)}}
Let $\widetilde{X}=(X,X^{(\infty)})$ be an $\omega$-invariant pair.
For $n\in\mathbb{N}$, set $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
Then we have the following.
\benu
\item $X^{(n)}$ is closed for all $n\in\mathbb{N}$.
\item $X=X^{(0)}$, $X^{(\infty)}=\bigcap_{n=1}^\infty X^{(n)}$.
\item For $0\leq n<m$, $X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^m(X+\omega_i)$.
\item For a positive integer $n$,
$$X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \bigg( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\bigg).$$
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proposition}f
\benu
\item Take $\gamma\in\overline{X^{(n)}}$ for a positive integer $n$.
If $U\cap X^{(\infty)}\neq\varepsilonmptyset$ for all neighborhood $U$ of $\gamma$,
then $\gamma\in X^{(\infty)}\subset X^{(n)}$ because $X^{(\infty)}$ is closed.
Otherwise, we can find a positive integer $i_U$ grater than $n$ with
$U\cap (X+\omega_{i_U})\neq\varepsilonmptyset$ for any neighborhood $U$ of $\gamma$.
If there exists $i$ such that $i_U=i$ eventually,
then $\gamma\in X+\omega_i\subset X^{(n)}$ because $X+\omega_i$ is closed.
If there are no such $i$, then we can see that
$\gamma\in\overline{\bigcup_{i=m}^\infty(X+\omega_i)}$ for any $m$ with $m>n$.
Hence $\gamma\in H_{X}\subset X^{(\infty)}\subset X^{(n)}$.
Thus we have proved that $\gamma\in X^{(n)}$,
from which it follows that $X^{(n)}$ is closed.
\item Since
$X\setminus\bigcup_{i=1}^\infty(X+\omega_i)\subset X^{(\infty)}\subset X$,
we have $X=X^{(0)}$.
We see that
\begin{align*}
\bigcap_{n=1}^\infty X^{(n)}
&=\bigcap_{n=1}^\infty
\bigg(X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg)
=X^{(\infty)}\cup\bigcap_{n=1}^\infty
\bigg(\bigcup_{i=n+1}^\infty(X+\omega_i)\bigg).
\varepsilonnd{align*}
Since $\bigcap_{n=1}^\infty\left(\bigcup_{i=n+1}^\infty(X+\omega_i)\right)
\subset H_{X}\subset X^{(\infty)}$,
we have $\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
\item It is obvious by the definition.
\item For a positive integer $n$, we have
$X=X^{(n)}\cup\bigcup_{i=1}^n(X+\omega_i)$ by (iii).
Recursively, we get
$X=\bigcup_{m=0}^{k-1}\big(\bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X^{(n)}+\omega_\mu)\big)
\cup\bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)$ for any positive integer $k$.
Hence $X=\bigcup_{\mu\in{\mathcal W}_n}(X^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \big( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X+\omega_\mu)\big)$.
\varepsilonnd{proposition}f
\varepsilonnd{enumerate}
\begin{definition}\rm
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
we define $I_{\widetilde{X}}\subset{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by
$$I_{\widetilde{X}}=\cspa\{S_\mu p_n fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X^{(n)}),\ n\in\mathbb{N}\},$$
where $X^{(n)}=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i)$.
\varepsilonnd{definition}
\begin{proposition}\label{I_X,Xinfty}
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$,
the set $I_{\widetilde{X}}$ becomes a gauge invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\varepsilonnd{proposition}
\begin{proposition}f
Clearly $I_{\widetilde{X}}$ is a $*$-invariant closed linear space,
and is invariant under the gauge action $\beta$ because
$\beta_t(S_\mu p_n fS_\nu^*)=t^{|\mu|-|\nu|}S_\mu p_n fS_\nu^*$
for $t\in\mathbb{T}$.
To prove that $I_{\widetilde{X}}$ is an ideal,
it suffices to show that for any $\mu_1,\nu_1,\mu_2,\nu_2\in{\mathcal W}_\infty$
and any $f\in C_0({\mathcal G}amma\setminus X^{(n)}),\ g\in C_0({\mathcal G}amma)$,
the product $xy$ of $x=S_{\mu_1}p_n fS_{\nu_1}^*\in I_{\widetilde{X}}$
and $y=S_{\mu_2}gS_{\nu_2}^*\in{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is in $I_{\widetilde{X}}$.
If $S_{\nu_1}^*S_{\mu_2}=0$ or $S_{\nu_1}^*S_{\mu_2}=S_\mu^*$
for some $\mu\in{\mathcal W}_\infty$,
then it is easy to see that $xy\in I_{\widetilde{X}}$.
Otherwise $S_{\nu_1}^*S_{\mu_2}=S_\mu$
for some $\mu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_\infty$ with $\mu\neq\varepsilonmptyset$.
When $i_1\leq n$, we have $p_n fS_\mu=p_n S_\mu\sigma_{\omega_\mu}f=0$.
Hence $xy=0\in I_{\widetilde{X}}$.
When $i_1> n$, we have
$p_n fS_\mu=p_n S_\mu\sigma_{\omega_\mu}f=S_\mu\sigma_{\omega_\mu}f$.
Now, $f\in C_0({\mathcal G}amma\setminus X^{(n)})$ implies
$\sigma_{\omega_\mu}f\in C_0({\mathcal G}amma\setminus X)$ because
$X+\omega_\mu\subset X+\omega_{i_1}\subset X^{(n)}$.
Hence we have $xy\in I_{\widetilde{X}}$.
It completes the proof.
\varepsilonnd{proposition}f
\begin{proposition}\label{exist}
Let $\widetilde{X}=(X,X^{(\infty)})$ be an $\omega$-invariant pair,
and set $I=I_{\widetilde{X}}$.
Then $\widetilde{X}_I=\widetilde{X}$.
\varepsilonnd{proposition}
\begin{proposition}f
By the definition of $I$, we get $X_I^{(n)}\subset X^{(n)}$ for any $n\in\mathbb{N}$.
We will first prove that $X_I=X$.
To the contrary, assume that $X_I\subsetneqq X$.
Then there exists $f\in I\cap C_0({\mathcal G}amma)$
such that $f(\gamma_0)=1$ for some $\gamma_0\in X$.
Since $f\in I$, there exist $n_l\in\mathbb{N}$,
$f_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ and
$\mu_l,\nu_l\in{\mathcal W}_\infty\ (l=1,2,\ldots,L)$ such that
$$\bigg\| f-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg\|<\frac12.$$
Take a positive integer $n$ so large that
$n_l\leq n$ and $\mu_l,\nu_l\in{\mathcal W}_n$ for $l=1,2,\ldots,L$.
For any $\mu_0\in{\mathcal W}_n$,
we have $p_nS_{\mu_0}^*fS_{\mu_0}p_n=p_n\sigma_{\omega_{\mu_0}}f$
and $\sigma_{\omega_{\mu_0}}f(\gamma_0-\omega_{\mu_0})=1$.
For $l$ with $\mu_l=\nu_l=\mu_0$, we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n=p_n f_l$.
For $l$ with $\mu_l\nu=\nu_l\nu=\mu_0$ for some
$\nu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_n$ with $i_1>n_l$,
we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n
=p_n\sigma_{\omega_{\nu}}f_l$.
We have $\sigma_{\omega_{\nu}}f_l\in C_0({\mathcal G}amma\setminus X)$,
because $X+\omega_{\nu}\subset X+\omega_{i_1}\subset X^{(n_l)}$.
For other $l$, we have
$p_nS_{\mu_0}^*(S_{\mu_l}p_{n_l}f_lS_{\nu_l}^*)S_{\mu_0} p_n=0$.
Hence we get
$$\bigg\|\sigma_{\omega_{\mu_0}}f-\sum_{l=1}^L g_l\bigg\|=
\bigg\|p_n\bigg(\sigma_{\omega_{\mu_0}}f-\sum_{l=1}^L g_l\bigg)\bigg\|=\bigg\|p_nS_{\mu_0}^*\bigg(f-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg)
S_{\mu_0}p_n\bigg\|<\frac12,$$
where $g_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ when $\mu_l=\nu_l=\mu_0$,
and $g_l\in C_0({\mathcal G}amma\setminus X)$ when $\mu_l\nu=\nu_l\nu=\mu_0$ for some
$\nu=(i_1,i_2,\ldots,i_k)\in{\mathcal W}_n$ with $i_1>n_l$, and $g_l=0$ otherwise.
To derive a contradiction, it suffices to find $\mu_0\in{\mathcal W}_n$ such that
$g_l(\gamma_0-\omega_{\mu_0})=0$ for any $l$.
By Lemma \ref{X^{(n)}} (iv), we have either $\gamma_0\in \bigcap_{m=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X+\omega_\mu)\big)$
or $\gamma_0\in X^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$.
When $\gamma_0\in \bigcap_{m=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(m)}}(X+\omega_\mu)\big)$,
take $\mu_0\in{\mathcal W}_n$ so that $|\mu_0|>|\mu_l|,|\nu_l|$ for $l=1,2,\ldots,L$
and $\gamma_0\in X+\omega_{\mu_0}$.
Then $\mu_l=\nu_l=\mu_0$ never occurs.
Hence $g_l\in C_0({\mathcal G}amma\setminus X)$ for any $l$.
We get $g_l(\gamma_0-\omega_{\mu_0})=0$ because $\gamma_0-\omega_{\mu_0}\in X$.
When $\gamma_0\in X^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$, take $\mu_0=\mu$.
Since $\gamma_0-\omega_{\mu_0}\in X^{(n)}\subset X^{(n_l)}\subset X$,
we have $g_l(\gamma_0-\omega_{\mu_0})=0$
either if $g_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ or
if $g_l\in C_0({\mathcal G}amma\setminus X)$.
Hence $g_l(\gamma_0-\omega_{\mu_0})=0$ for any $l$.
Therefore we have $X_I=X$.
Next we will show that $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
To derive a contradiction, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist $n_l\in\mathbb{N}$,
$f_l\in C_0({\mathcal G}amma\setminus X^{(n_l)})$ and
$\mu_l,\nu_l\in{\mathcal W}_\infty\ (l=1,2,\ldots,L)$ such that
$$\bigg\| p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that $\mu_l,\nu_l\in{\mathcal W}_{m}$, $n_l\leq m$
for $l=1,2,\ldots,L$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
When $\gamma_0\in X^{(m)}$, we have $f_l(\gamma_0)=0$ for any $l$.
On the other hand, we get $\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}f_l\|<1/2$
because
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}p_{n_l}f_lS_{\nu_l}^*
\bigg)p_{m}=p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}f_l.$$
This is a contradiction.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
We have shown that $\widetilde{X}_I=\widetilde{X}$.
\varepsilonnd{proposition}f
By Proposition \ref{exist}, the map $I\mapsto \widetilde{X}_I$
from the set of gauge invariant ideals $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
to the set of $\omega$-invariant pairs is surjective.
Now, we turn to showing that this map is injective (Proposition \ref{unique}).
To do so, we investigate the quotient $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by an ideal $I$
which is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Since $I\cap C_0({\mathcal G}amma)=C_0({\mathcal G}amma\setminus X_I)$, a $C^*$-subalgebra
$C_0({\mathcal G}amma)/(I\cap C_0({\mathcal G}amma))$ of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ is isomorphic to $C_0(X_I)$.
We will consider $C_0(X_I)$ as a $C^*$-subalgebra of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
We will use the same symbols
$S_1,S_2,\ldots\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$ as the ones in $M({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ for denoting
the isometries of ${\mathcal O}_\infty$ which is naturally embedded into $M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
For an $\omega$-invariant set $X$,
we can define a $*$-homomorphism $\sigma_{\omega_\mu}:C_0(X)\to C_0(X)$
for $\mu\in{\mathcal W}_\infty$.
This map $\sigma_{\omega_\mu}$ is always surjective,
but it is injective only in the case that $X\subset X+\omega_\mu$,
which is equivalent to $X=X+\omega_\mu$.
One can easily verify the following.
\begin{lemma}\label{cp/I}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For $\mu,\nu\in{\mathcal W}_\infty$ and $f\in C_0(X_I)\subset({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
the following hold.
\benu
\item $S_\mu fS_\nu^*=0$ if and only if $f=0$.
\item For $n\in\mathbb{N}$, $p_nf=0$ if and only if $f\in C_0(X_I\setminus X_I^{(n)})$.
\item $fS_\mu=S_\mu \sigma_{\omega_\mu}f$.
\item $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I=\cspa\{S_\mu fS_\nu^*\mid\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)\}$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
We define a $C^*$-subalgebra of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
which corresponds to the AF-core for Cuntz algebras.
\begin{definition}\rm\label{AFcore}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
We define $C^*$-subalgebras of $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ by
\begin{align*}
{\mathcal G}_I^{(n,k)}&=\spa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_n^{(k)},\
f\in C_0(X_I)\},\\
{\mathcal F}_I^{(n,k)}&=\spa\{S_\mu p_nfS_\nu^*\mid \mu,\nu\in{\mathcal W}_n^{(k)},\
f\in C_0(X_I)\},\\
{\mathcal F}_I^{(n)}&=\spa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_n,\
0\leq |\mu|=|\nu|\leq n,\ f\in C_0(X_I)\},\\
{\mathcal F}_I&=\cspa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
|\mu|=|\nu|,\ f\in C_0(X_I)\},
\varepsilonnd{align*}
for $n\in\mathbb{Z}_+, 0\leq k\leq n$.
\varepsilonnd{definition}
\begin{lemma}\label{F_n}
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
For $n\in\mathbb{Z}_+, 0\leq k\leq n$, we have the following.
\benu
\item ${\mathcal G}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I)$.
\item ${\mathcal F}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I^{(n)})$.
\item ${\mathcal F}_I^{(n)}\cong\bigoplus_{k=0}^{n-1}{\mathcal F}_I^{(n,k)}\oplus {\mathcal G}_I^{(n,n)}$.
\item $\bigcup_{n=1}^\infty{\mathcal F}_I^{(n)}$ is dense in ${\mathcal F}_I$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proposition}f
\benu
\item Since the set ${\mathcal W}_n^{(k)}$ has $n^k$ elements, we may use
$\{e_{\mu,\nu}\}_{\mu,\nu\in{\mathcal W}_n^{(k)}}$ for denoting the matrix units of
$\mathbb{M}_{n^k}$.
One can easily see that
$$\mathbb{M}_{n^k}\otimes C_0(X_I)\ni e_{\mu,\nu}\otimes f
\mapsto S_\mu fS_\nu^*\in {\mathcal G}_I^{(n,k)}$$
gives us an isomorphism from $\mathbb{M}_{n^k}\otimes C_0(X_I)$ to ${\mathcal G}_I^{(n,k)}$.
\item We can define a surjective map from ${\mathcal G}_I^{(n,k)}$ to ${\mathcal F}_I^{(n,k)}$ by
$${\mathcal G}_I^{(n,k)}\ni S_\mu fS_\nu^*\mapsto S_\mu p_n fS_\nu^*\in {\mathcal F}_I^{(n,k)}.$$
Its kernel is $\mathbb{M}_{n^k}\otimes C_0(X_I\setminus X_I^{(n)})$
under the isomorphism ${\mathcal G}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I)$
by Lemma \ref{cp/I} (ii).
Hence we have ${\mathcal F}_I^{(n,k)}\cong \mathbb{M}_{n^k}\otimes C_0(X_I^{(n)})$.
\item It can be done just by computation.
\item Obvious by the definitions of ${\mathcal F}_I^{(n)}$ and ${\mathcal F}_I$.
\varepsilonnd{proposition}f
\varepsilonnd{enumerate}
We will often identify ${\mathcal G}_I^{(n,n)}$ with $C_0(X_I,\mathbb{M}_{n^n})$.
The following lemma essentially appeared in \cite{C}.
\begin{lemma}\label{isom}
For $i=1,2$, let $E_i$ be a conditional expectation
from a $C^*$-algebra $A_i$ onto a $C^*$-subalgebra $B_i$ of $A_i$.
Let $\varphi:A_1\to A_2$ be a $*$-homomorphism
with $\varphi\circ E_1=E_2\circ\varphi$.
If the restriction of $\varphi$ on $B_1$ is injective and $E_1$ is faithful,
then $\varphi$ is injective.
\varepsilonnd{lemma}
For an ideal $I$ which is invariant under the gauge action $\beta$,
we can extend the gauge action on ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ to one on $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$,
which is also denoted by $\beta$.
The following lemma is standard.
\begin{lemma}\label{cond.exp1}
Let $I$ be a gauge invariant ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Then,
$$E_{I}:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\ni x\mapsto\int_{\mathbb{T}}\beta_t(x)dt\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$$
is a faithful conditional expectation onto ${\mathcal F}_{I}$,
where $dt$ is the normalized Haar measure on $\mathbb{T}$.
\varepsilonnd{lemma}
\begin{proposition}\label{unique}
For any gauge invariant ideal $I$, we have $I_{\widetilde{X}_I}=I$.
\varepsilonnd{proposition}
\begin{proposition}f
When $I={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have $X_I=X_I^{(\infty)}=\varepsilonmptyset$.
Thus $I_{\widetilde{X}_I}={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Let $I$ be a gauge invariant ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and set $J=I_{\widetilde{X}_I}$.
By the definition, $J\subset I$.
Hence there exists a surjective $*$-homomorphism $\pi:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
By Proposition \ref{exist} and Lemma \ref{F_n},
the restriction of $\pi$ on ${\mathcal F}_{J}^{(k)}$
is an isomorphism from ${\mathcal F}_{J}^{(k)}$ onto ${\mathcal F}_{I}^{(k)}$
and so the restriction of $\pi$ on ${\mathcal F}_{J}$ is an isomorphism
from ${\mathcal F}_{J}$ onto ${\mathcal F}_{I}$.
By Lemma \ref{cond.exp1}, there are faithful conditional expectations
$E_J:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to {\mathcal F}_{J}$ and $E_I:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{I}$
with $E_I\circ \pi=\pi\circ E_{J}$.
By Lemma \ref{isom}, $\pi$ is injective.
Therefore $I_{\widetilde{X}_I}=I$.
\varepsilonnd{proposition}f
\begin{theorem}\label{OneToOne}
The maps $I\mapsto \widetilde{X}_I$ and
$\widetilde{X}\mapsto I_{\widetilde{X}}$ induce
a one-to-one correspondence
between the set of gauge invariant ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma$.
\varepsilonnd{theorem}
\begin{proposition}f
Combine Proposition \ref{exist} and Proposition \ref{unique}.
\varepsilonnd{proposition}f
\section{Primeness for $\omega$-invariant pairs}
In this section, we give a necessary condition for an ideal to be primitive
in terms of $\omega$-invariant pairs.
We will use it after in order to determine all primitive ideals.
An ideal of a $C^*$-algebra is called primitive
if it is a kernel of some irreducible representation.
A $C^*$-algebra is called primitive if $0$ is a primitive ideal.
When a $C^*$-algebra $A$ is separable, an ideal $I$ of $A$ is primitive
if and only if $I$ is prime, i.e.
for two ideals $I_1,I_2$ of $A$,
$I_1\cap I_2\subset I$ implies either $I_1\subset I$ or $I_2\subset I$.
We define primeness for $\omega$-invariant pairs.
For two $\omega$-invariant pair $\widetilde{X}_1=(X_1,X_1^{(\infty)}),\
\widetilde{X}_2=(X_2,X_2^{(\infty)})$,
we write $\widetilde{X}_1\subset \widetilde{X}_2$
if $X_1\subset X_2, X_1^{(\infty)}\subset X_2^{(\infty)}$
and denote by $\widetilde{X}_1\cup\widetilde{X}_2$
the $\omega$-invariant pair $(X_1\cup X_2,X_1^{(\infty)}\cup X_2^{(\infty)})$.
\begin{definition}\rm
An $\omega$-invariant pair $\widetilde{X}$ is called {\varepsilonm prime}
if $\widetilde{X}_1\cup \widetilde{X}_2\supset\widetilde{X}$
implies either $\widetilde{X}_1\supset\widetilde{X}$
or $\widetilde{X}_2\supset\widetilde{X}$
for two $\omega$-invariant pairs
$\widetilde{X}_1,\ \widetilde{X}_2$.
\varepsilonnd{definition}
\begin{proposition}\label{prime}
If an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive,
then $\widetilde{X}_I$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}
\begin{proposition}f
Let $I$ be a primitive ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Take two $\omega$-invariant pairs
$\widetilde{X}_1$, $\widetilde{X}_2$ with
$\widetilde{X}_1\cup \widetilde{X}_2\supset \widetilde{X}_I$.
Set $I_1=I_{\widetilde{X}_1}$ and $I_2=I_{\widetilde{X}_2}$.
Then
$$I_1\cap I_2=I_{\widetilde{X}_1\cup \widetilde{X}_2}\subset
I_{\widetilde{X}_I}\subset I.$$
Since $I$ is prime, we have either $I_1\subset I$ or $I_2\subset I$.
Hence we get either $\widetilde{X}_1\supset \widetilde{X}_I$
or $\widetilde{X}_2\supset\widetilde{X}_I$.
Thus $\widetilde{X}_I$ is prime.
\varepsilonnd{proposition}f
In general, the converse of Proposition \ref{prime} is not true
(see Corollary \ref{primitive1} and Proposition \ref{primitive2}).
The ideal $I$ is prime if and only if
the equality $I_1\cap I_2=I$ implies either $I_1=I$ or $I_2=I$
for two ideals $I_1,I_2$
(see the proof of (iii)$\mathbb{R}ightarrow$(iv) of Proposition \ref{primepair}).
The following is the counterpart of this fact
for prime $\omega$-invariant pairs.
\begin{proposition}\label{primepair}
For an $\omega$-invariant pair $\widetilde{X}$,
the following are equivalent.
\benu
\item $\widetilde{X}$ is prime.
\item
For two $\omega$-invariant pairs $\widetilde{X}_1$,
$\widetilde{X}_2$,
the equality $\widetilde{X}_1\cup \widetilde{X}_2=\widetilde{X}$
implies either $\widetilde{X}_1=\widetilde{X}$
or $\widetilde{X}_2=\widetilde{X}$.
\item For two gauge invariant ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the equality $I_1\cap I_2=I_{\widetilde{X}}$ implies either
$I_1=I_{\widetilde{X}}$ or $I_2=I_{\widetilde{X}}$.
\item For two gauge invariant ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the inclusion $I_1\cap I_2\subset I_{\widetilde{X}}$ implies
either $I_1\subset I_{\widetilde{X}}$ or $I_2\subset I_{\widetilde{X}}$.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
\begin{proposition}f
(i)$\mathbb{R}ightarrow$(ii):
Take two $\omega$-invariant pairs $\widetilde{X}_1$,
$\widetilde{X}_2$ with $\widetilde{X}_1\cup \widetilde{X}_2=\widetilde{X}$.
By (i), we have either $\widetilde{X}_1\supset \widetilde{X}$
or $\widetilde{X}_2\supset \widetilde{X}$.
Hence we get either $\widetilde{X}_1=\widetilde{X}$
or $\widetilde{X}_2=\widetilde{X}$.
(ii)$\mathbb{R}ightarrow$(iii):
Take two gauge invariant ideals $I_1,I_2$
with $I_1\cap I_2=I_{\widetilde{X}}$.
We have $\widetilde{X}_{I_1}\cup \widetilde{X}_{I_2}=\widetilde{X}$.
By (ii), we have either $\widetilde{X}_{I_1}=\widetilde{X}$ or
$\widetilde{X}_{I_2}=\widetilde{X}$.
By Proposition \ref{unique},
we have either $I_1=I_{\widetilde{X}}$ or $I_2=I_{\widetilde{X}}$.
(iii)$\mathbb{R}ightarrow$(iv):
Take two gauge invariant ideals $I_1,I_2$
with $I_1\cap I_2\subset I_{\widetilde{X}}$.
Then we have
$$(I_1+I_{\widetilde{X}})\cap (I_2+I_{\widetilde{X}})=
(I_1\cap I_2) +I_{\widetilde{X}}=I_{\widetilde{X}}.$$
By (iii), either $I_1+I_{\widetilde{X}}=I_{\widetilde{X}}$ or
$I_2+I_{\widetilde{X}}=I_{\widetilde{X}}$ holds.
Hence we get
either $I_1\subset I_{\widetilde{X}}$ or $I_2\subset I_{\widetilde{X}}$.
(iv)$\mathbb{R}ightarrow$(i):
Similarly as the proof of Proposition \ref{prime}.
\varepsilonnd{proposition}f
We will use the implication (ii)$\mathbb{R}ightarrow$(i)
to determine which $\omega$-invariant pair is prime.
We also need a notion of primeness for $\omega$-invariant sets.
\begin{definition}\rm
An $\omega$-invariant set $X$ is called {\varepsilonm prime}
if $X_1\cup X_2\supset X$ implies either $X_1\supset X$ or $X_2\supset X$,
for any $\omega$-invariant sets $X_1,X_2$.
\varepsilonnd{definition}
We set ${\rm{sg}}(\omega)=\{\omega_\mu\mid \mu\in{\mathcal W}_\infty\}$
which is the semigroup generated by $\omega_1,\omega_2,\ldots$ and
denote by $\overline{\rm{sg}}(\omega)$ its closure.
Note that a closed subset $X$ of ${\mathcal G}amma$ is $\omega$-invariant
if and only if $X+\overline{\rm{sg}}(\omega)=X$.
For any $\gamma\in{\mathcal G}amma$, it is easy to see that
the set $\gamma+\overline{\rm{sg}}(\omega)$ is a prime $\omega$-invariant set.
The following is a necessary and sufficient condition
for an $\omega$-invariant set to be prime,
which can be considered as an analogue of maximal tails in \cite{BHRS}.
\begin{proposition}\label{Xprime}
An $\omega$-invariant set $X$ of ${\mathcal G}amma$ is prime if and only if for any
$\gamma_1,\gamma_2\in X$ and
any neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively,
there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
\varepsilonnd{proposition}
\begin{proposition}f
Suppose $X$ is a prime $\omega$-invariant set.
Take $\gamma_1,\gamma_2\in X$ and
neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively.
Set $X_j={\mathcal G}amma\setminus\bigcup_{\mu\in{\mathcal W}_\infty}(U_j-\omega_\mu)$ for $j=1,2$.
Then $X_1$ and $X_2$ are $\omega$-invariant sets satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
Since $X$ is prime, we have $X_1\cup X_2\not\supset X$.
Hence there exists $\gamma\in X$ with $\gamma\notin X_1\cup X_2$.
By the definition of $X_1$ and $X_2$, there exist $\mu_1,\mu_2$ such that
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Conversely assume that for any $\gamma_1,\gamma_2\in X$ and
any neighborhoods $U_1$, $U_2$ of $\gamma_1,\gamma_2$ respectively,
there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\in U_1$ and $\gamma+\omega_{\mu_2}\in U_2$.
Take $\omega$-invariant sets $X_1$ and $X_2$ satisfying
$X_1\not\supset X$ and $X_2\not\supset X$.
There exist $\gamma_1,\gamma_2\in X$
with $\gamma_1\notin X_1$ and $\gamma_2\notin X_2$.
Hence there exist $\gamma\in X$ and $\mu_1,\mu_2\in{\mathcal W}_\infty$ with
$\gamma+\omega_{\mu_1}\notin X_1$ and $\gamma+\omega_{\mu_2}\notin X_2$.
Since $X_1$ and $X_2$ are $\omega$-invariant, we have $\gamma\notin X_1$
and $\gamma\notin X_2$.
Therefore, $X_1\cup X_2\not\supset X$.
Thus, $X$ is prime.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair0}
If an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$ is prime,
then $X^{(\infty)}=H_X$
or $X^{(\infty)}=H_X\cup\{\gamma\}$ for some $\gamma\notin H_X$.
\varepsilonnd{lemma}
\begin{proposition}f
Let $\widetilde{X}=(X,X^{(\infty)})$ be a prime $\omega$-invariant pair.
To derive a contradiction, assume $X^{(\infty)}\setminus H_X$ has
two points $\gamma_1,\gamma_2$.
Take open sets $U_1\ni\gamma_1$, $U_2\ni\gamma_2$ with $U_1\cap U_2=\varepsilonmptyset$,
$U_1\cap H_X=\varepsilonmptyset$ and $U_2\cap H_X=\varepsilonmptyset$.
Then $\widetilde{X}_i=(X,X^{(\infty)}\setminus U_i)\ (i=1,2)$
are $\omega$-invariant pairs satisfying
$\widetilde{X}=\widetilde{X}_1\cup \widetilde{X}_2$.
However, we have $\widetilde{X}\not\subset \widetilde{X}_1$ and
$\widetilde{X}\not\subset \widetilde{X}_2$.
This contradicts the primeness of $\widetilde{X}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair1}
An $\omega$-invariant pair $(X,H_X)$ is prime if and only if
$X$ is a prime $\omega$-invariant set.
\varepsilonnd{lemma}
\begin{proposition}f
Suppose that $(X,H_X)$ is a prime $\omega$-invariant pair.
Take $\omega$-invariant sets $X_1,X_2$ with $X\subset X_1\cup X_2$.
We have $(X,H_X)\subset (X_1,X_1)\cup (X_2,X_2)$.
Since $(X,H_X)$ is prime,
either $(X,H_X)\subset (X_1,X_1)$ or $(X,H_X)\subset (X_2,X_2)$ holds.
Therefore $X$ is a prime $\omega$-invariant set.
Conversely assume that $X$ is a prime $\omega$-invariant set.
Take two $\omega$-invariant pairs $(X_1,X_1^{(\infty)})$,
$(X_2,X_2^{(\infty)})$ with
$(X_1,X_1^{(\infty)})\cup (X_2,X_2^{(\infty)})=(X,H_X)$.
Since $X$ is prime, either $X\subset X_1$ or $X\subset X_2$.
We may assume $X\subset X_1$.
Then $X=X_1$.
Hence $H_X=H_{X_1}\subset X_1^{(\infty)}\subset H_X$.
We get $(X_1,X_1^{(\infty)})=(X,H_X)$.
By Proposition \ref{primepair}, $(X,H_X)$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}f
\begin{lemma}\label{primepair2}
An $\omega$-invariant pair $(X,H_X\cup\{\gamma\})$ is prime
for some $\gamma\notin H_X$ if and only if $X=\gamma+\overline{\rm{sg}}(\omega)$.
\varepsilonnd{lemma}
\begin{proposition}f
Suppose that an $\omega$-invariant pair $(X,H_X\cup\{\gamma\})$ is prime.
Then $(X,H_X\cup\{\gamma\})\subset (X,H_X)\cup (\gamma+\overline{\rm{sg}}(\omega),\gamma+\overline{\rm{sg}}(\omega))$
implies $(X,H_X\cup\{\gamma\})\subset(\gamma+\overline{\rm{sg}}(\omega),\gamma+\overline{\rm{sg}}(\omega))$
because $H_X\cup\{\gamma\}\not\subset H_X$.
Hence $\gamma+\overline{\rm{sg}}(\omega)\subset X\subset\gamma+\overline{\rm{sg}}(\omega)$.
Thus, we get $X=\gamma+\overline{\rm{sg}}(\omega)$.
Conversely, assume $X=\gamma+\overline{\rm{sg}}(\omega)$.
Take two $\omega$-invariant pairs $(X_1,X_1^{(\infty)})$,
$(X_2,X_2^{(\infty)})$ with
$(X_1,X_1^{(\infty)})\cup (X_2,X_2^{(\infty)})=(X,H_X\cup\{\gamma\})$.
We may assume $\gamma\in X_1^{(\infty)}$.
Then we have $X=\gamma+\overline{\rm{sg}}(\omega)\subset X_1^{(\infty)}+\overline{\rm{sg}}(\omega)\subset X_1\subset X$.
Hence $X_1=X$.
We have $H_X\cup\{\gamma\}=H_{X_1}\cup\{\gamma\}\subset
X_1^{(\infty)}\subset H_X\cup\{\gamma\}$.
Therefore $(X_1,X_1^{(\infty)})=(X,H_X\cup\{\gamma\})$.
By Proposition \ref{primepair}, $(X,H_X\cup\{\gamma\})$ is a prime $\omega$-invariant pair.
\varepsilonnd{proposition}f
\begin{proposition}\label{pp}
An $\omega$-invariant pair $(X,X^{(\infty)})$ is prime if and only if
either $X$ is prime and $X^{(\infty)}=H_X$ or $X=\gamma+\overline{\rm{sg}}(\omega)$ and
$X^{(\infty)}=H_X\cup\{\gamma\}$ for some $\gamma\notin H_X$.
\varepsilonnd{proposition}
\begin{proposition}f
Combine Lemma \ref{primepair0}, Lemma \ref{primepair1}
and Lemma \ref{primepair2}.
\varepsilonnd{proposition}f
\section{The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ (part 1)}
In this section and the next section,
we completely determine the ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
(Theorem \ref{idestr1}, Theorem \ref{idestr2}).
The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ depends on whether $\omega\in{\mathcal G}amma^\infty$ satisfies
the following condition:
\begin{corollary}n\label{cond}
For each $i\in\mathbb{Z}_+$, one of the following two conditions is satisfied:
\benu
\item For any positive integer $k$, $k\omega_i\neq 0$.
\item There exists a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$ such that
$S_{\mu_k}^*S_i=0$ for any $k$ and $\lim_{k\to\infty}\omega_{\mu_k}=0$.
\varepsilonnd{enumerate}
\varepsilonnd{corollary}n
This condition is an analogue of Condition (K) in the case of
graph algebras \cite{BHRS}.
In this section,
we deal with the case that $\omega$ satisfies Condition \ref{cond}.
\begin{proposition}\label{cond.exp2}
If $\omega$ satisfies Condition \ref{cond},
then for an ideal $I$ that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
there exists a unique conditional expectation $E_I$ from $({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ onto
${\mathcal F}_I$ such that
$E_I(S_\mu fS_\nu^*)=\delta_{|\mu|,|\nu|}S_\mu fS_\nu^*$
for $\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)$.
\varepsilonnd{proposition}
\begin{proposition}f
Take $x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ where
$\mu_l,\nu_l\in{\mathcal W}_\infty$ and $f_l\in C_0(X_I)$ for $l=1,2,\ldots,L$.
Set $x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*$
and we will prove that $\|x_0\|\leq\|x\|$.
If we choose a positive integer $n$ so that $|\mu_l|,|\nu_l|\leq n$ and
$\mu_l,\nu_l\in{\mathcal W}_n$ for $l=1,2,\ldots,L$,
then $x_0\in{\mathcal F}_I^{(n)}$.
By Lemma \ref{F_n},
there exist $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
First we consider the case that $\|x_0\|=\|x_0^{(k)}\|$
for some $k\leq n-1$.
If we set $q_k=\sum_{\mu\in{\mathcal W}_n^{(k)}}S_\mu p_nS_\mu^*\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$,
then $q_k$ is a projection satisfying that
$q_kS_{\mu_l}S_{\nu_l}^*q_k=0$ if $|\mu_l|\neq |\nu_l|$.
Hence $q_kxq_k=q_kx_0q_k=x_0^{(k)}$.
We get $\|x_0\|=\|x_0^{(k)}\|=\|q_kxq_k\|\leq\|x\|$.
Next we consider the case that $\|x_0\|=\|x_0^{(n)}\|$.
Then there exists $\gamma_0\in X_I$
such that $\|x_0^{(n)}\|=\|x_0^{(n)}(\gamma_0)\|$.
By Lemma \ref{X^{(n)}} (iv), we have
$$X_I=\bigcup_{\mu\in{\mathcal W}_n}(X_I^{(n)}+\omega_\mu)\cup
\bigcap_{k=1}^\infty \bigg( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X_I+\omega_\mu)\bigg).$$
When $\gamma_0\in X_I^{(n)}+\omega_\mu$ for some $\mu\in{\mathcal W}_n$,
set $u=\sum_{\nu\in{\mathcal W}_n^{(n)}}S_\nu S_\mu p_nS_\nu^*\in M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
Then $u$ is a partial isometry.
We have $u^*xu=u^*x_0u=u^*x_0^{(n)}u=\pi_n(\sigma_{\omega_\mu}(x_0^{(n)}))$
where $\pi_n$ is the natural surjection
from ${\mathcal G}_I^{(n,n)}$ onto ${\mathcal F}_I^{(n,n)}$.
Since $\gamma_0-\omega_\mu\in X_I^{(n)}$, we have
$$\|\pi_n(\sigma_{\omega_\mu}(x_0^{(n)}))\|
\geq\|\sigma_{\omega_\mu}(x_0^{(n)})(\gamma_0-\omega_\mu)\|
=\|x_0^{(n)}(\gamma_0)\|=\|x_0^{(n)}\|=\|x_0\|.$$
Therefore $\|x_0\|\leq\|u^*xu\|\leq\|x\|$.
When $\gamma_0\in\bigcap_{k=1}^\infty
\big( \bigcup_{\mu\in{\mathcal W}_n^{(k)}}(X_I+\omega_\mu)\big)$,
we can find $i\in\{1,2,\ldots,n\}$ such that
$\gamma_0-k\omega_i\in X_I$ for all $k\in\mathbb{N}$.
Since $\omega$ satisfies Condition \ref{cond},
either $k\omega_i\neq 0$ for any $k\in\mathbb{Z}_+$ or there exists a sequence
$\{\mu_k\}_{k\in\mathbb{Z}_+}\subset{\mathcal W}_n$ with $\lim_{k\to\infty}\omega_{\mu_k}=0$
and $S_{\mu_k}^*S_i=0$ for any $k$.
In the case that $k\omega_i\neq 0$ for any $k\in\mathbb{Z}_+$,
we can find a neighborhood $U$ of $\gamma_0-n\omega_i\in X_I$ such that
$U\cap (U+k\omega_i)=\varepsilonmptyset$ for $k=1,2,\ldots,n$.
Choose a function $f$ with $0\leq f\leq 1$ satisfying that
the support of $f$ is contained in $U$ and $f(\gamma_0-n\omega_i)=1$.
Set $u=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_i^n f^{1/2} S_\mu^*\in({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$.
Since
$$u^*u=\sum_{\mu,\nu\in{\mathcal W}_n^{(n)}}S_\mu f^{1/2} {S_i^*}^n S_\mu^*
S_\nu S_i^n f^{1/2}S_\nu^*=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu f S_\mu^*,$$
$u^*u$ corresponds to $1\otimes f$
under the isomorphism ${\mathcal G}_I^{(n,n)}\cong\mathbb{M}_{n^n}\otimes C_0(X_I)$.
Thus we have $\|u^*u\|=\sup_{\gamma\in X_I}|f(\gamma)|=1$,
and so $\|u\|=1$.
A routine computation shows that
$u^*xu=u^*x_0^{(n)}u=f\sigma_{n\omega_i}x_0^{(n)}\in C_0(X_I,\mathbb{M}_{n^n})$.
Since $\gamma_0-n\omega_i\in X_I$, we have
$$\|u^*xu\|\geq\|f(\gamma_0-n\omega_i)\sigma_{n\omega_i}x_0^{(n)}(\gamma_0-n\omega_i)\|=\|x_0^{(n)}(\gamma_0)\|=\|x_0\|.$$
Hence $\|x_0\|\leq\|u^*xu\|\leq\|x\|$.
Finally, we consider the case that there exists a sequence
$\{\mu_k\}_{k\in\mathbb{Z}_+}\subset{\mathcal W}_n$
with $\lim_{k\to\infty}\omega_{\mu_k}=0$
and $S_{\mu_k}^*S_i=0$ for any $k\in\mathbb{Z}_+$.
For $k\in\mathbb{Z}_+$, define a partial isometry $u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_i^nS_{\mu_k}S_\mu^*\in{\mathcal O}_\infty\subset M(({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I)$.
A routine computation shows that
$u_k^*xu_k=u_k^*x_0^{(n)}u_k
=\sigma_{n\omega_i+\omega_{\mu_k}}x_0^{(n)}\in C_0(X_I,M_{n^n})$.
Since $\gamma_0-n\omega_i\in X_I$, we have
$$\|u_k^*xu_k\|\geq\|\sigma_{n\omega_i+\omega_{\mu_k}}x_0^{(n)}(\gamma_0-n\omega_i)\|=\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|.$$
Hence we have
$\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|\leq\|u_k^*xu_k\|\leq\|x\|$
for any $k\in\mathbb{Z}_+$.
Therefore $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|=\lim_{k\to\infty}\|x_0^{(n)}(\gamma_0+\omega_{\mu_k})\|\leq\|x\|$.
Hence the map
\begin{align*}
\spa\{S_\mu fS_\nu^*&\mid\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0(X_I)\}\ni x\\
&\mapsto\ x_0\in \spa\{S_\mu fS_\nu^*\mid\mu,\nu\in{\mathcal W}_\infty,\ |\mu|=|\nu|,\ f\in C_0(X_I)\}.
\varepsilonnd{align*}
is well-defined and norm-decreasing.
The extension $E_I$ of the map above is the desired conditional expectation
onto ${\mathcal F}_I$.
Uniqueness is easy to verify.
\varepsilonnd{proposition}f
By uniqueness, the conditional expectation $E_{I}$ above coincides
with the one in Lemma \ref{cond.exp1} when $I$ is gauge invariant.
Actually an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is gauge invariant
if there exists such a conditional expectation,
as we see in the proof of the following theorem.
\begin{theorem}\label{idestr1}
Suppose that $\omega$ satisfies Condition \ref{cond}.
Then for any ideal $I$ we have $I_{\widetilde{X}_I}=I$,
and so $I$ is gauge invariant.
Hence there is a one-to-one correspondence between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma$.
\varepsilonnd{theorem}
\begin{proposition}f
If $X_I=\varepsilonmptyset$, then $I={\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ so $I_{\widetilde{X}_I}=I$.
Let $I$ be an ideal that is not ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, and set $J=I_{\widetilde{X}_I}$.
By the same way as in the proof of Proposition \ref{unique},
we can find a surjective $*$-homomorphism $\pi:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/J\to({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$
whose restriction on ${\mathcal F}_{J}$ is an isomorphism from
${\mathcal F}_{J}$ onto ${\mathcal F}_{I}$.
By Proposition \ref{cond.exp2},
there exists a conditional expectation $E_I:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{I}$
satisfying $E_I\circ \pi=\pi\circ E_{J}$,
where $E_J:({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I\to{\mathcal F}_{J}$ is a faithful conditional expectation
defined in Lemma \ref{cond.exp1}.
By Lemma \ref{isom}, $\pi$ is injective.
Therefore $I=I_{\widetilde{X}_I}$.
The last part follows from Theorem \ref{OneToOne}.
\varepsilonnd{proposition}f
\begin{corollary}\label{primitive1}
When $\omega$ satisfies Condition \ref{cond},
an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive
if and only if the $\omega$-invariant pair $\widetilde{X}_I$ is prime.
\varepsilonnd{corollary}
\begin{proposition}f
It follows from Proposition \ref{primepair} and Theorem \ref{idestr1}.
\varepsilonnd{proposition}f
\section{The ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ (part 2)}
In this section, we investigate the ideal structure of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
when $\omega$ does not satisfy Condition \ref{cond}
i.e.\ there exists $i\in\mathbb{Z}_+$ such that
$k\omega_i=0$ for some positive integer $k$,
and that there exist no sequences $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$ such that
$S_{\mu_k}^*S_i=0$ for any $k$ and $\lim_{k\to\infty}\omega_{\mu_k}=0$.
Note that such $i$ is unique.
Without loss of generality, we may assume $i=1$.
Let $K$ be the smallest positive integer satisfying $K\omega_1=0$.
Denote by ${\mathcal G}amma'$ the quotient of ${\mathcal G}amma$
by the subgroup generated by $\omega_1$, which is isomorphic to $\mathbb{Z}/K\mathbb{Z}$.
We denote by $[\gamma]$ and $[U]$ the images in ${\mathcal G}amma'$
of $\gamma\in{\mathcal G}amma$ and $U\subset{\mathcal G}amma$ respectively.
We use the symbol $([\gamma],\theta)$
for denoting elements of ${\mathcal G}amma'\times\mathbb{T}$.
Define $A=\cspa\{S_1^kf{S_1^*}^l\mid f\in C_0({\mathcal G}amma), k,l\in\mathbb{N}\}$
which is a $C^*$-subalgebra of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
In \cite{Ka1},
we defined a $C^*$-algebra $T_K$ and a continuous family of $*$-homomorphisms
$\varphi_\gamma:A\to T_K$ for $\gamma\in{\mathcal G}amma$.
Note that $\varphi_\gamma(x)=0$
if and only if $\varphi_{\gamma+\omega_1}(x)=0$ for $x\in A$.
We also defined $\psi_{\gamma,\theta}=\pi_\theta\circ \varphi_\gamma$
for $(\gamma,\theta)\in{\mathcal G}amma\times\mathbb{T}$,
where $\pi_\theta:T_K\to\mathbb{M}_{K}$ is a continuous family of $*$-homomorphisms.
\begin{definition}\rm
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
we define the closed subset $Y_I$ of ${\mathcal G}amma'\times\mathbb{T}$ by
$$Y_I=\{([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}\mid \psi_{\gamma,\theta}(x)=0
\mbox{ for all }x\in A\cap I\}.$$
We denote by $\widetilde{Y}_I$ the pair $(Y_I,X_I^{(\infty)})$ of
a subset $Y_I$ of ${\mathcal G}amma'\times\mathbb{T}$
and a subset $X_I^{(\infty)}$ of ${\mathcal G}amma$.
\varepsilonnd{definition}
\begin{definition}\rm\label{omegaY}
For a pair $\widetilde{Y}=(Y,X^{(\infty)})$ of
a subset $Y$ of ${\mathcal G}amma'\times\mathbb{T}$
and a subset $X^{(\infty)}$ of ${\mathcal G}amma$,
we define subsets $X$ and $X^{(n)}$ of ${\mathcal G}amma$ by
\begin{align*}
X&=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y
\mbox{ for some } \theta\in\mathbb{T}\},\\
X^{(n)}&=X^{(\infty)}\cup\bigcup_{i=n+1}^\infty(X+\omega_i).
\varepsilonnd{align*}
With this notation,
a pair $\widetilde{Y}=(Y,X^{(\infty)})$ is called {\varepsilonm $\omega$-invariant}
if $(X,X^{(\infty)})$ is an $\omega$-invariant pair of subsets of ${\mathcal G}amma$
and if $Y$ is a closed set satisfying that $[X^{(1)}]\times\mathbb{T}\subset Y$.
\varepsilonnd{definition}
\begin{proposition}
For an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$,
the pair $\widetilde{Y}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}
\begin{proposition}f
By \cite[Proposition 5.15]{Ka1}, we have
$$X_I=\{\gamma\in{\mathcal G}amma\mid
([\gamma],\theta)\in Y_I\mbox{ for some } \theta\in\mathbb{T}\}.$$
By the argument in the proof of \cite[Lemma 5.21]{Ka1}, we have
$$X_I^{(1)}=\{\gamma\in{\mathcal G}amma\mid
\varphi_{\gamma}(x)=0\mbox{ for any } x\in A\cap I\}.$$
Therefore $[X_I^{(1)}]\times\mathbb{T}\subset Y_I$.
Thus the pair $\widetilde{Y}_I$ is $\omega$-invariant.
\varepsilonnd{proposition}f
We get the $\omega$-invariant pair $\widetilde{Y}_I$
from an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Conversely, from an $\omega$-invariant pair $\widetilde{Y}$ ,
we can construct the ideal $I_{\widetilde{Y}}$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\begin{definition}\rm
For an $\omega$-invariant pair $\widetilde{Y}=(Y,X^{(\infty)})$,
we define $J_{\widetilde{Y}}\subset A$
and $I_{\widetilde{Y}}\subset{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ by
\begin{align*}
J_{\widetilde{Y}}&=\{x\in A\mid
\psi_{\gamma,\theta}(x)=0\mbox{ for }([\gamma],\theta)\in Y,
\mbox{ and } \varphi_\gamma(x)=0\mbox{ for }\gamma\in X^{(1)}\},\\
I_{\widetilde{Y}}
&=\cspa\big(\{S_\mu xS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
x\in J_{\widetilde{Y}}\}\\
&\hspace*{2cm}\cup\{S_\mu p_nfS_\nu \mid \mu,\nu\in{\mathcal W}_\infty,\
n\in\mathbb{Z}_+,\ f\in C_0({\mathcal G}amma\setminus X^{(n)})\}\big),
\varepsilonnd{align*}
with the notation in Definition \ref{omegaY}.
\varepsilonnd{definition}
\begin{proposition}
For an $\omega$-invariant pair $\widetilde{Y}$,
$I_{\widetilde{Y}}$ is an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\varepsilonnd{proposition}
\begin{proposition}f
Once noting that
$J_{\widetilde{Y}}\cap C_0({\mathcal G}amma)=C_0({\mathcal G}amma\setminus X)$ and
$J_{\widetilde{Y}}\cap p_1 C_0({\mathcal G}amma)=p_1 C_0({\mathcal G}amma\setminus X^{(1)})$,
we can prove that $I_{\widetilde{Y}}$ is an ideal
in a similar way to Proposition \ref{I_X,Xinfty}
with the help of the computation in \cite[Proposition 5.20]{Ka1}.
\varepsilonnd{proposition}f
\begin{lemma}\label{Y_I_Y}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair.
For any $([\gamma],\theta)\not\in Y$, there exists
$x\in J_{\widetilde{Y}}$ such that $\psi_{\gamma,\theta}(x)\neq 0$.
\varepsilonnd{lemma}
\begin{proposition}f
The proof goes exactly the same as in the proof of \cite[Lemma 5.22]{Ka1},
once noting that $([\gamma],\theta)\not\in Y$ implies $\gamma\not\in X^{(1)}$.
\varepsilonnd{proposition}f
\begin{proposition}\label{exist2}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair,
and set $I=I_{\widetilde{Y}}$.
Then we have $\widetilde{Y}_I=\widetilde{Y}$.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{Y_I_Y}, we get $Y_I\subset Y$.
To prove the other inclusion, it is sufficient to see
that $\psi_{\gamma,\theta}(x)=0$
for $([\gamma],\theta)\in Y$ and $x\in I\cap A$.
Take $\varepsilon>0$ arbitrarily.
Since $x\in I$,
there exist $\mu_l,\nu_l\in{\mathcal W}_\infty$, $x_l\in J_{\widetilde{Y}}$
for $l=1,2,\ldots,L$ and $\mu_k',\nu_k'\in{\mathcal W}_\infty$, $n_k\in\mathbb{Z}_+$,
$f_k\in C_0({\mathcal G}amma\setminus X^{(n_k)})$ for $k=1,2,\ldots,K$
such that
$$\bigg\|x-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*-\sum_{k=1}^K
S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg\|<\varepsilon.$$
Take a positive integer $m$ such that $m\geq |\mu_l|,|\nu_l|$
for any $l$ and $m> |\mu_k'|,|\nu_k'|$ for any $k$.
Then, $\left\|{S_1^*}^mxS_1^m-\sum_{l=1}^Lx_l'\right\|<\varepsilon$
where $x_l'={S_1^*}^mS_{\mu_l}x_lS_{\nu_l}^*S_1^m$ for $l=1,2,\ldots,L$.
Since $x_l'\in J_{\widetilde{Y}}$,
we have $\|\psi_{\gamma,\theta}({S_1^*}^mxS_1^m)\|<\varepsilon$.
Since $\psi_{\gamma,\theta}(S_1)$ is a unitary,
we have $\|\psi_{\gamma,\theta}(x)\|<\varepsilon$ for arbitrary $\varepsilon>0$.
Hence, we have $\psi_{\gamma,\theta}(x)=0$.
Therefore we get $Y_I=Y$.
From $Y_I=Y$, we have $X_I=X$.
By the definition of $I$,
we see that $X_I^{(n)}\subset X^{(n)}$ for $n\in\mathbb{Z}_+$.
To the contrary, assume that $X_I^{(n)}\subsetneqq X^{(n)}$.
Then there exists $f\in C_0({\mathcal G}amma)$
such that $p_nf\in I$ and $f(\gamma_0)=1$ for some $\gamma_0\in X^{(n)}$.
Since $p_nf\in I$, there exist
$\mu_l,\nu_l\in{\mathcal W}_\infty$, $x_l\in J_{\widetilde{Y}}$
for $l=1,2,\ldots,L$ and $\mu_k',\nu_k'\in{\mathcal W}_\infty$, $n_k\in\mathbb{Z}_+$,
$f_k\in C_0({\mathcal G}amma\setminus X^{(n_k)})$ for $k=1,2,\ldots,K$
such that
$$\bigg\|p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*-\sum_{k=1}^K
S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg\|<\frac12.$$
Take a positive integer $m$ so large that
$\mu_l,\nu_l,\mu_k',\nu_k'\in{\mathcal W}_{m}$, $n_k\leq m$
for any $l,k$ and $n\leq m$.
By Lemma \ref{X^{(n)}} (iii), we have
$X^{(n)}=X^{(m)}\cup\bigcup_{i=n+1}^{m}(X+\omega_i)$.
We first consider the case that $\gamma_0\in X^{(m)}$.
By \cite[Lemma 5.4]{Ka1}, there exists
$g_l\in C_0({\mathcal G}amma\setminus X^{1})$ with $p_1x_lp_1=p_1g_l$ for any $l$.
Hence we have $p_{m}x_lp_{m}=p_{m}p_1x_lp_1p_{m}=p_{m}g_l$ for any $l$.
Since
$$p_{m}\bigg(p_nf-\sum_{l=1}^LS_{\mu_l}x_lS_{\nu_l}^*
-\sum_{k=1}^K S_{\mu_k'}p_{n_k}f_kS_{\nu_k'}^*\bigg)p_{m}=
p_{m}f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}p_{m}g_l
-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}p_{m}f_k,$$
we get
$\|f-\sum_{\mu_l=\nu_l=\varepsilonmptyset}g_l-\sum_{\mu_k'=\nu_k'=\varepsilonmptyset}f_k\|<1/2$.
This contradicts the fact that $f(\gamma_0)=1$, $g_l(\gamma_0)=0$ and
$f_k(\gamma_0)=0$ for any $l,k$.
When $\gamma_0\in X+\omega_i$ for some $i$ with $n<i\leq m$,
we have $\sigma_{\omega_i}f=S_i^*(p_nf)S_i\in I$ and
$\sigma_{\omega_i}f(\gamma_0-\omega_i)=1$.
This contradicts the fact that $X_I=X$.
Therefore $X_I^{(n)}=X^{(n)}$ for a positive integer $n$.
Hence $X_I^{(\infty)}=\bigcap_{n=1}^\infty X_I^{(n)}=
\bigcap_{n=1}^\infty X^{(n)}=X^{(\infty)}$.
Thus we have $\widetilde{Y}_I=\widetilde{Y}$.
\varepsilonnd{proposition}f
\begin{corollary}\label{Ysub}
For two $\omega$-invariant pairs $\widetilde{Y}_1=(Y_1,X_1^{(\infty)})$,
$\widetilde{Y}_2=(Y_2,X_2^{(\infty)})$,
we have $I_{\widetilde{Y}_1}\subset I_{\widetilde{Y}_2}$
if and only if $Y_1\supset Y_2$ and $X_1^{(\infty)}\supset X_2^{(\infty)}$.
\varepsilonnd{corollary}
A relation between $I_{\widetilde{Y}}$ and $I_{\widetilde{X}}$
can be described as follows.
\begin{proposition}\label{rotation}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant pair.
For $t\in\mathbb{T}$, set $\widetilde{Y}_t=(Y_t,X^{(\infty)})$ where
$Y_t=\{([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}\mid ([\gamma],t\theta)\in Y\}$.
Then $\widetilde{Y}_t$ is $\omega$-invariant
and $\beta_t(I_{\widetilde{Y}})=I_{\widetilde{Y}_{t^K}}$
where $\beta$ is the gauge action.
We also have $I_{\widetilde{X}}
=\bigcap_{t\in\mathbb{T}}I_{\widetilde{Y}_t}$
where $\widetilde{X}=(X,X^{(\infty)})$ and
$X=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y
\mbox{ for some }\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
\begin{proposition}f
See \cite[Proposition 5.24]{Ka1}.
\varepsilonnd{proposition}f
\begin{proposition}
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
of subsets of ${\mathcal G}amma$,
the pair $\widetilde{Y}=([X]\times\mathbb{T},X^{(\infty)})$ is $\omega$-invariant
and $I_{\widetilde{Y}}=I_{\widetilde{X}}$.
\varepsilonnd{proposition}
\begin{proposition}f
Obvious by Proposition \ref{rotation}.
\varepsilonnd{proposition}f
Now, we turn to showing that $I_{\widetilde{Y}_I}=I$ for any ideal $I$
(Theorem \ref{idestr2}).
To see this, we examine the primitive ideal space of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Set $\overline{\rm{sg}}_1(\omega)=\overline{\rm{sg}}(\omega)\setminus\{0,\omega_1,\ldots,(K-1)\omega_1\}$.
\begin{lemma}\label{isolate}
We have $\overline{\rm{sg}}_1(\omega)=\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$ and
$\overline{\rm{sg}}_1(\omega)$ is an $\omega$-invariant set.
\varepsilonnd{lemma}
\begin{proposition}f
For $\gamma\in\overline{\rm{sg}}(\omega)$, we can find $\mu_k\in{\mathcal W}_\infty$ such that
$\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
If $\mu_k=(1,1,\ldots,1)$ for sufficiently large $k$,
then $\gamma=m\omega_1$ for some $m\in\mathbb{N}$.
Hence for $\gamma\in\overline{\rm{sg}}_1(\omega)$, we can find $\mu_k\in{\mathcal W}_\infty$
with $\omega_{\mu_k}\in\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)$ such that
$\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
Thus $\overline{\rm{sg}}_1(\omega)\subset\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
To prove the other inclusion,
suppose $m\omega_1\in\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$
for some $0\leq m<K$
and we will derive a contradiction.
In this case, $0$ is also in $\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
Hence there exists a sequence $\{\mu_k\}$ in ${\mathcal W}_\infty$
with $S_{\mu_k}^*S_1=0$
such that $0=\lim_{k\to\infty}\omega_{\mu_k}$.
This contradicts the fact that $\omega$ does not satisfy Condition \ref{cond}.
Therefore $\overline{\rm{sg}}_1(\omega)=\overline{\bigcup_{i=2}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
From this equality, it is easy to see that
$\overline{\rm{sg}}_1(\omega)$ is an $\omega$-invariant set.
\varepsilonnd{proposition}f
\begin{corollary}\label{cptnbhd}
For any $\gamma_0\in{\mathcal G}amma$,
there exists a compact neighborhood $X$ of $\gamma_0$ satisfying that
$X\cap(X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$.
\varepsilonnd{corollary}
\begin{proposition}f
Since
$\overline{\rm{sg}}(\omega)\setminus\{0\}=\overline{\rm{sg}}_1(\omega)\cup\{\omega_1,2\omega_1,\ldots,(K-1)\omega_1\}$
is closed by Lemma \ref{isolate}, there exists a neighborhood $U$
of $0$ with $U\cap (\overline{\rm{sg}}(\omega)\setminus\{0\})=\varepsilonmptyset$.
If we take a compact neighborhood $V$ of $0$ such that $V-V\subset U$,
then $X=\gamma_0+V$ becomes a desired compact neighborhood of $\gamma_0$.
\varepsilonnd{proposition}f
\begin{lemma}\label{H_X}
For an $\omega$-invariant set $X$, we have
$H_X=\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}$.
If two $\omega$-invariant sets $X_1$ and $X_2$ satisfy $X_1\subset X_2$,
then $H_{X_1}\subset H_{X_2}$.
\varepsilonnd{lemma}
\begin{proposition}f
The former part follows from $X=X+\omega_1$, and this implies the latter part.
\varepsilonnd{proposition}f
\begin{proposition}
For any $\gamma\in{\mathcal G}amma$, we have $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}
\begin{proposition}f
By Lemma \ref{H_X}, we have
$$H_{\gamma+\overline{\rm{sg}}(\omega)}=
\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
\subset\overline{\bigcup_{i=2}^\infty(\gamma+\overline{\rm{sg}}(\omega)+\omega_i)}
=\gamma+\overline{\rm{sg}}_1(\omega).$$
Hence $\gamma\notin H_{\gamma+\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}f
For $\gamma\in{\mathcal G}amma$, we set $P_{\gamma}=I_{\widetilde{X}}$ where
$\widetilde{X}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$
which is a prime $\omega$-invariant pair.
We will show that $P_{\gamma}$ is the unique primitive ideal satisfying that
$\widetilde{X}_{P_{\gamma}}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$.
To see this, we need the following lemma.
\begin{lemma}\label{Pgamma0}
Let $I$ be an ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$.
Then $I=I_{\widetilde{X}_I}$.
\varepsilonnd{lemma}
\begin{proposition}f
By the argument in the proof of
Proposition \ref{cond.exp2} and Theorem \ref{idestr1},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*$.
If we choose a positive integer $n$
so that $|\mu_l|,|\nu_l|\leq n$ and $\mu_l,\nu_l\in{\mathcal W}_n$ for any $l$,
then $x_0\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
Since $X_I=\overline{X_I^{(\infty)}+{\rm{sg}}(\omega)}$,
there exist a sequence $\mu_1,\mu_2,\ldots\in{\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots,\in X_I^{(\infty)}$
such that $\gamma_0=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
We can find sequences $\mu_1',\mu_2',\ldots\in{\mathcal W}_\infty$ and
$\nu_1,\nu_2,\ldots\in{\mathcal W}_n$ such that
$\omega_{\mu_k}=\omega_{\mu_k'}+\omega_{\nu_k}$ and
none of $1,2,\ldots,n$ appears in the word $\mu_k'$ for any $k$.
For $k\in\mathbb{Z}_+$, define a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_{\nu_k}p_n S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k
=\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})$,
where $\pi_n$ is the natural surjection
from ${\mathcal G}_I^{(n,n)}$ onto ${\mathcal F}_I^{(n,n)}$.
Since $\gamma_k\in X_I^{(\infty)}$,
we have $\gamma_k+\omega_{\mu_k'}\in X_I^{(n)}$.
Hence
$$\|\pi_n(\sigma_{\omega_{\nu_k}}x_0^{(n)})\|
\geq\|\sigma_{\omega_{\nu_k}}x_0^{(n)}(\gamma_k+\omega_{\mu_k'})\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma_0)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k'}+\omega_{\nu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
\begin{proposition}\label{Pgamma}
For any $\gamma\in{\mathcal G}amma$, the ideal $P_{\gamma}$ is the unique primitive
ideal satisfying that
$\widetilde{X}_{P_{\gamma}}=(\gamma+\overline{\rm{sg}}(\omega),H_{\gamma+\overline{\rm{sg}}(\omega)}\cup\{\gamma\})$.
\varepsilonnd{proposition}
\begin{proposition}f
To prove that $P_{\gamma}$ is primitive,
it suffices to show that it is prime because ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is separable.
Let $I_1,I_2$ be ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
with $I_1\cap I_2= P_\gamma$.
Then we get
$\widetilde{X}_{I_1}\cup\widetilde{X}_{I_2}=\widetilde{X}_{P_\gamma}$.
Since $\widetilde{X}_{P_\gamma}$ is a prime $\omega$-invariant pair,
we have either
$\widetilde{X}_{I_1}=\widetilde{X}_{P_\gamma}$ or
$\widetilde{X}_{I_2}=\widetilde{X}_{P_\gamma}$.
By Lemma \ref{Pgamma0}, we have either $I_1=P_\gamma$ or $I_2=P_\gamma$.
Therefore $P_\gamma$ is primitive.
The uniqueness follows from Lemma \ref{Pgamma0}.
\varepsilonnd{proposition}f
We denote by $\varDelta$ the set of prime $\omega$-invariant sets
which are not of the form $\gamma+\overline{\rm{sg}}(\omega)$.
For $X\in\varDelta$, we denote by $P_X$ the ideal $I_{\widetilde{X}}$
for $\widetilde{X}=(X,H_X)$ which is a prime $\omega$-invariant pair.
We will show that for any $X\in\varDelta$,
$P_X$ is the unique primitive ideal satisfying $\widetilde{X}_{P_X}=(X,H_X)$.
\begin{lemma}\label{PX0}
Let $X\in\varDelta$ and $\gamma\in X$.
Then there exist a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X$ such that
$S_{\mu_k}^*S_1=0$ for any $k$
and $\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
\varepsilonnd{lemma}
\begin{proposition}f
Since $X\in\varDelta$, there exists $\gamma'\in X\setminus (\gamma+\overline{\rm{sg}}(\omega))$.
Since $X$ is prime, Proposition \ref{Xprime} gives us
two sequences $\mu_1,\mu_2,\ldots$, $\nu_1,\nu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X$ with
$\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$
and $\gamma'=\lim_{k\to\infty}(\gamma_k+\omega_{\nu_k})$.
We will show that we can choose such $\mu_k$ satisfying $S_{\mu_k}^*S_1=0$.
If not so, then $\mu_k=(1,1,\ldots,1)$ for sufficiently large $k$.
This implies $\gamma'=\lim_{k\to\infty}(\gamma-|\mu_k|\omega_1+\omega_{\nu_k})$
which contradicts the fact that $\gamma'\notin\gamma+\overline{\rm{sg}}(\omega)$.
Therefore we can find desired sequences.
\varepsilonnd{proposition}f
\begin{lemma}\label{PX1}
If an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ satisfies $X_I\in\varDelta$,
then $I=I_{\widetilde{X}_I}$.
\varepsilonnd{lemma}
\begin{proposition}f
Similarly as the proof of Lemma \ref{Pgamma0},
it suffices to show that $\|x_0\|\leq\|x\|$ for
$x=\sum_{l=1}^LS_{\mu_l}f_lS_{\nu_l}^*\in ({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)/I$ and
$x_0=\sum_{|\mu_l|=|\nu_l|}S_{\mu_l}f_lS_{\nu_l}^*\in {\mathcal F}_I^{(n)}$.
We can find $x_0^{(k)}\in{\mathcal F}_{I}^{(n,k)}\ (0\leq k\leq n-1)$
and $x_0^{(n)}\in{\mathcal G}_{I}^{(n,n)}$ such that $x_0=\sum_{k=0}^nx_0^{(k)}$.
We have $\|x_0\|=\max\{\|x_0^{(0)}\|,\ldots,\|x_0^{(n)}\|\}$.
In the case that $\|x_0\|=\|x_0^{(k)}\|$ for some $k\leq n-1$,
we can prove $\|x_0\|\leq\|x\|$ in a similar way to the proof of
Proposition \ref{cond.exp2}.
In the case that $\|x_0\|=\|x_0^{(n)}\|$,
there exists $\gamma_0\in X_I$
such that $\|x_0\|=\|x_0^{(n)}(\gamma_0)\|$.
By Lemma \ref{PX0}, we have a sequence $\mu_1,\mu_2,\ldots$ in ${\mathcal W}_\infty$
and a sequence $\gamma_1,\gamma_2,\ldots$ in $X_I$ such that
$S_{\mu_k}^*S_1=0$ and $\gamma=\lim_{k\to\infty}(\gamma_k+\omega_{\mu_k})$.
For $k\in\mathbb{Z}_+$, set a partial isometry
$u_k=\sum_{\mu\in{\mathcal W}_n^{(n)}}S_\mu S_1^{Kn}S_{\mu_k}S_\mu^*$.
We have
$u_k^*xu_k=u_k^*x_0u_k=u_k^*x_0^{(n)}u_k=\sigma_{\omega_{\mu_k}}x_0^{(n)}$.
Since $\gamma_k\in X_I$, we have
$$\|u_k^*xu_k\|\geq\|\sigma_{\omega_{\mu_k}}x_0^{(n)}(\gamma_k)\|
=\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|.$$
Therefore we get
$$\|x_0\|=\|x_0^{(n)}(\gamma)\|=\lim_{k\to\infty}
\|x_0^{(n)}(\gamma_k+\omega_{\mu_k})\|\leq\|x\|.$$
We are done.
\varepsilonnd{proposition}f
\begin{proposition}\label{PX}
For $X\in\varDelta$, the ideal $P_{X}$ is the unique primitive ideal
satisfying $\widetilde{X}_{P_X}=(X,H_X)$.
\varepsilonnd{proposition}
\begin{proposition}f
With the help of Lemma \ref{PX1},
the proof goes similarly as the one in Proposition \ref{Pgamma}.
\varepsilonnd{proposition}f
By Proposition \ref{pp}, the remaining candidates for primitive ideals
are ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
for some $\gamma_0\in{\mathcal G}amma$.
We will determine such primitive ideals.
\begin{definition}\rm\label{DefP}
For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$,
we set $Y_{([\gamma],\theta)}=\big\{([\gamma],\theta)\big\}\cup
\big([\gamma+\overline{\rm{sg}}_1(\omega)]\times\mathbb{T}\big)$.
Then $\widetilde{Y}=(Y_{([\gamma],\theta)},H_{\gamma+\overline{\rm{sg}}(\omega)})$
is an $\omega$-invariant pair.
We write $P_{([\gamma],\theta)}$
for denoting $I_{\widetilde{Y}}$.
\varepsilonnd{definition}
We can show that $P_{([\gamma],\theta)}$ is a primitive ideal
for any $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$
by using the technique in \cite{Ka1}.
To do so, we need Proposition \ref{local}, which
will also be used to determine the topology of primitive ideal space of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
\begin{lemma}\label{I_X=}
For an $\omega$-invariant set $X$,
the pair $\widetilde{X}=(X,X)$ is $\omega$-invariant and we have
$$I_{\widetilde{X}}=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}.$$
\varepsilonnd{lemma}
\begin{proposition}f
Clearly, $\widetilde{X}=(X,X)$ is $\omega$-invariant.
Set $I=\cspa\{S_\mu fS_\nu^*\mid
\mu,\nu\in{\mathcal W}_\infty,\ f\in C_0({\mathcal G}amma\setminus X)\}$.
In a similar way to the proof of Proposition \ref{I_X,Xinfty},
we can see that $I$ is a gauge-invariant ideal of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
We also see that $X_I^{(n)}=X$ for any $n\in\mathbb{N}$
by arguing as in the proof of Proposition \ref{exist}.
Hence $I_{\widetilde{X}}=I$ by Theorem \ref{OneToOne}.
\varepsilonnd{proposition}f
\begin{proposition}\label{local}
Let $X$ be a compact subset of ${\mathcal G}amma$ such that
$X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$,
and set $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$.
Then we have that $\widetilde{X}_0=(X_1,X_1)$, $\widetilde{X}_1=(X_1,X_2)$ and
$\widetilde{X}_2=(X_2,X_2)$ are $\omega$-invariant pairs,
and that
\begin{align*}
I_{\widetilde{X}_2}/I_{\widetilde{X}_1}&\cong\mathbb{K}\otimes C(X\times\mathbb{T}),&
I_{\widetilde{X}_1}/I_{\widetilde{X}_0}&\cong\mathbb{K}\otimes C(X_1\setminus X_2).
\varepsilonnd{align*}
\varepsilonnd{proposition}
\begin{proposition}f
Since $X$ is compact and $\overline{\rm{sg}}(\omega)$ is closed, $X_1=X+\overline{\rm{sg}}(\omega)$ becomes closed.
The same reason shows that $X_2$ is closed.
By Lemma \ref{isolate}, both $X_1$ and $X_2$ are $\omega$-invariant and
$X_2=\overline{\bigcup_{i=2}^\infty(X_1+\omega_i)}$.
Therefore $\widetilde{X}_0,\widetilde{X}_1,\widetilde{X}_2$ are
$\omega$-invariant pairs.
Since $I_{\widetilde{X}_1}\cap p_1 C_0({\mathcal G}amma)=p_1 C_0({\mathcal G}amma\setminus X_2)$,
we have $p_1f=0$
for any $f\in C_0(X_1\setminus X_2)\subset I_{\widetilde{X}_2}/I_{\widetilde{X}_1}$.
Note that $X_1\setminus X_2$ is a disjoint union of compact sets
$X,X+\omega_1,\ldots,X+(K-1)\omega_1$.
For $f\in C(X+m\omega_1)\subset I_{\widetilde{X}_2}/I_{\widetilde{X}_1}$
with $0<m<K$,
we have $\sigma_{m\omega_1}f\in C(X)$ and
\begin{align*}
S_1^m\sigma_{m\omega_1}f{S_1^*}^m
&=S_1^{m-1}S_1S_1^*\sigma_{(m-1)\omega_1}f{S_1^*}^{m-1}
=S_1^{m-1}\sigma_{(m-1)\omega_1}f{S_1^*}^{m-1}\\
&=\cdots=f.
\varepsilonnd{align*}
Hence, we have
$I_{\widetilde{X}_2}/I_{\widetilde{X}_1}
=\cspa\{S_\mu fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty, f\in C(X)\}$
by Lemma \ref{I_X=}.
Set ${\mathcal W}_\infty^+={\mathcal W}_\infty\setminus\{\mu 1^K\in{\mathcal W}_\infty\mid\mu\in{\mathcal W}_\infty\}$
and denote by $\chi$ the characteristic function of $X$.
Then $\{S_\mu\chi S_\nu^*\}_{\mu,\nu\in {\mathcal W}_\infty^+}$
satisfies the relation of matrix units
and $\sum_{\mu\in{\mathcal W}_\infty^+}S_\mu\chi S_\mu^*=1$
(strictly).
Hence we have $I_{\widetilde{X}_2}/I_{\widetilde{X}_1}\cong\mathbb{K}\otimes B$
where $B=\chi(I_{\widetilde{X}_2}/I_{\widetilde{X}_1})\chi$.
We have
$$B=\cspa\{\chi S_\mu fS_\nu^* \chi\mid \mu,\nu\in{\mathcal W}_\infty, f\in C(X)\}
=\cspa\{(S_1^K)^m f\mid m\in\mathbb{Z}, f\in C(X)\}.$$
Since $B$ is generated by $C(X)$ and a unitary $S_1^K\chi$ which commute
with each other and since $B$ is globally invariant under the gauge action,
we have $B\cong C(X\times\mathbb{T})$.
Therefore we get
$I_{\widetilde{X}_2}/I_{\widetilde{X}_1}\cong\mathbb{K}\otimes C(X\times\mathbb{T})$.
By the definition,
$$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}
=\cspa\{S_\mu p_nfS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty, n\geq 1,\
f\in C(X_1\setminus X_2)\}.$$
For $f\in C(X_1\setminus X_2)\subset
I_{\widetilde{X}_1}/I_{\widetilde{X}_0}$ and $i\geq 2$,
we have $S_iS_i^*f=S_i\sigma_{\omega_i}fS_i^*=0$.
Hence $p_nf=p_1f$ for any $n\geq 1$
and any $f\in C(X_1\setminus X_2)$.
Thus $I_{X_1,X_2}/I_{X_1,X_1}
=\cspa\{S_\mu p_2fS_\nu^*\mid \mu,\nu\in{\mathcal W}_\infty,\
f\in C(X_1\setminus X_2)\}.$
We can show that
$\{S_\mu p_2\chi'S_\nu^*\}_{\mu,\nu\in{\mathcal W}_\infty}$
satisfies the relation of matrix units
and $\sum_{\mu\in{\mathcal W}_\infty}S_\mu p_2\chi'S_\mu^*=1$ (strictly),
where $\chi'$ is the characteristic function of $X_1\setminus X_2$.
Hence we have $I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes B'$
where
$$B'=p_2\chi'(I_{\widetilde{X}_1}/I_{\widetilde{X}_0})p_2\chi'
=\cspa\{p_2f\mid f\in C(X_1\setminus X_2)\}\cong C(X_1\setminus X_2).$$
Therefore we get
$I_{\widetilde{X}_1}/I_{\widetilde{X}_0}\cong\mathbb{K}\otimes C(X_1\setminus X_2)$.
\varepsilonnd{proposition}f
With the help of Proposition \ref{local},
we have the following proposition
by exactly the same argument as the proof of \cite[Proposition 5.41]{Ka1}.
\begin{proposition}\label{Primsurj}
For $\gamma_0\in{\mathcal G}amma$, the set of all primitive ideals $P$ satisfying
$\widetilde{X}_P=(\gamma_0+\overline{\rm{sg}}(\omega),H_{\gamma_0+\overline{\rm{sg}}(\omega)})$
is $\{P_{([\gamma_0],\theta)}\mid\theta\in\mathbb{T}\}$.
\varepsilonnd{proposition}
Now, we can describe the primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
as follows.
\begin{proposition}\label{primitive2}
We have
$\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=\{P_z\mid z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\}$,
where $\sqcup$ means a disjoint union.
\varepsilonnd{proposition}
The primitive ideal space $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is a topological space
whose closed sets are given by
$\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I\subset P\}$ for ideals $I$.
We will investigate which subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
corresponds to a closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
By Corollary \ref{Ysub}, the following is easy to verify.
\begin{lemma}\label{Primlem1}
Let $\widetilde{Y}=(Y,X^{(\infty)})$ be an $\omega$-invariant set.
\benu
\item For $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$, we have
$I_{\widetilde{Y}}\subset P_{([\gamma],\theta)}$ if and only if
$([\gamma],\theta)\in Y$.
\item For $\gamma\in{\mathcal G}amma$, we have
$I_{\widetilde{Y}}\subset P_{\gamma}$ if and only if
$\gamma\in X^{(\infty)}$.
\item For $X\in\varDelta$, we have
$I_{\widetilde{Y}}\subset P_{X}$ if and only if
$[X]\times\mathbb{T}\subset Y$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{lemma}\label{Primlem2}
Let $X$ be a compact subset of ${\mathcal G}amma$
such that $X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$,
and set $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$, which are $\omega$-invariant sets.
If $X_0\in\varDelta$ satisfies $X_1\supset X_0$, then $X_2\supset X_0$.
\varepsilonnd{lemma}
\begin{proposition}f
To the contrary, assume $X_0\in\varDelta$ satisfies
$X_1\supset X_0$ and $X_2\not\supset X_0$.
Then $X_0\cap X\neq\varepsilonmptyset$ and
$(X_0\cap X)+\overline{\rm{sg}}(\omega)$ is an $\omega$-invariant set
satisfying $(X_0\cap X)+\overline{\rm{sg}}(\omega)\subset X_0$.
Since $((X_0\cap X)+\overline{\rm{sg}}(\omega)) \cup X_2\supset X_0$ and $X_0$ is prime,
we have $(X_0\cap X)+\overline{\rm{sg}}(\omega)\supset X_0$.
Hence $X_0=(X_0\cap X)+\overline{\rm{sg}}(\omega)$.
If $X_0\cap X$ has two points $\gamma_1,\gamma_2$,
then we can take open sets $U_1,U_2$ such that $\gamma_1\in U_1$,
$\gamma_2\in U_2$, $U_1\cap U_2=\varepsilonmptyset$.
Two $\omega$-invariant sets $X_1'=(X_0\cap X\setminus U_1)+\overline{\rm{sg}}(\omega)$,
$X_2'=(X_0\cap X\setminus U_2)+\overline{\rm{sg}}(\omega)$ satisfies
$X_1'\not\supset X_0$, $X_2'\not\supset X_0$ and $X_1'\cup X_2'=X_0$.
This contradicts the primeness of $X_0$.
Hence $X_0\cap X$ is just a point.
However, this contradicts the fact that $X_0\in\varDelta$.
Therefore $X_2\supset X_0$ when $X_0\in\varDelta$ satisfies $X_1\supset X_0$.
\varepsilonnd{proposition}f
\begin{lemma}\label{Primlem3}
Let $\widetilde{Y}_\lambda=(Y_\lambda,X_\lambda^{(\infty)})$
be an $\omega$-invariant pair for each $\lambda\in\Lambda$.
Set $I=\bigcap_{\lambda\in\Lambda}I_{\widetilde{Y}_\lambda}$.
Then $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{lemma}
\begin{proposition}f
For any $\lambda\in\Lambda$, we have $Y_I\supset Y_\lambda$
because $I\subset I_{\widetilde{Y}_\lambda}$.
Hence we get $Y_I\supset \overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Take
$([\gamma_0],\theta_0)\notin
\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
Then
there exists a neighborhood $U$ of $([\gamma_0],\theta_0)$ satisfying
$U\cap\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}=\varepsilonmptyset$.
By the same argument as in the proof of \cite[Lemma 5.22]{Ka1},
we can find $x_0\in A$ such that
$\psi_{([\gamma_0],\theta_0)}(x_0)\neq 0$ and
$\psi_{([\gamma],\theta)}(x_0)=0$ if $([\gamma],\theta)\notin U$
and $\varphi_{\gamma}(x_0)=0$ if $([\gamma]\times\mathbb{T})\cap U=\varepsilonmptyset$.
Therefore we have $x_0\in I$,
and it implies that
$([\gamma_0],\theta_0)\notin Y_I$.
Thus $Y_I=\overline{\bigcup_{\lambda\in\Lambda}Y_\lambda}$.
\varepsilonnd{proposition}f
\begin{lemma}\label{Primlem4}
For any $X\in\varDelta$, we have $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{lemma}
\begin{proposition}f
By Lemma \ref{Primlem1},
we have $P_{X}\subset\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
By Lemma \ref{Primlem3},
we have $Y_{\bigcap_{z\in [X]\times\mathbb{T}}P_z}=[X]\times\mathbb{T}$.
Hence we have $\bigcap_{z\in [X]\times\mathbb{T}}P_z\subset P_{X}$ by
Lemma \ref{Primlem1}.
Thus $P_{X}=\bigcap_{z\in [X]\times\mathbb{T}}P_z$.
\varepsilonnd{proposition}f
In the proof of the following proposition,
we use the fact that
the subset $\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I_1\subset P, I_2\not\subset P\}$
of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is homeomorphic to $\Prim(I_2/I_1)$,
for two ideals $I_1,I_2$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\subset I_2$.
\begin{proposition}\label{closed}
Let $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ be a subset of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$.
The set $P_Z=\{P_z\mid z\in Z\}$ is closed in $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$
if and only if $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
\varepsilonnd{proposition}
\begin{proposition}f
Let us take a subset $Z=Y\sqcup X^{(\infty)}\sqcup \varLambda$ of
$({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta$
satisfying that $(Y,X^{(\infty)})$ is an $\omega$-invariant set and
$\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$.
Then the set $P_Z=\{P_z\mid z\in Z\}$ coincides
with the closed subset defined by the ideal $I_{\widetilde{Y}}$
by Lemma \ref{Primlem1}.
Conversely, assume $P_Z$ is closed, that is,
there exists an ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with
$Z=\{z\in Y\sqcup X^{(\infty)}\sqcup \varLambda\mid I\subset P_z\}$.
We first show that $Y$ and $X^{(\infty)}$ is closed.
Take $\gamma_0\in{\mathcal G}amma$ arbitrarily.
By Corollary \ref{cptnbhd},
there exists a compact neighborhood $X$ of $\gamma_0$
such that $X\cap (X+\gamma)=\varepsilonmptyset$ for any $\gamma\in\overline{\rm{sg}}(\omega)\setminus\{0\}$.
Set $\widetilde{X}_0=(X_1,X_1)$, $\widetilde{X}_1=(X_1,X_2)$ and
$\widetilde{X}_2=(X_2,X_2)$ where $X_1=X+\overline{\rm{sg}}(\omega)$ and $X_2=X+\overline{\rm{sg}}_1(\omega)$.
Note that $X\ni\gamma\mapsto[\gamma]\in [X_1\setminus X_2]$ is a homeomorphism.
By Lemma \ref{Primlem1} and Lemma \ref{Primlem2}, we have
\begin{align*}
\big\{z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\mid
I_{\widetilde{X}_1}\subset P_z, I_{\widetilde{X}_2}\not\subset P_z\big\}
&=[X_1\setminus X_2]\times\mathbb{T}\subset {\mathcal G}amma'\times\mathbb{T},\\
\big\{z\in ({\mathcal G}amma'\times\mathbb{T})\sqcup{\mathcal G}amma\sqcup\varDelta\mid
I_{\widetilde{X}_0}\subset P_z, I_{\widetilde{X}_1}\not\subset P_z\big\}
&=X_1\setminus X_2\subset{\mathcal G}amma.
\varepsilonnd{align*}
By Proposition \ref{local}, the map
$[X_1\setminus X_2]\times\mathbb{T}\ni z\mapsto P_z\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ is a homeomorphism
from $[X_1\setminus X_2]\times\mathbb{T}$,
whose topology is the relative topology of ${\mathcal G}amma'\times\mathbb{T}$,
to the subset
$\{P\in\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)\mid I_{\widetilde{X}_1}\subset P,
I_{\widetilde{X}_2}\not\subset P\}$
of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
The set $Y\cap ([X_1\setminus X_2]\times\mathbb{T})\subset {\mathcal G}amma'\times\mathbb{T}$ is closed
in $[X_1\setminus X_2]\times\mathbb{T}$ because $P_Y$ is closed.
Hence, the subset $Y$ is closed in ${\mathcal G}amma'\times\mathbb{T}$.
Similarly $X^{(\infty)}$ is closed in ${\mathcal G}amma$.
Set $X=\{\gamma\in{\mathcal G}amma\mid ([\gamma],\theta)\in Y \mbox{ for some }\theta\in\mathbb{T}\}$, which is closed because $Y$ is closed.
Set $J=\bigcap_{([\gamma],\theta)\in Y}P_{([\gamma],\theta)}$.
We have $I\subset J$.
By Lemma \ref{Primlem3}, we have $Y_J=Y$.
Hence $H_X\subset X_J^{(\infty)}$.
We have $J\subset P_\gamma$ for any $\gamma\in H_X$ by Lemma \ref{Primlem1}.
Therefore $H_X\subset X^{(\infty)}$.
We have $([\gamma+\omega_i],\theta')\in Y$
for any $([\gamma],\theta)\in Y$, any $i\geq 2$ and any $\theta'\in\mathbb{T}$
because $P_{([\gamma],\theta)}\subset P_{([\gamma+\omega_i],\theta')}$.
Hence we get $[X+\omega_i]\times\mathbb{T}\subset Y$.
We also have $[X^{(\infty)}]\times\mathbb{T}\subset Y$
because $P_\gamma\subset P_{([\gamma],\theta)}$
for any $([\gamma],\theta)\in{\mathcal G}amma'\times\mathbb{T}$.
Therefore we have proved that $(Y,X^{(\infty)})$ is an $\omega$-invariant set.
Finally, we have $\varLambda=\{X\in\varDelta\mid [X]\times\mathbb{T}\subset Y\}$
by Lemma \ref{Primlem4}.
It completes the proof.
\varepsilonnd{proposition}f
By the proposition above, we get the following.
\begin{theorem}\label{idestr2}
When $\omega$ does not satisfy Condition \ref{cond},
there is a one-to-one correspondence between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
and the set of $\omega$-invariant pairs of subsets of ${\mathcal G}amma'\times\mathbb{T}$
and subsets of ${\mathcal G}amma$.
Hence for any ideal $I$ of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$, we have $I=I_{\widetilde{Y}_I}$.
\varepsilonnd{theorem}
\begin{proposition}f
There is a one-to-one correspondence
between the set of ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ and the closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$.
By Proposition \ref{closed},
the closed subset of $\Prim({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)$ corresponds bijectively
to the set of $\omega$-invariant pairs.
\varepsilonnd{proposition}f
\section{More about ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$}
In this section, we gather some general results on ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
First we compute the strong Connes spectrum of the action
$\alpha^{\omega}:G\curvearrowright{\mathcal O}_\infty$.
We need the following lemma.
\begin{lemma}\label{H_csg}
For any $\omega\in{\mathcal G}amma^\infty$, we have
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}
=\{0\}\cup\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
\varepsilonnd{lemma}
\begin{proposition}f
It suffices to show that
$$\overline{\rm{sg}}(\omega)\setminus\big(\{0\}\cup\bigcup_{i=1}^\infty(\overline{\rm{sg}}(\omega)+\omega_i)\big)\subset
\overline{\bigcup_{i=n+1}^\infty({\rm{sg}}(\omega)+\omega_i)}$$
for any $n\in\mathbb{Z}_+$.
Take $\gamma\in\overline{\rm{sg}}(\omega)\setminus
\big(\{0\}\cup\bigcup_{i=1}^\infty(\overline{\rm{sg}}(\omega)+\omega_i)\big)$
and $n\in\mathbb{Z}_+$.
Since $\gamma\in\overline{\rm{sg}}(\omega)$, there exists a sequence $\{\mu_k\}\subset{\mathcal W}_\infty$
such that $\gamma=\lim_{k\to\infty}\omega_{\mu_k}$.
We will show that we can find an integer grater than $n$ in the word $\mu_k$
for infinitely many $k$, from which it follows that
$\gamma\in\overline{\bigcup_{i=n+1}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
To the contrary, assume that $\mu_k\in{\mathcal W}_n$ for sufficiently large $k$.
Then there exists $i\in\{1,2,\ldots,n\}$ which appears in $\mu_k$ eventually.
We have $\gamma-\omega_i=\lim_{k\to\infty}(\omega_{\mu_k}-\omega_i)\in\overline{\rm{sg}}(\omega)$.
This contradicts the fact that $\gamma\notin\overline{\rm{sg}}(\omega)+\omega_i$.
Hence
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}
=\{0\}\cup\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}$.
\varepsilonnd{proposition}f
\begin{proposition}\label{SCS}
The strong Connes spectrum $\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
of the action $\alpha^{\omega}$ is $\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}
\begin{proposition}f
By \cite[Lemma 3.4]{Ki}, we have
$$\widetilde{{\mathcal G}amma}(\alpha^{\omega})
=\{\gamma\in{\mathcal G}amma\mid \widehat{\alpha^{\omega}}_\gamma(I)\subset I,
\mbox{for any ideal } I \mbox{ of } {\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G\},$$
where $\widehat{\alpha^{\omega}}:{\mathcal G}amma\curvearrowright{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is the dual
action of $\alpha^{\omega}$.
For an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
and $\gamma\in{\mathcal G}amma$,
we see that $\widehat{\alpha^{\omega}}_\gamma(I_{\widetilde{X}})
=I_{\widetilde{X}-\gamma}$ where
$\widetilde{X}-\gamma=(X-\gamma,X^{(\infty)}-\gamma)$.
Hence $\widehat{\alpha^{\omega}}_\gamma(I_{\widetilde{X}})\subset
I_{\widetilde{X}}$ is equivalent
to say that $X+\gamma\subset X$ and $X^{(\infty)}+\gamma\subset X^{(\infty)}$
for an $\omega$-invariant pair $\widetilde{X}=(X,X^{(\infty)})$
and $\gamma\in{\mathcal G}amma$.
Considering the case that $\widetilde{X}=(\overline{\rm{sg}}(\omega),\{0\}\cup H_{\overline{\rm{sg}}(\omega)})$,
we have $(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})+\gamma\subset\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$
for $\gamma\in\widetilde{{\mathcal G}amma}(\alpha^{\omega})$.
Hence $\widetilde{{\mathcal G}amma}(\alpha^{\omega})\subset\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
Let $(X,X^{(\infty)})$ be an $\omega$-invariant pair.
For $\gamma\in X$, we get
$$\gamma+\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty({\rm{sg}}(\omega)+\omega_i)}
=\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(\gamma+{\rm{sg}}(\omega)+\omega_i)}
\subset\bigcap_{n=1}^\infty\overline{\bigcup_{i=n}^\infty(X+\omega_i)}
\subset H_X\subset X^{(\infty)}.$$
By Lemma \ref{H_csg}, we have
$X^{(\infty)}+(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})\subset X^{(\infty)}$.
Since $\{0\}\cup H_{\overline{\rm{sg}}(\omega)}\subset\overline{\rm{sg}}(\omega)$,
we have $X+(\{0\}\cup H_{\overline{\rm{sg}}(\omega)})\subset X$.
Hence when $\omega$ satisfies Condition \ref{cond}, we have
$\widetilde{{\mathcal G}amma}(\alpha^{\omega})\supset \{0\}\cup H_{\overline{\rm{sg}}(\omega)}$
by Theorem \ref{idestr1},
and so $\widetilde{{\mathcal G}amma}(\alpha^{\omega})=\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
Next we consider the case
that $\omega$ does not satisfy Condition \ref{cond}.
For an $\omega$-invariant pair $(Y,X^{(\infty)})$,
we have $X+(H_{\overline{\rm{sg}}(\omega)}\setminus\{0\})\subset X^{(\infty)}$
by the former part of this proof, where
$X=\{\gamma\in{\mathcal G}amma\mid
([\gamma],\theta)\in Y \mbox{ for some }\theta\in\mathbb{T}\}$.
Hence for any $([\gamma_0],\theta_0)\in Y$
and $\gamma\in H_{\overline{\rm{sg}}(\omega)}\setminus\{0\}$,
we have $\gamma_0+\gamma\in X^{(\infty)}$
because $\gamma_0\in X$.
Since $[X^{(\infty)}]\times\mathbb{T}\subset Y$, we have
$([\gamma_0+\gamma],\theta_0)\in Y$.
Therefore we also have
$\{0\}\cup H_{\overline{\rm{sg}}(\omega)}\subset\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
by Theorem \ref{idestr2}.
Thus $\widetilde{{\mathcal G}amma}(\alpha^{\omega})=\{0\}\cup H_{\overline{\rm{sg}}(\omega)}$.
\varepsilonnd{proposition}f
Next we give necessary and sufficient conditions for $\omega\in{\mathcal G}amma^\infty$
that the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ becomes simple or primitive.
\begin{lemma}\label{0}
Let $I$ be an ideal of the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$.
Then $I=0$ if and only if $X_I={\mathcal G}amma$.
\varepsilonnd{lemma}
\begin{proposition}f
The ``only if'' part is trivial.
One can easily prove the ``if'' part
by the same arguments as in the proofs of
Proposition \ref{cond.exp2} and Theorem \ref{idestr1}.
\varepsilonnd{proposition}f
\begin{proposition}
For $\omega\in{\mathcal G}amma^\infty$, the following are equivalent:
\benu
\item The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is simple.
\item There are no $\omega$-invariants sets
other than ${\mathcal G}amma$ and $\varepsilonmptyset$.
\item ${\mathcal G}amma=\overline{\rm{sg}}(\omega)$.
\varepsilonnd{enumerate}
If ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is simple, then it is purely infinite.
\varepsilonnd{proposition}
\begin{proposition}f
The equivalence between (i) and (ii) follows from Lemma \ref{0}.
(ii) implies (iii) because $\overline{\rm{sg}}(\omega)$ is $\omega$-invariant.
(iii) implies (ii) because $X=X+\overline{\rm{sg}}(\omega)$ if $X$ is $\omega$-invariant.
For the last statement, see \cite[Proposition 5.2]{Ka2}.
\varepsilonnd{proposition}f
The equivalence between (i) and (iii) was already proved
by A. Kishimoto \cite{Ki} by using strong Connes spectrum.
Note that the strong Connes spectrum $\widetilde{{\mathcal G}amma}(\alpha^{\omega})$
is equal to ${\mathcal G}amma$ if and only if $\overline{\rm{sg}}(\omega)={\mathcal G}amma$ by Proposition \ref{SCS}.
\begin{proposition}
The following conditions for $\omega\in{\mathcal G}amma^\infty$ are equivalent:
\benu
\item The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is primitive.
\item ${\mathcal G}amma$ is a prime $\omega$-invariant set.
\item The closed group generated by $\omega_1,\omega_2,\ldots$
is equal to ${\mathcal G}amma$.
\varepsilonnd{enumerate}
\varepsilonnd{proposition}
\begin{proposition}f
(i)$\mathbb{R}ightarrow$(ii): This follows from Proposition \ref{prime}.
(ii)$\mathbb{R}ightarrow$(i): It suffices to show that $0$ is prime.
Let $I_1,I_2$ be ideals of ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ with $I_1\cap I_2=0$.
We have $X_{I_1}\cup X_{I_2}=X_{I_1\cap I_2}={\mathcal G}amma$.
Since ${\mathcal G}amma$ is prime, either $X_{I_1}\supset{\mathcal G}amma$ or $X_{I_2}\supset{\mathcal G}amma$.
If $X_{I_1}\supset{\mathcal G}amma$ hence $X_{I_1}={\mathcal G}amma$, then $I_1=0$ by Lemma \ref{0}.
Similarly if $X_{I_2}\supset{\mathcal G}amma$, then $I_2=0$.
Thus $0$ is prime and so ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a primitive $C^*$-algebra.
(ii)$\iff$(iii): This follows from Proposition \ref{Xprime}.
\varepsilonnd{proposition}f
One can prove the equivalence between (i) and (iii) in the above theorem
by characterization of primitivity of crossed products in terms of
the Connes spectrum due to D. Olesen and G. K. Pedersen \cite{OP} and
the computation of the Connes spectrum of our actions $\alpha^{\omega}$
due to A. Kishimoto \cite{Ki}.
\begin{proposition}\label{CP}
The crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to
the Cuntz-Pimsner algebra ${\mathcal O}_E$ of $C_0({\mathcal G}amma)$-bimodule
$E=C_0({\mathcal G}amma)^\infty$, whose left module structure is given by
$$f\cdot(f_1,f_2,\ldots,f_n,\ldots)=(\sigma_{\omega_1}(f)f_1,\sigma_{\omega_2}(f)f_2,\ldots,\sigma_{\omega_n}(f)f_n,\ldots)\in E$$
for $f\in C_0({\mathcal G}amma)$ and $(f_1,f_2,\ldots,f_n,\ldots)\in E$.
\varepsilonnd{proposition}
\begin{proposition}f
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ and
$E\ni (0,\ldots,0,f_n,0\ldots)\mapsto S_nf_n\in{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$
satisfies the conditions in \cite[Theorem 3.12]{Pi}.
Hence there exists a $*$-homomorphism
$\varphi:{\mathcal O}_E\to{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ which is surjective
since ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is generated by $\{S_nf\mid n\in\mathbb{Z}_+,\ f\in C_0({\mathcal G}amma)\}$.
One can show that $\varphi$ is injective by using Lemma \ref{isom}.
Thus ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is isomorphic to ${\mathcal O}_E$.
\varepsilonnd{proposition}f
\begin{corollary}
The inclusion $C_0({\mathcal G}amma)\hookrightarrow{\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is a KK-equivalence.
Hence for $i=0,1$, we have $K_i({\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G)=K_i(C_0({\mathcal G}amma))$.
\varepsilonnd{corollary}
\begin{proposition}f
See \cite[Corollary 4.5]{Pi}.
\varepsilonnd{proposition}f
\begin{proposition}\label{embedinfty}
If $\omega\in{\mathcal G}amma^\infty$ satisfies
$-\omega_i\notin\overline{\{\omega_{\mu}\mid\mu\in{\mathcal W}_n\}}$
for any $i,n\in\mathbb{Z}_+$,
then the crossed product ${\mathcal O}_\infty{\rtimes_{\alpha^\omega}}G$ is AF-embeddable.
\varepsilonnd{proposition}
\begin{proposition}f
See \cite[Proposition 5.1]{Ka2}.
\varepsilonnd{proposition}f
\begin{thebibliography}{BHRS}
\bibitem[BHRS]{BHRS}
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\bibitem[C]{C}
Cuntz, J. {\it Simple $C\sp*$-algebras generated by isometries.} Comm. Math. Phys. {\bf 57} (1977), no. 2, 173--185.
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Deaconu, V. {\it Continuous graphs and C*-algebras.} Operator theoretical methods, 137--149, Theta Found., Bucharest, 2000.
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Evans, D. E. {\it On $O\sb{n}$.} Publ. Res. Inst. Math. Sci. {\bf 16} (1980), no. 3, 915--927.
\bibitem[HS]{HS}
Hong, J. H.; Szymanski, W. {\it The primitive ideal space of the $C^*$-algebras of infinite graphs.} Preprint.
\bibitem[Ka1]{Ka1}
Katsura, T. {\it The ideal structures of crossed products of Cuntz algebras by quasi-free actions of abelian groups.} Preprint.
\bibitem[Ka2]{Ka2}
Katsura, T. {\it AF-embeddability of crossed products of Cuntz algebras.} Preprint.
\bibitem[Ki]{Ki}
Kishimoto, A. {\it Simple crossed products of $C\sp{*} $-algebras by locally compact abelian groups.} Yokohama Math. J. {\bf 28} (1980), no. 1-2, 69--85.
\bibitem[KK1]{KK1}
Kishimoto, A.; Kumjian, A. {\it Simple stably projectionless $C\sp *$-algebras arising as crossed products.} Canad. J. Math. {\bf 48} (1996), no. 5, 980--996.
\bibitem[KK2]{KK2}
Kishimoto, A.; Kumjian, A. {\it Crossed products of Cuntz algebras by quasi-free automorphisms.} Operator algebras and their applications, 173--192, Fields Inst. Commun., {\bf 13}, Amer. Math. Soc., Providence, RI, 1997.
\bibitem[OP]{OP}
Olesen, D.; Pedersen, G. K. {\it Applications of the Connes spectrum to $C\sp{*} $-dynamical systems.} J. Funct. Anal. {\bf 30} (1978), no. 2, 179--197.
\bibitem[Pi]{Pi}
Pimsner, M. V. {\it A class of $C\sp *$-algebras generalizing both Cuntz-Krieger algebras and crossed products by ${Z}$.} Free probability theory, 189--212, Fields Inst. Commun., {\bf 12}, Amer. Math. Soc., Providence, RI, 1997.
\varepsilonnd{thebibliography}
\varepsilonnd{document} |
\begin{document}
\title{A toy model for Macroscopic Quantum Coherence}
\author{R. Mu\~{n}oz-Vega}\email{rodrigo.munoz@uacm.edu.mx}
\affiliation{Universidad Aut\'{o}noma de la Ciudad de M\'{e}xico,
Centro Hist\'{o}rico, Fray Servando Teresa de Mier 92,
Col. Centro, Del. Cuauht\'{e}moc, M\'{e}xico D.F, C.P. 06080}
\author{Jos\'{e} Job Flores-Godoy}\email{ e-mail:job.flores@ibero.mx}
\author{ G. Fern\'{a}ndez-Anaya}\email{guillermo.fernandez@ibero.com}
\affiliation{Departamento de F\'isica y Matem\'aticas, Universidad Iberoamericana, Prol. Paseo de la Reforma 880, Col. Lomas de Santa Fe, Del. A. Obreg\'on, M\'exico, D. F. C.P. 01219}
\author{Encarnaci\'{o}n Salinas-Hern\'{a}ndez}
\email{ esalinas@ipn.mx}
\affiliation{ESCOM-IPN,
Av. Juan de Dios B\'{a}tiz s/n, Unidad Profesional Adolfo L\'{o}pez Mateos
Col. Lindavista, Del. G. A. Madero, M\'{e}xico, D. F, C.P. 07738}
\begin{abstract}
The present article deals with Macroscopic Quantum Coherence resorting only to basic quantum mechanics. A square double well is used to illustrate the Leggett-Caldeira oscillations. The effect of thermal-radiation on two-level systems is discussed to some extent. The concept of decoherence is introduced at an elementary level. Handles are deduced for the energy, temperature and time scales involved in Macroscopic Quantum Coherence.
\end{abstract}
\date{\today}
\pacs{01.40.Ha, 03.67.-a, 03.65.Fd, 03.65.Ge, 02.10.Ud, 03.65.Ca}
\maketitle
\section{Introduction}
Triggered by a seminal article \cite{Leggett} written by A J Leggett in 1980, research into Macroscopic Quantum Coherence (MQC) has yielded impressive experimental,\cite{Nakamuraetal, Makhlinetal, Friedmanetal, WS} theoretical\cite{CaldeiraLeggett, Leggett1, RevDiss, Tesche} and even technological achievements\cite{Carellietal, Manucharyanetal01, Manucharyanetal02}. The ideas developed in the last thirty so years by Leggett and his collaborators have not only changed the way we understand the relation between quantum and classical behaviours, but are also crucial in the future development of quantum computing. The present article aims at explaining the basic phenomenology of QMC resorting only to basic quantum mechanics. Thus, we believe this article can be of interest for any student who has attended at least a one-year course in quantum physics, and for faculty members committed to introducing students into contemporary research.
In order to explain briefly what MQC is, let us consider a particle in a symmetric double well potential (SDWP). Figure 1 depicts an example of such a potential. In freshmen courses we have been told what to expect when the particle is in a high-lying energy level in a nice, analytical, potential such as this: for states for which the change in potential energy within a de Broglie wavelength is much smaller than the mean kinetic energy, the specifically quantum features of the behavior result negligible and the classical description becomes adequate.\cite{Bohm} In that sense, classical behaviour can be considered as a limiting case of quantum mechanics \cite{LandauLif}. Suppose, nonetheless, the central barrier in the SDWP of Fig. 1 to be of macroscopic width. Then, the predictions of quantum mechanics and classical mechanics certainly clash for this system. A classical viewpoint would demand two distinct localized states of stable equilibrium, situated at $-x_{0}$ and $x_{0}$, while quantum mechanics predicts an even probability distribution for the (non-degenerate\cite{LandauA}) ground level state (which is, of course, the more stable stationary state.) In fact, the ground statefunction for such a potential would have to look something like Figure 2.
Moreover, the (odd) eigenfunction of the first excited level (Figure 3), and indeed each one of the stationary solutions of a SDWP, necessarily has an even probability distribution.
\begin{figure}
\caption{An example of a SDWP potential, with characteristic double minima and central peak.}
\label{fig:figure1}
\end{figure}
What Leggett predicted more than thirty wears ago, and what actually happens in experiments carried out in SDWPs of micrometric and nanometric typical lengths, is the appearance of a two-fold degenerate ground level $E^{\prime}$, with the system oscillating in an harmonic fashion between two eigenstates, $\vert L\rangle$ (Figure 4) and $\vert R \rangle$ (Figure 5) localized, respectively, at the left and right of the central barrier. At ground level, the position expectancy value oscillates in the accordance with:
\begin{equation}\label{I.1}
\langle x\rangle_{0}(t)=\langle x\rangle_{0}(0)\cos \omega t\textrm{.}
\end{equation}
This phenomenon, the so called Leggett-Caldeira oscillations, is closely related with the Rabi oscillations of atomic physics. It is explained as the result of the purported ground level $E^{\prime}$ resolving into a true ground level
\begin{equation}\label{I.2}
E_{+}=E^{\prime}-\hbar \omega/2\textrm{,}
\end{equation}
endowed with an even non-localized eigensolution $\vert +\rangle$, and a first excited level
\begin{equation}\label{I.3}
E_{-}=E^{\prime}+\hbar \omega/2\textrm{,}
\end{equation}
endowed with an odd non-localized eigensolution $\vert -\rangle$. When a quantum system tunnels periodically trough the barrier of a SDWP with a central barrier of macroscopic length, we have Macroscopic Quantum Coherence.
The states $\vert R\rangle$ and $\vert L\rangle$ have, each one on its own, a definite value of a macroscopic property (namely, the property of being localized at the left or the right of the barrier). At the same time, $\vert R \rangle$ and $\vert L\rangle$ are linear combinations of the states $\vert +\rangle$ and $\vert -\rangle$, which cannot be said to be localized. In order to understand Leggett's original motivation, notice the analogy between macroscopic SDWPs and Schroedinger's cat: the celebrated pet can be in any of two different \emph{macroscopically distinguishable} states (let us say, $\Psi_{1}$ for a live cat and $\Psi_{0}$ for a dead one) just as a particle in a SDWP. If any of these macroscopic systems obeys the laws of quantum mechanics, then it could be prepared in linear combinations that lack a sharp, well defined, value of the macroscopic property. Examples of these linear combinations are the $\vert\pm\rangle$ states of the SDWPs, and the ``neither dead nor alive" states
\begin{equation}
\Psi_{\pm}=\frac{1}{\sqrt{2}}\Big(\Psi_{0}\pm\Psi_{1}\Big)
\end{equation}
of the cat.
\begin{figure}
\caption{A rendering of what a ground-level eigenfunction (solid curve) should look like for a SDWP. The potential is shown as a dashed curve.}
\label{fig:B}
\label{fig:figure1}
\end{figure}
Thus, a more general definition of MQC is simply: the quantum superposition of distinct macroscopic states. Long time before the year of 1980, macroscopic quantum phenomena had been discovered: superconductivity in 1911, and superfluidity in 1937. Yet it remained for Leggett to identify the conditions necessary for a quantum system to present macroscopically distinguishable states.\cite{Leggett}
Some twenty years elapsed between Leggett's proposal and a credible experimental confirmation\cite{Friedmanetal, WS} of MQC. One of the main reasons for this delay lies in the fact that the phase coherence of the $\vert\pm\rangle$ states is rapidly lost due to the interaction of the system with its surroundings, so the system collapses into one of the localized states before one period of the Leggett-Caldeira oscillation is completed.\cite{Leggett, RevDiss}
MQC is relevant not only from the purely theoretical point of view. A physical qubit is a two-level system considered as a piece of hardware. And, as we shall see in the following pages, at least some SDWPs can behave as effective two-level systems at sufficiently low temperatures. Quantum computing (an area with impressive software development, but little hardware to show) requires qubits to interact with one another without loss of coherence, for fairly long times, even at fairly high temperatures. Thus, the study of two-level dissipative systems, to which Leggett and collaborators made far reaching contributions when delving in the foundations of quantum physics, has revealed itself crucial for people in the vanguard of technological development.\cite{Makhlinetal, WS}
\begin{figure}
\caption{Th first excited level eigenfunction (solid) of a SDWP (dashed).}
\label{fig:C}
\end{figure}
The rest of this article is structured as follows: in section~\ref{sec:2} we discuss the spectra of a family of symmetric double square well potentials, and the conditions under which a member of this family can be considered as an effective two-level system. Next, the properties of two-states systems arising from SDWP´s are discussed in section ~\ref{sec:3}. We then go on to examine in ~\ref{sec:4} how thermal radiation, by throwing the system into higher energy levels, renders the two-level model inapplicable. In section ~\ref{sec:5} decoherence is introduced in elementary terms, and its relation with dissipation is briefly discussed. Handles for the time, energy and temperature scales involved in MQC are derived from our toy model in section ~\ref{sec:6}. Finally, conclusions are laid down in section ~\ref{sec:7}.
\begin{figure}
\caption{The $\langle x\vert L\rangle$ state (shown solid) localized at the left of the SDWP (shown dashed) barrier.}
\label{fig:D}
\end{figure}
\begin{figure}
\caption{The $\langle x\vert R\rangle$ state (shown solid) localized at the right of the SDWP (shown dashed) barrier.}
\label{fig:E}
\end{figure}
\section{Symmetric Double Square Wells}\label{sec:2}
Leggett resorted to quasi-classical considerations when stating his original proposal.\cite{Leggett} Also, the WKB approximation has been applied to double well potentials by Landau and Lifshitz,\cite{LLandau} and more recently, in this Journal, by others.\cite{JelicMarsiglio} Here will take a different point of view, avoiding all together quasi-classical approximations, by considering a particular family of double infinite square well potentials as approximations to actual, analytic SDWPs. Our procedure will later allow us to get some reference values on the energies, temperatures and times involved in MQC. The following family of piece-wise-constant potentials will be considered:
\begin{equation}\label{ucases}
U_{b}(x)=\left \{ \begin{array}{c l}
\infty &\textrm{if } x \leq -a-b\textrm{,}\\
0 &\textrm{if } -b > x > -a-b\textrm{,}\\
k &\textrm{if } b\geq x \geq -b\textrm{,}\\
0 &\textrm{if } b+a> x > b\textrm{,}\\
\infty &\textrm{if } x \geq b+a\textrm{.}\\
\end{array}\right .
\end{equation}
Potentials of this kind have previously been studied in a different context, and it has been shown\cite{Munoz} that, if all other parameters held fixed, levels $E_{2n+1}$ and $E_{2n}$ coalesce as $k\rightarrow \infty$. Here we shall consider the barrier height $k>0$ as a fixed number, although ``big" in a sense that will be readily clarified. This, in order to keep the gap between the ground and first excited levels sufficiently small. We shall also take the width of each one of the lateral valleys, $a>0$, as a fixed value unless otherwise stated, leaving free the only other parameter, that is the barrier half-width $b>0$.
One of the two main objectives of this Section is to obtain a global lower bound on the energy gap between the first and second excited levels in the $U_{b}$ potentials. Just as important to our ends, we will learn on this Section that there is a ``running" upper bound (dependent on the value of $b$) on the gap between the ground and first excited levels. The consequences of this to facts, which are vital to the rest of the article, are explored in Sections III, IV and VI.
To be sure, non of the $U_{b}$ is continuous, yet they share the most prominent features of a SDWP, namely, they are even potentials with completely bounded, non-degenerate, spectra, as can be shown from boundary conditions. If instead of two minima, the $U_{b}$ have two non-overlapping regions of minima (\emph{viz.} $(-a-b,-b)$ and $(b,a+b)$), this distinction will prove to be quite unimportant.
\begin{figure}
\caption{A typical member of the $U_b(x)$ family of potentials}
\label{msbs1}
\label{fig:figure5r}
\end{figure}
Also from boundary conditions (or from more abstract, symmetry considerations) it is readily seen that the levels in the spectrum of any of the $U_{b}$ are classified according with parity, just like it happens for a continuous even potential:
\begin{equation}\label{II.1}
\psi_{2n,b}(-x)=\psi_{2n,b}(x)\textrm{,}\quad\quad n=0,1,\ldots\ \textrm{,}\quad b\in (0, \infty )
\end{equation}
and
\begin{equation}\label{II.2}
\psi_{2n+1,b}(-x)=-\psi_{2n+1,b}(x)\textrm{,}\quad\quad n=0,1,\ldots\ \textrm{,}\quad b\in (0, \infty )\textrm{.}
\end{equation}
Let us focus on the discretization conditions below the level of the central barrier ($E<k$). From the boundary conditions we get, for even states:
\begin{equation}\label{coneven}
-\sqrt{E_{2n}}\cot a\frac{\sqrt{2mE_{2n}}}{\hbar}=\sqrt{k-E_{2n}}\tanh b\frac{\sqrt{2m(k-E_{2n})}}{\hbar}\quad\textrm{,}
\end{equation}
while odd levels below the barrier level have to comply with
\begin{equation}\label{conodd}
-\sqrt{E_{2n+1}}\cot a\frac{\sqrt{2mE_{2n+1}}}{\hbar}=\sqrt{k-E_{2n+1}}\coth b\frac{\sqrt{2m(k-E_{2n+1})}}{\hbar}\quad\textrm{.}
\end{equation}
Notice how the first of these two conditions can be written in the form:
\begin{equation}
g(E_{2n})=h_{b}(E_{2n}),
\end{equation}
and the second can be rendered as:
\begin{equation}
g(E_{2n+1})=j_{b}(E_{2n+1}),
\end{equation}
with the meaning of $g, h_{b}$ and $j_{b}$ being obvious from the context.
Both of these two last equations are depicted in Figure 7, from which it can be seen that there exists an upper bound $B$, given by
\begin{equation}\label{inequalityA}
B=\frac{\pi^{2}\hbar ^{2}}{2m a^{2}},
\end{equation}
such that the ground and first excited states have to comply with
\begin{equation}\label{funnyeq}
\frac{B}{4}<E_{0}<E_{1}<B ,
\end{equation}
no matter the value of $b.$ Obviously, there can be no levels below the barrier unless $k>B/4$. We shall only consider potentials for which the condition:
\begin{equation}\label{bigenough}
k>>B
\end{equation}
is met, so that we will always have at least two levels below the barrier. Indeed, the number of levels below the barrier increases with increasing quotient $k/B$ and, more importantly, as $B$ is independent of $k$, condition (\ref{bigenough}) warrants that the gap between the first two level is always small.
It is not difficult to generalize (\ref{funnyeq}) starting from (\ref{coneven}) and (\ref{conodd}) and definition (\ref{inequalityA}). The result is that:
\begin{equation}\label{III.1.1}
(n+\frac{1}{2})^{2}B<E_{2n,k}<E_{2n+1,k}<(n+1)^{2}B\textrm{,}\quad n=0,1,2,\ldots N,
\end{equation}
if the level $2N+1$ is still below the barrier.
From inequality (\ref{III.1.1}) it follows that
\begin{equation}\label{III.1.2}
E_{2n+2}-E_{2n+1}>(n+5/4)B\textrm{,}\quad n=0,1,2,\ldots, N
\end{equation}
if level $2N+1$ is below the barrier. We then have that the gap between the ground and first excited levels will always be less than the gap between the first and second excited levels:
\begin{equation}\label{inek01}
E_{2}-E_{1}>\frac{5}{4}B>\frac{3}{4}B> E_{1}-E_{0}.
\end{equation}
But we can do much more better than that. Indeed, in Appendix A it is formally proven that for any given number $\delta>0$ there exist a value $b(\delta)>0$ such the gap between the ground and the first excited level of a $U_{b}$ potential will be less than $\delta$, that is
\begin{equation}
E_{1}-E_{0}<\delta ,
\end{equation}
if $b\geq b(\delta)$. In other words, if we choose the barrier length big enough, then we can make $E_{0}$ and $E_{1}$ as proximate as we want, while there is a lower bound for the gap between $E_{2}$ and $E_{1}$ which is independent of the value of this length. This will allow us to find examples of $U_{b}$ that will work as effective two-state systems for the lowest-lying energy levels, as illustrated in Figure 8.
\begin{figure}
\caption{Graphical solutions of transcendental equations (\ref{coneven}
\end{figure}
\begin{figure}
\caption{Doubling the barrier width produces a dramatic decrease in the gap between the ground and first excited levels. Shown: function $h_{b}
\end{figure}
Finally, there is one more inequality that can be derived from (\ref{III.1.1}) and that will prove useful in section IV. This inequality is:
\begin{equation}\label{2NF}
E_{2}-E_{1}<\frac{15}{4}B\ .
\end{equation}
Let us stress that relations (\ref{funnyeq}), (\ref{inek01}) and (\ref{2NF}) are verified for each $U_{b}$ regardless of the value of $b.$
\begin{figure}
\caption{Solid curve: the $\psi_{+}
\label{fig:figure1}
\end{figure}
\section{Two-level systems with reflection symmetry}\label{sec:3}
In the preceding section we have proven that there are $U_{b}$ potentials for which the gap between the ground and first excited energy levels is much more narrow than the one between the first and second excited levels. Consequently, for low energy expectancy values, a particle in one of such potentials acts as an effective two-level systems.\cite{Feynman, Cohen}
In the rest of this section we shall consider a fixed $U_{b}$ that behaves as a two-level system, and drop the $b$.
Consider now the non-stationary solutions $\psi_{L}$ and $\psi_{R}$ that one obtains from the linear combinations
\begin{equation}\label{II.3}
\psi_{L} (x,t)=\frac{1}{\sqrt{2}}\Big [\exp \left(-\imath\frac{E_{0}t}{\hbar}\right) \psi_{0}(x) + \exp\left(-\imath\frac{E_{1}t}{\hbar}\right)\psi_{1}(x)\Big ]
\end{equation}
\begin{figure}
\caption{ Solid: the (odd) eigenfunction of the first excited level. Dashed: the $U_{b}
\end{figure}
and
\begin{equation}\label{II.4}
\psi_{R} (x,t)=\frac{1}{\sqrt{2}}\Big [ \exp\left(-\imath\frac{E_{0}t}{\hbar}\right) \psi_{0}(x) - \exp\left(-\imath\frac{E_{1}t}{\hbar}\right) \psi_{1}(x)\Big]\textrm{.}
\end{equation}
These states have no definite parity, but instead one is the specular image of the other:
\begin{equation}\label{II.A.1}
\psi_{L}(-x,t)=\psi_{R}(x,t)\textrm{,}
\end{equation}
as can be seen from equations (\ref{II.1}), (\ref{II.2}), (\ref{II.3}) and (\ref{II.4}).
The position expectancy value for this states is calculated from (\ref{II.1}) in a straightforward manner:
\begin{equation}\label{II.5}
\langle x\rangle_{L}(t) =\ -\ \langle x\rangle_{R}(t) =\ \langle\psi_{0}\vert x\vert \psi_{1}\rangle\cos\frac{E_{1}-E_{0}}{\hbar}t,
\end{equation}
as is the energy expectancy value:
\begin{equation}\label{II.6}
\langle H\rangle_{L}=\langle H \rangle_{R}=\frac{E_{0}+E_{1}}{2}\textrm{.}
\end{equation}
Comparing (\ref{II.5}) with (\ref{I.1}) and (\ref{II.6}) with (\ref{I.2}) one may be tempted to make the identifications
\begin{equation}\label{II.7}
E^{\prime}=\langle H\rangle_{L}\quad\textrm{ and }\quad \omega=\frac{E_{1}-E_{2}}{\hbar}\textrm{,}
\end{equation}
from which (\ref{I.3}) would follow, so that the states of (\ref{II.3}) and (\ref{II.4}) could be interpreted as the localized states observed in the experiments, and $\psi_{0}$ and $\psi_{1}$ would correspond to the true ground level $E_{+}$ and the first excited state $E_{-}$. That is, it would be cogent that
\begin{equation}
\langle x\vert L\rangle=\psi_{L}(x)\textrm{,}\quad\langle x\vert R\rangle=\psi_{R}(x)\textrm{,}\quad
\langle x\vert +\rangle=\psi_{0}(x)\textrm{,}\quad\langle x\vert -\rangle=\psi_{1}(x)\textrm{.}
\end{equation}
\begin{figure}
\caption{The localized $\psi_{L}
\end{figure}
In this interpretation, however, there is no room for transitions. Indeed, the complete Schroedinger equation for a $U$ potential, which reads:
\begin{equation}\label{SchU}
-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}(x,t)+U(x)\psi(x,t)=i\hbar\frac{\partial\psi}{\partial t}(x,t) \textrm{,}
\end{equation}
predicts that if the system is initially prepared in the state $\psi_{L}(x,t=0)$ at time $t=0$, then it will remain in the $\psi_{L}(x,t)$ state for $t\in[0,\infty)$ (which is a sophisticated way to say: \emph{forever}). This is just consequence of PDE's theory.
Instead of periodical transitions between two different states, our equations predict the existence of a unique ``oscillating" state, because $\psi_{R}(x,t)$ is a time-displaced replica of $\psi_{L}(x,t)$:
\begin{figure}
\caption{The localized $\psi_{R}
\end{figure}
\begin{equation}\label{II.F}
\psi_{L}\Big(x,\ t+\frac{\pi}{\omega}\Big)=\imath\exp\Big(-\imath\frac{\pi\Omega}{\omega}\Big)\ \psi_{R}(x,t) \quad\textrm{ ,}
\end{equation}
where $\Omega$ stands for
\begin{equation}\label{II.G}
\Omega=(E_{1}+E_{2})/2\hbar
\end{equation}
and $\omega$ is as in (\ref{II.7}). One arrives at this result directly from (\ref{II.3}) and (\ref{II.4}) after some algebra.
\subsection{Flip-flops and Leggett-Caldeira oscillations}
Let us start from what we know happens in actual experiments (\emph{i. e.} the existence of an observable degenerate ground level) and proceed to deduce from there the perturbation needed to achieve such degeneracy. The SDWP Hamiltonian $H$ is represented by the matrix
\begin{equation}\label{IV.A.a}
\mathbb{H}=
\left(\begin{array}{cc}
E_{0}&0\\
0&E_{1}\\
\end{array}\right)
\end{equation}
in the symmetry-respecting basis formed by the eigenfunctions $\psi_{0}$ and $\psi_{1}$. Let us consider another Hamiltonian, $H^{\prime}$, represented by the matrix
\begin{equation}\label{IV.B.4}
\tilde{\mathbb{H}}^{\prime}=\mathbb{O}\mathbb{H}^{\prime}\mathbb{O}^{-1}=
\left(\begin{array}{cc}
E^{\prime}&0\\
0&E^{\prime}\\
\end{array}\right)
\end{equation}
in the symmetry-violating basis spanned by $\psi_{L}$ and $\psi_{R}$. Here, $\mathbb{O}$ stands for the unitary operator which transforms $\psi_{0}$ into $\psi_{L}$ and $\psi_{1}$ into $\psi_{R}$, thus:
\begin{equation}\label{IV.A.5}
\mathbb{O}\left(\begin{array}{c}
1\\
0\\
\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
1\\
1\\
\end{array}\right)
\quad\textrm{ and }\quad
\mathbb{O}\left(\begin{array}{c}
0\\
1\\
\end{array}\right)=
\frac{1}{\sqrt{2}}\left(\begin{array}{c}
1\\
-1\\
\end{array}\right).
\end{equation}
Notice that, as the rhs of (\ref{IV.B.4}) is proportional to the identity matrix, then:
\begin{equation}
\mathbb{H}^{\prime}=\left(\begin{array}{cc}
E^{\prime}&0\\
0&E^{\prime}\\
\end{array}\right).
\end{equation}
Thus, the perturbation is represented by:
\begin{equation}
\mathbb{W}=\mathbb{H}^{\prime}-\mathbb{H}=\left(\begin{array}{cc}
\hbar \omega/2&0\\
0&-\hbar \omega/2\\
\end{array}\right).
\end{equation}
Now, in the basis spanned by $\psi_{L}$ and $\psi_{R}$ the things look quite different. Indeed, we have that:
\begin{equation}
\tilde{\mathbb{H}}=\mathbb{OHO}^{-1}=\left(\begin{array}{cc}
E^{\prime}&-\hbar \omega/2\\
-\hbar \omega/2&E^{\prime}\\
\end{array}\right)
\end{equation}
and most importantly:
\begin{equation}
\tilde{\mathbb{W}}=\mathbb{OWO}^{-1}=\left(\begin{array}{cc}
0&\hbar \omega/2\\
\hbar \omega/2&0\\
\end{array}\right).
\end{equation}
So that the perturbation has no diagonal elements. This means that zeroth order corrections are strictly null for the perturbation, and, further more, that the off-diagonal elements are equal.
To be very clear, let us write the eigen-equations for each one of this distinct systems. For $H$ we have
\begin{equation}\begin{array}{cc}
H\psi_{0}(x,t)=i\hbar\frac{\partial\psi_{0}}{\partial t}(x,t)=E_{0}\psi_{0}(x,t),&H\psi_{1}(x,t)=i\hbar\frac{\partial\psi_{1}}{\partial t}(x,t)=E_{1}\psi_{1}(x,t),\\
\end{array}
\end{equation}
while $H^{\prime}$ responds to
\begin{equation}\label{IV.Z}
\begin{array}{cc}
H^{\prime}\psi_{L}(x,t)=i\hbar\frac{\partial\psi_{L}}{\partial t}(x,t)=E^{\prime}\psi_{L}(x,t),&H^{\prime}\psi_{R}(x,t)=i\hbar\frac{\partial\psi_{R}}{\partial t}(x,t)=E^{\prime}\psi_{R}(x,t).\\
\end{array}
\end{equation}
Now, let us consider $\tilde{\mathbb{H}}^{\prime}$ as the initial, unperturbed, Hamiltonian matrix, and
\begin{equation}
-\tilde{\mathbb{W}}=-\mathbb{OWO}^{-1}
\end{equation}
as the perturbation, so that $\tilde{\mathbb{H}}$ is the final, perturbed, Hamiltonian matrix. Then we can show that the $\psi_{L}(x,t)$ and $\psi_{R}(x,t)$ states transit from one another in Rabi style. Indeed, resorting to the time-dependent perturbation formalism,\cite{Landau2, Fitzpatrick1} we write, for a general state $\psi(x,t)$ of $\tilde{\mathbb{H}},$
\begin{equation}\label{IV.A.7}
\psi(x,t)=c_{L}(t)\psi_{L}(x,t)+c_{R}(t)\psi_{R}(x,t),
\end{equation}
in order to obtain the equation
\begin{equation}\label{mtdpC}
i\hbar\frac{d}{dt}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t)=\tilde{\mathbb{W}}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t),
\end{equation}
which is equivalent to the $2\times 2$ system of coupled linear equations:
\begin{equation}\label{IV.A.8}
i\hbar\frac{dc_{L}}{dt}=-\frac{\hbar \omega}{2}c_{R}\quad , i\hbar\frac{dc_{R}}{dt}=-\frac{\hbar \omega}{2}c_{L}.\\
\end{equation}
By uncoupling this system we get the harmonic oscillator equation
\begin{equation}
\frac{d^{2}c_{L}}{dt^{2}}=-\frac{\omega^{2}}{4}c_{L}
\end{equation}
and a similar equation for $c_{R}$, so that
\begin{equation}\label{Rabires}
c_{L}(t)=\sin\big(\omega t/2+\phi\big)\quad c_{R}(t)=\cos\big(\omega t/2+\phi\big),
\end{equation}
where $\phi$ is a constant that can be elucidate from initial conditions. The probability of finding the particle in the state $\psi_{L}$ is given, according to this last equations, by
\begin{equation}\label{probell}
P_{L}(t)=\sin^{2}\big(\omega t/2+\phi\big),
\end{equation}
and the probability of finding the the particle in the $\psi_{R}$ state is
\begin{equation}\label{probar}
P_{R}(t)=1-P_{L}(t).
\end{equation}
This is a particular instance of the Rabi oscillation, and this case is resonant due to the degeneracy of the ``initial" Hamiltonian $\tilde{H}^{\prime}$. But the ``perturbed" Hamiltonian $H$ is nothing else than the SDWP Hamiltonian of equation (\ref{SchU}).
Now, equations (\ref{probell}) and (\ref{probar}) predict the ``flip-flop" between the stationary states $\psi_{R}(x)$ and $\psi_{L}(x)$, so that, if the system is initial prepared in the state
\begin{equation}
\psi(x,t=0)=\psi_{L}(x),
\end{equation}
then we will have a 100\% certainty to find it in state $\psi_{R}(x)$ at times $t=\frac{\pi}{\omega},\frac{3\pi}{\omega},\frac{5\pi}{\omega}\ldots$ and a 100\% certainty to find it in state $\psi_{L}(x)$ at times $t=\frac{2\pi}{\omega},\frac{4\pi}{\omega},\frac{6\pi}{\omega}\ldots$. And this last result is consistent with equation (\ref{II.F}). Thus, we are in the presence of two different (yet not contradictory) descriptions of one and the same phenomenon: if $H$ is considered an unperturbed Hamiltonian, with complete stationary solutions $\psi_{0}(x,t)$ and $\psi_{1}(x,t)$, then we have an ``oscillating" non-stationary solution $\psi_{L}(x,t)$. If, on the other hand, $H$ is considered to be the result of a perturbation acting on the degenerate Hamiltonian $H^{\prime}$, we then get flip-flops between the complete stationary solutions of $H^{\prime}$, that is: $\psi_{L}(x,t)$ and $\psi_{R}(x,t)$.
In this manner, we obtain the periodic transitions (the zero point Leggett-Caldeira oscillations) observed in so many experiments. Notice that this transitions occur in the absence of external fields, thus without emission or absorption.
\section{Thermal radiation}\label{sec:4}
Due to the fact that no quantum system can be completely isolated from its environment, in any realistic description the Schroedinger equation must be supplemented with terms that describe the interaction between the system and its surroundings. But there are very different ways to describe this interaction and its results, depending on the time and energy scales involved, and the complexity of the analysis. Here we shall discuss the absorption-induced transitions by which the system is thrown into high-lying energy levels, rendering the two-level model inapplicable. The main result from this discussion will be a limit on the temperature at which Caldeira-Leggett oscillations can be observed.
\subsection{Oscillations near resonance}
Now, oscillatory behavior is to be expected not only for the the resonant, exactly degenerate, Hamiltonian matrix $\mathbb{H}^{\prime}$. Indeed, it would not be realistic to expect Leggett-Caldeira oscillations only in perfectly isolated systems. Consider a harmonic perturbation of the SDWP matrix Hamiltonian $\mathbb{H}$ of equation (\ref{IV.A.a}), that is, a perturbative term of the general form
\begin{equation}
\mathbb{V}=\mathbb{A}\exp(i\omega^{\prime}t)+\mathbb{A}^{\dagger}\exp(-i\omega^{\prime}t),
\end{equation}
and let us focus on the particularly simple case for which
\begin{equation}\label{simple.1}
\mathbb{A}=A\left(\begin{array}{cc}
0&1\\
0&0\\
\end{array}\right),
\end{equation}
so that the perturbative term can be written down as
\begin{equation}
\mathbb{V}=A\left(\begin{array}{cc}
0&\exp(i\omega^{\prime}t)\\
\exp(-i\omega^{\prime}t)&0\\
\end{array}\right).
\end{equation}
We then again resort to the time-dependent perturbation formalism, and write
\begin{equation}
\psi(x,t)=c_{0}(t)\psi_{0}(x,t)+c_{1}(t)\psi_{1}(x,t)
\end{equation}
in order to obtain the equation
\begin{equation}\label{mtdpC}
i\hbar\frac{d}{dt}\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t)=\mathfrak{V}(t)\left(\begin{array}{c}
c_{0}\\
c_{1}\\
\end{array}\right)(t),
\end{equation}
where $\mathfrak{V}$, defined by
\begin{equation}
\mathfrak{V}(t)=\exp\Big(i\mathbb{H} t/\hbar\Big)\mathbb{V}(t)\exp\Big(-i\mathbb{H} t/\hbar\Big)
\end{equation}
represents the perturbation in the interaction picture, and in our particularly simple case reduces to
\begin{equation}
\mathfrak{V}(t)=A\left(\begin{array}{ccc}
0&\quad&\exp \left(i(\omega^{\prime}-\omega)t \right)\\
& \\
\exp \left(-i(\omega^{\prime}-\omega)t \right) &\quad&0\\
\end{array}\right),
\end{equation}
so that equation (\ref{mtdpC}) is equivalent to the $2 \times 2$ system of coupled ODEs
\begin{equation}
i\hbar\frac{dc_{0}}{dt}=A\exp\Big[ i\big(\omega^{\prime}-\omega\big)t\Big] c_{1}\textrm{,}\quad
i\hbar\frac{dc_{1}}{dt}=A\exp\Big[ -i\big(\omega^{\prime}-\omega\big)t\Big] c_{0}\textrm{.}
\end{equation}
It can be checked by hand that
\begin{equation}
c_{0}(t)=\exp(i\Omega^{\prime} t/2)\Bigg\{\cos (R_{0} t)-\frac{i\Omega^{\prime}}{2R_{0}}\sin (R_{0}t)\Bigg\}
\end{equation}
and
\begin{equation}
c_{1}(t)=\frac{-iR_{1}}{R_{0}}\exp(-i\Omega^{\prime} t/2)\sin (R_{0}t)
\end{equation}
provide a solution for the initial conditions $c_{0}(t=0)=1 , c_{1}(t=0)=0$. Here we have used the following shorthand
\begin{equation}\begin{array}{cccc}
R_{0}=\sqrt{(A/\hbar)^{2}+\Big(\frac{\omega^{\prime}-\omega}{2}\Big)^{2}},&\ \Omega^{\prime}=\omega^{\prime}-\omega &
\textrm{ and }&
R_{1}=A/\hbar,\
\end{array}
\end{equation}
which lead to what is known as Rabi's formula,\cite{Fitzpatrick} namely:
\begin{equation}\label{Rah.1}
P_{1}(t)=\bigg(\frac{R_{1}}{R_{0}}\bigg)^{2}\sin^{2}\big(R_{0}t\big),
\end{equation}
\begin{equation}\label{Rah.2}
P_{0}(t)=1-P_{1}(t).
\end{equation}
It is not difficult to find the expressions for $P_{L}(t)$ and $P_{R}(t)$ for this particular choice of $\mathbb{A}$. We omit these, as they are not particularly illuminating. Let us just point out that in all instances $P_{R}$ and $P_{L}$ are oscillating functions of time, although they are generally not periodic. If one wishes to describe periodic Rabi oscillations in the $R$ and $L$ states, one should take, instead of (\ref{simple.1}),
\begin{equation}
\mathbb{A}=\frac{1}{2}\left(\begin{array}{cc}
1&-1\\
1&-1\\
\end{array}\right)
\end{equation}
as the natural choice for $\mathbb{A}$. By doing this one obtains expressions completely analogous to (\ref{Rah.1}) and (\ref{Rah.2}) for $P_{L}$ and $P_{R}$:
\begin{equation}
P_{L}(t)=\bigg(\frac{R_{1}}{R_{0}^{\prime}}\bigg)^{2}\sin^{2}\big(R_{0}^{\prime} t\big)
\end{equation}
\begin{equation}
P_{R}(t)=1-P_{L}(t),
\end{equation}
where the new frequency of the oscillation is now given by
\begin{equation}
R_{0}^{\prime}=\sqrt{\big(A/\hbar\big)^{2}+\big(\omega^{\prime}/2\big)^{2}}
\end{equation}
The main conclusion of this subsection is thus, that the Leggett-Caldeira can survive the influence of an environment on the particle in a SDWP under certain circumstances.
\subsection{A limit on temperature}
An important result from perturbation theory tells us that for harmonic perturbations the time-dependent transition amplitude, $c_{n\rightarrow m}(t)$, between two given eigenstates of the complete Hamiltonian $H$ is given by:\cite{EspositoMarmoSudarshan}
\begin{equation}\label{III.6}
c_{n\rightarrow m}(t)=\langle \psi_{m}\vert A\vert\psi_{n}\rangle\frac{1-\exp i(\frac{E_{m}-E_{n}}{\hbar}-\omega^{\prime})t}{E_{m}-E_{n}-\hbar\omega^{\prime}}+\langle \psi_{m}\vert A^{*}\vert\psi_{n}\rangle\frac{1-\exp i(\frac{E_{m}-E_{n}}{\hbar}+\omega^{\prime})t}{E_{m}-E_{n}+\hbar\omega^{\prime}}.
\end{equation}
As a consequence we get that, if the system is to stay in the two lowest lying levels, then the perturbation must meet the condition:
\begin{equation}
\omega^{\prime}<\frac{E_{2}-E_{1}}{\hbar}.
\end{equation}
Otherwise, the perturbation would excite the system to higher levels with non-negligible probability. This gives a limit on the temperature at which the system behaves like a low-lying two-level system. Indeed, recalling Wien's law for blackbody radiation, we get that thermal radiation at a temperature $T$ will have a maximal contribution of frequency $\omega^{\prime}$ when condition
\begin{equation}
\omega^{\prime}=\frac{2\pi c}{b_{W}}T
\end{equation}
is met. (Here, $T$ stands for the temperature of the radiation, $b_{W}$ is Wien's constant, and $c$ the velocity of light) Thus, if $\mathbb{V}(x,t)$ is somehow to represent thermal radiation, and if the perturbed Hamiltonian $H^{\prime}$ is to be described as a low-lying two-level system, then we must have:
\begin{equation}\label{temp.1}
T<\frac{b_{W}}{2\pi c}\ \frac{E_{2}-E_{1}}{\hbar}.
\end{equation}
In so many words: for each system there is a limit temperature above which the two-level system description is inapplicable, and zero-point Leggett-Caldeira oscillations become overshadowed by other transitions. Moreover, from inequalities (\ref{inek01}) and (\ref{2NF}) we get :
\begin{equation}\label{temp.2}
T_{B}(a,m)<\frac{b_{W}}{2\pi c}\ \frac{E_{2}-E_{1}}{\hbar}<3T_{B}(a,m)
\end{equation}
with this global bound given by:
\begin{equation}\label{temp.3}
T_{B}(a,m)=\frac{5\pi\hbar b_{W}}{16mca^{2}}
\end{equation}
The meaning of expressions (\ref{temp.2}) and (\ref{temp.3}) is the following: consider a family of double rectangular barriers, with a fixed $m$, $a$ and $k$, but free barrier width. When exposed to thermal radiation, there is a temperature $T_{B}$ for the radiation above which the Leggett-Caldeira oscillations are overshadowed by other transitions in least some the systems, and at temperature $3T_{B}$ the Calderia-Legget oscillations are surpassed by other transitions in all of the systems.
\section{Decoherence and dissipation}\label{sec:5}
We begin this section with a simplified exposition of mixed and pure states and the density matrix formalism as found in Landau and Lifshitz,\cite{LandauZ} to move on next to an also simplified rendering of some of Leggett's original argumentation. After that, decoherence is defined, an its relation with dissipation is briefly discussed.
The interaction of a system ($\mathfrak{S}$) with its surroundings ($\mathfrak{E}$) can be taken into account by considering a bigger \emph{isolated} system ($\mathfrak{U}$) which encompasses both $\mathfrak{S}$ and $\mathfrak{E}$ (that is: $\mathfrak{U}=\mathfrak{S}\bigcup\mathfrak{E}$). The state of this new, all including, system, $\mathfrak{U}$ is described by a state function $\Psi(j,\xi)$ that depends both on the coordinates of $\mathfrak{S}$ (the $j$) and the the coordinates of its environment (the $\xi$). The total Hamiltonian $H_{T}$ acting on $\mathfrak{U}$ can always be written in the form:
\begin{equation}
H_{T}=H+H_{\mathfrak{E}}+\lambda H_{I}
\end{equation}
where $H$ depends only on the $j$ and their generalized momenta, $H_{\mathfrak{E}}$ depends only on the $\xi$ and its momenta, and $H_{i}$ depends on both types of coordinates. We shall take the approximation, that $H$ is the Hamiltonian of $\mathfrak{S}$ when isolated, and that $H_{I}$ alone models the interaction between $\mathfrak{S}$ and $\mathfrak{E}$.
In principle, there can happy instances in which $\Psi(j,\xi)$ could be written as the product of two states functions:
\begin{equation}\label{pure}
\Psi(j,\xi)=\psi(j)\phi(\xi)
\end{equation}
but this does not need to be the case. States that can be written in the form (\ref{pure}) are called \emph{pure states} in the literature. States that are not pure are said to be \emph{mixed.}
In order to illustrate this let us consider the case in which both the original system and its surroundings can be represented as two-level systems. If the isolated Hamiltonian $H$ has eigenfunctions $\psi_{+}$ and $\psi_{-}$:
\begin{equation}
H\psi_{\pm}=E_{\pm}\psi_{\pm}
\end{equation}
and if $\phi_{\alpha}$ and $\phi_{\beta}$ are the eigenfunctions of $H_{e}$, \emph{i. e.}
\begin{equation}
H_{e}\phi_{\alpha}=E_{\alpha}\phi_{\alpha}\ , \ H_{e}\phi_{\beta}=E_{\beta}\phi_{\beta},
\end{equation}
then some examples of pure states are:
$$\begin{array}{ c }
\frac{1}{\sqrt{2}}(\psi_{+}\phi_{\beta}+\psi_{-}\phi_{\beta})=\frac{1}{\sqrt{2}}(\psi_{+}+\psi_{-})\phi_{\beta}, \ \frac{1}{2}\psi_{-}\phi_{\beta}+\frac{\sqrt{3}}{2}\psi_{-}\phi_{\alpha}=\psi_{-}(\frac{1}{2}\phi_{\beta}+\frac{\sqrt{3}}{2}\phi_{\alpha})\\
\textrm{and}\\
\frac{1}{4}\psi_{-}\phi_{\beta}+\frac{\sqrt{3}}{4}\psi_{-}\phi_{\alpha}-\frac{\sqrt{3}}{4}\psi_{-}\phi_{\beta}-\frac{3}{4}\psi_{+}\phi_{\alpha}=(\frac{1}{2}\psi_{-}-\frac{\sqrt{3}}{2}\psi_{+})(\frac{1}{2}\psi_{\beta}+\frac{\sqrt{3}}{2}\phi_{\alpha}).\\
\end{array}$$
On the other hand, as instances of mixed states, we can provide the following:
\begin{equation}\label{mixed}
\frac{1}{\sqrt{2}}(\psi_{-}\phi_{\alpha}+\psi_{+}\phi_{\beta})\ , \ \frac{1}{\sqrt{2}}(\psi_{-}\phi_{\beta}+\psi_{+}\phi_{\alpha})\ , \ \textrm{ and } \frac{1}{\sqrt{3}}(\psi_{+}\phi_{\alpha}+\psi_{+}\phi_{\beta}+\psi_{-}\psi_{\alpha}).
\end{equation}
The density matrix formalism was developed to treat systems that (like $\mathfrak{U}$) can present mixed states. The density matrix $\rho$ allows us to calculate the expected value $\langle f \rangle$ of any observable $f(x,p_{x})$ that depends only on the coordinates and momenta of $\mathfrak{S}$:
\begin{equation}\label{Tr}
\langle f \rangle\ =\ \textrm{Tr}\Big( f\rho\Big).
\end{equation}
The elements of the density matrix $\rho$ of a state $\Psi(j,\xi)$ of are defined as:
\begin{equation}\label{rho}
\rho_{j,j^{\prime}}=S_{\xi}\Psi^{*}(j,\xi)\Psi(j^{\prime},\xi),
\end{equation}
where $S_{\xi}$ stands for the sum over the discrete $\xi$ (if any) plus an integral over the continuous $\xi$ (if any). In the case of our $2\times 2$-level system, expression (\ref{rho}) reduces to
\begin{equation}\label{rho}
\rho_{j,j^{\prime}}=\Psi_{j,\alpha}^{*}\Psi_{j^{\prime},\alpha}+\Psi_{j,\beta}^{*}\Psi_{j^{\prime},\beta}, \quad j,j^{\prime}=\pm\ .
\end{equation}
The diagonal elements of density matrix, of the form $\rho_{j,j}$, are called \emph{populations,} while the off-diagonal elements (\emph{i. e.} the elements with $j\neq j^{\prime}$) are known as \emph{coherences.}
Suppose now that the 50-50 linear combinations
\begin{equation}
\psi_{L}=\frac{1}{\sqrt{2}}\Big(\psi_{+}+\psi_{-}\Big), \ \psi_{R}=\frac{1}{\sqrt{2}}\Big(\psi_{+}-\psi_{-}\Big)
\end{equation}
are eigenfunctions of a macroscopic observable $M$, let us say:
\begin{equation}
M\psi_{L,R}=\mu_{L,R} \psi_{L,R},
\end{equation}
and take then the mixed state given by
\begin{equation}\label{momix}
\Psi=c_{L}\psi_{L}\phi_{\alpha}+c_{R} \psi_{R}\phi_{\beta}.
\end{equation}
The density matrix associated with (\ref{momix}) is written as
\begin{equation}\label{diag}
\rho=\left(\begin{array}{cc}
\vert c_{L}\vert^{2}& 0\\
0&\vert c_{R}\vert^{2}\\
\end{array}\right)
\end{equation}
in the $\{\psi_{L}, \psi_{R}\}$ basis, as can be seen from (\ref{rho}), so that according to (\ref{Tr}) the expected value of any observable $f$ pertaining to $\mathfrak{S}$ yields the value
\begin{equation}\label{expvalue}
\langle f\rangle=\vert c_{L}\vert^{2}f_{L}+\vert c_{R}\vert^{2}f_{R},
\end{equation}
where $f_{L}$ and $f_{R}$ are the expected values of $f$ in the pure states
\begin{equation}
\Psi_{L}=\psi_{L}\phi_{\alpha}\ \textrm{ and }\ \Psi_{R}=\psi_{R}\phi_{\beta}.
\end{equation}
The point of this discussion is that the same result (\ref{expvalue}) is obtained if we make measurements on an ensemble of $\mathfrak{U}$ systems all in state $\Psi$, or if the same measurements are made with an ensemble of $\mathfrak{U}$ made up of a statistical mixture of the pure states $\Psi_{L}$ and $\Psi_{R}$, in proportions $\vert c_{L}\vert^{2}$ and $\vert c_{R}\vert^{2}$. If it were to be held true that only ensembles of the type (\ref{momix}) could be prepared for $\mathfrak{U}$, then it could be argued that property $M$ has a sharp value for each element of the ensemble, and that a measurement done on a particular element only removes our ignorance on its value for that particular system. Clearly, this opens the door for hidden variable theories. To put it succinctly: in this interpretation each one of the Schroedinger's cats in an ensemble of such felines would be either dead or alive, and never in superpositions composed of both dead and alive states. Only the behaviour of the ensemble would be quantal, its individual elements being essentially classical.
It is patent, on the other hand, that the pure state
\begin{equation}
\Psi_{+}=\psi_{+}\phi_{\alpha}
\end{equation}
cannot be written as a mixed state of the form (\ref{momix}) and that its corresponding density matrix cannot be diagonal in the $(L,R)$ basis, unlike (\ref{diag}). The impossibility of the simultaneous diagonalization of the density matrices of all possible states of a system $\mathfrak{S}$ is then a strong evidence of the true quantal behaviour of such system, as opposed to the behaviour required by hidden variable theories. Thus, for a system $S$ to be classical in any sense of the word, the coherences, \emph{i. e.} the off-diagonal elements, must be absent from the density matrix for each one of its possible states. This conclusion is generally valid, even if we resorted to the most trivial case in order to illustrate it.\cite{Leggett}
Decoherence can be defined as the decay of the off-diagonal elements in the density matrix as a result of the interaction of the system with its environment. Therefore decoherence allows a system to behave as quantal when isolated and as classical when the coupling with its environment is ``sufficiently effective." This is nowadays considered a plausible mechanism for the emergence of classical reality from a quantal substratum.
In most practical applications, the environment $\mathfrak{E}$ has a very large number of degrees of freedom (say of the order of the Avogadro number) and not just one, as in the example we have used. Thus $\mathfrak{U}$ is usually a thermodynamic system, so that the full toolbox of quantum statistical mechanics needs to be marshalled in order to describe its behaviour. In this case the interaction between $\mathfrak{S}$ and $\mathfrak{E}$ (interaction known as quantum dissipation in this context) involves the relaxation of the thermodynamical variables of $\mathfrak{U}$ towards thermal equilibrium, and not only decoherence.
Various models have been proposed over the years for the environment (or \emph{bath}) but one of first and most successful is the \emph{spin-boson Hamiltonian} approach, in which $H_{\mathfrak{E}}$ is taken as a collection of harmonic oscillators with various frequencies and the interaction term $H_{I}$ is linear both in the $j$ and in the $\xi$ coordinates. One important result from this approach is that a two-level system $\mathfrak{S}$ will describe damped oscillations between the localized states $\vert R\rangle$ and $\vert L\rangle$. Depending on the frequency distribution of the environment, $\mathfrak{S}$ may be localized at $T=0^{o}$K (the overdamped case, known as ``subohmic"), it may present critical damping (the ``ohmic case") or it may undergo underdamped coherent oscillations (the ``superohmic case.") The last one of these three instances is the most interesting for the present discussion, as it allows the observation of MQC before the complete relaxation of the system. The possibility of experimental MQC in the superohmic case depends in the interplay between a \emph{decoherence time} defined only by the bath parameters, and the period of the Leggett-Caldeira oscillation for system $\mathfrak{S}$.
\section{The scales of MQC}\label{sec:6}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
$b$ & $E_0$ & $E_1$ & $\Delta E$ & $\tau$ \\
( nm )& $( \times 10^{-26} $ J )& ( $\times 10^{-26}$ J ) & $ ( \times 10^{-28}$J )& ( $\mu$s )\\
\hline \hline
100.00000& 5.3753895& 5.4382093& 6.3 & 1.0\\ \hline
116.65290& 5.3899569& 5.4246062& 3.5 & 2.9 \\ \hline
136.07900& 5.3987829& 5.4160961& 1.7 & 3.8\\ \hline
158.74011& 5.4036276& 5.4113353& 0.77 & 8.6\\ \hline
185.17494& 5.4059909& 5.4089897& 0.30 & 22.0\\ \hline
216.01195& 5.4069931& 5.4079902& 0.10 & 66.0\\ \hline
251.98421& 5.4073539& 5.4076298& 2.7$\times 10^{-2}$ & 240\\ \hline
\hline
\end{tabular}
\caption{Period $\tau$ increases exponentially as the barrier width is augmented. $a=1.0\ \mu$m, $k=2\times10^{-20}$J. This table, as well as all figures, was generated with Matlab \textsuperscript{\textregistered} R2012a.}
\label{tab:sample}
\end{table}
Let us start by fixing the width of the lateral wells at:
\begin{equation}\label{VII.1}
a=1\mu\textrm{m,}
\end{equation}
a value typical of contemporary lithographic circuitry, and take $m$ to be the rest mass of an electron:
\begin{equation}\label{VII.2}
m=m_{e}=9.1\times 10^{-31}\textrm{kg}.
\end{equation}
With this, $B$ takes the value:
\begin{equation}\label{5.A}
B=0.6\times 10^{-25}\textrm{J}=0.36\ \mu\textrm{eV,}
\end{equation}
and $T_{B}$ is fixed at:
\begin{equation}
T_{B}\approx 1.1\ \textrm{mK.}
\end{equation}
From equation (\ref{II.7}), that gives the fundamental frequency of the Caldeira-Leggett oscillations, we get the corresponding period
\begin{equation}
\tau=\frac{2\pi\hbar}{E_{1}-E_{0}}.
\end{equation}
A global lower bound for this period is found from expressions (\ref{inequalityA}) and (\ref{funnyeq}):
\begin{equation}\label{5.B}
\tau >\frac{2\pi\hbar}{B}=\frac{4ma^{2}}{\pi\hbar}.
\end{equation}
For values (\ref{VII.1}) and (\ref{VII.2}) this gives
\begin{equation}
\tau>11\textrm{ns}.
\end{equation}
From table I (obtained through computer assisted numerical analysis) we get that as we sweep the barrier width from 0.2 to 0.5 $\mu$m the period of the Leggett -Caldeira oscillations for our square double well increases from 1.0 to 240 $\mu$s. Based on general considerations it has been estimated\cite{Leggett} that, for all practical purposes, MQC is lost if the period of the Legget-Caldiera oscillation is of the order $\tau\gtrsim100\mu$s. Thus, the last row of the table corresponds to a localized system. All the other tabulated values could in principle correspond to observable MQC.
\subsection{Some of the many things we have left out}
MQC experiments are carried out in superconducting quantum interference devices (SQUIDs) with low capacitance tunneling Josephson junctions,\cite{Leggett, Friedmanetal} and the relevant coordinate (\emph{i. e.} the analogous of coordinate $x$) is not of a geometric character (like a position) but is in most cases the phase difference between the states functions of the electrons in a Cooper pair (so that $m$ is not really the mass of the electron.) Thus our toy model is in reality a simplification of a mechanical analogy used to discuss experimental MQC.
\section{Conclusions}\label{sec:7}
Contemporary quantum mechanics, both experimental and theoretical, provides examples of basic concepts and techniques such as: tunneling, stationary states, two-level systems, perturbation theory, the density matrix and the WKB approximation. Classroom presentations of current areas of research, such as MQC, help to improve the understanding of quantum physics at university level, as they connect the simplified textbook models with the actual state of the field, and thus with the future professional activity of the student. Moreover, MQC illustrates in a beautiful way the interplay between theory and experiment, and between concepts and techniques arising in different areas of quantum physics.
We believe to have achieved in the present paper a level of exposition that makes it both clear and interesting for senior university students and recent graduates. In order to do so, we had to glide over the more technical aspects of experimental MQC and the intricate relation between MQC and the epistemology and the philosophy of physics. We hope that the present paper will encourage the interested reader to delve further into this facets of contemporary research.
\section{Appendix}
Consider condition (\ref{coneven}) for the ground level ($n=0$), that is:
\begin{equation}\label{app01}
E_{0}\cot^{2} a\frac{\sqrt{2mE_{0}}}{\hbar}=(k-E_{0})\tanh^{2}b\frac{\sqrt{2m(k-E_{0})}}{\hbar}
\end{equation}
We will now establish a lower bound for $E_{0}$ starting from (\ref{app01}), but we have to take some precautions in doing so because $E_{0}$ depends implictly on $b.$ In order to proceed, note that
\begin{equation}\label{app01A}
\forall b\in(0,\infty)\ , \ \quad\frac{\sqrt{2m(k-E_{0})}}{\hbar}<\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
so that
\begin{equation}\label{app02}
\forall b\in(0,\infty) \ , \ \quad \tanh^{2}b\frac{\sqrt{2m(k-E_{0})}}{\hbar}>\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
The dependence of the rhs of inequality (\ref{app02}) is explicit, so that the usual procedures of calculus can be applied. In particular as we now from elementary theorems that the limit
\begin{equation}
\lim_{b\rightarrow \infty}\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}=1
\end{equation}
holds true, we can affirm that: for given $\delta>0$ there exists a $b_{0}( \delta)$ such that any $b>b_{0}(\delta)$
\begin{equation}\label{app03}
\tanh^{2} b\frac{\sqrt{2m(k-B/4)}}{\hbar}>1-\frac{\delta}{2k}
\end{equation}
From (\ref{app01}) (\ref{app02}) and (\ref{app03}) we deduce that for any $b$ above a certain value $b_{0}(\delta)$, the ground energy of $U_{b}$ satisfies:
\begin{equation}\label{app04}
E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}>(k-E_{0})\Big(1-\frac{\delta}{2k}\Big)
\end{equation}
Turning our attention to the condition for $E_{1}$, \emph{i. e.}
\begin{equation}\label{app01B}
E_{1}\cot^{2} a\frac{\sqrt{2mE_{1}}}{\hbar}=(k-E_{1})\coth^{2}b\frac{\sqrt{2m(k-E_{1})}}{\hbar}
\end{equation}
we now find an upper bound for $E_{1}$, by noting that, because of (\ref{app01A}) and the known properties of the hyperbolic functions, the inequality
\begin{equation}
\coth^{2}b\frac{\sqrt{2m(k-E_{1})}}{\hbar}<\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}
\end{equation}
is verified for all strictly positive $b$. Furthermore,
\begin{equation}
\lim_{b\rightarrow\infty}\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}=1
\end{equation}
so that for every $\delta>0$ there exist a $b_{1}(\delta)$ such that, if $b>b_{1}(\delta)$, then inequality
\begin{equation}\label{app07A}
\coth^{2}b\frac{\sqrt{2m(k-B/4)}}{\hbar}<1+\frac{\delta}{2k}
\end{equation}
is satisfied for all strictly positive $b.$ And from (\ref{app01B}) and (\ref{app07A}) we get that, for all $b$ above a certain thershold value $b_{1}(\delta)$, the inequality
\begin{equation}\label{app08}
E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}<(k-E_{1})\Big(1+\frac{\delta}{2k}\Big)
\end{equation}
is satisfied.
Taking both (\ref{app04}) and (\ref{app08}) into consideration, we have that for every $\delta >0$ there exists a number $b(\delta)=\max \{b_{0}(\delta) b_{1}(\delta)\}$ such that for any $b>b(\delta)$ the inequality
\begin{equation}\label{app09}
E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<(E_{0}-E_{1})+\delta \Big(1-\frac{E_{0}+E_{1}}{2k}\Big)
\end{equation}
is satisfied. Now, it is not difficult to see that
\begin{equation}
w(E)=E\cot^{2}a\frac{\sqrt{2mE}}{\hbar}
\end{equation}
is a monotonically increasing function of $E$ in the range $B/4<E<B$, so that
\begin{equation}\label{app010}
0<E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}
\end{equation}
and in the other hand, we deduce
\begin{equation}\label{app011}
(E_{0}-E_{1})+\delta \Big(1-\frac{E_{0}+E_{1}}{2k}\Big)<\Big(1-\frac{E_{0}+E_{1}}{2k}\Big)\delta <\delta
\end{equation}
from and . From ( \ref{app09}), (\ref{app010}) and (\ref{app011}) we get:
\begin{equation}
0<E_{1}\cot^{2}a\frac{\sqrt{2mE_{1}}}{\hbar}-E_{0}\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\delta
\end{equation}
\begin{equation}
E_{1}\cot^{2} a\frac{\sqrt{2mE_{1}}}{\hbar}>k-E_{1}
\end{equation}
Finally, we notice that, as
\begin{equation}
v(E)=\cot^{2}a\frac{\sqrt{2mE}}{\hbar}
\end{equation}
is a monotonically increasing function of $E$ in the range $B/4<E<B$, then
\begin{equation}\label{app013}
(E_{1}-E_{0})\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\delta
\end{equation}
Now we just need to find a lower bound on $\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}$. This is obtained by turning back to condition (\ref{app01}) from which we get
\begin{equation}\label{app014}
\cot^{2}a\frac{\sqrt{2mE_{0}}}{\hbar}<\frac{k-B/4}{B}
\end{equation}
Finally, from (\ref{app013}) and (\ref{app014}) we arrive at
\begin{equation}\label{app015}
E_{1}-E_{0}<\delta\frac{B}{k-B/4}
\end{equation}
Let us stress that $k$ and $B$ are independent of $b$. In this manner, we have arrived at the following lemma:
For each strictly positive real number $ \delta$ there exists a
\begin{equation}
b^{\prime}(\delta)=b(\delta\frac{k-B/4}{B})
\end{equation}
such that for any $b>b^{\prime}(\delta)$ the gap between the ground and first excited levels of of $U_{b}$ is less than $\delta$, that is, such that:
\begin{equation}
E_{1}-E_{0}<\delta \ .
\end{equation}
And this is what we set out to prove in this appendix.
\end{document} |
\begin{document}
\title{On the Schr\"{o}dinger equation with singular potentials}
\author{{{Jaime Angulo Pava}}\\{\small IME-USP, Rua do Matao 1010, Cidade Universitaria}\\{\small {CEP 05508-090, Sao Paulo, SP, Brazil.}}\\{\small \texttt{E-mail:angulo@ime.usp.br}}
\\{{Lucas C. F. Ferreira}}\\{\small Universidade Estadual de Campinas, IMECC - Departamento de
Matem\'{a}tica,} \\{\small {Rua S\'{e}rgio Buarque de Holanda, 651, CEP 13083-859, Campinas-SP,
Brazil.}}\\{\small \texttt{E-mail:lcff@ime.unicamp.br}}}
\date{}
\maketitle
\begin{abstract}
We study the Cauchy problem for the non-linear Schr\"odinger equation with singular
potentials. For point-mass potential and nonperiodic case, we prove existence
and asymptotic stability of global solutions in weak-$L^{p}$ spaces. Specific interest is give to the point-like $\delta$ and $\delta'$ impurity and for two $\delta$-interactions in one dimension. We also consider the
periodic case which is analyzed in a functional space based on Fourier transform and
local-in-time well-posedness is proved.
\end{abstract}
{\small {\quad\textbf{AMS subject classification:} 35Q55, 35A05, 35A07, 35C15, 35B40, 35B10 }}
{\small
\quad\textbf{Keywords} NLS-Dirac equation, Singular
potential, Existence, Asymptotic behavior}
\section{Introduction}
We are interested in this paper in the the Cauchy problem for the following Schr\"odinger model
\begin{equation}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+\mu(x)u & = F(u),\quad x\in \mathbb R^n, \;\;t\in \mathbb R\ \label{SCH0}\\
u(x,0) & =u_{0}(x),
\end{aligned}\right.
\end{equation}
in weak-$L^p$ spaces (Marcinkiewicz spaces) and in a space based on Fourier transform. In the weak-$L^p$ spaces we consider the case $n=1$ and $\mu(x)=\sigma \delta$, $\mu(x)=\sigma(\delta(\cdot-a)+\delta(\cdot+a))$ (two Dirac's $\delta$ potentials place at the points $\pm a \in \mathbb R$) or $\mu(x)=\sigma \delta'$ where $\delta$ represents the delta function in the origin and $\sigma\in \mathbb R$, $F(u)= \lambda\left\vert u\right\vert ^{\rho
-1}u$, where $\lambda=\pm1$ and $\rho>1$. In the space based on Fourier transform we consider $n$ arbitrary and $\mu(x)$ being a bounded continuous function with a Fourier transform being a finite Radon measure and $F(u)= \lambda u^{\rho}$, where $\lambda=\pm1$ and $\rho\in \mathbb N$. The case $F(u)= \lambda\left\vert u\right\vert ^{\rho
-1}u$ is also commented.
The non-linear Schr\"odinger model (\ref{SCH0}) in the case $\mu(x)=\sigma\delta(x)$ (called the non-linear Schr\"odinger equation with a $\delta$-type impurity, the NLS-$\delta$ equation henceforth) arise in different areas of quantum field theory and are essential for understanding a number of phenomena in condensed matter physics. At the experimental side, the recent interest in point-like impurities (defects) is triggered by the great progress in building nanoscale devises. More exactly, the NLS-$\delta$ model with a impurity at the origin in the repulsive ($\sigma <0$) case and in the attractive ($\sigma >0$) is described by the following boundary problem (see Caudrelier\&Mintchev\&Ragoucy \cite{CMR})
\begin{equation}
\left \{
\begin{aligned}
i\partial_{t}u(x,t)+ u_{xx}(x,t) & = \lambda \left\vert u(x,t)\right\vert ^{\rho
-1}u(x,t),\quad x\neq0 \\
\lim_{x\to 0^+}[u(x,t)-u(-x, t)]&=0, \\
\lim_{x\to 0^+}[\partial_x u(x,t)-\partial_x u(-x, t)]&=\sigma u(0,t) \label{SCH02}\\
\lim_{x\to \pm \infty} u(x,t)=0,
\end{aligned}\right.
\end{equation}
hence $u(x,t)$ must be solution of the non-linear Schr\"odinger equation on $\mathbb R^{-}$ and $\mathbb R^{+}$, continuous at $x=0$ and satisfy a ``jump condition'' at the origin and it also vanishes at infinity.
The equations in (\ref{SCH02}) are a particular case of a more general model considering that the impurity is localized at $x=0$; in fact the equation of motion
$$
i\partial_{t}u(x,t)+ u_{xx}(x,t) = \lambda \left\vert u(x,t)\right\vert ^{\rho
-1}u(x,t),\quad x\neq0,
$$
with the impurity boundary conditions
\begin{equation}\label{bc}
\left(\begin{array}{c}u(0+,t) \\ \partial_xu(0+,t)\end{array}\right)=\alpha
\left(\begin{array}{cc} a & b\ \\c & d\end{array}\right)\left(\begin{array}{c} u(0-,t) \\ \partial_xu(0-,t) \end{array}\right)
\end{equation}
with
\begin{equation}\label{para}
\{a,b,c,d\in \mathbb R, \alpha\in \mathbb C: ad-bc=1, |\alpha |=1\}.
\end{equation}
The equation (\ref{bc}) captures the interaction of the ``field'' $u$ with the impurity \cite{CMR2}. The parameters
in (\ref{para}) label the self-adjoint extensions of the (closable) symmetric operator $H_0=-\frac{d^2}{dx^2}$ defined on the space $C_0^{\infty}(\mathbb R-\{0\})$ of smooth functions with compact support separated from the origin $x=0$. In fact, by von Neumann-Krein's theory of self-adjoint extensions for symmetric operators on Hilbert spaces, it is not difficult to show that there is a 4-parameter family of self-adjoint operators which describes all one point interactions in one-dimension of the second derivative operator $H_0$. Such a family
can be equivalently described through the family of boundary conditions at the origin
\begin{equation}\label{bc1}
\left(\begin{array}{c}\psi(0+) \\ \psi'(0+)\end{array}\right)=\alpha
\left(\begin{array}{cc} a & b\ \\c & d\end{array}\right)\left(\begin{array}{c} \psi(0-) \\ \psi'(0-)\end{array}\right)
\end{equation}
with $a,b,c,d$ and $\alpha$ satisfying the conditions in (\ref{para}) (see Theorem 3.2.3 in \cite{ak}).
Here we are interested in two specific choices of the parameters in (\ref{para}), which are relevant in physics applications (see \cite{CMR2}-\cite{CMR}). The first choice $\alpha=a=d=1$, $b=0$, $c=\sigma\neq 0$ corresponds to the case of a pure Dirac $\delta$ interaction of strength $\sigma$ (see Theorem \ref{self} below). The second one $\alpha=a=d=1$, $c=0$, $b=\beta\neq 0$ corresponds to the case of the so-called $\delta'$ interaction of strength $\beta$ (see Theorem \ref{selfd} below).
In section 2 below for convenience of the reader we present a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop0}
-\Delta_{\sigma}=-\frac{d^2}{dx^2}+\sigma \delta,
\end{equation}
which will be match with the singular boundary condition in (\ref{SCH02}) at every time $t$.
Existence and uniqueness of local and global-in-time solutions of problem (\ref{SCH0}) with $\mu(x)=0$ and $F(u)= \lambda\left\vert u\right\vert ^{\rho-1}u$ have been much studied in the framework of the Sobolev spaces $H^s(\mathbb R^n)$, $s\geqq 0$, i.e, the solutions and their derivatives have finite energy (see Cazenave's book \cite{C1} and the reference therein). In the case of $\delta$-interaction, namely, $\mu(x)=\sigma \delta$ the existence of global solution in $H^1(\mathbb R)$ and $L^2(\mathbb R)$ has been addressed in Adami\&Noja \cite{Adami} (we can also to apply Theorem 3.7.1 in \cite{C1} for obtaining a local-in-time well-posedness theory in $H^1(\mathbb R)$).
The first study of infinite $L^2$-norm solutions for $\mu(x)=0$ and $F(u)=\lambda\left\vert u\right\vert ^{\rho-1}u$ was addressed by Cazenave\&Weissler in \cite{Cazenave1} where they consider the space
$$
X_\rho=\{u:\mathbb{R}\rightarrow L^{\rho+1}(\mathbb R^n) \text{ \ Bochner meas.}: \sup_{-\infty<t<\infty}|t|^{\vartheta }\|u(t)\|_{L^{\rho+1}}<\infty\},
$$
where $\vartheta =\frac{1}{\rho-1}-\frac{n}{2(\rho+1)}$ and $\|\cdot\|_{L^{\rho+1}}$ denotes the usual ${L^{\rho+1}}$ norm. Under a suitable smallness condition on the initial data, they prove the existence of global solution in $X_\rho$, for $\rho_{0}(n)<\rho<\gamma(n)$ where $\rho_{0}(n)=\frac{n+2+\sqrt{n^{2}+12n+4}}{2n}>1$ is the positive root of the equation $n\rho^{2}-(n+2)\rho-2=0$ and $\gamma(n)=\infty$ if $n=1,2$ and $\gamma(n)=$ $\frac{n+2}{n-2}$ in otherwise.
Later on, in Cazenave\&Vega\&Vilela \cite{Cazenave2} the Cauchy problem was studied in the framework of weak-$L^p$ spaces. Using a Strichartz-type inequality, the authors obtained existence of solutions in the class $L^{(p,\infty)}(\mathbb R^{n+1})\equiv L_t^{(p,\infty)}(L_x^{(p,\infty)})$, where $(x,t)\in \mathbb R^{n}\times \mathbb R$ and $p=\frac{(\rho-1)(n+2)}{2\rho}$, for $\rho$ in the range
$$
\rho_0<\frac{4(n+1)}{n(n+2)}<\rho-1<\frac{4(n+1)}{n^2}<\frac{4}{n-2}.
$$
More recently, in Braz e Silva\&Ferreira\&Villamizar-Roa \cite{BFV} the Cauchy problem was studied in the Marcinkiewicz space $L^{(\rho+1,\infty)}$. Using bounds for the Schr\"odinger linear group in the context of Lorentz spaces, the authors showed existence and uniqueness of local-in-time solutions in the class
$$
\{u:\mathbb{R}\rightarrow L^{(\rho+1,\infty)} \text{ \ Bochner
meas.}: \sup_{-T<t<T}|t|^{\zeta}\|u(t)\|_{L^{(\rho+1,\infty)}}<\infty\},
$$
where $1<\rho<\rho_{0}(n)$ and $\frac{n(\rho-1)}{2(\rho+1)}=\zeta_{0}<\zeta<\frac{1}{\rho}$.
Since $\rho_{0}(n)<\frac{4}{n}$, the range for $\rho$ is different from the ones in Cazenave\&Weissler \cite{Cazenave1} and Cazenave {\it et al.} \cite{Cazenave2}. The existence of global solutions is showed in norms of type $\sup_{|t|>0}|t|^{\vartheta }\|u(t)\|_{L^{(\rho+1,\infty)}}$, where $\vartheta =\frac{1}{\rho-1}-\frac{n}{2(\rho+1)}$ and $\rho_0(n)<\rho<\gamma(n)$.
Our approach is based in some ideas in \cite{BFV}, so via real interpolation techniques we establish bounds for the Schr\"odinger linear group $G_\sigma(t)=e^{i(\partial_x^2+\sigma \delta)t}$ in the context of Lorentz spaces in the one-dimensional case. The cases $n=2,3$ remain open. The fundamental solution of the corresponding linear time-dependent Schr\"odinger equation, namely
$$
iu_t=-(\Delta+\sigma \delta)u,
$$
is now well know for $ n=1,2,3$; see Albeverio {\it et al.} \cite{ABD}- for instance. However, surprisingly, a \textquotedblleft good formula\textquotedblright of the unitary group $G_\sigma(t)\phi=e^{i(\Delta+\sigma \delta)t}\phi$ depending of the free linear propagator $e^{i\Delta t}\phi$ was found explicitly only for the one-dimensional case (see Holmer {\it et al.} \cite{Holmer5}-\cite{Holmer3}). In fact, by using scattering techniques, it was established in \cite{Holmer5} the convenient formula (for the case $\sigma\geqq 0$)
\begin{equation}\label{pospro1}
G_\sigma(t) \phi(x)= e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{-},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x),
$$
with $\chi^0_{+}$ the characteristic function of $[0,+\infty)$ and $\chi^0_{-}$ the characteristic function of $(-\infty, 0]$. For the case $\sigma<0$ see \cite{Holmer3} and Theorem \ref{expli} below. Here we show in the Appendix how to obtain the formula (\ref{pospro1}) based in the fundamental solution found in Albeverio {\it et al.} \cite{ABD}, which does not use scattering ideas. Nice formulas as that in (\ref{pospro1}) are not known for the cases $n=2,3$. Now, in the case $\sigma\geqq 0$, one can show from (\ref{pospro1}) the dispersive estimate in Lorentz spaces (see Lemma \ref{grupint1} below)
\begin{equation}\label{grupint02}
\left\Vert G_\sigma(t)f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac
{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p,d)},
\end{equation}
for $1\leqq d\leqq \infty$, $p'\in (2,\infty)$, $p\in (1,2)$ and $p'$ satisfying $\frac{1}{p}+\frac{1}{p'}=1$, where $C>0$ is independent of $f$ and $t\neq 0$. Then, under a suitable smallness condition on the initial data $u_0$, the existence of global solutions for (\ref{SCH0}) is proved in the space (see Theorem \ref{GlobalTheo})
$$
\mathcal{L}_{\vartheta }^{\infty}=\{u:\mathbb{R}\rightarrow L^{(\rho+1,\infty)} \text{ \ Bochner
meas.}: \sup_{-\infty<t<\infty}|t|^{\vartheta }\Vert u\Vert_{(\rho+1,\infty)}<\infty\},
$$
for $\sigma\geq0$, where $\vartheta =\frac{1}{\rho-1}-\frac{1}{2(\rho+1)}$ and $\rho_{0}=\frac{3+\sqrt{17}}{2}>1$ is the positive root of the equation $\rho^{2}-3\rho-2=0$. We also analyze the asymptotic stability of the global solutions (see Theorem \ref{TeoAssin}). For $\sigma <0$ our approach in general is not applicable because in this case the operator $-\Delta_\sigma$ has a non-trivial negative point spectrum. But, in this case it is possible to show the existence of a invariant manifold of periodic orbits in Lorentz spaces (see Section 6).
With regard to the more two singular cases: two Dirac's $\delta$ potentials placed at the points $\pm a \in \mathbb R$, $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$, and $\mu(x)=\beta \delta'$, i.e., the derivative of a $\delta$, a similar analysis to that above for the case of a $\delta$-potential can be established. In these cases, it is not known an explicit expression for the associated time propagator as that in (\ref{pospro1}) for the case of $\mu(x)=\sigma\delta$. However, by using a formula for the integral kernel of the time propagator associated (see \cite{KS} and \cite{AGHH}), we obtain an estimate similar to (\ref{grupint02}).
For the case $n\geqq 4$ we do not have Schr\"odinger operators with point interactions. In fact, the Schr\"odinger operators with point interactions, namely, perturbations of the Laplace operators by ``measures'' supported on a discrete set (supported at zero for simplicity, namely, by the Dirac delta measure $\delta$ centered at zero) are usually defined by means of von Neumann\&Krein theory of self-adjoint extensions of symmetric operators, and so as one of a whole family of self-adjoint (in $L^2(\mathbb R^n)$) extensions of an operator $A$, $D(A)=C^\infty_0(\mathbb R^n-\{0\})$, $Au=-\Delta u$, $u\in D(A)$. In the case $n\geqq 4$ it is well known that the theory trivializes where there is only one self-adjoint extension of $A$ (see Albeverio {\it et al.} \cite{AGHH}).
Next, let $\mathcal M$ be the set of finite Radon measure endowed with the norm of total variation, that is, $\|\omega\|_{\mathcal M}=|\omega|(\mathbb R^n)$ for $\omega\in \mathcal M$, $n\geq1$. Then, by considering $\mu(x)$ in (\ref{SCH0}) being a bounded continuous function with a Fourier transform such that $\widehat{\mu}\in \mathcal M$, we show a local-in-time well-posedness result in the Banach space
\begin{equation}
\mathcal{I}=[\mathcal{M}(\mathbb{R}^{n})]^{\vee}=\{f\in\mathcal{S}^{\prime
}(\mathbb{R}^{n}):\widehat{f}\in\mathcal{M}(\mathbb{R}^{n})\}
\end{equation}
whose norm is given by $\|f\|_\mathcal I=\|\widehat{f}\|_{\mathcal M}$. We also obtain a similar result in the periodic case (see Section 7).
\section{The one-center $\delta$-interaction in one dimension }
In this subsection for convenience of the reader we establish initially a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop}
-\Delta_{\sigma}=-\frac{d^2}{dx^2}+\sigma \delta,
\end{equation}
defined on functions on the real line. The parameter $\sigma$ represents the coupling constant or strength attached to the point source located at $x=0$. We note that there are many approaches for studying the operator in (\ref{deltaop}), for instance, by the use of quadratic forms or by the self-adjoint extensions of symmetric operators. We also note that the quantum mechanics model in (\ref{deltaop}) has been studied into a more general framework when it is associated with the Kronig-Penney model in solid state physics (see Chapter III.2 in Albeverio {\it et al.} \cite{AGHH}) or when it is associated to singular rank one perturbations (Albeverio {\it et al.} \cite{ak}).
By following \cite{ak}, we consider the operator $A=-\frac{d^2}{dx^2}$ with domain $D(A)= H^2(\mathbb R)$ and the (closeable) symmetric restriction $A^0\equiv A|_{D(A^0)}$ with dense domain $D(A^0)=\{\psi\in D(A) : (\delta, \psi)\equiv\psi(0)=0\}$. Then we obtain that the {\it deficiency subspaces} of $A^0$,
\begin{equation}\label{sa1}
\mathcal D_+=\text{Ker}({A^0}^*-i),\quad{\rm{and}}\quad\mathcal D_-=\text{Ker}({A^0}^*+i),
\end{equation}
have dimension ({\it deficiency indexes}) equal to $1$. It is no difficult to see that these subspaces are generated, respectively, by $g_{+i}\equiv (A-i)^{-1}\delta$ and $g_{-i}\equiv (A+i)^{-1}\delta$, called {\it deficiency elements} and given explicitly by (see \cite{ak}),
\begin{equation}\label{def}
g_{\pm i}(x)=\frac{i}{2\sqrt{\pm i}}e^{i\sqrt{\pm i} |x|},\qquad Im \sqrt{\pm i} >0.
\end{equation}
We note that the Fourier transform of $g_{\pm i}$ are given by $\widehat{g_{\pm i}}(\xi)=\frac{1}{\xi^2\mp i}$
Next we present explicitly all the self-adjoint extensions of the symmetric operator $A^0$, which will be parameterized by the strength $\sigma$. By normalizing the deficiency elements $\widetilde{g}_{\pm i}=\frac{g_{\pm i}}{\|g_{\pm i\|}}$ and for convenience of notation we will continue to use $g_{\pm i}$, we have from the von Neumann's theory of self-adjoint extensions for symmetric operators (see \cite{RS}) that all the closed symmetric extensions of $A^0$ are self-adjoint and coincides with the restriction of the operator ${A^0}^*$. Moreover, for $\theta\in[0,2\pi)$ the self-adjoint extension $A^0(\theta)$ of $A^0$ is defined as follows:
\begin{equation}\label{sa4}
\left\{
\begin{aligned}
&D(A^0(\theta))=\{\psi+\lambda g_i+\lambda e^{i\theta}g_{-i} :\psi\in D(A^0), \lambda\in \mathbb C\},\\
&A^0(\theta)(\psi+\lambda g_i+\lambda e^{i\theta}g_{-i})={A^0}^*(\psi+\lambda g_i+\lambda e^{i\theta}g_{-i})=A^0\psi+i\lambda g_i-i\lambda e^{i\theta}g_{-i}.
\end{aligned}\right.
\end{equation}
Thus from \eqref{sa4} and \eqref{def} we obtain that for $\zeta\in D(A^0(\theta))$, in the form $\zeta=\psi+\lambda g_i+\lambda e^{i\theta}g_{-i}$, we have the basic expression
\begin{equation}\label{sa5}
\zeta'(0+)-\zeta'(0-)=-\lambda(1+e^{i\theta}).
\end{equation}
Next we find $\sigma$ such that $\sigma\zeta(0)=-\lambda(1+e^{i\theta})$. Indeed, $\sigma$ is given by the formula
\begin{equation}\label{sa6}
\sigma(\theta)=\frac{-2\cos(\theta/2)}{cos(\frac{\theta}{2}-\frac{\pi}{4})}.
\end{equation}
So, from now on we parameterize all self-adjoint extensions of $A^0$ with the help of $\sigma$. Thus we get:
\begin{theorem}\label{self} All self-adjoint extensions of $A^0$ are given for $-\infty<\sigma\leqq +\infty$ by
\begin{equation}\label{sa8}
\left\{
\begin{aligned}
-\Delta_{\sigma}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\sigma})&=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{0\}): \zeta'(0+)-\zeta'(0-)=\sigma \zeta(0)\}.
\end{aligned}\right.
\end{equation}
The special case $\sigma=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\sigma=+\infty$ yields a Dirichlet boundary condition at zero,
\begin{equation}\label{sa9a}
D(-\Delta_{+\infty})=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{0\}): \zeta(0)=0\}.
\end{equation}
\end{theorem}
\textbf{Proof.} By the arguments sketched above we obtain easily that $A^0(\theta)\subset -\Delta_{\sigma}$,
with $\sigma=\sigma(\theta)$ given in \eqref{sa6}. But $-\Delta_{\sigma}$ is symmetric in the corresponding domain $D(-\Delta_{\sigma})$ for all $-\infty<\sigma\leqq +\infty$, which implies the relation $A^0(\theta)\subset -\Delta_{\sigma}\subset (-\Delta_{\sigma})^* \subset A^0(\theta)$.
It completes the proof of the Theorem.
\fin
Next, we recall the basic spectral properties of $-\Delta_\sigma$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol5b} Let $-\infty<\sigma\leqq +\infty$. Then the essential spectrum of $-\Delta_\sigma$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\sigma)=[0,+\infty)$.
If $-\infty<\sigma<0$, $-\Delta_\sigma$ has exactly one negative, simple eigenvalue, i.e., its discrete spectrum $\Sigma_{dis}(-\Delta_\sigma)$ is $\Sigma_{dis}(-\Delta_\sigma)=\{{-\sigma^2/4}\}$, with a strictly (normalized) eigenfunction
$$
\Psi_\sigma(x)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{2}|x|}.
$$
If $\sigma\geqq 0$ or $\sigma=+\infty$, $-\Delta_\sigma$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\sigma)=\emptyset$.
\end{theorem}
\section{Two symmetric $\delta$-interaction in one dimension }
The one-dimensional Schr\"odinger operator with two symmetric delta interactions of strength $\alpha$ and placed at the point $\pm a$ is given formally by the linear differential operator
\begin{equation}\label{deltaop1}
-\Delta_{\alpha}=-\frac{d^2}{dx^2}+\alpha( \delta(\cdot-a)+\delta(\cdot+a)),
\end{equation}
defined on functions on the real line. By using the same notations as in last section, the symmetric operator $A^1=A|_{D(A^1)}$ with dense domain
$$
D(A^1)=\{\psi\in H^2(\mathbb R): \psi(\pm a)=0\},
$$
has deficiency indices (2, 2), and so from the Von Neumann-Krein theory we have that all self-adjoint extensions of $A^1$ are given by a four-parameter family of self-adjoint operators. Here we restrict to the case of so-called separated boundary conditions at each point $\pm a$. More specifically, we have the following theorem (see \cite{ABD}).
\begin{theorem}\label{self3} There is a family of self-adjoint extensions of $A^1$ given for $-\infty<\alpha\leqq +\infty$ by
\begin{equation}\label{sa8}
\left\{
\begin{aligned}
-\Delta_{\alpha}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\alpha})&=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{\pm a\}): \zeta'(\pm a+)-\zeta'(\pm a-)=\alpha \zeta(\pm a)\},
\end{aligned}\right.
\end{equation}
The special case $\alpha=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\alpha=+\infty$ yields a Dirichlet boundary condition at the point $\pm a$,
\begin{equation}\label{sa9a}
D(-\Delta_{+\infty})=\{\zeta\in H^1(\mathbb R)\cap H^2(\mathbb R-\{\pm a\}): \zeta( \pm a+)=\zeta( \pm a-)=0\}.
\end{equation}
\end{theorem}
Next, we establish the basic spectral properties of $-\Delta_\alpha$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol2de} Let $-\infty<\alpha\leqq +\infty$. Then the essential spectrum of $-\Delta_\alpha$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\alpha)=[0,+\infty)$.
I) If $-\infty<\alpha<0$, then the discrete spectrum of $-\Delta_\alpha$, $\Sigma_{dis}(-\Delta_\alpha)$, consists of negative eigenvalues $\gamma$ given by the implicit equation
$$
(-2i \eta +\alpha)^2=\alpha^2e^{4i\eta a},\quad \eta=\sqrt{\gamma},\;\; Im \;\eta>0.
$$
Moreover, we have that:
\begin{enumerate}
\item[1)] if $a\leqq -\frac{1}{\alpha}$, then $\Sigma_{dis}(-\Delta_\alpha)=\{\gamma_1(a,\alpha)\}$, where $\gamma_1(a,\alpha)$ is defined by
$$
\gamma_1(a,\alpha)=-\frac{1}{4a^2}[W(-a\alpha e^{a\alpha})-a\alpha]^2,
$$
where $W(\cdot)$ is the Lambert special function (or product logarithm) defined by the equation $W(x)e^{W(x)}=x$.
\item[2)] if $a> -\frac{1}{\alpha}$, then $\Sigma_{dis}(-\Delta_\alpha)=\{\gamma_1(a,\alpha), \gamma_2(a,\alpha)\}$, where $\gamma_2(a,\alpha)$ is defined by
$$
\gamma_2(a,\alpha)=-\frac{1}{4a^2}[W(a\alpha e^{a\alpha})-a\alpha]^2.$$
\end{enumerate}
II) If $\alpha\geqq 0$ or $\alpha=+\infty$, $-\Delta_\alpha$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\alpha)=\emptyset$.
\end{theorem}
\section{The $\delta'$-interaction in one dimension }
In this subsection for convenience of the reader we establish a precise formulation for the point interaction determined by the formal linear differential operator
\begin{equation}\label{deltaop1}
-\Delta_{\beta}=-\frac{d^2}{dx^2}+\beta \delta',
\end{equation}
defined on functions on the real line. The parameter $\beta$ represents the coupling constant or strength attached to the point source located at $x=0$ and $ \delta'$ is the derivative of the $\delta$. By following Albeverio {\it et al.} \cite{AGHH}-\cite{ak}, the elements in the domain of the operator $-\Delta_{\beta}$ are characterized by suitable bilateral singular boundary conditions at the singularity (see (\ref{bc1})), while the real true action coincides with the laplacian out the singularity. At variance with the $\delta$ interaction, whose domain is contained in $H^1(\mathbb R)\cap H^2(\mathbb R-\{0\})$ (in particular in a continuous function set, see (\ref{sa8})), the latter has a domain contained only in $H^2(\mathbb R-\{0\})$ and so by allowing discontinuities of the elements at the position of the defect. More precisely, for $A^2=A|_{D(A^2)}$ being considered with dense domain
$$
D(A^2)=\{\psi\in H^2(\mathbb R): \psi(0)=\psi'(0)=0\}.
$$
$A^2$ has deficiency indices $(2,2)$ and hence it has a four-parameter family of self-adjoint. We are interested in the following one-parameter family of self-adjoint extensions (see \cite{AGHH}-\cite{ak}).
\begin{theorem}\label{selfd} There is a family of self-adjoint extensions of $A^2$ given for $-\infty<\beta\leqq +\infty$ by
\begin{equation}\label{sa8d}
\left\{
\begin{aligned}
-\Delta_{\beta}&=-\frac{d^2}{dx^2}\\
D(-\Delta_{\beta})&=\{\zeta\in H^2(\mathbb R-\{0\}): \zeta'(0+)=\zeta'(0-), \zeta(0+)-\zeta(0-)=\beta\zeta'(0-)\},
\end{aligned}\right.
\end{equation}
The special case $\beta=0$ just leads to the operator $-\Delta$ in $L^2(\mathbb R)$,
\begin{equation}\label{sa9d}
-\Delta=-\frac{d^2}{dx^2},\qquad D(-\Delta)= H^2(\mathbb R),
\end{equation}
whereas the case $\beta=+\infty$ yields a Neumann boundary condition at zero and decouples $(-\infty, 0)$ and $(0,+\infty)$, i.e.,
\begin{equation}\label{sa9d}
D(-\Delta_{+\infty})=\{\zeta\in H^2(\mathbb R-\{0\}): \zeta'(0+)=\zeta'(0-)=0\}.
\end{equation}
\end{theorem}
Note that the functions in the domain of $\delta'$ have a jump at the origin, and the left and right derivatives coincide. Next, we establish the basic spectral properties of $-\Delta_\beta$ which will be relevant for our results (see \cite{AGHH}).
\begin{theorem}\label{resol5d} Let $-\infty<\beta\leqq +\infty$. Then the essential spectrum of $-\Delta_\beta$ is the nonnegative real axis, $\Sigma_{ess}(-\Delta_\beta)=[0,+\infty)$.
If $-\infty<\beta<0$, $-\Delta_\beta$ has exactly one negative simple eigenvalue, i.e., its discrete spectrum $\Sigma_{dis}(-\Delta_\beta)$ is $\Sigma_{dis}(-\Delta_\beta)=\{-\frac{4}{\beta^2}\}$, with a (normalized) eigenfunction
$$
\Phi_\beta(x)=\Big(-\frac{2}{\beta}\Big)^{\frac12}\text{sign}(x) e^{\frac{2}{\beta}|x|}.
$$
If $\beta\geqq 0$ or $\beta=+\infty$, $-\Delta_\beta$ has not discrete spectrum, $\Sigma_{dis}(-\Delta_\beta)=\emptyset$.
\end{theorem}
\subsection{The linear propagator }
\subsubsection{The case $\mu(x)=\sigma \delta$.}
Next we determine the linear propagator $G_\sigma=e^{i(\Delta+\sigma\delta)t}$(unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\sigma\delta)u\equiv H_\sigma u, \label{SCH1}\\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $H_\sigma=-\Delta_{-\sigma}$.
We will use the representation of the propagator in terms of the eigenfunctions (associated to discrete eigenvalues) and generalized eigenfunctions (see Iorio \cite{Io}, Holmer {\it et al.} \cite{Holmer5} and Duch\^ene {\it et al.} \cite{DMW}). Indeed, the family of generalized eigenfunctions $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ will be such that satisfy
\begin{equation}
\left\{
\begin{aligned}\label{eigengen}
&H_\sigma \psi_\lambda=\lambda^2 \psi_\lambda,\qquad \psi_\lambda\;\text{continuous and}\\\
& \psi'_\lambda(0+)-\psi'_\lambda(0-)=\sigma\psi_\lambda(0).
\end{aligned}\right.
\end{equation}
Hence we obtain the following family of special solutions, $e_{\pm}(x,\lambda)$ to (\ref{eigengen}), as follows
\begin{equation}\label{efun}
e_{\pm}(x,\lambda)=t_\sigma(\lambda)e^{\pm i\lambda x}\chi^0_{\pm}+(e^{\pm i\lambda x}+r_\sigma(\lambda)e^{\mp i\lambda x})\chi^0_{\mp},
\end{equation}
where $\chi^0_{+}$ is the characteristic function of $[0,+\infty)$ and $\chi^0_{-}$ is the characteristic function of $(-\infty, 0]$. $t_\sigma$ and $r_\sigma$ are the transmission and reflection coefficients:
\begin{equation}\label{tr}
t_\sigma(\lambda)=\frac{2i\lambda}{2i\lambda-\sigma},\quad r_\sigma(\lambda)=\frac{\sigma}{2i\lambda-\sigma}.
\end{equation}
They satisfy the following two equations:
\begin{equation}\label{tr2}
| t_\sigma(\lambda)|^2+| r_\sigma(\lambda)|^2=1,\quad r_\sigma(\lambda)+1=t_\sigma(\lambda).
\end{equation}
Next, by defining the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ as
$$
\psi_\lambda (x)=\left\{
\begin{aligned}
& e_+(x,\lambda)\;\; \;\;for \;\; \lambda\geqq 0\\
&e_-(x, -\lambda)\;\; for \;\; \lambda < 0
\end{aligned} \right.
$$
we obtain from Theorem \ref{resol5b} the following relations (see \cite{DMW}, \cite{Io});
\begin{enumerate}\label{orthorela}
\item[1)] $\int_{\mathbb R} \Psi_\sigma(x) \overline{\psi_\lambda(x)}dx=0$, \;\; for all $\lambda \in \mathbb R$ and $ \sigma <0$,\\
\item[2)] $ \int_{\mathbb R} \psi_\mu(x) \overline{\psi_\lambda (x)}dx=\delta (\lambda-\mu)$, \;\; for all $\mu, \lambda \in\mathbb R$,\\
\item[3)] $\Psi_\sigma(x)\Psi_\sigma(y)+ \int_{\mathbb R} \psi_\lambda(x) \overline{\psi_\lambda (y)}d\lambda=\delta (x-y)$, \;\; $ \lambda \in \mathbb R$, $\sigma <0$.
\end{enumerate}
We recall that the relation 3) above, called the {\it completeness relations}, in the case $\sigma >0$ is reads as $\int_{\mathbb R} \psi_\lambda(x) \overline{\psi_\lambda}(y)d\lambda=\delta (x-y)$ (the proof of 3) for the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ can be showed by following the ideas in the proof of Theorem \ref{expli} below). Moreover, the family $\{\psi_\lambda\}_{\lambda\in \mathbb R}$ allows us to define the {\it generalized Fourier transform}
\begin{equation}
\mathcal F (f)(\lambda)=\int_{\mathbb R} f(x)\overline{\psi_\lambda (x)}dx,
\end{equation}
and its formal adjoint $\mathcal G (g)(x)=\int_{\mathbb R} \psi_\lambda (x) g(\lambda)d\lambda$. Hence, from 2) we obtain immediately that $\mathcal G$ is the inverse Fourier transform, namely,
$$
f(\lambda)=f\ast \delta (\lambda)=\int_{\mathbb R} \overline{\psi_\lambda (x)}\int_{\mathbb R} f(\mu)\psi_\mu (x)d\mu dx=\mathcal F(\mathcal G g)(\lambda).
$$
Moreover, from the completeness relations 3) we obtain for every $f\in L^2(\mathbb R)$ the following (orthogonal) expansion in eigenfunctions of $H_\sigma$,
\begin{equation}
f= \langle f, \Phi_\sigma\rangle \Phi_\sigma + \int_{\mathbb R} \mathcal F( f)(\lambda)\psi_\lambda(x)d\lambda.
\end{equation}
Thus for $u\in C(\mathbb R; L^2(\mathbb R))$ being a solution of (\ref{SCH1}), the method of separation of variables implies that
\begin{equation}\label{solu1}
u(x,t)=e^{-it H_\sigma} u_0(x)=
e^{i\frac{\sigma^2}{4}t} \langle u_0, \Phi_\sigma\rangle \Phi_\sigma (x) +\int_{\mathbb R}e^{-i\lambda^2 t}\mathcal F (u_0)(\lambda)\psi_\lambda (x)d\lambda.
\end{equation}
In the next Theorem we describe explicitly the propagator $e^{-it H_\sigma}$ in terms of the free propagator of the Schr\"odinger equation $e^{it\Delta }$ (see Holmer {\it et al.} \cite{Holmer5}, Datchev\&Holmer \cite{Holmer3}). In the Appendix we present a different proof based in the fundamental solution associated to (\ref{SCH1}).
\begin{theorem} \label{expli} Suppose that $ \phi\in L^1(\mathbb R)$ with $supp\; \phi \subset (-\infty, 0]$. Then,
\begin{enumerate}
\item[1)] Para $\sigma \geqq 0$, we have
\begin{equation}\label{pospro}
e^{-it H_\sigma} \phi(x)= e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{-},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x).
$$
\item[ 2)] Para $\sigma < 0$, we have
\begin{equation}\label{negpro}
e^{-it H_\sigma} \phi(x)= e^{i\frac{\sigma^2}{4} t} P\phi (x)+ e^{it \Delta} (\phi\ast \tau_\sigma) (x) \chi^0_{+} + \Big[e^{it \Delta} \phi(x) + e^{it \Delta} (\phi\ast \rho_\sigma) (-x) \Big ]\chi^0_{-}
\end{equation}
where
$$
\rho_\sigma(x)=\frac{\sigma}{2} e^{\frac{\sigma}{2} x}\chi^0_{+},\;\; \tau_\sigma(x)=\delta (x)+ \rho_\sigma(x),
$$
and $P$ is the $L^2(\mathbb R)$-orthogonal projection onto the eigenfunction $\Phi_\sigma$, $P\phi=\langle \phi, \Phi_\sigma\rangle \Phi_\sigma $.
\end{enumerate}
\end{theorem}
\begin{remark} We observe the following:
\begin{itemize}
\item[1) ] The Fourier transform de $\rho_\sigma$ for every sign of $\sigma$ is given by $\widehat{\rho_\sigma}(\lambda)=r_\sigma(\lambda)$ and so $\widehat{\tau_\sigma}(\lambda)=1+r_\sigma(\lambda)=t_\sigma(\lambda)$.
\item[2) ] Formula (\ref{pospro}) and (\ref{negpro}) will allow us to estimate the operator norm of $e^{-it H_\sigma}$ using $e^{it \Delta}$.
\end{itemize}
\end{remark}
\textbf{Proof.} We only consider the case $\sigma\geqq 0$. From (\ref{solu1}), without the term of projection, we have from the definition of the family $\{\psi_\lambda\}$ and a change of variable that
\begin{equation}\label{grupo1}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}\phi(y)\int_{0}^\infty e^{-it\lambda^2}\Big (e_+(x,\lambda)\overline{e_+(y,\lambda)}+e_-(x,\lambda)\overline{e_-(y,\lambda)}\Big)d\lambda dy.
\end{equation}
Next, we compute first
\begin{equation}
\begin{aligned}
& \int_{\mathbb R}\phi(y)\overline{e_+(y,\lambda)}dy=\int_{-\infty}^0 \phi(y)e^{-i\lambda y}dy +\overline{r_\sigma(\lambda)}\int_{-\infty}^0 \phi(y)e^{i\lambda y}dy=\widehat{\phi}(\lambda)+r_\sigma(-\lambda)\widehat{\phi}(-\lambda),\\
&\qquad\qquad\qquad \qquad\int_{\mathbb R}\phi(y)\overline{e_-(y,\lambda)}dy=t_\sigma(-\lambda)\widehat{\phi}(-\lambda),
\end{aligned}
\end{equation}
so for $x>0$ we have from (\ref{grupo1}) and the fact $r_\sigma(-\lambda)t_\sigma(\lambda)+r_\sigma(\lambda)t_\sigma(-\lambda)=0$ that
\begin{equation}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}e^{-it\lambda^2}t_\sigma(\lambda)\widehat{\phi}(\lambda)e^{i\lambda x}d\lambda=e^{it \Delta}(\tau_\sigma\ast\phi)(x),
\end{equation}
where $\widehat{\tau_\sigma}(\lambda)=t_\sigma(\lambda)$. Similarly, since $r_\sigma(-\lambda)r_\sigma(\lambda)+t_\sigma(-\lambda)t_\sigma(\lambda)=1$, we have for $x<0$
\begin{equation}
e^{-it H_\sigma} \phi(x)=\int_{\mathbb R}e^{-it\lambda^2}(\widehat{\phi}(\lambda)e^{i\lambda x}+r_\sigma(\lambda)\widehat{\phi}(\lambda)e^{-i\lambda x})d\lambda=e^{it \Delta}\phi(x)+e^{it \Delta}(\rho_\sigma\ast\phi)(-x),
\end{equation}
where $\widehat{\rho_\sigma}(\lambda)=r_\sigma(\lambda)$.
\fin
\subsubsection{The case $\mu(x)=\alpha (\delta(\cdot-a)+\delta(\cdot+a))$.}
Next we determine the linear propagator $M_\alpha(t)=e^{-itU_\alpha} $ (unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}\label{beta3}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\alpha (\delta(\cdot-a)+\delta(\cdot+a))u\equiv U_\alpha u, \\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $U_\alpha=-\Delta_{-\alpha}$.
We will use the fundamental solution $F_\alpha(x,y;t)$ to the Schr\"odinger equation (\ref{beta}) for obtaining the propagator (unitary group). Then we have the representation
\begin{equation}\label{uni}
e^{-itU_\alpha}f(x)=\int_{\mathbb R} F_\alpha(x,y;t)f(y)dy.
\end{equation}
Indeed, from \cite{ABD} and \cite{KS} we have for $S(x;t)$ denoting the free propagator in $\mathbb R$, i.e.
\begin{equation}\label{free}
S(x,t)= \frac{e^{-x^2/{4it}}}{(4i\pi t)^{1/2}},\quad t>0
\end{equation}
and so $e^{it\Delta}f(x)=S(x;t)\ast_x f(x)$, the following expression for $a\alpha \neq -1$:
\begin{enumerate}
\item[1)] For $\alpha>0$
\begin{equation}\label{ker51}
F_\alpha(x,y;t)=S(x-y;t)- \frac{1}{2\pi i}\int_{\mathbb R}e^{-i\xi ^2t}\frac{f_\alpha(x,y;\xi)}{(2\xi+i\alpha)^2+\alpha^2e^{i4\xi a}}d\xi
\end{equation}
with $f_\alpha(x,y;\xi)=\sum_{j=1}^4 L_\alpha^j(x,y;\xi)$ and
\begin{align}
L_\alpha^1(x,y;\xi)&=-\alpha(2\xi+i\alpha)e^{i\xi|x+a|}e^{i\xi|y+a|},\qquad L_\alpha^4(x,y;\xi)=L_\alpha^1(-x,-y;\xi)\\
L_\alpha^2(x,y;\xi)&=i\alpha^2e^{2i\xi a}e^{i\xi|x+a|}e^{i\xi|y-a|},\qquad\;\;\;\;\;\;\; L_\alpha^3(x,y;\xi)=L_\alpha^2(-x,-y;\xi).
\end{align}
\item[2)] For $\alpha<0$,
\begin{equation}\label{ker6}
F_\alpha(x,y;t)=e^{-it\gamma_1} \Gamma_1(x) \Gamma_1(y)+e^{-it\gamma_2} \Gamma_2(x) \Gamma_2(y) +F_{-\alpha}(x,y;t)
\end{equation}
where $ \Gamma_1$ and $ \Gamma_2$ are the normalized eigenfunction associated with the eigenvalues $\gamma_1$ and $\gamma_2$.
\end{enumerate}
\begin{remark} We observe the following:
\begin{itemize}
\item[1) ] The case $a\alpha=-1$ is assumed for technical reasons (see \cite{KS})
\item[2) ] For $a\alpha=-1$ we obtain that only $\gamma_1$ remains as an eigenvalue in the discrete spectrum.
\end{itemize}
\end{remark}
\subsubsection{The case $\mu(x)=\beta \delta'$.}
Next we determine the linear propagator $J_\beta(t)=e^{i(\Delta+\beta\delta')t}$ (unitary group) determined by the linear system associated with (\ref{SCH0}),
\begin{equation}\label{beta}
\left \{
\begin{aligned}
i\partial_{t}u&=-(\Delta u+\beta\delta')u\equiv K_\beta u, \\
u(0) & =u_{0},
\end{aligned} \right.
\end{equation}
where we are using the notation $K_\beta=-\Delta_{-\beta}$.
We will use the fundamental solution $S_\beta(x,y;t)$ to the Schr\"odinger equation (\ref{beta}) for obtaining the propagator (unitary group). Then we have the representation
\begin{equation}\label{uni}
e^{-itK_\beta}f(x)=\int_{\mathbb R} S_\beta(x,y;t)f(y)dy.
\end{equation}
Indeed, from \cite{AGHH} we have for $S(x;t)$ denoting the free propagator in (\ref{free}) the following:
\begin{enumerate}
\item[1)] For $\beta>0$
\begin{equation}\label{ker5}
S_\beta(x,y;t)=S(x-y;t)+\text{sgn}(xy) S(|x|+|y|;t)+ \frac{2}{\beta}\int_0^\infty \text{sgn}(xy)e^{-\frac{2}{\beta} s} S(s+|x|+|y|;t)ds
\end{equation}
\item[2)] For $\beta<0$,
\begin{equation}\label{ker6}
\begin{aligned}
S_\beta(x,y;t)=&S(x-y;t)+\text{sgn}(xy) S(|x|+|y|;t)+ e^{i\frac{4}{\beta^2} t} \Phi_\beta(x) \Phi_\beta(y)\\
&-\frac{2}{\beta} \int_0^\infty \text{sgn}(xy)e^{\frac{2}{\beta} s} S(s-|x|-|y|;t)ds
\end{aligned}
\end{equation}
where $ \Phi_\beta$ is defined in Theorem \ref{resol5d}.
\end{enumerate}
\subsubsection{Dispersive Estimates}
The following proposition extends the well known estimates for the free propagator $e^{it\Delta}$,
\begin{equation}\label{stric0}
\left\Vert e^{it\Delta}f(t)\right\Vert _{p^{\prime}}\leq C_0t^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
to the the groups $G_\sigma(t)=e^{-it H_\sigma}$, $M_\alpha(t)=e^{-it U_\alpha}$ and $J_\beta(t)=e^{i(\Delta+\beta\delta')t} $ in the one-dimensional case. We denote by $W_1$ the group $G_\sigma$, $W_2$ the group $M_\alpha$ and by $W_3$ the group $J_\beta$ .
\begin{proposition} \label{strich}
Let $p\in [1, 2]$ and $p'$ be such that $\frac{1}{p}+\frac{1}{p'}=1$. Then we have:
Suppose that $u(x,t)=W_i(t)f(x)$, $i=1,2,3$, is the solution of the linear equation (\ref{SCH1}), (\ref{beta3}) and (\ref{beta}), respectively. Then:
\begin{itemize}
\item[1)] for $\sigma\geqq 0$, $\alpha \geqq 0$ and $\beta\geqq 0$,
\begin{equation}\label{stric2}
\left\Vert W_i(t)f\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $C>0$ is independent of $f$ and $t\neq0$.
\item[2)] for $\sigma< 0$, $a\leqq -\frac{1}{\alpha}$ ($\alpha<0$) and $\beta<0$,
\begin{equation}\label{stric3}
\left\Vert W_i(t)f-e^{i\alpha_i t} P_if\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $\alpha_1=\frac{\sigma^2}{4}$, $P_1f= \langle f, \Psi_\sigma\rangle \Psi_\sigma$, $\alpha_2=-\gamma_1(a,\alpha)$, $P_2f= \langle f, \Gamma_1\rangle \Gamma_1$ and $\alpha_3=\frac{4}{\beta^2}$, $P_3f= \langle f, \Phi_\beta\rangle \Phi_\beta$, and $C>0$ is independent of $f$ and $t\neq0$.
\item[3)] for $\alpha<0$, $a> -\frac{1}{\alpha}$,
\begin{equation}\label{stric4}
\left\Vert W_2(t)f-\sum_{j=1}^2e^{-i\gamma_j t} Q_jf\right\Vert _{p^{\prime}}\leq C|t|^{-\frac{1}{2}
(\frac{2}{p}-1)}\left\Vert f\right\Vert _{p},
\end{equation}
where $\gamma_j=\gamma_j(a,\alpha)$, $j=1,2$,
$Q_1f= \langle f, \Gamma_1\rangle \Gamma_1$, $Q_2f= \langle f, \Gamma_2\rangle \Gamma_2$, and $C>0$ is independent of $f$ and $t\neq0$.
\end{itemize}
\end{proposition}
\textbf{Proof.} i) We consider $\sigma>0$. Initially $G_{\sigma}(t)$ is a unitary group on $L^2(\Bbb R)$, $\|G_{\sigma}\phi(t)\|_2=\|\phi\|_2$ for all $t\in \mathbb R$. Let $\phi\in L^1(\mathbb R)$ and $R\phi(x)=\phi(-x)$. Then for $\phi^-=\phi \chi^0_-$ and $\phi^+=\phi R\chi^0_+$ we have the decomposition $\phi=\phi^-+R\phi^+$. Hence since $supp\; \phi^+\subset (-\infty, 0]$, $RG_{\sigma}=G_{\sigma}R$ and $R(f\ast Rg)=(Rf)\ast g$
we obtain from the following equality,
\begin{align}
G_{\sigma}\phi(t)&=[e^{it\Delta}\phi^-+e^{it\Delta}(\phi^-\ast \rho_\sigma)]\chi^0_-+e^{it\Delta}(\phi^-\ast \tau_\sigma)\chi^0_+\\
&+[e^{it\Delta}R\phi^+ +e^{it\Delta}(\phi^+\ast \rho_\sigma)]\chi^0_ + +e^{it\Delta}(R\phi^+\ast R\tau_\sigma)\chi^0_-.
\end{align}
Therefore from (\ref{stric0}) and applying Young's inequality we obtain for $t\neq 0$
\begin{align}
\|G_{\sigma}\phi(t)\|_\infty&\leqq \frac{C_0}{\sqrt{|t|}}(\|\phi^-\|_1+\|\phi^-\ast \rho_\sigma\|_1+\|\phi^-\ast \tau_\sigma\|_1+\|R\phi^+\|_1+\|\phi^+\ast \rho_\sigma\|_1 +\|R\phi^+\ast R\tau_\sigma\|_1)\\
& \leqq \frac{C_0}{\sqrt{|t|}}(3+4\|\rho_\sigma\|_1)\|\phi\|_1=\frac{C}{\sqrt{|t|}}\|\phi\|_1
\end{align}
where $C=C(\sigma)$. By the Riesz-Thorin interpolation theorem we obtain (\ref{stric2}). The case $\sigma<0$ follows similarly from the expression (\ref{negpro}).
ii) Let $\beta>0$. From (\ref{ker5}) we obtain immediate for $x,y\in \mathbb R$ that
$$
|S_\beta(x,y,t)|\leqq C|t|^{-\frac12}.
$$
So, from (\ref{uni}) we have $\|J_\beta(t)\phi\|_\infty\leqq \frac{C}{\sqrt{|t|}}\|\phi\|_1$. By following a similar analysis as in the later case we obtain (\ref{stric2}).
iii) From the representations in (\ref{negpro}) and (\ref{ker6}) we get immediately (\ref{stric3}).
iv) The case of the group $W_2$ for $a\alpha \neq -1$, it follows of the estimate
$$
\|e^{-it U_\alpha}P_c f\|_\infty\leqq C t^{-1/2}\|f\|_1,
$$
for $t>0$ and $P_c$ being the spectral projector of $U_\alpha$ on its continuous spectrum.
\fin
\section{
Weak-$L^{p}$ Solutions}
In this section we focus our study of global solutions for the Cauchy problem
\begin{equation}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+ \mu(x)u& =\lambda\left\vert u\right\vert
^{\rho-1}u,\ x\in\mathbb{R},\ t\in\mathbb{R},\label{SCHd}\\
u(x,0) & =u_{0},
\end{aligned}\right.
\end{equation}
for $\mu(x)=\sigma \delta$ in the spaces $L^{(p, \infty)}(\mathbb R)$, which are called weak-$L^p$ or Marcinkiewicz spaces. The cases $\mu(x)=\alpha(\delta(\cdot-a)+ \delta(\cdot+a))$ and $\mu(x)=\beta \delta'$ are treatment similarly.
We start by recalling some facts about the weak spaces $L^{(p, \infty)}(\mathbb R)$. For $1<r\leq
\infty,$ we recall that a measurable function $f$ defined on $\mathbb{R}$
belongs to $L^{(r,\infty)}(\mathbb{R})$ if the norm
\[
\Vert f\Vert_{(r,\infty)}=\sup_{t>0}t^{\frac{1}{r}}f^{\ast\ast}(t)
\]
is finite, where
\[
f^{\ast\ast}(t)=\frac{1}{t}\int_{0}^{t}f^{\ast}(s)\mbox{ }ds,
\]
and $ f^{\ast}$ is the {\it decreasing rearrangement} of $f$ with regard to the Lebesgue measure $\nu$, namely,
\[
f^{\ast}(t)=\inf\{s>0:\nu(\{x\in\mathbb{R}:|f(x)|>s\})\leq t\},\text{ }t>0,
\]
The space $L^{(r,\infty)}$ with the norm $\Vert f\Vert_{(r,\infty)}$ is a Banach space. We have the continuous inclusion $L^{r}(\mathbb{R})\subset$ $L^{(r,\infty)}(\mathbb{R})$. Moreover, the H\"{o}lder's inequality holds true in this framework, namely
\begin{equation}
\Vert fg\Vert_{(r,\infty)}\leq C\Vert f\Vert_{(q_{1},\infty)}\Vert
g\Vert_{(q_{2},\infty)}, \label{Holder1}
\end{equation}
for $1<q_{1},q_{2}<\infty$, $\frac{1}{q_{1}}+\frac{1}{q_{2}}<1$ and $\frac
{1}{r}=\frac{1}{q_{1}}+\frac{1}{q_{2}}$, where $C>0$ depends only on $r$. Lastly, we have the Lorentz spaces $L^{(p,q)}(\mathbb R)$ that can be constructed via real interpolation; indeed, $L^{(p,q)}(\mathbb R)=(L^{1}(\mathbb R),L^{\infty}(\mathbb R))_{1-\frac{1}{p},q}$, $1<p<\infty$. They have the interpolation property
\begin{equation}\label{inter}
(L^{(p_0, q_0)}(\mathbb R), L^{(p_1, q_1)}(\mathbb R))_{\theta, q}=L^{(p,q)}(\mathbb R),
\end{equation}
provided $0<p_0<p_1<\infty$, $0<\theta<1$, $\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}$, $1\leqq q_0, q_1, q\leqq \infty$, where $(\cdot,\cdot)_{\theta, q}$ stands for the real interpolation spaces constructed via the $K$-method. For
further details about weak-$L^{r}$ and Lorentz spaces see \cite{BL} and Grafakos \cite{Gra}.
From (\ref{inter}) we obtain our main estimate for the group $G_\sigma$ in Lorentz spaces. A similar result is obtained for the groups $M_\alpha$ and $J_\beta$.
\begin{lemma}\label{grupint1}
Let $1\leqq d\leqq \infty$, $p' \in (2,\infty)$, and $p\in (1,2)$. If $p'$ satisfies $\frac{1}{p}+\frac{1}{p'}=1$, then there exists a constant $C>0$ such that:
\begin{enumerate}
\item[1)] for $\sigma\geqq 0$,
\begin{equation}\label{grupint2}
\left\Vert G_{\sigma}(t)f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac
{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p, d)},
\end{equation}
for all $f\in L^{(p,d)}(\mathbb R)$ and all $t\neq 0$.
\item[2)] for $\sigma< 0$,
\begin{equation}\label{grupint3}
\left\Vert G_{\sigma}(t)f-e^{i\frac{\sigma^2}{4}t} P_1f\right\Vert _{(p^{\prime},d)}\leq C|t|^{-\frac{1}{2}(\frac{2}{p}-1)}\left\Vert f\right\Vert _{(p, d)},
\end{equation}
for all $f\in L^{(p,d)}(\mathbb R)$ and all $t\neq 0$.
\end{enumerate}
\end{lemma}
\textbf{Proof.} We only consider the case $\sigma\geqq 0$ because for $\sigma<0$ the analysis is similar. Let fixed $t\neq 0$ and let $1<p_0<p<p_1<2$. From the $L^p=L^{(p,p)}$ estimate of the Schr\" odinger group in Proposition \ref{strich}, we have that $G_\sigma (t):L^{p_0}\to L^{p_0'}$ and $G_\sigma (t):L^{p_1}\to L^{p_1'}$ satisfy
$$
\|G_\sigma (t)\|_{p_0\to p_0'}\leqq C|t|^{-\frac
{1}{2}(\frac{2}{p_0}-1)},\qquad \|G_\sigma (t)\|_{p_1\to p_1'}\leqq C|t|^{-\frac
{1}{2}(\frac{2}{p_1}-1)}
$$
with $\frac{1}{p_0}+\frac{1}{p_0'}=1$ and $ \frac{1}{p_1}+\frac{1}{p_1'}=1$. Hence, for $\lambda\in (0,1)$, $\frac{1}{p}=\frac{1-\lambda}{p_0}+\frac{\lambda}{p_1}$, and $\frac{1}{p' }=\frac{1-\lambda}{p'_0}+\frac{\lambda}{p'_1}$, we obtain from (\ref{inter}) that
$$
\|G_\sigma (t)\|_{(p,d) \to (p',d)}\leqq \|G_\sigma (t)\|_{p_0\to p_0'}^\lambda\|G_\sigma (t)\|_{p_1\to p_1'}^{(1-\lambda)}\leqq C |t|^{-\frac{1}{2}(\frac{2}{p}-1)},
$$
which gives (\ref{grupint2}).
\fin
Next we establish the main results of this section. From now on we focus in the case of $\mu(x)=\sigma\delta$ in (\ref{SCHd}), since for the case of the potential being the two symmetric deltas and the derivative of the Dirac- delta we have a similar analysis. We start by defining $\mathcal{L}_{\vartheta
}^{\infty}$ as the Banach space of all Bochner measurable functions
$u:\mathbb{R}\rightarrow$ $L^{(\rho+1,\infty)}$ endowed with the norm
\begin{equation}
\Vert u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}=\sup_{-\infty<t<\infty
}|t|^{\vartheta }\Vert u(t)\Vert_{(\rho+1,\infty)}, \label{Norma1}
\end{equation}
where
\begin{equation}
\vartheta =\frac{1}{\rho-1}-\frac{1}{2(\rho+1)}. \label{exponent1}
\end{equation}
Let us also define the initial data space $\mathcal{E}_{0}$ as the set of all
$u\in\mathcal{S}^{\prime}(\mathbb{R})$ such that the norm
\[
\Vert u_{0}\Vert_{\mathcal{E}_{0}}=\sup_{-\infty<t<\infty}|t|^{\vartheta }\Vert
G_{\sigma}(t)u_{0}\Vert_{(\rho+1,\infty)}
\]
is finite. Throughout this paper we stand for $\rho_{0}=\frac
{3+\sqrt{17}}{2}>1$ the positive root of the equation $\rho
^{2}-3\rho-2=0$.
From Duhamel's principle, (\ref{SCHd}) is formally equivalent to
the integral equation
\begin{equation}
u(t)=G_{\sigma}(t)u_{0}-i\lambda\int_{0}^{t}G_{\sigma}(t-s)[|u(s)|^{\rho-1}u(s)]ds,
\label{int1}
\end{equation}
where $G_{\sigma}(t)=e^{i(\Delta+\sigma\delta)t}$ is the group determined
by the linear system associated with { (\ref{SCHd})}.
\begin{definition} A mild solution of the initial value problem
(\ref{SCHd}) is a complex-valued function $u\in\mathcal{L}_{\vartheta }^{\infty}$ satisfying (\ref{int1}).
\end{definition}
Our main results of this section read as follows.
\begin{theorem}
\label{GlobalTheo} Let $\sigma\geqq 0$, $\rho_{0}<\rho<\infty$ and $u_{0}\in
\mathcal{E}_{0}.$ There is $\varepsilon>0$ such that {if }$\left\Vert
u_{0}\right\Vert _{\mathcal{E}_{0}}\leq\varepsilon${\ then the IVP
(\ref{SCHd}) has a unique global-in-time mild solution
$u\in\mathcal{L}_{\vartheta }^{\infty}$ satisfying }$\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}\leq2\varepsilon.${ } Moreover, the data-solution map
$u_{0}\mapsto u$ from $\mathcal{E}_{0}$ into $\mathcal{L}_{\vartheta }^{\infty}$
is locally Lipschitz.
\end{theorem}
\begin{remark} The proof of Theorem \ref{GlobalTheo} is based in an argument of fixed point, so by using the implicit function theorem is not difficult to show that the data-solution map
$u_{0}\mapsto u$ from $\mathcal{E}_{0}$ into $\mathcal{L}_{\vartheta }^{\infty}$
is smooth.
\end{remark}
\begin{remark}
\label{rem-local}(Local-in-time solutions) Let $1<\rho<\rho_{0}$, $d_{0}=\frac{1}{2}(\frac{\rho
-1}{\rho+1}),$ and $d_{0}<\zeta<\frac{1}{\rho}$. For $0<T<\infty$, consider
the Banach space $\mathcal{L}_{\zeta}^{T}$ of all Bochner measurable functions
$u:(-T,T)\rightarrow$ $L^{(\rho+1,\infty)}$ endowed with the norm
\[
\Vert u\Vert_{\mathcal{L}_{\zeta}^{T}}=\sup_{-T<t<T}|t|^{\zeta}\Vert
u(\cdot,t)\Vert_{(\rho+1,\infty)}.
\]
A local-in-time existence result in $\mathcal{L}_{\zeta}^{T}$ could be proved
for (\ref{SCHd}) by considering $u_{0}\in L^{(\frac{\rho+1}{\rho},\infty)}
(\mathbb{R})$ and small $T>0$ (see \cite{BFV}).
\end{remark}
In the sequel we give an asymptotic stability result for the obtained solutions.
\begin{theorem}
\label{TeoAssin}(Asymptotic Stability) Under the hypotheses of Theorem \ref{GlobalTheo} let $u$ and
$v$ be two solutions of (\ref{int1}) obtained through Theorem \ref{GlobalTheo}
with initial data ${u}_{0}$ and $v_{0},$ respectively. We have that
\begin{equation}
\lim_{\left\vert t\right\vert \rightarrow\infty}\left\vert t\right\vert
^{\vartheta }\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{(\rho+1,\infty)}=0
\label{as1}
\end{equation}
if only if
\begin{equation}
\lim_{\left\vert t\right\vert \rightarrow\infty}\left\vert t\right\vert
^{\vartheta }\left\Vert G_{\sigma}(t)(u_{0}-v_{0})\right\Vert _{(\rho+1,\infty)}=0
\label{as2}
\end{equation}
The condition (\ref{as2}) holds, in particular, for $u_{0}-v_{0}\in
L^{(\frac{\rho+1}{\rho},\infty)}.$
\end{theorem}
\subsection{
Nonlinear Estimate}
In this subsection we give the nonlinear estimate essential in the proof of Theorem \ref{GlobalTheo}. We start by recalling the Beta function
$$
B(\nu,\eta)=\int_{0}^{1}(1-s)^{\nu-1}
s^{\eta-1}ds,
$$
which is finite for all $\nu>0$ and $\eta>0.$ So, for $k_{1}
,k_{2}<1$ and $t>0,$ the change of variable $s\rightarrow st$ yields
\begin{equation}
\int_{0}^{t}(t-s)^{-k_{1}}s^{-k_{2}}ds=t^{1-k_{1}-k_{2}}\int_{0}
^{1}(1-s)^{-k_{1}}s^{-k_{2}}ds=t^{1-k_{1}-k_{2}}B(1-k_{1},1-k_{2})<\infty.
\label{Beta}
\end{equation}
Next we denote by
\begin{equation}
\mathcal{N}(u)=-i\lambda \int_{0}^{t}G_{\sigma}(t-s)[\left\vert u(s)\right\vert ^{\rho-1
}u(s)]ds \label{non1}
\end{equation}
the nonlinear part of the integral equation (\ref{int1}). We have the following estimate in order to apply a point fixed argument.
\begin{lemma}
\label{lem8}Let $\sigma\geqq 0$ and $\rho_{0}<\rho<\infty$. There is a
constant $K>0$ such that
\begin{equation}
\Vert\mathcal{N}(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\leq
K \Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}(\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}^{\rho-1}+\Vert v\Vert_{\mathcal{L}_{\vartheta }^{\infty}
}^{\rho-1}) \label{est-non1}
\end{equation}
for all $u,v\in\mathcal{L}_{\vartheta }^{\infty}$.
\end{lemma}
\textbf{Proof.} Without loss of generality, we assume $t>0.$ It
follows from (\ref{grupint2}) with $p=\frac{{\rho+1}}{\rho}$, $d=\infty$ and
H\"older inequality (\ref{Holder1}) and $\||f|^r\|_{(p, \infty)}=\|f\|^r_{(rp, \infty)}$ that
\begin{align}
\Vert\mathcal{N}(u)-\mathcal{N}(v)\Vert_{(\rho+1,\infty)} & \leq\int_{0}
^{t}\Vert G_{\sigma}(t-s)(\left\vert u\right\vert ^{\rho-1}u-\left\vert
v\right\vert ^{\rho-1}v)\Vert_{(\rho+1,\infty)}ds\nonumber\\
& \hspace{-3.3cm}\leq C\int_{0}^{t}(t-s)^{-\frac{1}{2}(\frac{2\rho}{\rho
+1}-1)}\Vert(\left\vert u-v\right\vert )(\left\vert u\right\vert ^{\rho
-1}+\left\vert v\right\vert ^{\rho-1})\Vert_{(\frac{{\rho+1}}{\rho}{,\infty)}
}ds\nonumber\\
& \hspace{-3.3cm}\leq C\int_{0}^{t}(t-s)^{-\zeta}\Vert u-v\Vert
_{(\rho+1{,\infty)}}\left( \Vert u\Vert_{(\rho+1{,\infty)}}^{\rho-1}+\Vert
v\Vert_{(\rho+1{,\infty)}}^{\rho-1}\right) ds. \label{aux2}
\end{align}
Next, notice that $\zeta=\frac{1(\rho-1)}{2(\rho+1)}<1$ and $\vartheta \rho<1$ when
$\rho_{0}<\rho<\infty.$ Thus, by using (\ref{Beta}), the r.h.s of
(\ref{aux2}) can be bounded by
\begin{align*}
& \leq C\left( \sup_{0<t<\infty}t^{\vartheta }\Vert u-v\Vert_{(\rho+1{,\infty)}
}\sup_{0<t<\infty}\left( t^{\vartheta (\rho-1)}\Vert u\Vert_{(\rho+1{,\infty)}
}^{\rho-1}+t^{\vartheta (\rho-1)}\Vert v\Vert_{(\rho+1{,\infty)}}^{\rho-1}\right)
\right) \times\int_{0}^{t}(t-s)^{-\zeta}s^{-\vartheta \rho}ds\\
& =CB(1-\zeta, 1-\vartheta \rho) t^{1-\vartheta \rho-\zeta}\left( \Vert u-v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}}(\Vert u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1}+\Vert
v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1})\right) ,
\end{align*}
which implies (\ref{est-non1}), because $\zeta+\rho\vartheta =-\vartheta -1.$ \fin
\subsection{Proof of Theorem \ref{GlobalTheo} }
Consider the map $\Phi$ defined on $\mathcal{L}_{\vartheta }^{\infty}$ by
\begin{equation}
\Phi(u)=G_{\sigma}(t)u_{0}+\mathcal{N}(u)\label{map1}
\end{equation}
where $\mathcal{N}(u)$ is given in (\ref{non1}). Let $\mathcal{B}
_{\varepsilon}=\{u\in\mathcal{L}_{\vartheta }^{\infty};\Vert u\Vert_{\mathcal{L}
_{\vartheta }^{\infty}}\leq2\varepsilon\}$ where $\varepsilon>0$ will be chosen
later. Lemma \ref{lem8} implies that
\begin{align}\label{aux}
\Vert\Phi(u)-\Phi(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & =\Vert
\mathcal{N}(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\nonumber\leq K \Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}}(\Vert u\Vert
_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho-1}+\Vert v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}}^{\rho-1})\\
& \leq2^{\rho}\varepsilon^{\rho-1}K \Vert u-v\Vert_{\mathcal{L}_{\vartheta
}^{\infty}},
\end{align}
for all $u,v\in \mathcal{B}_{\varepsilon}.$ Since
\[
\Vert G_{\sigma}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}=\Vert u_{0}
\Vert_{\mathcal{E}_{0}}\leq\varepsilon,
\]
and by using inequality (\ref{est-non1}) with $v=0$ we obtain
\begin{align}
\Vert\Phi(u)\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & \leq\Vert G_{\sigma
}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+\Vert\mathcal{N}(u)\Vert
_{\mathcal{L}_{\vartheta }^{\infty}}\nonumber \leq\Vert G_{\sigma}(t)u_{0}\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+K \Vert
u\Vert_{\mathcal{L}_{\vartheta }^{\infty}}^{\rho}\nonumber\\
& \leq\varepsilon+2^{\rho}\varepsilon^{\rho}K \leq2\varepsilon, \label{aux6}
\end{align}
provided that $2^{\rho}\varepsilon^{\rho-1}K <1$ and $u\in \mathcal{B}_{\varepsilon}.$ It follows that $\Phi:\mathcal{B}_{\varepsilon}\rightarrow \mathcal{B}_{\varepsilon}$ is a contraction, and then it has a fixed point $u\in \mathcal{B}_{\varepsilon},$ $\Phi(u)=u$, which is the unique solution for the integral equation (\ref{int1}) satisfying $\Vert u\Vert
_{\mathcal{\mathcal L}^\infty_{\vartheta }}\leq2\varepsilon.$
In view of (\ref{aux}), if $u,v$ are two integral solutions with respective
data $u_{0},v_{0}$, then
\begin{align*}
\Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}} & =\Vert G_{\sigma}
(t)(u_{0}-v_{0})\Vert_{\mathcal{L}_{\vartheta }^{\infty}}+\Vert\mathcal{N}
(u)-\mathcal{N}(v)\Vert_{\mathcal{L}_{\vartheta }^{\infty}}\\
& \leq\Vert u_{0}-v_{0}\Vert_{\mathcal{E}_{0}}+2^{\rho}\varepsilon^{\rho
-1}K\Vert u-v\Vert_{\mathcal{L}_{\vartheta }^{\infty}},
\end{align*}
which, due to $2^{\rho}\varepsilon^{\rho-1}K <1,$ yields the Lipschitz
continuity of the data-solution map \fin
\subsection{Proof of Theorem \ref{TeoAssin}}
We will only prove that (\ref{as2}) implies (\ref{as1}). The converse follows
similarly (in fact it is easier) and it is left to the reader. For that matter, we subtract the
integral equations verified by $u$ and $v$ in order to obtain
\begin{align}
t^{\vartheta }\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{(\rho+1,\infty)} &
\leq t^{\vartheta }\left\Vert G_{\sigma}(t)(u_{0}-v_{0})\right\Vert _{(\rho
+1,\infty)}+\nonumber\\
& +t^{\vartheta }\left\Vert \int_{0}^{t}G_{\sigma}(t-s)(\left\vert u\right\vert
^{\rho-1}u-\left\vert v\right\vert ^{\rho-1}v)ds\right\Vert _{(\rho+1,\infty
)}. \label{diff2}
\end{align}
Since $\Vert u\Vert_{\mathcal{L}^\infty_{\vartheta }},\Vert v\Vert_{\mathcal{L}^\infty_{\vartheta }
}\leq2\varepsilon,$ we can estimate the second term in R.H.S. of (\ref{diff2})
as follows.
\begin{align}
& I(t)=t^{\vartheta }\left\Vert\int_{0}^{t}G_{\sigma}(t-s)[\left\vert u\right\vert
^{\rho-1}u-\left\vert v\right\vert ^{\rho-1}v]ds\right\Vert_{(\rho+1,\infty
)}\nonumber\\
& \leq Ct^{\vartheta }\int_{0}^{t}(t-s)^{-\frac{1}{2}(\frac{2\rho}{\rho+1}
-1)}s^{-\vartheta \rho}s^{\vartheta }\Vert u(\cdot,s)-v(\cdot,s)\Vert_{(\rho
+1,\infty)}ds\left( \Vert u\Vert_{\mathcal{L}^\infty_{\vartheta }}^{\rho-1}+\Vert
v\Vert_{\mathcal{L}^\infty_{\vartheta }}^{\rho-1}\right) \nonumber\\
& \leq C2^{\rho}\varepsilon^{\rho-1}t^{\vartheta }\int_{0}^{t}(t-s)^{-\zeta
}s^{-\vartheta \rho}s^{\vartheta }\Vert u(\cdot,s)-v(\cdot,s)\Vert_{(\rho+1,\infty
)}ds, \label{aux9}
\end{align}
where $\zeta=\frac{1}{2}(\frac{2\rho}{\rho+1}-1).$ Now, recalling that
$\zeta+\vartheta \rho-\vartheta -1=-\vartheta ,$ the change of variable $s\longmapsto ts$
in (\ref{aux9}) leads us to
\begin{equation}
I(t)\leq C2^{\rho}\varepsilon^{\rho-1}\int_{0}^{1}(1-s)^{-\zeta}s^{-\vartheta
\rho}(ts)^{\vartheta }\Vert u(\cdot,ts)-v(\cdot,ts)\Vert_{(\rho+1,\infty)}ds.
\label{aux10}
\end{equation}
Set
\begin{equation}
L=\limsup_{t\rightarrow\infty}t^{\vartheta }\Vert u(\cdot,t)-v(\cdot
,t)\Vert_{(\rho+1,\infty)}<\infty\label{defM}
\end{equation}
and recall from proofs of Lemma \ref{lem8} and Theorem \ref{GlobalTheo} that
\[
K =C\int_{0}^{1}(1-s)^{-\zeta}s^{-\vartheta \rho
}ds\quad\text{ and }\quad 2^{\rho}\varepsilon^{\rho-1}K <1.
\]
Then, computing $\limsup_{t\rightarrow\infty}$ in (\ref{diff2}) and using
(\ref{aux10}), we get
\begin{align*}
L & \leq\left( C2^{\rho}\varepsilon^{\rho-1}\int_{0}^{1}(1-s)^{-\zeta
}s^{-\vartheta \rho}ds\right) L\\
& =2^{\rho}\varepsilon^{\rho-1}KL
\end{align*}
and therefore $L=0$, as required. \fin
\section{
Existence of a invariant manifold of periodic orbits}
It is not clear for us whether the approach applied in the proof of Theorem \ref{GlobalTheo} for the case $\mu(x)=\sigma \delta$ in (\ref{SCHd}) with $\sigma\geqq 0$ can be applied for the case $\sigma<0$. Similar situation is happening for the cases $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$ and $\mu(x)=\beta \delta'$ with $\alpha<0$ and $\beta<0$, respectively.
But, for instance, in the case $\sigma<0$ we can establish a nice qualitative behavior associated to the linear flow generated by equation (\ref{SCHd}). In fact, it follows from Theorem \ref{resol5b} that the linear part of the NLS-$\delta$ equation (\ref{SCHd}) has a two-dimensional manifold of periodic orbits, namely,
$$
E^p=\{\gamma e^{i\theta} \Phi_\sigma(x):\gamma\geqq 0\;\;\text{and}\;\;\theta\in [0, 2\pi]\}.
$$
So, the estimate (\ref{grupint3}) will imply immediately that all solutions $u(t)$ of (\ref{SCHd}) with $\lambda=0$ and with initial conditions $u_0\in L^{(p,d)}(\mathbb R)$ will approach to one of the periodic orbits $\gamma e^{i(\frac{\sigma^2}{4}t+\theta)}\Phi_\sigma\in E^p$. More exactly, we have the following theorem.
\begin{theorem} \label{manifold} Let $\sigma< 0$. For $ d\in [1,\infty]$, $p'\in [1, \infty]$ and $p\in[1,2]$, we have that for $p'$ satisfying $\frac{1}{p}+\frac{1}{p'}=1$, the solution $u(t)$ of the linear equation associated to (\ref{SCHd}) with initial data $u(0)=u_0\in L^{(p,d)}(\mathbb R)$ satisfies
$$
\lim_{t\to \pm \infty} \|u(t)-\gamma_0 e^{i(\frac{\sigma^2}{4}t+\theta)}\Phi_\sigma\|_{(p',d)}=0,
$$
for $\gamma_0=|\langle u_0, \Phi_\sigma\rangle|$ and some $\theta\in [0,2\pi]$.
\end{theorem}
\begin{remark} 1) A similar result to that in Theorem \ref{manifold} can be obtained for the linear equation associated to (\ref{SCHd}) in the case of $\mu(x)=\beta \delta'$ with $\beta<0$ and for the linear equation (\ref{beta3}) in the case of $\mu(x)=\alpha(\delta(\cdot-a)+\delta(\cdot+a))$ with $a\leqq -\frac{1}{\alpha}$ and $\alpha<0$.
2) Note that $\gamma_0<\infty$. In fact, it is not difficult to see that for $
\Psi_\sigma(x)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{2}|x|}$ we have for $s\geqq 0$ that $\Psi^*_\sigma(s)=\sqrt{\frac{-\sigma}{2}}e^{\frac{\sigma}{4}s}$. So for all $p, q\in (0,\infty)$ we obtain that
$$
\| \Psi_\sigma\|^q_{L^{(p,q)}}=\int_0^\infty \Big(t^{\frac{1}{p}}\Psi^*_\sigma(t)\Big)^q\frac{dt}{t}= \Big(\frac{-\sigma}{2}\Big)^{\frac{q}{2}}\Big(\frac{-4}{q\sigma}\Big)^{\frac{q}{p}} \Gamma \Big(\frac{q}{p}\Big),
$$
where $\Gamma$ represents the Gamma function. The case $q=\infty$ is immediate. Next, by the Hardy-Littlewood inequality for decreasing rearrangements and the H\"older inequality in the classical $L^p(d\nu)$ spaces, we obtain for $p\in [1,2]$, $p'$ such that $\frac{1}{p}+\frac{1}{p'}=1$ and for $r$ such that $\frac{1}{d}+\frac{1}{r}=1$, with $d\geqq 1$, the estimate
\begin{align*}
\beta_0\leqq \int_{\mathbb R} |u_0(x)|| \Psi_\sigma (x)|dx&\leqq \int_0^\infty u^*_0(t) \Psi^*_\sigma (t)dt=\int_0^\infty t^{\frac{1}{p}}u^*_0(t) t^{\frac{1}{p'}}\Psi^*_\sigma (t)\frac {dt}{t}\\
&\leqq \|u_0\|_{(p,d)}\|\Psi_\sigma\|_{(p',r)}<\infty.
\end{align*}
\end{remark}
\section{
Spaces based on Fourier transform}
\ In this section we consider the nonlinear Schr\"odinger equation
\begin{equation}\label{SCH-F1}
\left\{
\begin{aligned}
i\partial_{t}u+\Delta u+\mu(x)u & =\lambda u^\rho,\ x\in\Omega,\ t\in \mathbb{R},\\
u(x,0) & =u_{0},\text{ }x\in\Omega
\end{aligned}\right.
\end{equation}
where $\mu\in
BC(\mathbb{R}^{n})$ (the space of all bounded continuous functions on $\mathbb{R}^{n})$, $\lambda=\pm1$ \ and $\rho\in\mathbb{N}$. Here we will consider
$\Omega=\mathbb{T}^{n}$ and $\Omega=\mathbb{R}^{n}$, i.e. the periodic and
nonperiodic cases, respectively. The nonlinearity $\lambda\left\vert u\right\vert ^{\rho-1}u$ could be considered in (\ref{SCH-F1}), however we prefer $u^{\rho}$ for the sake of simplicity of
the exposition. For more details, see Remark \ref{rem-Fourier2} below.
We start by defining the spaces for the nonperiodic case. We recall that if $\mathcal{M}(\mathbb{R}^{n})$ denotes the space of complex Radon measures on $\mathbb{R}^{n}$, then it is a vector space and for $\nu\in\mathcal{M}(\mathbb{R}^{n})$, $\|\nu\|_{\mathcal{M}}=|\nu|(\mathbb{R}^{n})$ is a norm on it, where $|\nu|$ is the total variation of $\nu$ (we note that every measure in $\mathcal{M}(\mathbb{R}^{n})$ is automatically a finite Radon measure. Moreover, we can embed $L^1(\mathbb{R}^{n}, dm)$ into $\mathcal{M}(\mathbb{R}^{n})$ by identifying $f\in L^1(\mathbb{R}^{n}, dm)$ with the complex measure $d\nu= fdm$, and $\|\nu\|_{\mathcal{M}}=\int |f|dm$. Next, every $\nu\in \mathcal{M}(\mathbb{R}^{n})$ defines a tempered distribution by $T_\nu(\varphi)=\int_{\mathbb{R}^{n}} \varphi(x)d\nu$, thereby identifying $ \mathcal{M}(\mathbb{R}^{n})$ with a subspace of $\mathcal S'$.
The Fourier transform on $L^1(\mathbb{R}^{n})$ can be extended of a natural form to $\mathcal{M}(\mathbb{R}^{n})$; if $\nu\in \mathcal{M}(\mathbb{R}^{n})$, the Fourier transform of $\nu$ is the function $\widehat{\nu}$ defined by
\begin{equation}\label{transf}
\widehat{\nu}(\xi)=\int e^{-2\pi i \xi\cdot x} d\nu(x),\quad \xi\in \mathbb{R}^{n}.
\end{equation}
Using that $ e^{-2\pi i \xi\cdot x} $ is uniformly continuous in $x$, it is not difficult to check that $\widehat{\nu} \in BC(\mathbb{R}^{n})$ and that $\|\widehat{\nu}\|_{\infty}\leqq \|\nu\|_{\mathcal{M}}$ (see Folland \cite{FO}). Similarly, we define the inverse Fourier transform $\check{\nu}$ of $\nu$ by $\check{\nu}(\xi)=\widehat{\nu}(-\xi)$ for $\xi\in \mathbb{R}^{n}$. Moreover, if $\mathcal F$ represents the Fourier transform on $\mathcal S'$, then for every $\nu\in \mathcal{M}(\mathbb{R}^{n})$ we have $\mathcal F (\nu)=\widehat{\nu}$. Similarly, $\mathcal F^{-1} (\nu)=\check{\nu}$.
Recall that the space $\mathcal{M}(\mathbb{R}^{n})$ can be identified with a subspace of $\mathcal S'$. Hence, if we assume that $ \mu\in \mathcal S'$ and $\mathcal F(\mu)$ is a finite Radon measure, then for $\nu=\mathcal F(\mu)$ we have
$$
\mu=\mathcal F^{-1} \mathcal F(\mu)=\mathcal F^{-1} (\nu)=\check{\nu}\in BC(\mathbb{R}^{n}).
$$
Next, by using the above identification between $\mathcal F$ and\; $\widehat{}$\; \;, we define the Banach space
\begin{equation}
\mathcal{I=}[\mathcal{M}(\mathbb{R}^{n})]^{\vee}=\{f\in\mathcal{S}^{\prime
}(\mathbb{R}^{n}):\widehat{f}\in\mathcal{M}(\mathbb{R}^{n})\}\subset
BC(\mathbb{R}^{n}), \label{esp2}
\end{equation}
with norm
\begin{equation}
\left\Vert f\right\Vert _{\mathcal{I}}=\Vert\widehat{f}\;\Vert
_{\mathcal{M}}.\text{ } \label{norm1}
\end{equation}
We note that in general $\mu\in \mathcal{I}$ may not to belong to $L^{p}(\mathbb{R}^{n})$, nor to $L^{p,\infty}(\mathbb{R}^{n}),$ with $p\neq\infty.$ In particular, $\mu\in \mathcal{I}$ may have infinite $L^{2}$-mass; for instance, if $\mu\equiv1$ then $\mathcal F(\mu)=\delta_{0}\in \mathcal{M}(\mathbb{R}^{n})$.
In the following we will consider $\mu, u_0\in \mathcal{I}$. The Cauchy problem (\ref{SCH-F1}) is formally equivalent to the
functional equation
\begin{equation}
u(t)=S(t)u_{0}+B(u)+L_{\mu}(u), \label{int2}
\end{equation}
where $S(t)=e^{it\Delta}$ is the Schr\"odinger group in $\mathbb{R}^{n}$, and the operators $L_{\mu}(u), B(u)$ are defined via Fourier transform by
\begin{equation}
\widehat{L_{\mu}(u)}=\int_{0}^{t}e^{-i\left\vert \xi\right\vert ^{2}
(t-s)}(\widehat{\mu}\ast\widehat{u})(\xi,s)ds,\label{op1}
\end{equation}
and
\begin{equation}
\widehat{B(u)}=\lambda\int_{0}^{t}e^{-i\left\vert \xi\right\vert ^{2}(t-s)}(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
}_{\rho-times})
(\xi,s)ds, \label{op3}
\end{equation}
for $\mu\in \mathcal{I}$ and $u\in L^{\infty}((-T,T);\mathcal{I})$. We recall that for arbitrary $\mu, \nu\in \mathcal{M}(\mathbb{R}^{n})$ their convolution $\widehat{\mu}\ast\widehat{\nu}\in \mathcal{M}(\mathbb{R}^{n})$ is defined by
$$
\widehat{\mu}\ast\widehat{\nu}(E)=\int\int\chi_E(x+y)d\mu(x)d\nu(y),
$$
for every Borel set $E$.
Let $\mathbb{T}^{n}=\mathbb{R}^{n}/{\mathbb{Z}^{n}}$ stand for the $n$-torus. We say that functions on $\mathbb{T}^{n}$ are functions $f:\mathbb{R}^{n}\to \mathbb{C}$ that satisfy $f(x+m)=f(x)$ for all $x\in \mathbb{R}^{n}$ and $m\in \mathbb{Z}^{n}$, which are called $1$-periodic in every coordinate. Let $\mathcal P=C^\infty_{per}=\{f:\mathbb{R}^{n}\to \mathbb{C}: f \;\text{is}\; C^\infty \;\text{and periodic with period}\; 1\}$. So $\mathcal{D}^{\prime}(\mathbb{T}^{n})$ is the set of all periodic distributions on $\mathcal P$. We say that $\mathcal T:\mathcal P\to \mathbb{C}$ is a {\it periodic distribution} if there exists a sequence $(\Psi_j)_{j\geqq 1}\subset \mathcal P$ such that
$$
\mathcal T(f)=\langle \mathcal T, f\rangle=\lim_{j\to \infty}\int_{[-1/2,1/2]^n} \Psi_j(x)f(x)dx,\qquad \; f\in \mathcal P.
$$
Above we have identify $\mathbb{T}^{n}$ with $[-1/2,1/2]^n$. For a complex-valued function $f\in L^1(\mathbb{T}^{n})$ and $m\in \mathbb{Z}^{n}$, we define
$$
\widehat{f}(m)=\int_{[-1/2,1/2]^n}f(x)e^{-2\pi i x\cdot m}dx.
$$
We call $\widehat{f}(m)$ the $m$-th Fourier coefficient of $f$. The Fourier series of $f$ at $x\in \mathbb{T}^{n}$ is the series
$$
\sum_{m\in \mathbb{Z}^{n}} \widehat{f}(m) e^{2\pi i x\cdot m}.
$$
The Fourier transform of $\mathcal T\in \mathcal{D}^{\prime}(\mathbb{T}^{n})$ is the function $\widehat{\mathcal T}: \mathbb{Z}^{n}\to \mathbb C$ defined by the formula
$$
\widehat{\mathcal T}(m)=\langle \mathcal T, e^{-2\pi i x\cdot m}\rangle,\quad m\in \mathbb{Z}^{n}.
$$
In the periodic case, we are going to study (\ref{SCH-F1}) in the space $\mathcal{I}_{per}$ which is defined by
\begin{equation}
\mathcal{I}_{per}=\{f\in\mathcal{D}^{\prime}(\mathbb{T}^{n}):\widehat{f}\in
l^{1}(\mathbb{Z}^{n})\} \label{peri1}
\end{equation}
endowed with the norm
\begin{equation}
\left\Vert f\right\Vert _{\mathcal{I}_{per}}=\Vert\widehat{f}\;\Vert
_{l^{1}(\mathbb{Z}^{n})}\text{.} \label{peri2}
\end{equation}
Here the IVP (\ref{SCH-F1}) is formally converted to
\begin{equation}
u(t)=S_{per}(t)u_{0}+B_{per}(u)+L_{\mu,per}(u), \label{int-per}
\end{equation}
where, similarly to above, we define the operators in (\ref{int-per}) via
Fourier transform in $\mathcal{D}^{\prime}(\mathbb{T}^{n})$. Precisely, $S_{per}(t)$ is the Schrodinger group in $\mathbb{T}^{n}$
\begin{equation}
S_{per}(t)u_{0}=\sum_{m\in\mathbb{Z}^{n}}\widehat{u}_{0}(m)e^{-4\pi
^{2}i\left\vert m\right\vert ^{2}t}e^{2\pi ix\cdot m}, \label{sch-per}
\end{equation}
\begin{equation}
\widehat{L_{\mu,per}(u)}(m,t)=\int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds \label{Op-per1}
\end{equation}
and
\begin{equation}
\widehat{B_{per}(u)}(m,t)=\int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
}_{\rho-times})(m,s)ds,
\label{Op-per2}
\end{equation}
for $u_{0},\mu\in \mathcal{I}_{per}$ and $u\in L^{\infty}((-T,T);\mathcal{I}_{per})$, where now the symbol $\ast$ denotes the discrete convolution
\[
\widehat{f}\ast\widehat{g}(m)=\sum_{\xi\in\mathbb{Z}^{n}}\widehat{f}
(m-\xi)\widehat{g}(\xi).
\]
Throughout this section, solutions of (\ref{int2}) or (\ref{int-per}) will be
called mild solutions for the IVP (\ref{SCH-F1}), according the
respective case.
In the above framework, our local-in-time well-posedness result reads as follows.
\begin{theorem}
\label{teoF1}Let $1\leq\rho<\infty.$
\end{theorem}
\begin{itemize}
\item[(1)] (Periodic case) Let $u_{0}\in\mathcal{I}_{per}$ and $\mu$
$\in\mathcal{I}_{per}$. There is $T>0$ such that the IVP (\ref{SCH-F1}) has a unique mild solution $u\in L^{\infty}((-T,T);\mathcal{I}_{per})$ satisfying
\[
\sup_{t\in(-T,T)}\left\Vert u(\cdot,t)\right\Vert _{\mathcal{I}_{per}}
\leq 2\left\Vert u_{0}\right\Vert _{\mathcal{I}_{per}}.
\]
Moreover, the data-map solution $u_{0}\rightarrow u$ is locally Lipschitz continuous
from $\mathcal{I}_{per}$ to {$L^{\infty}((-T,T);$}$\mathcal{I}_{per}\mathcal{)}.$
\item[(2)] (Nonperiodic case) Let $u_{0}\in\mathcal{I}$ and $\mu$
$\in\mathcal{I}$. The same conclusion of item (1) holds true by
replacing $\mathcal{I}_{per}$ by $\mathcal{I}$.
\end{itemize}
\begin{remark}
\label{rem-Fourier}In item (2) of the above theorem, one can show that the
solution $u(x,t)$ verifies $\widehat{u}(\xi,t)\in L^{1}(\mathbb{R}^{n}),$ for
all $t\in(-T,T),$ provided that $\widehat{u}_{0}\in L^{1}(\mathbb{R}^{n})$ and
$\widehat{\mu}\in L^{1}(\mathbb{R}^{n}).$ Then, due to Riemann-Lebesgue lemma,
it follows that
\[
u(x,t)\rightarrow0\text{ as }\left\vert x\right\vert \rightarrow\infty,\text{
for each }t\in(-T,T).
\]
\end{remark}
\subsection{
Nonlinear Estimate}
Before proceeding with the proof of Theorem \ref{teoF1}, let us recall the Young inequality for measures and
discrete convolutions (see Folland \cite{FO} and Iorio\&Iorio \cite{Ior}). For $\mu, \nu \in \mathcal{M}(\mathbb R^n)$ and $f, g\in l^{1}=l^{1}(\mathbb Z^n)$, we have the respective estimates
\begin{align}
\left\Vert \mu\ast \nu\right\Vert _{\mathcal{M}} & \leq\left\Vert \mu\right\Vert
_{\mathcal{M}}\left\Vert \nu\right\Vert _{\mathcal{M}}\label{Young}\\
\left\Vert f\ast g\right\Vert _{l^{1}} & \leq\left\Vert f\right\Vert
_{l^{1}}\left\Vert g\right\Vert _{l^{1}}. \label{Young-per}
\end{align}
\begin{lemma}
\label{lem-est1}Let $1\leq\rho<\infty$ and $0<T<\infty.$
\begin{itemize}
\item[(i)] There exists a positive constant $K>0$ such that
\begin{align}
\sup_{t\in(-T,T)}\left\Vert S_{per}(t)u_{0}\right\Vert _{\mathcal{I}_{per}}
& \leq\left\Vert u_{0}\right\Vert _{\mathcal{I}_{per}}\label{est1}\\
\sup_{t\in(-T,T)}\left\Vert L_{\mu,per}(u-v)\right\Vert _{\mathcal{I}_{per}}
& \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\sup_{t\in
(-T,T)}\left\Vert u(\cdot,t)-v(\cdot,t)\right\Vert _{\mathcal{I}_{per}
}\label{est2}\\
\sup_{t\in(-T,T)}\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{\mathcal{I}
_{per}} & \leq KT\sup_{t\in(-T,T)}\left\Vert u(\cdot,t)-v(\cdot
,t)\right\Vert _{\mathcal{I}_{per}}\label{est3}\\
& \times\left( \sup_{t\in(-T,T)}\left\Vert u(\cdot,t)\right\Vert
_{\mathcal{I}_{per}}^{\rho-1}+\sup_{t\in(-T,T)}\left\Vert v(\cdot
,t)\right\Vert _{\mathcal{I}_{per}}^{\rho-1}\right) ,\nonumber
\end{align}
for all $u_{0},\mu\in\mathcal{I}_{per}$ and $u,v\in${$L^{\infty}((-T,T);$}
$\mathcal{I}_{per}).$
\item[(ii)] The above estimates still hold true with $S(t),L_{\mu}(u),B(u)$
and $\mathcal{I}$ in place of $S_{per}(t),L_{\mu,per}(t),B_{per}(u)$ and
$\mathcal{I}_{per}$, respectively.
\end{itemize}
\end{lemma}
\textbf{Proof.} \ We will only prove the item (i) because (ii) follows similarly
by using (\ref{Young}) instead of (\ref{Young-per}). From definition of
$S_{per}(t),$ we have that
\[
\sup_{t\in(-T,T)}\left\Vert S_{per}(t)u_{0}\right\Vert _{\mathcal{I}_{per}
}=\left\Vert \left( \widehat{u}_{0}(m)e^{-4\pi^{2}i\left\vert m\right\vert
^{2}t}\right) _{m\in\mathbb{Z}^{n}}\right\Vert _{l^{1}(\mathbb{Z}^{n})}
\leq\left\Vert \hat{u}_{0}\right\Vert _{l^{1}(\mathbb{Z}^{n})}.
\]
The operator $L_{\mu,per}$ can be estimated as
\begin{align*}
\left\Vert L_{\mu,per}(u)\right\Vert _{\mathcal{I}_{per}} & =\left\Vert
\widehat{L_{\mu,per}(u)}\right\Vert _{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert \left( \int_{0}^{t}e^{-4\pi^{2}i\left\vert m\right\vert
^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds\right) _{m\in\mathbb{Z}^{n}
}\right\Vert _{l_{1}(\mathbb{Z}^{n})}\\
& \leq\sum_{m\in\mathbb{Z}^{n}}\left\vert \int_{0}^{t}e^{-4\pi^{2}i\left\vert
m\right\vert ^{2}(t-s)}(\widehat{\mu}\ast\widehat{u})(m,s)ds\right\vert \\
& \leq\int_{0}^{t}\sum_{m\in\mathbb{Z}^{n}}\left\vert (\widehat{\mu}
\ast\widehat{u})(m,s)\right\vert ds\\
& \leq\int_{0}^{t}\left\Vert \widehat{\mu}\right\Vert _{l^{1}(\mathbb{Z}
^{n})}\left\Vert \widehat{u}(\cdot,s)\right\Vert _{l^{1}(\mathbb{Z}^{n})}ds=\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\left\Vert u\right\Vert
_{L^{1}(0,T;\mathcal{I}_{per})} \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\left\Vert u\right\Vert
_{L^{\infty}(0,T;\mathcal{I}_{per})}.
\end{align*}
By elementary convolution properties and Young inequality (\ref{Young-per}),
it follows that
\begin{align*}
& \left\Vert (\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u}
)}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat{v}\ast...\ast\widehat{v}
)}_{\rho-times}\right\Vert _{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert \lbrack(\hat{u}-\hat{v})\ast\widehat{u}\ast...\ast
\widehat{u}+\hat{v}\ast(\widehat{u}-\widehat{v})\ast...\ast\widehat{u}+\hat
{v}\ast\widehat{v}\ast(\widehat{u}-\widehat{v})\ast...\ast\widehat{u}
+...+\hat{v}\ast\widehat{v}\ast...\ast(\widehat{u}-\widehat{v})\right\Vert
_{l^{1}(\mathbb{Z}^{n})}\\
& \leq\left\Vert (\hat{u}-\hat{v})\right\Vert _{l^{1}}\left\Vert \widehat
{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{u}\right\Vert _{l^{1}}^{\rho-2}\left\Vert
\widehat{v}\right\Vert _{l^{1}}+...+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{u}\right\Vert _{l^{1}}\left\Vert \widehat
{v}\right\Vert _{l^{1}}^{\rho-2}+\left\Vert (\hat{u}-\hat{v})\right\Vert
_{l^{1}}\left\Vert \widehat{v}\right\Vert _{l^{1}}^{\rho-1}\\
& \leq K\left\Vert (\hat{u}-\hat{v})\right\Vert _{l^{1}}\left( \left\Vert
\widehat{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert \widehat{v}\right\Vert
_{l^{1}}^{\rho-1}\right)
\end{align*}
Therefore
\begin{align*}
\left\Vert B_{per}(u)(t)-B_{per}(v)(t)\right\Vert _{\mathcal{I}_{per}} & =\left\Vert
\widehat{B_{per}(u)}-\widehat{B_{per}(v)}\right\Vert _{l^{1}}\\
& \leq\left\Vert \int_{0}^{t}e^{-4\pi^{2}i\left\vert \xi\right\vert
^{2}(t-s)}\left[ (\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast
\widehat{u})}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat{v}\ast
...\ast\widehat{v})}_{\rho-times}\right] ds\right\Vert _{l^{1}}\\
& \leq\int_{0}^{t}\left\Vert \left[ (\underbrace{\widehat{u}\ast\widehat
{u}\ast...\ast\widehat{u})}_{\rho-times}-\underbrace{(\widehat{v}\ast\widehat
{v}\ast...\ast\widehat{v})}_{\rho-times}\right] \right\Vert _{l^{1}}ds\\
& \leq K\int_{0}^{t}\left\Vert \hat{u}-\hat{v}\right\Vert _{l^{1}}\left(
\left\Vert \widehat{u}\right\Vert _{l^{1}}^{\rho-1}+\left\Vert \widehat
{v}\right\Vert _{l^{1}}^{\rho-1}\right) ds\\
& \leq KT\left\Vert u-v\right\Vert _{L^{\infty}(0,T;\mathcal{I}_{per
}\mathcal{)}}\left( \left\Vert u\right\Vert _{L^{\infty}(0,T;\mathcal{I}
_{per}\mathcal{)}}^{\rho-1}+\left\Vert v\right\Vert _{L^{\infty}
(0,T;\mathcal{I}_{per}\mathcal{)}}^{\rho-1}\right) ,
\end{align*}
as required. \fin
\begin{remark}
\label{rem-Fourier2}
The approach employed here could be used to treat (\ref{SCH-F1}) with
the nonlinearity $\left\vert u\right\vert ^{\rho-1}u$ instead of $u^{\rho}.$
For $\rho$ odd, it would be enough to write $\left\vert u\right\vert
^{\rho-1}u$ (in the above proof) as
\begin{align*}
\lbrack\left( \left\vert u\right\vert ^{2}\right) ^{\frac{\rho-1}{2}
}u]^{\wedge} & =[(u\cdot\overline{u})^{\frac{\rho-1}{2}}u]^{\wedge}\\
& =(\underbrace{\widehat{u}\ast\widehat{u}\ast...\ast\widehat{u})}_{\frac
{\rho-1}{2}-times}\ast(\underbrace{\widehat{\overline{u}}\ast\widehat
{\overline{u}}\ast...\ast\widehat{\overline{u}})}_{\frac{\rho-1}{2}-times}\ast
u
\end{align*}
and to note that $\widehat{\overline{u}}(\xi)=\overline{\widehat{u}}(-\xi)$
and $\left\Vert \overline{u}(\xi)\right\Vert _{\mathcal{I}_{per}}=\left\Vert u(\xi)\right\Vert _{\mathcal{I}_{per}}$.
\end{remark}
\subsection{Proof of Theorem \ref{teoF1}}
\textbf{Proof of (1).} \noindent Consider the ball $\mathcal{B}
_{\varepsilon}=\{u\in L^{\infty}(-T,T;\mathcal{I}_{per});\Vert u\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}\leq2\varepsilon\}$ endowed with the complete metric
$Z(\cdot,\cdot)$ defined by
$$
Z(u,v)=\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}
_{per})}.
$$
Let $\varepsilon=\Vert u_{0}\Vert_{\mathcal{I}_{per}}$ and $T>0$
such that
\begin{equation}
T(2\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho}\varepsilon^{\rho
-1}K)<1. \label{cond-small}
\end{equation}
Notice that $\varepsilon$ can be large. We shall show that the map
\begin{equation}
\Phi(u)=S_{per}(t)u_{0}+L_{\mu,per}(u)+B_{per}(u)\nonumber
\end{equation}
\newline is a contraction on $(\mathcal{B}_{\varepsilon},Z).$ Lemma
\ref{lem-est1} with $v=0$ yields
\begin{align}
\Vert\Phi(u)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})} & \leq\Vert
S_{per}(t)u_{0}\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+\Vert L_{\mu,per}(u)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)\right\Vert
_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& \leq\Vert u_{0}\Vert_{\mathcal{I}_{per}}+T\left\Vert \mu\right\Vert
_{\mathcal{I}_{per}}\Vert u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+TK\Vert
u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}^{\rho}\nonumber\\
& \leq\varepsilon+2\varepsilon T\left\Vert \mu\right\Vert _{\mathcal{I}
_{per}}+2^{\rho}\varepsilon^{\rho}TK\nonumber\\
& =\varepsilon+T(2\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho
}\varepsilon^{\rho-1}K)\varepsilon<2\varepsilon, \label{aux30}
\end{align}
for all $u\in\mathcal{B}_{\varepsilon}$ and thus $\Phi(\mathcal{B}
_{\varepsilon})\subset\mathcal{B}_{\varepsilon}.$ From Lemma \ref{lem-est1},
we\ also have
\begin{align}
\left\Vert \Phi(u)-\Phi(v)\right\Vert _{L^{\infty}(-T,T;\mathcal{I}_{per})}
& =\Vert L_{\mu,per}(u)-L_{\mu,per}(v)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& \leq T\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}\Vert u-v\Vert
_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
& +KT\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\left( \Vert
u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}^{\rho-1}+\Vert v\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}^{\rho-1}\right) \nonumber\\
& \leq T\left( \left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+K2^{\rho
}\varepsilon^{\rho-1}\right) \Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}, \label{aux3}
\end{align}
for all $u,v\in\mathcal{B}_{\varepsilon}.$ In view of (\ref{cond-small}),
(\ref{aux30}) and (\ref{aux3}), the map $\Phi$ is a contraction in
$\mathcal{B}_{\varepsilon}$ and then, the Banach fixed point theorem assures
the existence of a unique solution $u\in L^{\infty}(-T,T;\mathcal{I}_{per})$ for
(\ref{int2}) such that $\Vert u\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\leq2\Vert u_0\Vert_{\mathcal{I}_{per}}$.
On the other hand if $u,v$ are two solutions with respective initial data
$u_{0},v_{0}$ then, similarly in deriving (\ref{aux3}), one obtains
\begin{align*}
\Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})} & \leq\Vert u_{0}-v_{0}\Vert_{L^{\infty
}(-T,T;\mathcal{I}_{per})}+\Vert L_{\mu,per}(u)-L_{\mu,per}(v)\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}\nonumber\\
&+\left\Vert B_{per}(u)-B_{per}(v)\right\Vert _{L^{\infty
}(-T,T;\mathcal{I}_{per})}\\
& \leq\Vert u_{0}-v_{0}\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})}+T\left(
\left\Vert \mu\right\Vert _{\mathcal{I}_{per}}+2^{\rho}\varepsilon^{\rho
-1}K\right) \Vert u-v\Vert_{L^{\infty}(-T,T;\mathcal{I}_{per})},
\end{align*}
which, in view of (\ref{cond-small}), implies the desired local Lipschitz continuity.
\textbf{Proof of (2). }It follows by proceeding entirely parallel to the
proof of item (1) by replacing $\mathcal{I}_{per}$ by $\mathcal{I}.$ \fin
\section{Appendix}
Next we present a different proof of Theorem \ref{expli}. For $\sigma>0$, from Theorem 3.1 in Albeverio {\it et al.} \cite{AGHH} , the fundamental solution $S_\sigma(x, y;t)$ to the Schr\"odinger equation (\ref{SCH1}) is given by
\begin{equation}\label{kern}
S_\sigma(x, y,t)=S(x-y;t)-\frac{\sigma}{2}\int_0^\infty e^{-\frac{\sigma}{2} u} S(u+|x|+|y|;t)du
\end{equation}
where $S(x;t)$ is the free propagator in $\mathbb R$, i.e.
$$
S(x,t)= \frac{e^{-x^2/{4it}}}{(4i\pi t)^{1/2}},\quad t>0
$$
and $e^{it\Delta}f(x)=S(x;t)\ast_x f(x)$. Then we have the representation
\begin{equation}\label{propa}
e^{-itH_\sigma}f(x)=\int_{\mathbb R} S_\sigma(x, y,t)f(y)dy.
\end{equation}
Next, we consider $x>0$ and $f\in L^1(\mathbb R)$ with $supp f\subset (-\infty, 0]$. Then, since
$$
S(u+x-y;t)= (e^{-it\xi^2})^{\vee}(u+x-y)=(e^{-it\xi^2}e^{i\xi(x+u)})^{\vee}(y)
$$
we obtain via Parseval identity that (\ref{propa}) can be re-write for $x>0$ in the form
\begin{equation}\label{propa1}
\begin{aligned}
e^{-itH_\sigma}f(x)&=e^{it\Delta}f(x)\chi^0_+(x) +\int_{\mathbb R}e^{-ity^2}\Big[\int_{\mathbb R} -\frac{\sigma}{2} e^{\frac{\sigma}{2} s}\chi^0_-(s)e^{-iys}ds\Big] \widehat{f}(y)e^{iyx}dy\\
&=e^{it\Delta}f(x)\chi^0_+(x) +\int_{\mathbb R}e^{-ity^2}\widehat{\rho_\sigma\ast f}(y)e^{iyx}dy\\
&=e^{it\Delta}f(x)\chi^0_+(x) +e^{it\Delta}(\rho_\sigma\ast f)(x)\chi^0_+(x)
\end{aligned}
\end{equation}
where $\rho_\sigma(x)=-\frac{\sigma}{2} e^{\frac{\sigma}{2} s}\chi^0_-(s)$. Similarly, for $x<0$ we obtain the representation
\begin{equation}\label{propa2}
e^{itH_\sigma}f(x)=e^{it\Delta}f(x)\chi^0_-(x) +e^{it\Delta}(\rho_\sigma\ast f)(-x)\chi^0_-(x).
\end{equation}
From (\ref{propa1})-(\ref{propa2}) we obtain the formula (\ref{pospro}).
For $\sigma<0$, Theorem 3.1 in \cite{AGHH} establishes that the fundamental solution $S_\sigma(x, y;t)$ to the Schr\"odinger equation (\ref{SCH1}) is given by
$$
S_\sigma(x, y,t)=S(x-y;t) +e^{i\frac{\sigma^2 t}{4}}\Psi_\sigma(x)\Psi_\sigma(y)+\frac{\sigma}{2}\int_0^\infty e^{\frac{\sigma}{2} u} S(u-|x|-|y|;t)du.
$$
where $\Psi_\sigma$ is the (normalized) eigenfunction defined in Theorem \ref{resol5b}. Then, a similar analysis as above produces the formula (\ref{negpro}).
We note that since $|S(x;t)|\leqq C_0 t^{-1/2}$ for every $x\in \mathbb R$ and $t>0$ we obtain from (\ref{kern}) that $|S_\sigma(x,y;t)|\leqq 2C_0 t^{-1/2}$ for all $x, y\in \mathbb R$ and $t>0$. Therefore from (\ref{propa}) we obtain the dispersive estimate
$$
\|e^{-itH_\sigma}f\|_{\infty}\leqq 2C_0 t^{-1/2} \|f\|_1,
$$
which implies the estimate (\ref{stric2}) for $\sigma >0$.
\vskip0.2in
\textbf{ACKNOWLEDGEMENTS:} J. Angulo was partially supported by Grant CNPq/Brazil. L.C.F. Ferreira was supported by FAPESP-SP and CNPq/Brazil.
\end{document} |
\begin{document}
\title[On the dimension group of unimodular $\mathcal{S}$-adic subshifts]{On the dimension group\\ of unimodular $\mathcal{S}$-adic subshifts}
\author{V. Berth\'e}
\address{Universit\'e de Paris, IRIF, CNRS, F-75013 Paris, France}
\thanks{This work was supported by the Agence Nationale de la Recherche through the project ``Codys'' (ANR-18-CE40-0007).}
\email{berthe@irif.fr}
\author{P. Cecchi Bernales}
\address{Centro de Modelamiento Matemático, Universidad de Chile, Chile}
\thanks{The second author was supported by the PhD grant CONICYT - PFCHA / Doctorado Nacional / 2015-21150544.}
\email{pcecchi@ing.uchile.cl}
\author{F. Durand}
\address{LAMFA, UMR 7352 CNRS, Universit\'e de Picardie Jules Verne,
33, rue Saint-Leu, 80039 Amiens, France}
\email{fabien.durand@u-picardie.fr}
\author{J. Leroy}
\address{D\'epartement de math\'ematique, Universit\'e de Li\`ege,
12, All\'ee de la d\'ecouverte (B37), 4000 Li\`ege, Belgique}
\email{j.leroy@uliege.be}
\author{D. Perrin}
\address{Laboratoire d'Informatique Gaspard-Monge, Universit\'e Paris-Est, France}
\email{Dominique.Perrin@u-pem.fr}
\author{S. Petite}
\address{LAMFA, UMR 7352 CNRS, Universit\'e de Picardie Jules Verne,
33, rue Saint-Leu, 80039 Amiens, France}
\email{samuel.petite@u-picardie.fr}
\date{\today}
\begin{abstract}
Dimension groups are complete invariants of strong orbit equivalence
for minimal Cantor systems.
This paper studies a natural family of minimal Cantor systems having a finitely
generated dimension group, namely the primitive unimodular proper $\mathcal{S}$-adic subshifts. They are generated by iterating sequences
of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter.
This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations.
We compute their dimension group and investigate the relation between
the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a
topological characterization of balancedness for primitive unimodular proper S-adic subshifts.
\end{abstract}
\maketitle
\section{Introduction}
Two dynamical systems are topologically orbit equivalent if there is a homeomorphism between
them preserving the orbits. Originally, the notion of orbit equivalence was studied
in the measurable context (see for instance \cite{Dye,OrnsteinWeiss}), motivated by the classification of von Neumann algebras. In contrast
with the measurable case, Giordano, Putnam and Skau showed that, in the topological setting, uncountably
many classes appear by providing a dimension group as a complete invariant of strong
orbit equivalence \cite{GPS:95}.
Dimension groups are ordered direct limit groups defined by sequences of positive homomorphisms $(\theta _n : \mathbb{Z}^{d_n} \to \mathbb{Z}^{d_{n+1}})_n$,
where $ {\mathbb Z}^d$ is given the standard or simplicial order, and were defined by Elliott \cite{Elliott:76} to study approximately finite dimensional $C^*$-algebras.
In fact, an ordered group is a dimension group if and only if it is a Riesz group \cite{EffrosHandelmanShen:1980}.
They have been widely studied in the late 70's and at the beginning of the 80's \cite{Effros},
in particular when the dimension group is a direct limit given by unimodular matrices \cite{Effros&Shen:1979,Effros&Shen:1980,Effros&Shen:1981,Riedel:1981,Riedel:1981b}.
The present paper studies dynamical and ergodic properties of subshifts having dimension groups
with a group part of the form ${\mathbb Z}^d$.
We focus on the class of primitive unimodular proper $\mathcal{S}$-adic subshifts.
Such subshifts are generated by iterating sequences of substitutions. They have recently attracted much attention in symbolic dynamics
\cite{Berthe&Delecroix:14} and in tiling theory \cite{GM:13,Fusion:14}. Proper substitutions are such that images of letters start with a same letter and also
end with a same letter.
Proper minimal proper $\mathcal{S}$-adic systems have played an important role for the characterization of linearly recurrent subshifts \cite{Durand:2000, Durand:03}.
The term unimodular refers to the unimodularity of the incidence matrices of the associated substitutions.
Sturmian subshifts, subshifts generated by natural codings of interval exchange transformations or Arnoux-Rauzy subshifts are prominent examples
of unimodular proper $\mathcal{S}$-adic subshifts.
They also belong to a recently defined family of subshifts, called dendric subshifts, and considered in \cite{BDFDLPRR:15,BDFDLPRR:15bis,BFFLPR:2015,BDFDLPR:16,Rigidity} (see also Section~\ref{subsec:tree}).
In this series of papers, their elements have been studied under the name of tree words. We have chosen to use the terminology dendric subshift in order to avoid any ambiguity with respect to shifts defined on trees (see, e.g.,~\cite{AubrunBeal}) and also to avoid the puzzling term ``tree word''.
Minimal dendric subshifts are defined with respect to combinatorial properties of their language expressed in terms of extension graphs. For precise definitions, see Section~\ref{subsec:tree}. In particular,
they have linear factor complexity. Focusing on extension properties of factors is a combinatorial viewpoint that allows to highlight the common features shared by dendric subshifts, even if the corresponding symbolic systems have very distinct dynamical, ergodic and spectral properties.
For instance, a coding of a generic interval exchange is topologically weakly mixing for irreducible permutations not of rotation class \cite{NogRud}, whereas an Arnoux-Rauzy subshift is generically not topologically weakly mixing \cite{Cassaigne-Ferenczi-Messaoudi:08,BST:2019}.
Even though one can disprove, in some cases, whether two given minimal dendric subshifts are topologically conjugate by using e.g. asymptotic pairs (see for instance Section \ref{ex:balance}), the question of orbit equivalence is more subtle and is one of the motivations for the present work.
The aim of this paper is to study topological orbit equivalence and strong orbit equivalence for minimal unimodular proper $\mathcal{S}$-adic
subshifts. Let $(X,S)$ be such a subshift over a $d$-letter alphabet ${\mathcal A}$ and
let ${\mathcal M} (X,S)$ stand for its set of shift-invariant probability measures.
One of our main results states that any continuous integer-valued function defined on $X$ is cohomologous to some integer linear combination of characteristic functions of letter cylinders
(Theorem \ref{theo:cohoword}).
This relies on the fact that such subshifts being aperiodic (see Proposition \ref{prop:minimalSSIprimitif}) and recognizable by~\cite{BSTY}, they have a sequence of Kakutani-Rohlin tower partitions
with suitable topological properties.
We then deduce an explicit computation of their dimension group (Theorem \ref{theo:dg}).
Indeed, the dimension group $K^{0}(X,S)$ with ordered unit is isomorphic to
$\left( {\mathbb Z} ^d, \, \{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}\cup \{{\mathbf 0}\},\, \bf{1}\right )$,
where $\boldsymbol{ \mu}$ denotes the vector of measures of letter cylinders.
In other words,
strong orbit equivalence can be characterized by means of letter measures, i.e., by measures of letter cylinders.
In particular,
two shift-invariant probability measures on $(X,S)$ coinciding on the letter cylinders are proved to be equal (see Corollary \ref{coro:measures}).
This result extends a statement initially proved for interval exchanges in~\cite{FerZam:08}; see also \cite{Putnam:89,Putnam:92,GjerdeJo:02} and \cite{BHL:2019,BHL:2020}.
Moreover, two primitive unimodular proper ${\mathcal S}$-adic subshifts are proved to be strong orbit equivalent if and only if their simplexes of letter measures coincide up to a unimodular matrix (see Corollary \ref{cor:oe}), with the simplex of letter measures being the $d$-simplex generated by the vectors $(\nu[a])_ {a \in \mathcal{A}}$, for $\nu$ in ${\mathcal M}(X,S)$.
We also investigate in Section \ref{sec:saturation} the triviality of the infinitesimal subgroup and relate it to the notion of balance.
We provide
a characterization of the triviality of the infinitesimal subgroup for minimal unimodular proper $\mathcal{S}$-adic
subshifts in terms of rational independence of
measures of letters (see Proposition \ref{theo:saturated}).
Inspired by the classical notion of balance in word combinatorics (see e.g. references in \cite{BerCecchi:2018}),
we also introduce the notion of balanced functions and provide a topological characterization of this balance property for primitive unimodular proper $\mathcal{S}$-adic subshifts (see Corollary \ref{cor:balanced}).
We briefly describe the contents of this paper.
Definitions and basic notions are recalled in Section~\ref{sec:def}, including,
in particular, the notions of dimension group and orbit equivalence in Section~\ref{subsec:dim}, and of image subgroup in Section~\ref{subsec:cyl}. Primitive unimodular $\mathcal{S}$-adic subshifts are introduced in Section~\ref{sec:Sadic}, with dendric subshifts being discussed in more details in Section~\ref{subsec:tree}.
Their dimension groups are studied in Section~\ref{sec:proofs2}. Section~\ref{sec:saturation} is devoted to the study of infinitesimals
and their
connections with the notion of balance. Some examples are handled in Section \ref{sec:examples}.
\paragraph{\bf Acknowledgements}
We would like to thank M. I. Cortez and F. Dolce for stimulating discussions. We also thank warmly the referees of this paper for their careful reading and their
very useful comments, concerning in particular the formulation of Corollary \ref{coro:freq}.
\section{First definitions and background }\label{sec:def}
\subsection{Topological dynamical systems}
By a {\it topological dynamical system}, we mean a pair $(X,T)$
where $X$ is a compact metric space and $T: X \to X$ is a homeomorphism.
It is a {\it Cantor system} when $ X $ is a Cantor space,
that is, $ X $ has a countable basis of its topology which consists of closed and open sets (clopen sets) and does not have isolated points.
This system is {\em aperiodic} if it does not have periodic points, i.e., points $x$ such that $T^n(x) = x$ for some $n >0$.
It is {\em minimal} if it does not contain any non-empty proper closed $T$-invariant subset.
Any minimal Cantor system is aperiodic.
Two topological dynamical systems $(X_1,T_1)$, $(X_2,T_2)$ are {\em conjugate} when there is a {\em conjugacy} between them, i.e., a homeomorphism $\varphi:X_1 \to X_2$ such that $\varphi \circ T_1 = T_2 \circ \varphi$.
A complex number $\lambda$ is a {\em continuous eigenvalue} of
$(X,T)$ if there exists a non-zero continuous function $f \colon X \rightarrow {\mathbb C}$ such that
$f\circ T= \lambda f$.
An {\em additive eigenvalue} is a real number $\alpha$ such that
$\exp(2i \pi \alpha)$ is a continuous eigenvalue.
Let $E(X,T)$ be the (additive) group of additive eigenvalues of $(X,T)$.
We consider its rank over ${\mathbb Q}$, i.e., the maximal number of rationally independent elements of
$E(X,T)$.
Note that $1$ is always an additive eigenvalue and thus ${\mathbb Z} $ is included in $ E(X,T)$.
A probability measure $\mu$ on $X$ is said to be {\em $T$-invariant} if $\mu(T^{-1} A) = \mu(A)$ for every measurable subset $A$ of $ X$.
Let ${\mathcal M} (X,T)$ be the set of all $T$-invariant probability measures on $(X,T)$.
It is a convex set and any extremal point is called an {\em ergodic} $T$-invariant measure. It is well known that any topological dynamical system admits an ergodic invariant measure.
The set of ergodic $T$-invariant probability measures is denoted ${\mathcal M}_e (X,T)$.
Observe that if $(X,T)$ is a minimal Cantor system, then for all clopen $E$ and all $T$-invariant probability measures $\mu$, one has $\mu(E)>0$.
The topological dynamical system $(X,T)$ is {\em uniquely ergodic} if there exists a unique $T$-invariant probability measure on $X$.
It is said to be {\em strictly ergodic} if it is minimal and uniquely ergodic.
The notation $\chi_{E}$ stands for the characteristic function of $E$;
$\mathbb{N}$ stands for the set of non-negative integers ($0 \in \mathbb{N}$).
\subsection{Subshifts} \label{subsec:SD}
Let $\mathcal{A}$ be a finite alphabet of cardinality $d \geq 2$.
Let us denote by $\varepsilon$ the empty word of the free monoid $\mathcal{A}^*$ (endowed with concatenation), and by $\mathcal{A}^{\mathbb{Z}}$ the set of bi-infinite words over $\mathcal{A}$.
For a bi-infinite word $x \in \mathcal{A}^\mathbb{Z}$, and for $i,j \in \mathbb{Z}$ with $i \leq j$, the notation $x_{[i,j)}$ (resp., $x_{[i,j]}$) stands for $x_i \cdots x_{j-1}$ (resp., $x_i \cdots x_{j}$) with the convention $x_{[i,i)} = \varepsilon$.
For a word $w= w_{1} \cdots w_{\ell} \in \mathcal{A}^\ell$,
its {\em length} is denoted $|w|$ and equals $\ell$.
We say that a word $u$ is a {\em factor} of a word $w$ if there exist words $p,s$ such that $w = pus$.
If $p = \varepsilon$ (resp., $s = \varepsilon$) we say that $u$ is a {\em prefix} (resp., {\em suffix}) of $w$.
For a word $u \in \mathcal{A}^{*}$, an index $ 1 \le j \le \ell$ such that $w_{j}\cdots w_{j+|u|-1} =u$ is called an {\em occurence} of $u$ in $w$ and we use the same term for bi-infinite word in $\mathcal{A}^{\mathbb{Z}}$.
The number of occurrences of a word $u \in \mathcal{A}^*$ in a finite word $w$ is denoted as $|w|_u$.
The set $\mathcal{A}^{\mathbb{Z}}$ endowed with the product topology of the discrete topology on each copy of $\mathcal{A}$ is topologically a Cantor set.
The {\em shift map} $S$ defined by $S \left( (x_n)_{n \in \mathbb{Z}} \right) = (x_{n+1})_{n \in \mathbb{Z}}$ is a homeomorphism of $\mathcal{A}^{\mathbb{Z}}$.
A {\em subshift} is a pair $(X,S)$ where $X$ is a closed shift-invariant subset of some $\mathcal{A}^{\mathbb{Z}}$.
It is thus a {\em topological dynamical system}.
Observe that a minimal subshift is aperiodic whenever it is infinite.
The set of factors of a sequence $x \in \mathcal{A}^{\mathbb{Z}}$ is denoted $\mathcal{L}(x)$.
For a subshift $(X,S)$ its {\em language} $\mathcal{L}(X)$ is $\cup_{x\in X} \mathcal{L}(x)$. The {\em factor complexity} $p_X$ of the subshift $(X,S)$ is the function that with $n \in \mathbb{N}$ associates the number $p_X(n)$ of factors of length $n$ in $\mathcal{L}(X)$.
Let $w^-, w^+$ be two words. The cylinder $[w^-.w^+]$ is defined as the set
$\{ x \in X \mid x_{[-|w^-| , |w^+|)} = w^- w^+ \}$.
It is a clopen set.
When $w^-$ is the empty word $\varepsilon$, we set $[\varepsilon .w^+] = [w^+]$.
For $\mu$ a $S$-invariant probability measure, the measure of a factor $w \in \mathcal{L}(X)$ is defined as the measure of the cylinder $[w]$.
The notation $\boldsymbol{ \mu}$ stands for the vector $(\mu([a])_{ a \in {\mathcal A} } \in {\mathbb R}^{\mathcal A}$.
The {\em simplex of letter measures} is defined as the $d$-simplex consisting in all the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M} (X,S)$, i.e., it consists of all the convex combinations of the vectors $\boldsymbol{ \mu}$ with $\mu \in {\mathcal M}_e (X,S)$.
\subsection{Dimension groups and orbit equivalence}\label{subsec:dim}
Two minimal Cantor systems $(X_1,T_1)$ and $(X_2,T_2)$ are {\em orbit equivalent } if there exists a homeomorphism $\Phi \colon X_1 \rightarrow X_2$
mapping orbits onto orbits, i.e., for all $x \in X_1$, one has
$$\Phi ( \{ T_1^n x \mid n \in {\mathbb Z} \})= \{ T_2^n \Phi (x) \mid n \in {\mathbb Z}\}.$$
This implies that there exist two maps $n_1 \colon X_1 \rightarrow {\mathbb Z}$ and $n_2 \colon X_2 \rightarrow {\mathbb Z}$ (uniquely defined by aperiodicity) such that, for all $x \in X_1$,
$$\Phi \circ T_1 (x)= T_2^{n_1(x)} \circ \Phi(x) \quad \mbox { and } \quad \Phi \circ T_1^{n_2(x)} (x)= T_2 \circ \Phi (x).$$
The minimal Cantor systems $(X_1,T_1)$ and $(X_2,T_2)$ are {\em strongly orbit equivalent} if $n_1$ and $n_2$ both have at most one point of discontinuity.
For more on the subject, see e.g. \cite{GPS:95}.
There is a powerful and convenient way to characterize orbit and strong orbit equivalence in terms of ordered groups and dimension groups due to \cite{GPS:95}.
An {\em ordered group} is a pair $(G,G^+)$, where $G$ is a countable abelian group and $G^+$ is a subset of $G$, called the {\em positive cone}, satisfying
$$
G^+ + G^+ \subset G^+, \quad G^+ \cap (-G^+) = \{ 0 \}, \quad G^+ - G^+ = G.
$$
We write $a\leq b$ if $b-a \in G^+$, and $ a<b$ if $b-a \in G^+$ and $b \neq a$.
An \emph{order ideal}\index{order! ideal} $J$ of an ordered group $(G,G^+)$
is a subgroup $J$ of $G$ such that $J=J^+-J^+$ (with $J^+=J\cap G^+$)
and such that $0\le a\le b\in J$ implies $a\in J$.
An ordered group is \emph{simple}\index{simple ordered group}
\index{ordered group!simple} if it has no nonzero
proper order ideals.
An element $u $ in $G^+$ such that, for all $a $ in $G$, there exists some non-negative integer $n$ with $a \leq nu$ is called an {\em order unit} for $(G,G^+)$.
Two ordered groups with order unit $(G_1,G_1^+,u_1)$ and $(G_2,G_2^+,u_2)$ are {\em isomorphic} when there exists a group isomorphism
$\phi \colon G_1 \rightarrow G_2$ such that $ \phi(G_1^+)=G_2^+$ and $\phi (u_1)=u_2$.
We say that an ordered group is {\em unperforated} if for all $a\in G$, if $na\in G^+$ for some $n\in\mathbb{N}\setminus\{0\}$, then $a\in G^+$.
Observe that this implies in particular that $G$ has no torsion element.
A {\em dimension group} is an unperforated ordered group with order unit $(G,G^+,u)$ satisfying the {\em Riesz interpolation property}: given $a_1, a_2, b_1, b_2\in G$ with $a_i\leq b_j$ ($i,j=1,2$), there exists $c\in G$ with $a_i\leq c\leq b_j$.
Most examples of dimension groups we will deal with in this paper are of the following type:
$(G, G^+,u ) = ( {\mathbb Z} ^d, \, \{ {\bf x}\in {\mathbb Z} ^d \mid \theta_i ( {\bf x}) > 0, \, 1 \leq i \leq e \} \cup\{\mathbf{0}\},u ) $,
where the $\theta_i$'s are independent linear forms such that $\theta_i(u)=1$.
Let $(X,T)$ be a Cantor system.
Let $C(X, {\mathbb R})$ and $C(X, {\mathbb Z})$ respectively stand for the group of continuous functions from $X$ to ${\mathbb R}$ and ${\mathbb Z}$, and let $C(X, {\mathbb N})$ stand for the monoid of continuous functions from $X$ to $\mathbb{N}$, with the group and monoid operation being the addition.
Let
$$
\begin{array}{lrcl}
\beta \colon & C(X, {\mathbb Z}) & \rightarrow & C(X, {\mathbb Z}) \\
& f & \mapsto & f \circ T - f.
\end{array}
$$
A map $f$ is called a {\em coboundary} (resp., a {\em real coboundary}) if there exists a map $g$ in $C(X, {\mathbb Z})$ (resp. in $C(X, {\mathbb R})$) such that $f = g \circ T -g$.
Two maps $f,g \in C(X,\mathbb{Z})$ are said to be {\em cohomologous} whenever $f-g$ is a coboundary.
We consider the quotient group $H(X,T)=C(X, {\mathbb Z})/ \beta C(X,{\mathbb Z})$.
We denote $[f]$ the class of a function $f$ in $H$, and $\pi$ the natural projection $\pi \colon C(X, {\mathbb Z}) \rightarrow H(X,T)$.
We define $H^+ (X, T) = \pi (C(X, {\mathbb N}))$ as the set of classes of functions in $C(X, {\mathbb N})$.
The ordered group with order unit
$$
K^0(X,T) := (H(X,T), H^+ (X,T), [1]),
$$
where $1$ stands for the one constant valued function,
is a dimension group according to \cite{Putnam:89}, called the {\em dynamical dimension group } of $(X,T)$. We will use in this paper
the abbreviated terminology {\em dimension group of $(X,T)$}.
The next result shows that any simple dimension group can be realized as the dimension group of a minimal Cantor system.
\begin{theorem}
\cite[Theorem 5.4 and Corollary 6.3]{HermanPS:92}
\label{theo:simpleminimal}
Let $(G, G^+ , u)$ be a dimension group with order unit.
It is simple if and only if there exists a minimal Cantor system $(X,T)$ such that $(G, G^+ , u)$ is isomorphic to $K^0(X,T)$.
\end{theorem}
We also define the set of {\em infinitesimals} of $K^0 (X,T)$ as
$$
\mbox{Inf} (K^0 (X,T)) = \left \{ [f] \in H(X,T) : \int f d\mu = 0 \mbox { for all } \mu \in {\mathcal M} (X,T)\right\}.
$$
Note that $H(X,T)/\mbox{Inf}(K^0 (X,T))$ with the induced order also determines a dimension group \cite{GPS:95}. We denote it $K^{0} (X,T)/\mbox{\rm Inf} (K^0 (X,T))$.
The dimension groups $K^{0} (X,T)$ and $K^{0} (X,T)/\mbox{Inf} (K^0 (X,T))$ are complete invariants of strong orbit equivalence and orbit equivalence, respectively.
\begin{theorem}\cite{GPS:95}\label{oe}
Let $(X_1,T_1)$ and $(X_2,T_2)$ be two minimal Cantor systems.
The following are equivalent:
\begin{itemize}
\item $(X_1,T_1)$ and $(X_2,T_2)$ are strong orbit equivalent;
\item $K^{0} (X_1,T_1)$ and $K^0 (X_2,T_2)$ are isomorphic.
\end{itemize}
Similarly, the following are equivalent:
\begin{itemize}
\item $(X_1,T_1)$ and $(X_2,T_2)$ are orbit equivalent;
\item $K^{0} (X_1,T_1)/\mbox{\rm Inf} (K^0 (X_1,T_1))$ and $K^{0} (X_2,T_2)/\mbox{\rm Inf} (K^0 (X_2,T_2))$ are isomorphic.
\end{itemize}
\end{theorem}
\subsection{Image subgroup}
\label{subsec:cyl}
A {\em trace} (also called state) of a dimension group $(G,G^+,u)$ is a group homomorphism $p:G\to \mathbb{R}$ such that $p$ is non-negative ($p(G^+)\geq 0$) and $p(u)=1$.
The collection of all traces of $(G,G^+,u)$ is denoted by $\T(G,G^+,u)$.
It is known \cite{Effros} that $\T(G,G^+,u)$ completely determines the order on $G$, if the dimension group is simple.
In fact, one has
\[
G^+=\{a\in G: p(a)>0, \forall p\in \T(G,G^+,u)\}\cup \{0\}.
\]
For more on the subject, see e.g. \cite{Effros}.
Let $(X,T)$ be a minimal Cantor system.
Given $\mu \in {\mathcal M} (X,T)$, we define the {trace} $\tau_{\mu}$ on $K^0 (X,T)$ as $\tau_{\mu} ([f]):= \int f d\mu$.
It is shown in \cite{HermanPS:92} that the correspondence $\mu\mapsto \tau_\mu$ is an affine isomorphism from ${\mathcal M} (X,T)$ onto $\T(K^0(X,T))$.
Thus it sends the extremal points of ${\mathcal M} (X,T)$, i.e., the ergodic measures, to the extremal points of $\T(K^0(X,T))$,
called {\em pure traces}.
The {\em image subgroup} of $K^0 (X,T)$ is defined as the
ordered group with order unit
$$
(I(X,T), I(X,T) \cap {\mathbb R} ^+, 1),
$$
where
$$
I(X,T) = \bigcap_{ \mu \in {\mathcal M} (X,T)} \left\{ \int f d\mu \,: \, f \in C(X,{\mathbb Z})\right\}.
$$
Actually, $E(X,T)$ is a subgroup of $I(X,T)$ (see~\cite[Proposition 11]{CortDP:16} and also \cite[Corollary 3.7]{GHH:18}.
If $(X,T)$ is uniquely ergodic with unique $T$-invariant probability measure $\mu$, then $K^0 (X,T)/\mbox{Inf} (K^0 (X,T))$ is isomorphic to $(I(X,T), I(X,T) \cap {\mathbb R} ^+, 1)$, via the correspondence $$[f]+\mbox{Inf} (K^0 (X,T))\mapsto \int fd\mu.$$
Let us recall the following description of $I(X,T)$.
\begin{proposition}{ \cite[Corollary 2.6]{GHH:18}, \cite[Lemma 12]{CortDP:16}.}
\label{prop:GW}
Let $(X,T)$ be a minimal Cantor system.
Then
$$
I(X,T) = \left\{ \alpha : \exists g \in C(X,\mathbb{Z} ) , \alpha = \int g d \mu \ \forall \mu \in \mathcal{M} (X,T) \right\}.
$$
\end{proposition}
We give an other description of $I(X,T)$ using the following well known lemma.
\begin{lemma}{\cite[Lemma 2.4]{GW}}
\label{lemma:GW}
Let $(X,T)$ be a minimal Cantor system.
Let $f\in C (X, \mathbb{Z} )$ such that $\int_X f d \mu $ belongs to $]0,1[$ for every $\mu \in \mathcal{M} (X , T)$.
Then, there exists a clopen set $U$ such that $\int_X f d \mu = \mu (U)$ for every $\mu \in \mathcal{M} (X , T)$.
\end{lemma}
For a family of real
numbers $N = \{\alpha_{i} : i \in J\}$, we let $\langle N \rangle$ denote the abelian additive group generated by these real numbers.
\begin{proposition}\label{prop:cob}
Let $(X,T)$ be a minimal Cantor system.
Then,
\begin{equation}\label{eq:I}
I(X,T)
= \left\langle \{ \alpha : \exists \ \hbox{\rm clopen set } U \subset X , \ \alpha = \mu (U) \ \forall \mu \in \mathcal{M}(X,T)\}\right\rangle.
\end{equation}
\end{proposition}
\begin{proof}
There is just one inclusion to prove.
Let $\beta$ be in $I(X , T )$.
If $\beta$ is an integer then using that $\beta = \beta\mu (X)$ for all $\mu \in \mathcal{M} (X,T)$, it implies that $\beta$ belongs to the right member of the equality in \eqref{eq:I}.
Otherwise, let $n\in \mathbb{Z}$ be such that $\beta-n$ belongs to $]0,1[$.
From Proposition \ref{prop:GW} and Lemma \ref{lemma:GW} there exists a clopen set $U$ such that $\mu (U) = \beta-n$ for all $\mu \in \mathcal{M} (X,T)$.
It follows $\mu (U)$ and $n$ belong to
$\left\langle \{ \alpha : \exists \ \hbox{\rm clopen set } U \subset X , \ \alpha = \mu (U) \ \forall \mu \in \mathcal{M}(X,T)\}\right\rangle $. So it is also the case for $\beta$.
\end{proof}
One can obtain a more explicit description of the set $I (X,S)$ for minimal
subshifts.
\begin{proposition}\label{prop:cob}
Let $(X,S)$ be a minimal subshift.
Then,
$$\begin{array}{ll}
I(X,S)&=\bigcap_{\mu\in \mathcal{M}(X,S)} \left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle .
\end{array}$$
In particular, if $(X,S)$ is uniquely ergodic with $\mu$ its unique $S$-invariant probability measure, then
$I(X,S)=\left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle.$
\end{proposition}
\begin{proof}
The proof of the first equality is a direct consequence of the fact that every function belonging to $C(X,\mathbb{Z})$ is cohomologous to some cylinder function in $C(X,\mathbb{Z})$, i.e.,
to some function $h$ in $ C(X,\mathbb{Z})$ for which there exists $n>0$ such that for all $x\in X$, $h(x)$ depends only on $x_{[0,n)}$.
Indeed, let $f\in C(X,\mathbb{Z})$. Since $f$ is integer-valued, it is locally constant, and by compactness of $X$, there exists $k\in\mathbb{N}$ such that for all $x\in X$, $f(x)$ depends only on $x_{[-k,k]}$. Therefore, $g(x)=f\circ S^k(x)$ belongs to $C(X,\mathbb{Z})$ and depends only on $x_{[0,2k]}$ for all $x\in X$.
Finally, $f-g=f-f\circ S^k(x)= f- f\circ S+ f\circ S - f \circ S^2 + \cdots + f\circ S^{k-1}(x)+ f\circ S^{k}(x) $ is a coboundary.
Hence, $\int fd\mu=\int gd\mu$. Since $g$ is a cylinder function, $g$ can be written as a finite sum of the form
$g=\sum\ell_u\chi_{[u]},$ $u\in \mathcal{L}(X)$ and $\ell_u\in\mathbb{Z}$. Thus,
$\int f d\mu=\sum_{}\ell_u\mu([u])\in \left\langle \{\mu([w]): w\in\mathcal{L}(X)\}\right\rangle.$
\end{proof}
\section{Primitive unimodular proper $\mathcal{S}$-adic subshifts}\label{sec:Sadic}
In this section we first recall the notion of primitive unimodular proper $\mathcal{S}$-adic subshift in Section~\ref{subsec:sadic}.
We then illustrate it with the class of minimal dendric subshifts in Section \ref{subsec:tree}.
\subsection{$\mathcal{S}$-adic subshifts} \label{subsec:sadic}
Let $\mathcal{A},\, \mathcal{B}$ be finite alphabets and let $\tau:\, \mathcal{A}^*\to \mathcal{B}^*$ be a \emph{non-erasing} morphism (also called a \emph{substitution} if $\mathcal{A} = \mathcal{B}$). Let us note that a morphism is uniquely determined by its values on the alphabet ${\mathcal A}$
and this will be the way we will define them (see e.g. Example \ref{ex:fibo}). By non-erasing, we mean that the image of any letter is a non-empty word. We stress the fact that all morphisms are assumed to be non-erasing in the following.
Using concatenation, we extend $\sigma$ to~$\mathcal{A}^\mathbb{N}$ and~$\mathcal{A}^\mathbb{Z}$.
With a morphism $\tau :\mathcal{A}^* \to \mathcal{B}^*$, where $\mathcal{A}$ and $\mathcal{B}$ are finite alphabets, we classically associate an {\em incidence matrix} $M_\tau$ indexed by $\mathcal{B} \times \mathcal{A}$ such that for every $(b,a)\in \mathcal{B} \times \mathcal{A}$, its entry at position $(b,a)$ is the number of occurrences of $b$ in $\tau(a)$.
Alphabets are always assumed to have cardinality at least 2.
The morphism $\tau$ is said to be {\em left proper} (resp. {\em right proper}) when there exist a letter $b \in \mathcal{B}$ such that for all $a \in \mathcal{A}$, $\tau(a)$ starts with $b$ (resp., ends with $b$).
It is said to be {\em proper} if it is both left and right proper.
We recall the definition of an $\mathcal{S}$-adic subshift as stated in \cite{BSTY}, see also \cite{Berthe&Delecroix:14} for more on $\mathcal{S}$-adic subshifts.
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$ be a sequence of morphisms such that
$
\max_{a \in \mathcal{A}_n} |\tau_1 \circ \cdots \circ \tau_{n-1}(a)|
$
goes to infinity when $n$ increases. By non-erasing, we mean that the image of any letter is a non-empty word.
For $1\leq n<N$, we define $\tau_{[n,N)} = \tau_n \circ \tau_{n+1} \circ \dots \circ \tau_{N-1}$ and $\tau_{[n,N]} = \tau_n \circ \tau_{n+1} \circ \dots \circ \tau_{N}$.
For $n\geq 1$, the \emph{language $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$ of level $n$ associated with $\boldsymbol{\tau}$} is defined~by
\[
\mathcal{L}^{(n)}({\boldsymbol{\tau}}) = \big\{w \in \mathcal{A}_n^* \mid \mbox{$w$ occurs in $\tau_{[n,N)}(a)$ for some $a \in\mathcal{A}_N$ and $N>n$}\big\}.
\]
As $\max_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases, $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$ defines a non-empty subshift $X_{\boldsymbol{\tau}}^{(n)}$ that we call the {\em subshift generated by $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$}.
More precisely, $X_{\boldsymbol{\tau}}^{(n)}$ is the set of points $x \in \mathcal{A}_n^\mathbb{Z}$ such that $\mathcal{L} (x) \subseteq \mathcal{L}^{(n)}({\boldsymbol{\tau}})$. Note that it may happen that $\mathcal{L}(X_{\boldsymbol{\tau}}^{(n)})$ is strictly contained in $\mathcal{L}^{(n)}({\boldsymbol{\tau}})$.
We set $\mathcal{L}({\boldsymbol{\tau}}) = \mathcal{L}^{(1)}({\boldsymbol{\tau}})$,$X_{\boldsymbol{\tau}} = X_{\boldsymbol{\tau}}^{(1)}$ and call $(X_{\boldsymbol{\tau}},S)$ the \emph{$\mathcal{S}$-adic subshift} generated by the \emph{directive sequence}~$\boldsymbol{\tau}$.
We say that $\boldsymbol{\tau}$ is {\em primitive} if, for any $n\geq 1$, there exists $N>n$ such that
$M_{\tau_{[n,N)}} >0$, {\em i.e.}, for all $a \in \mathcal{A}_N$, $\tau_{[n,N)}(a)$ contains occurrences of all letters of $\mathcal{A}_{n}$.
Of course, $M_{\tau_{[n,N)}}$ is equal to $M_{\tau_n} M_{ \tau_{n+1}}\cdots M_{\tau_{N-1}}$.
Observe that if $\boldsymbol{\tau}$ is primitive, then $\min_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
In the primitive case $\mathcal{L}(X_{\boldsymbol{\tau}}^{(n)})= \mathcal{L}^{(n)}({\boldsymbol{\tau}})$, and $X_{\boldsymbol{\tau}}^{(n)}$ is a minimal subshift (see for instance ~\cite[Lemma 7]{Durand:2000}).
We say that $\boldsymbol{\tau}$ is ({\em left, right}) {\em proper} whenever each morphism $\tau_n$ is (left, right) proper.
We also say that $\boldsymbol{\tau}$ is {\em unimodular} whenever, for all $n \geq 1$, $\mathcal{A}_{n+1} = \mathcal{A}_n$ and the matrix $M_{\tau_n}$ has determinant of absolute value 1.
By abuse of language, we say that a subshift is a (unimodular, left or right proper, primitive) $\mathcal{S}$-adic subshift if there exists a (unimodular, left or right proper, primitive) sequence of morphisms $\boldsymbol{\tau}$ such that $X = X_{\boldsymbol{\tau}}$.
Let us give another way to define $X_{\boldsymbol{\tau}}$ when $\boldsymbol{\tau}$ is primitive and proper.
For an endomorphism $\tau$ of $\mathcal{A}^*$, let $\Omega (\tau ) = \overline{\bigcup_{k\in \mathbb{Z}} S^k \tau ( \mathcal{A}^\mathbb{Z} )}$.
\begin{lemma}
\label{lemma:omega}
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$ be a sequence of morphisms such that $\min_{a \in \mathcal{A}_n} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Then,
$$
X_{\boldsymbol{\tau}} \subset \bigcap_{n\in \mathbb{N}} \Omega (\tau_{[1,n]} ).
$$
Furthermore, when $\boldsymbol{\tau}$ is primitive and proper, then the equality $
X_{\boldsymbol{\tau}} = \bigcap_{n\in \mathbb{N}} \Omega (\tau_{[1,n]} )
$ holds.
\end{lemma}
\begin{proof}
The proof is left to the reader.
\end{proof}
With a left proper morphism $\sigma:\mathcal{A}^* \to \mathcal{B}^*$ such that $b \in \mathcal{B}$ is the first letter of all images $\sigma(a)$, $a \in \mathcal{A}$, we associate the right proper morphism $\overline{\sigma}:\mathcal{A}^* \to \mathcal{B}^*$ defined by
$b\overline{\sigma}(a) = \sigma(a)b$ for all $a \in \mathcal{A}$.
For all $x \in A^\mathbb{Z}$, we thus have $\bar \sigma(x) = S \sigma(x)$.
The next result is a weaker version of~\cite[Corollary 2.3]{Durand&Leroy:2012}.
\begin{lemma}
\label{lemma:proper}
Let $(X,S)$ be an $\mathcal{S}$-adic subshift generated by the
primitive and left proper directive sequence $\boldsymbol{\tau} = (\tau_n:\mathcal{A}_{n+1}^* \to \mathcal{A}_n^*)_{n \geq 1}$.
Then $(X,S)$ is also generated by
the primitive and proper directive sequence $\tilde{\boldsymbol{\tau}} = (\tilde{\tau}_n)_{n \geq 1}$, where for all $n$, $\tilde{\tau}_n = \tau_{2n-1} \overline{\tau}_{2n}$.
In particular, if $\boldsymbol{\tau}$ is unimodular, then so is $\tilde{\boldsymbol{\tau}}$.
\end{lemma}
\begin{proof}
Each morphism $\tilde{\tau}_n$ is trivially proper.
It is also clear that the unimodularity and the primitiveness of $\boldsymbol{\tau}$ are preserved in this process.
Using the relation $\bar \sigma(x) = S \sigma(x)$ and Lemma~\ref{lemma:omega}, we then get
\[
X_{\boldsymbol{\tau}}
\subset \bigcap_{n \in \mathbb{N}} \Omega (\tau_{[1,n]})
= \bigcap_{n \in \mathbb{N}} \Omega (\tilde{\tau}_{[1,n]})
= X_{\tilde{\boldsymbol{\tau}}}.
\]
Since both $\boldsymbol{\tau}$ and $\tilde{\boldsymbol{\tau}}$ are primitive, the subshifts $X_{\boldsymbol{\tau}}$ and $X_{\tilde{\boldsymbol{\tau}}}$ are minimal, hence they are equal.
\end{proof}
\begin{lemma}
\label{lemma:aperiodicity}
All primitive unimodular proper $\mathcal{S}$-adic subshifts are aperiodic.
\end{lemma}
\begin{proof}
Let $\boldsymbol{\tau}$ be a primitive unimodular proper directive sequence on the alphabet $\mathcal{A}$ of cardinality $d \geq 2$.
Suppose that it has a periodic point $x$ of period $p$, where $p$ is the smallest period of $x$ ($p>0$).
By primitiveness, all letters of $\mathcal{A}$ occur in $x$, so $p \geq d$.
We have $x = \cdots uu.uu\cdots $ for some word $u$ with $|u| = p$.
There exists some $n$ such that, for all $a $, one has $\tau_{[1,n]} (a) = s(a) u^{q(a)}p(a)$, where $s(a), p(a)$ are a strict suffix and a strict prefix of $u$ and $q(a) >1$.
Let $b\in \mathcal{A}$ and set $\tau_{n+1} (b) = b_0 b_1 \cdots b_k$.
As the directive sequence $\boldsymbol{\tau}$ is proper, $b_0 b_1 \cdots b_k b_0$ is also a word in $\mathcal{L}^{(n+1)} (\boldsymbol{\tau})$.
By a classical argument due to Fine and Wilf \cite{Fine:65}, one has $p(b_0)s(b_1) = p(b_i)s(b_{i+1})= p(b_k)s(b_0) = u$ for $1\leq i\leq k-1$.
Hence
$$
|\tau_1 \cdots \tau_{n+1} (b) |
\equiv
|s(c)p(c)s(b_1)p(b_1) \cdots s(b_k)p(b_k)|
\equiv
0 \mbox{ modulo } |u|,
$$
which contradicts the unimodularity of $\boldsymbol{\tau}$.
\end{proof}
The next two results will be important for the computation of the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifs.
The first one is a weaker version of~\cite[Theorem 3.1]{BSTY}.
\begin{theorem}[\cite{BSTY}]
\label{theo:BSTY}
Let $\tau :\mathcal{A}^* \to \mathcal{B}^*$ be such that its incidence matrix $M_\tau$ is unimodular.
Then, for any aperiodic $y \in \mathcal{B}^\mathbb{Z}$, there exists at most one $(k,x) \in \mathbb{N} \times A^\mathbb{Z}$ such that $y = S^k \tau(x)$, with $0 \leq k < |\tau(x_0)|$.
\end{theorem}
\begin{proposition}
\label{prop:minimalSSIprimitif}
Let $\boldsymbol{\tau} = (\tau_n : \mathcal{A}^* \to \mathcal{A}^*)_{n \geq 1}$ be a unimodular proper sequence of morphisms such that $\max_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Then $(X_{\boldsymbol{\tau}},S)$ is aperiodic and minimal if and only if $\boldsymbol{\tau}$ is primitive.
\end{proposition}
\begin{proof}
Recall that any $\mathcal{S}$-adic subshift with a primitive directive sequence is minimal (see, e.g.~\cite[Lemma 7]{Durand:2000}) and that aperiodicity is proved in Lemma \ref{lemma:aperiodicity}.
We only have to show that the condition is necessary.
We assume that $(X_{\boldsymbol{\tau}},S)$ is aperiodic and minimal.
For all $n \geq 1$, $(X_{\boldsymbol{\tau}}^{(n)},S)$ is trivially aperiodic.
Let us show that it is minimal.
Assume by contradiction that for some $n \geq 1$, $(X_{\boldsymbol{\tau}}^{(n)},S)$ is minimal, but not $(X_{\boldsymbol{\tau}}^{(n+1)},S)$.
There exist $u \in \mathcal{L}(X_{\boldsymbol{\tau}}^{(n+1)})$ and $x \in X_{\boldsymbol{\tau}}^{(n+1)}$ such that $u$ does not occur in $x$.
By Theorem~\ref{theo:BSTY}, $\{\tau_n([v]) \mid v \in \mathcal{L}(X_{\boldsymbol{\tau}}^{(n+1)}) \cap \mathcal{A}^{|u|}\}$ is a finite clopen partition of
$\tau_n(X_{\boldsymbol{\tau}}^{(n+1)})$.
Thus, considering $y = \tau_n(x)$, by minimality of $(X_{\boldsymbol{\tau}}^{(n)},S)$, there exists $k \geq 0$ such that $S^k y$ is in $\tau_n([u])$.
Take $z \in [u]$ such that $S^ky = \tau_n(z)$.
Since $y$ is aperiodic and since we also have $S^k y = S^{k'} \tau_n(S^\ell x)$ for some $\ell \in \mathbb{N}$ and $0 \leq k' < |\tau_n(x_\ell)|$,
we obtain that $\tau_n(z) = S^{k'} \tau_n(S^\ell x)$ with $z \in [u]$, $S^\ell x \notin [u]$ and $0 \leq k' < |\tau_n(x_\ell)|$; this contradicts Theorem~\ref{theo:BSTY}.
We now show that $\lim_{n \to +\infty} \min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)| = +\infty$.
We again proceed by contradiction, assuming that $(\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|)_{n \geq 1}$ is bounded.
Then there exists $N > 0$ and a sequence $(a_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that for all $n \geq N$, $\tau_n(a_{n+1}) = a_n$.
We claim that there are arbitrary long words of the form $a_N^k$ in $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$ which contradicts the fact that $(X_{\boldsymbol{\tau}}^{(N)},S)$ is minimal and aperiodic.
Since $\boldsymbol{\tau}$ is proper, for all $n \geq N$ and all $b \in \mathcal{A}$, $\tau_n(b)$ starts and ends with $a_n$.
As $\max_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity, there exists a sequence $(b_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that $|\tau_{[N,n)}(b_n)|$ goes to infinity and for all $n \geq N$, $b_n$ occurs in $\tau_n(b_{n+1})$.
This implies that there exists $M \geq N$ such that for all $n \geq M$, $b_n \neq a_n$ and, consequently, that $\tau_n(b_{n+1}) = a_n u_n$ for some word $u_n$ containing $b_n$.
It is then easily seen that, for all $k \geq 1$, $a_M^k$ is a prefix of $\tau_{[M,M+k)}(b_{M+k})$, which proves the claim.
We finally show that $\boldsymbol{\tau}$ is primitive.
If not, there exist $N \geq 1$ and a sequence $(a_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that for all $n > N$, $a_N$ does not occur in $\tau_{[N,n)}(a_n)$.
As $(|\tau_{[N,n)}(a_n)|)_n$ goes to infinity, this shows that there are arbitrarily long words in $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$ in which $a_N$ does not occur.
Since $\boldsymbol{\tau}$ is unimodular, there is also a sequence $(a'_n)_{n \geq N}$ of letters in $\mathcal{A}$ such that $a_N = a_N'$ and for all $n \geq N$, $a_n'$ occurs in $\tau_n(a_{n+1}')$.
Again using the fact that $|\tau_{[N,n)}(a'_n)|$ goes to infinity, this shows that $a_N$ belongs to $\mathcal{L}(X_{\boldsymbol{\tau}}^{(N)})$.
We conclude that $(X_{\boldsymbol{\tau}}^{(N)},S)$ is not minimal, a contradiction.
\end{proof}
\subsection{Dendric subshifts} \label{subsec:tree}
We now describe a subclass of the family of primitive unimodular proper $\mathcal{S}$-adic subshifts, namely
the class of dendric subshifts, that encompasses Sturmian subshifts, Arnoux-Rauzy subshifts (see Section \ref{subsec:AR}), as well as subshifts generated by interval exchanges (see~\cite{BDFDLPRR:15}). The ternary words generated by the Cassaigne-Selmer multidimensional continued fraction algorithm also provide dendric subshifts~\cite{ArnouxLa:2018,CLL:17}.
Dendric subshifts are defined with respect to combinatorial properties of their language expressed in terms of extension graphs.
We recall the notion of dendric words and subshifts, studied in \cite{BDFDLPRR:15,BDFDLPRR:15bis,BFFLPR:2015,BDFDLPR:16,Rigidity}.
Let $(X,S)$ be a minimal subshift defined on the alphabet $\mathcal{A}$.
For $w \in {\mathcal L}_X$, let
$$
\begin{array}{lcl}
L(w) = \{ a \in {\mathcal A} \mid aw \in {\mathcal L}_X\}, & & \ell(w) = \Card(L(w)), \\
R(w) = \{ a \in {\mathcal A} \mid wa \in {\mathcal L}_X\}, & & r(w) = \Card(R(w)).
\end{array}
$$
A word $w \in \mathcal{L}_X$ is said to be {\em right special} (resp. {\em left special}) if $r(w)\geq 2$ (resp. $\ell(w)\geq 2$). It is {\em bispecial} if it is both left and right special.
For a word $w \in \mathcal{L}(X)$, we consider the undirected bipartite graph $\E(w)$ called its \emph{extension graph} with respect to $X$ and defined as follows:
its set of vertices is the disjoint union of $L(w)$ and $R(w)$ and its edges are the pairs $(a,b) \in L(w) \times R(w)$ such that $awb \in \mathcal{L}(X)$.
For an illustration, see Example~\ref{ex:fibo} below.
We then say that a subshift $X$ is a \emph{dendric subshift} if, for every word $w \in {\mathcal L}(X)$, the graph $\E(w)$ is a tree.
Note that the extension graph associated with every non-bispecial word is trivially a tree.
We will consider here only minimal dendric subshifts.
The factor complexity of a dendric subshift over a $d$-letter alphabet is $(d-1)n + 1$ (see~\cite{BDFDLPR:16}),
and on a two-letter alphabet, the minimal dendric subshifts are the Sturmian subshifts.
Thus minimal dendric subshift are aperiodic when $d$ is greater or equal to $2$.
\begin{example}\label{ex:fibo}
\rm
Let $\sigma$ be the Fibonacci substitution defined over the alphabet $\{a,b\}$ by $\sigma \colon a \mapsto ab, b \mapsto a$ and consider the subshift generated by $\sigma$ (i.e., the set of bi-infinite words over $\mathcal{A}$ whose factors
belong to some $\sigma^n (a)$).
The extension graphs of the empty word and of the two letters $a$ and $b$ are represented in Figure~\ref{fig:fibo-ext}.
\begin{figure}
\caption{The extension graphs of $\varepsilon$ (on the left), $a$ (on the center) and $b$ (on the right) are trees.}
\label{fig:fibo-ext}
\end{figure}
\end{example}
The following theorem states a structural property
for return words of minimal dendric subshifts, from which a description as primitive unimodular proper $\mathcal{S}$-adic subshifts can be deduced (Proposition~\ref{prop:DendricareSadic} below).
Let $(X,S)$ be a minimal subshift over the alphabet $\mathcal{A}$ and let $w \in \mathcal{L}(X)$.
A {\em return word} to $w$ is a word $v$ in $\mathcal{L}(X)$ such that $w$ is a prefix of $vw$ and $vw$ contains exactly two occurrences of $w$.
We recall below a corollary of \cite[Theorem 4.5]{BDFDLPR:16}.
\begin{theorem}\label{theo:return}
Let $(X,S)$ be a minimal dendric subshift defined on the alphabet $\mathcal{A}$. Then, for any $w \in \mathcal{L}(X)$, the set of return words to $w$ is a basis of the free group on $\mathcal{A}$.
\end{theorem}
In particular, dendric subshifts have bounded topological rank.
The next result shows that minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
Similar results are proved with the same method in~\cite{BFFLPR:2015,Rigidity,BSTY} but not highlighting all the properties stated below, so we provide a proof for the sake of self-containedness.
It relies on $\mathcal{S}$-adic representations built from return words \cite{Durand:2000,Durand:03} together with the remarkable property of return words of dendric subshifts stated in Theorem \ref{theo:return}.
We also provide in
Section \ref{subsection:vs} an example of a primitive unimodular proper subshift which is not dendric and whose strong orbit equivalence class contains no dendric subshift.
\begin{proposition}\label{prop:DendricareSadic}
Minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{proposition}
\begin{proof}
Let $(X,S)$ be a minimal dendric subshift over the alphabet ${\mathcal A}= \{1,2, \dots,d\}$ and take any $x\in X$. For every $n\geq 1$, let $V_n(x):=\{v_{1,n},\cdots,v_{d,n}\}$ be the set of return words to $x_{[0,n)}$ and $V_0 (x) = {\mathcal A}$.
We stress the fact that $V_n(x)$ has cardinality $d$ for all $n$, according to Theorem~\ref{theo:return}.
Let $(n_i)_{i \geq 1}$ be a strictly increasing integer sequence such that $n_1 = 1$ and such that each $v_{j,n_i}x_{[0,n_i)}$ occurs in $x_{[0,n_{i+1})}$ and in each $v_{k,n_{i+1}}$.
Let $\theta_i $ be an endomorphism of ${\mathcal A}^*$ such that $\theta_i ({\mathcal A}) = V_{n_i} (x)$.
Since $x_{[0,n_i)}$ is a prefix of $x_{[0,n_{i+1})}$, any $v_{j,n_{i+1}}\in V_{n_{i+1}}(x)$ has a unique decomposition as a concatenation of elements $v_{k,n_i}\in V_{n_i}(x)$.
More precisely, for any $v_{j,n_{i+1}}\in V_{n_{i+1}}(x)$, there is a unique sequence $(v_{k_j(1),n_i},v_{k_j(2),n_i},\dots,v_{k_j(\ell_j),n_i})$ of elements of $V_{n_i}(x)$ such that $v_{k_j(1),n_i} \cdots v_{k_j(\ell_j),n_i} = v_{j,n_{i+1}}$ and for all $m \in \{1,\dots,\ell_j\}$, $v_{k_j(1),n_i} \cdots v_{k_j(m),n_i} x_{[0,n_i)}$ is a prefix of $v_{j,n_{i+1}} x_{[0,n_{i+1})}$.
This induces a unique endomorphism $\lambda_i$ of ${\mathcal A}^*$ defined by $\theta_{i+1} = \theta_i \circ \lambda_i$.
From the choice of the sequence $(n_i)_{i\geq 1}$, the matrices $M_{\lambda_i}$ have positive coefficients, so the sequence of morphisms $(\lambda_i)_{i \geq 1}$ is primitive.
Furthermore, as $x_{[0,n_{i+1}]}$ is prefix of each $v_{j,n_{i+1}}x_{[0,n_{i+1)}}$, there exists some $v \in V_{n_i}(x)$ such that $v_{k_j(1)}=v$ for all $j$.
In other words, the morphisms $\lambda_i$ are left proper.
Finally, from Theorem~\ref{theo:return}, the matrices $M_{\lambda_i}$ are unimodular.
Hence $(X,S)$ is $\mathcal{S}$-adic generated by the primitive directive sequence of unimodular left proper endomorphisms $\boldsymbol{\lambda} = (\lambda_i)_{i\geq 1}$.
We deduce from Lemma~\ref{lemma:proper} that minimal dendric subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{proof}
Observe that using Lemma~\ref{lemma:aperiodicity} we recover that minimal dendric subshifts on at least two letters are aperiodic.
\section{Dimension groups of primitive unimodular proper $\mathcal{S}$-adic subshifts} \label{sec:proofs2}
In this section we first prove a key result of this paper, namely Theorem \ref{theo:cohoword}, which states that $H(X,T)= C (X, \mathbb{Z})/ \beta C (X, \mathbb{Z})$ is generated as an additive group by the classes of the characteristic functions of letter cylinders.
We then deduce a simple expression for the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifts.
\subsection{From letters to factors}
We recall that $\chi_U$ stands for the characteristic function of the set $U$.
\begin{theorem}
\label{theo:cohoword}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift.
Any function $f\in C (X, \mathbb{Z})$ is cohomologuous to some integer linear combination of the form $\sum_{a\in \mathcal{A}} \alpha_a \chi_{[a]}\in C (X,\mathbb{Z})$. Moreover, the classes $[\chi_{[a]}]$, $a\in \mathcal{A}$, are $\mathbb{Q}$-independent.
\end{theorem}
\begin{proof}
Let $\boldsymbol{\tau} = (\tau_n:\mathcal{A}^* \to \mathcal{A}^*)_{n\geq 1}$ be a primitive unimodular proper directive sequence of $(X,S)$, hence $X = X_{\boldsymbol{\tau}}$.
Using Proposition~\ref{prop:minimalSSIprimitif}, all subshifts $(X_{\boldsymbol{\tau}}^{(n)},S)$ are minimal and aperiodic and $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity when $n$ increases.
Let us show that the group $H(X,S )=C(X, {\mathbb Z})/ \beta C(X,{\mathbb Z}) $ is spanned by the set of classes of characteristic functions of letter
cylinders
$\{ [\chi_{[a]}] \mid a \in \mathcal{A} \}$.
From Theorem~\ref{theo:BSTY} and using the fact that $(X,S)$ is minimal and aperiodic, one has, for all positive integer $n$, that
$$
\mathcal{P}_n = \{ S^k \tau_{[1,n]} ([a]) \mid 0\leq k < |\tau_{[1,n]} (a)| , a \in \mathcal{A} \}
$$
is a finite partition of $X$ into clopen sets. This provides a family of nested Kakutani-Rohlin tower partitions.
The morphisms of the directive sequence $\boldsymbol{\tau}$ being proper, for all $n$, there are letters $a_n, b_n$ such that all images $\tau_n (c)$, $c \in \mathcal{A}$, start with $a_n$ and end with $b_n$.
From this, it is classical to check that $(\mathcal{P}_n)_n$ generates the topology of $X$ (the proof is the same as~\cite[Proposition 14]{DHS:99} that is concerned with the particular case $\tau_{n+1}=\tau_n$ for all $n$).
We first claim that $H(X,S ) $ is spanned by the set of classes $\cup_n \Omega_n $, where
$$\Omega_n = \{ [\chi_{\tau_{[1,n]} ([a])}] \mid a \in \mathcal{A} \} \quad n \geq 1.$$
In other words, $H(X,S)$ is spanned by the set of classes of characteristic functions of bases of the sequence of partitions $({\mathcal P})_n$.
It suffices to check that, for all $u^- u^+\in \mathcal{L} (X)$, the class $[\chi_{[u^-.u^+]}]$ is a linear integer combination of elements belonging
to some $\Omega_n$.
Let us check this assertion. Let $u^-u^+$ belong to $\mathcal{L} (X)$.
Since $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity, there exists $n$ such that $|u^-|,|u^+| < \min_{a\in \mathcal{A}} |\tau_{[1,n)} (a) |$.
The directive sequence $ \boldsymbol{\tau}$ being proper, there exist words $w,w'$ with respective lengths $|w|=|u^-|$ and $|w'|=|u^+|$
such that all images $\tau_{[1,n]} (a)$ start with $w$ and end with $w'$.
Let $x \in [u^-. u^+]$.
Let $a \in \mathcal{A}$ and $k \in \mathbb{N}$, $0\leq k < |\tau_{[1,n]} (a)|$, such that $x$ belongs to the atom $S^k \tau_{[1,n]} ([a])$.
Observing that $\tau_{[1,n]} ([a])$ is included in $[w'.\tau_{[1,n]} (a)w]$, this implies that the full atom $S^k \tau_{[1,n]} ([a])$ is included in $[u^-. u^+]$.
Consequently $[u^-. u^+]$ is a finite union of atoms in $\mathcal{P}_n$.
But each characteristic function of an atom of the form $ S^k \tau_{[1,n]} ([a]) $ is cohomologous to $\chi_{\tau_{[1,n]} ([a])}$. The proof works as in the proof of Proposition \ref{prop:cob}. This thus proves the claim.
Now we claim that each element of $\Omega_n $ is a linear integer combination of elements in $\{ [\chi_{[a]}] \mid a \in \mathcal{A} \}$.
More precisely, let us show that $\chi_{\tau_{[1,n]}([b])}$ is cohomologous to
$$
\sum_{a\in \mathcal{A}} (M_{\tau_{[1,n]}}^{-1})_{b,a}\chi_{[a]} .
$$
Let $a\in \mathcal{A}$ and $n\geq 1$.
One has $[a] = \cup_{B \in \mathcal{P}_n } (B \cap [a]) $ and thus $\chi_{[a]} $ is cohomologous to the map
$$
\sum_{b\in \mathcal{A}} (M_{\tau_{[1,n]}})_{a,b}\chi_{\tau_{[1,n]}([b])} ,
$$
by using the fact that the maps $ \chi_{S^k \tau_{[1,n]} ([a]) }$ are cohomologous to $\chi_{\tau_{[1,n]} ([a])}$.
This means that for $U = ([\chi_{[a]} ] ) _{a\in \mathcal{A}} \in H (X,S)^\mathcal{A}$ and $V = ([\chi_{\tau_{[1,n]}([a])}])_{a\in \mathcal{A}} \in H (X,S)^\mathcal{A}$, one has
$$
U = M_{\tau_{[1,n]}}V
$$
and as a consequence $V = M_{\tau_{[1,n]}}^{-1}U$.
This proves the claim and the first part of the theorem.
To show the independence, suppose that there exists some row vector $\alpha = (\alpha_a )_{a\in \mathcal{A}} \in \mathbb{Z}^\mathcal{A}$ such that $\sum_a \alpha_a [\chi_{[a]}] = 0$.
Hence there is some $f\in C (X, \mathbb{Z} )$ such that $\sum_a \alpha_a \chi_{[a]} = f\circ S - f$.
We now fix some $n$ for which $f$ is constant on each atom of $\mathcal{P}_n$.
Observe that for all $x\in X$ and all $k \in \mathbb{N}$, one has $f (S^k x ) - f(x) = \sum_{j=0}^{k-1} \alpha_{x_j}$.
Let $c\in \mathcal{A}$ and $x\in \tau_{[1,n+1]} ([c])$.
Then, $x$ and $S^{ |\tau_{[1,n+1]} (c)|} (x)$ belong to $\tau_{[1,n]} ([a_{n+1}])$.
Hence, $f (S^{ |\tau_{[1,n+1]} (c)|} x ) - f(x) = 0 $, and thus
$$
(\alpha M_{\tau_{[1,n]}})_c = \sum_{j=0}^{|\tau_{[1,n+1]} (c)|-1} \alpha_{x_j} = 0 .
$$
This holds for all $c$, hence $\alpha M_{\tau_{[1,n]}} = 0$, which yields $\alpha = 0$, by invertibility of the matrix $ M_{\tau_{[1,n]}}$.
\end{proof}
Observe that in the previous result, we can relax the assumption of minimality. Indeed,
one checks that the same proof works if we assume that $(X,S)$ is aperiodic (recognizability then holds by \cite{BSTY})
and that $\min_{a \in \mathcal{A}} |\tau_{[1,n)}(a)|$ goes to infinity.
We now derive two corollaries from Theorem~\ref{theo:cohoword} dealing respectively with invariant measures and with the image subgroup.
Note that Corollary \ref{coro:measures} extends a statement initially proved for interval exchanges \cite{FerZam:08}.
See also \cite{BHL:2019} for a similar result in the framework of automorphisms of the free group and \cite{BHL:2020} for subshifts with finite rank.
\begin{corollary}\label{coro:measures}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over the alphabet $\mathcal{A}$ and let $\mu, \mu' \in \mathcal M (X,S)$.
If $\mu$ and $\mu'$ coincide on the letters, then they are equal,
that is, if $\mu([a]) = \mu'([a])$ for all a in $\mathcal{A}$, then $\mu(U) = \mu'(U)$, for any clopen subset $U$ of $X$.
\end{corollary}
\begin{corollary}\label{coro:freq}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over the alphabet ${\mathcal A}$.
The image subgroup of $(X,S)$ satisfies
\begin{align*}
I(X,S) & = \bigcap_{\mu\in\mathcal{M}(X,S)} \langle {\mu([a]) : a\in A} \rangle \\
& = \left\lbrace \alpha : \exists (\alpha_a)_{a\in \mathcal{A}} \in \mathbb{Z}^\mathcal{A} , \alpha
= \sum_{a\in\mathcal{A}} \alpha_a \mu([a]) \ \forall \mu \in \mathcal{M} (X,S) \right\rbrace .
\end{align*}
\end{corollary}
The proof of Corollary \ref{coro:freq} uses the two descriptions of the image subgroup given in Proposition \ref{prop:GW} and Proposition \ref{prop:cob}.
In both corollaries, the assumption of being proper can be dropped. The proof then uses the measure-theoretical Bratteli-Vershik representation of the primitive unimodular $\mathcal{S}$-adic subshift given in \cite[Theorem 6.5]{BSTY}.
\subsection{An explicit description of the dimension group}
Theorem~\ref{theo:cohoword} allows a precise description of the dimension group of primitive unimodular proper $\mathcal{S}$-adic subshifts.
Note that in the case of interval exchanges, one recovers the results obtained in \cite{Putnam:89}; see also \cite{Putnam:92,GjerdeJo:02}.
We first need the following Gottschalk-Hedlund type statement \cite{GotHed:55}.
\begin{lemma}[\cite{Host:95}, Lemma 2, \cite{DHP}, Theorem 4.2.3]
\label{lemma:positivecohomologous}
Let $(X,T)$ be a minimal Cantor system and let $f \in C(X,\mathbb{Z})$.
There exists $g \in C(X,\mathbb{N})$ that is cohomologous to $f$ if and only if for every $x \in X$, the sequence $\left( \sum_{k=0}^n f \circ T^k(x)\right)_{n \geq 0}$ is bounded from below.
\end{lemma}
We now can deduce one of our main statements.
\begin{theorem}\label{theo:dg}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift over a $d$-letter alphabet.
The linear map $\Phi : H (X,S) \to \mathbb{Z}^d$ defined by $\Phi ( [\chi_{[a]}] ) = e_a$, where $\{ e_a \mid a \in \mathcal{A} \}$ is the canonical base of $ \mathbb{Z}^d$,
defines an isomorphism of dimension groups from $K^{0}(X,S)$ onto
\begin{align}
\label{align:ZdGD}
\left( {\mathbb Z} ^d, \, \{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}\cup \{{\mathbf 0}\},\, \bf{1}\right ),
\end{align}
where the entries of $\bf{1}$ are equal to $1$.
\end{theorem}
\begin{proof}
From Theorem \ref{theo:cohoword}, $\Phi$ is well defined and is a group isomorphism from $H (X,S)$ onto $\mathbb{Z}^d$.
We obviously have $\Phi ( [1] ) = \Phi(\sum_{a \in \mathcal{A}} [\chi_{[a]}]) = \bf{1}$ and it remains to show that
\[
\Phi(H^+(X,S))
=
\{ {\bf x} \in {\mathbb Z}^d \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S) \}
\cup
\{{\mathbf 0}\}.
\]
Any element of $H^+(X,S)$ is of the form $[f]$ for some $f \in C(X,\mathbb{N})$.
From Theorem \ref{theo:cohoword}, there exists a unique vector ${\bf x} = (x_a)_{a \in \mathcal{A}}$ such that $[f] = \sum_{a \in \mathcal{A}} x_a [\chi_{[a]}]$.
As $f$ is non-negative, we have, for any $\mu \in {\mathcal M} (X,S)$,
\[
\langle \Phi([f]),\boldsymbol{\mu} \rangle
=
\sum_{a \in \mathcal{A}} x_a \mu([a])
=
\int f d\mu
\geq 0,
\]
with equality if and only if $f=0$ (in which case ${\bf x} = {\bf 0}$).
For the other inclusion, assume that ${\bf x} = (x_a)_{a \in \mathcal{A}} \in {\mathbb Z}^d$ satisfies $\langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \mbox{ for all }
\mu \in {\mathcal M} (X,S)$ (the case ${\bf x} = {\bf 0}$ is trivial).
We consider the function $f = \sum_{a \in \mathcal{A}} x_a \chi_{[a]}$. According to Lemma~\ref{lemma:positivecohomologous}, the existence of $f' \in [f]$ such that $f'$ is non-negative is equivalent to the existence of a lower bound for ergodic sums.
Assume by contradiction that there exists a point $x \in X$ such that the sequence $\left( \sum_{k=0}^n f \circ S^k(x)\right)_{n \geq 0}$ is not bounded from below.
Thus there is a an increasing sequence of positive integers $(n_i)_{i \geq 0}$ such that
\[
\lim_{i \to +\infty} \sum_{k=0}^{n_i-1} f \circ S^k(x) = -\infty.
\]
Passing to a subsequence $(m_i)_{i \geq 0}$ of $(n_i)_{i \geq 0}$ if necessary, there exists $\mu \in \mathcal{M}(X,S)$ satisfying
\[
\langle {\bf x}, \boldsymbol{\mu} \rangle = \int f d\mu = \lim_{i \to +\infty} \frac{1}{m_i} \sum_{k=0}^{m_i-1} f \circ S^k(x) \leq 0,
\]
which contradicts our hypothesis.
The sequence $\left( \sum_{k=0}^n f \circ S^k(x)\right)_{n \geq 0}$ is thus bounded from below and we conclude by using Lemma~\ref{lemma:positivecohomologous}.
\end{proof}
\begin{remark}
We cannot remove the hypothesis of being left or right proper in Theorem \ref{theo:dg}.
Consider indeed the subshift $(X,S)$ defined by the primitive unimodular non-proper substitution $\tau$ defined over $\{a,b\}^*$ as $\tau \colon a \mapsto aab, b \mapsto ba$.
According to \cite[p.114] {DurandThese}, the dimension group of $(X,S)$ is isomorphic to
$\left( {\mathbb Z}^3 , \left\{ {\mathbf x} \in {\mathbb Z}^3 : \langle {\mathbf x}, {\mathbf v} \rangle > 0 \right\} , (2,0,-1) \right)$
where ${\mathbf v}= ( \frac{1+ \sqrt 5}{2}, 2,1)$.
\end{remark}
\subsection{Ergodic measures}\label{subsection:ergodic}
We now focus on further consequences of Theorem \ref{theo:cohoword} for invariant measures of primitive unimodular proper $\mathcal{S}$-adic subshifts.
\begin{corollary}\label{cor:oe}
Two primitive unimodular proper $\mathcal{S}$-adic subshifts $(X_1,S)$ and $(X_2,S)$ are strong orbit equivalent
if and only if
there is a unimodular matrix $M$ such that $M {\bf 1} = {\bf 1}$ and
\[
\{\boldsymbol{\nu} \mid \nu \in \mathcal M(X_2,S)\}
=
\{M^{\rm T} \boldsymbol{\mu} \mid \mu \in \mathcal M(X_1,S)\}.
\]
In particular, $(X_1,S)$ and $(X_2,S)$ are defined on alphabets with the same cardinality.
\end{corollary}
\begin{proof}
For $i = 1,2$, let $\Phi_i:H(X_i,S) \to \mathbb{Z}^{d_i}$ be the map given in Theorem~\ref{theo:dg}, where $d_i$ is the cardinality of the alphabets $\mathcal{A}_i$ of $X_i$.
Let us also write ${\bf 1}_i$ the vector of dimension $d_i$ only consisting in $1$'s and
\[
C_i
=
\Phi_i(H^+(X_i,S))
=
\{ {\bf x} \in {\mathbb Z}^{d_i}
\mid
\langle {\bf x}, \boldsymbol{ \mu} \rangle > 0
\mbox{ for all } \mu \in {\mathcal M} (X_i,S) \}
\cup \{{\mathbf 0}\},
\]
so that $\Phi_i$ defines an isomorphism of dimension groups from $K^0(X_i,S)$ onto $(\mathbb{Z}^{d_i},C_i,{\bf{1}}_i)$.
First assume that $(X_1,S)$ and $(X_2,S)$ are strong orbit equivalent.
Theorem~\ref{oe} implies that there is an isomorphism of dimension group from $(\mathbb{Z}^{d_2},C_2,{\bf{1}}_2)$ onto $(\mathbb{Z}^{d_1},C_1,{\bf{1}}_1)$.
Hence $d_1 = d_2 = d$ and this isomorphism is given by a unimodular matrix $M$ of dimension $d$ satisfying $M{\bf 1} = {\bf 1}$ (where ${\bf 1} = {\bf 1}_1 = {\bf 1}_2$) and $MC_2 = C_1$.
We also denote by $M$ the map ${\bf x} \in \mathbb{Z}^d \mapsto M {\bf x}$.
Recall from Section~\ref{subsec:cyl} that the map
\[
\mu \in \mathcal M(X_i,S) \mapsto \left(\tau_\mu: [f] \in H(X_i,S) \mapsto \int f d\mu\right)
\]
is an affine isomorphism from $\mathcal M(X_i,S)$ to $\T(K^0(X_i,S))$.
Observing that for all $\mu \in \mathcal M(X_1,S)$, $\tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2$ is a trace of $K^0(X_2,S)$, it defines an affine isomorphism
$\mu \in \mathcal M(X_1,S) \mapsto \nu \in \mathcal M(X_2,S)$, where $\nu$ is such that $\tau_\nu = \tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2$.
Since $\boldsymbol{\mu} = (\tau_\mu([\chi_{[a]}]))_{a \in \mathcal{A}_1}$, we have, for all $a \in \mathcal{A}_2$,
\[
\nu([a])
= \tau_\nu( [\chi_{[a]}])
= \tau_\mu \circ \Phi_1^{-1} \circ M \circ \Phi_2 ( [\chi_{[a]}])
= \boldsymbol{\mu}^{\rm T} M e_a
= e_a^{\rm T} M^{\rm T} \boldsymbol{\mu},
\]
so that $\boldsymbol{\nu} = M^{\rm T} \boldsymbol{\mu}$.
Now assume that we are given a unimodular matrix $M$ satisfying $M {\bf 1} = {\bf 1}$ and
\[
\{\boldsymbol{\nu} \mid \nu \in \mathcal M(X_2,S)\}
=
\{M^{\rm T} \boldsymbol{\mu} \mid \mu \in \mathcal M(X_1,S)\}.
\]
In particular, this implies that $d_1 = d_2=d$.
Let us show that the map $M: {\bf x} \in \mathbb{Z}^d \mapsto M {\bf x}$ defines an isomorphism of dimension groups from $(\mathbb{Z}^{d},C_1,{\bf{1}})$ to $(\mathbb{Z}^{d},C_2,{\bf{1}})$.
We only need to show that $MC_1 = C_2$.
The matrix $M$ being unimodular, we have $M {\bf x} = {\bf 0}$ if and only if ${\bf x} = {\bf 0}$.
For ${\bf x} \neq {\bf 0}$, we have
\begin{align*}
{\bf x} \in C_2
& \Leftrightarrow \langle {\bf x},\boldsymbol{\nu} \rangle >0 \text{ for all } \nu \in \mathcal{M}(X_2,S) \\
& \Leftrightarrow \langle {\bf x},M^{\rm T}\boldsymbol{\mu} \rangle >0 \text{ for all } \mu \in \mathcal{M}(X_1,S) \\
& \Leftrightarrow \langle M {\bf x},\boldsymbol{\mu} \rangle >0 \text{ for all } \mu \in \mathcal{M}(X_1,S) \\
& \Leftrightarrow M {\bf x} \in C_1,
\end{align*}
which ends the proof.
\end{proof}
According to Theorem \ref{theo:dg}, dimension groups of primitive unimodular proper subshifts have rank $d$.
This implies that the number $e$ of ergodic measures satisfies $e\leq d$.
In fact, we have even more from the following result.
\begin{proposition}\label{prop:effrosshen81}
\cite[Proposition 2.4]{Effros&Shen:1981}
Finitely generated simple dimension groups of rank $d$ have at most $d-1$ pure traces.
\end{proposition}
Dimension groups of minimal Cantor systems $(X,T)$ are simple dimension groups (Theorem \ref{theo:simpleminimal}) and, since the Choquet simplex of traces is affinely isomorphic to the simplex of ergodic measures, we derive the following.
\begin{corollary}\label{cor:d-1}
Primitive unimodular proper $\mathcal{S}$-adic subshifts over a $d$-letter alphabet have at most $d-1$ ergodic measures.
\end{corollary}
If the primitive unimodular proper $\mathcal{S}$-adic subshift $(X,S)$ has some extra combinatorial properties, then the number of ergodic measures can be smaller.
Suppose indeed that $(X,S)$ is a minimal dendric subshift on a $d$-letter alphabet.
As its factor complexity equals $(d-1)n + 1$, one has a priori $e \leq d-2$ for $d \geq 3$ according to \cite[Theorem 7.3.4]{CANT}.
One can even have more as a direct consequence of \cite{Damron:2019} and \cite{DolPer:2019}. Note that this statement encompasses the case of interval exchanges handled in \cite{Katok:73,Veech:78}.
\begin{theorem} \label{theo:ergodic}
Let $(X,S)$ be a minimal dendric subshift over a $d$-letter alphabet.
One has
$$\mbox{\rm Card}({\mathcal M}_e (X,S)) \leq \frac{d}{2}.$$
\end{theorem}
\begin{proof}
According to \cite{Damron:2019}, a minimal subshift is said to satisfy the
regular bispecial condition if any large enough bispecial word $w$ has only one left extension $aw \in \mathcal{L}(X)$, $a \in \mathcal{A}$, that is right special and only one right extension $wa \in \mathcal{L}(X)$, $a \in \mathcal{A}$, that is left special. Now we use the fact that minimal dendric subshifts satisfy the regular bispecial condition according to \cite{DolPer:2019}.
We conclude by using the upper bound on the number ergodic measures from \cite{Damron:2019}.
\end{proof}
\section{Infinitesimals and balance property} \label{sec:saturation}
When the infinitesimal subgroup $ \mbox{\rm Inf} (K^0 (X,T))$ of a minimal Cantor system $(X,T)$ is trivial, the system is called {\em saturated}. This property is proved in \cite{TetRob:2016} to hold for primitive, aperiodic, irreducible substitutions for which images of letters have a common prefix.
At the opposite, an example of a dendric subshift with non-trivial infinitesimal subgroup is provided in Example \ref{ex:nontrivial}.
A formulation of saturation in terms of the topological full group is given in \cite{BezKwia:00}. Recall also that for saturated systems, the quotient group $I(X,T)/E(X,T)$ is torsion-free by \cite[Theorem 1]{CortDP:16} (see also \cite{GHH:18}).
We first state a characterization of the triviality of the infinitesimal subgroup $\mbox{\rm Inf} (K^0 (X,S))$ for minimal unimodular proper $\mathcal{S}$-adic subshifts (see Proposition \ref{theo:saturated}). We then relate the saturation property with a combinatorial notion called {\em balance property} and we provide a topological characterization of primitive
unimodular proper $\mathcal{S}$-adic subshifts that are balanced (see Corollary \ref{cor:balanced}).
\begin{proposition}\label{theo:saturated}
Let $(X,S)$ be a minimal unimodular proper $\mathcal{S}$-adic subshift on a $d$-letter alphabet $A$.
The infinitesimal subgroup $\mbox{Inf} (K^0 (X,S))$ is non-trivial
if and only if there is a non-zero vector ${\mathbf x} \in {\mathbb Z}^d$
orthogonal to any element of the simplex of letter measures.
\noindent In particular, if there exists some invariant measure $\mu \in {\mathcal M}(X,S)$ for which the frequencies of letters $\mu([a])$, $a \in A$, are rationally independent, then the infinitesimal subgroup $\mbox{Inf} (K^0 (X,S))$ is trivial.
\end{proposition}
\begin{proof}
According to Theorem \ref{theo:dg}, the elements of $ \mbox{\rm Inf} (K^0 (X,S))$ are the classes of functions that are represented by vectors ${\mathbf x} \in {\mathbb Z}^d$ such that
$\langle {\mathbf x}, \boldsymbol{ \mu} \rangle =0$ for every $\mu \in {\mathcal M} (X,S) $.
Recall also that coboundaries are represented by the vector ${\mathbf 0}$.
Hence $\mbox{Inf} (K^0 (X,S))$ is not trivial if and only if there exists $ {\mathbf x} \in {\mathbb Z}^d$, with ${\mathbf x} \neq {\mathbf 0}$,
such that $\langle {\bf x}, \boldsymbol{ \mu} \rangle = 0$, for every $\mu \in \mathcal{M}(X,S)$.
Assume now that there exists some invariant measure $\mu \in {\mathcal M}(X,S)$ for which the frequencies of letters are rationally independent.
Hence, for any vector ${\mathbf x} \in {\mathbb Z}^d$, $\langle {\bf x}, \boldsymbol{ \mu} \rangle = 0$ implies that ${\mathbf x}={\mathbf 0}$.
From above, this implies that $\mbox{Inf} (K^0 (X,S))$ is trivial.
\end{proof}
See Example \ref{ex:nontrivial} for an example of a dendric subshift with non-trivial infinitesimals.
We now introduce a notion of balance for functions. Let $(X,T)$ be a minimal Cantor system.
We say that $f \in C (X , \mathbb{R} )$ is {\em balanced} for $(X,T)$ whenever there exists a constant $C_f>0$
such that $$|\sum_{i=0}^n f(T^ix) - f(T^iy) | \leq C_f \mbox { for all }x, y \in X \mbox{ and for all } n .$$
Balance property is usually expressed for letters and factors (see for instance
\cite{BerCecchi:2018}). Indeed a minimal subshift $(X,S)$ is said to be {\em balanced on the factor} $v \in {\mathcal L} (X)$ if $\chi_{[v]}: X \to \{0,1\}$ is balanced, or, equivalently,
if there exists a constant $C_v$ such that for all $w,w'$ in $\mathcal{L}_X$ with $|w|=|w'|$, then $||w|_v-|w'|_v|\leq C_v.$
It is {\em balanced on letters} if it is balanced on each letter, and it is {\em balanced on factors} if it is balanced on all its factors.
More generally, we say that a system $(X,T)$ is balanced on a subset $H \subset C(X, \mathbb{R})$ whenever it is balanced for all $f$ in $H$.
It is standard to check that any system $(X,T)$ is balanced on the (real) coboundaries.
Of course, a subshift $(X,S)$ is balanced on a generating set of $C(X,\mathbb{Z})$ if and only if it is balanced on factors or, equivalently,
if every $f\in C(X,\mathbb{Z})$ is balanced.
One can observe that the balance property on letters is not necessarily preserved under topological conjugacy whereas the balance property on factors is.
Indeed, consider the shift generated by the Thue--Morse substitution $\sigma \colon a \mapsto ab, b \mapsto ba$.
It is clearly balanced on letters.
It is conjugate to the shift generated by the substitution
$\tau \colon a \mapsto bb$, $b\mapsto bd$, $c \mapsto ca$, $d\mapsto cb$ via the sliding block code
$00 \mapsto a,$ $01 \mapsto b,$ $10 \mapsto c,$ $11 \mapsto d$ (see \cite[p.149]{Queffelec:2010}).
The subshift generated by $\tau$ is not balanced on letters (see \cite{BerCecchi:2018}).
Next proposition will be useful to characterize balanced functions of a system $(X,T)$.
\begin{proposition}\label{prop:carBalanced}
Let $(X,T)$ be a minimal dynamical system. An integer valued continuous function $f \in C(X, {\mathbb Z})$ is balanced for $(X,T)$ if and only if there exists $\alpha \in {\mathbb R}$
such that the map $f-\alpha$ is a real coboundary.
In this case, $\alpha = \int f d\mu$, for any $T$-invariant probability measure $\mu$ in $X$.
\end{proposition}
\begin{proof}
If the function $f-\alpha$ is a real coboundary, one easily checks that $f$ is balanced.
Moreover, the integral with respect to any $T$-invariant probability measure is zero, providing the last claim.
Assume that $f \in C(X, {\mathbb R})$ is balanced for $(X,T)$. Let $C>0$ be a constant such that
$ |\sum_{i=0}^n f\circ T^{i}(x)-f\circ T^{i}(y)| \le C$ holds uniformly in $x,y \in X$ for all $n\ge 0$.
Thus, for any non-negative integer $p \in \mathbb{N}$, there exists $N_{p}$ such that, for any $x\in X$, one has the following inequalities:
$$N_{p} \le \sum_{i=0}^p f\circ T^i(x) \le N_{p} +C.$$
Moreover, one checks that, for any $p,q \in \mathbb{N}$:
$$
qN_{p} \le \sum_{i=0}^{pq} f\circ T^i(x) \le qN_{p} +qC \textrm{ and } pN_{q} \le \sum_{i=0}^{pq} f\circ T^i(x) \le pN_{q} +pC.
$$
It follows that $-qC \le qN_{p} -pN_{q} \le pC$ and thus $-C/p \le N_{p}/p -N_{q}/q \le C/q $. Hence the sequence $(N_{p}/p)_{p}$ is a Cauchy sequence.
Let $\alpha = \lim_{p\to \infty} N_{p}/p$. By letting $q$ going to infinity, we get $ -C \le N_{p}-p\alpha \le 0$, so that $-C \le \sum_{i=0}^p f\circ T^i(x) -p\alpha \le C$ for any $x\in X$. By the classical Gottschalk--Hedlund's Theorem \cite{GotHed:55}, the function $f-\alpha$ is a real coboundary.
\end{proof}
As a corollary, we deduce that a minimal Cantor system $(X,T)$ balanced on $C(X,{\mathbb Z})$ is uniquely ergodic.
It also follows that for a minimal subshift $(X,S)$ balanced on the factor $v$, the frequency $\mu_{v} \in {\mathbb R}_{+}$ of $v$ exists,
i.e.,
for any $x \in X$,
$\lim_{ n \rightarrow \infty} \frac{ | x_{-n} \cdots x_0 \cdots x_{n}| _v }{ 2n+1}= \mu_v$, and even, the quantity $\sup _{n \in \mathbb{N}} | | x_{-n} \cdots x_0 \cdots x_{n}| _v - (2n+1) \mu_v| $ is finite (see also \cite{BerTij:2002}).
Actually, integer-valued continuous functions that are balanced for a minimal Cantor system $(X,T)$ are related to the continuous eigenvalues of the system as illustrated by the following folklore lemma. We recall that $E(X,T)$ stands for the set of additive continuous eigenvalues.
\begin{lemma}\label{lem:BalancedEigenvalue}
Let $(X,T)$ be a minimal Cantor system and let $\mu$ be a $T$-invariant measure. If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then $\int f d\mu $ belongs to $ E(X,T)$.
\end{lemma}
\begin{proof}
If $f \in C(X,\mathbb{Z})$ is balanced for $(X,T)$, then so is $-f$ and there exists $g \in C(X,\mathbb{R})$ such that
$ -f+ \int f d\mu = g \circ T-g$ (by Proposition \ref{prop:carBalanced}).
This yields $\exp({2i\pi g\circ T})= \exp ({2i\pi \int f d \mu }) \exp ({ 2i \pi g})$ by noticing that $\exp({- 2i\pi f (x)})=1$ for any $x\in X$.
Hence $\exp ({ 2i \pi g})$ is a continuous eigenfunction associated with the additive eigenvalue $\int f d \mu$.
\end{proof}
We first give a statement valid for any minimal Cantor system that will then be applied below to primitive unimodular proper $\mathcal{S}$-adic subshifts.
We recall from \cite[Theorem 3.2, Corollary 3.6]{GHH:18} that there exists a one-to-one homomorphism $\Theta $ from $I (X,T)$ to $K^0 (X,T)$ such that, for $\alpha \in (0,1)\cap E (X,T)$, $\Theta (\alpha ) = [\chi_{U_{\alpha}}]$ where $U_\alpha$ is a clopen set, sucht that $\mu (U_\alpha)=\alpha$ for every
invariant measure $\mu$, and $ \chi_{U_{\alpha}} -\mu (U_{\alpha}) $ is a real coboundary. Hence $\chi_{U_{\alpha}}$ is balanced for $(X,T)$ (by Proposition \ref{prop:carBalanced}).
\begin{proposition}
\label{prop:balanced}
Let $(X,T)$ be a minimal Cantor system.
The following are equivalent:
\begin{enumerate}
\item
\label{item:1}
$(X,T)$ is balanced on some $H\subset C(X,\mathbb{Z})$ and $\{ [h] : h \in H \}$ generates $K^0 (X,T)$,
\item
\label{item:2}
$(X,T)$ is balanced on $C( X, \mathbb{Z})$,
\item
\label{item:3}
$\Theta (E(X,T))$ generates $K^0 (X,T)$.
\end{enumerate}
In this case $(X,T)$ is uniquely ergodic, then $I(X,T) = E(X,T)$ and ${\rm Inf } (K^0(X,T))$ is trivial.
\end{proposition}
\begin{proof}
Let us prove that \eqref{item:1} implies \eqref{item:2}.
Let $f\in C (X,\mathbb{Z} )$.
One has $[f] = \sum_{i=1}^{n} z_i [h_i]$ for some integers $z_i$ and some functions $h_i\in H$.
Hence $f = g\circ T - g + \sum_{i=1}^{n} z_i h_i$ for some $g\in C(X,\mathbb{Z} )$ and $f$ is balanced.
Consequently, $(X,T)$ is balanced on $C(X,\mathbb{Z})$.
It is immediate that \eqref{item:3} implies \eqref{item:1}.
Let us show that \eqref{item:2} implies \eqref{item:3}. Unique ergodicity holds by Proposition~\ref{prop:carBalanced}.
Let $\mu$ be the unique shift invariant probability measure of $(X,T)$.
For any $f\in C( X,\mathbb{Z})$, there are clopen sets $U_i$ and integers $z_{i}$ such that $f = \sum_{i=1}^n z_i \chi_{U_i}$.
From Lemma \ref{lem:BalancedEigenvalue} the values $ \mu(U_{i})\in I(X,T)$ are additive continuous eigenvalues in $E(X,T)$.
We get $[\chi_{U_{i}}] = \Theta(\mu(U_{i}))$ and $[f] = \sum_{i=1}^n z_i \Theta (\mu(U_{i}) )$.
This shows the claim \eqref{item:3}.
Assume that one of the three equivalent conditions holds.
Let $\mu$ denote the unique shift invariant probability measure.
Then, any map $f - \int f d \mu $, with $f\in C (X,\mathbb{Z} )$, is a real coboundary (by Proposition \ref{prop:carBalanced}). Hence, since any integer valued continuous function that is a real coboundary is a coboundary (\cite[Proposition 4.1]{Ormes:00}), the infinitesimal subgroup $ \mbox{\rm Inf} (K^0 (X,T))$ is trivial.
Moreover, Lemma \ref{lem:BalancedEigenvalue} implies that $I(X,T) \subset E(X,T)$.
The reverse implication $E(X,T) \subset I(X,T)$ comes from \cite[Proposition 11]{CortDP:16}, see also \cite[Corollary 3.7]{GHH:18}.
\end{proof}
Observe that when $(X,S)$ is a minimal subshift, by taking $H$
to be the set of classes of characteristic functions of cylinder sets, the balance
property is equivalent to the algebraic condition \eqref{item:3} of Proposition \ref{prop:balanced}.
We now provide a topological proof of the fact that the balance property on letters implies the balance property on factors
for primitive unimodular proper $\mathcal{S}$-adic subshifts.
For minimal dendric subshifts, this was already proved in \cite[Theorem 1.1]{BerCecchi:2018} using a combinatorial proof.
\begin{corollary} \label{cor:balanced}
Let $(X,S)$ be a primitive unimodular proper $\mathcal{S}$-adic subshift on a $d$-letter alphabet.
The following are equivalent:
\begin{enumerate}
\item
\label{item:balancedmaps}
$(X,S)$ is balanced for all integer valued continuous maps in $C (X , \mathbb{Z} )$,
\item
\label{item:balancedfactors}
$(X,S)$ is balanced on factors,
\item
\label{item:balancedletters}
$(X,S)$ is balanced on letters,
\item
\label{item:balancedrank}
$\mbox{\rm rank} (E(X,S)) =d$,
\end{enumerate}
and in this case $(X,S)$ is uniquely ergodic, $I(X,S) = E(X,S)$ and $\mbox{\rm Inf} (X,S)$ is trivial.
\end{corollary}
\begin{proof}
The implications $\eqref{item:balancedmaps} \Rightarrow \eqref{item:balancedfactors}\Rightarrow \eqref{item:balancedletters}$ are immediate.
Let us prove the implication $\eqref{item:balancedletters} \Rightarrow \eqref{item:balancedrank}$.
We deduce from Theorem \ref{theo:cohoword}, by taking $H$
to be the set of classes of characteristic functions of cylinder sets, that the conditions of Proposition \ref{prop:balanced} hold.
We deduce from Proposition \ref{theo:saturated} that \eqref{item:balancedrank} holds.
It remains to prove the implication $\eqref{item:balancedrank} \Rightarrow \eqref{item:balancedmaps}$. Suppose that $E(X,S)$ has rank $d$.
Let $\alpha_1 , \dots , \alpha_d\in E(X,S)$ be rationally independent.
There is no restriction to assume that they are all in $(0,1)$.
Consider, for $i =1\cdots,d$, $\Theta (\alpha_i ) = [\chi_{U_{\alpha_i}}]$ where $U_{\alpha_{i}}$ is a clopen set such that $\mu (U_{\alpha_{i}} ) = \alpha_i$ for any $S$-invariant measure $\mu \in {\mathcal M}(X,S)$ and $\chi_{U_{\alpha_{i}}}$ is balanced for $(X,S)$.
The classes $\Theta (\alpha_i)$'s are rationally independent because
the image of $\Theta (\alpha_i)$ by any trace is $\alpha_{i}$ and these values are assumed to be rationally independent.
As $K^0 (X,S)$ has rank $d$, by Theorem \ref{theo:cohoword}, and since it has no torsion (as recalled in Section~\ref{subsec:dim}), any element $[f]\in K^0 (X,S)$ is a rational linear combination of the $\Theta (\alpha_i )$'s. By Proposition \ref{prop:balanced}, any $f\in C(X, \mathbb{Z} )$ is balanced for $(X,S)$.
\end{proof}
\begin{remark}We deduce that primitive unimodular proper $\mathcal{S}$-adic subshifts that are balanced on letters have the maximal continuous eigenvalue group property, as defined in \cite{Durand&Frank&Maass:2019}, i.e., $E(X,S)= I(X,S)$. This implies in particular that non-trivial additive eigenvalues are irrational.
Indeed, by Corollary \ref{cor:balanced}, $\mbox{Inf} (K^0 (X,S))$ is trivial and thus, by Proposition \ref{theo:saturated}, the frequencies of letters (for the unique
shift-invariant measure) are rationally independent, which yields that $I(X,S)$ and thus $E(X,S)$ contain no rational non-trivial elements.
More generally, the fact that non-trivial additive eigenvalues are irrational hold for minimal dendric subshifts (even without
the balance property) \cite{Rigidity}. Note also that the triviality of $\mbox{Inf} (K^0 (X,S))$ says nothing about the balance property (see Example \ref{ex:balance}),
but the existence of non-trivial infinitesimals indicates that some letter is not balanced.
Lastly, the Thue--Morse substitution $\sigma \colon a \mapsto ab, b \mapsto ba$ generates
a subshift that is balanced on letters but not on factors \cite{BerCecchi:2018}. This substitution is neither unimodular, nor proper.
\end{remark}
\section{Examples and observations }\label{sec:examples}
\subsection{Brun subshifts}\label{subsec:Brun}
We provide a family of primitive unimodular proper $\mathcal{S}$-adic subshifts which are not dendric.
We consider the set of endomorphisms $S_\mathrm{Br} = \{\beta_{ab} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\}\}$ over $d$~letters defined by
\[
\beta_{ab}:\ b \mapsto ab,\ c\mapsto c\ \text{for}\ c \in \mathcal{A} \setminus \{ b\}.
\]
A subshift $(X,S)$ is a \emph{Brun subshift} if it is generated by a primitive directive sequence $\boldsymbol{\tau } = (\tau_n)_n \in S_\mathrm{Br}^\mathbb{N}$
such that for all $n$ the endomorphism $\tau_n\tau_{n+1}$ belongs to
\begin{align*}
\big\{\beta_{ab}\beta_{ab} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\}\big\}
\cup
\big\{\beta_{ab}\beta_{bc} \mid a \in \mathcal{A},\, b \in \mathcal{A} \setminus \{a\},\, c \in \mathcal{A} \setminus \{b\}\big\}.
\end{align*}
Observe that primitiveness of $\boldsymbol{\tau }$ is equivalent to the fact that for each $a \in \mathcal{A}$ there is $b \in \mathcal{A}$ such that $\beta_{ab}$ occurs infinitely often in $\boldsymbol{\tau }$.
Brun subshifts are not dendric in general: on a three-letter alphabet, they may contain strong and weak bispecial factors, hence that have an extension graph which is not a tree~\cite{Labbe&Leroy:16}.
However, we show below that they are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\begin{lemma}
\label{lemma:brunproper}
Let $\mathcal{A}$ be a finite alphabet and $\gamma_{ab} : \mathcal{A} \to \mathcal{A}$, $a\not = b$, be the letter-to-letter map defined by $\gamma_{ab} (a) = \gamma_{ab} (b) = a$ and $\gamma_{ab} (c) =c$ for $c\in \mathcal{A}\setminus \{ a,b\}$.
Let $(a_n)_{1\leq n\leq N}$ be such that $\{ a_n \mid 1\leq n \leq N \} = \mathcal{A}$.
Then, $\gamma_{a_1 a_{2}} \gamma_{a_2 a_{3}} \cdots \gamma_{a_{N-1} a_{N}}$ is constant.
\end{lemma}
\begin{proof}
It suffices to observe that $\gamma_{a_m a_{m+1}} \gamma_{a_{m+1} a_{m+2}} \cdots \gamma_{a_{n-1} a_{n}}$ identifies the letters $a_m, a_{m+1}, a_{m+2} \dots , a_n$ to $a_m$.
\end{proof}
\begin{lemma}
Brun subshifts are primitive unimodular proper $\mathcal{S}$-adic subshifts.
\end{lemma}
\begin{proof}
Let $(X,S)$ be a Brun subshift over the alphabet $\mathcal{A}$, generated by the directive sequence $\boldsymbol{\beta } = (\beta_{a_nb_n})_{n\geq 1} \in S_\mathrm{Br}^\mathbb{N}$.
With each endomorphism $\beta_{ab}$ one can associate the map $\gamma : \mathcal{A} \to \mathcal{A}$ defined by $\gamma (c) = \beta (c)_0$.
Clearly $\gamma $ is equal to $\gamma_{ab}$.
By primitiveness, there exists an increasing sequence of integers $(n_k)_k$, with $n_0 =0$, such that $\{ a_i \mid n_k \leq i < n_{k+1} \} = \mathcal{A}$ for all $k$.
Hence from Lemma \ref{lemma:brunproper} the morphisms $\boldsymbol{\beta}_{[n_k , n_{k+1})}$ are left proper.
We conclude by using Lemma~\ref{lemma:proper}.
\end{proof}
As a corollary, we recover the following result (which also follows from~\cite[Theorem~5.7]{Berthe&Delecroix:14}).
\begin{proposition}\label{prop:UE}
Brun subshifts are uniquely ergodic.
\end{proposition}
\begin{proof}
This follows from Corollary~\ref{coro:measures} and from the fact that Brun subshifts have a simplex of letter measures generated by a single vector (see~\cite[Theorem~3.5]{Brentjes:81}).
\end{proof}
Brun subshifts have been introduced in \cite{BST:2019} in order to provide symbolic models for two-dimensional toral translations.
In particular, they are proved to have generically pure discrete spectrum in \cite{BST:2019}.
\subsection{Arnoux-Rauzy subshifts} \label{subsec:AR}
A minimal subshift $(X,S)$ over $ \mathcal{A}= \{1,2,\ldots,d\}$ is an \emph{Arnoux-Rauzy subshift} if for all~$n$ it has $(d-1)n+1$ factors of length~$n$, with exactly one left special and one right special factor of length~$n$.
Consider the following set of endomorphisms defined on the alphabet $\mathcal{A} = \{1, \dots , d \}$, namely
$S_\mathrm{AR} = \{ \alpha_a \mid a \in \mathcal{A}\}$ with
\[
\alpha_a:\ a \mapsto a,\ b \mapsto ab\ \mbox{for}\ b \in \mathcal{A} \setminus \{a\}.
\]
A subshift $(X,S)$ generated by a primitive directive sequence $\boldsymbol{\tau } \in S_\mathrm{AR}^\mathbb{N}$ is called an \emph{Arnoux-Rauzy subshift}.
It is standard to check that primitiveness of $\boldsymbol{\alpha }$ is equivalent to the fact that each morphism $\alpha_a$ occurs infinitely often in $\boldsymbol{\alpha }$.
Arnoux-Rauzy subshifts being dendric subshifts, they are in particular primitive unimodular proper $\mathcal{S}$-adic subshifts.
We similarly recover, as in Proposition~\ref{prop:UE}, that Arnoux-Rauzy subshifts are uniquely ergodic (see~\cite[Lemma 2]{Delecroix&Hejda&Steiner:2013} for
the fact that Arnoux-Rauzy subshifts have a simplex of letter measures generated by a single vector).
\subsection{A dendric subshift with non-trivial infinitesimals} \label{ex:nontrivial}
Let us provide an example of a minimal dendric subshift with non-trivial infinitesimal subgroup, and thus with rationally dependent letter measures according to
Proposition \ref{theo:saturated}.
We take the interval exchange $T$ with permutation $(1,3,2)$ with intervals $[0,1-2 \alpha) , [1-2 \alpha, 1- \alpha), $ and $[1-\alpha,1)$, with $\alpha=(3-\sqrt{5})/2$.
The transformation $T$ is represented in Figure~\ref{figure3interval}, with
$I_{1}=[0,1-2\alpha), $ $I_{2}=[1-2\alpha,1-\alpha)$, $ I_{3}=[1-\alpha,1)$
and $
J_{1}=[0,\alpha)$, $J_{2}=[\alpha,2\alpha)$, $J_{3}=[2\alpha,1).$
\begin{figure}
\caption{The transformation $T$.}
\label{figure3interval}
\end{figure}
Measures of letters are rationally dependent and the natural coding of this interval exchange is a strictly ergodic dendric subshift $(X,S,\mu)$ by Theorem \ref{theo:ergodic}.
It is actually a representation on $3$ intervals of the rotation of angle $2\alpha$
(the point $1-\alpha$ is a separation point which is not a singularity of this interval exchange).
One has
$\mu([2])=\mu([3])$.
The class of the function $\chi_{[2]}- \chi_{[3]}$ is thus a non-trivial infinitesimal, according to Theorem \ref{theo:cohoword}.
\subsection{Dendric subshifts having the same dimension group and different spectral properties } \label{ex:balance}
It is well known that within any given class of strong orbit equivalence (i.e., by Theorem \ref{oe}, within any family of minimal Cantor systems sharing the same dimension group $(G,G^+,u)$), all minimal Cantor systems share the same set of rational additive continuous eigenvalues $E(X,T)\cap \mathbb{Q}$ \cite{Ormes:97}.
When this set is reduced to $\{ 0 \}$, then, in the strong orbit equivalence class of $(X,T)$, there are many weakly mixing systems, see \cite[Theorem 6.1]{Ormes:97}, \cite[Theorem 5.4]{GHH:18} or \cite[Corollary 23]{Durand&Frank&Maass:2019}.
We provide here an example of a strong orbit equivalence class that contains two minimal dendric subshifts, one being weakly mixing and the other one having pure discrete spectrum.
Both systems are saturated (they have no non-trivial infinitesimals) but they have different balance properties.
They are defined on a three-letter alphabet and have factor complexity $2n + 1$.
According to Theorem \ref{theo:ergodic}, they are uniquely ergodic.
From Corollary~\ref{cor:oe}, two minimal dendric subshifts on a three-letter alphabet are strong orbit equivalent if and only if there is a unimodular row-stochastic matrix $M$ sending the vector of letter measures of one subshift to the vector of letter measures of the other.
In particular, any Arnoux-Rauzy subshift is strong orbit equivalent to any natural coding of an i.d.o.c. exchange of three intervals for which the length of the intervals are given by the letter measures of the Arnoux-Rauzy subshift (recall that an interval exchange transformation satisfies the {\it infinite distinct orbit condition}, i.d.o.c. for short, if the negative trajectories of the discontinuity points are infinite disjoint sets; this condition implies minimality \cite{Keane:75}). We thus consider the subshift $(X,S)$ generated by the Tribonacci substitution
$\sigma \colon a \mapsto ab, b \mapsto ac, c \mapsto a$ which is uniquely ergodic, dendric, balanced and has discrete spectrum \cite{Rauzy:1982}.
Let $\mu$ be its unique invariant measure.
We also consider the natural coding $(Y,S)$ of the three-letter interval exchange defined on intervals of length $\mu[a]$, $\mu[b]$, $\mu[c]$ with permutation $(13)(2)$.
It is uniquely ergodic, topologically weakly mixing \cite{KatokStepin,FerHol:2004} and strong orbit equivalent to $(X,S)$ by Proposition \ref{prop:DendricareSadic}.
Hence, for spectral reasons, $(X,S)$ and $(Y,S)$
are not topologically conjugate, even if they are strong orbit equivalent.
We provide a further proof of non-conjugacy for the systems $(X,S)$ and $(Y,S)$ based on asymptotic pairs.
We first recall a few definitions.
Two points $x,y $ in a given subshift are said to be {\em right asymptotic} if they have a common tail, i.e.,
there exists $n $ such that $(x_k)_{k \geq n}= (y_k)_{k \geq n}$.
This defines an equivalence relation on the collection of orbits:
two $S$-orbits ${\mathcal O}_S(x) = \{ S^n x \mid n\in \mathbb{Z} \} $ and ${\mathcal O}_S(y)$ are asymptotically equivalent
if for any $x' \in {\mathcal O}_S(x)$, there is $y' \in {\mathcal O}_S(y)$ that is right asymptotic to $x'$.
We call {\em asymptotic component} any equivalence class under the asymptotic equivalence.
We say that it is {\em non-trivial} whenever it is not reduced to one orbit.
An Arnoux-Rauzy subshift $(X,S)$ has a unique non-trivial asymptotic component formed of three distincts orbits as, for all $n$, there is a unique word $w$ of length $n$ such that $\ell(w) \geq 2$ and this word is such that $\ell(w)=3$ (see Section \ref{subsec:tree} for the notation).
On the other side, any i.d.o.c. exchange of three-intervals $(Y,S)$ has 2 asymptotic components and thus cannot be conjugate to $(X,S)$.
Indeed, suppose that it has a unique non-trivial asymptotic component.
As a natural coding of an i.d.o.c interval exchange transformation has two left special factors for each large enough length,
this component should contain three sequences $x'x$, $ x''ux$ and $x''' ux$ belonging to $Y$ where $u$ is a non-empty word.
This would imply that the interval exchange transformation is not i.d.o.c.
Next statement illustrates the variety of spectral behaviours within strong orbit equivalence classes of dendric subshifts.
\begin{proposition}\label{prop:realization}
For Lebesgue a.e. probability vector $\boldsymbol{ \mu}$ in ${\mathbb R}^3_+$,
there exist two strictly ergodic proper unimodular $\mathcal{S}$-adic subshifts, one with pure discrete spectrum and another one which is weakly mixing, both having the same dimension group
$\left( {\mathbb Z} ^3, \, \{ {\bf x} \in {\mathbb Z}^3 \mid \langle {\bf x}, \boldsymbol{ \mu} \rangle > 0 \}\cup \{{\mathbf 0}\},\, \bf{1}\right ).$
\end{proposition}
\begin{proof}
Brun subshifts such as introduced in Section \ref{subsec:Brun} are proved to have generically pure discrete spectrum
in
\cite{BST:2019}.
See \cite{KatokStepin,FerHol:2004} for the genericity of weak mixing for subshifts generated by three-letter interval exchanges.
\end{proof}
\subsection{Dendric vs. primitive unimodular proper $\mathcal{S}$-adic subshifts }\label{subsection:vs}
In this section, we give an example of a primitive unimodular proper $\mathcal{S}$-adic subshift whose strong orbit equivalence class contains no minimal dendric subshift.
Theorem~\ref{theo:dg} provides a description of the dimension group of any primitive unimodular proper $\mathcal{S}$-adic subshift.
It is natural to ask whether a strong orbit equivalence class represented by such a dimension group includes a primitive unimodular proper $\mathcal{S}$-adic subshift.
This was conjectured in different terms in \cite{Effros&Shen:1979}.
It was shown to be true when the dimension group has a unique trace~\cite{Riedel:1981b} (or, equivalently, when all minimal systems in this class are uniquely ergodic) but shown to be false in general~\cite{Riedel:1981}. In the same spirit, one may ask if the strong orbit equivalence class of any primitive unimodular proper $\mathcal{S}$-adic subshift contains a dendric subshift.
Inspired from \cite{Effros&Shen:1979}, we negatively answer to that question below.
Indeed, this example provides a family of examples of primitive unimodular $\mathcal{S}$-adic subshifts on a three-letter alphabet with two ergodic invariant probability measures.
They thus cannot be dendric by Theorem~\ref{theo:ergodic} and their strong orbit equivalence class contains no minimal dendric subshift.
Let ${\mathcal A}=\{1,2,3\}$ and consider the directive sequence $\boldsymbol{\tau } = (\tau_n : {\mathcal A}^* \to {\mathcal A}^*)_{n \geq 1}$ defined by
\begin{align*}
\tau_{2n}:
&
\
1 \mapsto 2^{a_n}3, \quad
2 \mapsto 1, \quad
3 \mapsto 2 \\
\tau_{2n+1}:
&
\
1 \mapsto 32^{a_n}, \quad
2 \mapsto 1, \quad
3 \mapsto 2
\end{align*}
where $(a_n)_{n \geq 1}$ is an increasing sequence of positive integers satisfying $\sum_{n \geq 1} 1/a_n <1$.
The incidence matrix of each morphism $\tau_n$ is the unimodular matrix
$$
A_n =
\begin{pmatrix}
0 & 1 & 0 \\
a_n & 0 & 1 \\
1 & 0 & 0 \\
\end{pmatrix}.
$$
It is easily checked that any morphism $\tau_{[n,n+5)}$ is proper and has an incidence matrix with positive entries.
Therefore, the $\mathcal{S}$-adic subshift $(X_{\boldsymbol{\tau}} , S)$ is primitive, unimodular and proper.
Let us show that $(X_{\boldsymbol{\tau}} , S)$ has two ergodic measures. In fact, we prove that $(X_{\boldsymbol{\tau}} , S)$ has at least two ergodic measures.
This will imply that it has exactly two ergodic measures by Theorem \ref{prop:effrosshen81}.
For all $n \geq 1$, let $C_n$ be the first column vector of $A_{[1,n]} = A_1 \cdots A_n$ and $J_n = C_n/\Vert C_n\Vert_1$ where $\Vert\cdot\Vert_1$ stands for the L1-norm.
If $(X_{\boldsymbol{\tau}} , S)$ had only one ergodic measure $\mu$, then $(J_n)_{n \geq 1}$ would converge to $\boldsymbol{\mu}$.
Hence it suffices to show that $(J_n)_{n \geq 1}$ does not converge.
Observe that, using the shape of the matrices $A_n$, that for $n\geq 1$, and setting $C_0 = e_1, C_{-1} = e_2, C_{-2} = e_3$, one has
$$
C_n = a_n C_{n-2} + C_{n-3},
$$
where $e_1,e_2,e_3$ are the canonical vectors.
Hence we have $J_n = b_n J_{n-2} + c_nJ_{n-3}$ with
$b_n = a_n \frac{\Vert C_{n-2}\Vert_1}{\Vert C_{n}\Vert_1}$ and
$c_n = \frac{\Vert C_{n-3}\Vert_1}{\Vert C_{n}\Vert_1}$.
In particular, $b_n + c_n = 1$.
As $(\Vert C_n\Vert_1)_n$ is non-decreasing, we have
$1 \geq b_n \geq a_n c_n$
and thus $c_n \leq a_n^{-1}$.
Moreover, we have
\[
\Vert J_n - J_{n-2} \Vert_1 = \Vert(b_n - 1) J_{n-2} -c_n J_{n-3} \Vert_1 = c_n \Vert J_{n-2} - J_{n-3} \Vert_1 \leq \frac{2}{a_n},
\]
hence, for $0\leq m\leq n$,
$$
\Vert J_{2n}-J_{2m}\Vert_1 \leq 2 \sum_{k=m+1}^{n} \frac{1}{a_{2k}}.
$$
This shows that $(J_{2n})_{n \geq 1}$ is a Cauchy sequence. Let $\beta$ stand for its limit.
For $n \geq m=0$, we obtain
$\Vert J_{2n}-e_1\Vert_1 \leq 2 \sum_{k=1}^{n} \frac{1}{a_{2k}}$ and thus
$\Vert\beta-e_1\Vert_1 \leq 2 \sum_{k=1}^{\infty } \frac{1}{a_{2k}}$.
We similarly show that $(J_{2n+1})_{n \geq 1}$ is a Cauchy sequence. Let $\alpha $ stand for its limit. We have $\Vert\alpha-e_2\Vert_1 \leq 2 \sum_{k=0}^{\infty } \frac{1}{a_{2k+1}}$.
Consequently,
$$
\Vert\alpha - \beta \Vert_1 =
\Vert(\alpha -e_2) + (e_1- \beta ) +e_2 -e_1 \Vert_1 \geq 2 - 2 \sum_{k=1}^{\infty } \frac{1}{a_{k}}
$$
and $(J_n)_{n \geq 1}$ does not converge.
Consequently, $(X_{\boldsymbol{\tau}} , S)$ has exactly two ergodic measures.
\section{Questions and further works}
According to \cite{Riedel:1981} (see also Section \ref{subsection:vs}), not all strong orbit equivalence classes represented by dimension groups of the type \eqref{align:ZdGD} in Theorem~\ref{theo:dg} contain primitive unimodular proper $\mathcal{S}$-adic subshifts.
The description of the dynamical dimension group in Theorem~\ref{theo:dg} is not precise enough to explain the restrictions that occur for instance for the measures,
so that a complete characterization of the dynamical dimension groups of primitive unimodular proper $\mathcal{S}$-adic subshifts is still missing.
Similarly, we address the question of characterizing the strong orbit equivalence classes containing minimal dendric subshifts.
The combinatorial properties of these subshifts imply constraints, especially for the invariant measures, such as stated in Theorem \ref{theo:ergodic}.
For example, the question arises as to whether dimension groups of rank $d$ having at most $d/2$ extremal traces are dimension groups of minimal dendric subshifts.
Another question is about the properness assumption.
For dendric or Brun subshifts, we were able to find a primitive unimodular proper $\mathcal{S}$-adic representation.
One can easily define $\mathcal{S}$-adic subshifts by a primitive unimodular directive sequence that is not proper.
The question now is whether
a primitive unimodular proper $\mathcal{S}$-adic representation (up to conjugacy) of this subshift can be found.
Even in the substitutive case, we do not know whether such a representation exists.
The factor complexity of dendric subshifts is affine.
It is well known~\cite{Boyle&Handelman:94,Ormes:97,Sugisaki:2003} that inside the strong orbit equivalence class of any minimal Cantor system one can find another minimal Cantor systems with any other prescribed topological entropy.
Primitive unimodular proper $\mathcal{S}$-adic subshifts being of finite topological rank, they have zero topological entropy.
It would be interesting to exhibit a variety of asymptotic behaviours for complexity functions within a strong orbit equivalence class.
\end{document} |
\begin{equation}gin{document}
\title{Recurrence of biased quantum walks on a line}
\author{M. \v Stefa\v n\'ak$^{(1)}$, T. Kiss$^{(2)}$ and I. Jex$^{(1)}$}
\address{$^{(1)}$ Department of Physics, FJFI \v CVUT v Praze, B\v
rehov\'a 7, 115 19 Praha 1 - Star\'e M\v{e}sto, Czech Republic}
\address{$^{(2)}$ Department of Nonlinear and Quantum Optics, Research
Institute for Solid State Physics and Optics, Hungarian Academy of
Sciences, Konkoly-Thege u.29-33, H-1121 Budapest, Hungary}
\pacs{03.67.-a,05.40.Fb,02.30.Mv}
\date{\today}
\begin{equation}gin{abstract}
The P\'olya number of a classical random walk on a regular lattice is known to
depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is
extremely sensitive to the directional symmetry, any deviation
from the equal probability to travel in each direction results in a
change of the character of the walk from recurrent to transient.
Applying our definition of the P\'olya number to quantum walks on a
line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased
quantum walks which are recurrent.
\end{abstract}
\maketitle
\section{Introduction}
\label{sec1}
Random walks are a popular topic in physics \cite{overview,hughes}. The popularity
stems from several sources. First, random walks are rather simple in their
formulation yet powerful in their application and allow to pinpoint the essential
physics involved in the studied processes. Next, the random walks are
one of the tools which allow to connect the microdynamics with the
macrobehaviour of large systems. Finally, random walks are quite
flexible and popular also outside physics to describe various phenomena. Hence it comes not as a surprise
that the first random walks have not been formulated within physics but to describe the alternation of share prices on the stock
exchange or the spreading of insects in a forest \cite{bachelier,chandrasekhar:1943}.
The study of random walks obtained a new stimulus when they were combined with quantum mechanics \cite{aharonov,konno:book}. Here the walker is thought to be a non-classical object enriched with wave attributes. The novel features of
quantum walks have been shown to be not only of theoretical interest
but to also have practical implications, especially for quantum algorithms \cite{kempe,shenvi:2003,aurel:2007,santha}. An important concept is the hitting time \cite{kempe:2005,krovi:2006a,krovi:2006b,magniez} which helps to point out the fundamental difference between classical and quantum walks allowing for algorithmic speed-up. One of the simplest non-trivial examples for a quantum walk is the one on a line \cite{nayak} which is closely related to the so called optical Galton board \cite{optical:galton}. Various aspects of one dimensional quantum walks have been analyzed \cite{tregenna,wojcik,knight,carteret,chandrashekar:2007,konno:2002}. Additional interesting effects, e.g. localisation, arise when one considers multi-state quantum walks \cite{1dloc1,1dloc,miyazaki,sato}.
One of the characteristics of the random walk on an infinite lattice is
expressed by the probability of the walker to return to its starting position, called the P\'olya number \cite{polya}. If the P\'olya number equals one
the walk is called recurrent, otherwise there is a non-zero probability that the walker never returns to its starting position. Such walks are called transient. The recurrence nature of the random walk is determined by the asymptotic behaviour of the probability at the origin \cite{revesz}. One finds that a random walk is transient if the probability at the origin decays faster than $t^{-1}$. The recurrent behaviour has been studied in great detail for classical random walks in dependence on the dimension and the topology of the lattice \cite{domb:1954,montroll:1964}.
Recently, we have extended the concept of P\'olya number to quantum walks \cite{prl}. In our definition we proposed a particular measurement scheme to minimize disturbance: each measurement in a series is carried out on a different member of an ensemble of equally prepared quantum systems. We have shown that the recurrence nature of the quantum walk, according to the above definition, is determined by the asymptotic behaviour of the probability at the origin in a similar way as in classical random walks. However, due to interference the asymptotics of the probability at the origin does not depend solely on the dimension of the lattice, but also on the coin operator and the initial coin state. Hence, one can find strikingly different recurrence behaviour for quantum walks compared to their classical counterpart \cite{pra}. Note that recurrence is meant here as the return to the origin which can be considered as a fractional recurrence from the point of view of the whole quantum state \cite{peres,chandra:recurrence} So far we have considered balanced walks, i.e. ones where there is no preference in direction for the walker and the step lengths are equal. For a large class of quantum walks this assumption does not hold and we wish to study the implications of unbalanced coins and unequal step lengths for the recurrence properties.
In the present paper we study biased quantum walks on the line
and compare their properties with their classical counterparts. As we briefly review in \ref{app:a}, recurrence of classical random walks is a consequence of the walk's symmetry. They are recurrent if and only if the mean value
of the position of the particle vanishes. This is due to the fact
that the spreading of the probability distribution of the
position is diffusive while the mean
value of the position propagates with a constant velocity. In contrast, for
quantum walks both the spreading of the probability distribution and
the propagation of the mean value are ballistic. We show that this allows for maintaining recurrence even when the symmetry is broken.
Our paper is organized as follows: In Section~\ref{sec2} we describe
the biased quantum walk on a line. In Section~\ref{sec3} we solve
the time evolution equations with the help of the Fourier
transformation. We find that the probability amplitudes can be
expressed in terms of integrals where time enters only in the
rapidly oscillating phase factor. This fact allows a straightforward
asymptotic analysis of the probability amplitudes by means of the
method of stationary phase. We perform this analysis in
Section~\ref{sec4}. Since the recurrence of the quantum walk is
determined by the asymptotics of the probability at the origin we
find a condition under which the biased quantum walk on a line is
recurrent. In Section~\ref{sec5} we analyze the recurrence of biased
quantum walks from a different perspective. We find that the
recurrence is related to the velocities of the peaks of the
probability distribution generated by the quantum walk. The explicit
form of the velocities leads us to the same condition derived in
Section~\ref{sec4}. Finally, in Section~\ref{sec6} we analyze the
formula for the mean value of the position of the particle derived
in \ref{app:b} in dependence of the parameters of the walk
and the initial state. We find that there exist genuine biased
quantum walks which are recurrent. Conclusions and outlook are
left for Section~\ref{sec7}.
\section{Description of the walk}
\label{sec2}
Let us consider biased quantum walks on a line where the particle has two possibilities --- jump to the right or to the left. Without loss of generality we restrict ourselves to biased quantum walks where the jump to the right is of the length $r$ and the jump to
the left has a unit size. We depict the biased quantum walk schematically in \fig{fig1}.
\begin{equation}gin{figure}
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig1.eps}
\caption{Schematics of the biased quantum walk on a line. If the
coin is in the state $|R\rangle$ the particle moves to the right
to a point at distance $r$. With the coin state $|L\rangle$
the particle makes a unit length step to the left. Before the step
itself the coin state is rotated according to the coin operator
$C(\rho)$.}
\label{fig1}
\end{center}
\end{figure}
The Hilbert space of the particle has the form of the tensor product
\begin{equation}gin{equation}
{\cal H} = {\cal H}_P\otimes{\cal H}_C
\end{equation}
of the position space
\begin{equation}gin{equation}
{\cal H}_P = \ell^2(\mathds{Z}) = \textrm{Span}\left\{|m\rangle|\ m\in\mathds{Z}\right\},
\end{equation}
and the two dimensional coin space
\begin{equation}gin{equation}
{\cal H}_C = \mathds{C}^2 = \textrm{Span}\left\{|R\rangle,|L\rangle\right\}.
\end{equation}
A single step of the quantum walk is given by the propagator
\begin{equation}gin{equation}
U = S \left(I_P\otimes C\right).
\label{qw:time}
\end{equation}
Here $I_P$ denotes the unit operator acting on the position space $\mathcal{H}_P$. The displacement operator $S$ has the form
\begin{equation}gin{equation}
S = \sum\limits_{m=-\infty}^{+\infty}|m+r\rangle\langle m|\otimes|R\rangle\langle R|+\sum\limits_{m=-\infty}^{+\infty}|m-1\rangle\langle m|\otimes|L\rangle\langle L|.
\end{equation}
The coin flip $C$ is in general an arbitrary unitary operator acting on the coin space $\mathcal{H}_C$ and is applied on the
coin state before the displacement $S$ itself. However, as has been discussed in \cite{tregenna} the probability distribution is not affected by the complex phases of the coin operator. Hence, it is sufficient to consider the one-parameter
family of coins
\begin{equation}gin{equation}
C(\rho) = \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho} & \sqrt{1-\rho} \\
\sqrt{1-\rho} & -\sqrt{\rho} \\
\end{array}
\right).
\label{coins}
\end{equation}
From now on we restrict ourselves to this family
of coins. The value of $\rho=1/2$ corresponds to the well known case of the Hadamard walk.
We write the initial state of the particle in the form
\begin{equation}gin{equation}
|\psi(0)\rangle \equiv
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,0)|m\rangle\otimes|i\rangle.
\end{equation}
The state of the walker after $t$ steps is given by successive
application of the time evolution operator given by Eq.
(\ref{qw:time}) on the initial state
\begin{equation}gin{equation}
|\psi(t)\rangle \equiv
U^t|\psi(0)\rangle =
\sum\limits_{m=-\infty}^{+\infty}\sum_{i=R}^L\psi_i(m,t)|m\rangle\otimes|i\rangle.
\label{time:evol}
\end{equation}
The state of the particle is fully
determined by the set of two-component vectors
\begin{equation}gin{equation}
\psi(m,t)\equiv{\left(\psi_R(m,t),\psi_L(m,t)\right)}^T.
\label{prob:ampl}
\end{equation}
Here $\psi_{R(L)}(m,t)$ is the probability
amplitude to find the particle at position $m$ after $t$ steps with
the coin state $|R(L)\rangle$. The probability distribution
generated by the quantum walk is given by
\begin{equation}gin{eqnarray}
\nonumber P(m,t) & = & |\langle m,R|\psi(t)\rangle|^2+|\langle m,L|\psi(t)\rangle|^2 \\
\nonumber & = & |\psi_R(m,t)|^2+|\psi_L(m,t)|^2 = \|\psi(m,t)\|^2.\\
\end{eqnarray}
\section{Time evolution of the walk}
\label{sec3}
To obtain explicit and closed form expressions for the time
dependent state vector we rewrite the time evolution equation
(\ref{time:evol}) for the state vector $|\psi(t)\rangle$ into a set
of difference equations
\begin{equation}gin{eqnarray}
\nonumber \psi(m,t) & = & C_+(\rho)\psi(m-r,t-1)+\\
& & +C_-(\rho)\psi(m+1,t-1)
\label{time:evol2}
\end{eqnarray}
for the probability amplitude vectors $\psi(m,t)$. The form of the
matrices $C_\pm(\rho)$ follows from the matrix $C(\rho)$
\begin{equation}gin{equation}
C_+(\rho) = \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho} & \sqrt{1-\rho} \\
0 & 0 \\
\end{array}
\right),\qquad
C_-(\rho) = \left(
\begin{equation}gin{array}{cc}
0 & 0 \\
\sqrt{1-\rho} & -\sqrt{\rho} \\
\end{array}
\right).
\end{equation}
The time evolution equations (\ref{time:evol2}) are greatly
simplified with the help of the Fourier transformation
\begin{equation}gin{equation}
\tilde{\psi}(k,t)\equiv\sum\limits_{m=-\infty}^{+\infty}\psi(m,t)e^{i mk},
\label{qw:ft}
\end{equation}
where the momentum $k$ is a continuous parameter ranging from $-\pi$ to $\pi$. The new function
$\tilde{\psi}(k,t)$ is square integrable on a unit circle.
The time evolution in the Fourier picture turns into a single difference equation
\begin{equation}gin{equation}
\tilde{\psi}(k,t) = \widetilde{U}(k)\tilde{\psi}(k,t-1),
\label{qw:te:fourier}
\end{equation}
where the propagator has the form
\begin{equation}gin{equation}
\widetilde{U}(k) \equiv \left(
\begin{equation}gin{array}{cc}
\sqrt{\rho}e^{ikr} & \sqrt{1-\rho}e^{ikr} \\
\sqrt{1-\rho}e^{-ik} & -\sqrt{\rho}e^{-ik} \\
\end{array}
\right).
\label{teopF}
\end{equation}
The solution of (\ref{qw:te:fourier}) is
straightforward. We find
\begin{equation}gin{equation}
\tilde{\psi}(k,t) =\widetilde{U}^t(k)\tilde{\psi}(k,0),
\end{equation}
where $\tilde{\psi}(k,0)$ is the Fourier transformation of the initial
state. We restrict ourselves to the situation where the particle is
initially localized at the origin as dictated by the nature of
the problem we wish to study. As follows from (\ref{qw:ft}) the
Fourier transformation $\tilde{\psi}(k,0)$ of such an initial
condition is equal to the initial state of the coin
\begin{equation}gin{equation}
\tilde{\psi}(k,0) = \left(
\begin{equation}gin{array}{c}
\psi_R(0,0) \\
\psi_L(0,0) \\
\end{array}
\right),
\end{equation}
which we denote by $\psi$. Since $\psi$ can be an arbitrary
normalized complex two-component vector we parameterize it
by two parameters $a\in[0,1]$ and $\varphi\in[0,2\pi)$ in the
form
\begin{equation}gin{equation}
\psi = \left(
\begin{equation}gin{array}{c}
\sqrt{a} \\
\sqrt{1-a}e^{i\varphi} \\
\end{array}
\right).
\label{psi:init}
\end{equation}
To evaluate the powers of the propagator $\widetilde{U}(k)$
it is convenient to diagonalize it. Since the propagator is
unitary its eigenvalues have the form $e^{i\omega_{1,2}}$ where the
phases read
\begin{equation}gin{eqnarray}
\nonumber \omega_1(k) & = & \frac{r-1}{2}k+\arcsin\left(\sqrt{\rho}\sin\left(\frac{r+1}{2}k\right)\right),\\
\omega_2(k) & = & \frac{r-1}{2}k -\pi-\arcsin\left(\sqrt{\rho}\sin\left(\frac{r+1}{2}k\right)\right)
\label{omega}
\end{eqnarray}
We denote the corresponding eigenvectors by $v_{1,2}(k)$. We
give their explicit form in the \ref{app:b}. With this notation we write
the solution of the time evolution equation in the Fourier picture
in the form
\begin{equation}gin{equation}
\widetilde{\psi}(k,t) = \sum_{j=1}^2
e^{i\omega_j(k)t}\left(v_j(k),\psi\right)v_j(k).
\end{equation}
Here $( , )$ means scalar product in the coin space. Finally, we obtain the solution in position representation by
performing the inverse Fourier transformation
\begin{equation}gin{eqnarray}
\nonumber \psi(m,t) = \int_{-\pi}^\pi\frac{dk}{2\pi}\ \widetilde{\psi}(k,t)\ e^{-imk} = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i(\omega_j(k)t-mk)}\ \left(v_j(k),\psi\right)v_j(k).\\
\label{inv:f}
\end{eqnarray}
\section{Asymptotics of the quantum walk and recurrence}
\label{sec4}
To determine the recurrence nature of the biased quantum walk we
have to analyze the asymptotic behaviour of the probability at the
origin \cite{prl}. Exploiting (\ref{inv:f}) the amplitude at the origin
reads
\begin{equation}gin{equation}
\psi(0,t) = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i\omega_j(k)t}\
\left(v_j(k),\psi\right)v_j(k),
\label{psi:0}
\end{equation}
which allows us to find the asymptotics of the probability at the origin
with the help of the method of stationary phase \cite{statphase}.
The important contributions to the integrals in (\ref{psi:0}) arise
from the stationary points of the phases (\ref{omega}). We find that
the derivatives of the phases $\omega_{1,2}(k)$ are
\begin{equation}gin{eqnarray}
\nonumber \omega_1'(k) & = & \frac{r-1}{2}+\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}},\\
\nonumber \omega_2'(k) & = & \frac{r-1}{2}-\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}}.\\
\label{phase:der}
\end{eqnarray}
Using the method of stationary phase we find that the
amplitude will decay slowly - like $t^{-\frac{1}{2}}$, if at least
one of the phases has a vanishing derivative inside the integration
domain. Solving the equations $\omega_{1,2}'(k) = 0$ we find
that the possible saddle points are
\begin{equation}gin{equation}
k_0 = \pm\frac{2}{r+1}\arccos\left(\pm\sqrt{\frac{(1-\rho)(r-1)^2}{4\rho r}}\right).
\label{k0}
\end{equation}
The saddle points are real valued
provided the argument of the arcus-cosine in (\ref{k0}) is less or
equal to unity
\begin{equation}gin{equation}
\frac{(1-\rho)(r-1)^2}{4\rho r} \leq 1.
\end{equation}
This inequality leads us to the condition for the biased quantum
walk on a line to be recurrent
\begin{equation}gin{equation}
\rho_R(r) \geq \left(\frac{r-1}{r+1}\right)^2.
\label{crit:rec}
\end{equation}
We illustrate this result in \fig{fig2} for a particular choice of the walk parameter $r=3$.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{The existence of stationary points of the phases
$\omega_{1,2}(k)$ in dependence on the parameter $\rho$ and
a fixed step length $r$. We plot the implicit functions $\omega_{1,2}'(k)\equiv
0$ for $r=3$. The plot indicates that for
$\rho<\rho_R(3)=\frac{1}{4}$ the phases $\omega_{1,2}(k)$ do
not have any saddle points. Consequently, the probability amplitude
at the origin decays fast and such biased quantum walk on a line is
transient. For $\rho\geq\rho_R(3)$ the saddle points exist and the
quantum walk is recurrent.}
\label{fig2}
\end{center}
\end{figure}
Our simple result proves that there is an intimate nontrivial
link between the length of the step of the walk and the bias of the
coin. The parameter of the coin $\rho$ has to be at least equal to
a factor determined by the size of the step to the right $r$ for the
walk to be recurrent. We note that the recurrence nature of the
biased quantum walk on a line is determined only by the parameters
of the walk itself, i.e. the coin and the step, not by the initial
conditions. The parameters of the initial state $a$ and $\varphi$
have no effect on the rate of decay of the probability at the
origin.
\section{Recurrence of a quantum walk and the velocities of the peaks}
\label{sec5}
We can determine the recurrence nature of the biased quantum walk on
a line from a different point of view. This approach is based
on the following observation. The well known shape of the
probability distribution generated by the quantum walk consists of
two counter-propagating peaks. In between
the two dominant peaks the probability is roughly independent of $m$ and decays like $t^{-1}$.
On the other hand, outside the decay is exponential as we depart from the peaks. As it has been found in \cite{nayak} the position of the peaks varies linearly with the number of steps. Hence, the peaks propagate with constant velocities, say $v_L$ and $v_R$. For the
biased quantum walk to be recurrent the origin of the walk has to
remain in between the two peaks for all times. In other words, the
biased quantum walk on a line is recurrent if and only if the
velocity of the left peak is negative and the velocity of the right
peak is positive.
The velocities of the left and right peaks are easily determined. We
rewrite the formula (\ref{inv:f}) for the probability amplitude
$\psi(m,t)$ into the form
\begin{equation}gin{equation}
\psi(m,t) = \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}e^{i(\omega_j(k)-\alpha
k)t}\ \left(v_j(k),\psi\right)v_j(k),
\end{equation}
where we have introduced $\alpha = \frac{m}{t}$. Due to the fact that we now
concentrate on the amplitudes at the positions $m\sim t$ we have to
consider modified phases
\begin{equation}gin{equation}
\widetilde{\omega}_j(k) = \omega_j(k)-\alpha k.
\end{equation}
The peak occurs at such a position
$m_0$ where both the first and the second derivatives of
$\widetilde{\omega}_j(k)$ vanishes. The velocity of the peak is
thus $\alpha_0 = \frac{m_0}{t}$. Hence, solving the equations
\begin{equation}gin{eqnarray}
\nonumber \widetilde{\omega}_1'(k) & = & \frac{r-1}{2}+\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}} - \alpha = 0 ,\\
\nonumber \widetilde{\omega}_2'(k) & = & \frac{r-1}{2}-\frac{\sqrt{\rho}(r+1)\cos\left(k\frac{r+1}{2}\right)}{\sqrt{4+2\rho\left[\cos(k(r+1))-1\right]}} - \alpha = 0,\\
\nonumber \widetilde{\omega}_1''(k) & = & -\widetilde{\omega}_2''(k) = \frac{(\rho-1)\sqrt{\rho}(r+1)^2\sin\left(k\frac{r+1}{2}\right)}{\sqrt{2}\left[2-\rho+\rho\cos(k(r+1))\right]^\frac{3}{2}} = 0,\\
\end{eqnarray}
for $\alpha$ determines the velocities of the left and right peak $v_{L,R}$. The third equation is independent of $\alpha$ and we easily find the
solution
\begin{equation}gin{equation}
k_0 = \frac{4n\pi}{r+1},\ k_0=\frac{2\pi(2n+1)}{r+1},\ n\in\mathds{Z}.
\end{equation}
Inserting this $k_0$ into the first two equations we find the velocities of the left and right peak
\begin{equation}gin{eqnarray}
\nonumber v_L & = & \frac{r-1}{2}-\frac{r+1}{2}\sqrt{\rho}\\
v_R & = & \frac{r-1}{2}+\frac{r+1}{2}\sqrt{\rho}.
\label{velocities}
\end{eqnarray}
We illustrate this result in \fig{fig3} where we show the probability distribution generated by the quantum walk for the particular choice of the parameters $r = 3,\ \rho = \frac{1}{\sqrt{2}}$. The initial state was chosen according to $a = \frac{1}{\sqrt{2}}$ and $\varphi = \pi$. Since the velocity of the left peak $v_L$ is negative this biased quantum walk is recurrent.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.7\textwidth]{fig3.eps}
\caption{Velocities of the left and right peak of the probability distribution generated by the biased quantum walk on a line and the recurrence.
We have chosen the parameters $r=3$, $a=\rho=\frac{1}{\sqrt{2}}$ and $\varphi = \pi$. The left peak propagates with the velocity $v_L\approx -0.68$,
the velocity of the right peak is $v_R\approx 2.68$. In between the two peaks the probability distribution behaves like $t^{-1}$ while outside the
decay is exponential. Since the velocity $v_L$ is negative the origin of the walk remains in between the left and right peak. Consequently, this quantum
walk is recurrent.}
\label{fig3}
\end{center}
\end{figure}
The peak velocities have two contributions. One is identical
and independent of $\rho$, the second is a product of $r$ and $\rho$
and differs in sign for the two velocities. The obtained results indicate that biasing the walk by having
the size of the step to the right equal to $r$ results in dragging
the whole probability distribution towards the direction of
the larger step. This is manifested by the term $\frac{r-1}{2}$
which appears in both velocities $v_{L,R}$ with the same sign. On
the other hand the parameter of the coin $\rho$ does not bias the
walk. As we can see from the second terms entering the velocities it
rather influences the rate at which the walk spreads.
As we have discussed above the biased quantum walk on a line is recurrent if and only if $v_L$ is negative and $v_R$ is positive. The form of the
velocities (\ref{velocities}) implies that this condition is satisfied if and only if the criterion (\ref{crit:rec}) is fulfilled.
\section{Mean value of the biased quantum walk and recurrence}
\label{sec6}
As we discuss in the \ref{app:a} the classical random
walks are recurrent if and only if the mean value of the position
vanishes. We now show that this is not true for biased quantum
walks, i.e. there exist biased quantum walks on a line which are
recurrent but cannot produce probability distribution with zero mean
value. This is another unique feature of quantum walks
compared to the classical ones.
In the \ref{app:b} we derive the following formula
for the position mean value
\begin{equation}gin{eqnarray}
\nonumber \left\langle\frac{x}{t}\right\rangle & \approx & (1-\sqrt{1-\rho})(a(r+1)-1)+\\
\nonumber & & +\frac{\sqrt{a(1-a)}(1-\sqrt{1-\rho})(1-\rho)(r+1)\cos\varphi}{\sqrt{\rho(1-\rho)}}\\
& & +\frac{r-1}{2}\sqrt{1-\rho}+O(t^{-1}).
\label{mean}
\end{eqnarray}
We see that for quantum walks the mean value is affected by both the
fundamental walk parameters through $r$ and $\rho$ and the
initial state parameters $a$ and $\varphi$. The mean value is
typically non-vanishing even for unbiased quantum walks (
with $r=1$ ). However, one easily finds \cite{tregenna} that the initial
state with the parameters $a=1/2$ and $\varphi=\pi/2$ results in a
symmetric probability distribution with zero mean independent of the
coin parameter $\rho$. Indeed, the quantum walks with $r=1$, i.e.
with equal steps to the right and left, do not intrinsically
distinguish left from right. On the other hand the quantum walks
with $r>1$ treat the left and right direction in a different way.
Nevertheless, one can always find for a given $r$ a coin parameter
$\rho_0$ such that for all $\rho\geq\rho_0$ the quantum walk can
produce a probability distribution with zero mean value. This is
impossible for quantum walks with $\rho<\rho_0$ and we will call
such quantum walks genuine biased.
Let us now determine the minimal value of $\rho$ for a given $r$ for
which mean value vanishes. We first find the parameters of the
initial state $a$ and $\varphi$ which minimizes the mean value.
Clearly the term on the second line in (\ref{mean}) reaches the
minimal value for $\varphi_0=\pi$. Differentiating the resulting
expression with respect to $a$ and setting the derivative equal to
zero gives us the condition
\begin{equation}gin{equation}
2 + \frac{(2a-1)
\sqrt{\rho(1-\rho)}}{\rho\sqrt{a(1-a)}} = 0
\end{equation}
on the minimal mean value with respect to $a$. This relation is satisfied for
$a_0=\frac{1}{2}(1-\sqrt{\rho})$. The resulting formula for the mean
value reads
\begin{equation}gin{equation}
\left\langle\frac{x}{t}\right\rangle_{a_0,\varphi_0}
= \frac{r-1}{2}+\frac{\left(1-\sqrt{1-\rho}-\rho\right) (1+r)}{2
\sqrt{(1-\rho) \rho}}.
\label{mean:min}
\end{equation}
This expression vanishes for
\begin{equation}gin{equation}
\rho_0(r) = \left(\frac{r^2 - 1}{r^2 + 1}\right)^2.
\label{rho:0}
\end{equation}
Since (\ref{mean:min}) is a decreasing function of
$\rho$ the mean value is always positive for $\rho<\rho_0$
independent of the choice of the initial state. For $\rho>\rho_0$
one can achieve zero mean value for different combination of the
parameters $a$ and $\varphi$.
The formula (\ref{rho:0}) is reminiscent of the condition
(\ref{crit:rec}) for the biased quantum walk on a line to be
recurrent. However, $r$ is in (\ref{rho:0}) replaced by $r^2$.
Therefore we find the inequality $\rho_R<\rho_0$. Hence, the quantum
walks with the coin parameter $\rho_R<\rho<\rho_0$ are recurrent but
cannot produce a probability distribution with zero mean value. We
conclude that there are genuine biased quantum walks which are
recurrent in contrast to situations found for classical walks.
\section{Conclusions}
\label{sec7}
We have analyzed one dimensional biased quantum walks. Classically, the bias leading to a non-zero mean value of the particle's position can be introduced in two ways --- unequal step lengths or unfair coin. In contrast, for quantum walks on a line the initial state can introduce bias for any coin. On the other hand, for symmetric initial state modifying only the unitary coin operator while keeping the equal step lengths will not introduce bias. Finally, the bias due to unequal step lengths may be compensated for by the choice of the coin operator for some initial conditions. For this reason we have introduced the concept of the genuinely biased quantum walk for which there does not exists any initial state leading to vanishing mean value of the position.
We have determined the conditions under which one dimensional biased quantum walks are recurrent. This together with the condition of being genuinely biased give rise to three different regions in the parameter space which we depict as a "phase diagram" in \fig{fig4}.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig4.eps}
\caption{"Phase diagram" of biased quantum walks on a line. The horizontal axis represents the length of the step to the right $r$ and the vertical axis shows the coin parameter $\rho$. The dotted line corresponds to the recurrence criterion (\ref{crit:rec}), while the squares represent the condition (\ref{rho:0}) on the zero mean value of the particle's position. The quantum walks in the white area are transient and genuine biased. In between the two curves (light gray area) we find quantum walks which are recurrent but still genuine biased. The quantum walks in the dark gray area are recurrent and for a particular choice of the initial state they can produce probability distribution with vanishing mean value.}
\label{fig4}
\end{center}
\end{figure}
The presented results allow for generalization to biased quantum walks in higher dimensions assuming we keep the coin operator
in a tensorial form. For non-factorizable coin operators in higher dimensions it remains an open question when they are recurrent or transient.
\ack
The financial support by MSM 6840770039, M\v SMT LC 06002, the Czech-Hungarian cooperation project (KONTAKT,CZ-10/2007) and by the Hungarian
Scientific Research Fund (T049234) is gratefully acknowledged.
\begin{equation}gin{appendix}
\section{Recurrence of classical biased random walk on a line}
\label{app:a}
Classical random walks on a line can be biased in two ways - the step in one direction is greater than in the other one and the probability of
the step to the right is different from the probability of the step to the left (see \fig{fig5}).
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig5.eps}
\caption{Schematics of the biased random walk on a line. The particle can move to the right by a distance $r$ with the probability $p$.
The length of the step to the left is unity and the probability of this step is $1-p$.}
\label{fig5}
\end{center}
\end{figure}
Consider a random walk on a line such that the particle can make a
jump of length $r$ to the right with probability $p$ or make a unit
size step to the left with probability $1-p$. The random walk is
recurrent if and only if the probability to find the particle at the
origin at any time instant $t$ does not decays faster than $t^{-1}$.
This probability is easily found to be expressed by the binomial
expression
\begin{equation}gin{equation}
P_0(t) = (1-p)^{\frac{t r}{r+1}}p^{\frac{t}{r+1}}{t\choose \frac{t r}{r+1}}.
\end{equation}
With the help of the Stirling's formula
\begin{equation}gin{equation}
n! \approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n
\end{equation}
we find the asymptotic behaviour of the probability at the origin
\begin{equation}gin{equation}
P_0(t)\approx \frac{r+1}{\sqrt{2\pi r t}}\left[(1-p)^{\frac{r}{r+1}}p^{\frac{1}{r+1}}\frac{r+1}{r^\frac{r}{r+1}}\right]^t.
\end{equation}
The asymptotics of the probability $P_0(t)$ therefore depends on
the value of
\begin{equation}gin{equation}
q = (1-p)^{\frac{r}{r+1}}p^{\frac{1}{r+1}}\frac{r+1}{r^\frac{r}{r+1}}.
\end{equation}
Since $q\leq 1$ the probability $P_0(t)$ decays exponentially
unless the inequality is saturated. Hence, the random walk is
recurrent if and only if $q$ equals unity. This condition is
satisfied for
\begin{equation}gin{equation}
\label{rw:cond}
p = \frac{1}{r+1},
\end{equation}
i.e. the probability of the step to the right has to be inversely
proportional to the length of the step.
This result can be well understood from a different point of view, as we illustrate in \fig{fig6}.
The spreading of the probability distribution is diffusive, i.e.
$\sigma\sim\sqrt{t}$. The probability in the $\sigma$ neighborhood
of the mean value $\langle x\rangle$ behaves like $t^{-\frac{1}{2}}$
while outside this neighborhood the probability decays
exponentially. Therefore for the random walk to be recurrent the
origin must lie in this $\sigma$ neighborhood for all times $t$.
However, if the random walk is biased the mean value of the position
$\langle x\rangle$ varies linearly in time, thus it is a faster
process than the spreading of the probability distribution. In such
a case the origin would lie outside the $\sigma$ neighborhood of the mean
value after a finite number of steps leading to the exponential
asymptotic decay of the probability at the origin $P_0(t)$. Hence,
the random walk is recurrent if and only if the mean value of the
position equals zero. Since the individual steps are independent of
each other the mean value after $t$ steps is simply a $t$ multiple
of the mean value after single step, i.e.
\begin{equation}gin{equation}
\langle x (t)\rangle = t \langle x(1)\rangle = t\left[p(r+1)-1\right].
\end{equation}
We find that the mean value equals zero if and only if the condition (\ref{rw:cond})
holds.
\begin{equation}gin{figure}[h]
\begin{equation}gin{center}
\includegraphics[width=0.6\textwidth]{fig6.eps}
\caption{Spreading of the probability distribution versus the motion
of the mean value of a biased classical random walk on a line. While the
spreading is diffusive ($\sigma\sim\sqrt{t}$) the mean value
propagates with a constant velocity ($\langle x\rangle\sim t$). The
probability inside the $\sigma$ neighborhood of the mean value
$\langle x\rangle$ behaves like $t^{-\frac{1}{2}}$. On the other
hand, as we go away from the $\sigma$ neighbourhood the decay is exponential.
Hence, if the mean value $\langle x\rangle$ does not vanish the
origin of the walk leaves the $\sigma$ neighborhood of the mean
value. In such a case the probability at the origin decays
exponentially and the walk is transient.}
\label{fig6}
\end{center}
\end{figure}
\section{Mean value of the particle's position for a quantum walk on a line}
\label{app:b}
In this Appendix we find the explicit form of the position mean
value of the particle. With the help of the weak limit theorem
\cite{Grimmett} we express the mean value after $t$ steps in the
form
\begin{equation}gin{equation}
\left\langle \frac{x}{t}\right\rangle \approx \sum_{j=1}^2\int_{-\pi}^\pi\frac{dk}{2\pi}\ \omega_j'(k)\ \left(v_j(k),\psi\right)v_j(k),
\end{equation}
up to the corrections of the order $O(t^{-1})$. Here $v_j(k)$ are eigenvectors of the
unitary propagator $\widetilde{U}(k)$, $\omega_j'(k)$ are the
derivatives of the phases of the corresponding eigenvalues and
$\psi$ is the initial state expressed in (\ref{psi:init}). The
derivatives of the phases are given in (\ref{phase:der}). We express
the eigenvectors in the form
\begin{equation}gin{eqnarray}
\nonumber v_1(k) & = & n_1(\rho,k)\left(\sqrt{1-\rho}, -\sqrt{\rho} + e^{i(\omega_1(k)-rk)}\right)^T,\\
\nonumber v_2(k) & = & n_2(r,k)\left(\sqrt{1-\rho}, -\sqrt{\rho} + e^{i(\omega_2(k)-rk)}\right)^T.\\
\end{eqnarray}
The normalization factors of the eigenvectors read
\begin{equation}gin{eqnarray}
\nonumber n_1(u) & = & 2-2\sqrt{\rho}\cos\left(u-\arcsin\left[\sqrt{\rho}\sin u\right]\right),\\
\nonumber n_2(u) & = & 2+2\sqrt{\rho}\cos\left(u+\arcsin\left[\sqrt{\rho}\sin u\right]\right),\\
\end{eqnarray}
where we denote $u=\frac{k(r+1)}{2}$ to shorten the notation. The mean value is thus given by the following integral
\begin{equation}gin{equation}
\left\langle \frac{x}{t}\right\rangle \approx \int\limits_0^{(r+1)\pi}\frac{f(a,\varphi,\rho,r,u)du}{2(r+1)\pi \left[1 +\sqrt{\rho}\cos u_1\right] \left[1-\sqrt{\rho} \sin u_2\right]}+O(t^{-1}),
\end{equation}
where
\begin{equation}gin{equation}
u_1 = u + \arcsin(\sqrt{\rho}\sin u),\quad u_2 = u + \arccos(\sqrt{\rho}\sin u),
\end{equation}
and the numerator reads
\begin{equation}gin{eqnarray}
\nonumber f(a,\varphi,\rho,r,u) & = & (1-\rho) \left[r-1 + \rho \left(a + r (a-1)\right)\left(1+\cos(2u)\right)+\right.\\
\nonumber & & \left.+ \sqrt{a(1-a)}\sqrt{\rho(1-\rho)} (r+1) \left(\cos{\varphi}+\cos(\varphi+2u)\right)\right].\\
\end{eqnarray}
Performing the integrations we arrive at the result
\begin{equation}gin{eqnarray}
\nonumber \left\langle\frac{x}{t}\right\rangle & \approx & (1-\sqrt{1-\rho})(a(r+1)-1)+\\
\nonumber & & +\frac{\sqrt{a(1-a)}(1-\sqrt{1-\rho})(1-\rho)(r+1)\cos\varphi}{\sqrt{\rho(1-\rho)}}\\
& & +\frac{r-1}{2}\sqrt{1-\rho}+O(t^{-1}).
\end{eqnarray}
\end{appendix}
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\end{document} |
\begin{document}
\title{ A Linear Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization }
\author{E. A. Papa Quiroz\thanks{ Mayor de San Marcos National University, Department of Mathematical Sciences, Lima, Per\'{u} and Federal University of Rio de Janeiro, Computing and Systems Engineering Department, post office box 68511,CEP 21945-970, Rio de Janeiro, Brazil(erikpapa@gmail.com).}
\and{H. C. F. Apolinario\thanks{Federal University of Tocantins, Undergraduate Computation Sciences Course, ALC NO 14 (109 Norte) AV.NS.15 S/N , CEP 77001-090, Tel: +55 63 8481-5168; +55 63 3232-8027; FAX +55 63 3232-8020, Palmas, Brazil (hellena@uft.edu.br).}}\\
\and{K.D.V. Villacorta\thanks{Federal University of Paraíba, Campus V-Mangabeira, João Pessoa-Paraíba, Brazil. CEP: 58.055-000 }}
\and{P. R. Oliveira\thanks{Federal University of Rio de Janeiro, Computing and Systems Engineering Department, post office box 68511,CEP 21945-970, Rio de Janeiro, Brazil(poliveir@cos.ufrj.br).}}}
\date{\today}
\maketitle
\ \\[-0.5cm]
\begin{center}
{\bf Abstract}
\end{center}
In this paper we propose a linear scalarization proximal point algorithm for solving arbitrary lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and using the condition that the proximal parameters are bounded we prove the convergence of the sequence generated by the algorithm and when the objective functions are continuous, we prove the convergence to a generalized critical point. Furthermore, if each iteration minimize the proximal regularized function and the proximal parameters converges to zero we prove the convergence to a weak Pareto solution. In the continuously differentiable case, it is proved the global convergence of the sequence to a Pareto critical point and we introduce an inexact algorithm with the same convergence properties. We also analyze particular cases of the algorithm obtained finite convergence to a Pareto optimal point when the objective functions are convex and a sharp minimum condition is satisfied.
\\\\
\noindent{\bf Keywords:} Multiobjective minimization, lower semicontinuous quasiconvex functions, proximal point methods, Fejér convergence, Pareto-Clarke critical point, finite convergence.
\section{Introduction}
\noindent
In this work we consider the multiobjective minimization problem:
\begin{eqnarray}
\textrm{min}\lbrace F(x): x \in \mathbb{R}^n\rbrace
\label{prob}
\end{eqnarray}
where $F=(F_1,F_2,...,F_m): \mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ is a lower semicontinuous and
quasiconvex vector function on the Euclidean space $ \mathbb{R}^n.$ The above notation means that each $F_i$ is an extended function, that is, $F_i:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}$ .
The main motivation to study this problem are the consumer demand theory in economy, where the quasiconvexity of the objective vector function is a natural condition associated to diversification of the consumption, see Mas Colell et al. \cite{Colell}, and the quasiconvex optimization models in location theory, see \cite{Gromicho}.
Recently Apolinario et al. \cite{apo} has been introduced an exact linear scalarization proximal point algorithm to solve the above class of problems when the vector function $F$ is locally Lipschitz and $\textnormal{dom}(F)=\mathbb{R}^n$. The proposed iteration was the following:
given $p^{k} \in \mathbb{R}^n$, find $p^{k+1}\in \Omega_k=\left\{ x\in \mathbb{R}^n: F(x) \preceq F(p^k)\right\}$ such that:
$$
0 \in \partial^o\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - p^k \Vert ^2 \right) (p^{k+1}) + \mathcal{N}_{\Omega_k}(p^{k+1})
$$
where ${\partial}^o$ is the Clarke subdifferential, see Subsection 2.5 of \cite{apo}, $\alpha_k > 0 $, $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$, $\left\|z_k\right\| = 1$ and $\mathcal{N}_{\Omega_k}(p^{k+1})$ the normal cone to $\Omega_k$ at $x^{k+1},$ see Definition \ref{normal} in Section \ref{Prelimin} of this paper. The authors proved, under some natural assumptions, that the sequence generated by the above algorithm is well defined and converges globally to a Pareto-Clarke critical point.
Unfortunately, the algorithm proposed in that paper can not be applied to a general class of proper lower semicontinuous quasiconvex functions, and thus can not be applied to solve constrained multiobjective problems nor continuous quasiconvex functions which are not locally Lipschitz. Moreover, for a future implementation and application for example to costly improving behaviors of strongly averse agents in economy (see Sections 5 and 6 of Bento et al. \cite{Bento}), that paper did not provide an inexact version of the proposed algorithm .
Thus we have two motivations in the present paper: the first motivation is to extend the convergence properties of the linear scalarization proximal point method introduced in \cite{apo} to solve more general, probabily constrained, quasiconvex multiobjective problems of the form (\ref{prob}) and the second ones is to introduce an inexact algorithm when $F$ is continuously differentiable on $\mathbb{R}^m$.
The main iteration of the proposed algorithm is: Given $x^k,$ find $x^{k+1} $ such that
\begin{eqnarray}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\label{subdiferencial3i}
\end{eqnarray}
where $\hat{\partial}$ is the Fréchet subdifferential, see Subsection \ref{frechet}, \ $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$, $\alpha_k > 0 $, $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$ and $\left\|z_k\right\| = 1$.
Some works related to this paper are the following:
\begin{itemize}
\item Bento et al. \cite{Bento} introduced the nonlinear scalarized proximal iteration:
$$
y^{k+1}\in \arg \min \left\{f\left( F(x)+ \delta_{\Omega_k}(x)e+ \dfrac{\alpha_k}{2} \Vert\ .\ - y^k \Vert ^2e \right): x\in \mathbb{R}^n \right\}
$$
where $f:\mathbb{R}^n\longrightarrow \mathbb{R}$ is a function defined by $f(y):=\max_{i\in I}\{ \langle y,e_i \rangle \}$
with $e_i$ is the canonical base of the space $\mathbb{R}^n,$ $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(y^k)\right\}$ and $e=(1,1,...,1)\in \mathbb{R}^n.$ Assuming that $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is quasiconvex and continuously differentiable and under some natural assumptions the authors proved that the sequence $\{y^k\}$ converges to a Pareto Critical point of $F.$ Furthermore, assuming that $F$ is convex, the weak Pareto optimal set is weak sharp for the multiobjective problem and that the sequence is generated by the following unconstrained iteration
$$
y^{k+1}:=\arg \min \left\{f(F(x)) + \dfrac{\alpha_k}{2} \Vert\ x\ - y^k \Vert ^2:x\in \mathbb{R}^n \right\}
$$
then the above iteration obtain a Pareto optimal point after a finite number de iterations.
The difference between our work and the paper of Bento et al., \cite{Bento}, is that in the present paper we consider a linear scalararization of $F$ instead of a nonlinear ones proposed in \cite{Bento}, another difference is that our assumptions are more weak, in particular, we obtain convergence results for nondifferentiable quasiconvex functions.
\item Makela et al., \cite{makela}, developed a multiobjective proximal bundle method for nonsmooth optimization where the objective functions are locally Lipschitz (not necessarily smooth nor convex). The proximal method is not directly based on employing any scalarizing function but based on a improvement function $H:\mathbb{R}^n\times \mathbb{R}^n \longrightarrow \mathbb{R}$ defined by $H(x,y)=\max \{ F_i(x)-F_i(y),g_j(x):i=1,...,m, j=1,...,r \}$ with $\textnormal{dom} F=\{x\in \mathbb{R}^n: g_j(x)\geq 0, j=1,...,r\}.$ If $F_i$ and $g_j$ are pseudoconvex and weakly semismooth functions and certain constraint qualification is valid, the authors proved that any accumulation point of the sequence is a weak Pareto solution and without the assumption of pseudoconvex, they obtained that any accumulation point is a substationary point, that is, $0\in \partial H(\bar x, \bar x),$ where $\bar x$ is an accumulation point.
\item Chuong et al., \cite{chuong}, developed three algorithms of the so-called hybrid approximate proximal type to find Pareto optimal points for general class of convex constrained problems of vector optimization in finite and infinite dimensional spaces, that is, $\min_C \{F(x): x\in \Omega\},$ where $C$ is a closed convex and pointed cone and the minimization is understood with respect to the ordering relation given by $y\preceq_{C}x$ if and only if $x-y\in C$. Assuming that the set $\left(F(x^0) - C\right) \cap F(\Omega)$ is $C$ - quasi-complete for $\Omega,$ that is, for any sequence $\{u_l\}\subset \Omega$ with $u_0=x_0$ such that $F(u_{l+1})\preceq_{C}F(u_l)$ there exists $u\in VI(\Omega, A)$ satisfying $F(u)\preceq_{C}F(u_l),$ for every $l\in \mathbb{N};$ and the assumption that $F$ is $C^{+}-$ uniformly semicontinuous on $\Omega,$ the authors proved the convergence of the sequence generates by its algorithm.
\end{itemize}
Under the assumption that $F$ is a proper lower semicontinuous quasiconvex vector function and the assumption that the set $\left(F(x^0) - \mathbb{R}^m_+\right)\cap F(\mathbb{R}^n)$ is $\mathbb{R}^m_+$ - complete we prove the global convergence of the sequence $\{x^k\},$ generated by (\ref{subdiferencial3i}), to the set
$$E = \left\{x \in \mathbb{R}^n: F\left(x\right)\preceq F\left(x^k\right),\ \ \forall\ k \in \mathbb{N}\right\}.$$
Additionally, if $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ is continuous, and $0<\alpha_k<\bar{\alpha},$ for some $\bar{\alpha}>0,$ we prove that
$\lim \limits_{k\rightarrow +\infty}g^{k} = 0,$ where $g^k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1}).$ In the particular case when $ \lim \limits_{k\rightarrow +\infty}\alpha_k= 0$ and the iterations are given by
\begin{eqnarray}
x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\},
\label{recursao0i}
\end{eqnarray}
then the sequence $\lbrace x^k\rbrace$ converges to a weak pareto solution of the problem $(\ref{prob})$.
When the vector function $F: \mathbb{R}^n\longrightarrow \mathbb{R}^m$ is continuously differentiable and $0<\alpha_k<\bar{\alpha},$ for some $\bar{\alpha}>0,$ we prove that the sequence $\{x^k\},$ generated by (\ref{subdiferencial3i}), converges to a Pareto critical point of the problem (\ref{prob}). Then, we introduce an inexact proximal algorithm given by
\begin{equation}
0 \in \hat{\partial}_{\epsilon_k} \left( \langle F(x), z_k \rangle \right) (x) + \alpha_k\left(x - x^k\right) + \mathcal{N}_{\Omega_k}(x),
\label{diferenciali}
\end{equation}
\begin{equation}
\label{deltai}
\displaystyle \sum_{k=1}^{\infty} \delta_k < \infty,
\end{equation}
where $\delta_k = \textnormal{max}\left\lbrace \dfrac{\varepsilon_k}{\alpha_k}, \dfrac{\Vert \nu_k\Vert}{\alpha_k}\right\rbrace,$ $\varepsilon_k\geq 0,$ and $\hat{\partial}_{\varepsilon_k}$ is the Fréchet $\varepsilon_k$-subdifferential.
We prove the convergence of $\{x^k\},$ generated by (\ref{diferenciali}) and (\ref{deltai}) to a Pareto critical point of the problem (\ref{prob}).
We also analyze some conditions to obtain finite convergence of a particular case of the proposed algorithm.
The paper is organized as follows: In Section 2 we recall some concepts and basic results on multiobjective optimization, descent direction, scalar representation, quasiconvex and convex functions, Fr\'echet and Limiting subdiferential, $\epsilon-$Subdifferential and Fej\'{e}r convergence. In Section 3 we present the problem and we give an example of a quasiconvex model in demand theory. In Section 4 we introduce an exact algorithm and analyze its convergence. In Section 5 we present an inexact algorithm for the differentiable case and analyze its convergence. In Section 6, we introduce an inexact algorithm for nonsmooth proper lower semicontinuous convex multiobjective minimization and using some concepts of weak sharp minimum we prove the convergence of the iterations in a finite number of steps to a Pareto optimal point. In Section 7 give a numerical example of the algorithm and in Section 8 we give our conclusions.
\section{Preliminaries}
\label{Prelimin}
In this section, we present some basic concepts and results that are of fundamental importance for the development of our work. These facts can be found, for example, in Hadjisavvas \cite{Had}, Mordukhovich \cite{Mordukhovich} and, Rockafellar and Wets \cite{Rockafellar}.
\subsection{Definitions, notations and some basic results}
Along this paper $ \mathbb{R}^n$ denotes an Euclidean space, that is, a real vectorial space with the canonical inner product $\langle x,y\rangle=\sum\limits_{i=1}^{n} x_iy_i$ and the norm given by $||x||=\sqrt{\langle x, x\rangle }$.\\
Given a function {\small $f :\mathbb{R}^n\longrightarrow \mathbb{R}\cup\left\{+\infty\right\}$}, we
denote by $\textnormal{dom}(f)= \left\{x \in \mathbb{R}^n: f(x) < + \infty \right\},$ the {\it effective domain } of $f$.
If $\textnormal{dom}(f) \neq \emptyset$, $f $ is called proper.
If {\footnotesize $\lim \limits_{\left\|x\right\|\rightarrow +\infty}f(x) = +\infty$}, $f$ is called coercive. We denote by arg min $\left\{f(x): x \in \mathbb{R}^n\right\}$ the set of minimizer of the function $f$ and by $f * $, the optimal value of problem: $\min \left\{f(x): x \in \mathbb{R}^n\right\},$ if it exists.
The \ function \ $f$ is {\it lower semicontinuous} at $\bar{x}$ if for all sequence $\left\{x_k\right\}_{k \in \mathbb{N}} $ such that $\lim \limits_{k \rightarrow +\infty}x_k = \bar{x}$ we obtain that $f(\bar{x}) \leq \liminf \limits_{k \rightarrow +\infty}f(x_k)$.\\
The next result ensures that the set of minimizers of a function, under some assumptions, is nonempty.
\begin{proposicao}{\bf (Rockafellar and Wets \cite{Rockafellar}, Theorem 1.9)}\\
Suppose that {\small $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup\left\{+\infty\right\}$} is {\it proper, lower semicontinuous} and coercive, then the optimal value $ f^*$\ is finite and the set $\textnormal{arg min}$ $\left\{f(x): x \in \mathbb{R}^n\right\}$ is nonempty and compact.
\label{coercivaesemicont}
\end{proposicao}
\begin{Def}
Let $D \subset \mathbb{R}^n$ and $\bar{x} \in D$. The normal cone to $D$ at $\bar{x} \in D$ is given by $\mathcal{N}_{D}(\bar{x}) = \left\{v \in \mathbb{R}^n: \langle v, x - \bar{x}\rangle \leq 0, \forall \ x \in D\right\}$.
\label{normal}
\end{Def}
It follows an important result that involves sequences of non-negative numbers which will be useful in Section 5.
\begin{lema}
Let $\{w_k\}$, $\{p_k\}$ and $\{q_k\}$ sequences of non-negative real numbers. If
\begin{equation*}
w_{k+1} \leq \left( 1 + p_k\right)w_k + q_k, \ \ \ \ \displaystyle \sum_{i=1}^{\infty} p_k < +\infty \ \ \textnormal{and} \ \ \displaystyle \sum_{i=1}^{\infty} q_k < +\infty,
\end{equation*}
then the sequence $\{w_k\}$ is convergent.
\label{p}
\end{lema}
\begin{proof}
See Polyak \cite{Polyak}, Lema 2.2.2.
\end{proof}
\subsection{Multiobjective optimization}
In this subsection we present some properties and notation on multiobjective optimization. Those basic facts can be seen, for example, in Miettinen \cite{Kaisa} and Luc \cite{Luc}.\\
Throughout this paper we consider the cone $\mathbb{R}^m_+ = \{ y\in \mathbb{R}^m : y_i\geq0, \forall \ i = 1, ... , m \}$, which induce a partial order $\preceq$ in $\mathbb{R}^m$ given by, for $y,y'\in \mathbb{R}^m$,
$y\ \preceq\ y'$ if, and only if, $ y'\ - \ y$ $ \in \mathbb{R}^m_+$, this means that $ y_i \leq \ y'_i,$ for all $ i= 1,2,...,m $ . Given $ \mathbb{R}^m_{++}= \{ y\in \mathbb{R}^m : y_i>0, \forall \ i = 1, ... , m \}$ the above relation induce the following one $\prec$, induced by the interior of this cone, given by, $y\ \prec\ y'$, if, and only if, $ y'\ - \ y$ $ \in \mathbb{R}^m_{++}$, this means that $ y_i < \ y'_i$ for all $ i= 1,2,...,m$. Those partial orders establish a class of problems known in the literature as Multiobjective Optimization.\\ \\
Let us consider the unconstrained multiobjective optimization problem (MOP) :
\begin{eqnarray}
\textrm{min} \left\{G(x): x \in \mathbb{R}^n \right\}
\label{POM}
\end{eqnarray}
where $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$, with $G = \left(G_1, G_2, ... , G_m\right)$ and $G_i:\mathbb{R}^n \longrightarrow \mathbb{R}, \forall i=1,...,m$.
\begin{Def} {\bf (Miettinen \cite{Kaisa}, Definition 2.2.1)}
A point $x^* \in \mathbb{R}^n$ is a Pareto optimal point or Pareto solution of the problem $\left(\ref{POM}\right)$, if there does not exist $x \in \mathbb{R}^n $ such that $ G_{i}(x) \leq G_{i}(x^*)$, for all $i \in \left\{1,...,m\right\}$ and $ G_{j}(x) < G_{j}(x^*)$, for at least one index $ j \in \left\{1,...,m\right\}$ .
\end{Def}
\begin{Def}{\bf (Miettinen \cite{Kaisa},Definition 2.5.1)}
A point $x^* \in \mathbb{R}^n$ is a weak Pareto solution of the problem $\left(\ref{POM}\right)$, if there does not exist $x \in \mathbb{R}^n $ such that $ G_{i}(x) < G_{i}(x^*)$, for all $i \in \left\{1,...,m\right\}$.
\end{Def}
We denote by arg min$\left\{G(x):x\in \mathbb{R}^n \right\}$ and by arg min$_w$ $\left\{G(x):x\in \mathbb{R}^n \right\}$ the set of Pareto solutions and weak Pareto solutions to the problem $\left(\ref{POM}\right)$, respectively. It is easy to check that\\ arg min$\left\{G(x):x\in \mathbb{R}^n \right\} \subset$ arg min$_w$ $\left\{G(x):x\in \mathbb{R}^n \right\}$.
\subsection{Pareto critical point and descent direction}
Let $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ be a differentiable function and $x \in \mathbb{R}^n$, the jacobian of $G$ at $x$, denoted by $JG(x)$, is a matrix of order $m \times n$ whose entries are defined by $\left(JG(x)\right)_{i,j} = \frac{\partial G_i}{\partial x_j}(x)$. We may represent it by,
\begin{center}
$JG\left(x\right) := \left[\nabla G_1(x) \nabla G_2(x)... \nabla G_m(x) \right]^T$, $x \in \mathbb{R}^n$.
\end{center}
The image of the jacobian of $G$ at $x$ we denote by
\begin{center}
$\footnotesize{Im \left(JG\left(x\right)\right) := \lbrace JG\left(x\right)v = \left(\langle \nabla G_1(x) , v\rangle, \langle \nabla G_2(x) , v\rangle, ..., \langle \nabla G_m(x) , v\rangle\right): v \in \mathbb{R}^n \rbrace }$.
\end{center}
A necessary but not sufficient first order optimality condition for the problem
$(\ref {POM})$ at $x \in \mathbb {R}^n $, is
\begin{eqnarray}
Im \left(JG\left(x\right)\right)\cap\left(-\mathbb{R}^m_{++}\right)=\emptyset.
\label{cond}
\end{eqnarray}
Equivalently, $\forall \ v \in \mathbb{R}^n$, there exists $i_0 = i_0(v) \in \lbrace 1,...,m\rbrace$ such that
\begin{center}
$\langle \nabla G_{i_0}(x) , v \rangle \geq 0$.
\end{center}
\begin{Def}
Let $G:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ be a differentiable function. A point $x^* \in \mathbb{R}^n$ satisfying $(\ref {cond})$ is called a Pareto critical point.
\end{Def}
Follows from the previous definition, if a point $x$ is not Pareto critical point, then there exists a direction
$v \in \mathbb {R}^ n$ satisfying
\begin{center}
$JG\left(x\right)v \in \left(-\mathbb{R}^m_{++}\right)$,
\end{center}
i.e, $\langle \nabla G_i( x ) , v \rangle < 0, \ \forall \ i \in \lbrace 1,..., m \rbrace$. As $G$ is continuously differentiable, then
\begin{center}
$\displaystyle \lim_{t \rightarrow 0}\dfrac{G_i(x + tv) - G_i(x)}{t}= \langle \nabla G_i(x) ,v\rangle < 0, \ \forall \ i \in \lbrace 1,..., m \rbrace $.
\end{center}
This implies that $v$ is a {\it descent direction} for the function $G_i$, i.e, there exists $\varepsilon > 0 $, such that
\begin{center}
$G_i(x + tv) < G_i(x), \forall \ t \in (0 , \varepsilon ], \forall \ i \in \lbrace 1,..., m \rbrace $.
\end{center}
Therefore, $v$ is a {\it descent direction} for $G$ at $x$, i.e, there exists $ \varepsilon > 0 $ such that
\begin{center}
$ G(x + tv) \prec G(x), \ \forall \ t \in (0 , \varepsilon]$.
\end{center}
\subsection{Scalar representation}
In this subsection we present a useful technique in multiobjective optimization which allows to replace the original optimization problem into a scalar optimization problem or a family of scalar problems.
\begin{Def}{\bf (Luc \cite{Luc}, Definição 2.1)}
A function
$f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ is said to be a strict scalar representation of a map $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ when given $x,\bar{x}\in \mathbb{R}^n :$
\begin{center}
$F(x)\preceq F(\bar{x}) \Longrightarrow f(x)\leq f(\bar{x})$\ \ and \ \
$F(x)\prec F(\bar{x}) \Longrightarrow f(x)<f(\bar{x}).$
\end{center}
Furthermore, we say that $f$ is a weak scalar representation of $F$ if
\begin{center}
$F(x)\prec F(\bar{x})\Longrightarrow f(x)<f(\bar{x}).$
\end{center}
\label{escalarizacao}
\end{Def}
\begin{proposicao}
\label{rep}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a proper function. Then $f$ is a strict scalar representation of $F$ if, and only if, there exists a strictly increasing function $g:F\left(\mathbb{R}^n\right)\longrightarrow \mathbb{R}$ such that $f = g \circ F.$
\end{proposicao}
\begin{proof}
See Luc \cite{Luc}, Proposition 2.3.
\end{proof}
\begin{proposicao}
\label{inclusao}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$ be a weak scalar representation of a vector function $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ and $\textnormal{argmin}\left\{f(x):x\in \mathbb{R}^n\right\}$ the set of minimizer points of $f$. Then, we have
\begin{equation*}
\textnormal{argmin}\left\{f(x): x\in \mathbb{R}^n\right\}\subseteq \textnormal{argmin}_w \{F(x):x\in \mathbb{R}^n\}.
\end{equation*}
\end{proposicao}
\begin{proof}
It is immediate.
\end{proof}
\subsection{Quasiconvex and Convex Functions}
In this subsection we present the concept and characterization of quasiconvex functions and quasiconvex multiobjective function. This theory can be found in Bazaraa et al. \cite{Bazaraa}, Luc \cite{Luc}, Mangasarian \cite{Mangasarian}, and references therein.
\begin{Def}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, f is
called quasiconvex if for all $x,y\in \mathbb{R}^n$, and for all $ t \in \left[0,1\right]$, it holds
that $f(tx + (1-t) y)\leq \textnormal{max}\left\{f(x),f(y)\right\}$.
\end{Def}
\begin{Def}
Let $f:\mathbb{R}^n\longrightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, f is
called convex if for all $x,y\in \mathbb{R}^n$, and for all $ t \in \left[0,1\right]$, it holds
that $f(tx + (1-t) y)\leq tf(x) + (1 - t)f(y)$.
\end{Def}
Observe that if $f$ is a quasiconvex function then $\textnormal{dom}(f)$ is a convex set. On the other hand, while a convex function can be characterized by the convexity of its epigraph, a quasiconvex function can
be characterized by the convexity of the lower level sets:
\begin{Def}
Let \ \ $F= (F_1,...,F_m):\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function, then $F$ is $\mathbb{R}^m_+$ - quasiconvex if every component function of $F$, $F_i: \mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$, is quasiconvex.
\end{Def}
\begin{Def}
Let \ \ $F= (F_1,...,F_m):\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function, then $F$ is $\mathbb{R}^m_+$ - convex if every component function of $F$, $F_i: \mathbb{R}^n\longrightarrow \mathbb{R}\cup \{+ \infty \}$, is convex.
\end{Def}
\subsection{Fréchet and Limiting Subdifferentials}
\label{frechet}
\begin{Def}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{ +\infty \}$ be a proper function.
\begin{enumerate}
\item [(a)]For each $x \in \textnormal{dom}(f)$, the set of regular subgradients (also called Fréchet subdifferential) of $f$ at $x$, denoted by $\hat{\partial}f(x)$, is the set of vectors $v \in \mathbb{R}^n$ such that
\begin{center}
$f(y) \geq f(x) + \left\langle v,y-x\right\rangle + o(\left\|y - x\right\|)$, where $\lim \limits_{y \rightarrow x}\frac{o(\left\|y - x\right\|)}{\left\|y - x\right\|} =0$.
\end{center}
Or equivalently, $\hat{\partial}f (x) := \left\{ v \in \mathbb{R}^n : \liminf \limits_{y\neq x,\ y \rightarrow x} \dfrac{f(y)- f(x)- \langle v , y - x\rangle}{\lVert y - x \rVert} \geq 0 \right \}$.\\ If $x \notin \textnormal{dom}(f)$ then $\hat{\partial}f(x) = \emptyset$.
\item [(b)]The set of general subgradients (also called limiting subdifferential) $f$ at $x \in \mathbb{R}^n$, denoted by $\partial f(x)$, is defined as follows:
\begin{center}
$\partial f(x) := \left\{ v \in \mathbb{R}^n : \exists\ x_l \rightarrow x, \ \ f(x_l) \rightarrow f(x), \ \ v_l \in \hat{\partial} f(x_l)\ \textnormal{and}\ v_l \rightarrow v \right \}$.
\end{center}
\end{enumerate}
\label{fech}
\end{Def}
\begin{proposicao}{\bf (Fermat’s rule generalized)}
If a proper function $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ has a local minimum at $\bar{x} \in \textnormal{dom}(f)$, then $0\in \hat{\partial} f\left(\bar{x}\right)$.
\label{otimo}
\end{proposicao}
\begin{proof}
See Rockafellar and Wets \cite{Rockafellar}, Theorem 10.1.
\end{proof}
\begin{proposicao}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper function. Then, the following properties are true
\begin{enumerate}
\item[(i)]$\hat{\partial}f(x) \subset \partial f(x)$, for all $x \in \mathbb{R}^n$.
\item[(ii)]If $f$ is differentiable at $\bar{x}$ then $\hat{\partial}f(\bar{x}) = \{\nabla f (\bar{x})\}$, so $\nabla f (\bar{x})\in \partial f(\bar{x})$.
\item[(iii)] If $f$ is continuously differentiable in a neighborhood of $x$, then $\hat{\partial}f(x) = \partial f(x) = \{\nabla f (x)\}$.
\item[(iv)] If \ $ g = f + h $ with $f$ finite at $\bar{x}$ and $h$ is continuously differentiable in a neighborhood of $\bar{x}$, then $\hat{\partial}g(\bar{x}) = \hat{\partial}f(\bar{x}) + \nabla h(\bar{x})$ and\
$\partial g(\bar{x}) = \partial f(\bar{x}) + \nabla h(\bar{x})$.
\end{enumerate}
\label{somafinita}
\end{proposicao}
\begin{proof}
See Rockafellar and Wets \cite{Rockafellar}, Exercise $8.8$, page $304$.
\end{proof}
\subsection{{\bf$\varepsilon$}-Subdiffential}
We present some important concepts and results on $\varepsilon$-subdifferential. The theory of these facts can be found, for example, in Jofre et al. \cite{Jofre} and Rockafellar and Wets \cite {Rockafellar}.
\begin{Def}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+ \infty \}$ be a proper lower semicontinuous function and let $\varepsilon$ be an arbitrary nonnegative real number. The Fréchet $\varepsilon$-subdifferential of $f$ at $x \in \textnormal{dom}(f)$ is defined by
\begin{eqnarray}
\hat{\partial}_{\varepsilon}f (x) := \left\{ x^* \in \mathbb{R}^n : \liminf \limits_{\lVert h \rVert \rightarrow 0} \dfrac{f(x + h)- f(x)- \langle x^* , h\rangle}{\lVert h \rVert} \geq - \varepsilon \right \}
\label{frechet1}
\end{eqnarray}
\label{frechet3}
\end{Def}
\begin{obs}
When $\varepsilon = 0$, $(\ref{frechet1})$ reduces to the well known Fréchet subdifferential, wich is denoted by $\hat{\partial}f(x)$, according to \textnormal{Definition} $\ref{fech}$. More precisely,
\begin{center}
$x^* \in \hat{\partial}f(x)$, if and only if, for each $\eta > 0 $ there exists $\delta > 0$ such that\\
$\langle x^* , y - x\rangle \leq f(y) - f(x) + \eta\Vert y - x \Vert$, for all $y \in x + \delta \textnormal{B}$,
\end{center}
where $B$ is the closed unit ball in $\mathbb{R}^n$ centered at zero. Therefore $\hat{\partial}f(x)=\hat{\partial}_{0}f(x)\subset\hat{\partial}_{\varepsilon}f(x).$
\label{frechet2}
\end{obs}
From Definition 5.1 of Treiman, \cite{Treiman},
\begin{center}
$x^* \in \hat{\partial}_{\epsilon}f (x)\Leftrightarrow x^* \in \hat{\partial}(f + \epsilon\Vert . - x\Vert)(x)$.
\end{center}
Equivalently, $x^* \in \hat{\partial}_{\epsilon}f (x)$, if and only if, for each $\eta > 0$, there exists $\delta > 0$ such that
\begin{center}
$\langle x^* , y - x\rangle \leq f(y) - f(x) + (\epsilon + \eta)\Vert y - x \Vert$, for all $y \in x + \delta \textnormal{B}$.
\end{center}
We now defined a new kind of approximate subdifferential.
\begin{Def}
The limiting Fréchet $\varepsilon$-subdifferential of $f$ at $x \in \textnormal{dom} (f)$ is defined by
\begin{eqnarray}
\partial_\varepsilon f(x) := \limsup \limits_{y \stackrel{f}{\longrightarrow} x} \hat{\partial}_{\varepsilon}f (y)
\end{eqnarray}
where $$\limsup \limits_{y \stackrel{f}{\longrightarrow} x} \hat{\partial}_{\varepsilon}f (y):=\lbrace x^* \in \mathbb{R}^n: \exists\ x_l \longrightarrow x, f(x_l)\longrightarrow f(x), x^*_l \longrightarrow x^* \ \textnormal{with}\ x^*_l \in \hat{\partial}_{\varepsilon}f (x_l)\, \rbrace$$
\end{Def}
In the case where $f$ is continuously differentiable, the limiting Fréchet $\varepsilon$-subdifferential takes a very simple form, according to the following proposition
\begin{proposicao}
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuously differentiable function at $x$ with derivative
$\nabla f (x)$. Then
\begin{center}
$\partial_\varepsilon f(x) = \nabla f (x) + \varepsilon B$.
\end{center}
\label{fdif}
\end{proposicao}
\begin{proof}
See Jofré et al., \cite{Jofre}, Proposition 2.8.
\end{proof}
\subsection{Fejér convergence}
\begin{Def}
A seguence $\left\{y_k\right\} \subset \mathbb{R}^n$ is said to be Fejér convergent to a set $U\subseteq \mathbb{R}^n$ if,
$\left\|y_{k+1} - u \right\|\leq\left\|y_k - u\right\|, \forall \ k \in \mathbb{N},\ \forall \ u \in U$.
\end{Def}
The following result on Fejér convergence is well known.
\begin{lema}
If $\left\{y_k\right\}\subset \mathbb{R}^n$ is Fejér convergent to some set $U\neq \emptyset$, then:
\begin{enumerate}
\item [(i)]The sequence $\left\{y_k\right\}$ is bounded.
\item [(ii)]If an accumulation point $y$ of $\left\{y_k\right\}$ belongs to $ U$, then $\lim \limits_{k\rightarrow +\infty}y_k = y$.
\end{enumerate}
\label{fejerlim1}
\end{lema}
\begin{proof}
See Schott \cite{Schott}, Theorem $2.7$.
\end{proof}
\section{The Problem}
We are interested in solving the multiobjective optimization problem (MOP):
\begin{eqnarray}
\textrm{min}\lbrace F(x): x \in \mathbb{R}^n\rbrace
\label{pom3}
\end{eqnarray}
where $F=\left( F_1, F_2,..., F_m\right): \mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ is a vector function satisfying the following assumption:
\begin{description}
\item [$\bf (C_{1.1})$] $F$ is a proper lower semicontinuous vector function on $\mathbb{R}^n$, i.e, each $F_i:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}$, $i=1,...,m$, is a proper lower semicontinuous function.
\item [$\bf (C_{1.2})$] $0 \preceq F.$
\end{description}
\subsection{A quasiconvex model in demand theory}
\noindent
Let $n$ be a finite number of consumer goods. A consumer is an agent who must choose how much to consume of each good. An ordered set of numbers representing the amounts consumed of each good set is called vector of consumption, and denoted by $ x = (x_1, x_2, ..., x_n) $ where $ x_i $ with $ i = 1,2, ..., n $, is the quantity consumed of good $i$. Denote by $ X $, the feasible set of these vectors which will be called the set of consumption, usually in economic applications we have $ X \subset \mathbb{R} ^ n_ + $.
In the classical approach of demand theory, the analysis of consumer behavior starts specifying a preference relation over the set $X,$ denoted by $\succeq$. The notation: $ "x \succeq y " $ means that "$ x $ is at least as good as $ y $" or "$ y $ is not preferred to $x$". This preference relation $ \succeq $ is assumed rational, i.e, is complete because the consumer is able to order all possible combinations of goods, and transitive, because consumer preferences are consistent, which means if the consumer prefers $\bar{x}$ to $\bar{y} $ and $\bar{y}$ to $\bar{z}$, then he prefers $\bar{x}$ to $\bar{z} $ (see Definition 3.B.1 of Mas-Colell et al. \cite{Colell}).
The quasiconvex model for a convex preference relation $\succeq,$ is ${\ max}\{ \mu(x) :x \in X\},
$ where $\mu$ is the utility function representing the preference, see Papa Quiroz et al. \cite{PapaLenninOliveira} for more detail. Now consider a multiple criteria, that is, consider $ m $ convex preference relations denoted by $\succeq_i, i=1,2,...,m.$ Suppose that for each preference $\succeq_i,$ there exists an utility function, $ \mu_i,$ respectively, then the problem of maximizing the consumer preference on $ X $ is equivalent to solve the quasiconcave multiobjective optimization problem
\begin{eqnarray*}
\textnormal{(P')\ max}\{ (\mu_{1}(x), \mu_{2}(x), ..., \mu_{m}(x)) \in \mathbb{R}^m :x \in X\}.
\end{eqnarray*}
Since there is not a single point which maximize all the functions simultaneously the concept of optimality is established in terms of Pareto optimality or efficiency. Taking F = $ (- \mu_1, - \mu_2, ..., - \mu_m) $, we obtain a minimization problem with quasiconvex multiobjective function, since each component function is quasiconvex one.
\section{Exact algorithm}
In this section, to solve the problem $(\ref {pom3}),$ we propose a linear scalarization proximal point algorithm with quadratic regularization using the Fréchet subdifferential, denoted by {\bf SPP} algorithm. \\ \\
{\bf SPP Algorithm }
\begin{description}
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio3}
\end{eqnarray}
\item [Main Steps:] Given $x^k$ finding $x^{k+1} $ such that
\begin{eqnarray}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\label{subdiferencial3}
\end{eqnarray}
where $\hat{\partial}$ is the Fréchet subdifferential, $\Omega_k= \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$, $\alpha_k > 0 $,\\ $\left\{z_k\right\} \subset \mathbb{R}^m_+\backslash \left\{0\right\}$ and $\left\|z_k\right\| = 1$.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto critical point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\subsection{Existence of the iterates}
\begin{teorema}
\label{existe0}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a vector function satisfying $\bf (C_{1.1}),$ and $\bf (C_{1.2}),$ then the sequence $\left\{x^k\right\}$, generated by the ${\bf SPP}$ algorithm, is well defined.
\end{teorema}
\begin{proof}
Let $x^0 \in \mathbb{R}^n $ be an arbitrary point given in the initialization step. Given $x^k$, define $\varphi_k(x)=\left\langle F(x), z_k\right\rangle + \frac{\alpha_k}{2}\left\|x - x^k\right\|^2 +\delta_{\Omega_k}(x)$, where $\delta_{\Omega_k}(.)$ is the indicator function of ${\Omega_k}$. Then we have that min$\{\varphi_k(x): x \in \mathbb{R}^n\}$ is equivalent to min$\{\left\langle F(x), z_k\right\rangle + \frac{\alpha_k}{2}\left\|x - x^k\right\|^2: x \in \Omega_k\}$. As $\varphi_k$ is lower semicontinuous and coercive then, using Proposition \ref{coercivaesemicont}, we obtain that there exists $x^{k+1} \in \mathbb{R}^n$ which is a global minimum of $\varphi_k.$ From Proposition \ref{otimo}, $x^{k+1}$ satisfies:
$$ 0 \in \hat{ \partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.)\right) (x^{k+1})$$
\end{proof}
\subsection{Fejér convergence Property}
To obtain some desirable properties it is necessary to assume the following assumptions on the function $F$ and the initial point $x^0$ :
\begin{description}
\item [$\bf (C_2)$] $F$ is $\mathbb{R}^m_+$-quasiconvex;
\item [${\bf (C_3)}$] The set $\left(F(x^0) - \mathbb{R}^m_+\right)\cap F(\mathbb{R}^n)$ is $\mathbb{R}^m_+$ - complete, meaning that for all sequences $\left\{a_k\right\}\subset\mathbb{R}^n$, with $a_0 = x^0$, such that $F(a_{k+1}) \preceq F(a_k)$, there exists $ a \in \mathbb{R}^n$ such that $F(a)\preceq F(a_k), \ \forall \ k \in \mathbb{N}$.
\end{description}
\begin{obs}
The assumption $ {\bf (C_3)}$ is cited in several works involving the proximal point method for convex functions, see Bonnel et al. \cite{Iusem}, Ceng and Yao \cite {Ceng} and, Villacorta and Oliveira \cite {Villacorta}.
\end{obs}
\begin{proposicao}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a function that satisfies the assumptions $\bf (C_{1.1})$ and $\bf (C_2)$. If $g \in \hat{\partial}\left( \left\langle F(.), z\right\rangle + \delta_{\Omega} \right)(x)$, with $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, and $F(y) \preceq F(x)$, with $y \in \Omega$, and $\Omega \subset \mathbb{R}^n$ a closed and convex set, then
$\left\langle g , y - x\right\rangle \leq 0$.
\label{propfejer2}
\end{proposicao}
\begin{proof}
Let $t \in \left( 0, 1\right]$, then from the $\mathbb{R}^m_+$-quasiconvexity of $F$ and the assumption that $F(y) \preceq F(x)$, we have: $F_i(ty + (1-t)x) \leq \textrm{max}\left\{F_i(x), F_i(y)\right\} = F_i(x),\ \forall \ i \in \lbrace 1,...m\rbrace$. It follows that for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, we have
\begin{equation}
\left\langle F(ty + (1-t)x) , z\right\rangle \leq \left\langle F(x) , z\right\rangle.
\label{Fz2}
\end{equation}
As $g \in \hat{\partial}\left( \left\langle F(.), z\right\rangle + \delta_{\Omega} \right)(x)$, we obtain
\begin{equation}
\left\langle F(ty + (1-t)x) , z\right\rangle + \delta_{\Omega}(ty + (1-t)x)\geq \left\langle F(x) , z\right\rangle + \delta_{\Omega}(x) + t\left\langle g, y - x\right\rangle + o(t\left\|y - x\right\|)
\label{soma2}
\end{equation}
From $(\ref{Fz2})$ and $(\ref{soma2})$, we conclude
\begin{equation}
t\left\langle g, y - x\right\rangle + o(t\left\|y - x\right\|) \leq 0
\label{somafim2}
\end{equation}
\\
On the other hand, we have
$\lim\limits_{t \rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t\left\|y - x\right\|}= 0$. Thus,
$\lim\limits_{t\rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t}=\lim\limits_{t\rightarrow 0}\frac{o(t\left\|y - x\right\|)}{t\left\|y - x\right\|}\left\|y - x\right\|=0$.
Therefore, dividing $(\ref{somafim2})$ by $t$ and taking $t \rightarrow 0$, we obtain the desired result.
\end{proof}
Observe that if the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm satisfies the assumption ${\bf (C_3)}$ then the set
\begin{center}
$E = \left\{x \in \mathbb{R}^n: F\left(x\right)\preceq F\left(x^k\right),\ \ \forall\ k \in \mathbb{N}\right\}$
\end{center}
is nonempty.
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}$, ${\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)}$ the sequence $\left\{x^k\right\}$, generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ is Fejér convergent to $E$.
\label{fejer0}
\end{proposicao}
\begin{proof} Observe that $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x^k - x\right\|^2 = \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x\right\|^2 +
2\left\langle x^k - x^{k+1}, x^{k+1} - x\right\rangle.
\label{norma2}
\end{eqnarray}
From Theorem \ref{existe0}, $(\ref{subdiferencial3})$ and from Proposition \ref{somafinita}, $(iv)$, we have that there exists $g_k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$ such that:
\begin{equation}
x^k - x^{k+1} = \dfrac{1}{\alpha_k}g_k
\label{xk}
\end{equation}
Now take $x^* \in E$, then $x^* \in \Omega_k$ for all $k \in \mathbb{N}$. Combining $(\ref{norma2})$ with $x = x^*$ and $(\ref{xk})$, we obtain:
{\footnotesize
\begin{eqnarray}
\left\|x^k - x^*\right\|^2 = \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x^* \right\|^2 +
\frac{2}{\alpha_k}\left\langle g_k ,\ x^{k+1} -x^*\right\rangle
\geq \left\|x^k - x^{k+1}\right\|^2 + \left\|x^{k+1} - x^* \right\|^2\nonumber \\
\label{desigualdade0}
\end{eqnarray}
}
where the last inequality follows from Proposition \ref{propfejer2}. From $(\ref{desigualdade0})$, it implies that
\begin{eqnarray}
0\leq \left\|x^{k+1} - x^k\right\|^2 \leq \left\|x^k - x^*\right\|^2 - \left\|x^{k+1} - x^*\right\|^2.
\label{desigual0}
\end{eqnarray}
Thus,
\begin{equation}
\left\|x^{k+1} - x^*\right\| \leq \left\|x^k - x^*\right\|
\label{fejer70}
\end{equation}
\end{proof}
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}, {\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)},$ the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ satisfies
\begin{center}
$\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$.
\label{decrescente00}
\end{center}
\end{proposicao}
\begin{proof}
It follows from $(\ref{fejer70})$ that $ \forall\ x^* \in E$, $\left\{\left\|x^k - x^*\right\|\right\}$ is a nonnegative and nonincreasing sequence, and hence is convergent. Thus, the right-hand side of $(\ref{desigual0})$ converges to 0 when $k \rightarrow +\infty$, and the result is obtained.
\end{proof}
\subsection{Convergence Analysis I: non differentiable case }
In this subsection we analyze the convergence of the proposed algorithm when $F$ is a non differentiable vector function.
\begin{proposicao}
Under assumptions ${\bf(C_{1.1})}, {\bf(C_{1.2})},$ ${\bf(C_2)}$ and ${\bf(C_3)},$ the sequence $\left\{x^k\right\}$ generated by the {\bf SPP} algorithm converges to some point of $E$.
\label{acumulacao10}
\end{proposicao}
\begin{proof}
From Proposition $\ref{fejer0}$ and Lemma $\ref{fejerlim1}$, $(i)$, $\left\{x^k\right\}$ is bounded, then there exists a subsequence $\left\{x^{k_j}\right\}$ such that $\lim \limits_{j\rightarrow +\infty}x^{k_j} = \widehat{x}$. Since $\left\langle F(.), z\right\rangle$ is lower semicontinuos function for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$ then
$\left\langle F(\widehat x),z\right\rangle \leq \liminf \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle $. On the other hand, $x^{k+1} \in \Omega_k$ so $\left\langle F(x^{k+1}) , z\right\rangle \leq \left\langle F(x^{k}) , z\right\rangle $. Furthermore, from assumption ${\bf(C_{1.2})}$ the function $\left\langle F(.), z\right\rangle$ is bounded below for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\},$ then, the sequence $\left\{\left\langle F(x^k),z\right\rangle\right\}$ is nonincreasing and bounded below, hence convergent. Therefore
{\small
\begin{center}
$\left\langle F(\widehat {x}),z\right\rangle \leq \liminf \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = inf_{k\in \mathbb{N}}\left\{\left\langle F(x^k),z\right\rangle\right\}\leq \left\langle F(x^k),z\right\rangle$.
\end{center}
}
It follows that
$\left\langle F(x^k)-F(\widehat{x}),z\right\rangle \geq 0, \forall \ k \in \mathbb{N}, \forall \ z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$. We conclude that $F(x^k) - F(\widehat{x}) \in \mathbb{R}^m_+$, i.e, $F(\widehat{x})\preceq F(x^k), \forall \ k \in \mathbb{N}$. Therefore $\widehat{x}\in E,$ and by Lemma $\ref{fejerlim1}$, $(ii)$, we get the result.
\end{proof}
\subsubsection{Convergence to a weak Pareto solution}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a continuous vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_3)$. If $ \lim \limits_{k\rightarrow +\infty}\alpha_k= 0$ and the iterations are given in the form
\begin{eqnarray}
x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\},
\label{recursao0}
\end{eqnarray}
then the sequence $\lbrace x^k\rbrace$ converges to a weak Pareto solution of the problem $(\ref{pom3})$.
\end{teorema}
\begin{proof}
Let\ \ $x^{k+1}\in \textnormal{arg min} \left\{\left\langle F(x), z_k\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\Omega_k\right\}$, this implies that
{\small
\begin{eqnarray}
\left\langle F(x^{k+1}), z_k\right\rangle +\frac{\alpha_k}{2}\left\|x^{k+1}-x^k\right\|^2 \leq \left\langle F(x), z_k\right\rangle +
\frac{\alpha_k}{2}\left\|x -x^k\right\|^2,
\label{des0}
\end{eqnarray}
}
$\forall \ x \in \Omega_k$. Since the sequence $\left\{x^k\right\}$ converges to some point of $E$, then exists $x^* \in E$ such that $\lim \limits_{k\rightarrow +\infty}x^{k}= x^*$. Since that $\left\{z_k\right\}$ is bounded, there exists a subsequence $\left\{z_{k_l}\right\}_{l\in \mathbb{N}}$ such that $\lim \limits_{l\rightarrow +\infty}z_{k_l}=\bar{z}$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. Taking $k=k_l$ in $(\ref{des0})$, we have
{\small
\begin{eqnarray}
\left\langle F(x^{k_l+ 1}), z_{k_l}\right\rangle +\frac{\alpha_{k_l}}{2}\left\|x^{k_l+1}-x^{k_l}\right\|^2 \leq \left\langle F(x), z_{k_l}\right\rangle + \frac{\alpha_{k_l}}{2}\left\|x - x^{k_l}\right\|^2.
\label{desi0}
\end{eqnarray}
}
$\forall \ x \in E.$ As
\begin{center}
$ \frac{\alpha_{k_l}}{2}\left\|x^{k_{l+1}}-x^{k_l}\right\|^2 \rightarrow 0$ and $\frac{\alpha_{k_l}}{2}\left\|x - x^{k_l}\right\|^2 \rightarrow0$ when $l \rightarrow +\infty$
\end{center}
and from the continuity of $F$, taking $l \rightarrow +\infty$ in $(\ref{desi0})$, we obtain
\begin{eqnarray}
\left\langle F(x^*), \overline{z}\right\rangle \leq \left\langle F(x), \overline{z}\right\rangle, \forall \ x \in E
\label{minfi0}
\end{eqnarray}
Thus $x^* \in \textrm{arg min} \left\{\left\langle F(x), \overline{z}\right\rangle: x \in E\right\}$.
Now, $\left\langle F(.), \overline{z}\right\rangle$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$ is a strict scalar representation of $F$, so a weak scalar representation, then by Proposition $\ref{inclusao}$ we have that $x^* \in \textrm{arg min}_w \left\{F(x):x \in E \right\}$.\\
We shall prove that $x^* \in \textrm{arg min}_w \left\{F(x):x \in \mathbb{R}^n \right\}$. Suppose by contradiction that $x^* \notin \textrm{arg min}_w \left\{F(x):x \in \mathbb{R}^n \right\}$ then there exists $\widetilde{x} \in \mathbb{R}^n$ such that
\begin{equation}
F(\widetilde{x})\prec F(x^*)
\label{pareto0}
\end{equation}
So for $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$ it follows that
\begin{equation}
\left\langle F(\widetilde{x}), \bar{z}\right\rangle < \left\langle F(x^*), \bar{z}\right\rangle
\label{d0}
\end{equation}
Since $x^* \in E$, from $(\ref{pareto0})$ we conclude that $\widetilde{x} \in E$. Therefore from $(\ref{minfi0})$ and $(\ref{d0})$ we obtain a contradiction.
\end{proof}\\
\subsubsection{Convergence to a generalized critical point}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a continuous vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_3)$. If $0 < \alpha_k < \tilde{\alpha}$ then the sequence $\lbrace x^k\rbrace$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3})$ satisfies
$$\lim \limits_{k\rightarrow +\infty}g^{k} = 0,$$
where $g^k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$.
\label{geral}
\end{teorema}
\begin{proof}
From Theorem $\ref{existe0}$, $(\ref{subdiferencial3})$ and from Proposition $\ref{somafinita}$, $(iv)$, there exists a vector $g_k \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \delta_{\Omega_k}\right)(x^{k+1})$ such that $g^k = \alpha_k(x^k - x^{k+1})$. Since $0 < \alpha_k < \tilde{\alpha}$ then
\begin{equation}
0 \leq \Vert g^k\Vert \leq \tilde{\alpha}\left\|x^k - x^{k+1} \right\|
\label{beta0}
\end{equation}
From Proposition $\ref{decrescente00}$, $\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$, and from $(\ref{beta0})$ we have $\lim \limits_{k\rightarrow +\infty}g^k = 0$.
\end{proof}
\subsubsection{Finite Convergence to a Pareto Optimal Point}
\noindent
Following the paper of Bento et al, \cite{Bento} subsection 4.3, it is possible to prove the convergence of a special particular case of the proposed algorithm to a Pareto optimal point of the problem (\ref{pom3}). Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function and consider the following particular iteration of (\ref{subdiferencial3}):
\begin{eqnarray}
x^{k+1}= \textnormal{arg min} \left\{\left\langle F(x), z\right\rangle+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 : x\in\mathbb{R}^n\right\},
\label{recursao0f}
\end{eqnarray}
where $z\in \mathbb{R}^m_+\backslash \left\{0\right\}$ such that $||z||=1.$
\begin{Def}
Consider the set of Pareto optimal points of $(\ref{pom3})$, denoted by $Min(F)$ and let $\bar x\in Min(F)$. We say that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ if there exists a constant $\tau>0$ such that
$$
F(x)-F(\bar x)\notin B(0,\tau d(x,W_{F(\bar x)}))-\mathbb{R}^m_{+},\;\; x\in \mathbb{R}^n\backslash \,W_{F(\bar x)},
$$
where $d(x,Z)=\inf \{d(x,z):z\in Z\}$ and $W_{p}=\{x\in \mathbb{R}^n: F(x)=F(p)\}.$
\end{Def}
\begin{teorema}
Let $F$ be a proper lower semicontinuous convex vector function satisfying the assumptions $\bf (C_{1.2}),$ and $\bf (C_3).$ Assume that $\{x^k\}$ is a sequence generated from the {\bf SPP} algorithm with $x^{k+1}$ being generates from $(\ref{recursao0f})$. Consider also that the set of Pareto optimal points of $(\ref{pom3})$ is nonempty and assume that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ with constant $\tau>0$ for some $\bar x\in Min(F).$ Then the sequence $\{x^k\}$ converges, in a finite number of iterations, to a Pareto optimal point.
\end{teorema}
\begin{proof}
Simmilar to the proof of Theorem 4.3 of Bento et al., \cite{Bento}.
\end{proof}
\subsection{Convergence analysis II: Differentiable Case}
In this subsection we analyze the convergence of the method when $F$ satisfies the following assumption:
\begin{description}
\item [$\bf (C_4)$] $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m$ is a continuously differentiable vector function on $\mathbb{R}^n$.
\end{description}
The next proposition characterizes a quasiconvex differentiable vector functions.
\begin{proposicao}
Let $F :\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a differentiable function satisfying the assumption ${\bf(C_2)}$, ${\bf(C_3)}.$ If $x \in E,$ then $\left\langle \nabla F_i(x^k) , x - x^k\right\rangle \leq 0$, $\forall \ k \in \mathbb{N}$ and $\forall \ i \in \left\{1,...,m\right\}$.
\label{caracterizacaodif}
\end{proposicao}
\begin{proof}
Since \ $F$ is $\mathbb{R}^m_+$-quasiconvex each $F_i,$ $ i = 1,..., m,$ is quasiconvex. Then the result follows from the classical characterization of the scalar differentiable quasiconvex functions, see
$\textrm{see Mangasarian \cite{Mangasarian}, p.134}$.
\end{proof}
\begin{teorema}
\label{conv3}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a function satisfying the assumptions $\bf (C_2)$, $\bf (C_3)$ and $\bf (C_4)$. If $0 < \alpha_k < \tilde{\alpha}$, then the sequence $\lbrace x^k\rbrace$ generated by the {\bf SPP} algorithm, $(\ref{inicio3})$ and $(\ref{subdiferencial3}),$ converges to a Pareto critical point of the problem $(\ref{pom3})$.
\label{teoparetocri}
\end{teorema}
\begin{proof}
In Proposition \ref{acumulacao10} we prove that there exists $\widehat{x} \in E$ such that $\lim \limits_{k\rightarrow +\infty}x^{k}= \widehat{x}$. From Theorem $\ref{existe0}$ and $(\ref{subdiferencial3})$, we have
\begin{equation*}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2}\Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.) \right) (x^{k+1})
\end{equation*}
Due to Proposition $\ref{somafinita}$, $(iv)$, we have
\begin{center}
$0 \in \nabla\left(\left\langle F(.) , z_k\right\rangle\right)(x^{k+1})+ \alpha_k \left(x^{k+1} - x^k \right) + \mathcal{N}_{\Omega_k}(x^{k+1})$
\end{center}
where $\mathcal{N}_{\Omega_k}(x^{k+1})$ is the normal cone to $\Omega_k$ at $x^{k+1}\in \Omega_k$.\\
So there exists $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ such that:
\begin{equation}
0 = \sum_{i=1}^m \nabla F_i(x^{k+1})(z_k)_i + \alpha_k\left(x^{k+1} - x^k \right) + \nu_k.
\label{otimalidade0}
\end{equation}
Since $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ then
\begin{equation}
\left\langle \nu_k \ ,\ x - x^{k+1}\right\rangle \leq\ 0,\ \forall \ x \in \Omega_k.
\label{cone0}
\end{equation}
Take $\bar{x} \in E$. By definition of $E$, $\bar{x} \in \Omega_k$ for all $k \in \mathbb{N}$. Combining $(\ref{cone0})$ with $x = \bar{x}$ and $(\ref{otimalidade0})$, we have
{\small
\begin{eqnarray}
\left\langle \sum_{i=1}^m \nabla F_i(x^{k+1})(z_k)_i , \bar{x} - x^{k+1}\right\rangle + \alpha_k\left\langle x^{k+1} - x^k , \bar{x} - x^{k+1}\right\rangle\geq 0.&
\label{cone00}
\end{eqnarray}
}
Since that $\left\{z_k\right\}$ is bounded, then there exists a subsequence $ \left\{z_{k_j}\right\}_{j \in \mathbb{N}}$ such that $\lim \limits_{j\rightarrow +\infty}z_{k_j}= \bar{z}$ with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. Thus the inequality in $(\ref{cone00})$ becomes
{\small
\begin{eqnarray}
\left\langle \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i , \bar{x} - x^{{k_j}+1}\right\rangle + \alpha_{k_j}\left\langle x^{{k_j}+1} - x^{k_j} , \bar{x} - x^{{k_j}+1}\right\rangle\geq 0.&
\label{cone000}
\end{eqnarray}
}
Since $\left\{x^k\right\}$ and $\left\{ \alpha_k\right\}$ are bounded, $\lim \limits_{k\rightarrow +\infty}\left\|x^{k+1} - x^k\right\| = 0$ and $F$ is continuously differentiable, the inequality in $(\ref{cone000})$, for all $\bar{x}\in E$, becomes:
{\small
\begin{eqnarray}
\left\langle\sum_{i=1}^m \nabla F_i(\widehat{x})\bar{z}_i \ ,\ \bar{x} - \widehat{x}\right\rangle \geq 0
\mathbb{R}ightarrow \sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle\geq 0.
\label{somai0}
\end{eqnarray}
}
From the quasiconvexity of each component function $F_i$, for each $i \in \left\{1,...,m\right\}$, we have that\\
$\left\langle \nabla F_i(\widehat{x})\ ,\ \bar{x} - \widehat{x} \right\rangle\leq 0$ and because $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$, from $(\ref{somai0})$, we obtain
\begin{eqnarray}
\sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle = 0.
\label{somai00}
\end{eqnarray}
Without loss of generality consider the set $J = \left\{i \in I: \bar{z}_i > 0 \right\}$, where $I = \left\{1,...,m\right\}$. Thus, from $(\ref{somai00})$, for all $\bar{x} \in E$ we have
\begin{eqnarray}
\left\langle \nabla F_{i}(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle = 0,\ \forall \ \ i \in J.
\label{ecritico0}
\end{eqnarray}
Now we will show that $\widehat{x}$ is a Pareto critical point. \\
Suppose by contradiction that $\widehat{x}$ is not a Pareto critical point, then there exists a direction $v \in \mathbb{R}^n$ such that $JF(\widehat{x})v \in -\mathbb{R}^m_{++}$, i.e,
\begin{eqnarray}
\left\langle \nabla F_i(\widehat{x}) , v \right\rangle < 0, \forall \ i \in \left\{1,...,m\right\}.
\label{direcao10}
\end{eqnarray}
Therefore $v$ is a descent direction for the multiobjective function $F$ in $\widehat{x}$, so, $\exists \ \varepsilon > 0$ such that
\begin{eqnarray}
F(\widehat{x} + \lambda v) \prec F(\widehat{x}),\ \forall \ \lambda \in (0, \varepsilon].
\label{descida10}
\end{eqnarray}
Since \ $\widehat{x}\ \in\ E$, then from $(\ref{descida10})$ we conclude that $\widehat{x} + \lambda v \in E$. Thus, from $(\ref{ecritico0})$ with $\bar{x} = \widehat{x} + \lambda v $, we obtain: {\small $\left\langle \nabla F_{i}(\widehat{x})\ ,\ \widehat{x} + \lambda v - \widehat{x} \right\rangle = \left\langle \nabla F_{i}(\widehat{x})\ ,\ \lambda v \right\rangle = \lambda\left\langle \nabla F_{i}(\widehat{x})\ ,\ v \right\rangle = 0$}.\\
It follows that $\left\langle \nabla F_{i}(\widehat{x})\ ,\ v \right\rangle = 0$ for all $ i \in J,$ contradicting $(\ref{direcao10})$. Therefore $\widehat{x}$ is Pareto critical point of the problem $(\ref{pom3})$.
\end{proof}
\section{An inexact proximal algorithm}
In this section we present an inexact version of the {\bf SPP} algorithm, which we denote by {\bf ISPP} algorithm.
\subsection{{\bf ISPP} Algorithm}
Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a vector function satisfying the assumptions ${(\bf C_2)}$ and ${(\bf C_4)}$, and consider two sequences: the proximal parameters $\left\{\alpha_k\right\}$ and the sequence $ \left\{z_k \right\}\subset \mathbb{R}^m_+\backslash \left\{0\right\}$ with $\left\|z_k\right\| = 1$.
\begin{description}
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio2}
\end{eqnarray}
\item [Main Steps:] Given $x^k,$ define the function $\Psi_k : \mathbb{R}^n \rightarrow \mathbb{R} $ such that $ \Psi_k (x) = \left\langle F(x), z_k\right\rangle $ and consider $\Omega_k = \left\{ x\in \mathbb{R}^n: F(x) \preceq F(x^k)\right\}$. Find $x^{k+1}$ satisfying
\begin{equation}
0 \in \hat{\partial}_{\epsilon_k}\Psi_k (x^{k+1}) + \alpha_k\left(x^{k+1} - x^k\right) + \mathcal{N}_{\Omega_k}(x^{k+1}),
\label{diferencial}
\end{equation}
\begin{equation}
\label{delta}
\displaystyle \sum_{k=1}^{\infty} \delta_k < +\infty,
\end{equation}
where $\delta_k = \textnormal{max}\left\lbrace \dfrac{\varepsilon_k}{\alpha_k}, \dfrac{\Vert \nu_k\Vert}{\alpha_k}\right\rbrace,$ $\varepsilon_k\geq 0,$ and $\hat{\partial}_{\varepsilon_k}$ is the Fréchet $\varepsilon_k$-subdifferential.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto critical point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\subsubsection{Existence of the iterates}
\begin{proposicao}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m $ be a vector function satisfying the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_4)$. Then the sequence $\left\{x^k\right\}$ generated by the {\bf ISPP} algorithm, is well defined.
\label{iteracao2}
\end{proposicao}
\begin{proof}
Consider $x^0 \in \mathbb{R}^n $ given by (\ref{inicio2}). Given $x^k$, we will show that there exists $x^{k+1}$ satisfying the condition $(\ref{diferencial})$. Define the function $\varphi_k(x) = \Psi_k (x)+\frac{\alpha_k}{2}\left\|x - x^k\right\|^2 + \delta_{\Omega_k}(x)$. Analogously to the proof of Theorem\ $\ref{existe0}$ there exists $x^{k+1} \in \Omega_k$ which is a global minimum of $\varphi_k (.),$ so, from Proposition $\ref{otimo}$, $x^{k+1}$ satisfies
\begin{center}
$ 0 \in \hat{ \partial}\left( \Psi_k(.) + \dfrac{\alpha_k}{2}\Vert\ .\ - x^k \Vert ^2 + \delta_{\Omega_k}(.)\right) (x^{k+1})$.
\end{center}
From Proposition $\ref{somafinita},$ $\ (iii)$ and $\ (iv)$, we obtain
\begin{center}
$0 \in \hat{\partial}\Psi_k(x^{k+1})+ \alpha_k \left(x^{k+1} - x^k \right) + \mathcal{N}_{\Omega_k}(x^{k+1})$.
\end{center}
From Remark $\ref{frechet2}$, $x^{k+1}$ satisfies $(\ref{diferencial})$ with $\varepsilon_k = 0$.
\end{proof}
\begin{obs}
\label{bregman}
From the inequality $(a-1/2)^2\geq 0, \forall a\in \mathbb{R},$ we obtain the following relation
\begin{center}
$\Vert x - z\Vert^2 + \frac{1}{4} \geq \Vert x - z\Vert, \ \forall x,z \in \mathbb{R}^n$
\end{center}
\end{obs}
\begin{proposicao}
\label{fejer3}
Let $\left\{x^k\right\}$ be a sequence generated by the {\bf ISPP} algorithm. If the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$, ${\bf(C_4)}$ and $(\ref{delta})$ are satisfied, then for each $\hat{x} \in E$, $\{ \left\|\hat{x} - x^k\right\|^ 2 \}$ converges and $\{x^k\}$ is bounded.
\end{proposicao}
\begin{proof}
From $(\ref{diferencial})$, there exist $g_k \in \hat{\partial}_{\varepsilon_k}\Psi_k (x^{k+1})$ and $\nu_k \in \mathcal{N}_{\Omega_k}(x^{k+1})$ such that
\begin{center}
$0 = g_k + \alpha_k\left(x^{k+1} - x^k \right) + \nu_k$.
\end{center}
It follows that for any $x \in \mathbb{R}^n$, we obtain
\begin{equation*}
\langle - g_k , x - x^{k+1}\rangle + \alpha_k\langle x^k - x^{k+1}, x - x^{k+1}\rangle = \langle\nu_k, x - x^{k+1}\rangle \leq \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert
\label{diferenca2}
\end{equation*}
Therefore
\begin{equation}
\langle x^k - x^{k+1}, x - x^{k+1} \rangle \leq \dfrac{1}{\alpha_k}\left( \langle g_k , x - x^{k+1}\rangle + \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert\right) .
\label{a}
\end{equation}
Note that $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x - x^{k+1}\right\|^2 &-&\left\| x - x^k \right\|^2 \leq \ \ 2\left\langle x^k - x^{k+1}, x - x^{k+1} \right\rangle .
\label{norma5}
\end{eqnarray}
From $(\ref{a})$ and $(\ref{norma5})$, we obtain
\begin{equation}
\left\|x - x^{k+1}\right\|^2 -\left\| x - x^k \right\|^2 \leq \dfrac{2}{\alpha_k}\left( \langle g_k , x - x^{k+1}\rangle + \Vert \nu_k\Vert \Vert x - x^{k+1}\Vert\right) .
\label{b}
\end{equation}
On the other hand, let $\Psi_k (x) = \left\langle F(x), z_k\right\rangle$, where $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuously differentiable vector function, then $ \Psi_k : \mathbb{R}^n \rightarrow \mathbb{R}$ is continuously differentiable with gradient denoted by $\nabla \Psi_k $. From Proposition $\ref{fdif}$, we have
\begin{equation}
\partial_{\varepsilon_{k}}\Psi_k (x) = \nabla \Psi_k (x) + \varepsilon_{k} B,
\end{equation}
where $B$ is the closed unit ball in $\mathbb{R}^n$ centered at zero. Futhermore, $\hat{\partial}_{\varepsilon_k} \Psi_k(x) \subset \partial_{\varepsilon_k} \Psi_k(x)$, (see (2.12) in Jofré et al. \cite{Jofre}). As $g_k \in \hat{\partial}_{\epsilon_k}\Psi_k(x^{k+1})$, we have that $g_k \in \partial_{\epsilon_k}\Psi_k(x^{k+1})$, then $$g_k = \nabla\Psi_k(x^{k+1}) + \varepsilon_k h_k,$$ with $\Vert h_k \Vert \leq 1 $. Now take $\hat{x}\in \textnormal{E},$ then
\begin{eqnarray}
\langle g_k,\hat{x} - x^{k+1} \rangle & =& \left\langle \nabla \Psi_k(x^{k+1}) + \varepsilon_k h_k\ , \ \hat{x} - x^{k+1}\right\rangle \nonumber \\
&=&\sum_{i=1}^m \left\langle \nabla F_i(x^{k+1})\ ,\ \hat{x} - x^{k+1}\right\rangle (z_k)_i +\varepsilon_k\left\langle h_k\ ,\ \hat{x} -x^{k+1} \right\rangle \nonumber\\
\label{fe1}
\end{eqnarray}
From Proposition $\ref{caracterizacaodif}$, we conclude that $(\ref{fe1})$ becomes
\begin{eqnarray}
\langle g_k,\hat{x} - x^{k+1} \rangle \leq \varepsilon_k\left\langle h_k\ ,\ \hat{x} -x^{k+1} \right\rangle \leq \varepsilon_k \Vert\hat{x} - x^{k+1} \Vert \nonumber\\
\label{f2}
\end{eqnarray}
From Remark $\ref{bregman}$ with $x = \hat{x}$ and $z = x^{k+1}$, follows
\begin{eqnarray}
\Vert \hat{x} - x^{k+1}\Vert \leq \left( \Vert \hat{x} - x^{k+1}\Vert ^2 + \frac{1}{4}\right).
\label{c}
\end{eqnarray}
Consider $x = \hat{x}$ in $(\ref{b})$, using $(\ref{f2})$, $(\ref{c})$ and the condition (\ref{delta}) we obtain
\begin{eqnarray*}
\left\|\hat{x} - x^{k+1}\right\|^2 -\left\| \hat{x} - x^k \right\|^2 & \leq & \dfrac{2}{\alpha_k}\left( \varepsilon_k + \Vert \nu_k\Vert\right)\Vert \hat{x} - x^{k+1}\Vert \\
&\leq & 4 \delta_k \left\|\hat{x} - x^{k+1}\right\|^2 + \delta_k.
\end{eqnarray*}
Thus
\begin{eqnarray}
\left\|\hat{x} - x^{k+1}\right\|^2 \leq \left(\frac{1}{1-4\delta_k}\right)\left\|\hat{x} - x^{k}\right\|^2+ \frac{\delta_k}{1-4\delta_k}.
\label{d}
\end{eqnarray}
The condition (\ref{delta}) guarantees that
\begin{center}
$\delta_k < \dfrac{1}{4}, \ \ \forall k > k_0, $\\
\end{center}
where $k_0$ is a natural number sufficiently large, and so,
\begin{equation*}
1 \leq \frac{1}{1-4\delta_k} \leq 1+2\delta_k <2,\ \ \ \textnormal{for}\ \ k \geq k_0,
\end{equation*}
combining with $(\ref{d})$, results in
\begin{equation}
\left\|\hat{x} - x^{k+1}\right\|^2 \leq \left(1 + 2\delta_k \right)\left\|\hat{x} - x^{k}\right\|^2 + 2\delta_k .
\label{e2}
\end{equation}
Since $\displaystyle \sum_{i=1}^{\infty} \delta_k < \infty,$ applying Lemma $\ref{p}$ in the inequality $(\ref{e2})$, we obtain the convergence of $\{ \|\hat{x} - x^k\|^2 \}$, for each $\hat{x} \in E,$ which implies that there exists $M \in \mathbb{R}_+$, such that $\left\|\hat{x} - x^k \right\| \leq M, \ \ \forall \ k \in \mathbb{N}.
$
Now, since that $\Vert x ^k\Vert \leq \Vert x ^k - \hat{x} \Vert + \Vert \hat{x} \Vert,$ we conclude that $\{x^k\}$ is bounded, and so, we guarantee that the set of accumulation points of this sequence is nonempty. \end{proof}
\subsubsection{Convergence of the {\bf ISPP} algorithm}
\begin{proposicao}{(\bf Convergence to some point of E)}\\
If the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$ and ${\bf(C_4)}$ are satisfied, then the sequence $\left\{x^k\right\}$ generated by the {\bf ISPP} algorithm converges to some point of the set $E$.
\label{acumulacao100}
\end{proposicao}
\begin{proof}
As $\left\{x^k\right\}$ is bounded, then there exists a subsequence $\left\{x^{k_j}\right\}$ such that $\lim \limits_{j\rightarrow +\infty}x^{k_j} = \widehat{x}$. Since $F$ is continuous in $\mathbb{R}^n$, then the function $\left\langle F(.), z\right\rangle$ is also continuous in $\mathbb{R}^n$ for all $z \in \mathbb{R}^m$, in particular, for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$, and
$\left\langle F(\widehat x),z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle $. On the other hand, we have that $F(x^{k+1}) \preceq F(x^{k})$, and so, $\left\langle F(x^{k+1}) , z\right\rangle \leq \left\langle F(x^{k}) , z\right\rangle $ for all $z \in \mathbb{R}^m_+ \backslash \left\{0\right\}$. Furthermore the function $\left\langle F(.), z\right\rangle$ is
bounded below, for each $z \in \mathbb{R}^m_+\backslash \left\{0\right\}$, then the sequence $\left\{\left\langle F(x^k),z\right\rangle\right\}$ is nonincreasing and bounded below, thus convergent. So,
{\small
\begin{center}
$\left\langle F(\widehat {x}),z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k_j}) , z\right\rangle = \lim \limits_{j\rightarrow +\infty}\left\langle F(x^{k}) , z\right\rangle = inf_{k\in \mathbb{N}}\left\{\left\langle F(x^k),z\right\rangle\right\}\leq \left\langle F(x^k),z\right\rangle$.
\end{center}
}
It follows that $F(x^k) - F(\widehat{x}) \in \mathbb{R}^m_+$, i.e, $F(\widehat{x})\preceq F(x^k), \forall \ k \in \mathbb{N}$. Therefore $\widehat{x}\in E$. Now, from Proposition $\ref{fejer3}$, we have that the sequence $\{ \left\|\widehat{x} - x^k\right\|\}$ is convergent, and since $\lim \limits_{k\rightarrow +\infty}\left\|x^{k_j} - \hat{x}\right\| = 0$, we conclude that $\lim \limits_{k\rightarrow +\infty}\left\|x^{k} - \widehat{x}\right\| = 0$, i.e, $\lim \limits_{k\rightarrow +\infty}x^{k} = \widehat{x}.$
\end{proof}
\begin{teorema}
\label{conv4}
Suppose that the assumptions ${\bf(C_{1.2})},$ ${\bf(C_2)}$, ${\bf(C_3)}$ and ${\bf(C_4)}$ are satisfied. If $0 < \alpha_k < \widetilde{\alpha}$, then the sequence $\lbrace x^k\rbrace$ generated by the {\bf ISPP} algorithm , $(\ref{inicio2})$, $(\ref{diferencial})$ and $(\ref{delta}),$ converges to a Pareto critical point of the problem $(\ref{pom3})$.
\end{teorema}
\begin{proof}
From Proposition $\ref{acumulacao100}$ there exists $\widehat{x}\in E$ such that $\lim \limits_{j\rightarrow +\infty}x^{k}= \widehat{x}$. Furthermore, as the sequence $\left\{z^k\right\}$ is bounded, then there exists $ \left\{z^{k_j}\right\}_{j \in \mathbb{N}}$ such that $\lim \limits_{j\rightarrow +\infty}z^{k_j}= \bar{z}$, with $\bar{z} \in \mathbb{R}^m_+\backslash \left\{0\right\}$. From $(\ref{diferencial})$ there exists $g_{k_j} \in \hat{\partial}_{\varepsilon_{k_j}}\Psi_{k_j} (x^{{k_j}+1})$, with $g_{k_j} = \nabla\Psi_{k_j}(x^{{k_j}+1}) + \varepsilon_{k_j} h_{k_j}$ with $\Vert h_{k_j} \Vert \leq 1 $, and $\nu_{k_j} \in \mathcal{N}_{\Omega_{k_j}}(x^{{k_j}+1})$, such that:
\begin{equation}
0 = \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i + \varepsilon_{k_j} h_{k_j} + \alpha_{k_j} \left(x^{{k_j}+1} - x^{k_j} \right) + \nu_{k_j}
\label{otimalidade2}
\end{equation}
Since $\nu_{k_j} \in \mathcal{N}_{\Omega_{k_j}}(x^{{k_j}+1})$ then,
\begin{equation}
\left\langle \nu_{k_j} \ ,\ x - x^{{k_j}+1}\right\rangle \leq\ 0,\ \forall \ x \in \Omega_{k_j}
\label{cone6}
\end{equation}
Take $\bar{x} \in E$. By definition of $E$, $\bar{x} \in \Omega_k$, for all $ k \in \mathbb{N}$, so $\bar{x} \in \Omega_{k_j}$. Combining $(\ref{cone6})$ with $x = \bar{x}$ and $(\ref{otimalidade2})$, we have
{\footnotesize
\begin{eqnarray}
0 &\leq & \left\langle \sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle + \varepsilon_{k_j}\left\langle h_{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle +
+\alpha_{k_j}\left\langle x^{{k_j}+1} - x^{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle \nonumber \\
&\leq & \left\langle\sum_{i=1}^m \nabla F_i(x^{{k_j}+1})(z_{k_j})_i\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle + \varepsilon_{k_j}M + \tilde{\alpha}\left\langle x^{{k_j}+1} - x^{k_j}\ ,\ \bar{x} - x^{{k_j}+1}\right\rangle \nonumber \\
\label{aaa}
\end{eqnarray}
}Observe that, $\forall \ x \in \mathbb{R}^n$:
\begin{eqnarray}
\left\|x^{k+1} - x^k \right\|^2 &=& \left\| x - x^k \right\|^2 - \left\| x - x^{k+1}\right\|^2 + 2\left\langle x^k - x^{k+1} , x - x^{k+1} \right\rangle
\label{consecutiva}
\end{eqnarray}
Now, from $(\ref{a})$ with $x = \bar{x} \in E$, and $(\ref{f2})$, we obtain
\begin{eqnarray*}
\left\langle x^k - x^{k+1} ,\bar{x} - x^{k+1} \right\rangle\leq \left\| \bar{x} - x^{k+1}\right\|\left( \dfrac{\varepsilon_k}{\alpha_k} + \dfrac{\Vert \nu_k \Vert}{\alpha_k}\right) \leq 2M\delta_k
\end{eqnarray*}
Thus, from $(\ref{consecutiva})$, with $x = \bar{x}$, we have
\begin{eqnarray}
0 \leq \left\|x^{k+1} - x^k \right\|^2 \leq \left\| \bar{x} - x^k \right\|^2 - \left\| \bar{x} - x^{k+1}\right\|^2 +4M\delta_k
\label{consecutiva2}
\end{eqnarray}
Since that the sequence $\left\{ \left\|\bar{x} - x^k\right\|\right\}$ is convergent and $\displaystyle\sum_{i=1}^{\infty}\delta_k < \infty$, from $(\ref{consecutiva2})$ we conclude that $\displaystyle \lim_{k \to +\infty} \left\|x^{k +1} - x^k \right\| = 0$. Furthermore, as
\begin{eqnarray}
0 \leq \left\|x^{{k_j}+1} - \bar{x} \right\| \leq \Vert x^{{k_j}+1} - x^{k_j} \Vert + \Vert x^{k_j} - \bar{x}\Vert,
\label{zero}
\end{eqnarray}
we obtain that the sequence $\{ \left\|\bar{x} - x^{{k_j}+1}\right\|\}$ is bounded.\\
Thus returning to $(\ref{aaa})$, since $\lim \limits_{k\rightarrow+\infty}\varepsilon_k = 0 $, $\lim \limits_{j\rightarrow +\infty}x^{k}= \widehat{x}$ and $\lim \limits_{j\rightarrow +\infty}z^{k_j}= \bar{z}$, taking $j \rightarrow + \infty$, we obtain
\begin{equation}
\sum_{i=1}^m \bar{z}_i\left\langle \nabla F_i(\widehat{x}) \ ,\ \bar{x} - \widehat{x} \right\rangle\geq 0.
\label{cc}
\end{equation}
Therefore, analogously to the proof of Theorem $\ref{teoparetocri}$, starting in $(\ref{somai0})$, we conclude that $\widehat{x}$ is a Pareto critical point to the problem $(\ref{pom3})$.
\end{proof}
\section{Finite convergence to a Pareto optimal point}
\noindent
In this section we prove the finite convergence of a particular inexact scalarization proximal point algorithm for proper lower semicontinuous convex functions, which we call Convex Inexact Scalarization Proximal Point algorithm, {\bf CISPP} algorithm.
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function and consider $ z \in \mathbb{R}^m_+\backslash \left\{0\right\}$ with $\left\|z\right\| = 1$ and the sequences of the proximal parameters $\left\{\alpha_k\right\}$ such that $0<\alpha_k<\bar{\alpha}.$
\begin{description}
\item [{\bf CISPP algorithm}]
\item [\bf Initialization:] Choose an arbitrary starting point
\begin{eqnarray}
x^0\in\mathbb{R}^n
\label{inicio2c}
\end{eqnarray}
\item [Main Steps:] Given $x^k,$ and find $x^{k+1}$ satisfying
\begin{equation}
e^k\in \partial \left( \langle F(.), z\rangle+ \frac{\alpha_k}{2}\|. - x^k\|^2 \right)(x^{k+1})
\label{recursao0f2c}
\end{equation}
\begin{equation}
\label{deltac}
\displaystyle \sum_{k=1}^{\infty} ||e^k||<+\infty,
\end{equation}
where $\partial$ is the classical subdifferential for convex functions.
\item [Stop criterion:] If $x^{k+1}=x^{k} $ or $x^{k+1}$ is a Pareto optimal point, then stop. Otherwise to do $k \leftarrow k + 1 $ and return to Main Steps.
\end{description}
\begin{teorema}
Let $F:\mathbb{R}^n\longrightarrow \mathbb{R}^m\cup \{+ \infty \}^m$ be a proper lower semicontinuous convex function, $0\preceq F$ and assume that $\{x^k\}$ is a sequence generated by the {\bf CISPP} algorithm, $(\ref{inicio2c})$, $(\ref{recursao0f2c})$ and $(\ref{deltac})$. Consider also that the set of Pareto optimal points of $(\ref{pom3}),$ denoted by $Min(F),$ is nonempty and assume that $Min(F)$ is $W_{F(\bar x)}$-weak sharp minimum for the problem $(\ref{pom3})$ with constant $\tau>0$ for some $\bar x\in Min(F).$ Then the sequence $\{x^k\}$ converges, in a finite number of iterations, to a Pareto optimal point.
\end{teorema}
\begin{proof}
Denote by $g(x)=\langle F(.), z\rangle$ and $$U=\textnormal{arg min}\{g(x): x\in \mathbb{R}^n\}.$$
As $Min(F)$ is nonempty and it is $W_{F(\bar{x})}$-weak sharp minimum then $Min(F)=WMin(F),$ where $WMin (F)$ denotes the weak Pareto solution of the problem (\ref{pom3}). From Theorem 4.2 of \cite{Bento} it follows that $U$ is nonempty.\\
On the other hand, it is well known that the above {\bf CISPP} algorithm is well defined and converges to some point of $U,$ see
Rockafellar \cite{Rocka}. We will prove that this convergence is obtained to Pareto optimal point in a finite number of iterations.\\
Suppose, by contradiction, that the sequence $\{x^k\}$ is infinite and take $x^*\in U.$
From the iteration $(\ref{recursao0f2c})$ we have that
$$
g(x^{k+1})-g(x^*)\leq \frac{\alpha_k}{2} \left( ||x^k-x^*||^2-||x^{k+1}-x^k||^2\right)+||e^k|||x^{k+1}-x^*||
$$
From (\ref{norma2}) the above inequality implies
$$
g(x^{k+1})-g(x^*)\leq \frac{\alpha_k}{2} \left( ||x^{k+1}-x^*||^2 + 2||x^{k+1}-x^k||||x^{k+1}-x^*||\right)+||e^k|||x^{k+1}-x^*||
$$
Taking $x^*\in U$ such that $||x^{k+1}-x^*||=d(x^{k+1}, W_{F(\bar x)})$ and using the condition of $0<\alpha_k<\bar{\alpha},$ we obtain
$$
\frac{2\tau}{\bar{\alpha}}\leq d(x^{k+1},W_{F(\bar x)})+2||x^{k+1}-x^k|| +\frac{2}{\bar{\alpha}}||e^k||
$$
Letting $k$ goes to infinite in the above inequality we obtain that
$$
\frac{2\tau}{\bar{\alpha}}\leq 0,
$$
which is a contradiction. Thus the {\bf CISPP} algorithm converges to a some point $\hat{x}\in U$ in a finite number of steps.\\
Finally, we will prove that the point of convergence of $\{x^k\},$ denoted by $\hat{x}\in U,$ is a Pareto optimal point of the problem (\ref{pom3}). In fact, as $g$ is weak scalar of the vector function $F,$ then from Proposition \ref{inclusao}, we have $\hat{x}\in WMin(F)$ anf from the equality $Min(F)=WMin(F),$ we obtain that $\hat{x}\in U,$ is a Pareto optimal point of the problem.
\end{proof}
\section{A Numerical Result}
\noindent
In this subsection we give a simple numerical example showing the functionally of the proposed method. For that we use a Intel Core i5 computer 2.30 GHz, 3GB of RAM, Windows 7 as operational system with SP1 64 bits and we implement our code using MATLAB software 7.10 (R2010a).
\begin{exem}
Consider the following multiobjective minimization problem
$$
\min \left\{(F_1(x_1,x_2),F_2(x_1,x_2)): (x_1,x_2)\in \mathbb{R}^2 \right\}
$$
where $F_1(x_1,x_2)=-e^{-x_1^2-x_2^2}+1$ and $F_2(x_1,x_2)=(x_1-1)^2+(x_2-2)^2.$ This problem satisfies the assumptions $\bf (C_{1.2}),$ $\bf (C_2)$ and $\bf (C_4).$ We can easily verify that the points $\bar x=(0,0)$ and $\hat{x}=(1,2)$ are Pareto solutions of the problem.\\
We take $x^0=(-1,3)$ as an initial point and given $x^k\in \mathbb{R}^2,$ the main step of the {\bf SPP} $\mbox{algorithm}$ is to find a critical point ( local minimum, local maximum or a saddle point) of the following problem
$$
\label{e}
\left\{\begin{array}{l}
\min g(x_1,x_2)=(-e^{-x_1^2-x_2^2}+1)z_1^k + \left((x_1-1)^2+(x_2-2)^2 \right)z_2^k+\frac{\alpha_k}{2}\left((x_1-x_1^k)^2+(x_2-x_2^k)^2\right)
\\
s.to:\\
\hspace{0.5cm} x_1^2+x_2^2\leq (x_{1}^k)^2+(x_{2}^k)^2\\
\hspace{0.5cm} (x_1-1)^2+(x_2-2)^2\leq (x_1^k-1)^2+(x_2^k-2)^2
\end{array}\right.
$$
In this example we consider $z_k=\left(z_1^k,z_2^k\right)=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ and $\alpha_k=1,$ for each $k.$
We take $z^0=(2,3)$ as the initial point to solve all the subproblems using the MATLAB function fmincon (with interior point algorithm) and we consider the stop criterion $||x^{k+1}-x^k||<0.0001$ to finish the algorithm. The numerical results are given in the following table:
{\scriptsize $$ \begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
$k$& $ N[x^{k}] $& $x^k=(x^{k}_{1}, x^{k}_{2}) $ &$ ||x^{k}-x^{k-1} || $& $\sum F_i(x^k)z_i^k$& $ F_1(x_1^k,x_2^k)$ & $F_2(x_1^k,x_2^k)$ \\ \hline
1 & 10 & (0.17128, 2.41010)& 1.31144& 1.30959& 0.99709 &0.85496 \\
2 & 10 & (0.65440, 2.16217) &0.54302 & 0.80586 & 0.99392 & 0.14574 \\
3 & 9 & (0.85337, 2.05877 ) &0.22423 & 0.71983 & 0.99303 & 0.02496 \\
4 & 7 & (0.93534, 2.01588 ) & 0.09251 & 0.70518 & 0.99284 & 0.00443 \\
5 & 7 & (0.96912, 1.99814) & 0.03816 & 0.70268 & 0.99279 & 0.00096 \\
6 & 7 & (0.98305, 1.99080) & 0.01574 & 0.70226 & 0.99277 & 0.00037 \\
7 & 7 & (0.98879, 1.98776) & 0.00649 & 0.70219 & 0.99277 & 0.00028 \\
8 & 7 & (0.99115,1.98651) & 0.00268 & 0.70217 & 0.99276 & 0.00026 \\
9 & 7 & (0.99213, 1.98599) & 0.00110 & 0.70217& 0.99276 & 0.00026 \\
10 & 7 & (0.99253, 1.98578) & 0.00046 & 0.70217 & 0.99276 & 0.00026 \\
11 & 7 & (0.99270, 1.98569) & 0.00019 & 0.70217 & 0.99276 & 0.00026 \\
12 & 7 & (0.99277,1.98565) & 0.00008 & 0.70217 & 0.99276 & 0.00026 \\ \hline
\end{tabular}
$$}
The above table show that we need $k=12$ iterations to solve the problem, $N[x^{k}]$ denotes the inner iterations of each subproblem to obtain the point $x^{k},$ for example to obtain the point $x^3=(0.85337, 2.05877 )$ we need $N[x^3]=9$ inner iterations. Observe also that in each iteration we obtain $F(x^k)\succeq F(x^{k+1})$ and the function $\langle F(x^k),z^k\rangle$ is non increasing.
\end{exem}
\section{Conclusion}
\noindent
This paper introduce an exact linear scalarization proximal point algorithm, denoted by {\bf SPP} algorithm, to solve arbitrary extended multiobjective quasiconvex minimization problems. In the differentiable case it is presented an inexact version of the proposed algorithm and for the (not necessary differentiable) convex case, we present an inexact algorithm and we introduced some conditions to obtain finite convergence to a Pareto optimal point.
To reduce considerably the computational cost in each iteration of the {\bf SPP} algorithm it is need to consider the unconstrained iteration
\begin{equation}
\label{subdiferencialintF}
0 \in \hat{\partial}\left( \left\langle F(.), z_k\right\rangle + \dfrac{\alpha_k}{2} \Vert\ .\ - x^k \Vert ^2 \right) (x^{k+1})
\end{equation}
which is more practical than (\ref{subdiferencial3}). One natural condition to obtain (\ref{subdiferencialintF}) is that $x ^{k +1} \in (\Omega_k)^0$ (interior of $\Omega_k$). So we believe that a variant of the {\bf SPP} algorithm may be an interior variable metric proximal point method.
A future research may be the extension of the proposed algorithm for more general constrained vector minimization problems using proximal distances. Another future research may be to obtain a finite convergence of the {\bf SPP} algorithm for the quasiconvex case.
{\footnotesize
}
\end{document} |
\begin{document}
\title{Axiomatizing rectangular grids with no extra non-unary relations}
\begin{abstract}
We construct a formula $\phi$ which axiomatizes non-narrow rectangular grids
without using any binary relations other than the grid neighborship relations.
As a corollary, we prove that a set $A \subseteq {\mathbb N}$ is a spectrum of a formula
which has only planar models if numbers $n \in A$ can be recognized by a
non-deterministic Turing
machine (or a one-dimensional cellular automaton)
in time $t(n)$ and space $s(n)$, where $t(n)s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$.
\end{abstract}
\section{Introduction}
The \emph{spectrum} of a $\phi$, denoted ${\rm{spec}}(\phi)$ is the set of cardinalities of
models of $\phi$. Let ${\rm{SPEC}}$ be the set of $A \subseteq {\mathbb N}$ such that $A$ is
a spectrum of some formula $\phi$ is an interesting research area \cite{scholz,fiftyyears};
it is known that SPEC=NE, i.e., $A$ is a spectrum of a first order formula iff the set
of binary representations of the elements of $A$ is in the complexity class NE
\cite{fagin74,JonesS74}.
However, the characterization of spectra remains open if we require our formula, or
our models, to have additional properties. In \cite{planarspectra} we study the
complexity class ${\rm{FPSPEC}}$ (Forced Planar Spectra), which is the set of
$A \subseteq {\mathbb N}$ such that there exists a formula $\phi$ such that
${\rm{spec}}(\phi)=S$ and all models of $\phi$ are planar. It is shown there that
${\rm{FPSPEC}} \supseteq {\rm{NTISP}}(n^{1-\epsilon}, \log(n))$, where ${\rm{NTISP}}(t(n), s(n))$
the set fo $A \subseteq {\mathbb N}$ such that there exists a non-deterministic
Turing machine which
recognizes the binary representation of $n$ in time $t(n)$ and space $s(n)$.
However, this result is not satisfying, since space $\log(n)$ is very low;
a construction of which allows more space is left as an open problem.
In this paper we construct a formula $\phi$ over a signature consisting of only binary relations
$U,D,L,R$ (neighbors in the grid in all directions) and unary relations, and which axiomatizes rectangular grids which
are not {\it narrow}, i.e., grids of dimensions $x^* \times y^*$ where $x^* = \Omega(\log(y^*))$ and $y^* = \Omega(\log(x^*))$.
We show that it is impossible to give a similar axiomatization of rectangular grids which includes the narrow ones.
Non-narrow rectangular grids are planar graphs of bounded degree, and they can be used to simulate Turing machines, and thus we obtain the
following corollary: ${\rm{FPSPEC}} \supseteq {\rm{NTISP}}(t(n), s(n))$ for every
pair of functions $t(n), s(n)$ such that $t(n) \cdot s(n) \leq n$ and $t(n),s(n) = \Omega(\log(n))$.
In fact, we
get a bit more -- we can actually simulate a non-deterministic one-dimensional cellular automaton (1DCA)
working in the given time and memory. While 1DCAs are less commonly taught than Turing machines,
they are simpler to define and more powerful, since they can perform computations on the whole tape at once \cite{ccotb}.
\section{Axiomatizing a rectangular grid}\label{gridax}
We obtain our goal by showing a first-order formula whose all finite models are rectangular grids.
A {\bf rectangular grid} is a relational structure $G=(V(G),L,R,U,D)$ such that $V(G) = \{0..x^*\} \times \{0..y^*\}$, and
the relations $L$, $R$, $U$, $D$ hold only in the following situations:
$L((x,y), (x-1,y))$, $R((x,y), (x+1),y)$, $U((x,y), (x,y-1))$, $D((x,y), (x,y+1))$, as long as these vertices exist.
\paragraph{Geometry}
We will use four binary relations $L$, $R$, $U$, $D$, which correspond to Left, Right, Up, Down,
respectively. We will need axioms to specify that these four relations work according to
the Euclidean square grid geometry.
\begin{itemize}
\item {\bf Partial injectivity.}
Our relations $X \in \{L, R, U, D\}$ are partial injective functions.
That is, we have an axiom $\forall x \forall y X(x,y) \wedge X(x,z) \Rightarrow y=z$. For
$X \in \{L, R, U, D\}$, we will write $X(x)$ for the element $y$ such that $X(x,y)$
(if it exists).
\item {\bf Inverses.} $\forall x \forall y R(x,y) \iff L(y,x) \wedge U(x,y) \iff D(y,x)$. This axiom
formalizes our interpretation of directions (that Left is inverse to Right and Up is inverse to Down).
\item {\bf Commutativity.} Let $H \in \{L,R\}$ and $V \in \{U,D\}$. Then
$\forall x \forall y \forall z H(x,y) \wedge V(x,z) \Rightarrow \exists t H(z,t) \wedge H(y,t)$.
This axiom axiomatizes the Euclidean geometry of our grid: horizontal and vertical movements commute.
Additionally, it enforces that whenever we can go horizontally and vertically from the given $x$,
we can also combine these two movements and move diagonally.
\end{itemize}
\paragraph{Binary Counters}
We will require our grid to know its number of rows. To this end, we will introduce an extra
relation $B_V$. Intuitively, replace every vertex $v$ in the row $r=(x, R(x), R^2(x), \ldots)$,
where $L(x)$ is not defined, with
1 if $B_V(v)$, and 0 otherwise. The axioms in this section will enforce that the obtained number
(written in the little endian binary notation) is the index of our row.
\begin{itemize}
\item {\bf Horizontal Zero.} $\forall x (\neg \exists y U(x,y)) \Rightarrow (\neg B_V(v))$. The
binary number encoded in the first row is zero.
\item {\bf Horizontal Increment.}
To increment a (little endian) binary number, we change every bit which is either the leftmost one,
or such that its left neighbor changed from 1 to 0. This can be written as the following formula:
$\forall x (\exists y U(x,y)) \Rightarrow ((B_v(x) \not \iff B_v(U(x))) \iff C(x)$
where $C(x) = ((\neg\exists y L(x,y)) \vee (\neg B_v(L(x)) \wedge B_v(U(L(x))))$.
\item {\bf No Horizontal Overflow.} $\forall x (\neg \exists y R(x,y)) \Rightarrow (\neg B_V(v))$. This axiom
makes sure that our binary counter does not overflow.
\end{itemize}
We also have analogous axioms for vertical binary counters, using an extra unary relation $B_H$, counting
from right to left, with the least significant bit on the bottom. See Figures \ref{fig}a and \ref{fig}c, where
the vertices of the grid satisfying respectively $B_V$ and $B_H$ are shown (ignore the small white circles and
dark grey boxes for now -- they will be essential for our further construction).
Let $\phi_1$ be the conjunction of all axioms above.
\begin{theorem}\label{tgeo}
If $G$ is a connected finite model of $\phi_1$ and there exists an $v \in V(G)$ and a relation $X \in \{L,R,U,D\}$ such that
$X(v)$ is not defined, then $G$ is a rectangular grid.
\end{theorem}
\begin{proof}
Take $X$ and $v$ such that $X(v)$ be not defined. Without loss of generality we can assume that $X \in \{L,R\}$ (horizontal and
vertical axioms are symmetrical). Furthermore, we can also assume that $X = L$ (since $R$ is the
inverse of $L$, if $R$ is not defined for some element, then so is $L$).
Let $v+(0,y) = D^y(v)$, where $x \geq 0$ and $y \geq 0$. From the commutativity axiom, $L(v+(0,y))$ is not defined for any
$y$. Indeed, if $L(v+(0,y))$ was defined for $y>0$, we have $L(v+(0,y))$ and $U(v+(0,y)) = v+(0,y-1)$ defined, hence
$L(v+(0,y-1))$ is defined too.
Let $b_y = \sum 2^x [B_V(R^x(v+(0,y)))]$. From the Horizontal Increment and No Horizontal Overflow axioms, it is easy to show that $b_{y+1} = b_y+1$.
Furthermore, we have that $b_y < 2^{|V|}$. Therefore, there must exist $y$ such that $D(v+(0,y))$ is not defined. Let $v' = v+(0,y)$.
Let $x^*$ be the greatest x such that $R^x(v')$ is defined, and $y$ be the greatest $y^*$ such that $U^y(v')$ is defined. Let $G = \{0,\ldots,x^*\} \times \{0,\ldots,y^*\}$,
and for $(x,y) \in G$, let $m(x,y) = R^x(U^y(v'))$. It is straightforward that $m$ gives an isomorphism between the rectangular grid $G$ and $V$.
$\rule{2mm}{2mm}$\par
\end{proof}
\section{Forbidding Tori}
However, rectangular grids are not the only models of $\phi_1$. Consider the torus $T = \{0, \ldots, x^*\} \times \{0, \ldots, y^*\}$, where
$(x^*,y)$ is additionally connected (with the $R$ relation) to $(0,y$), and $(x,y^*)$ is additionally connected to $(x,0)$ (with the $U$ relation),
and we add the respective inverses to $L$ and $D$. If $B_V$ and $B_H$ are empty relations, the torus $T$ satisifes all of our axioms. Additionally,
if $G$ is a model of $\phi_1$, then the disjoint union $G \cup T$ is also a model of $\phi_1$.
To prevent this, we use the following result of Berger \cite{berger}.
\begin{theorem}\label{wangth}
There exists a finite set of Wang tiles $K = \{k_1, \ldots, k_t\}$ and relations $T_R, T_D \subseteq K \times K$ such that there exists a tiling
$C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ such that the following property holds:
\begin{equation}
T_R(C(x,y), C(x+1,y)) \wedge T_D(C(x,y), C(x,y+1)) \mbox{\ for each\ }x,y \in {\mathbb Z}. \label{tiling}
\end{equation}
However, no periodic tiling satisfying \ref{tiling} holds. A tiling $C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ is {\bf periodic} iff there exists
$(x_0, y_0) \neq (0,0)$ such that $C(x,y) = C(x+x_0, y+y_0)$ for each $x,y \in {\mathbb Z}$.
\end{theorem}
The original coloring by Berger used 20426 tiles. It is sufficient to use 11 tiles \cite{jeandel}.
We add a new relation $C$ for every tile $C \in K$. We also add the following axioms:
\begin{itemize}
\item {\bf Full tiling.} $\forall v \bigvee^!_{C \in K} C(v).$ Everything needs to have a color.
\item {\bf Correct tiling.}
For every pair of tiles $C_1, C_2 \in K$ such that $\neg T_R(C_1,C_2)$, we have $\neg \exists v C_1(v) \wedge C_2(R(v))$.
For every pair of tiles $C_1, C_2 \in K$ such that $\neg T_D(C_1,C_2)$, we have $\neg \exists v C_1(v) \wedge C_2(D(v))$.
\end{itemize}
Let $\phi_2$ be the conjuction of $\phi_1$ and the axioms above.
\begin{theorem}
If $G$ is a finite, connected model of $\phi_2$ then $G$ is a rectangular grid.
\end{theorem}
\begin{proof}
Take $v \in V$. If one of the relations $L$, $R$, $U$, $D$ is not defined for some $v \in V$, then $V$ is a rectangular grid by Theorem \ref{tgeo}.
Otherwise, let $C(x,y)$, for $x,y \geq 0$, be the relation $C \in K$ which is satisfied by $R^x(D^y(v))$. For $x<0$ or $y<0$, replace $R^x$ by $L^{-x}$ or $D^y$ by $U^{-y}$.
According to the correct tiling axiom, the property (\ref{tiling}) holds.
Since $V$ is finite, we must have $C(x_1,y_1)$ and $C(x_2,y_2)$ refer to the same element of our structure, even though $(x_1,y_1) \neq (x_2,y_2)$. It is easy to show that
$(x_1-x_2, y_1-y_2)$ is then the period of the tiling $C$, which contradicts Theorem \ref{wangth}.
$\rule{2mm}{2mm}$\par
\end{proof}
\begin{theorem}\label{allgrids}
There exists a formula $\phi_3$ such that the models of $\phi$, restricted to relations $L$, $R$, $U$, $D$, are precisely the rectangular grids $x^* \times y^*$ such that $y^* \leq 2^{x^*-1}$ and $x^* \leq 2^{y^*-1}$.
\end{theorem}
\begin{proof}
By adding an axiom that there exists exactly one element $v^*$ such that $L(v^*)$ and $U(v^*)$ are not defined, we obtain a formula $\phi_3$ whose all finite models are rectangular grids.
Now, take an $x^* \times y^*$ rectangular grid G. From Theorem \ref{wangth} there exists a tiling $C: {\mathbb Z} \times {\mathbb Z} \rightarrow K$ satisfying \ref{tiling}. Assign the relation $C(x,y)$
to each $(x,y)\in G$. If $y^* \leq 2^{x^*-1}$ and $x^* \leq 2^{y^*-1}$, we can also set $B_H(x,y)$ iff $x$-th bit of $y$ is 1, and $B_V(x,y)$ iff $y$-th bit of $x$ is 1.
Such a model will satisfy $\phi_3$. Note that if $x^* > 2^{y^*-1}$ or $y^* > 2^{x^*-1}$, the respective overflow axiom will not be satisfied.
$\rule{2mm}{2mm}$\par
\end{proof}
The number 2 in the theorem above can be changed to an integer $b \geq 2$ by using $b$-ary counters instead of the binary ones. However:
\begin{theorem} \label{srem} \rm
There is no formula $\phi$ over a signature consisting of $L$, $R$, $U$, $D$, and possibly extra unary relations whose all models restricted to relations $L$, $R$, $U$, $D$ are
precisely all rectangular grids. Furthermore, there is no such $\phi$ such that all models of $\phi$ are rectangular grids, and
there exists $y*$ such that for every $x*$ a rectangular model $x^* \times y^*$ of $\phi$ exists.
\end{theorem}
\begin{proof}
We will be using Hanf's locality lemma \cite{hanf}.
Let a $r$-neighborhood of the vertex $v \in V$, $N_r(v)$ be the set of all vertices whose distance from $v$ is at most $r$.
Let a {\bf $r$-type} of the vertex $v$, $\tau(v)$, be the isomorphism type of $N_r(v)$. When we restrict to models of degree bounded by $d$,
there are only finitely many such types. Let $T_r$ be the set of all types.
Let $f_{r,M}(G): T \rightarrow \{0..M\}$ be the function that assigns to each type $\tau \in T$ the minimum of $M$ and the number of vertices of type $\tau$ in $G$.
\begin{theorem}[Hanf's locality lemma\cite{hanf}]\label{hanf}
Let $\phi$ be a FO formula. Then there exist numbers $r$ and $M$ such that, for each graph $G=(V,E)$, $G \models \phi$ depends only on $f_{r,M}$.
\end{theorem}
Let $\phi$ be a FO formula such that all models of $\phi$ are rectangular grids. Take $r$ and $M$ from Theorem \ref{hanf}.
Let the rectangular grid $G$ be a model of $phi$, where
$V(G) = \{0,\ldots,x^*\} \times \{0,\ldots,y^*\}$. Let $\tau(x)$ be the type of column $x$, i.e., $\tau(x) = (\tau(x,0), \ldots, \tau(x,y*))$.
For sufficiently large $x^*$ there will be $x_1$ and $x_2$ such that $\tau(x_1+i) = \tau(x_2+i)$ for $i=-r, \ldots, r$ and
such that $f_{r,M}(G)(\tau(x,y)) \geq M$ for every $x \in \{x_1, \ldots, x_2\}$. Construct a new structure $G'$ by adding a cylinder of dimensions
$(x_2-x_1) \times y*$ to $G$, i.e.,
$V(G') = V(G) \cup \{(1,x,y): x \in \{x_1, \ldots, x_2-1\}, y \in \{0, \ldots, y*\}$, $U(1,x,y) = (1,x,y-1)$, $D(1,x,y) = (1,x,y+1)$,
$R(1,x,y) = (1,x+1,y)$, $L(1,x,y) = (1,x-1,y)$, $R(1,x_2-1,y) = (1,x_1,y)$, $L(1,x_1,y) = (1,x_2-1,y)$, whenever the point on the right hand side exists,
and undefined otherwise. For every unary relation $U$ we have $U(1,x,y)$ iff $U(x,y)$. It is easy to verify that $\tau(1,x,y) = \tau(x,y)$,
and every of these types already appeared at least $M$ times, and thus from Theorem \ref{hanf}, $G' \models \phi$.
$\rule{2mm}{2mm}$\par
\end{proof}
\section{Forced Planar Spectra}
\begin{corollary}
Let $S \subseteq \mathbb{N}$ be a set such that there exists a non-deterministic Turing machine (or 1DCA) recognizing the set of binary representations of
elements of $S$ in time $t(n)$ and memory $s(n)$, where $t(n) \cdot s(n) \leq n$ and $t(n),s(n) \geq \Omega(log(n))$. Then there exists a first-order formula $\phi$ such
that all models of $\phi$ are planar graphs, and the set of cardinalities of models of $\phi$ is $S$.
\end{corollary}
\begin{proof}
Let $A \subseteq {\mathbb N}$, and let $M$ be a non-deterministic Turing machine or a non-deterministic 1DCA recognizing $A$ in time $t(n)$ and space $s(n)$ such that $t(n) \cdot s(n) \leq n$.
A non-deterministic 1DCA is $M = (\Sigma, R, F)$ where $R \subseteq \Sigma^4$, and the final symbol $F \in \Sigma$. It is defined similar to a Turing machine,
but where computations are performed in parallel on all the tape cells: if $t(x,y)$ is the content of the tape at position $x$ and time $y$, then
the relation $R(t(x-1,y), t(x,y), t(x+1,y), t(x,y+1))$ must hold. The 1DCA accepts when it writes the symbol $F$.
Let $u(n) = n-t(n)s(n)$; without loss of
generality we can assume $u(n) < t(n)$.
It is well known that a first order formula on a grid can be used to simulate a Turing machine (or 1DCA):
the bottom row is the initial tape, and our formula ensures that each other row above it is a correct successor of the row below it.
Let $n \in A$. We will construct a formula $\phi$ which will have a model consisting of:
\begin{itemize}
\item A rectangular grid $G' = \{0,\ldots,s\} \times \{0,\ldots,t\}$, where $t = t(n)-1$ and $s = s(n)-1$. The structure of the grid is given by relations $L$, $R$, $U$ and $D$
just as in Section \ref{gridax}; we also have all the auxiliary relations required by Theorem \ref{allgrids}.
\item $u(n)$ elements which are not in the grid. The relation $P$ will hold for all the extra elements and only for them. The relation $Q$ will hold only
for the elements $(0,t-i) \in G'$ where $i \leq u(n)-1$. The relation $B$ gives a bijection between elements $x$ such that $P(x)$, and the elements $x$ such that $Q(x)$.
\item Encoding of the number $n$.
We encode the number $n$ in the leftmost cells in the initial tape using two relations $D_n$ and $E_n$ in the following way:
$D_a(x,t)$ is the $x$-th digit of $n$, and the relation $E_a$ signifies the end of the encoding: $E_a(x,y) \Rightarrow E_a(x+1,t) \wedge \neg D_a(x,t)$ (if $(x+1,t)$ exists).
\item Similarly we encode the numbers $s$, $t$ and $u$.
\item An encoded run of $M$ which accepts the encoded value $n$ as the input.
\item An encoded run of an one-dimensional cellular automaton $M_2$ which verifies that the relation $n = (s+1) \times (t+1) + u$ holds for the encoded numbers.
A one-dimensional cellular automaton can add and multiply $k$-digit numbers in time $O(k)$ \cite{atrubin}, hence our space $s$ will be sufficient.
\item Our grid already has the binary representations of $s$ and $t$ computed as the relations $B_H$ and $B_V$. In the case of $B_V$ the
computed $t$ is already where we need it (we only need to define the relation $E_{t}$ in the straightforward way). In the case of $B_H$ the
computed $s$ is in the rightmost column, so we add extra wiring relations $W$ to move it to the beginning of the initial tape. In the case of $u$,
we need to compute the binary representation of the number of rows $i$ such that $Q(0,i)$; this can be computed in the same way as we have computed
the number of all rows (using the relation $B_U$ similar to $B_V$).
\end{itemize}
\def\subfig#1#2{
\begin{minipage}[b]{.5\textwidth}\includegraphics[width=\textwidth]{#1}\begin{center}#2\end{center}
\end{minipage}
}
\begin{figure}
\caption{Computing the size of our model.\label{fig}
\label{fig}
\end{figure}
Figure \ref{fig} shows the elements of our construction. In all the pictures, the small circles are the extra elements (where $P$ holds), and the other elements are
the grid; the thin lines represent the relation $B$, the thick lines represent the relations $U$, $D$, $L$ and $R$. In \ref{fig}a the black circles
represent $B_H \equiv D_{t}$ and gray boxes represent $E_{t}$. In \ref{fig}b the gray circles represent $Q$, black circles represent $B_U$ and gray circles
represent $E_U$. In \ref{fig}c the black circles represent $B_V$, while in \ref{fig}d the extra thick lines represent $W$, black circles represent $E_{s}$,
and gray boxes represent $E_{s}$.
The formula $\phi$ will be the conjuction of the following axioms:
\begin{itemize}
\item (1) $\phi_3$, restricted to elements for which $P$ does not hold. This requires that we indeed have a rectangular grid.
\item (2) Axiomatiziations of the Turing machine $M$.
\item (3) $B$ is a bijection.
\item (4) The set of elements satisfying $Q$ has the correct shape: $Q(v) \Rightarrow \neg P(v) \wedge (\neg\exists y L(x,y)) \wedge (\exists y D(x,y) \Rightarrow Q(y))$,
\item (5) Axiomatiziation of $B_U$, similar to the axiomatization of $B_V$, but where we add 1 only in the rows $y$ where $Q(0,y)$ holds.
\item (6) Axiomatiziation of the wiring $W$ moving $s$. The axioms are as follows:
$W(v,w) \wedge D_t(v) \Rightarrow D_t(w)$; $W(v,w) \Rightarrow W(w,v)$; every v is connected to either (a) only $R(v)$ and $L(v)$ is undefined, (b) only $L(v)$ and $R(v)$;
(c) only $L(v)$ and $D(v)$; (d) only $D(v)$ and $U(v)$; (e) only $U(v)$ and $D(v)$ is not defined; (f) nothing. Furthermore, in case (c), $L(D(v))$ must either
be also case (c) or the bottom left corner; $D_t(v) \iff B_H(v)$ whenever $L(v)$ is undefined; and the case (a) holds whenever $L(v)$ undefined, $D_t(v)$,
and $D(v)$ defined.
\item (7) $\forall v D_{t}(v) \iff V_H(y)$.
\item (8) For every encoded number $a$, $E_a(v) \rightarrow (\neg D_a(v) \wedge \exists w R(v,w) \rightarrow E_a(w)$.
\item (9) Axiomatiziations of the automaton $M_2$.
\end{itemize}
Our model satisfies all these axioms.
On the other hand, suppose that $\phi$ has a model $G$ of size $n$. By (1) this model constists of a rectangular grid and
a number of $u$ extra elements. By (3) and (4) the relation $Q$ is satisfied only for $u$ bottommost elements in the leftmost column.
By (5) the encoded number $u$ equals the number of these elements. By (6) and (7) the encoded numbers $s$ and $t$ equal the dimensions of
the grid. By (8) and (9) we know that the encoded number $n$ indeed equals the size of $G$. By (2) we know that $M$ accepts $n$, therefore
$n \in A$.
\end{proof}
\end{document} |
\begin{document}
\title[Fractional Choquard equations with magnetic fields]{Concentration phenomena for a fractional Choquard equation with magnetic field}
\author[V. Ambrosio]{Vincenzo Ambrosio}
\address{Vincenzo AmbrosioH^{s}_{\e}fill\break\indent
Department of Mathematics H^{s}_{\e}fill\break\indent
EPFL SB CAMA H^{s}_{\e}fill\break\indent
Station 8 CH-1015 Lausanne, Switzerland}
\varepsilonmail{vincenzo.ambrosio2@unina.it}
\subjclass{Primary 35A15, 35R11; Secondary 45G05}
\date{}
\keywords{Fractional Choquard equation, fractional magnetic Laplacian, penalization method.}
\begin{abstract}
We consider the following nonlinear fractional Choquard equation
$$
\varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N},
$$
where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $0<\mu<2s$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a smooth magnetic potential, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a positive potential with a local minimum and $f$ is a continuous nonlinearity with subcritical growth.
By using variational methods we prove the existence and concentration of nontrivial solutions for $\varepsilon>0$ small enough.
\varepsilonnd{abstract}
\maketitle
\section{Introduction}
\noindent
In this paper we investigate the existence and concentration of nontrivial solutions for the following nonlinear fractional Choquard equation
\begin{equation}\label{P}
\varepsilon^{2s}(-\Delta)^{s}_{A/\varepsilon} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \,\mbox{ in } \,\mathbb{R}^{N},
\varepsilonnd{equation}
where $\varepsilon>0$ is a parameter, $s\in (0,1)$, $N\geq 3$, $0<\mu<2s$, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous potential and
$A:\mathbb{R}^{N}\rightarrow \mathbb{R}^{N}$ is a $C^{0, \alpha}$ magnetic potential, with $\alpha\in (0,1]$.
The nonlocal operator $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian which may be defined for any $u:\mathbb{R}^{N}\rightarrow \mathbb{C}$ smooth enough by setting
$$
(-\Delta)^{s}_{A}u(x)= c_{N,s} P.V. \int_{\mathbb{R}^{N}} \frac{u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}}{|x-y|^{N+2s}} \,dy \quad (x\in \mathbb{R}^{N}),
$$
where $c_{N, s}$ is a normalizing constant. This operator has been introduced in \cite{DS, I} with motivations falling into the framework of the general theory of L\'evy processes.
As showed in \cite{SV}, when $s\rightarrow 1$, the operator $(-\Delta)^{s}_{A}$ reduces to the magnetic Laplacian (see \cite{LaL, LL}) defined as
$$
\left(\frac{1}{\imath}\nabla-A\right)^{2}\!u= -\Delta u -\frac{2}{\imath} A(x) \cdot \nabla u + |A(x)|^{2} u -\frac{1}{\imath} u \dive(A(x)),
$$
which has been widely investigated by many authors: see \cite{AF, AFF, AS, Cingolani, CS, EL, K}.
Recently, many papers dealt with different fractional problems involving the operator $(-\Delta)^{s}_{A}$.
d'Avenia and Squassina \cite{DS} studied the existence of ground states solutions for some fractional magnetic problems via minimization arguments.
Pinamonti et al. \cite{PSV1, PSV2} obtained a magnetic counterpart of the Bourgain-Brezis-Mironescu formula and the MazÕya-Shaposhnikova formula respectively; see also \cite{NPSV} for related results.
Zhang et al. \cite{ZSZ} proved a multiplicity result for a fractional magnetic Schr\"odinger equation with critical growth.
In \cite{MPSZ} Mingqi et al. studied existence and multiplicity of solutions for a subcritical fractional Schr\"{o}dinger-Kirchhoff equation involving an external magnetic potential.
Fiscella et al. \cite{FPV} considered a fractional magnetic problem in a bounded domain proving the existence of at least two nontrivial weak solutions under suitable assumptions on the nonlinear term.
In \cite{AD} the author and d'Avenia used variational methods and Ljusternick-Schnirelmann theory to prove existence and multiplicity of nontrivial solutions for a fractional Schr\"{o}dinger equation with subcritical nonlinearities. \\
We note that when $A=0$, the operator $(-\Delta)^{s}_{A}$ becomes the celebrated fractional Laplacian $(-\Delta)^{s}$ which arises in the study of several physical phenomena like phase transitions, crystal dislocations, quasi-geostrophic flows, flame propagations and so on. Due to the extensive literature on this topic, we refer the interested reader to \cite{DPMV, DPV, MBRS} and the references therein.\\
In absence of the magnetic field, equation \varepsilonqref{P} is a fractional Choquard equation of the type
\begin{equation}\label{FChE}
(-\Delta)^{s} u + V(x)u = \left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \,\mbox{ in } \,\mathbb{R}^{N}.
\varepsilonnd{equation}
d'Avenia et al. \cite{DSS} studied the existence, regularity and asymptotic behavior of solutions to \varepsilonqref{FChE} when $f(u)=u^{p}$ and $V(x)\varepsilonquiv const$. If $V(x)=1$ and $f$ satisfies Berestycki-Lions type assumptions, the existence of ground state solutions for a fractional Choquard equation has been established in \cite{SGY}.
The analyticity and radial symmetry of positive ground state for a critical boson star equation has been considered by Frank and Lenzmann in \cite{FL}.
Recently, the author in \cite{Apota} studied the multiplicity and concentration of positive solutions for a fractional Choquard equation under local conditions on the potential $V(x)$. \\
When $s=1$, equation \varepsilonqref{FChE} reduces to the generalized Choquard equation:
\begin{equation}\label{GCE}
-\Delta u + V(x) u = \left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \,\mbox{ in } \,\mathbb{R}^{N}.
\varepsilonnd{equation}
If $p=\mu=2$, $V(x)\varepsilonquiv 1$, $F(u)=\frac{u^{2}}{2}$ and $N=3$, \varepsilonqref{GCE} is called the Choquard-Pekar equation which goes back to the 1954's work by Pekar \cite{Pek} to the description of a polaron at rest in Quantum Field Theory and to 1976's model of Choquard of an electron trapped in its own hole as an approximation to Hartree-Fock theory for a one-component plasma \cite{LS}. The same equation was proposed by Penrose \cite{Pen} as a model of self-gravitating matter and is known in that context as the Schr\"odinger-Newton equation.
Lieb in \cite{Lieb} proved the existence and uniqueness of positive solutions to a Choquard-Pekar equation.
Subsequently, Lions \cite{Lions} established a multiplicity result via variational methods. Ackermann in \cite{Ack} proved the existence and multiplicity of solutions for \varepsilonqref{GCE} when $V$ is periodic.
Ma and Zhao \cite{MZ} showed that, up to translations, positive solutions of equation \varepsilonqref{GCE} with $f(u)=u^{p}$, are radially symmetric and monotone decreasing for suitable values of $\mu$, $N$ and $p$. This results has been improved by Moroz and Van Schaftingen in \cite{MVS1}.
The same authors in \cite{MVS2} obtained the existence of ground state solutions with a general nonlinearity $f$.
Cingolani et al. \cite{CSS} showed the existence of multi-bump type solutions for a Schro\"odinger equation in presence of electric and magnetic potentials and Hartree-type nonlinearities.
Alves et al. \cite{AFY}, inspired by \cite{AFF, CSS}, studied the multiplicity and concentration phenomena of solutions for \varepsilonqref{GCE} in presence of a magnetic field. For a more detailed bibliography on the Choquard equation we refer to \cite{MVS3}.
Motivated by \cite{AFY, Apota, AD}, in this paper we focus our attention on the existence and concentration of solutions to \varepsilonqref{P} under local conditions on the potential $V$. Before stating our main result, we introduce the assumptions on $V$ and $f$.
Along the paper, we assume that the potential $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function verifying the following conditions introduced in \cite{DPF}:
\begin{enumerate}
\item [$(V_1)$] $V(x)\geq V_{0}>0$ for all $x\in \mathbb{R}^{N}$;
\item [$(V_2)$] there exists a bounded open set $\Lambda\subset \mathbb{R}^{N}$ such that
$$
V_{0}=\inf_{x\in \Lambda} V(x)<\min_{x\in \partial \Lambda} V(x),
$$
\varepsilonnd{enumerate}
and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(t)=0$ for $t<0$ and satisfies the following assumptions:
\begin{enumerate}
\item [($f_1$)] $\displaystyle{\lim_{t\rightarrow 0} f(t)=0}$;
\item [($f_2$)] there exists $q\in (2, \frac{2^{*}_{s}}{2}(2-\frac{\mu}{N}))$, where $2^{*}_{s}=\frac{2N}{N-2s}$, such that $\displaystyle{\lim_{t\rightarrow \infty} \frac{f(t)}{t^{\frac{q-2}{2}}}=0}$;
\item [($f_3$)] the map $\displaystyle{t \mapsto f(t)}$ is increasing for every $t>0$.
\varepsilonnd{enumerate}
We point to that the restriction on $q$ in $(f_2)$ is related to the Hardy-Littlewood-Sobolev inequality:
\begin{theorem}\label{HLS}\cite{LL}
Let $r, t>1$ and $0<\mu<N$ such that $\frac{1}{r}+\frac{\mu}{N}+\frac{1}{t}=2$. Let $f\in L^{r}(\mathbb{R}^{N})$ and $h\in L^{t}(\mathbb{R}^{N})$. Then there exists a sharp constant $C(r, N, \mu, t)>0$ independent of $f$ and $h$ such that
$$
\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{f(x)h(y)}{|x-y|^{\mu}}\, dx dy\leq C(r, N, \mu, t)\|f\|_{L^{r}(\mathbb{R}^{N})}\|h\|_{L^{t}(\mathbb{R}^{N})}.
$$
\varepsilonnd{theorem}
Indeed, by $(f_1)$ and $(f_2)$ it follows that $|F(|u|^{2})|\leq C(|u|^{2}+|u|^{q})$, so it is easy to check that
the term
\begin{equation}\label{chiara}
\left|\int_{\mathbb{R}^{N}} \left( \frac{1}{|x|^{\mu}}* F(|u|^{2})\right) F(|u|^{2}) dx\right|<\infty \quad \forall u\in H^{s}_{\e},
\varepsilonnd{equation}
where $H^{s}_{\e}$ is defined in Section $2$, when $F(|u|^{2})\in L^{t}(\mathbb{R}^{N})$ for all $t>1$ such that
$$
\frac{2}{t}+\frac{\mu}{N}=2, \, \mbox{ that is } t=\frac{2N}{2N-\mu}.
$$
Therefore, if $q\in (2, \frac{2^{*}_{s}}{2}(2-\frac{\mu}{N}))$ and $\mu\in (0, 2s)$ we can use the fractional Sobolev embedding $H^{s}(\mathbb{R}^{N}, \mathbb{R})\subset L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in [2, 2^{*}_{s}]$, to deduce that $tq\in (2, 2^{*}_{s})$ and then \varepsilonqref{chiara} holds true. \\
Now, we can state the main result of this paper:
\begin{theorem}\label{thm1}
Suppose that $V$ verifies $(V_1)$-$(V_2)$, $0<\mu<2s$ and $f$ satisfies $(f_1)$-$(f_3)$ with $q\in (2, 2\frac{(N-\mu)}{N-2s} )$. Then there exists $\varepsilon_{0}>0$ such that, for any $\varepsilon\in (0, \varepsilon_{0})$, problem \varepsilonqref{P} has a nontrivial solution. Moreover, if $|u_{\varepsilon}|$ denotes one of these solutions and $x_{\varepsilon}\in \mathbb{R}^{N}$ its global maximum, then
$$
\lim_{\varepsilon\rightarrow 0} V(x_{\varepsilon})=V_{0},
$$
and
\begin{align*}
|u_{\varepsilon}(x)|\leq \frac{\tilde{C} \varepsilon^{N+2s}}{\varepsilon^{N+2s}+|x-x_{\varepsilon}|^{N+2s}} \quad \forall x\in \mathbb{R}^{N}.
\varepsilonnd{align*}
\varepsilonnd{theorem}
\begin{remark}
Assuming $f\in C^{1}$, one can use Ljusternick-Schnirelmann theory and argue as in \cite{Apota, AD} to relate the number of nontrivial solutions to \varepsilonqref{P} with the topology of the set where the potential attains its minimum value.
\varepsilonnd{remark}
The proof of Theorem \ref{thm1} is inspired by some variational arguments used in \cite{AFF, AFY, AM, Apota}. Anyway, the presence of the fractional magnetic Laplacian and nonlocal Hartree-type nonlinearity does not permit to easily adapt in our setting the techniques developed in the above cited papers and, as explained in what follows, a more intriguing and accurate analysis will be needed. Firstly, after a change of variable, it is easy to check that problem (\ref{P}) is equivalent to the following one:
\begin{equation}\label{R}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N}
\varepsilonnd{equation}
where $A_{\varepsilon}(x):=A(\varepsilon x)$ and $V_{\varepsilon}(x):=V(\varepsilon x)$.
In the spirit of \cite{DPF} (see also \cite{AFF, AM}), we modify the nonlinearity in a suitable way and we consider an auxiliary problem. We note that the restriction imposed on $\mu$ allows us to use the penalization technique.
Without loss of generality, along the paper we will assume that $0\in \Lambda$ and $V_{0}=V(0)=\inf_{x\in\mathbb{R}^{N}} V(x)$.
Now, we fix $\varepsilonll>0$ large enough, which will be determined later on, and let $a>0$ be the unique number such that $f(a)=\frac{V_{0}}{\varepsilonll}$. Moreover, we introduce the functions
$$
\tilde{f}(t):=
\begin{cases}
f(t)& \text{ if $t \leq a$} \\
\frac{V_{0}}{\varepsilonll} & \text{ if $t >a$},
\varepsilonnd{cases}
$$
and
$$
g(x, t):=\chi_{\Lambda}(x)f(t)+(1-\chi_{\Lambda}(x))\tilde{f}(t),
$$
where $\chi_{\Lambda}$ is the characteristic function on $\Lambda$, and we write $G(x, t)=\int_{0}^{t} g(x, \tau)\, d\tau$.\\
From assumptions $(f_1)$-$(f_3)$, it is easy to verify that $g$ fulfills the following properties:
\begin{enumerate}
\item [($g_1$)] $\displaystyle{\lim_{t\rightarrow 0} g(x, t)=0}$ uniformly in $x\in \mathbb{R}^{N}$;
\item [($g_2$)] $\displaystyle{\lim_{t\rightarrow \infty} \frac{g(x, t)}{t^{\frac{q-2}{2}}}=0}$ uniformly in $x\in \mathbb{R}^{N}$;
\item [($g_3$)] $(i)$ $0\leq G(x, t)< g(x, t)t$ for any $x\in \Lambda$ and $t>0$, and \\
$(ii)$ $0<G(x, t)\leq g(x, t)t\leq \frac{V_{0}}{\varepsilonll}t$ for any $x\in \mathbb{R}^{N}\setminus \Lambda$ and $t>0$,
\item [($g_4$)] $t\mapsto g(x, t)$ and $t\mapsto \frac{G(x, t)}{t}$ are increasing for all $x\in \mathbb{R}^{N}$ and $t>0$.
\varepsilonnd{enumerate}
Thus, we consider the following auxiliary problem
\begin{equation*}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)g(\varepsilon x, |u|^{2})u \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation*}
and in view of the definition of $g$, we are led to seek solutions $u$ of the above problem such that
\begin{equation}\label{ue}
|u(x)|<a \mbox{ for all } x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon}, \quad \mbox{ where } \Lambda_{\varepsilon}:=\{x\in \mathbb{R}^{N}: \varepsilon x\in \Lambda\}.
\varepsilonnd{equation}
By using this penalization technique and establishing some careful estimates on the convolution term, we are able to prove that the energy functional associated with the auxiliary problem has a mountain pass geometry and satisfies the Palais-Smale condition; see Lemma \ref{MPG}, \ref{lemK} and \ref{PSc}. Then we can apply the Mountain Pass Theorem \cite{AR} to obtain the existence of a nontrivial solution $u_{\varepsilon}$ to the modified problem.
The H\"older regularity assumption on the magnetic field $A$ and the fractional diamagnetic inequality \cite{DS}, will be properly exploited to show an interesting and useful relation between the mountain pass minimax level $c_{\varepsilon}$ of the modified functional and the minimax level $c_{V_{0}}$ associated with the limit functional; see Lemma \ref{AMlem1}.
In order to verify that $u_{\varepsilon}$ is also solution of the original problem \varepsilonqref{P}, we need to check that $u_{\varepsilon}$ verifies \varepsilonqref{ue} for $\varepsilon>0$ sufficiently small.
To achieve our goal, we first use an appropriate Moser iterative scheme \cite{Moser} to show that $\|u_{\varepsilon}\|_{L^{\infty}(\mathbb{R}^{N})}$ is bounded uniformly with respect to $\varepsilon$. In these estimates, we take care of the fact that the convolution term
is a bounded term in view of Lemma \ref{lemK}.
After that, we use these informations to develop a very clever approximation argument related in some sense to the following fractional version of Kato's inequality \cite{Kato}
$$
(-\Delta)^{s}|u|\leq \mathbb{R}e(sign(u)(-\Delta)^{s}_{A}u),
$$
to show that $|u_{\varepsilon}|$ is a weak subsolution to the problem
$$
(-\Delta)^{s}|u|+V(x)|u|= h(|u|^{2})|u| \mbox{ in } \mathbb{R}^{N},
$$
for some subcritical nonlinearity $h$, and then we prove that $|u_{\varepsilon}(x)|\rightarrow 0$ as $|x|\rightarrow \infty$, uniformly in $\varepsilon$; see Lemma \ref{moser}. We point out that our arguments are different from the ones used in the classical case $s=1$ and the fractional setting $s\in (0,1)$ without magnetic field. Indeed, we don't know if a Kato's inequality is available in our framework, so we can not proceed as in \cite{CS, K} in which the Kato's inequality is combined with some standard elliptic estimates to obtain informations on the decay of solutions. Moreover, the appearance of magnetic field $A$ and the nonlocal character of $(-\Delta)^{s}_{A}$ do not permit to adapt the iteration argument developed in \cite{AFF, AFY} where $s=1$ and $A\not \varepsilonquiv 0$, and we can not use the well-known estimates based on the Bessel kernel (see \cite{AM, FQT}) established for fractional Schr\"odinger equations with $A=0$.
However, we believe that the ideas contained here can be also applied to deal with other fractional magnetic problems like \varepsilonqref{P}. Finally, we also give an estimate on the decay of modulus of solutions to \varepsilonqref{P} which is in clear accordance with the results in \cite{FQT}.
To the best of our knowledge, this is the first time that the penalization method is used to study nontrivial solutions for fractional Choquard equations with magnetic fields, and this represents the novelty of this work.\\
The paper is organized as follows: in Section $2$ we present some preliminary results and we collect some useful lemmas.
The Section $3$ is devoted to the proof of Theorem \ref{thm1}.
\section{Preliminaries and functional setting}
\noindent
For any $s\in (0,1)$, we denote by $\mathcal{D}^{s, 2}(\mathbb{R}^{N}, \mathbb{R})$ the completion of $C^{\infty}_{0}(\mathbb{R}^{N}, \mathbb{R})$ with respect to
$$
[u]^{2}=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}} \, dx \, dy =\|(-\Delta)^{\frac{s}{2}} u\|^{2}_{L^{2}(\mathbb{R}^{N})},
$$
that is
$$
\mathcal{D}^{s, 2}(\mathbb{R}^{N}, \mathbb{R})=\left\{u\in L^{2^{*}_{s}}(\mathbb{R}^{N},\mathbb{R}): [u]_{H^{s}(\mathbb{R}^{N})}<\infty\right\}.
$$
Let us introduce the fractional Sobolev space
$$
H^{s}(\mathbb{R}^{N}, \mathbb{R})= \left\{u\in L^{2}(\mathbb{R}^{N}) : \frac{|u(x)-u(y)|}{|x-y|^{\frac{N+2s}{2}}} \in L^{2}(\mathbb{R}^{2N}) \right \}
$$
endowed with the natural norm
$$
\|u\| = \sqrt{[u]^{2} + \|u\|_{L^{2}(\mathbb{R}^{N})}^{2}}.
$$
Let us denote by $L^{2}(\mathbb{R}^{N}, \mathbb{C})$ the space of complex-valued functions with summable square, endowed with the real
scalar product
$$
\langle u, v\rangle_{L^{2}}=\mathbb{R}e\left(\int_{\mathbb{R}^{N}} u \bar{v} dx\right)
$$
for all $u, v\in L^{2}(\mathbb{R}^{N}, \mathbb{C})$.
We consider the space
$$
\mathcal{D}^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})=\{u\in L^{2^{*}_{s}}(\mathbb{R}^{N}, \mathbb{C}) : [u]_{A}<\infty\}
$$
where
$$
[u]_{A}^{2}=\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}} dx dy.
$$
Then, we define the following fractional magnetic Sobolev space
$$
H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})=\{u\in L^{2}(\mathbb{R}^{N}, \mathbb{C}): [u]_{A}<\infty\}.
$$
It is easy to check that $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ is a Hilbert space with the real scalar product
\begin{align*}
\langle u, v\rangle_{s, A}&=\mathbb{R}e\iint_{\mathbb{R}^{2N}} \frac{(u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})\overline{(v(x)-v(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy\\
&\quad +\langle u, v\rangle_{L^{2}}
\varepsilonnd{align*}
for any $u, v\in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$. Moreover, $C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{C})$ is dense in $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ (see \cite{AD}).
Now, we recall the following useful results:
\begin{theorem}\label{Sembedding}\cite{DS}
The space $H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$ is continuously embedded into $L^{r}(\mathbb{R}^{N}, \mathbb{C})$ for any $r\in [2, 2^{*}_{s}]$ and compactly embedded into $L^{r}(K, \mathbb{C})$ for any $r\in [1, 2^{*}_{s})$ and any compact $K\subset \mathbb{R}^{N}$.
\varepsilonnd{theorem}
\begin{lemma}\label{DI}\cite{DS}
For any $u\in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$, we get $|u|\in H^{s}(\mathbb{R}^{N},\mathbb{R})$ and it holds
$$
[|u|]\leq [u]_{A}.
$$
We also have the following pointwise diamagnetic inequality
$$
||u(x)|-|u(y)||\leq |u(x)-u(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)}| \mbox{ a.e. } x, y\in \mathbb{R}^{N}.
$$
\varepsilonnd{lemma}
\begin{lemma}\label{aux}\cite{AD}
If $u\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and $u$ has compact support, then $w=e^{\imath A(0)\cdot x} u \in H^{s}_{A}(\mathbb{R}^{N}, \mathbb{C})$.
\varepsilonnd{lemma}
\noindent
For any $\varepsilon>0$, we denote by
$$
H^{s}_{\e}=\left\{u\in \mathcal{D}^{s}_{A_{\varepsilon}}(\mathbb{R}^{N}, \mathbb{C}): \int_{\mathbb{R}^{N}} V_{\varepsilon}(x) |u|^{2}\, dx<\infty\right\}
$$
endowed with the norm
$$
\|u\|^{2}_{\varepsilon}=[u]^{2}_{A_{\varepsilon}}+\|\sqrt{V_{\varepsilon}} |u|\|^{2}_{L^{2}(\mathbb{R}^{N})}.
$$
\noindent
From now on, we consider the following auxiliary problem
\begin{equation}\label{Pe}
(-\Delta)^{s}_{A_{\varepsilon}} u + V_{\varepsilon}(x)u = \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)g(\varepsilon x, |u|^{2})u \mbox{ in } \mathbb{R}^{N}
\varepsilonnd{equation}
and we note that if $u$ is a solution of (\ref{Pe}) such that
\begin{equation*}
|u(x)|<a \mbox{ for all } x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon},
\varepsilonnd{equation*}
then $u$ is indeed solution of the original problem (\ref{R}).
It is clear that weak solutions to (\ref{Pe}) can be found as critical points of the Euler-Lagrange functional $J_{\varepsilon}: H^{s}_{\varepsilon}\rightarrow \mathbb{R}$ defined by
$$
J_{\varepsilon}(u)=\frac{1}{2}\|u\|^{2}_{\varepsilon}-\frac{1}{4}\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx.
$$
We begin proving that $J_{\varepsilon}$ possesses a mountain pass geometry \cite{AR}.
\begin{lemma}\label{MPG}
$J_{\varepsilon}$ has a mountain pass geometry, that is
\begin{enumerate}
\item [$(i)$] there exist $\alpha, \rho>0$ such that $J_{\varepsilon}(u)\geq \alpha$ for any $u\in H^{s}_{\varepsilon}$ such that $\|u\|_{\varepsilon}=\rho$;
\item [$(ii)$] there exists $e\in H^{s}_{\varepsilon}$ with $\|e\|_{\varepsilon}>\rho$ such that $J_{\varepsilon}(e)<0$.
\varepsilonnd{enumerate}
\varepsilonnd{lemma}
\begin{proof}
By using $(g_1)$ and $(g_2)$ we know that for any $\varepsilonta>0$ there exists $C_{\varepsilonta}>0$ such that
\begin{equation}\label{g-estimate}
|g(\varepsilon x, t)|\leq \varepsilonta +C_{\varepsilonta} |t|^{\frac{q-2}{2}}.
\varepsilonnd{equation}
In view of Theorem \ref{HLS} and \varepsilonqref{g-estimate}, we can deduce that
\begin{align}\label{a1}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right| \leq C\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}},
\varepsilonnd{align}
where $\frac{1}{t}=\frac{1}{2}(2-\frac{\mu}{N})$. Since $2<q<\frac{2^{*}_{s}}{2}(2-\frac{\mu}{N})$ we have $t q\in (2, 2^{*}_{s})$ and by using Theorem \ref{Sembedding} we can see that
\begin{align}\label{a2}
\left(\int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q}\, dx)^{t}\right)^{\frac{2}{t}}\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}.
\varepsilonnd{align}
Putting together \varepsilonqref{a1} and \varepsilonqref{a2} we get
\begin{align*}
\left|\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2})\right)G(\varepsilon x, |u|^{2})\, dx\right|\leq C(\|u\|^{2}_{\varepsilon}+\|u\|^{q}_{\varepsilon})^{2}\leq C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}).
\varepsilonnd{align*}
Hence
$$
J(u)\geq \frac{1}{2}\|u\|^{2}_{\varepsilon}-C(\|u\|^{4}_{\varepsilon}+\|u\|^{2q}_{\varepsilon}),
$$
and recalling that $q>2$ we can infer that $(i)$ is satisfied.\\
Now, take a nonnegative function $u_{0}\in H^{s}(\mathbb{R}^{N}, \mathbb{R})\setminus\{0\}$ with compact support such that $supp(u_{0})\subset \Lambda_{\varepsilon}$. Then, by Lemma \ref{aux} we know that $u_{0}(x)e^{\imath A(0)\cdot x}\in H^{s}_{\varepsilon}\setminus\{0\}$. Set
$$
h(t)=\mathfrak{F}\left(\frac{t u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \mbox{ for } t>0,
$$
where
$$
\mathfrak{F}(u)=\frac{1}{4}\int_{\mathbb{R}^{N}} \left( \frac{1}{|x|^{\mu}}*F(|u|^{2}) \right) F(|u|^{2})\,dx.
$$
From $(f_3)$ we know that $F(t)\leq f(t)t$ for all $t>0$.
Then, being $G(\varepsilon x, |u_{0}|^{2})=F(|u_{0}|^{2})$, we deduce that
\begin{align}\label{a3}
\frac{h'(t)}{h(t)}\geq \frac{4}{t} \quad \forall t>0.
\varepsilonnd{align}
Integrating \varepsilonqref{a3} over $[1, t\|u_{0}\|_{\varepsilon}]$ with $t>\frac{1}{\|u_{0}\|_{\varepsilon}}$, we get
$$
\mathfrak{F}(t u_{0})\geq \mathfrak{F}\left(\frac{u_{0}}{\|u_{0}\|_{\varepsilon}}\right) \|u_{0}\|_{\varepsilon}^{4}t^{4}.
$$
Summing up
$$
J_{\varepsilon}(t u_{0})\leq C_{1} t^{2}-C_{2}t^{4} \mbox{ for } t>\frac{1}{\|u_{0}\|_{\varepsilon}}.
$$
Taking $e=t u_{0}$ with $t$ sufficiently large, we can see that $(ii)$ holds.
\varepsilonnd{proof}
\noindent
Denoting by $c_{\varepsilon}$ the mountain pass level of the functional $J_{\varepsilon}$ and recalling that $supp(u_{0})\subset \Lambda_{\varepsilon}$, we can find $\kappa>0$ independent of $\varepsilon, l, a$ such that
$$
c_{\varepsilon}= \inf_{u\in H^{s}_{\varepsilon}\setminus \{0\}}\max_{t\geq 0} J_{\varepsilon}(tu) <\kappa
$$
for all $\varepsilon>0$ small.
Now, let us define
$$
\B=\{u\in H^{s}(\mathbb{R}^{N}): \|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)\}
$$
and we set
$$
\tilde{K}_{\varepsilon}(u)(x)=\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u|^{2}).
$$
The next lemma is very useful because allows us to treat the convolution term as a bounded term.
\begin{lemma}\label{lemK}
Assume that $(f_1)$-$(f_3)$ hold and $2<q<\frac{2(N-\mu)}{N-2s}$. Then there exists $\varepsilonll_{0}>0$ such that
$$
\frac{\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}<\frac{1}{2} \mbox{ for any } \varepsilon>0.
$$
\varepsilonnd{lemma}
\begin{proof}
Let us prove that there exists $C_{0}>0$ such that
\begin{equation}\label{a6}
\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}\leq C_{0}.
\varepsilonnd{equation}
First of all, we can observe that
\begin{equation}\label{a5}
|G(\varepsilon x, |u|^{2})|\leq |F(|u|^{2})|\leq C(|u|^{2}+|u|^{q}) \mbox{ for all } \varepsilon>0.
\varepsilonnd{equation}
Hence, by using \varepsilonqref{a5}, we can see that
\begin{align}\label{a7}
|\tilde{K}_{\varepsilon}(u)(x)|
&\leq \Bigl| \int_{|x-y|\leq 1} \frac{F(|u|^{2})}{|x-y|^{\mu}} \,dy\Bigr|+\Bigl| \int_{|x-y|>1} \frac{F(|u|^{2})}{|x-y|^{\mu}} \,dy\Bigr| \nonumber\\
&\leq C \int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy+C \int_{\mathbb{R}^{N}} (|u|^{2}+|u|^{q})\, dy \nonumber\\
&\leq C \int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy+C
\varepsilonnd{align}
where in the last line we used Theorem \ref{Sembedding} and $\|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)$.\\
Now, we take
$$
t\in \Bigl(\frac{N}{N-\mu}, \frac{N}{N-2s}\Bigr] \mbox{ and } r\in \Bigl(\frac{N}{N-\mu}, \frac{2N}{q(N-2s)}\Bigr].
$$
By applying H\"older inequality, Theorem \ref{Sembedding} and $\|u\|^{2}_{\varepsilon}\leq 4(\kappa+1)$ we get
\begin{align}\label{a8}
\int_{|x-y|\leq 1} \frac{|u(y)|^{2}}{|x-y|^{\mu}}\, dy&\leq \Bigl(\int_{|x-y|\leq 1} |u|^{2t}\, dy \Bigr)^{\frac{1}{t}} \Bigl(\int_{|x-y|\leq 1} \frac{1}{|x-y|^{\frac{t\mu}{t-1}}}\, dy \Bigr)^{\frac{t-1}{t}}\nonumber \\
&\leq C_{*}(4(\kappa+1))^{2} \Bigl(\int_{\rho\leq 1} \rho^{N-1-\frac{t \mu}{t-1}}\, d\rho \Bigr)^{\frac{t-1}{t}}<\infty,
\varepsilonnd{align}
because of $N-1-\frac{t \mu}{t-1}>-1$.
In similar fashion we can prove
\begin{align}\label{a9}
\int_{|x-y|\leq 1} \frac{|u(y)|^{q}}{|x-y|^{\mu}}\, dy&\leq \Bigl(\int_{|x-y|\leq 1} |u|^{rq}\, dy \Bigr)^{\frac{1}{r}} \Bigl(\int_{|x-y|\leq 1} \frac{1}{|x-y|^{\frac{r\mu}{r-1}}}\, dy \Bigr)^{\frac{r-1}{r}}\nonumber \\
&\leq C_{*}(4(\kappa+1))^{q} \Bigl(\int_{\rho\leq 1} \rho^{N-1-\frac{r \mu}{r-1}}\, d\rho \Bigr)^{\frac{r-1}{r}}<\infty
\varepsilonnd{align}
in view of $N-1-\frac{r \mu}{r-1}>-1$.
Putting together \varepsilonqref{a8} and \varepsilonqref{a9} we obtain
$$
\int_{|x-y|\leq 1} \frac{|u(y)|^{2}+|u(y)|^{q}}{|x-y|^{\mu}}\, dy\leq C \mbox{ for all } x\in \mathbb{R}^{N}
$$
which together with \varepsilonqref{a7} implies \varepsilonqref{a6}.
Then we can find $\varepsilonll_{0}>0$ such that
$$
\frac{\sup_{u\in \B} \|\tilde{K}_{\varepsilon}(u)(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}\leq \frac{C_{0}}{\varepsilonll_{0}}< \frac{1}{2}.
$$
\varepsilonnd{proof}
\noindent
Let $\varepsilonll_{0}$ be as in Lemma \ref{lemK} and $a>0$ be the unique number such that
$$
f(a)=\frac{V_{0}}{\varepsilonll_{0}}.
$$
From now on we consider the penalized problem \varepsilonqref{Pe} with these choices.
\noindent
In what follows, we show that $J_{\varepsilon}$ verifies a local compactness condition.
\begin{lemma}\label{PSc}
$J_{\varepsilon}$ satisfies the $(PS)_{c}$ condition for all $c\in [c_{\varepsilon}, \kappa]$.
\varepsilonnd{lemma}
\begin{proof}
Let $(u_{n})$ be a Palais-Smale sequence at the level $c$, that is $J_{\varepsilon}(u_{n})\rightarrow c$ and $J_{\varepsilon}'(u_{n})\rightarrow 0$. Let us note that $(u_{n})$ is bounded and there exists $n_{0}\in \mathbb{N}$ such that $\|u_{n}\|^{2}_{\varepsilon}\leq 4(\kappa+1)$ for all $n\geq n_{0}$. Indeed, by using $(g_3)$ and Lemma \ref{lemK}, we can see that
\begin{align*}
c+o_{n}(1)\|u_{n}\|_{\varepsilon}\geq J_{\varepsilon}(u_{n})-\frac{1}{4}\langle J'_{\varepsilon}(u_{n}), u_{n}\rangle\geq \frac{1}{4} \|u_{n}\|^{2}_{\varepsilon}
\varepsilonnd{align*}
which implies the thesis.\\
Now, we divide the proof in two main steps.\\
{\bf Step $1$}: For any $\varepsilonta>0$ there exists $R=R_{\varepsilonta}>0$ such that
\begin{equation}\label{DF}
\limsup_{n\rightarrow \infty}\int_{\mathbb{R}^{N}\setminus B_{R}} \int_{\mathbb{R}^{N}} \frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}dx dy+\int_{\mathbb{R}^{N}\setminus B_{R}}V(\varepsilon x)|u_{n}|^{2}\, dx<\varepsilonta.
\varepsilonnd{equation}
Since $(u_{n})$ is bounded in $H^{s}_{\e}$, we may assume that $u_{n}\rightharpoonup u$ in $H^{s}_{\e}$ and $|u_{n}|\rightarrow |u|$ in $L^{r}_{loc}(\mathbb{R}^{N})$ for any $r\in [2, 2^{*}_{s})$.
Moreover, by Lemma \ref{lemK}, we can deduce that
\begin{equation}\label{Kbound}
\frac{\sup_{n\geq n_{0}}\|\tilde{K}_{\varepsilon}(u_{n})(x)\|_{L^{\infty}(\mathbb{R}^{N})}}{\varepsilonll_{0}}\leq \frac{1}{2}.
\varepsilonnd{equation}
Fix $R>0$ and let $\psi_{R}\in C^{\infty}(\mathbb{R}^{N})$ be a function such that $\psi_{R}=0$ in $B_{R/2}$, $\psi_{R}=1$ in $B_{R}^{c}$, $\psi_{R}\in [0, 1]$ and $|\nabla \varepsilonta_{R}|\leq C/R$.
Since $\langle J'_{\varepsilon}(u_{n}), \varepsilonta_{R}u_{n}\rangle =o_{n}(1)$ we have
\begin{align*}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})\overline{((u_{n}\varepsilonta_{R})(x)\!-\!(u_{n}\varepsilonta_{R})(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy \Bigr)\\
&+\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)\varepsilonta_{R} |u_{n}|^{2}\, dx=\int_{\mathbb{R}^{N}} \Bigl(\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u_{n}|^{2})\Bigr) g(\varepsilon x, |u_{n}|^{2})u_{n}\psi_{R}+o_{n}(1).
\varepsilonnd{align*}
Taking into account
\begin{align*}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}}\!\!\! \frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})\overline{((u_{n}\varepsilonta_{R})(x)\!-\!(u_{n}\varepsilonta_{R})(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}}{|x-y|^{N+2s}} dx dy \Bigr)\\
&=\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr)\\
&+\iint_{\mathbb{R}^{2N}} \!\!\!\varepsilonta_{R}(x)\frac{|u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}} dx dy,
\varepsilonnd{align*}
and choosing $R>0$ large enough such that $\Lambda_{\varepsilon}\subset B_{\frac{R}{2}}$, we can use $(g_3)$-$(ii)$ and \varepsilonqref{Kbound} to get
\begin{align}\label{PS1}
&\iint_{\mathbb{R}^{2N}} \varepsilonta_{R}(x)\frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}\, dx dy+\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)\varepsilonta_{R} |u_{n}|^{2}\, dx\nonumber\\
&\leq -\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr) \nonumber\\
&+\frac{1}{2}\int_{\mathbb{R}^{N}} V_{\varepsilon}(x) |u_{n}|^{2} \varepsilonta_{R} \, dx+o_{n}(1).
\varepsilonnd{align}
From the H\"older inequality and the boundedness of $(u_{n})$ in $H^{s}_{\e}$ it follows that
\begin{align}\label{PS2}
&\Bigl|\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \!\!\!\overline{u_{n}(y)}e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\frac{(u_{n}(x)\!-\!u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})(\varepsilonta_{R}(x)\!-\!\varepsilonta_{R}(y))}{|x-y|^{N+2s}} dx dy\Bigr)\Bigr| \nonumber\\
&\leq \Bigl(\iint_{\mathbb{R}^{2N}} \frac{|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}}{|x-y|^{N+2s}}dxdy \Bigr)^{\frac{1}{2}} \times \nonumber\\
&\quad \times \Bigl(\iint_{\mathbb{R}^{2N}} |\overline{u_{n}(y)}|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} dxdy\Bigr)^{\frac{1}{2}} \nonumber\\
&\leq C \Bigl(\iint_{\mathbb{R}^{2N}} |u_{n}(y)|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} \, dxdy\Bigr)^{\frac{1}{2}}.
\varepsilonnd{align}
By using Lemma 2.1 in \cite{A6} we can see that
\begin{equation}\label{PS3}
\lim_{R\rightarrow \infty}\limsup_{n\rightarrow \infty} \iint_{\mathbb{R}^{2N}} |u_{n}(y)|^{2}\frac{|\varepsilonta_{R}(x)-\varepsilonta_{R}(y)|^{2}}{|x-y|^{N+2s}} \, dxdy=0.
\varepsilonnd{equation}
Then, putting together \varepsilonqref{PS1}, \varepsilonqref{PS2} and \varepsilonqref{PS3} we can deduce that \varepsilonqref{DF} holds true.
\noindent
{\bf Step $2$}: Let us prove that $u_{n}\rightarrow u$ in $H^{s}_{\varepsilon}$ as $n\rightarrow \infty$.\\
Since $u_{n}\rightharpoonup u$ in $H^{s}_{\e}$ and $\langle J'_{\varepsilon}(u_{n}),u_{n}\rangle=\langle J'_{\varepsilon}(u_{n}),u\rangle=o_{n}(1)$
we can note that
$$
\|u_{n}\|^{2}_{\varepsilon}-\|u\|^{2}_{\varepsilon}=\|u_{n}-u\|^{2}_{\varepsilon}+o_{n}(1)=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx+o_{n}(1).
$$
Therefore, being $H^{s}_{\e}$ be a Hilbert space, it is enough to show that
$$
\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx=o_{n}(1).
$$
By Lemma \ref{lemK} we know that $|\tilde{K}_{\varepsilon}(u_{n})|\leq C$ for all $n\in \mathbb{N}$. Since $|u_{n}|\rightarrow |u|$ in $L^{r}(B_{R})$ for all $r\in [2, 2^{*}_{s})$ and $R>0$, we obtain
\begin{align}
\left|\int_{B_{R}} \tilde{K}_{\varepsilon}(u_{n})g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})dx\right|\leq C\int_{B_{R}} |g_{\varepsilon}(x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2})|dx\rightarrow 0.
\varepsilonnd{align}
By the Step $1$ and Theorem \ref{Sembedding}, for any $\varepsilonta>0$ there exists $R_{\varepsilonta}>0$ such that
$$
\limsup_{n\rightarrow \infty} \int_{\mathbb{R}^{N}\setminus B_{R}} \tilde{K}_{\varepsilon}(u_{n}) |g(\varepsilon x, |u_{n}|^{2})|u_{n}|^{2}| \, dx\leq C\varepsilonta.
$$
In similar way, from H\"older inequality, we can see that
$$
\limsup_{n\rightarrow \infty} \int_{\mathbb{R}^{N}\setminus B_{R}} \tilde{K}_{\varepsilon}(u_{n}) |g(\varepsilon x, |u_{n}|^{2})|u|^{2}| \, dx\leq C\varepsilonta.
$$
Taking into account the above limits we can infer that
$$
\lim_{n\rightarrow \infty} \int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n}) g(\varepsilon x, |u_{n}|^{2})(|u_{n}|^{2}-|u|^{2}) \, dx=0.
$$
This ends the proof of Lemma \ref{PSc}.
\varepsilonnd{proof}
\section{Concentration of solutions to \varepsilonqref{P}}
In this section we give the proof of the main result of this paper.
Firstly, we consider the limit problem associated with \varepsilonqref{R}, that is
\begin{equation}\label{APe}
(-\Delta)^{s} u + V_{0}u = \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)f(|u|^{2})u \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation}
and the corresponding energy functional $J_{0}: H^{s}_{0}\rightarrow \mathbb{R}$ given by
$$
J_{0}(u)=\frac{1}{2}\|u\|^{2}_{V_{0}}-\mathfrak{F}(u),
$$
where $H_{0}^{s}$ is the space $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ endowed with the norm
$$
\|u\|^{2}_{V_{0}}=[u]^{2}+\int_{\mathbb{R}^{N}} V_{0} u^{2}\,dx,
$$
and
$$
\mathfrak{F}(u)=\frac{1}{4}\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(|u|^{2})\right)F(|u|^{2})\, dx.
$$
As in the previous section, it is easy to see that $J_{0}$ has a mountain pass geometry and we denote by $c_{V_{0}}$ the mountain pass level of the functional $J_{0}$.
Let us introduce the Nehari manifold associated with (\ref{Pe}), that is
\begin{equation*}
\mathcal{N}_{\varepsilon}:= \{u\in H^{s}_{\e} \setminus \{0\} : \langle J_{\varepsilon}'(u), u \rangle =0\},
\varepsilonnd{equation*}
and we denote by $\mathcal{N}_{0}$ the Nehari manifold associated with \varepsilonqref{APe}.
It is standard to verify (see \cite{W}) that $c_{\varepsilon}$ can be characterized as
$$
c_{\varepsilon}=\inf_{u\in H^{s}_{\e}\setminus\{0\}} \sup_{t\geq 0} J_{\varepsilon}(t u)=\inf_{u\in \mathcal{N}_{\varepsilon}} J_{\varepsilon}(u).
$$
In the next result we stress an interesting relation between $c_{\varepsilon}$ and $c_{V_{0}}$.
\begin{lemma}\label{AMlem1}
The numbers $c_{\varepsilon}$ and $c_{V_{0}}$ satisfy the following inequality
$$
\limsup_{\varepsilon\rightarrow 0} c_{\varepsilon}\leq c_{V_{0}}.
$$
\varepsilonnd{lemma}
\begin{proof}
In view of Lemma $3.3$ in \cite{Apota}, there exists a ground state $w\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ to the autonomous problem \varepsilonqref{APe}, so that $J'_{0}(w)=0$ and $J_{0}(w)=c_{V_{0}}$. Moreover, we know that $w\in C^{0, \mu}(\mathbb{R}^{N})$ and $w>0$ in $\mathbb{R}^{N}$.
In what follows, we show that $w$ satisfies the following useful estimate:
\begin{equation}\label{remdecay}
0<w(x)\leq \frac{C}{|x|^{N+2s}} \mbox{ for large } |x|.
\varepsilonnd{equation}
By using $(f_1)$, $\lim_{|x|\rightarrow\infty}w(x)=0$ and the boundedness of the convolution term (see proof of Lemma \ref{lemK}) we can find $R>0$ such that $\left(\frac{1}{|x|^{\mu}}*F(w^{2})\right)f(w^{2})w\leq \frac{V_{0}}{2}w$ in $B_{R}^{c}$. In particular we have
\begin{equation}\label{BBMP1}
(-\Delta)^{s}w+\frac{V_{0}}{2}w=\left(\frac{1}{|x|^{\mu}}*F(w^{2})\right)f(w^{2})w-\left(V_{0}-\frac{V_{0}}{2}\right)w\leq 0 \mbox{ in } B_{R}^{c}.
\varepsilonnd{equation}
In view of Lemma $4.2$ in \cite{FQT} and by rescaling, we know that there exists a positive function $w_{1}$ and a constant $C_{1}>0$ such that for large $|x|>R$ it holds that $w_{1}(x)=C_{1}|x|^{-(N+2s)}$ and
\begin{equation}\label{BBMP2}
(-\Delta)^{s}w_{1}+\frac{V_{0}}{2}w_{1}\geq 0 \mbox{ in } B^{c}_{R}.
\varepsilonnd{equation}
Taking into account the continuity of $w$ and $w_{1}$ there exists $C_{2}>0$ such that $w_{2}(x)=w(x)-C_{2}w_{1}(x)\leq 0$ on $|x|=R$ (taking $R$ larger if necessary). Moreover, we can see that $(-\Delta)^{s}w_{2}+\frac{V_{0}}{2}w_{2}\leq 0$ for $|x|\geq R$ and by using the maximum principle we can infer that $w_{2}\leq 0$ in $B_{R}^{c}$, that is $w\leq C_{2}w_{1}$ in $B_{R}^{c}$. This fact implies that \varepsilonqref{remdecay} holds true.
Now, fix a cut-off function $\varepsilonta\in C^{\infty}_{c}(\mathbb{R}^{N}, [0,1])$ such that $\varepsilonta=1$ in a neighborhood of zero $B_{\frac{\delta}{2}}$ and $\supp(\varepsilonta)\subset B_{\delta}\subset \Lambda$ for some $\delta>0$.
Let us define $w_{\varepsilon}(x):=\varepsilonta_{\varepsilon}(x)w(x) e^{\imath A(0)\cdot x}$, with $\varepsilonta_{\varepsilon}(x)=\varepsilonta(\varepsilon x)$ for $\varepsilon>0$, and we observe that $|w_{\varepsilon}|=\varepsilonta_{\varepsilon}w$ and $w_{\varepsilon}\in H^{s}_{\e}$ in view of Lemma \ref{aux}. Let us prove that
\begin{equation}\label{limwr}
\lim_{\varepsilon\rightarrow 0}\|w_{\varepsilon}\|^{2}_{\varepsilon}=\|w\|_{0}^{2}\in(0, \infty).
\varepsilonnd{equation}
Clearly, $\int_{\mathbb{R}^{N}} V_{\varepsilon}(x)|w_{\varepsilon}|^{2}dx\rightarrow \int_{\mathbb{R}^{N}} V_{0} |w|^{2}dx$. Now, we show that
\begin{equation}\label{limwr*}
\lim_{\varepsilon\rightarrow 0}[w_{\varepsilon}]^{2}_{A_{\varepsilon}}=[w]^{2}.
\varepsilonnd{equation}
We note that, in view of Lemma $5$ in \cite{PP}, we have
\begin{equation}\label{PPlem}
[\varepsilonta_{\varepsilon} w]\rightarrow [w] \mbox{ as } \varepsilon\rightarrow 0.
\varepsilonnd{equation}
On the other hand
\begin{align*}
&[w_{\varepsilon}]_{A_{\varepsilon}}^{2}\nonumber \\
&=\iint_{\mathbb{R}^{2N}} \frac{|e^{\imath A(0)\cdot x}\varepsilonta_{\varepsilon}(x)w(x)-e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}e^{\imath A(0)\cdot y} \varepsilonta_{\varepsilon}(y)w(y)|^{2}}{|x-y|^{N+2s}} dx dy \nonumber \\
&=[\varepsilonta_{\varepsilon} w]^{2}
+\iint_{\mathbb{R}^{2N}} \frac{\varepsilonta_{\varepsilon}^2(y)w^2(y) |e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx dy\\
&+2\mathbb{R}e \iint_{\mathbb{R}^{2N}} \frac{(\varepsilonta_{\varepsilon}(x)w(x)-\varepsilonta_{\varepsilon}(y)w(y))\varepsilonta_{\varepsilon}(y)w(y)(1-e^{-\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)})}{|x-y|^{N+2s}} dx dy \\
&=: [\varepsilonta_{\varepsilon} w]^{2}+X_{\varepsilon}+2Y_{\varepsilon}.
\varepsilonnd{align*}
Taking into account
$|Y_{\varepsilon}|\leq [\varepsilonta_{\varepsilon} w] \sqrt{X_{\varepsilon}}$ and \varepsilonqref{PPlem}, we need to prove that $X_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ to deduce that \varepsilonqref{limwr*} holds true.
Let us observe that for $0<\beta<\alpha/({1+\alpha-s})$ we get
\begin{equation}\label{Ye}
\begin{split}
X_{\varepsilon}
&\leq \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|\geq\varepsilon^{-\beta}} \frac{|e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx\\
&+\int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|e^{\imath [A_{\varepsilon}(\frac{x+y}{2})-A(0)]\cdot (x-y)}-1|^{2}}{|x-y|^{N+2s}} dx \\
&=:X^{1}_{\varepsilon}+X^{2}_{\varepsilon}.
\varepsilonnd{split}
\varepsilonnd{equation}
Since $|e^{\imath t}-1|^{2}\leq 4$ and $w\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$, we can see that
\begin{equation}\label{Ye1}
X_{\varepsilon}^{1}\leq C \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{\varepsilon^{-\beta}}^\infty \rho^{-1-2s} d\rho\leq C \varepsilon^{2\beta s} \rightarrow 0.
\varepsilonnd{equation}
Now, by using $|e^{\imath t}-1|^{2}\leq t^{2}$ for all $t\in \mathbb{R}$, $A\in C^{0,\alpha}(\mathbb{R}^N,\mathbb{R}^N)$ for $\alpha\in(0,1]$, and $|x+y|^{2}\leq 2(|x-y|^{2}+4|y|^{2})$, we obtain
\begin{equation}\label{Ye2}
\begin{split}
X^{2}_{\varepsilon}&
\leq \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|A_{\varepsilon}\left(\frac{x+y}{2}\right)-A(0)|^{2} }{|x-y|^{N+2s-2}} dx \\
&\leq C\varepsilon^{2\alpha} \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{|x+y|^{2\alpha} }{|x-y|^{N+2s-2}} dx \\
&\leq C\varepsilon^{2\alpha} \left(\int_{\mathbb{R}^{N}} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{1 }{|x-y|^{N+2s-2-2\alpha}} dx\right.\\
&\qquad\qquad+ \left. \int_{\mathbb{R}^{N}} |y|^{2\alpha} w^{2}(y) dy \int_{|x-y|<\varepsilon^{-\beta}} \frac{1}{|x-y|^{N+2s-2}} dx\right) \\
&=: C\varepsilon^{2\alpha} (X^{2, 1}_{\varepsilon}+X^{2, 2}_{\varepsilon}).
\varepsilonnd{split}
\varepsilonnd{equation}
Then
\begin{equation}\label{Ye21}
X^{2, 1}_{\varepsilon}
= C \int_{\mathbb{R}^{N}} w^{2}(y) dy \int_0^{\varepsilon^{-\beta}} \rho^{1+2\alpha-2s} d\rho
\leq C\varepsilon^{-2\beta(1+\alpha-s)}.
\varepsilonnd{equation}
On the other hand, using \varepsilonqref{remdecay}, we can infer that
\begin{equation}\label{Ye22}
\begin{split}
X^{2, 2}_{\varepsilon}
&\leq C \int_{\mathbb{R}^{N}} |y|^{2\alpha} w^{2}(y) dy \int_0^{\varepsilon^{-\beta}}\rho^{1-2s} d\rho \\
&\leq C \varepsilon^{-2\beta(1-s)} \left[\int_{B_1(0)} w^{2}(y) dy + \int_{B_1^c(0)} \frac{1}{|y|^{2(N+2s)-2\alpha}} dy \right] \\
&\leq C \varepsilon^{-2\beta(1-s)}.
\varepsilonnd{split}
\varepsilonnd{equation}
Taking into account \varepsilonqref{Ye}, \varepsilonqref{Ye1}, \varepsilonqref{Ye2}, \varepsilonqref{Ye21} and \varepsilonqref{Ye22} we have $X_{\varepsilon}\rightarrow 0$, and then \varepsilonqref{limwr} holds.
Now, let $t_{\varepsilon}>0$ be the unique number such that
\begin{equation*}
J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})=\max_{t\geq 0} J_{\varepsilon}(t w_{\varepsilon}).
\varepsilonnd{equation*}
As a consequence, $t_{\varepsilon}$ satisfies
\begin{align}\label{AS1}
\|w_{\varepsilon}\|_{\varepsilon}^{2}&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon x, t_{\varepsilon}^{2} |w_{\varepsilon}|^{2})\right)g(\varepsilon x, t_{\varepsilon}^{2} |w_{\varepsilon}|^{2}) |w_{\varepsilon}|^{2}dx \nonumber \\
&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(t_{\varepsilon}^{2} |w_{\varepsilon}|^{2})\right) f(t_{\varepsilon}^{2} |w_{\varepsilon}|^{2}) |w_{\varepsilon}|^{2}dx
\varepsilonnd{align}
where we used $supp(\varepsilonta)\subset \Lambda$ and $g=f$ on $\Lambda$.\\
Let us prove that $t_{\varepsilon}\rightarrow 1$ as $\varepsilon\rightarrow 0$. Since $\varepsilonta=1$ in $B_{\frac{\delta}{2}}$, $w$ is a continuous positive function, and recalling that $f(t)$ and $F(t)/t$ are both increasing, we have
$$
\|w_{\varepsilon}\|_{\varepsilon}^{2}\geq \frac{F(t_{\varepsilon}^{2}\alpha_{0}^{2})}{\alpha_{0}^{2}} f(t_{\varepsilon}^{2}\alpha^{2}_{0})\int_{B_{\frac{\delta}{2}}}\int_{B_{\frac{\delta}{2}}}
\frac{|w(x)|^{2} |w(y)|^{2}}{|x-y|^{\mu}}dx dy,
$$
where $\alpha_{0}=\min_{\bar{B}_{\frac{\delta}{2}}} w>0$. \\
Let us prove that $t_{\varepsilon}\rightarrow t_{0}\in (0, \infty)$ as $\varepsilon\rightarrow 0$. Indeed, if $t_{\varepsilon}\rightarrow \infty$ as $\varepsilon\rightarrow 0$ then we can use $(f_3)$ to deduce that $\|w\|_{0}^{2}= \infty$ which gives a contradiction due to \varepsilonqref{limwr}.
When $t_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ we can use $(f_1)$ to infer that $\|w\|_{0}^{2}= 0$ which is impossible in view of \varepsilonqref{limwr}.\\
Then, taking the limit as $\varepsilon\rightarrow 0$ in \varepsilonqref{AS1} and using \varepsilonqref{limwr}, we can deduce that
\begin{equation}\label{AS2}
\|w\|_{0}^{2}=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*F(t_{0}^{2}|w|^{2})\right)f(t_{0}^{2} |w|^{2}) |w|^{2}dx.
\varepsilonnd{equation}
Since $w\in \mathcal{N}_{0}$ and using $(f_3)$, we obtain $t_{0}=1$. Hence, from the Dominated Convergence Theorem, we can see that $\lim_{\varepsilon\rightarrow 0} J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})=J_{0}(w)=c_{V_{0}}$.
Recalling that $c_{\varepsilon}\leq \max_{t\geq 0} J_{\varepsilon}(t w_{\varepsilon})=J_{\varepsilon}(t_{\varepsilon} w_{\varepsilon})$, we can infer that
$\limsup_{\varepsilon\rightarrow 0} c_{\varepsilon}\leq c_{V_{0}}$.
\varepsilonnd{proof}
\noindent
Arguing as in \cite{Apota}, we can deduce the following result for the autonomous problem:
\begin{lemma}\label{FS}
Let $(u_{n})\subset \mathcal{N}_{0}$ be a sequence satisfying $J_{0}(u_{n})\rightarrow c_{V_{0}}$. Then, up to subsequences, the following alternatives holds:
\begin{enumerate}
\item [$(i)$] $(u_{n})$ strongly converges in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$,
\item [$(ii)$] there exists a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that, up to a subsequence, $v_{n}(x)=u_{n}(x+\tilde{y}_{n})$ converges strongly in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
\varepsilonnd{enumerate}
In particular, there exists a minimizer for $c_{V_{0}}$.
\varepsilonnd{lemma}
\noindent
Now, we prove the following useful compactness result.
\begin{lemma}\label{prop3.3}
Let $\varepsilon_{n}\rightarrow 0$ and $(u_{n})\subset H^{s}_{\varepsilon_{n}}$ such that $J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}$ and $J'_{\varepsilon_{n}}(u_{n})=0$. Then there exists $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that $v_{n}(x)=|u_{n}|(x+\tilde{y}_{n})$ has a convergent subsequence in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$. Moreover, up to a subsequence, $y_{n}=\varepsilon_{n} \tilde{y}_{n}\rightarrow y_{0}$ for some $y_{0}\in \Lambda$ such that $V(y_{0})=V_{0}$.
\varepsilonnd{lemma}
\begin{proof}
Taking into account $\langle J'_{\varepsilon_{n}}(u_{n}), u_{n}\rangle=0$, $J_{\varepsilon_{n}}(u_{n})= c_{\varepsilon_{n}}$, Lemma \ref{AMlem1} and arguing as in Lemma \ref{PSc}, it is easy to see that $(u_{n})$ is bounded in $H^{s}_{\varepsilon_{n}}$ and $\|u_{n}\|^{2}_{\varepsilon_{n}}\leq 4(\kappa+1)$ for all $n\in \mathbb{N}$. Moreover, from Lemma \ref{DI}, we also know that $(|u_{n}|)$ is bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.\\
Let us prove that there exist a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ and constants $R>0$ and $\gamma>0$ such that
\begin{equation}\label{sacchi}
\liminf_{n\rightarrow \infty}\int_{B_{R}(\tilde{y}_{n})} |u_{n}|^{2} \, dx\geq \gamma>0.
\varepsilonnd{equation}
Otherwise, if \varepsilonqref{sacchi} does not hold, then for all $R>0$ we have
$$
\lim_{n\rightarrow \infty}\sup_{y\in \mathbb{R}^{N}}\int_{B_{R}(y)} |u_{n}|^{2} \, dx=0.
$$
From the boundedness $(|u_{n}|)$ and Lemma $2.2$ in \cite{FQT} we can see that $|u_{n}|\rightarrow 0$ in $L^{q}(\mathbb{R}^{N}, \mathbb{R})$ for any $q\in (2, 2^{*}_{s})$.
By using $(g_1)$-$(g_2)$ and Lemma \ref{lemK} we can deduce that
\begin{align}\label{glimiti}
\lim_{n\rightarrow \infty}\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|^{2} \,dx=0= \lim_{n\rightarrow \infty}\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) G(\varepsilon_{n} x, |u_{n}|^{2}) \, dx.
\varepsilonnd{align}
Since $\langle J'_{\varepsilon_{n}}(u_{n}), u_{n}\rangle=0$, we can use \varepsilonqref{glimiti} to deduce that $\|u_{n}\|_{\varepsilon_{n}}\rightarrow 0$ as $n\rightarrow \infty$. This gives a contradiction because $u_{n}\in \mathcal{N}_{\varepsilon_{n}}$ and by using $(g_1)$, $(g_2)$ and Lemma \ref{lemK} we can find $\alpha_{0}>0$ such that $\|u_{n}\|^{2}_{\varepsilon_{n}}\geq \alpha_{0}$ for all $n\in \mathbb{N}$.\\
Set $v_{n}(x)=|u_{n}|(x+\tilde{y}_{n})$. Then $(v_{n})$ is bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, and we may assume that
$v_{n}\rightharpoonup v\not\varepsilonquiv 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$.
Fix $t_{n}>0$ such that $\tilde{v}_{n}=t_{n} v_{n}\in \mathcal{N}_{0}$. By using Lemma \ref{DI}, we can see that
$$
c_{V_{0}}\leq J_{0}(\tilde{v}_{n})\leq \max_{t\geq 0}J_{\varepsilon_{n}}(tv_{n})= J_{\varepsilon_{n}}(u_{n})=c_{\varepsilon_{n}}
$$
which together with Lemma \ref{AMlem1} implies that $J_{0}(\tilde{v}_{n})\rightarrow c_{V_{0}}$. In particular, $\tilde{v}_{n}\nrightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
Since $(v_{n})$ and $(\tilde{v}_{n})$ are bounded in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and $\tilde{v}_{n}\nrightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, we obtain that $t_{n}\rightarrow t^{*}> 0$.
From the uniqueness of the weak limit, we can deduce that $\tilde{v}_{n}\rightharpoonup \tilde{v}=t^{*}v\not\varepsilonquiv 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$.
This together with Lemma \ref{FS} gives
\begin{equation}\label{elena}
\tilde{v}_{n}\rightarrow \tilde{v} \mbox{ in } H^{s}(\mathbb{R}^{N}, \mathbb{R}),
\varepsilonnd{equation}
and as a consequence $v_{n}\rightarrow v$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$.
Now, we set $y_{n}=\varepsilon_{n}\tilde{y}_{n}$. We aim to prove that $(y_{n})$ admits a subsequence, still denoted by $y_{n}$, such that $y_{n}\rightarrow y_{0}$ for some $y_{0}\in \Lambda$ such that $V(y_{0})=V_{0}$. Firstly, we prove that $(y_{n})$ is bounded. Assume by contradiction that, up to a subsequence, $|y_{n}|\rightarrow \infty$ as $n\rightarrow \infty$. Take $R>0$ such that $\Lambda \subset B_{R}(0)$. Since we may suppose that $|y_{n}|>2R$, we have that $|\varepsilon_{n}z+y_{n}|\geq |y_{n}|-|\varepsilon_{n}z|>R$ for any $z\in B_{R/\varepsilon_{n}}$.
Taking into account $(u_{n})\subset \mathcal{N}_{\varepsilon_{n}}$, $(V_{1})$, Lemma \ref{DI} and the change of variable $x\mapsto z+\tilde{y}_{n}$ we get
\begin{align*}
&[v_{n}]^{2}+\int_{\mathbb{R}^{N}} V_{0} v_{n}^{2}\, dx \\
&\leq C_{0}\int_{\mathbb{R}^{N}} g(\varepsilon_{n} z+y_{n}, |v_{n}|^{2}) |v_{n}|^{2} \, dz \\
&\leq C_{0}\int_{B_{\frac{R}{\varepsilon_{n}}}(0)} \tilde{f}(|v_{n}|^{2}) |v_{n}|^{2} \, dz+C_{0}\int_{\mathbb{R}^{N}\setminus B_{\frac{R}{\varepsilon_{n}}}(0)} f(|v_{n}|^{2}) |v_{n}|^{2} \, dz,
\varepsilonnd{align*}
where we used $u_{n}\in \B$ for all $n$ big enough and Lemma \ref{lemK}.
By using $v_{n}\rightarrow v$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ as $n\rightarrow \infty$ and $\tilde{f}(t)\leq \frac{V_{0}}{\varepsilonll_{0}}$ we obtain
$$
\min\left\{1, \frac{V_{0}}{2} \right\} \left([v_{n}]^{2}+\int_{\mathbb{R}^{N}} |v_{n}|^{2}\, dx\right)=o_{n}(1).
$$
Then $v_{n}\rightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$ and this is impossible. Therefore, $(y_{n})$ is bounded and we may assume that $y_{n}\rightarrow y_{0}\in \mathbb{R}^{N}$. If $y_{0}\notin \overline{\Lambda}$, we can argue as before to deduce that $v_{n}\rightarrow 0$ in $H^{s}(\mathbb{R}^{N}, \mathbb{R})$, which gives a contradiction. Therefore $y_{0}\in \overline{\Lambda}$, and in view of $(V_2)$, it is enough to verify that $V(y_{0})=V_{0}$ to conclude the proof of lemma. Assume by contradiction that $V(y_{0})>V_{0}$.
Then, by using (\ref{elena}), Fatou's Lemma, the invariance of $\mathbb{R}^{N}$ by translations, Lemma \ref{DI} and Lemma \ref{AMlem1}, we get
\begin{align*}
&c_{V_{0}}=J_{0}(\tilde{v})<\frac{1}{2}[\tilde{v}]^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}} V(y_{0})\tilde{v}^{2} \, dx-\mathfrak{F}(\tilde{v}) \\
&\leq \liminf_{n\rightarrow \infty}\Bigl[\frac{1}{2}[\tilde{v}_{n}]^{2}+\frac{1}{2}\int_{\mathbb{R}^{N}} V(\varepsilon_{n}z+y_{n}) |\tilde{v}_{n}|^{2} \, dz-\mathfrak{F}(\tilde{v}_{n}) \Bigr] \\
&\leq \liminf_{n\rightarrow \infty}\Bigl[\frac{t_{n}^{2}}{2}[|u_{n}|]^{2}+\frac{t_{n}^{2}}{2}\int_{\mathbb{R}^{N}} V(\varepsilon_{n}z) |u_{n}|^{2} \, dz-\mathfrak{F}(t_{n} u_{n}) \Bigr] \\
&\leq \liminf_{n\rightarrow \infty} J_{\varepsilon_{n}}(t_{n} u_{n}) \leq \liminf_{n\rightarrow \infty} J_{\varepsilon_{n}}(u_{n})\leq c_{V_{0}}
\varepsilonnd{align*}
which gives a contradiction.
\varepsilonnd{proof}
\noindent
The next lemma will be fundamental to prove that the solutions of \varepsilonqref{Pe} are also solutions of the original problem \varepsilonqref{P}. We will use a suitable variant of the Moser iteration argument \cite{Moser}.
\begin{lemma}\label{moser}
Let $\varepsilon_{n}\rightarrow 0$ and $u_{n}\in H^{s}_{\varepsilon_{n}}$ be a solution to \varepsilonqref{Pe}.
Then $v_{n}=|u_{n}|(\cdot+\tilde{y}_{n})$ satisfies $v_{n}\in L^{\infty}(\mathbb{R}^{N},\mathbb{R})$ and there exists $C>0$ such that
$$
\|v_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C \mbox{ for all } n\in \mathbb{N},
$$
where $\tilde{y}_{n}$ is given by Lemma \ref{prop3.3}.
Moreover
$$
\lim_{|x|\rightarrow \infty} v_{n}(x)=0 \mbox{ uniformly in } n\in \mathbb{N}.
$$
\varepsilonnd{lemma}
\begin{proof}
For any $L>0$ we define $u_{L,n}:=\min\{|u_{n}|, L\}\geq 0$ and we set $v_{L, n}=u_{L,n}^{2(\beta-1)}u_{n}$ where $\beta>1$ will be chosen later.
Taking $v_{L, n}$ as test function in (\ref{Pe}) we can see that
\begin{align}\label{conto1FF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{((u_{n}u_{L,n}^{2(\beta-1)})(x)-(u_{n}u_{L,n}^{2(\beta-1)})(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber \\
&=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{n}) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|^{2}u_{L,n}^{2(\beta-1)} \,dx-\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x) |u_{n}|^{2} u_{L,n}^{2(\beta-1)} \, dx.
\varepsilonnd{align}
Let us observe that
\begin{align*}
&\mathbb{R}e\left[(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})\overline{(u_{n}u_{L,n}^{2(\beta-1)}(x)-u_{n}u_{L,n}^{2(\beta-1)}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}\right] \\
&=\mathbb{R}e\Bigl[|u_{n}(x)|^{2}v_{L}^{2(\beta-1)}(x)-u_{n}(x)\overline{u_{n}(y)} u_{L,n}^{2(\beta-1)}(y)e^{-\imath A(\frac{x+y}{2})\cdot (x-y)}\\
&\quad-u_{n}(y)\overline{u_{n}(x)} u_{L,n}^{2(\beta-1)}(x) e^{\imath A(\frac{x+y}{2})\cdot (x-y)} +|u_{n}(y)|^{2}u_{L,n}^{2(\beta-1)}(y) \Bigr] \\
&\geq (|u_{n}(x)|^{2}u_{L,n}^{2(\beta-1)}(x)-|u_{n}(x)||u_{n}(y)|u_{L,n}^{2(\beta-1)}(y) \\
&\quad -|u_{n}(y)||u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)+|u_{n}(y)|^{2}u^{2(\beta-1)}_{L,n}(y) \\
&=(|u_{n}(x)|-|u_{n}(y)|)(|u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)-|u_{n}(y)|u_{L,n}^{2(\beta-1)}(y)),
\varepsilonnd{align*}
which implies that
\begin{align}\label{realeF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{((u_{n}u_{L,n}^{2(\beta-1)})(x)-(u_{n}u_{L,n}^{2(\beta-1)})(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber\\
&\geq \iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)}{|x-y|^{N+2s}} (|u_{n}(x)|u_{L,n}^{2(\beta-1)}(x)-|u_{n}(y)|u_{L,n}^{2(\beta-1)}(y))\, dx dy.
\varepsilonnd{align}
As in \cite{Apota}, for all $t\geq 0$, we define
\begin{equation*}
\gamma(t)=\gamma_{L, \beta}(t)=t t_{L}^{2(\beta-1)}
\varepsilonnd{equation*}
where $t_{L}=\min\{t, L\}$.
Since $\gamma$ is an increasing function we have
\begin{align*}
(a-b)(\gamma(a)- \gamma(b))\geq 0 \quad \mbox{ for any } a, b\in \mathbb{R}.
\varepsilonnd{align*}
Let
\begin{equation*}
\Lambda(t)=\frac{|t|^{2}}{2} \quad \mbox{ and } \quad \Gamma(t)=\int_{0}^{t} (\gamma'(\tau))^{\frac{1}{2}} d\tau.
\varepsilonnd{equation*}
Since
\begin{equation}\label{Gg}
\Lambda'(a-b)(\gamma(a)-\gamma(b))\geq |\Gamma(a)-\Gamma(b)|^{2} \mbox{ for any } a, b\in\mathbb{R},
\varepsilonnd{equation}
we get
\begin{align}\label{Gg1}
|\Gamma(|u_{n}(x)|)- \Gamma(|u_{n}(y)|)|^{2} \leq (|u_{n}(x)|- |u_{n}(y)|)((|u_{n}|u_{L,n}^{2(\beta-1)})(x)- (|u_{n}|u_{L,n}^{2(\beta-1)})(y)).
\varepsilonnd{align}
Putting together \varepsilonqref{realeF} and \varepsilonqref{Gg1}, we can see that
\begin{align}\label{conto1FFF}
&\mathbb{R}e\Bigl(\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber\\
&\quad \times \overline{(u_{n}u_{L,n}^{2(\beta-1)}(x)\!-\!u_{n}u_{L,n}^{2(\beta-1)}(y)e^{\imath A(\frac{x+y}{2})\cdot (x-y)})} dx dy\Bigr) \nonumber\\
&\geq [\Gamma(|u_{n}|)]^{2}.
\varepsilonnd{align}
Since $\Gamma(|u_{n}|)\geq \frac{1}{\beta} |u_{n}| u_{L,n}^{\beta-1}$ and using the fractional Sobolev embedding $\mathcal{D}^{s,2}(\mathbb{R}^{N}, \mathbb{R})\subset L^{2^{*}_{s}}(\mathbb{R}^{N}, \mathbb{R})$ (see \cite{DPV}), we can infer that
\begin{equation}\label{SS1}
[\Gamma(|u_{n}|)]^{2}\geq S_{*} \|\Gamma(|u_{n}|)\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}\geq \left(\frac{1}{\beta}\right)^{2} S_{*}\||u_{n}| u_{L,n}^{\beta-1}\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}.
\varepsilonnd{equation}
Then \varepsilonqref{conto1FF}, \varepsilonqref{conto1FFF} and \varepsilonqref{SS1} yield
\begin{align}\label{BMS}
&\left(\frac{1}{\beta}\right)^{2} S_{*}\||u_{n}| u_{L,n}^{\beta-1}\|^{2}_{L^{2^{*}_{s}}(\mathbb{R}^{N})}+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|^{2}u_{L,n}^{2(\beta-1)} dx\nonumber\\
&\leq \int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon_{n}}(u_{n}) g(\varepsilon_{n}x, |u_{n}|^{2}) |u_{n}|^{2} u_{L,n}^{2(\beta-1)} dx.
\varepsilonnd{align}
By $(g_1)$ and $(g_2)$, we know that for any $\xi>0$ there exists $C_{\xi}>0$ such that
\begin{equation}\label{SS2}
g(x, t^{2})t^{2}\leq \xi |t|^{2}+C_{\xi}|t|^{2^{*}_{s}} \mbox{ for any } (x, t)\in \mathbb{R}^{N}\times\mathbb{R}.
\varepsilonnd{equation}
Hence, using \varepsilonqref{BMS}, \varepsilonqref{SS2}, $u_{n}\in \B$, Lemma \ref{lemK} and choosing $\xi>0$ sufficiently small, we can see that
\begin{equation}\label{simo1}
\|w_{L,n}\|_{L^{2^{*}_{s}}(\mathbb{R}^{N})}^{2}\leq C \beta^{2} \int_{\mathbb{R}^{N}} |u_{n}|^{q}u_{L,n}^{2(\beta-1)},
\varepsilonnd{equation}
for some $C$ independent of $\beta$, $L$ and $n$. Here we set $w_{L,n}:=|u_{n}| u_{L,n}^{\beta-1}$.
Arguing as in the proof of Lemma $5.1$ in \cite{Apota} we can see that
\begin{equation}\label{UBu}
\|u_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq K \mbox{ for all } n\in \mathbb{N}.
\varepsilonnd{equation}
Moreover, by interpolation, $(|u_{n}|)$ strongly converges in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in (2, \infty)$, and in view of the growth assumptions on $g$, also $g(\varepsilon_{n} x, |u_{n}|^{2})|u_{n}|$ strongly converges in the same Lebesgue spaces. \\
In what follows, we show that $|u_{n}|$ is a weak subsolution to
\begin{equation}\label{Kato0}
\left\{
\begin{array}{ll}
(-\Delta)^{s}v+V(\varepsilon_{n} x) v=\left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, v^{2})\right)g(\varepsilon_{n} x, v^{2})v &\mbox{ in } \mathbb{R}^{N} \\
v\geq 0 \quad \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{array}
\right.
\varepsilonnd{equation}
Fix $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ such that $\varphi\geq 0$, and we take $\psi_{\delta, n}=\frac{u_{n}}{u_{\delta, n}}\varphi$ as test function in \varepsilonqref{Pe}, where $u_{\delta,n}=\sqrt{|u_{n}|^{2}+\delta^{2}}$ for $\delta>0$. We note that $\psi_{\delta, n}\in H^{s}_{\varepsilon_{n}}$ for all $\delta>0$ and $n\in \mathbb{N}$.
Indeed, it is clear that
$$
\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x) |\psi_{\delta,n}|^{2} dx\leq \int_{\supp(\varphi)} V(\varepsilon_{n} x)\varphi^{2} dx<\infty.
$$
Now, we show that $[\psi_{\delta, n}]_{A_{\varepsilon}}$ is finite. Let us observe that
\begin{align*}
\psi_{\delta,n}(x)-\psi_{\delta,n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}&=\Bigl(\frac{u_{n}(x)}{u_{\delta,n}(x)}\Bigr)\varphi(x)-\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(y)}\Bigr)\varphi(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\\
&=\Bigl[\Bigl(\frac{u_{n}(x)}{u_{\delta,n}(x)}\Bigr)-\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}\Bigr)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\Bigr]\varphi(x) \\
&+\Bigl[\varphi(x)-\varphi(y)\Bigr] \Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}\Bigr) e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \\
&+\Bigl(\frac{u_{n}(y)}{u_{\delta,n}(x)}-\frac{u_{n}(y)}{u_{\delta,n}(y)}\Bigr)\varphi(y) e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}.
\varepsilonnd{align*}
Then, by using $|z+w+k|^{2}\leq 4(|z|^{2}+|w|^{2}+|k|^{2})$ for all $z,w,k\in \mathbb{C}$, $|e^{\imath t}|=1$ for all $t\in \mathbb{R}$, $u_{\delta,n}\geq \delta$, $|\frac{u_{n}}{u_{\delta,n}}|\leq 1$, \varepsilonqref{UBu} and $|\sqrt{|z|^{2}+\delta^{2}}-\sqrt{|w|^{2}+\delta^{2}}|\leq ||z|-|w||$ for all $z, w\in \mathbb{C}$, we obtain that
\begin{align*}
&|\psi_{\delta,n}(x)-\psi_{\delta,n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2} \\
&\leq \frac{4}{\delta^{2}}|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}\|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} +\frac{4}{\delta^{2}}|\varphi(x)-\varphi(y)|^{2} \|u_{n}\|^{2}_{L^{\infty}(\mathbb{R}^{N})} \\
&+\frac{4}{\delta^{4}} \|u_{n}\|^{2}_{L^{\infty}(\mathbb{R}^{N})} \|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} |u_{\delta,n}(y)-u_{\delta,n}(x)|^{2} \\
&\leq \frac{4}{\delta^{2}}|u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}|^{2}\|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} +\frac{4K^{2}}{\delta^{2}}|\varphi(x)-\varphi(y)|^{2} \\
&+\frac{4K^{2}}{\delta^{4}} \|\varphi\|^{2}_{L^{\infty}(\mathbb{R}^{N})} ||u_{n}(y)|-|u_{n}(x)||^{2}.
\varepsilonnd{align*}
Since $u_{n}\in H^{s}_{\varepsilon_{n}}$, $|u_{n}|\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ (by Lemma \ref{DI}) and $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$, we conclude that $\psi_{\delta,n}\in H^{s}_{\varepsilon_{n}}$.
Then we get
\begin{align}\label{Kato1}
&\mathbb{R}e\Bigl[\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber \\
&\quad \times \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)\!-\!\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr) dx dy\Bigr]
+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx\nonumber\\
&=\int_{\mathbb{R}^{N}} \Bigl(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\Bigr) g(\varepsilon_{n} x, |u_{n}|^{2})\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx.
\varepsilonnd{align}
Now, we aim to pass to the limit as $\delta\rightarrow 0$ in \varepsilonqref{Kato1} to deduce that \varepsilonqref{Kato0} holds true.
Since $\mathbb{R}e(z)\leq |z|$ for all $z\in \mathbb{C}$ and $|e^{\imath t}|=1$ for all $t\in \mathbb{R}$, we have
\begin{align}\label{alves1}
&\mathbb{R}e\Bigl[(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}) \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)-\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr)\Bigr] \nonumber\\
&=\mathbb{R}e\Bigl[\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-\frac{u_{n}(x)\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \nonumber \\
&-\frac{u_{n}(y)\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)}\Bigr] \nonumber \\
&\geq \Bigl[\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(x)|\frac{|u_{n}(y)|}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(y)|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\varphi(x) \Bigr].
\varepsilonnd{align}
Let us note that
\begin{align}\label{alves2}
&\frac{|u_{n}(x)|^{2}}{u_{\delta,n}(x)}\varphi(x)+\frac{|u_{n}(y)|^{2}}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(x)|\frac{|u_{n}(y)|}{u_{\delta,n}(y)}\varphi(y)-|u_{n}(y)|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\varphi(x) \nonumber\\
&= \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(x)-\frac{|u_{n}(y)|}{u_{\delta,n}(y)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&=\Bigl[\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(x)-\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)\varphi(y)\Bigr] \nonumber\\
&\quad +\Bigl(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \Bigr) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&=\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y)) \nonumber \\
&\quad +\Bigl(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \Bigr) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y) \nonumber\\
&\geq \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))
\varepsilonnd{align}
where in the last inequality we used the fact that
$$
\left(\frac{|u_{n}(x)|}{u_{\delta,n}(x)}-\frac{|u_{n}(y)|}{u_{\delta,n}(y)} \right) (|u_{n}(x)|-|u_{n}(y)|)\varphi(y)\geq 0
$$
because
$$
h(t)=\frac{t}{\sqrt{t^{2}+\delta^{2}}} \mbox{ is increasing for } t\geq 0 \quad \mbox{ and } \quad \varphi\geq 0 \mbox{ in }\mathbb{R}^{N}.
$$
Since
\begin{align*}
&\frac{|\frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))|}{|x-y|^{N+2s}}\\
&\quad \leq \frac{||u_{n}(x)|-|u_{n}(y)||}{|x-y|^{\frac{N+2s}{2}}} \frac{|\varphi(x)-\varphi(y)|}{|x-y|^{\frac{N+2s}{2}}}\in L^{1}(\mathbb{R}^{2N}),
\varepsilonnd{align*}
and $\frac{|u_{n}(x)|}{u_{\delta,n}(x)}\rightarrow 1$ a.e. in $\mathbb{R}^{N}$ as $\delta\rightarrow 0$,
we can use \varepsilonqref{alves1}, \varepsilonqref{alves2} and the Dominated Convergence Theorem to deduce that
\begin{align}\label{Kato2}
&\limsup_{\delta\rightarrow 0} \mathbb{R}e\Bigl[\iint_{\mathbb{R}^{2N}} \frac{(u_{n}(x)-u_{n}(y)e^{\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)})}{|x-y|^{N+2s}} \times \nonumber \\
&\quad \times \Bigl(\frac{\overline{u_{n}(x)}}{u_{\delta,n}(x)}\varphi(x)-\frac{\overline{u_{n}(y)}}{u_{\delta,n}(y)}\varphi(y)e^{-\imath A_{\varepsilon}(\frac{x+y}{2})\cdot (x-y)} \Bigr) dx dy\Bigr] \nonumber\\
&\geq \limsup_{\delta\rightarrow 0} \iint_{\mathbb{R}^{2N}} \frac{|u_{n}(x)|}{u_{\delta,n}(x)}(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y)) \frac{dx dy}{|x-y|^{N+2s}} \nonumber\\
&=\iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))}{|x-y|^{N+2s}} dx dy.
\varepsilonnd{align}
On the other hand, from the Dominated Convergence Theorem (we note that $\frac{|u_{n}|^{2}}{u_{\delta, n}}\leq |u_{n}|$, $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ and $\tilde{K}_{\varepsilon}(u_{n})$ is bounded in view of Lemma \ref{lemK}) we can infer that
\begin{equation}\label{Kato3}
\lim_{\delta\rightarrow 0} \int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx=\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|\varphi dx
\varepsilonnd{equation}
and
\begin{align}\label{Kato4}
&\lim_{\delta\rightarrow 0} \int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2})\frac{|u_{n}|^{2}}{u_{\delta,n}}\varphi dx\nonumber \\
&=\int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|\varphi dx.
\varepsilonnd{align}
Taking into account \varepsilonqref{Kato1}, \varepsilonqref{Kato2}, \varepsilonqref{Kato3} and \varepsilonqref{Kato4} we can see that
\begin{align*}
&\iint_{\mathbb{R}^{2N}} \frac{(|u_{n}(x)|-|u_{n}(y)|)(\varphi(x)-\varphi(y))}{|x-y|^{N+2s}} dx dy+\int_{\mathbb{R}^{N}} V(\varepsilon_{n} x)|u_{n}|\varphi dx\\
&\leq \int_{\mathbb{R}^{N}} \left(\frac{1}{|x|^{\mu}}*G(\varepsilon_{n} x, |u_{n}|^{2})\right) g(\varepsilon_{n} x, |u_{n}|^{2}) |u_{n}|\varphi dx
\varepsilonnd{align*}
for any $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N}, \mathbb{R})$ such that $\varphi\geq 0$. Then $|u_{n}|$ is a weak subsolution to \varepsilonqref{Kato0}.
By using $(V_{1})$, $u_{n}\in \B$ for all $n$ big enough, and Lemma \ref{lemK}, it is clear that $v_{n}=|u_{n}|(\cdot+\tilde{y}_{n})$ solves
\begin{equation}\label{Pkat}
(-\Delta)^{s} v_{n} + V_{0}v_{n}\leq C_{0} g(\varepsilon_{n} x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n} \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{equation}
Let us denote by $z_{n}\in H^{s}(\mathbb{R}^{N}, \mathbb{R})$ the unique solution to
\begin{equation}\label{US}
(-\Delta)^{s} z_{n} + V_{0}z_{n}= C_{0}g_{n} \mbox{ in } \mathbb{R}^{N},
\varepsilonnd{equation}
where
$$
g_{n}:=g(\varepsilon_{n} x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n}\in L^{r}(\mathbb{R}^{N}, \mathbb{R}) \quad \forall r\in [2, \infty].
$$
Since \varepsilonqref{UBu} yields $\|v_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C$ for all $n\in \mathbb{N}$, by interpolation we know that $v_{n}\rightarrow v$ strongly converges in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ for all $r\in (2, \infty)$, for some $v\in L^{r}(\mathbb{R}^{N}, \mathbb{R})$, and by the growth assumptions on $f$, we can see that also $g_{n}\rightarrow f(v^{2})v$ in $L^{r}(\mathbb{R}^{N}, \mathbb{R})$ and $\|g_{n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq C$ for all $n\in \mathbb{N}$.
Since $z_{n}=\mathcal{K}*(C_{0}g_{n})$, where $\mathcal{K}$ is the Bessel kernel (see \cite{FQT}), we can argue as in \cite{AM} to infer that $|z_{n}(x)|\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly with respect to $n\in \mathbb{N}$.
On the other hand, $v_{n}$ satisfies \varepsilonqref{Pkat} and $z_{n}$ solves \varepsilonqref{US} so a simple comparison argument shows that $0\leq v_{n}\leq z_{n}$ a.e. in $\mathbb{R}^{N}$ and for all $n\in \mathbb{N}$. This means that $v_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$.
\varepsilonnd{proof}
\noindent
At this point we have all ingredients to give the proof of Theorem \ref{thm1}.
\begin{proof}
By using Lemma \ref{prop3.3} we can find a sequence $(\tilde{y}_{n})\subset \mathbb{R}^{N}$ such that $\varepsilon_{n}\tilde{y}_{n}\rightarrow y_{0}$ for some $y_{0} \in \Lambda$ such that $V(y_{0})=V_{0}$.
Then there exists $r>0$ such that, for some subsequence still denoted by itself, we have $B_{r}(\tilde{y}_{n})\subset \Lambda$ for all $n\in \mathbb{N}$.
Hence $B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n})\subset \Lambda_{\varepsilon_{n}}$ for all $n\in \mathbb{N}$, which gives
$$
\mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}\subset \mathbb{R}^{N} \setminus B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n}) \mbox{ for any } n\in \mathbb{N}.
$$
In view of Lemma \ref{moser}, we know that there exists $R>0$ such that
$$
v_{n}(x)<a \mbox{ for } |x|\geq R \mbox{ and } n\in \mathbb{N},
$$
where $v_{n}(x)=|u_{\varepsilon_{n}}|(x+ \tilde{y}_{n})$.
Then $|u_{\varepsilon_{n}}(x)|<a$ for any $x\in \mathbb{R}^{N}\setminus B_{R}(\tilde{y}_{n})$ and $n\in \mathbb{N}$. Moreover, there exists $\nu \in \mathbb{N}$ such that for any $n\geq \nu$ and $r/\varepsilon_{n}>R$ it holds
$$
\mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}\subset \mathbb{R}^{N} \setminus B_{\frac{r}{\varepsilon_{n}}}(\tilde{y}_{n})\subset \mathbb{R}^{N}\setminus B_{R}(\tilde{y}_{n}),
$$
which gives $|u_{\varepsilon_{n}}(x)|<a$ for any $x\in \mathbb{R}^{N}\setminus \Lambda_{\varepsilon_{n}}$ and $n\geq \nu$. \\
Therefore, there exists $\varepsilon_{0}>0$ such that problem \varepsilonqref{R} admits a nontrivial solution $u_{\varepsilon}$ for all $\varepsilon\in (0, \varepsilon_{0})$. Then $H^{s}_{\e}at{u}_{\varepsilon}(x)=u_{\varepsilon}(x/\varepsilon)$ is a solution to (\ref{P}).
Finally, we study the behavior of the maximum points of $|u_{\varepsilon_{n}}|$. In view of $(g_1)$, there exists $\gamma\in (0,a)$ such that
\begin{align}\label{4.18HZ}
g(\varepsilon x, t^{2})t^{2}\leq \frac{V_{0}}{\varepsilonll_{0}}t^{2}, \mbox{ for all } x\in \mathbb{R}^{N}, |t|\leq \gamma.
\varepsilonnd{align}
Using a similar discussion above, we can take $R>0$ such that
\begin{align}\label{4.19HZ}
\|u_{\varepsilon_{n}}\|_{L^{\infty}(B^{c}_{R}(\tilde{y}_{n}))}<\gamma.
\varepsilonnd{align}
Up to a subsequence, we may also assume that
\begin{align}\label{4.20HZ}
\|u_{\varepsilon_{n}}\|_{L^{\infty}(B_{R}(\tilde{y}_{n}))}\geq \gamma.
\varepsilonnd{align}
Otherwise, if \varepsilonqref{4.20HZ} does not hold, then $\|u_{\varepsilon_{n}}\|_{L^{\infty}(\mathbb{R}^{N})}< \gamma$ and by using $J_{\varepsilon_{n}}'(u_{\varepsilon_{n}})=0$, \varepsilonqref{4.18HZ}, Lemma \ref{DI} and
$$
\left\|\frac{1}{|x|^{\mu}}*G(\varepsilon x, |u_{n}|^{2})\right\|_{L^{\infty}(\mathbb{R}^{N})}<C_{0},
$$
we have
\begin{align*}
[|u_{\varepsilon_{n}}|]^{2}+\int_{\mathbb{R}^{N}}V_{0}|u_{\varepsilon_{n}}|^{2}dx\leq \|u_{\varepsilon_{n}}\|^{2}_{\varepsilon_{n}}&=\int_{\mathbb{R}^{N}} \tilde{K}_{\varepsilon}(u_{\varepsilon_{n}}) g_{\varepsilon_{n}}(x, |u_{\varepsilon_{n}}|^{2})|u_{\varepsilon_{n}}|^{2}\,dx\\
&\leq \frac{C_{0}V_{0}}{\varepsilonll_{0}}\int_{\mathbb{R}^{N}}|u_{\varepsilon_{n}}|^{2}\, dx
\varepsilonnd{align*}
and being $\frac{C_{0}}{\varepsilonll_{0}}<\frac{1}{2}$ we deduce that $\||u_{\varepsilon_{n}}|\|_{H^{s}(\mathbb{R}^{N})}=0$ which is impossible.
From \varepsilonqref{4.19HZ} and \varepsilonqref{4.20HZ}, it follows that the maximum points $p_{n}$ of $|u_{\varepsilon_{n}}|$ belong to $B_{R}(\tilde{y}_{n})$, that is $p_{n}=\tilde{y}_{n}+q_{n}$ for some $q_{n}\in B_{R}$. Since $H^{s}_{\e}at{u}_{n}(x)=u_{\varepsilon_{n}}(x/\varepsilon_{n})$ is a solution to \varepsilonqref{P}, we can see that the maximum point $\varepsilonta_{\varepsilon_{n}}$ of $|H^{s}_{\e}at{u}_{n}|$ is given by $\varepsilonta_{\varepsilon_{n}}=\varepsilon_{n}\tilde{y}_{n}+\varepsilon_{n}q_{n}$. Taking into account $q_{n}\in B_{R}$, $\varepsilon_{n}\tilde{y}_{n}\rightarrow y_{0}$ and $V(y_{0})=V_{0}$ and the continuity of $V$, we can infer that
$$
\lim_{n\rightarrow \infty} V(\varepsilonta_{\varepsilon_{n}})=V_{0}.
$$
Finally, we give a decay estimate for $|H^{s}_{\e}at{u}_{n}|$. We follow some arguments used in \cite{A6}.\\
Invoking Lemma $4.3$ in \cite{FQT}, we can find a function $w$ such that
\begin{align}\label{HZ1}
0<w(x)\leq \frac{C}{1+|x|^{N+2s}},
\varepsilonnd{align}
and
\begin{align}\label{HZ2}
(-\Delta)^{s} w+\frac{V_{0}}{2}w\geq 0 \mbox{ in } \mathbb{R}^{N}\setminus B_{R_{1}}
\varepsilonnd{align}
for some suitable $R_{1}>0$. Using Lemma \ref{moser}, we know that $v_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$, so there exists $R_{2}>0$ such that
\begin{equation}\label{hzero}
h_{n}=C_{0}g(\varepsilon_{n}x+\varepsilon_{n}\tilde{y}_{n}, v_{n}^{2})v_{n}\leq \frac{C_{0}V_{0}}{\varepsilonll_{0}}v_{n}\leq \frac{V_{0}}{2}v_{n} \mbox{ in } B_{R_{2}}^{c}.
\varepsilonnd{equation}
Let us denote by $w_{n}$ the unique solution to
$$
(-\Delta)^{s}w_{n}+V_{0}w_{n}=h_{n} \mbox{ in } \mathbb{R}^{N}.
$$
Then $w_{n}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $n\in \mathbb{N}$, and by comparison $0\leq v_{n}\leq w_{n}$ in $\mathbb{R}^{N}$. Moreover, in view of \varepsilonqref{hzero}, it holds
\begin{align*}
(-\Delta)^{s}w_{n}+\frac{V_{0}}{2}w_{n}=h_{n}-\frac{V_{0}}{2}w_{n}\leq 0 \mbox{ in } B_{R_{2}}^{c}.
\varepsilonnd{align*}
Take $R_{3}=\max\{R_{1}, R_{2}\}$ and we define
\begin{align}\label{HZ4}
a=\inf_{B_{R_{3}}} w>0 \mbox{ and } \tilde{w}_{n}=(b+1)w-a w_{n}.
\varepsilonnd{align}
where $b=\sup_{n\in \mathbb{N}} \|w_{n}\|_{L^{\infty}(\mathbb{R}^{N})}<\infty$.
We aim to prove that
\begin{equation}\label{HZ5}
\tilde{w}_{n}\geq 0 \mbox{ in } \mathbb{R}^{N}.
\varepsilonnd{equation}
Let us note that
\begin{align}
&\lim_{|x|\rightarrow \infty} \tilde{w}_{n}(x)=0 \mbox{ uniformly in } n\in \mathbb{N}, \label{HZ0N} \\
&\tilde{w}_{n}\geq ba+w-ba>0 \mbox{ in } B_{R_{3}} \label{HZ0},\\
&(-\Delta)^{s} \tilde{w}_{n}+\frac{V_{0}}{2}\tilde{w}_{n}\geq 0 \mbox{ in } \mathbb{R}^{N}\setminus B_{R_{3}} \label{HZ00}.
\varepsilonnd{align}
We argue by contradiction, and we assume that there exists a sequence $(\bar{x}_{j, n})\subset \mathbb{R}^{N}$ such that
\begin{align}\label{HZ6}
\inf_{x\in \mathbb{R}^{N}} \tilde{w}_{n}(x)=\lim_{j\rightarrow \infty} \tilde{w}_{n}(\bar{x}_{j, n})<0.
\varepsilonnd{align}
From (\ref{HZ0N}) it follows that $(\bar{x}_{j, n})$ is bounded, and, up to subsequence, we may assume that there exists $\bar{x}_{n}\in \mathbb{R}^{N}$ such that $\bar{x}_{j, n}\rightarrow \bar{x}_{n}$ as $j\rightarrow \infty$.
In view of (\ref{HZ6}) we can see that
\begin{align}\label{HZ7}
\inf_{x\in \mathbb{R}^{N}} \tilde{w}_{n}(x)= \tilde{w}_{n}(\bar{x}_{n})<0.
\varepsilonnd{align}
By using the minimality of $\bar{x}_{n}$ and the integral representation formula for the fractional Laplacian \cite{DPV}, we can see that
\begin{align}\label{HZ8}
(-\Delta)^{s}\tilde{w}_{n}(\bar{x}_{n})=\frac{C(N, s)}{2} \int_{\mathbb{R}^{N}} \frac{2\tilde{w}_{n}(\bar{x}_{n})-\tilde{w}_{n}(\bar{x}_{n}+\xi)-\tilde{w}_{n}(\bar{x}_{n}-\xi)}{|\xi|^{N+2s}} d\xi\leq 0.
\varepsilonnd{align}
Putting together (\ref{HZ0}) and (\ref{HZ6}), we have $\bar{x}_{n}\in \mathbb{R}^{N}\setminus B_{R_{3}}$.
This fact combined with (\ref{HZ7}) and (\ref{HZ8}) yields
$$
(-\Delta)^{s} \tilde{w}_{n}(\bar{x}_{n})+\frac{V_{0}}{2}\tilde{w}_{n}(\bar{x}_{n})<0,
$$
which gives a contradiction in view of (\ref{HZ00}).
As a consequence (\ref{HZ5}) holds true, and by using (\ref{HZ1}) and $v_{n}\leq w_{n}$ we can deduce that
\begin{align*}
0\leq v_{n}(x)\leq w_{n}(x)\leq \frac{(b+1)}{a}w(x)\leq \frac{\tilde{C}}{1+|x|^{N+2s}} \mbox{ for all } n\in \mathbb{N}, x\in \mathbb{R}^{N},
\varepsilonnd{align*}
for some constant $\tilde{C}>0$. Recalling the definition of $v_{n}$, we can obtain that
\begin{align*}
|H^{s}_{\e}at{u}_{n}(x)|&=\left|u_{\varepsilon_{n}}\left(\frac{x}{\varepsilon_{n}}\right)\right|=v_{n}\left(\frac{x}{\varepsilon_{n}}-\tilde{y}_{n}\right) \\
&\leq \frac{\tilde{C}}{1+|\frac{x}{\varepsilon_{n}}-\tilde{y}_{\varepsilon_{n}}|^{N+2s}} \\
&=\frac{\tilde{C} \varepsilon_{n}^{N+2s}}{\varepsilon_{n}^{N+2s}+|x- \varepsilon_{n} \tilde{y}_{\varepsilon_{n}}|^{N+2s}} \\
&\leq \frac{\tilde{C} \varepsilon_{n}^{N+2s}}{\varepsilon_{n}^{N+2s}+|x-\varepsilonta_{\varepsilon_{n}}|^{N+2s}}.
\varepsilonnd{align*}
\varepsilonnd{proof}
\noindent {\bf Acknowledgements.}
The author thanks Claudianor O. Alves and Hoai-Minh Nguyen for delightful and pleasant discussions about the results of this work.
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\varepsilonnd{document} |
\begin{document}
\title{Quantum Dissipation and Decoherence via Interaction with
Low-Dimensional Chaos: a Feynman-Vernon Approach}
\author{M.V.S. Bonan\c{c}a and M.A.M. de Aguiar}
\affiliation{Instituto de F\'isica 'Gleb Wataghin',
Universidade Estadual de Campinas, \\
Caixa Postal 6165, 13083-970 Campinas, S\~ao Paulo, Brazil}
\begin{abstract}
We study the effects of dissipation and decoherence induced on a
harmonic oscillator by the coupling to a chaotic system with two
degrees of freedom. Using the Feynman-Vernon approach and treating
the chaotic system semiclassically we show that the effects of the
low dimensional chaotic environment are in many ways similar to
those produced by thermal baths. The classical correlation and
response functions play important roles in both classical and
quantum formulations. Our results are qualitatively similar to the
high temperature regime of the Caldeira-Leggett model.
\end{abstract}
\maketitle
\section{Introduction.}
The relation between chaos and the phenomena of quantum dissipation
and decoherence has attracted a lot of attention in the last ten
years
\cite{cohen1,cohen2,cohen3,wilkinson,berry1,jarzynski,tulio,zurek1,zurek2,zurek3,cohen4,paz}.
The problem considered in most works involves the weak interaction
of a chaotic system with an external oscillator. Various points of
view have been considered by different authors. One approach
concentrates on the chaotic system itself, focusing on the
dissipation and treating the external oscillator as a time dependent
parameter that perturbs the chaotic system
\cite{cohen1,cohen2,cohen3}. The basic assumption is that the
external system is slow and sufficiently heavy not to be affected
significantly by the coupling
\cite{wilkinson,berry1,jarzynski,tulio}. The effect on the chaotic
system, on the other hand, is that of an adiabatic perturbation.
Under these conditions, a dissipative force, acting on the external
system, may result.
A different view of the same problem focus on the semiclassical
limit of chaotic systems. It has been shown
\cite{zurek1,zurek2,zurek3,cohen4,paz} that the coupling of a
chaotic system with an external environment, represented implicitly
by a small diffusion constant in the classical and quantum versions
of the Focker-Planck equation, leads to a very close correspondence
between the classical and quantum evolutions. The coupling causes
destruction of quantum interference and, at the same time, it washes
out the fine structures of the classical distributions, bringing the
two dynamics together.
In this paper we consider the chaotic system and the external
oscillator explicitly, as a single globally conservative Hamiltonian
system. We use the Feynman-Vernon approach to trace out the chaotic
system variables and construct an effective dynamics for the
oscillator, in close analogy with the treatment of the Brownian
motion considered by Caldeira and Leggett
\cite{caldeira1,caldeira2}. While we focus on the oscillator,
looking at dissipation and decoherence, the effects of the coupling
on the chaotic system are also taken into account consistently. This
characteristic of the Feynman-Vernon method turns out to be very
important in this problem, since both the oscillator and the chaotic
system have small number of degrees of freedom and are both affected
by the mutual interaction. However, whereas the treatment of
Caldeira and Leggett is phenomenological, in the sense that the
spectral properties of the reservoir are not derived from its
Hamiltonian, the case of a small chaotic environment has to be
treated dynamically. In other words, dissipation and decoherence
have to come out directly from correlations and response functions.
Our purpose here is to understand under what conditions a chaotic
system with only two degrees of freedom can produce dissipation and
decoherence, phenomena usually related to many body thermal baths.
In a previous paper \cite{bonanca} we have considered the
interaction of an oscillator with a chaotic system from a classical
point of view. We showed that the effects of the oscillator on the
environment cannot be neglected. Here we consider the quantum
version of the same problem, assuming the chaotic system to be in
the semiclassical regime.
In treatment of the Brownian motion by Caldeira and Leggett, the
degrees of freedom playing the role of the environment are averaged
with a canonical ensemble, since the reservoir is kept at the
constant temperature. Here, since the environment is small, the
microcanonical ensemble is more adequate. A similar approach was
recently considered by Esposito and Gaspard using random matrix
theory to model the chaotic environment
\cite{esposito1,esposito2,esposito3}.
The paper is organized as follows: in Sec.II, we review some aspects
of the classical formulation that are useful for the quantum
analysis. In Sec.III, we present the quantum formulation using
Feynman-Vernon approach \cite{feynman1}. The formal development
leads to quantum correlation and response functions, that we
calculate semiclassically. In Sec.IV, we consider two basic
applications: first, the propagation of an Gaussian state, where we
characterize quantum dissipation. Second, we calculate the evolution
of a superposition of two Gaussian states, focusing on the
decoherence due to the chaotic environment. In section V we present
our conclusions.
\section{Classical Formulation}
In this section we describe the behavior of a system of interest
coupled to a small chaotic environment from the classical point of
view. Although the formalism presented here can be extended to more
general systems, we particularize right away to the case of a
harmonic oscillator interacting with the so called Nelson system.
The discussion outlined here is a summary of the detailed results
presented in ref.\cite{bonanca} (see also
\cite{wilkinson,berry1,jarzynski,tulio}). The Hamiltonian of the
system is given by
\begin{eqnarray}
H = H_o(z) + H_c(x,y) + V_I(x,z), \label{eq1}
\end{eqnarray}
where
\begin{equation}
H_o(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2},
\end{equation}
represents the system of interest,
\begin{equation}
V_I(x,z) = \gamma xz,
\end{equation}
is the interaction potential and
\begin{equation}
H_c(x,y) = \frac{p_x^2}{2} + \frac{p_y^2}{2} + \left(y - \frac{x^2}
{2}\right)^2 + 0.1\frac{x^2}{2}
\end{equation}
is the chaotic Hamiltonian, known as Nelson system (NS)
\cite{baranger}. The NS exhibits soft chaos and is fairly regular
for $E_c \,_{\sim}^<\,0.05$, strongly chaotic for $E_c \,_{\sim}^>\,
0.3$ and mixed for intermediate values of the energy.
In order to investigate the situation where $H_c$ plays the role of
an external environment for the oscillator, we assume that detailed
information about the chaotic system is not available. If the
environment were modeled by a heat bath, the only macroscopic
relevant information would be its temperature. In the present case
we assume that the only information available is the initial energy
$E_c(0)$ of the chaotic system. For the oscillator this implies that
only averages of its observables (over the chaotic system variables)
are accessible.
When the coupling between the chaotic system and the oscillator is
turned on, the overall conserved energy flows from one system to the
other. The oscillator's energy, in particular, fluctuates as a
function of the time for each specific trajectory. The oscillator's
average energy is calculated by taking an ensemble of initial
conditions uniformly distributed over the chaotic energy surface
$E_c(0)$. For the oscillator we fix only one initial condition,
which we choose to be $z(0) = 0$ and $p_z(0) = \sqrt{2mE_o (0)}$.
The microcanonical ensemble of chaotic initial conditions plus the
fixed oscillator's initial condition are propagated numerically and,
at each instant, $H_o$ is calculated for each trajectory and its
average value is computed. We have shown in \cite{bonanca} that the
oscillator's average energy $\langle E_o (t)\rangle$ tends to a
constant value for long times, indicating a 'thermalization' of the
systems.
The short time behavior of $\langle E_o(t)\rangle$ can be obtained
from the Linear Response Theory \cite{kubo}. From the equations of
motion for $z$ and $p_z$ we find
\begin{eqnarray}
z(t)=z_d (t)-\frac{\gamma}{m}\int_0^t
\mathrm{d}s\Gamma(t-s)x(s),\label{eq3} \\
p_z (t) = p_{z_d} (t)-\gamma\int_0^t \mathrm{d}s\chi(t-s)x(s), \label{eq4}
\end{eqnarray}
where $z_d (t)$ and $p_{z_d} (t)$ are the decoupled solutions, given
by,
\begin{equation}
z_d (t)=\frac {p_z (0)}{m\omega_0}\sin{(\omega_0t)} \qquad p_{z_d}
(t)=p_z (0)\cos{(\omega_0t)}\;,
\end{equation}
where $\Gamma(t-s) = \sin{[\omega_0(t-s)]}/\omega_0$ and $\chi(t-s)
= \cos{[\omega_0 (t-s)]}$.
Thus,
\begin{eqnarray}
\label{eq5}
\langle p_z^2 (t)\rangle = p^2_{z_d}(t)-2\gamma
p_{z_d} (t)\int_0^t
\mathrm{d}s\chi(t-s)\langle x(s)\rangle+\nonumber \\
\gamma^2\int_0^t\mathrm{d}s\int_0^t\mathrm{d}u
\chi(t-s)\chi(t-u)\langle x(s)x(u)\rangle
\end{eqnarray}
and
\begin{eqnarray}
\label{eq6} \langle z^2 (t)\rangle=z^2_{d}(t)-\frac{2\gamma}{m}
z_d (t)\int_0^t
\mathrm{d}s\Gamma(t-s)\langle x(s)\rangle+\nonumber \\
\frac{\gamma^2}{m^2}\int_0^t\mathrm{d}s\int_0^t\mathrm{d}u \Gamma(t-s)
\Gamma(t-u)\langle x(s)x(u)\rangle.
\end{eqnarray}
The oscillator's average energy can now be obtained from
\begin{equation}
\langle E_o (t)\rangle=\frac{\langle p_z^2 (t)\rangle}{2m}+\frac{m
\omega_0^2 \langle z^2 (t)\rangle}{2} \label{eq2}
\end{equation}
Equations (\ref{eq5}) and (\ref{eq6}) show that we need $\langle
x(t)\rangle$ and $\langle x(0)x(t)\rangle$ in order to calculate
$\langle E_o(t)\rangle$. The calculation of such averages involve
the distribution function $\rho(q,p;t)$ whose initial value is
$\rho(q,p;0) = \delta(H_c(q,p)-E_c(0))/\Sigma(E_c(0))$, with
$\Sigma(E_c(0))=\int \mathrm{d}q\mathrm{d}p\, \delta
(H_c(q,p)-E_c(0))$. Here we are using $q=(x,y)$ and $p=(p_x,p_y)$
for the coordinates and momenta of the chaotic system. If the
chaotic system were isolated, $\rho$ would be an invariant
distribution and $\rho(q,p;t)=\rho(q,p;0)$. The coupling, however,
causes the value of $H_c(q(t),p(t))$ to fluctuate in time,
distorting the energy surface $H_c=E_c(0)$. Linear Response Theory
provides the first order corrections to this distribution in the
limit of weak coupling \cite{kubo}. Keeping in (\ref{eq5}) and
(\ref{eq6}) only terms up to order $\gamma^2$, we find
\begin{eqnarray}
\langle x(t)\rangle=\langle x(t)\rangle_e -
\gamma\int_0^t\mathrm{d}s \phi_{xx} (t-s)z(s) \label{eq7}
\end{eqnarray}
and
\begin{eqnarray}
\langle x(0)x(t)\rangle=\langle x(0)x(t)\rangle_e, \label{eq8}
\end{eqnarray}
where $\langle A(q,p)\rangle_e=\int\mathrm{d}q\mathrm{d}p\,A(q,p)
\rho_e(q,p)$ and $\rho_e=\delta(H_c-E_c(0))/\Sigma(E_c(0))$ is the
microcanonical distribution of the isolated chaotic system.
$\phi_{xx} (t)$ is the response function, given by \cite{kubo}
\begin{eqnarray}
\phi_{xx}(t)=\langle \{x(0),x(t)\}\rangle_e=\int\int\mathrm{d}V\,
\rho_e\,\{x(0),x(t)\} \label{eq9}
\end{eqnarray}
where $\mathrm{d}V = \mathrm{d}x(0) \mathrm{d}y(0) \mathrm{d}
p_x(0)\mathrm{d}p_y(0)$ and $\{.,.\}$ is the Poisson bracket with
respect to the initial conditions. Since $H_c(-x)=H_c(x)$, $\langle
x(t)\rangle_e=0$. Substituting (\ref{eq7}) and (\ref{eq8}) into
(\ref{eq5}) and (\ref{eq6}) we obtain
\begin{equation}
\begin{array}{ll}
\langle p^2_z (t)\rangle &=
p^2_{z_d}(t)+2\gamma^2 p_{z_d}(t)\int_0^t\mathrm{d}s\chi(t-s)
\int_0^s\mathrm{d}u\phi_{xx}(s-u)z_d(u) \\
& + \gamma^2\int_0^t \mathrm{d}s\int_0^t
\mathrm{d}u\chi(t-s)\chi(t-u) \langle x(s)x(u)\rangle_e,
\label{eq10}
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ll}
\langle z^2(t)\rangle & =
z^2_d(t)+\frac{2\gamma^2}{m}z_d(t)\int_0^t\mathrm{d}s
\Gamma(t-s)\int_0^s\mathrm{d}u\phi_{xx}(s-u)z_d(u) \\
&+\frac{\gamma^2}{m^2}\int_0^t \mathrm{d}s \int_0^t
\mathrm{d}u\Gamma(t-s)\Gamma(t-u)\langle x(s)x(u) \rangle_e.
\label{eq11}
\end{array}
\end{equation}
Eqs. (\ref{eq10}) and (\ref{eq11}) show that all the influence of
the chaotic system is contained in the functions $\langle
x(0)x(t)\rangle_e$ and $\phi_{xx}(t)$. For NS, the response function
is given by \cite{bonanca}
\begin{equation}
\phi_{xx}(t)=\frac{2}{E_c(0)}\langle p_x(0)x(t)\rangle_e.
\label{eq-resp}
\end{equation}
The correlation functions $\langle p_x(0)x(t)\rangle_e$ and $\langle
x(0)x(t)\rangle_e$ are obtained numerically with a fixed time step
symplectic integration algorithm \cite{forest} applied to the
isolated chaotic system. Fig. 1 shows the numerical correlation
functions for $E_c(0)=0.38$. These numerical results can be well
fitted by the expressions
\begin{equation}
\begin{array}{ll}
\langle x(0)x(t)\rangle_e &=\sigma e^{-\alpha t} \cos{\omega t},\\
\langle p_x (0) x(t)\rangle_e &=\mu e^{-\beta t} \sin{\Omega t},
\label{eq12}
\end{array}
\end{equation}
with decay rates $\alpha=0.0418$ and $\beta=0.0456$, amplitudes
$\sigma=1.865$ and $\mu=0.409$ and frequencies of oscillation
$\omega=0.1963$ and $\Omega=0.2043$ with $\chi^2\,\sim\,10^{-4}$
(see Fig.1). Notice that the exponents $\alpha$ and $\beta$ and the
frequencies $\omega$ and $\Omega$ are very similar. If the
interacting system were integrable, these functions would exhibit
quasi-periodic oscillations.
Considering the expressions (\ref{eq12}) and assuming
$\Omega\approx\omega$ and $\beta\approx\alpha$, we obtain the
following result for $\langle E_o(t)\rangle$
\begin{eqnarray}
\langle E_o(t)\rangle = E_o(0)+\frac{\gamma^2}{m}(B+At+f(t)+g(t)),
\label{eq13}
\end{eqnarray}
where $B$ is a constant, $f(t)$ is an oscillatory function and
$g(t)$ is proportional to $e^{-\alpha t}$. The important result is
the coefficient $A$
\begin{equation}
A=4\mu\omega\alpha \frac{\left[\frac{\sigma}{4\mu\omega}
(\omega^2_0+\omega^2+\alpha^2)-\frac{E_o(0)}
{E_c(0)}\right]}{[(\omega_0-\omega)^2+\alpha^2][(\omega_0+
\omega)^2+\alpha^2]}. \label{eq14}
\end{equation}
For fixed oscillator frequency $\omega_0$ and a given chaotic
energy shell $E_c(0)$ (and, consequently, for given $\sigma$,
$\mu$, $\omega$ and $\alpha$), the ratio $E_o(0)/E_c(0)$ is the
responsible for the average increase or decrease of $\langle
E_o(t)\rangle$. The short time equilibrium in the energy flow is
given by the condition $A=0$, or
\begin{equation}
\frac{E_o(0)}{E_c(0)}=\frac{\sigma}{4\mu\omega}(\omega^2_0+\omega^2
+\alpha^2) \;. \label{eq15}
\end{equation}
The equation of motion of $z(t)$ under the average effect of the
interaction with the chaotic system can also be written in terms of
the response function as
\begin{equation}
\langle\ddot{z}(t)\rangle+\omega^2_0 \langle z(t)\rangle
=-\frac{\gamma}{m}\langle x(t)\rangle
=\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, \phi_{xx} (t-s)
\langle z(s)\rangle.
\end{equation}
Integrating by parts yields
\begin{eqnarray}
\langle\ddot{z}(t)\rangle+\left(\omega^2_0-\frac{\gamma^2
F(0)}{m}\right) \langle z(t)\rangle+
\frac{\gamma^2}{m}\int_0^t\mathrm{d}s \, F(t-s) \langle
\dot{z}(s)\rangle+ \frac{\gamma^2}{m}z(0)F(t)=0, \label{eq16}
\end{eqnarray}
where
\begin{equation}
F(t-s) =\int\mathrm{d}s\,\phi_{xx}(t-s)=\frac{2\mu
e^{-\alpha(t-s)}}{E_c(0)(\alpha^2+\omega^2)}\{\omega
\cos{[\omega(t-s)]} +\alpha\sin{[\omega(t-s)]}\}. \label{eq17}
\end{equation}
Eq.(\ref{eq16}) shows that the interaction produces a harmonic
correction to the original potential, a dissipative term with memory
and an external force proportional to $z(0)$. The choice $z(0)=0$
simplifies (\ref{eq16}) and turns it into an average Langevin
equation.
Fig.2 shows a comparison between the numerically calculated `bare'
oscillator energy $\langle E_o(t)\rangle$, where
\begin{equation}
E_o(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2},
\end{equation}
the `re-normalized' oscillator energy $\langle E_{or}(t)\rangle$,
where
\begin{equation}
E_{or}(z) = \frac{p_z^2}{2m} + \frac{m\omega_0^2 z^2}{2}
-\frac{\gamma^2}{2}F(0)z^2, \label{eq18}
\end{equation}
and the expression (\ref{eq13}) without the oscillating term $f(t)$.
We have chosen $\gamma$ and $m$ so that $\omega^2_0 - \gamma^2
F(0)/m>0$. We also have chosen $\omega_0$ so that
$e^{-\alpha/\omega_0}\approx 10^{-4}$ and $g(t)$ decreases very
fast. In this case only the linear and the oscillating terms in
Eq.(\ref{eq13}) are important. We have subtracted the oscillating
part of Eq.(\ref{eq13}) in Fig.2 to highlight the linear increase or
decrease in the average energy. In the time scale of Fig.2, which
corresponds to several periods of the decoupled oscillator, the
linear behavior describes very well the numerical results. Fig.2(b)
shows the equilibrium situation according to Eq.(\ref{eq15}). Notice
that $F(t)$ decays very fast in the time scale $1/\omega_0$, leading
to the dissipative force in Eq.(\ref{eq16}).
In the next sections we consider the quantum counterpart of these
classical calculations. The chaotic system will be treated
semiclassically and the quantum versions of the response and
correlation functions will play important roles.
\section{Quantum Formulation}
\subsection{The Feynman-Vernon Approach}
In this section we describe the dynamics of the coupled oscillator
from a quantum point of view. In order to do that we need, like in
the classical case, a systematic way to eliminate the detailed
information we don't need about the chaotic system. We will do that
using the Feynman-Vernon approach \cite{feynman1}. Because of the
non-linear chaotic system, we will not be able to perform exact
calculations. Instead, we will resort to semiclassical
approximations.
We consider the quantum version of the full Hamiltonian,
Eq.(\ref{eq1}), and again we denote by $q=(x,y)$ the pair of
coordinates of the chaotic system. The density matrix operator can
be written as
\begin{eqnarray}
\hat{\rho}(T)=|\psi(T)\rangle \langle\psi(T)| = e^{-i\hat{H}T/\hbar}
|\psi(0)\rangle \langle\psi(0)|e^{i\hat{H}T/\hbar} \nonumber
\end{eqnarray}
where $\psi(T)$ is the wave function of the whole system. In the
position representation
\begin{eqnarray}
\lefteqn{\rho(z(T),q(T),z'(T),q'(T))=\langle z,q|\psi(T)\rangle
\langle\psi(T)|z',q'\rangle} \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{d}q(0)\mathrm{d}q'(0)
\langle z,q|e^{-i\hat{H}T/\hbar}|z(0),q(0)\rangle\langle z(0),q(0)|
\psi(0)\rangle \times \nonumber \\
& &\langle\psi(0)|z'(0),q'(0)\rangle\langle z'(0),q'(0)|
e^{i\hat{H}T/\hbar}|z',q'\rangle \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{d}q(0)\mathrm{d}q'(0)
K(z(T),q(T),z(0),q(0))\psi(z(0),q(0)) \times \nonumber \\
& &K^*(z'(T),q'(T),z'(0),q'(0))\psi^*(z'(0),q'(0)),
\end{eqnarray}
where the propagators can be written in terms of Feynman path
integrals as \cite{feynman2}
\begin{eqnarray}
K(z(T),q(T),z(0),q(0))=\int\mathrm{D}z(t)\mathrm{D}q(t)\exp{\left[
\frac{i}{\hbar}S[z(t),q(t)]\right]},
\end{eqnarray}
with action
\begin{eqnarray}
S[z(t),q(t)]&=& \int_0^T\mathrm{d}t
(L_0(z(t))+L_c(q(t))+L_I(z(t),q(t))) \nonumber \\
&\equiv &S_o+S_c+S_I.
\end{eqnarray}
Thus,
\begin{eqnarray}
\lefteqn{\rho(z(T),q(T),z'(T),q'(T))=} \nonumber \\
&=&\int\mathrm{d}z(0)\mathrm{d}z'(0)
\mathrm{d}q(0)\mathrm{d}q'(0)\mathrm{D}z(t)\mathrm{D}z'(t)\mathrm{D}q(t)
\mathrm{D}q'(t)\nonumber \\
&
&\exp\left[\frac{i}{\hbar}\left(S[z(t),q(t)]-S[z'(t),q'(t)]\right)\right]
\rho(z(0),q(0),z'(0),q'(0)),
\end{eqnarray}
where
\begin{equation}
\rho(z(0),q(0),z'(0),q'(0))=\psi(z(0),q(0))\psi^*(z'(0),q'(0))
\end{equation}
is the initial state. As usual we use $z(T)$ and $q(T)$ just as
labels for the positions $z$ and $q$ at $T$. $z(T)$ and $q(T)$ are
not functions of $T$.
We assume that the initial state can be written as
\begin{equation}
\rho(z(0),q(0),z'(0),q'(0))=\rho_o(z(0),z'(0))\rho_c(q(0),q'(0))
\end{equation}
and define the reduced density matrix by
\begin{equation}
\rho_o(z(T),z'(T))=\int\mathrm{d}q(T)\rho(z(T),q(T),z'(T),q(T)).
\end{equation}
We obtain
\begin{eqnarray}
\lefteqn{\rho_o(z(T),z'(T))=\int\mathrm{d}z(0)\mathrm{d}z'(0)\mathrm{D}z(t)
\mathrm{D}z'(t)} \nonumber \\
& &\left\{\int\mathrm{d}q(0)\mathrm{d}q'(0)\mathrm{d} q(T)
\mathrm{d}q'(T)\mathrm{D}q(t)
\mathrm{D}q'(t)\delta(q(T)-q'(T))\right.
\nonumber \\
& &\exp{\left[\frac{i}{\hbar}(S_c[q(t)]-S_c[q'(t)]+S_I[z(t),
q(t)] -S_I[z'(t),q'(t)\right]} \nonumber\\
& &\rho_c(q(0),q'(0)) \bigg\} \exp{\left[\frac{i}
{\hbar}\left(S_o[z(t)]-S_o[z'(t)]\right)\right]\rho_o(z(0),z'(0))},
\end{eqnarray}
which can be written as
\begin{equation}
\rho_o(z(T),z'(T))=\int\mathrm{d}z(0)\mathrm{d}z'(0)J(z(T),z'(T),z(0),
z'(0))\rho_o(z(0),z'(0)), \label{eq19}
\end{equation}
where
\begin{equation}
J(z(T),z'(T),z(0),z'(0))=\int\mathrm{D}z(t)\mathrm{D}z'(t)\mathcal{F}
[z(t),z'(t)]\exp[\frac{i}{\hbar}(S_o[z(t)]-S_o[z'(t)])]\label{eq20}
\end{equation}
is the `superpropagator' and
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)]=\int\mathrm{d}q(0)\mathrm{d}q'(0)
\mathrm{d}q(T)
\mathrm{d}q'(T)\mathrm{D}q(t)\mathrm{D}q'(t)\delta(q(T)-q'(T))}
\nonumber \\ &
&\exp\left[\frac{i}{\hbar}\left(S_c[q(t)]-S_c[q'(t)]+S_I[z(t),q(t)]-
S_I[z'(t),q'(t)]\right)\right] \; \rho_c(q(0),q'(0)),
\end{eqnarray}
is the so called `influence functional', which contains all the
information about the chaotic system.
Equation (\ref{eq19}) is the equation of motion for the reduced
density matrix. For $\mathcal{F}=1$, $J$ becomes the propagator of
the isolated oscillator. Our goal is to get an approximate
expression for $J$ that includes the effects of the chaotic system.
We take as the initial state for the chaotic system one of its
energy eigenstate $\phi_a(q)=\langle q|a\rangle$. Thus,
\begin{equation}
\rho_c(q(0),q'(0))=\phi_a(q(0))\phi_a^*(q'(0)).
\end{equation}
This is the quantum version of the classical microcanonical
distribution we considered in section II.
The difficulty in the calculation of $J$ is that the chaotic
Lagrangian $L_c$ is not quadratic. Therefore, $\mathcal{F}$ has to
be treated in a perturbative manner. Rewriting $\mathcal{F}$ as
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)]=\int\mathrm{d}q(0)\ldots
\mathrm{D}q'(t)\delta(q(T)-
q'(T))\exp\left[\frac{i}{\hbar}(S_c[q(t)]-S_c[q'(t)])\right]}\nonumber\\
&
&\exp\left[-\gamma\frac{i}{\hbar}\left(\int_0^T\mathrm{d}t(z(t)x(t)-
z'(t)x'(t))\right)\right]\phi_a(q(0))\phi_a^*(q'(0))
\end{eqnarray}
we assume that $\gamma$ is small enough so that the exponential in
the second line can be expanded to second order in its argument.
These terms can be calculated by inserting complete sets of energy
eigenstates of $H_c$. The result, following \cite{feynman1}, is
\begin{eqnarray}
\lefteqn{\mathcal{F}[z(t),z'(t)] \approx
1-\left(\frac{i\gamma}{\hbar}\right)x_{aa}
\int_0^T\mathrm{d}t[z(t)-z'(t)]} \nonumber \\
& &-\left(\frac{\gamma^2}{\hbar}\right)
\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s[z(t)-z'(t)][z(s)F_a^*(t-s)
-z'(s)F_a(t-s)],
\end{eqnarray}
where
\begin{eqnarray}
F_a(t-s) = \sum_b\frac{|x_{ba}|^2}{\hbar}\exp[i\omega_{ba}(t-s)],
\qquad\omega_{ba}=\frac{E_b-E_a}{\hbar},\label{eq21}
\end{eqnarray}
\begin{eqnarray}
x_{ba}=\int\mathrm{d}q\phi_b^*(q)x\phi_a(q),
\end{eqnarray}
and $E_b$ are the eigen-energies of chaotic system ($x$ is the
coordinate of $H_c$ in $V_I$ ). For the NS $H_c(-x)=H_c(x)$ and
$x_{aa}=0$. Thus,
\begin{eqnarray}
\mathcal{F}[z(t),z'(t)] \approx 1-\frac{1}{\hbar}\Phi[z(t),z'(t)]
\approx \exp\left[-\frac{1}{\hbar}\Phi[z(t),z'(t)]\right]
\end{eqnarray}
where
\begin{eqnarray}
\Phi[z,z'] = \frac{\gamma^2}{\hbar}
\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s[z(t)-z'(t)][z(s)F_a^*(t-s)-
z'(s)F_a(t-s)].
\end{eqnarray}
With these approximations the superpropagator can be written as
\begin{eqnarray}
J(z(T),z'(T),z(0),z'(0))=\int\mathrm{D}z(t)\mathrm{D}z'(t) \, e^{
\frac{i}{\hbar}(\tilde{S}_{ef}[z(t),z'(t)])},
\end{eqnarray}
where we have defined the effective action
\begin{equation}
\tilde{S}_{ef}[z(t),z'(t)]=S_o[z(t)]-S_o[z'(t)]+i\Phi[z(t),z'(t)].
\label{sefect}
\end{equation}
Since $\tilde{S}_{ef}$ is quadratic in $z$ and $z'$ the path
integral can be solved exactly by the stationary phase method. It is
convenient to define the new variables $r(t)=(z(t)+z'(t))/2$ and
$y(t)=z(t)-z'(t)$ \cite{weiss} and to separate
$F_a(t)=F_a^{'}(t)+iF_a^{''}(t)$ into real and imaginary parts. This
allows us to write
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))= \int\mathrm{D}r(t)\mathrm{D}y(t)\, e^{
\frac{i}{\hbar}\tilde{S}[r(t),y(t)] \, - \, \frac{1}{\hbar}
\phi[r(t),y(t)]} \label{pathint}.
\end{eqnarray}
where
\begin{eqnarray}
\tilde{S}[r(t),y(t)]\equiv\int_0^T\mathrm{d}t\left\{m[\dot{r}(t)\dot{y}(t)
-\omega_0^2r(t)y(t)]+2\gamma^2y(t)\int_0^t\mathrm{d}sF_a^{''}(t-s)
r(s)\right\}, \label{sefreal}
\end{eqnarray}
and
\begin{eqnarray}
\phi[r(t),y(t)]\equiv \gamma^2\int_0^T\mathrm{d}t\int_0^t\mathrm{d}s
y(t)y(s)F_a^{'}(t-s), \label{sefimag}
\end{eqnarray}
are the real and imaginary parts of $\tilde{S}_{ef}$. In Appendix A
we show that
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))=G(T,0)\exp\left\{\frac{i}{\hbar}\tilde{S}[r_e,y_e]
\right\}\exp\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\},
\label{superprop}
\end{eqnarray}
where $G(T,0)$ can be obtained by the normalization condition of
reduced density matrix and $r_e(t)$ and $y_e(t)$ are the extremum
paths of $\tilde{S}$, which satisfy
\begin{eqnarray}
\ddot{r}_e(t)+
\omega_0^2r_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(t-s)
r_e(s)=0,\label{eq22} \\
\ddot{y}_e(t)+
\omega_0^2y_e(t)-\frac{2\gamma^2}{m}\int_0^t\mathrm{d}sF_a{''}(s-t)
y_e(s)=0.\label{eq23}
\end{eqnarray}
Therefore we need $F_a^{''}$ to solve (\ref{eq22}) and (\ref{eq23})
and we also need $F_a^{'}$ to calculate $\phi[y_e,y_e]$.
From Eq.(\ref{eq21}) it follows that
\begin{eqnarray}
F_a(t)=\frac{\langle
a|\hat{x}(0)\hat{x}(t)|a\rangle}{\hbar},\label{eq24}
\end{eqnarray}
where $\hat{x}(t)$ is the Heisenberg representation of $\hat{x}$.
The real and imaginary parts of $F_a$ are
\begin{eqnarray}
F_a^{'}(t)=\frac{\langle a| \{\hat{x}(0),\hat{x}(t)\}| a
\rangle}{2\hbar},
\end{eqnarray}
and
\begin{eqnarray}
F_a^{''}(t)=\frac{\langle
a|[\hat{x}(0),\hat{x}(t)]|a\rangle}{2i\hbar},
\end{eqnarray}
where $\{.\}$ is the anticomutator and $[.]$ is the comutator. Thus,
$F_a^{'}$ and $F_a^{''}$ are, respectively, the quantum analogs of
the classical correlation and response functions of Section II.
\subsection{Semiclassical Expressions for Correlation Functions.}
In this section we obtain semiclassical formulas for $F_a^{'}(t)$
and $F_a^{''}(t)$. We write
\begin{eqnarray}
F_a^{'}(t) = \frac{1}{2\hbar} \sum_b\langle
b|\{\hat{x}(0),\hat{x}(t)\}\hat{\rho}|b\rangle =
\frac{1}{2\hbar}Tr(\hat{f}(t)\hat{\rho}),\label{eq25}
\end{eqnarray}
where $\hat{\rho}=|a\rangle\langle a|$ is the microcanonical
distribution of the chaotic system and $\hat{f} =
\{\hat{x}(0),\hat{x}(t)\}$. To calculate $F''_a$ we take
$\hat{f}=-i[\hat{x}(0),\hat{x}(t)]$. Using the Wigner-Weyl
representation \cite{wigner} the trace can be written as
\begin{eqnarray}
Tr[\hat{f}(t)\hat{\rho}]=\int\frac{\mathrm{d}^2 q\mathrm{d}^2 p}
{(2\pi\hbar)^2}f(q,p;t)W(q,p) \label{eq26}
\end{eqnarray}
where
\begin{eqnarray}
f(q,p;t)=\int_{-\infty}^{\infty}\mathrm{d}u \, e^{
\frac{i}{\hbar}p\cdot u}\langle
q-u/2|\hat{f}(t)|q+u/2\rangle.\label{eq27}
\end{eqnarray}
is the Weyl transformation (or symbol) of $\hat{f}(t)$ and $W(q,p)$
is the Wigner function of $\hat{\rho}$. For
$\hat{f}(t)=-i[\hat{x}(0),\hat{x}(t)]$ we have
\begin{eqnarray}
f(q,p;t)&=&-i\int_{-\infty}^{\infty}\mathrm{d}u \, e^{ ip\cdot
u/\hbar} \, \langle
q-u/2|(\hat{x}\hat{x}(t)-\hat{x}(t)\hat{x})|q+u/2\rangle
\nonumber \\
&=&-i\int_{-\infty}^{\infty}\mathrm{d}u \, e^{ ip\cdot u/\hbar} \,
(-u_x)\langle q-u/2|\hat{x}(t)|q+u/2\rangle
\nonumber \\
&=&\hbar\frac{\partial}{\partial
p_x}\left\{\int_{-\infty}^{\infty}\mathrm{d} u \, e^{ ip\cdot
u/\hbar} \,\langle
q-u/2|\hat{x}(t)|q+u/2\rangle\right\} \nonumber \\
&=& \hbar\frac{\partial}{\partial p_x} \left\{
\int_{-\infty}^{\infty}\mathrm{d} u\, e^{ ip\cdot u/\hbar} \,
\int_{-\infty}^{\infty}\mathrm{d} v\, v_x
K^*(v,q-u/2;t)K(v,q+u/2;t)\right\} \label{eq28}
\end{eqnarray}
where $u$ and $v$ represent coordinates of the chaotic system and
$K(v,v';t)=\langle v|e^{-i \hat{H}_c t/\hbar}|v'\rangle$ is the
propagator.
We now replace the propagators by their semiclassical expressions
$\tilde{K}$ and do the integrals by the stationary phase
approximation. The stationary phase condition shows that the most
important contributions come from the trajectories starting at
$q-u/2$ (for $K^*$) and $q+u/2$ (for $K$) and arriving at $v$ in the
time $t$ such that
\begin{eqnarray}
\nabla_{v}R_k(v,q-u/2)-\nabla_{v}R_l(v, q+u/2)=0,\label{eq30}
\end{eqnarray}
where $R_k$ and $R_l$ are Hamilton's principal functions coming from
the phases in $\tilde{K}$ and $\tilde{K}^*$. Since $\nabla_{v}
R_i(v,v')$ gives the final momentum, (\ref{eq30}) imposes that the
final momenta of the two trajectories must be equal. Since the final
positions are also equal, the two trajectories must be identical.
Thus,
\begin{eqnarray}
\int_{-\infty}^{\infty}\mathrm{d}v\, v_x \tilde{K}^*(
v,q-u/2;t)\tilde{K}( v,q+ u/2;t)\approx x(q,p;t)\delta(u),
\label{eq31}
\end{eqnarray}
where $x(q,p;t)$ is the coordinate $x$ of the stationary trajectory.
Using (\ref{eq31}) in (\ref{eq28}) we find
\begin{eqnarray}
f(q,p;t)=\hbar\frac{\partial}{\partial p_x} x (q,p;t) =
\hbar\{x(0),x(t)\}.\label{eq32}
\end{eqnarray}
since
\begin{eqnarray}
\{x(0),x(t)\}=\frac{\partial x(0)}{\partial x(0)}\frac{\partial
x(t)}{\partial p_x(0)}-\frac{\partial x(0)}{\partial
p_x(0)}\frac{\partial x(t)}{\partial x(0)}=\frac{\partial
x(t)}{\partial p_x(0)}.\label{eq33}
\end{eqnarray}
For $\hat{f}(t)=\{\hat{x}(0),\hat{x}(t)\}$, we find
\begin{eqnarray}
f(q,p;t)&=&
\int_{-\infty}^{\infty}\mathrm{d}u\exp\left(\frac{i}{\hbar} p\cdot
u\right)2q_x\langle q-u/2|\hat{X}(t)|q+u/2\rangle \nonumber
\\ &\approx& 2q_x\, x(q,p;t)=2x(0)x(t). \label{eq35}
\end{eqnarray}
The semiclassical limit of the Wigner function
\begin{eqnarray}
W(q,p;E)=\frac{1}{\hbar}\int_{-\infty}^{\infty}\mathrm{d}t
e^{iEt/\hbar}\int\mathrm{d}u\exp\left(\frac{i}{\hbar}p\cdot u
\right)K(q+u/2,q-u/2;t).\label{eq37}
\end{eqnarray}
was obtained by Berry \cite{berry2} and can be written as
\begin{eqnarray}
W(q,p;E)\approx\delta(E-H(q,p))+W_1(q,p;E). \label{eq38}
\end{eqnarray}
The first term is the classical micro-canonical distribution and the
second, $W_1$, is given by classical periodic orbits corrections to
the classical function. These periodic orbits have energy $E=E_a$,
corresponding to the eigenstate $|a\rangle$ of the microcanonical
quantum distribution. Using (\ref{eq38}) we write
\begin{eqnarray}
Tr[\hat{f}(t)\hat{\rho}] &\approx
\displaystyle{\frac{1}{(2\pi\hbar)^2}} & \left[ \int\mathrm{d}q
\mathrm{d} p f(q,p;t)\delta(E-H(q,p)) \, + \int\mathrm{d}q
\mathrm{d} p
f(q,p;t)W_1(q,p;E) \right]\nonumber \\
& & \equiv \frac{1}{(2\pi\hbar)^2} \left[f^0 + f^1\right].
\label{eq39}
\end{eqnarray}
When $f(q,p;t)$ is replaced by the semiclassical expressions for the
anticomutator and comutator, the first term of (\ref{eq39}) becomes,
except for a normalization, the classical expressions for the
response and correlation functions respectively. Following
\cite{berry2} the second term of (\ref{eq39}) becomes
\begin{eqnarray}
f^1(E,t) \approx \sum_jA_j\cos{(S_j(E)/\hbar+\gamma_j)}\oint
\mathrm{d}\tau f(q_j(\tau),p_j(\tau);t).\label{eq40}
\end{eqnarray}
where, $A_j$ depends on the stability of $j$-periodic orbit,
$S_j(E)$ is its action, $\gamma_j$ is the Maslov index and the
integral is calculated over a period of the $j$-orbit. Analogous
results can be found, for example, in \cite{eckart}.
Furthermore \cite{berry2},
\begin{equation}
\begin{array}{ll}
Tr(\hat{\rho})&=\displaystyle{\frac{1}{(2\pi\hbar)^2}} \left[
\int\mathrm{d}q\mathrm{d}p
\,\delta(E-H(q,p))+n_q(E;\hbar) \right] \\ \\
&=\displaystyle{\frac{1}{(2\pi\hbar)^2}}[n_c(E)+ n_q(E;\hbar)]\\
\label{eq42}
\end{array}
\end{equation}
where the first term is the classical density of states and the
second term is known as Gutzwiller's trace formula. Thus,
\begin{equation}
\begin{array}{ll}
\displaystyle{\frac{Tr[\hat{f}(t)\hat{\rho}]}{Tr[\hat{\rho}]}} &
\approx \displaystyle{\frac{f^0(E;t)+f^1(E;t)}{n_c(E)+n_q(E;\hbar)}}
\\ \\ & \approx \langle f(q,p;t)\rangle_{cl.}\left[1-
\frac{n_q(E;\hbar)}{n_c(E)}\right]+\frac{f^1(E;t)}{n_c(E)}\left[1-
\frac{n_q(E;\hbar)}{n_c(E)}\right] \label{eq47}
\end{array}
\end{equation}
where $\langle f(q,p;t)\rangle_{cl.}=f^0(E,t)/n_c(E)$.
Finally, we can calculate the semiclassical expressions for
$F_a^{''}(t)$ and $F_a^{'}(t)$. From (\ref{eq32}), (\ref{eq35}) and
(\ref{eq47})
\begin{eqnarray}
F_a^{'}(t)&\approx& \frac{\langle
x(0)x(t)\rangle_{cl.}}{\hbar}
\left[1-\frac{n_q(E;\hbar)}{n_c(E)}\right]
\nonumber \\
&+&\frac{1}{\hbar n_c(E)}\sum_j A_j
\cos\left(S_j(E)/\hbar+\gamma_j\right) \oint\mathrm{d}\tau
x_j(\tau)x_j(\tau+t).\label{eq49}
\end{eqnarray}
and
\begin{eqnarray}
F_a^{''}(t)=\frac{\langle\hat{O}(t)\rangle}{2\hbar}&\approx&\frac{\langle
\{x(0),x(t)\}\rangle_{cl.}}{2}
\left[1-\frac{n_q(E;\hbar)}{n_c(E)}\right]
\nonumber \\
&+&\frac{1}{2 n_c(E)}\sum_j A_j \cos\left(S_j(E)/\hbar+\gamma_j\right)
\oint\mathrm{d}\tau \{x_j(\tau),x_j(\tau+t)\}.\nonumber \\
\,\label{eq50}
\end{eqnarray}
Both these semiclassical expressions are given by their classical
counterparts multiplied by a correction to their amplitudes, given
by $n_q(E;\hbar)/n_c(E)$, plus a correction from periodic orbits.
The temporal dependence of the first is given solely by the
classical dynamics and it decays exponentially. The second term,
however, is a sum of oscillating functions and carries the temporal
dependence characteristic of the chaotic system. As a final remark
we note in \cite{esposito1} these functions were calculated using
random matrix theory.
\subsection{The Superpropagator}
The semiclassical expressions for $F_a^{'}(t)$ and $F_a^{''}(t)$
allows us to solve the equations of motion (\ref{eq22}) and
(\ref{eq23}) and to calculate the superpropagator,
Eq.(\ref{superprop}). In order to have explicit formula, we shall
consider only the zero order approximation $F_a^{''}(t) \approx
\phi_{xx}(t)/2$. Substituting it in (\ref{eq22}) and (\ref{eq23})
and integrating by parts we get
\begin{eqnarray}
\ddot{r}_e(t)+\Omega^2_0\, r_e(t)+\frac{\gamma^2}{m}\int_0^t\mathrm{d}s
F(t-s)\dot{r}_e(s)&=&-\frac{\gamma^2}{m}r_e(0)F(t), \label{eq-3}\\
\ddot{y}_e(t)+\chi^2_0\, y_e(t)-\frac{\gamma^2}{m}\int_0^t\mathrm{d}s
F(t-s)\dot{y}_e(s)&=&\frac{\gamma^2}{m}y_e(0)F(t), \label{eq-4}
\end{eqnarray}
where $\Omega_0^2=\omega_0^2-\gamma^2F(0)/m$, $\chi^2_0=\omega_0^2+
\gamma^2F(0)/m$ and
\begin{eqnarray}
F(t)=\int\mathrm{d}s\phi_{xx}(t-s),\label{eq-5}
\end{eqnarray}
as in the classical case (eq.(\ref{eq17})).
In Appendix B we solve these equations by the method of Laplace
transforms. We find that the solutions involve two very different
time scales. The shortest time scale is relevant only for times much
smaller than $1/\omega_0$, the period of the decoupled oscillator.
For times of the order of $1/\omega_0$ these terms can be discarded
as transients. Here we shall adopt this approximation and keep only
the terms that are significant for times of $1/\omega_0$. In this
case we show that (\ref{eq-3}) and (\ref{eq-4}) can be written as
\begin{eqnarray}
\ddot{r}_e(t)+2\Lambda\dot{r}_e(t)+\Omega_0^2r_e(t)=0, \quad
\Omega_0^2\gg\Lambda^2
,\label{eq-6} \\
\ddot{y}_e(t)-2\Lambda\dot{y}_e(t)+\chi_0^2y_e(t)=0,\quad
\chi_0^2\gg\Lambda^2, \label{eq-7}
\end{eqnarray}
where
\begin{eqnarray}
\Lambda=\frac{\gamma^2}{2m}\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{eq-8}
\end{eqnarray}
Approximating $\Omega_0 \approx \chi_0 \approx \omega_0$ we find
\begin{eqnarray}
r_e(t)=e^{-\Lambda
t}\left\{\frac{\sin{[\omega_0(T-t)]}}{\sin{(\omega_0 T)}}r(0)+
e^{\Lambda T} \frac{\sin{(\omega_0 t)}}{\sin{(\omega_0
T)}}r(T)\right\},\nonumber \\
y_e(t)=e^{\Lambda
t}\left\{\frac{\sin{[\omega_0(T-t)]}}{\sin{(\omega_0 T)}}y(0)+
e^{-\Lambda T} \frac{\sin{(\omega_0 t)}}{\sin{(\omega_0
T)}}y(T)\right\}.\label{eq-10}
\end{eqnarray}
Within this approximation we can calculate the real part $\tilde{S}$
of the effective action, Eq.(\ref{sefreal}). We obtain
\begin{eqnarray}
\tilde{S}[r_e,y_e]&=&[m\omega_0^2
K(T)-m\Lambda]r(T)y(T)+[m\omega_0^2 K(T)
+m\Lambda]r(0)y(0) \nonumber \\
& &-m\omega_0^2L(T)r(0)y(T)-m\omega_0^2N(T)
r(T)y(0),\label{eq-13}
\end{eqnarray}
where,
\begin{eqnarray}
K(T)&=&\frac{1}{\omega_0}\frac{\cos(\omega_0 T)}
{\sin(\omega_0 T)}, \label{eq-14}\\
L(T)&=&\frac{1}{\omega_0}\frac{e^{-\Lambda T}}{\sin(\omega_0 T)},
\label{eq-15}\\
N(T)&=&\frac{1}{\omega_0}\frac{e^{\Lambda T}}{\sin(\omega_0 T)}.\label{eq-16}
\end{eqnarray}
For the imaginary part $\phi$ of the effective action,
Eq.(\ref{sefimag}), we get
\begin{eqnarray}
\phi[y_e,y_e]=\gamma^2\int_0^T\mathrm{d}t\,y_e(t)\int_0^t\mathrm{d}s\,
F_a^{'}(t-s)y_e(s).\label{eq-17}
\end{eqnarray}
Using again only the zero order term of the semiclassical expression
(\ref{eq49}) for $F_a^{'}(t)$ and the fact that the classical
correlation function decays exponentially, as in (\ref{eq-8}), we
can approximate, for $t \sim 1/\omega_0$
\begin{eqnarray}
\int_0^t\mathrm{d}s\,F_a^{'}(t-s)y_e(s)\approx y_e(t)
\lim_{t\to\infty}\int_0^{t}\mathrm{d}s\,F_a^{'}(t-s) \equiv y_e(t)
\frac{B'}{\hbar}, \label{eq-20}
\end{eqnarray}
which implies that
\begin{equation}
\begin{array}{ll}
\phi[y_e,y_e] &= \displaystyle{\frac{\gamma^2B'}{\hbar}}
\int_0^T\mathrm{d}t\,y^2_e(t) \\
&=\displaystyle{\frac{\gamma^2B'}{\hbar}}[A(T)y^2(T)+B(T)y(T)y(0) +
C(T)y^2(0)],\label{eq-22}
\end{array}
\end{equation}
where
\begin{eqnarray}
A(T)&=&\frac{[-\omega_0^2e^{-2\Lambda
T}+(\Lambda^2+\omega_0^2)-\Lambda^2\cos{(2\omega_0
T)-\Lambda\omega_0\sin{(2\omega_0
T)}}]}{4\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}},
\label{eq-x1} \\
B(T)&=&\frac{[-\omega_0^2\sinh{(\Lambda
T)}\cos{(\omega_0 T)}+\Lambda\omega_0\cosh{(\Lambda T)}\sin{(\omega_0
T)}]}{\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}},
\label{eq-x2} \\
C(T)&=&\frac{[\omega_0^2e^{2\Lambda
T}-(\Lambda^2+\omega_0^2)+\Lambda^2\cos{(2\omega_0
T)-\Lambda\omega_0\sin{(2\omega_0
T)}}]}{4\Lambda(\omega_0^2+\Lambda^2)\sin^2{(\omega_0 T)}}.
\label{eq-x3}
\end{eqnarray}
Finally, putting everything together we get
\begin{eqnarray}
\lefteqn{J(r(T),y(T),r(0),y(0))=G(T,0)\exp{\left\{\frac{i}{\hbar}\tilde{S}
[r_e,y_e]\right\}}\exp{\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\}}
}\nonumber \\
&=&G(T,0)\exp{\left\{\frac{i}{\hbar}\tilde{K}_2(T)r(T)y(T)\right\}}
\exp{\left\{-\frac{1}{\hbar}\tilde{A}(T)y^2(T)\right\}}\times \nonumber \\
&
&\exp{\left\{\frac{i}{\hbar}\left[\tilde{K}_1(T)r(0)y(0)-\tilde{L}(T)r(0)y(T)
-\tilde{N}(T)r(T)y(0)\right]\right\}}\times \nonumber \\
& &\exp{\left\{-\frac{1}{\hbar}\tilde{B}(T)y(T)y(0)-\frac{1}{\hbar}
\tilde{C}(T)y^2(0)\right\}} \label{eq-24}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{K}_{1,2}(T)=m\omega_0^2K(T)\pm m\Lambda
\qquad&\tilde{L}(T)=m\omega_0^2L(T)&\qquad
\tilde{N}(T)=m\omega_0^2N(T) \nonumber \\ \nonumber \\
\tilde{A}(T)=\frac{\gamma^2B'}{\hbar}A(T)\qquad&\tilde{B}(T)=
\frac{\gamma^2B'}{\hbar}B(T)&\qquad\tilde{C}(T)=\frac{\gamma^2B'}
{\hbar}C(T). \nonumber \\
\label{eq-23}
\end{eqnarray}
We finish this section with a comment about the physical situation
described by these calculations. Since we have considered only the
first terms of the semiclassical expressions for $F_a^{'}$ and
$F_a^{''}$ our results are valid only in the Ehrenfest time scale of
the chaotic system given by \cite{beenaker}
\begin{eqnarray}
t_E\sim\frac{1}{\lambda_L}\ln{\left(\frac{S_c}{\hbar}\right)},
\end{eqnarray}
where $\lambda_L$ is the Lyapunov exponent and $S_c$ is a typical
action of the chaotic system, for example the action of the shortest
periodic orbit. Approximating $\lambda_L$ by $\alpha$, we see that,
in order to observe effects for $t\sim 1/\omega_0$, we must have
\begin{eqnarray}
t_E\sim\frac{1}{\omega_0}=\frac{1}{\alpha}\ln{\left(\frac{S_c}{\hbar}\right)}
\Rightarrow
\frac{S_c}{\hbar}=\exp{\left(\frac{\alpha}{\omega_0}\right)}.
\label{eq-25}
\end{eqnarray}
For NS, $\alpha/\omega_0\sim 8$ and $S_c \approx 10$ which means
that $\hbar$ must be smaller than $10^{-3}$ for our results to be
valid.
\section{Applications}
The superpropagator allows us to study the time evolution of the
oscillator under the influence of the chaotic system. The reduced
density matrix satisfies
\begin{eqnarray}
\rho_o(r(T),y(T))=\int\mathrm{d}r(0)\mathrm{d}y(0)\,J(r(T),y(T),r(0),y(0))
\,\rho_o(r(0),y(0)). \label{eq-26}
\end{eqnarray}
In the following we calculate explicitly the propagation of two
different oscillator's states. These two applications are similar to
the ones presented by Caldeira and Leggett to study dissipation and
decoherence.
\subsection{Propagation of a Gaussian State.}
For a Gaussian state
\begin{eqnarray}
\psi(z(0))=\frac{1}{(2\pi\sigma^2)^{1/4}}\,e^{\frac{i}{\hbar}pz(0)/\hbar}
e^{-z^2(0)/4\sigma^2} \label{eq-28}
\end{eqnarray}
the density matrix $\rho_o(z(0),z'(0))=\psi^*(z'(0))\psi(z(0))$ can
be written in terms of $r=(z+z')/2$ and $y=z-z'$ as
\begin{eqnarray}
\rho_o(r(0),y(0))=\frac{1}{(2\pi\sigma^2)^{1/2}}\,e^{\frac{i}{\hbar}py(0)}
e^{-r^2(0)/2\sigma^2}e^{-y^2(0)/8\sigma^2}. \label{eq-29}
\end{eqnarray}
Substituting (\ref{eq-24}) and (\ref{eq-29}) in (\ref{eq-26}) and
performing the integrals, we get
\begin{eqnarray}
\lefteqn{\rho_o(r(T),y(T))=G(T,0)\left[\frac{2\pi\hbar^2}{2\hbar
\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)}\right]^{1/2}}\nonumber \\
&\times \exp{\left\{-
\frac{\tilde{N}^2(T)}{2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}
\left(r(T)-\frac{p}{\tilde{N}(T)}\right)^2\right\}}&
\nonumber \\
&\times\exp{\left\{
-\left[\frac{\tilde{A}(T)}{\hbar}+\frac{\sigma^2
\tilde{L}^2(T)}{2\hbar^2}-\frac{(\sigma^2\tilde{K}_1(T)\tilde{L}(T)-\hbar
\tilde{B}(T))^2}{2\hbar^2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}
\right]y^2(T)\right\}}& \nonumber \\
&\times\exp\left\{\frac{i}{\hbar}\tilde{K}_2(T)r(T)y(T)
-\frac{i}{\hbar}\frac{(\sigma^2\tilde{K}_1(T)\tilde{L}(T)
-\hbar\tilde{B}(T))}{(2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T))}
\tilde{N}(T)
\left(r(T)-\frac{p}{\tilde{N}(T)}\right)y(T)\right\}&, \nonumber \\
\label{eq-30}
\end{eqnarray}
where,
\begin{eqnarray}
\tilde{C}_1(T)=\tilde{C}(T)+\frac{\hbar}{8\sigma^2}. \label{eq-31}
\end{eqnarray}
For $y=z-z'=0$, $\rho_o$ becomes the probability density. After
normalizing we obtain
\begin{eqnarray}
\rho_o(r(T),0)&=&\left[\frac{\tilde{N}^2(T)}{2\pi[2\hbar\tilde{C}_1(T)+
\sigma^2\tilde{K}_1^2(T)]}\right]^{1/2}\nonumber \\
&\times&\exp{\left\{-\frac{\tilde{N}^2(T)}
{2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)]}\left(r(T)-\frac{p}
{\tilde{N}(T)}\right)^2\right\}}. \label{eq-33}
\end{eqnarray}
Eq.(\ref{eq-33}) represents a Gaussian packet whose center follows
the trajectory
\begin{eqnarray}
r(T)=\frac{p}{\tilde{N}(T)}=\frac{p}{m\omega_0}e^{-\Lambda
T}\sin{(\omega_0 T)}. \label{eq-34}
\end{eqnarray}
The dissipative effect due to the interaction with the chaotic
system is explicit. The same behavior was obtained by Caldeira and
Leggett \cite{caldeira1} using a thermal bath with many degrees of
freedom. Eq.(\ref{eq-34}) represents the trajectory of a weakly
damped harmonic oscillator. The critical and strongly damped cases
cannot be described by this formalism because of the weak coupling
regime adopted.
The width of the evolved packet is given by
\begin{eqnarray}
\sigma^2(T)&=&\frac{\sigma^2\tilde{K}_1^2(T)+2\hbar\tilde{C}_1(T)}
{\tilde{N}^2(T)} .
\end{eqnarray}
After some algebra we can show that
\begin{eqnarray}
\sigma^2(T)&=& \sigma^2\bigg\{\frac{(1+\epsilon^2)e^{-2\Lambda
T}}{1+ \epsilon^2} \nonumber \\
&+&\frac{\Gamma[1-e^{-2\Lambda T}(1+2\epsilon\sin{(\omega_0
T)}\cos{(\omega_0 T)}+2\epsilon^2 \sin^2{(\omega_0
T)})]}{1+\epsilon^2}\bigg\}, \label{eq-35}
\end{eqnarray}
where
\begin{eqnarray}
\epsilon=\frac{\Lambda}{\omega_0}\qquad\mathrm{and}\qquad\Gamma=\frac{E_c(0)}
{\hbar\omega_0}. \label{eq-36}
\end{eqnarray}
The expression above for $\Gamma$ comes from the following
considerations: from $\tilde{C}_1(T)$ it follows that
\begin{eqnarray}
\Gamma=\frac{\gamma^2B'}{\hbar m\omega_0\Lambda}.
\end{eqnarray}
Using Eqs.(\ref{eq-8}) and (\ref{eq17}) and the relation
\cite{weiss}
\begin{eqnarray}
\langle p_x(0)x(t)\rangle_e=-\frac{\partial}{\partial t}\langle
x(0)x(t)\rangle_e
\end{eqnarray}
we obtain
\begin{eqnarray}
\Lambda=\frac{\gamma^2 B'}{m E_c(0)},
\end{eqnarray}
which leads directly to (\ref{eq-36}). Notice that, due to
(\ref{eq-25}), $\Gamma\gg 1$ is the only possibility.
Fig.3 shows that $\sigma^2(T)$ for $\Gamma=1$, $0.5$ and $2.0$.
These curves can be well fitted by the simpler expression
\begin{eqnarray}
\sigma^2(T)=\sigma^2[e^{-2\Lambda T}+\Gamma(1-e^{-2\Lambda T})],
\label{eq-37}
\end{eqnarray}
which, for $t\sim 1/\omega_0$, can be written as
\begin{eqnarray}
\sigma^2(T)=\sigma^2[1+(\Gamma-1)2\Lambda T]. \label{eq-38}
\end{eqnarray}
We see that $\sigma^2(\Gamma-1)\Lambda$ plays the role of a
diffusion constant. Fig.3 also shows that $\Gamma$ controls the
increase or decrease of $\sigma^2(T)$. In the present case,
$\sigma^2(T)$ can only increase because of the constraint $\Gamma\gg
1$. In the Caldeira-Leggett model, on the other hand, the width can
also decrease if the the temperature is very low.
\subsection{Superposition of Two Gaussian States.}
We now consider an initial state consisting of two Gaussian
wave-packets, one at the origin and one centered at $z(0)=q_0$:
\begin{eqnarray}
\psi(z(0))&=&N^{1/2}[\psi_1(z(0))+\psi_2(z(0))]\nonumber \\
&=&N^{1/2}\left\{\exp{\left[
-\frac{z^2(0)}{4\sigma^2}\right]}+\exp{\left[-\frac{(z(0)-q_0)^2}
{4\sigma^2}\right]}\right\}. \label{eq-39}
\end{eqnarray}
The density matrix is given by
\begin{eqnarray}
\rho_o(z(0),z'(0))=&N&[\rho_{11}(z(0),z'(0))+\rho_{22}(z(0),z'(0))\nonumber \\
&+&\rho_{12}(z(0),z'(0))+\rho_{21}(z(0),z'(0))]. \label{eq-40}
\end{eqnarray}
with $\hat{\rho}_{ij}=|\psi_i\rangle\langle\psi_j|$. The time
evolution of $\rho_o$ can again be calculated with Eq.(\ref{eq-26}).
The result, for $y=z-z'=0$ is
\begin{eqnarray}
\rho_{11}(r(T),0)&=&\frac{1}{2[1+h(T)]}\left(\frac{\tilde{N}^2(T)}
{\pi\tilde{f}(T)}\right)^{1/2}
\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}r^2(T)\right\}},
\label{eq-41} \\
\rho_{22}(r(T),0)&=&\frac{1}{2[1+h(T)]}\left(\frac{\tilde{N}^2(T)}
{\pi\tilde{f}(T)}\right)^{1/2}
\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}\left[r(T)-Q(T)\right]^2
\right\}}, \label{eq-42} \\
\rho_{12}(r(T),0)&+&\rho_{21}(r(T),0) \, = \nonumber \\
& &\frac{1}{2[1+h(T)]}\left(
\frac{\tilde{N}^2(T)}{\pi\tilde{f}(T)}\right)^{1/2}\exp{
\left[-\frac{q^2_0}{8\sigma^2}g(T)\right]}\exp{\left\{-\frac
{\tilde{N}^2(T)}{\tilde{f}(T)}r^2(T)\right\}} \nonumber \\
&\times&\exp{\left\{-\frac{\tilde{N}^2(T)}{\tilde{f}(T)}\left[r(T)-Q(T)
\right]^2\right\}}\nonumber \\
&\times&2\cos\left\{\frac{\hbar\tilde{N}^2(T)}
{4\sigma^2\tilde{f}(T)\tilde{K}_1(T)}
\left[\left(r(T)-Q(T)\right)^2-r^2(T)\right]\right\}, \label{eq-43}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{f}(T)&=&2[2\hbar\tilde{C}_1(T)+\sigma^2\tilde{K}_1^2(T)],
\label{eq-44} \\
h(T)&=&\exp{\left\{-\frac{q^2_0}{8\sigma^2}\left[1+g(T)\right]\right\}},
\label{eq-45} \\
Q(T)&=&\frac{\tilde{K}_1(T)}{\tilde{N}(T)}q_0, \label{eq-46} \\
g(T)&=&\frac{2\hbar\tilde{C}(T)}{2\hbar\tilde{C}_1(T)+\sigma^2
\tilde{K}_1^2(T)}.
\label{eq-47}
\end{eqnarray}
The interference term can also be rewritten as
\begin{eqnarray}
\rho_{12}(r(T),0)&+&\rho_{21}(r(T),0)=2\cos{\left[a(T)((r(T)-Q(T))^2-r^2(T))
\right]}\nonumber \\
&\times&\rho_1^{1/2}(r(T),0)\rho_2^{1/2}(r(T),0)\exp{\left[-\frac{q^2_0}
{8\sigma^2}g(T)\right]}. \label{eq-48}
\end{eqnarray}
Eq.(\ref{eq-48}) shows that the interference is attenuated by
$\exp{[-(q^2_0/8\sigma^2)g(T)]}$. Eq.(\ref{eq-48}) is very similar
to the expression obtained by Caldeira and Leggett \cite{caldeira2},
although there is no temperature dependence in $g(T)$, which can be
written as
\begin{eqnarray}
g(T)=\frac{\Gamma\,b(T)}{(1+\epsilon^2)+\Gamma\,b(T)}, \label{eq-49}
\end{eqnarray}
with
\begin{eqnarray}
b(T)=e^{2\Lambda T}-1-2\epsilon\sin{(\omega_0 T)}\cos{(\omega_0
T)}-2\epsilon^2\sin^2{(\omega_0 T)} \label{eq-50}
\end{eqnarray}
and $\epsilon=\Lambda/\omega_0$. We note that the asymptotic limits
\begin{eqnarray}
g(T=0)=0\qquad\mathrm{and}\qquad g(T\rightarrow\infty)\rightarrow 1
\label{eq-51}
\end{eqnarray}
are the same as those in the Caldeira-Leggett model.
Fig.4 shows $g(T)$ for $\Gamma=10$. In the regime $\Gamma\gg 1$, we can
approximate $g(T)$ by
\begin{eqnarray}
g(T)=\frac{2\Gamma\Lambda T}{1+2\Gamma\Lambda T}. \label{eq-52}
\end{eqnarray}
This simplified expression helps to estimate of the decoherence
time. For example, with (\ref{eq-52}), we can estimate the time $T'$
such that
\begin{eqnarray}
\exp{\left[-\frac{q^2_0}{8\sigma^2}g(T')\right]}\sim
10^{-3}. \label{eq-53}
\end{eqnarray}
Defining $n\equiv q^2_0/8\sigma^2$ (the number of quanta
$\hbar\omega_0$ of the wave packet centered at $q_0$), we get
\begin{eqnarray}
\left[\frac{n-\ln(10)}{3\ln(10)}\right]2\Gamma\Lambda
T'=2\tilde{n}\Gamma\Lambda T'=1\Rightarrow
T'=\frac{1}{2\tilde{n}\Gamma\Lambda}. \label{eq-54}
\end{eqnarray}
Since we are interested in the situation where $n \gg 1$ and $\Gamma
\gg 1$, we find that the decoherence time is much smaller than the
time scale where dissipation takes place, i.e., $T' \ll 1/\Lambda$.
\section{Discussion and Conclusions}
We have made two important assumptions in our calculation of the
superpropagator. The first of these assumptions, the weak coupling
regime, was important to reduce the path integral to a quadratic
form in the oscillator variables. The second assumption was the
semiclassical regime of the chaotic system. This was essential to
establish the connection between the coupling in the influence
functional and the classical correlation and response functions that
enter in the classical description of the system. The use of these
classical functions make the importance of the chaotic dynamics
explicit and show that the time scales obtained classically are
important ingredients to describe dissipation. In particular, the
exponential decay of correlations happens in a time scale much
shorter than the natural period of the oscillator. The time of
correlation loss plays the role of the microscopic time scale in the
Brownian motion, which is much shorter than the macroscopic one
\cite{reif}. Moreover, the exponential decay of the classical
correlations is what makes dissipation possible in the present
treatment. The corrections due to periodic orbits have not been
explored here and the importance of their contribution to
dissipation and decoherence is not clear at this point.
The effective dynamics we obtained, expressed in (\ref{eq-24}), is
analogous to the Caldeira-Leggett theory in the limit of high
temperatures and weak damping \cite{caldeira1,caldeira2}. For
example, the diffusion constant in (\ref{eq-38}) can be written,
for $\Gamma\gg1$, as
\begin{eqnarray}
\sigma^2\Gamma\Lambda=\frac{E_c(0)}{2m\omega_0^2}\Lambda,
\end{eqnarray}
which should be compared with
\begin{eqnarray}
D=\frac{k_B T}{m\omega_0^2}\Lambda
\end{eqnarray}
for the Brownian motion. Therefore, $E_c(0)$ plays the role of $k_B
T$. From Fig.4, $\Gamma$ seems to play the role of $k_B
T/\hbar\omega_k$ since it controls the behavior of $\sigma^2(T)$.
However, despite this close analogy between the two models, our
results are valid only for short times since they are limited by
Ehrenfest time and perturbation theory.
In summary, we have shown, using Feynman-Vernon approach, that a
chaotic system with two degrees of freedom can induce dissipation
and decoherence in a simple quantum system when weakly coupled to
it. The formalism we have chosen allows us a close analogy with the
many body formulation of the Caldeira-Leggett model. The most
important quantities in the formalism, the correlation and response
functions, are obtained directly from the dynamics, and not from
phenomenological assumptions as in the Caldeira-Legget model. In our
approach we have used simple classical approximations and discarded
all periodic orbits corrections. The effects of these corrections
are certainly worth studying.
\begin{appendix}
\section{The Stationary Phase Approximation}
In this appendix we solve the path integral Eq.(\ref{pathint}) by
the stationary phase approximation. Let $(r_e(t),y_e(t))$ be the
stationary path and
\begin{eqnarray}
&r(t)=r_e(t)+\delta r(t)=r_e(t)+\epsilon_1\tilde{r}(t)& \\
&y(t)=y_e(t)+\delta y(t)=y_e(t)+\epsilon_2\tilde{y}(t)&,
\end{eqnarray}
be a neighboring path with $\tilde{r}(T)=\tilde{r}(0)=0$ and
$\tilde{y}(T)=\tilde{y}(0)=0$.
The stationary path is obtained from the condition
\begin{equation}
\begin{array}{ll}
\Delta \tilde{S} & \equiv
\tilde{S}[r_e(t)+\epsilon_1\tilde{r}(t),y_e(t)+\epsilon_2
\tilde{y}(t)]-\tilde{S}[r_e(t),y_e(t)] \\
&= \epsilon_1\frac{d\Delta\tilde{S}}{d\epsilon_1}+\epsilon_2\frac{d\Delta
\tilde{S}}{d\epsilon_2} = 0
\end{array}
\end{equation}
We find
\begin{eqnarray}
\frac{d\Delta\tilde{S}}{d\epsilon_1}
=-\int_0^T\mathrm{d}t\tilde{r}(t)\left\{m[\ddot{y}_e(t)+\omega_0^2
y_e(t)]-2\gamma^2\int_0^t\mathrm{d}sF_a{''}(s-t)y_e(s)\right\},
\end{eqnarray}
and
\begin{eqnarray}
\frac{d\Delta\tilde{S}}{d\epsilon_2}
=-\int_0^T\mathrm{d}t\,\tilde{y}(t)\left\{m[\ddot{r}_e(t)+\omega_0^2
r_e(t)]-2\gamma^2\int_0^t\mathrm{d}sF_a{''}(t-s)r_e(s)\right\},
\end{eqnarray}
where we used $\tilde{r}(T)=\tilde{r}(0)=0$,
$\tilde{y}(T)=\tilde{y}(0)=0$ and
$\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s=2\int_0^T\mathrm{d}t\int_0^t
\mathrm{d}s$.
Therefore, the equations of motion for the stationary path are given
by
\begin{equation}
\ddot{y}_e(t)+\omega_0^2y_e(t)-\frac{2\gamma^2}{m}
\int_0^t\mathrm{d}sF_a{''}(s-t) y_e(s)=0\label{yclas}
\end{equation}
and
\begin{equation}
\ddot{r}_e(t)+\omega_0^2r_e(t)-\frac{2\gamma^2}{m}
\int_0^t\mathrm{d}sF_a{''}(t-s) r_e(s)=0.\label{rclas}
\end{equation}
Expanding $\phi$, Eq.\ref{sefimag}, around the stationary path we
find
\begin{eqnarray}
\lefteqn{\phi[r_e(t)+\epsilon_1\tilde{r}(t),y_e(t)+\epsilon_2
\tilde{y}(t)]=} \nonumber \\
& &=\frac{1}{2}\gamma^2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s[y_e(t)+
\epsilon_2
\tilde{y}(t)][y_e(s)+\epsilon_2\tilde{y}(s)]F_a{'}(t-s) \nonumber \\
& &=\frac{1}{2}\gamma^2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}sy_e(t)y_e(s)
F_a{'}(t-s) + \nonumber \\
& & \; \frac{1}{2} \gamma^2
\epsilon_2\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s
[\tilde{y}(t)y_e(t)+y_e(s)\tilde{y}(s)]F_a^{'}(t-s) + \nonumber \\
& & \; \epsilon_2^2 \frac{1}{2} \gamma^2
\int_0^T\mathrm{d}t\int_0^T\mathrm{d}s
\tilde{y}(t)\tilde{y}(s)F_a^{'}(t-s)=\phi[y_e,y_e]+2\varphi[\tilde{y},
y_e]+\varphi[\tilde{y},\tilde{y}].
\end{eqnarray}
Therefore, from (\ref{pathint}), we have
\begin{eqnarray}
\lefteqn{J(r(T),y(T),r(0),y(0))=} \nonumber \\
& &e^{\frac{i}{\hbar}\tilde{S}[r_e,y_e]}
e^{-\frac{1}{\hbar}\phi[y_e,y_e]}\int_0^0\mathrm{D}\delta y(t)
\mathrm{D}\delta r(t) e^{\frac{i}{\hbar}\tilde{S}[\delta r,\delta y]}
e^{-\frac{2}{\hbar}\varphi[\delta y,y_e]}e^{-\frac{1}{\hbar}
\varphi[\delta y,\delta y]}. \label{superpro2}
\end{eqnarray}
We are now going to show that (\ref{superpro2}) is a function of the
initial and final times only, which is not obvious because of the
functional dependence on $y_e(t)$. In order to do this we discretize
the paths and re-write (\ref{superpro2}) in the form
\cite{feynman2}:
\begin{eqnarray}
\lefteqn{\exp\left\{\frac{i}{\hbar}\tilde{S}[\delta r,\delta
y]\right\}\approx} \nonumber \\
& &\exp\bigg\{\frac{i}{\hbar}\bigg[\sum_{j=1}^N \epsilon \,m \bigg(\frac
{(\delta r_j-\delta r_{j-1})(\delta y_j-\delta y_{j-1})}{\epsilon^2}
-\omega_0^2\delta r_{j-1}\delta y_{j-1}\bigg) \nonumber \\
& &+\gamma^2\epsilon^2\sum_{j=1}^N\sum_{k=1}^N \delta y_j\delta r_k
F_{a_{(j-k)}}^{''}\bigg]\bigg\},
\end{eqnarray}
where $\delta r_j=\delta r(t_j)$,
$F_{a_{(j-k)}}^{''}=F_a^{''}(t_j-t_k)$,
\begin{eqnarray}
\exp\left\{-\frac{1}{\hbar}\varphi[\delta y,\delta y]\right\}\approx
\exp\left\{-\frac{1}{\hbar}\sum_{j=1}^N
2\gamma^2\epsilon^2\sum_{k=1} ^N \delta y_j\delta y_k
F_{a_{(j-k)}}^{'}\right\}
\end{eqnarray}
and
\begin{eqnarray}
\exp\left\{-\frac{2}{\hbar}\varphi[\delta y,y_e]\right\}\approx
\exp\left\{-\frac{1}{\hbar}\sum_{j=1}^N
4\gamma^2\epsilon^2\sum_{k=1} ^N \delta y_j y_{e_{k}}
F_{a_{(j-k)}}^{'}\right\}.
\end{eqnarray}
Grouping the exponents we obtain
\begin{eqnarray}
\exp{\left\{\frac{i}{\hbar}\tilde{S}[\delta r,\delta y]
-\frac{1}{\hbar}\varphi[\delta y,\delta y]
-\frac{2}{\hbar}\varphi[\delta y,y_e]\right\}} \approx
\exp{\left\{-\frac{i}{2}U^T M U -A^T U \right\}},
\end{eqnarray}
with
\begin{eqnarray}
U^T\equiv(\delta r_1 \ldots \delta r_N \, \delta y_1 \ldots \delta y_N)
\qquad M\equiv \left( \begin{array}{cc}
0 & p \\
p & r
\end{array} \right),
\end{eqnarray}
and where $p$ and $r$ are $N$x$N$ matrices and $A^T=(0 \; a)$ and
$a$ are $N$-dimensional vectors. To solve the path integral we need
to integrate this exponent over $\mathrm{d}U=\mathrm{d}\delta r_1
\ldots \mathrm{d}\delta r_N \, \mathrm{d}\delta y_1 \ldots \delta
y_N$. The result is \cite{swanson}
\begin{eqnarray}
\frac{1}{(\mathrm{det}\, M)^{1/2}}\exp\left[-\frac{1}{4}A^T M^{-1} A \right].
\end{eqnarray}
Because $M$ has a zero upper left block, its inverse has a zero
lower right block and, therefore, $A^T M^{-1}A = 0$. Since all the
dependence on the initial and final positions is contained in $A$,
(\ref{superpro2}) is indeed a function only of the initial and final
times. Therefore we may write the superpropagator as
\begin{eqnarray}
J(r(T),y(T),r(0),y(0))=G(T,0)\exp\left\{\frac{i}{\hbar}\tilde{S}[r_e,y_e]
\right\}\exp\left\{-\frac{1}{\hbar}\phi[y_e,y_e]\right\},
\end{eqnarray}
and $G(T,0)$ can be calculated by imposing the normalization of the
reduced density operator.
\section{Solution of the Equations of Motion.}
Taking the Laplace transform of (\ref{eq22}), we get (with
$F^{''}_a(t)\approx\phi_{xx}(t)/2$)
\begin{eqnarray}
\left[(s^2+\Omega_0^2)-\frac{\gamma^2}{m}\tilde{\phi}_{xx}(s)\right]
\tilde{r}_e(s)=sr(0)+\dot{r}(0).\label{a2-1}
\end{eqnarray}
where $\tilde{f}(s)=\mathcal{L}\{f(t)\}$ is the Laplace transform of
$f(t)$. Using
\begin{eqnarray}
\phi_{xx}(t)=\frac{2}{E_c(0)}\langle
p_x(0)x(t)\rangle_e=A\,e^{-\alpha|t|}\sin{(\omega t)}, \label{a2-3}
\end{eqnarray}
(\ref{a2-1}) becomes
\begin{eqnarray}
\tilde{r}_e(s)=\frac{s[(s+\alpha)^2+\omega^2]r(0)+[(s+\alpha)^2+\omega^2]
\dot{r}(0)}{\{(s^2+\Omega_0^2)[(s+\alpha)^2+\omega^2]-
\frac{\gamma^2}{m}A\omega\}}.\label{a2-4}
\end{eqnarray}
The Heaviside's theorem establishes that if $P(s)$ and $Q(s)$ are
polynomials such that the order of $P(s)$ is smaller than the order
of $Q(s)$, then
\begin{eqnarray}
\mathcal{L}^{-1}\left[\frac{P(s)}{Q(s)}\right]=\sum_{i=1}^{n}
\frac{P(s_i)}{Q'(s_i)} e^{s_it},\label{a2-5}
\end{eqnarray}
where $s_i$ are the roots of $Q(s)=0$ and $Q'(s)$ is the
$s$-derivative of $Q(s)$. Therefore we need the roots of
\begin{eqnarray}
\left[x^2+\left(\frac{\omega_0}{\alpha}\right)^2\right]\left[(x+1)^2+
\left(\frac{\omega}{\alpha}\right)
^2\right]-\frac{\gamma^2}{m}\frac{A\omega}{\alpha^4}=0,
\label{a2-7}
\end{eqnarray}
where $x=s/\alpha$ and $\Omega_0\approx \omega_0$. From Section II
we have
\begin{eqnarray}
\left(\frac{\omega_0}{\alpha}\right)^2\approx 1.6\times 10^{-2}\qquad
\left(\frac{\omega}{\alpha}\right)^2\approx 25 \qquad
\frac{\gamma^2}{m}\frac{A\omega}{\alpha^4}\approx 3\times 10^{-2}.\label{a2-8}
\end{eqnarray}
and the roots of (\ref{a2-7}) are
\begin{eqnarray}
x_1=-1.00-i 5.00 &\qquad& x_2=-1.00+i5.00
\nonumber \\
x_3=-4\times10^{-5}-i0.12 &\qquad& x_4=-4\times10^{-5}+i0.12,\label{a2-9}
\end{eqnarray}
Multiplying these roots by $\alpha$, we get
\begin{eqnarray}
s_1\approx-\alpha-i \omega &\qquad& s_2\approx-\alpha+i\omega
\nonumber \\
s_3\approx-\Lambda-i\omega_0 &\qquad& s_4\approx
-\Lambda+i\omega_0.\label{a2-10}
\end{eqnarray}
The same procedure is applied to (\ref{eq23}). The Laplace transform
of (\ref{eq23}) is written as
\begin{eqnarray}
\tilde{y}_e(s)=\frac{s[(s+\alpha)^2+\omega^2]y(0)+[(s+\alpha)^2+\omega^2]
\dot{y}(0)}{\{(s^2+\Omega_0^2)[(s+\alpha)^2+\omega^2]+
\frac{\gamma^2}{m}A\omega\}}\label{a2-11}
\end{eqnarray}
and the roots are
\begin{eqnarray}
s_1\approx-\alpha-i \omega &\qquad& s_2\approx-\alpha+i\omega
\nonumber \\
s_3\approx \Lambda-i\omega_0 &\qquad& s_4\approx
\Lambda+i\omega_0.\label{a2-12}
\end{eqnarray}
Since we are interested on time scales such that $t\sim 1/\omega_0$,
$s_1$ and $s_2$ are transient solutions and only $s_3$ and $s_4$ are
important. Therefore, turning to the equations (\ref{eq-3}) and
(\ref{eq-4}) and considering times on the scale $t\sim
1/\omega_0$, we see that those equations can be rewritten
approximately as
\begin{eqnarray}
\ddot{r}_e(t)+2\Lambda\dot{r}_e(t)+\Omega_0^2r_e(t)=0,\label{a2-13} \\
\ddot{y}_e(t)-2\Lambda\dot{y}_e(t)+\chi_0^2y_e(t)=0,\label{a2-14}
\end{eqnarray}
where terms proportional to $F(t)$ were disregarded (since they go to
zero for $t\sim 1/\omega_0$) and the convolutions terms were
approximated in the following way
\begin{eqnarray}
\int_0^t\mathrm{d}s\,F(t-s)\dot{r}_e(s)\approx\dot{r}_e(t)
\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{a2-15}
\end{eqnarray}
Thus, $\Lambda$ is given by
\begin{eqnarray}
\Lambda=\frac{\gamma^2}{2m}\lim_{t\to\infty}\int_0^{t}\mathrm{d}s
F(t-s).\label{a2-16}
\end{eqnarray}
Indeed, applying the Laplace transform in (\ref{a2-13}) and
(\ref{a2-14}), we get the roots
\begin{eqnarray}
s_1=-\Lambda-i\omega_0 &\qquad& s_2=-\Lambda+i\omega_0,\label{a2-17}
\end{eqnarray}
for $r_e(t)$ and
\begin{eqnarray}
s_1=\Lambda-i\omega_0 &\qquad& s_2=\Lambda+i\omega_0,\label{a2-18}
\end{eqnarray}
for $y_e(t)$ since $\Omega_0^2,\chi_0^2\gg\Lambda$ and
$\Omega_0^2\approx\chi_0^2\approx\omega_0^2$. Comparing
(\ref{a2-17}) and (\ref{a2-18}) with (\ref{a2-10}) and
(\ref{a2-12}), we conclude that the equations (\ref{a2-13}) and
(\ref{a2-14}) give a good description of the behavior given by
(\ref{eq-3}) and (\ref{eq-4}) for $t\sim 1/\omega_0$.
\end{appendix}
\centerline{\bf Acknowledgements} \noindent This paper was partly
supported by the Brazilian agencies {\bf FAPESP}, under contracts
number 02/04377-7 and 03/12097-7, and {\bf CNPq}. Especial thanks to
S.M.P.
\begin{figure}
\caption{Correlation functions for the NS for
$E_c=0.38$: (a) $\langle p_x (0)x(t)\rangle_e$; (b) $\langle
x(0)x(t)\rangle_e$. The full line shows the numerical results and
the dashed line shows the fitting. The averages were computed using
35000 initial conditions.}
\label{fig2}
\end{figure}
\begin{figure}
\caption{Average oscillator energy at short times
with the NS as chaotic system. $T_0=1/\omega_0$. The dashed line
shows $\langle
E_o(t)\rangle$ and the doted line shows $\langle
E_{or}
\label{fig3}
\end{figure}
\begin{figure}
\caption{Squared width of wave packet
$\sigma^2(T)/\sigma^2$ as given by Eq.(\ref{eq-35}
\label{fig4}
\end{figure}
\begin{figure}
\caption{The full line shows $g(T)$ as given by
Eq.(\ref{eq-49}
\label{fig5}
\end{figure}
\end{document} |
\begin{document}
\title{Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations}
\section{Introduction}
Fractional calculus, as a generalization of ordinary calculus, has been an intriguing topic for many famous mathematicians since the end of the 17th century. During the last four decades, many scholars have been working on the development of theory for fractional derivatives and integrals, found their way in the world of fractional calculus and their applications. For more detailed information on the historical background, we refer the interested reader to the following books: \cite{Oldhamspanier:1974, Samko:1993, MillerB:1993, Podlubny:1999, Hilfer:2000, Kilbas:2006, Baleanu:2011} and \cite{Herrmann:2014}. Differential equations possessing terms with fractional derivatives in the space- or time- or space-time direction have become very important in many application areas. Particularly, in recent years a huge amount of interesting and surprising fractional models have been proposed. Here, we mention just a few typical applications: in the theory of Hankel transforms \cite{Erdelyi:1940}, in financial models \cite{Scalas:2000, Wyss:2000}, in elasticity theory \cite{Bagleytorvik:1983}, in medical applications \cite{Santamaria:2006, Langlands:2009}, in geology \cite{Benson:2000, Liu:2003}, in physics \cite{Carpinteri:1997,Barkai:2000, Metzler:2000} and many more.
Similar to the work for ordinary differential equations, that has started more than a century earlier, research on numerical methods for time fractional differential equations (tfDEs) started its development. In this paper we consider approximations to tfDEs involving Caputo fractional derivatives of order $0 < \alpha < 1$ in the form of
\begin{equation}
\label{eq:nolinfode}
{^{C}}D^{\alpha}u(t)=f(t,u(t)),\hspace{0.618cm} t\in (0, T]
\end{equation}
with prescribed initial condition $u(0)=u_{0}$. According to \cite{DiethelmK:2004}, it holds that if function $f(t, u(t))$ is continuous and satisfies the Lipschitz condition with respect to the second variable, the problem \eqref{eq:nolinfode} then possesses a unique solution $u(t)\in C([0, T])$. In terms of the numerical approximation of formula \eqref{eq:nolinfode}, we mainly aim at the numerical discretisation to the Caputo fractional derivative, the definition of which we refer to Definition \ref{def:Caputoderiv} in the next section. It is observed that the Caputo fractional derivative of a well-behaved function is an operator combined with the integer-order derivative and the fractional integral, which can be regarded as a convolution of the weakly singular kernel $t^{-\beta} (0<\beta<1)$ and a function. The research on numerical approximations to fractional integral was developed in numerically solving a type of Volterra integral equation
\begin{equation}
\label{eq:nolinfode1}
u(t)=u_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\xi)^{\alpha-1}f(\xi, u(\xi))\mathrm{d}\xi,
\end{equation}
which is an equivalent form of \eqref{eq:nolinfode} when $f$ satisfies the Lipschitz continuous condition. With respect to numerical approximation for \eqref{eq:nolinfode1}, two general approaches are proposed respectively, i.e, the product integration method and fractional linear multistep methods. In these cases, a general discretized formula to \eqref{eq:nolinfode1} is written as
\begin{equation}
u_{n}=u_{0}+(\Delta t)^{\alpha}\sum_{j=0}^{n}\omega_{n-j}f(t_{j}, u_{j})+(\Delta t)^{\alpha}\sum_{j=0}^{k-1}w_{n,j}f(t_{j}, u_{j}), \hspace{0.5cm} n\ge k.
\end{equation}
The fractional linear multistep methods was originally proposed in \cite{LubichC:1986a} in the mid eighties of the last century. This type of methods is devoted to constructing the power series generated by the convolution quadrature weights $\{\omega_{j}\}_{j=0}^{\infty}$ based on classical implicit linear multistep formulae $(\rho, \sigma)$ with the following relationship
\begin{equation*}
\sum_{j=0}^{\infty}\omega_{j}\xi^{j}=\left(\frac{\sigma(1/\xi)}{\rho(1/\xi)}\right)^{\alpha}.
\end{equation*}
For the motivation behind this idea we refer to \cite{LubichC:1988a}. For this type of methods, the accuracy and stability properties highly benefit from those of the corresponding multistep methods \cite{LubichC:1986b, Lubich:1985}. Another more straightforward approach to generate the weights $\{\omega_{j}\}$ and $\{w_{n,j}\}$ is based on product quadrature method applied to the underlying fractional integral, for examples, to replace the integrand $f(\xi, u(\xi))$ by the piecewise Lagrange interpolation polynomials of degree $k (k\ge 0)$, and approximate the corresponding fractional integral in \eqref{eq:nolinfode1}. On the accuracy and efficiency of this class of methods to a type of nonlinear Volterra integral equation with irregular kernel, we can refer to \cite{Linz:1969, HoogR:1974, CameronS:1984, DixonJ:1985} and recently \cite{Baleanu:2011, LiZeng:2015}.
Other useful approaches such as collocation methods on non-smooth solution are discussed in \cite{Brunner:1985, BrunnerH:1986}. In addition, a series of other approaches were developed such as in \cite{Garappa:2013}, exponential integrators are applied to fractional order problems. Generalized Adams methods and so-called $m$-steps methods are utilized by \cite{Aceto:2014, Aceto:2015}.
In contrast to previous methods, the numerical approach to solve tfDEs in this paper is arrived at through directly approximating fractional derivative combining with numerical differentiation and integration. Recently, a new type of numerical schemes was designed to approximate the Caputo fractional derivative for solving time fractional partial differential equations, such as $L1$ method \cite{LinX:2007}, $L$1-2 method \cite{GaoSZ:2014}, $L$2-$1_\sigma$ method \cite{Alikhanov:2015}. These methods are all based on piecewise linear or quadratic interpolating polynomials approximation. It is natural to generalise the approach by improving the degree of the piecewise polynomial to approximate function that possesses suitable smoothness, in which situation the higher order of accuracy can be obtained. In the next section, we will devote to deriving a series of numerical schemes for formula \eqref{eq:nolinfode} based on constructing piecewise interpolation polynomials on interval $[0, t]$ as the approximations to solution $u(t)$, and consequently, the $\alpha$ order Caputo derivative of the polynomials as the approximation to ${^{C}}D^{\alpha}u(t)$. The local truncation errors of the numerical schemes are discussed correspondingly. The flexibility via choosing different interpolating points on subintervals to construct the piecewise polynomials will produce various schemes under the similar restriction of accuracy order.
In order to study the numerical stability of such methods applying to problem \eqref{eq:nolinfode}, we will examine the behaviour of the numerical method on the linear scalar equation
\begin{equation}
\label{eq:testeqC}
{^{C}}D^{\alpha}u(t)=\lambda u(t), \hspace{0.5cm} \lambda\in\mathbb{C}
\end{equation}
with initial value $u(0)=u_{0}$. It is already shown that the solution of \eqref{eq:testeqC} satisfies that $u(t)\to 0$ as $t\to +\infty$ provided that $|\mathrm{arg}(\lambda)|>\frac{\alpha\pi}{2}~ (-\pi\le \mathrm{arg}(\lambda)\le \pi)$ for arbitrary bounded initial value \cite{LubichC:1986b, MatignonD:1996},
accordingly, it can be studied in seeking those $\lambda$ for which the corresponding numerical solutions preserve the same property as true solution. In fact, several classical numerical stability theories have been constructed on solving problem \eqref{eq:testeqC} in the case of $\alpha=1$ \cite{HairerWN:1991,HairerWN:1987}. Furthermore, there are some efforts on generalising the numerical stability theory on linear multistep methods to integral equations, such as Volterra-type integral equation \cite{LubichC:1983, LubichC:1986b}. It is known that, for example, in the case of all those $\lambda$ satisfying $|\mathrm{arg}(\lambda)|>\frac{\alpha\pi}{2}~(0<\alpha\le 1)$, if the numerical solution has the same asymptotical stability property as true solution, the numerical method is called $A$-stable, and in other case of $\lambda$ lying in the sector with $|\mathrm{arg}(\lambda)|\ge \theta~(\frac{\alpha\pi}{2}<|\theta|\le \pi)$, it is referred to as $A(\theta)$-stable. Inspired by the successive work, we make use of the technique pioneered in \cite{LubichC:1986b}, specialise and refine the results to the fractional case in this paper. We confirm the stability regions and strong stability of the proposed numerical methods, and provide the rigorous analysis on the $A(\frac{\pi}{2})$-stability of some methods. Actually, it can be observed from the numerical experiments that the class of methods possesses the property of $A(\theta)$-stability uniformly for $0<\alpha<1$, and for some specific $\alpha\in (0,1)$, the $A$-stability can be obtained.
The paper is organized as follows. Section \ref{piecewise} introduces the continuous piecewise polynomial approximations of the Caputo derivative. We also derive some useful properties of the weight coefficients and discuss the local truncation errors. Section \ref{stabsection} and Section \ref{conversection} respectively treat the stability and convergence aspects of the numerical schemes when applied to time fractional differential equations. In section \ref{numexpsection} numerical experiments confirm our theoretical considerations with respect to order of convergence and stability restrictions.
\section{Continuous piecewise polynomial approximation to the Caputo fractional derivative}
\label{piecewise}
We first introduce the fractional derivative in the Caputo sense:
\begin{definition}[\cite{DiethelmK:2004}]\label{def:Caputoderiv}
Let $\alpha>0$, and $n=\lceil \alpha \rceil$, the $\alpha$ order Caputo derivative of function $u(t)$ on $[0,T]$ is defined by
\begin{equation}
{^{C}}D^{\alpha}u(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}\frac{u^{(n)}(\xi)}{(t-\xi)^{\alpha-n+1}}\mathrm{d}\xi
\end{equation}
whenever $u^{(n)}(t)\in L^{1}[0, T]$. In particular, the Caputo derivative of order $\alpha\in(0,1)$ is defined by
\begin{equation}
\label{eq:Cderiv}
{^C}D^{\alpha} u(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha}u^{(1)}(\xi)\mathrm{d}\xi
\end{equation}
whenever $u^{(1)}(t)\in L^{1}[0, T]$.
\end{definition}
In the sequel, we will derive a class of piecewise polynomial approximations to the Caputo fractional derivative of order $\alpha\in(0,1)$. The main idea is as follows.
Let $I=[0, T]$ be an interval, the $M+1$ nodes $\{t_{i}\}_{i=0}^{M}$ define a partition
\begin{equation}\label{par}
0=t_{0}<t_{1}\cdots<t_{M-1}<t_{M}=T.
\end{equation}
If the solution $u(t)$ in \eqref{eq:Cderiv} is assumed to be continuous on the interval $I$, when we think about an piecewise polynomial approximation to $u(t)$, it is reasonable to find the approximate solution at least in the continuous piecewise polynomial space, which is defined by
\begin{equation*}
C_{p}(I)=\{v(t)\in C(I): ~~v(t) \text{ is a polynomial on each subinterval } I_{j}=[t_{j-1}, t_{j}]\}.
\end{equation*}
Specifically, denoting the space of continuous piecewise polynomial of degree at most $k$ by
\begin{equation*}
C_{p}^{k}(I)=\{v(t)\in C(I):~~ v(t)=\sum_{l=0}^{k}a_{j,l}t^{l} \text{ on } I_{j}\},
\end{equation*}
we then construct a class of approximate solutions of $u(t)$ in the space $C_{p}^{k}(I)$. Here the Lagrange interpolation technique by prescribing interpolation conditions on distinct $k+1$ nodes is made use of such that the coefficients $\{a_{j,l}\}$ for $0\le l\le k$ on each $I_{j}$ are uniquely determined. In addition, suppose that $u(t)$ is not a constant function, we only need to focus on the cases of $k\ge 1$ in view of the continuity restriction. The choice of the interpolating points is provided in the following way.
We define a class of polynomials $p_{j, q}^{k}(t)$ which are of degree $k\ge 1$ and have a compact support $I_{j}$. The coefficients of the polynomials are uniquely determined by the following $k+1$ interpolation conditions
\begin{equation}\label{eq:interpcond}
p_{j, q}^{k}(t_{n})=u(t_{n}),\hspace{0.618cm} n=j+q-1, j+q-2,\cdots, j+q-k-1.
\end{equation}
Here the index $q$ records the number of shifts of the $k+1$ interpolating nodes $\{t_{n}\}_{n=j-1-k}^{j-1}$, and the sign of $q$ indicates the direction of the shift. Based on \eqref{eq:interpcond}, the polynomials can be represented by a Lagrange form
\begin{equation}\label{eq:pjqk}
p_{j, q}^{k}(t)=\sum_{n=j+q-k-1}^{j+q-1}\prod_{m=j+q-k-1 \atop m\ne n}^{j+q-1}\frac{t-t_{m}}{t_{n}-t_{m}} u(t_{n}).
\end{equation}
In particular, if the partition \eqref{par} is equidistant, i.e., $t_{n}=n\Delta t$ and $\Delta t=\frac{T}{M}$ as $M\in \mathbb{N}^{+}$, the alternative Newton expression is given by
\begin{equation}\label{eq:piqk}
p_{j, q}^{k}(t)=\sum_{n=0}^{k}\frac{\nabla^{n}u(t_{j+q-1})}{n!(\Delta t)^{n}}\prod_{l=0}^{n-1}(t-t_{j+q-1-l}).
\end{equation}
For the convenience of notation, we then rewrite \eqref{eq:piqk} by changing the variable $t=t_{j-1}+s\Delta t$ to obtain
\begin{equation}
\label{eq:pk}
p_{j,q}^{k}(t)=p_{j,q}^{k}(t_{j-1}+s\Delta t)=\sum_{r=0}^{k}\binom{s-q+r-1}{r}\nabla^{r}u(t_{j+q-1}),
\end{equation}
where the $r$-th order backward difference operator $\nabla^{r}$ is commonly defined by
\begin{equation*}
\nabla^{0} u(t_{i})=u(t_{i}),\hspace{0.4cm} \nabla^{r}u(t_{i})=\nabla^{r-1}u(t_{i})-\nabla^{r-1}u(t_{i-1})
\end{equation*}
and $\binom{s-q+r-1}{r}$ is the binomial coefficient.
In the following, we construct a class of approximate solution $P_{i}^{k}(t)\in C_{p}^{k}(I)$ to $u(t)$ on the uniform grid for $1\le i\le k\le 6$. The general representations are proposed by
\begin{equation}\label{eq:Pik}
P_{i}^{k}(t)=\sum_{j=1}^{k-i}p_{j,k-j}^{k-1}(t)+\sum_{j=k}^{n}p_{j-i+1,i}^{k}(t)+\sum_{j=n-i+2}^{n}p_{j,n+1-j}^{k}(t)
\end{equation}
for $t\in (t_{n-1}, t_{n}]$ and $1\le n\le M$, where $\sum_{j=1}^{k-i}p_{j,k-j}^{k-1}(t)=0$ and $\sum_{j=n-i+2}^{n}p_{j,n+1-j}^{k}(t)=0$ if $k-i<1$ and $n-i+2>n$, respectively.
\begin{remark}\label{re:1}
The construction of polynomials $P_{i}^{k}(t)$ is mainly based on the continuity requirement on interval $I$, i.e., the interpolation conditions
\begin{equation}
\label{eq:2.4}
p_{j,q}^{k}(t_{n})=u(t_{n}),\hspace{0.618cm} n=j-1,\hspace{0.1cm} j
\end{equation}
should be satisfied. It yields that on each subinterval $I_{j}$, according to \eqref{eq:interpcond}, both conditions $j+q-1\ge j$ and $j+q-k-1\le j-1$ should be satisfied, which indicates $1\le q\le k$. Therefore, in the case of $k=1$, there is a unique continuous piecewise linear polynomial, denoted by $P_{1}^{1}(t)$, in the space $C_{p}^{1}(I)$. According to \eqref{eq:Pik}, it is expressed by
\begin{equation*}\label{eq:P11}
P_{1}^{1}(t)=\sum_{j=1}^{n}p_{j,1}^{1}(t),
\end{equation*}
In the other case of $k=2$, there are three options on each $I_{j}$, that is, $p_{j,1}^{1}(t)$, $p_{j,1}^{2}(t)$ and $p_{j,2}^{2}(t)$ to constitute the interpolating polynomial that belongs to space $C_{p}^{2}(I)$. It is known that the construction of $P^{2}(t)$ is therefore not unique. In order to preserve the convolution property as much as possible, here we propose three available continuous piecewise polynomials in the forms of
\begin{equation}\label{eq:P2}
P_{1}^{2}(t)=p_{1,1}^{1}(t)+\sum_{j=2}^{n}p_{j,1}^{2}(t),\hspace{0.618cm} P_{2}^{2}(t)=\sum_{j=1}^{n-1}p_{j,2}^{2}(t)+p_{n,1}^{2}(t)
\end{equation}
and
\begin{equation*}\label{eq:P23}
P_{3}^{2}(t)=p_{1,2}^{2}(t)+\sum_{j=2}^{n}p_{j,1}^{2}(t)
\end{equation*}
when $t\in (t_{n-1}, t_{n}]$. In addition, as shown in \eqref{eq:Pik}, we restrict our further discussion to the case $i\le k$, and it is because in that situation, the corresponding discretized operators $D_{k,i}^{\alpha}u_{n}$ defined in \eqref{Dki1} can be computed recursively when the least starting values are prescribed.
\end{remark}
As a consequence, the operator
\begin{equation}
\label{Dki}
D_{k,i}^{\alpha}u(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha}\frac{\mathrm{d} P_{i}^{k}}{\mathrm{d} \xi}\mathrm{d}\xi
\end{equation}
is proposed on $t\in I$ as the approximations to ${^{C}}D^{\alpha}u(t)$. In the case of $t=t_{n}$, formula \eqref{Dki} can also be rewritten as
\begin{equation}
\label{Dki1}
\begin{split}
D_{k,i}^{\alpha}u_{n}&=\frac{(\Delta t)^{-\alpha}}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha}\frac{\mathrm{d} P_{i}^{k}(t_{j-1}+s\Delta t)}{\mathrm{d}s}\mathrm{d}s \\
&=(\Delta t)^{-\alpha}\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}u_{j}+(\Delta t)^{-\alpha}\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}u_{j},
\end{split}
\end{equation}
where $u_{n}:=u(t_{n})$.
In the following part, we will present the explicit representations of the weight coefficients $\{w_{n,j}^{(k,i)}\}$ and $\{\omega_{j}^{(k,i)}\}$ when $1\le i\le k\le 3$ as examples. First, define that
\begin{equation}
\label{ljqr}
I_{n,q}^{r}=\left\{
\begin{array}{cl}
\frac{1}{\Gamma(1-\alpha)}\int\limits_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r},& n\ge 0, \\
0, & n<0, \\
\end{array}
\right.
\end{equation}
where $q, r\in\mathbb{N}^{+}$ and $n\in\mathbb{Z}$. In addition, note that
\begin{equation*}
\begin{split}
& I_{n}:=I_{n,q}^{1},\hspace{0.3cm} \forall~q=1,2,\cdots, \\
&\nabla^{k}I_{n,q}^{r}=\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r}, \hspace{0.3cm} \forall~k\in\mathbb{N}^{+}.
\end{split}
\end{equation*}
Then the weight coefficients can be expressed in terms of integrals $I_{n,q}^{r}$ by
\begin{equation*}
\left\{
\begin{split}
(k,i)=(1,1): &\hspace{0.2cm}w_{m,0}=-I_{m},\hspace{0.3cm} m\ge 1, \hspace{0.4cm} \omega_{n}=\nabla I_{n},\hspace{0.3cm}n\ge 0, \\
(k,i)=(2,1): &\hspace{0.2cm}w_{m,0}=2I_{m-1,1}^{2}-I_{m,1}^{2}-I_{m}, \hspace{0.4cm} w_{m,1}=-I_{m-1,1}^{2},\hspace{0.2cm}m\ge 2, \\
&\hspace{0.2cm}\omega_{n}=\nabla I_{n}+\nabla^{2} I_{n,1}^{2},\hspace{0.3cm} n\ge 0, \\
(k,i)=(2,2): &\hspace{0.2cm}w_{m,0}=-\nabla I_{m+1,1}^{2}+I_{m,2}^{2}, \hspace{0.4cm}w_{m,1}=-I_{m,1}^{2},\hspace{0.3cm} m\ge 2, \\
&\hspace{0.2cm}\omega_{0}=I_{0}+I_{1}+I_{0,1}^{2}+I_{1,2}^{2},\hspace{0.4cm}\omega_{1}=\nabla I_{2}-I_{0}+I_{2,2}^{2}-2I_{0,1}^{2}-2I_{1,2}^{2}, \\
&\hspace{0.2cm}\omega_{2}=\nabla I_{3}+\nabla^{2}I_{3,2}^{2}+I_{0,1}^{2}, \hspace{0.4cm} \omega_{n}=\nabla I_{n+1}+\nabla^{2} I_{n+1,2}^{2},\hspace{0.2cm}n\ge 3, \\
\end{split}
\right.
\end{equation*}
and by more complicated formulae of forms
\begin{description}
\item [i).] $(k,i)=(3,1)$,
\begin{equation}\label{eq:ki31}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m}-I_{m,1}^{2}+2I_{m-1,1}^{2}+I_{m-1,2}^{2}-I_{m,1}^{3}+3I_{m-1,1}^{3}-3I_{m-2,1}^{3},\\
&w_{m,1}=-2I_{m-1}-2I_{m-1,2}^{2}-I_{m-1,1}^{2}-I_{m-1,1}^{3}+3I_{m-2,1}^{3},\\
&w_{m,2}=I_{m-1}+I_{m-1,2}^{2}-I_{m-2,1}^{3},\hspace{0.5cm} m\ge 3, \\
&\omega_{n}=\nabla I_{n}+\nabla^{2}I_{n,1}^{2}+\nabla^{3}I_{n,1}^{3}, \hspace{0.5cm} n\ge 0, \\
\end{split}
\right.
\end{equation}
\item [ii).] $(k,i)=(3,2)$,
\begin{equation}\label{eq:ki32}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m+1}-I_{m+1,2}^{2}+2I_{m,2}^{2}-I_{m+1,2}^{3}+3I_{m,2}^{3}-3I_{m-1,2}^{3},\\
&w_{m,1}=-I_{m}-I_{m,2}^{2}-I_{m,2}^{3}+3I_{m-1,2}^{3},\\
&w_{m,2}=-I_{m-1,2}^{3},\hspace{0.3cm}m\ge 3, \\
&\omega_{0}=I_{0}+I_{1}+I_{1,2}^{2}+I_{0,1}^{2}+I_{1,2}^{3}+I_{0,1}^{3}, \\
&\omega_{1}=\nabla I_{2}-I_{0}+I_{2,2}^{2}-2I_{1,2}^{2}-2I_{0,1}^{2}+I_{2,2}^{3}-3I_{1,2}^{3}-3I_{0,1}^{3}, \\
&\omega_{2}=\nabla I_{3}+\nabla^{2}I_{3,2}^{2}+I_{0,1}^{2}+I_{3,2}^{3}-3I_{2,2}^{3}+3I_{1,2}^{3}+3I_{0,1}^{3}, \\
&\omega_{3}=\nabla I_{4}+\nabla^{2}I_{4,2}^{2}+\nabla^{3}I_{4,2}^{3}-I_{0,1}^{3}, \\
&\omega_{n}=\nabla I_{n+1}+\nabla^{2}I_{n+1,2}^{2}+\nabla^{3}I_{n+1,2}^{3}, \hspace{0.3cm} n\ge 4,\\
\end{split}
\right.
\end{equation}
\item [iii).]$(k,i)=(3,3)$,
\begin{equation}\label{eq:ki33}
\left\{
\begin{split}
&w_{m,0}=-\nabla I_{m+2}-\nabla^{2} I_{m+2,3}^{2}-I_{m+2,3}^{3}+3I_{m+1,3}^{3}-3I_{m,3}^{3},\\
&w_{m,1}=-\nabla I_{m+1}-I_{m+1,3}^{2}+2I_{m,3}^{2}-I_{m+1,3}^{3}+3I_{m,3}^{3},\\
&w_{m,2}=-I_{m}-I_{m,3}^{2}-I_{m,3}^{3},\hspace{0.3cm}m\ge 3, \\
&\omega_{0}=I_{0}+I_{1}+I_{2}+I_{0,1}^{2}+I_{1,2}^{2}+I_{2,3}^{2}+I_{0,1}^{3}+I_{1,2}^{3}+I_{2,3}^{3}, \\
&\omega_{1}=\nabla I_{3}-I_{0}-I_{1}+I_{3,3}^{2}-2I_{2,3}^{2}-2I_{1,2}^{2}-2I_{0,1}^{2}+I_{3,3}^{3}-3I_{2,3}^{3}-3I_{1,2}^{3}-3I_{0,1}^{3},\\
&\omega_{2}=\nabla I_{4}+\nabla^{2}I_{4,3}^{2}+I_{1,2}^{2}+I_{0,1}^{2}+I_{4,3}^{3}-3I_{3,3}^{3}+3I_{2,3}^{3}+3I_{0,1}^{3}+3I_{1,2}^{3}, \\
&\omega_{3}=\nabla I_{5}+\nabla^{2} I_{5,3}^{2}+\nabla^{3} I_{5,3}^{3}-I_{1,2}^{3}-I_{0,1}^{3},\\
&\omega_{n}=\nabla I_{n+2}+\nabla^{2}I_{n+2,3}^{2}+\nabla^{3}I_{n+2,3}^{3}, \hspace{0.3cm}n\ge 4.\\
\end{split}
\right.
\end{equation}
\end{description}
In addition, it is observed that when $\alpha\to1$, the difference operator $D_{k,i}^{\alpha}u_{n}$ in $\eqref{Dki1}$ recovers to the $k$-step BDF method.
\begin{remark}
The construction process of operator \eqref{Dki} can be extended to the case of $\alpha>1$ as well. In a general case of $\lceil \alpha \rceil-1<\alpha<\lceil \alpha \rceil$, the interpolating polynomials $P_{i}^{k}(t)\in C_{p}^{k}(I)$ could be constructed as the approximations to $u(t)$ under the condition that $k\ge \lceil \alpha \rceil$, and the $\alpha$ order Caputo derivative of $P_{i}^{k}(t)$ are proposed in an analogous way by
\begin{equation*}
D_{k,i}^{\alpha}u(t)=\frac{1}{\Gamma(\lceil \alpha \rceil-\alpha)}\int_{0}^{t}(t-\xi)^{-\alpha+\lceil \alpha \rceil-1}\frac{\mathrm{d^{\lceil \alpha \rceil}} P_{i}^{k}}{\mathrm{d} \xi^{\lceil \alpha \rceil}}\mathrm{d}\xi
\end{equation*}
as the approximation to ${^{C}}D^{\alpha}u(t)$. Here the condition of $k\ge \lceil \alpha \rceil$ is required such that the $\lceil \alpha \rceil$ order derivative of $P_{i}^{k}(t)$ is nonzero a.e.. We take the case of $\lceil \alpha \rceil=2$ as an example. Assume that $\beta=\alpha-1\in (0,1)$, the polynomials $P_{i}^{2}(t)$ denoted by \eqref{eq:P2} are in the space $C_{p}^{2}(I)$, and it follows
\begin{equation*}
\begin{split}
D_{2,i}^{\alpha}u(t_{n})=&\frac{1}{\Gamma(1-\beta)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-\xi)^{-\beta}\frac{\mathrm{d^2} P_{i}^{2}}{\mathrm{d} \xi^2}\mathrm{d}\xi \\
=&\frac{(\Delta t)^{-\alpha}}{\Gamma(1-\beta)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\beta}\frac{\mathrm{d}^{2} P_{i}^{2}(t_{j-1}+s\Delta t)}{\mathrm{d}s^{2}}\mathrm{d}s,
\end{split}
\end{equation*}
where the last equality holds based on the relation $\frac{\mathrm{d}^{2} P_{i}^{k}(\xi(s))}{\mathrm{d}s^{2}}=\frac{\mathrm{d^2} P_{i}^{k}(\xi)}{\mathrm{d} \xi^2}\left(\frac{\mathrm{d}\xi}{\mathrm{d}s}\right)^{2}+\frac{\mathrm{d} P_{i}^{k}(\xi)}{\mathrm{d}\xi}\frac{\mathrm{d}^{2}\xi}{\mathrm{d}s^{2}}$.
Moreover, it can be rewritten as a form analogous to \eqref{Dki1}, and the corresponding weights coefficients $\{w_{n,i}\}$ and $\{\omega_{n}\}$ are therefore derived by
\begin{equation}\left\{
\begin{split}
&\omega_{j}^{(2,1)}=\nabla^{2} I_{j},\hspace{0.4cm} j\ge 0,\hspace{0.4cm} w_{n,0}^{(2,1)}=-\left(I_{n}-2I_{n-1}\right), \hspace{0.4cm}w_{n,1}^{(2,1)}=-I_{n-1},\hspace{0.4cm}n\ge 2, \\
&\omega_{0}^{(2,2)}=I_{0}+I_{1},\hspace{0.5cm}\omega_{1}^{(2,2)}=I_{2}-2I_{1}-2I_{0}, \hspace{0.5cm} \omega_{2}^{(2,2)}=\nabla^{2}I_{3}+I_{0}, \\
&\omega_{j}^{(2,2)}=\nabla^{2}I_{j+1}, \hspace{0.4cm}j\ge 3, \hspace{0.5cm} w_{n,0}^{(2,2)}=-I_{n+1}+2I_{n}, \hspace{0.4cm} w_{n,1}^{(2,2)}=-I_{n},\hspace{0.4cm}n\ge 2.
\end{split}
\right.
\end{equation}
Here the integrals $I_{n,q}^{r}=I_{n,q}^{r}(\beta)$ are defined by \eqref{ljqr} where the index $\alpha$ is replaced by $\beta$.
\end{remark}
\subsection{Complete monotonicity and error analysis}
First, we explore the completely monotonic property of the sequence $\{I_{n,q}^{r}\}_{n=0}^{\infty}$.
\begin{lemma}
\label{le:Ijqr}
Assume that $I_{n,q}^{r}$ is defined by \eqref{ljqr}, then for $n\ge k$ with $k\in \mathbb{N}$, it holds that
\begin{equation}
\label{eq:r<q}
(-1)^{k+r+1}\nabla^{k} I_{n,q}^{r}\ge 0
\end{equation}
in the case of $r\le q$, and
\begin{equation}
\label{eq:r>q}
(-1)^{k+q+1}\nabla^{k} I_{n,q}^{r}\ge 0
\end{equation}
in the case of $ r>q$.
\end{lemma}
\begin{proof}
We begin with the case of $r\le q$, according to the definition of $I_{n,q}^{r}$ in \eqref{ljqr}, it holds that
\begin{equation}\label{eq:inqr}
\begin{split}
I_{n,q}^{r}=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\sum_{n=0}^{r-1}\frac{1}{(s-q+n)}\binom{s-q+r-1}{r}\mathrm{d}s,
\end{split}
\end{equation}
since $(s-q+n)\le 0$ for $0\le s\le 1$ and $n=0,\cdots,r-1$, it yields that $(-1)^{r}\binom{s-q+r-1}{r}\ge 0$,
and consequently $(-1)^{r+1}\frac{\mathrm{d}}{\mathrm{d}s}\binom{s-q+r-1}{r}\ge 0$, combined with $(n+1-s)^{-\alpha}>0$ for any $n\ge 0$ and $\alpha>0$, it leads to $(-1)^{r+1}I_{n,q}^{r}\ge 0$. In addition, by definition, we may see that
\begin{equation}\label{eq:nablainqr}
\begin{split}
\nabla I_{n,q}^{r}=&\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}\Big((n+1-s)^{-\alpha}-(n-s)^{-\alpha}\Big)\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{n}^{n+1}(\xi-s)^{-\alpha-1}\mathrm{d}\xi\mathrm{d}\binom{s-q+r-1}{r} \\
=&\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1}(\xi+n-s)^{-\alpha-1}\mathrm{d}\xi\mathrm{d}\binom{s-q+r-1}{r},
\end{split}
\end{equation}
with $(\xi+n-s)^{-\alpha-1}\ge 0$ for $n\ge 1$ and $0\le \xi, s\le 1$, then $(-1)^{r+2}\nabla I_{n,q}^{r}\ge 0$.
Assume that for $k\ge 2$, it holds
\begin{equation*}
\nabla^{k-1}I_{n,q}^{r}=\frac{(-\alpha)_{k-1}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}(\sum_{i=1}^{k-1}\xi_{i}+n-k+2-s)^{-\alpha-k+1}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r},
\end{equation*}
where denote that $(\alpha)_{k-1}=\alpha(\alpha-1)\cdots(\alpha-k+2)$ and $\mathrm{d}^{k-1}\mathbf{\xi}=\mathrm{d}\xi_{1}\cdots\mathrm{d}\xi_{k-1}$, then
\begin{equation}\label{eq:nablakinqr}
\begin{split}
\nabla^{k}I_{n,q}^{r}&=\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r} \\
&=\frac{(-\alpha)_{k-1}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}\nabla(\sum_{i=1}^{k-1}\xi_{i}+n-k+2-s)^{-\alpha-k+1}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k}}\int_{n+1}^{n+2}(\sum_{i=1}^{k}\xi_{i}-k-s)^{-\alpha-k}\mathrm{d}\xi_{k}\mathrm{d}^{k-1}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r}.
\end{split}
\end{equation}
Since $(\sum\limits_{i=1}^{k}\xi_{i}+n-k+1-s)\ge 0$ for $n\ge k\ge 1$ and $0\le\xi_{i}, s\le 1$, then \eqref{eq:r<q} holds.
In the other case of $r\ge q+1$, integrating by part yields that
\begin{equation}\label{eq:inqr1}
\begin{split}
I_{n,q}^{r}&=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\mathrm{d}\binom{s-q+r-1}{r} \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha-1}\binom{s-q+r-1}{r}\mathrm{d}s,
\end{split}
\end{equation}
since $\binom{s-q+r-1}{r}$ includes a factor $s(s-1)$ for $r\ge q+1, q\in\mathbb{N}^{+}$. The sign of $\binom{s-q+r-1}{r} $ is the same with that of $\prod\limits_{i=1}^{q}(s-i)$, thus $(-1)^{q}\binom{s-q+r-1}{r}\ge 0$, and it holds that $(-1)^{q+1}I_{n,q}^{r}\ge 0$ for $n\ge 0$.
Furthermore, the induction process demonstrates that
\begin{equation}\label{eq:nablakinqr1}
\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k+1}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k-1}\binom{s-q+r-1}{r}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}s
\end{equation}
for $n\ge k\ge 1$, which arrives at \eqref{eq:r>q}.
\end{proof}
Moreover, we discuss the complete monotonicity of a general class of sequences.
\begin{lemma}
\label{le:1.5}
The sequence $\{s_{n}\}_{n=0}^{\infty}$ is defined by
\begin{equation*}
s_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}\varphi(s)\mathrm{d}s, \hspace{0.5cm} n\ge 0,
\end{equation*}
where $\varphi(s)\ge 0$ for $0\le s\le 1$. Then for $n\ge k$, it holds that $(-1)^{k}\nabla^{k}s_{n}\ge 0$.
\end{lemma}
\begin{proof}
It is easy to check that $s_{n}\ge 0$ for all $n\ge 0$, since for $n\ge 0$, $0\le s\le 1$, it holds that $(n+1-s)^{-\alpha}>0$ and $\varphi(s)\ge0$ by assumption. The definition of $s_{n}$ implies that
\begin{equation*}
\begin{split}
\nabla s_{n}&=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}\left((n+1-s)^{-\alpha}-(n-s)^{-\alpha}\right)\varphi(s)\mathrm{d}s \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1}(n+\xi-s)^{-\alpha-1}\varphi(s)\mathrm{d}\xi\mathrm{d}s, \\
\end{split}
\end{equation*}
since $(n+\xi-s)^{-\alpha-1}>0$ and $\varphi(s)\ge 0$ for $n\ge 1$ and $0\le s, \xi\le 1$, thus $\nabla s_{n}\le 0$ holds.
Therefore an induction process yields that
\begin{equation*}
\nabla^{k} s_{n}=\frac{(-\alpha)_{n}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}(\sum_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\mathrm{d}^{k}\xi\mathrm{d}s.
\end{equation*}
Since for $n\ge k$ and $0\le s,\xi_{i}\le 1$, it holds that $\sum\limits_{i=1}^{k}\xi_{i}+n-k+1-s)^{-\alpha-k}\varphi(s)\ge 0$, thus we can obtain that $(-1)^{k}\nabla^{k}s_{n}\ge 0$ for $n\ge k$.
\end{proof}
Next, we construct the numerical scheme
\begin{equation}
\label{eq:nnonliode}
D_{k,i}^{\alpha} u_{n}=f(t_{n},u_{n}), \hspace{0.5cm} n\ge k,
\end{equation}
as the approximation to problem \eqref{eq:nolinfode} with prescribed starting values, and define the local truncation error of the $n$-th step by
\begin{equation}
\label{eq:trunc}
\tau_{n}^{(k,i)}=D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n}), \hspace{0.5cm} n\ge k,
\hspace{0.2cm}n\in\mathbb{N}^{+},
\end{equation}
where $u(t)$ is the exact solution of problem \eqref{eq:nolinfode}.
\begin{theorem}
\label{th:errorestmat}
Assume that $u(t)\in C^{k+1}[0, T]$ and $0<\alpha<1$, it holds that
\begin{equation}\label{eq:truncki}
D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})=O\left((t_{n-k+i})^{-\alpha-1}\Delta t^{k+1}+\Delta t^{k+1-\alpha}\right),\hspace{0.4cm}
\end{equation}
for $n\ge k$ in the cases of $1\le i<k\le 6$. In particular
\begin{equation}\label{eq:trunckk}
D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})=O(\Delta t^{k+1-\alpha}), \hspace{0.4cm} k=1,\cdots,6
\end{equation}
holds uniformly for $n\ge k$.
\end{theorem}
\begin{proof}
According to \eqref{eq:pk}, it holds that
\begin{equation}
\label{eq:2.9}
p_{j,q}^{k}(t)-u(t)=u^{(k+1)}(\xi_{j})\binom{s-q+k}{k+1}(\Delta t)^{k+1},
\end{equation}
where $t=t_{j-1}+s\Delta t$ with $0\le s\le 1$ and $t_{j+q-k-1}\le \xi_{j}\le t_{j+q-1}$.
Inspired by \cite{GaoSZ:2014}, making use of the integration by part technique, we arrive at
\begin{equation}\label{eq:2.23}
\begin{split}
D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})&=\frac{1}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-t)^{-\alpha}
\left(\frac{\mathrm{d} P_{i}^{k}(t)}{\mathrm{d}t}-\frac{\mathrm{d} u(t)}{\mathrm{d} t}\right)\mathrm{d}t \\
&=\frac{-\alpha}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{t_{j-1}}^{t_{j}}(t_{n}-t)^{-\alpha-1}\left(P_{i}^{k}(t)-u(t)\right)\mathrm{d}t \\
&=\frac{-\alpha (\Delta t)^{-\alpha}}{\Gamma(1-\alpha)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\left(P_{i}^{k}(t_{j-1}+s\Delta t)-u(t_{j-1}+s\Delta t)\right)\mathrm{d}s
\end{split}
\end{equation}
for $n\ge k$, which is based on the conditions of \eqref{eq:2.4} and \eqref{eq:2.9}.
From the general representation of $P_{i}^{k}(t)$ in \eqref{eq:Pik}, it is known that for $k-i\ge 1$, the polynomials of degree $(k-1)$ are used on subinterval $\cup_{j=1}^{k-i}I_{j}$ to construct $P_{i}^{k}(t)$, in the other case of $k=i$, the polynomials of degree $k$ are chosen on each subinterval $I_{j}$ instead. Therefore, we next consider the two cases seperately.
Substituting \eqref{eq:Pik} and \eqref{eq:2.9} into the last equivalent formula of \eqref{eq:2.23}, if $k=i$, one obtains
\begin{equation*}
\begin{split}
|D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}\max_{\xi\in I}|u^{(k+1)}(\xi)|\Big(\sum_{j=1}^{n-k+1}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s}{k+1}\mathrm{d}s\Big| \\
&+\sum_{j=n-k+2}^{n}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-n-1+j}{k+1}\mathrm{d}s\Big|\Big),
\end{split}
\end{equation*}
and if $1\le i\le k-1$, one has
\begin{equation*}
\begin{split}
|D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le \frac{\alpha\left(\Delta t\right)^{-\alpha} }{\Gamma(1-\alpha)}\Big((\Delta t)^{k}\max_{\xi\in \cup_{j=1}^{k-i}I_{j}}|u^{(k)}(\xi)|\sum_{j=1}^{k-i}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+j-1}{k}\mathrm{d}s\Big| \\
&+(\Delta t)^{k+1}\max_{\xi\in I}|u^{(k+1)}(\xi)|\sum_{j=k-i+1}^{n-i+1}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-i}{k+1}\mathrm{d}s\Big| \\
&+(\Delta t)^{k+1}\max_{\xi\in I}|u^{(k+1)}(\xi)|\sum_{j=n-i+2}^{n}\Big|\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\binom{s+k-n-i+j}{k+1}\mathrm{d}s\Big|\Big).
\end{split}
\end{equation*}
Since for any $q\le k$ and $q, k\in \mathbb{N}^{+}$, the factor $(1-s)$ is included in $\binom{s-q+k}{k+1}$ and $\frac{1}{1-s}\binom{s-q+k}{k+1}$ is bounded for $0\le s
\le 1$, thus we can obtain that
\begin{equation*}
\begin{split}
|D_{k,k}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|\le&\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}C^{(k)}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}(1-s)\mathrm{d}s \\
\le&\frac{\alpha\left(\Delta t\right)^{k+1-\alpha} }{\Gamma(1-\alpha)}C^{(k)}\Big(\sum_{j=1}^{n-1}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\mathrm{d}s+\int_{0}^{1}(1-s)^{-\alpha}\mathrm{d}s\Big) \\
\le&\left(\Delta t\right)^{k+1-\alpha}C^{(k)}
\Big(\frac{1}{\Gamma(1-\alpha)}+\frac{1}{\Gamma(2-\alpha)}\Big),
\end{split}
\end{equation*}
where $C^{(k)}$ is bounded relevant to $u^{(k+1)}$ and $k$.
Moreover, for $i<k$, it holds
\begin{equation*}
\begin{split}
|D_{k,i}^{\alpha}u(t_{n})-{^{C}}D^{\alpha}u(t_{n})|&\le \frac{\alpha}{\Gamma(1-\alpha)}C^{(k,i)}\Big((\Delta t)^{k-\alpha}\sum_{j=1}^{k-i}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}\mathrm{d}s \\
&+(\Delta t)^{k+1-\alpha}\sum_{j=1}^{n}\int_{0}^{1}(n-j+1-s)^{-\alpha-1}(1-s)\mathrm{d}s\Big) \\
&\le C^{(k,i)}\Big(\frac{\alpha}{\Gamma(1-\alpha)}(\Delta t)^{k+1}(k-i)(t_{n-k+i})^{-\alpha-1} \\
&+(\Delta t)^{k+1-\alpha}\big(\frac{1}{\Gamma(1-\alpha)}+\frac{1}{\Gamma(2-\alpha)}\big)\Big),
\end{split}
\end{equation*}
where $C^{(k,i)}$ is a constant depending on $u^{(k)}$, $u^{(k+1)}$ and $k, i$.
\end{proof}
\begin{remark}
It is shown from formula \eqref{eq:truncki} that the order accuracy isn't uniform for all $n\ge k$. In the case of $t_{n}$ being near the origin, the accuracy order of the local truncation error reduced to the $(k-\alpha)$ order, in view that the $(k-1)$ degree polynomials as shown in \eqref{eq:P23} are chosen on the subinterval $\cup_{j=1}^{k-i} I_{j}$. However, replacing polynomials of degree $k$ on the corresponding subinterval can avoid this drawback, which is shown in \eqref{eq:trunckk}.
\end{remark}
\begin{remark}
There is need to point out that the local truncation error estimations \eqref{eq:truncki} and \eqref{eq:trunckk} holds only in the case of the solution $u(t)$ possessing proper smoothness on the closed interval $[0, T]$. In order to check the convergence rate of the global error when $f(t, u(t))$ is smooth with respect to $t$ and $u$, we apply the methods \eqref{Dki1} in the cases of $1\le i\le k\le 3$ on the test equation
\begin{equation}\label{eq:duf}
{^{C}}D^{\alpha}u(t)=f(t),\hspace{0.618cm} t\in (0, 1],
\end{equation}
such that the exact solution is $u(t)=E_{\alpha,1}(-t^{\alpha})\in C[0,1]\cap C^{\infty}(0,1]$. In Table \ref{ta:1} and \ref{ta:2}, the accuracy and the convergence order of the error $|u(t_{M})-u_{M}|$ are shown for different timestep and order $\alpha$. According to the numerical experiment, the high order convergence seems to reduce to the first order in the cases of $1\le i\le k\le 3$.
It is because that the solution of the problem \eqref{eq:nolinfode} only possess continuity on interval $I$ if function $f(t, u(t))$ is smooth on $I$. On the other hand, the integer order derivative of any smooth function on a compact domain $I$ still preserves to be smooth on $I$, in contrast, the $\alpha$ order fractional derivative of the smooth function isn't smooth any more, which implies there exist some continuous functions $f(t,u)$ such that the solution is smooth on $I$.
\end{remark}
\begin{table}
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in problem \eqref{eq:duf}.}
\label{ta:1}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l@{\extracolsep{1cm}}l @{\extracolsep{1.4cm}}l@{\extracolsep{1cm}}l@{\extracolsep{1cm}} l@{\extracolsep{1cm}}l @{\extracolsep{1cm}}l@{\extracolsep{1cm}}l@{\extracolsep{\fill}} }
\toprule
$\alpha$& $M$ & \multicolumn{2}{l}{$(k, i)=(1,1)$} & \multicolumn{2}{l}{$(k,i)=(2, 1)$}&\multicolumn{2}{l}{$(k,i)=(2, 2)$} \\
\cline{3-8}
&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.1 & 20 & 4.70994E-03 & - & 9.55476E-04 & - & 9.67511E-04 & - \\
& 40 & 2.37818E-03 & 0.99 & 4.93426E-04 & 0.95 & 4.96390E-04 & 0.96 \\
& 80 & 1.21118E-03 & 0.97 & 2.53608E-04 & 0.96 & 2.54352E-04 & 0.96 \\
& 160 & 6.19034E-04 & 0.97 & 1.30008E-04 & 0.96 & 1.30197E-04 & 0.97 \\
& 320 & 3.16763E-04 & 0.97 & 6.65299E-05 & 0.97 & 6.65780E-05 & 0.97 \\
\midrule
0.5 & 20 & 3.59879E-02 & - & 1.27599E-03 & - & 1.24693E-03 & - \\
& 40 & 1.87445E-02 & 0.94 & 5.84522E-04 & 1.13 & 5.76449E-04 & 1.11 \\
& 80 & 9.67807E-03 & 0.95 & 2.79573E-04 & 1.06 & 2.77440E-04 & 1.06 \\
& 160 & 4.95832E-03 & 0.96 & 1.36716E-04 & 1.03 & 1.36166E-04 & 1.03 \\
& 320 & 2.52435E-03 & 0.97 & 6.76067E-05 & 1.02 & 6.74666E-05 & 1.01 \\
\midrule
0.9 & 20 & 6.28955E-02 & - & 2.95957E-03 & - & 2.96453E-03 & - \\
& 40 & 3.23694E-02 & 0.96 & 1.33551E-03 & 1.15 & 1.33646E-03 & 1.15 \\
& 80 & 1.64619E-02 & 0.98 & 6.25412E-04 & 1.09 & 6.25601E-04 & 1.10 \\
& 160 & 8.31769E-03 & 0.98 & 3.00737E-04 & 1.06 & 3.00775E-04 & 1.06 \\
& 320 & 4.18769E-03 & 0.99 & 1.47049E-04 & 1.03 & 1.47057E-04 & 1.03 \\
\bottomrule
\end{tabular*}
\end{table}
\begin{table}[ht]
\centering
\footnotesize
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in problem \eqref{eq:duf}.}
\label{ta:2}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}l@{\extracolsep{1cm}}l @{\extracolsep{1.4cm}}l@{\extracolsep{1cm}}l@{\extracolsep{1cm}} l@{\extracolsep{1cm}}l @{\extracolsep{1cm}}l@{\extracolsep{1cm}}l@{\extracolsep{\fill}} }
\toprule
$\alpha$& $M$ & \multicolumn{2}{l}{$(k, i)=(3,1)$} & \multicolumn{2}{l}{$(k,i)=(3,2)$}&\multicolumn{2}{l}{$(k,i)=(3,3)$} \\
\cline{3-8}
&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.1 & 20 & 9.24206E-04 & - & 9.17788E-04 & - & 9.23060E-04 & - \\
& 40 & 4.71821E-04 & 0.97 & 4.70329E-04 & 0.96 & 4.71636E-04 & 0.97 \\
& 80 & 2.41265E-04 & 0.97 & 2.40900E-04 & 0.97 & 2.41229E-04 & 0.97 \\
& 160 & 1.23379E-04 & 0.97 & 1.23288E-04 & 0.97 & 1.23371E-04 & 0.97 \\
& 320 & 6.30607E-05 & 0.97 & 6.30376E-05 & 0.97 & 6.30591E-05 & 0.97 \\
\midrule
0.5 & 20 & 2.87149E-03 & - & 2.85982E-03 & - & 2.86650E-03 & - \\
& 40 & 1.42015E-03 & 1.02 & 1.41730E-03 & 1.01 & 1.41912E-03 & 1.01 \\
& 80 & 7.06991E-04 & 1.01 & 7.06289E-04 & 1.00 & 7.06757E-04 & 1.01 \\
& 160 & 3.52818E-04 & 1.00 & 3.52644E-04 & 1.00 & 3.52763E-04 & 1.00 \\
& 320 & 1.76251E-04 & 1.00 & 1.76208E-04 & 1.00 & 1.76237E-04 & 1.00 \\
\midrule
0.9 & 20 & 1.30920E-03 & - & 1.30931E-03 & - & 1.30919E-03 & - \\
& 40 & 6.18111E-04 & 1.08 & 6.18099E-04 & 1.08 & 6.18098E-04 & 1.08 \\
& 80 & 3.00274E-04 & 1.04 & 3.00268E-04 & 1.04 & 3.00270E-04 & 1.04 \\
& 160 & 1.47969E-04 & 1.02 & 1.47967E-04 & 1.02 & 1.47968E-04 & 1.02 \\
& 320 & 7.34347E-05 & 1.01 & 7.34342E-05 & 1.01 & 7.34345E-05 & 1.01 \\
\bottomrule
\end{tabular*}
\end{table}
\section{Stability analysis}\label{stabsection}
To consider the numerical stability of schemes \eqref{eq:nnonliode} with initial value $u(0)=u_{0}$,
the analysis on the linear difference equation
\begin{equation}
\label{DkiTest}
D_{k,i}^{\alpha}u_{n}=\lambda u_{n}, \hspace{0.618cm} n\ge k
\end{equation}
is given as follows. Using formulae \eqref{DkiTest}, we construct the equivalent relationship with respect to the generating power series
\begin{equation*}
\sum_{n=0}^{\infty}D_{k,i}^{\alpha}u_{n+k}\xi^{n}=\lambda\sum_{n=0}^{\infty}u_{n+k}\xi^{n}.
\end{equation*}
Replacing \eqref{Dki1}, one hence has
\begin{equation}
\label{DkiTest1}
\omega^{(k,i)}(\xi)u(\xi)=zu(\xi)+g^{(k,i)}(\xi),
\end{equation}
where $z:=\lambda(\Delta t)^{\alpha}$, the formal power series are denoted by
\begin{equation}\label{eq:fps}
\begin{split}
& u(\xi)=\sum\limits_{n=0}^{\infty}u_{n+k}\xi^{n}, \hspace{0.2cm} \omega^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n}, \\
&g^{(k,i)}(\xi)=-\sum_{j=0}^{k-1}u_{j}\sum_{n=0}^{\infty}\big(w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)}\big)\xi^{n}.
\end{split}
\end{equation}
Inspired by stability anlyses in \cite{LubichC:1983,LubichC:1986b}, we therefore present the following preliminary conclusions.
\begin{lemma}[\cite{LubichC:1986b}]
\label{le:3.4}
Assume that the coefficient sequence of $a(\xi)$ is in $l^{1}$. Let $|\xi_{0}|\le 1$. Then the coefficient sequence of
\begin{equation*}
b(\xi)=\frac{a(\xi)-a(\xi_{0})}{\xi-\xi_{0}}
\end{equation*}
converges to zero.
\end{lemma}
\begin{theorem}[\cite{Rudin:1987, ZygmundA:2002}]
\label{th:pw}
Suppose that
\begin{equation*}
f(z)=\sum_{n=0}^{\infty}c_{n}z^{n}, \hspace{0.4cm} \sum_{n=0}^{\infty}|c_{n}|<\infty,
\end{equation*}
and $f(z)\ne 0$ for every $|z|\le 1$. Then
\begin{equation*}
\frac{1}{f(z)}=\sum_{n=0}^{\infty}a_{n}z^{n}\hspace{0.2cm}\text{with}\hspace{0.1cm} \sum_{n=0}^{\infty}|a_{n}|<\infty.
\end{equation*}
\end{theorem}
\begin{theorem}[\cite{AkhiezerN:1965, ShohatJJ:1970}]
\label{th:cm}
For the moment problem
\begin{equation*}
s_{k}=\int_{0}^{1}u^{k}\mathrm{d}\sigma(u),\hspace{0.5cm} k=0, 1, \cdots
\end{equation*}
to be soluble within the class of non-decreasing functions iff the inequalities
\begin{equation*}
(-1)^{m}\nabla^{m}s_{k}\ge 0
\end{equation*}
hold for $k\ge m$.
\end{theorem}
\begin{lemma}
\label{le:3.1}
The coefficient sequences of series $g^{(k,i)}(\xi)$ converge to zero.
\end{lemma}
\begin{proof}
According to the expression of $\nabla^{k}I_{n,q}^{r}$ in Lemma \ref{le:Ijqr}, it yields that
\begin{equation*}
\lim_{n\to\infty}\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k}}{\Gamma(1-\alpha)}\int_{[0,1]^{k+1}}\lim_{n\to\infty}\Big(\sum_{i=1}^{k}\xi_{i}+n-k+1-s\Big)^{-\alpha-k}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}\binom{s-q+r-1}{r}=0
\end{equation*}
or
\begin{equation*}
\lim_{n\to\infty}\nabla^{k}I_{n,q}^{r}=\frac{(-\alpha)_{k+1}}{\Gamma(1-\alpha)}\int_{[0, 1]^{k+1}}\lim_{n\to\infty}\Big(\sum_{i=1}^{k}\xi_{i}+n-k+1-s\Big)^{-\alpha-k-1}\binom{s-q+r-1}{r}\mathrm{d}^{k}\mathbf{\xi}\mathrm{d}s=0
\end{equation*}
for $k, q, r\in\mathbb{N}^{+}$ that are independent of $n$ and $\alpha>0$. Note that $g_{n}^{(k,i)}=-\sum\limits_{j=0}^{k-1}u_{j}(w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)})$ is the finite linear combination of $\nabla^{k}I_{j,q}^{r}$ for finite $k$, thus it deduces $g_{n}^{(k,i)}\to 0$ as $n\to\infty$ if $\{u_{j}\}_{j=0}^{k-1}$ are bounded.
\end{proof}
\begin{lemma}
\label{le:3.2}
For $1\le i\le k\le 6$, the coefficient sequence of $\omega^{(k,i)}(\xi)$ belongs to $l^{1}$ space.
\end{lemma}
\begin{proof}
As indicated in Lemma \ref{le:Ijqr} and Lemma \ref{le:3.1}, the following relationship
\begin{equation}
\label{eq:3.3a}
\sum_{n=p}^{\infty}|\nabla^{k}I_{n,q}^{r}|=|\sum_{n=p}^{\infty} (\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r})|=|\nabla^{k-1}I_{p-1,q}^{r}|
\end{equation}
holds for $p\ge k\ge 1$. Therefore, according to the definition of sequence $\{\omega_{n}^{(k,i)}\}_{n=0}^{\infty}$, there exists finite positive integer $M=M(k,i)$, such that
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}|\omega_{n}^{(k,i)}|&\le \sum_{n=0}^{M}|\omega_{n}^{(k,i)}|+\sum_{m=1}^{k}\sum_{n=m}^{\infty}|\nabla^{m}I_{n,i}^{m}| \\
&\le \sum_{n=0}^{M}|\omega_{n}^{(k,i)}|+\sum_{m=1}^{k}|\nabla^{m-1} I_{m-1,i}^{m}|,
\end{split}
\end{equation*}
which implies the result.
\end{proof}
\begin{lemma}
\label{le:3.3}
For $1\le i\le k\le 6$ and $|\xi_{0}|\le 1$, the coefficient sequence of $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ belongs to $l^{1}$ space.
\end{lemma}
\begin{proof}
According to the expression of $\omega^{(k,i)}(\xi)$, the following series can be rewritten to
\begin{equation*}
\begin{split}
(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}&=(1-\xi)\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\frac{\xi^{n}-\xi_{0}^{n}}{\xi-\xi_{0}} \\
&=(1-\xi)\sum_{n=1}^{\infty}\omega_{n}^{(k,i)}\sum_{m=0}^{n-1}\xi_{0}^{n-1-m}\xi^{m} \\
&=(1-\xi)\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\omega_{n+m+1}^{(k,i)}\xi_{0}^{n}\xi^{m} \\
&=\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}+\sum_{m=1}^{\infty}\big(\sum_{n=0}^{\infty}\nabla \omega_{n+m+1}^{(k,i)}\xi_{0}^{n}\big)\xi^{m}.
\end{split}
\end{equation*}
On one hand, from Lemma \ref{le:3.2}, we have
\begin{equation*}
|\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}|\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}||\xi_{0}|^{n}\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}|<+\infty.
\end{equation*}
On the other hand, by the definition of $\{\nabla^{k+1}I_{n,q}^{r}\}_{n=k+1}^{\infty}$ in Lemma \ref{le:Ijqr}, it can be verified that
\begin{equation*}
\begin{split}
\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}|\nabla^{k+1} I_{m+n+1,q}^{r}|&=|\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}\big(\nabla^{k} I_{m+n+1,q}^{r}-\nabla^{k} I_{m+n,q}^{r}\big)| \\
&=|\sum_{m=p}^{\infty}\big(\nabla^{k-1} I_{m,q}^{r}-\nabla^{k-1} I_{m-1,q}^{r}\big)| \\
&=|\nabla^{k-1}I_{p-1,q}^{r}|
\end{split}
\end{equation*}
for $p\ge k\ge 1$. Therefore there exists $M_{1}=M_{1}(k,i)\ge 1$ and $M_{2}=M_{2}(k,i)\ge 0$ such that
\begin{equation*}
\begin{split}
\sum_{m=1}^{\infty}\sum_{n=0}^{\infty}|\nabla \omega_{n+m+1}^{(k,i)}|&\le \sum_{m=1}^{M_{1}}\sum_{n=0}^{M_{2}}|\nabla \omega_{n+m+1}^{(k,i)}|+\sum_{p=1}^{k}\sum_{m=p}^{\infty}\sum_{n=0}^{\infty}|\nabla^{p+1}I_{m+n+1,i}^{p}| \\
&\le \sum_{m=1}^{M_{1}}\sum_{n=0}^{M_{2}}|\nabla \omega_{n+m+1}^{(k,i)}|+\sum_{p=1}^{k}|\nabla^{p-1}I_{p-1,i}^{p}|.
\end{split}
\end{equation*}
Combining with
\begin{equation*}
|\sum_{n=0}^{\infty}\omega_{n+1}^{(k,i)}\xi_{0}^{n}|+\sum_{m=1}^{\infty}|\sum_{n=0}^{\infty}\nabla \omega_{n+m+1}^{(k,i)}\xi_{0}^{n}|\le \sum_{n=0}^{\infty}|\omega_{n+1}^{(k,i)}|+\sum_{m=1}^{\infty}\sum_{n=0}^{\infty} |\nabla \omega_{n+m+1}^{(k,i)}|,
\end{equation*}
we arrive at the conclusion.
\end{proof}
\begin{corollary}
\label{co:1}
For any $|\xi_{0}|\le 1$ and $1\le i\le k\le 6$, it holds that the sequence
\begin{equation*}
(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}
\end{equation*}
belongs to $l^{1}$ space, where series $\varphi^{(k,i)}(\xi)$ is defined to satisfy the relation $
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi)$.
\end{corollary}
\begin{proof}
Based on the definition of $\varphi^{(k,i)}(\xi)$, it yields that
\begin{equation*}
\begin{split}
(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}&=(1-\xi)\frac{(1-\xi)\varphi^{(k,i)}(\xi)-(1-\xi_{0})\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}} \\
&=(1-\xi)(1-\xi_{0})\frac{\varphi^{(k,i)}(\xi)-\varphi^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}-\omega^{(k,i)}(\xi),
\end{split}
\end{equation*}
because of the absolute convergence of sequences $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ and $\omega^{(k,i)}(\xi)$ shown in Lemma \ref{le:3.2} and Lemma \ref{le:3.3}, we shall arrive at the result.
\end{proof}
\begin{theorem}
\label{the:stabilityr}
The stability region of method $D_{k,i}^{\alpha}u_{n}=\lambda u_{n}$ is $\mathrm{S}^{(k,i)}=\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ in the cases of $1\le i\le k\le 6$.
\end{theorem}
\begin{remark}
The definition of stability region $\mathrm{S}^{(k,i)}$ of method $D_{k,i}^{\alpha}u_{n}=\lambda u_{n}$ is the set of $z=\lambda (\Delta t)^{\alpha}\in\mathbb{C}$ with $\Delta t>0$ for which there is $u_{n}\to 0$ as $n\to\infty$ whenever the starting values $u_{0}, \cdots, u_{k-1}$ are bounded.
\end{remark}
\begin{proof}
The provement of $S^{(k,i)}=\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ is equivalent with proving both $S^{(k,i)}\supseteq\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ and $S^{(k,i)}\subseteq\mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, i.e., to prove that for any $z\in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, there is $z\in S^{(k,i)}$ and for any $z\not \in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$, there is $z\not\in S^{(k,i)}$.
On one hand, if $z\in \mathbb{C}\backslash\{\omega^{(k,i)}(\xi): |\xi|\le 1\}$ and $|z|\le 1$, there is $z-\omega^{(k,i)}(\xi)\neq 0$ for $|\xi|\le 1$, thus according to Lemma \ref{le:3.1}, Lemma \ref{le:3.2} and Theorem \ref{th:pw}, it yields that the coefficient sequence of reciprocal of $z-\omega^{(k,i)}(\xi)$ is in $l^{1}$ and coefficient sequence of series $g^{(k,i)}(\xi)$ tends to zero.
If $|z|>1$, formula \eqref{DkiTest1} can be rewritten to
\begin{equation*}
u(\xi)=\frac{\frac{g^{(k,i)}(\xi)}{z}}{\frac{\omega^{(k,i)}(\xi)}{z}-1},
\end{equation*}
in which case the coefficient sequence of reciprocal of $\frac{\omega^{(k,i)}(\xi)}{z}-1$ is in $l^{1}$, and the coefficient sequence of series $\frac{g^{(k,i)}(\xi)}{z}$ converges to zero. In addition, assume that $\lim\limits_{n\to\infty}\sum\limits_{j=0}^{n}|l_{i}|=L<+\infty$ and $\lim\limits_{j\to\infty}c_{j}=0$, it holds that $\lim\limits_{n\to\infty}\sum\limits_{j=0}^{n}l_{n-j}c_{j}=0$, thus, implies that $u_{n}\to 0$ as $n\to\infty$.
On the other hand, assume that for any $z=\omega^{(k,i)}(\xi_{0})$ with $|\xi_{0}|\le 1$, according to \eqref{DkiTest1} the solution satisfies that
\begin{equation}
\label{eq:3.3b}
\left(\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})\right)u(\xi)=g^{(k,i)}(\xi).
\end{equation}
Note that method \eqref{Dki1} is exact for constant function, which leads to
\begin{equation*}
\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}=0,\hspace{0.5cm} n\ge k,
\end{equation*}
and a corresponding formal power series satisfies that
\begin{equation*}
\begin{split}
&\sum_{n=k}^{\infty}\left(\sum_{j=0}^{k-1}w_{n,j}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}\right)\xi^{n-k} \\
=&\sum_{n=0}^{\infty}\left(\sum_{j=0}^{k-1}w_{n+k,j}^{(k,i)}+\sum_{j=0}^{n+k}\omega_{n+k-j}^{(k,i)}\right)\xi^{n} \\
=&\sum_{n=0}^{\infty}\sum_{j=0}^{k-1}\left( w_{n+k,j}^{(k,i)}+\omega_{n+k-j}^{(k,i)} \right)\xi^{n}+\frac{\omega^{(k,i)}(\xi)}{1-\xi}=0. \\
\end{split}
\end{equation*}
Assume that $u_{0}=\cdots=u_{k-1}\neq0$, then according to the expression of $g^{(k,i)}(\xi)$, it holds that
$
g^{(k,i)}(\xi)=u_{0}\frac{\omega^{(k,i)}(\xi)}{1-\xi}$. In the case of $\omega^{(k,i)}(\xi_{0})=0$, it yields that $u(\xi)=\frac{u_{0}}{1-\xi}$, which means that $u_{n}=u_{0}$ for any $n\in \mathbb{N}$. And for the rest case, there is
\begin{equation*}
u(\xi)(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}=u_{0}\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}+u_{0}\frac{\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}.
\end{equation*}
If assume that $u_{n}\to0$ as $n\to\infty$, since according to Lemma \ref{le:3.3}, the coefficient sequence of $(1-\xi)\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ is in $l_{1}$, which derives that the coefficient sequence of $u(\xi)(1-\xi)\frac{\omega^{(1,1)}(\xi)-\omega^{(1,1)}(\xi_{0})}{\xi-\xi_{0}}$ tends to zero, in addition, according to Lemma \ref{le:3.4}, it yields that the coefficient sequence of $\frac{\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})}{\xi-\xi_{0}}$ converges to zero, however, the divergence of the coefficient sequence of $\frac{1}{\xi-\xi_{0}}$ for $|\xi_{0}|\le 1$ leads to the contradiction. Thus, it holds that there exist some nonzero bounded initial values $\{u_{i}\}_{i=0}^{k-1}$ such that $u_{n} \not\to 0$ as $n\to\infty$, which indicates that $z\not\in \mathrm{S}^{(k,i)}$.
\end{proof}
According to the definition of $A(\theta)$-stability \cite{HairerWN:1991} in usual case, we define the $A(\theta)$-stability in the following sense of $0<\alpha<1$.
\begin{definition}
A method is said to be $A(\theta)$-stable for $\theta\in[0, \pi-\frac{\alpha\pi}{2})$, if the sector
\begin{equation*}
S_{\theta}=\{z: |\mathrm{arg}(-z)|\le \theta, \hspace{0.2cm}z\neq 0\}
\end{equation*}
is contained in the stability region.
\end{definition}
\begin{theorem}\label{th:Api}
The methods \eqref{DkiTest} are $A(\frac{\pi}{2})$-stable in the cases of $1\le i\le k\le 2$.
\end{theorem}
\begin{proof}
In view of the definition of $A(\theta)$-stability, in particular, when $\theta=\frac{\pi}{2}$, it suffices to prove that $\mathrm{S}_{\frac{\pi}{2}}\subseteq S^{(k,i)}$ for $1\le i\le k\le 2$, i.e., to prove $\omega^{(k,i)}(\xi)=0$ for some $|\xi|\le 1$ and $\mathrm{Re}(\omega^{(k,i)}(\xi))>0$ otherwise.
First of all, it can be readily checked that $\omega^{(k,i)}(1)=0$, which implies $0\not \in \mathrm{S}_{\frac{\pi}{2}}$. In the case of $(k,i)=(1,1)$, resulting from the expression of $\omega^{(1,1)}(\xi)$, there is
\begin{equation}
\label{eq:omega11}
\omega^{(1,1)}(\xi)=I_{0}+\sum_{j=1}^{\infty}\nabla I_{j}\xi^{j}=(1-\xi)I(\xi),
\end{equation}
where $I(\xi)=\sum\limits_{n=0}^{\infty}I_{n}\xi^{n}$.
Since, according to Lemma \ref{le:Ijqr} and Theorem \ref{th:cm}, we have
\begin{equation}
\label{eq:ln}
I_{n}=\int_{0}^{1}r^{n}\mathrm{d}\sigma(r), \hspace{0.4cm} n\in \mathbb{N},
\end{equation}
where $\sigma(r)$ is a non-decreasing function,
then suppose that $|\xi|<1$, substituting \eqref{eq:ln} into \eqref{eq:omega11} yields that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(1,1)}(\xi)\Big)=\mathrm{Re}\Big((1-\xi)\sum_{n=0}^{\infty}\int_{0}^{1}r^{n}\mathrm{d}\sigma(r)\xi^{n}\Big)=\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\sigma(r).
\end{equation*}
Let $\xi=|\xi|(\cos\theta+i\sin\theta)$, there is
\begin{equation*}
\frac{1-\xi}{1-r\xi}
=\frac{\left(1-(r+1)|\xi|\cos\theta+r|\xi|^{2}\right)+i\left((r-1)|\xi|\sin\theta\right)}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}},
\end{equation*}
and for $0\le r\le1$ and $|\xi|<1$, it holds that
\begin{equation*}
\begin{split}
&1-(r+1)|\xi|\cos\theta+r|\xi|^{2}\ge \min\left((1-|\xi|\cos\theta)^{2},1-|\xi|\cos\theta\right), \\
&1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}\le (1+r|\xi|)^{2}\le 4,
\end{split}
\end{equation*}
which arrives at
\begin{equation*}
\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\sigma(r)
\ge\frac{\min\left((1-|\xi|\cos\theta)^{2},1-|\xi|\cos\theta\right)}{4}I_{0}.
\end{equation*}
In other case of $(k,i)=(2,1)$, from the definition of $\omega^{(2,1)}(\xi)$, it induces that
\begin{equation}\label{eq:fomega21}
\begin{split}
\omega^{(2,1)}(\xi)=&\sum_{n=0}^{\infty}\left(\nabla I_{n}+\nabla^{2}I_{n,1}^{2}\right)\xi^{n} \\
=&(1-\xi)I(\xi)+(1-\xi)^{2}I_{1}^{2}(\xi) \\
=&(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)+(3-\xi)I_{1}^{2}(\xi)\right),
\end{split}
\end{equation}
where
\begin{equation*}
I(\xi)=\sum_{n=0}^{\infty}I_{n}\xi^{n},\hspace{0.7cm} I_{1}^{2}(\xi)=\sum_{n=0}^{\infty}I_{n,1}^{2}\xi^{n}.
\end{equation*}
According to Lemma \ref{le:Ijqr}, Corollary \ref{co:s} and Theorem \ref{th:cm},
there exist non-decreasing functions $\upsilon$ and $\gamma$, respectively, such that
\begin{equation}
\label{eq:l12}
I_{n}-2I_{n,1}^{2}=\int_{0}^{1}r^{n}\mathrm{d}\upsilon(r), \hspace{0.3cm}n=0,1,\cdots,
\end{equation}
and
\begin{equation}
\label{eq:ln12}
I_{n,1}^{2}=\int_{0}^{1}r^{n}\mathrm{d}\gamma(r),\hspace{0.5cm}n=0,1,\cdots.
\end{equation}
Then for $|\xi|<1$, in place of $\omega^{(2,1)}(\xi)$, we can get
\begin{equation*}
\mathrm{Re}\Big(\omega^{(2,1)}(\xi)\Big)=\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\upsilon(r)+\int_{0}^{1}\mathrm{Re}\Big(\frac{(1-\xi)(3-\xi)}{1-r\xi}\Big)\mathrm{d}\gamma(r).
\end{equation*}
Moreover, it indicates
\begin{equation*}
\begin{split}
\frac{(1-\xi)(3-\xi)}{1-r\xi}
=&\frac{(3-4|\xi|\cos\theta+|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+(4-2|\xi|\cos\theta)r|\xi|^{2}\sin^{2}\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}} \\
&\hspace{0.5cm}+i\frac{(3r-|\xi|^{2}r-4+2|\xi|\cos\theta)|\xi|\sin\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}},
\end{split}
\end{equation*}
since
\begin{equation*}
\begin{split}
3-4|\xi|\cos\theta+|\xi|^{2}\cos2\theta=3-4|\xi|\cos\theta+2|\xi|^{2}\cos^{2}\theta-|\xi|^{2}\ge 2(1-|\xi|\cos\theta)^{2},
\end{split}
\end{equation*}
there is
\begin{equation*}
\int_{0}^{1}\mathrm{Re}\Big(\frac{(1-\xi)(3-\xi)}{1-r\xi}\Big)\mathrm{d}\gamma(r)
\ge\frac{\min\Big((1-|\xi|\cos\theta)^{3},(1-|\xi|\cos\theta)^{2}\Big)}{2}I_{0,1}^{2}.
\end{equation*}
For the rest case of $(k,i)=(2,2)$, we begin with the equivalent form of $\omega^{(2,2)}(\xi)$, which satisfies that
\begin{equation}\label{eq:fomega22}
\begin{split}
\omega^{(2,2)}(\xi)=&I_{0}(1-\xi)+I_{0,1}^{2}(1-\xi)^{2}+(1-\xi)\sum_{n=0}^{\infty}I_{n+1}\xi^{n}+(1-\xi)^{2}\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n} \\
=&I_{0,1}^{2}(1-\xi)(3-\xi)+(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)\right),
\end{split}
\end{equation}
since for any $n\ge 0$, because of the relation $I_{n}+I_{n,2}^{2}=I_{n,1}^{2}$, there is
\begin{equation*}
\begin{split}
&(1-\xi)\sum_{n=0}^{\infty}I_{n+1}\xi^{n}+(1-\xi)^{2}\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n} \\
=&(1-\xi)\Big(\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}-\xi\sum_{n=0}^{\infty}I_{n+1,2}^{2}\xi^{n}\Big) \\
=&(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\sum_{n=0}^{\infty}\left(I_{n+1}-2I_{n+1,1}^{2}\right)\xi^{n+1}\\
=&(1-\xi^{2})\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+(1-\xi)\left(I(\xi)-2I_{1}^{2}(\xi)-(I_{0}-2I_{0,1}^{2})\right).
\end{split}
\end{equation*}
Consequently, suppose that $|\xi|<1$, substituting conclusions \eqref{eq:l12} and \eqref{eq:ln12} into $\omega^{(2,2)}(\xi)$, we have
\begin{equation*}
\begin{split}
\mathrm{Re}\Big(\omega^{(2,2)}(\xi)\Big)=&\int_{0}^{1}\mathrm{Re}\Big((1-\xi)(3-\xi)\Big)\mathrm{d}\gamma(r) \\
&+\int_{0}^{1}r\mathrm{Re}\Big(\frac{1-\xi^{2}}{1-r\xi}\Big)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\Big(\frac{1-\xi}{1-r\xi}\Big)\mathrm{d}\upsilon(r),
\end{split}
\end{equation*}
furthermore, there is
\begin{equation*}
\begin{split}
\frac{1-\xi^{2}}{1-r\xi}
=&\frac{(1-|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+r|\xi|^{3}\sin\theta\sin2\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}} \\
&+i\frac{(1-|\xi|^{2}\cos2\theta)r|\xi|\sin\theta-(1-r|\xi|\cos\theta)\rho^{2}\sin2\theta}{(1-r|\xi|\cos\theta)^{2}+(r|\xi|\sin\theta)^{2}}.
\end{split}
\end{equation*}
Since for $0\le r\le 1$, it holds that
\begin{equation*}
\begin{split}
&(1-|\xi|^{2}\cos2\theta)(1-r|\xi|\cos\theta)+r|\xi|^{3}\sin\theta\sin2\theta \\
=&1-|\xi|^{2}\cos2\theta-r|\xi|\cos\theta+r|\xi|^{3}\cos\theta \\
\ge&(1-|\xi|^{2})(1-|\xi||\cos\theta|),
\end{split}
\end{equation*}
then, we may see that
\begin{equation*}
\begin{split}
\int_{0}^{1}r\mathrm{Re}\Big(\frac{1-\xi^{2}}{1-r\xi}\Big)\mathrm{d}\gamma(r)
\ge& \frac{(1-|\xi|^{2})(1-|\xi||\cos\theta|)}{4}\int_{0}^{1}r\mathrm{d}\gamma(r) \\
=&\frac{(1-|\xi|^{2})(1-|\xi||\cos\theta|)}{4}I_{1,1}^{2}.
\end{split}
\end{equation*}
As a result, for $1\le i\le k\le 2$, it demonstrates that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(k,i)}(\xi)\Big)\ge\frac{\min\left((1-|\xi|\cos\theta)^{2}, 1-|\xi|\cos\theta\right)}{4}I_{0}>0,\hspace{0.6cm}|\xi|<1.
\end{equation*}
In addition, according to Lemma \ref{le:3.3}, there exists constant $M^{(k,i)}>0$ such that
\begin{equation*}
|\omega^{(k,i)}(\xi)-\omega^{(k,i)}(\xi_{0})| \le \frac{M^{(k,i)}}{|1-\xi |} |\xi-\xi_{0}|,\hspace{0.5cm} \xi\ne 1,
\end{equation*}
which yields the pointwise continuity of $\omega^{(k,i)}(\xi)$ for $|\xi|\le 1$ with the exception of $\xi=1$. Therefore for any fixed $\xi$ lying on the unit circle, the angle of which satisfying $\mathrm{arg}(\xi)=\theta_{\xi}\neq 0$, correspondingly, there exists a sequence $\xi_{n}=(1-\frac{1}{n})\xi$ with $|\xi_{n}|<1$ for any $n=1, 2, \cdots$, such that
\begin{equation*}
\mathrm{Re}\Big(\omega^{(k,i)}(\xi)\Big)=\lim\limits_{n\to\infty}\mathrm{Re}\Big(\omega^{(k,i)}(\xi_{n})\Big)\ge \frac{I_{0}}{4}\min\left( (1-\cos\theta_{\xi})^{2}, 1-\cos\theta_{\xi}\right)>0.
\end{equation*}
\end{proof}
Theorem \ref{th:Api} shows that all the rest zeros of series $\omega^{(k,i)}(\xi)~(1\le i\le k\le2)$ are outside the unit disc besides $\xi=1$, followed by which we next consider the location of zeros of series $\omega^{(k,i)}(\xi)$ in the examples of $1\le i\le k\le 3$ and confirm that $\xi=1$ is a simple zero. The following result can be considered as a generalisation of the strong root condition.
\begin{theorem}
\label{coro:1}
For $1\le i\le k\le 3$, the series $\omega^{(k,i)}(\xi)$ satisfies the following conditions:
\begin{description}
\item[i).] $\omega^{(k,i)}(\xi)\ne 0$ within the unit circle $|\xi|\le 1$ and $\xi\ne 1$;
\item[ii).] $\xi=1$ is the simple zero.
\end{description}
\end{theorem}
\begin{proof}
It can be easily checked that $\omega^{(k,i)}(1)=0$, yielding that $\xi=1$ is a zero, if we rewrite the series $\omega^{(k,i)}(\xi)$ in the form of
\begin{equation}\label{eq:fomega}
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi),
\end{equation}
it remains to prove that $\varphi^{(k,i)}(\xi)\ne 0$ for $|\xi|\le 1$, which is suffice to prove that $\mathrm{Re}(\varphi^{(k,i)}(\xi))>0$ for all $|\xi|\le 1$. First of all, assuming $\xi=|\xi|e^{i\theta}$
with $|\xi|<1$. Then in the case of $(k,i)=(1,1)$, the rewritten form \eqref{eq:ln} deduces that
\begin{equation*}
\varphi^{(1,1)}(\xi)=I(\xi)=\int_{0}^{1}\frac{1}{1-r\xi}\mathrm{d\sigma(r)},
\end{equation*}
and furthermore,
\begin{equation*}
\mathrm{Re}\left(\varphi^{(1,1)}(\xi)\right)=\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d\sigma(r)}:=\int_{0}^{1}f(r,|\xi|,\theta)\mathrm{d\sigma(r)},
\end{equation*}
where $f(r,|\xi|,\theta)=\frac{1-r|\xi|\cos\theta}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}}$. Since a calculation deduces
\begin{equation*}
\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)=\frac{-r|\xi|\sin\theta (1-r^{2}|\xi|^{2})}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}
\end{equation*}
possesses the same sign as $(-\sin\theta)$ for $0\le r\le 1$ and $0\le |\xi|<1$, it hence follows that $0<f(r,|\xi|,\pi)\le f(r,|\xi|,\theta)\le f(r,|\xi|,0)$ and thus
\begin{equation*}
\mathrm{Re}\left(\varphi^{(1,1)}(\xi)\right)\ge \int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d\sigma(r)}\ge \frac{I_{0}}{2}.
\end{equation*}
In the case of $(k,i)=(2,1)$, according to formulae \eqref{eq:fomega21}, \eqref{eq:l12} and \eqref{eq:ln12}, one obtains
\begin{equation*}
\begin{split}
\varphi^{(2,1)}(\xi)=&I(\xi)-2I_{1}^{2}(\xi)+(3-\xi)I_{1}^{2}(\xi) \\
=&\int_{0}^{1}\frac{1}{1-r\xi}\mathrm{d}\upsilon(r)+\int_{0}^{1}\frac{3-\xi}{1-r\xi}\mathrm{d}\gamma(r)
\end{split}
\end{equation*}
and thus
\begin{equation*}
\begin{split}
\mathrm{Re}\left(\varphi^{(2,1)}(\xi)\right)=&\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\upsilon(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{3-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r) \\
\ge&\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\upsilon(r)+2\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r) >\frac{I_{0}}{2}.
\end{split}
\end{equation*}
In the case of $(k,i)=(2,2)$, it can be obtained from \eqref{eq:fomega22} that
\begin{equation*}
\varphi^{(2,2)}(\xi)=I_{0,1}^{2}(3-\xi)+(1+\xi)\sum_{n=0}^{\infty}I_{n+1,1}^{2}\xi^{n}+I(\xi)-2I_{1}^{2}(\xi),
\end{equation*}
and consequently, the real part of the series can be expressed by
\begin{equation*}
\mathrm{Re}\left(\varphi^{(2,2)}(\xi)\right)=I_{0,1}^{2}\mathrm{Re}(3-\xi)+\int_{0}^{1}r\mathrm{Re}\left(\frac{1+\xi}{1-r\xi}\right)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\upsilon(r)\ge\frac{I_{0}}{2}+I_{0,1}^{2}.
\end{equation*}
Since
\begin{equation*}
\mathrm{Re}\left(\frac{1+\xi}{1-r\xi}\right)=\frac{1-r|\xi|\cos\theta+|\xi|\cos\theta-r|\xi|^{2}}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}}:=f(r,|\xi|,\theta),
\end{equation*}
the relation $\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)=\frac{-|\xi|\sin\theta(r+1)(1-r^{2}|\xi|^{2})}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}$ induces that $f(r,|\xi|,\theta)\ge f(r,|\xi|,\pi)>0$ for $0\le r\le 1$ and $0\le |\xi|<1$.
In the case of $(k,i)=(3,1)$, based on \eqref{eq:ki31}, the rewritten form of series $\omega^{(3,1)}(\xi)$ is
\begin{equation*}
\omega^{(3,1)}(\xi)=(1-\xi)I(\xi)+(1-\xi)^{2}I_{1}^{2}(\xi)+(1-\xi)^{3}I_{1}^{3}(\xi),
\end{equation*}
and consequently $\varphi^{(3,1)}(\xi)$ is given by
\begin{equation*}
\begin{split}
\varphi^{(3,1)}(\xi)&=I(\xi)+(1-\xi)I_{1}^{2}(\xi)+(1-\xi)^{2}I_{1}^{3}(\xi) \\
&=I(\xi)-3I_{1}^{3}(\xi)+(1-\xi)I_{1}^{2}(\xi)+(4-2\xi+\xi^{2})I_{1}^{3}(\xi). \\
\end{split}
\end{equation*}
According to Lemma \ref{le:1.5}, it yields that
\begin{equation*}
\begin{split}
I_{n}-3I_{n,1}^{3}&=\frac{3}{2}\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}(1-s^{2})\mathrm{d}s, \hspace{0.5cm} n\ge 0\\
\end{split}
\end{equation*}
is a completely monotonic sequence, from Theorem \ref{th:cm} there exists a non-decreasing function $\eta$ such that
\begin{equation*}
I_{n}-3I_{n,1}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\eta(r), \hspace{0.5cm} n=0,1,\cdots.
\end{equation*}
In addition, we have from Lemma \ref{le:Ijqr} that sequence $\{I_{n,1}^{3}\}_{n=0}^{\infty}$ is a complete monotonic sequence, therefore it can be represented by
\begin{equation}\label{eq:ln13}
I_{n,1}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\beta(r), \hspace{0.5cm} n=0,1,\cdots,
\end{equation}
where the function $\beta(r)$ is non-decreasing on $[0,1]$. We thus represent the series into the integral form and take the real part,
\begin{equation*}
\begin{split}
\mathrm{Re}\left(\varphi^{(3,1)}(\xi)\right)=&
\int_{0}^{1}\mathrm{Re}\left(\frac{1}{1-r\xi}\right)\mathrm{d}\eta(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{1-\xi}{1-r\xi}\right)\mathrm{d}\gamma(r)+\int_{0}^{1}\mathrm{Re}\left(\frac{4-2\xi+\xi^{2}}{1-r\xi}\right)\mathrm{d}\beta(r) \\
\ge&\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\eta(r)+\frac{5}{2}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\beta(r)
\ge\frac{1}{2}I_{0}-\frac{1}{4}I_{0,1}^{3},
\end{split}
\end{equation*}
since there holds following estimate
\begin{equation*}
\mathrm{Re}\left(\frac{\frac{3}{2}-2\xi+\xi^{2}}{1-r\xi}\right)=\frac{(1-r|\xi|\cos\theta)(\frac{3}{2}-2|\xi|\cos \theta+|\xi|^{2}\cos2\theta)+2r|\xi|^{2}\sin^{2}\theta(1-|\xi|\cos\theta)}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}},
\end{equation*}
and there is
\begin{equation*}
\frac{3}{2}-2|\xi|\cos \theta+|\xi|^{2}\cos2\theta=\frac{1}{2}(1-2|\xi|\cos\theta)^{2}+(1-|\xi|^{2})\ge 0
\end{equation*}
for $|\xi|\le 1$ and $\theta\in\mathbb{R}$.
In the case of $(k,i)=(3,2)$, according to the representation of $\{\omega_{n}^{(2,2)}\}_{n=0}^{\infty}$ in \eqref{eq:ki32}, we derive the expression of $\varphi^{(3,2)}(\xi)$ by
\begin{equation}\label{eq:phi32}
\begin{split}
\varphi^{(3,2)}(\xi)=&I_{0}+\sum_{j=0}^{\infty}I_{j+1}\xi^{j}+I_{0,1}^{2}(1-\xi)+(1-\xi)\sum_{j=0}^{\infty}I_{j+1,2}^{2}\xi^{j}+(1-\xi)^{2}I_{0,1}^{3}+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j},
\end{split}
\end{equation}
Substituting the relations $
I_{n,2}^{2}=I_{n,1}^{2}-I_{n}$ and $
I_{n,2}^{3}=I_{n,1}^{3}-I_{n,1}^{2}$ into \eqref{eq:phi32} deduces that
\begin{equation*}
\begin{split}
\varphi^{(3,2)}(\xi)
=&I(\xi)+(1-\xi)I_{1}^{2}(\xi)+(1-\xi^{2})\sum_{j=0}^{\infty}
I_{j+1,1}^{3}\xi^{j} \\
&-2(1-\xi)I_{1}^{3}(\xi)+(3-\xi)(1-\xi)I_{0,1}^{3} \\
=&(\frac{5}{6}+\frac{1}{6}\xi)I(\xi)+(1-\xi)(I_{1}^{2}(\xi)-2I_{1}^{3}(\xi)+\frac{1}{6}I(\xi))
\\
&+(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+1,1}^{3}\xi^{j}+(3-\xi)(1-\xi)I_{0,1}^{3}.
\end{split}
\end{equation*}
Based on Lemma \ref{le:1.5}, one hence obtains that
\begin{equation*}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{1}(n+1-s)^{-\alpha}s(1-s)\mathrm{d}s, \hspace{0.5cm}n\ge 0
\end{equation*}
is a completely monotonic sequence. Therefore the sequence can be expressed by
\begin{equation}\label{eq:lnmix32}
I_{n,1}^{2}-2I_{n,1}^{3}+\frac{1}{6}I_{n}=\int_{0}^{1}r^{n}\mathrm{d}\mu(r),\hspace{0.5cm} n=0,1,\cdots
\end{equation}
together with the function $\mu(r)$ being non-decreasing on interval $[0, 1]$. Therefore, from formulae \eqref{eq:ln}, \eqref{eq:lnmix32} and \eqref{eq:ln13}, it follows
\begin{equation*}
\begin{split}
\mathrm{Re}(\varphi^{(3,2)}(\xi))=&\int_{0}^{1}\mathrm{Re}(\frac{\frac{5}{6}+\frac{1}{6}\xi}{1-r\xi})\mathrm{d}\sigma(r)
+\int_{0}^{1}\mathrm{Re}(\frac{1-\xi}{1-r\xi})\mathrm{d}\mu(r) \\
+&\int_{0}^{1}r\mathrm{Re}(\frac{1-\xi^{2}}{1-r\xi})\mathrm{d}\beta(r)+\mathrm{Re}((3-\xi)(1-\xi))I_{0,1}^{3} \\
\ge& \frac{2}{3}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\sigma(r)\ge \frac{I_{0}}{3}.
\end{split}
\end{equation*}
In the case of $(k,i)=(3,3)$, the definition of series $\omega^{(3,3)}(\xi)$ coincides with
\begin{equation}
\begin{split}
\omega^{(3,3)}(\xi)&=(1-\xi)(I_{0}+I_{1})+(1-\xi)\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)^{2}(I_{0,1}^{2}+I_{1,2}^{2}) \\
&+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j}+(1-\xi)^{3}(I_{0,1}^{3}+I_{1,2}^{3})+(1-\xi)^{3}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j},
\end{split}
\end{equation}
therefore it follows that
\begin{equation}
\label{eq:phi33}
\begin{split}
\varphi^{(3,3)}(\xi)=&I_{0}+I_{1}+\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)\left(I_{0,1}^{2}+I_{1,2}^{2}\right)+(1-\xi)\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j} \\
&+(1-\xi)^{2}\left(I_{0,1}^{3}+I_{1,2}^{3}\right)+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j}. \\
\end{split}
\end{equation}
In addition, substituting the relations
\begin{equation*}
\begin{split}
&I_{n,3}^{2}=I_{n,2}^{2}-I_{n}=I_{n,1}^{2}-2I_{n}, \\
&I_{n,3}^{3}=I_{n,2}^{3}-I_{n,2}^{2}=I_{n,1}^{3}-2I_{n,1}^{2}+I_{n}, \hspace{0.5cm} n\ge 0
\end{split}
\end{equation*}
into \eqref{eq:phi33} yields that
\begin{equation*}
\begin{split}
\varphi^{(3,3)}(\xi)
=&I_{0}+I_{1}+\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)\left(I_{0,1}^{2}+I_{1,1}^{2}-I_{1}\right)+(1-\xi)\sum_{j=0}^{\infty}\left(I_{j+2,1}^{2}-2I_{j+2}\right)\xi^{j} \\
&+(1-\xi)^{2}\left(I_{0,1}^{3}+I_{1,
1}^{3}-I_{1,1}^{2}\right)+(1-\xi)^{2}\sum_{j=0}^{\infty}\left(I_{j+2,1}^{3}-2I_{j+2,1}^{2}+I_{j+2}\right)\xi^{j} \\
=&I(\xi)+(1-\xi)I_{1}^{2}(\xi)-2(1-\xi)I_{1}^{3}(\xi)+(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+2,2}^{3}\xi^{j}-(3-4\xi+\xi^{2})\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j} \\
&+(3-\xi)(1-\xi)\left(I_{0,1}^{3}+I_{1,2}^{3}\right)+\left(1-\xi^{2}\right)\sum_{j=0}^{\infty}I_{j+1,1}^{3}\xi^{j}.
\end{split}
\end{equation*}
In view of Lemma \ref{le:Ijqr}, sequence $(-I_{n,2}^{3})_{n=0}^{\infty}$ is completely monotonic, thus there exists a non-decreasing function $\vartheta(r)$ on $[0, 1]$ such that
\begin{equation*}
-I_{n,2}^{3}=\int_{0}^{1}r^{n}\mathrm{d}\vartheta(r), \hspace{0.5cm} n=0, 1, \cdots,
\end{equation*}
which yields
\begin{equation*}
(1-\xi^{2})\sum_{j=0}^{\infty}I_{j+2,2}^{3}\xi^{j}-(3-4\xi+\xi^{2})\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j}
=\int_{0}^{1}\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\mathrm{d}\vartheta(r)
\end{equation*}
for $|\xi|<1$. It follows
\begin{equation*}
\begin{split}
&\mathrm{Re}\left(\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\right) \\
=&\frac{(3r-r^{2})(1-r|\xi|\cos\theta)+4r^{2}|\xi|^{2}-4r|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\cos2\theta-(r^{3}+r^{2})|\xi|^{3}\cos\theta}{1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}} \\
:=&f(r,|\xi|,\theta).
\end{split}
\end{equation*}
A calculation yields that
\begin{equation}\label{eq:pf}
\begin{split}
\frac{\partial f}{\partial \theta}(r,|\xi|,\theta)
&=\frac{r|\xi|\sin\theta}{(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2})^{2}}g(r,|\xi|,\theta),
\end{split}
\end{equation}
where
\begin{equation*}
\begin{split}
g(r,|\xi|,\theta)=&\left(4+3r-r^{2}-4(r+1)|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\right)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}) \\
-&2\left((3r-r^{2})(1-r|\xi|\cos\theta)+4r^{2}|\xi|^{2}-4r|\xi|\cos\theta+(r^{2}+r)|\xi|^{2}\cos2\theta-(r^{3}+r^{2})|\xi|^{3}\cos\theta\right),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\frac{\partial g}{\partial \theta}(r,|\xi|,\theta)=4|\xi|\sin\theta(r+1)(1-2r|\xi|\cos\theta+r^{2}|\xi|^{2}).
\end{split}
\end{equation*}
We thus know that $g(r,|\xi|,0)\le g(r,|\xi|,\theta)\le g(r,|\xi|,\pi)$. In addition, there is
\begin{equation*}
g(r,|\xi|,0)=(4-3r+r^{2})-4(1+r)|\xi|+(7r+3r^{2}+3r^{3}-r^{4})|\xi|^{2}-4(r^{3}+r^{2})|\xi|^{3}+(r^{3}+r^{4})|\xi|^{4},
\end{equation*}
and
\begin{equation}\label{eq:pg}
\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)=-4(1+r)+2(7r+3r^{2}+3r^{3}-r^{4})|\xi|-12(r^{3}+r^{2})|\xi|^{2}+4(r^{3}+r^{4})|\xi|^{3}.
\end{equation}
In view of
\begin{equation*}
\frac{\partial^{2} g}{\partial |\xi|^{2}}(r,|\xi|,0)=r\left(12(r+1)(r|\xi|-1)^{2}+2(1-r)^{3}\right)\ge 0
\end{equation*}
for all $0\le r\le 1$ and $0\le|\xi|<1$, we can therefore obtain that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)< \frac{\partial g}{\partial |\xi|}(r,1,0)$. In addition, formula \eqref{eq:pg} shows that
\begin{equation*}
\frac{\partial g}{\partial |\xi|}(r,1,0)=2(r^{3}-3r+2)(r-1)\le 0
\end{equation*}
for all $0\le r\le 1$, it derives that $\frac{\partial g}{\partial |\xi|}(r,|\xi|,0)<\frac{\partial g}{\partial |\xi|}(r,1,0)\le 0$ for all $0\le r\le 1$ and $0\le|\xi|<1$. We can finally get that
\begin{equation*}
g(r,|\xi|,0)>g(r,1,0)=0,\hspace{0.5cm} 0\le r\le 1, \hspace{0.2cm} 0\le|\xi|<1.
\end{equation*}
Hence, it holds that $g(r, |\xi|, \theta)\ge g(r,|\xi|,0)>0$ in the cases of $0\le r\le 1$ and $0\le |\xi|<1$. According to formula \eqref{eq:pf}, we have that $f(r, |\xi|, 0)\le f(r, |\xi|, \theta)\le f(r, |\xi|, \pi)$ for all $0\le r\le 1$ and $0\le |\xi|<1$. The definition of $f(r,|\xi|,\theta)$ states
\begin{equation*}
f(r,|\xi|,0)=\frac{3r-r^{2}-4r|\xi|+r^{2}|\xi|^{2}+r|\xi|^{2}}{1-r|\xi|}.
\end{equation*}
Taking the derivative with respect to $|\xi|$ obtains
\begin{equation*}
\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)=\frac{r}{(1-r|\xi|)^{2}}\left(-4+3r-r^{2}+2(r+1)|\xi|-(r^{2}+r)|\xi|^{2}\right)=\frac{r}{(1-r|\xi|)^{2}}h(r,|\xi|).
\end{equation*}
It can be easily checked that $\frac{\partial h}{\partial |\xi|}(r,|\xi|)\ge 2(1-r^{2})\ge 0$ for $0\le r\le 1$, in combination with $h(r,1)=-2(1-r)^{2}\le 0$, we have that $h(r,|\xi|)\le h(r,1)\le 0$ for $0\le r\le 1$. And consequently, result $\frac{\partial f}{\partial |\xi|}(r,|\xi|,0)\le 0$ suggests that $f(r,|\xi|,0)\ge f(r,1,0)$ for
$0\le r\le 1$ and $0\le |\xi|<1$. In combination with $f(r,1,0)=0$, we can obtain that
\begin{equation*}
f(r,|\xi|,\theta)\ge f(r,|\xi|,0)\ge 0, \hspace{0.5cm}\forall~ 0\le r\le 1, \hspace{0.2cm} |\xi|<1,\hspace{0.2cm} \theta\in\mathbb{R}.
\end{equation*}
Therefore, it follows
\begin{equation*}
\begin{split}
\mathrm{Re}(\varphi^{(3,3)}(\xi))=&\int_{0}^{1}\mathrm{Re}(\frac{\frac{5}{6}+\frac{1}{6}\xi}{1-r\xi})\mathrm{d}\sigma(r)
+\int_{0}^{1}\mathrm{Re}(\frac{1-\xi}{1-r\xi})\mathrm{d}\mu(r)+\int_{0}^{1}r\mathrm{Re}(\frac{1-\xi^{2}}{1-r\xi})\mathrm{d}\beta(r) \\
+&\int_{0}^{1}\mathrm{Re}\left(\frac{(3r-r^{2})-4r\xi+(r^{2}+r)\xi^{2}}{1-r\xi}\right)\mathrm{d}\vartheta(r)+\mathrm{Re}\left((3-\xi)(1-\xi)\right)(I_{0,1}^{3}
+I_{1,2}^{3}) \\
\ge& \frac{2}{3}\int_{0}^{1}\frac{1}{1+r|\xi|}\mathrm{d}\sigma(r)\ge \frac{I_{0}}{3},
\end{split}
\end{equation*}
since there holds that $I_{0,1}^{3}+I_{1,2}^{3}=\frac{2^{1-\alpha}(\alpha^{2}+\alpha)}{3\Gamma(4-\alpha)}\ge0$ for all $0\le \alpha\le 1$.
In addition, for $\xi=1$, assume that $\varphi^{(k,i)}(1)=0$, we know from the definition of $\varphi^{(k,i)}(\xi)$ that
\begin{equation}
\label{eq:id1.39}
\varphi^{(k,i)}(\xi)=I(\xi)+l^{(k,i)}(\xi),
\end{equation}
where the coefficients of series $l^{(k,i)}(\xi)$ is absolutely convergent. The definition of coefficients of $I(\xi)$ yields that $\sum_{i=0}^{n}I_{i}$ is arbitrary large as increasing $n$. However, the boundedness of $l^{(k,i)}(1)$ contradicts identity \eqref{eq:id1.39} for $\xi=1$, which obtains $\varphi^{(k,i)}(1)\ne 0$.
In the rest case of $|\xi|=1$ and $\xi\ne 1$, we can get from Corollary \ref{co:1} that
series $\varphi^{(k,i)}(\xi)$ is pointwise continuous on $|\xi|\le 1$ except $\xi=1$, then for the sequence $\xi_{n}=(1-\frac{1}{n})\xi$ satisfying $|\xi_{n}|<1$ for all $n\in \mathbb{N}^{+}$, $\varphi^{(k,i)}(\xi)$ is the limit point of sequence $\varphi^{(k,i)}(\xi_{n})$, thus
\begin{equation*}
\mathrm{Re}( \varphi^{(k,i)}(\xi))=\lim_{n\to +\infty}\mathrm{Re}(\varphi^{(k,i)}(\xi_{n}))\ge c^{(k,i)}>0,
\end{equation*}
where constants $c^{(k,i)}$ are independent of $n$.
\end{proof}
\section{Convergence analysis}\label{conversection}
In this section, we consider the global error estimation for the problem \eqref{eq:nolinfode} when the numerical approximations \eqref{eq:nnonliode} are employed. Assume that $u(t_{n})$ is the exact solution of \eqref{eq:nolinfode} at $t=t_{n}$, then it satisfies
\begin{equation}\label{eq:exac}
D_{k,i}^{\alpha}u(t_{n})=f(t_{n}, u(t_{n}))+\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N,
\end{equation}
where the difference operator $D_{k,i}^{\alpha}$ is defined by \eqref{Dki1} and the local truncation error $\tau_{n}^{(k,i)}$ is denoted by \eqref{eq:trunc}. Suppose that $u_{n}^{(k,i)}$ is the solution of \eqref{eq:nnonliode} for each pair of $(k,i)$, we denote the global error by
\begin{equation}
e_{n}^{(k,i)}=u(t_{n})-u_{n}^{(k,i)} \hspace{0.618cm}\text{for} \hspace{0.2cm}0\le n\le N,
\end{equation}
where $e_{0}^{(k,i)}=0$. Thus subtracting \eqref{eq:nnonliode} by \eqref{eq:exac} implies that
\begin{equation}\label{eq:Dkie}
D_{k,i}^{\alpha}e_{n}^{(k,i)}=\delta f_{n}^{(k,i)}+\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N,
\end{equation}
where the notation $\delta f_{n}^{(k,i)}$ is denoted by $f(t_{n}, u(t_{n}))-f(t_{n}, u_{n}^{(k,i)})$. In addition, substituting \eqref{Dki1} into \eqref{eq:Dkie} yields
\begin{equation}\label{eq:node}
\sum_{m=0}^{k-1}w_{n,m}^{(k,i)}e_{m}^{(k,i)}+\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}e_{j}^{(k,i)}
=(\Delta t)^{\alpha}\delta f_{n}^{(k,i)}+(\Delta t)^{\alpha}\tau_{n}^{(k,i)},\hspace{0.618cm} k\le n\le N.
\end{equation}
Multiplying $\xi^{n-k}$ on both sides of \eqref{eq:node} and summing up for all $n\ge k$, one obtains
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}\sum_{m=0}^{k-1}&\left(w_{n+k,m}^{(k,i)}+\omega_{n+k-m}^{(k,i)}\right)e_{m}^{(k,i)}\xi^{n}+\sum_{n=0}^{\infty}\sum_{j=k}^{n+k}\omega_{n+k-j}^{(k,i)}e_{j}^{(k,i)}\xi^{n}\\
&=(\Delta t)^{\alpha}\sum_{n=0}^{\infty}\delta f_{n+k}^{(k,i)}\xi^{n}+(\Delta t)^{\alpha}\sum_{n=0}^{\infty}\tau_{n+k}^{(k,i)}\xi^{n},
\end{split}
\end{equation*}
since
\begin{equation*}
\sum_{n=0}^{\infty}\sum_{j=k}^{n+k}\omega_{n+k-j}^{(k,i)}e_{j}^{(k,i)}\xi^{n}=
\sum_{n=0}^{\infty}\sum_{j=0}^{n}\omega_{n-j}^{(k,i)}e_{j+k}^{(k,i)}\xi^{n}
=\sum_{j=0}^{\infty}e_{j+k}^{(k,i)}\xi^{j}\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n},
\end{equation*}
it follows
\begin{equation}\label{eq:omegakie}
\omega^{(k,i)}(\xi)e^{(k,i)}(\xi)
=\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi),
\end{equation}
where
\begin{equation}\label{eq:denote1}
\begin{split}
&s_{m}^{(k,i)}(\xi):=\sum_{n=0}^{\infty}s_{n,m}^{(k,i)}\xi^{n}=-\sum_{n=0}^{\infty}\left(w_{n+k,m}^{(k,i)}+\omega_{n+k-m}^{(k,i)}\right)\xi^{n},\hspace{0.618cm} e^{(k,i)}(\xi)=\sum_{n=0}^{\infty}e_{n+k}^{(k,i)}\xi^{n}, \\
&\omega^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\omega_{n}^{(k,i)}\xi^{n},\hspace{0.718cm}
\delta f^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\delta f_{n+k}^{(k,i)}\xi^{n},\hspace{0.618cm}
\tau^{(k,i)}(\xi)=\sum_{n=0}^{\infty}\tau_{n+k}^{(k,i)}\xi^{n}.
\end{split}
\end{equation}
\begin{lemma}
For $1\le i\le k\le3$ and $0\le m\le k-1$, the coefficients $s_{n,m}^{(k,i)}$ are denoted by \eqref{eq:denote1}. Then for all $n \ge 0$, it holds that $s_{n,m}^{(k,i)}$ is bounded, and there exist some bounded constants $c_{m}^{(k,i)}>0$ , which are independent of $n$ and $\alpha$, such that
\begin{equation}\label{eq:ski}
|s_{n,0}^{(k,i)}|\le \frac{c_{0}^{(k,i)}n^{-\alpha}}{\Gamma(1-\alpha)},\hspace{0.9cm}
|s_{n,m}^{(k,i)}|\le \frac{c_{m}^{(k,i)}n^{-\alpha-1}}{|\Gamma(-\alpha)|}
\end{equation}
for $n\ge 1$ and $m\ge 1$.
\end{lemma}
\begin{proof}
It is known from \eqref{ljqr} that for any finite $q, r\in\mathbb{N}^{+}$, $I_{n,q}^{r}$ is bounded for all $n\in\mathbb{Z}$. Since the coefficients $s_{n,m}^{(k,i)}$ are denoted as the linear combinations of $I_{n,q}^{r}$, we can immediately obtain the boundedness of $s_{n,m}^{(k,i)}$ for all integer $n\ge 0$.
Moreover, in the cases of $1\le i\le k\le 3$, each $s_{n,0}^{(k,i)}$ can be expressed as a linear combination of $I_{l}$ and $I_{l,1}^{r}$ with $l\ge n$ and $1\le r\le 3$. Based on formulae \eqref{eq:inqr} and \eqref{eq:inqr1}, it yields $I_{n}=O\left(\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right)$ and $I_{n,1}^{r}=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)=o\left(\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right)$ for $r\ge 2$ and $n\ge 1$, respectively. Therefore, it implies that there is a uniform bound with respect to $n$ and $\alpha$, denoted by $c_{0}^{(k,i)}>0$, such that $|s_{n,0}^{(k,i)}|\le \frac{c_{0}^{(k,i)}n^{-\alpha}}{\Gamma(1-\alpha)}$ as $n\ge 1$.
In terms of $m\ge 1$, observe that $s_{n,m}^{(k,i)}$ are the linear combinations of $\nabla I_{l}$, $I_{l,1}^{r}$ and $\nabla^{p}I_{l,1}^{r}$, for $l\ge n+1$, $r\ge 2$ and $1\le p\le 3$. According to formulae \eqref{eq:nablainqr} and \eqref{eq:nablakinqr1}, we know that
$\nabla I_{n}=O\left(\frac{(n-1)^{-\alpha-1}}{\Gamma(-\alpha)}\right)=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$ and
$\nabla^{p}I_{n,1}^{r}=O\left(\frac{(n-p)^{-\alpha-p-1}}{\Gamma(-\alpha-p+1)}\right)=o\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$, therefore it holds $s_{n,m}^{(k,i)}=O\left(\frac{n^{-\alpha-1}}{\Gamma(-\alpha)}\right)$, and hence there exist constants $c_{m}^{(k,i)}>0$ such that the last inequality of \eqref{eq:ski} is satisfied.
\end{proof}
According to the definition of the series $\omega^{(k,i)}(\xi)$, it is important to notice the decompositions of the form
\begin{equation}\label{eq:rela}
\omega^{(k,i)}(\xi)=(1-\xi)\varphi^{(k,i)}(\xi)=(1-\xi)^{\alpha}\psi^{(k,i)}(\xi),\hspace{0.5cm}\text{for}~~0<\alpha<1,
\end{equation}
where series $\varphi^{(k,i)}(\xi)$ is defined by \eqref{eq:fomega} and denote that
\begin{equation}\label{eq:psiki}
\psi^{(k,i)}(\xi)=(1-\xi)^{1-\alpha}\varphi^{(k,i)}(\xi).
\end{equation}
Formula \eqref{eq:rela} indicates a relationship between the proposed method and the fractional Euler method mentioned in \cite{LubichC:1986a}.
In the following part, we would like to discuss some relevant properties of the series $\psi^{(k,i)}(\xi)$ as preliminaries.
\begin{lemma}\label{le:gn}
Assume that sequences $\{g_{n}^{(\beta)}\}_{n=0}^{\infty}$ are generated by the power series $(1-\xi)^{\beta}$ for $\beta\in\mathbb{R}$, i.e.,
\begin{equation}\label{eq:gn}
(1-\xi)^{\beta}=\sum_{n=0}^{\infty}(-1)^{n}\binom{\beta}{n}\xi^{n}=\sum_{n=0}^{\infty}g_{n}^{(\beta)}\xi^{n}.
\end{equation}
Therefore, in the cases of $\beta\in(-1,1)$, based on \eqref{eq:gn}, there holds
\begin{equation}\label{eq:gn1}
\left\{
\begin{split}
\beta\in (-1,0): &\hspace{0.2cm}g_{0}^{(\beta)}=1,\hspace{0.4cm}g_{0}^{(\beta)}>g_{1}^{(\beta)}>\cdots>0,\\
&\hspace{0.2cm}\sum_{i=0}^{n}g_{i}^{(\beta)}=g_{n}^{(\beta-1)}, \hspace{0.2cm} n\ge 0; \\
\beta\in (0,1):\hspace{0.3cm} &\hspace{0.2cm}g_{0}^{(\beta)}=1,\hspace{0.4cm}g_{n}^{(\beta)}<0,\hspace{0.2cm} n\ge 1, \\
&\hspace{0.2cm}1>|g_{1}^{(\beta)}|>|g_{2}^{(\beta)}|>\cdots>0, \\
&\hspace{0.2cm}\sum_{i=0}^{\infty}g_{i}^{(\beta)}=0,\hspace{0.4cm}\sum_{i=0}^{n}g_{i}^{(\beta)}=g_{n}^{(\beta-1)}, \hspace{0.2cm} n\ge 0. \\
\end{split}
\right.
\end{equation}
\end{lemma}
\begin{lemma}\label{le:le1}
In the cases of $1\le i\le k\le 6$, the coefficients of the power series $\psi^{(k,i)}(\xi)$ belong to $l_{1}$ space.
\end{lemma}
\begin{proof}
According to the expression of $\varphi^{(k,i)}(\xi)$ presented in Theorem \ref{coro:1}, there holds
\begin{equation}\label{rela:1}
\varphi^{(k,i)}(\xi)=I(\xi)+l^{(k,i)}(\xi), \hspace{0.9cm}\text{with}\hspace{0.3cm} \sum_{n=0}^{\infty}|l_{n}^{(k,i)}|<\infty.
\end{equation}
In addition, together with \eqref{eq:psiki}, it follows
\begin{equation*}
\psi^{(k,i)}(\xi)=(1-\xi)^{(1-\alpha)}I(\xi)+(1-\xi)^{(1-\alpha)}l^{(k,i)}(\xi),
\end{equation*}
therefore, it suffices to prove that the coefficients of series $(1-\xi)^{1-\alpha}I(\xi)$ belong to $l_{1}$ space.
From the gamma function's definition of the form
\begin{equation*}
\Gamma(\beta)=\lim_{n\to\infty}\frac{n^{\beta}}{(-1)^{n}\binom{-\beta}{n}(n+\beta)},\hspace{0.618cm} \beta\ne 0, -1, -2, \cdots,
\end{equation*}
one obtains the asymptotically equal relation
\begin{equation}\label{eq:gammaasym}
\frac{n^{\beta-1}}{\Gamma(\beta)}\cong(-1)^{n}\binom{-
\beta}{n},\hspace{0.618cm}\text{as}\hspace{0.209cm}n\to\infty,
\end{equation}
where the notation $\cong$ means the ratio $\left(n^{\beta-1}/\Gamma(\beta)\right)\big/(-1)^{n}\binom{-
\beta}{n}\to 1$ as $n\to\infty$.
Furthermore, it is known from \cite{Erdelyi:1953, LubichC:1986b} that
\begin{equation}\label{eq:asymptgamma}
(-1)^{n}\binom{-\beta}{n}=\frac{n^{\beta-1}}{\Gamma(\beta)}\left(1+O\left(\frac{\beta-1}{n}\right)\right).
\end{equation}
On the other hand, based on the definition of $I_{n}$, it yields that $
I_{n}\cong \frac{n^{-\alpha}}{\Gamma(1-\alpha)}$ as $n\to\infty$, furthermore,
\begin{equation}
\begin{split}
\sum_{n=1}^{\infty}\left|I_{n}-\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\right|&=\frac{1}{\Gamma(1-\alpha)}\sum_{n=1}^{\infty}\int_{0}^{1}\left(n^{-\alpha}-(n+1-s)^{-\alpha}\right)\mathrm{d}s \\
&=\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{1}\int_{0}^{1-s}\sum_{n=1}^{\infty}(n+t)^{-\alpha-1}\mathrm{d}t\mathrm{d}s \\
&\le\frac{\alpha}{\Gamma(1-\alpha)}\sum_{n=1}^{\infty}n^{-\alpha-1} \\
&\le \frac{\alpha}{\Gamma(1-\alpha)}\left(1+\int_{1}^{\infty}x^{-\alpha-1}\mathrm{d}x\right)\\
&=\frac{\alpha+1}{\Gamma(1-\alpha)}<+\infty.
\end{split}
\end{equation}
Therefore, in combination with \eqref{eq:asymptgamma}, it holds that
\begin{equation}\label{rela:2}
I_{n}=g_{n}^{(\alpha-1)}+v_{n},\hspace{0.409cm}\text{with}\hspace{0.209cm}\sum_{n=0}^{\infty}|v_{n}|<\infty,
\end{equation}
hence one has
\begin{equation*}
(1-\xi)^{1-\alpha}I(\xi)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}I_{k}\right)\xi^{n}
\end{equation*}
with the relationship that
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}I_{k}|
&= \sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}\left(g_{k}^{(\alpha-1)}+v_{k}\right)|\\
&\le \sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}g_{k}^{(\alpha-1)}|+\sum_{n=0}^{\infty}|\sum_{k=0}^{n}g_{n-k}^{(1-\alpha)}v_{k}| \\
&\le 1+\sum_{n=0}^{\infty}|g_{n}^{(1-\alpha)}|\sum_{k=0}^{\infty}|v_{k}|<\infty,
\end{split}
\end{equation*}
which yields the desired result.
\end{proof}
\begin{lemma}\label{le:le2}
In the cases of $1\le i\le k\le 3$, it holds that $\psi^{(k,i)}(\xi)\ne 0$ for any $|\xi|\le 1$.
\end{lemma}
\begin{proof}
In the provement process of the Theorem \ref{coro:1}, it implies that $\varphi^{(k,i)}(\xi)\ne 0$ for all $|\xi|\le 1$ and $1\le i\le k\le 3$. The corresponding complex numbers $(1-\xi)^{1-\alpha}$ for any prescribed $|\xi|\le 1$ are located within the sector $S_{\alpha}=\{z: \hspace{0.2cm}|\mathrm{arg}(z)|\le \frac{(1-\alpha)\pi}{2}\}$. In addition, notice that $(1-\xi)^{1-\alpha}=0$ if and only if $\xi=1$. Thus, it remains to confirm the
value of the series $(1-\xi)^{1-\alpha}\varphi^{(k,i)}(\xi)$ at $\xi=1$. In fact, according
to the formulae \eqref{rela:1} and \eqref{rela:2}, it follows
\begin{equation*}
\begin{split}
\sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\varphi_{l}^{(k,i)}
&= \sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\left(g_{l}^{(\alpha-1)}+v_{l}+l_{l}^{(k,i)}\right)\\
&= \sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}g_{l}^{(\alpha-1)}+\sum_{n=0}^{\infty}\sum_{l=0}^{n}g_{n-l}^{(1-\alpha)}\left(v_{l}+l_{l}^{(k,i)}\right) \\
&= 1+\sum_{n=0}^{\infty}g_{n}^{(1-\alpha)}\sum_{l=0}^{\infty}\left(v_{l}+l_{l}^{(k,i)}\right)=1,
\end{split}
\end{equation*}
where the last equality holds in view of the Lemma \ref{le:gn}.
\end{proof}
Therefore, according to the statements from Theorem \ref{th:pw}, Lemma \ref{le:le1} and Lemma \ref{le:le2}, we can immediately obtain the following result.
\begin{proposition}\label{pro:1}
For $1\le i\le k\le 3$ and $0<\alpha<1$, let
\begin{equation}\label{eq:rnki}
\frac{1}{\psi^{(k,i)}(\xi)}=r^{(k,i)}(\xi)=\sum_{n=0}^{\infty}r_{n}^{(k,i)}\xi^{n},
\end{equation}
then there exist bounded positive constants, denoted by $M_{\alpha}^{(k,i)}$, such that $\sum\limits_{n=0}^{\infty}
|r_{n}^{(k,i)}|=M_{\alpha}^{(k,i)}$ holds for each $k, i$.
\end{proposition}
\begin{theorem}
Let $u(t)$ and $\{u_{n}\}_{n=k}^{N}$ be the solutions of equations \eqref{eq:nolinfode} and \eqref{eq:node}, respectively. The function $f(t, u(t))$ in \eqref{eq:nolinfode} is assumed to satisfy the Lipschitz continuous condition with respect to the second variable $u$, and chosen properly such that the solution of \eqref{eq:nolinfode} is sufficiently smooth. Then
\begin{description}
\item [i)] in the cases of $1\le k\le 3$, it holds
\begin{equation}
|e_{n}^{(k,k)}|\le C^{(k,k)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,k)}|+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right),\hspace{0.618cm} k\le n\le N
\end{equation}
\item [ii)]in the cases of $1\le i<k\le 3$, it holds
\begin{equation}
|e_{n}^{(k,i)}|\le C^{(k,i)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,i)}|+\left(\Delta t\right)^{k}+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right),\hspace{0.618cm} k\le n\le N
\end{equation}
\end{description}
for sufficiently small $\Delta t>0$,
where $N\Delta t=T$ is fixed and constant $C^{(k,i)}>0$ is independent of $N$ and $n$.
\end{theorem}
\begin{proof}
Substituting formula \eqref{eq:rela} into \eqref{eq:omegakie} and using \eqref{eq:rnki}, one has
\begin{equation}\label{eq:nonlinear11}
\begin{split}
e^{(k,i)}(\xi)=\frac{r^{(k,i)}(\xi)}{(1-\xi)^{\alpha}}\left(\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi)\right),
\end{split}
\end{equation}
which can also be written into a matrix-vector form
\begin{equation}\label{eq:MD12}
\begin{split}
\left[\begin{array}{l} e_{k}^{(k,i)} \\ e_{k+1}^{(k,i)} \\ \vdots \\ e_{N}^{(k,i)}\end{array}\right]
&=\left[\begin{array}{llll}
r_{0}^{(k,i)} & & & \\
r_{1}^{(k,i)} & r_{0}^{(k,i)} & & \\
\vdots & \ddots & \ddots & \\
r_{N-k}^{(k,i)} & \cdots & r_{1}^{(k,i)} & r_{0}^{(k,i)}
\end{array}\right]
\left[\begin{array}{llll}
g_{0}^{(-\alpha)} & & & \\
g_{1}^{(-\alpha)} & g_{0}^{(-\alpha)} & & \\
\vdots & \ddots & \ddots & \\
g_{N-k}^{(-\alpha)} & \cdots & g_{1}^{(-\alpha)} & g_{0}^{(-\alpha)}
\end{array}\right] \\
&\left(e_{0}^{(k,i)}
\left[\begin{array}{l} s_{0,0}^{(k,i)} \\ s_{1,0}^{(k,i)} \\ \vdots \\ s_{N-k,0}^{(k,i)}\end{array}\right]+\cdots
+e_{k-1}^{(k,i)}\left[\begin{array}{l} s_{0,k-1}^{(k,i)} \\ s_{1,k-1}^{(k,i)} \\ \vdots \\ s_{N-k,k-1}^{(k,i)}\end{array}\right]
+(\Delta t)^{\alpha}\left[\begin{array}{l} \delta f_{k}^{(k,i)} \\ \delta f_{k+1}^{(k,i)} \\ \vdots \\ \delta f_{N}^{(k,i)}\end{array}\right]+(\Delta t)^{\alpha}\left[\begin{array}{l} \tau_{k}^{(k,i)} \\ \tau_{k+1}^{(k,i)} \\ \vdots \\ \tau_{N}^{(k,i)}\end{array}\right]\right)
\end{split}
\end{equation}
with arbitrary $N\in\mathbb{N}$. Therefore for any $k\le n\le N$, it holds that
\begin{equation}\label{eq:}
\begin{split}
e_{n}^{(k,i)}&=\sum_{m=0}^{k-1}e_{m}^{(k,i)}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}s_{i,m}^{(k,i)}+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}\delta f_{i+k}^{(k,i)} \\
&\hspace{0.918cm}+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}r_{n-k-j}^{(k,i)}\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}\tau_{i+k}^{(k,i)},
\end{split}
\end{equation}
where the coefficients $\{g_{n}^{(-\alpha)}\}$ are provided in
Lemma \ref{le:gn}. Since it is assumed that function $f(t, u(t))$ satisfies the Lipschitz continuous condition, there exists constant $L^{(k,i)}>0$ such that $|\delta f_{n}^{(k,i)}|\le L^{(k,i)}|e_{n}^{(k,i)}|$ for $k\le n\le N$. It follows
\begin{equation}\label{eq:enki}
\begin{split}
|e_{n}^{(k,i)}|&\le \sum_{m=0}^{k-1}|e_{m}^{(k,i)}|\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|s_{i,m}^{(k,i)}| \\
&+(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,i)}|\left(L^{(k,i)}|e_{i+k}^{(k,i)}|+|\tau_{i+k}^{(k,i)}|\right). \\
\end{split}
\end{equation}
On one hand, based on the relations \eqref{eq:ski} and \eqref{rela:2}, there exist constant $\tilde{c}_{k,i}>0$, such that $
|s_{n,0}^{(k,i)}|\le c_{k,i}\frac{n^{-\alpha}}{\Gamma(1-\alpha)}\le \tilde{c}_{k,i}g_{n}^{(\alpha-1)}$, one hence obtains
\begin{equation}\label{eq:ski0}
\begin{split}
\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|s_{i,0}^{(k,i)}|&\le \tilde{c}_{k,i}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}g_{i}^{(\alpha-1)} \\
&\le \tilde{c}_{k,i}\sum_{j=0}^{\infty}|r_{j}^{(k,i)}|=\tilde{c}_{k,i}M_{\alpha}^{(k,i)},
\end{split}
\end{equation}
where $\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}g_{i}^{(\alpha-1)}=1$ for any $j\ge 0$ in view of the equality $(1-\xi)^{-\alpha}(1-\xi)^{\alpha-1}=(1-\xi)^{-1}$. On the other hand, there exist constant $\tilde{c}_{m}^{(k,i)}>0$, such that $|s_{n,m}^{(k,i)}|\le c_{m}^{(k,i)}\frac{n^{-\alpha-1}}{|\Gamma(-\alpha)|}\le \tilde{c}_{m}^{(k,i)}|g_{n}^{(\alpha)}|$, and for $m\ge 1$, it yields
\begin{equation}\label{eq:skim}
\begin{split}
\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}|&\le \tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| \\
&\le 2\tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)}, \\
\end{split}
\end{equation}
since it is known that $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=0$ for any $j\ge 1$
according to the equality $(1-\xi)^{-\alpha}(1-\xi)^{\alpha}=1$, in combination with Lemma \ref{le:gn}, there yields $\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}|=g_{j}^{(-\alpha)}g_{0}^{(\alpha)}-\sum_{l=1}^{j}g_{j-l}^{(-\alpha)}g_{l}^{(\alpha)}=2g_{j}^{(-\alpha)}$. In addition, it is known that sequences $\{r_{n}^{(k,i)}\}$ belong to $l^{1}$ space,
and $g_{n}^{(-\alpha)}\to 0$ as $n\to \infty$, therefore, we know that the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)}\to 0$ as $n\to \infty$. Then, at least, the sequences $\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}|$ can be bounded by $2\tilde{c}_{m}^{(k,i)}M_{\alpha}^{(k,i)}$.
In the cases of $1\le k\le 3$, recall that $|\tau_{n}^{(k,k)}|\le C_{\alpha}^{(k)}\left(\Delta t\right)^{k+1-\alpha}$ uniformly for $n\ge k$ in Theorem \ref{th:errorestmat}, it follows from \eqref{eq:gammaasym} that
\begin{equation}\label{eq:taukk}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,k)}|\sum_{i=0}^{j}g_{j-i}^{(-\alpha)}|\tau_{i+k}^{(k,k)}|
&=(\Delta t)^{\alpha}\sum_{j=0}^{n-k}g_{n-k-j}^{(-\alpha)}\sum_{i=0}^{j}|r_{j-i}^{(k,k)}||\tau_{i+k}^{(k,k)}|\\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\sum_{j=0}^{n-k}g_{j}^{(-\alpha)} \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+C\sum_{j=1}^{n-k}\frac{j^{\alpha-1}}{\Gamma(\alpha)}\big) \\
&\le\left(\Delta t\right)^{k+1}C_{\alpha}^{(k)}M_{\alpha}^{(k,k)}\big(1+
\frac{C}{\Gamma(\alpha)}\int_{0}^{n-k}t^{\alpha-1}\mathrm{d}t\big) \\
&\le\tilde{C}_{\alpha}^{(k,k)}\left((\Delta t)^{k+1}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
In the other case of $1\le i<k\le 3$, according to Theorem \ref{th:errorestmat}, there exists constant $C_{\alpha}^{(k,i)}>0$, such that
\begin{equation*}
|\tau_{n}^{(k,i)}|\le C_{\alpha}^{(k,i)}\left((\Delta t)^{k-\alpha}\frac{(n-k)^{-\alpha-1}}{|\Gamma(-\alpha)|}+\frac{(\Delta t)^{k+1-\alpha}}{\Gamma(1-\alpha)}\right),
\end{equation*}
if $n\ge k$,
together with \eqref{eq:gammaasym}, it follows
\begin{equation}\label{eq:tauki}
\begin{split}
(\Delta t)^{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|\tau_{l+k}^{(k,i)}|&\le C_{\alpha}^{(k,i)}\Big((\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{n-k}g_{l}^{(-\alpha)}\Big) \\
&\le C_{\alpha}^{(k,i)}\Big(2(\Delta t)^{k}\tilde{c}_{\alpha}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)} \\
&\hspace{1.236cm}+(\Delta t)^{k+1}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{n-k}^{(-\alpha-1)}\Big)\\
&\le \tilde{C}_{\alpha}^{(k,i)}\left((\Delta t)^{k}+(\Delta t)^{k+1-\alpha}t_{n-k}^{\alpha}\right).
\end{split}
\end{equation}
Therefore formula \eqref{eq:enki} becomes
\begin{equation*}
\begin{split}
|e_{n}^{(k,i)}|&\le(\Delta t)^{\alpha}L^{(k,i)}\Big(\sum_{j=0}^{n-k-1}g_{n-k-j}^{(-\alpha)}\sum_{l=0}^{j}|r_{j-l}^{(k,i)}||e_{l+k}^{(k,i)}|+g_{0}^{(-\alpha)}\sum_{l=0}^{n-k-1}|r_{n-k-l}^{(k,i)}||e_{l+k}^{(k,i)}| \\
&\hspace{2.618cm}+g_{0}^{(-\alpha)}|r_{0}^{(k,i)}||e_{n}^{(k,i)}|\Big)+\delta_{n}^{(k,i)}, \hspace{0.618cm}n\ge k.
\end{split}
\end{equation*}
If the time step $\Delta t>0$ is chosen sufficiently small, there exists bounded constants $c_{k,i}^{*}$ such that $0<\frac{1}{1-(\Delta t)^{\alpha}L^{(k,i)}g_{0}^{(-\alpha)}|r_{0}^{(k,i)}|}\le c_{k,i}^{*}$, and it follows
\begin{equation}\label{eq:enki}
\left\{
\begin{split}
|e_{k}^{(k,i)}|\le&\tilde{\delta}_{k}^{(k,i)}, \\
|e_{n}^{(k,i)}|\le&\tilde{\delta}_{n}^{(k,i)}+(\Delta t)^{\alpha}c_{k,i}^{*}L^{(k,i)}\Big(\sum_{j=0}^{n-k-1}g_{n-k-j}^{(-\alpha)}\sum_{l=0}^{j}|r_{j-l}^{(k,i)}||e_{l+k}^{(k,i)}|\\
&\hspace{0.618cm}+g_{0}^{(-\alpha)}\sum_{l=0}^{n-k-1}|r_{n-k-l}^{(k,i)}||e_{l+k}^{(k,i)}|\Big), \hspace{0.618cm}n\ge k+1,
\end{split}
\right.
\end{equation}
where denote by $\tilde{\delta}_{n}^{(k,i)}=c_{k,i}^{*}\delta_{n}^{(k,i)}$, and in the cases of $1\le k\le 3$, one has from \eqref{eq:ski0}, \eqref{eq:skim} and \eqref{eq:taukk} that
\begin{equation*}
\delta_{n}^{(k,k)}=C_{\alpha}^{(k,k)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,k)}|+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right), \hspace{0.618cm}n\ge k,
\end{equation*}
and in the cases of $1\le i<k\le 3$, from formulae \eqref{eq:ski0}, \eqref{eq:skim} and \eqref{eq:tauki}, it yields
\begin{equation*}
\delta_{n}^{(k,i)}=C_{\alpha}^{(k,i)}\left(\sum_{m=0}^{k-1}|e_{m}^{(k,i)}|+\left(\Delta t\right)^{k}+\left(\Delta t\right)^{k+1-\alpha}t_{n-1}^{\alpha}\right), \hspace{0.618cm}n\ge k.
\end{equation*}
The constants satisfy $C_{\alpha}^{(k,i)}=\max\{\tilde{c}_{k,i}M_{\alpha}^{(k,i)}, 2\tilde{c}_{m}^{(k,i)}M_{\alpha}^{(k,i)}, \tilde{C}_{\alpha}^{(k,i)} \}$. Then assume that the non-negative sequence $\{p_{n}^{(k,i)}\}_{n\ge 0}$ satisfies
\begin{equation}\label{eq:pnki}
\left\{
\begin{split}
&p_{0}^{(k,i)}=\tilde{\delta}_{k}^{(k,i)}, \\
&p_{n}^{(k,i)}=\tilde{\delta}_{n+k}^{(k,i)}+\frac{(\Delta t)^{\alpha}\tilde{L}^{(k,i)}}{\Gamma(\alpha)}\sum_{j=0}^{n-1}(n-j)^{\alpha-1}p_{j}^{(k,i)}, \hspace{0.618cm}n\ge 1,
\end{split}
\right.
\end{equation}
where the coefficient $\tilde{L}^{(k,i)}$ is chosen such that
\begin{equation*}
\tilde{L}^{(k,i)}=\max\{c_{k,i}^{*}L^{(k,i)}M_{\alpha}^{(k,i)}\left(1+\Gamma(\alpha)g_{1}^{(-\alpha)}\right), c_{k,i}^{*}L^{(k,i)}M_{\alpha}^{(k,i)}g_{n}^{(-\alpha)}n^{1-\alpha}\Gamma(\alpha)\}.
\end{equation*}
Therefore, according to the weakly singular discrete Gronwall inequality shown in \cite{DixonM:1986}, the monotonic increasing property of sequence $\{\tilde{\delta}_{n}^{(k,i)}\}_{n\ge 0}$ yields the sequence $\{p_{n}^{(k,i)}\}_{n\ge 1}$ is monotonic increasing with respect to $n$ for each $1\le i\le k\le 3$, and
correspondingly, it follows
\begin{equation*}
p_{n}^{(k,i)}\le \tilde{\delta}_{n+k}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}(n\Delta t)^{\alpha}\right), \hspace{0.618cm} n\ge 1,
\end{equation*}
where $E_{\alpha}(\cdot)$ is denoted by the Mittag-Leffler function.
In addition, according to \eqref{eq:enki} and \eqref{eq:pnki}, an induction process yields that $|e_{n}^{(k,i)}|\le p_{n-k}^{(k,i)}$ as $n\ge k$, thus, one obtains that
\begin{equation*}
|e_{n}^{(k,i)}|\le \tilde{\delta}_{n}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}(n-k)^{\alpha}\Delta t^{\alpha}\right)\le \tilde{\delta}_{n}^{(k,i)}E_{\alpha}\left(\tilde{L}^{(k,i)}T^{\alpha}\right)
\end{equation*}
in the case of $k\le n\le N$.
\end{proof}
\begin{remark}
The provided convergence order is uniform for all $n\ge k$, especially suitable for the step $t_{n}$ near the origin. On the other hand, for those $t_{n}$ away from origin, the convergence result can be better. For example, it can be observed numerically if the computed starting values satisfy $u_{m}=u(t_{m})+O((\Delta t)^{k})$ for $1\le m\le k-1$, there holds that $|u(t_{M})-u_{M}|=O((\Delta t)^{k+1-\alpha})$ in the cases of $1\le i\le k\le3$.
\end{remark}
\section{Numerical experiments}\label{numexpsection}
In this section, we utilize formula \eqref{Dki1} to approximate the following equations in Example \ref{ex:4} and Example \ref{ex:5}, and we prescribe the starting values exactly. Since in practical computation, the starting values are normally obtained by numerical computation in advance.
\begin{example}
\label{ex:4}
We consider the linear equation
\begin{equation}
\label{eq:4.1}
\begin{cases}
&{^C}D^{\alpha}u(t)=\lambda u(t)+f(t), \hspace{0.3cm} t\in(0,1], \\
&u(0)=u_{0}
\end{cases}
\end{equation}
with $0<\alpha<1$. The exact solution is $u(t)=e^{-t}\in C^{\infty}[0,1]$, and $
f(t)=-t^{1-\alpha}E_{1,2-\alpha}(-t)-\lambda e^{-t}\in C[0,1]\cap C^{\infty}(0,1]$, where the Mittag-Leffler function \cite{Podlubny:1999} is defined by
\begin{equation*}
E_{\alpha, \beta}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\alpha k+\beta)}, \hspace{0.5cm} \alpha>0,\hspace{0.1cm}\beta>0.
\end{equation*}
\end{example}
\begin{figure}
\caption{The boundary of the stability region for different $\alpha$ and $\lambda$.}
\label{F1:subfig1}
\label{F1:subfig2}
\label{F1:subfig3}
\label{F1:subfig4}
\label{fig:F1}
\end{figure}
\begin{table}[ht!]
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in Example \ref{ex:4} for different $\alpha$ and $\lambda$.}
\label{ta:3}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{0.1cm}}c@{\extracolsep{0.4cm}}c@{\extracolsep{0.5cm}}l@{\extracolsep{0.5cm}}l@{\extracolsep{0.8cm}}l@{\extracolsep{0.8cm}} l@{\extracolsep{0.8cm}}l @{\extracolsep{0.8cm}}l@{\extracolsep{0.8cm}}l@{\extracolsep{0.1cm}} }
\toprule
$\alpha$&$\lambda$ & $M$ &\multicolumn{2}{l}{$(k, i)=(1,1)$} & \multicolumn{2}{l}{$(k,i)=(2, 1)$}&\multicolumn{2}{l}{$(k,i)=(2, 2)$} \\
\cline{4-9}
&&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.5 & -1 & 128 & 1.59038E-04 & - & 1.60073E-07 & - & 1.34983E-07 & - \\
& & 256 & 5.53407E-05 & 1.52 & 2.86635E-08 & 2.48 & 2.37399E-08 & 2.51 \\
& & 512 & 1.93502E-05 & 1.52 & 5.11315E-09 & 2.49 & 4.18230E-09 & 2.50 \\
& & 1024 & 6.78837E-06 & 1.51 & 9.09687E-10 & 2.49 & 7.37621E-10 & 2.50 \\
& & 2048 & 2.38698E-06 & 1.51 & 1.61541E-10 & 2.49 & 1.30187E-10 & 2.50 \\
\midrule
0.3 & $20\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 1.34901E-06 & - & 2.69029E-09 & - & 1.08970E-09 & - \\
& & 256 & 3.73401E-07 & 1.85 & 4.44355E-10 & 2.60 & 1.62447E-10 & 2.75 \\
& & 512 & 1.04588E-07 & 1.84 & 7.14993E-11 & 2.64 & 2.44242E-11 & 2.73 \\
& & 1024 & 2.96205E-08 & 1.82 & 1.13426E-11 & 2.66 & 3.69238E-12 & 2.73 \\
& & 2048 & 8.47625E-09 & 1.81 & 1.78666E-12 & 2.67 & 5.57796E-13 & 2.73 \\
\midrule
0.9 & $1000\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 7.84215E-07 & - & 3.94997E-09 & - & 3.83098E-09 & - \\
& & 256 & 3.63985E-07 & 1.11 & 9.18845E-10 & 2.10 & 8.91101E-10 & 2.10 \\
& & 512 & 1.69345E-07 & 1.10 & 2.14033E-10 & 2.10 & 2.07571E-10 & 2.10 \\
& & 1024 & 7.88889E-08 & 1.10 & 4.98901E-11 & 2.10 & 4.83852E-11 & 2.10 \\
& & 2048 & 3.67748E-08 & 1.10 & 1.16334E-11 & 2.10 & 1.12827E-11 & 2.10 \\
\midrule
0.98 & $500 i$ & 128 & 2.55224E-06 & - & 1.32455E-08 & - & 1.31778E-08 & - \\
& & 256 & 1.25624E-06 & 1.02 & 3.25627E-09 & 2.02 & 3.23963E-09 & 2.02 \\
& & 512 & 6.18899E-07 & 1.02 & 8.01689E-10 & 2.02 & 7.97599E-10 & 2.02 \\
& & 1024 & 3.05047E-07 & 1.02 & 1.97518E-10 & 2.02 & 1.96512E-10 & 2.02 \\
& & 2048 & 1.50388E-07 & 1.02 & 4.86819E-11 & 2.02 & 4.84347E-11 & 2.02 \\
\bottomrule
\end{tabular*}
\end{table}
\begin{table}[ht!]
\centering
\caption{The error accuracy and convergence rate of $|u(t_{M})-u_{M}|$ in Example \ref{ex:4} for different $\alpha$ and $\lambda$.}
\label{ta:4}
\footnotesize
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}c@{\extracolsep{0.4cm}}c@{\extracolsep{0.5cm}}l@{\extracolsep{0.5cm}}l@{\extracolsep{0.7cm}}c@{\extracolsep{0.7cm}}l@{\extracolsep{0.7cm}}c @{\extracolsep{0.7cm}}l@{\extracolsep{0.7cm}}c@{\extracolsep{0.1cm}} }
\toprule
$\alpha$&$\lambda$ & $M$ &\multicolumn{2}{l}{$(k, i)=(3,1)$} & \multicolumn{2}{l}{$(k,i)=(3, 2)$}&\multicolumn{2}{l}{$(k,i)=(3, 3)$} \\
\cline{4-9}
&&&$|u(t_{M})-u_{M}|$&rate&$|u(t_{M})-u_{M}|$ &rate &$|u(t_{M})-u_{M}|$ &rate \\
\midrule
0.5 & -1 & 128 & 1.04028E-09 & - & 9.41107E-10 & - & 9.99698E-10 & - \\
& & 256 & 9.23186E-11 & 3.49 & 8.25515E-11 & 3.51 & 8.86817E-11 & 3.49 \\
& & 512 & 8.18229E-12 & 3.50 & 7.25575E-12 & 3.51 & 7.85577E-12 & 3.50 \\
& & 1024 & 7.25420E-13 & 3.50 & 6.37490E-13 & 3.51 & 6.92002E-13 & 3.50 \\
& & 2048 & 6.82232E-14 & 3.41 & 5.34572E-14 & 3.58 & 5.85643E-14 & 3.56 \\
\midrule
0.3 & $20\times e^{\frac{i\pi\alpha}{2}}$& 128 & 1.73101E-11 & - & 1.03070E-11 & - & 1.53161E-11 & - \\
& & 256 & 1.33740E-12 & 3.69 & 7.55245E-13 & 3.77 & 1.18503E-12 & 3.69 \\
& & 512 & 1.03673E-13 & 3.69 & 5.58325E-14 & 3.76 & 9.13054E-14 & 3.70 \\
& & 1024 & 6.13266E-15 & 4.08 & 4.40825E-15 & 3.66 & 7.52355E-15 & 3.60 \\
& & 2048 & 1.20505E-15 & 2.35 & 6.86635E-16 & 2.68 & 1.12983E-15 & 2.74 \\
\midrule
0.9 & $1000\times e^{\frac{i\pi\alpha}{2}}$ & 128 & 2.28884E-11 & - & 2.24089E-11 & - & 2.26564E-11 & - \\
& & 256 & 2.65645E-12 & 3.11 & 2.60061E-12 & 3.11 & 2.62969E-12 & 3.11 \\
& & 512 & 3.09069E-13 & 3.10 & 3.02593E-13 & 3.10 & 3.05970E-13 & 3.10 \\
& & 1024 & 9.71032E-07 & -21.58 & 4.20376E-06 & -23.73 & 4.42542E-07 & -20.46 \\
& & 2048 & 6.28199E+34 & -135.57 & 1.41894E+35 & -134.63 & 1.09390E+34 & -134.18 \\
\midrule
0.98 & $500 i$ & 128 & 7.76488E-11 & - & 7.73754E-11 & - & 7.75095E-11 & - \\
& & 256 & 9.52734E-12 & 3.03 & 9.49401E-12 & 3.03 & 9.51047E-12 & 3.03 \\
& & 512 & 9.64788E-07 & -16.63 & 1.77544E-06 & -17.51 & 1.37083E-06 & -17.14 \\
& & 1024 & 3.88686E-14 & 24.57 & 1.83022E-13 & 23.21 & 1.31363E-13 & 23.31 \\
& & 2048 & 1.75624E-14 & 1.15 & 1.70830E-14 & 3.42 & 1.66951E-14 & 2.98 \\
\bottomrule
\end{tabular*}
\end{table}
Figures \ref{F1:subfig1}-\ref{F1:subfig4} plot the truncated boundary locus curves $\sum_{n=0}^{6000}\omega_{n}^{(k,i)}e^{in\theta}~(0\le \theta\le 2\pi)$ in the cases of $1\le i\le k\le 3$ for different $\alpha\in (0,1)$. It is known from Theorem \ref{the:stabilityr} that the stability regions of methods \eqref{DkiTest} lie outside the corresponding curves. We introduce the points $z_{n}:=\lambda (\Delta t_{n})^{\alpha}$ for $1\le n\le 5$, where $\Delta t_{n}=1/2^{n+6}$ correspond to different time stepsize. Table \ref{ta:3} and Table \ref{ta:4} list the global error $e_{M}=u(t_{M})-u_{M}$ in Example \ref{ex:4}, where $t_{M}=M\Delta t=1$ is fixed and $M=2^{j}$ for $7\le j\le 11$, $u(t_{M})$ and $u_{M}$ are the exact solution and the computed solution, respectively.
By the comparison of Figures \ref{F1:subfig1}-\ref{F1:subfig4} and Tables \ref{ta:3}-\ref{ta:4}, we can see the influence of the stability of the numerical methods on the error accuracy. In Figure \ref{F1:subfig1}, the points $z_{n}$ for $1\le n\le 5$ all lie in the stability regions in the case of $\alpha=0.5$ and $\lambda=-50$, in which situation the reliable accuracy is obtained, and it is observed that $|e_{M}|=O(\Delta t^{k+1-\alpha})$ in Table \ref{ta:3}-\ref{ta:4}. In Figure \ref{F1:subfig2}-\ref{F1:subfig3}, $z_{n}$ are choose on the half line with angle $\frac{\pi\alpha}{2}$ and different $\lambda$, it is observed that when all $\{z_{n}\}_{n=1}^{5}$ fall out of the instability region (cf. Figure \ref{F1:subfig2}), correspondingly, as shown in Table \ref{ta:3}-\ref{ta:4}, the the global error $e_{M}$ agrees with the expectation of the accuracy. On the other hand, due to points $z_{4}$ and $z_{5}$
outside the stability regions of $k=3$ (cf. Figure \ref{F1:subfig3}), the perturbation errors are magnified and accumulated significantly, which are shown in Table \ref{ta:3}-\ref{ta:4} as well. In Figure \ref{F1:subfig4}, $z_{n}$ are choose on the imaginary axis with imaginary number $\lambda$, according to Theorem \ref{th:Api}, all the $z_{n}$ belongs to the stability region of the method methods \eqref{DkiTest} in the case of $k=1, 2$, and the error accuracy and the convergence order are obtained (cf. Table \ref{ta:3}). As a counter example, when $z_{3}$ doesn't belong to the stability region for $\alpha=0.98$ in Figure \ref{F1:subfig4}, the corresponding error $e_{M}$ shown in Table \ref{ta:4} can't ensure the desirable accuracy. In fact, it can be observed that for $k=3$, methods \eqref{DkiTest} don't possess $A(\frac{\pi}{2})$-stability when $\alpha$ tends to $1$, which appears to be predictable, since it is well known that BDF3 method for ODEs is not $A(\frac{\pi}{2})$-stable.
\begin{example}
\label{ex:5}
Consider the nonlinear equation
\begin{equation}\label{eq:4.2}
\begin{cases}
&{^C}D^{\alpha}u(t)=-u^{2}+f(t), \hspace{0.3cm}t\in (0,1] \\
&u(0)=u_{0}
\end{cases}
\end{equation}
with exact solution $u(t)=e^{\mu t}$ and source function
$f(t)=\mu t^{1-\alpha}E_{1,2-\alpha}(\mu t)+e^{2\mu t}$.
\end{example}
\begin{figure}
\caption{Errors and convergence orders of $|e_{M}
\label{F4:subfig1}
\label{F4:subfig2}
\label{F4:subfig3}
\label{F4:subfig4}
\label{fig:F4}
\end{figure}
\begin{figure}
\caption{Errors and convergence orders of $|e_{M}
\label{F5:subfig1}
\label{F5:subfig2}
\label{F5:subfig3}
\label{F5:subfig4}
\label{fig:F5}
\end{figure}
Figure \ref{fig:F4} and \ref{fig:F5} plot the global error $|e_{M}|=|u(t_{M})-u_{M}|$ in Example \ref{ex:5} for different $\mu$ and $\alpha$, where $t_{M}=1$ is fixed and $\Delta t=1/M$ with $M=2^{j},~2\le j\le 11$. It is observed that $|e_{M}|=O(\Delta t^{k+1-\alpha})$ in the cases of $1\le i\le k\le 3$, when using discretisation formula \eqref{eq:nnonliode} in combination with Newton's method for the nonlinear equation behind the implicit method.
\section{Conclusions}\label{concl}
We have proposed a new higher-order approximation method for solving time-fractional initial value models of order $0<\alpha<1$. Furthermore, the local truncation error in terms of solution possessing sufficient smoothness is derived and demonstrated. Additionally, the stability and convergence analyses of the proposed method are provided in details, which also motivate the relevant research on applications to time-fractional partial differential equations.
\end{document} |
\begin{document}
\title{Monitoring Hyperproperties with Circuits hanks{The authors were supported by the projects `Open Problems in the Equational Logic of Processes’ (OPEL) (grant No 196050-051) and `Mode(l)s of Verification and Monitorability' (MoVeMent) (grant No~217987) of the Icelandic Research Fund, and `Runtime and Equational Verification of Concurrent Programs' (ReVoCoP) (grant No 222021), of the Reykjavik University Research Fund. Luca Aceto's work was also partially supported by the Italian MIUR PRIN 2017 project FTXR7S IT MATTERS `Methods and Tools for Trustworthy Smart Systems'.}
\begin{abstract}
This paper presents an extension of the safety fragment of Hennessy-Milner Logic with recursion over sets of traces, in the spirit of Hyper-LTL. It then introduces a novel monitoring setup that employs circuit-like structures to combine verdicts from regular monitors.
The main contribution of this study is the definition of the monitors and their semantics, as well as a monitor-synthesis procedure from formulae in the logic that yields ‘circuit-like monitors’ that are sound and violation complete over a finite set of infinite traces.
\end{abstract}
\section{Introduction}
The field of runtime verification concerns itself with providing methods for checking whether a system satisfies its intended specification at runtime. This runtime analysis is done through a computing device called a \textit{monitor} that observes the current run of a system in the form of a trace \cite{Bartocci2018,francalanza_found_Run_Moni}. Runtime verification has recently been extended to the setting of concurrent systems
\cite{AcetoPOPL19,Bocchi_Tuosto_DistInteraction,Cassar2017ReliabilityAF,choreogr_monitors_as_mem}
with several attempts to specify properties over sets of traces, and to introduce novel monitoring setups \cite{Agrawal2016RuntimeVO,complex_monitor_hyper,monitor_hyper}. A centerpiece in this line of work has been the specification logic Hyper-\ltl \cite{hyperproperties}. Intuitively Hyper-\ltl allows for existential and universal quantification over a set of traces (which describes the set of observed system runs). The properties over one trace are stated in \ltl, with free trace variables, and then made dependent on properties of other traces via the quantification that binds the trace variables.
We define the linear-time specification logic \hyperhml, as a counterpart to Hyper-\ltl, building on
previous studies of monitorability and monitor synthesis for \muhml \cite{AcetoPOPL19,hml_monitors}, which are necessary for the kind of correctness and complexity guarantees we aim to achieve in this work.
However, just like Hyper-\ltl, \hyperhml can define dependencies over different traces, which intuitively causes extra delays in the processing of traces as the properties observed on one of them can impact what is expected for another. For example, if a property requires that an event of a trace is compared against an event occurring in all other traces then the processing cost of this event becomes dependent on the number of traces.
In this approach, we keep the processing-at-runtime cost (as defined in \cite{rabin_real_time_comp}) minimal by restricting the type of properties verified to a natural fragment of \hyperhml, but applying no assumptions on the system under scrutiny.
This comes in contrast with the existing research, where the runtime verification of such properties is dealt with via a plethora of modifications and assumptions made over the monitoring setup, such as being able to restart an execution or having access to all executions of a system.
Our monitor setup is engineered for the studied fragment of the specification language, by utilizing circuit-like structures to combine verdicts over different traces. The fragment of the logic restricts the amount of quantification that can be applied to the properties of individual traces and thus limits the dependencies between them. This naturally induces circuits with monitors from \cite{AcetoPOPL19} as input nodes and simple kinds of gates at the higher levels, with the resulting structure having constant depth with respect to the corresponding formula, which is considered efficient in the field of parallel computation \cite{complexity_small_depth_circuits}. Thus, each step taken by such a monitor in response to an event of the system under scrutiny takes constant time, which makes the monitors `real time' in the sense of \cite{rabin_real_time_comp}.
\section{The logic}
Our logic is defined in the style of Hyper-\ltl as presented in \cite{hyperproperties}. The quantification among traces remains the same, but the language in which local trace properties are stated is \muhml.
We consider the following restriction to a multi-trace \shml logic (the \textit{safety} fragment of \muhml \cite{AcetoPOPL19}), with no alternating quantifiers, called \ellihml. We can similarly define the \chml (co-safety) fragment, and the \hml fragment.
\begin{definition}
Formulae in $\textsc{Hyper}^1\text{-}\textsc{sHML}\xspace$ are constructed by the following grammar:
\begin{align*}
\varphi \in \textsc{Hyper}^1\text{-}\textsc{sHML}\xspace& ::= \exists_{\pi} \psi &\mid&~ \forall_{\pi} \psi &\mid&~ \varphi \sqcup \varphi &\mid&~ \varphi \sqcap \varphi
\end{align*}
where
$\psi$ stands for a formula in \shml and $\pi$ is a trace variable from an infinite suppy of trace variables $\mathcal{V}$.
$\sqcup \text{ and }\sqcap$ stand for the regular $\vee$ and $\wedge$ boolean connectives, only usable at the top syntax level.
Although the syntactic distinction is cosmetic, it allows us to keep the synthesis function in Definition \ref{def:synthesis} clearer.
\end{definition}
\textit{Semantics} The semantics of \hyperhml is given over a finite set of infinite traces $T$ over \act and it is a natural extension of the linear-time semantics of \muhml. The existential and universal quantification happens via the trace variable $\pi$ which ranges over the traces in $T$. The extension of the \muhml linear-time semantics from \cite{AcetoPOPL19} to the \hyperhml semantics is done in the style of Hyper-\ltl.
This semantics applies to \ellihml, which is a fragment of \hyperhml.
We only consider \textit{closed} formulae in \ellihml and for these we use the standard notation $T \models \varphi$ to mean that a set of traces $T$ satisfies $\varphi$ (and similarly for $ T \not \models \varphi$).
\begin{example}\label{ex:1} The \ellihml formula $ \forall_{\pi} [a]\mathtt{ff} \sqcap \exists_{\pi} [b] ( max~x. ([a]\mathtt{ff} ~\wedge ~[b] x))$, over the set of actions $\{a,b\}$, states that for any set of traces $T$, none of the traces in $T$ start with $a$, and $b^\omega \in T$.
\end{example}
\section{The monitors}
The intuition behind our monitor design is the following (we recommend following this intuition along with the example given in Figure \ref{fig:circuit_monitor}).
Over a finite set of traces $T$ we instrument a circuit-like structure.
Each trace $t \in T$ is assigned a fixed set of regular monitors that correspond to the properties in \shml to be verified.
These regular monitors are connected with simple gates which evaluate to $yes$, $no$ or $end$ based on the verdicts produced by their associated regular monitors. Once some of these gates start evaluating to verdicts, they communicate with more complex gates, connected in a circuit-like graph, which propagate input verdicts though logic operations until the root node of the circuit reaches a verdict as well. The formal definition of a circuit monitor is given in the style of computational complexity circuits \cite[Definition~1.10]{circuit_complexity}.
\begin{definition}
The language $\text{C}\textsc{mon}_k$ of $k$-ary monitors, for $k >0$, is given through the following grammar:
\begin{align*}
&M \in \text{C}\textsc{mon}_k
::= \bigvee [m]_k &&|~~ \bigwedge [m]_k &&| ~~ M \vee M &&|~~ M \wedge M
\\
& m ::=~~~ yes ~| ~no ~|~end &&|~~ a.m,~ a\in \act &&|~~ m +n &&|~~ rec ~x.m &&|~~ x
\end{align*}
\text{C}\textsc{mon}\xspace is the collection of infinite \textit{sequences} $(M_i)_{i \in \nat}$ of terms that are generated by substituting $k = i, \forall i \in \nat$, in a term $M
$ in $\text{C}\textsc{mon}_k$.
\end{definition}
We use $M,M' \ldots$ to denote the monitors (infinite sequences of terms generated by the first line of this grammar), and refer to them as circuit monitors, and $m_1, m_2 \ldots$ to denote the regular monitors described by the second line.
The notation $[m]_k$ corresponds to the parallel dispatch of $k$ identical regular monitors $m$, where $k = |T|$, with $T =\{t_1, \ldots ,t_k \}$.
Given a monitor $M \in \text{C}\textsc{mon}\xspace$, for some $k >1$, we will call each syntactic sub-monitor of $M$ a gate. For example, we have inductively that over the monitor $M' \vee M''$ we have the gates $M' \vee M''$ and all gates contained in monitors $M'$, and $M''$, while for the monitor $\bigvee [m]_k$ we have the gates $\bigvee [m]_k$ and gates $m_{[i]}$ for $i \in \{ 1, \ldots, k \}$.
For $M \in \text{C}\textsc{mon}\xspace$ we define a set of \textit{program variables} $G_M$, where one variable $g_{M'}$ is assigned to each gate $M'$ of $M$.
For readability purposes we will be omitting the naming $g$ of the program variables and call them by the name of the gate they represent.
We use
$m_{[i]}$ to mean the regular monitor $m$ instrumented over the trace $t_i$.
It is important here to see that $g_{m_{[i]}}$ will be the \textit{name} of the gate assigned to one such monitor and stays unchanged while the actual monitor advances its computation as trace events are read.
This will be clarified later, through the instrumentation rules.
A program variable related to gate $M$, can be assigned the following values: $yes$, $no$, $end$, and $j$, with $j \in \{ 0, \dots, 2^{(\ell+1)}-1 \}$, $\ell$ being the number of immediate syntactical sub-monitors of gate $M$.
Number $j$ is encoded in binary, and is used to carry the information of which sub-gates have given some verdict (this means that the encoding of $j$ has $\ell +1$ bits).
The value of the $\ell +1$-th bit of $j$ is reserved to encode that one of the sub gates has outputted an $end$. The information that $j$ carries is very important for the evaluation of a gate, as often this evaluation depends on the verdicts of more that one sub-gate, as well as what these verdicts are (see Figure \ref{fig:circuit_monitor}). A variable $g_m$ can only take the values $yes$, $no$ and $end$, produced by the relevant monitor instrumented over a trace.
A \textbf{\textit{configuration}} of monitor $M \in \text{C}\textsc{mon}\xspace$, for some $k>1$, is an array $s_M$ containing a value for all program variables $g$ of $M$. We denote the set of all configurations for a monitor $M$ as $\mathcal{S}_M$. We use the notation $s[M\backslash i]$ to denote the update of a configuration $s$ where gate $M$ stores some value $j$ to one where the $i$-th coordinate of $j$ is $0$, while all other variables have the value they had in configuration $s$. Similarly, we use the notation $s[M\backslash end_i]$ to refer to a configuration where the update $s[M\backslash i]$ has taken place \textit{and} the value of the $\ell +1$-th bit of $j$ is set to $1$. We also use the notation $s[\sfrac{v}{M}]$ with $v \in \{yes,no,end\}$, to mean a configuration where the value of the variable for gate $M$ is updated to $v$.
All gate variables in a circuit monitor are initialized to $2^{\ell}-1$ (a sequence of $\ell$-many zeros), to represent that all sub-gates are waiting to give some output and $s_{M_{init}}$ stands for the initial configuration of $M$. Since $M$ is a family of circuits, we have that the initial configuration of each monitor $M_i$ in the family corresponds to a different initial configuration $s_{M_{i-init}}$.
\begin{example}
Figure \ref{fig:circuit_monitor}, is an example of a circuit monitor and its evaluation.
\end{example}
\begin{figure}
\caption{The circuit monitor for the formula from Example \ref{ex:1}
\label{fig:circuit_monitor}
\end{figure}
\begin{figure}
\caption{\label{tab:sos_rules}
\label{tab:sos_rules}
\end{figure}
\textit{Semantics}
The semantics of a regular monitors is as presented in \cite{AcetoPOPL19}. Each regular monitor corresponds to an LTS, and a transition labeled with $a \in \act$ corresponds to a regular monitor observing the event $a$ when instrumented with a system $p$ that produces it.
The semantics of a circuit monitor is given as a transition relation $\xrightarrow[\text{}]{} \subseteq$ $\mathcal{S}_M \times \mathcal{S}_M$ and the instrumentation $\triangleleft$ takes place over a set of regular monitors $\overrightarrow{m}$ instrumented over a set of traces $T$, denoted $M(T)$.
We define $M(T): = s_{M_{|T|-init}} \triangleleft \overrightarrow{m}_{[i]} \triangleleft T$, where $\overrightarrow{m}$ is the set of regular monitors that occur in $M$, and $\overrightarrow{m}_{[i]}$ is $ \overrightarrow{m}$, instrumented over the trace $t_i \in T.$ When $m$ is a regular monitor then $\triangleleft$ stands for the existing instrumentation relation from \cite{AcetoPOPL19}.
The transition and instrumentation relations are defined as the least ones that satisfy the axioms and rules in Figure \ref{tab:sos_rules}. Due to lack of space, we only include the rules giving the semantics of the $\bigvee [m]_k$ monitor. Those for the other operators follow the same structure. The proof in Appendix \ref{appendix:proof_viol_comp} could help with the understanding of the more intricate instrumentation rules.
A monitor is required to be \textit{correct} with respect to some specification formula $\varphi$. The notions of correctness we use in this work are defined below.
\begin{definition}
Given a monitor $M \in$ \text{C}\textsc{mon}\xspace , and a set of traces $T$.
\begin{itemize}
\item $M$ \textbf{rejects} $T$ (resp. \textbf{accepts} $T$) denoted $rej(M,T)$ (resp. $acc(M,T)$) iff $M(T) \rightarrow^* s \triangleleft \overrightarrow{n}\triangleleft T'$ for some $s,\overrightarrow{n},~ T'$, where $s[M]=no$ (resp. $s[M]=yes$).
\item Given a formula $\varphi \in$ \hyperhml, $M$ is \textbf{sound} for $\varphi$ if $\forall T$, $acc(M,T) \implies T \models \varphi$, and $rej(M,T) \implies T \not \models \varphi$.
\item $M$ is \textbf{violation complete} for $\varphi$ if $\forall T$, $T\not \models \varphi \implies rej(M,T)$.
\end{itemize}
\end{definition}
\textit{Synthesis:}
Given a formula $\varphi$ in \ellihml, We synthesize a circuit monitor $M$ through the following recursive function $Syn(-) :$ \ellihml $\rightarrow$ \text{C}\textsc{mon}\xspace.
\begin{definition}[Circuit Monitor Synthesis]\label{def:synthesis}
\begin{align*}
&Syn(\exists_{\pi} \varphi)= \bigvee [m(\varphi)]_{k} &&~~ Syn(\forall_{\pi} \varphi)= \bigwedge [m(\varphi)]_{k} \\
&Syn( \varphi_1 \sqcup \varphi_2)= Syn(\varphi_1) \vee Syn(\varphi_2)
&&~~ Syn( \varphi_1 \sqcap \varphi_2)= Syn(\varphi_1) \wedge Syn(\varphi_2)
\end{align*}
Where $m(-)$ is the monitor synthesis function for \shml defined in \cite{AcetoPOPL19}.
\end{definition}
\begin{proposition}\label{{prop:viol_comp}} Given a formula $\varphi$ in \ellihml, we have $Syn(\varphi)$ is a sound and violation-complete monitor for $\varphi$.
\end{proposition}
\begin{proof}
The proof is by induction on the structure of $\varphi$. We present here a characteristic case and give more details for some of them in the Appendix \ref{appendix:proof_viol_comp}.
Assume that $ \varphi = \exists_{\pi} \psi$, with $ \psi \in \shml$ and that we have a set of traces $T$ s.t. $T \not \models \varphi$.
From the semantics of \ellihml, we have that $t_i \not\models \psi$, for all traces $t_i$ in $T$.
However $\psi \in \shml$ and thus from \cite{AcetoPOPL19} we get that $m_{\psi}$ is a violation complete monitor for $\psi$.
This means that for all $t_i \in T$, there exist $t_i' \in \act^*$ and $t_i'' \in \act^{\omega}$, such that $t_i = t_i'.t_i''$, such that the monitor $m_{\psi}$ rejects $t_i'$.
From the rules in Figure \ref{tab:sos_rules} we see that each gate $g_{m_{\psi[i]}}$ will reach the value $no$ as enough events over the trace $t_i'$ will occur. I.e. $s_M \triangleleft \overrightarrow{m_{\psi_{[i]}}} \triangleleft T \rightarrow^* s_M \triangleleft \overrightarrow{m}_{[i]}[\sfrac{no}{m_{[i]}}] \triangleleft T[\sfrac{t_i''}{t_i}]$, witch propagates to the evaluation of $g_{m_{[i]}}$ to $no$, for all $i$. We now study the transitions $s_M[\sfrac{no}{g_{m_{\psi}[i]}}]$ since those can be then composed with this instrumentation via the fourth instrumentation rule. Applying the SOS rules yields that the update $\backslash i$ takes place for all $i$ at the gate $\bigvee [m]_k$ which means that the value of $j$ stored in it becomes $0$. This finally yields that the value of the final gate $\bigvee [m]_k$ becomes $0$, i.e. $s_M [\sfrac{no}{g_{m_{[i]}}}~ \forall i] \rightarrow s_M [\sfrac{no}{\bigvee [m]_k}]$. Since this transition can be composed with the discussed instrumentation we have that $s_M \triangleleft \overrightarrow{m_{\psi_{[i]}}} \triangleleft T \rightarrow s_M[\sfrac{no}{g_{\bigvee [m_{\psi}]_{[i]}}}]\triangleleft \overrightarrow{n} \triangleleft T'$ for some $\overrightarrow{n}$ and $T$ and we are done. \qed
\end{proof}
\subsection{Runtime costs}
The monitor synthesis in Definition \ref{def:synthesis} provides a family of circuits that can be instrumented appropriately on an arbitrary set of traces to analyze the events occurring in them. Ideally, the runtime cost of monitoring resulting from our constructions should be bounded by a constant that does not depend on the parameters of the system (such as the number of available traces, or of the events observed so far) \cite{rabin_real_time_comp}. In this way, if a monitor is launched along with the system components, it will only induce a feasible computational overhead.
We already know that the regular monitors instrumented with individual traces analyze the system events they observe with a constant overhead \cite{hml_monitors}. Regarding the computational cost of the circuit part, since we are given $k$ many traces, it must be that the necessary computation performed from a circuit monitor can be performed in parallel, distributed over the components that produced the traces in the first place. This means that we can only concern ourselves with the \textit{circuit complexity} \cite{circuit_complexity} of a given monitor, which encapsulates the parallel processing power necessary for its evaluation.
We now observe the synthesis function. There, a formula $\varphi$ in \ellihml will be turned into a family of circuit monitors where, for each connective of the original formula $\varphi$, the output monitor increases in size based on the size for the monitors of the sub-formulae of $\varphi$. However, for each connective of the formula, the \textit{depth} of the circuit is only increased by $1$ which means that the output circuit monitor has a depth bounded by the size of the formula $\varphi$. Since the gates of the output monitor can have either a fixed amount of sub-gates ($\vee, \wedge$), or $k$ many ($\bigvee, \bigwedge$), we have that the output circuit is in the complexity class AC$^0$ \cite{circuit_complexity}. Thus, the monitor only adds a constant computational overhead when executed over the computational resources of the distributed components of the system.
\section{Conclusion and future work}
We expect that the fragment \ellihml is maximal with respect to violation completeness, which means that any monitor in \text{C}\textsc{mon}\xspace is monitoring for a formula in \ellihml.
However, the ultimate goal of this work is to extend the collection of monitorable properties by allowing alternating quantifiers in the syntax.
This is a very important aspect of any work in this field, as the more interesting hyperproperties, such as the property ``at all times, if one trace encounters the event $p$ then all traces do so as well'' which is a necessary component for the expression of properties such as noninference \cite{temp_logic_hyperprop,possibilistic_prop}, require alternation of quantifiers.
A way to tackle this would be to project such properties into the \ellihml fragment. However this procedure is not formally yet defined, or trivial and one could argue that since every hyperproperty has been shown (\cite{hyperproperties}) to be the intersection of a liveness and a safety hyperproperty, (and since liveness and safety properties are widely accepted as independent \cite{alpernBowenSchneiderLiveandSafe}), an elimination of alternating quantifiers can only take place in very few cases.
Thus, our main purpose is to extend the logic and the consequent monitors in order to express and monitor for the most general class of such properties.
The main objective of the logical fragment we give here is to establish a formal baseline which we will attempt to extend in future work.
Our approach to an extension would be to allow a notion of synchronization rounds among the regular monitors (or equivalently a round of communication). This would enable more complex dependencies between traces, as now the properties required of a given trace can be impacted by the state of the ones monitored for on a different one. However, the analysis of communications among the monitors is a complicated extension, as their exact content plays a significant role to our insight over the system, as well as the processing at runtime cost. We plan to implement this therefore by utilizing dynamic epistemic logic \cite{dyn_epist_logic} in order to perform this extension formally and soundly.
\appendix
\section{Appendix: cases for the proof of violation completeness}\label{appendix:proof_viol_comp}
Here we give some more insight on the remaining cases of the violation completeness proof. First we highlight that the second base case of our proof, for formulae of the form $\forall_{\pi} \psi$ is completely analogous to the one we give and thus omitted.
We will here give an important lemma necessary for analyzing both remaining cases, and then present the high level details for the case of $\sqcap$. The intuition of the importance of the lemma is that the monitors $Syn(\varphi_1)$ and $Syn(\varphi_2)$ should not have their computation affected from the fact that they are run in parallel over a set of traces $T$.
\begin{lemma}
If
\begin{itemize}
\item $s_{M_1} \triangleleft \overrightarrow{m_1}[i] \triangleleft T\rightarrow s_{M_1}'\triangleleft \overrightarrow{m_1}[i]' \triangleleft T'$, and
\item $s_{M_2} \triangleleft \overrightarrow{m_2}[i] \triangleleft T\rightarrow s_{M_2}'\triangleleft \overrightarrow{m_2}[i]' \triangleleft T'$
\end{itemize}
then
\begin{itemize}
\item $s_{M_1 \vee M_2} \triangleleft \overrightarrow{m_{12}}[i] \triangleleft T\rightarrow s_{M_1 \wedge M_2}' \triangleleft \overrightarrow{m_{12}}[i]' \triangleleft T'$, and
\item $s_{M_1 \wedge M_2} \triangleleft \overrightarrow{m_{12}}[i] \triangleleft T\rightarrow s_{M_1 \wedge M_2}' \triangleleft \overrightarrow{m_{12}}[i]' \triangleleft T'$,
\end{itemize}
where $\overrightarrow{m_{12}} = \overrightarrow{m_{2}} \cup \overrightarrow{m_{2}}$ and $\overrightarrow{m_{12}}' = \overrightarrow{m_{2}}' \cup \overrightarrow{m_{2}}'$ respectively.
\end{lemma}
\begin{proof}
We note here that a configuration for $s_{M_1 \vee M_2}$ is identical to one for $s_{M_1 \wedge M_2}$ except the root variable, as all other variables they both contain are $s_{M_1}' \cup s_{M_2}'$.
The key aspect of this proof is the third rule of the instrumentation relation. There we can see that in order for a configuration instrumented over a set of regular monitors, instrumented over a set of traces, can only advance its computation, if all monitors instrumented over the same trace progress with their computation synchronously by reading the next trace event.
Thus, form the assumptions of this lemma we get that for all $j = \{1,\ldots r\}$, where $r$ is the total amount of different regular monitors occurring in $M_1$ and $M_2$ the premise of our rule is satisfied and thus the cumulative configuration of variables amounting for the union of variables of the two circuit monitors $M_1$ and $M_2$ (including the root variable), can perform the necessary transition to the new state, where all regular monitors (those both from $M_1$ and $M_2$) assigned to trace $t_i$ have processed the event $a$, and we are done. \qed
\end{proof}
Having the above lemma streamlines our inductive step for the rest of the cases.
Assuming a non-base-case formula in \ellihml we can clearly see that it must be of the form $\varphi = \varphi_1 \sqcap \varphi_2$ or $\varphi = \varphi_1 \sqcap \varphi_2$. We only analyze one of the two cases as they are symmetrical.
For any set of traces $T$, such that $T \not\models \varphi$, from the semantics of \ellihml, we have that $T \not\models \varphi_1$ and $T \not\models \varphi_2$. Since the synthesized monitor for $\varphi_1 \sqcap \varphi_2$ can reach a configuration where the values of the gates for $Syn(\varphi_1)$ and $Syn(\varphi_2)$ are the same as they would be for the individual monitors instrumented over $T$, and by inductive hypothesis (which guarantees that $Syn(\varphi_1)$ and $Syn(\varphi_2)$ are violation-complete) we have necessary conclusion by combining the two negative verdicts of the individual monitors via the semantics. \qed
\end{document} |
\begin{document}
\title[$\text{\sc Duality of Bochner spaces}$]
{Duality of Bochner spaces}
\author[S. Hiltunen]{Seppo\ I\. Hiltunen}
\address{Aalto University \vskip0mm$\hspace{2mm}$
Department of Mathematics and Systems Analysis \vskip0mm$\hspace{2mm}$
P.O.\ Box 11100 \vskip0mm$\hspace{2mm}$
FI-00076 Aalto \vskip0mm
Finland}
\email{seppo.i.hiltunen\,@\,aalto.fi}
\subjclass[2010]{Primary 46E40\kern0.37mm, 46A20\kern0.37mm, 46G10\kern0.37mm, 46E30\,; Secondary
28A20\kern0.37mm, 28B05\kern0.37mm, 46A16\kern0.37mm, 28C20}
\keywords{Bochner space, Banach space, duality, positive measure, positive
Radonian, vector measure, Lebesgue space, topological vector space, suitable
space, measurability, bounded variation, Dunford\,--\,Pettis.}
\begin{abstract} \renewcommand\lower1.05mm\hbox{$^+$}\infty{\lower.82mm\hbox{$^+$}\infty}\def\AbsLrs#1#2^#3{\raise.95mm\hbox{\font\≈=cmtt5\≈m\font\≈=cmssq5\≈v}\kern-.3mm\lower.25mm\hbox{\font\Â=cmitt10\ÂL}\kern0.25mm\kern0.#2mm\lower.#1mm\hbox{\raise.4mm\hbox{$^{#3}$}}\kern0.25mm}
We construct the generalized Lebesgue\,--\,Bochner spaces \math{
\AbsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } for positive measures \math{\mu} and for
suitable real or complex topological vector spaces \math{\vPi} so that for \linebreak
\ú$
1<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and Banachable \math{\vPi} with separable topology the
strong dual of the classical Bochner space \math{
\AbsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } becomes canonically represented by \mathss37{
\AbsLrs23^p
\LHB{.2}{\KN{.2}^{^*}}\kern-0.3mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi^{\kern0.37mm\prime}_\sigma\kern0.07mm) }. Hence we
need no separability assumption of the norm topology of the strong dual \math{
\vPi^{\kern0.37mm\prime}_
{
\hbox{\font\≈=cmssi5\≈\char'031}}} of \mathss30{\vPi}. For \linebreak
\ú$
p=1\kern0.37mm$ and for suitably restricted positive measures \math{\mu} we even get
a similar result without any separability of the norm topology of the target
space \mathss30{\vPi}. For positive Radon measures on locally compact
topological spaces these results are essentially contained on pages
588\,--\,606 in R.\ E.\ Edwards' classical {\sl Functional Analysis\kern0.15mm}.
\end{abstract}
\maketitle
\Fsubhead Introduction and some preliminaries
Our main objective in this article is the following
\begin{Atheorem}\label{main Th}
Let \ú$\, 1 \le p < \lower1.05mm\hbox{$^+$}\infty ${\,\rm, }and let \ú$\,q =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} $ if \ú$\, p \not= 1
${\kern0.37mm\rm, }and \ú$\,q=\lower1.05mm\hbox{$^+$}\infty$ if \ú$\,p=1\kern0.37mm$.
Further{\kern0.15mm\rm, }
let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega${\,\rm, }and with \ú$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \linebreak
\ú$\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be such that {\,\rm(1)} or {\,\rm(2)} or
{\,\rm(3)} or {\,\rm(4)} or {\,\rm(5)} or {\,\rm(6)} below holds when
{\,\rm(\kern0.15mm\erm D\kern0.15mm)} means that $\,\mu$ is almost decomposable.
Also
let \ú$\, F = \mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) ${\KP1\rm, }and \ú$\, F\aar 1
= {\kern-0.63mm} $ \ú$\mvLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) $ if {\,\rm(1)} or
{\,\rm(2)} or {\,\rm(5)} or {\,\rm(6)} below holds,
otherwise letting \ú$\,
F\aar 1 = {\kern-0.63mm} $ \linebreak
$ \mvsLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)
\KP1 $. \ For \KP1 \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \seq{ \KP{1.2}
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} :
\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\, } $} \inskipline{.5}0
then $\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in \Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds. In addition $\,
F\aar 1=\mvLrs03^q(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ if {\,\rm(1)} or
{\,\rm(5)} holds. {\rm \inskipline14
(1)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(2)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable
topology{\kern0.37mm\rm, \inskipline{.5}4
(3)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mLrs42^1(\kern0.37mm\mu\kern0.15mm)$ is
a separable topology{\kern0.37mm\rm, \inskipline{.5}4
(4)} \ $p=1$ and {\,\rm(\kern0.15mm\erm D\kern0.15mm)} and a choice function $\,c\in
\Cal L\,(\kern0.37mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm)\,,\kern0.15mm
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)) $ exists{\kern0.37mm\rm, \inskipline{.5}4
(5)} \ $p\not=1$ and $\,\vPi$ is reflexive{\kern0.37mm\rm, \inskipline{.5}4
(6)} \ $p\not=1$ and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology.
\end{Atheorem}
The proof is given on pages \pageref{Sec D}\,--\,\pageref{endmpf} below. Here
we first explain the notation appearing above, mentioning that we generally
utilize the notational convention explained in \cite[pp.\ 4\,--\,8]{HiDim}\,,
\cite[pp.\ 4\,--\,9]{SeBGN} and \cite[p.\ 1]{FKBGN}\,, and further to be
\q{polished} in \cite{Hif}\,.
Having \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} means that \math{\bosy K} is either the
topological field of real numbers or that of the complex ones. The underlying
sets of these fields are \math{\mathbb R} \nolinebreak and \nolinebreak \mathss38{
\mathbb C}, \linebreak
respectively. Then \math{\vPi\in\roman{BaS}\kern0.4mmps0(K)} means that \math{\vPi}
is a \erm Banach{\sl able\kern0.15mm}, i.e.\ a complete norm{\sl able\kern0.15mm} real or
complex topological vector space. Thus there is a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}
on the underlying vector space \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} such that \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\kern0.15mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a norm{\sl ed\kern0.37mm} Banach space. Being
{\it compatible\kern0.37mm} here means that \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP{1.1} 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] :
n\in\rbb Z^+\kern0.37mm\big\} } is a filter base for the filter $\neiBoo\vPi$ of
zero neighbourhoods. Above \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP{1.1} 0\,,\kern0.07mm
n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] } is the image of the closed interval \math{
[\KP{1.1} 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm\big] } under the relational inverse \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm\inve} of \mathss31{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}. Here \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ is a function \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi
\to\lbb R_+} where \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} is the underlying set of vectors of \mathss31{
\vPi}. For \linebreak
$\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm$ one may sometimes write \math{\|\,\xi\,\| }
for the value \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi} of \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} at \mathss34{\xi}.
For \mathss37{\vPi\aar 1\in\tvsps0(K)}, \,i.e.\ having \math{\vPi\aar 1} a
real or complex topological vector space with possibly non\kern0.37mm-\kern0.15mm Hausdorff
topology \mathss34{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 1}, \,the exact construction of the space
$E=
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1)$ is given in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) on page \pageref{simpL^p}
below. Here we informally explain the basic ideas
under the additional assumption that
$\vPi\aar 1$ is {\sl suitable\kern0.15mm} in the sense of
Definitions \ref{df suit} on page \pageref{df suit} below. Then
it suffices to consider one fixed
{\sl dominating\kern0.15mm} norm $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ for $\vPi\aar 1\kern0.37mm$.
We consider functions
$x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ such that on every set $A$ of finite measure, i.e.\
for $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ it holds that outside some set $N$ of
measure zero, i.e.\ with $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ we have $x$
a pointwise limit of a sequence of simple functions, with convergence in the
sense of the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$. In the case $p<\lower1.05mm\hbox{$^+$}\infty$ we then take
the subset of those $x$ such that the generally nonmeasurable
function \vskip.3mm\centerline{$
\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x:{}^{}\Cal Omega\owns\eta\mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\KP1\RHB{.3}{^p}\in\lbb R_+ $} \inskipline{.3}0
is dominated by some integrable function
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm:{}^{}\Cal Omega\to[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \,$. With the pointwise
vector operations from $\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ the set of these $x$ becomes a
vector substructure $X$ of $\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{vs} \,$.
Then we take $E=(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\,,\kern0.07mm\scrmt T\,)$ when $
N\aar 0$ is the set of all $x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms X$ such that for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ and
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ we have
$\int_{\,A\,}u\circ x\rmdss11\mu=0 \,$. Here
$\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ is the set of all
continuous linear maps $\vPi\to\bosy K\kern0.37mm$. Furthermore, we take the
topology $\scrmt T$ so that
a filter of zero neighbourhoods is formed by the sets \vskip.3mm\centerline{$
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\kern0.15mm)
\capss31\{\,\smb X:
\eexi{x\in\smb X}\,
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu < \varepsilon\KPt9\} $} \inskipline{.3}0
for $\varepsilon\in\rbb R^+$. Here we have the upper integral
of the
not\kern0.37mm-\kern0.15mm necessarily measurable function
$\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x$ that is defined
as the infimum of the set of all
$\int_{\KP{1.1}{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu$ with $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm$ as above.
For $p=\lower1.05mm\hbox{$^+$}\infty$ we take the \q{obvious} modification.
The space \math{E\ar 1=
\mvsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1)
} is constructed otherwise similarly except that
we instead require the functions $x$ to be such that
$u\circ x\KP1|\KP1(\kern0.15mm A\kern0.15mm\setminus N\kern0.37mm)$ is measurable, that is,
for every
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ we require existence of
some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ it holds that
$u\circ x\KP1|\KP1(\kern0.15mm A\kern0.15mm\setminus N\kern0.37mm)$ is a measurable real or
complex valued function on $A\kern0.15mm\setminus N$. Then every vector of $E$ is
contained in some vector of \mathss34{E\ar 1}, \,but we need not have \mathss34{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 1 }. Note above that \mathss39{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms(\kern0.15mm X\kern0.15mm/\vsquotient N\aar 0\kern0.07mm) }, \,and that \mathss03{
X\kern0.15mm/\vsquotient N\aar 0 } is the quotient vector space structure of \math{X}
by the linear subspace \mathss36{N\aar 0}.
Having now informally explained the general construction of our generalized
Bochner spaces, we note that if \math{\vPi} is \erm Banachable, then \math{
\vPi\dualsigma0} is its weak dual space, and that \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\kern0.15mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a Banach space for any compatible norm \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss31{\vPi}. At least for \rsigma6finite positive measures \math{
\mu} then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\,,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 2\kern0.15mm) } is a
classical Bochner space when \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 2} is defined by \math{ \smb X
\mapsto\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}\Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x
\rmdss11\mu\kern0.37mm)\KP1\RHB{.2}{^p}{^{^{-1}}} } for any \mathss31{x\in\smb X}.
The appearing \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is the function \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 \owns \smb Y
\mapsto\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y} with \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss02\smb Y } given by \vskip.3mm\centerline{$
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \owns \smb X \mapsto \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $} \inskipline{.3}0
for any \math{x\in\smb X} and \mathss30{y\in\smb Y}. Here \math{y\,.\KPt8 x}
is the function \vskip.3mm\centerline{$
{}^{}\Cal Omega\owns\eta\mapsto y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) =
(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) \in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K \in
\{\ssbb97 R,\kern0.07mm\ssbb08 C\} \KP1 $.} \inskipline{.3}0
The message of Theorem \nfss A\,\ref{main Th} is then that \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in
\Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm) } holds, i.e.\ that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is
a linear homeomorphism \math{F\aar 1\to F\dlbetss10} where \math{F\dlbetss10}
is the normable, hence \erm Banachable strong dual of \math{F} with \mathss38{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm(\kern0.15mm F\dlbetss10\kern0.15mm)=\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }.
For
the spaces appearing in (3) and \linebreak
(4) note that we define \mathss39{
\mLrs03^p(\kern0.37mm\mu\kern0.15mm) = \mvLrs03^p(\kern0.37mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) }. Below note that
in the usual man- ner we have \math{ p\,^* =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} } for \math{
1 < p < \lower1.05mm\hbox{$^+$}\infty} and \math{p\,^*=\lower1.05mm\hbox{$^+$}\infty} for \mathss34{p=1}, \,and also $
p\,^*=1\kern0.37mm$ in the case where \math{p=\lower1.05mm\hbox{$^+$}\infty} holds.
The last part in condition (4) means that
there is a continuous linear map
$c:\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)\to
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)$
such that $c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\in\smb X$ holds for all $
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) \, $.
Continuity here being equivalent to the property that
for some $\smb M\in\lbb R_+$ it holds that
for $x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm)$ and for all $
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+
$ there is $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$
|\KP1 c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|
\le\smb M\KP1|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|$ holds for all $\eta\in
A\kern0.15mm\setminus N$, our condition is
weaker than the requirement (b) in
\cite[Theorem 8.18.2\kern0.37mm, p.\ 588]{Edw} that \math{
\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.37mm) } can be \q{lifted}.
\begin{remarks}\label{Rem Rad-deco}
At first sight, it may seem that Theorem \nfss A\,\ref{main Th} is less
general than the results contained in
\cite[Theorems 8.18.2\kern0.37mm, 8.18.3\kern0.37mm, pp.\ 588\kern0.37mm, 590]{Edw} when \math{p=1}
since in Edwards' presentation there is stated no assumption on any kind of
\q{decompos- ability}. However, one should note that in \cite{Edw} one
considers only positive measures that are {\sl positive \esl Radonian\kern0.15mm} in
the sense of \kern0.15mm Definitions \ref{df top deco}\,(4) on page \pageref{df pos Radon}
below, and that by Proposition \ref{Propo top-deco} these are \q{automatically}
almost decomposable. See also \cite[Proposition 4.14.9\kern0.37mm, p.\ 229]{Edw}\,.
We also remark the main ideas of the proof of \kern0.15mm Theorem \nfss A\,\ref{main Th}
are essentially, at least implicitly, contained in \cite[pp.\ 573\,--\,607]{Edw}
although it is not quite straight- forward to see the exact details from the
presentation there.
Note that in \cite{Edw} positive measures are obtained from positive linear
functionals in the vector spaces \math{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\kern-0.3mm\sbi{\rm c}\kern0.37mm(\kern0.37mm\scrmt T\,) } of compactly supported continuous
functions for locally compact Hausdorff topologies \mathss30{\scrmt T}, \,
cf.\ \cite[4.3\kern0.37mm, pp.\ 177\,--\,179]{Edw}\,. Furthermore, in \cite{Edw}
measurability of functions is defined by the Lusin property which is
meaningless for general measures.
\end{remarks}
\begin{remark}
Using Theorem \nfss A\,\ref{main Th} one is able to prove
\cite[5.22\kern0.37mm, p.\ 27]{Am97} in the more general case where only separability
of the topology \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} is required instead of having \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } separable. Then for example in the case \math{
\vPi=\LLrs42^1(\ssbb44 I) } the strong dual of the Besov space \math{
\Besovrss600_q^{\emath s\kern0.15mm,\,p}\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\kern0.37mm) } is seen
to be canonically represented by \linebreak
\ú$
\Besovrss300_{q\sast}^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s\kern0.15mm,\,p\sast}\kern-0.63mm \sbig(2\yi N\ssbb67 R,\kern0.07mm
\vPi\dlbetss01\kern0.15mm) \kern0.37mm $ when \math{s\in\mathbb R} and \math{1\le p<\lower1.05mm\hbox{$^+$}\infty }
and \math{1\le q<\lower1.05mm\hbox{$^+$}\infty} and \math{\smb N\in\mathbb N} hold. \linebreak
This is in
constrast with the case of Bessel potential spaces where the strong dual \linebreak
of \math{
\HBsrss606^{\emath s\kern0.15mm,\,p}\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\kern0.37mm) } is
represented only by \mathss38{\HBsrss300^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s\kern0.15mm,\,p\sast}
\sbig(2\yi N\ssbb67 R,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }. We hope to have the
opportunity to give the details of the proof in a future publication.
\end{remark}
We shortly review the {\font\≈=cmss10\≈it contents\kern0.37mm} which is organized according
to the scheme: \vskip.5mm {\newcommand\nrm[1]{$\null$\kern2.8mm{\font\≈=cmr9\≈#1\kern.4mm}}\newcommand\ntrm[2]{$\null$\kern2.8mm\ref{#1}\,{\font\≈=cmr8\≈#2\kern.4mm}}
\parindent6mm \inskipline07
1. Some special constructions \dotfill \ p.\KP{2.95} \pageref{Ss spec ctrs} \KP4 \inskipline07
2. Suitable locally convex spaces \dotfill \ p.\ \pageref{Ss suit lcs} \KP4
A \ Measurability and integration \dotfill \ p.\ \pageref{Sec A} \KP4 \inskipline07
1. Measurability of measure\kern0.37mm-\kern0.15mm vector maps \dotfill \ p.\ \pageref{Ss C1} \KP4 \inskipline07
2. Decomposable positive measures \dotfill \ p.\ \pageref{Ss decos} \KP4 \inskipline07
3. Integration of scalar functions \dotfill \ p.\ \pageref{Ss int scal} \KP4 \inskipline07
4. Pettis integration of vector functions \dotfill \ p.\ \pageref{Ss Pettis} \KP4
B \ Generalized Bochner spaces \dotfill \ p.\ \pageref{Sec B} \KP4
C \ Lifting and integral representations \dotfill \ p.\ \pageref{Sec C} \KP4 \inskipline07
1. Dunford\,--\,Pettis property of $\,\mLrs42^1(\kern0.37mm\mu\kern0.37mm)$ \dotfill \ p.\ \pageref{Ss Dun-Pet} \KP4 \inskipline07
2. Absolutely continuous vector measures \dotfill \ p.\ \pageref{Ss abs conti} \KP4
D \ Duality of Bochner spaces \dotfill \ p.\ \pageref{Sec D} \KP4
E \ Examples and open problems \dotfill \ p.\ \pageref{Sec E} \KP4 \inskipline0{9.2}
References \dotfill \ p.\ \pageref{Sec Bib} \KP4
\par} \vskip.5mm
In subsection 1 of this introductory section we give some special
constructions in order to be able to express certain matters concisely and
precisely at the same time. In 2 we give the basic definitions associated with
suitable spaces. We also establish some lemmas that are needed in the sequel.
In section A we present our approach to measurability and integration of
scalar and vector valued functions, or put more precisely,
{\sl mv\kern0.37mm-\kern0.15mm map\kern0.37mm}s. These are triplets \math{
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where \math{\vPi} is a real or complex
topological vector space and \math{\mu} is a positive measure on some set \math{
{}^{}\Cal Omega} and \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is a function.
In section B we first give the formal construction of our generalized
Lebesgue\,--\,Bochner spaces of equivalence classes \math{\smb X} of
measurable functions \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } when a positive measure \math{
\mu} on \math{{}^{}\Cal Omega} is given. Then we prove several results associated with
these spaces that are needed in the proof of our main theorem.
Section C contains several auxiliary results that are needed to prove that a
given continuous linear functional \math{ \smb U \kern-0.3mm :
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) = F \to \bosy K } can be represented by
some vector \math{\smb Y} of the space \math{F\aar 1} in the sense that for \math{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{y\in\smb Y} we have the equality \mathss36{
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu}.
At the beginning of section D we note how other assertions
of \kern0.15mm Theorem \nfss A\,\ref{main Th} except surjectivity of \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
follow from results that have already been established in section B\kern0.37mm. Then
we prove the surjectivity in Lemmas \nfss A\,\ref{LeA(1)}$\,,\ldots\KPt8
$\nfss A\,\ref{final lemma} separately in the cases (1)$\,,\ldots\KPt8$(6)
with (5) and (6) being treated together in \nfss A\,\ref{final lemma}\kern0.37mm.
In section E we have collected examples to make more concrete some points of
the general theory. We also present some related open problems. \vskip.5mm
Above we already indicated that \math{\tvsps0(K)} is the class of all
topological vector spaces over \math{\bosy K} when \math{\bosy K} is a
topological field. We put \vskip.3mm\centerline{$
\roman{TVS}\kern0.4mmps0(K) = \tvsps0(K)\capss31\{\, E :
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm\text{ is a Hausdorff topology } \} \KP1$,} \inskipline{.3}0
and we let \math{\roman{LCS}\kern0.4mmps0(K)} be the subclass of \math{\roman{TVS}\kern0.4mmps0(K)} formed by
the locally convex spaces. For \math{E\in\tvsps0(K)} we have \math{\bouSet E}
the set of all bounded sets in \mathss35{E}, also cal- led the {\sl von Neumann
bornology\kern0.15mm} of \mathss35{E}. For \math{E\,,\kern0.15mm F\in\tvsps0(K) } we let \math{
E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} mean \linebreak
that the identity \math{\kern.3mm\roman{id}\kern.7mmv F} is a continuous linear map \mathss35{
F\to E}.
If \math{E} is a real or complex topological vector space, then \math{
\Cal S\sbi{\kern0.15mm\emath r\,}E } and \math{\BSnorm E} and \linebreak
\ú$\Bqnorm E\kern0.37mm$ are the
sets of continuous \mathss35{r}--\,seminorms, bounded seminorms and bounded
quasi\kern0.37mm-\kern0.37mm seminorms, respectively, the formal constructions being given
in (1)$\,,\ldots\,$(3) below. We also put \math{ \Cal S_{_N}\kern0.15mm E =
\Cal S\LHB{.3}{_{\,1\,}} E} thus getting the set of continuous seminorms.
Note the implication \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm \in \Cal S\sbi{\kern0.15mm\emath r\,}E \impss33
0 < r \le 1 } and that a quasi\kern0.37mm-\kern0.37mm seminorm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} being
{\sl bounded\kern0.37mm} means that \math{ \sup \KP1 (\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\kern0.37mm) <
\lower1.05mm\hbox{$^+$}\infty } holds for every \mathss35{B\in\bouSet E}. We generally have \mathss35{
\Cal S_{_N}\kern0.15mm E\subseteq\BSnorm E}, \,and the converse inclusion holds if \math{E} is
normable.
\begin{enumerate}\begin{myLeftskip}{-2}{.5}{.3}
\item \ $\Cal S\sbi{\kern0.15mm\emath r\,}E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : 0 < r \le 1\kern0.37mm\text{ and }\kern0.37mm \label{df r-semin E}
\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, $ \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E\impss33
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \kern-0.3mm \le \kern0.15mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}\kern-0.3mm} +
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\kern0.37mm)\RHB{.2}{\KPt8^{\emath r}} \KPp1.4\big]$ \newfline
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} is continuous \mathss39{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\to\nsTbb_R \, \} }, \KP{10}
\item \ $\Bqnorm E = \kern0.15mm^{\svecs E}\KP1\lbb R_+\kern-0.2mm\cap\kern0.37mm \{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm : \label{defi bqnor E}
\eexi{\smb A}\,\aall{t\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\,
\smb A\in\rbb R^+\kern0.15mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y = |\,t\,|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x\kern0.37mm)\KP{1.4}] \kern0.37mm$ and \newskline{22}
$[\KP{1.4}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E\impss33 \label{defi bqnor E p}
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z \le \smb A\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\kern0.37mm)
\KP{1.4}]$ \newfline
and $\kern0.37mm \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt7\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\bouSet E\subseteq\bouSet\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R \KP1 \} \KP1 $, \KP{10}
\item \ $\BSnorm E = \Bqnorm E\capss30\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm:\aall{x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z}\, \label{defi bsnor E}
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)\in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm E$ \newfline
$\impss03 \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 z \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 x + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss30 y\KPt9\}\KP1$. \KP{10}
\end{myLeftskip}\end{enumerate}
\insubsubhead Some special constructions \label{Ss spec ctrs}
We are working within a Kelley\,--\,Morse\,--\,G\"odel\,--\,Bernays\,--\,von
Neumann type approach to set theory, like for example the one introduced in
\cite[pp.\ 250\,--\,281]{Ky}\,. Then with \math{ x\kern0.37mm\lower1mm\hbox{$^{^+}$} =
x\cupss21\{\kern0.37mm x\kern0.37mm\} } putting \mathss38{ \mathbb No = \kern0.37mm
\bigcap\KPt8\{\,N\kern-0.3mm:\emptyset\in N\kern0.37mm\text{ and }\kern0.37mm\aall{k\in N}\,
k\kern0.37mm\lower1mm\hbox{$^{^+}$}\in N\KPt9\} }, \,we may call \math{\mathbb No} the set of {\sl natural
number\kern0.15mm}s. It equals the set of finite cardinals, as well as the set of
finite ordinals. Let \math{\infty=\mathbb No} and \mathss38{ \mathbb N =
\mathbb No\kern-0.3mm\setminus\{\kern0.37mm\emptyset\kern0.37mm\} }.
We assume that the set \math{\mathbb H} of {\sl quaternion\kern0.15mm}s is constructed
in a certain manner so that we have \math{\mathbb H\subseteq
\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1\kern-0.3mm\times(\kern0.37mm\ovbbR\ar 1\kern-0.3mm\times\ovbbR\ar 1) }
for some set \math{\ovbbR\ar 1} with \vskip.3mm\centerline{$
\ovbbR\ar 1 \subseteq \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)\times
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.37mm\mathbb No\kern-0.2mm\times\mathbb N\kern0.37mm)) $} \inskipline{.3}0
where the {\sl power class\kern0.15mm} \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A} of \math{A} is defined in
Definitions \ref{misc defs}\,(14) below. Then for some set \math{ 0\ar 1 \in
\ovbbR\ar 1} we have \math{\mathbb R \subseteq \ovbbR\ar 1\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern-0.2mm
\times\sbig(2\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 } and \inskipline{.2}{41.7}
$\mathbb C \subseteq \ovbbR\ar 1\kern-0.3mm\times \ovbbR\ar 1\kern-0.3mm\times\sbig(2
\{\,0\ar 1\kern-0.2mm\}\kern-.2mm\times\kern-.2mm\{\,0\ar 1\kern-0.2mm\}\kern0.15mm\sbig)0 \KP1 $. \inskipline{.2}0
The definitions of the sets \math{\mathbb Z} and \math{\lbb Z_+} of
{\sl integers\kern0.15mm} and {\sl nonnegative\kern0.15mm} integers, respectively, being given
in \ref{misc defs}\,(5) and \ref{misc defs}\,(7) below, we have a bijection \math{
\mathbb No\to\lbb Z_+} given by \mathss03{i\mapsto n=i\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}} with inverse \math{
n\mapsto i=n\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} and now for example \mathss30{ i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} =
(\kern0.37mm i + 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm)\kern0.07mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} = n + 1 } and \math{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} = (\kern0.37mm i + 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm)\kern0.07mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}=n + 2 } and \mathss37{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} = (\kern0.37mm n + 1\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} }. Also \math{
\emptyset = 0\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} holds.
Having \mathss38{ \ovbbR = [\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KP1] = \{\,t :
\kern.2mm\lower1.05mm\hbox{$^-$}infty\le t\le\lower1.05mm\hbox{$^+$}\infty\KPt9\} }, \,we assume the formal definitions
having been arranged so that for all \math{u\kern0.37mm,\kern0.15mm v} we have \math{u\le v}
if{}f \math{\kern.2mm\lower1.05mm\hbox{$^-$}infty\le u\le v\le\lower1.05mm\hbox{$^+$}\infty} or
\math{u\kern0.37mm,\kern0.15mm v} are functions with
\math{u\cupss22 v\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} v\times\ovbbR } and \mathss38{{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq
\{\,\eta:u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KPt8\} }. Hence if \math{u} and \math{v}
are extended real valued functions, then \math{u\le v} means that
we have \math{{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} v} and that
\math{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for all
\mathss34{\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}.
Furthermore \math{\emptyset\le v} is equivalent to having \math{v} a
function with \mathss38{{}^{}{\rm rng}\,{}_{{}^{}} v\subseteq\ovbbR}.
In order to specify some set theoretic notation already utilized above that
also has largely been explained in \cite[pp.\ 4\,--\,8]{HiDim} and
\cite[pp.\ 4\,--\,9]{SeBGN}\,, ending on page \pageref{END set th extract}
below, we next present an extract from \cite{Hif}\,.
We assume that the {\sl intuitive class\kern0.15mm} of all {\sl variable symbol\kern0.15mm}s
of our set theory is implicitly intuitively well\kern0.37mm-\kern0.37mm ordered so that it
makes sense to speak of the first variable (\kern0.15mm symbol\kern0.37mm) not possessing
some property.
\begin{def:al schemas}[set notation]\label{defi {F:...:P}}
Let \math{\mfrk F} be any term and \math{\mfrk P} a formula and $\afr x_1\kern0.15mm,
\ldots\,\mfrk x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern0.37mm$
distinct variable symbols such that \math{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}
are precisely the variable symbols which have a free occurrence both in \math{
\mfrk F} and \math{\mfrak P} and are not in the list \math{\afr y_1\kern0.15mm,\ldots\,
\mfrk y\kern0.37mm\ai l}. Also let \math{\mfrk x} be the first variable symbol not
occurring free in \math{\mfrk F} or \math{\mfrk P}. Then we let \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\} =
\{\,\mfrk x:\exi{\afr x_1\kern0.15mm,\ldots\,\mfrk x\kern0.37mm\ai k}\,\mfrk x=\mfrk F\kern0.37mm$
and $\kern0.37mm\mfrk P\,\}\yxbtext{15}1b \kern0.15mm}.
In the case where \math{
\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l} is an empty list, we further let \math{
\{\,\mfrk F:\mfrk P\,\} = \{\,\mfrk F:\ :\mfrk P\,\} \kern0.15mm}.
The variable symbols which are free in the term \math{
\{\,\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrk y\kern0.37mm\ai l\kern-0.63mm:\mfrk P\,\}} are (\kern0.37mm by
re- cursive definition\kern0.15mm) exactly those which are free either in \math{
\mfrk F} or \math{\mfrk P}, and are not in the list \math{\afr x_1\kern0.15mm,\ldots\,
\mfrk x\kern0.37mm\ai k}. The free variables of \math{\{\,\mfrk F:\mfrk P\,\}} are
precisely those which are free in \math{\mfrk F} or \math{\mfrk P} but not in
both of them.
\end{def:al schemas}
The above schemata, which we introduced to overcome the notational problem
presented in \cite[4 Notes, pp.\ 5\,--\,6]{Ky}\kern0.37mm, only provide reduction of
\PouN$\kern0.37mm\{\KPt8\mfrk F:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:\mfrak P\,\}$
and \math{\{\KPt8\mfrk F:\mfrak P\,\} } to \math{
\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \kern0.15mm}. In order to be able to prove
something nontrivial about $\{\KPt8\mfrk x:\mfrk Q\,\}\yxbtext{15}1b \,$, we
need some {\font\≈=cmss10\≈it axioms\kern0.37mm}. As such, we accept all the formulas
\begin{enumerate}\begin{myLeftskip}{-4}{.3}{.1}
\itemb0_ax $u=v\equivss22\aall x\,x\in u\equivss22 x\in v\,$, \label{ax of extent}
\itemb0_ax $u\in v\impss22\eexi{w\kern0.37mm,z}\,w\in z\kern0.37mm$ and $\,
\aall x\,x\subseteq u\impss22 x\in w\kern0.37mm$, \label{ax of subsets}
\itemb0_ax $x\in u\kern0.37mm$ and $\kern0.37mm y\in v\impss22\eexi w\,
x\subseteq w\kern0.37mm$ and $\kern0.37mm y\subseteq w\kern0.37mm$, \label{ax of union}
\itemb0_ax $u\in z\kern0.37mm$ and $\,[\KP{1.4}\aall{x\kern0.37mm,y\kern0.37mm,z}\,
(\kern0.37mm x\kern0.37mm,y\kern0.15mm)\kern0.37mm,(\kern0.37mm x\kern0.37mm,z\kern0.15mm)\in f \impss22
y=z \KP{1.4} ] \impss22 \eexi{v\kern0.37mm,w}$ \newfline
$v\in w\kern0.37mm$ and $\,\aall y\,y\in v\equivss22\eexi x\,x\in u\kern0.37mm$
and $\kern0.37mm(\kern0.37mm x\kern0.37mm,y\kern0.15mm)\in f\kern0.15mm$, \KP{17.9} \label{ax of substi}
\itemb0_ax $z\in w\impss22\eexi{u\kern0.37mm,v}\,u\in v\kern0.37mm$ and $\,\aall x\,
x\in u\equivss22\eexi y\,x\in y\in z\,$, \label{ax of amalg}
\itemb0_ax $v\in u\impss22\eexi x\,x\in u\kern0.37mm$ and not $\eexi z\,
z\in x\kern0.37mm$ and $\kern0.37mm z\in u\,$, \label{ax of regularity}
\itemb0_ax $\eexi{e\kern0.37mm,\kern0.07mm N\kern0.15mm,\kern0.07mm S}\,e\in N\in S\kern0.37mm$ and \,[\ not $
\eexi x\,x\in e \KP{1.4} ]\kern0.37mm$ and $\,\aall{n\kern0.37mm,\kern0.07mm m}$ \newfline
$n\in N\kern0.37mm$ and $\,[\KP{1.4}\aall x\,x\in m\equivss22
x\in n\kern0.37mm$ or $\kern0.37mm x=n \KP{1.4} ]\impss22 m\in N\kern0.37mm$, \KP{17.9} \label{ax of infin}
\itemb0_ax $\eexi C\,[ \KP{1.4} \aall{x\kern0.37mm,z\kern0.37mm,u}\,(\kern0.15mm u\kern0.37mm,x\kern0.15mm)\kern0.37mm,
(\kern0.15mm u\kern0.37mm,z\kern0.15mm)\in C\impss22 x=z\in u \KP{1.4} ]\kern0.37mm$ and \newfline
$\aall{z\kern0.37mm,u\kern0.37mm,w}\,z\in u\in w\impss22\eexi x\,
(\kern0.15mm u\kern0.37mm,x\kern0.15mm)\in C\kern0.37mm$, \KP{17.9} \label{ax of choice}
\end{myLeftskip}\end{enumerate}
\noindent and also all the formulas (\kern0.15mm s\kern0.15mm) given in the next
\begin{axiom schema}[classification]\label{class axi}
Let \math{\mfrk x} be any variable symbol and \math{\mfrk P} any formula. Let
\math{\mfrk y} be the first variable symbol distinct from \math{\mfrk x} and
not occurring free in \math{\mfrk P}. Then we accept as an axiom the formula
(\kern0.15mm s\kern0.15mm) \ $\mfrk x\in\{\KPt8\mfrk x:\mfrk P\,\}\yxbtext{15}1b\equivss22
\eexi{\mfrk y}\,\mfrk x\in\mfrk y\kern0.37mm$ and $\kern0.37mm\mfrk P\,$.
\end{axiom schema}
Above (\ref{ax of choice})$\ar{ax}$ is the {\sl global axiom of choice\kern0.15mm} and
(\ref{ax of infin})$\ar{ax}$ is the {\sl axiom of infinity\kern0.15mm}.
\begin{remark}\label{rem about class sch}
Among others, we accept as logical axioms the formulas \inskipline{.3}3
(1)$\ar{az}\,$ \ $\mfrk P\impss22\aall{\mfrk x}\,\mfrk P\,$, \KP{20}
(2)$\ar{az}\,$ \ $[ \KP{1.4} \aall{\mfrk x}\,\mfrk Q \KP{1.4} ]\impss22
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\lleftarrow\kern-0.3mm\mfrak F\kern0.37mm) \,$, \inskipline{.3}0
when \math{\mfrk x} is any variable symbol and \math{\mfrk F} is any term and
\math{\mfrk P\kern0.37mm,\mfrk Q} are any formulas such that for any variable symbol \math{
\mfrk y} having a free occurrence in \math{\mfrk F} the bound (\kern0.37mm i.e.\
non\kern0.37mm-\kern0.37mm free\kern0.15mm) occurrences of \math{\mfrk y} in \math{\mfrk Q} and \math{
\mfrk Q\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak y\kern0.37mm)} are the same. Having these
logical axioms, we could give Axiom schema \ref{class axi} above a simpler
formulation than has the corresponding \cite[\erm{II}\kern0.37mm, p.\ 253]{Ky} which
in our notation would (\kern0.15mm as already a bit corrected\kern0.37mm) read as follows.
For any variable symbols \math{\mfrk x\kern0.37mm,\kern0.15mm\mfrk y\kern0.37mm,\kern0.15mm\mfrk z} and for
any formula \mathss30{\mfrk P} such that \math{\mfrk y} is the first one
distinct from \math{\mfrk x} and \mathss34{\mfrk z}, \,and not occurring free
in \mathss34{\mfrk P}, we accept as an axiom the formula \inskipline{.3}3
(t) \ $\aall{\mfrk z}\,\mfrk z\in\{\,\mfrk x:\mfrk P\,\}\yxbtext{15}1b\equivss22
\eexi{\mfrk y}\,\mfrk z\in\mfrk y\kern0.37mm$ and $\kern0.37mm
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak z\kern0.37mm)\,$. \inskipline{.3}0
However, this would make the system contradictory as shown in Example \ref{exa about Kelley's class sch}
below. One should put the additional restriction that the bound occurrences of \math{
\mfrk z} in $\kern0.37mm\mfrk P$ \linebreak
and \math{
\mfrk P\,(\kern0.37mm\mfrk x\kern-0.3mm\leftarrow\kern-0.3mm\mfrk z\kern0.37mm) } are the same.
\end{remark}
\begin{example}\label{exa about Kelley's class sch}
It follows from (\ref{ax of infin})$\ar{ax}$ and (\ref{ax of extent})$\ar{ax}$
and Proposition \ref{Pro basic}\,(18) below that there are \math{a\kern0.37mm,\kern0.15mm b
\kern0.37mm,\kern0.15mm c} with \math{b\not=a} and \mathss36{a\kern0.37mm,\kern0.15mm b\in c}. For \mathss36{
A = \{\,x:\eexi y\,x=y\kern0.37mm\text{ and }\kern0.37mm y=a\,\}\yxbtext{15}1b }, we then
get from Remark \ref{rem about class sch}\,(t) and
Proposition \ref{Pro basic}\,(17) that for all \math{x\kern0.37mm,\kern0.15mm y} we have \vskip.4mm \centerline{$
x\in A\equivss22[ \KP{1.4} x\kern0.37mm$ set and $\kern0.37mm\eexi y\, x = y \kern0.37mm$ and $\kern0.37mm
y = a \KP{1.4} ]\equivss22 x = a \,$,} \inskipline{.2}0
and $\KP{13.7} y \in A \equivss22[ \KP{1.4} y \kern0.37mm$ set and $\kern0.37mm \eexi y\,
y = y\kern0.37mm $ and $\kern0.37mm y = a \KP{1.4} ]\equivss22 y\kern0.37mm$ set\kern0.37mm, \inskipline{.4}0
whence taking \math{x=y=b\kern0.15mm}, we obtain \mathss38{[ \KPp1.4 b \kern0.37mm\text{ set }
\impss03 b\in A\impss33 b = a \KPp1.4 ] }, \,a {\sl contradiction\kern0.15mm}. The
formula \q{\math{\eexi y\, y = y\kern0.37mm\text{ and }\kern0.37mm y = a }} contains four
occurrences of \math{\Symboo yœ}. They are all bound and the second of them is
not present in \q{\math{\eexi y\, x = y\kern0.37mm$ and $\kern0.37mm y = a }}.
\end{example}
When we write a formula \mathss34{\mfrak P}, \,for example \q{\mathss00{ x =
\int_{\,\ssmb A}^{\KPt8\ssmb B}f\fvalss21 t\rmdss11 t}}, associated with the
writing appearance of \math{\mfrak P} we assume that there is an implicitly
understood well\kern0.15mm-\kern0.37mm order between the occurring variable symbols so that
e.g.\ it makes sense to refer to the first variable symbol occurring free in
the {\sl writing appearance\kern0.15mm} of \mathss34{\mfrak P}. This has nothing to do
with the intuitive \q{overall} well\kern0.15mm-\kern0.37mm order of all variable symbols of
our set theoretic language.
For example in the above formula the variable symbols \math{\Symboo xœ,
\Symboo\kern-0.63mm\smb A\kern-0.3mmœ,\Symboo\kern-0.2mm\smb B\kern-0.2mmœ,\Symboo\kern-0.63mm fœ} occur free, and \math{
\Symboo tœ} has two bound occurrences. We may assume that the order of the
free variable\kern0.37mm( symbol\kern0.37mm)\kern0.37mm s is precisely the one given above,
although it may not be perfectly clear which one of \math{
\Symboo\kern-0.63mm\smb A\kern-0.3mmœ} and \math{\Symboo\kern-0.2mm\smb B\kern-0.2mmœ} is before the other.
To avoid confusion, in such vague cases we refrain from referring to that
\q{implicit order}. In the above case we may then say that \math{\Symboo xœ}
is the first one, whereas in the case of the formula \q{\kern-0.63mm\mathss01{
\int_{\,\ssmb A}^{\KPt8\ssmb B}f\fvalss21 t\rmdss11 t = x }} we would not
speak of the free variable symbol that is in the first place in the writing
appearance.
Having the above preparative explanation, in order to have available a
convenient means of specifying functions, we give the following
\begin{def:al schema}\label{<T:F>}
Let \math{\mfrak T} be a term and \math{\mfrak F} a formula and \math{
\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,\mfrak x\kern0.37mm\ai k\kern0.37mm,\kern0.15mm\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} distinct variable symbols such that \math{\afr x_1\kern0.15mm,
\ldots\,\mfrak x\kern0.37mm\ai k} are precisely the variable symbols which have a
free occurrence both in \math{\mfrak T} and \math{\mfrak F} and are distinct
from any of $\,\mfrak x\,,\kern0.15mm\afr y_1\kern0.15mm,$ $\ldots\,\mfrak y\kern0.37mm\ai l\kern0.37mm$.
Also assume that \math{\mfrak F} is of the form \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} or \math{
\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'074}\kern.2mm\,\mfrak k\kern0.15mm\,\mfrak p\kern0.15mm\,\mfrak x\kern0.37mm\kern.2mm\raise1.5mm\hbox{\font\SweD =cmmi5\SweD \char'076}\kern0.15mm\,\mfrak E} where \math{
\mfrak p} is some predicate symbol and \math{\mfrak k} is a connective such
that in the writing appearance of $\kern0.37mm\mfrak F$ we have \math{\mfrak x} in
the first place. Then we let \vskip.2mm\centerline{$
\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\afr y_1\kern0.15mm,\ldots\,\mfrak y\kern0.37mm\ai l\kern-0.63mm:
\mfrak F\kern0.15mm\,}=\{\,\mfrak z:\exi{\mfrak x\kern0.37mm,\afr x_1\kern0.15mm,\ldots\,
\mfrak x\kern0.37mm\ai k}\,\mfrak z=(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm
\mfrak F\,\}$} \vskip.2mm
\noindent where \math{\mfrak z} is the first variable symbol not occurring free in
\math{\mfrak T} or \math{\mfrak F\kern0.15mm}.
We also put \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:\mfrak F\kern0.15mm\,}=\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\ :\mfrak F\kern0.15mm\,}} in the case where \math{\afr y_1\kern0.15mm,\ldots\,
\mfrak y\kern0.37mm\ai l} is an empty list, and further \math{\seq{\,\kern0.15mm\mfrak T\kern-0.3mm:
\mfrak x\in\mfrak U\kern0.15mm\,}\subtext{old}=\{\,\mfrak z:\exi{\mfrak x}\,\mfrak z =
(\kern0.37mm\mfrak x\,,\mfrak T\kern0.37mm)\kern0.37mm$ and $\kern0.37mm\mfrak x\in\mfrak U\,\}} when \math{
\mfrak U} is any term not containing a free occurrence of \math{\mfrak x\kern0.15mm},
and \math{\mfrak z} is the first variable symbol distinct from \math{\mfrak x}
and not occurring free in \math{\mfrak T} or \math{\mfrak U\kern0.37mm}.
\end{def:al schema}
\begin{def:al schema}\label{uniqset}
We let \math{\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,
\mfrak P\kern0.15mm\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\kern0.15mm}, when \math{\mfrak x\kern0.37mm,
\mfrak z} are any distinct variable symbols and \math{\mfrak P} is any formula
where \math{\mfrak z} does not occur free. To get $\mfrak z$ uniquely chosen,
we may take as \math{\mfrak z} the first admissible w.r.t\ the intuitive
well\kern0.37mm-\kern0.37mm ordering of the variable symbols of our set theoretic language.
\end{def:al schema}
Under the agreement of unique choice of $\mfrak z$ above, for any formula $
\mfrak P$ and any distinct variable symbols $\mfrak x\kern0.37mm,\mfrak z$ with $
\mfrak z$ not occurring free in $\mfrak P\kern0.37mm$, now the formula $
\uniqset\mfrak x:\mfrak P=\bigcap\,\{\,\mfrak z:\all{\mfrak x}\,\mfrak P\kern0.15mm
\Leftrightarrow\kern0.15mm\mfrak x=\mfrak z\,\}\,$ is a theorem. \vskip.2mm
One quickly deduces that if a unique {\it set\kern0.37mm} \math{\mfrak x} exists with
\math{\mfrak P}, then \math{\uniqset\mfrak x:\mfrak P=\mfrak x\kern0.15mm}. In all
other cases, i.e.\ when there is no \math{\mfrak x\in\hbox{\font\SweD =cmssbx10\SweD U}{}} with \math{
\mfrak P}, or if (with $\mfrak y$ being a variable symbol not occurring in \math{
\mfrak P}) there are \math{\mfrak x\kern0.37mm,\mfrak y\in\hbox{\font\SweD =cmssbx10\SweD U}{}} with \math{
\mfrak x\not=\mfrak y} and \math{\mfrak P} and \math{
\mfrak P\,(\kern0.37mm\mfrak x\kern-0.3mm\leftarrow\kern-0.3mm\mfrak y\kern0.37mm)\kern0.15mm}, substitution in
places of free occurrence, then \math{ \label{END set th extract}
\uniqset\mfrak x:\mfrak P = \bigcap\kern0.37mm\emptyset = \hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm}. \vskip.4mm
Below in Definitions \ref{misc defs}\,(9) we have \math{\raise1.35mm\hbox{\font\SweD =cmr5\SweD c}\kern-.15mm\infty =
(\kern0.37mm 0\ar 1\kern0.15mm,\lower1.05mm\hbox{$^+$}\infty\ar 1\kern0.37mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1) } the
{\sl complex infinity\kern0.15mm} for some \math{\lower1.05mm\hbox{$^+$}\infty\ar 1\in\ovbbR\ar 1} whose
exact construction we here omit. Also omitting the precise definition, note
that \math{|\,\zeta\,|\suba} is the standard \erm Euclidean absolute value of
any quaternion \math{\zeta} and that we below usually have \math{\zeta} a real
or complex number.
\begin{definitions}\label{misc defs}
(1) \ $\bbI=[\KPp1.1 0\,,\kern0.07mm 1\KPt9] \KP1 $, \KP{
8.3}
(2) \ $\mathbb J=\openIval{\KPt5 0\,,\kern0.07mm 1\kern0.15mm} \KP1 $, \inskipline{.5}2
(3) \ $\rbb R^+=\mathbb R\capss41\{\,t:0<t\KPt9\} \KP1 $, \KP{7.5}
(4) \ $\lbb R_+=\mathbb R\capss41\{\,t:0\le t\KPt9\} \KP1 $, \inskipline{.5}2
(5) \ $
\mathbb Z=
\bigcap\KPt8\{\,N\kern-0.3mm:
0\in N\subseteq\mathbb R\text{ and }\kern0.37mm
\aall{n\in N}\,
\{\,n-1\kern0.37mm,\kern0.15mm n+1\,\}\subseteq N\KP1\}
\KP1 $, \inskipline{.5}2
(6) \ $\rbb Z^+=\mathbb Z\capss41\{\,n:0<n\KPt9\} \KP1 $, \KP{6.5}
(7) \ $\lbb Z_+=\mathbb Z\capss41\{\,n:0\le n\KPt9\} \KP1 $, \inskipline{.5}2
(8) \ $p\,^* = \uniqset t:[\KPp1.4 1 < p < \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t =
(\kern0.37mm 1 - p\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big){}^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\KP1\big]\kern0.37mm$ or \inskipline0{31.6}
$[\KPp1.4 p=1\kern0.37mm$ and $\kern0.37mm t=\lower1.05mm\hbox{$^+$}\infty\KPp1.4]\kern0.37mm$ or $\kern0.37mm
[\KPp1.4 p=\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm t=1\KPp1.4] \KP1 $, \inskipline{.5}2
(9) \ $\Abrs33^p = \uniqset\chi:p\in\rbb R^+\kern0.15mm$ and \inskipline{.2}{28.65}
$\chi \kern0.37mm = \kern0.37mm \{\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\raise1.35mm\hbox{\font\SweD =cmr5\SweD c}\kern-.15mm\infty\,\}\kern-.2mm\times\kern-.2mm
\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} \cupss21 \big\langle\KP{1.2}
|\,\zeta\,|\suba\RHB{.25}{^p} \kern-0.3mm : \zeta\in\mathbb H\KPp1.2 \rangle \KP1 $, \inskipline{.5}2
(10) \ $\scrb8 T\,$ is a {\it topology\kern0.37mm}
$\equivss33 \emptyset\not=\scrb8 T\in\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\Cal A\subseteq\scrb8 T\impss33
\bigcup\,\Cal A\in\scrb8 T\kern0.37mm\text{ and }\kern0.37mm
[\KP{1.2} \Cal A\not=\emptyset\kern0.37mm\text{ and }\kern0.37mm
\Cal A\kern0.37mm\text{ is finite }\Rightarrow\kern0.37mm\bigcap\,\Cal A\in
\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(11) \ $\scrb8 T\,$ is a {\it separable\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\eexi D\,D\kern0.37mm$ is countable and
$\kern0.37mm\aall U\,U\in\scrb8 T\impss33 D\capss33 U\not=\emptyset\kern0.37mm\text{ or }\kern0.37mm
U=\emptyset\KPt8$, \inskipline{.5}2
(12) \ $\scrb8 T\,$ is a {\it compact\kern0.37mm} topology $\equivss33
\scrb8 T\,$ is a topology and \inskipline{.2}{10.5}
$\aall{\Cal A}\,\eexi{\Cal B}\,
\Cal A\subseteq\scrb8 T\impss33
\Cal B\subseteq\Cal A\kern0.37mm$ and $\kern0.37mm\Cal B\kern0.37mm$ is finite and \inskipline{.2}{10.5}
$\kern0.37mm[\KP{1.6}
\bigcup\,\Cal A\subseteq\bigcup\,\Cal B\kern0.37mm$ or $\kern0.37mm
\bigcup\,\Cal A\not=\bigcup\,\scrb8 T \KP{1.5} ] \KP1$, \inskipline{.5}2
(13) \ $\scrmt A\kern0.37mm$ is {\it disjoint\kern0.37mm} $\equivss33\aall{A\,,\kern0.15mm B}\,
A\,,\kern0.15mm B\in\scrmt A\impss33 A=B\kern0.37mm$ or $\kern0.37mm A\capss32 B=\emptyset \,$, \inskipline{.5}2
(14) \ $\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A=\{\,B:B\subseteq A\KP1\} \KP1 $, \KP{7.2}
(15) \ $\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 B=\{\,A\capss32 B : A\in\scrmt A\KP1\} \KP1 $, \inskipline{.5}2
(16) \ $\nsTbb_R=\big\{\kern0.37mm\bigcup\,\scrmt A:\scrmt A\subseteq\big\{\,
\openIval{\kern0.37mm s\kern0.37mm,\kern0.07mm t\kern0.37mm}:\kern.2mm\lower1.05mm\hbox{$^-$}infty < s < t < \lower1.05mm\hbox{$^+$}\infty\KPt9\}
\kern0.15mm\} \KP1 $, \inskipline{.5}2
(17) \ $\barscTbb_R=
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm[\kern0.37mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KP1]\capss51\{\KPt8 U\kern-0.3mm:
U\cap\ssbb40 R\in\nsTbb_R\kern0.37mm$ and $\kern0.37mm\eexi{s\kern0.37mm,\kern0.15mm r\in\mathbb R}\,$
\inskipline{.2}{21}
$
[\KPp1.4 \lower1.05mm\hbox{$^+$}\infty\in U\impss33
{\kern0.37mm]}\KP1 s\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\KPt9]
\subseteq U\KPp1.4]\kern0.37mm$ and $\kern0.37mm[\KPp1.4 \kern.2mm\lower1.05mm\hbox{$^-$}infty\in U\impss33
[\kern0.37mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\kern0.07mm r\KP1{[\kern0.37mm}
\subseteq U\KPp1.4]\KP1\big\} \KP1 $, \inskipline{.5}2
(18) \ $f\fvalss20 x=
\bigcap\KPt8\{\,y:\aall z\,
(\kern0.37mm x\kern0.37mm,\kern0.07mm z\kern0.37mm)\in f\equivss33 y=z\,\} \KP1 $, \inskipline{.5}2
(19) \ $f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A=\{\,y:
\eexi{x\in A}\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \KP4
(20) \ $f\KPt8[\KPt8 A\KPt9]=f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A \KPt8 $, \inskipline{.5}2
(21) \ $f\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern-0.3mm\scrmt A=
\{\,f\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A:A\in\scrmt A\KP1\} \KP1 $, \KP{18.2}
(22) \ $f\invss40=\{\,(\kern0.37mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(23) \ ${{}^{}{\rm dom}\,{}_{{}^{}}} f=\{\,x:\eexi y\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \KP8
(24) \ ${}^{}{\rm rng}\,{}_{{}^{}} f=\{\,y:\eexi x\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in f\KP1\} \KP1 $, \inskipline{.5}2
(25) \ $E\Reit3=\uniqset F:\eexi{a\kern0.37mm,\kern0.15mm c\,,\kern0.15mm\scrmt S}\, E =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\kern0.37mm$ and \mathss38{F =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\KPp1.1|\KP1(\ssbb40 R\times\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm)\,,\kern0.07mm\scrmt S\kern0.37mm) }, \inskipline{.5}2
(26) \ $^A\,B = A\times B\capss31\{\,f:f\kern0.37mm\text{ is a function and }\kern0.37mm
A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} f\KP1\} \KP1 $, \inskipline{.5}2
(27) \ $\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.15mm\bmii8 A \kern0.15mm = \kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii6 A}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,
x:\aall{i\kern0.37mm,\kern0.15mm\xi}\,(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in x\impss33\xi\in
\bmii8 A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\KPt9\} \KP1 $, \inskipline{.5}2
(28) \ $\bosy x\to x\kern0.37mm$ in top \math{\scrmt T\equivss33\bosy x \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bigcup\,\scrmt T } and \math{x\in\bigcup\,\scrmt T } and \inskipline0{24.2}
$\aall{\kern0.37mm U}\,\eexi{\smb N}\,x\in\kern0.15mm U\kern-0.3mm\in\scrmt T \impss33 \smb N \in
\mathbb No \kern0.37mm$ and \mathss30{\bosy x\KP1[\KPp1.1\mathbb No\kern-0.3mm\setminus\smb N\KP1]
\subseteq \kern0.15mm U}, \inskipline{.5}2
(29) \ $E\subsigrs04=\uniqset F:\eexi{\bosy K}\,\bosy K\kern0.37mm$ is a
topological division ring and \math{\Bnull_{\bosy K} = 0 } and \inskipline0{14}
$E\in\tvsps0(K) \kern0.37mm $ and \math{\aall{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm,\kern0.15mm I\kern0.15mm,\kern0.15mm\scrmt T}\,
I = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } and \inskipline0{31.5}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seqss03{\kern-0.3mm\seqss33{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x:u\in I}:x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} \kern0.37mm$ and \mathss39{
\scrmt T=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm(\kern0.37mm
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.15mm I\kern0.15mm]_{ti}\big) } \inskipline0{90.5}
$\impss03 F = (\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\scrmt T\,) \KP1 $.
\end{definitions}
About the {\sl weakening\kern0.15mm} \math{E\subsigrs04} of \math{E} in
Definitions \ref{misc defs}\,(29) above we note the following. If \math{E} is
a topological vector space over a topological division ring \mathss32{\bosy K
}, \,there may exist another topological division ring \math{\bosy K\kern-0.3mm\ar 1}
with \mathss38{E\in\tvsps9(K\kern-0.3mm\ar 1\kern-0.3mm) }. In every case then \math{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\aR 1} holds, but \math{\bosy K} and \math{
\bosy K\kern-0.3mm\ar 1} may possess different zero elements if \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E =
\{\,\Bnull_E\} } holds. Then the condition \math{\Bnull_{\bosy K}=0=
\Bnull_{\kern0.15mm\aars{\bosy K\kern-0.2mm}_1} } excludes this possibility. If \math{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E=\{\,\Bnull_E\} } holds, for \math{ I =
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } and \mathss30{ n =
\{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } then necessarily \math{ I =
\{\kern0.37mm n\kern0.37mm\} } holds, and we get \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \{\KPt8(\,\Bnull_E\kern0.37mm,\kern0.15mm
\{\,(\kern0.37mm n\kern0.37mm,\kern0.07mm 0\kern0.37mm)\,\}\kern0.37mm\sbig)2\kern0.37mm\big\} } and further \mathss06{
\scrmt T=\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E }. Hence in this case \math{\scrmt T} is uniquely
determined although \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\not=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\kern-0.3mm\ar 1 } may
hold. If \math{I\not=\{\kern0.37mm n\kern0.37mm\} } holds, then one deduces from the
postulates in the definition of a topological vector space that we necessarily
have \math{\bosy K=\bosy K\kern-0.3mm\ar 1 } and consequently again \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\scrmt T\,) } is uniquely determined.
Thus the above definition of \math{E\subsigrs04} is meaningful for precisely
those topological vector spaces \math{E} that are \q{over} some topological
division ring whose zero element is the same as that of the quaternionic one.
For more general cases one has to use a more complicated notation e.g.\ from \math{
E\subsigrs04\kern0.37mm\langle\,\bosy K=E\subsigrs04\kern0.37mm(\kern0.15mm I\kern0.37mm) } for \math{I} as
above, once the appropriate additional definition is specified.
In \ref{misc defs}\,(18) above \math{f\fvalss20 x} is the function value of \math{
f} at \math{x} which usually is written in a more complicated manner
\q{$f\kern0.37mm(x)$}, and possibly having a different formal definition as for
example in \cite[Definition 68\kern0.37mm, p.\ 261]{Ky}\,. We further state some
basic definitions and their simple consequences without proofs in the
following
\begin{proposition}\label{Pro basic}
$\null$ {\rm \inskipline{.5}2
(1) \ }$\emptyset=\{\,x:x\not=x\,\} \KP1 ${\rm, \KP{24}
(2) \ }$\hbox{\font\SweD =cmssbx10\SweD U}{}=\{\,x:x=x\,\} \KP1 ${\rm, \inskipline{.5}2
(3) \ $ \roman{pr}\ar 1 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm x\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1 $, \KP5
(4) \ }$\roman{pr}\ar 2 =
\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm y\kern0.37mm):x\kern0.37mm,\kern0.15mm y\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \KP1
${\rm, \inskipline{.5}2
(5) \ }$\roman{ev}=\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm u\kern0.37mm,\kern0.07mm y\kern0.37mm):u\kern0.37mm\text{ is a
function and }\kern0.37mm(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in u\,\} \KP1 ${\rm, \inskipline{.5}2
(6) \ }$\roman{ev}\sbi{\kern0.15mm\emath x} =
\seqss33{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x:u\kern0.37mm\text{ is a function}}$ \inskipline{.2}{14.45}
${} = \{\,(\kern0.37mm u\,,\kern0.07mm y\kern0.37mm):u\kern0.37mm\text{ is a function and }\kern0.37mm
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in u\KPt8\} \KP1 ${\rm, \inskipline{.5}2
(7) \ }$\scrmt A\,,\kern0.15mm\scrmt B$ disjoint iff $\,\scrmt A$ and $\,\scrmt B$
disjoint iff $\,\scrmt A$ and $\,\scrmt B$ are disjoint iff \inskipline0{52.2}
$\scrmt A$ is disjoint and $\,\scrmt B$ is disjoint{\kern0.15mm\rm, \inskipline{.5}2
(8) \ }$ \newcommand\opair[2]{\hbox{\kern.#1mm\kern-.2mm\font\≈=cmtt10\≈,\kern-.2mm\kern.#2mm}}
x \opair14 y = (\kern0.15mm x\kern0.37mm,\kern0.07mm y\kern0.37mm) =
\{\kern0.15mm\{\,x\kern0.37mm,\kern0.15mm y\,\}\,,\kern0.15mm\{\kern0.37mm y\kern0.37mm\}\kern0.15mm\} \KP1 ${\rm, \KP{3.9}
(9) \ }$(\kern0.15mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm) =
((\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\kern0.37mm,\kern0.07mm z\kern0.37mm) \KP1 ${\rm, \inskipline{.5}2
(10) \ }$(\kern0.37mm x\,;\kern0.07mm y\kern0.37mm,\kern0.07mm z\kern0.37mm) =
(\kern0.37mm x\kern0.37mm,\kern0.07mm(\kern0.37mm y\kern0.37mm,\kern0.07mm z\kern0.37mm)) \KP1 ${\rm, \KP{15}
(11) \ }$(\kern0.15mm x\kern0.37mm,\kern0.07mm y\,;\kern0.07mm u\kern0.37mm,\kern0.07mm v\kern0.37mm) =
(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm(\kern0.37mm u\kern0.37mm,\kern0.07mm v\kern0.37mm)) \KP1 ${\rm, \inskipline{.5}2
(12) \ }$\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z = \bigcup\bigcup\kern0.15mm z\setminus\bigcup\bigcap\kern0.15mm z
\kern0.15mm\cup\kern0.15mm \bigcap\bigcup\kern0.15mm z ${\,\rm, \KP{3.8}
(13) \ }$\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z=\bigcap\bigcap\kern0.15mm z ${\,\rm, \inskipline{.5}2
(14) \ }$\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm z = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z\kern0.37mm) ${\KP1\rm, \KP{22.8}
(15) \ }$\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z = \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z\kern0.37mm) ${\KP1\rm, \inskipline{.5}2
(16) \ }$z = (\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\in\hbox{\font\SweD =cmssbx10\SweD U}{}\impss33 x = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z$ and $\kern0.37mm
y = \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm z ${\,\rm, \inskipline{.5}2
(17) \ }$x\kern0.37mm$ is a set $\equivss22 x\kern0.37mm$ a set $\equivss22
x\kern0.37mm$ set $\equivss22 \eexi y\,x\in y ${\kern0.37mm\rm, \inskipline{.5}2
(18) \ }$x \not= y\equivss22$ not $[\KP{1.4} x=y \KP{1.4} ] \KP1 $.
\end{proposition}
Observe for example that if \math{E=(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt S\kern0.37mm)\not=
\hbox{\font\SweD =cmssbx10\SweD U}{} } with \math{c} a function \mathss34{R\times S\to S}, \,then \
${{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm)))=
{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.37mm a\kern0.37mm,\kern0.15mm c\kern0.37mm)))$ \inskipline0{28}
${}={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}} c\kern0.37mm)={{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.15mm R\times S\kern0.37mm)=R\kern0.37mm$ {\sl if\kern0.15mm} \math{
S \not= \emptyset } holds. \vskip.5mm
To see that the above given convention of \q{\mathss00{u\le v}} having a
meaning for both extended real numbers and extended real number valued
functions \math{u\kern0.37mm,\kern0.15mm v} does not create any contradiction in our logical
system, we need the following
\begin{lemma}
For every function $\,u$ with $\,{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq\ovbbR$ it holds that $\,u\not\in
\ovbbR \, $.
\end{lemma}
\begin{proof} The {\sl regularity axiom\kern0.15mm} (\ref{ax of regularity})$\ar{ax}$
on page \pageref{ax of regularity} above, cf.\ \cite[\erm{VII}\kern0.15mm, p.\ 266]{Ky}
or \cite[\erm{ZF\,}9\kern0.37mm, p.\ 401]{Du}\,, has the simple consequence that
there {\sl do not exist\kern0.15mm} any \mathss30{x\ar 0\,,\kern0.15mm x\ar 1\kern0.15mm,\kern0.15mm x\ar 2}
such that \math{x\ar 0\in x\ar 1\in x\ar 2\in x\ar 0} holds. We show that this
will be contradicted if there exists a function \math{u} with \math{{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq
\ovbbR} and \mathss37{u\in\ovbbR}. Indeed, then there is \math{r} with \inskipline{.42}{17.72}
$u = (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.07mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)
= (\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1\kern0.15mm,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))$ \inskipline{.2}{19.8}
${}= ((\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))
= \{\kern0.15mm\{\,(\kern0.37mm r\kern0.15mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\,,\kern0.15mm
\{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\kern0.15mm\} \KP1 $. \inskipline{.4}0
Since \math{u} is a function with \math{{}^{}{\rm rng}\,{}_{{}^{}} u\subseteq\ovbbR} there are \math{
x\kern0.37mm,\kern0.15mm s} with \inskipline{.4}{22}
$ \{\kern0.07mm\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\}\kern0.07mm\}
= \{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}
= (\kern0.37mm x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1))$ \inskipline{.2}{36.8}
${}= \{\kern0.15mm\{\,x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,
\}\,,\kern0.15mm\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,
\}\kern0.15mm\} \KP1 $, \inskipline{.4}0
and hence \mathss38{
\{\,x\kern0.37mm,\kern0.15mm(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\} =
\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\} = \{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm
0\ar 1) \, \} }, \,whence further \mathss30{\{\,0\ar 1\} } \mathss08{{\KN{.99}}
= (\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1\kern0.15mm;\kern0.15mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)
= \{\kern0.15mm\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm 0\ar 1)\,,\kern0.15mm(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\,,\kern0.15mm
\{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\}\kern0.15mm\} }. Then we get \inskipline{.2}{7.7}
$0\ar 1\in\{\,0\ar 1\} \in \{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\} \in
\{\kern0.07mm\{\kern0.07mm\{\,0\ar 1\}\kern0.07mm\}\kern0.07mm\} = \{\,(\kern0.37mm 0\ar 1\kern0.15mm,\kern0.07mm 0\ar 1)\,\} =
0\ar 1 \kern0.37mm $, \,a {\sl contradiction\kern0.15mm}.
\end{proof}
For \math{0<q<\lower1.05mm\hbox{$^+$}\infty} we assume that \math{ \lower1.05mm\hbox{$^+$}\infty\RHB{.25}{\KPt8^q}
= \lower1.05mm\hbox{$^+$}\infty} in the following
\begin{constructions}[of Lebesgue quasi\kern0.37mm-\kern0.37mm norms]\label{Ctr |x|_lL^p} $\null$ \inskipline{.7}2
(1) \ $\|\,x\,\|\lllnor_p=\uniqset s : [\KPp1.4 0<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm
s=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\emath x} \kern0.37mm | \KP1
x\fvalss01 i\KP1|\suba\RHB{.25}{^p}\kern0.37mm\sbig)0\,^{p^{-1}} \KPp 1.4 \big ] \kern0.37mm $ \inskipline{.2}{40}
or $\kern0.37mm [\KPp1.4 p = \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm s = \sup\kern0.15mm\big\{\,
|\,t\,|\suba\kern-0.3mm:t\in{}^{}{\rm rng}\,{}_{{}^{}} x\KPt8\} \KPp 1.4 \big ] \KP1 $, \inskipline{.5}2
(2) \ $\|\,x\,\|\Lnorss33^p_\mu=\uniqset s : [\KPp1.4 0<p<\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm \label{ctr L^p-norm}
\aall{}^{}\Cal Omega\,{}^{}\Cal Omega = \bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\impss30{}$ \inskipline{.2}{35.8}
$s = \inf\kern0.15mm\big\{\,\big(\kern0.15mm\int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\rmdss11\mu\kern0.37mm)\KP1
^{p^{-1}}\kern-0.63mm:\varphi\in\kern0.15mm^{}^{}\Cal Omega\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9]\kern0.37mm$
and \inskipline{.2}{23.4}
$\varphi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
\aall{\eta\,,\kern0.15mm t}\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm t\kern0.37mm)\in x\impss33
|\,t\,|\suba\RHB{.25}{^p} \le \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1\} \KPp 1.4 \big ] $ \inskipline{.2}{7.5}
or $\kern0.37mm[\KPp1.4 p=\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and $\kern0.37mm s=\inf\,\{\,\smb M:\smb M\in
\rbb R^+\kern0.15mm$ and $\kern0.37mm\aall A\,\eexi N\,A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ \inskipline{.2}{23.5}
${}\impss03 N\in\mu\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm\sup\kern0.15mm\big\{\,
|\,t\,|\suba\kern-0.3mm:t\in x\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\KPt8\}\le\smb M\KP1\}
\KPp 1.4 \big ] \KP1 $.
\end{constructions}
For completeness' sake, in Constructions \ref{defi $L^p$} below of the
generalized Lebesgue\,--\,Bochner spaces we have included items (7) and (11)
where we define \math{\suptext{vc}0\Lrs03^p(\vcal Q\kern0.15mm) } and \mathss38{
\LLrs03^p(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) }. There we utilize the concepts of
{\sl quasi\kern0.37mm-\kern0.15mm\esl Euclidean vector column\kern0.15mm} and
{\sl quasi\kern0.37mm-\kern0.15mm usual space\kern0.15mm}. To make matters precise, we give the
following
\begin{definitions}
(1) \ Say that \math{\vcolQ} is a {\it quasi\kern0.37mm-\kern0.15mm\eit Euclidean \mathss36{
\bosy K}--\,vector column\kern0.37mm} if{}f there are $Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mm,\kern0.15mm\vPi\kern0.37mm$
with \math{\vcolQ = (\kern0.37mm Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mmp,\kern0.07mm\vPi\kern0.37mm) } and such that \math{
Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} and \math{\Yps\in\roman{LCS}\kern0.4mmps5(\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R) } and $\vPi\in\tvsps0(K)$
hold with \math{{\rm dim_{_{\kern.2mm Ha}}}\Yps\in\mathbb No} and \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and for every
\math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi
} there is \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{\xi=\Bnull_\vPi} or \mathss06{
u\fvalss02\xi \not= 0 }. \inskipline{.5}2
(2) \ Say that \math{\ebit F} {\it usualizes} \math{F} over \math{\bosy K}
if{}f \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and there is \math{k\in\mathbb No} with \math{
(\kern0.37mm\emptyset\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.15mm(\kern0.37mm k\kern0.37mm,\kern0.07mm F\kern0.37mm)\in
\ebit F\in\kern0.15mm^{k\kern0.37mm +\kern0.37mm 1.}\,\hbox{\font\SweD =cmssbx10\SweD U}{} } and for every \math{i\in k} there are \math{
i\ar 1\kern0.37mm,\kern0.15mm i\ar 2\in i\kern0.37mm\lower1mm\hbox{$^{^+}$} } and \math{l\in\mathbb N} and \math{
\ebit E\in\kern0.15mm^l\,(\kern0.37mm\ebit F\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} i\kern0.37mm\lower1mm\hbox{$^{^+}$})} with \mathss38{
\ebit F\fvalss61 i\kern0.37mm\lower1mm\hbox{$^{^+}$} \in \{\KPt8\bmii8 F\fvalss61 i\ar 1\sqcap\kern0.15mm
(\kern0.07mm\ebit F\fvalss61 i\ar 2\kern0.07mm)\,,\kern0.07mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\text{\,-\kern0.15mm}
\prodsubtext{tvs}\ebit E\KP1\} }. \inskipline{.5}2
(3) \ Say that \math{F} is {\it quasi\kern0.37mm-\kern0.15mm usual\,} over \math{\bosy K}
if{}f \inskipline0{18.8}
there is \math{\ebit F} such that \math{\ebit F} usualizes \math{F} over \math{
\bosy K}.
\end{definitions}
A quasi\kern0.37mm-\kern0.15mm usual space necessarily has finite nonzero dimension. For
example \linebreak the space \math{ F =
\bosy R\kern0.15mm\sqcap(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}} is
quasi\kern0.37mm-\kern0.15mm usual over \math{\bosy R} when \math{ \bosy R \in \{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and $
\smb N\in\mathbb N\kern0.15mm\,$, with for example \math{
\seqss44{\bosy R\,,\kern0.15mm\bosy R\kern0.15mm\sqcap\bosy R\,,
(\kern0.07mm\bosy R\kern0.15mm\sqcap\bosy R\kern0.37mm)\expnota^\ssmb N]_{tvs}\kern0.07mm,\kern0.15mm F} }
usualizing \mathss31{F}.
\begin{lemma}\label{Le for q-usu}
For \PouN$\kern0.37mm\sbi{\iota\kern0.37mm=\kern0.37mm\sixroman{1\kern0.15mm,\kern0.37mm 2}}\,,$ let $\,
\varUpsilon\kern-0.63mm\sbi\iota\kern0.15mm$ be quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm
\bosy K\kern-0.2mm\sbi\iota$ with $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\varUpsilon\kern-0.63mm\sbi\iota\kern0.37mm$. If
also $\,[\ Q\not=\emptyset\kern0.15mm$ and $\,\bosy K\kern-0.3mm\ar 1=\bosy K\kern-0.2mm\ar 2\ ]$ or
$\,\Int_taurd{\varUpsilon_\iota} Q\not=\emptyset\,,$ \,then $\,
\varUpsilon\aar 1=\varUpsilon\aar 2\,$.
\end{lemma}
Thus for example quasi\kern0.37mm-\kern0.15mm usual spaces \math{\Yps} over \math{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} are
such that every single point \math{\eta\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} uniquely determines the
whole algebraic and topological structure \mathss30{\Yps}. The proof of
Lemma \ref{Le for q-usu} is given in \cite{Hif}\,. It is quite long and
requires delving in the set theoretic formal construction of the complex
number system starting from the set \math{\mathbb No} of natural numbers, and so we
omit it here.
For \math{Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps} this allows us to define a structured vector space
\math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } based on a set of functions \math{
Q\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} without explicit reference to the structure \math{\Yps} by
putting \math{\roman S\,(\kern0.37mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\roman S\,(\kern0.37mm Q\,\sbi\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } when the latter is already
defined. So we just get a bit simpler notation for the same space.
\insubsubhead Suitable locally convex spaces \label{Ss suit lcs}
Suitable locally convex spaces are those that are obtained from some
\erm Banachable space by weakening the topology so that we {\sl do not\kern0.15mm}
get more bounded sets. Our basic important examples of suitable spaces are the
weak$^*$ duals \math{E\dlsigss22} of \erm Banachable spaces \mathss35{E}. We
put the following
\begin{definitions}\label{df suit}
Say that\inskipline{.5}2
(1) \ $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ is a {\it dominating norm\kern0.37mm} for \math{E} if{}f
there is \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} with \inskipline09
$E\in\tvsps0(K)\kern0.37mm$ and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} a norm on \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} with \inskipline09
$\bouSet E = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\capss31\{\,B:\eexi{n\in\rbb Z^+}\,B \subseteq
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP{1.1} 0\,,\kern0.07mm n\KPt9]\KP1\big\} \KP1 $, \inskipline{.5}2
(2) \ $E\kern0.37mm$ is {\it almost suitable\kern0.37mm} over \math{\bosy K} if{}f \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \inskipline09
there is a normable \math{F\in\roman{LCS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$, \inskipline{.5}2
(3) \ $E\kern0.37mm$ is {\it suitable\kern0.37mm} over \math{\bosy K} if{}f \math{ \bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \inskipline09
there is \math{F\in\roman{BaS}\kern0.4mmps0(K)} with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and $\kern0.37mm
\bouSet E\subseteq\bouSet F\kern0.15mm$. \inskipline{.5}2
For \q{almost suitable} or \q{suitable} in place of \kern0.15mm X also say that \inskipline0{38.6}
$E\kern0.37mm$ is X if{}f \math{E} is X over \math{\bosy K} for some \mathss31{
\bosy K}.
\end{definitions}
If \math{ \bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{E\in\roman{LCS}\kern0.4mmps0(K)} and \math{F\in\roman{BaS}\kern0.4mmps0(K)}
with \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} and \ú$\kern0.37mm\bouSet E$ \linebreak
\ú${\kern-0.63mm}\subseteq\bouSet F\kern0.15mm$, then \math{
\bouSet E=\bouSet F} holds since from \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm F} we get \mathss35{
\bouSet F\subseteq\bouSet E}. If we \linebreak
also have \math{E\kern1.4mm\raise1.7mm\hbox{\font\SweD =cmr5\SweD v}\kern-1.2mm\lower.25mm\hbox{\font\SweD =cmr5\SweD t}\kern-1.4mm\hbox{\font\SweD =cmsy10\SweD \char'026}\kern1mm G\in\roman{BaS}\kern0.4mmps0(K)} and \mathss32{
\bouSet E\subseteq\bouSet G}, \,then \math{F=G} holds. This is seen by noting that
\erm Banachable spaces are bornological, and hence have the strongest locally
convex topology with the same bounded sets. Thus \math{F} is the unique
\erm Banachable space from which \math{E} is obtained by weakening the
topology. The dominating norms \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ for \math{E} are precisely the
compatible norms for \mathss30{F}, \,and then \math{
(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\kern0.37mm,\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm) } is a corresponding (\kern0.15mm norm{\sl ed\kern0.37mm})
Banach space.
One should observe that the bornology of a suitable space {\sl does not\kern0.15mm}
determine the dual, i.e.\ there exist suitable spaces obtained by weakening
the same \erm Banachable space but with different duals. This is seen by
considering \math{\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\subw0 } and \ú$\kern0.37mm
\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\subsigrs03$ \linebreak
which both are obtained by weakening \mathss38{
\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) }. The former has the initial topolo- gical
vector structure from \math{(\kern0.37mm\roman I\,A\,,\kern-0.3mm\tvbbR4^A\kern0.15mm\big) } for \math{
A=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\kern0.37mm\mathbb No\kern0.07mm) } and the latter for \mathss30{ A = \kern-0.3mm} \mathss03{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) } when we let \mathss39{ \roman I\,A =
\big\langle\big\langle\kern0.37mm\sum\KP1(\kern0.37mm x\cdot y\kern0.37mm) : y\in A\KP1\rangle :
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm)\KP1\rangle }.
\begin{lemma}\label{Le suit dom}
Let $\,E$ be almost suitable with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a dominating norm.
Then for
every $\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\in\Bqnorm E$ there is $\,\smb M\in\rbb R^+$ with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\le\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm)$ for all $\,
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$.
\end{lemma}
\begin{proof} We have \mathss35{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 20 I\in\bouSet E
}, \,and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\in\Bqnorm E} then \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\,[\KP{1.1}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 26 I\,] \in {\kern-0.63mm}} \mathss03{
\bouSet\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R } and hence for \math{ \smb M =
\sup\KPt8(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\,[\KP{1.1}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 26 I\,]\kern0.37mm\sbig)0 }
we have \mathss35{\smb M < \lower1.05mm\hbox{$^+$}\infty }. Considering \ú$\kern0.37mm x\in {\kern-0.63mm}$ $
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$, if \math{x=\Bnull_E} holds, we trivially have \mathss38{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=0\le 0=\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm) }. Otherwise
taking \math{\smb A=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x} we have \mathss34{
(\kern0.15mm\smb A\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1\,}x\kern0.37mm)\svs E \in \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb 20 I}, \,and
hence \mathss30{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm\smb A\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1\,}x\kern0.37mm)\svs E \le
\smb M} and further \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x \le \smb M\,\smb A =
\smb M\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10 x\kern0.37mm) }.
\end{proof}
\begin{lemma}\label{Le E'' adher}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ and \ú$\, w \in
\Cal L\,(\kern0.15mm E\dlbetss12\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) \KP1 $. Then there is some \ú$\,
B\in\bouSet E$ such that for every finite \ú$\, A \subseteq
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) $ there is \ú$\, x \in B $ with $\,
|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x - w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u\KP1|\suba\le 1 $ for all $\,u\in A\,$.
\end{lemma}
\begin{proof} Putting \mathss38{D\ar 1=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K\capss21\{\,t:|\,t\,|\suba
\le 1 \KPt8\} }, \,from \math{ w \in
\Cal L\,(\kern0.15mm E\dlbetss12\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } we first get existence of
some nonempty absolutely convex bounded set \math{B} in \math{E} such that for \mathss03{
U = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) \capss31 \{\, u : u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B
\subseteq D\ar 1\kern0.37mm\} } we have \mathss34{w\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm U \subseteq D\ar 1 }. Then
for the canonical evaluation \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \seqss33{
\roman{ev}\sbi{\kern0.15mm\emath x}\,|\KP1\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) :
x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } and for \math{\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlbetss12{\kern-0.63mm}\dlsigss02) }
from the {\sl bipolar theorem\kern0.15mm} \cite[3.3.1\kern0.15mm, p.\ 192]{Ho} or
\cite[8.2.2\kern0.37mm, p.\ 149]{Jr} we see \math{ w \in
\roman{Cl}\sbi{\KPt8\scrm7 T\KP1}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\kern0.37mm) } to hold whence
the assertion follows.
\end{proof}
The content of \cite[Lemma 8.17.8 \erm B\kern0.37mm, p.\ 585]{Edw} is in the
following
\begin{lemma}\label{Le 8.17.8 B}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.37mm\rm, }and
let \ú$\,F=E\dlbetss12\,$. Also let $\,S\ar 1$ be a linear subspace in $\,
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$ such that \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1$ is a separable
topology{\kern0.37mm\rm, }and let \ú$\, w \in
\Cal L\,(\kern0.15mm F_{\kern0.15mm/\kern0.37mm\aars S_1},\kern0.07mm\bosy K\kern0.37mm) \,$. Then there is \ú$\,
\bosy x\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E$ with \ú$\,{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\in\bouSet E$ and such that $\,
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \lim\,(\kern0.37mm u\circss01\bosy x\kern0.37mm)$ holds for every $\,u\in
S\ar 1\kern0.37mm$.
\end{lemma}
\begin{proof} By Hahn\,--\,Banach there is \math{\bar w\in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } with \mathss35{w\subseteq\bar w}.
Furthermore, for \linebreak
\mathss03{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2} the canonical embedding \mathss36{E\to
E\dlbetss12{\!}\dlbetss02}, \,i.e.\ for \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2 =
\seqss33{\roman{ev}\sbi{\kern0.15mm\emath x}\,|\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F:x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } and for \mathss03{
\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlbetss12{\!}\dlsigss04) } by \kern0.15mm Lemma \ref{Le E'' adher}
above, there is some \math{B\in\bouSet E} such that \mathss03{\bar w\in B\ar 2}
holds for \mathss38{B\ar 2 =
\roman{Cl}\sbi{\KPt8\scrm7 T\KPt8}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm B\kern0.37mm) }. Now
letting \math{\scrmt T\aR 1} be the initial topology from \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm \kern0.15mm F_{\kern0.15mm/\kern0.37mm\aars S_1}\sbig)0\dlsigss12\kern0.07mm) } under \math{
\seqss33{z\KP1|\KP1 S\ar 1\kern-0.3mm:z\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)} } we
have \math{\scrmt T\aR 1\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\ar 2\subseteq \scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 B\ar 2 } with \math{
\scrmt T\aR 1\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\ar 2 } semimet- rizable. Hence there is \math{\bosy x
\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,B} with \mathss30{w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \bar w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u =
\lim\,(\kern0.37mm u\circss01\bosy x\kern0.37mm) } for \mathss34{u\in S\ar 1}.
\end{proof}
\begin{lemma}\label{Le qtvs}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\tvsps0(K)$ and let $\,S$ be a vector
subspace in $\,\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\,$.
Also let \ú$\,F=E\,/\tvsquotient S$ and \ú$\,
\Cal V = \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X:U\capss02\smb X\not=\emptyset\KP1\} :
U\kern-0.3mm\in\Cal U\KP1\}$ \linebreak
where $\,\Cal U$ is a filter base for $\,\neiBoo E\,$.
Then $\,\Cal V$ is a filter base for $\,\neiBoo F\kern0.15mm$.
\end{lemma}
\begin{proof} With \mathss38{ \tweq = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \capss31 \{\,
(\kern0.37mm x\kern0.37mm,\kern0.07mm\smb X\kern0.15mm):x\in\smb X\,\} } we know from the discussion in
\cite[p.\ 104]{Ho} that \math{\tweq} is continuous and open \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\to\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}, \,and consequently \linebreak
\ú$\Cal V=\tweq\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm\Cal U\subseteq
\neiBoo F\kern0.37mm$ holds. Moreover, for every \math{V\in\neiBoo F} we have \ú$\kern0.37mm
\tweq\invss64\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm V\in\neiBoo E$ \linebreak
and hence there is \math{U\in\Cal U}
with $\kern0.37mm U\subseteq\tweq\invss64\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm V\kern0.07mm$, but then \math{
\tweq\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} U\subseteq V} holds.
\end{proof}
\Ssubhead A Measurability and integration \label{Sec A}
In this section, we first explain what it means for functions \math{ x :
{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} to be measurable when \math{\vPi} is a topological vector
space and \math{{}^{}\Cal Omega} is a set equipped with a positive measure \mathss36{
\mu}. In the next section, for \math{0\le p\le\lower1.05mm\hbox{$^+$}\infty} we construct the
spaces $\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm$ and \math{
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } of certain equivalence classes \math{
\smb X} of such \mathss34{x}.
By saying that \math{\mu} is a {\it positive measure\kern0.37mm} on \math{{}^{}\Cal Omega} we
mean that \math{\mu} is a function with \math{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu={}^{}\Cal Omega} and \math{
{}^{}{\rm rng}\,{}_{{}^{}}\mu\subseteq[\KP1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} a
\rsigma3algebra and such that \linebreak
\ú$ \sum\,(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) =
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\big(\kern0.15mm\bigcup\,\scrmt A\kern0.37mm) \kern0.37mm$ holds for any countable
disjoint \mathss36{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Here we assume that the definitions
associated with sum conventions are arranged so that \linebreak
\ú$ \sum\,\emptyset = 0\kern0.37mm$
holds. Further, by a \rsigma0{\it algebra\kern0.37mm}, usually written
\q{$\sigma\,$-\,algebra}, we mean any \linebreak
$\scrmt A\kern0.37mm$ such that \math{
\bigcup\,\scrmt A\kern0.15mm\setminus\kern0.37mm\bigcup\,\scrmt B\in\scrmt A} holds for any
countable \mathss36{\scrmt B\subseteq\scrmt A}. A positive measure $\mu\kern0.37mm$ is
\rsigma1{\it finite\kern0.37mm} if{}f \math{
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq\bigcup\,\scrmt A } holds for some countable \mathss35{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+ }.
To compact language in some, quite rare cases, we introduce the concept of
measure space as follows. Say that \math{P} is a {\it measure space\kern0.37mm} if{}f
there are \math{\mu} and \math{{}^{}\Cal Omega} such \linebreak
that \math{\mu} is a positive
measure on \math{{}^{}\Cal Omega} with \mathss38{P = (\kern0.37mm{}^{}\Cal Omega\kern0.37mm,\kern0.07mm\mu\kern0.37mm) }. We
also say that \math{\mu} is a positive measure if{}f \math{\mu} is a positive
measure on \mathss36{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and a measure space \math{P} we \linebreak
say
to be \rsigma6finite in the case where \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm P} is such. \vskip.5mm
\insubsubhead Measurability of measure\kern0.37mm-\kern0.15mm vector maps \label{Ss C1}
We consider {\it mv\kern0.37mm-\kern0.15mm map\,}s, short for \q{measure\kern0.37mm-\kern0.15mm vector},
which are triplets, i.e.\ ordered pairs \math{ \tilde x =
(\kern0.37mm x\kern0.37mm,\kern0.07mm\varXi\kern0.37mm) = (\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where in
turn \math{\varXi = (\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is an {\it mv\kern0.37mm-\kern0.15mm pair\kern0.15mm}.
This means that \math{\vPi} is a real or complex topological vector space and \math{
\mu} is a positive measure on some \math{{}^{}\Cal Omega} and \math{ x : {}^{}\Cal Omega \to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} is a function. In order to introduce some concepts of
measurability for such mv\kern0.37mm-\kern0.15mm maps we first put the following
\begin{definitions}\label{df simple}
(1) \ Say that \math{\sigma} is {\it simple\kern0.37mm} in \math{\varXi} if{}f
there are \math{\bosy K\kern0.15mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi} with \math{\mu}
a po- sitive
measure on \math{{}^{}\Cal Omega} and \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and
\math{\vPi\in\tvsps0(K)} and
\math{\varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
and \mathss03{\sigma\in\kern0.15mm^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and
\math{{}^{}{\rm rng}\,{}_{{}^{}}\sigma} finite and
\math{\{\,\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}:\xi\in
{}^{}{\rm rng}\,{}_{{}^{}}\sigma\kern0.07mm\setminus\{\,\Bnull_\vPi\}\kern0.15mm\}\subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \inskipline{.5}2
(2) \ Say that \math{\bosy\sigma} is a {\it simple sequence\kern0.37mm} in \math{\varXi}
if{}f \math{\bosy\sigma\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}} and \inskipline0{41.5}
$\sigma\kern0.37mm$ is simple in \math{\varXi} for all \math{\sigma\in
{}^{}{\rm rng}\,{}_{{}^{}}\bosy\sigma}.
\end{definitions}
Let \math{\text{\efss R\KPt8}\tilde x\,A=
\uniqset\tilde z:\eexi{x\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm\vPi}\,
\tilde x=(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \mathss38{
\tilde z=(\kern0.37mm x\KP1|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) }.
\begin{def:al schemas}\label{df meas}
For any mv\kern0.37mm-\kern0.15mm map \math{\tilde x=
(\kern0.37mm x\kern0.37mm,\kern0.07mm\varXi\kern0.37mm)=(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } with \math{
{}^{}\Cal Omega={{}^{}{\rm dom}\,{}_{{}^{}}} x}
assuming $\vPi\in\tvsps0(K)$ , first say that \inskipline12
(1) \ $\tilde x\kern0.37mm$ is {\it measurable\kern0.37mm} if{}f \math{
\{\,x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm U\kern-0.3mm:U\in\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\KP1\}\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds, \inskipline{.5}2
(2) \ $\tilde x\kern0.37mm$ is {\it simply measurable\kern0.37mm} if{}f there is \math{
\bosy\sigma} with \math{\bosy\sigma} a simple sequence in \math{\varXi} \inskipline09
and \math{\bosy\sigma\to x} in top \mathss35{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{ti} }, $
\bosy\sigma\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.37mm\big(\,^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm) $ \KP6 \inskipline{.5}2
(3) \ $\tilde x\kern0.37mm$ is {\it scalarly measurable\kern0.37mm} if{}f
$(\kern0.37mm u\circ x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is measurable for all
$u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \,$.\KN9 \inskipline10
Then for any of \q{measurable}, \q{simply measurable} or
\q{scalarly measurable} in place of \,X\kern0.37mm, say that \inskipline12
(4) \ $\tilde x\kern0.37mm$ is {\it almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\KP1(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus\kern0.15mm N\kern0.37mm) } is X
for some \mathss39{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \inskipline{.5}2
(5) \ $\tilde x\kern0.37mm$ is {\it finitely\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}, \inskipline{.5}2
(6) \ $\tilde x\kern0.37mm$ is {\it finitely almost\kern0.37mm} X if{}f
\math{\text{\efss R\KPt8}\tilde x\,A} is almost X for
every \mathss36{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}.
\end{def:al schemas}
In loose speach, we may express the content of \ref{df meas}\,(2) by saying
that \math{\bosy\sigma} is a {\sl sequence of simple functions converging
pointwise\kern0.15mm} to \mathss34{x}. Then for the \math{\sigma\in\bosy\sigma} there \linebreak
we may also say that \math{(\kern0.37mm\sigma\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is a
{\it simple\kern0.37mm} mv\kern0.37mm-\kern0.15mm map, and we may loosely say that \linebreak
$\sigma\kern0.37mm$ is a
{\sl simple function\kern0.15mm}.
Note that by our definitions above we may also say e.g.\ that \math{\tilde x}
is {\sl measurable\kern0.15mm} if{}f there are \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm
x} with \math{{}^{}\Cal Omega\times\kern-0.2mm\{\,\Bnull_\vPi\} } simple in \math{
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \mathss03{ \tilde x =
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{x\in\kern0.15mm^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } and \mathss36{
x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.15mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}.
\begin{proposition}\label{pro-mea-equ}
Let \PouN$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}${\,\rm, }and let \PouN$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be
normable with $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Also let $\,\mu$ be a \kern0.15mm
\rsigma1finite positive measure on $\,{}^{}\Cal Omega\,$. If in addition $ \tilde x =
(\kern0.37mm x\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\, x \in\kern0.15mm^{}^{}\Cal Omega\,
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \KP1 ${\rm, \,}then $\
(1) \equivss22 (2) \equivss22 (3) $ \,where {\rm \inskipline{.9}4
(1)} \ $(\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\,;\kern0.15mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is
measurable for all $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi${\kern0.15mm\rm, \inskipline{.4}4
(2)} \ $\tilde x$ is simply measurable{\kern0.37mm\rm, \inskipline{.4}4
(3)} \ $\tilde x$ is measurable\kern0.37mm. \end{proposition}
\begin{proof} Since \math{ \{\KP1 \roman{ev}\kern0.15mm\sbi\xi \, | \KP1
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) : \xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\} \subseteq
\Cal L\,(\kern0.07mm\vPi\dualsigma0\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } holds, we trivially have
the implication \mathss36{(3)\impss22(1)}. Likewise, we trivially have \mathss36{
(2)\impss22(1)}. It now suffices to prove that the implications \math{
(1)\impss22(3)} and \math{(3)\impss22(2)} hold.
For \mathss36{(1)\impss22(3)}, letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} be some compatible norm for \mathss33{
\vPi}, \,let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 1} be the corresponding dual norm, i.e.\ put \mathss39{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} }.
\newline$\null
$
For \mathss36{
i\in\mathbb No}, \,then put \mathss38{ \roman B\,i =
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aar 1\!\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1 0\,,\kern0.15mm i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,] }. Assuming
(1)\,, if
\linebreak
now \mathss37{U\in\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm) }, \,for any
fixed \math{i\in\mathbb No} and for \mathss35{\roman U\,i=U\capss14\roman B\,i }, \,
it suffices to prove that \math{x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\roman U\,i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}
holds. For \mathss35{ \scriptm9 T\ar 1 =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33\roman B\,i }, \,now \math{
\scriptm9 T\aR 1} is the uniform $\kern0.37mm\scriptm{10}U\,$-- \linebreak
topology where \math{\scriptm{10}U } is the uniformity generated by \PouN$\kern0.37mm
\{\KP1\roman V\,\xi\,n:\xi\in D\kern0.37mm\text{ and }\kern0.37mm n\in\rbb Z^+\,\}$ where \math{
\roman V\,\xi\,n = \roman B\,i\times\roman B\,i \capss31 \{\,
(\kern0.37mm u\kern0.37mm,\kern0.07mm v\kern0.37mm) :
|\KP1 u\fvalss01\xi - v\fvalss01\xi\KP1| < n^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,\} } and \math{
D} is any countable $
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22(\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss43\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPt9 0\,,1\KPt8]\kern0.37mm)\,$--\,dense
set. By Alaoglu's theorem, we have \math{\scriptm9 T\aR 1} a compact topology.
Since \math{\scriptm{10}U } is generated by a countable set, we see that there
is some \mathss33{\scriptm9 T\aR 1}-- dense and countable set \mathss34{
D\aar 1}. Using (1) and noting that \
$x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\}\kern0.15mm) = {} $ \vskip.4mm\centerline{$
\bigcap\KP1\{\,\{\KP1 t:|\KP1 x\fvalss01 t\fvalss21\xi -
u\fvalss01\xi\KP1| < n^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}
\kern0.37mm\text{ and }\,
|\KP1 x\fvalss01 t\fvalss21\zeta\KP1|\le i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,\} :
\zeta\in D\KP1\}\KP1$,} \inskipline{.4}0
we see that \math{
x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\}\kern0.15mm)\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds
when \mathss31{ (\kern0.37mm\xi\,,\kern0.07mm u\kern0.37mm,\kern0.07mm n\kern0.37mm) \in
D\times D\ar 1\kern-0.3mm\times\kern0.15mm\rbb Z^+}. It is left as an exercise to the reader
to show that \math{\roman U\,i} can be expressed as a union of finite
intersections of the sets \mathss37{\roman V\,\xi\,n\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\kern0.15mm u\kern0.15mm\} }.
Then \math{x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\roman U\,i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} follows.
For \mathss36{(3)\impss22(2)}, we need the assumption that \math{\mu} be
\rsigma6finite. It is an easy exercise to show that if the implication to be
established holds for bounded measures, then it holds also for \rsigma6finite
ones. So we assume that \mathss36{\mu\fvalss01{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty}. Further,
if we can show that for any fixed \mathss37{i\in\mathbb No}, \,and for \math{\bar x
=x\capss33(\kern0.37mm\hbox{\font\SweD =cmssbx10\SweD U}{}\times\roman B\,i\kern0.37mm)} with \mathss34{B={{}^{}{\rm dom}\,{}_{{}^{}}}\bar x}, \,
the required implication holds for \math{
(\kern0.37mm\bar x\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm)} in place of \mathss34{
\tilde x}, \,then it easily follows also for \mathss34{\tilde x}, observing
that \mathss36{ B = \bigcap \KP1 \{\, \{ \KP1 t :
|\KP1 x\fvalss01 t\fvalss21\zeta\KP1|\le i\,\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,\} :
\zeta\in D\KP1\}\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}.
As just explained, for \mathss36{(3)\impss22(2)}, assuming (3) and making the
additional assumptions that \math{\mu\fvalss01{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty} and \math{
{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq B\ar 1=\roman B\,i\ar 0 } for some fixed \mathss36{i\ar 0\in\mathbb No},
we should establish (2)\kern0.37mm. For this, we construct \math{ \bosy s \in
\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.37mm\big(\,^{}^{}\Cal Omega\kern0.37mm B\ar 1\kern0.07mm) } with \math{
(\kern0.37mm\smb S\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } a simple mv\kern0.37mm-\kern0.37mm map for
every \mathss36{\smb S\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s}, \,and such that \math{\bosy s\to x} in
top \math{\scriptm9 T\expnota^\KPt8{}^{}\Cal Omega\kern0.15mm]_{ti} } when we take \linebreak \PouN$
\scriptm9 T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 B\ar 1\,$. Since \math{
\scriptm9 T} is a compact topology, and also the uniform $\kern0.37mm\scriptm{10}U\,
$-- topology with \math{\scriptm{10}U} being countably generated, we may first
choose some decreasing \linebreak \PouN$
\bosy w:\mathbb No\to\scriptm{10}U\kern0.37mm$ with \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy w} a symmetric base
for \mathss36{\scriptm{10}U}, \,and then some \math{\bosy u:\mathbb No\to\hbox{\font\SweD =cmssbx10\SweD U}{}}
with the following property. For every \math{i\in\mathbb No} there is \math{
k\in\mathbb No} with \math{\bosy u\fvalss01 i\in B\ar 1\kern0.15mm^k } and $ B\ar 1 \subseteq
\bigcup\KP1\{\,\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 j\,\}
:j\in k\,\}\KPt8$. For short writing \inskipline{.8}{6.75}
$\roman U\,i\,j =
\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 j\,\}
\kern0.15mm\setminus\, \bigcup \KPt8 \{\,
\bosy w\fvalss01 i\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\{\,\bosy u\fvalss01 i\fvalss21 l\,\}
: l\in j\,\} $ \ and \inskipline{.4}{6.75}
$\roman A\,i\,j = x\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.37mm\roman U\,i\,j\,$, \,and taking \vskip.4mm
$ \null
\bosy s = \big\langle\, \bigcup \KPt8 \{\,
\roman A\,i\,j\times\{\,\bosy u\fvalss01 i\fvalss21 j\,\} :
\bosy u\kern0.37mm,\kern0.07mm i : j\in{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm\bosy u\fvalss01 i\kern0.37mm)\,\} :
i\in\mathbb No \,\big\rangle \KP1$, \,we are done, \inskipline{.5}0
leaving the required straightforward verifications as exercises to the reader.
\end{proof}
\begin{example}
Without separability of the topology \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}, \,the implication \mathss30{
(1)\impss11(2)} \linebreak
in Proposition \ref{pro-mea-equ} need not hold. Indeed, with \math{
1<p<\lower1.05mm\hbox{$^+$}\infty} and \math{\vPi=\lll^p(\ssbb44 I) } and letting \math{\mu} be
the Lebesgue measure defined for all Lebesgue measurable sets \mathss35{A\subseteq
\bbI}, \,taking \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
where \mathss38{x = \seqss34{\roman{ev\kern0.15mm}\sbi t\,|\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-0.2mm:t\in\bbI}
}, \,we trivially have (1) since \linebreak
\mathss02{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\fvalss01 t =
x\fvalss01 t\fvalss21\xi = \xi\fvalss11 t }, \,and so \math{
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\fvalss01 t \not = 0 } only for countably many \math{
t\in\bbI} for \linebreak
each fixed \mathss30{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. However \math{\tilde x}
cannot be simply measurable since otherwise there would exist some countable
set \math{N\aar 0\subseteq\bbI} such that \math{x\fvalss01 t\fvalss21\xi\not=0 }
holds only for vec- \linebreak
tors \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \math{
\xi\invss46[\KP{1.2}\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.15mm\setminus\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\KP1]\capss42 N\aar 0
\not=\emptyset} when \mathss34{t\in\bbI}.
\end{example}
\begin{proposition}\label{Pro rfx si mea}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive{\kern0.15mm\rm, }
and let \inskipline06
$(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\subsigrs03\kern0.07mm)$ be simply measurable. Then $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ is simply measurable.
\end{proposition}
\begin{proof} Putting \mathss35{{}^{}\Cal Omega=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,let \math{
\bosy\sigma} be some simple sequence in \math{
(\kern0.37mm\mu\,,\kern0.07mm\vPi\subsigrs03\kern0.07mm) } with \math{ \bosy\sigma\to x} in top \mathss35{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\subsigrs03\kern0.07mm)\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{ti} }, \,and let \math{
S} be the closed linear span of \math{{}^{}{\rm rng}\,{}_{{}^{}}\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bosy\sigma} in \mathss31{
\vPi}. Then trivially \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is scalarly
measurable, and by Hahn\,--\,Banach also \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S} is a
separable topology with \mathss34{{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq S}. Consequently by
Pettis' theorem the assertion follows.
\end{proof}
\begin{lemma}\label{Le Nu_1 ci y meas}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Also let \vskip.5mm\centerline{$
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1 = \seqss44{ \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) : u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} $} \inskipline{.5}0
and let $\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ be scalarly
measurable. Then $\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)$ is measurable.
\end{lemma}
\begin{proof} Putting \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and
taking any countable \math{D} such that \math{D}
is
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)}--\,dense,
for every \math{\eta\in{}^{}\Cal Omega} we
have \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta= \label{Nu_1 = sup ...}
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
=\sup\KPt8\{\KPt8\Abrs00^1\circ\KPt2
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\xi\in D\KP1\} }. Noting that
by our assumption for every fixed \math{\xi\in D} we have
that
\math{
(\kern0.37mm\Abrs00^1\circ\KPt2
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)
} is
measurable, the assertion immediately follows.
\end{proof}
\begin{lemma}\label{Le Nu_1 = sup ...}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive with $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and $\,S$ a separable linear subspace in
$\,\vPi\dlbetss01\,$. Also let \vskip.5mm\centerline{$
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1 = \seqss44{ \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44
\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) : u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} \KP1 $.} \inskipline{.5}0
Then there is a countable $\,D\subseteq\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb10 I$ with
$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u=
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)$ for $\,u\in S\,$.
\end{lemma}
\begin{proof} Putting \math{E=\vPi\dlbetss01\kern0.37mm_{/\,S} } let \math{A} be
countable and \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense, and let \math{\scrmt R} be the set
of all pairs \math{(\kern0.37mm u\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in A\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi\le 1} and \mathss35{u\fvalss01\xi=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u
}. By reflexivity and Hahn\,--\,Banach we then have \math{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt R}
and hence by {\kern0.15mm countable choice\kern0.15mm} there is a function \math{\scrmt P\subseteq
\scrmt R} with \mathss31{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt P}. Let \mathss31{D={}^{}{\rm rng}\,{}_{{}^{}}\scrmt P}.
Now for all \math{u\in S} we trivially have \mathss38{\sup \KPt8(\kern0.37mm
\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u }, \,and hence
assuming that \math{\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
< \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u } holds for some \mathss34{u\in S}, \,it suffices to
get a contradiction. Taking \math{ \varepsilon = \frac 12\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}
u - \sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)) } we first find
some \math{v\in A} with \mathss34{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) < \varepsilon}.
Then for \math{\xi=\scrmt P\fvalss30 v} we have \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi\le 1}
and \mathss34{v\fvalss01\xi=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} v}, \,and hence \ $
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
= \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u - 2\KPt8\varepsilon
< \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u - \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) - \varepsilon$ \inskipline{.25}{35.7}
${}\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} v - \varepsilon = v\fvalss01\xi - \varepsilon =
(\kern0.37mm v - u\kern0.37mm)\fvalss01\xi + u\fvalss01\xi - \varepsilon $ \inskipline{.25}{35.7}
${}\le|\KP1 u\fvalss01\xi\KP1| + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u - v\kern0.37mm) -\varepsilon
< |\KP1 u\fvalss01\xi\KP1| \KP1 $, \,a contradiction.
\end{proof}
\insubsubhead Decomposable positive measures \label{Ss decos}
Decomposability, as well as being {\sl almost decomposable\kern0.15mm}, is a property
for a positive measure $\mu$ that is weaker than the usual
\q{$\sigma\,$-\,finiteness} which we call \rsigma6finiteness, and that is
sufficiently strong still to have \mathss38{\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) }
canonically represent the strong dual of \mathss38{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) }.
For example Haar measures of suitably \q{large} locally compact topological
groups are almost decomposable but not \rsigma6finite. See
Example \ref{Exa Haar} on page \pageref{Exa Haar} below for some details
concerning this assertion.
\begin{definitions}\label{df decomp}
(1) \ Say that \math{N\kern0.15mmrim1} is \mathss37{\mu}{\it--\,negligible\kern0.37mm} if{}f \math{
\mu} is a positive measure with \inskipline09
$N\kern0.15mmrim1\subseteq \kern0.15mm\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and \mathss38{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern-0.63mm\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}\kern0.15mm N\kern0.15mmrim1 \subseteq \kern0.37mm
\bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\kern-0.3mm:N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} }, \inskipline{.5}2
(2) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it almost decomposable\kern0.37mm} if{}f there are \mathss30{\scrmt A \subseteq \label{decos A}
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+} and \math{N\kern0.15mmrim1} with \math{
\scrmt A\cupss42\{\kern0.37mm N\kern0.15mmrim1\kern0.15mm\} } disjoint and \math{ {}^{}\Cal Omega =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and such that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, and such that also \math{N\kern0.15mmrimm1} is \mathss37{\mu
}--\,negligible whenever \ú$\kern0.37mm N\kern0.15mmrimm1\subseteq{}^{}\Cal Omega$ \linebreak
is such that for every \math{
A\in\scrmt A} there is \math{N} with \mathss38{A \capss31 N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \inskipline{.5}2
(3) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it decomposable\kern0.37mm} if{}f there is some disjoint \math{\scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A } and such
that every \math{N\kern0.15mmrim1\subseteq{}^{}\Cal Omega} is \mathss37{\mu}--\,negligible whenever \math{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N\kern0.15mmrim1 \subseteq \kern0.37mm \bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N \kern-0.3mm :
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } holds, \inskipline{.5}2
(4) \ For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} say that \math{\mu}
is {\it truly decomposable\kern0.37mm} if{}f there is some disjoint \math{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A }
and such that \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } holds for every
\math{N\subseteq{}^{}\Cal Omega} with \mathss38{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{definitions}
Trivially \rsigma6finite positive measures are truly decomposable, and these
in turn are decomposable by
Proposition \ref{Propo top-deco} below. If
\math{\mu} is a positive measure on \math{{}^{}\Cal Omega} such that \mathss03{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+=\emptyset} holds, \,then trivially every \math{
A\subseteq{}^{}\Cal Omega} is \mathss37{\mu}--\,negligible, and hence \math{\mu} is
almost decomposable. A positive measure \math{\mu} on \math{ {}^{}\Cal Omega =
\mathbb R\kern-.2mm\times\kern-.2mm\mathbb R} that is decomposable but not truly decomposable is given in
Example \ref{Exa not trul deco} on page \pageref{Exa not trul deco} below. It
seems to be quite difficult to find positive measures that {\sl are not\kern0.15mm}
almost decomposable.
See Problem \ref{Prblm z-z mea} on page \pageref{Prblm z-z mea}
as well as the subsequent examples and problems.
\begin{definitions}\label{df top deco}
(1) \ Say that \math{\scrmt T} {\it positively almost \eit Radonizes\kern0.37mm} \math{
\mu} if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a \linebreak
positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \math{\scrmt K=\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} } it holds that \math{ \scrmt K \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\KP1\} }, \inskipline{.5}2
(2) \ Say that \math{\scrmt T} {\it positively \eit Radonizes\kern0.37mm} \math{\mu \label{df pos rdz} }
if{}f there is \math{{}^{}\Cal Omega} with \math{\mu} a positive measure on \math{
{}^{}\Cal Omega} and \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff
topological space such that for \ú$\kern0.37mm\scrmt K={\kern-0.63mm}$ \linebreak
\ú$\{\,K:K\kern0.37mm\text{ is \mathss37{
\scrmt T}--\,compact } \} \kern0.37mm$ it holds that \math{\scrmt T\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \math{
\scrmt K\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } and also for all \math{ U \kern-0.3mm\in
\scrmt T} it holds that \mathss38{\mu\fvalss01 U = \sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:
K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm U\KP1\} } and for all \ú$\kern0.37mm A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern-0.63mm$ \linebreak
it holds that \mathss38{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A
= \inf\,\{\,\mu\fvalss01 U:A\subseteq U\in\scrmt T\KP1\} }, \inskipline{.5}2
(3) \ Say that \math{\mu} is {\it positive almost \eit Radonian\kern0.37mm} if{}f \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively almost
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it positive \eit Radonian\kern0.37mm} if{}f \label{df pos Radon} \inskipline0{11}
there is \math{\scrmt T} such that \math{\scrmt T} positively
\erm Radonizes \mathss35{\mu}, \inskipline{.5}2
(5) \ Say that \math{\mu} is {\it topologically almost decomposable\kern0.37mm} if{}f
there are \math{\scrmt A\,,\kern0.15mm\scrmt T\kern0.07mm,\kern0.15mm N\kern0.15mmrim1} such that \math{
\scrmt T} positively almost \erm Radonizes \mathss35{\mu} and for \math{
\scrmt K = \{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T}--\,compact } \} } it holds
that \math{\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{
\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq \kern0.37mm \bigcup\KPt8\{\, \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N : N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } and \math{
\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\} } is disjoint with \math{\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is
countable for all \math{K\in\scrmt K} and also \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for all \math{
A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}, \inskipline{.5}2
(6) \ Say that \math{\mu} is {\it topologically decomposable\kern0.37mm} if{}f there
are \math{\scrmt A\,,\kern0.15mm\scrmt T\kern0.07mm,\kern0.15mm N\kern0.15mmrim1} such that \math{
\scrmt T} positively \erm Radonizes \mathss35{\mu} and such that for
\math{\scrmt K=
\{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T}--\,compact } \} } it
holds that $\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ and
$\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ and
$\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\}$ is disjoint with
$\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu=\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1$
and \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \math{K\in\scrmt K}
and \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
\end{definitions}
Note that the condition \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } in (5) and (6) of
Definitions \ref{df top deco} above means that we have \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} or \math{A\capss33 U \in 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} =
\{\kern0.37mm\emptyset\kern0.37mm\} } which in turn is equivalent to having \math{
0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) < \lower1.05mm\hbox{$^+$}\infty} or \mathss35{ A\capss33 U =
\emptyset}. Note also the impli- cations \mathss35{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \in \rbb R^+ \impss13
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm) \not= \hbox{\font\SweD =cmssbx10\SweD U}{} \impss33
A\capss33 U\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. That positive (\kern0.15mm almost\kern0.15mm) \erm Radonian measures
are topologically (\kern0.15mm almost\kern0.15mm) decomposable, and that these in turn are
almost decomposable is seen from the next
\begin{proposition}\label{Propo top-deco}
For the properties given below the implications \ú$\,(5)\impss11(6)\impss11(7)$
and $\,(3)\impss11(4)\impss11(7)$ and $\,(1)\impss11[\KPp1.4(2)\text{ and }(3)\KPp1.4
]$ and $\,(2)\impss11(4)$ hold. {\rm\inskipline{.6}4
(1) \ }$\mu$ is positive \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(2) \ }$\mu$ is positive almost \eit Radonian{\kern0.15mm\rm,\inskipline{.2}4
(3) \ }$\mu$ is topologically decomposable{\kern0.15mm\rm,\inskipline{.2}4
(4) \ }$\mu$ is topologically almost decomposable{\kern0.15mm\rm,\inskipline{.2}4
(5) \ }$\mu$ is truly decomposable{\kern0.15mm\rm,\inskipline{.2}4
(6) \ }$\mu$ is decomposable{\kern0.15mm\rm,\inskipline{.2}4
(7) \ }$\mu$ is almost decomposable.
\end{proposition}
\begin{proof} For \math{(1)\impss11(2)} letting \math{\scrmt K\,,\kern0.07mm\scrmt T\kern0.15mm
,\kern0.15mm\mu} be as in Definitions \ref{df top deco}\,(2) above, we need to verify
that for \math{ A \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } we have \mathss38{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A = \sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\KP1\}
}. Thus for given \math{\varepsilon\in\rbb R^+} it suffices to find some \math{K\aR 1
\in\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A} with \mathss34{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A - \varepsilon <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1}. Now, we first find some set \math{U\kern-0.3mm\in\scrmt T} with \math{
A\subseteq U} and \mathss35{\mu\fvalss01 U < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + \frac 13\KP1\varepsilon}.
Then we find some \math{V\kern-0.3mm\in\scrmt T} with \math{U\kern0.37mm\setminus A\subseteq V}
and \mathss35{\mu\fvalss01 V <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm U\kern0.37mm\setminus A\kern0.37mm) + \frac 13\KP1\varepsilon}. We further find
some \math{K\in\scrmt K} with \math{K\subseteq U} and \mathss31{
\mu\fvalss01 U - \frac 13\KP1\varepsilon < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K}, \,and taking \math{K\aR 1 =
K\kern0.37mm\setminus V} we now see that \math{K\aR 1\in\scrmt K} holds with \mathss36{
K\aR 1\subseteq A}. Furthermore, we have \inskipline{.5}{14.6}
$\mu\fvalss01 V <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm U\kern0.37mm\setminus A\kern0.37mm) + \frac 13\KP1\varepsilon
=\mu\fvalss01 U - \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + \frac 13\KP1\varepsilon < \frac 23\KP1\varepsilon
$ \ and hence \vskip.2mm\centerline{$
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A - \varepsilon \le \mu\fvalss01 U - \varepsilon
= \mu\fvalss01 U - \frac 13\KP1\varepsilon - \frac 23\KP1\varepsilon
< \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K - \mu\fvalss01 V \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1\,$.} \vskip.5mm
Having the above, for the proofs of \math{(1)\impss11(3)} and \math{
(2)\impss11(4)} letting \math{\scrmt K\,,\kern0.07mm\scrmt T\kern0.15mm,\kern0.15mm\mu} be as in
Definitions \ref{df top deco}\,(5) above, it suffices to show existence of \math{
\scrmt A} and \math{N\kern0.15mmrim1} such that we have \math{ \scrmt A \subseteq
\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and such that \math{
\scrmt A\cupss42\{\,N\kern0.15mmrim1\kern0.15mm\} } is disjoint with \math{ \bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu =
\bigcup\,\scrmt A\cupss42 N\kern0.15mmrim1 } and for all \math{K\in\scrmt K} we have
that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable and also \math{
A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for
all \math{A\in\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}.
To get such \math{\scrmt A\,,\kern0.15mm N\kern0.15mmrim1} we let \math{\Cal K} be the set of
all disjoint \math{\scrmt A\subseteq\scrmt K\kern0.15mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } with the
property that \math{ A\capss33 U \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } holds for all \math{A\in
\scrmt A} and \mathss30{U\kern-0.3mm\in\scrmt T}. Since trivi- ally \math{\emptyset
\in\Cal K} holds, by {\sl Zorn's lemma\kern0.15mm} there is some \math{\scrmt A} that
is maximal in \mathss34{\Cal K}. Then we take \mathss36{ N\kern0.15mmrim1 =
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.07mm\setminus\kern0.15mm\bigcup\,\scrmt A }.
We first show that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K} is countable for all \mathss35{
K\in\scrmt K}. So we fix \math{K} and using local compactness of \math{
\scrmt T} find a relatively \mathss37{\scrmt T}--\,compact \math{ U \kern-0.3mm \in
\scrmt T} with \mathss30{K\subseteq U}. Then we have \math{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} U < \lower1.05mm\hbox{$^+$}\infty } from which it follows that the set \vskip.3mm
$\scrmt A\capss31\{\,A:\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\not=0\KP1\} \kern0.37mm$ is countable. Since we have
an injection $\kern0.37mm \scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K\kern0.37mm\setminus 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}
\to
\scrmt A\capss31\{\,A:\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss33 U\kern0.37mm)
\not=0\KP1\} $
\noindent
given by \mathss35{A\capss31 K\mapsto A}, \,the assertion follows.
To establish \math{ \scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1 \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } indirectly, suppose that we have some \math{
K\aar 0\in\scrmt K} with \mathss38{K\aar 0\capss02 N\kern0.15mmrim1 \not \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Since \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 K\aar 0}
is countable, we first see
that \math{K\aar 0\capss02 N\kern0.15mmrim1\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds, and hence we have
\mathss30{K\aar 0\capss02 N\kern0.15mmrim1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}. Then
by \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm K\aar 0\capss02 N\kern0.15mmrim1\kern0.15mm) =
\sup\KPt8\{\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K:K\in\scrmt K\capss22
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm K\aar 0\capss02 N\kern0.15mmrim1\kern0.15mm)\,\} } we find
some \math{K\aR 1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\scrmt K} with
\mathss30{K\aR 1\subseteq K\aar 0\capss02 N\kern0.15mmrim1}, \,and we take
$A = K\aR 1\kern-0.63mm\setminus\kern0.15mm\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1)) $
having now $A\in\scrmt K$ with
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$
for all $U\kern-0.3mm\in\scrmt T$.
Indeed, with \math{U\kern-0.3mm\in\scrmt T} supposing that \math{ \emptyset \not =
A\capss33 U\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } holds, we then have
$A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ , and
{\sl for the moment supposing\kern0.15mm} (\kern0.15mm$*$\kern0.15mm) that also
$K\aR 1\kern-0.63mm\setminus A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$
holds, we
obtain \math{\emptyset\not=A\capss33 U\subseteq
K\aR 1\kern-0.3mm\cap\kern0.37mm U\subseteq\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
} and
hence \mathss36{A\capss34\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
\not=\emptyset }, \,a {\sl contradiction\kern0.15mm}.
Since now \math{A\capss35\bigcup\,\scrmt A\subseteq K\aR 1\capss05\bigcup\,\scrmt A
\subseteq N\kern0.15mmrim1\capss05\bigcup\,\scrmt A = \emptyset} holds, by the maximality
of \linebreak
$\scrmt A\kern0.37mm$ we have \math{A=\emptyset} and hence \mathss38{ K\aR 1 =
\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1)) }.
From this we see existence of some set \math{V} with \math{K\aR 1\subseteq \kern0.15mm
V\kern-0.3mm\in\scrmt T} and \mathss36{ \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} K\aR 1 =
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm V\capss01 K\aR 1)=0 }. Hence we obtain \mathss38{ K\aR 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 }, \,a {\sl contradiction\kern0.15mm}.
So, to finish the indirect proof of
\mathss39{\scrmt K\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 N\kern0.15mmrim1\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
}, \,we must show that (\kern0.15mm$*$\kern0.15mm) above holds. Indeed,
in the contrary case we have \math{
K\aR 1\kern-0.63mm\setminus A\capss33 U\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
and then, as above, we find
some \math{K\aar 2\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\scrmt K}
with
\mathss38{K\aar 2\subseteq
K\aR 1\kern-0.63mm\setminus A\capss33 U
\subseteq K\aR 1\kern-0.63mm\setminus A\subseteq
\bigcup\KP1(\kern0.37mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 K\aR 1))
}, \,and further
some
\math{V\kern-0.3mm\in\scrmt T} with
\math{K\aar 2\subseteq V\capss01 K\aR 1} and
\mathss36{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm V\capss01 K\aR 1)=0}.
Hence we obtain
\mathss38{K\aar 2\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,a
{\sl contradiction\kern0.15mm}.
For \math{(5)\impss11(6)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm N\kern0.15mmrim1} be as
in Definitions \ref{df decomp}\,(3) above and letting $\kern0.37mm\scrmt A$ \linebreak
be as in
(4) there, using the {\sl axiom of choice\kern0.15mm} we find \math{N} with \math{
N\kern0.15mmrim1\subseteq N\subseteq{}^{}\Cal Omega } and such that \math{\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } holds. Now it easily follows that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible.
For \math{(6)\impss11(7)} letting \math{\mu\,,\kern0.15mm{}^{}\Cal Omega} be as
in Definitions \ref{df decomp}\,(2) and letting \math{\scrmt A\kern0.15mmrim0} stand
for the \math{\scrmt A} in (3) there, we take \math{ \scrmt A =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern-0.2mm\cap\KPt5\scrmt A\kern0.15mmrim0 } and \mathss38{
N\kern0.15mmrim1=\kern0.37mm\bigcup\KP1(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\capss22
\scrmt A\kern0.15mmrim0\kern0.15mm) }. It is now a simple exercise to show that \math{
N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible, and that also the condition
concerning \math{N\kern0.15mmrimm1} there holds.
By \math{(1)\impss11(2)} we trivially have \mathss37{(3)\impss11(4)}, \,and
for \math{(4)\impss11(7)} letting \ú$\kern0.37mm\scrmt A\,,\kern0.15mm\scrmt K\,,\kern0.15mm\scrmt T\kern0.07mm
,$ \linebreak
\ú$N\kern0.15mmrim1,\kern0.15mm\mu\kern0.37mm$ be as in Definitions \ref{df top deco}\,(5) it
suffices to show that \math{\scrmt A \subseteq \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+ } holds
and that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, and that also \math{N\kern0.15mmrimm1} is \mathss37{\mu
}--\,negligible whenever \mathss30{N\kern0.15mmrimm1\subseteq{}^{}\Cal Omega \kern-0.3mm} \mathss03{
\kern-0.3mm=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } is
such that for \math{
A\kern0.07mm\ar 1\in\scrmt A} there is \math{N} with \mathss38{
A\kern0.07mm\ar 1\kern-0.2mm\cap\kern0.15mm N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
Now for \math{\scrmt A \subseteq \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+ } taking \math{ A \in
\scrmt A} we have \math{A\not=\emptyset } and hence \mathss30{ A =
A\capss32{}^{}\Cal Omega\in\kern-0.3mm} \mathss03{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cup\kern0.15mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } whence
\mathss30{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ \kern-0.3mm}. To show that \math{N\kern0.15mmrim1}
is \mathss37{\mu}--\,negligible, given any \mathss03{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } we must find some \math{N} with \mathss38{
A\capss31 N\kern0.15mmrim1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Using
{\sl countable choice\kern0.15mm}, we first find an increasing \math{ \bmii8 K \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.37mm) } with \mathss36{ \mu\circ\bmii8 K
\to\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A }. Then we find \math{ \bmii8 N \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 } with \math{
\bmii8 K\fvalss51 i\capss42 N\kern0.15mmrim1\subseteq\bmii8 N\fvalss41 i } for all \mathss36{
i\in\mathbb No}. Now it suffices to take \mathss32{ N =
A\kern0.07mm\setminus\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\cupss54\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 N }.
To get the assertion concerning \math{N\kern0.15mmrimm1} given \math{A} we first take \math{
\bmii8 K} as above. Then noting that \math{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\kern0.37mm\bmii8 K\fvalss51 i\kern0.37mm) } is countable for every \math{
i\in\mathbb No } by {\sl countable choice\kern0.15mm} we get \mathss03{
\bmii8 K\aR 1\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\scrmt K\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.37mm) } with \math{
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\kern0.37mm\setminus N\kern0.15mmrim1\subseteq
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\aR 1 } and such that for every \mathss30{i\in\mathbb No }
there is some \math{A\kern0.07mm\ar 1\in\scrmt A } with \mathss34{
\bmii8 K\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\subseteq A\kern0.07mm\ar 1 }. Then again by {\sl countable
choice\kern0.15mm} we get \mathss03{ \bmii8 N \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0 } with \math{
\bmii8 K\aR 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\capss42 N\kern0.15mmrimm1\subseteq\bmii8 N\fvalss41 i } for
all \mathss36{i\in\mathbb No }. Since we already know that \math{N\kern0.15mmrim1} is \mathss37{
\mu}--\,negligible, we find some \math{N\aar 1} with \mathss38{
A\capss31 N\kern0.15mmrim1\subseteq N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Now it
suffices to take \mathss32{ N = A\kern0.07mm\setminus\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 K\cupss54
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 N\cupss52 N\aar 1 } to get some \math{N} such that we
have \mathss38{ A\capss32 N\kern0.15mmrimm1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{proof}
The idea for the proofs of \math{(1)\impss11(3)} and \math{(2)\impss11(4)}
above is taken from \cite[Proposition 4.14.9\kern0.37mm, p.\ 229]{Edw}\,. Note that
the logical structure of these proofs of the implication \math{(\kern0.37mm i\kern0.37mm)
\impss33(\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.3mm\lower1mm\hbox{$^{^+}$}\kern0.07mm) } is basically the following: \
{\sl Axioms\kern0.15mm} \ $\vdash\KN{1.64}\RHB{.35}-$ \inskipline{.5}5
$[\KPp1.4(\kern0.37mm i\kern0.37mm)\impss33\eexi{\scrmt Z}\,\mfrak P\kern0.37mm$ and \math{[\KPp1.4
\mfrak Q\kern0.37mm\text{ or }\neg\KP1\mfrak Q\KPp1.4]\KP1]} and \inskipline{.3}5
$[\KP1[\KPp1.4\eexi{\scrmt Z}\,\mfrak P\kern0.37mm\text{ and }\kern0.37mm\mfrak Q\KPp1.4]
\impss33(\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.3mm\lower1mm\hbox{$^{^+}$}\kern0.07mm)\KPp1.4] \kern0.37mm$ and \math{[\KP1[\KPp1.4
\aall{\scrmt Z}\,\mfrak P\kern0.37mm\text{ and }\kern0.37mm\neg\KP1 \mfrak Q\KPp1.4]\impss33
\mfrak R\kern0.37mm\text{ or }\neg\KP1\mfrak R\KPp1.4]} \inskipline0{26}
and \math{ [\KPp1.4 \mfrak R \impss33 \mfrak S\ar 0\kern0.37mm\text{ and }\kern0.37mm \neg\KP1
\mfrak S\ar 0 \KP1 ] } and \math{ [\KP1 \neg\KP1\mfrak R \impss33 \mfrak R^*\kern0.15mm\text{
or }\kern0.37mm \neg\KP1\mfrak R^* \KP1 ] } \inskipline0{26}
and \math{ [\KPp1.4 \mfrak R^* \impss33 \mfrak S\ar 1\kern0.15mm\text{ and }\kern0.37mm \neg\KP1
\mfrak S\ar 1 \, ] } and \mathss39{ [\KP1 \neg\KP1 \mfrak R^* \impss33
\mfrak S\ar 2\kern0.37mm\text{ and }\kern0.37mm \neg\KP1\mfrak S\ar 2 \KP1 ] }.
\begin{lemma}[schema]\label{Le deco meas}
Let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega${\,\rm, }and with \ú$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\tvsps0(K)$ hold. Also let \ú$\, x \in \kern0.15mm
^{}^{}\Cal Omega\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ and let {\,\rm X} stand for any of {\rm\,\q{almost}} or
{\rm\,\q{almost scalarly}} or {\rm\,\q{almost simply}}. Further{\kern0.15mm\rm, }let $\,
\scrmt A\,,\kern0.15mm N\kern0.15mmrim1$ be as in {\,\rm Definitions \ref{df decomp}\,(2)} on
page {\,\rm\pageref{decos A}} above. If \ú$\,
(\kern0.37mm x\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm)$ is {\,\rm X}
measurable for all \ú$\,A\in\scrmt A${\,\rm, }then $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ is finitely {\,\rm X} measurable.
\end{lemma}
\begin{proof} Noting that we can write \q{X measurable} in the form \q{almost
Z}, assuming that \math{
(\kern0.37mm x\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) } is X
measurable for every \mathss36{A\in\scrmt A}, \,for given \linebreak
\mathss03{
A\kern0.15mm\ar 0\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} it suffices to find some \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that for \math{ B =
A\kern0.15mm\ar 0\kern-0.63mm\setminus N } and \math{\mu\ar 0=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B } and \math{
x\ar 0=x\KP1|\KP1 B} it holds that \math{
(\kern0.37mm x\ar 0\,;\kern0.07mm\mu\ar 0\,,\kern0.07mm\vPi\kern0.37mm)} is Z\kern0.37mm. For this letting \math{
\scrmt N} be the set of all pairs \math{(\kern0.15mm A\,,\kern0.07mm N\aar 1)} with \math{
A\in\scrmt A} and \math{N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
such that for \math{A\kern0.07mm\ar 1=A\kern0.07mm\setminus N\aar 1 } it holds that \math{
(\kern0.37mm x\KP1|\KP1 A\kern0.07mm\ar 1\kern0.37mm;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.07mm\ar 1\kern0.15mm,\kern0.07mm
\vPi\kern0.37mm) } is Z\kern0.37mm, by our assumption \math{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N}
holds, and hence by the {\sl axiom of choice\kern0.15mm} there is a function \math{
\scrmt N\kern0.07mm\ar 0\subseteq\scrmt N } with \mathss36{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N\kern0.07mm\ar 0}. Then taking \math{ \scrmt A\kern0.15mm\ar 0 =
\scrmt A\capss41\{\,A : A\capss31 A\kern0.15mm\ar 0 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\} } we have \math{\scrmt A\kern0.15mm\ar 0}
countable, and for \math{N\kern0.15mmrimm1 = N\kern0.15mmrim1 \cupss04 \bigcup \KP1 (\kern0.37mm
\scrmt N\kern0.07mm\ar 0\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm\scrmt A\kern0.15mm\ar 0\kern0.07mm) \cupss24 \bigcup\KP1 (\kern0.37mm
\scrmt A\setminus\scrmt A\kern0.15mm\ar 0\kern0.07mm) \capss21 A\kern0.15mm\ar 0} it holds that \math{
N\kern0.15mmrimm1} is \mathss37{\mu}--\,negligible. Since \math{N\kern0.15mmrim1\subseteq A\kern0.15mm\ar 0}
holds, for some \math{N} we have \mathss38{N\kern0.15mmrim1 \subseteq N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,and taking \math{B} as above, it is
straightforward to verify that \math{
(\kern0.37mm x\ar 0\,;\kern0.07mm\mu\ar 0\,,\kern0.07mm\vPi\kern0.37mm)} is Z\kern0.37mm.
\end{proof}
\insubsubhead Integration of scalar functions \label{Ss int scal}
Since for arbitrary functions \math{ u \subseteq
{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } we need to consider upper
inte- grals \math{\upint u\rmdss11\mu } where \math{\mu} is a positive measure
on \mathss36{{}^{}\Cal Omega}, \,we here shortly give the asso- ciated formal
definitions in order to make things precise. Note that in the definition of
the Lebesgue quasi\kern0.37mm-\kern0.37mm norm \math{\|\,x\,\|\Lnorss33^p_\mu } in
Constructions \ref{Ctr |x|_lL^p}\,(2) on page \pageref{ctr L^p-norm} above we
already implicitly used the concept of upper integral.
\begin{constructions}[\kern0.15mm positive, real and pseudo\kern0.37mm-\kern0.15mm usual integrals]\label{defi re scal int} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.6}
\item \ $\loint u\rmdss20\mu = \uniqset t: \mu \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and \newskline{32}
$ u \in\kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath u}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm$ and \newskline{28}
$ t = \sup\kern0.15mm\big\{\kern0.37mm\sum\,\seq{\KPt8
t\kern-.2mm\cdot\kern-.2mm(\kern0.37mm\mu\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm t\kern0.37mm\}\kern0.15mm))
: \sigma : t\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma\KPt8} : \mu : $ \newskline{25.5}
$\sigma\in\kern0.15mm^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath\sigmaa}\KPt8\lbb R_+ \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}}\sigma\subseteq\{\,\eta:\sigma\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}$ \newfline
and $\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\sigma\kern0.37mm$ is finite and $\kern0.37mm \{\,\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm
t\kern0.37mm\} : t \in \hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\mu \,\big\} \KP1 $, \KP9
\item \ $\upint u\rmdss20\mu = \uniqset t : \mu \in \kern0.15mm \label{ctr upint}
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and \newskline{31.9}
$u \in\kern0.15mm ^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath u}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1] \kern0.37mm$ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm$ and \newskline{11.5}
$ t = \inf\kern0.15mm\big\{\kern0.15mm\loint v\rmdss20\mu:\mu:v\in\kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath v}\KP1[\KP{1.1} 0\,,\lower1.05mm\hbox{$^+$}\infty\KP1]\kern0.37mm$ and \newfline
$v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu \kern0.37mm $ and $\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}} u\subseteq\{\KPt8\eta:u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \KPt9 \}\,\big\} \KP1 $, \KP9
\item \ $\plusint x\rmdss20\mu = \uniqset t : t \label{def +int}
= \loint x\rmdss20\mu = \upint x\rmdss20\mu\,$,
\item \ $\Reint x\rmdss10\mu=\uniqset\smb I\kern-0.3mm\ar 1\kern-0.3mm:x\kern0.37mm$ a function \label{defi Reint}
and $\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}} x\subseteq[\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\,]\kern0.37mm$ and $\,
\aall{\smb I\kern0.15mm,\kern0.07mm\smb J}$ \newskline{25.7}
$\smb I = \plusint\,\seq{\KP1\sup\,\{\,0\,,x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}:\eta=\eta\KP1}
\rmdss20\mu\kern0.37mm$ and \newfline
$\smb J=\plusint\,\seq{\KP1\sup\,\{\,0\,,\kern.2mm\lower1.05mm\hbox{$^-$}(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\,\} :
\eta=\eta\KP1}\rmdss20\mu\impss22\smb I\kern-0.3mm\ar 1=\smb I-\smb J\,$, \KP9
\item \ $\int_{\,A\,}x\rmdss10\mu=\uniqset\smb I:A\subseteq\bigcup\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ \label{defi ps-usual int}
and $\,\eexi{\vPi\kern0.15mm,\kern0.07mm S}\,x \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\eightmath x}\kern0.37mm S \kern0.37mm $ and \newskline{23}
$\vPi\kern0.37mm$ is complex pseudo\kern0.37mm-\kern0.15mm usual and $\kern0.37mm\vPi\not=\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm$
and $\,\big[\ \big[\KP{1.5} S = $ \newskline{9.5}
$\mathbb C\cupss31\{\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\,\}\kern0.37mm$ and $\kern0.37mm
\smb I = \Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Re}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu\kern0.15mm + \kern0.37mm\roman i\,
\Reint\,(\kern0.15mm\kern.1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f\kern.1mm}\roman{Im}\kern.9mm x\,|\,A\kern0.15mm)\rmdss10\mu $ \newskline{13}
$\in S \KP{1.5}\big]\kern0.37mm$ or $\kern0.37mm\big[\KP{1.5} x \not= \emptyset\kern0.37mm$ and $\kern0.37mm
\smb I \in S = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm$ and $\, \aall \ell\,
\ell\in\Cal L\,(\kern0.07mm\vPi\Reit2\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)$ \newfline
$\impss02 \ell\kern0.37mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb I = \Reint\,
(\kern0.37mm\ell\kern0.15mm\circ\kern0.07mm x\KP1|\KP1 A\kern0.37mm)\rmdss10\mu\KP{1.5}\big]\ \big]\ $.
\end{myLeftskip}\end{enumerate}
\end{constructions}
From Constructions \ref{defi re scal int}\,(1) we get the {\it lower integral\kern0.37mm}
of a\q{positive} valued function \math{u} with respect to a positive measure \mathss35{
\mu}, \,and \ref{defi re scal int}\,(2) and \ref{defi re scal int}\,(3) give
the corresponding {\it upper\kern0.37mm} and {\it positive\kern0.37mm} integral. The
{\it real\kern0.37mm} integral of an extended real valued function w.r.t.\ a positive
measure is given in \ref{defi re scal int}\,(\ref{defi Reint})\,, and item (5)
defines the {\it pseudo\kern0.37mm-\kern0.15mm usual\,} integral. Without delving in the
relevant formal definition given in \cite{Hif} we shortly remark that
pseudo\kern0.37mm-\kern0.15mm usual spaces \math{E} are such structured vector spaces over
some subfield \math{\bold K} of the complex field \math{\raise1.23mm\hbox{\font\SweD =cmr5\SweD f}\kern.1mm\mathbb C} that e.g.\ we
have unambiguously \math{(\kern0.37mm x + y\kern0.37mm)\svs E = x + y } and \math{
(\kern0.37mm t\,x\kern0.37mm)\svs E = t\,y } for all \math{x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } and \mathss36{
t\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bold K}. If \linebreak
\ú$\vPi\kern0.37mm$ is pseudo\kern0.37mm-\kern0.15mm usual and \math{I} is any
set with \math{I\in\{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.15mm\} } or \mathss31{
3\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\subseteq{}^{}{\rm Card}{}^{}\,\kern0.15mm I}, \,then \ú$\kern0.37mm(\kern0.15mm X\kern0.15mm,\kern0.07mm S\kern0.37mm)$ \linebreak
is
pseudo\kern0.37mm-\kern0.15mm usual for any set \math{S} and any vector substructure \math{X}
of \math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.15mm I\kern0.15mm]_{vs} }.
\begin{definitions}
(1) \ Say that \math{u} is {\it positive \mathss37{\mu}--\,measurable\kern0.37mm}
if{}f there is \math{{}^{}\Cal Omega} such that \math{\mu} is a positive measure on \math{
{}^{}\Cal Omega} and \math{u} is a function with \math{ u \subseteq
{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and such that \math{
u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1 r\kern0.15mm,\lower1.05mm\hbox{$^+$}\infty\KPt9]\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} holds for all \mathss30{
r\in\rbb R^+}, \inskipline{.5}2
(2) \ Say that \math{u} is {\it fully positive \mathss37{\mu}--\,measurable\kern0.37mm}
if{}f \inskipline0{23.6}
$u\kern0.37mm$ is positive \mathss37{\mu}--\,measurable with \mathss36{ {{}^{}{\rm dom}\,{}_{{}^{}}} u = \kern0.15mm
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \inskipline{.5}2
(3) \ Say that \math{\sigma} is {\it positive \mathss37{\mu}--\,simple\kern0.37mm}
if{}f
there is \math{{}^{}\Cal Omega} such that \math{\mu} is a positive measure on \math{
{}^{}\Cal Omega} and \math{\sigma} is a function with \math{\sigma\subseteq{}^{}\Cal Omega\times
\lbb R_+ } and such that \math{{}^{}{\rm rng}\,{}_{{}^{}}\sigma} is finite and also \math{
\{\,\sigma\invss33\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\} : t \in \hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\mu }
holds.
\end{definitions}
Thus in the case where \math{\mu} is a positive measure, in
Constructions \ref{defi re scal int}\,(\ref{ctr upint}) above we have \math{t}
the infimum of the set of lower integrals of all positive \mathss37{\mu
}--\,measurable functions \math{v} dominating \math{u} in the sence that \math{
u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for all \mathss34{\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}.
\begin{lemma}\label{Le +int}
Let $\,\mu$ be a positive measure. Then for all $\,x$ the equivalences \vskip.5mm\centerline{$
\plusint x\rmdss20\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} \equivss33 0\le\plusint x\rmdss20\mu =
\loint x\rmdss20\mu = \upint x\rmdss20\mu\le\lower1.05mm\hbox{$^+$}\infty $} \inskipline{.5}{20}
and $ \KPp18.6 0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty \equivss33 (\kern0.15mm*\kern0.15mm)
\null
$ hold when $\,(\kern0.15mm*\kern0.15mm)$ \inskipline{.5}0
means that there exist positive $\,\mu\,$--\,measurable functions $\,u\kern0.37mm,\kern0.15mm
v$ with \ú$\,u\le x\le v$ and $\,\loint u\rmdss20\mu \not= \lower1.05mm\hbox{$^+$}\infty$ and $\,
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\cupss31\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 $.
\end{lemma}
\begin{proof} Assuming \math{\plusint x\rmdss20\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} } we have \math{
\plusint x\rmdss20\mu =\loint x\rmdss20\mu = \upint x\rmdss20\mu = t } for
some \mathss36{t\in\hbox{\font\SweD =cmssbx10\SweD U}{}}. Then taking \math{\sigma = \emptyset } in
Constructions \ref{defi re scal int}\,(1) we see that \math{t = \sup\kern0.37mm A }
for some \math{A} with \math{0\in A\subseteq[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } whence \math{
0\le t\le\lower1.05mm\hbox{$^+$}\infty } follows. Conversely, if we have \mathss35{ 0 \le
\plusint x\rmdss20\mu\le\lower1.05mm\hbox{$^+$}\infty }, \,then \math{ \plusint x\rmdss20\mu \in
\hbox{\font\SweD =cmssbx10\SweD U}{} } and hence \mathss35{\plusint x\rmdss20\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} }.
Assuming \math{0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty } from (1) and
(\ref{ctr upint}) in Constructions \ref{defi re scal int} we see existence of
sequences \math{\bosy u} of positive \mathss37{\mu}--\,simple and \math{
\bosy v} of positive \mathss37{\mu}--\,measurable functions with \math{
\lim\sbi{\kern0.15mm i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm\loint\bosy w\fvalss01 i\rmdss11\mu=0 } for \math{
\bosy w=\seq{\seqss33{
\bosy v\fvalss01 i\fvalss10\eta - \bosy u\fvalss01 i\fvalss10\eta:
\eta=\eta}:i\in\mathbb No\,} } and
\math{\loint\bosy v\fvalss01\emptyset\rmdss11\mu < \lower1.05mm\hbox{$^+$}\infty} and such that
\math{\bosy u\fvalss01 i\le\bosy u\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le x\le
\bosy v\fvalss01 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\le\bosy v\fvalss01 i} holds for all
\mathss36{i\in\mathbb No}. Taking then
\math{u=\seqss43{\sup\KPt8\{\KPt8\bosy u\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy u:\eta\in\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\bosy u} }
and
\mathss39{v=\langle\KP1\inf\,\{\KPt8\bosy v\fvalss01 i\fvalss10\eta:
i\in\mathbb No\kern0.37mm\}:\bosy v:\eta\in\bigcap\KPt8\{\,{{}^{}{\rm dom}\,{}_{{}^{}}} v:
v\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy v\KPt8\} \,\big\rangle }, \,now
$u$ and $v$ are positive $\mu\,$--\,measurable and hence
$\{\KPt8
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm,\kern0.15mm\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\}\kern0.15mm\}
\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ holds. If we
have \mathss38{
v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u
\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,we see that
$\loint u\rmdss11\mu < \loint v\rmdss11\mu$ holds, leading to
a contradiction. Similarly we see that \math{\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\}
\not\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } is impossible. So we have
\mathss38{v\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.2mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\cupss31\{\,\eta :
v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\not=u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\in\hbox{\font\SweD =cmssbx10\SweD U}{}\KP1\} \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
The implication \math{(\kern0.15mm*\kern0.15mm)\impss33 0\le\plusint x\rmdss20\mu < \lower1.05mm\hbox{$^+$}\infty }
is straightforward.
\end{proof}
Assuming that \math{\mu} is a positive measure on \math{{}^{}\Cal Omega} and that \math{
x} is a function \mathss36{{}^{}\Cal Omega\to\mathbb C}, \,from Lemma \ref{Le +int} via
inspection of items (\ref{defi Reint}) and (\ref{defi ps-usual int}) in
Constructions \ref{defi re scal int} above we see that \math{ \smb I =
\int_{\KPp1.1{}^{}\Cal Omega}\kern0.37mm x\rmdss11\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} } implies that \math{\smb I
\in\mathbb C} holds together with \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) \label{int fin a.e. meas} }
being finitely almost \mathss37{\mu}--\,measurable. Thus in the case of an
incomplete probability measure an {\sl integrable function need not be \label{int not meas}
measurable\kern0.15mm} according to our conventions.
\begin{proposition}\label{Pro upint}
Let $\,p\in\rbb R^+$ and
let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega\,$. Also let
$\,w=
\seqss43{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta=\eta}$
where $\,u$ and $\,v$ are any functions with
$\,u\cupss22 v\subseteq{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \KP1 $.
Then $\,\|\,w\,\|\Lnorss33^p_\mu
\le
\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\}\KP1\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0$ holds.
\end{proposition}
\begin{proof} Let $\,\roman M\,x=
\kern0.15mm^{}^{}\Cal Omega\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9]
\capss51\{\,\varphi:\varphi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\kern0.07mm\barscTbb_R\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ \inskipline{.2}{56}
$\text{ and }\kern0.37mm\aall{\eta\,,\kern0.15mm t}\,
(\kern0.37mm\eta\,,\kern0.07mm t\kern0.37mm)\in x\impss33
t\le\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
\KP1\} $ . \inskipline{.4}0
From \cite[pp.\ 49\,--\,50]{Jr} we know that the assertion
holds under the additional restriction that we have
\math{u\in\roman M\,u} and \math{v\in\roman M\,v} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm u\cupss22 v\kern0.37mm)\subseteq\ssbb05 R}, \,noting that
it is trivial if \math{
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu=\lower1.05mm\hbox{$^+$}\infty} holds. From
this one easily extends the result to the case
where the restriction $\lower1.05mm\hbox{$^+$}\infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm u\cupss22 v\kern0.37mm)$ is
removed.
Now putting
\math{\smb A=\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\} } for
the general case, to proceed indirectly,
suppose that we have
$\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
<\|\,w\,\|\Lnorss33^p_\mu$ and take any $\varepsilon\in\rbb R^+$ with
$2\KPt8\smb A\KPt8\varepsilon < \|\,w\,\|\Lnorss33^p_\mu -
\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0$ .
Then there are $\varphi\in\roman M\,u$ and $\psi\in\roman M\,v$ with
$
\|\,\varphi\,\|\Lnorss33^p_\mu
< \|\,u\,\|\Lnorss33^p_\mu + \varepsilon \,$ and $\,
\|\,\psi\,\|\Lnorss33^p_\mu < \|\,v\,\|\Lnorss33^p_\mu + \varepsilon \,$,
whence with
$\chi=\seqss43{\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta=\eta}$
we obtain $\chi\in\roman M\,w$ and consequently \inskipline{.7}{8.5}
$\|\,\chi\,\|\Lnorss33^p_\mu\le\smb A\,\big(\kern0.37mm
\|\,\varphi\,\|\Lnorss33^p_\mu + \|\,\psi\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
$ \inskipline{.4}{21}
${}<\smb A\,\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
+2\KPt8\smb A\KPt8\varepsilon
< \|\,w\,\|\Lnorss33^p_\mu \,$, \,a contradiction.
\end{proof}
\begin{remark}\label{Rem +*}
According to our updated definitional conventions in \cite{Hif} concerning
sums and products of elements in a {\sl pseudo\kern0.37mm-\kern0.15mm usual algebroid\kern0.15mm}, if
in Proposition \ref{Pro upint} for \mathss03{ \scrmt A =
\{\,{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm,{{}^{}{\rm dom}\,{}_{{}^{}}} v\,\} } we have \math{\bigcap\,\scrmt A \not= \emptyset }
or \mathss36{\bigcap\,\scrmt A = \emptyset = \bigcup\,\scrmt A }, \,then also \mathss30{w =
u + v } holds. Hence under this additional assumption we could have written
the expression for \math{w} a bit more simply. However, if \math{
\bigcap\,\scrmt A = \emptyset \not= \bigcup\,\scrmt A } holds and we also
have \mathss06{\|\,u\,\|\Lnorss33^p_\mu \not= \lower1.05mm\hbox{$^+$}\infty \not=
\|\,v\,\|\Lnorss33^p_\mu }, \,then \math{u + v = \hbox{\font\SweD =cmssbx10\SweD U}{} } and for this in
place of \math{w} we would \linebreak \vskip-3.3mm\noindentdent get $ \KP{3.3}
\lower1.05mm\hbox{$^+$}\infty = \inf\kern0.37mm\emptyset = \|\KPt8\hbox{\font\SweD =cmssbx10\SweD U}{}\KPt8\|\Lnorss33^p_\mu
= \|\KP1 u + v\KP1\|\Lnorss33^p_\mu = \|\,w\,\|\Lnorss33^p_\mu$ \inskipline{.2}{15}
${} \le \sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\}\KP1\big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss33^p_\mu\kern0.15mm\sbig)0
< \lower1.05mm\hbox{$^+$}\infty \,$, \,a contradiction. \inskipline{.6}0
A similar remark applies to \math{\varphi\,,\kern0.15mm\psi\kern0.37mm,\kern0.15mm\chi} in the proof,
thus having \math{\chi = \varphi + \psi } the function given by \math{{}^{}\Cal Omega
\owns\eta\mapsto\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } since here \math{
{}^{}\Cal Omega={{}^{}{\rm dom}\,{}_{{}^{}}}\varphi={{}^{}{\rm dom}\,{}_{{}^{}}}\psi } holds.
We also suggest the reader to see \cite{A+Mo} for another kind of treatment of
the notational \q{plus\kern0.37mm-\kern0.15mm times} problem referred to above.
\end{remark}
We also extend H\"older's inequality to upper integrals in the next
\begin{proposition}\label{Pro Hˆlder}
Let $\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega
\,$. Also let
$\,w=\seqss43{u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot(\kern0.37mm v\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm):\eta=\eta}$ where $\,u$
and $\,v$ are any functions with
$\,u\cupss22 v\subseteq{}^{}\Cal Omega\times[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] \KP1 $.
Then $\,\upint w\rmdss11\mu
\le \|\,u\,\|\Lnorss33^p_\mu \KP1 \|\,v\,\|\Lnorss40^{p\sast}_\mu$ holds.
\end{proposition}
\begin{proof} Let \math{\roman M\,x} be as in the proof of Proposition \ref{Pro upint}
above. For the indirect verification, suppose now that we have \mathss37{
\|\,u\,\|\Lnorss33^p_\mu \KP1 \|\,v\,\|\Lnorss40^{p\sast}_\mu <
\upint w\rmdss11\mu}, \,and with \mathss03{ A =
\big\{\,\|\,u\,\|\Lnorss33^p_\mu \,,\kern0.15mm \|\,v\,\|\Lnorss40^{p\sast}_\mu\kern0.37mm\} }
then put \mathss36{\smb M=\sup\,A}. We cannot have \math{0\in A} since by a
simple exercise this would force \mathss36{ \upint w\rmdss11\mu = 0
}, \,contradicting our assumption. It follows that \math{\smb M < \lower1.05mm\hbox{$^+$}\infty}
holds, and we then take any \math{\varepsilon} with \vskip.5mm\centerline{$
0 < \varepsilon < \inf\,\{\,1\kern0.37mm,\kern0.15mm(\kern0.37mm 2\KPt8\smb M + 1\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.37mm
\big(\kern0.15mm\upint w\rmdss11\mu - \|\,u\,\|\Lnorss33^p_\mu \KP1
\|\,v\,\|\Lnorss40^{p\sast}_\mu \kern0.15mm \sbig) 0 \,\big\} \KP1 $.} \inskipline{.5}0
Now there are functions \math{\varphi\in\roman M\,u} and \math{ \psi \in
\roman M\,v} with \math{ \|\,\varphi\,\|\Lnorss33^p_\mu <
\|\,u\,\|\Lnorss33^p_\mu + \varepsilon } and \mathss05{
\|\,\psi\,\|\Lnorss40^{p\sast}_\mu < \|\,v\,\|\Lnorss40^{p\sast}_\mu + \varepsilon
}, \,whence taking \math{ \chi =
\seqss43{\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \cdot(\kern0.37mm\psi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm):\eta=\eta} } we
then have \math{\chi\in\roman M\,w} and consequently by the usual H\"older's
inequality extended to measurable functions with values in \math{
[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } we obtain \inskipline{.7}{7.78}
$ \upint w\rmdss11\mu \le \int_{\KPp1.1{}^{}\Cal Omega\,}\chi\rmdss11\mu
\le \|\,\varphi\,\|\Lnorss33^p_\mu \KP1 \|\,\psi\,\|\Lnorss40^{p\sast}_\mu $ \inskipline{.4}{20}
${}< \big(\kern0.37mm\|\,u\,\|\Lnorss33^p_\mu + \varepsilon\kern0.37mm)\KP1
\big(\kern0.37mm\|\,v\,\|\Lnorss40^{p\sast}_\mu + \varepsilon\kern0.37mm) $ \inskipline{.4}{20}
${}= \|\,u\,\|\Lnorss33^p_\mu\KP1\|\,v\,\|\Lnorss40^{p\sast}_\mu + \big(\kern0.37mm
\|\,u\,\|\Lnorss33^p_\mu + \|\,v\,\|\Lnorss40^{p\sast}_\mu\kern0.37mm)\KP1\varepsilon
+ \varepsilon\KPt8^2 $ \inskipline{.4}{20}
${}\le \|\,u\,\|\Lnorss33^p_\mu\KP1\|\,v\,\|\Lnorss40^{p\sast}_\mu +
(\kern0.37mm 2\KPt8\smb M + 1\kern0.37mm)\KP1\varepsilon
< \upint w\rmdss11\mu \,$, \,a contradiction.
\end{proof}
\begin{constructions}[standard Lebesgue measures]\label{defi Leb mea} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.4}
\item \ $\upCth\mu = \mu\KP1|\KP1\{\,A:\aall{B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}\, \label{ctr Carath}
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss21 B\kern0.37mm) +
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm B\kern0.15mm\setminus A\kern0.37mm) \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm B\KP1\} \KP1 $,
\item \ $\mu\meatimes\nu = \uniqset\mu\ar 1\kern-0.3mm:\aall{\mu\ar 2\kern0.37mm,\kern0.15mm\mu\ar 3}$ \newskline8
$\mu\ar 2 = \{\,(\kern0.07mm A\times B\kern0.37mm,\kern0.15mm s\kern-.2mm\cdot\kern-.2mm t\,):
(\kern0.07mm A\, ,\kern0.07mm s\kern0.37mm)\in\mu \kern0.37mm$ and $\kern0.37mm
(\kern0.07mm B\kern0.37mm,\kern0.07mm t\kern0.37mm)\in\nu\,\}\kern0.37mm$ and \vskip-.3mm
$\nKP{12.7} \mu\ar 3 =
\big\langle\,\inf\kern0.15mm\big\{\,\sum\,(\kern0.37mm\mu\ar 2\circ\ebit B\kern0.37mm)
: \mu\ar 2\kern-0.2mm : \ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 2)\kern0.37mm$ and \newfline
$ A\subseteq\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit B\KPt8\} : \mu\ar 2\kern-0.2mm :
A\subseteq\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 2\,\big\rangle \impss22
\mu\ar 1 = \upCth\mu\ar 3 \KPt8 $, \KP{8.5}
\item \ $\Lebmef^{\kern0.37mm\ssmb N} = \uniqset\mu\ar 1\kern-0.3mm:\smb N\in\mathbb No\kern0.37mm$ and $\, \label{ctr Lebmea}
[\ [\KP{1.3}\smb N = \emptyset\kern0.37mm$ and $\kern0.37mm
\mu\ar 1 = \seq{\,0\,,1\,} \KP{1.3}]\kern0.37mm$ or \newskline{15}
$[\KP{1.3} \smb N \not= \emptyset\kern0.37mm$ and $\,\aall{
\Cal B\kern0.37mm,\kern0.15mm\Cal J\kern0.07mm,\kern0.07mm\mu\,,\kern0.07mm\nu\kern0.37mm,\kern0.07mm\nu\ar 1}\,
\Cal J = \{\,\openIval{\smb A\kern0.37mm,\smb B} :
\smb A\kern0.37mm,\kern0.07mm\smb B\in\ssbb09 R\}\kern0.37mm$ and \newskline{6.5}
$\nu\ar 1 = \seq{\,\smb B-\smb A : J = \openIval{\smb A\kern0.37mm,\smb B}
\in\Cal J\kern0.37mm$ and $\kern0.37mm\smb A\le\smb B\,}\kern0.37mm$ and \newskline{8}
$\nu = \Seq{\,\prod\,(\kern0.37mm\nu\ar 1\kern-0.2mm\circ\kern0.07mm\ebit I\kern0.37mm) : B =
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\ebit I\kern0.37mm$ and $\kern0.37mm\ebit I\in\kern0.15mm\yi N\kern0.15mm\Cal J\KP1}\kern0.37mm$ and $\kern0.37mm
\Cal B={{}^{}{\rm dom}\,{}_{{}^{}}}\nu$
$\nKP{8} \mu = \Seq{\,\inf\kern0.15mm\big\{\,\sum\,(\kern0.37mm\nu\circ\ebit B\kern0.37mm) :
\ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\Cal B\kern0.37mm$ and $\kern0.37mm A\subseteq\bigcup\kern0.15mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit B\,\} :
A\subseteq\kern0.15mm\yi N\ssbb69 R}$ \newfline
$\impss02 \mu\ar 1 = \upCth\mu \KP{1.3}]\ ]\ $, \KP{8.5}
\item \ $\Lebmef^{} = \Lebmef^{\kern0.15mm 1.} \circ\kern0.15mm
\big\langle\KP1^{1.}A:A\subseteq\ssbb09 R\kern0.15mm\rangle \KP1 $, \label{def Lebm on R} \vskip.3mm
\item \ $\int_{\,\ssmb A}^{\,\ssmb B\,}x = \uniqset\smb I : \label{df Leb int}
\eexi{A\,,\kern0.07mm\sigma}\,\big[\ \big[\KP{1.5} \sigma = 1\kern0.37mm$ and $\kern0.37mm
\smb A < \smb B\kern0.37mm$ and $\kern0.37mm
A = \openIval{\smb A\kern0.37mm,\kern0.07mm\smb B} \KP{1.5} \big]$ \inskipline{-.1}{43.5}
or $\kern0.37mm\big[\KP{1.5} \sigma = \kern.2mm\lower1.05mm\hbox{$^-$} 1\kern0.37mm$ and $\kern0.37mm \smb B \le \smb A\kern0.37mm$
and $\kern0.37mm A = \openIval{\smb B\kern0.37mm,\kern0.07mm\smb A} \KP{1.5} \big]\ \big]$ \inskipline{-.1}{44.33}
and $\, \smb I = \sigma\kern0.07mm\int_{\,A}\kern0.37mm x\rmdss10\Lebmef^{} \KP1 $.
\end{myLeftskip}\end{enumerate}
\end{constructions}
\newcommand\sPows{{\lower.22mm\hbox{\font\≈=eusm8\≈P}\kern-.3mm\lower.7mm\hbox{\font\≈=cmss5\≈s}\kern.65mm}}
Saying that \math{\mu} is an {\it outer measure\kern0.37mm} on \math{{}^{}\Cal Omega} if{}f \math{
\mu\in\kern0.15mm^{\sPows{}^{}\Cal Omega}\KP1[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] } and \mathss30{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\le\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm B } and \math{
\mu\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\,\scrmt A\le\sum\KP1(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) }
hold whenever we have \math{A\subseteq B\subseteq{}^{}\Cal Omega } and \math{ \scrmt A \subseteq
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{}^{}\Cal Omega } with \linebreak
$\scrmt A\kern0.37mm$ countable, essentially from
\cite[Lemma 3.1.8\kern0.37mm, Proposition 3.1.9\kern0.37mm, pp.\ 67\,--\,68]{Du} we get the
proof of the following
\begin{proposition}\label{Pro Cth}
If $\,\mu$ is an outer measure on $\,{}^{}\Cal Omega${\,\rm, }then $\,\upCth\mu$ is a
complete positive measure on $\,{}^{}\Cal Omega\,$.
\end{proposition}
Thus by Proposition \ref{Pro Cth} in \ref{defi Leb mea}\,(\ref{ctr Carath}) we
have the standard {\sl Carath\'eodory construc- tion\kern0.15mm} associating a
complete positive measure with any outer measure. For \mathss36{\smb N\in\mathbb N
}, \,the function \math{\Lebmef^{\kern0.37mm\ssmb N}} is the standard complete
Lebesgue measure on \math{\yi N\ssbb60 R} defined on the class of Lebesgue
measurable subsets. The corresponding measure on \math{\mathbb R} is \math{
\Lebmef^{}\kern0.15mm}. Note that if we had not separately defined \math{
\Lebmef^{\kern0.15mm 0.} = \seqss20{0\,,\kern0.07mm 1} = \{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm
(\kern0.37mm 1\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\} } by inserting \q{\mathss03{ \smb N =
\emptyset\kern0.37mm\text{ and }\kern0.37mm\mu\ar 1=\seqss20{0\,,\kern0.07mm 1} }} in
\ref{defi Leb mea}\,(\ref{ctr Lebmea})\,, then it would have given \mathss30{
\Lebmef^{\kern0.15mm 0.}=\kern-0.3mm} \mathss08{2\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\} }. We also
put \math{ \LeBmef^{\kern0.37mm\ssmb N} =
\Lebmef^{\kern0.37mm\ssmb N}\,|\KP1\sigmAlg3\nsTbb_R\kern-0.63mm\yi N } and \math{\LeBmef^{}
= \Lebmef^{}\,|\KP1\sigmAlg3\nsTbb_R } get- ting the restrictions of the
Lebesgue measures to the standard Borel \rsigma3algebras.
\insubsubhead Pettis integration of vector functions \label{Ss Pettis}
In some places of the proof of Theorem \nfss A\,\ref{main Th} we refer to
something being {\sl Pettis\kern0.15mm}. In order to make the meaning of this
explicit, we give the following
\begin{definitions}[for Pettis integration]\label{df Petti}
(1) \ Say that \math{\tilde c} is {\it scalar integrable\kern0.37mm} to \math{x}
if{}f \math{\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \math{\bosy K\kern0.37mm,\kern0.15mm
E\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\kern0.37mm,\kern0.15mm x\kern0.37mm,\kern0.15mm u} from \math{ \tilde c =
(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) } and \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{{{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K } and \math{ u \in
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } it follows that \newline
$u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=\int_{\KPp1.1{}^{}\Cal Omega}\,u\circss01 c\rmdss21\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{}$ holds, \inskipline{.5}2
(2) \ $E\vPettis3int_A\,c\rmdss21\mu=\uniqset x:c\kern0.37mm$ a function and \math{
\mu} is a positive measure and \inskipline{.2}{8.5}
$A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} c\capss43\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and \math{
(\,c\KP{1.2}|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm E\kern0.37mm)} is scalar
integrable to \mathss34{x}, \inskipline{.5}2
(3) \ Say that \math{\tilde c} is {\it Pettis\kern0.37mm} if{}f
\math{\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all
\math{A\,,\kern0.15mm E\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm c} from
$\kern0.37mm\tilde c={}$ \inskipline{.2}{8.5}
$(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) \kern0.37mm$ and
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it follows that \math{
E\vPettis3int_A\,c\rmdss21\mu\not=\hbox{\font\SweD =cmssbx10\SweD U}{} } holds.
\end{definitions}
Because of the manner we have put the definitions, from the discussion after
the proof of Lemma \ref{Le +int} on page \pageref{int not meas} above, it
follows that \math{\tilde c} being Pettis implies it being {\sl finitely
almost scalarly measurable\kern0.15mm}. Also from \math{(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\kern0.37mm) }
being Pettis with \mathss03{{}^{}{\rm rng}\,{}_{{}^{}}\mu\capss34\rbb R^+\not=\emptyset} it follows
that \math{{}^{}{\rm Card}{}^{}\,\kern0.15mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\not=1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}={}^{}{\rm Card}{}^{}\,\kern0.15mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm(\kern0.15mm E\dlsigss11\kern0.15mm) }
cannot hold. That is, if \math{\mu} and \math{E} are nontrivial, then also the
dual of \math{E} must be such. For example \mathss03{
(\,c\KPt8;\Lebmef^{}\kern0.37mm,\kern0.07mm\LLrs42^{\frac 12}(\ssbb44 I)) } cannot be
Pettis, whereas \math{
(\,c\KPt8;\Lebmef^{}\kern0.37mm,\kern0.15mm\ell\KPt8^{\frac 12\kern0.37mm}(\kern0.37mm\mathbb No\kern0.07mm)) } can. \vskip.3mm
To have at our disposal also some partially weaker and more general notions of
integrability of mv\kern0.37mm-\kern0.15mm maps, we put the following
\begin{definitions}\label{df sc+Gel-int}
(1) \ Say that \math{\tilde c} is {\it scalarly integrable\kern0.37mm} if{}f \math{
\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \mathss03{c\,,\kern0.15mm\mu\,,\kern0.15mm
\bosy K\kern0.15mm,\kern0.15mm\vPi} from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi } it follows that
for all \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus\{\,\Bnull_\vPi\} } there is \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{u\fvalss01\xi\not=0 }
and for all \math{u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and \math{ A \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it holds that \mathss36{
\int_{\,A\,}u\circss01 c\rmdss21\mu \not= \hbox{\font\SweD =cmssbx10\SweD U}{} }, \inskipline{.5}2
(2) \ Say that \math{\tilde c} is {\it finitely scalarly integrable\kern0.37mm} if{}f \math{
\tilde c} is an mv\kern0.37mm-\kern0.15mm map and for all \mathss30{c\,,\kern0.15mm\mu\,,} $
\vPi\kern0.37mm$ from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} it follows that \math{
(\,c\KPp1.2|\KP1 A\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\kern0.37mm) } is scalarly
integrable, \inskipline{.5}2
(3) \ Say that \math{\tilde c} is {\it Gelfand\,} if{}f \math{\tilde c} is
scalarly integrable and for all \mathss30{c\,,\kern0.15mm\mu\,,\kern0.15mm A\,,\kern0.15mm\bosy K\kern0.15mm
,\kern0.15mm\vPi} from \math{\tilde c=(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K={{}^{}{\rm dom}\,{}_{{}^{}}}m\tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi } and \mathss30{A\in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} it follows that \math{
\big\langle\kern0.15mm\int_{\,A\,}u\circss01 c\rmdss21\mu : u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\KP1\rangle } is continuous \mathss32{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\to\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}.
\end{definitions}
A simple example of a Banach space valued mv\kern0.37mm-\kern0.15mm map that is Gelfand but
not Pettis is given in the following
\begin{example}\label{Exa Gel /= Pet}
Let \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } where \math{\mu=
\seqss30{\card3 A:A\subseteq\mathbb No\kern-0.2mm} } and \math{\vPi=\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\kern0.37mm\mathbb No\kern0.07mm) }
and \mathss08{x=\seqss30{(\kern0.37mm\mathbb No\kern-0.3mm\setminus\{\kern0.37mm i\kern0.37mm\}\kern0.15mm\sbig)0\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22\{\,(\kern0.37mm i\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\}:i\in\mathbb No\kern-0.2mm} }. For
every \math{A\subseteq\mathbb No } and \mathss30{ \zeta \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) } then \mathss38{
\int_{\,A\,}x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot\zeta\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm) =
\sum\KP1(\kern0.37mm\zeta\KP1|\KP1 A\kern0.37mm) }, \,and hence \math{\tilde x} is scalarly
integrable. It is also Gelfand since for \math{ \lambda =
\mathbb No\kern-.2mm\times\kern-.2mm\{\kern0.37mm 1\kern0.37mm\} } we have \math{ \lambda \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) } with \mathss30{
\sum\KP1(\kern0.37mm\lambda\cdot\zeta\KP1|\KP1 A\kern0.37mm) = \kern-0.3mm} \mathss03{
\sum\KP1(\kern0.37mm\zeta\KP1|\KP1 A\kern0.37mm) =
\int_{\,A\,}x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot\zeta\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm) } for all \math{
A\subseteq\mathbb No} and \mathss38{\zeta\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\ell\KPt8^1\kern0.07mm(\kern0.37mm\mathbb No\kern0.07mm) }. Since
we here have \mathss31{\lambda\not\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}, \,we see that \math{\tilde x}
is not Pettis.
\end{example}
\Ssubhead B Generalized Bochner spaces \label{Sec B}
In this section, we first give the formal construction of the generalized
Lebesgue\,--\,Bochner spaces spaces \math{ F =
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{ F\aar 1 =
\mvsLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } of equivalence classes of order \math{p}
integrable functions \math{x:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} when \math{ 0 \le p \le
\lower1.05mm\hbox{$^+$}\infty} and \math{\mu} is a positive measure on some set \math{{}^{}\Cal Omega}
and \math{\vPi} is a real or complex topological vector space. Then we
establish the basic relevant properties of these spaces under the additional
assumption that the space \math{\vPi} is suitable.
For the construction of the space $\kern0.37mm F$, the functions \math{x} are
required to be such that \math{\tilde x=(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)} is
{\sl finitely almost simply measurable\kern0.15mm} in the sense of
Definitional schemata \ref{df meas} on page \pageref{df meas} above. For \math{
F\aar 1} we instead require \math{\tilde x} to be only {\sl finitely almost
scalarly measurable\kern0.15mm}.
The integrability condition is formulated so that in the case \math{p\not=0}
for any bounded quasi\kern0.37mm-\kern0.37mm seminorm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} in \math{\vPi} we should
have \math{\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu<\lower1.05mm\hbox{$^+$}\infty } which in the
case \math{p\in\rbb R^+} is equivalent to the function \math{
\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x : {}^{}\Cal Omega \owns \eta \mapsto
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\fvalss11\eta\kern0.37mm)\,^p } \nolinebreak pos- sessing a
dominating \mathss37{
\mu}--\,integrable function \mathss38{ \varphi : {}^{}\Cal Omega \to
[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] }. Then
$\null$
\math{
\upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 x\rmdss11\mu<\lower1.05mm\hbox{$^+$}\infty} holds, and
this determines
by $x\mapsto\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu$ a corresponding
quasi\kern0.37mm-\kern0.37mm seminorm
.
For \math{p=0} no integrability is required, and in this case the topology is
determined by the quasi\kern0.37mm-\kern0.37mm semimetrics \math{ \roman d\,A\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm :
(\kern0.15mm x\kern0.37mm,\kern0.15mm y\kern0.37mm)\mapsto\upint\kern0.37mm\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 z\rmdss11\mu}
where with given \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+} we have \mathss38{
z = (\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\seqss33{(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta-y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi\kern-0.2mm:\eta\in A} }, \,and \math{
\rmmd = \seqss30{ (\kern0.37mm 1 + t\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t : t \in \lbb R_+ \kern-0.3mm } }
hence \math{\lbb R_+\to[\KPp1.1 0\,,\kern0.07mm 1\KPt9{[\,} } given by \mathss34{
t \mapsto (\kern0.37mm 1 + t\kern0.37mm)^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,t }.
To get Hausdorff topologies for the spaces \math{F} and \mathss32{F\aar 1
}, \,we finally take the quotient space by the vector subspace \math{N\aar 0}
of functions \math{x} with \math{\int_{\,A}\kern0.15mm u\circss01 x\rmdss11\mu=0} for
all \linebreak
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and for all \mathss38{
u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }.
Observe that if with \math{\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} we have for example \mathss38{\vPi=
\LLrs42^{\frac 12}(\ssbb44 I) }, \,then the dual set \math{
\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) =
\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} } and hence the spaces \math{F}
and \math{F\aar 1 \label{triv L^p exa}} become trivial. For $ \vPi =
\ell\KPt8^{\frac 12}\kern0.15mm(\kern0.37mm\mathbb No\kern0.07mm) \kern0.37mm $ the situation is different
since then \math{\vPi} has nontrivial dual.
\begin{constructions}[of generalized Lebesgue\,--\,Bochner spaces]\label{defi $L^p$} $\null$ \vskip.5mm
\begin{enumerate}\begin{myLeftskip}{-4}{.6}{.6}
\item \ $\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M=\uniqset\Cal V:0\le p\le\lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and \math{ \label{L^p nbhs}
\eexi{\bosy K\kern0.37mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi} } \newskline{20}
$\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} \kern0.37mm$ and $\kern0.37mm
\mu\kern0.37mm$ is a positive measure on $\kern0.37mm{}^{}\Cal Omega \kern0.37mm$ and \newfline
$ \vPi\in\tvsps0(K)\kern0.37mm$ and $\kern0.37mm \varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$
and $ [ \KP1 [ \KP{1.4} p = 0 \kern0.37mm$ and \KP{11.1} \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm,\kern0.07mm A\,,\kern0.07mm\varepsilon}\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi \kern0.37mm$
and $\kern0.37mm A\in\mu\invss23\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+\kern0.37mm$
and $\kern0.37mm \varepsilon\in\rbb R^+ \kern0.37mm$ and \newfline
$V=M\capss21\{\,x:
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\rmdss41\mu<\varepsilon\,\}
\kern0.15mm\} \KP{1.4} ] \kern0.37mm$ or $\kern0.37mm [ \KP{1.4} p\in\rbb R^+\kern0.37mm$ and \KP{8.3} \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\,V = M\capss21\{\, x :
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu<1\,\}\kern0.15mm\} \KP{1.4} ] $ \newfline
or $\kern0.37mm [ \KP{1.4} p = \lower1.05mm\hbox{$^+$}\infty\kern0.37mm$ and \KP7 \newskline6
$\Cal V=\{\KPt8 V\kern-0.63mm:\eexi{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\, V = M\capss21\{\,x:
\aall{A\in\mu\invss23\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}$ \newskline{16}
$\eexi{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }\,
\sup\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1 [\KP1 A\setminus N\KP1]\kern0.37mm) < 1 \KPt8\}\,\}
\KP{1.4} ] \KP{1.2} ] \KP{1.4} $,
\item \ $\raise1.7mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p\varXi\kern0.15mm\sbi{M\kern0.37mm\aars N_0}=\uniqset F: \label{preL^p_{MN_0}}
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and \newskline6
$
\aall{\mu\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm S\kern0.37mm,\kern0.15mm\scrmt T\kern0.15mm,\kern0.15mm\Cal V\kern0.15mm,
\kern0.15mm\scrmt V\aar 0\,,\kern0.15mm X\kern0.15mm,\kern0.15mm Y}\,
\varXi=(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$
and \newskline6
$X=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and $\kern0.37mm\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
\varXi\sbi M\kern0.15mm$ and \newskline6
$S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} \kern0.37mm$ and $\kern0.37mm Y=
X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0\kern0.37mm$ and \newskline6
$\scrmt V\aar 0 = \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms Y\capss01\{\,\smb X : \smb X\capss02 V\aar 1
\not= \emptyset \KP1 \} : V\aar 1\in\Cal V\KP1\} \kern0.37mm $ and \newskline6
$\scrmt T=
\{\KPt8 U\kern-0.3mm:\aall{\smb X\in U}\,\eexi{V\in\scrmt V\aar 0}\,
[\KP{1.1}\{\kern0.37mm\smb X\kern0.37mm\} + V\KP{1.2}]\vvs Y\subseteq U\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm Y\KP1\}$ \newskline6
$
\impss33 M\kern0.37mm$ is a vector subspace in $\kern0.37mm X\kern0.37mm$ and \newskline6
$N\aar 0\kern0.37mm$ is a vector subspace in $\kern0.37mm X_{\kern0.37mm|\,S}\kern0.37mm$ and
$\kern0.37mm F=
(\kern0.37mm Y\kern0.07mmp,\kern0.07mm\scrmt T\,) \KPt9 $,
\item \ $\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)=\uniqset F: \eexi{\bosy K}\, \label{simpL^p}
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\kern0.37mm$ and $\kern0.37mm\vPi\in\tvsps0(K)\kern0.37mm$ and \newskline{11}
$\aall{M\kern0.15mm,\kern0.15mm N\aar 0\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm S\kern0.37mm,\kern0.15mm
\Cal V\kern0.15mm,\kern0.15mm X}\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
X = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and \newskline{14}
$ M = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm $ is
finitely \newskline{50.5}
almost simply measurable $ \} \kern0.37mm $ and \newskline6
$\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M\kern0.15mm$ and $\kern0.37mm S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\}\kern0.37mm$ and \newskline6
$N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30}
$\impss03
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm
u\circ x\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\subseteq\{\kern0.37mm 0\kern0.37mm\}\,\} $ \newfline
$\impss03 \Cal V\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and $\kern0.37mm F=
\raise1.8mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi{M\kern0.37mm\aars N_0} \,$, \KP9
\item \ $\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)=\uniqset F: \eexi{\bosy K}\, \label{ctr mvL_s^p}
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\kern0.37mm$ and $\kern0.37mm\vPi\in\tvsps0(K)\kern0.37mm$ and \newskline{11}
$\aall{M\kern0.15mm,\kern0.15mm N\aar 0\,,\kern0.15mm{}^{}\Cal Omega\,,\kern0.15mm S\kern0.37mm,\kern0.15mm
\Cal V\kern0.15mm,\kern0.15mm X}\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\kern0.37mm$ and $\kern0.37mm
X = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.15mm$ and \newskline{14}
$ M = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm $ is
finitely \newskline{49}
almost scalarly measurable $ \} \kern0.37mm $ and \newskline6
$\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M\kern0.15mm$ and $\kern0.37mm S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\}\kern0.37mm$ and \newskline6
$N\aar 0=S\capss21\{\,x:\aall{A\,,\kern0.15mm u}\,\eexi N\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.15mm$ and $\kern0.37mm u\in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)$ \newskline{30}
$\impss03
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.37mm$ and $\kern0.37mm
u\circ x\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.15mm A\setminus N\kern0.37mm)\subseteq\{\kern0.37mm 0\kern0.37mm\}\,\} $ \newfline
$\impss03 \Cal V\not=\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm$ and $\kern0.37mm F=
\raise1.8mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi{M\kern0.37mm\aars N_0} \,$, \KP9
\item \ $\mLrs03^p(\kern0.37mm\mu\kern0.15mm) =
\mvLrs03^p(\kern0.37mm\mu\,,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(6) \ $\mLrs03^p(\kern0.37mm\mu\kern0.15mm)\lfbb_C =
\mvLrs02^p(\,\mu\,,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $, \KP{13} \inskipline0{-2}
(7) \ $\suptext{vc}0\Lrs03^p(\vcal Q\kern0.15mm) = \uniqset F:\eexi{\bosy K}\,\vcal Q\kern0.37mm$
is a quasi\kern0.37mm-\kern0.37mm\erm Euclidean \mathss37{\bosy K}--\,vector column \newskline{17}
and $\,\aall{\ell\,,\kern0.15mm\smb N\kern0.15mm,\kern0.15mm\mu\,,Q\kern0.37mm,\kern0.15mm\Yps\kern0.07mm,\kern0.07mm\vPi}\,
\vcal Q = (\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm$ and $\kern0.37mm
\smb N \in \mathbb No \kern0.37mm$ and \newfline
$\ell\in\Lis(\,\Yps\Reit0\kern0.37mm,\tvbbR5^{\ssmb N}\kern0.37mm\big)\kern0.37mm$ and $\kern0.37mm
\mu = \seq{\KP1 r : A \subseteq Q \kern0.37mm$ and $\kern0.37mm
r = \Lebmef^{\kern0.37mm\ssmb N}\!\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\ell\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,) \KP1 } $ \KP{8.95} \newfline
$\impss02 \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} Q \not = \hbox{\font\SweD =cmssbx10\SweD U}{} \kern0.37mm$ and $\kern0.37mm
F = \mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP9 \inskipline0{-2}
(8) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) =
\suptext{vc}0\Lrs03^p(\kern-0.3mm(\kern0.15mm Q\kern0.37mm,\Yps\kern0.15mm,\kern0.07mm\vPi\kern0.37mm)\kern-0.3mm) \KP1 $, \inskipline0{-2}
(9) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps) =
\LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(10) \ $\LLrs03^p(\kern0.15mm Q\,\sbi\Yps)\lfbb_C =
\LLrs03^p(\kern0.37mm Q\,\sbi\Yps\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $, \KP{11.6} \inskipline0{-3.75}
(11) \ $\LLrs03^p(\kern0.15mm Q\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) = \uniqset F:\eexi\Yps\,\Yps\kern0.37mm$ is
quasi\kern0.37mm-\kern0.15mm usual over $\kern0.37mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm$ \newfline
and $\kern0.37mm Q\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\Yps\kern0.37mm$ and $\kern0.37mm
F = \LLrs03^p(\kern0.15mm Q\,\sbi\Yps\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) \KP1 $, \KP{17.4} \inskipline0{-3.75}
(12) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm) =
\LLrs03^p(\kern0.15mm Q\kern0.37mm,\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KP1 $,
(13) \ $\LLrs03^p(\kern0.07mm Q\kern0.15mm)\lfbb_C = \label{df $L^p(Q)_C$}
\LLrs03^p(\kern0.37mm Q\kern0.37mm,(\kern0.15mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\Reit1) \KP1 $. \KP{16.9}
\end{myLeftskip}\end{enumerate}
\end{constructions}
\begin{theorem}\label{L^p in TVS}
Let \ú$\,0\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure. With \ú$\,
\bosy K \in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} $ also let \ú$\,\vPi\in\tvsps0(K)$ and either \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ or \ú$ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KPt8 $. Then \ú$\,F\in{\kern-0.63mm}$ $
\roman{TVS}\kern0.4mmps0(K)$ holds. If in addition \ú$\,1\le p$ and $\,\vPi$ is almost
suitable{\kern0.15mm\rm, }then \ú$\,F\in\roman{LCS}\kern0.4mmps0(K)$ holds with $\,F$ normable.
Furthermore{\kern0.15mm\rm, }for $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ any dominating norm for $\,\vPi$ it holds
that $\,\seqss43{\inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\kern-0.2mm
: x\in\smb X\,\} : \smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} $ is a compatible norm for $\,F\kern0.15mm$.
\end{theorem}
\begin{proof} Let \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs} } and \vskip.5mm\centerline{$
M=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm X\capss01\{\,x : (\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \kern0.37mm \text{ is
finitely almost S measurable } \} $} \inskipline{.5}0
where S stands for either \q{simply} or \q{scalarly}. Then \math{X} is a
vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}, \,and it is a straightforward
standard exercise (\kern0.15mm to the reader\kern0.15mm) to verify that \math{M} is a vector
subspace in \mathss31{X}. So \math{X_{\kern0.37mm|\,M}} is a vector structure over \mathss32{
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. Now for \math{\Cal V=
\roman{Leb}\sbi{\sixroman n\fiveroman{bh}}\!\RHB{.3}{^p}\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.37mm\sbi M} and \math{S = \kern0.37mm
\bigcap\kern0.37mm\big\{\KP{1.1}[\KP{1.2}\mathbb Z\KP{1.1}V\KP{1.1}]\vvs X\kern-0.3mm:
V\in\Cal V\KP1\} } we first see that \ú$\kern0.37mm S\subseteq{\kern-0.63mm}$ \linebreak
$M\kern0.37mm$ holds and
that \math{S} is a vector subspace in \mathss31{X}. Hence \math{X_{\kern0.37mm|\,S}}
is a vector structure over \mathss32{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K}. For the set \math{
\Cal V\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 S} in \math{X_{\kern0.37mm|\,S}} one verifies that the properties
(\kern0.15mm\erm{NB}\,1\kern0.15mm) and (\kern0.15mm\erm{NB}\,2\kern0.15mm) given in \cite[p.\ 33]{Jr} hold.
Indeed, for given \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} utilizing the short- \linebreak
hands \math{
\|\,x\,\|\subnu=\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu } and \mathss38{
\trN03\smb X\trNu2=\inf\kern0.15mm\big\{\KPt8\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\}
}, \,in the case \ú$\kern0.37mm p\not=0$ \linebreak
from Proposition \ref{Pro upint} on page \pageref{Pro upint}
above, putting \math{ \smb M = \smb A \kern-0.3mm \cdot
\sup\KPt8\{\,1\kern0.37mm,\kern0.07mm 2\KP1^{p^{-1}-\kern0.37mm 1\kern0.37mm}\big\} }
$\null
$ where \math{\smb A} is as on line 3 in (\ref{defi bqnor E}) on
page \pageref{defi bqnor E p} above, we first see \linebreak
that \math{
\|\KP1(\kern0.37mm x + y\kern0.37mm)\svs X\,\|\subnu \kern0.15mm \le \kern0.37mm
\smb M\KPt8\big(\kern0.37mm\|\,x\,\|\subnu + \|\,y\,\|\subnu\kern0.15mm) } holds for all \mathss31{
x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mms X}. This gives (\kern0.15mm\erm{NB}\,2\kern0.15mm) and
(\kern0.15mm\erm{NB}\,1\kern0.15mm) follows trivially from the property given on line 2 in
(\ref{defi bqnor E}) above. In the case \math{p=0} again (\kern0.15mm\erm{NB}\,1\kern0.15mm)
is trivial, and (\kern0.15mm\erm{NB}\,2\kern0.15mm) is seen by observing that we have
\math{1\le\smb A} and hence for all
\math{x\kern0.37mm,\kern0.15mm y\in S} and \math{\eta\in{}^{}\Cal Omega} it holds that \inskipline{.5}{19}
$ \rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm x + y\kern0.37mm)\vvs X\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
\le \rmmd\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb A\KPt8(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta + \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)) $ \inskipline{.2}{53}
${} \le \smb A\KPt8(\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta +
\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) $ \inskipline{.5}0
whence further $\,\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00
(\kern0.37mm x + y\kern0.37mm)\vvs X\,|\KP1 A\rmdss41\mu$ \inskipline{.4}{25}
${} \le \smb A\,\big(\kern0.37mm
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\rmdss41\mu\kern0.15mm + \kern-0.3mm
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 y\KP1|\KP1 A\rmdss41\mu\kern0.37mm) \KP1 $. \inskipline{.5}0
Consequently, we see that there is a unique vector topology \math{
\scrmt T\aR 1} for \math{X_{\kern0.37mm|\,S}} such that with \math{E =
(\kern0.15mm X_{\kern0.37mm|\,S}\kern0.37mm,\kern0.07mm\scrmt T\aR 1)} we have \math{\Cal V\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 S} a
filter base for \mathss35{\neiBoo E}.
Now letting \math{N\aar 0} be as on lines 6--7 in
Constructions \ref{defi $L^p$}\,(\ref{simpL^p}) or (\ref{ctr mvL_s^p})\,, it
is a simple matter to verify that \math{N\aar 0} is a vector subspace in \mathss37{
X_{\kern0.37mm|\,S}}. So for \math{Y= X_{\kern0.37mm|\,S}\,/\vsquotient N\aar 0} and \linebreak
\ú$F\aar 1=E\,/\tvsquotient N\aar 0\kern0.37mm$ we have \math{F\aar 1} a topological
vector space over \math{\bosy K} with \ú$\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\aar 1=Y\kern0.07mmp$. Let- ting \math{
\scrmt V\aar 0} and \math{\scrmt T} be as on lines 5--6 in
Constructions \ref{defi $L^p$}\,(\ref{preL^p_{MN_0}})\,, we have \math{ F =
(\kern0.37mm Y\kern0.07mmp,\kern0.07mm\scrmt T\,)} and from Lemma \ref{Le qtvs} on
page \pageref{Le qtvs} above we see that \math{\scrmt V\aar 0} is a filter
base for \mathss34{\neiBoo F\aar 1}, \,and hence \math{ F = F\aar 1 \in
\tvsps0(K) } holds.
To prove that \math{F\in\roman{TVS}\kern0.4mmps0(K)} holds, we need to show that \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}
is a Hausdorff topology. For this,
arbitrarily fixing \mathss38{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\setminus\{\,\Bnull_F\}
}, \,in the case \math{p\not=0} it
suffices to show existence of some \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi} such that \math{
\trN03\smb X\trNu2\not=0 } holds.
To proceed, fixing any \mathss30{x\ar 0\in\smb X}, \,there are some
\math{u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and
\math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ }
with $\int_{\,A}\kern0.37mm u\circss00 x\ar 0\rmdss01\mu\not=0$ and
hence also
$\int_{\,A\,}|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\rmdss11\mu\not=0$ .
Consequently for \math{A\kern0.15mm\ar 1=
A\capss31\{\,\eta:
|\KP1 u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1| \not= 0 \KP1\} }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm=\kern0.37mm
\big\langle\KP1|\KP1 u\fvalss02\xi\KP1|:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\KP1\rangle }
now \math{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1 > 0} and
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S_{_N}\kern0.15mm\vPi\subseteq\Bqnorm\vPi } hold.
For every \math{x\in\smb X} and
\math{B\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} we have
$\int_{\KPt8 B\,}u\circss00 x\rmdss11\mu
=\int_{\KPt8 B\,}u\circss00 x\ar 0\rmdss01\mu$
and hence there is some $N
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$u\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
=u\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ and hence also
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$
holds for all $\eta\in A\kern0.15mm\ar 1\kern-0.63mm\setminus N$ .
In the case
\math{p\not=0} we hence
get \math{
0<\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\ar 0\,|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
=\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
\le\|\,x\,\|\subnu } .
Since this holds for arbitrarily
given \math{x\in\smb X} we
consequently
obtain
\mathss38{0<\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\ar 0\,|\KP1 A\kern0.15mm\ar 1\,\|\Lnorss33^p_\mu
\le\inf\kern0.15mm\big\{\KPt8\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\}=
\trN03\smb X\trNu2 }.
In the case \math{p=0} the above deduction gives \vskip.5mm\centerline{$
0 < \int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\rmdss01\mu
= \int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss11\mu
= \upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\kern0.15mm\ar 1\rmdss01\mu $} \inskipline{.3}0
for all \math{x\in\smb X} and hence taking \math{ \varepsilon =
\int_{\,\aars A_1\kern-0.3mm}\rmmd\circ\kern0.07mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\ar 0\rmdss01\mu } and \inskipline{.2}{7.3}
$V\aar 1 = M\capss21\{\,x :
\upint\kern0.37mm\rmmd\kern-0.2mm\circ\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 A\kern0.15mm\ar 1\rmdss01\mu
< \varepsilon \KPt8 \} $ \inskipline{.4}0
and \math{V\aar 0 = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X : \smb X\capss02 V\aar 1
\not= \emptyset \KP1 \} } we have \mathss36{ \smb X \kern-0.3mm\not\in\kern0.15mm V\aar 0
\in \scrmt V\aar 0 }. \vskip.3mm
Finally assuming that also \math{1\le p} holds and that \math{\vPi} is almost
suitable, we fix any dominating norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss31{\vPi}. Then by
Lemma \ref{Le suit dom} on page \pageref{Le suit dom} above, we see that the
set \math{ \{\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X\kern-0.3mm:n\KPt8\trN03\smb X\trNu2 < 1\KPt8
\} : n\in\rbb Z^+\kern0.37mm\big\} } a filter base for \mathss31{\neiBoo F}.
Consequently $\kern0.37mm F$ \linebreak
is locally convex and normable with a compatible norm as
asserted. Note that we get the triangle inequality \math{
\trN04(\kern0.37mm\smb X + \smb Y\,)\vvs Y\trNu2
\le \trN03\smb X\trNu2 + \trN03\smb Y\trNu4 } for \math{\smb X\kern0.15mm,\kern0.15mm\smb Y\kern0.15mm
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F } from \vskip.4mm\centerline{$
\inf\,\{\,\|\,z\,\|\subnu\kern-0.2mm:z\in(\kern0.37mm\smb X+\smb Y\,)\vvs Y\,\}
\le \inf\,\{\,\|\,x\,\|\subnu\kern-0.2mm:x\in\smb X\,\} +
\inf\,\{\,\|\,y\,\|\subnu\kern-0.2mm:y\in\smb Y\KP1\} \KP1 $,} \inskipline{.4}0
and that the implication \math{\trN03\smb X\trNu2=0\impss33 \smb X=\Bnull_F}
holds for all \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} since we already know that \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} is a Hausdorff topology.
\end{proof}
\begin{lemma}\label{Le 0_{L^p}}
Let \ú$\,0 \le p \le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure. With \ú$\,
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ also let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.15mm\rm, }and
let \ú$\,F=\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and \ú$\,x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$
and \ú$\,y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$. Then \ú$\,y\in\smb X$ holds if and only
if for every $\,A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ there is some $\,
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ with $\,
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y \, $.
\end{lemma}
\begin{proof} The asserted sufficiency being trivial, we only verify
necessity. So letting \math{y\in\smb X} and \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+ } we need to get some \math{N\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y }. Now we first find some \math{
N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and simple sequences
\math{\bosy\sigma\aR 1} and \math{\bosy\sigma\aR 2} in
\newline
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm A\kern0.07mm\setminus N\aar 1)\,,\kern0.07mm\vPi\kern0.37mm) }
with \math{
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy\sigma\aR 1\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } and \math{
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy\sigma\aR 2\to y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \mathss34{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \math{\eta\in A\kern0.07mm\setminus N\aar 1}. Then
letting \math{S} be the linear
\mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,span of \math{
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm\bosy\sigma\aR 1\kern-0.2mm\cup\kern0.37mm\bosy\sigma\aR 2\kern0.07mm) } we
have
\math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S } a separable topology, and for \math{B\ar 1}
the closed unit dual ball corresponding to some fixed compatible norm for
\math{\vPi} and for \math{\scrmt T=
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.07mm\vPi_{\kern0.37mm/\,S}\kern0.07mm)\dlsigss00\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 B\ar 1 } hence
by \cite[Proposition 8.5.3\kern0.37mm, p.\ 157]{Jr} we see that
\math{\scrmt T} is a
metrizable and separable topology. Let then
\math{D} be countable and \mathss37{\scrmt T}--\,dense. Now
by {\sl Hahn\,--\,Banach\kern0.15mm} for
every fixed \math{u\in D} and for all \math{
B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm(\kern0.15mm A\kern0.07mm\setminus N\aar 1) } we have \math{
\int_{\KP1 B\,}u\circss00 x\rmdss11\mu =
\int_{\KP1 B\,}u\circss00 y\rmdss11\mu } and hence there is some
\math{N\kern0.15mmrim1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
u\circss00 x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.15mmrim1\kern0.15mm)\subseteq
u\circss00 y }. By {\sl countable choice\kern0.15mm} taking the union of these
\math{N\kern0.15mmrim1} for
\math{u\in D} we obtain \math{N} with
\math{N\aar 1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and \mathss34{
u\circss00 x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq
u\circss00 y } for all \math{u\in D}. Then to get \math{
x\KP1|\KP1(\kern0.15mm A\kern0.07mm\setminus N\kern0.37mm)\subseteq y } arbitrarily fixing \math{\eta\in
A\kern0.07mm\setminus N } and \math{v\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }
by {\sl Hahn\,--\,Banach\kern0.15mm} it suffices to have
\mathss34{v\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=v\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta}. Now we find some
\math{\bosy u\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,D } with
\math{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\bosy u \to v\fvalss01\xi }
for all
\newline
\math{\xi\in\{\,x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm,\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KPt8\} } and then we get
\inskipline{.2}{21}
$ v\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
= \lim\,(\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm}\eta}\kern-0.2mm\circ\kern0.15mm\bosy u\kern0.37mm)
= \lim\,(\kern0.37mm\roman{ev}\kern0.15mm\sbi{y\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm}\eta}\kern-0.2mm\circ\kern0.15mm\bosy u\kern0.37mm)
= v\circss00 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta \, $.
\end{proof}
From Lemma \ref{Le 0_{L^p}} we see in particular that in the case where \math{
\mu} is \rsigma5finite, elements \mathss03{x\kern0.37mm,\kern0.15mm y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
represent the same vector of \math{F} if and only if they are equal almost
everywhere in the classical sense. In the case \math{p\not=0} even without
\rsigma5finiteness we also see that corresponding to any given compatible norm \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \math{\vPi} we have the equality \math{
\inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 z\KP1\|\Lnorss33^p_\mu \kern-0.2mm : \label{discus inf N = N}
z\in\smb X\,\} = \|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu } for \mathss31{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. \vskip.3mm
On page \pageref{triv L^p exa} above we noted that for \math{ 0 \le p \le
\lower1.05mm\hbox{$^+$}\infty } and e.g.\ for \ú$\kern0.37mm F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ \linebreak
with \math{\vPi=\LLrs42^{\frac 12}(\ssbb44 I) } we have \math{F} trivial in
the sense that \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\{\,\Bnull_F\} } holds. However, in
Constructions \ref{defi $L^p$}\,(\ref{preL^p_{MN_0}}) taking \math{ M =
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\Bnull_F} and \vskip.5mm\centerline{$
N\aar 0 = M\capss21\{\,x:\aall{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm\vPi}\,
\upint\kern0.37mm \Abrs33^p\circss00\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\rmdss41\mu = 0\KP1\} $} \inskipline{.5}0
we generally get a nontrivial space \math{ E =
\raise1.7mm\hbox{\font\≈=cmssi5\≈pr}\kern-.3mm
\LLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\kern0.15mm\sbi{M\kern0.37mm\aars N_0} } such that
e.g.\ for \ú$\kern0.37mm p=\frac 12$ \linebreak
and \math{\mu=\Lebmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI } the
spaces \math{E} and \math{\LLrs03^p(\ssbb40 I\times\ssbb04 I) } become
naturally linearly homeomorphic. We leave the proof as an exercise to the
reader.
\begin{theorem}\label{Th L_s^p Ba}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in
\roman{LCS}\kern0.4mmps0(K)$ be suitable. Let $\,\mu$ be a positive measure such that in the
case \ú$\,p=\lower1.05mm\hbox{$^+$}\infty$ it holds that $\,\mu$ is almost decomposable. Then $\,
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \in \roman{BaS}\kern0.4mmps0(K) $ holds. If in addition $\,
\vPi$ is \eit Ba- nachable{\kern0.15mm\rm, }then also $\,
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \in \roman{BaS}\kern0.4mmps0(K) $ holds.
\end{theorem}
\begin{proof} We give the proof for \math{\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
and leave it as an exercise to the reader to make the slight modifications
that are needed to get the assertion related to \math{
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } that is classical in the cases where \math{
p\not=\lower1.05mm\hbox{$^+$}\infty} holds or \math{\mu} is \rsigma5finite. For hint we only
mention that \cite[Theorem 4.2.2\kern0.37mm, p.\ 95]{Du} and
\cite[Corollary 4.2.7, p.\ 97]{Du} together can be utilized to deduce that for
the obtained \math{y} it then holds that \mathss03{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is finitely almost simply measurable.
Now we put \math{{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \math{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.37mm]_{vs} } and \mathss38{ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)}, \,and let \linebreak
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.37mm$ be any dominating
norm for \mathss31{\vPi}. By Theorem \ref{L^p in TVS} only completeness of \math{
F} has to be verified. For this, it suffices to show that
for any \PouN$\sevib X \in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ with
\PouN$\trN05\sevib X\fvalss32 i\trNu6 <
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}$ for all $i\in\mathbb No\,$, the sequence
\math{\sevib Y=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm F\text{\,-\kern0.15mm}\sum_{\,k\kern0.37mm\in\kern0.37mm i^+\kern0.37mm}(\kern0.15mm
\sevib X\fvalss32 k\kern0.37mm):i\in\mathbb No} } converges in the
topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$. In order to get this, we first take some $\bosy x \in
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.15mm\sevib X$ with
$\|\KP1\bosy x\fvalss12 i\KP1\|\subnu <
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}$ for all $i\in\mathbb No\,$,
and put $\bosy y=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\vPi\text{\,-\kern0.15mm}\sum_{\,k\kern0.37mm\in\kern0.37mm i^+\kern0.37mm}(\kern0.37mm
\bosy x\fvalss12 k\kern0.37mm):i\in\mathbb No}\,$.
First considering the case \mathss35{p=\lower1.05mm\hbox{$^+$}\infty}, \,letting \math{\scrmt A}
and \math{N\kern0.15mmrim1} be as in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above, let \math{\scrmt N\ar 1} be the set of all \math{(\kern0.15mm A\,,\kern0.07mm N\aar 1) }
with \math{A\in\scrmt A} and \ú$\kern0.37mm N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} $ \linebreak
and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for all \math{i\in\mathbb No} and \math{
\eta\in A\kern0.15mm\setminus N\aar 1}. Then we have \mathss32{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N\ar 1}, \,and hence by the {\sl axiom of choice\kern0.15mm} there is a
function \ú$\kern0.37mm\scrmt N\subseteq\scrmt N\ar 1$ \linebreak
with \mathss35{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt N}. Now taking \mathss35{N\kern0.15mmrimm1 =
N\kern0.15mmrim1\cupss24\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\scrmt N }, \,we see that \math{N\kern0.15mmrimm1} is \mathss37{
\mu}--\,negli- gible and such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)} } holds for \math{i\in\mathbb No} and \mathss32{
\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1}.
Letting \math{\vPi\ar 0} be the Banach{\sl able\kern0.15mm} space determined by the
norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss30{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}, \,from the above we see that for
every fixed \math{\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1 } the sequence \math{
\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y } converges in the topology \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0} and hence also in the weaker topology \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}.
Taking \vskip.3mm\centerline{$
y = N\kern0.15mmrimm1\kern-0.3mm\times\kern-0.2mm\{\,\Bnull_\vPi\} \cupss22 \{ \,
(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm):\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mmrimm1\kern0.15mm\text{ and }\kern0.37mm
\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y\to\xi\kern0.37mm\text{ in top }\kern0.37mm
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0\,\} \KP1 $,} \inskipline{.3}0
by Lemma \ref{Le deco meas} on page \pageref{Le deco meas} above \math{
(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } is finitely almost scalarly measurable. It
is also a simple exercise to see that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} holds, and
that for the unique class \math{\smb Y} with \math{y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} we
indeed have \math{\ebit Y\to\smb Y} in top \mathss30{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}.
Next, for the case \mathss35{p<\lower1.05mm\hbox{$^+$}\infty}, \,we choose a sequence \math{
\bosy u} of fully positive \mathss37{\mu}--\,mea- surable functions such that
for all \math{i\in\mathbb No} and \math{\eta\in{}^{}\Cal Omega} we have
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\eta\le
\bosy u\fvalss01 i\fvalss10\eta$ and $
\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$. Putting
$\roman A\,i=\{\,\eta:
\bosy u\fvalss01 i\fvalss10\eta
\ge 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}\,\big\}$ and
$\roman B\,i=
\bigcup\,\{\,\roman A\,j:i\subseteq j\in\mathbb No\,\}\,$, for $i\in\mathbb No$ we
have $
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\roman A\,i\cdot 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}
\le\int_{\KP1\roman A\kern0.37mm i}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
\le\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\aabs99^p\circss01(\kern0.37mm
\bosy u\fvalss01 i\kern0.37mm)\rmdss11\mu
< 4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$
and hence $
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\roman A\,i < 2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)\,p}\,$, whence
further $
\mu\fvalss13\roman B\,i < 2\KP1^{(\kern0.15mm 1\kern0.37mm-\kern0.37mm(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm))\,p}\,$,
and
consequently for
$N=\bigcap\,\{\KP1\roman B\,i:i\in\mathbb No\,\}\,$, we get $
N\in\mu\invss34\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm 0\kern0.15mm\}\,$. For each fixed $\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ there
is $i\ar 0\in\mathbb No$ with $\eta\not\in
\roman A\,i$ for all $i\in\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$. Hence for $i\in
\mathbb No\kern-0.3mm\setminus\kern0.07mm i\ar 0\,$, we have
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ(\kern0.37mm\bosy x\fvalss01 i\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\eta\le
\bosy u\fvalss01 i\fvalss10\eta <
2^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern0.15mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.07mm)}\kern0.15mm$, and consequently the sequence $
\,\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y$
converges in the topology $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\aar 0\,$. It follows that
there is a function $y:{}^{}\Cal Omega\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ with $y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\Bnull_\vPi$ for $
\eta\in N$ and
$\roman{ev}\sbi\eta\circ\kern0.07mm\bosy y\to y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ in top
$\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ for $\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.15mm$. This immediately
gives that
$(\kern0.37mm u\circss01 y\,;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is finitely
measurable for every $u\in\Cal L\,(\kern0.07mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\,$.
To show that $y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$, we must verify that
$\upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 y\rmdss11\mu < \lower1.05mm\hbox{$^+$}\infty$
holds.
For each fixed $\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ we have
\vskip1mm
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\text{\KP1-}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm))) $ \inskipline{.7}{15.5}
${}
= \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\text{\KP1-}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm)))$ \inskipline{.7}{15.5}
${}
\le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circ\kern0.15mm\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy x\KP1|\KP1 i\kern0.37mm)
\le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)$
\vskip1mm
\noindent
and hence by Fatou's lemma we get \vskip1mm
$ \upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10 y\rmdss11\mu
\le
\int_{\KP{1.1}{}^{}\Cal Omega\,}(\,
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}=\int_{\KP{1.1}{}^{}\Cal Omega\,}
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty}\int_{\KP{1.1}{}^{}\Cal Omega\kern0.15mm}
\big(\kern0.15mm\sum\KP1(\kern0.37mm
\roman{ev}\kern0.07mm\sbi\eta\kern-0.2mm\circ\kern0.15mm\bosy u\KP1|\KP1 i\kern0.37mm)
\sbig)3\RHB{.3}{^{\,p}}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{30.2}
${}\le
\liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\big(\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}
(\kern0.37mm\bosy u\fvalss01 k\fvalss10\eta\kern0.37mm)\RHB{.3}{\KP1^p}
\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm))\KP1^{p^{-1}}\big)
\RHB{.7}{\,^p}$ \inskipline{.7}{30.2}
${} \le \liminf\sbi{i\kern0.37mm\to\kern0.37mm\infty\,}\big(\kern0.15mm
\sum_{\,k\kern0.37mm\in\kern0.37mm i\,}4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern0.07mm\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\big)
\RHB{.7}{\,^p}
= \big(\kern0.15mm\frac43\kern0.15mm\big)\RHB{.7}{\,^p}<\lower1.05mm\hbox{$^+$}\infty$ . \inskipline10
So we have $y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.15mm$, and hence there is
$\smb Y$ with $y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$. It remains to show that
$\sevib Y\to\smb Y$ in top $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F$. For this, similarly as above, we
compute \vskip1mm
$ \upint\kern0.37mm\aabs99^p\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss10
(\kern0.37mm\bosy y\fvalss01 i - y\kern0.37mm)\vvs X\rmdss11\mu
\le \int_{\KP1{}^{}\Cal Omega}\,(\,\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta
\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline1{30}
${}=\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}\bosy u\fvalss01 k\fvalss10\eta\,)
\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\int_{\KP1{}^{}\Cal Omega}\kern0.37mm\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
\bosy u\fvalss01 k\fvalss10\eta\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+}\big(\int_{\KP1{}^{}\Cal Omega}\kern0.37mm(\kern0.37mm
\bosy u\fvalss01 k\fvalss10\eta\,)\RHB{.3}{\KPt8^p}\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
)\RHB{.3}{\KPt8^{p^{-1}}}\kern0.15mm\big)^{\,p}$ \inskipline1{15}
${}\le\liminf_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}
= \lim_{\,j\kern0.37mm\to\kern0.37mm\infty}\big(\sum_{\KPt8
k\kern0.37mm\in\kern0.37mm j\kern0.37mm\setminus\kern0.37mm i\kern0.15mm^+\,}
4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm k\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}$ \inskipline1{15}
${}=(\kern0.37mm 3^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,4^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}(\kern0.15mm i\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm)}\kern0.15mm\big)\RHB{.7}{\,^p}
\to 0 \,$ as $\, i\to\infty \KPt7 $, \,whence the assertion.
\end{proof}
\begin{corollary}\label{Cor L^p Ban}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ with \ú$\,\bosy K\in
\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\KPt7$. Let $\,\mu$ be a positive measure such that in the case \ú$\,p=
\lower1.05mm\hbox{$^+$}\infty$ it holds that $\,\mu$ is decomposable. Also let \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) ${\KP1\rm, }or let \ú$\, F =
\mvLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\,\vPi$ reflexive or $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ a separable topology. Then \ú$\,F\in\roman{BaS}\kern0.4mmps0(K)$ holds. Furthermore \ú$\,
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) =
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ holds when the space $\,\vPi$ is
reflexive.
\end{corollary}
\begin{proof} The first alternative is immediate. For the second in the
separable case we note that by Proposition \ref{pro-mea-equ} on page \pageref{pro-mea-equ}
for \math{\vPi\aar 1=\vPi\dualsigma0} we have \mathss37{ F =
\mvsLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\aar 1\kern0.07mm) }. Since the conditions of Theorem \ref{Th L_s^p Ba}
for \math{\vPi\aar 1} in place of \math{\vPi} hold true, consequently the
assertion follows.
For the reflexive case putting \mathss37{ F\aar 0 =
\LLrs02^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }, \,it suffices to verify that \math{
F=F\aar 0} holds, and this in turn follows if \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} can
be established. Trivially every \math{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} is contained in
some \mathss30{\smb Y\aR 1\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. For the converse, letting \mathss30{ y
\in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \mathss30{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+
}, \,there is \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
(\kern0.37mm y\KP1|\KP1 B\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is
simply measurable for \mathss30{B=A\kern0.07mm\setminus N}. Hence by Proposition \ref{Pro rfx si mea}
on page \pageref{Pro rfx si mea} above also \mathss03{
(\kern0.37mm y\KP1|\KP1 B\,;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is
simply measurable, and so \math{
(\kern0.37mm y\KP1|\KP1 A\KPt8;\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is
almost simply measurable. Having here \math{A} arbitrary, consequently \math{
y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 0} holds, and we are done.
\end{proof}
Note that by Banach\,--\,Steinhaus for the second alternative in \label{non Ban but Bai}
Corollary \ref{Cor L^p Ban} we could weaken the assumption that \math{
\vPi\in\roman{BaS}\kern0.4mmps0(K)} hold to requiring \ú$\kern0.37mm\vPi\in\roman{LCS}\kern0.4mmps0(K)$ \linebreak
with \math{\vPi}
normable and barrelled. For an example of an incomplete normable barrelled
space, see e.g.\ \cite[5.7.\kern0.15mm\erm B\kern0.37mm, p.\ 97]{Jr}\,. \vskip3mm
Since in \cite[10.7, p.\ 214]{Jr} the term {\sl quasi\kern0.37mm-\kern0.15mm normable\kern0.15mm} is
reserved for a different meaning, for Proposition \ref{Pro simp Lp dense}
below we here agree to say that a real or complex topological vector space \math{
E} is {\it pseudonormable\kern0.37mm} if{}f there is some \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Bqnorm E} with \mathss03{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss34{\neiBoo E}.
Now the Hausdorff quotients of pseudonormable spaces correspond to the locally
bounded spaces in the following sense. If \math{E} is pseudonormable, then for \linebreak \mathss03{
F=E\,/\tvsquotient\kern-0.3mm\bigcap\KPt8\neiBoo E} we have that \math{F} is locally
bounded, and from \cite[Theorem 6.8.3\kern0.37mm, p.\ 114]{Jr} it follows existence
of \math{r\kern0.15mm,\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0} with \math{0 < r \le 1} and \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0 \in
\Cal S\sbi{\kern0.15mm\emath r\,}F } with \linebreak
\mathss03{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\subseteq\{\,\Bnull_F\} } and \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm}
: n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss30{\neiBoo F}. Now with \math{
\tweq=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm\smb X\kern0.15mm) :
x\in\smb X\,\} } taking \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 0\circ\kern0.15mm\tweq} we see that \mathss30{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S\sbi{\kern0.15mm\emath r\,}E } \linebreak
with also \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } a filter base for \mathss34{\neiBoo E}. Thus the
{\sl zero neighbourhoods of a pseudonormable space \math{E} are given by a
single continuous \mathss35{r}--\,seminorm\kern0.15mm} which is an \mathss35{r
}--\,norm if \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} is a Hausdorff topogy. In particular, the
Hausdorff pseudonormable spaces are precisely the locally bounded ones.
\begin{proposition}\label{Pro simp Lp dense}
Let $\,p\in\rbb R^+$ and let $\,\mu$ be a positive measure{\kern0.15mm\rm, }and with
$\,\bosy K\in{}$
\newline
$\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,\vPi\in\tvsps0(K)$ be pseudonormable. Also let
$\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and
\newline
$\,D=
{\kern-0.63mm}$ $ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,\smb X:
\eexi{x\in\smb X}\,x\kern0.37mm\text{ is simple in }\kern0.37mm
(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)\KPt8\} \KPt8 $. Then $\,D$ is $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\,
$--\,dense.
\end{proposition}
\begin{proof} Put \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and let \mathss31{x\in
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\in\Cal S\sbi{\kern0.15mm\emath r\,}\vPi} be such
that \math{
\{\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm{\big[\kern0.15mm} :
n\in\rbb Z^+\kern0.15mm\big\} } is a filter base for \mathss31{\neiBoo\vPi}. Then
we take
some fully positive
\mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\le\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm}
and \mathss36{
\int_{\KPp1.1{}^{}\Cal Omega\,}
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty}. Putting
\newline
\math{\roman A\,n=
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KP1 2\KP1^{\emath n\kern0.37mm - \kern0.37mm 1}\kern0.07mm,\kern0.07mm
2\KPt9^{\emath n}\kern0.37mm{[\kern0.15mm} } we have \math{\{\,\roman A\,n:n\in\ssbb03 Z\,\}
\subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \,and by
{\sl countable choice\kern0.15mm} we find \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
\math{
\bmii8 S\in\kern0.15mm^{\ssbb05 Z}\big(\KPt6^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm) } such that for
every \math{n\in\mathbb Z} we have \math{\bmii8 S\fvalss20 n} a
simple sequence in
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\roman A\,n\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm(\kern0.15mm\bmii8 S\fvalss20 n\kern0.37mm)
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}
for all \mathss30{\eta\in\roman A\,n\setminus N}. Then let
\math{\bmii8 S\ar 1\in\kern0.15mm^{\ssbb05 Z}\big(\KPt6^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm) } be
the unique one such that for all
\math{n\in\mathbb Z} and \math{i\in\mathbb No} and \math{
\sigma=\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i} we have \math{
\sigma\in\kern0.15mm^{\roman A\,\emath n}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and such that
for all \math{\eta\in
\roman A\,n} and \math{\xi\ar 1=
\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } and \math{\xi=
\bmii8 S\fvalss20 n\fvalss01 i\fvalss10\eta } we have
\math{\xi\ar 1=\xi} if \math{\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss11\xi <
2\KPt9^{\emath n} } holds, otherwise having \mathss32{
\xi\ar 1=\Bnull_\vPi}. Then for
every \math{n\in\mathbb Z} we have \math{\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n} a
simple sequence in
\math{(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\roman A\,n\kern0.37mm,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm(\kern0.15mm\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\kern0.37mm)
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for
all \mathss30{\eta\in\roman A\,n\setminus N}, \,and in addition \math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.15mm
\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i\kern0.37mm)
\le 2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \math{n\in\mathbb Z} and \mathss36{i\in\mathbb No}.
Letting \math{\roman E\,\sigma=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus\kern-0.3mm{{}^{}{\rm dom}\,{}_{{}^{}}}\sigma\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\sigma } we now take
\newline
\math{\bosy\sigma=
\seqss40{\roman E\KPt8\bigcup\KPt8\{\,
\bmii8 S\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} n\fvalss01 i:
i:n\in\mathbb Z\kern0.37mm\text{ and }\kern0.37mm|\,n\,|\suba\le i\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm\}:
i\in\mathbb No\kern-0.2mm} } thus obtaining a
simple sequence \math{\bosy\sigma} in
\math{(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } with
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \mathss30{\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N} with
\mathss36{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\fvalss10\eta\not=\lower1.05mm\hbox{$^+$}\infty}, \,and such that also
\math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\le
2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No}. Noting
\math{\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty } and that
from \math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} we get
\newline
\math{\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\,}(\kern0.37mm
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(
\kern0.37mm\bosy\sigma\fvalss01 i\fvalss10\eta - x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi\kern0.07mm
) = 0 } it now follows from the dominated convergence theorem
that \math{
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm
\int_{\KPp1.1{}^{}\Cal Omega\,}\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(
\kern0.37mm\bosy\sigma\fvalss01 i\fvalss10\eta - x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\svs\vPi
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) = 0 } holds, giving the conclusion.
\end{proof}
\begin{proposition}\label{Pro LpLp* dual}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,
{}^{}\Cal Omega${\,\rm, }and with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ and $\,
F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KPt8 $. Also let \inskipline0{23.4}
$\,F\aar 1=\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ or $\,
F\aar 1=\mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) \, $. For \vskip.5mm\centerline{$
\beta=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times\kern0.15mm\ssbb00 C\capss41\{\,
(\kern0.15mm\smb X\kern0.15mm,\kern0.15mm\smb Y\kern0.07mmp,\kern0.07mm t\kern0.37mm):\aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t=\int_{\KP1{}^{}\Cal Omega}\,y\,.\KPt8 x\rmdss11\mu\KPt9\} \KP1 $,} \inskipline{.5}0
then $\,\beta$ is a continuous bilinear map $\,F\kern0.37mm\sqcap\kern0.15mm F\aar 1\to
\bosy K$ \inskipline0{17}
with $\,\seqss40{\beta\,(\,\cdot\,,\smb Y\,) : \smb Y\kern0.15mm\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm}$
an injection.
\end{proposition}
\begin{proof} First we note that \math{\beta} is trivially a function since
the vectors of \math{F} and $\kern0.37mm F\aar 1$ \linebreak
are nonempty sets. Further, if we
know ($\kern0.15mm*\kern0.15mm$) that \math{\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\beta } holds,
then bilinearity is readily seen. So we only need to prove ($\kern0.15mm*\kern0.15mm$)
together with continuity and the last nondegeneracy assertion. For short let \mathss36{
\roman I\KP1 x\KPt8 y = \int_{\KP1{}^{}\Cal Omega}\,y\,.\KPt8 x\rmdss11\mu}.
For ($\kern0.15mm*\kern0.15mm$) arbitrarily given \math{x\kern0.37mm,\kern0.15mm x\ar 1\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
and \mathss31{y\kern0.37mm,\kern0.15mm y\ar 1\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}, \,we need to verify
that \math{\roman I\KP1 x\KPt8 y=\roman I\KP1 x\ar 1\,y\ar 1 \in \mathbb C}
holds. For this we first note that \math{\roman I\KP1 x\KPt8 y \in\mathbb C }
under the additional assumption that \math{(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm) }
is a simple mv\kern0.37mm-\kern0.15mm map. Indeed, in this case \mathss03{
\roman I\KP1 x\KPt8 y} a finite sum of expressions of the type \math{
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } where \math{\xi
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \mathss04{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+}. Noting that we
here have \vskip.5mm\centerline{$
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 A \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs24^{p\sast}\kern-0.63mm(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm
\bosy K\kern0.37mm) \subseteq
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs42^1(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm
\bosy K\kern0.37mm) \KP1 $,} \inskipline{.5}0
the assertion follows. Directly from the definition we then see that \mathss30{
\roman I\KP1 x\KPt8 y=\roman I\KP1 x\KPt8 y\ar 1} holds. Then considering the
general \math{x} first with \math{p\not=\lower1.05mm\hbox{$^+$}\infty} and taking a compatible
norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \math{\vPi} and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1} be the
corresponding dual norm, similarly as in the proof of Proposition \ref{Pro simp Lp dense}
above we find
a
simple sequence \math{\bosy\sigma} in
\math{(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and
some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
and a positive \mathss37{\mu}--\,measurable
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu <
\lower1.05mm\hbox{$^+$}\infty } and such that
\math{
\Abrs33^p\kern-0.2mm\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\le
2\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No} and also
\newline
\math{\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy\sigma
\to x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} for all \mathss30{\eta\in
{}^{}\Cal Omega\kern0.07mm\setminus N}. Then with
\newline
\math{A=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\invss46[\KPp1.1\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.07mm\setminus\{\kern0.37mm 0\kern0.37mm\}\KP1]
\kern0.37mm\setminus N } we take
a positive \mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} with
\newline
\math{\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^{\kern0.15mm p\sast\kern-0.3mm}_\mu < \lower1.05mm\hbox{$^+$}\infty }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm z\KP1|\KP1 A \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} for
\mathss38{z\in\{\KPt8 y\kern0.37mm,\kern0.15mm y\ar 1\kern0.15mm\} }.
For
\newline
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2=2\KP1\RHB{.3}{^p}\LHB{.2}{\kern0.15mm^{^{-1}}\kern0.37mm}
\Abrs33^p\LHB{.2}{\kern0.15mm^{^{-1}}}\KN{.5}\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\cdot\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} now
\math{
\seqss40{z\,.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)
\KP1|\KP1 A
:i\in\mathbb No} } converges
pointwise to
\newline
\math{z\,.\KPt8 x\KP1|\KP1 A } and is dominated by
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2}
for which H\"older's inequality gives
\newline
\mathss36{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 2\,\|\Lnorss33^1_\mu
\le
(\,2\kern0.15mm
\int_{\KPp1.1{}^{}\Cal Omega\,}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu
\kern0.37mm)\KP1\RHB{.3}{^p}\LHB{.2}{\kern0.15mm^{^{-1}}\kern0.37mm}
\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^{\kern0.15mm p\sast\kern-0.3mm}_\mu < \lower1.05mm\hbox{$^+$}\infty
}. Consequently, by the dominated convergence theorem we obtain \inskipline{.5}{14.8}
$\roman I\KP1 x\KPt8 y=
\int_{\,A}\kern0.37mm y\,.\KPt8 x\rmdss11\mu=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}
\int_{\,A}\kern0.37mm y\,.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\rmdss11\mu$ \inskipline{.5}{21.75}
${}=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}
\int_{\,A}\kern0.37mm y\ar 1\kern0.37mm.\KPt8(\kern0.37mm\bosy\sigma\fvalss01 i\kern0.37mm)\rmdss11\mu
=
\int_{\,A}\kern0.37mm y\ar 1\kern0.37mm.\KPt8 x\rmdss11\mu
=
\roman I\KP1 x\KPt8 y\ar 1\in\mathbb C \KPt9 $. \vskip.5mm
In the case \math{p=\lower1.05mm\hbox{$^+$}\infty} we modify the above deduction as follow.
Indeed, now we have \math{p^{\,*}=1} and taking
a positive \mathss37{\mu}--\,measurable \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} with
\math{\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern0.37mm\|\Lnorss33^1_\mu < \lower1.05mm\hbox{$^+$}\infty }
and \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm z \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1} for
\mathss38{z\in\{\KPt8 y\kern0.37mm,\kern0.15mm y\ar 1\kern0.15mm\} } we let \mathss38{\scrmt A=
\{\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\ar 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}
\kern0.07mm[\KP1 2\KP1^{\emath n\kern0.37mm - \kern0.37mm 1}\kern0.07mm,\kern0.07mm
2\KPt9^{\emath n}\kern0.37mm{[\kern0.15mm}:n\in\ssbb03 Z\,\} }. Then we find
\math{\smb M\in\lbb R_+} and \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
such that for \math{B=\bigcup\,\scrmt A\kern0.07mm\setminus N} we have
\newline
\mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 x\KP1|\KP1 B\le B\times\kern-0.2mm\{\kern0.37mm\smb M\,\} }. Now
with the notation \math{\roman E\KP1 x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\KPt8\Bnull_\vPi\}\cupss22
(\kern0.37mm x\KP1|\KP1 A\kern0.37mm) } a slight modification of
the above deduction gives us
\math{\roman I\KP1\roman E\KP1 x\,A\KP1 y=
\roman I\KP1\roman E\KP1 x\,A\KP1 y\ar 1 \in\mathbb C }
for all \mathss36{A\in\scrmt A}. Then again
by dominated convergence we obtain \inskipline{.4}{22}
$\roman I\KP1 x\KPt8 y = \roman I\KP1\roman E\KP1 x\,B\KP1 y =
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman I\KP1\roman E\KP1 x\,A\KP1 y$ \inskipline{.2}{29}
${}= \sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman I\KP1\roman E\KP1 x\,A\KP1 y\ar 1
= \roman I\KP1\roman E\KP1 x\,B\KP1 y\ar 1
= \roman I\KP1 x\KPt8 y\ar 1 \in \mathbb C \KPt9 $. \inskipline{.5}0
Now for the general case by the above we get \math{ \roman I\KP1 x\KPt8 y =
\roman I\KP1 x\KP1 y\ar 1 = \roman I\KP1 x\ar 1\,y\ar 1} by noting that for
some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } we have \math{
y\,.\KPt8 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=y\,.\KPt8 x\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta } for all \mathss30{
\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N}. \vskip.3mm
For continuity putting \math{ \trN03\smb X\trNu2 = \inf\kern0.15mm \big\{ \KPt8
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\kern-0.2mm:x\in\smb X\,\} } and \inskipline{.25}{40.1}
$\trN03\smb Y\trNun5 = \inf\kern0.15mm\big\{\KPt8 \|\KP1 \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y \KP1
\|\Lnorss40^{p\sast}_\mu\kern-0.2mm : y\in\smb Y\KP1\} \KP1 $, \inskipline{.25}0
by Theorem \ref{L^p in TVS} on page \pageref{L^p in TVS} above it suffices
that we have \vskip.3mm\centerline{$
|\KPp1.2\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\KP1|
\le\trN03\smb X\trNu2\,\trN03\smb Y\trNun5 $} \inskipline{.3}0
for \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. By Proposition \ref{Pro Hˆlder}
on page \pageref{Pro Hˆlder} for \math{x\in\smb X} and \math{y\in\smb Y} we
have \math{ |\KPp1.2\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\KP1| \le
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1\|\Lnorss33^p_\mu\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu} trivially giving the
result.
Finally, letting \math{\Bnull_{\aars F_1}\not=\smb Y\kern0.15mm\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} we
need to show existence of some \math{\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} with \linebreak \mathss03{
\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,)\not=0}. Now, by \math{ \smb Y \not =
\Bnull_{\aars F_1}} there are \math{y\in\smb Y} and \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} and \mathss30{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \mathss36{
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu\not=0}. Then taking \math{
x=(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}\kern0.15mm\sbig)0 } there is \linebreak \mathss03{
\smb X} with \mathss30{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,and we now have \mathss36{
\beta\fvalss10(\kern0.15mm\smb X\kern0.07mm,\smb Y\,) =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \not = 0 }.
\end{proof}
\begin{corollary}\label{Coro Io inj etc}
Let \ú$\,1\le p\le\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive
measure on $\,{}^{}\Cal Omega$ and with
\newline
$\,\bosy K
\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\vPi\in\roman{BaS}\kern0.4mmps0(K) \KP1 $. Also let
$\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) $ and let
\newline
$\,F\aar 1\in\{\,
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)\,,
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)\,,
\mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)\KPt8\}$ and
$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,(\kern0.37mm\smb Y\kern0.07mmp,\kern0.07mm\smb U
\kern0.15mm):\aall{y\in\smb Y}\,$ \inskipline{0}{37.7}
$\smb U=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\}\kern0.15mm\}
\KP1 $.
\noindent
Then
$\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds with
$\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm$ an injection.
\end{corollary}
\begin{proof} Note that although written differently, the \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} above
is precisely the same as in Theorem \nfss A\,\ref{main Th} above. Now we first
see that the assertion directly follows from Proposition \ref{Pro LpLp* dual}
above in the cases where \math{ F\aar 1 =
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } or \mathss30{
F\aar 1 = \mvsLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } holds.
For the case \math{ F\aar 1 =
\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } putting \math{
F\aar 0=\mvLrs14^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } and
letting \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 0} be the corresponding \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
in the corollary, taking
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\kern-0.3mm\times F\aar 0\capss01\{\,(\kern0.15mm\smb X\kern0.07mm,\kern0.07mm\smb Z\kern0.15mm):
\smb X\subseteq\smb Z\,\} } we then have
\mathss34{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 0\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 }. Trivially having
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\aar 0\kern0.07mm) } we get
\math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\in\Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm) } and we only
need to show that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is injective. Indeed, supposing that we
have \math{x\in\smb X} and \mathss34{
(\kern0.15mm\smb X\kern0.07mm,\kern0.07mm\Bnull_{\aars F_0}\sbig)0\in
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 }, \,for arbitrarily given \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
and \math{w\in\Cal L\,(\kern0.15mm\vPi\dlbetss01\KPt2,\kern0.07mm\bosy K\kern0.37mm) } we then must
show that \math{
\int_{\,A}\kern0.37mm w\circss00 x\rmdss11\mu = 0 } holds. In order to get this, we
first note that there are some \math{
N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and a
separable linear subspace
\math{S\ar 1} in \math{\vPi\dlbetss01} with \mathss34{
x\KP1[\KP1 A\setminus N\KP1]\subseteq S\ar 1 }. Then from
Lemma \ref{Le 8.17.8 B} on page \pageref{Le 8.17.8 B} above we get
existence of some \math{\bosy\xi\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi\in\bouSet\vPi } and such that \math{
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u = \lim\,(\kern0.37mm u\circss11\bosy\xi\kern0.37mm) } holds for every
\mathss34{u\in S\ar 1}. Now for all
\math{\eta\in A\setminus N} we have \vskip.25mm\centerline{$
w\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\lim\,(\kern0.37mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\circss11\bosy\xi\kern0.37mm)
=
\lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\,}(\KPt5
\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) \KP1 $.} \vskip.25mm
Since \math{x\in\smb X\subseteq\Bnull_{\aars F_0} } holds, for any fixed \math{ \xi
\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we have \mathss36{\int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.07mm x\rmdss11\mu = 0 }. From \math{ x \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} we see that \math{ x\KP1|\KP1 A \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs42^1(\kern0.37mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
holds, and taking into account \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi\in\bouSet\vPi } we get
existence of some positive \mathss37{\mu}--\,measurable \mathss30{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm} with \math{
\|\KPt8\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\,\|\Lnorss33^1_\mu < \lower1.05mm\hbox{$^+$}\infty } and such that \math{
\Abrs03^1\kern-0.2mm\circ\kern0.15mm\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x
\KP1|\KP1 A \le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm } holds for all \mathss36{i\in\mathbb No}. Then by dominated
convergence we obtain \inskipline{.5}{11}
$ \int_{\,A}\kern0.37mm w\circss00 x\rmdss11\mu
= \int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}w\circss00 x\rmdss11\mu
= \lim_{\KPt8 i\kern0.37mm\to\kern0.37mm\infty\kern0.15mm}\int_{\,A\kern0.37mm\setminus\kern0.37mm N\,}
\roman{ev}\KPt2\sbi{\bosy\xi\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i}\circ\kern0.15mm x\rmdss11\mu
= 0 \KPt8 $.
\end{proof}
In the next lemma we utilize the formal definitions \inskipline{.6}{18}
$^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A =
(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\} \cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm\xi\kern0.37mm\}\kern0.07mm\sbig)0
$ and \KP{18} \inskipline{.4}{18}
$\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A = \uniqset\smb X : {}
^{\bigcup\kern0.15mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\mu\kern0.15mm,\KPt3\vPi}\kern0.37mm\xi\kern0.15mm\sbi A \in \smb X \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) \KP1 $. \vskip.5mm
If \math{\mu} is a positive measure on \mathss36{{}^{}\Cal Omega}, \,for all \math{
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} and \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} thus \mathss30{
^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } is the simple function \math{{}^{}\Cal Omega\to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} that has the value \math{\xi} at points \math{\eta\in A} and \math{
\Bnull_\vPi} else- where. Then \math{
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } is the unique vector of \math{
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } having \math{
^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A } as one of its representatives.
\begin{lemma}\label{Le-first}
Let $\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on
$\,{}^{}\Cal Omega\,$. Also with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let
$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm and
$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)}$
\noindent
and $\,F=\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ and
$\,\smb U\in\Cal L\,(\kern0.15mm F\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) ${\KP1\rm, }and let
$\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ be finitely almost
scalarly measurable with $\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\ar 1\kern-0.2mm\circ\kern0.15mm y\KP1 \|\Lnorss50^{p^*}_\mu < \lower1.05mm\hbox{$^+$}\infty$
and such that
$\,
\smb U\fvalss11\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
=\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu
$ holds for all $\,A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and
$\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
Then $\,y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ holds with
$\,\smb U=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} \KP1 $.
\end{lemma}
\begin{proof} We get \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } directly from the
definition, and hence
only the last formula has to be verified. To get this, we note that for \math{
\smb X} and \math{x} with \math{{}^{}{\rm rng}\,{}_{{}^{}} x} finite and \mathss30{
x \in \smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,i.e.\ for some finite function \math{ \scrmt S
\subseteq (\kern0.37mm\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.07mm\sbig)0\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } with \math{
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt S} disjoint and \math{x=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.15mm]_{vs}\,\text{-}\sum_{\,A\kern0.37mm\in\kern0.37mm
{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S\,}
{}^{{}^{}\Cal Omega\kern0.15mm,\kern0.15mm\vPi}\kern0.37mm
(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\kern0.15mm\sbi A } we trivially have \inskipline{.5}{20}
$ \smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
= \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu $ \inskipline{.7}0
and by Proposition \ref{Pro Hˆlder} with \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm = \seqss33{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\cdot(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss11 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm) :
\eta = \eta} } we get \inskipline1{19.1}
$ \big|\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KP1|
= \big|\,\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm
\int_{\,A}\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)\KP1|$ \inskipline{.7}{41}
${}\le\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm\int_{\,A\,}|\KP{1.2}
y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) \KP{1.1} |
\rmdss21\mu\,(\kern0.15mm\eta\kern0.15mm)$ \inskipline{.7}{41}
${}\le\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.2mm\scrm7 S}\kern0.15mm\upint\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss10(\kern0.37mm\scrmt S\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1|\KP1 A\kern0.37mm)\rmdss11\mu$ \inskipline{.7}{41}
${}\le \upint\kern0.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'013}\kern0.15mm\rmdss11\mu
\le \|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\,
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss11 x\KP1\|\Lnorss33^p_\mu \KP1 $. \inskipline10
Since by Proposition \ref{Pro simp Lp dense} on page \pageref{Pro simp Lp dense}
above the set of vectors with simple representatives is \mathss35{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F
}--\,dense, from Corollary \ref{Coro Io inj etc} it follows that \math{
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X = \int_{\KP{1.1}{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu }
holds for all \math{x\kern0.37mm,\kern0.15mm\smb X} with \math{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}, \,and
this is precisely what we needed.
\end{proof}
\begin{proposition}\label{Pro L^1'=L^i}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let $\,F=\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$
and $\,F\aar 1=\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ and \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1=\seq{ \KP{1.2} \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,
(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) : \aall{x\in\smb X\kern0.15mm,\kern0.15mm y\in\smb Y}\,
t = \int_{\KP{1.1}{}^{}\Cal Omega\,}x\cdot y\rmdss11\mu\KPt9\} :
\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1\, } \KP1 $.} \inskipline{.5}0
Then $\,\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\in\Lis(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm F\dlbetss10\kern0.15mm)$ holds.
\end{proposition}
\begin{proof} Taking \math{p=1} and \math{\vPi=\bosy K} in Corollary \ref{Coro Io inj etc}
above, we see that $\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1$ is a continuous linear injection \mathss35{
F\aar 1\to F^{\kern0.4mm\prime}_{\kern-.2mm\raise.95mm\hbox{$_{_\beta}$}}}. Since \math{F} is normable by Theorem \ref{L^p in TVS}
above, by Corollary \ref{Cor L^p Ban} the spaces \math{F\aar 1} and \math{
F\dlbetss10} are \erm Banachable, and so by the open mapping theorem we only
need to prove that \math{ \Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) \subseteq
{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1 } holds. To establish this, arbitrarily fixing \mathss37{
\smb U\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,let
$\smb M=\sup\kern0.15mm\big\{\KPt8|
\KP{1.1}\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\KPt9|:
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\text{ and }\kern0.37mm\aall{x\in\smb X}\,
\int_{\KP{1.1}{}^{}\Cal Omega\,}
|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\rmdss11\mu\,(\kern0.07mm\eta\kern0.07mm)
\le 1\KPt9\}$ ,
and let $\scrmt A$ and $N\kern0.15mmrim1$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above.
Then $\smb M\in\lbb R_+$ holds, and we let $\scrmt Y\ar 1$ be the set of all
pairs $(\kern0.15mm A\ar 1\kern0.15mm,\kern0.07mm y\ar 1)$ with $A\ar 1\in\scrmt A$ and
$y\ar 1\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu
\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) \,$ and $\,
\sup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}(\kern0.37mm\Abrs00^1\circ\kern0.15mm y\ar 1) \le \smb M \,$ and
such that
$\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X=
\int_{\,\aars A_1}x\cdot y\ar 1\rmdss01\mu$
holds for all $x\kern0.37mm,\kern0.15mm\smb X$ with $x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ and
$x\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\,\ssbb52 C\setminus\{\kern0.37mm 0\kern0.37mm\}\KP{1.1}]\subseteq A\ar 1
\kern0.37mm$.
Then from \cite[Theorem 6.4.1\kern0.15mm, p.\ 162]{Du} we know that
$\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1$ holds, and hence by the {\sl axiom of choice\kern0.15mm}
there is a function $\scrmt Y\subseteq\scrmt Y\ar 1$ with
$\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\,$. Taking \math{y=
N\kern0.15mmrim1\kern-0.3mm\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss24\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y}, \,by
Lemma \ref{Le deco meas} on page \pageref{Le deco meas}
above \math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } now
\linebreak
is finitely almost measurable, and hence
\math{y\in\smb Y} holds for some \mathss31{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}.
Then for given \math{x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} letting
\mathss38{\scrmt A\kern0.15mm\ar 0=\scrmt A\capss31\{\,A:
\int_{\,A\kern0.37mm}\Abrs00^1\circ\kern0.15mm x\rmdss11\mu\not=0\KP1\} }, \,we have
\math{\scrmt A\kern0.15mm\ar 0} countable. If \math{\scrmt A\kern0.15mm\ar 0} is infinite,
we take any bijection \mathss36{\ebit A:\mathbb No\to\scrmt A\kern0.15mm\ar 0}, \,and if
it is finite, for some
\math{\smb N\in\mathbb No} we first take a
bijection \mathss36{\ebit A\ar 0\kern-0.2mm:\smb N\to\scrmt A\kern0.15mm\ar 0} and then
put \math{\ebit A=(\kern0.37mm\mathbb No\kern-0.3mm\setminus\smb N\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm
\emptyset\kern0.37mm\}\cupss21\ebit A\ar 0}.
Let now
\math{\ebit B=\kern0.15mm\big\langle\,\bigcup\KP1(
\kern0.15mm\ebit A\KPt8|\KP1 i\kern0.37mm):i\in\mathbb No\,\rangle} and
$\roman x\,i=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.37mm\ebit B\fvalss01 i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.37mm\ebit B\fvalss01 i
\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0 \,$ and
$\bosy x=
\seqss30{(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm))\times\kern-0.2mm
\{\kern0.37mm 0\kern0.37mm\}\cupss22(\kern0.37mm x\KP1|\KP1(\kern0.15mm\ebit A\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)):
i\in\mathbb No}$
and \math{\ebit X=
\seqss30{\uniqset\smb X\kern-0.3mm:\bosy x\fvalss01 i\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.3mm:
i\in\mathbb No} }
and
\mathss39{\ebit Y\kern0.37mm=
\seqss33{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\KPt8\text{-\kern0.15mm}\sum\KP1(\kern0.15mm\ebit X\KPt8|\KP1 i\kern0.37mm)
:i\in\mathbb No} }.
Then we have
\math{\ebit Y\kern0.37mm\to\smb X} in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} and hence also
\math{\smb U\circ\ebit Y\to
\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X}. Consequently, by
dominated convergence we obtain \inskipline1{11}
$\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\,(\kern0.37mm\smb U\circ\ebit Y\fvalss81 i\kern0.37mm)
=
\lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\,}
(\kern0.37mm\smb U\circ\ebit X\fvalss21 k\kern0.37mm) $ \inskipline1{18}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\sum_{\KPt8 k\kern0.37mm\in\kern0.37mm i\kern0.15mm}
\int_{\,\bmii6 A\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} k\,}x\cdot y\rmdss11\mu
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}
\int_{\KPt8\bmii6 B\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\,}x\cdot y\rmdss11\mu $ \inskipline1{18}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}
\int_{\KPp1.1{}^{}\Cal Omega\,}x\cdot y\cdot\roman x\,i\rmdss11\mu
= \int_{\KPp1.1{}^{}\Cal Omega\,}x\cdot y\rmdss11\mu
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb Y\KPt8\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb X\kern0.37mm$.
\end{proof}
For a topology \math{\scrmt T} we say that that \math{\scrmt T} is
{\it separably metrizable\kern0.37mm} if{}f \math{\scrmt T} is a metrizable topology
and there is a countable \math{D\subseteq\bigcup\,\scrmt T} with \mathss34{
\bigcup\,\scrmt T\subseteq\roman{Cl\KPt8}\sbi{\scrm7 T\KPt8}D }. In particular then \math{
D} is \mathss37{\scrmt T}--\,dense. Now, for the purpose of Lemma \ref{Le |int| < M imp ...}
below we put the following
\begin{definitions}\label{df sep cnv metr}
(1) \ Say that \math{C} is {\it separably uniform metrizable\kern0.37mm} in \math{E}
if{}f \math{E} is a real or complex topological vector space and there are
some nonempty countable sets \mathss03{D\kern0.37mm,\kern0.15mm\scrmt U} with \math{D\subseteq C
\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} and \math{\scrmt U\subseteq\neiBoo E} and such that \math{D} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C}--\,dense and for every \math{x\in C} it holds that \math{
\big\{\,[\KP1\{\kern0.37mm x\kern0.37mm\} +\kern0.15mm U\KPp1.1]\svs E\capss13 C : \kern0.15mm U \kern-0.3mm \in
\scrmt U\KP1\} } is a filter base for \mathss38{
\Cal N_{\font\SweD =cmmi6\lower.15mm\hbox{\kern.1mm\SweD bh\kern.15mm}}(\kern0.37mm x\kern0.37mm,\kern0.07mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\kern0.37mm) }, \inskipline{.5}2
(2) \ Say that \math{E} is {\it countably separably convex metrizable\kern0.37mm}
if{}f \math{E} is a real or complex Hausdorff locally convex space and
there is a countable \math{\scrmt C} with \math{ \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E = \bigcup\,\scrmt C}
and such that \math{C} is separably uniform metrizable in \mathss03{E} for
every \mathss34{C\in\scrmt C}.
\end{definitions}
Examples of countably separably convex metrizable spaces are all locally
convex spaces \math{E} with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E} separably metrizable as well as
countable strict inductive limits of such spaces. In particular, for example \math{
\mathscr D\,(\ssbb43 R) } and \math{C\kern.6mm\raise.15mm\hbox{$^\infty$}\kern.07mm(\ssbb43 R) } are countably
separably convex metrizable. Also \math{\vPi\dlsigss00\kern0.07mm} is countably
separably convex metrizable when \math{\vPi} is normable with \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} a separable topology.
Note that by the metrization theorem \cite[6.13\kern0.37mm, p.\ 186]{Ky} the
\q{uniform} filter base condition in Definitions \ref{df sep cnv metr}\,(1)
implies that \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C} is a metrizable topology. We leave it
as an {\sl open problem\kern0.15mm} whether we would have obtained an equivalent
definition if in \ref{df sep cnv metr}\,(1) instead of that uniformity
condition we had just required \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C} to be a metrizable
topology. We also remark that the definition given above is precisely what we
need in the next
\begin{lemma}\label{Le |int| < M imp ...}
Let $\,\vPi$ be countably separably convex metrizable{\kern0.15mm\rm, }and let $\,C$
be closed and convex in $\,\vPi\kern0.15mm$. Also let $\,
(\kern0.37mm x\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\kern0.37mm)$ be finitely scalarly integrable and such
that \vskip.4mm\centerline{$
\int_{\,A\,}u\circss00 x\rmdss11\mu\in\{\KPt8\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\cdot t :
t\in u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KP1\} $} \inskipline{.4}0
for $\, A \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and $\, u \in
\Cal L\,(\kern0.15mm\vPi\Reit2\kern0.15mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) \KPt8 $. Then $\,
x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1]$ is $\,\mu\,$--\,negligible.
\end{lemma}
\begin{proof} Let \math{\scrmt C} be as in
Definitions \ref{df sep cnv metr}\,(2) above when in place of \math{E} we have
taken the \math{\vPi} in the lemma. Then taking into account
(\kern0.15mm\erm{NB}\,2\kern0.15mm) in \cite[p.\ 33]{Jr} by {\sl dependent choice\kern0.15mm} we find
countable sets \math{D\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \math{\scrmt P\subseteq
\scrmt C\times(\kern0.37mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\capss12\neiBoo\vPi\kern0.37mm)} with
\math{\scrmt C\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt P } and such that for \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.15mm U\kern0.37mm)\in\scrmt P } and
\math{\scrmt U=\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } it holds that
\math{U} is absolutely \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,convex and there is
\math{V\in\scrmt U} with
\math{[\KPp1.1 V\kern-0.3mm + \kern0.15mm V\KPp1.1]\svs\vPi\subseteq U}
and
also \math{C\ar 1\kern-0.2mm\cap\KPt3 D} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\ar 1}--\,dense and
for every \math{\xi\in C\ar 1} it holds that
\newline
\math{
\big\{\,[\KP1\{\kern0.37mm\xi\kern0.37mm\} +\kern0.15mm V\KPp1.1]\svs\vPi\capss13 C\ar 1\kern-0.3mm :
\kern0.15mm V \kern-0.3mm \in
\scrmt U\KP1\} } is a filter base for \mathss38{
\Cal N_{\font\SweD =cmmi6\lower.15mm\hbox{\kern.1mm\SweD bh\kern.15mm}}(\,\xi\,,\kern0.07mm\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss33 C\ar 1\kern0.07mm) }.
Now we let \math{\scrmt R} be the countable set of all triplets \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1) } such that there is
\math{U} with \math{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.15mm U\kern0.37mm)\in\scrmt P } and
\math{\xi\in C\ar 1\kern-0.2mm\cap\KPt3 D } and
\math{U\aar 1=[\KPp1.1\{\kern0.37mm\xi\kern0.37mm\} + \kern0.15mm U\KPp1.1]\svs\vPi } and
\mathss36{C\capss23 U\aar 1=\emptyset}. Then by
{\sl Hahn\,--\,Banach\kern0.15mm} \cite[7.3.2\kern0.37mm, p.\ 130]{Jr} in conjunction
with {\sl countable choice\kern0.15mm} we get existence of a function
\math{\scrmt R\to\Cal L\,(\kern0.15mm\vPi\Reit2\kern0.15mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) } with the
property that
\newline
\math{\sup\,(\kern0.37mm u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm) < u\fvalss01\xi\ar 1}
holds for
\math{(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1\kern0.15mm,\kern0.07mm u\kern0.37mm)\in
\scrmt S} and \mathss32{\xi\ar 1\in\kern0.15mm U\aar 1}.
Now taking \mathss38{ \scrmt O = \{\,u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\openIval{\sup\,(\kern0.37mm
u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm)\,,\lower1.05mm\hbox{$^+$}\infty\kern0.15mm}:u\in{}^{}{\rm rng}\,{}_{{}^{}}\scrmt S\KPt8\} }, \,we have \mathss30{
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C={\kern-0.63mm}} \mathss04{\bigcup\,\scrmt O}. Indeed,
trivially \math{\bigcup\,\scrmt O\subseteq\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C } holds, and for
the converse inclusion arbitrarily fixing \math{ \xi\ar 0 \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C} we first find some \math{C\ar 1}
with \mathss34{\xi\ar 0\in C\ar 1\in \scrmt C}. Then we find
\math{U\in\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } such that for
\math{U\aar 0=[\KPp1.1\{\,\xi\ar 0\kern0.15mm\} + \kern0.15mm U\KPp1.1]\svs\vPi }
we have \mathss36{C\capss23 U\aar 0=\emptyset}. We further find
\math{V\in\scrmt P\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,C\aar 1\} } with
\math{[\KPp1.1 V\kern-0.3mm + \kern0.15mm V\KPp1.1]\svs\vPi\subseteq U}
and then there is some \math{\xi\in C\ar 1\kern-0.2mm\cap\KPt3 D }
with \mathss30{(\,\xi - \xi\ar 0\kern0.07mm)\svs\vPi\in\kern0.15mm V}. Now
putting \math{U\aar 1=[\KPp1.1\{\kern0.37mm\xi\kern0.37mm\} + \kern0.15mm V\KPp1.1]\svs\vPi }
we have \math{(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1)
\in\scrmt R={{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt S } and hence there is
\math{u} with \mathss34{
(\kern0.37mm C\ar 1\kern0.15mm,\kern0.07mm\xi\,,\kern0.07mm U\aar 1\kern0.15mm,\kern0.07mm u\kern0.37mm)\in\scrmt S }. Noting
that now \math{\xi\ar 0\in U\aar 1\subseteq
u\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\openIval{\sup\,(\kern0.37mm
u\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} C\kern0.37mm)\,,\lower1.05mm\hbox{$^+$}\infty\kern0.15mm}
} holds,
we obtain \mathss34{\xi\ar 0\in\bigcup\,\scrmt O }.
Now, to prove that \math{x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1] } is \mathss37{
\mu}--\,negligible, arbitrarily fixing
\newline
\mathss30{A\kern0.15mm\ar 0\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}, \,by
{\sl countable choice\kern0.15mm} and the discussion after
the proof of Lemma \ref{Le +int} on page \pageref{int not meas} we find
\math{N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and
a function \math{\varPhi:{}^{}{\rm rng}\,{}_{{}^{}}\scrmt S\to
\kern0.15mm^{\aars A_0}\,\mathbb R} such that \math{
(\kern0.37mm\varphi\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)
} is measurable and such that
\math{u\circss00 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta} holds for
\math{(\kern0.37mm u\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\varPhi} and
\mathss32{\eta\in A\kern0.15mm\ar 0\kern-0.3mm\setminus N\aar 1}. For
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0} we then also have
\newline
\mathss38{
\int_{\,A}\kern0.37mm\varphi\rmdss21\mu=
\int_{\,A\,}u\circss00 x\rmdss11\mu\in
\{\KPt8
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\cdot t:t\in u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KP1\} }. Now
\math{u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C} is a real interval and hence for
\math{N\kern0.15mmrim1=\varphi\invss44
[\,\ssbb42 R\setminus u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C\KPp1.1] } we have
\mathss36{N\kern0.15mmrim1\in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\kern0.15mm\ar 0}. Since \math{0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\kern0.15mmrim1}
would
trivially give a contradiction, we in fact have
\mathss38{N\kern0.15mmrim1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
Then {\sl countable choice\kern0.15mm} gives us existence of
\math{N} with
\math{N\aar 1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } and such that
\newline
\math{\varphi\KPp1.1[\KP1 A\kern0.15mm\ar 0\kern-0.3mm\setminus N\KP1]\subseteq
u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm C} holds for
\mathss34{(\kern0.37mm u\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\varPhi}. It being a trivial
exercise to check that now \math{
x\invss46[\KP1\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm\setminus C\KP1]\capss41 A\kern0.15mm\ar 0
\subseteq N} holds, we are done.
\end{proof}
From Lemma \ref{Le |int| < M imp ...} we obtain the following immediate
\begin{corollary}\label{Coro |f|<M}
Let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega$ with \ú$\,\mu\fvalss01{}^{}\Cal Omega <
\lower1.05mm\hbox{$^+$}\infty${\,\rm, }and with \ú$\,\smb M\in\lbb R_+$
and
\ú$\,\varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs42^1(\kern0.37mm\mu\,,\kern-0.2mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) $ let \ú$\,
\big|\kern0.15mm\int_{\,A}\kern0.37mm\varphi\rmdss11\mu\KP1|\le\smb M
\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)$ hold for all
\ú$\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then there is
$\,N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that $\,
|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le\smb M$
holds for all $\,\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm$.
\end{corollary}
\begin{lemma}\label{Le L^1_si-compa}
Let
$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and let $\,\mu$ be a positive measure with
$\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty\,$. Also let
$\,K\in
\bouSet
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \KP1$. Then $\,K$ is relatively
$\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\sbig(3\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\subsigma\kern0.07mm)\,
$--\,compact if and only if
for every $\,\varepsilon\in\rbb R^+$ there is $\,\delta\in\rbb R^+$
such that $\,\|\KP1\varphi\KP1|\KP1 A\KP1\|\Lnorss33^1_\mu
<\varepsilon$ holds for all
$\,\varphi\in\bigcup\,K$ and $\,
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[}
\KP1$.
\end{lemma}
\begin{proof} \newcommand\sFsigmaprime{F^{\kern.2mm\prime}_{\kern-.2mm\sigma}}
The assertion is already in
\cite[Theorem 3.2.1\kern0.15mm, p.\ 376]{Du-pe}\,, although one should note that
\q{weakly compact} there means \q{relatively weakly sequentially compact}. To
get a proper proof, suitably adapt the proof of
\cite[Theorem 4.21.2\kern0.37mm, pp.\ 274\,--\,275]{Edw}\,. Since we shall below need
the \q{if\kern0.15mm} part, we here give an explicit proof of it. Indeed, letting
(\kern0.15mm$*$\kern0.15mm) denote the asserted sufficient condition, and putting \mathss03{E=
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and \mathss38{F=
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }, \,let \mathss30{ \twEps =
\seqss43{\roman{ev}\kern0.37mm\sbi{\ssmb\Phii}\KPt8|\KP1\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm
\bosy K\kern0.37mm):\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} } \linebreak
and \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\circ\kern0.15mm\twEps}
where \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2:E\dlbetss12\kern-0.63mm\dlbetss01\to F\dlbetss10} is the
transpose of \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.2mm:F\to E\dlbetss11} when \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is
as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i} above.
Then \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is a strict morphism \math{E\subsigrs04\to F\dlsigss10} in
the sense of \cite[Definition 2.5.1\kern0.37mm, p.\ 100]{Ho}\,. Now assuming that
(\kern0.15mm$*$\kern0.15mm) holds, since by Alaoglu's theorem from \math{K\in\bouSet E} we
know that \math{\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm) } is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\dlsigss00\kern0.15mm)}--\,compact, it suffices to prove that \math{
\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } holds.
Thus arbitrarily given \math{w\in
\Cl_taurd{(\sFsigmaprime)}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K\kern0.37mm) } with \math{{}^{}\Cal Omega=
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \vskip.3mm\centerline{$
\roman x\,A=(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0$} \inskipline{.3}0
and \math{\eightroman X\,A=\uniqset\smb\Psii:\roman x\,A\in\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}
putting \math{\lambda=\seqss33{w\fvalss02\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} } we
see that now \math{\lambda} is a real or complex measure that is absolutely \mathss37{
\mu}--\,continuous.
Indeed, given \math{\varepsilon\in\rbb R^+} by (\kern0.15mm$*$\kern0.15mm) there is \math{ \delta \in
\rbb R^+} such that for all \math{ A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[} } we have \math{
|\KP1 z\fvalss02\text{\erm X\,}A\KP1|\le\varepsilon } for all \math{ z \in
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} and hence also \math{|\KP1\lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1| =
|\KP1 w\fvalss02\text{\erm X\,}A\KP1| \le \varepsilon } holds.
Note that \math{\lambda} is trivially finitely additive, and that countable
additivity then follows from the established absolute continuity. Now by
Radon\,--\,Nikodym there is some \mathss03{\varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} with \math{
w\fvalss02\text{\erm X\,}A = \lambda\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\int_{\,A\kern0.15mm}\varphi\rmdss11\mu } for all \mathss36{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Then there
is \mathss30{\smb\Phii} with \mathss35{\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E}, \,and
noting that the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,span of \math{
\{\KPt8\text{\erm X\,}A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} } is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,dense,
we first see that \mathss30{w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii=
\int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\cdot\psi\rmdss11\mu} holds for \mathss31{\psi\in
\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}. Then we get \math{w=\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
from \inskipline{.5}{19.6}
$ w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \int_{\KPp1.1{}^{}\Cal Omega\,}\varphi\cdot\psi\rmdss11\mu
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii
= \twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\kern0.15mm) $ \inskipline{.4}{26}
${}
= \twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\circss00\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2\circ\twEps\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Psii\,$.
\end{proof}
\Ssubhead C Lifting and integral representations \label{Sec C}
As auxiliary results for the proof of Theorem \nfss A\,\ref{main Th} we
reformulate some forms of the Dunford\,--\,Pettis theorem
\cite[8.17.6\,--\,8\kern0.37mm, p.\ 584]{Edw} in Propositions \ref{Pro Edw 8.17.6}
and \ref{Pro Edw 8.17.8} below. The essential content of
\cite[Lemma 8.17.1\,(a)\,, p.\ 579]{Edw} is in the following
\begin{proposition}\label{Pro lift}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega
${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and \ú$\,
G=\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ let $\,S$ be a
vector subspace in $\,\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G$ such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S$ is a
separable topology. \vskip.2mm\centerline{
Then a choice function $\,c\in
\Cal L\,(\kern0.37mm G_{\kern0.15mm/\kern0.37mm S}\kern0.37mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm))$ exists.}
\end{proposition}
\begin{proof} Letting
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G_{\kern0.37mm|\kern0.37mm S}=(\kern0.37mm a\,,\kern0.15mm c\kern0.37mm)} and
\math{R=
\{\,s + t\KP1\imag:s\kern0.37mm,\kern0.15mm t\in\ssbb04 Q\,\} } we first put
\ú$\kern0.37mm X={\kern-0.63mm}$
\linebreak
\ú$(\kern0.37mm a\,,\kern0.15mm c\KP1|\KP1(\kern0.15mm R\times S\kern0.37mm)) \kern0.37mm$ and
consider vector subspaces in the
possibly complex rational vector space \mathss31{X}.
Thus, letting \math{D} be countable and \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm G\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S
}--\,dense,
let $S\ar 1$ be the linear
\mathss37{X}--\,span of $D$. Then let $B$ be a linear basis of $
X_{\kern0.37mm|\kern0.37mm\aars S_1}$.
By {\sl countable choice\kern0.15mm} there is a choice function
$c\ar 0$ of $B$, and we let $\bar c\ar 0$ be its unique linear extension $
X_{\kern0.37mm|\kern0.37mm\aars S_1}\to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\,{}^{}\Cal Omega\kern0.37mm]_{vs}\kern0.37mm_{|
\,\bigcup\,S} \,$. Letting $\scrmt A$ and $N$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above,
for every fixed $A\in\scrmt A$ we then see existence of some $N\aar 1\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$|\KP{1.2}\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in A\KP1\}$ holds for $\varphi\in\smb\Phii\in S\ar 1$ and
$\eta\in A\kern0.15mm\setminus N\aar 1\kern0.37mm$. Then by
the {\sl axiom of choice\kern0.15mm} from the property of being almost
decomposable we see existence of a \mathss37{\mu}--\,negligible
$N\kern0.07mm{}^{{}_{{}^{'\!}}}$ such that
$|\KP{1.2}\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in{}^{}\Cal Omega\KP1\}$ holds for
$\varphi\in\smb\Phii\in S\ar 1$ and
$\eta\in{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern0.15mm$. Now taking
\mathss38{c\ar 1=
\seqss40{
\bar c\ar 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\KP1|\KP1(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern0.07mm
)\cupss22(\kern0.15mm N\kern0.07mm{}^{{}_{{}^{'\!}}}\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\sbig)0
:\smb\Phii\in S\ar 1} }, \,we have $c\ar 1$ a
linear map $X_{\kern0.37mm|\kern0.37mm\aars S_1}\to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm) $
and also
\math{|\KP{1.2} c\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le
\sup\kern0.37mm\big\{\,|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|:
\eta\in{}^{}\Cal Omega\KP1\} } holds for $\varphi\in\smb\Phii\in S\ar 1$ and all
$\eta\in{}^{}\Cal Omega\,$. Then by density of $S\ar 1$ and completeness of
$\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)$ letting $c$ be the unique continuous
extension of $c\ar 1$ we first get
$c\in
\Cal L\,(\kern0.37mm G_{\kern0.15mm/\kern0.37mm S}\kern0.37mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm
(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) \,$, and further using classical
convergence results for sequences of measurable functions we see that also
$c$ is a choice function.
\end{proof}
For a positive measure \math{\mu} on \math{{}^{}\Cal Omega} and for \mathss38{ X =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) }, \,by a {\it lift\kern0.37mm} of a linear
subspace \math{S} in \math{X} one means a linear map \math{c:X_{\kern0.37mm|\,S} \to
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm) } that is also a choice function such
that for \math{(\kern0.37mm\smb\Phii,\kern0.07mm\varphi\kern0.37mm)\in c} and for every \math{
\varphi\ar 1\in\smb\Phii } and \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+} we \linebreak
have \mathss38{ \sup\KP1(\kern0.37mm\varphi\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) \le \sup\KP1(\kern0.37mm
\varphi\ar 1\kern-0.3mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) }. So from the proof of
Proposition \ref{Pro lift} we see that we could have more specifically stated
that a lift exists. However, below we shall have no essential use of this
additional information encoded in the definition of lift. \vskip.3mm
Essentially the content of \cite[Theorem 8.17.2\kern0.37mm, p.\ 582]{Edw} is in the
following
\begin{proposition}\label{Pro Edw 8.17.2}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable and
such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology. Also let \ú$\,\smb U \in
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm))$ be such
that there is a choice function \ú$\, c \in
\Cal L\,(\kern0.37mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)
_{\kern0.37mm/\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern-0.2mm\sixmath U} \, , \kern0.37mm
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) \, $. Then there
is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\, \smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Phii\kern0.07mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Phii\,\} \KP1 $.
\end{proposition}
\begin{proof} With \math{ y = \seqss33{
\roman{ev}\kern0.15mm\sbi\eta\kern-0.2mm\circ\kern0.15mm c\circss11\smb U:\eta\in{}^{}\Cal Omega} } we have \math{
y} a function \mathss38{{}^{}\Cal Omega\to\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }, \,and
for fixed \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} noting that \math{c} is a choice function, we
obtain \inskipline{.5}2
(\kern0.15mm$*$\kern0.15mm) $ \KP{10.55}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y
= \seqss33{\roman{ev}\kern0.15mm\sbi\eta\kern-0.2mm\circ\kern0.15mm c\circss11\smb U\fvalss10\xi :
\eta\in{}^{}\Cal Omega}
= \seqss33{c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb U\fvalss10\xi\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta:\eta\in{}^{}\Cal Omega}$ \inskipline{.5}{29}
${}
= c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb U\fvalss10\xi\kern0.37mm)
\in \smb U\fvalss10\xi
\in \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \KP1 $. \KP6 From
this, \inskipline{.5}0
we see that \math{\smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Phii\kern0.07mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Phii\,\} } holds.
It remains to verify that \math{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } holds. First, to
prove that $(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable, let
$\scrmt A$ and $N\kern0.15mmrim1$ be as in
Definitions \ref{df decomp}\,(2) on page \pageref{decos A} above, and let
$D$ be countable and $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\,$--\,dense.
Then for every fixed
$A\in\scrmt A$ and $\xi\in D$ from (\kern0.15mm$*$\kern0.15mm) we see existence of
$N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
with
\linebreak
\ú$B=A\kern0.15mm\setminus N\kern0.37mm$ we have
\math{(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } measurable.
By countability of $D$ we can here take $N$ independent of
\mathss34{\xi\in D}.
Then by the {\sl axiom of choice\kern0.15mm}
in conjunction with the decomposability property we get
existence of a $\mu\,$--\,negligible $N\kern0.15mmrimm1$ such that
for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ and $\xi\in D$
with $B=A\kern0.15mm\setminus N\kern0.15mmrimm1$ we have
$(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\KP1|\KP1 B\,;
\kern0.15mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)$
\noindent
almost measurable.
By countability and
density of $D$ we then see that
$(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$
is finitely almost scalarly measurable.
To complete the proof of \mathss38{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }, \,by \math{
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\subseteq\bouSet(\kern0.15mm\vPi\dlsigss00\kern0.07mm)} it suffices
to show that \math{{}^{}{\rm rng}\,{}_{{}^{}} y\in\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } holds. For this,
we first note that there is some \mathss03{\smb A\in\lbb R_+} such that for \math{
\varphi\in\smb\Phii\in{}^{}{\rm rng}\,{}_{{}^{}}\smb U} we have \mathss36{
\|\KP1 c\fvalss11\smb\Phii\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le \smb A \KP1
\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu}. Then with \math{ B\ar 1 =
{}^{}{\rm rng}\,{}_{{}^{}}\smb U\capss21\{\KPt8\smb\Phii \kern-0.2mm : \aall{\varphi\in\smb\Phii}\,
\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu\le 1\KPt8\} } taking \math{ U =
\smb U\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm B\ar 1 } we have \mathss03{U\in\neiBoo\vPi } and
hence \vskip.3mm\centerline{$
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1|
= |\KPp1.1 c\circss00\smb U\fvalss11\xi\fvalss10\eta\KP1|
\le \|\KP1 c\circss00\smb U\fvalss11\xi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu
\le \smb A\KP1\|\KP1\varphi\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu
\le \smb A $} \inskipline{.5}0
for \math{\eta\in{}^{}\Cal Omega} and \math{\xi\in U} and \mathss35{ \varphi \in
\smb U\fvalss11\xi}. Consequently \math{ {}^{}{\rm rng}\,{}_{{}^{}} y \in
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } holds.
\end{proof}
The essential content of \cite[Theorem 8.17.6\kern0.37mm, p.\ 584]{Edw} is in the
following
\begin{proposition}\label{Pro Edw 8.17.6}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{LCS}\kern0.4mmps0(K)$ be normable and
such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable topology. Then for every \ú$\,\smb V
\in\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$
there is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ such that $\,
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ holds for $\,
\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
\end{proposition}
\begin{proof} We first get a continuous bilinear map \math{ \beta :
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\kern0.37mm\sqcap\kern0.15mm\vPi\to\bosy K } defined by \mathss35{
(\kern0.37mm\smb\Phii\kern0.07mm,\kern0.07mm\xi\kern0.37mm)\mapsto\smb V\fvalss70\smb\Phii\fvalss01\xi }, \,
and then a continuous linear map \math{ \smb U : \vPi \to
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } de- fined by \math{\xi
\mapsto\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\beta\,(\,\cdot\,,\kern0.07mm\xi\kern0.37mm)) }
where \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} is as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i}
above. Then by Propositions \ref{Pro lift} and \ref{Pro Edw 8.17.2} there is \math{
y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } with \vskip.4mm\centerline{$
\smb U =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\capss31
\{\,(\kern0.37mm\xi\,,\kern0.15mm\smb\Psii\kern0.15mm) : \roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Psii\,\} \KP1 $.} \inskipline{.4}0
Now for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} with \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V }
and \math{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y \in
\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } we
have \mathss03{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V\,) =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\beta\,(\,\cdot\,,\kern0.07mm\xi\kern0.37mm)) =
\smb U\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\xi = \smb\Psii } and hence \math{
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb\Psii } whence finally \mathss36{
\smb V\fvalss60\smb\Phii\fvalss00\xi =
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm\smb V\,\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.37mm\smb\Psii\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii =
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu}.
\end{proof}
The content of \cite[Lemma 8.17.8 \erm A\kern0.37mm, p.\ 584]{Edw} is in the
following
\begin{lemma}\label{Le 8.17.8 A}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable{\kern0.37mm\rm, }and
let \ú$\,F=E\dlbetss12\,$. Also let $\,S\ar 1$ be a closed linear subspace in $\,
F$ such that \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1$ is a separable topology. Then there
is a closed linear subspace $\,S$ in $\,E$ with \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S$ a
separable topology and such that \ú$\,\seq{\KP1 u\KPt9|\KP1 S:u\in S\ar 1\,}$
is a strict morphism \ú$\, F_{\kern0.15mm/\kern0.37mm\aars S_1} \to
(\kern0.15mm E_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12 $ in the sense of
{\,\rm\cite[Definition 2.5.1\kern0.15mm, p.\ 100]{Ho}\,.}
\end{lemma}
\begin{proof} Fixing a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} for \mathss35{E}, \,let \math{
\bosy u\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,S\ar 1} be such that ${}^{}{\rm rng}\,{}_{{}^{}}\bosy u$ is \mathss34{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22 S\ar 1}--\,dense, and let $\bosy x\in\kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I)$ be such that $
(\kern0.37mm 1 + i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\kern0.15mm\big)\KP1
(\kern0.37mm\bosy u\,.\KPt8\bosy x\fvalss01 i\kern0.37mm)
=\sup\kern0.37mm\big\{\KPt8|\KP1\bosy u\fvalss01 i\fvalss10 x\KP1|:
x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}$ holds for all $i\in\mathbb No\,$. Then we let
$S$ be the closed linear span of ${}^{}{\rm rng}\,{}_{{}^{}}\bosy x$ in $E\,$, and take \mathss38{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seqss40{u\KPt9|\KP1 S:u\in S\ar 1} }.
One easily verifies that \math{
\sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}
\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\KPt9\} } holds
for every fixed $u\in S\ar 1\kern0.37mm$, and hence we get \vskip.6mm
$
\sup\kern0.37mm\big\{\KPt8|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss10 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
= \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
$ \inskipline{.4}{10.6}
${}\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb23 I\,\}
\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\KPt9\}
$ \inskipline{.4}{10.6}
${}\le \sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\}
= \sup\kern0.37mm\big\{\KPt8|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss10 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb20 I\capss32 S\KP1\} \KP1 $, \inskipline{.6}0
from which the assertion easily follows.
\end{proof}
The content of \cite[8.17.8\kern0.37mm, pp.\ 584\,--\,586]{Edw} is in the following
\begin{proposition}\label{Pro Edw 8.17.8}
Let $\,\mu$ be an almost decomposable positive measure on $\,{}^{}\Cal Omega${\,\rm, }
and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ and \ú$\,\smb V\in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$
be such that $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi^{\kern0.4mm\prime}_{\kern-.2mm\raise.95mm\hbox{$_{_\beta}$}}\kern0.07mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\smb V$ is a
separable topology. Then there is \ú$\, y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with $\,
{}^{}{\rm rng}\,{}_{{}^{}} y\subseteq\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V$ and such that \vskip.1mm\centerline{$
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu $} \inskipline{.5}0
holds for $\,\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
\end{proposition}
\begin{proof} Taking \math{S\ar 1=\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V } and \math{E=
\vPi} in Lemma \ref{Le 8.17.8 A} above, there is a closed linear subspace \math{
S} in \math{\vPi} with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S } a separable topology and
such that for \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm=\seq{\KP1 u\KPt9|\KP1 S:u\in S\ar 1\,} } we have \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} a strict morphism \mathss37{\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1} \to
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12 }.
Now we have \mathss38{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.15mm\smb V \in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) \, , \kern0.15mm
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) }, \,and by separability of the
topol- \linebreak
ogy \math{ \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm) =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32 S } we can apply Proposition \ref{Pro Edw 8.17.6} to
deduce existence of some \linebreak
$y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \kern0.37mm$
with \inskipline{.5}2
(\kern0.15mm r\kern0.15mm) $ \KP{12}
\smb V\fvalss60\smb\Phii\fvalss00\xi
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\smb V\fvalss60\smb\Phii\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi
= \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.07mm\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern-0.63mm\cdot\varphi\rmdss21\mu$ \inskipline{.5}0
whenever \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \math{\xi\in S} hold.
Then taking \math{ y = \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\kern-0.2mm\circ\kern0.15mm y\ar 1 } we get a function \mathss38{
y:{}^{}\Cal Omega\to S\ar 1\subseteq\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) }, \,and it remains
to establish \ú$\kern0.37mm y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) $ and to show that \inskipline{.5}2
(\kern0.15mm s\kern0.15mm) $ \KP{12}
\smb V\fvalss60\smb\Phii\fvalss00\xi = \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ \inskipline{.5}0
holds for \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}.
Noting that \math{\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) } by Banach\,--\,Steinhaus,
we see that \vskip.5mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlsigss12) \subseteq
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}s\bouSet((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm) \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1}\kern0.07mm\sbig)0 \subseteq
\bouSet(\kern0.15mm\vPi\dlbetss01\kern0.15mm) \subseteq \bouSet(\kern0.15mm\vPi\dlsigss12) \KP1 $,} \inskipline{.5}0
and hence \math{y} is similarly \q{finitely almost bounded} as \math{y\ar 1}
is. So, in order to establish \linebreak
\ú$y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) \kern0.37mm$ we only need
to show that \math{(\KPt5 y\,;\kern0.15mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is finitely
almost scalarly measurable. For this, arbitrarily fixing \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,we first observe that \linebreak
$\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1 \in
\Cal L\,(\kern0.15mm\vPi\dlbetss01{}_{\kern0.15mm/\kern0.37mm\aars S_1},\kern0.07mm\bosy K\kern0.37mm) \kern0.37mm$ and
hence also \mathss38{
\roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1\kern-0.2mm\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve \in
\Cal L\,(
(\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12{}_{\kern0.37mm/\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern-0.2mm\emath\iotaa}\kern0.37mm , \kern0.07mm
\bosy K\kern0.37mm) }.
By separability of the topology \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm\vPi_{\kern0.37mm/\kern0.37mm S}\kern0.07mm)\dlbetss12\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } we are
able to apply Lemma \ref{Le 8.17.8 B} on page \pageref{Le 8.17.8 B} above to
get existence of \math{ \bosy\xi \in \kern0.15mm ^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,S } with \math{{}^{}{\rm rng}\,{}_{{}^{}}\bosy\xi
\in\bouSet\vPi } and \vskip.4mm\centerline{$
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\fvalss02\zeta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi
= \roman{ev}\kern0.15mm\sbi\xi\,|\KP1 S\ar 1\kern-0.2mm\circ\kern0.07mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\inve\fvalss02\zeta\ar 1
=\lim\,(\kern0.37mm\zeta\ar 1\kern-0.3mm\circ\kern0.37mm\bosy\xi\kern0.37mm) $} \inskipline{.4}0
for all \math{\zeta\ar 1\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm }, \,and hence \math{\zeta\fvalss11\xi =
\lim\,(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\fvalss11\zeta\circss11\bosy\xi\kern0.37mm) } for all \mathss34{
\zeta\in S\ar 1}. In particular for \linebreak
\ú$(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\zeta\kern0.37mm)\in y \kern0.37mm $
we obtain \mathss38{\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta =
\lim\,(\kern0.37mm y\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\circss11\bosy\xi\kern0.37mm) }, \,giving the
required measurability. To get (\kern0.15mm s\kern0.15mm) we note that by the above we also
have \inskipline{.7}{8.5}
$ \smb V\fvalss60\smb\Phii\fvalss00\xi
= \lim\,(\kern0.37mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\circ\kern0.07mm\smb V\fvalss60\smb\Phii\circ\kern0.15mm\bosy\xi\kern0.37mm)
= \lim\,(\kern0.37mm\smb V\fvalss60\smb\Phii\circ\kern0.15mm\bosy\xi\kern0.37mm)$ \inskipline{.5}{19}
${}
= \lim\kern0.15mm\sbi{i\kern0.37mm\to\kern0.37mm\infty}\kern0.15mm\int_{\KP{1.1}{}^{}\Cal Omega\,}
y\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\KP1(\kern0.37mm
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
= \int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu \,$, \inskipline{.7}0
where we used (\kern0.15mm r\kern0.15mm) with dominated convergence, noting that it is
legitimate by the above established boundedness and measurability properties.
\end{proof}
\insubsubhead Dunford\,--\,Pettis property of $\,\mLrs42^1(\kern0.37mm\mu\kern0.15mm)$ \label{Ss Dun-Pet}
When treating the reflexive case in the proof of
Theorem \nfss A\,\ref{main Th}\kern0.15mm, we need to know that \mathss03{
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } has the Dunford\,--\,Pettis property,
or in our terminology introduced below, is a {\eightsl DP}{\sl\,--\,space\kern0.15mm}.
This is equivalent to \math{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } being a
\erm{DP}\,--\,space, and for this reason we here consider this matter to some
extent. Although we shall need the result only for positive measures \math{\mu}
with \mathss36{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty}, \,we anyhow
consider the situation for general positive measures.
In what follows, note
that \math{\pi} is said to be a {\it probability measure\kern0.37mm} if{}f \math{
\pi} is a positive measure with \mathss34{\pi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\pi=1}.
\begin{lemma}\label{Le L^p(m)=L^p(p)}
Let $\,1\le p\le\lower1.05mm\hbox{$^+$}\infty${\,\rm, }and let $\,\mu$ be a \rsigma0finite
positive measure with
$\,{}^{}{\rm rng}\,{}_{{}^{}}\mu\not=\{\kern0.37mm 0\kern0.37mm\}\KPt8$. Then
there is a probability measure $\,\pi$ with
$\,\mLrs03^p(\kern0.37mm\mu\kern0.15mm)$ and $\,\mLrs03^p(\kern0.15mm\pi\kern0.15mm)$
linearly homeomorphic.
\end{lemma}
\begin{proof} Letting \math{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} be a
finite or countably infinite partition of \mathss36{{}^{}\Cal Omega}, \,let \math{
\bosy a:\scrmt A\to\rbb R^+} be any function with \math{\sum\,\bosy a=1} and
take \vskip.5mm\centerline{$
\pi =
\kern0.37mm\big\langle\,\sum_{\KPt8 B\kern0.37mm\in\kern0.37mm\scrm7 A\,}
(\kern0.37mm\bosy a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\kern0.37mm))) : A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KP1\rangle \KP1 $.} \inskipline{.5}0
Then \math{\pi} is a probability measure with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\pi={{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and we
define
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm:\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs03^p(\kern0.37mm\mu\kern0.15mm)
\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs03^p(\kern0.15mm\pi\kern0.15mm)$
by $\smb\Phii\mapsto\smb\Psii$ when $x\in\smb\Phii$ and
$\bigcup\KPt8\{\KPt9\roman b\,B\KP1(\kern0.37mm x\KP1|\KP1 B\kern0.37mm)
:B\in\scrmt A\KP1\}\in\smb\Psii$
where $\roman b\,B=
(\kern0.37mm\bosy a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\kern0.37mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p^{-1}}
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\KP1^{p^{-1}}$
for $1\le p < \lower1.05mm\hbox{$^+$}\infty$
and \math{\roman b\,B=1} for
\mathss36{p = \lower1.05mm\hbox{$^+$}\infty}.
\end{proof}
\begin{definitions}\label{df D-P}
(1) \ Say that \math{E} is a {\it\eit{DP\,}--\,space over\kern0.37mm} \math{\bosy K}
if{}f \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \mathss30{E \in \kern-0.3mm} \mathss03{
\roman{LCS}\kern0.4mmps0(K)} and for all \math{F\in\roman{BaS}\kern0.4mmps0(K) } and \math{ \smb U \in
\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm F\kern0.37mm) } and for \inskipline07
$\scrmt K=\{\,A:A\kern0.37mm\text{ is absolutely \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,convex and \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) }--\,compact }\}
$ from \KP3 \inskipline0{11}
$\smb U\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\bouSet E\subseteq\{\,B:B\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\subsigrs00) }--\,compact }\}
$ it follows that \KP3 \inskipline0{14.8}
$\smb U\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\scrmt K\subseteq\{\,B:B\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,compact }\} \KP1 $ holds, \inskipline{.5}2
(2) \ Say that \math{E} is a {\it\eit{DP\,}--\,space\kern0.37mm} if{}f there is \math{
\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} \inskipline0{7.7}
such that \math{E} is a \erm{DP\,}--\,space over \mathss32{\bosy K}.
\end{definitions}
Instead of saying that \math{E} is a \erm{DP}\,--\,space, we may also say that
it is a {\it Dunford\,--\,Pettis\kern0.37mm} space. For application of the
Dunford\,--\,Pettis property one should note that by
\cite[Remarks 9.4.1\,(3)\,, p.\ 634]{Edw} for \math{E} complete in
Definitions \ref{df D-P}\,(1) we get an equivalent condition if we instead
take \vskip.2mm\centerline{$
\scrmt K = \{\,A:A\kern0.37mm\text{ is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) }--\,compact }\} \KP1 $.} \inskipline{.4}0
In particular, this holds if with \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}} and \math{\mu} a
positive measure we take \mathss08{E=\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }.
For the proof in the general case one possibly uses {\sl Krein's theorem\kern0.15mm}
\cite[9.8.5\kern0.37mm, p.\ 192]{Jr}\,. The Lebesgue case with \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu < \lower1.05mm\hbox{$^+$}\infty } also follows from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above, and only this will be needed in the
sequel.
\begin{proposition}\label{Pro DP-seq}
Let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$ with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}\KP1$. Then $\,E$ is a
Dunford\,--\,Pettis space if and only if \ú$\,
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm\to 0$ holds for all $\,
\bosy x\kern0.37mm,\kern0.15mm\bosy y$ with $\,\bosy x\to\Bnull_E$ in top $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm
E\subsigrs04)$ and $\,\bosy y\to\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ in top $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm E\dlbetss22\kern0.15mm)\subsigrs03) \KP1 $.
\end{proposition}
\begin{proof} See \cite[Proposition 20.7.1\kern0.15mm, p.\ 473]{Jr} or
\cite[p.\ 636]{Edw}\,.
\end{proof}
\begin{lemma}\label{Le E_{/S} DP imp ...}
With \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$ be such that for every $\,
\bosy x$ with \ú$\,\bosy x\to\Bnull_E$ in top $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)$
there is a closed linear subspace $\,S$ in $\,E$ with \ú$\,{}^{}{\rm rng}\,{}_{{}^{}}\bosy x\subseteq S$
and such that $\,E_{\,/\,S}$ is a \eit{DP}\,--\,space. Then $\,E$ is a
\eit{DP}\,--\,space.
\end{lemma}
\begin{proof} Given \math{\bosy x\kern0.37mm,\kern0.15mm\bosy y } with \math{\bosy x\to
\Bnull_E } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) } and \math{\bosy y\to
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } in top \mathss08{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm
E\dlbetss22\kern0.15mm)\subsigrs03) }, \,by Proposition \ref{Pro DP-seq} it suffices
to show that \math{\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm\to 0 }
holds. To get this, putting \math{F=E_{\,/\,S} } and \math{\bosy z=\seqss30{
\bosy y\fvalss01 i\KP1|\KP1 S:i\in\mathbb No} } we now have \mathss30{
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy y\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm = \kern-0.3mm} \mathss03{
\roman{ev}\circ\kern0.07mm[\KP1\bosy x\kern0.37mm,\kern0.07mm\bosy z\kern.9mm]\kern.2mm\lower.8mm\hbox{\font\SweD =cmr5\SweD f}\kern.2mm } and hence we are done
{\sl if\kern0.15mm} we can show (\kern0.15mm a\kern0.15mm) that \math{\bosy x\to\Bnull_E } in top \mathss30{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm F\subsigrs00) } holds, and (\kern0.15mm b\kern0.15mm) that \math{\bosy z\to
S\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm
F\dlbetss20\kern0.15mm)\subsigrs03) } holds. Now (\kern0.15mm a\kern0.15mm) follows trivially from
Hahn\,--\,Banach since given \math{v\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }
there is \math{u} with \mathss30{v\subseteq u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) }
and hence \math{ v\circss00\bosy x = u\circss00\bosy x\to 0 } holds. For
(\kern0.15mm b\kern0.15mm) taking the annihilators \inskipline{.2}{16.5}
$N\aar 0 = \Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)\capss31\{\,u :
u\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm S\subseteq\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} \KP{13.5} $ and \inskipline0{17}
$S\ar 1 = \Cal L\,(\kern0.15mm E\dlbetss22\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\capss31\{\,w :
w\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm N\aar 0\subseteq\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} \KP9 $ and putting \inskipline{.2}0
\ú$F\aar 1=E\dlbetss22\kern0.37mm/\tvsquotient N\aar 0 \KPt7 $, \,for \math{ w \ar 1
\in \Cal L\,(\kern0.15mm F\dlbetss20,\kern0.07mm\bosy K\kern0.37mm) } from
\cite[3.13\kern0.37mm, pp.\ 261\,--\,263]{Ho} we first get ex- istence of \math{
w\ar 2 \in \Cal L\,(\kern0.15mm F\aar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{
w\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb U = w\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u\KP1|\KP1 S\kern0.37mm) } for \mathss34{
u\in\smb U\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 }. Then we get existence of \math{w\in S\ar 1} with \math{
w\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} u=w\ar 2\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb U=w\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm u\KP1|\KP1 S\kern0.37mm) }
for \math{u\kern0.37mm,\kern0.15mm\smb U} as above. Hence we finally get \math{
w\ar 1\kern-0.3mm\circ\kern0.07mm\bosy z=w\circss00\bosy y\to 0 } as required.
\end{proof}
\begin{proposition}\label{Pro L^(p) is DP}
If $\,\pi$ is any probability measure{\kern0.15mm\rm, }then $\,\mLrs42^1(\kern0.15mm\pi\kern0.15mm)$
is a \eit{DP}\,--\,space.
\end{proposition}
\begin{proof} See \cite{Brg} and \cite{Sch}\,, and take \math{T=1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} in
the latter.
\end{proof}
\begin{corollary}\label{Cor L^1 is D-S}
For $\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and for any \inskipline0{21.5}
positive measure $\,\mu$ it holds that $\,
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ is a \eit{DP}\,--\,space.
\end{corollary}
\begin{proof} From Proposition \ref{Pro L^(p) is DP} we first see that also \math{
\mvLrs42^1(\kern0.37mm\pi\kern0.15mm,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm) } is a \erm{DP\,}--\,space for any
probability measure \mathss32{\pi}. Indeed, for \math{ G =
\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\kern0.37mm\sqcap\kern0.15mm\mLrs42^1(\kern0.15mm\pi\kern0.15mm) } from
\cite[9.4.3\,(a)\,, p.\ 635]{Edw} we first see that \math{G} is a
\erm{DP\,}--\,space, and since \math{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C} and \math{G}
are naturally linearly homeomorphic, also \math{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C}
is a \erm{DP\,}--\,space. If now \math{\smb U} is a continuous linear map \mathss30{
\mvLrs42^1(\kern0.37mm\pi\kern0.15mm,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\kern0.37mm)\to F}, \,it is also a continuous real
linear map \mathss30{\mLrs42^1(\kern0.15mm\pi\kern0.15mm)\flbb_C\to F\Reit0} whence the
assertion follows by noting that the equality \math{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\RHB{.15}{\subsigma}\kern0.07mm) =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\Reit1\kern-0.2mm\RHB{.15}{\subsigma}\kern0.07mm) } holds for every \mathss38{
E\in\roman{LCS}\kern0.4mmps5(\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C) }.
By Lemma \ref{Le L^p(m)=L^p(p)} from the above we know that \math{
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } is a \erm{DP\,}--\,space for any
\rsigma5finite positive measure \mathss36{\mu}. Then by Lemma \ref{Le E_{/S} DP imp ...}
we get the general case as follows. Putting \math{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and letting \math{
\bmii8\Phii\to\Bnull_E} in top \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04) } we first find
some countable \math{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\lbb R_+} such that \math{
\|\KP1\varphi\KP1|\KP1 A\KP1\|\Lnorss33^1_\mu=0 } holds for all \mathss30{A\in
\scrmt A} and \math{\varphi\in\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\kern0.15mm\bmii8\Phii}. Then taking \math{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\kern-0.3mm\sbig(1\bigcup\,\scrmt A\kern0.37mm) } we have \math{
\mvLrs42^1(\kern0.37mm\mu\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } a \erm{DP\,}--\,space. Moreover,
we have an obvious strict morphism \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm :
\mvLrs42^1(\kern0.37mm\mu\ar 1\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)\to E } with \mathss03{
{}^{}{\rm rng}\,{}_{{}^{}}\kern0.15mm\bmii8\Phii\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} whence Lemma \ref{Le E_{/S} DP imp ...} gives
the conclusion.
\end{proof}
To fill the gap \q{exists $\ldots\,\eightroman M_{\kern0.37mm\roman t}\,\ldots$} in
\cite[p.\ 3]{Sch} we give the following
\begin{lemma}
With $\,\pi$ a probability measure let \ú$\,E=\mLrs42^1(\kern0.15mm\pi\kern0.15mm)$ and also
let \ú$\,\bmii8\Phii\to\Bnull_E$ in top $\, \tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)\KP1$.
Further let $\,\varepsilon\in\rbb R^+$ and $\, \bosy\varphi \in
\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\kern0.37mm\bmii8\Phii\kern0.37mm$ with \vskip.3mm\centerline{$
{}^{}{\rm rng}\,{}_{{}^{}}\bosy\varphi\subseteq\{\,\varphi:(\KPt5\varphi\KPt8;\kern0.07mm\pi\kern0.15mm,\kern-0.63mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\kern0.37mm\text{
is measurable }\} \KP1 $.} \inskipline{.5}0
Then there is some \ú$\,\smb M\in\rbb R^+$ such that \ú$\,\|\KP1\varphi\KP1|\KP1
((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1\smb M\kern0.15mm,
\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)3\,
\big\|\LHB{.4}{\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033}} < \varepsilon$ holds for all $\,
\varphi \in {}^{}{\rm rng}\,{}_{{}^{}}\bosy\varphi \, $.
\end{lemma}
\begin{proof} If the assertion is false, by {\sl dependent choice\kern0.15mm} there is
a stricly increasing \mathss03{\bosy n:\mathbb No\to\mathbb No } such that \math{\varepsilon\le
\|\KP1\varphi\KP1|\KP1((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm
[\KP1 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056},\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)3\,
\big\|\LHB{.4}{\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033}} } holds for \mathss30{
(\kern0.37mm i\kern0.37mm,\kern0.07mm\varphi\kern0.37mm) \in \kern-0.3mm} \mathss06{\bosy\varphi\circss00\bosy n
}. Noting that \math{{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8\Phii } is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact, then from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above it follows indirectly that there is \math{
\delta\in\rbb R^+ } with the property that for \math{
(\kern0.37mm i\kern0.37mm,\kern0.07mm\varphi\kern0.37mm)\in\bosy\varphi\circss00\bosy n } we have \mathss38{
\delta\le\pi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}((\kern0.37mm\Abrs03^1\kern-0.2mm\circ\kern0.15mm\varphi\kern0.37mm)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KP1
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056},\lower1.05mm\hbox{$^+$}\infty\KP1{[\KPt8}\sbig)0 }. This implies that \mathss03{
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,\delta\le\|\,\varphi\,\|\Lnorss33^1_{\emath\char'031}\def\rhoo{\char'032}\def\sigmaa{\char'033} } holds, giving
a {\sl contradiction\kern0.15mm} with \mathss36{{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8\Phii\in\bouSet E }.
\end{proof}
\insubsubhead Absolutely continuous vector measures \label{Ss abs conti}
We here give some basic definitions for vector measures in order to be able to
present a decent proof for Proposition \ref{Pro Phi5.4} below that is needed
as an auxiliary result for the proof of Theorem \nfss A\,\ref{main Th} above.
\begin{definitions}\label{df vec mea}
(1) \ Say that \math{E} is a {\it topologized conoid\,} if{}f there are \math{
a\kern0.37mm,\kern0.15mm c\,,\kern0.15mm o\kern0.37mm,\kern0.15mm R\,,\kern0.15mm S\kern0.37mm,\kern0.15mm\scrmt T} with \math{
\lbb R_+\subseteq R\subseteq\mathbb C } and \math{o\in S} and \math{
(\kern0.15mm S\kern0.37mm,\kern0.07mm\scrmt T\,) } a Hausdorff topological space and \mathss30{ E = {\kern-0.63mm}} \mathss03{
(\kern0.37mm a\kern0.37mm,\kern0.07mm c\,,\kern0.07mm\scrmt T\,) } and \math{a} a function \math{S\times S
\to S} and \math{c} a function \math{R\times S\to S} and such that for all \math{
x\kern0.37mm,\kern0.15mm y\kern0.37mm,\kern0.15mm z\in S } and for all \math{s\kern0.37mm,\kern0.15mm t\in R} it holds that \math{
a\,(\kern0.37mm x\kern0.37mm,\kern0.15mm\cdot\,) } and \math{c\KPt8(\kern0.37mm t\kern0.37mm,\kern0.15mm\cdot\,) } are
continuous \math{\scrmt T\to\scrmt T } and in addition \inskipline09
$a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)\,,\kern0.15mm z\kern0.37mm)=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm y\kern0.37mm,\kern0.07mm z\kern0.37mm))\,$ and
$\,a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm)=a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm y\kern0.37mm,\kern0.07mm x\kern0.37mm)\,$ and \inskipline09
$a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm o\kern0.37mm)=c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm 1\kern0.15mm,\kern0.07mm x\kern0.37mm)=x\,$ and
$\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\KPt8 t\kern0.37mm,\kern0.07mm x\kern0.37mm)=
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm ))\,$ and \inskipline09
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s + t\kern0.37mm,\kern0.07mm x\kern0.37mm)=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm x\kern0.37mm)\,,\kern0.15mm
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm t\kern0.37mm,\kern0.07mm x\kern0.37mm)) \,$ and \inskipline09
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm))=
a\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\,c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm x\kern0.37mm)\,,\kern0.07mm
c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm s\kern0.37mm,\kern0.07mm y\kern0.37mm)) \KP1 $, \inskipline{.5}2
(2) \ Say that \math{m} is an \mathss35{E}{\it--\,measure\kern0.37mm} if{}f \math{E}
is a topologized conoid and \math{(\kern0.37mm\emptyset\,,\kern0.07mm\Bnull_E) \in {\kern-0.63mm}} \inskipline09
$m\in\kern0.15mm^{\roman{dom}\KPt8 m}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\kern0.37mm$ and for all \math{ A\,,\kern0.15mm B \in
{{}^{}{\rm dom}\,{}_{{}^{}}} m} it holds that \math{\{\,A\cupss31 B\kern0.37mm,\kern0.15mm A\setminus B\,\}\subseteq {\kern-0.63mm}} \inskipline0{8.7}
${{}^{}{\rm dom}\,{}_{{}^{}}} m \kern0.37mm $ and \mathss36{A\capss31 B=\emptyset\impss33
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss31 B\kern0.37mm) = (\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\svs E}, \inskipline{.5}2
(3) \ Say that \math{m} is {\it countably \mathss37{E}--\,additive\kern0.37mm} if{}f \math{
m} is an \mathss37{E}--\,measure and \inskipline09
for all countable disjoint \math{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} with \math{
\bigcup\,\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} \inskipline0{36.5}
it holds that \mathss38{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\bigcup\,\scrmt A =
E\vtopsum3\KP1(\kern0.37mm m\KP1|\KP1\scrmt A\kern0.37mm) }, \inskipline{.5}2
(4) \ Say that \math{m} is {\it absolutely \mathss57{\mu}--\,continous\kern0.37mm} in
\math{E} if{}f \math{\mu} is a positive measure \inskipline09
and \math{m} is an \mathss37{E}--\,measure and for every \math{ U\kern-0.3mm \in
\neiBoo E} there is some \math{\delta\in\rbb R^+} \inskipline0{76}
with \mathss30{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\delta\KP1{[\kern0.15mm} \subseteq
m\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022} U}, \inskipline{.5}2
(5) \ Say that \math{m} has {\it bounded \mathss57{\mu}--$\KP2^p\,
$variation\kern0.37mm} in \math{E} if{}f \math{1\le p < \lower1.05mm\hbox{$^+$}\infty} and \math{\mu}
is a positive measure and \math{m} is an \mathss35{E}--\,measure with \math{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} and for ev- ery \math{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm \in
\Bqnorm E} there is \math{\smb M\in\lbb R_+} such that \mathss30{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss00 m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)
\RHB{.3}{\KP1^p}\kern0.37mm\sbig)0\le\smb M } holds for all finite disjoint \mathss30{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}.
\end{definitions}
In Example \ref{Exa sign mea} on page \pageref{Exa sign mea} we demonstrate
how also the concepts of positive measure and of {\sl signed measure\kern0.15mm} in
the sense of \cite[5.6\kern0.37mm, p.\ 137]{Du} can be subsumed in Definitions \ref{df vec mea}
above. By a {\it real measure\kern0.37mm} we mean any countably \mathss37{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R
}--\,additive \linebreak
$m\kern0.37mm$ such that \math{{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra. The
definition of {\it complex measure\kern0.37mm} is obtained by taking here \math{
\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} in place of \mathss36{\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}. \vskip.3mm
The essential content of \cite[Lemma 5.3\kern0.37mm, p.\ 133]{Phil} is reformulated
in the next
\begin{lemma}\label{Le Phi5.3}
Let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a
compatible norm for $\,E${\,\rm, }and let $\,m$ be absolutely $\,\mu\,$--\,
continuous in $\,E$ with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and \ú$\,\mu\fvalss01{}^{}\Cal Omega <
\infty$ for the set \ú$\,{}^{}\Cal Omega=\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then for every \ú$\,
\smb M\in\rbb R^+$ there exist some \ú$A\ar 0\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and a countable set \ú$\,
\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,\scrmt A\cupss31\{\,A\ar 0\kern0.15mm\}$ disjoint and
also with \ú${}^{}\Cal Omega = \bigcup\,\scrmt A\cupss31 A\ar 0$ and such that for all $\,
A\,,\kern0.15mm B$ {\rm\inskipline{.7}2
(1) \ }$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\le\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1
${\rm,\inskipline{.4}2
(2) \ }$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,A\subseteq B\in\scrmt A \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1
$.
\end{lemma}
\begin{proof} We first note that in the case \math{\bosy K=\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C} we have \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} also a compatible norm for the realification \math{E\Reit4} of \math{E}
and hence we may without loss of generality assume that \math{\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R}
holds. Now we let \math{\roman P\,B\,u} mean that \math{B\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \ú$\kern0.37mm
u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) $ \linebreak
with \math{
\sup\kern0.37mm\big\{\KPt8|\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1|:x\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I\,\}
\le 1 } and that for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B } we have \linebreak
\ú$
u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \kern0.37mm $
and that \math{ u\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) } holds for all \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}
with \linebreak
\ú$A\capss31 B=\emptyset \,$. Also let \mathss38{ \scrmt A\kern0.15mm\ar 0 =
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm\{\,A:\eexi{B\kern0.37mm,\kern0.15mm u}\,A\subseteq B\kern0.37mm
\text{ and }\kern0.37mm\roman P\,B\,u\KPt8\} }. By con- sidering the set \math{
\scrmt P} of disjoint subsets \math{\scrmt A} of \math{\scrmt A\kern0.15mm\ar 0}
partially ordered by inclusion, from \linebreak
{\sl Zorn's lemma\kern0.15mm} we get existence of some
maximal \math{\scrmt A} of \mathss30{\scrmt P}. Then by \math{
\mu\fvalss02{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty } we \linebreak
see that \math{\scrmt A} is countable,
and we take \mathss36{ A\ar 0 = {}^{}\Cal Omega\kern0.07mm\setminus\bigcup\,\scrmt A }.
Now, for the proof (1) letting \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 } we
first note (\kern0.15mm$*$\kern0.15mm) that by \linebreak
maximality of \math{\scrmt A} there cannot
exist \math{B\kern0.37mm,\kern0.15mm u} with \math{\roman P\,B\,u} and \mathss31{A\capss31 B
\in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ }. Fur- \linebreak
thermore, by Hahn\,--\,Banach it
suffices for arbitrarily fixed \math{u\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) }
with norm \math{ \sup \kern0.37mm \big\{ \KPt8 |\KP1 u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x\KP1| : x \in
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb24 I\,\} \le 1 } to verify that \math{
u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le \smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) } holds. \linebreak
Since \math{u\circss01 m\KP1|\KP1{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } is absolutely \mathss37{\mu
}--\,continuous, there is a Radon\,--\,Nikodym deri- vative of it, and by
considering one such we see existence of \math{B} with \mathss36{
\roman P\,B\,u}. Then by (\kern0.15mm$*$\kern0.15mm) we have \math{ A\capss31 B \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } whence with \math{ A\ar 1 =
A\kern0.07mm\setminus B } we finally get \vskip.3mm\centerline{$
u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A = u\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ar 1 \le
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ar 1) =
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $.} \vskip.3mm
For the proof of (2) letting \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} with \mathss36{ A \subseteq B \in
\scrmt A\subseteq\scrmt A\kern0.15mm\ar 0 }, \,there are some \linebreak
\ú$B\ar 1\kern0.37mm$ and \math{u} with \math{
\roman P\,B\ar 1\kern0.37mm u} and \mathss34{B\subseteq B\ar 1}. Then we also have \math{
A\subseteq B\ar 1} and consequently \linebreak
$u\circ m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \ge
\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \kern0.37mm $ whence further \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\ge\smb M\KP1(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) }
trivially follows.
\end{proof}
The essential content of \cite[Lemma 5.4\kern0.37mm, p.\ 133]{Phil} is reformulated
in the next
\begin{proposition}\label{Pro Phi5.4}
Let \ú$\,E\in\roman{LCS}\kern0.4mmps0(K)$ be normable with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ and $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a
compatible norm for $\,E${\,\rm, }and let $\,m$ be absolutely $\,\mu\,$--\,
continuous in $\,E$ with \ú$\,\mu\fvalss01{}^{}\Cal Omega < \infty$ for \ú$\, {}^{}\Cal Omega =
\bigcup\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Also let \ú$\,1\le p < \lower1.05mm\hbox{$^+$}\infty$ and let $\,m$ have
bounded $\,\mu\,$--$\RHB{.3}{\KP{1.5} ^p \kern0.37mm}$variation in $\,E\,$. Then
there is a decreasing \ú$\,\bmii8 A\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 A\kern0.37mm)=0$ and such that $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\le i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)$ holds
for $\,i\in\mathbb No$ and $\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with $\,
A\capss32(\kern0.37mm\bmii8 A\fvalss01 i\kern0.37mm)=\emptyset\,$.
\end{proposition}
\begin{proof} We first note that the requirement \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m} in
Lemma \ref{Le Phi5.3} holds since from (4) and (5) in Definitions \ref{df vec mea}
we get \math{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m } and \mathss30{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} \mathss04{\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m }. Now, for each fixed \math{
i\in\mathbb No} taking \math{ \smb M = i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056} } in Lemma \ref{Le Phi5.3}
above, by {\sl countable choice\kern0.15mm} we get existence of \math{ \scrb8 A \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } with the property that for every \math{i\in\mathbb No} we
have \math{\scrb8 A\fvalss11 i} countable and disjoint and such that for all \math{
A\,,\kern0.15mm B} and for \math{ A\kern0.15mm\ar 0 =
{}^{}\Cal Omega\kern0.07mm\setminus\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } we have \inskipline{.7}3
(a) \ $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 0 \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $, \inskipline{.4}3
(b) \ $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,A\subseteq B\in\scrb8 A\fvalss11 i \impss33
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \ge
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,(\kern0.37mm\mu\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) \KP1 $. \inskipline{.7}0
Then we take \ $\bmii8 B \kern0.37mm = \kern0.37mm
\big\langle\KPt8\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) : i\in\mathbb No\,\rangle \,$
and \inskipline{.5}{23}
$ \bmii8 A \kern0.37mm = \kern0.37mm
\big\langle\KPt8\bigcup\KP1(\kern0.37mm\bmii8 B\KP1|\KP1(\kern0.37mm
\mathbb No\kern-0.3mm\setminus i\kern0.37mm)) : i\in\mathbb No\,\rangle \KP1 $. \inskipline{.4}0
It is now clear that \math{\bmii8 A} is decreasing with \mathss36{\bmii8 A \in
\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and for the proof of the remaining required
properties we proceed as follows.
By the bounded variation property there is \math{\smb M\aR 1\in\rbb R^+ } such
that for all \math{i\in\mathbb No} and for all finite \math{\scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.15mm(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } in view
of (b) above we have \vskip.5mm\centerline{$
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^p}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\bigcup\,\scrmt A\kern0.37mm)
\le \sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}((\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\KP1^{1\kern0.37mm-\kern0.37mm p}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm
)\RHB{.3}{\KP1^p}\kern0.15mm\big)
\le \smb M\aR 1 \, $,} \inskipline{.5}0
and hence \mathss36{\mu\circss11\bmii8 B\fvalss21 i \le
\smb M\aR 1\,i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\RHB{.3}{\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p}} }, \,whence further \mathss36{
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 B\kern0.37mm) = 0 }. Next considering \mathss03{B\ar 1=
\bmii8 B\fvalss21 i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\setminus(\kern0.37mm\bmii8 B\fvalss21 i\kern0.37mm) } for
all \math{A\in\scrb8 A\fvalss11 i\kern0.37mm\lower1mm\hbox{$^{^+}$} } by both (a) and (b) above we
have \vskip.5mm\centerline{$
(\kern0.37mm i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm + 1\kern0.37mm)\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
A\capss31 B\ar 1))
\le \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)
\le i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1)) $} \inskipline{.5}0
and hence \math{\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\capss31 B\ar 1) = 0 } whence further \mathss36{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\ar 1 = 0 }. Now for every \math{i\in\mathbb No} we have \math{
\mu\kern0.07mm\circ\bmii8 A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i
\le \mu\kern0.07mm\circ\bmii8 B\fvalss21 i +
\sum_{\,j\kern0.37mm\in\KPt5{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.15mm\setminus\kern0.37mm i\,}(\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
\bmii8 B\fvalss20 j\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern-0.2mm\setminus(\kern0.37mm\bmii8 B\fvalss20 j\kern0.37mm)
))
= \mu\kern0.07mm\circ\bmii8 B\fvalss21 i} and hence we obtain \mathss36{
\lim\,(\kern0.37mm\mu\kern0.07mm\circ\bmii8 A\kern0.37mm) = 0 }. For the remaining property letting \math{
i\in\mathbb No} and \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} with \mathss38{ \emptyset =
A\capss32(\kern0.37mm\bmii8 A\fvalss01 i\kern0.37mm) =
A\capss34\bigcup\KP1(\kern0.37mm\bmii8 B\KP1|\KP1(\kern0.37mm\mathbb No\kern-0.3mm\setminus i\kern0.37mm))
}, \,we hence also have \mathss30{ \emptyset =
A\capss32(\kern0.37mm\bmii8 B\fvalss21 i\kern0.37mm) } \mathss03{{\KN{.7}} =
A\capss34\bigcup\KP1(\kern0.15mm\scrb8 A\fvalss11 i\kern0.37mm) } and consequently by (a) we
obtain \mathss38{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
i\kern0.37mm\lower1mm\hbox{$^{^+}$}\,(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) }.
\end{proof}
\begin{proposition}\label{Pro mA=int ev_x c mu}
Let $\,\mu$ be a positive measure with \ú$\,\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu
< \lower1.05mm\hbox{$^+$}\infty${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\, \vPi \in
\roman{BaS}\kern0.4mmps0(K)$ be such that either $\,\vPi$ is reflexive or $\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a
separable topology. Also let \ú$\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)}$ where $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ is a compatible norm
for $\,\vPi${\kern0.15mm\rm, }and let $\,m$ be a $\,
\vPi\dlbetss01\,$--\,measure with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}} m={{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and such that
\ú$\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ holds
for all \ú$\,A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. Then there are $\,y\kern0.37mm,\kern0.15mm S$ such that
$\,S$ is a separable closed linear subspace in $\,\vPi\dlbetss01$ and
\ú$\,
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ is
simply measurable and Pettis with
\ú$\, m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu$ for all
\ú$\, A
\in{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and \ú$\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi${\kern0.15mm\rm, }and in addition such that also
\ú$\,{}^{}{\rm rng}\,{}_{{}^{}} y\subseteq S$ holds and
\ú$\,(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ is simply
measurable in the case where $\,\vPi$ is reflexive.
\end{proposition}
\begin{proof} Let \math{{}^{}\Cal Omega=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} and \mathss38{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }. Putting
\newline
\math{\roman x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0} and
\math{\eightroman X\,A=\uniqset\smb\Phii:
\roman x\,A\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} let \math{D}
be the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span of
\mathss38{\{\KPt8\eightroman X\,A:
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }. Thus
\math{D} is the set of all \math{\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} such that
there is \math{\varphi\in\smb\Phii} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\varphi} finite. We know that
\math{D} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense. Then
we let \math{\smb V\aR 0} be the unique linear extension
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E_{\KPt8|\,D}\to\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm) } of
\newline
\mathss38{\{\,(\kern0.37mm\eightroman X\,A
\,,\kern0.07mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm):
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }, \,noting
that by the assumptions on \math{m} we indeed get a linear map.
Now for finite functions \math{\bosy s\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\bosy s} disjoint and
\newline
\math{\varphi=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.37mm{}^{}\Cal Omega\kern0.15mm]_{vs}\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1\roman x\,A
\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E } we obtain \vskip.5mm\centerline{$
|\KP1\smb V\aR 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\fvalss01\xi\KP1|=
\big|\,
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\kern0.37mm)\KP1|\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1|\KPp1.2
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi\KP1| $} \inskipline1{21.6}
${}\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)=
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss01\xi\kern0.37mm)\KP1\|\,\varphi\,\|\Lnorss33^1_\mu \, $. \inskipline{.7}0
Consequently \math{\smb V\aR 0} has a unique continuous extension \mathss38{
\smb V\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }.
Now assuming that \math{\vPi} is reflexive and taking
\mathss38{K=\{\KPt8\eightroman X\,A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} }, \,from
Lemma \ref{Le L^1_si-compa} on page \pageref{Le L^1_si-compa} above
we see that
\math{K}
is relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact. Since by
Corollary \ref{Cor L^1 is D-S} on page \pageref{Cor L^1 is D-S} above
\math{E} is a \erm{DP\,}--\,space, noting that by reflexivity
of \math{\vPi} all bounded sets in \math{\vPi\dlbetss01} are relatively
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm((\kern0.15mm\vPi\dlbetss01\kern0.15mm)\subsigma)}--\,compact, we
see that \math{\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm K} is
relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)}--\,compact.
Since the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span of
\math{K} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense, it follows that
\math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.07mm\vPi\dlbetss01\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32{}^{}{\rm rng}\,{}_{{}^{}}\smb V } is a
separable topology. Taking \math{S=
\CltaurdvPidualbeta{}^{}{\rm rng}\,{}_{{}^{}}\smb V} hence by
Proposition \ref{Pro Edw 8.17.8} on page \pageref{Pro Edw 8.17.8} above
there is $y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with \math{
{}^{}{\rm rng}\,{}_{{}^{}} y\ar 1\subseteq S} and (\kern0.15mm$*$\kern0.15mm) that \math{
\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern-0.63mm\cdot\kern0.07mm\varphi\rmdss21\mu
} holds for \math{
\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. By Pettis'
theorem and reflexivity of \math{\vPi} in fact \math{
y\ar 1 \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)
} holds. Hence
there is some \math{N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that
for \math{B={}^{}\Cal Omega\kern0.07mm\setminus N} we have \math{
(\kern0.37mm y\ar 1\kern0.37mm|\KP1 B\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
simply measurable, and so taking
\newline
\math{y=N\kern-.2mm\times\kern-.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss22
(\kern0.37mm y\ar 1\kern0.37mm|\KP1 B\kern0.37mm) } we get
\math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } simply measurable. To
conclude the proof in the reflexive case, it suffices to take
\math{\varphi=\roman x\,A} in (\kern0.15mm$*$\kern0.15mm) above.
In the separable case we instead apply Proposition \ref{Pro Edw 8.17.6} on
page \pageref{Pro Edw 8.17.6} above to get existence of \math{y\ar 1} with
(\kern0.15mm$*$\kern0.15mm) above. To see that \math{y\ar 1} can be modified on a set of
measure zero to get some \math{y} with \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } simply measurable, we proceed as
follows. We take a countable \math{D\ar 1} such that \math{D\ar 1} is \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi}--\,dense. For every fixed \linebreak
\mathss03{\xi\in D\ar 1} we now know
that \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\kern0.37mm;\kern0.07mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }
is almost measurable, and hence there is some \math{ N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\,|\KP1 B\,;\kern0.07mm
\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } for \math{ B =
{}^{}\Cal Omega\kern0.07mm\setminus N\aar 1 } is measurable. Since \math{D\ar 1} is
countable, by {\sl countable choice\kern0.15mm} we then find \mathss30{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } \linebreak
such that with \math{ B =
{}^{}\Cal Omega\kern0.07mm\setminus N } we have \math{
(\,\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\,|\KP1 B\,;\kern0.07mm
\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) } measurable for all \linebreak \mathss04{
\xi\in D\ar 1 }. By density, we can extend this to hold for all \mathss31{
\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. By Proposition \ref{pro-mea-equ} on page \pageref{pro-mea-equ}
above, this gives that \math{
(\kern0.37mm y\ar 1\,|\KP1 B\,;\kern0.07mm\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm B\kern0.37mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
is simply measurable, and so it suffices to take \math{y} as in the reflexive
case above.
\end{proof}
From the logical point of view, note that in the nonreflexive case in
Proposition \ref{Pro mA=int ev_x c mu} we may trivially take for example \mathss38{
S=\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\} }. We give below an alternative
proof for the existence of \math{y} above. It has the drawback of not giving
existence of the separable \math{S} that allowed us to deduce the stronger
measurability in the reflexive case. The underlying argument of applying
Alaoglu's theorem is already shortly sketched in \cite[p.\ 131]{Phil}\,, and
in a more explicit manner it is also utilized in
\cite[pp.\ 594\,--\,595]{Edw}\,. This alternative in fact was our first
approach but then we noticed that using Propositions \ref{Pro Edw 8.17.8} and
\ref{Pro Edw 8.17.6} offers a more uniform way to treating the cases (5) and
(6) in Theorem \nfss A\,\ref{main Th} together.
\vskip1mm
{\it$\null
$
Let $\,\mu$ be a positive measure on
$\,{}^{}\Cal Omega$ with $\,\mu\fvalss02{}^{}\Cal Omega < \lower1.05mm\hbox{$^+$}\infty${\,\rm, }and
with
\linebreak
$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,E\in\roman{BaS}\kern0.4mmps0(K)$
with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1$ a compatible dual norm for
$\,F=E\dlbetss12\,$.
Also let $\,m$ be an $\,F\,$--\,measure with $\,{{}^{}{\rm dom}\,{}_{{}^{}}} m={{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and
such that $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$
holds for all $\,A \in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$.
Then there is $\, c \in \kern0.15mm ^{}^{}\Cal Omega\,\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm)$ such
that $\,(\,c\KPt8;\kern0.15mm\mu\,,\kern0.07mm E\dlsigss12\kern0.07mm)$ is Pettis and such that
$\, m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x =
\int_{\,A}\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm
\circ\kern0.15mm c\rmdss11\mu$ holds for all \ú$\, A
\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ and $\,x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\,$.
}
\begin{proof} The assertion being trivial if \math{\mu\fvalss02{}^{}\Cal Omega=0}
holds, assuming \math{\mu\fvalss02{}^{}\Cal Omega > 0} we consider the net \math{
(\kern0.15mm\varDelta\,,\kern0.07mm\bosy c\kern0.37mm) } obtained as follows.
Let $\varDelta$ be the set of all pairs
$(\kern0.15mm\scrmt A\,,\kern0.07mm\scrmt B\kern0.37mm)$ where
\math{\scrmt A\,,\kern0.15mm\scrmt B\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} m\setminus\kern0.15mm
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } are finite partitions of
${}^{}\Cal Omega$ such that for every $B\in\scrmt B$ there is some $A\in\scrmt A$
with $B\subseteq A \,$. Then $\varDelta$ is a direction, and we take \vskip.3mm\centerline{$
\bosy c=\langle\KP1
{}^{}\Cal Omega\times\hbox{\font\SweD =cmssbx10\SweD U}{}\capss31\{\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm u\kern0.37mm):
\all A\,\eta\in A\in\scrmt A\impss33
u=(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KPt9\}:\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta
\KP{1.3}\rangle $} \inskipline{.3}0
thus obtaining a function ${{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta\to
\kern0.15mm^{}^{}\Cal Omega\,\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.15mm\bosy K\kern0.37mm)$
such that for every $\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta$ we have
$\bosy c\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\scrmt A$ the function ${}^{}\Cal Omega\owns\eta\mapsto
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) $
when $\eta\in A\in\scrmt A$ holds.
We further let \math{\varLambda} be the set of all pairs \math{
(\kern0.37mm\eta\kern0.37mm,\kern0.07mm u\kern0.37mm)\in{}^{}\Cal Omega\times\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm\bosy K\kern0.37mm) }
such that \math{u} is a \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\dlsigss12\kern0.07mm) }--\,limit point
of the net \mathss38{ (\kern0.15mm \varDelta \, , \kern0.07mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\kern0.37mm) }. Then by {\sl Alaoglu's theorem\kern0.15mm}
we have \mathss36{{}^{}\Cal Omega\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\varLambda }, \,and hence by the {\sl axiom of
choice\kern0.15mm} there is a function \math{c\subseteq\varLambda} with \mathss06{ {{}^{}{\rm dom}\,{}_{{}^{}}} c =
{}^{}\Cal Omega}. Arbitrarily fixing \mathss34{x\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E}, \,it remains to show that
\math{\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c } is inte- grable over every \mathss36{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}, \,and that \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x =
\int_{\,A}\kern0.37mm\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\rmdss11\mu } holds.
To see this, we let \math{\varphi} be a Radon\,--\,Nikodym derivative
with respect to $\mu$ of
${{}^{}{\rm dom}\,{}_{{}^{}}}\mu\owns A\mapsto
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x \,$,
noting that some such exist since by our assumption
for some $\smb M\in\rbb R^+$ we have
$|\KP1 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x\KP1|
\le\smb M\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)$ for all
$A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\,$. For the same reason we may assume that
$|\KP1\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\le\smb M$ holds for all $
\eta\in {}^{}\Cal Omega\,$. We now have
$m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss10 x=
\int_{\,A}\kern0.37mm\varphi\rmdss11\mu$
for $A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\, $, and it suffices to show
existence of some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ such that
$\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ holds for all
$\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N\kern0.37mm$.
By taking inverse images under $\varphi$ of partitions of
$\mathbb C\capss31\{\,z:|\,z\,|\le\smb M\KPt8\}$ into sets of diameter
${}<i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$, we obtain a sequence
$\bosy s\ar 1$ of simple functions such that
$|\KP{1.1}\bosy s\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\fvalss10\eta - \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP{1.2}|
< i\kern0.37mm\lower1mm\hbox{$^{^+}$}\kern0.15mm\kern.2mm\raise1.9mm\hbox{\font\SweD =cmb10\SweD \char'056}\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}$ holds for all
$i\in\mathbb No$ and $\eta\in {}^{}\Cal Omega\,$.
If $\sigma\ar 1\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s\ar 1$ is such that $
\sigma\ar 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ holds for
some $s\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma\ar 1\,$, on a set of measure zero we can modify
$\sigma\ar 1$ to get another simple function $\sigma$
such that for every
$s\in{}^{}{\rm rng}\,{}_{{}^{}}\sigma$ we have $\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\,$.
Using this observation in conjunction with
{\sl countable choice\kern0.15mm} we obtain another sequence $\bosy s$ of
simple functions and some $N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
$ such that for all $\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N$ and $i\in\mathbb No$ we have
$\bosy s\fvalss01 i\fvalss10\eta=
\bosy s\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\fvalss10\eta\,$.
Now arbitrarily given \math{\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N} and \math{\varepsilon\in
\rbb R^+} we pick some $\sigma\in{}^{}{\rm rng}\,{}_{{}^{}}\bosy s$ such that for all $\eta\ar 1\in
{}^{}\Cal Omega\kern0.07mm\setminus N$ we have
$|\KP1\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.2mm - \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\,| < \varepsilon \,$. Then
with $A=\sigma\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,\sigma\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,\}$ we take either
$\scrmt A=\{\,A\,,\kern0.07mm {}^{}\Cal Omega\kern0.07mm\setminus A\KPt9\}$
or $\scrmt A=\{\kern0.37mm A\,\}$
according to whether
$A\not={}^{}\Cal Omega$ or $A={}^{}\Cal Omega$ holds, getting then
$\scrmt A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\varDelta$ by construction.
If now $\eta\in B\in\scrmt B\in\varDelta\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\,\scrmt A\,\}$
holds, we have $B\subseteq A$ and hence \vskip.5mm\centerline{$
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\fvalss02\scrmt B
=\bosy c\fvalss02\scrmt B\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss02\xi=
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\fvalss11\xi\kern0.37mm)=
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\int_{\KP1 B}\kern0.37mm\varphi\rmdss11\mu $} \inskipline{.5}0
further giving
$|\KP{1.2}
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\fvalss02\scrmt B
- \varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP{1.2}| < 2\KP1\varepsilon \,$.
Since $c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x$ is a $\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\,$--\,limit point of
the net
$(\kern0.15mm\varDelta\,,\kern0.07mm
\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm
\roman{ev}\sbi\eta\kern-0.2mm\circ\kern0.07mm\bosy c\kern0.37mm) \KPt8 $, this gives
$c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} x=\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\,$, and having
here $\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N$ arbitrarily fixed, we see that
$\roman{ev}\kern0.15mm\sbi{\emath x}\kern-0.2mm\circ\kern0.15mm c\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=
\varphi\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta$ holds for all
$\eta\in {}^{}\Cal Omega\kern0.07mm\setminus N\kern0.37mm$.
\end{proof}
\begin{corollary}\label{Coro q-var}
Let \ú$\,1\le q<\lower1.05mm\hbox{$^+$}\infty$ and let $\,\mu$ be a positive measure on $\,{}^{}\Cal Omega
${\,\rm, }and with \ú$\,\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,\vPi\in\roman{BaS}\kern0.4mmps0(K)$ be such
that either \ú$\,\vPi$ is reflexive or \ú$\,\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi$ is a separable
topology. Also let $\,m$ be a $\,\vPi\dlbetss01\,$--\,measure with \ú$\,{{}^{}{\rm dom}\,{}_{{}^{}}} m
= \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+$ and such that $\,m$ is absolutely $\,\mu\,
$--\,continuous in $\,\vPi\dlbetss01$ with $\,m$ having bounded $\,\mu\,$--$
\RHB{.3}{\KP{1.5} ^q \kern0.37mm}$variation in $\,\vPi\dlbetss01\,$. Then there are
some countable disjoint $\,\scrmt A$ and $\,y$ with \ú$\, \scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ and \ú$
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ simply measurable and such that
{\,\rm(1)} and {\,\rm(2)} and {\,\rm(3)} and {\,\rm(4)} below hold for all $\,
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m$ and $\,A\kern0.07mm\ar 1\in\scrmt A$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi$ and $\,\eta
\in{}^{}\Cal Omega\,$. {\rm\inskipline14
(1)} \ $\eta\not\in\bigcup\,\scrmt A\impss33
y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 ${\rm, \inskipline{.5}4
(2)} \ $\bigcup\,\scrmt A\capss31 A=\emptyset\impss33 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A =
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \KP1 ${\rm, \inskipline{.5}4
(3)} \ $A\subseteq A\kern0.07mm\ar 1\impss33 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \KPt6 ${\rm, \inskipline{.5}4
(4)} \ $\vPi$ is reflexive $\impss33
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm)$ is simply measurable. \vskip1mm
\end{corollary}
\begin{proof} Let \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1=\seqss44{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
:u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)} } where
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} is some fix- ed compatible norm
for \mathss31{\vPi}. We first show that there is a countable disjoint
\math{\scrmt C\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ } such that
\math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A=\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } for all
\math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} with
\mathss36{\bigcup\,\scrmt C\capss31 A=\emptyset}. To see
this, with \math{
\scrmt A\ar 1=
{{}^{}{\rm dom}\,{}_{{}^{}}} m\capss21\{\,A:\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm
m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\not=0\KPt9\} }
\newline
we let \mathss38{\Cal P=\{\,(\kern0.15mm\scrmt A\,,\kern0.07mm\scrmt B\kern0.37mm)
:\scrmt A\,,\kern0.15mm\scrmt B\kern0.37mm\text{ are disjoint and }\kern0.37mm
\scrmt A\subseteq\scrmt B\subseteq\scrmt A\ar 1\,\} }.
Then $\Cal P$ is a nonempty partial order, and if
$\Cal C$ is a $\Cal P\,$--\,chain, then
$\bigcup\KP1\Cal C$ is an upper $\Cal P\,$--\,bound. Hence by
{\sl Zorn's lemma\kern0.15mm} there exists some $\Cal P\,$--\,maximal $\scrmt C\kern0.37mm$.
Clearly $\scrmt C$ is as required {\sl if it is countable\kern0.15mm}. To verify this,
we note that $\scrmt C
=\{\KP1\roman C\,n:n\in\rbb Z^+\kern0.15mm\big\}$ when
$\roman C\,n$ is the set of all $A\in\scrmt C$
with \mathss31{n^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1} <
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^{1\kern0.37mm-\,q}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm
m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)\RHB{.3}{\KPt8^q} }. If $\roman C\,n$
is finite for every $n\in\rbb Z^+$, then $\scrmt C$ is countable.
If $\roman C\,n$ is
infinite for some $n\in\rbb Z^+$, we
get a contradiction with the assumption that
\math{m} has bounded \mathss37{\mu}--$
\RHB{.3}{\KP{1.5} ^q \kern0.37mm}$variation in \mathss34{\vPi\dlbetss01}.
Next, using Proposition \ref{Pro Phi5.4} on page \pageref{Pro Phi5.4}
above we find a countable disjoint
$\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ with
$\bigcup\,\scrmt A=\bigcup\,\scrmt C$ and such that
$\sup\KPt8\{\KPt8(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) :
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.37mm\} < \lower1.05mm\hbox{$^+$}\infty$
holds for every fixed \mathss36{A\ar 1\in\scrmt A}.
Indeed, we just apply Proposition \ref{Pro Phi5.4} separately to
\math{\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm C}
for every fixed \math{C\in\scrmt C} and then take
the union of the thus obtained partitions.
Finally we let \math{\scrmt Y} be the set of all
pairs \math{
(\kern0.15mm A\ar 1\kern0.15mm,\kern0.07mm y\ar 1) } with \mathss03{ A\ar 1 \in
\scrmt A} and such that for \mathss03{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and
\math{m\ar 1 = m\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} we have \math{
(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlsigss00\kern0.07mm) }
simply measurable and
Pettis with \math{m\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\rmdss01\mu }
for all \math{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }, \,and such that also
\math{(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is simply
measurable if \math{\vPi} is reflexive. Then
considering arbitrarily fixed \math{
A\ar 1\in\scrmt A} and with \vskip.3mm\centerline{$
\smb M=\sup\KPt8\{\KPt8(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)^{\,\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm) :
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern-0.3mm\cap\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1\kern0.37mm\} $} \inskipline{.3}0
taking \mathss03{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mmrim2 =
\{\,(\kern0.37mm\xi\,,\kern0.07mm\smb M\KPt8 t\kern0.37mm) :
(\kern0.37mm\xi\,,\kern0.07mm t\kern0.37mm)\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt8\} } in place of
\math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm} in
Proposition \ref{Pro mA=int ev_x c mu}
on page \pageref{Pro mA=int ev_x c mu} above,
we see that \mathss03{ \scrmt A
\subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y } holds, and hence by {\sl countable choice\kern0.15mm} there
is
a function \mathss30{\scrmt Y\ar 1\subseteq\scrmt Y} with \mathss34{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1}. Now taking \vskip.3mm\centerline{$
y = (\kern0.37mm{}^{}\Cal Omega\setminus\bigcup\,\scrmt A\,)
\times\kern-0.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss24
\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y\ar 1 \kern0.37mm $,} \inskipline{.3}0
it is clear that all the asserted properties hold.
\end{proof}
Although we shall not need below the result, as an application of the
Dunford\,--\,Pettis property of \math{\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } we reformulate
the a bit mysterious looking assertion \vskip.5mm\centerline{
\q{if $|\tau_0|\tau<\infty$, then $[x(\tau)|\tau\subset\tau_0]$ is compact
valued}} \inskipline{.5}0
from \cite[p.\ 131]{Phil} in the following
\begin{proposition}
Let $\,\mu$ be a positive measure with \ú$\,\sup{}^{}{\rm rng}\,{}_{{}^{}}\mu < \lower1.05mm\hbox{$^+$}\infty${\,\rm, }
and with \ú$\,\bosy K\in{\kern-0.63mm}$ $\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\}$ let \ú$\,F\in\roman{BaS}\kern0.4mmps0(K)$ be reflexive
with $\,\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm$ a compatible norm. Also let \ú$\,m \in{\kern-0.63mm}$ \ú$
^{\roman{dom\,}\mu}\,\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E$ be such that \ú$\,m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss31 B\kern0.37mm)
= (\kern0.37mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A + m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} B\kern0.37mm)\svs E$ and \ú$\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A \le \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ hold for all \ú$\,A\,,\kern0.15mm B
\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ with \ú$\,A\capss31 B=\emptyset\,$. Then $\,{}^{}{\rm rng}\,{}_{{}^{}} m$ is relatively $\,
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\,$--\,compact.
\end{proposition}
\begin{proof} Let \math{{}^{}\Cal Omega=\kern0.15mm\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \mathss38{ E =
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }. Also
putting \math{\roman x\,A=
(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22
(\kern0.15mm A\times\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0}
and \math{\eightroman X\,A=\uniqset\smb\Phii:
\roman x\,A\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} let \math{S}
be the linear \mathss37{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,span
of
\mathss38{\{\KPt8\eightroman X\,A:
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }. Thus
\math{S} is the set of all \math{\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} such that
there is \math{\varphi\in\smb\Phii} with
\math{{}^{}{\rm rng}\,{}_{{}^{}}\varphi} finite. We know that
\math{S} is \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E}--\,dense. Then
we let \math{\smb V\aR 0} be the unique linear extension
\math{\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm E_{\KPt8|\,S}\to\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F } of
\mathss38{\{\,(\kern0.37mm\eightroman X\,A
\,,\kern0.07mm m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm):
A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} }, \,noting that by the assumptions
on \math{m} we indeed get a linear map.
For finite functions \math{\bosy s\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 1\kern-0.3mm\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\bosy K} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\bosy s} disjoint
and
\newline
\math{\varphi=
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\bosy K\expnota^\kern0.15mm\aars A_1]_{vs}\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1\roman x\,A
\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E }
and for \math{u\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } with \math{
\sup\KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm u\circss01\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I)
\le 1 } we obtain \vskip.5mm
$|\KP1 u\circss00 \smb V\aR 0\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb\Phii\KP1|=
\big|\,
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}
(\kern0.37mm\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1
(\kern0.37mm u\circss00 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1|\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1|\KP1
u\circss00 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1| $ \inskipline1{11}
${}\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\le
\sum_{\,A\kern0.37mm\in\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii8 s\,}|\KP1
\bosy s\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\KP1|\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)=
\|\,\varphi\,\|\Lnorss33^1_{\aars\mu_1} \, $. \inskipline{.7}0
Consequently \math{\smb V\aR 0} has a unique continuous extension \mathss38{
\smb V\in\Cal L\,(\kern0.15mm E\kern0.37mm,\kern0.07mm F\kern0.37mm) }.
Taking \mathss38{K=\{\KPt8\eightroman X\,A:A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\KPt8\} }, \,from Lemma \ref{Le L^1_si-compa}
on page \pageref{Le L^1_si-compa} above we see that \math{K} is relatively \mathss37{
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm E\subsigrs04)}--\,compact. Since by
Corollary \ref{Cor L^1 is D-S} on page \pageref{Cor L^1 is D-S} above
\math{E} is a \erm{DP\,}--\,space, noting that by reflexivity of
\math{F} all bounded sets in \math{F} are relatively
\mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(F\!\RHB{.25}{\subsigma})}--\,compact, we
see that \math{\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} is
relatively \mathss37{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F}--\,compact. Noting that also
\math{{}^{}{\rm rng}\,{}_{{}^{}} m\subseteq\smb V\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm K} holds,
we are done.
\end{proof}
\Ssubhead D Duality of Bochner spaces \label{Sec D}
Proceeding by a sequence of lemmas, we here give the proof of
Theorem \nfss A\,\ref{main Th} on page \pageref{main Th} above. From now on
untill the end of the proof of Lemma \nfss A\,\ref{final lemma} on
page \pageref{endmpf} below, without further mention we let \math{p\,,\kern0.15mm
\bosy K\kern0.37mm,\kern0.15mm\vPi\kern0.15mm,\kern0.15mm\mu\,,\kern0.15mm{}^{}\Cal Omega
\,,\kern0.15mm F\kern0.15mm,\kern0.15mm F\aar 1\kern0.37mm,\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } be as in
Theorem \nfss A\,\ref{main Th}\kern0.15mm. For short, we call this assumption
together with the temporary shorthands below
{\it Assumptions
\hbox{\font\≈=cmssi9\≈A}\kern0.15mm}. From Corollary \ref{Coro Io inj etc}
on page \pageref{Coro Io inj etc} above, we see that \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is an
injective continuous linear map \mathss34{F\aar 1\to F\dlbetss10 }. Since by
Theorem \ref{Th L_s^p Ba} and Corollary \ref{Cor L^p Ban} the spaces \math{
F\aar 1} and \math{F\dlbetss10} are \erm Banachable, by the open mapping
theorem we only need to verify the surjectivity \mathss34{
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \label{page surj}
}. This we shall do separately for \mathss30{p=1} under (1) or (2) or (3) or
(4) and for \math{1 < p < \lower1.05mm\hbox{$^+$}\infty} under (5) or (6)\,.
Fixing a compatible norm \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm}
for \math{\vPi} and letting \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1} be the dual norm, we introduce the
following shorthands \vskip.6mm
$\|\,\smb X\kern0.37mm\|\sNorF = \inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1
\|\Lnorss33^p_\mu\kern-0.2mm:x\in\smb X\,\}
$ and \KP7 \vskip.4mm
$\|\,\smb Y\KPt8\|\sNorFp = \inf\kern0.15mm\big\{\KPt8\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\ar 1\kern-0.2mm\circ\kern0.15mm y\KP1
\|\Lnorss50^{p^*}_\mu\kern-0.2mm:y\in\smb Y\KPp1.1\}
$ and \KP7 \vskip.4mm
$\|\KPt8\smb U\,\| = \sup\kern0.37mm\big\{\KP1|\KP{1.1}\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\smb X\KPt9| :
\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern0.37mm\text{ and }\kern0.37mm \|\,\smb X\kern0.37mm\|\sNorF\le 1\KPt9\}
$
and \KP7 \vskip.4mm
$\roman f\,u\,\xi=\uniqset\smb X:{}$ \inskipline0{18.5}
$(\kern0.37mm{}^{}\Cal Omega\setminus\kern-0.2mm{{}^{}{\rm dom}\,{}_{{}^{}}} u\kern0.37mm)\times\kern-0.2mm\{\,\Bnull_\vPi\}\cupss22
\seqss33{((\kern0.37mm u\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\KP1\xi\kern0.37mm)\svs\vPi\kern-0.3mm:\eta\in{{}^{}{\rm dom}\,{}_{{}^{}}} u}
\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \kern0.15mm $. \inskipline{.6}0
Note that by the discussion after the proof of Lemma \ref{Le 0_{L^p}} on page \pageref{discus inf N = N}
above we in fact have \math{\|\,\smb X\kern0.37mm\|\sNorF =
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\circss01 x\KP1 \|\Lnorss33^p_\mu } for \mathss30{x\in\smb X\in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F}.
\begin{Alemma}\label{LeA(1)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(1)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Arbitrarily fix \math{ \smb U \in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } and let \math{\scrmt A} and \math{
N\kern0.15mmrim1} be as in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above. Then let \math{\scrmt Y} be the set of all pairs \math{
(\kern0.15mm A\ar 1\KPt2;\kern0.15mm y\ar 1\kern0.15mm,\kern0.07mm S\ar 1) } with \mathss03{ A\ar 1 \in
\scrmt A} and \math{{}^{}{\rm rng}\,{}_{{}^{}} y\ar 1\subseteq S\ar 1} and \math{S\ar 1} a separable
closed linear subspace in \math{\vPi\dlbetss01} and such that for \math{
\mu\ar 1=\mu\KP1|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and \mathss30{m = \langle\kern0.37mm\seqss33{
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\aars\mu_1\kern0.07mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A \kern-0.2mm
: \xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} : A \in \mu\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.37mm
\big\rangle } we have \math{
(\kern0.37mm y\ar 1\KPt2;\kern0.07mm\mu\ar 1\kern0.15mm,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } measurable and
Pettis with \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\ar 1\rmdss01\mu } for all \linebreak \mathss03{
A\in{{}^{}{\rm dom}\,{}_{{}^{}}} m} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. Now considering arbitrarily fixed \math{
A\ar 1\in\scrmt A} and choosing \mathss03{ \hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\kern0.15mmrim2 =
\{\,(\kern0.37mm\xi\,,\kern0.15mm t\KPt8\|\KPt8\smb U\,\|\kern0.15mm\sbig)0 :
(\kern0.37mm\xi\,,\kern0.07mm t\kern0.37mm)\in\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\KPt8\} } by Proposition \ref{Pro mA=int ev_x c mu}
on page \pageref{Pro mA=int ev_x c mu} above we see that \mathss03{ \scrmt A
\subseteq {{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y } holds, and hence by the {\sl axiom of choice\kern0.15mm} there is
a function \mathss30{\scrmt Y\ar 1\subseteq\scrmt Y} with \mathss34{ \scrmt A \subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\scrmt Y\ar 1}. Let \mathss34{ y = (\kern0.37mm{}^{}\Cal Omega\setminus N\kern0.15mmrim1\kern0.15mm)
\times\kern-0.2mm\{\KPt8\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}\cupss24
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm{}^{}{\rm rng}\,{}_{{}^{}}\scrmt Y\ar 1}.
To verify that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} holds, it suffices to get \mathss38{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu\le
\|\KPt8\smb U\,\| }. This in turn follows if for every fixed \math{A\ar 1\in
\scrmt A} we show existence of some \math{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\le\|\KPt8\smb U\,\| } holds for \mathss30{
\eta\in A\ar 1\kern-0.63mm\setminus N}. Now for \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss22\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1 }
and \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we have \math{
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
= m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss01\xi
= \int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } and hence \vskip.2mm\centerline{$
\big|\kern0.15mm\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu\KP1|
\le \|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm)\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm) \KP1 $.} \inskipline{.6}0
Then by Corollary \ref{Coro |f|<M} on page \pageref{Coro |f|<M}
above for every \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} there is \math{ N\aar 1 \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } holds for \mathss34{ \eta
\in A\ar 1\kern-0.63mm\setminus N\aar 1 }.
Now taking \math{S\ar 1=
\roman{pr}\ar 2\circ\scrmt Y\ar 1\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.07mm\ar 1 } in place of \math{
S} in Lemma \ref{Le Nu_1 = sup ...} on page \pageref{Le Nu_1 = sup ...} above,
let \math{D} be as given there. Then considering fixed \math{\xi\in D} we find \math{
N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } for all \mathss34{ \eta \in
A\ar 1\kern-0.63mm\setminus N\aar 1}. By {\sl countable choice\kern0.15mm} taking as \math{N}
the union of these \math{N\aar 1} we get \math{
|\KPp1.1 y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi\KP1| \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss02\xi\kern0.37mm) } for all \math{ \eta \in
A\ar 1\kern-0.63mm\setminus N } and \mathss34{\xi\in D}. Now having \mathss38{
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta =
\sup \KPt8(\kern0.37mm\Abrs00^1\circ\kern0.15mm(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.3mm D\kern0.37mm)
\le\|\KPt8\smb U\,\|} for all \mathss30{\eta\in A\ar 1\kern-0.63mm\setminus N}, \,the
assertion follows.
Thus having \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} there is \math{\smb Y} with \mathss34{
y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. To proceed, we first note that we now have \math{
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi = \label{Le A2 final ded}
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } for all \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+} and \mathss31{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. To see this,
let \math{\scrmt C=\scrmt A\capss31\{\,A\kern0.07mm\ar 1\kern-0.63mm:A\kern0.07mm\ar 1\kern-0.2mm\cap\kern0.15mm A
\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm\big\} } and \mathss36{N = N\kern0.15mmrim1\cup\kern0.37mm
\bigcup\KP1(\kern0.15mm\scrmt A\kern0.15mm\setminus\scrmt C\kern0.37mm)\capss21 A }. Then \math{
\scrmt C} is countable since otherwise \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}
would be contradicted. In addition \math{ N \in \bigcup\KPt8\{\KPt8
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\aar 1\kern-0.3mm:N\aar 1\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} \kern0.15mm\} } holds
with \mathss30{ A = \bigcup\KP1(\kern0.15mm\scrmt C\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss31 A\kern0.37mm)\cupss21 N}.
Now by do- \linebreak minated convergence we obtain \inskipline{.6}{20.2}
$ m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi
= \smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm
\smb U\fvalss11\lfloor\,^{1\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi
{\aars A_1\capss25 A}\kern0.15mm\sbig)0 $ \inskipline{.6}{31}
${}
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\,}
m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\kern0.07mm\ar 1\kern-0.3mm\cap\kern0.15mm A\kern0.37mm)\fvalss01\xi $ \inskipline{.4}{31}
${}
= \sum_{\,\aars A_1\kern0.15mm\in\kern0.37mm\scrm7 A\kern0.37mm}\int_{\,\aars A_1\capss25 A}\kern0.37mm
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu
= \int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu \,$. \inskipline{.8}0
Then by Lemma \ref{Le-first} on page \pageref{Le-first} above we have \vskip.3mm\centerline{$
U=
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\kern-0.2mm\times\mathbb C\capss31\{\,(\kern0.37mm\smb X\kern0.15mm,\kern0.07mm t\kern0.37mm) :
\aall{x\in\smb X}\,
t = \int_{\KPp1.1{}^{}\Cal Omega\,}y\,.\KPt8 x\rmdss11\mu\KPt9\} $} \inskipline{.5}0
and hence \math{ \smb U = \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb Y} holds and so \math{
\smb U\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is established.
\end{proof}
\begin{Alemma}\label{LeA(2)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(2)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Given \mathss38{\smb U
\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,letting \vskip.5mm\centerline{$
\smb V=
\kern0.37mm\big\langle\KPt8
\{\,(\kern0.37mm\xi\,,\kern0.07mm\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
\kern0.37mm):\vPi:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.37mm\text{ and }\kern0.37mm\varphi\in\smb\Phii\,\}
:
\smb\Phii \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)
\KP1\big\rangle \KP1 $,} \inskipline{.5}0
we easily see \math{\smb V \in
\Cal L\,(\kern0.37mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) }
to hold. Hence by
Proposition \ref{Pro Edw 8.17.6} on page \pageref{Pro Edw 8.17.6} above there
exists some \math{y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } such that \vskip.5mm\centerline{$
\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
=\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu $} \inskipline{.5}0
holds for \math{\varphi\in\smb\Phii\in{{}^{}{\rm dom}\,{}_{{}^{}}}\smb V} and \mathss31{ \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi }. Noting that from \mathss30{ y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } we directly get \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu < \lower1.05mm\hbox{$^+$}\infty } now
Lemma \ref{Le-first} gives the conclusion similarly as in the proof of Lemma \nfss A\,\ref{LeA(1)}
above.
\end{proof}
\begin{Alemma}\label{LeA(3)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(3)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Given \mathss38{\smb U
\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }, \,define $\smb V:
\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)\to\vPi\dlbetss01$ by
$\smb V\fvalss60\smb\Phii\fvalss00\xi=
\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi$ for
$\varphi \in \smb\Phii \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm)$ and $\,\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi\kern0.15mm$.
Then by
Proposition \ref{Pro Edw 8.17.8} on page \pageref{Pro Edw 8.17.8} above there
is $y \in \bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\kern-0.3mm
\mvsLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm)$ with
$\smb U\fvalss22\roman f\KPt8\varphi\KP1\xi
=\smb V\fvalss60\smb\Phii\fvalss00\xi=
\int_{\KP{1.1}{}^{}\Cal Omega\,}
\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\cdot\varphi\rmdss21\mu$ .
The rest proceeds as in the proof of Lemma \nfss A\,\ref{LeA(1)} above.
\end{proof}
\begin{Alemma}\label{LeA(4)}
If under {\,\rm Assumptions \nfss A} also {\,\rm(4)} holds{\kern0.37mm\rm, }then $\,
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} \newcommand\erU{\hbox{\font\≈=cmr8\≈U}\kern.7mm}
Arbitrarily fix \mathss38{\smb U \in
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) }. Putting \math{ G =
\mvLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) } and letting \math{c} be as
given by (4) in Theorem \nfss A\,\ref{main Th} let \math{c\ar 1 = c } if \math{
\bosy K=\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R} holds and in the complex case let \math{ c\ar 1 =
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 3\circ\kern0.15mm(\kern0.37mm c\hbox{${}\times\kern-2.7mm\lower.9mm\hbox{\font\SweD =cmr5\SweD f}\kern1.93mm$} c\kern0.37mm)\circ\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2} where with \math{
S = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) \times
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\mLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\kern0.37mm\mu\kern0.15mm) } we have \inskipline{.5}{10}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 3 = \{\,(\kern0.37mm x\kern0.37mm,\kern0.07mm y\kern0.37mm,\kern0.07mm x + \imag\KPt8 y\kern0.37mm) :
x\kern0.37mm,\kern0.15mm y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.15mm{}^{}\Cal Omega\kern0.15mm)\KPt8\} \KP{33} $ and \inskipline{.2}{10}
$\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 2 = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G\times S\capss31\{\,(\kern0.37mm\smb\Psii\,;\kern0.15mm
\smb\Psii\ar 1\kern0.15mm,\kern0.07mm\smb\Psii\aR 2\kern0.07mm) : \aall{\psi\ar 1\in\smb\Psii\ar 1\KPt2
,\kern0.15mm\psi\ar 2\in\smb\Psii\ar 2}\,
\psi\ar 1\kern-0.2mm + \imag\KPt8\psi\ar 2\in\smb\Psii\,\} \KP1 $. \inskipline{.5}0
Then \math{c\ar 1} is a continuous linear choice function \math{G\to
\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm) } and hence there is some \math{
\smb A\in\lbb R_+} with the property that \math{
\|\KP1 c\fvalss01\smb\Psii\KP1\|\lllnor_{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}
\le \smb A\KP1\|\,\psi\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu } holds for \mathss30{
\psi\in\kern-0.63mm} \mathss02{\smb\Psii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G}. Further let \math{\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1} be
as in Proposition \ref{Pro L^1'=L^i} on page \pageref{Pro L^1'=L^i} above.
Then for \mathss30{ E = \kern-0.63mm } \mathss03{\mvLrs42^1(\kern0.37mm\mu\,,\kern0.07mm\bosy K\kern0.37mm) }
with \math{\erU\xi = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\ssbb00 C\capss41\{\,
(\kern0.37mm\smb\Phii,\kern0.07mm t\kern0.37mm) : \aall{\varphi\in\smb\Phii}\, t =
\smb U\fvalss11\roman f\,\varphi\KP1\xi\KPt9\} } we obtain \math{ \smb V \in \kern-0.63mm } \mathss03{
\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.15mm\lll^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}\kern0.15mm(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\bosy K\kern0.37mm)) }
by taking \mathss38{\smb V = c\ar 1\kern-0.3mm\circ\kern0.15mm\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\circ\kern0.15mm
\seqss33{\erU\xi:\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} }.
Now for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} and \math{\varphi\in\smb\Phii\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E} we
have \math{\smb V\fvalss50\xi\in\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\erU\xi } and
hence \mathss30{\smb U\fvalss11\roman f\,\varphi\KP1\xi } \mathss03{ =
\erU\xi\kern0.15mm\fvalss10\smb\Phii =
\int_{\KP{1.1}{}^{}\Cal Omega}\kern0.15mm\smb V\fvalss50\xi\cdot\varphi\rmdss11\mu }. Taking \inskipline{.5}{12}
$\smb B = \inf\,\{\KPt8\sup\kern0.37mm\big\{\,\big|\kern0.15mm\int_{\KPp1.1{}^{}\Cal Omega\,}
\psi\cdot\varphi\rmdss21\mu\KP1| : \varphi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\kern0.37mm\text{
and }\kern0.37mm \|\,\varphi\,\|\Lnorss33^1_\mu\le 1\KPt8\}$ \inskipline{.5}{58.6}
${} : \psi\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm G\kern0.37mm\text{ and }\kern0.37mm
\|\,\psi\,\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu = 1\KPt8\} \KP1 $, \inskipline{.5}0
we have \math{\smb B\in\rbb R^+} unless \math{{}^{}\Cal Omega} is \mathss37{\mu
}--\,negligible in which case the assertion of the lemma to be proved
trivially holds. Then for \math{\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} we get \inskipline{.4}4
(\kern0.15mm$*$\kern0.15mm) \ $\|\KP1\smb V\fvalss50\xi\KP1\|\lllnor_{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty} \le
\smb A\KP1\|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\ar 1\kern-0.3mm\inve\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\erU\xi
\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le \smb A\,\smb B^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\fvalss12\xi\kern0.37mm) \KP1 $. \vskip.5mm
Now taking \mathss38{ y =
\seqss33{\roman{ev}\kern0.15mm\sbi{\eta}\kern-0.2mm\circ
\smb V
:\eta\in{}^{}\Cal Omega} }, \,trivially \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is finitely almost sca- larly
measurable, and by (\kern0.15mm$*$\kern0.15mm) above having \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.2mm\circ\kern0.15mm y\KP1\|\Lnorss33^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}_\mu \le
\smb A\,\smb B^{\kern0.37mm\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} 1}\,\|\KPt8\smb U\,\| } we get \mathss02{ y \in
\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1 }. Then the conclusion follows from Lemma \ref{Le-first}
similarly as above.
\end{proof}
As opposed to the case (1) in Lemma \nfss A\,\ref{LeA(1)} above, note that in
the cases (2) and (3) and (4) in Lemmas \nfss A\,\ref{LeA(2)} and
\nfss A\,\ref{LeA(3)} and \nfss A\,\ref{LeA(4)} we only got \math{
\|\,\smb Y\KPt8\|\sNorFp\le\smb A\KP1\|\KP1\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\smb Y\KPp1.2\| }
for all \mathss03{\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} for some \math{\smb A} with \math{
1\le\smb A<\lower1.05mm\hbox{$^+$}\infty} and possibly \math{1<\smb A}.
\begin{Alemma}\label{final lemma}
If under {\,\rm Assumptions \nfss A} also
{\,\rm(5)} or {\,\rm(6)} holds{\kern0.37mm\rm, } \inskipline{.2}{54.3}
then $\,\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm\kern0.37mm$.
\end{Alemma}
\begin{proof} Since the verification is quite long ending on page \pageref{endmpf}
below, we devide it into Steps 1\kern0.37mm$,\ldots\,$4\kern0.37mm. Now, arbitrarily fixing \math{
\smb U \in \Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm) } let \vskip.5mm\centerline{$
m = \langle\kern0.37mm\seqss33{ \smb U\fvalss11
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm\xi\kern0.15mm\sbi A\kern-0.2mm : \xi \in
\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi} : A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+\kern0.37mm\big\rangle \KP1 $.}
\Step 1.0 We first show that \math{m} has bounded \mathss37{
\mu}--$\KP{1.5} ^{p\sast}\kern0.37mm$variation in \mathss34{\vPi\dlbetss01
}. Indeed, we show that \math{
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p\sast}\kern0.37mm(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A\kern0.37mm)
\KP1^{p\sast}\kern0.15mm\sbig)0
\le\|\KPt8\smb U\,\|\KP1^{p\sast} } holds for arbitrarily given finite
disjoint \mathss30{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+}. In order to get
this, we first note that for arbitrarily given \math{
\bosy\xi\in\kern0.15mm^{\scrm7 A}\,(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\ssbb15 I) } it suffices to
show that \vskip.0mm\centerline{$
\big(\kern0.15mm
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}( (\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{1\kern0.37mm-\,p\sast}\kern0.37mm
|\KPp1.1 m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A
\KP1|\KP1^{p\sast}\kern0.37mm\sbig)0\sbig)0\KP1^{p\sast\kern0.15mm^{-1}}\kern-0.3mm\le\kern0.15mm
\|\KPt8\smb U\,\| $} \inskipline{.5}0
holds since otherwise we could easily get a contradiction.
$\null
$
In order to get this, taking
\math{s=p^{\,*\kern-0.3mm}-1} and with the short-
\linebreak
hand \math{
\roman v\kern0.37mm A=(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\KP1^{p\sast\kern0.15mm^{-1}\kern0.07mm -\kern0.37mm 1}\,(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm) } putting \math{v=
\seqss33{\roman v\kern0.37mm A:A\in\scrmt A} } we have
\math{v\in\kern0.15mm^{\scrm7 A}\,\ssbb10 C } and we need to show that \math{
\|\,v\,\|\lllnor_{p\sast}\le\|\KPt8\smb U\,\| } holds. We may assume that
\math{\|\,v\,\|\lllnor_{p\sast}\not=0 } holds, and then taking \math{u=
\seqss33{\roman u\,A:A\in\scrmt A} } where
\math{\roman u\,A=
\|\,v\,\|\lllnor_{p\sast}\kern-0.2mm^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000}\emath s}\,
|\KP1\roman v\kern0.37mm A\KP1|\KPt8^{\emath s\kern0.37mm - \kern0.37mm 1}\,
\overline{\roman v\kern0.37mm A\RHB{.0}{\KN{.99}\phantom{'}}}
} if \math{\roman v\kern0.37mm A\not=0} holds, otherwise having
\mathss36{\roman u\,A=0}, \,we now have \math{
\|\,u\,\|\lllnor_p=1 } and \vskip.5mm\centerline{$
\big|\kern0.15mm\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)\KP1|=
\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)=
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm\roman u\,A\KP1\roman v\kern0.37mm A\kern0.37mm)=
\|\,v\,\|\lllnor_{p\sast} \KP1 $.} \inskipline{.3}0
Furthermore, with the shorthand \math{ \roman t\,A = \roman u\,A\KP1(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p}\LHB{.2}{^{^{-1}}} } we have \inskipline1{13}
$ \big|\kern0.15mm\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm)\KP1|
= \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\kern0.37mm
\roman u\,A\KP1\roman v\kern0.37mm A\kern0.37mm) \KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^p\LHB{.2}{^{^{-1}}} (\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KP1^{p\sast\kern0.15mm^{-1}\kern0.07mm -\kern0.37mm 1}\,(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 | $ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
m\,.\KPt9\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm )\kern0.37mm )\KP1 |$ \inskipline{.6}{31.5}
${} = \big|\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}(\,\roman t\,A\KP1(\kern0.37mm
\smb U\fvalss20
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\kern0.37mm))\KP1|$ \inskipline{.6}{31.5}
${} = |\KP1\smb U\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,|$ \inskipline{.6}{31.5}
${}\le\|\KPt8\smb U\,\|\KP1\|\KP1\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}
\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A\,}\roman t\,A\KP1
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm(\kern0.37mm
\bosy\xi\KPt4\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\KN{.4}A\kern0.37mm)\kern0.15mm\sbi A\,\|\sNorF$ \inskipline{.6}{31.5}
${}\le\|\KPt8\smb U\,\|\KP1\big(\kern0.15mm\sum_{\,A\kern0.37mm\in\kern0.37mm\scrm7 A}\kern0.15mm\sbig(3
|\KP1\roman t\,A\KP1|\RHB{.3}{\KP1^p}\,(\kern0.37mm
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)))\KP1^p\LHB{.2}{^{^{-1}}} $ \inskipline{.6}{31.5}
${} = \|\KPt8\smb U\,\|\KP1\|\,u\,\|\lllnor_p=\|\KPt8\smb U\,\| \KP1
$, \,giving the assertion.
\Step 2.0 Noting that the requirement of absolute continuity holds since we
trivially have \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm m\kern0.07mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm A \le
\|\KPt8\smb U\,\|\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)\KPt8\RHB{.2}{^p}{^{^{\kern0.15mm-1}}} }
for any \math{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}, \,now let \math{\scrmt A}
and \math{y} be as given by Corollary \ref{Coro q-var} on page \pageref{Coro q-var}
above. Then we have \math{(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } simply
measurable and such that \math{y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta=\vPi\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }
holds for \math{\eta\in{}^{}\Cal Omega\kern0.15mm\setminus\bigcup\,\scrmt A } and such that we
also have \math{m\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\fvalss12\xi =
\int_{\,A}\kern0.37mm\roman{ev}\KPt2\sbi\xi\kern-0.2mm\circ\kern0.15mm y\rmdss11\mu } for \math{
A\kern0.07mm\ar 1 \in\scrmt A} and \math{A\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\capss33\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm A\ar 1} and \mathss31{
\xi\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi}. In addition \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlbetss01\kern0.15mm) } is simply measurable if \math{
\vPi} is reflexive.
\Step 3.0 Under (5) or (6) to prove that \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\le\|\KPt8\smb U\,\| }
holds, noting that in the reflexive case now \math{
(\kern0.37mm\Abrs33^{p\sast}\KN1\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\,;\kern0.07mm
\mu\,,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm) } is trivially measurable, and that by Lemma \ref{Le Nu_1 ci y meas}
on page \pageref{Le Nu_1 ci y meas} above the same holds also in the separable
case, it suffices to show that \math{\int_{\KPp1.1{}^{}\Cal Omega\,}\Abrs33^{p\sast}\KN1
\circ\KPt2\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\rmdss11\mu \le
\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} } holds. For this in turn for every
fixed $A\kern0.15mm\ar 0\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+\kern0.15mm$ it suffices to show that \math{
\int_{\,\aars A_0\kern0.15mm}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} } holds.
Now we can express \math{A\kern0.15mm\ar 0} as the union of an increasing sequence of
$A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+$ such that
$\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y$ is bounded on every $A\,$,
say \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq
[\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9] } with \mathss30{\smb M\in\rbb R^+}, \,and
it further suffices to show that for every such $A$ with
$0 < \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A$ we have
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}} \kern0.15mm$.
To proceed indirectly, supposing that
$\|\KPt8\smb U\,\|\RHB{.2}{\KP1^{p\sast}}
<\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)$ holds, we let
$\varepsilon=
\frac 14\KP1
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\,^{\raise.18mm\hbox{\font\SweD =cmsy5\SweD \char'000} p^{\kern0.15mm-1}}}
\big(\kern0.15mm
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}
\kern-0.2mm - \|\KPt8\smb U\,\|\kern0.37mm\sbig)0 \KP1 $. \vskip1mm
Since \math{\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1|\KPt9 A } is positive \mathss37{\mu
}--\,measurable with \mathss35{
\sup\KPt8(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\kern0.37mm) < \lower1.05mm\hbox{$^+$}\infty}, \,we
can find a finite partion \math{\scrmt A\kern0.15mm\ar 0\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} of \math{A} such
that \math{ |\KP{1.2}\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta -
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\,| < \varepsilon } holds
for all \mathss36{\eta\kern0.37mm,\kern0.15mm\eta\ar 1\in A\ar 1\in\scrmt A\kern0.15mm\ar 0 }. Taking \math{
S=\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } and \vskip.4mm\centerline{$
P =
A\times S\capss31\{\,(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm):
0 \le y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi \kern0.37mm \text{ and }\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta
< y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\fvalss01\xi + \varepsilon \KPt9\} \KP1 $,} \inskipline{.4}0
we first see that \math{A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}} P} holds. In the reflexive case letting \math{
S\ar 0} be the closed linear span in \math{\vPi\dlbetss01} of \math{{}^{}{\rm rng}\,{}_{{}^{}} y} we
take \math{\scrmt T\aR 1=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlbetss01\kern0.15mm)\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[\KPp1.1 0\,,\kern0.07mm\smb M\KPt9]\capss42
S\ar 0\kern0.07mm) } whereas in the separable case we put \mathss38{\scrmt T\aR 1 =
\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm(\kern0.15mm\vPi\dlsigss00\kern0.07mm)
\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm[
\KP{1.1} 0\,,\kern0.07mm\smb M\KPt9]\kern0.15mm
\sbig)0 }. Noting that in both cases now
\math{\scrmt T\aR 1} is a separable and metrizable and hence
second countable topology, we find some
$\bmii8 U\in\kern0.15mm^{{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}}\,\scrmt T\aR 1$ with
$y\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\subseteq\bigcup\,{}^{}{\rm rng}\,{}_{{}^{}}\kern0.37mm\bmii8 U$ and
such that for every
$U\in{}^{}{\rm rng}\,{}_{{}^{}}\bmii8 U$ there are $\xi\,,\kern0.15mm\eta$ with
$(\kern0.37mm\eta\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in P$ and
\mathss38{U\subseteq\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm)
\capss31\{\,\zeta:
|\KP{1.1}(\kern0.37mm\zeta - y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\fvalss01\xi\KP{1.1}|
< \varepsilon\KPt8\} }.
We next fix some bijection $
\bmii8 A\kern0.15mm\ar 0:k\to\scrmt A\kern0.15mm\ar 0$ with $k\in\mathbb N\,$ and construct the
countable finite or infinite sequence $\bmii8A$ as follows. Indeed, we first
let $\bmii8 A\ar 1$ be the infinite
sequence of possibly empty finite sequences obtained as follows. For
every fixed $i\in\mathbb No$ with
$B=y\invss46[\KP{1.2}
\bmii8 U\fvalss51 i\kern0.15mm\setminus\bigcup\KP1(\kern0.37mm\bmii8 U\KPt8\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm i\kern0.37mm)
\KP{1.1}]$ let
$\bmii8 A\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i$ with
$l\in\mathbb No$ be the unique bijection
$l\to\scrmt A\kern0.15mm\ar 0\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 B\kern0.15mm\setminus 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ ordered by
$\bmii8 A\ar 0 \,$. Then let
$\bmii8 A$ be the infinite concatenation of
$\bmii8 A\ar 1 \kern0.37mm $. Now
$\bmii8 A$ is injective with ${}^{}{\rm rng}\,{}_{{}^{}}\bmii8 A\subseteq
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\setminus 1\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}$ and such that
${}^{}{\rm rng}\,{}_{{}^{}}\bmii8 A$ is a partition of $A$ refining $\scrmt A\kern0.15mm\ar 0\,$, i.e.\ for
every $i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A$ there is $A\ar 1\in\scrmt A\kern0.15mm\ar 0$ with
$\bmii8 A\fvalss51 i\subseteq A\ar 1 \kern0.37mm$. Possibly by
{\sl countable choice\kern0.15mm} we
take any
$\bosy\eta\in\prod{_{_{\kern-.3mm\bold c\kern.15mm}}}\bmii8 A$ and any $\bosy\xi\in\kern0.15mm^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\bmii6 A}\,S$
such that
$(\kern0.37mm\bosy\eta\fvalss01 i\kern0.37mm,\kern0.07mm\bosy\xi\fvalss01 i\kern0.37mm)
\in P$ holds for all $i\in{{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A \,$. Now by construction
$|\KP{1.1}(\kern0.37mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta - y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1)\fvalss01\xi\KP{1.1}|
< \varepsilon \,$ and $\,
\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1
< y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi + \varepsilon
\,$
and $0\le y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm\xi\,$ hold
whenever we have
$(\kern0.37mm i\kern0.37mm,\kern0.07mm A\ar 1)\in\bmii8 A$ and $\eta\in A\ar 1$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\eta\ar 1)\in\bosy\eta$ and
$(\kern0.37mm i\kern0.37mm,\kern0.07mm\xi\kern0.37mm)\in\bosy\xi \,$.
With \math{N\aar 0={{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A} we next compute \vskip1mm
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
=\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm) $ \inskipline{.7}{27}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\circss00\bosy\eta\fvalss01 i + \varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \inskipline{.7}{20}
${}\le
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\aars N_0}\kern0.37mm
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)$ \vskip.7mm\centerline{$
{}=\lim\sbi{\kern0.15mm\ssmb N\kern0.37mm\to\kern0.37mm\infty\kern0.37mm}
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm) \KP1 $,} \inskipline10
where the last limit expression is valid and needed only in the case where
\math{\bmii8 A} is infinite. According to whether
\math{\bmii8 A} is finite or infinite, with
\math{\smb N=N\aar 0} or
for arbitrarily fixed \math{\smb N\in\mathbb N} considering
\math{u\in\kern0.15mm^{\ssmb N}\KPt8\lbb R_+} given by \vskip.5mm\centerline{$
u=\seqss33{
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}\kern-0.63mm:i\in\smb N} \KP1 $,} \inskipline{.5}0
we know that for some \math{v\in\kern0.15mm^{\ssmb N}\KPt8\lbb R_+} with \math{
\|\,v\,\|\lllnor_p=1} we have \mathss30{ \|\,u\,\|\lllnor_{p\sast} = \label{express p-norm}
\sum\KP1(\kern0.37mm u\cdot v\kern0.37mm) } where \math{u\cdot v} is the pointwise product \mathss39{
\smb N\owns i\mapsto u\fvalss01 i\cdot(\kern0.37mm v\fvalss01 i\kern0.37mm) }. Using this,
we get \vskip1mm
$
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern0.37mm
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
= \big(\kern0.15mm\sum\KP1
(\kern0.37mm u\cdot v\kern0.37mm))\,^{\,p\sast}$ \vskip.7mm
${}=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,(
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1}}}
(\kern0.37mm v\fvalss01 i\kern0.37mm)))\RHB{.2}{\KP1^{p\sast}}$ \vskip.7mm
${}=\big(\kern0.15mm\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(
(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm) + 2\KPt8\varepsilon\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}
(\kern0.37mm v\fvalss01 i\kern0.37mm))
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm))\RHB{.2}{\KP1^{p\sast}}$ \vskip.7mm
${}=(\kern0.37mm\smb I\aR 1\kern-0.2mm + 2\KP1\varepsilon\KP1\smb I\aR 2\kern0.07mm
)\RHB{.2}{\KP1^{p\sast}}$ where we have \vskip.7mm
$\smb I\aR 1 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\KP1
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \,$ and
$\smb I\aR 2 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(
\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}
(\kern0.37mm v\fvalss01 i\kern0.37mm)\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \KP1 $. \inskipline10
Now with \math{A\kern0.15mm\ar 1=\bigcup\KP1(\kern0.15mm\bmii8 A\kern0.07mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\smb N\kern0.37mm) }
a direct computation using H\"older's inequality
gives
\math{\smb I\aR 2\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.15mm\ar 1)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}\le
(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}} }, \,and to estimate \math{\smb I\aR 1}, \,taking
$\bosy\xi\ar 1=
\seqss33{((\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)\KP1
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm))\svs\vPi\kern-0.3mm
:i\in\smb N} \,$ and
$\smb X
=\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F\text{\KPt8-}\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\,
\lfloor\,^{p\kern0.15mm,\kern0.37mm\mu\kern0.15mm,\kern0.37mm\vPi}\kern0.37mm
(\kern0.37mm\bosy\xi\fvalss11 i\kern0.37mm)\sbi{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i} \KP1 $, we get \vskip1mm
$\smb I\aR 1 = \sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}y\circss00\bosy\eta\fvalss01 i
\fvalss20(\kern0.37mm\bosy\xi\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)
=\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X + \smb I\aR 3 $ where \vskip.7mm
$\smb I\aR 3 =
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}(\kern0.37mm y\circss00\bosy\eta\fvalss01 i
- y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.37mm\bosy\xi\ar 1\kern-0.3mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern0.15mm i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm) \KP1 $. \inskipline10
A direct computation gives \math{\|\,\smb X\kern0.37mm\|\sNorF \le 1} whence we get \mathss38{
|\KP1\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X\,|\le\|\KPt8\smb U\,\|}, \,and further \vskip1mm
$|\KP1\smb I\aR 3\,| \le \varepsilon\,
\sum_{\KPt8 i\kern0.37mm\in\kern0.37mm\ssmb N}\kern0.07mm
\int_{\,\bmii6 A\kern-0.2mm\hbox{\kern.2mm\font\SweD =cmr7\SweD \char'022\kern-.2mm} i\KPt8}
(\kern0.37mm\mu\circss10\bmii8 A\fvalss01 i\kern0.37mm)
\RHB{.2}{\KP1^{p\sast\kern0.37mm^{-1} - \kern0.37mm 1\,}}(\kern0.37mm v\fvalss01 i\kern0.37mm)
\rmdss11\mu\,(\kern0.15mm\eta\kern0.15mm)=
\varepsilon\KP1\smb I\aR 2
\le\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}} $. \inskipline10
Putting these results together, and letting
\math{\smb N\to\infty} or taking \math{\smb N={{}^{}{\rm dom}\,{}_{{}^{}}}\bmii8 A} if
\math{\bmii8 A} is finite, \,we finally obtain \inskipline1{11}
$
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\le
\big(\kern0.37mm \|\KPt8\smb U\,\| + 3\KP1
\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}
\kern0.15mm\big)\RHB{.2}{\KP1^{p\sast}} $ \inskipline{.7}{47}
${}<
\big(\kern0.37mm \|\KPt8\smb U\,\| + 4\KP1
\varepsilon\KP1(\kern0.37mm\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A\kern0.37mm)
\RHB{.2}{\KP1^{p^{\kern0.15mm-1}}}
\kern0.15mm\big)\RHB{.2}{\KP1^{p\sast}} $ \inskipline{.7}{47}
${}=
\int_{\,A\KPt8}(\kern0.37mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\kern0.37mm)
\RHB{.2}{\KP1^{p\sast}}\kern-0.63mm\rmdss01\mu\,(\kern0.15mm\eta\kern0.15mm)
\KP1 $, \,a {\sl contradiction\kern0.15mm}. \vskip1mm
\Step 4.0 Now having obtained \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu\le\|\KPt8\smb U\,\| }
we know that \math{y\in\bigcup\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1} holds, and hence there is
some \math{\smb Y} with \math{y\in\smb Y\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\aar 1}. Then we get \math{
\smb U \in {}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} from Lemma \ref{Le-first} similarly as in the proof of
Lemma \nfss A\,\ref{LeA(1)} on page \pageref{Le A2 final ded} above. \label{endmpf}
\end{proof}
We have now established Theorem \nfss A\,\ref{main Th} since as noted at the
beginning of this section on page \pageref{page surj} above, only the
surjectivity \mathss34{\Cal L\,(\kern0.15mm F\kern0.07mm,\kern0.07mm\bosy K\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm }
remained to be verified, and this is done in the various cases in
Lemmas \nfss A\,\ref{LeA(1)}$\,,\ldots\KPt8$\nfss A\,\ref{final lemma} above.
Note also that as opposed to the treatments in \cite{Phil} and \cite{Edw}\,,
we succeeded to handle the cases (5) and (6) simultaneously. In \cite{Phil}
only the case (5) is considered, and the text also contains some quite obscure
passages. In \cite{Edw} the case (6) is treated under the additional
assumption that \math{\mu} be at least positive \erm Radonian.
\Ssubhead E Examples and open problems \label{Sec E}
Below, we have collected some examples in order to make more concrete some
points of the abstract theory given above. We also point out some related open
problems. In the first example we demonstrate that in Theorem \nfss A\,\ref{main Th}
the case (3) {\sl does not\kern0.15mm} cover (1) and (2) even when \math{\mu} is a
probability measure.
\begin{example}\label{Exa big compact} \renewcommand\sNorF{\sNor{\fivemath F}}\renewcommand\erm[1]{\hbox{\font\≈=cmr8\≈#1}}
For \mathss37{{}^{}\Cal Omega=\kern0.15mm^\bbI\ssbb70 I}, \,we construct a probability measure \math{
\mu} on \math{{}^{}\Cal Omega} such that for the space \math{ F =
\mLrs42^1(\kern0.37mm\mu\kern0.37mm) } the topology \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm F} is not separable.
Indeed, for details referring to \cite[199\,--\,203]{Du} let \math{ \mu =
\otimes_{\fiveroman{mea}\,}(\ssbb60 I\times\kern-0.2mm\{\,\LeBmef^{}\,|\KP1
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI\KP1\}\kern0.15mm\sbig)0 } be the uncountable product measure of the
Borel\,--\,Lebesgue measure on the closed unit interval.
Now with
$\roman A\,s=
{}^{}\Cal Omega\capss41\{\,\eta:\frac 12 \le \eta\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s \le 1\KPt9\} \, $ let
$\roman x\,s=
(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus\roman A\,s\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}
\cupss22(\kern0.15mm\roman A\,s\times\kern-0.2mm\{\kern0.37mm 2\kern0.37mm\}\kern0.15mm\sbig)0 \, $ and
$\, \erm X\,s=\uniqset\smb X:\roman x\,s\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F \kern0.15mm $.
For $\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F$ letting
$\|\,\smb X\kern0.37mm\|\sNorF = \uniqset s:\aall{x\in\smb X}\, s =
\int_{\KP{1.1}{}^{}\Cal Omega\,}|\KP1 x\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\eta\KP1|\suba
\rmdss01\mu\,(\kern0.07mm\eta\kern0.07mm)$
and
$\|\KPt9\smb X - \smb Y\KP{1.1}\|\sNorF=
\|\KP1(\kern0.15mm\smb X - \smb Y\,)\svs F\,\|\sNorF$ ,
\noindent
then $\{\KPt8\erm X\,s:s\in\bbI\KP1\}$ is uncountable, and
for $s\kern0.37mm,\kern0.15mm t\in\bbI$ with $s\not=t$ by a simple computation we get
$\|\KP1\erm X\,s - \erm X\,t\KP1\|\sNorF=1 \,$, \,giving the
assertion on nonseparability.
\end{example}
Equally well in Example \ref{Exa big compact} above we could have taken the
uncountable \q{coin tossing} measure \math{ \mu =
\otimea3(\kern0.37mm I\times\kern-0.2mm\{\kern0.37mm\pi\kern0.37mm\}\kern0.15mm\sbig)0 } for any uncountable set \math{
I} when \vskip.3mm\centerline{$
\pi = \{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.15mm\{\,1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm\}\kern0.15mm\}\kern-.2mm\times\kern-.2mm\big\{\kern0.37mm
\frac 12\kern0.37mm\big\}\cupss22\{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm
(\kern0.37mm 2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm 1\kern0.37mm)\,\} \KP1 $.}
\begin{problem}
{\it Does {\,\rm(4)} hold\,} in Theorem \nfss A\,\ref{main Th} when \math{\mu}
is the probability measure constructed in Example \ref{Exa big compact} above?
Observe that \cite[Lemma 8.17.1\,(\kern0.15mm b\kern0.07mm)\,, p.\ 580]{Edw} would give a
positive answer only if \math{
(\kern0.37mm\nsTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02\ssbb05 I)\expnota^\ssbb44 I]_{ti} } were a metrizable
topology, thus requiring the set \math{\mathbb I=[\KPp1.1 0\,,\kern0.07mm 1\KPt9] }
to be countable.
\end{problem}
\begin{example}\label{Exa not trul deco}
For \math{{}^{}\Cal Omega=\mathbb R\times\mathbb R} we construct a {\sl decomposable\kern0.15mm} positive
measure
\math{\mu} on \math{{}^{}\Cal Omega} that is {\sl not truly decomposable\kern0.15mm}. We also
get a
function \math{u:{}^{}\Cal Omega\to\{\KPt8 0\,,\kern0.07mm 1\KPt5\} } with \math{
\upint u\rmdss11\mu=
\lower1.05mm\hbox{$^+$}\infty}
but \math{
\int_{\,A}\kern0.37mm u\rmdss11\mu=
0} for all \mathss34{A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\lbb R_+}. Indeed,
let $\mu$ be the set of all pairs
$(\kern0.15mm A\,,\kern0.07mm s\kern0.37mm)$
with $A\subseteq{}^{}\Cal Omega$ and such that there are
$B\in\{\,A\,,\kern0.07mm{}^{}\Cal Omega\kern0.15mm\setminus A\KPt8\}$ and a
countable $C\subseteq\mathbb R$ such that
$B\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}\in{{}^{}{\rm dom}\,{}_{{}^{}}}\Lebmef^{}$ holds
for all $t\in C$, and that
$B\,\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}=\emptyset$ for
$t\in\ssbb02 R\setminus C$, and that $s=\sum\KP1
\seqss33{\Lebmef^{}\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm
A\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm t\kern0.37mm\}\kern0.37mm\big):t\in\mathbb R} \KP1 $. For $N=
\mathbb R\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}$ then $N\not\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu$ but
$A \capss31 N \in \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} $
for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+$.
It follows that $\mu$ cannot be truly decomposable. To see that $\mu$ is
decomposable, just take \mathss39{\scrmt A=
\big\{\kern0.37mm\{\kern0.37mm t\kern0.37mm\}\kern-0.2mm\times
{]}\KP{1.2} n\kern0.37mm,\kern0.07mm n + 1\KP1]:
t\in\mathbb R\kern0.37mm$ and $\kern0.37mm n\in\mathbb Z\KP1 \} }. One also
observes that for \math{u\ar 0=N\kern-.2mm\times\kern-.2mm\{\kern0.37mm 1\kern0.37mm\} } and \math{u
=(\kern0.37mm{}^{}\Cal Omega\kern0.15mm\setminus N\kern0.37mm)\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22 u\ar 0 }
we have $\upint u\rmdss11\mu=\upint u\ar 0\rmdss01\mu=
\lower1.05mm\hbox{$^+$}\infty$
but $\int_{\,A}\kern0.37mm u\rmdss11\mu=\int_{\,A}\kern0.37mm u\ar 0\rmdss01\mu=
0$ for all $A\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\lbb R_+$.
\end{example}
Decomposable but not \rsigma6finite positive measures are given in the next
\begin{example}\label{Exa Haar}
Let \math{g\in\kern0.15mm^{S\kern0.37mm\times\kern0.37mm S}\,S } be a group operation with \math{S}
uncountable. Then with \mathss03{{}^{}\Cal Omega=S\kern-0.2mm\times\mathbb R } and \math{ \scrmt T
= \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm S\kern0.15mm\hbox{\kern-.2mm${}\times\kern-2.5mm\lower.8mm\hbox{\font\SweD =cmr5\SweD t}\kern1.8mm$}\nsTbb_R } and \inskipline{.2}{4.4}
$a=\{\,(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm t\ar 1\KPt2;\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm t\ar 2\,;\kern0.07mm
s\ar 3\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) :
(\kern0.37mm s\ar 1\kern0.15mm,\kern0.07mm s\ar 2\kern0.37mm,\kern0.07mm s\ar 3\kern0.07mm) \in g\kern0.37mm\text{ and }\kern0.37mm
(\kern0.37mm t\ar 1\kern0.15mm,\kern0.07mm t\ar 2\kern0.37mm,\kern0.07mm t\ar 3\kern0.07mm) \in\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f}\kern.3mm\mathbb R\KP1\}
$
putting \inskipline{.5}{4}
$\mu \kern0.15mm = \kern0.37mm \big\langle\kern0.37mm\sum_{\KPt8\emath s\kern0.37mm\in\kern0.37mm S\KPt8}
(\kern0.37mm\Lebmef^{}\kern-0.63mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}\kern0.15mm\sbig)0\big) :
A\subseteq{}^{}\Cal Omega\kern0.37mm\text{ and }\kern0.37mm\aall{s\in S}\,A\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm s\kern0.37mm\}
\in {{}^{}{\rm dom}\,{}_{{}^{}}}\Lebmef^{} \KPt8 \rangle $ \inskipline{.7}0
and \mathss38{\mu\ar 1=\mu\KP1|\KP1\{\,A:{{}^{}{\rm dom}\,{}_{{}^{}}} A\kern0.37mm\text{ is countable or
\math{{{}^{}{\rm dom}\,{}_{{}^{}}}(\kern0.37mm{}^{}\Cal Omega\kern0.07mm\setminus A\kern0.37mm) } is countable }\} },
we
have \linebreak \mathss03{
(\kern0.37mm a\,,\kern0.07mm\scrmt T\,) } a locally compact Hausdorff topological group
with \math{\mu} a modified {\sl Haar me- asure\kern0.15mm} for it and \mathss38{
\mu\ar 1 = \mu\KP1|\KP1\sigmAlg3\{\,K:K\kern0.37mm\text{ is \mathss37{\scrmt T
}--\,compact }\} }. With \vskip.2mm\centerline{$
\scrmt A = \{\kern0.15mm\{\kern0.37mm s\kern0.37mm\}\kern-0.2mm\times[\KP1 n\kern0.37mm,\kern0.07mm n + 1 \KP1 {[\kern0.15mm} :
s\in S\kern0.37mm\text{ and }\kern0.37mm n\in\ssbb04 Z\,\} $} \inskipline{.2}0
one checks that \math{\mu} is {\sl truly decomposable\kern0.15mm} and that \math{
\mu\ar 1} is {\sl decomposable\kern0.15mm}. Note that for \math{ g = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\kern-0.2mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD f}\kern.3mm\mathbb R =
\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R } we have \math{\mu\ar 1} precisely the \math{\mu} given in
Example \ref{Exa not trul deco} above.
\end{example}
\begin{problem}\label{Prblm z-z mea}
{\it Is $\,\mu$ almost decomposable\kern0.37mm} in the following situation\kern0.15mm? Let \math{
{}^{}\Cal Omega=\kern0.15mm^{2.}\ssbb60 R } and with \math{\roman S\,a\,b=\{\,a + t\KP1(\kern0.37mm
b - a\kern0.37mm):0\le t\le 1\KPt8\} } let \mathss30{\mu\ar 0=\{\,(\KPt5
\roman S\,a\,b\,,\kern0.07mm\|\KPt8 a - b\KP1\|\lllnor_2\kern0.15mm):a\kern0.37mm,\kern0.15mm b\in{}^{}\Cal Omega
\KP1\} } and \ $\mu\kern0.15mm = \kern0.15mm \upCth\kern0.37mm \seqss43{ \inf \kern0.15mm \big \{\kern0.37mm \sum\KP1
(\kern0.37mm\mu\ar 0\circ\bmii8 A\kern0.37mm) : \mu\ar 0\kern-0.2mm : \ebit A \in \kern0.15mm
^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0\kern0.37mm\text{ and }$ \inskipline0{39}
$ A\subseteq\bigcup\kern0.37mm{}^{}{\rm rng}\,{}_{{}^{}}\ebit A\,\} :
A\subseteq{}^{}\Cal Omega } \KP1|\KP1\sigmAlg3{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0 \KPt8 $. \inskipline{0.2}0
Since from \cite[Proposition 3.2.4\kern0.37mm, p.\ 72]{Du} we know that \math{\mu} is
a positive measure, the problem is whether there exist \math{\scrmt A\,,\kern0.15mm
N\kern0.15mmrim1 } as required in Definitions \ref{df decomp}\,(2) on page \pageref{decos A}
above. An appeal to intuition suggests that \math{\mu} is {\sl not\kern0.15mm} almost
decomposable, but a possible proof does not seem to be simple.
Note that if we above instead had written \vskip.25mm\centerline{$
\mu=\uniqset m:m\kern0.37mm$ is a positive measure and
\math{{{}^{}{\rm dom}\,{}_{{}^{}}} m=\sigmAlg3{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 0 } and \mathss36{\mu\ar 0\subseteq m },} \inskipline{.25}0
then it might have happened that \math{\mu=\hbox{\font\SweD =cmssbx10\SweD U}{}} holds, and hence the answer
to the above question would trivially have been \q{no}, noting that by the
lacking \rsigma5finiteness the uniqueness in \cite[Theorem 3.1.10\kern0.37mm, p.\ 68]{Du}
is not applicable in this situation.
\end{problem}
{ \newcommand\elvrB{\lower.2mm\hbox{\font\≈=cmr11\≈B\kern.7mm}}
Similarly as in Problem \ref{Prblm z-z mea} above we might ask whether with \math{
k\kern0.37mm,\smb N\in\mathbb N } and \mathss30{ {}^{}\Cal Omega = \kern-0.3mm} \mathss30{\kern0.15mm
^{k\kern0.37mm+\KPt4\ssmb N}\ssbb80 R } and suitably fixed \math{\lambda\in\rbb R^+ }
and \vskip.25mm\centerline{$
\elvrB r=\{\KPt8{}^{}\Cal Omega\capss31\{\,\eta:\|\KP1\eta - \eta\ar 0\,\|\lllnor_2 <
\smb R\,\}:\eta\ar 0\in{}^{}\Cal Omega\kern0.37mm\text{ and }\kern0.37mm
0 < \smb R \le r \, \} $} \inskipline{.25}0
and \math{ \alpha = \seqss30{ t\KPt8^{\ssmb N\,(\kern0.15mm k \kern0.37mm + \kern0.37mm\ssmb N\kern0.37mm)
\kern.2mm\raise1.35mm\hbox{\font\SweD =cmb7\SweD \char'056}\kern.2mm\kern0.15mm^{-1}}\KN{.8}:t\in\lbb R_ +\kern-0.63mm} } the \mathss36{\smb N
}--\,dimensional {\sl Hausdorff measure\kern0.15mm} \inskipline{.3}{24}
$\upCth\kern0.37mm\seqss43{\lim_{\KP1\emath r\kern0.37mm\to\,0^+}\kern0.15mm\inf\kern0.15mm\big\{\,\lambda\kern0.15mm
\sum\,(\kern0.37mm\alpha\circss00\Lebmef^{\kern0.37mm k\kern0.37mm+\kern0.37mm\ssmb N}\kern-0.3mm\circ\ebit B\kern0.37mm
) : {} $ \inskipline0{38.3}
$ \ebit B\in\kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\elvrB r\kern0.37mm\text{ and }\kern0.37mm A \subseteq \bigcup \kern0.15mm {}^{}{\rm rng}\,{}_{{}^{}}
\ebit B\,\} : A\subseteq{}^{}\Cal Omega}$ \inskipline{.3}0
is almost decomposable. }
In search for an example of a positive measure that would not be almost
decomposable we noticed the positive measure \math{\mu} in the following
\begin{example}\label{Exa non-Rad?}
Let \math{\mu\ar 1=\scrmt N\times\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\cupss22\{\,(\kern0.37mm
\varOmega\aar 1\kern-0.63mm\setminus N\kern0.15mm,\kern0.07mm 1\kern0.37mm):N\in\scrmt N\KP1\} } where \math{
\varOmega\aar 1} is an uncountable set and \math{\scrmt N} is a \rsigma7ideal
in \math{\varOmega\aar 1} with \math{\varOmega\aar 1\not\in\scrmt N } and \mathss34{
\{\kern0.15mm\{\kern0.37mm\eta\kern0.37mm\}:\eta\in\varOmega\aar 1\kern0.37mm \}\subseteq\scrmt N }. For
example, we might have \math{\varOmega\aar 1=\bbI } and \mathss36{ \scrmt N =
\Lebmef^{}\KN1\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss22\bbI }, \,or \vskip.2mm\centerline{$
\scrmt N=\kern0.15mm\big\{\kern0.37mm\bigcup\,\scrmt A:\scrmt A\kern0.37mm\text{ is countable and }\kern0.37mm
\scrmt A\subseteq\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\varOmega\aar 1\cap\kern0.15mm\{\,A:\roman{Int\,}\sbi{\scrm7 T\KP1}
\roman{Cl\,}\sbi{\scrm7 T\KPt8}A = \emptyset \KPt9 \} \kern0.15mm \} $} \inskipline{.2}0
where \math{\scrmt T} is a regular locally compact or completely metrizable
topology for \mathss34{\varOmega\aar 1}.
With a fixed \math{s\ar 0\in\varOmega\aar 1 } for \math{ \eta\ar 0 =
(\kern0.37mm s\ar 0\KPt5,\kern0.07mm s\ar 0) } and for \math{ {}^{}\Cal Omega =
\varOmega\aar 1\kern-0.3mm\times\varOmega\aar 1 } we then construct a positive
measure \math{\mu} on \math{{}^{}\Cal Omega} such that \math{\mu} is {\sl decomposable\kern0.15mm}
but not \rsigma5finite and such that \math{ \bigcap\,\scrmt A\kern0.15mm\ar 0 =
\{\,\eta\ar 0\kern0.07mm\}\not\in{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } and \math{ \bigcap\,\scrmt A \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } hold for \vskip.2mm\centerline{$
\scrmt A\kern0.15mm\ar 0 = \mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\capss22\{\, A :
\eta\ar 0 \in A \KP1 \} $} \inskipline{.2}0
and for all nonempty countable \mathss36{\scrmt A\subseteq\scrmt A\kern0.15mm\ar 0 }.
Indeed, with \math{ \roman P\kern0.37mm\ebit A = \bigcup\KPt8\{\kern0.15mm\{\kern0.37mm s\kern0.37mm\}
\kern-.2mm\times\kern-.2mm A\kern0.07mm\ar 1\kern-0.63mm:(\kern0.37mm s\kern0.37mm,\kern0.07mm A\kern0.07mm\ar 1)\in\ebit A\,\} } and \math{
\scrmt P} the set of all countable functions \math{ \ebit A \subseteq
\varOmega\aar 1\kern-0.3mm\times{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\ar 1 } with \math{[\KPp1.4 s\ar 0 \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\ebit A\kern0.37mm\text{ and }\kern0.37mm\ebit A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s\ar 0\not\in\scrmt N\impss33
s\ar 0\in\ebit A\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm} s\ar 0 \KP1 ] } we let \math{ \mu =
\{\KPt8(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm):\eexi{\ebit A\in\scrmt P}\,[\KPp1.4 A =
\roman P\kern0.37mm\ebit A\kern0.37mm\text{ and }\kern0.37mm t =
\sum\KP1(((\kern0.37mm\mu\ar 1\kern-0.3mm\circ\kern0.07mm\ebit A\kern0.37mm
)\invss24\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0\times\kern-0.2mm
\{\kern0.37mm 1\kern0.37mm\}\kern0.15mm\sbig)0\KPp1.4\big] } \inskipline0{71.4}
or $\kern0.37mm[\KPp1.4 A = {}^{}\Cal Omega\kern0.07mm\setminus\roman P\kern0.37mm\ebit A\kern0.37mm\text{ and }\kern0.37mm
t = \lower1.05mm\hbox{$^+$}\infty\KPp1.4]\KP1
\lower.1mm\hbox{\font\≈=cmsy11\≈\char'147}
\KP1 $. \inskipline{.25}0
Note that \math{\{\KPt9\roman P\kern0.37mm\ebit A:\ebit A\in\scrmt P\KP1\} } is a
\rsigma3ring and hence that \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} is a \rsigma3algebra.
One sees that for \math{0 < p\le\lower1.05mm\hbox{$^+$}\infty } and for \math{\vPi\in\roman{LCS}\kern0.4mmps0(K) }
with \math{\bosy K\in\{\kern0.15mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\,,\kern-0.3mm\raise1.23mm\hbox{\font\SweD =cmr5\SweD tf}\kern.1mm\mathbb C\,\} } the spaces \math{
\mvLrs03^p(\kern0.37mm\mu\,,\kern0.07mm\vPi\kern0.37mm) } and \math{
\lll^p\kern0.07mm(\kern0.15mm\varOmega\aar 1\kern0.15mm,\kern0.07mm\vPi\kern0.37mm) } are linearly homeomorphic
under \math{\smb X\mapsto y } when \mathss03{ y \in \kern0.15mm
^{\aars\varOmega_1}\kern0.37mm\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm\vPi } is such that for \math{x\in\smb X} and \math{
u\in\Cal L\,(\kern0.15mm\vPi\kern0.15mm,\kern0.07mm\bosy K\kern0.37mm) } and for finite \math{ \ebit A \in
\scrmt P } with \mathss03{ {}^{}{\rm rng}\,{}_{{}^{}}\ebit A \subseteq
\mu\ar 1\kern-0.63mm\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } we have \mathss38{
\int_{\KPp1.1\roman P\KPt2\bmii6 A}\kern0.15mm u\circss00 x\rmdss11\mu
= \sum\KP1(\kern0.37mm u\circss00 y\KP1|\KP1{{}^{}{\rm dom}\,{}_{{}^{}}}\ebit A\kern0.37mm) }.
\end{example}
From Example \ref{Exa non-Rad?} above we arrive at the following
\begin{problem}
{\it Is $\,\mu$ positive \eit Radonian\kern0.37mm} when \math{\mu =
\Lebmef^{}\,|\KP1(\kern0.37mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI\capss42(\kern0.37mm
\Lebmef^{}\KN1\inve\kern0.37mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\KPt8 0\,,\kern0.07mm 1\,\}\kern0.15mm\sbig)0\sbig)0 }
holds\kern0.37mm? Note that if there is \math{\scrmt T} that
positively \erm Radonizes \math{\mu} above, then necessarily \mathss30{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm K\in\kern-0.3mm} \mathss03{
\{\KPt8 0\,,\kern0.07mm 1\,\} } holds when \math{K} is \mathss37{
\scrmt T}--\,compact. Furthermore, there is some \mathss30{ N \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } such that \math{
\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\ssbb62 I\setminus N\kern0.37mm) } is a compact topology. Then we
get \math{\mu\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}(\kern0.37mm\scrmt T\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss32(\ssbb62 I\setminus N\kern0.37mm)) \subseteq \kern-0.3mm} \mathss03{
\{\KPt8 0\,,\kern0.07mm 1\,\} } and hence \mathss38{ \mu\kern0.15mm\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.37mm\scrmt T \subseteq
\{\KPt8 0\,,\kern0.07mm 1\,\} }.
\end{problem}
Observe that if we take the trivially positive \erm Radonian \mathss38{
\mu\ar 0 = \{\KPt8(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm,\kern0.07mm
1\kern0.37mm)\KPt8 \} }, then for \math{ q = \bbI\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056} } and \math{
\mu\ar 2 = \{\,(\,q\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,,\kern0.07mm t\kern0.37mm):(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm)\in
\mu\ar 0\,\} } and for \math{0 \le p \le \lower1.05mm\hbox{$^+$}\infty } and \mathss03{ E =
\mLrs03^p(\kern0.37mm\mu\kern0.15mm) } and \math{F = \mLrs03^p(\kern0.37mm\mu\ar 2\kern0.07mm) } and \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F\capss21\{\,(\kern0.37mm\smb\Phii\kern0.07mm,\kern0.07mm\smb\Psii\kern0.15mm) :
\smb\Phii\capss12\smb\Psii\not=\emptyset \KPt9 \} } we \linebreak
have \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm:E\to F} a linear homeomorphism. This leads us to the following
\begin{definitions}\label{Df Leb equ}
(1) \ $q\meastss33\mu = \{\KPt7(\,q\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.63mm A\,,\kern0.07mm t\kern0.37mm) :
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm)\in\mu\KPt8\} \KP1 $, \inskipline{.5}2
(2) \ Say that \math{N\kern0.15mmrim1} is {\it finitely \mathss37{\mu}--\,negligible\kern0.37mm}
if{}f \math{N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible \inskipline0{37.5}
and \math{\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.15mm\lower1.05mm\hbox{$^+$}\infty\,\}\capss13\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N\kern0.15mmrim1 =
\emptyset} holds, \inskipline{.5}2
(3) \ Say that \math{\mu\ar 1\kern0.15mm,\kern0.15mm\mu\ar 2} are {\it Lebesgue equal\,} if{}f \math{
\mu\sbi{\sixmath\nuu} } for \math{\sbi{ \sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm
1\kern0.07mm,\kern0.37mm 2}}} is a positive measure and there are \math{N\aar 1\kern0.15mm,\kern0.15mm
N\aar 2\kern0.37mm,\kern0.15mm q\ar 1\kern0.15mm,\kern0.15mm q\ar 2\kern0.37mm,\kern0.15mm Q } with \math{
N\kern-0.3mm\sbi{\sixmath\nuu} } finitely \mathss37{\mu\sbi{\sixmath\nuu}
}--\,negligible and \math{q\sbi{\sixmath\nuu} } is a surjection \mathss03{Q\to
\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\sbi{\sixmath\nuu}\kern-0.3mm\setminus N\kern-0.3mm\sbi{\sixmath\nuu} } for \math{\sbi{
\sixmath\nuu \sixroman{\kern0.37mm =\kern0.37mm 1\kern0.07mm,\kern0.37mm 2}} } and \math{ \kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm \in
\Lis(\kern0.15mm E\ar 1\kern0.15mm,\kern0.07mm E\ar 2\kern0.07mm) } holds for \mathss30{E\sbi{\sixmath\nuu}
= \mLrs42^1(\,q\sbi{\sixmath\nuu}\kern-0.63mm\meastss03\mu\sbi{\sixmath\nuu}) } and \mathss38{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm = \upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 1\kern-0.3mm\times\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm E\ar 2\capss01\{\,(\kern0.37mm\smb\Phii\kern0.07mm , \kern0.07mm
\smb\Psii\kern0.15mm) : \smb\Phii\capss12\smb\Psii\not=\emptyset \KPt9 \} }, \inskipline{.5}2
(4) \ Say that \math{\mu} is {\it essentially positive \eit Radonian\kern0.37mm}
if{}f \inskipline0{9}
there is a positive \erm Radonian \math{\mu\ar 0} such that \math{
\mu\,,\kern0.15mm\mu\ar 0} are Lebesgue equal.
\end{definitions}
Note that if \math{\mu} is a positive measure and \math{q} is a small function
such that for the set \math{N\kern0.15mmrim1=\bigcup\kern0.37mm{{}^{}{\rm dom}\,{}_{{}^{}}}\mu\setminus{}^{}{\rm rng}\,{}_{{}^{}} q } it
holds that \math{N\kern0.15mmrim1} is \mathss37{\mu}--\,negligible, then \math{
q\meastss33\mu } {\sl need not\kern0.15mm} be a positive measure since it may even
fail to be a function. For example, taking \mathss03{q=\emptyset} and \math{
\mu = \{\KPt8(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,,\kern0.15mm(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm ,
\lower1.05mm\hbox{$^+$}\infty\kern0.37mm)\KPt8\} } we get \mathss38{ q\meastss33\mu =
1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\KPt7 0\,,\lower1.05mm\hbox{$^+$}\infty\,\} }. However, if we know that \math{
q\meastss33\mu } is a function, then it is also a positive measure as one
quickly verifies. A sufficient condition to guarantee that \math{
q\meastss33\mu } be a function is that \mathss30{N\kern0.15mmrim1} be finitely \mathss37{
\mu}--\,negligible as we have required in Definitions \ref{Df Leb equ}\,(3)
above.
Note also that our Definition \ref{Df Leb equ}\,(3) is not entirely
satisfactory since for example taking \math{ \mu\ar 0 =
\{\,(\kern0.37mm\emptyset\,,\kern0.07mm 0\kern0.37mm)\,\} } we have both \math{
\mLrs42^1(\kern0.37mm\mu\kern0.15mm) } and \math{\mLrs42^1(\kern0.37mm\mu\ar 0\kern0.07mm) } linearly
homeomorphic to the trivial space \math{
(\kern0.37mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm , \kern0.07mm
\mathbb R\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-0.2mm\times 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.37mm , \kern0.07mm\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern0.07mm) }
but \math{\mu\,,\kern0.15mm\mu\ar 0} are not Lebesgue equal by the above. We further
remark that the relation of being Lebesgue equal is not an equivalence since
for example taking the positive \erm Radonian \math{ \mu\ar 1 =
2\kern0.15mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}\kern-.2mm\times\kern-.2mm\{\kern0.37mm 0\kern0.37mm\} } that is positively \erm Radonized by \math{
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm 1\kern0.07mm\kern.2mm\hbox{\font\SweD =cmb10\SweD \char'056}} we have both \math{\mu\,,\kern0.15mm\mu\ar 1} Lebesgue equal and \mathss30{
\mu\ar 0\,,\kern0.15mm\mu\ar 1} Lebesgue equal. Now we can pose the following
\begin{problem}\label{Prblm all Rado?}
{\sl Is every positive measure essentially positive \eit Radonian\kern0.37mm}?
\end{problem}
If the answer to the question in Problem \ref{Prblm all Rado?} is positive,
then one might be able to remove from Theorem \nfss A\,\ref{main Th} the
assumption on \math{\mu} being almost decomposable in the case where \math{p=1}
holds. It seems that possibly by using {\sl Kakutani's theorem\kern0.15mm}
\cite[4.23.2\kern0.37mm, p.\ 287]{Edw} one might be able to prove that this indeed is
the case. However, we leave these matters open here.
For example the {\sl Wiener\kern0.15mm} probability {\sl measure\kern0.15mm} in \cite{Pi-Po}
on a non\kern0.37mm-\kern0.37mm locally compact separably metrizable topological space
{\sl is essentially positive \esl Radonian\kern0.15mm} directly by its construction
since it is obtained by restricting a \erm Radonian probability measure on a
compact topological space to a subset of measure unity. More specifically, one
first constructs a probability measure \math{\pi} that is positively
\erm Radonized by a compact topology \mathss30{\scrmt T}. Then for a certain
separably metrizable topological space \math{(\kern0.37mm{}^{}\Cal Omega\,,\kern0.07mm\scrmt U\kern0.37mm) }
one shows that \math{\sigmAlg3\scrmt U={{}^{}{\rm dom}\,{}_{{}^{}}}\pi\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss23{}^{}\Cal Omega } and \math{
{}^{}\Cal Omega\in\pi\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } hold, and finally one defines \linebreak
$\kern.3mm\roman{id}\kern.7mm\kern0.15mm{}^{}\Cal Omega\meastss33\pi\kern0.37mm$ to be the Wiener measure.
We remark that there is some confusion in \cite[pp.\ 12\,--\,25]{Pi-Po} and
that the above is not a review but rather an interpretation of how it could
have been done.
\begin{example}\label{Exa pos Rad C(I)}
It holds that \math{\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm{}^{}\Cal Omega} {\sl positively almost \esl Radonizes\kern0.15mm} \math{
\pi} in the following situation. Let \math{(\kern0.37mm{}^{}\Cal Omega\kern0.37mm,\kern0.07mm\scrmt T\,) }
be a separably metrizable and not locally compact topological space with \math{
{}^{}\Cal Omega} uncountable, and let \math{D} countable and \mathss37{\scrmt T
}--\,dense with \math{\bosy a\in\kern0.15mm^D\KPt8\rbb R^+ } and \mathss04{
\sum\,\bosy a=1}. Then let \mathss38{\pi = \kern0.15mm \big\langle\kern0.37mm\sum\KP1(\kern0.37mm
\bosy a\KPt9|\KPt8 A\kern0.37mm) : A \in \sigmAlg3\scrmt T\KPp1.2\rangle }. For
example, we might have \mathss03{\scrmt T=\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm C\,(\ssbb55 I) } and \math{D}
the set of all polynomial functions with rational coefficients, or the set of
all piecewise affine functions with rational \q{break} points.
\end{example}
\begin{example}\label{Exa Sum Lebm^N}
For \math{{}^{}\Cal Omega = \kern0.15mm^{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\,\bbI } we obtain a
{\sl decomposable\kern0.15mm}
non\kern0.37mm-\kern0.15mm\rsigma5finite positive mea- \linebreak
sure \math{\mu} on \math{{}^{}\Cal Omega} by
taking \math{\mu=\sum\KP1\seqss30{\roman m\KPt8\alpha:\alpha\subseteq\mathbb No} } where
with \math{m\ar 1 = \Lebmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\bbI } and \mathss03{\delta\ar 0 =
{{}^{}{\rm dom}\,{}_{{}^{}}} m\ar 1\kern-0.2mm\times\{\KPt7 0\,,\kern0.07mm 1\KPt6\}\capss22\{\,
(\kern0.15mm A\,,\kern0.07mm t\kern0.37mm) : t = 1 \equivss33 0 \in A \KP1 \} } we have \vskip.3mm\centerline{$
\roman m\KPt8\alpha=
\otimea0((\kern0.37mm\mathbb No\kern-0.3mm\setminus\alpha\kern0.37mm)\times\kern-0.2mm\{\KPt8 m\ar 1\}\cupss22
(\kern0.37mm\alpha\times\kern-0.2mm\{\,\delta\ar 0\kern0.07mm\}\kern0.15mm\sbig)0\sbig)0 \KP1 $.} \inskipline{.3}0
Indeed, with \math{\roman A\KPt8\alpha = {}^{}\Cal Omega\capss41\{\,\eta :
\eta\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}=\alpha\KPt8\} } taking \math{ \scrmt A =
\{\,\roman A\KPt8\alpha:\alpha\subseteq\mathbb No\,\} } we have \mathss03{\scrmt A}
uncountable and disjoint with \math{{}^{}\Cal Omega=\bigcup\,\scrmt A} and \mathss38{
\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} }. Furthermore, for \mathss03{
\alpha\KPt5,\kern0.15mm\kappa\in\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No} we have \math{
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\roman A\KPt8\alpha = 1 } and \mathss36{
\alpha\not=\kappa\impss33\roman m\KPt8\kappa\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\roman A\KPt8\alpha=0}.
To see these, by straightforward inspection one first verifies the last
assertion which then directly implies that \math{ \scrmt A \subseteq
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 1\kern0.37mm\} } holds. Consequently \math{\mu} is not
\rsigma5finite.
To show that \math{\mu} is decomposable, let \math{A\in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+} and \math{N\kern0.15mmrim1\subseteq{}^{}\Cal Omega} with \mathss30{
\scrmt A\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss42 N\kern0.15mmrim1 \subseteq \kern-0.3mm} \mathss08{
\bigcup\KPt8\{\KPt8\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm N \kern-0.3mm : N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\}\kern0.15mm\}
}. Then taking \mathss30{ \varLambda\kern0.15mm\ar 0 = \Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No\capss01\{\,\alpha :
A\capss32\roman A\KPt8\alpha\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.15mm\rbb R^+\kern0.15mm\big\} } we
have \math{\varLambda\kern0.15mm\ar 0} countable, and also putting \math{ N\aar 1 =
A\kern0.07mm\setminus\bigcup\KPt8\{\,\roman A\KPt8\alpha:\alpha\in\varLambda\kern0.15mm\ar 0
\,\} } we get \mathss30{N\aar 1 \in \kern-0.3mm } \mathss08{
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }. Indeed, trivially \math{ N\aar 1 \in
{{}^{}{\rm dom}\,{}_{{}^{}}}\mu } holds, and if we have \mathss38{ N\aar 1 \not \in
\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }, \,then \math{0 <
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1 = \sum\KP1\seqss30{
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1\kern-0.3mm : \alpha\in\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No} =
\sum\KP1\seqss30{\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1\kern-0.3mm : \alpha \in
\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb No\kern-0.3mm\setminus\varLambda\kern0.15mm\ar 0} } when- ce there is some \math{
\alpha\in\mathbb No\kern-0.3mm\setminus\varLambda\kern0.15mm\ar 0 } with \math{ 0 <
\roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm N\aar 1 \le \roman m\KPt8\alpha\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.63mm A }
and hence \mathss36{\alpha\in\varLambda\kern0.15mm\ar 0}, \,a {\sl contradiction\kern0.15mm}.
Now by {\sl countable choice\kern0.15mm} there is some countable \mathss30{ \scrmt N
\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} } with \mathss34{
\bigcup\KPt8\{\,N\kern0.15mmrim1\cap\kern0.15mm\roman A\KPt8\alpha : \alpha \in
\varLambda\kern0.15mm\ar 0\,\}\subseteq\bigcup\,\scrmt N }. Then taking \math{ N =
\bigcup\,\scrmt N\cupss42 N\aar 1} we finally get \mathss08{
A\capss31 N\kern0.15mmrim1\subseteq N\in\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern-0.2mm\{\kern0.37mm 0\kern0.37mm\} }.
\end{example}
Observe that if in Example \ref{Exa Sum Lebm^N} in place of \math{\mathbb No} we
take any uncountable set, then we obtain a trivial measure \math{\mu} in the
sense that \math{{}^{}{\rm rng}\,{}_{{}^{}}\mu=\{\KPt7 0\,,\lower1.05mm\hbox{$^+$}\infty\,\} } holds.
\begin{problem}\label{Prblm Sum Lebm^N}
{\it Is $\,\mu$ positive \eit Radonian\kern0.37mm} in Example \ref{Exa Sum Lebm^N}
above\kern0.15mm? Note that at least we cannot take \math{ \scrmt T =
(\kern0.37mm\nsTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}\ssbb25 I)\expnota^\kern0.15mm{\mathbb N\kern.07mm\lower.45mm\hbox{\font\SweD =cmr5\SweD 0}\kern.1mm}\kern0.15mm]_{ti} } in order to
positively \erm Radonize \math{\mu} since \math{{}^{}\Cal Omega} is \mathss37{\scrmt T
}--\,compact with \mathss35{\mu\fvalss01{}^{}\Cal Omega=\lower1.05mm\hbox{$^+$}\infty}.
\end{problem}
\begin{example}\label{Exa sign mea}
We say that \math{\mu} is a {\it signed measure\kern0.37mm} if{}f \math{ \mu \in \kern0.15mm
^{{{}^{}{\rm dom}\,{}_{{}^{}}}\kern-0.3mm\mu}\,\ovbbR} with \math{{{}^{}{\rm dom}\,{}_{{}^{}}}\mu} being a \rsigma3algebra and \math{
\mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.3mm\bigcup\,\scrmt A = \sum\KP1(\kern0.37mm\mu\KP1|\KP1\scrmt A\kern0.37mm) } for
all countable disjoint \mathss35{\scrmt A\subseteq{{}^{}{\rm dom}\,{}_{{}^{}}}\mu}. Then \linebreak
we cannot have \math{
\{\kern0.15mm\kern.2mm\lower1.05mm\hbox{$^-$}infty\,,\lower1.05mm\hbox{$^+$}\infty\KPt8\}\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\mu } since otherwise we could
find \math{A\,,\kern0.15mm B} with \mathss03{
\{\KPt8(\kern0.15mm A\,,\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm)\,,\kern0.15mm(\kern0.15mm B\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm)\KPt8\} \subseteq
\mu} and \math{A\capss32 B=\emptyset} whence we would get \vskip.3mm\centerline{$
\hbox{\font\SweD =cmssbx10\SweD U}{} = \sum\KP1(\,\mu\KP1|\KPt8\{\,A\,,\kern0.15mm B\KPt8\}\kern0.15mm\sbig)0
= \mu\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}(\kern0.15mm A\cupss32 B\kern0.37mm) \in \ovbbR \KP1 $,} \inskipline{.3}0
a contradiction following from our sum conventions in \cite{Hif}\,. Now the
positive measures are precisely the signed measures \math{\mu} with \mathss38{
{}^{}{\rm rng}\,{}_{{}^{}}\mu\subseteq[\KPp1.1 0\,,\lower1.05mm\hbox{$^+$}\infty\KPt9] }, \,and a signed measure \math{\mu}
we say to be {\it positively signed\,} if{}f \math{\kern.2mm\lower1.05mm\hbox{$^-$}infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}\mu}
holds. Similarly the con- dition \math{\lower1.05mm\hbox{$^+$}\infty\not\in{}^{}{\rm rng}\,{}_{{}^{}}\mu} defines being
{\it negatively signed\,}. Real measures are now those that are both
positively and negatively signed\kern0.15mm.
We next construct the topologized conoid \math{ \tcbbR_+ =
(\kern0.37mm a\kern0.37mm,\kern0.15mm c\,,\kern0.07mm\scrmt T\,) } so that \math{m} is positively signed
if{}f \math{m} is countably \mathss37{\tcbbR_+}--\,additive and such that \math{
{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra. Indeed, taking \math{R=\lbb R_+} and \math{ S =
{\kern0.37mm]}\,\kern.2mm\lower1.05mm\hbox{$^-$}infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\KP1] } let \math{ \scrmt T =
\barscTbb_R\hbox{\kern.45mm$_{^\downarrow}\kern-1.280mm\cap\kern.85mm$}ss02 S} and \inskipline{.3}{10.3}
$a = \sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern-1.7mm\raise1.25mm\hbox{\font\SweD =cmr6\SweD 2}\kern1mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\cupss31\{\,(\kern0.37mm s\kern0.37mm,\kern0.07mm t\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm) :
\lower1.05mm\hbox{$^+$}\infty\in\{\,s\kern0.37mm,\kern0.15mm t\,\}\subseteq S\KP1\} \KP{26.5} $ and \inskipline{.3}{10.7}
$c = \tau\sigma\kern-.2mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\KP1|\KP1(\kern0.15mm R\times\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.37mm)\cupss21\{\,
(\kern0.37mm 0\,,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\kern0.07mm 0\kern0.37mm)\,\}\cupss22\{\,
(\kern0.37mm s\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm,\lower1.05mm\hbox{$^+$}\infty\kern0.37mm):s\in\rbb R^+\kern0.15mm\big\} \KP1 $. \inskipline{.3}0
Making the obvious modifications we similarly get the topologized conoid \math{
\tcbbR_-} so that \math{m} is negatively signed if{}f \math{m} is countably \mathss37{
\tcbbR_-}--\,additive and such that \mathss30{{{}^{}{\rm dom}\,{}_{{}^{}}} m} is a \rsigma3algebra.
Likewise, we can construct the topologized conoid \math{\tcovbbRplus}
characterizing the positive measures.
\end{example}
\begin{problem}\label{Prblm Io sur}
In Theorem \nfss A\,\ref{main Th} taking for example \math{ \mu =
\LeBmef^{}\,|\KP1\Cal P\kern-.7mm\lower.15mm\hbox{$_s$}\kern.3mm\mathbb I } and \vskip.5mm\centerline{$
\vPi \in \{\KP1\hbox{\font\≈=cmmi12\≈c}\lower.8mm\hbox{\font\≈=cmr6\≈o}\kern.4mm(\ssbb44 I) \, , \kern0.15mm \ell\KPt8^1\kern0.15mm(\ssbb44 I) \, , \kern0.15mm
\LLrs23^{\raise.18mm\hbox{\font\SweD =cmr5\SweD \char'053}\infty}(\ssbb44 I) \KPt9 \} \KP1 $,} \inskipline{.3}0
hence \math{\vPi} being nonreflexive with \math{\tau\kern-.4mm\lower.7mm\hbox{\font\SweD =cmr6\SweD r\font\SweD =cmr5\SweD d}\kern.6mm\vPi} a nonseparable
topology, for \math{p=1} we see \linebreak
that (3) holds, and then with \math{ F\aar 1 =
\mvsLrs23^{p\sast}\kern-0.63mm(\kern0.37mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } we obtain that \math{
\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} is a linear ho- \linebreak
meomorphism \mathss30{F\aar 1\to F\dlbetss10}, \,and
hence in particular that \math{\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}
holds. However, if we instead take \math{1 < p < \lower1.05mm\hbox{$^+$}\infty} for example with \mathss35{
p=2}, \,then we {\sl do not\kern0.15mm} know whether \math{
\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\subseteq{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm } holds. So under these
circumstances we may ask\kern0.15mm: {\it Is there\kern0.37mm} \math{\smb U} such that \math{
\smb U\in\Cal L\,(\kern0.15mm F\kern0.07mm,\kern-0.3mm\raise1.2mm\hbox{\font\SweD =cmr5\SweD tf}\kern.3mm\mathbb R\kern0.37mm)\setminus{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm} holds\,{\bf?}
We remark that by suitably adapting the proof of Corollary \ref{Coro q-var}
above it seems \linebreak
to be possible to deduce existence of some \math{y} and a
countable disjoint \mathss30{\scrmt A\subseteq\mu\invss44\hbox{\font\SweD =cmr10\SweD \char'022\kern-1mm\char'022}\kern0.07mm\rbb R^+ \kern-0.63mm}
with \mathss30{\bigcup\,\scrmt A = \mathbb I } and such that \math{
(\KPt5 y\,;\kern0.07mm\mu\,,\kern0.07mm\vPi\dlsigss00\kern0.07mm) } is scalarly measurable and such
that we have \math{\smb U\kern0.15mm\hbox{\kern.2mm\font\SweD =cmr10\SweD \char'022\kern-.2mm}\kern-0.2mm\smb X=
\int_{\,A}\kern0.37mm y\,.\KPt8 x\rmdss11\mu } for all \math{x\kern0.37mm,\kern0.15mm\smb X} with \math{
x\in\smb X\in\upsilon\kern-0.3mm\lower.15mm\hbox{$_s$}\kern0.3mm F} and \math{{}^{}{\rm rng}\,{}_{{}^{}} x} finite and such that for some \math{
A\in\scrmt A} we have \mathss36{
x\invss46[\KPp1.1\hbox{\font\SweD =cmssbx10\SweD U}{}\kern0.15mm\setminus\{\,\Bnull_\vPi\}\KP1]\subseteq A }. However, we
do not know whether \math{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu < \lower1.05mm\hbox{$^+$}\infty } holds.
If we could get \math{y} with these properties together with \mathss36{
\|\KP1\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'027}\kern0.25mm\aR 1\kern-0.3mm\circ\kern0.15mm y\KP1\|\Lnorss40^{p\sast}_\mu < \lower1.05mm\hbox{$^+$}\infty }, \,then
we would also get \mathss34{\smb U\in{}^{}{\rm rng}\,{}_{{}^{}}\kern.15mm\hbox{\font\SweD =cmmi10 scaled\magstep1\SweD \char'023}\kern0.2mm}.
\end{problem}
\end{document} |
\begin{document}
\begin{abstract}
This paper is devoted to investigate the singular directions of
meromorphic functions in some angular domains. We will confirm the
existence of Hayman $T$ directions in some angular domains. This is
a continuous work of Yang [Yang L., Borel directions of meromorphic
functions in an angular domain, Science in China, Math.
Series(I)(1979), 149-163.] and Zheng [Zheng, J.H., Value
Distribution of Meromorphic Functions, preprint.].
\end{abstract}
\maketitle \keywords{ Keywords and phases: Hayman $T$ direction,
Angular domain, P\'{o}lya peaks, Order}
\maketitle
\section{Introduction and Main Results}
\setcounter{equation}{0} Let $f(z)$ be a meromorphic function on the
whole complex plane. We will use the standard notation of the
Nevanlinna theory of meromorphic functions, such as $T(r,f), N(r,f),
m(r,f), \delta(a,f)$. For the detail, see \cite{Yang}. The order and
lower order of it are defined as follows
$$\lambda(f)=\limsup\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{\log r}$$
and
$$\mu(f)=\liminf\limits_{r\rightarrow\infty}\frac{\log T(r,f)}{\log r}.$$
In view of the second fundamental theorem of Nevanlinna, Zheng
\cite{Zheng} introduced a new singular direction, which is named $T$
direction.\\
\begin{defi}\label{def2} A direction $L: \operatorname{arg} z=\theta$ is called a
$T$ direction of $f(z)$ if for any $\varepsilon>0$, we have
\begin{equation*}
\limsup\limits_{r\rightarrow\infty}\frac{N(r,Z_\varepsilon(\theta),f=a)}{T(r,f)}>0
\end{equation*}
for all but at most two values of $a$ in the extended complex plane
$\widehat{\mathbb{C}}$. Here
$$N(r,\Omega,f=a)=\int_1^r{n(t,\Omega,f=a)\over t}dt,$$
where $n(t,\Omega,f=a)$ is the number of the roots of $f(z)=a$ in
$\Omega\cap\{1<|z|<t\},$ counted according to multiplicity. And
through out this paper, we denote
$Z_\varepsilon(\theta)=\{z:\theta-\varepsilon<\operatorname{arg}
z<\theta+\varepsilon\}$ and $\Omega(\alpha,\beta)=\{z:\alpha<\operatorname{arg}
z<\beta\}$.
\end{defi}
The reason about the name is that we use the Nevanlinna's
characteristic $T(r,f)$ as comparison body. Under the growth
condition
\begin{equation}\limsup\limits_{r\rightarrow\infty}\frac{T(r,f)}{(\log
r)^2}=+\infty. \end{equation} Guo, Zheng and Ng \cite{GuoZhengNg}
confirmed the existence of this type direction and they pointed out
the growth condition (1.1) is sharp. Later, Zhang \cite{zhang}
showed that $T$ directions are different from Borel directions whose
definition can be found in \cite{Hayman}.
In 1979, Yang \cite{Yang01} showed the following theorem, which says that the condition for an angular domain to contain at least one Borel direction.\\
\textbf{Theorem A.} \emph{Let $f(z)$ be a meromorphic function on
the whole complex plane, with $\mu<\infty,0<\lambda\leq\infty$. Let
$\rho$ be a finite number such that $\lambda\geq\rho\geq\mu$ and
$\rho>1/2$. If $f^{(k)}(z)(k\geq0)$ has $p$ distinct deficient
values $a_1,a_2,\cdots,a_p$, then in any angular domain
$\Omega(\alpha,\beta)$ such that
$$\beta-\alpha>\max\{\frac{\pi}{\rho},2\pi-\frac{4}{\rho}\sum\limits_{i=1}^p\arcsin\sqrt{\frac{\delta(a_i,f^{(k)})}{2}}\},$$
$f(z)$ has a Borel direction with order $\geq\rho$.}
Recently, Zheng \cite{JH01} discussed the problem of $T$ directions
of a meromorphic function in one angular domain by proving.\\
\textbf{Theorem B.}\emph{ Let $f(z)$ be a transcendental meromorphic
function with finite lower order $\mu$ and non-zero order $\lambda$
and $f$ has a Nevanlinna deficient value $a\in\widehat{\mathbb{C}}$
with $\delta=\delta(a,f)>0$. For any positive and finite $\tau$ with
$\mu\leq\tau\leq\lambda$, consider the angular domain
$\Omega(\alpha,\beta)$ with
$$\beta-\alpha>\max\{\frac{\pi}{\tau},2\pi-\frac{4}{\tau}\arcsin\sqrt{\frac{\delta}{2}}\}.$$
Then $f(z)$ has a T direction in $\Omega=\Omega(\alpha,\beta)$.}
Following Yang \cite{Yang01} and Zheng \cite{JH01}, we will
continue the discussion of singular directions of $f(z)$ in some
angular domains. The following three questions will be mainly
investigated in this paper.
\begin{que}Can we extend Theorem B to some angular domains
$$X=\bigcup\limits_{j=1}^q\{z:\alpha_j\leq\operatorname{arg} z\leq\beta_j\},$$
where the $q$ pair of real numbers $\{\alpha_j,\beta_j\}$ satisfy
\begin{equation}\label{05}
-\pi\leq\alpha_1<\beta_1\leq\alpha_2<\beta_2\leq\cdots\leq\alpha_q<\beta_q\leq\pi?
\end{equation}
\end{que}
\begin{que} Can $f(z)$ in Theorem B be replaced by any derivative
$f^{(p)}(z)(p\geq0)$?
\end{que}
\begin{que}What can we do if $f(z)$ has many deficient values $a_1, a_2, a_3, \cdots,
a_l$ in Theorem B?
\end{que}
According to the Hayman inequality (see \cite{Hayman}) on the
estimation of $T(r,f)$ in terms of only two integrated counting
functions for the roots of $f(z)=a$ and $f^{(k)}(z)=b$ with
$b\not=0$, Guo, Zheng and Ng proposed in \cite{GuoZhengNg} a
singular direction named Hayman $T$ direction as follows.
\begin{defi}\label{def2}\ Let $f(z)$ be a transcendental meromorphic
function. A direction $L: \operatorname{arg} z=\theta$ is called a Hayman $T$
direction of $f(z)$ if for any small $\varepsilon>0$, any positive
integer $k$ and any complex numbers $a$ and $b\not=0$, we have
\begin{equation*}\label{4}
\limsup\limits_{r\longrightarrow\infty}\frac{N(r,Z_{\varepsilon}(\theta),f=a)
+N(r,Z_{\varepsilon}(\theta),f^{(k)}=b)}{ T(r,f)}>0.
\end{equation*}\end{defi}
Recently, Zheng and the first author \cite{Zheng02} confirmed the
existence of Hayman $T$ direction under the condition that
\begin{equation}\limsup\limits_{r\rightarrow+\infty}\frac{T(r,f)}{(\log r)^3}=+\infty
\end{equation}\\
In the same paper, the authors pointed out the Hayman $T$ direction
is different from the $T$ direction and they gave an example to
show the growth condition (1.3) is sharp. Can we discuss the
problem in some angular domains in the viewpoint of Question 1.1-1.3
? Though out this paper, we define
$$\omega=\max\{\frac{\pi}{\beta_1-\alpha_1},\cdots,\frac{\pi}{\beta_q-\alpha_q}\}.$$
Now, we state our theorems as follows.
\begin{thm}\label{thm1.1}
Let $f(z)$ be a transcendental meromorphic function with finite
lower order $\mu<\infty$, $0<\lambda\leq\infty$. There is an integer
$p\geq0$, such that $f^{(p)}$ has a Nevanlinna deficient value
$a\in\widehat{\mathbb{C}}$ with $\delta(a,f^{(p)})>0$. For $q$ pairs
of real numbers satisfies \eqref{05}. $f$ has at least one Hayman
$T$ direction in $X$ if
\begin{equation}\label{02}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j)<\frac{4}{\sigma}\arcsin\sqrt{\frac{\delta(a,f^{(p)})}{2}},
\end{equation}
where $\mu\leq\sigma\leq\lambda$, and $\omega<\sigma$.
\end{thm}
\begin{thm}\label{thm1.2}
Let $f(z)$ be a transcendental meromorphic function with finite
lower order $\mu<\infty$, $0<\lambda\leq\infty$. There is an integer
$p\geq0$, such that $f^{(p)}$ has $l\geq1$ distinct deficient values
$a_1, a_2, \cdots, a_l$ with the corresponding deficiency
$\delta(a_1,f^{(p)})$, $\delta(a_2,f^{(p)}),
\cdots,\delta(a_l,f^{(p)})$. For $q$ pair of real numbers
$\{\alpha_j,\beta_j\}$ satisfying \eqref{05} and
\begin{equation}\label{02}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j)<\sum\limits_{j=1}^l\frac{4}{\sigma}\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}},
\end{equation}
where $\mu\leq\sigma\leq\lambda$. If $\omega<\sigma$, then $f$ has
at least one Hayman $T$ direction in $X$.
\end{thm}
We will only prove Theorem \ref{thm1.2}, and Theorem \ref{thm1.1}
is a special case of Theorem \ref{thm1.2}.
\section{Primary knowledge and some lemmas}
In order to prove the theorems, we give some lemmas. The following
result is from \cite{Zheng}.
\begin{lem}
Let $f(z)$ be a transcendental meromorphic function with lower order
$\mu<\infty$ and order $0<\lambda\leq\infty$, then for any positive
number $\mu\leq\sigma\leq\lambda$ and a set $E$ with finite measure,
there exist a sequence $\{r_n\}$, such that
(1) $r_n\notin E$,
$\lim\limits_{n\rightarrow\infty}\frac{r_n}{n}=\infty$;
(2) $\liminf\limits_{n\rightarrow\infty}\frac{\log T(r_n,f)}{\log
r_n}\geq\sigma$;
(3) $T(t,f)<(1+o(1))(\frac{2t}{r_n})^\sigma
T(r_n/2,f),t\in[r_n/n,nr_n]$;
(4)$T(t,f)/t^{\sigma-\varepsilon_n}\leq2^{\sigma+1}T(r_n,f)/r_n^{\sigma-\varepsilon_n},1\leq
t\leq nr_n, \varepsilon_n=[\log n]^{-2}.$
\end{lem}
We recall that $\{r_n\}$ is called the P\'{o}lya peaks of order
$\sigma$ outside $E$. Given a positive function $\Lambda(r)$
satisfying $\lim_{r\rightarrow\infty}\Lambda(r)=0$. For $r>0$ and
$a\in\mathbb{C}$, define
$$D_\Lambda(r,a)=\{\theta\in[-\pi,\pi):\log^+\frac{1}{|f(re^{i\theta})-a|}>\Lambda(r)T(r,f)\},$$
and
$$D_\Lambda(r,\infty)=\{\theta\in[-\pi,\pi):\log^+|f(re^{i\theta})|>\Lambda(r)T(r,f)\}.$$
The following result is called the generalized spread relation, and
Wang in \cite{Wang} proved this.
\begin{lem}\label{lem03}
Let $f(z)$ be transcendental and meromorphic in $\mathbb{C}$ with
the finite lower order $\mu<\infty$ and the positive order
$0<\lambda\leq\infty$ and has $l\geq1$ distinct deficient values
$a_1, a_2, \cdots, a_l$. Then for any sequence of P\'{o}lya peaks
$\{r_n\}$ of order $\sigma>0,\mu\leq\sigma\leq\lambda$ and any
positive function $\Lambda(r)\rightarrow0$ as $r\rightarrow+\infty$,
we have
$$\liminf\limits_{n\rightarrow\infty}\sum\limits_{j=1}^l \operatorname{meas} D_\Lambda(r_n,a_j)\geq\min\{2\pi, \frac{4}{\sigma}\sum\limits_{j=1}^l \arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}}\}.$$
\end{lem}
From \cite{Yang01}, we know that for $a\neq b$ are two deficient
values of $f$, then we have $D_\Lambda(r,a)\bigcap
D_\Lambda(r,b)=\emptyset$.
Nevanlinna theory on the angular domain plays an important role in
this paper. Let us recall the following terms:
\begin{equation*}
\begin{split}
A_{\alpha,\beta}(r,f)&=\frac{\omega}{\pi}\int_1^r(\frac{1}
{t^\omega}-\frac{t^\omega}{r^{2\omega}})\{\log^+|f(te^{i\alpha})|+\log^+|f(te^{i\beta})|\}\frac{dt}{t},\\
B_{\alpha,\beta}(r,f)&=\frac{2\omega}{\pi
r^\omega}\int_{\alpha}^\beta\log^+|f(re^{i\theta})|\sin\omega(\theta-\alpha)d\theta,\\
C_{\alpha,\beta}(r,f)&=2\sum\limits_{1<|b_n|<r}(\frac{1}{|b_n|^\omega}-
\frac{|b_n|^\omega}{r^{2\omega}})\sin\omega(\theta_n-\alpha),
\end{split}
\end{equation*}
where $\omega=\frac{\pi}{\beta-\alpha}$, and
$b_n=|b_n|e^{i\theta_n}$ is a pole of $f(z)$ in the angular domain
$\Omega(\alpha,\beta)$, appeared according to the multiplicities.
The Nevanlinna's angular characteristic is defined as follows:
\begin{equation*}
S_{\alpha,\beta}(r,f)=A_{\alpha,\beta}(r,f)+B_{\alpha,\beta}(r,f)+C_{\alpha,\beta}(r,f).
\end{equation*}
From the definition of $B_{\alpha,\beta}(r,f)$, we have the
following inequality, which will be used in the next.
\begin{equation}\label{06}
B_{\alpha,\beta}(r,f)\geq\frac{2\omega\sin(\omega\varepsilon)}{\pi
r^\omega}\int_{\alpha+\varepsilon}^{\beta-\varepsilon}\log^+|f(re^{i\theta})|d\theta
\end{equation}
The following is the Nevanlinna first and second fundamental
theorem on the angular domains.
\begin{lem}\label{notation01}
Let $f$ be a nonconstant meromorphic function on the angular domain
$\Omega(\alpha,\beta)$. Then for any complex number $a$,
\begin{equation*}
S_{\alpha,\beta}(r,f)=S_{\alpha,\beta}(r,\frac{1}{f-a})+O(1),
r\rightarrow\infty,
\end{equation*}
and for any $q(\geq3)$ distinct points $a_j\in\widehat{\mathbb{C}}\
(j=1,2,\ldots,q)$,
\begin{equation*}\label{nevan01}
\begin{split}
(q-2)S_{\alpha,\beta}(r,f)&\leq\sum\limits_{j=1}^q\overline{C}_{\alpha,\beta}(r,\frac{1}{f-a_j})+Q_{\alpha,\beta}(r,f),
\end{split}
\end{equation*}
where
$$Q_{\alpha,\beta}(r,f)=(A+B)_{\alpha,\beta}(r,\frac{f'}{f})
+\sum\limits_{j=1}^q(A+B)_{\alpha,\beta}(r,\frac{f'}{f-a_j})+O(1).$$
\end{lem}
The key point is the estimation of error term
$Q_{\alpha,\beta}(r,f)$, which can be obtained for our purpose of
this paper as follows. And the following is true(see
\cite{Goldberg}). Write
$$Q(r,f)=A_{\alpha,\beta}(r,\frac{f^{(p)}}{f})+B_{\alpha,\beta}(r,\frac{f^{(p)}}{f}).$$
Then
(1)$Q(r,f)=O(\log r)$ as $r\rightarrow\infty$, when
$\lambda(f)<\infty$.
(2)$Q(r,f)=O(\log r+\log T(r,f))$ as $r\rightarrow\infty$ and
$r\notin E$ when $\lambda(f)=\infty$, where $E$ is a set with finite
linear measure.
The following result is useful for our study, the proof of which is
similar to the case of the characteristic function $T(r,f)$ and
$T(r,f^{(k)})$ on the whole complex plane. For the completeness, we
give out the proof.
\begin{lem}\label{lem01}
Let $f(z)$ be a meromorphic function on the whole complex plane.
Then for any angular domain $\Omega(\alpha,\beta)$, we have
$$S_{\alpha,\beta}(r,f^{(p)})\leq(p+1)S_{\alpha,\beta}(r,f)+O(\log r+\log T(r,f)),$$
possibly outside a set of $r$ with finite measure.
\end{lem}
\begin{proof}
In view of the definition of $S_{\alpha,\beta}(r,f)$ and Lemma
\ref{notation01}, we get the following
\begin{equation*}
\begin{split}
S_{\alpha,\beta}(r,f^{(p)})&\leq
C_{\alpha,\beta}(r,f^{(p)})+(A+B)_{\alpha,\beta}(r,f)+(A+B)_{\alpha,\beta}(r,\frac{f^{(p)}}{f})\\
&=p\overline{C}_{\alpha,\beta}(r,f)+S_{\alpha,\beta}(r,f)+(A+B)_{\alpha,\beta}(r,\frac{f^{(p)}}{f})\\
&\leq(p+1)S_{\alpha,\beta}(r,f)+Q(r,f).
\end{split}
\end{equation*}
\end{proof}
Recall the definition of Ahlfors-Shimizu characteristic in an angle
(see \cite{Tsuji}). Let $f(z)$ be a meromorphic function on an angle
$\Omega=\{z:\alpha\leq\operatorname{arg} z\leq \beta\}$. Set
$\Omega(r)=\Omega\cap\{z:1<|z|<r\}$. Define
$$\mathcal{S}(r,\Omega,f)=\frac{1}{\pi}\int\int_{\Omega(r)}{\left(|f'(z)|\over
1+|f(z)|^2\right)^2}d\sigma$$ and
$$\mathcal{T}(r,\Omega,f)=\int_1^r{\mathcal{S}(t,\Omega,f)\over
t}dt.$$
The following lemma is a theorem in \cite{Zheng02}, which is to
controll the term $\mathcal {T}(r,\Omega_\varepsilon)$ using the
counting functions $N(r,\Omega,f=a)$ and $N(r,\Omega,f^{(k)}=b)$.
\begin{lem}\label{04}
Let $f(z)$ be meromorphic in an angle $\Omega=\{z:\alpha\leq\operatorname{arg} z
\leq\beta\}$. Then for any small $\varepsilon>0$, any positive
integer $k$ and any two complex numbers $a$ and $b\not=0$, we have
\begin{equation}\label{9}
\mathcal{T}(r,\Omega_\varepsilon,f) \leq K\{N(2r,\Omega,f=a)+
N(2r,\Omega,f^{(k)}=b)\}+O(\log^3 r)
\end{equation} for a positive constant $K$ depending only on $k$, where $\Omega_\varepsilon=\{z:\alpha+\varepsilon<\operatorname{arg}
z<\beta-\varepsilon\}$.
\end{lem} In order to prove our theorem, we have to use the following
lemma, which is a consequent result of Theorem 3.1.6 in \cite{JH01}.
\begin{lem}
Let $f(z)$ be a transcendental meromorphic function in the whole
plane, and satisfies the conditions of Theorem \ref{thm1.2} or
Theorem \ref{thm1.1}. Take a sequence of P\'{o}lya peak $\{r_n\}$ of
$f(z)$ of order $\sigma>\omega=\frac{\pi}{\beta-\alpha}$. If $f(z)$
has no Hayman T direction in the angular domain
$\Omega(\alpha,\beta)$, then the following real function satisfy
$\lim\limits_{r\rightarrow\infty}\Lambda(r)=0$, which $\Lambda(r)$
is defined as follows
$$\Lambda(r)^2=\max\{\frac{\mathcal
{T}(r_n,\Omega_\varepsilon,f)}{T(r_n,f)},
\frac{r_n^{\omega}}{T(r_n,f)}\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega_\varepsilon,f)}{t^{\omega+1}}dt,\frac{r_n^\omega[\log
r_n+\log T(r_n,f)]}{T(r_n,f)}\},$$ for $r_n\leq r<r_{n+1}.$
\end{lem}
\begin{proof} We should treat two cases.\\
Case (I). If there is no Hayman $T$ direction on $\Omega$, then from
Lemma \ref{04}, we have
$$\mathcal {T}(r,\Omega_\varepsilon,f)=o(T(2r,f))+O(\log^3r), \ as\ r\rightarrow\infty.$$
Combining Lemma 2.1 and $\sigma>\omega$, we have
\begin{equation*}
\begin{split}
\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega_\varepsilon,f)}{t^{\omega+1}}dt&=o(\int_1^{r_n}\frac{T(2t,f)}{t^{\omega+1}}dt)+\int_1^{r_n}\frac{O(\log^3
t)}{t^{\omega+1}}dt\\
&\leq
o(\int_1^{r_n}\frac{T(r_n,f)}{t^{\omega+1}}(\frac{2t}{r_n})^\sigma
dt)+O(\log^3r_n)\\
&=o(\frac{T(r_n,f)}{r_n^\omega})+O(\log^3r_n)\\
\end{split}
\end{equation*}
Then
$$\frac{r_n^\omega}{T(r_n,f)}\int_1^{r_n}\frac{\mathcal {T}(t,\Omega_\varepsilon)}{t^{\omega+1}}dt\rightarrow0, \ as \ n\rightarrow\infty. $$
Case (II). If $$\limsup\limits_{n\rightarrow\infty}\frac{\mathcal
{T}(r_n,\Omega_\varepsilon,f)}{T(r_n,f)}>0,$$ then by \eqref{9}, we
have
$$\limsup\limits_{n\rightarrow\infty}\frac{N(2r_n,\Omega,f=a)+N(2r_n,\Omega,f^{(k)}=b)}{T(r_n,f)}>0.$$
Since $\{r_n\}$ is a sequence of P\'{o}lya peaks of order $\sigma$,
then we have $$T(2r_n,f)\leq2^\sigma T(r_n,f).$$ Then $\Omega$ must
contain a Hayman $T$ direction of $f(z)$. This is contradict to the
hypothesis.
From Case (I) and Case (II) and notice that $r_n^\omega[\log
r_n+\log T(r_n,f)]/T(r_n,f)\rightarrow0,(n\rightarrow\infty)$, we
have proved that $\limsup_{r\rightarrow\infty}\Lambda(r)=0$.
\end{proof}
The following result was firstly established by Zheng
\cite{JH01}(Theorem 2.4.7), it is crucial for our study.
\begin{lem}
Let $f(z)$ be a function meromorphic on
$\Omega=\Omega(\alpha,\beta)$. Then
$$S_{\alpha,\beta}(r,f)\leq2\omega^2\frac{\mathcal {T}(r,\Omega,f)}{r^\omega}+\omega^3\int_1^r\frac{\mathcal {T}(t,\Omega,f)}{t^{\omega+1}}dt+O(1),\ \ \omega=\frac{\pi}{\beta-\alpha}.$$
\end{lem}
We also have to use the following lemma, which is due to Hayman and
Miles \cite{Miles}.
\begin{lem}\label{lem02}
Let $f(z)$ be meromorphic in the complex plane. Then for a given
$K>1$, there exists a set $M(K)$ with $\overline{\log
dens}M(K)\leq\delta(K)$,
$\delta(K)=\min\{(2e^{K-1}-1)^{-1},(1+e(K-1)exp(e(1-K)))\}$, such
that
$$\limsup\limits_{r\rightarrow+\infty,r\notin M(K)}\frac{T(r,f)}{T(r,f^{(p)})}\leq3eK.$$
\end{lem}
\section{Proof of theorem \ref{thm1.2}}
\begin{proof}
Case(I). $\lambda(f)>\mu$. Then we choose $\sigma$ such that
$\lambda(f^{(p)})=\lambda(f)>\sigma\geq\mu=\mu(f^{(p)}),
\sigma>\omega$. From the inequality \eqref{02}, we can take a real
number $\varepsilon>0$ such that
\begin{equation}\label{01}
\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j+4\varepsilon)+\varepsilon<\sum\limits_{j=1}^l\frac{4}{\sigma+2\varepsilon}\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}},
\end{equation}
and $$\lambda(f^{(p)})>\sigma+2\varepsilon>\mu.$$ Then there exists
a sequence of P\'{o}lya peaks $\{r_n\}$ of order
$\sigma+2\varepsilon$ of $f^{(p)}$ such that $\{r_n\}$ are not in
the set of Lemma \ref{lem01} and Lemma \ref{lem02}.
We define $q$ real functions $\Lambda_j(r)(j=1,2,\cdots,q)$ as
follows.
\begin{equation*}
\begin{split}
\Lambda_j(r)^2=\max\{&\frac{\mathcal
{T}(r_n,\Omega(\alpha_j+\varepsilon,\beta_j-\varepsilon),f)}{T(r_n,f)},\\
&\frac{r_n^{\omega_j}}{T(r_n,f)}
\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega(\alpha_j+\varepsilon,\beta_j-\varepsilon),f)}{t^{\omega_j+1}}dt,\frac{r_n^{\omega_j}[\log
r_n+\log T(r_n,f)]}{T(r_n,f)}\},
\end{split}
\end{equation*}
for $r_n\leq r<r_{n+1},
\omega_{j}=\frac{\pi}{\beta_{j}-\alpha_{j}}$. By using Lemma 2.5, we
have $\Lambda_j(r)\rightarrow0$, as $r\rightarrow\infty$, if $f(z)$
has no Hayman $T$ directions on $X$. Set $\Lambda(r)=\max_{1\leq
j\leq q}\{\Lambda_j(r)\}$, we have
$\lim_{r\rightarrow\infty}\Lambda(r)=0$. Therefore for large enough
$n$, by Lemma \ref{lem03} we have
\begin{equation}\label{03}
\sum\limits_{j=1}^l \operatorname{meas} D_\Lambda(r_n,a_j)>\min\{2\pi,
\frac{4}{\sigma+2\varepsilon}\sum\limits_{j=1}^l
\arcsin\sqrt{\frac{\delta(a_j,f^{(p)})}{2}}\}-\varepsilon.
\end{equation}
We note that $\sigma+2\varepsilon>1/2$, we suppose for any $n$
\eqref{03} holds. Set
$$K_n=\operatorname{meas}
((\bigcup\limits_{j=1}^lD_\Lambda(r_n,a_j))\bigcap(\bigcup\limits_{j=1}^q(\alpha_j+2\varepsilon,\beta_j-2\varepsilon))).$$
Combining \eqref{01} with \eqref{03}, we obtain
\begin{equation*}
\begin{split}
K_n&\geq \sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\operatorname{meas}([-\pi,\pi)\backslash\bigcup\limits_{j=1}^q(\alpha_j+2\varepsilon,\beta_j-2\varepsilon))\\
&= \sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\operatorname{meas}(\bigcup\limits_{j=1}^q(\beta_j-2\varepsilon,\alpha_{j+1}+2\varepsilon))\\
&=\sum\limits_{j=1}^l\operatorname{meas} (D_\Lambda(r_n,
a_j))-\sum\limits_{j=1}^q(\alpha_{j+1}-\beta_j+4\varepsilon)>\varepsilon>0.
\end{split}
\end{equation*} It is easy to see that, there exists a $j_0$ such that for
infinitely many $n$, we have
\begin{equation*}
\operatorname{meas}
(\bigcup\limits_{j=1}^lD_\Lambda(r_n,a_j)\bigcap(\alpha_{j_0}+2\varepsilon,\beta_{j_0}-2\varepsilon))>\frac{K_n}{q}>\frac{\varepsilon}{q}.
\end{equation*}
We can assume that the above holds for all the $n$.
Set
$E_{nj}=D(r_n,a_j)\bigcap(\alpha_{j_0}+2\varepsilon,\beta_{j_0}-2\varepsilon)$.
Thus we have
\begin{equation}\label{07}
\begin{split}
\sum\limits_{j=1}^l\int_{\alpha_{j_0}+2\varepsilon}^{\beta_{j_0}-2\varepsilon}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta&\geq
\sum\limits_{j=1}^l\int_{E_{nj}}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta\\
&\geq\sum\limits_{j=1}^l\operatorname{meas}(E_{nj})\Lambda(r_n)T(r_n,f^{(p)})\\&>\frac{\varepsilon}{q}\Lambda(r_n)T(r_n,f^{(p)})\\
&>\frac{\varepsilon}{3eqK}\Lambda(r_n)T(r_n,f).
\end{split}
\end{equation}
The last inequality uses Lemma 2.8.
On the other hand, we have
\begin{equation}\label{08}
\begin{split}
&\sum\limits_{j=1}^l\int_{\alpha_{j_0}+2\varepsilon}^{\beta_{j_0}-2\varepsilon}\log^+\frac{1}{|f^{(p)}(r_ne^{i\theta})-a_j|}d\theta\leq\sum\limits_{j=1}^l
\frac{\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}B_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,\frac{1}{f^{(p)}-a_j})\\
&<\sum\limits_{j=1}^l\frac{\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,\frac{1}{f^{(p)}-a_j})\\
&=\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,f^{(p)})+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[(p+1)S_{\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon}(r_n,f)+\log r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}(p+1)[2\omega_{j_0}^2\mathcal
{T}(r_n,\Omega(\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon),f)\\&+\omega_{j_0}^3r_n^{\omega_{j_0}}\int_1^{r_n}\frac{\mathcal
{T}(t,\Omega(\alpha_{j_0}+\varepsilon,\beta_{j_0}-\varepsilon),f)}{t^{\omega_{j_0}+1}}dt]\\
&+\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[\log r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}})\\
&\leq\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}(p+1)[2\omega_{j_0}^2\Lambda(r_n)^2T(r_n,f)
+\omega_{j_0}^3\Lambda(r_n)^2T(r_n,f)]\\
&+\frac{l\pi}{2\omega_{j_0}\sin(\varepsilon\omega_{j_0})}r_n^{\omega_{j_0}}[\log
r_n+\log T(r_n,f)]+O(r_n^{\omega_{j_0}}),\ \ \ \
\omega_{j_0}=\frac{\pi}{\beta_{j_0}-\alpha_{j_0}-2\varepsilon}. \
\end{split}
\end{equation}
\eqref{07} and \eqref{08} imply that
$$\Lambda(r_n)\leq O(\Lambda(r_n)^2).$$
A contradiction is derived because $\Lambda(r_n)\rightarrow0$ as
$n\rightarrow\infty$.
Case (II). $\lambda(f)=\mu$. By the same argument as in Case1 with
all the $\sigma+2\varepsilon$ replaced by $\sigma=\mu$, we can
derive the same contradiction.
\end{proof}
\end{document}
\end{document} |
\begin{document}
\title{Introducing a new concept of distance on a topological space by generalizing the definition of quasi-pseudo-metric
}
\author{Hamid Shobeiri\footnote{Email address: shobeiri@mail.kntu.ac.ir (A.H. Shobeiri)}\\
\footnotesize{{\it Department of Mathematics, K.N. Toosi
University of Technology,}} \\
\footnotesize{{\it P.O.Box 16315-1618, Tehran, Iran}}\\
}
\date{}
\maketitle
\begin{abstract}
In this paper, a new structure is defined on a topological space that equips the space with a concept of distance in order to do that firstly, a generalization of quasi-pseudo-metric space named R.O-metric space is introduced, and some of its basic properties is studied. Afterwards the concept of generalized R.O-metric space is defined .Finally, we establish that every topological space is generalized R.O-metrizable.
\end{abstract}
\textbf{Keywords:} Topological space, Quasi-pseudo-metric space, R.O-metric space, Generalized R.O-metric space. \\
\textbf{AMS Subject Classifications:} 54D80, 54D35, 54E35, 54E99, 54D65 .
\section{Introduction}
Topological spaces are extension of metric spaces. It is well known that each arbitrary topological space is not necessary metrizable (see \cite{munkres1975topology} or \cite{simmons1963introduction}). Therefore despite of the beauty and simplicity of such extension, it involves some limitations. For example, size of neighborhoods of two distinct points are not comparable in topological spaces. In addition, uniform continuity, Cauchy sequence and complete space are no more definable in arbitrary topological spaces. These limitations may raise the idea of defining topological spaces through a new concept of distance, in order to simplify working in these spaces.Defining a new concept of distance, will be useful. In this direction, some mathematicians introduced some structures weaker than metric spaces.
A metric on a set $X$ is a function $d:X\times X\to [0,\infty)$ such that for all $x,y,z\in X$, the following conditions are satisfied:
\begin{align}
d(x,y)=0\Leftrightarrow x=y,\label{I.1}\\
d(x,y)=d(y,x)\label{I.2},\\
d(x,z)\leq d(x,y)+d(y,z). \label{I.3}
\end{align}
One of the generalized metric spaces is semi-metric space that is introduced by Frechet and Menger which satisfies conditions \eqref{I.1} and \eqref{I.2} of definition of metric space (see \cite{cicchese1976questioni}, \cite{wilson1931semi}, \cite{frechet1906quelques} and \cite{menger1928untersuchungen}). In the last few years, the study of non-symmetric topology has received a new derive as a consequence of it's applications to the study of several problems in theoretical computer science and applied physics.
One of such structures is quasi-metric space that is introduced by W.A.Wilson (see \cite{wilson1931quasi}) which has conditions \eqref{I.1} and \eqref{I.3}. One other generalization of metric spaces is called pseudo-metric space, which satisfies conditions $(\displaystyle d(x,x)=0)$, (2) , (3) (see \cite{simmons1963introduction}). Quasi-pseudo-metric space is introduced by Kelly (see \cite{kelly1963bitopological}) which satisfies conditions $( d(x,x)=0)$ and (3). $T_0$-quasi-metric space is quasi-pseudo-metric space that satisfies condition $(d(x,y)=0=d(y,x)\Rightarrow x=y)$ that is presented in paper \cite{kemajou2012isbell}.
Multi-metric space is defined by Smarandache $($see \cite{smarandache2000mixed}, \cite{smarandache2001unifying}$)$, which is a union $\tilde{M} =\bigcup_{i=1}^{n}M_i$, such that each $M_i$ is a space
with metric $d_i$ for all $1\le i\le n$.
The above mentioned structures can not describe all topological spaces. In this paper, it is aimed to present a new structure to be able to describe all topological spaces. It is started by definition of structure that is called R.O-metric space (Right-Oriented-metric space: this terminology comes from non-symmetric meter) which is a generalization of quasi-pseudo-metric space. Then generalized R.O-metric is defined which reforms the definition of topological space. In the first section, the concept of R.O-metric space is defined. In the second section, R.O-metric space is generalized and improved by adding some conditions.
\section{Preliminaries}
\begin{defn}
Let $X$ be a non empty set; a function $\overrightarrow{d}:X\times X\to [0,\infty)$ is called a R.O-metric on $X$ iff for every $x,y,z\in X$, the following conditions hold:
\begin{enumerate}
\item$\overrightarrow{d}(x,x)=0$,
\item $\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)\neq 0 \Rightarrow \overrightarrow{d}(x,y)\le \overrightarrow{d}(x,z)+\overrightarrow{d}(z,y),$
\end{enumerate}
and then $(X,\overrightarrow{d})$ is called a R.O-metric space.
\end{defn}
\begin{ex}
Every metric space is a R.O-metric space.
\end{ex}
\begin{ex}
As another example , for $X=\{a,b,c\}$ consider $$\overrightarrow{d}(a,b)=\overrightarrow{d}(b,c)=1\; , \overrightarrow{d}(a,c)=2\; ,$$$$\overrightarrow{d}(a,a)=\overrightarrow{d}(b,b)=\overrightarrow{d}(c,c)=\overrightarrow{d}(b,a)=\overrightarrow{d}(c,a)=\overrightarrow{d}(c,b)=0,$$ then $(X,\overrightarrow{d})$ is a R.O-metric space.\ref{fig:55}
\begin{figure}\label{fig:55}
\end{figure}
\end{ex}
\begin{defn}
In a R.O-metric space $(X,\overrightarrow{d})$,
the set, $V_{r}(p)=\{x\;|\;\overrightarrow{d}(p,x)<r\}$ is called a $r$-ball of a point $p$ with radius $r>0$.
\end{defn}
\begin{n}
Let $(X,\overrightarrow{d})$ be a R.O-metric space ,
then the set $S_{\overrightarrow{d}}=\{V_{r}(p)\;|\;r>0,p\in X\}$ is a subbasis for a topology on X, which is called the generated topology by $\overrightarrow{d}$ and is shown by $\tau_{\overrightarrow{d}}.$
\end{n}
\begin{defn}
Topological space $(X,\tau)$ is called R.O-metrizable iff there exists a R.O-metric $\overrightarrow{d}$ such that $\tau_{\overrightarrow{d}}=\tau$.
\end{defn}
It can be shown that many of the most familiar topological spaces are R.O-metric spaces, here are some examples of non metrizable topological spaces which are R.O-metric spaces:
\begin{ex}
Let $X$ be a set and $\phi \neq A\subseteq X$, then $\tau_{A}=\{B\subseteq X \;|\; A\subseteq B\}\cup \{\emptyset\}$ is a topology on $X$; we define R.O-metric $\overrightarrow{d}$ on $X$ as follows:
\begin{enumerate}
\item $\forall x\in X;\;\overrightarrow{d}(x,x)=0,$ \item $\forall a \in X , \forall b\in A, \; \overrightarrow{d}(a,b)=0,$\item $\forall a\in X , \forall b\in A^{c} ;\; \overrightarrow{d}(a,b)=1.$
\end{enumerate}
The topology $\tau_{\overrightarrow{d}}$ is generated by subbasis $$S_{\overrightarrow{d}}=\{V_{r}(p)\;|\;r>0,p\in X\}=\{V_{1}(a)\;|\;a\in A\}\cup{\{V_{1}(b)\;|\;b\in A^{c}\}}\cup{\{V_{2}(x)\;|\;x\in X\}}$$
$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\{A\}\cup{\{\{b\}\cup{A}\;|\;b\in A^{c}\}}\cup{\{X\}}.$$
Thus $\tau_{\overrightarrow{d}}=\tau_{A}$.
\end{ex}
\begin{ex}\label{13}
Let $\tau$ be cofinite topology on infinite set $X$, that means topology in which the open sets are the subset of $X$ with finite complements. It is known that (see \cite{hrbacek1999introduction}) the set $X$ can be written as $X=\bigcup_{\alpha\in I}{A_{\alpha}}$ such that for all $\alpha\in I$, $A_{\alpha}$ is countable and $A_{\alpha}=\{x_{\alpha,1},x_{\alpha,2},...\}$. Now define R.O-metric $\overrightarrow{d}$ on $X$ as follows:
\begin{enumerate}
\item$\forall x\in X;\;\overrightarrow{d}(x,x):=0,$\item$\forall\alpha\in I,\;\forall n\in O;(O\;is\;the\;odd\;natural\;numbers)\\\;\overrightarrow{d}(x_{\alpha,n},x_{\alpha,n+1}):=1=\overrightarrow{d}(x_{\alpha,n+1},x_{\alpha,n})$,\item$\overrightarrow{d}(x_{\alpha,n},y):=0=\overrightarrow{d}(x_{\alpha,n+1},z),\;\forall y\ne x_{\alpha,n+1},\;\forall z\ne x_{\alpha,n}$
\end{enumerate}
the induced topology by $\overrightarrow{d}$ which is generated by subbasis $$S_{\overrightarrow{d}}=\{V_{1}(x_{\alpha,n})\;|\;\alpha\in I\;,\;n\in O\}\cup \{V_{1}(x_{\alpha,n})\;|\;\alpha\in I\;,\;n\in E\}\cup\{V_{2}(p)\;|\;p\in X\}$$ $$\;\;\;\;\;\;=\{X-\{x_{\alpha,n+1}\}\;|\;n\in O\;,\;\alpha\in I\}\cup \{X-\{x_{\alpha,n-1}\}\;|\;n\in E\;,\;\alpha\in I\}\cup\{X\},$$
in which E is the even natural numbers. Thus $\tau=\tau_{\overrightarrow{d}}$.
\end{ex}
\begin{ex}
Let $\tau$ be K-topology on $\Bbb{R}$,that means the topology generated by the basis $\{(a,b)\;|\;a,b\in \Bbb{R}\}\cup \{(a,b)-\{\frac{1}{n}\}_{n\in \Bbb{N}}\;|\;a,b\in \Bbb{R}\}$, then we define R.O-metric $\overrightarrow{d}$ as follows:
\begin{enumerate}
\item$\forall x\in X;\;\overrightarrow{d}(x,x):=0,$\item$\forall x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}},\;\overrightarrow{d}(x,\frac{1}{n}):=\mid{x-\frac{1}{n}}\mid+1,$\item$\; Otherwise\; \overrightarrow{d}(x,y):=|x-y|.$
\end{enumerate}
Then it is easy to check that
$$S_{\overrightarrow{d}}=\{V_{r}(x)\;|\;x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;,\;0< r\le 1\}\cup \{V_{r}(x)\;|\;x\notin \{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;,\;r>1\}$$$$\cup\{V_r(\frac{1}{n})\;|\;n\in\Bbb{N}\;,\;r>0\}=\{(x-r,x+r)-\{\frac{1}{n}\}_{n\in {\Bbb{N}}}\;|\;0<r\le 1\}$$$$\cup\{(x-r,x+r)\;|\;r>1\}\cup\{(\frac{1}{n}-r,\frac{1}{n}+r)\;|\;n\in\Bbb{N}\;,\;r>0\},$$
generates topology of K-topology on $\Bbb{R}$.
\end{ex}
\begin{ex}
For lower limit topology $\Bbb{R}_{l}$, which is a topology on $\Bbb{R}$ that has a basis as $\{[a,b)\;|\;a,b\in \Bbb{R}\}$, define R.O-metric $\overrightarrow{d}$ as follows:
\begin{enumerate}
\item$\forall a\in\Bbb{R};\;\overrightarrow{d}(a,a)=0,$\item$\overrightarrow{d}(a,b)=\begin{cases}a-b+1&b<a\\b-a&a\le b\end{cases}.$
\end{enumerate}
Then $$S_{\overrightarrow{d}}=\{V_r(a)\;|\;0<r\le 1\;,\;a\in\Bbb{R}\}\cup\{V_r(a)\;|\;r>1\;,\;a\in\Bbb{R}\}$$$$=\{[a,a+r)\;|\;0<r\le 1\;,\;a\in\Bbb{R}\}\cup\{(a-r,a+r)\;|\;r>1\;,\;a\in\Bbb{R}\}.$$
Simply we can see $\tau_{\overrightarrow{d}}$ is the lower limit topology.
\end{ex}
Now, we give an example which shows that there can be fined R.O-metric spaces that are NOT qusi-metrizable, pseudo-metrizable and NOT quasi-pseudo-metrizable.
\begin{ex}
Suppose $(X,\tau_c)$ be a cofinite topological space and $Card(X)\ge Card(\Bbb{R})$. By Example \ref{13}, $(X,\tau_c)$ is R.O-metrizable. Now we prove that $(X,\tau_c)$ is not quasi-pseudo-metrizable. If it is quasi-pseudo-metrizable, then there exists a quasi-pseudo-metric $d$, such that $\tau_d=\tau_c$, thus $B=\{V_r(x)\;|\;r>0\;,\;x\in X\}$ is a basis, and for each $x\in X$ and $U$ open set containing $x$, there exists $t>0$ such that $V_t(x)\subseteq U$. Thus $B_x=\{V_r(x)\;|\;r>0\}$ is a local base at $x$, since $V_r(x)^{c}$ is finite, thus $B_x$ is at most countable. Therefore $(X,\tau_c)$ is first countable and it is a contradiction, because $Card(X)\ge Card(\Bbb{R})$ and cofinite topological spaces like $(X,\tau)$ with $Card(X)>Card(\Bbb{N})$ are not first countable $($see \cite{simmons1963introduction}$)$. Since quasi-metrizable space is quasi-pseudo-metrizable, so $(X,\tau_c)$ is not quasi-metrizable. Also if $(X,\tau_c)$ is pseudo-metrizable, then $B$ is a basis for this topology. In addition, for each $x\in X$ and $U$ open set containing $x$, there exist $t>0$ such that $V_t(x)\subseteq U$ and by the same procedure as above, it causes a contradiction.
\end{ex}
\begin{p}
Suppose $(X,\overrightarrow{d})$ is a R.O-metric space and for each $x,y\in X$, define $\overrightarrow{\bar{d}}(x,y)=\frac{\overrightarrow{d}(x,y)}{\overrightarrow{d}(x,y)+1}$. Then $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}$.
\end{p}
\begin{proof}
Obviously condition (1) in the definition of R.O-metric holds. Now for all $x,y\in X$, if $\overrightarrow{d}(x,y)\le \overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)$, then $$\frac{1}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}\le \frac{1}{\overrightarrow{d}(x,y)+1}$$ $$\Rightarrow 1-\frac{1}{\overrightarrow{d}(x,y)+1}\le 1-\frac{1}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}=\frac{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)}{\overrightarrow{d}(x,z)+\overrightarrow{d}(z,y)+1}$$ $$\Rightarrow \frac{\overrightarrow{d}(x,y)}{\overrightarrow{d}(x,y)+1}\le \frac{\overrightarrow{d}(x,z)}{\overrightarrow{d}(x,z)+1}+\frac{\overrightarrow{d}(z,y)}{\overrightarrow{d}(z,y)+1}.$$
Therefore $\overrightarrow{\bar{d}}$ is R.O-metric on $X$. To prove $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}$, assume $V_r(x)\in S_{\overrightarrow{d}}$ and $z\in V_r(x)$, so by definition $\overrightarrow{d}(x,z)<r$. If $\overrightarrow{d}(x,z)\ne 0$, then $$\overrightarrow{\bar{d}}(x,z)<\frac{r}{r+1}\Rightarrow z\in \bar{V}_{\frac{r}{r+1}}(x)\in S_{\overrightarrow{\bar{d}}},$$in which $\bar{V}_{\frac{r}{r+1}}(x)$ is a $r$-ball with respect to $\overrightarrow{\bar{d}}$, and if $$\overrightarrow{d}(x,z)=0,\;then\; \overrightarrow{\bar{d}}(x,z)=0<\frac{r}{r+1},$$thus $z\in\bar{V}_{\frac{r}{r+1}}(x)$ and we have $V_r(x)\subseteq \bar{V}_{\frac{r}{r+1}}(x).$ It is easy to check that $\bar{V}_{\frac{r}{r+1}}(x)\subseteq V_r(x),$ for all $x\in X$ and all non negative real numbers. Thus $$S_{\overrightarrow{d}}\subseteq S_{\overrightarrow{\bar{d}}}\;\;\;(*).$$
Now let $\bar{V}_r(x)\in S_{\overrightarrow{\bar{d}}}$ and $z\in \bar{V}_r(x),\;r<1$, then $\overrightarrow{\bar{d}}(x,z)<r$. If $\overrightarrow{\bar{d}}(x,z)\ne 0$, then $\overrightarrow{d}(x,z)<\frac{r}{1-r}$, and if $\overrightarrow{\bar{d}}(x,z)=0$, then $\overrightarrow{d}(x,z)=0<\frac{r}{1-r}$, thus $z\in V_{\frac{r}{1-r}}(x)\in S_{\overrightarrow{d}}$, that implies $\bar{V}_r(x)\subseteq V_{\frac{r}{1-r}}(x)$. It is easy to check that $V_{\frac{r}{1-r}}(x)\subseteq \bar{V}_r(x)$, thus we get $ S_{\overrightarrow{\bar{d}}}\subseteq S_{\overrightarrow{d}}$, and by virtue of $(*)$, $S_{\overrightarrow{\bar{d}}}= S_{\overrightarrow{d}}$, which implies $\tau_{\overrightarrow{d}}=\tau_{\overrightarrow{\bar{d}}}.$
\end{proof}
\begin{n}\label{1}
It is well-known that every finite topological space $(X,\tau)$ has a subbasis $S$ such that $Card(S)\le Card(X)$, since for every point $p$ in $X$ there is the smallest open set with respect to $(\subseteq)$ containing $p$ and the set of these open sets is a subbasis $S$ for $(X,\tau)$ , obviously $Card(S)\le Card(X)$. In the following example we show that this property does not necessarily hold for infinite topological spaces.
\end{n}
\begin{ex} \label{4}
Let $Y=\Bbb{N}\times\Bbb{N}$ , $p$ be a point not in $Y$ and $X=\{p\}\cup Y$. For each function $ f:\Bbb{N}\to\Bbb{N}$, let
$$B_f:=\{p\}\cup\{(k,\ell)\in Y\;|\;\ell\ge f(k)\}\;.$$
Topologize $X$ by making each point of $Y$ isolated and taking $\left\{B_f:f\in{\Bbb{N}^{\Bbb{N}}}\right\}$ as a local subbasis at $p$.
We show that $X$ has no countable subbase.
Let $S=\left\{B_f:f\in{\Bbb{N}^{\Bbb{N}}}\right\}\cup\{(n,m)\;|\;n,m\in\Bbb{N}\}$ and
$$\mathscr{B}=\left\{\bigcap\mathscr{F}:\mathscr{F}\subseteq S\text{ and }\mathscr{F}\text{ is finite}\right\}\;.$$
Thus $\mathscr{B}$ is the base generated by the subbasis $S$. $S$ is infinite and has $|S|$ finite subsets, and therefore $|\mathscr{B}|=|S|$. If $S$ is countable, $\mathscr{B}$ is also countable, and $X$ is second countable and hence first countable. But we show that there is no countable local base at $p$.
Suppose that $\mathscr{U}=\{U_n:n\in\Bbb N\}$ is a countable family of open neighbourhoods of $p$. For each $n\in\Bbb{N}$ there are $f_{n_1},f_{n_2},\ldots,f_{n_{m_n}}\in \Bbb {N}^{\Bbb{ N}}$ such that $\bigcap_{n_1\le i\le n_{m_n}} B_{f_i}\subseteq U_n$. Define
$$g:\Bbb N\to\Bbb N:k\mapsto 1+\max\{f_{k_i}(\ell):\ell\le k,\;1\le i\le m_k\}\;;$$
then $B_g\nsupseteq B_{f_{n_i}}$, because it is evident that $(k,f_{k_i}(k))\in B_{f_{k_i}}$, but $(k,f_{k_i}(k))\notin B_g$ . Therefore for all $n\in \Bbb{N}$, $B_g\nsupseteq U_n$, so $\mathscr{U}$ is not a local base at $p$.
\end{ex}
\begin{n}
If $X$ is infinite and $\overrightarrow{d}$ is a R.O-metric on $X$, by definition of R.O-metric and $S_{\overrightarrow{d}}$, we can see that $Card(S_{\overrightarrow{d}})\le Card(X)$.
Since for every $p$ in $X$, $Card(\{V_{r}(p)\;|\;r\in \Bbb{R}^{+}\})\le Card(X)$ ,
hence $Card(S_{\overrightarrow{d}})=Card(\bigcup_{p\in X}\{V_{r}(p)\;|\;r\in \Bbb{R}^{+}\})\le Card{X}.$ This shows that the topological space in Example \ref{4} is not R.O-metrizable.
\end{n}
\begin{p}
Let $(X,\tau)$ be a topological space. If it has a subbasis $S$ such that $|S|\le |X|$, and there is a function $f:X \to S$ such that $x\in f(x)$, for every $x\in X$, then $(X,\tau)$ is R.O-metrizable.
\end{p}
\begin{proof}
By the hypothesis $S=\{f(x)\;|\;x\in X\}$. For every $x,y\in X$ define $$\overrightarrow{d}(x,y):=\begin{cases} 0&y\in f(x) \\1&otherwise \end{cases}$$
It is easy to check that $S_{\overrightarrow{d}}=S$, hence $\tau_{\overrightarrow{d}}=\tau.$
\end{proof}
\begin{co}
Every finite topological space is R.O-metrizable since by Note \ref{1} we can define $f:X\to S$ such that $f(x)$ is the smallest open set containing $x$.
\end{co}
\begin{cun}
Let $(X,\tau)$ and $S$ be a subbasis of it, such that $Card\;(S)\;\le Card\;(X)$, then $(X,\tau)$ is a R.O-metrizable.
\end{cun}
Now we mention three lemmas that will be useful for the last section.
\begin{lem}\label{2}
Let $(X,\tau)$ be finite $T_0$-topological space, then there exists $a\in X$ such that $\{a\}\in\tau$.
\end{lem}
\begin{proof}
Suppose that $U\in\tau$ be a minimal open set by relation $\subseteq$. If $Card\;(U)>1$, then there exists at least two points $a,b\in U$ and since $(X,\tau)$ is $T_0$, there exists $V\in\tau$ such that $a\in V,\;b\notin V$. Therefore $\emptyset\ne U\cap V\in\tau$ and $Card\;(U\cap V)<Card\;(U)$, this is a contradiction with minimality of $U$.
\end{proof}
\begin{lem}\label{15}
Let $(X,\tau)$ be a topological space and $(\sim)$ be a relation on $X$ by :$$x\sim y\Leftrightarrow \forall U\in\tau,\;(x\in U\Leftrightarrow y\in U).$$
Then $(\sim)$ is an equivalence relation on $X$ and $\tau^{'}=\{\;[U]\;|\;U\in \tau\}$ is a $T_0$-topology on $\mathcal{A}_{\tau}=\{\;[x]\;|\;x\in X\}$.
\end{lem}
\begin{proof}
Obviuosely $(\sim)$ is an equivalence relation on $X$. Suppose $[z]\in [U]\cap [V]$, in which $[U],[V]\in \tau^{'}$, then iff $[z]\in [U\cap V]$. Thus $[U]\cap [V]=[U\cap V]$. Also if $[z]\in\bigcup_{\alpha\in I}[U_{\alpha}]$, then iff $[z]\in [\bigcup_{\alpha\in I}U_{\alpha}]$. Thus $\bigcup_{\alpha\in I}[U_{\alpha}]=[\bigcup_{\alpha\in I}U_{\alpha}]$. Therefore $\tau^{'}$ is a topology on $\mathcal{A}_{\tau}$. Assume $[x]\ne [y]$, thus without losing the quality, there exist $U\in \tau$ such that $x\in U$ and $y\notin U$. Thus $[x]\in [U]$ and $\bar{[y]}\subseteq [U]^c$, so $\bar{[x]}\ne \bar{[y]}$. Therefore $\tau^{'}$ is $T_0$-topology on $\mathcal{A}_{\tau}$.
\end{proof}
\begin{lem}\label{3}
Let $(X,\tau)$ be a topological space. If $(\mathcal{A}_{\tau},\tau^{'})$ is a R.O-metrizable, then $(X,\tau)$ is R.O-metrizable.
\end{lem}
\begin{proof}
There exist R.O-metric $\overrightarrow{d}$ such that $\tau_{\overrightarrow{d}}=\tau^{'}$. Now define for all $x,y\in X$, $$\overrightarrow{D}(x,x)=0,\;\overrightarrow{D}(x,y)=\begin{cases}0&[x]=[y]\\\overrightarrow{d}([x],[y])&[x]\ne [y]\end{cases}.$$ Obviuosly $\overrightarrow{D}$ is a metric on $X$. Assume that $[U]=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}([x_i]))$. We clame that $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. So suppose that $z\in U$. Since $U=\bigcup_{[x]\in [U]}[x]$, then there exist $\alpha_0\in I$ such that $[z]\in \bigcap_{i=1}^{n_{\alpha_0}}U_{r_i}([x_i])$. Hence for all $1\le i\le n_{\alpha_0}$, $[z]\in U_{r_i}([x_i])$ which means $\overrightarrow{d}([x_i],[z])<r_i$. By definition of $\overrightarrow{D}$, we have $\overrightarrow{D}(x_i,z)<r_i$, thus for all $1\le i\le n_{\alpha_0}$, $z\in U_{r_i}(x_i)$. Therefore $U\subseteq \bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$. Checking $\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))\subseteq U$ is easy. So $U=\bigcup_{\alpha\in I}(\cap_{i=1}^{n_{\alpha}}U_{r_i}(x_i))$, thus $\tau_{\overrightarrow{D}}=\tau$.
\end{proof}
\section{Generalized R.O-metric space}
\begin{defn}
Suppose $(X,\overrightarrow{d})$ is a R.O-metric space. $(X,\overrightarrow{d},\beta)$ is called a generalized R.O-metric space if and only if there exists a collection $\beta=\{f_{\alpha}:X\to X\;|\;\alpha \in I\}$ such that $Id_X \in \beta$ and $$\forall y\in V_{r,\alpha}(x),\;\exists t>0\;,\;\exists \eta\in I \;s.t\; y\in f_{\eta}(X)\;,\;V_{t,\eta}(y)\subseteq V_{r,\alpha}(x)$$
where $V_{r,\alpha}(x)=\{f_{\alpha}(y)\in X\;|\;d(x,f_{\alpha}(y))<r\; ,\; y\in X\; ,\; x\in f_{\alpha}(X)\}$.
\end{defn}
\begin{n}
If $(X,\overrightarrow{d},\beta)$ is a generalized R.O-metric space the set $S_{\overrightarrow{d,\beta}}=\{V_{r,\alpha}(x)\;|\;x\in X\; ,\; r>0\; ,\; x\in f_{\alpha}(X)\}$ is a basis of a topology on $X$.
Topology generated by $S_{\overrightarrow{d},\beta}$ is denoted by $\tau_{(\overrightarrow{d},\beta)}$.
\end{n}
\begin{defn}
Topological space $(X,\tau)$ is called generalized R.O-metrizable iff there exists $\overrightarrow{d}$ such that $\tau_{(\overrightarrow{d},\beta)}=\tau$.
\end{defn}
\begin{ex} \label{5}
Let $X$ be the set from Example \ref{4} and $\tau$ be the topology on $X$ from the same example,for every $(m,n)$ and $(s,r)$ in $\Bbb{N}^{2}$ such that $(m,n)\ne (s,r)$ define $$\overrightarrow{d}((m,n),(m,n))=0=\overrightarrow{d}(p,(m,n))=\overrightarrow{d}(p,p)$$
$$\overrightarrow{d}((m,n),(s,r))=1=\overrightarrow{d}((m,n),p)$$
And for every $f\in \Bbb{N}^{\Bbb{N}}$ define $F_{f}:X\to X$ as follows : $$F_{f}(p)=p \;,\; F_{f}((m,n))=G((m,n))$$ Where $G:\Bbb{N}^{2} \to B_f$ is a surjective function, let $\beta=\{F_{f}\;|\;f\in \Bbb{N}^{\Bbb{N}}\} \cup \{Id_X\} $.
Now $$V_{(\frac{1}{2} ,f)}(p)$$ $$=\{F_{f}(a)\in X\;|\;\overrightarrow{d}(p,F_{f}(a))<\frac{1}{2} \}$$ $$=\{F_{f}(a) \;|\;a\in X \}=B_f $$ and $$V_{(\frac{1}{2} ,f)}((m,n))$$ $$=\{F_{f}(a)\in X\;|\;\overrightarrow{d}((m,n),F_{f}(a))<\frac{1}{2} \}=\{(m,n)\}$$ But if $r>1$ , $x\in X$ and $f\in \beta$ then $V_{(r,f)}(x)=X$ therefore $\tau_{(\overrightarrow{d},\beta)}=\tau$. So $(X,\tau)$ is generalize R.O-metric space which is NOT R.O-metrizable space.
\end{ex}
\begin{ex} \label{6}
Let $X$ be a infinite set, $p\in X$ be fixed and $B\subset X$ which is $Card\;(B)=Card\;(B^{c})=Card\;(X)$ containing $p$. Suppose $\tau=\{A\subset X\;|\;B\subset A\;,\;A\ne B \}\cup \{\{x\}\;|\;x\in X\;,\;x\ne p\}$ then $\tau$ is a topology on $X$. Let $f_{A}:X\to X$ be surjective function such that $f_{A}(p)=p$ and $f_{A}(x)\ne p \; \forall x\ne p $ and let $\beta=\{f_{A}\;|\;A\in \tau \}\cup \{Id_X\}$. Now For distinct $x$ and $y$ in $X$ such that $x,y\ne p$ define $$\overrightarrow{d}(x,y)=0=\overrightarrow{d}(p,x)=\overrightarrow{d}(p,p)$$ $$\overrightarrow{d}(x,y)=1=\overrightarrow{d}(x,p).$$ It is easy to check that $\tau_{(\overrightarrow{d},\beta)}=\tau$ thus $(X,\tau)$ is a generalized R.O-metrizable space.
\end{ex}
\begin{theorem}
Every topological space $(X,\tau)$ is generalized R.O metrizable
\end{theorem}
\begin{proof}
Suppose $(X,\tau^{'})$ is an arbitrary $T_0$ topological space. Now let $(Y,\tau^{''})$ be a topological space where $Y=\{0,1\}$ and $\tau^{''}=\{\emptyset ,\;\{1\},\;Y\}$ and $\overrightarrow{d}(0,1)=0=\overrightarrow{d}(0,0)=\overrightarrow{d}(1,1),\;\overrightarrow{d}(1,0)=1$ (obviously $\tau_{\overrightarrow{d}}=\tau^{''}$). Assume that $C$ is a proper closed subset of $(X,\tau)$ , Define $f_{C}:X\to Y$ as follows :
$f_{C}(C)=0$ and $f_{C}(X-C)=1$
$f$ is obviously continuous, Let $J=\{f_{C}\;|\;X-C\in \tau\}$. $J$ is a family of continuous maps that separates points from closed sets. Define $$F:X\to Y^J\;\;\;,F(x)=(f_{\alpha}(x))_{\alpha\in J}$$ obviously $F$ is an embedding when $Y^J$ is equipped with the product topology $(F(X)\cong X)$.
Let $ (<)$ be a well-ordering on $J$ define $$\overrightarrow{D}((x_{\alpha}),(y_{\alpha}))=\overrightarrow{d}(1,y_{\beta})$$ where $\beta$ is the smallest index in $(J,<)$ which $x_{\beta}=1$. $\overrightarrow{D}$ is a R.O metric that generates the product topology of $Y^J$ therefore $(F(X),\overrightarrow{D}|_{F(X)})$ is a R.O-metric space.We know that topology induced of $ (Y^J,\tau_{\overrightarrow{D}})$ on $F(X)$ is equal to $\{U\cap F(X)\;|\; U\in \tau_{\overrightarrow{D}}\}$ which is equal to the topology generated by $\{V_{r}(y)\cap F(X)\;|\;r>0\;,\;y\in Y^J\}$. Now let $A=\{V_{r}(y)\cap F(X)\;|\;r>0\;,\;y\in Y^J-F(X)\}$, consider $B=\{f_{r,y}:X\to X\;|\; y\in Y^J-F(X) \;,\; r>0\;,f_{r,y}(X)=V_{r}(y)\cap F(X)\}\cup\{Id_X\}$ we claim that $(F(X),\overrightarrow{D}|_{F(X)},B)$ is a generalized R.O-metric space, it suffices to prove that $\tau_{\overrightarrow{D}|_{F(X)},B}$ generates the induced topology of the product topology of $Y^J$ on $F(X)$. For $y\in Y^J$ and $r>0$ if $y\in F(X)$ then $V_{r}(y)\cap F(X) \in \tau_{\overrightarrow{D}|_{F(X)}}$. Also suppose $y\notin F(X)$, $r>0$ and $z\in V_{r,y}(y)=f_{r,y}(X)=V_{r}(y)\cap F(X)$, then $V_{r,y}(z)\subseteq V_{r,y}(y)$.Thus $\tau_{\overrightarrow{D}|_{F(X)},B}$ generates the induced topology of the product topology of $Y^J$ on $F(X)$. Since $F(X)\cong X$, therefore $(X,\tau^{'})$ is a generalized R.O-metrizable. By Lemma \ref{3} $(A,\tau^{'})$ is a $T_0$-topological space, thus there exist $\overrightarrow{d}$ and $\beta=\{f_{\alpha}:A\to A\}\cup \{Id_X\}$ such that $(A,\overrightarrow{d},\beta)$ is a generalized R.O-metric space. Define $$\overrightarrow{D}(x,y)=\begin{cases}0&[x]=[y]\\\overrightarrow{d}(x,y)&[x]\ne [y]\end{cases}$$ and $$F_{\alpha}:X\to X\;\;,\;\;F_{\alpha}(x)=g_{[x]}(x)$$where $g_{[x]}$ is a map between $[x]$ and $f_{\alpha}([x])$. Therefore $(X,\tau)$ is a generalized R.O-metrizable.
\end{proof}
\noindent
At the end, it is useful to see the figure \ref{fig:16} for understanding the subject.
\begin{figure}
\caption{diagram for topological spaces}
\label{fig:16}
\end{figure}
\end{document} |
\betagin{document}
\title{A relative mass cocycle and the mass of asymptotically hyperbolic manifolds}
\author{Andreas \v Cap, and A.\ Rod Gover}
\circperatorname{ad}dress{A.\v C.: Faculty of Mathematics\\
University of Vienna\\
Oskar--Morgenstern--Platz 1\\
1090 Wien\\
Austria\\
A.R.G.:Department of Mathematics\\
The University of Auckland\\
Private Bag 92019\\
Auckland 1142\\
New Zealand}
\email{Andreas.Cap@univie.ac.at}
\email{r.gover@auckland.ac.nz}
\betagin{abstract}
We construct a cocycle that, for a given $n$-manifold, maps pairs of
asymptotically locally hyperbolic (ALH) metrics to a tractor-valued
$(n-1)$-form field on the conformal infinity. This describes, locally
on the boundary, a relative mass difference between the pairs of ALH
metrics, where the latter are required to be asymptotically related to
a given order that depends on the dimension. It is distinguished as a
geometric object by its property of being invariant under suitable
diffeomorphisms fixing the boundary, and that act on
(either) one of the argument metrics.
Specialising to the case of an ALH metric $h$ that is suitably
asymptotically related to a locally hyperbolic conformally compact
metric, we show that the cocycle detemines an absolute invariant
$c(h)$. This tractor-valued $(n-1)$-form field on the conformal
infinity is canonically associated to $h$ (i.e. is not dependent on
other choices) and is equivariant under the appropriate
diffeomorphisms.
Finally specialising further to the case that the boundary is a sphere and
that a metric $h$ is asymptotically related to a hyperbolic metric on
the interior, we show that the invariant $c(h)$ can be integrated
over the boundary. The result pairs with solutions of the KID (Killing
initial data) equation to recover the known description of hyperbolic
mass integrals of Wang, and Chru\'{s}ciel--Herzlich.
\end{abstract}
\subjclass[2020]{primary: 53A55; secondary: 53C18, 53C25, 53C80, 83C30, 83C60}
\keywords{mass in GR, mass aspect, asymptotically hyperbolic manifolds, geometric
invariants, tractor calculus, conformally compact manifolds}
\thetaanks{A.\v C.\ gratefully acknowledges support by the Austrian Science Fund (FWF):
P33559-N and the hospitality of the University of Auckland. A.R.G.\ gratefully
acknowledges support from the Royal Society of New Zealand via Marsden Grants
16-UOA-051 and 19-UOA-008. We thank P.\ Chru\'sciel for very helpful discussions.}
\maketitle
\pagestyle{myheadings} \markboth{\v Cap, Gover}{Mass and Tractors}
\section{Introduction}\lambdabel{1}
In general relativity, and a number of related mathematical studies,
the notion of a ``mass'' invariant for relevant geometric manifolds is
extremely important and heavily studied \cite{Bartnik,LP,S-Yau}. In
general defining and interpreting a suitable notion of mass is not
straightforward. For so-called asymptotically flat manifolds the
Arnowitt-Deser-Misner (ADM) energy-momentum, is well established and
is usually accepted as the correct definition. Motivated by the
desire to define and study mass in other settings X.\ Wang \cite{Wang}
and P.T.\ Chru\'sciel and M.\ Herzlich \cite{Chrusciel-Herzlich}
introduced a notion of mass integrals and ``energy-momentum'' for
Riemannian manifolds that, in a suitable sense, are asymptotically
hyperbolic. These have immediate applications for a class of static
spacetimes.
The aim of this article is to construct new invariants that capture a notion of mass
density, in the setting of asymptotically hyperbolic metrics. These invariants are
local as quantities on the boundary at infinity and they specialise to recover (by an
integration procedure) the mass as introduced by Wang and Chru\'sciel--Herzlich. The
setting we work in is rather restrictive in some aspects but very general in other
aspects. The main restriction is that we are working in a conformally compact
setting, so we need strong assumptions on the order of asymptotics. On the other
hand, underlying this is an arbitrary manifold with boundary with no restrictions on
the topology of the boundary. We also allow a fairly general ``background metric'';
the core of our results only require a background metric that is asymptotically
locally hyperbolic (ALH), as in Definition \ref{def2.2}, part (1). The main
invariant we construct is associated to a pair of ALH metrics (that are asymptotic to
sufficient order), so it should be thought of as a \textit{relative local mass} or as
a \textit{local mass difference}.
Our basic setting looks as follows. We start with an arbitrary
manifold $\circverline{M}$ of dimension $n\geq 3$, with boundary $\partial M$
and interior $M$. Given two conformally compact metrics $g$ and $h$ on
$M$, there is a well defined notion of $g$ and $h$ approaching each
other asymptotically to certain orders towards the boundary. The
actual order we need depends on the dimension and specializes to the
order required in \cite{Wang} on hyperbolic space. This defines an
equivalence relation on conformally compact metrics and we consider
one equivalence class $\mathcal G$ of such metrics. The only additional
requirement at this point is that $\mathcal G$ consists of
ALH-metrics. Since $\mathcal G$ consists of conformally compact metrics,
each $g\in\mathcal G$ gives rise to a conformal structure on $\partial M$,
called the conformal infinity of $g$. Moreover, the asymptotic
condition used to define $\mathcal G$ is strong enough to ensure that all
metrics in $\mathcal G$ lead to the same conformal infinity. Thus $\mathcal G$
canonically determines a conformal structure $[\mathcal G]$ on $\partial
M$.
This last fact is crucial for the
further development, since the invariants we construct are geometric
objects for this conformal structure on $\partial M$ (and a slightly
stronger structure in case that $\dim(\circverline{M})=3$). Indeed, the conformal
structure $[\mathcal G]$ canonically determines the so-called
\textit{standard tractor bundle} $\mathcal T\partial M\to\partial M$. This
is a vector bundle of rank $n+1$ endowed with a Lorentzian bundle
metric and a metric linear connection \cite{BEG}. We construct
\textit{cocycles} $c$ that associate to each pair of metrics
$g,h\in\mathcal G$ a tractor valued $(n-1)$-form
$c(g,h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, with the property
that $c(h,g)=-c(g,h)$ and $c(g,k)=c(g,h)+c(h,k)$ for all $g,h,k\in\mathcal
G$. These cocycles are the basic ``relative local masses'' we
consider.
The idea of the construction is that any metric $g\in\mathcal G$ determines a conformal
class on $\circverline{M}$ (since the conformal class of $g$ on $M$ extends to the boundary by
conformal compactness) and this restricts to $[\mathcal G]$ at the boundary. Working with
this conformal structure, we can use standard techniques of tractor calculus, and an
instance of what is called a BGG splitting operator, to associate to any $h\in\mathcal G$
a one-form with values in the tactor bundle $\mathcal T\circverline{M}$. The Hodge dual (with
respect to $g$) of this one-form is then shown to admit a smooth extension of the
boundary whose boundary value is defined to be $c(g,h)$, see Propositions
\ref{prop3.1} and \ref{prop3.2}. The proof that this actually defines a cocycle
needs some care because of the use of different conformal structures on $\circverline{M}$, but
is otherwise straigthforward.
Initially, this leads to a two parameter family of cocyles since there
are two constructions of the above type, one depending on the trace of
the difference $g-h$, the other on its trace-free part. Our
constructions do not require charts or coordinates, so, in that sense,
are automatically geometric in nature. The constructions also
readily imply equivariancy with respect to the appropriate diffeomorphisms. If
$\Phi$ is a diffeomorphism of $\circverline{M}$ which preserves the class $\mathcal
G$ (in an obvious sense) then by definition the restriction
$\Phi|_{\partial M}$ is a conformal isometry for $[\mathcal G]$. Therefore,
it naturally acts on sections of $\mathcal T\partial M$ (and of course on
forms on $\partial M$) and for any of the cocycles
$c(\Phii^*g,\Phii^*h)=(\Phii|_{\partial M})^*c(g,h)$, see Proposition
\ref{prop3.3}.
There is a much more subtle compatibility condition with
diffeomorphisms, however. Indeed, consider a diffeomorphism $\Psii$
that is compatible with $[\mathcal G]$ and suppose that $\Psii|_{\partial
M}=\circperatorname{id}_{\partial M}$. Then it turns out that $\Psii$ is asymptotic to
the identity of order $n+1$ in a well-defined sense, see Section
\ref{3.4} and Theorem \ref{thm3.5}. The main technical result of our
article then is that there is a unique ratio of the two parameters for
our cocyles, which ensures that $c(g,\Psii^*h)=c(g,h)$ for any $\Psii$
which is asymptotic to the identity of order $n+1$. Hence, up to an
overall normalization we obtain a unique cocylce $c$ which has this
invariance property in addition to the equivariancy property mentioned
above. This proof is based on an idea in \cite{CDG} that shows that
the action of diffeomorphisms that are asymptotic to the identity can
be absorbed into a geometric condition relating the two metrics (and
an adapted defining function for one of them). The key feature of this
property is that it holds in the general setting of a class of ALH
metrics on a manifold with boundary. In the special cases for which
we obtain invariants of single metrics, this property enables us to
prove equivariancy of such invariants under diffeomorphisms preserving
$\mathcal G$, see below.
To pass from our cocycles to invariants of a single metric, one has to
go to specific situations in which $\mathcal G$ constains particularly
nice metrics. We only discuss the case that $\mathcal G$ locally contains
metrics that are hyperbolic, i.e.\ have constant sectional curvature
$-1$. This of course implies that $(\partial M,[\mathcal G])$ is
conformally flat, but it does not impose further restrictions on the
topology of $\partial M$, see Section \ref{3.8}. Under this
assumption, we show that, for a cocycle from our one-parameter family
and a metric $h\in\mathcal G$, all local hyperbolic metrics $g\in\mathcal G$
locally lead to the same tractor-valued form $c(g,h)$. These local
forms then piece together to define an object
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$ that is canoncially
associated to $h$. We prove that this is equivariant under
diffeomorphisms preserving the class $\mathcal G$, see Theorem
\ref{thm3.8}.
As a last step, we prove that, in the conformally compact setting, the AH mass as
introduced in \cite{Chrusciel-Herzlich} can be obtained by integrating our
invariant. Thus we have to specialize to the case that $\circverline{M}$ is an open
neighborhood of the boundary sphere in the closed unit ball and $\mathcal G$ contains the
restriction of the Poincar\'e metric. This implies that $\partial M=S^{n-1}$ and
$[\mathcal G]$ is the round conformal structure and thus the standard tractor bundle
$\mathcal TS^{n-1}$ can be globally trivialized by parallel sections. Hence $\mathcal
TS^{n-1}$--valued $(n-1)$-forms can be integrated to global parallel sections of
$\mathcal TS^{n-1}$ (say by expanding in any globally parallel frame and then integrating
the coefficient forms). Now it is well known how to make the trivialization of $\mathcal
TS^{n-1}$ explicit, and we show that there is a particularly nice way to do this
using the conformal class $[g]$ on $\circverline{M}$ determined by the Poincar\'e metric. Via
boundary values of parallel tractors in the interior, the parallel sections of $\mathcal
TS^{n-1}$ turn out to be parametrised by the solutions of the KID (Killing initial
data) equation (\ref{KID}) on $M$. But these solutions exactly parametrise the mass
integrals used to define the AH mass in the style of \cite{Chrusciel-Herzlich}, see
\cite{Michel}. This last fact was our original motivation to look for a tractor
interpretation of the AH mass.
Now a solution $V$ of the KID equation determines a parallel section
$s_V$ of $\mathcal TS^{n-1}$ and we can proceed as follows. Given
$h\in\mathcal G$, we can form the invariant $c(h)\in
\Omega^{n-1}(S^{n-1},\mathcal TS^{n-1})$ and integrate it to a parallel
section $\int_{S^{n-1}}c(h)$ of $\mathcal TS^{n-1}$. This can then be
paired, via the tractor metric, with $s_V$. Analyzing the boundary
behavior of the mass integral determined by $V$, we show in Theorem
\ref{thm3.9} that, after appropriate normalization, this pairing
exactly recovers the mass integral, of \cite{Chrusciel-Herzlich},
determined by $V$.
Throughout all manifolds, tensor fields, and related objects, will be
taken to be smooth in the sense of $C^\infty$. For most results lower
regularity would be sufficient, but we do not take that up here.
\section{Setup and tractors}\lambdabel{2}
\subsection{Conformally compact metrics}\lambdabel{2.1}
Let $\circverline{M}$ be a smooth manifold with boundary $\partial M$ and
interior $M$. Recall that a \textit{local defining function} for
$\partial M$ is a smooth function $\rho:U\to\mathbb R_{\geq 0}$ defined
on some open subset $U\subset\circverline{M}$ such that $U\cap\partial
M=\rho^{-1}(\{0\})$ and such that $d\rho|_{U\cap\partial M}$ is
nowhere vanishing. For two local defining functions $\rho$ and
$\hat\rho$ defined on the same open set $U$, there is a smooth
function $f:U\to\mathbb R_{>0}$ such that $\hat\rho=f\rho$. It is often
convenient to write such positive function as $f=e^{\tilde f}$ for
some smooth function $\tilde f$. This notion of definining functions
extends, without problem, from functions to smooth sections of line
bundles. One just has to replace $d\rho$ by the covariant derivative
with respect to any linear connection, which is independent of the
connection along the zero set of the section. In particular, taking the line bundle
concerned to be a density bundle (as discussed above Proposition \ref{prop2.1} below)
leads to \textit{defining densities} \cite{Proj-comp}.
A pseudo-Riemannian metric $g$ on $M$ is called \textit{conformally
compact} if, for any point $x\in\partial M$, there is a local defining
function $\rho$ for $\partial M$ defined on some neighborhood $U$ of
$x$ such that $\rho^2g$ admits a (sufficiently) smooth extension from
$U\cap M$ to all of $U$. It is easy to see that this property is
independent of the choice of defining function, so if $g$ is
conformally compact, then for any local defining function $\hat\rho$
for $\partial M$, the metric $\hat\rho^2 g$ admits a smooth extension
to the boundary. While the metric on the boundary, induced by such an
extension, depends on the chosen defining function, such extensions
are always conformally related. Hence a conformally compact metric on
$M$ gives rise to a well-defined conformal class on $\partial M$,
which is called the \textit{conformal infinity} of $g$. The model
example of a conformally compact metric is the Poincar\'e ball model
of hyperbolic space. Here $\circverline{M}$ is the closed unit ball in $\mathbb
R^{n}$ and one defines a metric $g$ on the open unit ball as
$\tfrac{4}{(1-|x|^2)^2}$ times the Euclidean metric, see
\cite{Graham:Srni}. The resulting conformal infinity is the conformal class of
the round metric on $S^{n-1}$.
There is a conceptual description of conformally compact metrics in
the language of conformal geometry: Since $\rho^2g$ is conformal to
$g$ away from the boundary, it provides a (sufficiently) smooth
extension of the conformal structure on $M$ defined by $g$ to all of
$\circverline{M}$. Conformally compact metrics can be neatly characterized in
this picture via their volume densities. Recall that the volume
density of $g$ is nowhere vanishing, so one can form powers with
arbitrary real exponents to obtain nowehere vanishing densities of all
(non-zero) weights. Each of these densities is parallel for the
connection on the appropriate density bundle induced by the
Levi-Civita connection, and up to constant multiples, this is the only
parallel section.
In the usual conventions of conformal geometry, see \cite{BEG}, the
square of the top exterior power ot the tangent bundle, that is
$\circtimes^2(\Lambdambda^nTM)$, is identified with a line bundle of {\em
densities} of weight $2n$ that we denote $\mathcal{E}[2n]$. This is
oriented and thus there is a standard notion of its roots. With these
conventions, on an $n$-manifold a metric $g$ determines a volume
density $\circperatorname{vol}_g$ that has conformal weight $-n$,
meaning $\circperatorname{vol}_g\in\Gamma(\mathcal E[-n])$. Rescaling $g$ to $\rho^2g$, we get
$\circperatorname{vol}_{\rho^2g}=\rho^n\circperatorname{vol}_g$ and thus
$\circperatorname{vol}_{g}^{-1/n}=\rho\circperatorname{vol}_{\rho^2g}^{-1/n}\in\Gamma(\mathcal E[1])$. Assuming
that $g$ is conformally compact, $\circperatorname{vol}_{\rho^2g}^{-1/n}$ is smooth up
to the boundary and nowhere vanishing, so this equation shows that
$\circperatorname{vol}_{g}^{-1/n}\in\Gamma(\mathcal E[1])$ is a defining density for $\partial
M$. Similar arguments prove the converse, which leads to the following
result.
\betagin{prop}\lambdabel{prop2.1}
Let $\circverline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$, which
is endowed with a conformal structure $c$. Then a metric $g$ on $M$ which lies in
$c|_M$ is conformally compact if an only if any non-zero section of $\sigma\in\Gamma(\mathcal
E[1]|_M)$ which is parallel for the Levi-Civita connection of $g$ extends by $0$ to a
defining density for $\partial M$.
\end{prop}
\subsection{ALH-metrics and adapted defining functions}\lambdabel{2.2}
Consider a conformally compact metric $g$ and for a local defining
function $\rho$ put $\circverline{g}:=\rho^2g$, and note that this is
smooth and non-degenerate up to the boundary. Hence also the inverse metric
$\circverline{g}^{-1}$ is smooth up to the boundary, so
$\circverline{g}^{-1}(d\rho,d\rho)$ a smooth function on the domain of
definition of $\rho$. Moreover, since $d\rho$ is nowhere vanishing
along the boundary $\circverline{g}^{-1}(d\rho,d\rho)$ has the same
property. Replacing $\rho$ by $\hat\rho:=e^f\rho$, we obtain
$d\hat\rho=\hat\rho df+e^fd\rho$, which easily implies that the
restriction of $\circverline{g}^{-1}(d\rho,d\rho)$ to $\partial M$ is
independent of the defining function $\rho$.
So $\circverline{g}^{-1}(d\rho,d\rho)$ is an invariant of the metric $g$,
and this is related to the asymptotic behavior of the curvature of
$g$, see e.g.\ \cite{Graham:Srni}. In particular, if
$(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}\equiv 1$, then the
sectional curvature of $g$ is asymptotically constant $-1$. This
justifies the terminology in the first part of the following
definition and leads to a subclass of defining functions:
\betagin{definition}\lambdabel{def2.2}
Consider a smooth manifold $\circverline{M}=M\cup\partial M$ with boundary and a conformally
compact metric $g$ on $M$.
(1) The metric $g$ is called \textit{asymptotically locally hyperbolic} (or an
ALH-metric) if $(\rho^2g)^{-1}(d\rho,d\rho)|_{\partial M}$ is identically one.
(2) Assume that $g$ is an ALH-metric on $M$ and that $U\subset\circverline{M}$ (with
$U\cap\partial M\neq\emptyset$ to be of interest). Then a local defining function
$\rho$ for $\partial M$ defined on $U$ is called \textit{adpated to $g$} if the
function $(\rho^2 g)^{-1}(d\rho,d\rho)$ is identically one on some open
neighborhood of $U\cap\partial M$.
\end{definition}
\betagin{remark} Note that in the literature the terminology
``asymptotically locally hyperbolic'' is sometimes used for (various) more
restrictive classes of geometry. Also the condition we have (1) is
sometimes referred to as simply ``asymptotically
hyperbolic''. However the latter is also used for rather special
settings where in particular the boundary is necessarily a sphere,
hence our use here.
\end{remark}
The existence of adpated defining functions, as in part (2) of
Definition \ref{def2.2}, can be established by solving an appropriate
non-characteristic first order PDE. The following precise description
of adapted defining functions is given in Lemma 2.1 of
\cite{Graham:Srni}.
\betagin{prop}\lambdabel{prop2.2}
Consider $\circverline{M}=M\cup\partial M$ and an ALH metric $g$ on $M$. Then
for any choice of a representative metric $h$ in the conformal
infinity of $g$, there exists an adapted defining function $\rho$ for
$g$, defined on an open neighborhood of $\partial M$ in $\circverline{M}$,
such that $\rho^2g$ induces the metric $h$ on $\partial
M$. Moreover, for fixed $h$, the germ of $\rho$ along $\partial M$
is uniquely determined.
\end{prop}
\subsection{The basic setup}\lambdabel{2.3}
Defining functions can be used to measure the asymptotic growth (or fall-off) of
functions and more general geometric objects on the interior of a manifold with
boundary. A fundamental property of defining functions is that for a function $f$
that is smooth up to the boundary $f|_{\partial M}\equiv 0$ if and only if for any
local defining function $\rho$ for $\partial M$, we obtain, on the domain of
definition of $\rho$, $f=\rho f_1$ for a function $f_1$ that is smooth up to the
boundary. We say that $f$ is $\mathcal O(\rho)$ in this case, observing that this notion
is actually independent of the specific defining function $\rho$. Similarly if, in
such an expansion, $f_1$ also vanishes along the boundary then this fact does not
depend on the choice of $\rho$, and in that case we say that $f$ is $\mathcal
O(\rho^2)$. Inductively, one obtains the notion that $f$ is $\mathcal O(\rho^N)$ for any
integer $N>0$, which again does not depend on the specific choice of defining
function.
Given a smooth function $f:M\to\mathbb R$ and an integer $N>0$, we then say that $f$ is
${\mathcal O} (\rho^{-N})$ if locally around each $x\in \partial M$ we find a defining
function $\rho$ such that $\rho^Nf$ admits a smooth extension to the boundary. Again,
the fact that such an extension exists is independent of the choice of defining
function, as is vanishing of the boundary value of $\rho^N f$ in some point. In
points where the boundary value of $\rho^Nf$ is nonzero, the actual value does depend
on the choice of $\rho$, however.
For two functions $f_1,f_2:\circverline{M}\to\mathbb R$ and $N>0$, we write $f_1\sigmam_N f_2$ if
$f_1-f_2$ is $\mathcal O(\rho^N)$. By definition, this means that, on the domain of a
local defining function $\rho$, we can write $f_2=f_1+\rho^Nf$ for some function $f$
that is smooth up to the boundary. Observe that this defines an equivalence relation.
All this extends without problems to tensor fields of arbitrary
(fixed) type. This can be seen immediately from looking at coordinate
functions in (boundary) charts. So for $N>0$, a tensor field $t$ on
$M$ is $\mathcal O(\rho^N)$ if can be written as $\rho^N\tilde t$ for a
tensor field $\tilde t$ that is smooth up to the boundary. Likewise,
$t$ is $\mathcal O(\rho^{-N})$ if $\rho^Nt$ admits a smooth extension to
the boundary. In the obvious way we extend the notation $t_1\sigmam_N
t_2$ with $N>0$ to tensor fields $t_1,t_2$ that are smooth up to the
boundary. Observe also that these concepts are compatible in an
obvious sense with tensorial operations, like inserting vector fields
into metrics, etc.
In this langauge, a conformally compact metric $g$ is $\mathcal
O(\rho^{-2})$ and, writing $g=\rho^{-2}\circverline g$, it satisfies that
$\circverline g$ is nowhere vanishing along the boundary. Oberserve that
this implies that the inverse metric $g^{-1}$ is $\mathcal O(\rho^2)$ and,
in particular, vanishes along $\partial M$. For conformally compact
metrics $g$ and $h$ we can consider the metrics $\rho^2g$ and
$\rho^2h$ that are smooth up to the boundary and require that
$\rho^2g\sigmam_N\rho^2h$, which again is independent of the choice of
defining function $\rho$. This defines an equivalence relation on the
set of conformally compact metrics, and to simplify notation, we
formally write this as $g\sigmam_{N-2}h$. In the current article we will
mainly be interested in the case that $N=n=\dim(M)$ but we carry out
most computations for general integers $N>0$, since this does not lead
to difficulties and as a preparation for later extensions.
We will start with an equivalence class $\mathcal G$ of conformally compact metrics on
$M$ with respect to the equivalence relation $\sigmam_{N-2}$ for some $N>0$. Observe
that for two metrics $g,h\in\mathcal G$ and a local defining fuction $\rho$, the metrics
$\rho^2g$ and $\rho^2h$, by definition, admit a smooth extension to the boundary with
the same boundary value. In particular, they induce the same conformal infinity on
$\partial M$. Thus the class $\mathcal G$ of metrics gives rise to a well defined
conformal structure on $\partial M$ that we will denote by $[\mathcal G]$. As we shall
see below, if one metric $g\in\mathcal G$ is ALH, then the same holds for all metrics in
$\mathcal G$. We shall always assume that this is the case from now on. Let us also
remark here, that in the case that $\dim(\circverline{M})=3$, $[\mathcal G]$ induces a stronger
structure that just a conformal structure on $\partial M$ which will be needed in
what follows. This will be discussed in more detail below.
From now on, we will sometimes use abstract index notation. In that
notation we write $g_{ij}$ for the metric $g$, $g^{ij}$ for its
inverse and so on (even though no coordinates or frame field is
chosen). Given $g_{ij},h_{ij}\in\mathcal G$, and fixing a local defining
function $\rho$ for $\partial M$, by definition there is a section
$\mu_{ij}$ that is smooth up to the boundary such that
\betagin{equation}\lambdabel{h-g}
h_{ij}=g_{ij}+\rho^{N-2}\mu_{ij}.
\end{equation}
Since $g^{ij}$ is $\mathcal O(\rho^2)$, we see that $g^{ij}\mu_{ij}=\rho^2\mu$ for some
function $\mu$ that is smooth up to the boundary, whence
$g^{ij}(h_{ij}-g_{ij})=\rho^N\mu$. Using this, we next compute the relation between
the defining densities $\sigma,\tau\in\Gamma(\mathcal E[1])$ determined by $g_{ij}$ and
$h_{ij}$, respectively. Writing
\betagin{equation}\lambdabel{h-g2}
h_{ij}=g_{ik}(\deltalta^k_j+\rho^{N-2}g^{k\ell}\mu_{\ell j}),
\end{equation}
we can take determinants to find that
$\deltat(h_{ij})=\deltat(g_{ik})(1+\rho^N\mu+O(\rho^{N+1}))\in\Gamma(\mathcal
E[-2n])$. (Formally, the determinants are formed by using two
copies of the canonical section of $\Lambda^nTM[-n]$
that expresses the
isomorphism between volume forms and
densities of conformal weight $-n$ on an oriented manifold. Since two
copies of the forms are used, this is well defined even in the
non-orientable case, but this will not be needed here.) To obtain
$\sigma$ and $\tau$, we have to take $\tfrac{-1}{2n}$th powers, and
taking into account that $\sigma$ is $\mathcal O(\rho)$, we get
\betagin{equation}\lambdabel{tau-si}
\tau-\sigma=-\sigma\tfrac{\rho^N}{2n}\mu+\mathcal O(\rho^{N+2}),
\end{equation}
so this is $\mathcal O(\rho^{N+1})$. Moreover, contracting \eqref{h-g2} with $g^{ai}$, we
get $g^{ai}h_{ij}=\deltalta^a_j+\rho^{N-2}g^{ai}\mu_{ij}$, which in turn easily
implies that
\betagin{equation}\lambdabel{hinv}
h^{ij}=g^{ij}-\rho^{N-2}g^{ik}\mu_{k\ell}g^{\ell j}+\mathcal O(\rho^{N+3}).
\end{equation}
Hence $h^{ij}-g^{ij}$ is $\mathcal O(\rho^{N+2})$, which in particular shows that, as
claimed above, $h$ is ALH provided that $g$ has this property.
\subsection{Tractors}\lambdabel{2.4}
For a manifold $K$ of dimension $n\geq 3$ which is endowed with a conformal
structure, the \textit{standard tractor bundle} \cite{BEG,CG-TAMS} is a vector bundle
$\mathcal TK=\mathcal E^A\to K$ of rank $n+2$ endowed with the following data.
\betagin{itemize}
\item A Lorentzian bundle metric, called the \textit{tractor metric}, which we
denote by $\lambdangle\ ,\ \rangle$.
\item A distinguished isotropic line subbundle $\mathcal T^1K\subset \mathcal TK$ that is
isomorphic to the density bundle $\mathcal E[-1]$.
\item A canonical linear connection, called the \textit{tractor connection}, that
preserves the tractor metric and satisfies a non-degeneracy condition.
\end{itemize}
Since $\mathcal T^1K$ is isotropic it is contained in $(\mathcal T^1K)^\perp$, and
$\lambdangle\ ,\ \rangle$ induces a positive definite bundle metric on $(\mathcal
T^1K)^\perp/\mathcal T^1K$. Via the tractor connection, this quotient gets identified
with $\mathcal E^a[-1]$, so the tractor metric gives rise to a section of $\mathcal
E_{ab}[2]$, which is exactly the \textit{conformal metric} $\mathbf{g}_{ab}$ that
defines the conformal structure on $K$. These properties together determine the data
uniquely up to isomorphism. Observe that the inclusion of $\mathcal T^1K\cong\mathcal E[-1]$
into $\mathcal TK=\mathcal E^A$ can be viewed as defining a canonical section
$\mathbf{X}^A\in\Gamma(\mathcal E^A[1])$. Moreover, pairing with $\mathbf{X}^A$, with
respect to the tractor metric, defines an isomorphism $\mathcal TK/(\mathcal
T^1K)^\perp\to\mathcal E[1]$.
As mentioned in \ref{2.3}, we will use these tools in a slightly unusual setting,
since we will deal with different conformal structures on $\circverline{M}$ at the same time,
and only the induced conformal structure on the boundary $\partial M$ will be the
same for all structures in question. Hence we have to carefully discuss how to form
boundary values of sections of the standard tractor bundles.
For our current purposes the ``naive'' approach to tractors (which avoids the
explicit use of Cartan connections or similar tools) is most appropriate and we'll
describe this next. A crucial feature in all approaches to tractors is that the
standard tractor bundle admits a simple description depending on the choice of a
metric $g$ in the conformal class. Such a choice gives an isomorphism $\mathcal E^A\cong
\mathcal E[1]\circplus\mathcal E_a[1]\circplus \mathcal E[-1]$, with the last summand corresponding to
$\mathcal T^1K$ and the last two summands corresponding to $(\mathcal T^1K)^\perp$. The
resulting elements are usually written as column vectors, with the first
component in the top, and there are simple explicit formulae for the tractor metric
and the tractor connection in these terms, namely:
\betagin{equation}\lambdabel{trac-met}
\left\lambdangle\betagin{pmatrix}\sigma \\ \mu_a\\ \nu
\end{pmatrix},\betagin{pmatrix}\tilde\sigma \\ \tilde\mu_a\\ \tilde\nu
\end{pmatrix}\right\rangle=\sigma\tilde\nu+\nu\tilde\sigma+\mathbf{g}^{ab}\mu_a\tilde\mu_b
\end{equation}
with $\mathbf{g}^{ab}$ denoting the inverse of the conformal metric, and
\betagin{equation}\lambdabel{trac-conn}
\qquad \nabla_a \betagin{pmatrix}\sigma\\ \mu_b\\ \nu\end{pmatrix}=
\betagin{pmatrix}\nabla_a\sigma-\mu_a\\ \nabla_a\mu_b+\mathbf{g}_{ab}\nu+
\mbox{\textsf{P}}_{ab}\sigma\\
\nabla_a\nu-\mathbf{g}^{ij}\mbox{\textsf{P}}_{ai}\mu_j\end{pmatrix}.
\end{equation}
In the right hand side of this, we the Levi-Civita connection and the Schouten tensor
$\mbox{\textsf{P}}_{ab}$ of $g$. This is a trace-modification of the Ricci tensor
$\text{Ric}_{ab}$ of $g$ characterized by $\text{Ric}_{ab}=(n-2)\mbox{\textsf{P}}_{ab}+\mbox{\textsf{P}}
\mathbf{g}_{ab}$, where $\mbox{\textsf{P}}=\mathbf{g}^{ij}\mbox{\textsf{P}}_{ij}$.
Changing from $g$ to a metric $\widehat{g}=e^{2f}g$ for
$f\in C^\infty(K,\mathbb R)$ there is an explicit formula for the change of the
identification in terms of $\Upsilon_a=df$, namely
\betagin{equation}\lambdabel{trac-transf}
\betagin{pmatrix}\widehat{\sigma}\\ \widehat{\mu_a}\\ \widehat{\nu}
\end{pmatrix}=\betagin{pmatrix} \sigma\\ \mu_a+\Upsilon_a\sigma
\\ \nu-\mathbf{g}^{ij}(\Upsilon_i\mu_j+\frac{1}{2}\Upsilon_i\Upsilon_j\sigma) \end{pmatrix}
\end{equation}
Now one may turn around the line of argument and define sections of
the tractor bundle as equivalence classes of quadruples consisting of
a metric in the conformal class and sections of $\mathcal E[1]$, $\mathcal
E_a[1]$ and $\mathcal E[-1]$ with respect to the equivalence relation
defined in \eqref{trac-transf}. Recall that the behavior of
the Schouten tensor under a conformal change is given by
\betagin{equation}\lambdabel{Rho-transf}
\widehat{\mbox{\textsf{P}}}_{ab}=\mbox{\textsf{P}}_{ab}-\nabla_a\Upsilon_b+\Upsilon_a\Upsilon_b-\tfrac12\Upsilon_i\Upsilon^ig_{ab}.
\end{equation}
Using this, direct computations show that the definitions in \eqref{trac-met} and
\eqref{trac-conn} are independent of the choice of metric, so they give rise to a
well defined bundle metric and linear connection on the resulting bundle. That this
cannot directly done in a point-wise manner is a consequence of the fact that
tractors are more complicated geometric objects than tensors, since the action of
conformal isometries in a point depends on the two-jet of the isometry in that
point. To come to a point-wise construction, one would have to use 1-jets of metrics
in the conformal class instead.
In the above discussion we have assumed that $n\geq 3$. Indeed, it is well known that
conformal structures in dimension two behave quite differently from higher
dimensions. In particular, they do not allow an equivalent description in term of a
normal Cartan geometry or of tractors. Still we can obtain boundary tractors as
follows. Note first, that associating to a conformal structure a tractor bundle and a
tractor metric via formulae \eqref{trac-transf} and \eqref{trac-met} works without
problems in dimension two. This observation is already sufficient for most of our
results, where we just need a vector bundle canonically associated to some structure
on the boundary. Now one view into the different behavior in dimension two is seen by
the fact that the definition of the Schouten tensor $\mbox{\textsf{P}}_{ab}$ via the Ricci
curvature breaks down. While there are other ways to understand the Schouten tensor
it is nevertheless true that on a 2 dimensional Riemannian manifold there is no
natural tensor that transforms conformally according to (\ref{Rho-transf}). Thus one
cannot use \eqref{trac-conn} to associate to a conformal structure a canonical
connection on the tractor bundle.
However, in the computations needed to verify that \eqref{trac-conn} leads to a well
defined connection only the transformation law \eqref{Rho-transf} for the Schouten
tensor under conformal changes is needed, the relation to the Riemann curvature does
not play a role. (In fact this computation only involves single covariant
derivatives, so there is no chance for curvature terms to arise.) Consequently, the
construction of a canonical connection on the tractor bundle extends to dimension
two, provided that in addition to a conformal class one associates to each metric in
that class a symmetric tensor $\mbox{\textsf{P}}_{ab}$, such that the tensors associated to
conformally related metrics satisfy the transformation law \eqref{Rho-transf}.
The observation just made, for constructing a tractor bundle in
dimension 2, is close to the idea of a M\"obius structure, but
actually it is a slight generalization of the concept of a M\"obius
structure in the sense of \cite{Calderbank:Moebius}; compare in
particular with the MR review \cite{Eastwood:Moebius} of that
article. To define a M\"obius structure, one requires, in addition,
that the trace of the tensor $\mbox{\textsf{P}}_{ab}$ associated to $g_{ab}$ is one
half times the scalar curvature of $g_{ab}$. This can be expressed as
a normalization condition on the curvature of the tractor connection,
but we will not need this. Having said that in the cases in which we will need the
tractor connection in dimension two, we actually will deal with M\"obius
structures, since we get flat tractor connections.
\subsection{Boundary values of tractors}\lambdabel{2.5}
In our usual setting of $\circverline{M}=M\cup \partial M$, equipped with the
conformal class defined by a conformally compact metric, we will deal
with standard-tractor-valued differential forms on $\circverline{M}$. The
strategy is to associate to suitable forms, of this type, a boundary
value, and interpret this as a form taking values in the standard
tractor bundle $\mathcal T\partial M$ of the conformal infinity. As
discussed at the end of Section \ref{2.4} above, such a bundle is
available in all relevant dimensions $\dim(\circverline{M})=n\geq 3$. Moreover,
since this infinity is the same for all the metrics in a class $\mathcal
G$, as discussed in Section \ref{2.2}, this allows us to relate the
boundary values obtained from different metrics in $\mathcal G$. However,
even for $n\geq 4$, it is not obvious how to relate $\mathcal T\circverline{M}$ and
$\mathcal T\partial M$, so we discuss this next. This discussion will also
show how to canonically obtain the ``abstract Schouten tensors''
needed to define a tractor connection on $\mathcal T\partial M$ for
$n=3$. Hence we obtain a uniform description of boundary values in all
dimensions.
Provided that one works with metrics that are smooth up to the
boundary, the discussion of boundary values of tractors can be reduced
to the case of hypersurfaces in conformal manifolds. Observe first
that in our setting, it is no problem to relate density bundles on
$\circverline{M}$ and on $\partial M$ of any conformal weight. This is based on
the fact that $\mathcal E[2]$ can always be viewed as the line subbundle
spanned by the conformal class and one can form boundary values for
metrics in the conformal class (that are smooth up to the
boundary). So the densities of weight $w$ on $\partial M$ are simply
the restriction of the ambient densities of weight $w$ i.e.\ sections
of $\mathcal{E}[w]|_{\partial M}$.
Now the conformal metric and its inverse define inner products on
$\mathcal E^a[-1]$ and its dual $\mathcal E_a[1]$. In the case of a boundary,
there thus is a unique inward pointing unit normal normal
$n^i\in\Gamma(T\circverline{M}[-1]|_{\partial M})$ to the subbundle $T\partial M$,
and we put $n_i:=\mathbf{g}_{ij}n^j\in \Gamma(T^*\circverline{M}[1]|_{\partial
M})$. We will assume that $n^i$ and $n_i$ are (arbitrarily) extended
off the boundary, if needed. For a choice of metric $\bar{g}$ (which
is smooth up to the boundary) in the conformal class, we observe that
the restriction of $\nabla^{\bar g}_in_j$ to $T\partial M\times T\partial
M$ is independent of the chosen extension. This is the (weighted)
second fundamental form of $\partial M$ in $\circverline{M}$ with respect to
$\bar g$, and we can decompose it into a trace-free part and a
trace-part with respect to $\mathbf{g}_{ab}$. A short computation
shows that, along $\partial M$, the tracefree part is conformally
invariant. The trace part can be encoded into the mean curvature
$H^{\bar g}\in \Gammamma(\mathcal{E} [-1])$ of $\partial M$ in $\circverline{M}$
with respect to $\bar g$. Its behaviour under a conformal change
corresponding to $\Upsilon_a$ is given by $H^{\widehat{\bar g}}=H^{\bar
g}+\Upsilon_in^i$.
It is a classical fact, see Section 2.7 of \cite{BEG}, that these
ingredients can be encoded into a conformally invariant normal tractor
$N^A\in\Gamma(\mathcal E^A|_{\partial M})$. In the splitting corresponding to
$\bar g$, the tractor $N^A$ corresponds to the triple $(0,n_i,-H^{\bar
g})$, which shows that $N^A$ is orthogonal to $\mathbf{X}^B$ and has
norm $1$. Conformal invariance of $N^A$ follows immediately from the
change of $H^{\bar g}$ described above. It then turns out that, for
$n\geq 4$, there is a conformally invariant isomorphism between the
orthocomplement ${N^A}^\perp\subset \mathcal T\circverline{M}|_{\partial M}$ and the
intrinsic tractor bundle $\mathcal T\partial M$ of the conformal infinity
\cite{Gover:P-E}. In a scale $\bar g$ that is smooth up to the
boundary, a triple $(\sigma,\mu_a,\nu)$ is orthogonal to $N^A$ if and
only if $\mu_jn^j=H^{\bar g}\sigma $. Such a triple then gets mapped to
$$(\sigma,\mu_a-H^{\bar g} n_a\sigma,\nu+\tfrac12(H^{\bar g})^2\sigma)$$ in the splitting
corresponding to $\bar g|_{\partial M}$, see Section 6.1 of \cite{Curry-Gover}. In
particular, in the case that $\partial M$ is minimal with respect to $\bar g$
(i.e.\ that $H^{\bar g}$ vanishes identically), we simply get the na\"{\i}ve
identification of triples.
As mentioned above, these considerations also show how to obtain a
tractor connection on $\mathcal T\partial M$ in the case that $n=3$. We
can do this in the setting of hypersurfaces and as discussed in
Section \ref{2.4}, we have to associate an ``abstract Schouten
tensor'' to the metrics in the conformal class $\partial M$. The idea
here is simply that for metrics $\bar g$ such that $H^{\bar g}=0$, we
associate the restriction of the Schouten tensor of $\bar g$ to
$\partial M$ as an ``abstract Schouten tensor'' for the metric $\bar
g|_{\partial M}$. If $\bar g$ and $\widehat{\bar g}$ are two such
metrics, then for the change $\Upsilon_a$, we get $\Upsilon_in^i=0$, which
implies that the restriction of $\Upsilon_i\Upsilon^i$ to $\partial M$ coincides
with the trace of the restriction of $\Upsilon_a$ to $T\partial M$. From
this and the Gauss formula we conclude that restrictions of the
Schouten tensors to $\partial M$ satisfy the correct transformation
law \eqref{Rho-transf}. This is already sufficient to obtain a tractor
connection on $\mathcal T\partial M$. Since we know the behavior of all
objects under conformal rescalings, one can deduce a description of
$\mbox{\textsf{P}}_{ab}$ for general metrics, but we won't need that here. A
different approach to induced M\"obius structures on hypersurfaces and
more general submanifolds in conformal manifolds can be found in
\cite{Belgun}.
In any case, it is clear from this description that the above discussion of boundary
values now extends to $n=3$. Finally, consider a class $\mathcal G$, as discussed in
Section \ref{2.2}, and metrics $g,h\in\mathcal G$. Then for the conformal classes $[g]$
and $[h]$ the metrics $\rho^2g$ and $\rho^2h$, for a local defining function $\rho$,
admit a smooth extension to the boundary. Now by definition $\rho^2g\sigmam_N\rho^2h$
and $N\geq 3$. In particular, the difference of their curvatures is $\mathcal
O(\rho^{N-2})$ and $N-2\geq 1$, which implies that the restrictions of their Schouten
tensors to the boundary agree. This implies that, in dimension $n=3$, all metrics in
$\mathcal G$ lead to the same tractor connection on $\mathcal T\partial M$. In higher
dimensions the tractor connection on $\mathcal T\partial M$ is determined by the conformal
structure $[\mathcal G]$, and so the equivalent result holds trivially.
\subsection{The scale tractor}\lambdabel{2.6}
We now combine the ideas about boundary tractors with the conformally
compact situation. This needs one more basic tool of tractor calculus,
the so-called tractor $D$-operator (also called the Thomas
$D$-operator). In the simplest situation, which is all that we need
here, this is an operator $D^A:\mathcal E[w]\to\mathcal E^A[w-1]$, which in
triples is defined by
\betagin{equation}\lambdabel{D-def}
D^A\tau=\left(w(n+2w-2)\tau, (n+2w-2)\nabla_a\tau,
-\mathbf{g}^{ij}(\nabla_i\nabla_j+\mbox{\textsf{P}}_{ij})\tau\right).
\end{equation}
Again, a direct computation shows that this is conformally invariant. We will mainly
need the case that $w=1$, so that $D^A$ maps sections of the quotient bundle $\mathcal
E[1]$ of $\mathcal E^A$ to sections of $\mathcal E^A$. Since the first component of
$\tfrac1nD^A\sigma$ is $\sigma$, this operator is referred to as a \textit{splitting
operator}. In particular, given a metric in a conformal class, we can apply this
splitting operator to the canonical section of $\mathcal E[1]$ obtained from the volume
density of the metric. The resulting section of $\mathcal E^A$ is called the
\textit{scale tractor} associated to the metric.
Computing in the splitting determined by the metric itself, the associated section
$\sigma$ of $\mathcal E[1]$ satisfies $\nabla_a\sigma=0$. Hence in this splitting,
$\tfrac1nD^A\sigma$ corresponds to $(\sigma,0,-\tfrac1n\mbox{\textsf{P}}\sigma)$, where
$\mbox{\textsf{P}}:=\mathbf{g}^{ij}\mbox{\textsf{P}}_{ij}$. Applying the tractor connection to this, we get
$(0,(\mbox{\textsf{P}}_{ab}-\frac{1}{n}\textbf{g}_{ab}\mbox{\textsf{P}})\sigma,-\frac{1}{n}\sigma\nabla_c\mbox{\textsf{P}})$. Observe
that the middle slot of this vanishes iff $\mbox{\textsf{P}}_{ab}$ is pure trace, i.e.\ iff the
metric is Einstein. In that case, $\mbox{\textsf{P}}$ which is just a multiple of the scalar
curvature, is constant, hence a metric is Einstein iff its scale tractor is parallel.
Now we move to the case of $\circverline{M}=M\cup\partial M$ and a conformally compact metric
$g$ on $M$. By Proposition \ref{prop2.1}, the canonical section $\sigma\in\Gamma(\mathcal
E[1])$ which is parallel for $\nabla^g$ admits a smooth extension to the boundary (as
a defining density). Since we have a conformal structure on all of $\circverline{M}$ also the
scale tractor $I^A:=\tfrac{1}{n}D^A\sigma$ and its covariant derivative $\nabla_aI^A$
are smooth up to the boundary. On $M$, we can compute in the splitting determined by
$g$, which shows that $\lambdangle
I,I\rangle=-\tfrac2n\sigma^2\mbox{\textsf{P}}=-\tfrac2ng^{ij}\mbox{\textsf{P}}_{ij}$. Now the relation between the
Schouten tensor $\mbox{\textsf{P}}_{ij}$ and the Ricci-tensor shows that the scalar curvature
$R=g^{ij}\text{Ric}_{ij}$ of $g$ can be written as $2(n-1)g^{ij}\mbox{\textsf{P}}_{ij}$ and thus
$\lambdangle I,I\rangle=-\tfrac1{n(n-1)}R$. In particular, if $g$ is ALH, then this is
identically $1$ along the boundary.
Under slightly stronger assumptions, the restriction of $I^A$ to the boundary is
nicely related to the objects discussed in Section \ref{2.5}. To formulate this,
recall from Section \ref{2.5} that the trace-free part of the second fundamental form
is conformally invariant along $\partial M$. This vanishes identically if and only
if $\partial M$ is totally umbilic in $\circverline{M}$, which thus is a conformally invariant
condition.
\betagin{prop}\lambdabel{prop2.6}
Consider a manifold $\circverline{M}$ with boundary $\partial M$ and interior $M$ and a
conformally compact metric $g$ on $M$; let $\sigma\in\Gamma(\mathcal E[1])$ be the
corresponding density and $I^A:=\tfrac1nD^A\sigma$ the scale tractor.
(1) If $\lambdangle I,I\rangle=1+\mathcal O(\rho^2)$ near to $\partial M$, then the
restriction of $I^A$ to $\partial M$ coincides with the normal tractor $N^A$.
(2) If in addition $g$ is asymptotically Einstein in the sense that
$\nabla_aI^A|_{\partial M}$ vanishes in tangential directions, then the boundary
$\partial M$ is totally umbilic.
\end{prop}
\betagin{proof}
For (1), see Proposition 7.1 of \cite{Curry-Gover} (or Proposition 6
\cite{G-sigma}).
(2) There is a nice characterization for a hypersurface to be totally
umbilic, see \cite{BEG} and e.g.\ Lemma 6.2 of \cite{Curry-Gover}:
Extending the normal tractor $N_A$ arbitrarily off $\partial M$ one
can apply the tractor connection to obtain a tractor valued one
form. The restriction of this derivative to tangential directions
along $\partial M$ is independent of the chosen extension and
$\partial M$ is totally umbilic in $\circverline{M}$ if and only if this
restrictions vanishes. This implies the claim.
\end{proof}
\section{The tractor mass cocycle}\lambdabel{3}
We consider the tractor version of the classical almost hyperbolic
mass here, so the order of asymptotics we need corresponds to
$N=n=\dim(M)$ in the notation of Section \ref{2}.
\subsection{The contribution from the trace}\lambdabel{3.1}
Most of the theory we develop applies in the
general setting of an oriented manifold $\circverline{M}$ with boundary
$\partial M$, interior $M$, and equipped with a class $\mathcal G$ of
metrics on $M$, as introduced in Section \ref{2.3} with $N=n$. The
only additional assumption is that the metrics in $\mathcal G$ are ALH in
the sense of Definition \ref{def2.2}. Given two metrics $g,h\in\mathcal
G$, we denote by $\sigma,\tau\in \Gamma(\mathcal E[1])$ the corresponding powers
of the volume densities of $g$ and $h$. Recall that the class $\mathcal G$
gives rise to a well defined standard tractor bundle $\mathcal T\partial
M$ over $\partial M$. Our aim is to associate to $g$ and $h$ a form
$c(g,h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, i.e.~a top-degree
form on $\partial M$ with values in $\mathcal T\partial M$. Further, we
want this to satisfy a cocyle property, i.e.~that $c(h,g)=-c(g,h)$ and
that $c(g,k)=c(g,h)+c(h,k)$ for $g,h,k\in\mathcal G$.
The first ingredient for this is rather simple: Given $g,h\in\mathcal G$, we use the
conformal structure $[g]$ on $\circverline{M}$, and consider the $\mathcal T\circverline{M}$-valued one-form
$\tfrac{1}n\nabla_bD^A(\tau-\sigma)$. We already know that $\tau$ and $\sigma$ admit a smooth
extension to the boundary, so this is well defined and smooth up to the boundary. Now
on $M$, we can apply the Hodge-$*$-operator determined by $g$ to convert this into a
$\mathcal T\circverline{M}$-valued $(n-1)$-form. The following result shows that this is smooth up to
the boundary and that its boundary value is orthogonal to the normal tractor $N^A$
(and non-zero in general). Thus this boundary value defines a form in
$\Omega^{n-1}(\partial M,\mathcal T\partial M)$.
\betagin{prop}\lambdabel{prop3.1}
In the setting $\circverline{M}=M\cup\partial M$ and $g,h\in\mathcal G$ as described above, let
$\rho$ be a local defining function for the boundary. Put $\circverline{g}:=\rho^2g$,
let $\circverline{\sigma}\in\Gamma(\mathcal E[1])$ be the corresponding density and let
$\circverline{g}_{\infty}$ be the boundary value of $\circverline{g}$.
(1) In terms of the canonical section $\textbf{X}^A\in\mathcal T\circverline{M}[1]$ from Section
\ref{2.4} and the function $\mu$ from \eqref{tau-si} and writing $\rho_a$ for
$d\rho$, we get
\betagin{equation}\lambdabel{c11}
\nabla_bD^A(\tau-\sigma)=
\tfrac{n^2-1}{2}\rho^{n-2}\rho_b\mu\circverline{\sigma}^{-1}\mathbf{X}^A+ \mathcal
O(\rho^{n-1}).
\end{equation}
(2) The form $\star_g\nabla_bD^A(\tau-\sigma)$ is smooth up to the boundary and
its boundary value is given by
\betagin{equation}\lambdabel{c12}
\tfrac{n^2-1}{2}\circperatorname{vol}_{\circverline{g}_{\infty}}\mu_\infty\circverline{\sigma}_{\infty}^{-1}\mathbf{X}^A.
\end{equation}
Here $\circperatorname{vol}_{\circverline{g}_{\infty}}$ is the volume form of $\circverline{g}_\infty$,
$\circverline{\sigma}_{\infty}$ is the corresponding $1$-density on $\partial M$, and
$\mu_{\infty}$ is the boundary value of $\mu$. In particular, this is perpendicular
to $N^A|_{\partial M}$ and thus defines an $(n-1)$-form with values in $\mathcal T\partial
M$.
\end{prop}
\betagin{proof}
Observe first that for a connection $\nabla$, a section $s$ that both are smooth
up to the boundary, and an integer $k>0$, we get $\nabla_a(\rho^k
s)=k\rho^{k-1}\rho_as+\mathcal O(\rho^k)$. This can be applied both to the tractor
connection and to the Levi-Civita connection of $\circverline{g}$, which will both be written
as $\nabla$ in this proof.
Using that $\circverline{\sigma}=\tfrac{\sigma}{\rho}$, we can write formula \eqref{tau-si}
(still for $N=n$) as $\tau-\sigma=-\circverline{\sigma}\tfrac{\rho^{n+1}}{2n}\mu+\mathcal
O(\rho^{n+2})$. Now the defining formula \eqref{D-def} for $D^A$ shows that the first
two slots of $D^A(\tau-\sigma)$ in the splitting determined by $\circverline{g}$ are $\mathcal
O(\rho^{n+1})$ and $\mathcal O(\rho^n)$, respectively, while in the last slot the only
contribution which is not $\mathcal O(\rho^n)$ comes from the double derivative. This
shows that
$$
D^A(\tau-\sigma)=-\tfrac{n(n+1)}{2n}\rho^{n-1}\circverline{\sigma}\mu(-
\textbf{g}^{ij}\rho_i\rho_j)\mathbf{X}^A+\mathcal O(\rho^n).
$$ Using that $\circverline{g}^{ij}=\circverline{\sigma}^2\textbf{g}^{ij}$ and that $g$ is
ALH, we conclude that $\circverline{\sigma}
\textbf{g}^{ij}\rho_i\rho_j=\circverline{\sigma}^{-1}+\mathcal O(\rho)$, so we get
$$
D^A(\tau-\sigma)=\tfrac{n+1}{2}\rho^{n-1}\mu\circverline{\sigma}^{-1}\mathbf{X}^A
+\mathcal O(\rho^n).
$$
From this \eqref{c11} and hence part (1) follows immediately.
(2) Since $\circverline{M}$ is oriented we have an isomorphism
$ \mathcal E[-n]\stackrel{\cong}{\longrightarrow} \Lambda^nT^*\circverline{M} $
that can be interpreted as a canonical section $\betap_{a_1\dots a_n}\in
\Gammamma (\Lambda^nT^*\circverline{M} [n])$. In terms of this, the volume form of $g$ is given by
$\sigma^{-n}\betap_{a_1\dots a_n}$. Now by definition,
$\star_g\rho^{n-2}\rho_a$ is given by contracting $\rho^{n-2}\rho_a$ into the volume
form of $g$ via $g^{-1}$. So this is given by
\betagin{equation}\lambdabel{sigma-tech}
\sigma^{-n}\rho^{n-2}g^{ij}\rho_i\betap_{ja_1\dots a_{n-1}}.
\end{equation}
Now $\sigma^{-2}g^{ij}=\textbf{g}^{ij}$, while
$\sigma^{2-n}\rho^{n-2}=\circverline{\sigma}^{2-n}$. Together with part (1), this shows that
$$
\star_g\nabla_aD^A(\tau-\sigma)=\tfrac{n^2-1}2\circverline{\sigma}^{2-n}\mathbf{g}^{ij}\rho_i\betap_{ja_1\dots
a_{n-1}}\mu\circverline{\sigma}^{-1}\mathbf{X}^A+O(\rho).
$$
This is evidently smooth up to the boundary and its boundary value is a multiple
of $\mathbf{X}^A$ and thus perpendicular to $N^A$. To obtain the interpretation of
the boundary value, we can rewrite \eqref{sigma-tech} as
$\circverline{\sigma}^{-n}\circverline{g}^{ij}\rho_i\betap_{ja_1\dots a_{n-1}}$. Since the first and
last terms combine to give the volume form of $\circverline{g}$ and, along the boundary,
$\circverline{g}^{ib}\rho_i$ gives the unit normal with respect to $\circverline{g}$, we
conclude that the boundary value of \eqref{sigma-tech} is the volume form of
$\circverline{g}_\infty$. From this, \eqref{c12} and thus part (2) follows immediately.
\end{proof}
There actually is a simpler way to write out the boundary value of
$\star_g\nabla_bD^A(\tau-\sigma)$ than \eqref{c12} that needs less choices. The function
$\mu$ defined in \eqref{tau-si} of course depends on the choice of the defining
function $\rho$, and there is no canonical choice of defining function. However,
fixing the metric $g\in\mathcal G$, we of course get the distinguished defining density
$\sigma$, and we can get a more natural version of \eqref{tau-si} by phrasing things
in terms of densities. Namely, for the current setting with $N=n$, we can define
$\nu\in\Gamma(\mathcal E[-n])$ to be the unique density such that
\betagin{equation}\lambdabel{tau-si-dens}
\tau-\sigma=-\tfrac{1}{2n}\sigma^{n+1}\nu+\mathcal O(\rho^{n+2}).
\end{equation}
Then of course $\nu$ is uniquely determined by $g$ and $h$. In terms of a defining
function $\rho$ and the corresponding function $\mu$, we get
$\nu=(\tfrac{\rho}{\sigma})^n\mu$, which shows that $\nu$ is smooth up to the boundary
and non-zero wherever $\mu$ is non-zero. Let us write the boundary value of $\nu$ as
$\nu_\infty$, which by Section \ref{2.5} can be interpreted as a density of weight
$-n$ on $\partial M$. In the setting of Proposition \ref{prop3.1}, we then have
$\circverline{g}=\rho^2g$, so $\circverline{\sigma}=\tfrac{\sigma}{\rho}$. The latter is smooth
up to the boundary and from Section \ref{2.5} we know that its boundary value is the
$1$-density $\circverline{\sigma}_\infty$ on $\partial M$ corresponding to
$\circverline{g}_\infty$. Hence our construction implies that $\circperatorname{vol}_{\circverline{g}_\infty}$
corresponds to the density $\circverline\sigma_{\infty}^{1-n}$, so \eqref{c12} simplifies
$\tfrac{n^2-1}2\circverline{\sigma}_\infty^{-n}\mu_\infty\mathbf{X}^A=
\tfrac{n^2-1}2\nu_\infty\mathbf{X}^A$. Using this we can easily prove that we have
constructed a cocyle.
\betagin{cor}\lambdabel{cor3.1}
Let us denote the section of $\mathcal T\partial M[-n+1]$ associated to $g,h\in\mathcal G$
via formula \eqref{c12} by $c_1(g,h)$. Then $c_1$ is a cocycle in the sense that
$c_1(h,g)=-c_1(g,h)$ and that $c_1(g,k)=c_1(g,h)+c_1(h,k)$ for $g,h,k\in\mathcal G$.
\end{cor}
\betagin{proof}
From $\tau-\sigma=-\tfrac{1}{2n}\sigma^{n+1}\nu+\mathcal O(\rho^{n+2})$, we get
$\tau=\sigma+\mathcal O(\rho^{n+1})$ and hence
$\sigma-\tau=-\tfrac{1}{2n}\tau^{n+1}(-\nu)+\mathcal O(\rho^{n+2})$, so
$c_1(h,g)=-c_1(g,h)$. The second claim follows similarly.
\end{proof}
\subsection{The contribution from the trace-free part}\lambdabel{3.2}
We next need another element of tractor calculus that, again, concerns one-forms with
values in the standard tractor bundle. Returning to the setting of a general
conformal manifold $K$, for $k=1,\dots,n$, there is a natural bundle map
$\partial^*:\Lambda^kT^*K\circtimes \mathcal TK\to\Lambda^{k-1}T^*K\circtimes\mathcal TK$, which is
traditionally called the \textit{Kostant codifferential}. This has the crucial
feature that $\partial^*\circ\partial^*=0$, so for each $k$, one obtains nested natural
subbundles $\circperatorname{im}(\partial^*)\subset\ker(\partial^*)$ and hence there is the
subquotient $\mathcal H_k=\ker(\partial^*)/\circperatorname{im}(\partial^*)$. To make this explicit in low
degrees let us write the spaces $\Lambdambda^kT^*K\circtimes\mathcal TK$ for $k=0,1,2$ in the
obvious extension of the vector notation for standard tractors:
$$
\betagin{pmatrix} \mathcal E[1] \\ \mathcal E_c[1] \\ \mathcal E[-1]\end{pmatrix}
\circverset{\partial^*}{\longleftarrow}
\betagin{pmatrix} \mathcal E_a[1] \\ \mathcal E_{ac}[1] \\ \mathcal E_a[-1]\end{pmatrix}
\circverset{\partial^*}{\longleftarrow}
\betagin{pmatrix} \mathcal E_{[ab]}[1] \\ \mathcal E_{[ab]c}[1] \\ \mathcal E_{[ab]}[-1]\end{pmatrix} .
$$
Here $\mathcal E_{[ab]}$ is the abstract index notation for $\Lambda^2T^*K$.
By general facts, $\partial^*$ maps each row to the row below, so in particular the
bottom row is contained in the kernel of $\partial^*$. Moreover, Kostant's version of
the Bott-Borel-Weil Theorem implies that $\mathcal H_0\cong \mathcal E[1]$ and $\mathcal H_1\cong
\mathcal E_{(ab)_0}[1]$ (symmetric trace-free part). In particular, for $k=0$, we
conclude that $\partial^*$ has to map onto the two bottom rows, so
$\circperatorname{im}(\partial^*)=(\mathcal T^1K)^\perp$. Hence $\mathcal H_0$ coincides with the natural
quotient bundle $\mathcal E[1]$ of $\mathcal TK$ considered above.
In degree $k=1$, we conclude that $\partial^*$ maps the top
slot isomorphically onto the middle slot of $\mathcal TK$, while its restriction to the
middle slot must be a non-zero multiple of the trace. Hence $\ker(\partial^*)$
consists exactly of those elements for which the top slot vanishes and the middle
slot is trace-free. Similarly, we conclude that $\partial^*$ has to map the top slot
of the degree-two-component injectively into the middle slot of the degree-one part,
while the middle slot has to map onto the bottom slot of the degree-one
component. This shows how $\mathcal H_1\cong \mathcal E_{(ab)_0}[1]$ naturally arises as the
subquotient $\ker(\partial^*)/\circperatorname{im}(\partial^*)$ of $T^*K\circtimes\mathcal TK$.
Now similarly to the tractor-D operator, the machinery of BGG sequences constructs a
conformally invariant \textit{splitting operator}
$$
S:\Gamma(\mathcal H_1K)\to \Gamma(\ker(\partial^*))\subset \Omega^1(K,\mathcal TK).
$$
Apart from the fact that for the projection $\pi_H:\ker(\partial^*)\to\mathcal H_1$, one
obtains $\pi_H(S(\alpha))=\alpha$, this operator is characterized by the single property
that, for the covariant exterior derivative $d^\nabla$ induced by the tractor
connection, one gets $\partial^*\circ d^\nabla(S(\alpha))=0$ for any $\alpha\in\Gamma(\mathcal
H_1)$.
To compute the explicit expression for $S$, we again use the notation of triples.
\betagin{lemma}\lambdabel{lem3.2}
Let $K$ be a conformal manifold and let $g$ be a metric in the conformal
class. Then for $\phi=\phi_{ab}\in\Gamma(\mathcal E_{(ab)_0}[1])$, the section
$S(\phi)\in\Omega^1(K,\mathcal TK)$ is, in the splitting determined by $g$, given by
\betagin{equation}\lambdabel{S-formula}
(0,\phi_{ab}, \tfrac{-1}{n-1}\mathbf{g}^{ij}\nabla_i\phi_{aj}).
\end{equation}
\end{lemma}
\betagin{proof}
From above, we know that $S$ has values in $\ker(\partial^*)$ and that this implies
that the first component of $S(\phi)$ has to be zero. The fact that $\pi_H\circ S$ is the
identity map then shows that the middle component has to coincide with
$\phi_{ab}$. Thus it remains to determine the last component, which we temporarily
denote by $\psi=\psi_a\in\Gamma(\mathcal E_a[-1])$. This can be determined by exploiting the
fact that $\partial^*\circ d^\nabla(S(\phi))=0$. To compute $d^\nabla(0,\phi_{bc},\psi_b)$,
we first have to use formula \eqref{trac-conn} to compute
$\nabla_a(0,\phi_{bc},\psi_b)$ viewing the form index $b$ as a mere ``passenger
index''. This leads to $(-\phi_{ba},\nabla_a\phi_{bc}+\mathbf{g}_{ac}\psi_b,*)$ where we
don't compute the last component, which will not be needed in what follows. Then we
have to apply twice the alternation in $a$ and $b$, which kills the first component
by symmetry of $\phi$ and leads to $2(\nabla_{[a}\phi_{b]c}-\psi_{[a}\mathbf{g}_{b]c})$
in the middle component. From the description of $\partial^*$ above we know that
$\partial^*\circ d^\nabla(S(\phi))=0$ is equivalent to the fact that this middle
component lies in the kernel of a surjective natural bundle map to $\mathcal E_a[-1]$.
By naturality, this map has to be a nonzero multiple of the contraction by
$\mathbf{g}^{bc}$. Using trace-freeness of $\phi$, we conclude that $\partial^*\circ
d^\nabla(S(\phi))=0$ is equivalent to
$$
0=\mathbf{g}^{ij}(-\nabla_i\phi_{aj})-(n-1)\psi_a,
$$
which gives the claimed formula.
\end{proof}
Now we return to our setting $\circverline{M}=M\cup\partial M$, and a class $\mathcal G$ of metrics
with $N=n$ as before. Given two metrics $g,h\in\mathcal G$, we now consider the
trace-free part $(h_{ij}-g_{ij})^0$ of $h_{ij}-g_{ij}$ with respect to $g$, which
defines a smooth section of $\mathcal E_{(ab)_0}$. Thus for the density $\sigma\in\Gamma(\mathcal
E[1])$ determined by $g$, we can apply the splitting operator $S$ to
$\sigma(h_{ij}-g_{ij})^0$, to obtain a $\mathcal T\circverline{M}$-valued one-form $\phi_a^B$. We next
prove that this has the right asymptotic behavior to apply $\star_g$ and construct a
boundary value which lies in $\Omega^{n-1}(\partial M,\mathcal T\partial M)$, as we did for
the trace part in Section \ref{3.1} above.
Choosing a local defining function $\rho$ for the boundary, we get the function
$\mu_{ij}$ defined in \eqref{h-g}, and then
\betagin{equation}\lambdabel{h-g0}
(h_{ij}-g_{ij})^0=\rho^{n-2}(\mu_{ij}-\tfrac1n\rho^2\mu g_{ij})+\mathcal O(\rho^{n-1}),
\end{equation}
and clearly $\mu^0_{ij}:=\mu_{ij}-\tfrac1n\rho^2\mu g_{ij}$ defines a section of
$\mathcal E_{(ab)_0}$ that is smooth up to the boundary.
\betagin{prop}\lambdabel{prop3.2}
In the setting and notation of Proposition \ref{prop3.1} and using $\mu_{ij}^0$ as
defined above, we get:
(1) The form $S(\sigma(h_{ij}-g_{ij})^0)\in\mathcal E_a^A$ is given by
\betagin{equation}\lambdabel{c21}
-\rho^{n-2}\circverline{g}^{ij}\rho_i\mu^0_{aj}\circverline{\sigma}^{-1}\mathbf{X}^A+\mathcal
O(\rho^{n-1}).
\end{equation}
(2) The form $\star_gS(\sigma(h_{ij}-g_{ij})^0)$ is smooth up to the boundary and its
boundary value is given by
\betagin{equation}\lambdabel{c22}
-\circverline{g}^{ij}\circverline{g}^{k\ell}\rho_i\rho_k\mu^0_{j\ell}\circperatorname{vol}_{\circverline{g}_\infty}
\circverline{\sigma}_{\infty}^{-1}\mathbf{X}^A.
\end{equation}
This is perpendicular
to $N^A|_{\partial M}$ and thus defines an $(n-1)$-form with values in $\mathcal T\partial
M$.
\end{prop}
\betagin{proof}
(1) As before, we will work in the splitting determined by $\circverline{g}_{ij}$
throughout the proof. By construction
$\sigma(h_{ij}-g_{ij})^0=\sigma\rho^{n-2}\mu_{ij}^0=\circverline{\sigma}\rho^{n-1}\mu_{ij}^0$
is $\mathcal O(\rho^{n-1})$. Using Lemma \ref{lem3.2}, we see that the first slot of
$S(\sigma(h_{ij}-g_{ij})^0)$ vanishes and its middle slot is $\mathcal
O(\rho^{n-1})$. Using the observation from the beginning of the proof of
Proposition \ref{3.1} we see that the covariant derivative of
$\circverline{\sigma}\rho^{n-1}\mu_{ij}^0$, with respect to the Levi-Civita connection of $\circverline{g}_{ij}$, is given
by $(n-1)\circverline{\sigma}\rho^{n-2}\rho_k\mu_{ij}^0+\mathcal O(\rho^{n-1})$. Using this,
the claimed formula follows immediately from Lemma \ref{lem3.2} and the fact that
$\circverline{g}^{ij}=\circverline{\sigma}^2\mathbf{g}^{ij}$.
\noindent (2) Proceeding as in the proof of Proposition \ref{prop3.1}, we now show that
\mbox{$\star_gS((h_{ij}-g_{ij})^0)$} is given by
$$
-\circverline{g}^{ij}\rho_i\mu^0_{kj}\circverline{g}^{k\ell}\betap_{\ell a_1\dots
a_{n-1}}\circverline{\sigma}^{-n-1}\mathbf{X}^A.
$$
This is evidently smooth up to the boundary and, as observed there,
$\circverline{\sigma}^{-n}\betap_{a_1\dots a_n}$ is the volume form of
$\circverline{g}_{ij}$. Writing $\circperatorname{vol}_{\circverline{g}}|_{\partial M}$ as
$d\rho\wedge\circperatorname{vol}_{\circverline{g}_\infty}$ and using that the image of $d\rho$ in
$\Omega^1(\partial M)$ vanishes, we directly get the claimed formula for the boundary
value. The final statement follows the same argument as in Proposition \ref{prop3.1}.
\end{proof}
Similarly as in Section \ref{3.1} above, this admits a more natural interpretation
when working with densities. Again fixing $N=n$, instead of \eqref{h-g} we can start
from
\betagin{equation}\lambdabel{h-g-dens}
h_{ij}=g_{ij}+\sigma^{n-2}\nu_{ij},
\end{equation}
where $\nu_{ij}\in\Gamma(\mathcal E_{(ij)}[-n+2])$ now is a weighted symmetric two-tensor
that is smooth up to the boundary. For a choice of local defining function $\rho$,
the relation to \eqref{h-g} is described by
$\nu_{ij}=(\frac{\rho}{\sigma})^{n-2}\mu_{ij}$. This immediately implies that
$$
\mathbf{g}^{ij}\nu_{ij}=(\tfrac{\rho}{\sigma})^{n-2}\tfrac1{\sigma^2}g^{ij}\mu_{ij}=(\tfrac{\rho}{\sigma})^n\mu,
$$ so this coincides with the density $\nu\in\Gamma(\mathcal E[-n])$ used in Section
\ref{3.1}. The tracefree part $\nu^0_{ij}$, then of course is $\nu_{ij}-\tfrac1n
\mathbf{g}_{ij}\nu=(\frac{\rho}{\sigma})^{n-2}\mu^0_{ij}$. On the other hand, the fact
that $g_{ij}$ is ALH shows that $\tfrac{1}{\rho^2}g^{ij}\rho_i\rho_j$ is identically
one along the boundary. Using $g^{ij}=\sigma^2\mathbf{g}^{ij}$, we conclude that
$\tfrac{\sigma}{\rho}\mathbf{g}^{ij}\rho_i\in\mathcal E^a[-1]$ coincides, along $\partial M$,
with the conformal unit normal $n^j$ from Section \ref{2.5}. Hence
$\circverline{g}^{ij}\rho_j=\circverline{\sigma}n^i$ and hence we can rewrite formula
\eqref{c21} as $-\sigma^{n-2}\nu^0_{ij}n^j\mathbf{X}_A+\mathcal O(\rho^{n-1})$, where we
have extended $n^j$ arbitrarily off the boundary.
To rewrite the formula \eqref{c22} for the boundary value in a similar
way, we use the observation that $\circperatorname{vol}_{\circverline{g}_\infty}$
corresponds to $(\tfrac{\rho}{\sigma})^{n-1}$, as discussed in Section
\ref{3.1}. Using this and the above, see that \eqref{c22} equals
$$
-n^in^j(\nu^0_{ij})_\infty\mathbf{X}_A,
$$
where $(\nu^0_{ij})_\infty$ indicates the boundary value of $\nu^0_{ij}$. Using this
formulation, it is easy to prove that we obtain another cocyle.
\betagin{cor}\lambdabel{cor3.2}
Let us denote the section of $\mathcal T\partial M[-n+1]$ associated to $g,h\in\mathcal G$
via formula \eqref{c22} by $c_2(g,h)$. Then $c_2$ is a cocycle in the sense that
$c_2(h,g)=-c_2(g,h)$ and that $c_2(g,k)=c_2(g,h)+c_2(h,k)$ for $g,h,k\in\mathcal G$.
\end{cor}
\betagin{proof}
Suppose that the tracefree part of $h_{ij}-g_{ij}$ with respect to $g$ is given by
$\sigma^{n-2}\nu^0_{ij}$ for $\nu^0_{ij}\in\Gamma(\mathcal E_{ij}[2-n])$ and similarly the
tracefree part of $(g_{ij}-h_{ij})^0$ with respect to $h$ corresponds to
$\tilde\nu^0_{ij}$. Then one immediately verifies that
$\tilde\nu^0_{ij}=-\nu^0_{ij}+\mathcal O(\rho^{n-2})$. Together with the above formula,
this readily implies that $c_2(h,g)=-c_2(g,h)$. The second claim follows similarly.
\end{proof}
\subsection{Diffeomorphisms}\lambdabel{3.3}
We next start to study the compatibility of the cocycles we have
constructed above with diffeomorphisms. There are various concepts of
compatibility here, and we have to discuss some background
first. Recall that a diffeomorphism $\Psii:\circverline{M}\to\circverline{M}$ of a manifold
with boundary maps $M$ to $M$ and $\partial M$ to $\partial M$. This
also shows that for $x\in\partial M$, the linear isomorphism
$T_x\Psii:T_xM\to T_{\Psii(x)}M$ maps the subspace $T_x\partial M$ to
$T_{\Psii(x)}\partial M$. This readily implies that for a defining
function $\rho$ for $\partial M$, also $\Psii^*\rho=\rho\circ\Psii$ is a
defining function for $\partial M$. This also works for a local
defining function defined on a neighborhood of $\Psii(x)$, which gives
rise to a local defining function defined on a neighborhood of $x$.
This in turn readily implies that the relation $\sigmam_N$ on tensor
fields defined in Section \ref{2.3} is compatible with diffeomorphisms
in the sense that $t\sigmam_N\tilde t$ implies $\Psii^*t\sigmam_N\Psii^*\tilde
t$ for each $N>0$. In particular, given an equivalence class $\mathcal G$
of conformally compact metrics, also $\Psii^*\mathcal G$ is such an
equivalence class. We are particularly interested in the case that
$\Psii^*\mathcal G=\mathcal G$, in which we say that \textit{$\Psii$ preserves
$\mathcal G$}. The diffeomorphisms with this property clearly form a
subgroup of the diffeomorphism group $\circperatorname{Diff}(\circverline{M})$ which we denote by
$\circperatorname{Diff}_{\mathcal G}(\circverline{M})$. From our considerations it follows
immediately that this is equivalent to the fact that there is one
metric $g\in\mathcal G$ such that $\Psii^*g\in\mathcal G$ or equivalently
$\Psii^*g\sigmam_{N-2}g$. (In \cite{CDG} an analogous property is phrased
by saying that $\Psii$ is an ``asymptotic isometry'' of $g$. We don't
use this terminology since $\Psii$ is not more compatible with $g$
than with any other metric in $\mathcal G$.)
Recall from Section \ref{2.3} that all metrics in $\mathcal G$ give rise to the same
conformal infinity on $\partial M$. This implies that for $\Psii\in\circperatorname{Diff}_{\mathcal
G}(\circverline{M})$ the restriction $\Psii_\infty:=\Psii|_{\partial M}$ is not only a
diffeomorphism, but actually a conformal isometry of the conformal infinity of $\mathcal
G$. In particular, it induces a well defined bundle automorphism on the standard
tractor bundle $\mathcal T\partial M$ and hence we can pullback sections of $\mathcal
T\partial M$ along $\Psii_\infty$. This also works for $n=3$ without
problems. Using this, we can prove the first and
simpler compatibility condition of our cocylces with diffeomorphisms.
\betagin{prop}\lambdabel{prop3.3}
Consider a manifold $\circverline{M}=M\cup\partial M$ with boundary and a class
$\mathcal G$ of metrics, in the case $N=n$. Then the cocycles constructed
in Propositions \ref{prop3.1} and \ref{prop3.2} are compatible with
the action of a diffeomorphism $\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$ in the
sense that for each such cocycle
$c(\Psii^*g,\Psii^*h)=(\Psii_\infty)^*c(g,h)$. Here
$\Psii_\infty=\Psii|_{\partial M}$, and on the right hand side we have
the action of a conformal isometry of the conformal infinity of $\mathcal
G$ (on $\partial M$) on tractor valued forms.
\end{prop}
\betagin{proof}
This basically is a direct consequence of the invariance properties of the
constructions we use. If $g$ corresponds to $\sigma\in\Gamma(\mathcal E[1])$, then of course
$\Psii^*g$ corresponds to $\Psii^*\sigma$. Moreover, $\Psii_\infty$ defines an conformal
isometry between the conformal structures on $\partial M$ induced by $\Psii^*g$ and
$g$, respectively. Similarly, $\Psii^*h$ corresponds to $\Psii^*\tau$ and naturality of
the tractor constructions
implies that
$\nabla_aD_B(\Psii^*\tau-\Psii^*\sigma)$ (computed in the conformal structure $[\Psii^*g]$)
equals $\Psii^*(\nabla_aD_B(\tau-\sigma))$. Since $\Psii|_M$ is an isometry from $\Psii^*g$
to $g$, we get $\Psii^*\circperatorname{vol}_g=\circperatorname{vol}_{\Psii^*g}$, which implies compatibility with the
Hodge-star. Hence on $M$, we get
$$
\star_{\Psii^*g}\nabla_aD_B(\Psii^*\tau-\Psii^*\sigma)=\Psii^*(\star_g\nabla_aD_B(\tau-\sigma))
$$
and since both sides admit a smooth extension to the boundary, the boundary values
have to coincide, too. But these than are exactly $c_1(\Psii^*g,\Psii^*h)$ and the
pullback induced by the conformal isometry $\Psii_\infty$ of $c_1(g,h)$. This
completes the proof for $c_1$.
For $c_2$, we readily get that $(\Psii^*h-\Psii^*g)^0$ (tracefree part with respect to
$\Psii^*g$) coincides with $\Psii^*(h-g)^0$ (tracefree part with respect to $g$). Using
naturality of the splitting operator $S$, the proof is completed in the
same way as for $c_1$.
\end{proof}
\subsection{Diffeomorphisms asymptotic to the identity}\lambdabel{3.4}
To move towards a more subtle form of compatibility of our cocycles with
diffeomorphisms, we need a concept of asymptotic relation between diffeomorphisms.
\betagin{definition}\lambdabel{def3.4}
Let $\circverline{M}=M\cup\partial M$ be a manifold with boundary and let
$\Psi,\tilde\Psi:\circverline{M}\to\circverline{M}$ be diffeomorphisms.
(1) We say that $\Psi$ and $\widetilde{\Psi}$ are asymptotic of order $N>0$ and write
$\Psii\sigmam_N\widetilde{\Psi}$ if and only if for any function $f\in C^\infty(\circverline{M}, \mathbb R)$
we get $f\circ\Psii\sigmam_Nf\circ\widetilde{\Psi}$ in the sense of Section \ref{2.3}.
(2) For $N>0$, we define $\circperatorname{Diff}_0^N(\circverline{M})$ to be the set of diffeomorphisms which are
asymptotic to the identity $\circperatorname{id}_{\circverline{M}}$ of order $N$.
\end{definition}
Since $\sigmam_N$ clearly defines an equivalence relation on functions,
we readily see that it is an equivalence relation on
diffeomorphisms. Moreover, since the pullback of a local defining
function for $\partial M$ along a diffeomorphism of $\circverline{M}$ again is a
local defining function, we conclude that $\Psii\sigmam_N\widetilde{\Psi}$ implies
$\Psii\circ\Phi\sigmam_N\widetilde{\Psi}\circ\Phi$ and $\Phi\circ\Psii\sigmam_N\Phii\circ\widetilde{\Psi}$ for any
diffeomorphism $\Phii$ of $\circverline{M}$. In particular, this shows either of
$\widetilde{\Psi}^{-1}\circ\Psii\sigmam_N\circperatorname{id}$ and $\widetilde{\Psi}\circ\Psii^{-1}\sigmam_N\circperatorname{id}$ is
equivalent to $\Psii\sigmam_N\widetilde{\Psi}$.
On the other hand, we need some observations on charts. Given a manifold
$\circverline{M}=M\cup\partial M$ with boundary, take a point $x\in\partial M$. Then by
definition, there is a chart $(U,u)$ around $x$, so $U$ is an open neighborhood of
$x$ in $\circverline{M}$ and $u:U\to u(U)$ is a diffeomorphism onto an open subset of an
$n$-dimensional half space. Then $u$ restricts to a diffeomorphism between the open
neighborhood $U\cap\partial M$ of $x$ in $\partial M$ and the open subspace
$u(U)\cap\mathbb R^{n-1}\times\{0\}$ of $\mathbb R^{n-1}$. Then by definition, the last
coordinate function $u^n$ is a local defining function of $\partial M$. Conversely,
any local defining function can locally be used as such a coordinate function in a
chart.
If $\Psii\in\circperatorname{Diff}(\circverline{M})$ is a diffeomorphism, then for a chart $(U,u)$ also
$(\Psii^{-1}(U),u\circ\Psii)$ is a chart. If $\widetilde{\Psi}$ is another diffeomorphism such that
$\widetilde{\Psi}|_{\partial M}=\Psii|_{\partial M}$, then $V:=\Psii^{-1}(U)\cap\widetilde{\Psi}^{-1}(U)$ is
an open subset in $\circverline{M}$ which contains $\Psii^{-1}(U\cap\partial M)$. For any tensor
field $t$ defined on $U$, both $\Psii^*t$ and $\widetilde{\Psi}^*t$ are defined on $V$, and can
be compared asymptotically there. Using these observations, we start by proving a
technical lemma.
\betagin{lemma}\lambdabel{lem3.4}
Let $\circverline{M}=M\cup\partial M$ be a smooth manifold with boundary,
let $\Psii,\widetilde{\Psi}\in\circperatorname{Diff}(\circverline{M})$ be diffeomorphisms, and fix
$N>0$. Then the following conditions are equivalent:
\betagin{itemize}
\item[(i)] $\Psii\sigmam_{N+1}\widetilde{\Psi}$
\item[(ii)] $\Psii|_{\partial M}=\widetilde{\Psi}|_{\partial M}$ and for any tensor field $t$
on $\circverline{M}$, we get $\Psii^*t\sigmam_N\widetilde{\Psi}^*t$.
\item[(iii)] $\Psii|_{\partial M}=\widetilde{\Psi}|_{\partial M}$ and for each $x\in\partial M$, there
is an chart $(U,u)$ for $\circverline{M}$ with $x\in U$, whose coordinate functions $u^i$
satisfy $\Psii^*u^i\sigmam_{N+1}\widetilde{\Psi}^*u^i$ locally around $\Psii^{-1}(x)$.
\end{itemize}
\end{lemma}
\betagin{proof}
Replacing $\Psii$ by $\widetilde{\Psi}^{-1}\circ\Psii$ we may without loss of generality assume
that $\widetilde{\Psi}=\circperatorname{id}_{\circverline{M}}$, which we do throughout the proof.
(i)$\Rightarrow$(iii): We first claim that $\Psii|_{\partial M}=\circperatorname{id}_{\partial
M}$. For $x\in\partial M$, take an open neighborhood $W$ of $x$ in $\partial
M$. Then there is a bump function $f\in C^\infty(\circverline{M},\mathbb R)$ with values in
$[0,1]$ such that $f(x)=1$ and such that $f^{-1}(\{1\})\cap\partial M\subset W$. By
assumption $f\circ\Psii\sigmam_{N+1}f$, so in particular, these functions have to agree on
$\partial M$ and hence at $x$. Since $\Psii(x)\in \partial M$, by construction, we
get $\Psii(x)\in W$. Since $W$ was arbitrary, this implies that $\Psii(x)=x$ and
hence the claim. Having this at hand, we take any chart $(U,u)$ with $x\in U$,
extend the coordinate functions $u^i$ to globally defined functions on $M$ without
changing them locally around $x$ and then (i) immediately implies that
$\Psii^*u^i\sigmam_{N+1}u^i$ locally around $x$.
(iii)$\Rightarrow$(ii): For any tensor field $t$, it suffices to verify
$\Psii^*t\sigmam_N t$ locally around each boundary point $x\in\partial M$. Fixing $x$,
we take a chart $(U,u)$ as in (iii) and its coordinate functions $u^i$ and we work
on $V=\Psii^{-1}(U)\cap U$. Taking a vector field $\timesi\in\mathfrak X(U)$ we can compare
$\Psii^*\timesi$ and $\timesi$ on $V$. We can do this via coordinate expressions with
respect to the chart $(U,u)$ and we denote by $\timesi^i$ and $(\Psii^*\timesi)^i$ the
component functions. By assumption, $u^i\circ\Psii=u^i+\mathcal O(\rho^{N+1})$ and
differentiating this with $\Psii^*\timesi$, we obtain $(\Psi^*\timesi)(u^i)+\mathcal
O(\rho^N)$. Thus we conclude that $(\Psii^*\timesi)(u^i\circ\Psii)\sigmam_N(\Psii^*\timesi)^i$. But
by definition of the pullback, we get
$(\Psii^*\timesi)(u^i\circ\Psii)=\timesi(u^i)\circ\Psii\sigmam_{N+1}\timesi^i$. Overall, we conclude that
$(\Psii^*\timesi)^i\sigmam_N\timesi^i$ on $V$, which implies that $\Psii^*\timesi\sigmam_N\timesi$ on $V$
and hence we get condition (ii) for vector fields.
In particular, this implies that the coordinate vector fields $\partial_i$ of the
chart $(U,u)$ satisfy $\Psii^*\partial_i\sigmam_N\partial_i$ on $V$. On the other hand,
applying the exterior derivative to $u^i\circ\Psii\sigmam_{N+1}u^i$, we conclude that
$\Psii^*du^i=d(u^i\circ\Psii)\sigmam_Ndu^i$. Of course, on $V$ the $du^i$ coincide with the
coordinate one-forms of the chart $(U,u)$. Now given a tensor field $t$ of any
type, we can take $\Psii^*t$ and hook in vector fields $\Psii^*\partial_{i_a}$ and
one-forms $\Psii^*du^{j_b}$. On $V$ this by construction produces one of the
component functions of $t$ up to $\mathcal O(\rho^N)$. On the other hand, by definition
of the pullback, this coincides with the composition of the corresponding coordinate
function of $t$ with $\Psii$, and hence with that coordinate function up to $\mathcal
O(\rho^{N+1})$, so (ii) is satisfied in general.
(ii)$\Rightarrow$(i): Let $f\in C^\infty(\circverline{M},\mathbb R)$ be a smooth function. Then
by assumption we know that $f\circ\Psii\sigmam_Nf$, so choosing a local defining function
$\rho$ for $\partial M$, we get $f\circ\Psii=f+\rho^N\tilde f$ for some smooth function
$\tilde f\in C^\infty(\circverline{M},\mathbb R)$. But then
$\Psii^*df=d(f\circ\Psii)=df+N\rho^{N-1}\tilde fd\rho+\mathcal O(\rho^N)$. However,
condition (ii) also says that $\Psii^*df\sigmam_Ndf$ and since $d\rho|_{\partial M}$ is
nowhere vanishing, this implies that $\tilde f|_{\partial M}=0$. But this implies
that $f\circ\Psii\sigmam_{N+1}f$ and hence condition (i) follows.
\end{proof}
\subsection{The relation to adapted defining functions}\lambdabel{3.5}
In a special case and in quite different language, it has been observed in \cite{CDG}
that there is a close relation between diffeomorphisms asymptotic to the identity and
adapted defining functions. We start discussing this with the following lemma.
\betagin{lemma}\lambdabel{lem3.5}
In our usual setting, of $\circverline{M}=M\cup\partial M$, let $\mathcal G$ be an
equivalence class of ALH metrics on $M$ for the relation $\sigmam_{N-2}$
for some $N\geq 3$. Take two metrics $g,h\in\mathcal G$, and let $\rho$
and $r$ be local defining functions for $\partial M$ defined on the
same open subset $U\subset\circverline{M}$. If $\rho$ is adapted to $g$ and $r$
is adapted to $h$ in the sense of Definition \ref{def2.2} and if
$\rho^2g_{ij}$ and $r^2h_{ij}$ induce the same metric on the boundary,
then $\rho\sigmam_{N+1}r$.
\end{lemma}
\betagin{proof}
We have to analyze the asymptotics of solutions to the PDE that governs the change to
an adapted defining function. Replacing $\circverline{M}$ by an appropriate open subset, we
may assume that $\rho$ and $r$ are defined on all of $\circverline{M}$. Then we can write
$r=\rho e^v$ for some smooth function $v\in C^\infty(\circverline{M},\mathbb R)$ which gives
$dr=rdv+e^vd\rho$. In abstract index notation, this reads as
$r_i=rv_i+e^v\rho_i$. The fact that $r$ is adapted to $h_{ij}$ says that
$r^{-2}h^{ij}r_ir_j$ is identically $1$ on a neighborhood of the boundary, and
inserting we conclude that
\betagin{equation}\lambdabel{adap-PDE}
1\equiv \rho^{-2}h^{ij}\rho_i\rho_j+2\rho^{-1}h^{ij}\rho_iv_j+h^{ij}v_iv_j.
\end{equation}
Observe that $\rho^{-2}h^{ij}$, $\rho_i$ and $v_i$ are all smooth up to the boundary,
so the terms in the right hand side are $\mathcal O(1)$, $\mathcal O(\rho)$, and $\mathcal
O(\rho^2)$, respectively.
Now on the one hand, since $g,h\in\mathcal G$, we know from \eqref{hinv} that
$\rho^{-2}h^{ij}=\rho^{-2}g^{ij}+\mathcal O(\rho^N)$. Since $\rho$ is adapted to $g$,
this means that the first term in the right hand side of \eqref{adap-PDE} is $1+\mathcal
O(\rho^N)$. Inserting into \eqref{adap-PDE}, we conclude that
\betagin{equation}\lambdabel{adap-PDE2}
2\rho^{-1}h^{ij}\rho_iv_j+h^{ij}v_iv_j=\mathcal O(\rho^N).
\end{equation}
On the other hand, $r^2h_{ij}=e^{2v}\rho^2h_{ij}$, and $r^2h_{ij}$, by assumption, is
smooth up to the boundary with the same boundary value as $\rho^2g_{ij}$. Hence our
assumption on the induced metrics on the boundary imply that $e^{2v}|_{\partial
M}=1$, so $v$ has to vanish identically along the boundary and hence $v=\mathcal
O(\rho)$. Inductively, putting $v=\rho^\ell\tilde v$ for $\ell\geq 1$, we get
$v_i=\ell\rho^{\ell-1}\tilde v\rho_i+\mathcal O(\rho^\ell)$, which implies that the left
hand side of \eqref{adap-PDE2} becomes $2\ell\rho^\ell\tilde
v\rho^{-2}h^{ij}\rho_i\rho_j+\mathcal O(\rho^{\ell+1})$. As long as $\ell<N$, this shows
that $\tilde v= \mathcal O (\rho)$, so we conclude that we can write $v=\rho^N\tilde v$
where $\tilde v$ is smooth up to the boundary. But then
$$r=\rho e^v=\rho(1+\rho^N\tilde v+\mathcal O(\rho^{N+1}))=\rho+\mathcal O(\rho^{N+1}),$$
which completes the proof.
\end{proof}
Using this, we can now establish several important properties of
diffeomorphisms that are asymptotic to the identity.
\betagin{thm}\lambdabel{thm3.5}
Let $\circverline{M}=M\cup\partial M$ be a smooth manifold with boundary and, for some $N\geq
3$, let $\mathcal G$ be an equivalence class of ALH metrics on $M$ for the relation
$\sigmam_{N-2}$. Let us denote by $[\mathcal G]$ the conformal structure on $\partial M$
defined by the conformal infinity of $\mathcal G$ and by $\circperatorname{Conf}(\partial M,[\mathcal G])$
the group of its conformal isometries. Then we have
(1) $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is a normal subgroup in $\circperatorname{Diff}(\circverline{M})$ and
is contained in $\circperatorname{Diff}_{\mathcal G}(\circverline{M})$.
(2) Restriction of diffeomorphisms to the boundary induces a homomorphism
$\circperatorname{Diff}_{\mathcal G}(\circverline{M})\to\circperatorname{Conf}(\partial M,[\mathcal G])$ with kernel
$\circperatorname{Diff}^{N+1}_0(\circverline{M})$.
\end{thm}
\betagin{proof}
(1) The observations on the relation $\sigmam_N$ for diffeomorphisms we have made after
Definition \ref{def3.4} readily imply that $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is stable under
inversions as well as compositions, and conjugations by arbitrary elements of
$\circperatorname{Diff}(\circverline{M})$. Hence $\circperatorname{Diff}^{N+1}_0(\circverline{M})$ is a normal subgroup of
$\circperatorname{Diff}(\circverline{M})$. Taking $g\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$, we know that $\rho^2g_{ij}$ admits a smooth extension to the
boundary. Thus, given $\Psii\in \circperatorname{Diff}^{N+1}_0(\circverline{M})$, we may apply part (2) of
Lemma \ref{lem3.4} to conclude that $\Psii^*(\rho^2g_{ij})\sigmam_N\rho^2g_{ij}$. Now
$\Psii^*(\rho^2)=(\rho\circ\Psii)^2$ and $\rho\circ\Psii\sigmam_{N+1}\rho$. Hence
$\rho^2\Psii^*g_{ij}\sigmam_N \Psii^*(\rho^2g_{ij})\sigmam_N\rho^2g_{ij}$ and restricting
to $M$ we conclude that $\Psii^*g_{ij}\in\mathcal G$. This completes the proof of (1).
(2) It follows readily from the definitions that, for
$\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$, the diffeomorphism $\Psii|_{\partial
M}$ of $\partial M$ preserves the conformal structure $[\mathcal
G]$. Thus we get a homomorphism $\circperatorname{Diff}_{\mathcal
G}(\circverline{M})\to\circperatorname{Conf}(\partial M,[\mathcal G])$ as claimed and it remains
to prove the claim about the kernel. So let us take a diffeomorphism
$\Psii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$ such that $\Psii|_{\partial
M}=\circperatorname{id}_{\partial M}$ and we want to show that
$\Psii\sigmam_{N+1}\circperatorname{id}$. To prove this, we can apply condition (iii) of
Lemma \ref{lem3.4} and work locally around a boundary point $x$. Let
us choose $g\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$ which is adapted to $g$ and defined on some open
neighborhood $U$ of $x$ in $\circverline{M}$. Now we consider the normal field
(to $\rho $ level sets) determined by $g$ and $\rho$, i.e.~we put
$\timesi^i:=\rho^{-2}g^{ij}\rho_j$. This admits a smooth
extension to the boundary, and the fact that $\rho$ is adapted to $g$
exactly says that $\rho^2g_{ij}\timesi^i\timesi^j$ is identically one on a
neighborhood of the boundary.
For $x\in\partial M$, we now work on an open neighborhood $W$ of $x$
in $\circverline{M}$ such that $W\subset\Psii^{-1}(U)\cap U$. On $W$, we can
pull back all our data by $\Psii$, thus obtaining
$h_{ij}:=\Psii^*g_{ij}$, $r:=\rho\circ\Psii$, and $\eta:=\Psii^*\timesi$. By
assumption, $h_{ij}\in\mathcal G$ and pulling back the defining equation
for $\timesi^i$ we get $\eta^i=r^{-2}h^{ij}r_j$. Also by pulling back,
we readily see that the $r^2h_{ij}\eta^i\eta^j$ is identically one
on a neighborhood of the boundary. Hence we conclude that the local
defining function $r$ is adapted to $h_{ij}$ and hence Lemma
\ref{lem3.5} shows that $r\sigmam_{N+1}\rho$. Observe that this implies
that $r_j\sigmam_N\rho_j$ and together with $g^{ij}\sigmam_{N+2}h^{ij}$
the defining equations for $\timesi$ and $\eta$ show that
$\timesi\sigmam_N\eta$.
Next, we pass to an appropriate collar of the boundary. We can
choose an open neighborhood $V$ of $x$ in $\partial M$ and a
positive number $\epsilonsilon$ such that the flow map
$(y,t)\mapsto\circperatorname{Fl}^\eta_t(y)$ defines a diffeomorphism $\phi$ from $V\times
[0,\epsilonsilon)$ onto an open subset contained in $\Psii^{-1}(W)\cap W$,
and on which $\rho$ satisfies $(\rho^2 g)^{-1}(d\rho,d\rho)=1$.
Now let us define $\partial_t:=\phi^*\eta$. Note that this is the
coordinate vector field for any product chart on $V\times
[0,\epsilonsilon)$ induced by some chart on $V$. Since
$\eta=\Psii^*\timesi$, the fact the $\Psii$-related vector fields have
$\Psii$-related flows together with $\Psii|_{\partial M}=\circperatorname{id}$
readily implies that $(\Psii\circ\phi)(y,t)=\circperatorname{Fl}^\timesi_t(y)$. Using
Section \ref{3.4}, we see that we can complete the proof by
showing that $\Psii\circ\phi\sigmam_{N+1}\phi$.
By Lemma \ref{lem3.4} it suffices to consider the pullbacks of coordinate functions
of local charts along these diffeomorphisms. We apply this to charts which are
obtained by composing a product chart for $V\times [0,\epsilonsilon)$ with
$(\Psii\circ\phi)^{-1}$. Now the fact that $d\rho(\timesi)\equiv 1$ (and that $\Psii\circ\phi$
maps $V\times \{0\}$ into $\partial M$) shows that applying this construction to
the coordinate $t$ on $[0,\epsilonsilon)$, we obtain $\rho$. As observed above,
$\rho\circ\Psii\sigmam_{N+1}\rho$ and thus $\rho\circ\Psii\circ\phi\sigmam_{N+1}\rho\circ\phi$. Thus
it remains to consider functions $f$ such that $f\circ(\Psii\circ\phi)$ is one of the
boundary coordinate functions, and hence $\partial_t\cdot (f\circ \Psi\circ\phi)\equiv
0$ or, equivalently, $\timesi\cdot f\equiv 0$ on an appropriate neighborhood of the
boundary. We have to compare $f\circ(\Psii\circ\phi)$ to $f\circ\phi$, and of course
$\phi^*\timesi\cdot (f\circ\phii)\equiv 0$.
As we have observed above, we get $\eta\sigmam_N\timesi$
and hence
$\phi^*\timesi\sigmam_N\phi^*\eta=\partial_t$. Thus we get
$\phi^*\timesi=\partial_t+t^N\tilde\timesi$ for some vector field
$\tilde\timesi\in\mathfrak(V\times [0,\epsilonsilon))$. By construction $f\circ\phi$ and
$f\circ(\Psi\circ\phi)$ agree on $V\times\{0\}$, so
$f\circ\phi\sigmam_1f\circ(\Psi\circ\phi)$. Assuming that $f\circ\phi\sigmam_kf\circ(\Psi\circ\phi)$ for
some $k\geq 1$, we get $f\circ\phi=f\circ(\Psi\circ\phi)+t^k\tilde f$ for some $\tilde
f\in C^\infty(V\times[0,\epsilon),\mathbb R)$. Then we compute
$$
0=(\partial_t+t^N\tilde\timesi)\cdot (f\circ(\Psi\circ\phi)+t^k\tilde
f)=0+kt^{k-1}\tilde f+\mathcal O(t^{\text{min}(k,N)})
$$
If $k\leq N$, then this equation shows that $\tilde f$ vanishes along $V\times\{0\}$
and hence $(f\circ\phi)\sigmam_{k+1}f\circ(\Psi\circ\phi)$. Inductively, this gives
$(f\circ\phi)\sigmam_{N+1}f\circ(\Psi\circ\phi)$, which completes the proof.
\end{proof}
\subsection{Aligned metrics}\lambdabel{3.6}
Following an idea in \cite{CDG} we next show that the freedom under diffeomorphisms
asymptotic to the identity can be absorbed in a geometric relation between the
metrics. The analogous condition in \cite{CDG} is phrased as ``transversality''.
\betagin{definition}\lambdabel{def3.6}
Let $\circverline{M}=M\cup\partial M$ be a manifold with boundary and let $\mathcal G$ be an
equivalence class of ALH metrics for the relation $\sigmam_{N-2}$ for some $N\geq
3$. Consider two metrics $g,h\in\mathcal G$ and a local defining function $\rho$ for
$\partial M$ defined on some open subset $U\subset\circverline{M}$. Then we say that $h$
\textit{is aligned with $g$ with respect to $\rho$} if
$$
\rho_ig^{ij}(h_{jk}-g_{jk})\equiv 0
$$
on some open neighborhood of $U\cap\partial M$ in $U$.
\end{definition}
Observe that the condition in Definition \ref{def3.6} can be rewritten
as $\rho_ig^{ij}h_{jk}=\rho_k$. This in turn implies that
$\rho_ih^{ij}=\rho_ig^{ij}$ on a neighborhood of the boundary. This
shows that in the case that $\rho$ is adapted to $g_{ij}$ and $h_{ij}$ is
aligned to $g_{ij}$ with respect to $\rho$, then $\rho$ is also
adapted to $h_{ij}$.
\betagin{thm}\lambdabel{thm3.6}
In our usual setting, of $\circverline{M}=M\cup\partial M$ and a class $\mathcal G$ of metrics,
assume that $g\in\mathcal G$ and $\rho$ is a local defining function for $\partial M$
that is adapted to $g$. Then for any $h\in\mathcal G$ and locally around any point
$x\in\partial M$ in the domain of definition of $\rho$, there exist a diffeomorphism
$\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ and such that $\Psii^*h$ is aligned to
$g$ with respect to $\rho$. Moreover, the germ of $\Psii$ along the intersection of
its domain of definition with $\partial M$ is uniquely determined by this condition.
\end{thm}
\betagin{proof}
Since $\rho$ is adapted to $g$ the function
$\rho^{-2}g^{ij}\rho_i\rho_j$ is identically one on some neighborhood
of the boundary. Now given $h_{ij}\in\mathcal G$, we can use Proposition
\ref{prop2.2} to modify $\rho$ to a local defining function $r$
adapted to $h_{ij}$ in such a way that $\rho^2g_{ij}$ and $r^2h_{ij}$
induce the same metric on the boundary, compare also to the proof of
Lemma \ref{lem3.5}. Now we define vector fields $\timesi=\timesi^i$ and
$\eta=\eta^i$ by $\timesi^i:=\rho^{-2}g^{ij}\rho_j$ and
$\eta^i:=r^{-2}h^{ij}r_j$ where as usual we write $\rho_j$ for $d\rho$
and similarly for $r$. Recall from the proof of Theorem \ref{thm3.5} that this implies
that $\timesi\sigmam_N\eta$.
Now, via the construction of collars from the proof of Theorem \ref{thm3.5}, we
obtain a diffeomorphism $\Psii$ which has the property that for any $y$ in an
appropriate neighborhood of $x$ in $\partial M$, we get
$\Psii\circ\circperatorname{Fl}^\timesi_t(y)=\circperatorname{Fl}^\eta_t(y)$. Differentiating this equation shows that
$\timesi=\Psii^*\eta$. By construction the derivative of the function $t\mapsto
r(\circperatorname{Fl}^\eta_t(y))$ is given by
$$
dr(\eta)=r^{-2}r_ih^{ij}r_j\equiv 1,
$$ so $r(y)=0$ shows that $r(\circperatorname{Fl}^\eta_t(y))=t$. In the same way
$\rho(\circperatorname{Fl}^\timesi_t(y))=t$, which shows that $\Psii^*r=\rho$.
Knowing this and $\timesi\sigmam_N\eta$, the last part of the proof of
Theorem \ref{thm3.5} shows that $\Psii\sigmam_{N+1}\circperatorname{id}$.
To show that $\Psii$ has the required property, observe that by
construction $r^2h_{ij}\eta^j=r_i$. That is $i_\eta r^2h=dr$ and in the
same way $i_\timesi \rho^2g=d\rho$. Using this and the above, we now
obtain
$$
i_\timesi\rho^2\Psii^*h=i_{\Psii^*\eta}\Psii^*r^2h=\Psii^*(i_\eta r^2h)=\Psii^*(dr)=d\rho=i_\timesi
\rho^2g.
$$
This shows that inserting $\timesi$ into the bilinear form
$\rho^2(\Psii^*h-g)$, which by construction is smooth up to the
boundary, the result vanishes identically on a neighborhood of the
boundary. This exactly shows that $\Psii^*h$ is aligned with $g$ with
respect to $\rho$, so the proof of existence is complete.
To prove uniqueness, assume that $h$ is aligned to $g$ with respect to $\rho$ and
that a diffeomorphism $\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ has the property that also
$\Psii^*h$ is aligned to $g$ with respect to $\rho$. Observe that $\Psii\sigmam_{N+1}\circperatorname{id}$
implies that $\Psii^*h\in\mathcal G$. As observed above, the fact that both $h$ and
$\Psii^*h$ are aligned with $g$ with respect to $\rho$ implies that $\rho$ is adapted
both to $h$ and to $\Psii^*h$. But on the other hand, the fact that $\rho$ is adapted
to $h$ of course implies that $\Psii^*\rho$ is adapted to $\Psii^*h$. Now by
construction $\rho\circ\Psii\sigmam_{N+1}\rho$ and hence $(\rho\circ\Psii)\Psii^*h$ and
$\rho\Psii^*h$ induce the same metric on the boundary. Hence the uniqueness part in
Proposition \ref{prop2.2} implies that $\rho\circ\Psii=\rho$ and hence
$\Psii^*\rho_i=\rho_i$ on a neighborhood of the boundary.
Now of course the inverse metric to $\Psii^*h_{ij}$ is $\Psii^*(h^{ij})$
and $\Psii^*\rho_i\Psii^*h^{ij}=\rho_i\Psii^*h^{ij}$. Since
$\Psii^*h_{ij}$ is aligned with $g_{ij}$ with respect to $\rho$, we get
$ \rho_i\Psii^*h^{ij}=\rho_ig^{ij}$ and since also $h_{ij}$ is aligned
with $g$ with respect to $\rho$, it equals $\rho_ih^{ij}$.
Hence putting $\timesi^j:=\rho_ih^{ij}$ we conclude that $\Psii^*\timesi=\timesi$.
Since $\Psii$ is the identity on the boundary, this implies that
$\Psii(\circperatorname{Fl}^\timesi_t(y))=\circperatorname{Fl}^\timesi_t(y)$ for $y$ in the boundary and $t$
sufficiently small, so $\Psii=\circperatorname{id}$ locally around the boundary.
\end{proof}
\subsection{The action of the aligning diffeomorphism}\lambdabel{3.7}
Fix $g\in\mathcal G$ and a local defining function $\rho$ for the boundary which is
adapted to $g$. Consider another metric $h\in\mathcal G$ and the corresponding tensor
field $\mu_{ij}$ defined by \eqref{h-g}. From Theorem \ref{thm3.6} we then know that
locally around each boundary point, we find an essentially unique diffeomorphism
$\Psii$ such that $\Psii\sigmam_{N+1}\circperatorname{id}$ and such that $\Psii^*h$ is aligned with $g$ with
respect to $\rho$. Since $\Psii^*h\in\mathcal G$, the analog of \eqref{h-g} defines a
tensor field $\tilde\mu_{ij}$ that describes the difference between $\Psii^*h$ and
$g$. We now prove that the boundary value of $\tilde\mu_{ij}$ can be explicitly
computed from the boundary value of $\mu_{ij}$. This will be the crucial step towards
finding combinations of the two cocycles constructed in Sections \ref{3.1} and
\ref{3.2} which are invariant under diffeomorphisms asymptotic to the
identity. Observe that this formally looks like the coordinate formula in Proposition
2.16 of \cite{CDG}, but the actual meaning is different: Our description
does not involve any choice of local coordinates, but uses only abstract indices.
\betagin{thm}\lambdabel{thm3.7}
In the setting of Theorem \ref{thm3.6} for some fixed order $N$, let
$\mu_{ij}$ be the tensor field relating $h_{ij}$ and $g_{ij}$
according to \eqref{h-g}, and let $\tilde\mu_{ij}$ be the
corresponding tensor field relating $\Psii^*(h_{ij})$ and
$g_{ij}$. Then putting $\timesi^i:=\rho^{-2}g^{ij}\rho_j$, we obtain
\betagin{equation}\lambdabel{alignment-diffeo}
\tilde\mu_{ij}=\mu_{ij}-\rho_i\mu_{j\ell}\timesi^\ell-\rho_j\mu_{i\ell}\timesi^\ell+
\tfrac{\timesi^k\mu_{k\ell}\timesi^\ell}{N}(\rho^2g_{ij}+(N-1)\rho_i\rho_j)+\mathcal O(\rho).
\end{equation}
\end{thm}
\betagin{proof}
We use the quantities introduced in the proof of Theorem \ref{thm3.6}:
We denote by $r$ the defining function adapted to $h_{ij}$ such that
$r^2h_{ij}$ and $\rho^2g_{ij}$ induce the same metric on the
boundary. Further we put $\timesi^i:=\rho^{-2}g^{ij}\rho_j$ and
$\eta^i:=r^{-2}h^{ij}r_j$. In terms of these, we know from the proof
of Theorem \ref{thm3.6} that the alignment diffeomorphism $\Psii$
satisfies $r\circ\Psii=\rho$ and $\Psii^*\eta=\timesi$, and hence
$\Psii\circ\circperatorname{Fl}^\timesi_t=\circperatorname{Fl}^\eta_t$ wherever the flows are defined. Moreover,
writing $r=e^v\rho$, we know from the proof of Lemma \ref{lem3.5} that
$v=\rho^N\tilde v$ for a function $\tilde v$ that admits a smooth
extension to the boundary. Moreover, we can compute the boundary value
of $\tilde v$ from that proof: In equation \eqref{adap-PDE} we can
bring the first term on the right hand side to the left hand side and
then use use fact that $\rho$ is adapted to $g$ to rewrite
\eqref{adap-PDE} as
$$
(\rho^{-2}g^{ij}-\rho^{-2}h^{ij})\rho_i\rho_j=2\rho\rho^{-2}h^{ij}\rho_iv_j+\rho^2\rho^{-2}h^{ij}v_iv_j.
$$ In the left hand side, we can insert \eqref{hinv} and use the definition of $\timesi$
to obtain $\rho^N\timesi^k\mu_{k\ell}\timesi^\ell$. On the other hand $v_j=N\rho^{N-1}\tilde
v\rho_j+\mathcal O(\rho^N)$. Inserting this in the right hand side and using that
$\rho^{-2}h^{ij}\rho_i\rho_j=1+\mathcal O(\rho)$, we obtain $2N\rho^N\tilde v+\mathcal
O(\rho^{N+1})$, which shows that
\betagin{equation}\lambdabel{tildev}
\tilde v=\tfrac{1}{2N}\timesi^k\mu_{k\ell}\timesi^\ell+\mathcal O(\rho).
\end{equation}
The basis for the further computation will be the fact that for vector fields
$\zeta_1,\zeta_2$ (which we assume to be smooth up to the boundary), we get
$\Psii^*h(\Psii^*\zeta_1,\Psii^*\zeta_2)=h(\zeta_1,\zeta_2)\circ\Psii$. Multiplying by $\rho^2$, we
obtain
\betagin{equation}\lambdabel{PB}
\rho^2\Psii^*h(\Psii^*\zeta_1,\Psii^*\zeta_2)=(r^2h(\zeta_1,\zeta_2))\circ\Psii,
\end{equation}
and both sides admit a smooth extension to the boundary. Hence the right hand side
equals $r^2h(\zeta_1,\zeta_2)+\mathcal O(\rho^{N+1})$. Inserting $r=\rho e^v$ and
\eqref{h-g}, we conclude that this equals
$$
e^{2v}\rho^2g(\zeta_1,\zeta_2)+e^{2v}\rho^N\mu(\zeta_1,\zeta_2)+\mathcal O(\rho^{N+1}).
$$
Of course, $e^{2v}=1+2\rho^N\tilde v+\mathcal O(\rho^{N+1})$ and we conclude that the
right hand side of \eqref{PB} equals
\betagin{equation}\lambdabel{PB-RHS}
\rho^2g(\zeta_1,\zeta_2)+\rho^N\big(2\tilde v\rho^2g(\zeta_1,\zeta_2)+
\mu(\zeta_1,\zeta_2)\big)+\mathcal O(\rho^{N+1}).
\end{equation}
The left hand side of \eqref{PB}, by definition, can be written as
$\rho^2g(\Psii^*\zeta_1,\Psii^*\zeta_2)+\rho^N\tilde\mu(\Psii^*\zeta_1,\Psii^*\zeta_2)$. Now we
know that $\Psii^*\zeta_1\sigmam_N\zeta_1$ and hence $\Psii^*\zeta_1=\zeta_1+\rho^N\tilde\zeta_1$,
where $\tilde\zeta_1$ admits a smooth extension to the boundary, and similarly for
$\zeta_2$. Inserting this, we obtain
\betagin{equation}\lambdabel{PB-LHS}
\rho^2g(\zeta_1,\zeta_2)+\rho^N\big(\rho^2g(\zeta_1,\tilde\zeta_2)+\rho^2g(\tilde\zeta_1,\zeta_2)+
\tilde\mu(\zeta_1,\zeta_2)\big)+\mathcal O(\rho^{N+1}).
\end{equation}
Since this has to equal \eqref{PB-RHS}, we conclude that
\betagin{equation}\lambdabel{tilde-mu-main}
\tilde\mu(\zeta_1,\zeta_2)=\mu(\zeta_1,\zeta_2)-\rho^2g(\zeta_1,\tilde\zeta_2)-
\rho^2g(\tilde\zeta_1,\zeta_2)+2\tilde v\rho^2g(\zeta_1,\zeta_2)+\mathcal O(\rho).
\end{equation}
The key observation now is that the computation of $\tilde\zeta_1$ and $\tilde\zeta_2$
essentially reduces to the computation of the vector field $\tilde\eta$ which has the
property that $\eta=\timesi+\rho^N\tilde\eta$. As a first step, we claim that for a
vector field $\zetata$ which is tangent to the boundary along the boundary, we have
$\Psii^*\zetata\sigmam_{N+1}\zetata$. This can of course be proved locally, so we can use
local charts obtained from a collar construction as in the proof of Theorem
\ref{thm3.5}. These have $r$ as one coordinate and $\eta$ as the corresponding
coordinate vector field. We first consider the case that $\zeta$ is the coordinate
vector field $\partial_i$ associated to one of the boundary coordinates. Of course,
$0=[\eta,\partial_i]$ and pulling back along $\Psii$, we conclude that
$0=[\timesi,\Psii^*\partial_i]$. Now by Lemma \ref{lem3.4}, we know that $\timesi\sigmam_N\eta$
and $\Psii^*\partial_i\sigmam_N\partial_i$, and we express this via
$\timesi=\eta+r^N\tilde\eta$ and $\Psii^*\partial_i=\partial_i+r^N\tilde\zetata$, where
$\tilde\eta$ and $\tilde\zeta$ admit a smooth extension to the boundary. Plugging these
expressions into the Lie bracket and using that $\partial_i\cdot r=0$ and $\eta\cdot
r=1$, we conclude that
$$
0=[\timesi,\Psii^*\partial_i]=[\eta,\partial_i]+Nr^{N-1}\tilde\zeta+\mathcal O(r^N).
$$
This shows that $\tilde\zeta$ vanishes along the boundary and hence
$\Psii^*\partial_i\sigmam_{N+1}\partial_i$. Now a general vector field $\zetata$ that is
tangent to the boundary along the boundary can be be written as $f\eta+\sum
f_i\partial_i$ for arbitrary smooth functions $f_i$ and a smooth function $f$ which
is $\mathcal O(r)$. Thus the general version of our claim follows readily since
$\Psii^*\eta\sigmam_N\eta$, $f_i\circ\Psii\sigmam_{N+1}f_i$ and $f\circ\Psii\sigmam_{N+1}f$.
Now for any vector field $\zeta$ that is smooth up to the boundary, the difference
$\zeta-d\rho(\zeta)\eta$ is smooth up to the boundary and tangent to the boundary along
the boundary. Of course $\Psii^*(d\rho(\zeta)\eta)=(d\rho(\zeta)\circ\Psii)\timesi$ and so this
equals $d\rho(\zeta)\timesi+\mathcal O(\rho^{N+1})$. Thus, writing
$\zeta=d\rho(\zeta)\eta+(\zeta-d\rho(\zeta)\eta)$ and pulling back, we get
\betagin{equation}\lambdabel{tilde-zeta}
\rho^N\tilde\zeta=\Psii^*\zeta-\zeta=d\rho(\zeta)(\timesi-\eta)+\mathcal O(\rho^{N+1}).
\end{equation}
To compute the difference $\timesi-\eta$, we first use $r=e^v\rho$ to conclude that
$r_j=e^v\rho_j+e^v\rho v_j$ and $v_j=N\rho^{N-1}\tilde v\rho_j+\mathcal O(\rho^N)$. This
shows that $r_j=\rho_j(1+(N+1)\rho^N\tilde v)+\mathcal O(\rho^{N+1})$. Next, by
definition $\eta^i=e^{-2v}\rho^{-2}h^{ij}r_j$ and
$$
e^{-2v}(1+(N+1)\rho^N\tilde v)=1+(N-1)\rho^N\tilde v+\mathcal O(\rho^{N+1}).
$$
Now \eqref{hinv} shows that
$$
\rho^{-2}h^{ij}=\rho^{-2}g^{ij}-\rho^N(\rho^{-2}g^{ik}\mu_{k\ell}\rho^{-2}g^{\ell
j})+\mathcal O(\rho^{N+1}).
$$
Putting all this together, we see that
\betagin{equation}\lambdabel{eta-xi}
\eta^j-\timesi^j=\rho^N((N-1)\tilde v\timesi^j-\rho^{-2}g^{jk}\mu_{k\ell}\timesi^\ell)+\mathcal
O(\rho^{N+1}).
\end{equation}
Dividing the negative of the right hand side by $\rho^N$ and contracting with
$\rho^2g_{ij}$, we obtain $-(N-1)\tilde v\rho_j+\mu_{j\ell}\timesi^{\ell}$. Using this
and \eqref{tilde-zeta}, we can write \eqref{tilde-mu-main} in abstract index notation
as
\betagin{equation}\lambdabel{tilde-mu-main2}
\tilde\mu_{ij}=\mu_{ij}+2(N-1)\tilde v\rho_i\rho_j-\rho_i\mu_{j\ell}\timesi^\ell
-\rho_j\mu_{i\ell}\timesi^\ell +2\tilde v\rho^2g_{ij}+\mathcal O(\rho),
\end{equation}
which together with \eqref{tildev} exactly gives the claimed formula.
\end{proof}
Using this, we can easily deduce that appropriate combinations of the cocycles
constructed in Sections \ref{3.1} and \ref{3.2} remain unchanged if one of the two
metrics involved is pulled back by a diffeomorphisms that is asymptotic to the
identity. Since we are dealing with the situation of the classical mass here, we have
to specialize to the case that $N=n$.
\betagin{cor}\lambdabel{cor3.7}
In our usual setting, of $\circverline{M}=M\cup\partial M$, let $\mathcal G$ be an equivalence
class of ALH metrics on $M$ for the relation $\sigmam_{n-2}$. Let $c_1$ and $c_2$ be the
cocycles constructed in Sections \ref{3.1} and \ref{3.2}, respectively, and let $c$
be a constant multiple of $\tfrac1nc_1+\tfrac12c_2$. Then $c$ defines a cocyle on
$\mathcal G$ that has the property that for metrics $g,h\in\mathcal G$ and any diffeomorphism
$\Phii\in \circperatorname{Diff}_0^{n+1}(\circverline{M})$, we get $c(g,h)=c(g,\Phii^*h)$.
\end{cor}
\betagin{proof}
We fix $g\in\mathcal G$ and a local defining function $\rho$ that is adapted to
$g$. Then we show that for $c=\tfrac1nc_1+\tfrac12c_2$ and the alignment
diffeomorphism $\Psii$ obtained from Theorem \ref{thm3.7}, we get
$c(g,\Psii^*h)=c(g,h)$. The last part of Theorem \ref{thm3.6} shows that $\Psii^*h$
is the unique metric in the the orbit of $h$ under $\circperatorname{Diff}_0^{n+1}(\circverline{M})$ which is
aligned to $g$ with respect to $\rho$. Hence applying the construction of Theorem
\ref{thm3.7} to $\Phii^*h$ for arbitrary $\Phii\in \circperatorname{Diff}_0^{n+1}(\circverline{M})$, we also
have to arrive at $\Psii^*h$, which then implies the result.
Using the formulae in parts (2) of Propositions \ref{prop3.1} and \ref{prop3.2}, we
see that to prove our claim it suffices to show that the boundary value of
$$
\tfrac{n^2-1}{2n}\rho^{-2}g^{ij}\mu_{ij}-\tfrac12\timesi^i\mu^0_{ij}\timesi^j
$$
coincides with the boundary value of the analogous expression formed from
$\tilde\mu_{ij}$. Inserting
$\mu^0_{ij}=\mu_{ij}-\tfrac{1}{n}\rho^{-2}g^{k\ell}\mu_{k\ell}\rho^2g_{ij}$ and
using that $\timesi^i\rho^2g_{ij}\timesi^j=1$ on a neighborhood of $\partial M$, we see that
our expression equals
$$
\tfrac{n}2\rho^{-2}g^{ij}\mu_{ij}-\tfrac12\timesi^i\mu_{ij}\timesi^j.
$$
Contracting $\rho^{-2}g^{ij}$ into formula \eqref{alignment-diffeo} (for the case
$N=n$) and multiplying by $\tfrac{n}2$, we obtain
$$
\tfrac{n}2\rho^{-2}g^{ij}\tilde\mu_{ij}=\tfrac{n}2\rho^{-2}g^{ij}\mu_{ij}-
\tfrac12\timesi^i\mu_{ij}\timesi^j.
$$
By alignment, $\timesi^i\tilde\mu_{ij}=0$, so this proves our claim.
\end{proof}
\subsection{From relative to absolute invariants}\lambdabel{3.8}
So far, we have not imposed any restriction on the equivalence class
$\mathcal G$ of metrics beyond the fact that it consists of
ALH-metrics. We next show that assuming that $\mathcal G$ locally contains
metrics that are hyperbolic (i.e.~have constant sectional curvature
$-1$), one can use our construction to obtain an invariant for
(single) metrics in $\mathcal G$. This assumption of course implies that
the conformal infinity $[\mathcal G]$ on $\partial M$ is conformally flat,
but as we shall see below, it does not impose further restrictions on
the topology of $\circverline{M}$.
The key step toward this are results on the uniqueness of hyperbolic
metrics with prescribed infinity that are discussed in Chapter 7 of
\cite{FeffGr}. These build on results in \cite{SkenSol} and are
related to the work in \cite{Epstein}.
\betagin{thm}\lambdabel{thm3.8}
Consider our usual setting, of $\circverline{M}=M\cup\partial M$, and an equivalence class
$\mathcal G$ of ALH metrics on $M$ for the relation $\sigmam_{n-2}$. Assume that for each
$x\in\partial M$ there is an open neighborhood $U$ of $x$ in $\circverline{M}$ and a metric $g$
in $\mathcal G$ that is hyperbolic (i.e.\ has constant sectional curvature $-1$) on
$U$. Let $c$ be any constant multiple of the cocycle $\tfrac1n c_1+\tfrac12 c_2$ from
Corollary \ref{cor3.7}.
Then for an open subset $U$ as above, and two metrics $g_1,g_2\in\mathcal
G$ that are hyperbolic on $U$, we get $c(g_1,h)|_{U\cap\partial
M}=c(g_2,h)|_{U\cap\partial M}$ for any $h\in\mathcal G$. Hence these
quantities fit together to a well defined section
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$, thus defining a map
$c$ from $\mathcal G$ to tractor valued differential forms. This is equivariant under
diffeomorphisms preserving $\mathcal G$ in the sense that for
$\Phii\in\circperatorname{Diff}_{\mathcal G}(\circverline{M})$, we obtain
$$
c(\Phii^*h)=(\Phii_\infty)^*c(h).
$$
Here $\Phii_{\infty}:=\Phii|_{\partial M}\in\circperatorname{Conf}(\partial M)$ and in the right hand
side we use the standard action of conformal isometries on tractor valued differential forms.
\end{thm}
\betagin{proof}
Suppose that $g_1,g_2\in\mathcal G$ are hyperbolic on $U$. Then we can apply
Proposition 7.4 of \cite{FeffGr} (see also the discussion of the proof of this
result in \cite{FeffGr}) to their restrictions to $U$. This implies that there is a
neighborhood $V$ of $U\cap\partial M$ in $U$ and a diffeomorphism $\Psii:V\to V$
which restricts to the identity on $U\cap\partial M$ such that
$g_1|_{V}=\Psii^*(g_2|_V)$. Theorem \ref{thm3.5} and Corollary \ref{cor3.7} then
immediately imply that $c(g_1,h)$ and $c(g_2,h)$ coincide on $U\cap\partial M$. It
is then clear that we obtain the map $c$ as claimed.
The equivariancy of $c$ can be proved locally. So we take $\Phii\in\circperatorname{Diff}_{\mathcal
G}(\circverline{M})$ and let $\Phii_\infty$ be its restriction to the boundary. Given
$x\in\partial M$ we find an open neighborhood $U$ of $x$ in $\circverline{M}$ and a metric
$g\in\mathcal G$ such that $g|_U$ is hyperbolic. Now $\Phii^{-1}(U)$ is an open
neighborhood of $\Phii^{-1}(x)$ in $\circverline{M}$ and $\Phii^*g|_{\Phii^{-1}(U)}$ is
hyperbolic on $\Phii^{-1}(U)$. Thus we can compute $c(\Phii^*h)$ as
$c(\Phii^*g,\Phii^*h)$ on $\Phii^{-1}(U)$, and by Proposition \ref{prop3.3} this
coincides with $(\Phii_\infty)^*(c(g,h)|_U)$. Since $c(g,h)|_U=c(h)|_U$, this implies the
claim.
\end{proof}
Suppose that $x\in\partial M$ and $U$ is an open neighborhood of $x$
in $\circverline{M}$ such that $\mathcal G$ contains a metric $g$ which is
hyperbolic on $U$. Then of course the conformal class $[\mathcal G]$ has
to be flat on $U\cap\partial M$. In particular, the assumptions of
Theorem \ref{thm3.8} imply that $(\partial M,[\mathcal G])$ is conformally
flat, which in turn imposes restrictions on $\partial M$. However, if
we are given a manifold $\circverline{M}$ with boundary $\partial M$ and a flat
conformal structure on $\partial M$, then there always is a class $\mathcal G$
of conformally compact metrics on $M$, for which the assumptions of
Theorem \ref{thm3.8} are satisfied, and hence we obtain an invariant for
single metrics in $\mathcal G$.
Indeed, Proposition 7.2 of \cite{FeffGr} (see also the discussion on
p.\ 72 of that reference) shows that there is a hyperbolic metric $g$
on some open neighborhood of $\partial M$ in $\circverline{M}$ which induces the
given boundary structure. Then of course $g$ determines an equivalence
class $\mathcal G$ of conformally compact ALH-metrics on $M$ for which all
the assumptions of Theorem \ref{thm3.8} are satisfied.
We want to point out that it is not clear whether the condition of
conformal flatness in Theorem \ref{thm3.8} is of a fundamental
nature. What one would need in more general situations is a class of
``model metrics'' in $\mathcal G$ which can be characterized well enough
to obtain ``uniqueness up to diffeomorphism'' in a form as used in the
proof of Theorem \ref{thm3.8}. An obvious idea is to assume that $\mathcal
G$ contains at least one Einstein metric (which is a condition that is
stable under diffeomorphism) and then look at appropriate classes of
Einstein metrics in $\mathcal G$. In general, the Einstein condition is
certainly not enough to pin down a metric up to diffeomorphism,
compare with the non-uniqueness issues for the ambient
metric. However, it is well possible that there are situations in
which additional (geometric) conditions can be imposed to ensure
uniqueness.
\subsection{Recovering mass}\lambdabel{3.9}
We now show that in the special case of hyperbolic space that, by an
integration process of our cocycles, we can recover the mass for
asymptotically hyperbolic metrics as introduced by Wang \cite{Wang}
and Chru\'{s}iel-Herzlich \cite{Chrusciel-Herzlich}. To do this, we
specialize to the case that $\circverline{M}$ is an open neighbourhood of the
boundary $S^{n-1}$ in the closed unit ball and that $\mathcal G$ is the
equivalence class of (the restriction to $M$ of) the Poincar\'e metric
which we denote by $g$ here. This of course implies that $[\mathcal G]$ is
the round conformal structure on $S^{n-1}$ and that $\mathcal G$ satisfies
the conditions of Theorem \ref{thm3.8}. Hence we get a map $c:\mathcal
G\to \Omega^{n-1}(S^{n-1},\mathcal TS^{n-1})$ as described there.
If $n\geq
4$, conformal flatness of the round metric on $S^{n-1}$ implies that
the tractor connection $\nabla^{\mathcal T}$ is flat. Moreover, since
$(\partial M,[\mathcal G])$ is the homogeneous model of conformal
structures, the tractor bundle $\mathcal T\partial M$ admits a global
trivialisation by parallel sections. This extends to the case $n=3$
with the tractor connection on $S^2$ constructed as discussed in
Section \ref{2.5}. Indeed, since $g$ is conformally flat and Einstein,
the ambient tractor connection is flat and the scale tractor $I^A$ is
parallel on all of $\circverline{M}$. By Proposition \ref{2.5}~ $\partial M$ is
totally umbilic in $\circverline{M}$, and so the second fundamental form with
respect to any metric conformal to $g$ is pure trace. Using a scale
with vanishing mean curvature, as in Section \ref{2.5}, the second
fundamental form actually vanishes. Hence the ambient Levi Civita
connection restricts to the Levi Civita connection on the boundary and
by definition we use the restriction of the ambient Schouten tensor in
the construction of the tractor connection on the boundary in Section
\ref{2.5}. Hence formula \eqref{trac-conn} directly implies that the
boundary tractor connection coincides with the restriction of the
ambient tractor connection, so it is flat since the normal tractor is
parallel. The trivialisation by parallel sections then works exactly
as in higher dimensions.
This easily implies that, fixing an orientation on $\partial M$, there
is a well defined integral that associates to each form
$\circmega\in\Omega^{n-1}(\partial M, \mathcal T \partial M)$ a parallel section of
$\mathcal T\partial M$. Indeed, on $\partial M$ we can take a global frame
$\{s_i\}$ of $\mathcal T \partial M$ consisting of parallel sections,
expand $\circmega$ as $\sum_i\circmega_is_i$ with $\circmega_i\in\Omega^{n-1}(\partial M)$
and then define $\int_{\partial M}\circmega:=\sum_i(\int_{\partial M}\circmega_i)
s_i$. Of course, any other parallel frame consists of linear
combinations of the $s_i$ with constant coefficients, so the result is
independent of the choice of parallel frame.
Now the boundary tractor metric induces a tensorial map $\Omega^{n-1}(\partial
M,\mathcal T \partial M)\times\Gamma(\mathcal T \partial M )|\to\Omega^{n-1}(\partial M)$,
which we write as $(\circmega,s)\mapsto\lambdangle \circmega,s\rangle$. Observe that
the definition of the integral readily implies that for any parallel
section $s\in\Gamma(\mathcal T \partial M)$ on $\partial M$, we obtain $\lambdangle
\int_{\partial M}\circmega,s\rangle=\int_{\partial
M}\lambdangle\circmega,s\rangle$. In particular, the coefficients of
$\int_{\partial M}\circmega$ with respect to a parallel frame can be
computed as an ordinary integral over an $(n-1)$-form. Given a metric
$h\in\mathcal G$, we can in particular apply this to $c(h)$ and we will to
show that, after appropriate normalization, $\int_{\partial M}c(h)$
recovers the mass of $h$.
We will work on $\circverline{M}$ with the extension of the conformal class of
$g$, which we again denote by $[g]$. Since $g$ is conformally flat,
this leads to a flat tractor connection and $[g]$ restricts to $[\mathcal
G]$ on $\partial M$. Hence any parallel section $s\in\Gamma(\mathcal
T\partial M)$ can be extended to a parallel section of $\mathcal T\circverline{M}$
on a neighborhood of $\partial M$. (In fact, also $\mathcal T\circverline{M}$ is
globally trivialized by parallel sections, but we don't really need
this here.) It is well known that parallel sections of the standard
tractor bundle always are in the image of the tractor
$D$-operator. Denoting by $\sigma\in\Gamma(\mathcal E[1])$ the density
determined by $g$, $D^A(\sigma)$ is parallel since $g$ is Einstein. We
know from Proposition \ref{prop2.6}, that this parallel tractor spans
the tractor normal bundle along the boundary.
On the other hand, on $M$, $\sigma$ is nowhere vanishing. Thus on $M$, we can write any
parallel section of $\mathcal T\circverline{M}$ as $D^A(V\sigma)$ for some smooth function $V:M\to\mathbb
R$. Now $\nabla^{\mathcal T}_aD^A(V\sigma)=0$ is a differential equation on $V$ that can be
easily computed from formulae \eqref{D-def} and \eqref{trac-conn} in the scale $g$,
and using that $\mbox{\textsf{P}}_{ab}$ is a multiple of $g_{ab}$. This shows that
$\nabla_a\nabla_b V$ must have vanishing trace-free part, and it is well known that,
since $D^A(\sigma)$ is parallel, this is also sufficient for $D^A(V\sigma)$ being
parallel (see e.g. \cite{BEG}).
To obtain parallel sections that lead to boundary tractors along
$\partial M$, we can require in addition that $D^A(V\sigma)$ is orthogonal to
$D^A(\sigma)$. One immediately verifies that this condition is equivalent to requiring
that $V$, in addition, satisfies $\Delta V=-2\mbox{\textsf{P}} V=nV$. The two required properties
then can be equivalently encoded as a single equation, the KID (``Killing initial
data'') equation
\betagin{equation}\lambdabel{KID}
\nabla_a\nabla_b V-g_{ab}\Delta V+(n-1)g_{ab}V=0.
\end{equation}
Hence we see that, via $\tfrac1n D^A(V\sigma)$ on $M$, solutions to this
equations parametrise those parallel tractors on $\circverline{M}$, which lie in
$\mathcal T\partial M$ along $\partial M$. But on the other hand,
solutions to this equation parametrise the mass integrals in the
classical approach to the AH version of mass, which was our original
motivation for looking for a tractor description.
\betagin{thm}\lambdabel{thm3.9}
Let $\circverline{M}$ be an open neighborhood of $\partial M=S^{n-1}$ in the closed unit ball,
let $\mathcal G$ be the equivalence class of the restriction of the hyperbolic
(Poincar\'e) metric $g$ to $M$. For a metric $h\in\mathcal G$ consider
$c(h)\in\Omega^{n-1}(\partial M,\mathcal T\partial M)$ as in Theorem \ref{thm3.8}. For a
solution $V$ of the KID equation \eqref{KID}, let $s_V\in\Gamma(\mathcal T\partial M)$ be
the parallel section obtained as the boundary value of $\frac{1}{n}
D^A(V\sigma)\in\Gamma(\mathcal T\circverline{M})$ (with respect to $[g]$).
Then $\lambdangle\int_{\partial M} c(h),s_V\rangle$ coincides with the mass integral
associated to $V$ in \cite{Chrusciel-Herzlich}.
\end{thm}
\betagin{proof}
Using the conformal class $[g]$ on $\circverline{M}$ we have constructed $c(h)=c(g,h)$ as the
boundary value of $\star_g\alpha$ for a certain $\mathcal T\circverline{M}$-valued one-form $\alpha$ on
$M$. Likewise, for a solution $V$ of \eqref{KID}, the parallel boundary tractor $s_V$
is the restriction to $\partial M$ of a parallel section $\tilde s_V$ of $\mathcal
T\circverline{M}$. As we have noted already, $\lambdangle\int_{\partial M}
c(h),s_V\rangle=\int_{\partial M}\lambdangle c(h),s_V\rangle$. This integrand is the
boundary value of $\lambdangle\star_g\alpha,\tilde s_V\rangle$, which equals
$\star_g\lambdangle\alpha,\tilde s_V\rangle$ by definition. Since this form is smooth up to
the boundary, its integral over $\partial M$ equals the limit as $\epsilon\to 0$ of the
integrals over the level sets $S_{\epsilon}=\{x:\rho(x)=\epsilon\}$. Since the mass integral
associated to $V$ is also expressed via a one-form, it suffices to compare that
one-form to $\lambdangle\alpha,\tilde s_V\rangle$. In this comparison, we may work up to
terms that vanish along the boundary after application of $\star_g$ and hence up to
$\mathcal O(\rho^{n-1})$, cf.\ the proof of Proposition \ref{prop3.1}.
We use the description of the mass integral associated to $V$ from
\cite{Michel}, it is shown in that reference that this agrees with the
original mass integral introduces in \cite{Chrusciel-Herzlich}. The
mass integrand associated to a solution $V$ of the KID equation
\eqref{KID} is given by
$$
V(\nabla^i\lambdambda_{ia}-\nabla_a\circperatorname{tr}(\lambdambda))-g^{ij}\lambdambda_{ia}\nabla_jV+
\circperatorname{tr}(\lambdambda)\nabla_aV,
$$
where $\lambda_{ij}=h_{ij}-g_{ij}$, the hyperbolic metric $g$ is used to raise and lower
indices and to form traces, and $\nabla$ is the Levi-Civita connection of $g$.
Decomposing $\lambdambda_{ij}=\lambdambda^0_{ij}+\frac{1}{n}\circperatorname{tr}(\lambdambda)g_{ij}$, this becomes
\betagin{equation}\lambdabel{Michel}
(V\nabla^i\lambdambda^0_{ia}-\lambdambda^0_{ia}\nabla^iV)+
\tfrac{n-1}n(\circperatorname{tr}(\lambdambda)\nabla_aV-V\nabla_a\circperatorname{tr}(\lambdambda)).
\end{equation}
Choosing a defining function $\rho$ adapted to $g_{ij}$, we then obtain
$\lambda^0_{ij}=\rho^{n-2}\mu^0_{ij}$ and $\circperatorname{tr}(\lambda)=\rho^n\mu$ for the quantities
introduced in and below equation \eqref{h-g} (for $N=n$). Note that $\mu^0_{ij}$ and
$\mu$ are smooth up to the boundary. We also know from above that $\sigma V$ is the
projection of the parallel tractor $\tilde s_V=\frac{1}{n}D(\sigma V)$, so this is
smooth up to the boundary. Since $\sigma$ is a defining density for $\partial M$, it
follows that $\rho V$ is smooth up to the boundary.
Now we analyse the two parts of \eqref{Michel} separately, starting with the part
involving $\circperatorname{tr}(\lambda)$. Here we have the advantage that covariant derivatives are only
applied to smooth functions (and not to tensor fields), so the fact that $\nabla$ is
not smooth up to the boundary does not matter. Since $\rho V$ is smooth up to the
boundary,
$\rho\nabla_a\rho V$ is $\mathcal
O(\rho)$. Writing $\rho_a$ for $d\rho$ as before, we compute this as $\rho_a\rho
V+\rho^2\nabla_aV$. Thus $\rho^2\nabla_aV=-\rho_a\rho V+\mathcal O(\rho)$ and in
particular is smooth up to the boundary. Using this, we obtain
$$
\circperatorname{tr}(\lambdambda)\nabla_aV=\rho^n\mu\nabla_aV=-\rho^{n-2}\rho_a\rho V\mu +\mathcal
O(\rho^{n-1}).
$$
Similarly, $V\nabla_a\circperatorname{tr}(\lambda)=V\nabla_a\rho^n\mu=n\rho^{n-2}(\rho
V)\rho_a\mu+O(\rho^{n-1})$. Hence the second part in \eqref{Michel} simply gives
$-\frac{n^2-1}{n}\rho^{n-2}(V\rho)\rho_a\mu+O(\rho^{n-1})$.
Analyzing the second summand in the first part of \eqref{Michel} is similarly
easy. This writes as
$$ -\rho^{-2} g^{ij}\rho^n\mu^0_{ia}\nabla_jV=\rho^{n-2} \rho^{-2}
g^{ij}\rho_j\mu^0_{ia}(\rho V)+\mathcal O(\rho^{n-1}).
$$
For the first summand in \eqref{Michel}, the analysis is slightly more
complicated. This can be written
$V\rho\rho^{-2}g^{ij}\rho\nabla_i\rho^{n-2}\mu^0_{ja}$ and hence equals
\betagin{equation}\lambdabel{tech}
V\rho \rho^{-2}g^{ij}\rho^{n-2}((n-2)\rho_i\mu^0_{ja}+\rho\nabla_i\mu^0_{ja}).
\end{equation}
Since in the last summand, we apply a covariant derivative to a tensor
field, we have to change to a connection that admits a smooth
extension to the boundary in order to analyse the boundary
behavior. Hence we change from the Levi-Civita connection $\nabla$ of
$g_{ij}$ to the Levi-Civita connection $\bar\nabla$ of $\bar
g_{ij}:=\rho^2g_{ij}$, which has this property. For the usual
conventions, as used in \cite{BEG}, the one form $\Upsilon_a$ associated to
this conformal change is given by $\Upsilon_a=\tfrac{\rho_a}{\rho}$. The
relevant formula for the change of connection is then given by
$$
\nabla_i\mu^0_{ja}=\bar\nabla_i\mu^0_{ja}+2\Upsilon_i\mu^0_{ja}+\Upsilon_j\mu^0_{ia}+
\Upsilon_a\mu^0_{ij}-\Upsilon_k\bar g^{k\ell}\mu^0_{\ell a}\bar g_{ij}-\Upsilon_k\bar
g^{k\ell}\mu^0_{j\ell}\bar g_{ia},
$$
This immediately shows that $\rho\nabla_i\mu^0_{ja}$ admits a
smooth extension to the boundary and its boundary value can be
obtained by dropping the first summand in the right hand side of this
formula and replacing each occurrence of $\Upsilon$ in the remaining terms
by $d\rho$, so $\Upsilon_i$ becomes $\rho_i$ and so on. Inserting this back
into \eqref{tech}, we get a contraction with $\bar g^{ij}$. This kills
the term involving $\mu^0_{ij}$ by trace-freeness, while all other
terms become multiples of $\bar g^{ij}\rho_i\mu^0_{ja}$. The factors
of the individual terms are $2$, $1$, $-n$, and $-1$, respectively, so
we'll get a total contribution of $(2-n)\bar
g^{ij}\rho_i\mu^0_{ja}$. This actually implies that \eqref{tech} is
$\mathcal O(\rho^{n-1})$. Hence we finally conclude that \eqref{Michel}
equals
\betagin{equation}\lambdabel{Michel-asymp}
\rho^{n-2}(\rho V)\left(\rho^{-2}g^{ij}\rho_j\mu^0_{ia}-
\tfrac{n^2-1}n\rho_a\mu\right)+O(\rho^{n-1}).
\end{equation}
Now recall the formula for $\nabla_bD^A(\tau-\sigma)$ from part (1) of Proposition
\ref{prop3.1}, taking into account the definition of $\mathbf{X}^A$. This shows that,
up to $\mathcal O(\rho^{n-1})$, $\nabla_bD^A(\tau-\sigma)$ is given by inserting
$\tfrac{n^2-1}2\rho^{n-2}\rho_b\mu\rho\sigma^{-1}$ into the bottom slot of a
tractor. Pairing this with $\frac{1}{n}D^B(\sigma V)$ using the tractor metric, we
simply simply obtain the product of $\sigma V$ with this bottom slot, i.e.\
$$
\tfrac{n^2-1}2\rho^{n-2}\rho_b\mu(\rho V)+\mathcal O(\rho^{n-1}).
$$
Analyzing the formula for $S(\sigma(h_{ij}-g_{ij})^0)$ from part (1) of Proposition
\ref{prop3.2} we similarly see that the pairing of this with $\frac{1}{n}D^B(\sigma V)$
gives
$$
-\rho^{n-2}\rho^{-2}g^{ij}\rho_j\mu^0_{ia}(\rho V)+\mathcal O(\rho^{n-1}).
$$
But this exactly tells us that, up to $\mathcal O(\rho^{n-1})$, \eqref{Michel-asymp}
equals $\lambdangle\alpha,\tilde s_V\rangle$, where $\alpha$ corresponds to
$c=-\tfrac{2}nc_1-c_2$, which is one of the cocycles identified in Theorem
\ref{thm3.8}.
\end{proof}
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\textcolor{blue}egin{eqnarray}gin{document}
\title{Density analysis of BSDEs}
\textcolor{blue}egin{eqnarray}gin{abstract}
In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution $(Y,Z)$ to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin-Viens formula, we provide estimates on these densities.
\end{abstract}
{\noindentindent \textit{Key words:} BSDEs; Malliavin Calculus; Density analysis; Nourdin-Viens' Formula; PDEs.
}
\noindentindent
{\noindentindent \textit{AMS 2010 subject classification:} Primary: 60H10; Secondary: 60H07.
\noindentrmalsize
}
\tableofcontents
\section{Introduction}
In recent years the field of Backward Stochastic Differential Equations (BSDEs) has been a subject of growing interest in stochastic calculus, as these equations naturally arise in stochastic control problems in Finance, and as they provide Feynman-Kac type formulae for semi-linear PDEs (\cite{PardouxPeng_92}). Before going further let us recall that a solution to a BSDE is a pair of \textit{regular enough} (in a sense to be made precise) predictable processes $(Y,Z)$ such that
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintro}
Y_t=\timesi+\int_t^T h(s,Y_s,Z_s)ds -\int_t^T Z_s dW_s, \quad t\in [0,T],
\end{equation}
where $W$ is a one-dimensional Brownian motion, $h$ is a predictable process and $\timesi$ is a $\mathcal{F}_T$-measurable random variable (with $(\mathcal{F}_t)_{t \in [0,T]}$ the natural completed and right-continuous filtration generated by $W$). Since it is generally not possible to provide an explicit solution to \eqref{eq:BSDEintro}, except for instance when $h$ is a linear mapping of $(y,z)$, one of the main issues, especially regarding the applications is to provide a numerical analysis for the solution of a BSDE. This calls for a deep understanding of the regularity of the solution processes $Y$ and $Z$. The classical regularity related to obtaining a numerical scheme for the solution $(Y,Z)$ is the so-called \textit{path regularity} for the $Z$ component originally studied in \cite{Ma_Zhang_PTRF02}. In this paper, we aim at studying another type of regularity namely, we focus on the law of the marginals of the random variables $Y_t$, $Z_t$ at a given time $t$ in $(0,T)$. More precisely, we are interested in providing sufficient conditions which ensure the existence of a density (with respect to the Lebesgue measure) for these marginals on the one hand, and in deriving some estimates on these densities on the other hand. This type of information on the solution is of theoretical and of practical interest since the description of the tails of the (possible) density of $Z_t$ would provide more accurate estimates on the convergence rates of numerical schemes for quadratic growth BSDEs (qgBSDEs in short), that is when $h$ in \eqref{eq:BSDEintro} has quadratic growth in the $z$-variable, as noted in \cite{DosReisPhD}.
Before reviewing the results available in the literature and the one we derive in this paper, we would like to illustrate with the two following simple examples that the existence and the estimate of densities issues for BSDEs are very different from the one concerning the classical (forward) SDEs. For instance consider the following very particular case of \eqref{eq:BSDEintro} given by:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintrobis}
Y_t=W_1+\int_t^T (s-W_s) ds -\int_t^T Z_s dW_s, \quad t\in [0,1], \; (T=1).
\end{equation}
This equation should be extremely simple in the sense that the driver $h$ does not depend on $(Y,Z)$, and indeed it can be solved explicitly to get that:
$$ Y_t = W_t \left(-\frac12 + 2t -\frac{t^2}{2}\right), \quad t\in [0,1]. $$
Hence $Y_t$ is a Gaussian random variable for every time $t$ in $(0,2-\sqrt{3})$, then $Y_{2-\sqrt{3}} = 0$, and for $t$ in $(2-\sqrt{3},1]$, $Y_t$ is Gaussian distributed once again. This illustrates the difficulty of the problem and somehow shows how it is different from the study of forward SDEs. This example, even though it is very simple is pretty insightful and will be studied as Example \ref{exemple} in Section \ref{section:lip}.
Concerning the density estimates, the backward case brings here also, significant differences with the forward case as the following example illustrates. Consider the following equation:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:BSDEintroter}
Y_t=W_1^3+\int_t^T 3 W_s ds -\int_t^T Z_s dW_s, \quad t\in [0,1], \; (T=1),
\end{equation}
which can be solved explicitly:
$$ Y_t = W_t^3 +6 W_t(1-t), \quad Z_t = 3 W_t^2 + 6(1-t), \; t\in [0,1],$$
from which we deduce that both $Y_t$ and $Z_t$ admits a density with respect to the Lebesgue's measure for $t$ in $(0,1]$. However, it is clear that neither the law of $Y_t$ nor the one of $Z_t$ admits Gaussian tails. This example will be considered in Section \ref{section:densY} as Example \ref{rem.toostringent}.
Coming back to the
general problem of existence of densities for the marginal laws of $Y$ and $Z$, it is worth mentioning that this issue has been pretty few studied in the literature, since up to our knowledge only references \cite{AntonelliKohatsu,AbouraBourguin} address this question. The first results about this problem have been derived in \cite{AntonelliKohatsu}, where the authors provide existence and smoothness properties of densities for the marginals of the $Y$ component only and when the driver $h$ is Lipschitz continuous in $(y,z)$. Note that two kinds of sufficient conditions for the existence of a density for $Y$ are derived in \cite{AntonelliKohatsu}: the so-called \textit{first-order} (cf. \cite[Theorem 3.1]{AntonelliKohatsu}) and \textit{second-order} (see \cite[Theorem 3.6]{AntonelliKohatsu}) conditions. Concerning the $Z$ component, much less is known since existence of a density for $Z$ has been established in \cite{AbouraBourguin} only under the condition that the driver is linear in $z$. This constitutes, to our point of view, a major restriction since up to a Girsanov transformation this case basically reduces to the situation where the driver does not depend on $z$. Nonetheless, in \cite{AbouraBourguin}, estimates on the densities of the laws of $Y_t$ and $Z_t$ are given using the Nourdin-Viens formula.
In this paper we revisit and extend the results of \cite{AntonelliKohatsu,AbouraBourguin} by providing sufficient conditions for the existence of densities for the marginal laws of the solution $Y_t,Z_t$ (with $t$ an arbitrary time in $(0,T)$) of a qgBSDE with a terminal condition $\timesi$ in \eqref{eq:BSDEintro} given as a deterministic mapping of the value at time $T$ of the solution to a one-dimensional SDE, together with some estimates on these densities. The results concerning the Lipschitz case, \textit{i.e.} when the generator $h$ is Lipschitz, are presented in Section \ref{section:lip}. As recalled above, the case where $h$ is Lipschitz continuous in $(y,z)$ has been investigated in \cite{AntonelliKohatsu} for the $Y$ component only, where the authors have derived two types of sufficient conditions. However, we provide as Example \ref{exemple} a counter-example to \cite[Theorem 3.6]{AntonelliKohatsu} which is devoted to the second-order conditions. This is due to an inefficiency in the proof that can be easily fixed by making a small change in a key quantity in the statement of the result. Hence, we propose a new version of this result as Theorem \ref{AKmodifie}. Then, we gather in Section \ref{section:lip:z} the first existence results of a density for the $Z$ component for Lipschitz BSDEs. Concerning the quadratic case, studied in Section \ref{section:quadratic}, we propose sufficient conditions for the existence of a density first for the $Y$ component of qgBSDEs (in Section \ref{section:quadratic:y}), then for the $Z$ component of qgBSDEs (in Section \ref{section:quadratic:z}). We would like to stress once more at this stage that concerning the existence of a density for the $Y$ component, only the Lipschitz case was known and concerning the control variable $Z$, only the case of linear drivers in $z$ was studied (see \cite[Theorem 4.3]{AbouraBourguin}) up to now, which makes our result a major improvement on the existing literature. Finally, we derive in Section \ref{section:densY}, density estimates for the marginal laws of $Y$ and $Z$ using the Nourdin-Viens formula, and taking advantage of the connection between the solution to a Markovian BSDE and the solution to its associated semi-linear PDE. Note that contrary to \cite{AbouraBourguin}, we do not assume that the Malliavin derivative of $Y$ (or $Z$) to be bounded which is, from our point of view, a too stringent assumption (as illustrated in Example \ref{rem.toostringent}) both from the theoretical and practical point of view. Indeed, such an assumption leads to Gaussian tails for the densities of $Y$ or $Z$.
However, even in seemingly benign situations, we will see that it is not generally the case for BSDEs, and unlike most of the literature, we have obtained tail estimates which are {\it not} Gaussian. This might be seen as a significant difference between BSDEs and diffusive equations (i.e. with an initial condition) like SDEs or SPDEs for instance \cite{Kohatsu_03,Kohatsu_2003b,Nualart_Quer}.
\noindentindent
Before going further, we would like to explain why our results are quite relevant for financial applications and some stochastic control problems. Most of problems in portfolio management, utility maximization or risk sensitive control (see \textit{e.g.} \cite[Section 4.2]{elk_hamadene_matoussi}) can be essentially reduced to study a qgBSDE. Let us present two examples.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[1.] Assume that a financial agent wants to maximize her utility under constraints, \textit{i.e.} her investment strategies are restricted to a specific closed set $C$, it was proved in \cite{elk_rouge} and \cite{HIM} that her optimal strategies are essentially given through the $Z$ component of a qgBSDE of the form
$$Y_t=\timesi +\int_t^T h(s,Z_s)ds -\int_t^T Z_s dW_s, \; \forall t\in [0,T]\; \mathbb{P}-a.s.$$
with
$$h(s,z):= -z\theta_s - \frac{|\theta_s|^2}{2\alpha}+\frac\alpha2 \text{dist}_{C}^2\left(z+\frac{\theta_s}{\alpha}\right),$$ where $\alpha$ denotes the risk aversion of the investor and $\theta$ is the market price of risk, and where $\text{dist}_C(x)$ denotes the Euclidean distance between $x$ and $C$. Hence, if one obtains a criterion providing density existence for the $Z$ component solution to a qgBSDE with estimates on its tails, then one gets crucial information to study the behaviors of optimal strategies for utility maximization problems. For example, since $Z$ essentially gives the optimal quantity of money which should be invested in the risky asset, being able to estimate the probability that $Z$ becomes large is particularly meaningful in risk management. Besides, the control of the tails of the density of $Z$ could give important information concerning the rate of convergence for numerical schemes to solve numerically BSDEs, so as to compute optimal strategies (see \cite{ImkellerDosreis, ChassagneuxRichou}). For instance, one can check directly that if $\theta$ above is deterministic, $C$ is smooth (that is its boundary is a $C^2$ Jordan arc), and $\timesi=g(W_T)$, where $g$ is any bounded function such that its second-order derivative is non-negative almost everywhere and positive on a set of positive Lebesgue measure (for instance a smoothed butterfly spread), then Theorem \ref{thm_density_z_quadra} below applies and $Z_t$ admits a density for all $t\in(0,T]$.
\item[2.] Assume now that a controller, sensitive to risk, wants to maximize on the control set $ \mathcal{U}$
\textcolor{blue}egin{eqnarray}gin{equation}\label{RSP} J(u):= \mathbb{E}^u\left[ \exp\left(\theta \int_0^T H(s,X_{\cdot},u_s)ds + g(X_T)\right)\right], \; u \in \mathcal{U},\end{equation}
where $\theta$ denotes the sensitiveness of the controller with respect to risk and $X$ denotes a solution to a classical SDE. This is the classical risk sensitive control problem introduced in \cite{jacobson}. Hence, this risk sensitive control problem can be rewritten in term of the well-known risk entropic measure (see \cite{barrieu_elk} for more details). Then, according to \cite[Theorem 4.3]{elk_hamadene_matoussi}, one can find a maximizer $u^\star$ of \eqref{RSP} which is essentially given by a process $Z^\star$ which is the second component of the solution to the following qgBSDE
$$ Y_t^\star= g(X_T)+\int_t^T h(s,x_{\cdot}, Z_s^\star, u^\star_s)+\frac12 |Z^\star_s|^2ds -\int_t^T Z_s^\star dW_s, \;\forall t\in [0,T], \; \mathbb{P}-a.s.,$$
where $h$ is the Hamiltonian process (which is given explicitly in terms of $H$), which is such that $z\longmapsto h(s,x_{\cdot}, z, u_s)+\frac12 |z|^2$ has a quadratic growth for every $s\in [0,T]$ and $u\in \mathcal{U}$. Again, our results give information on the density of $Z^\star$ and thus on the law of the optimal control which is important for obtaining qualitative properties of this optimal control as well as for numerical approximations.
\end{itemize}
\section{Preliminaries}
\subsection{General notations}
In this paper we fix $T \in (0,\infty)$. Let $W:=(W_t)_{t\in [0,T]}$ be a standard one-dimensional Brownian motion on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, and we denote by $\mathbb{F}:=(\mathcal{F}_t)_{t\in [0,T]}$ the natural (completed and right-continuous) filtration generated by $W$. We denote by $\lambda$ the Lebesgue measure on $\mathbb{R}$ and we set for any $p\in [1,+\infty]$, $L^p(\mathbb{P}):=L^p(\Omega,\mathcal{F}_T,\mathbb{P})$ and denote by $\No{\cdot}_p$ the associated norm.
We denote by $\mathcal{C}_b(\mathbb{R}^n)$ ($n \geq 1$) the set of functions from $\mathbb{R}^n$ to $\mathbb{R}$ which are infinitely differentiable with bounded partial derivatives. Similarly, for any $n\geq 1$ and any $p\in\N^*$, we denote by $\mathcal C^p(\R^n)$ the set of functions $f:\R^n\rightarrow \R$ which are $p$-times continuously differentiable. For $f$ in $\mathcal{C}_b(\mathbb{R}^n)$, we set $f_{x_{i_1}\cdots x_{i_n}}$ the $n$-th partial derivative with respect to the variables $x_{i_1},\ldots,x_{i_k}$ with $i_1+\ldots+i_k=n$. For a differentiable mapping $f:\mathbb{R} \longrightarrow \mathbb{R}$, we denote $f'$ its derivative in place of $f_x$. Let us denote, for any $(p,q)\in\N^2$, by ${\cal C}^{p,q}$ the space of functions $f:[0,T]\times\R\rightarrow \R$ which are $p$-times differentiable in $t$ and $q$-times differentiable in space with partial derivatives continuous (in $(t,x)$).
Finally, we introduce the following norms and spaces for any $p\geq 1$. $\mathbb S^p$ is the space of $\mathbb R$-valued, continuous and $\mathbb F$-progressively measurable processes $Y$ s.t.
$$\No{Y}^p_{\mathbb S^p}:=\mathbb E\left[\underset{0\leq t\leq T}{\sup} |Y_t|^p\right]<+\infty.$$
$\mathbb S^\infty$ is the space of $\mathbb R$-valued, continuous and $\F$-progressively measurable processes $Y$ s.t.
$$\No{Y}_{\mathbb S^\infty}:=\underset{0\leq t\leq T}{\sup}\No{Y_t}_\infty<+\infty.$$
$\mathbb H^p$ is the space of $\mathbb R$-valued and $\F$-predictable processes $Z$ such that
$$\No{Z}^p_{\mathbb H^p}:=\mathbb E\left[\left(\int_0^T\abs{Z_t}^2dt\right)^{\frac p2}\right]<+\infty.$$
$\rm{BMO}$ is the space of square integrable, continuous, $\mathbb R$-valued martingales $M$ such that
$$\No{M}_{\rm{BMO}}:=\underset{\tau\in\mathcal T_0^T}{{\rm ess \, sup}}\No{\mathbb E_\tau\left[\left(M_T-M_{\tau}\right)^2\right]}_{\infty}<+\infty,$$
where for any $t\in[0,T]$, $\mathcal T_t^T$ is the set of $\F$-stopping times taking their values in $[t,T]$. Accordingly, $\mathbb H^2_{\rm{BMO}}$ is the space of $\mathbb R$-valued and $\F$-predictable processes $Z$ such that
$$\No{Z}^2_{\mathbb H^2_{\rm{BMO}}}:=\No{\int_0^.Z_sdB_s}_{\rm{BMO}}<+\infty.$$
\subsection{Elements of Malliavin calculus and density analysis}
In this section we introduce the basic material on the Malliavin calculus that we will use in this paper. Set $\mathfrak{H}:=L^2([0,T],\mathcal B([0,T]),\lambda)$, where $\mathcal B([0,T])$ is the Borel $\sigma$-algebra on $[0,T]$, and let us consider the following inner product on $\mathfrak{H}$
$$\langle f,g\rangle :=\int_0^Tf(t)g(t)dt, \quad \forall (f,g) \in \mathfrak{H}^2,$$
with associated norm $\No{\cdot}_{\mathfrak{H}}$. Let $\mathcal{S}$ be the set of cylindrical functionals, that is the set of random variables $F$ in $L^2(\mathbb{P})$ of the form
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:cylindrical}
F=f(W_{t_1},\ldots,W_{t_n}), \quad (t_1,\ldots,t_n) \in [0,T]^n, \; f \in \mathcal{C}_b(\mathbb{R}^n), \; n\geq 1.
\end{equation}
For any $F$ in $\mathcal{S}$ of the form \eqref{eq:cylindrical}, the Malliavin derivative $D F$ of $F$ is defined as the following $\mathfrak{H}$-valued random variable:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:DF}
D F:=\sum_{i=1}^n f_{x_i}(W_{t_1},\ldots,W_{t_n}) \textbf{1}_{[0,t_i]}.
\end{equation}
It is then customary to identify $DF$ with the stochastic process $(D_t F)_{t\in [0,T]}$. Denote then by $\mathbb{D}^{1,2}$ the closure of $\mathcal{S}$ with respect to the Sobolev norm $\|\cdot\|_{1,2}$, defined as:
$$ \|F\|_{1,2}:=\mathbb{E}\left[|F|^2\right] + \mathbb{E}\left[\int_0^T |D_t F|^2 dt\right]. $$
In an iterative way, one may define $D^n F$ (for $n\geq 1$) as the following $\mathfrak{H}^{\odot n}$-valued random variable:
$$ D^nF:=D (D^{n-1} F), $$
where $\mathfrak{H}^{\odot n}$ denotes the $n$-times symmetric tensor product of $\mathfrak{H}$. We refer to \cite{Nualartbook} for more details.
We recall the following criterion for absolute continuity of the law of a random variable $F$ with respect to the Lebesgue measure.
\textcolor{blue}egin{eqnarray}gin{Theorem}[Bouleau-Hirsch, see e.g. Theorem 2.1.2 in \cite{Nualartbook}]\label{BH}
Let $F$ be in $\mathbb{D}^{1,2}$. Assume that $\|DF\|_{\mathfrak{H}} >0$, $\mathbb{P}-$a.s. Then $F$ has a probability distribution which is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.
\end{Theorem}
Let $F$ such that $\|DF\|_{\mathfrak{H}} >0$, $\mathbb{P}-$a.s., then the previous criterion implies that $F$ admits a density $\rho_F$ with respect to the Lebesgue measure. Assume there exists in addition a measurable mapping ${\mathfrak{H}at P}i_F$ with ${\mathfrak{H}at P}i_F : \mathbb{R}^{\mathfrak{H}} \rightarrow \mathfrak{H}$, such that $DF={\mathfrak{H}at P}i_F(W)$, then we set:
\textcolor{blue}egin{eqnarray}gin{equation}\label{gzt}
g_F(x):=\int_0^\infty e^{-u} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_F(W),\widetilde{{\mathfrak{H}at P}i_F^u}(W)\rangle_{\mathfrak{H}}] \Big{|} F-\mathbb{E}(F)=x\right] du, \ x\in \mathbb{R},
\end{equation}
where $\widetilde{{\mathfrak{H}at P}i_F^u}(W):={\mathfrak{H}at P}i_F(e^{-u}W+\sqrt{1-e^{-2u}}W^*)$ with $W^*$ an independent copy of $W$ defined on a probability space $(\Omega^*,\mathcal{F}^*,\mathbb{P}^*)$, and $\mathbb{E}^*$ denotes the expectation under $\mathbb{P}^*$ (${\mathfrak{H}at P}i_F$ being extended on $\Omega\times \Omega^*$). We recall the following result from \cite{NourdinViens}.
\textcolor{blue}egin{eqnarray}gin{Theorem}[Nourdin-Viens' formula]\label{thm_NourdinViens}
$F$ has a density $\rho$ with the respect to the Lebesgue measure if and only if the random variable $g_F(F-\mathbb{E}[F])$ is positive a.s.. In this case, the support of $\rho$, denoted by $\text{supp}(\rho)$, is a closed interval of $\mathbb{R}$ and for all $x \in \text{supp}(\rho)$:
\textcolor{blue}egin{eqnarray}gin{equation*}
\rho(x)=\frac{\mathbb{E}(|F-\mathbb{E}[F]|)}{2g_{F}(x-\mathbb{E}[F])}\exp{\left( -\int_0^{x-\mathbb{E}[F]} \frac{udu}{g_F(u)} \right)}.
\end{equation*}
\end{Theorem}
\subsection{The FBSDE under consideration}
\label{sub:X}
In this paper, we consider a FBSDE of the form:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle X_t= X_0+\int_0^t b(s,X_s) ds +\int_0^t \sigma(s,X_s)dW_s,\ t\in[0,T],\ \mathbb{P}-a.s.\\
\displaystyle Y_t = g(X_T) +\int_t^T h (s,X_s,Y_s,Z_s) ds -\int_t^T Z_s dW_s, \ t\in [0,T],\ \mathbb{P}-a.s.,
\end{cases}
\end{equation} with $X_0$ a given real constant. We denote by $\mathfrak S(X_t)$ the support of the law of $X_t$ under $\mathbb P$, that is to say the smallest closed subset $A$ of $\mathbb{R}$ such that $\mathbb{P}(X_t\in A)=1$.
Throughout this paper we will make the following standing assumption on the process $X$ in \eqref{edsr}.
\textbf{Standing assumptions on $X$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(X)]
$b,\sigma : [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ are continuous in time and continuously differentiable in space for any fixed time $t$ and such that there exist $k_b,k_\sigma >0$ with
$$|b_x(t,x)|\leq k_b,\ |\sigma_x(t,x)|\leq k_\sigma, \text{ for all $x\in\R$}.$$
Besides $b(t,0), \sigma(t,0)$ are bounded functions of $t$ and there exists $c>0$ such that for all $t\in [0,T]$ $$0<c\leq |\sigma(t,\cdot)|, \ \lambda(dx)-a.e.$$
\end{itemize}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{densite_x}According to Theorem 2.1 in \cite{FournierPrintems}, $(X)$ implies that for all $t\in(0,T]$, the law of $X_t$, denoted by $\mathcal{L}(X_t)$, has a density with respect to the Lebesgue measure.
\end{Remark}
Our results will obviously need conditions on the parameters $g$, $h$ which appear in the backward component of \eqref{edsr}. More precisely, one can distinguish between two regimes which call for two different analyses: the case where $h$ exhibits Lipschitz growth in its variables (developed in Section \ref{section:lip}), and the case where $h$ has quadratic growth in the $z$ variable (studied in Section \ref{section:quadratic}). We start with the Lipschitz situation.
\section{The Lipschitz case}
\label{section:lip}
In this section, we focus on the solution $(Y,Z)$ of FBSDE \eqref{edsr} under a Lipschitz-type assumption on the driver $h$. The problem of existence of a density for the marginal laws of $Y$ has been first studied in \cite{AntonelliKohatsu}, when the generator $h$ is assumed to be uniformly Lipschitz continuous in $y$ and $z$. We first recall in Section \ref{sub:lipprel} some general results on Lipschitz FBSDEs, then we review in Section \ref{section:lip:y} the results from \cite{AntonelliKohatsu}. Next, we point out an inefficiency in \cite[Theorem 3.6]{AntonelliKohatsu} by providing a counter example to this result, and we make precise how this small flaw can be corrected, and propose a precised version of it as Theorem \ref{AKmodifie}. Finally, in Section \ref{section:lip:z}, we study the existence of a density for the marginal laws of $Z$ when the generator $h$ of the BSDE satisfies Assumption $($L$)$.
\subsection{Generalities on Lipschitz FBSDEs}
\label{sub:lipprel}
We start by making precise as Assumption $($L$)$ the Lipschitz condition on $h$ and the associated condition on the terminal condition $g$. We set:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(L)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is such that $\mathbb{E}[g(X_T)^2]<+\infty$.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that there exist $(k_x,k_y,k_z)\in(\R_+^*)^3$ such that for all $(t,x_1,x_2,y_1,y_2,z_1,z_2) \in [0,T]\times \mathbb{R}^6$,
$$ |h(t,x_1,y_1,z_1)-h(t,x_2,y_2,z_2)|\leq k_x|x_1-x_2|+k_y|y_1-y_2|+k_z|z_1-z_2|.$$
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
Before going to the density analysis of the $Y$ and $Z$ components we recall briefly well-known facts about existence, uniqueness and Malliavin differentiability for the system \eqref{edsr} which can be found in \cite{PardouxPeng,ElkarouiPengQuenez}.
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{PardouxPeng,ElkarouiPengQuenez}]$($Existence and uniqueness$)$\label{propex}
Under Assumptions $(X)$ $($that we recall is given in Section \ref{sub:X}$)$ and $(L)$, there exists a unique solution $(X,Y,Z)$ in $\mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{H}^2$ to the FBSDE \eqref{edsr}.
\end{Proposition}
Concerning the Malliavin differentiability of $(X,Y,Z)$, it can obtained (see \cite{PardouxPeng} and \cite[Remark of Proposition 5.3]{ElkarouiPengQuenez}) under the following assumptions:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(D1)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is differentiable, $\mathcal{L}(X_T)-$a.e., $g$ and $g'$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\item[(D2)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is twice differentiable, $\mathcal{L}(X_T)-$a.e., $g$, $g'$ and $g''$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is twice continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\end{itemize}
Note that (D1) ensures that $Y$ is Malliavin differentiable, whereas $(D2)$ ensures it is twice Malliavin differentiable. As it will be made more clear below, since $Z$ can be represented as a Malliavin trace of $Y$, the fact that $Y$ is twice Malliavin differentiable entails that $Z$ is Malliavin differentiable.
\textcolor{blue}egin{eqnarray}gin{Proposition}$($Malliavin differentiabiliy$)$ \label{MD}
Under $(X)$, $(L)$ and $(D1)$, we have for any $t\in[0,T]$ that $(X_t,Y_t) \in (\mathbb{D}^{1,2})^2$, $Z_t \in \mathbb{D}^{1,2}$ for almost every $t$, and for all $0<r\leq t \leq T$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_derive}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle D_r X_t=\sigma(r,X_r) + \int_r^t b_x(s,X_s) D_r X_s ds + \int_r^t \sigma_x(s,X_s) D_r X_s dW_s\\
\displaystyle D_r Y_t=g'(X_T)D_rX_T+\int_t^T H(s,D_r X_s,D_r Y_s, D_r Z_s)ds -\int_t^T D_r Z_s dW_s,
\end{cases}
\end{equation}
where $H(s,x,y,z):=h_x(s,X_s,Y_s,Z_s)x+h_y(s,X_s,Y_s,Z_s)y+h_z(s,X_s,Y_s,Z_s)z.$
\end{Proposition}
Notice that BSDE \eqref{edsr_derive} is a linear BSDE, whose solution can be computed using the linearization method (see \cite{ElkarouiPengQuenez}).
We will need extra properties on the Malliavin derivative of $Y$ and $Z$ for which the following result will be crucial. These results rely heavily on the Markovian framework we are working with.
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{MaZhang,IRR}]\label{prop:Markov}
Let Assumptions $(X)$, $(L)$ and $(D1)$ hold, then there exists a map $u:[0,T] \times \mathbb{R} \longrightarrow \mathbb{R}$ in ${\cal C}^{1,2}$ such that
$$Y_t =u(t,X_t), \quad t\in [0,T], \; \mathbb{P}-a.s.$$
In addition, $Z$ admits a continuous version given by
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:Zu'}
Z_t = u_x(t,X_t) \sigma(t,X_t), \quad t\in [0,T], \; \mathbb{P}-a.s.
\end{equation}
\end{Proposition}
In view of Proposition \ref{prop:Markov}, the chain rule formula implies that $Y_t$ belongs to $\mathbb{D}^{2,2}$ and
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:D2Y}
D^2 Y_t = u_x(t,X_t) D^2 X_t + u_{xx}(t,X_t) (D X_t)^{\otimes 2}, \quad \mathbb{P}-a.s.
\end{equation}
Note that by definition, $Z$ is an element of $\mathbb{H}^2$. As a consequence, for any fixed element $t$ in $[0,T]$, the random variable $Z_t$ is not uniquely defined, which makes the density analysis ill-posed. However, by the previous proposition, $Z$ admits in our framework a continuous version. From now on, we will always consider this version.
The following Lemma is due to Ma and Zhang in \cite[Lemma 2.4]{MaZhang} and to Pardoux and Peng \cite{PardouxPeng_92} for the representation of $Z$ as a Malliavin trace of $Y$ (see \eqref{prop:PP92} below).
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{lemma_gradient_malliavin}
Let Assumptions $(X)$, $(L)$, $(D1)$ and $(D2)$ hold. Then, there exists a version of $(D_r X_t, D_r Y_t, D_r Z_t)$ for all $0<r\leq t\leq T$ which satisfies:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r X_t&=\nabla X_t (\nabla X_r)^{-1}\sigma(r,X_r),\ D_r Y_t&=\nabla Y_t (\nabla X_r)^{-1}\sigma(r,X_r),\ D_r Z_t&=\nabla Z_t (\nabla X_r)^{-1}\sigma(r,X_r),
\end{align*}
\textcolor{blue}egin{eqnarray}gin{equation}
\label{prop:PP92}
Z_t=D_t Y_t:=\lim_{s \nearrow t} D_s Y_t,\ \mathbb{P}-a.s., \textrm{ for } a.e. \; t \in [0,T],
\end{equation}
where $(\nabla X,\nabla Y,\nabla Z)$ is the solution to the following FBSDE:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_gradient}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \nabla X_t= \int_0^t b_x(s,X_s) \nabla X_s ds +\int_0^t \sigma_x(s,X_s)\nabla X_sdW_s,\\
\displaystyle \nabla Y_t = g'(X_T)\nabla X_T +\int_t^T \nabla h(s,{\mathfrak{H}at T}eta_s)\cdot \nabla {\mathfrak{H}at T}eta_s ds -\int_t^T \nabla Z_s dW_s.
\end{cases}
\end{equation}
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{Remark}
Assumptions $(D1)$ and $(D2)$ are linked to the existence of first and second-order Malliavin derivatives for the $Y$ component of the solution of \reff{edsr}. We would like to point out to the reader that we only require the differentiability of $g$, $\mathcal{L}(X_T)-$a.e. Such a relaxation will be particularly useful in the quadratic case $($\textit{i.e.} in Section \ref{section:quadratic}$)$. We emphasize that when we work under Assumption $(X)$, the law of $X_T$ is absolutely continuous with respect to the Lebesgue measure and $X_T$ has finite moments of any order. Thus, thanks to standard approximation arguments, we can show that the usual chain rule formula of Malliavin calculus $($see Proposition 1.2.3. in \cite{Nualartbook}$)$ still holds for the random variable $g(X_T)$, under Assumptions $(D1)$ or $(D2)$.
\end{Remark}
Finally, set the following assumption
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(M)] There exists a function $f\in \mathcal{C}^2(\mathbb{R})$ such that for all $t\in [0,T]$: $X_t=f(t,W_t)$.
\end{itemize}
We obtain the following proposition
\textcolor{blue}egin{eqnarray}gin{Proposition}\label{prop_dy_dz_r}
Under Assumptions $(M)$, $(L)$ and $(D2)$, for all $0<r,s\leq t\leq T$ we have $D_r Y_t=D_s Y_t=Z_t$ and $D_r Z_t=D_s Z_t$, $\mathbb{P}-$a.s.
\end{Proposition}
\textcolor{blue}egin{eqnarray}gin{proof}
Once again we set ${\mathfrak{H}at T}eta_s:=(X_s,Y_s,Z_s)$. We know that for all $0<r\leq t\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r Y_t&=g'(X_T) f'(T,W_T)+\int_t^T (h_x(s,{\mathfrak{H}at T}eta_s)f'(s,W_s)+h_y(s,{\mathfrak{H}at T}eta_s)D_r Y_s+h_z(s,{\mathfrak{H}at T}eta_s)D_r Z_s)ds\\
&\mathfrak{H}space{1cm}- \int_t^T D_r Z_s dW_s. \end{align*}
Then $(D_r Y, D_r Z)$ satisfies a linear BSDE which does not depend on $r$ and by the uniqueness of the solution we deduce that for all $0<r,s\leq t\leq T$ we have $D_r Y_t=D_s Y_t$ and $D_r Z_t=D_s Z_t$, $\mathbb{P}-$a.s. Finally, $D_r Y_t=Z_t$ by \eqref{prop:PP92}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\subsection{Existence of a density for the $Y$ component}
\label{section:lip:y}
We focus in this section on the existence of a density for the marginal laws of the process $Y$ in the Lipschitz case, pursuing the study started in \cite{AntonelliKohatsu}. Towards this goal, we recall first the so-called \textit{first order conditions} introduced in \cite{AntonelliKohatsu}, which are only sufficient, as illustrated in Example \ref{exemple}. We then turn our attention to the \textit{second-order conditions} of Theorem 3.6 in \cite{AntonelliKohatsu}. We point out a (small) inefficiency in the proof of \cite[Theorem 3.6]{AntonelliKohatsu} and provide a corrected version of this result as Theorem \ref{AKmodifie}.
As in \cite{AntonelliKohatsu}, we set for any $A\in{\cal B}(\R)$ (i.e. the Borel $\sigma$-algebra on $\R$), and $t$ in $[0,T]$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:barg}
\underline{g}:= \inf\limits_{x \in \mathbb{R}} g'(x), \quad \underline{g}^A:=\inf\limits_{x\in A} g'(x),\quad \overline{g}:= \sup\limits_{x \in \mathbb{R}} g'(x), \quad \overline{g}^A:=\sup\limits_{x\in A} g'(x),
\end{equation}
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:barh}
\underline{h}(t):=\inf\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),\quad \quad \overline{h}(t):=\sup\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z).
\end{equation}
\textcolor{blue}egin{eqnarray}gin{Theorem}$($First-order conditions \cite[Theorem 3.1]{AntonelliKohatsu}$)$\label{thm_H+H-}
Assume that $(X)$, $(L)$ and $(D1)$ hold. Fix some $t\in(0,T]$ and set $K:=k_b+k_y+k_{\sigma}k_z$. If there exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds
\textcolor{blue}egin{eqnarray}gin{align*}
&(H+)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds\geq0 \\
\displaystyle \underline{g}^Ae^{-\text{sgn}(\underline{g}^A)KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds>0
\end{cases}\\[0.3em]
&(H-)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds\leq 0 \\
\displaystyle \overline{g}^Ae^{-\text{sgn}(\overline{g}^A)KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds<0,
\end{cases}
\end{align*}
then $Y_t$ has a law absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{Remark}
Notice that $\underline{g}$ $($resp. $\overline{g})$ could be equal to $-\infty$ $($resp. $+\infty)$. Then Assumption $(H+)$ $($resp. $(H-))$ cannot be satisfied. Therefore, there is no problem if we allow the extrema of $g$ to take the values $\pm \infty$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{positivite_dy}
In view of the proof of \cite[Theorem 3.1]{AntonelliKohatsu}, one can show that under $(X)$, $(L)$, and $(D1)$ and if $g'\geq 0$ and $\underline{h}(t)\geq 0$ $($resp. $g'\leq 0$ and $\overline{h}(t)\leq 0)$ for $t \in [0,T]$, then for all $0<r\leq t \leq T$, $D_r Y_t \geq 0$ $($resp. $D_r Y_t \leq 0)$ and the inequality is strict if there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T \in A|\mathcal{F}_t)>0$ and $g'_{|A}>0$ $($resp. $g'_{|A}<0)$.
\end{Remark}
Note that neither Condition $(H+)$ nor Condition $(H-)$ are necessary for getting existence of a density as illustrated in the following example.
\textcolor{blue}egin{eqnarray}gin{Example}\label{exemple}
Let $T=1$, $g(x)=x$, $X=W$, $h(s,x,y,z)=(s-2)x$. In this case, $K=0$ and $h_x(s,x,y,z)=s-2$ for all $(x,y,z)\in \mathbb{R}^3$. For any $t$ in $(0,1]$, we have:
\[ \overline{g}=\underline{g}=1,\ \underline{h}(t)=t-2,\ \overline{h}(t)=-1, \]
so that Assumption $(H-)$ is not satisfied. Indeed,
$$ \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds=1-(1-t)=t>0.$$
Similarly, $(H+)$ is not satisfied for any $t \in \left(0, (3-\sqrt{5})/2\right)$ since:
$$ \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds=1+(t-2)(1-t)=-t^2+3t-1,$$ which is negative for $t \in \left(0, (3-\sqrt{5})/2\right)$. We deduce that for $t\in \left(0, (3-\sqrt{5})/2\right)$ neither Assumption $(H+)$ nor Assumption $(H-)$ is satisfied. However, we know that:
\textcolor{blue}egin{eqnarray}gin{align}
\label{eq:counterex}
Y_t &= \mathbb{E}\left[\left.W_1+\int_t^1 (s-2)W_s ds \right| \mathcal{F}_t\right]\noindentnumber\\
&=W_t\left(1+\int_t^1 (s-2)ds\right)=W_t\left(-\frac12+2t-\frac{t^2}{2}\right), \ \forall t\in [0,1],\ \mathbb{P}-a.s.,
\end{align}
which admits a density with respect to the Lebesgue measure except when $t=0$ and $t=2-\sqrt{3}$.
\end{Example}
Notice that in the previous example, the generator does not depend on $z$. In that setting, another result is derived \cite{AntonelliKohatsu}, involving so-called \textit{second order conditions}. There, the authors of \cite{AntonelliKohatsu} benefit from the absence of $z$ in the driver to make a higher order expansion of the Malliavin norm $\int_0^T |D_r Y_t|^2 dr$. The price to pay is that the condition involves a mapping $\tilde{h}$ (see \eqref{eq:htilde} below), which is essentially a sum of derivatives of the driver $h$, which goes beyond the simple derivative $h_x$. However, Example \ref{exemple} provides a counter-example to \cite[Theorem 3.6]{AntonelliKohatsu}. Indeed, the second-order conditions proposed in \cite[Theorem 3.6]{AntonelliKohatsu} entails that $Y_t$ admits a density, when $t\neq \frac12$, so in particular at $t=2-\sqrt{3}$. However from \eqref{eq:counterex}, $Y_{2-\sqrt{3}}=0$. This example proves that \cite[Theorem 3.6]{AntonelliKohatsu} has to be modified. The proof of \cite[Theorem 3.6]{AntonelliKohatsu} is essentially correct, except that in their proof the original Brownian motion $W$ is not a Brownian motion any more under the new measure $\mathbb Q$ defined in \cite[page 275]{AntonelliKohatsu} and need to be replaced by the process $W'_\cdot:=W_\cdot-\int_0^\cdot \sigma_x(s,X_s)ds$ which is a $\mathbb Q$-Brownian motion. This leads to the two extra terms $-(\sigma \sigma_xh_{xx}+z\sigma_xh_{xy})$ in the expression of the mapping \eqref{eq:htilde} below, compareLd to the original expression of $\tilde{h}$ in the statement of \cite[Theorem 3.6]{AntonelliKohatsu}. We refer the reader to Example \ref{counterexample} below and we propose a corrected version of \cite[Theorem 3.6]{AntonelliKohatsu} as Theorem \ref{AKmodifie} (whose proof exactly follows the original one up to the introduction of $W'$), in which the modified second-order conditions are sufficient, and necessary in the special situation of Example \ref{exemple}.
Consider the FBSDE \eqref{edsr} when $h$ does not depend on $z$ and define:
\textcolor{blue}egin{eqnarray}gin{align}
\label{eq:htilde}
\tilde{h}(s,x,y,z):=& -\left( h_{xt}+b h_{xx}-hh_{xy}+\frac12(\sigma^2 h_{xxx}+2z\sigma h_{xxy}+z^2h_{xxy})\right)(s,x,y)\noindentnumber\\
&-\left((h_y+b_x)h_x+\sigma \sigma_xh_{xx}+z\sigma_xh_{xy}\right)
(s,x,y).
\end{align}
$$ \tilde{g}(x):=g'(x)+(T-t)h_x(T,x,g(x)),$$
$$ \underline{\tilde{g}}:=\min\limits_{x\in \mathbb{R}} \tilde{g}(x), \quad \overline{\tilde{g}}:=\max\limits_{x\in \mathbb{R}} \tilde{g}(x),\quad \underline{\tilde{g}}^A:=\min\limits_{x\in A} \tilde{g}(x), \quad \overline{\tilde{g}}^A:=\max\limits_{x\in A} \tilde{g}(x),$$
$$\underline{\tilde{h}}(t):=\min\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z), \ \overline{\tilde{h}}(t):=\max\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z).$$
The following theorem corrects Theorem 3.6 in \cite{AntonelliKohatsu}.
\textcolor{blue}egin{eqnarray}gin{Theorem}$($Second-order conditions \cite[Theorem 3.6]{AntonelliKohatsu}$)$\label{AKmodifie}
Fix some $t\in(0,T]$, assume that $h$ does not depend on $z$, that Assumptions $(X)$, $(L)$ and $(D1)$ hold and set $K:=k_y+k_b$. If there exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds
\textcolor{blue}egin{eqnarray}gin{align*}
&\widetilde{(H+)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{\tilde{g}}e^{-\text{sgn}(\underline{\tilde{g}})KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds\geq0 \\
\displaystyle \underline{\tilde{g}}^Ae^{-\text{sgn}(\underline{\tilde{g}}^A)KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds>0,
\end{cases}\\[0.3em]
&\widetilde{(H-)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{\tilde{g}}e^{-\text{sgn}(\overline{\tilde{g}})KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds\leq 0 \\
\displaystyle \overline{\tilde{g}}^Ae^{-\text{sgn}(\overline{\tilde{g}}^A)KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds<0,
\end{cases}
\end{align*}
then the first component $Y_t$ of the solution of BSDE \eqref{edsr} has a law which is absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{Example}\label{counterexample}
We go back to Example \ref{exemple} with $g\equiv Id.$ and $h(s,x,y,z)=(s-2)x$ which does not depend on $z$. On the one hand, we know from \eqref{eq:counterex} that for all $t\in (0,1]$, the law of $Y_t$ has a density except when $t=0$ or $t=2-\sqrt{3}$. On the other hand, our conditions in Theorem \ref{AKmodifie} read:
\[ \overline{\tilde{g}}=\tilde{\underline{g}}=\tilde{g}(x)=t, \quad \tilde{h}(t,x,y)=\overline{\tilde{h}}(t)=\underline{\tilde{h}}(t)=-1, \quad K=0,
\]
from which $\widetilde{(H+)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 >0,\]
and $\widetilde{(H-)}$ becomes:
\[ t-\int_t^1 (1-s) ds = t-(1-t)+\frac12-\frac{t^2}{2}=-\frac{t^2}{2}+2t-\frac12 <0.\]
We hence conclude, in view of Theorem \ref{AKmodifie}, that the law of $Y_t$ has a density with respect to the Lebesgue measure for every $t\in (0,1] \textcolor{blue}ackslash \{2-\sqrt{3}\}$.
In this particular example, notice that Theorem \ref{AKmodifie} is more accurate than Theorem \ref{thm_H+H-} since Condition $\widetilde{(H+)}$ and Condition $\widetilde{(H-)}$ are sufficient \underline{and} necessary to obtain the existence of a density for $Y$. Finally, we emphasize once more that the counterpart of Condition $\widetilde{(H-)}$ in \cite[Theorem 3.6]{AntonelliKohatsu} gives that whenever $2t-1<0$, $Y_t$ admits a density, which is clearly satisfied for $t=2-\sqrt{3}$. However we know that $Y_{2-\sqrt{3}}=0$.
\end{Example}
\subsection{Existence of a density for the control variable $Z$}
\label{section:lip:z}
We now turn to the problem of existence of a density for the marginal laws of $Z$. This question was studied in \cite{AbouraBourguin} when the generator is linear in $z$, that is to say $h(t,x,y,z)=\tilde{h}(t,x,y)+\alpha z$, which is from our point of view a too stringent assumption since by a Girsanov transformation this equation basically reduces to a BSDE with a generator which does not depend on $z$. We focus here on a general function $h$ satisfying Assumption (L). Consider the two following assumptions
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(C+)] $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\geq 0$ and $h_{xz}= h_{yz}= 0$,
\item[(C-)] $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\leq 0$ and $h_{xz}= h_{yz}= 0$.
\end{itemize}
Let $t\in (0,T]$ and $A\in{\cal B}(\R)$. We set:
$$ \underline{g''}:= \min\limits_{x \in \mathfrak{S}(X_T)} g''(x), \quad \underline{g''}^A:=\min\limits_{x\in \mathfrak{S}(X_T) \cap A} g''(x),\quad \underline{g'}:= \min\limits_{x \in \mathfrak{S}(X_T)} g'(x), \quad \underline{g'}^A:=\min\limits_{x\in \mathfrak{S}(X_T) \cap A} g'(x),$$
$$ \underline{h_{xx}}(t):=\min\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_{xx}(s,x,y,z).$$
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_density_z_lip}
Let Assumption $(X)$, $(L)$ and $(D2)$ hold. Let $0< t \leq T$ and assume moreover
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)\in(0,+\infty)$, such that $\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$,
\item There exists $\overline b\geq 0$, such that $0\leq D_{r,t}^2 X_u\leq \overline{b}$, for all $0<r,t<u\leq T$,
\item $(C+)$ holds
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)-a.e.)$.
\end{itemize}
If there exists a set $A\in \mathcal B(\mathbb{R})$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and such that
$$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0, $$
then, the law of $Z_t$ has a density with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof} Under the assumptions of Theorem \ref{thm_density_z_lip}, we obtain for $0<r,s< t\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}
D_{r,s}^2 Y_t=&\ g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T -\int_t^T D_{r,s}^2 Z_u dW_u\\
&+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_{xy}(u,{\mathfrak{H}at T}eta_u) D_s X_u D_r Y_u\\
&+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Y_u +h_{xy}(u,{\mathfrak{H}at T}eta_u) D_r X_u D_s Y_u+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u\\
&+h_z(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Z_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du.
\end{align*}
Let $\tilde{\mathbb{P}}$ be the probability equivalent to $\mathbb{P}$ such that
\textcolor{blue}egin{eqnarray}gin{equation} \label{tilde_p}
\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}=\exp\left(\int_0^T h_z(s,{\mathfrak{H}at T}eta_s) dW_s-\frac12\int_0^T\abs{h_z(s,{\mathfrak{H}at T}eta_s)}^2ds\right),
\end{equation}
where $h_z$ is bounded thanks to Assumption (L).
Under $\tilde{\mathbb{P}}$ defined by \eqref{tilde_p}, we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T\\
& \mathfrak{H}space{1cm}+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_{xy}(u,{\mathfrak{H}at T}eta_u) D_s X_u D_rY_u\\
&\mathfrak{H}space{2cm}+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,s}^2 Y_u +h_{xy}(u,{\mathfrak{H}at T}eta_u) D_r X_u D_s Y_u+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u\\
&\mathfrak{H}space{2cm}+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|{\cal F}_t\Big].
\end{align*}
By standard linearization techniques, we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T) \\
&\mathfrak{H}space{1cm}+\int_t^T e^{\int_t^u h_y(v,{\mathfrak{H}at T}eta_v)dv}[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u\\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm} + h_{xy}(u,{\mathfrak{H}at T}eta_y) (D_r X_u D_s Y_u +D_s X_u D_r Y_u)\\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm}+h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|\mathcal F_t\Big]. \end{align*}
Then, using Remark \ref{positivite_dy}, Lemma \ref{lemma_gradient_malliavin} and our assumptions we obtain:
\textcolor{blue}egin{eqnarray}gin{align*}
&e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T)\\
&\quad +\int_t^T e^{\int_t^u h_y(v,{\mathfrak{H}at T}eta_v)dv}[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u\\
&\quad+ h_{xy}(u,{\mathfrak{H}at T}eta_y) (D_r X_u D_s Y_u +D_s X_u D_r Y_u)\\
&\quad +h_{yy}(u,{\mathfrak{H}at T}eta_u) D_r Y_u D_s Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\\
&\geq e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du} \left(\mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\right)\geq 0.
\end{align*}
We deduce that:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t\geq&\ \mathbb{E}^{\tilde{\mathbb{P}}}\Big[e^{\int_t^T h_y(u,{\mathfrak{H}at T}eta_u)du}\mathbf{1}_{X_T\in A}(g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T) \\
&+\mathbf{1}_{X_T\in A}\int_t^T e^{-K(u-t)}[h_{xx}(u,{\mathfrak{H}at T}eta_u)\underline{a}^2] du\Big|\mathcal{F}_t\Big]\\
\geq &\ e^{-KT}\left(\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}\right)\mathbb{\tilde{\mathbb{P}}}(X_T\in A | \mathcal{F}_t)\\
& +e^{-KT}\left(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t)\right)\underline{a}^2\mathbb{\tilde{\mathbb{P}}}(X_T\in A | \mathcal{F}_t).
\end{align*}
Using the fact that $D^2 Y_t$ is symmetric, the chain rule formula, \eqref{eq:Zu'} and \eqref{eq:D2Y} and the fact that $\lim_{s \nearrow t} D_{r,s}^2 X_t = \sigma'(t,X_t) D_r X_t$, we have that $ \lim_{s \nearrow t} D^2_{r,s} Y_t =D_r Z_t,$ from which we deduce that $D_r Z_t>0$, $\mathbb{P}-a.s.$ Then according to Bouleau and Hirsch's Theorem, we conclude that the law of $Z_t$ has a density with respect to the Lebesgue measure.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{ab}
Notice that the sign assumption on $D^2X$ can be obtained under the following sufficient conditions.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(X+)$] For any $t\in[0,T]$, the maps $x\longmapsto b(t,x)$ and $x\longmapsto \sigma(t,x)$ are respectively in $\mathcal C^2(\R)$ and $\mathcal C^3(\R)$, and there exists $c>0$ such that $$\sigma\geq c>0,\quad \sigma' \geq 0,\quad \sigma'',\sigma'''\leq 0 \; \text{ and } \; [\sigma,[\sigma,b]]\geq 0,$$ where $[b,\sigma]$ denotes the Lie bracket between $b$ and $\sigma$ defined by $[b,\sigma]:=b'\sigma+\sigma' b$.
\item[$(X-)$] For any $t\in[0,T]$, the maps $x\longmapsto b(t,x)$ and $x\longmapsto \sigma(t,x)$ are respectively in $\mathcal C^2(\R)$ and $\mathcal C^3(\R)$, and there exists $c<0$ such that
$$\sigma\leq c<0,\ \sigma' \leq 0,\ \sigma'',\sigma'''\geq 0\text{ and }[\sigma,[\sigma,b]]\leq 0.$$
\end{itemize}
Indeed, according to the first step of the proof of Theorem 4.3 in \cite{AbouraBourguin}, Condition $(X+)$ $($resp. $(X-))$ ensures that $D^2 X$ is non-negative $($resp. non-positive$)$.\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}
\label{rem:X+-}
One can provide an alternative version of the previous result, whose proof follows the same lines as the one of Theorem \ref{thm_density_z_lip}. Fix $t$ in $(0,T]$, let Assumptions $(L)$, $(X)$ and $(D2)$ hold and assume that there exists $A\in{\cal B}(\mathbb{R})$ such that $\mathbb{P}(\left.X_T\in A \right| \mathcal{F}_t)>0$, and such that one of the two following conditions is satisfied:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] $(X+)$ and $(C+)$ hold true and $g''\geq0, \; g''_{\vert A}>0 \text{ and } g' \geq0, \quad \mathcal{L}(X_T)\text{-a.e.}$
\item[$b)$] $(X-)$ and $(C-)$ hold true and $g''\leq0, \; g''_{\vert A}<0 \text{ and } g' \leq0, \quad \mathcal{L}(X_T)\text{-a.e.},$
\end{itemize}
then, for all $t \in (0,T]$, the law of $Z_t$ has a density with the respect to Lebesgue measure.
\end{Remark}
When Assumption (M) holds, Theorem \ref{thm_density_z_lip} takes a different form as shown below in Theorem \ref{densite_z_f}, mainly because of Proposition \ref{prop_dy_dz_r}. Indeed, consider the following assumptions:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[($\tilde{C}+$)] $h_{zz}\geq 0$ and $h_{xz}=h_{yz}\equiv 0$.
\item[($\tilde{C}-)$] $h_{zz}\leq 0$ and $h_{xz}=h_{yz}\equiv 0$.
\end{itemize}
Under Assumption ($\tilde{C}+)$ or ($\tilde{C}-)$, we recall that:
\textcolor{blue}egin{eqnarray}gin{align*}\label{drz}
D_r Z_t=&\ g''(X_T)|f'(T,W_T)|^2+g'(X_t)f''(T,W_T)\\
&+\int_t^T \Big[h_x(u,{\mathfrak{H}at T}eta_u)f''(u,W_u)+h_y(u,{\mathfrak{H}at T}eta_u) D_{r,t}^2 Y_u \Big]du\\
&+\int_t^T\Big[|f'(u,W_u)|^2 h_{xx}(u,{\mathfrak{H}at T}eta_u)+\left(h_{xy}(u,{\mathfrak{H}at T}eta_u)D_r Y_u+ D_t Y_uh_{xy}(u,{\mathfrak{H}at T}eta_u) \right) f'(u,W_u)\Big]du\\
&+\int_t^T\Big[h_{yy}(u,{\mathfrak{H}at T}eta_u) \underbrace{D_t Y_uD_r Y_u}_{=|Z_u|^2}+\underbrace{D_t Z_u D_r Z_u}_{=|D_r Z_u|^2} h_{zz}(u,{\mathfrak{H}at T}eta_u) \Big]du - \int_t^T D_{r,t}^2 Z_u d\tilde{W}_u,
\end{align*}
with $\tilde{W}:=W-\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) ds$.
We set $\theta=(x,y,z),$ and
\textcolor{blue}egin{eqnarray}gin{align*}
\tilde{h}(t,w,x,y,z,\tilde{z}):= &\ h_{xx}(t,\theta)|f'(t,w)|^2+h_x(t,\theta)f''(t,w)+(h_{yy}(t,\theta)z+2h_{xy}(t,\theta)f'(t,w))z\\
&+h_y(t,\theta)\tilde{z},
\end{align*}
$$ \underline{\tilde{h}}(t)=\min\limits_{(s,w,x,y,z,\tilde{z})\in [t,T]\times \mathbb{R}^5} \tilde{h}(s,w,x,y,z,\tilde{z}), \quad \overline{\tilde{h}}(t)=\max\limits_{(s,w,x,y,z,\tilde{z})\in [t,T]\times \mathbb{R}^5} \tilde{h}(s,w,x,y,z,\tilde{z}).$$
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{densite_z_f}
Assume that $(M)$, $(L)$ and $(D2)$ are satisfied and that there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] Assumption $(\tilde{C}+)$, $ \underline{((g'\circ f) f')'}+(T-t)\underline{\tilde{h}}(t) \geq 0$ and
$\underline{((g'\circ f) f')'_A} +(T-t)\underline{\tilde{h}}(t)>0.$
\item[$b)$] Assumption $(\tilde{C}-)$, $\overline{((g'\circ f) f')'} +(T-t)\overline{\tilde{h}}(t)\leq 0$ and $\overline{((g'\circ f) f')'_A}+(T-t)\overline{\tilde{h}}(t) <0.$
\end{itemize}
Then, the law of $Z_t$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}$.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof}
Using Proposition \ref{prop_dy_dz_r}, we recall that:
\textcolor{blue}egin{eqnarray}gin{align*}
D_r Z_t=&\ g''(X_T)|f'(T,W_T)|^2+g'(X_t)f''(T,W_T)\\
&+\int_t^T \tilde{h}(u,W_u,X_u,Y_u,Z_u,D_r Z_u)+|D_r Z_u|^2 h_{zz}(u) du - \int_t^T D_{r,t}^2 Z_u d\tilde{W}_u,
\end{align*}
where $\tilde{W}:=W-\int_0^\cdot h_z(u,{\mathfrak{H}at T}eta_u) du$.
Then the proof follows exactly the same line as the one of Theorem \ref{thm_density_z_lip}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\section{The quadratic case}
\label{section:quadratic}
We now turn to the quadratic case and provide an extension of both Theorem \ref{thm_H+H-} and Theorem \ref{thm_density_z_lip}. Note however that the assumptions of these theorems do not find immediate counterparts in the quadratic setup since the latter involves the Lipschitz constant of $h$ with respect to the $z$ variable (see Remark \ref{rk:condi}). We also emphasize that existence of densities for the $Y$ and $Z$ components in the quadratic case that we consider here was open until now. We first make precise the quadratic growth setting together with existence, uniqueness and Malliavin differentiability results for these equations in the next section. Then, we investigate respectively in Sections \ref{section:quadratic:y} and \ref{section:quadratic:z} the existence of density for respectively $Y$ and $Z$.
\subsection{Generalities on quadratic FBSDEs}
\label{sub:quadprel}
In contadistinction to the previous section, we will now assume that $h$ exhibits quadratic growth in the $z$ variable. As noted in the introduction, this case is particularly useful for applications, especially in Finance where any pricing and hedging problem on an incomplete market which can be translated into a BSDE analysis will lead to a quadratic BSDE. The precise assumption for dealing with quadratic BSDEs is given as:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(Q)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is bounded.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\triangleright$] There exists $(K,K_z,K_y)\in(\R_+^*)^3$ such that for all $(t,x,y,z) \in [0,T]\times \mathbb{R}^3$
$$\mathfrak{H}space{-3em}|h(t,x,y,z)|\leq K(1+|y|+|z|^2),\ \abs{h_z}(t,x,y,z)\leq K_z(1+|z|),\ \abs{h_y}(t,x,y,z)\leq K_y.$$
\item[$\triangleright$] There exists $C>0$ such that for all $(t,x,y,z_1,z_2) \in [0,T] \times \mathbb{R}^4$
\textcolor{blue}egin{eqnarray}gin{equation}\label{condition_h} |h(t,x,y,z_1)-h(t,x,y,z_2)| \leq C(1+|z_1|+|z_2|) |z_1-z_2|.\end{equation}
\end{itemize}
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
Existence and uniqueness of a solution triplet $(X,Y,Z)$ under Assumption $(Q)$ has been obtained in \cite{Kobylanski}. More precisely:
\textcolor{blue}egin{eqnarray}gin{Proposition}[\cite{Kobylanski}]$($Existence and uniqueness of BSDEs$)$\label{propexq}
Under Assumptions $(X)$ and $(Q)$, there exists a unique solution $(X,Y,Z)$ in $\mathbb{S}^2 \times \mathbb S^\infty\times\mathbb H^2_{\rm BMO}$.
\end{Proposition}
Note that Condition \eqref{condition_h} on the generator $h$ in Assumption $(Q)$ in the one that ensures uniqueness of the solution. Hence, it can be dropped and one can then consider the maximal solution $Y$ of the BSDE, for which our proofs still apply.
Concerning the Malliavin differentiability of the processes $(X,Y,Z)$ it has been obtained in the quadratic case in \cite{AIDR} under the Assumptions $(D1)$ and $(D2)$ (that are defined in Section \ref{sub:lipprel}). Note that Proposition \ref{prop:Markov} still holds true if Assumption $(L)$ is replaced by Assumption $(Q)$. However, although the above proposition is completely proved in \cite{MaZhang} in the Lipschitz case, we did not find a proper reference in the quadratic case, except for \cite{IRR} which proves the result under Assumption $(Q)$, with the exception that $u$ is only shown to be in ${\cal C}^{1,1}$. Nonetheless, one can still obtain the required result by proving that Theorem $3.1$ of \cite{MaZhang} still holds for a BSDE with a driver which is uniformly Lipschitz in $y$ and stochastic Lipschitz in $z$ with a Lipschitz process in $\mathbb H^2_{\rm BMO}$ $($which is exactly the case of the BSDE satisfied by the Malliavin derivative of $Y)$. This can be achieved by following exactly the steps of the proof of Theorem 3.1 in \cite{MaZhang}, where the a priori estimates of their Lemma $2.2$ have to be replaced by those given in Lemma $A.1$ of \cite{IRR}. As in the Lipschitz case, Relation \eqref{eq:D2Y} still holds true under $(Q)$. In addition, as for Proposition \ref{prop:Markov}, the proof of Lemma \ref{lemma_gradient_malliavin} can be extended to the quadratic setting. Finally, Propositions \ref{MD} and \ref{prop_dy_dz_r} are valid if one replaces Assumption $(L)$ by Assumption $(Q)$.
\textcolor{blue}egin{eqnarray}gin{Proposition}$($Malliavin differentiabiliy$)$ \label{MDq}
Under $(X)$, $(Q)$ and $(D1)$, we have for any $t\in[0,T]$ that $(X_t,Y_t) \in (\mathbb{D}^{1,2})^2$, $Z_t \in \mathbb{D}^{1,2}$ for almost every $t$, and for all $0<r\leq t \leq T$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{edsr_derive1}
\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle D_r X_t=\sigma(r,X_r) + \int_r^t b_x(s,X_s) D_r X_s ds + \int_r^t \sigma_x(s,X_s) D_r X_s dW_s\\
\displaystyle D_r Y_t=g'(X_T)D_rX_T+\int_t^T H(s,D_r X_s,D_r Y_s, D_r Z_s)ds -\int_t^T D_r Z_s dW_s,
\end{cases}
\end{equation}
where $H(s,x,y,z):=h_x(s,X_s,Y_s,Z_s)x+h_y(s,X_s,Y_s,Z_s)y+h_z(s,X_s,Y_s,Z_s)z.$
\end{Proposition}
\subsection{Existence of a density for the $Y$ component}
\label{section:quadratic:y}
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_H+H-_quadra}
Fix $t\in (0,T]$ and assume that $(X)$, $(Q)$ and $(D1)$ hold. If there is $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T\in A \ | \ \mathcal{F}_t)>0$ and one of the following assumptions holds $($see Definitions \eqref{eq:barg}-\eqref{eq:barh}$)$
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(Q+)$] $g'\geq 0$ and $g'_{\vert A} >0$, $\mathcal{L}(X_T)-$a.e. and $\underline{h}(t)\geq 0$,
\item[$(Q-)$] $g'\leq 0,\ g'_{\vert A}<0$, $\mathcal{L}(X_T)-$a.e. and $\overline{h}(t)\leq 0$,
\end{itemize}
then $Y_t$ has a law absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof}
To simplify the notations for any $s$ in $[0,T]$, we set ${\mathfrak{H}at T}eta_s:=(X_s,Y_s,Z_s)$. We set $K:=k_b\vee k_y\vee k_\sigma$. We assume that (Q+) is satisfied (the proof with (Q-) follows the same lines, so we omit it). According to Bouleau-Hirsch's criterion, it is enough to show that $\gamma_{Y_t}:=\int_0^T |D_r Y_t|^2 dr >0$, $\mathbb{P}$-a.s. As in the proof of \cite[Theorem 3.6]{AntonelliKohatsu}, we have for $0\leq r \leq t \leq T$, that $D_r Y_t$ writes down as:
\textcolor{blue}egin{eqnarray}gin{equation}\label{eq:y}
D_r Y_t=g'(X_T)D_r X_T+\int_t^T
h_x(s,{\mathfrak{H}at T}eta_s)D_r X_s+h_y(s,{\mathfrak{H}at T}eta_s)D_r Y_s ds +\int_t^T D_r Z_s dW_s.
\end{equation}
From \reff{eq:y}, and following the expression of $\gamma_{Y_t}$ given in \cite[page 271]{AntonelliKohatsu}, we deduce that
$$ \gamma_{Y_t}=\left(\mathbb{E}\left[ g'(X_T)\zeta_T \psi_T +\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds | \mathcal{F}_t \right]\right)^2 (\psi_t^{-1})^2 \int_0^t (\zeta_r^{-1}\sigma(r,X_r))^2 dr,$$
with
$$ \psi_t\zeta_t=\underbrace{e^{\int_0^t (b_x(s,X_s)+h_y(s,{\mathfrak{H}at T}eta_s)+\sigma_x(s,X_s)h_z(s,{\mathfrak{H}at T}eta_s))ds}}_{=:E_t} \underbrace{e^{\int_0^t (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s)) dW_s - \frac12 \int_0^t (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s))^2 ds}}_{=:M_t}.$$
Let $\mathbb{Q}$ the probability measure equivalent to $\mathbb{P}$ with density $\frac{d\mathbb{Q}}{d\mathbb{P}}:=M_T$. Indeed, $M$ is a martingale as $\int_0^\cdot (\sigma_x(s,X_s)+h_z(s,{\mathfrak{H}at T}eta_s)) dW_s$ is a BMO martingale due to the boundedness of $\sigma_x$ (by (X)) and the fact that $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|)$ (by (Q)) and from the BMO property of $\int_0^\cdot Z_s dW_s$ (by Proposition \ref{propex}). We therefore have:
$$ \mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]=M_t\mathbb{E}^{\mathbb{Q}}\left[ g'(X_T) E_T +\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds \Big| \mathcal{F}_t\right].$$
Using (Q+), we know that:
$$g'(X_T)E_T+\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds \geq \underline{g}E_T+\underline{h}(t)\int_t^T E_sds\geq 0. $$
Thus,
\textcolor{blue}egin{eqnarray}gin{align*}
&\mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]\\
&\geq M_t \mathbb{E}^{\mathbb{Q}}\left[ \mathbf{1}_{X_T \in A}\left(g'(X_T) E_T +\int_t^T h_x(s,{\mathfrak{H}at T}eta_s) E_s ds\right) \Big| \mathcal{F}_t\right]\\
&\geq M_t\Big(\underline{g}^Ae^{-2KT}\mathbb{E}^\mathbb{Q}\left[ \mathbf{1}_{X_T\in A} e^{-K\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|ds}\textcolor{blue}ig| \mathcal{F}_t\right]\\
&\mathfrak{H}space{0.9em}+\underline{h}(t)e^{-2KT}(T-t)\mathbb{E}^\mathbb{Q}\left[ \mathbf{1}_{X_T\in A} e^{-K\int_0^T|h_z(s,{\mathfrak{H}at T}eta_s)|ds} \textcolor{blue}ig| \mathcal{F}_t \right]\Big)\\
&\geq M_t\Big(\underline{g}^Ae^{-2KT}\mathbb{E}^\mathbb{Q}\Big[ \mathbf{1}_{X_T\in A} e^{-K\sqrt{T}\sqrt{\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|^2ds}}\textcolor{blue}ig| \mathcal{F}_t\Big]\\
&\mathfrak{H}space{0.9em}+\underline{h}(t)e^{-2KT}(T-t)\mathbb{E}^\mathbb{Q}\Big[ \mathbf{1}_{X_T\in A} e^{-K\sqrt{T}\sqrt{\int_0^T|h_z(s,{\mathfrak{H}at T}eta_s)|^2ds} }\textcolor{blue}ig| \mathcal{F}_t \Big]\Big),
\end{align*}
where the last inequality is due to Cauchy-Schwarz inequality. Besides, according to Assumption (Q), $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|).$ Then, we deduce that $\int_0^T |h_z(s,{\mathfrak{H}at T}eta_s)|^2ds<+\infty, \ \mathbb{P}-$a.s., since $Z\in \mathbb{H}^2$. Hence, $M_t>0$, $\mathbb{P}-$a.s.
Given that the law of $X_T$ is absolutely continuous with respect to the Lebesgue measure, we deduce that $\mathbb{E}\left[ g'(X_T)\psi_T\zeta_T+\int_t^T \psi_s h_x(s,{\mathfrak{H}at T}eta_s)\zeta_s ds \Big|\mathcal{F}_t\right]>0, \ \mathbb{P}-\text{a.s.}$
We conclude using Theorem \ref{BH}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Remark}
Similarly to Remark \ref{positivite_dy}, the proof of Theorem \ref{thm_H+H-_quadra} shows that under $(X)$, $(Q)$, $(D1)$ and if $g'\geq 0$ and $\underline{h}(t)\geq 0$ $($resp. $g'\leq 0$ and $\overline{h}(t)\leq 0)$ for $t \in [0,T]$, then for all $0<r\leq t \leq T$, $D_r Y_t \geq 0$ $($resp. $D_r Y_t \leq 0)$ and the inequality is strict if there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T \in A|\mathcal{F}_t)>0$ and $g'_{|A}>0$ $($resp. $g'_{|A}<0)$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}\label{rk:condi}
Conditions $(Q+)$ and $(Q-)$ are stronger than $(H+)$ and $(H-)$, due to the unboundedness of $h_z$, which prevents us from reproducing the same proof than in \cite{AntonelliKohatsu}. Indeed, in this framework the quantity appearing for instance in $(H+)$ becomes:
$$ \underline{g}e^{-2K\text{sgn}(\underline{g})T}e^{-K\text{sgn}(\underline{g})\int_0^T |h_z(s)|ds}+\underline{h}(t)e^{-2K\text{sgn}(\underline{h}(t))T}\int_t^T e^{-K\text{sgn}(\underline{h}(t))\int_0^s |h_z(s)|ds},$$
whose sign for every $K\geq 0$ depends strongly on those of $g'$ and $h_x$. This is why we must use the stronger conditions $(Q+)$ and $(Q-)$.
\end{Remark}
\textcolor{blue}egin{eqnarray}gin{Remark}
In \cite[Corollary 3.5]{dosdos} comonotonicty conditions on the data of a BSDE under Assumption $(Q)$ are given so that $Z_t \geq 0$, $\mathbb{P}-$a.s., $\forall t \in [0,T]$. In addition, the authors claim that strict comonotonicity entails that $Z_t>0$, which implies by Bouleau-Hirsch criterion that the law of $Y_t$ has a density with respect to the Lebesgue measure. However, we do not understand their proof and it is not true that an increasing mapping which is differentiable has a positive derivative everywhere $($even if one relaxes it by asking for a positive derivative Lebesgue-almost everywhere$)$ and one needs an extra assumption to prove that the derivative does not vanish. Indeed, take any closed set of positive Lebesgue measure with empty interior $($for instance the Smith-Volterra-Cantor set on $\R)$. By Whitney's extension Theorem, there exists a differentiable increasing map whose derivative vanishes on this set.
\end{Remark}
\subsection{Existence of a density for the control variable $Z$}
\label{section:quadratic:z}
In this section, we obtain existence results for the density of $Z$ under Assumption (Q). We actually have exactly the same type of results as in the Lipschitz case with similar proofs, which highlights the robustness and flexibility of our approach. Let us detail first the changes that we have to make.
Under (Q), using the fact that for all $s\in [0,T]$ $|h_z(s,{\mathfrak{H}at T}eta_s)|\leq C(1+|Z_s|)$ and according to Proposition \ref{propex} we deduce that $\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) dW_s$ is a BMO-martingale. Then, according to Theorem 2.3 in \cite{Kazamaki}, the stochastic exponential of $\int_0^\cdot h_z(s,{\mathfrak{H}at T}eta_s) dW_s$ is a uniformly integrable martingale and we can apply Girsanov's Theorem. We also emphasize that in (Q), $g$ is not assumed to be twice continuously differentiable. Indeed, to recover the BMO properties linked to quadratic BSDEs (and thus in order to be able to apply the above reasoning), $g$ needs to be bounded, which is incompatible with g convex (or concave). Nevertheless, there exist terminal conditions $g$ which are twice differentiable almost everywhere on the support of the law of $X_T$ (which is some closed subset of $\R$), such that their second-order derivative have a given sign there. As an example, take $X=W$ and $g(x):= f(x)\mathbf{1}_{x\in[a,b]}+f(a)\mathbf{1}_{x\leq a} + f(b) \mathbf{1}_{x\leq b}$ with $f$ a twice differentiable convex function and $a,b \in \mathbb{R}$.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_density_z_quadra}
Let Assumptions $(X)$, $(Q)$ and $(D2)$ hold. Let $0< t \leq T$ and assume moreover
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)$ s.t., $0<\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$.
\item There exists $\overline b$ s.t., $0\leq D_{r,s}^2 X_u\leq \overline{b}$, for all $0<r,s<u\leq T$.
\item $(C+)$ holds and $h_y\geq 0$.
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)$-a.e.$)$.
\end{itemize}
If there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and such that:
$$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0,$$
then, the law of $Z_t$ has a density with respect to the Lebesgue measure.
\end{Theorem}
\textcolor{blue}egin{eqnarray}gin{proof} As in the proof of Theorem \ref{thm_density_z_lip}, we notice that for all $0<r,t\leq s\leq T$:
\textcolor{blue}egin{eqnarray}gin{align*}D_{r,s}^2 Y_t&=\mathbb{E}^{\tilde{\mathbb{P}}}\Big[g''(X_T)D_r X_T D_s X_T+g'(X_T)D_{r,s}^2 X_T\\
& \mathfrak{H}space{1cm}+\int_t^T[h_x(u,{\mathfrak{H}at T}eta_u)D_{r,s}^2 X_u+h_{xx}(u,{\mathfrak{H}at T}eta_u)D_r X_u D_s X_u+h_y(u,{\mathfrak{H}at T}eta_u)D_r Y_u D_s Y_u \\
&\mathfrak{H}space{1cm}\mathfrak{H}space{1cm}+h_{yy}(u,{\mathfrak{H}at T}eta_u)D^2_{r,s}Y_u+h_{zz}(u,{\mathfrak{H}at T}eta_u)D_r Z_u D_s Z_u] du\Big|{\cal F}_t\Big], \end{align*}
where $\tilde{\mathbb{P}}$ is the equivalent probability measure to $\mathbb{P}$ with density $$\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}:=\exp\left(\int_0^T h_z(u,{\mathfrak{H}at T}eta_u)dW_u-\frac12\int_0^T \abs{h_z(u,{\mathfrak{H}at T}eta_u)}^2du\right),$$ given that $\int_0^\cdot h_z(u,{\mathfrak{H}at T}eta_u)dW_u$ is a BMO-martingale and using Theorem 2.3 in \cite{Kazamaki}. Then the proof is similar to that of Theorem \ref{thm_density_z_lip}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Remark} In order to satisfy the condition in Theorem \ref{thm_density_z_quadra}, there are basically two types of sufficient conditions
\textcolor{blue}egin{eqnarray}gin{itemize}
\item First of all, if the support of the law of $X_T$ is bounded from above, then one can take $g$ to continuously differentiable everywhere, non-decreasing, convex and bounded on this support. Then it suffices to take $h$ to be convex in $x$ as well.
\item However, when the support of the law of $X_T$ is no longer bounded from above, then it is no longer possible to find $g$ which is non-decreasing, bounded and convex on this support. We must therefore allow $g''$ to become non-positive, and the role of $h_{xx}$ becomes then crucial, as it has to be sufficiently positive in order to balance $g''$. As an example, take $X:=W$. Then $\overline a=\underline a =1$ and $\overline b=0$. One can choose $g(x):= \frac{1}{1+x^2}$. Then, there exists a positive constant $M$ such that $-2\leq g''(x)\leq M$ and by choosing $h$ such that $h$ satisfies the assumptions in Theorem \ref{thm_density_z_quadra} and $t\in(0,T)$ such that $\underline{h_{xx}}(t)(T-t)\geq 2$, we deduce that $Z_t$ admits a density.
\end{itemize}
\end{Remark}
We give also a theorem under Assumption (M):
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{densite_z_f_quadra}
Assume that $(M)$, $(Q)$ and $(D2)$ are satisfied and that there exists $A\in{\cal B}(\R)$ such that $\mathbb{P}(X_T\in A | \mathcal{F}_t)>0$ and one of the two following assumptions holds:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$a)$] Assumption $(\tilde{C}+)$, $\underline{((g'\circ f) f')'}+(T-t)\underline{\tilde{h}}(t) \geq 0$ and
$ \underline{((g'\circ f) f')'_A} +(T-t)\underline{\tilde{h}}(t)>0.$
\item[$b)$] Assumption $(\tilde{C}-)$, $\overline{((g'\circ f) f')'} +(T-t)\overline{\tilde{h}}(t)\leq 0$ and $\overline{((g'\circ f) f')'_A}+(T-t)\overline{\tilde{h}}(t) <0.$
\end{itemize}
Then, the law of $Z_t$ is absolutely continuous with respect to the Lebesgue measure.
\end{Theorem}
The proof is the same as the proof of Theorem \ref{densite_z_f} using the BMO property of $\int_0^\cdot Z_s dW_s$, we therefore omit it. We now turn to the simplest case of quadratic growth BSDE and verify that it is covered by our result.
\textcolor{blue}egin{eqnarray}gin{Example}
Let us consider the following BSDE
\[ Y_t=g(W_T)+\int_t^T \frac12 |Z_s|^2ds -\int_t^T Z_s dW_s,\]
where $g$ is bounded. According to Theorem \ref{thm_density_z_quadra} with $\overline{a}=\underline{a}=1$, $\overline{b}=0$ and $h_{xx}=0$, we deduce that for all $t\in (0,T]$, the law of $Z_t$ has a density with respect to the Lebesgue measure if $g''\geq 0$, $\lambda(dx)$-a.e. and if there exists $A\in{\cal B}(\R)$ with positive Lebesgue measure such that $g''_{|A}>0$.
We emphasize that, as a sanity check, this can be verified by direct calculations. Indeed, using the fact that if $F \in \mathbb{D}^{1,2}$ then $D_r (\mathbb{E}[F|\mathcal{F}_t])=\mathbb{E}[D_r F|\mathcal{F}_t] \mathbf{1}_{[0,t)} (r)$ $($see \cite[Proposition 1.2.4]{Nualartbook}$)$ we deduce that if $0\leq r<t\leq T$ then:
$$D_r Y_t = \frac{\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]},$$which does not depend on $r$. Then according to Proposition \ref{prop:PP92},
$Z_t= \frac{\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]}.$
Take $0<r<t\leq T$, then:
$$D_r Z_t= \frac{\mathbb{E}[g''(W_T)e^{g(W_T)}+ |g'(W_T)|^2 e^{g(W_T)} | \mathcal{F}_t]\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]-|\mathbb{E}[g'(W_T)e^{g(W_T)}| \mathcal{F}_t]|^2}{\mathbb{E}[e^{g(W_T)}| \mathcal{F}_t]}.$$
Using Cauchy-Schwarz inequality, if $g''\geq 0$, $\lambda(dx)$-a.e. and if there exists $A\in{\cal B}(\R)$ with positive Lebesgue measure such that $g''_{|A}>0$, we deduce that for all $t\in (0,T]$, $Z_t$ has a density with respect to the Lebesgue measure by Theorem \ref{BH}.
\end{Example}
\section{Density estimates for the marginal laws of $Y$ and $Z$}
\label{section:densY}
Up to now, the density estimates obtained in the literature relied mainly on the fact that the framework considered implied that the Malliavin derivative of $Y$ was bounded. Hence, using the Nourdin-Viens' formula (or more precisely their Corollary 3.5 in \cite{NourdinViens}), it could be showed that the law of $Y$ has Gaussian tails. Although such an approach is perfectly legitimate from the theoretical point of view, let us start by explaining why, as pointed out in the introduction, we think that this is not the natural framework to work with when dealing with BSDEs. Consider indeed the following example.
\textcolor{blue}egin{eqnarray}gin{Example}\label{rem.toostringent}
Let us consider the FBSDE \reff{edsr}, with $T=1$, $g(x):=x^3$, $h(t,x,y,z):=3x$, $b(t,x)=0$, $\sigma(t,x)=1$ and $X_0=0$. Then, simple computations show that the unique solution is given by
$$X_t=W_t,\quad Y_t=W_t^3+6W_t(1-t),\quad Z_t=3W_t^2+6(1-t).$$
Then, both $Y_t$ and $Z_t$ have a law which is absolutely continuous with respect to the Lebesgue measure, for every $t\in(0,1]$, but neither $Y_t$ nor $Z_t$ has Gaussian tails.
\end{Example}
Moreover, when it comes to applications dealing with generators with quadratic growth, assuming that the Malliavin derivative of $Y$ is bounded implies that the process $Z$ itself is bounded as $Z_t=D_t Y_t$, which is seldom satisfied in applications, since in general, one only knows that $Z\in\mathbb{H}^2_{\rm BMO}$.
One of the main applications of the results we obtain in this section is the precise analysis of the error in the truncation method in numerical schemes for quadratic BSDEs, introduced in \cite{ImkellerDosreis} and studied in \cite{ChassagneuxRichou}. We recall that according to Proposition \ref{prop:Markov} there exists a function $v : [0,T]\times \mathbb{R}\longmapsto \mathbb{R}$ in ${\cal C}^{1,2}$ such that $Y_t=v(t,X_t)$ and $Z_t=v_x(t,X_t)\sigma(t,X_t)$. Since we want to study the tails of the laws of $Y$ and $Z$, we will assume from now on that the support of these laws is $\mathbb{R}$, which implies that neither $v$ nor $v'$ is bounded from below or above. Moreover, we emphasize that throughout this section, we will assume that $Y_t$ and $Z_t$ do have a law which is absolutely continuous, so as to highlight the conditions needed to obtain the estimates. Throughout this section we assume that $X_t=W_t$ in \eqref{edsr} (that is $X_0=0, \; \sigma\equiv 1, \; b \equiv 0$).
\subsection{Preliminary results}
We will have to study the asymptotic growth of $v$ and $v_x$ in the neighborhood of $\pm \infty$. To this end, we introduce for any measurable function $f:\R\longrightarrow \R$ the following two kinds of growth rates:
\[ \overline{\alpha_f}:=\inf\left\{\alpha>0,\ \limsup\limits_{|x| \to +\infty} \abs{\frac{f(x)}{x^\alpha}}<+\infty\right\}, \quad \underline{\alpha_f}:=\inf\left\{\alpha>0,\ \liminf\limits_{|x| \to +\infty} \abs{\frac{f(x)}{x^\alpha}}<+\infty\right\}. \]
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{lemme_v-1}
Let $ f \in \mathcal{C}^1(\mathbb{R})$. Assume that for all $x \in \mathbb{R}$, $f'(x)>0$. If $0<\underline{\alpha_f}<+\infty$ then for all positive constant $0<\eta<\underline{\alpha_f}$:
$$\overline{\alpha_{f^{(-1)}}}\leq \frac{1}{\underline{\alpha_f}-\eta},$$
where $f^{(-1)}$ is the inverse function of $f$.
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{proof}
Using the definition of $\underline{\alpha_f}$, we deduce that for all $\eta>0$,
\textcolor{blue}egin{eqnarray}gin{equation*}
\liminf\limits_{|x| \to + \infty} \abs{\dfrac{f(x)}{x^{\underline{\alpha_f}-\eta}}}=\lim\limits_{|x| \to + \infty} \abs{\dfrac{f(x)}{x^{\underline{\alpha_f}-\eta}}}=+\infty.
\end{equation*}
Since $f$ and $f^{(-1)}$ are increasing and unbounded from above and below, we deduce that there exists $\overline{x}>0$ such that for all $x\geq \overline{x}$, $f(x)$ and $f^{(-1)}$ are positive. Then, for all $M>0$, there exists $x_0 \geq \overline{x}$ such that for all $x\geq x_0>0$ and for all $y \geq M x_0^{\underline{\alpha_f}-\eta}\vee \overline x$
\textcolor{blue}egin{eqnarray}gin{align*}
f(x)\geq M x^{\underline{\alpha_f}-\eta}\Longleftrightarrow &\; f\left((y M^{-1})^\frac{1}{\underline{\alpha_v}-\eta}\right)\geq y\Longleftrightarrow(y M^{-1})^{\frac{1}{\underline{\alpha_f}-\eta}} \geq f^{(-1)}(y).
\end{align*}
This implies directly that $\limsup\limits_{y \to +\infty} \abs{\frac{f^{(-1)}(y)}{y^{\frac{1}{\underline{\alpha_f}-\eta}}}} <+\infty.$ The proof is similar when $y$ goes to $ -\infty$.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
It is rather natural to expect that for well-behaved functions $f \in \mathcal{C}^1(\mathbb{R})$, $\overline{\alpha_f}=\underline{\alpha_f}$ and $\overline{\alpha_f}=\overline{\alpha_{f'}}+1$. However, the situation is unfortunately not that clear. First of all, this may not be true if $f$ is not monotone. Indeed, let $f(x):=x^2\sin(x)$, then $\overline{\alpha_f}=\underline{\alpha_f}=2$. Furthermore, the strict monotonicity of $f$ is not sufficient either. Without being completely rigorous, let us describe a counterexample. Consider a function $f$ defined on $\R_+$, equal to the identity on $[0,1]$, which then increases as $x^4$ until it crosses $x\longmapsto x^2$ for the first time, which then increases as $x^{1/2}$ until it crosses $x\longmapsto x$ for the first time and so on. Finally, extend it by symmetry to $\R_-$. Then, it can be checked that $\overline{\alpha_f}=2$, $\underline{\alpha_f}=1$, $\overline{\alpha_{f'}}=3$, $\underline{\alpha_{f'}}=0$.
A nice sufficient condition for the aforementioned result to hold is that $f'$ is a \textit{regularly varying function} (see \cite{BinghamGoldieTeugels} and \cite{Seneta}).
\textcolor{blue}egin{eqnarray}gin{Lemma}\label{prop_regularly}
Assume that $f'$ is equivalent in $+ \infty$ $($resp. in $-\infty)$ to a regularly varying function with Karamata's decomposition $x^\textcolor{blue}egin{eqnarray}ta L_1(x)$ where $L_1$ is slowly varying $($resp. $x^\textcolor{blue}egin{eqnarray}ta L_2(x)$ where $L_2$ is slowly varying$)$ and where $\textcolor{blue}egin{eqnarray}ta>0$. Then
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\rm{(i)}$] $f$ is equivalent in $+ \infty$ $($resp. in $-\infty)$ to a regularly varying function with Karamata's decomposition $x^{\textcolor{blue}egin{eqnarray}ta+1} \widetilde{L_1}(x)$ where $\widetilde{L_1}$ is slowly varying $($resp. $x^{\textcolor{blue}egin{eqnarray}ta+1} \widetilde{L_2}(x)$ where $\widetilde{L_2}$ is slowly varying$)$.
\item[$\rm{(ii)}$] $ \overline{\alpha_f}=\underline{\alpha_f}=\underline{\alpha_{f'}}+1=\overline{\alpha_{f'}}+1.$
\end{itemize}
\end{Lemma}
\textcolor{blue}egin{eqnarray}gin{proof} By Karamata's Theorem (see Theorem 1.5.11 in \cite{BinghamGoldieTeugels} with $\sigma=1$), for any $x_0 \in \mathbb{R}$:
\textcolor{blue}egin{eqnarray}gin{equation}
\label{eq:cvreg}
\frac{xf'(x)}{f(x)-f(x_0)} \longrightarrow \textcolor{blue}egin{eqnarray}ta +1, \ \text{ when } x\longrightarrow +\infty.
\end{equation}
In addition, $f'$ is equivalent to a regularly varying function with Karamata's decomposition $x^\textcolor{blue}egin{eqnarray}ta L_1(x)$ when $x\longrightarrow +\infty$, hence in view of \eqref{eq:cvreg}, there exists a function $\widetilde{L_1}$ (equivalent to a constant times $L_1$ at $+\infty$) slowly varying such that $f$ is equivalent when $x\longrightarrow +\infty$ to a regularly varying function with Karamata's decomposition $x^{\textcolor{blue}egin{eqnarray}ta+1}\widetilde{L_1}(x)$. The same result holds when $x\longrightarrow -\infty$.
We now show (ii). According to Proposition 1.3.6 (v) in \cite{BinghamGoldieTeugels} and (i), we deduce that:
$$ \overline{\alpha_{f}}=\textcolor{blue}egin{eqnarray}ta+1=\underline{\alpha_{f}} \ \text{ and } \ \overline{\alpha_{f'}}=\textcolor{blue}egin{eqnarray}ta=\underline{\alpha_{f'}} .$$
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
\subsection{A general estimate}
From now on, for a map $(t,x)\longmapsto v(t,x)$, $v'(t,x)$ will denote for simplicity the derivative of $v$ with respect to the space variable. Before enonciating a general theorem which gives us density estimates for the tails of the law of random variables of the form $v(t,W_t)$ and will be used to obtain estimates for the laws of $Y_t$ and $Z_t$, we set some
constants in order to simplify the notations in Theorem \ref{thm_estime_y} below.
\paragraph*{List of constants} Let $\alpha\in (0,+\infty)$, $\alpha' \in\R_+$ and $\tilde\alpha>0$. For $\varepsilon >0$, we set
$$C_{\varepsilon,v,\alpha}:=\sup\limits_{x\in \R,\; t\in [0,T]} \frac{\abs{v(t,x)}}{1+|x|^{\alpha+\varepsilon}},\ \delta_{\alpha'}:= \max(1,2^{\alpha'}), \ \Xi_{\alpha'}:=\frac{\alpha'\Gamma\left(\frac{1+\alpha'}{2}\right)}{2\sqrt{\pi}},\ \mu(\tilde\alpha):=\int_\R\frac{\phi(z)}{1+|z|^{\tilde{\alpha}}} dz,$$
$$ D_{\alpha'}:=\max\left(1+\delta_{\alpha'}\Xi_{\alpha'} +\frac{\delta_{\alpha'}^2}{2}\left( \Xi_{\alpha'} + (1+\alpha')^{-1}\right)^2, \frac12 + \frac{\delta_{\alpha'}}{1+\alpha'}\right),$$
where $\Gamma$ is the usual Euler function and $\phi$ the distribution function of the normal law, defined by
$$\Gamma(x):=\int_0^{+\infty} e^{-t}t^{x-1}dt,\ x> 0, \ \text{and }\phi(x):=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}},\ x\in\mathbb R.$$
We emphasize that the following theorem can be applied in much more general cases, and it is clearly not limited to the context of BSDEs. It could for instance be used to provide non-Gaussian tail estimates for the law of solutions to some SDEs. Therefore, it has an interest of its own.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{thm_estime_y}
Fix $t\in (0, T]$. Let $v: [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ in $\mathcal{C}^{1,1}$ and let $P_t:=v(t,W_t)$. Assume furthermore that $P_t\in L^1(\mathbb{P})$, that $v$ is unbounded in $x$ both from above and from below, that $v'>0$, $\underline{\alpha_v}\in (0,+\infty)$, $\overline{\alpha_{v'}}<+\infty$ and that there exist $\tilde{\alpha}> 0$ and $K>0$ such that:
\textcolor{blue}egin{eqnarray}gin{equation}\label{ineg_v'}
\frac{1}{v'(t,x)}\leq K(1+|x|^{\tilde{\alpha}}),\text{ for all $x\in\R$}.
\end{equation}
Then, the law of $P_t$ has a density with respect to the Lebesgue measure, denoted by $\rho_t$, and for all $\varepsilon, \varepsilon' >0$ and for every $y\in\R$
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha}
\rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon')t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\right)\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{(M'(\varepsilon,\varepsilon')t)^{-1}xdx}{1+|x+\mathbb{E}[P_t]|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\right),
\end{equation} and
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha_bis}
\rho_t(y)\geq \frac{(2M'(\varepsilon,\varepsilon')t)^{-1}\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{1+|y|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{v^{(-1)}}}+\varepsilon')}}\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon')}}\right)dx}{M(\varepsilon')t}\right),
\end{equation}
with $$M'(\varepsilon, \varepsilon'):=C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 D_{\overline{\alpha_{v'}}+\varepsilon}\left(1+C_{\varepsilon',v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}^{2(\overline{\alpha_{v'}}+\varepsilon)}\right) \delta_{2(\overline{\alpha_{v'}}+\varepsilon)} ,$$
and $$ M(\varepsilon'):= \frac{\mu(\tilde\alpha)}{K^2 \left(1+C^{2\tilde \alpha}_{\varepsilon', v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}\delta_{2\tilde\alpha}\right)} ,$$ using the aforementioned definitions of the constants.
\end{Theorem}
\textbf{Proof.} Notice immediately that since the map $x\longmapsto v(t,x)$ is in $\mathcal C^1(\R)$ and increasing, the law of $P_t$ clearly has a density. We prove inequalities \eqref{estime_y_alpha} and \eqref{estime_y_alpha_bis} using Nourdin and Viens' formula (see Theorem \ref{thm_NourdinViens}).The rest of the proof is divided into three steps.
{\textcolor{blue}f Step $1$:} Given that for all $0<r\leq t \leq T$, $D_r P_t=v'(t,W_t)$, the function $g_{P_t}$ defined by \eqref{gzt} becomes
$$ g_{P_t}(y):= \int_0^\infty e^{-a} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_{P_t}(W),\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W)\rangle_{\mathfrak{H}}] | P_t-\mathbb{E}[P_t]=y\right] da, \quad y\in \mathbb{R}, $$
with\footnote{Knowing that $D_r P_t$ does not depend on $r$, ${\mathfrak{H}at P}i_{P_t}(W):[0,T]\longrightarrow L^2(\Omega,{\cal F},\mathbb{P})$ is a random process which is actually constant on $[0,t]$.} ${\mathfrak{H}at P}i_{P_t}(W):=v'(t,W_t)$ and where $\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W):={\mathfrak{H}at P}i_{P_t}(e^{-a}W+\sqrt{1-e^{-2a}}W^*)$ with $W^*$ an independent copy of $W$ defined on a probability space $(\Omega^*,\mathcal{F}^*,\mathbb{P}^*)$ where $\mathbb{E}^*$ is the expectation under $\mathbb{P}^*$ (${\mathfrak{H}at P}i_{P_t}$ being extended on $\Omega\times \Omega^*$). Letting $\phi(z):=\frac{1}{\sqrt{2\pi t}} e^{-\frac{z^2}{2t}}$, we get that
\textcolor{blue}egin{eqnarray}gin{align} \label{expression_gy}
\noindentnumber g_{P_t}(y)&= \int_0^\infty e^{-a} \mathbb{E}\left[\mathbb{E}^*[\langle {\mathfrak{H}at P}i_{P_t}(W),\widetilde{{\mathfrak{H}at P}i_{P_t}^a}(W)\rangle_{\mathfrak{H}}] | W_t=v^{(-1)}(t,y+\mathbb{E}[P_t])\right] da, \quad y\in \mathbb{R},\\
&=tv'(t,v^{(-1)}(t,y+\mathbb{E}[P_t]))\int_0^\infty e^{-a} \int_\mathbb{R} v'\Big(t,e^{-a} v^{(-1)}(t,y+\mathbb{E}[P_t])+\sqrt{1-e^{-2a}}z\Big)\phi(z) dz da.
\end{align}
{\textcolor{blue}f Step $2$: Upper bound for $g_{P_t}$}
Recall that for all $\varepsilon >0$:
$$0<v'(t,x)\leq C_{\varepsilon,v',\overline{\alpha_{v'}}}\left(1+|x|^{\overline{\alpha_{v'}}+\varepsilon}\right), \quad \forall x \in \mathbb{R}.$$
Then, using \eqref{expression_gy} we get:
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)\leq &\ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+\abs{v^{(-1)}\left(y+\mathbb{E}[P_t]\right)}^{\overline{\alpha_{v'}}+\varepsilon}\right)\\
&\times \int_0^{+\infty} e^{-a}\int_\mathbb{R}\left(1+ \abs{e^{-a} v^{(-1)}(t,y+\mathbb{E}[P_t])+\sqrt{1-e^{-2a}}z}^{\overline{\alpha_{v'}}+\varepsilon}\right)\phi(z) dz da\\
\leq &\ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+|v^{(-1)}(y+\mathbb{E}[P_t])|^{(\overline{\alpha_{v'}}+\varepsilon)}\right)\\
&\times \int_0^{+\infty} e^{-a}\int_\mathbb{R}\left(1+\delta_{\overline{\alpha_{v'}}+\varepsilon}\left(e^{-a(\overline{\alpha_{v'}}+\varepsilon)} \abs{ v^{(-1)}(t,y+\mathbb{E}[P_t])}^{\overline{\alpha_{v'}}+\varepsilon}+\abs{z}^{\overline{\alpha_{v'}}+\varepsilon}\right)\right)\phi(z) dz da\\
\leq & \ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t \left(1+|v^{(-1)}(y+\mathbb{E}[P_t])|^{\overline{\alpha_{v'}}+\varepsilon}\right)\\
&\times \left(1+ \frac{\delta_{\overline{\alpha_{v'}}+\varepsilon}}{1+\overline{\alpha_{v'}}+\varepsilon} |v^{(-1)}(y+\mathbb{E}[P_t])|^{\overline{\alpha_{v'}}+\varepsilon} +\delta_{\overline{\alpha_{v'}}+\varepsilon} \Xi _{\overline{\alpha_{v'}}+\varepsilon}\right)\\
\leq & \ C_{\varepsilon,v',\overline{\alpha_{v'}}}^2 t D_{\overline{\alpha_{v'}} +\varepsilon}\left( 1+ |v^{(-1)}(y+\mathbb{E}[P_t])|^{2(\overline{\alpha_{v'}}+\varepsilon)}\right).
\end{align*}
By Lemma \ref{lemme_v-1}, $\overline{\alpha_{v^{(-1)}}}$ belongs to $(0,+\infty)$, hence by the definition of $\overline{\alpha_{v^{(-1)}}}$ it holds for all $\varepsilon'>0$ that
\textcolor{blue}egin{eqnarray}gin{equation}\label{upper_bound_gy}g_{P_t}(y)\leq M'(\varepsilon,\varepsilon')t\left(1+|y+\mathbb{E}[P_t]|^{2(\overline{\alpha_{v'}}+\varepsilon)(\overline{\alpha_{(v)^{-1}}}+\varepsilon')}\right).
\end{equation}
{\textcolor{blue}f Step $3$: Lower bound for $g_{P_t}$}
Using Assumption \eqref{ineg_v'} and \eqref{expression_gy} we have that
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)\geq &\ \frac{t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}})}\\
&\times\int_0^{+\infty} e^{-a}\int_\mathbb{R} \frac{1}{1+|e^{-a} (v)^{-1}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}}+|\sqrt{1-e^{-2a}}z|^{\tilde{\alpha}}}\phi(z) dz da.
\end{align*}
Noticing that $|\sqrt{1-e^{-2a}}z|^{\tilde{\alpha}}\leq |z|^{\tilde{\alpha}}$, and that
$$\int_\mathbb{R} \frac{(1+|x|^{\tilde{\alpha}})\phi(z)}{1+|x|^{\tilde{\alpha}}+|z|^{\tilde{\alpha}}} dz\geq \mu(\tilde\alpha), \quad \forall x\in \mathbb{R}$$
we deduce that:
\textcolor{blue}egin{eqnarray}gin{align*}
g_{P_t}(y)&\geq \frac{\mu(\tilde\alpha)t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}})}\int_0^{+\infty} e^{-a}\frac{1}{1+e^{-a\tilde{\alpha}}|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{\tilde{\alpha}}}da.
\end{align*}
Hence:
$
g_{P_t}(y)\geq \frac{\mu(\tilde\alpha)t}{K^2(1+|v^{(-1)}(t,y+\mathbb{E}[P_t])|^{2\tilde{\alpha}})}.
$
We finally get Relation \eqref{estime_y_alpha_bis} for
$$M(\varepsilon'):= \frac{\mu(\tilde\alpha)}{K^2 \left(1+C^{2\tilde \alpha}_{\varepsilon', v^{(-1)}, \overline{\alpha_{v^{(-1)}}}}\delta_{2\tilde\alpha}\right)}.$$
We conclude using Nourdin and Viens' formula.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\textcolor{blue}egin{eqnarray}gin{Corollary}\label{cor_estime_y}
Let the assumptions in Theorem \ref{thm_estime_y} hold, with the same notations. Assume moreover that $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $. Then there exist $\varepsilon_0,\varepsilon'_0>0$, $y_0>0$ and $\gamma\in(0,1)$ such that for any $|y|> y_0$:
\textcolor{blue}egin{eqnarray}gin{equation}\label{estime_y_alpha_sansint}
\rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon'_0)t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}}\right)\exp\left(-\frac{\abs{y-\mathbb{E}[P_t]}^{2(1-\gamma)}-\abs{y_0-\mathbb{E}[P_t]}^{2(1-\gamma)}}{4(1-\gamma)tM'(\varepsilon_0,\varepsilon'_0)}\right),
\end{equation}
and
\textcolor{blue}egin{eqnarray}gin{align}\label{estime_y_alpha_bis_sansint}
\noindentnumber\rho_t(y)\geq &\ \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M'(\varepsilon_0,\varepsilon'_0)t\left(1+|y|^{\gamma}\right)}\exp\left(-\frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{M(\varepsilon_0')t(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}\right)\\
&\times \exp\left(-\frac{\abs{y_0-\mathbb{E}[P_t]}^2}{M(\varepsilon_0')t}\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right)\right).
\end{align}
\end{Corollary}
\textcolor{blue}egin{eqnarray}gin{proof}
Let us define for any $\varepsilon,\varepsilon'>0$
$$\gamma(\varepsilon,\varepsilon'):=(\overline{\alpha_{v'}}+\varepsilon) (\overline{\alpha_{v^{(-1)}}}+\varepsilon').$$
Since we assumed that $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $, we can deduce using Lemma \ref{lemme_v-1} that there exist some $\varepsilon_0,\varepsilon_0'>0$ such that
$$\gamma:=\gamma(\varepsilon_0,\varepsilon_0')<1.$$
We start with \reff{estime_y_alpha_sansint}. We have from Theorem \ref{thm_estime_y}
$$ \rho_t(y)\leq \frac{\mathbb{E}[|P_t-\mathbb{E}[P_t]|]}{2M(\varepsilon'_0)t}\left(1+|y|^{2\tilde{\alpha}{(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}}\right)\exp \left(-\int_0^{y-\mathbb{E}[P_t]} \frac{xdx}{M'(\varepsilon_0,\varepsilon'_0)t\left(1+|x+\mathbb{E}[P_t]|^{2\gamma}\right)}\right).$$
We notice that $$\lim\limits_{\abs{x}\to +\infty}\frac{x}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})} \times \frac{1}{\frac{x}{M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}}=1,$$ so that there exists $x_0$ large enough such that $\frac{x}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})} \geq \frac{x}{2M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}$ when $\abs{x}\geq x_0$. Hence, since $\gamma\in(0,1)$, we know that we can find some $y_0>0$ large enough such that if $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
& \int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} \frac{xdx}{M'(\varepsilon_0,\varepsilon_0')t(1+|x+\mathbb{E}[P_t]|^{2\gamma})}\\&\geq \int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} \frac{xdx}{2M'(\varepsilon_0,\varepsilon_0')t|x|^{2\gamma}}\\
&=\frac{1}{4(1-\gamma)tM'(\varepsilon_0,\varepsilon'_0)}\left(\abs{y-\mathbb{E}[P_t]}^{2(1-\gamma)}-\abs{y_0-\mathbb{E}[P_t]}^{2(1-\gamma)}\right),
\end{align*}
from which \reff{estime_y_alpha_sansint} follows directly. Similarly, increasing $y_0$ if necessary, we have that for $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
&\int_0^{y-\mathbb{E}[P_t]}x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx\\
&=\underbrace{\int_0^{y_0-\mathbb{E}[P_t]} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx}_{:=I_1}+\underbrace{\int_{y_0-\mathbb{E}[P_t]}^{y-\mathbb{E}[P_t]} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx}_{:=I_2}.\\
\end{align*}
Using the fact that the function $x\longmapsto 1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)} $ is convex, we deduce that for $y_0$ large enough
$$I_1\leq \abs{y_0-\mathbb{E}[P_t]}^2\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right). $$
Moreover, since $\lim\limits_{x\to +\infty} x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon'_0)} +\varepsilon'_0)}\right) \times \frac{1}{x^{2{\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}}}=1 $, we obtain for $x$ large enough
$$ x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}+\varepsilon'_0)} +\varepsilon'_0)}\right)\leq 2x^{2{\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}}. $$
Then, we have that for $|y|\geq y_0$
$$ I_2\leq \frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1}.$$
Hence,
\textcolor{blue}egin{eqnarray}gin{align*}
&\int_0^{y-\mathbb{E}[P_t]}x\left(1+|x+\mathbb{E}[P_t]|^{2\tilde{\alpha}(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)}\right)dx\\
&\leq \abs{y_0-\mathbb{E}[P_t]}^2\left(1+y_0^{2\tilde\alpha\left(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0\right)}\right) +\frac{\abs{y-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}-\abs{y_0-\mathbb{E}[P_t]}^{2(\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1)}}{\tilde\alpha(\overline{\alpha_{v^{(-1)}}}+\varepsilon'_0)+1},
\end{align*}
from which the second inequality \reff{estime_y_alpha_bis_sansint} follows directly using \reff{estime_y_alpha_bis}.
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\end{proof}
Finally, we have the following theorem, which is a simple application of the results obtained above in the special cases where we take the random variables $(Y_t,Z_t)$ solutions to the BSDE \reff{edsr} when they can be written $Y_t=v(t,W_t)$ and $Z_t=v'(t,W_t)$.
\textcolor{blue}egin{eqnarray}gin{Theorem}\label{estim.dens}
Let $(Y,Z)$ be the solution to the BSDE \reff{edsr} $($which is assumed to exist and to be unique$)$. Assume that there exists a map $v\in\mathcal C^{1,2}$ such that $Y_t=v(t,W_t)$.
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\rm{(i)}$] If in addition, $v'>0$, $0\leq \overline{\alpha_{v'}}<\underline{\alpha_v}<+\infty $ and there exist $K>0$, $\tilde\alpha>0$ such that $v'(t,x)\geq 1/(K(1+\abs{x}^{\tilde\alpha}))$ then, denoting $\rho_{Y_t}$ the density of the law of $Y_t$, there exist $y_0>0$, $C_1,C_2>0$, $p_1\in(0,2)$ and $p_2>0$ $($which are given explicitly in Theorem \ref{thm_estime_y}$)$ such that for any $\abs{y}>y_0$
\textcolor{blue}egin{eqnarray}gin{align*}
\rho_{Y_t}(y)\geq&\ \frac{\mathbb{E}[\abs{Y_t-\mathbb{E}[Y_t]}}{C_2t\left(1+\abs{y}^{1-p_1/2}\right)}\exp\left(-\frac{\abs{y-\mathbb{E}[Y_t]}^{2(p_2+1)}-\abs{y_0-\mathbb{E}[Y_t]}^{2(p_2+1)}}{(p_2+1)C_2t}\right)\\
\rho_{Y_t}(y)\leq&\ \frac{\mathbb{E}[\abs{Y_t-\mathbb{E}[Y_t]}}{C_1t}\left(1+\abs{y}^{2p_2}\right)\exp\left(-\frac{2\abs{y_0-\mathbb{E}[Y_t]}^2}{C_2t}\left(1+y_0^{2p_2}\right)\right)\\
&\times\exp\left(-\frac{\abs{y-\mathbb{E}[Y_t]}^{p_1}-\abs{y_0-\mathbb{E}[Y_t]}^{p_1}}{p_1C_2t}\right).
\end{align*}
\item[$\rm{(ii)}$] If in addition, $v''>0$, $0\leq \overline{\alpha_{v''}}<\underline{\alpha_{v'}}<+\infty $ and there exist $K>0$, $\tilde\alpha>0$ such that $v''(t,x)\geq 1/(K(1+\abs{x}^{\tilde\alpha}))$ then, denoting $\rho_{Z_t}$ the density of the law of $Z_t$, there exists $Z_0>0$, $C_1,C_2>0$, $p_1\in(0,2)$ and $p_2>0$ $($which are given explicitly in Theorem \ref{thm_estime_y}$)$ such that for any $\abs{z}>z_0$
\textcolor{blue}egin{eqnarray}gin{align*}
\rho_{Z_t}(z)\geq&\ \frac{\mathbb{E}[\abs{Z_t-\mathbb{E}[Z_t]}}{C_2t\left(1+\abs{z}^{1-p_1/2}\right)}\exp\left(-\frac{\abs{z-\mathbb{E}[Z_t]}^{2(p_2+1)}-\abs{z_0-\mathbb{E}[Z_t]}^{2(p_2+1)}}{(p_2+1)C_2t}\right)\\
\rho_{Z_t}(y)\leq&\ \frac{\mathbb{E}[\abs{Z_t-\mathbb{E}[Z_t]}}{C_1t}\left(1+\abs{z}^{2p_2}\right)\exp\left(-\frac{2\abs{z_0-\mathbb{E}[Z_t]}^2}{C_2t}\left(1+z_0^{2p_2}\right)\right)\\
&\times\exp\left(-\frac{\abs{z-\mathbb{E}[Z_t]}^{p_1}-\abs{z_0-\mathbb{E}[Z_t]}^{p_1}}{p_1C_2t}\right).
\end{align*}
\end{itemize}
\end{Theorem}
\subsection{Verifying the assumptions of Theorem \ref{estim.dens}}
In this subsection, we give some conditions which ensure that the assumptions in Corollary \ref{cor_estime_y} hold. We recall that under Assumptions (X), (L) or (Q), (D1) and according to Proposition \ref{prop:Markov}, there exists a map $u:[0,T] \times \mathbb{R} \longrightarrow \mathbb{R}$ in $\mathcal{C}^{1,2}$ such that $Y_t =u(t,W_t), \ t\in [0,T], \ \mathbb{P}-$a.s., and $Z$ admits a continuous version given by $ Z_t = u'(t,W_t), \ t\in [0,T], \ \mathbb{P}-$a.s., assuming that $\sigma\equiv1$ and $b\equiv 0$ in the studied FBSDE \eqref{edsr}. Moreover we suppose for simplicity that the generator $h$ of BSDE \eqref{edsr} depends only on $z$, and that $u'$ and $u''$ are\footnote{This assumption is satisfied if $g$ and $h$ are smooth enough.} in $\mathcal{C}^{1,2}$. By a simple application of the non-linear Feynman-Kac formula (see for instance \cite{PardouxPeng_92}), and by differentiating it repeatedly, it can be shown that $u$, $u'$ and $u''$ are respectively classical solutions of the following PDEs:
\textcolor{blue}egin{eqnarray}gin{align}\label{PDE}
&\textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle -u_t(t,x)-\frac12 u_{xx}(t,x)-h(t, u_x(t,x))=0,\ (t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u(T,x)=g(x),\ x\in \mathbb{R},
\end{cases}\\
\label{PDE'}
& \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle - u'_t(t,x)-\frac12 u'_{xx}(t,x)-h_z(t, u'(t,x)) u'_x(t,x)=0,\ (t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u'(T,x)=g'(x),\ x\in \mathbb{R},\end{cases}\\
\label{PDE''}
& \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle - u''_t(t,x)-\frac12 u''_{xx}(t,x)-h_z(t, u'(t,x)) u''_x(t,x)-h_{zz}(t,u'(t,x))|u''(t,x)|^2=0,\; \mathfrak{H}space{-0.2em}(t,x)\in [0,T)\times \mathbb{R} \\
\displaystyle u''(T,x)=g''(x),\ x\in \mathbb{R}.
\end{cases}
\end{align}
We show in the following proposition and its corollary that under some conditions on $g,g',g''$ and $h, h_z$, the assumptions in Theorem \ref{estim.dens} are satisfied. We emphasize that this is only one possible set of assumptions, and that the required properties of $u$ and its derivatives can be checked on a case by case analysis.
\textcolor{blue}egin{eqnarray}gin{Proposition}\label{prop_illustration_lipy}
Let $u$, $u'$ and $u''$ be respectively the solution to \reff{PDE}, \reff{PDE'} and \reff{PDE''} and assume that a comparison theorem holds for classical super and sub-solutions of these PDEs, in the class of functions with polynomial growth. Assume that there exist $(\varepsilon,\underline C,\overline C)\in(0,1)\times(0,+\infty)^3$, such that for all $x\in \mathbb{R}$
$$\underline{C}(1+|x|^{1-\varepsilon})\leq g(x)\leq \overline{C}(1+|x|^{1+\varepsilon}).$$
Assume moreover that $h$ is non-positive and that there exist $(\varepsilon',\underline{D},\overline{D})\in(0,\varepsilon)\times(0,+\infty)^2$ s.t.
$$\underline{D}(1+|x|^{\varepsilon'})\leq g'(x)\leq \overline{D}(1+|x|^{\varepsilon}).$$
Assume that there exist $(\underline{B},\overline{B}) \in (0,+\infty)^2$ such that for all $x\in \mathbb{R}$
$$\underline{B}\leq g''(x)\leq \overline{B}, \text{ and } 0\leq h_{zz}(t,x)<\frac{1}{4\overline{B}T}.$$
Assume finally that there exist $\lambda\in(0,\varepsilon^{-1}-1]$ and $C>0$ such that $|h_z(t,z)|\leq C(1+|z|^\lambda)$, then for all $(t,x)\in[0,T]\times\R$,
$$\underline{\alpha_u}\in [1-\varepsilon,1+\varepsilon], \ \overline{\alpha_{u'}},\underline{\alpha_{u'}}\in [\varepsilon',\varepsilon],\ \overline{\alpha_{u''}}=0, \ u'(t,x)\geq \underline D \text{ and } u''(t,x)\geq \underline{B}.$$
\end{Proposition}
\textcolor{blue}egin{eqnarray}gin{proof}
Let $\varphi(t,x):=\tilde{C}(T-t)+\overline{C}k_\varepsilon(x)$, where $k_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{1+\varepsilon})$ outside some closed interval centered at $0$ and is always greater than $(1+\abs{x}^{1+\varepsilon})$. We show that $\varphi$ is a (classical) super-solution to \reff{PDE} for some positive constant $\tilde{C}$ large enough. Indeed we can choose $\tilde{C}>0$ such that for any $(t,x)\in[0,T)\times\R$
$$ -\varphi_t(t,x)-\frac12 \varphi_{xx}(t,x)-h(t, \varphi_x(t,x))=\tilde{C}-\frac12 \overline{C}k_\varepsilon''(x)-h(t,\varphi_x(t,x))\geq 0,$$
since $h\leq 0$ and $\lim\limits_{|x|\to \infty }\frac12 k_\varepsilon''(x)=0$.
Moreover, by the assumption made on $g$, we clearly have for all $x\in\R$, $g(x)\leq \overline{C}k_\varepsilon(x)$, so that we deduce by comparison that for all $(t,x)\in [0,T] \times \mathbb{R}$:
$$ u(t,x)\leq \overline{C}k_\varepsilon(x)+\tilde{C}(T-t).$$
Now, we let $\phi(t,x):=-\tilde{C}_1(T-t)+\underline{C}\kappa_\varepsilon(x)$ for $(t,x)\in [0,T)\times \mathbb{R}$, where $\kappa_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{1-\varepsilon})$ outside some closed interval centered at $0$ and is always smaller than $(1+\abs{x}^{1-\varepsilon})$. We show that $\phi$ is a classical subsolution to \reff{PDE} for some positive constant $\tilde{C}_1$ large enough. We have
\textcolor{blue}egin{eqnarray}gin{equation}\label{truc} - \phi_t(t,x)-\frac12 \phi_{xx}(t,x)-h(t,\phi_x(t,x))=-\tilde{C}_1+\frac12\underline{C}\kappa_\varepsilon''(x)-h(t, \phi_x(t,x)).
\end{equation}
Given that the quantity $h(t,\phi_x(t,x))=h(t,\underline C\kappa_\varepsilon'(x))$ is bounded because $\lim\limits_{|x|\to \infty}\kappa_\varepsilon'(x)=0$ and $h$ is continuous, we can always choose $\tilde C_1$ so that \reff{truc} is non-positive. Then, since we clearly have for all $x\in\R$, $g(x)\geq \underline{C}\kappa_\varepsilon(x)$, we deduce by comparison that for all $(t,x)\in [0,T] \times \mathbb{R}$:
$$ u(t,x)\geq \overline{C}\kappa_\varepsilon(x)+\tilde{C}_1(T-t).$$
To sum up, we have showed that for all $(t,x)\in [0,T]\times \mathbb{R}$:
$$ \underline{C}\kappa_\varepsilon(x)-\tilde{C}_1(T-t)\leq u(t,x)\leq \overline{C}k_\varepsilon(x)+\tilde{C}(T-t).$$ In other words $[\underline{\alpha_u},\overline{\alpha_u}]\subset [1-\varepsilon,1+\varepsilon]. $
We now study \reff{PDE'}. Define for some constant $\tilde C_2>0$ to be fixed later
\textcolor{blue}egin{eqnarray}gin{align*}
\psi(t,x):=\tilde C_2(T-t)+\overline D\Upsilon_\varepsilon(x),
\end{align*}
where $\Upsilon_\varepsilon(x)$ is in $\mathcal C^\infty(\R)$, coincides with the function $(1+\abs{x}^{\varepsilon})$ outside some closed interval centered at $0$ and is always greater than $(1+\abs{x}^{\varepsilon})$. We then have
\textcolor{blue}egin{eqnarray}gin{align*}
-\psi_t(t,x)-\frac12 \psi_{xx}(t,x)-h_z(t,\psi(t,x)) \psi_x(t,x)=\tilde{C}_2-\frac12\overline{D}\Upsilon_\varepsilon''(x)-h_z(t,\psi(t,x))\overline D\Upsilon_\varepsilon'(x).
\end{align*}
Next, for some constant $C>0$ which may vary from line to line
$$|h_z(t,\psi(t,x))|\leq C(1+\abs{\psi(t,x)}^{\lambda})\leq C(1+\abs{x}^{\lambda\varepsilon}),$$ and since $\lambda \leq\frac1\varepsilon-1$ we deduce that:
$$\abs{h_z(t,\psi(t,x))\overline D\Upsilon_\varepsilon'(x)}\leq C(1+|x|^{\lambda \varepsilon+\varepsilon-1}) \text{, which is bounded. }$$
Since in addition we have $\Upsilon_\varepsilon''(x)\longrightarrow 0$ as $\abs{x}$ goes to $+\infty$, we can always choose $\tilde C_2$ large enough so that
$$- \psi_t(t,x)-\frac12 \psi_{xx}(t,x)-h_z(t,\psi(t,x))\psi_x(t,x)\geq 0.$$
By the assumption we made on $g$, we can use once more the comparison theorem to obtain
$$u'(t,x)\leq \psi(t,x).$$ Similarly, we show that $\underline{D}\Upsilon_{\varepsilon'}(x)-\tilde{C}_3(T-t)$ is a sub-solution of \reff{PDE'} for some positive constant $\tilde{C}_3$, since $\lambda\leq \varepsilon^{-1}-1\leq \varepsilon'^{-1}-1$. Then, by comparison, we deduce that $\overline{\alpha_{u'}}, \underline{\alpha_{u'}} \in [\varepsilon', \varepsilon]$. Moreover, we notice that $\underline{D}\leq g'(x)$ for all $x\in \mathbb{R}$, so $\underline{D}$ is a sub-solution of \reff{PDE'}. Thus, using once more the comparison theorem $u'(t,x)\geq \underline{D}$ for all $(t,x)\in [0,T]\times \mathbb{R}$.
We now study \reff{PDE''}. Given that $h_{zz}$ is non negative and $\underline{B}\leq g''(x)$ for all $(t,x)\in [0,T]\times \mathbb{R}$, we deduce directly that $\underline{B}$ is a sub-solution of \reff{PDE''}. Next, let $\varpi(t,x)=\overline{B}+\frac{\overline{B}}{T^{1-\eta}}(T-t)^{1-\eta}$ where $\eta \in (0,1)$ is chosen small enough so that $h_{zz}(t,x)\leq \frac{1-\eta}{4T\overline{B}}$. Thus,
\textcolor{blue}egin{eqnarray}gin{align*}
& - \varpi_t(t,x)-\frac12 \varpi_{xx}(t,x)-h_z(t, u'(t,x)) \varpi_x(t,x)-h_{zz}(t,u'(t,x))|\varpi(t,x)|^2\\
&=(1-\eta)\frac{\overline{B}}{T^{1-\eta}}(T-t)^{-\eta}-h_{zz}(t,u'(t,x))\overline{B}^2\left(1+\frac{(T-t)^{1-\eta}}{T^{1-\eta}} \right)^2\\
&\geq (1-\eta)\frac{\overline{B}}{T^{1-\eta}}(T-t)^{-\eta}-\frac{1-\eta}{4T}\overline{B}\left(1+\frac{(T-t)^{1-\eta}}{T^{1-\eta}} \right)^2\\
&\geq 0.
\end{align*}
We deduce that $\varpi$ is a super solution of \reff{PDE''}, which by comparison, implies that $u''$ is bounded, so $\overline{\alpha_{u''}}=0$.
\end{proof}
\textcolor{blue}egin{eqnarray}gin{Corollary}
Consider the FBSDE \eqref{edsr} and assume that for all $t\in [0,T]$ $X_t=W_t$ and $h$ depends only on $z$. Let $u(t,X_t):=Y_t$ and assume that $u\in {\cal C}^{1,2}$, $u'\in {\cal C}^{1,2}$ and $u''\in {\cal C}^{1,2}$. Let the assumptions of Proposition \ref{prop_illustration_lipy} hold, and assume moreover that $\varepsilon \in (0,\frac12)$. Then, the assumptions of Theorem \ref{estim.dens} hold.
\end{Corollary}
\textcolor{blue}egin{eqnarray}gin{proof}
According to Proposition \ref{prop_illustration_lipy}, $\underline{\alpha_u}\geq 1-\varepsilon$, $\overline{\alpha_{u'}}\leq \varepsilon$ and $u'(t,x)\geq \underline D,\ (t,x)\in[0,T]\times\R$. From the fact that $\varepsilon$ is smaller than $1/2$, we deduce that $0\leq\overline{\alpha_{u'}}<\underline{\alpha_{u}}<+\infty$. Moreover, $0=\overline{\alpha_{u''}}<\varepsilon'\leq\underline{\alpha_{u'}}$.
\end{proof}
\textcolor{blue}egin{eqnarray}gin{flushright}
$\qed$
\end{flushright}
\section*{Acknowledgments}
Thibaut Mastrolia is grateful to R\'egion Ile-De-France for financial support. The authors thank an Associate Editor and two anonymous Referees for their careful reading of this paper and for insightful suggestions which have greatly improve its presentation.
\section{Table of assumptions-results}
In this appendix we recall the different assumptions made within this paper and we give a summary table of some most significant results on BSDEs including ours.
\paragraph*{Assumption for $X$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(X)]$b,\sigma : [0,T]\times \mathbb{R} \longrightarrow \mathbb{R}$ are continuous in time and continuously differentiable in space for any fixed time $t$ and such that there exist $k_b,k_\sigma >0$ with
$$|b_x(t,x)|\leq k_b,\ |\sigma_x(t,x)|\leq k_\sigma, \text{ for all $x\in\R$}.$$
Besides $b(t,0), \sigma(t,0)$ are bounded functions of $t$ and there exists $c>0$ such that for all $t\in [0,T]$ $$0<c\leq |\sigma(t,\cdot)|, \ \lambda(dx)-a.e.$$
\end{itemize}
\textbf{List of assumptions for BSDEs:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(L)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is such that $\mathbb{E}[g(X_T)^2]<+\infty$.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that there exist $(k_x,k_y,k_z)\in(\R_+^*)^3$ such that for all $(t,x_1,x_2,y_1,y_2,z_1,z_2) \in [0,T]\times \mathbb{R}^6$,
$$ |h(t,x_1,y_1,z_1)-h(t,x_2,y_2,z_2)|\leq k_x|x_1-x_2|+k_y|y_1-y_2|+k_z|z_1-z_2|.$$
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\item[(Q)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g : \mathbb{R} \longrightarrow \mathbb{R}$ is bounded.
\item[(ii)] $h : [0,T]\times \mathbb{R}^3 \longrightarrow \mathbb{R}$ is such that:
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$\triangleright$] There exists $(K,K_z,K_y)\in(\R_+^*)^3$ such that for all $(t,x,y,z) \in [0,T]\times \mathbb{R}^3$
$$\mathfrak{H}space{-3em}|h(t,x,y,z)|\leq K(1+|y|+|z|^2),\ \abs{h_z}(t,x,y,z)\leq K_z(1+|z|),\ \abs{h_y}(t,x,y,z)\leq K_y.$$
\item[$\triangleright$] There exists $C>0$ such that for all $(t,x,y,z_1,z_2) \in [0,T] \times \mathbb{R}^4$
\textcolor{blue}egin{eqnarray}gin{equation*} |h(t,x,y,z_1)-h(t,x,y,z_2)| \leq C(1+|z_1|+|z_2|) |z_1-z_2|.\end{equation*}
\end{itemize}
\item[(iii)] $\int_0^T |h(s,0,0,0)|^2ds<+\infty$.
\end{itemize}
\end{itemize}
\paragraph*{List of assumptions for Malliavin differentiability of $(X,Y,Z)$:}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(D1)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is differentiable, $\mathcal{L}(X_T)-$a.e., $g$ and $g'$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\item[(D2)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[(i)] $g$ is twice differentiable, $\mathcal{L}(X_T)-$a.e., $g$, $g'$ and $g''$ have polynomial growth.
\item[(ii)] $(x,y,z)\mapsto h(t,x,y,z)$ is twice continuously differentiable for every $t$ in $[0,T]$.
\end{itemize}
\end{itemize}
\paragraph*{List of assumptions for the existence of densities for $Y$ and $Z$:}
$$\underline{g}:= \inf\limits_{x \in \mathbb{R}} g'(x), \quad \underline{g}^A:=\inf\limits_{x\in A} g'(x),\quad \overline{g}:= \sup\limits_{x \in \mathbb{R}} g'(x), \quad \overline{g}^A:=\sup\limits_{x\in A} g'(x),$$
$$\underline{h}(t):=\inf\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),\quad \quad \overline{h}(t):=\sup\limits_{ s\in [t,T], (x,y,z) \in \mathbb{R}^3} h_x(s,x,y,z),$$
and $K:=k_b+k_y+k_{\sigma}k_z$. There exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ and such that:
\textcolor{blue}egin{eqnarray}gin{align*}
&(H+)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{g}e^{-\text{sgn}(\underline{g})KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds\geq0 \\
\displaystyle \underline{g}^Ae^{-\text{sgn}(\underline{g}^A)KT}+\underline{h}(t)\int_t^T e^{-\text{sgn}(\underline{h}(s))Ks}ds>0
\end{cases}\\[0.3em]
&(H-)\quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{g}e^{-\text{sgn}(\overline{g})KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds\leq 0 \\
\displaystyle \overline{g}^Ae^{-\text{sgn}(\overline{g}^A)KT}+\overline{h}(t)\int_t^T e^{-\text{sgn}(\overline{h}(s))Ks}ds<0,
\end{cases}
\end{align*}
Set
\textcolor{blue}egin{eqnarray}gin{align*}
\tilde{h}(s,x,y,z):=& -\left( h_{xt}+b h_{xx}-hh_{xy}+\frac12(\sigma^2 h_{xxx}+2z\sigma h_{xxy}+z^2h_{xxy})\right)(s,x,y)\noindentnumber\\
&-\left((h_y+b_x)h_x+\sigma \sigma_xh_{xx}+z\sigma_xh_{xy}\right)
(s,x,y).\\
\tilde{g}(x):=&\ g'(x)+(T-t)h_x(T,x,g(x)),
\end{align*}
and
$$ \underline{\tilde{g}}:=\min\limits_{x\in \mathbb{R}} \tilde{g}(x), \quad \overline{\tilde{g}}:=\max\limits_{x\in \mathbb{R}} \tilde{g}(x),\quad \underline{\tilde{g}}^A:=\min\limits_{x\in A} \tilde{g}(x), \quad \overline{\tilde{g}}^A:=\max\limits_{x\in A} \tilde{g}(x),$$
$$\underline{\tilde{h}}(t):=\min\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z), \ \overline{\tilde{h}}(t):=\max\limits_{[t,T]\times \mathbb{R}^3} \tilde{h}(s,x,y,z),$$
and set $K:=k_y+k_b$. There exists $A\in\mathcal B(\R)$ such that $\mathbb{P}(X_T \in A | \mathcal{F}_t)>0$ \textcolor{blue}egin{eqnarray}gin{align*}
&\widetilde{(H+)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \underline{\tilde{g}}e^{-\text{sgn}(\underline{\tilde{g}})KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds\geq0 \\
\displaystyle \underline{\tilde{g}}^Ae^{-\text{sgn}(\underline{\tilde{g}}^A)KT}+\underline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\underline{\tilde{h}}(s))Ks}(T-s)ds>0,
\end{cases}\\[0.3em]
&\widetilde{(H-)} \quad \textcolor{blue}egin{eqnarray}gin{cases}
\displaystyle \overline{\tilde{g}}e^{-\text{sgn}(\overline{\tilde{g}})KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds\leq 0 \\
\displaystyle \overline{\tilde{g}}^Ae^{-\text{sgn}(\overline{\tilde{g}}^A)KT}+\overline{\tilde{h}}(t)\int_t^T e^{-\text{sgn}(\overline{\tilde{h}}(s))Ks}(T-s)ds<0.
\end{cases}
\end{align*}
\textcolor{blue}egin{eqnarray}gin{itemize}
\item[$(Q+)$] $g'\geq 0$ and $g'_{\vert A} >0$, $\mathcal{L}(X_T)-$a.e. and $\underline{h}(t)\geq 0$,
\item[$(Q-)$] $g'\leq 0,\ g'_{\vert A}<0$, $\mathcal{L}(X_T)-$a.e. and $\overline{h}(t)\leq 0$,
\item[(Z+)]
\textcolor{blue}egin{eqnarray}gin{itemize}
\item There exist $(\underline a,\overline a)$ s.t., $0<\underline{a}\leq D_r X_u \leq \overline{a}$, for all $0<r<u\leq T$.
\item There exists $\overline b$ s.t., $0\leq D_{r,s}^2 X_u\leq \overline{b}$, for all $0<r,s<u\leq T$.
\item $h_x,h_{xx},h_{yy},h_{zz},h_{xy}\geq 0$ and $h_{xz}= h_{yz}= 0$ (and $h_y\geq 0$ under $(Q)$)
\item $h_{xy}= 0$ or $(h_{xy}\geq 0$ and $g'\geq 0$, $\mathcal{L}(X_T)$-a.e.$)$.
\item We have $$ \mathbf{1}_{\{\underline{g''}<0\}}\underline{g''}\overline{a}^2+\underline{g'}\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}+(\mathbf{1}_{\{\underline{g''}\geq 0\}}\underline{g''}+\underline{h_{xx}}(t)(T-t))\underline{a}^2\geq 0,$$
and
$$ (\mathbf{1}_{\{\underline{g''}^A<0\}}\underline{g''}^A\overline{a}^2+\underline{g'}^A\mathbf{1}_{\{\underline{g'}<0\}}\overline{b}) +(\mathbf{1}_{\{\underline{g''}^A\geq 0\}}\underline{g''}^A+\underline{h_{xx}}(t)(T-t))\underline{a}^2>0,$$
\end{itemize}
\end{itemize}
We give the following summary table which sums up significant results for BSDEs in both the Lipschitz case and the quadratic case with assumptions made and references.
\small
\textcolor{blue}egin{eqnarray}gin{center}
\textcolor{blue}egin{eqnarray}gin{tabular}{|c|c|c|}
\mathfrak{H}line \diagbox{Results}{Cases} & Lipschitz case (L) & Quadratic case (Q) \\
\mathfrak{H}line Existence and uniqueness &\multirow{2}{*}{Prop. \ref{propex} (X)} & \multirow{2}{*}{Prop. \ref{propexq} (X) }\\
of solutions of BSDEs& & \\
\mathfrak{H}line Malliavin differentiability & \multirow{2}{*}{Prop. \ref{MD} (X) and (D1) } & \multirow{2}{*}{Prop. \ref{MDq} (X) and (D1)} \\
of $(X,Y,Z)$ & &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Y$} & Th. \ref{thm_H+H-} (X), (D1) and (H+) or (H-)& \multirow{2}{*}{Th. \ref{thm_H+H-_quadra} (X), (D2) and (Q+) or (Q-)} \\
& Th. \ref{AKmodifie} (X), (D1) and ($\widetilde{H+}$) or ($\widetilde{H-}$) &\\
\mathfrak{H}line \multirow{2}{*}{Density existence for $Z$} & \multirow{2}{*}{Th. \ref{thm_density_z_lip} (X), (D2) and (Z+)} & \multirow{2}{*}{Th. \ref{thm_density_z_quadra} (X), (D2) and (Z+)}\\
& &\\
\mathfrak{H}line
\end{tabular}
\end{center}
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\end{document} |
\begin{document}
\date{Draft}
\title{The polyharmonic heat flow of closed plane curves}
\author[S. Parkins]{Scott Parkins}
\address{Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics\\
University of Wollongong\\
Northfields Ave, Wollongong, NSW $2500$,\\
Australia}
\email{srp854@uow.edu.au}
\author[G. Wheeler]{Glen Wheeler}
\address{Institute for Mathematics and its Applications, School of Mathematics and Applied Statistics\\
University of Wollongong\\
Northfields Ave, Wollongong, NSW $2500$,\\
Australia}
\email{glenw@uow.edu.au}
\thanks{The research of the first author was supported by an Australian
Postgraduate Award. The research of the second author was supported in part by
Discovery Projects DP120100097 and DP150100375 of the Australian Research Council.}
\subjclass{53C44}
\begin{abstract}
In this paper we consider the polyharmonic heat flow of a closed curve in the
plane.
Our main result is that closed initial data with initially small normalised
oscillation of curvature and isoperimetric defect flows exponentially fast in
the $C^\infty$-topology to a simple circle.
Our results yield a characterisation of the total amount of time during which
the flow is not strictly convex, quantifying in a sense the failure of the
maximum principle.
\end{abstract}
\maketitle
\begin{section}{Introduction}
Let $\gamma_0:\mathbb{S}\rightarrow\mathbb{R}^2$ be a smooth, closed regular immersed plane curve.
Let $p\in\mathbb{N}_0$.
A one-parameter family $\gamma:\mathbb{S}\times\left[0,T\right)\rightarrow\mathbb{R}^2$ satisfying
\begin{equation}
\frac{\partial}{\partial
t}\gamma=\oo{-1}^{p}\kappa_{s^{2p}}\nu\label{FlowDefinition}\tag{$PF_p$}
\end{equation}
is called the $(2p+2)$-th order polyharmonic heat flow of $\gamma_0$, or the
\emph{polyharmonic flow} for short.
Here $s$ is the regular Euclidean arc length
$s\oo{u}=\int_{0}^{u}{\norm{\gamma_{v}}\,du}$ and $\kappa_{s^{2p}}$ is
$2p$ derivatives of the Euclidean curvature $\kappa$ with respect to
arc length:
\[
\kappa_{s^{2p}} := \partial_s^{2p}\kappa := \frac{\partial^{2p}\kappa}{\partial
s^{2p}}.
\]
We take $\nu$ to be a unit normal vector field to $\gamma$ such that $\kappa\nu =
\partial_s^2\gamma$.
If we take $p=0$, then \eqref{FlowDefinition} is the well-studied \emph{curve
shortening flow} made famous by Hamilton, Gage and Grayson \cite{Hamilton2,grayson1989}:
\[
\partial_t\gamma = \kappa\nu\,.
\]
The curve shortening flow is second-order, and, being a nonlinear geometric
heat equation for the immersion $\gamma$, enjoys the maximum principle and its
standard variations (Harnack inequality, comparison/avoidance principles). This
allows for trademark characteristics such as moving immediately from weak
convexity to strong convexity, preservation of convexity, preservation of
embeddedness, and preservation of graphicality.
The curve shortening flow is the $W^{-0,2} = L^2$ gradient flow for length.
Taking the $H^{-1} = W^{-1,2}$ gradient flow for length yields the fourth order
flow termed the \emph{curve diffusion flow}, whose origins lie in material
science \cite{Mullins1}.
Its gradient flow structure was only later discovered by Fife \cite{Fife}.
The qualitative properties mentioned above for the curve shortening flow do not
hold for the curve diffusion flow (and in fact do not hold for any of the flows
\eqref{FlowDefinition} for $p>0$).
We refer the reader to \cite{Blatt2010,EllPaa2001,EscherIto2005,GI1998,GI1999}
for an overview of these interesting phenomena.
We additionally mention numerical examples contributed by Mayer
\cite{privatecommsmayer} of finite-time singularities arising from embedded
initial data (the resolution of this is an open conjecture that to our
knowledge is due to Giga \cite{privatecommsgiga}).
While local well-posedness belongs by now to standard theory (see for example
\cite{Baker,Mantegazza2}), global analysis and qualitative properties of the
flow remain largely unresolved.
Recently, there have been advances in understanding the stability of the curve
diffusion flow about circles, with work of Elliott-Garcke \cite{EG97}
strengthened by the second author in \cite{Wkosc}.
The result of \cite{Wkosc} relies on the blowup criterion discovered by
Dziuk-Kuwert-Sch\"atzle \cite{DKS}.
The core idea of \cite{Wkosc} is to analyse the normalised oscillation of
curvature:
\[
K_{osc}:=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\,.
\]
The key observation for the curve diffusion flow is that $K_{osc}$ is a natural
energy, being both integrable (in time) for any allowable initial data and
whose blowup characterises finite-time blowup in general.
In this article, we prove that $K_{osc}$ remains a natural energy for every
polyharmonic flow, regardless of how large $p$ is.
In the theorem below and for the remainder of the article we assume $p\in\mathbb{N}$.
\begin{theorem}
\label{TM1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then there exists a constant
$\varepsilon_{0}>0$ depending only on $p$ such that if
\begin{equation}
\label{EQsmallness}
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\end{equation}
then $\gamma\oo{\mathbb{S}^{1}}$ approaches a round circle exponentially fast
with radius $\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}$.
\end{theorem}
Although there is a plethora of negative results on the curve diffusion flow
violating positivity over time, there are relatively few results guaranteeing
preservation.
Theorem \ref{TM1} implies that after some \emph{waiting time}, the flow is
uniformly convex and remains forever so.
An estimate for the waiting time for the curve diffusion flow was given in
\cite{Wkosc}.
Here we extend this to each of the \eqref{FlowDefinition} flows.
\begin{proposition}
\label{PN1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. If $\gamma(\cdot,0)$ satisfies \eqref{EQsmallness}, then
\[
\mathcal{L}\{t\in[0,\infty)\,:\,k(\cdot,t)\not>0\}
\le \frac{2}{p+1}\bigg[
\bigg( \frac{L(\gamma_0)}{2\pi}\bigg)^{2(p+1)}
- \bigg( \frac{A(\gamma_0)}{\pi} \bigg)^{p+1}
\bigg]
\,.
\]
\end{proposition}
Above we have used $k(\cdot,t) \not> 0$ to mean $k(s_0,t) \le 0$ for at least
one $s_0$.
This estimate is optimal in the sense that the right hand side is zero for a
simple circle.
One may wish to compare this with the case for classical PDE of higher-order,
where exciting progress on eventual positivity continues to be made
\cite{DGK15,FGG08,GG08,GG09}.
The remainder of the present paper is devoted to proving Theorem \ref{TM1} and
Proposition \ref{PN1}.
We cover some basic definitions and integral formulae in Section 2, before
moving on to essential evolution equations for length, area, and curvature in
Section 3. We study $K_{osc}$ directly in Section 4, obtaining precise control
over $K_{osc}$ in the case where the initial data is sufficiently close in a
weak isoperimetric sense to a circle and has $K_{osc}$ initially smaller than
an explicit constant.
We continue by adapting Dziuk-Kuwert-Sch\"atzle's blowup criterion argument to
\eqref{FlowDefinition} flows (Lemma \ref{LongTimeLemma1}), yielding in Section
5 global existence.
Further analysis gives exponentially fast convergence to a circle with specific
radius dependent on the initial enclosed area.
We finish Section 5 by giving the proof of Proposition \ref{PN1}.
\end{section}
\begin{section}{Preliminaries}
\begin{lemma}\label{PrelimLemma1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$, and
$f:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}$ is a periodic
function with the same period as $\gamma$. Then
\begin{equation*}
\frac{d}{dt}\intcurve{f}=\intcurve{f_{t}+\oo{-1}^{p+1}f\cdot\kappa\cdot \kappa_{s^{2p}}}.
\end{equation*}
\end{lemma}
\begin{proof}
We first calculate the evolution of arc length. Because $\nu\perp\tau$, it follows from the Frenet-Serret equations that
\begin{align}
\partial_{s}ds&=\partial_{t}\norm{\gamma_{u}}\,du=\partial_{t}\inner{\gamma_{u},\gamma_{u}}^{\frac{1}{2}}\,du\nonumber\\
&=\norm{\gamma_{u}}^{-1}\inner{\partial_{ut}\gamma,\gamma_{u}}\,du=\inner{\partial_{s}\gamma_{t},\tau}\,ds\nonumber\\
&=\inner{\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu},\tau}\,ds=\oo{-1}^{p}\kappa_{s^{2p}}\inner{\partial_{s}\nu,\tau}\,ds\nonumber\\
&=\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}\,ds.\label{PrelimLemma1,0}
\end{align}
Next, using the fundamental theorem of calculus and $\oo{\ref{PrelimLemma1,0}}$ we have
\begin{align}
\frac{d}{dt}\int_{\gamma}{f\,ds}&=\frac{d}{dt}\int_{0}^{P\oo{t}}{f\oo{u,t}\left|\gamma_{u}\oo{u,t}\right|\,du}\nonumber\\
&=\intcurve{f_{t}}+\intcurve{f\partial_{t}}+P'\oo{t}\cdot\frac{d}{dP\oo{t}}\int_{0}^{P\oo{t}}{f\oo{u,t}\left|\gamma_{u}\oo{u,t}\right|\,du}\nonumber\\
&=\intcurve{f_{t}+\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}}+P'\oo{t}f\oo{P\oo{t},t}\left|\gamma_{u}\oo{P\oo{t},t}\right|\nonumber\\
&=\intcurve{f_{t}+\oo{-1}^{p+1}\kappa\cdot\kappa_{s^{2p}}}.\label{PrelimLemma1,1}
\end{align}
Here the last line follows from the fact that
\[
P'(t)\norm{\gamma_{u}\oo{P\oo{t},t}}=\Big(\partial_{t}\big(\gamma\oo{P\oo{t},t}-\gamma\oo{0,t}\big)\Big)^\top=0
\]
because $\partial_{t}\gamma$ is purely normal to $\gamma$.
\end{proof}
\end{section}
\begin{section}{Fundamental evolution equations}
\begin{corollary}\label{PrelimCor1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ Then
\[
\frac{d}{dt}L=-\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\text{ and }\frac{d}{dt}A=0.
\]
In particular, the isoperimetric ratio decreases in absolute value with velocity
\[
\frac{d}{dt}I=-\frac{2I}{L}\intcurve{\kappa_{s^{p}}^{2}}\leq0.
\]
\end{corollary}
\begin{proof}
Applying Lemma $\ref{PrelimLemma1}$ with $f\equiv1$ gives the statement for $L$:
\[
\frac{d}{dt}L=\frac{d}{dt}\intcurve{}=\oo{-1}^{p+1}\intcurve{\kappa\cdot \kappa_{s^{2p}}}=-\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\leq0.
\]
Here we have performed integration by parts $p$ times. For the statement regarding area, we first state the Frenet-Serret formulas with no torsion:
\begin{equation}
\tau_{s}=\kappa\nu\text{ and }\nu_{s}=-\kappa\tau.\label{PrelimCor1,1}
\end{equation}
Using the equations in $\oo{\ref{PrelimCor1,1}}$, we wish to derive a formula for the time derivative of the unit normal $\nu$. We first work out the commutator:
\begin{align}
\partial_{ts}&=\partial_{t}\oo{\partial_{s}}=\partial_{t}\oo{\norm{\gamma_{u}}^{-1}\partial_{u}}=\left|\gamma_{u}\right|^{-1}\partial_{t}\partial_{u}-\left|\gamma_{u}\right|^{-2}\oo{\partial_{t}\left|\gamma_{u}\right|}\partial_{u}\nonumber\\
&=\partial_{st}-\norm{\gamma_{u}}^{-3}\inner{\partial_{u}\gamma_{t},\gamma_{u}}\partial_{u}=\partial_{st}-\inner{\partial_{s}\gamma_{t},\tau}\partial_{s}\nonumber\\
&=\partial_{st}-\inner{\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu},\tau}\partial_{s}\nonumber\\
&=\partial_{st}+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\partial_{s}.\label{PrelimCor1,2}
\end{align}
We then use $\oo{\ref{PrelimCor1,1}},\oo{\ref{PrelimCor1,2}}$ and the identity $\gamma_{s}=\tau$ to calculate:
\begin{align}
\partial_{t}\tau&=\partial_{ts}\gamma=\partial_{st}\gamma+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\partial_{s}\gamma\nonumber\\
&=\oo{-1}^{p}\cc{\kappa_{s^{2p+1}}\cdot\nu-\kappa\cdot\kappa_{s^{2p}}\cdot\tau}+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\cdot\tau\nonumber\\
&=\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu.\label{PrelimCor1,3}
\end{align}
Using the fact that $\nu\perp\tau$ and $\norm{\nu}^{2}=1\implies\partial_{t}\nu\perp\nu$, it then follows from $\oo{\ref{PrelimCor1,3}}$ that
\begin{align}
\partial_{t}\nu&=\inner{\partial_{t}\nu,\tau}\tau=-\inner{\nu,\partial_{t}\tau}\tau\nonumber\\
&=-\inner{\nu,\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu}\tau=\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau.\label{PrelimCor1,4}
\end{align}
Applying Lemma $\ref{PrelimLemma1}$ with $f=\inner{\gamma,\nu}$ then gives
\begin{align}
\frac{d}{dt}A&=-\frac{1}{2}\frac{d}{dt}\int_{\gamma}{\inner{\gamma,\nu}\,ds}=-\frac{1}{2}\intcurve{\partial_{t}\inner{\gamma,\nu}+\oo{-1}^{p+1}\inner{\gamma,\nu}\cdot\kappa\cdot\kappa_{s^{2p}}}\nonumber\\
&=-\frac{1}{2}\intcurve{\inner{\oo{-1}^{p}\kappa_{s^{2p}}\cdot\nu,\nu}+\inner{\gamma,\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau}+\oo{-1}^{p+1}\inner{\gamma,\nu}\cdot\kappa\cdot\kappa_{s^{2p}}}\nonumber\\
&=-\frac{1}{2}\intcurve{\oo{-1}^{p}\kappa_{s^{2p}}+\oo{-1}^{p+1}\kappa_{s^{2p+1}}\inner{\gamma,\tau}+\oo{-1}^{p+1}\inner{\gamma,\tau_{s}}\kappa_{s^{2p}}}\nonumber\\
&-\frac{1}{2}\intcurve{\oo{-1}^{p}\kappa_{s^{2p}}+\oo{-1}^{p+1}\kappa_{s^{2p+1}}\inner{\gamma,\tau}+\oo{-1}^{p}\cc{\inner{\gamma_{s},\tau}\kappa_{s^{2p}}+\inner{\gamma,\tau}\kappa_{s^{2p+1}}}}\nonumber\\
&=\oo{-1}^{p+1}\intcurve{\kappa_{s^{2p}}}=\oo{-1}^{p+1}\kappa_{s^{2p-1}}\Bigg|_{s=0}^{s=L\oo{\gamma}}\nonumber\\
&=0.\nonumber
\end{align}
Here we have used integration by parts in the third last line. The last step follows from the divergence theorem, and using the periodicity of $\gamma$.
To establish the evolution equaiton for the isoperimetric ratio we simply combine the two established results for $L$ and $A$:
\begin{align}
\frac{\partial}{\partial t}I&=\frac{\partial}{\partial t}\oo{\frac{L^{2}}{4\pi A}}=\frac{1}{4\pi A^{2}}\cc{2AL\frac{\partial}{\partial t}L-L^{2}\frac{\partial}{\partial t}A}\nonumber\\
&=-\frac{2L}{4\pi A}\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\nonumber\\
&=-\frac{2I}{L}\int_{\gamma}{\kappa_{s^{p}}^{2}\,ds}\leq0.\nonumber
\end{align}
This completes the proof.
\end{proof}
\begin{lemma}\label{CurvatureLemma1}
Suppose that
$\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$ solves
$\eqref{FlowDefinition}$ and
\[
\intcurve{\kappa}\Big|_{t=0}=2\omega\pi.
\]
Then
\[
\intcurve{\kappa}=2\omega\pi
\]
for $t\in\left[0,T\right)$. Moreover, the average curvature $\overline{\kappa}=\frac{1}{L}\intcurve{\kappa}$ increases in absolute value with velocity
\[
\frac{d}{dt}\overline{\kappa}=\frac{2\omega\pi}{L^2}\llll{\kappa_{s^{p}}}_{2}^{2}\geq0.
\]
\end{lemma}
\begin{proof}
We first need to calculate the evolution equation for curvature. Using the definition $\kappa=\inner{\nu,\gamma_{ss}}$ along with previous identities, we have
\begin{align}
\frac{\partial}{\partial t}\kappa&=\frac{\partial}{\partial t}\inner{\nu,\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{t}\gamma_{ss}}=\inner{\nu_{t},\gamma_{ss}}+\inner{\nu,\partial_{ts}\tau}\nonumber\\
&=\inner{\oo{-1}^{p+1}\kappa_{s^{2p+1}}\cdot\tau,\kappa\cdot\nu}+\inner{\nu,\partial_{st}\tau+\oo{-1}^{p}\kappa\cdot\kappa_{s^{2p}}\cdot\tau_{s}}\nonumber\\
&=\inner{\nu,\partial_{s}\oo{\oo{-1}^{p}\kappa_{s^{2p+1}}\cdot\nu}+\oo{-1}^{p}\kappa^{2}\cdot\kappa_{s^{2p}}\cdot\nu}\nonumber\\
&=\oo{-1}^{p}\oo{\kappa_{s^{2p+2}}+\kappa^{2}\cdot\kappa_{s^{2p}}}.\label{CurvatureLemma1,1}
\end{align}
Then, applying Lemma $\ref{PrelimLemma1}$ with $f=\kappa$ gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa}=\intcurve{\kappa_{t}+\oo{-1}^{p+1}\kappa^{2}\cdot\kappa_{s^{2p}}}=0.\label{CurvatureLemma1,3}
\end{equation}
It follows from $\oo{\ref{CurvatureLemma1,3}}$ that the integral $\intcurve{\kappa}$ stays constant on $\left[0,T\right)$. This gives the first assertion of the lemma. For the second assertion, we simply use $\oo{\ref{CurvatureLemma1,3}}$ and Corollary $\ref{PrelimCor1}$ and compute:
\begin{align*}
\frac{d}{dt}\overline{\kappa}&=\frac{d}{dt}\oo{\frac{1}{L}\intcurve{\kappa}}=\frac{1}{L^2}\cc{L\cdot\frac{d}{dt}\intcurve{\kappa}-\intcurve{\kappa}\cdot\frac{d}{dt}L}\\
&=-\frac{2\omega\pi}{L^{2}}\cdot-\intcurve{\kappa_{s^{p}}^{2}}=\frac{2\omega\pi}{L^{2}}\llll{\kappa_{s^{p}}}_{2}^{2}\\
&\geq0.
\end{align*}
This completes the proof.
\end{proof}
\end{section}
\begin{section}{The Normalised Oscillation of Curvature}
We now introduce a scale-invariant quantity
\[
K_{osc}:=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}
\]
which we call the \emph{normalised oscillation of curvature}.
One can deduce from our previous calculations that this quantity is a natural
one, being that for a one parameter family of curves $\gamma_{t}$ that solves
$\eqref{FlowDefinition}$, $K_{osc}\oo{t}$ is a bounded quantity in $L^{1}$
(and in fact is bounded by a quantity that depends on the initial data,
$\gamma_{0}$ and so can be controlled a priori). Indeed, The fact that
$\intcurve{\oo{\kappa-\bar{\kappa}}}=0$ means that we can apply Lemma
$\ref{AppendixLemma1}$, giving
\[
K_{osc}=L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\leq L\oo{\frac{L}{2\pi}}^{2}\intcurve{\kappa_{s}^{2}}.
\]
Now the periodicity of $\kappa$ implies that for every $i\geq 1$, $\intcurve{\kappa_{s^{i}}}=0$, so we can apply Lemma $\ref{AppendixLemma1}$ to the right hand side of the above inequality $p$ more times, yielding
\begin{equation}
K_{osc}\leq L\oo{\frac{L}{2\pi}}^{2p}\intcurve{\kappa_{s^{p}}^{2}}=-\frac{L^{2p+1}}{\oo{2\pi}^{2p}}\frac{d}{dt}L=-\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}\frac{d}{dt}\oo{L^{2\oo{p+1}}}.\label{OscillationOfCurvature1}
\end{equation}
Here we have utilised the evolution of the length functional. We conclude that for any $t\in\left[0,T\right)$
\begin{align}
\int_{0}^{t}{K_{osc}\oo{\tau}\,d\tau}&\leq-\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}\oo{L^{2\oo{p+1}}\oo{\gamma_{t}}-L^{2p+1}\oo{\gamma_{0}}}\nonumber\\
&\leq\frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{\gamma_{0}}.\label{OscillationCurvature2}
\end{align}
We deduce from $\oo{\ref{OscillationCurvature2}}$ that the normalised oscillation of curvature is a priori controlled in $L^{1}$ over the time of existence of the flow. Furthermore, by repeatedly using Lemma $\ref{AppendixLemma2}$ in a similar fashion, one can easily obtain an $L^{1}$ bound for $\llll{\kappa-\bar{\kappa}}_{\infty}^{2}$ over the interval $\left[0,T\right)$. Firstly
\begin{align*}
\llll{\kappa-\bar{\kappa}}_{\infty}^{2}&\leq\frac{L}{2\pi}\intcurve{\kappa_{s}^{2}}\leq \frac{L}{2\pi}\oo{\frac{L}{2\pi}}^{2}\intcurve{\kappa_{s^{2}}^{2}}\\
&\vdots\\
&\leq \frac{L}{2\pi}\oo{\frac{L}{2\pi}}^{2\oo{p-1}}\intcurve{\kappa_{s^{p}}^{2}}=-\frac{L^{2p-1}}{\oo{2\pi}^{2p-1}}\frac{d}{dt}L\\
&=-\frac{1}{2p\oo{2\pi}^{2p-1}}\frac{d}{dt}L^{2p}.
\end{align*}
Hence for any $t\in\left[0,T\right)$,
\begin{equation}
\int_{0}^{t}{\llll{\kappa-\bar{\kappa}}_{\infty}^{2}\,d\tau}\leq-\frac{1}{2p\oo{2\pi}^{2p-1}}\oo{L^{2p}\oo{\gamma_{t}}-L^{2p}\oo{\gamma_{0}}}\leq\frac{1}{2p\oo{2\pi}^{2p-1}}L^{2p}\oo{\gamma_{0}}.\label{OscillationCurvature3}
\end{equation}
Next we formulate the evolution equation for $K_{osc}$.
\begin{lemma}\label{CurvatureLemma2}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then
\begin{align*}
&\frac{d}{dt}\oo{K_{osc}+8\omega^{2}\pi^{2}\ln{L}}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+2L\llll{\kappa_{s^{p+1}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}.
\end{align*}
\end{lemma}
\begin{proof}
We have
\begin{align*}
&\frac{d}{dt}K_{osc}=\frac{d}{dt}L\cdot\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}+L\cdot\frac{d}{dt}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\\
&=-\llll{\kappa_{s^{p}}}_{2}^{2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}+L\Biggl[2\intcurve{\oo{\kappa-\bar{\kappa}}\kappa_{t}}+\oo{-1}^{p+1}\intM{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot\kappa_{s^{2p}}}\Biggr]\\
&=-\frac{\llll{k_{s^{p}}}_{2}^{2}}{L}K_{osc}+2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\oo{\kappa_{s^{2p+2}}+\kappa^{2}\cdot\kappa_{s^{2p}}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot\kappa_{s^{2p}}}\\
&=-\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}-2L\llll{\kappa_{s^{p+1}}}_{2}^{2}+2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\cdot\kappa^{2}\cdot\kappa_{s^{2p}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\kappa\cdot \kappa_{s^{2p}}}.
\end{align*}
Hence
\begin{align*}
&\frac{d}{dt}K_{osc}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+2L\llll{\kappa_{s^{p+1}}}_{2}^{2}\\
&=2\oo{-1}^{p}L\intcurve{\oo{\kappa-\bar{\kappa}}\cdot\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\cdot \kappa_{s^{2p}}}\\
&+\oo{-1}^{p+1}L\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\cdot\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\cdot\kappa_{s^{2p}}}\\
&=\oo{-1}^{p}L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}^{2}\oo{\kappa-\bar{\kappa}}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}+2\bar{\kappa}^{2}L\llll{\kappa_{s^{p}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}+\frac{8\omega^{2}\pi^{2}}{L}\llll{\kappa_{s^{p}}}_{2}^{2}\\
&=L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}-8\omega^{2}\pi^{2}\frac{d}{dt}\ln{L}
\end{align*}
Here we have used Corollary $\ref{PrelimCor1}$ and Lemma $\ref{CurvatureLemma1}$ in the penultimate step. Rearranging then yields the desired result.
\end{proof}
\begin{lemma}\label{CurvatureLemma3}
\begin{equation}
L\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{3}+\bar{\kappa}\oo{\kappa-\bar{\kappa}}^{2}}_{s^{p}}\oo{\kappa-\bar{\kappa}}_{s^{p}}}\leq L\oo{c_{1}K_{osc}+c_{2}\sqrt{K_{osc}}}
\end{equation}
for some universal constants $c_{1},c_{2}>0$. Here $c_{i}=c_{i}\oo{p}$.
\end{lemma}
\begin{proof}
The proof follows from an application of a number of interpolation inequalities which can be found in \cite{DKS}. It has been included in the Appendix for the convenience of the reader.
\end{proof}
\begin{corollary}\label{CurvatureCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then
\[
\frac{d}{dt}\oo{K_{osc}+8\omega^{2}\pi^{2}\ln{L}}+\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}K_{osc}+L\oo{2-c_{1}K_{osc}-c_{2}\sqrt{K_{osc}}}\llll{\kappa_{s^{p+1}}}_{2}^{2}\leq0.
\]
Here $c_{1}\oo{p},c_{2}\oo{p}$ are the universal constants given in Lemma $\ref{CurvatureLemma3}$. Moreover, if there exists a $T^{*}$ such that for $t\in\left[0,T^{*}\right)$
\begin{equation}
K_{osc}\oo{t}\leq\frac{8c_{1}+2c_{2}^{2}-2c_{2}\sqrt{8c_{1}+c_{2}^{2}}}{4c_{1}^{2}}=2K^{*},\label{CurvatureCorollary1,0}
\end{equation}
then during this time the estimate
\begin{equation}
K_{osc}+8\omega^{2}\pi^{2}\ln{L}+\int_{0}^{t}{K_{osc}\frac{\llll{\kappa_{s}^{p}}_{2}^{2}}{L}\,d\tau}\leq K_{osc}\oo{0}+8\omega^{2}\pi^{2}\ln{L\oo{0}}\label{CurvatureCorollary1,1}
\end{equation}
holds.
\end{corollary}
\begin{proof}
Combining Lemma $\ref{CurvatureLemma2}$ and Lemma $\ref{CurvatureLemma3}$ immediately gives the first result. Using the assumed smallness of $K_{osc}$ then gives the second.
\end{proof}
Note that although Corollary $\ref{CurvatureCorollary1}$ implies that the
normalised oscillation of curvature remains bounded if initially sufficiently
small, it does not seem to give tight control of the quantity per se, because
we already know that $\ln{L}$ (on the left hand side of
$\oo{\ref{CurvatureCorollary1,1}}$) is decreasing, and so without further
analysis, one might think that $K_{osc}$ could be static in time (or even
worse, \emph{increasing}).
However, note by the isoperimetric inequality that for any closed curve solving
$\eqref{FlowDefinition}$ we have
\[
\frac{L^{2}\oo{\gamma}}{4\pi A\oo{\gamma}}\geq1,\text{ and so }\frac{1}{L\oo{\gamma}}\leq\frac{1}{\sqrt{4\pi A\oo{\gamma}}}.
\]
It follows that for any $t\in\left[0,T\right)$,
\begin{equation}
\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}\leq\frac{L\oo{\gamma_{0}}}{\sqrt{4\pi A\oo{\gamma_{t}}}}=\frac{L\oo{\gamma_{0}}}{\sqrt{4\pi A\oo{\gamma_{0}}}}=\sqrt{I\oo{\gamma_{0}}}.\label{IsoperimetricConsequence}
\end{equation}
Here we have used the fact that by Corollary $\ref{PrelimCor1}$, the enclosed area of our family of immersed curves is static in time.
So, the quantity $\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}$ can be controlled over $\left[0,T\right)$ a priori by assuming that $\gamma_{0}$ is ``sufficiently circular''. In particular, since we may choose $\gamma_{0}$ such that $I\oo{\gamma_{0}}$ is as close to $1$ as we wish (and so $\frac{L\oo{\gamma_{0}}}{L\oo{\gamma_{t}}}$ remains close to $1$ as well), equation $\oo{\ref{CurvatureCorollary1,1}}$ becomes much more appealing because it can be rearranged to give
\begin{equation}
K_{osc}+\int_{0}^{t}{K_{osc}\frac{\llll{\kappa_{s^{p}}}_{2}^{2}}{L}\,d\tau}\leq K_{osc}\oo{0}+8\omega^{2}\pi^{2}\ln{\sqrt{I\oo{0}}}=K_{osc}\oo{0}+4\omega^{2}\pi^{2}\ln\oo{{I\oo{0}}}.\label{OscillationOfCurvature4}
\end{equation}
This of course is an improvement upon Corollary $\ref{CurvatureCorollary1}$ because it tells us that $K_{osc}$ can not get larger than the right hand side of the inequality. One problem is that this inequality as it stands is only valid whilst $K_{osc}$ satisfies $\oo{\ref{CurvatureCorollary1,0}}$, and it is not clear from $\oo{\ref{OscillationOfCurvature4}}$ that this smallness condition should hold for the duration of the flow.
A little bit of tweaking will give us tighter control over $K_{osc}$ for the duration of the flow, and we present this result in the following proposition.
\begin{proposition}\label{FlowProp2}
Let $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solve $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is a
simple closed curve with $\omega=1$, satisfying
\[
K_{osc}\oo{0}\leq K^{\star}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}.
\]
Then
\[
K_{osc}\oo{t}\leq 2K^{\star}\,\,\text{for}\,\,t\in\left[0,T\right).
\]
\end{proposition}
\begin{proof}
Suppose for the sake of contradiction that $K_{osc}$ does \emph{not} remain bounded by $2K^{\star}$ for the duration of the flow. Then we can find a maximal $T^{\star}<T$ such that
\[
K_{osc}\oo{t}\leq 2K^{\star}\text{ for }t\in\left[0,T^{\star}\right).
\]
Then, by $\oo{\ref{OscillationOfCurvature4}}$, the following identity holds for $t\in\left[0,T^{\star}\right)$:
\begin{equation}
K_{osc}\oo{t}\leq K_{osc}\oo{0}+4\pi^{2}\ln\oo{{I\oo{0}}}\leq K^{\star}+4\pi^{2}\ln\oo{e^{\frac{K^{\star}}{8\pi^{2}}}}=\frac{3K^{\star}}{2}\text{ for }t\in\left[0,T^{\star}\right).\label{FlowProp2,1}
\end{equation}
We have also used the fact that Lemma $\ref{CurvatureLemma1}$ ensures that $\omega=1$ for the duration of the flow.
Taking $t\nearrow T$ in inequality $\oo{\ref{FlowProp2,1}}$ gives $K_{osc}\leq\frac{3K^{\star}}{2}<2K^{\star}$, meaning that by continuity, $K_{osc}\leq 2K^{\star}$ on some larger time interval $\left[0,T^{\star}+\delta\right)$. But $\left[0,T^{\star}\right)$ was chosen to be the largest time interval containing $0$ such that $K_{osc}$ remains bounded by $2K^{\star}$ and so we have arrived at a contradiction. Thus our assumption that $T<T^{\star}$ must have been false, and the result of the proposition follows.
\end{proof}
\begin{corollary}\label{CurvatureCorollary2}
Let $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solve $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is a
simple embedded closed curve satisfying
\[
K_{osc}\oo{0}\leq K^{\star}<\epsilon_{0}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}\leq e^{\frac{\varepsilon_{0}}{8\pi^{2}}},
\]
where $\epsilon_{0}<32-2\pi^{2}$ is a sufficiently small constant. Then $\gamma$ remains embedded on $\left[0,T\right)$.
\end{corollary}
\begin{proof}
Suppose $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ is a smooth immersed curve with winding number $\omega=1$. From the Gauss-Bonnet theorem and Lemma $\ref{CurvatureLemma1}$, we know that for $t\in\left[0,T\right)$ the winding number of $\gamma_{t}$ remains the same. Therefore the hypothesis of the corollary implies that $\omega=1$ for the duration of the flow. Define $m\oo{\gamma}$ to be the maximum number of times that $\gamma$ intersects itself in any one point. That is,
\[
m\oo{\gamma}:=\sup_{x\in\mathbb{R}^{2}}\norm{\gamma^{-1}\oo{x}}.
\]
By Theorem $16$ from \cite{Wkosc}, $m$ satisfies the following inequality:
\begin{equation}
K_{osc}\oo{\gamma}\geq16m^{2}-4\omega^{2}\pi^{2}=16m^{2}-4\pi^{2}.\nonumber
\end{equation}
Hence
\begin{equation}
m^{2}\leq\frac{1}{16}\oo{K_{osc}\oo{\gamma}+4\pi^{2}}.\label{CurvatureCorollary2,1}
\end{equation}
Proposition $\ref{FlowProp2}$ then tells us that by the hypothesis of the corollary, $K_{osc}$ remains bounded above by $2K^{\star}$ for the duration of the flow. We can assume without loss of generality that $K^{\star}<32-2\pi^{2}\approx12.26$, and so we have $K_{osc}\oo{\gamma}<64-4\pi^{2}$ on $\left[0,T\right)$. Therefore by $\oo{\ref{CurvatureCorollary2,1}}$ we have
\[
m^{2}<\frac{1}{16}\oo{64-4\pi^{2}+4\pi^{2}}=4\,\,\text{for}\,\,t\in\left[0,T\right),
\]
and embeddedness follows immediately.
\end{proof}
\begin{lemma}\label{LongTimeLemma1}
Suppose $\gamma:\mathbb{S}^{1}\times \left[0,T\right)\rightarrow\mathbb{R}^{2}$
is a maximal solution to $\eqref{FlowDefinition}$. If $T<\infty$ then
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+1}}
\]
for a universal constant $c>0$.
\end{lemma}
\begin{proof}
Deriving an evolution equation for $\intcurve{\kappa^{2}}$ in the same manner as Lemma $\ref{CurvatureLemma2}$ and using an interpolation inequality in the same vein as \cite{DKS} gives us
\begin{equation}
\frac{d}{dt}\intcurve{\kappa^{2}}+\intcurve{\kappa_{s^{p+1}}^{2}}\leq c\oo{p}\oo{\intcurve{\kappa^{2}}}^{2\oo{m+p}+3},\nonumber
\end{equation}
which implies that
\begin{equation}
-\frac{1}{2\oo{p+1}}\cc{\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{1}}}^{-1/2\oo{p+1}}-\oo{\intcurve{\kappa^{2}}\Big|_{t=t_{0}}}^{-1/2\oo{p+1}}}\leq c\oo{t_{1}-t_{0}}.\label{LongTimeLemma1,1}
\end{equation}
for any times $t_{0}\leq t_{1}$. Note that if $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}=\infty$, then taking $t_{1}\nearrow T$ in $\oo{\ref{LongTimeLemma1,1}}$ and rearranging will prove the lemma. Assume for the sake of contradiction that $\intcurve{\kappa^{2}}\leq\varrho$ for all $t<T$. By using an argument similar to Theorem $3.1$ of \cite{Wkosc}, we are able to show that the inequality
\[
\llll{\partial_{u}^{m}\gamma}_{\infty}\leq c_{m}\oo{\varrho,\gamma_{0},T}<\infty
\]
holds for every $m\in\mathbb{N}$ up until time $T$. By short time existence we are then able to extend the life of the flow, contradicting the maximality of $\gamma$. Hence our assumption that $\limsup_{t\rightarrow T}\intcurve{\kappa^{2}}<\infty$ must have been incorrect, and so the limit must diverge. We then conclude the desired result of the lemma from $\oo{\ref{LongTimeLemma1,1}}$.
\end{proof}
\end{section}
\begin{section}{Global analysis}
\begin{corollary}\label{LongTimeCorollary1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Additionally, suppose that $\gamma_{0}$ is
a simple closed curve satisfying
\[
K_{osc}\oo{0}\leq K^{\star}\,\,\text{and}\,\,I\oo{0}\leq e^{\frac{K^{\star}}{8\pi^{2}}}.
\]
Then $T=\infty$.
\end{corollary}
\begin{proof}
Suppose for the sake of contradiction that $T<\infty$. Then by Lemma $\ref{LongTimeLemma1}$ we have
\[
\intcurve{\kappa^{2}}\geq c\oo{T-t}^{-1/2\oo{p+3}}
\]
and so in particular,
\begin{equation}
\intcurve{\kappa^{2}}\rightarrow\infty\text{ as }t\rightarrow T.\label{LongTimeCorollary1,1}
\end{equation}
Next note that $\oo{\ref{IsoperimetricConsequence}}$ gives us an absolute lower bound on the length of $\gamma$:
\[
L\oo{\gamma_{t}}\geq\sqrt{4\pi A\oo{\gamma_{0}}}.
\]
Hence we establish the following following bound on $K_{osc}$:
\[
K_{osc}=L\intcurve{\kappa^{2}}-4\pi^{2}\geq\sqrt{4\pi A\oo{\gamma_{0}}}\intcurve{\kappa^{2}}-4\pi^{2}.
\]
Hence it follows from $\oo{\ref{LongTimeCorollary1,1}}$ that
\[
K_{osc}\oo{t}\rightarrow\infty\,\,\text{as}\,\,t\rightarrow T.
\]
But this directly contradicts the results of Proposition $\ref{FlowProp2}$, and so we conclude that our assumption that $T$ was finite must have been incorrect. Thus $T=\infty$.
\end{proof}
Recall we know that if $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$ satisfies the hypothesis of Corollary $\ref{LongTimeCorollary1}$ then $T=\infty$, and then identity $\oo{\ref{OscillationCurvature2}}$ tells us that
\begin{equation}
K_{osc}\in L^{1}\oo{\left[0,\infty\right)},\,\,\text{with}\,\,\int_{0}^{\infty}{K_{osc}\oo{\tau}\,d\tau}\leq \frac{1}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{0}.\label{OscillationOfCurvatureL1}
\end{equation}
So we can conclude that the ``tail'' of the function $K_{osc}\oo{t}$ must get small as $t\nearrow\infty$. However, at the present time we have not ruled out the possibility that $K_{osc}$ gets smaller and smaller as $t$ gets large, whilst vibrating with higher and higher frequency, remaining in $L^{1}\oo{\left[0,\infty\right)}$ whilst never actually fully dissipating to zero in a smooth sense. To rule out this from happening, it is enough to show that $\norm{\frac{d}{dt}K_{osc}}$ remains bounded by a universal constant for all time. To do so we will need to first show that $\llll{\kappa_{s^{p}}}_{2}^{2}$ remains bounded. We will address this issue with the following proposition.
\begin{proposition}\label{LongTimeProp1}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ and is simple. There exists a
$\varepsilon_{0}>0$ (with $\varepsilon_{0}\leq K^{\star}$) such that if
\[
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\]
then $\llll{k_{s^{p}}}_{2}^{2}$ remains bounded for all time. In particular,
\[
\intcurve{\kappa_{s^{p}}^{2}}\leq\tilde{c}\oo{\gamma_{0}}
\]
for some constant $\tilde{c}\oo{\gamma_{0}}$ depending only upon the initial immersion.
\end{proposition}
\begin{proof}
We first derive the evolution equation for the quantity $\intcurve{\kappa_{s^{p}}^{2}}$. Applying Lemma $\ref{PrelimLemma1}$ along with repeated applications of the formula for the commutator $\cc{\partial_{t},\partial_{s}}$, we have
\begin{align}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}&=2\intcurve{\kappa_{s^{p}}\partial_{t}\kappa_{s^{p}}}+\intcurve{\kappa\cdot\kappa_{s^{p}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\oo{-1}^{p}\sum_{j=0}^{p}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{p-j}}\cdot\kappa_{s^{p+j}}\cdot\kappa_{s^{2p}}}\nonumber\\
&+\oo{-1}^{p+1}\intcurve{\kappa\cdot\kappa_{s^{p}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=0}^{p-1}\oo{-1}^{j}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+j}}\oo{\kappa_{s^{2p}}}}\nonumber\\
&+\oo{-1}^{p+1}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&\leq-2\intcurve{\kappa_{s^{2p+1}}^{2}}+2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}\nonumber\\
&+c\oo{p}\intcurve{\norm{P_{4}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}+c\oo{p}L^{-1}\intcurve{\norm{P_{3}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}.\label{LongTimeProp1,1}
\end{align}
Here $P_{i}^{j,k}\oo{\cdot}$ stands for a polynomial in $\phi$ of the form
\[
P_{i}^{j,k}\oo{\phi}=\sum_{\mu_{1}+\dots+\mu_{i}=j,\mu_{l}\leq k}\partial_{s}^{\mu_{1}}\phi\star\partial_{s}^{\mu_{2}}\phi\star\cdots\star\partial_{s}^{\mu_{i}}\phi.
\]
(See, for example \cite{DKS} for more details).
Using Lemma $\ref{AppendixLemma5}$, it follows that
\[
\intcurve{\norm{P_{4}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{4p,2p}\oo{\kappa-\bar{\kappa}}}}\leq c\oo{p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{2p+1}}^{2}},
\]
Hence inequality $\oo{\ref{LongTimeProp1,1}}$ can be rearranged to read
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}+\oo{2-c\oo{p}\oo{K_{osc}+\sqrt{K_{osc}}}}\intcurve{\kappa_{s^{2p+1}}^{2}}\leq2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}.\label{LongTimeProp1,2}
\end{equation}
Next we expand the right hand side of $\ref{LongTimeProp1,2}$ and use the Cauchy-Schwarz inequality on the result:
\begin{align}
2\intcurve{\kappa^{2}\cdot\kappa_{s^{2p}}^{2}}&=2\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\kappa_{s^{2p}}^{2}}\nonumber\\
&\leq4\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\kappa_{s^{2p}}^{2}}+4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}.\label{LongTimeProp1,5}
\end{align}
The first term in $\oo{\ref{LongTimeProp1,5}}$ can be estimated easily, using
Lemma \ref{AppendixLemma2} with $f=\kappa_{s^{2p}}$:
\begin{align}
4\intcurve{\oo{\kappa-\bar{\kappa}}^{2}\kappa_{s^{2p}}^{2}}&\leq4\llll{\kappa_{s^{2p}}}_{\infty}^{2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}\nonumber\\
&\leq 4\oo{\frac{L}{2\pi}\intcurve{\kappa_{s^{2p+1}}^{2}}}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}=\frac{2}{\pi}K_{osc}\intcurve{\kappa_{s^{2p+1}}^{2}}.\label{LongTimeProp1,6}
\end{align}
The second term is dealt with by using Lemma $\ref{AppendixLemma0}$
with $m=2p$:
\begin{align}
4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}&=16\pi^{2}L^{-2}\intcurve{\kappa_{s^{2p}}^{2}}\nonumber\\
&\leq16\pi^{2}L^{-2}\oo{\varepsilon L^{2}\intcurve{\kappa_{s^{2p+1}}^{2}}+\frac{1}{4\varepsilon^{2p}}L^{-\oo{4p+1}}K_{osc}}\nonumber\\
&=16\pi^{2}\varepsilon\intcurve{\kappa_{s^{2p+1}}^{2}}+\frac{16\pi^{2}}{4\varepsilon^{2p}}L^{-\oo{4p+3}}K_{osc}.\nonumber
\end{align}
Here, of course $\varepsilon>0$ can be made as small as desired. Letting $\varepsilon^{\star}=16\pi^{2}\varepsilon$ yields
\begin{equation}
4\bar{\kappa}^{2}\intcurve{\kappa_{s^{2p}}^{2}}\leq\varepsilon^{\star}\intcurve{\kappa_{s^{2p+1}}^{2}}+4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}L^{-\oo{4p+3}}K_{osc}.\label{LongTimeProp1,7}
\end{equation}
Substituting $\oo{\ref{LongTimeProp1,6}}$ and $\oo{\ref{LongTimeProp1,7}}$ into $\oo{\ref{LongTimeProp1,2}}$ gives
\begin{align}
&\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}+\oo{2-\oo{c\oo{p}+\frac{2}{\pi}+\varepsilon^{\star}}K_{osc}-c\oo{p}\sqrt{K_{osc}}}\intcurve{\kappa_{s^{2p+1}}^{2}}\nonumber\\
&\leq4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}L^{-\oo{4p+3}}K_{osc}\leq4\pi^{2}\oo{\frac{16\pi^{2}}{\varepsilon^{\star}}}^{2p}\oo{4\pi A\oo{\gamma_{0}}}^{-\oo{4p+3}/2}K_{osc}.\label{LongTimeProp1,8}
\end{align}
Here we have used the inequality $\oo{\ref{IsoperimetricConsequence}}$ in the last step. Hence Proposition $\ref{FlowProp2}$ tells us that choosing choosing $K_{osc}\oo{0}<\varepsilon_{0}$ for $\varepsilon_{0}>0$ sufficiently small yields the following inequality
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{p}}^{2}}\leq c\oo{\gamma_{0}}K_{osc}\label{LongTimeProp1,9}
\end{equation}
for some constant $c\oo{\gamma_{0}}$ which only depends upon our initial immersion. This inequality is valid over $\left[0,T\right)$. Note that we have chosen $\varepsilon^{\star}$ to be sufficiently small so that the absorption process is valid in the last step.
Integrating $\oo{\ref{LongTimeProp1,9}}$ while using our $L^{1}$ bound for $K_{osc}$ from $\oo{\ref{OscillationOfCurvatureL1}}$ then yields for any $t\in\left[0,T\right)$ the following inequality:
\[
\intcurve{\kappa_{s^{p}}^{2}}\leq\intcurve{\kappa_{s^{p}}^{2}}\Big|_{t=0}+\frac{c\oo{\gamma_{0}}}{2\oo{p+1}\oo{2\pi}^{2p}}L^{2\oo{p+1}}\oo{\gamma_{0}}\leq \tilde{c}\oo{\gamma_{0}}
\]
for some new constant $\tilde{c}\oo{\gamma_{0}}$ that only depends on the initial immersion. This completes the proof.
\end{proof}
\begin{corollary}\label{LongTimeCorollary2}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$. Then there exists a constant
$\varepsilon_{0}>0$ (with $\varepsilon_{0}\leq K^{\star}$) such that if
\[
K_{osc}\oo{0}<\varepsilon_{0}\text{ and }I\oo{0}<e^{\frac{\varepsilon_{0}}{8\pi^{2}}}
\]
then $\gamma\oo{\mathbb{S}^{1}}$ approaches a round circle with radius $\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}$.
\end{corollary}
\begin{proof}
Recall from a previous discussion that to show $K_{osc}\searrow0$, it will be enough to show that $\norm{K_{osc}'}$ is bounded for all time.
Firstly, by Corollary $\ref{CurvatureCorollary1}$ and Corollary $\ref{LongTimeCorollary1}$ we know that for $\epsilon_{0}>0$ sufficiently small $T=\infty$ and for all time we have the estimate
\[
\norm{\frac{d}{dt}K_{osc}}\leq\oo{\frac{8\pi^{2}-K_{osc}}{L}}\llll{\kappa_{s^{p}}}_{2}^{2}\leq\frac{8\pi^{2}}{\sqrt{4\pi A\oo{\gamma_{0}}}}\llll{\kappa_{s^{p}}}_{2}^{2}\leq\frac{8\pi^{2}}{\sqrt{4\pi A\oo{\gamma_{0}}}}\cdot\tilde{c}\oo{\gamma_{0}}<\infty.
\]
Here we have used also the results of Proposition $\ref{LongTimeProp1}$.
This immediately tells us that $K_{osc}\searrow0$ as $t\nearrow\infty$.
We will denote the limiting immersion by $\gamma_{\infty}$. That is,
\[
\gamma_{\infty}:=\lim_{t\to\infty}\gamma_{t}\oo{\mathbb{S}^{1}}=\lim_{t\to\infty}\gamma\oo{\cdot,t}.
\]
Our earlier equations imply that $K_{osc}\oo{\gamma_{\infty}}\equiv0$. Note that because the isoperimetric inequality forces $L\oo{\gamma_{\infty}}\geq\sqrt{4\pi A\oo{\gamma_{\infty}}}=\sqrt{4\pi A\oo{\gamma_{0}}}>0$, we can not have $L\searrow0$ and so we may conclude that
\begin{equation}
\int_{\gamma_{\infty}}{\oo{\kappa-\bar{\kappa}}^{2}\,ds}=0.\label{LongTimeCorollary2,1}
\end{equation}
It follow from $\oo{\ref{LongTimeCorollary2,1}}$ that $\kappa\oo{\gamma_{\infty}}\equiv C$ for some constant $C>0$ (note that we know $C$ must be positive because it is impossible for a closed curve with constant curvature to possess negative curvature). That is to say, $\gamma_{t}\oo{\mathbb{S}^{1}}$ approaches a round circle as $t\nearrow\infty$. The final statement of the Corollary regarding the radius of $\gamma_{\infty}$ (which we denote $r\oo{\gamma_{\infty}}$) then follows easily because the enclosed area $A\oo{\gamma_{t}}$ is static in time:
\[
r\oo{\gamma_{\infty}}=\sqrt{\frac{A\oo{\gamma_{\infty}}}{\pi}}=\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}.
\]
\end{proof}
Since the previous corollary tells us that $\gamma_{t}\oo{\mathbb{S}^{1}}\rightarrow\mathbb{S}_{\sqrt{\frac{A\oo{\gamma_{0}}}{\pi}}}^{1}$, we can conclude that for every $m\in\mathbb{N}$ there exists a sequence of times $\left\{t_{j}\right\}$ such that
\[
\int{\kappa_{s^{m}}^{2}}\Big|_{t=t_{j}}\searrow0.
\]
Unfortunately, this is only subconvergence, and does not allow us to rule out the possibility of short sharp ``spikes'' (oscillations) in time. Indeed, even if we were to show that for every $m\in\mathbb{N}$ we have $\llll{\kappa_{s^{m}}}_{2}^{2}\in L^{1}\oo{\left[0,\infty\right)}$ (which is true), this would not be enough because these aforementioned ``spikes'' could occur on a time interval approaching that of (Lebesgue) measure zero. To overcome this dilemma, we attempt to control $\norm{\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}}$, and show that his quantity can be bounded by a multiple of $K_{osc}\oo{0}$ (which can be fixed to be as small as desired a priori). We will see this allows to strengthen the subconvergences result above to one of classical exponential convergence.
\begin{corollary}[Exponential Convergence]\label{LongTimeCorollary3}
Suppose $\gamma:\mathbb{S}^{1}\times\left[0,T\right)\rightarrow\mathbb{R}^{2}$
solves $\eqref{FlowDefinition}$ and satisfies the assumptions of Corollary
$\ref{LongTimeCorollary2}$. Then for each $m\in\mathbb{N}$ there are constants
$c_{m},c_{m}^{\star}$ such that we have the estimates
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq c_{m}e^{-c_{m}^{\star}t}\text{ and }\llll{\kappa_{s^{m}}}_{\infty}\leq \sqrt{\frac{L\oo{\gamma_{0}}c_{m+1}}{2\pi}}e^{-\frac{c_{m+1}^{\star}}{2}t}.
\]
\end{corollary}
\begin{proof}
We first derive the evolution equation for $\intcurve{\kappa_{s^{m}}^{2}},m\in\mathbb{N}$ in a similar manner to Proposition $\ref{LongTimeProp1}$:
\begin{align}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}&=-2\intcurve{\kappa_{s^{m+p+1}}^{2}}+2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}\nonumber\\
&+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}.\label{LongTimeCorollary3,1}
\end{align}
We need to be careful in dealing with the extraneous terms in $\oo{\ref{LongTimeCorollary3,1}}$. We wish to apply Lemma $\ref{AppendixLemma5}$
but to do so must consider the cases $p\geq m$ and $p\leq m$ separately.
If $p\geq m$, with $p=m+l,l\in\mathbb{N}_{0}$, then we can perform integration by parts on each term in $\oo{\ref{LongTimeCorollary3,1}}$ $l$ times:
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{m-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m-j}}\oo{\kappa-\bar{\kappa}}_{s^{m+j}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{2m}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&+\oo{-1}^{p}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{m-1}\oo{-1}^{j+l}\intcurve{\partial_{s}^{l}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m-j}}\oo{\kappa-\bar{\kappa}}_{s^{m+j}}}\oo{\kappa-\bar{\kappa}}_{s^{p+p}}}\nonumber\\
&+2\intcurve{\partial_{s}^{l}\cc{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{2m}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p}}}\nonumber\\
&+\oo{-1}^{p+l}\intcurve{\partial_{s}^{l}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{m}}^{2}}\oo{\kappa-\bar{\kappa}}_{s^{m+p}}}\nonumber\\
&\leq c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\cdot L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+2\bar{\kappa}^{2}\intcurve{\kappa_{s^{2m+l}}\cdot\kappa_{s^{m+p}}}\nonumber\\
&\leq4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}+c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,2}
\end{align}
Here we have used the energy inequality Lemma \ref{AppendixLemma5} on the $P$-style terms, as well as Lemma $\ref{CurvatureLemma1}$ which tells us that $\bar{\kappa}=\frac{2\pi}{L}$.
If $p<m$ (say with $m=p+l,l\in\mathbb{N}$), then we must proceed slightly differently. In this case, identity $\oo{\ref{LongTimeCorollary3,1}}$ becomes
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa^{2}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&=2\oo{-1}^{p}\sum_{j=1}^{p}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+2\oo{-1}^{m+p}\intcurve{\cc{\oo{\kappa-\bar{\kappa}}^{2}+2\bar{\kappa}\oo{\kappa-\bar{\kappa}}+\bar{\kappa^{2}}}\oo{\kappa-\bar{\kappa}}_{s^{m+p+l}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&+\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l}}^{2}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&\leq 2\oo{-1}^{m+p}\oo{\frac{2\pi}{L}}^{2}\intcurve{\kappa_{s^{m+p+l}}\cdot\kappa_{s^{2p}}}+c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&+2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\label{LongTimeCorollary3,3}
\end{align}
Here $M:=\max\left\{m,2p\right\}<m+p$.
The second and third terms of $\oo{\ref{LongTimeCorollary3,3}}$ are identical to those in our calculation of $\oo{\ref{LongTimeCorollary3,2}}$, in which we established the identity
\begin{align}
&\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,4}
\end{align}
The fourth and fifth terms in $\oo{\ref{LongTimeCorollary3,3}}$ are estimated in a similar way. Because $M<m+p$, we are free to utilise Lemma $\ref{AppendixLemma5}$ wth $K=m+p+1$ and then the terms are estimatable in the same way as the $P$-style terms in $\oo{\ref{LongTimeCorollary3,2}}$. We conclude that
\begin{align}
&\intcurve{\norm{P_{4}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}+L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},M}\oo{\kappa-\bar{\kappa}}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,5}
\end{align}
Finally, the last part of $\oo{\ref{LongTimeCorollary3,3}}$ involving the summation can be estimated by applying integrating by parts by parts $j-p$ times to each term in $j$ and then estimating in the same way as above:
\begin{align}
&2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{j}\intcurve{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{p+l+j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\nonumber\\
&2\oo{-1}^{p}\sum_{j=p+1}^{p+l-1}\oo{-1}^{2j-p}\intcurve{\partial_{s}^{j-p}\cc{\oo{\kappa-\bar{\kappa}+\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{p+l-j}}\oo{\kappa-\bar{\kappa}}_{s^{2p}}}\oo{\kappa_{s^{2p+l}}}}\nonumber\\
&\leq c\oo{m,p}\intcurve{\norm{P_{4}^{2\oo{m+p},m+p}\oo{\kappa-\bar{\kappa}}}}+c\oo{m,p}L^{-1}\intcurve{\norm{P_{3}^{2\oo{m+p},m+p}}\oo{\kappa-\bar{\kappa}}}\nonumber\\
&\leq c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,6}
\end{align}
Combining $\oo{\ref{LongTimeCorollary3,4}}$,$\oo{\ref{LongTimeCorollary3,5}}$ and $\oo{\ref{LongTimeCorollary3,6}}$ and substituting into $\oo{\ref{LongTimeCorollary3,3}}$ then gives
\begin{align}
&2\oo{-1}^{p}\sum_{j=1}^{m}\oo{-1}^{j}\intcurve{\kappa\cdot\kappa_{s^{m-j}}\cdot\kappa_{s^{m+j}}\cdot\kappa_{s^{2p}}}+\oo{-1}^{p}\intcurve{\kappa\cdot\kappa_{s^{m}}^{2}\cdot\kappa_{s^{2p}}}\nonumber\\
&\leq 4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}+c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,7}
\end{align}
We can clearly see from $\oo{\ref{LongTimeCorollary3,2}}$ and $\oo{\ref{LongTimeCorollary3,7}}$ that the estimates of the extraneous terms in $\oo{\ref{LongTimeCorollary3,1}}$ are of the same form, regardless of the sign of $p-m$. We can conclude that
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}+\oo{2-c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq 4\pi^{2}L^{-2}\intcurve{\kappa_{s^{m+p}}^{2}}.\label{LongTimeCorollary3,8}
\end{equation}
Let us step back for a moment and forget about our time parameter, assuming without loss of generality that we are looking at a fixed time slice.
We claim that for any smooth closed curve $\gamma$ and any $l\in\mathbb{N}$, there exists a universal, bounded constant $c_{l}>0$ such that
\begin{equation}
\intcurve{\kappa_{s^{l}}^{2}}\leq c_{l}L^{2}K_{osc}\intcurve{\kappa_{s^{l+1}}^{2}}.\label{LongTimeCorollary3,9}
\end{equation}
Let us assume for the sake of contradiction that we can not find a suitable constant $c_{l}<\infty$ such that inequality $\oo{\ref{LongTimeCorollary3,9}}$ holds. Then, there exists a sequence of immersions $\left\{\gamma_{j}\right\}$ such that
\begin{equation}
R_{j}:=\frac{\llll{\kappa_{s^{l}}}_{2,\gamma_{j}}^{2}}{L^{2}\oo{\gamma_{j}}K_{osc}\oo{\gamma_{j}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}\nearrow\infty\text{ as }j\rightarrow\infty.\label{LongTimeCorollary3,10}
\end{equation}
Now, Theorem $\ref{AppendixTheorem1}$ implies that for any $j\in\mathbb{N}$ we have
\[
R_{j}\leq \frac{\frac{L^{2}\oo{\gamma_{j}}}{4\pi^{2}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}{L^{2}\oo{\gamma_{j}}K_{osc}\oo{\gamma_{j}}\llll{\kappa_{s^{l+1}}}_{2,\gamma_{j}}^{2}}=\frac{1}{4\pi^{2}K_{osc}\oo{\gamma_{j}}},
\]
and so the only way that $\oo{\ref{LongTimeCorollary3,10}}$ can occur is if we have
\begin{equation}
K_{osc}\oo{\gamma_{j}}\searrow0\text{ as }j\rightarrow\infty.\label{LongTimeCorollary3,11}
\end{equation}
Then, as each $\gamma_{j}$ satisfies the criteria of Theorem $\ref{AppendixTheorem1}$, we conclude there is a subsequence of immersions $\left\{\gamma_{j_{k}}\right\}$ and an immersion $\gamma_{\infty}$ such that $\gamma_{j_{k}}\rightarrow\gamma_{\infty}$ in the $C^{1}$-topology. Moreover, by $\oo{\ref{LongTimeCorollary3,11}}$ we have $K_{osc}\oo{\gamma_{\infty}}=0$. But this implies that $\gamma_{\infty}$ must be a circle, in which case both sides of inequality $\oo{\ref{LongTimeCorollary3,9}}$ are zero. Hence the inequality holds trivially with the immersion $\gamma_{\infty}$ for \emph{any} $c_{l}$ we wish, and so we can not in fact have $R_{j}\nearrow\infty$. This contradicts $\oo{\ref{LongTimeCorollary3,10}}$, and so the assumption that we can not find a constant $c_{l}$ such that the inequality $\oo{\ref{LongTimeCorollary3,9}}$ holds, must be false.
Next, combining $\oo{\ref{LongTimeCorollary3,9}}$ with $\oo{\ref{LongTimeCorollary3,8}}$ gives us
\[
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}+\oo{2-c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}-4\tilde{c}_{m}\pi^{2}K_{osc}}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq0.
\]
Here $\tilde{c}_{m}$ is our new interpolative constant, which we take to be the largest of all optimal constants in inequality $\oo{\ref{LongTimeCorollary3,9}}$ for closed simple curves with length bounded by $L\oo{\gamma_{0}}$. Then, because $K_{osc}\rightarrow0$, we know that there exists a time, say $t_{m}$, such that for $t\geq t_{m}$, $c\oo{m,p}\oo{K_{osc}+\sqrt{K_{osc}}}+4\tilde{c}_{m}\pi^{2}K_{osc}\leq1$. Hence for $t\geq t_{m}$ the previous inequality implies that
\begin{equation}
\frac{d}{dt}\intcurve{\kappa_{s^{m}}^{2}}\leq-\intcurve{\kappa_{s^{m+p+1}}^{2}}.\label{LongTimeCorollary3,12}
\end{equation}
Next, applying inequality Lemma $\ref{AppendixLemma1}$ $p+1$ times and using the monotonicity of $L\oo{\gamma_{t}}$ gives
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\oo{\frac{L^{2}\oo{\gamma_{t}}}{4\pi^{2}}}^{p+1}\intcurve{\kappa_{s^{m+p+1}}^{2}}\leq\oo{\frac{L^{2}\oo{\gamma_{0}}}{4\pi^{2}}}^{p+1}\intcurve{\kappa_{s^{m+p+1}}^{2}}.
\]
Hence if we define $c_{m}^{\star}:=\oo{\frac{4\pi^{2}}{L^{2}\oo{\gamma_{0}}}}^{p+1}$ then we conclude from $\oo{\ref{LongTimeCorollary3,12}}$ that for any $t\geq t_{m}$ we have the estimate
\[
\frac{d\intcurve{\kappa_{s^{m}}^{2}}}{\intcurve{\kappa_{s^{m}}^{2}}}\leq-c_{m}^{\star}\,dt.
\]
Integrating over $\left[t_{m},t\right]$ and exponentiating yields
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\int_{\gamma_{t_{m}}}{\kappa_{s^{m}}^{2}\,ds}\cdot e^{-c_{m}^{\star}\oo{t-t_{m}}}=\oo{e^{c_{m}^{\star}t_{m}}\int_{\gamma_{t_{m}}}{\kappa_{s^{m}}^{2}\,ds}}\cdot e^{-c_{m}^{\star}t},
\]
which is the first statement of the corollary. For the second statement, we simply combine the first statement and Lemma $\ref{AppendixLemma2}$ with $f=\kappa_{s^{m}}$:
\[
\llll{\kappa_{s^{m}}}_{\infty}^{2}\leq \frac{L\oo{\gamma_{0}}}{2\pi}\intcurve{\kappa_{s^{m+1}}^{2}}\leq \frac{L\oo{\gamma_{0}}c_{m+1}}{2\pi}e^{-c_{m+1}^{\star}t}.
\]
The pointwise exponential convergence result follows immediately from taking the square root of both sides.
\end{proof}
Let us finish by proving Proposition \ref{PN1}.
\begin{proof}
We follow \cite{Wkosc}.
Rearranging $\gamma$ in time if necessary, we may assume that
\begin{align*}
k(\cdot,t) \not> 0 ,\qquad &\text{ for all }t\in[0,t_0)\\
k(\cdot,t) > 0, \qquad &\text{ for all }t\in[t_0,\infty)
\end{align*}
where
$t_0 > \frac{2}{p+1}\bigg[
\bigg( \frac{L(\gamma_0)}{2\pi}\bigg)^{2(p+1)}
- \bigg( \frac{A(\gamma_0)}{\pi} \bigg)^{p+1}
\bigg]$,
otherwise we have nothing to prove. However in this case we have
\begin{align*}
\frac{d}{dt}L
&= -\vn{k_{s^p}}_2^2
\le -\frac{4\pi^2}{L^2}\vn{k_{s^{p-1}}}_2^2
\le \cdots
\le -\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1}\vn{k_s}_2^2
\\
&\le -\frac{\pi^2}{L^2}\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1}\vn{k}_2^2
\\
&\le -\frac{4\pi^4}{L^3}\bigg(\frac{4\pi^2}{L^2}\bigg)^{p-1},&\text{for}\ t\in[0,t_0),
\intertext{where we used the fact that $\gamma$ is closed and that the curvature has a zero. This implies}
L^{2p+2}(t) &\le -\frac{p+1}{2}(2\pi)^{2p+2}t + L^{2p+2}(\gamma_0),& \text{for}\ t\in[0,t_0),
\end{align*}
and thus $L^{2p+2}(t_0) < (4\pi A(\gamma_0))^{2p+2}$.
This is in contradiction with the isoperimetric inequality.
\end{proof}
\end{section}
\begin{section}{Appendix}
\begin{lemma}\label{AppendixLemma0}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve with Euclidean curvature $\kappa$ and arc length element $ds$. Then for any $m\in\mathbb{N}$ we have
\[
\intcurve{\kappa_{s^{m}}^{2}}\leq\varepsilon L^{2}\intcurve{\kappa_{s^{m+1}}^{2}}+\frac{1}{4\varepsilon^{m}}L^{-\oo{2m+1}}K_{osc},
\]
where $\varepsilon>0$ can be made as small as desired.
\end{lemma}
\begin{proof}
We will prove the lemma inductively. The case $m=1$ can be checked quite easily, by applying integration by parts and the Cauchy-Schwarz inequality:
\begin{align}
\intcurve{\kappa_{s}^{2}}&=\intcurve{\oo{\kappa-\bar{\kappa}}_{s}^{2}}=-\intcurve{\oo{\kappa-\bar{\kappa}}\oo{\kappa-\bar{\kappa}}_{s^{2}}}\nonumber\\
&\leq\oo{\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}}^{\frac{1}{2}}\oo{\intcurve{\kappa_{s^{2}}^{2}}}^{\frac{1}{2}}\nonumber\\
&\leq\varepsilon L^{2}\intcurve{\kappa_{s^{2}}^{2}}+\frac{1}{4\varepsilon^{1}}L^{-2}\intcurve{\oo{\kappa-\bar{\kappa}}^{2}}.\nonumber
\end{align}
Next assume inductively that the statement is true for $j=m$. That is, assume that
\begin{equation}
\intcurve{\kappa_{s^{j}}^{2}}\leq\varepsilon L^{2}\intcurve{\kappa_{s^{j+1}}^{2}}+\frac{1}{4\varepsilon^{j}}L^{-\oo{2j+1}}K_{osc}\label{AppendixLemma0,1}
\end{equation}
where $\varepsilon>0$ can be made as small as desired.
Again performing integration by parts and the Cauchy-Schwarz inequality, we have for any $\varepsilon>0$:
\begin{align}
\intcurve{\kappa_{s^{j+1}}^{2}}&=-\intcurve{\kappa_{s^{j}}\cdot\kappa_{s^{j+2}}}\leq\oo{\intcurve{\kappa_{s^{j}}^{2}}}^{\frac{1}{2}}\oo{\intcurve{\kappa_{s^j+2}^{2}}}^{\frac{1}{2}}\nonumber\\
&\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2\varepsilon}L^{-2}\intcurve{\kappa_{s^{j}}^{2}}.\label{AppendixLemma0,2}
\end{align}
Substituting the inductive assumption $\oo{\ref{AppendixLemma0,1}}$ into $\oo{\ref{AppendixLemma0,2}}$ then gives
\begin{align}
\intcurve{\kappa_{s^{j+1}}^{2}}&\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2\varepsilon}L^{-2}\cc{\varepsilon L^{2}\intcurve{\kappa_{s^{j+1}}^{2}}+\frac{1}{4\varepsilon^{j}}\oo{\varepsilon}L^{-\oo{2j+1}}K_{osc}},\nonumber
\end{align}
meaning that
\[
\frac{1}{2}\intcurve{\kappa_{s^{j+1}}^{2}}\leq\frac{\varepsilon}{2} L^{2}\intcurve{\kappa_{s^{j+2}}^{2}}+\frac{1}{2}\cdot\frac{1}{4\varepsilon^{j+1}}L^{-\oo{2\oo{j+1}+1}}K_{osc}.
\]
Multiplying out by $2$ then gives us the inductive step, completing the lemma.
\end{proof}
\begin{lemma}\label{AppendixLemma1}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an absolutely continuous and periodic function of period $P$. Then, if $\int_{0}^{P}{f\,dx}=0$ we have
\[
\int_{0}^{P}{f^{2}\,dx}\leq\frac{P^{2}}{4\pi^{2}}\int_{0}^{P}{f_{x}^{2}\,dx},
\]
with equality if and only if
\[
f\oo{x}=A\cos\oo{\frac{2\pi}{P}x}+B\sin\oo{\frac{2\pi}{P}x}
\]
for some constants $A,B$.
\end{lemma}
\begin{proof}
We will use the calculus of variations. Essentially, we wish to find $f$ that maximises the integral $\int_{0}^{P}{f^{2}\,dx}$, given a fixed value of $\int_{0}^{P}{f_{x}^{2}\,dx}$. We will show that combining this with the requirement that $\int_{0}^{P}{f\,dx}=0$ forces the extremal function to satisfy
\[
\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}\leq\frac{P^{2}}{4\pi^{2}}.
\]
For the constrained problem, the associated Euler-Lagrange equation is
\[
L=f^{2}+\lambda f_{x}^{2},
\]
with extremal functions satisfying
\[
\frac{\partial L}{\partial f}-\frac{d}{dx}\oo{\frac{\partial F}{\partial f_{x}}}=2f-2\lambda f_{xx}=0.
\]
That is to say,
\begin{equation}
f_{xx}-\frac{1}{\lambda}f=0.\label{AppendixLemma1,1}
\end{equation}
This means that
\[
0\leq\int_{0}^{P}{f_{x}^{2}\,dx}=-\int_{0}^{P}{ff_{xx}\,dx}=-\frac{1}{\lambda}\int_{0}^{P}{f^{2}\,dx},
\]
which forces $\lambda<0$. By standard arguments, we conclude from $\oo{\ref{AppendixLemma1,1}}$ that our extremal function is
\begin{equation}
f\oo{x}=A\cos\oo{\frac{x}{\sqrt{\norm{\lambda}}}}+B\sin\oo{\frac{x}{\sqrt{\norm{\lambda}}}}.\label{AppendixLemma1,2}
\end{equation}
Here $A,B$ are constants. The periodicity of $f$ forces $f\oo{0}=f\oo{P}$, so
\begin{equation}
A=A\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}+B\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}.\label{AppendixLemma1,3}
\end{equation}
Also, the requirement that $\int_{0}^{P}{f\,dx}=0$ forces
\begin{equation}
A\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}-B\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}=-B.\label{AppendixLemma1,4}
\end{equation}
Combining $\oo{\ref{AppendixLemma1,3}}$ and $\oo{\ref{AppendixLemma1,4}}$,
\[
A^{2}=A^{2}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}+AB\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}}\text{ and }B^{2}=B^{2}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}-AB\sin\oo{\frac{P}{\sqrt{\norm{\lambda}}}},
\]
meaning that
\[
A^{2}+B^{2}=\oo{A^{2}+B^{2}}\cos\oo{\frac{P}{\sqrt{\norm{\lambda}}}}.
\]
We conclude
\[
\frac{P}{\sqrt{\norm{\lambda}}}=2n\pi
\]
for some $n\in\mathbb{Z}\backslash\left\{0\right\}$ to be determined.
Hence
\begin{equation}
f\oo{x}=A\cos\oo{\frac{2n\pi x}{P}}+B\sin\oo{\frac{2n\pi x}{P}}.\label{AppendixLemma1,5}
\end{equation}
A quick calculation yields
\[
\int_{0}^{P}{f^{2}\,dx}=\oo{\frac{A^{2}+B^{2}}{2}}P,\,\,\text{and}\,\,\int_{0}^{P}{f_{x}^{2}\,dx}=\oo{\frac{2n\pi}{P}}^{2}\oo{\frac{A^{2}+B^{2}}{2}}P.
\]
Hence for any of our extremal functions $f$,
\[
\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}=\oo{\frac{P}{2n\pi}}^{2}\leq\frac{P^{2}}{4\pi^{2}},
\]
with equality if and only if $n=1$. Thus our constrained function $f$ that maximises the ratio $\frac{\int_{0}^{P}{f^{2}\,dx}}{\int_{0}^{P}{f_{x}^{2}\,dx}}$ is given by
\[
f\oo{x}=A\cos\oo{\frac{2\pi}{P}x}+B\sin\oo{\frac{2\pi}{P}x},
\]
with
\[
\int_{0}^{P}{f^{2}\,dx}\leq\frac{P^{2}}{4\pi^{2}}\int_{0}^{P}{f_{x}^{2}\,dx}
\]
amongst all continuous and $P-$periodic functions with $\int_{0}^{P}{f\,dx}=0$.
\end{proof}
\begin{lemma}\label{AppendixLemma2}
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an absolutely continuous and periodic function of period $P$. Then, if $\int_{0}^{P}{f\,dx}=0$ we have
\[
\llll{f}_{\infty}^{2}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx}.
\]
\end{lemma}
\begin{proof}
Since $\int_{0}^{P}{f\,dx}=0$ and $f$ is $P-$periodic we conclude that there exists distinct $0\leq p<q<P$ such that
\[
f\oo{p}=f\oo{q}=0.
\]
Next, the fundamental theorem of calculus tells us that for any $x\in\oo{0,P}$,
\[
\frac{1}{2}\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}=\int_{q}^{x}{ff_{x}\,dx}.
\]
Hence
\begin{align*}
&\cc{f\oo{x}}^{2}=\int_{p}^{x}{ff_{x}\,dx}-\int_{qx}^{q}{ff_{x}\,dx}\leq\int_{p}^{q}{\norm{ff_{x}}\,dx}\leq\int_{0}^{P}{\norm{ff_{x}}\,dx}\\
&\leq\oo{\int_{0}^{P}{f^{2}\,dx}\cdot\int_{0}^{P}{f_{x}^{2}\,dx}}^{\frac{1}{2}}\leq\frac{P}{2\pi}\int_{0}^{P}{f_{x}^{2}\,dx},
\end{align*}
where the last step follows from Lemma $\ref{AppendixLemma1}$. We have also utilised H\"{o}lder's inequality with $p=q=2$.
\end{proof}
\begin{lemma}[\cite{DKS}, Lemma $2.4$]\label{AppendixLemma3}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve. Let $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a sufficiently smooth function. Then for any $l\geq 2,K\in\mathbb{N}$ and $0\leq i<K$ we have
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{\phi}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}L^{\frac{1-\alpha}{2}}\oo{\intcurve{\phi^{2}}}^{\frac{1-\alpha}{2}}\llll{\phi}_{K,2}^{\alpha}.\label{AppendixLemma3,1}
\end{equation}
Here $\alpha=\frac{i+\frac{1}{2}-\frac{1}{l}}{K}$, and
\[
\llll{\phi}_{K,2}:=\sum_{j=0}^{K}L^{j+\frac{1}{2}}\oo{\intcurve{\oo{\phi}_{s^{j}}^{2}}}^{\frac{1}{2}}.
\]
In particular, if $\phi=\kappa-\bar{\phi}$, then
\begin{equation}
L^{i+1-\frac{1}{l}}\oo{\intcurve{\oo{k-\bar{k}}_{s^{i}}^{2}}}^{\frac{1}{l}}\leq c\oo{K}\oo{K_{osc}}^{\frac{1-\alpha}{2}}\llll{k-\bar{k}}_{K,2}^{\alpha}.\label{AppendixLemma3,2}
\end{equation}
\end{lemma}
\begin{proof}
The proof is identical to that of Lemma $2.4$ from \cite{DKS} and is of a
standard interpolative nature. Note that although we use $k-\bar{k}$ in the
identity (as opposed to \cite{DKS} where $k\nu$ is used).
\end{proof}
\begin{lemma}[Proposition $2.5$, \cite{DKS}]\label{AppendixLemma4}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve. Let $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be a sufficiently smooth function. Then for any term $P_{\nu}^{\mu}\oo{\phi}$ (where $P_{\nu}^{\mu}\oo{\cdot}$ denotes the same $P$-style notation used in for example \cite{DKS}) with $\nu\geq2$ which contains only derivatives of $\kappa$ of order at most $K-1$, we have
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu}\oo{\phi}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2}}\llll{\phi}_{K,2}^{\eta}.\label{AppendixLemma4,1}
\end{equation}
In particular, for $\phi=\kappa-\bar{\kappa}$ we have the estimate
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu}\oo{\kappa-\bar{\kappa}}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{K_{osc}}^{\frac{\nu-\eta}{2}}\llll{\kappa-\bar{\kappa}}_{K,2}^{\eta}\label{AppendixLemma4,2}
\end{equation}
where $\eta=\frac{\mu+\frac{\nu}{2}-1}{K}$.
\end{lemma}
\begin{proof}
Using H\"{o}lder's inequality and Lemma $\ref{AppendixLemma3}$ with $K=\nu$, if $\sum_{j=1}^{\nu}i_{j}=\mu$ we have
\begin{align}
&\intcurve{\norm{\phi_{s^{i_{1}}}\star\cdots\star\phi_{s^{i_{\nu}}}}}\nonumber\\
&\leq\prod_{j=1}^{\nu}\oo{\intcurve{\phi_{s^{i_{j}}}^{\nu}}}^{\frac{1}{\nu}}=L^{1-\mu-\nu}\prod_{j=1}^{\nu}L^{i_{j}+1-\frac{1}{\nu}}\oo{\intcurve{\phi_{s^{i_{j}}}^{\nu}}}^{\frac{1}{\nu}}\nonumber\\
&\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\prod_{j=1}^{\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{1-\alpha_{j}}{2}}\llll{\phi}_{K,2}^{\alpha_{j}}\label{AppendixLemma4,3}
\end{align}
where $\alpha_{j}=\frac{i_{j}+\frac{1}{2}-\frac{1}{\nu}}{K}$. Now
\[
\sum_{j=1}^{\nu}\alpha_{j}=\frac{1}{K}\sum_{j=1}^{\nu}\oo{i_{j}+\frac{1}{2}-\frac{1}{\nu}}=\frac{\mu+\frac{\nu}{2}-1}{K}=\eta,
\]
ans so substituting this into $\oo{\ref{AppendixLemma4,3}}$ gives the first inequality of the lemma. It is then a simple matter of substituting $\phi=\kappa-\bar{\kappa}$ into this result to prove statement $\oo{\ref{AppendixLemma4,2}}$.
\end{proof}
\begin{lemma}[\cite{DKS}]\label{AppendixLemma5}
Let $\gamma:\mathbb{S}^{1}\rightarrow\mathbb{R}^{2}$ be a smooth closed curve and $\phi:\mathbb{S}^{1}\rightarrow\mathbb{R}$ a sufficiently smooth function.
Then for any term $P_{\nu}^{\mu}\oo{\phi}$ with $\nu\geq2$ which contains only derivatives of $\kappa$ of order at most $K-1$, we have for any $\varepsilon>0$
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu,K-1}\oo{\phi}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{L\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2}}\oo{L^{2K+1}\intcurve{\phi_{s^{K}}^{2}}+L\intcurve{\phi^{2}}}^{\frac{\eta}{2}}.\label{AppendixLemma5,1}
\end{equation}
Moreover if $\mu+\frac{1}{2}\nu<2K+1$ then $\eta<2$ and we have for any $\varepsilon>0$
\begin{equation}
\intcurve{\norm{P_{\nu}^{\mu,K-1}\oo{\phi}}}\leq\varepsilon\intcurve{\phi_{s^{K}}^{2}}+c\cdot\varepsilon^{-\frac{\eta}{2-\eta}}\oo{\intcurve{\phi^{2}}}^{\frac{\nu-\eta}{2-\eta}}+c\oo{\intcurve{\phi^{2}}}^{\mu+\nu-1}.\label{AppendixLemma5,2}
\end{equation}
In particular, for $\phi=\kappa-\bar{\kappa}$, we have the estimate
\[
\intcurve{\norm{P_{\nu}^{\mu}\oo{k-\bar{k}}}}\leq c\oo{K,\mu,\nu}L^{1-\mu-\nu}\oo{K_{osc}}^{\frac{\nu-\eta}{2}}\oo{L^{2K+1}\intcurve{\oo{k-\bar{k}}_{s^{K}}^{2}}}^{\frac{\eta}{2}}.
\]
Here, as before, $\eta=\frac{\mu+\frac{\nu}{2}-1}{K}$.
\end{lemma}
\begin{proof}
Combining the previous lemma with the following standard interpolation inequality from that follows from repeated applications of Lemma $\ref{AppendixLemma0}$ (and is also found in \cite{Aubin1})
\[
\llll{\phi}_{K,2}^{2}\leq c\oo{K}\oo{L^{2K+1}\intcurve{\phi_{s^{K}}^{2}}+L\intcurve{\phi^{2}}}
\]
yields the identity $\oo{\ref{AppendixLemma5,1}}$ immediately. To prove $\oo{\ref{AppendixLemma5,2}}$ we simply combine $\oo{\ref{AppendixLemma5,1}}$ with the Cauchy-Schwarz identity. The final identity of the Lemma follow by letting $\phi=\kappa-\bar{\kappa}$ in $\oo{\ref{AppendixLemma5,1}}$ and combining this with the identity
\begin{equation}
K_{osc}\leq L\oo{\frac{L^{2}}{4\pi^{2}}}^{K}\intcurve{\oo{\kappa-\bar{\kappa}}_{s^{K}}^{2}}= c\oo{K}L^{2K+1}\intcurve{\oo{\kappa-\bar{\kappa}}_{s^{K}}^{2}},\label{AppendixLemma5,3}
\end{equation}
which is a direct consequence of applying Lemma $\ref{AppendixLemma1}$ $\oo{p+1}$ times repeatedly.
\end{proof}
\begin{theorem}[\cite{Breuning1}, Theorem $1.1$]\label{AppendixTheorem1}
Let $q\in\mathbb{R}^{n}$, $m,p\in\mathbb{N}$ with $p>m$. Additionally, let $\mathcal{A},\mathcal{V}>0$ be some fixed constants. Let $\mathfrak{T}$ be the set of all mappings $f:\mathbb{S}igma:\rightarrow\mathbb{R}^{n}$ with the following properties:
\begin{itemize}
\item $\mathbb{S}igma$ is an $m$-dimensional, compact manifold (without boundary)
\item $f$ is an immersion in $W^{2,p}\oo{\mathbb{S}igma,\mathbb{R}^{n}}$ satisfying
\begin{align*}
\llll{A\oo{f}}_{p}&\leq\mathcal{A},\\
\text{vol}\oo{\mathbb{S}igma}&\leq\mathcal{V},\text{ and }\\
q&\in f\oo{\mathbb{S}igma}.
\end{align*}
\end{itemize}
Then for every sequence $f^{i}:\mathbb{S}igma^{i}\rightarrow\mathbb{R}^{n}$ in $\mathfrak{T}$ there is a subsequence $f^{j}$, a mapping $f:\mathbb{S}igma\rightarrow\mathbb{R}^{n}$ in $\mathfrak{T}$ and a sequence of diffeomorphisms $\phi^{j}:\mathbb{S}igma\rightarrow\mathbb{S}igma^{j}$ such that $f^{j}\circ\phi^{j}$ converges in the $C^{1}$-topology to $f$.
\end{theorem}
\end{section}
\end{document} |
\begin{document}
\title{Error rates and resource overheads of encoded three-qubit gates}
\author{Ryuji Takagi, Theodore J. Yoder and Isaac L. Chuang}
\affiliation{Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA}
\date{\today}
\begin{abstract}
A non-Clifford gate is required for universal quantum computation, and, typically, this is the most error-prone and resource intensive logical operation on an error-correcting code. Small, single-qubit rotations are popular choices for this non-Clifford gate, but certain three-qubit gates, such as Toffoli or controlled-controlled-$Z$ (CCZ), are equivalent options that are also more suited for implementing some quantum algorithms, for instance, those with coherent classical subroutines.
Here, we calculate error rates and resource overheads for implementing logical CCZ with pieceable fault-tolerance, a non-transversal method for implementing logical gates.
We provide a comparison with a non-local magic-state scheme on a concatenated code and a local magic-state scheme on the surface code. We find the pieceable fault-tolerance scheme particularly advantaged over magic states on concatenated codes and in certain regimes over magic states on the surface code.
Our results suggest that pieceable fault-tolerance is a promising candidate for fault-tolerance in a near-future quantum computer.
\end{abstract}
\mathcal{A}ketitle
\section{Introduction}
Quantum error-correcting codes are the most promising route to scalable quantum computation. However, some of their limitations are well-known. For instance, a major problem is that a single code cannot support a full set of universal, transversal operations \cite{Zeng2007,Chen2008,Eastin2009a}. Often, and always for 2D designs \cite{Bravyi2013a}, the missing gate is not in the normalizer of the Pauli group; that is, it is non-Clifford.
The techniques of gate-teleportation \cite{Gottesman1999b} and magic-states \cite{Bravyi2005a} can overcome the lack of a non-Clifford gate. Different magic-states can be created to implement small $Z$-rotations such as the $T$-gate or 3-qubit operations, like Toffoli or controlled-controlled-$Z$ (CCZ).
However, the process to create a magic-state occurs post-selectively and recursively and leads to large resource overheads. Although improving consistently \cite{Jones2013a,Campbell2017a} approaching believed fundamental limits \cite{Bravyi2012a}, large resource demands remain a serious obstacle for near-future architectures.
Certain other approaches exist in the literature for implementing a universal gate-set while circumventing the use of magic-states. A popular approach is gauge-fixing \cite{Paetznick2013a,Anderson2014a,Bombin2015a}, in which a subsystem code can implement complementary sets of transversal logical gates depending on the settings of the gauge qubits. Another approach \cite{Jochym-OConnor2014a,Nikahd2016a,Nikahd2016b} concatenates different codes with complementary transversal gate sets to achieve the same effect in one larger code. Recently, this approach was shown to lead to asymptotic thresholds around $\sim10^{-3}$ albeit using more physical qubits than, for example, surface code magic-state distillation \cite{Chamberland2016d,Chamberland2017}.
Any fault-tolerant, universal computing scheme operating without magic states is expected to be a promising candidate for near-future architectures where fairly accurate physical components are supplied but space-time resources, like qubit count and circuit depth, are limited.
The primary goal in this near-future regime is to achieve some desired target error rate after a finite-sized computation with small resource overheads.
Such constraints imply that the logical error rates of encoded gates and the first-level pseudothreshold \cite{Svore2005b} (called just pseudothreshold hereafter) are more important measures than asymptotic threshold, which only becomes meaningful with access to huge amounts of resources.
To evaluate near-future fault-tolerant computation, we focus on another magic-less alternative that allows for a logical implementation of three-qubit gates, the pieceable fault-tolerance scheme \cite{Yoder2016c}. In this approach, a logical gate is done non-transversally through the ``round-robin" construction, and made fault-tolerant via partial error-correction performed throughout the circuit.
This construction has recently been used in \cite{Chao2017a} to perform fault-tolerant, universal computing on seven logical qubits requiring only four ancillary qubits and 15 code qubits. The circuit volume metric, a space-time resource measure that counts all gates weighted by the number of qubits involved, was used in \cite{Yoder2016c} to argue that pieceable fault-tolerance reduces logical gate overhead by nearly a factor of two over magic-state creation and injection. However, little was said about error rates of pieceable gates.
In this paper, we calculate these error rates and compare to magic-state schemes for implementing three-qubit non-Clifford gates. Our contenders are (1) a non-local magic-state scheme: magic-states created postselectively on Steane's 7-qubit code (also known as the smallest color code), (2) a local magic-state scheme: surface code magic-state distillation, and (3) pieceable fault-tolerance on the (a) 5-qubit \cite{Yoder2016c}, (b) 7-qubit \cite{Yoder2016c}, (c) $3\times3$ Bacon-Shor \cite{Yodera}, and (d) $3\times9$ Bacon-Shor \cite{Yodera} codes. Our metrics are (I) error rate of the logical gate and (II) circuit volume. Among concatenated schemes (1) and (3), we can definitively declare pieceable $3\times3$ Bacon-Shor the winner in both metrics (I) and (II). When comparing to (2), the picture is more complicated and interesting.
The pieceable $3\times3$ Bacon-Shor beats the surface code in error rate at low distance and in circuit volume when the physical error rate is sufficiently low compared with the desired target logical error.
On the other hand, asymptotically in code distance, the surface code outperforms pieceable $3\times 3$ Bacon-Shor due to better scaling of logical error rate and volume with distance.
\section{Methods}
We first describe our method to evaluate the logical error rates. Evaluating the surface code scheme (2) draws on the extensive literature on the topic \cite{Fowler2012a}.
Our calculations of the logical error rates of schemes (1) and (3) at code distance $d=3$ are done by exact enumeration of all combinations of up to two faults in the circuit extended-rectangle (exREC) \cite{Aliferis2005a} under the standard depolarizing noise model (which serves as a model of average-case noise).
In \cite{Aliferis2005a}, a rigorous upper bound on the logical error rate under depolarizing noise is given.
In contrast, we provide formulas giving a rigorous lower bound as well as a tighter rigorous upper bound.
The lower and upper bounds on logical error rate also determine lower and upper bounds on the pseudothreshold. Having both bounds allows us to definitively prove a separation between two different schemes when it exists.
Our method also confers some advantages over a Monte Carlo simulation. First, we can rigorously verify our circuits are fault-tolerant under the chosen noise model by checking that all single faults are correctable. Second, once the counting is complete, we can independently vary noise for each type of gate.
Our standard noise breakdown assigns single-qubit gates, two-qubit gates, and three-qubit gates each their own failure probabilities $p_1,p_2,$ and $p_3$, respectively.
In the circuit depolarization noise model, an $r$-qubit gate fails with one of the $4^r-1$ $r$-qubit Pauli errors with probability $p_r/(4^r-1)$. In principle, preparation and measurement could be treated separately as well, though we will assign them failure probabilities also equal to $p_1$. Bounds on the error rate can always be written as polynomials in $p_1,p_2,p_3$ as we discuss below.
Our ultimate goal in error-rate estimation is to find the probability the exREC is incorrect given that all ancillas pass verification. Denote this $P_{\text{fail}|\text{acc}}=\text{Pr}\left[\text{fail}|\text{acc}\right]$. Our counting gives the exact values of
\begin{align}
P^{(2)}_{\text{fail},\text{acc}}&=\text{Pr}\left[\text{fail},\text{acc},\le2\text{ faults}\right],\label{eq:fail_acc}\\
P^{(2)}_{\text{succ},\text{acc}}&=\text{Pr}\left[\neg\text{fail},\text{acc},\le2\text{ faults}\right],\label{eq:succ_acc}\\
P^{(2)}_{\text{rej}}&=\text{Pr}\left[\neg\text{acc},\le2\text{ faults}\right],\label{eq:rej}
\end{align}
as polynomials in $p_1,p_2,p_3$ with degree equal to the number of potentially faulty components in the entire exREC. These exactly calculated quantities are enough to bound $P_{\text{fail},\text{acc}}=\text{Pr}\left[\text{fail},\text{acc}\right]$, $P_{\text{succ},\text{acc}}=\text{Pr}\left[\neg\text{fail},\text{acc}\right]$, and $P_{\text{acc}}=\text{Pr}\left[\text{acc}\right]$ as
\begin{comment}
$P^{(2)}_{\text{fail},\text{acc}}\le P_{\text{fail},\text{acc}},
P^{(2)}_{\text{succ},\text{acc}}\le P_{\text{succ},\text{acc}},
P_{\text{acc}} \le 1-P^{(2)}_{\text{rej}}$.
\end{comment}
\begin{align}
P^{(2)}_{\text{fail},\text{acc}}&\le P_{\text{fail},\text{acc}},\\
P^{(2)}_{\text{succ},\text{acc}}&\le P_{\text{succ},\text{acc}},\\
P_{\text{acc}} &\le 1-P^{(2)}_{\text{rej}}.
\end{align}
Thus,
\begin{equation}
\frac{P^{(2)}_{\text{fail},\text{acc}}}{1-P^{(2)}_{\text{rej}}}\le P_{\text{fail}|\text{acc}}=1-P_{\text{succ}|\text{acc}}\le1-\frac{P^{(2)}_{\text{succ},\text{acc}}}{1-P^{(2)}_{\text{rej}}}.
\label{eq:bound}
\end{equation}
More details on the simulation including the description on how to obtain these polynomials can be found in Appendix~\ref{app:sim_details}.
Next, we consider evaluating the resource overhead.
There exist various resource measures such as qubit count, circuit volume, gate counts and so on. The number of reusable physical qubits is often taken as a physical resource measure in the literature.
However, it is not the best, especially when we would like to compare resource overheads between different codes, because there is ambiguity that comes with the level of parallelization we assume.
In this paper, we mainly focus on circuit volume, a space-time resource measure that counts all gates weighted by the number of qubits involved.
Unlike physical qubit count, circuit volume takes into account the trade-off between space and time resources. The circuit volume is a space-time metric in the same vein as the ``quantum volume'' \cite{qvolume}, except for evaluating specific circuits rather than a universal quantum computer.
The circuit volume at a high concatenation level is easy to compute using the volume of the logical construction at the first level of encoding.
Let $V^{(k)}_G$ be the volume for implementing circuit component $G$ at the $k^{\text{th}}$ level of concatenation.
Then, there is a recursion relation between two concatenation levels, $V^{(k+1)}_G=\sum_{G'} N_G^{G'} V^{(k)}_{G'}$ where $N_G^{G'}$ is the number of the circuit component $G'$ in the logical construction of component $G$.
We can understand this as evolution of a vector of circuit volumes of each component via a transformation matrix determined by the logical gate constructions. Namely, we get
\begin{equation}
{\bf V^{(k)}}=A^k {\bf V^{(0)}},
\label{eq:higher_concate}
\end{equation}
where $A$ is the matrix $A_{ij}=N_{G_i}^{G_j}$, ${\bf V^{(k)}}_i = V^{(k)}_{G_i}$, and $V^{(0)}_G$ is the volume of an unencoded component.
We set ${\bf V^{(k)}}=(V^{(k)}_3,V^{(k)}_2,V^{(k)}_1,V^{(k)}_{prep},V^{(k)}_{meas})^T$, where the components refer to the circuit volume of three qubit gates, two qubit gates, single qubit gates, $\ket{0}$ or $\ket{+}$ preparation, and measurement respectively.
Note that $(V^{(0)}_3,V^{(0)}_2,V^{(0)}_1,V^{(0)}_{prep},V^{(0)}_{meas})=(3,2,1,1,1)$.
\section{Logical constructions}\label{circuit_details}
Here, we describe the logical constructions used in the simulation. Explicit descriptions of the circuits at the gate level can be found in \cite{circuit}. All of our constructions begin with a round of syndrome measurement and recovery (the leading error correction) and end with the same (the trailing error correction), in accordance with the exREC formalism \cite{Aliferis2005a}. The rest of the circuit may also include rounds of error correction, called intermediate, in accordance with pieceable fault-tolerance \cite{Yoder2016c}.
For the 5-qubit code, we implement a logical CCZ gate by the round-robin construction \cite{Yoder2016c} with three intermediate error corrections.
The leading error correction and trailing error correction are done by Steane's error correction \cite{Steane1997}.
Since the 5-qubit code is non-CSS, a 10-qubit ancilla is needed to extract the entire syndrome simultaneously.
We actually find that the circuit in \cite{Steane1997} needs some modification for non-CSS codes, which we discuss in the Appendix~\ref{app:generalized_steane} in detail.
For intermediate error corrections, we use Shor-type error correction with CAT states \cite{Shor2011a}.
The size of the CAT states is always four for measuring constant stabilizers (those that commute with the preceeding circuitry), but it varies for measuring non-constant stabilizers because their weight changes as they go through the CCZ gates.
For our circuit, we need to use 9-CAT, 13-CAT, 9-CAT at maximum for the first, second and third intermediate error correction respectively.
For the 7-qubit code, we consider the construction that requires only one intermediate error correction \cite{Yoder2016c}.
All of the error corrections are done by Steane's error correction.
Since the 7-qubit code is a CSS code, correction of $Z$ type errors can be done separately from that of $X$ type errors, and only the encoded states $\ket{\bar{0}}$ and $\ket{\bar{+}}$ are needed. The state
$\ket{\bar{0}}$($\ket{\bar{+}}$) is verified by applying CNOT gates transversally to another noisy $\ket{\bar{0}}$($\ket{\bar{+}}$) and measuring it transversally (a Steane ancilla factory \cite{Steane1998a}).
If some error is detected, we discard the state and start again.
For estimating the circuit volume, we consider a more resource-efficient state preparation method proposed by Goto \cite{Goto2016}.
Although we did not estimate the logical error rate using the Goto's method, we suspect that the change in the logical error rate between different verification methods would be small as indicated in \cite{Goto2016}.
Since intermediate $Z$-type error correction is not needed, we just apply the $X$-type error correction in the middle and notify the trailing error correction about possible locations of $Z$-type errors as described in \cite{Yoder2016c}.
Logical CCZ on the Bacon-Shor code is implemented as proposed in \cite{Yodera}. On the $3\times 3$ Bacon-Shor we need no intermediate correction although we do use a non-Pauli recovery at the end. Furthermore, since the ancilla for the error correction is a tensor product of 3-CAT states, there is no need for verification since, modulo its stabilizers, an error on a 3-CAT is equivalent to a weight one error.
In contrast, the $3\times 9$ Bacon-Shor implements logical CCZ transversally, but it comes with a substantially larger overhead \cite{Yodera}.
For the non-local magic-state scheme, we use magic state injection on the 7-qubit code to implement a logical CCZ gate.
The CCZ magic state is defined by the stabilizers $\left< X_1\text{CZ}(2,3), X_2\text{CZ}(1,3), X_3\text{CZ}(1,2)\right>$.
The protocol consists of two parts, a state preparation circuit and a teleportation circuit.
The state preparation starts with the +1 eigenstate of the second and the third stabilizer, $\ket{\bar{0}}\ket{\bar{+}}\ket{\bar{+}}$, and measures the first stabilizer \cite{Zhou2000}.
Our circuit is a variant of the circuit in \cite{Monroe2014} which we modify to create the CCZ state instead. Two measurements of $X_1\text{CZ}(2,3)$ are done with complete error-correction in between. This makes the circuit fault-tolerant (to one fault).
If the two measurement results do not match, we discard the created state and start over again. If they match and they both show the result -1, we apply $\bar{Z}$ on the first code block to put it back to the desired magic state. If both show +1, we do not need to apply a correction. Like the pieceable 7-qubit case, all the error corrections are done using Steane's method \cite{Steane1997}.
\section{Comparison of concatenated schemes}
We compute the logical error rates and resource overheads of pieceably fault-tolerant CCZ gates on the 5-qubit code, 7-qubit code \cite{Yoder2016c}, $3\times 3$ Bacon-Shor code and $3\times 9$ Bacon-Shor code \cite{Yodera}, and compare them to a magic-state scheme on the 7-qubit code.
Fig.~\fig{logical_error} shows the obtained logical error rates for these cases using two different settings of physical error rate, $p_1=p_2=p_3=p$ and $10p_1=p_2=0.1p_3=p$.
Lower and upper bounds on pseudothresholds are the crossing points of ``break-even" line and the upper and lower bounds for logical error rates. For both settings of physical error rate, the $3\times 3$ Bacon-Shor code has lower logical error rate than the magic-state scheme below pseudothreshold.
For the 7-qubit code, whether the pieceable scheme has a lower rate than the magic-state scheme depends on the physical error rate setting.
The 5-qubit code has a large logical error rate due to a large number of pieces in the round-robin construction.
Similarly, the $3\times 9$ Bacon-Shor code has a higher logical error rate than the $3\times 3$ Bacon-Shor code because the size of the logical code block is obviously much bigger. Moreover, the $3\times 9$ Bacon-Shor needs to implement verification for 9-qubit CAT states.
We now compare the resource overheads. Table~\tbl{resource} shows the resource overheads to implement a logical CCZ gate with these constructions.
We assume that the ancillas are not reusable.
Due to a finite ancilla verification rejection rate, the effective resource count is slightly higher than the values in the table.
However, the rejection rate of the verification is $\mathcal{O}(p)$, and the effective resource count is obtained by multiplying $(1-n_{rej}p)^{-1}$ where $n_{rej}$ is the number of error locations that lead to rejection.
Since we are interested in the region $p<10^{-4}$, and the largest module involving verification is the magic state preparation circuit, which has $n_{rej}\sim 100$, increase in the resource due to verification is within 1\%.
Thus, it is safe to ignore the effects of verification.
Besides using more 3-qubit gates, pieceable constructions on the 7-qubit and $3\times 3$ Bacon-Shor code have smaller resource overheads compared to the magic-state scheme.
In particular, they have a significant reduction in circuit volume.
Fig.~\fig{volume} shows circuit volumes for the pieceable 7-qubit code, $3\times 3$ Bacon-Shor code, and magic-state scheme.
Transformation matrices $A$ (see Eq.~\eqref{eq:higher_concate}) for these codes are given in Appendix~\ref{app:volume_calc}.
Combining the results for the logical error rates and circuit volume, we conclude that the pieceable construction on the $3\times 3$ Bacon-Shor code beats the magic-state scheme on the 7-qubit code in both the criteria. The pieceable construction on the 7-qubit code also beats magic state injection in circuit volume, and in logical error rate when $p_1=p_2=p_3$.
\begin{figure}
\caption{(Color online.) Logical error rates of 3-qubit gate on (a,b) pieceable 7-qubit code (green, dot-dashed), pieceable $3\times 3$ Bacon-Shor code (blue, dashed), (c,d) pieceable 5-qubit code (green, dot-dashed), pieceable $3\times 9$ Bacon-Shor code (blue, dashed), and magic state injection on 7-qubit code (orange, solid) with (a,c) $p_1=p_2=p_3=p$ and (b,d) $10p_1=p_2=0.1p_3=p$ where $p_i$ refers to physical error rate of $i$-qubit gate. Initialization of single-qubit states $\ket{0}
\label{fig:logical_error}
\end{figure}
\begin{table}[htbp]
\centerline{\begin{tabular}{|c|c|c|c|c|}
\hline
& Volume & Qubits & 2-qubit gates & 3-qubit gates\\
\hline\hline
Pieceable 5-qubit & 3841 & 364 & 445 & 46\\
\hline
Pieceable 7-qubit & 771 & 93 & 162 & 21\\
\hline
$3\times 3$ Bacon-Shor & 414 & 81 & 90 & 27\\
\hline
$3\times 9$ Bacon-Shor & 1350 & 252 & 306 & 27\\
\hline
Magic state & 1352 & 154 & 267 & 14\\
\hline\hline
$3\times 3$ BS/Magic & 0.31 & 0.59 & 0.34 & 1.9\\
\hline
\end{tabular}}
\caption{Resource overheads to implement logical CCZ. Volume refers to the circuit volume, which counts all gates weighted by the number of qubits involved. Qubits are the number of physical qubits including data qubits and ancilla qubits where ancilla qubits are assumed to be not reusable. Numbers for the 5-qubit code include all the resources for the adaptive measurements.}
\label{tbl:resource}
\end{table}
\section{Comparison to surface codes}
\text{Next,} we compare logical error rate and resource overheads to a local magic-state scheme on surface codes.
We find that the pieceable construction can have a significant advantage in circuit volume in a certain region in terms of physical error rate and target logical error rate.
\subsection{Logical error rates}
Surface codes are known to have high asymptotic threshold, which is 0.1\%-1\% depending on assumptions and error model \cite{Fowler2012a,Fowler2009,Wang2009b,Wang2011,Wootton2012}, and thus they have attracted attention as a candidate for a scalable quantum computer.
However, having a high asymptotic threshold does not automatically imply that logical error rate is always low for reasonably sized codes.
Firstly, as can be seen in \cite{Fowler2012a}, in the low distance regime the pseudothreshold of the surface code is much smaller than the asymptotic threshold.
Thus, if the physical error rate is lower than the asymptotic threshold but not below the relevant pseudothreshold, encoding at low distance does not help to reduce the error rate.
Secondly, the logical error rate of a logical gate can be large even if the error rate for one surface code {\it cycle} is small, because a logical gate is made up of many cycles. Each cycle consists of measuring the complete error syndrome once via measurement qubits, one per stabilizer generator, as in \cite{Fowler2012a}.
Let $\bar{p}_{cycle}$ be the logical error rate for surface code per surface code cycle.
Let $C_G$ be the number of surface code cycles it takes to implement a logical version of gate $G$.
Then, logical error rate of gate $G$ is $\bar{p}_G\approx C_G \bar{p}_{cycle}$.
Since $C_G \propto d$ and $\bar{p}_{cycle}\propto p^{(d+1)/2}$ where $d$ is the surface code distance, $\bar{p}_{cycle}$ dominates for large distance.
However, when $d$ is small, the contribution to $\bar p_G$ from $C_G$ is not negligible.
In Appendix~\ref{app:surface_calc}, we find a specific form of $C_G$ for the logical Toffoli gate for two different implementations.
Fig.~\fig{error_concate_surface} shows logical error rates of a 3-qubit gate on the surface code using a Toffoli state, and upper bounds of logical error rate of pieceable $3\times 3$ Bacon-Shor code and pieceable 7-qubit code in terms of code distance with three different physical error rates.
Upper bounds are obtained by concatenating the function upper bounding the actual rate in Eq.~\eq{bound}.
Since the 3-qubit gate is the largest component among the components that appear in the logical construction of 3-qubit gate, concatenating the upper bounding error function for the 3-qubit gate upper bounds its error rates at higher concatenation level.
However, because logical 3-qubit gates have an order of magnitude higher error rate than 2-qubit gates and the logical constructions of 3-qubit gates mostly consist of single gates and 2-qubit gates, this upper bound is highly pessimistic.
A careful analysis taking into account error functions for other types of components and possibly even using better decoding algorithm \cite{Poulin2006,Fern2008a} at a higher levels may greatly reduce estimates of logical error rates.
Nevertheless, in Fig.~\fig{error_concate_surface}, we can see that surface codes have better scaling with distance than pieceable concatenated codes, which should be attributed to the high threshold. However, for small $d$, $C_{\mbox{Toffoli}}$ has a significant contribution, and when $d=3$ the logical error rate of the pieceable constructions is two orders of magnitude lower than that of the surface code.
\begin{figure}
\caption{Logical error rates for 3-qubit gate on the surface codes and (a) pieceable $3\times 3$ Bacon-Shor code (b) pieceable 7-qubit code in terms of code distance. Shown rates for pieceable codes are upper bounds obtained by concatenating the upper bounding function from Eq.~\eq{bound}
\label{fig:error_concate_surface}
\end{figure}
\subsection{Resource overheads}
We also count the circuit volume for implementing logical Toffoli on surface codes.
This allows us to compare the circuit volume between pieceable codes and the surface codes, shown in Fig.~\fig{volume}.
Although surface codes have better scaling with distance, pieceable constructions have a significant advantage until three concatenations.
This is especially true at distance three, where the difference is three orders of magnitude.
\begin{figure}
\caption{Circuit volume for logical 3-qubit gate on pieceable 7-qubit code(circles), pieceable $3\times 3$ Bacon-Shor code(squares), and magic-state scheme on 7-qubit code(diamonds) in terms of code distance. The dots correspond to every concatenation level in the range. Although it may be hard to see the data for pieceable 7-qubit code because they are close to the data for the magic-state scheme, the pieceable 7-qubit has slightly lower volume than the magic-state scheme for every distance shown.
}
\label{fig:volume}
\end{figure}
Consider now the space consisting of pairs (physical error rate, target logical error rate)$\equiv (p,p_T)$. Combining volume and error rate estimates, the region of this space where concatenated pieceable constructions require less circuit volume for implementing Toffoli than the surface code can be obtained. Fig.~\fig{error_volume} shows this region for the pieceable $3\times 3$ Bacon-Shor code and the 7-qubit code.
It shows that in large range, pieceable $3\times 3$ Bacon-Shor code has advantage in circuit volume over surface code, and the difference can be significant as can be seen in Fig.~\fig{volume}.
This region is actually determined by the upper bound of error rates at the third level concatenation.
It is because surface code with distance five has already larger volume than $3\times 3$ Bacon-Shor code with three concatenations as can be seen in Fig.~\fig{volume}.
For the 7-qubit code, Fig.~\fig{volume} shows that a 3-qubit logical gate at two concatenations of the 7-qubit code has less circuit volume than the surface code of any size. Thus, whenever two concatenations are sufficient to achieve the target logical error rate, the 7-qubit code will be advantaged, as is represented by the region in Fig.~\ref{fig:error_volume}.
Fig.~\fig{volume} also shows that the volume for the 7-qubit code with three concatenations is slightly larger than that for the surface code with distance seven.
Thus, the surface code is advantaged for the region where distance seven is enough for the surface code but three concatenations are needed for the 7-qubit code, which corresponds to the region between the upper purple region and the lower purple region in Fig.~\ref{fig:error_volume}.
The 7-qubit code again starts to have advantage over surface code for the region where the surface code needs distance nine whereas the 7-qubit code only needs to be concatenated three times, which corresponds to the lower purple region in Fig.~\ref{fig:error_volume}.
\begin{figure}
\caption{(color online.) The region where pieceable $3\times 3$ Bacon-Shor code (orange and purple) and pieceable 7-qubit code (purple) use less volume than surface codes to implement 3-qubit gate to achieve fixed target logical error rate, $p_T$, with fixed physical error rate, $p$. Dashed lines labeled by ${\bf ST_j}
\label{fig:error_volume}
\end{figure}
\section{Conclusions}
In this paper, we calculated logical error rates and resource overheads of 3-qubit gates using pieceable fault-tolerant constructions, a non-local magic-state scheme (on the 7-qubit code), and a local magic-state scheme (on the surface code).
In comparison with the non-local magic-state scheme, we found that while pieceable constructions have comparable, or even lower logical error rate to the magic-state scheme, the required circuit volume can be as little as 30\%. This suggests that the pieceable construction is a promising complement to schemes relying on magic states.
We also compared the pieceable construction to the surface codes and found that in quite a large region in terms of physical error rates and target logical error rates, pieceable constructions can have significantly lower circuit volume than surface codes.
Although realizing physical components with a small physical error rate such that pieceable constructions have a great advantage is challenging, one should notice that surface codes also have as hard a challenge as this in terms of resource overheads.
Just as surface codes are good candidates given access to large overheads, the pieceable construction appears to be a good candidate given access to small physical error rates.
Another difference between pieceable constructions and the surface code is locality, i.e.~the constraint that physical gates involved act only between qubits that are neighboring in some chosen low-dimensional layout.
Although the locality property is desirable in many experimental setups, some systems allow non-local interaction too \cite{Monroe2014}.
Our result indicates that such a non-local techniques can lead to significant reduction of resource use for quantum error correcting codes.
\section{Steane's error correction for arbitrary stabilizer codes}\label{app:generalized_steane}
Here, we describe the error correction used for the leading error correction and trailing error correction of the 5-qubit code. Since the 5-qubit code is not CSS, one might think Steane's error-correction is inappropriate. However, in \cite{Steane1997}, Steane proposes a circuit to do just that for the 5-qubit code. Unfortunately, Steane's construction as written is not quite correct. We present the correct method that works for any stabilizer code. We will also see that this method gives a conceptually simple way to prepare the necessary ancilla state in line with Steane's original proposal \cite{Steane1997}.
Consider a $\llbracket n,k\rrbracket$ stabilizer code $\mathcal{A}thcal{C}$ with stabilizer
\begin{equation}
S=\left(\begin{array}{c|c}S_x&S_z\end{array}\right),
\end{equation}
and logical operators
\begin{equation}
N=\left(\begin{array}{c|c}N_x&N_z\end{array}\right),
\end{equation}
written in symplectic matrix form. That is, ${S_x,S_z\in\mathcal{A}thbb{F}_2^{n-k}\times\mathcal{A}thbb{F}_2^n}$ and ${N_x,N_z\in\mathcal{A}thbb{F}_2^{2k}\times\mathcal{A}thbb{F}_2^{n}}$. Also, if we define ${\Lambda=\left(\begin{smallmatrix}0&I\{\mathbb I}&0\end{smallmatrix}\right)\in\mathcal{A}thbb{F}_2^{2n}\times\mathcal{A}thbb{F}_2^{2n}}$ using $k\times k$ block notation, then the canonical commutation relations are expressed by ${S\Lambda S^T=S\Lambda N^T}$ and ${N\Lambda N^T=A}$ for the ${2k\times 2k}$ matrix $A$ with $1$s on only the antidiagonal.
Following Steane, we propose the circuit in Fig.~\ref{fig:Steane_EC} to extract the syndrome of $\mathcal{A}thcal{C}$. The ancilla state used is twice the size of the code $\mathcal{A}thcal{C}$. The stabilizer of the ancilla state $\ket{\overline{a}}$ can be written
\begin{equation}\label{eq:anc_stabilizer}
S_a=\left(\begin{array}{cc|cc}
S_z&S_x&0&S_z\\
0&0&S_x&S_z\\
0&0&N_x&N_z
\end{array}\right).
\end{equation}
We show that this ancilla state and the circuit in Fig.~\ref{fig:Steane_EC} successfully extract the syndrome without giving information about the logical operators by propagating the observables of the code $\mathcal{A}thcal{C}$ and the stabilizer $S_a$ through the circuit. Begin with,
\begin{align}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&0&0\\bN_x&0&0&N_z&0&0\\0&S_z&S_x&0&0&S_z\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right),
\end{align}
where the syndrome is $a\in\mathcal{A}thbb{F}_2^{n-k}$ and logical operator values are $b=\mathcal{A}thbb{F}_2^{2k}$. After the controlled-$Z$ gates,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&S_x&0\\bN_x&0&0&N_z&N_x&0\\0&S_z&S_x&S_z&0&S_z\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right).
\end{equation}
After the controlled-$X$ gates,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&S_x&S_z\\bN_x&0&0&N_z&N_x&N_z\\S_x&S_z&S_x&S_z&0&0\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right).
\end{equation}
This is equivalent to the stabilizer,
\begin{equation}
\left(\begin{array}{ccc|ccc}aS_x&0&0&S_z&0&0\\bN_x&0&0&N_z&0&0\\a0&S_z&S_x&0&0&0\\0&0&0&0&S_x&S_z\\0&0&0&0&N_x&N_z\end{array}\right),
\end{equation}
and so we see that measuring all ancilla qubits in the $X$-basis results in a bitstring $m\in\mathcal{A}thbb{F}_2^{2n}$ such that $S\Lambda m=a$.
We note that $\ket{\overline{a}}$ is simply related to a Bell pair $\ket{\Phi}=(\ket{00}+\ket{11})/\sqrt{2}$ encoded in $\mathcal{A}thcal{C}$. If $\text{CX}_{tb}$ denotes $n$ CX gates transversally acting from the top $n$ qubits of the ancilla to the bottom $n$ and $H_t$ denotes $n$ $H$ gates applied to the top $n$ qubits, then $\ket{\overline{a}}=H_t\text{CX}_{tb}\ket{\overline{\Phi}}$. Thus, we can think of $\ket{\overline{a}}$ as an encoded Bell pair that has been ``transversally disentangled". Circuit identities can be used to rearrange Fig.~\ref{fig:Steane_EC} to Knill's error-correction \cite{Knill2005a}. Also, if $\mathcal{A}thcal{C}$ is CSS, Fig.~\ref{fig:Steane_EC} reduces to Steane's error-correction for CSS codes \cite{Steane1998a}.
Steane's original proposal for non-CSS error-correction \cite{Steane1997} omitted the $S_z$ on the right side of the first row of Eq.~\eqref{eq:anc_stabilizer}. Doing the same calculation as above shows that this will not succeed in measuring the syndrome. Steane's proposal suggested that the ancilla state would always be CSS for any code. This, unfortunately, is not true. Indeed, the 7-qubit code from \cite{Yoder2017} has an ancilla that is not even local-Clifford (LC) equivalent to a CSS state.
However, there are non-CSS codes for which $S_a$ is LC equivalent to a CSS code. The 5-qubit code with stabilizer
\begin{equation}
S_5=\left(\begin{array}{ccccc|ccccc}
1&0&1&0&0&0&0&0&1&1\\
0&1&0&0&1&0&0&1&1&0\\
1&0&0&1&0&0&1&1&0&0\\
0&0&1&0&1&1&1&0&0&0
\end{array}\right)
\end{equation}
is one of these. Indeed, $S_a$ can be written using only $Y$-type and $Z$-type generators. This allows us to prepare the ancilla using Fig.~\ref{fig:Steane_prep}, and verify the ancilla against single circuit faults using Fig.~\ref{fig:Steane_verification}, which are both standard constructions for CSS states \cite{Steane1998a,Cross2009}.
\begin{figure}
\caption{Circuit for Steane's error correction on a non-CSS code. $\ket{\bar{\psi}
\label{fig:Steane_EC}
\end{figure}
\begin{figure}
\caption{Preparing the error-correction ancilla state for the 5-qubit code for use in Fig.\fig{Steane_EC}
\label{fig:Steane_prep}
\end{figure}
\begin{figure}
\caption{Verification circuit for the ancilla state prepared by the circuit in Fig.\fig{Steane_prep}
\label{fig:Steane_verification}
\end{figure}
\section{Details of the simulation for logical error estimation}\label{app:sim_details}
Here, we describe some techniques used in the estimation of logical error rates.
For reasons of simulation efficiency, only errors originating from at most two faults are considered, but all such errors are counted. For Clifford circuits, propagating the Pauli errors resulting from circuit depolarizing noise can be done simply using the Gottesman-Knill theorem \cite{Gottesman1998a}. However, some of our circuits are built from non-Clifford CCZ gates. In this case, a tracked error is modified to include controlled-$Z$ (CZ) terms. A Pauli error that propagates through $m$ CCZ gates picks up at most $m$ CZ terms (some may cancel). Upon measurement (e.g.~in the error-correction circuits), the CZ terms must be broken down into a sum of Paulis, only some of which flip measurement bits to cause a signal.
We treat each term as different error element with the probability equal to the square of the amplitude of the term.
There is a subtlety in breaking down CZ errors.
As a sum of Pauli terms, a CZ error is written $(II+ZI+IZ-ZZ)/2$.
If there are multiple CZ errors, this Pauli sum has every possible combination of $I$ and $Z$ on the qubits on which CZ errors are applied, each with a plus or minus sign.
Thus, $m$ CZ errors applied on different qubit pairs are decomposed into a Pauli sum with $4^m$ terms.
If we treated each term as different error element at this point, each term would be assigned the probability square of the amplitude.
However, some terms may be equivalent to other terms up to stabilizers.
Such terms should interfere coherently.
In the simulation of pieceable CCZ on $3\times 3$ Bacon-Shor code, all the terms in the Pauli sum are rewritten in an unambiguous way up to stabilizers, and terms interfere before assigning them a probability.
Now that we recognize the subtle issue as the coherent addition of the Pauli terms, we argue that it does not affect the logical error rate except of the $3\times 3$ Bacon-Shor code case.
Firstly, note that the coherent addition can only happen when the number of qubits in one block on which CZ errors are applied is more than or equal to the weight of stabilizers.
This is because if stabilizers have higher weight, multiplying a stabilizer necessarily gives extra Paulis on the qubits that are not affected by CZ errors.
It prevents the term multiplied by a stabilizer being the same as another term in the Pauli sum.
In pieceable CCZ circuit on the 5-qubit code, CZ errors only occur on the three qubits, the support of logical $Z$. Since stabilizers are weight four, the coherent addition will not happen for the above reason.
For the pieceable CCZ circuit on the 7-qubit code, we argue that although the coherent addition may happen, it will not affect logical error rate.
Since stabilizers for the 7-qubit code are weight four, the coherent addition could happen only when two $X$ errors go through in the same block in the first piece.
However, these $X$ errors cannot be corrected because the 7-qubit code is a perfect CSS code.
Thus, all the error elements where the coherent addition could happen end up with logical errors regardless, and it does not matter whether we accurately interfere the terms.
For the pieceable CCZ circuit on the $3\times 9$ Bacon-Shor code, the situation is similar to the 7-qubit code case; the coherent addition could happen, but will not affect the logical error rate.
Since CCZs on the $3\times 9$ Bacon-Shor code are transversal, the number of qubits in the same code block on which CZ errors are applied is at most two.
Since weight-two $Z$-gauge operators are aligned along a row, the coherent addition could only happen when two CZ errors are applied on the two qubits in the same row in some code block.
However, all the terms in the Pauli sum for the CZ errors in that block are $Z$-type errors whose weight is less than or equal to two, and whose support is in the same row.
Since weight-one errors can be corrected by the standard error correction, and weight-two errors in the same row are equivalent to the identities up to stabilizers for the $Z$-gauge Bacon-Shor code, the terms in the Pauli sum are all correctable when the concerned coherent addition could happen.
Thus, it will not affect the logical error rate.
Considering CZ errors as a Pauli sum is inefficient -- $m$ CZ terms lead to $4^m$ Pauli addends.
However, in the simulation, we do not actually break down all the CZ errors.
Under certain cases, we definitely know that the final error correction will succeed to correct the CZ error.
One of such cases is that the CZ error is applied over different code blocks and those code blocks do not have any $Z$ errors.
The other case is that the CZ error is applied in one code block, there are no $Z$ errors in the block, and an intermediate error correction notifies the correct locations that the CZ error is applied over.
Also, we can reduce the number of CZ errors by removing harmless CZ errors before the measurements in the final error correction take place.
A harmless error is one that does not affect encoded states.
When errors are only Paulis, like in the circuits that only consist of Clifford gates, such errors are just stabilizers.
The following theorem generalizes the condition for the harmless errors to non-Pauli case.
\begin{thm}\label{thm:harmless}
Let $E$ be an error operator, $S=\left<g_1,\dots ,g_{n-k}\right>$ be the stabilizers, $\left<g_{n-k+1},\dots,g_{n+k}\right>$ be the logical operators of the code, and $\ket{\bar{\psi}}$ be an encoded state. If $g_i^{\dagger}E^{\dagger}g_iE\in S$ for all $i=1,\dots,n+k$, then $E\ket{\bar{\psi}}=\ket{\bar{\psi}}$ up to global phase.
\end{thm}
\begin{proof}
By the assumption, there exists a stabilizer $s_l$ such that $g_iE=Eg_is_l, \forall i$.
For $i=1,\dots,n-k$, since
\begin{eqnarray}
g_iE\ket{\bar{\psi}}=Eg_is_l\ket{\bar{\psi}}=E\ket{\bar{\psi}},
\end{eqnarray}
$E$ preserves the codeword space. Now for $i=n-k+1\dots n+k$, let $\ket{g_i^{(\pm)}}$ be the eigenstate of the logical operator $g_i$ with eigenvalue $\pm 1$, then
\begin{eqnarray}
g_i E\ket{g_i^{(\pm)}}=E g_i s_l\ket{g_i^{(\pm)}}=\pm E\ket{g_i^{(\pm)}}.
\end{eqnarray}
Thus, $E$ also preserves the logical space.
\end{proof}
\nonumber\\indent This theorem allows us to ignore the CZ errors that satisfy the above condition, which greatly reduces the computational task.
When intermediate error corrections are present, CZ errors need to be broken down according to the Pauli sum in the intermediate error corrections, and need to be propagated until the error correction at the end.
If the number of intermediate corrections is zero or one, it is rather easy to deal with, because the number of error elements due to the CZ errors that need to be propagated until the end is limited.
Actually, except the pieceable 5-qubit code, all the CZ errors that do not satisfy the condition of Theorem~\ref{thm:harmless} were broken down upon measurement and tracked to see if they end with a logical error.
For the 5-qubit code, to reduce the computational demand, we take the rule where we declare an error to be a logical error as soon as some CZ errors are measured in an intermediate error correction.
Although this strategy would cause some overestimation of the logical error rate, we argue that the probability that CZ errors are measured in an intermediate error correction is rather small.
CZ errors are measured in an intermediate error correction in the following two cases.
The first case is that an $X$ or $Y$-type error is caught by a CCZ gate in the adaptive nonconstant-stabilizer measurement.
It is described in \cite{Yoder2016c} that the adaptive nonconstant-stabilizer measurement is only triggered when some constant stabilizer measurements click due to an $X$ or $Y$-type error only for a single code block.
The adaptive measurement may contain CCZ gates connected between the ancilla block and the code blocks whose constant stabilizers did {\it not} click.
Thus, an $X$ or $Y$-type error is caught by a CCZ gate in the adaptive measurement only when an $X$ or $Y$ error triggers the adaptive measurement, the constant measurement in different code blocks fail with $X$ or $Y$ type error, and it goes to a CCZ gate in the adaptive measurement.
The second case is that a CZ error is caught by a CNOT gate in the adaptive measurement.
Note that CZ error only happens when an $X$ or $Y$-type error propagates through the CCZ gates in the code blocks. CZ errors are then present in code blocks other than the one in which the $X$ or $Y$-type error exists. Also, CNOT gates in the adaptive measurement could be only applied to the code block whose constant stabilizers click.
Thus, a CZ error is caught by a CNOT gate in the adaptive measurement only when an $X$ or $Y$-type error generates CZ errors in different code blocks, a later CCZ gate fails to cancel the first $X$ or $Y$-type error and generate another $X$ or $Y$-type error in the other code block that will make the constant measurement click, and the CZ error goes into CNOT gate in the adaptive measurement.
These two cases are realized in very restricted situations, so the contribution to the total logical error rate from these cases would be rather small.
Another situation arises with two or more intermediate corrections.
The pieceable construction on the 5-qubit code has multiple intermediate corrections, and they detect $X$ errors and notify possible error locations to the final error correction so that the final error correction can correct up to weight two located errors.
However, multiple faults can cause two intermediate error corrections to incorrectly notify more than two locations to the final error correction.
We declare those elements to be logical errors.
\section{Details on the error polynomials}\label{app:error_polys}
Here, we describe how to obtain Eq.~\eq{fail_acc}-\eq{rej} from the exact counting.
We first consider Eq.~\eq{fail_acc}, the probability that one or two faults occur and that pattern is accepted by all the verification modules through the propagation, but ends up with a logical error.
Due to the fault-tolerant property, a single fault never causes a logical error. Thus, it suffices to consider the cases when two faults occur.
In the simulation, each combination of two-fault patterns is assigned a probability $\left(\frac{p_{r}}{4^{r}-1}\right)\left(\frac{p_{s}}{4^{s}-1}\right)$ if the faulty components are an $r$-qubit gate and an $s$-qubit gate.
We propagate all the errors until the end and sum up the probabilities of the errors that lead to logical errors.
During the propagation, these errors may encounter verification processes.
If they are accepted by the verification, we keep propagating them. Otherwise, we stop propagating them so that they do not contribute to the logical error rate.
Let $Q_{\rm fail, acc}$ denote the estimated logical error rate. Since each physical error rate is either $p_1, p_2$ or $p_3$, it looks like
\begin{equation}
Q_{\rm fail,acc}=\sum_{r=1}^3\sum_{s\geq r}^3F_{rs}^{(2)} p_r p_s.
\end{equation}
Let $n_r$ be the total number of $r$-qubit gates.
Since we assume that different components fail independently, Eq.~\eq{fail_acc} is obtained as
\begin{equation}
P_{\rm fail,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\left(\sum_{r=1}^3\sum_{s\geq r}^3F_{rs}^{(2)} \frac{p_r}{1-p_r} \frac{p_s}{1-p_s} \right).
\label{eq:fail_acc_formula}
\end{equation}
Similarly, $Q_{\rm succ, acc}$, the sum of the assigned probability of the patterns that are accepted by all the verification modules and do not cause a logical error, looks like
\begin{equation}
Q_{\rm succ,acc}=\sum_{r=1}^3 S_{r}^{(1)}p_r+\sum_{r=1}^3\sum_{s\geq r}^3S_{rs}^{(2)} p_r p_s
\end{equation}
and Eq.~\eq{succ_acc} is obtained as
\begin{eqnarray}
&&P_{\rm succ,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot \nonumber\\number\\
&&\left(1+\sum_{r=1}^3 \frac{S_{r}^{(1)} p_r}{1-p_r}+\sum_{r=1}^3\sum_{s\geq r}^3 \frac{S_{rs}^{(2)} p_r p_s}{(1-p_r)(1-p_s)} \right)
\label{eq:succ_acc_formula}
\end{eqnarray}
The patterns that are not counted in either $Q_{\rm fail,acc}$ or $Q_{\rm succ,acc}$ are rejected in some verification module. Thus, we obtain Eq.~\eq{rej} as
\begin{eqnarray}
&&P_{\rm rej}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot \nonumber\\number\\
&&\left(\sum_{r=1}^3 \frac{A_r^{(1)} p_r}{1-p_r}+\sum_{r=1}^3\sum_{s> r}^3 \frac{A^{(2)}_{rs} p_r p_s}{(1-p_r)(1-p_s)}\right)
\end{eqnarray}
where
\begin{eqnarray}
A^{(1)}_r&=&n_r-S_{r}^{(1)}\\
A^{(2)}_{rs}&=&
\begin{cases}
n_r n_s - F_{rs}^{(2)}-S_{rs}^{(2)} & (r\neq s)\\
\binom{n_r}{2} - F_{rr}^{(2)}-S_{rr}^{(2)} &(r=s)
\end{cases}
\end{eqnarray}
Special care is required for 5-qubit code because $n_r$ cannot be definitely determined because of the adaptive measurements.
Note that at most two adaptive measurements are triggered when one or two faults occur.
Thus, taking $n_r$ that includes two largest adaptive measurements, which are the ones with 13-CAT and 9-CAT, the lower bound in Eq.~\eq{bound} still holds.
Instead of Eq.~\eq{succ_acc_formula}, we take
\begin{eqnarray}
&&P_{\rm succ,acc}^{(2)}=\left[\Pi_{t=1}^3 (1-p_t)^{n_t'}\right]\cdot \left(1+\sum_{r=1}^3 \frac{S_{r}^{(1)} p_r}{1-p_r}\right)\nonumber\\number \\
&&+\left[\Pi_{t=1}^3 (1-p_t)^{n_t}\right]\cdot\left(\sum_{r=1}^3\sum_{s\geq r}^3 \frac{S_{rs}^{(2)} p_r p_s}{(1-p_r)(1-p_s)} \right)
\end{eqnarray}
where $n_r'$ is the number of fault locations not including adaptive measurements.
For the 5-qubit code we also use
\begin{equation}
P_{\rm acc}=\Pi_j P_{{\rm acc},j}=\Pi_j (1-P_{{\rm rej},j})<\Pi_{j'} (1-P_{{\rm rej},j'}^{(2)})
\end{equation}
where $j$ is taken over all the verification modules and $j'$ is taken over all the verification modules except adaptive measurements.
The following are the obtained values for the parameters for each construction.
\begin{itemize}
\item $3\times 3$ Bacon-Shor
\begin{equation}
n_1=252,n_2=180,n_3=27
\end{equation}
\begin{equation}
S_{r}^{(1)}=(252,180,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
4216.8& 4271.9 & 783.5 \\
& 1194.5 & 461.5 \\
& & 34.9
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
27409.2& 41088.1 & 6020.5\\
& 14915.5 & 4398.5 \\
& & 316.1
\end{pmatrix}
\end{eqnarray}
\item Pieceable 7-qubit
\begin{equation}
n_1=648,n_2=480,n_3=21
\end{equation}
\begin{equation}
S_{r}^{(1)}=(383,224,21)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
13258.4& 12722.6 & 3581.4 \\
& 3077.3 & 1855.3 \\
& & 176.7
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
56460.9& 68953.7 & 4461.6\\
& 20748.8 & 2848.7 \\
& & 33.3
\end{pmatrix}
\end{eqnarray}
\item $3\times 9$ Bacon-Shor
\begin{equation}
n_1=2736,n_2=864,n_3=27
\end{equation}
\begin{equation}
S_{r}^{(1)}=(1524,566.4,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
52074 & 43098.4 & 7049.2 \\
& 8663.0 & 2968.1 \\
& & 183.3
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
1013640 & 748636 & 34098.8\\
& 138296 & 12324.7 \\
& & 167.7
\end{pmatrix}
\end{eqnarray}
\item Pieceable 5-qubit
\begin{eqnarray}
n_1&=&3365,n_2=1228,n_3=41\\
n_1'&=&2967, n_2'=1152, n_3'=27
\end{eqnarray}
\begin{equation}
S_{r}^{(1)}=(1475,457.6.,27)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
113030.0 & 85261.6 & 14679.2 \\
& 16067.4 & 5551.4 \\
& & 332.5
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
639301.0 & 482043.0 & 20392.4\\
& 90716.0 & 7554.7 \\
& & 59.3
\end{pmatrix}
\end{eqnarray}
\item 7-qubit with magic state
\begin{equation}
n_1=1138,n_2=743,n_3=14
\end{equation}
\begin{equation}
S_{r}^{(1)}=(612,324.3,6.9)
\end{equation}
\begin{eqnarray}
F_{rs}^{(2)}&=&
\begin{pmatrix}
25436.5 & 24565.9 & 1078.5 \\
& 6232.4 & 521.1 \\
& & 26.9
\end{pmatrix}\\
S_{rs}^{(2)}&=&
\begin{pmatrix}
154650 & 166625 & 3921.4\\
& 44308.3 & 2178.5 \\
& & 18.6
\end{pmatrix}
\end{eqnarray}
\end{itemize}
\section{Transformation matrix for volume calculation}\label{app:volume_calc}
As explained in the main text, the circuit volume for concatenated codes at higher concatenation level is described by a transformation matrix $A$ where $A_{ij}=N^{G_j}_{G_i}$.
We show the matrices for pieceable $3\times 3$ Bacon-Shor code, pieceable 7-qubit code, and 7-qubit with magic state, which are denoted by $A_{pBS}$,$A_{p7}$, $A_{m7}$ respectively.
We take the following order for gates; {\bf G}=\{3-qubit gate, 2-qubit gate, single qubit gate, $\ket{0}$ and $\ket{+}$ preparation, $X$ basis and $Z$ basis measurement\}.
For preparation of $\ket{\bar{0}}$ and $\ket{\bar{+}}$ on 7-qubit code, we use the method proposed by Goto \cite{Goto2016}, which requires just one additional ancilla.
\begin{equation*}
A_{pBS}=
\begin{pmatrix}
27 & 90 & 45 & 54 & 54 \\
0 & 69 & 30 & 36 & 36 \\
0 & 30 & 24 & 18 & 18 \\
0 & 6 & 3 & 9 & 0 \\
0 & 0 & 0 & 0 & 9
\end{pmatrix}
\label{eq:pBS_trans}
\end{equation*}
\begin{equation*}
A_{p7}=
\begin{pmatrix}
21 & 162 & 240 & 72 & 72 \\
0 & 79 & 104 & 32 & 32 \\
0 & 36 & 59 & 16 & 16 \\
0 & 11 & 22 & 8 & 1 \\
0 & 0 & 0 & 0 & 7
\end{pmatrix}
\label{eq:p7_trans}
\end{equation*}
\begin{equation*}
A_{m7}=
\begin{pmatrix}
14 & 267 & 504 & 136 & 136 \\
0 & 79 & 104 & 32 & 32 \\
0 & 36 & 59 & 16 & 16 \\
0 & 11 & 22 & 8 & 1 \\
0 & 0 & 0 & 0 & 7
\end{pmatrix}
\label{eq:m7_trans}
\end{equation*}
\section{Detailed resource analysis for surface code}\label{app:surface_calc}
We describe the detailed resource analysis to implement logical Toffoli gate on the surface code.
There are mainly two ways to do it, synthesizing a Toffoli gate using Clifford gates and $T$ gates, and injecting a logical Toffoli state by gate teleportation.
Consider the first method, in the context of the Toffoli implementation proposed by Jones \cite{Jones2013} using four $T$ gates. The
$T$ gates are implemented by $\ket{T}$ state and gate teleportation where $\ket{T}$ state is purified by a distillation protocol.
We use the 15-1 protocol~\cite{Bravyi2005a,Fowler2012a} which reduces error rates of $\ket{T}$ from $\mathcal{O}(p)$ to $\mathcal{O}(p^3)$, because it requires the smallest circuit volume compared to others \cite{Bravyi2012a,Meier2013,Reichardt2004a}.
Since the region of the physical error rate that pieceable construction helps to reduce error rate is $p<10^{-4}$ as can be seen in Fig.\fig{logical_error}, the logical error rate of the magic state distilled once is $<10^{-12}$.
Although the reduction in error rate may not be sufficiently low depending on the goal logical error rate, one distillation already gives large overheads.
Thus, we consider the circuit volume for one distillation as a lower bound and proceed the discussion.
It may come as a surprise that other distillation protocols with better conversion rate between noisy magic state and purified magic state have larger circuit volume.
It comes from that Hadamard gate and phase gate are not transversal on the surface code.
For implementing the Hadamard gate or phase gate fault-tolerantly, some non-trivial techniques, such as state injection, lattice surgery \cite{Horsman2012a}, code deformation \cite{Bombin2009a}, or surface folding \cite{Moussa2016}, are required.
These take many surface code steps, which affect the circuit volume.
Even though conversion rate between noisy $T$ state and purified $T$ state is high, if it requires many costly Clifford gates, the circuit volume will be large.
Especially in the case when only one distillation is required, a poor conversion rate does not hurt circuit volume that much.
Let us analyze the number of surface code cycles and circuit volume for each gate that are necessary to implement the logical Toffoli gate.
Let $C_G$ and $V_G$ be surface code cycles and circuit volume it takes to implement $G$.
We discuss circuit volume in units of [qubit$\cdot$cycle] and then convert it to [qubit$\cdot$step] using the fact that one surface code cycle consists of six steps \cite{Fowler2012a}.
Also, let $d$ be surface code distance, and $n=(2d-1)^2$ be the number of physical qubits on a surface.
Necessary components here are \{$\ket{\bar{0}}$ and $\ket{\bar{+}}$ preparation, CNOT, Hadamard, Phase\}.
For logical state preparation, we initialize a surface with physical $\ket{0}$ for $\ket{\bar{0}}$ preparation, and $\ket{+}$ for $\ket{\bar{+}}$ preparation.
After $d$ rounds of error correction, an appropriate recovery can be determined to prepare the desired logical state fault-tolerantly.
Thus, we find $C_{prep}=d$, $V_{prep}=nd$.
The CNOT gate can be transversally implemented if we allow non-locality or a 3D layered architecture.
However, since one of the striking features of surface codes is local interactions in a 2D architecture, we use lattice surgery to implement the CNOT gate \cite{Horsman2012a}.
First, prepare a surface with $\ket{\bar{+}}$ state between the control surface and the target surfaces.
The control surface and the intermediate surface are merged while obtaining measurement syndromes.
This corresponds to $\bar{Z}\bar{Z}$ measurement.
After that, the surface is split into two original surfaces and the intermediate surface is merged to target surface, which corresponds to $\bar{X}\bar{X}$ measurement.
It ends with splitting it into the two original surfaces.
Since merger and splitting each take $d$ rounds of error correction to stabilize the surface,
\begin{equation}
C_{CNOT}=C_{prep}+4d=5d
\end{equation}
and
\begin{eqnarray}
V_{CNOT}&=&V_{prep}+(3n+2(2d-1))(C_{prep}+4d)\nonumber\\number \\
&=& 6 d - 44 d^2 + 64 d^3.
\end{eqnarray}
The Hadamard gate is also implemented by the lattice surgery.
In the lattice surgery technique, firstly Hadamard gates are applied transversally.
To correct the orientation of the boundary, additional qubits are merged to the boundary and some qubits are split out so that it restores the original boundary orientation. The protocol ends with moving the surface back to the original position.
It takes $d$ cycles to stabilize the original surface after applying transversal $H$, $d$ cycles for lattice merger, $d$ cycles for lattice splitting, and $d$ cycles for SWAP operations to move the lattice back to the original position. Thus, $C_H=4d$.
For circuit volume, we need a bigger surface to carry out merger and split by one more column and row of qubits. Thus, $V_H=(2d)^2 C_H=16d^3$.
For implementing phase gate, we use the circuit in Fig.\fig{S_synthesis}.
A good thing about this circuit is that the ancilla state $\ket{S}=S\ket{+}=(\ket{0}+i\ket{1})/\sqrt{2}$ is preserved.
Thus, once a purified $\ket{S}$ state is prepared at the beginning of the computation, it can be reused whenever a phase gate needs to be applied. After averaging over a whole computation, the volume use for the distillation process at the beginning will be negligible per one logical gate construction. Note that if only local interactions are allowed, it may take additional circuit volume when the qubit to which the phase gate should be applied is far from the stored $\ket{S}$ state.
Thus, our estimation should be considered as a lower bound of the actual circuit volume under the setting in which only local interactions are allowed.
It gives $C_S=2C_{CNOT}+2C_H=18d$ and $V_S=2V_{CNOT}+2V_H+2nC_H=20 d - 120 d^2 + 192 d^3$.
Combining these building blocks, we find the number of cycles and volume required to implement a $T$ gate and a Toffoli gate.
For distilling a $T$ state, $\ket{T}=T\ket{+}$, we use the circuit in \cite{Fowler2012a} which takes 15 $\ket{T}$ states and output 1 $\ket{T}$ with lower error rate.
It takes 7 surface code cycles for CNOTs and 2 steps for transversal $T$ and measurements, which is 1/4 surface code cycle. Ignoring the last 1/4 cycles, we get $C_{\ket{T}}=7C_{CNOT}=35d$.
With some parallelization, we get $V_{\ket{T}}=16V_{prep}+V_{CNOT}+7(5V_{CNOT}+6nC_{CNOT})+\frac{1}{4}\cdot16n d=446 d - 2504 d^2 + 3224 d^3$
For implementing a $T$ gate, we use the usual gate teleportation technique \cite{Nielsen2010}.
The $S$ gate correction is applied with probability $1/2$.
We get $C_T = C_{CNOT}+\frac{1}{2}C_S=14d$ and $V_T =V_{\ket{T}}+V_{CNOT}+\frac{1}{2}V_S=462 d - 2608 d^2 + 3384 d^3$.
Since the surface code is CSS, we can transversally make measurements on all the data qubits, and extract eigenvalue for measurement operator.
Thus, measurement is done with only one time step, which is 1/8 of one surface code cycle, we ignore the volume due to the measurement.
Toffoli gate synthesis in \cite{Jones2013} consists of two steps.
In the first part, one constructs the $\mbox{Toffoli}^*$ gate, which is Toffoli gate followed by controlled-$S^{\dagger}$ gate, where four $T$ gates and two $H$ gates are used. Also, note that one logical ancilla block is used.
The second part takes the $\mbox{Toffoli}^*$ gate to the usual Toffoli gate with help of one additional ancilla block.
By construction of the synthesis circuit, we get
\begin{equation}
C_{\mbox{Toffoli}^*}=2C_H+4C_{CNOT}+C_T=42d
\end{equation}
and
\begin{eqnarray}
V_{\mbox{Toffoli}^*} &=&V_{prep}+6n C_H+2V_H+8V_{CNOT}+4 V_T\nonumber\\number \\
&=&1921 d - 10884 d^2 + 14180 d^3
\end{eqnarray}
Second part of the circuit gives
\begin{equation}
C_{\mbox{Toffoli}}=C_{\mbox{Toffoli}^*}+C_{S}+C_{CNOT}+C_{H}+C_{\ket{T}}=104d
\end{equation}
and
\begin{eqnarray}
V_{\mbox{Toffoli}}&=& V_{\mbox{Toffoli}^*}+V_{prep}+nC_{\mbox{Toffoli}^*}+V_S+V_{CNOT}\nonumber\\number \\
&+&V_H +3n(C_S+C_{CNOT}+C_H)\nonumber\\number \\
&+&n(C_{\mbox{Toffoli}^*}+C_{CNOT}+C_H)\nonumber\\number \\
&=& 2118 d - 11732 d^2 + 15136 d^3
\label{eq:VTof_1st}
\end{eqnarray}
where the unit for the volume is [qubit $\cdot$ cycle].
We included $C_{\ket{T}}$ in $C_{\mbox{Toffoli}}$ because cycles in the distillation circuit also contribute increasing in the final logical error rate.
Note that it includes the ancilla qubits for keeping $\ket{S}$ state that is kept during the whole computation.
Another way to implement logical Toffoli gate on the surface code is to use Toffoli state.
To locally prepare the Toffoli state, we use the protocol that takes eight $\ket{H}$ states and outputs one Toffoli state \cite{Eastin2013}.
In the preparation circuit, there are two $Y(\pi/4)$ gates and four $Y(-\pi/4)$ gates, which are rotations with respect to $Y$ axis.
These gates are implemented using $\ket{H}$ state with $Y$ basis measurement, controlled-$Y$ gate, and $Y(\pm \pi/2)$ gate. To implement these gates on the surface code, we use phase gates and Hadamard gates to rotate them to $X$ basis measurement, CNOT gate, and phase gate.
We then obtain
\begin{eqnarray}
C_{Y(\pi/4)}&=&C_S+C_{CNOT}+C_S+(2C_H+C_S)/2\nonumber\\number \\
&=&54d
\end{eqnarray}
and
\begin{eqnarray}
V_{Y(\pi/4)}&=& V_{prep}+V_S+V_{CNOT}+2V_S+(2V_H+V_S)/2\nonumber\\number \\
&=&77 d - 468 d^2 + 756 d^3
\end{eqnarray}
Using these, we obtain
\begin{eqnarray}
C_{\ket{\mbox{Toffoli}}}&=&7C_{CNOT} + (15/2)C_S + 3C_H\nonumber\\number \\
&=&182d
\end{eqnarray}
and
\begin{eqnarray}
V_{\ket{\mbox{Toffoli}}}&=&4V_{prep} + 3(2V_{CNOT} + 2 V_{Y(\pi/4)} + 2nC_{Y(\pi/4)})\nonumber\\number \\
&&+ V_{CNOT} + 2nC_{CNOT}\nonumber\\number \\
&=&842 d - 4468 d^2 + 6336 d^3
\end{eqnarray}
where $\ket{\mbox{Toffoli}}$ refers to the Toffoli state.
Cycles and volume for the teleportation circuit, which we write $C_{tele}$ and $V_{tele}$ are
\begin{eqnarray}
C_{tele}&=&C_{CNOT}+1/2(3C_{CNOT}+2C_H)\nonumber\\number\\
&=&16.5d
\end{eqnarray}
\begin{eqnarray}
V_{tele}&=&3V_{CNOT}+\{4nC_H+ 3(V_{CNOT}+nC_{CNOT})\}/2 \nonumber\\number \\
&=&42.5 d - 260 d^2 + 350 d^3
\end{eqnarray}
Combining all of them, we get
\begin{eqnarray}
C_{\mbox{Toffoli}}=198.5d
\end{eqnarray}
and
\begin{eqnarray}
V_{\mbox{Toffoli}}=7076 d - 37824 d^2 + 53488 d^3
\label{eq:VTof_2nd}
\end{eqnarray}
where the unit for the volume is [qubit $\cdot$ cycle].
Fig.~\fig{volume_Tofstate_T} shows circuit volume with unit [qubit$\cdot$step] in terms of code distance for both ways of implementation.
We can see that the the scheme with Toffoli state has lower circuit volume.
It is the reason why the scheme with Toffoli state is discussed in the main text.
\begin{figure}
\caption{Circuit identity used for implementing $S$ gate. Here $\ket{S}
\label{fig:S_synthesis}
\end{figure}
\begin{figure}
\caption{Circuit volume for two different implementations of Toffoli gate. Dashed: gate synthesis using $T$ gate. Solid: Toffoli state scheme}
\label{fig:volume_Tofstate_T}
\end{figure}
\end{document} |
\begin{document}
\title{ space{-0.7cm}
\begin{abstract}
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$.
One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree.
It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$.
In this paper, we prove this conjecture for large $n$.
\epsilonilonnd{abstract}
\section{Introduction}\label{intro}
The study of decomposition problems for graphs and hypergraphs has a very long history, going back more than two hundred years to the work of Euler on Latin squares. Latin squares are $n \times n$ arrays filled with $n$ symbols such that each symbol appears once in every row and column.
In 1782, Euler asked for which values of $n$ there is a Latin square which can be decomposed into $n$ disjoint transversals, where a transversal is a collection of cells of the Latin square which do not share the same row, column or symbol.
This problem has many equivalent forms. In particular, it is equivalent to a \epsilonilonmph{graph decomposition problem}.
We say that a graph $G$ has a \epsilonilonmph{decomposition into copies of a graph $H$} if the edges of $G$ can be partitioned into edge-disjoint subgraphs isomorphic to $H$. Euler's problem is equivalent to asking for which values of $n$ does the balanced complete $4$-partite graph $K_{n,n,n,n}$ have a decomposition into copies of the complete graph on $4$ vertices, $K_4$.
In 1847, Kirkman studied decompositions of complete graphs $K_n$ and showed that they can be decomposed into copies of a triangle if, and only if, $n\epsilonilonquiv 1 \text{ or } 3 \pmod 6$. Wilson~\cite{wilson1975decompositions} generalized this result by proving necessary and sufficient conditions for a complete graph $K_n$ to be decomposed into copies of \epsilonilonmph{any} graph, for large $n$.
A very old problem in this area, posed in 1853 by Steiner, says that, for every $k$, modulo an obvious divisibility condition every sufficiently large complete $r$-uniform hypergraph can be decomposed into edge-disjoint copies of a complete $r$-uniform hypergraph on $k$ vertices. This problem was the so-called ``existence of designs'' question and has practical relevance to experimental designs. It was resolved only very recently in spectacular
work by Keevash \cite{keevash} (see the subsequent work of \cite{glock-all} for an alternative proof of this result). Over the years graph and hypergraph decomposition problems have been extensively studied and
by now this has become a vast topic with many exciting results and conjectures (see, for example, \cite{gallian2009dynamic,wozniak2004packing,yap1988packing}).
In this paper, we study decompositions of complete graphs into large trees, where a tree is a connected graph with no cycles. By large we mean that the size of the tree is comparable with the size of the complete graph (in contrast with the existence of designs mentioned above, where the decompositions are into small subgraphs).
The earliest such result was obtained more than a century ago by Walecki. In 1882 he proved that a complete graph $K_n$ on an even number of vertices can be partitioned into edge-disjoint Hamilton paths.
A Hamilton path is a path which visits every vertex of the parent graph exactly once. Since paths are a very special kind of tree it is natural to ask which other large trees can be used to decompose a complete graph. This question was raised by Ringel \cite{ringel1963theory}, who in 1963 made the following
appealing conjecture on the decomposition of complete graphs into edge-disjoint copies of a tree with roughly half the size of the complete graph.
\begin{conjecture} \label{ringelconj}
The complete graph $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges.
\epsilonilonnd{conjecture}
Ringel's conjecture is one of the oldest and best known open conjectures on graph decompositions. It has been established for many very special classes of trees such as caterpillars, trees with $\leq 4$ leaves, firecrackers, diameter $\leq 5$ trees, symmetrical trees, trees with $\leq 35$ vertices, and olive trees (see Chapter 2 of \cite{gallian2009dynamic} and the references therein).
There have also been some partial general results in the direction of Ringel's conjecture. Typically, for these results, an extensive technical method is developed which is capable of almost-packing any appropriately-sized collection of certain sparse graphs, see,
e.g., \cite{bottcher2016approximate, messuti2016packing, ferber2017packing, kim2016blow}. In particular, Joos, Kim, K{\"u}hn and Osthus~\cite{joos2016optimal} have proved Ringel's conjecture for very large bounded-degree trees. Ferber and Samotij~\cite{ferber2016packing} obtained an almost-perfect packing of almost-spanning trees with maximum degree $O(n/\log n)$, thus giving an approximate version of Ringel's conjecture for trees with maximum degree $O(n/\log n)$. A different proof of this was obtained by
Adamaszek, Allen, Grosu, and Hladk{\'y}~\cite{adamaszek2016almost}, using graph labellings. Allen, B\"ottcher, Hladk{\'y} and Piguet~\cite{allen2017packing} almost-perfectly packed arbitrary spanning graphs with maximum degree $O(n/ \log n)$ and constant degeneracy\footnote{A graph is $d$-degenerate if each induced subgraph has a vertex of degree $\leq d$. Trees are exactly the $1$-degenerate, connected graphs.} into large complete graphs.
Recently Allen, B\"ottcher, Clemens, and Taraz~\cite{allen2019perfectly} found perfect packings of complete graphs into specified graphs with maximum degree $o(n/\log n)$, constant degeneracy, and linearly many leaves.
To tackle Ringel's conjecture, the above mentioned papers developed many powerful techniques based on the application of probabilistic methods and
Szemer\'edi's regularity lemma. Yet, despite the variety of these techniques, they all have the same limitation, requiring that the maximum degree of the tree should be much smaller than $n$.
A lot of the work on Ringel's Conjecture has used the \epsilonilonmph{graceful labelling} approach.
This is an elegant approach proposed by R\'osa~\cite{rosa1966certain}. For an $(n+1)$-vertex tree $T$ a bijective labelling of its vertices $f: V(T) \rightarrow \{0, \dots, n\}$ is called graceful
if the values $|f(x)-f(y)|$ are distinct over the edges $(x,y)$ of $T$.
In 1967 R\'osa conjectured that every tree has a graceful labelling. This conjecture has attracted a lot of attention in the last 50 years but has only been proved for some special classes of trees, see e.g., \cite{gallian2009dynamic}.
The most general result for this problem was obtained by Adamaszek, Allen, Grosu, and Hladk{\'y}~\cite{adamaszek2016almost} who proved it asymptotically for trees with maximum degree $O(n/\log n)$.
The main motivation for studying graceful labellings is that one can use them to prove Ringel's conjecture. Indeed,
given a graceful labelling $f: V(T) \rightarrow \{0, \dots, n\}$, think of it as an embedding of $T$ into $\{0, \dots, 2n\}$. Using addition modulo $2n+1$, consider $2n+1$ cyclic shifts $T_0, \ldots, T_{2n}$ of $T$, where the tree $T_i$ is an isomorphic copy of $T$ whose vertices are
$V(T_i)=\{f(v)+i~|~ v \in V(T)\}$ and whose edges are $E(T_i)=\{(f(x)+i,f(y)+i)~|~(x,y)\in E(T)\}$. It is easy to check that the fact that $f$ is graceful implies that the trees $T_i$ are edge disjoint and therefore
decompose $K_{2n+1}$.
R\'osa also introduced a related proof approach to Ringel's conjecture called ``$\rho$-valuations''. We describe it using the language of ``rainbow subgraphs'', since this is the language which we ultimately use in our proofs.
A \epsilonilonmph{rainbow} copy of a graph $H$ in an edge-coloured graph $G$ is a subgraph of $G$ isomorphic to $H$ whose edges have different colours. Rainbow subgraphs are important because many problems in combinatorics can be rephrased as problems asking for rainbow subgraphs (for example the problem of Euler on Latin squares mentioned above).
Ringel's conjecture is implied by the existence of a rainbow copy of every $n$-edge tree $T$ in the following edge-colouring of the complete graph $K_{2n+1}$, which we call the \epsilonilonmph{near distance (ND-)colouring}.
Let $\{0,1,\dots,2n\}$ be the vertex set of $K_{2n+1}$. Colour the edge $ij$ by colour $k$, where $k\in [n]$, if either $i=j+k$ or $j=i+k$ with addition modulo $2n+1$.
Kotzig~\cite{rosa1966certain} noticed that if the ND-coloured $K_{2n+1}$ contains a rainbow copy of a tree $T$, then $K_{2n+1}$ can be decomposed into copies of $T$ by taking $2n+1$ cyclic shifts of the original rainbow copy, as explained above (see also Figure~\ref{FigureIntro}). Motivated by this and Ringel's Conjecture, Kotzig conjectured that the ND-coloured $K_{2n+1}$ contains a rainbow copy of every tree on $n$ edges.
To see the connection with graceful labellings, observe that such a labelling of the tree $T$ is equivalent to a rainbow copy of this tree in the ND-colouring whose vertices are $\{0, \dots, n\}$. Clearly, specifying exactly the vertex set of the tree adds an additional restriction which makes it harder to find such a rainbow copy.
\begin{figure}[b]
\centering
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}
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\epsilonilonnd{scope}
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}
\draw ($0.8*(A4)+0.8*(A5)$) node {\large $K_9:$};
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}
\epsilonilonnd{tikzpicture}
\caption{The ND-colouring of $K_9$ and a rainbow copy of a tree $T$ with four edges. The colour of each edge corresponds to its Euclidean length. By taking cyclic shifts of this tree around the centre of the picture we obtain $9$ disjoint copies of the tree decomposing $K_9$ (and thus a proof of Ringel's Conjecture for this particular tree). To see that this gives 9 disjoint trees, notice that edges must be shifted to other edges of the same colour (since shifts are isometries).}\label{FigureIntro}
\epsilonilonnd{figure}
In \cite{MPS} we gave a new approach to embedding large trees (with no degree restrictions) into edge-colourings of complete graphs, and used this to prove Conjecture
\ref{ringelconj} asymptotically. Here, we further develop and refine this approach, combining it with several critical new ideas to prove Ringel's conjecture for large complete graphs.
\begin{theorem}\label{main}
For every sufficiently large $n$ the complete graph $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges.
\epsilonilonnd{theorem}
The proof of Theorem \ref{main} uses the last of the three approaches mentioned above. Instead of working directly with tree decompositions, or studying graceful labellings, we instead prove for large $n$ that every ND-coloured $K_{2n+1}$ contains a rainbow copy of every $n$-edge tree (see Theorem~\ref{Theorem_Ringel_proof}).
Then, we obtain a decomposition of the complete graph by considering cyclic shifts of one copy of a given tree (as in Figure~\ref{FigureIntro}). The existence of such a cyclic decomposition was separately conjectured by Kotzig~\cite{rosa1966certain}. Therefore, this also gives a proof of the conjecture by Kotzig for large $n$.
Our proof approach builds on ideas from the previous research on both graph decompositions and graceful labellings.
From the work on graph decompositions, our approach is inspired by randomized decompositions and the absorption technique. The rough idea of absorption is as follows. Before the embedding of $T$ we prepare a template which has some useful properties. Next we find a partial embedding of the tree $T$ with some vertices removed such that we did not use the edges of the template.
Finally we use the template to embed the remaining vertices. This idea was introduced as a general method by R\"odl, Ruci\'nski and Szemer\'edi \cite{RRS} and has been used extensively since then. For example, the proof of Ringel's Conjecture for bounded degree trees is based on this technique~\cite{joos2016optimal}.
We are also inspired by graceful labellings. When dealing with trees with very high degree vertices, we use a completely deterministic approach for finding a rainbow copy of the tree. This approach heavily relies on features of the ND-colouring and produces something very close to a graceful labelling of the tree.
Our theorem is the first general result giving a perfect decomposition of a graph into subgraphs with arbitrary degrees. As we mentioned, all previous comparable results placed a bound on the maximum degree of the subgraphs into which they decomposed the complete graph. Therefore, we hope that further development of our techniques can help overcome this ``bounded degree barrier'' in other problems as well.
\section{Proof outline}\label{Section_proof_outline}
From the discussion in the introduction, to prove Theorem~\ref{main} it is sufficient to prove the following result.
\begin{theorem}\label{Theorem_Ringel_proof}
For sufficiently large $n$, every ND-coloured $K_{2n+1}$ has a rainbow copy of every $n$-edge tree.
\epsilonilonnd{theorem}
That is, for large $n$, and each $(n+1)$-vertex tree $T$, we seek a rainbow copy of $T$ in the ND-colouring of the complete graph with $2n+1$ vertices, $K_{2n+1}$. Our approach varies according to which of 3 cases the tree $T$ belongs to. For some small $\delta>0$, we show that, for every large $n$, every $(n+1)$-vertex tree falls in one of the following 3 cases (see Lemma~\ref{Lemma_case_division}), where a bare path is one whose internal vertices have degree 2 in the parent tree.
\begin{enumerate}[label = \Alph{enumi}]
\item $T$ has at least $\delta^6 n$ non-neighbouring leaves.
\item $T$ has at least $\delta n/800$ vertex-disjoint bare paths with length $\delta^{-1}$.
\item Removing leaves next to vertices adjacent to at least $\delta^{-4}$ leaves gives a tree with at most
$n/100$ vertices.
\epsilonilonnd{enumerate}
As defined above, our cases are not mutually disjoint. In practice, we will only use our embeddings for trees in Case A and B which are not in Case C. In~\cite{MPS}, we developed methods to embed any $(1-\epsilonilonps)n$-vertex tree in a rainbow fashion into any 2-factorized $K_{2n+1}$, where $n$ is sufficiently large depending on $\epsilonilonps$. A colouring is a \epsilonilonmph{2-factorization} if every vertex is adjacent to exactly 2 edges of each colour. In this paper, we embed any $(n+1)$-vertex tree $T$ in a rainbow fashion into a specific 2-factorized colouring of $K_{2n+1}$, the ND-colouring, when $n$ is large. To do this, we introduce three key new methods, as follows.
\begin{enumerate}[label = {\bfseries M\arabic{enumi}}]
\item We use our results from \cite{montgomery2018decompositions} to suitably randomize the results of \cite{MPS}. This allows us to randomly embed a $(1-\epsilonilonps)n$-vertex tree into any 2-factorized $K_{2n+1}$, so that the image is rainbow and has certain random properties. These properties allow us to apply a case-appropriate \epsilonilonmph{finishing lemma} with the uncovered colours and vertices.\label{T2}
\item We use a new implementation of absorption to embed a small part of $T$ while using some vertices in a random subset of $V(K_{2n+1})$ and exactly the colours in a random subset of $C(K_{2n+1})$. This uses different \epsilonilonmph{absorption structures} for trees in Case A and in Case B, and in each case gives the finishing lemma for that case.\label{T1}
\item We use an entirely new, deterministic, embedding for trees in Case C.\label{T3}
\epsilonilonnd{enumerate}
For trees in Cases A and B, we start by finding a random rainbow copy of most of the tree using~\ref{T2}, as outlined in Section~\ref{sec:T2}. Then, we embed the rest of the tree using uncovered vertices and exactly the unused colours using~\ref{T1}, which gives a finishing lemma for each case. These finishing lemmas are discussed in Section~\ref{sec:T1}. We use \ref{T3} to embed trees in Case C, which is essentially independent of our embeddings of trees in Cases A and B. This method is outlined in Section \ref{sec:T3}. In Section~\ref{sec:overhead}, we state our main lemmas and theorems, and prove Theorem~\ref{Theorem_Ringel_proof} subject to these results.
The rest of the paper is structured as follows. Following details of our notation, in Section~\ref{sec:prelim} we recall and prove various preliminary results. We then prove the finishing lemma for Case A in Section~\ref{sec:finishA} and the finishing lemma for Case B in Section~\ref{sec:finishB} (together giving \ref{T1}). In Section~\ref{sec:almost}, we give our randomized rainbow embedding of most of the tree (\ref{T2}). In Section~\ref{sec:lastC}, we embed the trees in Case C with a deterministic embedding (\ref{T3}). Finally, in Section~\ref{sec:conc}, we make some concluding remarks.
\subsection{\ref{T2}: Embedding almost all of the tree randomly in Cases A and B}\label{sec:T2}
For a tree $T$ in Case A or B, we carefully choose a large subforest, $T'$ say, of $T$, which contains almost all the edges of $T$. We find a rainbow copy $\hat{T}'$ of $T'$ in the ND-colouring of $K_{2n+1}$ (which exists due to~\cite{MPS}), before applying a finishing lemma to extend $\hat{T}'$ to a rainbow copy of $T$. Extending to a rainbow copy of $T$ is a delicate business --- we must use exactly the $n-e(\hat{T})$ unused colours. Not every rainbow copy of $T'$ will be extendable to a rainbow copy of $T$. However, by combining our methods in \cite{montgomery2018decompositions} and \cite{MPS}, we can take a random rainbow copy $\hat{T}'$ of $T'$ and show that it is likely to be extendable to a rainbow copy of $T$. Therefore, some rainbow copy of $T$ must exist in the ND-colouring of $K_{2n+1}$.
As $\hat{T}'$ is random, the sets $\bar{V}:=V(K_{2n+1})\setminus V(\hat{T})$ and $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T})$ will also be random. The distributions of $\bar{V}$ and $\bar{C}$ will be complicated, but we will not need to know them. It will suffice that there will be large (random) subsets $V\subseteq \bar{V}$ and $C\subseteq \bar{C}$ which each do have a nice, known, distribution.
Here, for example, $V\subseteq V(K_{2n+1})$ has a nice distribution if there is some $q$ so that each element of $V(K_{2n+1})$ appears independently at random in $V$ with probability $q$ --- we say here that $V$ is \epsilonilonmph{$q$-random} if so, and analogously we define a $q$-random subset $C\subseteq C(K_{2n+1})$ (see Section~\ref{sec:prob}). A natural combination of the techniques in~\cite{montgomery2018decompositions,MPS} gives the following.
\begin{theorem}\label{Sketch_Near_Embedding} For each $\epsilonilonpsilon>0$, the following holds for sufficiently large $n$.
Let $K_{2n+1}$ be $2$-factorized and let $T'$ be a forest on $(1-\epsilonilonpsilon)n$ vertices. Then, there is a randomized subgraph $\hat{T}'$ of $K_{2n+1}$ and random subsets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that the following hold for some $p:=p(T')$ (defined precisely in Theorem~\ref{nearembedagain}).
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $\hat{T}'$ is a rainbow copy of $T'$ with high probability.\label{woo1}
\item $V$ is $(p+\epsilonilonps)/6$-random and $C$ is $(1-\epsilonilonpsilon)\epsilonilonps$-random. ($V$ and $C$ may depend on each other.)\label{woo2}
\epsilonilonnd{enumerate}
\epsilonilonnd{theorem}
We will apply a variant of Theorem~\ref{Sketch_Near_Embedding} (see Theorem~\ref{nearembedagain}) to subforests of trees in Cases A and B, and there we will have that $p\gg \epsilonilonps$. Note that, then, $C$ will likely be much smaller than $V$. This reflects that $\hat{T}'\subseteq K_{2n+1}$ will contain fewer than $n$ out of $2n+1$ vertices, while $C(K_{2n+1})\setminus C(\hat{T}')$ contains exactly $n-e(\hat{T}')$ out of $n$ colours.
As explained in~\cite{MPS}, in general the sets $V$ and $C$ cannot be independent, and this is in fact why we need to treat trees in Case C separately. In order to finish the embedding in Cases A and B, we need, essentially, to find \epsilonilonmph{some} independence between the sets $V$ and $C$ (as discussed below). The embedding is then as follows for some small $\delta$ governing the case division, with $\epsilonilonps=\delta^6$ in Case A and $\bar{\epsilonilonps}\oldgg \delta$ in Case B. Given an $(n+1)$-vertex tree $T$ in Case A or B we delete either $\epsilonilonpsilon n$ non-neighbouring leaves (Case A) or $\bar{\epsilonilonpsilon} n/k$ vertex-disjoint bare paths with length $k=\delta^{-1}$ (Case B) to obtain a forest $T'$. Using (a variant of) Theorem~\ref{Sketch_Near_Embedding}, we find a randomized rainbow copy $\hat{T}'$ of $T'$ along with some random vertex and colour sets and apply a finishing lemma to extend this to a rainbow copy of $T$.
After a quick note on the methods in~\cite{montgomery2018decompositions} and~\cite{MPS}, we will discuss the finishing lemmas, and explain why we need some independence, and how much independence is needed.
\subsubsection*{Randomly embedding nearly-spanning trees}
In~\cite{MPS}, we embedded a $(1-\epsilonilonpsilon)n$-vertex tree $T$ into a 2-factorization of $K_{2n+1}$ by breaking it down mostly into large stars and large matchings. For each of these, we embedded the star or matching using its own random set of vertices and random set of colours (which were not necessarily independent of each other). In doing so, we used almost all of the colours in the random colour set, but only slightly less than one half of the vertices in the random vertex set. (This worked as we had more than twice as many vertices in $K_{2n+1}$ than in $T$.) For trees not in Case C, a substantial portion of a large subtree was broken down into matchings. By embedding these matchings more efficiently, using results from~\cite{montgomery2018decompositions}, we can use a smaller random vertex set. This reduction allows us to have, disjointly from the embedded tree, a large random vertex subset $V$.
More precisely, where $q,\epsilonilonpsilon\oldgg n^{-1}$, using a random set $V$ of $2qn$ vertices and a random set $C$ of $qn$ colours, in~\cite{MPS} we showed that, with high probability, from any set $X\subseteq V(K_{2n+1})\setminus V$ with $|X|\leq (1-\epsilonilonpsilon)qn$, there was a $C$-rainbow matching from $X$ into $V$. Dividing $V$ randomly into two sets $V_1$ and $V_2$, each with $qn$ vertices, and using the results in~\cite{montgomery2018decompositions}, we can use $V_1$ to find the $C$-rainbow matching (see Lemma~\ref{Lemma_MPS_nearly_perfect_matching}).
Thus, we gain the random set $V_2$ of $qn$ vertices which we do not use for the embedding of $T$, and we can instead use it to extend this to an $(n+1)$-vertex tree in $K_{2n+1}$.
Roughly speaking, if in total $pn$ vertices of $T$ are embedded using matchings, then we gain altogether a random set of around $pn$ vertices.
If a tree is not in Case C, then the subtree/subforest we embed using these techniques has plenty of vertices embedded using matchings, so that in this case we will be able to take $p\geq 10^{-3}$ when we apply the full version of Theorem~\ref{Sketch_Near_Embedding} (see Theorem~\ref{nearembedagain}). Therefore, we will have many spare vertices when adding the remaining $\epsilonilonps n$ vertices to the copy of $T$. Our challenges are firstly that we need to use exactly all the colours not used on the copy of $T'$ and secondly that there can be a lot of dependence between the sets $V$ and $C$. We first discuss how we ensure that we use every colour.
\subsection{\ref{T1}: Finishing the embedding in Cases A and B}\label{sec:T1}
To find trees using every colour in an ND-coloured $K_{2n+1}$ we prove two \epsilonilonmph{finishing lemmas} (Lemma~\ref{lem:finishA} and~\ref{lem:finishB}). These lemmas say that, for a given randomized set of vertices $V$ and a given randomized set of colours $C$, we can find a rainbow matching/path-forest which uses exactly the colours in $C$, while using some of the vertices from $V$. These lemmas are used to finish the embedding of the trees in Cases A and B, where the last step is to (respectively) embed a matching or path-forest that we removed from the tree $T$ to get the forest $T'$. Applying (a version of) Theorem~\ref{Sketch_Near_Embedding} we get a random rainbow copy $\hat{T'}$ of $T'$ and random sets $\bar{V}=V(K_n)\setminus V(\hat{T}')$ and $\bar{C}=C(K_n)\setminus C(\hat{T}')$.
In order to apply the case-appropriate finishing lemma, we need some independence between $\bar{V}$ and $\bar{C}$, for reasons we now discuss for Case A, and then Case B. Next, we discuss the independence property we use and how we achieve this independence. (Essentially, this property is that $\bar{V}$ and $\bar{C}$ contain two small random subsets which are independent of each other.) Finally, we discuss the absorption ideas for Case A and Case B.
\subsubsection*{Finishing with matchings (for Case A)}
In Case A, we take the $(n+1)$-vertex tree $T$ and remove a large matching of leaves, $M$ say, to get a tree, $T'$ say, that can be embedded using Theorem~\ref{Sketch_Near_Embedding}. This gives a random copy, $\hat{T}'$ say, of $T'$ along with random sets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$
which are $(p+\epsilonilonps)/6$-random and $(1-\epsilonilonps)\epsilonilonps$-random respectively. For trees not in Case C we will have $p\oldgg \epsilonilonps$.
Let $X\subseteq V(\hat{T}')$ be the set of vertices we need to add neighbours to as leaves to make $\hat{T}'$ into a copy of $T$.
We would like to find a perfect matching from $X$ to $\bar{V}:=V(K_{2n+1})\setminus V(\hat{T}')$ with exactly the colours in $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T}')$, using that $V\subseteq \bar{V}$ and $C\subseteq \bar{C}$. (A perfect matching from $X$ to $\bar{V}$ is such a matching covering every vertex in $X$.)
Unfortunately, there may be some $x\in X$ with no edges with colour in $\bar{C}$ leading to $\bar{V}$. (If $C$ and $V$ were independent, then this would not happen with high probability.) If this happens, then the desired matching will not exist.
In Case B, a very similar situation to this may occur, as discussed below, but in Case A there is another potential problem. There may be some colour $c\in \bar{C}$ which does not appear between $X$ and $\bar{V}$, again preventing the desired matching existing. This we will avoid by carefully embedding a small part of $T'$ so that every colour appears between $X$ and $\bar{V}$ on plenty of edges.
\subsubsection*{Finishing with paths (for Case B)}
In Case B, we take the $(n+1)$-vertex tree $T$ and remove a set of vertex-disjoint bare paths to get a forest, $T'$ say, that can be embedded using Theorem~\ref{Sketch_Near_Embedding}. This gives a random copy, $\hat{T}'$ say, of $T'$ along with random sets $V\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C\subseteq C(K_{2n+1})\setminus C(\hat{T}')$
which are $(p+\epsilonilonps)/6$-random and $(1-\epsilonilonps)\epsilonilonps$-random respectively. For trees not in Case C we will have $p\oldgg \epsilonilonps$.
Let $\epsilonilonll$ and $X=\{x_1,\ldots,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}\subseteq V(\hat{T}')$ be such that to get a copy of $T$ from $\hat{T}'$ we need to add vertex-disjointly a suitable path between $x_i$ and $y_i$, for each $i\in [\epsilonilonll]$.
We would like to find these paths with interior vertices in $\bar{V}:= V(K_{2n+1})\setminus V(\hat{T}')$ so that their edges are collectively rainbow with exactly the colours in $\bar{C}:=C(K_{2n+1})\setminus C(\hat{T}')$, using that $V\subseteq \bar{V}$ $C\subseteq \bar{C}$. Unfortunately, there may be some $x\in X$ with no edges with colour in $\bar{C}$ leading to $V$. (If $C$ and $V$ were independent, then, again, this would not happen with high probability.) If this happens, then the desired paths will not exist.
Note that the analogous version of the second problem in Case A does not arise in Case B. Here, it is likely that every colour appears on many edges within $V$, so that we can use any colour by putting an appropriate edge within $V$ in the middle of one of the missing paths.
\subsubsection*{Retaining some independence}
To avoid the problem common to Cases A and B, when proving our version of Theorem~\ref{Sketch_Near_Embedding} (that is, Theorem~\ref{nearembedagain}), we set aside small random sets $V_0$ and $C_0$ early in the embedding, before the dependence between colours and vertices arises. This gives us a version of Theorem~\ref{Sketch_Near_Embedding} with the additional property that, for some $\mu\oldll \epsilonilonps$, there are additional random sets $V_0\subseteq V(K_{2n+1})\setminus (V(\hat{T}')\cup V)$ and $C_0\subseteq C(K_{2n+1})\setminus (C(\hat{T}')\cup C)$ such that the following holds in addition to \ref{woo1} and \ref{woo2}.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item $V_0$ is a $\mu$-random subset of $V(K_{2n+1})$, $C_0$ is a $\mu$-random subset of $C(K_{2n+1})$, and they are independent of each other.\label{propq2}
\epsilonilonnd{enumerate}
\noindent
Then, by this independence, with high probability, every vertex in $K_{2n+1}$ will have $\mu^2 n/2$ adjacent edges with colour in $C_0$ going into the set $V_0$ (see Lemma~\ref{Lemma_high_degree_into_random_set}).
To avoid the problem that only arises in Case A, consider the set $U\subseteq V(T')$ of vertices which need leaves added to them to reach $T$ from $T'$. By carefully embedding a small subtree of $T'$ containing plenty of vertices in $U$, we ensure that, with high probability, each colour appears plenty of times between the image of $U$ and $V_0$. That is, we have the following additional property for some $1/n\oldll \xi\oldll \mu$.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item With high probability, if $Z$ is the copy of $U$ in $\hat{T}'$, then every colour in $C(K_{2n+1})$, has at least $\xi n$ edges between $Z$ and $V_0$.\label{propq1}
\epsilonilonnd{enumerate}
Of course, \ref{propq2} and \ref{propq1} do not show that our desired matching/path-collection exists, only that (with high probability) there is no single colour or vertex preventing its existence. To move from this to find the actual matching/path-collection we use \epsilonilonmph{distributive absorption}.
\subsubsection*{Distributive absorption}
To prove our finishing lemmas, we use an \epsilonilonmph{absorption strategy}. Absorption has its origins in work by Erd\H{o}s, Gy\'arf\'as and Pyber~\cite{EP} and Krivelevich~\cite{MKtri}, but was codified by R\"odl, Ruci\'nski and Szemer\'edi~\cite{RRS} as a versatile technique
for extending approximate results into exact ones. For both Case A and Case B we use a new implementation of \epsilonilonmph{distributive absorption}, a method introduced by the first author in~\cite{montgomery2018spanning}.
To describe our absorption, let us concentrate on Case A. Our methods in Case B are closely related, and we comment on these afterwards. To recap, we have a random rainbow tree $\hat{T}'$ in the ND-colouring of $K_{2n+1}$ and a set $X\subseteq V(\hat{T})$, so that we need to add a perfect matching from $X$ into $\bar{V}=V(K_{2n+1})\setminus V(\hat{T}')$ to make $\hat{T}'$ into a copy of $T$. We wish to add this matching in a rainbow fashion using (exactly) the colours in $\bar{C}= C(K_{2n+1})\setminus C(\hat{T})$.
To use distributive absorption, we first show that for any set $\hat{C}\subseteq C(K_{2n+1})$ of at most 100 colours, we can find a set $D\subseteq \bar{C}\setminus C$ and sets $X'\subseteq X$ and $V'\subseteq \bar{V}$ with $|D|\leq 10^3$, $|V'|\leq 10^4$ and $|X'|=|D|+1$, so that the following holds.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item Given any colour $c\in\hat{C}$, there is a perfect $(D\cup \{c\})$-rainbow matching from $X'$ to $V'$.\label{switchprop}
\epsilonilonnd{enumerate}
We call such a triple $(D,X',V')$ a \epsilonilonmph{switcher} for $\hat{C}$. AS $|\bar{C}|=|X|$, a perfect $(\bar{C}\setminus D)$-rainbow matching from $X\setminus X'$ into $V\setminus V'$ uses all but 1 colour in $\bar{C}\setminus D$. If we can find such a matching whose unused colour, $c$ say, lies in $\hat{C}$, then using \ref{switchprop}, we can find a perfect $(D\cup\{c\})$-rainbow matching from $X'$ to $V'$. Then, the two matchings combined form a perfect $\bar{C}$-rainbow matching from $X$ into $V$, as required.
The switcher outlined above only gives us a tiny local variability property, reducing finding a large perfect matching with exactly the right number of colours to finding a large perfect matching with one spare colour so that the unused colour lies in a small set (the set $\bar{C}$). However, by finding many switchers for carefully chosen sets $\hat{C}$, we can build this into a global variability property. These switchers can be found using different vertices and colours (see Section~\ref{sec:switcherspaths}), so that matchings found using the respective properties \ref{switchprop} can be combined in our embedding.
We choose different sets $\hat{C}$ for which to find a switcher by using an auxillary graph as a template. This template is a \epsilonilonmph{robustly matchable bipartite graph} --- a bipartite graph, $K$ say, with vertex classes $U$ and $Y\cup Z$ (where $Y$ and $Z$ are disjoint), with the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $Z^\ast\subseteq Z$ with size $|U|-|Y|$, there is a perfect matching in $K$ between $U$ and $Y\cup Z^\ast$.
\epsilonilonnd{enumerate}
Such bipartite graphs were shown to exist by the first author~\cite{montgomery2018spanning}, and, furthermore, for large $m$ and $\epsilonilonll\leq m$, we can find such a graph with maximum degree at most 100, $|U|=3m$, $|Y|=2m$ and $|Z|=m+\epsilonilonll$ (see Lemma~\ref{Lemma_H_graph}).
To use the template, we take disjoint sets of colours, $C'=\{c_v:v\in Y\}$ and $C''=\{c_v:v\in Z\}$ in $\bar{C}$. For each $u\in U$, we find a switcher $(D_u,X_u,V_u)$ for the set of colours $\{c_v:v\in N_K(u)\}$. Furthermore, we do this so that the sets $D_u$ are disjoint and in $\bar{C}\setminus (C'\cup C'')$, and the sets $X_u$, and $V_u$, are disjoint and in $X$, and $\bar{V}$, respectively. We can then show we have the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any set $C^*\subseteq C''$ of $m$ colours, there is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$.\label{Kprop}
\epsilonilonnd{enumerate}
Indeed, to see this, take any set of $C^*\subseteq C''$ of $m$ colours, let $Z^*=\{v:c_v\in C^*\}$ and note that $|Z^*|=m$. By \ref{Kprop}, there is a perfect matching in $K$ from $U$ into $Y\cup Z^\ast$, corresponding to the function $f:U\to Y\cup Z^*$ say. For each $u\in U$, using that $(D_u,X_u,V_u)$ is a switcher for $\{c_v:v\in N_K(u)\}$ and $uf(u)\in E(K)$, find a perfect $(D_u\cup \{c_{f(u)}\})$-rainbow matching $M_u$ from $X_u$ to $V_u$.
As the sets $D_u$, $X_u$, $V_u$, $u\in U$, are disjoint, $\cup_{u\in U}M_u$ is a perfect $(C^*\cup C'\cup(\cup_{u\in U}D_u))$-rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$, as required.
Thus, we have a set of colours $C''$ from which we are free to use any $\epsilonilonll$ colours, and then use the remaining colours together with $C'\cup(\cup_{u\in U}D_u)$ to find a perfect rainbow matching from $\cup_{u\in U}X_u$ into $\cup_{u\in U}V_u$. By letting $m$ be as large as allowed by our construction methods, and $C''$ be a random set of colours, we have a useful \epsilonilonmph{reservoir} of colours, so that we can find a structure in $T$ using $\epsilonilonll$ colours in $C''$, and then finish by attaching a matching to $\cup_{u\in U}X_u$.
We have two things to consider to fit this final step into our proof structure, which we discuss below. Firstly, we can only absorb colours in $C''$, so after we have covered most of the colours, we need to cover the unused colours outside of $C''$ (essentially achieved by \ref{propp2} below). Secondly, we find the switchers greedily in a random set. There are many more unused colours from this set than we can absorb, and the unused colours no longer have good random properties, so we also need to reduce the unused colours to a number that we can absorb (essentially achieved by \ref{propp1} below).
\subsubsection*{Creating our finishing lemmas using absorption}
To recap, we wish to find a perfect $\bar{C}$-rainbow matching from $X$ into $\bar{V}$. To do this, it is sufficient to find partitions $X=X_1\cup X_2\cup X_3$, $\bar{V}=V_1\cup V_2\cup V_3$ and $\bar{C}=C_1\cup C_2\cup C_3$ with the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item There is a perfect $C_1$-rainbow matching from $X_1$ into $V_1$. \label{propp1}
\item Given any set of colours $C'\subseteq C_1$ with $|C'|\leq |C_1|-|X_1|$, there is a perfect $(C_2\cup C')$-rainbow matching from $X_2$ into $V_2$ which uses each colour in $C'$.\label{propp2}
\item Given any set of colours $C''\subseteq C_2$ with size $|X_3|-|C_3|$, there is a perfect $(C''\cup C_3)$-rainbow matching from $X_3$ into $V_3$.\label{propp3}
\epsilonilonnd{enumerate}
Finding such a partition requires the combination of all our methods for Case A. In brief, however, we develop \ref{propp1} using a result from~\cite{montgomery2018decompositions} (see Lemma~\ref{Lemma_MPS_nearly_perfect_matching}), we develop \ref{propp2} using the condition \ref{propq1}, and we develop \ref{propp3} using the distributive absorption strategy outlined above.
If we can find such a partition, then we can easily show that the matching we want must exist. Indeed, given such a partition, then, using \ref{propp1}, let $M_1$ be a perfect $C_1$-rainbow matching from $X_1$ into $V_1$, and let $C'=C_1\setminus C(M_1)$. Using~\ref{propp2}, let $M_2$ be a perfect $(C_2\cup C')$-rainbow matching from $X_2$ into $V_2$ which uses each colour in $C'$, and let $C''=(C_2\cup C')\setminus C(M)=C_2\setminus C(M)$. Finally, noting that $|C''|+|C_3|=|\bar{C}|-|X_1|-|X_2|=|X_3|$, using~\ref{propp3}, let $M_3$ be a perfect $(C''\cup C_3)$-rainbow matching from $X_3$ into $V_3$. Then, $M_1\cup M_2\cup M_3$ is a $\bar{C}$-rainbow matching from $X$ into $\bar{V}$.
The above outline also lies behind our embedding in Case B, where we finish instead by embedding $\epsilonilonll$ paths vertex-disjointly between certain vertex pairs, for some $\epsilonilonll$. Instead of the partition $X_1\cup X_2\cup X_3$ we have a partition $[\epsilonilonll]=I_1\cup I_2\cup I_3$, and, instead of each matching from $X_i$ to $V_i$, $i\in [3]$, we find a set of vertex-disjoint $x_j,y_j$-paths, $j\in I_i$, with interior vertices in $V_i$ which are collectively $C_i$-rainbow. The main difference is how we construct switchers using paths instead of matchings (see Section~\ref{sec:switchers}).
\subsection{\ref{T3}: The embedding in Case C}\label{sec:T3}
After large clusters of adjacent leaves are removed from a tree in Case C, few vertices remain. We remove these large clusters, from the tree, $T$ say, to get the tree $T'$, and carefully embed $T'$ into the ND-colouring using a deterministic embedding. The image of this deterministic embedding occupies a small interval in the ordering used to create the ND-colouring. Furthermore, the embedded vertices of $T'$ which need leaves added to create a copy of $T$ are well-distributed within this interval. These properties will allow us to embed the missing leaves using the remaining colours. This is given more precisely in Section~\ref{sec:caseC}, but in order to illustrate this in the easiest case, we will give the embedding when there is exactly one vertex with high degree.
Our embedding in this case is rather simple. Removing the leaves incident to a very high degree vertex, we embed the rest of the tree into $[n]$ so that the high degree vertex is embedded to 1. The missing leaves are then embedded into $[2n+1]\setminus [n]$ using the unused colours.
\begin{theorem}[One large vertex]\label{Theorem_one_large_vertex}
Let $n\geq 10^6$. Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be an $(n+1)$-vertex tree containing a vertex $v_1$ which is adjacent to $\geq 2n/3$ leaves.
Then, $K_{2n+1}$ contains a rainbow copy of $T$.
\epsilonilonnd{theorem}
\begin{proof}
See Figure~\ref{Figure_1_vertex} for an illustration of this proof.
Let $T'$ be $T$ with the neighbours of $v_1$ removed and let $m=|T'|$. By assumption, $|T'|\leq n/3+1$. Order the vertices of $T'-v_1$ as $v_2, \dots, v_{m}$ so that $T[v_1, \dots, v_i]$ is a tree for each $i\in [m]$. Embed $v_1$ to $1$ in $K_{2n+1}$, and then greedily embed $v_2, \dots, v_m$ in turn to some vertex in $[n]$ so that the copy
of $T'$ which is formed is rainbow in $K_{2n+1}$. This is possible since at each step at most $|T'|\leq n/3$ of the vertices in $[n]$ are occupied, and at most $e(T')\leq n/3-1$ colours are used. Since the $ND$-colouring has 2 edges of each colour adjacent to each vertex,
this forbids at most $n/3+2(n/3-1)=n-2$ vertices in $[n]$. Thus, we can embed each $v_{i}$, $2\leq i\leq m$ using an unoccupied vertex in $[n]$ so that the edge from $v_i$ to $v_1, \ldots, v_{i-1}$ has a colour that we have not yet used. Let $S'$ be the resulting rainbow copy of $T'$, so that $V(S')\subseteq [n]$.
Let $S$ be $S'$ together with the edges between $1$ and $2n+2-c$ for every $c\in [n]\setminus C(S')$. Note that the neighbours added are all bigger than $n$, and so the resulting graph is a tree. There are exactly $n-e(T')$ edges added, so $S$ is a copy of $T$. Finally, for each $c\in [n]\setminus C(S')$, the edge from $1$ to $2n+2-c$ is colour $c$, so the resulting tree is rainbow.
\epsilonilonnd{proof}
\begin{figure}[h]
\begin{center}
{
\input{caseConevertex}
}
\epsilonilonnd{center}
\caption{Embedding the tree in Case C when there is 1 vertex with many leaves as neighbours.}\label{Figure_1_vertex}
\epsilonilonnd{figure}
The above proof demonstrates the main ideas of our strategy for Case C. Notice that the above proof has two parts --- first we embed the small tree $T'$, and then we find the neighbours of the high degree vertex $v_1$. In order to ensure that the final tree is rainbow we choose the neighbours of $v_1$ in some interval $[n+1, 2n]$ which is disjoint from the copy of $T'$, and to which every colour appears from the image of $v_1$. This way, we were able to use every colour which was not present on the copy of $T'$.
When there are multiple high degree vertices $v_1, \dots, v_{\epsilonilonll}$ the strategy is the same --- first we embed a small rainbow tree $T'$ containing $v_1, \dots, v_{\epsilonilonll}$, then we embed the neighbours of $v_1, \dots, v_{\epsilonilonll}$. This is done in Section~\ref{sec:caseC}.
\subsection{Proof of Theorem~\ref{Theorem_Ringel_proof}}\label{sec:overhead}
Here we will state our main theorems and lemmas, which are proved in later sections, and combine them to prove Theorem~\ref{Theorem_Ringel_proof}. First, we have our randomized embedding of a $(1-\epsilonilonpsilon)n$-vertex tree, which is proved in Section~\ref{sec:almost}. For convenience we use the following definition.
\begin{definition}
Given a vertex set $V\subseteq V(G)$ of a graph $G$, we say $V$ is \epsilonilonmph{$\epsilonilonll$-replete in $G$} if $G[V]$ contains at least $\epsilonilonll$ edges of every colour in $G$. Given, further, $W\subseteq V(G)\setminus V$, we say $(W,V)$ is \epsilonilonmph{$\epsilonilonll$-replete in $G$} if at least $\epsilonilonll$ edges of every colour in $G$ appear in $G$ between $W$ and $V$. When $G=K_{2n+1}$, we simply say that $V$ and $(W,V)$ are \epsilonilonmph{$\epsilonilonll$-replete}.
\epsilonilonnd{definition}
\begin{theorem}[Randomised tree embeddings]\label{nearembedagain} Let $1/n\ll \xi\ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll 1$ and $\xi\ll 1/k\ll \log^{-1}n$.
Let $K_{2n+1}$ be ND-coloured, let $T'$ be a $(1-\epsilonilonpsilon)n$-vertex forest and let $U\subseteq V(T')$ contain $\epsilonilonps n$ vertices. Let $p$ be such that removing leaves around vertices next to $\geq k$ leaves from $T'$ gives a forest with $pn$ vertices.
Then, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and disjoint random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that the following hold.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item With high probability, $\hat{T}'$ is a rainbow copy of $T'$ in which, if $W$ is the copy of $U$, then $(W,V_0)$ is $(\xi n)$-replete,\label{propa1}
\item $V_0$ and $C_0$ are $\mu$-random and independent of each other, and\label{propa2}
\item $V$ is $(p+\epsilonilonpsilon)/6$-random and $C$ is $(1-\epsilonilonta)\epsilonilonpsilon$-random.\label{propa3}
\epsilonilonnd{enumerate}
\epsilonilonnd{theorem}
Next, we have the two finishing lemmas, which are proved in Sections~\ref{sec:finishA} and~\ref{sec:finishB} respectively.
\begin{lemma}[The finishing lemma for Case A]\label{lem:finishA} Let $1/n\ll \xi\ll \mu \ll \epsilonilonta \ll\epsilonilonps\ll p\leq 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V,V_0$ are disjoint subsets of $V(K_{2n+1})$ which are $p$- and $\mu$-random respectively. Suppose that $C,C_0$ are disjoint subsets in $C(K_{2n+1})$, so that $C$ is $(1-\epsilonilonta)\epsilonilonps$-random, and
$C_0$ is $\mu$-random and independent of $V_0$. Then, with high probability, the following holds.
Given any disjoint sets $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ with $|X|=\epsilonilonps n$, so that $(X,Z)$ is $(\xi n)$-replete, and any set $D\subseteq C(K_{2n+1})$ with $|D|=\epsilonilonps n$ and $C_0\cup C\subseteq D$, there is a perfect $D$-rainbow matching from $X$ into $V\cup V_0\cup Z$.
\epsilonilonnd{lemma}
Note that in the following lemma we implicitly assume that $m$ is an integer. That is, we assume an extra condition on $n$, $k$ and $\epsilonilonps$. We remark on this further in Section~\ref{sec:not}.
\begin{lemma}[The finishing lemma for Case B]\label{lem:finishB} Let $1/n\ll 1/k\ll \mu\ll \epsilonilonta \ll\epsilonilonps\ll p\leq 1$ be such that $k = 7\mod 12$ and $695|k$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V,V_0$ are disjoint subsets of $V(K_{2n+1})$ which are $p$- and $\mu$-random respectively. Suppose that $C,C_0$ are disjoint subsets in $C(K_{2n+1})$, so that $C$ is $(1-\epsilonilonta)\epsilonilonps$-random, and $C_0$ is $\mu$-random and independent of $V_0$. Then, with high probability, the following holds with $m=\epsilonilonps n/k$.
For any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$, and any set $D\subseteq C(K_{2n+1})$ with $|D|=mk$ and $C\cup C_0\subseteq D$, the following holds. There is a set of vertex-disjoint $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow.
\epsilonilonnd{lemma}
Finally, the following theorem, proved in Section~\ref{sec:caseC}, will allow us to embed trees in Case C.
\begin{theorem}[Embedding trees in Case C]\label{Theorem_case_C}
Let $n\geq 10^6$. Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be a tree on $n+1$ vertices with a subtree $T'$ with $\epsilonilonll:=|T'|\leq n/100$ such that $T'$ has vertices $v_1, \dots, v_{\epsilonilonll}$ so that adding $d_i\geq \log^4n$ leaves to each $v_i$ produces $T$.
Then, $K_{2n+1}$ contains a rainbow copy of $T$.
\epsilonilonnd{theorem}
We can now combine these results to prove Theorem~\ref{Theorem_Ringel_proof}. We also use a simple lemma concerning replete random sets, Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random}, which is proved in Section~\ref{sec:prelim}. As used below, it implies that if $V_0$ and $V_1$, with $V_1\subseteq V_0\subseteq V(K_{2n+1})$, are $\mu$- and $\mu/2$-random respectively, then, given any randomised set $X$ such that $(X,V_0)$ is with high probability replete (for some parameter), then $(X,V_1)$ is also with high probability replete (for some suitably reduced parameter).\begin{proof}[Proof of Theorem~\ref{Theorem_Ringel_proof}]
Choose $\xi, \mu,\epsilonilonta,\delta,\bar{\mu},\bar{\epsilonilonta}$ and $\bar{\epsilonilonps}$ such that $1/n \ll\xi \ll\mu\ll \epsilonilonta\ll \delta\ll\bar{\mu}\ll\bar{\epsilonilonta}\ll\bar{\epsilonilonps} \ll\log^{-1} n$ and $k=\delta^{-1}$ is an integer such that $k = 7\mod 12$ and $695|k$. Let $T$ be an $(n+1)$-vertex tree and let $K_{2n+1}$ be ND-coloured.
By Lemma~\ref{Lemma_case_division}, $T$ is in Case A, B or C for this $\delta$. If $T$ is in Case C, then Theorem~\ref{Theorem_case_C} implies that $K_{2n+1}$ has a rainbow copy of $T$. Let us assume then that $T$ is not in Case C. Let $k=\delta^{-4}$, and note that, as $T$ is not in Case $C$, removing from $T$ leaves around any vertex adjacent to at least $k$ leaves gives a tree with at least $n/100$ vertices.
If $T$ is in Case A, then let $\epsilonilonps=\delta^6$, let $L$ be a set of $\epsilonilonps n$ non-neighbouring leaves in $T$, let $U=N_T(L)$ and let $T'=T-L$. Let $p$ be such that removing from $T'$ leaves around any vertex adjacent to at least $k$ leaves gives a tree with $pn$ vertices. Note that each leaf of $T'$ which is not a leaf of $T$ must be adjacent to a vertex in $L$ in $T$. Note further that, if a vertex in $T$ is next to fewer than $k$ leaves in $T$, but at least $k$ leaves in $T'$, then all but at most $k-1$ of those leaves in $T'$ must have a neighbour in $L$ in $T$. Therefore, $p\geq 1/100-(k+1)\epsilonilonps\geq 1/200$.
By Theorem~\ref{nearembedagain}, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that \ref{propa1}--\ref{propa3} hold.
Using \ref{propa2}, let $V_1,V_2\subseteq V_0$ be disjoint $(\mu/2)$-random subsets of $V(K_{2n+1})$.
By Lemma~\ref{lem:finishA} (applied with $\xi'=\xi/4, \mu'=\mu/2, \epsilonilonta=\epsilonilonta, \epsilonilonpsilon=\epsilonilonpsilon, p'=(p+\epsilonilonpsilon)/6$, $V=V, C=C, V_0'=V_1,$ and $C_0'$ a $(\mu/2)$-random subset of $C_0$)
and \ref{propa2}--\ref{propa3}, and by \ref{propa1} and Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item Given any disjoint sets $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_1)$ so that $|X|=\epsilonilonps n$ and $(X,Z)$ is $(\xi n/4)$-replete, and any set $D\subseteq C(K_{2n+1})$ with $|D|=\epsilonilonps n$ and $C_0\cup C\cup D$, there is a perfect $D$-rainbow matching from $X$ into $V\cup V_1\cup Z$.\label{fine1}
\item $\hat{T}'$ is a rainbow copy of $T'$ in which, letting $W$ be the copy of $U$, $(W,V_2)$ is $(\xi n/4)$-replete.\label{fine2}
\epsilonilonnd{enumerate}
Let $D=C(K_{2n+1})\setminus C(\hat{T}')$, so that $C_0\cup C\subseteq D$, and, as $\hat{T}'$ is rainbow by \ref{fine2}, $|D|=\epsilonilonps n$. Let $W$ be the copy of $U$ in $\hat{T}'$. Then, using \ref{fine1} with $Z=V_2$ and \ref{fine2}, let $M$ be a perfect $D$-rainbow matching from $W$ into $V\cup V_1\cup V_2\subseteq V\cup V_0$. As $V\cup V_0$ is disjoint from $V(\hat{T}')$, $\hat{T}'\cup M$ is a rainbow copy of $T$. Thus, a rainbow copy of $T$ exists with high probability in the ND-colouring of $K_{2n+1}$, and hence certainly some such rainbow copy of $T$ must exist.
If $T$ is in Case B, then recall that $k=\delta^{-1}$ and let $m=\bar{\epsilonilonps} n/k$. Let $P_1,\ldots,P_m$ be vertex-disjoint bare paths with length $k$ in $T$. Let $T'$ be $T$ with the interior vertices of $P_i$, $i\in [m]$, removed.
Let $p$ be such that removing from $T'$ leaves around any vertex adjacent to at least $k$ leaves gives a forest with $pn$ vertices. Note that (reasoning similarly to as in Case A) $p\geq 1/100-2(k+1)m\geq 1/200$.
By Theorem~\ref{nearembedagain}, there is a random subgraph $\hat{T}'\subseteq K_{2n+1}$ and disjoint random subsets $V,V_0\subseteq V(K_{2n+1})\setminus V(\hat{T}')$ and $C,C_0\subseteq C(K_{2n+1})\setminus C(\hat{T}')$ such that \ref{propa1}--\ref{propa3} hold with $\xi=\xi$, $\mu=\bar{\mu}$, $\epsilonilonta=\bar{\epsilonilonta}$ and $\epsilonilonps=\bar{\epsilonilonps}$. By Lemma~\ref{lem:finishB} and \ref{propa1}--\ref{propa3}, and by \ref{propa1}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ and any set $D\subseteq C(K_{2n+1})$ with $|D|=mk$ and $C_0\cup C\subseteq D$, there is a set of vertex-disjoint paths $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow.\label{fine11}
\item $\hat{T}'$ is a rainbow copy of $T'$.\label{fine12}
\epsilonilonnd{enumerate}
Let $D=C(K_{2n+1})\setminus C(\hat{T}')$, so that $C_0\cup C\subseteq D$, and, as $\hat{T}'$ is rainbow by \ref{fine12}, $|D|=\epsilonilonps n$.
For each path $P_i$, $i\in [m]$, let $x_i$ and $y_i$ be the copy of the endvertices of $P_i$ in $\hat{T}'$. Using \ref{fine11}, let $Q_i$, $i\in [m]$, be a set of vertex-disjoint $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $D$-rainbow. As $V\cup V_0$ is disjoint from $V(\hat{T}')$,
$\hat{T}'\cup (\cup_{i\in [m]}Q_i)$ is a rainbow copy of $T$. Thus, a rainbow copy of $T$ exists with high probability in the ND-colouring of $K_{2n+1}$, and hence certainly some such rainbow copy of $T$ must exist.
\epsilonilonnd{proof}
\section{Preliminary results and observations}\label{sec:prelim}
\subsection{Notation}\label{sec:not}
For a coloured graph $G$, we denote the set of vertices of $G$ by $V(G)$, the set of edges of $G$ by $E(G)$, and the set of colours of $G$ by $C(G)$.
For a coloured graph $G$, disjoint sets of vertices $A,B\subseteqeq V(G)$ and a set of colours $C\subseteqeq C(G)$ we use $G[A,B,C]$ to denote the subgraph of $G$ consisting of colour $C$ edges from $A$ to $B$, and $G[A,C]$ to be the graph of the colour $C$ edges within $A$.
For a single colour $c$, we denote the set of colour $c$ edges from $A$ to $B$ by $E_c(A,B)$.
A coloured graph is \epsilonilonmph{globally $k$-bounded} if every colour is on at most $k$ edges.
For a set of colours $C$, we say that a graph $H$ is ``$C$-rainbow'' if $H$ is rainbow and $C(H)\subseteqeq C$. We say that a collection of graphs $H_1, \dots, H_k$ is \epsilonilonmph{collectively} rainbow if their union is rainbow.
A star is a tree consisting of a collection of leaves joined to a single vertex (which we call the \epsilonilonmph{centre}).
A \epsilonilonmph{star forest} is a graph consisting of vertex disjoint stars.
For any reals $a,b\in \mathbb{R}$, we say $x=a\pm b$ if $x\in [a-b,a+b]$.
\subsubsectionme{Asymptotic notation}
For any $C\geq 1$ and $x,y\in (0,1]$, we use ``$x\oldll_C y$'' to mean ``$x\leq \frac{y^C}{C}$''. We will write ``$x{\ll} y$'' to mean that there is some absolute constant $C$ for which the proof works with ``$x{\ll}{} y$'' replaced by ``$x\oldll_{C} y$''.
In other words the proof works if $y$ is a small but fixed power of $x$.
This notation compares to the more common notation $x\oldll y$ which means ``there is a fixed positive continuous function $f$ on $(0,1]$ for which the remainder of the proof works with ``$x\oldll y$'' replaced by ``$x\leq f(y)$''. (Equivalently, ``$x\oldll y$'' can be interpreted as ``for all $x\in (0,1]$, there is some $y\in (0,1]$ such that the remainder of the proof works with $x$ and $y$''.) The two notations ``$x{\ll}{} y$'' and ``$x\oldll y$'' are largely interchangeable --- most of our proofs remain correct with all instances of ``${\ll}$'' replaced by ``$\oldll$''. The advantage of using ``${\ll}$'' is that it proves polynomial bounds on the parameters (rather than bounds of the form ``for all $\epsilonilonpsilon>0$ and sufficiently large $n$''). This is important towards the end of this paper, where the proofs need polynomial parameter bounds.
While the constants $C$ will always be implicit in each instance of ``$x{\ll}{} y$'', it is possible to work them out explicitly. To do this one should go through the lemmas in the paper and choose the constants $C$ for a lemma after the constants have been chosen for the lemmas on which it depends. This is because an inequality $x\oldll_C y$ in a lemma may be needed to imply an inequality $x\oldll_{C'} y$ for a lemma it depends on. Within an individual lemma we will often have several inequalities of the form $x{\ll} y$. There the constants $C$ need to be chosen in the reverse order of their occurrence in the text. The reason for this is the same --- as we prove a lemma we may use an inequality $x\oldll_C y$ to imply another inequality $x\oldll_{C'} y$ (and so we should choose $C'$ before choosing $C$).
Throughout the paper, there are four operations we perform with the ``$x{\ll}{} y$'' notation:
\begin{enumerate}[label = (\alph{enumi})]
\item We will use $x_1{\ll} x_2{\ll}\dots{\ll} x_k$ to deduce finitely many inequalities of the form ``$p(x_1, \dots, x_k)\leq q(x_1, \dots, x_k)$'' where $p$ and $q$ are {monomials} with non-negative coefficients and $\min\{i: p(0, \dots, 0, x_{i+1}, \dots, x_k)=0\}< \min\{j: q(0, \dots, 0, x_{j+1}, \dots, x_k)=0\}$ e.g.\ $1000x_1\leq x_2^5x_4^2x_5^3$ is of this form.
\item We will use $x{\ll} y$ to deduce finitely many inequalities of the form ``$x\oldll_C y$'' for a fixed constant $C$.
\item For $x{\ll} y$ and fixed constants $C_1, C_2$, we can choose a variable $z$ with $x\oldll_{C_1} z\oldll_{C_2}y$.
\item For $n^{-1}{\ll} 1$ and any fixed constant $C$, we can deduce $n^{-1}\oldll_C \log^{-1} n\oldll_C 1$.
\epsilonilonnd{enumerate}
See \cite{montgomery2018decompositions} for a detailed explanation of why the above operations are valid.
\subsubsectionme{Rounding}
In several places, we will have, for example, constants $\epsilonilonps$ and integers $n,k$ such that $1/n\ll \epsilonilonps,1/k$ and require that $m=\epsilonilonps n/k$ is an integer, or even divisible by some other small integer. Note that we can arrange this easily with a very small alteration in the value of $\epsilonilonps$. For example, to apply Lemma~\ref{lem:finishB} we assume that $m$ is an integer, and therefore in the proof of Theorem~\ref{Theorem_Ringel_proof}, when we choose $\bar{\epsilonilonps}$ we make sure that when this lemma is applied with $\epsilonilonps=\bar{\epsilonilonps}$ the corresponding value for $m$ is an integer.
\subsection{Probabilistic tools}\label{sec:prob}
For a finite set $V$, a $p$-random subset of $V$ is a set formed by choosing every element of $V$ independently at random with probability $p$. If $V$ is not specified, then we will implicitly assume that $V$ is $V(K_{2n+1})$ or $C(K_{2n+1})$, where this will be clear from context.
If $A,B\subseteqeq V$ with $A$ $p$-random and $B$ $q$-random, we say that $A$ and $B$ are \epsilonilonmph{disjoint} if every $v\in V$ is in $A$ with probability $p$, in $B$ with probability $q$, and outside of $A\cup B$ with probability $1-p-q$ (and this happens independently for each $v\in V$). We say that a $p$-random set $A$ is independent from a $q$-random set $B$ if the choices for $A$ and $B$ are made independently, that is, if $\mathbb{P}(A=A'\land B=B')=\mathbb{P}(A=A')\mathbb{P}(B=B')$ for any outcomes $A'$ and $B'$ of $A$ and $B$.
Often, we will have a $p$-random subset $X$ of $V$ and divide it into two disjoint $(p/2)$-random subsets of $V$. This is possible by choosing which subset each element of $X$ is in independently at random with probability $1/2$ using the following simple lemma.
\begin{lemma}[Random subsets of random sets]\label{Lemma_mixture_of_p_random_sets}
Suppose that $X,Y:\Omega\to 2^V$ where $X$ is a $p$-random subset of $V$ and $Y|X$ is a $q$-random subset of $X$ (i.e. the distribution of $Y$ conditional on the event ``$X=X'$'' is that of a $q$-random subset of $X'$). Then $Y$ is a $pq$-random subset of $V$.
\epsilonilonnd{lemma}
\begin{proof}
First notice that, to show a set $X\subseteqeq V$ is $(pq)$-random, it is sufficient to show that $\mathbb{P}(S\subseteqeq X)=(pq)^{|S|}$ for all $S\subseteqeq V$ (for example, by using inclusion-exclusion).
Now, we prove the lemma.
Since $X$ is $p$-random we have $\mathbb{P}(S\subseteqeq X)=p^{|S|}$. Since $Y|X$ is $q$-random, we have $\mathbb{P}(S\subseteqeq Y|S\subseteqeq X)=q^{|S|}$. This gives $\mathbb{P}(S\subseteqeq Y)= \mathbb{P}(S\subseteqeq Y|S\subseteqeq X)\mathbb{P}(S\subseteqeq X)=(pq)^{|S|}$ for every set $S\subseteqeq V$.
\epsilonilonnd{proof}
We will use the following standard form of Azuma's inequality and a Chernoff Bound. For a probability space $\Omega=\prod_{i=1}^n \Omega_i$, a random variable $X:\Omega\to \mathbb{R}$ is \epsilonilonmph{$k$-Lipschitz} if changing $\omega\in \Omega$ in any one coordinate changes $X(\omega)$ by at most $k$.
\begin{lemma}[Azuma's Inequality]\label{Lemma_Azuma}
Suppose that $X$ is $k$-Lipschitz and influenced by $\leq m$ coordinates in $\{1, \dots, n\}$. Then{, for any $t>0$,}
$$\mathbb{P}\left(|X-\mathbb{E}(X)|>t \right)\leq 2e^{\frac{-t^2}{mk^2}}$$.
\epsilonilonnd{lemma}
Notice that the bound in the above inequality can be rewritten as $\mathbb{P}\left(X\neq \mathbb{E}(X)\pm t \right)\leq 2e^{\frac{-t^2}{mk^2}}$.
\begin{lemma}[Chernoff Bound] \label{Lemma_Chernoff}
Let $X$ be a binomial random variable with parameters $(n,p)$.
Then, for each $\epsilonilonpsilon\in (0,1)$, we have
$$\mathbb P\big(|X-pn|> \epsilonilonpsilon pn\big)\leq
2e^{-\frac{pn\epsilonilonpsilon^2}{3}}.$$
\epsilonilonnd{lemma}
For an event $X$ in a probability space depending on a parameter $n$, we say ``$X$ holds with high probability'' to mean ``$X$ holds with probability $1-o(1)$'' where $o(1)$ is some function $f(n)$ with $f(n)\to 0$ as $n\to \infty$.
We will use this definition for the following operations.
\begin{itemize}
\item \textbf{Chernoff variant:} For $\epsilonilonpsilon \gg n^{-1}$, if $X$ is a $p$-random subset of $[n]$, then, with high probability, $|X|=(1\pm \epsilonilonpsilon)pn$.
\item \textbf{Azuma variant:} For $\epsilonilonpsilon\gg n^{-1}$ and fixed $k$, if $Y$ is a $k$-Lipschitz random variable influenced by at most $n$ coordinates, then, with high probability, $Y=\mathbb{E}(Y)\pm \epsilonilonpsilon n$.
\item \textbf{Union bound variant:} For fixed $k$, if $X_1, \dots, X_{k}$ are events which hold with high probability then they simultaneously occur with high probability.
\epsilonilonnd{itemize}
The first two of these follow directly from Lemmas~\ref{Lemma_Azuma} and~\ref{Lemma_Chernoff}, the latter from the union bound.
\subsection{Structure of trees}\label{Section_structure_of_trees}
Here, we gather lemmas about the structure of trees. Most of these lemmas say something about the leaves and bare paths of a tree. It is easy to see that a tree with few leaves must have many bare paths. The most common version of this is the following well known lemma.
\begin{lemma}[\cite{montgomery2018spanning}]\label{split} For any integers $n,k>2$, a tree with~$n$ vertices either has at least $n/4k$ leaves or a collection of at least $n/4k$ vertex disjoint bare paths, each with length $k$.
\epsilonilonnd{lemma}
As a corollary of this lemma we show that every tree either has many bare paths, many non-neighbouring leaves, or many large stars. This lemma underpins the basic case division for this paper. The rest of the proofs focus on finding rainbow copies of the three types of trees.
\begin{lemma}[Case division]\label{Lemma_case_division}
Let $1\gg \delta \gg n^{-1}$. Every $n$-vertex tree satisfies one of the following:
\begin{enumerate}[label= \Alph{enumi}]
\item There are at least $\delta n/800$ vertex-disjoint bare paths with length at least $\delta^{-1}$.
\item There are at least $\delta^6 n$ non-neighbouring leaves.
\item Removing leaves next to vertices adjacent to at least $\delta^{-4}$ leaves gives a tree with at most
$n/100$ vertices.
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
\begin{proof}
Take $T$ and remove leaves around any vertex adjacent to at least $\delta^{-4}$ leaves, and call the resulting tree $T'$. If $T'$ has at most $n/100$ vertices then we are in Case C. Assume then, that $T'$ has at least $n/100$ vertices.
By Lemma~\ref{split} applied with $n'=|T'|$ and $k=\lceil \delta^{-1}\rceil$, the tree $T'$ either has at least $\delta n/600$ vertex disjoint bare paths with length at least $\delta^{-1}$ or at least $\delta n/600$ leaves. In the first case, as vertices were deleted next to at most $\delta^4 n$ vertices to get $T'$ from $T$, $T$ has at least $\delta n/600-\delta^4n\geq \delta n/800$ vertex disjoint bare paths with length at least $\delta^{-1}$, so we are in Case B.
In the second case, if there are at least $\delta n/1200$ leaves of $T'$ which are also leaves of $T$, then, as there are at most $\delta^{-4}$ of these leaves around each vertex in $T'$, $T$ has at least $(\delta n/1200)/\delta^{-4}\geq \delta^{6}n$ non-neighbouring leaves, so we are in Case A. On the other hand, if there are not such a number of leaves of $T'$ which are leaves of $T$, then $T'$ must have at least $\delta n/1200$ leaves which are not leaves of $T$, and which therefore are adjacent to a leaf in $T$. Thus, $T$ has at least $\delta n/1200\geq \delta^{6}n$ non-neighbouring leaves, and we are also in Case A.
\epsilonilonnd{proof}
We say a set of subtrees $T_1,\ldots, T_\epsilonilonll\subseteq T$ divides a tree $T$ if $E(T_1)\cup\ldots \cup E(T_\epsilonilonll)$ is a partition of $E(T)$. We use the following lemma.
\begin{lemma}[\cite{montgomery2018spanning}]\label{littletree} Let $n,m\in \mathbb N$ satisfy $1\leq m\leq n/3$. Given any tree~$T$ with~$n$ vertices and a vertex $t\in V(T)$, we can find two trees $T_1$ and $T_2$ which divide~$T$ so that $t\in V(T_1)$ and $m\leq |T_2|\leq 3m$.
\epsilonilonnd{lemma}
Iterating this, we can divide a tree into small subtrees.
\begin{lemma}\label{dividetree} Let $T$ be a tree with at least $m$ vertices, where $m\geq 2$. Then, for some $s$, there is a set of subtrees $T_1,\ldots, T_s$ which divide $T$ so that $m\leq |T_i|\leq 4m$ for each $i\in [s]$.
\epsilonilonnd{lemma}
\begin{proof}
We prove this by induction on $|T|$, noting that it is trivially true if $|T|\leq 4m$.
Suppose then $|T|>4m$ and the statement is true for all trees with fewer than $|T|$ vertices and at least $m$ vertices. By Lemma~\ref{littletree}, we can find two trees $T_1$ and $S$ which divide $T$ so that $m\leq |T_1|\leq 3m$. As $|T|>4m$, we have $m<|S|<|T|$, so there must be a set of subtrees $T_2,\ldots,T_s$, for some $s$, which divide $S$ so that $m\leq |T_i|\leq 4m$ for each $2\leq i\leq s$.
The subtrees $T_1,\ldots,T_s$ then divide $T$, with $m\leq |T_i|\leq 4m$ for each $i\in [s]$.
\epsilonilonnd{proof}
For embedding trees in Cases A and B, we will need a finer understanding of the structure of trees. In fact, every tree can be built up from a small tree by successively adding leaves, bare paths, and stars, as follows.
\begin{lemma}[\cite{MPS}]\label{Lemma_decomp} Given integers $d$ and $n$, $\mu>0$ and a tree $T$ with at most $n$ vertices, there are integers $\epsilonilonll\leq 10^4 d\mu^{-2}$ and $j\in\{2,\ldots,\epsilonilonll\}$ and a sequence of subgraphs $T_0\subseteq T_1\subseteq \ldots \subseteq T_\epsilonilonll=T$ such that
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item for each $i\in [\epsilonilonll]\setminus \{1,j\}$, $T_{i}$ is formed from $T_{i-1}$ by adding non-neighbouring leaves,\label{cond1}
\item $T_j$ is formed from $T_{j-1}$ by adding at most $\mu n$ vertex-disjoint bare paths with length $3$,\label{cond2}
\item $T_1$ is formed from $T_0$ by adding vertex-disjoint stars with at least $d$ leaves each, and\label{cond3}
\item $|T_0|\leq 2\mu n$.\label{cond4}
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
The following variation will be more convenient to use here. It shows that an arbitrary tree $T$ can be built out of a preselected, small subtree $T_1$ by a sequence of operations. It is important to control the starting tree as it allows us to choose which part of the tree will form our absorbing structure.
For forests $T'\subseteqeq T$, we say that $T$ is obtained from $T'$ by \epsilonilonmph{adding a matching of leaves} if all the vertices in $V(T)\setminus V(T')$ are non-neighbouring leaves in $T$.
\begin{lemma}[Tree splitting]\label{Lemma_tree_splitting}
Let $1\geq d^{-1} \gg n^{-1}$.
Let $T$ be a tree with $|T|=n$ and $U\subseteqeq V(T)$ with $|U|\geq n/d^3$.
Then, there are forests $T_1^{\mathrm{small}}\subseteqeq T_2^{\mathrm{stars}}\subseteqeq T_3^{\mathrm{match}}\subseteqeq T_4^{\mathrm{paths}}\subseteqeq T_5^{\mathrm{match}}=T$ satisfying the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $|T_{1}^{\mathrm{small}}|\leq n/d$ and $|U\cap T_1^{small}|\geq n/d^6$.\label{rush1}
\item $T_2^{\mathrm{stars}}$ is formed from $T_{1}^{\mathrm{small}}$ by adding vertex-disjoint stars of size at least $d$.\label{rush2}
\item $T_3^{\mathrm{match}}$ is formed from $T_2^{\mathrm{stars}}$ by adding a sequence of $d^8$ matchings of leaves.\label{rush3}
\item $T_4^{\mathrm{paths}}$ is formed from $T_3^{\mathrm{match}}$ by adding at most $n/d$ vertex-disjoint paths of length $3$.\label{rush4}
\item $T_5^{\mathrm{match}}$ is formed from $T_4^{\mathrm{paths}}$ by adding a sequence of $d^8$ matchings of leaves.\label{rush5}
\epsilonilonnd{enumerate}
\epsilonilonnd{lemma}
\begin{proof}
First, we claim that there is a subtree $T_0$ of order $\leq n/2d^2$ containing at least $n/d^6$ vertices of $U$. To find this, use Lemma~\ref{dividetree} to find $s\leq 32d^2$ subtrees $T_1,\ldots, T_{s}$ which divide $T$ so that $n/16d^2\leq |T_i|\leq n/2d^2$ for each $i\in [s]$. As each vertex in $U$ must appear in some tree $T_i$, there must be some tree $T_k$ which contains at least $|U|/32d^2\geq n/d^6$ vertices in $U$, as required.
Let $T'$ be the $n$-vertex tree formed from $T$ by contracting $T_k$ into a single vertex $v_0$ and adding $e(T_k)$ new leaves at $v_0$ (called ``dummy'' leaves).
Notice that Lemma~\ref{Lemma_decomp} applies to $T'$ with $d=d, \mu= d^{-3}, n=n$ which gives a sequence of forests $T_0', \dots, T_{\epsilonilonll}'$ for $\epsilonilonll\leq 10^4d^7\leq d^8$.
Notice that $v_0\in T_0'$. Indeed, by construction, every vertex which is not in $T_0'$ can have in $T_{\epsilonilonll}'=T'$ at most $\epsilonilonll$ leaves. Since $v_0$ has $\geq e(T_k)>\epsilonilonll$ leaves in $T'$, it must be in $T'_0$.
For each $i=0, \dots, {\epsilonilonll}$, let $T_i''$ be $T_i'$ with $T_k$ uncontracted and any dummy leaves of $v_0$ deleted.
Let $T_2^{\mathrm{stars}}=T_1''$, $T_3^{\mathrm{match}}=T_{j-1}''$, $T_4^{\mathrm{paths}}=T_j''$, $T_5^{\mathrm{match}}=T_{\epsilonilonll}''$.
Let $T_1^{\mathrm{small}}$ be $T_0''$ together with the $T_1''$-leaves of any $v\in T_k$ for which $|N_{T_1''}(v)\setminus N_{T_0''}(v)|<d$. We do this because when we uncontract $T_k$ the leaves which were attached to $v_0$ in $T_0'$ are now attached to vertices of $T_k$.
If they form a star of size less than $d$ we cannot add them when we form $T_2^{\mathrm{stars}}$ without violating \ref{rush2} so we add them already when we form $T_1^{\mathrm{small}}$.
Since we are adding at most $d$ leaves for every vertex of $T_k$, we have $|T_1^{\mathrm{small}}|\leq d|T_k|+|T_0'|\leq n/d$ so \ref{rush1} holds. This ensures that at least $d$ leaves are added to vertices of $T_1^{\mathrm{small}}$ to form $T_2^{\mathrm{stars}}$ so \ref{rush2} holds. The remaining conditions \ref{rush3} -- \ref{rush5} are immediate from the application of Lemma~\ref{Lemma_decomp}.
\epsilonilonnd{proof}
\subsection{Pseudorandom properties of random sets of vertices and colours}\label{Section_pseudorandomness}
Suppose that $K_{2n+1}$ is $2$-factorized. Choose a $p$-random set of vertices $V\subseteqeq V(K_{2n+1})$ and a $q$-random set of colours $C\subseteqeq C(K_{2n+1})$. What can be said about the subgraph $K_{2n+1}[V,C]$ consisting of edges within the set $V$ with colour in $C$? What ``pseudorandomness'' properties is this subgraph likely to have? In this section, we gather lemmas giving various such properties.
The setting of the lemmas is quite varied, as are the properties they give. For example, sometimes the sets $V$ and $C$ are chosen independently, while sometimes they are allowed to depend on each other arbitrarily. We split these lemmas into three groups based on the three principal settings.
\subsubsectionme{Dependent vertex/colour sets}
In this setting, our colour set $C\subseteqeq C(K_{2n+1})$ is $p$-random, and the vertex set is $V(K_{2n+1})$ (i.e., it is 1-random). Our pseudorandomness condition is that the number of edges between any two sizeable disjoint vertex sets is close to the expected number. Though a lemma of this kind was first proved in~\cite{alon2016random}, the precise pseudorandomness condition we will use here is in the following version from~\cite{MPS}. A colouring is \epsilonilonmph{locally $k$-bounded} if every vertex is adjacent to at most $k$ edges of each colour.
\begin{lemma}[\cite{MPS}]\label{MPScolour}\label{Lemma_MPS_boundrandcolour} Let $k\in\mathbb N$ be constant and let $\epsilonilonpsilon, p\geq n^{-1/100}$. Let $K_n$ have a {locally} $k$-bounded colouring and suppose $G$ is a subgraph of $K_n$ chosen by including the edges of each colour independently at random with probability $p$. Then, with probability $1-o(n^{-1})$, for any disjoint sets $A,B\subseteq V(G)$, with $|A|,|B|\geq n^{3/4}$,
\[
\big|e_G(A,B)-p|A||B|\big|\leq \epsilonilonpsilon p|A||B|.
\]
\epsilonilonnd{lemma}
If $V\subseteqeq V(K_{2n+1})$ is a $p$-random set of vertices, then the edges going from $V$ to $V(K_{2n+1})\setminus V$ are typically pseudorandomly coloured. The lemma below is a version of this. Here ``pseudorandomly coloured'' means that most colours have at most a little more than the expected number of colours leaving $V$.
\begin{lemma}[\cite{MPS}]\label{MPSvertex}\label{Corollary_MPS_randsetcor}
Let $k$ be constant and $\epsilonilonpsilon,p\geq n^{-1/10^3}$. Let $K_n$ have a {locally} $k$-bounded colouring and let $V$ be a $p$-random subset of $V(K_n)$. Then, with probability $1-o(n^{-1})$, for each $A\subseteq V(K_n)\setminus V$ with $|A|\geq n^{1/4}$, for all but at most $\epsilonilonpsilon n$ colours there are at most $(1+\epsilonilonpsilon)pk|A|$ edges of that colour between $V$ and $A$.
\epsilonilonnd{lemma}
A random vertex set $V$ likely has the property from Lemma~\ref{MPSvertex}. A random colour set $C$ likely has the property from Lemma~\ref{MPScolour}. If we combine these two properties, we can get a property involving $C$ and $V$ that is likely to hold. Importantly, this will be true even if $C$ and $V$ are not independent of each other. Doing this, we get the following lemma.
\begin{lemma}[Nearly-regular subgraphs]\label{Lemma_nearly_regular_subgraph}
Let $p, \gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$, and $C\subseteqeq C(K_{2n+1})$ with $V$ $p/2$-random and $C$ $p$-random (possibly depending on each other). The following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|= pn$, there are subsets $U'\subseteqeq U, V'\subseteqeq V, C'\subseteqeq C$ with $|U'|=|V'|= (1\pm\gamma)|U|$
so that $G=K_{2n+1}[U',V',C']$ is globally $(1+\gamma)p^2n$-bounded, and every vertex $v\in V(G)$ has $d_{G}(v)=(1\pm\gamma)p^2n$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ and $\epsilonilonpsilon$ so that $p, \gamma\gg \alpha\gg \epsilonilonpsilon\gg n^{-1}$. With high probability, by Lemma~\ref{MPScolour}, Lemma~\ref{MPSvertex} (with $k=2$, $n'=2n+1$, and $p'=p/2$) and Chernoff's bound, we can assume the following occur simultaneously.
\begin{enumerate}[label = (\roman{enumi})]
\item \label{prop1} For any disjoint $A,B\subseteq V(K_{2n+1})$ with $|A|,|B|\geq (2n+1)^{3/4}$, $|E_C(A,B)|=(1\pm \epsilonilonps)p|A||B|$.
\item \label{prop2} For any $A\subseteqeq V(K_{2n+1})\setminus V$ with $|A|\geq n^{1/4}$, for all but at most $\epsilonilonpsilon n$ colours there are at most $(1+\epsilonilonpsilon)p|A|$ edges of that colour between $A$ and $V$.
\item \label{prop3} $|V|=(1\pm \epsilonilonps)pn$.
\epsilonilonnd{enumerate}
Let $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|= pn$ be arbitrary. Let $\hat C\subseteqeq C$ be the subset of colours $c\in C$ with $|E_c(U,V)|> (1+ \epsilonilonpsilon)p|U|$. Note that, from \ref{prop2}, we have $|\hat C|\leq \epsilonilonps n$.
Let $\hat U^+\subseteqeq U$ and $\hat V^+\subseteqeq V$ be subsets of vertices $v$ with $d_{K_{2n+1}[U,V,C]}(v)> (1+ \alpha)p^2 n$, and let $\hat U^-\subseteqeq U$ and $\hat V^-\subseteqeq V$ be subsets of vertices $v$ with $d_{K_{2n+1}[U,V,C]}(v)< (1- \alpha)p^2 n$.
Now, $|E_C(U^+,V)|> |U^+|(1+\alpha)p^2 n\geq (1+\epsilonilonps)p|U^+||V|$ by the definition of $U^+$ and \ref{prop3}. Therefore, as, by \ref{prop3}, $|V|\geq (1-\epsilonilonps)pn>(2n+1)^{3/4}$, from \ref{prop1} we must have $|U^+|\leq (2n+1)^{3/4}\leq \epsilonilonps n$. Similarly, we have $|U^-|,|V^+|,|V^-|\leq \epsilonilonps n$.
Let $\hat U=U^+\cup U^-$ and Let $\hat V=V^+\cup V^-$. We have $|U\setminus \hat U|, |V\setminus \hat V| \geq (1\pm \epsilonilonpsilon)pn\pm 2\epsilonilonpsilon n$. Therefore, we can choose subsets $U'\subseteqeq U\setminus \hat U$ and $V'\subseteqeq V\setminus \hat V$ with $|U'|=|V'|= pn-3\epsilonilonpsilon n=(1\pm \gamma)pn$. Note that, by \ref{prop3} and as $|U|=pn$, $|U\setminus U'|,|V\setminus V'|\leq 4\epsilonilonpsilon n$.
Let $C'=C\setminus \hat C$ and set $G=K_{2n+1}[U',V',C']$.
For each $v\in U'\cup V'$ we have $d_G(v)= (1\pm \alpha)p^2 n \pm 2|\hat C|\pm |U\setminus U'|\pm |V\setminus V'|=(1\pm \gamma)p^2 n$. For each $c\in C'$ we have $|E_c(U,V)|\leq (1+ \epsilonilonpsilon)p|U|\leq (1+ \gamma)p^2n$.
\epsilonilonnd{proof}
\subsubsectionme{Deterministic colour sets and random vertex sets}
The following lemma bounds the number of edges each colour typically has within a random vertex set.
\begin{lemma}[Colours inside random sets]\label{Lemma_number_of_colours_inside_random_set}
Let $p,\gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p$-random. With high probability, every colour has $(1\pm \gamma)2p^2n$ edges inside $V$.
\epsilonilonnd{lemma}
\begin{proof}
For an edge $e\in E(K_{2n+1})$ we have $\mathbb{P}(e \in E(K_n[V]))=p^2$.
By linearity of expectation, for any colour $c\in C(K_{2n+1})$, we have $\mathbb{E}(|E_c(V)|)=p^2(2n+1)$. Note that $|E_c(V)|$ is $2$-Lipschitz and affected by $\leq 2n+1$ coordinates.
By Azuma's inequality, we have $\mathbb{P}(|E_c(V)|\neq (1\pm \gamma)2p^2n )\leq e^{-\gamma^2p^4n/100}= o(n^{-1})$.
The result follows by a union bound over all the colours.
\epsilonilonnd{proof}
Recall that for any two sets $U,V\subseteqeq V(G)$ inside a coloured graph $G$, we say that the pair $(U,V)$ is \epsilonilonmph{$k$-replete} if every colour of $G$ occurs at least $k$ times between $U$ and $V$.
We will use the following auxiliary lemma about how this property is inherited by random subsets.
\begin{lemma}[Repletion between random sets]\label{Lemma_inheritence_of_lower_boundedness_random}
Let $q, p\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Suppose that $A,B\subseteqeq V(K_{2n+1})$ are disjoint randomized sets with the pair $(A,B)$ $pn$-replete with high probability.
Let $V\subseteqeq V(K_{2n+1})$ be $q$-random and independent of $A,B$.
Then with high probability the pair $(A, B\cap V)$ is $(qpn/2)$-replete.
\epsilonilonnd{lemma}
\begin{proof}
Fix some choice $A'$ of $A$ and $B'$ of $B$ for which the pair $(A',B')$ is $pn$-replete. As $V$ is independent of $A,B$, for each edge $e$ between $A$ and $B$ we have $\mathbb{P}(e\cap B\cap V\neq \epsilonilonmptyset|A=A', B=B')=q$.
Therefore, for any colour $c$, conditional on ``$A=A', B=B'$'', we have $\mathbb{E}(|E_c(A,B\cap V)|)=q|E_c(A,B)|\geq qpn$. Note that $|E_c(A,B\cap V)|$ is $2$-Lipschitz and affected by $\leq 2n+1$ coordinates.
By Azuma's inequality, we have $\mathbb{P}(|E_c(A,B\cup V)|< qpn/2| A=A', B=B')\leq e^{-q^2p^2n/100}= o(1)$. Thus, with probability $1-o(1)$, conditioned on $A=A', B=B'$ we have that $(A,B\cap V)$ is $(qn/2)$-replete.
This was all under the assumption that $A=A'$ and $B=B'$.
Therefore using that $(A,B)$ is $pn$-replete with high probability, we have
\begin{align*}
\mathbb{P}(\text{$(A,B\cap V)$ is $(qn/2)$-replete})&\geq \sum_{\substack{(A',B') \\ \text{ $pn$-replete}}}\mathbb{P}(\text{$(A,B\cap V)$ is $(qn/2)$-replete}|A=A', B=B')\cdot\mathbb{P}(A=A', B=B')\\
&\geq \sum_{\substack{(A',B') \\ \text{ $pn$-replete}}} (1-o(1))\cdot \mathbb{P}(A=A', B=B')=1-o(1).\qedhere
\epsilonilonnd{align*}
\epsilonilonnd{proof}
\subsubsectionme{Independent vertex/colour sets}
The setting of the next three lemmas is the same: we independently choose a $p$-random set of vertices $V$ and a $q$-random set of colours $C$. For such a pair $V,C$ we expect all vertices of the vertices $v$ in $K_{2n+1}$ to have many $C$-edges going into $V$. Each of the following lemmas is a variation on this theme.
\begin{lemma}[Degrees into independent vertex/colour sets]\label{Lemma_high_degree_into_random_set}
Let $p, q\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p$-random, and let $C\subseteqeq C(K_{2n+1})$ be $q$-random and independent of $V$. With probability $1-o(n^{-1})$, every vertex $v\in V(K_{2n+1})$ has $|N_C(v)\cap V|\geq pq n$.
\epsilonilonnd{lemma}
\begin{proof} Let $v\in V(K_{2n+1})$.
For any vertex $x\neq v$, we have $\mathbb{P}(x\in N_C(v)\cap V)= pq$ and so $\mathbb{E}(|N_C(v)\cap V|)= 2pq n$.
Also $|N_C(v)\cap V|$ is $2$-Lipschitz and affected by $3n$ coordinates.
By Azuma's Inequality, we have that $\mathbb{P}(|N_C(v)\cap V|\leq pq n)\leq 2e^{-p^2q^{2}n/1000}= o(n^{-2})$. The result follows by taking a union bound over all $v\in V(K_{2n+1})$.
\epsilonilonnd{proof}
\begin{lemma}\label{lem:goodpairs} Let $1/n\ll \epsilonilonta \ll \mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$-random subsets which are independent. With high probability, for each distinct $u,v\in V(K_{2n+1})$, there are at least $\epsilonilonta n$ colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$.
\epsilonilonnd{lemma}
\begin{proof} Let $u,v\in V(K_{2n+1})$ be distinct, and let $X_{u,v}$ be the number of colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$. Note that $\mathbb{E} X_{u,v} \geq \mu^3 n$, $X_{u,v}$ is
$2$-Lipschitz and affected by $3n-1$ coordinates.
By Azuma's Inequality, we have that $\mathbb{P}(X_{u,v}\leq \mu^3 n/2)\leq 2e^{-\mu^6 n/1000}= o(n^{-2})$. The result follows by taking a union bound over all distinct pairs $u,v\in V(K_{2n+1})$.
\epsilonilonnd{proof}
Lemma~\ref{Lemma_high_degree_into_random_set} says that, with high probability, every vertex $v$ has many colours $c\in C$ for which there is a $c$-edge into $V$. The following lemma is a strengthening of this. It shows that, for any set $Y$ of $100$ vertices, there are many colours $c\in C$ for which \epsilonilonmph{each $v\in Y$} has a $c$-edge into $V$.
\begin{lemma}[Edges into independent vertex/colour sets]\label{Lemma_edges_into_independent_vertexcolour_sets}
Let $p\gg q\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ be $p$-random and independent.
Then, with high probability, for any set $Y$ of $100$ vertices, there are $qn$ colours $c\in C$ for which each $y\in Y$ has a $c$-neighbour in $V$.
\epsilonilonnd{lemma}
\begin{proof} Fix $Y\subseteq V(K_{2n+1})$ with $|Y|=100$.
Let $C_Y=\{c\in C: \mbox{each $y\in Y$ has a $c$-neighbour in $V$}\}$.
For any colour $c$ without edges inside $Y$, we have $\mathbb{P}(c\in C_Y)\geq p^{101}$ and so $\mathbb{E}(|C_Y|)\geq p^{101}(n-\binom{|Y|}2)\geq 2qn$. Notice that $|C_Y|$ is $100$-Lipschitz and affected by $3n+1$ coordinates.
By Azuma's inequality, we have that $\mathbb{P}(|C_Y|\leq qn)\leq e^{-q^2n/10^6}= o(n^{-100})$. The result follows by taking a union bound over all sets $Y\subseteq V(K_{2n+1})$ with $|Y|=100$.
\epsilonilonnd{proof}
\subsection{Rainbow matchings}\label{Section_matchings}
We now gather lemmas for finding large rainbow matchings in random subsets of coloured graphs, despite dependencies between the colours and the vertices that we use. Simple greedy embedding strategies are insufficient for this, and instead we will use a variant of R\"odl's Nibble proved by the authors in~\cite{montgomery2018decompositions}.
\begin{lemma}[\cite{montgomery2018decompositions}]\label{Lemma_MPS_nearly_perfect_matching}
Suppose that we have $n, \delta, \gamma, p, \epsilonilonll$ with $1\geq \delta \gg p \gg \gamma \gg n^{-1}$ and $n\gg \epsilonilonll$.
Let $G$ be a locally $\epsilonilonll$-bounded, globally $(1+\gamma) \delta n$-bounded, coloured, balanced bipartite graph with $|G|=(1\pm \gamma)2n$ and $d_G(v)=(1\pm \gamma)\delta n$ for all $v\in V(G)$. Then $G$ has a random rainbow matching $M$ which has size $\geq (1-2p)n$ where
\begin{align}
\mathbb{P}(e\in E(M))&\geq(1- 9p)\frac{1}{\delta n}\hspace{0.5cm}\text{ for each $e\in E(G)$.} \label{Eq_Near_Matching_Edge_Probability_Lower_Bound}
\epsilonilonnd{align}
\epsilonilonnd{lemma}
We remark that in the statement of this lemma \cite{montgomery2018decompositions}, the conditions
``$|G|=(1\pm \gamma)2n$ and $d_G(v)=(1\pm \gamma)\delta n$ for all $v\in V(G)$'' are referred to collectively as ``$G$ is $(\gamma, \delta, n)$-regular''.
The following lemma is at the heart of the proofs in this paper. It shows there is typically a nearly-perfect rainbow matching using random vertex/colour sets. Moreover, it allows \epsilonilonmph{arbitrary} dependencies between the sets of vertices and colours. As mentioned before, when embedding high degree vertices such dependencies are unavoidable. Because of this, after we have embedded the high degree vertices, the remainder of the tree will be embedded using variants of this lemma.
\begin{lemma}[Nearly-perfect matchings]\label{Lemma_nearly_perfect_matching}
Let $p\in [0,1]$, $\beta \gg n^{-1}$, and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $p/2$-random and let $C\subseteqeq C(K_{2n+1})$ be $p$-random (possibly depending on each other). Then, with probability $1-o(n^{-1})$, for every $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|\leq pn$, $K_{2n+1}$ has a $C$-rainbow matching of size $|U|-\beta n$ from $U$ to $V$.
\epsilonilonnd{lemma}
\begin{proof}
The lemma is vacuous when $p<\beta$, so suppose $p\geq \beta$. We will first prove the lemma in the special case when $p\leq 1-\beta$.
Choose $p\geq \beta\gg \alpha \gg \gamma\gg n^{-1}$.
With probability $1-o(n^{-1})$, $V$ and $C$ satisfy the conclusion of Lemma~\ref{Lemma_nearly_regular_subgraph} with $p, \gamma, n$.
Using Chernoff's bound and $p\leq 1-\beta$, with probability $1-o(n^{-1})$ we have $|V|\leq n$. By the union bound, both of these simultaneously occur.
Notice that it is sufficient to prove the lemma for sets $U$ with $|U|=pn$ (since any smaller set $U$ is contained in a set of this size which is disjoint from $V$ as $|V| \leq n$).
From Lemma~\ref{Lemma_nearly_regular_subgraph}, we have that, for $U$ of order $pn$, there are subsets $U'\subseteqeq U, V'\subseteqeq V, C'\subseteqeq C$ with $|U'|=|V'|= (1\pm\gamma)pn$
so that $G=K_{2n+1}[U',V',C']$ is globally $(1+\gamma)p^2n$-bounded, and every vertex $v\in V(G)$ has $d_{G}(v)=(1\pm\gamma)p^2n$. Now $G$ satisfies the assumptions of Lemma~\ref{Lemma_MPS_nearly_perfect_matching} (with $n'=pn$, $\delta =p$, $p'= \alpha$, $\gamma'=2\gamma$ and $\epsilonilonll=2$), so it has a rainbow matching of size $(1-2\alpha)pn\geq pn-\beta n$.
Now suppose that $p\geq 1-\beta$. Choose a $(1-\beta/2)p/2$-random subset $V'\subseteqeq V$ and a $(1-\beta/2)p$-random subset $C'\subseteqeq C$. Fix $p'=(1-\beta/2)p$ and $\beta'=\beta/2$ and note that $p' \leq 1-\beta'$. By the above argument again, with high probability the conclusion of the ``$p'\leq (1-\beta')$'' version of the lemma applies to $V',C',p',\beta'$. Let $U$ be a set with $|U| \leq pn$. Choose $U'\subseteqeq U$ with $|U'|=|U|-\beta n/2$. Then $|U'| \leq pn - \beta n/2 \leq p'n$.
From the ``$p'\leq (1-\beta')$'' version of the lemma we get a $C'$-rainbow matching $M$ from $U'$ to $V'$ of size $|U'|-\beta n/2= |U|-\beta n$.
\epsilonilonnd{proof}
The following variant of Lemma~\ref{Lemma_nearly_perfect_matching} finds a rainbow matching which completely covers the deterministic set $U$. To achieve this we introduce a small amount of independence between the vertices/colours which are used in the matching.
\begin{lemma}[Perfect matchings]\label{Lemma_sat_matching_random_embedding}
Let $1\geq \gamma \gg n^{-1}$, let $p\in[0,1]$, and let $K_{2n+1}$ be $2$-factorized.
Suppose that we have disjoint sets $V_{dep}, V_{ind}\subseteqeq V(K_{2n+1})$, and $C_{dep}, C_{ind}\subseteqeq C(K_{2n+1})$ with $V_{dep}$ $p/2$-random, $C_{dep}$ $p$-random, and $V_{ind}, C_{ind}$ $\gamma$-random. Suppose that $V_{ind}$ and $C_{ind}$ are independent of each other. Then, the following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus (V_{dep}\cup V_{ind})$ of order $\leq p n$, there is a perfect $(C_{dep}\cup C_{ind})$-rainbow matching from $U$ into $(V_{dep}\cup V_{ind})$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\beta$ such that $1\geq \gamma \gg \beta\gg n^{-1}$.
With probability $1-o(n^{-1})$, we can assume the conclusion of
Lemma~\ref{Lemma_nearly_perfect_matching} holds for $V=V_{dep}, C=C_{dep}$ with $p=p, \beta=\beta, n=n$, and, by Lemma~\ref{Lemma_high_degree_into_random_set} applied to $V=V_{ind}, C=C_{ind}$ with $p=q=\gamma, n=n$ that the following holds. For each $v\in V(K_{2n+1})$, we have $|N_{C_{ind}}(v) \cap V_{ind}| \geq \gamma^2n>\beta n$. We will show that the property in the lemma holds.
Let then $U\subseteq V(K_{2n+1})\setminus (V_{dep}\cup V_{ind})$ have order $\leq p n$.
From the conclusion of Lemma~\ref{Lemma_nearly_perfect_matching}, there is a $C_{dep}$-rainbow matching $M_1$ of size $|U|-\beta n$ from $U$ to $V_{dep}$.
Since $|N_{C_{ind}}(v) \cap V_{ind}| >\beta n$ for each $v\in U\setminus V(M_1)$ we can construct a $C_{ind}$-rainbow matching $M_2$ into $V_{ind}$ covering $U\setminus V(M_1)$ (by greedily choosing this matching one edge at a time). The matching $M_1\cup M_2$ then satisfies the lemma.
\epsilonilonnd{proof}
We will also use a lemma about matchings using an exact set of colours.
\begin{lemma}[Matchings into random sets using specified colours]\label{Lemma_matching_into_random_set_using_specified_colours}
Let $p\gg q\gg\beta \gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $V\subseteqeq V(K_{2n+1})$ be $(p/2)$-random. With high probability, for any $U\subseteqeq V(K_{2n+1})\setminus V$ with $|U|\geq pn$, and any $C\subseteqeq C(K_{2n+1})$ with $|C|\leq q n$, there is a $C$-rainbow matching of size $|C|-\beta n$ from $U$ to $V$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\gamma$ such that $\beta\gg\gamma \gg n^{-1}$.
By Lemma~\ref{MPSvertex} (applied with $n'=2n+1$), with high probability, we have that, for any set $A\subseteq V(K_{2n+1})\setminus V$ with $|A|\geq pn\geq (2n+1)^{1/4}$, for all but at most $\gamma n$ colours there are at most $(1+\gamma)p|A|$ edges of that colour between $A$ and $V$. By Chernoff's bound, with high probability $|V|=(1\pm \gamma)pn$. We will show that the property in the lemma holds.
Fix then an arbitrary pair $U$, $C$ as in the lemma. Without loss of generality, $|U|=pn$. Let $M$ be a maximal $C$-rainbow matching from $U$ to $V$. We will show that $|M|\geq |C|-\beta n$, so that the matching required in the lemma must exist (by removing edges if necessary). Now, let $C'=C\setminus C(M)$. For each $c\in C'$, any edge between $U$ and $V$ with colour $c$ must have a vertex in $V(M)$, by maximality. Therefore, there are at most $2|V(M)|\leq 4qn$ edges of colour $c$ between $U$ and $V$. From the property from Lemma~\ref{MPScolour}, there are at most $\gamma n$ colours with more than $(1+\gamma)p|U|$ edges between $U$ and $V$. Therefore, using $|V|=(1\pm \gamma)pn$,
\[
|U||V|\leq 4qn|C'|+(2n+1)\gamma n+(1+\gamma)p|U|(n-|C'|)\leq (4qn-p^2n)|C'|+3\gamma n^2+ (1+\gamma)(1+2\gamma)|U||V|,
\]
and hence
\[
|C'|p^2n/2\leq |C'|(p^2-4q)n\leq ((1+\gamma)(1+2\gamma)-1)|U||V|+3\gamma n^2\leq 7\gamma n^2.
\]
It follows that $|C'|\leq 14\gamma n/p^2\leq \beta n$. Thus, $|M|= |C|-|C'|\geq |C|-\beta n$, as required.
\epsilonilonnd{proof}
\subsection{Rainbow star forests}\label{Section_stars}
Here we develop techniques for embedding the high degree vertices of trees, based on our previous methods in~\cite{MPS}. We will do this by proving lemmas about large star forests in coloured graphs. In later sections, when we find rainbow trees, we isolate a star forest of edges going through high degree vertices, and embed them using the techniques from this section.
We start from the following lemma.
\begin{lemma}[\cite{MPS}]\label{Corollary_MPS_kdisjstars} Let $0<\epsilonilonpsilon<1/100$ and $\epsilonilonll\leq \epsilonilonpsilon^{2}n/2$. Let $G$ be an $n$-vertex graph with minimum degree at least $(1-\epsilonilonpsilon)n$ which contains an independent set on the distinct vertices $v_1,\ldots,v_\epsilonilonll$. Let $d_1,\ldots,d_\epsilonilonll\geq 1$ be integers satisfying $\sum_{i\in [\epsilonilonll]}d_i\leq (1-3\epsilonilonpsilon)n/k$,
and suppose $G$ has a locally $k$-bounded edge-colouring.
Then, $G$ contains disjoint stars $S_1,\ldots,S_\epsilonilonll$ so that, for each $i\in [\epsilonilonll]$, $S_i$ is a star centered at $v_i$ with $d_i$ leaves, and $\cup_{i\in [\epsilonilonll]}S_i$ is rainbow.
\epsilonilonnd{lemma}
The following version of the above lemma will be more convenient to apply.
\begin{lemma}[Star forest]\label{Lemma_star_forest}
Let $1\gg \epsilonilonta \gg\gamma\gg n^{-1}$ and let $K_{2n+1}$ be $2$-factorized.
Let $F$ be a star forest with degrees $\geq 1$ whose set of centers is $I=\{i_1, \dots, i_{\epsilonilonll}\}$ with $e(F)\leq (1-\epsilonilonta)n$. Suppose we have disjoint sets $J, V\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ with $|V|\geq (1-\gamma)2n$, $|C|\geq (1-\gamma)n$ and $J=\{j_1, \dots, j_{\epsilonilonll}\}$.
Then, there is a $C$-rainbow copy of $F$ with $i_t$ copied to $j_t$, for each $t\in [\epsilonilonll]$, whose vertices outside of $I$ are copied to vertices in $V$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\epsilonilonpsilon$ such that $\epsilonilonta \gg \epsilonilonpsilon\gg\gamma$.
Let $G$ be the subgraph of $K_n$ consisting of edges touching $V$ with colours in $C$. Notice that $\delta(G)\geq (1-4\gamma)2n\geq (1-\epsilonilonpsilon)(2n+1)$. Since $V$ and $J$ are disjoint, $J$ contains no edges in $G$. Notice that, as $J$ and $V$ are disjoint, $\epsilonilonpsilon\gg \gamma$, and $|V|\geq (1-\gamma)2n$, we have $\epsilonilonll=|J|\leq 2\gamma n+1\leq \epsilonilonpsilon^2 (2n+1)/2$. Let $k=2$ and let $d_1, \dots, d_{\epsilonilonll}\geq 1$ be the degrees of $i_1,\ldots,i_\epsilonilonll$ in $F$. Notice that $\epsilonilonta \gg\epsilonilonpsilon$ implies $\sum_{i=1}^{\epsilonilonll}d_i=e(F)\leq (1-\epsilonilonta)n\leq (1-3\epsilonilonpsilon)(2n+1)/k$.
Applying Lemma~\ref{Corollary_MPS_kdisjstars} to $G$ with $\{v_1, \dots, v_{\epsilonilonll}\}=J$, $n'=2n+1$, $k=2$, $\epsilonilonll=\epsilonilonll$, $\epsilonilonpsilon=\epsilonilonpsilon$ we find the required rainbow star forest.
\epsilonilonnd{proof}
The above lemma can be used to find a rainbow copy of any star forest $F$ in a $2$-factorization as long as there are more than enough colours for a rainbow copy of $F$. However, we also want this rainbow copy to be suitably randomized. This is achieved by finding a star forest larger than $F$ and then randomly deleting each edge independently.
The following lemma is how we embed rainbow star forests in this paper. It shows that we can find a rainbow copy of any star forest so that the unused vertices and colours are $p$-random sets. Crucially, and unavoidably, the sets of unused vertices/colours depend on each other. This is where the need to consider dependent sets arises.
\begin{lemma}[Randomized star forest]\label{Lemma_randomized_star_forest}
Let $1\geq p, \alpha\gg \gamma\gg d^{-1}, n^{-1}$ and $\log^{-1} n\gg d^{-1}$. Let $K_{2n+1}$ be $2$-factorized.
Let $F$ be a star forest with degrees $\geq d$ with $e(F)= (1-p)n$ whose set of centers is $I=\{i_1, \dots, i_{\epsilonilonll}\}$. Suppose we have disjoint sets $V,J\subseteqeq V(K_{2n+1})$ and $C\subseteqeq C(K_{2n+1})$ with $|V|\geq (1-\gamma)2n$, $|C|\geq (1-\gamma)n$ and $J=\{j_1, \dots, j_{\epsilonilonll}\}$
Then, there is a randomized subgraph $F'$ which is with high probability a $C$-rainbow copy of $F$, with $i_t$ copied to $j_t$ for each $t$ and whose vertices outside $I$ are copied to vertices in $V$. Additionally there are randomized sets $U\subseteqeq V\setminus V(F'), D\subseteqeq C\setminus C(F')$ such that $U$ is a $(1-\alpha)p$-random subset of $V$ and $D$ is a $(1-\alpha)p$-random subset of $C$ (with $U$ and $D$ allowed to depend on each other).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\epsilonilonta$ such that $p, \alpha \gg \epsilonilonta\gg \gamma$.
Let $\hat F$ be a star forest obtained from $F$ by replacing every star $S$ with a star $\hat S$ of size $e(\hat S)=e(S)(1-\epsilonilonta)/(1-p)$. Notice that $e(\hat F)=(1-\epsilonilonta)n$, and, for each vertex $v$ at the centre of a star, $S$ say, in $F$, we have $d_{\hat F}(v)=e(\hat S)=e(S)(1-\epsilonilonta)/(1-p)=d_F(v)(1-\epsilonilonta)/(1-p)$. By Lemma~\ref{Lemma_star_forest}, there is a $C$-rainbow embedding $\hat F'$ of $\hat F$ with $i_t$ copied to $j_t$ for each $t$ and whose vertices outside $I$ are contained in $V$.
Let $U$ be a $(1-\alpha)p$-random subset of $V$. Let $F'=\hat F' \setminus U$. By Chernoff's Bound and $\log^{-1}n, p, \gamma \gg d^{-1}$, we have $\mathbb{P}(d_{F'}(v)< (1-\gamma)(1-p+\alpha p)d_{\hat F'}(v))\leq e^{-(1-p+\alpha p)\gamma^2d/3}\leq e^{-\log^5 n}$ for each center $v$ in $F$. Note that, for each centre $v$, $(1-\gamma)(1-p+\alpha p)d_{\hat F'}(v)=(1-\gamma)(1-p+\alpha p) d_F(v)(1-\epsilonilonta)/(1-p) \geq d_F(v)$, where this holds as $p,\alpha\gg \epsilonilonta\gg \gamma$. Taking a union bound over all the centers shows that, with high probability, $F'$ contains a copy of $F$.
Since $\hat F'$ was rainbow, we have that $C(\hat F')\setminus C(F')$ is a $(1-\alpha)p$-random subset of $C(\hat F')$.
Let $\hat{C}$ be a $(1-\alpha)p$-random subset of $C\setminus C(\hat F')$ and set $D=\hat{C}\cup (C(\hat F')\setminus C(F'))$. Now $D$ and $U$ are both $(1-\alpha)p$-random subsets of $C$ and $V$ respectively.
\epsilonilonnd{proof}
\subsection{Rainbow paths}\label{Section_paths}
Here we collect lemmas for finding rainbow paths and cycles in random subgraphs of $K_{2n+1}$. First we prove two lemmas about short paths between prescribed vertices. These lemmas are later used to incorporate larger paths into a tree.
\begin{lemma}[Short paths between two vertices]\label{Lemma_short_paths_between_two_vertices} Let $p\gg \mu \gg n^{-1}>0$ and suppose $K_{2n+1}$ is $2$-factorized. Let $V\subseteq V(K_{2n+1})$ and $C\subseteq C(K_{2n+1})$ be $p$-random and independent. Then, with high probability, for each pair of distinct vertices $u,v\in V(K_{2n+1})$ there are at least
$\mu n$ internally vertex-disjoint $u,v$-paths with length $3$ and internal vertices in $V$ whose union is $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof}
Choose $p\gg \mu \gg n^{-1}>0$.
Randomly partition $C=C_1\cup C_2\cup C_{3}$ into three $p/3$-random sets and $V=V_1\cup V_2$ into two $p/2$-random sets. With high probability the following simultaneously hold.
$\bullet$ By Lemma~\ref{Lemma_MPS_boundrandcolour} applied to $C_3$ with $p=p/3, \epsilonilonpsilon=1/2, n=2n+1$, for any disjoint vertex sets $U,V$ of order at least $p^2n/10\geq (2n+1)^{3/4}$, we have $e_{C_3}(U,V)\geq p|U||V|/6\geq 10^{-3}p^5n^2$.
$\bullet$ By Lemma~\ref{Lemma_high_degree_into_random_set} applied to $C_i, V_j$ with $p=p/2, q=p/3, n=n$ we have $|N_{C_{i}}(v)\cap V_j| \geq p^2n/6$ for every $v\in V(K_{2n+1})$, $i\in [3]$, and $j\in [2]$.
We claim the property holds. Indeed, pick an arbitrary distinct pair of vertices $u,v\in V(K_{2n+1})$. Let $M$ be a maximum $C_3$-rainbow matching between $N_{C_{1}}(v)\cap (V_1\setminus \{u\})$ and $N_{C_{2}}(u)\cap (V_2\setminus \{v\})$ (these sets have size at least $p^2n/24$). Each of the $2e(M)$ vertices in $M$ has $2n$ neighbours in $K_{2n+1}$, and each colour in $M$ is on $2n+1$ edges in $K_{2n+1}$. The number of edges of $K_{2n+1}$ sharing a vertex or colour with $M$ is thus $\leq 7ne(M)$. By maximality, and the property from Lemma~\ref{Lemma_MPS_boundrandcolour}, we have $7ne(M)\geq e_{C_3}(U,V) \geq 10^{-3}p^5n^2$, which implies that $e(M)\geq 10^{-4}p^5n \geq 4\mu n$.
For any edge $v_1v_2$ in the matching $M$, the path $uv_1v_2v$ is a rainbow path. In the union of these paths, the only colour repetitions can happen at $u$ or $v$. Since $K_{2n+1}$ is $2$-factorized, there is a subfamily of $\mu n$ paths which are collectively rainbow.
\epsilonilonnd{proof}
We can use this lemma to find many disjoint length $3$ connecting paths.
\begin{lemma}[Short connecting paths]\label{Lemma_few_connecting_paths} Let $p\gg q \gg n^{-1}>0$ and let $K_{2n+1}$ be $2$-factorized. Let $V$ be a $p$-random set of vertices, and $C$ a $p$-random set of colours independent from $V$.
Then, with high probability, for any set of $\{x_1, y_1, \dots, x_{q n}, y_{q n}\}$ of vertices,
there is a collection $P_1, \dots, P_{q n}$ of vertex-disjoint paths with length $3$, having internal vertices in $V$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [qn]$, and $P_1\cup \dots\cup P_{q n}$ is $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof}
By Lemma~\ref{Lemma_short_paths_between_two_vertices} applied to $C,V$ with $p=p, \mu=10q, n=n$, with high probability, between any $x_i$ and $y_i$, there is a collection of $10q n$ internally vertex disjoint $x_i,y_i$-paths, which are collectively $C$-rainbow, and internally contained in $V$. By choosing such paths greedily one by one, making sure never to repeat a colour or vertex, we can find the required collection of paths.
\epsilonilonnd{proof}
\section{The finishing lemma in Case A}\label{sec:finishA}
The proof of our finishing lemmas uses \epsilonilonmph{distributive} absorption, a technique introduced by the first author in \cite{montgomery2018spanning}. For the finishing lemma in Case A, we start by constructing colour switchers for sets of $\leq 100$ colours of $C$. These are $|C|$ perfect rainbow matchings, each from the same small set $X$ into a larger set $V$ which use the same colour except for one different colour from $C$ per matching (see Lemma~\ref{absorbA}). This gives us a small amount of local variability, but we can build this into a global variability property (see Lemma~\ref{absorbAmacro}). This will allow us to choose colours to use from a large set of colours, but not all the colours, so we will need to find matchings which ensure any colours outside of this are used (see Lemma~\ref{colourcover}). To find the switchers we use a small proportion of colours in a random set. We will have to cover the colours not used in this, with no random properties for the colours remaining (see Lemma~\ref{Lemma_saturating_matching_lemma}). We put this all together to prove Lemma~\ref{lem:finishA} in Section~\ref{sec:finishAfinal}.
\subsection{Colour switching with matchings}\label{sec:switcherspaths}
We start by constructing colour switchers using matchings.
\begin{lemma}\label{absorbA} Let $1/n\ll \beta \ll\xi, \mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$-random subsets which are independent. With high probability, the following holds.
Let $C,\bar{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},Z\subseteq V(K_{2n+1})$ satisfy $|\bar{{C}}|,|\bar{V}|\leq \beta n$, $|C|\leq 100$ and suppose that $(X,Z)$ is $(\xi n)$-replete.
Then, there are sets $X'\subseteq X\setminus \bar{V}$, $C'\subseteq D_0\setminus \bar{C}$ and $V'\subseteq (V_0\cup Z)\setminus \bar{V}$ with sizes $|C|$, $|C|-1$ and $\leq 3|C|$ respectively such that, for every $c\in C$, there is a perfect $(C'\cup \{c\})$-matching from $X'$ to $V'$.
\epsilonilonnd{lemma}
\begin{proof}
With high probability, by Lemma~\ref{lem:goodpairs} we have the following property.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}}]
\item For each distinct $u,v\in V(K_{2n+1})$, there are at least $100\beta n$ colours $c\in D_0$ for which there are colour-$c$ neighbours of both $u$ and $v$ in $V_0$.\label{prop:goodpairs}
\epsilonilonnd{enumerate}
We will show the property in the lemma holds. Let $C,\bar{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},Z\subseteq V(K_{2n+1})$ be sets with $|\bar{{C}}|,|\bar{V}|\leq \beta n$, $|C|\leq 100$ and suppose that $(X,Z)$ is $(\xi n)$-replete.
Let $\epsilonilonll=|C|\leq 100$ and label $C=\{c_1,\ldots,c_\epsilonilonll\}$.
For each $i\in [\epsilonilonll]$, using that there are at least $\xi n$ edges with colour $C$ between $X$ and $Z$, and $\xi\gg \beta$, $\epsilonilonll\leq 100$ and $|\bar{V}|\leq \beta n$, pick a vertex $x_i\in X\setminus (\bar{V}\cup \{x_1,\ldots,x_{i-1}\})$ which has a $c_i$-neighbour $y_i\in Z\setminus (\bar{V}\cup \{y_1,\ldots,y_{i-1}\})$.
For each $1\leq i\leq \epsilonilonll-1$, using~\ref{prop:goodpairs}, pick a colour $d_i\in D_0\setminus (\bar{C}\cup C\cup \{d_1,\ldots,d_{i-1}\})$
and (not necessarily distinct) vertices $z_i,z'_i\in V_0\setminus (\bar{V}\cup \{y_1,\ldots,y_{\epsilonilonll},z_1,\ldots,z_{i-1},z_1',\ldots,z_{i-1}'\})$ such that $x_iz_i$ and $x_{i+1}z'_{i}$ are both colour $d_i$. Let $C'=\{d_1,\ldots,d_{\epsilonilonll-1}\}$, $X'=\{x_1,\ldots,x_{\epsilonilonll}\}$ and $V'= \{y_1,\ldots,y_{\epsilonilonll},z_1,\ldots,z_{\epsilonilonll-1},z_1',\ldots,z_{\epsilonilonll-1}'\}$. See Figure~\ref{Figure_Matching_Switching} for an example of the edges that we find.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{matchingswitching.pdf}
\caption{An example of the graph we find in Lemma~\ref{absorbA} when $\epsilonilonll=5$. The key property it has is that for any colour $c_i$, there is a matching covering $x_1, \dots, x_{\epsilonilonll}$ using $c_i$ and all the colours $d_{1}, \dots, d_{\epsilonilonll-1}$.}
\label{Figure_Matching_Switching}
\epsilonilonnd{figure}
Now, for each $j\in [\epsilonilonll]$, let $M_j=\{x_jy_j\}\cup \{x_iz_i:i<j\}\cup \{x_{i+1}z_i':j\leq i\leq \epsilonilonll-1\}$. Note that $M_j$ is a perfect $(C\cup \{c_j\})$-matching from $X'$ to $V'$, and thus $X'$, $C'$ and $V'$ have the property required.
\epsilonilonnd{proof}
\subsection{Distributive absorption with matchings}
We now put our colour switchers together to create a global flexibility property. To do this, we find certain disjoint colour switchers, governed by a suitable \epsilonilonmph{robustly matchable bipartite graph}.
This is a bipartite graph which has a lot of flexibility in how one of its parts can be covered by matchings. It exists by the following lemma.
\begin{lemma}[Robustly matchable bipartite graphs, \cite{montgomery2018spanning}]\label{Lemma_H_graph}
There is a constant $h_0 \in \mathbb N$ such that, for every $h \geq h_0$, there exists
a bipartite graph $H$ with maximum degree at most $100$ and vertex classes $X$ and $Y \cup Y'$, with
$|X| = 3h$, and $|Y| = |Y'| = 2h$, so that the following is true. If $Y_0 \subseteqeq Y'$ and $|Y_0 | = h$, then
there is a matching between $X$ and $Y \cup Y_0$.
\epsilonilonnd{lemma}
\begin{lemma}\label{absorbAmacro} Let $1/n\ll \beta \ll \epsilonilonta\ll \xi,\mu$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $D_0\subseteq C(K_{2n+1})$ are $\mu$- and $\epsilonilonta$-random respectively, and suppose that they are independent. With high probability, the following holds.
Suppose $X,Z\subseteq V(K_{2n+1})$ are disjoint subsets such that $(X,Z)$ is $(\xi n)$-replete, $\alpha\leq \beta$ and $C\subseteq C(K_{2n+1})\setminus D_0$ is a set of at most $2\beta n$ colours. Then, there is a set $X_0$ of $|D_0|+\alpha n$ vertices in $X$ such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect $(D_0\cup C')$-rainbow matching from $X_0$ into $V_0\cup Z$.
\epsilonilonnd{lemma}
\begin{proof}
With high probability, by Lemmas~\ref{Lemma_Chernoff} and~\ref{absorbA} (applied with $\beta'=10^3\beta, \mu'=\epsilonilonta$, $\xi'=\xi/2$, and an $\epsilonilonta$-random subset of $V(K_{2n+1})$ contained in $V_0$) we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $\epsilonilonta n/2\leq |D_0|\leq 2\epsilonilonta n$.\label{aa2}
\item Let $\bar{C},\hat{C}\subseteq C(K_{2n+1})$ and $X,\bar{V},\bar{Z}\subseteq V(K_{2n+1})$ satisfy $|\bar{{C}}|,|\bar{V}|\leq 10^3\beta n$, $|\hat{C}|\leq 100$ and suppose that $(X,\bar{Z})$ is $(\xi n/2)$-replete.
Then, there are sets $X'\subseteq Y\setminus \bar{V}$, $C'\subseteq D_0\setminus \bar{C}$ and $V'\subseteq V\cup \bar Z \setminus \bar{V}$ with sizes $|\hat{C}|$, $|\hat{C}|-1$ and $\leq 3|\hat{C}|$ such that, for every $c\in \hat{C}$, there is a perfect $(C'\cup \{c\})$-matching from $Y'$ to $V'$.\label{aa3}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. For this, let $X,Z\subseteq V(K_{2n+1})$ be disjoint subsets such that $(X,Z)$ is $(\xi n)$-replete, let $\alpha\leq \beta$ and let $C\subseteq C(K_{2n+1})\setminus D_0$ be a set of at most $2\beta n$ colours.
Let $h=2\beta n$, and, using \ref{aa2}, pick a set $D_1\subseteq D_0$ of $(3h-\alpha n)$ colours.
Noting that $|D_1|\geq 2h$ and $|D_1\cup C|\leq 4h$, we can define sets $Y,Y'$ of order $2h$ with $Y\subseteqeq D_1$ and $D_1\cup C\subseteqeq Y\cup Y'$ (where any extra elements of $Y'$ are arbitrary dummy colours which will not be used in the arguments).
Using Lemma~\ref{Lemma_H_graph}, let $H$ be a bipartite graph with vertex classes $[3h]$ and $Y\cup Y'$, which has maximum degree at most 100, and is such that for any $\bar{Y}\subseteqeq Y'$ with $|\bar{Y}|=h$, there is a perfect matching between $[3h]$ and $Y\cup \bar{Y}$. In particular, since $|D_1|=3h-\alpha n$ and $Y\subseteqeq D_1$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\addtocounter{enumi}{2}
\item For each set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect matching between $[3h]$ and $D_1\cup C'$ in $H$. \label{Hmatch}
\epsilonilonnd{enumerate}
For each $i\in [3h]$, let $D_i=N_H(i)\cap C(K_{2n+1})$ (i.e.\ $D_i$ is the set of non-dummy colours in $N_H(i)$). Using \ref{aa3} repeatedly, find sets $X_i\subseteq X\setminus (X_1\cup\ldots\cup X_{i-1})$, $C_i\subseteq D_0\setminus (D_1\cup C_1\cup\ldots\cup C_{i-1})$ and $V_i\subseteq (V_0\cup Z)\setminus (V_1\cup \ldots\cup V_{i-1})$,
with sizes $|D_i|$, $|D_i|-1$ and at most $300$, such that the following holds.
(To see that we can repeat this application of \ref{aa3} this many times, note that $h =2\beta n$ and $\xi \gg \beta$ and so at each application we delete $O(h)$ vertices from $X$ or $Z$, leaving a pair which is still $(\xi n/2)$-replete.)
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\addtocounter{enumi}{3}
\item For every $c\in D_i$, there is a perfect $(C_i\cup\{c\})$-rainbow matching from $X_i$ into $V_i$.
\label{propabsorb}
\epsilonilonnd{enumerate}
Greedily, using \ref{aa2}, that $h=2\beta n$, $\beta,\epsilonilonta\ll \xi$ and $(X,Z)$ is $(\xi n)$-replete, find a $(D_0\setminus (D_1\cup C_1\cup\ldots\cup C_{3h}))$-rainbow matching $M$ with $|D_0\setminus (D_1\cup C_1\cup \ldots\cup C_{3h})|$ edges from $X\setminus (X_1\cup\ldots\cup X_{3h})$ into $Z\setminus (X_1\cup\ldots\cup X_h)$.
Let $X_0=(X\cap V(M))\cup X_1\cup \ldots\cup X_{3h}$. Note that $|X_0|=|D_0|-|D_1|-\sum_{i=1}^{3h}(|X_i|-|C_i|)=|D_0|-|D_1|+3h=|D_0|+\alpha n$.
We claim this has the property required. Indeed, suppose $C'\subseteq C$ is a set of $\alpha n$
colours. Using~\ref{Hmatch}, let $M'$ be a perfect matching between $[3h]$ and $D_1\cup C'$ in $H$, and label $D_1\cup C'=\{c_1,\ldots,c_{3h}\}$ so that, for each $i\in [3h]$, $c_i$ is matched to $i$ in $M'$. Note that the colours $c_i$ are not dummy colours.
By the definition of each $D_i$, $c_i\in D_i$ for each $i\in [3h]$.
By \ref{propabsorb}, for each $i\in [3h]$, there is a perfect $(C_i\cup \{c_i\})$-matching, $M_i$ say, from $X_i$ to $V_i$. Then, $M\cup M_1\cup \ldots\cup M_{3h}$ is a perfect matching from $X_0$ into $V_0\cup Z$ which is $(C(M)\cup D_1\cup C'\cup C_1\cup\ldots\cup C_{3h})$-rainbow, and hence $(D_0\cup C')$-rainbow, as required.
\epsilonilonnd{proof}
\subsection{Covering small colour sets with matchings}
We find perfect rainbow matchings covering a small set of colours using the following lemma.
\begin{lemma}\label{colourcover} Let $1/n\ll \nu\ll \lambda\ll 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $X,V\subseteq V(K_{2n+1})$ are disjoint, $(X,V)$ is $(10\nu n)$-replete and $|X|\leq \lambda n$. Suppose that $C\subseteq C(K_{2n+1})$ is such that every vertex $v\in V(K_{2n+1})$ has at least $3\lambda n$ colour-$C$ neighbours in $V$.
Then, given any set $C'\subseteq C(K_{2n+1})$ with at most $\nu n$ colours, there is a perfect $(C'\cup C)$-rainbow matching from $X$ to $V$ which uses every colour in $C'$.
\epsilonilonnd{lemma}
\begin{proof} Using that $(X,V)$ is $(10\nu n)$-replete, greedily find a matching $M_1$ with $|C'|$ edges from $X$ to $V$ which is $C'$-rainbow. Then, using that every vertex $v\in V(K_{2n+1})$ has at least $3\lambda n$ colour-$C$ neighbours in $V$, greedily find a perfect $C$-rainbow matching $M_2$ from $X\setminus V(M_1)$ to $V\setminus V(M_1)$. This is possible as, when building $M_2$ greedily, each vertex in $X\setminus V(M_1)$ has at most $2|X|\leq 2\lambda n$ neighbouring edges with colour used in $C\cap C(M_1\cup M_2)$ (as the colouring is $2$-bounded) and at most $|X|\leq \lambda n$ colour-$C$ neighbours in $V(M_1)\cup V(M_2)$.
The matching $M_1\cup M_2$ then has the required property.
\epsilonilonnd{proof}
\subsection{Almost-covering colours with matchings}
We find rainbow matchings using almost all of a set of colours using the following lemma.
\begin{lemma}\label{Lemma_saturating_matching_lemma}
Let $1\geq p\gg q\gg \gamma,\epsilonilonta \gg \nu\gg 1/n$ and let $K_{2n+1}$ be $2$-factorized.
Suppose we have disjoint sets $V, V_0\subseteqeq V(K_{2n+1})$, and $D,D_0\subseteqeq C(K_{2n+1})$ with $V$ $p$-random, $D$ $q$-random, and $V_0, D_0$ $\gamma$-random. Furthermore, suppose that $V_0$ and $D_0$ are independent of each other. Then, the following holds with probability $1-o(n^{-1})$.
For every $U\subseteqeq V(K_{2n+1})\setminus (V\cup V_0)$ with at most $(q+\gamma+\epsilonilonta-\nu)n$ vertices, and any $C\subseteq C(K_{2n+1})$ with $D\cup D_0\subseteq C$ with $|C|\geq |U|+\nu n$, there is a perfect $C$-rainbow matching from $U$ into $V\cup V_0$.
\epsilonilonnd{lemma}
\begin{proof}
Choose $\beta$ such that $\nu\gg \beta\gg n^{-1}$.
Partition $V=V_1\cup V_2$ so that $V_1$ and $V_2$ are $(p/2)$-random. Partition $D_0=D_1\cup D_2$ with $D_1$ $(\gamma-\nu/4)$-random and $D_2$ $(\nu/4)$-random. We now find properties \ref{grape1} -- \ref{grape4}, which all hold with probability $1-o(n^{-1})$, as follows.
By Lemma~\ref{Lemma_matching_into_random_set_using_specified_colours} with $V=V_1$, $p'=q/2$, $q=\gamma+2\epsilonilonta$, $\beta=\beta$, and $n=n$, we have the following.
\addtocounter{propcounter}{1}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any $U\subseteq V(K_{2n+1})\setminus V_1$ with $|U|\geq qn/2$ and any $C'\subseteq C(K_{2n+1})$ with $|C'|\leq (\gamma +2\epsilonilonta)n$, there is a $C'$-rainbow matching of size $|C'|-\beta n$ from $U$ into $V_1$. \label{grape1}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_nearly_perfect_matching} with $V=V_2$, $C=D$, $p=q$, $\beta=\beta$, and $n=n$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\stepcounter{enumi}
\item For any $U\subseteq V(K_{2k+1})\setminus V_2$ with $|U|\leq q n$, there is a $D$-rainbow matching from $U$ into $V_2$ with size $|U|-\beta n$.\label{grape2b}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_high_degree_into_random_set} with $V=V_0$, $C=D_2$, $p=\gamma$, $q=\nu/4$, and $n=n$ we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item Every vertex $v\in V(K_{2n+1})$ has $|N_{D_2}(v)\cap V_0|\geq \gamma\nu n/4$.\label{grape3}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_Chernoff}, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $|D_1|=(\gamma-\nu/4\pm \beta)n\leq \gamma n$, $|D|=(q\pm \beta)n$ and $|D_2| \leq \nu n/3$.\label{grape4}
\epsilonilonnd{enumerate}
We now show that the property in the lemma holds. For this, let $U\subseteqeq V(K_{2n+1})\setminus (V\cup V_0)$ have at most $(q+\gamma+\epsilonilonta-\nu)n$ vertices, and let $C\subseteq C(K_{2n+1})$ satisfy $D\cup D_0\subseteq C$ and $|C|\geq |U|+\nu n$.
Note that we can assume that $|C|=|U|+\nu n$, so that, using \ref{grape4}, $|C\setminus (D \cup D_2)|\leq |U|+\nu n -|D| \leq (q+\gamma+\epsilonilonta-\nu)n+\nu n-(q-\beta)n= (\gamma+\epsilonilonta +\beta)n\leq (\gamma+2\epsilonilonta)n$. Notice that, by \ref{grape4} and as $D\subseteq C$, we have $|U|\geq |C|-\nu n\geq q n/2$.
Therefore, by \ref{grape1}, there is a $(C\setminus (D \cup D_2))$-rainbow matching $M_1$ of size $|C\setminus (D \cup D_2)|-\beta n$ from $U$ into $V_1$. Now, using \ref{grape4}, $|U\setminus V(M_1)|=|U|-|C\setminus (D \cup D_2)|+\beta n\leq |D|+|D_2|-\nu n+\beta n\leq (q+\beta)n+\nu n/3-\nu n+\beta n\leq q n$. By \ref{grape2b}, there is a $D$-rainbow matching $M_2$ of size $|U\setminus V(M_1)| -\beta n$ from $U\setminus V(M_1)$ into $V_2$.
Notice that $|U\setminus (V(M_1)\cup V(M_2))|\leq \beta n$.
From \ref{grape3}, we have that $|N_{D_2}(v)\cap V_0|\geq \gamma \nu n/4> 3 \beta n \geq 3|U\setminus (V(M_1)\cup V(M_2))|$ for every vertex $v\in V(K_{2n+1})$. By greedily choosing neighbours of $u\in U\setminus (V(M_1)\cup V(M_2))$ one at a time (making sure to never repeat colours or vertices) we can find a $D_2$-rainbow matching $M_3$ into $V_0$ covering $U\setminus (V(M_1)\cup V(M_2))$. Indeed, when we seek a new colour-$D_2$ neighbour of a vertex $u\in U\setminus (V(M_1)\cup V(M_2))$ there are at most 2 neighbours of $u$ of each colour used from $D_2$ in $V_0$, ruling out at most $2|U\setminus (V(M_1)\cup V(M_2))|$ colour-$D_2$ neighbours of $u$ in $V_0$, while at most $|U\setminus (V(M_1)\cup V(M_2))|$ vertices have been used in $V_0$ in the matching.
Now the matching $M_1\cup M_2\cup M_3$ satisfies the lemma.
\epsilonilonnd{proof}
\subsection{Proof of the finishing lemma in Case A}\label{sec:finishAfinal}
Finally, we can put all this together to prove Lemma~\ref{lem:finishA}.
\begin{proof}[Proof of Lemma~\ref{lem:finishA}]
Pick $\nu,\lambda,\beta$ and $\alpha$ so that $1/n\ll \nu \ll\lambda\ll\beta\ll \alpha\ll \xi$ and recall that $\xi \ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll p\leq 1$. Let $V_1',V_2',V_3'$ be disjoint $(p/3)$-random subsets in $V$ and let $W_1,W_2,W_3$ be disjoint $(\mu/3)$-random subsets in $V_0$.
Let $D_1$, $D_2$, $D_3$ be disjoint $(\mu/3)$-, $\beta$- and $\alpha$-random disjoint subsets in $C_0$ respectively.
We now find properties \ref{mouse11}--\ref{mouse44}, which collectively hold with high probability as follows.
First note that $(1-\epsilonilonta)\epsilonilonps+(\mu/3)+\epsilonilonta-\nu \geq \epsilonilonps$. Thus, by Lemma~\ref{Lemma_saturating_matching_lemma} applied with $V=V_1'$, $V_0=W_1$, $D=C$, $D_0=D_1$, $p'=p/3$, $q=(1-\epsilonilonta)\epsilonilonpsilon$, $\gamma=\mu/3$, $\epsilonilonta=\epsilonilonta$ and $\nu=\nu$ we get the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any $U\subseteq V(K_{2n+1})\setminus (V'_1\cup W_1)$ with $|U|\leq \epsilonilonps n$ and any set of colours $C'\subseteq C(K_{2n+1})$ with $C\cup D_1\subseteq C'$, and $|C'| \geq |U|+\nu n$, there is a perfect $C'$-rainbow matching from $U$ into $V_1'\cup W_1$.\label{mouse11}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_high_degree_into_random_set}, we get the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item Each $v\in V(K_{2n+1})$ has at least $\beta^2 n$ colour-$D_2$ neighbours in $W_2$.\label{mouse2}
\epsilonilonnd{enumerate}
By Lemma~\ref{absorbAmacro} applied with $V_0=W_3$,
$D_0=D_3$, $\mu'=\mu/3$, $\xi'=\xi/3$, $\epsilonilonta'=\alpha, \alpha'=\gamma$ and $\beta'=2\beta$, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any disjoint vertex sets $Y,Z\subseteq V(K_{2n+1})\setminus W_3$ for which $(Y,Z)$ is $(\xi n/3)$-replete, any $\gamma\leq 2\beta$ and any set $\hat{C}\subseteq C(K_{2n+1})\setminus D_3$ of at most $4\beta n$ colours, the following holds. There is a subset $Y'\subseteq Y$ of $|D_3|+\gamma n$ vertices so that, for any $C'\subseteq \hat{C}$ with $|C'|=\gamma n$, there is a perfect $(D_3\cup C')$-rainbow matching from $Y'$ into $Z\cup W_3$.\label{mouse3}
\epsilonilonnd{enumerate}
Finally, by Lemma~\ref{Lemma_Chernoff}, the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $2\beta n\geq |D_2|\geq \beta n/2$ and $2\alpha n\geq |D_3|\geq \alpha n/2$.\label{mouse44}
\epsilonilonnd{enumerate}
We will now show that the property in the lemma holds.
Let then $X,Z\subseteq V(K_{2n+1})\setminus (V\cup V_0)$ be disjoint and let $D\subseteq C(K_{2n+1})$, so that $|X|=|D|=\epsilonilonps n$, $C_0\cup C\subseteq D$ and $(X,Z)$ is $(\xi n)$-replete. Let $Z_1$, $Z_2$ be disjoint $(1/2)$-random subsets of $Z$. Note that, by Lemma~\ref{Lemma_Chernoff}, $(X,Z_1)$ and $(X,Z_2)$ are with high probability $(\xi/3)$-replete. Thus, we can pick an instance of $Z_1$ and $Z_2$ for which this holds. Let $V_1=V_1'\cup W_1$, $V_2=V_2'\cup W_2\cup Z_1$ and $V_3=V_3'\cup W_3\cup Z_2$.
Set $\gamma=(|D_2|/n)-\lambda$, so that, by \ref{mouse44}, $2\beta\geq \gamma>0$. Then, by \ref{mouse44} and \ref{mouse3} applied with $\hat{C}=D_2$, $Y=X_1, Z=Z_2$ there is a set $X_3\subseteq X$ with size $|D_3|+\gamma n=|D_3|+|D_2|-\lambda n$ and the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{4}
\item For any $C'\subseteq D_2$ with $|C'|=\gamma n=|D_2|-\lambda n$, there is a perfect $(D_3\cup C')$-rainbow matching from $X_3$ into $V_3$.\label{mouse4}
\epsilonilonnd{enumerate}
Let $C_1=D\setminus (D_2\cup D_3)$, $C_2=D_2$ and $C_3=D_3$, and note that $C\cup D_1\subseteq C_1$.
Let $X'=X\setminus X_3$, so that $|X'|=(\epsilonilonps +\lambda)n-|D_3|-|D_2|=|C_1|+\lambda n$. Using \ref{mouse44}, we have $|X_3|\leq |D_2|+|D_3|\leq \xi n/10$, and hence $(X',Z_1)$ is $(\xi n/6)$-replete.
Let $X_2'\subseteq X_2$ be a $(\lambda n/|X'|)$-random subset of $X'$. By Lemma~\ref{Lemma_Chernoff}, $|X_2'|\leq (\lambda+\nu)n$ and $(X_2',Z_1)$ is $(10\nu n)$-replete. Choose disjoint sets $X_1$ and $X_2$ of $X'$ with size $|X'|-(\lambda+\nu)n=|C_1|-\nu n$ and $(\lambda+\nu)n$ respectively, and so that $X_2'\subseteq X_2$. Note that $(X_2,Z_1)$ is $(10\nu n)$-replete. Therefore, by Lemma~\ref{colourcover} (with $\nu=\nu$, $\lambda'=\lambda+\nu \ll \beta$, $X=X_2, V=V_2$, and $C=C_2=D_2$) and \ref{mouse2}, the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{5}
\item For any set $C'\subseteq C(K_n)$ of at most $\nu n$ colours, there is a perfect $(C'\cup C_2)$-rainbow matching from $X_2$ to $V_2$ which uses every colour in $C'$.\label{mouse5}
\epsilonilonnd{enumerate}
We thus have partitions $X=X_1\cup X_2\cup X_3$, $C=C_1\cup C_2\cup C_3$ and $V=V_1\cup V_2\cup V_3$, for which, by \ref{mouse11}, \ref{mouse5} and \ref{mouse4}, we have that
\ref{propp1}--\ref{propp3} hold. Therefore, by the discussion after \ref{propp1}--\ref{propp3}, we have a perfect $C$-rainbow matching from $X$ into $V$.
\epsilonilonnd{proof}
\section{The finishing lemma in Case B}\label{sec:finishB}
Our proof of the finishing lemma in Case B has the same structure as the finishing lemma in Case A, except we first construct the colour switchers in the ND-colouring in two steps, before showing this can be done with random vertices and colours. In overview, in this section we do the following, noting the lemmas in which the relevant result is given and the comparable lemmas in Section~\ref{sec:finishA}.
\begin{itemize}
\item Lemma~\ref{lem-switchpath}: We construct colour switchers in the ND-colouring for any pair of colours $(c_1,c_2)$ -- that is, two short rainbow paths with the same length between the same pair of vertices, whose colours are the same except that one uses $c_1$ and one uses $c_2$.
\item Lemma~\ref{lem-absorbpath}: We use Lemma~\ref{lem-switchpath} to construct a similar colour switcher in the ND-colouring that can use 1 of a set of 100 colours.
\item Lemma~\ref{lem-randabsorbpath}: We show that, given a pair of independent random vertex and colour sets, many of these switchers use only vertices and (non-switching) colours in the subsets (cf.\ Lemma~\ref{absorbA}).
\item Lemma~\ref{absorbBmacro}: We use distributive absorption to convert this into a larger scale absorption property (cf.\ Lemma~\ref{absorbAmacro}).
\item Lemma~\ref{colourcoverB}: We embed paths while ensuring that an (arbitrary) small set of colours is used (cf.\ Lemma~\ref{colourcover}).
\item Lemma~\ref{finishingB}: We embed paths using almost all of a set of colours, in such a way that this can reduce the number of `non-random' colours (cf.\ Lemma~\ref{Lemma_saturating_matching_lemma}).
\item Finally, we put this all together to prove Lemma~\ref{lem:finishB}, the finishing lemma in Case B (cf.\ the proof of Lemma~\ref{lem:finishA}).
\epsilonilonnd{itemize}
\subsection{Colour switching with paths}\label{sec:switchers}
We start by constructing colour switchers in the ND-colouring capable of switching between 2 colours.
We use switchers consisting of two paths with length 7 between the same two vertices. By using one path or the other we can choose which of two colours $c_1$ and $c_2$ are used, as the other colours used appear on both paths.
\begin{defn}
In a complete graph $K_{2n+1}$ with vertex set $[2n+1]$, let $(x_0,a_1,\ldots,a_\epsilonilonll)$ denote the path with length $\epsilonilonll$ with vertices $x_0x_1\ldots x_\epsilonilonll$ where, for each $i\in [\epsilonilonll]$, $x_i=x_{i-1}+a_i\pmod 2n+1$.
\epsilonilonnd{defn}
\begin{figure}[h]
\begin{center}
{
\begin{tikzpicture}[scale=0.7]
\draw [red,thick] (-1,0) -- (0,0);
\draw (-0.5,-0.3) node {$i$};
\draw [blue,thick] (4,1) -- (0,0);
\draw (2,0.75) node {$c_1$};
\draw [green,thick] (4,1) -- (6,1);
\draw (5,1.25) node {$d_1$};
\draw [magenta,thick] (13,1) -- (6,1);
\draw (9.5,1.25) node {$d_2$};
\draw [purple,thick] (13,1) -- (10,0.5);
\draw (11.5,0.45) node {$d_3$};
\draw [orange,thick] (5,0) -- (10,0.5);
\draw (7.5,0.55) node {$d_4$};
\draw [brown,thick] (5,0) -- (18,0);
\draw (15,-0.3) node {$y-i-c_1-k-x$};
\draw [green,thick] (5,0) -- (7,-0.5);
\draw (6,-0.55) node {$d_1$};
\draw [magenta,thick] (14,-1) -- (7,-0.5);
\draw (10.5,-0.45) node {$d_2$};
\draw [purple,thick] (14,-1) -- (11,-1);
\draw (12.5,-1.25) node {$d_3$};
\draw [orange,thick] (6,-1) -- (11,-1);
\draw (8.5,-1.25) node {$d_4$};
\draw [cyan,thick] (6,-1) -- (0,0);
\draw (3,-0.75) node {$c_2$};
\draw [fill=black] (-1,0) circle [radius = 0.1cm];
\draw [fill=black] (18,0) circle [radius = 0.1cm];
\draw [fill=black] (0,0) circle [radius = 0.1cm];
\draw [fill=black] (4,1) circle [radius = 0.1cm];
\draw [fill=black] (6,1) circle [radius = 0.1cm];
\draw [fill=black] (13,1) circle [radius = 0.1cm];
\draw [fill=black] (10,0.5) circle [radius = 0.1cm];
\draw [fill=black] (5,0) circle [radius = 0.1cm];
\draw [fill=black] (7,-0.5) circle [radius = 0.1cm];
\draw [fill=black] (6,-1) circle [radius = 0.1cm];
\draw [fill=black] (11,-1) circle [radius = 0.1cm];
\draw [fill=black] (14,-1) circle [radius = 0.1cm];
\draw (18.5,0) node {$y$};
\draw (-1.5,0) node {$x$};
\epsilonilonnd{tikzpicture}
}
\epsilonilonnd{center}
\caption{Two $x,y$-paths used to switch colours between $c_1$ and $c_2$.}\label{colourswitcher}
\epsilonilonnd{figure}
\begin{lemma}[1 in 2 colour switchers]\label{lem-switchpath}
Let $1\gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. Suppose we have a pair of distinct vertices $x,y\in V(K_{2n+1})$, a pair of distinct colours $c_1,c_2\in C(K_{2n+1})$, and sets $X\subseteq V(K_{2n+1})$ and $C\subseteq C(K_{2n+1})$ with $|X|\leq n/25$ and $|C|\leq n/25$.
Then, there is some set $C'\subseteq C(K_{2n+1})\setminus (C\cup\{c_1,c_2\})$ with size $6$ so that, for each $i\in \{1,2\}$, there is a $(C'\cup\{c_i\})$-rainbow $x,y$-path with length 7 and interior vertices in $V(K_{2n+1})\setminus X$.
\epsilonilonnd{lemma}
\begin{proof}
Since $2n+1$ is odd we can relabel $c_1$ and $c_2$ such that $c_1+2k=c_2$ for some $k\in [n]$ (here and later in this proof all the additions are$\pmod{2n+1}$). We will construct a switcher as depicted in Figure~\ref{colourswitcher}.
Find distinct $d_1,d_2,d_3,d_4\in [n]\setminus (C\cup \{c_1,c_2\})$ such that $d_1+d_2=d_3+d_4+k$ and $(1,c_1,d_1,d_2,-d_3,-d_4)$ and $(1,c_1+k,d_1,d_2,-d_3,-d_4)$ are both valid paths (that is, they have distinct vertices) starting at $1$. This is possible, as follows.
Pick $d_1,d_2,d_3\in [n]\setminus (C\cup \{c_1,c_2\})$ distinctly in turn so that $(1,c_1,d_1,d_2,-d_3)$ and $(1,c_1+k,d_1,d_2,-d_3)$ are both valid paths with the first one avoiding $1+c_1+k$ and the second one avoiding $1+c_2$. Note that, as we choose each $d_i$, we add one more vertex to each of these paths, which have at most 4 vertices already, so at most 12 colours are ruled out by the requirement these paths are valid and avoid the mentioned vertices. Furthermore, there are at most $|C|+4\leq n/20$ colours in $C\cup \{c_1,c_2\}$ or already chosen as some $d_j$, $j<i$. Thus, there are at least $n/2$ choices for each $d_i$, and therefore at least $(n/2)^3$ choices in total.
Now, let $d_4:=d_1+d_2-d_3-k$. Note that there are at most $n^2(|C|+5)\leq n^3/20$ choices for $\{d_1,d_2,d_3\}$ so that $d_4\in C\cup\{c_1,c_2,d_1,d_2,d_3\}$. Therefore, we can choose distinct $d_1,d_2,d_3\in [n]\setminus (C\cup \{c_1\cup c_2\})$ so that $d_4\notin C\cup\{c_1,c_2,d_1,d_2,d_3\}$, in addition to $(1,c_1,d_1,d_2,-d_3)$ and $(1,c_1+k,d_1,d_2,-d_3)$ both being valid paths which avoid $1+c_1+k$ and $1+c_2$, respectively. Noting that $1+c_1+d_1+d_2-d_3-d_4=1+c_1+k$, $(1,c_1,d_1,d_2,-d_3,-d_4)$ is therefore a valid path. Noting that $1+c_1+k+d_1+d_2-d_3-d_4=1+c_1+2k=1+c_2$, $(1,c_1+k,d_1,d_2,-d_3,-d_4)$ is therefore a valid path, with endvertex $1+c_2$. Therefore, its reverse, $(1+c_2,d_4,d_3,-d_2,-d_1,-c_1-k)$, is a valid path that ends with $1$. Moving the vertex $1$ to the start of the path, we get the valid path $(1,c_2,d_4,d_3,-d_2,-d_1)$.
Let $I$ be the set of $i\in [n]\setminus (C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\})$ such that $x$ and $y$ are not on $(x+i,c_1,d_1,d_2,-d_3,-d_4)$, or $(x+i,c_2,d_4,d_3,-d_2,-d_1)$, and that $y-i-c_1-k-x\notin C\cup\{c_1,c_2,d_1,d_2,d_3,d_4\}$. Note that these conditions rule out at most 12, 12, and $|C|+6$ values of $i$ respectively, so that $|I|\geq n-2|C|-12-24\geq n/2$.
For each $i\in I$, add $x$ and $y$ as the start and end of
$(x+i,c_1,d_1,d_2,-d_3,-d_4)$ respectively, giving, as $k=d_1+d_2-d_3-d_4$ the path
\[
P_i:=(x,i,c_1,d_1,d_2,-d_3,-d_4,y-i-c_1-k-x).
\]
Then, add $x$ and $y$ as the start and end of
$(x+i,c_2,d_4,d_3,-d_2,-d_1)$ respectively, giving, as $d_4+d_3-d_1-d_2=-k=c_1-c_2+k$ the path
\[
Q_i:=(x,i,c_2,d_4,d_3,-d_2,-d_1,y-i-c_1-k-x).
\]
Let $C_i=\{i,d_1,d_2,d_3,d_4,y-i-c_1-k\}$. Note that, $P_i$ and $Q_i$ are both rainbow $x,y$-paths with length 7, with colour sets $(C_i\cup\{c_1\})$ and $(C_i\cup\{c_1\})$ respectively (indeed, when we add $-d_i, 1 \leq i \leq 4$ to get the next vertex of the path the colour of this edge is $d_i$ and ``$-$" just indicates in which direction we are moving).
Each vertex in $X$ can appear as the interior vertex of at most 6 different paths $P_i$ and 6 different paths $Q_i$. As $|I|\geq n/2 > 12|X|$, there must be some $j\in I$ for which the interior vertices of $P_j$ and $Q_j$ avoid $X$. Then, $C'=C_j$ is a colour set as required by the lemma, as demonstrated by the paths $P_j$ and $Q_j$.
\epsilonilonnd{proof}
The following lemma uses this to find colour switchers for an arbitary set of 100 colours $\{c_1,\ldots,c_{100}\}$ in the ND-colouring between an arbitrary vertex pair $\{x,y\}$. A sketch of its proof is as follows. First, we select a vertex $x_1$ and colours $d_1,\ldots,d_{100}$ so that $c_i+d_i=x_1-x$ for each $i\in [100]$. By choosing the vertex between $x$ and $x_1$ appropriately, we can find a $\{c_i,d_i\}$-rainbow $x,x_1$-path with length 2 for each $i\in [100]$. This allows us to use any pairs of colours $\{c_i,d_i\}$, so we need only construct a path which can switch between using any set of 99 colours from $\{d_1,\ldots,d_{100}\}$. This we do by constructing a sequence of $(d_i,d_{i+1})$-switchers for each $i\in [99]$ and putting them together between $x_1$ and $y$.
\begin{lemma}[1 in 100 colour absorbers]\label{lem-absorbpath}
Let $1\gg n^{-1}$. Suppose $K_{2n+1}$ is ND-coloured.
Suppose we have a pair of distinct vertices $x,y\in V(K_{2n+1})$, a set $C\subseteq C(K_{2n+1})$ of $100$ colours, and sets $X\subseteq V(K_{2n+1})$ and $C'\subseteq C(K_{2n+1})$ with $|X|\leq n/10^3$ and $|C'|\leq n/10^3$.
Then, there is a set $\bar{C}\subseteq C(K_{2n+1})\setminus (C\cup C')$ of 694 colours and a set $\bar{X}\subseteq V(K_{2n+1})\setminus X$ of at most $1500$ vertices so that, for each $c\in C$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $\bar{X}$.
\epsilonilonnd{lemma}
\begin{proof}
Let $\epsilonilonll=100$, $x_0=x$, and $x_\epsilonilonll=y$, and label $C=\{c_1,\ldots,c_\epsilonilonll\}$. Pick $x_1\in [2n+1]\setminus (X\cup\{x_0,x_\epsilonilonll\})$ so that $x_1-x_0+c_i\in [n]\setminus (C\cup C')$ and $x_1-c_i\in [2n+1]\setminus X$
for each $i\in [\epsilonilonll]$. Each such condition forbids at most $n/10^3+100$ points and we have at most $2\epsilonilonll=200$ conditions so we can indeed find $x_1$ satisfying all of them.
For each $i\in [\epsilonilonll]$, let $d_i=x_1-x_0+c_i$ and $y_i= x_1-c_i=x_0+d_i$. Then,
\begin{itemize}
\item $c_1,\ldots,c_\epsilonilonll,d_1,\ldots,d_\epsilonilonll$ are distinct colours in $C\cup ([n]\setminus C')$,
\item $\{x_0,x_1,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}$ are distinct vertices in $\{x,y\}\cup ([2n+1]\setminus X)$, and
\item for each $i\in [\epsilonilonll]$, $x_0y_ix_1$ is a $\{c_i,d_i\}$-rainbow path.
\epsilonilonnd{itemize}
Let $C''=\{d_1,\ldots,d_\epsilonilonll\}$.
Pick distinct vertices $x_2,\ldots,x_{\epsilonilonll-1}\in [2n+1]\setminus (X\cup\{x_0,x_1,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\})$ and let $X'=\{x_0,x_1,\ldots,x_\epsilonilonll,y_1,\ldots,y_\epsilonilonll\}$.
Next, iteratively, for each $1\leq i\leq \epsilonilonll-1$, using Lemma~\ref{lem-switchpath} find a set $X_i$ of at most 12 vertices in $[2n+1]\setminus (X\cup X'\cup(\cup_{j<i}X_j))$ and a set $C_i$ of 6 colours in $[n]\setminus (C\cup C'\cup C''\cup(\cup_{j<i}C_j))$ so that there is a $(C_i\cup\{d_{i}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$ and a $(C_i\cup\{d_{i+1}\})$-rainbow
$x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$.
Let $\bar{C}=C''\cup(\cup_{i\in [\epsilonilonll-1]}C_i)$ and $\bar{X}=X'\cup (\cup_{i\in [\epsilonilonll-1]}X_i)$, and note that $|\bar{C}|=\epsilonilonll+6(\epsilonilonll-1)=694$ and
$|\bar{X}|\leq 2\epsilonilonll+1+12(\epsilonilonll-1) \leq 1500$. We will show that $\bar{C}$ and $\bar{X}$ satisfy the condition in the lemma.
Let then $j\in [\epsilonilonll]$. For each $1\leq i< j$, let $P_i$ be a $(C_i\cup\{d_{i}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$. For each $j\leq i\leq \epsilonilonll-1$, let $P_i$ be a $(C_i\cup\{d_{i+1}\})$-rainbow $x_i,x_{i+1}$-path with length 7 and internal vertices in $X_i$. Thus, the paths $P_i$, $i\in [\epsilonilonll-1]$, cover all the colours in $C''$ except for $d_j$, as well as the colours in each set $C_i$.
Therefore, as $x_0=x$ and $x_\epsilonilonll=y$,
\[
x_0y_jx_1P_1x_2P_2x_3P_3\ldots P_{\epsilonilonll-1}x_{\epsilonilonll}
\]
is a $(\bar{C}\cup \{c_j\})$-rainbow $x,y$-path with length $2+7\times 99=695$ whose interior vertices are all in $\bar{X}$, as required.
\epsilonilonnd{proof}
The following corollary of this will be convenient to apply.
\begin{corollary}\label{cor-absorbpath}
Let $1\gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. For each pair of distinct vertices $x,y\in V(K_{2n+1})$ and each set $C\subseteq C(K_{2n+1})$ of 100 colours, there are $\epsilonilonll=n/10^7$ disjoint vertex sets $X_1,\ldots,X_\epsilonilonll\subseteq V(K_{2n+1})\setminus \{x,y\}$ with size at most 1500 and disjoint colour sets $C_1,\ldots,C_\epsilonilonll\subseteq [n]\setminus C$ with size 694
such that the following holds.
For each $i\in [\epsilonilonll]$ and $c\in C$, there is a $C_i\cup\{c\}$-rainbow $x,y$-path with length 695 and interior vertices in $X_i$.
\epsilonilonnd{corollary}
\begin{proof}
Iteratively, for each $i=1, \dots, \epsilonilonll$, choose sets $X_i$ and $C_i$ using Lemma~\ref{lem-absorbpath} (at the $i$th iteration letting $X=X_1\cup \dots \cup X_{i-1}$, $C'=C_1\cup \dots \cup C_{i-1}$).
\epsilonilonnd{proof}
The following lemma finds $1$-in-$100$ colour switchers in a random set of colours and vertices.
\begin{lemma}[Colour switchers using random vertices and colours]\label{lem-randabsorbpath}
Let $p,q\gg \mu \gg n^{-1}$ and suppose that $K_{2n+1}$ is ND-coloured. Let $X\subseteq V_{2k+1}$ be $p$-random and $C\subseteq C(K_{2n+1})$ $q$-random, and such that $X$ and $C$ are independent. With high probability the following holds.
For every distinct $x,y\in V(K_{2n+1})$, $C'\subseteq C(K_{2n+1})$ with $|C'|=100$, and $X'\subseteq X$, $C''\subseteq C$ with $|X'|,|C''|\leq\mu n$, there is a set $\bar{C}\subseteq C\setminus (C'\cup C'')$ of 694 colours and a set $X''\subseteq X\setminus X'$ of at most $1500$ vertices with the following property. For each $c\in C'$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $X''$.
\epsilonilonnd{lemma}
\begin{proof}
We will show that for any distinct pair $x,y\in V(K_{2n+1})$ of vertices and set $C'\subseteq C(K_{2n+1})$ of 100 colours the property holds with probability $1-o(n^{-102})$ so that the lemma holds by a union bound.
Fix then distinct $x,y\in V(K_{2n+1})$ and a set $C'\subseteq C(K_{2n+1})$ with size 100. Fix $\epsilonilonll= n/10^7$ and use Corollary~\ref{cor-absorbpath} to find disjoint vertex sets $X_i\subseteq V(K_{2n+1})$, $i\in [\epsilonilonll]$, with size at most 1500 and disjoint colours sets $C_i\subseteq C(K_{2n+1})$, $i\in [\epsilonilonll]$, with size $694$ so that for each $c\in C'$ and $i\in [\epsilonilonll]$ there is a $C_i\cup \{c\}$-rainbow $x,y$-path with length 695 and internal vertices in $X_i$.
Let $I\subseteq [\epsilonilonll]$ be the set of $i\in [\epsilonilonll]$ for which $X_i\subseteq X$ and $C_i\subseteq C$. Note that $|I|$ is $1$-Lipschitz, and, for each $i\in [\epsilonilonll]$, we have $\mathbb{P}(X_i\subseteqeq X, C_i\subseteqeq C)\geq p^{1500}q^{695}\gg \mu$.
By Azuma's inequality, with probability $1-o(n^{-102})$ we have $|I|\geq 10^4\mu n$.
Take then any $X'\subseteq X$ and $C''\subseteq C$ with size at most $\mu n$ each. There must be some $j\in I$ for which $X'\cap X_j=\epsilonilonmptyset$ and $C''\cap C_j=\epsilonilonmptyset$. Let $\bar{C}=C_j\subseteq C\setminus (C'\cup C'')$ and $X''=X_j\subseteq X\setminus X'$. Then, as required, for each $c\in C'$ there is a $(\bar{C}\cup\{c\})$-rainbow $x,y$-path with interior vertices in $X''$.
\epsilonilonnd{proof}
\subsection{Distributive absorption with paths}
We now use Lemma~\ref{lem-randabsorbpath} and distributive absorption to get a larger scale absorption property, as follows.
\begin{lemma}\label{absorbBmacro} Let $1/n\ll \epsilonilonta \ll \mu\ll \epsilonilonps$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0$ is an $\epsilonilonps$-random subset of $V(K_{2n+1})$ and $C_0$ is an $\epsilonilonps$-random subset of $C(K_{2n+1})$, which is independent of $V_0$. Suppose $\epsilonilonll=\mu n/695$ and that $\epsilonilonll/3\in \mathbb{N}$. With high probability, the following holds.
Suppose that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})\setminus C_0$ is a set of at most $\epsilonilonta n$ colours. Then, there is a set $\hat{C}\subseteq C_0$ of $695\epsilonilonll-\alpha n$ colours such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C')$-rainbow, have length 695, and internal vertices in $V_0$.
\epsilonilonnd{lemma}
\begin{proof} By Lemma~\ref{lem-randabsorbpath} (applied to $X=V_0, C=C_0$ with $p=q=\epsilonilonps$ and $\mu'=3\mu$), we have the following property with high probability.
\stepcounter{propcounter}
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For each distinct $x,y\in V(K_{2n+1})$, and every $C'\subseteq C(K_{2n+1})$ with $|C'|=100$, and $X'\subseteq V_0$, $C''\subseteq C_0$ with $|X'|,|C''|\leq 3\mu n$, there is a set $\bar{C}\subseteq C_0\setminus (C'\cup C'')$ of 694 colours and a set $X''\subseteq V_0\setminus X'$ of at most $1500$ vertices with the following property.
For each $c\in C'$,
there is a $(\bar{C}\cup \{c\})$-rainbow $x,y$-path with length 695 and internal vertices in $X''$.\label{bbb2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Suppose then that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colors.
Let $h=\epsilonilonll/3$. By Lemma~\ref{Lemma_Chernoff}, with high probability
$|C_0|\geq \epsilonilonpsilon n/2> \mu n$. Therefore we can pick a set $\hat C_0 \subseteq C_0$ of $(3h-\alpha n)$ colours. Noting that $|\hat C_0|\geq 2h$ and $|\hat C_0\cup C|\leq 4h$.
By the same reasoning as just before \ref{Hmatch}, by Lemma~\ref{Lemma_H_graph} there is a bipartite graph, $H$ with maximum degree 100 say, with vertex classes $[3h]$ and $\hat C_0\cup C$ such that the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For each set $C'\subseteq C$ of $\alpha n$ colours, there is a perfect matching between $[3h]$ and $\hat C_0\cup C'$ in $H$.\label{Hmatch2}
\epsilonilonnd{enumerate}
Iteratively, for each $1\leq i\leq 3h$, let $D_i=N_H(i)$ and, using \ref{bbb2}, find sets $C_i\subseteq C_0\setminus (C_1\cup\ldots\cup C_{i-1})$ and $V_i\subseteq V_0\setminus (V_1\cup \ldots\cup V_{i-1})$,
with sizes $694$ and at most $1500$, such that the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For each $c\in D_i$, there is a $(C_i\cup\{c\})$-rainbow $x_i,y_i$-path with length 695 with internal vertices in $V_i$.\label{propabsorb2}
\epsilonilonnd{enumerate}
Note that in the $i$th application of \ref{bbb2}, we have $X'=V_1\cup \ldots\cup V_{i-1}$ and $C''=C_1\cup\ldots \cup C_{i-1}$, so that $|X'|\leq 1500\epsilonilonll\leq 3\mu n$ and $|C'|\leq 694\epsilonilonll\leq 3\mu n$, as required.
Let $\hat{C}=\hat C_0\cup C_1\cup \ldots\cup C_{3h}$. We claim $\hat{C}$ has the property required. Indeed, suppose $C'\subseteq C$ is a set of $\alpha n$
colours. Using~\ref{Hmatch2}, let $M$ be a perfect matching between $[3h]$ and $\hat{C}_0\cup C'$ in $H$, and label $\hat C_0\cup C'=\{c_1,\ldots,c_{3h}\}$ so that, for each $i\in [3h]$, $c_i$ is matched to $i$ in $M$.
By \ref{propabsorb2}, for each $i\in [3h]$, there is an $x_i,y_i$-path, $P_i$ say, with length 695 which is $(C_i\cup\{c\})$-rainbow with internal vertices in $V_i$. Then, $P_1,\ldots, P_{3h}$ are the paths required.
\epsilonilonnd{proof}
The following corollary of Lemma~\ref{absorbBmacro} will be convenient to apply.
\begin{corollary}\label{cor-absorbBmacro} Let $1/n\ll \epsilonilonta \ll \mu\ll \epsilonilonps$ and $1/k\ll 1$. Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0$ is an $\epsilonilonps$-random subset of $V(K_{2n+1})$ and $C_0$ is an $\epsilonilonps$-random subset of $C(K_{2n+1})$, which is independent of $V_0$. Suppose that $\epsilonilonll=\mu n/k$ and $695|k$. With high probability, the following holds.
Suppose that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colours. Then, there is a set $\hat{C}\subseteqeq C_0$ of $k\epsilonilonll-\alpha n$ colours such that, for every set $C'\subseteq C$ of $\alpha n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C')$-rainbow, have length k, and internal vertices in $V_0$.
\epsilonilonnd{corollary}
\begin{proof} Let $V_1,V_2\subseteq V_0$ be disjoint and $(\epsilonilonps/2)$-random and let $C_0'\subseteq C_0$ be $(\epsilonilonps/2)$-random. Let $\epsilonilonll'=\mu n/695=\epsilonilonll \cdot k/695$. We can assume that $3|\epsilonilonll'$, as discussed in Section~\ref{sec:not}.
By Lemma~\ref{absorbBmacro} (applied with $\epsilonilonps'=\epsilonilonps/2$, $\mu=\mu$, $\epsilonilonta=\epsilonilonta$, $C_0'=C_0'$ and $V_0'=V_1$) and Lemma~\ref{Lemma_Chernoff}, with high probability we have the following properties.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item
Let $\{x_1,\ldots,x_{\epsilonilonll'},y_1,\ldots,y_{\epsilonilonll'}\}\subseteq V(K_{2n+1})$, $\alpha'\leq \epsilonilonta$ and let $C'\subseteq C(K_{2n+1})$ be a set of at most $\epsilonilonta n$ vertices. Then, there is a set $\hat{C}$ of $695\epsilonilonll'-\alpha' n$ colours such that, for every set $C''\subseteq C$ of $\alpha' n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths which are collectively $(\hat{C}\cup C'')$-rainbow, have length 695, and internal vertices in $V_1$.\label{argh1}
\item $|V_2|\geq \mu n/4$.\label{argh2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Suppose therefore that $\{x_1,\ldots,x_{\epsilonilonll},y_1,\ldots,y_\epsilonilonll\}\subseteq V(K_{2n+1})$, $\alpha\leq \epsilonilonta$ and $C\subseteq C(K_{2n+1})$ is a set of at most $\epsilonilonta n$ colours. Let $k'=k/695\in \mathbb{N}$ and, let $\{z_{i,j}:i\in [\epsilonilonll],j\in [k']\}$ be a set of vertices in $V_2$, using \ref{argh2}. Apply \ref{argh1} to the pairs $(x_i,z_{i,1})$, $(z_{i,j},z_{i,j+1})$ and $(z_{i,k'},y_i)$, $i\in [\epsilonilonll]$, $1\leq j<k'$, and take their union, to get the required paths.
\epsilonilonnd{proof}
\subsection{Covering small colour sets with paths}
We now show that paths can be found using every colour in an arbitrary small set of colours.
\begin{lemma}\label{colourcoverB} Let $1/n\ll \xi \ll\beta$, let $1/n\ll 1/k\ll 1$ with $k= 3 \mod 4$.
Let $K_{2n+1}$ be 2-factorized. Suppose that $V_0\subseteq V(K_{2n+1})$ and $C_0\subseteq C(K_n)$ are $\beta$-random and independent of each other. With high probability, the following holds with $m=5\xi n/k$.
For any set $C\subseteq C(K_{n+1})\setminus C_0$ of at most $\xi n$ colours, and any set $\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus V_0$, there is a set of vertex-disjoint paths $x_i,y_i$-paths, $i\in [m]$, each with length $k$ and interior vertices in $V_0$, which are collectively $(C\cup C_0)$-rainbow and use all the colours in $C$.
\epsilonilonnd{lemma}
\begin{proof} Let $V_1,V_2\subseteq V_0$ be disjoint and $(\beta/2)$-random. Let $C_1,C_2\subseteq C_0$ be disjoint and $(\beta/2)$-random.
By Lemma~\ref{Lemma_Chernoff}, and Lemma~\ref{Lemma_number_of_colours_inside_random_set}, with high probability, the following properties hold.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $|C_1|\geq \beta n/3$.\label{beep1}
\item $V_1$ is $(\beta^2 n/4)$-replete. \label{beep1prime}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_few_connecting_paths} applied with $p=\beta /2$, $q=4\xi$, $V=V_2$ and $C=C_2$, with high probability, we have the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any $m'\leq 4\xi n$ and set $\{x_1, y_1, \dots, x_{m'}, y_{m'}\}\subseteq V(K_{2n+1})$
there is a collection $P_1, \dots, P_{m'}$ of vertex-disjoint paths with length $3$, with internal vertices in $V_2$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [m']$, and $P_1\cup \dots\cup P_{m'}$ is $C_2$-rainbow.\label{beep2}
\epsilonilonnd{enumerate}
We will now show that the property in the lemma holds. Let then $C\subseteq C(K_{n+1})\setminus C_0$ have at most $\xi n$ colours and let $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus V_0$. Take $\epsilonilonll=(k-3)/4$, and note that this is an integer and $\epsilonilonll m\geq \xi n$. Using \ref{beep1}, take an order $\epsilonilonll m$ set $C'\subseteq C\cup C_1$ with $C\subseteq C'$ and label $C'=\{c_{i,j}:i\in [m],j\in [\epsilonilonll]\}$.
Using \ref{beep1prime}, greedily find independent edges $s_{i,j}t_{i,j}$, $i\in [m],j\in [\epsilonilonll]$, with vertices in $V_1$, so that each edge $s_{i,j}t_{i,j}$ has colour $c_{i,j}$. Note that, when the edge $s_{i,j}t_{i,j}$ is chosen at most $2m\epsilonilonll\leq 2mk\leq 10\xi n$ vertices in $V_1$ are in already chosen edges.
Using \ref{beep2} with $m'=m(\epsilonilonll+1)$, find vertex-disjoint paths $P_{i,j}$, $i\in [m]$, $0\leq j\leq \epsilonilonll$, with length 3 and internal vertices in $V_2$ so that these paths are collectively $C_2$-rainbow and the following holds for each $i\in [m]$.
\begin{itemize}
\item $P_{i,0}$ is a $x_{i}s_{i,1}$-path.
\item For each $1\leq j<\epsilonilonll$, $P_{i,j}$ is a $t_{i,j},s_{i,j+1}$-path.
\item $P_{i,\epsilonilonll}$ is a $t_{i,\epsilonilonll},y_i$-path.
\epsilonilonnd{itemize}
Then, the paths $P_i=\cup_{0\leq j\leq \epsilonilonll}P_{i,j}$, $i\in [m]$, use each colour in $C'\subseteq C\cup C_1$, and hence $C$, and otherwise use colours in $C_2$, have interior vertices in $V_0$ and, for each $i\in [m]$, $P_i$ is a length $3(\epsilonilonll+1)+\epsilonilonll=k$ path from $x_i$ to $y_i$. That is, the paths $P_i$, $i\in[m]$, satisfy the condition in the lemma. \epsilonilonnd{proof}
\subsection{Almost-covering colour sets with paths}
We now show that paths can be found using almost every colour in a set of mostly-random colours.
\begin{lemma}\label{finishingB}
Let $1\geq p\gg q\gg \gamma,\epsilonilonta\gg 1/k \gg 1/n$ with $k=1\mod 3$, and let $m\leq 1.01q n/k$.
Let $K_{2n+1}$ be $2$-factorized.
Suppose we have disjoint sets $V, V_0\subseteqeq V(K_{2n+1})$, and $D,D_0\subseteqeq C(K_{2n+1})$ with $V$ $p$-random, $D$ $q$-random, and $V_0, D_0$ $\gamma$-random. Suppose further that $V_0$ and $D_0$ are independent of each other. Then, the following holds with high probability.
For any set $C\subseteq C(K_{n+1})$ with $D\cup D_0\subseteq C$ of $mk+\epsilonilonta n$ colours, and any set $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V\cup V_0)$, there is a set of vertex-disjoint paths $x_i,y_i$-paths with length $k$, $i\in [m]$, which have interior vertices in $V\cup V_0$ and which are collectively $C$-rainbow.
\epsilonilonnd{lemma}
\begin{proof} Let $\epsilonilonll=(k-4)/3$ and note that this is an integer. Let $m'=m+\epsilonilonta n/6k$. Using that $p\gg q,\epsilonilonta$, take in $V$ vertex disjoint sets $V',V_1,\ldots,V_{\epsilonilonll}$, such that $V'$ is $(p/2)$-random and, for each $i\in [\epsilonilonll]$, $V_i$ is $(m'/2n)$-random. Take an $(\epsilonilonta\gamma)$-random subset $D'_0\subseteq D_0$. Take in $D$ vertex disjoint sets $C_1,\ldots,C_{2\epsilonilonll}$ so that, and, for each $1\leq i\leq 2\epsilonilonll$, $C_i$ is $(m'/n)$-random.
Note that this later division is possible as $2\epsilonilonll\cdot m'/n\leq 2\epsilonilonll \cdot 1.02q n/k\leq q$.
By Lemma~\ref{Lemma_number_of_colours_inside_random_set}, with high probability the following holds.
\addtocounter{propcounter}{1}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item $V'$ is $(p^2 n/4)$-replete.\label{mice1}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_few_connecting_paths}, applied with $p'=\gamma\epsilonilonta$ and $q'=2q/k$ to $D'_0$ and a $(\gamma\epsilonilonta)$-random subset of $V_0$, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $\{x_1, y_1, \dots, x_{2m}, y_{2m}\}\subseteq V(K_{2n+1})$
there is a collection $P_1, \dots, P_{2m}$ of vertex-disjoint paths with length $3$, having internal vertices in $V_0$, where $P_i$ is an $x_i,y_i$-path, for each $i\in [2m]$, and $P_1\cup \dots\cup P_{2m}$ is $D'_0$-rainbow.\label{mice1a}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_Chernoff}, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item $|D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll}|\leq 2m'\epsilonilonll +\epsilonilonta n/2$. \label{mice1b}
\item For each $j\in [\epsilonilonll]$, $|V_j|\leq m'+\epsilonilonta^2 n/k^2$.\label{mice1bb}
\epsilonilonnd{enumerate}
By Lemma~\ref{Lemma_nearly_perfect_matching}, applied for each $i\in [2\epsilonilonll]$ and $j\in [\epsilonilonll]$ with $p'=m'/n$ and $\beta=\epsilonilonta^2/k^2$, with high probability the following holds.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{5}
\item For each $i\in [2\epsilonilonll]$ and $j\in [\epsilonilonll]$ and any vertex set $Y\subseteq V(K_{2n+1})\setminus V$ with $|Y|\leq m'$ there is a $C_i$-rainbow matching from $Y$ into $V_j$ with at least $|Y|-\epsilonilonta^2 n/k^2$ edges.\label{mice2}
\epsilonilonnd{enumerate}
We will show that the property in the lemma holds. Set then $C\subseteq C(K_{n+1})$ with $D\cup D_0\subseteq C$ so that $|C|=mk+\epsilonilonta n$ and let $X:=\{x_1,\ldots,x_m,y_1,\ldots,y_m\}\subseteq V(K_{2n+1})\setminus (V_0\cup V)$.
Note that, using \ref{mice1b},
\begin{align*}
|C\setminus (D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll})|&\geq mk+\epsilonilonta n-2m'\epsilonilonll-\epsilonilonta n/2=
(3\epsilonilonll+4)m+\epsilonilonta n/2-2m'\epsilonilonll\geq 3\epsilonilonll (m-m')+\epsilonilonta n/2+m'\epsilonilonll\\
&= -3\epsilonilonll \epsilonilonta n/6k+\epsilonilonta n/2+m'\epsilonilonll
\geq m'\epsilonilonll,
\epsilonilonnd{align*}
and take disjoint sets $C'_1,\ldots,C_\epsilonilonll'\subseteq C\setminus (D_0'\cup C_1\cup \ldots\cup C_{2\epsilonilonll})$ with size $m'$.
Let $C'=C'_1\cup\ldots \cup C'_\epsilonilonll$. Greedily, using~\ref{mice1}, take vertex disjoint edges $x_cy_c$, $c\in C'$, with vertices in $V'$ so that the edge $x_cy_c$ has colour $c$. Note that these edges have $2m'\epsilonilonll \leq qn$ vertices in total, so that this greedy selection is possible.
Let $M$ be the matching $\{x_cy_c:c\in C'\}$.
For each $i\in [\epsilonilonll]$, let $Z_i=\{x_c:c\in C'_i\}$ and $Y_i=\{y_c:c\in C'_i\}$.
For each $i\in [\epsilonilonll-1]$, use~\ref{mice2} to find a $C_{i}$-rainbow matching $M_i$ with $m'-\epsilonilonta^2 n/k^2$ edges from $Z_i$ into $V_i$, and a $C_{i+\epsilonilonll}$-rainbow matching, $M'_i$ say, with $m'-\epsilonilonta^2 n/k^2$ edges from $Y_i$ into $V_{i-1}$. Note that, by \ref{mice1bb} these matchings overlap in at least $m'-4\epsilonilonta^2 n/k^2$ vertices.
Therefore, putting together $M$ with the matchings $M_i$, $M_i'$, $i\in [\epsilonilonll]$ gives at least $m'-\epsilonilonll \cdot 4\epsilonilonta^2 n/k^2\geq m$ vertex disjoint paths with length $3\epsilonilonll-2$. Furthermore, these paths are collectively rainbow with colours in $(C\setminus D'_0)$. Take $m$ such paths, $Q_i$, $i\in [m]$. Apply \ref{mice1a}, to connect one endpoint of $Q_i$ to $x_i$ and another endpoint to $y_i$ using two paths of length 3 and new vertices and colours in $V_0$ and $D'_0$ respectively to get the paths with length $k=3\epsilonilonll+4$ as required.
\epsilonilonnd{proof}
\subsection{Proof of the finishing lemma in Case B}\label{subsec:finishB}
We can now prove Lemma~\ref{lem:finishB}.
\begin{proof}[Proof of Lemma~\ref{lem:finishB}]
Pick $\xi,\beta,\lambda,\alpha$ so that $1/k\ll \xi\ll\beta\ll\lambda\ll \alpha\ll \mu\ll \epsilonilonta\ll \epsilonilonps\ll p\leq 1$. Let $V_1',V_2',V_3'\subseteq V$ be disjoint sets which are each $(p/3)$-random. Let $W_1,W_2,W_3\subseteq V_0$ be disjoint sets which are each $(\mu/3)$-random.
Let $D_1,D_2,D_3\subseteq C_0$ be disjoint and $(\mu/3)$-, $\beta$- and $\alpha$-random respectively.
Set $m_1=(\epsilonilonps-5\xi-\lambda)n/k$, $m_2=5\xi n/k$ and $m_3=\lambda n/k$. By Lemma~\ref{Lemma_Chernoff}, with high probability we have that $|D_2|\leq 2\beta n$.
By Lemma~\ref{finishingB}, applied with $p'=p/3$, $q=(1-\epsilonilonta)\epsilonilonps$, $\gamma=\mu/3$, $\epsilonilonta'=\xi$, $n=n$, $k=k$, $m=m_1$, $V'=V_1'$, $V_0=W_1$, $D=C$, $D_0=D_1$, with high probability we have the following.
\stepcounter{propcounter}
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item For any set $\bar{C}\subseteq C(K_{n+1})$ with $C\cup D_1\subseteq \bar{C}$ of $m_1k+\xi n=(\epsilonilonps-4\xi-\lambda)n$ colours, and any collection of vertices $\{x_1,\ldots,x_{m_1},y_1,\ldots,y_{m_1}\}\subseteq V(K_{2n+1})\setminus (V_1'\cup W_1)$, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [m_1]$, each with length $k$, which have interior vertices in $V_1'\cup W_1$ and which are collectively $\bar{C}$-rainbow.
\label{mouseb11}
\epsilonilonnd{enumerate}
By Lemma~\ref{colourcoverB}, applied with $m=m_2$, $V_0\subseteq W_2$ a $\beta$-random subset and $C_0=D_2$, with high probability, we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{1}
\item For any set $\bar{C}\subseteq C(K_{n+1})\setminus D_2$ of at most $\xi n$ colours, and any set $\{x_1,\ldots,x_{m_2},y_1,\ldots,y_{m_2}\}\subseteq V(K_{2n+1})\setminus W_2$, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [{m_2}]$, each with length $k$ and interior vertices in $W_2$, which are collectively $(\bar{C}\cup D_2)$-rainbow and use all the colours in $\bar{C}$.\label{mouseb2}
\epsilonilonnd{enumerate}
By Corollary~\ref{cor-absorbBmacro}, applied with $\epsilonilonta'=2\beta$, $\mu'=\lambda$, $\epsilonilonps'=\alpha$, $n=n$, $k=k$, $\epsilonilonll=m_3$, $V_0\subseteq W_3$ an $\alpha$-random subset and $C_0=D_3$, with high probability we have the following.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{2}
\item For any $\{x_1,\ldots,x_{m_3},y_1,\ldots,y_{m_3}\}$, $\bar{\beta}\leq 2\beta$ and $\bar{C}\subseteq C(K_{2n+1})$ with $|\bar{C}|\leq 2\beta n$, there is a set $D_3'\subseteq D_3$ of $m_3k-\bar{\beta}n$ colours
such that, for every set $C'\subseteq \bar{C}$ of $\bar{\beta} n$ colours, there is a set of vertex-disjoint $x_i,y_i$-paths, $i\in [m_3]$, which are collectively $(D_3'\cup C')$-rainbow, have length $k$, and internal vertices in $W_3$.\label{mouseb3}
\epsilonilonnd{enumerate}
Let $m=\epsilonilonps n/k$, so that $m=m_1+m_2+m_3$. We will show that the property in the lemma holds.
Suppose then that $x_1,\ldots,x_m,y_1,\ldots,y_m$ are distinct vertices in $V(K_{2n+1})\setminus (V\cup V_0)$ and $D\subseteq C(K_{2n+1})$ so that $|D|=\epsilonilonps n$ and $C\cup C_0\subseteq D$. Let $V_1=V_1'\cup W_1$, $V_2=V_2'\cup W_2$ and $V_3=V_3'\cup W_3$.
Let $[m]=I_1\cup I_2\cup I_3$ be a partition with $|I_i|=m_i$ for each $i\in [3]$.
By \ref{mouseb3} (applied with $\bar C=D_2$ and $\bar{\beta}=(|D_2|/n)-4\xi$), there is a set $D_3'\subseteq D_3$ of $m_3k-|D_2|+4\xi n$ colours with the following property.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]\addtocounter{enumi}{3}
\item For every set $C'\subseteq D_2$ of $|D_2|-4\xi n$ colours, there is a set of vertex disjoint $x_i,y_i$-paths, $i\in I_3$, which are collectively $(D_3'\cup C')$-rainbow, have length $k$, and internal vertices in $W_3$.\label{mouseb4}
\epsilonilonnd{enumerate}
Let $C_1=D\setminus (D_2\cup D_3')$, $C_2=D_2$ and $C_3=D_3'$. We now reason analogously to the discussion after \ref{propp1}--\ref{propp3}.
Note that $C\cup D_1\subseteq C_1$ and
\begin{equation}
|C_1|=|D|-|D_2|-(m_3k-|D_2|+4\xi n)=mk-m_3k-4\xi n=(m_1+m_2)k-4\xi n=m_1k+\xi n.
\label{coldrain}
\epsilonilonnd{equation}
Therefore, using \ref{mouseb11}, we can find $m_1$ paths $\{P_1, \dots, P_{m_1}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_1$, so that these paths are vertex disjoint with internal vertices in $V_1=V_1'\cup W_1$ and are collectively $C_1$-rainbow.
Let $C'=C_1\setminus (\cup_{i\in X_1}C(P_i))$, so that, by \epsilonilonqref{coldrain}, $|C'|=\xi n$.
Using \ref{mouseb2}, we can find $m_2$ paths $\{P_1, \dots, P_{m_2}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_2$, so that these paths are vertex disjoint with internal vertices in $V_2=V_2'\cup W_2$ and are collectively $(C_2\cup C')$-rainbow and use every colour in $C'$.
Let $C''=C_2\setminus (\cup_{i\in X_2}C(P_i))$, and note that $|C''|=|C_2\cup C'|-m_2k=|D_2|+\xi n-m_2k=|D_2|-4\xi n$.
Using \ref{mouseb4} and that $C_3=D_3'$, we can find $m_3$ paths $\{P_1, \dots, P_{m_3}\}$ such that $P_i$ is an $x_i,y_i$-path with length $k$, for each $i\in I_3$, so that the paths are vertex disjoint with internal vertices in $V_3=V_3'\cup W_3$ which are collectively $(C_3\cup C'')$-rainbow.
Then, for each $i\in [\epsilonilonll]$, the path $P_i$ is an $x_i,y_i$-path with length $k$, so that all the paths are vertex disjoint with internal vertices in $V_1\cup V_2\cup V_3\subseteq V_0\cup V$ and which are collectively $D$-rainbow, as required.
\epsilonilonnd{proof}
\section{Randomized tree embedding}\label{sec:almost}
In this section, we prove Theorem~\ref{nearembedagain}. We start by formalising what we mean by a random rainbow embedding of a tree.
\begin{definition}
For a probability space $\Omega$, tree $T$ and a coloured graph $G$, a randomized rainbow embedding of $T$ into $G$ is
a triple $\phi=(V_\phi, C_\phi, T_\phi)$ consisting of a random set of vertices $V_\phi:\Omega\to V(G)$, a random set of colours $C_\phi:\Omega\to C(G)$, and a random subgraph $T_\phi:\Omega\to E(G)$ such that:
\begin{itemize}
\item $V(T_\phi)\subseteqeq V_\phi$ and $C(T_\phi)\subseteqeq C_\phi$ always hold.
\item With high probability, $T_\phi$ is a rainbow copy of $T$.
\epsilonilonnd{itemize}
\epsilonilonnd{definition}
We will embed a tree bit by bit, starting with a randomized rainbow embedding of a small tree and then extending it gradually. For this, we need a concept of one randomized embedding \epsilonilonmph{extending} another.
\begin{definition}\label{Definition_embedding_extension}
Let $G$ be a graph and $T_1\subseteqeq T_2$ two nested trees. Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, S_{\phi_1})$ and $\phi_2=(V_{\phi_2}, C_{\phi_2}, S_{\phi_2})$ be randomized rainbow embeddings of $T_1$ and $T_2$ respectively into $G$. We say that $\phi_2$ extends $\phi_1$ if
\begin{itemize}
\item $T_{\phi_1}\subseteqeq T_{\phi_2}$, $C_{\phi_1}\subseteqeq C_{\phi_2}$, and $V_{\phi_1}\subseteqeq V_{\phi_2}$ always hold.
\item $V(T_{\phi_2})\setminus V(T_{\phi_1})\subseteq V_{\phi_2}\setminus V_{\phi_1}$ and $C(T_{\phi_2})\setminus C(T_{\phi_1})\subseteq C_{\phi_2}\setminus C_{\phi_1}$ always hold.
\epsilonilonnd{itemize}
\epsilonilonnd{definition}
The above definition implicitly assumes that the two randomized embeddings are defined on the same probability space, which is the case in our lemmas, except for Lemma~\ref{Lemma_extending_with_large_stars}. When $\phi_1$ and $\phi_2$ are defined on different probability spaces $\Omega_{\phi_1}$ and $\Omega_{\phi_2}$ respectively, we use the following definition. We say that an extension of $\phi_1$ is a measure preserving transformation $f:\Omega_{\phi_2}\to \Omega_{\phi_1}$ (i.e.\ $\mathbb{P}(f^{-1}(A))=\mathbb{P}(A)$ for any $A\subseteqeq \Omega_{\phi_1}$). We say that $\phi_2$ extends $\phi_1$ if ``for every $\omega\in \Omega_{\phi_2}$ we have $T_{\phi_1}( f(\omega))\subseteqeq T_{\phi_2}$, $C_{\phi_1}( f(\omega))\subseteqeq C_{\phi_2}$, $V_{\phi_1}( f(\omega))\subseteqeq V_{\phi_2}$, and $V(T_{\phi_2})\setminus V(T_{\phi_1}( f(\omega)))$ is
contained in $V_{\phi_2}\setminus V_{\phi_1}( f(\omega))$ and $C(T_{\phi_2})\setminus C(T_{\phi_1}(f(\omega)))$ is contained in
$C_{\phi_2}\setminus C_{\phi_1}( f(\omega))$''. Such a measure preserving transformation ensures that $\phi_1'=(T_{\phi_1}\circ f, C_{\phi_1}\circ f, V_{\phi_1}\circ f)$ is a randomized embedding of $T_1$ defined on $\Omega_{\phi_2}$ which is equivalent to $\phi_1$ (i.e.\ the probability of any outcomes $\phi_1$ and $\phi_1'$ are the same). It also ensures that $\phi_2$ is an extension of $\phi_1'$ as in Definition~\ref{Definition_embedding_extension}.
The following lemma extends randomized embeddings of trees by adding a large star forest. Recall that a
$p$-random subset of some finite set is formed by choosing every element of it independently
with probability $p$.
\begin{lemma}[Extending with a large star forest]\label{Lemma_extending_with_large_stars}
Let $p\gg \beta\gg \gamma \gg d^{-1}, n^{-1}$ and $\log^{-1} n\gg d^{-1}$.
Let $T_1\subseteqeq T_2$ be forests such that $T_2$ is formed by adding stars with $\geq d$ leaves to vertices of $T_1$. Let $K_{2n+1}$ be $2$-factorized and suppose that $|T_2|=(1-p)n$.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$ where $V_{\phi_1}$ and $C_{\phi_1}$ are both $\gamma$-random.
Then, $\phi_1$ can be extended into a randomized rainbow embedding $\phi_2=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2$ so that $V_{\phi_2}$ and $C_{\phi_2}$ are $(1-p+\beta)$-random sets (with $V_{\phi_2}$ and $C_{\phi_2}$ allowed to depend on each other).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ such that $\beta\gg \alpha \gg \gamma $. Let $\theta =|T_1|/n$ and $F=T_2\setminus T_1$, noting that $F$ is a star forest with degrees $\geq d$ and $e(F)=(1-p-\theta)n$. Let $I\subseteqeq V(T_1)$ be the vertices to which stars are added to get $T_2$ from $T_1$.
Let $\Omega$ be the probability space for $\phi_1$.
We call $\omega\in \Omega$ \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and if we have $|V_{\phi_1}(\omega)|, |C_{\phi_1}(\omega)|\leq 4\gamma n$.
Using Chernoff's bound and the fact that $\phi_1$ is a randomized embedding of $T_1$ we have that with high probability a random $\omega \in \Omega$ is succesful.
For every successful $\omega$, let $J^{\omega}$ be the copy of $I$ in $T_{\phi_1}(\omega)$.
We can apply Lemma~\ref{Lemma_randomized_star_forest} with $F=F$, $J=J^{\omega}$,
$V=V(K_{2n+1})\setminus V_{\phi_1}(\omega)$, $C=C(K_{2n+1})\setminus C_{\phi_1}(\omega)$, $p'=p+\theta, \alpha=\alpha, \gamma'=4\gamma, d=d$ and $n=n$. This gives a probability space $\Omega^\omega$, and a randomized subgraph $F^{\omega}$, and randomized sets
$U^\omega\subseteqeq V(K_{2n+1})\setminus (V_{\phi_1}(\omega)\cup V(F^{\omega})),$ $D^{\omega}\subseteqeq C(K_{2n+1})\setminus (C_{\phi_1}(\omega)\cup C(F^{\omega}))$ (so $F^{\omega}$, $U^\omega$, $D^{\omega}$ are functions from $\Omega^{\omega}$ to the families of subgraphs/subsets of vertices/sets of colours of $K_{2n+1}$ respectively). From Lemma~\ref{Lemma_randomized_star_forest} we know that, for each $\omega$, $F^{\omega}$ is with high probability a copy of $F$, and that $U^{\omega}$ and $D^{\omega}$ are $(1-\alpha)(p+\theta)$-random subsets of $V(K_{2n+1})\setminus V_{\phi_1}(\omega)$ and $C(K_{2n+1})\setminus C_{\phi_1}(\omega)$ respectively. Setting $T^{\omega}=F^{\omega}\cup T_{\phi_1}(\omega)$ gives a subgraph which is a copy of $T_2$ with high probability (for successful $\omega$).
For every unsuccessful $\omega$, set $T^{\omega}=T_{\phi_1}(\omega)$, and choose $U^\omega\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}(\omega),$ $D^{\omega}\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}(\omega)$ to be independent $(1-\alpha)(p+\theta)$-random subsets (in this case letting $\Omega^{\omega}$ be an arbitrary probability space on which such $U^{\omega}, D^{\omega}$ are defined).
Let $\Omega_2=\{(\omega, \omega'): \omega\in \Omega, \omega'\in \Omega^{\omega}\}$ and set $\mathbb{P}((\omega, \omega'))=\mathbb{P}(\omega)\mathbb{P}(\omega')$. Notice that $\Omega_2$ is a probability space.
Let $T_{\phi_2}$ be a random subgraph formed by choosing $(\omega, \omega')\in \Omega_2$, and setting $T_{\phi_2}= T^\omega(\omega')$. Similarly define $U=U^{\omega}(\omega')$ and $D=D^{\omega}(\omega')$. Notice that $V\setminus V_{\phi_1}$ is $(1-\gamma)$-random and that $U|V_{\phi_1}$ is a $(1-\alpha)(p+\theta)$-random subset of $V\setminus V_{\phi_1}$. By Lemma~\ref{Lemma_mixture_of_p_random_sets}, $U$ is a $(1-\alpha)(1-\gamma)(p+\theta)$-random subset of $V(K_{2n+1})$. Similarly, $D$ is a $(1-\alpha)(1-\gamma)(p+\theta)$-random subset of $C(K_{2n+1})$. Since $\beta\gg \alpha,\gamma$, we have $(1-\alpha)(1-\gamma)(p+\theta)\geq p-\beta$, and therefore can choose $(p-\beta)$-random subsets $U'$ and $D'$ such that $U'\subseteqeq U$ and $D'\subseteqeq D$.
Now $V_{\phi_2}:=V(K_{2n+1})\setminus U'$ and $C_{\phi_2}=C(K_{2n+1})\setminus D'$ are $(1-p+\beta)$-random sets of vertices/colours.
We also have that with high probability $T_{\phi_2}$ is a copy of $T_2$ since
$$\mathbb{P}(\text{$T_{\phi_2}(\omega, \omega')$ is not a copy of $T_2$})\leq \mathbb{P}(\text{$\omega$ unsuccessful})+\mathbb{P}(\text{$\omega$ successful and $T_{\omega}$ not a copy of $T_2$}).$$
Thus the required extension of $\phi_1$ is $\phi_{2}=(T_{\phi_2}, V_{\phi_2}, C_{\phi_2})$ together with the measure preserving transformation $f:\Omega_2\to \Omega$ with $f:(\omega, \omega')\to\omega$.
\epsilonilonnd{proof}
The following lemma extends randomized embeddings of trees by adding connecting paths.
\begin{lemma}[Extending with connecting paths]\label{Lemma_extending_with_connecting_paths}
Let $p\gg q\gg n^{-1}$.
Let $T_1\subseteqeq T_2$ be forests such that $T_2$ is formed by adding $qn$ paths of length $3$ connecting different components of $T_1$. Let $K_{2n+1}$ be $2$-factorized.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$ and let $U\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $p$-random, independent subsets. Let $V_{\phi_2}=V_{\phi_1}\cup U$ and $C_{\phi_2}=C_{\phi_1}\cup D$.
Then, $\phi_1$ can be extended into a randomized rainbow embedding $\phi_2=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2$.
\epsilonilonnd{lemma}
\begin{proof}
Let $\Omega$ be the probability space for $\phi_1$.
We say that $\omega\in \Omega$ is \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and the conclusion of Lemma~\ref{Lemma_few_connecting_paths} holds with $V=U$, $C=D$, $p=p$, $q=q$ and $n=n$. As $\phi_1$ is a randomized embedding of $T_1$, and by Lemma~\ref{Lemma_few_connecting_paths}, we have that, with high probability, $\omega$ is successful.
For each successful $\omega$, let $x_1^{\omega}, y_1^{\omega}, \dots, x_{qn}^{\omega}, y_{qn}^{\omega}$ be the vertices of $T_{\phi_1}(\omega)$ which need to be joined by paths of length $3$ to get a copy of $T_2$. From the conclusion of Lemma~\ref{Lemma_few_connecting_paths}, for each successful $\omega$, we can find paths $P_i^\omega$, $i\in [qn]$, so that the paths are vertex disjoint and collectively $D$-rainbow, and each path $P_i^\omega$ is an $x_i,y_i$-path with length 3 and internal vertices in $U$.
Letting $T_{\phi_2}=T_{\phi_1}(\omega)\cup P_1^{\omega}\cup \dots\cup P_{qn}^{\omega}$, $(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ gives the required randomized embedding of $T_2$.
\epsilonilonnd{proof}
The following lemma extends randomized embeddings of trees by adding a sequence of matchings of leaves.
\begin{lemma}[Extending with matchings]\label{Lemma_extending_with_matchings}
Let $\epsilonilonll^{-1},p, q\gg n^{-1}$.
Let $T_1\subseteqeq T_{\epsilonilonll}$ be forests such that $T_{\epsilonilonll}$ is formed by adding a sequence of $\epsilonilonll$ matchings of leaves to $T_1$. Let $K_{2n+1}$ be $2$-factorized. Suppose $|T_{\epsilonilonll}|-|T_1|\leq pn$.
Let $\phi_1=(V_{\phi_1}, C_{\phi_1}, T_{\phi_1})$ be a randomized rainbow embedding of $T_1$ into $K_{2n+1}$. Let $U_{ind}\subseteqeq V(K_{2n+1})\setminus V_{\phi_1}$, $D_{ind}\subseteqeq C(K_{2n+1})\setminus C_{\phi_1}$ be $q$-random, independent subsets. Let $U_{dep}\subseteqeq V(K_{2n+1})\setminus (V_{\phi_1}\cup U_{ind})$ be $p/2$-random, and $D_{dep}\subseteqeq C(K_{2n+1})\setminus(C_{\phi_1}\cup D_{ind})$ $p$-random (possibly depending on each other).
Then $\phi_1$ can be extended into a randomized rainbow embedding $\phi_{\epsilonilonll}=(V_{\phi_{\epsilonilonll}}, C_{\phi_{\epsilonilonll}}, T_{\phi_{\epsilonilonll}})$ of $T_{\epsilonilonll}$ into $V_{\phi_{\epsilonilonll}}=V_{\phi_1}\cup U_{ind}\cup U_{dep}$ and $C_{\phi_{\epsilonilonll}}=C_{\phi_1}\cup D_{ind}\cup D_{dep}$.
\epsilonilonnd{lemma}
\begin{proof}
Without loss of generality, suppose that $|T_{\epsilonilonll}|-|T_1|= pn$.
Define forests $T_1, \dots, T_{\epsilonilonll}$, and $p_1, \dots, p_{\epsilonilonll}\in [0,1]$ such that each $T_{i+1}$ is constructed from $T_i$ by adding a matching of $p_in$ leaves.
Randomly partition $U_{dep}=\{U_{dep}^1, \dots, U_{dep}^{\epsilonilonll}\}$ and $D_{dep}=\{D_{dep}^1, \dots, D_{dep}^{\epsilonilonll}\}$ so that each set $U_{dep}^i\subseteqeq V(K_{2n+1})$ is $p_i/2$-random and each set $D_{dep}^i\subseteqeq C(K_{2n+1})$ is $p_i$-random.
Randomly partition $U_{ind}=\{U_{ind}^1, \dots, U_{ind}^{\epsilonilonll}\}, D_{ind}=\{D_{ind}^1, \dots, D_{ind}^{\epsilonilonll}\}$ so that each set $U_{ind}^i$ and $D_{ind}^i$ is $(q/\epsilonilonll)$-random.
Let $\Omega$ be the probability space for $\phi_1$.
We say that $\omega\in \Omega$ is \epsilonilonmph{successful} if $T_{\phi_1}(\omega)$ is a copy of $T_1$ and, for each $i\in [\epsilonilonll]$, the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds for $U_{dep}^{i}$, $U_{ind}^{i}$, $D_{dep}^{i}$, $D_{ind}^{i}$, $p=p_i$, $\gamma = q/\epsilonilonll$ and $n=n$.
As $\phi_1$ is a randomized embedding of $T_1$, and by Lemma~\ref{Lemma_sat_matching_random_embedding}, we have that, with high probability, $\omega$ is successful. Note that, for this, we take a union bound over $\epsilonilonll$ events, using that the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds
with probability $1-o(n^{-1})$ in each application.
For each successful $\omega$, define a sequence of trees $T_0^{\omega},T_1^{\omega}, \dots, T_{\epsilonilonll}^{\omega}$. Fix $T_0^{\omega}=T_{\phi_1}(\omega)$ and recursively construct $T_{i+1}^{\omega}$ from $T_i^{\omega}$ by adding an appropriate $(D_{dep}^i\cup D_{ind}^i)$-rainbow size $p_in$ matching into $U_{dep}^i\cup U_{ind}^i$ to make $T_i^\omega$ into a copy of $T_{i+1}$ (which exists as the conclusion of Lemma~\ref{Lemma_sat_matching_random_embedding} holds because $\omega$ is successful).
Letting $T_{\phi_{\epsilonilonll}}(\omega)= T_{\epsilonilonll}^{\omega}$ gives the required randomized embedding of $T_{\epsilonilonll}$.
\epsilonilonnd{proof}
The following lemma gives a randomized embedding of any small tree.
\begin{lemma} [Small trees with a replete subset]\label{Lemma_embedding_small_tree}
Let $q\gg \gamma\geq \nu \gg \xi \gg n^{-1}$.
Let $T$ be a tree with $|T|= \gamma n$ containing a set $U\subseteq V(T)$ with $|U|= \nu n$. Let $K_{2n+1}$ be $2$-factorized with $V_\phi, C_\phi$ $q$-random sets of vertices and colours respectively.
Then, there is a randomized rainbow embedding $\phi=(T_\phi, V_\phi, C_\phi)$ along with independent $q/2$-random sets $V_0\subseteqeq V_{\phi}\setminus V(T_{\phi})$, $C_0\subseteqeq C_{\phi}\setminus C(T_{\phi})$ with $(V_0, U_\phi)$ $\xi n$-replete with high probability (where $U_\phi$ is the copy of $U$ in $T_\phi$).
\epsilonilonnd{lemma}
\begin{proof}
Choose $\alpha$ such that $q\gg \alpha\gg \gamma, \nu$.
Inside $V_{\phi}$ choose disjoint sets $V_0, V_1, V_2$ which are $q/2$-random, $\alpha$-random, and $100\nu/q$-random respectively.
Inside $C_{\phi}$ choose disjoint sets $C_0, C_1, C_2$ which are $q/2$-random, $q/4$-random, and $q/4$-random respectively. Notice that the total number of edges of any particular color is $2n+1$
and the probability that any such edge is going from $V_2$ to $V_1 \cup V_2$ is
$2(100\nu/q)(100\nu/q+\alpha)$. Therefore by Lemmas~\ref{Lemma_high_degree_into_random_set} and \ref{Lemma_Azuma} the following hold with high probability.
\begin{enumerate}[label = {\bfseries \Alph{propcounter}\arabic{enumi}}]
\item \label{eq_repletetree_1} Every vertex $v$ has $|N_{C_1}(v)\cap V_1|\geq \alpha q n/4\geq 3|T|$ and $|N_{C_2}(v)\cap V_2|\geq (100\nu/q)\cdot qn/4\geq 4|U|$.
\item \label{eq_repletetree_3} Every colour has at most $400\alpha (\nu/q) n\leq 0.1|U|$ edges from $V_2$ to $V_1\cup V_2$.
\epsilonilonnd{enumerate}
Using \ref{eq_repletetree_1}, we can greedily find a rainbow copy $T_\phi$ of $T$ in $V_1\cup V_2\cup C_1\cup C_2$ with $U$ embedded into $V_2$. Indeed, embed one vertex at a time, always embedding a vertex which has one neighbour into preceding vertices. Moreover, put a vertex from $T\setminus U$ into $V_1$ using an edge of colour from $C_1$ and put a vertex from $U$ into $V_2$ using an edge of colour from $C_2$.
Let $U_\phi$ be the copy of $U$ in $T_\phi$. By~\ref{eq_repletetree_3}, and as $|U_\phi|=\nu n$, $(U_\phi, V(K_{2n+1})\setminus (V_1\cup V_2))$ is $0.9\nu n$-replete. Indeed, for every color there are at least $|U_\phi|$ edges of this color incident to $U_\phi$ and at most $0.1|U_\phi|$ of them going to $V_1 \cup V_2$.
Consider a set $V_0'\subseteqeq V(K_{2n+1})$ which is $\big(\frac{1}{1-\alpha-100\nu/q}\big)\cdot (q/2)$-random and independent of $V_1, V_2, C_0, C_1, C_2$. Notice that $V_0'\setminus (V_1\cup V_2)$ is $q/2$-random. Thus the joint distributions of the families of random sets $\{V_0'\setminus (V_1\cup V_2), V_1, V_2, C_0, C_1, C_2\}$ and $\{V_0, V_1, V_2, C_0, C_1, C_2\}$ are exactly the same. By Lemma~\ref{Lemma_inheritence_of_lower_boundedness_random} applied with $A=U_{\phi}, B=V(K_{2n+1})\setminus (V_1\cup V_2), V=V_0'$, the pair $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ is $\xi$-replete with high probability. Since $(U_{\phi}, V_0'\setminus (V_1\cup V_2))$ has the same distribution as $(U_{\phi}, V_0)$, the latter pair is also $\xi$-replete with high probability.
\epsilonilonnd{proof}
The following is the main result of this section. It finds a rainbow embedding of every nearly-spanning tree. Given the preceding lemmas, the proof is quite simple. First, we decompose the tree into star forests, matchings of leaves, and connecting paths using Lemma~\ref{Lemma_tree_splitting}. Then, we embed each of these parts using the preceding lemmas in this section.
\begin{proof}[Proof of Theorem~\ref{nearembedagain}]
Choose $\beta$ and $d$ such that $1\geq \epsilonilonps\gg \epsilonilonta\gg \beta \gg \mu\gg d^{-1}\gg \xi$ with $k^{-1}\gg d^{-1}$.
By Lemma~\ref{Lemma_tree_splitting}, there are forests $T_1^{\mathrm{small}}\subseteqeq T_2^{\mathrm{stars}}\subseteqeq T_3^{\mathrm{match}}\subseteqeq T_4^{\mathrm{paths}}\subseteqeq T_5^{\mathrm{match}}=T'$ satisfying \ref{rush1} -- \ref{rush5} with $d=d, n=(1-\epsilonilonps)n$ and $U=U$. Fix $U'=U\cap T_1^{\mathrm{small}}$.
Fix $p'=1-|T_2^{\mathrm{stars}}|/n=(|T'|-|T_2^{\mathrm{stars}}|)/n+\epsilonilonps$. We claim that $p'\geq (p+\epsilonilonps)/2$. To see this, let $T''$ be the tree with $|T''|=pn$ formed by deleting leaves of vertices with $\geq k$ leaves in $T'$. Let $A\subseteq V(T_1^{\mathrm{small}})$ be the vertices to which leaves need added to get $T_2^{\mathrm{stars}}$ from $T_1^{\mathrm{small}}$, and, for each $v\in A$, let $d_v\geq d$ be the number of such leaves that are added to $v$. Then we have that $\sum_{v\in A} d_v=|T_2^{\mathrm{stars}}|-|T_1^{\mathrm{small}}|$.
Let $d'_v$ be the number of these leaves added to $v$ which are not themselves leaves in $T'$.
Note that, for each $v\in A$, at least $d_v'$ neighbours of $v$ lie in $V(T')\setminus V(T_2^{\mathrm{stars}})$. Thus, we have
$\sum_{v\in A}d_v'\leq |T'|-|T_2^{\mathrm{stars}}|$. Moreover, for each $v\in A$ with $d_v-d'_v \geq k$ we deleted $d_v-d'_v$ leaves attached to $v$ when we formed $T''$ from $T'$, otherwise we deleted nothing. In both cases we deleted at least $d_v-d'_v-k$ leaves for each $v \in A$.
Therefore, using that $|A| \leq|T_1^{\mathrm{small}}|\leq n/d$, we have
$$|T'|-|T''| \geq \sum_{v\in A}(d_v-d_v'-k)=|T_2^{\mathrm{stars}}|-|T_1^{\mathrm{small}}|-\sum_{v\in A}(d_v'+k) \geq |T_2^{\mathrm{stars}}|-(k+1)n/d-\sum_{v\in A}d_v'\,.$$
Thus,
$$
pn= |T''| \leq |T'|-|T_2^{\mathrm{stars}}|+(k+1)n/d+\sum_{v\in A}d_v'\leq 2(|T'|-|T_2^{\mathrm{stars}}|)+\epsilonilonps n= 2p'n-\epsilonilonps n,
$$
implying, $p'\geq (p+\epsilonilonps)/2$.
For each $i\in \{2, \dots, 5\}$ (and the label $*$ meaning ``$\mathrm{small},\, \mathrm{stars},\, \mathrm{match}$ or $\mathrm{paths}$" appropriately), let $p_i=(|T_i^*|-|T_{i-1}^*|)/n$. Set $p_6=\epsilonilonpsilon-\beta.$
These constants represent the proportion of colours which are used for embedding each of the subtrees (with $p_6$ representing the proportion of colours left over for the set $C$ in the statement of the lemma).
Notice that $p'=p_3+p_4+p_5+p_6+\beta$.
For each $i\in [6]$, choose disjoint, independent $2\mu$-random sets $V^{i}_{ind}, C_{ind}^i$. Do the following.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_embedding_small_tree} to $T_1^{\mathrm{small}}$, $U=U'$, $V_\phi=V_{1}^{ind}$, $C_\phi=C_{1}^{ind}$, $q=2\mu$, $\gamma=|T_1^{small}|/n$, $\nu=|U'|/n$ and $\xi=\xi$.
This gives a randomized rainbow embedding $\phi_1=(T_{\phi_1}, V_{1}^{ind}, C_{1}^{ind})$ of $T_1^{small}$ and $\mu$-random, independent sets $V_0\subseteqeq V_{1}^{ind}\setminus V(T_{\phi_1})$, $C_0\subseteqeq C_{1}^{ind}\setminus C(T_{\phi_1})$ with $(U'_{\phi_1},V_0)$ $\xi n$-replete with high probability (where $U'_{\phi_1}$ is the copy of $U'$ in $T_{\phi_1}$).
\epsilonilonnd{itemize}
Notice that $\phi_1'=(V_1^{ind}\cup \dots \cup V_6^{ind}, C_1^{ind}\cup \dots\cup C_6^{ind}, T_{\phi_1} )$ is also a randomized embedding of $T_1^{\mathrm{small}}$.
Then, do the following.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_extending_with_large_stars} with $T_1=T_1^{\mathrm{small}}$, $T_2=T_2^{\mathrm{stars}}$, $\phi_1=\phi_1'$, $p=p', \beta=\beta, \gamma=12\mu, d=d, n=n$. This gives a randomized embedding $\phi_{2}=(V_{\phi_2}, C_{\phi_2}, T_{\phi_2})$ of $T_2^{\mathrm{stars}}$ extending $\phi_{1}'$, with $V_{dep}=V(K_{2n+1})\setminus V_{\phi_2}$, $C_{dep}=C(K_{2n+1})\setminus C_{\phi_2}$ $(p'-\beta)$-random sets of vertices/colours.
\epsilonilonnd{itemize}
Notice that $\phi_2'=(V_{\phi_2}\setminus (V_3^{ind}\cup \dots \cup V_6^{ind}), C_{\phi_2}\setminus (C_3^{ind}\cup \dots \cup C_6^{ind}), T_{\phi_2})$ is also a randomized embedding of $T_2^{\mathrm{stars}}$. Indeed, recall that by our definition of extension we embed vertices of $T_2^{\mathrm{stars}}\setminus T_1^{small}$ outside of the set $V_1^{ind}\cup \dots \cup V_6^{ind}$ using the edges whose colors are also outside of $C_1^{ind}\cup \dots\cup C_6^{ind}$. Moreover, by our constriction, $V(T_{\phi_1})$ is disjoint from
$V_2^{ind}\cup \dots \cup V_6^{ind}$ and $C(T_{\phi_1})$ is disjoint from
$C_2^{ind}\cup \dots\cup C_6^{ind}$.
Using that $p_3+p_4+p_5+p_6 = p'-\beta$ and Lemma~\ref{Lemma_mixture_of_p_random_sets}, randomly partition $C_{dep}= C_{dep}^3\cup C_{dep}^4\cup C_{dep}^5\cup C_{dep}^6$ with the sets $C_{dep}^i$ $p_i$-random subsets of $C(K_{2n+1})$.
Using that $p'-\beta= (p_3+p_4+p_5+p_6)/2 + (p'-\beta)/2\geq p_3/2+p_4/2+p_5/2+ (p+\epsilonilonps)/6$, inside $V_{dep}$ we can choose disjoint sets $V_{dep}^3, V_{dep}^4, V_{dep}^5, V_{dep}^6$ which are $\rho$-random for $\rho=p_3/2, p_4/2, p_5/2, (p+\epsilonilonps)/6$ respectively. We remark that some of the random sets ($V_{ind}^2, C_{ind}^2, V_{ind}^6, C_{ind}^6, V_{dep}^4,$ and $C_{dep}^4$) will not actually used in the embedding. They are only there because it is notationally convenient to allocate equal amounts of vertices/colours for the different parts of the embedding.
\begin{itemize}
\item Apply Lemma~\ref{Lemma_extending_with_matchings} with $T_{\epsilonilonll}=T_3^{\mathrm{match}}, T_1=T_{2}^{\mathrm{stars}}$, $V_{dep}=V_{dep}^3$, $C_{dep}=C_{dep}^3$, $V_{ind}=V_{ind}^3$, $C_{ind}=C_{ind}^3$, $\phi_1=\phi_2'$, $\epsilonilonll=d, p=p_3, q=2\mu, n=n$. This gives we get a randomized embedding $\phi_3=(V_{\phi_3}, C_{\phi_3}, T_{\phi_3})$ of $T_3^{\mathrm{match}}$ into $V^3_{ind}\cup V^3_{dep}\cup V_{\phi_2'}$, $V^3_{dep}\cup V^3_{ind}\cup C_{\phi_2'}$ extending $\phi_2$.
\item Apply Lemma~\ref{Lemma_extending_with_connecting_paths} with $T_2=T_4^{\mathrm{paths}}, T_1=T_{3}^{\mathrm{match}}$, $V_{ind}=V_{ind}^4$, $C_{ind}=C_{ind}^4$, $\phi_1=\phi_3$, $p=2\mu, q=(2k)^{-1}, n=n$. This gives a randomized embedding $\phi_4=(V_{\phi_4}, C_{\phi_4}, T_{\phi_4})$ of $T_4^{\mathrm{paths}}$ into $V^4_{ind}\cup V_{\phi_3}, C^4_{ind}\cup C_{\phi_3}$ extending $\phi_3$.
\item By Lemma~\ref{Lemma_extending_with_matchings} with $T_{\epsilonilonll}=T_5^{\mathrm{match}}, T_1=T_{4}^{\mathrm{paths}}$, $V_{dep}=V_{dep}^5$, $C_{dep}=C_{dep}^5$, $V_{ind}=V_{ind}^5$, $C_{ind}=C_{ind}^5$, $\phi_1=\phi_4$, $\epsilonilonll=d, p=p_5, q=2\mu, n=n$, we get a randomized embedding $\phi_5=(V_{\phi_5}, C_{\phi_5}, T_{\phi_5})$ of $T_5^{\mathrm{match}}$ into $V^5_{ind}\cup V^5_{dep}\cup V_{\phi_4}$, $C^5_{dep}\cup C^5_{ind}\cup V_{\phi_4}$ extending $\phi_4$.
\epsilonilonnd{itemize}
Now the lemma holds with $\hat T'=T_{\phi_5}$, $V_0=V_0$, $C_0=C_0$ (recall that these sets are from the application of Lemma~\ref{Lemma_embedding_small_tree} above), $V=V_{dep}^6$, and $C\subseteqeq C_{dep}^6$ a $(1-\epsilonilonta)\epsilonilonps$-random subset.
\epsilonilonnd{proof}
\section{The embedding in Case C}\label{sec:lastC}\label{sec:caseC}
For our embedding in Case C, we distinguish between those trees with one very high degree vertex, and those trees without. The first case is covered by Theorem~\ref{Theorem_one_large_vertex}. In this section, we prove Lemma~\ref{Lemma_small_tree}, which covers the second case and thus completes the proof of Theorem~\ref{Theorem_case_C}.
For trees with no very high degree vertex, we start with the following lemma which embeds a small tree in the $ND$-colouring in a controlled fashion. In particular, we wish to embed a prescribed small portion of the tree into an interval in the cyclic ordering of $K_{2n+1}$ so that the distance between any two consecutive vertices in the image of the embedding is small.
\begin{lemma}[Embedding a small tree into prescribed intervals]\label{Lemma_small_tree}
Let $n\geq 10^5$.
Let $K_{2n+1}$ be $ND$-coloured, and let $T$ be a tree with at most $n/100$ vertices whose vertices are partitioned as $V(T)=V_0\cup V_1\cup V_2$ with $|V_1|, |V_2|\leq 2n/\log^4 n$.
Let $I_0, I_1, I_2$ be disjoint intervals in $V(K_{2n+1})$ with $|I_0|\geq 7|V_0|$, $|I_1|\geq 8|V_1|\log^3 n $ and $|I_2|\geq 8|V_2|\log^3 n $.
Then, there is a rainbow copy $S$ of $T$ in $K_{2n+1}$ such that, for each $0\leq i\leq 2$, the image of $V_i$ is contained in $I_i$, and, furthermore, any consecutive pair of vertices of $S$ in $I_1$, or in $I_2$, are within distance $16\log^3 n$ of each other.
\epsilonilonnd{lemma}
\begin{proof}
Let $k\in \{\lfloor 2\log n\rfloor,\lfloor 2\log n\rfloor-1\}$ be odd.
Choose consecutive subintervals $I_1^{1}, \dots, I_1^{|V_1|}$ in $I_1$ of length $k^3$, and choose consecutive subintervals $I_2^{1}, \dots, I_2^{|V_2|}$ in $I_2$ of length $k^3$.
Partition $C(K_{2n+1})$ as $C_0\cup C_1^{1}\cup \dots\cup C_1^{k}\cup C_2^{1}\cup \dots\cup C_2^{k}$ so that, for each $m\in [n]$ and $i\in [k]$,
\[
m\in \left\{\begin{array}{ll}
C_0 & \text{ if }m\text{ is odd},\\
C_1^i & \text{ if }m\text{ is even and }m\epsilonilonquiv i\mod{2k+1},\\
C_2^i & \text{ if }m\text{ is even and }m\epsilonilonquiv i+k\mod{2k+1}.
\epsilonilonnd{array}
\right.
\]
Note that, for every $p\in [2]$, $i\in [k]$, $j\in [|V_p|]$, and $v\notin I_p^j$, since $C_p^i$ contains only even colours and $k$ is odd, $v$ has at least $\lfloor |I_p^j|/2(2k+1)\rfloor\geq k$ colour-$C_p^i$ neighbours in $I_p^j$ (we call a vertex $u$ a colour-$C_p^i$ neighbour of $v$ if the colour of the edge $vu$ belongs to $C_p^i$).
Order $V(T)$ arbitrarily as $v_1, \dots, v_{m}$ such that $\{v_1, \dots, v_i\}$ forms a subtree of $T$ for each $i\in [m]$.
We embed the vertices $v_1, \dots, v_{m}$ one by one, in $m$ steps, so that, in step $i$, we embed $v_i$ to some $u_i\in V(K_{2n+1})$. Since $T[v_1, \dots, v_i]$ is a tree, for each $i$, there is a unique $f(i)<i$ for which $v_{f(i)}v_i\in E(T)$. We will maintain that the edges $u_iu_{f(i)}$ have different colours for each $i$.
At step $i$ (the step in which $u_1, \dots, u_{i-1}$ have already been chosen and we are choosing $u_i$) we say that a vertex $x$ is \epsilonilonmph{free} if $x\not\in\{u_1, \dots, u_{i-1}\}$. We say that an interval $I_s^t$ is \epsilonilonmph{free}, if it contains no vertices from $\{u_1, \dots, u_{i-1}\}$. We say that a colour is \epsilonilonmph{free} if it does not occur on $\{u_{f(j)}u_j:2\leq j\leq i-1\}$. The procedure to choose $u_i$ at step $i$ is as follows.
\begin{itemize}
\item If $v_i\in V_0$, then $u_i$ is embedded to any free vertex in $I_0$ so that, if $i\geq 2$, $u_{f(i)}u_i$ is any free colour in $C_0$.
\item For $p\in [2]$, if $v_i\in V_p$, then $u_i$ is embedded into any free interval $I_p^i$ so that, if $i\geq 2$, $u_{f(i)}u_i$ uses any free colour in $C_p^j$, for $j$ as small as possible.
\epsilonilonnd{itemize}
The lemma follows if we can embed all the vertices in this way. Indeed, as we have exactly the right number of intervals to embed one vertex per interval, consecutive vertices in $I_1$ (or $I_2$) are in consecutive intervals $I_1^i$, and hence at most $16\log^3n$ apart. Since $C_0$ contains all the odd colours,
vertex $u_{f(i)}$ has $\geq \lfloor |I_0|/2\rfloor-1 > 3|V_0|$ colour-$C_0$ neighbours in $I_0$. At each step at most $|V_0|$ of the vertices in $I_0$ are occupied, and at most $|V_0|$ colours are used. Since the $ND$-colouring is locally $2$-bounded,
this forbids at most $3|V_0|$ vertices. Therefore, there is always room to embed each $v_i \in V_0$ into $I_0$.
To finish the proof of the lemma, it is sufficient to show that, throughout the process, for each $p\in [2]$, there will always be sets $C_p^j$ which are never used.
\begin{claim}
Let $p\in [2]$, $i\in [m]$ and $s\in [k]$.
At step $i$, if there are $\geq |V_p|/2^{s}$ free intervals $I_p^j$, then no colour from $C_p^s$ has been used up to step $i$.
\epsilonilonnd{claim}
\begin{proof}
The proof is by induction on $i$. The initial case $i=0$ is trivial.
Suppose it is true for steps $1,\ldots,i-1$ and suppose that there are still at least $|V_p|/2^s$ free intervals $I_p^j$. Now, by the induction hypothesis for $p'=p$, $i'=i-1$ and $s'=s-1$, colours from $C_p^{s-1}$ were only used when there were fewer than $|V_p|/2^{s-1}$ free intervals $I_p^j$, and hence at most $|V_p|/2^{s-1}$ vertices were embedded using colours from $C_p^{s-1}$. In each free interval, $u_{f(i)}$ has at least $k$ colour-$C_p^{s-1}$ neighbours.
and hence at least $k|V_p|/2^s$ colour-$C_p^{s-1}$ neighbours in the free intervals in total. As the $ND$-colouring is locally 2-bounded, this gives at least
$k|V_p|/2^{s+1}> |V_p|/2^{s-1}$ colour-$C_p^{s-1}$ neighbours adjacent to $u_{f(i)}$ by edges of different colours. Therefore $u_{f(i)}$ has some colour $C_p^{s-1}$-neighbour $u_i$ in a free interval so that $u_{f(i)}u_i$ is a free colour. Thus, no colour from $C_p^s$ is used in step $i$.
\epsilonilonnd{proof}
Taking $s=k$, we see that colours from $C_1^k$ and $C_2^k$ never get used while there is at least one free interval. As there are exactly as many intervals as vertices we seek to embed, the colours from $C_1^k$ and $C_2^k$ are never used, and hence the procedure successfully embeds $T$.
\epsilonilonnd{proof}
\begin{figure}[h]
\begin{center}
{
\begin{tikzpicture}[scale=0.7,define rgb/.code={\definecolor{mycolor}{rgb}{#1}},
rgb color/.style={define rgb={#1},mycolor}]]
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\draw [thick,rgb color={0,0,1}, rotate = {4.9315068493*-35}] (0:4) to [in={4.9315068493*25+180},out=180] ({4.9315068493 *25}:4);
\draw [thick,rgb color={0.166666666666667,0,1}, rotate = {4.9315068493*-36}] (0:4) to [in={4.9315068493*26+180},out=180] ({4.9315068493 *26}:4);
\draw [thick,rgb color={0.333333333333333,0,1}, rotate = {4.9315068493*-37}] (0:4) to [in={4.9315068493*27+180},out=180] ({4.9315068493 *27}:4);
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\draw [thick,rgb color={0.666666666666667,0,1}, rotate = {4.9315068493*-39}] (0:4) to [in={4.9315068493*29+180},out=180] ({4.9315068493 *29}:4);
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\draw [thick,rgb color={1,0,0.833333333333333}, rotate = {4.9315068493*-42}] (0:4) to [in={4.9315068493*32+180},out=180] ({4.9315068493 *32}:4);
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\draw [thick,rgb color={1,0,0.5}, rotate = {4.9315068493*-44}] (0:4) to [in={4.9315068493*34+180},out=180] ({4.9315068493 *34}:4);
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\draw [thick,rgb color={1,0,0.166666666666667}, rotate = {4.9315068493*-46}] (0:4) to [in={4.9315068493*36+180},out=180] ({4.9315068493 *36}:4);
\foreach \x in {0,...,72}
{
\draw [fill=black] (4.9315068493*\x:4) circle [radius=0.05cm];
}
\def4.25{4.25}
\def4.5{4.5}
\def4.5{4.5}
\def4.9{4.9}
\def5.1{5.1}
\def5.6{5.6}
\def5.35{5.35}
\def5.7{5.7}
\def6.25{6.25}
\def\small{\small}
\draw [rotate=4.9315068493*-4] (0:4.25) -- (0:4.5);
\draw [rotate=4.9315068493*0] (0:4.25) -- (0:4.5);
\draw [rotate=4.9315068493*-4] (4.5,0) arc (0:4.9315068493*4:4.5);
\draw (4.9315068493*-2:4.9) node {\small $V_0$};
\draw [rotate=4.9315068493*-4,black] (0:5.1) -- (0:5.6);
\draw [rotate=4.9315068493*-4,black] (0:6.25) node {\small $0.9n$};
\draw [rotate=4.9315068493*3,black] (0:5.1) -- (0:5.6);
\draw [rotate=4.9315068493*3,black] (0:6.25) node {\small $0.83n$};
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\draw [black] (4.9315068493*-0.5:5.7) node {\small $I_0$};
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\draw [rotate=4.9315068493*9] (4.5,0) arc (0:4.9315068493*2:4.5);
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\draw [rotate=4.9315068493*8,black] (5.35,0) arc (0:4.9315068493*4:5.35);
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\draw [rotate=4.9315068493*24,black] (0:5.1) -- (0:5.6);
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\epsilonilonnd{tikzpicture}
}
\epsilonilonnd{center}
\caption{Embedding the tree in Case C.}\label{Figure_many_vertex}
\epsilonilonnd{figure}
Now we are ready to prove Case C. Letting $T'$ be a tree $T$ in Case C with the neighbouring leaves of vertices with $\geq \log^4 n$ leaves removed, we embed $T'$ using the above lemma.
Then, much like in the proof Theorem~\ref{Theorem_one_large_vertex}, leaves are added to high degree vertices one at a time. In order to make sure we use all the colours, these leaves are located in particular intervals of the ND-colouring. In the proof, for technical reasons, we have three different kinds of high degree vertices (called type 1, type 2 and type 3 vertices), with vertices of a particular type having its neighbours chosen by a particular rule. (We note that we may have no type 2 vertices.) We embed the vertices of different types into 3 disjoint intervals.
\begin{proof}[Proof of Theorem~\ref{Theorem_case_C}]
See Figure~\ref{Figure_many_vertex} for an illustration of this proof. If $d_i\geq 2n/3$ for some $i$, then the result follows from Theorem~\ref{Theorem_one_large_vertex}.
Assume then that $v_1, \dots, v_{\epsilonilonll}$ are ordered so that $2n/3\geq d_1\geq \dots\geq d_{\epsilonilonll}$. Note that $d_1+\dots+d_{\epsilonilonll}= n+1-|T'|\geq 99n/100+1$. Choose the smallest $m$ with $0.05n< d_1+ \dots+ d_m$. From minimality, we have $d_1+\dots+ d_m\leq d_1+0.05n$. Let $V_1=\{v_1, \dots ,v_m\}$, $V_2=\{v_{m+1}, \dots, v_{\epsilonilonll}\}$, and $V_0=V(T')\setminus (V_1\cup V_2)$.
Let $I_0=[0.83n, 0.9n]$, $I_1=[0.7n, 0.71n]$, and $I_2=[0.91n, 0.92n]$. Then, it is easy to check that $|I_0| \geq 7|V_0|$, $|I_1|\geq 8|V_1|\log^3 n$ and $|I_2|\geq 8|V_2|\log^3 n$.
By Lemma~\ref{Lemma_small_tree}, there is a rainbow embedding of $T'$ in $I_0\cup I_1\cup I_2$ with $V_i$ embedded to $I_i$, for each $0 \leq i\leq 2$, and consecutive embedded vertices in $I_1$, and in $I_2$, within distance $16\log^3 n$ of each other. Say the copy of $T'$ from this is $S'$.
For odd $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m},u_{m-1},u_{m-3},\ldots,u_2.
\]
For even $m$, relabel vertices in $V_1$ so that, if $u_i$ is the image of $v_i$ under the embedding, then, for each $i\in [m]$, the vertices in the image of $V_1$ appear in $I_1$ in the following order:
\[
u_1,u_3,u_5,\ldots,u_{m-1},u_{m},u_{m-2},\ldots,u_2.
\]
Relabel the vertices in $V_2$ so that, if $u_i$ is the image of $v_i$ under the embedding, then for each $i\in [\epsilonilonll]\setminus [m]$, the vertices in the image of $V_2$ appear in $I_2$ in the order $u_{m+1},\ldots,u_\epsilonilonll$. Finally, relabel the integers $d_i$, $i\in [\epsilonilonll]$, to match this relabelling of the vertices $v_i$, $i\in [\epsilonilonll]$. We uses this new labelling to embed neighbours of $u_i, i \leq \epsilonilonll$ into three disjoint intervals (that are also disjoint from $V(S')$): $[1,u_1)$ if $i\leq m$ is odd, $(u_2,0.82n]$ if $i\leq m$ is even and $[0.96n,1.92n]$ if $m < i \leq \epsilonilonll$.
For each $1\leq i\leq m$, if $i$ is odd we say that $u_i$ is \epsilonilonmph{type 1}, and if $i$ is even we say that $u_i$ is \epsilonilonmph{type 2}. For each $m<i\leq \epsilonilonll$, we say that $u_i$ is \epsilonilonmph{type 3}. Note that, if $m=1$, then there are no type 2 vertices.
Furthermore, note that, if $m=1$, then $V(S')\subseteq \{u_1\}\cup I_0\cup I_2\subseteq \{u_1\}\cup [0.83n,0.92n]$, while, if $m>1$, then $V(S')\subseteq [u_1,u_2]\cup I_0\cup I_2\subseteq [u_1,u_2]\cup [0.83n,0.92n]$.
Let $C(K_{2n+1})\setminus C(S')=C_1\cup \ldots \cup C_\epsilonilonll$ so that, for each $i\in [\epsilonilonll]$, $|C_i|=d_i$, and, if $i<j$, then all the colours in $C_i$ are smaller than all the colours in $C_j$. Note that this is possible as $|C(K_{2n+1})\setminus C(S')|=n-|E(T')|=d_1+\ldots+d_\epsilonilonll$.
Futhermore, note that, if $i,j\in [\epsilonilonll]$ with $i\leq j-2$, then $\max(C_i)\leq \min (C_j)-d_{i+1}\leq \min(C_j)-\log^4 n$.
For each $i\in [\epsilonilonll]$, do the following.
\begin{itemize}
\item If $u_i$ is a type 1 vertex, embed the vertices in $U_i:=\{u_i-c:c\in C_{i}\}$ as neighbours of $u_i$.
\item If $u_i$ is a type 2 or 3 vertex, embed the vertices in $U_i:=\{u_i+c:c\in C_{i}\}$ as neighbours of $u_i$.
\epsilonilonnd{itemize}
Notice that, from the definition of the ND-colouring, we use edges with colour in $C_i$ to attach the neighbours of $u_i$, for each $i\in [\epsilonilonll]$. Therefore, by design, the graph formed is rainbow. We need to show that the sets $U_i$, $i\in [\epsilonilonll]$, are disjoint from each each other and from $V(S')$, so that we have a copy of $T$.
\begin{claim} If $m=1$, then $\max(C_m)\leq 0.68n$, while, if $m>1$, then $\max(C_m)\leq 0.11n$.
\epsilonilonnd{claim}
\begin{proof}
If $m=1$, then, as at most $0.01n$ colours are used to embed $T'$ and $d_1\leq 2n/3$, we have $\max(C_m)\leq 0.67n+0.01n=0.68n$.
Suppose then that $m>1$, and hence $d_1<0.05n$. By the minimality in the choice of $m$, we have that $d_1+\ldots+d_m\leq 0.05n+0.05n=0.1n$. Therefore, $\max(C_m)\leq 0.1n+0.01n= 0.11n$.
\epsilonilonnd{proof}
From the claim, as the vertices $u_1,\ldots,u_m$ are in $[0.7n,0.71n]$ and $\max(C_i) \leq \max (C_m)$, for each $i\in [m]$, for each type 1 vertex $u_{i}$, we have $U_{i}\subseteq [u_{i}-\max (C_i),u_i-\min(C_i)]\subseteq [0.02n,0.71n] \subseteq [1,0.71n]$. For each type 2 vertex $u_{i}$, we have $U_{i}\subseteq [u_{i}+\min (C_i),u_{i}+\max(C_i)]\subseteq [0.7n,0.82n]$.
Now, for each type 3 vertex $u_{i}$, we have, as $i>m$, that $\min(C_i)\geq 0.05n$. Therefore, as $\max(C_i)\leq n$, we have $U_{i}\subseteq [u_{i}+\min (C_i),u_{i}+\max(C_i)]\subseteq [0.96n,1.92n]$. Furthermore, by the ordering of $u_{m+1},\ldots,u_\epsilonilonll$, for each $m+1\leq i<j\leq \epsilonilonll$, we have $u_i<u_j$ and $\max(C_i)<\min(C_j)$ so that
$u_{i}+\max (C_i)< u_{j}+\min (C_j)$. Thus, the sets $U_i$, $m+1\leq i\leq \epsilonilonll$ are disjoint sets in $[0.96n,1.92n]$.
\begin{claim}
If $u_{i}$ is a type 1 vertex with $i\leq m-2$, then $\min(U_i)< \max (U_{i+2})$.
If $u_{i}$ is a type 2 vertex with $i\leq m-2$, then $\max(U_i) < \min (U_{i+2})$.
\epsilonilonnd{claim}
\begin{proof}
If $u_i$ is a type 1 vertex with $i\leq m-2$, then $u_{i+2}$ is also a type 1 vertex, which appears consecutively with $u_i$ in the interval $I_1$ in the embedding of $T'$, and hence $u_{i+2}-u_i\leq 16 \log^3 n<\log^4 n$. Therefore,
\[
\min(U_i)=u_{i}-\max (C_i)\geq u_i-\min(C_{i+2})+\log^4 n > u_{i+2}-\min(C_{i+2})=\max(U_{i+2}).
\]
On the other hand, if $u_i$ is a type 2 vertex with $i\leq m-2$, then $u_{i+2}$ is also a type 2 vertex, which appears consecutively with $u_i$ in the interval $I_1$ in the embedding of $T'$, and hence $u_{i}-u_{i+2}\leq 16 \log^3 n <\log^4 n$. Therefore,
\[
\max(U_i)=u_{i}+\max (C_i)\leq u_i+\min(C_{i+2})-\log^4 n < u_{i+2}+\min(C_{i+2})=\min(U_{i+2}).\qedhere
\]
\epsilonilonnd{proof}
By the claim, the sets $U_i$, $i\leq m$ and $i$ is odd, are disjoint and lie in the interval $[1,\max(U_1)]\subseteq [1,u_1)$. If $m\geq 2$, then, by the claim again, the sets $U_i$, $i\leq m$ and $i$ is even, are disjoint and lie in the interval $[\min(U_2),0.82n]\subseteq (u_2,0.82n]$.
In summary, we have shown the following.
\begin{itemize}
\item The sets $U_i$, $i\leq m$ and $i$ is odd are disjoint sets in $[1,u_1)$.
\item If $m\geq 2 $, then the sets $U_i$, $i\leq m$ and $i$ is even, are disjoint sets in $(u_2,0.82n]$.
\item The sets $U_i$, $m<i\leq \epsilonilonll$, are disjoint sets in $[0.96n,1.92n]$.
\item If $m=1$, then $V(S')\subseteq \{u_1\}\cup [0.83n,0.92n]$, while, if $m\geq 2$, then $V(S')\subseteq [u_1,u_2]\cup [0.83n,0.92n]$.
\epsilonilonnd{itemize}
Thus, both when $m=1$ and when $m\geq 2$, we have that $U_i$, $i\in [\epsilonilonll]$, are disjoint sets in $[2n+1]\setminus V(S')$, as required.
\epsilonilonnd{proof}
\section{Concluding remarks}\label{sec:conc}
Many powerful techniques have been developed to decompose graphs into bounded degree graphs~\cite{bottcher2016approximate, messuti2016packing, ferber2017packing, kim2016blow,joos2016optimal}.
On the other hand, all these techniques encounter some barrier when dealing with trees with arbitrarily large degrees.
Having overcome this ``bounded degree barrier'' for Ringel's Conjecture, we hope that our ideas might be useful for other problems as well. Here we mention two such questions.
The closest relative of Ringel's conjecture is the following conjecture on graceful labellings, which is also mentioned in the introduction. This is a very natural problem to apply our techniques to.
\begin{conjecture}[K\"otzig-Ringel-Rosa, \cite{rosa1966certain}]\label{Conjecture_Graceful}
The vertices of every $n$ vertex tree $T$ can be labelled by the numbers $1, \dots, n$, such that the differences $|u-v|$, $uv\in E(T)$, are distinct.
\epsilonilonnd{conjecture}
\noindent
This conjecture was proved for many isolated classes of trees like caterpillars, trees with $\leq 4$ leaves, firecrackers, diameter $\leq 5$ trees, symmetrical trees, trees with $\leq 35$ vertices, and olive trees (see Chapter 2 of \cite{gallian2009dynamic} and the references therein).
Conjecture~\ref{Conjecture_Graceful} is also known to hold asymptotically for trees of maximum degree at most $n/\log n$ \cite{adamaszek2016almost} but solving it for general trees, even asymptotically, is already wide open.
Another very interesting related problem is the G\'yarf\'as Tree Packing Conjecture. This also concerns decomposing a complete graph into specified trees, but the trees are allowed to be different from each other.
\begin{conjecture}[Gy\'arf\'as, \cite{gyarfas1978packing}]\label{Conjecture_Gyarfas}
Let $T_1, \dots, T_{n}$ be trees with $|T_i|=i$ for each $i\in [n]$. The edges of $K_n$ can be decomposed into $n$ trees which are isomorphic to $T_1, \dots, T_{n}$ respectively.
\epsilonilonnd{conjecture}
\noindent
This conjecture has been proved for bounded degree trees by Joos, Kim, K{\"u}hn and Osthus~\cite{joos2016optimal} but in general it is wide open.
It would be interesting to see if any of our techniques can be used here to make further progress.
\noindent
{\bf Acknowledgements.} Parts of this work were carried out when the first two authors visited the Institute for Mathematical Research (FIM) of ETH Zurich. We would like to thank FIM for its hospitality and for creating a stimulating research environment.
\epsilonilonnd{document} |
\begin{document}
\title[Notes on the codimension one conjecture]{Notes on the codimension one conjecture in the operator corona theorem}
\author{Maria F. Gamal'}
\address{
St. Petersburg Branch\\ V. A. Steklov Institute
of Mathematics\\
Russian Academy of Sciences\\ Fontanka 27, St. Petersburg\\
191023, Russia
}
\email{gamal@pdmi.ras.ru}
\thanks{Partially supported by RFBR grant No. 14-01-00748-a}
\subjclass[2010]{ Primary 30H80; Secondary 47A45, 47B20.}
\keywords{Operator corona theorem, contraction, similarity to an isometry}
\begin{abstract}
Answering on the question of S.R.Treil \cite{23}, for every $\delta$, $0<\delta<1$,
examples of contractions are constructed such that their characteristic functions
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ satisfy the conditions $$\|F(z)x\|\geq\delta\|x\| \ \text{ and } \
\dim\mathcal E_\ast\ominus F(z)\mathcal E =1 \ \text{ for every } \ z\in\mathbb D, \ \ x\in\mathcal E,$$
but $F$ are not
left invertible. Also, it is shown that the condition
$$\sup_{z\in\mathbb D}\|I-F(z)^\ast F(z)\|_{\frak S_1}<\infty,$$
where $\frak S_1$ is the trace class of operators, which is sufficient for the
left invertibility
of the operator-valued function $F$ satisfying the estimate $\|F(z)x\|\geq\delta\|x\|$
for every $z\in\mathbb D$, $x\in\mathcal E$, with some $\delta>0$ (S.R.Treil, \cite{22}), is necessary
for the left invertibility of an inner function $F$ such that $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$
for some $z\in\mathbb D$.
\end{abstract}
\maketitle
\section{Introduction}
Let $\mathbb D$ be the open unit disk, let $\mathbb T$ be the unit circle,
and let $\mathcal E$ and $\mathcal E_\ast$ be separable Hilbert spaces.
The space $H^\infty(\mathcal E\to\mathcal E_\ast)$ is the space of bounded analytic functions on $\mathbb D$
whose values are (linear, bounded) operators acting from $\mathcal E$ to $\mathcal E_\ast$.
If $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$, then $F$ has nontangential boundary values
$F(\zeta)$ for a.e. $\zeta\in\mathbb T$ with respect to the Lebesgue measure $m$ on $\mathbb T$.
Every function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ has the inner-outer
factorization, that is, there exist an auxiliary Hilbert space $\mathcal D$ and two functions
$\Theta\in H^\infty(\mathcal E\to\mathcal D)$ and $\Omega\in H^\infty(\mathcal D\to\mathcal E_\ast)$ such that
$F=\Theta\Omega$, $\Theta$ is inner, that is, $\Theta(\zeta)$ is an isometry for a.e. $\zeta\in\mathbb T$,
and $\Omega$ is outer. The definition of outer function is not recalled here,
but it need to mentioned that
for an outer function $\Omega$, $\operatorname{clos}\Omega(z)\mathcal D=\mathcal E_\ast$
for all $z\in\mathbb D$ and for a.e. $z\in\mathbb T$. Recall that a function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is called $\ast$-inner
($\ast$-outer), if the function $F_\ast\in H^\infty(\mathcal E_\ast\to\mathcal E)$,
$F_\ast(z)=F^\ast(\overline z)$, $z\in\mathbb D$, is inner (outer) (see {\cite[Ch.V]{16}}, also {\cite[\S A.3.11.5]{14}}).
Every analytic operator-valued function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ such that $\|F\|\leq 1$ can be represented as an orthogonal sum of a unitary constant and a purely contractive function $F_0$, that is,
there exist representations of Hilbert spaces $\mathcal E=\mathcal E'\oplus\mathcal E_0$ and
$\mathcal E_\ast=\mathcal E_\ast'\oplus\mathcal E_{\ast 0}$ and a unitary operator
$W\colon\mathcal E'\to\mathcal E_\ast'$ such that $F(z)\mathcal E'\subset\mathcal E_\ast'$,
$F(z)|_{\mathcal E'}=W$, $F(z)\mathcal E_0\subset\mathcal E_{\ast 0}$, $F_0(z)=F(z)|_{\mathcal E_0}$ for every
$z\in\mathbb D$, and $\|F_0(0)x\|<\|x\|$ for every $x\in\mathcal E_0$, $x\neq 0$
({\cite[Proposition V.2.1]{16}}). In all questions considered in this note it can be supposed that $\|F\|\leq 1$ and $F$ is purely contractive.
The Operator Corona Problem is to find necessary and sufficient condition
for a function
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ to be left invertible, that is, to exist a function
$G\in H^\infty(\mathcal E_\ast\to\mathcal E)$ such that $G(z)F(z)=I_{\mathcal E}$ for all $z\in\mathbb D$.
If $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is left invertible, then there exists $\delta>0$
such that
\begin{equation}\label{1.1}\|F(z)x\|\geq \delta \|x\| \ \ \ \ \text{ for all } \ x\in\mathcal E, \ \ z\in\mathbb D. \end{equation}
It is easy to see that if $F$ satisfies \eqref{1.1} and $F=\Theta\Omega$ is the inner-outer factorization of $F$,
then the outer function $\Omega$ is invertible, and the inner function $\Theta$ satisfies \eqref{1.1}
(may be with another $\delta$).
The condition \eqref{1.1} is sufficient for left invertibility, if $\dim\mathcal E<\infty$ \cite{19}, but
is not sufficient in general \cite{20}, \cite{21}. Also, \eqref{1.1} is not sufficient under additional assumption
\begin{equation}\label{1.2}\dim\mathcal E_\ast\ominus F(z)\mathcal E=1 \ \ \text{ for all } \ z\in\mathbb D. \end{equation}
In \cite{23}, for every $\delta$, $0<\delta<1/3$, two functions $F_1$, $F_2\in H^\infty(\mathcal E\to\mathcal E_\ast)$
are constructed such that
\begin{equation}\label{1.3} \|x\|\geq\|F_k(z)x\|\geq \delta \|x\| \ \ \ \ \text{ for all } \ x\in\mathcal E, \ \ z\in\mathbb D,\end{equation}
\eqref{1.2} is fulfilled for $F_k$, $k=1,2$, $$F_1(\zeta)\mathcal E=\mathcal E_\ast \ \text{ for a.e. }\ \zeta\in\mathbb T,
\ \ \ \
\dim\mathcal E_\ast\ominus F_2(\zeta)\mathcal E=1 \ \text{ for a.e. }\ \zeta\in\mathbb T,$$
but $F_1$ and $F_2$ are not
left invertible. It is mentioned in \cite{23} that the method from \cite{20}, \cite{21} gives examples of such functions for
$\delta<1/\sqrt 2$, and a question was posed if for every $\delta$, $0<\delta<1$, there exists
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ such that $F$ satisfies \eqref{1.2} and \eqref{1.3}, and $F$ is not
left invertible.
In this note, it is shown that such function $F$ exists for every $0<\delta<1$, and
any from the following cases can be realized:
\begin{equation}\label{1.4}F(\zeta)\mathcal E=\mathcal E_\ast \ \ \text{ for a.e. } \ \zeta\in\mathbb T, \end{equation}
or
\begin{equation}\label{1.5}\dim\mathcal E_\ast\ominus F(\zeta)\mathcal E=1 \ \ \text{ for a.e. } \ \zeta\in\mathbb T, \end{equation}
or
\begin{equation}\label{1.6}\dim\mathcal E_\ast\ominus F(\zeta)\mathcal E=1 \ \ \text{ for a.e. } \ \zeta\in E, \ \text{ and }
\ \
F(\zeta)\mathcal E=\mathcal E_\ast \ \ \text{ for a.e. } \zeta\in\mathbb T\setminus E, \end{equation}
where
$E\subset\mathbb T$ is a closed set satisfying the Carleson condition with $0<m(E)<1$ (see Sec. 5
of this note where the definition is recalled).
Actually, not operator-valued functions, but contractions are constructed, and the required
functions are the characteristic functions of these contractions {\cite[Ch. VI]{16}}, see Sec. 3 of this note.
In \cite{22}, some sufficient conditions are given,
which imply the left invertibility of functions, in particular,
it is proved in \cite{22}, that if $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$
satisfies to \eqref{1.1} and
\begin{equation}\label{1.7}\sup_{z\in\mathbb D}\|I_{\mathcal E}-F(z)^\ast F(z)\|_{\frak S_1}<\infty,
\end{equation}
where $\frak S_1$ is the trace class of operators, then $F$ is left invertible.
In this note, it is shown that the condition \eqref{1.7} is necessary for left invertibility of $F$, if $F$ is inner
and $\dim\mathcal E_\ast\ominus F(z)\mathcal E<\infty$ for some $z\in\mathbb D$. Actually, an appropriate fact
is proved for contractions with such characteristic functions, and the statement on function follows from
the fact for contractions.
The function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ has a left scalar multiple if there exist
$G\in H^\infty(\mathcal E_\ast\to\mathcal E)$ and a function $\rho\in H^\infty$, where $H^\infty$ is the algebra of all
bounded analytic functions on $\mathbb D$, such that
$\rho(z)I_{\mathcal E}= G(z)F(z)$ for all $z\in\mathbb D$.
The left invertibility of $F$ means that $F$ has a scalar multiple, which is invertible in $H^\infty$.
Functions $F_1$ and $F_2$ from \cite{23} mentioned above do not have scalar multiple,
the proof is actually the same as the proof that $F_1$ and $F_2$ are not left invertible.
In this note, it is shown that the existence of the left scalar multiple of $F$ with \eqref{1.1} is not sufficient
for the left invertibility of $F$, even if $F$ is inner and $F$ satisfies \eqref{1.2}. In this case,
$I_{\mathcal E}-F(z)^\ast F(z)\in\frak S_1$ for every $z\in\mathbb D$, but
$\sup_{z\in\mathbb D}\|I_{\mathcal E}-F(z)^\ast F(z)\|_{\frak S_1}=\infty$. Again,
an appropriate
contraction is constructed, and $F$ is the characteristic function of this contraction.
We shall use the following notation: $\mathbb D$ is the open unit disk,
$\mathbb T$ is the unit circle, $m$ is the normalized Lebesgue measure on $\mathbb T$,
and $H^2$ is
the Hardy space in $\mathbb D$.
For a positive integer $n$, $1\leq n<\infty$, $H^2_n$ and $L^2_n$
are orthogonal sums of $n$ copies of spaces $H^2$ and $L^2 =L^2(\mathbb T,m)$,
respectively.
The unilateral shift $S_n$ and the bilateral shift $U_n$ of multiplicity $n$
are the operators of multiplication by the independent variable in spaces
$H^2_n$ and $L^2_n$, respectively. For a Borel set $\sigma\subset\mathbb T$, by
$U(\sigma)$ we denote the operator of multiplication by the
independent variable on the space $L^2(\sigma,m)$ of functions from $L^2$
that are equal to zero a.e. on $\mathbb T\setminus\sigma$.
For a Hilbert space $\mathcal H$, by $I_{\mathcal H}$ and $\mathbb O_{\mathcal H}$ the identity and the zero operators
acting on $\mathcal H$ are denoted, respectively.
Let $T$ and $R$ be operators on spaces $\mathcal H$ and $\mathcal K$,
respectively, and let $X:\mathcal H\to\mathcal K$ be a (linear, bounded) operator which
intertwines $T$ and $R$: $XT=RX$. If $X$ is unitary, then $T$ and $R$
are called {\it unitarily equivalent}, in notation:
$T\cong R$. If $X$ is invertible (the inverse $X^{-1}$ is bounded), then $T$ and $R$
are called {\it similar}, in notation: $T\approx R$.
If $X$ a quasiaffinity, that is, $\ker X=\{0\}$ and
$\operatorname{clos}X\mathcal H=\mathcal K$, then
$T$ is called a {\it quasiaffine transform} of $R$,
in notation: $T\prec R$. If $T\prec R$ and $R\prec T$,
then $T$ and $R$ are called {\it quasisimilar}, in notation: $T\sim R$.
Let $\mathcal H$ be a Hilbert space,
and let $T\colon\mathcal H\to\mathcal H$ be a (linear, bounded) operator. $T$ is called a {\it contraction}, if $\|T\|\leq 1$.
Let $T$ be a contraction
on a space $\mathcal H$. $T$ is {\it of class} $C_{1\cdot}$
($T\in C_{1\cdot}$), if $\lim_{n\to\infty}\|T^nx\|>0$ for
each $x\in \mathcal H$, $x\neq 0$, $T$ is {\it of class} $C_{0\cdot}$
($T\in C_{0\cdot}$), if
$\lim_{n\to\infty}\|T^nx\|=0$ for each $x\in\mathcal H$, and $T$ is
of class $C_{\cdot a}$, $a=0,1$, if
$T^\ast$ is of class $C_{a\cdot}$.
It is easy to see that if a contraction $T$
is a quasiaffine transform of an isometry, then $T$ is of class $C_{1\cdot}$,
and if $T$ is a quasiaffine transform of a unilateral shift,
then $T$ is of class $C_{10}$.
The paper is organized as follows.
In Sec. 2 and 3, the known facts about contractions, their relations to isometries,
and their characteristic functions are collected. In Sec. 4, the necessity of \eqref{1.7}
to the left invertibility of some operator-valued functions is proved. Sec. 5 is the main section
of this paper, where for any $\delta$, $0<\delta<1$, examples of subnormal contractions are constructed such that
their characteristic functions satisfy \eqref{1.2}, and \eqref{1.3} with $\delta$, and are not
left invertible.
\section{Isometric and unitary asymptotes of contractions}
For a contraction $T$ the {\it isometric asymptote} $(X_{T,+},T_+^{(a)})$, and
the {\it unitary asymptote} $(X_T,T^{(a)})$ was defined,
see, for example, {\cite[Ch. IX.1]{16}}. There are some ways to construct
the isometric asymptote of the contraction, for our purpose,
it is convenient to use the following, see \cite{11}.
Let $(\cdot,\cdot)$ be the inner
product on the Hilbert space $\mathcal H$,
and let $T\colon\mathcal H\to\mathcal H$ be a contraction.
Define a new semi-inner product on
$\mathcal H$ by the formula $\langle x,y\rangle=\lim_{n\to\infty}(T^nx,T^ny)$,
where $x,y\in \mathcal H$. Set
$$\mathcal H_0=\mathcal H_{T,0}=\{x\in\mathcal H:\ \langle x, x\rangle=0\}.$$
Then the factor space $\mathcal H/\mathcal H_0$ with the inner product
$\langle x+\mathcal H_0, y+\mathcal H_0\rangle=\langle x,y\rangle$ will
be an inner product space.
Let $\mathcal H_+^{(a)}$ denote the resulting Hilbert space obtained
by completion, and let $X_{T,+}\colon\mathcal H\to\mathcal H_+^{(a)}$
be the natural imbedding,
$X_{T,+}x=x+\mathcal H_0$. Clearly, $X_{T,+}$ is a (linear, bounded) operator, and $\|X_{T,+}\|\leq 1$.
Clearly, $\langle Tx,Ty\rangle=\langle x,y\rangle$ for every $x,y\in \mathcal H$.
Therefore, $T_1: x+\mathcal H_0\mapsto Tx+\mathcal H_0$ is a well-defined isometry on
$\mathcal H/\mathcal H_0$. Denote by $T_+^{(a)}$ the continuous extension of $T_1$ to the space
$\mathcal H_+^{(a)}$. Clearly, $X_{T,+}T=T_+^{(a)}X_{T,+}$. The pair $(X_{T,+},T_+^{(a)})$ is called
the {\it isometric asymptote} of a contraction $T$. The operator $X_{T,+}$ is called the {\it canonical intertwining mapping}.
A contraction $T$ is similar
to an isometry $V$ if and only if $X_{T,+}$ is an invertible operator,
that is, $\ker X_{T,+}=\{0\}$ and $X_{T,+}\mathcal H= \mathcal H_+^{(a)}$, and in this case, $V\cong T_+^{(a)}$
(see {\cite[Theorem 1]{11}}). In particular, if $T$ is a contraction of class $C_{10}$, and $T_+^{(a)}$
is a unitary operator, then $T$ is not similar to an isometry.
Denote by $T^{(a)}$ the minimal unitary extension of $T_+^{(a)}$, by $\mathcal H^{(a)}\supset\mathcal H_+^{(a)}$
the space on which $T^{(a)}$ acts, and by $X_T$ the imbedding of $\mathcal H$ into $\mathcal H^{(a)}$.
Clearly, $X_TT=T^{(a)}X_T$ and $X_{T,+}x=X_Tx$ for every $x\in\mathcal H$.
The pair $(X_T,T^{(a)})$ is called the {\it unitary asymptote} $(X,T^{(a)})$ of a contraction $T$.
\section{Contractions and their characteristic functions}
All statement of this section are well-known and can be found in {\cite[Ch. VI]{16}}, see also
{\cite[Ch. C.1]{14}}.
Let $\mathcal H$ be a separable Hilbert space, and let $T\colon\mathcal H\to\mathcal H$ be
a contraction.
A contraction $T$ is called {\it completely nonunitary}, if $T$ has no invariant subspace such that
the restriction of $T$ on this subspace is unitary.
For a contraction $T$ put $\mathcal D_T=\operatorname{clos}(I_{\mathcal H}-T^\ast T)\mathcal H$.
It is easy to see that \begin{equation}\label{3.1}\text{ if } \ x\in\mathcal H\ominus\mathcal D_T, \ \
\text{ then } \ \ x=T^\ast Tx \ \ \text{ and } \ \ \|Tx\|=\|x\|. \end{equation}
Also, $T\mathcal D_T\subset\mathcal D_{T^\ast}$ and $T(\mathcal H\ominus\mathcal D_T)=
\mathcal H\ominus\mathcal D_{T^\ast}$ (see {\cite[Ch. I.3.1]{16}}), therefore,
\begin{equation}\label{3.2} \dim \mathcal D_{T^\ast}\ominus T\mathcal D_T = \dim\mathcal H\ominus T\mathcal H . \end{equation}
Since $I_{\mathcal H} - T^\ast T=(I_{\mathcal D_T} - T^\ast T|_{\mathcal D_T})\oplus\mathbb O_{\mathcal H\ominus\mathcal D_T}$,
\begin{equation}\label{3.3}\|I_{\mathcal H} - T^\ast T\|_{\frak S_1}=\|I_{\mathcal D_T} - T^\ast T|_{\mathcal D_T}\|_{\frak S_1}. \end{equation}
\begin{lemma}\label{lem3.1} Let $T\colon\mathcal H\to\mathcal H$ be a contraction, and let $0<\delta\leq 1$.
Then $\|Tx\|\geq\delta\|x\|$ for every $x\in\mathcal H$ if and only if
$\|Tx\|\geq\delta\|x\|$ for every $x\in\mathcal D_T$.\end{lemma}
\begin{proof} Indeed, it need to prove the ``if" part only. Let $x\in\mathcal D_T$, and let $y\in\mathcal H\ominus\mathcal D_T$.
Then, by \eqref{3.1}, $(Tx,Ty)=(x,T^\ast Ty)=(x,y)=0$ and $\|Ty\|=\|y\|\geq\delta\|y\|$.
Therefore, $\|T(x+y)\|^2 = \|Tx\|^2+\|Ty\|^2\geq\delta^2\|x\|^2+\delta^2\|y\|^2=\delta^2\|x+y\|^2$. \end{proof}
The characteristic function $\Theta_T$ of the contaction $T$ is the analytic
operator-valued function acting by the formula
\begin{equation}\label{3.4}
\Theta_T(z)=\big(-T+z (I-TT^\ast)^{1/2}(I-zT^\ast)^{-1}(I-T^\ast T)^{1/2}\big)|_{\mathcal D_T},
\ \ z\in\mathbb D. \end{equation}
For every $z\in\mathbb D$ the inclusion $\Theta_T(z)\mathcal D_T\subset\mathcal D_{T^\ast}$ holds,
the mapping $z\mapsto\Theta_T(z)$ is an analytic function from $\mathbb D$ to
the space of all (linear, bounded) operators from $\mathcal D_T$ to $\mathcal D_{T^\ast}$,
and $\|\Theta_T(z)\|\leq 1$ for every $z\in\mathbb D$.
That is, $\Theta_T\in H^\infty(\mathcal D_T\to\mathcal D_{T^\ast})$, and $\|\Theta_T\|\leq 1$. It is easy to see
that $\Theta_T$ is purely contractive.
Conversely, for every analytic operator-valued function $F\in H^\infty(\mathcal E\to\mathcal E_\ast)$
such that $\|F\|\leq 1$ and $F$ is purely contractive there exists a contraction $T$ such that $F=\Theta_T$ {\cite[Ch. VI]{16}}.
The following theorem was proved in \cite{15}, see also {\cite[C.1.5.5]{14}}.
\begin{theoremcite}\cite{15}\label{tha}
The contraction $T$ is similar to an isometry if and only if
$\Theta_T$ is left invertible.\end{theoremcite}
Let $T$ be a completely nonunitary contraction. Then $T$ is
of class $C_{1\cdot}$ if and only if $\Theta_T$ is $\ast$-outer,
and $T$ is of class $C_{\cdot 0}$,
if and only if $\Theta_T$ is inner {\cite[VI.3.5]{16}}.
Recall that the {\it multiplicity} of an operator is the minimum dimension
of its reproducing subspaces. An operator is called {\it cyclic} if its multiplicity is equal to $1$.
The following theorem was proved in \cite{7}, \cite{10}, \cite{17}, \cite{24}, \cite{25}.
\begin{theoremcite}\label{thb} Let $T$ be a contraction, and let
$1\leq n <\infty$. The following are equivalent:
$(1)$ $T\prec S_n$;
$(2)$ $T$ is of class $C_{10}$, $\dim\ker T^\ast =n$, and
$I-T^\ast T \in\frak S_1$;
$(3)$ $\Theta_T$ is an inner $\ast$-outer function, $\Theta_T$ has a left
scalar multiple, and $\dim\mathcal D_{T^\ast}\ominus\Theta_T(\lambda)\mathcal D_T=n$
for some $\lambda\in\mathbb D$.
Moreover, if $T$ is a contraction such that $T\prec S_n$, $1\leq n <\infty$, then the following are equivalent:
$(4)$ $T\sim S_n$;
$(5)$ multiplicity of $T$ is equal to $n$;
$(6)$ $\Theta_T$ has an outer left scalar multiple.\end{theoremcite}
{\bf Remark.} If $T$ is a contraction and $T\prec S_n$, $1\leq n <\infty$, then $b_\lambda(T)$ is a contraction and
$b_\lambda(T)\prec b_\lambda(S_n)\cong S_n$. Therefore, $I-b_\lambda(T)^\ast b_\lambda(T)\in\frak S_1$ for
every $\lambda\in\mathbb D$. Here $b_\lambda(T)=(T-\lambda)(I-\overline\lambda T)^{-1}$.
Let $T$ be a completely nonunitary contraction.
Put \begin{equation}\label{3.5}\Delta_\ast(\zeta)=(I_{\mathcal D_{T^\ast}}-
\Theta_T(\zeta)\Theta_T(\zeta)^\ast)^{1/2}, \ \ \zeta\in\mathbb T, \ \
\text{ and }\ \ \omega_T=\{\zeta\in\mathbb T:
\Delta_\ast(\zeta) \neq \mathbb O\}.\end{equation}
Then the unitary asymptote $T^{(a)}$ of a completely nonunitary contraction $T$
is unitarily equivalent to the operator of multiplication by the independent variable $\zeta$ on
$\operatorname{clos}\Delta_\ast L^2(\mathcal D_{T^\ast})$. In particular, $T^{(a)}$ is cyclic if and only if
\begin{equation}\label{3.6}\dim \Delta_\ast(\zeta) \mathcal D_{T^\ast}\leq 1 \ \ \text{ for a.e. } \ \zeta\in\mathbb T,
\end{equation} and in this case
$T^{(a)}\cong U(\omega_T)$ (see {\cite[Ch. IX.2]{16}}).
Also, if the function $\Theta_T$ is inner and \eqref{3.6} holds, then
\begin{equation}\label{3.7}\begin{aligned}\omega_T & = \{ \zeta\in\mathbb T: \
\dim \mathcal D_{T^\ast}\ominus\Theta(\zeta)\mathcal D_T = 1 \} \\
\text{ and } \ \ \mathbb T\setminus\omega_T & =\{ \zeta\in\mathbb T:
\ \Theta(\zeta)\mathcal D_T=\mathcal D_{T^\ast}\}.\end{aligned}\end{equation}
For $\lambda\in\mathbb D$ put $b_\lambda(z)=\frac{z-\lambda}{1-\overline\lambda z}$, $z\in\mathbb D$.
Then $b_\lambda(T)=(T-\lambda)(I-\overline\lambda T)^{-1}$ is a contraction. For every $\lambda\in\mathbb D$ there exists
unitary operators
$$V_\lambda\colon\mathcal D_T\to\mathcal D_{b_\lambda(T)} \ \ \ \text{ and } \ \ \
V_{\lambda\ast}\colon\mathcal D_{b_\lambda(T)^\ast}\to\mathcal D_{T^\ast}$$
such that
\begin{equation}\label{3.8} V_{\lambda\ast}\Theta_{b_\lambda(T)}(z)V_\lambda=
\Theta_T(b_{-\lambda}(z)). \end{equation}
Setting $z=0$ in \eqref{3.4} and \eqref{3.8}, we conclude that
\begin{equation}\label{3.9}\Theta_T(\lambda)=-V_{\lambda\ast}b_\lambda(T)V_\lambda \ \ \text{ for every } \
\lambda\in\mathbb D \end{equation}
({\cite[Ch. VI.1.3]{16}}).
The following lemma is a straightforward consequence of \eqref{3.2}, \eqref{3.3}, \eqref{3.9},
and Lemma \ref{lem3.1}.
\begin{lemma}\label{lem3.2} Suppose $T\colon\mathcal H\to\mathcal H$ is a completely
nonunitary contraction.
$\ \ \text{\rm (i)}$ Let $0<\delta\leq1$. Then $\|\Theta_T(\lambda)x\|\geq\delta\|x\|$
for every $x\in\mathcal D_T$, $\lambda\in\mathbb D$,
if and only if $\|b_\lambda(T)x\|\geq\delta\|x\|$ for every
$x\in\mathcal H$, $\lambda\in\mathbb D$.
$\ \text{\rm (ii)}$ $\dim\mathcal D_{T^\ast}\ominus\Theta_T(\lambda)\mathcal D_T=
\dim\mathcal H\ominus b_\lambda(T)\mathcal H$
for every $\lambda\in\mathbb D$.
$\text{\rm (iii)}$ $\|I_{\mathcal D_T}-
\Theta_T^\ast(\lambda)\Theta_T(\lambda)\|_{\frak S_1} =
\|I_{\mathcal H}-b_\lambda(T)^\ast b_\lambda(T) \|_{\frak S_1}$
for every $\lambda\in\mathbb D$.\end{lemma}
\section{On contractions similar to an isometry}
The following theorem is actually proved in {\cite[Theorem 2.1]{7}} (see also enlarged version on arXiv).
\begin{theorem}\label{th4.1} \cite{7} Suppose $T$ is a contraction
with finite multiplicity,
and $T$ is similar to an isometry.
Then \begin{equation}\label{4.1}\sup_{\lambda\in\mathbb D}\|I-b_\lambda(T)^\ast b_\lambda(T)\|_{\frak S_1}
<\infty. \end{equation} \end{theorem}
\begin{corollary}\label{cor4.2} Suppose $\mathcal E$, $\mathcal E_\ast$ are Hilbert spaces,
$F\in H^\infty(\mathcal E\to\mathcal E_\ast)$ is an inner function, and
$\dim \mathcal E_\ast\ominus F(\lambda)\mathcal E<\infty$ for some $\lambda \in\mathbb D$.
If $F$ is left invertible, then $F$ satisfies \eqref{1.7}.\end{corollary}
\begin{proof} Let $\mathcal H$ be a Hilbert space, and let $T\colon\mathcal H\to\mathcal H$ be
a contraction such that $\Theta_T=F$, where $\Theta_T$ is the characteristic function
of $T$ (see {\cite[Ch. VI.3]{16}}). Since $F$ is inner, $T\in C_{\cdot 0}$, see {\cite[VI.3.5]{16}}.
By Lemma \ref{lem3.2}(ii),
$$\dim \mathcal H\ominus b_\lambda(T)\mathcal H = \dim \mathcal E_\ast\ominus F(\lambda)\mathcal E<\infty.$$
Now suppose that $F$ is left invertible. Then, by Theorem \ref{tha}, there exist a Hilbert space
$\mathcal K$ and an isometry $V\colon\mathcal K\to\mathcal K$ such that $T\approx V$. Since $T\in C_{\cdot 0}$,
$V$ is a unilateral shift, and the multiplicity of $V$ is equal to
$$\dim \mathcal K\ominus b_\lambda(V)\mathcal K = \dim \mathcal H\ominus b_\lambda(T)\mathcal H <\infty.$$
By Theorem \ref{th4.1}, $T$ satisfies \eqref{4.1}. By Lemma \ref{lem3.2}(iii), $F$ satisfies \eqref{1.7}. \end{proof}
\section{Subnormal contractions}
Operators that are considered in this sections are subnormal ones, and are studied by
many authors, the reader can consult with the book \cite{6}.
Let $\nu$ be
a positive finite Borel measure on the closed unit disk $\overline{\mathbb D}$. Denote by
$P^2(\nu)$ the closure of analytic polynomials in $L^2(\nu)$, and by $S_\nu$
the operator of multiplication by the independent variable in $P^2(\nu)$, i.e.
\begin{align*} & S_\nu: P^2(\nu)\to P^2(\nu), \\ & (S_\nu f)(z)=zf(z) \text{ for a.e. } z\in \overline{\mathbb D}
\ \text{ with respect to } \nu , \ \ f\in P^2(\nu).\end{align*}
Clearly, $S_\nu$ is a contraction.
Recall that $m$ is the Lebesgue measure on $\mathbb T$. If $\nu=m$, then $S_\nu$
is the unilateral shift of multiplicity 1, it is denoted by $S$ in this section.
The following lemma is a straightforward consequence of the construction of the
isometric asymptote of a contraction from \cite{11}, see Sec. 2 of this paper, so its proof is omitted.
\begin{lemma}\label{lem5.1} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, $\mathcal H=P^2(\nu)$, and $T=S_\nu$.
Then $\mathcal H_{0,T}=\{f\in P^2(\nu): \ f=0 \ \text {a.e. on}\ \ \mathbb T
\text { with respect to}\ \nu\}$, $\mathcal H_+^{(a)}=P^2(\nu|_{\mathbb T})$,
$$X_{T,+}\colon P^2(\nu)\to P^2(\nu|_{\mathbb T}), \ \ \ X_{T,+}f=f|_{\mathbb T}, \ \ f\in P^2(\nu),$$
is the natural imbedding, and $T_+^{(a)}=S_{\nu|_{\mathbb T}}$.\end{lemma}
The proof of the following lemma is obvious and omitted.
\begin{lemma}\label{lem5.2} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, and $f\in P^2(\nu)$. Then there exists $\lambda\in \mathbb D$
such that $\|b_\lambda f\|=\|f\|$ if and only if $f(z)=0$ for a.e. $z\in\mathbb D$
with respect to $\nu$.\end{lemma}
\begin{corollary}\label{cor5.3} Suppose $\nu$ is a positive finite Borel measure on
$\overline{\mathbb D}$, and $\lambda\in \mathbb D$. Then
$$ P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)}\subset
\{f\in P^2(\nu):\ f(z)=0 \ \text{ for a.e. }\ z\in\mathbb D
\ \text{ with respect to } \nu\}$$
and
$$P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)^\ast}\subset
\{f\in P^2(\nu):\ f(z)=0 \ \text{ for a.e. }\ z\in\mathbb D
\ \text{ with respect to } \nu\}.$$\end{corollary}
\begin{proof} Let $P_+:L^2(\nu)\to P^2(\nu)$ be the orthogonal projection.
It is easy to see that $(b_\lambda(S_\nu)f)(z)=b_\lambda(z)f(z)$ for a.e. $z\in \overline{\mathbb D}$
with respect to $\nu$, and $b_\lambda(S_\nu)^\ast f=P_+(\overline b_\lambda f)$, $f\in P^2(\nu)$.
If $f\in P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)}$, then, by \eqref{3.1}, $\|f\|=\| b_\lambda f\|$,
and, by Lemma \ref{lem5.2}, $f(z)=0$ for a.e. $z\in\mathbb D$ with respect to $\nu$.
If $f\in P^2(\nu)\ominus\mathcal D_{b_\lambda(S_\nu)^\ast}$, then, by \eqref{3.1},
$\|f\|=\| P_+(\overline b_\lambda f)\|\leq\|\overline b_\lambda f\|=\|b_\lambda f\|$,
and, by Lemma \ref{lem5.2}, $f(z)=0$ for a.e. $z\in\mathbb D$ with respect to $\nu$. \end{proof}
Denote by $m_2$ the normalized Lebesgue measure on the unit disk $\mathbb D$, for $-1<\alpha<\infty$
put $\text{\rm d}A_\alpha(z)=(\alpha+1)(1-|z|^2)^\alpha\text{\rm d}m_2(z)$. It is well known that
the Bergman space $P^2(A_\alpha)$ has the following properties:
$f\in P^2(A_\alpha)$ if and only $f$ is an analytic function in $\mathbb D$ and $f\in L^2(A_\alpha)$,
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha)$ for every $z\in\mathbb D$,
and there exists a constant $C_\alpha>0$ (which depends on $\alpha$) such that
\begin{equation}\label{5.1} |f(z)|\leq C_\alpha\frac{\|f\|_{P^2(A_\alpha)}}{(1-|z|^2)^{1+\alpha/2}},
\ \ \ z\in\mathbb D, \end{equation}
(see, for example, {\cite[Sec. 1.1 and 1.2]{9}}). It is easy to see that
$S_{A_\alpha}\in C_{00}$.
\begin{lemma}\label{lem5.4} {\cite[Lemma 4.2]{3}} Let $-1<\alpha<\infty$.
Then for every $f\in P^2(A_\alpha)$ and $\lambda\in\mathbb D$
$$\int_{\mathbb D}|b_\lambda f|^2\text{\rm d}A_\alpha\geq
\frac{1}{\alpha+2}\int_{\mathbb D}|f|^2\text{\rm d}A_\alpha.$$\end{lemma}
\begin{corollary}\label{cor5.5} Let $-1<\alpha<\infty$, and let $\mu$ be a positive finite Borel measure on $\mathbb T$.
Then for every $f\in P^2(A_\alpha+\mu)$ and $\lambda\in\mathbb D$
$$\int_{\overline{\mathbb D}}|b_\lambda f|^2\text{\rm d}(A_\alpha+\mu)\geq
\frac{1}{\alpha+2}\int_{\overline{\mathbb D}}|f|^2\text{\rm d}(A_\alpha+\mu).$$\end{corollary}
\begin{proof} Clearly, $P^2(A_\alpha+\mu)\subset P^2(A_\alpha)$. Let
$f\in P^2(A_\alpha+\mu)$, and let $\lambda\in\mathbb D$. We have
\begin{align*}\int_{\overline{\mathbb D}} |b_\lambda & f|^2\text{\rm d}(A_\alpha+\mu) =
\int_{\mathbb D}|b_\lambda f|^2\text{\rm d}A_\alpha +
\int_{\mathbb T}|b_\lambda f|^2\text{\rm d}\mu \\ &\geq
\frac{1}{\alpha+2}\int_{\mathbb D}|f|^2\text{\rm d}A_\alpha +
\int_{\mathbb T}|f|^2\text{\rm d}\mu
\geq\frac{1}{\alpha+2}\int_{\overline{\mathbb D}}|f|^2\text{\rm d}(A_\alpha+\mu),\end{align*}
because of $|b_\lambda|=1$ on $\mathbb T$ and $1> 1/(\alpha+2)$ for $-1<\alpha<\infty$. \end{proof}
Recall the following definition.
{\bf Definition.} Let $E$ be a closed subset of $\mathbb T$, and let $\{J_k\}_k$
be the collection of open arcs of $\mathbb T$ such that $J_k\cap J_\ell=\emptyset$
for $k\neq\ell$ and
$\mathbb T=E\cup\bigcup_kJ_k$. The set $E$ satisfies the {\it Carleson condition}
if $\sum_km(J_k)\log m(J_k)>-\infty$.
Let $w\in L^1(\mathbb T,m)$, $w\geq 0$ a.e. on $\mathbb T$. Then $P^2(wm)=L^2(wm)$ if and only if
$\log w\not\in L^1(\mathbb T,m)$, and then $S_{wm}\cong U(\sigma)$, where $\sigma\subset\mathbb T$ is
a measurable set such that $wm$ and $m|_\sigma$ are mutually absolutely continuous.
If $ \log w\in L^1(\mathbb T,m)$, then there exists an outer function $\psi\in H^2$ such that
$|\psi|^2=w$ a.e. on $\mathbb T$. Then
\begin{equation}\label{5.2} \begin{aligned} P^2(wm) & =\frac{H^2}{\psi}=\Big\{\frac{h}{\psi}: \ h\in H^2\Big\},
\ \ \Big\|\frac{h}{\psi}\Big\|_{P^2(wm)}=\|h\|_{H^2}, \ h\in H^2,\\
\text{ and } \ \ S_{wm} & \cong S \end{aligned} \end{equation}
(see, for example, {\cite[Ch. III.12]{6}} or {\cite[A.4.1.5]{14}}).
In Theorems \ref{th5.6} and \ref{th5.7}, we consider nontangential boundary values of functions from
$P^2(\mu)$ for some measures $\mu$.
Nontangential boundary values of functions from
$P^t(\mu)$ with $1\leq t<\infty$ are considered in \cite{4} in relation to another questions, see also references therein,
especially \cite{1}, \cite{12}, \cite{13}, \cite{18}, and \cite{2}. In Theorems \ref{th5.6} and \ref{th5.7} we formulate particular cases of these results in the form convenient to our purpose.
Theorems \ref{th5.6} and the main part of Theorem \ref{th5.7} were proved in {\cite[Sec. 2]{8}} for $\alpha=0$,
but the proofs are the same in
the case of $-1<\alpha\leq 0$ (because the estimate \eqref{5.1} involves the estimate
$|f(z)|\leq C_\alpha\frac{\|f\|_{P^2(A_\alpha)}}{1-|z|^2}$ for
$-1<\alpha\leq 0$, which is used in {\cite[Sec. 2]{8}}), therefore,
the proofs of Theorem \ref{th5.6} and of the main part of Theorem \ref{th5.7} are omitted.
In addition, to prove Theorems \ref{th5.6} and \ref{th5.7}, one needs to apply the notion of isometric asymptote (see Sec. 2 of this paper and references therein).
\begin{theorem}\label{th5.6} \cite{8} Let $-1<\alpha\leq 0$, and let $E\subset\mathbb T$ be a closed set
such that $0<m(E)<1$ and $E$ satisfies the Carleson condition. Then
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha+m|_E)$ for every $z\in\mathbb D$.
Furthermore,
for
$f\in P^2(A_\alpha+m|_E)$ the restriction $f|_{\mathbb D}$ is analytic on $\mathbb D$,
$f|_{\mathbb D}$ has nontangential boundary values
a.e. on $E$ with respect to $m$, which coincide with $f|_E$.
Therefore, $S_{A_\alpha+m|_E}\in C_{10}$.
Also, $I-S_{A_\alpha+m|_E}^\ast S_{A_\alpha+m|_E}$ is compact, and
$(S_{A_\alpha+m|_E})^{(a)}_+=U(E)$. Thus, $S_{A_\alpha+m|_E}$
is not similar to an isometry.\end{theorem}
\begin{theorem}\label{th5.7} \cite{8} Let $-1<\alpha\leq 0$, and let
$w\in L^1(\mathbb T,m)$.
Suppose that for every closed arc $J\subset\mathbb T\setminus\{1\}$
there exist two constants
$0<c_J<C_J<\infty$ such that $c_J\leq w\leq C_J$ a.e. on $J$
(with respect to $m$). Then
the functional $f\mapsto f(z)$ is bounded on $P^2(A_\alpha+wm)$
for every $z\in\mathbb D$. Furthermore, for
$f\in P^2(A_\alpha+wm)$ the restriction $f|_{\mathbb D}$ is analytic on
$\mathbb D$, $f|_{\mathbb D}$ has nontangential boundary values
a.e. on $\mathbb T$ with respect to $m$, which coincide with $f|_{\mathbb T}$.
Therefore, $S_{A_\alpha+wm}\in C_{10}$.
Also, $I-S_{A_\alpha+wm}^\ast S_{A_\alpha+wm}$ is compact, $(S_{A_\alpha+wm})^{(a)}_+=S_{wm}$,
and the canonical mapping which intertwines $S_{A_\alpha+wm}$ with $S_{wm}$
is the natural imbedding
$$P^2(A_\alpha+wm)\to P^2(wm), \ \ f\mapsto f|_{\mathbb T}, \ f\in P^2(A_\alpha+wm).$$
Therefore,
$\ \ \text{\rm (i)}$ $S_{A_\alpha+wm}\sim S$ if and only if $\log w\in L^1(\mathbb T,m)$;
$\ \text{\rm (ii)}$ $S_{A_\alpha+wm}$ is similar to an isometry if and only if $S_{A_\alpha+wm}\approx S$;
$\text{\rm (iii)}$ $S_{A_\alpha+wm}\approx S$ if and only if
$\log w\in L^1(\mathbb T,m)$ and for every $h\in H^2$ there exists $f \in P^2(A_\alpha+wm)$
such that $f|_{\mathbb T}=h/\psi$ a.e. on $\mathbb T$ (with respect to $m$), where $\psi\in H^2$ is
an outer function such that
$|\psi|^2=w$ a.e. on $\mathbb T$. \end{theorem}
{\bf Remark.} In the conditions of Theorem \ref{th5.7}, let $h,\psi\in H^2$, $\psi(z)\neq 0$ for every $z\in\mathbb D$,
$f\in P^2(A_\alpha+wm)$, and $f|_{\mathbb T} =h/\psi$ a.e. on $\mathbb T$ (with respect to $m$). Then
$f(z)=h(z)/\psi(z)$ for every $z\in\mathbb D$. Indeed, set $g(z)=h(z)/\psi(z)$, $z\in\mathbb D$. Then
$g$ is a function analytic on $\mathbb D$, and $g$ has nontangential boundary values $h(\zeta)/\psi(\zeta)$
for a.e. $\zeta\in\mathbb T$. Then $f-g$ is a function analytic on $\mathbb D$, and $f-g$ has zero nontangential
boundary values a.e. on $\mathbb T$. By Privalov's theorem (see, for example, {\cite[Theorem 8.1]{5}}),
$f(z)=g(z)$ for every $z\in\mathbb D$. Therefore, if the conditions (iii) of Theorem \ref{th5.7} are fulfilled,
then $ P^2(A_\alpha+wm)=H^2/\psi$ as the set, and the norms on these spaces are equivalent.
{\it Proof of Theorem \ref{th5.7}.} The main part of Theorem \ref{th5.7} is proved in {\cite[Sec. 2]{8}}.
Let $X$ be the imbedding,
$$X\colon P^2(A_\alpha+wm)\to P^2(wm), \ \ Xf=f|_{\mathbb T}, \ \ f\in P^2(A_\alpha+wm).$$
Then $XS_{A_\alpha+wm}=S_{wm}X$. Since $S_{A_\alpha+wm}\in C_{1\cdot}$, $\ker X=\{0\}$,
therefore, $X$ is a quasiaffinity which realizes the relation $S_{A_\alpha+wm}\prec S_{wm}$.
If $\log w\in L^1(\mathbb T,m)$, then $S_{wm}\cong S$, therefore, $S_{A_\alpha+wm}\prec S$.
Since $S_{A_\alpha+wm}$ is a cyclic contraction, $S_{A_\alpha+wm}\sim S$ by Theorem \ref{thb}(5).
The ``if" part of (i) is proved.
The assumptions of the ``if" part of (iii) mean that $X P^2(A_\alpha+wm) = P^2(wm)$, see \eqref{5.2}.
Thus, $X$ realizes the relation $S_{A_\alpha+wm}\approx S_{wm}$, and, since $S_{wm}\cong S$,
the relation $S_{A_\alpha+wm}\approx S$ is proved.
Now suppose that $S_{A_\alpha+wm}\approx V$, where $V$ is an isometry. Then, by {\cite[Theorem 1]{11}}
(see Sec. 2 of this paper), $X P^2(A_\alpha+wm) = P^2(wm)$ and $V\cong S_{wm}$. Since
$S_{A_\alpha+wm}\in C_{10}$, $S_{wm}\in C_{10}$. By {\cite[A.4.1.5]{14}} (see the description of $S_{wm}$
before \eqref{5.2} in this paper), $\log w\in L^1(\mathbb T,m)$ and $S_{wm}\cong S$. The parts (ii) and (iii)
are proved.
Now suppose that $S_{A_\alpha+wm}\sim S$. By {\cite[Theorem 1]{11}}, see also {\cite[Ch. IX.1]{16}}, there exists
an operator $Y\colon P^2(wm)\to H^2$ such that $YS_{wm}=SY$ and $\operatorname{clos}Y P^2(wm)= H^2$.
If $\log w\not\in L^1(\mathbb T,m)$, then $S_{wm}$ is unitary, and from the relations $Y^\ast S^\ast=S_{wm}^\ast Y^\ast$
and $\ker Y^\ast=\{0\}$ we conclude that $S^\ast\in C_{1\cdot}$, a contradiction. Therefore,
$\log w\in L^1(\mathbb T,m)$. The ``only if" part of (i) is proved. \qed
The following lemma is a variant of {\cite[Theorem 1.7]{9}}.
\begin{lemma}\label{lem5.8} Let $-1<\alpha<\infty$, and let $\beta\in\mathbb R$.
Put $\varphi_\beta(z)=1/(1-z)^\beta$, $z\in\mathbb D$. Then
$\varphi_\beta\in H^2$ if and only if $\beta<1/2$, and
$\varphi_\beta\in P^2(A_\alpha)$ if and only if $\beta<1+\alpha/2$.\end{lemma}
\begin{proof} Put $v_n=2(\alpha+1)\int_0^1r^{2n+1}(1-r^2)^\alpha\text{\rm d}r$, $n\geq 0$.
Then $v_n=\frac{n!\Gamma(\alpha+2)}{\Gamma(\alpha+n+2)}$, where $\Gamma$ is the Gamma function,
and
for every function $f$ analytic on $\mathbb D$
$$\int_{\mathbb D}|f|^2\text{\rm d} A_\alpha=\sum_{n=0}^\infty |\widehat f(n)|^2 v_n.$$
If $\beta\leq 0$, then $\varphi_\beta\in P^2(A_\alpha)$ for every $\alpha$, $-1<\alpha<\infty$. Suppose $\beta>0$.
Since $\widehat \varphi_{\beta}(n)=\frac{\Gamma(\beta+n)}{n!\Gamma(\beta)}$, $n\geq 0$, we have that
$\varphi_\beta\in P^2(A_\alpha)$ if and only if the series
$\sum_{n=0}^\infty
\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}$ converges. By Stirling's formula,
$$\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}\sim (n+1)^{2\beta-\alpha-3} \ \ \text { as } \ n\to\infty.$$
Therefore, the series
$\sum_{n=0}^\infty
\frac{\Gamma(\beta+n)^2}{n!\Gamma(\alpha+n+2)}$ converges if and only if $\beta<1+\alpha/2$.
The first statement of the lemma can be proved similarly. \end{proof}
\begin{lemma}\label{lem5.9} Let $-1<\alpha\leq 0$, and let $\beta<-1-\alpha$. Then $S_{A_\alpha+|\varphi_\beta| m}\sim S$,
but $S_{A_\alpha+|\varphi_\beta| m}$ is not similar to an isometry.\end{lemma}
\begin{proof}
By Theorem \ref{th5.7}(i),
$S_{A_\alpha+|\varphi_\beta|m}\sim S$. Put $\psi=\varphi_{\beta/2}$. Clearly, $|\psi|^2=|\varphi_\beta|$.
By \eqref{5.2}, $P^2(|\varphi_\beta|m)=H^2/\psi$. By Theorem \ref{th5.7}(iii), if $S_{A_\alpha+|\varphi_\beta|m}$
is similar to an isometry, then $H^2/\psi = P^2(A_\alpha+|\varphi_\beta|m)\subset P^2(A_\alpha)$.
Take $\gamma$, $1+\alpha/2+\beta/2\leq\gamma<1/2$. Put $h=\varphi_\gamma$. Then $h\in H^2$,
and $h/\psi=\varphi_{\gamma-\beta/2}\not\in P^2(A_\alpha)$
by Lemma \ref{lem5.8}. Therefore, $S_{A_\alpha+|\varphi_\beta|m}$ is not similar to an isomertry. \end{proof}
\begin{corollary}\label{cor5.10} Let $0<\delta<1$, and let $E\subset\mathbb T$ be a closed set
satisfying the Carleson condition and such that $0<m(E)<1$. Then there exist operator-valued
inner functions $F_k$,
such that $F_k$ satisfy $\eqref{1.2}$, and $\eqref{1.3}$ with $\delta$, and $F_k$ are not
left invertible,
$k=1,2,3$. Also, $F_1$, $F_2$, $F_3$ satisfy $\eqref{1.4}$, $\eqref{1.6}$, $\eqref{1.5}$, respectively,
$F_3$ has an outer left scalar multiple, and $I-F_3(z)^\ast F_3(z)\in\frak S_1$ for every $z\in\mathbb D$.
\end{corollary}
{\bf Remark.} Since $F_3$ is not left invertible, $F_3$ does not satisfy \eqref{1.7}, see \cite{22}.
{\it Proof of Corollary \ref{cor5.10}.} Put $\alpha=1/\max(\delta^2, 1/2)-2$, then $-1<\alpha\leq 0$.
Take $\beta<-1-\alpha$. Put
$$\mathcal H_1=P^2(A_\alpha), \ \ \mathcal H_2=P^2(A_\alpha+m|_E), \ \ \mathcal H_3=P^2(A_\alpha+|\varphi_\beta|m), $$
$$ T_1=S_{A_\alpha}, \ \ T_2=S_{A_\alpha+m|_E}, \ \ T_3=S_{A_\alpha+|\varphi_\beta|m}.$$
Clearly, $T_k$ are cyclic contractions, $k=1,2,3$, $T_1\in C_{00}$, and $T_k\in C_{10}$ by
Theorems \ref{th5.6} and \ref{th5.7},
$k=2,3$. By Corollary \ref{cor5.5}, \begin{equation}\label{5.3} \|b_\lambda(T_k)f\|^2\geq \delta^2\|f\|^2 \ \text{ for every } \ \lambda\in\mathbb D, \
\ f\in\mathcal H_k, \ \ k=1,2,3. \end{equation}
By Theorems \ref{th5.6} and \ref{th5.7}, the functionals
$$ f\mapsto f(z), \ \ \mathcal H_k\to\mathbb C,$$
are bounded for every $z\in\mathbb D$, $k=1,2,3$.
Therefore, \begin{equation}\label{5.4} \dim \mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k =1 \ \text{ for every } \ \lambda\in\mathbb D, \
\ \ k=1,2,3. \end{equation}
Indeed, let $k$ be fixed, and let $\lambda\in\mathbb D$. Then there exists $g_\lambda\in \mathcal H_k$ such that
$f(\lambda)=(f,g_\lambda)$ for every $f\in \mathcal H_k$. Since $(b_\lambda(T_k)f)(z)=b_\lambda(z)f(z)$ for every
$z\in\mathbb D$, $f\in \mathcal H_k$, it is clear that $g_\lambda\in\mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k$.
Since $T_k$ is cyclic, $T_k-\lambda I$ is cyclic, too, therefore,
$\dim \mathcal H_k\ominus b_\lambda(T_k)\mathcal H_k = \dim \mathcal H_k\ominus (T_k-\lambda I)\mathcal H_k\leq 1$.
The equality \eqref{5.4} is proved.
Now find the unitary asymptotes of $T_k$, $k=1,2,3$, see Sec. 2 of this paper and references therein.
Since $T_1\in C_{00}$, $T_1^{(a)}=\mathbb O$. By Theorem \ref{th5.6}, $T_2^{(a)}=U(E)$. By Lemma \ref{lem5.9},
$T_3\sim S$, therefore, $T_3^{(a)}=U(\mathbb T)$, the bilateral shift of multiplicity 1.
Since $T^{(a)}\cong U(\omega_T)$ for every cyclic completely nonunitary contraction $T$, where
$\omega_T$ is defined in \eqref{3.5}, we conclude that
\begin{equation}\label{5.5} \omega_{T_1}=\emptyset, \ \ \ \omega_{T_2}=E, \ \ \ \omega_{T_3}=\mathbb T. \end{equation}
Also, $T_1$ is not similar to an isometry, because $T_1\in C_{00}$, and $T_2$ and $T_3$ are
not similar to an isometry by Theorem \ref{th5.6} and Lemma \ref{lem5.9}, respectively.
Now put $F_k=\Theta_{T_k}$, that is, $F_k$ is the characteristic function of the contraction $T_k$,
$k=1,2,3$, see Sec. 3 of this paper and references therein. Since $T_k\in C_{\cdot 0}$, $F_k$ are inner.
By \eqref{5.3} and Lemma \ref{lem3.2}(i), $F_k$ satisfy \eqref{1.3} with $\delta$. By \eqref{5.4} and
Lemma \ref{lem3.2}(ii), $F_k$ satisfy \eqref{1.2}. $F_k$ are not left invertible, because of $T_k$ are not
similar to an isometry, see Theorem \ref{tha}. $F_1$, $F_2$, $F_3$ satisfy \eqref{1.4}, \eqref{1.6}, \eqref{1.5}, respectively,
because of \eqref{3.7} and \eqref{5.5}. Since $T_3\sim S$ (by Lemma \ref{lem5.9}), $F_3$ has an outer
left scalar multiple
by Theorem \ref{thb}(6), and $I-F_3(z)^\ast F_3(z)\in\frak S_1$ for every $z\in\mathbb D$
by Lemma \ref{lem3.2}(iii) and Theorem \ref{thb}(2).
\qed
\end{document} |
\begin{document}
\title{Minimum number of additive tuples in groups of prime order}
\begin{abstract} For a prime number $p$ and a sequence of integers $a_0,\dots,a_k\in \{0,1,\dots,p\}$, let $s(a_0,\dots,a_k)$ be the minimum number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots + x_k$, over subsets $A_0,\dots,A_k\subseteq\Z{p}$ of sizes $a_0,\dots,a_k$ respectively. An elegant argument of Lev (independently rediscovered by Samotij and Sudakov) shows that there exists an extremal configuration with all sets $A_i$ being intervals of appropriate length, and that the same conclusion also holds for the related problem, reposed by Bajnok, when $a_0=\dots=a_k=:a$ and $A_0=\dots=A_k$, provided $k$ is not equal 1 modulo~$p$. By applying basic Fourier analysis, we show for Bajnok's problem that if $p\geqslant 13$ and $a\in\{3,\dots,p-3\}$ are fixed while $k\varepsilonquiv 1\pmod p$ tends to infinity, then the extremal configuration alternates between at least two affine non-equivalent sets.
\varepsilonnd{abstract}
\section{Introduction}
Let $\Gamma$ be a given finite Abelian group, with the group operation written additively.
For $A_0,\dots,A_k\subseteq\Gamma$, let $s(A_0,\dots,A_k)$ be the number of $(k+1)$-tuples $(x_0,\dots,x_k)\in A_0\times\dots\times A_k$ with $x_0=x_1+\dots+x_k$. If $A_0=\dots=A_k:=A$, then we use the shorthand $s_k(A):=S(A_0,\dots,A_k)$. For example, $s_2(A)$ is the number of \varepsilonmph{Schur triples} in $A$, that is, ordered triples $(x_0,x_1,x_2)\in A^3$ with $x_0=x_1+x_2$.
For integers $n\geqslant m\geqslant 0$, let $[m,n]:=\{m,m+1,\dots,n\}$ and
$[n]:=[0,n-1]=\{0,\dots,n-1\}$.
For a sequence $a_0,\dots,a_k\in [\,|\Gamma| +1\,] = \lbrace 0,1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}, |\Gamma|\rbrace$, let $s(a_0,\dots,a_k;\Gamma)$ be the minimum of $s(A_0,\dots,A_k)$ over subsets $A_0,\dots,A_k\subseteq \Gamma$ of sizes $a_0,\dots,a_k$ respectively. Additionally, for $a\in [0,p]$, let $s_k(a;\Gamma)$ be the minimum of $s_k(A)$ over all $a$-sets $A\subseteq \Gamma$.
The question of finding the maximal size of a sum-free subset of $\Gamma$ (i.e.\ the maximum $a$ such that $s_2(a;\Gamma)=0$) originated in a paper of Erd\H os~\cite{Erdos65} in 1965 and took 40 years before it was resolved in full generality by Green and Ruzsa~\cite{GreenRuzsa05}.
In this paper, we are interested in the case where $p$ is a fixed prime and the underlying group $\Gamma$ is taken to be $\Z p$, the cyclic group of order $p$,
which we identify with the additive group of residues modulo $p$ (also using the multiplicative structure on it when this is useful).
Lev~\cite{Lev01duke} solved the problem of finding $s_k(a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k;\Z p)$, where $p$ is prime (in the equivalent guise of considering solutions to $x_1+\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} + x_k=0$).\footnote{We learned of Lev's work after the publication of this paper. For completeness, we still provide a proof of Theorem~\ref{th:main} in Section~\ref{knot=1}, which is essentially the same as the original proof of Lev's more general result, and which was rediscovered in~\cite{SamotijSudakov16pmcps}.}
For $I\subseteq\Z p$ and $x,y\in\Z p$, write $x\cdot I+y:=\{x\cdot z+y: z\in I\}$.
\begin{theorem}\cite{Lev01duke} \label{th:main} For arbitrary $k\geqslant 1$ and $a_0,\dots,a_k\in [0,p]$, there is $t\in\Z p$ such that
$$
s(a_0,\dots,a_k;\Z p)=s([a_0]+t,[a_1],\dots,[a_k]; \Z p).\qed
$$
\varepsilonnd{theorem}
Huczynska, Mullen and Yucas~\cite{HuczynskaMullenYucas09jcta} found a new proof of the $s_2(a; \Z p)$-problem, while also addressing some extensions. Samotij and Sudakov~\cite{SamotijSudakov16pmcps} rediscovered Lev's proof of the $s_2(a ; \Z p)$-problem and showed that, when $s_2(a ,\Z p)>0$, then the $a$-sets that achieve the minimum
are exactly those of the form $\xi\cdot I$ with $\xi\in\Z{p}\setminus\{0\}$, where $I$ consists of the residues modulo $p$ of $a$ integers closest to $\frac{p-1}2\in\I Z$. Each such set is an arithmetic progression; its difference can be any non-zero value but the initial element has to be carefully chosen.
(By an \varepsilonmph{$m$-term arithmetic progression} (or \varepsilonmph{$m$-AP} for short) we mean a set of the form $\{x,x+d,\dots,x+(m-1)d\}$ for some $x,d\in\Z p$ with $d\not=0$. We call $d$ the \varepsilonmph{difference}.)
Samotij and Sudakov~\cite{SamotijSudakov16pmcps} also solved the $s_2(a)$-problem for various groups $\Gamma$.
Bajnok~\cite[Problem~G.48]{Bajnok18acmrp} suggested the more general problem of considering $s_k(a;\Gamma)$.
This is wide open in full generality.
This paper concentrates on the case $\Gamma = \Z p$, for $p$ prime, and the sets which attain equality in Theorem~\ref{th:main}.
In particular, we write $s(a_0,\dots,a_k):= s(a_0,\dots,a_k;\Z p)$ and $s_k(a):=s_k(a;\Z p)$. Since the case $p=2$ is trivial, let us assume that $p\geqslant 3$.
Since
\begin{equation}\label{eq:equiv}
s(A_0,\dots,A_k)=s(\xi\cdot A_0+\varepsilonta_0,\dots,\xi\cdot A_k+\varepsilonta_k),\quad\mbox{for $\xi\not=0$ and $\varepsilonta_0=\varepsilonta_1+\dots+\varepsilonta_k$},
\varepsilonnd{equation}
Theorem~\ref{th:main} shows that, for any difference $d$, there is at least one extremal configuration consisting of $k+1$ arithmetic progressions with the same difference $d$.
In particular, if $a_0=\dots=a_k=:a$, then one extremal configuration consists of $A_1=\dots=A_k=[a]$ and $A_0=[t,t+a-1]$ for some $t\in\Z p$.
Given this, one can write down some formulas for $s(a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k)$ in terms of $a_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k$ involving summation (based on~\varepsilonqref{eq:s} or a version of~\varepsilonqref{eq:sk(A)}) but there does not seem to be a closed form in general.
If $k\not\varepsilonquiv1\pmod p$, then by taking $\xi:=1$, $\varepsilonta_1:=\dots:=\varepsilonta_k:=-t(k-1)^{-1}$, and $\varepsilonta_0:=-kt(k-1)^{-1}$ in~\varepsilonqref{eq:equiv}, we can get another extremal configuration where all sets are the same: $A_0+\varepsilonta_0=\dots=A_k+\varepsilonta_k$. Thus Theorem~\ref{th:main} directly implies the following corollary.
\begin{corollary}\label{cr:main} For every $k\geqslant 2$ with $k\not\varepsilonquiv1\pmod p$ and $a\in [0,p]$, there is $t\in\Z p$ such that $s_k(a)=s_k([t,t+a-1])$.\qed\varepsilonnd{corollary}
Unfortunately, if $k\geqslant 3$, then there may be sets $A$ different from APs that attain equality in Corollary~\ref{cr:main} with $s_k(|A|)>0$ (which is in contrast to the case $k=2$). For example, our (non-exhaustive) search showed that this happens already for $p=17$, when
$$
s_3(14)=2255=s_3([-1,12])=s_3([6,18]\cup\{3\}).
$$
Also, already the case $k=2$ of the more general Theorem~\ref{th:main} exhibits extra solutions. Of course, by analysing the proof of Theorem~\ref{th:main} or Corollary~\ref{cr:main} one can write a necessary and sufficient condition for the cases of equality. We do this in Section~\ref{knot=1}; in some cases this condition can be simplified.
The first main result of this paper is
to describe the extremal sets for Corollary~\ref{cr:main} when $k \not\varepsilonquiv 1 \pmod p$ is sufficiently large.
The proof uses basic Fourier analysis on $\Z p$.
\begin{theorem}\label{th:knot1} Let a prime $p\geqslant 7$ and an integer $a\in [3,p-3]$ be fixed, and let $k\not\varepsilonquiv1\pmod p$ be sufficiently large. Then
there exists $t \in \Z p$ for which the only $s_k(a)$-extremal sets are $\xi\cdot[t,t+a-1]$ for all non-zero $\xi \in \Z p$.
\varepsilonnd{theorem}
\begin{problem} Find a `good' description of all extremal families for Corollary~\ref{cr:main} (or perhaps Theorem~\ref{th:main}) for $k\geqslant 3$.\varepsilonnd{problem}
While Corollary~\ref{cr:main} provides an example of an $s_k(a)$-extremal set for $k\not\varepsilonquiv1\pmod p$, the case $k\varepsilonquiv1\pmod p$ of the $s_k(a)$-problem turns out to be somewhat special. Here, translating a set $A$ has no effect on the quantity $s_k(A)$. More generally, let $\mathcal{A}$ be the group of all invertible affine transformations of $\Z p$, that is, it consists of maps $x\mapsto \xi\cdot x+\varepsilonta$, $x\in\Z p$, for $\xi,\varepsilonta\in \Z p$ with $\xi\not=0$. Then \begin{equation}\label{eq:equiv1}
s_k(\alpha(A))=s_k(A),\quad \mbox{for every $k\varepsilonquiv1\!\!\pmod p$\ \ and\ \ $\alpha\in\mathcal{A}$}.
\varepsilonnd{equation}
Let us call two subsets $A,B\subseteq \Z p$ \varepsilonmph{(affine) equivalent} if there is $\alpha\in \mathcal{A}$ with $\alpha(A)=B$. By~\varepsilonqref{eq:equiv1}, we need to consider sets only up to this equivalence.
Trivially, any two subsets of $\Z p$ of size $a$ are equivalent if $a \leqslantq 2$ or $a \geqslantq p-2$.
Our second main result, is to describe the extremal sets when $k \varepsilonquiv 1 \pmod p$ is sufficiently large, again using Fourier analysis on $\Z p$.
\begin{theorem}\label{th:k1} Let a prime $p\geqslant 7$ and an integer $a\in [3,p-3]$ be fixed, and let $k\varepsilonquiv1\pmod p$ be sufficiently large. Then the following statements hold for the $s_k(a)$-problem.
\begin{enumerate}
\item If $a$ and $k$ are both even, then $[a]$ is the unique (up to affine equivalence) extremal set.
\item If at least one of $a$ and $k$ is odd, define $I':=[a-1]\cup\{a\}=\{0,\dots,a-2,a\}$. Then
\begin{enumerate}
\item $s_k(a)<s_k([a])$ for all large $k$;
\item $I'$ is the unique extremal set for infinitely many $k$;
\item $s_k(a)<s_k(I')$ for infinitely many $k$, provided there are at least three non-equivalent $a$-subsets of $\Z p$.
\varepsilonnd{enumerate}
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
It is not hard to see that there are at least three non-equivalent $a$-subsets of $\Z p$ if and only if $p\geqslant 13$ and $a\in [3,p-3]$, or $p\geqslant 11$ and $a\in [4,p-4]$. Thus Theorem~\ref{th:k1} characterises pairs $(p,a)$ for which there exists an $a$-subset $A$ which is $s_k(a)$-extremal for \varepsilonmph{all} large $k\varepsilonquiv1\pmod p$.
\begin{corollary} Let $p$ be a prime and $a\in[0,p]$. There is an $a$-subset $A\subseteq \Z p$ with $s_k(A)=s_k(a)$ for all large $k\varepsilonquiv1\pmod p$ if and only if $a\leqslant 2$, or $a\geqslant p-2$, or $p\in\{7,11\}$ and $a=3$.\qed
\varepsilonnd{corollary}
As is often the case in mathematics, a new result leads to further open problems.
\begin{problem} Given $a\in[3,p-3]$, find a `good' description of all $a$-subsets of $\Z p$ that are $s_k(a)$-extremal for at least one (resp.\ infinitely many) values of $k\varepsilonquiv1\pmod p$.\varepsilonnd{problem}
\begin{problem} Is it true that for every $a\in[3,p-3]$ there is $k_0$ such that for all $k\geqslant k_0$ with $k\varepsilonquiv 1\pmod p$, any two $s_k(a)$-extremal sets are affine equivalent?\varepsilonnd{problem}
\section{Proof of Theorem~\ref{th:main}}\label{knot=1}
For completeness, here we prove Theorem~\ref{th:main}, which is a special case of Theorem~1 in~\cite{Lev01duke}.
Let $A_1,\dots,A_k$ be subsets of $\Z p$. Define $\sigma(x;A_1,\dots,A_k)$ as the number of $k$-tuples $(x_1,\dots,x_k)\in A_1\times\dots\times A_k$ with $x=x_1+\dots+x_k$. Also, for an integer $r\geqslant 0$, let
\begin{eqnarray*}
N_r(A_1,\dots,A_k)&:=&\{x\in\Z p: \sigma(x;A_1,\dots,A_k)\geqslant r\},\\
n_r(A_1,\dots,A_k)&:=&|N_r(A_1,\dots,A_k)|.
\varepsilonnd{eqnarray*}
These notions are related to our problem because of the following easy identity:
\begin{equation}\label{eq:s}
s(A_0,\dots,A_k)=\sum_{r=1}^\infty |A_0\cap N_r(A_1,\dots,A_k)|.
\varepsilonnd{equation}
Let an \varepsilonmph{interval} mean an arithmetic progression with difference $1$, i.e.\ a subset $I$ of $\Z p$ of form $\{x,x+1,\dots,x+y\}$. Its \varepsilonmph{centre} is $x+y/2\in \I Z_p$; it is unique if $I$ is \varepsilonmph{proper} (that is, $0<|I|<p$).
Note the following easy properties of the sets $N_r$:
\begin{enumerate}
\item These sets are nested:
\begin{equation}\label{eq:nested}
N_0(A_1,\dots,A_k)=\Z p\supseteq N_1(A_1,\dots,A_k)\supseteq N_2(A_1,\dots,A_k)\supseteq \dots
\varepsilonnd{equation}
\item If each $A_i$ is an interval with centre $c_i$, then $N_r(A_1,\dots,A_k)$ is an interval with centre $c_1+\dots+c_k$.
\varepsilonnd{enumerate}
We will also need the following result of Pollard~\cite[Theorem~1]{Pollard75}.
\begin{theorem}\label{th:Pollard} Let $p$ be a prime, $k\geqslant 1$, and $A_1,\dots,A_k$ be subsets of $\Z{p}$ of sizes $a_1,\dots,a_k$. Then for every integer $r\geqslant 1$, we have
$$
\sum_{i=1}^r n_i(A_1,\dots,A_k)\geqslant \sum_{i=1}^r n_i([a_1],\dots,[a_k]).\qed
$$
\varepsilonnd{theorem}
\bpf[Proof of Theorem~\ref{th:main}] Let $A_0,\dots,A_k$ be some extremal sets for the $s(a_0,\dots,a_k)$-problem. We can assume that $0<a_0<p$, because
$s(A_0,\dots,A_k)$ is $0$ if $a_0=0$ and $\prod_{i=1}^ka_i$
if $a_0=p$, regardless of the choice of the sets $A_i$.
Since $n_0([a_1],\dots,[a_k])=p>p-a_0$ while $n_r([a_1],\dots,[a_k])=0<p-a_0$ when, for example, $r>\prod_{i=1}^{k-1} a_i$, there is a (unique) integer $r_0\geqslant 0$ such that
\begin{eqnarray}
n_{r}([a_1],\dots,[a_k])&>&p-a_0,\quad\mbox{all $r\in [0,r_0]$,}\label{eq:r01}\\
n_{r}([a_1],\dots,[a_k])&\leqslant &p-a_0,\quad\mbox{all integers $r\geqslant r_0+1$.}\label{eq:r02}
\varepsilonnd{eqnarray}
The nested intervals $N_1([a_1],\dots,[a_k])\supseteq N_2([a_1],\dots,[a_k])\supseteq\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}$ have the same centre $c:=((a_1-1)+\dots+(a_k-1))/2$. Thus there is a translation $I:=[a_0]+t$ of $[a_0]$, with $t$ independent of $r$, which has as small as possible intersection with each $N_r$-interval above given their sizes, that is,
\begin{equation}\label{eq:intersection}
|I\cap N_r([a_1],\dots,[a_k])|=\max\{\,0,\,n_r([a_1],\dots,[a_k])+a_0-p\,\},\quad \mbox{for all $r\in\I N$}.
\varepsilonnd{equation}
This and Pollard's theorem give the following chain of inequalities:
\begin{eqnarray*}
s(A_0,\dots,A_k)&\stackrel{\varepsilonqref{eq:s}}{=}& \sum_{i=1}^\infty |A_0\cap N_i(A_1,\dots,A_k)|\\
&\geqslant & \sum_{i=1}^{r_0} |A_0\cap N_i(A_1,\dots,A_k)|\\
&\geqslant & \sum_{i=1}^{r_0} (n_i(A_1,\dots,A_k)+a_0-p)\\
&\stackrel{\mathrm{Thm~\ref{th:Pollard}}}{\geqslant} & \sum_{i=1}^{r_0} (n_i([a_1],\dots,[a_k])+a_0-p)\\
&\stackrel{\varepsilonqref{eq:r01}-\varepsilonqref{eq:r02}}{=} & \sum_{i=1}^{\infty} \max\{\,0,\, n_i([a_1],\dots,[a_k])+a_0-p\,\}\\
&\stackrel{\varepsilonqref{eq:intersection}}=& \sum_{i=1}^{\infty}|I\cap N_i([a_1],\dots,[a_k])|\\
&\stackrel{\varepsilonqref{eq:s}}=& s(I,[a_1],\dots,[a_k]),
\varepsilonnd{eqnarray*}
giving the required.\varepsilonpf
Let us write a necessary and sufficient condition for equality in Theorem~\ref{th:main} in the case $a_0,\dots,a_k\in [1,p-1]$. Let $r_0\geqslant 0$ be defined by \varepsilonqref{eq:r01}--\varepsilonqref{eq:r02}. Then, by~\varepsilonqref{eq:nested}, a sequence $A_0,\dots,A_k\subseteq \Z p$ of sets of sizes respectively $a_0,\dots,a_k$ is extremal if and only if
\begin{eqnarray}
A_0\cap N_{r_0+1}(A_1,\dots,A_k)&=&\varepsilonmptyset,\label{eq:empty}\\
A_0\cup N_{r_0}(A_1,\dots,A_k)&=& \Z p,\label{eq:whole}\\
\sum_{i=1}^{r_0} n_i(A_1,\dots,A_k)&=& \sum_{i=1}^{r_0} n_i([a_1],\dots,[a_k]).\label{eq:PollEq}
\varepsilonnd{eqnarray}
Let us now concentrate on the case $k=2$, trying to simplify the above condition. We can assume that no $a_i$ is equal to 0 or $p$ (otherwise the choice of the other two sets has no effect on $s(A_0,A_1,A_2)$ and every triple of sets of sizes $a_0$, $a_1$ and $a_2$ is extremal). Also, as in~\cite{SamotijSudakov16pmcps}, let us exclude the case $s(a_0,a_1,a_2)=0$, as then there are in general many extremal configurations. Note that $s(a_0,a_1,a_2)=0$ if and only if $r_0=0$; also,
by the Cauchy-Davenport theorem (the special case $k=2$ and $r=1$ of Theorem~\ref{th:Pollard}), this is equivalent to $a_1+a_2-1\leqslant p-a_0$.
Assume by symmetry that $a_1\leqslant a_2$.
Note that~\varepsilonqref{eq:r01} implies that $r_0\leqslant a_1$.
The condition in~\varepsilonqref{eq:PollEq} states that we have equality in Pollard's theorem. A result of Nazarewicz, O'Brien, O'Neill and Staples~\cite[Theorem~3]{NazarewiczObrienOneillStaples07} characterises when this happens (for $k=2$), which in our notation is the following.
\begin{theorem}\label{th:NazarewiczObrienOneillStaples07} For $k=2$ and $1\leqslant r_0\leqslant a_1\leqslant a_2<p$, we have equality in~\varepsilonqref{eq:PollEq} if and only if at least one of the following conditions holds:
\begin{enumerate}
\item\label{it:1} $r_0=a_1$,
\item\label{it:2} $a_1+a_2\geqslant p+r_0$,
\item\label{it:3} $a_1=a_2=r_0+1$ and $A_2=g-A_1$ for some $g\in \Z{p}$,
\item\label{it:4} $A_1$ and $A_2$ are arithmetic progressions with the same difference.
\varepsilonnd{enumerate}
\varepsilonnd{theorem}
Let us try to write more explicitly each of these four cases, when combined with~\varepsilonqref{eq:empty} and~\varepsilonqref{eq:whole}.
First, consider the case $r_0=a_1$. We have $N_{a_1}([a_1],[a_2])=[a_1-1,a_2-1]$ and thus $n_{a_1}([a_1],[a_2])=a_2-a_1+1>p-a_0$, that is, $a_2-a_1\geqslant p-a_0$. The condition~\varepsilonqref{eq:empty} holds automatically since $N_i(A_1,A_2)=\varepsilonmptyset$ whenever $i>|A_1|$. The other condition~\varepsilonqref{eq:whole} may be satisfied even when none of the sets $A_i$ is an arithmetic progression (for example, take $p=13$, $A_1=\{0,1,3\}$, $A_2=\{0,2,3,5,6,7,9,10\}$ and let $A_0$ be the complement of $N_3(A_1,A_2)=\{3,6,10\}$). We do not see any better characterisation here, apart from stating that~\varepsilonqref{eq:whole} holds.
Next, suppose that $a_1+a_2\geqslant p+r_0$. Then, for any two sets $A_1$ and $A_2$ of sizes $a_1$ and $a_2$, we have $N_{r_0}(A_1,A_2)=\Z p$; thus~\varepsilonqref{eq:whole} holds automatically. Similarly to the previous case, there does not seem to be a nice characterisation of~\varepsilonqref{eq:empty}. For example,~\varepsilonqref{eq:empty} may hold even when none of the sets $A_i$ is an AP: e.g.\ let $p=11$, $A_1=A_2=\{0,1,2,3,4,5,7\}$, and let $A_0=\{0,2,10\}$ be the complement of $N_4(A_1,A_2)=\{1,3,4,5,6,7,8,9\}$ (here $r_0=3$).
\comment{Let us verify that indeed $r_0=3$. Indeed, $n_3([7],[7])=11$ by above while $N_4([7],[7])=[3,9]$ has $7\leqslant 11-a_0=8$ elements.
}
Next, suppose that we are in the third case. The primality of $p$ implies that $g\in\Z p$ satisfying $A_2=g-A_1$ is unique and thus $N_{r_0+1}(A_1,A_2)=\{g\}$. Therefore~\varepsilonqref{eq:empty} is equivalent to $A_0\not\ni g$. Also, note that if $I_1$ and $I_2$ are intervals of size $r_0+1$, then $n_{r_0}(I_1,I_2)=3$. By the definition of $r_0$, we have
$p-2\leqslant a_0\leqslant p-1$.
Thus we can choose any integer $r_0\in [1,p-2]$ and $(r_0+1)$-sets $A_2=g-A_1$, and then let $A_0$ be obtained from $\Z p$ by removing $g$ and at most one further element of $N_{r_0}(A_1,A_2)$. Here, $A_0$ is always an AP (as a subset of $\Z p$ of size $a_0\geqslant p-2$) but $A_1$ and $A_2$ need not be.
Finally, let us show that if $A_1$ and $A_2$ are arithmetic progressions with the same difference $d$ and we are not in Case~1 nor~2 of Theorem~\ref{th:NazarewiczObrienOneillStaples07}, then $A_0$ is also an arithmetic progression whose difference is~$d$. By~\varepsilonqref{eq:equiv}, it is enough to prove this when $A_1=[a_1]$ and $A_2=[a_2]$ (and $d=1$).
Since $a_1+a_2\leqslant p-1+r_0$ and $r_0+1\leqslant a_1\leqslant a_2$, we have that
\begin{eqnarray*} N_{r_0}(A_1,A_2)&=&[r_0-1,a_1+a_2-r_0-1]\\
N_{r_0+1}(A_1,A_2)&=& [r_0,a_1+a_2-r_0-2]
\varepsilonnd{eqnarray*}
have sizes respectively $a_1+a_2-2r_0+1<p$ and $a_1+a_2-2r_0-1>0$. We see that $N_{r_0+1}(A_1,A_2)$ is obtained from the proper interval $N_{r_0}(A_1,A_2)$ by removing its two endpoints. Thus $A_0$, which is sandwiched between the complements of these two intervals by~\varepsilonqref{eq:empty}--\varepsilonqref{eq:whole}, must be an interval too. (And, conversely, every such triple of intervals is extremal.)
\section{The proof of Theorems~\ref{th:knot1} and~\ref{th:k1}}
Let us recall the basic definitions and facts of Fourier analysis on $\Z p$. For a more detailed treatment of this case, see e.g.~\cite[Chapter~2]{Terras99faofg}.
Write $\omega := e^{2\pi i /p}$ for the $p^{\mathrm{th}}$ root of unity.
Given a function $f : \Z p \rightarrow \mathbb{C}$, we define its \varepsilonmph{Fourier transform} to be the function $\fourier{f}:\Z p\to \mathbb{C}$ given by
$$
\fourier{f}(\gamma) := \sum_{x=0}^{p-1} f(x)\, \omega^{-x\gamma}, \qquad\text{for } \gamma \in \Z p.
$$
Parseval's identity states that
\begin{equation}\label{eq:Parseval}
\sum_{x=0}^{p-1} f(x)\,\overline{g(x)} = \frac{1}{p}\sum_{\gamma=0}^{p-1} \fourier{f}(\gamma)\,\overline{\fourier{g}(\gamma)}.
\varepsilonnd{equation}
The \varepsilonmph{convolution} of two functions $f,g : \Z p \rightarrow \mathbb{C}$ is given by
$$
(f * g)(x) := \sum_{y=0}^{p-1}f(y)\,g(x-y).
$$
It is not hard to show that the Fourier transform of a convolution equals the product of Fourier transforms, i.e.
\begin{equation}\label{convolution}
\fourier{f_1 * \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} * f_k} = \fourier{f_1} \cdot\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} \cdot \fourier{f_k}.
\varepsilonnd{equation}
We write $f^{*k}$ for the convolution of $f$ with itself $k$ times. (So, for example, $f^{* 2} = f * f$.)
Denote by $\mathbbm{1}_A$ the \varepsilonmph{indicator function} of $A \subseteq \Z p$ which assumes value 1 on $A$ and $0$ on $\Z p\setminus A$. We will call $\fourier{{\mathbbm{1}}}_A(0)=|A|$ the \varepsilonmph{trivial Fourier coefficient of $A$}.
Since the Fourier transform behaves very nicely with respect to convolution, it is not surprising that our parameter of interest, $s_k(A)$, can be written as a simple function of the Fourier coefficients of $\mathbbm{1}_A$.
Indeed,
let $ A \subseteq \mathbb{Z}_p$ and $x \in \Z p$.
Then the number of tuples $(a_1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a_k) \in A^k$ such that $a_1+\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}+a_k=x$ (which is $\sigma(x;A,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},A)$ in the notation of Section~\ref{knot=1}) is precisely $\mathbbm{1}^{* k}_A(x)$. The function $s_k(A)$ counts such a tuple if and only if its sum $x$ also lies in $A$.
Thus,
\begin{equation}\label{eq:sk(A)}
s_k(A) = \sum_{x = 0}^{p-1}\mathbbm{1}^{* k}_A(x)\, \mathbbm{1}_A(x) \stackrel{(\ref{eq:Parseval})}= \frac{1}{p}\sum_{\gamma=0}^{p-1}\fourier{\mathbbm{1}^{* k}_A}(\gamma)\,\overline{\fourier{\mathbbm{1}_A}(\gamma)} \stackrel{(\ref{convolution})}{=} \frac{1}{p} \sum_{\gamma=0}^{p-1} \leqslantft(\fourier{\mathbbm{1}_A}(\gamma)\right)^k\, \overline{\fourier{\mathbbm{1}_A}(\gamma)}.
\varepsilonnd{equation}
Since every set $A \subseteq \Z p$ of size $a$ has the same trivial Fourier coefficient (namely $\fourier{\mathbbm{1}_A}(0)=a$), let us re-write~\varepsilonqref{eq:sk(A)} as
\beq{eq:SkF1}
p s_k(A)-a^{k+1}= \sum_{\gamma=1}^{p-1} (\fourier{\mathbbm{1}_A}(\gamma))^k\,
\O{\fourier{\mathbbm{1}_A}(\gamma)} =: F(A).
\varepsiloneq
Thus we need to minimise $F(A)$ (which is a real number for any $A$) over $a$-subsets $A\subseteq\Z p$.
To do this when $k$ is sufficiently large, we will consider the largest in absolute value non-trivial Fourier coefficient $\fourier{\mathbbm{1}_{A}}(\gamma)$ of an $a$-subset $A$. Indeed, the term $(\fourier{\mathbbm{1}_A}(\gamma))^k\overline{\fourier{\mathbbm{1}_A}(\gamma)}$ will dominate $F(A)$, so if it has strictly negative real part, then $F(A)<F(B)$ for all $a$-subsets~$B\subseteq \Z p$ with $\max_{\delta\not=0}|\fourier{\mathbbm{1}_B}(\delta)|<|\fourier{\mathbbm{1}_A}(\gamma)|$.
Given $a \in [p-1]$, let
$$
I := [a]=\lbrace 0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a-1\rbrace\quad\text{and}\quad I' := [a-1]\cup\lbrace a \rbrace = \lbrace a,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},a-2,a\rbrace.
$$
In order to prove Theorems~\ref{th:knot1} and~\ref{th:k1}, we will make some preliminary observations about these special sets.
The set of $a$-subsets which are affine equivalent to $I$ is precisely the set of $a$-APs.
Next we will show that
\begin{equation}\label{skI}
F(I) = 2\sum_{\gamma=1}^{(p-1)/2} (-1)^{\gamma(a-1)(k-1)} \leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1}\quad\text{if }k \varepsilonquiv 1 \pmod p.
\varepsilonnd{equation}
Note that $(-1)^{\gamma(a-1)(k-1)}$ equals $(-1)^\gamma$ if both $a,k$ are even and 1 otherwise.
To see~\varepsilonqref{skI}, let $\gamma \in \lbrace 1,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},\frac{p-1}{2}\rbrace$ and write $\fourier{\mathbbm{1}_I}(\gamma) = re^{\theta i}$ for some $r >0$ and $0 \leqslantq \theta < 2\pi$.
Then $\theta$ is the midpoint of $0,-2\pi \gamma/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt}, -2(a-1)\gamma\pi/p$,~i.e.
$
\theta = -\pi(a-1)\gamma/p
$.
Choose $s \in \mathbb{N}$ such that $k = sp+1$.
Then
\begin{equation}\label{termx}
(\fourier{\mathbbm{1}_I}(\gamma))^k \overline{\fourier{\mathbbm{1}_I}(\gamma)} = \leqslantft(r e^{-\pi i (a-1)\gamma/p}\right)^k r e^{\pi i (a-1)\gamma/p} = r^{k+1} e^{-\pi i(a-1)\gamma s},
\varepsilonnd{equation}
and $e^{-\pi i(a-1)s}$ equals $1$ if $(a-1)s$ is even, and $-1$ if $(a-1)s$ is odd.
Note that, since $p$ is an odd prime, $(a-1)s$ is odd if and only if $a$ and $k$ are both even.
So~(\ref{termx}) is real, and the fact that $\fourier{\mathbbm{1}_I}(p-\gamma) = \overline{\fourier{\mathbbm{1}_I}(\gamma)}$ implies that the corresponding term for $p-\gamma$ is the same as for $\gamma$.
This gives~\varepsilonqref{skI}.
A very similar calculation to~(\ref{termx}) shows that
\begin{equation}\label{skI2}
F(I+t) = \sum_{\gamma=1}^{p-1} e^{-\pi i (2t+a-1)(k-1)\gamma/p}|\fourier{\mathbbm{1}_{I+t}}(\gamma)|^{k+1}\quad\text{for all }k \geqslantq 3.
\varepsilonnd{equation}
Given $r>0$ and $0 \leqslantq \theta < 2\pi$, we write $\arg(re^{\theta i}) := \theta$.
\begin{proposition}\label{angleprop}
Suppose that $p \geqslantq 7$ is prime and $a \in [3,p-3]$.
Then
$\arg\leqslantft(\fourier{\mathbbm{1}_{I'}}(1)\right)$ is not an integer multiple of $\pi/p$.
\varepsilonnd{proposition}
\bpf
Since $\fourier{\mathbbm{1}_A}(\gamma)=-\fourier{\mathbbm{1}_{\Z p\setminus A}}(\gamma)$ for all $A \subseteq \Z p$ and non-zero $\gamma \in \Z p$, we may assume without loss of generality that $a \leqslantq p-a$. Since $p$ is odd, we have $a \leqslantq (p-1)/2$.
Suppose first that $a$ is odd.
Let $m := (a-1)/2$. Then $m \in [1,\frac{p-3}{4}]$.
Observe that translating any $A \subseteq \Z p$ changes the arguments of its Fourier coefficients by an integer multiple of $2\pi/p$.
So, for convenience of angle calculations, here we may redefine $I := [-m,m]$ and $I' := \lbrace -m-1\rbrace\cup[-m+1,m]$.
Also let $I^- := [-m+1,m-1]$, which is non-empty.
The argument of $\fourier{\mathbbm{1}_{I^-}}(1)$ is $0$.
Further, $\fourier{\mathbbm{1}_{I'}}(1) = \fourier{\mathbbm{1}_{I^-}}(1) + \omega^{m+1}+\omega^{-m}$.
Since $\omega^{m+1},\omega^{-m}$ lie on the unit circle, the argument of $\omega^{m+1}+\omega^{-m}$ is either $\pi/p$ or $\pi+\pi/p$. But the bounds on $m$ imply that it has positive real part, so $\arg(\omega^{m+1}+\omega^{-m})=\pi/p$.
By looking at the non-degenerate parallelogram in the complex plane with vertices $0,\fourier{\mathbbm{1}_{I^-}}(1),\omega^{m+1}+\omega^{-m},\fourier{\mathbbm{1}_{I'}}(1)$, we see that the argument of $\fourier{\mathbbm{1}_{I'}}(1)$ lies strictly between that of $\fourier{\mathbbm{1}_{I^-}}(1)$ and $\omega^{m+1}+\omega^{-m}$, i.e.~strictly between $0$ and $\pi/p$, giving the required.
\begin{figure}[h]\label{figure}
\centering
\begin{tikzpicture}
\clip (-3.1,-3.1) rectangle (9,3.1);
\draw[thick,gray,dotted] (-3.5,0) -- (9,0);
\foreach \x in {0,15.65217,31.30435,...,350}
{
\draw[gray!50] (0,0) -- (\x:3);
}
\draw[black,thick] (0,0) circle (3cm);
\draw (0,0) node[circle,inner sep=2,fill = black,label=left:{$0$}] (0) {};
\draw (46.9563:3) node[circle,inner sep=2,fill = black,label=right:{$\omega^{m+1}$}] (m+1) {};
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=right:{$\omega^{-m}$}] (-m) {};
\draw[red,very thick] (31.3043:2.7) arc (31.3043:-31.3043:2.7);
\draw[] (2.5,-0.75) node[draw=none,label=above:{\textcolor{red}{$I$}}] () {};
\draw[red,thick] (31.3043:2.6) -- (31.3043:2.8);
\draw[red,thick] (-30.9:2.6) -- (-30.9:2.8);
\begin{scope}[shift=(m+1)]
\draw (-31.3043:3) node[circle,inner sep=2, fill=black,label=above:{~~~~$\omega^{m+1}+\omega^{-m}$}] (pip) {};
\draw[gray!50] (0,0) -- (pip);
\varepsilonnd{scope}
\draw (46.9563:3) node[circle,inner sep=2,fill = black] (m+1) {};
\draw[black,thick] (0,0) -- (m+1);
\draw[black,thick] (0,0) -- (-m);
\draw[black,thick] (0,0) -- (pip);
\draw (0:4.5) node[circle,inner sep=2, fill=black,label=below:{$\fourier{\mathbbm{1}_{I^-}}(1)$}] (I-) {};
\draw[thick] (0,0) -- (I-);
\begin{scope}[shift=(I-)]
\draw[] (7.8261:4) node[circle,inner sep=2, fill=black,label=above:{$\fourier{\mathbbm{1}_{I'}}(1)$}] (pipshift) {};
\draw[gray!50] (0,0) -- (pipshift);
\varepsilonnd{scope}
\draw (0,0) -- (pipshift);
\draw[gray!50] (pipshift) -- (pip);
\draw[] (I-) node[circle,inner sep=2, fill=black] () {};
\varepsilonnd{tikzpicture}
\varepsilonnd{figure}
Suppose now that $a$ is even and let $m := (a-2)/2 \in [1,\frac{p-5}{4}]$.
Again without loss of generality we may redefine $I := [-m,m+1]$ and $I' := \lbrace -m-1\rbrace \cup [ -m+1,m+1]$.
Let also $I^- := [-m+1,m]$, which is non-empty.
The argument of $\fourier{\mathbbm{1}_{I^-}}(1)$ is $-\pi/p$.
Further, $\fourier{\mathbbm{1}_{I'}}(1) = \fourier{\mathbbm{1}_{I^-}}(1) + \omega^{m+1}+\omega^{-(m+1)}$.
The argument of $\omega^{m+1}+\omega^{-(m+1)}$ is $0$, so as before the argument of $\fourier{\mathbbm{1}_{I'}}(1)$ is strictly between $-\pi/p$ and $0$, as required.
\varepsilonpf
We say that an $a$-subset $A$ is a \varepsilonmph{punctured interval} if $A=I'+t$ or $A = -I'+t$ for some $t \in \Z p$.
That is, $A$ can be obtained from an interval of length $a+1$ by removing a penultimate point.
\begin{lemma}\label{int-equivalence}
Let $p \geqslantq 7$ be prime and let $a \in \lbrace 3,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},p-3\rbrace$.
Then the sets $I,I'\subseteq \Z p$ are not affine equivalent.
Thus no punctured interval is affine equivalent to an interval.
\varepsilonnd{lemma}
\bpf Suppose on the contrary that there is $\alpha \in \mathcal{A}$ with $\alpha(I')=I$. Let a \varepsilonmph{reflection} mean an affine map $R_c$ with $c\in\Z p$ that maps $x$ to $-x+c$.
Clearly, $I=[a]$ is invariant under the reflection $R:=R_{a-1}$. Thus $I'$ is invariant under the map $R':=\alpha^{-1}\circ R\circ \alpha$. As is easy to see, $R'$ is also some reflection and thus preserves the cyclic distances in $\Z p$. So $R'$ has to fix $a$, the unique element of $I'$ with both distance-1 neighbours lying outside of $I'$. Furthermore, $R'$ has to fix $a-2$, the unique element of $I'$ at distance 2 from $a$. However, no reflection can fix two distinct elements of $\Z p$, a contradiction.
\varepsilonpf
We remark that the previous lemma can also be deduced from Proposition~\ref{angleprop}. Indeed, for any $A \subseteq \Z p$, the multiset of Fourier coefficients of $A$ is the same as that of $x\cdot A$ for $x \in \Z p\setminus\lbrace 0 \rbrace$, and translating a subset changes the argument of Fourier coefficients by an integer multiple of $2\pi/p$. Thus for every subset which is affine equivalent to $I$, the argument of each of its Fourier coefficients is an integer multiple of $\pi/p$.
Let
$$
\rho(A) := \max_{\gamma \in \Z p \setminus \lbrace 0 \rbrace}|\fourier{\mathbbm{1}}_A(\gamma)|\quad\text{and}\quad R(a) := \leqslantft\lbrace \rho(A) : A \in \binom{\Z p}{a}\right\rbrace = \lbrace m_1(a) > m_2(a) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} \rbrace.
$$
Given $j \geqslantq 1$, we say that $A$ \varepsilonmph{attains} $m_j(a)$, and specifically that \varepsilonmph{$A$ attains $m_j(a)$ at $\gamma$} if $m_j(a) = \rho(A)=|\fourier{\mathbbm{1}_A}(\gamma)|$.
Notice that, since $\fourier{\mathbbm{1}_A}(-\gamma)=\overline{\fourier{\mathbbm{1}_A}(\gamma)}$, the set $A$ attains $m_j(a)$ at $\gamma$ if and only if $A$ attains $m_j(a)$ at $-\gamma$ (and $\gamma,-\gamma \neq 0$ are distinct values).
As we show in the next lemma, the $a$-subsets which attain $m_1(a)$ are precisely the affine images of $I$ (i.e.~arithmetic progressions), and the $a$-subsets which attain $m_2(a)$ are the affine images of the punctured interval~$I'$.
\begin{lemma}\label{lm:MaxNontriv} Let $p\geqslantq 7$ be prime and let $a\in [3,p-3]$. Then $|R(a)| \geqslantq 2$ and
\begin{itemize}
\item[(i)] $A \in \binom{\Z p}{a}$ attains $m_1(a)$ if and only if $A$ is affine equivalent to $I$, and every interval attains $m_1(a)$ at $1$ and $-1$ only;
\item[(ii)] $B\in\binom{\Z p}{a}$ attains $m_2(a)$ if and only if $B$ is affine equivalent to $I'$, and every punctured interval attains $m_2(a)$ at $1$ and $-1$ only.
\varepsilonnd{itemize}
\varepsilonnd{lemma}
\bpf
Given $D \in \binom{\Z p}{a}$, we claim that there is some $D_{\rm{pri}} \in \binom{\Z p}{a}$ with the following properties:
\begin{itemize}
\item $D_{\rm{pri}}$ is affine equivalent to $D$;
\item $\rho(D) = |\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)|$; and
\item $-\pi/p < \arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) \leqslantq \pi/p$.
\varepsilonnd{itemize}
Call such a $D_{\rm{pri}}$ a \varepsilonmph{primary image} of $D$.
Indeed, suppose that $\rho(D) = |\fourier{\mathbbm{1}_D}(\gamma)|$ for some non-zero $\gamma \in \Z p$, and let $\fourier{\mathbbm{1}_D}(\gamma) = r'e^{\theta' i}$ for some $r' > 0$ and $0 \leqslantq \theta' < 2\pi$.
(Note that we have $r'>0$ since $p$ is prime.)
Choose $\varepsilonll \in \lbrace 0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},p-1\rbrace$ and $-\pi/p < \phi \leqslantq \pi/p$ such that $\theta' = 2\pi \varepsilonll/p + \phi$.
Let $D_{\rm{pri}} := \gamma\cdot D + \varepsilonll$. Then
$$
|\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)| = \leqslantft|\sum_{x \in D}\omega^{-\gamma x - \varepsilonll}\right| = |\omega^{-\varepsilonll} \fourier{\mathbbm{1}_D}(\gamma)| = |\fourier{\mathbbm{1}_D}(\gamma)| = \rho(D),
$$
and
$$
\arg\leqslantft(\fourier{\mathbbm{1}_{D_{\rm{pri}}}}(1)\right) = \arg(e^{\theta' i}\omega^{-\varepsilonll}) = 2\pi \varepsilonll/p + \phi - 2\pi \varepsilonll/p = \phi,
$$
as required.
Let $D \subseteq \Z p$ have size $a$ and write $\fourier{\mathbbm{1}_D}(1) = re^{\theta i}$.
Assume by the above that $-\pi/p < \theta \leqslantq \pi/p$.
For all $j \in \Z p$, let
$$
h(j) := \Re(\omega^{-j}e^{-\theta i}) = \cos\leqslantft(\frac{2\pi j}{p}+\theta\right),
$$
where $\Re(z)$ denotes the real part of $z\in\mathbb{C}$.
Given any $a$-subset $E$ of $\Z p$, we have
\begin{equation}\label{hmbound}
H_D(E) := \sum_{j \in E}h(j) = \Re\leqslantft(e^{-\theta i}\sum_{j \in E}\omega^{-j}\right) = \Re\leqslantft(e^{-\theta i} \fourier{\mathbbm{1}_E}(1)\right) \leqslantq |\fourier{\mathbbm{1}_E}(1)|.
\varepsilonnd{equation}
Then
\begin{equation}\label{HAA}
H_D(D) = \sum_{j \in D}h(j) = \Re(e^{-\theta i} \fourier{\mathbbm{1}_D}(1)) = r = |\fourier{\mathbbm{1}_D}(1)|.
\varepsilonnd{equation}
Note that $H_D(E)$ is the (signed) length of the orthogonal projection of $\fourier{\mathbbm{1}_E}(1)\in\mathbb{C}$ on the 1-dimensional
line $\{xe^{i\theta}: x\in\I R\}$. As stated in~\varepsilonqref{hmbound} and~\varepsilonqref{HAA}, $H_D(E)\leqslant |\fourier{\mathbbm{1}_E}(1)|$
and this is equality for $E=D$. (Both of these facts are geometrically obvious.) If $|\fourier{\mathbbm{1}_D}(1)|=m_1(a)$ is maximum, then no
$H_D(E)$ for an $a$-set $E$ can exceed $m_1(a)=H_D(D)$. Informally speaking, the main idea of the proof is that if we fix the direction $e^{i\theta}$, then the projection length is maximised if we take $a$ distinct elements $j\in \I Z_p$ with the $a$ largest values of $h(j)$, that is, if we take some interval (with the runner-up being a punctured interval).
Let us provide a formal statement and proof of this now.
\begin{claim}\label{claim}
Let $\mathcal{I}_a$ be the set of length-$a$ intervals in $\Z p$.
\begin{itemize}
\item[(i)]
Let $M_1(D) \subseteq \binom{\Z p}{a}$ consist of $a$-sets $E\subseteq \Z p$ such that $H_D(E) \geqslantq H_D(C)$ for all $C \in \binom{\Z p}{a}$.
Then $M_1(D) \subseteq \mathcal{I}_a$.
\item[(ii)] Let $M_2(D) \subseteq \binom{\Z p}{a}$ be the set of $E \notin \mathcal{I}_a$ for which $H_D(E) \geqslantq H_D(C)$ for all $C \in \binom{\Z p}{a} \setminus \mathcal{I}_a$.
Then every $E \in M_2(A)$ is a punctured interval.
\varepsilonnd{itemize}
\varepsilonnd{claim}
\bpf
Suppose that $0 < \theta < \pi/p$.
Then $h(0) > h(1) > h(-1) > h(2) > h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} > h(\frac{p-1}{2}) > h(-\frac{p-1}{2})$.
In other words, $h(j_\varepsilonll) > h(j_k)$ if and only if $\varepsilonll < k$, where $j_m := (-1)^{m-1}\lceil m/2\rceil$.
Letting $J_{a-1} := \lbrace j_0,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},j_{a-2}\rbrace$, we see that
$$
H_D(J_{a-1} \cup \lbrace j_{a-1}\rbrace) > H_D(J_{a-1} \cup \lbrace j_{a}\rbrace) > H_D(J_{a-1} \cup \lbrace j_{a+1}\rbrace), H_D(J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace) > H_D(J)
$$
for all other $a$-subsets $J$.
But $J_{a-1} \cup \lbrace j_{a-1}\rbrace$ and $J_{a-1} \cup \lbrace j_a\rbrace$ are both intervals, and $J_{a-1} \cup \lbrace j_{a+1}\rbrace$ and $J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace$ are both punctured intervals.
So in this case $M_1(D) := \lbrace J_{a-1}\cup\lbrace j_{a-1}\rbrace\rbrace$ and $M_2(D) \subseteq \lbrace J_{a-1}\cup\lbrace j_{a+1}\rbrace, J_{a-2} \cup \lbrace j_{a-1},j_a\rbrace\rbrace$, as required.
The case when $-\pi/p < \theta < 0$ is almost identical except now $j_\varepsilonll := (-1)^\varepsilonll\lceil \varepsilonll/2\rceil$ for all $0 \leqslantq \varepsilonll \leqslantq p-1$.
If $\theta=0$ then $h(0) > h(1) = h(-1) > h(2) = h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} > h(\frac{p-1}{2}) = h(-\frac{p-1}{2})$.
If $\theta=-\pi/p$ then $h(0)=h(-1) > h(1)=h(-2) > \hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt} = h(-\frac{p-1}{2}) > h(\frac{p-1}{2})$.
\varepsilonpf
{\mathrm e}dskip
\noindent
We can now prove part~(i) of the lemma.
Suppose $A \in \binom{\Z p}{a}$ attains $m_1(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Then the primary image $D$ of $A$ satisfies $|\fourier{\mathbbm{1}_D}(1)|=m_1(a)=|\fourier{\mathbbm{1}_A}(\gamma)|$.
So, for any $E \in M_1(D)$,
$$
|\fourier{\mathbbm{1}_A}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| \stackrel{(\ref{HAA})}{=} H_D(D) \leqslantq H_D(E) \stackrel{(\ref{hmbound})}{\leqslantq} |\fourier{\mathbbm{1}_E}(1)|,
$$
with equality in the first inequality if and only if $D \in M_1(D)$.
Thus, by Claim~\ref{claim}(i), $D$ is an interval, and so $A$ is affine equivalent to an interval, as required.
Further, if $A$ is an interval then $D$ is an interval if and only if $\gamma=\pm 1$.
This completes the proof of (i).
{\mathrm e}dskip
\noindent
For~(ii), note that $m_2(a)$ exists since by Lemma~\ref{int-equivalence}, there is a subset (namely $I'$) which is not affine equivalent to $I$. By~(i), it does not attain $m_1(a)$, so $\rho(I') \leqslantq m_2(a)$.
Suppose now that $B$ is an $a$-subset of $\Z p$ which attains $m_2(a)$ at $\gamma \in \Z p \setminus \lbrace 0 \rbrace$.
Let $D$ be the primary image of $B$.
Then $D$ is not an interval.
This together with Claim~\ref{claim}(i) implies that $H_D(D) < H_D(E)$ for any $E \in M_1(D)$.
Thus, for any $C \in M_2(D)$, we have
$$
m_2(a) = |\fourier{\mathbbm{1}_B}(\gamma)| = |\fourier{\mathbbm{1}_D}(1)| = H_D(D) \leqslantq H_D(C) \leqslantq |\fourier{\mathbbm{1}_C}(1)|.
$$
with equality in the first inequality if and only if $D \in M_2(D)$.
Since $C$ is a punctured interval, it is not affine equivalent to an interval. So the first part of the lemma implies that $|\fourier{\mathbbm{1}_C}(1)|\leqslant m_2(a)$.
Thus we have equality everywhere and so $D \in M_2(D)$.
Therefore $B$ is the affine image of a punctured interval, as required.
Further, if $B$ is a punctured interval, then $D$ is a punctured interval if and only if $\gamma=\pm 1$.
This completes the proof of (ii).
\varepsilonpf
We will now prove Theorem~\ref{th:knot1}.
\bpf[Proof of Theorem~\ref{th:knot1}.]
Recall that $p\geqslant 7$, $a\in [3,p-3]$ and $k > k_0(a,p)$ is sufficiently large with $k\not\varepsilonquiv 1\pmod p$. Let $I = [a]$.
Given $t \in \Z p$, write $\rho_t := (\fourier{\mathbbm{1}_{I+t}}(1))^k\overline{\fourier{\mathbbm{1}_{I+t}}(1)}$ as $r_te^{\theta_t i}$, where $\theta_t \in [0,2\pi)$ and $r_t > 0$. Then~(\ref{skI2}) says that $\theta_t$ equals $-\pi(2t+a-1)(k-1)/p$ modulo $2\pi$.
Increasing $t$ by $1$ rotates $\rho_t$ by $-2\pi(k-1)/p$. Using the fact that $k-1$ is invertible modulo $p$, we have the following.
If $(a-1)(k-1)$ is even, then the set of $\theta_t$ for $t \in \Z p$ is precisely $0,2\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-2)\pi/p$, so there is a unique $t$ (resp.\ a unique $t'$) in $\Z p$ for which $\theta_t=\pi+\pi/p$ (resp.\ $\theta_{t'} = \pi-\pi/p$). Furthermore, $t' = -(a-1)-t$ and
$I+t' = -(I+t)$; thus $I+t$ and $I+t'$ have the same set of dilations.
If $(a-1)(k-1)$ is odd, then the set of $\theta_t$ for $t \in \Z p$ is precisely $\pi/p,3\pi/p,\hspace{0.9pt}.\hspace{0.3pt}.\hspace{0.3pt}.\hspace{1.5pt},(2p-1)\pi/p$, so there is a unique $t \in \Z p$ for which $\theta_t = \pi$.
We call $t$ (and $t'$, if it exists) \varepsilonmph{optimal}.
Let $t$ be optimal.
To prove the theorem, we will show that $F(\xi\cdot(I+t)) < F(A)$ (and so $s_k(\xi\cdot(I+t))<s_k(A)$) for any $a$-subset $A\subseteq\Z p$ which is not a dilation of $I+t$.
We will first show that $F(I+t)<F(A)$ for any $a$-subset $A$ which is not affine equivalent to an interval.
By Lemma~\ref{lm:MaxNontriv}(i), we have that $|\fourier{\mathbbm{1}_{I+t}}(\pm 1)|=m_1(a)$ and $\rho(A) \leqslantq m_2(a)$.
Let $m_2'(a)$ be the maximum of $\fourier{\mathbbm{1}_J}(\gamma)$ over all length-$a$ intervals $J$ and $\gamma \in [2,p-2]$.
Lemma~\ref{lm:MaxNontriv}(i) implies that $m_2'(a)<m_1(a)$.
Thus
\begin{eqnarray}\label{knot1eq2}
\leqslantft|F(I+t)-2(m_1(a))^{k+1}\cos(\theta_t) - F(A)\right| \leqslantq (p-1)(m_2(a))^{k+1} + (p-3)\leqslantft(m_2'(a)\right)^{k+1}.
\varepsilonnd{eqnarray}
Now $\cos(\theta_t) \leqslantq \cos(\pi-\pi/p) < -0.9$ since $p \geqslantq 7$.
This together with the fact that $k\geqslant k_0(a,p)$ and Lemma~\ref{lm:MaxNontriv} imply that the absolute value of $2(m_1(a))^{k+1}\cos(\theta_t)<0$ is greater than the right-hand size of~(\ref{knot1eq2}).
Thus $F(I+t) < F(A)$, as required.
The remaining case is when $A=\zeta\cdot(I+v)$ for some non-optimal $v \in \Z p$ and non-zero $\zeta \in \Z p$. Since $s_k(A)=s_k(I+v)$, we may assume that $\zeta=1$.
Note that $\cos(\theta_t) \leqslantq \cos(\pi-\pi/p) < \cos(\pi-2\pi/p) \leqslantq \cos(\theta_v)$.
Thus
\begin{align*}
F(I+t)-F(I+v) &\leqslantq 2(m_1(a))^{k+1}(\cos(\theta_t)-\cos(\theta_v)) + (2p-4)(m_2'(a))^{k+1}\\
&\leqslantq 2(m_1(a))^{k+1}(\cos(\pi-\pi/p)-\cos(\pi-2\pi/p)) + (2p-4)(m_2'(a))^{k+1} <0
\varepsilonnd{align*}
where the last inequality uses the fact that $k$ is sufficiently large.
Thus $F(I+t)<F(I+v)$, as required.
\varepsilonpf
Finally, using similar techniques, we prove Theorem~\ref{th:k1}.
\bpf[Proof of Theorem~\ref{th:k1}.]
Recall that $p\geqslant 7$, $a\in [3,p-3]$ and $k > k_0(a,p)$ is sufficiently large with $k\varepsilonquiv1\pmod p$. Let $I:=[a]$ and $I'=[a-1]\cup\{a\}$.
Suppose first that $a$ and $k$ are both even. Let $A\subseteq\Z p$ be
an arbitrary $a$-set not affine equivalent to the interval~$I$.
By Lemma~\ref{lm:MaxNontriv}, $I$ attains $m_1(a)$ (exactly at $x=\pm 1$), while $\rho(A)<m_1(a)$. Also, $m_2'(a)<m_1(a)$, where $
m_2'(a):=\max_{\gamma\in [2,p-2]}|\fourier{\mathbbm{1}_I}(\gamma)|$. Thus
\begin{eqnarray*}
F(I) - F(A) &\stackrel{(\ref{eq:SkF1}),(\ref{skI})}{\leqslantq}& 2\sum_{\gamma=1}^{\frac{p-1}{2}} (-1)^\gamma\leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1} + \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_A}(\gamma)\right|^{k+1}\\
&\leqslantq& -2(m_1(a))^{k+1} + (2p-4) (\max\{m_2(a),m_2'(a)\})^{k+1}\ <\ 0,
\varepsilonnd{eqnarray*}
where the last inequality uses the fact that $k$ is sufficiently large.
So $s_k(a)=s_k(I)$.
Using Lemma~\ref{lm:MaxNontriv}, the same argument shows that, for all $B \in \binom{\Z p}{a}$, we have $s_k(B)=s_k(a)$ if and only if $B$ is an affine image of $I$.
This completes the proof of Part 1 of the theorem.
Suppose now that at least one of $a,k$ is odd. Let $A$ be an $a$-set not equivalent to~$I$. Again by Lemma~\ref{lm:MaxNontriv}, we have
\begin{eqnarray*}
F(I) - F(A) &\geqslantq& \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_I}(\gamma)\right|^{k+1} - \sum_{\gamma=1}^{p-1} \leqslantft|\fourier{\mathbbm{1}_A}(\gamma)\right|^{k+1}\\
&\geqslantq& 2(m_1(a))^{k+1} - (p-1)(m_2(a))^{k+1} \ >\ 0.
\varepsilonnd{eqnarray*}
So the interval $I$ and its affine images have in fact the largest number of additive $(k+1)$-tuples among all $a$-subsets of $\Z p$. In particular, $s_k(a) < s_k(I)$.
Suppose that there is some $A \in \binom{\Z p}{a}$ which is not affine equivalent to $I$ or~$I'$. (If there is no such $A$, then the unique extremal sets are affine images of $I'$ for all $k > k_0(a,p)$, giving the required.)
Write $\rho := re^{\theta i} = \fourier{\mathbbm{1}_{I'}}(1)$.
Then by Lemma~\ref{lm:MaxNontriv}(ii), we have $r=m_2(a)$, and $\rho(A) \leqslantq m_3(a)$.
Given $k \geqslantq 2$, let $s \in \mathbb{N}$ be such that $k=sp+1$. Then
\begin{equation}\label{FI'}
\Big|F(I') - 2m_2(a)^{k+1}\cos (sp\theta)-F(A)\Big|\leqslant (p-1)m_3(a)^{k+1}+(p-3)\leqslantft(m_2'(a)\right)^{k+1}.
\varepsilonnd{equation}
Proposition~\ref{angleprop} implies that there is an even integer $\varepsilonll \in \I N$
for which $c := p\theta - \varepsilonll\pi \in (-\pi,\pi)\setminus\{0\}$. Let $\varepsilon := \frac{1}{3}\min\lbrace |c|,\pi-|c|\rbrace > 0$.
Given an integer $t$, say that $s\in\I N$ is \varepsilonmph{$t$-good} if $sc \in ((t-\frac{1}{2})\pi+\varepsilon,(t+\frac{1}{2})\pi-\varepsilon)$.
This real interval has length $\pi-2\varepsilon > |c|>0$, so must contain at least one integer multiple of $c$.
In other words, for all $t \in \mathbb{Z}\setminus\{0\}$ with the same sign as $c$, there exists a $t$-good integer $s> 0$.
As $sp\theta\varepsilonquiv sc\pmod{2\pi}$, the sign of $\cos(sp\theta)$ is $(-1)^{t}$. Moreover, Lemma~\ref{lm:MaxNontriv} implies that
$m_2(a)> m_3(a), m'_2(a)$.
Thus, when $k=sp+1>k_0(a,p)$, the absolute value of $2m_2(a)^{k+1}\cos(sp\theta)$ is greater than the right-hand side of~(\ref{FI'}).
Thus, for large $|t|$, we have $F(A) > F(I')$ if $t$ is even and $F(A)<F(I')$ if $t$ is odd, implying the theorem by~\varepsilonqref{eq:SkF1}.
\varepsilonpf
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\varepsilonnd{thebibliography}
\varepsilonnd{document} |
\begin{document}
\title{Duality results for a general trigonometric approximation problem}
\begin{abstract}
Let \(\alpha\in\iva\) and \(\mu\) be a regular finite Borel measure on a locally compact abelian group.
The paper deals with a general trigonometric approximation problem in \(\Lam\), which arises in prediction theory of harmonizable symmetric \tas{} processes.
To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors.
The novelty of the present paper is that we do not make any additional assumption on \(\mu\).
Moreover, for \(\alpha=2\), multivariate extensions are obtained.
\end{abstract}
\begin{description}
\item[Keywords:] Regular Borel measure, space of \tai{} functions, trigonometric approximation, duality.
\end{description}
\section{Introduction}
Let \(w\) be a weight function on \((-\pi,\pi]\), \(S\) a nonvoid subset of the set \(\Z\) of integers, \(s\in S\), and \(\TZS\) the linear space of trigonometric polynomials with frequencies from \(\ZS\).
An important task in prediction theory of weakly stationary or, more generally, harmonizable symmetric \tas{} sequences is to compute the prediction error, \tie{}, the distance of the function \(\ec^{\iu s\cdot}\) to the set \(\TZS\) with respect to the metric of the Banach space \(\Law\), \(\alpha\in\iva\).
In 1984 Nakazi~\zita{MR766702} introduced a new idea into the study of this problem.
Among other things, his method opened a way for him to give an elegant proof of the celebrated Szeg\H{o} infimum formula.
Miamee and Pourahmadi~\zita{MR949088} pointed out that the essence of Nakazi's technique is a certain duality between the spaces \(\Law\) and \(\Ldaw\), \(\da \defeq\alpha/(\alpha-1)\), presumed that \(w^\inv\) exists and is integrable.
In the sequel this duality relation turned out to be rather fruitful.
It was applied to a variety of sets \(S\), modified, and extended to more general prediction problems, \tcf{}\ the papers~\zitas{MR1443377,MR1230571,MR1289507,MR2262930}, as well as~\zita{MR1170451} for fields on \(\Z^2\),~\zita{MR772194} for processes on discrete abelian groups,~\zita{MR1928917} for multivariate sequences.
Urbanik~\zita{MR1808671} defined a notion of a dual stationary sequence, \tcf{}~\zitaa{MR1849562}{\csec{8.5}} and~\zita{MR1899439}.
Kasahara, Pourahmadi and Inoue~\zita{MR2547424} used a modified duality method to obtain series representations of the predictor.
It should also be mentioned that \cthm{24} of~\zita{MR0009098} was, perhaps, the first published duality result of the type in question, \tcf{} its extension by Yaglom~\zitaa{MR0031214}{\cthm{2}}.
Many of the preceding results were obtained under the rather strong additional condition that \(w^\inv\) exists and is integrable, although for some special sets authors succeeded in weakening this condition, \tcf{}~\zitas{MR1443377,MR772194,MR2262930}.
In our paper we study the above described problem for spaces \(\Lam\), where the Borel measure \(\mu\) is not assumed to be absolutely continuous.
Moreover, motivated by Weron's paper~\zita{MR772194} we shall be concerned with regular finite Borel measures on locally compact abelian groups.
Since in the literature there exists differing definitions of a regular measure, \rsec{3} deals with the definition and some basic facts of regular measures.
The results of \rsecss{2}{4} show that under a condition, which is satisfied by many sets \(S\) occurring in application, one can assume that the measure \(\mu\) is absolutely continuous.
Establishing this result, we introduce a class of sets, which we call class of \tACs{s} and which, as it seems to us, deserves further investigation.
\rsec{5} gives a solution to the problem if \(\mu\) is absolutely continuous.
Unlike most of the authors above we do not make any condition on the corresponding Radon-Nikodym derivative.
In~\zita{MR1462266} there were defined Banach spaces of matrix-valued functions \tai{} with respect to a positive semidefinite matrix-valued measure, see \rsec{2} for the definition and basic facts.
Since part of our results can be easily generalized, we state and prove them in this more general framework.
If \(S\) is a singleton, the corresponding results can be used to obtain minimality criteria for multivariate stationary sequences.
We shall not go into detail but refer to the recent paper~\zita{KM15}, where various minimality notions were discussed.
If \(X\) is a matrix, denote by \(X^\ad\), \(X^\mpi\), and \(\ran{X}\) its adjoint, Moore-Penrose inverse, and range, \tresp{} The symbol \(\Oe\) stands for the zero element in an arbitrary linear space.
\section{The space $\LaM$}\label{2}
Let \(q\in\N\), \(\Mggq\) be the cone of all \tpsd{} (hence, \tH{}) \tqqa{matrices} with complex entries.
Let \(\OA\) be a measurable space and \(M\) an \taval{\Mggq} measure on \(\A\).
If \(\tau\) is a \tsf{} measure on \(\A\) such that \(M\) is absolutely continuous with respect to \(\tau\), denote by \(\dif M/\dif\tau\) the corresponding Radon-Nikodym derivative and by \(P(\omega)\) the orthoprojection in \(\Cq\) onto \(\ran{\rk{\dif M/\dif\tau}(\omega)}\), \(\omega\in\Omega\).
Let \(\no{\cdot}\) be the euclidean norm on \(\Cq\) and write the vectors of \(\Cq\) as column vectors.
Two \tAm{} \taval{\Cq} functions \(f\) and \(g\) are called \tMe{} if \(Pf=Pg\) \tae{\tau} For \(\alpha\in\iva\), denote by \(\LaM\) the space of all (\tMec{es} of) \tAm{} functions \(f\) such that \(\noa{f}\defeq\ek{\int_\Omega\no{\rk{\dif M/\dif\tau}^{1/\alpha}f}^\alpha\dif\tau}^{1/\alpha}<\infty\).
Recall that the definition of \(\LaM\) does not depend on the choice of \(\tau\) and that \(\LaM\) is a Banach space with respect to the norm \(\noa{\cdot}\).
Note that \(\LhM\) is a Hilbert space with inner product \(\int_\Omega g^\ad\rk{\dif M/\dif\tau}f\dif\tau\), \(f,g\in\LhM\), and that for \(q=1\), the space \(\LaM\) is the well known space of (equivalence classes of) \tAm{} \taval{\C} functions \tai{} with respect to \(M\).
The space \(\LhM\) was introduced by I.~S.~Kats~\zita{MR0080280} and in a somewhat more general form by Rosenberg~\zita{MR0163346}.
Both notions were applied in the theory of weakly stationary processes with the same success, \tcf{}~\zitas{MR0159363,MR0279952} for an application of Kats' and Rosenberg's definitions, \tresp{} An extension to \(\alpha\neq2\) was given by Duran and Lopez-Rodriguez~\zita{MR1462266}, \tcf{}~\zita{MR1160966} for a more general setting of operator-valued measures.
To simplify the presentation slightly we shall be concerned with the space \(\LaM\) as defined above, which can be considered as a generalization of Kats' definition to the case \(\alpha\neq2\).
\begin{lem}[\tcf{}~\zitaa{MR1462266}{\cthm{2.5}},~\zitaa{MR1160966}{\cthm{9}}]\label{L2.1}
Let \(\alpha\in\iva\) and \(\da \defeq\alpha/\rk{\alpha-1}\).
If \(\ell\) is a bounded linear functional on \(\LaM\), then there exists \(g\in\LdaM\) such that \(\ell(f)=\int_\Omega g^\ad\rk{\dif M/\dif\tau}f\dif\tau\) for all \(f\in\LaM\).
The correspondence \(\ell\mapsto g\) establishes an isometric isomorphism between the dual space of \(\LaM\) and the space \(\LdaM\).
\end{lem}
Let \(M_1\) and \(M_2\) be \taval{\Mggq} measures on \(\A\) such that \(M=M_1+M_2\) and \(M_1(A)=M_2(\Omega\setminus A)=\Oe\) for some set \(A\in\A\).
Let \(\ind{C}\) denote the indicator function of a set \(C\).
Identifying \(\LaMa\) with the space \(\indOA \LaM=\setaa{\indOA f}{f\in\LaM}\) and \(\LaMb\) with \(\indA \LaM\), one obtains a direct sum decomposition \(\LaM=\LaMa\dotplus\LaMb\).
For a linear subset \(\Lc\) of \(\LaM\), denote by \(\Lca\) its closure in \(\LaM\)
\begin{lem}\label{L2.2}
If
\begin{equation}\label{e2.1}
\LaMb
\subseteq\Lca,
\end{equation}
then
\begin{equation}\label{e2.2}
\cl{\rk{\indA \Lc}}
=\LaMb
\end{equation}
and
\begin{equation}\label{e2.3}
\Lca
=\cl{\rk{\indOA \Lc}}\dotplus\cl{\rk{\indA \Lc}}
=\cl{\rk{\indOA \Lc}}\dotplus\LaMb.
\end{equation}
\end{lem}
\begin{proof}
The continuity of the map \(f\mapsto\indA f\), \(f\in\LaM\), and condition \eqref{e2.1} yield \(\LaMb=\indA\LaM\subseteq\indA\Lca\subseteq\cl{\rk{\indA\Lc}}\).
Since the inclusion \(\cl{\rk{\indA\Lc}}\subseteq\LaMb\) is obvious, equality \eqref{e2.2} follows.
For the proof of \eqref{e2.3}, note first that \(\Lca\subseteq\cl{\rk{\indOA\Lc}}\dotplus\cl{\rk{\indA\Lc}}\).
To prove the opposite inclusion, let \(f\in\Lca\).
By \eqref{e2.2} and \eqref{e2.1}, we have \(\indA f\in\indA\Lca\subseteq\cl{\rk{\indA\Lc}}=\LaMb\subseteq\Lca\), which gives \(\cl{\rk{\indA\Lc}}\subseteq\cl{\rk{\indA\Lca}}\subseteq\Lca\).
The relation \(\indOA f=f-\indA f\in\Lca\) implies that \(\cl{\rk{\indOA\Lc}}\subseteq\Lca\), hence, \(\cl{\rk{\indOA \Lc}}\dotplus\cl{\rk{\indA \Lc}}\subseteq\Lca\).
\end{proof}
\begin{lem}\label{L2.3}
Let \(f\in\LaM\) and \(\rho\defeq\inf\setaa{\noa{f-g}}{g\in\Lc}\), \(\rho_1\defeq\inf\setaa{\noa{\indOA(f-g)}}{g\in\Lc}\).
If \eqref{e2.1} is satisfied, then \(\rho=\rho_1\).
\end{lem}
\begin{proof}
For \(\epsilon>0\), there exist \(g_1,g_2\in\Lc\) such that \(\noa{\indOA(f-g_1)}<\rho_1+\epsilon\) and \(\noa{\indA(f-g_2)}<\epsilon\) according to \eqref{e2.2}.
By \rlem{L2.2}, \(\indOA g_1+\indA g_2\in\Lca\).
Therefore, \(\rho\leq\noa{f-\rk{\indOA g_1+\indA g_2}}\leq\noa{\indOA(f-g_1)}+\noa{\indA(f-g_2)}<\rho_1+2\epsilon\), hence, \(\rho\leq\rho_1\).
Since the inequality \(\rho_1\leq\rho\) is trivial, the result follows.
\end{proof}
\section{Regular Borel measures on locally compact abelian groups}\label{3}
Let \(\Gamma\) be a locally compact abelian group, \(\BsaG\) the \tsa{} of Borel sets of \(\Gamma\), and \(\lambda\) a Haar measure on \(\BsaG\).
We recall the definition and some elementary properties of regular measures, however, we mention that we do not know an example of a non-regular finite Borel measure on a locally compact abelian group.
Therefore, some of the following assertions might be redundant.
A finite non-negative measure \(\mu\) on \(\BsaG\) is called
\begin{itemize}
\item[a)] \emph{outer regular} if \(\mu(B)=\inf\setaa{\mu(U)}{U\text{ is open and }B\subseteq U\subseteq\Gamma}\) for all \(B\in\BsaG\),
\item[b)] \emph{inner regular} if \(\mu(B)=\sup\setaa{\mu(K)}{K\text{ is compact and }K\subseteq B}\) for all \(B\in\BsaG\),
\item[c)] \emph{regular} if it is outer regular and inner regular,
\end{itemize}
\tcf{}~\zitaa{MR578344}{\cpage{206}} and~\zitaa{MR0156915}{(11.34)}.
\begin{lem}\label{L3.1}
\begin{aeqi}{0}
\item\label{L3.1.i} Let \(\mu\) be a finite non-negative measure on \(\BsaG\).
If it is inner regular, then it is outer regular.
\item\label{L3.1.ii} Any positive linear combination of regular finite non-negative measures is regular.
\item\label{L3.1.iii} Let \(\mu\) and \(\nu\) be finite non-negative measures on \(\BsaG\).
If \(\mu\) is regular and \(\nu\) is absolutely continuous with respect to \(\mu\), then \(\nu\) is regular.
\end{aeqi}
\end{lem}
\begin{proof}
To prove~\ref{L3.1.i} let \(B\in\BsaG\) and \(\co{B}\defeq\Gamma\setminus B\).
If \(\mu\) is inner regular, for \(\epsilon>0\) there exists a compact set \(K\subseteq\co{B}\) such that \(\mu(\co{B}\setminus K)<\epsilon\).
Since \(\Gamma\) is a Hausdorff space, the compact set \(K\) is closed, hence, the set \(\co{K}\) is open, and it satisfies \(B\subseteq\co{K}\) and \(\mu(\co{K}\setminus B)=\mu(\co{B}\setminus K)<\epsilon\), which implies that \(\mu\) is outer regular.
Assertion~\ref{L3.1.ii} is clear and assertion~\ref{L3.1.iii} is an immediate consequence of the following fact.
If \(\nu\) is absolutely continuous with respect to \(\mu\), for \(\epsilon>0\) there exists \(\delta>0\) such that for all \(B\in\BsaG\), the inequality \(\mu(B)<\delta\) yields \(\nu(B)<\epsilon\), \tcf{}~\zitaa{MR578344}{\clem{4.2.1}}.
\end{proof}
A \taval{\C} measure \(\mu\) on \(\BsaG\) is called \emph{regular} if its variation \(\var{\mu}\) is regular.
It is called \emph{absolutely continuous} or \emph{singular}, \tresp{}, if its variation is absolutely continuous or singular with respect to \(\lambda\).
\begin{lem}\label{L3.2}
Let \(\mu\) be an absolutely continuous and regular \taval{\C} measure on \(\BsaG\).
Then there exists a Radon-Nikodym derivative of \(\mu\) with respect to \(\lambda\).
\end{lem}
\begin{proof}
Since \(\mu\) is assumed to be regular, there exists a sequence \(\gk{K_n}_{n\in\N}\) of compact sets such that \(\var{\mu}(\Gamma\setminus K_n)<1/n\), \(n\in\N\).
If \(B\defeq\bigcup_{n=1}^\infty K_n\), then \(B\in\BsaG\), \(\var{\mu}(\Gamma\setminus B)=0\), and from \(\lambda(K_n)<\infty\), \(n\in\N\), we get that \(B\) is a set of \tsf{} \tam{\lambda}.
Thus, the assertion follows from the Radon-Nikodym theorem, \tcf{}~\zitaa{MR578344}{\cthm{4.2.3}}.
\end{proof}
By definition, an \taval{\Mggq} measure \(M\) on \(\BsaG\) is \emph{regular} if all its entries are regular.
Let \(\mu_{jk}\) be the measure at place \((j,k)\).
Since for all \(B\in\BsaG\), \(\abs{\mu_{jk}(B)}\leq\ek{\mu_{jj}(B)+\mu_{kk}(B)}/2\), hence \(\var{\mu_{jk}}(B)\leq\ek{\mu_{jj}(B)+\mu_{kk}(B)}/2\), from \rlem{L3.1} one can conclude that \(M\) is regular if and only if each measure on the principal diagonal is regular.
Taking into account \rlemp{L3.1}{L3.1.iii} and \rlem{L3.2}, we arrive at the following result.
\begin{lem}\label{L3.3}
If \(\ac{M}\) is the absolutely continuous part of a regular \taval{\Mggq} measure \(M\) on \(\BsaG\), then there exists the Radon-Nikodym derivative \(\dif\ac{M}/\dif\lambda\eqdef W\) of \(\ac{M}\) with respect to \(\lambda\).
\end{lem}
\section{A general trigonometric approximation problem in $\LaM$}\label{4}
For \(k\in\mn{1}{q}\) denote by \(\eu{k}\) the \tth{k} vector of the standard orthonormal basis of \(\Cq\).
Let \(G\) be a locally compact abelian group and \(\Gamma\) its dual.
The value of \(\gamma\in\Gamma\) at \(x\in G\) is denoted by \(\inner{\gamma}{x}\).
For \(x\in G\), define a function \(\chu{x}\) by \(\chua{x}{\gamma}\defeq\inner{\gamma}{x}\), \(\gamma\in\Gamma\), and set \(\chuu{x}{k}\defeq\chu{x}\eu{k}\).
Let \(S\) be a nonvoid subset of \(G\).
If \(\GsS\) is not empty, denote by \(\T{\GsS}\) the linear space of all \taval{\Cq} trigonometric polynomials on \(\Gamma\) with frequencies from \(\GsS\), \tie{}, \(t\in\T{\GsS}\) if and only if it has the form \(t=\sum_{j=1}^n\chu{x_j}u_j\), \(x_j\in\GsS\), \(u_j\in\Cq\), \(j\in\mn{1}{n}\), \(n\in\N\).
If \(\GsS\) is empty, let \(\T{\GsS}\) be the space consisting of the zero function on \(\Gamma\).
Let \(M\) be a regular \taval{\Mggq} measure on \(\BsaG\).
Motivated by problems of prediction theory we are interested in computing the distance
\begin{align*}
d&\defeq\inf\setaa*{\noa*{\chuu{s}{k}-t}}{t\in\T{\GsS}},&s&\in S,&k&\in\mn{1}{q}.
\end{align*}
\begin{thm}\label{T4.1}
Let \(M=\ac{M}+\si{M}\) be the Lebesgue decomposition of \(M\) into its absolutely continuous part \(\ac{M}\) and singular part \(\si{M}\).
If
\begin{equation}\label{T4.1.1}
\Loa{\alpha}{\si{M}}
\subseteq\cl{\ek*{\T{\GsS}}},
\end{equation}
then for \(s\in S\), \(k\in\mn{1}{q}\),
\[
d
=\inf\setaa*{\ek*{\int\no*{\rk*{\frac{\dif\ac{M}}{\dif\lambda}}^{1/\alpha}\rk{\chuu{s}{k}-t}}^\alpha\dif\lambda}^{1/\alpha}}{t\in\T{\GsS}},
\]
where here and in what follows the domain of integration is \(\Gamma\) if it is not indicated explicitly.
\end{thm}
The preceding theorem immediately follows from \rlemss{L2.3}{L3.3} and it shows that under condition \eqref{T4.1.1} it is enough to solve the approximation problem for the absolutely continuous part of \(M\).
The next section deals with its partial solution and the rest of the present section is devoted to subsets of \(G\) with the property that \eqref{T4.1.1} is satisfied for each regular \taval{\Mggq} measure.
For a regular \taval{\Cq} measure \(\mu\) on \(\BsaG\), denote by \(\fsinv{\mu}\) its inverse Fourier-Stieltjes transform, \tie{}\ \(\fsinv{\mu}(x)\defeq\int\inner{\gamma}{x}\mu(\dif\gamma)=\int\chu{x}\dif\mu\), \(x\in G\).
Let \(\LlCq\) be the Banach space of \tam{\BsaG}able \taval{\Cq} functions integrable with respect to \(\lambda\).
If \(f\in\LlCq\), the symbol \(\fsinv{f}\) stands for its inverse Fourier transform.
\begin{defn}\label{D4.2}
A subset \(S\) of \(G\) is called an \emph{\tACs{}}, if for all regular \taval{\Cq} measures \(\mu\) on \(\BsaG\), the equality \(\fsinv{\mu}(x)=0\) for all \(x\in S\) implies that \(\mu\) is absolutely continuous.
\end{defn}
\begin{lem}\label{L4.3}
If \(\GsS\) is an \tACs{}, then \eqref{T4.1.1} is satisfied.
\end{lem}
\begin{proof}
Assume that there exists \(f\in\Loa{\alpha}{\si{M}}\), which does not belong to \(\cl{\ek{\T{\GsS}}}\).
Then by \rlem{L2.1} there exists \(g\in\LdaM\) such that
\begin{align}\label{L4.3.1}
\int\chuu{x}{k}^\ad\frac{\dif M}{\dif\tau}g\dif\tau&=0&\text{for all }x&\in\GsS\text{, }k\in\mn{1}{q},
\end{align}
and
\begin{equation}\label{L4.3.2}
\int f^\ad\frac{\dif M}{\dif\tau}g\dif\tau
\neq0.
\end{equation}
From \eqref{L4.3.1} we can derive that the \taval{\Cq} measure \((\dif M/\dif\tau)g\dif\tau\) is absolutely continuous if \(\GsS\) is an \tACs{}.
Since \(f\in\Loa{\alpha}{\si{M}}\), it follows \(\int f^\ad(\dif M/\dif\tau)g\dif\tau=0\), a contradiction to \eqref{L4.3.2}.
\end{proof}
We conclude the section with some examples of \tACs{s}.
A deeper study of this class of subsets would be of interest.
\begin{exam}\label{E4.4}
\begin{itemize}
\item[a)] If \(S\) is a compact subset of \(G\), then \(\GsS\) is an \tACs{}.
For if \(\mu\) is a regular \taval{\Cq} measure on \(\BsaG\) and \(\fsinv{\mu}(x)=0\), \(x\in\GsS\), then the continuous function \(\fsinv{\mu}\) has compact support and, hence, is integrable with respect to a Haar measure on \(G\).
Applying the inversion theorem, \tcf{}~\zitaa{MR0152834}{\cthm{1.5.1}}, and a uniqueness property of the inverse Fourier-Stieltjes transform, \tcf{}~\zitaa{MR0152834}{\cthm{1.3.6}}, we obtain that \(\mu\) is absolutely continuous.
\item[b)] Each subset of a compact abelian group is an \tACs{}.
\item[c)] If \(S\) is an \tACs{}, then \(-S\) and \(x+S\), \(x\in G\), are \tACs{s}, and if \(S\subseteq S_1\subseteq G\), then \(S_1\) is an \tACs{}.
\item[d)] If \(G=\Z\) and \(S=\N\) or \(G=\R\) and \(S=[0,\infty)\), \tresp{}, then the Theorem of F.~and M.~Riesz implies that \(S\) is an \tACs{}, \tcf{}~\zitaa{MR0152834}{\cthmss{8.2.1}{8.2.7}}.
\item[e)] A theorem of Bochner claims that the set of points of \(\Z^2\) which belong to a closed sector of the plane whose opening is larger than \(\pi\), is an \tACs{}, \tcf{}~\zitaa{MR0152834}{\cthm{8.2.5}}.
Note that for the measure \(\mu\defeq\lambda\otimes\delta_0\) on \(\Bsa{(-\pi,\pi]\times(-\pi,\pi]}\), where \(\delta_0\) denotes the Dirac measure at \(0\), one has \(\fsinv{\mu}((m,n))=0\) for all \(m\in\Z\setminus\set{0}\), \(n\in\Z\).
This shows that the lattice points of a half-plane do not form an \tACs{}.
\end{itemize}
\end{exam}
\section{The case of an absolutely continuous measure}\label{5}
Let \(\dif M=W\dif\lambda\), where \(W\) is a \tamable{\BsaG} \taval{\Mggq} function on \(\Gamma\) integrable with respect to \(\lambda\).
For brevity, set \(\LaM\eqdef\LaW\).
It is clear that the value of \(d\) does not depend on how large the \tMec{es} are chosen.
Recall that \(P(\gamma)\) is the orthoprojection in \(\Cq\) onto \(\ran{W(\gamma)}\), \(\gamma\in\Gamma\).
If \(f\) is a \tam{\BsaG}able \taval{\Cq} function, then \(f\) and \(Pf\) are \tMe{}.
Therefore, each \tMec{} contains a function \(h\) such that
\begin{align}\label{E5.1}
h(\gamma)&\in\Ran{P(\gamma)}&\text{for all }\gamma&\in\Gamma,
\end{align}
and shrinking the \tMec{} we can and shall assume that for all functions \(h\) of the equivalence classes of \(\LaW\) relation \eqref{E5.1} is satisfied.
The goal of the present section is to derive expressions for \(d\) if \(q=1\) and \(\alpha\in\iva\) or if \(q\in\N\) and \(\alpha=2\).
Let us assume first that \(q=1\) and let us denote the scalar-valued weight function \(W\) by \(w\).
Setting \(B\defeq\setaa{\gamma\in\Gamma}{w(\gamma)\neq0}\), one has
\[
w^\mpi(\gamma)
=
\begin{cases}
1/w(\gamma),&\text{if }\gamma\in B\\
0,&\text{if }\gamma\in\Gamma\setminus B
\end{cases}
\]
and condition \eqref{E5.1} implies that if \(f\in\Law\), then \(f=0\) on \(\Gamma\setminus B\).
The distance \(d\) can be written as \[\begin{split}
d
&=\inf\setaa*{\noa{\chu{s}-t}}{t\in\T{\GsS}}\\
&=\inf\setaa*{\rk*{\int\abs{\chu{s}-t}^\alpha w\dif\lambda}^{1/\alpha}}{t\in\T{\GsS}}\\
&=\inf\setaa*{\rk*{\int\abs{\ind{B}\chu{s}-t}^\alpha w\dif\lambda}^{1/\alpha}}{t\in\ind{B}\T{\GsS}},
\end{split}\]
\(s\in S\).
For \(\alpha\in\iva\), let \(\da \defeq\alpha/(\alpha-1)\) and \(\beta\defeq1/(\alpha-1)\).
\begin{lem}\label{L5.1}
For any bounded linear functional \(\ell\) on \(\Law\), there exists \(h\in\Ldawpb \) such that
\begin{align*}
\ell(f)&=\int f h^\ad\dif\lambda,&f&\in\Law.
\end{align*}
The mapping \(\ell\mapsto h\) establishes an isometric isomorphism between the dual space of \(\Law\) and the space \(\Ldawpb \).
\end{lem}
\begin{proof}
If \(g\in\Loa{\da}{w}\), then \(\int\abs{gw}^\da \rk{w^\mpi}^\beta\dif\lambda=\int\abs{g}^\da w\dif\lambda\), which shows that the correspondence \(g\mapsto gw\), \(g\in\Loa{\da}{w}\), is an isometry from \(\Loa{\da}{w}\) into \(\Ldawpb \).
Moreover, if \(h\in\Loa{\da}{\rk{w^\mpi}^\beta}\), then \(hw^\mpi w=h\) by \eqref{E5.1} and \(\int\abs{hw^\mpi}^\da w\dif\lambda=\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda<\infty\).
Therefore, the correspondence \(g\mapsto gw\) maps \(\Loa{\da}{w}\) onto \(\Loa{\da}{\rk{w^\mpi}^\beta}\), and the lemma follows from the well known description of the dual space of \(\Law\), \tcf{}~\rlem{L2.1}.
\end{proof}
The preceding lemma shows that if \(f\in\Law\) and \(h\in\Ldawpb \) satisfies \eqref{E5.1}, then
\[
\int\abs*{fh^\ad}\dif\lambda
\leq\noa{f}\ek*{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{1/\da}.
\]
It follows that under condition \eqref{E5.1} the set
\[
\D
\defeq\setaa*{h\in\LoA{\da}{\rk{w^\mpi}^\beta}}{\fsinv{\rk{h^\ad}}(x)=0\text{ for all }x\in\GsS}
\]
is defined correctly and that \(\int fh^\ad\dif\lambda=0\) if \(f\in\cl{\T{\GsS}}\) and \(h\in\D\).
Taking into account a general approximation result in Banach spaces, \tcf{}~\zitaa{MR0268655}{\cthm{7.2}}, we can conclude that the following assertion is true.
\begin{lem}\label{L5.2}
For any \(s\in S\), the distance \(d\) is equal to \(\sup\setaa{\abs{\fsinv{\rk{h^\ad}}(s)}}{h\in\D\text{ and } \int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda\leq1}\).
\end{lem}
Let \(\Ds\defeq\setaa{h\in\D}{\fsinv{\rk{h^\ad}}(s)=1}\), \(s\in S\).
\begin{thm}\label{T5.3}
Let \(q=1\), \(\alpha\in\iva\), and \(s\in S\).
Then
\begin{equation}\label{E5.2}
d
=\inf\setaa*{\noa{\chu{s}-t}}{t\in\T{\GsS}}
=\sup\setaa*{\ek*{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{-1/\da}}{h\in\Ds},
\end{equation}
with the convention, that the right-hand side of \eqref{E5.2} is assumed to be \(0\) if \(\Ds\) is empty.
\end{thm}
\begin{proof}
Define on the linear space \(\D\) two positive homogeneous and non-negative functionals \(\F\) and \(\G\) by \(\F(h)\defeq\ek{\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda}^{1/\da}\) and \(\G(h)\defeq\abs{\fsinv{\rk{h^\ad}}(s)}\), \(h\in\D\).
If \(\G(h)=0\) for all \(h\in\D\), then the left-hand side of \eqref{E5.2} is \(0\) by \rlem{L5.2} and the right-hand side equals \(0\) by the convention made.
If \(\G(h)\neq0\) for some \(h\in\D\), then \rlem{L5.2} and a well known duality relation, \tcf{}~\zitaa{MR0224158}{\clem{7.1}}, imply that the distance $d$ is equal to \(d=\sup\setaa{\G(h)}{h\in\D\text{ and }\F(h)\leq1}=\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)\geq1}}^\inv\).
Since \(\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)\geq1}}^\inv={\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\G(h)=1}}^\inv}=\ek{\inf\setaa{\F(h)}{h\in\D\text{ and }\fsinv{\rk{h^\ad}}(s)=1}}^\inv=\sup\setaa{\ek{\F(h)}^\inv}{h\in\Ds}\), the theorem is proved.
\end{proof}
\begin{cor}\label{C5.4}
The set \(\T{\GsS}\) is dense in \(\Law\) if and only if there does not exist a function \(h\in\D\setminus\set{0}\) such that \(\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda<\infty\).
\end{cor}
\begin{proof}
Note that \(h\in\D\setminus\set{0}\) if and only if \(h\in\D\) and there exists \(s\in S\) such that \(\fsinv{\rk{h^\ad}}(s)\neq0\), which gives \(\D\setminus\set{0}=\bigcup\setaa{a\Ds}{a\in\C\setminus\set{0}\text{, }s\in S}\).
If \(\delta\) denotes the right-hand side of \eqref{E5.2}, we obtain the following chain of equivalences: \(\T{\GsS}\) is dense in \(\Law\)\tarlr{}\(d=0\) for all \(s\in S\)\tarlr{}\(\delta=0\) for all \(s\in S\)\tarlr{}\(\int\abs{h}^\da\rk{w^\mpi}^\beta\dif\lambda=\infty\) for all \(h\in\D\setminus\set{0}\).
\end{proof}
If \(q\) is an arbitrary positive integer, an analogous result to that of \rlem{L5.1} is not true in general.
However, for \(\alpha=2\), the above method can be extended.
We briefly sketch the main steps.
\begin{description}
\item[Step 1:] The correspondence \(\ell\mapsto h\), defined by \(\ell(f)=\int h^\ad f\dif\lambda\), \(f\in\Loa{2}{W}\), establishes an isometric isomorphism between the dual space of \(\Loa{2}{W}\) and the space \(\Loa{2}{W^\mpi}\).
\item[Step 2:] \(\int\abs{h^\ad f}\dif\lambda\leq\noq{f}\ek{\int h^\ad W^\mpi h\dif\lambda}^{1/2}\) for all \(f\in\Loa{2}{W}\) and \(h\in\Loa{2}{W^\mpi}\) satisfying \eqref{E5.1}.
\item[Step 3:] The set \(\Dt\defeq\setaa{h\in\Loa{2}{W^\mpi}}{\fsinv{\rk{h^\ad}}(x)=0\text{ for }x\in\GsS}\) is defined correctly and \(\int h^\ad f\dif\lambda=0\) for all \(f\in\Loa{2}{W}\) and \(h\in\Loa{2}{W^\mpi}\) satisfying \eqref{E5.1}.
\item[Step 4:] For all \(s\in S\) and \(k\in\mn{1}{q}\), the distance \(d\) is equal to \(\sup\setaa{\abs{\fsinv{\rk{h^\ad}}(s)\eu{k}}}{h\in\Dt\text{ and } \int h^\ad W^\mpi h\dif\lambda\leq1}\).
\item[Step 5:] For \(s\in S\) and \(k\in\mn{1}{q}\), set \(\Dtsk\defeq\setaa{h\in\Dt}{\fsinv{\rk{h^\ad}}(s)\eu{k}=1}\).
Introducing two positive homogeneous and non-negative functionals \(\Ft\) and \(\Gt\) on \(\Dt\) by \(\Ft(h)\defeq\ek{\int h^\ad W^\mpi h\dif\lambda}^{1/2}\) and \(\Gt(h)\defeq\abs{\fsinv{\rk{h^\ad}}(s)\eu{k}}\), \(h\in\Dt\), similarly to the proof of \rthm{T5.3} one can derive the following assertion.
\end{description}
\begin{thm}\label{T5.5}
Let \(q\in\N\) and \(s\in S\).
Then
\begin{equation}\label{E5.3}
d
=\inf\setaa*{\noq{\chuu{s}{k}-t}}{t\in\T{\GsS}}
=\sup\setaa*{\ek*{\int h^\ad W^\mpi h\dif\lambda}^{-1/2}}{h\in\Dtsk},
\end{equation}
where the right-hand side of \eqref{E5.3} is to be interpreted as \(0\) if \(\Dtsk\) is empty.
\end{thm}
\begin{cor}[\tcf{}~\zitaa{MR0426126}{\ccor{3.16}}]\label{C5.6}
Let \(q\in\N\).
The set \(\T{\GsS}\) is dense in \(\Loa{2}{W}\) if and only if there does not exist a function \(h\in\Dt\setminus\set{0}\) such that \(\int h^\ad W^\mpi h\dif\lambda<\infty\).
\end{cor}
\noindent
\begin{minipage}{0.5\textwidth}
Universit\"at Leipzig\\
Fakult\"at f\"ur Mathematik und Informatik\\
PF~10~09~20\\
D-04009~Leipzig
\end{minipage}
\begin{minipage}{0.49\textwidth}
\begin{flushright}
\texttt{
klotz@math.uni-leipzig.de\\
maedler@math.uni-leipzig.de
}
\end{flushright}
\end{minipage}
\end{document} |
\begin{document}
\title{Condition Number of\ Full Rank Linear least-squares Solutions}
\begin{abstract}
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares coefficient matrix and 2-norms. It depends on three quantities each of which can contribute ill-conditioning. The literature presents several estimates for this condition number with varying results; even standard reference texts contain serious overestimates. The use of the nuclear norm affords a single derivation of the best known lower and upper bounds on the condition number and shows why there is unlikely to be a closed formula.
\end{abstract}
\begin{keywords}
linear least-squares, condition number, applications of functional analysis, nuclear norm, trace norm
\end{keywords}
\begin{AMS}
65F35, 62J05, 15A60
\end{AMS}
\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{JOSEPH F. GRCAR}{LINEAR LEAST SQUARES SOLUTION}
\section {Introduction}
\subsection {Purpose}
Linear least-squares problems, in the form of statistical regression analyses, are a basic tool of investigation in both the physical and the social sciences, and consequently they are an important computation.
This paper develops a single methodology that determines tight lower and upper estimates of condition numbers for several problems involving linear least-squares. The condition numbers are with respect to the matrices in the problems and scaled $2$-norms. The problems are: orthogonal projections and least-squares residuals \citep {Grcar2010f}, minimum $2$-norm solutions of underdetermined equations \citep {Grcar2010b}, and in the present case, the solution of overdetermined equations
\begin {equation}
\label {eqn:LLS}
x = \arg \min_u \| b - A u \|_2 \quad \Rightarrow \quad A^t A x = A^t b \, ,
\end {equation}
where $A$ is an $m \times n$ matrix of full column rank (hence, $m \ge n$). Some presentations of error bounds contain formulas that can severely overestimate the condition number, including the SIAM documentation for the LAPACK software.
This introduction provides some background material. Section \ref {sec:definition} discusses definitions of condition numbers. Section \ref {sec:brief} describes the estimate and provides an example; this material is appropriate for presentation in class. Section \ref {sec:derivation} proves that the condition number varies from the estimate within a factor of $\sqrt 2$. The derivation relies on a formula for the nuclear norm of a matrix. (This norm is the sum of the singular values including multiplicities, and is also known as the trace norm.) Section \ref {sec:comparison} examines overestimates in the literature. Section \ref {sec:advanced} evaluates the nuclear norm of rank $2$ matrices (lemma \ref {lem:rank2}).
\subsection {Prior Work}
Ever since \citet {Legendre1805-HM} and \citet {Gauss1809-HM} invented the method of least-squares, the problems had been solved by applying various forms of elimination to the normal equations, $A^t A x = A^t b$ in equation (\ref {eqn:LLS}). Instead, \citet {Golub1965} suggested applying Householder transformations directly to $A$, which removed the need to calculate $A^t A$. However, \citet [p.\ 144] {GolubWilkinson1966} reported that $A^tA$ was still ``relevant to some extent'' to the accuracy of the calculation because they found that $A^tA$ appears in a bound on perturbations to $x$ that are caused by perturbations to $A$. Their discovery was ``something of a shock'' \citep [p.\ 241] {vanderSluis1975}.
The original error bound of \citet [p.\ 144, eqn.\ 43] {GolubWilkinson1966} was difficult to interpret because of an assumed scaling for the problem. \citet [pp.\ 15, 17, top] {Bjorck1967a} derived a bound by the augmented matrix approach that was suggested to him by Golub. \citet [pp.\ 224--226] {Wedin1973} re-derived the bound from his study of the matrix pseudoinverse and exhibited a perturbation to the matrix that attains the leading term. Van der Sluis (1975, p.\ 251, eqn.\ 5.8) also derived Bj\"orck's bound and introduced a simplification of the formula and a geometric interpretation of the leading term. \citet [p.\ 31, eqn.\ 1.4.28] {Bjorck1996} later followed Wedin in basing the derivation of his bound on the pseudoinverse. \citet [p.\ 1189, eqn.\ 2.4 and line --6] {Malyshev2003} derived a lower bound for the condition number thereby proving that his formula and the coefficient in Bj\"orck's bound are quantifiably tight estimates of the spectral condition number. In contrast, condition numbers with respect to Frobenius norms have exact formulas that have been given in various forms by \citet {Geurts1982}, \citet {Gratton1996}, and \citet {Malyshev2003}.
\section {Condition numbers}
\label {sec:definition}
\subsection {Error bounds and definitions of condition numbers}
The oldest way to derive perturbation bounds is by differential calculus. If $y = f(x)$ is a vector valued function of the vector $x$ whose partial derivatives are continuous, then the partial derivatives give the best estimate of the change to $y$ for a given change to $x$
\begin {equation}
\label {eqn:approximation-1}
\Dy = f(x + \Dx) - f(x) \approx J_f (x) \, \Dx
\end {equation}
where $J_f (x)$ is the Jacobian matrix of the partial derivatives of $y$ with respect to $x$. The magnitude of the error in the first order approximation (\ref {eqn:approximation-1}) is bounded by Landau's little $o ( \| \Dx \| )$ for all sufficiently small $\| \Dx \|$.\footnote {The continuity of the partial derivatives establishes the existence of the Fr\'echet derivative and its representation by the Jacobian matrix. The definition of the Fr\'echet derivative is responsible for the error in equation (\ref {eqn:approximation-1}) being $o ( \| \Dx \| )$. The order of the error terms is independent of the norm because all norms for finite dimensional spaces are equivalent \citep [p.\ 54, thm.\ 1.7] {Stewart1990}.} Thus $J_f (x) \, \Dx$ is the unique linear approximation to $\Dy$ in the vicinity of $x$.\footnote {Any other linear function added to $J_f (x) \, \Dx$ differs from $\Dy$ by ${\mathcal O} (\| \Dx \|)$ and therefore does not provide a $o ( \| \Dx \| )$ approximation.} Taking norms produces a perturbation bound,
\begin {equation}
\label {eqn:calculus-1}
\| \Dy \| \le \| J_f (x) \| \, \| \Dx \| + o (\| \Dx \|) \, .
\end {equation}
Equation (\ref {eqn:calculus-1}) is the smallest possible bound on $\| \Dy \|$ in terms of $\| \Dx \|$ provided the norm for the Jacobian matrix is induced from the norms for $\Dy$ and $\Dx$. In this case for each $x$ there is some $\Dx$, which is nonzero but may be chosen arbitrarily small, so the bound (\ref {eqn:calculus-1}) is attained to within the higher order term, $o (\| \Dx \|)$. There may be many ways to define condition numbers, but because equation (\ref {eqn:calculus-1}) is the smallest possible bound, any definition of a condition number for use in bounds equivalent to (\ref {eqn:calculus-1}) must arrive at the same value, $\chi_y (x) = \| J_f (x) \|$.\footnote {A theory of condition numbers in terms of Jacobian matrices was developed by \citet [p.\ 292, thm.\ 4] {Rice1966}. Recent presentations of the formula $\chi_y (x) = \| J_f (x) \|$ are given by \citet [p.\ 44] {Chatelin1996}, \citet [p.\ 27] {Deuflhard2003}, \citet [p.\ 39] {Quarteroni2000}, and \citet [p.\ 90] {Trefethen1997}.} The matrix norm may be too complicated to have an explicit formula, but tight estimates can be derived as in this paper.
\subsection {One or separate condition numbers}
\label {sec:separate}
Many problems depend on two parameters $u$, $v$ which may consist of the entries of a matrix and a vector (for example). In principle it is possible to treat the parameters altogether.\footnote {As will be seen in table \ref {tab:LLS}, \citet {Gratton1996} derived a joint condition number of the least-squares solution with respect to a Frobenius norm of the matrix and vector that define the problem.} A condition number for $y$ with respect to joint changes in $u$ and $v$ requires a common norm for perturbations to both. Such a norm is
\begin {equation}
\label {eqn:joint-norm}
\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \, .
\end {equation}
A single condition number then follows that appears in an optimal error bound,
\begin {equation}
\label {eqn:single}
\| \Dy \| \le \| J_f (u, v) \| \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) .
\end {equation}
The value of the condition number is again $\chi_y(u, v) = \| J_f (u, v) \|$ where the matrix norm is induced from the norm for $\Dy$ and the norm in equation (\ref {eqn:joint-norm}).
Because $u$ and $v$ may enter into the problem in much different ways, it is customary to treat each separately. This approach recognizes that the Jacobian matrix is a block matrix
\begin {displaymath}
J_f (u, v) = \onetwo {J_{f_1} (u)} {J_{f_2} (v)}
\end {displaymath}
where the functions $f_1 (u) = f(u, v)$ and $f_2(v) = f(u, v)$ have $v$ and $u$ fixed, respectively.
The first order differential approximation (\ref {eqn:approximation-1}) is unchanged but is rewritten with separate terms for $u$ and $v$,
\begin {equation}
\label {eqn:approximation-2}
\Dy \approx J_{f_1} (u) \, \Du + J_{f_2} (v) \, \Dv \, .
\end {equation}
Bound (\ref {eqn:single}) then can be weakened by applying norm inequalities,
\begin {eqnarray}
\nonumber
\| \Dy \|& \le& \| J_{f_1} (u) \Du + J_{f_2} (v) \Dv \| + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right)\\
\noalign {
}
\label {eqn:double}
& \le& \left( \strut \| J_{f_1} (u) \| + \| J_{f_2} (v) \| \right) \, \max \big\{ \| \Du \|, \, \| \Dv \| \big\}\\
\nonumber
&& \hspace* {12em} {} + o \left(\max \big\{ \| \Du \|, \, \| \Dv \| \big\} \right) \, .
\end {eqnarray}
The coefficients $\chi_y(u) = \| J_{f_1} (u) \|$ and $\chi_y(v) = \| J_{f_2} (v) \|$ are the separate condition numbers of $y$ with respect to $u$ and $v$, respectively.
These two different approaches lead to error bounds (\ref {eqn:single}, \ref {eqn:double}) that differ by at most a factor of $2$ because it can be shown \citep {Grcar2010f}
\begin {equation}
\label {eqn:sum-6}
{\chi_y (u) + \chi_y (v) \over 2} \le \chi_y (u, v) \le \chi_y (u) + \chi_y (v) \, .
\end {equation}
Thus, for the purpose of deriving tight estimates of joint condition numbers, it suffices to consider $\chi_y (u)$ and $\chi_y (v)$ separately.
\section {Conditioning of the least-squares solution}
\label {sec:brief}
\subsection {Reason for considering matrices of full column rank}
\label {sec:reason}
The linear least-squares problem (\ref {eqn:LLS}) does not have an unique solution when $A$ does not have full column rank. A specific $x$ can be chosen such as the one of minimum norm. However, small changes to $A$ can still produce large changes to $x$.\footnote {If $A$ does not have full column rank, then for every nonzero vector $z$ in the right null space of the matrix, $(A + b z^t) (z / z^t z) = b$. Thus, a change to the matrix of norm $\| b \|_2 \, \|z \|_2$ changes the solution from $A^\dag b$ to $z /\| z \|_2^2$.} In other words, a condition number of $x$ with respect to rank deficient $A$ does not exist or is ``infinite.'' Perturbation bounds in the rank deficient case can be found by restricting changes to the matrix, for which see \citet [p.\ 30, eqn.\ 1.4.26] {Bjorck1996} and \citet [p.\ 157, eqn.\ 5.3] {Stewart1990}. That theory is beyond the scope of the present discussion.
\subsection {The condition numbers}
\label {sec:brief-1}
This section summarizes the results and presents an example. Proofs are in section \ref {sec:derivation}. It is assumed that $A$ has full column rank and the solution $x$ of the least-squares problem (\ref {eqn:LLS}) is not zero. The solution is proved to have a condition number $\chi_x(A)$ with respect to $A$ within the limits,
\begin {equation}
\label {eqn:chixA}
\fbox {$\boldkappa \, \sqrt { \strut [\vds \tan (\boldtheta)]^2 + 1} $} \; \le \; \chi_x (A) \; \le \; \fbox {$\boldkappa [\vds \tan (\boldtheta) + 1]$} \, ,
\end {equation}
where $\boldkappa$, $\vds$, and $\boldtheta$ are defined below; they are bold to emphasize they are the values in the tight estimates of the condition number. There is also condition number with respect to $b$,
\begin {equation}
\label {eqn:chixb}
\chi_x (b) = \fbox {$\vds \sec (\boldtheta)$} \, .
\end {equation}
These condition numbers $\chi_x(A)$ and $\chi_x (b)$ are for measuring the perturbations to $A$, $b$, and $x$ by the following scaled $2$-norms,
\begin {equation}
\label {eqn:specific-scaled-norms}
{\| \DA \|_2 \over \| A \|_2} \, ,
\qquad
{\| \Db \|_2 \over \| b \|_2} \, ,
\qquad
{\| \Dx \|_2 \over \| x \|_2} \, .
\end {equation}
Like equation (\ref {eqn:double}), the two condition numbers appear in error bounds of the form,\footnote {The constant denominators $\| A \|_2$ and $\| b \|_2$ could be discarded from the $o$ terms because only the order of magnitude of the terms is pertinent.}
\begin {equation}
\label {eqn:error-bound}
{\| \Dx \|_2 \over \| x \|_2} \le \chi_x (A) {\| \DA \|_2 \over \| A \|_2} + \chi_x (b) {\| \Db \|_2 \over \| b \|_2} + o \left( \max \left\{ {\| \DA \|_2 \over \| A \|_2}, \, {\| \Db \|_2 \over \| b \|_2} \right\} \right) ,
\end {equation}
where $x + \Dx$ is the solution of the perturbed problem,
\begin {equation}
\label {eqn:perturbed-problem}
x + \Dx = \arg \min_u \| (b + \Db) - (A + \DA) u \|_2 \, .
\end {equation}
The quantities $\boldkappa$, $\vds$, and $\boldtheta$ in the formulas (\ref {eqn:chixA}, \ref {eqn:chixb}) are
\begin {equation}
\label {eqn:three}
\boldkappa = {\| A \|_2 \over \sigmamina} \qquad
\vds = {\| A x \|_2 \over \| x \|_2 \, \sigmamina} \qquad
\tan (\boldtheta) = {\| r \|_2 \over \| A x \|_2}
\end {equation}
where $\boldkappa$ is the spectral matrix condition number of $A$ ($\sigmamin$ is the smallest singular value of $A$), $\vds$ is van der Sluis's ratio between $1$ and $\boldkappa$,\footnote {The formulas of \citet [p.\ 251] {vanderSluis1975} contain in his notation $R(x_0) / \sigma_n$, which is the present $\vds$.} $\boldtheta$ is the angle between $b$ and $\col (A)$,\footnote {The notation $\col (A)$ is the column space of $A$.} and $r = b - Ax$ is the least-squares residual.
\begin {enumerate}
\item $\boldkappa$ depends only on the extreme singular values of $A$.
\item $\boldtheta$ depends only on the ``angle of attack'' of $b$ with respect to $\col (A)$.
\item If $A$ is fixed, then $\vds$ depends on the orientation of $b$ to $\col (A)$ but not on $\boldtheta$.\footnote {Because $A$ has full column rank, $Ax$ and $x$ can only vary proportionally when their directions are fixed.}
\end {enumerate}
Please refer to Figure \ref {fig:schematic} as needed.
If $\col (A)$ is fixed, then $\boldkappa$ and $\vds$ depend only on the singular values of $A$, and $\boldtheta$ depends only on the orientation of $b$. Thus, $\boldkappa$ and $\boldtheta$ are separate sources of ill-conditioning for the solution. If $Ax$ has comparatively large components in singular vectors corresponding to the largest singular values of $A$, then $\vds \approx \boldkappa$ and the condition number $\chi_x (A)$ depends on $\boldkappasquared$ which was the discovery of \citet {GolubWilkinson1966}. Otherwise, $\boldkappasquared$ ``plays no role'' \citep [p.\ 251] {vanderSluis1975}.
\begin {figure}
\centering
\includegraphics [scale=1] {schematic}
\caption {Schematic of the least-squares problem, the projection $Ax$, and the angle $\boldtheta$ between $Ax$ and $b$.}
\label {fig:schematic}
\end {figure}
\subsection {Conditioning example}
\label {sec:example}
This example illustrates the independent effects of $\boldkappa$, $\vds$, and $\boldtheta$ on $\chi_x (A)$. It is based on the example of \citet [p.\ 238] {Golub1996}. Let
\begin {displaymath}
A = \left[ \begin {array} {c c} 1& 0\\ 0& \alpha\\ 0& 0\\ \end {array}
\right] ,
\quad
b = \left[ \begin {array} {c} \beta \cos (\phi)\\ \beta \sin (\phi) \\ 1 \end {array} \right] \, ,
\quad
\DA = \left[ \begin {array} {c c} 0& 0\\ 0& 0\\ 0& \epsilon \end {array}
\right] .
\end {displaymath}
where $0 < \epsilon \ll \alpha, \beta < 1$.
In this example,
\begin {displaymath}
x = \left[ \begin {array} {c} \beta \cos (\phi) \\ {\beta \vphantom {(} \over \vphantom {(} \alpha} \sin (\phi) \\ \end {array} \right] ,
\quad
r = \left[ \begin {array} {c} 0\\ 0\\ 1 \end {array} \right] ,
\quad
x + \Dx = \left[ \begin {array} {c} \beta \cos (\phi) \\[0.1ex] {\epsilon + \alpha \beta \sin (\phi) \vphantom {(} \over \vphantom {(}\alpha^2 + \epsilon^2} \\ \end {array} \right] \, .
\end {displaymath}
The three terms in the condition number are
\begin {displaymath}
\boldkappa = {1 \over \alpha} \, , \qquad
\vds = {1 \over \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} \, , \qquad
\tan (\boldtheta) = {1 \over \beta} \, .
\end {displaymath}
These values can be independently manipulated by choosing $\alpha$, $\beta$, and $\phi$. The tight upper bound for the condition number is
\begin {displaymath}
\chi_x (A) \le {1 \over \alpha} \left ( {1 \over \beta \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + 1 \right) \, .
\end {displaymath}
The relative change to the solution of the example
\begin {displaymath}
{\| \Dx \|_2 \over \| x \|_2}
= {\epsilon \over \alpha \, \beta \, \sqrt {[\alpha \cos (\phi)]^2 + [\sin (\phi)]^2}} + {\mathcal O} (\epsilon^2) \, .
\end {displaymath}
is close to the bound given by the condition number estimate and the relative change to $A$.
\section {Derivation of the condition numbers}
\label {sec:derivation}
\subsection {Notation}
The formula for the Jacobian matrix $J_x (b)$ of the solution $x = (A^t A)^{-1} b$ with respect to $b$ is clear.\footnote {The notation of section \ref {sec:separate} would introducing a name, $f_2$, for the function by which $x$ varies with $b$ when $A$ is held fixed, $x = f_2 (b)$, so that the notation for the Jacobian matrix is then $J_{f_2} (b)$. This pedantry will be discarded here to write $J_x(b)$ with $A$ held fixed; and similarly for $J_x(A)$ with $b$ held fixed.} For derivatives with respect to the entries of $A$, it is necessary to use the ``$\Vector$'' construction to order the matrix entries into a column vector; $\Vector (B)$ is the column of entries $B_{i,j}$ with $i,j$ in co-lexicographic order.\footnote {The alternative to placing the entries of matrices into column vectors is to use more general linear spaces and the Fr\'echet derivative. That approach seems unnecessarily abstract because the spaces have finite dimension.} The approximation is then
\begin {equation}
\label {eqn:total-differential}
\Dx = \JxA \, \Vector (\DA) + \Jxb \, \Db + \mbox {higher order terms}
\end {equation}
and upon taking norms
\begin {equation}
\label {eqn:differential-bound}
\| \Dx \| \le \underbrace {\| \JxA \|}_{\displaystyle \chi_x (A)} \, \| \DA \| + \underbrace {\displaystyle \| \Jxb \|}_{\displaystyle \chi_x (b)} \, \| \Db \| + o( \max \{ \| \DA \|, \, \| \Db \| \} ) \, ,
\end {equation}
where it is understood the norms on the two Jacobian matrices are induced from the following norms for $\DA$, $\Db$, and $\Dx$.
\subsection {Choice of Norms}
\label {sec:norms}
Equation (\ref {eqn:differential-bound}) applies for any choice of norms. In theoretical numerical analysis especially for least-squares problems the spectral norm is preferred. For $2$-norms the matrix condition number of $A^t A$ is the square of the matrix condition number of $A$. The norms used in this paper are thus,
\begin {equation}
\label {eqn:norms}
\| \Vector (\DA) \|_\scaleA = {\| \DA \|_2 \over \scaleA} \, ,
\qquad
\| \Db \|_\scaleB = {\| \Db \|_2 \over \scaleB} \, ,
\qquad
\| \Dx \|_\scaleX = {\| \Dx \|_2 \over \scaleX} \, ,
\end {equation}
where $\scaleA$, $\scaleB$, $\scaleX$ are constant scale factors. These formulas define norms for $m \times n$ matrices, for $m$ vectors, and for $n$ vectors. The scaling makes the size of the changes relative to the problem of interest. The scaling used in equations (\ref {eqn:chixA}--\ref {eqn:specific-scaled-norms}) is
\begin {equation}
\label {eqn:scale-factors}
\scaleA = \| A \|_2 \, ,
\qquad
\scaleB = \| b \|_2 \, ,
\qquad
\scaleX = \| x \|_2 \, .
\end {equation}
\subsection {Condition number of {\itshape x\/} with respect to {\itshape b\/}}
\label {sec:conditionwrtb}
From $x = (A^t A)^{-1} A^t b$ follows $\Jxb = (A^t A)^{-1} A^t$ and then for the scaling of equation (\ref {eqn:scale-factors})
\begin {equation}
\label {eqn:conditionwrtb}
\renewcommand {\arraycolsep} {0.125em}
\begin {array} {r c l}
\| \Jxb \|&
=&
\displaystyle \max_{\Db} {\| \Jxb \, \Db \|_\scaleA \over \| \Db \|_\scaleB}
= \displaystyle \max_{\Db} {\displaystyle \left( {\| (A^t A)^{-1} A^t \Db \|_2 \over \scaleX} \right) \over \displaystyle \left( {\| \Db \|_2 \over \scaleB} \right)}
= {\scaleB \over \scaleX \, \sigmamina}\\
&
=&
\displaystyle {\| b \|_2 \over \| A x \|_2} {\| A x \|_2 \over \| x \|_2 \, \sigmamina} = \sec (\boldtheta) \, \vds \, .
\end {array}
\end {equation}
\subsection {Condition number of {\itshape x\/} with respect to {\itshape A\/}}
\label {sec:begin}
The Jacobian matrix $\JxA$ is most easily calculated from the total differential of the identity $F = A^t (b - A x) = 0$ with respect to $A$ and $x$, which is $J_F [\Vector (A)] \, \Vector (dA) + J_F (x) \, dx = 0$. Hence
\begin {equation}
\label {eqn:hence}
dx = \underbrace {- [J_F (x)]^{-1} J_F [\Vector (A)]}_{\displaystyle J_x[\Vector (A)]} \Vector (dA)
\end {equation}
where $J_F (x) = - A^t A$ and where
\begin {equation}
\label {eqn:where}
J_F [\Vector (A)]
=
\left[
\setlength {\arraycolsep} {0.25em}
\begin {array} {c c c}
r^t\\
& \ddots\\
&& \hspace {0.5em} r^t
\end {array}
\right]
-
\left[
\setlength {\arraycolsep} {0.33em}
\begin {array} {c c c c c}
x_1 A^t& \cdots& x_n A^t
\end {array}
\right] ,
\end {equation}
in which $r = b - A x$ is the least-squares residual, and $x_i$ is the $i$-th entry of $x$.
\subsection {Transpose formula for condition numbers}
\label {sec:transpose}
The desired condition number is the norm induced from the norms in equation (\ref {eqn:norms}).
\begin {equation}
\label {eqn:induced}
\setlength {\arraycolsep} {0.25em}
\begin {array} {r c l}
\| \JxA \|
& =
& \displaystyle \max_{\DA} {\| \JxA \, \Vector (\DA) \|_\scaleX \over \| \Vector (\DA) \|_\scaleA}
\\ \noalign {
}
& =
& \displaystyle {\scaleA \over \scaleX} \max_{\DA} {\| \JxA \, \Vector (\DA) \|_2 \over \| \DA \|_2}
\end {array}
\end {equation}
The numerator and denominator are vector and matrix $2$-norms, respectively. If $A$ is an $m \times n$ matrix, then the maximization in equation (\ref {eqn:induced}) has many degrees of freedom. An identity for operator norms can be applied to avoid this large optimization problem.
Suppose $\reals^M$ and $\reals^N$ have the norms $\| \cdot \|_M$ and $\| \cdot \|_N$, respectively. If a problem with data $d \in \reals^N$ has a solution function $s = f(d) \in \reals^M$, then the condition number is the induced norm of the $M \times N$ Jacobian matrix,
\vspace {-1ex}
\begin {equation}
\label {eqn:condition}
\| J_f (d) \| = \max_{\Dd} {\| J_f(d) \, \Dd \|_M \over \| \Dd \|_N} \, .
\end {equation}
This optimization problem has $N$ degrees of freedom. An alternate expression is the norm for the transposed operator represented by the transposed matrix,\footnote {Equation (\ref {eqn:transpose}) is stated by
\citet [chp.\ IV, p.\ 7, eqn.\ 4] {BourbakiTVS},
\citet [p.\ 478, lem.\ 2] {DunfordSchwartz1958},
\citet [p.\ 93, thm.\ 4.10, eqn.\ 2] {Rudin1973},
and
\citet [p.\ 195, thm.\ 2$^\prime$, eqn.\ 3] {Yosida1974}.
The name of the transposed operator varies. See \citet [chp.\ IV, p.\ 6, top] {BourbakiTVS} for ``transpose,'' Dunford and Schwartz (loc.\ cit.)\ and Rudin (loc.\ cit.)\ for ``adjoint,'' and \citet [p.\ 194, def.\ 1] {Yosida1974} for ``conjugate'' or ``dual.'' Some parts of mathematics use ``adjoint'' in the restricted context of Hilbert spaces, for example in linear algebra see \citet [pp.\ 168--174, sec.\ 5.1] {Lancaster1985}. That concept is actually a ``slightly different notion'' \citep [p.\ 479] {DunfordSchwartz1958} from the Banach space transpose used here.
}
\begin {equation}
\label {eqn:transpose}
\| J_f (d) \| = \| [J_f (d)]^t \|^* = \max_{\Ds} {\| [J_f (d)]^t \Ds \|_N^* \over \| \Ds \|_M^*} \, .
\end {equation}
The norm is induced from the dual norms $\| \cdot \|_M^*$ and $\| \cdot \|_N^*$ which must be determined. This optimization problem has $M$ degrees of freedom. Equation (\ref {eqn:transpose}) might be easier to evaluate, especially when the problem has many more data values than solution variables, $N \gg M$, as is often the case.
Applying the formula (\ref {eqn:transpose}) for the norm of the transpose matrix to the equation (\ref {eqn:induced}) results in the simpler optimization problem,
\begin {equation}
\label {eqn:simpler}
\| \JxA \| =
{\scaleA \over \scaleX} \max_{\Dx} {\| \{ \JxA \}^t \Dx \|^*_2 \over \| \Dx \|^*_2}
\end {equation}
The norm for the transposed Jacobian matrix is induced from the duals of the $2$-norms for matrices and vectors. The vector $2$-norm in the denominator is its own dual. So as not to interrupt the present discussion, some facts needed to evaluate the numerator are proved in section \ref {sec:advanced}: the dual of the matrix $2$-norm is the nuclear norm (section \ref {app:spectral}), and a formula for the nuclear norm is given (section \ref {app:rank2}).
\subsection {Condition number of {\itshape x\/} with respect to {\itshape A\/}, continued}
\label {sec:conditionwrtAcontinued}
The application of equation (\ref {eqn:simpler}) requires evaluating the matrix-vector product in the numerator. Continuing the derivation of section \ref {sec:begin} from equation (\ref {eqn:where}), the vector-matrix product $v^t J_F [\Vector (A)]$ for some $v$ can be evaluated by straightforward multiplication,
\begin {displaymath}
v^t J_F [\Vector (A)]
=
\left[
\setlength {\arraycolsep} {0.25em}
\begin {array} {c c c}
v_1 r^t
& \cdots
& v_n r^t
\end {array}
\right]
-
\left[
\setlength {\arraycolsep} {0.33em}
\begin {array} {c c c c c}
x_1 v^t A^t & \cdots& x_n v^t A^t
\end {array}
\right] .
\end {displaymath}
This row vector, when transposed, is expressed more simply using $\Vector$ notation: the first part is $r$ scaled by each entry of $v$, $\Vector (r v^t)$, the second part is $A v$ scaled by each entry of $x$, $\Vector (A v x^t) $. Altogether
\begin {displaymath}
J_F [\Vector (A)]^t v
=
\Vector (rv^t - Avx^t) \, .
\end {displaymath}
Substituting $v = \{- [J_F (x)]^{-1}\}^t \Dx = (A^t A)^{-1} \Dx$ for some $\Dx$ gives, by equation (\ref {eqn:hence}),
\begin {displaymath}
\JxA^t \Dx
=
\Vector \left\{ r \, [(A^t A)^{-1} \Dx]^t - A \, [(A^t A)^{-1} \Dx] \, x^t \right\} \, ,
\end {displaymath}
or equivalently,
\begin {equation}
\label {eqn:JxADx}
\Matrix \{ \JxA^t \Dx \}
=
u_1^{} v_1^t + u_2^{} v_2^t
\end {equation}
where ``$\Matrix$'' is the inverse of ``$\Vector$,'' and
\begin {equation}
\label {eqn:vectors}
\setlength {\arraycolsep} {0.125em}
\begin {array} {r c l r c l}
u_1& =& r = b - Ax,&
v_1& =& (A^t A)^{-1} \Dx,\\ \noalign {
}
u_2& =& - A (A^t A)^{-1} \Dx, \qquad&
v_2& =& x .
\end {array}
\end {equation}
The matrix on the right side of equation (\ref {eqn:JxADx}) has rank $2$. Moreover, the two rank $1$ pieces are mutually orthogonal because the least-squares residual $r$ is orthogonal to the coefficient matrix $A$. With these replacements equation (\ref {eqn:simpler}) becomes
\begin {equation}
\label {eqn:inducedtransposed}
\big \| \JxA \big\|
=
{\scaleA \over \scaleX} \max_{\| \Dx \|_2 = 1} \| u_1^{} v_1^t + u_2^{} v_2^t \|^*_2 \, .
\end {equation}
Lemma \ref {lem:kahan} shows that the dual of the spectral matrix norm is the matrix norm that sums the singular values of the matrix, which is called the nuclear norm. Lemma \ref {lem:rank2} then evaluates this norm for rank $2$ matrices to find that the objective function of equation (\ref {eqn:inducedtransposed}) is
\begin {equation}
\label {eqn:numerator}
\sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2 \|_2 \, \cos (\theta_u - \theta_v)} \, ,
\end {equation}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and $\theta_v$ is the angle between $v_1$ and $v_2$, and both angles should be taken from $0$ to $\pi$. Since $u_1$ is orthogonal to $u_2$ therefore $\theta_u = \pi / 2$ and then $| \theta_u - \theta_v | \le \pi / 2$ so $\cos (\theta_u - \theta_v)$ is not negative. This means the maximum lies between the lower and upper limits
\begin {equation}
\label {eqn:lowerandupper}
\sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2} \quad \mbox {and} \quad \| u_1 \|_2 \, \| v_1 \|_2 + \| u_2 \|_2 \, \| v_2 \|_2 \, .
\end {equation}
With $\| \Dx \|_2$ restricted to $1$, the lower bound and also the upper bound attain their maxima when $\Dx$ is a right singular unit vector for the smallest singular value of $A$,
\begin {equation}
\label {eqn:maxima}
\| u_1 \|_2 \, \| v_1 \|_2 = {\| r \|_2 \over (\sigmamina)^2} \quad \mbox {and} \quad \| u_2 \|_2 \, \| v_2 \|_2 = {\| x \|_2 \over \sigmamina} \, .
\end {equation}
Some value of $\| u_1^{} v_1^t + u_2^{} v_2^t \|^*_2$ lies between the limits when $\Dx$ is a right singular unit vector for the smallest singular value of $A$. Because these are the largest possible limits, the maximum value must lie between them as well. These limits must be multiplied by the coefficient $\scaleA / \scaleX$ in equation (\ref {eqn:inducedtransposed}) to obtain bounds for the norm of the Jacobian matrix.
\subsection {Summary of condition numbers}
\label {sec:conditionwrtAfinished}
\ \par
\begin {theorem}
[\scshape Spectral condition numbers]
\label {thm:condition-numbers}
For the full column rank linear least-squares problem with solution $x = (A^tA)^{-1} A^t b$, and for the scaled norms of equation (\ref {eqn:norms}) with scale factors $\scaleA$, $\scaleB$, and $\scaleX$,
\begin {displaymath}
\chi_x (b) = {\scaleB \over \scaleX \sigmamin}
\qquad
\chi_x (A) = {\scaleA \over \scaleX} \max_{\| \Delta x \|_2 = 1} \sigma_1 + \sigma_2 \, ,
\end {displaymath}
where $\sigma_1$ and $\sigma_2$ are the singular values of the rank $2$ matrix $u_1^{} v_1^t + u_2^{} v_2^t$ \,for
\begin {displaymath}
\setlength {\arraycolsep} {0.125em}
\begin {array} {r c l r c l}
u_1& =& r = b - Ax,&
v_1& =& (A^t A)^{-1} \Dx,\\ \noalign {
}
u_2& =& - A (A^t A)^{-1} \Dx, \qquad&
v_2& =& x .
\end {array}
\end {displaymath}
The value of $\chi_x (A)$ lies between the lower limit of Malyshev and the upper limit of Bj\"orck,
\begin {displaymath}
{\scaleA \over \scaleX \sigmamin} \sqrt { \left( {\| r \|_2 \over \sigmamin} \right)^2 + \| x \|_2^2}
\, \le \,
\chi_x (A)
\, \le \,
{\scaleA \over \scaleX \sigmamin} \left( {\| r \|_2 \over \sigmamin} + \| x \|_2 \right) \, .
\end {displaymath}
The upper bound exceeds $\chi_x (A)$ by at most a factor $\sqrt 2$. The formula for $\chi_x(b)$ and the limits for $\chi_x(A)$ simplify to equations (\ref {eqn:chixA}, \ref {eqn:chixb}) for the scale factors in equation (\ref {eqn:scale-factors}).
\end {theorem}
\begin {proof}
Section \ref {sec:conditionwrtb} derives $\chi_x(b)$, and sections \ref {sec:begin}--\ref {sec:conditionwrtAfinished} derive $\chi_x(A)$ and the bounds.
\end {proof}
\section {Discussion}
\label {sec:comparison}
\subsection {Example of strict limits}
The condition number $\chi_x (A)$ in theorem \ref {thm:condition-numbers} can lie strictly between the limits of Bj\"orck and Malyshev. For the example of section \ref {sec:example}, the rank $2$ matrix in the theorem is
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
u_1^{} v_1^t + u_2^{} v_2^t = \left[ \begin {array} {c c}
- \beta \cos (\phi) \Delta x_1& - \beta \sin (\phi) \Delta x_1 / \alpha\\
- \beta \cos (\phi) \Delta x_2 / \alpha& - \beta \sin (\phi) \Delta x_2 / \alpha^2\\
\Delta x_1& \Delta x_2 / \alpha^2 \\ \end {array} \right] .
\end {displaymath}
For the specific values $\alpha = 1/10$, $\beta = 1$, $\phi = \pi / 10$, the sum of the singular values of this matrix can be numerically maximized over $\| \Delta x \|_2 = 1$ to evaluate the condition number with the following results.
\begin {displaymath}
\renewcommand {\arraystretch} {1.25}
\renewcommand {\tabcolsep} {0.25em}
\begin {tabular} {r c l l}
$\displaystyle \boldkappa (\vds \tan (\boldtheta) + 1)$& $=$& 40.928\ldots& upper limit of Bj\"orck\\
$\displaystyle \chi_x (A)$& $=$& 35.193\ldots& condition number\\
$\displaystyle \boldkappa \, \big( \strut [\vds \tan (\boldtheta)]^2 + 1\big)^{1/2}$& $=$& 32.505\ldots& lower limit of Malyshev\\
\end {tabular}
\end {displaymath}
These calculations were done with Mathematica \citep {Wolfram2003}.
\subsection {Exact formulas for some condition numbers}
\label {sec:exact}
\begin {table}
\caption {\it Cases for which condition numbers have been determined for the full column rank least-squares problem, $\min_x \| b - A x \|_2$. All the formulas are for $\chi_x(A)$ except Grattan's formula is for $\chi_x(A,b)$. Notation: $r$ is the residual, $\sigmamin$ is the smallest singular value of $A$. In column 5, ``approx'' means the value in column 4 approximates the condition number, ``exact'' means it is the condition number for the chosen norms.}
\label {tab:LLS}
\newcommand {\textstrut} {\vrule depth1.125ex height2.375ex width\strutwidth}
\newcommand {\leftbox} [3] {\begin {minipage}{#1} \vrule depth0ex height4.0ex width\strutwidth #2\\ \scriptsize #3\vrule depth3.0ex height0ex width\strutwidth \end {minipage}}
\setlength {\arraycolsep} {0.6em}
\vspace {-4ex}
\begin {displaymath}
\small
\begin {array} {| l | c | c | c | c |}
\cline {2-3}
\multicolumn {1} { c |} {\textstrut}
& \multicolumn {2} { c |} {\mbox {norms}}
\\
\hline
\multicolumn {1} {| c |} {\textstrut \mbox {source}}
& \mbox {data}
& \mbox {solution}
& \mbox {formula}
& \mbox {status}
\\ \hline \hline
\leftbox {10em}
{Bj\"orck and theorem \ref {thm:condition-numbers}}
{\nocite {Bjorck1996}(1996, p.\ 31, eqn.\ 1.4.28)}
& \displaystyle {\| \DA \|_2 \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \displaystyle {\| A \|_2 \over \strut \sigmamin} \left( {\| r \|_2 \over \strut \sigmamin \, \| x \|_2} \, + 1 \right)
& \mbox {approx}
\\
\leftbox {10em}
{Geurts}
{\nocite {Geurts1982}(1982, p.\ 93, eqn.\ 4.3)}
& \displaystyle {\| \DA \|\sub{F} \over \| A \|\sub{F}}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \displaystyle {\| A \|\sub{F} \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \sigmamin^2 \, \| x \|_2^2} + 1}
& \mbox {exact}
\\
\multicolumn {1} {| l} {\leftbox {10em} {Gratton} {\nocite {Gratton1996}(1996, p.\ 525, eqn.\ 2.1)}}
& \hspace {-4em} \displaystyle \left\| \strut [ \alpha \, \DA, \beta \, \Db \, ] \right \|\sub{F}
& \| \Dx \|_2
& \displaystyle {1 \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \alpha^2 \sigmamin^2} + {\| x \|_2^2 \over \alpha^2} + {1 \over \beta^2}}
& \mbox {exact}
\\
\leftbox {10em}
{Malyshev}
{\nocite {Malyshev2003}(2003, p.\ 1187, eqn.\ 1.8)}
& \displaystyle {\| \DA \|\sub{F} \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
&& \mbox {exact}
\\
\leftbox {11.5em}
{Malyshev and theorem \ref {thm:condition-numbers}}
{\nocite {Malyshev2003}(2003, p.\ 1189, eqn.\ 2.4 and line --6)}
& \displaystyle {\| \DA \|_2 \over \| A \|_2}
& \displaystyle {\| \Dx \|_2 \over \| x \|_2}
& \mbox{\hspace {-1.5em} \raisebox{4.0ex}[0pt][0pt]{$\displaystyle \left.
\vrule depth7ex height0ex width 0pt \right\} \hspace {0.5em} {\| A \|_2 \over \sigmamin^{}} \sqrt{ {\| r \|_2^2 \over \sigmamin^2 \, \| x \|_2^2} +
1}$}}
& \mbox {approx}
\\ \hline
\end {array}
\end {displaymath}
\vspace*{-2ex}
\end {table}
Table \ref {tab:LLS} lists several condition numbers or approximations to condition numbers for least-squares solutions. The three exact values measure changes to $A$ by the Frobenius norm, while the two approximate values are for the spectral norm. The difference can be attributed to the ease or difficulty of solving the maximization problem in equation (\ref {eqn:inducedtransposed}). The dual spectral norm of the rank $2$ matrix involves a trigonometric function, $\cos (\pi / 2 - \theta_v)$ in equation (\ref {eqn:numerator}), whose value only can be estimated. If a Frobenius norm were used instead, then lemma \ref {lem:rank2} shows the dual norm of the rank $2$ matrix involves an expression, $\cos (\pi / 2) \cos (\theta_v)$, whose value is zero, which greatly simplifies the maximization problem.
\subsection {Overestimates of condition numbers}
\label {sec:overestimate}
Many error bounds in the literature combine $\chi_x(A) + \chi_x(b)$ in the manner of equation (\ref {eqn:double}),
\begin {eqnarray}
\nonumber
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \chi_x (A,b) \, \epsilon + o (\epsilon) \qquad \mbox {attainable}
\\
\label {eqn:bestbound-1}
& \le
& \big[ \chi_x(A) + \chi_x(b) \big] \epsilon + o (\epsilon) \qquad \mbox {overestimate by at most $\times 2$}
\\ \noalign {
}
\label {eqn:bestbound-2}
& \le
& \big[ \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta) \big] \epsilon + o (\epsilon) \quad \mbox {further at most $\times \sqrt 2$}
\\ \noalign {
}
\label {eqn:bestbound-3}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| b \|_2 \over \sigmamin \| x \|_2} \right) \epsilon +o (\epsilon) \, ,
\end {eqnarray}
where $\epsilon = \max \{ \| \DA \|_2 / \| A \|_2, \| \Db \|_2 / \| b \|_2 \}$. Bounds (\ref {eqn:bestbound-1}, \ref {eqn:bestbound-2}) are larger than the attainable bound by at most factors $2$ and $2 \sqrt 2$, respectively, by equation (\ref {eqn:sum-6}) and theorem \ref {thm:condition-numbers}.
Some bounds are yet larger. \citet [p.\ 382, eqn.\ 20.1] {Higham2002} reports
\begin {eqnarray*}
{\| \Dx \|_2 \over \| x \|_2}
& \le
& \boldkappa \epsilon \left( 2 + ( \boldkappa + 1) {\| r \|_2 \over \| A \|_2 \| x \|_2} \right) + {\mathcal O} (\epsilon^2)
\\ \noalign {
}
& =
& \left( {\| A \|_2 \| r \|_2 \over \sigmamin^2 \| x \|_2} + {\| A \|_2 \over \sigmamin} + {\| A \|_2 \| x \|_2 + \| r \|_2 \over \sigmamin \| x \|_2} \right) \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {eqnarray*}
This bound is an overestimate in comparison to equation (\ref {eqn:bestbound-3}).
An egregious overestimate occurs in an error bound that appears to have originated in the 1983 edition of the popular textbook of \citet [p.\ 242, eqn.\ 5.3.8] {Golub1996}. The overestimate is restated by \citet [p.\ 50] {Anderson1992} in the LAPACK documentation, and by \citet [p.\ 117] {Demmel1997},
\begin {equation}
\label {eqn:GVL}
{\| \Dx \|_2 \over \| x \|_2}
\le
\left[ 2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \right] \epsilon + {\mathcal O} (\epsilon^2) \, .
\end {equation}
In comparison with equation (\ref {eqn:bestbound-2}) this bound replaces $\vds$ by $\boldkappa$ and replaces $1$ by $\sec (\boldtheta)$. An overestimate by a factor of $\boldkappa$ occurs for the example of section \ref {sec:example} with $\alpha \ll 1$, $\beta = 1$, and $\phi = {\pi \over 2}$. In this case the ratio of equation (\ref {eqn:GVL}) to equation (\ref {eqn:bestbound-2}) is
\begin {displaymath}
{2 \sec (\boldtheta) \, \boldkappa + \tan (\boldtheta) \, \boldkappasquared \over \boldkappa (\vds \tan (\boldtheta) + 1) + \vds \sec (\boldtheta)} = {\displaystyle {1 + 2 \alpha \sqrt {1 + \beta^2} \over \alpha^2 \beta} \over \displaystyle {1 + \beta + \alpha \sqrt {1 + \beta^2} \over \alpha \beta}} \approx {1 \over 2 \, \alpha} = {\boldkappa \over 2} \, .
\end {displaymath}
\section {Norms of operators on normed linear spaces of finite dimension}
\label {sec:advanced}
\subsection {Introduction}
This section describes the dual norms in the formulas of sections \ref {sec:transpose} and \ref {sec:conditionwrtAcontinued}. The actual mathematical concept is a norm for the dual space. However, linear algebra ``identifies'' a space with its dual, so the concept becomes a ``dual norm'' for the same space. This point of view is appropriate for Hilbert spaces, but it omits an important level of abstraction. As a result, the linear algebra literature lacks a complete development of finite dimensional normed linear (Banach) spaces. Rather than make functional analysis a prerequisite for this paper, here the identification approach is generalized to give dual norms for spaces other than column vectors (which is needed for data in matrix form), but only as far as the dual norm itself in section \ref {app:duals}. Section \ref {app:spectral} gives the formula for the dual of the spectral matrix norm. Section \ref {app:rank2} evaluates the norm for matrices of rank $2$.
{\it Banach spaces are needed in this paper because the norms used in numerical analysis are not necessarily those of a Hilbert space.} The space of $m \times n$ matrices viewed as column vectors has been given the spectral matrix norm in equation (\ref {eqn:norms}). If the norm were to make the space a Hilbert space, then the norm would be given by an inner product. There would be an $mn \times mn$ symmetric matrix, $S$, so that for every $m \times n$ matrix $B$,
\begin {displaymath}
\| B \|_2
=
\sqrt {\strut [\Vector (B)]^t \; S \; \Vector (B)} \, ,
\end {displaymath}
which is impossible.
\subsection {Duals of normed spaces}
\label {app:duals}
If $\X$ is a finite dimensional vector space over $\reals$, then the dual space $\X^*$ consists of all linear transformations $f : \X \rightarrow \reals$, called functionals. If $\X$ has a norm, then $\X^*$ has the usual operator norm given by
\begin {equation}
\label {eqn:operatorNorm}
\| f \| = \sup_{\| x \| = 1} f (x) \, .
\end {equation}
One notation is used for both norms because whether a norm is for $\X$ or $\X^*$ can be decided by what is inside.
For a finite dimensional $\X$ with a basis $e_1$, $e_2$, $\ldots\,$, $e_n$, the dual space has a basis $g_1$, $g_2$, $\ldots\,$, $f_n$ defined by $f_i (e_j) = \delta_{i,j}$ where $\delta_{i,j}$ is Kronecker's delta function. In linear algebra for finite dimensional spaces, it is customary to represent the arithmetic of $\X^*$ in terms of $\X$ under the transformation $T : \X \rightarrow \X^*$ defined on the bases by $T(e_i) = f_i$. \textit {This transformation is not unique because it depends on the choices of bases.\/} Usually $\X$ has a favored or ``canonical'' basis whose $T$ is said to ``identify'' $\X^*$ with $\X$. Under this identification the norm for the dual space then is regarded as a norm for the original space.
\begin {definition}
[\scshape Dual norm]
Let $\X = \reals^m$ have norm $\| \cdot \|$ and let $T$ identify $\X$ with the dual space $\X^*$. The dual norm for $\X$ is
\begin {displaymath}
\| v \|^* = \| T(v) \|
\end {displaymath}
where the right side is the norm in equation (\ref {eqn:operatorNorm}) for the dual space.
\end {definition}
The notation $\| \cdot \|^*$ avoids confusing the two norms for $\X$. There seems to be no standard notation for the dual norm; others are $\| \cdot \|_D$, $\| \cdot \|^D$, and $\| \cdot \|_{\rm d}$ which are used respectively by \citet [p.\ 107, eqn.\ 6.2] {Higham2002}, \citet [p.\ 275, def.\ 5.4.12] {Horn1985}, and \citet [p.\ 381, eqn.\ 1] {Lancaster1985}.
\subsection {Dual of the spectral matrix norm}
\label {app:spectral}
The space $\reals^{m \times n}$ of real $m \times n$ matrices has a canonical basis consisting of the matrices $E^{(i,j)}$ whose entries are zero except the $i,j$ entry which is $1$. This basis identifies a matrix $A$ with the functional whose value at a matrix $B$ is $\sum_{i,j} A_{i,j} B_{i,j} = \tr (A^t B)$.
\begin {lemma}
[\scshape Dual of the spectral matrix norm]
\label {lem:kahan}
The dual norm of the spectral matrix norm with respect to the aforementioned canonical basis for $\reals^{m \times n}$ is given by $\| A \|_2^* = \| \sigma(A) \|_1$, where $\sigma (A) \in \reals^{\min \{m, n\}}$ is the vector of $A$'s singular values including multiplicities. That is, $\| A \|_2^*$ is the sum of the singular values of $A$ with multiplicities, which is called the nuclear norm or the trace norm.
\end{lemma}
\begin {proof}
{\it (Supplied by \citet {Kahan2003}.)} Let $A = U \Sigma V^t$ be a ``full'' singular value decomposition of $A$, where both $U$ and $V$ are orthogonal matrices, and where $\Sigma$ is an $m \times n$ ``diagonal'' matrix whose diagonal entries are those of $\sigma (A)$. The trace of a square matrix, $M$, is invariant under conjugation, $V^{-1} M V$, so
\begin {displaymath}
\| A \|_2^*
=
\sup_{\| B \|_2 = 1} \tr (A^t B)
=
\sup_{\| B \|_2 = 1} \tr (V \Sigma^t \, U^t B)
=
\sup_{\| B \|_2 = 1} \tr (\Sigma^t \, U^t B \, V) \, .
\end {displaymath}
Since $\| U^t B V \|_2 = \| B \|_2 = 1$, the entries of $U^t B V$ are at most $1$ in magnitude, and therefore $| \tr (\Sigma^t \, U^t B \, V) | \le \tr (\Sigma^t)$. This upper bound is attained for $B = U D V^t$ where $D$ is the $m \times n$ ``identity'' matrix. \end {proof}
An alternate proof is offered by the work of \nocite {Taub1963} von \citet {vonNeumann1937siam}. He studied a special class of norms for $\reals^{m \times n}$. A symmetric gauge function of order $p$ is a norm for $\reals^p$ that is unchanged by every permutation and sign change of the entries of the vectors. Such a function applied to the singular values of a matrix always defines a norm on $\reals^{m \times n}$. For example, $\| A \|_2 = \| \sigma (A) \|_\infty$ where as in lemma \ref {lem:kahan} $\sigma (A)$ is the length $\min \{ m, n \}$ column vector of singular values for $A$. The dual of this norm is given by the dual norm for the singular values vector, see \citet [p.\ 78, lem.\ 3.5] {Stewart1990}.
\begin {proof}
{\it (In the manner of John von Neumann.)} By the aforementioned lemma to von Neumann's gauge theorem, $\| A \|_2^* = \| \sigma (A) \|_\infty^* = \| \sigma (A) \|_1$.
\end {proof}
\subsection {Rank 2 Matrices}
\label {app:rank2}
This section finds singular values of rank $2$ matrices to establish some norms of the matrices that simplify the condition numbers in equation (\ref {eqn:inducedtransposed}).
\begin {lemma}
[\scshape Frobenius and nuclear norms of rank 2 matrices.]
\label {lem:rank2}
If $u_1, u_2 \in \reals^m$ and $v_1, v_2 \in \reals^n$, then (Frobenius norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F}^* =
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|\sub{F} =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u) \cos (\theta_v)} \, ,
\end {array}
\end {displaymath}
and (nuclear norm, or trace norm)
\begin {displaymath}
\begin {array} {l}
\left\| u_1^{} v_1^t + u_2^{} v_2^t \right\|_2^* =
\\
\noalign {
}
\hspace {2em} \sqrt {\strut \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2
\, \| v_2 \|_2^2 + 2 \, \| u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2
\|_2 \, \cos (\theta_u - \theta_v)} \, ,
\end {array}
\end {displaymath}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and $\theta_v$ is
the angle between $v_1$ and $v_2$. Both angles should be taken from $0$ to $\pi$.
\end{lemma}
\begin {proof}
If any of the vectors vanish, then the formulas are clearly true, so it may be assumed that the vectors are nonzero. The strategy of the proof is to represent the rank $2$ matrix as a $2 \times 2$ matrix whose singular values can be calculated. Since singular values are wanted, it is necessary that the bases for the $2 \times 2$ representation be orthonormal.
To that end, let $w_1$ and $w_2$ be orthogonal unit vectors with $u_1 = \alpha_1 w_1$ and $u_2 = \alpha_2 w_1 + \beta w_2$. The coefficients' signs are indeterminate, so without loss of generality assume $\alpha_1 \ge 0$ and $\beta \ge 0$, in which case
\begin {displaymath}
\alpha_1 = \| u_1 \|_2
\qquad
\alpha_2 = {u_1^t u_2^{} \over \| u_1 \|_2}
\qquad
\beta = \left\| u_2 - \left( {u_1^t u_2^{} \over \| u_1 \|_2} \right)
\left( {u_1 \over \| u_1 \|_2} \right) \, \right\|_2 \, .
\end {displaymath} Similarly, let $x_1$ and $x_2$ be mutually orthogonal
unit vectors with $v_1 = \gamma_1 x_1$ and $v_2 = \gamma_2 x_1 + \delta
x_2$. Again without loss of generality $\gamma_1 \ge 0$ and $\delta \ge 0$
so that
\begin {displaymath}
\gamma_1 = \| v_1 \|_2
\qquad
\gamma_2 = {v_1^t v_2^{} \over \| v_1 \|_2}
\qquad
\delta = \left\| v_2 - \left( {v_1^t v_2^{} \over \| v_1 \|_2} \right)
\left( {v_1 \over \| v_1 \|_2} \right) \, \right\|_2 \, .
\end {displaymath}
Notice that
\begin {displaymath}
\beta^2 = \| u_2 \|_2^2 - \left( {u_1^t u_2^{} \over \| u_1 \|_2} \right)^2
\qquad
\delta^2 = \| v_2 \|_2^2 - \left( {v_1^t v_2^{} \over \| v_1 \|_2}
\right)^2 \, .
\end {displaymath}
Let $G = u_1^{} v_1^t + u_2^{} v_2^t$. A straightforward calculation shows that, with respect to the orthonormal basis consisting of $x_1$ and $x_2$, the matrix $G^t G$ is represented by the matrix
\begin {displaymath}
M = \left[ \begin {array} {c c} {\beta }^2\,{{{\gamma }_2}}^2 +
{\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)
}^2& {\beta }^2\,\delta \,{{\gamma }_2} +
\delta \,{{\alpha }_2}\,
\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)\\ \noalign {
} {\beta
}^2\,\delta \,{{\gamma }_2} +
\delta \,{{\alpha }_2}\,
\left( {{\alpha }_1}\,{{\gamma }_1} +
{{\alpha }_2}\,{{\gamma }_2} \right)& {\beta }^2\,{\delta }^2 +
{\delta }^2\,{{{\alpha }_2}}^2 \end {array} \right] \, .
\end {displaymath}
The desired norms are now given in terms of the eigenvalues of $M$, $\lambda_\pm$,
\begin {displaymath}
\| G \|\sub{F}
=
\sqrt {\lambda_+ + \lambda_-} = \sqrt {\tr (M)}
\quad \mbox {and} \quad
\| G \|_2^*
=
\sqrt {\lambda_+} + \sqrt {\lambda_-} \, .
\end {displaymath}
The expression for $\| G \|_2^*$ requires further analysis. For any $2 \times 2$ matrix $M$,
\begin {displaymath}
\lambda_\pm = {\tr (M) \over 2} \pm \sqrt { \left( {\tr (M) \over 2}
\right)^2 - \det (M)} \, .
\end {displaymath}
In the present case these eigenvalues are real because the $M$ of interest is symmetric, and $\det (M) \ge 0$ because it is also positive semidefinite. Altogether $[\tr (M)]^2 \ge 4 \det (M) \ge 0$, so $\tr (M) \ge 2 \det (M) \ge 0$. These bounds prove the following quantities are real, and it can be verified they are the square roots of the eigenvalues of $M$,
\begin {displaymath}
\sqrt {\lambda_\pm} = \sqrt {{\tr (M) \over 4} + \sqrt {\det (M) \over 4}}
\pm \sqrt {{\tr (M) \over 4} - \sqrt {\det (M) \over 4}} \, ,
\end {displaymath}
thus
\begin {displaymath}
\| G \|_2^*
=
\sqrt {\lambda_+} + \sqrt {\lambda_-}
=
\sqrt {\tr (M) + 2 \sqrt {\det (M)}} \, .
\end {displaymath}
In summary, the desired quantities $\| G \|_2^*$ and $\| G \|\sub{F}^{}$ have been expressed in terms of $\det (M)$ and $\tr (M)$ which the expression for $M$ expands into formulas of $\alpha_i$, $\beta$, $\gamma_i$, and $\delta$. These in turn expand to expressions of $u_i$ and $v_i$. It is remarkable that the ultimate expressions in terms of $u_i$ and $v_i$ are straightforward,
\begin {eqnarray*}
\tr (M)
& =
& \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \,
(u_1^t u_2^{}) (v_1^t v_2^{})\\
\noalign {
}
& =
& \| u_1 \|_2^2 \, \| v_1 \|_2^2 + \| u_2 \|_2^2 \, \| v_2 \|_2^2 + 2 \, \|
u_1 \|_2 \, \| v_1 \|_2 \, \| u_2 \|_2 \, \| v_2 \|_2 \, \cos (\theta_u)
\cos (\theta_v)\\
\noalign {
}
\det (M)
& =
& \left( \| u_1 \|_2^2 \, \| u_2 \|_2^2 - (u_1^t u_2^{})^2 \right) \left(
\| v_1 \|_2^2 \, \| v_2 \|_2^2 - (v_1^t v_2^{})^2 \right)\\
\noalign {
}
& =
& \left( \| u_1 \|_2^2 \, \| u_2 \|_2^2 (\sin (\theta_u))^2 \right) \left(
\| v_1 \|_2^2 \, \| v_2 \|_2^2 (\sin (\theta_v))^2 \right) \, ,
\end {eqnarray*}
where $\theta_u$ is the angle between $u_1$ and $u_2$, and similarly for $\theta_v$. The formula for $\| G \|\sub{F}$ is established. The formula for $\| G \|_2^*$ simplifies, using the difference formula for cosine, to the one in the statement of the lemma. Since the positive root of $\sqrt {\det (M)}$ is wanted, the angles should be chosen from $0$ to $\pi$ so the squares of the sines can be removed without inserting a change of sign. These calculations have been verified with Mathematica \citep {Wolfram2003}.
\end {proof}
\raggedright
\bibliographystyle {plainnat}
\bibliographystyle {siam}
\end{document} |
\begin{document}
\title{Jumps, folds and hypercomplex structures}
\author{Roger Bielawski* \and Carolin Peternell}
\operatorname{ad}dress{Institut f\"ur Differentialgeometrie,
Leibniz Universit\"at Hannover,
Welfengarten 1, 30167 Hannover, Germany}
\email{bielawski@math.uni-hannover.de}
\thanks{Both authors are members of, and the second author is fully supported by the DFG Priority
Programme 2026 ``Geometry at infinity".}
\begin{abstract}
We investigate the geometry of the Kodaira moduli space $M$ of sections of $\pi:Z\to {\mathbb{P}}^1$, the normal bundle of which is allowed to jump from ${\mathcal{O}}(1)^{n}$ to ${\mathcal{O}}(1)^{n-2m}\oplus{\mathcal{O}}(2)^{m}\oplus{\mathcal{O}}^{m}$. In particular, we identify the natural assumptions which guarantee that the Obata connection of the hypercomplex part of $M$ extends to a logarithmic connection on $M$.
\end{abstract}
{\mathfrak s \mathfrak u}bjclass{53C26, 53C28}
\maketitle
\thispagestyle{empty}
\section{Introduction}
It is well known that a hyperk\"ahler or a hypercomplex structure on a smooth manifold $M$ can be encoded in the {\em twistor space}, which is a complex manifold $Z$ fibring over ${\mathbb{P}}^1$ and equipped with an antiholomorphic involution $\sigma$ covering the antipodal map. The manifold $M$ is recovered as the parameter space of $\sigma$-invariant sections with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$ ($n=\dim_{\mathbb C} M$). If we start with an arbitrary complex manifold $Z$ equipped with a holomorphic submersion $\pi:Z\to {\mathbb{P}}^1$ and an involution $\sigma$, then the corresponding component of the Kodaira moduli space of sections of $\pi$ will typically also contain sections with other normal bundles $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$. This kind of jumping normal bundle attracted recently attention in the case of $4$-dimensional hyperk\"ahler manifolds \cite{Hit2,Biq,Dun}, in the context of a phenomenon known as {\em folding} (one speaks then of {\em folded hyperk\"ahler metrics}).
\par
Folded hyperk\"ahler structures do not exhaust all geometric possibilities which arise when the normal bundle is allowed to jump. Even in four dimensions there are examples which are not folded (Example \operatorname{Re}f{proj} below). The aim of this paper is to investigate the natural extension of the hypercomplex geometry arising on such manifolds of sections (folded or not). More precisely, we are interested in the differential geometry of the (smooth) parameter space $ M$ of sections of $\pi:Z\to {\mathbb{P}}^1$ with normal bundle $N$ isomorphic $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$, where each $k_i{\mathfrak g}eq 0$. We shall discuss only the purely holomorphic case, i.e. we are interested in all sections, not just $\sigma$-invariant. Choosing an appropriate $\sigma$ allows one to carry over all results to hypercomplex or split hypercomplex manifolds.
\par
Our particular object of interest is the (holomorphic) {\em Obata connection} $\nabla$, i.e. the unique torsion free connection preserving the hypercomplex (or, rather, the biquaternionic, i.e. complexified hypercomplex) structure. This is defined on the open subset $U$ of $M$ corresponding to the sections with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$. The general twistor machinery (see, e.g. \cite{BE}) implies that $\nabla$ extends to a first order differential operator $D$ on sections of certain vector bundle defined over all of $M$.
Our point of view is to regard $D$ as a particular type of meromorphic connection with polar set ${\partial}elta=M\backslash U$. In general, this {\em meromorphic Obata connection} can have higher order poles along ${\partial}elta$. We show, however, that in the case when $M$ arises from a (partial) compactification of the twistor space of a hypercomplex manifold, the meromorphic Obata connection has a simple pole, and in fact it is then a logarithmic connection.
\section{Geometry of jumps}
{\mathfrak s \mathfrak u}bsection{Two examples}
We begin with two examples illustrating the different geometric possibilities occuring when the normal bundle of a twistor line jumps. The first example is the basic example of a {\em hyperk\"ahler fold}, as explained by Hitchin \cite{Hit2}.
\begin{example} (Calabi-Eguchi-Hanson gravitational instanton) The twistor space $Z$ of the Calabi-Eguchi-Hanson metric is the resolution of the variety
$$\{(x,y,z)\in {\mathcal{O}}(2)\oplus{\mathcal{O}}(2)\oplus{\mathcal{O}}(2); xy=(z-p_1)(z-p_2)\},$$
where $p_i(\zeta)=a_i\zeta^2+2b_i\zeta+c_i$, $i=1,2$, are two fixed sections of ${\mathcal{O}}(2)$. The sections of the projection $Z\to {\mathbb{P}}^1$ can be described as follows \cite{Hit1}: let $z=a\zeta^2+2b\zeta+c$ be a section of ${\mathcal{O}}(2)$ and let $\alpha_i,\beta_i$ be the roots of $z-p_i$, $i=1,2$. Then the sections are given by
$$ z=a\zeta^2+2b\zeta+c, \enskip x=A(\zeta-\alpha_1)(\zeta-\alpha_2),\enskip y=B(\zeta-\beta_1)(\zeta-\beta_2),$$
where $AB=(a-a_1)(a-a_2)$.
A computation by Hitchin in \cite{Hit1} determines the splitting type of the normal bundle and can be interpreted as follows. Elements $\tau$ of $GL_2({\mathbb C})$ with $\det\tau=-1$ and $\operatorname{tr}\tau =0$ satisfy $\tau^2=1$. For any pair $p_1,p_2$ of quadratic polynomials there exists such an $\tau$ exchanging $p_1$ and $p_2$, which, consequently, acts on $Z$.
The normal bundle of a $\tau$-invariant section splits as ${\mathcal{O}}(2)\oplus {\mathcal{O}}$; otherwise as ${\mathcal{O}}(1)\oplus{\mathcal{O}}(1)$.
\par
To see this directly, observe that modulo translations and the action of $GL_2({\mathbb C})$, $p_1(\zeta)=\zeta$ and $p_2(\zeta)=-\zeta$. The involution $\tau$ is then simply $\zeta\mapsto -\zeta$, and since the normal bundle $N$ of an invariant section satisfies $\tau^\ast N=N$, it must split into line bundles of even degrees. The $\tau$-invariant sections of $Z$ are given by
$z=a\zeta^2 +c$ and by
$$ x=A\left(\zeta+\frac{1+\sqrt{ac}}{a}\right)\left(\zeta-\frac{1-\sqrt{ac}}{a}\right), \enskip y=B\left(\zeta-\frac{1+\sqrt{ac}}{a}\right)\left(\zeta +\frac{1-\sqrt{ac}}{a}\right),$$
where $AB=a^2$. Consequently, for every $\zeta\neq 0,\infty$, the map given by intersecting a section with the fibre $\pi^{-1}(\zeta)$ remains surjective when restricted to sections with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$.
\label{CEH}\end{example}
\begin{example} (${\mathbb{R}} \rm P^4$) Let $Z={\mathbb{P}}\bigl({\mathcal{O}}(1)\oplus{\mathcal{O}}(1)\oplus{\mathcal{O}}\bigr)$ be the compactification of the twistor space of ${\mathbb{R}}^4$. Sections are described as projective equivalence classes:
$$\zeta\mapsto [a_1\zeta+b_1,a_2\zeta+b_2,c],\enskip a_i,b_i,c\in {\mathbb C},\enskip c\neq 0\operatorname{Im}plies \det\begin{pmatrix} a_1 & b_1\\a_2 & b_2\end{pmatrix}\neq 0.$$
The normal bundle of such a section is ${\mathcal{O}}(1)\oplus{\mathcal{O}}(1)$ if $c\neq 0$, and ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ if $c=0$. The manifold of all sections is an open subset of ${\mathbb C}\rm P^4$, and the manifold of real sections, i.e. satisfying $b_1=-\bar a_2$, $b_2=\bar a_1$, $c=\bar c$, is ${\mathbb{R}} \rm P^4$.
\par
Observe that the twistor lines with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ are all contained in their own minitwistor space ${\mathbb{P}}\bigl({\mathcal{O}}(1)\oplus{\mathcal{O}}(1)\oplus 0\bigr)\simeq {\mathbb{P}}^1\times {\mathbb{P}}^1$.
\label{proj}\end{example}
From the point of view of hyperk\"ahler geometry, the difference between the two examples is clear: in the first case, there is a well defined ${\mathcal{O}}(2)$-valued symplectic form along the fibres of $Z$. In the second example, this is not the case. The second example does not fit into Hitchin's theory of folded hyperk\"ahler manifolds: the $3$-dimensional submanifold of real twistor lines with normal bundle ${\mathcal{O}}(2)\oplus {\mathcal{O}}$ is ${\mathbb{R}} \rm P^3$, so it is not even a contact manifold.
\par
Our aim now is to investigate both the common features and the differences in the behaviour of the hypercomplex structure and of the Levi-Civita (i.e. Obata) connection.
{\mathfrak s \mathfrak u}bsection{$2$-Kronecker structures\label{Kr}}
Let $Z$ be a complex manifold of dimension $n+1$ and $\pi:Z\to {\mathbb{P}}^1$ a surjective holomorphic submersion. We are interested in the (necessarily smooth) parameter space $M$ of sections of $\pi$ with normal bundle $N$ isomorphic to $\bigoplus_{i=1}^n{\mathcal{O}}(k_i)$, where $k_i\in\{0,1,2\}$ and $n={\mathfrak s \mathfrak u}m k_i$. Its dimension (as long as it is nonempty) is $2n$ and we consider a connected component $M$ which contains a section with normal bundle isomorphic to ${\mathcal{O}}(1)^{\oplus n}$.
\par
The tangent space $T_m M$ at any $m\in M$ is canonically isomorphic to $H^0(s_m,N)$, where $s_m$ is the section corresponding to $m$. Similarly, we have a rank $n$ bundle $E$ over $M$, the fibre of which is $H^0(s_m,N(-1))$, where $N(-1)=N\otimes \pi^\ast{\mathcal{O}}_{{\mathbb{P}}^1}(-1)$. The multiplication map $H^0(N(-1))\otimes H^0({\mathcal{O}}(1))\to H^0(N)$ induces a homomorphism
$$ \alpha:E\otimes {\mathbb C}^2\to TM,$$
which is an isomorphism at any $m$ with $N_{s_m/Z}\simeq {\mathcal{O}}(1)^n$. It follows that the subset of $M$ consisting of sections with other normal bundles is a divisor ${\partial}elta$ in $M$. We shall assume throughout that the set of singular points of ${\partial}elta$ has codimension $2$ in $M$ (in particular ${\partial}elta$ is reduced). This means that the normal bundle of a section corresponding to a smooth point of ${\partial}elta$ is isomorphic to ${\mathcal{O}}(1)^{n-2}\oplus {\mathcal{O}}(2)\oplus{\mathcal{O}}$.
\par
Observe also that $\alpha$ is injective on each subbundle of the form $E\otimes v$, where $v$ is a fixed nonzero vector in ${\mathbb C}^2$. The image $D_v$ of the subbundle $E\otimes v$ is an integrable distribution on $TM$ (sections of $\pi$ vanishing at the zero of $v\in H^0({\mathcal{O}}(1))$) and we recover $Z$ as the space of leaves of the distribution $D$ on $M\times {\mathbb{P}}^1$ given by $D|_{M\times [v]}=D_v$.
\begin{remark} We can also define $E$ as the kernel of the evaluation map $H^0(N)\otimes{\mathcal{O}}_{{\mathbb{P}}^1}\to N$ (which is what we do in \cite{BP1}), i.e.
$$0\to E_m\otimes {\mathcal{O}}_{{\mathbb{P}}^1}(-1) \stackrel{A}{\longrightarrow} H^0(N)\otimes{\mathcal{O}}_{{\mathbb{P}}^1}\longrightarrow N\to 0.$$
We obtain again a map $\alpha:E\otimes {\mathbb C}^2\to TM$ by restricting $A$ to each subspace of the form $E\otimes\langle v\rangle$, $v\in {\mathbb C}^2$. But then the above sequence identifies $H^0(N(-1))$ with $E_m\otimes H^1({\mathcal{O}}_{{\mathbb{P}}^1}(-2)$. Thus, viewing $\alpha$ as the multiplication map $H^0(N(-1))\otimes H^0({\mathcal{O}}(1))\to H^0(N)$ means that we have implicitly identified $H^1({\mathcal{O}}_{{\mathbb{P}}^1}(-2))$ with
${\mathbb C}$. Such an identification yields also a choice of a symplectic form on $H^0({\mathcal{O}}(1))$ within its conformal class, i.e. an identification of ${\mathbb C}^2$ with $({\mathbb C}^2)^\ast$.
\label{subtle}\end{remark}
This geometric structure on $M$ was introduced in \cite{BP1} as an {\em integrable $2$-Kronecker structure}. We now want to present a different point of view, directly in terms of the tangent bundle of $M$.
\par
Let $M$ be a complex manifold and let ${\partial}elta$ be a divisor satisfying the above smoothness assumption. Suppose that we are given a codimension $1$ distribution ${\mathcal{V}}$ on the smooth locus ${\partial}elta_{\rm reg}$ of ${\partial}elta$. We define $TM(-{\mathcal{V}})$ to be the sheaf of germs of holomorphic vector fields $X$ on $M$ such that $X_x\in {\mathcal{V}}_x$ for any $x\in {\partial}elta_{\rm reg}$. If the sheaf $TM(-{\mathcal{V}})$ is locally free, i.e. a vector bundle $F$, then we obtain a homomorphism $\alpha:F\to TM$ from the inclusion $TM(-{\mathcal{V}}){\mathfrak h}ookrightarrow TM$ (and ${\mathcal{V}}=\operatorname{Im}\alpha$). In the case of $M$ arising as the parameter space of sections as at the beginning of the subsection, $F\simeq E\otimes {\mathbb C}^2$ and the action of $\operatorname{Mat}_{2}({\mathbb C})$ gives an action of complexified quaternions on $TM(-{\mathcal{V}})$. Moreover, for any ${\mathcal{J}}\in SL_2({\mathbb C})$ with $\operatorname{tr} {\mathcal{J}}=0$ (which implies ${\mathcal{J}}^2=-1$), the $i$-eigensubsheaf of $TM(-{\mathcal{V}})$ is closed under the Lie bracket.
Restricting to a real submanifold of $M$ (and corresponding real slices of ${\partial}elta$ and ${\mathcal{V}}$) describes the extension of the hypercomplex or split-hypercomplex geometry to manifolds of sections with jumping normal bundles.
{\mathfrak s \mathfrak u}bsection{Logarithmic hypercomplex structures}
In the setting of the above paragraph, the case of particular interest is ${\mathcal{V}}=T{\partial}elta_{\rm reg}$. Vector fields in $TM(-{\mathcal{V}})$ are called then {\em logarithmic} and $TM(-{\mathcal{V}})$ is denoted by $TM(-\log {\partial}elta)$ \cite{Saito}. Another way to characterise logarithmic vector fields is via the condition $X.z\in (z)$, where $z=0$ is the local equation of ${\partial}elta$. This shows, in particular, that the subsheaf $TM(-\log {\partial}elta)$ is closed under the Lie bracket.
\begin{definition} Let $M$ be a complex manifold and ${\partial}elta$ a divisor in $M$ such that its set of singular points is of codimension $2$ in $M$. A {\em logarithmic biquaternionic structure} on $M$ is an action of $\operatorname{Mat}_2({\mathbb C})$ on $TM(-\log {\partial}elta)$ such that the Nijenhuis tensor of each $A\in \operatorname{Mat}_2({\mathbb C})$ vanishes.\label{log}\end{definition}
\begin{remark} The same definition can be used for real manifolds and we can speak of logarithmic hypercomplex or logarithmic split hypercomplex structures.\end{remark}
Observe that for a logarithmic biquaternionic structure the leaves of the distribution $D_v=\alpha(E\otimes v)$ on ${\partial}elta$ are contained in ${\partial}elta$, i.e. the image of ${\partial}elta$ in each fibre of the twistor space has codimension $1$. In other words $Z$ is a (partial) compactification of the twistor space of a hypercomplex manifold.
More precisely:
\begin{proposition} The following two conditions are equivalent:
\begin{itemize}
\item[(i)] $\operatorname{Im}\alpha_x=T_x{\partial}elta$ for each $x\in {\partial}elta_{\rm reg}$;
\item[(ii)] for each $\zeta\in{\mathbb{P}}^1$, the map ${\partial}elta\to \pi^{-1}(\zeta)$, given by intersecting a section with the fibre, maps a neighbourhood of each point $x\in {\partial}elta_{\rm reg}$ onto an $(n-1)$-dimensional submanifold.
\end{itemize}
\label{Tw}\end{proposition}
\begin{proof} Let $f$ denote the map $M\to \pi^{-1}(\zeta)$, given by intersecting a section with the fibre. For an $x\in{\partial}elta_{\rm reg}$, we have
\begin{equation} \operatorname{Im}\alpha_x=H^0({\mathcal{O}}(1)^{n-2}\oplus {\mathcal{O}}(2)){\mathfrak s \mathfrak u}bset H^0(N)\simeq T_xM.\label{sD}\end{equation}
Thus $df(\operatorname{Im}\alpha_x)$ is an $n-1$-dimensional subspace for any $x\in {\partial}elta_{\rm reg}$.
Since $f$ is a submersion at $x$, the condition $\operatorname{Im}\alpha_x=T_x{\partial}elta$ implies now that the $f({\partial}elta_{\rm reg})$ is an immersed $(n-1)$-dimensional submanifold. Conversely, suppose that the condition (ii) holds. Then the image $f(U)$ of a neighbourhood $U$ of $x\in {\partial}elta_{\rm reg}$ is a codimension $1$ submanifold $Z_0$ of $Z$. It follows that $T_x U\simeq H^0(N_{s/Z_0})$, where $s$ is the section corresponding to $x$. Suppose that $N_{s/Z_0}\simeq \bigoplus_{i=1}^{n-1}{\mathcal{O}}(k_i)$. Given the injection $N_{s/Z_0}{\mathfrak h}ookrightarrow N_{s/Z}$, we have (after reordering the $k_i$) $k_1\leq 2$ and $k_2,\dots,k_{n-1}\leq 1$. Since $H^1(N_{s/Z_0})=0$, we have $\dim {\partial}elta_{\rm reg}=h^0(N_{s/Z_0})$ and therefore ${\mathfrak s \mathfrak u}m_{i=1}^{n-1}(k_i+1)=2n-1$. Thus $(2-k_1)+{\mathfrak s \mathfrak u}m_{i=2}^{n-1}(1-k_i)=0$ and since each summand is nonnegative, we conclude that $N_{s/Z_0}\simeq {\mathcal{O}}(2)\oplus {\mathcal{O}}(1)^{n-2}$.
Thus $T_{x}{\partial}elta=\operatorname{Im}\alpha_{x}$.
\end{proof}
\begin{remark} This is precisely the situation in Example \operatorname{Re}f{proj}. In Example \operatorname{Re}f{CEH}, the $2$-Kronecker structure is not logarithmic.
\end{remark}
\section{The meromorphic Obata connection}
The Obata connection of a hypercomplex manifold is the unique torsion-free connection with respect to which the hypercomplex structure is parallel. In the case of a hyperk\"ahler manifold, it coincides with the Levi-Civita connection. From the twistor point of view it is obtained via the Ward transform \cite{Ward,HM}. We now wish to discuss an extension of the Obata connection to a $2$-Kronecker manifold.
\par
Let $Z,M,{\partial}elta,E$ and $\alpha$ be all as in the previous section. We consider the double fibration
$$ M\stackrel{\tau}{\longleftarrow} M\times {\mathbb{P}}^1 \stackrel{\eta}{\longrightarrow} Z.$$
The normal bundle $N$ of any section of $\pi$ is isomorphic to the vertical tangent bundle $T_\pi Z=\operatorname{Ker} d\pi$ restricted to the section, and, consequently, the (holomorphic) tangent bundle $TM$ can be viewed as the Ward transform of $T_\pi Z$, i.e. $TM\simeq \tau_\ast\eta^\ast T_\pi Z$. Similarly, the bundle $E$ is the Ward transform of $T_\pi Z\otimes \pi^\ast{\mathcal{O}}(-1)$.
\par
In \cite[\S 2]{BP1} we have identified the algebraic condition satisfied by the differential operator produced by the Ward transform from any $M$-uniform vector bundle $F$ on $Z$.
In our situation, we can state the results for $F=T_\pi Z(-1)$ as:
\begin{proposition} The bundle $E$ is equipped with a first order differential operator $D:E\to E^\ast\otimes TM$ which satisfies
$D(fs)=\sigma(df\otimes s)+fDs$, where $\sigma$ (the principal symbol of $D$) is the composition of the following two maps
\begin{equation}\begin{CD}
T^\ast M\otimes E @> \alpha^\ast\otimes 1 >> E^\ast\otimes {\mathbb C}^2\otimes E @> 1\otimes\alpha>> E^\ast\otimes TM
\end{CD}\label{phi}\end{equation}
(where ${\mathbb C}^2\simeq ({\mathbb C}^2)^\ast$ as explained in Remark \operatorname{Re}f{subtle}).{\mathfrak h}fill ${\mathbb{B}}ox$
\label{D}\end{proposition}
\begin{remark} On $M\backslash{\partial}elta$\; $\sigma$ is invertible and $\sigma^{-1}\circ D$ is the standard hyperholomorphic connection on $E$, i.e. its tensor product with the standard flat connection on ${\mathbb C}^2$ is the (holomorphic) Obata connection on $M\backslash{\partial}elta$.
\end{remark}
\begin{remark} Given any first order differential operator $D:E\to F$ between (sections of) vector bundles on a manifold $M$, with symbol $\sigma:E\otimes T^\ast M \to F$, we can ``tensor" it with any connection $\nabla$ on a vector bundle $W$ over $M$:
$$ (D\otimes_\sigma \nabla) (e\otimes w)=D(e)\otimes w + (\sigma\otimes 1)(e\otimes \nabla w).$$
The symbol of this new operator is $\sigma\otimes 1$. We can do this for our operator $D$ and the flat connection on ${\mathbb C}^2$. We obtain a differential operator $\tilde D:E\otimes {\mathbb C}^2\to E^\ast\otimes TM\otimes {\mathbb C}^2$ which extends the Obata connection. \label{tensor}\end{remark}
\begin{remark} The results claimed by Pantillie \cite{Pant} would imply that the Obata connection extends to a differential operator satisfying $\tilde D(fs)=\alpha^\ast(df)\otimes s+f\tilde Ds$, but we have trouble following his arguments (in particular the second last paragraph in the proof of his Theorem 2.1).
\end{remark}
We can view $\sigma^{-1}\circ D$ as a meromorphic connection on $E$, with polar set ${\partial}elta$. Similarly the Obata connection on $M\backslash {\partial}elta$ can be viewed as a meromorphic connection on $TM$ with polar set ${\partial}elta$. It follows from Proposition \operatorname{Re}f{D} that $\sigma^{-1}$ generally has a double pole along ${\partial}elta$ and, hence, so does $\sigma^{-1}\circ D$. We shall now discuss conditions under which the pole becomes simple.
\par
Let $z=0$ be the local equation of ${\partial}elta$. The meromorphic connection $\sigma^{-1}\circ D$ has a simple pole if $\lim_{z\to 0} z^2\sigma^{-1}\circ D=0$. Let us trivialise locally $E$, so that $\alpha$ is an endomorphism of the trivial bundle. We can then write $z=\det\alpha$, and owing to Proposition \operatorname{Re}f{D}, we have:
$$ z^2\sigma^{-1}( De)=((\alpha^\ast)_{\rm adj}\otimes 1)(1\otimes \alpha_{\rm adj})(De),$$
where the subscript ``adj" denotes the classical adjoint. Thus, we can conclude:
\begin{lemma} The meromorphic connection $\sigma^{-1}\circ D$ has a simple pole along ${\partial}elta$ provided that $De|_x\in E^\ast_x\otimes\operatorname{Im}\alpha_x$ for any $x\in {\partial}elta_{\rm reg}$ and any local section $e$ of $E$. If this is the case, then the residue of $\sigma^{-1}\circ D$ belongs to $\operatorname{Ker}\alpha^\ast\otimes \operatorname{End} E$.{\mathfrak h}fill${\mathbb{B}}ox$\label{lemma}\end{lemma}
Returning to the description of a $2$-Kronecker structure given at the end of \S\operatorname{Re}f{Kr}, observe that the subsheaf $TM[-{\mathcal{V}}]$ is precisely the subsheaf $\operatorname{Im}\alpha$, and therefore the condition of the last lemma is equivalent to the existence of a differential operator
$$ D^\prime:E\to E^\ast \otimes {\mathbb C}^2\otimes E$$
such that $D=(1\otimes\alpha)\circ D^\prime$.
We shall now show that for a logarithmic hypercomplex structure (Definition \operatorname{Re}f{log}) the condition of the above lemma is automatically satisfied.
\begin{proposition} Suppose that $\operatorname{Im}\alpha_x=T_x{\partial}elta$ for each $x\in {\partial}elta_{\rm reg}$. Then the condition of Lemma \operatorname{Re}f{lemma} is satisfied.\end{proposition}
\begin{proof} Proposition \operatorname{Re}f{Tw} implies that points of ${\partial}elta_{\rm reg}$ correspond to sections of $\pi:Z\to {\mathbb{P}}^1$ contained in a codimension $1$ submanifold $Z^\prime$ of $Z$.
The differential operator $D$ is obtained by the push-forward of the flat relative connection $\nabla_\eta$ on $\eta^\ast T_\pi Z(-1)$, i.e. of the exterior derivative in the vertical directions of the projection $\eta:M\times {\mathbb{P}}^1\to Z$. It follows that, over ${\partial}elta_{\rm reg}$, $D$ restricts to an operator $D^\prime$ defined in the same way as $D$, but with $Z$ replaced by $Z^\prime$. This means that $D^\prime$ takes values in $E\otimes T{\partial}elta_{\rm reg}$.
\end{proof}
Recall that a meromorphic connection on a vector bundle $E$ is called logarithmic, if it has a simple pole along ${\partial}elta=\{z=0\}$ and its residue is of the form $Adz$, where
$A\in\operatorname{End} E$. Thus, under the assumption of the last proposition, $\sigma^{-1}\circ D$ is a logarithmic connection.
\par
We finish with some remarks about the meromorphic Obata connection. As remarked in \operatorname{Re}f{tensor}, we can tensor $D$ with the flat connection on ${\mathbb C}^2$ to obtain an operator $\tilde D:E\otimes {\mathbb C}^2\to E^\ast\otimes TM\otimes {\mathbb C}^2$. The meromorphic Obata connection is a meromorphic connection on $TM$ given
by $(\sigma\otimes 1)^{-1}\circ \tilde D\circ \alpha^{-1}$, where $\sigma$ is the symbol of $D$. Thus, in general we can expect the Obata connection to have a third order pole along the divisor ${\partial}elta$ ($\sigma^{-1}$ contributing two orders and $\alpha^{-1}$ another one). The next example shows that this is indeed the case.
\begin{example} Consider again the twistor space of the Calabi-Eguchi-Hanson gravitational instanton, described in Example \operatorname{Re}f{CEH}. Choose a family of real sections containing the (real) jumping lines. The resulting metric is given in the complex coordinates corresponding to the complex structure $I$, namely $z=-\bar{a}$, $u=\ln \bar A^2$ by the formula \cite[(4.6)]{Hit1}:
$$ {\mathfrak g}amma dzd\bar z+(du+\bar\delta dz)(d\bar u+\delta d\bar z),$$
where ${\mathfrak g}amma$ and $\delta$ are certain functions of the coordinates and the fold ${\partial}elta$ is given by ${\mathfrak g}amma=0$. The Hermitian matrix of this metric is then
$$\begin{pmatrix} {\mathfrak g}amma+{\mathfrak g}amma^{-1}|\delta|^2 & {\mathfrak g}amma^{-1}\bar\delta\\ {\mathfrak g}amma^{-1}\delta & {\mathfrak g}amma^{-1}\end{pmatrix},$$
and it follows that the Levi-Civita connection has poles of third order along ${\partial}elta$.
\end{example}
It is interesting to observe that if the assumption of Lemma \operatorname{Re}f{lemma} is satisfied, then the Obata connection still has a simple pole along ${\partial}elta$ (rather than a second order one, as one could expect). Indeed, as noted above, the operator $D$ is then of the form $D=(1\otimes\alpha)\circ D^\prime$, where $ D^\prime:E\to E^\ast\otimes {\mathbb C}^2 \otimes E$ has symbol $\alpha^\ast\otimes 1$. It follows that the meromorphic Obata connection as an operator $TM\to T^\ast M\otimes TM$ is of the form
$$ (1\otimes\alpha)\circ(\alpha^\ast\otimes 1)^{-1}\circ \widetilde{D^\prime}\circ \alpha^{-1}.$$
Since $(\alpha^\ast\otimes 1)^{-1}\circ \widetilde{D^\prime}$ is a meromorphic connection on $E\otimes{\mathbb C}^2$ with a simple pole along ${\partial}elta$, the meromorphic Obata connection also has a simple pole, as the conjugation by $\alpha$ does not increase the order of the pole.
In particular:
\begin{corollary} Suppose that the equivalent conditions of Proposition \operatorname{Re}f{Tw} are satisfied. Then the holomorphic Obata connection on $M\backslash{\partial}elta$ extends to a logarithmic connection on $M$.{\mathfrak h}fill ${\mathbb{B}}ox$
\end{corollary}
\end{document} |
\begin{document}
\title{A Weak Galerkin finite element method for second-order elliptic problems}
\begin{abstract}
In this paper, authors shall introduce a finite element method by
using a weakly defined gradient operator over discontinuous
functions with heterogeneous properties. The use of weak gradients
and their approximations results in a new concept called {\em
discrete weak gradients} which is expected to play important roles
in numerical methods for partial differential equations. This
article intends to provide a general framework for operating
differential operators on functions with heterogeneous properties.
As a demonstrative example, the discrete weak gradient operator is
employed as a building block to approximate the solution of a model
second order elliptic problem, in which the classical gradient
operator is replaced by the discrete weak gradient. The resulting
numerical approximation is called a weak Galerkin (WG) finite
element solution. It can be seen that the weak Galerkin method
allows the use of totally discontinuous functions in the finite
element procedure. For the second order elliptic problem, an optimal
order error estimate in both a discrete $H^1$ and $L^2$ norms are
established for the corresponding weak Galerkin finite element
solutions. A superconvergence is also observed for the weak Galerkin
approximation.
\end{abstract}
\begin{keywords}
Galerkin finite element methods, discrete gradient, second-order
elliptic problems, mixed finite element methods
\end{keywords}
\begin{AMS}
Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
\end{AMS}
\pagestyle{myheadings}
\section{Introduction}
The goal of this paper is to introduce a numerical approximation
technique for partial differential equations based on a new
interpretation of differential operators and their approximations.
To illustrate the main idea, we consider the Dirichlet problem for
second-order elliptic equations which seeks an unknown functions
$u=u(x)$ satisfying
\begin{eqnarray}
-\nabla\cdot (a \nabla u)+\nabla\cdot (b u)+cu &=& f\quad
\mbox{in}\;\Omega,\label{pde}\\
u&=&g\quad \mbox{on}\; \partial\Omega,\label{bc}
\end{eqnarray}
where $\Omega$ is a polygonal or polyhedral domain in
$\mathbb{R}^d\; (d=2,3)$, $a=(a_{ij}(x))_{d\times d}\in
[L^{\infty}(\Omega)]^{d^2}$ is a symmetric matrix-valued function,
$b=(b_i(x))_{d\times 1}$ is a vector-valued function, and $c=c(x)$
is a scalar function on $\Omega$. Assume that the matrix $a$
satisfies the following property: there exists a constant $\alpha
>0$ such that
\begin{equation}\label{matrix}
\alpha\xi^T\xi\leq \xi^T a\xi,\quad\forall \xi\in \mathbb{R}^d.
\end{equation}
For simplicity, we shall concentrate on two-dimensional problems
only (i.e., $d=2$). An extension to higher-dimensional problems is
straightforward.
The standard weak form for (\ref{pde}) and (\ref{bc}) seeks $u\in
H^1(\Omega)$ such that $u=g$ on $\partial\Omega$ and
\begin{eqnarray}\label{weakform}
(a\nabla u, \nabla v)-(b u,\nabla v)+(cu,v)=(f,v)\quad \forall v\in
H_0^1(\Omega),
\end{eqnarray}
where $(\phi,\psi)$ represents the $L^2$-inner product of
$\phi=\phi(x)$ and $\psi=\psi(x)$ -- either vector-valued or
scalar-valued functions. Here $\nabla u$ denotes the gradient of the
function $u=u(x)$, and $\nabla$ is known as the gradient operator.
In the standard Galerkin method (e.g., see \cite{ci, sue}), the
trial space $H^1(\Omega)$ and the test space $H_0^1(\Omega)$ in
(\ref{weakform}) are each replaced by properly defined subspaces of
finite dimensions. The resulting solution in the subspace/subset is
called a Galerkin approximation. A key feature in the Galerkin
method is that the approximating functions are chosen in a way that
the gradient operator $\nabla$ can be successfully applied to them
in the classical sense. A typical implication of this property in
Galerkin finite element methods is that the approximating functions
(both trial and test) are continuous piecewise polynomials over a
prescribed finite element partition for the domain, often denoted by
${\cal T}_h$. Therefore, a great attention has been paid to a
satisfaction of the embedded ``continuity" requirement in the
research of Galerkin finite element methods in existing literature
till recent advances in the development of discontinuous Galerkin
methods. But the interpretation of the gradient operator still lies
in the classical sense for both ``continuous" and ``discontinuous"
Galerkin finite element methods in current existing literature.
In this paper, we will introduce a weak gradient operator defined on
a space of functions with heterogeneous properties. The weak
gradient operator will then be employed to discretize the problem
(\ref{weakform}) through the use of a discrete weak gradient
operator as building bricks. The corresponding finite element method
is called {\em weak Galerkin} method. Details can be found in
Section \ref{section4}.
To explain weak gradients, let $K$ be any polygonal domain with
interior $K^0$ and boundary $\partial K$. A {\em weak function} on
the region $K$ refers to a vector-valued function $v=\{v_0, v_b\}$
such that $v_0\in L^2(K)$ and $v_b\in H^{\frac12}(\partial K)$. The
first component $v_0$ can be understood as the value of $v$ in the
interior of $K$, and the second component $v_b$ is the value of $v$
on the boundary of $K$. Note that $v_b$ may not be necessarily
related to the trace of $v_0$ on $\partial K$ should a trace be
defined. Denote by $W(K)$ the space of weak functions associated
with $K$; i.e.,
\begin{equation}\label{hi.888.new}
W(K) = \{v=\{v_0, v_b \}:\ v_0\in L^2(K),\; v_b\in
H^{\frac12}(\partial K)\}.
\end{equation}
Recall that the dual of $L^2(K)$ can be identified with itself by
using the standard $L^2$ inner product as the action of linear
functionals. With a similar interpretation, for any $v\in W(K)$, the
{\bf weak gradient} of $v$ can be defined as a linear functional
$\nabla_d v$ in the dual space of $H({\rm div},K)$ whose action on
each $q\in H({\rm div},K)$ is given by
\begin{equation}\label{weak-gradient-new}
(\nabla_d v, q) := -\int_K v_0 \nabla\cdot q dK+ \int_{\partial K}
v_b q\cdot{\bf n} ds,
\end{equation}
where ${\bf n}$ is the outward normal direction to $\partial K$. Observe
that for any $v\in W(K)$, the right-hand side of
(\ref{weak-gradient-new}) defines a bounded linear functional on the
normed linear space $H({\rm div}, K)$. Thus, the weak gradient
$\nabla_d v$ is well defined. With the weak gradient operator
$\nabla_d$ being employed in (\ref{weakform}), the trial and test
functions can be allowed to take separate values/definitions on the
interior of each element $T$ and its boundary. Consequently, we are
left with a greater option in applying the Galerkin to partial
differential equations.
Many numerical methods have been developed for the model problem
(\ref{pde})-(\ref{bc}). The existing methods can be classified into
two categories: (1) methods based on the primary variable $u$, and
(2) methods based on the variable $u$ and a flux variable (mixed
formulation). The standard Galerkin finite element methods
(\cite{ci, sue, baker}) and various interior penalty type
discontinuous Galerkin methods (\cite{arnold, abcm, bo, rwg, rwg-1})
are typical examples of the first category. The standard mixed
finite elements (\cite{rt, arnold-brezzi, babuska, brezzi, bf, bdm,
bddf, wang}) and various discontinuous Galerkin methods based on
both variables (\cite{ccps, cs, cgl, jp}) are representatives of the
second category. Due to the enormous amount of publications
available in general finite element methods, it is unrealistic to
list all the key contributions from the computational mathematics
research community in this article. The main intention of the above
citation is to draw a connection between existing numerical methods
with the one that is to be presented in the rest of the Sections.
The weak Galerkin finite element method, as detailed in Section
\ref{section4}, is closely related to the mixed finite element
method (see \cite{rt, arnold-brezzi, babuska, brezzi, bdm, wang})
with a hybridized interpretation of Fraeijs de Veubeke \cite{fdv1,
fdv2}. The hybridized formulation introduces a new term, known as
the Lagrange multiplier, on the boundary of each element. The
Lagrange multiplier is known to approximate the original function
$u=u(x)$ on the boundary of each element. The concept of {\em weak
gradients} shall provide a systematic framework for dealing with
discontinuous functions defined on elements and their boundaries in
a near classical sense. As far as we know, the resulting weak
Galerkin methods and their error estimates are new in many
applications.
\section{Preliminaries and Notations}\label{section2}
We use standard definitions for the Sobolev spaces $H^s(D)$ and
their associated inner products $(\cdot,\cdot)_{s,D}$, norms
$\|\cdot\|_{s,D}$, and seminorms $|\cdot|_{s,D}$ for $s\ge 0$. For
example, for any integer $s\ge 0$, the seminorm $|\cdot|_{s, D}$
is given by
$$
|v|_{s, D} = \left( \sum_{|\alpha|=s} \int_D |\partial^\alpha v|^2 dD
\right)^{\frac12},
$$
with the usual notation
$$
\alpha=(\alpha_1, \alpha_2), \quad |\alpha| = \alpha_1+\alpha_2,\quad
\partial^\alpha =\partial_{x_1}^{\alpha_1} \partial_{x_2}^{\alpha_2}.
$$
The Sobolev norm $\|\cdot\|_{m,D}$ is given by
$$
\|v\|_{m, D} = \left(\sum_{j=0}^m |v|^2_{j,D} \right)^{\frac12}.
$$
The space $H^0(D)$ coincides with $L^2(D)$, for which the norm and
the inner product are denoted by $\|\cdot \|_{D}$ and
$(\cdot,\cdot)_{D}$, respectively. When $D=\Omega$, we shall drop
the subscript $D$ in the norm and inner product notation. The space
$H({\rm div};\Omega)$ is defined as the set of vector-valued
functions on $\Omega$ which, together with their divergence, are
square integrable; i.e.,
\[
H({\rm div}; \Omega)=\left\{ {\bf v}: \ {\bf v}\in [L^2(\Omega)]^2,
\nabla\cdot{\bf v} \in L^2(\Omega)\right\}.
\]
The norm in $H({\rm div}; \Omega)$ is defined by
$$
\|{\bf v}\|_{H({\rm div}; \Omega)} = \left( \|{\bf v}\|^2 + \|\nabla
\cdot{\bf v}\|^2\right)^{\frac12}.
$$
\section{A Weak Gradient Operator and Its Approximation}\label{section3}
The goal of this section is to introduce a weak gradient operator
defined on a space of functions with heterogeneous properties. The
weak gradient operator will then be employed to discretize partial
differential equations. To this end, let $K$ be any polygonal domain
with interior $K^0$ and boundary $\partial K$. A {\em weak function}
on the region $K$ refers to a vector-valued function $v=\{v_0,
v_b\}$ such that $v_0\in L^2(K)$ and $v_b\in H^{\frac12}(\partial
K)$. The first component $v_0$ can be understood as the value of $v$
in the interior of $K$, and the second component $v_b$ is the value
of $v$ on the boundary of $K$. Note that $v_b$ may not be
necessarily related to the trace of $v_0$ on $\partial K$ should a
trace be well defined. Denote by $W(K)$ the space of weak functions
associated with $K$; i.e.,
\begin{equation}\label{hi.888}
W(K) = \{v=\{v_0, v_b \}:\ v_0\in L^2(K),\; v_b\in
H^{\frac12}(\partial K)\}.
\end{equation}
\begin{defi}
The dual of $L^2(K)$ can be identified with itself by using the
standard $L^2$ inner product as the action of linear functionals.
With a similar interpretation, for any $v\in W(K)$, the {\bf weak
gradient} of $v$ is defined as a linear functional $\nabla_d v$ in
the dual space of $H({\rm div},K)$ whose action on each $q\in H({\rm
div},K)$ is given by
\begin{equation}\label{weak-gradient}
(\nabla_d v, q) := -\int_K v_0 \nabla\cdot q dK+ \int_{\partial K}
v_b q\cdot{\bf n} ds,
\end{equation}
where ${\bf n}$ is the outward normal direction to $\partial K$.
\end{defi}
Note that for any $v\in W(K)$, the right-hand side of
(\ref{weak-gradient}) defines a bounded linear functional on the
normed linear space $H({\rm div}, K)$. Thus, the weak gradient
$\nabla_d v$ is well defined. Moreover, if the components of $v$ are
restrictions of a function $u\in H^1(K)$ on $K^0$ and $\partial K$,
respectively, then we would have
$$
-\int_K v_0 \nabla\cdot q dK+ \int_{\partial K} v_b q\cdot{\bf n} ds =
-\int_K u \nabla\cdot q dK+ \int_{\partial K} u q\cdot{\bf n} ds =
\int_K \nabla u \cdot q dK.
$$
It follows that $\nabla_d v= \nabla u$ is the classical gradient of
$u$.
Next, we introduce a discrete weak gradient operator by defining
$\nabla_d$ in a polynomial subspace of $H({\rm div}, K)$. To this
end, for any non-negative integer $r\ge 0$, denote by $P_{r}(K)$ the
set of polynomials on $K$ with degree no more than $r$. Let
$V(K,r)\subset [P_{r}(K)]^2$ be a subspace of the space of
vector-valued polynomials of degree $r$. A discrete weak gradient
operator, denoted by $\nabla_{d,r}$, is defined so that
$\nabla_{d,r} v \in V(K,r)$ is the unique solution of the following
equation
\begin{equation}\label{discrete-weak-gradient}
\int_K \nabla_{d,r} v\cdot q dK = -\int_K v_0 \nabla\cdot q dK+
\int_{\partial K} v_b q\cdot{\bf n} ds,\qquad \forall q\in V(K,r).
\end{equation}
It is not hard to see that the discrete weak gradient operator
$\nabla_{d,r}$ is a Galerkin-type approximation of the weak gradient
operator $\nabla_d$ by using the polynomial space $V(K,r)$.
The classical gradient operator $\nabla=(\partial_{x_1},
\partial_{x_2})$ should be applied to functions with certain smoothness in
the design of numerical methods for partial differential equations.
For example, in the standard Galerkin finite element method, such a
``smoothness" often refers to continuous piecewise polynomials over
a prescribed finite element partition. With the weak gradient
operator as introduced in this section, derivatives can be taken for
functions without any continuity across the boundary of each
triangle. Thus, the concept of weak gradient allows the use of
functions with heterogeneous properties in approximation.
Analogies of weak gradient can be established for other differential
operators such as divergence and curl operators. Details for weak
divergence and weak curl operators and their applications in
numerical methods will be given in forthcoming papers.
\section{A Weak Galerkin Finite Element Method}\label{section4}
The goal of this section is to demonstrate how discrete weak
gradients be used in the design of numerical schemes that
approximate the solution of partial differential equations. For
simplicity, we take the second order elliptic equation (\ref{pde})
as a model for discussion. With the Dirichlet boundary condition
(\ref{bc}), the standard weak form seeks $u\in H^1(\Omega)$ such
that $u=g$ on $\partial\Omega$ and
\begin{eqnarray}\label{weakform-new}
(a\nabla u, \nabla v)-(b u,\nabla v)+(cu,v)=(f,v)\quad \forall v\in
H_0^1(\Omega).
\end{eqnarray}
Let ${\cal T}_h$ be a triangular partition of the domain $\Omega$
with mesh size $h$. Assume that the partition ${\cal T}_h$ is shape
regular so that the routine inverse inequality in the finite element
analysis holds true (see \cite{ci}). In the general spirit of
Galerkin procedure, we shall design a weak Galerkin method for
(\ref{weakform-new}) by following two basic principles: {\em (1)
replace $H^1(\Omega)$ by a space of discrete weak functions defined
on the finite element partition ${\cal T}_h$ and the boundary of
triangular elements; (2) replace the classical gradient operator by
a discrete weak gradient operator $\nabla_{d,r}$ for weak functions
on each triangle $T$.} Details are to be presented in the rest of
this section.
For each $T\in {\cal T}_h$, Denote by $P_j(T^0)$ the set of
polynomials on $T^0$ with degree no more than $j$, and
$P_\ell(\partial T)$ the set of polynomials on $\partial T$ with
degree no more than $\ell$ (i.e., polynomials of degree $\ell$ on
each line segment of $\partial T$). A {\em discrete weak function}
$v=\{v_0, v_b\}$ on $T$ refers to a weak function $v=\{v_0, v_b\}$
such that $v_0\in P_j(T^0)$ and $v_b\in P_\ell(\partial T)$ with
$j\ge 0$ and $\ell \ge 0$. Denote this space by $W(T, j, \ell)$,
i.e.,
$$
W(T,j,\ell) := \left\{v=\{v_0, v_b\}:\ v_0\in P_j(T^0), v_b\in
P_\ell(\partial T)\right\}.
$$
The corresponding finite element space would be defined by patching
$W(T,j,\ell)$ over all the triangles $T\in {\cal T}_h$. In other
words, the weak finite element space is given by
\begin{equation}\label{weak-fes}
S_h(j,\ell) :=\left\{ v=\{v_0, v_b\}:\ \{v_0, v_b\}|_{T}\in
W(T,j,\ell), \forall T\in {\cal T}_h \right\}.
\end{equation}
Denote by $S_h^0(j,\ell)$ the subspace of $S_h(j,\ell)$ with
vanishing boundary values on $\partial\Omega$; i.e.,
\begin{equation}\label{weak-fes-homo}
S_h^0(j,\ell) :=\left\{ v=\{v_0, v_b\}\in S_h(j,\ell),
{v_b}|_{\partial T\cap \partial\Omega}=0, \ \forall T\in {\cal T}_h
\right\}.
\end{equation}
According to (\ref{discrete-weak-gradient}), for each $v=\{v_0,
v_b\} \in S_h(j,\ell)$, the discrete weak gradient of $v$ on each
element $T$ is given by the following equation:
\begin{equation}\label{discrete-weak-gradient-new}
\int_T \nabla_{d,r} v\cdot q dT = -\int_T v_0 \nabla\cdot q dT+
\int_{\partial T} v_b q\cdot{\bf n} ds,\qquad \forall q\in V(T,r).
\end{equation}
Note that no specific examples of the approximating space $V(T,r)$
have been mentioned, except that $V(T,r)$ is a subspace of the set
of vector-valued polynomials of degree no more than $r$ on $T$.
For any $w,v \in S_h(j,\ell)$, we introduce the following bilinear
form
\begin{equation}\label{linearform-a}
a(w,v)=(a\nabla_{d,r} w,\;\nabla_{d,r} v)-(b u_0,\nabla_{d,r}
v)+(cu_0,v_0),
\end{equation}
where
\begin{eqnarray*}
(a\nabla_{d,r} w,\;\nabla_{d,r} v)&=&\int_\Omega a\nabla_{d,r} w
\cdot\nabla_{d,r}v d\Omega,\\
(bw_0,\;\nabla_{d,r} v)&=&\int_{\Omega} bu_0\cdot \nabla_{d,r}v
d\Omega,\\
(cw_0,v_0)&=&\int_\Omega cw_0 v_0 d\Omega.
\end{eqnarray*}
\begin{algorithm}
A numerical approximation for (\ref{pde}) and (\ref{bc}) can be
obtained by seeking $u_h=\{u_0,u_b\}\in S_h(j,\ell)$ satisfying
$u_b= Q_b g$ on $\partial \Omega$ and the following equation:
\begin{equation}\label{WG-fem}
a(u_h,v)=(f,\;v_0), \quad\forall\ v=\{v_0, v_b\}\in S_h^0(j,\ell),
\end{equation}
where $Q_b g$ is an approximation of the boundary value in the
polynomial space $P_\ell(\partial T\cap \partial\Omega)$. For
simplicity, $Q_b g$ shall be taken as the standard $L^2$ projection
for each boundary segment; other approximations of the boundary
value $u=g$ can also be employed in (\ref{WG-fem}).
\end{algorithm}
\section{Examples of Weak Galerkin Method with Properties}
Although the weak Galerkin scheme (\ref{WG-fem}) is defined for
arbitrary indices $j, \ell$, and $r$, the method can be shown to
produce good numerical approximations for the solution of the
original partial differential equation only with a certain
combination of their values. For one thing, there are at least two
prominent properties that the discrete gradient operator
$\nabla_{d,r}$ should possess in order for the weak Galerkin method
to work well. These two properties are:
\begin{enumerate}
\item[\bf P1:] For any $v\in S_h(j,\ell)$, if $\nabla_{d,r} v=0$ on $T$, then
one must have $v\end{equation}uiv constant$ on $T$. In other words,
$v_0=v_b=constant$ on $T$;
\item[\bf P2:] Let $u\in H^m(\Omega) (m\ge 1)$ be a smooth function on
$\Omega$, and $Q_h u$ be a certain interpolation/projection of $u$
in the finite element space $S_h(j,\ell)$. Then, the discrete weak
gradient of $Q_h u$ should be a good approximation of $\nabla u$.
\end{enumerate}
The following are two examples of weak finite element spaces that
fit well into the numerical scheme (\ref{WG-fem}).
\begin{example}\label{wg-example1}
In this example, we take $\ell=j+1, r=j+1$, and
$V(T,j+1)=\left[P_{j+1}(T)\right]^2$, where $j\ge 0$ is any
non-negative integer. Denote by $S_h(j,j+1)$ the corresponding
finite element space. More precisely, the finite element space
$S_h(j,j+1)$ consists of functions $v=\{v_0, v_b\}$ where $v_0$ is a
polynomial of degree no more than $j$ in $T^0$, and $v_b$ is a
polynomial of degree no more than $j+1$ on $\partial T$. The space
$V(T,r)$ used to define the discrete weak gradient operator
$\nabla_{d,r}$ in (\ref{discrete-weak-gradient-new}) is given as
vector-valued polynomials of degree no more than $j+1$ on $T$.
\end{example}
\begin{example}\label{wg-example2}
In the second example, we take $\ell=j, r=j+1$, and
$V(T,r=j+1)=\left[P_{j}(T)\right]^2 + \widehat P_j(T) {\bf x}$, where
${\bf x}=(x_1,x_2)^T$ is a column vector and $\widehat P_j(T)$ is the
set of homogeneous polynomials of order $j$ in the variable ${\bf x}$.
Denote by $S_h(j,j)$ the corresponding finite element space. Note
that the space $V(T,r)$ that was used to define a discrete weak
gradient is in fact the usual Raviart-Thomas element \cite{rt} of
order $j$ for the vector component.
\end{example}
Let us demonstrate how the two properties {\bf P1} and {\bf P2} are
satisfied with the two examples given as above. For simplicity, we
shall present results only for {\bf WG Example \ref{wg-example1}}.
The following result addresses a satisfaction of the property {\bf
P1}.
\begin{lemma}\label{lemma-zero}
For any $v=\{v_0, v_b\}\in W(T, j, j+1)$, let $\nabla_{d,j+1} v$ be
the discrete weak gradient of $v$ on $T$ as defined in
(\ref{discrete-weak-gradient-new}) with
$V(T,r)=\left[P_{j+1}(T)\right]^2$. Then, $\nabla_{d,j+1} v =0$
holds true on $T$ if and only if $v=constant$ (i.e.,
$v_0=v_b=constant$).
\end{lemma}
\begin{proof}
It is trivial to see from (\ref{discrete-weak-gradient-new}) that if
$v=constant$ on $T$, then the right-hand side of
(\ref{discrete-weak-gradient-new}) would be zero for any $q\in
V(T,j+1)$. Thus, we must have $\nabla_{d,j+1} v =0$.
Now assume that $\nabla_{d,j+1} v =0$. It follows from
(\ref{discrete-weak-gradient-new}) that
\begin{equation}\label{discrete-weak-gradient-newv}
-\int_T v_0 \nabla\cdot q dT+ \int_{\partial T} v_b q\cdot{\bf n}
ds=0,\qquad \forall q\in V(T,j+1).
\end{equation}
Let $\bar{v}_0$ be the average of $v_0$ over $T$. Using the results
of \cite{bdm}, there exists a vector-valued polynomial $q_1\in
V(T,j+1)=[P_{j+1}(T)]^2$ such that $q_1\cdot{\bf n}=0$ on $\partial T$
and $\nabla\cdot q_1 = v_0 - \bar{v}_0$. With $q=q_1$ in
(\ref{discrete-weak-gradient-newv}), we arrive at $\int_T
(v_0-\bar{v}_0)^2 dT=0$. It follows that $v_0=\bar{v}_0$, and
(\ref{discrete-weak-gradient-newv}) can be rewritten as
\begin{equation}\label{discrete-weak-gradient-newvw}
\int_{\partial T} (v_b-v_0) q\cdot{\bf n} ds=0,\qquad \forall q\in
V(T,j+1).
\end{equation}
Now since $v_b-v_0\in P_{j+1}(\partial T)$, then one may select a
$q\in V(T,j+1)=[P_{j+1}(T)]^2$ such that
$$
\int_{\partial T} \phi q\cdot{\bf n} ds = \int_{\partial T} \phi
(v_b-v_0)ds,\qquad \forall \phi\in P_{j+1}(\partial T),
$$
which, together with (\ref{discrete-weak-gradient-newvw}) and
$\phi=v_b-v_0$ yields
$$
\int_{\partial T} (v_b-v_0)^2ds = 0.
$$
The last equality implies $v_b=v_0=constant$, which completes a
proof of the lemma.
\end{proof}
To verify property {\bf P2}, let $u\in H^1(T)$ be a smooth function
on $T$. Denote by $Q_h u=\{Q_0 u,\;Q_bu\}$ the $L^2$ projection
onto $P_j(T^0)\times P_{j+1}(\partial T)$. In other words, on each
element $T$, the function $Q_0 u$ is defined as the $L^2$ projection
of $u$ in $P_j(T)$ and on $\partial T$, $Q_b u$ is the $L^2$
projection in $P_{j+1}(\partial T)$. Furthermore, let $R_h$ be the
local $L^2$ projection onto $V(T,j+1)$. According to the definition
of $\nabla_{d,j+1}$, the discrete weak gradient function
$\nabla_{d,j+1}(Q_hu)$ is given by the following equation:
\begin{equation}\label{discrete-weak-gradient-hi}
\int_T \nabla_{d,j+1}(Q_h u) \cdot q dT = -\int_T (Q_0 u)
\nabla\cdot q dT+ \int_{\partial T} (Q_b u) q\cdot{\bf n} ds,\quad
\forall q\in V(K,j+1).
\end{equation}
Since $Q_0$ and $Q_b$ are $L^2$-projection operators, then the
right-hand side of (\ref{discrete-weak-gradient-hi}) is given by
\begin{eqnarray*}
-\int_T (Q_0 u) \nabla\cdot q dT+ \int_{\partial T} (Q_b u)
q\cdot{\bf n} ds &=& -\int_T u \nabla\cdot q dT+ \int_{\partial T} u
q\cdot{\bf n} ds \\
&=& \int_T (\nabla u)\cdot q dT = \int_T (R_h \nabla u)\cdot q dT.
\end{eqnarray*}
Thus, we have derived the following useful identity:
\begin{equation}\label{4.88}
\nabla_{d,j+1}(Q_h u) =R_h (\nabla u),\qquad \forall u\in H^1(T).
\end{equation}
The above identity clearly indicates that $\nabla_{d,j+1}(Q_h u)$ is
an excellent approximation of the classical gradient of $u$ for any
$u\in H^1(T)$. Thus, it is reasonable to believe that the weak
Galerkin finite element method shall provide a good numerical scheme
for the underlying partial differential equations.
\section{Mass Conservation of Weak Galerkin}
The second order elliptic equation (\ref{pde}) can be rewritten in a
conservative form as follows:
$$
\nabla \cdot q + cu = f, \quad q=-a\nabla u + bu.
$$
Let $T$ be any control volume. Integrating the first equation over
$T$ yields the following integral form of mass conservation:
\begin{equation}\label{conservation.01}
\int_{\partial T} q\cdot {\bf n} ds + \int_T cu dT = \int_T f dT.
\end{equation}
We claim that the numerical approximation from the weak Galerkin
finite element method for (\ref{pde}) retains the mass conservation
property (\ref{conservation.01}) with a numerical flux $q_h$. To
this end, for any given $T\in {\cal T}_h$, we chose in
(\ref{WG-fem}) a test function $v=\{v_0, v_b=0\}$ so that $v_0=1$ on
$T$ and $v_0=0$ elsewhere. Using the relation (\ref{linearform-a}),
we arrive at
\begin{equation}\label{mass-conserve.08}
\int_T a\nabla_{d,r} u_h\cdot \nabla_{d,r}v dT - \int_T b u_{0}
\cdot\nabla_{d,r}v dT + \int_T c u_{0} dT = \int_T f dT.
\end{equation}
Using the definition (\ref{discrete-weak-gradient-new}) for
$\nabla_{d,r}$, one has
\begin{eqnarray}
\int_T a\nabla_{d,r} u_h\cdot \nabla_{d,r}v dT &=& \int_T
R_h(a\nabla_{d,r} u_h)\cdot \nabla_{d,r}v dT \nonumber\\
&=& - \int_T \nabla\cdot R_h(a\nabla_{d,r} u_h) dT \nonumber\\
&=& - \int_{\partial T} R_h(a\nabla_{d,r}u_h)\cdot{\bf n} ds
\label{conserv.88}
\end{eqnarray}
and
\begin{eqnarray}
\int_T b u_{0} \cdot\nabla_{d,r}v dT &=& \int_T R_h(b u_{0})
\cdot\nabla_{d,r}v dT\nonumber\\
&=& -\int_T \nabla\cdot R_h(b u_{0})dT\nonumber\\
&=& -\int_{\partial T} R_h(b u_{0})\cdot{\bf n} ds\label{conserve.89}
\end{eqnarray}
Now substituting (\ref{conserve.89}) and (\ref{conserv.88}) into
(\ref{mass-conserve.08}) yields
\begin{equation}\label{mass-conserve.09}
\int_{\partial T} R_h\left(-a\nabla_{d,r}u_h + b u_{0}
\right)\cdot{\bf n} ds +\int_T c u_{0} dT = \int_T f dT,
\end{equation}
which indicates that the weak Galerkin method conserves mass with a
numerical flux given by
$$
q_h\cdot{\bf n} =R_h\left(-a\nabla_{d,r}u_h + b u_{0} \right)\cdot{\bf n}.
$$
The numerical flux $q_h\cdot{\bf n}$ can be verified to be continuous
across the edge of each element $T$ through a selection of the test
function $v=\{v_0,v_b\}$ so that $v_0\end{equation}uiv 0$ and $v_b$ arbitrary.
\section{Existence and Uniqueness for Weak Galerkin Approximations}
Assume that $u_h$ is a weak Galerkin approximation for the problem
(\ref{pde}) and (\ref{bc}) arising from (\ref{WG-fem}) by using the
finite element space $S_h(j,j+1)$ or $S_h(j,j)$. The goal of this
section is to derive a uniqueness and existence result for $u_h$.
For simplicity, details are only presented for the finite element
space $S_h(j,j+1)$; the result can be extended to $S_h(j,j)$ without
any difficulty.
First of all, let us derive the following analogy of G\aa{rding's}
inequality.
\begin{lemma}
Let $S_h(j,\ell)$ be the weak finite element space defined in
(\ref{weak-fes}) and $a(\cdot,\cdot)$ be the bilinear form given in
(\ref{linearform-a}). There exists a constant $K$ and $\alpha_1$
satisfying
\begin{equation}\label{garding}
a(v,v)+K(v_0, v_0)\ge \alpha_1(\|\nabla_{d,r}v\|^2+\|v_0\|^2),
\end{equation}
for all $v\in S_{h}(j,\ell)$.
\end{lemma}
\begin{proof}
Let $B_1=\|b\|_{L^\infty(\Omega)}$ and
$B_2=\|c\|_{L^\infty(\Omega)}$ be the $L^\infty$ norm of the
coefficients $b$ and $c$, respectively. Since
\begin{eqnarray*}
|(bv_0, \nabla_{d,r} v)|&\le& B_1\|\nabla_{d,r} v\|\ \|v_0\|,\\
|(cv_0, v_0)|&\leq & B_2 \|v_0\|^2,
\end{eqnarray*}
then it follows from (\ref{linearform-a}) that there exists a
constant $K$ and $\alpha_1$ such that
\begin{eqnarray*}
a(v,v)+K(v_0,v_0)&\ge& \alpha\|\nabla_{d,r}v\|^2-B_1\|\nabla_{d,r}v\|\|v_0\|+(K-B_2)\|v_0\|^2\\
&\ge& \alpha_1(\|\nabla_{d,r}v\|^2+\|v_0\|^2),
\end{eqnarray*}
which completes the proof.
\end{proof}
For simplicity of notation, we shall drop the subscript $r$ in the
discrete weak gradient operator $\nabla_{d,r}$ from now on. Readers
should bear in mind that $\nabla_d$ refers to a discrete weak
gradient operator defined by using the setups of either Example
\ref{wg-example1} or Example \ref{wg-example2}. In fact, for these
two examples, one may also define a projection $\Pi_h$ such that
$\Pi_h{\bf q}\in H({\rm div},\Omega)$, and on each $T\in {\cal T}_h$,
one has $\Pi_h{\bf q} \in V(T, r=j+1)$ and the following identity
$$
(\nabla\cdot{\bf q},\;v_0)_T=(\nabla\cdot\Pi_h{\bf q},\;v_0)_T, \qquad
\forall v_0\in P_j(T^0).
$$
The following result is based on the above property of $\Pi_h$.
\begin{lemma}
For any ${\bf q}\in H({\rm div},\Omega)$, we have
\begin{equation}\label{4.200}
\sum_{T\in {\cal T}_h}(-\nabla\cdot{\bf q}, \;v_0)_T=\sum_{T\in {\cal
T}_h}(\Pi_h{\bf q}, \;\nabla_dv)_T,
\end{equation}
for all $v=\{v_0,v_b\}\in S^0_h(j,j+1)$.
\end{lemma}
\begin{proof}
The definition of $\Pi_h$ and the definition of $\nabla_d v$ imply ,
\begin{eqnarray*}
\sum_{T\in {\cal T}_h}(-\nabla\cdot{\bf q}, \;v_0)_T&=&\sum_{T\in {\cal T}_h}(-\nabla\cdot \Pi_h{\bf q}, \;v_0)_T\\
&=&\sum_{T\in {\cal T}_h}(\Pi_h{\bf q}, \nabla_d v)_T - \sum_{T\in {\cal
T}_h} \langle v_b,
\Pi_h {\bf q}\cdot {\bf n}\rangle_{\partial T} \\
&=&\sum_{T\in {\cal T}_h}(\Pi_h {\bf q}, \nabla_d v)_T.
\end{eqnarray*}
Here we have used the fact that $\Pi_h {\bf q}\cdot {\bf n}$ is continuous
across each interior edge and $v_b=0$ on $\partial \Omega$. This
completes the proof.
\end{proof}
\begin{lemma}\label{approx}
For $u\in H^{1+s}(\Omega)$ with $s>0$, we have
\begin{eqnarray}
\|\Pi_h(a\nabla u)-a\nabla_d(Q_h u)\|&\le& Ch^s\|u\|_{1+s},\label{a2}\\
\|\nabla u-\nabla_d(Q_hu)\|&\le&Ch^s\|u\|_{1+s}.\label{a3}
\end{eqnarray}
\end{lemma}
\begin{proof} Since from (\ref{4.88}) we have $\nabla_d(Q_h u) =
R_h(\nabla u)$, then
$$
\|\Pi_h(a\nabla u)-a\nabla_d(Q_h u)\| = \|\Pi_h(a\nabla
u)-aR_h(\nabla u)\|.
$$
Using the triangle inequality and the definition of $\Pi_h$ and
$R_h$, we have
\begin{eqnarray*}
\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|&\le&\|\Pi_h(a\nabla u)-a\nabla u\|+\|a\nabla u-aR_h(\nabla u)\|\\
&\le& Ch^s\|u\|_{1+s}.
\end{eqnarray*}
The estimate (\ref{a3}) can be derived in a similar way. This
completes a proof of the lemma.
\end{proof}
We are now in a position to establish a solution uniqueness and
existence for the weak Galerkin method (\ref{WG-fem}). It suffices
to prove that the solution is unique. To this end, let $e\in
S_h^0(j,j+1)$ be a discrete weak function satisfying
\begin{equation}\label{uniq}
a(e,v)=0,\qquad\forall v=\{v_0, v_b\}\in S_h^0(j,j+1).
\end{equation}
The goal is to show that $e\end{equation}uiv 0$ by using a duality approach
similar to what Schatz \cite{schatz} did for the standard Galerkin
finite element methods.
\begin{lemma}\label{L2byH1}
Let $e=\{e_0, e_b\}\in S_h^0(j,j+1)$ be a discrete weak function
satisfying (\ref{uniq}). Assume that the dual of (\ref{pde}) with
homogeneous Dirichlet boundary condition has the $H^{1+s}$
regularity ($s\in (0,1]$). Then, there exists a constant $C$ such
that
\begin{equation}\label{dual-1}
\|e_0\|\le Ch^s\|\nabla_d e\|,
\end{equation}
provided that the mesh size $h$ is sufficient small, but a fixed
constant.
\end{lemma}
\begin{proof}
Consider the following dual problem: Find $w\in H^1(\Omega)$ such
that
\begin{eqnarray}
-\nabla\cdot (a \nabla w)-b\cdot\nabla w+ cw &=&e_0 \quad
\mbox{in}\;\Omega\label{dual1}\\
w&=&0\quad \mbox{on}\; \partial\Omega,\label{dual1-BC}
\end{eqnarray}
The assumption of $H^{1+s}$ regularity implies that $w\in
H^{1+s}(\Omega)$ and there is a constant $C$ such that
\begin{equation}\label{reg1}
\|w\|_{1+s}\le C\|e_0\|.
\end{equation}
Testing (\ref{dual1}) against $e_0$ and then using (\ref{4.200})
lead to
\begin{eqnarray*}
\|e_0\|^2&=&(-\nabla\cdot (a \nabla w),\;e_0)-(b\cdot\nabla w,\;e_0)+ (cw,\;e_0)\\
&=&(\Pi_h(a\nabla w),\;\nabla_d e)-(\nabla w,\;be_0)+ (cw,\;e_0)\\
&=&(\Pi_h(a\nabla w)-a\nabla_d(Q_h w),\;\nabla_d e)+(a\nabla_d(Q_h w),\;\nabla_d e)\\
& &-(\nabla w-\nabla_d(Q_hw),\;be_0))-(\nabla_d(Q_hw),\;be_0)\\
& &+(cw-c(Q_0w),\;e_0)+(Q_0w,\;ce_0).
\end{eqnarray*}
The sum of the second, forth and sixth term on the right hand side
of the above equation equals $a(e, Q_hw)=0$ due to (\ref{uniq}).
Therefore, it follows from Lemma \ref{approx} that
\begin{eqnarray*}
\|e_0\|^2&=&(\Pi_h(a\nabla w)-a\nabla_d(Q_h w),\;\nabla_d e)-(\nabla w-\nabla_d(Q_hw),\;be_0)\\
& &+(c(w-Q_0w),\;e_0)\\
&\le&Ch^s\|w\|_{1+s}\left(\|\nabla_de\|+\|e_0\|\right) + C h \|w\|_1
\; \|e_0\|.
\end{eqnarray*}
Using the $H^{1+s}$-regularity assumption (\ref{reg1}), we arrive at
$$
\|e_0\|^2 \leq C h^s\|e_0\|\left(\|\nabla_de\|+\|e_0\|\right),
$$
which leads to
$$
\|e_0\| \leq C h^s\left(\|\nabla_de\|+\|e_0\|\right).
$$
Thus, when $h$ is sufficiently small, one would obtain the desired
estimate (\ref{dual-1}). This completes the proof.
\end{proof}
\begin{theorem}
Assume that the dual of (\ref{pde}) with homogeneous Dirichlet
boundary condition has $H^{1+s}$-regularity for some $s\in (0,1]$.
The weak Gakerkin finite element method defined in (\ref{WG-fem})
has a unique solution in the finite element spaces $S_h(j,j+1)$ and
$S_j(j,j)$ if the meshsize $h$ is sufficiently small, but a fixed
constant.
\end{theorem}
\begin{proof} Observe that uniqueness is equivalent to existence for the solution of
(\ref{WG-fem}) since the number of unknowns is the same as the
number of equations. To prove a uniqueness, let $u^{(1)}_h$ and
$u^{(2)}_h$ be two solutions of (\ref{WG-fem}). By letting
$e=u^{(1)}_h- u^{(2)}_h$ we see that (\ref{uniq}) is satisfied. Now
we have from the G{\aa}rding's inequality (\ref{garding}) that
$$
a(e,e) + K \|e_0\| \ge \alpha_1\left( \|\nabla_d e\| +
\|e_0\|\right).
$$
Thus, it follows from the estimate (\ref{dual-1}) of Lemma
\ref{L2byH1} that
$$
\alpha_1\left( \|\nabla_d e\| + \|e_0\|\right)\leq CK h^s \|\nabla_d
e\|
$$
for $h$ being sufficiently small. Now chose $h$ small enough so that
$CKh^s\leq \frac{\alpha_1}{2}$. Thus,
$$
\|\nabla_d e\| + \|e_0\|\ = 0,
$$
which, together with Lemma \ref{lemma-zero}, implies that $e$ is a
constant and $e_0=0$. This shows that $e=0$ and consequently,
$u^{(1)}_h=u^{(2)}_h$.
\end{proof}
\section{Error Analysis}
The goal of this section is to derive some error estimate for the
weak Galerkin finite element method (\ref{WG-fem}). We shall follow
the usual approach in the error analysis: (1) investigating the
difference between the weak finite element approximation $u_h$ with
a certain interpolation/projection of the exact solution through an
error equation, (2) using a duality argument to analyze the error in
the $L^2$ norm.
Let us begin with the derivation of an error equation for the weak
Galerkin approximation $u_h$ and the $L^2$ projection of the exact
solution $u$ in the weak finite element space $S_h(j,j+1)$. Recall
that the $L^2$ projection is denoted by $Q_h u \end{equation}uiv \{Q_0u, Q_b
u\}$, where $Q_0$ denotes the local $L^2$ projection onto $P_j(T)$
and $Q_b$ is the local $L^2$ projection onto $P_{j+1}(\partial T)$
on each triangular element $T\in {\cal T}_h$. Let $v=\{v_0,v_b\}\in
S_h^0(j,j+1)$ be any test function. By testing (\ref{pde}) against
the first component $v_0$ and using (\ref{4.200}) we arrive at
\begin{eqnarray*}
(f, v_0) &=& \sum_{T\in {\cal T}_h}(-\nabla\cdot (a\nabla u), \;v_0)_T
+(\nabla\cdot (b u),\; v_0)+(cu, \;v_0)\nonumber\\
&=&(\Pi_h(a\nabla u),\; \nabla_dv)-(\Pi_h(bu),\; \nabla_d v)+(cu,\; v_0).
\end{eqnarray*}
Adding and subtracting the term $a(Q_hu, v)\end{equation}uiv
(a\nabla_d(Q_hu),\;\nabla_d
v)-(b(Q_0u),\;\nabla_dv)+(c(Q_0u),\;v_0)$ on the right hand side of
the above equation and then using (\ref{4.88}) we obtain
\begin{eqnarray}
(f, v_0)&=&(a\nabla_d(Q_hu),\;\nabla_d v)-(bQ_0u,\;\nabla_dv)+(cQ_0u,\;v_0)\label{true}\\
& &+(\Pi_h(a\nabla u)-aR_h(\nabla u),\;
\nabla_dv)\nonumber\\
& & -(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0),\nonumber
\end{eqnarray}
which can be rewritten as
\begin{eqnarray}
a(u_h, v)&=&a(Q_hu,v)+(\Pi_h(a\nabla u)-aR_h(\nabla u),\;
\nabla_dv)\nonumber\\
& & -(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0).\nonumber
\end{eqnarray}
It follows that
\begin{eqnarray}
a(u_h-Q_hu,\;v)&=&(\Pi_h(a\nabla u)-aR_h(\nabla u),\; \nabla_dv)\nonumber\\
& &-(\Pi_h(bu)-bQ_0u,\; \nabla_d v)+(c(u-Q_0u),\; v_0).\label{diff}
\end{eqnarray}
The equation (\ref{diff}) shall be called the {\em error equation}
for the weak Galerkin finite element method (\ref{WG-fem}).
\subsection{An estimate in a discrete $H^1$-norm}
We begin with the following lemma which provides an estimate for the
difference between the weak Galerkin approximation $u_h$ and the
$L^2$ projection of the exact solution of the original problem.
\begin{lemma}\label{h1-error}
Let $u\in H^{1}(\Omega)$ be the solution of (\ref{pde}) and
(\ref{bc}). Let $u_h\in S_h(j,j+1)$ be the weak Galerkin
approximation of $u$ arising from (\ref{WG-fem}). Denote by
$e_h:=u_h-Q_h u$ the difference between the weak Galerkin
approximation and the $L^2$ projection of the exaction solution
$u=u(x_1,x_2)$. Then there exists a constant $C$ such that
\begin{eqnarray}
\frac{\alpha_1}{2}(\|\nabla_d(e_h)\|^2+\|e_{h,0}\|^2) &\le&
C\left(\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|^2 +\|c(u-Q_0u)\|^2
\right.\nonumber\\
& & +\left.
\|\Pi_h(bu)-bQ_0u\|^2\right)+K\|u_0-Q_0u\|^2.\label{H1errorestimate}
\end{eqnarray}
\end{lemma}
\begin{proof} Substituting $v$ in (\ref{diff}) by $e_h:=u_h-Q_hu$
and using the usual Cauchy-Schwarz inequality we arrive at
\begin{eqnarray*}
a(e_h,\;e_h)&=&(\Pi_h(a\nabla u)-aR_h(\nabla u),\; \nabla_d (u_h-Q_hu))\\
& &-(\Pi_h(bu)-bQ_0u,\; \nabla_d (u_h-Q_hu))+(c(u-Q_0u),\; u_0-Q_0u)\\
&\le&\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|\;\|\nabla_d (u_h-Q_hu)\|\\
& &+\|\Pi_h(bu)-bQ_0u\|\;\|\nabla_d( u_h-Q_hu)\|
+\|c(u-Q_0u)\|\;\|u_0-Q_0u\|.
\end{eqnarray*}
Next, we use the G{\aa}rding's inequality (\ref{garding}) to obtain
\begin{eqnarray*}
\alpha_1(\|\nabla_d(e_h)\|^2+\|e_{h,0}\|^2)&\le&
\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|\;\|\nabla_d (u_h-Q_hu)\|\\
& &+\|\Pi_h(bu)-bQ_0u\|\;\|\nabla_d( u_h-Q_hu)\|\\
& &+\|c(u-Q_0u)\|\;\|u_0-Q_0u\|+K\|u_0-Q_0u\|^2\\
&\le& \frac{\alpha_1}{2} (\|\nabla_d(u_h-Q_hu)\|^2+\|u_0-Q_0u\|^2)\\
& &+C\left(\|\Pi_h(a\nabla u)-aR_h(\nabla u)\|^2 +
\|\Pi_h(bu)-bQ_0u\|^2\right.\\
& & +\left. \|c(u-Q_0u)\|^2\right)+K\|u_0-Q_0u\|^2,
\end{eqnarray*}
which implies the desired estimate (\ref{H1errorestimate}).
\end{proof}
\subsection{An estimate in $L^2(\Omega)$}
We use the standard duality argument to derive an estimate for the
error $u_h-Q_h u$ in the standard $L^2$ norm over domain $\Omega$.
\begin{lemma}\label{l2-error}
Assume that the dual of the problem (\ref{pde}) and (\ref{bc}) has
the $H^{1+s}$ regularity. Let $u\in H^{1}(\Omega)$ be the solution
(\ref{pde}) and (\ref{bc}), and $u_h$ be a weak Galerkin
approximation of $u$ arising from (\ref{WG-fem}) by using either the
weak finite element space $S_h(j,j+1)$ or $S_h(j,j)$. Let $Q_hu$ be
the $L^2$ projection of $u$ in the corresponding finite element
space (recall that it is locally defined). Then, there exists a
constant $C$ such that
\begin{eqnarray*}
\|Q_0u-u_0\|&&\le Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& +\|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|\right),
\end{eqnarray*}
provided that the meshsize $h$ is sufficiently small.
\end{lemma}
\begin{proof}
Consider the dual problem of (\ref{pde}) and (\ref{bc}) which seeks
$w\in H_0^1(\Omega)$ satisfying
\begin{eqnarray}
-\nabla\cdot (a \nabla w)-b\cdot\nabla w+ cw &=& Q_0u-u_0\quad
\mbox{in}\;\Omega\label{dual}
\end{eqnarray}
The assumed $H^{1+s}$ regularity for the dual problem implies the
existence of a constant $C$ such that
\begin{equation}\label{reg}
\|w\|_{1+s}\le C\|Q_0u-u_0\|.
\end{equation}
Testing (\ref{dual}) against $Q_0u-u_0$ element by element gives
\begin{eqnarray}
\|Q_0u-u_0\|^2&=&(-\nabla\cdot (a \nabla w),\;Q_0u-u_0)-(b\cdot\nabla w,\;Q_0u-u_0)
+ (cw,\;Q_0u-u_0)\nonumber\\
&=&I+II+III,\label{m1}
\end{eqnarray}
where $I, II,$ and $III$ are defined to represent corresponding
terms. Let us estimate each of these terms one by one.
For the term $I$, we use the identity (\ref{4.200}) to obtain
\begin{eqnarray*}
I&=&(-\nabla\cdot (a\nabla w),Q_0u-u_0)= (\Pi_h (a\nabla
w),\nabla_d(Q_hu-u_h)).
\end{eqnarray*}
Recall that $\nabla_d(Q_hu)=R_h (\nabla u)$ with $R_h$ being a local
$L^2$ projection. Thus,
\begin{eqnarray}\nonumber
I&=& (\Pi_h (a\nabla w),\nabla_d(Q_hu-u_h))=(\Pi_h (a\nabla w),R_h \nabla u-\nabla_d u_h)\\
&=& (\Pi_h (a\nabla w),\nabla u-\nabla_d u_h)\nonumber\\
&=& (\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h) + (a\nabla
w,\nabla u-\nabla_d u_h).\label{yes.888}
\end{eqnarray}
The second term in the above equation above can be handled as
follows. Adding and subtracting two terms $(a\nabla_d Q_h w,
\nabla_d u_h)$ and $(a(\nabla w - R_h \nabla w), \nabla u)$ and
using the fact that $\nabla_d (Q_h u) =R_h (\nabla u)$ and the
definition of $R_h$, we arrive at
\begin{eqnarray}
(a\nabla w,\nabla u-\nabla_d u_h) &=& (a\nabla w, \nabla u)
-(a\nabla w, \nabla_d u_h) \nonumber\\
&=& (a\nabla w, \nabla u) -(a\nabla_d Q_h w, \nabla_d u_h)- (a(\nabla w - R_h \nabla w), \nabla_d u_h)\nonumber\\
&=& (a\nabla w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h)- (a(\nabla w - R_h \nabla w),
\nabla_d u_h - \nabla u)\nonumber\\
& & - (a(\nabla w - R_h \nabla w), \nabla u)\nonumber\\
&=& (a\nabla w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h) -
(a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\label{yes.889}\\
& & - (\nabla w - R_h \nabla w, a\nabla u -R_h (a\nabla
u)).\nonumber
\end{eqnarray}
Substituting (\ref{yes.889}) into (\ref{yes.888}) yields
\begin{eqnarray}\label{yes.termI}
I&=&(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h)- (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\\
& &-(\nabla w - R_h \nabla w, a\nabla u -R_h (a\nabla u))+(a\nabla
w, \nabla u)- (a\nabla_d Q_h w, \nabla_d u_h).\nonumber
\end{eqnarray}
For the term $II$, we add and subtract $(\nabla_d(Q_h
w),\;b(Q_0u-u_0))$ from $II$ to obtain
\begin{eqnarray*}
II&=&-(b\cdot\nabla w,\;Q_0u-u_0)\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;b(Q_0u-u_0))\\
&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w),\;bQ_0u)+(\nabla_d(Q_h w),\;bu_0).
\end{eqnarray*}
In the following, we will deal with the second term on the right
hand side of the above equation. To this end, we use (\ref{4.88})
and the definition of $R_h$ and $Q_0$ to obtain
\begin{eqnarray*}
(\nabla_d(Q_h w),\;bQ_0u)&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u)+(\nabla w,\;bQ_0u)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(\nabla_d(Q_h w)-\nabla w,\;bu)\\
& &+(\nabla w,\;bQ_0u-bu)+(\nabla w,\;bu)\\
&=&(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)+(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))\\
& &+(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)+(\nabla w,\;bu).
\end{eqnarray*}
Combining the last two equations above, we arrive at
\begin{eqnarray}\label{yes.termII}
II&=&-(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\\
& &-(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))-(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\nonumber\\
& &-(\nabla w,\;bu)+(\nabla_d(Q_h w),\;bu_0).\nonumber
\end{eqnarray}
As to the term $III$, by adding and subtracting some terms and
using the fact that $Q_0$ is a local $L^2$ projection, we easily
obtain the following
\begin{eqnarray*}
III&=& (cw,\;Q_0u-u_0)=(cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u-u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w,\;Q_0u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(cQ_0w-cw,\;Q_0u)+(cw,\;Q_0u-u)\\
& &+(cw,\;u)-(cQ_0w,\;u_0)\\
&=& (cw-cQ_0w,\;Q_0u-u_0)+(Q_0w-w,\;cQ_0u-cu)+(Q_0w-w,\;cu-Q_0(cu))\\
& & +(cw-Q_0(cw),\;Q_0u-u)+(cw,\;u)-(cQ_0w,\;u_0).\\
\end{eqnarray*}
Note that the sum of the last two terms in $I$ (see
(\ref{yes.termI})), $II$ (see (\ref{yes.termII})), and $III$ (see
the last equation above) gives
\begin{eqnarray*}
(a\nabla w, \nabla u)- &&(a\nabla_d Q_h w, \nabla_d u_h)-(\nabla
w,\;bu)+(\nabla_d(Q_h w),\;bu_0)
+(cw,\;u)-(cQ_0w,\;u_0)\\
&&=a(u,w)-a(u_h, Q_hw)\\
&&=(f,\; w)-(f,\; Q_0w)\\
&&=(f-Q_0f,\;w-Q_0w).
\end{eqnarray*}
Thus, the sum of $I$, $II$, and $III$ can be written as follows:
\begin{eqnarray}
\|Q_0u-u_0\|^2=&& (f-Q_0f,w-Q_0w)+(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d u_h)\nonumber\\
&-& (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)- ((\nabla w - R_h \nabla w), a\nabla u -R_h (a\nabla u))\nonumber\\
&-&(\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0))-(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\nonumber\\
&-&(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))-(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\nonumber\\
&+&(cw-cQ_0w,\;Q_0u-u_0)+(Q_0w-w,\;cQ_0u-cu)\nonumber\\
&+&(Q_0w-w,\;cu-Q_0(cu))+(cw-Q_0(cw),\;Q_0u-u).\label{noname}
\end{eqnarray}
Using the triangle inequality, (\ref{4.88}) and (\ref{reg}), we can bound the second term
on the right hand side in the above equation by
\begin{eqnarray*}
\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_d
u_h)\right|&\leq&
\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla u-\nabla_dQ_h u)\right|\\
& & +\left|(\Pi_h (a\nabla w)-a\nabla w,\nabla_dQ_h u-\nabla_d u_h)\right|\\
&\le&Ch^{s}\left(\|\nabla u-R_h(\nabla
u)\|+\|\nabla_d(Q_hu-u_h)\|\right)\|Q_0u-u_0\|.
\end{eqnarray*}
The other terms on the right hand side of (\ref{noname}) can be
estimated in a similar fashion, for which we state the results as
follows:
\begin{eqnarray*}
\left| (a(\nabla w - R_h \nabla w), \nabla_d u_h - \nabla u)\right|
&\le&Ch^{s}\left(\|\nabla u-R_h(\nabla
u)\|+\|\nabla_d(Q_hu-u_h)\|\right)\|Q_0u-u_0\|,\\
\left|((\nabla w - R_h \nabla w), a\nabla u -R_h (a\nabla u))\right|
&\le&Ch^{s}\|a\nabla u-R_h(a\nabla
u)\|\ \|Q_0u-u_0\|,\\
\left| (\nabla w-\nabla_d(Q_hw),\;b(Q_0u-u_0)) \right|
&\le&Ch^{s} \|Q_0u-u_0\|^2,\\
\left|(\nabla_d(Q_h w)-\nabla w,\;bQ_0u-bu)\right|
&\le&Ch^{s}\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left|(R_h(\nabla w)-\nabla w,\;bu-R_h(bu))\right|
&\le&Ch^{s}\|bu-R_h(bu)\|\ \|Q_0u-u_0\|,\\
\left|(b\cdot\nabla w-Q_0(b\cdot\nabla w),\;Q_0u-u)\right|
&\le&Ch^{s}\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left| (cw-cQ_0w,\;Q_0u-u_0) \right|
&\le&Ch\|Q_0u-u_0\|^2,\\
\left|(Q_0w-w,\;cQ_0u-cu)\right|
&\le&Ch\|u-Q_0u\|\ \|Q_0u-u_0\|,\\
\left|(Q_0w-w,\;cu-Q_0(cu))\right|
&\le&Ch\|cu-Q_0(cu)\|\ \|Q_0u-u_0\|,\\
\left|(cw-Q_0(cw),\;Q_0u-u)\right| &\le&Ch\|u-Q_0u\|\ \|Q_0u-u_0\|.
\end{eqnarray*}
Substituting the above estimates into (\ref{noname}) yields
\begin{eqnarray*}
\|Q_0u&&-u_0\|^2\le Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& + \|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|+\|Q_0u-u_0\|\right)\|Q_0u-u_0\|.
\end{eqnarray*}
For sufficiently small meshsize $h$, we have
\begin{eqnarray*}
\|Q_0u-u_0\|&\le & Ch^s\left( h\|f-Q_0f\|+\|\nabla u- R_h(\nabla
u)\| + \|a\nabla u - R_h(a\nabla u)\| +\|u-Q_0 u\|\right.\\
&& +\|bu-R_h(bu)\|+\left.
\|cu-Q_0(cu)\|+\|\nabla_d(Q_hu-u_h)\|\right),
\end{eqnarray*}
which completes the proof.
\end{proof}
\subsection{Error estimates in $H^1$ and $L^2$}
With the results established in Lemma \ref{h1-error} and Lemma
\ref{l2-error}, we are ready to derive an error estimate for the
weak Galerkin approximation $u_h$. To this end, we may substitute
the result of Lemma \ref{l2-error} into the estimate shown in Lemma
\ref{h1-error}. If so, for sufficiently small meshsize $h$, we would
obtain the following estimate:
\begin{eqnarray*}
\|\nabla_d(u_h-Q_hu)\|^2+\|u_0-Q_0u\|^2 &\le& C\left(\|\Pi_h(a\nabla
u)-aR_h(\nabla u)\|^2 +\|c(u-Q_0u)\|^2
\right.\\
&& +\left. \|\Pi_h(bu)-bQ_0u\|^2\right) \\
&& + Ch^{2s}\left(h^2\|f-Q_0f\|^2+\|\nabla u- R_h(\nabla
u)\|^2\right.\\
&& + \left. \|a\nabla u - R_h(a\nabla u)\|^2 +\|u-Q_0 u\|^2\right.\\
&& +\left. \|bu-R_h(bu)\|^2+ \|cu-Q_0(cu)\|^2\right).
\end{eqnarray*}
A further use of the interpolation error estimate leads to the
following error estimate in a discrete $H^1$ norm.
\begin{theorem}\label{H1error-estimate}
In addition to the assumption of Lemma \ref{l2-error}, assume that
the exact solution $u$ is sufficiently smooth such that $u\in
H^{m+1}(\Omega)$ with $0\le m \le j+1$. Then, there exists a
constant $C$ such that
\begin{eqnarray}\label{trueH1error}
\|\nabla_d(u_h-Q_hu)\|+\|u_0-Q_0u\| \le C(h^{m} \|u\|_{m+1}
+h^{1+s}\|f-Q_0f\|).
\end{eqnarray}
\end{theorem}
Now substituting the error estimate (\ref{trueH1error}) into the
estimate of Lemma \ref{l2-error}, and then using the standard
interpolation error estimate we obtain
\begin{eqnarray*}
\|u_h-Q_hu\|&\le& C\left(h^{1+s}\|f-Q_0f\|+h^{m+s} \|u\|_{m+1} +
h^s(h^{m} \|u\|_{m+1} +h^{1+s}\|f-Q_0f\|)\right)\\
&\le&C\left(h^{1+s}\|f-Q_0f\|+h^{m+s} \|u\|_{m+1}\right).
\end{eqnarray*}
The result can then be summarized as follows.
\begin{theorem}\label{trueL2error} Under the assumption of Theorem
\ref{H1error-estimate}, there exists a constant $C$ such that
\begin{eqnarray*}
\|u_h-Q_hu\| \le C\left(h^{1+s}\|f-Q_0f\|+h^{m+s}
\|u\|_{m+1}\right), \quad s\in (0,1],\ m\in (0,j+1],
\end{eqnarray*}
provided that the mesh-size $h$ is sufficiently small.
\end{theorem}
If the exact solution $u$ of (\ref{pde}) and (\ref{bc}) has the
$H^{j+2}$ regularity, then we have from Theorem \ref{trueL2error}
that
\begin{eqnarray*}
\|u_h-Q_hu\| &\le& C\left(h^{1+s}h^{j}\|f\|_{j}+h^{j+s+1}
\|u\|_{j+2}\right)\\
&\le& C h^{j+s+1} \left(\|f\|_{j}+\|u\|_{j+2}\right)
\end{eqnarray*}
for some $0<s\leq 1$, where $s$ is a regularity index for the dual
of (\ref{pde}) and (\ref{bc}). In the case that the dual has a full
$H^2$ (i.e., $s=1$) regularity, one would arrive at
\begin{eqnarray}\label{superc}
\|u_h-Q_hu\| \le C h^{j+2} \left(\|f\|_{j}+\|u\|_{j+2}\right).
\end{eqnarray}
Recall that on each triangular element $T^0$, the finite element
functions are of polynomials of order $j\ge 0$. Thus, the error
estimate (\ref{superc}) in fact reveals a superconvergence for the
weak Galerkin finite element approximation arising from
(\ref{WG-fem}).
\end{document} |
\begin{document}
\title{
Speeding-up $q$-gram mining on grammar-based compressed texts
}
\begin{abstract}
We present an efficient algorithm for calculating
$q$-gram frequencies on strings represented in compressed form,
namely, as a straight line program (SLP).
Given an SLP $\mathcal{T}$ of size $n$ that represents string $T$,
the algorithm computes the occurrence frequencies of
{\em all} $q$-grams in $T$, by reducing the problem to the weighted
$q$-gram frequencies problem on a trie-like structure of size
$m = |T|-\mathit{dup}(q,\mathcal{T})$,
where $\mathit{dup}(q,\mathcal{T})$ is a quantity
that represents the amount of redundancy that the SLP captures with respect to
$q$-grams.
The reduced problem can be solved in linear time.
Since $m = O(qn)$, the running time of our algorithm is
$O(\min\{|T|-\mathit{dup}(q,\mathcal{T}),qn\})$,
improving our previous $O(qn)$ algorithm when
$q = \Omega(|T|/n)$.
\end{abstract}
\section{Introduction}
Many large string data sets are usually first compressed and
stored, while they are decompressed afterwards in order to be used and
analyzed.
Compressed string processing (CSP) is an approach that has been
gaining attention in the string processing community.
Assuming that the input is given in compressed form,
the aim is to develop methods where the string is processed or analyzed without
explicitly decompressing the entire string,
leading to algorithms with time and space complexities
that depend on the compressed size rather than the
whole uncompressed size.
Since compression algorithms inherently capture regularities
of the original string, clever CSP algorithms
can be
theoretically~\cite{NJC97,crochemore03:_subquad_sequen_align_algor_unres_scorin_matric,hermelin09:_unified_algor_accel_edit_distan,gawrychowski11:_LZ_comp_str_fast_}, and
even
practically~\cite{shibata00:_speed_up_patter_match_text_compr,goto11:_fast_minin_slp_compr_strin},
faster than algorithms which process the uncompressed string.
In this paper, we assume that the input string
is represented as a Straight Line Program (SLP),
which is a context free grammar in Chomsky normal form
that derives a single string.
SLPs are a useful tool when considering CSP algorithms, since
it is known that outputs of various
grammar based compression algorithms~\cite{SEQUITUR,LarssonDCC99},
as well as dictionary compression algorithms~\cite{LZ78,LZW,LZ77,LZSS}
can be modeled efficiently by SLPs~\cite{rytter03:_applic_lempel_ziv}.
We consider the $q$-gram frequencies problem on compressed text
represented as SLPs.
$q$-gram frequencies have profound applications in the field of string
mining and classification.
The problem was first considered for the CSP setting
in~\cite{inenaga09:_findin_charac_subst_compr_texts},
where an $O(|\Sigma|^2n^2)$-time $O(n^2)$-space algorithm
for finding the {\em most frequent} $2$-gram from an
SLP of size $n$ representing text $T$ over alphabet $\Sigma$ was presented.
In~\cite{claudear:_self_index_gramm_based_compr}, it is claimed
that the most frequent $2$-gram can be found in $O(|\Sigma|^2n\log n)$-time
and $O(n\log|T|)$-space, if the SLP is pre-processed and a self-index is built.
A much simpler and efficient $O(qn)$
time and space algorithm for general $q \geq 2$ was recently
developed~\cite{goto11:_fast_minin_slp_compr_strin}.
Remarkably, computational experiments on various data sets
showed that the $O(qn)$ algorithm is actually faster than
calculations on uncompressed strings, when $q$ is
small~\cite{goto11:_fast_minin_slp_compr_strin}.
However, the algorithm
slows down considerably
compared to the uncompressed approach when $q$ increases.
This is because the algorithm reduces the
$q$-gram frequencies problem on an SLP of size $n$,
to the weighted $q$-gram frequencies problem on a weighted string of
size at most $2(q-1)n$.
As $q$ increases, the length of the string becomes longer than the
uncompressed string $T$.
Theoretically $q$ can be as large as $O(|T|)$,
hence in such a case the algorithm requires $O(|T|n)$ time,
which is worse than a trivial $O(|T|)$ solution that first decompresses the given SLP and
runs a linear time algorithm for $q$-gram frequencies computation on $T$.
In this paper, we solve this problem, and improve the previous $O(qn)$ algorithm
both theoretically and practically.
We introduce a $q$-gram neighbor relation on SLP variables,
in order to reduce the redundancy
in the partial decompression of the string which is performed in the
previous algorithm.
Based on this idea, we are able to convert the problem to a
weighted $q$-gram frequencies problem
on a weighted trie, whose size is at most
$|T|-\mathit{dup}(q,\mathcal{T})$.
Here, $\mathit{dup}(q,\mathcal{T})$ is a quantity that represents
the amount of redundancy that the SLP captures with respect to
$q$-grams. Since the size of the trie is also bounded by $O(qn)$,
the time complexity of our new algorithm is
$O(\min\{qn,|T|-\mathit{dup}(q,\mathcal{T})\})$,
improving on our previous $O(qn)$ algorithm when
$q = \Omega(|T|/n)$.
Preliminary computational experiments show that our new approach achieves
a practical speed up as well, for all values of $q$.
\section{Preliminaries}
\subsection{Intervals, Strings, and Occurrences}
For integers $i \leq j$, let $[i:j]$ denote the interval of integers $\{i,\ldots, j\}$.
For an interval $[i:j]$ and integer $q > 0$,
let $\mathit{pre}([i:j],q)$ and $\mathit{suf}([i:j],q)$
represent respectively, the length-$q$ prefix and suffix interval,
that is,
$\mathit{pre}([i:j],q) = [i:\min(i+q-1,j)]$ and
$\mathit{suf}([i:j],q) = [\max(i,j-q+1):j]$.
Let $\Sigma$ be a finite {\em alphabet}.
An element of $\Sigma^*$ is called a {\em string}.
For any integer $q > 0$, an element of $\Sigma^q$ is called a \emph{$q$-gram}.
The length of a string $T$ is denoted by $|T|$.
The empty string $\varepsilon$ is a string of length 0,
namely, $|\varepsilon| = 0$.
For a string $T = XYZ$, $X$, $Y$ and $Z$ are called
a \emph{prefix}, \emph{substring}, and \emph{suffix} of $T$, respectively.
The $i$-th character of a string $T$ is denoted by $T[i]$, where $1 \leq i \leq |T|$.
For a string $T$ and interval $[i:j] (1 \leq i \leq j \leq |T|)$, let $T([i:j])$
denote the substring of $T$ that begins at position $i$ and ends at
position $j$.
For convenience, let $T([i:j]) = \varepsilon$ if $j < i$.
For a string $T$ and integer $q \geq 0$,
let $\mathit{pre}(T,q)$ and $\mathit{suf}(T,q)$
represent respectively, the length-$q$ prefix and suffix of $T$,
that is,
$\mathit{pre}(T,q) =T(\mathit{pre}([1:|T|],q))$ and $\mathit{suf}(T,q) = T(\mathit{suf}([1:|T|],q))$.
For any strings $T$ and $P$,
let $\mathit{Occ}(T,P)$ be the set of occurrences of $P$ in $T$, i.e.,
$\mathit{Occ}(T,P) = \{k > 0 \mid T[k:k+|P|-1] = P\}$.
The number of elements $|\mathit{Occ}(T,P)|$ is called
the \emph{occurrence frequency} of $P$ in $T$.
\subsection{Straight Line Programs}
\begin{wrapfigure}[11]{r}{0.5\textwidth}
\centerline{\includegraphics[width=0.45\textwidth]{slp.eps}}
\caption{
The derivation tree of
SLP $\mathcal T = \{ X_1 \rightarrow \mathtt{a}$, $X_2 \rightarrow \mathtt{b}$, $X_3 \rightarrow X_1X_2$,
$X_4 \rightarrow X_1X_3$, $X_5 \rightarrow X_3X_4$, $X_6
\rightarrow X_4X_5$, $X_7 \rightarrow X_6X_5 \}$,
representing string $T = \mathit{val}(X_7) = \mathtt{aababaababaab}$.
}
\label{fig:SLP}
\end{wrapfigure}
A {\em straight line program} ({\em SLP}) is a set of assignments
$\mathcal T = \{ X_1 \rightarrow expr_1, X_2 \rightarrow expr_2, \ldots, X_n \rightarrow expr_n\}$,
where each $X_i$ is a variable and each $expr_i$ is an expression, where
$expr_i = a$ ($a\in\Sigma$), or $expr_i = X_{\ell(i)} X_{r(i)}$~($i > \ell(i),r(i)$).
It is essentially a context free grammar in the Chomsky normal form, that derives a single string.
Let $\mathit{val}(X_i)$ represent the string derived from variable $X_i$.
To ease notation, we sometimes associate $\mathit{val}(X_i)$ with $X_i$ and
denote $|\mathit{val}(X_i)|$ as $|X_i|$,
and $\mathit{val}(X_i)([u:v])$ as $X_i([u:v])$ for any interval $[u:v]$.
An SLP $\mathcal{T}$ {\em represents} the string $T = \mathit{val}(X_n)$.
The \emph{size} of the program $\mathcal T$ is the number $n$ of
assignments in $\mathcal T$.
Note that $|T|$ can be as large as $\Theta(2^n)$. However, we assume
as in various previous work on SLP,
that the computer word size is at least $\log |T|$, and hence,
values representing lengths and positions of $T$
in our algorithms can be manipulated in constant time.
The derivation tree of SLP $\mathcal{T}$ is a labeled
ordered binary tree where each internal node is labeled with a
non-terminal variable in $\{X_1,\ldots,X_n\}$, and each leaf is labeled with a terminal character in $\Sigma$.
The root node has label $X_n$.
Let $\mathcal{V}$ denote the set of internal nodes in
the derivation tree.
For any internal node $v\in\mathcal{V}$,
let $\langle v\rangle$ denote the index of its label
$\variable{v}$.
Node $v$ has a single child which is a leaf labeled with $c$
when $(\variable{v} \rightarrow c) \in \mathcal{T}$ for some $c\in\Sigma$,
or
$v$ has a left-child and right-child respectively denoted $\ell(v)$ and $r(v)$,
when
$(\variable{v}\rightarrow \variable{\ell(v)}\variable{r(v)}) \in \mathcal{T}$.
Each node $v$ of the tree derives $\mathit{val}(\variable{v})$,
a substring of $T$,
whose corresponding interval $\mathit{val}Int(v)$,
with $T(\mathit{val}Int(v)) = \mathit{val}(\variable{v})$,
can be defined recursively as follows.
If $v$ is the root node, then $\mathit{val}Int(v) = [1:|T|]$.
Otherwise, if $(\variable{v}\rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{T}$,
then,
$\mathit{val}Int(\ell(v)) = [b_v:b_v+|\variable{\ell(v)}|-1]$
and
$\mathit{val}Int(r(v)) = [b_v+|\variable{\ell(v)}|:e_v]$,
where $[b_v:e_v] = \mathit{val}Int(v)$.
Let $\mathit{vOcc}(X_i)$ denote the number of times a variable $X_i$ occurs
in the derivation tree, i.e.,
$\mathit{vOcc}(X_i) = |\{ v \mid \variable{v}=X_i\}|$.
We assume that any variable $X_i$ is used at least once,
that is $\mathit{vOcc}(X_i) > 0$.
For any interval $[b:e]$ of $T (1\leq b \leq e \leq |T|)$,
let $\xi_\mathcal{T}(b,e)$ denote the deepest node $v$ in the derivation tree,
which derives an interval containing $[b:e]$, that is,
$\mathit{val}Int(v)\supseteq [b:e]$,
and no proper descendant of $v$
satisfies this condition.
We say that node $v$ {\em stabs} interval $[b:e]$,
and $\variable{v}$ is called the variable that stabs the interval.
If $b = e$, we have that
$(\variable{v} \rightarrow c) \in \mathcal{T}$ for some $c\in\Sigma$,
and $\mathit{val}Int(v) = b = e$.
If $b < e$, then
we have $(\variable{v} \rightarrow
\variable{\ell(v)}\variable{r(v)})\in\mathcal{T}$,
$b\in \mathit{val}Int(\ell(v))$, and $e\in\mathit{val}Int(r(v))$.
When it is not confusing, we will sometimes
use $\xi_\mathcal{T}(b,e)$ to denote the variable $\variable{\xi_\mathcal{T}(b,e)}$.
SLPs can be efficiently pre-processed to hold various information.
$|X_i|$ and $\mathit{vOcc}(X_i)$ can be computed for all variables $X_i
(1\leq i\leq n)$ in a total of $O(n)$ time by a simple dynamic
programming algorithm.
Also, the following Lemma is useful for partial decompression of
a prefix of a variable.
\begin{lemma}[\cite{gasieniec05:_real_time_traver_gramm_based_compr_files}]
\label{label:prefix_decompression}
Given an SLP $\mathcal{T} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$,
it is possible to pre-process $\mathcal{T}$ in $O(n)$ time and
space, so that for any variable $X_i$ and $1 \leq j \leq |X_i|$,
${X_i}([1:j])$ can be computed in $O(j)$ time.
\end{lemma}
The formal statement of the problem we solve is:
\begin{problem}[$q$-gram frequencies on SLP]
\label{problem:SLPqgramfreq}
Given integer $q\geq 1$ and an SLP $\mathcal{T}$ of size $n$ that represents string $T$,
output $(i, |\mathit{Occ}(T,P)|)$ for all $P\in\Sigma^q$ where
$\mathit{Occ}(T,P)\neq\emptyset$, and some $i\in\mathit{Occ}(T,P)$.
\end{problem}
Since the problem is very simple for $q = 1$,
we shall only consider the case for $q\geq 2$ for the rest of the paper.
Note that although the number of distinct $q$-grams in $T$ is bounded by $O(qn)$,
we would require an extra multiplicative $O(q)$ factor for the output if we output
each $q$-gram explicitly as a string.
In our algorithms to follow, we compute
a compact, $O(qn)$-size representation of the output,
from which each $q$-gram can be easily obtained in $O(q)$ time.
\section{$O(qn)$ Algorithm~\cite{goto11:_fast_minin_slp_compr_strin}}
\label{section:qn}
In this section, we briefly describe the $O(qn)$ algorithm
presented in~\cite{goto11:_fast_minin_slp_compr_strin}.
The idea is to count occurrences of $q$-grams with respect to
the variable that stabs its occurrence.
The algorithm reduces Problem~\ref{problem:SLPqgramfreq}
to calculating the frequencies of all $q$-grams in a weighted set of strings,
whose total length is $O(qn)$.
Lemma~\ref{lemma:qn_key}
shows the key idea of the algorithm.
\begin{lemma}
\label{lemma:qn_key}
For any SLP $\mathcal{T} = \{ X_i \rightarrow \mathit{expr}_i \}_{i=1}^n$
that represents string $T$, integer $q \geq 2$, and $P \in \Sigma^q$,
$|\mathit{Occ}(T,P)| = \sum_{i=1}^n\mathit{vOcc}(X_i)\cdot |\mathit{Occ}(t_i, P)|$,
where
$t_i = \mathit{suf}(\mathit{val}(X_{\ell(i)}),q-1)\mathit{pre}(\mathit{val}(X_{r(i)}),q-1)$.
\end{lemma}
\begin{proof}
For any $q \geq 2$, $v$ stabs the interval $[u:u+q-1]$
if and only if
$[u:u+q-1]\subseteq
[s_v:f_v] =
\mathit{suf}(\mathit{val}Int(\ell(v)),q-1)\cup\mathit{pre}(\mathit{val}Int(r(v)),q-1)$.
(See Fig.~\ref{fig:SLP-kgram}.)
Also, since an occurrence of $X_i$ in the derivation tree always
derives the same string $\mathit{val}(X_i)$,
$t_i = T([s_v:f_v])$ for any node $v$ such that $\variable{v} = X_i$.
Therefore,
\begin{eqnarray*}
\lefteqn{|\mathit{Occ}(T,P)| = \big|\{ u>0 \mid T([u:u+q-1]) = P\}\big|}\\
& = & \sum_{v\in \mathcal{V}} \big| \{ u>0 \mid \xi_{\mathcal{T}}(u,u+q-1)=v, j=u-s_v+1, \variable{v}([j:j+q-1]) = P \}\big| \\
& = & \sum_{i=1}^n \sum_{v\in \mathcal{V}: \variable{v}=X_i} \big|\{
u>0 \mid \xi_{\mathcal{T}}(u,u+q-1)=v,j=u-s_v+1, \variable{v}([j:j+q-1]) = P\}\big|\\
& = & \sum_{i=1}^n \sum_{v\in \mathcal{V}: \variable{v}=X_i} \mathit{Occ}(T([s_v:f_v]),P)
= \sum_{i=1}^n \mathit{vOcc}(X_i)\cdot \mathit{Occ}(t_i,P).\\
\end{eqnarray*}
\qed
\end{proof}
\begin{wrapfigure}[15]{r}{0.4\textwidth}
\begin{center}
\includegraphics[width=0.4\textwidth]{SLP_ngram.eps}
\end{center}
\caption{
Length-$q$ intervals where
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$,
and $(X_i\rightarrow X_{\ell(i)} X_{r(i)}) \in \mathcal{T}$.
}
\label{fig:SLP-kgram}
\end{wrapfigure}
From Lemma~\ref{lemma:qn_key}, we have that occurrence frequencies in
$T$ are equivalent to occurrence frequencies in $t_i$ weighted by $\mathit{vOcc}(X_i)$.
Therefore, the $q$-gram frequencies problem can be regarded as
obtaining the {\em weighted} frequencies of all $q$-grams in the set
of strings $\{t_1,\ldots,t_n\}$,
where each occurrence of a $q$-gram in $t_i$ is weighted by $\mathit{vOcc}(X_i)$.
This can be further reduced to a weighted $q$-gram frequency
problem for a single string $z$, where each position of $z$
holds a weight associated with the $q$-gram that starts at that position.
String $z$ is constructed by concatenating all $t_i$'s with length at
least $q$.
The weights of positions corresponding to the first $|t_i| - (q-1)$
characters of $t_i$ will be $\mathit{vOcc}(X_i)$, while the last $(q-1)$
positions will be $0$ so that superfluous $q$-grams generated by the
concatenation are not counted.
The remaining is a simple linear time algorithm
using suffix and lcp arrays on the weighted string,
thus solving the problem in $O(qn)$ time and space.
\section{New Algorithm}
\label{section:new_algorithm}
We now describe our new algorithm which solves the $q$-gram
frequencies problem on SLPs.
The new algorithm basically follows the previous $O(qn)$ algorithm,
but is an elegant refinement.
The reduction for the previous $O(qn)$ algorithm
leads to a fairly large amount of redundantly decompressed regions of
the text as $q$ increases.
This is due to the fact that
the $t_i$'s are considered independently for each variable $X_i$,
while {\em neighboring} $q$-grams that are stabbed by different
variables actually share $q-1$ characters.
The key idea of our new algorithm is to exploit this redundancy.
(See Fig.~\ref{fig:qgramneighbor}.)
In what follows, we introduce the concept of $q$-gram neighbors,
and reduce the $q$-gram frequencies problem on SLP to
a weighted $q$-gram frequencies problem on a weighted tree.
\subsection{$q$-gram Neighbor Graph}
We say that
$X_j$ is a {\em right $q$-gram neighbor} of $X_i$ $(i \neq j)$,
or equivalently,
$X_i$ is a {\em left $q$-gram neighbor} of $X_j$,
if for some integer $u \in [1:|T|-q]$,
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$ and
$\variable{\xi_\mathcal{T}(u+1,u+q)} = X_j$.
Notice that $|X_i|$ and $|X_j|$ are both at least $q$
if $X_i$ and $X_j$ are right or left $q$-gram neighbors of each other.
\begin{figure}
\caption{$q$-gram neighbors and redundancies.
(Left) $X_j$ is a right $q$-gram neighbor of
$X_i$, and $X_i$ is {\em a}
\label{fig:qgramneighbor}
\end{figure}
\begin{definition}
For $q\geq 2$, the right $q$-gram neighbor graph of SLP
$\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$ is the directed graph
$G_q = (V,E_r)$, where
\begin{eqnarray*}
V&=&\{ X_i \mid i \in \{1,\ldots, n\}, |X_i| \geq q \}\\
E_r &=& \{ (X_i,X_j) \mid X_j \mbox{ is a right $q$-gram neighbor of
$X_i$ } \}
\end{eqnarray*}
\end{definition}
Note that there can be multiple right $q$-gram neighbors for a given variable.
However,
the total number of edges in the neighbor graph
is bounded by $2n$, as will be shown below.
\begin{lemma}
\label{lemma:unique_neighbors}
Let $X_j$ be a right $q$-gram neighbor of $X_i$.
If, $|X_{r(i)}| \geq q$, then $X_j$ is the label of the deepest
variable on the left-most path of the derivation tree rooted at
a node labeled $X_{r(i)}$ whose length is at least $q$.
Otherwise, if $|X_{r(i)}| < q$, then
$X_i$ is the label of the deepest variable on the right-most path rooted at
a node labeled $X_{\ell(j)}$
whose length is at least $q$.
\end{lemma}
\begin{proof}
Suppose $|X_{r(i)}| \geq q$.
Let $u$ be a position, where
$\variable{\xi_\mathcal{T}(u,u+q-1)} = X_i$ and $\variable{\xi_\mathcal{T}(u+1,u+q)} = X_j$.
Then, since the interval $[u+1:u+q]$ is a prefix of
$\mathit{val}Int(X_{r(i)})$,
$X_j$ must be on the left most path rooted at $X_{r(i)}$.
Since $X_j = \variable{\xi_\mathcal{T}(u+1,u+q)}$,
the lemma follows from the definition of $\xi_\mathcal{T}$.
The case for $|X_{r(i)}| < q$ is symmetrical
and can be shown similarly.
\qed
\end{proof}
\begin{lemma}
For an arbitrary SLP
$\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$
and integer $q\geq 2$,
the number of edges in the right $q$-gram neighbor graph $G_q$ of $\mathcal{T}$ is at most $2n$.
\end{lemma}
\begin{proof}
Suppose $X_j$ is a right $q$-gram neighbor of $X_i$.
From Lemma~\ref{lemma:unique_neighbors}, we have that
if $|X_{r(i)}| \geq q$, the right $q$-gram neighbor of $X_i$ is uniquely determined
and that $|X_{\ell(j)}| < q$.
Similarly, if $|X_{r(i)}| < q$, $|X_{\ell(j)}|\geq q$ and the left $q$-gram neighbor of $X_j$
is uniquely $X_i$.
Therefore,
\begin{eqnarray*}
&&\sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| \geq q \}|
+ \sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| < q \}|\\
&=&
\sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{r(i)}| \geq q \}|
+ \sum_{i=1}^n |\{ (X_i,X_j) \in E_r \mid |X_{\ell(j)}| \geq q \}| \leq 2n.
\end{eqnarray*}
\qed
\end{proof}
\begin{lemma}
For an arbitrary SLP $\mathcal{T} = \{ X_i \rightarrow expr_i \}_{i=1}^n$ and
integer $q\geq 2$,
the right $q$-gram neighbor graph $G_q$ of $\mathcal{T}$ can be constructed in $O(n)$ time.
\end{lemma}
\begin{proof}
For any variable $X_i$, let
$\mathit{lm}_q(X_i)$ and $\mathit{rm}_q(X_i)$ respectively represent the
index of the label of the deepest
node with length at least $q$
on the left-most and right-most path
in the derivation tree rooted at $X_i$, or $\mathit{null}$ if $|X_i| < q$.
These values can be computed for all variables in
a total of $O(n)$ time based on the following recursion:
If $(X_i \rightarrow a)\in\mathcal{T}$ for some $a\in\Sigma$, then
$\mathit{lm}_q(X_i) = \mathit{rm}_q(X_i) = \mathit{null}$.
For $(X_i \rightarrow X_{\ell(i)}X_{r(i)})\in\mathcal{T}$,
\begin{equation*}
\mathit{lm}_q(X_i) = \begin{cases}
\mathit{null} & \mbox{if } |X_i| < q,\\
i & \mbox{if } |X_i| \geq q \mbox{ and } |X_{\ell(i)}| < q,\\
\mathit{lm}_q(X_{\ell(i)}) & \mbox{otherwise. }
\end{cases}
\end{equation*}
$\mathit{rm}_q(X_i)$ can be computed similarly.
Finally,
\begin{eqnarray*}
E_r & = &
\{ (X_i, X_{\mathit{lm}_q(X_{r(i)})}) \mid \mathit{lm}_q(X_{r(i)}) \neq \mathit{null},
i = 1,\ldots, n \}\\
&&\cup \{ (X_{\mathit{rm}_q(X_{\ell(i)})}, X_i) \mid \mathit{rm}_q(X_{\ell(i)}) \neq
\mathit{null},
i=1,\ldots, n\}.
\end{eqnarray*}
\qed
\end{proof}
\begin{lemma}
\label{lemma:connected}
Let $G_q = (V, E_r)$ be the right $q$-gram neighbor graph of
SLP $\mathcal{T} = \{ X_i = expr_i \}_{i=1}^n$ representing string $T$,
and let $X_{i_1} = \variable{\xi_\mathcal{T}(1,q)}$.
Any variable $X_j \in V (i_1 \neq j)$ is reachable from $X_{i_1}$,
that is, there exists a directed path from $X_{i_1}$ to $X_j$ in $G_q$.
\end{lemma}
\begin{proof}
Straightforward, since any $q$-gram of $T$ except for the left most
$T([1:q])$ has a $q$-gram on its left.\qed
\end{proof}
\subsection{Weighted $q$-gram Frequencies Over a Trie}
\label{section:weighted_q-gram_frequencies_over_a_trie}
From Lemma~\ref{lemma:connected}, we have that the right $q$-gram
neighbor graph is connected.
Consider an arbitrary directed spanning tree
rooted at $X_{i_1} = \variable{\xi_\mathcal{T}(1,q)}$ which can be obtained in linear time by a depth
first traversal on $G_q$ from $X_{i_1}$.
We define the label $\mathit{label}(X_i)$ of each node $X_i$ of the
$q$-gram neighbor graph, by \[\mathit{label}(X_i) = t_i[q:|t_i|] \]
where $t_i =
\mathit{suf}(\mathit{val}(X_{\ell(i)}),q-1)\mathit{pre}(\mathit{val}(X_{r(i)}),q-1)$ as before.
For convenience, let $X_{i_0}$ be a dummy variable
such that
$\mathit{label}(X_{i_0}) = T([1:q-1])$, and
$X_{r(i_0)} = X_{i_1}$ (and so $(X_{i_0},X_{i_1})\in E_r$).
\begin{lemma}
\label{lemma:path}
Fix a directed spanning tree on the right $q$-gram neighbor graph
of SLP $\mathcal{T}$, rooted at $X_{i_0}$.
Consider a directed path $X_{i_0}, \ldots, X_{i_m}$ on the spanning tree.
The weighted $q$-gram frequencies on the string obtained by
the concatenation $\mathit{label}(X_{i_0}) \mathit{label}(X_{i_1}) \cdots \mathit{label}(X_{i_m})$,
where each occurrence of a $q$-gram that ends in a position in
$\mathit{label}(X_{i_j})$ is weighted by
$\mathit{vOcc}(X_{i_j})$,
is equivalent to the
weighted $q$-gram frequencies of
strings $\{t_{i_1}, \ldots t_{i_m}\}$
where each $q$-gram in $t_{i_j}$ is weighted by $\mathit{vOcc}(X_{i_j})$.
\end{lemma}
\begin{proof}
Proof by induction:
for $m = 1$,
we have that $\mathit{label}(X_{i_0})\mathit{label}(X_{i_1}) = t_{i_1}$.
All $q$-grams in $t_{i_1}$ end in $t_{i_1}$
and so are weighted by $\mathit{vOcc}(X_{i_1})$.
When $\mathit{label}(X_{i_j})$ is added to
$\mathit{label}(X_{i_0}) \cdots \mathit{label}(X_{i_{j-1}})$,
$|\mathit{label}(X_{i_j})|$ new $q$-grams are formed, which correspond to
$q$-grams in $t_{i_j}$, i.e. $|t_{i_j}| = q - 1 + |\mathit{label}(X_{i_j})|$,
and $t_{i_j}$ is a suffix of $\mathit{label}(X_{i_{j-1}})\mathit{label}(X_{i_{j}})$.
All the new $q$-grams end in $\mathit{label}(X_{i_j})$ and are thus
weighted by $\mathit{vOcc}(X_{i_j})$.
\qed
\end{proof}
From Lemma~\ref{lemma:path}, we can construct a weighted trie $\Upsilon$
based on a directed spanning tree of $G_q$ and $\mathit{label}()$,
where
the weighted $q$-grams in $\Upsilon$ (represented as length-$q$ paths)
correspond to the
occurrence frequencies of $q$-grams in $T$.
\footnote{A minor technicality is that a node in $\Upsilon$ may have
multiple children with the same character label, but this
does not affect the time complexities of the algorithm.}
\begin{algorithm2e}[t]
\caption{Constructing weighted trie from SLP}
\label{algo:slp2trie}
Construct right $q$-gram neighbor graph $G=(V,E_r)$\;
Calculate $\mathit{vOcc}(X_i)$ for $i = 1,\ldots, n$\;
Calculate $|\mathit{label}(X_i)|$ for $i = 1,\ldots, n$\;
\lFor{$i = 0,\ldots, n$}{
$\mathsf{visited}[i] = \mathsf{false}$\;
}
$X_{i_1} = \variable{\xi_\mathcal{T}(1,q)} = \mathit{lm}_q(X_n)$\;
Define $X_{i_0}$ so that $X_{r(i_0)} = X_{i_1}$ and $|\mathit{label}(X_{i_0})| = q-1$\;
$\mathit{root} \leftarrow$ new node\tcp*[l]{root of resulting trie}
\ref{procedure:bdf}($i_0$, $\mathit{root}$)\;
\Return $\mathit{root}$
\end{algorithm2e}
\begin{procedure}[t]
\caption{BuildDepthFirst($i$, $\mathit{trieNode}$)}
\label{procedure:bdf}
\SetKw{KwAND}{and}
\SetKw{KwOR}{or}
\SetKw{KwBREAK}{break}
\SetKwFunction{BDF}{\ref{procedure:bdf}}
\tcp{add prefix of $r(i)$ to trieNode while right neighbors of $i$ are unique}
$l \leftarrow 0$; $k \leftarrow i$\;
\While{
{$\textsf{true}$}
}{
$l \leftarrow l + |\mathit{label}(X_k)|$\;
$\mathsf{visited}[k] \leftarrow \mathsf{true}$\;
\tcp{exit loop if right neighbor is possibly non-unique or is visited}
\lIf{$|X_{r(k)}| < q$ \KwOR $\mathsf{visited}[\mathit{lm}_q(X_{r(k)})] =
\mathsf{true}$}{
{\KwBREAK}\;
}
$k \leftarrow\mathit{lm}_q(X_{r(k)})$\;
}
add new branch from $\mathit{trieNode}$ with string $X_{r(i)}([1:l])$\;\label{algo:prefadd}
let end of new branch be $\mathit{newTrieNode}$\;
\tcp{If $|X_{r(k)}| < q$, there may be multiple right neighbors.}
\tcp{If $|X_{r(k)}|\geq q$, nothing is done because it has already been visited.}
\For{$X_c \in \{ X_j \mid (X_k,X_j) \in E_r \}$}{
\If{$\mathsf{visited}[c] = \mathsf{false}$}{
\BDF($X_c$, $\mathit{newTrieNode}$)\;
}
}
\end{procedure}
\begin{lemma}
$\Upsilon$ can be constructed in time linear in its size.
\end{lemma}
\begin{proof}
See Algorithm~\ref{algo:slp2trie}.
Let $G$ be the $q$-gram neighbor graph.
We construct $\Upsilon$ in a depth first manner starting at $X_{i_0}$.
The crux of the algorithm is that rather than
computing $\mathit{label}()$ separately for each variable,
we are able to aggregate the $\mathit{label}()$s and limit all partial
decompressions of variables to prefixes of variables, so that
Lemma~\ref{label:prefix_decompression} can be used.
Any directed acyclic path on $G$ starting at $X_{i_0}$
can be segmented into multiple sequences of variables,
where each sequence $X_{i_j}, \ldots, X_{i_k}$
is such that $j$ is the only integer in $[j:k]$ such that
$j = 0 $ or $|X_{r(i_{j-1})}| < q$.
From Lemma~\ref{lemma:unique_neighbors},
we have that $X_{i_{j+1}},\ldots,X_{i_k}$
are uniquely determined.
If $j>0$, $\mathit{label}(X_{i_j})$ is a prefix of $\mathit{val}(X_{r(i_j)})$
since $|X_{r(i_{j-1})}| < q$
(see Fig.~\ref{fig:qgramneighbor} Right),
and if $j=0$, $\mathit{label}(X_{i_0})$ is again a prefix of
$\mathit{val}(X_{r(i_0)}) = \mathit{val}(X_{i_1})$.
It is not difficult to see that $\mathit{label}(X_{i_j})\cdots\mathit{label}(X_{i_{k}})$ is
also a prefix of $X_{r(i_j)}$ since
$X_{i_{j+1}},\ldots,X_{i_k}$ are all descendants of $X_{r(i_j)}$, and
each $\mathit{label}()$ extends the
partially decompressed string to consider consecutive $q$-grams in $X_{r(i_j)}$.
Since prefixes of variables of SLPs can be decompressed in time
proportional to the output size with linear time pre-processing
(Lemma~\ref{label:prefix_decompression}), the lemma follows.
\qed
\end{proof}
We only illustrate how the character labels are determined
in the pseudo-code of Algorithm~\ref{algo:slp2trie}.
It is straightforward to assign a weight $\mathit{vOcc}(X_k)$ to
each node of $\Upsilon$ that corresponds to $\mathit{label}(X_k)$.
\begin{lemma}
\label{lemma:size_of_trie}
The number of edges in $\Upsilon$ is
$(q-1) + \sum \{ |t_i|-(q-1) \mid |X_i| \geq q, i=1,\ldots,n\} =
|T| - \mathit{dup}(q,\mathcal{T})$ where
\[
\mathit{dup}(q,\mathcal{T})
= \sum\{ (\mathit{vOcc}(X_i) - 1) \cdot (|t_i| - (q-1)) \mid |X_i| \geq q, i=1,\ldots,n \}\}
\]
\end{lemma}
\begin{proof}
$(q-1)+\sum \{ |t_i|-(q-1) \mid |X_i| \geq q, i=1,\ldots,n\}$ is straight
forward from the definition of $\mathit{label}(X_i)$ and the
construction of $\Upsilon$.
Concerning $\mathit{dup}$,
each variable $X_i$ occurs $\mathit{vOcc}(X_i)$ times
in the derivation tree, but only once in the directed spanning tree.
This means that for each occurrence after the first,
the size of $\Upsilon$ is reduced by $|\mathit{label}(X_i)|=|t_i| - (q-1)$ compared to $T$.
Therefore, the lemma follows.
\qed
\end{proof}
To efficiently count the weighted $q$-gram frequencies on $\Upsilon$,
we can use suffix trees.
A suffix tree for a trie is defined as a generalized suffix tree
for the set of strings represented in the trie as leaf to root paths.
\footnote{
When considering leaf to root paths on $\Upsilon$,
the direction of the string is the reverse of what is
in $T$. However, this is merely a matter of representation of the
output.
}
The following is known.
\begin{lemma}[\cite{shibuya03:_const_suffix_tree_tree_large_alphab}]
Given a trie of size $m$,
the suffix tree for the trie can be constructed in
$O(m)$ time and space.
\end{lemma}
With a suffix tree, it is a simple exercise to solve the weighted
$q$-gram frequencies problem on $\Upsilon$ in linear time.
In fact, it is known that the suffix array for the common suffix trie
can also be constructed in
linear time~\cite{ferragina09:_compr}, as well as its longest common prefix
array~\cite{kimura11}, which can also be used to solve the problem in
linear time.
\begin{corollary}
The weighted $q$-gram frequencies problem on a trie of size $m$
can be solved in $O(m)$ time and space.
\end{corollary}
From the above arguments, the theorem follows.
\begin{theorem}
The $q$-gram frequencies problem on an SLP $\mathcal{T}$ of size
$n$, representing string $T$ can be solved in
$O(\min\{qn,|T| - \mathit{dup}(q,\mathcal{T})\})$ time and space.
\end{theorem}
Note that since each $q\leq |t_i|\leq 2(q-1)$,
and $|\mathit{label}(X_i)| = |t_i| - (q-1)$, the
total length of decompressions made by the algorithm, i.e. the size
of the reduced problem, is at least halved and can be as small as $1/q$
(when all $|t_i|=q$, for example, in an SLP that represents LZ78
compression),
compared to the previous $O(qn)$ algorithm.
\section{Preliminary Experiments}
We first evaluate the size of the trie $\Upsilon$ induced from the
right $q$-gram neighbor graph, on which the running time of the new algorithm
of Section~\ref{section:new_algorithm} is dependent.
We used data sets obtained from Pizza \& Chili Corpus,
and constructed SLPs using the RE-PAIR~\cite{LarssonDCC99} compression algorithm.
Each data is of size 200MB.
Table~\ref{table:zsize} shows the sizes of $\Upsilon$ for different values of $q$,
in comparison with the total length of strings $t_i$,
on which the previous $O(qn)$-time algorithm of Section~\ref{section:qn} works.
We cumulated the lengths of all $t_i$'s only for those satisfying $|t_i| \geq q$,
since no $q$-gram can occur in $t_i$'s with $|t_i| < q$.
Observe that for all values of $q$ and for all data sets,
the size of $\Upsilon$ (i.e., the total number of characters in $\Upsilon$) is
smaller than those of $t_i$'s and the original string.
\begin{table}[t]
\caption{
A comparison of the size of $\Upsilon$ and the total length of strings $t_i$ for
SLPs that represent textual data from Pizza \& Chili Corpus.
The length of the original text is 209,715,200.
The SLPs were constructed by RE-PAIR~\cite{LarssonDCC99}.
}
\label{table:zsize}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{1pt}
\renewcommand{ptm}{ptm}
\renewcommand{phv}{phv}
\renewcommand{pcr}{pcr}
\normalfont
\input{table_zsize.tex}
\end{center}
\end{table}
The construction of the suffix tree or array for a trie,
as well as the algorithm for Lemma~\ref{label:prefix_decompression},
require various tools such as level ancestor
queries~\cite{dietz91:_findin,berkman94:_findin,bender04:_level_ances_probl}
for which we did not have an efficient implementation.
Therefore, we try to assess the practical impact of the reduced problem size
using a simplified version of our new algorithm.
We compared three algorithms ($\textrm{NSA}$, $\textrm{SSA}$, $\textrm{STSA}$) that count the occurrence frequencies of all
$q$-grams in a text given as an SLP.
$\textrm{NSA}$ is the $O(|T|)$-time algorithm which works on the uncompressed text,
using suffix and LCP arrays.
$\textrm{SSA}$ is our previous $O(qn)$-time algorithm~\cite{goto11:_fast_minin_slp_compr_strin},
and $\textrm{STSA}$ is a simplified version of our new algorithm.
$\textrm{STSA}$ further reduces the weighted $q$-gram frequencies problem on $\Upsilon$,
to a weighted $q$-gram frequencies problem on a single string as follows:
instead of constructing $\Upsilon$, each branch of $\Upsilon$ (on line~\ref{algo:prefadd} of~\ref{procedure:bdf})
is appended into a single string.
The $q$-grams that are represented in the branching edges of $\Upsilon$
can be represented in the single string, by redundantly adding
$\mathit{suf}(X_{r(i)}([1:l]),q-1)$ in front of the string corresponding to
the next branch.
This leads to some duplicate partial decompression, but the resulting
string is still always shorter than the string produced by our
previous algorithm~\cite{goto11:_fast_minin_slp_compr_strin}.
The partial decompression of $X_{r(i)}([1:l])$ is implemented using
a simple $O(h+l)$ algorithm, where $h$ is the height of the SLP
which can be as large as $O(n)$.
All computations were conducted on a Mac Pro (Mid 2010)
with MacOS X Lion 10.7.2,
and 2 x 2.93GHz 6-Core Xeon processors and 64GB Memory,
only utilizing a single process/thread at once.
The program was compiled using the GNU C++ compiler ({\tt g++}) 4.6.2
with the {\tt -Ofast} option for optimization.
The running times were measured in seconds, after reading
the uncompressed text into memory for $\textrm{NSA}$, and
after reading the SLP that represents the text into memory for $\textrm{SSA}$ and
$\textrm{STSA}$.
Each computation was repeated at least 3 times, and the average was taken.
Table~\ref{table:running_time} summarizes the running times of the three algorithms.
$\textrm{SSA}$ and $\textrm{STSA}$ computed weighted $q$-gram frequencies on $t_i$ and $\Upsilon$, respectively.
Since the difference between the total length of $t_i$ and
the size of $\Upsilon$ becomes larger as $q$ increases,
$\textrm{STSA}$ outperforms $\textrm{SSA}$ when the value of $q$ is not small.
In fact, in Table~\ref{table:running_time} SSA2 was faster than $\textrm{SSA}$
for all values of $q > 3$.
$\textrm{STSA}$ was even faster than $\textrm{NSA}$ on the XML data whenever $q \leq 20$.
What is interesting is that $\textrm{STSA}$ outperformed $\textrm{NSA}$ on the ENGLISH data
when $q = 100$.
\begin{table}[t]
\caption{
Running time in seconds for SLPs that represent textual data from Pizza \& Chili Corpus.
The SLPs were constructed by RE-PAIR~\cite{LarssonDCC99}.
Bold numbers represent the fastest time for each data and $q$.
$\textrm{STSA}$ is faster than $\textrm{SSA}$ whenever $q>3$.
}
\label{table:running_time}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{1pt}
\renewcommand{ptm}{ptm}
\renewcommand{phv}{phv}
\renewcommand{pcr}{pcr}
\normalfont
\input{table_time.tex}
\end{center}
\end{table}
\end{document} |
\begin{document}
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\begin{abstract}
Our aim in this paper, is to establish certain new integrals for the the $(p,q)-$Mathieu--power series. In particular, we investigate the Mellin-Barnes type integral representations
for a particular case of thus special function. Moreover, we introduce the notion of the $(p,q)-$Mittag-Leffler functions and we present a relationships between thus two functions. Some other applications are proved, in particular two Tur\'an type inequalities for the $(p,q)-$Mathieu series are proved.
\end{abstract}
\section{\bf Introduction}
\setcounter{equation}{0}
The following familiar infinite series
\begin{equation}
S(r)=\sum_{n=1}^\infty\frac{2n}{(n^2+r^2)^2},
\end{equation}
is called a Mathieu series. It was introduced and studied by \'Emile Leonard Mathieu in his
book \cite{18} devoted to the elasticity of solid bodies. Bounds for this series
are needed for the solution of boundary value problems for the biharmonic equations in a two--dimensional rectangular domain, see \cite[Eq. (54), p. 258]{13}.
Several interesting problems and solutions dealing with integral representations and
bounds for the following slight generalization of the Mathieu series with a fractional
power:
\begin{equation}\label{0t1}
S_\mu(r)=\sum_{n=1}^\infty\frac{2n}{(n^2+r^2)^{\mu+1}},\;(\mu>0,\;r>0),
\end{equation}
can be found in the recent works by Diananda \cite{D}, Tomovski and Tren\v{c}evski \cite{TT}, Srivatava et al. \cite{SKZ}. In \cite{SKZ}, the authors derived the following new Laplace type integral representation via Schlomilch series:
\begin{equation}
S_\mu(r)=\frac{\sqrt{\pi}}{2^{\mu-\frac{1}{2}}\Gamma(\mu+1)}\int_0^\infty e^{-rt} \mathcal{K}_\mu(t)dt, \;\mu>\frac{3}{2},
\end{equation}
where
$$\mathcal{K}_\mu(t)=t^{\mu+\frac{1}{2}}\sum_{k=1}^\infty\frac{J_{\mu+\frac{1}{2}}(kt)}{k^{\mu-\frac{1}{2}}}$$
with $J_\mu(z)$ is the Bessel function. Motivated essentially by the works of Cerone and Lenard \cite{C}, Srivastava and Tomovski in \cite{ZY} defined a family of generalized Mathieu series
\begin{equation}\label{;,}
S_\mu^{(\alpha,\beta)}(r; \textbf{a})=S_\mu^{(\alpha,\beta)}(r; \{a_k\}_{k=0}^\infty)=\sum_{k=1}^\infty\frac{2a_k^\beta}{(a_k^\alpha+r^2)^\mu},\;(\alpha,\beta,\mu,r>0),
\end{equation}
where it is tacitly assumed that the positive sequence
$$\textbf{a}=\{a_k\}=\{a_1,a_2,...\},\;\textrm{such \;that\;}\lim_{k\longrightarrow\infty}a_k=\infty,$$
is so chosen that the in?nite series in de?nition (\ref{;,}) converges, that is, that the following
auxiliary series
$$\sum_{k=1}^\infty \frac{1}{a_k^{\mu\alpha-\beta}},$$
is convergent.
\begin{define} $($see \cite[Eq. (6.1), p. 256]{SR}$)$ The extended beta function $B_{p,q}(x,y)$ is defined by
\begin{equation}\label{1}
B_{p,q}(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}E_{p,q}(t)dt,\;x,y,p,q\in\mathbb{C},\Re(p),\Re(q)>0,
\end{equation}
where $E_{p,q}(t)$ is defined by
$$E_{p,q}(t)=\exp\left(-\frac{p}{t}-\frac{q}{1-t}\right),\;p,q\in\mathbb{C},\Re(p),\Re(q)>0.$$
\end{define}
In particular, Chaudhry et al. \cite[p. 20, Eq. (1.7)]{CH}, introduced the $p–$extension of the Eulerian Beta function $B(x, y):$
$$B_p(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}e^{-\frac{p}{t(1-t)}}dt,\;\Re(p)>0,$$
whose special case when $p=0$ ( or $p=q=0$ in (\ref{1}) )we get the familiar beta integral
\begin{equation}
B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt,\;\Re(x),\Re(y)>0.
\end{equation}
\begin{define} $($see \cite[p. 4, Eq. 2.1]{23}$)$ Assume that $\lambda,\mu,s,p,q\in\mathbb{C}$ such that $\Re(p),\Re(q)\geq0$ and $\nu,a\in\mathbb{C}\setminus\mathbb{Z}_0^-.$ The extended Hurwitz-Lerch zeta function is defined by
\begin{equation}\label{2}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\sum_{n=0}^\infty\frac{(\lambda)_n}{n!}\frac{B_{p,q}(\mu+n,\nu-\mu)}{B(\mu,\nu-\mu)}\frac{z^n}{(a+n)^s},\:(|z|<1),
\end{equation}
where $(\lambda)_n$ denotes the Pochhammer symbol (or the shifted factorial) defined, in terms of Euler's Gamma function, by
\begin{displaymath}
(\lambda)_\mu=\frac{\Gamma(\lambda+\mu)}{\Gamma(\lambda)}=\left\{ \begin{array}{ll}
1& \textrm{$(\mu=0;\lambda\in\mathbb{C}\setminus\{0\})$}\\
\lambda(\lambda+1)...(\lambda+n-1)& \textrm{$(\mu=n\in\mathbb{N};\lambda\in\mathbb{C})$}
\end{array} \right.
\end{displaymath}
\end{define}
Upon setting $\lambda=1,$ (\ref{2}) reduces to
$$\Phi_{\mu,\nu}(z,s,a;p,q)=\sum_{n=0}^\infty\frac{B_{p,q}(\mu+n,\nu-\mu)}{B(\mu,\nu-\mu)}\frac{z^n}{(a+n)^s},\:(|z|<1).$$
It is easy to observe that
\begin{equation}\label{ZZ}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\frac{1}{\Gamma(\lambda)}D_z^{\lambda-1}\{z^{\lambda-1}\Phi_{\mu,\nu}(z,s,a;p,q)\},\;(\Re(\lambda)>0),
\end{equation}
where $D_z^{\lambda}$ denotes the well-known Riemann-Liouville fractional derivative operator defined by
\begin{equation}
D_z^{\lambda} f(z)=\left\{ \begin{array}{ll}
\frac{1}{\Gamma(-\lambda)}\int_0^z(z-t)^{-\lambda-1}f(t)dt& \textrm{$(\Re(\lambda)<0)$}\\
\frac{d^m}{dz^m}D_z^{\lambda-m} f(z)& \textrm{$(m-1\leq\Re(\lambda)<m,\;(m\in\mathbb{N}))$}
\end{array} \right.
\end{equation}
In \cite [Theorem 3.8]{LU1} Luo et al. proved the following integral representation for the extended Hurwitz-Lerch zeta funtion $\Phi_{\lambda,\mu,\nu}(z,s,a;p,q):$
\begin{equation}
\Phi_{\lambda,\mu,\nu}(z,s,a;p,q)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1} e^{-at}{}_2F_1\Big[^{\;\lambda,\;\mu}_{\;\;\nu};ze^{-t};p,q\Big]dt,\;|z|<1,
\end{equation}
$$\left(p,q,a,s>0,\lambda,\mu\in\mathbb{C},\nu\in\mathbb{C}\setminus\mathbb{Z}_0^-\right),$$
where ${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p,q\Big]$ is the extended Gauss hypergeometric function defined by
$${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p,q\Big]=\sum_{n=0}^\infty (a)_n\frac{B_{p,q}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\;|z|<1,$$
$$\Big(\Re(p),\Re(q)\geq0, a, b\in\mathbb{C}, c\in\mathbb{C}\setminus\mathbb{Z}_0^-,\;\Re(c)>\Re(b)>0\Big).$$
When $p=q$ we obtain the extended of the extended of the Gaussian hypergeometric function $F_p$ defined by \cite{CH}:
$${}_2F_1\Big[^{\;a,\;b}_{\;\;c};z;p\Big]=\sum_{n=0}^\infty (a)_n\frac{B_{p}(b+n,c-b)}{B(b,c-b)}\frac{z^n}{n!},\;|z|<1,$$
$$\Big(\Re(p)\geq0, a, b\in\mathbb{C}, c\in\mathbb{C}\setminus\mathbb{Z}_0^-,\;\Re(c)>\Re(b)>0\Big).$$
The Fox-Wright function ${}^p\Psi_q[.]$ with $p$ numerator parameters $\alpha_1,...,\alpha_p$ and $q$ denominator parameters $\beta_1,...,\beta_q$ which are defined by
\begin{equation}\label{3}
{}_p\Psi_q\Big[_{(\beta_1,B_1),...,(\beta_q,B_q)}^{(\alpha_1,A_1),...,(\alpha_p,A_p)}\Big|z \Big]={}_p\Psi_q\Big[_{(\beta_q,B_q)}^{(\alpha_p,A_p)}\Big|z \Big]=\sum_{k=0}^\infty\frac{\prod_{l=1}^p\Gamma(\alpha_l+kA_l)}{\prod_{j=1}^q\Gamma(\beta_l+kB_l)}\frac{z^k}{k!},
\end{equation}
The defining series in (\ref{3}) converges in the whole complex $z-$plane when
$$\Delta=\sum_{j=1}^q B_j-\sum_{j=1}^p A_j>-1;$$
when $\Delta= 0$, then the series in (\ref{3}) converges for $|z|<\nabla,$ where
$$\nabla=\left(\prod_{j=1}^p A_j^{-A_j}\right)\left(\prod_{j=1}^qB_j^{B_j}\right).$$
If, in the definition (\ref{3}), we set
$$A_1=...=A_p=1\;\;\;\textrm{and}\;\;\;B_1=...=B_q=1,$$
we get the relatively more familiar generalized hypergeometric function ${}_pF_q[.]$ given by
\begin{equation}\label{hyper}
{}_p F_q\left[^{\alpha_1,...,\alpha_p}_{\beta_1,...,\beta_q}\Big|z\right]=\frac{\prod_{j=1}^q\Gamma(\beta_j)}{\prod_{i=1}^p\Gamma(\alpha_i)}{}_p\Psi_q\Big[_{(\beta_1,1),...,(\beta_q,1)}^{(\alpha_1,1),...,(\alpha_p,1)}\Big|z \Big]
\end{equation}
In this paper we consider the $(p,q)-$Mathieu type power series defined by:
\begin{equation}\label{*}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n B_{p,q}(\tau+n,\omega-\tau)z^n}{n!B(\tau,\omega-\tau)(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\Re(p),\Re(q)\geq0,\;|z|\leq1\right).$$
In particular case when $p=q,$ we define the $p-$Mathieu type power series defined by:
\begin{equation}\label{**}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n B_{p}(\tau+n,\omega-\tau)z^n}{n!B(\tau,\omega-\tau)(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\Re(p)\geq0,\;|z|\leq1\right).$$
The function $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ has many other special cases. We set $p=q=0$ we get
\begin{equation}\label{**}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};z)=S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};0,0;z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n (\tau)_nz^n}{n!(\omega)_n(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu,\tau,\omega>0,\;|z|\leq1\right).$$
On the other hand, by letting $\tau=\omega$ in (\ref{**}) we obtain \cite[Eq. 5, p. 974]{ZK}:
\begin{equation}
S_{\mu,\nu}^{(\alpha,\beta)}(r;\textbf{a};z)=S_{\mu,\nu,\tau,\tau}^{(\alpha,\beta)}(r;\textbf{a};z)=\sum_{n=1}^\infty \frac{2a_n^\beta(\nu)_n z^n}{n!(a_n^\alpha+r^2)^\mu},
\end{equation}
$$\left(r,\alpha,\beta,\nu>0,\;|z|\leq1\right).$$
Furthermore, the special cases when $\nu=z=1$ we get the generalized Mathieu series (\ref{;,}).
The contents of our paper is organized as follows. In section 2, we present new integral representation for the $(p,q)-$Mathieu series. In particular, we derive the Mellin-Barnes type integral representations
for $(p,q)-$Mathieu series $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).$ As applications, In Section 3, we introduce the $(p,q)-$Mittag-Leffler functions and we derive some relationships between thus two special functions, in particular we derive new series representations for the $(p,q)-$Mathieu series. Relationships between the $(p,q)-$ and generalized Mathieu series are proved and two Tur\'an type inequalities are established.
\section{\bf Integral representation for the $(p,q)-$Mathieu types series}
In the course of our investigation, one of the main tools is the following result providing the integral representation for the $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z).$
\begin{theorem}\label{T1}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0$ such that $\gamma(\mu\alpha-\beta)>0.$ Then $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z)$ possesses the integral representation given by:
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p,q;z)=$$
\begin{equation}\label{a}
=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{theorem}
\begin{proof}By using the definition (\ref{*}), we can write the extended Mathieu types series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ in the following form:
\begin{equation}\label{mmm}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=1}^\infty\frac{(\nu)_n}{a_n^{(\mu+m)\alpha-\beta}}\frac{B_{p,q}(\tau+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{n!}.
\end{equation}
Therefore,
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{k^\gamma\}_{k=0}^\infty;p,q;z\Big)=$$
\begin{equation*}
\begin{split}
\;\;\;\;&=2z\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu)_{n+1}}{(n+1)!}\frac{B_{p,q}(\tau+1+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{(n+1)^{\gamma((\mu+m)\alpha-\beta})}\\
\;\;\;\;&=2\nu z\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu+1)_{n}}{n!}\frac{B_{p,q}(\tau+1+n,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^n}{(n+1)^{\gamma((\mu+m)\alpha-\beta)+1}}\\
\;\;\;\;&=\frac{2\nu zB(\tau+1,\omega-\tau)}{B(\tau,\omega-\tau)} \sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m\sum_{n=0}^\infty\frac{(\nu+1)_{n}B_{p,q}(\tau+1+n,\omega-\tau)z^n}{n!B(\tau+1,\omega-\tau)(n+1)^{\gamma((\mu+m)\alpha-\beta)+1}}\\
\;\;\;\;&=\frac{2\nu z B(\tau+1,\omega-\tau)}{B(\tau,\omega-\tau)} \sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \Phi_{\nu+1,\tau+1,\omega+1}(z,\gamma[(\mu+m)\alpha-\beta]+1,1;p,q)\\
\;\;\;\;\;\;\;\;\;\;\;\;&=\frac{2\nu \tau z}{\omega}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\left[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\right]\left(\sum_{m=0}^\infty\frac{\binom{\mu+m-1}{m}(-r^2t^{\gamma\alpha})^m }{\Gamma(\gamma[(\mu+m)\alpha-\beta]+1)}\right)dt\\
\;\;\;\;\;\;\;\;&=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p,q\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{split}
\end{equation*}
This completes the proof of Theorem \ref{T1}.
\end{proof}
Now, in ths case $p=q$, Theorem \ref{T1} reduces to the following corollary.
\begin{coro}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;\Re(p)\geq0$ such that $\gamma(\mu\alpha-\beta)>0.$ Then $(p,q)-$Mathieu types power series $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p;z)$ possesses the integral representation given by:
\begin{equation}\label{a}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;p;z)=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t};p\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{coro}
\begin{remark}
1. By letting $p=q=0$ in (\ref{a}), we deduce that the function $S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)$ possesses the following integral representation:
\begin{equation}\label{b}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)=\frac{2\nu \tau z}{\omega\Gamma(\mu)}\int_0^\infty t^{\gamma[(\mu\alpha-\beta]}e^{-t}{}_2F_1\Big[^{\;\nu+1,\tau+1}_{\;\omega+1};ze^{-t}\Big]{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
2. Setting $\tau=\omega$ in (\ref{b}) and using the fact that
$${}_2F_1\Big[^{\;a,\;b}_{\;\;b};z\Big]=(1-z)^{-a}$$
we obtain the following integral representation for the function $S_{\mu,\nu}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)$\cite[Theorem 1, Eq. 8]{ZK}
\begin{equation}
S_{\mu,\nu}^{(\alpha,\beta)}(r;\{k^\gamma\}_{k=0}^\infty;z)=\frac{2\nu z}{\Gamma(\mu)}\int_0^\infty \frac{t^{\gamma[(\mu\alpha-\beta]}e^{-t}}{(1-ze^{-t})^{\nu+1}}{}_1\Psi_1\Big[^{\;\;\;\;(\mu,1)}_{(\gamma(\mu\alpha-\beta)+1,\gamma\alpha)}\Big|-r^2t^{\gamma\alpha}\Big]dt.
\end{equation}
\end{remark}
In the next Theorem we present the Mellin-Barnes integral representation for the alternating Mathieu-series $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).$
\begin{theorem}\label{TTT222222222222}Let $r,\nu,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0.$ Then the following integral representation
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big)=$$
\begin{equation}
=-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p,q}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds,
\end{equation}
holds true for all $|\arg(-z)|<\pi$.
\end{theorem}
\begin{proof}The contour of integration extends from $c-i\infty$ to $c+i\infty,$ such that all the poles of the Gamma function $\Gamma(\nu-s+1)$ at the points $s=k+\nu+1,\;k\in\mathbb{N}$ are separated from the poles of the gamma function $\Gamma(s)$ at the points $s=-k,\; k\in\mathbb{N}.$ Suppose that the the poles of the integrand are simple and using the fact that
$$ \textrm{res}[\Gamma,-k] =\lim_{s\longrightarrow -k}(s+k)\Gamma(s)=\frac{(-1)^k}{k!},$$
we find that
$$\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p,q}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds$$
\begin{equation*}
\begin{split}
&=\frac{2z}{\Gamma(\nu)}\sum_{k=0}^\infty \lim_{s\longrightarrow -k}\frac{(s+k)\Gamma(s)B_{p,q}(\tau-s+1,\omega-\tau)\Gamma(\nu-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}\\
&=\frac{2z}{\Gamma(\nu)}\sum_{k=0}^\infty \frac{(-1)^k}{k!}\frac{B_{p,q}(\tau+k+1,\omega-\tau)\Gamma(\nu+k+1)}{B(\tau,\omega-\tau)((k+1)^2+r^2)^\mu}z^k\\
&=-2\sum_{k=1}^\infty\frac{(\nu)_k k B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)(k^2+r^2)^\mu}\frac{(-z)^k}{k!}\\
&=-S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;-z\Big).
\end{split}
\end{equation*}
This completes the proof of Theorem \ref{TTT222222222222}
\end{proof}
\begin{coro}\label{cccc}
Let $r,\nu,\mu,\tau,\omega>0,\;\Re(p)\geq0.$ Then the following integral representation
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p;-z\Big)=$$
\begin{equation}
=-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)B_{p}(\tau-s+1,\omega-\tau)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{B(\tau,\omega-\tau)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds,
\end{equation}
holds true for all $|\arg(-z)|<\pi$.
\end{coro}
\begin{remark}If we set $p=0$ in Corollary \ref{cccc}, then we get the Mellin-Barnes representation of the function $S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big):$
$$S_{\mu,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big)=$$
\begin{equation}
=-\frac{z\Gamma(\omega)}{i\pi\Gamma(\nu)\Gamma(\tau)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)\Gamma(\tau-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{\Gamma(\omega-s+1)\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds
\end{equation}
In particular, for $\tau=\omega$ we get
$$S_{\mu,\nu}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;-z\Big)=
-\frac{z}{i\pi\Gamma(\nu)}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(s)\Gamma(\nu-s+1)\left[\Gamma(-s+ir+1)\Gamma(-s-ir+1)\right]^\mu}{\left[\Gamma(-s+ir+2)\Gamma(-s-ir+2)\right]^\mu}z^{-s}ds.$$
Moreover, if we set $\mu=2$ and $\nu=1$ in the above equation we get the Mellin-Barnes for the alternating Mathieu-series proved by Saxena el al. \cite[Theorem 3.1]{SA}.
\end{remark}
\section{\bf Applications}
In our first application in this section we present the relationships between the $(p,q)-$Mathieu-type series $S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)$ and the Rieman-Liouvile operator.
\subsection{Relationships with $(p,q)-$Mathieu-type series and the Rieman-Liouvile operator}
Our first main application is asserted by the following Theorem.
\begin{theorem} Let $r,\mu,\tau,\omega>0,\;\Re(p),\Re(q)\geq0$ and $0\leq\nu<1$. Then
\begin{equation}
S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)=\frac{1}{2ir\Gamma(\nu)}\left\{D_z^{\nu-1}\left(z^{\nu-1}\Phi_{\tau,\omega}(z,2,-ir;p,q)\right)-D_z^{\nu-1}\left(z^{\nu-1}\Phi_{\tau,\omega}(z,2,ir;p,q)\right)\right\}.
\end{equation}
\end{theorem}
\begin{proof}By using the definition of the $(p,q)-$Mathieu-type series, we can write the Mathieu-type series\\ $S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)$ in the following form:
\begin{equation}
S_{2,\nu,\tau,\omega}^{(2,1)}\Big(r;\{k\}_{k=0}^\infty;p,q;z\Big)=\frac{1}{2ir}\left[\Phi_{\nu,\tau,\omega}(z,2,-ir;p,q)-\Phi_{\nu,\tau,\omega}(z,2,ir;p,q)\right].
\end{equation}
Combining the above equation with (\ref{ZZ}), we get the desired result.
\end{proof}
\subsection{Relationships with $(p,q)-$Mittag-Leffler function and $(p,q)-$Mathieu-type series} In this section, we introduce the definition of the $(p,q)-$Mittag-Leffler function and we establish an integral representation for this function and we present some relationships with the $(p,q)-$Mathieu-type series.
For $\lambda,\tau,\omega,\theta,\sigma, \delta>0$ and $\Re(p),\Re(q)\geq0$ we define the $(p,q)-$Mittag-Leffler function by
\begin{equation}\label{def1}
E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}, z\in\mathbb{C}.
\end{equation}
In the case $p=q$ we define the $p-$Mittag-Leffler function by
\begin{equation}
E_{\delta,\theta,\sigma;p}^{(\lambda,\tau,\omega)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}, z\in\mathbb{C},
\end{equation}
whose special case when $p=0$ reduces to the generalized Mittag-Leffler function, introduced by Tomovski and Mehrez in \cite{ZK}
\begin{equation}
E_{\delta,\theta,\sigma}^{(\lambda)}(z)=\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{z^k}{k!}, z\in\mathbb{C}.
\end{equation}
For $\lambda=1$ the above series was introduced by S. Gerhold \cite{GG}.
\begin{lemma}\label{LLLL}For $\tau,\omega,\theta,\sigma,\delta>0$ and $\Re(p),\Re(q)\geq0.$ Then we have
\begin{equation}\label{55555}
E_{\delta,\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z)=\frac{\omega}{z\tau}\left[E_{\delta,\theta,\sigma;p,q}^{(1,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\delta}B(\tau,\omega-\tau)}\right].
\end{equation}
\end{lemma}
\begin{proof}By computation, we get
\begin{equation*}
\begin{split}
E_{\delta,\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z)&=\sum_{k=0}^\infty \frac{B_{p,q}(\tau+k+1,\omega-\tau)z^k}{B(\tau+1,\omega-\tau)\left[\Gamma(\theta k+\sigma+\theta)\right]^\delta}\\
&=\frac{B(\tau,\omega-\tau)}{zB(\tau+1,\omega-\tau)}\sum_{k=1}^\infty \frac{B_{p,q}(\tau+k,\omega-\tau)z^k}{B(\tau,\omega-\tau)\left[\Gamma(\theta k+\sigma)\right]^\delta}\\
&=\frac{\omega}{z\tau}\left[E_{\delta,\theta,\sigma;p,q}^{(1,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\delta}B(\tau,\omega-\tau)}\right].
\end{split}
\end{equation*}
The proof of Lemma \ref{LLLL} is completes.
\end{proof}
\begin{theorem}\label{8888}Let $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$Then the $(p,q)-$Mathieu-type series admits the following series representation:
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)$$
\begin{equation}\label{--}
\begin{split}
\qquad\qquad\qquad\qquad=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma;p,q}^{(\nu,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\gamma[(\mu+m)\alpha-\beta]}B(\tau,\omega-\tau)}\right].
\end{split}
\end{equation}
Moreover, the following series representation
\begin{equation}\label{;;}
S_{\mu,1 ,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=\frac{2z\tau}{\omega}\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma+\theta;p,q}^{(1,\tau+1,\omega+1)}(z).
\end{equation}
holds true.
\end{theorem}
\begin{proof} In view of the definition of the $(p,q)-$Mittag-Leffler function (\ref{def1}) and the equation (\ref{mmm}) we obtain (\ref{--}). Finally, combining the equation (\ref{--}) with the following relation (\ref{55555}) we obtain the formula (\ref{;;}).
\end{proof}
Taking in (\ref{--}) the values $\theta=\sigma=1$ we obtain the following representation:
\begin{coro}Let $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$Then the $(p,q)-$Mathieu-type series admits the following series representations:
\begin{equation}
\begin{split}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{(k!)^\gamma\}_{k=0}^\infty;p,q;z\Big)=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[E_{\gamma[(\mu+m)\alpha-\beta],1,1;p,q}^{(\nu,\tau,\omega)}(z)-\frac{B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}\right],
\end{split}
\end{equation}
and
\begin{equation}
S_{\mu,1 ,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{k!^\gamma\}_{k=0}^\infty;p,q;z\Big)=\frac{2z\tau}{\omega}\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],1,2;p,q}^{(1,\tau+1,\omega+1)}(z).
\end{equation}
\end{coro}
\begin{lemma}\label{L1}For $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$ Then the the $(p,q)-$Mittag-Leffler function $E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)$ possesses the following integral representation:
\begin{equation}\label{int1}
E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z)=\frac{1}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\delta,\theta,\sigma}^{(\lambda)}(zt)dt,
\end{equation}
holds true.
\end{lemma}
\begin{proof}By using the definition of the $(p,q)-$Beta function we get
\begin{equation*}
\begin{split}
\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\delta,\theta,\sigma}^{(\lambda)}(zt)dt&=\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\left(\sum_{k=0}^\infty\frac{(\lambda)_k(zt)^k}{\left[\Gamma(\theta k+\sigma)\right]^\delta k!}\right)dt\\
&=\sum_{k=0}^\infty\frac{(\lambda)_k z^k}{\left[\Gamma(\theta k+\sigma)\right]^\delta k!}\int_0^1 t^{\tau+k-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)dt\\
&=B(\tau,\omega-\tau)\sum_{k=0}^\infty \frac{(\lambda)_k}{\left[\Gamma(\theta k+\sigma)\right]^\delta}\frac{B_{p,q}(\tau+k,\omega-\tau)}{B(\tau,\omega-\tau)}\frac{z^k}{k!}\\
&=B(\tau,\omega-\tau)E_{\delta,\theta,\sigma;p,q}^{(\lambda,\tau,\omega)}(z).
\end{split}
\end{equation*}
The proof of Lemma \ref{L1} is completes.
\end{proof}
\begin{theorem}\label{T5}For $\lambda,\tau,\omega,\theta,\sigma>0, \delta\in\mathbb{N}$ and $\Re(p),\Re(q)\geq0.$ Then the following integral representation
\begin{equation}\label{HHHHH}
\begin{split}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,\nu}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(\tau^2+[\Gamma(\sigma)]^\alpha)^\mu}
\end{split}
\end{equation}
holds true for all $|z|<1.$ Moreover, the following integral representation
\begin{equation}\label{HHHHHH}
\begin{split}
S_{\mu,1,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=
\end{split}
\end{equation}
$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{2z\tau}{\omega B(\tau+1,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,1}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt,$$
holds true for all $|z|<1.$
\end{theorem}
\begin{proof}By means of Lemma \ref{L1} and the integral representation (\ref{--}), we get
$$S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;p,q;z\Big)=$$
\begin{equation*}
\begin{split}
&=2\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m \left[\frac{1}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma}^{(\nu)}(zt)dt\right]\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\mu\alpha-\beta}B(\tau,\omega-\tau)}\sum_{m=0}^\infty\binom{\mu+m-1}{m} \left(\frac{-\tau^2}{[\Gamma(\sigma)]^\alpha}\right)^m\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\left(\sum_{m=0}^\infty\binom{\mu+m-1}{m}(-r^2)^m E_{\gamma[(\mu+m)\alpha-\beta],\theta,\sigma}^{(\nu)}(zt)\right)dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{[\Gamma(\sigma)]^{\mu\alpha-\beta}B(\tau,\omega-\tau)}.\frac{1}{(1+\frac{r^2}{[\Gamma(\sigma)]^\alpha})^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k}{[\Gamma(\theta k+\sigma)]^{\gamma(\mu\alpha-\beta)}}\left(\sum_{m=0}^\infty\frac{\binom{\mu+m-1}{m}(-r^2)^m}{[\Gamma(\theta k+\sigma)]^{\gamma m\alpha}}\right)\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k}{[\Gamma(\theta k+\sigma)]^{\gamma(\mu\alpha-\beta)}}\left(1+\frac{r^2}{[\Gamma(\theta k+\sigma)]^{\gamma\alpha}}\right)^{-\mu}\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)\sum_{k=0}^\infty\frac{(\nu)_k[\Gamma(\theta k+\sigma)]^{\gamma\beta}}{\left([\Gamma(\theta k+\sigma)]^{\gamma\alpha}+r^2\right)^\mu}\frac{(zt)^k}{k!}dt\\
&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(\tau^2+[\Gamma(\sigma)]^\alpha)^\mu}\\
&=\frac{2}{B(\tau,\omega-\tau)}\int_0^1 t^{\tau-1}(1-t)^{\omega-\tau-1}E_{p,q}(t)S_{\mu,\nu}^{(\alpha,\beta)}\Big(r;\{[\Gamma(\theta k+\sigma)]^\gamma\}_{k=0}^\infty;zt\Big)dt\\&-\frac{2B_{p,q}(\tau,\omega-\tau)}{B(\tau,\omega-\tau)}.\frac{[\Gamma(\sigma)]^\beta}{(r^2+[\Gamma(\sigma)]^\alpha)^\mu},
\end{split}
\end{equation*}
which evidently completes the proof of the representation (\ref{HHHHH}). Finally, combining (\ref{int1}) and (\ref{;;}) and repeating the same calculations as above we get (\ref{HHHHHH}). The proof of Theorem \ref{T5} is completes.
\end{proof}
\subsection{Tur\'an type inequalities for the $(p,q)-$Mathieu-type series }
\begin{theorem}Let $r,\alpha,\beta,\nu,\mu,\tau,\omega>0,\;p,q\geq0.$ Then the following assertions are true:
1. The $(p,q)-$Mathieu-type series considered as a function in $p$ $( $or $q)$ is completely monotonic and log-convex on $(0,\infty).$ Furthermore, the following Tur\'an type inequality
\begin{equation}\label{TURAN}
\left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p+1,q;z)\right]^2\leq S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p+2,q;z)
\end{equation}
holds true for all $z\in (0,1).$\\
2. Assume that $r^2+\textbf{a}\geq1.$ Then the $(p,q)-$Mathieu-type series considered as a function in $\mu$ is completely monotonic and log-convex on $(0,\infty).$ Furthermore, the following Tur\'an type inequality
\begin{equation}\label{TURAN1}
\left[S_{\mu+1,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)\right]^2\leq S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)S_{\mu+2,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)
\end{equation}
holds true for all $z\in (0,1)$ such that $r^2+\textbf{a}\geq1.$
\end{theorem}
\begin{proof}1. In \cite[Corollary 2.7]{LU1}, the authors proved that the extended beta function $p(\;\textrm{or}\;q)\mapsto B_{p,q} (x, y)$ is completely monotonic function on $(0,\infty)$ and using the fact that sums of completely monotonic functions are completely monotonic too, we deduce that the $p (\;\textrm {or}\;q)\mapsto S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ is completely monotonic and log-convex on $(0,\infty),$ since every completely monotonic function is log-convex ( see \cite[p.167]{WI}. Thus, for all $p_1,p_2>0,$ and $t\in[0,1],$ we obtain
\begin{equation*}
S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};tp_1+(1-t)p_2,q;z)\leq\left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p_1,q;z)\right]^t \left[S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p_2,q;z)\right]^{1-t}.
\end{equation*}
Letting $t=\frac{1}{2},\;p_1=p$ and $p_2=p+2$ in the above inequality we get the Tur\'an type inequality (\ref{TURAN}).\\
2. We note that the function $\mu\mapsto (r^2+\textbf{a})^{-\mu}$ is completely monotonic on $(0,\infty)$ such that $r^2+\textbf{a}\geq1,$ and consequently the function $\mu\mapsto S_{\mu,\nu,\tau,\omega}^{(\alpha,\beta)}(r;\textbf{a};p,q;z)$ is completely monotonic and log-convex on $(0,\infty).$
\end{proof}
\begin{remark}The condition $r^2+\textbf{a}$ is not necessary for proved the Tur\'an type inequality (\ref{TURAN1}), a similar proof of the Theorem 2 in \cite{SKZ}, we obtain (\ref{TURAN1}).
\end{remark}
\end{document} |
\begin{document}
\title{Filtering problems with exponential criteria for general Gaussian signals}
\begin{abstract} The explicit solution of the discrete time filtering problems with exponential criteria for a general
Gaussian signal is obtained through an approach based on a conditional Cameron-Martin type formula.
This key formula
is derived for conditional expectations
of exponentials of some quadratic forms of Gaussian sequences. The formula
involves conditional expectations
and conditional covariances in some auxiliary optimal risk-neutral filtering problem
which is used in the proof. Closed form recursions of Volterra
type for these ingredients are provided. Particular cases for which the results can be further
elaborated are investigated.
\end{abstract}
\vspace*{4mm}
\paragraph{Key words.}{Gaussian process, optimal
filtering, filtering error,
Riccati-Volterra equation, risk-sensitive filtering, exponential criteria }
\vspace*{0.25cm}
\paragraph{AMS subject classifications.} Primary 60G15. Secondary 60G44, 62M20.
\section{Introduction}
The linear exponential Gaussian (LEG for short) filtering problem,
\textit{i.e.}, with an exponential cost criteria (see the definition
(\ref{LEGdef}) below), and the so called risk-sensitive (RS for short) filtering
problem (see \cite{collings} and the statement (\ref{rde})
below) have been given a great deal of interest over the last
decades. Numerous results have been already reported in specific
models, specially around Markov models, but, as far as we know,
without exhibiting the relationship between these two problems.
See, \textit{e.g.}, Whittle \cite{whittle4}-\cite{whittle3}, Speyer \textit{et
al.} \cite{speyer}, Elliott \textit{et al}. \cite{elliott4}, \cite{dey1},
\cite{elliott1} and Bensoussan and van Schuppen
\cite{bensoussan} for contributions on this subject and related LEG and RS control problems. Therein
the notion of ``information state" has been introduced
without any clear
probabilistic meaning for auxiliary processes which are involved,
even in the Gauss-Markov case.
Moreover, the method proposed in \cite{elliott4} does not work in
a non Markovian situation.
In our paper \cite {AMM}, we have solved the LEG and RS filtering problems for general
Gaussian signal processes in continuous time and in the particular setting where the functional in the exponential is a \textit{singular} quadratic functional. Moreover we have proved that actually in this case the solutions coincide.
In our paper \cite{AMM2} we have solved the LEG and RS filtering problems for Gauss-Markov processes but with a \textit{nonsingular} quadratic functional in the exponential.
In this setting we have proposed an example to show that the solutions may be different.
On the other hand,
the general solution for the
optimal risk-neutral linear filtering problem and
a Cameron-Martin type formula for general Gaussian sequences have been obtained in \cite{mkalbmcr3}. It seems natural
to use the approach proposed in \cite{mkalbmcr3} and \cite{AMM} to derive the solution of the
LEG and RS filtering problems for general Gaussian signals in discrete time setting, to precise their link and also to give a
probabilistic interpretation for the ingredients of the ``information state".
In the present paper we are interested in the explicit solution of the Linear Exponential Gaussian (LEG) and Risk Sensitive (RS) filtering problems for general Gaussian signals. Namely we deal with a signal-observation model
$
(X_t,Y_t)_{t\ge 1},
$
where the signal $X=(X_t)_{t\ge 1}$ is an arbitrary Gaussian sequence with mean
$m=(m_t, t\geq 1)$ and covariance $K =(K(t,s), t\geq 1,
s\geq 1)$, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg X_t=m_t,\quad\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)(X_s-m_s)=K(t,s)\,, \quad
t\geq 1\,,\; s\geq 1\,,
$$
and, for some sequence $A=(A_t,\, t\ge 1)$ of the real numbers, the observation process $Y=(Y_t,\, t\ge 1)$ is given by
\begin{equation}\label{observ}
Y_t= A_t X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t,
\end{equation}
where $\mathop{\mathrm{var}}\nolimitsepsilon=(\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d. $\mathcal{N}(0,1)$ random variables and $\mathop{\mathrm{var}}\nolimitsepsilon$ and $X$ are independent.
Suppose that only $Y$ is observed and for a given real number $\mu$ and a fixed sequence $(Q_{t})_{t\ge 1}$ of nonnegative real numbers, one wishes
to minimize with respect to $h:\,h_t\in {\cal Y}_{t}, t\ge 1$ the quantity:
\begin{equation}\label{LEGcrit}
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\},
\end{equation}
where $({{\cal Y}}_t)$ is the natural filtration of $Y$, \textit{i.e.},
${{\cal Y}}_t=\sigma(\{Y_u\, ,\, 1\leq u\leq t\})$ and $h_t\in{{\cal Y}}_t$ means that $h_t$ is ${{\cal Y}}_t$-measurable.
Note
that, according to the sign of the real parameter $\mu$, there are two different
cases for this linear exponential Gaussian (LEG)
filtering problem (the terminology is taken from the linear exponential Gaussian optimal control problem) :
\begin{itemize}
\item$\mu < 0 ,$ called \textit{risk-preferring} filtering problem,
\item $\mu > 0, $ called the \textit{risk-averse} filtering problem.
\end{itemize}
It is well known (see, \textit{e.g.}, \cite{speyer} for the
Markov case) that the solution to this problem is
not the conditional expectation of $X_t$ given the $\sigma$-field
${\cal Y}_t$. Our first aim is to show that the solution
can be completely explicited~: the characteristics of the optimal
solution are obtained as the solution of a closed form system of
Volterra type equations which actually reduce to the
equations known also for the RS setting when the signal process $X$ is
Gauss-Markov (see, \textit{e.g.}, \cite{elliott2}). Our second aim is to give
the probabilistic interpretation of this optimal solution in terms of an
auxiliary \textit{risk-neutral } filtering problem. Actually, we extend the
filtering approach initiated in \cite{mkalbmcr3} and \cite{AMM} for
one-dimensional processes, to obtain a conditional Cameron-Martin
type formula for the {\em conditional Laplace transform} of a
quadratic functional of the involved process. Namely, we give an
explicit representation for the random variable
\begin{equation}\label{def:It}
\mathcal{I}_T= \mathop{\mathbb{E}}\nolimits \left(\left.\exp\left\{\frac{\mu}{2} \sum_{s=1}^T (X_s-h_s)^2 Q_s \right\} \right/ {{\cal Y}}_{T}\right),
\end{equation}
where $h_s\in {\cal Y}_{s}, \, s\ge 1$.
The paper is organized as follows. In Section \ref{LEG} we derive
the solution of the LEG filtering problem : explicit recursive
equations, involving the covariance function of
the filtered process, are obtained. In particular, in Section \ref{CMC}, an appropriate auxiliary risk-neutral filtering problem is
matched to that of deriving the key
Cameron-Martin type formula. The solution of this auxiliary filtering problem is discussed
in Section \ref{AFP}. In Section \ref{PC} we investigate some specific cases
where the results can be further elaborated. In Section \ref{disc} we discuss the relationship between LEG and
RS filtering problems. Section \ref{interpret}
is devoted to the interpretation for the ingredients of the ``information state''.
Finally, Sections \ref{complements} and \ref{againpart} are devoted to a more general case, namely when the particular structure of the observation sequences $(Y_t)_{t\ge 1}$ is not specified.
\section{Solution of the LEG filtering problem}\label{LEG}
Let us introduce the following condition $(C_{\mu})$:
\begin{enumerate}
\item[$(C_{\mu})$]
the equation
\begin{equation}\label{GAMMABAR}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} \overline{\gamma}(t,l)\overline{\gamma}(s,l) \, \frac{S_l}{1+S_l \overline{\gamma}_l},\quad S_l=A_l^2-\mu Q_l
\end{equation}
has a unique and bounded solution on $\{(t,s):1\le s \le t \le T\}$, such that $\overline{\gamma}_l=\overline{\gamma}(l,l) \ge 0,\, l\ge 1$ and moreover
$$ \displaystyle{1 +
S_{l}\overline{\gamma}_l}> 0,\, l\ge 1.
$$
\end{enumerate}
\begin{rem}
Notice that for all $\mu$ \textbf{negative} the condition $(C_{\mu})$ is satisfied and if $\mu $ is \textbf{positive}, the condition $(C_{\mu})$ is satisfied for $\mu$ sufficiently small, for example, those such that for any
$ t \le T \, A_{t}^{2}-\mu Q_{t}$ is nonnegative.
\end{rem}
The first result is the following
\begin{theorem}\label{LEGsol}
Suppose that the condition $(C_{\mu})$ is satisfied. Let $(\overline{h}_t)_{t\ge 1}$ be the solution of the following equation:
\begin{equation}\label{hbar}
\overline{h}_t=m_t + \sum_{l=1}^t A_l\overline{\gamma}(t,l) (Y_l-A_l \overline{h}_l),
\end{equation}
where $\overline{\gamma}=(\overline{\gamma}(t,s), 1\le s \le t \le T)$ is the unique solution of equation \eqref{GAMMABAR}.\\
Then $(\overline{h}_t)_{t\ge 1}$ is the solution of the LEG filtering problem,
\textit{i.e.},
\begin{equation}\label{LEGdef}
\overline{h}=\mathop{\mathrm{argmin}}_{h:\,h_t\in {\cal Y}_{t}, t\ge 1} \mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}.
\end{equation}
Moreover, the corresponding optimal risk is given by
$$
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-\overline{h}_t)^2 Q_t \right\}
=\mu \prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2}.
$$
\end{theorem}
Theorem \ref{LEGsol} is a direct consequence of
results of Section \ref{CMC}. Its proof will be given at the end of
Section \ref{CMC}.
\begin{rem}
\begin{itemize}
\item Note that equation \eqref{hbar} is really recursive equation and it can be rewritten in the equivalent form:
$$
\overline{h}_t=\frac{1}{1+A_t^2\overline{\gamma}_{t}}\left[m_t + \sum_{l=1}^{t-1} A_l\overline{\gamma}(t,l) (Y_l-A_l \overline{h}_l)+ A_t \overline{\gamma}_{t}Y_{t}\right],
$$
\item It is worth emphasizing that taking $\mu=0$ in equation
(\ref{GAMMABAR}), one gets through equation (\ref{hbar}) the solution $\bar{h}$ of the
\textit{risk-neutral} filtering problem of the signal $X$ given
the observation $ Y$, \textit{i.e.}, $\bar{h}_{t}= \mathop{\mathbb{E}}\nolimitsg ( X_{t} / {\cal{ Y }}_{t})$ (see, \textit{e.g.}, \cite{mkalbmcr3}).
\end{itemize}
\end{rem}
\subsection{Conditional version of a Cameron-Martin formula}\label{CMC}
The proof of Theorem~\ref{LEGsol} is based on the conditional version of the Cameron--Martin formula which provides
the conditional expectation $\mathcal{I}_t$ defined by \eqref{def:It}. Let
\begin{equation}\label{def:Jt}
J_t=\exp\left\{ -\frac{1}{2}\sum\limits_{s=1}^t (X_s- h_s)^2 Q_s \right\}.
\end{equation}
Then $\mathcal{I}_t=\pi_t (J_t)$,
where for any random variable $\eta$ such that
$\mathop{\mathbb{E}}\nolimitsg |\eta|<+\infty$, the notation
$\pi_t(\eta)$ is used for the conditional expectation of $\eta$ given the
$\sigma$-field
${\cal
Y}_t=\sigma(\{Y_s\, ,\, 1\leq s\leq t\}),$
$$
\pi_t(\eta) = \mathop{\mathbb{E}}\nolimitsg(\eta/{\cal Y}_t)\,.
$$
\begin{proposition}\label{p:CM}
Suppose that the condition $(C_{\mu})$ is satisfied. Let $(\overline{\gamma}(t,s),\, 1\le s \le t \le T)$ be the solution of equation \eqref{GAMMABAR} and $(Z_t^h,\,t\ge 1)$ be the solution of the following equation
\begin{equation}\label{eqzh}
Z_t^h = m_t -\sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{\mu Q_l}{1+S_l \overline{\gamma}_l} (h_l - Z_{l}^h) + \sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{A_l}{1+S_l \overline{\gamma}_l} (Y_l - A_l Z_{l}^{h}).
\end{equation}
Then the following representation of the random variable $\mathcal{I}_T$ defined by \eqref{def:It} holds for any $T\ge 1$:
$$
\mathcal{I}_T=\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_t - \frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t } \right]^{2}\right\} \times \mathcal{M}_T,
$$
where $(\mathcal{M}_T)_{T\ge 1}$ is a martingale defined by :
\begin{multline}\label{martin}
\mathcal{M}_T = \prod_{t=1}^T \left[\frac{(1+A_t^2 \gamma_t)}{1+A_t^{2} \overline{\gamma}_t}\right]^{1/2} \exp\left\{ \frac{A_t}{1+A_t^2 \overline{\gamma}_t} \, (Z_{t}^h - \pi_{t-1}(X_t) ) \nu_t - \right.
\\
\left. -\frac{1}{2} \cdot \frac{A_t^2}{1+A_t^2 \overline{\gamma}_t} \, (Z_{t}^h - \pi_{t-1}(X_t))^2
- \frac{1}{2} \cdot \frac{A_t^2 (\gamma_t - \overline{\gamma}_t)\cdot \nu_t^2}{(1+A_t^2 \overline{\gamma}_t) (1+A_t^2 \gamma_t)}
\right\},
\end{multline}
in terms of the innovation sequence $(\nu_t)_{t \ge 1}$:
$$
\nu_t=Y_t - A_{t}\pi_{t-1}(X_t);\quad \pi_{t-1}(X_t)=\mathop{\mathbb{E}}\nolimits (X_t / {\cal Y}_{t-1}),
$$
and of the variances of one-step prediction errors $(\gamma_t)_{t \ge 1}$:
$$
\quad \gamma_t = \mathop{\mathbb{E}}\nolimits (X_t-\pi_{t-1}(X_t))^2.
$$
\end{proposition}
\begin{rem}
\begin{enumerate}
\item
The probabilistic interpretation of the auxiliary processes $(Z_t^h)$ and $(\overline{\gamma}_t)_{t\ge 1}$ appearing in the Proposition~\ref{p:CM} will be clarified below.
\item Proposition~\ref{p:CM} reduces to the ordinary Cameron-Martin type formula (\textit{cf.} Theorem~1 \cite{mkalbmcr3}) for $h\equiv 0$ when $A_t=0,\,l\ge 1$ and hence $X$ and $Y$ are independent.
\end{enumerate}
\end{rem}
\paragraph{Proof of Proposition~\ref{p:CM}}
We will prove Proposition~\ref{p:CM} for $\mu<0$, namely $\mu=-1$. Then we can replace $Q$ by $-\mu Q$ and the statement of Proposition~\ref{p:CM} is still valid because of the analytical properties of the involved functions.
The proof of Proposition~\ref{p:CM} for $\mu=-1$ will be separated into two steps.
\textbf{I.} (Actually it is the discrete time analog for the general filtering theorem.) Since $h_t\in {\cal Y}_{t},\, t\ge 1,$ in the proof we can suppose that $h$ is a deterministic function.
First of all, we claim that for $J_t$, defined by \eqref{def:Jt}
\begin{equation}\label{pitJ}
\pi_t (J_t) = \left. \frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(\beta_t^y)} \right|_{y=Y_t},
\end{equation}
where $\beta_t^y = \exp (A_t X_t y - \frac{1}{2} A_t X_t^2)$.
Indeed, let us introduce the new probability measure $\hat{\Pg}$, defined by
$$
\frac{d\hat{\Pg}}{d\Pg} = \exp (-A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2).
$$
The classical Bayes formula gives that
$$
\pi_t(J_t) = \frac{\hat{\pi}_t (J_t \exp (A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t + \frac{1}{2} A_t^2 X_t^2))}{\hat{\pi}_t( \exp (A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t + \frac{1}{2} A_t^2 X_t^2))} =
\frac{\hat{\pi}_t (J_t \exp (A_t X_t Y_t - \frac{1}{2} A_t^2 X_t^2))}{\hat{\pi}_t (\exp (A_t X_t Y_t - \frac{1}{2} A_t^2 X_t^2))},
$$
where $\hat{\pi}_t(\cdot)$ denotes a conditional expectation with respect to ${\cal Y}_{t}$ under $\hat{\Pg}$.
Note that under $\hat{\Pg}$ the distribution of $(X_s, Y_r)_{s\le t, \, r\le t-1}$ is the same as under $\Pg$ and
$Y_t$ is a ${\cal N}(0,1)$ random variable independent of $(X_s, Y_r)_{s\le t, \, r\le t-1}$ .
To understand this point it is sufficient to write the following equality for the mutual characteristic function with arbitrary real numbers $(\alpha_{j}, \lambda_{j})$:
\begin{multline*}
\hat{\mathop{\mathbb{E}}\nolimits} \exp \left\{i\sum_{j=1}^t \alpha_j X_j + i\sum_{j=1}^t \lambda_j Y_j \right\} =
\\
=\mathop{\mathbb{E}}\nolimits \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} Y_t - A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2 \right\} = \\
= \mathop{\mathbb{E}}\nolimits \left( \mathop{\mathbb{E}}\nolimits \left. \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} Y_t - A_t X_t \mathop{\mathrm{var}}\nolimitsepsilon_t - \frac{1}{2} A_t^2 X_t^2 \right\} \right/ {\cal X }_{t}\right) =
\\
=\mathop{\mathbb{E}}\nolimits \exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j + i\lambda_{t} A_t X_t - \frac{1}{2} A_t^2 X_t^2 + \frac{1}{2} (i\lambda_{t} - A_t X_t)^2 \right\} =
\\
= e^{-\frac{1}{2}\lambda_{t}^2} \mathop{\mathbb{E}}\nolimits\exp \left\{i\sum_{j=1}^{t} \alpha_j X_j + i\sum_{j=1}^{t-1} \lambda_j Y_j \right\},
\end{multline*}
where ${\cal
X}_t$ is the $\sigma$-field
${\cal
X}_t=\sigma(\{X_s\, ,\, 1\leq s\leq t\})$.
Hence,
\begin{multline*}
\hat{\pi}_t (J_t \exp(A_t X_t Y_t - \frac{1}{2}A_t^2 X_t^2 )) =
\\
=\pi_{t-1} (J_t \exp (A_t X_t y - \frac{1}{2} A_t^2 X_t^2 ))|_{y=Y_t} =
\\
= \pi_{t-1} (J_t \beta_t^y)|_{y=Y_t}.
\end{multline*}
Similarly,
$$
\hat{\pi}_t \left(\exp (A_t X_t y - \frac{1}{2} A_t^2 X_t^2 )\right) = \left.\pi_{t-1} (\beta_t^y)\right|_{y=Y_t}\,,
$$
and hence \eqref{pitJ} holds.
\textbf{II.} In the second step we will calculate the ratio $\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}}$ which, due to \eqref{pitJ} can be rewritten as
\begin{equation}\label{ratio}
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})}=\left. \frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\beta_t^y)} \right|_{y=Y_t}.
\end{equation}
For this aim similarly to what we proposed in~\cite{mkalbmcr3} and \cite{AMM} we introduce the auxiliary processes $(Y_t^2)_{t\ge 1}$ and $(\xi_t)_{t\ge 1}$.
Let $\bar{\mathop{\mathrm{var}}\nolimitsepsilon} =(\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t\ge 1}$ be a sequence of i.i.d. $\mathcal{N}(0,1)$ random variables independent of $X$ and define $(Y_t^2, \xi_t)_{t\ge 1}$ by:
\begin{equation}\label{Yaux}
Y_t^2= Q_t(X_t-h_t) + \sqrt{Q_t}\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t,
\end{equation}
\begin{equation}\label{xi:eq}
\xi_t=\sum_{s=1}^{t}(X_s-h_s) Y_s^2.
\end{equation}
Now the following equality holds:
$$
\left.\frac{\pi_{t-1}(J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})}\right|_{y=Y_t} = \left.\frac{\overline{\pi}_{t-1}(\exp \{-\frac{1}{2}Q_t (X_t-h_t)^2 - \xi_{t-1}\} \beta_t^y)}{\overline{\pi}_{t-1}(\exp(-\xi_{t-1}))}\right|_{y=Y_t},
$$
where $\overline{\pi}_t(\cdot)$ stands for a conditional expectation w.r.t. to the $\sigma$-field $\bar{\cal Y}_t=\sigma(\{Y_s, Y_s^2, {s\le t}\})$ under the initial measure $\Pg$.
Again the proof of this equality is based on the Bayes formula. Namely, let $\tilde{\Pg}$ be the new probability measure defined by
\begin{equation}\label{ptilde}
\frac{d\tilde{\Pg}}{d\Pg} = \rho_{t-1}=\exp\left\{ -\frac{1}{2} \sum_1^{t-1} Q_s (X_s-h_s)^2 -\sum_1^{t-1} \sqrt{Q_s} (X_s-h_s) \bar{\mathop{\mathrm{var}}\nolimitsepsilon}_s \right\}.
\end{equation}
Then $J_t \rho_{t-1}= \exp \{-\xi_{t-1} -\frac{1}{2} Q_t (X_t-h_t)^2 \}$ and $J_{t-1} \rho_{t-1}= \exp \{-\xi_{t-1}\}$. Thus
\begin{multline*}
\left.\frac{\overline{\pi}_{t-1} (\exp(-\xi_t - \frac{1}{2}Q_t (X_t-h_t)^2 ) \beta_t^y)}{\overline{\pi}_{t-1} (\exp(-\xi_{t-1}))}\right|_{y=Y_t}=
\\
= \left.\frac{\mathop{\mathbb{E}}\nolimits (J_t \beta_t^y \rho_{t-1}/\bar{{\cal Y}}_{t-1})}{\mathop{\mathbb{E}}\nolimits (\rho_{t-1}/\bar{{\cal Y}}_{t-1})}\cdot
\frac{\mathop{\mathbb{E}}\nolimits (\rho_{t-1}/\bar{{\cal Y}}_{t-1})}{\mathop{\mathbb{E}}\nolimits \exp(J_{t-1}\rho_{t-1})/\bar{{\cal Y}}_{t-1})}\right|_{y=Y_t}=
\\
= \left.\frac{\tilde{\mathop{\mathbb{E}}\nolimits}(J_t \beta_t^y/\bar{{\cal Y}}_{t-1})}{\tilde{\mathop{\mathbb{E}}\nolimits}(J_{t-1}/\bar{{\cal Y}}_{t-1})}\right|_{y=Y_t} =\left. \frac{\pi_{t-1} (J_t \beta_t^y)}{\pi_{t-1}(J_{t-1})}\right|_{y=Y_t},
\end{multline*}
where the last equality holds because under the probability measure $\tilde\Pg$ the distribution of $(X_s, Y_s)_{s\le t}$
is the same as under the initial measure $\Pg$ and
$(X_s, Y_s)_{s\le t-1}$ is independent of $(Y_{s}^{2})_{s\leq t-1}$.
Finally we have proved the following:
\begin{equation}\label{derivlogar}
\frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})} = \left.\frac{\overline{\pi}_{t-1}(\exp\left[-\xi_{t-1}+A_t X_t y -\frac{1}{2} Q_t (X_t -h_t)^2 - \frac{1}{2}A_t^2 X_t^2\right])}{\overline{\pi}_{t-1}(\exp(-\xi_{t-1})) \pi_{t-1}(\beta_{t}^{y})}\right|_{y=Y_t}.
\end{equation}
At this point we will use the conditionally Gaussian properties of $(X_{t},\xi_{t-1})$ w.r.t. $\bar{{\cal Y}}_{t-1}$ and Lemma 11.6 \cite{lipshi1} which says that for a Gaussian pair $(U,V)$ with mean values $m_{_{U}},m_{_{V}}$, variances $\gamma_{_{U}},\gamma_{_{V}}$ and covariance $\gamma_{_{UV}}$
\begin{multline}\label{feqg}
\mathop{\mathbb{E}}\nolimits\exp\left\{-\frac{1}{2} DU^2 + \lambda_1 U - \lambda_2 V \right\} = (1+D\gamma_{_{U}})^{-1/2} \times
\\
\times \exp\left\{ -\lambda_2 m_{V} + \frac{\lambda_2^2}{2} \gamma_{_{V}} - \frac{1}{2} \cdot\frac{D}{1+D\gamma_{_{U}}} (m_{_{U}}-\lambda_2 \gamma_{_{UV}})^2 \right. +
\\
+\left. \frac{\lambda_1^2 \gamma_{_{U}} + 2\lambda_1 (m_{_{U}}-\lambda_2\gamma_{_{UV}})}{2(1+D\gamma_{_{U}})} \right\},
\end{multline}
for any real numbers $\lambda_{1},\lambda_{2}$ and $D\ge 0$.
Indeed, in \eqref{derivlogar} we will apply this formula to $(U,V)=(X_{t},\xi _{t-1})$ given $\bar{{\cal Y}} _{t-1}$ with
$$
D=S_{t}=Q_t+A_t^2, \quad \lambda_2=1, \quad \lambda_1=A_t y+ Q_t h_t,
$$
in the numerator and
$D=\lambda_1=0 ,\quad \lambda_2=1 $ in the first factor of the denominator and again to $(U,V)=(X_{t},\xi _{t-1})$ given ${{\cal Y}} _{t-1}$ with
$$
D=A_t^2, \quad \lambda_2=0, \quad \lambda_1=A_t y,
$$
in the second factor of the denominator.
Collecting the terms as coefficients for $h^{2}_{t}$ and $h_{t}$, we obtain that
\begin{multline*}
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{(1+S_{t}\overline{\gamma}_t)^{-1/2}}{(1+A_{t}^{2}\gamma_t)^{-1/2}} \cdot \exp \left\{-\frac{Q_{t}}{2} \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_{t}-\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+A_t^2 \overline{\gamma}_t} \right]^2 \right\} \times \\
\times \exp\left\{
-\frac{A_t^2 (Z_{t}^h)^2 - A_{t}^{2} \overline{\gamma}_{t} Y_t^2}{2(1+A_t^2 \overline{\gamma}_{t})} + \frac{Y_tZ_{t}^hA_t}{1+A_{t}^2 \overline{\gamma}_t} + \frac{1}{2} \cdot \frac{A_t^2 \pi_{t-1}^2(X_{t}) - 2 A_t \pi_{t-1}(X_{t}) Y_t - A_{t}^2 Y_t^2 \gamma_t}{1+A_{t}^2\gamma_t}
\right\},
\end{multline*}
where
$Z_{t}^h=\overline{\pi}_{t-1}(X_{t})-\overline{\gamma}_{_{X\xi}}(t)$ with
\begin{multline}\label{def gamma Xxi}
\overline{\gamma}_{_{X\xi}}(t)=\mathop{\mathbb{E}}\nolimitsg[(X_t-\overline{\pi}_{t-1}(X_t))(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t-1}],\,t\ge 2\,;\,
\overline{\gamma}_{_{X\xi}}(1)=0\,.
\end{multline}
To finish the proof we just replace $Y_t$ by $\nu_t+A_t \pi_{t-1}(X_{t})$. Thus in the last exponential term we find:
\begin{multline*}
\exp\left\{ - \frac{\nu_t^2 A_t^2 (\gamma_t-\overline{\gamma}_t)} {2(1+A_t^2 \overline{\gamma}_t)(1+A_t^2 \gamma_t)} + \frac{Z_{t}^{h}-\pi_{t-1}(X_{t})}{1+A_{t}^2\overline{\gamma}_t}A_t \nu_t \right. -
\\ - \left.\frac{1}{2}\cdot \frac{A_t^2}{1+A_t^2 \overline{\gamma}_t} (Z_{t}^h-\pi_{t-1}(X_{t}))^2 \right\},
\end{multline*}
which gives the Proposition.
\begin{rem}\label{probinterp}
\begin{enumerate}
\item
Note that now the probabilistic interpretation of the ingredients $\overline{\gamma}_t$ and $Z_t^{h}$ is clarified for \textbf{negative} $\mu$. Namely,
$\overline{\gamma}_t=\mathop{\mathbb{E}}\nolimits (X_t-\overline{\pi}_{t-1}(X_t))^2,$
and $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$, but when
$\mu$ is \textbf{positive}, there is no such connection anymore.
\item Observe that actually $\overline{\pi}_{t-1}(X_t)$ and $\overline{\gamma}_{_{X\xi}}(t)$ are $\bar{{\cal Y}}_{t-1}$-measurable, but the difference $Z_t^{h}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ is ${\cal Y}_{t-1}$ measurable.
\end{enumerate}
\end{rem}
\paragraph{Proof of Theorem \ref{LEGsol}} The statement of Theorem
\ref{LEGsol} is the direct consequence of
Proposition \ref{CMC}. Indeed, we claim that the following chain of inequalities holds for any $h:\,h_t\in {\cal Y}_{t}, t\ge 1$ :
$$
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
$$
$$
=\mathop{\mathbb{E}}\nolimits\left[\mathop{\mathbb{E}}\nolimits\mu \left(\left.\exp\left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\} \right/ {{\cal Y}}_{T}\right)\right]
$$
$$
=\mu\mathop{\mathbb{E}}\nolimits\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t} \times \left[ h_t - \frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t } \right]^{2}\right\} \times \mathcal{M}_T,
$$
$$
\stackrel{(a)}{\ge}\prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} \mu \mathop{\mathbb{E}}\nolimits \mathcal{M}_T
$$
$$
\stackrel{(b)}{=} \mu \prod_{t=1}^T \left[\frac{1+S_t \overline{\gamma}_t}{1+A_t^{2} \overline{\gamma}_t}\right]^{-1/2} .
$$
Of course under condition $(C_{\mu})$, since the term in the last line is finite, it is sufficient to consider the case:
\begin{equation}\label{expfin}
\mathop{\mathbb{E}}\nolimits \mu \exp \left\{\frac{\mu}{2} \sum_{t=1}^T (X_t-h_t)^2 Q_t \right\}
< \infty,
\end{equation}
which gives the first equality. Inequality $(a)$ follows directly from Proposition~\ref{CMC}. Equality $(b)$ is a direct consequence of \eqref{feqg} which gives that $\mathop{\mathbb{E}}\nolimits \mathcal{M}_T=1.$
Now, to obtain the lower bound
we must take
$$
\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,
$$
or equivalently
$$
\overline{h}_{t}= Z_{t}^{\overline{h}} + \frac{A_t \overline{\gamma}_{t}}{1+A_t^2 \overline{\gamma}_{t}} (Y_t - A_t Z_{t}^{\overline{h}}),\, t\ge 1,
$$
where $Z^{h}$ is the solution
of equation (\ref{eqzh}), which means that
$$
Z_{t}^{\overline{h}} =m_t +\sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l) A_l}{1+A_l^2 \overline{\gamma}_l} [Y_l -A_l Z_{l}^{\overline{h}}],
$$
and hence
$$
\overline{h}_{t}= m_t + \sum_{l=1}^{t} \frac{\overline{\gamma}(t,l) A_l}{1+A_l^2 \overline{\gamma}_l} [Y_l -A_l Z_{l}^{\overline{h}}]=m_t+\sum_{l=1}^{t} A_l \overline{\gamma}(t,l) (Y_l -A_l \overline{h}_l ).
$$
Thus
$\bar{h}$ is the unique solution of equation \eqref{hbar}.
Finally for $\bar{h}$ the lower bound is attained.
\begin{rem}\label{probinterp'}
\begin{enumerate}
\item It is worth emphasizing that the process $\displaystyle{\tilde{Z}^{h}_{t}=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t }}$ is the solution of the following recursive equation:
\begin{equation}\label{zhtild}
\tilde{Z}^{h}_{t}=m_t -\sum_{l=1}^{t-1}\overline{\gamma}(t,l) \frac{\mu Q_l}{1+S_l \overline{\gamma}_l} (h_l - \tilde{Z}_{l}^h) + \sum_{l=1}^{t}\overline{\gamma}(t,l) A_l (Y_l - A_l \tilde{Z}_{l}^{h}),
\end{equation}
and hence the equality $\overline{h}_t =\displaystyle{\widetilde{Z}^{\overline{h}}_{t}} $ implies immediately the equation \eqref{eqzh} for $\overline{h}$.
This process $\tilde{Z}^{h}$ also has
a probabilistic interpretation as well as $\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$. This interpretation will be given in Section~\ref{interpret}.
\end{enumerate}
\end{rem}
\subsection{Solution of the auxiliary filtering problems}\label{AFP}
Here, for an arbitrary Gaussian sequence $X$, we deal with the
one-step prediction
and filtering problems of the signals $X$
and $\xi$ given by \eqref{xi:eq} respectively from the observation of
$\bar{Y}=(Y,Y^{2})$ defined in \eqref{observ} and \eqref{Yaux}.
Actually, we follow the ideas proposed in our paper
\cite{mkalbmcr3}.
Recall that the solutions can
be reduced to equations for the conditional moments. The following
statement provides the equations for the characteristics which give
the solution
of the prediction problem and the equation for the other quantity
$\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ appearing in Proposition~\ref{p:CM} for $\mu=-1.$
\begin{theorem}\label{filter}
The conditional mean $\overline{\pi}_{t-1}(X_t)$ and the
variance of the one-step prediction error
$\overline{\gamma}_t=\mathop{\mathbb{E}}\nolimitsg[X_t-\overline{\pi}_{t-1}(X_t)]^{2}$ are given by the equations
\begin{multline}\label{eq1 filter}
\overline{\pi}_{t-1}(X_t)=m_{t}+\sum\limits_{s=1}^{t-1}
\frac{\overline{\gamma}(t,s)}{1+(A_s^2+Q_s)\overline{\gamma}_s}[A_s(Y_{s}-A_s\overline{\pi}_{s-1}(X_s)) \\
+Q_s(Y_s^2 - Q_s(\overline{\pi}_{s-1}(X_s)-h_s)]\,,\quad t\geq 1,
\end{multline}
\begin{equation}\label{eq2 filter}
\overline{\gamma}_t=\overline{\gamma}(t,t)\,,\quad t\geq 1\,.
\end{equation}
where $\overline{\gamma}=(\overline{\gamma}(t,s),\, 1\le s \le t)$ is the unique solution of equation \eqref{GAMMABAR}.
Moreover, with $\overline{\gamma}_{_{X\xi}}(t)$ defined by (\ref{def gamma Xxi}), the difference $\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$ is the
solution $Z^h_t$ of equation (\ref{eqzh}).
\end{theorem}
\paragraph{Proof} Note that since $h_t \in {\cal Y}_t$ and the joint
distribution of $(X_{r},Y_{s}, Y_s^2+Q_s h_s)$ for any $r\,,s$ is Gaussian, we
can apply the Note following
Theorem 13.1 in \cite{lipshi1bis}. For any
$k\le t$ we can write
\begin{equation}\label{piX}
\left\{
\begin{array}{l}
\overline{\pi}_k(X_t)=\overline{\pi}_{k-1}(X_t) + [\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_k)]^{\prime} \mathop{\mathrm{var}}\nolimits(\overline{\nu}_k)^{-1} \overline{\nu}_k,
\\
\overline{\pi}_{0}(X_{t})=m_{t},
\end{array}
\right.
\end{equation}
where
$$
\overline{\nu}_{k}=\bar{Y}_{k}-\mathop{\mathbb{E}}\nolimitsg(\bar{Y}_{k}/\overline{{\cal Y}}_{k-1})=\left(
\begin{array}{c}
Y_k-A_k\overline{\pi}_{k-1}(X_k) \\
Y_k^2 +Q_kh_k - Q_k\overline{\pi}_{k-1}(X_k)
\end{array}
\right)
$$
is the innovation with covariance matrices
\begin{equation}\label{varnu}
\mathop{\mathrm{var}}\nolimits(\overline{\nu}_k)=\left(
\begin{array}{cc}
1+A_k^2\overline{\gamma}_k & A_kQ_k\overline{\gamma}_k\\
A_kQ_k\overline{\gamma}_k & Q_k+Q_k^2\overline{\gamma}_k
\end{array}
\right),
\end{equation}
and
\begin{equation}\label{covarXnu}
\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_k) = \overline{\gamma} (t,k) \left( \begin{array}{c} \! A_k \! \\ \! Q_k \! \end{array} \right),
\end{equation}
with
\begin{equation}\label{def gamma t s}
\overline{\gamma}(t,k)=\mathop{\mathbb{E}}\nolimitsg (X_t-\overline{\pi}_{k-1}(X_t))(X_{k}-\overline{\pi}_{k-1}(X_k))\,.
\end{equation}
By the definition (\ref{def gamma t s}), we see for $k=t$ that the
variance $\overline{\gamma}_t$ is given by (\ref{eq2 filter}). Now,
equality (\ref{piX}) implies
\begin{multline}\label{pikt}
\overline{\pi}_k(X_t)=m_t+\sum_{l=1}^k \overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l=
\\
=m_{t}+\sum\limits_{s=1}^{k}
\frac{\overline{\gamma}(t,s)}{1+(A_s^2+Q_s)\overline{\gamma}_s}[A_s(Y_{s}-A_s\overline{\pi}_{s-1}(X_s))+
\\
+Q_s(Y_s^2 - Q_s(\overline{\pi}_{s-1}(X_s)-h_s)],
\end{multline}
and putting $k=t-1$ we get nothing but equation (\ref{eq1
filter}). Concerning the solution of the one-step prediction
problem, it just remains to show that the covariance $\overline{\gamma}(t,s)$
satisfies equation (\ref{GAMMABAR}).
Let us define
$$
\delta_{X} (t,l) = X_t - \overline{\pi}_l (X_t)\,.
$$
According to (\ref{piX}) we can write
$$
\delta_X (t,l) = \delta_X (t,l-1) - \overline{\gamma}(t,l)\left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l,
$$
and so
\begin{multline*}
\mathop{\mathbb{E}}\nolimits \delta_X(t_1, l) \delta_X(t_2,l) = \mathop{\mathbb{E}}\nolimits \delta_X(t_1, l-1) \delta_X(t_2, l-1) -
\\
-\overline{\gamma}(t_1,l) \overline{\gamma}(t_2,l) \left({A_l \atop Q_l}\right)^{\prime} \mathop{\mathrm{var}}\nolimits(\bar{\nu}_l)^{-1} \left({A_l \atop Q_l}\right),
\end{multline*}
or
\begin{multline}\label{deltadelta}
\mathop{\mathbb{E}}\nolimitsg \delta_{X}(t^1,l)\delta_{X}(t^2,l)=\mathop{\mathbb{E}}\nolimitsg
\delta_{X}(t^1,0)\delta_{X}(t^2,0)- \\
-\sum\limits_{r=1}^{l}\overline{\gamma}(t,r) \overline{\gamma}(s,r) \frac{A_r^2+Q_r}{1+ (A_r^2 +Q_r ) \overline{\gamma}_{r}}.
\end{multline}
Taking $t^1=t\,, t^2=s\,, l= s-1$ in (\ref{deltadelta}), it is
readily seen that
equation (\ref{GAMMABAR}) holds for $\overline{\gamma}(t,s)$.
Now we analyze the difference $\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$.
Using the representation $\xi_t=\sum_{s=1}^t (X_s-h_s) Y_s^2$
we can rewrite $\overline{\pi}_{t-1}(\xi_{t-1})$ in the following form
$$
\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (\pi_{t-1}(X_s)-h_s) Y_s^2,
$$
which implies that
$$
\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1})= \sum_{s=1}^{t-1} (X_s - \overline{\pi}_{t-1}(X_s)) Y_s^2.
$$
So we have
\begin{multline}\label{ga X xi}
\overline{\gamma}_{_{X\xi}}(t) = \sum_{s=1}^{t-1} \overline{\pi}_{t-1} [(X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))] Y_s^2 =
\\
= \sum_{s=1}^{t-1} \mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t)) Y_s^2
= \sum_{s=1}^{t-1} \widetilde{\gamma}(t,s) Y_s^2,
\end{multline}
where
\begin{equation}\label{ad gamma }
\widetilde{\gamma}(t,s)=\mathop{\mathbb{E}}\nolimits (X_s-\overline{\pi}_{t-1}(X_s)) (X_t-\overline{\pi}_{t-1}(X_t))= \overline{\gamma}(s,t).
\end{equation}
Using the definitions (\ref{def gamma t s}) and (\ref{ad gamma })
we can write
$$
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) = - \mathop{\mathbb{E}}\nolimits X_t (\overline{\pi}_{t-1}(X_s) - \overline{\pi}_{s-1}(X_s)).
$$
Again, applying the Note following Theorem 13.1 in \cite{lipshi1bis}, we
can write also
$$
\overline{\pi}_{l}(X_{r})= \overline{\pi}_{l-1}(X_{r})+\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l.
$$
This means that
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})= \sum\limits_{l=r}^{t-1}\overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right)
(\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l\,,
$$
or equivalently
$$
\pi_{t-1}(X_{r})- \pi_{r-1}(X_{r})
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,t) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}
\overline{\nu}_l\,.
$$
Then, multiplying by $X_t$ and taking expectations in both sides, we get
\begin{multline*}
\mathop{\mathbb{E}}\nolimitsg X_{t}
(\pi_{t-1}(X_{r})-\pi_{r-1}(X_{r}))
=\sum\limits_{l=r}^{t-1}\widetilde{\gamma}(l,r) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1}\mathop{\mathrm{cov}}\nolimits(X_t,\overline{\nu}_l)=
\\
=\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{multline*}
Hence we have proved the following relation
\begin{equation}\label{dif gamma}
\widetilde{\gamma}(t,s)-\overline{\gamma}(t,s) =-\sum_{l=s}^{t-1} \widetilde{\gamma}(l,s) \overline{\gamma}(t,l)\frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} .
\end{equation}
Now we can show that the difference $Z^h_{t}=\overline{\pi}_{t-1}(X_t)-\overline{\gamma}_{_{X\xi}}(t)$
satisfies the equation (\ref{eqzh}). Using (\ref{pikt}) and (\ref{ga X xi}),
we can write
\begin{multline}\label{zhbeg}
Z^h_{t} = m_t+\sum_{l=1}^{t-1} \overline{\gamma}(t,l) \left( \begin{array}{cc} \! A_l \! & \! Q_l \! \end{array} \right) (\mathop{\mathrm{var}}\nolimits \overline{\nu}_l)^{-1} \overline{\nu}_l - \sum_{s=1}^{t-1} \widetilde{\gamma}(t,s) Y_s^2 =
\\
= m_t +\sum_{l=1}^{t-1} \frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l}(Y_l - A_l \overline{\pi}_{l-1}(X_l)) +
\\
+
\sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l}(Y_l^2-Q_l (\overline{\pi}_{l-1}(X_l)-h_l))
-\sum_{l=1}^{t-1} \widetilde{\gamma}(t,l) Y_l^2 =
\\
= m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Q_l h_l - \\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l} \overline{\pi}_{l-1}(X_l) +
\\
+ \sum_{l=1}^{t-1} [\frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} - \widetilde{\gamma} (t,l)] Y_l^2.
\end{multline}
Now we can rewrite the last term in \eqref{zhbeg} using the equality \eqref{dif gamma}. We have
\begin{multline}\label{gammause}
\sum_{l=1}^{t-1} [\frac{\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} - \widetilde{\gamma} (t,l)] Y_l^2= \sum_{l=1}^{t-1} \overline{\gamma}(t,l) (\frac{1} {1+(A_l^2 +Q_l)\overline{\gamma}_l}-1) Y_l^2 +
\\
+ \sum_{l=1}^{t-1} \sum_{r=l}^{t-1} \overline{\gamma}(t,r) \widetilde{\gamma}(r,l)\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r} Y_l^2 = \\
= \sum_{r=1}^{t-1} \overline{\gamma}(t,r)
\left[\sum_{l=1}^{r-1} \widetilde{\gamma}(r,l) Y_l^2 \right]
\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r}=
\\
=\sum_{r=1}^{t-1} \overline{\gamma}(t,r)\overline{\gamma}_{_{X\xi}}(r)\frac{A_r^2 +Q_r}{1+(A_r^2 +Q_r)\overline{\gamma}_r}\,,
\end{multline}
where in the last step we have used equality (\ref{ga X xi}).
Finally \eqref{zhbeg}-\eqref{gammause} imply:
\begin{multline*}
Z^h_{t} = m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{Q_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} h_l -
\\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l}[ \overline{\pi}_{l-1}(X_l)-\overline{\gamma}_{_{X\xi}}(l)]=
\\
m_t +\sum_{l=1}^{t-1}\frac{A_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} Y_l + \sum_{l=1}^{t-1} \frac{Q_l\overline{\gamma}(t,l)} {1+(A_l^2 +Q_l)\overline{\gamma}_l} h_l -
\\
- \sum_{l=1}^{t-1} \overline{\gamma}(t,l) \frac{A_l^2 +Q_l}{1+(A_l^2 +Q_l)\overline{\gamma}_l}Z^h_{l}\,,
\end{multline*}
which is nothing else but equation (\ref{eqzh}) with $\mu=-1$.
\section{Particular cases and applications}\label{PC}
Here we deal with some specific cases where the results can be further elaborated. For two examples we can apply directly Theorem~\ref{LEGsol} and moreover
the special structure of the covariances allows to simplify the answer.
\subsection{LEG filtering of Gauss-Markov sequences }\label{GMS}
In this part we concentrate on the case of a
Gaussian AR(1) process $X$, {\em i.e.}, a Gauss-Markov process
driven by
\begin{equation}\label{model AR}
X_{t}= a_{t} X_{t-1}+D_t^\frac{1}{2}\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}\,,\; t\ge 1\,;
\quad X_{0}=x\,,
\end{equation}
where $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables and
$(D_t,\,t\ge 1)$ is a (deterministic)
sequence of real numbers such that $D_t\ge 0$ for $t\ge 1$. In this setting,
it is easy to check that the mean
and covariance functions of $X$ are given by
$$
m_{t}=[\prod_{u=1}^{t}a_u]x=\Lambda_{t}x\,;\quad
K(t,s)=[\prod_{u=s+1}^{t}a_u]k_s=\frac{\Lambda_{t}}{\Lambda_{s}}k_s\,,\;1\le s\le t\,,
$$
where $\Lambda_{t}=\prod\limits_{u=1}^{t}a_u$ and
$$
k_t=a_t^2k_{t-1} +D_t,\, t\ge 1,\,k_0=0.
$$
Suppose that the following the Riccati type equation
\begin{equation}\label{ricmar}
\overline{\gamma}_s=D_s+\frac{a_{s}^2 \overline{\gamma}_{s-1}}{1+(A_{s-1}^2-\mu Q_{s-1})\overline{\gamma}_{s-1}},\, s\ge 1,\,\overline{\gamma}_{0}=0,
\end{equation}
has a unique nonnegative solution.
From the classical filtering theory it is
well-known that (for $\mu<0$ ) $\overline{\gamma}_s$ is nothing but the variance of the error of the one-step prediction problem
of the signal $X$ given by the auxiliary observation $\bar{Y}$ defined by equations \eqref{observ} and \eqref{Yaux}. Then, it is readily
seen that the function $\overline{\gamma}(t,s)$, where
$\displaystyle{\overline{\gamma}(t,s)=\frac{\Lambda_{t}}{\Lambda_{s}}\overline{\gamma}_s}$ is the
solution of equation (\ref{GAMMABAR}) and that moreover equation
\eqref{hbar} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation}\label{rsmkk}
\overline{h}_t=\frac{a_t}{1+A_t^2\overline{\gamma}_t}\overline{h}_{t-1} +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t}Y_t, \, t\ge 1,\, \overline{h}_0=x,
\end{equation}
or, equivalently:
\begin{equation*}
\overline{h}_t=a_t \overline{h}_{t-1} +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t} [Y_t-a_tA_t \overline{h}_{t-1}], \, t\ge 1,\, \overline{h}_0=x.
\end{equation*}
Actually equation \eqref{rsmkk} can also be obtained directly from the general filtering theory (for $\mu=-1$ and replacing $Q$ by $-\mu Q$).
For arbitrary $(h_{t} \in {\cal Y}_{t}, t\ge 1)$ the Note following
Theorem 13.1 in \cite{lipshi1bis} gives the equation for $Z^{h}$:
\begin{multline*}
Z^{h}_{t}=a_t Z^{h}_{t-1} +a_t \overline{\gamma}_t\frac{Q_{t-1} }{1+S_{t-1}\overline{\gamma}_t} [h_{t-1}-Z^h_{t-1}] +
\\
+a_t \overline{\gamma}_t\frac{A_{t-1} }{1+S_{t-1}\overline{\gamma}_t} [Y_{t-1}-A_{t-1}Z^h_{t-1}], \, t\ge 1,\, Z^h_0=x.
\end{multline*}
Hence, again the solution
$\displaystyle{\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,
}$ of the LEG filtering problem (\ref{LEGdef}) is given by \eqref{rsmkk}.
Let us emphasize that these equations are nothing but those given in Speyer \textit{et al.}
\cite{speyer}.
It is interesting to note that in the case $a_t=0$ (i.i.d. signal) the solution of the LEG filtering problem is
nothing else but the solution of the risk neutral filtering problem \textit{i.e.} $\overline{h}_t=\pi_t(X_t)$.
\subsection{LEG filtering of moving averages of order 1}\label{MA1X}
Here we consider the case of a MA(1) process, {\em i.e}, a non
Markovian process $X$ defined by
$$
X_{t} = \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t} + \lambda \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t-1}\,; t\ge 1\,,
$$
where $( \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_0, \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_1,\dots)$~is a sequence of i.i.d. standard
Gaussian variables and $\lambda$ is a real number. Of course
$X$ is centered and has the
covariance function
$K(t,s)= 1+\lambda^{2}$ if $s=t$, $\lambda$ if $s=t-1$ and
$0$ if $s < t-1$.
In order to solve equation (\ref{GAMMABAR}) we can take
$$
\overline{\gamma}(t,s) =0\,,\; s < t-1\,; \quad \overline{\gamma}(t,t-1) = \lambda\,,\;
t\ge 1\,,
$$
and $\overline{\gamma}(t,t)=\overline{\gamma}_{t}$ where $\overline{\gamma}_{t}$ is
the solution of the equation:
$$
\overline{\gamma}_{t} =1+\lambda^{2}
- \lambda \frac{A_{t-1}^2-\mu Q_{t-1}}{1+(A_{t-1}^2-\mu Q_{t-1})\overline{\gamma}_{t-1}}\,,\;t\ge 1\,;\quad \overline{\gamma}_0=1+\lambda^{2},
$$
provided that this equation has a unique nonnegative solution.
Moreover equation
\eqref{hbar} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation*}
\overline{h}_t=\lambda\frac{A_{t-1}}{1+A_t^2\overline{\gamma}_t}[Y_{t-1}-A_{t-1}\overline{h}_{t-1}] +\frac{A_t \overline{\gamma}_t}{1+A_t^2\overline{\gamma}_t}Y_t, \, t\ge 1,\, \overline{h}_0=0.
\end{equation*}
Again, it is interesting to note that for $\lambda=0$ (i.i.d. signal) the solution of LEG filtering problem is
nothing else but the solution of the risk neutral filtering problem \textit{i.e.} $\overline{h}_t=\pi_t(X_t)$.
\section{LEG and RS filtering problems}\label{disc}
Here, at first we show that actually the LEG and RS filtering problems have the same solution. Then we give an example which shows that in a more general context similar problems may have different solutions.
\subsection{Equivalence of LEG and RS filtering problems}
Let $\bar{h}=(\overline{h}_s)_{s \ge 1}$ be the solution of the LEG filtering problem (\ref{LEGdef})
given by equation \eqref{hbar}.
For any fixed $t \le T$, let us denote by $\hat{g}_{t}:$
\begin{multline*}
\hat{g}_{_t}=
\displaystyle{\arg\min_{g \in {\cal Y}_t} }\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}(X_{t}-g)^{2}Q_{t}}
+\left.\displaystyle{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-\bar{h}(s))^{2}Q_{s} }\Big\} \right/
{\cal Y}_t\Big],
\end{multline*}
where $g\in {\cal Y}_t$ means that $g$ is a ${\cal Y}_t$-measurable variable.
It follows directly from
Proposition~\ref{p:CM}
that, provided that $\displaystyle{1 + S_t\overline{\gamma}_t}> 0$, the equality
$\displaystyle{\hat{g}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1}$ holds.
Since it was noted in the proof of Theorem~\ref{LEGsol} that $\displaystyle{\overline{h}_{t}= \displaystyle{\frac{Z_{t}^{\bar{h}}+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t}},\, t\ge 1,}$
hence we have also $\hat{g}_{t}=\bar{h}_t$. It means that for $ t\ge 1 $
the solution $\bar{h}$ of the LEG filtering problem satisfies the following recursive
equation:
\begin{multline}\label{rde}
\hat{g}_{_t}=
\displaystyle{\arg\min_{g \in {\cal Y}_t} }\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}(X_{t}-g)^{2}Q_{t}}
+\left.\displaystyle{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-\bar{h}(s))^{2}Q_{s} }\Big\} \right/
{\cal Y}_t\Big].
\end{multline}
Indeed, in the literature, the recursion (\ref{rde}) is the basic
\textbf{definition} of the so-called risk-sensitive (RS)
filtering problem which was introduced in \cite{elliott3}. Therefore we have also proved the following
statement
\begin{theorem}\label{RS}
Assume that the condition $(C_{\mu})$ is
satisfied. Let $\overline{h}=(\overline{h}_t)_{t\ge 1}$ be the unique solution of equation
(\ref{hbar}), \textit{i.e.}, $\bar{h}$ is the solution of the LEG
filtering problem \eqref{LEGdef}.
Then $\bar{h}$ is the solution of the RS filtering problem
\eqref{rde}.
\end{theorem}
\subsection{Discrepancy between LEG and RS type filtering problems: an example }\label{EX}
Actually, we did not find in the literature any trace of the
discussion
about the relationship between the LEG
filtering problem (\ref{LEGdef}) and the RS filtering
problem (\ref{rde}) even in a Gauss-Markov case. As a complement to our observation that these two problems
have the same solution, we propose an example to show that in a bit more general setting,
two similar problems may have different solutions.
For given positive symmetric deterministic $2\times 2$ matrices
$\Lambda_s,1\le s\le T$, let us set $\Phi_{t}(h)=(X_{t} \, h_t)\Lambda_{t}\left(
\begin{array}{c}
X_{t} \\
h_t \\
\end{array}
\right)$. We can define $\bar{h}_{t}\in {{\cal Y}}_{t},\, t\ge 1$ as a solution of a
\textit{LEG type filtering problem} :
\begin{equation}\label{defrssex}
\overline{h}= \arg\min_{h_{t}\in {\cal Y}_{t},\, t\ge 1}\mathop{\mathbb{E}}\nolimitsg \left[\mu \exp\left\{\frac{\mu}{2}
\sum_1^T \Phi_{s}(h)\right\}\right].
\end{equation}
We can also define
$\hat{h}$ as the solution of
the following recursive equation (\textit{RS type filtering problem}):
\begin{equation}\label{rdeex}
\hat{h}_{t}= \displaystyle{\arg\min_{g \in {\cal Y}_t}
}\,\mathop{\mathbb{E}}\nolimitsg\Big[\mu
\exp\Big\{\displaystyle{\frac{\mu}{2}\Phi_{t}(g)\,}+\left.\displaystyle{\frac{\mu}{2} \sum_{1}^{t-1}
\Phi_{s}(\hat{h})\, }\Big\} \right/ {\cal Y}_t\Big].
\end{equation}
The question which we discuss now is the following: does the equality
$\bar{h}=\hat{h}$ hold?
As we have just proved, the answer is positive for singular matrices
$\Lambda$, namely, when
$\Lambda_{11}=\Lambda_{22}=-\Lambda_{12}=Q$.
But in the general situation the answer may be negative. Actually it is sufficient to consider the following example:
$\Lambda =\left(
\begin{array}{cc}
2 & -1 \\
-1 & 1 \\
\end{array}
\right),\, A_{t}=1,\,\mu=-1 $ and $X_{t}= X_{t-1}+\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}$, where $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$
is a sequence of i.i.d.
standard Gaussian random variables. Even in this Markov
case $\hat{h}\ne \overline{h} $. More explicitly let us introduce the new probability measure $\hat{\Pg}:$
$$
\frac{d\hat{\Pg}}{d\Pg}= \frac{\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right]}{\mathop{\mathbb{E}}\nolimitsg\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right]}.
$$
One can check that with respect to $\hat{\Pg}$ the observation model $(X_t,Y_t)_{t \ge 1}$ can be written in the following form:
$$
X_t= a_tX_{t-1} + D_t^\frac{1}{2}\hat{\mathop{\mathrm{var}}\nolimitsepsilon}_{t}\,,\; t\ge 1\,;
\quad X_{0}=x\,,
$$
$$
Y_t=X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t\,,
$$
where $(\hat\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d.
standard Gaussian random variables independent of the sequence $\mathop{\mathrm{var}}\nolimitsepsilon$,
$$
a_t=D_t=\frac{1}{1+ \Gamma(T,t)}\,,
$$
and $\Gamma(T,\cdot)$ is the solution of the backward Riccati equation
$$
\Gamma(T,t) = 1 +\frac{\Gamma(T,t+1)}{1+\Gamma(T,t+1)},\,\Gamma(T,T)=0.
$$
It can be checked that
$$
\Gamma(T,t)=10\frac{\lambda^{T}-\lambda^{t}}{(1-\sqrt{5})\lambda^{T}-(1+\sqrt{5})\lambda^{t}},\, \lambda = \frac{(3-\sqrt{5})}{(3+\sqrt{5})}.
$$
Indeed to explain this change of the observation model it is sufficient to calculate the conditional characteristic function:
$$
\hat{\mathop{\mathbb{E}}\nolimitsg}\left[\left.\exp (i\lambda X_t)\right/{\cal
X}_{t-1}\right]=\frac{\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left[i\lambda X_t-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right] \right/{\cal
X}_{t-1}\right]}{\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left[-\frac{1}{2}\sum\limits_{i=1}^{T}X_i^2\right] \right/{\cal
X}_{t-1}\right]},
$$
where ${\cal
X}_{t-1}$ is the
$\sigma$-field
${\cal
X}_{t-1}=\sigma(\{X_s\, ,\, 1\leq s\leq t-1\})$.
But it follows directly from the equation (19)-(20) in \cite{mkalbmcr3} and from \eqref{feqg}
that
$$\hat{\mathop{\mathbb{E}}\nolimitsg}\left[\left.\exp (i\lambda X_t)\right/{\cal
X}_{t-1}\right]=\exp\left\{\frac{i\lambda}{1+ \Gamma(T,t)}X_{t-1} -\frac{\lambda^{2}}{2(1+ \Gamma(T,t))}\right\}.
$$
Since the density $\displaystyle{\frac{d\widehat{\Pg}}{d\Pg}}$ does not depend on $h$, the initial LEG filtering problem \eqref{defrssex} can be rewritten as:
$$
\overline{h}= \arg\min_{h_{t}\in {\cal Y}_{t},\, 1\le t\le T}\hat{\mathop{\mathbb{E}}\nolimitsg} \left[- \exp\left\{-\frac{1}{2}
\sum_1^T (X_s-h_s)^{2}\right\}\right].
$$
Hence we can apply Theorem~\ref{LEGsol} or in particular \eqref{ricmar} and \eqref{rsmkk}. Clearly, $\overline{h}$ depends on $T$ and $\hat{h}$ does not depend on $T$ by the definition. A bit more explicitly we have for example for $t=1$:
$
\displaystyle{\overline{h}_{1}=\frac{1+\Gamma(T,1)}{2+\Gamma(T,1)}Y_1}
$
and obviously
$
\displaystyle{\hat{h}_{1}=\frac{\pi_{1}(X_1)}{1+\gamma_{1}}=\frac{1}{4}Y_1}
$
and clearly they are different.
\section{Information state, interpretation}\label{interpret}
In this section we discuss the probabilistic interpretation of the ingredients of the ``information state'' which was introduced
in the context of RS filtering and LEG control problems.
By the definition, the ``information state'' contains all the information needed to describe the solution of the concerned optimization problem. In particular it takes into account the cost function but not only estimates of the signal and it should
give the total information about the model states available in the measurement.
\textit{Risk-Sensitive Filtering}
\noindent
In the context of the RS filtering problem the definition of the information state can be found for example in \cite{elliott1}.
It is the density $\lambda_{t}$, with respect to the Lebesgue measure, of the non normalized random measure
$\omega_{t}$:
\begin{equation}\label{mescond}
\omega_{t}(dx)=\displaystyle{\mathop{\mathbb{E}}\nolimitsg\left[\left.\mathbb{I}(X_t \in dx)\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-h(s))^{2}Q_{s}\right\} \right/ {\cal
Y}_t\right]},
\end{equation}
where $h_t\in {\cal Y}_t, t\ge 1 $
and the observation $Y$ is defined by the equation \eqref{observ}.\\
In a classical Gauss-Markov setting, an explicit representation of $\lambda_{t}$ can be obtained as the solution of some
recurrence equation (see, \textit{e.g.},
\cite{collings}).\\
We claim that for a general Gaussian signal $X$ the density $\lambda_{t}$
satisfies the following equality:
\begin{multline}\label{cdensg}
\lambda_{t}(x) =
\displaystyle{\frac{1}{\sqrt{2\pi \widetilde{\gamma}_t}}
\exp\left\{-\frac{(x-\widetilde{Z}_{t}^h)^{2}}{2\widetilde{\gamma}_{t}}
\right\}}\times
\\
\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r} \times \left[ h_r -\widetilde{Z}_{t}^h \right]^{2}\right\} \times \mathcal{M}_t,
\end{multline}
where $\displaystyle{ \widetilde{Z}_{t}^h=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+A_t^2\overline{\gamma}_t}}$ is the solution of the equation \eqref{zhtild}, $\displaystyle{ \widetilde{\gamma}_{t}= \frac{\overline{\gamma}_{t}}{1+A_t^2\overline{\gamma}_t}}$, $\overline{\gamma}, Z^h$ are the solutions of
equations (\ref{GAMMABAR}) and (\ref{eqzh}) respectively and the martingale $(\mathcal{M}_t)_{\ge 1}$ is defined by \eqref{martin}.
Indeed, to prove (\ref{cdensg}) it is sufficient to write the following:
\begin{equation}\label{bayesford}
\omega_{t}(dx) = \frac{\mathop{\mathbb{E}}\nolimitsg[\mathbb{I}(X_t \in dx)\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}{\mathop{\mathbb{E}}\nolimitsg[\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}\mathop{\mathbb{E}}\nolimitsg\left[\left.\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
(X_{s}-h(s))^{2}Q_{s}\right\}\right/ {\cal
Y}_t\right],
\end{equation}
where
$\sigma$-field
${\overline{{\cal Y}}}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Again, conditionally Gaussian
properties of the pair $(X,\xi)$ imply that
\begin{equation*}
\frac{\mathop{\mathbb{E}}\nolimitsg \left[\mathbb{I}(X_{t} \in dx)\exp\left\{-\xi_{t} \right\}/
{\cal \overline{Y}}_{t,t-1} \right]}{\mathop{\mathbb{E}}\nolimitsg[\exp(-\xi_{t-1})/
{\cal \overline{Y}}_{t,t-1}]}=
\displaystyle{[2\pi\widetilde{\gamma}_{t}]^{-\frac{1}{2}}}
\end{equation*}
\begin{equation}\label{gausden}
\times \exp \displaystyle{
\left\{-\frac{1}{2}(x-\widetilde{Z}^{h}_{t})^{2}\widetilde{\gamma}^{-1}_t
\right\}\,dx},
\end{equation}
where $\displaystyle{\widetilde{Z}^{h}_{t}=\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}]- \mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t,t-1}]}$ and $\displaystyle{\widetilde{\gamma}_{t}=\mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])]^{2}}$.
Now the desired equality \eqref{bayesford} follows directly from Proposition~\ref{p:CM}.\\
It is worth emphasizing that (for negative $\mu$) now we know the probabilistic interpretation of the involved processes
$(Z^h,\widetilde{Z}^{h}, \,\bar{\gamma},\widetilde{\gamma})$. Actually we have proved that $Z^{h}$
is the difference $\bar{\pi}_{t}(X)-\bar{\gamma}_{_{X\xi}}(t)$ and
$\bar{\gamma}$ is nothing but the covariance of the filtering
error of $X$ in view of auxiliary observations $\bar{Y}$.\\ For the pair $(\widetilde{Z}^{h}, \,\widetilde{\gamma})$ we have the same relations but with respect to the $\sigma$-field
${\cal
Y}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Of course, after a simple integration of $\lambda_{t}$, formula (\ref{cdensg}) gives
Proposition~\ref{CMC} and therefore the solution of the LEG and RS filtering problems.
Let us also observe that the relations
$\displaystyle{\tilde{Z}^{h}_{t}=\frac{Z_{t}^h+A_t \overline{\gamma}_t Y_t}{1+ A_t^2 \overline{\gamma}_t }},\,
\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$ which were announced in Remark~\ref{probinterp'} follow from the Note following
Theorem 13.1 in \cite{lipshi1bis}.
\textit{Linear Exponential Gaussian Control}
\noindent
In the context of the LEG control problem for a partially observed process, the information state is also defined (see, \textit{e.g.}, \cite{elliott1}) as the density $\lambda_{t}$, with respect to the Lebesgue measure,
of the non normalized random measure
$\omega_{t}$:
\begin{equation}\label{mescondcon}
\omega_{t}(dx)=\displaystyle{\mathop{\mathbb{E}}\nolimitsg\left[\left.\mathbb{I}(X_t \in dx)\exp\left\{\frac{\mu}{2} \sum_{s=1}^{t-1}
X_{s}^{2}Q_{s} \right\} \right/ {\cal
Y}_t\right]},
\end{equation}
where $X$ is the controlled state governed by the equation:
\begin{equation}\label{SDECONT}
X_t=a_tX_{t-1}+b_tu_t +\widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t\,,\;t\geq 1\,;\; X_0=0\,,
\end{equation}
$( \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t \ge 1}$ is a sequence of i.i.d. standard
Gaussian variables
and $u_{t}\in {\cal Y}_{t-1} $ corresponding to the available observation $Y$ defined by the equation \eqref{observ}.\\
By the same way that we have just explained, for the conditionally Gaussian pair $(X,Y)$, one can check that the density $\lambda_{t}$
satisfies the following equality:
\begin{multline}\label{cdensgcon}
\lambda_{t}(x) =
\displaystyle{\frac{1}{\sqrt{2\pi \widetilde{\gamma}_t}}
\exp\left\{-\frac{(x-\widetilde{Z}_{t})^{2}}{2\widetilde{\gamma}_{t}}
\right\}}\times
\\
\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2} \times \exp\left\{\frac{\mu}{2} Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r} \times \widetilde{Z}_{t}^{2}\right\} \times \mathcal{M}_t,
\end{multline}
where $\displaystyle{\tilde{\gamma}_{t}=\frac{ \overline{\gamma}_t}{1+ A_t^2 \overline{\gamma}_t }}$, $\bar{\gamma}$ is the solutions of
equation (\ref{GAMMABAR}), the martingale $(\mathcal{M}_t)_{\ge 1}$ is defined by \eqref{martin}
and $\widetilde{Z}$ is the solution of the equation
\begin{equation}\label{z:repres}
\widetilde{Z}_{t} = \frac{a_t}{1+ S_t \overline{\gamma}_t }\widetilde{Z}_{t-1}+\frac{b_t}{1+ S_t \overline{\gamma}_t }u_t
+ \bar{\gamma}_t A_tY_{t}.
\end{equation}
Actually it is the equation for the difference $\widetilde{Z}=\bar{\pi}_{t,t-1}(X)-\bar{\gamma}_{X\xi}(t,t-1)$, where the conditional expectations are taken with respect to the auxiliary observation process $\bar{Y}$ defined by the equations \eqref{observ} and \eqref{Yaux} with $h=0$.\\
Equality \eqref{cdensgcon} gives the possibility to rewrite the cost function in terms of the completely observable process $\widetilde{Z}$, namely:
\begin{equation}\nonumber
\begin{array}{ccl}
\mathop{\mathbb{E}}\nolimitsg\Big[
\exp\Big\{\displaystyle{\frac{\mu}{2} \sum_{s=1}^T
X_{s}^{2}Q_{s} }
\Big]
=\mathop{\mathbb{E}}\nolimitsg\Big\{ \mathop{\mathbb{E}}\nolimitsg\Big[\left.
\exp\Big\{\displaystyle{\frac{\mu}{2} \sum_{s=1}^T
X_{s}^{2}Q_{s} }\Big\} \right/
{\cal Y}_T\Big]\Big\}
\\
=\displaystyle{\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2}} \mathop{\mathbb{E}}\nolimitsg\Big[\displaystyle{\exp\left\{\frac{\mu}{2}\sum_{s=1}^T\widetilde{Z}_{s}^{2}\widetilde{Q}_{s}
\right\}}\Big]\times \mathcal{M}_T
\\
=\displaystyle{\prod_{r=1}^{t-1} \left[\frac{1+S_r \overline{\gamma}_r}{1+A_r^{2} \overline{\gamma}_r}\right]^{-1/2}} \widetilde{\mathop{\mathbb{E}}\nolimitsg}\Big[\displaystyle{\exp\left\{\frac{\mu}{2}\sum_{s=1}^T\widetilde{Z}_{s}^{2}\widetilde{Q}_{s}
\right\}}\Big],
\end{array}
\end{equation}
where $\displaystyle{\widetilde{Q}_{r} = Q_r \frac{1+A_r^2 \overline{\gamma}_r}{1+ S_r \overline{\gamma}_r}}$ and $\widetilde{\mathop{\mathbb{E}}\nolimitsg}$ stands for an expectation with respect to the new measure $\widetilde{\Pg}$ such that:
$$
\frac{d\widetilde{\Pg}}{d\Pg} =\mathcal{M}_T .
$$
With respect to this new measure the solution of equation \eqref{z:repres} can be represented as
\begin{equation}\label{z:repres'}
\widetilde{Z}_{t} = a_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t}\widetilde{Z}_{t-1}+b_t \frac{1+A_t^2 \overline{\gamma}_t}{1+ S_t \overline{\gamma}_t}u_t
+ \frac{\bar{\gamma}_tA_t}{1+A_r^{2} \overline{\gamma}_r}\bar{\mathop{\mathrm{var}}\nolimitsepsilon}_{t},
\end{equation}
where $( \bar{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t \ge 1}$ is a new sequence of i.i.d. standard
Gaussian variables.
Thus, the new process $\widetilde{Z}$ plays the role of the completely observed controlled state (see \cite{bensoussan} and \cite{elliott1}).
Now we emphasize that the probabilistic interpretation of the
``information state" $\widetilde{Z}$, used in \cite{elliott1}
is nothing but
$\widetilde{Z}_{t}=\bar{\pi}_{t,t-1}(X)-\bar{\gamma}_{X\xi}(t)$, where the conditional expectations are taken with respect to the auxiliary observation process $\bar{Y}$ defined by the equations \eqref{observ} and \eqref{Yaux} with $h=0$. Also, $\bar{\gamma}$ is the conditional covariance of $X$.
\section{Complementary part - More general case}\label{complements}
In this section we analyze LEG and RS filtering problems in a more general contexts when we do not suppose a special structure of the observation sequence $(Y_t)_{t\ge 1}$. We suppose only that the process $(X_t,\,Y_t)_{t\ge 1}$ is Gaussian (even conditionally Gaussian). Our goal is to reduce LEG (RS) filtering problems to an auxiliary risk-neutral filtering problem.
First of all we fix $\mu=-1$ and we will find the probabilistic interpretation of the solution. After to find the solution for $\mu \ne -1$ we shall have only to replace $Q$ by $-\mu Q$ in the answer.
So, let $(Y^{2}_{t},\, \xi_{t})$ be defined by equations \eqref{Yaux} - \eqref{xi:eq} and let us denote by
\begin{equation}\label{Ztildainterpr}
\widetilde{Z}^{h}_{t}=\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}]- \mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t,t-1}],
\end{equation}
\begin{equation}
\widetilde{\gamma}_{t}=\mathop{\mathbb{E}}\nolimitsg[(X_t-\mathop{\mathbb{E}}\nolimitsg[X_t/
{\bar{{\cal Y}}}_{t,t-1}])]^{2},
\end{equation}
where ${
\bar{{\cal Y}}}_{t,t-1}$ is the
$\sigma$-field
${
\bar{{\cal Y}}}_{t,t-1}=\sigma(\{(Y_s, Y^{2}_{r}) ,\, 1\leq s\leq t,\, 1\leq r\leq t-1\})$.
Again, let $\displaystyle {J_t= \exp\left\{-\frac{1}{2} \sum_{s=1}^t (X_s-h_s)^2 Q_s \right\}}$ and let us denote by $\mathcal{I}_t$ the conditional expectation
$\displaystyle {\mathcal{I}_t= \pi_t(J_t)}$, or
$$
\mathcal{I}_t= \mathop{\mathbb{E}}\nolimits \left(\left.\exp\left\{-\frac{1}{2} \sum_{s=1}^t (X_s-h_s)^2 Q_s \right\} \right/ {{\cal Y}}_{t}\right),
$$
where $h_s\in {\cal Y}_{s}, \, s\ge 1$.
We claim the following generalization of Proposition~\ref{p:CM}.
\begin{proposition}\label{p:CMG}
The following equality holds for any $T\ge 1$:
$$
\mathcal{I}_T=\prod_{t=1}^T \left[1+Q_t\widetilde{\gamma}_{t}\right]^{1/2} \times \exp\left\{-\frac{1}{2} \frac{Q_t}{1+ Q_t \widetilde{\gamma}_t} \times \left[ h_t -\widetilde{Z}^{h}_{t} \right]^{2}\right\} \times \mathcal{M}_T,
$$
where $(\mathcal{M}_T)_{T\ge 1}$ is a martingale defined by :
\begin{multline}\label{martin}
\mathcal{M}_T = \prod_{t=1}^T \left[\frac{\sigma_t^{2}}{\bar{\sigma}_t^{2} }\right]^{1/2} \exp\left\{ \frac{1}{2\sigma_t^{2}} \, (Y_t - \pi_{t-1}(Y_t) )^{2} -\frac{1}{2\bar{\sigma}_t^{2}} \, (Y_t -\bar{ V}_{t} )^{2}
\right\},
\end{multline}
where
$$
\sigma_t^{2}=\mathop{\mathbb{E}}\nolimitsg(Y_t-\pi_{t-1}(Y_t))^{2},\,
\bar{\sigma}_t^{2}=\mathop{\mathbb{E}}\nolimitsg(Y_t-\bar{\pi}_{t-1}(Y_t))^{2},
$$
$$
\bar{ V}_{t}=\bar{\pi}_{t-1}(Y_t) -\overline{\gamma}_{_{Y\xi}}(t),\, \overline{\gamma}_{_{Y\xi}}(t)=\mathop{\mathbb{E}}\nolimitsg[(Y_t-\bar{\pi}_{t-1}(Y_t))
(\xi_{t-1}-\overline{\pi}_{t-1}(\xi_{t-1}))/
{\bar{{\cal Y}}}_{t-1}].
$$
\end{proposition}
\paragraph{Proof}To prove Proposition~\ref{p:CMG}
let us again calculate the ratio
$$
\frac{\mathcal{I}_t}{\mathcal{I}_{t-1}} = \frac{\pi_t(J_t)}{\pi_{t-1}(J_{t-1})}= \frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}\frac{\pi_{t}(J_{t-1})}{\pi_{t-1}(J_{t-1})}=
$$
$$
=\frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}\frac{\mathcal{M}_t}{\mathcal{M}_{t-1}}
$$
with a martingale $\mathcal{M}_t$ such that:
\begin{equation}\label{martgen}
\mathcal{M}_{t}=\prod_{s=1}^{t}\frac{\pi_{s}(J_{s-1})}{\pi_{s-1}(J_{s-1})}.
\end{equation}
The same arguments that we used in the proof of Proposition~\ref{p:CM} show that
$$
\frac{\pi_t(J_t)}{\pi_{t}(J_{t-1})}= \frac{\overline{\pi}_{t,t-1}(\exp \{-\frac{1}{2}Q_t (X_t-h_t)^2 - \xi_{t-1}\} )}{\overline{\pi}_{t,t-1}(\exp(-\xi_{t-1}))}
$$
$$
=(1+Q_t\widetilde{\gamma}(t))^{-1/2} \exp\left\{ - \frac{1}{2} \cdot\frac{Q_t}{1+Q_t\widetilde{\gamma}(t)}(\widetilde{Z}^{h}_{t}-h_t)^2 \right\}.
$$
To finish the proof we turn to the representation of the martingale $\mathcal{M}_t$ defined by \eqref{martgen}.
First of all we claim that
\begin{equation}\label{martbayes}
\frac{\mathcal{M}_{t}}{\mathcal{M}_{t-1}}=\left.\frac{\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))}{\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t},
\end{equation}
where $\widetilde{\pi}$ stands for the conditional expectation with respect to the measure $\widetilde{\Pg}$ such that
$\displaystyle{\frac{d \widetilde{\Pg}}{d \Pg}={\mathcal{M}_{T}}}$. Indeed, it is the direct consequence of the classical Bayes formula
$$
\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))=\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy){\mathcal{M}_{T}})}{\pi_{t-1}({\mathcal{M}_{t-1}})}
=\pi_{t-1}(\mathbb{I}(Y_t \in dy){\mathcal{M}_{t}}).
$$
To finish the proof it is sufficient to note that representations \eqref{martgen} and \eqref{martbayes} imply that
$$
\frac{\mathcal{M}_t}{\mathcal{M}_{t-1}}=\left.\frac{\pi_{t}(J_{t-1})}{\pi_{t-1}(J_{t-1})}=\left.\frac{\widetilde{\pi}_{t-1}(\mathbb{I}(Y_t \in dy))}{\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}=\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)\pi_{t}(J_{t-1}))}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}=
$$
$$
\left.\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)J_{t-1})}{\pi_{t-1}(J_{t-1})\pi_{t-1}(\mathbb{I}(Y_t \in dy))}\right|_{y=Y_t}.
$$
Again, we can use the same arguments that we used in the proof of Proposition~\ref{p:CM}:
$$
\frac{\pi_{t-1}(\mathbb{I}(Y_t \in dy)J_{t-1})}{\pi_{t-1}(J_{t-1})}=\frac{\overline{\pi}_{t-1}(\mathbb{I}(Y_t \in dy)\exp\{-\xi_{t-1}\})}{\overline{\pi}_{t-1}(\exp\{-\xi_{t-1}\})}=
$$
$$
=\frac{1}{\sqrt{2\pi \bar{\sigma}^{2}}}\exp\left(-\frac{(Y_t-\bar{V}_t )^{2}}{2\bar{\sigma}^{2}}\right).
$$
A direct consequence of Proposition~\ref{p:CMG} is the following statement:
\begin{corollary}
Let $\overline{h}$ be the solution of LEG (and RS) filtering problem \eqref{LEGdef} (and \eqref{rde}). Then the following equality holds for any $t \ge 1$:
$$
\overline{h}_t= \widetilde{Z}^{\overline{h}}_{t}.
$$
\end{corollary}
\section{Particular cases - again}\label{againpart}
\subsection{Markov type observations}
Here we turn to the case when the observations $(Y_t)_{t\ge 1}$ are conditionally independent given $X$. More precisely,
we deal with a signal-observation model
$
(X_t,Y_t)_{t\ge 1},
$
where the signal $X=(X_t)_{t\ge 1},\, X_t \in \mathbb{R}^{n}$ is an arbitrary Gaussian sequence with mean vector
$m=(m_t, t\geq 1)$ and covariance matrix $K =(K(t,s), t\geq 1,
s\geq 1)$, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg X_t=m_t,\quad\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)(X_s-m_s)^{\prime}=K(t,s)\,.
t\geq 1\,,\; s\geq 1\,,
$$
The observation process $Y=(Y_t,\, t\ge 1)$ is given by
\begin{equation}\label{obser}
Y_t= A_t X_t +\mathop{\mathrm{var}}\nolimitsepsilon_t,
\end{equation}
for some sequence $A=(A_t,\, t\ge 1)$ of $m\times n$ matrices, where $\mathop{\mathrm{var}}\nolimitsepsilon=(\mathop{\mathrm{var}}\nolimitsepsilon_{t})_{t\ge 1}$ is a sequence of i.i.d. $\mathcal{N}(0,Id)$ random variables and $\mathop{\mathrm{var}}\nolimitsepsilon$ and $X$ are independent.
In this case we can write the multidimensional analogue of the equation \eqref{zhtild}, which is nothing else but the dynamic equation for the process $\widetilde{Z}^{h}$ defined by \eqref{Ztildainterpr}. We obtain:
$$
\tilde{Z}^{h}_{t}=m_t +\sum_{l=1}^{t-1}\overline{\gamma}(t,l)[Id+\overline{\gamma}_{l}(A^{\prime}_{l}A_{l}-\mu Q_{l})]^{-1}\mu Q_{l} (h_l - \tilde{Z}_{l}^h) + \sum_{l=1}^{t}\overline{\gamma}(t,l) A^{\prime}_{l} (Y_l - A_l \tilde{Z}_{l}^{h}),
$$
where the matrix $\overline{\gamma}(t,l)$ satisfies the following equation (which is the multidimensional analog of the equation \eqref{GAMMABAR}):
\begin{equation}\label{gammamultdim}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} \overline{\gamma}(t,l)\bar{A}_{l}^{\prime}[Id +\bar{A}_{l}\overline{\gamma}_l\bar{A}_{l}^{\prime}]^{-1}\bar{A}_{l}^{\prime} \overline{\gamma}^{\prime}(s,l),
\end{equation}
where $\bar{A}_{l}=\left(
\begin{array}{c}
A_l \\
-\mu Q_l \\
\end{array}
\right).$
Now the solution of the LEG (and RS) filtering problem $\bar{h}$ is nothing else but:
$$
\bar{h}_{t}=m_t + \sum_{l=1}^{t}\overline{\gamma}(t,l) A^{\prime}_{l} (Y_l - A_l\bar{h}_{l}).
$$
\subsection{Markov type observations, correlated signal and observation noises}
Let us drop the assumption that $X$ and $\mathop{\mathrm{var}}\nolimitsepsilon $ in the observation equation \eqref{obser} are independent. Denote by $K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s)$ the covariance matrix of the signal and the observation noise, \textit{i.e.},
$$
\mathop{\mathbb{E}}\nolimitsg (X_t-m_t)\mathop{\mathrm{var}}\nolimitsepsilon_s^{\prime}=K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s), \quad
t\geq 1\,,\; s\geq 1.
$$
It can be checked that the following slight modification of the previous statement holds. \\ Let the matrix $\overline{\gamma}(t,l)$ be the unique solution of the following equation
\begin{multline}\label{gammacorrel}
\overline{\gamma}(t,s)=K(t,s)-\sum_{l=1}^{s-1} [\overline{\gamma}(t,l)\bar{A}_{l}^{\prime}+\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)]
\\
[Id +\bar{A}_{l}\overline{\gamma}_l\bar{A}_{l}^{\prime} +\bar{A}_{l}\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)+\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)^{\prime}\bar{A}_{l}^{\prime}]^{-1}
\\
[\bar{A}_{l}^{\prime} \overline{\gamma}^{\prime}(s,l)+\bar{K^{\prime}}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(s,l)],
\end{multline}
with $\bar{A}_{l}=\left(
\begin{array}{c}
A_l \\
-\mu Q_l \\
\end{array}
\right),\quad \bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)=(K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)\quad \mathbf{0}).$\\
Then the solution of the LEG (and RS) filtering problem $\bar{h}$ satisfies the following equation
\begin{equation}\label{hbarcorrel}
\bar{h}_{t}=m_t + \sum_{l=1}^{t}[\bar{K}_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,l)+\overline{\gamma}(t,l) A^{\prime}_{l}][Id +A_{l}K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(l,l)]^{-1} (Y_l - A_l\bar{h}_{l}).
\end{equation}
\subsection{Observations containing Moving Averages of order~1}\label{MA1XY}
Now we consider the case of a MA(1) type process, {\em i.e.}, the following signal-observation model:
$$
X_{t} = \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t} + \lambda \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_{t-1}\,; t\ge 1\,,
$$
$$
Y_{t} =\alpha_t X_t + \mathop{\mathrm{var}}\nolimitsepsilon_{t} + \beta \mathop{\mathrm{var}}\nolimitsepsilon_{t-1}\,; t\ge 1\,,
$$where $( \mathop{\mathrm{var}}\nolimitsepsilon_t, \widetilde{\mathop{\mathrm{var}}\nolimitsepsilon}_t)_{t\ge 0}$~is a sequence of i.i.d.
Gaussian variables and $\lambda$ and $\beta$ are real numbers.\\
Let us denote by $A_t$ the row $\bar{A}_{t}=(\alpha_{t}\quad \beta)$ and by $\bar{X}_{t}$ the vector $\bar{X}_{t}=\left(
\begin{array}{c}
X_t \\
\mathop{\mathrm{var}}\nolimitsepsilon_{t-1} \\
\end{array}
\right).$
Of course
$\bar{X}$ is centered, has the
covariance matrix
$K(t,s) =\left(
\begin{array}{cc}
(1+\lambda^{2}) \mathbf{1}(s=t-1)+ \lambda \mathbf{1}(s=t)& 0 \\
0 & \mathbf{1}(s=t) \\
\end{array}
\right)$
and the covariance between $\bar{X}$ and $\mathop{\mathrm{var}}\nolimitsepsilon$ is $K_{_{X\mathop{\mathrm{var}}\nolimitsepsilon}}(t,s)=\left(
\begin{array}{c}
0 \\
\mathbf{1}(s=t-1) \\
\end{array}
\right).$
The solution $\overline{\gamma}$ to (\ref{gammacorrel}) then can be found as:
$$
\overline{\gamma}(t,s) =\mathbf{0}\,,\; s < t-1\,; \quad \overline{\gamma}(t,t-1) = \left(
\begin{array}{cc}
\lambda & 0 \\
0 & 0 \\
\end{array}
\right)\,,\;
t\ge 1\,,
$$
and $\overline{\gamma}(t,t)=\overline{\gamma}_{t}$ where $\overline{\gamma}_{t}$ is
the solution of the equation:
\begin{multline*}
\overline{\gamma}_{t} =\left(
\begin{array}{cc}
1+\lambda^{2} & 0 \\
0 & 1 \\
\end{array}
\right) +
\\
+\left(
\begin{array}{cc}
\lambda\alpha_{t-1} & -\lambda\mu Q_{t-1}\\
0 & 1 \\
\end{array}
\right)\left[ Id + \left(
\begin{array}{cc}
\alpha_{t-1} & \beta \\
-\mu Q_{t-1} & 0 \\
\end{array}
\right)\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
\alpha_{t-1} & -\mu Q_{t-1} \\
\beta & 0 \\
\end{array}
\right)\right]^{-1}
\\
\times \left(
\begin{array}{cc}
\lambda\alpha_{t-1} & 0 \\
-\lambda\mu Q_{t-1} & 1 \\
\end{array}
\right) \,,\;t\ge 1\,;
\quad \overline{\gamma}_0= \mathbf{0}.
\end{multline*}
provided that this equation has the unique nonnegative definite solution.
Moreover, equation
\eqref{hbarcorrel} for the solution $\overline{h}$ of the LEG filtering problem (\ref{LEGdef}) can be reduced to the
following one:
\begin{equation*}
\overline{h}_t=\Lambda_{t}^{-1} \left(
\begin{array}{c}
\lambda\alpha_{t} \\
1+\beta \\
\end{array}
\right) [Y_{t-1}-A_{t-1}\overline{h}_{t-1}] +\Lambda_{t}^{-1}\overline{\gamma}_t \left(
\begin{array}{c}
\alpha_{t} \\
\beta \\
\end{array}
\right)Y_t, \, t\ge 1,\, \overline{h}_0=0,
\end{equation*}
with $\Lambda_{t}= Id + \overline{\gamma}_{t}A_{t}^{\prime}A_{t}.$
\subsection{Observations containing Gaussian AR(1) process }\label{GMSXY}
In this part we concentrate on the case of a
Gaussian AR(1) type process $Y$, {\em i.e.},
\begin{equation}\label{model ARY}
Y_{t}= \alpha_{t} X_{t}+\mathop{\mathrm{var}}\nolimitsepsilon_{t}\,,\; t\ge 1\,;
\quad Y_{0}=0\,,
\end{equation}
where
$$
\mathop{\mathrm{var}}\nolimitsepsilon_{t}=b \mathop{\mathrm{var}}\nolimitsepsilon_{t-1}+\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t},
$$
and $(\widetilde\mathop{\mathrm{var}}\nolimitsepsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables independent of $X$.
We also suppose that the signal $X$ is a Gaussian AR(1) process,
{\em i.e.},
\begin{equation}\label{model ARY}
Y_{t}= a_{t} X_{t}+\epsilon_{t}\,,\; t\ge 1\,;
\quad X_{0}=0\,,
\end{equation}
and also $(\epsilon_{t}, \; t=1,2,\dots)$ is a sequence of i.i.d.
standard Gaussian random variables.
Proceeding as in Sections \ref{GMS} and \ref{MA1XY} we can write the dynamic equation for the solution of LEG and RS filtering problems $\bar{h}$. Namely, $\bar{h}$ is the first component $\bar{h}^{1} $ of the solution of the following recursive equation:
\begin{equation*}
\overline{h}_t=\overline{\gamma}_t A_t^{\prime}[Y_{t}-A_{t}\overline{h}_{t}] +\left(
\begin{array}{cc}
a_{t} & 0 \\
0 & b \\
\end{array}
\right) \overline{h}_{t-1} + \left(
\begin{array}{c}
0 \\
1 \\
\end{array}
\right)[Y_{t-1}-A_{t-1}\overline{h}_{t-1}] \, t\ge 1,\, \overline{h}_0=\mathbf{0},
\end{equation*}
where $A_t= (\alpha_{t} \quad \beta)$ and $\overline{\gamma}$ is the unique nonnegative defined solution of the Ricatti equation:
\begin{multline*}
\overline{\gamma}_t=\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right)\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right) -\left[\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right)\overline{\gamma}_{t-1}\bar{A}^{\prime}_{s-1} +\left(
\begin{array}{ccc}
0 & 0 & 0\\
0 & 1 & 0\\
\end{array}
\right)\right]
\\
\times\left[Id +\bar{A}_{t-1} \overline{\gamma}_{t-1}\bar{A}^{\prime}_{s-1}\right]^{-1}\left[\bar{A}_{t-1}\overline{\gamma}_{t-1}\left(
\begin{array}{cc}
a_t & 0 \\
0 & b\\
\end{array}
\right) +\left(
\begin{array}{cc}
0 & 0 \\
0 & 1 \\
0 & 0 \\
\end{array}
\right)\right],
\end{multline*}
with $\bar{A}_{t}=\left(
\begin{array}{cc}
\alpha_{t} & b \\
-\mu Q_t & 0 \\
0 & 0 \\
\end{array}
\right).$
\end{document} |
\begin{equation}gin{document}
\title{Quantum optical memory protocols in atomic ensembles}
\author{Thierry Chaneli\`ere}
\affiliation{Laboratoire Aim\'e Cotton, CNRS, Univ. Paris-Sud, ENS Paris-Saclay, Universit\'e Paris-Saclay, 91405 Orsay, France}
\author{Gabriel H\'etet}
\affiliation{Laboratoire Pierre Aigrain, Ecole normale sup\'erieure, PSL Research University, CNRS, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'es, Universit\'e Paris Diderot, Sorbonne Paris-Cit\'e, 24 rue Lhomond, 75231 Paris Cedex 05, France}
\author{Nicolas Sangouard}
\affiliation{Quantum Optics Theory Group, Department of Physics, University of Basel, CH-4056 Basel, Switzerland}
\begin{equation}gin{abstract}
We review a series of quantum memory protocols designed to store the {quantum} information carried by {light} into atomic ensembles. {In particular, we show how a simple semiclassical formalism allows to gain insight into various memory protocols and to highlight strong analogies between them. These analogies naturally lead to a classification of light storage protocols in{to} two categories, {namely} photon echo and {\it slow-light} memories.} {We focus on the storage and retrieval dynamics as a key step to map the optical information into the atomic excitation.} We finally {review} various criteria adapted for both continuous variables and photon-counting measurement techniques to {certify} the quantum nature of {these} memory {protocols}.
\end{abstract}
\maketitle
\section{Introduction}
{The potential of quantum information sciences for applied physics is currently highlighted by coordinated and voluntarist policies.} In a global scheme of {probabilistic} quantum information processing, {quantum} memory is a key element {to synchronize independent events \cite{bussieres2013prospective}}. {Memory for light} can be more generally considered as an interface between light (optical or radiofrequency) and a material medium \cite{hammerer2010quantum} where the quantum information is mapped from one form (optical for example) to the other (atomic excitation) and {\it vice versa}. {In this chapter, we review quantum protocols for light storage}. The objective is not to make a comparative and exhaustive review of the different systems {or applications} of interest. Analysis {along these lines} can be found in many review articles \cite{bussieres2013prospective, hammerer2010quantum, lvovsky2009optical, afzelius2010photon, review_Simon2010, review_Heshami_2016, review_Ma_2017} perfectly reflecting the state of the art. Instead, we focus on pioneering protocols {in atomic ensembles} that we analyze with the same formalism to extract the common {features} and differences.
{First,} we consider two representative classes of storage protocols, the photon echo in section \ref{sec:2PE} and the {\it slow-light} memories in section \ref{sec:SL}. In both cases, we first derive a minimalist semi-classical Schr\"odinger-Maxwell model to describe the propagation of a weak signal {in an atomic ensemble}. Two-level atoms are sufficient to characterize the photon echo protocols among which the standard two-pulse photon echo is the historical example (section \ref{sec:2PE}). On the contrary, as in the widely studied {\it stopped-light} by means of electromagnetically induced transparency (EIT), the minimal atomic structure consists of three levels (section \ref{sec:SL}). {In both cases}, however, the semi-classical Schr\"odinger-Maxwell formalism is sufficient to describe the optical storage dynamics and evaluate the theoretical efficiencies.
To fully replace our analysis in the context of the quantum storage, we finally derive a variety of criteria in section \ref{sec:certification} to {certify} the quantum nature of optical memories. Our approach is pragmatic in this section as we do not develop a fully quantized propagation model mirroring our semi-classical analysis in \ref{sec:2PE} and \ref{sec:SL}. Instead, we use an atomic chain quantum toy model to characterize the {noise of various storage protocols}. Criteria depending on experimentally accessible parameters are {reviewed} for both continuous and discrete variables.
\section{Photon echo memories}\label{sec:2PE}
The photon echo technique as the optical {\it alter ego} of the spin echo has been considered early as a spectroscopic tool \cite{Kopvillem,Hartmann64, Hartmann66, Hartmann68}. Its extensive description can be found in many textbooks as an example of a coherent transient light-atoms interaction \cite{allen2012optical}. Due to its coherent nature and many experimental realizations over the last decades, the photon echo has been reconsidered in the context of quantum storage \cite{afzelius2010photon}. In this section, we will first establish the formalism describing the propagation and the retrieval of week signals in a two-level inhomogeneous atomic medium. We then describe and evaluate the efficiency of the standard two-pulse photon echo from the point of view of a storage protocol. The latter is not immune to noise but has stimulated the design of noise free alternatives, namely the Controlled reversible inhomogeneous broadening and the Revival of silenced echo that we will describe using the same formalism.
The signal propagation and photon echo retrieval can be modeled by the Schr\"odinger-Maxwell equations in one dimension (along $z$) with an inhomogeneously broadened two-level atomic ensemble that we will first illustrate.
\subsection{Two-level atoms Schr\"odinger-Maxwell model}
On one side, the atomic evolution under the field excitation is given by the Schr\"odinger equation and on the other side, the field propagation is described by the Maxwell equation that we successively remind.
\subsubsection{Schr\"odinger equation for two-level atoms}
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.7\linewidth]{2level_3level.eps}}
\caption{Two-level (left) and three-level atoms (right) used to describe the photon echo (section \ref{sec:2PE}) and the {\it slow-light} memories (section \ref{sec:SL}).The signal $\mathcal{E}$ is applied on the $|g\rangle$ and $|e\rangle$ transition. For the three-level atoms, a control field $\Omega$ is applied on the $|s\rangle$ and $|e\rangle$ transition.}
\label{fig:2level_3level}
\end{figure}
For two-level atoms, labeled $|g\rangle$ and $|e\rangle$ for the ground and excited states (see fig.\ref{fig:2level_3level}, left), the rotating-wave probability amplitudes $C_g$ and $C_e$ respectively are governed by the time-dependent Schr\"odinger equation \cite[eq. (8.8)]{shore2011manipulating}:
\begin{equation}gin{align}
i \partial_t \left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
0 &\displaystyle \frac{\mathcal{E}^*}{2} \\
\displaystyle\frac{\mathcal{E}}{2} & -\Delta \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
\end{array}\right]
\label{bloch2}\end{align}
where $\mathcal{E}(z,t)$ is the complex envelope of the input signal expressed in units of Rabi frequency. $\Delta$ is the laser detuning.
The atomic variables $C_g$ and $C_e$ depend on $z$ and $t$ for a given detuning $\Delta$. The detunings can be made time-dependent
\cite{loy1974observation,vitanov2001laser}, position-dependent or both \cite{hetet2008electro} but this is not the case here.
Decay terms can be added {by-hand} by introducing a complex detuning $\Delta \rightarrow \Delta - i \Gamma$ where $\Gamma$ is the decay rate of the excited state $|e\rangle$ \footnote{We do not distinguish the decay terms for the population and the coherence. This is an intrinsic limitation of the Schr\"odinger model as opposed to the density matrix formalism (optical Bloch equations).}.
\subsubsection{Maxwell propagation equation}
The propagation of the signal $\mathcal{E}(z,t)$ is described by the Maxwell equation that can be simplified in the slowly varying envelope approximation \cite[eq. (21.15)]{shore2011manipulating}. This reads for an homogeneous ensemble whose linewidth is given by the decay term $\Gamma$:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle {i \alpha} \Gamma C_g^* C_e \label{MB_M_hom}
\end{equation}
The term $C_g C_e^* $ is the atomic coherence on the $|g\rangle \rightarrow |e\rangle$ transition directly proportional to the atomic polarization. The light coupling constant is included in the absorption coefficient $\alpha$ (inverse of a length unit), thus the right hand side represents the macroscopic atomic polarization.
The Maxwell equation can be generalized to an inhomogeneously broaden ensemble \cite{allen2012optical}:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) C_g^* C_e \mathrm{d}\Delta \label{MB_M_inhom}
\end{equation}
where $ g\left(\Delta\right) $ is the normalized inhomogeneous distribution.
Photon echo memories precisely rely on the inhomogeneous broadening as an incoming bandwidth. The set of equations (\ref{bloch2}\&\ref{MB_M_inhom}) are then relevant in that case. The resolution can be further simplified for weak $\mathcal{E}(z,t)$ signals as expected for quantum storage. This is the so-called perturbative regime. More importantly, the perturbative limit is necessary to ensure the linearity of the storage scheme and is then not only a formal simplification. The perturbative expansion should be used with precaution when photon echo protocols are considered. When strong (non-perturbative) $\pi$-pulses are used to trigger the retrieval as a coherence rephasing, they unavoidably invert the population. This interplay between rephasing and inversion is the essence of the photon echo technique. Population inversion should be avoided because spontaneous emission induces noise \cite{ruggiero}. We will nevertheless first consider the standard two-pulse photon echo scheme because this is the ancestor and an inspiring source for modified photon echo schemes adapted for quantum {storage}.
\subsubsection{Coherent transient propagation in an inverted or non-inverted medium}
The goal of the present section is to describe the propagation of a weak signal representing both the incoming signal and the echo. For the standard two-pulse photon echo (see section \ref{2PE}), the echo is emitted in an inverted medium so we will consider both an inverted and a non-inverted medium corresponding to the ideal storage scheme (see \ref{CRIB} and \ref{ROSE}).
The propagation is coherent in the sense that the pulse duration is much shorter than the coherence time. The decay term (that could be introduced with a complex detuning $\Delta$) is fully neglected in eq.\eqref{bloch2}.
The coherent propagation is defined in the perturbative regime. This latter should be defined with precaution if the medium is inverted or not. The coherence term $\mathcal{P}=C_g^* C_e$ appearing in the propagation equation (eq.\ref{MB_M_hom} or \ref{MB_M_inhom}) is described by rewriting the Schr\"odinger equation
as
\begin{equation}gin{equation}
\partial_t \mathcal{P} =i\Delta \mathcal{P} + \left( C_e^* C_e-C_g^* C_g \right) i \frac{\mathcal{E}}{2}
\end{equation}
The reader more familiar with the optical Bloch equations can directly recognize the evolution of the coherence term (non-diagonal element of the density matrix) where the term $ \left( C_e^* C_e-C_g^* C_g \right) $ is the population difference (diagonal element{s}).
For a non-inverted medium, the atoms are essentially in the ground state, so in the perturbative limit $ \left( C_e^* C_e-C_g^* C_g \right) \rightarrow -1$. The population goes as the second order in field excitation thus justifying the perturbative expansion where the coherences $\mathcal{P}$ goes as the first order. Along the same line, $ \left( C_e^* C_e-C_g^* C_g \right) \rightarrow 1$ for an inverted medium. The atomic evolution reads as
\begin{equation}gin{equation}
\partial_t \mathcal{P} =i\Delta \mathcal{P} \mp i \frac{\mathcal{E}}{2}
\end{equation}
where $\mp$ indicates if the medium is non-inverted (ground state) or inverted (excited state). This can be alternatively written in an integral form as
\begin{equation}gin{equation}
\mathcal{P}(z,t) =\mp \frac{i}{2} \int_{-\infty}^t \mathcal{E}\left(z,t^\prime\right) \exp \left(i\Delta\left(t-t^\prime\right)\right) \ensuremath{\mathrm{d}} t^\prime \label{integral_form}
\end{equation}
As given by eq.\eqref{MB_M_inhom}, the propagation in the inhomogeneous medium is described by
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}_\Delta(z,t) \ensuremath{\mathrm{d}} \Delta \label{MB_M_inhom_P}
\end{equation}
We remind by an index $ \mathcal{P}_\Delta$ that the coherence term depends on the detuning $\Delta$ as a parameter.
To avoid the signal temporal distortion, the incoming pulse bandwidth should be narrower than the inhomogeneous broadening given by the distribution $g\left(\Delta\right) $ so we can safely assume $g\left(\Delta\right) \rightarrow 1$. The double integral term $\displaystyle \int_\Delta \mathcal{P}_\Delta \ensuremath{\mathrm{d}} \Delta$ from eq.\eqref{integral_form} can be simplified by writing $\displaystyle \int_\Delta \exp \left(i\Delta\left(t-t^\prime\right)\right) \ensuremath{\mathrm{d}} \Delta \rightarrow 2 \pi \delta_{t^\prime=t} $ as a representation of the Dirac peak $\delta_0$
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
\mp\displaystyle\frac{\alpha}{2} \mathcal{E}(z,t)\label{bouguer0}
\end{equation}
Eq.\eqref{bouguer0} is the absorption law or gain if the medium is inverted. The absorption law was at first discovered by Bouguer \cite{bouguer1760traite}, today known as the Bouguer-Beer-Lambert law. The description can be even more simplified by noting that the pulse length is usually much longer the medium spatial extension. The term $\displaystyle \frac{1}{c}\partial_t$ can be dropped leading to the canonical version of the Bouguer-Beer-Lambert law \cite{allen2012optical}.
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)=
\mp\displaystyle\frac{\alpha}{2} \mathcal{E}(z,t)\label{bouguer}
\end{equation}
This form can be alternatively obtained by writing the equation in the moving frame at the speed of light. Introducing the moving frame may be a source of mistake when the backward retrieval configuration is considered (see section \ref{CRIB}). {Anyway,} the moving frame {does not need to be introduced} because the medium length $L$ is in practice much shorter than the pulse extension. In other words, the delay induced by the propagation $L/c$ is negligible with respect to the pulse duration. The term {\it propagation} is in that case arguable when the term $\displaystyle \frac{1}{c}\partial_t$ is absent. Propagation should be considered in the general sense. The absorption coefficient in eq.\eqref{bouguer} defines a propagation constant. This latter is real as opposed to a propagation delay which would appear as a complex (purely imaginary) constant.
The Bouguer-Beer-Lambert law can be obtained equivalently with an homogeneous medium including the coherence decay term. This is not the case here. We insist: there is no decay and the evolution is fully coherent. To illustrate this fundamental aspect of the coherent propagation, we can show that the field excitation is actually recorded into the medium. On the contrary, with a decoherence term, the field excitation would be lost in the environment. The complete field excitation to coherence mapping is a key ingredient of the photon echo memory scheme.
\subsubsection{Field excitation to coherence mapping}\label{mapping}
In the coherent propagation regime, the evolution of the atomic and optical variables is fully coherent. Let {us} restrict the discussion to the case of interest, namely the photon echo scheme of an initially non-inverted (ground state) medium. The field is absorbed following the Bouguer-Beer-Lambert law (eq.\ref{bouguer}). This disappearance of the field is not due to the atomic dissipative decay but to the inhomogeneous dephasing. For example, in an homogeneous sample, the absorption of the laser beam can be due to spontaneous emission: the beam is depleted because the photons are scattered in other modes. In an inhomogeneous sample, the beam depletion is due to dephasing and not dissipation. In other words, the forward scattered dipole emissions destructively interfere. Since the evolution is coherent, the field should be fully mapped into the atomic excitation. In that case, the expression \eqref{integral_form} can be reconsidered by noting that after the absorption process, the integral boundary can be pushed to $+\infty$ as
\begin{equation}gin{align}
\mathcal{P}_\Delta(z,t)& =- \frac{i}{2} \exp \left(i \Delta t \right) \int_{-\infty}^{+\infty} \mathcal{E}\left(z,t^\prime\right) \exp \left(- i\Delta t^\prime \right) \ensuremath{\mathrm{d}} t^\prime \\ &=- \frac{i}{2} \exp \left(i \Delta t \right) \tilde{\mathcal{E}}(z,\Delta) \label{mapping}
\end{align}
where $ \tilde{\mathcal{E}}(z,\omega)$ is the Fourier transform of the incoming pulse $\displaystyle \mathcal{E}\left(z,t^\prime\right)$ \footnote{ We define the Fourier transform pairs as \begin{equation}gin{align}
\tilde{f}(\omega)&=\int_t f(t) \exp( -i\omega t)\mathrm{d} t\\
{f}(t)&=\frac{1}{2\pi}\int_\omega \tilde{f}(\omega) \exp(i\omega t)\mathrm{d} \omega
\end{align}
}. This expression tells that the incoming spectrum is entirely mapped into the atomic excitation. More precisely, each class $\mathcal{P}_\Delta$ in the atomic distribution actually records the corresponding part in the incoming spectrum $\tilde{\mathcal{E}}(z,\Delta)$. The term $\exp \left(i \Delta t \right)$ simply reminds us that the coherence freely oscillates after the field excitation. An exponential decay term could be added by-hand by giving an imaginary part to the detuning $\Delta$.
This mapping stage when the field is recorded into the atomic coherences of an inhomogeneous medium is the initial step of the different photon echo memory schemes. Various techniques have been developed to retrieve the signal after the initial absorption stage. The inhomogeneous dephasing is the essence of the field to coherence mapping since the field spectrum is recorded in the inhomogeneous distribution. The retrieval is in that sense always associated to a rephasing or compensation of the inhomogeneous dephasing. This justifies the term photon echo used to classify this family of protocols. We will start by describing the standard two-pulse photon echo (2PE). Despite a clear limitation for quantum storage, this is an enlightening historical example. Its descendants as the so-called Controlled reversible inhomogeneous broadening (CRIB) and Revival of silenced echo (ROSE) have been precisely designed to avoid the deleterious effect of the $\pi$-pulse rephasing used in the 2PE sequence.
\subsection{Standard two-pulse photon echo}\label{2PE}
Inherited from the magnetic resonance technique \cite{hahn}, the coherence rephasing {and the subsequent field reemission} is triggered by applying a strong $\pi$-pulse (fig.\ref{fig:2PE}). The possibility to use the 2PE for pulse storage has been mentioned early in the context of optical processing \cite{Carlson:83}. The retrieval efficiency can indeed be remarkably high \cite{moiseev1987some,azadeh, SjaardaCornish:00}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{2PE.eps}}
\caption{Standard two-pulse photon echo sequence. Inherited from magnetic resonance, $\pi/2-\pi$ sequence (delayed by $\tau$, in magenta) produces an echo at $2\tau$ (in red). When considered for optical storage, the first pulse is weak (in blue) and longer than the rephasing $\pi$-pulse.}
\label{fig:2PE}
\end{figure}
This particularity has attracted a renewed curiosity in the context of quantum information \cite{rostovtsev2002photon, moiseev_echo_03}.
\subsubsection{Retrieval efficiency}
The retrieval efficiency can be derived analytically from the Schr\"odinger-Maxwell model. Following the sequence in fig.\ref{fig:2PE}, the signal absorption is first described by the Bouguer-Beer-Lambert law (eq.\ref{bouguer}). The initial stage is followed by a free evolution during a delay $\tau$. The $\pi$-pulse will trigger a retrieval. The action of a strong pulse on the atomic variables is described by the propagator
\begin{equation}gin{align}
\left[
\begin{equation}gin{array}{c}
C_g\left(\tau^+\right) \\
C_e\left(\tau^+\right) \\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
\displaystyle \cos(\theta/2) & \displaystyle -i \sin(\theta/2) \\
\displaystyle -i \sin(\theta/2) &\displaystyle \cos(\theta/2) \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \left(\tau^-\right) \\
C_e \left(\tau^-\right) \\
\end{array}\right]
\end{align}
which links the atomic variables just before ($\tau^-$) and after ($\tau^+$) a general $\theta$-area pulse. This solution of the canonical Rabi problem is only valid for a {very short} pulse (hard pulse). More precisely, in the atomic evolution eq.\eqref{bloch2}, the Rabi frequency {must be} much larger than the detuning. In the 2PE scheme, this means that the atoms excited by the signal (first pulse) are uniformly (spectrally) covered by the strong rephasing pulse. This translates in the time domain as a condition on the relative pulse durations: the $\pi$-pulse {must} be much shorter than signal. This aspect appears as an initial condition for the pulse durations but is also intimately related to the transient coherent propagation of strong pulses among which $\pi$-pulses are a particular case. This will be discussed in more details in the appendix \ref{strong_pulse}. Assuming the ideal situation of the uniform $\theta=\pi$ pulse area, the propagator takes the simple form
$$
\left[
\begin{equation}gin{array}{ccc}
0 & -i \\
-i & 0 \\
\end{array}\right]$$
fully defining the effect of the $\pi$-pulse on the stored coherence
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^+\right)=-\mathcal{P}_\Delta^*\left(\tau^-\right) =- \frac{i}{2} \exp \left(-i \Delta \tau \right) \tilde{\mathcal{E}}^*(z,\Delta)\label{P_tauplus}
\end{equation}
{T}he free evolution resumes by adding the inhomogeneous phase $\Delta \left( t- \tau\right)$
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau\right)=\mathcal{P}_\Delta\left(\tau^+\right) \exp \left(i \Delta \left( t- \tau\right) \right) =- \frac{i}{2} \exp \left(i \Delta \left( t- 2\tau\right) \right) \tilde{\mathcal{E}}^*(z,\Delta)\label{P_echo}
\end{equation}
In the expression \eqref{P_echo}, we see that the inhomogeneous phase $\Delta \left( t- 2\tau\right)$ is zero at the instant $t=2\tau$ of the retrieval thus justifying the term rephasing.
The propagation of the retrieved echo $\mathcal{E}^R$ follows eq.\eqref{MB_M_inhom_P}. The source term on the right-hand side has now two contributions. The first one gives the Bouguer-Beer-Lambert law (eq \ref{bouguer}) for the echo field $\mathcal{E}^R$ itself. A critical aspect of the 2PE is the population inversion induced by the $\pi$-pulse. The intuition can be confirmed by calculating from the propagator $ \left( C_e^*\left(\tau^+\right) C_e\left(\tau^+\right)-C_g^*\left(\tau^+\right) C_g\left(\tau^+\right) \right)$ to the first order by noting that $ C_g\left(\tau^-\right) \simeq 1$. The echo field $\mathcal{E}^R$ exhibits gain. The second one comes from the coherence initially excited by the signal freely oscillating after the $\pi$-pulse rephasing. In other words, the coherences at the instant of retrieval are the sum of the free running term due to the signal excitation from eq.\eqref{P_echo} and the contribution from the echo field itself.
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
+\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) -\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}_\Delta(z,t>\tau) \ensuremath{\mathrm{d}} \Delta
\end{equation}
The integral source term representing the build-up of the macroscopic polarisation at the instant of retrieval is directly related to the signal field excitation $\mathcal{E}$ which appears as the inverse Fourier transform of $\tilde{\mathcal{E}}^*(z,\Delta)$ from eq.\eqref{P_echo}, {that is}
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
+\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}^*(z,2\tau-t) \label{eq_echo}
\end{equation}
{Eq.\eqref{eq_echo} is simple but rich because it can be modified by-hand to describe the descendants of the 2PE protocol that are suitable for quantum storage as we will see in sections \ref{CRIB} and \ref{Rose}. Note that it can be adapted to account for rephasing pulse areas $\theta$ that are not $\pi.$ They lead to imperfect rephasing and incomplete medium inversion thus modifying the terms in eq.\eqref{eq_echo} \cite{ruggiero}. Very general expressions for the efficiency as a function of $\theta$ can be analytically derived \cite{moiseev1987some}.}
{Knowing that the incoming signal follows the Bouguer-Beer-Lambert law (eq.\ref{bouguer}) of absorption $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right),$ the efficiency of the 2PE can be obtained as a function of optical depth $d=\alpha L$ from the ratio between the output and input intensities
\begin{equation}gin{equation}
\eta=\frac{|\mathcal{E}^R(L,t)|^2}{|\mathcal{E}(0,2\tau-t)|^2}\label{efficiency}
\end{equation}
For a $\pi$-rephasing pulse, we find
\begin{equation}gin{equation}\label{etaPi}
\eta\left(d\right)=\left[\exp\left(d/2\right)-\exp\left(-d/2\right)\right]^2 =4~{\rm sinh}^2\left(d/2\right)
\end{equation}
}
{At large optical depth $d$, the efficiency scales as $\exp\left(d\right)$ resulting in an exponential amplification of the input field. This amplification prevents the 2PE to be used as a quantum storage protocol. The simplest but convincing argument uses the no-cloning theorem \cite{nocloning}. Alternatively, we can apply various criteria to certify the quantum nature of the memory on the echo and show that none of these criteria witnesses its non-classical feature, as wee will see section \ref{sec:certification}.}
In fig.\ref{fig:2PE_simul} (bottom), we have represented this efficiency scaling (eq.\ref{etaPi}) that we compare with a numerical simulation of a 2PE sequence solving the Schr\"odinger-Maxwell model. For a given inhomogeneous detuning $\Delta$, we calculate the atomic evolution eq.\eqref{bloch2} by using a fourth-order Runge-Kutta method. After summing over the inhomogeneous broadening, the output pulse is obtained by integrating eq.\eqref{MB_M_inhom} along $z$ using the Euler method.
In the numerical simulation, there is no assumption on the $\pi$-pulse duration with respect to the signal bandwidth (as needed to derive the analytical formula eq.\ref{etaPi}). The excitation pulses are assumed Gaussian as shown for the incoming and the outgoing pulses of a 2PE sequence after propagation though an optical depth $d= 2$ (fig.\ref{fig:2PE_simul}, top). We consider different durations for the $\pi$-pulse (of constant area) and a fixed signal duration.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.85\linewidth]{2PE_simul.eps}}
\caption{Top: Numerical simulation of a 2PE sequence with a weak incoming signal (area $\pi/20$). The signal and the echo fields are in blue and have been magnified by a factor 10 (shaded area). The incoming pulses are in solid lines. The outgoing pulses after propagation though $d = 2$ are in dashed line. The $\pi$-pulse is two times shorter than the signal. Bottom: Storage efficiencies (see text for the definition) as a function of the optical depth $d$ . The black line is the analytical solution eq.\eqref{etaPi}. Three simulations have been performed depending on the relative duration of the $\pi$-pulse with respect to the signal: when the $\pi$-pulse has the same duration than the signal (ratio 1), when it is 2 times (ratio 2) and 10 times shorter (ratio 10). The circle corresponds to the 2PE sequence on top.}
\label{fig:2PE_simul}
\end{figure}
From the numerical simulation, the efficiency is evaluated by integrating under the intensity curves of the echo (shades area). This latter reaches 152\% for the sequence of fig.\ref{fig:2PE_simul} (top), larger than 100\% as expected for an inverted medium. Still, this is much smaller than the 552\% {efficiency} expected from eq.\eqref{etaPi} with $d= 2$. This discrepancy is essentially explained by the $\pi$-pulse distortion through propagation (magenta dashed line in fig.\ref{fig:2PE_simul},top) than can be observed numerically. The $\pi$-pulse should stay shorter than the signal to properly ensure the coherence rephasing. This is obviously not the case because the pulse is distorted as we briefly analyze in appendix \ref{strong_pulse} with the energy and area conservation laws.
As a summary, we have evaluated numerically the efficiencies when the $\pi$-pulse has the same duration than the signal (ratio 1), when it is 2 times and 10 times shorter (ratio 2 and 10 respectively). We see in fig.\ref{fig:2PE_simul} (bottom) than the efficiencies deviates significantly from the prediction eq.\eqref{etaPi}. There is less discrepancy when the $\pi$-pulse is 10 times shorter than the signal (ratio 10), especially at low optical depth. Still, for larger $d$, the distortions are sufficiently important to reduce the efficiency significantly.
Despite a clear deviation from the analytical scaling (eq.\ref{etaPi}), the echo amplification is important (efficiency $>$ 100\%). This latter comes from the inversion of the medium. As a consequence, the amplified spontaneous emission mixes up with the retrieved signal then inducing noise. It should be noted that the signal to noise ratio only depends on the optical depth \cite{ruggiero,RASE,Sekatski}. This may be surprising at first sight because the coherent emission of the echo and the spontaneous emission seems to have completely different collection patterns offering a significant margin to the experimentalist to filter out the noise. This is not the case. The excitation volume is defined by the incoming laser focus. On the one hand, a tighter focus leads to a smaller number of inverted atoms thus reducing the number spontaneously emitted photons. On the other hand, a tight focus requires a larger collection angle of the retrieved echo. Less atoms are excited but the spontaneous emission collection angle is larger. The noise in the echo mode is unchanged. This qualitative argument which can be seen as a conservation of the optical etendue is quantitatively supported by a quantized version of the Bloch-Maxwell equations \cite{RASE,Sekatski}. This aspect will be discussed in sections \ref{CV_criterion} and \ref{counting_criterion} using a simplified quantum model.
In any case, population inversion should be avoided. This statement motivated many groups to conceive rephasing protocols by keeping the best of the 2PE but avoiding the deleterious effect of $\pi$-pulses as we will see now in sections \ref{CRIB} and \ref{Rose}.
\subsection{Controlled reversible inhomogeneous broadening}\label{CRIB}
The controlled reversible inhomogeneous broadening (CRIB) offers a solid alternative to the 2PE \cite{CRIB1,CRIB2,CRIB3,CRIB4, sangouard_crib}. The CRIB,,as represented in fig.\ref{fig:CRIB}, has been successfully implemented with large efficiencies \cite{hedges2010efficient, hosseini2011high} and low noise measurements \cite{lauritzen2010telecommunication} validating the protocol as a quantum memory in different systems, from atomic vapors to doped solids \cite{lauritzen2011approaches}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{CRIB.eps}}
\caption{CRIB echo sequence. As compared to the 2PE sequence, no rephasing pulse is applied, the inhomogeneous broadening is reversed using a controllable electric field for example \cite{CRIB3} (magenta line).}
\label{fig:CRIB}
\end{figure}
Fundamentally, an echo is generated by rephasing the coherences corresponding to the cancellation of the inhomogeneous phase. As indicated by eq.\eqref{mapping}, the accumulated phase is $\Delta t$. Taking control of the detuning $\Delta$ is sufficient to produce an echo without a $\pi$-pulse. This is the essence of the CRIB sequence, where the detuning is actively switched from $\Delta$ for $t<\tau$ to $-\Delta$ for $t>\tau$. We won't focus on the realization of the detuning inversion. This aspect has been covered already and we recommend the reading of the review papers \cite{lvovsky2009optical, afzelius2010photon}. We here focus on the coherence rephasing and evaluate the efficiency which can be compared to other protocols. It should be noted that the gradient echo memory scheme (GEM) \cite{hetet2008electro} is not covered by our description. We will assume that the coherences undergo the transform $+\Delta \rightarrow -\Delta$ independently of the atomic position $z$. This is not the case for the GEM where the detuning $\Delta$ goes linearly (or at least monotonically) with the position $z$. The GEM can be called the longitudinal CRIB. This specificity of the GEM makes it remarkably efficient \cite{hetet2008electro, hedges2010efficient, hosseini2011high}.
Assuming that $\Delta \rightarrow -\Delta$ for $t>\tau$, it should be first noted that at the switching time $\tau$, the coherence term is continuous
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^+\right)=\mathcal{P}_\Delta\left(\tau^-\right) =- \frac{i}{2} \exp \left(i \Delta \tau \right) \tilde{\mathcal{E}}(z,\Delta)
\end{equation}
but will evolve with a different detuning afterward, that is
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau\right)=\mathcal{P}_\Delta\left(\tau^+\right) \exp \left(-i \Delta \left( t- \tau\right) \right) =- \frac{i}{2} \exp \left(i \Delta \left(2\tau -t\right) \right) \tilde{\mathcal{E}}(z,\Delta)\label{P_crib}
\end{equation}
The latter gives the source term of the differential equation defining the efficiency similar to eq.\eqref{eq_echo} for the 2PE
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
-\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}(z,2\tau-t) \label{eq_crib}
\end{equation}
Eqs \eqref{eq_echo} and \eqref{eq_crib} are very similar. The first term on the right hand side is now negative (proportional to $-\displaystyle\frac{\alpha}{2}$) because the medium is not inverted in the CRIB sequence. This is a major difference. Again, the incoming signal follows the Bouguer-Beer-Lambert law of absorption $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right)$ but the efficiency defined by \eqref{efficiency} is now {given} after integration by
\begin{equation}gin{equation}\label{eta_crib}
\eta\left(d\right)=d^2 \exp\left(-d\right)
\end{equation}
The maximum efficiency is obtained for $d=\alpha L=2$ with $\eta\left(2\right)=54\%$ \cite{sangouard_crib} (see fig.\ref{fig:compar_eff}). There is no gain so the semi-classical efficiency is always smaller than one. The efficiency is limited in the so-called forward configuration because the echo is {\it de facto} emitted in an absorbing medium. The re-absorption of the echo limits the efficiency to $54\%$. Ideal echo emission with unit efficiency can be obtained in the backward configuration. This latter is implemented by applying auxiliary pulses, typically Raman pulses modifying the phase matching condition from forward to backward echo emission. The Raman pulses increase the storage time by shelving the excitation into nuclear spin state for example. This ensures the complete reversibility by flipping the apparent temporal evolution (as shown by eq.\eqref{P_crib}) and the wave-vector \cite{reversibility}.
Despite its simplicity, eq.\eqref{eq_crib} can be adapted to describe the backward emission without working out the exact phase matching condition. We consider the following equivalent situation. The signal is first absorbed: $\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(-\alpha z/2\right)$. We now fictitiously flip the atomic medium: the incoming slice $z=0$ becomes $z=L$ and {\it vice versa}. The atomic excitation would correspond to the absorption of a backward propagating field
$$\displaystyle \mathcal{E}(z,t)=\mathcal{E}(0,t) \exp\left(\alpha \left(z-L\right)/2\right)$$
Eq.\eqref{eq_crib} can be integrated with this new boundary condition, giving the backward efficiency of the CRIB
\begin{equation}gin{equation}\label{eta_crib_back}
\eta\left(d\right)=\left[1 - \exp\left(-d\right)\right]^2
\end{equation}
For a sufficiently large optical depth, the efficiency is close to unity. As a comparison, we have represented the forward (eq.\ref{eta_crib}) and backward (eq.\ref{eta_crib_back}) CRIB efficiencies in fig.\ref{fig:compar_eff}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.75\linewidth]{compar_eff.eps}}
\caption{Comparison of the forward (eq.\ref{eta_crib}, in blue) and backward (eq.\ref{eta_crib_back}, in red) CRIB efficiency scaling. The standard 2PE efficiency is represented as a reference (eq.\ref{etaPi}, in black)}
\label{fig:compar_eff}
\end{figure}
The practical implementation of the CRIB requires to control dynamically the detuning by Stark or Zeeman effects. The {\it natural} inhomogeneous broadening has a static microscopic origin and cannot be used {\it as it is}. The initial optical depth has to be sacrificed to obtain an effective controllable broadening. This statement motivates the reconsideration of the 2PE which precisely exploit the bare inhomogeneous broadening offering advantages in terms of available optical depth and bandwidth.
\subsection{Revival of silenced echo}\label{ROSE} \label{Rose}
The Revival of silenced echo (ROSE) is a direct descendant of the 2PE \cite{rose}. The ROSE is essentially a concatenation of two 2PE sequences as represented in fig.\ref{fig:ROSE}. In practice, the ROSE sequence advantageously replace{s} $\pi$-pulses by complex hyperbolic secant (CHS) pulses as we will specifically discuss in \ref{strong_pulse_rose}. For the moment, we assume that the rephasing pulses are simply $\pi$-pulses. This is sufficient to evaluate the efficiency and derive the phase matching conditions.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.85\linewidth]{ROSE.eps}}
\caption{A schematic ROSE echo sequence that can be seen as the concatenation of two 2PE sequences (fig.\ref{fig:2PE}). The first echo at $t=\tau$ (in dashed red) should be silenced by the phase matching conditions (see \ref{phase_match}). A second $\pi$-pulse at $t=\tau^\prime$ induces the emission of the ROSE echo at $t=2\left(\tau^\prime-\tau\right)$ (in red).}
\label{fig:ROSE}
\end{figure}
Concatenated with a 2PE sequence, a second $\pi$-pulse (at $t=\tau^\prime$ in fig.\ref{fig:ROSE}) triggers a second rephasing of the coherences at $t=2\left(\tau^\prime-\tau\right)$. This latter leaves the medium non-inverted avoiding the deleterious effect of a single 2PE sequence. This reasoning is only valid if the first echo is not emitted. In that case, the coherent free evolution continues after the first rephasing. The first echo is said to be silent (giving the name to the protocol) because the coherence rephasing is not associated to a field emission. The phase matching conditions are indeed designed to make the first echo silent but preserve the final retrieval of the signal. Along the same line with the same motivation, McAuslan {\it et al.} proposed to use the Stark effect to silence the emission of the first echo \cite{HYPER} by cunningly applying the tools developed for the CRIB to the 2PE, namely by inducing an artificial inhomogeneous reversible broadening. The AC-Stark shift (light shift) also naturally appeared as a versatile tool to manipulate the retrieval \cite{Chaneliere:15}. We will discuss the phase matching conditions latter. Before that, we will evaluate the retrieval efficiency applying the method developed for the 2PE and CRIB.
\subsubsection{Retrieval efficiency}
Following the procedure in section \ref{2PE}, we assume that a second $\pi$-pulse is applied at $t=\tau^\prime$. Starting from eq.\eqref{P_echo}, we can track the inhomogeneous phase at $t=\tau^\prime$ when the $\pi$-pulse is applied (similar to eq.\ref{P_tauplus}) as
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(\tau^{\prime+}\right)=-\mathcal{P}_\Delta^*\left(\tau^{\prime-}\right) =- \frac{i}{2} \exp \left(-i \Delta \left( \tau^\prime -2\tau\right) \right) \tilde{\mathcal{E}}^*(z,\Delta)
\end{equation}
freely evolving afterward as
\begin{equation}gin{equation}
\mathcal{P}_\Delta\left(t>\tau^\prime\right)=- \frac{i}{2} \exp \left(i \Delta \left( t- 2\tau^\prime+2\tau\right) \right) \tilde{\mathcal{E}}(z,\Delta)\label{P_rose}
\end{equation}
There is indeed a rephasing at $ t=2\left(\tau^\prime-\tau\right)$. The retrieval follows the common differential equation (as eqs. \eqref{eq_echo} and \eqref{eq_crib})
\begin{equation}gin{equation}
\partial_z\mathcal{E}^R(z,t)=
-\displaystyle\frac{\alpha}{2} \mathcal{E}^R(z,t) - \alpha \mathcal{E}(z,t- 2\tau^\prime+2\tau) \label{eq_rose}
\end{equation}
As compared to the 2PE, the ROSE echo is not emitted in an inverted medium. One can note that the signal is not time-reversed as in the 2PE and CRIB, so the efficiency is defined as
\begin{equation}gin{equation}
\eta=\frac{|\mathcal{E}^R(L,t)|^2}{|\mathcal{E}(0,t- 2\tau^\prime+2\tau)|^2}
\end{equation}
The ROSE efficiency is exactly similar to CRIB due to the similarity of eqs.\eqref{eq_crib} and \eqref{eq_rose}. It is limited to 54\% in the forward direction because the medium is absorbing. {Complete reversal can be obtained in the backward direction by precisely designing the phase matching condition, the latter being a critical ingredient of the ROSE protocol.}
Even if there is no population inversion at the retrieval, the use of strong pulses for the rephasing is a potential source of noise. First of all, any imperfection of the $\pi$-pulses may leave some population in the excited state leading to a partial amplification of the signal. Secondarily, the interlacing of strong and weak pulses within the same temporal sequence is like playing with fire. This is a common feature of many quantum memory protocols for which control fields may leak in the signal mode. Many experimental techniques are combined to isolate the weak signal: different polarization, angled beams (spatial selection) and temporal separation. Encouraging demonstrations of the ROSE down to few photons per pulses have been performed by combining theses techniques \cite{bonarota_few}, thus showing the {potentials} of the protocol.
\subsubsection{Phase matching conditions}\label{phase_match}
Phase matching can be considered in a simple manner by exploiting the spectro-spatial analogy. Each atom in the inhomogeneous medium is defined by its detuning (frequency) and position (space), both contributing to the inhomogeneous phase. In that sense, the instant of emission can be seen as a spectral phase matching condition. Following this analogy, the spatial phase matching condition can be derived from the photon echo time sequence \cite{mukamel}.
Let {us} take the 2PE as an example (fig.\ref{fig:2PE}). The 2PE echo is emitted at $t=t_1+2\tau=2t_2-t_1$ where $t_1$ is the arrival time of the signal (first pulse) and $t_2$ the $\pi$-pulse (second pulse). In fig.\ref{fig:2PE}, we have chosen $t_1=0$ and $\tau=t_2-t_1$ for simplicity . By analogy, the echo should be emitted in the direction $\overrightarrow{k}=2\overrightarrow{k_2}-\overrightarrow{k_1}$ where $\overrightarrow{k_1}$ and $\overrightarrow{k_2}$ are the wavevectors of the signal and $\pi$-pulse respectively. In that case, if $\overrightarrow{k_1}$ and $\overrightarrow{k_2}$ are not collinear ($\overrightarrow{k_1}\neq \overrightarrow{k_2}$), the phase matching cannot be fulfilled: there is no 2PE echo emission.
Following the same procedure, the ROSE echo is emitted at $t=t_1+2(\tau^\prime-\tau)=t_1+2(t_3-t_2)$ where $t_3$ is the arrival time of the second $\pi$-pulse (third pulse). The ROSE echo should be emitted if the $\overrightarrow{k}=\overrightarrow{k_1}+2(\overrightarrow{k_3}-\overrightarrow{k_2})$ direction ($\overrightarrow{k_3}$ is the direction of the second $\pi$-pulse). The canonical experimental situation satisfying the ROSE phase matching condition corresponds to $\overrightarrow{k_1}\neq \overrightarrow{k_2}$ (not collinear) but keeping $\overrightarrow{k_3}=\overrightarrow{k_2}$ \cite{Dajczgewand:14, Gerasimov2017}. There is no 2PE in that case because $\overrightarrow{k_1}\neq \overrightarrow{k_2}$ but the ROSE echo is emitted in the direction $\overrightarrow{k_1}$ of the signal as represented in fig.\ref{fig:phase_matching}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{phase_matching.eps}}
\caption{ROSE Phase matching conditions. Left: canonical experimental situation where the two $\pi$-pulses are on the same beam. The echo $\vec{k}$ is in the signal mode $\vec{k_1}$ (forward). The angle between the signal and the rephasing can be large for a good isolation of the echo as recently tested in the orthogonal configuration \cite{Gerasimov2017}. Right: backward retrieval of the ROSE echo $\vec{k}$. The signal $\vec{k_1}$ and two rephasing beams forms a equilateral triangle in that case: the echo is emitted backward.}
\label{fig:phase_matching}
\end{figure}
The backward retrieval configuration is illustrated as well in fig.\ref{fig:phase_matching} (right). The efficiency can reach 100\% because the reversibility of the process is ensured spatially and temporally.
\subsubsection{Adiabatic pulses}\label{strong_pulse_rose}
Even if the protocol can be understood with $\pi$-pulses, the rephasing pulses can be advantageously replaced by complex hyperbolic secant (CHS) in practice \cite{Dajczgewand:14, Gerasimov2017}. The CHS are another heritage from the magnetic resonance techniques \cite{Garwood2001155}. As representative of the much broad class of adiabatic and composite pulses, CHS produce a robust inversion because for example the final state weakly depends on the pulse shape and amplitude. Within a spin or photon echo sequence, they must be applied by pairs because each CHS adds an inhomogeneous phase due to the frequency sweep. This latter can be interpreted as a sequential flipping of the inhomogeneous ensemble. Two identical CHS produce a perfect rephasing because the inhomogeneous phases induced by the CHS cancel each other \cite{minar_chirped, PascualWinter}.
CHS additionally offer{s} an advantage that is somehow underestimated. As we have just said, CHS must be appl{ied} by pairs. It means that the first echo in the ROSE sequence is also silenced because it would follow the first CHS, as opposed to the second echo which follows a pair of CHS. How much the first echo is silenced depends on the parameters of the CHS, namely the Rabi frequency and the frequency sweep. This degree of freedom should not be neglected when the phase matching conditions cannot be modified as in the cavity case in the optical or RF domain \cite{Grezes}.
To conclude about the ROSE and because of its relationship with the 2PE, it is important to question the strong pulse propagation that we pointed out as an important efficiency limitation of the 2PE (with $\pi$-pulses) by analyzing fig.\ref{fig:2PE_simul} (see appendix \ref{strong_pulse} for a more detailled discussion). In that sense as well, the CHS are superior to $\pi$-pulses. CHS are indeed very robust to propagation in absorbing media so their preserve their amplitude and frequency sweep \cite{Warren, PhysRevLett.82.3984}. CHS are not constrained by the McCall and Hahn Area Theorem (eq.\ref{area}). The latter isn't valid for frequency swept pulses \cite{Eberly:98}. This robustness to propagation can be explained qualitatively by considering the energy conservation \cite{rose}.
The different advantages of the CHS as compared to $\pi$-pulses have been studied accurately using numerical simulations in \cite{Demeter}, confirming both their versatility and robustness.
\subsection{Summary and perspectives}
We have described the variations from the well-known photon echo technique adapted for quantum storage. We haven't discussed in details the gradient echo memory scheme (GEM) \cite{hetet2008electro} (sometimes called longitudinal CRIB) which can be seen as an evolution of the CRIB protocol. The GEM is remarkable for its efficiency \cite{hetet2008electro, hedges2010efficient, hosseini2011high} allowing demonstrations in the quantum regime of operation \cite{hosseini2011unconditional}. The scheme has been enriched {by} processing functions as {a} pulse sequencer \cite{hosseini2009coherent, hosseini_jphysb}. More importantly, the GEM has been considered for RF storage in an ensemble of spins thus covering different physical realities and frequency ranges \cite{wu2010storage, zhang2015magnon}. As previously mentioned, the GEM is not covered by our formalism because the scheme couples the detuning and the position $z$. An analytical treatment is possible but is beyond the scope of our paper \cite{LongdellAnalytic}.
The specialist reader may be surprised because we did not discussed the atomic frequency comb (AFC) protocol \cite{afc} despite an undeniable series of success. The early demonstration of {weak classical field and single photon} storage \cite{usmani2010mapping, saglamyurek2011broadband, clausen2011quantum, PhysRevLett.108.190505, gundogan, PhysRevLett.115.070502, Tiranov:15, maring2017photonic} has been pushed to a remarkable level of integration \cite{saglamyurek2015quantum, PhysRevLett.115.140501, Zhong1392}. The main advantage of the AFC is a high multimode capacity \cite{afc, bonarota2011highly} which has been identified as an critical feature of the deployment of quantum repeaters \cite{collins,simon2007}. Despite a clear filiation of the AFC with the photon echo technique \cite{Mitsunaga:91}, there are also fundamental differences. For the AFC, there is no direct field to coherence mapping as discussed in section \ref{mapping}. The AFC is actually based on a population grating. Without going to much into a semantic discussion, the AFC is a descendant of the three-pulse photon echo and not the two-pulse photon echo \cite{mukamel} that we analyze in this section \ref{sec:2PE}. As a consequence, the AFC can be surprisingly linked to the {\it slow-light} protocols \cite{afc_slow} that we will discuss in the next section \ref{sec:SL}
\section{Slow-light memories}\label{sec:SL}
Since the seminal work of Brillouin \cite{brillouin} and Sommerfeld \cite{sommerfeld}, {\it slow-light} is a fascinating subject whose impact has been significantly amplified by the popular science-fiction culture \cite{shaw}. The external control of the group velocity reappeared in the context of quantum information as a mean to store and retrieve optical {light while preserving its quantum features} \cite{EIT_Harris, fleischhauer2000dark, FLEISCHHAUER2000395}. The rest is a continuous success story that can only be embraced by review papers \cite{review_Ma_2017}.
We will start this section by deriving the Schr\"odinger-Maxwell equations used to describe the signal storage and retrieval. Our analysis is based on the following classification. We first consider the fast storage and retrieval scheme as introduced by Gorshkov {\it et al.} \cite{GorshkovII}. In other words, the storage is triggered by brief Raman $\pi$-pulses \cite{GorshkovII, legouet_raman}. We then consider the more established electromagnetically induced transparency (EIT) and the Raman schemes. In theses cases, the storage and retrieval are activated by a control field that is on or off. The difference between EIT and Raman is the control field detuning: on-resonance for the EIT scheme and off-resonance for the Raman. Both lead to very different responses of the atomic medium. In the EIT scheme, the presence of the control field produces the so-called dark atomic state. As a consequence, absorption is avoided and the medium is transparent. On the contrary, in the Raman scheme, the control beam generates an off-resonance absorption peak (Raman absorption): the medium is absorbing.
To give a common vision of the fast storage (Raman $\pi$-pulses) and the EIT/Raman schemes, we first introduce a Loren{tz}ian susceptibility response as an archetype for absorption and its counterpart the inverted-Lorentzian that describes a generic transparency window. We will define the different terms in \ref{archetypes}.
\subsection{Three-level atoms Schr\"odinger-Maxwell model}
Following the same approach as in section \ref{sec:2PE}, the pulse propagation and storage can be modeled by the Schr\"odinger-Maxwell equations in one dimension (along $z$). {We now give these equations for three level atoms.}
\subsubsection{Schr\"odinger equation for three-level atoms}
For three-level atoms, labeled $|g\rangle$, $|e\rangle$ and $|s\rangle$ for the ground, excited and spin states (see fig.\ref{fig:2level_3level}, right), the rotating-wave probability amplitudes $C_g$, $C_e$ and $C_s$ respectively are governed by the time-dependent Schr\"odinger equation similar to eq.\eqref{bloch2} \cite[eq. (13.29)]{shore2011manipulating}:
\begin{equation}gin{align}
i \partial_t \left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
C_s\\
\end{array}\right]
=
\left[
\begin{equation}gin{array}{ccc}
0 &\displaystyle \frac{\mathcal{E}^*}{2} & 0 \\
\displaystyle\frac{\mathcal{E}}{2} & -\Delta &\displaystyle \frac{\Omega}{2} \\
0 &\displaystyle \frac{\Omega^*}{2} & - \delta \\
\end{array}\right]
\left[
\begin{equation}gin{array}{c}
C_g \\
C_e \\
C_s\\
\end{array}\right]
\label{bloch3}\end{align}
where $\mathcal{E}(z,t)$ and $\Omega(t)$ are the complex envelopes of the input signal and the Raman field respectively (units of Rabi frequency). If we consider the spin level $|s\rangle$ as empty, the Raman field is not attenuated (nor amplified) by the propagation so $\Omega(t)$ doesn't depend on $z$. The parameters $\Delta$ and $\delta$ are the one-photon and two-photon detunings respectively (see fig.\ref{fig:2level_3level}, right).
The atomic variables $C_g$, $C_e$ and $C_s$ depend on $z$ and $t$ for given detunings $\Delta$ and $\delta$. As in section \ref{sec:2PE}, the detunings are chosen position and time independent. Again, decay terms can be added {\it by-hand} by introducing complex detunings for $\Delta$ and $\delta$.
\subsubsection{Maxwell propagation equation}
Eqs \eqref{MB_M_hom} (homogeneous ensemble) and \eqref{MB_M_inhom} (inhomogeneous) still describe the propagation of the signal in the slowly varying envelope approximation.
The two sets of equations (\ref{bloch3}\&\ref{MB_M_hom}) or (\ref{bloch3}\&\ref{MB_M_inhom}) depending if the ensemble is homogeneous or inhomogeneous are sufficient to describe the different situations that we will consider. As already mentioned in section \ref{sec:2PE}, the equations of motion can be further simplified for weak $\mathcal{E}(z,t)$ signals (perturbative regime).
\subsubsection{Perturbative regime}
The linearisation of the Schr\"odinger-Maxwell equations (\ref{bloch3}, \ref{MB_M_hom} \&\ref{MB_M_inhom}) corresponds to the so-called perturbative regime. To the first order in perturbation, the atoms stays in the ground, $C_g \simeq 1$ because the signal is weak. The atomic evolution (eq.\ref{bloch3}) is now only given by $C_e$ and $C_s$ that we write with $\mathcal{P}\simeq C_e$ and $\mathcal{S}\simeq C_s$ to describe the optical (polarization $\mathcal{P}$) and spin ($\mathcal{S}$) excitations \cite{GorshkovII}. The atoms dynamics from eq.\eqref{bloch3} becomes:
\begin{equation}gin{align}
\partial_t \mathcal{P} &= (i\Delta-\Gamma) \mathcal{P} - i \frac{\Omega}{2} \mathcal{S} - i \frac{\mathcal{E}}{2}\label{bloch_P}\\
\partial_t \mathcal{S} &= - i \frac{\Omega^*}{2} \mathcal{P} + i \delta \mathcal{S} \label{bloch_S}
\end{align}
We have introduced the optical homogeneous linewidth $\Gamma$ that will be used later. The decay of the spin is neglected which would correspond to an infinite storage time when the excitation in shelved into the spin coherence. This is an ideal case.
The Raman field $\Omega(t)$ is unaffected by the propagation if the spin state is empty. The Raman pulse keeps its initial temporal shape so there is no differential propagation equation governing $\Omega(t)$. This a major simplification especially when a numerical integration (along $z$) is necessary.
We will only consider real envelope $\Omega(t)$ for the Raman field. Nevertheless, a complex envelope can still be used if the Raman field is chirped for example \cite{minar_chirped}.
The exact same set of equations can alternatively be derived from the density matrix formalism in the perturbative regime, the terms $\mathcal{P}$ and $\mathcal{S}$ representing the off-diagonal coherences of the $|g\rangle$-$|e\rangle$ and $|g\rangle$-$|s\rangle$ transitions respectively owing to the $C_g\simeq1$ hypothesis.
Using the polarization $ \mathcal{P}(t,\Delta)$, the Maxwell equations \eqref{MB_M_hom} and \eqref{MB_M_inhom} are rewritten as:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle{i \alpha} \Gamma \mathcal{P}(t) \label{MB_M_hom_pert}
\end{equation}
or for inhomogeneous ensembles as:
\begin{equation}gin{equation}
\partial_z\mathcal{E}(z,t)+ \frac{1}{c}\partial_t\mathcal{E}(z,t)=
-\displaystyle\frac{i \alpha}{\pi}\int_\Delta g\left(\Delta\right) \mathcal{P}(t,\Delta) d\Delta \label{MB_M_inhom_pert}
\end{equation}
This formalism is sufficient to describe the different situations we will consider now. The simplified perturbative set of coupled equations (\ref{bloch_P}\&\ref{bloch_S}) cannot be solved analytically when $\Omega(t)$ is time-varying, {thus} acting as a parametric driving. A numerical integration is usually necessary to fully recover the outgoing signal shape after the propagation given by eqs.\eqref{MB_M_hom_pert} or \eqref{MB_M_inhom_pert}. Simpler situations can still be examined to discuss the dispersive properties of a {\it slow-light} medium. When $\Omega(t)=\Omega$ is static, the susceptibility describing the linear propagation of the signal field $\mathcal{E}(z,t)$ can be explicitly derived. This is a very useful guide for the physical intuition.
\subsection{Inverted-Lorentzian and Lorentzian responses: two archetypes of {slow-light}}\label{archetypes}
Before going into details, we would like to describe qualitatively two archetypal situations without specific assumption on the underlying level structure or temporal shapes of the field. From our point of view, {\it slow-light} propagation should be considered as the precursor of storage. We use the term precursor as an allusion to the work of Brillouin \cite{brillouin} and Sommerfeld \cite{sommerfeld}.
The first situation corresponds to the well-known {\it slow-light} propagation in a transparency window. More specifically, we will assume {that} the susceptibility {is} given by an inverted-Lorentzian shape. The Lorentzian should be inverted to obtain transparency and not absorption at the center. The susceptibility is defined as the proportionality constant between the frequency dependent polarization and electric field (including the vacuum permittivity $\epsilon_0$). This latter can be directly identified from the field propagation equation as we will see later in \ref{section:TW} and \ref{section:AW}.
The second situation is the complementary. A Lorentzian (non-inverted) can also be considered to produce a retarded response. This is useful guide to described certain storage protocols and revisit the concept of {\it slow-light}. The Lorentzian response naturally comes out of the Lorentz-Lorenz model when the electron is elastically bound to the nucleus when light-matter interaction is introduced to the undergraduate students. These two archetypes represent a solid basis to interpret the different protocols we will detail in section \ref{raman_stopped} and \ref{EIT_stopped}.
\subsubsection{Transparency window of an inverted-Lorentzian}\label{section:TW}
We assume that the susceptibility {is} given by an inverted-Lorentzian. This is {the} simplest case because a group delay can be explicitly derived. Whatever is the exact physical situation, the source term on the right-hand sides of eqs \eqref{MB_M_hom_pert} or \eqref{MB_M_inhom_pert} can be replaced by a linear response in the spectral domain (linear susceptibility) when $\Omega(t)$ is static. The propagation equation would read in the spectral domain \cite[p.12]{allen2012optical}
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right] \tilde{\mathcal{E}}(z,\omega) \label{propag_ILorentz} \end{equation}
where $\tilde{\mathcal{E}}(z,\omega)$ is the Fourier transform of $\mathcal{E}(z,t)$. The left-hand side simply describes the free-space propagation of the slowly varying envelope. The right-hand side is proportional to the inverted-Lorentzian susceptibility defining the complex propagation constant as:
\begin{equation}gin{equation}
\tilde{\alpha}\left(\omega\right)=-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]
\end{equation}
The different terms can be analyzed as follows. $ \frac{\alpha}{2} $ is the far-off resonance (or background) absorption coefficient for the amplitude $\tilde{\mathcal{E}}$ such as the intensity $|\tilde{\mathcal{E}}|^2$ decays exponentially with a coefficient $\alpha$ following the Bouguer-Beer-Lambert absorption law. The term $\displaystyle \left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]$ represents the Lorentzian shape of a transparency window (width $\Gamma_0$) that we choose as an archetype. With this definition, the susceptibility $\chi$ can be written as $\displaystyle \chi\left(\omega\right)=-\frac{2i}{k} \tilde{\alpha}\left(\omega\right)$ where $k$ is the wavevector \footnote{With our definitions, the real part of the propagation constant $\tilde{\alpha}$ gives the absorption and the imaginary part, the dispersion. For the susceptibility, this is the other way around.}.
At the center $\omega=0$, there is no absorption (complete transparency). We choose a complex Lorentzian $\displaystyle \frac{1}{1+i\omega/\Gamma_0}$ and not a real one $\displaystyle \frac{1}{1+\omega^2/\Gamma_0^2}$ because the complex Lorentzian satisfies {\it de facto} the Kramers-Kronig relation so we implicitly respect the causality. The propagation within the transparency window is given by a first-order expansion of the susceptibility when $\omega \ll \Gamma_0$ leading to
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega) \simeq \displaystyle-\frac{\alpha}{2} i\frac{\omega}{\Gamma_0} \tilde{\mathcal{E}}(z,\omega) \end{equation}
and after integration over the propagation distance $L$
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) \simeq \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{i \omega \alpha L }{2\Gamma_0} \right) \end{equation}
or equivalently in the time domain
\begin{equation}gin{equation} \label{eq:nunn2} \mathcal{E}(L,t) \simeq \mathcal{E}(0,t-\frac{L }{c}-\frac{\alpha L}{2 \Gamma_0}) \end{equation}
where $\displaystyle \frac{L }{c}+\frac{d}{2 \Gamma_0}$ is the group delay with the optical depth $d=\alpha L$. If the incoming pulse bandwidth fits the transparency window or in other words if the pulse is sufficiently long, the pulse is simply delayed by $\displaystyle \frac{d}{2 \Gamma_0}$. This latter defines the group delay.
Shorter pulses are distorted and partially absorbed when the bandwidth extends beyond the transparency window. In that case, eq.\eqref{propag_ILorentz} can be integrated analytically to give the general formal solution:
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) = \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{d}{2} \frac{i \omega}{\Gamma_0+i\omega}\right) \label{SL_TW} \end{equation}
The outgoing pulse shape $\mathcal{E}(L,t) $ is given by the inverse Fourier transform of $ \tilde{ \mathcal{E}}(L,\omega)$. As an example, we plot the outgoing pulse in fig.\ref{fig:SL_ILorentz} for a Gaussian input $\displaystyle \mathcal{E}(0,t)= \exp \left(-\frac{t^2}{2\sigma^2}\right)$. We choose $\Gamma_0=1$ and a pulse duration $\sigma =\displaystyle \frac{d}{2 \Gamma_0}$ corresponding to the expected group delay. We take $d=20$ for the optical depth, which corresponds to realistic experimental situations.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.65\linewidth]{SL_ILorentz.eps}}
\caption{{\it Slow-light} in a Lorentzian transparency window. Top: real and imaginary part of the propagation constant $\tilde{\alpha}\left(\omega\right)$. The real part represents the absorption and the imaginary part the refractive index (dispersion) whose slope is the group delay. The shaded area corresponds to the {\it slow-light} region, the positive slope of the imaginary part leads to a positive group delay. Bottom: {\it Slow-light} propagation of a Gaussian incoming pulse (in blue) producing a delayed output pulse (in red) calculated from eq.\eqref{SL_TW}.}
\label{fig:SL_ILorentz}
\end{figure}
The outgoing pulse is essentially delayed by $\displaystyle \frac{d}{2 \Gamma_0}=10$ and only weakly absorbed through the propagation. A longer pulse would lead to less absorption but the input and output would be much less separated. As we will see later, this point is critical for {\it slow-light} storage protocols.
\subsubsection{Dispersion of a Lorentzian}\label{section:AW}
We now consider a Lorentzian as a complementary situation. This may sound surprising for the reader familiar with the EIT transparency window. {However,} the Lorentzian is a useful reference to interpret the Raman memory {that will be discussed in section \ref{Raman}.
We {consider a} propagation constant given by
\begin{equation}gin{equation}
\tilde{\alpha}\left(\omega\right)=-\frac{\alpha}{2} \frac{1}{1+i\omega/\Gamma_0}
\end{equation}
This is a quite simple case corresponding to the transmission of an homogeneous ensemble of dipoles. To take the terminology of the previous case, one could speak of an absorption window as opposed to a transparency window. To follow up the analogy, there is no {\it slow-light} at the center of an absorption profile. The susceptibility is inverted thus leading to {\it fast-light} (negative group delay). A retarded response can still be expected but on the wings (off-resonance) of the absorption profile. As represented on fig.\ref{fig:SL_Lorentz}, the slope is negative at the center ({\it fast-light}) but it changes sign out of resonance leading to a distorted version of {\it slow-light}. Distortion are indeed expected because the dispersion cannot be considered as linear. Still, what comes out of the medium after the incoming pulse can be interpreted as a precursor for light storage.
By inverted analogy with the previous case, the propagation can be solved to the first order when the pulse bandwidth is much larger than the absorption profile (off-resonant excitation of the wings). The Lorentzian $\displaystyle \frac{1}{1+i\omega/\Gamma_0}$ simplifies to the first order in $\displaystyle \frac{\Gamma_0}{i\omega}$ leading to the solution in the spectral domain:
\begin{equation}gin{equation} \label{eq:propag3} \tilde{ \mathcal{E}}(L,\omega) \simeq \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{\alpha L \Gamma_0}{2i \omega} \right) \end{equation}
or alternatively in the time domain
\begin{equation}gin{equation} \mathcal{E}(L,t) \simeq \mathcal{E}(0,t) \ast F(L,t) \label{FID_convol} \end{equation}
where $F(L,t)$ is the impulse response convoluting ($\ast$) the incoming pulse shape and analytically given by \cite{bateman1954tables}:
\begin{equation}gin{equation}\label{eq:FID}
F\left(L,t\right) = \delta_{t=0} - {\alpha L \Gamma_0} \frac{J_1\left(\sqrt{2d \Gamma_0 t}\right)}{\sqrt{2d \Gamma_0 t}} \mbox{ for t$>$0 and 0 elsewhere} \end{equation}
$J_1$ is the Bessel function of the first kind of order 1 with the optical depth $d=\alpha L$. $\delta_{t=0}$ is the Dirac peak. The time $\displaystyle \frac{1}{d \Gamma_0}$ appears as a typical delay due to propagation. The output shape will be distorted by the strong oscillations of the Bessel function. This can be investigated by considering the following numerical example without first order expansion. The output shape is indeed more generally given by the inverse Fourier transform of the integrated form:
\begin{equation}gin{equation} \tilde{ \mathcal{E}}(L,\omega) = \tilde{ \mathcal{E}}(0,\omega) \exp \left(-\frac{i \omega L }{c} \right) \exp \left(-\frac{d }{2} \frac{\Gamma_0}{\Gamma_0+i\omega}\right) \label{SL_TA} \end{equation}
Again we plot the outgoing pulse in fig.\ref{fig:SL_Lorentz} for a Gaussian input $\displaystyle \mathcal{E}(0,t)= \exp \left(-\frac{t^2}{2\sigma^2}\right)$ whose duration is now $\sigma =\displaystyle\frac{1}{d \Gamma_0}$ ($\Gamma_0=1$) corresponding to the expected generalized group delay. As before, the optical depth is $d=20$.
Two lobes appear at the output (fig.\ref{fig:SL_Lorentz}) as expected from the approximated expression eq.\eqref{FID_convol} involving the oscillating Bessel function. Still, a significant part of the incoming pulse is retarded in the general sense whatever is the exact outgoing shape.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.65\linewidth]{SL_Lorentz.eps}}
\caption{{\it Slow-light} from a Lorentzian absorption window. Top: real and imaginary part of the propagation constant $\tilde{\alpha}\left(\omega\right)$. The shaded area corresponds to the {\it slow-light} region (positive group delay). Bottom: {\it Slow-light} propagation of a gaussian incoming pulse (in blue) producing a retarded output pulse (in red) calculated from eq.\eqref{SL_TA}.}
\label{fig:SL_Lorentz}
\end{figure}
As will see now, what is retarded can be stored.
\subsubsection{A retarded response as a precursor for storage}
{\it Slow-light} is a precursor of storage called {\it stopped-light} in that case. The transition from {\it slow} to {\it stopped-light} is summarized in fig.\ref{fig:SL_Lorentz_ILorentz_shaded}.
When input and output are well separated in time, storage is possible in principle. If we look at the standard situation of {\it slow-light} in a transparency window (fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, top), we choose a frontier between input and output at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$. At this given moment, most of the output pulse has entered the atomic medium. There is only a small fraction of the input pulse (blue shaded area) that leaks out. This part will be lost. Concerning the output pulse, the red shaded area (subtracted from the blue area) is essentially contained inside the medium and {\it de facto} stored into the atomic excitation \cite{Shakhmuratov, ChaneliereHBSM}. The same qualitative description also applies to the retarded response from a Lorentzian absorption window (fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, bottom). Storage can be expected as well but at the price of temporal shape distortion.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.7\linewidth]{SL_Lorentz_ILorentz_shaded.eps}}
\caption{Top: {\it Slow-light} in a transparency window as in fig. \ref{fig:SL_ILorentz}. The shaded area after half of the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$ represents the separation between the input and the outgoing pulses. Bottom: Retarded response from a Lorentzian absorption window as in fig. \ref{fig:SL_Lorentz}. We choose for the separation between input and output the expected generalized group delay $\displaystyle\frac{1}{d \Gamma_0}=.05$.}
\label{fig:SL_Lorentz_ILorentz_shaded}
\end{figure}
Following our interpretation, as soon as input and output are well separated, there is a moment when a f{r}action of the light is contained in the atomic excitation. This fraction defines the storage efficiency. The transition from {\it slow} to {\it stopped-light} requires to detail the specific storage protocols by giving a physical reality to the (inverted-)Lorenzian susceptibility. {\it Slow-light} ensures that the optical excitation is transiently contained in the atomic medium. For {permanent} storage {and on-demand readout}, it is necessary to act dynamically on the atomic excitation as we will see now. More precisely, the shelving of the excitation into the spin (by a brief Raman $\pi$-pulses or by switching off the control field as we will see in \ref{raman_stopped} and \ref{EIT_stopped} respectively) prevents the radiation of the retarded response. The excitation is trapped in the atomic ensemble. The evolution is resumed at the retrieval stage by the reversed operation (by a second brief Raman pulses or by switching on the control field).
Before going into details of the storage schemes, we briefly show that the correct orders of magnitude for the efficiencies can be derived from our simplistic vision. From fig.\ref{fig:SL_Lorentz_ILorentz_shaded}, we can roughly evaluate the efficiency by subtracting the blue from the red area assuming the incoming energy (integral of the incoming pulse) is one. We find for {\it slow-light} in a transparency window (inverted-Lorentzian profile) a potential efficiency of 43\% and for the retarded response of an absorption window (Lorentzian profile) 32\%.
We will keep these numbers as points of comparison for specific protocols that we will first explicitly connect to the {\it slow-light} propagation from an inverted-Lorentzian or a Lorentzian and then numerically simulate with the previously established Schr\"odinger-Maxwell equations.
\subsection{Fast storage and retrieval with brief Raman $\pi$-pulses}\label{raman_stopped}
Our approach is based on the fact that {\it slow-light} is associated with the transient storage of the incoming pulse into the atomic excitation. A simple method to store {more} permanently the excitation is to convert instantaneously the optical excitation into a spin wave. This can be done by a Raman $\pi$-pulse as proposed in different protocols. We will now go into details and properly define the level structure and the temporal sequence required to implement the previously discussed archetypes (sections \ref{section:TW} and \ref{section:AW}). We will consider two specific protocols: the spectral hole memory and the free induction decay memory proposed in \cite{Lauro1} and \cite{Vivoli} respectively.
\subsubsection{Spectral hole memory}\label{SHOME}
The spectral hole memory has been proposed by Lauro {\it et al.} in \cite{Lauro1} and partially investigated experimentally in \cite{Lauro2}. The protocol has been successfully implemented in \cite{SHOME} at the single photon level with a quite promising efficiency of 31\%. An inhomogeneously broaden ensemble is first considered. A spectral hole is then burnt into the inhomogeneous distribution. This situation is realistic and corresponds to rare-earth doped crystals for which the spectral hole burning mechanism, as spectroscopic tool, can be efficiently used to sculpt the absorption profile \cite{liu2006spectroscopic}. When the hole profile is Lorenztian, the propagation of a weak signal pulse precisely corresponds to the situation \ref{section:TW} as we will see now.
The atomic evolution is described by eqs.(\ref{bloch_P}\&\ref{bloch_S}) and the propagation by eq.\eqref{MB_M_inhom_pert}. The signal $\mathcal{E}(z,t)$ propagates initially through the atomic distribution described by
\begin{equation}gin{equation}
g\left(\Delta\right)=\left[ 1- \frac{1}{1+\left(\Delta/\Gamma_0\right)^2} \right]\label{g_shome}
\end{equation}
where $\Gamma_0$ is the spectral hole width.
The Raman field is initially off and is only applied for the rapid conversion into the spin wave. When the Raman field is off, the evolution eq.\eqref{bloch_P} reads as $\displaystyle \partial_t \mathcal{P} = (i\Delta-\Gamma) \mathcal{P} - i \frac{\mathcal{E}}{2}$. The coherence lifetime $1/\Gamma$ (inverse of the homogeneous linewidth) is assumed to be much longer than the time of the experiment such as in the spectral domain we write in the limit $\Gamma \rightarrow 0$
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ \tilde{\mathcal{E}}(z,\omega)}{2\left(\Delta-\omega\right)} \end{equation}
So the propagation reads as
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta \end{equation}
The term $\displaystyle \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta$ represents the susceptibility. The integral over $\Delta$ ensures that the Kramers-Kroning relations are satisfied. This last term is then given by the Hilbert transform of the distribution $g\left(\Delta\right)$ so we have $\displaystyle \frac{i}{\pi} \int_\Delta \frac{g\left(\Delta\right)}{\Delta-\omega} d\Delta =\left[ 1- \frac{1}{1+i\omega/\Gamma_0} \right]$. The propagation of the signal is indeed given by eq.\eqref{propag_ILorentz} as described in section \ref{section:TW} and as represented in figs.\ref{fig:SL_ILorentz} and \ref{fig:SL_Lorentz_ILorentz_shaded} (top). The delayed pulse (or at least the fraction which is sufficiently separated from the input) can be stored as represented by shaded areas in fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (top). As proposed in \cite{Lauro1}, a Raman $\pi$-pulse can be used to shelve the optical excitation into the spin. A second Raman $\pi$-pulse triggers the retrieval. They are applied on resonance ($|s\rangle$-$|e\rangle$ transition) so $\delta=0$ in eq.\eqref{bloch_S}.
When the input and the output overlap as in many realistic situations or in other words when the signal cannot be fully compressed spatially into the medium, the storage step cannot be solved analytically. A numerical simulation of the Schr\"odinger-Maxwell equations is necessary (eqs.\ref{bloch_P}\&\ref{bloch_S} with $\Gamma=0$ and $\delta=0$ and eq.\eqref{MB_M_inhom_pert} for the propagation). For a given inhomogeneous detuning $\Delta$, we calculate the atomic evolution eqs.(\ref{bloch_P}\&\ref{bloch_S}) by using a fourth-order Runge-Kutta method. After summing over the inhomogeneous broadening, the output pulse is given by integrating eq.\eqref{MB_M_inhom_pert} along $z$ using the Euler method. A good test for the numerical simulation is to calculate the output pulse without Raman pulses and compare it to the analytic expression from the Fourier transform of eq.\eqref{SL_TW}.
The Raman $\pi$-pulses defined by $\Omega(t)$ are taken as two Gaussian pulses whose area is $\pi$. Following the insight of fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (top), we choose to apply the first Raman pulse at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$. The second Raman pulse is applied later to trigger the retrieval. The result of the storage and retrieval sequence is presented in fig.\ref{fig:plot_outputIO_SHOME}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_SHOME.eps}}
\caption{Spectral hole memory protocol. Top: Absorption profile from the inhomogeneous distribution $g$ defined by eq.\eqref{g_shome}. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference when there is no Raman pulse. Bottom: Two Raman $\pi$-pulses. The first one is applied at half the group delay $\displaystyle \frac{d}{4 \Gamma_0}=5$ and the second later on to trigger the retrieval.
}
\label{fig:plot_outputIO_SHOME}
\end{figure}
As parameters for the simulation, we choose the same as in \ref{section:TW} meaning $\Gamma_0=1$, an optical depth of $d=20$ and $\sigma =\displaystyle \frac{d}{2 \Gamma_0}=10$ for the incoming pulse duration. The Raman pulses should be ideally short to uniformly cover the signal excitation bandwidth. In our case, we choose Gaussian Raman pulses with a duration $\sigma_\pi=1$ (ten times shorter than the signal).
In fig.\ref{fig:plot_outputIO_SHOME} (middle), we clearly see that the first Raman pulse somehow clips the {\it slow-light} pulse corresponding to the shelving of the optical excitation into the spin wave. At this moment, since part of the input pulse is still present, a small replica is generated leaving the medium at time $20$ in our units. The second Raman $\pi$-pulse (at time $60$) triggers the retrieval that we shaded in pale red. A realistic storage situation cannot be fully described by our qualitative picture in fig.\ref{fig:SL_Lorentz_ILorentz_shaded} where the {\it slow-light} signal would be clipped, frozen, delayed and retrieved later on. The complex propagation of clipped Gaussian excitations in the medium can only be accurately embraced by a numerical simulation. The naive picture gives nonetheless a qualitative guideline to understand the storage. A quantitative analysis can be performed by evaluating the stored energy corresponding to the pale red shaded area under the intensity curve. From the simulation, we obtain 36\% to be compared with the 43\% obtained from fig.\ref{fig:SL_Lorentz_ILorentz_shaded}. The agreement is satisfying given the numerical uncertainties and the complexity of the propagation process when the Raman pulses are applied. We now turn to the complementary situation described in section \ref{section:AW} by following the same procedure.
\subsubsection{Free induction decay memory} \label{FID}
Free induction decay memory to take the terminology of the article by Caprara Vivoli {\it et al.} \cite{Vivoli} has not been yet implemented in practice despite a connection with the extensively studied {\it slow-light} protocols. The situation actually corresponds to our description in section \ref{section:AW} where the response of a Lorenztian to a pulsed excitation is considered. This response has been analyzed as a generalization of the free induction decay phenomenon (FID) by Caprara Vivoli {\it et al.} \cite{Vivoli}. The FID is usually observed in low absorption sample after a brief excitation. The analysis in terms of FID is perfectly valid. The response that we considered with eq.\eqref{eq:FID} with a first order expansion of the susceptibility falls into this framework. We analyze the same situation in different terms recovering the same reality. The excitation produces a retarded response that we consider as a generalized version of {\it slow-light}. This semantically connects {\it slow-light} and FID in the context of optical storage.
For the FID memory, the transition can be inhomogeneously or homogeneously broaden{ed}. Both lead to the same susceptibility. We assume the medium homogenous with a linewidth $\Gamma$ thus simplifying the analysis and the numerical simulation, the propagation being given by eq.\eqref{MB_M_hom_pert}.
As in the spectral hole memory, the Raman field is initially off and serves as a rapid conversion into the spin wave by the application of a $\pi$-pulse. When the Raman field is off, the evolution (eq.\ref{bloch_P}) reads as $\displaystyle \partial_t \mathcal{P} = -\Gamma \mathcal{P} - i \frac{\mathcal{E}}{2}$. The signal is directly applied on resonance so $\Delta=0$. We then obtain for the polarization \begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega+\Gamma\right)} \end{equation}
and the propagation
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{\Gamma}{i\omega+\Gamma} \end{equation} whose solution is indeed given by eq.\eqref{SL_TA}. The output pulse is distorted and globally affected by a typical delay $\displaystyle \frac{1}{d \Gamma}$. A first Raman $\pi$-pulse can be applied at this moment. A second Raman $\pi$-pulse triggers the retrieval. As in the spectral hole memory (section \ref{SHOME}), they are applied on resonance so $\delta=0$ in eq.\eqref{bloch_S}.
The complete protocol (when Raman pulses are applied) can only be simulated numerically from the Schr\"odinger-Maxwell equations (eqs.\ref{bloch_P}\&\ref{bloch_S} with $\Delta=0$ and $\delta=0$ and eq.\eqref{MB_M_hom_pert} for the propagation in an homogeneous sample). Following fig.\ref{fig:SL_Lorentz_ILorentz_shaded} (bottom), we choose to apply the first Raman pulse at the generalized group delay $\displaystyle \frac{1}{d \Gamma}$. The second Raman pulse is applied later on to trigger the retrieval.
For the simulation, we again choose the parameters used in section \ref{section:AW} namely a linewdith $\Gamma=1$ and a signal pulse duration of $\sigma =\displaystyle\frac{1}{d \Gamma}=0.05$ corresponding to the expected generalized group delay for an optical depth $d=20$. The Raman pulses have a duration $\sigma_\pi=0.005$ (ten times shorter than the signal) and a $\pi$-area. The result is presented in fig.\ref{fig:plot_outputIO_FID}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_FID.eps}}
\caption{Free induction decay memory. Top: Lorentzian absorption profile from an homogeneous sample. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference when there is no Raman pulse. Bottom: Two Raman $\pi$-pulses. The first one is applied at $\displaystyle \frac{1}{d \Gamma}=0.05$ and the second later on to trigger the retrieval.}
\label{fig:plot_outputIO_FID}
\end{figure}
We retrieve the tendencies of the spectral hole memory. The first Raman pulse clips the {\it slow-light} pulse by storing the excitation into the spin state. As opposed to the propagation in the spectral hole, there is no replica after the first Raman pulse. This replica is strongly attenuated (slightly visible in fig.\ref{fig:plot_outputIO_FID}) because it propagates through the absorption window. We trigger the retrieval at time $0.8$ by a second Raman $\pi$-pulse. The temporal output shape cannot be compared to a clipped version of the input or the {\it slow-light} pulse. This situation is clearly more complex than the spectral hole memory. That being said, the resemblance of the output shape with a exponential decay somehow a posteriori justifies the term FID for this memory scheme. The red pale shaded area represents an efficiency of 42\% with respect to the input pulse energy. This numerical result has to be compared with 32\% obtained from fig.\ref{fig:SL_Lorentz_ILorentz_shaded}. The agreement is not satisfying even if it is difficult to have a clear physical vision of the pulse distortion induced by the propagation at large optical depth. The order of magnitude is nevertheless correct.
The FID protocol can be optimally implemented by using an exponential rising pulse for the incoming signal (instead of a Gaussian in fig.\ref{fig:plot_outputIO_FID}, middle) as analyzed in the reference paper \cite{Vivoli}. In that case, input (rising exponential) and output (decaying exponential) pulse shapes are time-reversed corresponding to the optimization procedures defined in \cite{GorshkovII, GorshkovPRL} and implemented in the EIT/Raman memories \cite{Novikova, nunnMultimode, zhou2012optimal}
Starting from two representative situations in \ref{section:TW} and \ref{section:AW} where the dispersion produces a retarded response from the medium, we have analyzed two related protocols in \ref{SHOME} and \ref{FID} that qualitatively corresponds to the storage of this delayed response. Except in a recent implementation \cite{SHOME}, these protocols have not been much considered in practice despite a clear connection with the archetypal propagation through the Lorentzian susceptibility of an atomic medium. On the contrary, electromagnetically induced transparency and Raman schemes are well-known and extensively studied experimentally. We will show now that they follow the exact same classification thus enriching our comparative analysis.
\subsection{Electromagnetically induced transparency and Raman schemes}\label{EIT_stopped}
Starting from two pioneer realizations \cite{phillips2001storage, liu2001observation}, the implementation of the electromagnetically induced transparency (EIT) scheme has been continuously active in the prospect of quantum storage. As opposed to the spectral hole (section \ref{SHOME}) and the free induction decay (section \ref{FID}) memories and recalling to the reader the main difference, EIT is not based on the transient excitation of the optical transition that is rapidly transfered into the spin by a Raman $\pi$-pulse. In EIT, the direct optical excitation is avoided by precisely using the so-called dark state in a $\Lambda$-system \cite{fleischhauer2000dark, FLEISCHHAUER2000395}. Practically, a control field is initially applied on the Raman transition to obtain {\it slow-light} from the $\Lambda$-system susceptibility\footnote{Inversely, for the spectral hole in \ref{SHOME} and the free induction decay in \ref{FID} memory, the Raman field is initially off.}. As a first cousin, the Raman memory scheme has been proposed and realized afterward \cite{nunnPRA,nunnNat}. EIT and Raman memories are structurally related by a common $\Lambda$-system which is weakly excited by the signal on one branch and controlled by a strong laser on the Raman branch (see fig.\ref{fig:2level_3level}). The main difference comes from the excited state detuning. For EIT scheme, the control field is on resonance. For the Raman scheme, the control field is off resonance. As we will see now, these two situations actually corresponds to the archetypal dispersive profiles described in \ref{section:TW} and \ref{section:AW} respectively.
\subsubsection{Electromagnetically induced transparency memory}\label{EIT}
The atomic susceptibility in a $\Lambda$-system is derived from eqs.(\ref{bloch_P}\&\ref{bloch_S}). Initially, the Raman field $\Omega$ is on and assumed constant in time. In EIT, the signal and control fields are on resonance so $\Delta=\delta=0$. The medium is assumed homogeneous even if the calculation can be extended to the inhomogeneously broaden systems \cite{kuznetsova2002atomic}. The propagation is here given by eq.\eqref{MB_M_hom_pert}.
In the spectral domain, eqs.(\ref{bloch_P}\&\ref{bloch_S}) read as
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega+\Gamma-\displaystyle i\frac{\Omega^2}{4\omega}\right)} \end{equation}
We have assumed the control field {to be} real so the intensity is written as $\Omega^2$ which can be generalized to $\Omega^*\Omega$ for complex values (chirped Raman pulses for example).
The linear susceptibility for the signal field is defined by the propagation equation in the spectral domain
\begin{equation}gin{equation} \frac{\ensuremath{\partial}\tilde{\mathcal{E}}(z,\omega)}{\ensuremath{\partial} z} +i\frac{\omega}{c}\tilde{\mathcal{E}}(z,\omega)= \displaystyle-\frac{\alpha}{2} \tilde{\mathcal{E}}(z,\omega) \frac{\Gamma}{i\omega+\Gamma - \displaystyle i\frac{\Omega^2}{4\omega} } \label{propag_EIT}\end{equation}
The term $i \displaystyle \frac{\Omega^2}{4\omega}$ induces the transparency when the control field is applied. Without control, the susceptibility is Lorentzian and the signal would be absorbed following the Bouguer-Beer-Lambert absorption law (eq.\ref{bouguer}). On the contrary, when the control field is on, the susceptibility is zero when $\omega \rightarrow 0$. This corresponds to the resonance condition because we assumed $\Delta=\delta=0$. The analysis can be further simplified by considering a first order expansion within the transparency window.
The width of the transparency window is $\ensuremath{\Gamma_\mathrm{EIT}}=\displaystyle \frac{\Omega^2}{4\Gamma}$ which is usually much narrower than $\Gamma$. So, in the limit $\omega \ll \ensuremath{\Gamma_\mathrm{EIT}} \ll \Gamma$, the propagation constant reads as
\begin{equation}gin{equation} \displaystyle-\frac{\alpha}{2} \frac{\Gamma}{i\omega+\Gamma - \displaystyle i\frac{\Omega^2}{4\omega} } \simeq \displaystyle-\frac{\alpha}{2} \left[ 1- \frac{1}{1+i\omega/\ensuremath{\Gamma_\mathrm{EIT}}} \right] \end{equation}
The EIT window is locally an inverted-Lorentzian that we have analyzed in \ref{section:TW}. The {\it slow-light} propagation is precisely due to the presence of the control field. The so-called dark state corresponds to a direct spin wave excitation whose radiation is mediated by the control field. The storage simply requires the extinction of the control field. The excitation is then frozen in the non-radiating Raman coherence because of the absence of control. The retrieval is triggered by switching the control back on.
The {\it stopped-light} experimental sequence can be simulated numerically from eqs.(\ref{bloch_P}\&\ref{bloch_S}) and eq.\eqref{MB_M_hom_pert}. For the parameters, we choose the same as in \ref{section:TW} and \ref{SHOME}, meaning $\ensuremath{\Gamma_\mathrm{EIT}}=1$ so the width of the inverted-Lorentzian is $1$. We opt for $\Omega=4$ and $\Gamma=4$ so the condition $\ensuremath{\Gamma_\mathrm{EIT}} \ll \Gamma$ is vaguely satisfied. Again the optical depth is $d=20$ and $\sigma =\displaystyle \frac{d}{2 \ensuremath{\Gamma_\mathrm{EIT}}}=10$ is the incoming pulse duration. At time $\displaystyle \frac{d}{4\ensuremath{\Gamma_\mathrm{EIT}}}=5$, half the group delay, the control field is switched off ($\Omega=0$). The result is plotted in fig.\ref{fig:plot_outputIO_EIT}.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_EIT.eps}}
\caption{Electromagnetically induced transparency memory. Top: EIT absorption profile. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulse (dashed red) as a reference from eq.\eqref{propag_EIT} when the control field is always on. Bottom: The control is initially on with $\Omega^2=16$. It is switched off at half of the group delay $\displaystyle \frac{d}{4 \ensuremath{\Gamma_\mathrm{EIT}}}=5$ and back on later to trigger the retrieval.}
\label{fig:plot_outputIO_EIT}
\end{figure}
Although the condition $\ensuremath{\Gamma_\mathrm{EIT}}=1 \ll \Gamma=4$ is only roughly satisfied so the absorption profile is not a pure inverted-Lorenzian, this has a minor influence on the {\it slow} and {\it stopped-light} pulses. The resemblance with fig.\ref{fig:plot_outputIO_SHOME} is striking even if the spectral hole and EIT memories cover different physical realities. From fig.\ref{fig:plot_outputIO_EIT}, we can estimate the efficiency (red pale area) to 42\% thus retrieving the same expected efficiency as the spectral hole memory. One difference between fig.\ref{fig:plot_outputIO_SHOME}(middle) and fig.\ref{fig:plot_outputIO_EIT}(middle) is worth being commented: there is not {\it slow-light} replica after the time $\displaystyle \frac{d}{4 \ensuremath{\Gamma_\mathrm{EIT}}}=5$ for the EIT situation. This replica is absorbed in that case because when the control field is switched off, the absorption is fully restored. The presence or the absence of replicas does not change the efficiency because they correspond to a fraction of the incoming pulse that is not compressed in the medium. This leaks out and is lost anyway.
We will now complete our picture by considering the Raman memory and emphasize the resemblance with the free induction decay discussed in \ref{FID}.
\subsubsection{Raman memory}\label{Raman}
The Raman memory scheme is based on the same $\Lambda$-structure when a control field is applied far off-resonance on the Raman branch \cite{nunnPRA,nunnNat, sheremet} (see fig.\ref{fig:2level_3level}). The condition $\Delta \gg \Gamma$ defines literally the Raman condition as opposed to EIT where the control is on resonance ($\Delta=0$). The absorption profile exhibits the so-called Raman absorption peak. This Lorentzian profile is the basis for a retarded response that we introduced in \ref{section:AW}. We first verify that the far off-resonance excitation of the control leads to a Lorenztian susceptibility for the signal. As in the EIT case (see \ref{EIT}), the atomic evolution in a $\Lambda$-system is given by eqs.(\ref{bloch_P}\&\ref{bloch_S}) and the propagation by eq.\eqref{MB_M_hom_pert}.
The polarization is
\begin{equation}gin{equation} \tilde{\mathcal{P}}(\Delta,\omega)= \displaystyle \frac{ -i \tilde{\mathcal{E}}(z,\omega)}{2\left(i\omega-i\Delta+\Gamma-\displaystyle i\frac{\Omega^2}{4(\omega-\delta)}\right)} \end{equation}
The two-photon detuning $\delta$ is not zero in that case because the Raman absorption peak in shifted by the AC-Stark shift (light shift). The signal pulse has to be detuned by $\delta=\displaystyle \frac{\Omega^2}{4\Delta}$, the light-shift, to be centered on the Raman absorption peak. Following the same approach as in the EIT case, the analysis can be simplified by a first order expansion is $\omega$. Assuming the incoming pulse bandwidth $\omega$ smaller than the light shift $\delta,$ the latter being smaller than the detuning $\Delta$, that is $\omega \ll \delta \ll \Delta$, the propagation constant reads to the first order in $\omega$ as
\begin{equation}gin{equation}
\displaystyle-\frac{\alpha}{2} \frac{\Gamma}{i\omega-i\Delta+\Gamma - \displaystyle i\frac{\Omega^2}{4(\omega-\delta)} } \simeq \displaystyle-\frac{\alpha}{2} \frac{\ensuremath{\Gamma_\mathrm{R}}}{\ensuremath{\Gamma_\mathrm{R}}+i\omega} \label{Raman_susceptibility}
\end{equation}
where $\displaystyle \ensuremath{\Gamma_\mathrm{R}}=\frac{\Omega^2\Gamma}{4\Delta^2}$ is the width of the Raman absorption profile. This Lorentzian absorption profile can be used for storage as discussed in \ref{section:AW} and \ref{FID}. As in the EIT case, the storage is triggered by the extinction of the Raman control field. To fully exploit the analogy with \ref{section:AW} and \ref{FID}, we will choose $\ensuremath{\Gamma_\mathrm{R}}=1$. To satisfy the far off resonance Raman condition, we choose $\Gamma=10$ and $\Delta=1000$ thus imposing $\Omega=200\sqrt{10}$ and $\delta=100$. We run a numerical simulation of eqs.(\ref{bloch_P}\&\ref{bloch_S}) and eq.\eqref{MB_M_hom_pert} with a Gaussian incoming pulse whose duration is again $\sigma =\displaystyle\frac{1}{d \Gamma_R}=0.05$ and with an optical depth $d=20$. The result is presented in fig.\ref{fig:plot_outputIO_Raman} where the control Raman field is switched off at time $\displaystyle\frac{1}{d \Gamma_R}=0.05$ (the typical delay) and switched back on later to trigger the retrieval.
\begin{equation}gin{figure}
\centering
\fbox{\includegraphics[width=.8\linewidth]{plot_outputIO_Raman.eps}}
\caption{Raman memory. Top: Raman absorption profile. Middle: Incoming signal (in blue) and outgoing stored pulse (in red). We have also represented the {\it slow-light} pulsed (dashed red) as a reference from eq. \eqref{Raman_susceptibility} when the Raman control field is always on. Bottom: The control is initially on with $\Omega=200\sqrt{10}$ and is switched later on to trigger the retrieval.}
\label{fig:plot_outputIO_Raman}
\end{figure}
The resemblance with fig.\ref{fig:plot_outputIO_FID} is noticeable. Transient rapid oscillations appears when the control is abruptly switched, this is a manifestation of the light-shift. Without surprise, the expected efficiency (red-pale area) is 42\% as the free induction decay memory with the same intensive parameters (see \ref{FID}).
\subsection{Summary and perspectives}
We have given in this section a unified vision of different {\it slow-light} based protocols. In this category, the ambassador is certainly the EIT scheme which has been particularly studied in the last decade with remarkable results in the quantum regime \cite{review_Ma_2017}. The linear dispersion associated with the EIT transparency window allows to define unambiguously a slow group velocity whose reduction to zero produces {\it stopped-light}. We have extended this concept to any retarded response that can be seen as a precursor for storage. This approach allows us to interpret the Raman scheme within the same framework. In that case, the group velocity cannot be defined {\it per se} but the dispersion profile still produces a retarded response that can be stored by shelving the excitation into a long lived spin state. The price to pay at the retrieval step is a significant pulse distortion even if the efficiency (input/output energy ratio) is quite satisfying. The pulse distortion at the retrieval is somehow a false problem. Distortions are more or less always present. Even in the more favorable EIT scheme, the pulse can be partially clipped because of a limited optical depth. It should be kept in mind that {quantum} repeater architectures {use} interference between outgoing photons \cite{simon2007, RevModPhys.83.33, bussieres2013prospective}. As soon as the different memories induce the same distortion, the retrieved outgoing {fields} can perfectly interfere. In that sense, the deformation can also be considered as an unitary transform between temporal modes without degrading the quantum information quality \cite{brecht2015photon, Thiel:17}.
The signal temporal deformation also raises the question of the waveform control through the storage step. We have used a simplistic model for the control field (on/off or $\pi$-pulses). A more sophisticated design of the control actually allows a build-in manipulation of the temporal and frequency modes of the stored qubit \cite{fisher2016frequency, conversion}. A quantum memory can be also considered a versatile light-matter interface with a enhanced panel of processing functions. Waveform shaping is not considered anymore as a detrimental experimental limitation but as new degree of freedom whose first benefit is the storage efficiency \cite{Novikova, zhou2012optimal} when specific optimization procedures are implemented \cite{GorshkovPRL, GorshkovII, nunnMultimode}. The optimization strategy by temporal shaping is beyond the scope of this chapter but would certainly {deserve} a review paper by itself.
The fast storage schemes and the EIT/Raman sequences that we analyzed in parallel in sections \ref{raman_stopped} and \ref{EIT_stopped} respectively, both rely on a Raman coupling field that control the storage and retrieval steps. The fast storage schemes depend on $\pi$-pulses and the EIT/Raman sequence on a control on/off switching. A three-level $\Lambda$-system seems to be necessary in that case. This is not rigorously true even if the $\Lambda$-structure is widely exploited for quantum storage. {\it Stopped-light} can indeed be obtained in two-level atoms by dynamically controlling the atomic properties \cite{simon_index, simon_dipole, chos}. Despite a lack of experimental demonstrations, these two-level alternative approaches conceptually extend the protocols away from the well-established atomic $\Lambda$-structure.
To close the loop with the previous section \ref{sec:2PE} on photon echo memories, we would like to discuss again the atomic frequency comb (AFC) protocol \cite{afc}. Despite its historical connection with the three-pulse photon echo sequence, {it has been argued that} the AFC falls in the {\it slow-light} memories \cite{afc_slow}. A judicious periodic shaping of the absorption profile, forming a comb, allows to produce an efficient echo. This latter can alternatively be interpreted as an undistorted retarded response using the terminology of section \ref{sec:SL}. This retarded part is a precursor that can be stored by shelving the excitation into spin states by a Raman $\pi$-pulse, thus definitely positioning the AFC in the fast storage schemes (section \ref{raman_stopped}).
\section{Certifying the quantum nature of light storage protocols}\label{sec:certification}
The question that we address in this section is how to prove that a storage protocol operates in the quantum regime. The most natural answer is: { by demonstrating that the quantum nature of a light beam is preserved after storage}. There are, however, several ways for a {memory to output light beams that show} quantum features. It can simply be a light pulse, like a single photon, that cannot be described by a coherent state or a statistical mixture of coherent state \cite{glauber}.
Alternatively, a state can be qualified as being quantum when it leads to correlations between measurement results that cannot be reproduced by classical strategies based on pre-agreements and communications, as some entanglement states do. What is thus the difference between showing the capability of a given memory to store and retrieve single photons and entangled states ?
{Faithful} storage and retrieval of single photons {demonstrates} that the noise {generated by} the memory is low enough to preserve the photon statistics, even when {these statistics} cannot be reproduced by classical light. It does not show, however, that the memory preserves coherence. Furthermore, it does not prove that the memory cannot be reproduced by a classical strategy, that is, a protocol which would first {measure} the incoming photon and create another photon when requested.
{On the other hand}, the storage and retrieval of entangled states can be {implemented} to show that the memory outperforms any classical {measure}-and-prepare strategy. This is true provided that the fidelity of the storage protocol is high enough. For example, if a memory is characterized by storing one part of a two-qubit entangled state, the fidelity threshold is given by the fidelity of copies that would be created by a cloning machine taking one qubit and producing infinitely many copies. This is known to be one of the optimal {strategies} for determining an unknown qubit state \cite{Gisin1}.
Note that the fidelity reference can also be taken as the fidelity that would be obtained by a cloning protocol producing only one copy of the output state \cite{Scarani}. In this case, the goal is to {ensure} that the memory delivers the state with the highest possible fidelity, that is, if a copy exists, it cannot have a higher overlap with the input state. This {condition} is relevant whenever one wants to show the suitability of the memory for applications related to secure communications, {where third parties should not obtain information about the stored state} \cite{BB84}.
The suitability of a memory for secure communications can ultimately be certified {by} Bell tests \cite{Bell:1964kc}. In this case, the quality of the memory can be {estimated} without {assumptions} on the input state or on the measurements performed on the {retrieved} state. This {ensures} that the memory can be used in networks where secure communications can be realized over long distances with security guarantees holding independently of the details of the actual implementation.
We show in the following sections how these criteria can be tested in practice, describing separately benchmarks based on continuous and discrete variables. Various memory protocols are {used} as examples, including protocols such as {the two-pulse photon echo (2PE)} \cite{ruggiero} or the classical teleporter, which are known to be classical. In order to prove it, we first show how to compute the noise {inherent to} classical protocols by moving away from the semi-classical picture. While a fully quantized propagation model can be found in the literature \cite{GorshkovII} mirroring the semi-classical Schrödinger-Maxwell equations that we use in the previous sections, we present a toy model using an atomic chain to characterize memory protocols together with their noise (section \ref{toy_model}). Criteria are then derived first for continuous (section \ref{CV_criterion}) and then for {discrete} variables (section \ref{counting_criterion}).
}
\subsection{Atomic chain quantum model}\label{toy_model}
The aim is to derive a simple quantum model allowing to characterize different storage protocols including the noise. Although quantum, the model is very simple and uses the basic tools of quantum optics.
\subsubsection{Jaynes-Cummings propagator}
We consider an electromagnetic field described by the bosonic operators $a$ and $a^\dag$ resonantly interacting with a single two-level atom (with levels $|g\rangle$ and $|e\rangle$) thought the Jaynes-Cummings Hamiltonian
\begin{equation}gin{equation}
\label{Jaynes_Cummings}
H_{\text{int}} = i \kappa (a^\dag \sigma_- - a \sigma_+).
\end{equation}
Here, $\sigma_\pm$ are atomic operators corresponding to the creation $\sigma_+ = |e\rangle\langle g| $ and annihilation $\sigma_- = |g\rangle\langle e|$ of an atomic excitation. The first term in \eqref{Jaynes_Cummings} is thus associated to the emission of a photon while the second term corresponds to its absorption. The corresponding propagator
\begin{equation}gin{equation}
U(\tau) = e^{\kappa \tau (a^\dag \sigma_- - a \sigma_+)} = \sum_{n \geq 0} \frac{(\kappa \tau)^n}{n!}(a^\dag \sigma_- - a \sigma_+)^n
\end{equation}
can be written as
\begin{equation}gin{align}
\nonumber
&& U(\tau) = \cos(\kappa \tau \sqrt{a^\dag a}) |g\rangle \langle g | - a \sin(\kappa \tau \sqrt{a^\dag a})/\sqrt{a^\dag a} |e\rangle\langle g| \\
\nonumber
&&\quad + \cos(\kappa \tau \sqrt{a a^\dag}) |e\rangle \langle e| + a^\dag \sin(\kappa \tau \sqrt{a a^\dag})/\sqrt{a a^\dag} |g\rangle\langle e|
\end{align}
by noting that
\begin{equation}gin{equation}
\nonumber
(a^\dag \sigma_- - a \sigma_+)^{2k} = (-1)^k \left(\left(a^\dag a \right)^k |g\rangle\langle g| + \left(a a^\dag \right)^k |e\rangle\langle e|\right),
\end{equation}
and
\begin{equation}gin{equation}
\nonumber
(a^\dag \sigma_- - a \sigma_+)^{2k+1} = (-1)^k \left(a^\dag \left(a a^\dag \right)^k \sigma_- - a \left(a^\dag a \right)^k \sigma_+\right).
\end{equation}
Hence, the following initial states read
\begin{equation}gin{align}
\nonumber
U(\tau) |g,0\rangle & \rightarrow |g,0\rangle,\\
\nonumber
U(\tau) |g,1\rangle& \rightarrow \cos(\kappa \tau) |g,1\rangle - \sin(\kappa \tau) |e,0\rangle,\\
\nonumber
U(\tau) |e,0\rangle &\rightarrow \cos(\kappa \tau) |e,0\rangle + \sin(\kappa \tau) |g,1\rangle.
\end{align}
\subsubsection{Absorption}
Let us now consider a collection of $N$ atoms, all prepared in the ground state $|g\rangle$ and each interacting with a single photon through the Jaynes-Cummings interaction. The state of the atoms associated with a successful absorption is given by
\begin{equation}gin{equation}
\nonumber
\rho_{\text{cond}} = \mathrm{Tr}_{\text{light}} \left[\ket{0}\bra{0} \otimes \id \-\ U_N \hdots U_1 |g\hdots g,1\rangle \langle g\hdots g,1 | U_1^\dag \hdots U_N^\dag\right]
\end{equation}
and takes the form $\rho_{\text{cond}} = |\Psi_{\text{cond}}\rangle\langle \Psi_{\text{cond}}|$ when applying explicitly the $N$ propagators, where
\begin{equation}gin{equation}
\nonumber
|\Psi_{\text{cond}}\rangle =c^{N-1}s |g\hdots g e\rangle + c^{N-2} s |g\hdots e g \rangle + \hdots + s |e g \hdots g\rangle.
\end{equation}
Note that we have introduced the shorthands $c=\cos(\kappa \tau)$ and $s=\sin(\kappa \tau).$ The normalization of $\Psi_{\text{cond}},$ that is $1-\cos^{2N}(\kappa \tau),$ gives the probability of a successful absorption. For a small absorption amplitude per atom $\kappa \tau = \sqrt{d /N} \ll 1$ where $d=\alpha L$ is the total optical depth of the atomic chain and a large atom number, we have
\begin{equation}gin{equation}
\lim_{N\rightarrow \infty} \cos^{2N}(\kappa \tau) \approx \lim_{N\rightarrow \infty} \left(1-\frac{d}{2N}\right)^{2N} \rightarrow e^{-d}
\end{equation}
which corresponds to the Bouguer-Beer-Lambert absorption law (eq.\ref{bouguer}). Similarly, the absorption probability $1- \cos^{2N}(\kappa \tau)$ tends to $1-e^{-d}.$
\subsubsection{Storage and retrieval probability}
The overall efficiency including the storage and retrieval probabilities is obtained by calculating
\begin{equation}gin{equation}
\nonumber
|\langle g\hdots g, 1| U_N \hdots U_1 |\Psi_{\text{cond}}\rangle|^2.
\end{equation}
Note that we here consider a forward emission in which the retrieved photon is emitted in the same direction that the input photon. We obtain
\begin{equation}gin{align}
\nonumber
&&|\langle g\hdots g, 1| U_N \hdots U_1 |\Psi_{\text{cond}}\rangle|^2 = N^2 s^4 c^{2N-2} \\
&& \approx d^2 \left(1-\frac{d}{N}\right)^{2N-2} \rightarrow d^2 e^{-d} \-\ \text{when} \-\ N \rightarrow \infty
\end{align}
and thus retrieve the semi-classical forward efficiency eq.\eqref{eta_crib}.
For a backward emission, we obtain
\begin{equation}gin{equation}
|\langle g\hdots g, 1| U_1 \hdots U_N |\Psi_{\text{cond}}\rangle|^2 \rightarrow (1-e^{-d})^2 \-\ \text{when} \-\ N \rightarrow \infty
\end{equation}
corresponding to the semi-classical backward efficiency eq.\eqref{eta_crib_back}.
\subsubsection{Amplification through an inverted atomic ensemble} This simple model allows {us} to compute the expected noise of protocols for which the excited states are significantly populated when the stored excitation is released as in the two-pulse photon echo (2PE) protocol described in section \ref{2PE}. Consider first the case where all the atoms are in $|e\rangle$ and the field is in the vacuum state $|0\rangle.$ The mean photon number after an interaction time $\tau$ is given by
\begin{equation}gin{equation}
\nonumber
\langle e \hdots e,0 | U_N^\dag(\tau) \hdots U_1^\dag(\tau) a^\dag a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle
\end{equation}
which can be seen as the square of the norm of $a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle.$ In the regime where the population remains essentially in the excited state, the atomic operators $\sigma_\pm$ verifies
\begin{equation}gin{equation}
[\sigma_+, \sigma_-] = |e\rangle\langle e| - |g\rangle \langle g| \approx 1.
\end{equation}
In this case, the Hamiltonian \eqref{Jaynes_Cummings} is a squeezing operator between two bosonic modes and the formula $e^B A e^{-B} = \sum_{n\geq 0} \frac{1}{n!} \underbrace{[B,\hdots [B, A]\hdots ]}_{\text{n times}}$ can be used to prove that
\begin{equation}gin{equation}
a U_1 = U_1 \left(\cosh(\kappa \tau) a + \sinh(\kappa \tau) \sigma_-^{(1)}\right).
\end{equation}
Commuting $a$ with $U_2 \hdots U_N,$ we obtain
\begin{equation}gin{align}
\label{transform_field_s}
&a U_1 \hdots U_N = U_1 \hdots U_N \Big(\cosh(\kappa \tau)^N a \\
\nonumber
&+ \cosh(\kappa \tau)^{N-1} \sinh(\kappa \tau) \sigma_-^{(N)} + \hdots +\sinh(\kappa \tau) \sigma_-^{(1)} \Big)
\end{align}
where $\sigma_-^{(i)}$ is the atomic operator $\sigma_-$ for the $i^{th}$ atom. This leads to
\begin{equation}gin{align}
\nonumber
& || a U_1(\tau) \hdots U_N(\tau) | e \hdots e,0\rangle ||^2 \\
\nonumber
&= \sinh(\kappa \tau)^2 \sum_{j=1}^N \cosh(\kappa \tau)^{2j-2} = \cosh(\kappa \tau)^{2N}-1.
\end{align}
Using $\kappa \tau = \sqrt{d/N} \ll 1$ and taking the limit of large $N$, the mean photon number is
\begin{equation}gin{equation}
e^d-1.
\end{equation}
This corresponds to the number of photons emitted in a single mode by an inverted ensemble \cite{RASE, Sekatski}.
More generally, eq.\eqref{transform_field_s} shows that in the regime where the atoms are mainly in the excited state, the atomic ensemble operates as a classical amplifier, the gain $G$ depending exponentially on the optical depth via $G=e^{d}.$ Such an amplifier transforms the field operators according to
\begin{equation}gin{align}
\label{a_out}
& \bar U^\dag a \bar U = \sqrt{G} a + \sqrt{G-1} \-\ \sigma_c^\dag \\
\label{adag_out}
& \bar U^\dag a^\dag \bar U = \sqrt{G} a^\dag + \sqrt{G-1} \-\ \sigma_c
\end{align}
where $\bar U^\dag a \bar U = U_N^\dag (\tau) \hdots U_1^\dag (\tau) a U_1(\tau) \hdots U_N(\tau).$
The bosonic operators $\sigma_c$ and $\sigma_c^\dag$ annihilates and creates collectively atoms in the ground state
$$
\sigma_c^\dag = \frac{1}{\sqrt{M}}(\cosh(\kappa \tau)^{N-1} \sinh(\kappa \tau) \sigma_-^{(N)} + \hdots +\sinh(\kappa \tau) \sigma_-^{(1)})
$$
with the normalization coefficient $M=\cosh(\kappa \tau)^{2N}-1.$\\
Equations \eqref{a_out}\&\eqref{adag_out} allow one to derive expectations values for the field when the atoms are mostly excited. Let us consider for example a 2PE where the first pulse is a single photon Fock state and the second pulse is a $\pi$-pulse. We can compute the response of a non-photon number resolving detector with dectection efficiency $\eta$ at the echo time. If the photon has been successfully absorbed, at the echo time, the atoms are well described by a single excitation $|1\rangle$ in the collective mode $\sigma_c$ and the field $a$ is in the vacuum state. The probability that the photon detector clicks is given by
\begin{equation}gin{align}
\nonumber
\langle 01| \bar U^\dag \big(1-(1-\eta)^{a^\dag a}\big) \bar U |01\rangle & = 1 - \langle 01| \bar U^\dag :e^{-\eta a^\dag a} : \bar U |01\rangle \\
\nonumber
& = 1- \langle 01| \bar U^\dag (1 -\eta aa^\dag + \frac{\eta^2}{2} a^\dag a ^\dag a a - \hdots) \bar U |01\rangle.
\end{align}
Using eqs.\eqref{a_out}\&\eqref{adag_out}, we easily show that
$$ \langle \bar U^\dag \underbrace{a^\dag \hdots a ^\dag}_{\text{k times}} \underbrace{a \hdots a}_{\text{k times}} \bar U\rangle = (k+1)! (G-1)^k.$$
Therefore
\begin{equation}gin{align}
\nonumber
\langle 01| \bar U^\dag \big(1-(1-\eta)^{a^\dag a} \big) \bar U |01\rangle & = 1- \sum_{k\geq 0} (-1)^k \eta^k (G-1)^k (k+1)\\
\label{clic_2pulse}
& = 1- \frac{1}{\big(1+\eta(G-1)\big)^2}.
\end{align}
This formula shows that no click is obtained when the detection efficiency is null while the detectors clicks with unit probability as long as $\eta G \gg 1.$
\subsubsection{Beamsplitter interaction in a non-inverted ensemble}
In the regime where the atoms are and remain essentially in the ground state, the atomic operators $\sigma_\pm$ verifies
\begin{equation}gin{equation}
[\sigma_+, \sigma_-] = |e\rangle\langle e| - |g\rangle \langle g| \approx -1.
\end{equation}
As in the previous paragraph, the formula $e^B A e^{-B} = \sum_{n\geq 0} \frac{1}{n!} [B, A]^n$ can thus be used to prove that in this regime
\begin{equation}gin{equation}
a U_N = U_N \left(\cos(\kappa \tau) a + \sin(\kappa \tau) \sigma_-^{(N)}\right).
\end{equation}
We thus have
\begin{equation}gin{align}
\label{transform_field}
&a U_N \hdots U_1 = U_N \hdots U_1 \Big(\cos(\kappa \tau)^N a \\
\nonumber
&+ \cos(\kappa \tau)^{N-1} \sin(\kappa \tau) \sigma_-^{(1)} + \hdots + \sin(\kappa \tau) \sigma_-^{(N)} \Big).
\end{align}
By introducing the collective operator
\begin{equation}gin{equation}
\bar{\sigma}_c = \frac{1}{\sqrt{\overline M}}(\cos(\kappa \tau)^{N-1} \sin(\kappa \tau) \sigma_-^{(N)} + \hdots +\sin(\kappa \tau) \sigma_-^{(1)})
\end{equation}
with $\overline M = 1-\cos(\kappa \tau)^{2N},$ the atom-light interaction can be seen as a standard beamsplitter-type interaction
\begin{equation}gin{align}
\label{a2}
& \bar U^\dag a \bar U = \sqrt{e^{-d}} a + \sqrt{1-e^{-d}} \-\ \bar{\sigma}_c \\
\label{adag2}
& \bar U^\dag a^\dag \bar U = \sqrt{e^{-d}} a^\dag + \sqrt{1-e^{-d}} \-\ \bar{\sigma}_c^\dag.
\end{align}
The formulas derived from this simple quantum model will be helpful to characterize the quantum nature of different storage protocols as we will see now.
\subsection{Continuous variable criterion}\label{CV_criterion}
Here, we study the propagation and read-out of a pulse with quantum noise through different memories and {review} a criterion to evaluate if {the output state is the best cloned copy of the input, that is, to guarantee that no better copy of the input state is available}. We analyze generic storage protocols in a continuous variable perspective to estimate the amount of noise and loss that can be tolerated {to fulfill this criterion}.
\subsubsection{The stored quantum states}
A quantum memory should be able to store and retrieve any state {while preserving its quantum features}. The state can be a classical state but its quantum statistics should be preserved. In continuous variable quantum information, the variables of interest are the field quadratures, defined as
\begin{equation}gin{equation}X^+=a+a^\dagger\end{equation} and
\begin{equation}gin{equation}X^-=-i(a-a^\dagger)\end{equation}
where $a$ and $a^\dagger$ are the creation and annihilation operators of the field, as in the previous section. As they satisfy the canonical commutation relations $[a,a^\dagger]=1$, it follows that $[X^+,X^-]=2i$ and that
\begin{equation}gin{equation}n=\frac{1}{4}\big[(X^+)^2+(X^-)^2\big]-\frac{1}{2}\end{equation}
where $n=a^\dagger a$ is the photon number operator.
The signal at the output of a quantum memory can be decomposed into a classical amplitude $\alpha$ and a fluctuating noise term $\delta\hat{X}^\pm$.
Formally, for a gaussian state, we write the amplitude and phase quadratures of the field as
\begin{equation}gin{equation}\hat{X}^\pm=\alpha^\pm+\delta\hat{X}^\pm \end{equation}
To avoid writing the propagators when describing the field at the output of the memory, we now introduce the subscript $_{\rm {out}}$ defined as $a_{\rm {out}}= \overline{U}^\dagger a\overline{U}$ for example. Similarly, the subscript $_{\rm {in}}$ is used to describe the input of the memory.
The measured output signal is generally the power spectral density, given by the Fourier transform of the autocorrelation function.
It reads as
\begin{equation}gin{equation}S_{\rm out}^\pm=\langle (\hat{X}_{\rm out}^\pm)^2 \rangle \end{equation}
and the noise as
\begin{equation}gin{equation}V_{\rm out}^\pm=\langle \delta (\hat{X}_{\rm out}^\pm)^2 \rangle \end{equation}
We thus obtain
\begin{equation}gin{equation}S_{\rm out}^\pm=(\alpha_{\rm out}^\pm)^2+V_{\rm out}^\pm.\end{equation}
We will estimate $\alpha_{\rm out}^\pm$ and $V_{\rm out}^\pm$ at the output of the optical memories and identify the values that enable entering the quantum memory regime.
\subsubsection{Quantum memory criterion}
Generally, optical memories are benchmarked against quantum information criteria. In particular, the performance of a given quantum memory can be {evaluated similarly to a quantum teleportation scheme} {by quantifying the quality of the output state with respect to the input.}
Figure \ref{carct} shows the schematics of the quantum memory benchmark. The optimal classical measure and prepare strategy for optical memory consists in measuring the input state jointly on two conjugate quadratures using two homodyne schemes \cite{Hammerer}. The measured information is stored before fed-forward onto an independent beam. In this classical scheme, the storage time can be arbitrarily long without additional degradation. However, two conjugate observables cannot be simultaneously measured and stored without paying a quantum of duty. Moreover, the encoding of information onto an independent beam will also introduce another quantum of noise. In total, the entire process will incur two additional quanta of noise onto the output optical state \cite{HetetPRA}.
Characterizing quantum memory using the state-dependent fidelity as a measure can be complicated for exotic mixed states. Alternatively, we use the signal transfer coefficients $T$ and the input-output conditional variances $V_{cv}$ to establish the efficiency of a process \cite{Grangier, Ralph}. The conditional variances and signal transfer coefficients are defined as
\begin{equation}gin{equation} V_{cv}^{\pm}=V_{\rm out}^{\pm} -\frac{ |\langle X^\pm_{\rm in} X^\pm_{\rm out} \rangle|^2}{V^\pm_{\rm in}} \end{equation}
and
\begin{equation}gin{equation} T^\pm = \frac{R^\pm_{\rm out}}{R^\pm_{\rm in}} \end{equation}
where $R^\pm_{\rm out/in}$ is defined as
\begin{equation}gin{equation} R^\pm_{\rm out/in}=\frac{4(\alpha^\pm_{\rm out/in})^2}{V^\pm_{\rm out/in}}. \end{equation}
We now define two parameters that take into account the performances of the system on both conjugate observables as
\begin{equation}gin{equation} V=\sqrt{V_{cv}^{+}V_{cv}^{-}} \end{equation}
and \begin{equation}gin{equation} T=T^++T^- \end{equation}
It can be shown that a classical memory {based on the measure and prepare scheme described before} cannot overcome the $T>1$ or $V<1$ limits \cite{Ralph}. With a pair of entangled beams, it is possible to have an output state with $V<1$ or $T>1$, hence demonstrating that the memory outperforms the optimal measure and prepare strategy. In case where the output state satisfies both $V<1$ and $T>1,$ the output is the best possible cloned copy of the input state \cite{Grosshans}.
A perfect quantum memory would satisfy both $T=2$ and $V=0$.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.3}{\includegraphics{Fig_CVcriterion.eps}}}
\caption{General scheme for characterizing an optical memory. A pair of EPR entangled beams with a mean signal amplitude is prepared. One of these beams is injected into, stored, and readout from the optical quantum memory (QM) while the other is being propagated in free space. A joint measurement with appropriate delay is then used to measure the quantum correlations between the quadratures of the two beams.
}\label{carct}
\end{figure}
\subsubsection{{\it Slow-light} memory}\label{CV_SL}
\begin{equation}gin{figure}[hb!]
\centerline{\scalebox{0.9}{\includegraphics{gain.eps}}}
\caption{Field propagating in a medium with gain $\alpha$ and loss $\begin{equation}ta$.
}\label{Setup}
\end{figure}
We now present a general theory for amplification and attenuation of a traveling wave and use the TV diagram to quantify the amount of excess noise that is tolerated.
This theory is well adapted to {\it slow-light} memories \cite{HetetPRA} but can be carried over to other memories, like Raman (section \ref{Raman}) or CRIB (section \ref{CRIB}) memories. Gain can indeed be present if, for instance, population has been transferred to other states during the mapping and read-out stages.
As discussed in the previous section, and in particular in eqs.\eqref{a_out}\&\eqref{adag_out}, the output of an ideal linear amplifier with a gain factor $G>1$, relates to the input field {\it via} this relation :
\begin{equation}gin{equation} a_{\rm out}=\sqrt{G} a_{\rm in}+\sqrt{G-1}\sigma_c^\dagger \end{equation}
where $\sigma_c^{\dagger}$ is a bosonic operator in the vacuum state. The power
spectrum at the output of an ideal phase-insensitive amplifier
is then given by \begin{equation}gin{equation}S_{\rm out}=GS_{\rm in} +G-1 \end{equation} where $S_{\rm in}$ is input spectrum.
By concatenating $m$ amplifying and attenuating infinitesimal slices with linear amplification $1+\alpha \delta z$ and attenuation
$1-\begin{equation}ta \delta z$ where $\delta z=z/m$, as represented in fig.\ref{Setup}, we will calculate the noise properties of the field.
The power spectrum of the field at a slice $m$ is
\begin{equation}gin{equation}
S_m=(1+\frac{(\alpha-\begin{equation}ta)z}{m})^m (S_{in}-1) +1+2\alpha\sum (1+\frac{(\alpha-\begin{equation}ta)z}{m})^{m-j}
\end{equation}
\\
In the infinitesimal slice width limit, we obtain
\begin{equation}gin{equation} S_{\rm out}=\eta S_{\rm in} +(1-\eta)(1+N_f) \end{equation}
where $N_f=2\alpha/(\begin{equation}ta-\alpha)$ and $\eta=\exp{((-\begin{equation}ta+\alpha)L)}$ where $L$ is the length of the medium.
Using standard memory protocols, one can find a relationship between $\alpha$, $\begin{equation}ta$ and the memory parameters.
One can then show that
\begin{equation}gin{equation} V=1-\eta+V_{\rm noise} \end{equation}
and \begin{equation}gin{equation} T=2\eta/(1+V_{\rm noise}),\end{equation} where $V_{\rm noise}=1+(1-\eta) N_f$.
Figure \ref{TV} shows a TV diagram for a memory with varying loss (arrows) and three different gain values.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.7}{\includegraphics{TV.eps}}}
\caption{TV diagram for a CRIB memory with varying gain and loss. The dashed line shows the evolution of the standard 2PE memory performance as a function of optical depth.
}\label{TV}
\end{figure}
If there is no mean intensity of the field at the input, the output field is simply the memory output noise. It reads as
\begin{equation}gin{equation} S_{\rm out}=\eta +(1-\eta) (1+N_f) \end{equation}
If we further assume that all the atoms are in the excited state, that is the atomic
medium operates as an amplifier ($\begin{equation}ta$=0), $N_f=-2$, now we obtain
\begin{equation}gin{equation} S_{\rm out}=1-2(1-\eta)=2\eta-1 \end{equation}
Assuming that the noise is the same on both quadratures, and the relation between the mean number of photons and the field quadratures leads to
\begin{equation}gin{equation} \langle n \rangle=\frac{1}{2}\big[\langle X ^2 \rangle-1\big]
= \frac{1}{2}\big[S_{\rm out}-1\big]=\eta-1
\end{equation}
The mean number of photon is thus
\begin{equation}gin{equation} \label{mean_photon_number}
e^{d}-1
\end{equation}
where $d=\alpha L$ in the optical depth, as was found in the previous section.
\begin{equation}gin{figure}[ht!]
\centerline{\scalebox{0.55}{\includegraphics{2PE_CV.eps}}}
\caption{Beam splitter description of photon echo memories.}\label{BS}
\end{figure}
\subsubsection{Photon echo memories}
Standard photon echo protocols that use long lived excited state transitions are generally not immune to noise. If the emission takes place while population remains in the excited state, gain will be present so the memory will not enter the quantum regime.
\paragraph{Controlled reversible inhomogeneous broadening}
The CRIB scheme can be modeled using arrays of beam-splitters. In its most efficient form, namely using the gradient echo memory scheme (GEM) \cite{hetet2008electro} (sometimes called longitudinal CRIB) or using (transverse-) CRIB with a backward write pulse
(section \ref{CRIB}), the write and read stages can be seen as two beam-splitters with a reflectivity that depends on the optical depth \cite{LongdellAnalytic}, as depicted in fig.\ref{BS}.
Let us note that without a backward pulse, more beam-splitters are needed to describe the output field of the CRIB memory \cite{LongdellAnalytic}.
In these scheme, the population remains mainly in the ground state so that gain, and thus noise, will be absent.
For the write stage, the transmitted pulse field intensity is attenuated according to the Bouguer-Beer-Lambert law by $e^{-d}$.
In terms of quadratures, including the "vacuum port" modeling atomic fluctuations, we deduce from eqs.\eqref{a2}\&\eqref{adag2} the expressions for the light and spin quadratures at the two output ports as defined in fig.\ref{BS}
\begin{equation}gin{equation} X_t=\sqrt{e^{-d}} X_{\rm in} - \sqrt{1-e^{-d}}X_{c1} \end{equation}
and
\begin{equation}gin{equation} X_s=\sqrt{1-e^{-d}} X_{\rm in} + \sqrt{e^{-d}}X_{c1} \end{equation}
The vacuum contribution ensures preservation of the commutation relations of the field and atomic operators.
In the case of CRIB with backward propagation or GEM (forward), the beam-splitter reflectivity is "inverted" and the output field can be written simply as
\begin{equation}gin{equation} X_{\rm out}=\sqrt{1-e^{-d}} X_{s} + \sqrt{e^{-d}}X_{c2}.\end{equation}
In the absence of signal, the input is in the vacuum state $V_{\rm in}=1$, so that $V_{\rm out}=1$.
We thus find a conditional variance
\begin{equation}gin{equation} V_{cv}^{\pm}=1-(1-e^{-d})^2 \end{equation}
and transfer coefficient.
\begin{equation}gin{equation} T=2 (1-e^{-d})^2.\end{equation}
So for the present case of CRIB and in the limit of large optical depth, $T\rightarrow 2$ and $V\rightarrow 0$, the CRIB memory is a quantum memory, as represented by the green area in fig.\ref{TV}.
\paragraph{Standard two-pulse photon echo}
Let us now consider the 2PE memory described in section \ref{2PE}. The difference between the 2PE memory and the CRIB is that the atoms are in the excited state during the read-out retrieval stage. This inversion implies that the input light will be amplified, which will invariably add noise.
Considering again the two-beam-splitter approach depicted in fig.\ref{BS}, the writing stage is the same as CRIB, so a fraction $\sqrt{\eta_R}=\sqrt{1-e^{-d}}$ of the field is written in the memory.
The output field however is amplified by a quantity $\sqrt{\eta_W}=\sqrt{e^d-1}$ as discussed in the {\it slow-light} section \ref{CV_SL}.
In total, the transmission thus reads \begin{equation}gin{equation}\eta_R \eta_W=(1-e^{-d})(e^d-1)=4~{\rm sinh}^2(d/2)\end{equation}We here retrieve the semi-classical 2PE efficiency eq.\eqref{etaPi}.
In terms of the field quadratures, we have
\begin{equation}gin{equation} X_{\rm s}=\sqrt{1-e^{-d}} X_{\rm in}+\sqrt{e^{-d}}X_{\rm c1}\end{equation}
and
\begin{equation}gin{equation} X_{\rm out}=\sqrt{e^{d}-1} X_{\rm s}+\sqrt{e^{d}}X_{\rm c2}\end{equation}
just like for a linear amplifier (eqs.\ref{a2}\&\ref{adag2}).
The product of the conditional variances is thus
\begin{equation}gin{equation}V=1-e^{-d}+e^d\end{equation}
and the sum of the two signal-to-noise transfer coefficients is
\begin{equation}gin{equation}T=\frac{4 {\rm sinh}(d/2)^2}{2 e^d-1}\end{equation}
These two quantities are plotted in fig.\ref{TV} (dotted line), showing the 2PE memory does not enter the quantum regime for any optical depth.
We have checked that a better performance (a lower V and larger T) can be obtained if the optical depth is lowered during the writing stage but the memory would still operate in the classical domain.
\subsection{Photon counting criteria}\label{counting_criterion}
We now present various criteria for certifying the quantum nature of storage protocols based on photon counting, including the autocorrelation measurement, the Cauchy-Schwarz criterion and the Bell test.
\subsubsection{Autocorrelation measurement}
Let us consider a single-mode of the electromagnetic field with bosonic operators $a$ and $a^\dag$ and described by the state $\rho_a.$ This state is said classical if it can be represented as a mixture of coherent states $|\alpha\rangle$, that is, one can find a quasi-probability distribution $P(\alpha) \geq 0$ such that
\begin{equation}gin{equation}
\rho_a=\int d^2 \alpha P(\alpha) |\alpha\rangle\langle \alpha|.
\end{equation}
The autocorrelation of this field defined as
\begin{equation}gin{equation}
g_a^{(2)}=\frac{\langle a^{\dag 2} a^2 \rangle}{\langle a^\dag a \rangle^2}
\end{equation}
is at least equal to 1 \cite{loudon2000quantum}. Conversely, if the result of an autocorrelation measurement is smaller than 1, one can conclude that the measured state is non-classical. A single photon Fock state for example, is a non-classical state because its autocorrelation is 0. A simple way to certify the quantum nature of a given memory is thus to store a single photon and to check that the result of an autocorrelation measurement after retrieval is smaller than 1. This shows that the memory preserves the non-classical feature of light.
Note that in practice, non photon-number resolving detectors can be used to certify the non-classical nature of a single-mode field: it is sufficient to put two of these detectors after a 50/50 beamsplitter and to check that the probability of a twofold coincidence is smaller than the product of probabilities of singles \cite{sekatski2012detector}. Let us thus consider the experiment represented in fig.\ref{Fig1} where a source produces a single photon that is subsequently stored in a memory. The photon is then released and an autocorrelation measured with non photon-number resolving detectors ($d_a$ and $\bar{d}_a$) with efficiency $\eta_d$ each. Let $\eta_m$ be the efficiency of the memory and $p_{{\text{dc}}}$ the probability to get a dark count, that is a click on one detector when the photon source is switched off. Obviously, $\eta_m$ can include the non-unit efficiency of the source and the loss from the source to the memory. $\eta_d$ also accounts for the loss between the 50/50 beamsplitter and each detector. $p_{{\text{dc}}}$ includes the detector dark counts and various sources of noise operating independently on each detector. We assume that $p_{{\text{dc}}}$ is the same for both detectors ($d_a$ and $\bar{d}_a$). To obtain twofold coincidences smaller than the product of singles, these parameters has to fulfill the following inequality (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{equation}
\label{autocorrelation}
g_a^{(2)}=\frac{1-2(1-p_{\text{dc}})(1-\eta_d\eta_m/2)+(1-p_{\text{dc}})^2(1-\eta_d\eta_m)}{\left(1-(1-p_{\text{dc}})(1-\eta_d\eta_m/2)\right)^2} < 1.
\end{equation}
Note that in the absence of noise ($p_{\text{dc}}=0$), this ratio is zero independently of the efficiency. In other words, for an ideal implementation of a memory protocol without noise, there is no constraint on the memory efficiency to prove that it can preserve the result of an autocorrelation measurement performed on a single photon. For unit efficiencies $\eta_d\eta_m =1,$ the ratio eq.\eqref{autocorrelation} tends to $1-\epsilon^2/4$ for $p_{\text{dc}} \approx 1-\epsilon.$ For low efficiencies $\eta_d\eta_m \ll 1,$ the inequality \eqref{autocorrelation} is fulfilled as long as ${p_\text{dc}} \leq 3\eta_d\eta_m.$
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig1_counts.eps}
\caption{Certifying the quantum nature of a memory by checking that it can preserve the non-classical property of a single-mode field. A single photon Fock state is stored and an autocorrelation measurement is performed on the retrieved photon.}
\label{Fig1}
\end{figure}
It is worth mentioning that the here proposed criterion does not allow to conclude that the memory outperforms classical strategies for storage as a device that would throw the photon emitted by the source and create a photon afterward would lead to a zero autocorrelation measurement. However, under the assumption that memory under test indeed operates as a storage/retrieval protocol, this criterion shows that this memory is in the quantum regime, in the sense that it preserves the non-classical nature of a single mode field.
It is interesting to compute the result of an autocorrelation measurement that would be obtained by storing a single photon in an atomic ensemble using the 2PE technique. In the ideal scenario where there is no loss before and inside the memory and the photon absorption is successful, we find for the autocorrelation
\begin{equation}gin{equation}
\label{g22pe}
\frac{1-\frac{2}{(1+\frac{\eta_d}{2}(e^d-1))^2} + \frac{1}{(1+\eta_d(e^d-1))^2}}{\left(1-\frac{1}{(1+\frac{\eta_d}{2}(e^d-1))^2}\right)^2}
\end{equation}
which tends to $3/2$ for small optical depth $d\ll 1$ and to 1 when $\eta/2 (e^{d}-1) \geq 1.$ Even in an ideal scenario, we conclude that a storage technique based on a two-pulse photon echo does not preserve the non-classical nature of a single photon.
\subsubsection{Cauchy-Schwarz criterion}
The Cauchy-Schwarz parameter R can be used to reveal non-classical correlations between two fields \cite{clauser1974experimental}. Consider two single-mode fields and their respective bosonic operators $a,$ $a^\dag$ and $b,$ $b^\dag.$ We say that these fields are classically correlated if their state $\rho_{ab}$ can be written as a mixture of coherent states $|\alpha\rangle$, $|\begin{equation}ta\rangle,$ that is, there exists a non negative function $P(\alpha, \begin{equation}ta)$ such that
\begin{equation}gin{equation}
\rho_{ab}=\int d^2\alpha d^2\begin{equation}ta P(\alpha, \begin{equation}ta) |\alpha, \begin{equation}ta\rangle\langle \alpha, \begin{equation}ta|.
\end{equation}
The Cauchy-Schwarz parameter defined as
\begin{equation}gin{equation}
R=\frac{\langle a^\dag b^\dag b a \rangle}{ \langle a^{\dag 2} a^2 \rangle \langle b^{\dag 2} b^2 \rangle}
\end{equation}
is at most equal to 1 when calculated on classically-correlated states. $R > 1$ is a witness of non-classical correlations.
As for the autocorrelation measurement, the Cauchy-Schwarz parameter can be measured with non photon-number resolving detectors \cite{sekatski2012detector}, see fig.\ref{Fig2}. It is sufficient to take the ratio between the square of twofold coincidences between detectors $d_a$\&$d_b$ and the product of coincidences between $d_a$\&$\bar{d}_a$ and $d_b$\&$\bar{d}_b.$ Let us consider the experiment shown in fig.\ref{Fig2} with a source producing two-mode vacuum squeezed states, that is
\begin{equation}gin{equation}
(1-p)^{\frac{1}{2}} e^{\sqrt{p} a^\dag b^\dag} |00\rangle.
\end{equation}
Further consider the storage and release of the mode $a$ into a memory with efficiency $\eta_m.$ Let $\eta_{d}^a$ ($\eta_{d}^b$) be the efficiency of detectors $d_a$ and $\bar{d}_a$ ($d_b$ and $\bar{d}_b$) and $p_{\text{dc}}^a$ ($p_{\text{dc}}^b$) the probability to get a click on the detector $d_a$ or $\bar d_a$ ($d_b$ or $\bar d_b$) when the source is switched off (dark counts). As before, the memory efficiency includes the loss from the source to the memory. The efficiency of detectors $d_a$ and $\bar{d}_a$ includes the loss from the beamsplitter to the detector while the efficiency of the detectors $d_b$ and $\bar{d}_b$ includes the loss from the source to the detector (without the transmission of the beamsplitter). $p_{\text{dc}}^a$ which we assume to be the same for the two detector $d_a$ and $\bar{d}_a,$ includes various source of noise that can be modeled as detector dark counts. In this scenario, the Cauchy-Schwarz parameter is given by (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{align}
\label{Cauchy_Schwarz}
R=&\Bigg[1-\frac{(1-p_{\text{dc}}^a)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})}-\frac{(1-p_{\text{dc}}^b)(1-p)}{1-p(1-\frac{\eta_d^b}{2})}
+\frac{(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})(1-\frac{\eta_d^b}{2})}\Bigg]^2/\nonumber \\
\nonumber
& \Bigg[\left(1-2\frac{\left(1-p_{\text{dc}}^{a}\right)(1-p)}{1-p(1-\frac{\eta_d^a\eta_m}{2})}+\frac{\left(1-p_{\text{dc}}^{a}\right)^2(1-p)}{1-p(1-\eta_d^a\eta_m)}\right)\times\\
&\left(1-2\frac{\left(1-p_{\text{dc}}^{b}\right)(1-p)}{1-p(1-\frac{\eta_d^b}{2})}+\frac{\left(1-p_{\text{dc}}^{b}\right)^2(1-p)}{1-p(1-\eta_d^b)}\right)\Bigg]
\end{align}
and has to be larger than $1$ to certify that the tested memory preserves non-classical correlations. In the ideal case with unit efficiencies and no dark count, the Cauchy-Schwarz parameter tends to $\frac{1}{4}(1+\frac{1}{p})^2$ for $p \ll 1$. Note that $p$ can be written as a function of the mean photon-number emitted in one mode ($a$ or $b$) as $p=n/(n+1).$
The Cauchy-Schwarz criterion leads to similar conclusions than the autocorrelation measurement. If the memory under test is a device that throws the incoming field away and produces a single photon at a later time, the Cauchy-Schwarz parameter would tend to infinity, independently of the state of mode $b$. However, assuming that the tested memory indeed operates as a storage/retrieval protocol, the Cauchy-Schwarz criterion allows to conclude that the memory preserves non-classical correlations between two fields.
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Fig2_counts.eps}
\caption{Setup to certify the quantum nature of a memory by checking that it can preserve the non-classical correlations between two fields. A photon pair source is used to produce two-mode squeezed vacuum states. One of the two modes (mode $a$) is stored in a memory and the Cauchy-Schwarz parameter is measured between the mode $a$ after retrieval and the mode $b.$}
\label{Fig2}
\end{figure}
\subsubsection{Bell test}
The two criteria presented previously do not test the capability of a memory to preserve the coherence. This can be done by storing a part of an entangled state and by checking that the entanglement is preserved using, for example, a Bell test. A possible realization would use a photon pair source emitting entangled photon pairs, for example, in polarization. The spatial mode $a$ is stored in a memory and subsequently released. Measurements are finally performed combining wave-plates, polarizing beamsplitters and one detector on each side. The twofold coincidences are recorded. Two interference patterns are obtained, one by rotating the analyzer on the left side, the other one by rotating the analyzer on the right side. If the mean visibility of this interference patterns is larger than $1/3,$ one can conclude about the presence of entanglement under the assumption that the state is a mixture between the singlet state and white noise. As the memory operates as a local operation, which cannot increase entanglement, a high interference visibility witnesses the presence of entanglement between the photon in $b$ and the excitation stored in the memory. Note that there is no need to close the detection and locality loopholes here as the Bell test is used as an entanglement witness, not as non-locality Bell test.
Let us consider the experimental realization shown in fig.\ref{Fig3} with a source based on spontaneous parametric down conversion, that is, photon pairs described by
\begin{equation}gin{equation}
|\psi^-_{a_h a_v b_h b_v}\rangle = (1-p) e^{\sqrt{p} (a_h^\dag b_v^\dag-a_v^\dag b_h^\dag)} |00\rangle.
\end{equation}
Let $\eta_a$ and $\eta_b$ be the detector efficiency on side $a$ and $b$ respectively and $p_{\text{dc}}^a$ and $p_{\text{dc}}^b$ the corresponding noise. As before, the memory efficiency is labeled $\eta_m.$ The visibility of the interference $V$ is given by (see appendix \ref{appendix:formulas_counts} for details)
\begin{equation}gin{align}
\label{Bell}
V=\Bigg[\frac{(1-p)(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{1-p(1-\eta_a)(1-\eta_b)} -
\frac{(1-p)^2(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{(1-p(1-\eta_a))(1-p(1-\eta_b))}\Bigg]/\nonumber \\
\Bigg[ 2-2\frac{(1-p_{\text{dc}}^a) (1-p)}{1-p(1-\eta_a)}-2\frac{(1-p_{\text{dc}}^b) (1-p)}{1-p(1-\eta_b)}+
\frac{(1-p)(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{1-p(1-\eta_a)(1-\eta_b)} +\nonumber \\
\frac{(1-p)^2(1-p_{\text{dc}}^a)(1-p_{\text{dc}}^b)}{(1-p(1-\eta_a))(1-p(1-\eta_b))}
\Bigg].
\end{align}
As before, $p$ can be written as a function of the mean photon number in one mode ($a_h,$ $a_v,$ $b_h$ or $b_v$) as $p=n/(n+1).$
\begin{equation}gin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Fig3_counts.eps}
\caption{Setup to certify the quantum nature of a memory by checking that it can preserve entanglement between two fields. A photon pair source is used to produce entangled photon pairs in polarization. Two of the four modes (mode $a_h$ and $a_v$) are stored in a memory and a Bell inequality violation can be inferred from the visibility of the interference that is obtained by recording the twofold coincidences while rotating the measurement settings locally.}
\label{Fig3}
\end{figure}
Interestingly, one can conclude from such a Bell test that the memory performs better that any possible classical strategies using for example a measure and prepare strategy or cloning followed by measurements in different basis. In this case, entanglement would be broken and the visibility would be limited to 1/3 assuming that the classical strategies introduce white noise on the singlet state.
Note that it has been shown recently that a device-independent certification is possible in the setup presented in fig.\ref{Fig3} \cite{sekatski_inprep}. In other words, it is possible to certify that the memory is a unitary operation and applies the identity on the qubits independently of the details and imperfections of the actual implementation by performing Bell tests with and without storage.
\\
We have derived and analyzed complementary criteria, both for continuous and discrete variables. They can be used as a benchmark to certify the quantum nature of the memory outcome. Our goal was to relate quantum optics measurements to experimentally accessible quantities that can be evaluated independently. This explicit criteria can be alternatively considered as a guide to anticipate the result of quantum measurements, to identify the limitations of an experimental setup and/or as an analytical tool for modeling a posteriori.
\section{Conclusion}
We have reviewed a series of quantum optical memory protocols conceived to store information in atomic ensembles. Without providing an exhaustive review of the different systems and techniques, we propose to put the protocols into two categories, namely photon echo and {\it slow-light} memories.
Our analysis is based on the significant differences of storage and retrieval dynamics. We have used a minimalist semi-classical Schr\"odinger-Maxwell model to describe the signal propagation and to evaluate the storage efficiency in atomic ensembles. The efficiency scaling allows to compare the different memory types but represents only one figure of merit.The applications in quantum information processing go beyond the simple analogy with classical memories where the signal is stored and retrieved. The different figures of merit should be considered in that prospect as the storage time, the bandwidth and the multimode capacity that we only superficially address when discussing the storage dynamics. In that sense, our contribution is mainly an introduction that can be pushed further to give a more complete comparison of the memories' performance.
Our objective was essentially to give a fundamental vision of few protocols that we consider as archetypes and hopefully stimulate the proposition of new architectures. We have finally replaced our analysis in the context of the quantum storage by deriving a variety of criteria adapted for both continuous and discrete variables. We have developed a toy model for the interaction of light with an atomic ensemble to evaluate the outcome of various quantum optics measurements that can serve as benchmarks to certify the quantum nature of optical memories.
We haven't insisted on the different material systems that physically represent the memory support. They all have in common to exhibit long lived (optical or spin) coherent states but they can cover different realities going from cold atomic vapors to luminescent impurities in solids as rare-earth doped insulators or excitons in semi-conductors holding a lot of promises in terms of integration. The portability of each protocol to a specific system would deserve a discussion by itself for which our analytic review of protocols can be seen as an introductory basis.
\section*{Acknowledgments}
Research at the University of Basel is supported by the Swiss National Science Foundation (SNSF) through the NCCR QSIT, the grant number PP00P2-150579 and the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet.
The work at Laboratoire Aimé Cotton received funding from the national grant ANR DISCRYS (ANR-14-CE26-0037-02), from Investissements d'Avenir du LabEx PALM ExciMol and ATERSIIQ (ANR-10-LABX-0039-PALM).
The work at Laboratoire Pierre Agrain has been partially funded by the French National Research Agency (ANR) through the project SMEQUI.
\appendix
\section{Strong pulse propagation}\label{strong_pulse}
Even if the standard 2PE presented in section \ref{2PE} is not appropriate for quantum storage, as an example, it illustrates a potential issue when strong pulses are used in echo sequences. The $\pi$-pulse as an element of the toolbox for quantum memories should be used with precaution. The best anticipated answer is certainly not to use the $\pi$-pulse as proposed in the controlled reversible inhomogeneous broadening protocol detailed section \ref{CRIB}. We here briefly discuss the propagation of strong $\pi$-pulses which appeared as critical element to understand the 2PE efficiencies as simulated in fig.\ref{fig:2PE_simul}.
We have assumed the $\pi$-pulse sufficiently short to have a well-defined action of the stored coherence. In practice, the rephasing pulse should be much shorter than the signal. This condition should be maintained all along the propagation which is far from guaranteed. $\pi$-pulses are very singular in that sense because they maximally invert the atoms irremediably associated to the lost energy from the pulse. The requirement of energy conservation actually imposes a distortion of the pulse. There is no analytical solution to the propagation of strong pulses in absorbing media. Numerical simulations are then necessary to predict the exact pulse shape. That being said, the qualitatively analysis can be reinforced by invoking the McCall and Hahn Area Theorem \cite{area67, allen2012optical, Eberly:98}. This latter gives a remarkable conservation law for the pulse area though propagation as
\begin{equation}gin{equation}
\partial_z\theta(z)=
-\displaystyle\frac{\alpha}{2} \sin\left(\theta(z)\right)\label{area}
\end{equation}
In the weak signal limit (small area \cite{crisp1970psa}), one retrieves the Bouguer-Beer-Lambert law for the area (eq.\ref{bouguer}) as expected in the perturbative regime. A $2\pi$-pulse typically undergo the so-called self-induced transparency (SIT) \cite{area67}. The shape preserving propagation \cite{allen2012optical} is not surprising in the light of the energy and area conservations of $2\pi$-pulses. Indeed, a $2\pi$ Rabi flopping of the atoms doesn't leave any energy in the population. Additionally, the $2\pi$-area is unaffected by the propagation as given by the singularities of eq.\eqref{area} (for any area as multiple of $\pi$).
Along the same lines, a $\pi$-pulse conserves its area but not its energy. Pulse distortions are then expected to satisfy two contradictory conditions on the energy and the area. The pulse amplitude is reduced as the duration increases to preserve the total area. The energy scales as the pulse amplitude (multiplied by the area which is constant in that case) is then reduced. Theses considerations should not be underestimated when strong and more specifically $\pi$-pulses are used. To illustration the pulse distortion, we question the expression \eqref{etaPi} by performing a numerical simulation with different $\pi$-pulse durations (see fig.\ref{fig:2PE_simul}). The deviation for the expected scaling precisely comes the $\pi$-pulse distortion as observed in fig.\ref{fig:2PE_simul} (top). This is an intrinsic limitation of $\pi$-pulse when used in absorbing ensemble. This latter is fundamental and cannot be avoided by using a cavity to enhance the interaction with a weakly absorbing sample. Same distortions are expected in cavities \cite{gti}. The real alternative is the complex hyperbolic secant (CHS) pulse as discussed in section \ref{strong_pulse_rose}. This latter is not only robust under the experimental imperfections (as power fluctuation) but is also much less sensitive to propagation distortions \cite{Demeter}. Following our analysis, there is no constraint, as the area theorem, on the CHS as frequency swept pulses.
\section{Photon counting measurements}\label{appendix:formulas_counts}
We here give the detailed derivation of the formulas \eqref{autocorrelation}, \eqref{Cauchy_Schwarz} and \eqref{Bell} used in section \ref{counting_criterion}. We consider non photon-number resolving detectors with noise. Let $D_a(\eta_d)$ be the POVM element (positive-operator valued measure) associated to the event click when such a detector operates on a single mode of the electromagnetic field characterized by the annihilation $a$ and creation $a^\dag$ operators. Let $\eta_d$ be the efficiency of the detector and $p_{\text{dc}}$ the probability of a dark count. We have
\begin{equation}gin{equation}
D_a(\eta_d) = \mathbb{1} - (1-p_{\text{dc}})(1-\eta_d)^{a^\dag a}.
\end{equation}
We first focus on the setup presented in fig.\ref{Fig1} by assuming that the noise and efficiency of the two detectors are the same. The ratio between the twofold coincidences and the product of singles is given by
\begin{equation}gin{equation}
g_a^{(2)} = \frac{\langle D_{d_a}(\eta_d) D_{\bar{d}_a}(\eta_d)\rangle}{\langle D_{d_a}(\eta_d) \rangle \langle D_{\bar{d}_a}(\eta_d) \rangle}.
\end{equation}
Basic algebra using the relation between the modes $a,$ $d_a$ and $\bar{d}_a$ shows that $D_{d_a}(\eta_d) D_{\bar{d}_a}(\eta_d) = \mathbb{1}-2(1-p_{\text{dc}})(1-\eta_d/2)^{a^\dag a}+(1-p_{\text{dc}})^2(1-\eta_d)^{a^\dag a}$ and $D_{d_a}(\eta_d)=D_{a}(\eta_d/2)$. By including the memory efficiency in the detector efficiency, the ratio $g_a^{(2)} $ can be computed from
\begin{equation}gin{equation}
\label{auxg2}
g_a^{(2)} =\frac{\langle 1| \mathbb{1}-2(1-p_{\text{dc}})(1-\eta/2)^{a^\dag a}+(1-p_{\text{dc}})^2(1-\eta)^{a^\dag a} |1 \rangle}{\langle 1| \mathbb{1}-2(1-p_{\text{dc}})(1-\eta/2)^{a^\dag a}|1 \rangle^2}
\end{equation}
with $\eta=\eta_d\eta_m.$ Using an exponential form for $(1-\eta)^{a^\dag a}$ and expanding as a Taylor series, we find
\begin{equation}gin{equation}
\label{meanvaluedet}
\langle n| (1-\eta)^{a^\dag a} |n\rangle = (1-\eta)^{n}.
\end{equation}
Eq. \eqref{autocorrelation} is obtained by combining \eqref{auxg2} and \eqref{meanvaluedet}.\\
The expression for the Cauchy-Schwarz parameter is obtained from
\begin{equation}gin{equation}
R=\frac{\langle D_{d_a}(\eta_d)D_{d_b}(\eta_d)\rangle^2}{\langle D_{d_a}(\eta_d)D_{\bar{d}_a}(\eta_d) \rangle \langle D_{d_b}(\eta_d)D_{\bar{d}_b}(\eta_d) \rangle}
\end{equation}
which leads to \eqref{Cauchy_Schwarz} by using the following results
\begin{equation}gin{align}
\nonumber
\text{tr} (\rho_a x^{a^\dag a})& = \frac{1-p}{1-px},\\
\text{tr} (\rho_{ab} x^{a^\dag a + b^\dag b}) &= \frac{1-p}{1-px^2}
\end{align}
where $\rho_{ab}$ is the density matrix associated to a two-mode vacuum squeezed state and $\rho_a = \text{tr}_b \rho_{ab}.$
The expression for the visibility of the interference pattern observed in the Bell test experiment is obtained by noting that twofold coincidences are maximum between orthogonal polarizations while the minimum is obtained between identical polarizations. Hence, the numerator of eq.\eqref{Bell} can be obtained by taking the difference between $$\langle \psi^-_{a_h a_v b_h b_v} | D_{a_h}(\eta_d) D_{b_v}(\eta_d) | \psi^-_{a_h a_v b_h b_v}\rangle$$ and $$\langle \psi^-_{a_h a_v b_h b_v} | D_{ah}(\eta_d) D_{bh}(\eta_d) | \psi^-_{a_h a_v b_h b_v}\rangle$$ while the denominator comes from the sum of these two expectation values. \\
\end{document} |
\begin{document}
\vskip 20pt
MSC 34C10
\vskip 20pt
\centerline{\bf Oscillatory and non oscillatory criteria for linear}
\centerline{\bf four dimensional hamiltonian systems }
\vskip 20 pt
\centerline{\bf G. A. Grigorian}
\centerline{\it Institute of Mathematics NAS of Armenia}
\centerline{\it E -mail: mathphys2@instmath.sci.am}
\vskip 20 pt
\noindent
Abstract. The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of linear four dimensional hamiltonian systems. An oscillatory and three non oscillatory criteria are proved. On examples the obtained results are compared with some well known ones.
\vskip 20 pt
Key words: Riccati equation, oscillation, non oscillation, conjoined (prepared, preferred) solution, Liuville's formula.
\vskip 20 pt
{\bf 1. Introduction.} Let $A(t) \equiv \bigl(a_{jk}(t)\bigr)_{j,k=1}^2,\phantom{a} B(t)\equiv \bigl(b_{jk}(t)\bigr)_{j,k=1}^2, \phantom{a} C(t)\equiv \phantom{a} \bigl(c_{jk}(t)\bigr)_{j,k=1}^2, \linebreak t\ge t_0$, be complex valued continuous matrix functions on $[t_0;+\infty)$ and let $B(t)$ and $C(t)$ be Hermitian, i.e., $B(t) = B^*(t), \phantom{a} C(t) = C^*(t), \phantom{a} t\ge t_0$. Consider the four dimensional hamiltonian system
$$
\sist{\phantom{a}i'= A(t)\phantom{a}i + B(t)\psi;}{\psi' = C(t)\phantom{a}i - A^*(t)\psi, \phantom{a}h t\ge t_0.} \eqno (1.1)
$$
Here $\phantom{a}i = (\phantom{a}i_1, \phantom{a}i_2), \phantom{a} \psi = (\psi_1, \psi_2)$ are the unknown continuously differentiable vector functions on $[t_0;+\infty)$. Along with the system (1.1) consider the linear system of matrix equations
$$
\sist{\Phi'= A(t)\Phi + B(t)\Psi;}{\Psi' = C(t)\Phi - A^*(t)\Psi, \phantom{a}h t\ge t_0,} \eqno (1.2)
$$
Where $\Phi(t)$ and $\Psi(t)$ are the unknown continuously differentiable matrix functions of dimension $2\times 2$ on $[t_0;+\infty)$.
{\bf Definition 1.1}. {\it A solution $(\Phi(t), \Psi(t))$ of the system (1.2) is called conjoined (or prepared, preferred) if $\Phi^*(t)\Psi(t) = \Psi^*(t)\Phi(t), \phantom{a} t\ge t_0$.}
{\bf Definition 1.2.} {\it A solution $(\Phi(t), \Psi(t))$ of the system (1.1) is called oscillatory if $\det \Phi(t)$ has arbitrary large zeroes.}
{\bf Definition 1.3} {\it The system (1.1) is called oscillatory if all conjoined solutions of the system (1.2) are oscillatory, otherwise it is called non oscillatory}.
Study of the oscillatory and non oscillatory behavior of hamiltonian systems (in particular of the system (1.1)) is an important problem of qualitative theory of differential equations and many works are devoted to it (see e.g., [1 - 10] and cited works therein). For any Hermitian matrix $H$ the nonnegative (positive) definiteness of it we denote by $H \ge 0, \phantom{a} (H>0$). In the works [1 - 9] the oscillatory behavior of general hamiltonian systems is studied under the condition that the coefficient corresponding to $B(t)$ is assumed to be positive definite. In this paper we study the oscillatory and non oscillatory behavior of the system (1.1) in the direction that the assumption $B(t) > 0, \phantom{a} t\ge t_0,$ may be destroyed.
{\bf 2. Auxiliary propositions}. Let $f(t), \phantom{a} g(t), \phantom{a} h(t), \phantom{a} h_1(t)$ be real valued continuous functions on $[t_0;+\infty)$. Consider the Riccati equations
$$
y' + f(t) y^2 + g(t) y + h(t) = 0, \phantom{a}h t\ge t_0; \eqno (2.1)
$$
$$
y' + f(t) y^2 + g(t) y + h_1(t) = 0, \phantom{a}h t\ge t_0; \eqno (2.2)
$$
{\bf Theorem 2.1}. {\it Let Eq. (2.2) has a real valued solution $y_1(t)$ on $[t_1;t_2) \phantom{a} (t_0\le t_1 < t_2 \le +\infty)$, and let $f(t) \ge 0, \phantom{a} h(t) \le h_1(t), \phantom{a} t\in [t_1;t_2)$. Then for each $y_{(0)} \ge y_1(t_0)$ Eq. (2.1)
has the solution $y_0(t)$ on $[t_1;t_2)$ with $y_0(t_0) = y_{(0)}$, and $y_0(t) \ge y_1(t), \phantom{a} t\in [t_1;t_2)$.}
A proof for a more general theorem is presented in [11] (see also [12]).
Denote: $I_{g,h}(\xi;t) \equiv \il{\xi}{t} \exp\biggl\{-\il{\tau}{t}g(s) d s\biggr\} h(\tau) d \tau, \phantom{a} t\ge \xi \ge t_0.$ Let $t_0 < \tau_0 \le + \infty$ and let $t_0 < t_1 < ... $ be a finite or infinite sequence such that $t_k \in [t_0;\tau_0], \phantom{a} k=1,2, ...$ We assume that if $\{t_k\}$ is finite then the maximum of $t_k$ is equal to $\tau_0$ and if $\{t_k\}$ is infinite then $\lim\limits_{k\to +\infty} t_k = \tau_0$.
{\bf Theorem 2.2.} {\it Let $f(t) \ge 0, \phantom{a} t\in [t_0; \tau_0), \phantom{a} t\in [t_0; \tau_0)$, and
$$
\il{t_k}{t}\exp\biggl\{\il{t_k}{\tau}\bigl[g(s) - I_{g,h}(t_k;s)\bigr]d s\biggr\} h(\tau) d \tau \le 0, \phantom{a} t\in [t_k;t_{k+1}), \phantom{a} k=0,1, ....
$$
Then for every $y_{(0)} \ge 0$ Eq. (2.1) has the solution $y_0(t)$ on $[t_0;\tau_0)$ satisfying the initial condition $y_0(t_0) = y_{(0)}$ and $y_0(t) \ge 0, \phantom{a} t\in [t_0; \tau_0)$.}
See the proof in [12].
Consider the matrix Riccati equation
$$
Z' + Z B(t) Z + A^*(t) Z + Z A(t) - C(t) = 0, \phantom{a}h t\ge t_0. \eqno (2.3)
$$
The solutions $Z(t)$ of this equation existing on an interval $[t_1; t_2) (t_0 \le t_1 < t_2 \le +\infty)$ are connected with solutions $(\phantom{a}i(t), \Psi(t))$ of the system (1.2) by relations (see [10]):
$$
\Phi'(t) = [A(t) + B(t) Z(t)] \Phi(t), \phantom{a} \Phi(t_1) \ne 0, \phantom{a} \Psi(t) = Z(t) \Phi(t), \phantom{a} t\in [t_1; t_2). \eqno (2.4)
$$
Let $Z_0(t)$ be a solution to Eq. (2.3) on $[t_1; t_2)$.
{\bf Definition.} {\it We will say that $[t_1; t_2)$ is the maximum existence interval for $Z_0(t)$ if $Z_0(t)$ cannot be continued to the right of $t_2$ as a solution of Eq. (2.3).}
{\bf Lemma 2.1}. {\t Let $Z_0(t)$ be a solution of Eq. (2.3) on $[t_1;t_2)$ and let $t_2 < +\infty$. Then $[t_1;t_2)$
cannot be the maximum existence interval for $Z_0(t)$ provided the function $G(t) \equiv \il{t_1}{t}tr [B(\tau) Z_0(\tau)]d\tau, \phantom{a} t\in [t_1; t_2)$, is bounded from below on $[t_1; t_2)$.}
Proof. By analogy of the proof of Lemma 2.1 from [10].
Assume $B(t) = diag \{b_1(t), b_2(t)\}, \phantom{a} t\ge t_0$. Then it
is not difficult to verify that for Hermitian unknowns $Z=\begin{pmatrix}z_{11} & z_{12}\\ \overline{z}_{12} & z_{22}\end{pmatrix}$ Eq. (2.3) is equivalent to the following nonlinear system
$$
\left\{
\begin{array}{l}
z'_{11} + b_1(t) z^2_{11} + 2 Re a_{11}(t) z_{11} + b_2(t)|z_{12}|^2 + a_{21}(t) z_{12} + \overline{a}_{21}(t) \overline{z}_{12} - c_{11}(t) = 0;\\
z'_{12} + [b_1(t) z_{11} + b_2(t) z_{22} + \overline{a}_{11}(t) + a_{22}(t)] z_{12} + \\ \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}+ a_{12}(t) z_{11} + a_{21}(t) z_{22} - c_{12}(t) = 0;\\
z'_{22} + b_2(t) z_{22}^2 + 2 Re a_{22}(t) z_{22} + b_1(t)|z_{12}|^2 + \overline{a}_{12}(t) z_{12} + a_{12}(t) \overline{z}_{12} - c_{22}(t) = 0,
\end{array}
\right.
\eqno (2.5)
$$
$t\ge t_0.$ If $b_2(t) \ne 0, \phantom{a} t\ge t_0,$ then it is not difficult to verify that the first equation of the system (2.5) can be rewritten in the form
$$
z'_{11} + b_1(t) z^2_{11} + 2 Re a_{11}(t) z_{11} + b_2(t)\left|z_{12} + \frac{\overline{a}_{21}(t)}{b_2(t)}\right|^2 - \frac{|a_{21}(t)|^2}{b_2(t)} - c_{11}(t) = 0, \phantom{a} t\ge t_0, \eqno (2.6)
$$
and if in addition $\overline{a}_{21}(t)/ b_2(t)$ is continuously differentiable on $[t_0; +\infty)$ then by the substitution
$$
z_{12} = y - \frac{\overline{a}_{21}(t)}{b_2(t)}, \phantom{a}h t \ge t_0, \eqno (2.7)
$$
in the first and second equations of the system (2.5) we get the subsystem
$$
\left\{
\begin{array}{l}
z'_{11} + b_1(t) z^2_{11} + 2 Re a_{11}(t) z_{11} + b_2(t)|y|^2 - \frac{|a_{21}(t)|^2}{b_2(t)} - c_{11}(t) = 0\\
y' + [b_1(t) z_{11} + b_2(t) z_{22} + \overline{a}_{11}(t) + a_{22}(t)]y + \bigl(a_{12}(t) - \frac{b_1(t)}{b_2(t)} \overline{a}_{21}(t)\bigr) z_{11}-\\ \phantom{a}antom{aaaaaaaaaaaaaaaaaa} - \bigl(\frac{\overline{a}_{21}(t)}{b_2(t)}\bigr)' - \frac{\overline{a}_{21}(t)}{b_2(t)}\bigl(\overline{a}_{11}(t) + a_{22}(t)\bigr) - c_{12}(t) = 0, \phantom{a} t\ge t_0.
\end{array}
\right.
\eqno (2.8)
$$
Analogously if $b_1(t) \ne 0, \phantom{a} t\ge t_0,$ then the third equation of the system (2.5) can be rewritten in the form
$$
z'_{22} + b_2(t) z^2_{22} + 2 Re a_{22}(t) z_{22} + b_1(t)\left|z_{12} + \frac{a_{12}(t)}{b_1(t)}\right|^2 - \frac{|a_{12}(t)|^2}{b_1(t)} - c_{22}(t) = 0, \phantom{a} t\ge t_0, \eqno (2.9)
$$
and if in addition $a_{12}(t)/ b_1(t)$ is continuously differentiable on $[t_0; +\infty)$ then by the substitution
$$
z_{12} = v - \frac{a_{12}(t)}{b_1(t)}, \phantom{a}h t \ge t_0, \eqno (2.10)
$$
in the second and third equations of the system (2.5) we obtain the subsystem
$$
\left\{
\begin{array}{l}
z'_{22} + b_2(t) z^2_{22} + 2 Re a_{22}(t) z_{22} + b_1(t)|v|^2 - \frac{|a_{12}(t)|^2}{b_1(t)} - c_{22}(t) = 0\\
y' + [b_1(t) z_{11} + b_2(t) z_{22} + \overline{a}_{11}(t) + a_{22}(t)]v + \bigl(\overline{a}_{21}(t) - \frac{b_2(t)}{b_1(t)} a_{12}(t)\bigr) z_{22}-\\ \phantom{a}antom{aaaaaaaaaaaaaaaaa} - \bigl(\frac{a_{12}(t)}{b_1(t)}\bigr)' - \frac{a_{12}(t)}{b_1(t)}\bigl(\overline{a}_{11}(t) + a_{22}(t)\bigr) - c_{12}(t) = 0, \phantom{a} t\ge t_0.
\end{array}
\right.
\eqno (2.11)
$$
If $(z_{11}(t), y(t))$ is a solution of the subsystem (2.8) on $[t_0;t_1) (t_0 < t_1 \le + \infty)$ with $y(t_0) = 0$ and $(z_{22}(t), v(t))$ is a solution of the subsystem (2.11) on $[t_0;t_1)$ with $v(t_0)=0$ then by Cauchi formula from the second equation of the subsystem (2.8) and from the second equation of the subsystem (2.11) we have respectively:
$$
y(t) = - \exp\biggl\{-\il{t_0}{t}b_1(\tau)z_{11}(\tau)d\tau\biggr\}\il{t_0}{t}\biggl[\exp\biggl\{\il{t_0}{\tau}b_1(s) z_{11}(s)d s\biggr\}\biggr]'\biggl(\frac{a_{12}(\tau)}{b_1(\tau)} - \frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\biggr)\times
$$
$$
\times\exp\biggl\{-\il{\tau}{t}\bigl(b_2(s) z_{22}(s) + \overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}d\tau +
$$
$$
+\il{t_0}{t}\exp\biggl\{-\il{\tau}{t}\bigl(b_1(s)z_{11}(s) + b_2(s) z_{22}(s) + \overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}\biggl[\biggl(\frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\biggr)' +\phantom{a}antom{aaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}+\frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr) + c_{12}(\tau)\biggr]d\tau,
$$
$$
v(t) = - \exp\biggl\{-\il{t_0}{t}b_2(\tau)z_{22}(\tau)d\tau\biggr\}\il{t_0}{t}\biggl[\exp\biggl\{\il{t_0}{\tau}b_2(s) z_{22}(s)d s\biggr\}\biggr]'\biggl(\frac{\overline{a}_{21}(\tau)}{b_2(\tau)} - \frac{a_{12}(\tau)}{b_1(\tau)} \biggr)\times
$$
$$
\times\exp\biggl\{-\il{\tau}{t}\bigl(b_1(s) z_{11}(s) + \overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}d\tau +
$$
$$
+\il{t_0}{t}\exp\biggl\{-\il{\tau}{t}\bigl(b_1(s)z_{11}(s) + b_2(s) z_{22}(s) + \overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}\biggl[\biggl(\frac{a_{12}(\tau)}{b_1(\tau)}\biggr)' +\phantom{a}antom{aaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaa}+\frac{a_{12}(\tau)}{b_1(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr) + c_{12}(\tau)\biggr]d\tau, \phantom{a} t\in [t_0;t_1).
$$
From here it is easy to derive
{\bf Lemma 2.2}. {\it Let $b_j(t) > 0, \phantom{a} j=1,2,$ the functions $a_{12}(t)/b_1(t), \phantom{a} \overline{a}_{21}(t)/b_2(t)$ be continuously differentiable on $[t_0; t_1) (t_0 < t_1 < + \infty))$ and let $(z_{11}(t), y(t))$ and $(z_{22}(t), v(t))$ be solutions of the subsystems (2.8) and (2.11) respectively on $[t_0; t_1)$ such that $z_{jj}(t) \ge~ 0, \linebreak t\in ~ [t_0;t_1), \phantom{a} j=1,2, \phantom{a} y(t_0) = v(t_0) = 0$. Then
$$
|y(t)| \le \mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{-\il{\tau}{t}\bigl(\overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}\biggl[\biggl(\frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\biggr)'+
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}+\frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr)+ + c_{12}(\tau)\biggr]\biggr|d\tau,
$$
$$
|v(t)| \le \mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{-\il{\tau}{t}\bigl(\overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}\biggl[\biggl(\frac{a_{12}(\tau)}{b_1(\tau)}\biggr)'+
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaa}+\frac{a_{12}(\tau)}{b_1(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr)+ + c_{12}(\tau)\biggr]\biggr|d\tau, \phantom{a} t\in [t_0;t_1),
$$
where
$$
\mathfrak{M}(t)\equiv \max\limits_{\tau\in [t_0; t]}\biggl|\exp\biggl\{-\il{\tau}{t}\bigl(\overline{a}_{11}(s) + a_{22}(s)\bigr)ds\biggr\}\biggl(\frac{a_{12}(\tau)}{b_1(\tau)} - \frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\biggr)\biggr|, \phantom{a} t\ge t_0.
$$
}
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}$\Box$
{\bf Lemma 2.3.} {\it For any two square matrices $M_1\equiv (m_{ij}^1)_{ij=1}^n, \phantom{a} M_2\equiv (m_{ij}^2)_{ij=1}^n$ the equality
$$
tr (M_1 M_2) = tr (M_2 M_1)
$$
is valid.}
Proof. We have: $tr (M_1 M_2) = \sum\limits_{j=1}^n(\sum\limits_{k=1}^n m_{jk}^1 m_{kj}^2) = \sum\limits_{k=1}^n(\sum\limits_{j=1}^n m_{jk}^1 m_{kj}^2) = \sum\limits_{k=1}^n(\sum\limits_{j=1}^n m_{kj}^2 m_{jk}^1) = tr (M_2 M_1).$ The lemma is proved.
{\bf 3. Main results}. Let $f_{jk}(t), \phantom{a} j,k =1,2, \phantom{a} t\ge t_0,$ be real valued continuous functions on $[t_0; +\infty)$. Consider the linear system of equations
$$
\sist{\phantom{a}i_1' = f_{11}(t) \phantom{a}i_1 + f_{12}(t) \psi_1;}{\psi_1' = f_{21}(t) \phantom{a}i_1 + f_{22}(t) \psi_1, \phantom{a} t\ge t_0,} \eqno (3.1)
$$
and the Riccati equation
$$
y' + f_{12}(t) y^2 + [f_{11}(t) - f_{22}(t)] y - f_{12}(t) = 0, \phantom{a}h t\ge t_0. \eqno (3.2)
$$
All solutions $y(t)$ of the last equation, existing on some interval $[t_1; t_2)\hskip 2pt (t_0 \le t_1 < t_2 \le + \infty)$ are connected with solutions $(\phantom{a}i_1(t), \psi_1(t))$ of the system (3.1) by relations (see [13]):
$$
\phantom{a}i_1(t) = \phantom{a}i_1(t_1)\exp\biggl\{\il{t_1}{t}\bigl[f_{12}(\tau) y(\tau) + f_{11}(\tau)\bigr]d\tau\biggr\}, \phantom{a} \phantom{a}i_1(t_1)\ne 0, \phantom{a} \psi_1(t) = y(t) \phantom{a}i_1(t), \eqno (3.3)
$$
$t\in [t_1; t_2).$
{\bf Definition 3.1.} {\it The system (3.1) is called oscillatory if for its every solution \linebreak $(\phantom{a}i_1(t), \psi_1(t))$ the function $\phantom{a}i_1(t)$ has arbitrary large zeroes.}
{\bf Remark 3.1.} {\it Some explicit oscillatory criteria for the system (3.1) are proved in [10] amd [14]}.
{\bf 3.1. The case when $B(t)$ is a diagonal matrix}. In this subsection we will assume that $B(t) = diag\{b_1(t), b_2(t)\}$.
Denote:
$$
\chi_j(t) \equiv \sist{c_{jj}(t) \phantom{a} if b_{3-j}(t) = 0;}{c_{jj}(t) + \frac{|a_{3-j,j}(t)|^2}{b_{3-j}(t)}, \phantom{a} if b_{3-j}(t) \ne 0,} \phantom{a}h t\ge t_0, \phantom{a} j=1,2.
$$
{\bf Theorem 3.1.} {\it Assume $b_j(t) \ge 0, \phantom{a} t\ge t_0,$ and if $b_j(t) = 0$ then $a_{3-j,j}(t) = 0, \phantom{a} j=1,2, \phantom{a} t\ge t_0.$
Under these restrictions the system (1.1) is oscillatory provided one of the systems
$$
\sist{\phantom{a}i_1'= 2 Re (a_{jj}(t)) \phantom{a}i_1 + b_j(t) \psi_1;}{\psi_1' = - \chi_j(t) \phantom{a}i_1, \phantom{a}h t\ge t_0,} \eqno (3.4_j)
$$
j=1,2, is oscillatory.
}
Proof. Suppose the system (1.1) is not oscillatory. Then for some conjoined solution $(\Phi(t), \Psi(t))$ of the system (1.2) there exists $t_1 \ge t_0$ such that $det \Phi(t) \ne 0, \phantom{a} t\ge t_1.$ Due to (2.4) from here it follows that $Z(t)\equiv \Psi(t) \Phi^{-1}(t), \phantom{a} t\ge t_1,$ is a Hermitian solution to Eq. (2.3) on $[t_1; +\infty)$.
Let $Z(t) = \begin{pmatrix}z_{11}(t) & z_{12}(t)\\ \overline{z}_{12}(t) & z_{22}(t)\end{pmatrix}, \phantom{a} t\ge t_1.$
Consider the Riccati equations
$$
y' + b_1(t) y^2 + 2 (Re a_{11}(t)) y + b_2(t)|z_{12}(t)|^2 + a_{21}(t) z_{12}(t) + \overline{a}_{21}(t) \overline{z}_{12}(t) - c_{11}(t) = 0, \eqno (3.5)
$$
$$
y' + b_2(t) y^2 + 2 (Re a_{22}(t)) y + b_1(t)|z_{12}(t)|^2 + \overline{a}_{12}(t) z_{12}(t) + a_{12}(t) \overline{z}_{12}(t) - c_{22}(t) = 0, \eqno (3.6)
$$
$$
y' + b_j(t) y^2 + 2 (Re a_{jj}(t) y + \chi_j(t) = 0, \eqno (3.7_j)
$$
$j=1,2, \phantom{a} t\ge t_1.$ By (2.6) and (2.9) from the conditions of the theorem it follows that
$$
\chi_1(t) \le b_2(t)|z_{12}(t)|^2 + a_{21}(t) z_{12}(t) + \overline{a}_{21}(t) \overline{z}_{12}(t) - c_{11}(t), \phantom{a} t\ge t_1,
$$
$$
\chi_2(t) \le b_1(t)|z_{12}(t)|^2 + \overline{a}_{12}(t) z_{12}(t) + a_{12}(t) \overline{z}_{12}(t) - c_{22}(t), \phantom{a} t\ge t_1.
$$
Using Theorem 2.1 to the pairs (3.5), $(3.7_1)$ and (3.6), $(3.7_2)$ of equations from here we conclude that the equations $(3.7_j), \phantom{a} j=1,2,$ have solutions on $[t_1; +\infty)$. By (3.1) - (3.3) from here it follows that the systems $(3.4_j), \phantom{a} j=1,2,$ are not oscillatory which contradicts the condition of the theorem. The obtained contradiction completes the proof of the theorem.
Denote:
$
I_j(\xi;t) \equiv \il{\xi}{t}\exp\biggl\{-\il{\tau}{t}2(Re a_{jj}(s))d s \biggr\}\chi_j(\tau) d\tau, \phantom{a} t\ge \xi \ge t_0, \phantom{a} j=1,2.
$
{\bf Theorem 3.2.} {\it Assume $b_1(t) \ge 0 (\le 0), \phantom{a} b_2(t) \le 0 (\ge 0)$ and if $b_j(t) = 0$ then $a_{j, 3-j}(t) = 0, \phantom{a} j=1,2, \phantom{a} t\ge t_0$; there exist infinitely large sequences $\xi_{j,0} = t_) < \xi_{j,1} < ... < \xi_{j,m} , ..., \phantom{a} j=1,2, $ such that
$$
1_j) \phantom{a} (-1)^j \il{\xi_{j,m}}{t} \exp\biggl\{\il{\xi_{j,m}}{\tau}\biggl[2 Re a_{jj}(s) - (-1)^j I_j(xi_{j,m},s)\biggr] d s\biggr\} \chi_j(\tau) d\tau \ge 0 \phantom{a} (\le 0),
$$
$\phantom{a} t\in [\xi_{j,m}; \xi_{j,m +1}), \phantom{a} m=1,2,3, ....., \phantom{a} j=1,2$. Then the system (1.1) is non oscillatory.}
Proof. Let us prove the theorem only in the case when $b_1(t) \ge 0, \phantom{a} b_2(t) \le 0, \phantom{a} t\ge~ t_0$. The case $b_1(t) \le 0, \phantom{a} b_2(t) \ge 0, \phantom{a} t\ge t_0$, can be proved by analogy. Let $(\Phi(t), \Psi(t))$ be a conjoined solution of the system (1.2) with $\Phi(t_0) = \begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$ and let $[t_0; T)$ be the maximum interval such that $det \Phi(t) \ne 0, \phantom{a} t\in [t_0; T)$. Then by (2.4) the matrix function $Z(t) \equiv \Psi(t) \phantom{a}i^{-1}(t), \phantom{a} t\in [t_0; T)$, is a Hermitian solution to Eq. (2.3) on $[t_0; T)$. By (2.5), (2.7), (2.8), (2.10), (2.11) from here it follows that the subsystems (2.8) and (2.11) have solutions $(z_{11}(t), y(t))$ and $(z_{22}(t), v(t))$ respectively on $[t_0; T)$ with $z_{11}(t_0) = 1, \linebreak z_{22}(t_0) =-1$. Show that
$$
z_{11}(t) \ge 0, \phantom{a}h t\in [t_0; T). \eqno (3.8)
$$
Consider the Riccati equations
$$
z' + b_1(t) z^2 + 2 (Re a_{11}(t)) z + b_2(t)|y(t)|^2 + \chi_1(t) = 0, \phantom{a}h t\in [t_0;T), \eqno (3.9)
$$
$$
z' + b_1(t) z^2 + 2 (Re a_{11}(t)) z + \chi_1(t) = 0, \phantom{a}h t\in [t_0;T), \eqno (3.10)
$$
By Theorem 2.2 from the conditions of the theorem it follows that the last equation has a nonnegative solution on $[t_0; T).$ Then using Theorem 2.1 to the pair of equations (3.9), (3.10) on the basis of the conditions of the theorem we conclude that Eq. (3.9) has a nonnegative solution $z_0(t)$ on $[t_0; T)$ with $z_0(t_0) = 0$. Then since $z_{11}(t)$ is a solution to Eq. (3.9) on $[t_0;T)$ and $z_{11}(t_0) =1$ we have (3.8). Show that
$$
z_{22}(t) \le 0, \phantom{a}h t\in [t_0; T). \eqno (3.11)
$$
Consider the Riccati equations
$$
z' - b_2(t) z^2 + 2 (Re a_{22}(t)) z - \chi_2(t) = 0, \phantom{a}h t\in [t_0;T), \eqno (3.12)
$$
$$
z' - b_2(t) z^2 + 2 (Re a_{22}(t)) z - b_1(t)|v(t)|^2 - \chi_2(t) = 0, \phantom{a}h t\in [t_0;T). \eqno (3.13)
$$
By Theorem 2.2 from the conditions of the theorem it follows that Eq. (3.12) has a nonnegative solution $z_1(t)$ on $[t_0; T)$ with $z_1(t_0) = 0$. Then using Theorem 2.1 to the pair of equations (3.12) and (3.13) we derive that Eq. (3.13) has a nonnegative solution $z_2(t)$ on $[t_0; T)$ whit $z_2(t_0) = 0$. Hence since obviously $-z_{22}(t)$ is a solution of Eq. (3.13) on $[t_0; T)$ and $-z_{11}(t_0) =1$ we have (3.11). Since $b_1(t) \ge 0, \phantom{a} b_2(t) \le 0, \phantom{a} t\in [t_0;T)$ from (3.8) and (3.11) it follows:
$$
\il{t_0}{t}\Bigl[b_1(\tau) z_{11}(\tau) + b_2(\tau) z_{22}(\tau)\Bigr]d\tau \ge 0, \phantom{a}h t\in [t_0; T). \eqno (3.14)
$$
To complete the proof of the theorem it remains to show that $T = + \infty$. Suppose $T < + \infty$. Then by virtue of Lemma 2.1 from (3.14) it follows that $[t_0; T)$ is not the maximum existence interval for $Z(t)$. By (2.4) from here it follows that $det \Phi(t) \ne 0, \phantom{a} t\in [t_0; T_1),$ for some $T_1 > T$. We have obtained a contradiction which completes the proof of the theorem.
{\bf Remark 3.2.} {\it The conditions $1_j), \phantom{a} j=1,2,$ are satisfied if in particular $(-1)^j\chi_j(t) \ge~ 0 \linebreak (\le 0), \phantom{a} t \ge t_0$.}
Denote:
$$
\chi_3(t) \equiv b_2(t)\biggl[\mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{- \il{\tau}{t}\bigl[\overline{a}_{11}(s) + a_{22}(s)\bigr]d s\biggr\}\times \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aa}\times \biggl[\biggl(\frac{\overline{a}_{21}(t)}{b_2(t)}\biggl)' + \frac{\overline{a}_{21}(\tau)}{b_2(\tau)}(\overline{a}_{11}(\tau) + a_{22}(\tau)\biggr) + c_{12}(\tau)\biggr]\biggr|d\tau\biggr]^2 - \frac{|a_{21}(t)|^2}{b_2(t)} - c_{11}(t),
$$
$$
\chi_4(t) \equiv b_1(t)\biggl[\mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{- \il{\tau}{t}\bigl[\overline{a}_{11}(s) + a_{22}(s)\bigr]d s\biggr\}\times \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aa}\times \biggl[\biggl(\frac{a_{12}(t)}{b_1(t)}\biggl)' + \frac{a_{12}(\tau)}{b_1(\tau)}(\overline{a}_{11}(\tau) + a_{22}(\tau)\biggr) + c_{12}(\tau)\biggr]\biggr|d\tau\biggr]^2 - \frac{|a_{12}(t)|^2}{b_1(t)} - c_{22}(t),
$$
$$
I_{j+2}(\xi;t) \equiv \il{\xi}{t}\exp\biggl\{-\il{\tau}{t}2(Re a_{jj}(s))d s \biggr\}\chi_{j+2}(\tau) d\tau, \phantom{a} t\ge \xi \ge t_0, \phantom{a} j=1,2.
$$
{\bf Theorem 3.3.} {\it Let the following conditions be satisfied
\noindent
1) $b_j(t)> 0, \phantom{a} t\ge t_0, \phantom{a} j=1,2;$
\noindent
2) the functions $a_{12}(t)/b_1(t)$ and $\overline{a}_{21}(t)/b_2(t)$ are continuously differentiable on $[t_0;+\infty)$;
\noindent
3) there exist infinitely large sequences $\xi_{j,0} = t_0 < \xi_{j,1} < ... < \xi_{j,m} , ..., \phantom{a} j=1,2, $ such that
$$
\il{\xi_{j,m}}{t} \exp\biggl\{\il{\xi_{j,m}}{\tau}\biggl[2 Re a_{jj}(s) - I_{j+2}(\xi_{j,m},s)\biggr] d s\biggr\} \chi_{j+2}(\tau) d\tau \le 0, \phantom{a} t\in [\xi_{j,m}; \xi_{j,m +1}),
$$
$m=1,2,3, ....., \phantom{a} j=1,2$. Then the system (1.1) is non oscillatory.}
Proof. Let $Z(t) \equiv \begin{pmatrix}z_{11}(t) & z_{12}(t)\\\overline{z}_{12}(t) & z_{22}(t)\end{pmatrix}$ be the Hermitian solution of Eq. (2.3) on $[t_0; T)$ satisfying the initial condition $Z(t_0) = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}$, where $[t_0;T)$ is the maximum existence interval for $Z(t)$. Due to (2.4) to prove the theorem it is enough to show that
$$
T= + \infty. \eqno (3.15)
$$
By (2.5), (2.7), (2.8), (2.10), (2.11) from the conditions 1) and 2) it follows that \linebreak $(z_{11}(t), z_{12}(t) + \overline{a}_{21}(t)/ b_2(t))$ and $(z_{22}(t), z_{12}(t) + a_{12}(t)/ b_1(t))$ are solutions of the subsystems (2.8) and (2.11) respectively on $[t_0; T)$. Show that
$$
z_{jj}(t) > 0, \phantom{a}h t\in [t_0; T). \eqno (3.16)
$$
Suppose it is not so. Then there exists $T_1 \in (t_0; T)$ such that
$$
z_{11}(t)z_{22}(t) > 0, \phantom{a} t\in [t_0; T_1), \phantom{a} z_{11}(T_1)z_{22}(T_1) = 0. \eqno (3.17)
$$
Without loss of generality we may take that $a_{12}(t_0) = a_{21}(t_0) = 0$. Then by virtue of Lemma 2.2 from (3.17) it follows that
$$
\biggl|z_{12}(t) + \frac{\overline{a}_{21}(t)}{b_2(t)}\biggr| \le \mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{ - \il{\tau}{t}\bigl(\overline{a}_{11}(s) + a_{22}(s)\biggr)d s\biggr\}\biggl[\biggl(\frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\biggr)' + \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} + \frac{\overline{a}_{21}(\tau)}{b_2(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr) - c_{12}(\tau)\biggr]\biggr|d\tau,
$$
$$
\biggl|z_{12}(t) + \frac{a_{12}(t)}{b_1(t)}\biggr| \le \mathfrak{M}(t) + \il{t_0}{t}\biggl|\exp\biggl\{ - \il{\tau}{t}\bigl(\overline{a}_{11}(s) + a_{22}(s)\biggr)d s\biggr\}\biggl[\biggl(\frac{a_{12}(\tau)}{b_1(\tau)}\biggr)' + \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaa} + \frac{a_{12}(\tau)}{b_1(\tau)}\bigl(\overline{a}_{11}(\tau) + a_{22}(\tau)\bigr) - c_{12}(\tau)\biggr]\biggr|d\tau, \phantom{a} t\in [t_0;T_1).
$$
Hence
$$
b_2(t)\biggl|z_{12}(t) + \frac{\overline{a}_{21}(t)}{b_2(t)}\biggr| - \frac{|a_{21}(t)|^2}{b_2(t)} - c_{11}(t) \le \chi_3(t), \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaa}b_1(t)\biggl|z_{12}(t) + \frac{a_{12}(t)}{b_1(t)}\biggr|^2 - \frac{|a_{12}(t)|^2}{b_2(t)} - c_{22}(t) \le \chi_4(t), \phantom{a}h t\in [t_0;T_1),
$$
By virtue of Theorem 2.1 and Theorem 2.2 from here and from the condition 3) it follows that the Riccati equations
$$
z' + b_1(t) z^2 + 2(Re a_{11}(t)) z + b_2(t)\biggl|z_{12}(t) + \frac{\overline{a}_{21}(t)}{b_2(t)}\biggr| - \frac{|a_{21}(t)|^2}{b_2(t)} - c_{11}(t) = 0, \eqno (3.18)
$$
$$
z' + b_2(t) z^2 + 2(Re a_{22}(t)) z + b_1(t)\biggl|z_{12}(t) + \frac{a_{12}(t)}{b_1(t)}\biggr|^2 - \frac{|a_{12}(t)|^2}{b_2(t)} - c_{22}(t) = 0, \eqno (3.19)
$$
$ t\in [t_0; T_1),$ have nonnegative solutions $z_1(t)$ and $z_2(t)$ respectively on $[t_0; T_1)$ with $z_1(t_0) = z_2(t_0) = 0$. Obviously $z_{11}(t)$ and $z_{22}(t)$ are solutions of Eq. (3.18) and (3.19) respectively on $[t_0; T_1]$. Therefore since $z_{jj}(t_0) = 1 > z_j(t_0) = 0, \phantom{a} j=1,2$ due to uniqueness theorem $z_{jj}(t) > 0, \phantom{a} t\in [t_0;T_1], \phantom{a} j=1,2,$ which contradicts (3.17). The obtained contradiction proves (3.16). From (3.16) and 1) it follows that
$$
\il{t_0}{t}\bigl[b_1(\tau) z_{11}(\tau) + b_2(\tau) z_{22}(\tau)\bigr] d \tau \ge 0, \phantom{a}h t\in [t_0; T). \eqno (3.20)
$$
Suppose $T < + \infty$. Then by Lemma 2.1 from (3.20) it follows that $[t_0; T)$ is not the maximum existence interval for $Z(t)$ which contradicts our assumption. The obtained contradiction proves (3.15). The theorem is proved.
{\bf Remark 3.3.} {\it The conditions 3) of Theorem 3.3 are satisfied if in particular $\chi_j(t) \le ~0, \linebreak t\ge t_0, \phantom{a} j=1,2.$}
{\bf 3.2. The case when $B(t)$ is nonnegative definite}.
In this subsection we will assume that $B(t)$ is nonnegative definite and $\sqrt{B(t)}$ is continuously differentiable on $[t_0;+ \infty)$. Consider the matrix equation
$$
\sqrt{B(t)} X [A(t) \sqrt{B(t)} - \sqrt{B(t)}'] = A(t) \sqrt{B(t)} - \sqrt{B(t)}', \phantom{a}h t\ge t_0. \eqno (3.21)
$$
Obviously this equation has always a solution on $[a;b] (\subset [t_0; + \infty))$ when $B(t) > 0, \linebreak t\in [a;b] \phantom{a} (X(t) = B^{-1}(t), \phantom{a} t\in [a;b])$. It may have also a solution on $[a;b]$ in some cases when $B(t) \ge 0, \phantom{a} t\in [a;b]$ (e.g., $A(t) = \begin{pmatrix} a_1(t) & a_2(t)\\ 0 & 0 \end{pmatrix}, \phantom{a} B(t) = \begin{pmatrix} b_1(t) & 0 \\ 0 & 0 \end{pmatrix}, \phantom{a} b_1(t) >~ 0, \linebreak t\in [a;b]).$ In this subsection we also will assume that Eq. (3.21) has always a solution on $[t_0; + \infty)$. Let $F(t)$ be a solution of Eq. (3.21) on $[t_0; + \infty)$. Denote:
$$P(t) \equiv F(t) [A(t)\sqrt{B(t)} - \sqrt{B(t)}'] = (p_{jk}(t))_{j,k =1}^2, \eqno (3.22)$$
$\phantom{a} Q(t) \equiv \sqrt{B(t)} C(t) \sqrt{B(t)}, \phantom{a} (q_{jk}(t))_{j,k =1}^2, \phantom{a} \widetilde{\chi}_j(t) \equiv q_{jj}(t) + |p_{3 -j, j}(t)|^2, \phantom{a} j=~1,2, \phantom{a} t\ge t_0$.
{\bf Corollary 3.1}. {\it The system (1.1) is oscillatory provided one of the equations
$$
\phantom{a}i_1'' + 2 [Re p_{jj}(t)] \phantom{a}i_1' + \widetilde{\chi}_j(t) \phantom{a}i_1 = 0, \phantom{a} j =1,2, \phantom{a} t\ge t_0. \eqno (3.23_j)
$$
is oscillatory.}
Proof. Multiply Eq. (2.3) at left and at right by $\sqrt{B(t)}$. Taking into account the equality $(\sqrt{B(t)}Z \sqrt{B(t)})' = \sqrt{B(t)}Z' \sqrt{B(t)} + \sqrt{B(t)}'Z \sqrt{B(t)} + \sqrt{B(t)}Z \sqrt{B(t)}' \phantom{a} t\ge t_0,$
we obtain
$$
V' + V^2 + P^*(t) V + V P(t) - Q(t) = 0, \phantom{a} t\ge t_0, \eqno (3.24)
$$
where $V \equiv \sqrt{B(t)}Z \sqrt{B(t)}$. To this equation corresponds the following matrix hamiltonian system
$$
\sist{\Phi'= P(t)\Phi + \Psi;}{\Psi' = Q(t)\Phi - P^*(t)\Psi, \phantom{a}h t\ge t_0.} \eqno (3.25)
$$
Suppose the system (1.1) is not oscillatory. Then by (2.4) Eq. (2.3) has a Hermitian solution $Z(t)$ on $[t_1; + \infty)$ for some $t_1 \ge t_0$. Therefore $V(t) \equiv \sqrt{B(t)} Z(t) \sqrt{B(t)}, \phantom{a} t\ge t_1,$ is a hermitian solution of Eq. (3.24) on $[t_1; + \infty)$ and hence the system (3.25) has a conjoined solution $(\Phi (t), \Psi(t))$ such that $det \Phi(t) \ne 0, \phantom{a} t\ge t_.$ It means that the hamiltonian system
$$
\sist{\phantom{a}i'= P(t)\phantom{a}i + \psi;}{\psi' = Q(t)\phantom{a}i - P^*(t)\psi, \phantom{a}h t\ge t_0,}
$$
is not oscillatory. By Theorem 3.1 from here it follows that the scalar systems
$$
\sist{\phantom{a}i_1'= 2 Re p_{jj}(t)\phantom{a}i_1 + \psi_1;}{\psi_1' = - \widetilde{\chi}_j(t) \phantom{a}i_1, \phantom{a}h t\ge t_0,}
$$
$j = 1,2,$ are not oscillatory. Therefore the corresponding equations $(3.23_j), \phantom{a} j=1,2,$ are not oscillatory, which contradicts the conditions of the corollary. The corollary is proved.
Denote:
$$
\widetilde{\mathfrak{M}}(t)\equiv \max\limits_{\tau\in [t_0;t]}\biggl|\exp\biggl\{-\il{\tau}{t}\bigl(\overline{p}_{11}(s) + p_{22}(s)\bigr)ds\biggr\}(p_{12}(\tau) - \overline{p}_{21}(\tau))\biggr|;
$$
$$
\widetilde{\chi}_3(t) \equiv \biggl[\widetilde{\mathfrak{M}}(t) + \il{t_0}{t}\biggl|\exp\biggl\{- \il{\tau}{t}[\overline{p}_{11}(s) + p_{22}(s)]d s\biggr\}\times \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaa}\times \Bigl[\overline{p}_{21} \hskip 0.1pt'(t) + \overline{p}_{21}(\tau)(\overline{p}_{11}(\tau) + p_{22}(\tau)) + q_{12}(\tau)\Bigr]\biggr|d\tau\biggr]^2 - |p_{21}(t)|^2 - q_{11}(t);
$$
$$
\widetilde{\chi}_4(t) \equiv \biggl[\widetilde{\mathfrak{M}}(t) + \il{t_0}{t}\biggl|\exp\biggl\{- \il{\tau}{t}\bigl[\overline{p}_{11}(s) + p_{22}(s)\bigr]d s\biggr\}\times \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
$$
$$
\phantom{a}antom{aaaaaa}\times \Bigl[p_{12}\hskip0.1pt'(t) + p_{12}(\tau)(\overline{p}_{11}(\tau) + p_{22}(\tau)) + q_{12}(\tau)\Bigr]\biggr|d\tau\biggr]^2 - |p_{12}(t)|^2 - q_{22}(t), \phantom{a} t\ge t_0;
$$
$$
\widetilde{I}_{j+2}(\xi,t) \equiv \il{\xi}{t}\exp\biggl\{-\il{\tau}{t}2(Re\hskip 2pt p_{jj}(s))d s \biggr\}\widetilde{\chi}_{j+2}(\tau) d\tau, \phantom{a} t\ge \xi \ge t_0, \phantom{a} j=1,2.
$$
{\bf Theorem 3.4.} {\it Let the following conditions be satisfied:
\noindent
$1') B(t) \ge 0, \phantom{a} t\ge t_0;$
\noindent
$2')$ Eq. (3.21) has a solution $F(t)$ on $[t_0; + \infty)$
\noindent
$3')$ the functions $p_{12}(t)$ and $p_{21}(t)$, defined by (3.22) are continuously differentiable on $[t_0; + \infty)$;
\noindent
4') there exist infinitely large sequences $\xi_{j,0} = t_0 < \xi_{j,1} < ... < \xi_{j,m}, ...$ such that
$$
\il{\xi_{j,m}}{t} \exp\biggl\{\il{\xi_{j,m}}{\tau}\biggl[2 Re \hskip 2pt a_{jj}(s) - \widetilde{I}_{j+2}(\xi_{j,m},s)\biggr] d s\biggr\} \widetilde{\chi}_{j+2}(\tau) d\tau \le 0, \phantom{a} t\in [\xi_{j,m}; \xi_{j,m +1}),
$$
$ \phantom{a} m=1,2,3, ....., \phantom{a} j=1,2$. Then the system (1.1) is non oscillatory.}
Proof. Let $Z(t) \equiv \begin{pmatrix} z_{11}(t) & z_{12}(t)\\ \overline{z}_{12}(t) & z_{22}(t)\end{pmatrix}$ be the Hermitian solution of Eq. (2.3) satisfying the initial condition $Z(t_0) = \begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}$, and let $[t_0;T)$ be the maximum existence interval for $Z(t)$. Then $V(t) \equiv \sqrt{B(t)}Z(t) \sqrt{B(t)}$ is a soluyion of Eq. (3.24) on $[t_0; T)$. Without loss of generality we may assume that $B(t_0) = \begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}$. Then $V(t_0) = \begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}$, and by analogy of the proof of Theorem 3.3 we can show that from the conditions of the theorem it follows that
$$
\il{t_0}{t} tr V(\tau) d \tau \ge 0, \phantom{a}h t\in [t_0;T). \eqno (3.26)
$$
By virtue of Lemma 2.3 we have: $tr V(t) = tr [B(t) Z(t)], \phantom{a} t\in [t_0;T)$. From here and from (3.26) it follows:
$$
\il{t_0}{t} tr [B(\tau) Z(\tau)] d \tau \ge 0, \phantom{a}h t\in [t_0;T). \eqno (3.27)
$$
To complete the proof of the theorem it remains to show that $T = + \infty$. Suppose $T < + \infty$. Then by virtue of Lemma 2.2 from (3.27) it follows that $[t_0;T)$ is not the maximum existence interval for $Z(t)$ which contradicts our assumption. The obtained contradiction shows that $T = + \infty$. The theorem is proved.
Example 3.1. Consider the second order vector equation
$$
\phantom{a}i'' + K(t)\phantom{a}i = 0, \phantom{a}h t\ge t_0, \eqno (3.28)
$$
where $K(t) \equiv \begin{pmatrix} \mu(t) & 10 i\\ -10 i & - t^2\end{pmatrix}, \phantom{a} \mu(t) \equiv p_1 \sin (\lambda_1 t + \theta_1) + p_2 \sin (\lambda_2 t + \theta_2), \phantom{a} t\ge t_0, \phantom{a} p_j, \phantom{a} \lambda_j\ne~ 0, \linebreak \theta_j, \phantom{a} j=1,2,$ are some real constants such that $\lambda_1$ and $\lambda_2$ are rational independent. This equation is equivalent to the system (1.1) with $A(t)\equiv~ 0, \phantom{a} B(t) \equiv \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}, \phantom{a} C(t) =~ - K(t), \linebreak t\ge t_0$. Hence by Theorem 3.1 Eq. (3.28) is oscillatory provided is oscillatory the following scalar system
$$
\sist{\phantom{a}i_1' = \psi_1;}{\psi_1' = - \mu(t) \phantom{a}i_1, \phantom{a} t\ge t_0.}
$$
This system is equivalent to the second order scalar equation
$$
\phantom{a}i_1'' + \mu(t) \phantom{a}i_1 = 0, \phantom{a}h t\ge t_0,
$$
which is oscillatory (see [15]). Therefore Eq. (3.28) is oscillatory.
It is not difficult to verify that the results of works [16 -20] are not applicable to Eq. (3.28).
Example 3.2.
Let
$$
B(t) = \begin{pmatrix} 1 & 1\\ 1 & 1\end{pmatrix}, \phantom{a} t\ge t_0. \eqno (3,29)
$$
Then $\sqrt{B(t)} = \frac{\sqrt{2}}{2} \begin{pmatrix} 1 & 1\\ 1 & 1\end{pmatrix}, \phantom{a} \sqrt{B(t)}' \equiv~ 0, \phantom{a} t\ge t_0,$ and $ F(t) = \sqrt{2}\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}, \phantom{a} t\ge t_0,$ is a solution of Eq. (3.21), on $[t_0;+\infty)$,
$$
P(t) = \begin{pmatrix} a_{11}(t) + a_{12}(t) & a_{11}(t) + a_{12}(t) \\ a_{21}(t) + a_{22}(t) & a_{21}(t) + a_{22}(t) \end{pmatrix}, \phantom{a}antom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} \eqno (3.30)
$$
$$
\phantom{a}antom{aaaaaaaaaaaaaaaaaaaaa} Q(t) = (c_{11}(t) + 2 Re\hskip 2pt c_{12}(t) + c_{22}(t))B(t), \phantom{a}h t\ge t_0.
\eqno (3.31)
$$
Assume
$$
a_{11}(t) + a_{12}(t) = a_{21}(t) + a_{22}(t) \equiv 0, \phantom{a}h t\ge t_0. \eqno (3.32)
$$
Then taking into account (3.30) and (3.31) we have: $\widetilde{\chi}_1(t) = \widetilde{\chi}_2(t)= - c_{11}(t) - 2 Re\hskip2pt c_{12}(t) - \linebreak - c_{22}(t), \phantom{a} t\ge t_0.$ Therefore by Corollary 3.1 under the restrictions (3.29) and (3.32) the system (1.1) is oscillatory provided the scalar equation
$$
\phantom{a}i_1''(t)- [c_{11}(t) + 2 Re \hskip2pt c_{12}(t) + c_{22}(t)] \phantom{a}i_1(t) = 0, \phantom{a}h t\ge t_0,
$$
is oscillatory.
Assume now:
$$
a_{11}(t) + a_{12}(t) = a_{21}(t) + a_{22}(t) = \frac{\alpha}{t}, \phantom{a} c_{11}(t) + 2 Re \hskip2pt c_{12}(t) + c_{22}(t) = \frac{\alpha - \alpha^2}{t^2}, \eqno (3.33)
$$
$0\le \alpha \le 1, \phantom{a} t\ge 1$. Then taking into account (3.30) and (3.31) it is not difficult to verify that $\widetilde{\chi}_3(t) = \widetilde{\chi}_4(t) = \frac{\alpha^2 - \alpha}{t^2} \le 0, \phantom{a} t\ge 1.$ Hence by Theorem 3.4 under the restrictions (3.29) and (3.33) the system (1.1) is non oscillatory.
Let now we assume:
\noindent
$\alpha_1) \phantom{a} a_{11}(t) + a_{12}(t) = a_{21}(t) + a_{22}(t) > 0, \phantom{a} t\ge t_0;$
\noindent
$\alpha_2) \phantom{a} a_{11}(t) + a_{12}(t)$ is increasing and continuously differentiable on $[t_0;+ \infty)$;
\noindent
$\alpha_3) \phantom{a} \frac{|(a_{11}(t) + a_{12}(t))' + c_{11}(t) + 2 Re \hskip2pt c_{12}(t) + c_{22}(t)|}{a_{11}(t) + a_{12}(t)} \le \lambda = const, \phantom{a} t\ge t_0.$
\noindent
Then taking into account (3.30) and (3.31) it is not difficult to verify that $\widetilde{\chi}_3(t) \le \lambda - [c_{11}(t) + 2 Re \hskip2pt c_{12}(t) + c_{22}(t)], \phantom{a} \widetilde{\chi}_4(t) \le \lambda - [c_{11}(t) + 2 Re \hskip 2pt c_{12}(t) + c_{22}(t)], \phantom{a} t\ge t_0.$ Therefore by virtue of Theorem 3.4 under the restrictions (3.29) and $\alpha_1) - \alpha_3)$ the system (1.1) is non oscillatory.
{\bf Remark 3.4.} {\it Since under the restriction (3.29) $det B(t) \equiv 0, \phantom{a} t\ge t_0,$ the results of works [1 -9] are not applicable to the system (1.1) with (3.29).}
\vskip 20 pt
\centerline{ \bf References}
\vskip 20pt
\noindent
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\noindent
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\end{document} |
\begin{document}
\title{Generalized solutions for the Euler-Bernoulli model
with Zener viscoelastic foundations and distributional
forces\thanks{Supported by
the Austrian Science Fund
(FWF) START program Y237 on 'Nonlinear distributional geometry',
and the Serbian Ministry of Science Project 144016}
}
\author{
G\"unther H\"ormann
\footnote{Faculty of Mathematics, University of Vienna,
Nordbergstr.\ 15, A-1090 Vienna, Austria,
Electronic mail: guenther.hoermann@univie.ac.at}\\
Sanja Konjik
\footnote{Faculty of Sciences, Department of Mathematics and Informatics, University of Novi Sad,
Trg D. Obradovi\'ca 4, 21000 Novi Sad, Serbia,
Electronic mail: sanja.konjik@dmi.uns.ac.rs}\\
Ljubica Oparnica
\footnote{Faculty of Education, University of Novi Sad,
Podgori\v cka 4, 25000 Sombor, Serbia,
Electronic mail: ljubica.oparnica@pef.uns.ac.rs}
}
\date{}
\maketitle
\begin{abstract}
We study the initial-boundary value problem for
an Euler-Bernoulli beam model with discontinuous bending stiffness
laying on a viscoelastic foundation and subjected to an axial
force and an external load both of Dirac-type. The
corresponding model equation is fourth order partial differential equation
and involves discontinuous and distributional coefficients
as well as a distributional right-hand side. Moreover the
viscoelastic foundation is of Zener type and described by
a fractional differential equation with respect to time.
We show how functional analytic methods for abstract variational
problems can be applied in combination with regularization
techniques to prove existence and uniqueness of generalized
solutions.
\vskip5pt
\noindent
{\bf Mathematics Subject Classification (2010):}
35D30, 46F30, 35Q74, 26A33, 35A15
\vskip5pt
\noindent
{\bf Keywords:} generalized solutions, Colombeau generalized functions,
fractional derivatives, functional analytic methods, energy estimates
\end{abstract}
\section{Introduction and preliminaries}
\label{ssec:intro}
We study existence and uniqueness of a generalized solution
to the initial-boundary value problem
\begin{align}
& \ensuremath{\partial}^2_tu + Q(t,x,\ensuremath{\partial}_x)u + g = h, \label{eq:PDE} \\
& D_t^\alpha u + u = \theta\, D_t^\alpha g + g, \label{eq:FDE}\\
& u|_{t=0} = f_1, \quad \ensuremath{\partial}_t u|_{t=0} = f_2, \nonumber \tag{IC}\\
& u|_{x=0} =u|_{x=1}=0, \quad
\ensuremath{\partial}_x u|_{x=0} = \ensuremath{\partial}_x u|_{x=1}=0, \nonumber \tag{BC}
\end{align}
where $Q$ is a differential operator of the form
$$
Q u := \ensuremath{\partial}_x^2(c(x)\ensuremath{\partial}_x^2 u) + b(x,t)\ensuremath{\partial}_x^2 u,
$$
$b,c,g,h,f_1$ and $f_2$ are generalized functions,
$\theta$ a constant, $0<\theta<1$, and
$D_t^\alpha$ denotes the left Riemann-Liouville fractional derivative
of order $\alpha$ with respect to $t$.
Problem (\ref{eq:PDE})-(\ref{eq:FDE}) is equivalent to
\begin{equation} \label{eq:IntegroPDE}
\ensuremath{\partial}^2_tu + Q(t,x,\ensuremath{\partial}_x)u + Lu = h,
\end{equation}
with $L$ being the (convolution) operator given by
($\ensuremath{{\cal L}}$ denoting the Laplace transform)
\begin{equation} \label{eq:operator_L}
Lu(x,t) =
\ensuremath{{\cal L}}^{-1} \left(\frac{1+s^\alpha}{1+\theta s^\alpha}\right)(t)
\ast_t u(x,t),
\end{equation}
with the same initial (IC) and boundary (BC) conditions
(cf.\ Section \ref{sec:EBmodel}).
The precise structure of the above problem is motivated by
a model from mechanics describing the displacement of a beam
under axial and transversal forces connected to the viscoelastic
foundation, which we briefly discuss in Subsection
\ref{ssec:motivation}. We then briefly introduce the theory of
Colombeau generalized functions which forms the framework
for our work.
Similar problems involving distributional and generalized
solutions to Euler-Bernoulli beam models have been studied in
\cite{BiondiCaddemi, HoermannOparnica07, HoermannOparnica09, YavariSarkani, YavariSarkaniReddy}.
The development of the theory in the paper is divided into
two parts. In Section \ref{sec:abstract} we consider
the initial-boundary value problem (\ref{eq:IntegroPDE})-(IC)-(BC)
on the abstract level. We prove, in Theorem \ref{lemma:m-a},
an existence result for the abstract variational problem
corresponding to (\ref{eq:IntegroPDE})-(IC)-(BC)
and derive energy estimates (\ref{eq:EE}) which guarantee uniqueness
and serve as a key tool in the analysis of Colombeau
generalized solutions. In Section \ref{sec:EBmodel},
we first show equivalence of the system (\ref{eq:PDE})-(\ref{eq:FDE})
with the integro-differential equation (\ref{eq:IntegroPDE}), and
apply the results from Section \ref{sec:abstract} to the
original problem in establishing weak solutions, if the
coefficients are in $L^\infty$. Afterwards we allow the
coefficients to be more irregular, set up the problem and
show existence and uniqueness of solutions in the space of
generalized functions.
\subsection{The Euler-Bernoulli beam with viscoelastic foundation}
\label{ssec:motivation}
Consider the Euler-Bernoulli beam positioned on the
viscoelastic foundation (cf.\ \cite{Atanackovic-book} for mechanical
background).
One can write the differential
equation of the transversal motion
\begin{equation} \label{eq:mot-trans motion}
\frac{\ensuremath{\partial}^2}{d x^2}\left(A(x)\frac{\ensuremath{\partial}^2u}{d x^2}\right)
+ P(t) \frac{\ensuremath{\partial}^2u}{\ensuremath{\partial} x^2}
+ R(x) \frac{\ensuremath{\partial}^2u}{\ensuremath{\partial} t^2}
+ g(x,t)= h(x,t),
\qquad x\in [0,1],\, t > 0,
\end{equation}
where
\begin{itemize}
\item $A$ denotes the bending stiffness and is given by
$A(x) = EI_1 + H(x-x_0)EI_2$. Here, the constant
$E$ is the modulus of elasticity,
$I_1$, $I_2$, $I_1\neq I_2$, are the moments of
inertia that correspond to the two parts of the beam,
and $H$ is the Heaviside jump function;
\item $R$ denotes the line density (i.e., mass per length)
of the material and is of the form
$ R(x)= R_0 + H(x-x_0)(R_1-R_2)$;
\item $P(t)$ is the axial force, and is assumed to be of
the form $P(t)=P_0 + P_1\delta(t-t_1)$, $P_0,P_1>0$;
\item $g=g(x,t)$ denotes the force terms coming from
the foundation;
\item $u=u(x,t)$ denotes the displacement;
\item $h=h(x,t)$ is the prescribed external load (e.g. when
describing moving load it is of the form
$h(x,t)=H_0\delta(x-ct)$, $H_0$ and $c$ are constants).
\end{itemize}
Since the beam is connected to the viscoelastic
foundation there is a constitutive equation describing relation
between the force of foundation and the displacement of the beam.
We use the Zener generalized model given by
\begin{equation} \label{eq:mot-const eq}
D_t^\alpha u(x,t) + u(x,t) = \theta\, D_t^\alpha g(x,t) + g(x,t),
\end{equation}
where $0<\theta<1$ and $D_t^\alpha$ denotes the left
Riemann-Liouville fractional derivative of order $\alpha$
with respect to $t$, defined by
$$
D_t^\alpha u (t) = \frac{1}{\Gamma (1-\alpha)} \frac{d}{dt}
\int_0^t \frac{u(\tau)}{(t-\tau)^\alpha} \,d\tau.
$$
System (\ref{eq:mot-trans motion})-(\ref{eq:mot-const eq}) is supplied
with initial conditions
$$
u(x,0) = f_1(x), \qquad \ensuremath{\partial}_t u (x,0) = f_2(x),
$$
where $f_1$ and $f_2$ are the initial displacement and the initial
velocity. If $f_1(x)=f_2(x)=0$ the only solution to
(\ref{eq:mot-trans motion})-(\ref{eq:mot-const eq})
should be $u\equiv g\equiv 0$.
Also, the beam is considered to be
fixed at both ends, hence boundary conditions take the form
$$
u(0,t) = u(1,t) = 0, \qquad \ensuremath{\partial}_x u (0,t)=\ensuremath{\partial}_x u (1,t) = 0.
$$
By a change of variables $t\mapsto \tau$ via
$t(\tau) = \sqrt{R(x)}\tau$ the problem
(\ref{eq:mot-trans motion})-(\ref{eq:mot-const eq}) is
transformed into the standard form given in
(\ref{eq:PDE})-(\ref{eq:FDE}). The function $c$ in
(\ref{eq:PDE}) equals $A$ and therefore is of Heaviside
type, and the function $b$ is then given by
$b(x,t)= P(R(x)t)$ and its regularity properties
depend on the assumptions on $P$ and $R$.
As we shall see in Section \ref{sec:EBmodel}, standard functional
analytic techniques reach as far as the following:
boundedness of $b$ together with sufficient (spatial Sobolev)
regularity of the initial values $f_1, f_2$ ensure existence
of a unique solution $u\in L^2(0,T; H^2_0((0,1)))$ (in fact
$u\in L^2(0,T; H^2_0((0,1)))$) to (\ref{eq:IntegroPDE}) with
(IC) and (BC). However, the prominent case
$b = p_0 + p_1 \delta (t-t_1)$ is clearly not covered by such
a result, so in order to allow for these stronger singularities
one needs to go beyond distributional solutions.
\subsection{Basic spaces of generalized functions}
\label{ssec:Colombeau}
We shall set up and solve Equation (\ref{eq:IntegroPDE}) subject
to the initial and boundary conditions (IC) and (BC)
in an appropriate space of Colombeau generalized functions on the
domain $X_T := (0,1) \times (0,T)$ (with $T > 0$)
as introduced in \cite{BO:92} and applied later on, e.g., also in
\cite{GH:04,HoermannOparnica09}. As a few standard references for
the general background concerning Colombeau algebras on arbitrary
open subsets of $\mathbb R^d$ or on manifolds we may mention \cite{CCR, c1, book, MOBook}.
We review the basic notions and facts about the kind of
generalized functions we will employ below: we start with
regularizing families $(u_{\varepsilon})_{\varepsilon\in (0,1]}$ of smooth functions
$u_{\varepsilon}\in H^{\infty}(X_T)$ (space of smooth functions on
$X_T$ all of whose derivatives belong to $L^2$). We will often
write $(u_{\varepsilon})_{\varepsilon}$ to mean $(u_{\varepsilon})_{\varepsilon\in (0,1]}$.
We consider the following subalgebras:
{\it Moderate families}, denoted by $\ensuremath{{\cal E}}_{M,H^{\infty}(X_T)} $, are
defined by the property
$$
\forall\,\alpha\in\mathbb N_0^n, \exists\,p\geq 0:
\|\ensuremath{\partial}^{\alpha}u_{\varepsilon}\|_{L^2(X_T)}= O(\varepsilon^{-p}), \quad \text{ as }
\varepsilon\to 0.
$$
{\it Null families}, denoted by $\ensuremath{{\cal N}}_{H^{\infty}(X_T)}$,
are the families in $\ensuremath{{\cal E}}_{M,H^{\infty}(X_T)}$ satisfying
$$
\forall\,q\geq 0: \|u_{\varepsilon}\|_{L^2(X_T)} = O(\varepsilon^q) \quad \text{ as }
\varepsilon\to 0.
$$
Thus moderateness requires $L^2$ estimates with at most
polynomial divergence as $\varepsilon\to 0$, together with all
derivatives, while null families vanish very rapidly as $\varepsilon \to 0$.
We remark that null families in fact have all derivatives satisfy
estimates of the same kind (cf. \cite[Proposition 3.4(ii)]{garetto_duals}).
Thus null families form a differential ideal in the collection
of moderate families and we may define the {\it Colombeau algebra}
as the factor algebra
$$
\ensuremath{{\cal G}}_{H^{\infty}(X_T)} =
\ensuremath{{\cal E}}_{M, H^{\infty}(X_T)}/\ensuremath{{\cal N}}_{H^{\infty}(X_T)}.
$$
A typical notation for the
equivalence classes in
$\ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ with representative $(u_\varepsilon)_\varepsilon$ will be
$[(u_\varepsilon)_\varepsilon]$. Finally, the algebra $\ensuremath{{\cal G}}_{H^{\infty}((0,1))}$
of generalized functions on the interval $(0,1)$ is defined
similarly and every element can be considered to be a member
of $\ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ as well.
We briefly recall a few technical remarks from \cite[Subsection 1.2]{HoermannOparnica09}:
If $(u_{\varepsilon})_{\varepsilon}$ belongs to $\ensuremath{{\cal E}}_{M,H^{\infty}(X_T)}$
we have smoothness up to the boundary for every $u_\varepsilon$, i.e.\
$u_{\varepsilon}\in C^{\infty}([0,1] \times [0,T])$ (which follows from
Sobolev space properties on the Lipschitz domain $X_T$; cf.\ \cite{AF:03})
and therefore
the restriction $u |_{t=0}$ of a generalized function
$u \in \ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ to $t=0$ is well-defined by
$u_{\varepsilon}(\cdot,0)\in \ensuremath{{\cal E}}_{M,H^{\infty}((0,1))}$.
If $v \in \ensuremath{{\cal G}}_{H^{\infty}((0,1))}$ and in addition
we have for some representative $(v_\varepsilon)_\varepsilon$ of $v$ that
$v_\varepsilon \in H_0^{2}((0,1))$, then $v_\varepsilon(0) = v_\varepsilon(1) = 0$
and $\ensuremath{\partial}_x v_\varepsilon (0) = \ensuremath{\partial}_x v_\varepsilon(1) = 0$. In particular,
$$
v(0) = v(1) = 0 \quad\text{and}\quad \ensuremath{\partial}_x v(0) = \ensuremath{\partial}_x v(1) = 0
$$
holds in the sense of generalized numbers.
Note that $L^2$-estimates for parametrized families
$u_\varepsilon \in H^\infty(X_T)$ always yield similar $L^\infty$-estimates
concerning $\varepsilon$-asymptotics
(since $H^\infty(X_T) \subset C^\infty(\overline{X_T})
\subset W^{\infty,\infty}(X_T)$).
The space $H^{-\infty}(\mathbb R^d)$, i.e.\ distributions of finite order,
is embedded (as a linear space) into $\ensuremath{{\cal G}}_{H^{\infty}(\mathbb R^d)}$ by
convolution regularization (cf.\ \cite{BO:92}). This embedding
renders $H^\infty(\mathbb R^d)$ a subalgebra of $\ensuremath{{\cal G}}_{H^{\infty}(\mathbb R^d)}$.
Certain generalized functions possess distribution aspects, namely
we call $u =[(u_{\varepsilon})_{\varepsilon}]\in \ensuremath{{\cal G}}_{H^{\infty}}$
{\it associated with the distribution} $w\in\ensuremath{{\cal D}}'$, notation
$u \approx w$, if for some (hence any) representative $(u_{\varepsilon})_{\varepsilon}$
of $u$ we have $u_{\varepsilon}\to w$ in $\ensuremath{{\cal D}}'$, as $\varepsilon\to 0$.
\section{Preparations: An abstract evolution problem in variational form and the convolution-type operator $L$}
\label{sec:abstract}
In this section we study an abstract background of equation
(\ref{eq:IntegroPDE}) subject to the initial and boundary
conditions (IC) and (BC) in terms of bilinear forms on
arbitrary Hilbert spaces.
First we shall repeat standard results and then extend them to a
wider class of problems.
We shall show existence of a unique solution, derive energy
estimates, and analyze the particular form of the operator
$L$ appearing in (\ref{eq:IntegroPDE}).
Let $V$ and $H$ be two separable Hilbert spaces,
where $V$ is densely embedded into $H$. We shall denote the norms
in $V$ and $H$ by $\|\cdot\|_V$ and $\|\cdot\|_H$ respectively.
If $V'$ denotes the dual of $V$, then $V \subset H \subset V'$
forms a Gelfand triple.
In the sequel we shall also make use of the Hilbert spaces
$E_V:=L^2(0,T;V)$ with the norm
$\|u\|_{E_V}:=(\int_0^T \|u(t)\|_V^2\,dt)^{1/2}$, and
$E_H:=L^2(0,T;H)$ with the norm
$\|u\|_{E_H}:=(\int_0^T \|u(t)\|_H^2\,dt)^{1/2}$.
Since, $\|v\|_H\leq \ensuremath{{\cal C}}\cdot\|v\|_V$, $v\in V$,
(without loss of generality we may assume that $\ensuremath{{\cal C}}=1$),
it follows that $\|u\|_{E_H}\leq \|u\|_{E_V}$, $u\in E_V$,
and $E_V\subset E_H$.
The bilinear forms we shall deal with will be of the following
type:
\begin{assumption} \label{Ass1}
Let $a(t,\cdot,\cdot)$, $a_0(t,\cdot,\cdot)$ and
$a_1(t,\cdot,\cdot)$, $t\in [0,T]$,
be (parametrized) families of continuous bilinear forms on $V$
with
$$
a(t,u,v)=a_0(t,u,v)+ a_1(t,u,v) \qquad \forall\, u, v \in V,
$$
such that the 'principal part' $a_0$ and the remainder $a_1$
satisfy the following conditions:
\begin{itemize}
\item[(i)] $t \mapsto a_0(t,u,v)$ is continuously
differentiable $[0,T] \to \mathbb R$, for all $u,v \in V$;
\item[(ii)] $a_0$ is symmetric, i.e., $a_0(t,u,v)=
a_0(t,v,u)$, for all $u,v\in V$;
\item[(iii)] there exist real constants $\lambda,\mu >0$ such that
\begin{equation} \label{eq:coercivity}
a_0(t,u,u) \geq \mu \|u\|_V^2 - \lambda \|u\|_H^2,
\qquad \forall\, u\in V,\, \forall\, t\in [0,T];
\end{equation}
\item[(iv)] $t \mapsto a_1(t,u,v)$ is continuous $[0,T] \to \mathbb R$,
for all $u,v\in V$;
\item[(v)] there exists $C_1 \geq 0$ such that for all $t\in [0,T]$ and
$u,v\in V$, $|a_1(t,u,v)|\leq C_1 \|u\|_V\, \|v\|_H$.
\end{itemize}
\end{assumption}
It follows from condition (i) that there exist nonnegative
constants $C_0$ and $C_0'$ such that for all $t\in [0,T]$ and
$u,v\in V$,
\begin{equation} \label{i_cons}
|a_0(t,u,v)| \leq C_0 \|u\|_V\,\|v\|_V
\quad \mbox{ and } \quad
|a'_0(t,u,v)| \leq C_0' \|u\|_V \,\|v\|_V,
\end{equation}
where $a'_0(t,u,v):=\frac{d}{dt} a_0(t,u,v)$.
It is shown in \cite[Ch.\ XVIII, p.\ 558, Th.\ 1]{DautrayLions-vol5}
(see also \cite[Ch.\ III, Sec.\ 8]{LionsMagenes})
that the above conditions guarantee unique solvability of the
abstract variational problem in the following sense:
\begin{theorem} \label{th:avp}
Let $a(t,\cdot,\cdot)$, $t\in[0,T]$, satisfy Assumption
\ref{Ass1}.
Let $u_0\in V$, $u_1\in H$ and $f\in E_H$.
Then there exists a unique $u \in E_V$ satisfying
the regularity conditions
\begin{equation} \label{eq:avp-reg}
u'= \frac{du}{dt}\in E_V
\quad \mbox{ and } \quad
u''=\frac{d^2u}{dt^2}\in L^2(0,T;V')
\end{equation}
(here time derivatives should be understood in distributional
sense),
and solving the abstract initial value problem
\begin{align}
&\dis{u''(t)}{v} + a(t,u(t),v)=\dis{f(t)}{v},
\qquad \forall\, v\in V, \,
\mbox{ for a.e. } t \in (0,T) \label{eq:avp}\\
& u(0)=u_0,\qquad u'(0)=u_1. \label{eq:avp-ic}
\end{align}
(Note that (\ref{eq:avp-reg}) implies that $u \in C([0,T],V)$ and
$u' \in C([0,T],V')$. Hence it makes sense to evaluate $u(0)\in V$
and $u'(0) \in V'$ and (\ref{eq:avp-ic}) claims that these equal
$u_0$ and $u_1$, respectively.)
\end{theorem}
\begin{remark} \label{rem:distr-intrpr}
The precise meaning of (\ref{eq:avp}) is the following:
$\forall\, \varphi\in\ensuremath{{\cal D}}((0,T))$,
$$
\dis{\dis{u''(t)}{v}}{\varphi}_{(\ensuremath{{\cal D}}',\ensuremath{{\cal D}})}
+ \dis{a(t,u(t),v)}{\varphi}_{(\ensuremath{{\cal D}}',\ensuremath{{\cal D}})}
= \dis{\dis{f(t)}{v}}{\varphi}_{(\ensuremath{{\cal D}}',\ensuremath{{\cal D}})},
$$
or equivalently,
$$
\int_0^T \dis{u(t)}{v}\varphi''(t)\,dt
+ \int_0^T a(t,u(t),v)\varphi(t)\,dt
= \int_0^T \dis{f(t)}{v}\varphi(t)\,dt.
$$
\end{remark}
The proof of this theorem proceeds by showing that $u$ satisfies
a priori (energy) estimates which immediately imply uniqueness of
the solution, and then using the Galerkin approximation method to
prove existence of a solution. An explicit form of the energy
estimate for the abstract variational problem
(\ref{eq:avp-reg})-(\ref{eq:avp-ic})
with precise dependence of all constants is derived in
\cite[Prop.\ 1.3]{HoermannOparnica09} in the form
\begin{equation} \label{eq:ee-ThmP1}
\|u(t)\|^2_V + \|u'(t)\|^2_H \leq
\left(
D_T\|u_0\|_V^2 + \|u_1\|_H^2 + \int_0^t \|f(\tau)\|_H^2\,d\tau
\right) \cdot e^{t\cdot F_T},
\end{equation}
where $D_T:=\frac{C_0 + \lambda (1+ T)}{\min\{1,\mu\}}$ and
$F_T := \max \{\frac{C_0' +C_1}{\min\{1,\mu\}},
\frac{C_1 +T+ 2)}{\min\{1,\mu\}}\}$.\\
\subsection{Existence of a solution to the abstract variational problem}
\label{ssec:existence}
We shall now prove a similar result for a slightly modified
abstract variational problem, which is to encompass our problem
(\ref{eq:IntegroPDE}). Here in addition to the bilinear forms we
shall consider "causal" operators $L:L^2(0,T_1;H)\to L^2(0,T_1;H)$,
$\forall\,T_1<T$, which satisfy the estimate: $\exists\, C_L>0$
such that
\begin{equation} \label{eq:L-estimate}
\|Lu\|_{L^2(0,T_1;H)} \leq C_L \|u\|_{L^2(0,T_1;H)},
\end{equation}
where $C_L$ is independent of $T_1$.
\begin{lemma} \label{lemma:m-a}
Let $a(t,\cdot,\cdot)$, $t\in[0,T]$, satisfy Assumption
\ref{Ass1}.
Let $f_1\in V$, $f_2\in H$ and $h\in E_H$.
Let $L:E_H\to E_H$ satisfy (\ref{eq:L-estimate}).
Then there exists a $u\in E_V$ satisfying the regularity conditions
$$
u'=\frac{du}{dt} \in E_V \quad \mbox{ and } \quad
u''=\frac{d^2 u}{dt^2} \in L^2(0,T;V')
$$
and solving the abstract initial value problem
\begin{align}
&\dis{u''(t)}{v} + a(t,u(t),v) + \dis{Lu(t)}{v}= \dis{h(t)}{v},
\qquad \forall\, v\in V,\,
\mbox{ for a.e. } t \in (0,T), \label{eq:avp-L}\\
& u(0)=f_1, \qquad u'(0) = f_2. \label{eq:avp-L-ic}
\end{align}
Moreover, we have $u\in\ensuremath{{\cal C}}([0,T];V)$ and $u'\in\ensuremath{{\cal C}}([0,T];H)$.
\end{lemma}
Here we give a proof based on an iterative procedure and employing
Theorem \ref{th:avp} and the energy estimate (\ref{eq:ee-ThmP1})
in each step. Notice that the precise meaning of (\ref{eq:avp-L})
(in distributional sense) is explained in Remark
\ref{rem:distr-intrpr}.
\begin{proof}
Let $u_0\in E_H$ be arbitrarily chosen and consider the initial
value problem for $u$ in the sense of Remark \ref{rem:distr-intrpr}
\begin{align}
& \dis{u''(t)}{v}+ a(t,u(t),v)+ \dis{Lu_0(t)}{v}= \dis{h(t)}{v},
\qquad \forall\, v\in V,\,
\mbox{ for a.e. } t \in (0,T), \label{eq:sol u1}\\
& u(0) = f_1,\quad u'(0) = f_2. \nonumber
\end{align}
By Theorem \ref{th:avp} there exists a unique
$u_1\in E_V$ satisfying $u_1'\in E_V$,
$u_1''\in L^2(0,T;V')$, and solving (\ref{eq:sol u1}).
Consider now (\ref{eq:sol u1}) with $Lu_1$ instead of
$Lu_0$. As above, by Theorem \ref{th:avp}, one obtains
a unique solution $u_2\in E_V$ with $u_2'\in E_V$
and $u_2''\in L^2(0,T;V')$.
Repeating this procedure we obtain a sequence of functions
$\{u_k\}_{k\in\mathbb N}\in E_V$,
satisfying $u_k'\in E_V$, $u_k''\in L^2(0,T;V')$, and
solving the following problems: for each $k\in\mathbb N$,
\begin{align*}
& \dis{u_k''(t)}{v} + a(t,u_k(t),v) + \dis{Lu_{k-1}(t)}{v}= \dis{h(t)}{v},
\qquad \forall\, v\in V,\,
\mbox{ for a.e. } t \in (0,T), \\
& u_k(0) = f_1,\quad u_k'(0) = f_2.
\end{align*}
Also, for all $k\in\mathbb N$, $u_k$ satisfies the energy estimate of
type (\ref{eq:ee-ThmP1}):
$$
\|u_k(t)\|_V^2 + \|u_k'(t)\|_H^2
\leq \left(
D_T \,\|f_1\|_V^2 + \|f_2\|_H^2
+ \int_0^t\|(h-Lu_{k-1})(\tau)\|_H^2 \,d\tau
\right) \cdot e^{t\cdot F_T},
$$
where the constants $D_T$ and $F_T$ are independent of $k$. We claim that $\{u_k\}_{k\in\mathbb N}$
converges in $E_V$.
To see this, we first note that $u_l-u_k$ solves
\begin{align*}
& \dis{(u_l-u_k)''(t)}{v} + a(t,(u_l-u_k)(t),v) +
\dis{L(u_{l-1}-u_{k-1})(t)}{v}= 0, \quad \forall\,v\in V,\,
\mbox{ for a.e. } t \in (0,T), \\
& (u_l-u_k)(0) = 0, \quad (u_l-u_k)'(0) = 0,
\end{align*}
and $u_l-u_k\in E_V$, with $u_l'-u_k'\in E_V$ and
$u_l''-u_k''\in L^2(0,T;V')$. Moreover, the corresponding energy
estimate is of the form
\begin{equation} \label{EE1}
\|(u_l-u_k)(t)\|_V^2 + \|(u_l-u_k)'(t)\|_H^2 \leq
e^{t\cdot F_T} \cdot
\int_0^t \|L(u_{l-1}-u_{k-1})(\tau)\|_H^2\,d\tau.
\end{equation}
Thus,
$$
\|(u_l-u_k)(t)\|_V^2
\leq e^{T\cdot F_T} \cdot \int_0^T
\|L(u_{l-1}-u_{k-1})(\tau)\|_H^2\,d\tau
= e^{T\cdot F_T} \cdot \|L(u_{l-1}-u_{k-1})\|_{E_H}^2.
$$
Integrating from $0$ to $T$ and using assumption
(\ref{eq:L-estimate}) on $L$, one obtains
\begin{equation} \label{uk-ul_estimates}
\|u_l-u_k\|_{E_V}
\leq \gamma_T \|u_{l-1}-u_{k-1}\|_{E_H},
\end{equation}
where $\gamma_T:=C_L \sqrt{T} e^{\frac{T\cdot F_T}{2}}$.
Taking now $l=k+1$ in (\ref{uk-ul_estimates}) successively, yields
$$
\|u_{k+1}-u_k\|_{E_V} \leq \gamma_T^k \|u_{1}-u_{0}\|_{E_H} \leq
\gamma_T^k \|u_{1}-u_{0}\|_{E_V},
$$
and hence
$$
\|u_l-u_k\|_{E_V} \leq \|u_l-u_{l-1}\|_{E_V}+\ldots
+\|u_{k+1}-u_{k}\|_{E_V}
\leq \sum_{i=l-1}^{k} \gamma_T^i \|u_{1}-u_{0}\|_{E_V}.
$$
We may choose $T_1<T$ such that $\gamma_{T_1}< 1$, hence
$\sum_{i=0}^\infty \gamma_{T_1}^i$ converges.
Note that $t\mapsto\gamma_t$ is increasing. By abuse of notation
we denote $L^2(0,T_1;V)$ again by $E_V$.
This further implies that $\{u_k\}_{k\in\mathbb N}$ is a Cauchy sequence
and hence convergent in $E_V$, say
$u:= \lim_{k\to\infty} u_k$. Similarly,
one can show convergence of $u_k'$ in $E_V$, i.e., existence of
$v:= \lim_{k\to\infty} u_k' \in E_V$.
In the distributional setting $\lim_{k\to\infty} u_k' = u'$,
and therefore $u'=v\in E_V$
(cf.\ \cite[Ch.\ XVIII, p.\ 473, Prop.\ 6]{DautrayLions-vol5}).
We also have to show that $u$ solves equation (\ref{eq:avp-L}).
Let $\varphi\in \ensuremath{{\cal D}}((0,T))$. Then
\begin{align*}
\dis{\dis{h(t)}{v}}{\varphi} & =
\dis{\dis{u_k''(t)}{v}}{\varphi} + \dis{a(t,u_k(t),v)}{\varphi}
+ \dis{\dis{Lu_{k-1}(t)}{v}}{\varphi} \\
& = \dis{\dis{u_k}{v}}{\varphi''} + \dis{a(t,u_k(t),v)}{\varphi}
+ \dis{\dis{Lu_{k-1}(t)}{v}}{\varphi} \\
& \to \dis{\dis{u}{v}}{\varphi''} + \dis{a(t,u(t),v)}{\varphi}
+ \dis{\dis{Lu(t)}{v}}{\varphi} \\
& = \dis{\dis{u''(t)}{v}}{\varphi} + \dis{a(t,u(t),v)}{\varphi}
+ \dis{\dis{Lu(t)}{v}}{\varphi}.
\end{align*}
Here we used that $\varphi''\in \ensuremath{{\cal D}}((0,T))$.
Therefore $u$ solves (\ref{eq:avp-L}) on the time interval
$[0,T_1]$. The initial conditions are satisfied by construction
of $u$.
It remains to extend this result on existence of a solution to
the whole interval $[0,T]$.
Since $T_1$ is independent on the initial conditions, if $T>T_1$
one needs at most $\frac{T}{T_1}$ steps to reach
convergence in $E_V$. In fact, one has to show
regularity at the end point $T_1$ of the interval $[0,T_1]$
on which the solution exists, i.e.,
$$
u(T_1)\in V \quad \mbox{ and } \quad
u'(T_1)\in H.
$$
To see this, it suffices to show that $u_k\to u$ in
$Y_V:=\ensuremath{{\cal C}}([0,T_1];V)$ and $u_k'\to u'$ in $Y_H:=\ensuremath{{\cal C}}([0,T_1];H)$.
From (\ref{EE1}) and assumption (\ref{eq:L-estimate}) on $L$
we obtain
$$
\|(u_l-u_k)(t)\|_V^2
\leq e^{T_1\cdot F_{T_1}} C_L^2 \int_0^{T_1} \|(u_l-u_k)(\tau)\|_H^2 \,d\tau
\leq e^{T_1\cdot F_{T_1}} C_L^2 \int_0^{T_1} \|(u_l-u_k)(\tau)\|_V^2 \,d\tau.
$$
Taking first the square root and then the supremum over all
$t\in[0,T]$ yields
$$
\|u_l-u_k\|_{Y_V} \leq \gamma_{T_1} \|u_l-u_k\|_{Y_V}.
$$
Since $\gamma_{T_1}<1$ this implies that $\{u_k\}_{k\in\mathbb N}$
is a Cauchy sequence in $Y_V$.
Similarly,
$$
\|(u_l-u_k)'(t)\|_H^2 \leq e^{T_1\cdot F_{T_1}} C_L^2
\int_0^{T_1} \|(u_l-u_k)(\tau)\|_V^2 \,d\tau,
$$
which upon taking the supremum gives
$$
\|(u_l-u_k)'\|_{Y_H} \leq \gamma_{T_1}
\|u_l-u_k\|_{Y_V},
$$
thus $u'_k\to u'$ in $Y_H$ (due to the already established
convergence of $u_k$ in $Y_V$),
and $u'(T_1)\in H$. This proves the claim.
\end{proof}
\subsection{Energy estimates}
\label{ssec:ee}
In Section \ref{sec:EBmodel} we shall need
a priori (energy) estimate for problem (\ref{eq:IntegroPDE}).
In fact, for the verification of moderateness in the Colombeau
setting it will be crucial to know all constants in the energy
estimate precisely. Therefore, we shall now derive it.
\begin{proposition} \label{prop:EnergyEstimates}
Under the assumptions of Lemma \ref{lemma:m-a},
let $u$ be a solution to the abstract
variational problem (\ref{eq:avp-L})-(\ref{eq:avp-L-ic}).
Then, for each $t\in [0,T]$,
\begin{equation}\label{eq:EE}
\|u(t)\|^2_V + \|u'(t)\|^2_H \leq
\left(
D_T\|f_1\|_V^2 + \frac{1}{\nu} \left(
\|f_2\|_H^2 + \int_0^{t} \|h(\tau)\|^2_H \,d\tau
\right) \right)
\cdot e^{t\cdot F_T},
\end{equation}
where $\nu:=\min\{1,\mu\}$,
$D_T:=\frac{C_0 + \lambda (1+ T)}{\nu}$ and
$F_T := \max \{\frac{C_0' +C_1+ C_L}{\nu},
\frac{C_1 +2+ \lambda (1+ T)}{\nu}\}$.
\end{proposition}
\begin{proof}
Setting $v:= u'(t)$ in (\ref{eq:avp-L})
we obtain (as an equality of integrable functions with
respect to $t$)
$$
\dis{u''(t)}{u'(t)} + a(t,u(t),u'(t)) + \dis{Lu(t)}{u'(t)}
= \dis{h(t)}{u'(t)}.
$$
Since $a(t,u,v)=a_0(t,u,v)+a_1(t,u,v)$ and
$\dis{u''(t)}{u'(t)}= \frac{1}{2}\frac{d}{dt}\dis{u'(t)}{u'(t)}
=\frac{1}{2}\frac{d}{dt}\|u'(t)\|^2_H$, we have
$$
\frac{d}{dt}\|u'(t)\|^2_H = - 2 a_0(t,u(t),u'(t))
- 2a_1(t,u(t), u'(t)) - 2 \dis{Lu(t)}{u'(t)}
+ 2 \dis{h(t)}{u'(t)}.
$$
Integration from $0$ to $t_1$, for arbitrary $0< t_1 \leq T$,
gives
\begin{eqnarray}s
\|u'(t_1)\|^2_H - \|f_2\|^2_H
&=&
- 2 \int_0^{t_1} a_0(t,u(t),u'(t))\,dt
-2 \int_0^{t_1} a_1(t,u(t), u'(t))\,dt \\
&& \quad
- 2 \int_0^{t_1}\dis{Lu(t)}{u'(t)}\,dt
+ 2 \int_0^{t_1} \dis{h(t)}{u'(t)}\,dt.
\end{eqnarray}s
Note that $\frac{d}{dt}a_0(t,u(t),u(t))= a_0'(t,u(t),u(t))
+a_0(t,u'(t),u(t))+a_0(t,u(t),u'(t))$ and hence,
by Assumption \ref{Ass1} (ii),
$2a_0(t,u'(t),u(t))=\frac{d}{dt}a_0(t,u(t),u(t))-a_0'(t,u(t),u(t))$.
This yields
\begin{align}
& LHS := \|u'(t_1)\|^2_H + a_0(t_1, u(t_1), u(t_1))
= \|f_2\|^2_H + a_0(0,u(0),u(0))
- \int_0^{t_1} a_0'(t,u(t),u(t))\,dt \nonumber \\
& \qquad
- 2\int_0^{t_1} a_1(t,u(t), u'(t)) \,dt
- 2\int_0^{t_1} \dis{Lu(t)}{u'(t)}\,dt
+ 2\int_0^{t_1} \dis{h(t)}{u'(t)}\,dt
= : RHS. \label{eq:srednja}
\end{align}
Further, by (\ref{i_cons}), Assumption \ref{Ass1} (v), the
Cauchy-Schwartz inequality, the inequality $2ab\leq a^2+b^2$,
and the assumption (\ref{eq:L-estimate}) on $L$ we have
\begin {align*}
|RHS|
& \leq \|f_2\|^2_H + C_0\|u(0)\|^2_V
+ C_0' \int_0^{t_1} \|u(t)\|^2_V \,dt \\
& \quad
+ 2 C_1 \int_0^{t_1} \|u(t)\|_V \| u'(t)\|_H \,dt
+ 2 \int_0^{t_1} \|Lu(t)\|_H \|u'(t)\|_H \,dt
+ 2 \int_0^{t_1} \|h(t)\|_H \|u'(t)\|_H \,dt \\
& \leq \|f_2\|^2_H + C_0\|f_1\|^2_V
+ (C_0' +C_1) \int_0^{t_1} \| u(t)\|^2_V \,dt \\
& \quad
+ (C_1 +2) \int_0^{t_1} \| u'(t)\|^2_H \,dt
+ \|Lu\|^2_{L^2(0,t_1;H)}
+ \int_0^{t_1} \|h(t)\|^2_H \,dt \\
& \leq \|f_2\|^2_H + C_0\|f_1\|^2_V
+ (C_0' +C_1+ C_L) \int_0^{t_1} \| u(t)\|^2_V \,dt \\
& \quad
+ (C_1 +2) \int_0^{t_1} \|u'(t)\|^2_H \,dt
+ \int_0^{t_1} \|h(t)\|^2_H \,dt.
\end{align*}
Further, it follows from (\ref{eq:coercivity}) that
$$
LHS = \|u'(t_1)\|^2_H + a_0(t_1, u(t_1), u(t_1))
\geq \|u'(t_1)\|^2_H + \mu \|u(t_1)\|^2_V - \lambda \|u(t_1)\|^2_H,
$$
and therefore (\ref{eq:srednja}) yields
\begin{align*}
\|u'(t_1)\|^2_H + \mu \|u(t_1)\|^2_V
& \leq
\lambda \|u(t_1)\|^2_H + C_0 \|f_1\|^2_V +\|f_2\|^2_H
+ \int_0^{t_1} \|h(t)\|^2_H \,dt \\
& \qquad
+ (C_0' +C_1+ C_L) \int_0^{t_1} \| u(t)\|^2_V \,dt
+ (C_1 +2) \int_0^{t_1} \|u'(t)\|^2_H \,dt.
\end{align*}
As shown in \cite{HoermannOparnica09} we have that
$\|u(t)\|^2_H \leq (1+t)(\|f_1\|^2_V + \int_0^{t} \|u'(s)\|^2_H \,ds)$,
hence
\begin{align*}
\|u(t_1)\|^2_V + \|u'(t_1)\|^2_H \leq D_T \|f_1\|^2_V +
\frac{1}{\nu} \left(
\|f_2\|^2_H + \int_0^{t_1} \|h(t)\|^2_H \,dt
\right)
+ F_T \int_0^{t_1} (\| u(t)\|^2_V + \|u'(t)\|^2_H )\,dt.
\end{align*}
where $\nu:=\min\{1,\mu\}$,
$D_T:=\frac{C_0 + \lambda (1+ T)}{\nu}$ and
$F_T := \max \{\frac{C_0' +C_1+ C_L}{\nu},
\frac{C_1 +2+ \lambda (1+ T)}{\nu} \}$.
The claim now follows from Gronwall's lemma.
\end{proof}
As a consequence of Proposition \ref{prop:EnergyEstimates},
one also has uniqueness of the solution in Lemma \ref{lemma:m-a}.
\begin{theorem}
Under the assumptions of Lemma \ref{lemma:m-a} there exists a
unique $u\in E_V$ satisfying the regularity conditions
$u'\in E_V$ and $u''\in L^2(0,T;V')$,
and solving the abstract initial value problem
(\ref{eq:avp-L})-(\ref{eq:avp-L-ic}).
Moreover, $u\in\ensuremath{{\cal C}}([0,T];V)$ and $u'\in\ensuremath{{\cal C}}([0,T];H)$.
\end{theorem}
\begin{proof}
Since existence of a solution is proved in Lemma \ref{lemma:m-a},
it remains to show uniqueness part of the theorem. Thus, let
$u$ and $w$ be solutions to the abstract initial value problem
(\ref{eq:avp-L})-(\ref{eq:avp-L-ic}), satisfying the regularity
conditions $u',w'\in E_V$ and $u'',w''\in L^2(0,T;V')$. Then
$u-w$ is a solution to the homogeneous abstract problem with
vanishing initial data
\begin{align*}
&\dis{(u-w''(t)}{v} + a(t,(u-w)(t),v) + \dis{L(u-w)(t)}{v}= 0,
\qquad \forall\, v\in V,\,
\mbox{ for a.e. } t \in (0,T), \\
& (u-w)(0)=0, \qquad (u-w)'(0) = 0.
\end{align*}
Moreover, according to Proposition \ref{prop:EnergyEstimates},
$u-w$ satisfies the energy estimates (\ref{eq:EE}) with
$f_1=f_2=h\equiv 0$. This implies uniqueness of the solution.
\end{proof}
\subsection{Basic properties of the operator $L$}
\label{ssec:L}
In this subsection we analyze our particular form of the operator
$L$, relevant to the problem described in the Introduction.
Therefore, we consider an operator of convolution type
and seek for conditions which guarantee estimate
(\ref{eq:L-estimate}).
\begin{lemma} \label{lemma:L_1}
Let $l\in L^2_{loc}(\mathbb R)$ with $\mathop{\rm supp}\nolimits l \subset [0,\infty)$. Then
for all $T_1\in[0,T]$, the operator $L$ defined by $Lu(x,t) :=
\int_0^t l(s) u(x,t-s)\,ds$ maps $L^2(0,T_1;H)$ into itself, and
(\ref{eq:L-estimate}) holds with $C_L = \|l\|_{L^2(0,T)}\cdot T $.
\end{lemma}
\begin{remark}
We may think of $u$ being extended by $0$ outside $[0,T]$ to a
function in $L^2(\mathbb R;H)$, and then identify $Lu$ with $l\ast_t u$.
\end{remark}
\begin{proof}
Integration of $\|Lu(t)\|_H^2 \leq \int_0^t |l(t-s)|\|u(s)\|_H \,ds$
from $0$ to $T_1$, $0<T_1\leq T$, yields
\begin{multline*}
\left(
\int_0^{T_1}\|Lu(t)\|_H^2 \,dt
\right)^{1/2}
\leq \left(\int_0^{T_1} (\int_0^t |l(t-s)|\|u(s)\|_H \,ds)^2 \,dt\right)^{1/2} \\
\leq \left(\int_0^{T_1} (\int_0^{T_1} |l(t-s)|\|u(s)\|_H \,ds)^2 \,dt\right)^{1/2}
\leq \int_0^{T_1} \left(\int_0^{T_1} |l(t-s)|^2 \|u(s)\|^2_H \,dt\right)^{1/2} \,ds \\
= \int_0^{T_1} \left( \int_0^{T_1} |l(t-s)|^2 \,dt\right)^{1/2} \|u(s)\|_H \,ds
= \|l\|_{L^2(0,T_1)}\cdot \|u\|_{L^1(0,T_1; H)} \\
\leq \|l\|_{L^2(0,T)}\cdot T \cdot \|u\|_{L^2(0,T_1; H)},
\end{multline*}
where we have used the support property of $l$, Minkowski's
inequality for integrals (c.f \cite[p.\ 194]{Folland}), and
the Cauchy-Schwartz inequality.
\end{proof}
In the following lemma we discuss a regularization of $L$, which
will be used in Section \ref{ssec:Colombeau sol}.
\begin{lemma} \label{lemma:reg L}
Let $l\in L^1_{loc}(\mathbb R)$ with $\mathop{\rm supp}\nolimits l \subset [0,\infty)$.
Let $\rho\in\ensuremath{{\cal D}}(\mathbb R)$ be a mollifier ($\mathop{\rm supp}\nolimits \rho\subset B_1(0)$,
$\int \rho =1$). Define $\rho_\varepsilon(t):=\gamma_\varepsilon
\rho(\gamma_\varepsilon t)$, with $\gamma_\varepsilon>0$ and
$\gamma_\varepsilon\to\infty$ as $\varepsilon\to 0$,
$l_\varepsilon := l*\rho_\varepsilon $ and
$\tilde{L}_\varepsilon u(t):= (l_\varepsilon *_t u)(t)$, for
$u\in E_H$. Then
$\forall\,p\in[1,\infty)$,
$l_\varepsilon\in L^p_{loc}(\mathbb R)$ and $l_\varepsilon\to l$
in $L^1_{loc}(\mathbb R)$.
\end{lemma}
\begin{proof}
Let $K$ be a compact subset of $\mathbb R$. Then
\begin{multline*}
\|l_\varepsilon\|_{L^p(K)}
= \|l*_t\rho_\varepsilon\|_{L^p(K)}
= \left(\int_K |\int_{-\infty}^\infty
l(\tau)\rho_\varepsilon(t-\tau)\,d\tau|^p \,dt\right)^{1/p} \\
\leq \left(\int_K \left(\int_{-\infty}^\infty
|l(\tau)||\rho_\varepsilon(t-\tau)|\,d\tau\right)^p \,dt\right)^{1/p}
\leq \left(\int_K \left( \int_{K+ B_1(0)}
|l(\tau)||\rho_\varepsilon(t-\tau)|\,d\tau \right)^p \,dt\right)^{1/p} \\
\leq \int_{K+ B_1(0)} \left(\int_K
|l(\tau)|^p|\rho_\varepsilon(t-\tau)|^p \,dt\right)^{1/p} \,d\tau
= \int_{K+ B_1(0)} |l(\tau)|\left(\int_K
|\rho_\varepsilon(t-\tau)|^p \,dt\right)^{1/p} \,d\tau \\
= \int_{K+ B_1(0)} |l(\tau)| \|\rho_\varepsilon\|_{L^p(B_1(0))} \,d\tau
= \|l\|_{L^1(K+B_1(0))} \|\rho_\varepsilon\|_{L^p(B_1(0))} \\
= \|l\|_{L^1(K+B_1(0))}
\cdot \gamma_\varepsilon^{1-\frac{1}{p}} \cdot
\|\rho\|_{L^p(B_1(0))}.
\end{multline*}
where the second inequality follows from the support properties
of $l$ and $\rho$ ($t-\tau\in B_1(0), t\in K$ implies
$\tau\in K+B_1(0)$), while for the third inequality we used
Minkowski's inequality for integrals.
Further, we shall show that $l_\varepsilon\to l$ in $L^1_{loc}(\mathbb R)$.
Let $K\subset\subset\mathbb R$. We claim that
$\int_K|l_\varepsilon-l|\to 0 $, as $\varepsilon\to 0$. Indeed,
\begin{multline*}
\int_K |\int_\mathbb R l(t-s)\rho_\varepsilon(s)\,ds -
l(t)\cdot\int_\mathbb R\rho_\varepsilon(s)\,ds|\,dt
= \int_K |\int_\mathbb R (l(t-s)-l(t))\rho_\varepsilon(s)\,ds|\,dt\\
\stackrel{[\gamma_\varepsilon s=\tau]}{=}
\int_K |\int_\mathbb R (l(t-\frac{\tau}{\gamma_\varepsilon}) -
l(t))\rho(\tau)\,d\tau|\,dt
\leq
\int_K \int_\mathbb R |l(t-\frac{\tau}{\gamma_\varepsilon}) -
l(t)||\rho(\tau)|\,d\tau \,dt\\
=
\int_\mathbb R |\rho(\tau)| \int_K |l(t-\frac{\tau}{\gamma_\varepsilon})-l(t)|\,dt \,d\tau.
\end{multline*}
By \cite[Prop.\ 8.5]{Folland}, we have that
$\|l(\cdot-\frac{\tau}{\gamma_\varepsilon})-l\|_{L^1(K)}\to 0$, as
$\varepsilon\to 0$ and therefore the integrand converges to 0 pointwise
almost everywhere. Since it is also bounded
by $2|\rho(\tau)|\|l\|_{L^1(K)}\in L^1(\mathbb R)$,
Lebesgue's dominated convergence theorem implies the result.
\end{proof}
\section{Weak and generalized solutions of the model equations}
\label{sec:EBmodel}
We now come back to the problem
(\ref{eq:PDE})-(\ref{eq:FDE})-(IC)-(BC) or
(\ref{eq:IntegroPDE})-(IC)-(BC), and hence
need to provide assumptions which guarantee
that it can be interpreted in the form (\ref{eq:avp-L}), in order
to the apply results obtained above.
For that purpose we need to prescribe the regularity
of the functions $c$ and $b$ which appear in $Q$.
In Section \ref{ssec:Colombeau sol} we shall use these
results on the level of representatives to prove existence of
solutions in the Colombeau generalized setting.
Thus, let
$H := L^2(0,1)$ with the standard scalar product
$\dis{u}{v}=\int_0^1u(x)v(x)\,dx$ and $L^2$-norm
denoted by $\|\cdot\|_H$. Let $V$ be the Sobolev space
$H^2_0((0,1))$, which is the completion of the space of compactly
supported smooth functions $C^\infty_c((0,1))$
with respect to the norm
$\|u\|_{2} = (\sum_{k= 0}^2 \|u^{(k)}\|^2)^{1/2}$ (and inner
product $(u,v) \mapsto \sum_{k=0}^2\dis{u^{(k)}}{v^{(k)}}$).
Then $V'= H^{-2}((0,1))$, which consists of distributional
derivatives up to second order of functions in $L^2(0,1)$,
and $V\hookrightarrow H \hookrightarrow V'$ forms a Gelfand
triple. With this choice of spaces $H$ and $V$ we also have that
$E_V=L^2(0,T; H^2_0((0,1)))$ and $E_H=L^2((0,1)\times (0,T))$.
Let
\begin{equation} \label{HypothesisOn_c_and_b}
c\in L^\infty(0,1) \mbox{ and real},
\qquad b\in C([0,T];L^{\infty}(0,1)),
\end{equation}
and suppose that there exist constants $c_1 > c_0 > 0$ such that
\begin{equation} \label{addHypothesisOn_c}
0 < c_0\leq c(x)\leq c_1, \qquad \mbox{ for almost every } x.
\end{equation}
For $t\in [0,T]$ we define the bilinear forms
$a(t,\cdot,\cdot)$, $a_0(t,\cdot,\cdot)$
and $a_1(t,\cdot,\cdot)$ on $V\times V$ by
\begin{equation} \label{sesforma}
a_0(t,u,v) = \dis{c(x)\, \ensuremath{\partial}_x^2u}{\ensuremath{\partial}_x^2v}, \qquad
a_1(t,u,v) = \dis{b(x,t)\, \ensuremath{\partial}_x^2u}{v},
\qquad u,v\in V,
\end{equation}
and
\begin{equation} \label{sesform2}
a(t,u,v) = a_0(t,u,v) + a_1(t,u,v).
\end{equation}
Properties (\ref{HypothesisOn_c_and_b}),
(\ref{addHypothesisOn_c}) imply that $a_0$, $a_1$ defined as in (\ref{sesforma})
satisfy the conditions of Assumption \ref{Ass1}
(cf.\ \cite[proof of Th.\ 2.2]{HoermannOparnica09}).
The specific form of the operator $L$ is designed to achieve
equivalence of the system (\ref{eq:PDE})-(\ref{eq:FDE}) with the
equation (\ref{eq:IntegroPDE}), which we show in the sequel.
Let $\ensuremath{{\cal S}}'_+$ denote the space of Schwartz' distributions
supported in $[0,\infty)$. It is known (c.f. \cite{Oparnica02})
that for given $z\in \ensuremath{{\cal S}}'_+$ there is a unique $y\in\ensuremath{{\cal S}}'_+$
such that $D_t^\alpha z + z = \theta D_t^\alpha y + y$. Moreover,
it is given by $y=\tilde{L}z$, where $\tilde{L}$ is linear
convolution operator acting on $\ensuremath{{\cal S}}'_+$ as
\begin{equation} \label{operatorLonEs}
\tilde{L}z: =\ILT \left(\frac{1 + s^{\alpha}}{1 + \theta s^{\alpha}}\right)
\ast_t z, \qquad z\in\ensuremath{{\cal S}}'_+.
\end{equation}
The following lemma extends the operator $\tilde{L}$ to the space
$E_H$.
\begin{lemma}
Let $\tilde{L}:\ensuremath{{\cal S}}'_+ \to \ensuremath{{\cal S}}'_+$ be defined as in (\ref{operatorLonEs}).
Then $\tilde{L}$ induces a continuous operator $L=\mathop{\rm Id}\nolimits+L_\alpha$
on $E_H$, where $L_\alpha$ corresponds to convolution in time
variable with a function $l_\alpha\in L^1_{loc}([0,\infty))$.
\end{lemma}
\begin{proof}
Recall that for the Mittag-Leffler function $e_\alpha(t,\lambda)$,
defined by
$$
e_\alpha(t,\lambda) =
\sum_{k=0}^\infty \frac{(-\lambda t^\alpha)^k}{\Gamma(\alpha k+1)},
$$
we have that
$\LT (e_{\alpha}(t,\lambda))(s) =\frac{s^{\alpha-1}}{s^{\alpha} + \lambda}$,
$e_{\alpha}\in C^\infty((0,\infty))\cap C([0,\infty))$ and
$e'_{\alpha} \in C^\infty((0,\infty))\cap L^1_{loc}([0,\infty))$
(cf.\ \cite{MainardiGorenflo2000}).
Also,
$$
\ILT \left(\frac{1+s^{\alpha}}{1+\theta s^{\alpha}}\right)(t)=
\ILT\left(
1 + \frac{(1-\theta)s^\alpha}{\theta (s^\alpha + \frac{1}{\theta})}
\right)(t)
= \delta(t) + \left(\frac{1}{\theta}-1\right)
e_\alpha'\left(t,\frac{1}{\theta}\right)
=: \delta(t)+l_\alpha(t).
$$
Let $ u\in E_H$. Then
\begin{equation}
\ILT \left(\frac{1+ s^{\alpha}}{1+ \theta s^{\alpha}}\right)(\cdot)
\ast_t u (x,\cdot)
= u(x,\cdot) + \left(\frac{1}{\theta}-1\right) e_\alpha'
\ast u(x,\cdot)
\end{equation}
is an element in $L^2(0,T)$ for almost all $x$
(use Fubini's theorem, $e'_{\alpha} \in L^1(0,T)$ and
$L^1\ast L^p\subset L^p$ (cf.\ \cite{Folland})).
Extend this to a measurable function on $(0,1)\times(0,T)$,
denoted by $Lu$.
By Young's inequality we have
\begin{align*}
\|(Lu)(x,\cdot)\|_{L^2(0,T)}
& \leq \|u(x,\cdot)\|_{L^2(0,T)}
+|\frac{1}{\theta}-1|\|e_\alpha'\ast u(x,\cdot)\|_{L^2(0,T)} \\
& \leq \|u(x,\cdot)\|_{L^2(0,T)} + |\frac{1}{\theta}-1|
\|e'_\alpha\|_{L^1(0,T)} \|u(x,\cdot)\|_{L^2(0,T)},
\end{align*}
hence,
\begin{equation} \label{Lbound}
\|Lu\|_{E_H} \leq (1 + |\frac{1}{\theta}-1|
\|e'_\alpha\|_{L^1(0,T)}) \|u\|_{E_H}.
\end{equation}
Thus, $Lu\in E_H$ and $L$ is continuous on $E_H$.
\end{proof}
We may write
\begin{equation} \label{eq:opL}
Lu := (\mathop{\rm Id}\nolimits+L_\alpha)u
= l\ast_t u
= (\delta + l_\alpha)\ast_t u
\quad \mbox{ with } \quad
L_\alpha u := l_\alpha\ast_t u,
\;
l_\alpha:= (\frac{1}{\theta} - 1)
e_\alpha' (t,\frac{1}{\theta}),
\end{equation}
and therefore the model system (\ref{eq:PDE})-(\ref{eq:FDE}) is
equivalent to Equation (\ref{eq:IntegroPDE}).
\subsection{Weak solutions for $L^{\infty}$ coefficients}
\label{ssec:weak sol}
Now we are in a position to apply the abstract results from the
previous section to the original problem.
\begin{theorem} \label{th:weak sol}
Let $b$ and $c$ be as in (\ref{HypothesisOn_c_and_b}) and
(\ref{addHypothesisOn_c}). Let the bilinear form
$a(t,\cdot,\cdot)$, $t\in [0,T]$, be defined by
(\ref{sesforma}) and (\ref{sesform2}), and the operator $L$ as in
(\ref{eq:opL}). Let $f_1\in H_0^2((0,1))$, $f_2\in L^2(0,1)$ and $h\in L^2((0,1)\times(0,T))$.
Then there exists a unique $u\in L^2(0,T;H_0^2(0,1))$ satisfying
\begin{equation} \label{sol_reg}
u' = \frac{du}{dt} \in L^2(0,T;H_0^2(0,1)),
\qquad
u'' = \frac{d^2 u}{dt^2}\in L^2(0,T;H^{-2}(0,1)),
\end{equation}
and solving the initial value problem
\begin{align}
& \dis{u''(t)}{v} + a(t,u(t),v) + \dis{Lu(t)}{v} = 0,
\qquad \forall\, v\in H_0^2((0,1)),\, t \in (0,T), \label{vf}\\
& u(0)=f_1, \qquad u'(0) = f_2. \label{vf_ini}
\end{align}
(Note that, as in the abstract version, since (\ref{sol_reg})
implies $u \in C([0,T],H^2_0((0,1)))$ and $u' \in C([0,T],H^{-2}((0,1)))$ it makes sense to
evaluate $u(0)\in H^2_0((0,1))$ and
$u'(0) \in H^{-2}((0,1))$ and (\ref{vf_ini}) claims that
these equal $f_1$ and $f_2$, respectively.)
\end{theorem}
\begin{proof}
We may apply Lemma \ref{lemma:m-a} because the bilinear form $a$
and the operator $L$ satisfy Assumption \ref{Ass1} and condition
(\ref{eq:L-estimate}).
The latter is true according to (\ref{Lbound}) with
$C_L=(1 + |\frac{1}{\theta}-1| \|e'_\alpha\|_{L^1(0,T)})
=1+\|l_\alpha\|_{L^2(0,T)}$.
As noted earlier, the bilinear forms $a$, $a_0$ and $a_1$ are as
in \cite[(20) and (21)]{HoermannOparnica09}.
Moreover, it follows as in the proof of \cite[Theorem\ 2.2]{HoermannOparnica09} that
$a$ satisfies Assumption \ref{Ass1} with
\begin{equation} \label{eq:constants}
C_0:=\|c\|_{L^\infty(0,1)},
\quad
C_0':=0,
\quad
C_1:=\|b\|_{L^\infty((0,1)\times(0,T))},
\quad
\mu:=\frac{c_0}{2},
\quad
\lambda:=C_{1/2}\cdot c_0,
\end{equation}
where $C_{1/2}$ is corresoponding constant form Ehrling's lemma.
\end{proof}
We briefly recall two facts about the solution $u$ obtained in
Theorem \ref{th:weak sol} (as noted similarly in
\cite[Section 2]{HoermannOparnica09}):
(i) Since $u(.,t) \in H^2_0((0,1))$ for all
$t \in [0,T]$ and $H^2_0((0,1))$ is continuously embedded in
$\{v \in C^1([0,1]): v(0,t) = v(1,t) = 0,
\ensuremath{\partial}_x v(0,t) = \ensuremath{\partial}_x v(1,t) = 0\}$ (\cite[Corollary 6.2]{Wloka})
the solution $u$ automatically satisfies the boundary conditions.
(ii) The properties in (\ref{sol_reg}) imply that $u$ belongs to
$C^1([0,T],H^{-2}((0,1))) \cap L^2((0,T)\times(0,1))$, which is
a subspace of $\ensuremath{{\cal D}}'((0,1)\times(0,T))$. Thus in case of smooth
coefficients $b$ and $c$ we obtain a distributional solution to the
``integro-differential'' equation
$$
\ensuremath{\partial}_t^2 u + \ensuremath{\partial}^2_x(c \,\ensuremath{\partial}^2_x)u + b \,\ensuremath{\partial}_x^2 u + l *_t u = h.
$$
\subsection{Colombeau generalized solutions}
\label{ssec:Colombeau sol}
We will prove unique solvability of Equation (\ref{eq:IntegroPDE})
(or equivalently, of Equations (\ref{eq:PDE})-(\ref{eq:FDE}))
with (IC) and (BC) for $u \in \ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ when
$b,c, f_1, f_2, g$ and $h$ are Colombeau generalized functions, where $X_T:=(0,1)\times (0,T)$.
In more detail, we find a unique solution
$u\in\ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ to the equation
$$
\ensuremath{\partial}^2_tu + Q(t,x,\ensuremath{\partial}_x)u + Lu = h,
\qquad \mbox{ on } X_T
$$
with initial conditions
$$
u|_{t=0} = f_1\in \ensuremath{{\cal G}}_{H^{\infty}((0,1))},
\qquad \ensuremath{\partial}_t u|_{t=0} = f_2\in \ensuremath{{\cal G}}_{H^{\infty}((0,1))}
$$
and boundary conditions
$$
u|_{x=0} =u|_{x=1}=0,
\qquad \ensuremath{\partial}_x u|_{x=0} = \ensuremath{\partial}_x u|_{x=1}=0.
$$
Here $Q$ is a partial differential operator on $\ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ with generalized
functions as coefficients, defined by
its action on representatives in the form
$$
(u_\varepsilon)_\varepsilon \mapsto \left(
\ensuremath{\partial}_x^2(c_\varepsilon(x)\ensuremath{\partial}_x^2 u_\varepsilon))
+ b_\varepsilon(x,t)\ensuremath{\partial}_x^2 (u_\varepsilon)
\right)_\varepsilon =: (Q_\varepsilon u_\varepsilon)_\varepsilon.
$$
Furthermore, the operator $L$ corresponds to convolution on
the level of representatives with regularizations of $l$
as given in Lemma \ref{lemma:reg L}:
$$
(u_\varepsilon)_\varepsilon \mapsto \left(
l_\varepsilon\ast_t u_\varepsilon (t)
\right)_\varepsilon =: (L_\varepsilon u_\varepsilon)_\varepsilon,
$$
where $l_\varepsilon=l\ast \rho_\varepsilon$, with $\rho_\varepsilon$ introduced in
Lemma \ref{lemma:reg L}.
\begin{lemma}
(i)\ If $l\in L^2_{loc}(\mathbb R)$ and $l$ is $\ensuremath{{\cal C}}^\infty$ in
$(0,\infty)$ then $L$ is a continuous operator on $H^{\infty}(X_T)$.
Thus $(u_\varepsilon)_\varepsilon\mapsto (Lu_\varepsilon)_\varepsilon$ defines a linear map
on $\ensuremath{{\cal G}}_{H^\infty(X_T)}$.
(ii)\ If $l\in L^1_{loc}(\mathbb R)$ then $\forall\,\varepsilon\in(0,1]$ the
operator $L_\varepsilon$ is continuous on $H^{\infty}(X_T)$ and
$(u_\varepsilon)_\varepsilon\mapsto (Lu_\varepsilon)_\varepsilon$ defines a linear map
on $\ensuremath{{\cal G}}_{H^\infty(X_T)}$.
\end{lemma}
\begin{proof}
(i)\ From Lemma \ref{lemma:L_1} with $H=L^2(0,1)$ we have that
$L$ is continuous on $L^2(X_T)$ with operator norm $\|L\|_{op}\leq
T\cdot \|l\|_{L^2(0,T)}$.
Let $u\in H^\infty(X_T)$ and $Lu(x,t)=\int_0^t l(s)u(x,t-s)\,ds$.
We have to show that all derivatives of $Lu$ with respect to both
$x$ and $t$ are in $L^2(X_T)$.
\begin{itemize}
\item $\ensuremath{\partial}_x^l Lu(x,t)=\int_0^t l(s) \ensuremath{\partial}_x^l u(x,t-s)\,ds$, and
hence, $\|\ensuremath{\partial}_x^l Lu\|_{L^2(X_T)}\leq T\cdot \|l\|_{L^2(0,T)}
\|\ensuremath{\partial}_x^l u\|_{L^2(X_T)}$.
\item $\ensuremath{\partial}_t^k \ensuremath{\partial}_x^l Lu=\ensuremath{\partial}_t^k L(\ensuremath{\partial}_x^l u)$, and since the
estimates for $L(\ensuremath{\partial}_x^l u)$ are known it suffices to consider
only terms $\ensuremath{\partial}_t^k Lu$. For the first order derivative we have
$$
\ensuremath{\partial}_t Lu(x,t) = l(t)u(x,0) + \int_0^t l(s)\ensuremath{\partial}_t u(x,t-s)\, ds\\
$$
and therefore
\begin{eqnarray}s
\|\ensuremath{\partial}_t Lu\|_{L^2(X_T)} &\leq&
\|l\|_{L^2(0,T)} \|u(\cdot,0)\|_{L^2(0,1)}
+ T\cdot \|l\|_{L^2(0,T)} \|\ensuremath{\partial}_t u\|_{L^2(X_T)} \\
&\leq& \|l\|_{L^2(0,T)} (\|u\|_{H^m(X_T)}
+ T\cdot \|u\|_{H^1(X_T)}),
\end{eqnarray}s
where we have used the fact that $\mathop{\rm Tr}\nolimits:H^\infty(X_T)\to
H^\infty((0,1))$, $u\mapsto u(\cdot,0)$ is continuous, and more
precisely, $\mathop{\rm Tr}\nolimits:H^m(X_T)\to H^{m-1}((0,1))$ with estimates
$\|\ensuremath{\partial}_x^l \ensuremath{\partial}_t^k u(\cdot,0)\|_{L^2(0,1)}\leq \|u\|_{H^m(X_T)}$,
$m=m(k,l)$.
Higher order derivatives involve terms $l^{(r)}(t) \ensuremath{\partial}_t^p u(x,0),
\ldots, \int_0^t l(s) \ensuremath{\partial}_t^p u(x,t-s)\, ds$, which can be
estimated as above.
\end{itemize}
(ii)\ From Lemma \ref{lemma:reg L} it follows that $l_\varepsilon\in
L^2_{loc}(\mathbb R)$, and $\|l_\varepsilon\|_{L^2(0,T)}\leq\gamma_\varepsilon^\frac12
\cdot \|l\|_{L^1(0,T)} \|\rho\|_{L^2(0,T)}$. From Lemma
\ref{lemma:L_1} we know that $L_\varepsilon$ is continuous $X_T\to X_T$,
with $\|L_\varepsilon\|_{op}\leq T\cdot \|l_\varepsilon\|_{L^2(0,T)} \leq
T\cdot \gamma_\varepsilon^\frac12 \cdot \|l\|_{L^1(0,T)}
\|\rho\|_{L^2(0,T)}$, which is moderate. We can now proceed as in
(i) to produce estimates of $\|L_\varepsilon u_\varepsilon\|_{H^r(X_T)}$,
$\forall\,r\in\mathbb N$, always replacing $\|l\|_{L^2(0,T)}$ by
$\gamma_\varepsilon^\frac12 \cdot \|l\|_{L^1(0,T)} \|\rho\|_{L^2(0,T)}$
factors. Since $\gamma_\varepsilon\leq \varepsilon^{-N}$ it follows that
$(L_\varepsilon u_\varepsilon)_\varepsilon \in \ensuremath{{\cal E}}_{H^\infty(X_T)}$.
\end{proof}
\begin{remark} The function $l$ as defined in (\ref{eq:opL})
belongs to $L^2_{loc}(\mathbb R)$, if $\alpha>1/2$, and to
$L^1_{loc}(\mathbb R)$, if $\alpha\leq 1/2$ (which follows from
the explicit form of $e_\alpha'(t,\frac{1}{\theta})$).
This means that in case $\alpha > 1/2$ we could in fact
define the operator $L$ without regularization of $l$.
\end{remark}
As in the classical case we also have to impose a
condition to ensure compatibility of initial with boundary
values, namely (as equation in generalized numbers)
\begin{equation} \label{compatibility}
f_1(0) = f_1(1) = 0.
\end{equation}
Note that if $f_1 \in \ensuremath{{\cal G}}_{H^\infty((0,1))}$ satisfies
(\ref{compatibility}) then there is some representative
$(f_{1,\varepsilon})_\varepsilon$ of $f_1$ with the property $f_{1,\varepsilon} \in
H^2_0((0,1))$ for all $\varepsilon \in\, (0,1)$ (cf.\ the
discussion right below Equation (28) in \cite{HoermannOparnica09}).
Motivated by condition (\ref{addHypothesisOn_c}) above
on the bending stiffness we assume the following about $c$:
There exist real constants $c_1 > c_0 > 0$ such that
$c\in \ensuremath{{\cal G}}_{H^{\infty}(0,1))}$
possesses a representative $(c_{\varepsilon})_\varepsilon$ satisfying
\begin{equation} \label{eq:c-1-2}
0 < c_0 \leq c_{\varepsilon}(x) \leq c_1
\qquad \forall\, x\in (0,1), \forall\, \varepsilon \in\, (0,1].
\end{equation}
(Hence any other representative of $c$ has upper and
lower bounds of the same type.)
As in many evolution-type problems with Colombeau generalized
functions we also need the standard assumption that $b$ is of
$L^\infty$-log-type (similar to \cite{MO-89}), which means that
for some (hence any) representative $(b_\varepsilon)_\varepsilon$ of
$b$ there exist $N\in\mathbb N$ and $\varepsilon_0 \in (0,1]$ such that
\begin{equation} \label{log_type}
\|b_\varepsilon\|_{L^\infty(X_T)}
\leq N\cdot \log(\frac{1}{\varepsilon}),
\qquad 0 < \varepsilon \leq \varepsilon_0.
\end{equation}
It has been noted already in \cite[Proposition 1.5]{MO-89}
that log-type regularizations of distributions are obtained
in a straight-forward way by convolution with logarithmically
scaled mollifiers.
\begin{theorem} \label{th:main}
Let $b\in \ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ be of $L^\infty$-log-type
and $c\in \ensuremath{{\cal G}}_{H^{\infty}(0,1))}$ satisfy (\ref{eq:c-1-2}).
Let $\gamma_\varepsilon=O(\log \frac{1}{\varepsilon})$.
For any $f_{1} \in \ensuremath{{\cal G}}_{H^{\infty}((0,1))}$ satisfying
(\ref{compatibility}), $f_{2}\in \ensuremath{{\cal G}}_{H^{\infty}((0,1))}$,
$h\in \ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ and
$l\in\ensuremath{{\cal G}}_{H^{\infty}((0,1))}$,
there is a unique solution $u \in \ensuremath{{\cal G}}_{H^{\infty}(X_T)}$ to
the initial-boundary value problem
\begin{align*}
& \ensuremath{\partial}^2_tu + Q(t,x,\ensuremath{\partial}_x)u + Lu = h, \\
& u|_{t=0} = f_1, \quad \ensuremath{\partial}_t u|_{t=0} = f_2, \\
& u|_{x=0} =u|_{x=1}=0, \quad
\ensuremath{\partial}_x u|_{x=0} = \ensuremath{\partial}_x u|_{x=1}=0.
\end{align*}
\end{theorem}
\begin{proof} Thanks to the preparations a considerable part of
the proof may be adapted from the corresponding proof in
\cite[Theorem 3.1]{HoermannOparnica09}.
Therefore we give details only for the first part and sketch
the procedure from there on.
{\bf Existence:} \enspace \enspace
We choose representatives $(b_\varepsilon)_\varepsilon$ of $b$ and
$(c_{\varepsilon})_\varepsilon$ of $c$ satisfying (\ref{eq:c-1-2})
and (\ref{log_type}). Further let
$(f_{1\varepsilon})_\varepsilon$, $(f_{2\varepsilon})_\varepsilon$, $(l_{\varepsilon})_\varepsilon$,
and $(h_{\varepsilon})_\varepsilon$ be
representatives of $f_1$,$f_2$, $l$, and $g$, respectively,
where we may assume $f_{1,\varepsilon} \in H^2_0((0,1))$ for all
$\varepsilon \in \,(0,1)$ (cf.\ (\ref{compatibility})).
For every $\varepsilon\in (0,1]$ Theorem
\ref{th:weak sol}
provides us with a unique
solution $u_{\varepsilon}\in H^1((0,T),H^2_0((0,1))) \cap
H^2((0,T),H^{-2}((0,1)))$ to
\begin{align}
P_{\varepsilon}u_{\varepsilon} &: = \ensuremath{\partial}^2_t u_{\varepsilon} +
Q_{\varepsilon}(t,x,\ensuremath{\partial}_x)u_{\varepsilon} + L_\varepsilon u_\varepsilon
= h_{\varepsilon}
\qquad \mbox{ on } X_T, \label{Peps}\\
& u_{\varepsilon}|_{t=0} = f_{1\varepsilon},
\qquad \ensuremath{\partial}_tu_{\varepsilon}|_{t=0} = f_{2\varepsilon}. \nonumber
\end{align}
In particular, we have $u_{\varepsilon}\in C^1([0,T],H^{-2}((0,1))) \cap
C([0,T],H^2_0((0,1)))$.
Proposition \ref{prop:EnergyEstimates} implies the energy estimate
\begin{equation}\label{eq:EEeps1}
\|u_{\varepsilon}(t)\|_{H^2}^2 + \|u_{\varepsilon}'(t)\|_{L^2}^2 \leq
\begin{itemize}g( D_T^\varepsilon\, \|f_{1\varepsilon}\|_{H^2}^2 + \|f_{2,\varepsilon}\|_{L^2}^2
+ \int_0^t \|h_{\varepsilon}(\tau)\|_{L^2}^2\,d\tau \begin{itemize}g)
\cdot \exp(t\, F_T^\varepsilon),
\end{equation}
where with some $N$ we have as $\varepsilon \to 0$
\begin{align}
D_T^\varepsilon &= (\|c_\varepsilon\|_{L^{\infty}} +
\lambda (1+T)) /\min(\mu,1)
= O(\|c_\varepsilon\|_{L^{\infty}}) = O(1)
\label{const_D_eps} \\
F_T^\varepsilon &= \frac{\max \{\|b_\varepsilon\|_{L^{\infty}}+C_{L,\varepsilon},
\|b_\varepsilon\|_{L^{\infty}} +2+\lambda(1+T) \}}{\min(\mu,1)}
= O(C_{L,\varepsilon}+\|b_\varepsilon\|_{L^{\infty}})
= O(\log(\varepsilon^{-N})), \label{const_F_eps}
\end{align}
since $\mu$ and $\lambda$ are independent of
$\varepsilon$, and $C_{L,\varepsilon} = O(\log \frac{1}{\varepsilon})$
(cf.\ (\ref{eq:constants})).
By moderateness of the initial data
$f_{1\varepsilon}$, $f_{2\varepsilon}$ and of the right-hand side $h_{\varepsilon}$
the inequality (\ref{eq:EEeps1}) thus implies that there
exists $M$ such that for small $\varepsilon > 0$ we have
\begin{equation}\label{EEeps}
\|u_{\varepsilon}\|^2_{L^2(X_T)} +
\|\ensuremath{\partial}_x u_{\varepsilon}\|^2_{L^2(X_T)}
+ \| \ensuremath{\partial}_x^2 u_{\varepsilon}\|^2_{L^2(X_T)}
+ \|\ensuremath{\partial}_t u_{\varepsilon}\|^2_{L^2(X_T)} =
O (\varepsilon^{-M}),
\qquad \varepsilon\to 0.
\end{equation}
From here on the remaining chain of arguments proceeds along the lines of the
proof in \cite[Theorem 3.1]{HoermannOparnica09}.
We only indicate a few key points requiring certain adaptions.
The goal is to prove the following properties:
\begin{itemize}
\item[1.)] For every $\varepsilon \in (0,1]$ we have $u_{\varepsilon}\in H^{\infty}(X_T)
\subseteq C^{\infty}(\overline{X_T})$.
\item[2.)] Moderateness, i.e.\ for all $l,k\in\mathbb N$ there is some
$M\in\mathbb N$ such that for small $\varepsilon > 0$
\begin{equation} \tag{$T_{l,k}$} \label{T_lk}
\|\ensuremath{\partial}^l_t\ensuremath{\partial}^k_xu_{\varepsilon}\|_{L^2(X_T)} = O(\varepsilon^{-M}).
\end{equation}
Note that (\ref{EEeps}) already yields (\ref{T_lk}) for
$(l,k) \in \{ (0,0),(1,0),(0,1),(0,2)\}$.
\end{itemize}
\noindent\emph{Proof of 1.)} Differentiating (\ref{Peps}) (considered
as an equation in $\ensuremath{{\cal D}}'((0,1)\times(0,T))$) with respect to $t$
we obtain
$$
P_{\varepsilon}(\ensuremath{\partial}_tu_{\varepsilon}) = \ensuremath{\partial}_th_{\varepsilon}
- \ensuremath{\partial}_t b_\varepsilon(x,t)\ensuremath{\partial}_x^2u_{\varepsilon}
- l_\varepsilon(t) f_{1,\varepsilon}
=: \tilde{h}_\varepsilon,
$$
where we used $\ensuremath{\partial}_t(L_\varepsilon u_\varepsilon)=
L_\varepsilon (\ensuremath{\partial}_t u_\varepsilon) + l_\varepsilon(t)u_\varepsilon(0)$.
We have $\tilde{h}_\varepsilon \in H^1((0,T),L^2(0,1))$
since
$\ensuremath{\partial}_t h_{\varepsilon}\in H^{\infty}(X_T)$,
$l_\varepsilon\in H^{\infty}((0,T))$,
$f_{1,\varepsilon}\in H^{\infty}((0,1))$,
$\ensuremath{\partial}_t b_\varepsilon(x,t)\in H^{\infty}(X_T) \subset W^{\infty,\infty}(X_T)$
and $\ensuremath{\partial}^2_xu_{\varepsilon}\in H^1((0,T),L^2(0,1))$.
Furthermore, since $Q_\varepsilon$ depends smoothly on $t$ as a
differential operator in $x$ and $u_\varepsilon(0) = f_{1,\varepsilon} \in
H^{\infty}((0,1))$ we have
\begin{align*}
(\ensuremath{\partial}_t u_{\varepsilon})(\cdot,0) & = f_{2,\varepsilon} =:
\tilde{f}_{1,\varepsilon} \in H^{\infty}((0,1)),\\
(\ensuremath{\partial}_t(\ensuremath{\partial}_tu_{\varepsilon}))(\cdot,0) & =
h_{\varepsilon}(\cdot,0) - Q_{\varepsilon}(u_{\varepsilon}(\cdot,0))-L_\varepsilon u_\varepsilon (\cdot,0)=
h_{\varepsilon}(\cdot,0) - (Q_{\varepsilon}+L_\varepsilon) f_{1,\varepsilon} :=
\tilde{f}_{2,\varepsilon} \in H^{\infty}((0,1)).
\end{align*}
Hence $\ensuremath{\partial}_tu_{\varepsilon}$ satisfies an initial value problem for the
partial differential operator $P_\varepsilon$ as in (\ref{Peps}) with
initial data
$\tilde{f}_{1,\varepsilon}$, $\tilde{f}_{2,\varepsilon}$ and right-hand side
$\tilde{h}_{\varepsilon}$ instead. However, this time we have to use $V = H^2((0,1))$
(replacing $H^2_0((0,1))$) and $H = L^2(0,1)$ in the abstract setting,
which still can serve to define a Gelfand triple
$V\hookrightarrow H \hookrightarrow V'$ (cf.\ \cite[Theorem 17.4(b)]{Wloka})
and thus allows for application of Lemma \ref{lemma:m-a} and
the energy estimate (\ref{eq:EE}) (with precisely the same constants).
Therefore we obtain
$\ensuremath{\partial}_tu_{\varepsilon}\in H^1([0,T],H^2((0,1)))$,
i.e.\ $u_{\varepsilon}\in H^2((0,T),H^2((0,1)))$ and from the
variants of (\ref{eq:EEeps1}) (with exactly the same constants
$D_T^\varepsilon$ and $F_T^\varepsilon$) and (\ref{EEeps})
with $\ensuremath{\partial}_t u_\varepsilon$ in place of $u_\varepsilon$ that for some $M$
we have
\begin{equation}
\| \ensuremath{\partial}_t u_{\varepsilon}\|^2_{L^2(X_T)} +
\|\ensuremath{\partial}_x \ensuremath{\partial}_t u_{\varepsilon}\|^2_{L^2(X_T)}
+ \| \ensuremath{\partial}_x^2 \ensuremath{\partial}_t u_{\varepsilon}\|^2_{L^2(X_T)}
+ \|\ensuremath{\partial}_t^2 u_{\varepsilon}\|^2_{L^2(X_T)} =
O (\varepsilon^{-M})\quad (\varepsilon\to 0).
\end{equation}
Thus we have proved (\ref{T_lk}) with $(l,k) = (2,0), (1,1), (1,2)$ in
addition to those obtained from (\ref{EEeps}) directly.
The remaining part of the proof of property 1.) requires the
exact same kind of adaptions in the corresponding parts in
Step 1 of the proof of \cite[Th.\ 3.1]{HoermannOparnica09}
and we skip its details here. In particular, along the way
one also obtains that
\begin{center}
($T_{l,k}$) holds for
derivatives of arbitrary $l$ and $k \leq 2$.
\end{center}
\noindent\emph{Proof of 2.)} From the estimates achieved in proving 1.)
and equation (\ref{Peps}) we deduce that
$$
k_\varepsilon := \ensuremath{\partial}_x^2(c_\varepsilon\, \ensuremath{\partial}_x^2 u_\varepsilon) = h_\varepsilon -
b_\varepsilon\, \ensuremath{\partial}_x^2 u_\varepsilon - \ensuremath{\partial}_t^2 u_\varepsilon - L_\varepsilon u_\varepsilon
$$
satisfies for all $l \in \mathbb N$ with some $N_l$ an estimate
\begin{equation}\label{mod22}
\| \ensuremath{\partial}_t^l k_\varepsilon \|_{L^2(X_T)}
= O(\varepsilon^{-N_l}) \qquad (\varepsilon\to 0).
\end{equation}
Here we are again in the same situation as in Step 2 of
the proof of \cite[Theorem 3.1]{HoermannOparnica09},
where now $k_\varepsilon$ plays the role of $h_\varepsilon$ there.
Skipping again details of completely analogous arguments
we arrive at the conclusion that the class of $(u_\varepsilon)_\varepsilon$
defines a solution to the initial value problem.
Moreover, we have by construction that $u_\varepsilon(t) \in H^2_0((0,1))$
for all $t \in [0,T]$, hence $u(0,t) = u(1,t) = 0$ and
$\ensuremath{\partial}_x u(0,t) = \ensuremath{\partial}_x u(1,t) = 0$ and thus $u$ also satisfies
the boundary conditions.
{\bf Uniqueness:}\enspace \enspace If $u =
[(u_\varepsilon)_\varepsilon]$ satisfies initial-boundary value problem with
zero initial values and right-hand side, then we have for all
$q\geq 0$
$$
\|f_{1,\varepsilon}\| = O(\varepsilon^q), \quad
\|f_{2,\varepsilon}\| = O(\varepsilon^q), \quad
\|h_{\varepsilon}\|_{L^2(X_T)} = O(\varepsilon^q) \qquad \text{as } \varepsilon\to 0.
$$
Therefore the energy estimate (\ref{eq:EEeps1}) in combination with
(\ref{const_D_eps})-(\ref{const_F_eps}) imply for all $q \geq 0$ an
estimate
$$
\|u_\varepsilon\|_{L^2(X_T)} = O(\varepsilon^q) \quad (\varepsilon \to 0),
$$
from which we conclude that $(u_\varepsilon)_\varepsilon \in \ensuremath{{\cal N}}_{H^{\infty}(X_T)}$, i.e., $u = 0$.
\end{proof}
\end{document} |
\begin{document}
\title{Semigroups in Stable Structures}
\author{Yatir Halevi\footnote{The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291111.}}
\date{}
\maketitle
\begin{abstract}
Assume $G$ is a definable group in a stable structure $M$. Newelski showed that the semigroup $S_G(M)$ of complete types concentrated on $G$ is an inverse limit of the $\infty$-definable (in $M^{eq}$) semigroups $S_{G,\Delta}(M)$. He also shows that it is strongly $\pi$-regular: for every $p\in S_{G,\Delta}(M)$ there exists $n\in\mathbb{N}$ such that $p^n$ is in a subgroup of $S_{G,\Delta}(M)$. We show that $S_{G,\Delta}(M)$ is in fact an intersection of definable semigroups, so $S_G(M)$ is an inverse limit of definable semigroups and that the latter property is enjoyed by all $\infty$-definable semigroups in stable structures.
\end{abstract}
\section{Introduction}
A Semigroup is a set together with an associative binary operation. Although the study of semigroups stems in the start of the 20th century not much attention has been given to semigroups in stable structures. One of the only facts known about them is
\begin{proposition*}{\cite{unidim}}
A stable semigroup with left and right cancellation, or with left cancellation and right identity, is a group.
\end{proposition*}
Recently $\infty$-definable semigroups in stable structures made an appearance in a paper by Newelski \cite{Newelski-stable-groups}:
Let $G$ be a definable group inside a stable structure $M$. Define $S_G(M)$ to be all the types of $S(M)$ which are concentrated on $G$. $S_G(M)$ may be given a structure of a semigroup by defining for $p,q\in S_G(M)$:
\[p\cdot q=tp(a\cdot b/M),\]
where $a\models p, b\models q$ and $a\forkindep[M]b$.
Newelski gives an interpretation of $S_{G,\Delta}(M)$ (where $\Delta$ is a finite set of invariant formulae) as an $\infty$-definable set in $M^{eq}$ and thus $S_G(M)$ may be interpreted as an inverse limit of $\infty$-definable semigroups in $M^{eq}$.
As a result he shows that for every local type $p\in S_{G,\Delta}(M)$ there exists an $n\in\mathbb{N}$ such that $p^n$ is in a subgroup of $S_{G,\Delta}(M)$. In fact he shows that $p^n$ is equal to a translate of a $\Delta$-generic of a $\Delta$-definable connected subgroup of $G(M)$.
\begin{definition*}
A semigroup $S$ is called \emph{strongly $\pi$-regular} or an \emph{epigroup} if for all $a\in S$ there exists $n\in\mathbb{N}$ such that $a^n$ is in a subgroup of $S$.
\end{definition*}
\begin{question}
Is this property enjoyed by all $\infty$-definable semigroups in stable structures?
\end{question}
Since we're dealing with $\infty$-definable semigroups, remembering that every $\infty$-definable group in a stable structure is an intersection of definable groups, an analogous questions arises:
\begin{question}
Is every $\infty$-definable semigroup in a stable structure an intersection of definable ones? Is $S_{G,\Delta}(M)$ an intersection of definable semigroups?
\end{question}
In this paper we answer both theses questions.
It is a classical result about affine algebraic semigroups that they are strongly $\pi$-regular. Recently Brion and Renner \cite{RenBrio} proved that this is true for all Algebraic Semigroups. In fact, we'll show that
\begin{propff}{T:inf-def-is-spr}
Let $S$ be an $\infty$-definable semigroup inside a stable structure. Then $S$ is strongly $\pi$-regular.
\end{propff}
At least in the definable case, this is a direct consequence of stability, the general case is not harder but a bit more technical.
One can ask if what happens in $S_{G,\Delta}(M)$ is true in general $\infty$-definable semigroups. That is, is every element a power away from a translation of an idempotent. However, this is already is not true in $M_2(\mathbb{C})$.
As for the second question, in Section \ref{sec:S_G(M)} we show that $S_{G,\Delta}(M)$ is an intersection of definable semigroups. In fact,
\begin{theoremff}{T:invlim}
$S_G(M)$ is an inverse limit of definable semigroups in $M^{eq}$.
\end{theoremff}
Unfortunately not all $\infty$-definable semigroups are an intersection of definable ones.
Milliet showed that every $\infty$-definable semigroup inside a small structure is an intersection of definable semigroups \cite{on_enveloping}. In particular this is true for $\omega$-stable structures, and so for instance in $ACF$. Already in the superstable case this is not true in general, see \Cref{E:counter-exam}.
However there are some classes of semigroups in which this does hold. We recall some basic definitions from semigroup theory we'll need. See \Cref{ss:semigroups} for more information.
\begin{definition*}
\begin{enumerate}
\item An element $e\in S$ in a semigroup $S$ is an \emph{idempotent} if $e^2=e$.
\item A semigroup $S$ is called an \emph{inverse semigroup} if for every $a\in S$ there exists a unique $a^{-1}\in S$ such that
\[aa^{-1}a=a,\quad a^{-1}aa^{-1}=a^{-1}.\]
\item A \emph{Clifford semigroup} is an inverse semigroup in which the idempotents are central. A \emph{surjective Clifford monoid} is a Clifford monoid in which for every $a\in S$ there exists $g\in G$ and idempotent $e$ such that $a=ge$, where $G$ is the unit group of $S$.
\end{enumerate}
\end{definition*}
These kinds of semigroups do arise in the context of $S_G(M)$.
It is probably folklore, but one may show (see \Cref{ss:S_G inverse}) that if $G$ is $1$-based then $S_G(M)$ is an inverse monoid. In \Cref{ss:S_G inverse} we give a condition on $G$ for $S_G(M)$ to be Clifford.
\begin{theoremff}{T:Cliff_with_surj}
Let $S$ be an $\infty$-definable surjective Clifford monoid in a stable structure. Then $S$ is contained in a definable monoid, extending the multiplication on $S$. This monoid is also a surjective Clifford monoid.
\end{theoremff}
As a result from the proof, every such monoid is an intersection of definable ones.
In the process of proving the above Theorem we show two results which might be interesting in their own right.
Since $\infty$-definable semigroups in stable structures are s$\pi$r, one may define a partial order on them given by
\[a\leq b \Leftrightarrow a=be=fb \text{ for some } e,f\in E(S^1),\]
where $S^1$ is $S\cup \{1\}$ where we define $1$ to be the identity element.
If for every $a,b\in S$, $a\cdot b\leq a,b$ one may show that there exists $n\in\mathbb{N}$ such that every product of $n+1$ elements is already a product of $n$ of them (\Cref{P:in negative_order_n+1_is_n}). As a result any such semigroup is an intersection of definable ones. In particular,
\begin{propff}{P:idemp_is_inside_definable}
Let $E$ be an $\infty$-definable commutative idempotent semigroup inside a stable structure, then $E$ is contained in a definable commutative idempotent semigroup. Furthermore, it is an intersection of definable ones.
\end{propff}
\section{Preliminaries}\label{S:Preliminaries}
\subsection{Notations}
We fix some notations.
We'll usually not distinguish between singletons and sequences thus we may write $a\in M$ and actually mean $a=(a_1,\dots,a_n)\in M^n$, unless a distinguishment is necessary. $A,B,C,\dots$ will denote parameter sets and $M,N,\dots$ will denote models. When talking specifically about semigroups, monoids and groups (either definable, $\infty$-definable or models) we'll denote them by $S$, $M$ and $G$, respectively. We use juxtaposition $ab$ for concatenation of sequences, or $AB$ for $A\cup B$ if dealing with sets. That being said, since we will be dealing with semigroups, when there is a chance of confusion we'll try to differentiate between the concatenation $ab$ and the semigroup multiplication $ab$ by denoting the latter by $a\cdot b$.
\subsection{Semigroups}\label{ss:semigroups}
Clifford and Preston \cite{Clifford1,Clifford2} is still a very good reference for the theory of semigroups, but Higgins \cite{Higgins} and Howie \cite{Howie} are much more recent sources.
A set $S$ with an associative binary operation is called a \emph{semigroup}.\\
An element $e\in S$ is an \emph{idempotent} if $e^2=e$. We'll denote by $E(S)$ the subset of all idempotents of $S$.\\
By a subgroup of $S$ we mean a subsemigroup $G\subseteq S$ such that there exists an idempotent $e\in G$ such that $(G,\cdot)$ is a group with neutral element $e$.\\
$S$ is \emph{strongly $\pi$-regular} (s$\pi$r) if for each $a\in S$ there exits $n>0$ such that $a^n$ lies in a subgroup of $S$.
\begin{remark}
These type of semigroups are also known as \emph{epigroups} and their elements as \emph{group-bound}.
\end{remark}
A semigroup with an identity element is called a \emph{monoid}. Notice that any semigroup can be extended to a monoid by artificially adding an identity element. We'll denote it by $S^1$. If $S$ is a monoid we'll denote by $G(S)$ its subgroup of invertible elements.\\
Given two semigroups $S,S^\prime$, a \emph{homomorphism of semigroups} is map \[\varphi :S\to S^\prime\] such that $\varphi(xy)=\varphi(x)\varphi(y)$ for all $x,y\in S$. If $S,S^\prime$ are monoids, then we say $\varphi$ is a \emph{homomorphism of monoids} if in addition $\varphi(1_S)=1_{S^\prime}$.\\
\begin{definition}
The \emph{natural partial order} on $E(S)$ is defined by \[e\leq f \Leftrightarrow ef=fe=e.\]
\end{definition}
\begin{proposition}\cite[Section 1.7]{Clifford1}\label{P:Max.subgroups}
For every $e,f\in E(S)$ we have the following:
\begin{enumerate}
\item $eSe$ is a subsemigroup of $S$. In fact, it is a monoid with identity element $e$;
\item $eSe\subseteq fSf \Leftrightarrow e\leq f;$
\item Every maximal subgroup of $S$ is of the form $G(eSe)$ (the unit group of $eSe$) for $e\in E(S)$;
\item If $e\neq f$ then $G(eSe)\cap G(fSf)=\emptyset$.
\end{enumerate}
\end{proposition}
There are various ways to extend the partial order on the idempotents to a partial order on the entire semigroup. See \cite[Section 1.4]{Higgins} for a discussion about them. We'll use the \emph{natural partial order} on $S$. It has various equivalent definitions, we present the one given in \cite[Proposition 1.4.3]{Higgins}.
\begin{definition}\label{D:general_def_of_order}
The relation
\[a\leq b \Leftrightarrow a=xb=by,xa=a \text{ for some } x,y\in S^1\]
is called the \emph{natural partial order} on $S$.
\end{definition}
Notice that this extends the partial order on $E(S)$. If $S$ is s$\pi$r this partial order takes a more elegant form:
\begin{proposition}\cite[Corollary 1.4.6]{Higgins}
On s$\pi$r semigroups there is a natural partial order extending the order on $E(S)$:
\[a\leq b \Leftrightarrow a=be=fb \text{ for some } e,f\in E(S^1).\]
\end{proposition}
\subsubsection{Clifford and Inverse Semigroups}\label{subsec:Clifford}
\begin{definition}
A semigroup $S$ is called $regular$ if for every $a\in S$ there exists at least one element $b\in S$ such that
\[aba=a,\quad bab=b.\]
Such an element $b$ is a called a \emph{pseudo-inverse} of $a$
\end{definition}
\begin{definition}
A semigroup $S$ is called an \emph{inverse semigroup} if for every $a\in S$ there exists a unique $a^{-1}\in S$ such that
\[aa^{-1}a=a,\quad a^{-1}aa^{-1}=a^{-1}.\]
\end{definition}
Basic facts about inverse semigroups:
\begin{proposition}\cite[Section V.1, Theorem 1.2 and Proposition 1.4]{Howie}
Let $S$ be an inverse semigroup.
\begin{enumerate}
\item For every $a,b\in S$, $\left( a^{-1}\right)^{-1}=a$ and $\left( ab\right)^{-1}=b^{-1}a^{-1}.$
\item For every $a\in S$, $aa^{-1}$ and $a^{-1}a$ are idempotents.
\item The idempotents commute. Thus $E(S)$ is a commutative subsemigroup, and hence a semilattice.
\end{enumerate}
\end{proposition}
The basic example for inverse semigroups is the set, $\mathscr{I}(X)$, of partial one-to-one mappings for a set $X$, that means that the domain is a (possibly empty) subset of $X$. The composition of two "incompatible" mappings will be the empty mapping. The first surprising fact is that this is in fact an \emph{inverse} semigroup, but one can say even more (a generalization of Cayley's theorem for groups):
\begin{theorem}[The Vagner-Preston Representation Theorem]\cite[Section V.1, Theorem 1.10]{Howie}
If $S$ is an inverse semigroup then there exists a set $X$ and a monomorphism $\phi: S\to \mathscr{I}(X)$.
\end{theorem}
If $S$ is an inverse semigroup, the partial order on $S$ gets the following form: $a\leq b$ if there exists $e\in E(S)$ such that $a=eb$.
\begin{proposition}
\begin{enumerate}
\item $\leq$ is a partial order relation.
\item If $a,b,c\in S$ such that $a\leq b$ then $ac\leq bc$ and $ca\leq cb$. Futhermore, $a^{-1}\leq b^{-1}$.
\end{enumerate}
\end{proposition}
\begin{definition}
A \emph{Clifford semigroup} is an inverse semigroup in which the idempotents are central.
\end{definition}
\begin{remark}
Different sources give different, but equivalent, definitions of a Clifford semigroup. For instance Howie defines a Clifford semigroup to be a regular semigroup $S$ in which the idempotents are central \cite[Section IV.2]{Howie}. One may show that $S$ is an inverse semigroup if and only if it is regular and the idempotents commute \cite[Section V.1, Theorem 1.2]{Howie}, so the definitions coincide.
\end{remark}
The following is well known, but we'll add a proof instead of adding another source.
\begin{proposition}
$S$ is a Clifford semigroup if and only if it is an inverse semigroup and $aa^{-1}=a^{-1}a$ for all $a\in S$.
\end{proposition}
\begin{proof}
Assume $S$ is a Clifford semigroup and let $a\in S$. Since $aa^{-1}$ and $a^{-1}a$ are idempotents and central,
\[aa^{-1}=a(a^{-1}a)a^{-1}=(a^{-1}a)aa^{-1}=a^{-1}a(aa^{-1})=a^{-1}(aa^{-1})a=a^{-1}a.\]
Conversely, we must show that the idempotents are central. For $a\in S$ and $e\in E(S)$, we'll show that $ea=(ea)(a^{-1}e)(ea)=(ea)(ea^{-1})(ea)$ and thus by the uniqueness of the pseudo-inverses, $ea=ae$. By our assumption
\[(ea)(ea^{-1})(ea)=eae(a^{-1}e)(ea)=eae(ea)(a^{-1}e)=eaeaa^{-1}e\]
Again, $(ea)(a^{-1}e)=(a^{-1}e)(ea)$ so
\[=eaa^{-1}eea=eaa^{-1}ea,\]
and by the commutativity of the idempotents ($e$ and $aa^{-1}$),
\[=eaa^{-1}a=ea.\]
\end{proof}
\begin{definition}\cite[Chapter IV]{Howie}
A semigroup $S$ is said to be a \emph{strong semilattice of semigroups} if there exists a semilattice $Y$, disjoint subsemigroups $\{S_{\alpha}:\alpha\in Y\}$ and homomorphisms $\{\phi_{\alpha,\beta}:S_\alpha\to S_\beta:\alpha,\beta\in Y, \alpha\geq \beta\}$ such that
\begin{enumerate}
\item $S=\bigcup_\alpha S_\alpha$.
\item $\phi_{\alpha,\alpha}$ is the identity.
\item For every $\alpha\geq \beta\geq\gamma$ in $Y$, $\phi_{\beta,\gamma}\phi_{\alpha,\beta}=\phi_{\alpha,\gamma}$.
\end{enumerate}
\end{definition}
\begin{theorem}\cite[Section IV.2, Theorem 2.1]{Howie}
$S$ is a Clifford semigroup if and only if it is a strong semilattice of groups. The semilattice is $E(S)$, the disjoint groups are \[\{G_e=G(eSe): e\in E(S)\}\] the maximal subgroups of $S$ and the homomorphism $\phi_{e,ef}$ is given by multiplication by $f$.
\end{theorem}
\section{$\infty$-definable Semigroups and Monoids}\label{S:big-wedge-defin}
Let $S$ be an $\infty$-definable semigroup in a stable structure. Assume that $S$ is defined by \[\bigwedge_i \varphi_i(x).\]
\begin{remark}
We assume that $S$ is defined over $\emptyset$ just for notational convenience. Moreover we assume that the $\varphi_i$s are closed under finite conjunctions.
\end{remark}
\subsection{Strongly $\pi$-regular}
Our goal is to prove that an $\infty$-definable semigroup inside a stable structure is s$\pi$r. To better understand what's going on we start with an easier case:
\begin{definition}
A \emph{stable semigroup} is a stable structure $S$ such that there is a definable binary function $\cdot$ which makes $(S,\cdot)$ into a semigroup.
\end{definition}
The following was already noticed in \cite{LoseyHans} for semigroups with chain conditions, but we give it in a "stable semigroup" setting.
\begin{proposition}\label{P:idem}
Any stable semigroup has an idempotent.
\end{proposition}
\begin{proof}
Let $a\in S$ and let \[\theta(x,y)= \exists u (u\cdot a=a\cdot u\wedge u\cdot x=y).\] Obviously, $S\models \theta(a^{3^m},a^{3^n})$ for $m<n$. $S$ is stable hence $\theta$ doesn't have the order property. Thus there exists $m<n$ such that $S\models \theta(a^{3^n},a^{3^m})$. Let $C\in S$ be such that $C\cdot a^{3^n}=a^{3^m}$ and commutes with $a$.
Since $3^n> 2\cdot3^m$ then multiplying by $a^{3^n-2\cdot 3^m}$ yields $Ca^{2(3^n-3^m)}=a^{3^n-3^m}$. Notice that since $C$ commutes with $a$, $Ca^{3^n-3^m}$ is an idempotent.
\end{proof}
\begin{proposition}\label{P:s pi r}
Any stable semigroup is $s\pi r$.
\end{proposition}
\begin{proof}
Let $a\in S$. From the proof of \Cref{P:idem} there exists $C\in S$ that commutes with $a$ and $n>0$ such that $Ca^{2n}=a^n$. Set $e:=Ca^n$. Indeed, $a^n=e\cdot a^n\cdot e$ and $a^n\cdot eCe=e$.
\end{proof}
\begin{remark}
Given $a\in S$, there exists a unique idempotent $e=e_a\in S$ such that $a^n$ belongs to the unit group of $eSe$ for some $n>0$. Indeed, for two idempotents $e\neq f$ the unit groups of $eSe$ and $fSf$ are disjoint (Proposition \ref{P:Max.subgroups}).
\end{remark}
Furthermore, we have
\begin{lemma}\cite{Munn}\label{L:higher-power-in-max-subgrp}
Let $S$ be a semigroup and $x\in S$. If for some $n$, $x^n$ lies in a subgroup of $S$ with identity $e$ then $x^m$ lies in the unit group of $eSe$ for all $m\geq n$.
\end{lemma}
\begin{corollary}\label{C:universal n}
There exists $n>0$ (depending only on $S$) such that for all $a\in S$, $a^n$ belongs to the unit group of $e_aSe_a$.
\end{corollary}
\begin{proof}
Let $\phi_i(x)$ be the formula '$x^i\in \text{ the unit group of } e_xSe_x$'. $\bigcup_i[\phi(x)]=S_1(S)$, since every elementary extension of $S$ is also stable and hence $s\pi r$. By compactness there exist $n_1,\dots,n_k>0$ such that $S_1(S)=[\phi_{n_1}\vee\dots\vee \phi_{n_k}]$. $n=n_1\cdots n_k$ is our desired integer.
\end{proof}
We return to the general case of $S$ being an $\infty$-definable semigroup inside a stable structure. The following is an easy consequence of stability:
\begin{proposition} \label{P:chain of idempotents}
Every chain of idempotents in $S$, with respect to the partial order on them, is finite and uniformly bounded.
\end{proposition}
Our goal is to show that for every $a\in S$ there exists an idempotent $e\in S$ and $n\in\mathbb{N}$ such that $a^n$ is in the unit group of $eSe$.
We'll want to assume that $S$ is a conjunction of countably many formulae. For that we'll need to make some observations, the following is well known but we add a proof for completion,
\begin{lemma}
Let $S$ be an $\infty$-definable semigroup. Then there exist $\infty$-definable semigroups $H_i$ such that each $H_i$ is defined by at most a countable set of formulae, and $S=\bigcap H_i$.
\end{lemma}
\begin{proof}
Let $S=\bigwedge_{i\in I} \varphi_i$ and assume that the $\varphi_i$s are closed under finite conjunctions. By compactness we may assume that for all $i$ and $x,y,z$
$$\varphi_i(x)\wedge\varphi_i(y)\wedge\varphi_i(z)\to (xy)z=x(yz).$$
Let $i^0\in I$. By compactness there exists $i^0_1\in I$ such that for all $x,y$:
$$\varphi_{i^0_1}(x)\wedge \varphi_{i^0_1}(y)\to \varphi_{i^0}(xy).$$
Thus construct a sequence $i^0,i^0_1,i^0_2,\dots$ and define
$$H_{i^0}=\bigwedge_j \varphi_{i^0_j}.$$
This is indeed a semigroup and
$$S=\bigcap_{i\in I} H_i.$$
\end{proof}
The following is also well known,
\begin{proposition}{\cite{unidim}}
An $\infty$-definable semigroup in a stable structure with left and right cancellation, or with left cancellation and right identity, is a group.
\end{proposition}
As a consequence,
\begin{lemma}\label{L:max_subgrps_are_inf_definable}
Let $S$ be an $\infty$-definable semigroup and $G_e\subseteq S$ a maximal subgroup (with idempotent $e\in E(S)$). $G_e$ is relatively definable in $S$.
\end{lemma}
\begin{proof}
Let $S=\bigwedge_i \varphi_i(x)$. By compactness, there exists a definable set $S\subseteq S_0$ such that for all $x,y,z\in S_0$ \[x(yz)=(xy)z.\]
Let $G_e(x)$ be
\[\bigwedge_i\varphi_i(x)\wedge (xe=ex=x)\wedge \bigwedge_i(\exists y\in S_0)(\varphi_i(y)\wedge ye=ey=y\wedge yx=xy=e).\]
This $\infty$-formula defines the maximal subgroup $G_e$. Indeed if $a\models G_e(x)$ and $b,b^\prime\in S_0$ are such that
\[\varphi_i(b)\wedge be=eb=b\wedge ba=ab=e\]
and
\[\varphi_j(b^\prime)\wedge b^\prime e=eb^\prime=b^\prime\wedge b^\prime a=ab^\prime=e,\]
then
\[b^\prime=b^\prime e=b^\prime(ab)=(b^\prime a)b=eb=b.\]
Hence there exists an inverse of $a$ in $S$.
Let $G_e\subseteq G_0$ be a definable group containing $G_e$ (see \cite{unidim}). $G_0\cap S$ is an $\infty$-definable subsemigroup of $S$ with cancellation, hence a subgroup. It is thus contained in the maximal subgroup $G_e$ and so equal to it.
\end{proof}
\begin{lemma}
Let $S$ be an $\infty$-definable semigroup and $S\subseteq S_1$ an $\infty$-definable semigroup containing it. If $S_1$ is s$\pi$r then so is $S$.
\end{lemma}
\begin{proof}
Let $a\in S$ and let $a^n\in G_e\subseteq S_1$ where $G_e$ is a maximal subgroup of $S_1$. Thus $a^n\in G_e\cap S$. Since $G_e\cap S$ is an $\infty$-definable subsemigroup of $S$ with cancellation, it is a subgroup.
\end{proof}
We may, thus, assume that $S$ is the conjunction of countably many formulae.
Furthermore, we may, and will, assume that $S$ is commutative. Indeed, let $a\in S$. By compactness we may find a definable set $S\subseteq S_0$ such that for all $x,y,z\in S_0$:
\[x(yz)=(xy)z.\]
Define $D_1=\{x\in S_0: xa=ax\}$ and then \[D_2=\{x\in S: (\forall c\in D_1)\: xc=cx\}.\]
$D_2$ is an $\infty$-definable commutative subsemigroup of $S$ with $a\in D_2$.
\begin{lemma}\label{L:technical-stuff-for-spr}
There exist definable sets $S_i$ such that $S=\bigcap S_i$, the multiplication on $S_i$ is commutative and that for all $1<i$ there exists $C_i\in S_i$ and $n_i,m_i\in \mathbb{N}$ such that
\begin{enumerate}
\item $n_i>2m_i$;
\item $e_i:=C_ia^{n_i-m_i}$ is an idempotent;
and furthermore for all $1<j\leq i$:
\item $n_j-m_j\leq n_i-m_i$;
\item $e_je_i=e_i$;
\item $e_ia^{n_i-m_i}=a^{n_i-m_i}$.
\end{enumerate}
\end{lemma}
\begin{proof}
By compactness we may assume that $S=\bigcap S_i$, where \[S_0\supseteq S_1\supseteq S_2\supseteq \dots\]
are definable sets such that for all $i>1$ we are allowed to multiply associatively and commutatively $\leq 20$ elements of $S_i$ and get an element of $S_{i-1}$.
Let $i>1$ and let $\theta(x,y)$ be
\[\exists u\in S_i \: ux=y.\]
Obviously, $\models \theta(a^{3^k},a^{3^l})$ for $k<l$. By stability $\theta$ doesn't have the order property. Thus there exist $k<l$ such that $\models \theta(a^{3^l},a^{3^k})$. Let $C_i\in S_i$ be such that $C_ia^{3^l}=a^{3^k}$.
Since $l>k$ we have $3^l> 2\cdot 3^k$ (this gives 1). Let $n_i=3^l$ and $m_i=3^k$. $e_i:=C_ia^{n_i-m_i}\in S_{i-1}$ is an idempotent (this gives 2), for that first notice that:
\[C_ia^{2n_i-2m_i}=C_ia^{n_i}a^{n_i-2m_i}=a^{m_i}a^{n_i-2m_i}=a^{n_i-m_i}.\]
Hence,
\[(C_ia^{n_i-m_i})(C_ia^{n_i-m_i})=C_i^2a^{2n_i-2m_i}=C_ia^{n_i-m_i}.\]
We may take $n_i-m_i$ to be minimal, but then since $S_i\subseteq S_j$ for $j<i$ we have $n_j-m_j\leq n_i-m_i$ (this gives 3).
As for 4, if $1<j<i$:
\[e_ie_j=C_ia^{n_i-m_i}C_ja^{n_j-m_j}=C_ia^{n_i-m_i+m_j}C_ja^{n_j-2m_j}\]
but $n_i-m_i+m_j\geq n_j$, so
\[C_ia^{n_i-m_i+m_j-n_j}C_ja^{2n_j-2m_j}=C_ia^{n_i-m_i+m_j-n_j}a^{n_j-m_j}=e_i.\]
5 follows quite similarly to what we've done.
\end{proof}
\begin{proposition}\label{T:inf-def-is-spr}
Let $S$ be an $\infty$-definable semigroup inside a stable structure. Then $S$ is strongly $\pi$-regular.
\end{proposition}
\begin{proof}
Let $a\in S$. For all $i>1$ let $S_i, C_i, n_i$ and $m_i$ be as in Lemma \ref{L:technical-stuff-for-spr}. Set $k_i=n_i-m_i$ and
\[e_{i-1}=C_ia^{k_i} \text{ and } \beta_{i-1}=e_iC_ie_i,\]
notice that these are both elements of $S_{i-1}$ (explaining the sub-index).
By Lemma \ref{L:technical-stuff-for-spr}{(4)} we get a descending sequence of idempotents \[e_1\geq e_2\geq \dots,\] with respect to the partial order on the idempotents. By stability it must stabilize.
Thus we may assume that $e:=e_1=e_2=\dots$ and is an element of $S$.
Moreover, for all $i>1$
\[\beta_1=\beta_1\cdot e=\beta_1a^{k_{i+1}}\cdot \beta_i=e\cdot a^{k_{i+1}-k_2}\beta_i.\]
So \[\beta_1=e\cdot a^{k_{i+1}-k_2}eC_{i+1}e\]
which is a product of $\leq 20$ elements of $S_{i+1}$ and thus $\in S_i$. Also $\beta_1\in S$.
In conclusion, by setting $k:=k_2$ and $\beta:=\beta_1$,
\[a^ke=ea^k=a^k, \text{ } a^k\beta=\beta a^k=e \text{ and } \beta e=e\beta=\beta.\]
So $a^k$ is in the unit group of $eSe$.
\end{proof}
\begin{corollary}
There exists an $n\in\mathbb{N}$ such that for all $a\in S$, $a^n$ is an element of a subgroup of $S$.
\end{corollary}
\begin{proof}
Compactness.
\end{proof}
\begin{corollary}
$S$ has an idempotent.
\end{corollary}
\begin{remark}
In the notations of \Cref{sec:S_G(M)}, Newelski showed in \cite{Newelski-stable-groups} that $S_{G,\Delta}(M)$ is an $\infty$-definable semigroup in $M^{eq}$ and that it is s$\pi$r. \Cref{T:inf-def-is-spr}, thus, gives another proof.
\end{remark}
\subsection{A Counter Example}
It is known that every $\infty$-definable group inside a stable structure is an intersection of definable ones. It would be even better if every such semigroup were an intersection of definable semigroups. Milliet showed that every $\infty$-definable semigroup inside a small structure is an intersection of definable semigroups \cite{on_enveloping}. In particular this is true for $\omega$-stable structures. So, for instance, any $\infty$-definable subsemigroup of $M_n(k)$ for $k\models ACF$ is an intersection of definable semigroups.
Unfortunately, this is not true already in the superstable case, as the following example will show.
\begin{example}\label{E:counter-exam}
Pillay and Poizat give an example of an $\infty$-definable equivalence relation which is not an intersection of definable ones \cite{Pillay-Poizat}. This will give us our desired semigroup structure.
Consider the theory of a model which consists of universe $\mathbb{Q}$ (the rationals) with the unary predicates:
\[U_a=\{x\in\mathbb{Q}: x\leq a\}\] for $a\in\mathbb{Q}$.
The equivalence relation, $E$, is defined by
\[\bigwedge_{a<b} ((U_a(x)\to U_b(y))\wedge (U_a(y)\to U_b(x)).\]
It is an equivalence relation and in particular a preorder (reflexive and transitive).
Notice that it also follows that $E$ can't be an intersection of definable preorders. For if $E=\bigwedge R_i$ (for preorders $R_i$) then we also have \[E=\bigwedge (R_i\wedge\overline{R}_i)\]
where $x\overline{R}_iy=yR_ix$ (since $E$ is symmetric). But $R_i\wedge\overline{R}_i$ is a definable equivalence relation (the symmetric closure) hence trivial. So the $R_i$ are trivial.
Milliet showed that in an arbitrary structure, every $\infty$-definable semigroup is an intersection of definable semigroups if and only if this is true for all $\infty$-definable preorders \cite{on_enveloping}. As a consequence in the above structure we can define an $\infty$-definable semigroup which will serve as a counter-example. Specifically, it will be the following semigroup:
If the preorder is on a set $X$, add a new element $0$ and add $0R0$ to the preorder. Define a semigroup multiplication on $R$:
\begin{equation*}
(a,b)\cdot (c,d) =
\begin{cases}
(a,d) & \text{if }b=c,\\
(0,0) & \text{else }
\end{cases}
\end{equation*}
\begin{remark}
This example also shows that even "presumably well behaved" $\infty$-definable semigroups need not be an intersection of definable ones. In the example at hand the maximal subgroups are uniformly definable (each of them is finite) and the idempotents form a commutative semigroup.
\end{remark}
\end{example}
\subsection{Semigroups with Negative Partial Order}\label{Subsec:inf-def-with-neg-part}
We showed in \Cref{T:inf-def-is-spr} that every $\infty$-definable semigroup in a stable structure is strongly $\pi$-regular, hence the natural partial order on it has the following form:
For any $a,b\in S$, $a\leq b$ if there exists $f,e\in E(S^1)$ such that $a=be=fb$.
\begin{remark}
Notice that this order generalizes the order on the idempotents.
\end{remark}
In a similar manner to what was done with the order of the idempotents, we have the following:
\begin{proposition}
$ $
\begin{enumerate}[(i)]
\item Every chain of elements with regard to the natural partial order is finite.
\item By compactness, the length of the chains is bounded.
\end{enumerate}
\end{proposition}
\begin{definition}
We'll say that a semigroup $S$ is \emph{negatively ordered} with respect to the partial order if \[a\cdot b\leq a,b\]
for all $a,b\in S$.
\end{definition}
\begin{example}
A commutative idempotent semigroup (a (inf)-semilattice) is negatively ordered.
\end{example}
Negativily ordered semigroups were studied by Maia and Mitsch \cite{NegPart}. We'll only need the definition.
\begin{proposition}\label{P:in negative_order_n+1_is_n}
Let $S$ be a negatively ordered semigroup. Assume that the length of chains is bounded by $n$, then any product of $n+1$ elements is a product of $n$ of them.
\end{proposition}
\begin{proof}
Let $a_1\cdot\dotsc\cdot a_{n+1}\in S$. Since $S$ is negatively ordered,
\[a_1\cdot \dotsc \cdot a_{n+1}\leq a_1\cdot\dotsc\cdot a_n\leq\dots\leq a_1\cdot a_2\leq a_1.\]
Since $n$ bounds the length of chains, we must have
\[a_1\cdot\dotsc\cdot a_i=a_1\cdot\dotsc\cdot a_{i+1}\]
for a certain $1\leq i\leq n$.
\end{proof}
This property is enough for an $\infty$-definable semigroup to be contained inside a definable one.
\begin{proposition}\label{P:product_of_n+1_is_n_is_inside_def}
Let $S$ be an $\infty$-definable semigroup (in any structure). If every product of $n+1$ elements in $S$ is a product of $n$ of them, then $S$ is contained inside a definable semigroup. Moreover, $S$ is an intersection of definable semigroups.
\end{proposition}
\begin{proof}
Let $S\subseteq S_0$ be a definable set where the multiplication is defined. By compactness, there exists a definable subset $S\subseteq S_1\subseteq S_0$ such that
\begin{itemize}
\item Any product of $\leq 3n$ elements of $S_1$ is an element of $S_0$;
\item Associativity holds for products of $\leq 3n$ elements of $S_1$;
\item Any product of $n+1$ elements of $S_1$ is already a product of $n$ of them.
\end{itemize}
Let \[S_1\subseteq S_2=\{x\in S_0: \exists y_1,\dots,y_n\in S_1\quad \bigvee_{i=1}^n x=y_1\cdot \dotsc\cdot y_i\}.\]
We claim that if $a\in S_1$ and $b\in S_2$ then $ab\in S_2$, indeed this follows from the properties of $S_1$. Define
\[S_3=\{x\in S_2:xS_2\subseteq S_2\}.\]
$S_3$ is our desired definable semigroup.
\end{proof}
As a consequence of these two Propositions, we have
\begin{proposition}
Every $\infty$-definable negatively ordered semigroup inside a stable structure is contained inside a definable semigroup. Furthermore, it is an intersection of definable semigroups.
\end{proposition}
Since every commutative idempotent semigroup is negatively ordered we have the following Corollary,
\begin{corollary}\label{P:idemp_is_inside_definable}
Let $E$ be an $\infty$-definable commutative idempotent semigroup inside a stable structure, then $E$ is contained in a definable commutative idempotent semigroup. Furthermore, it is an intersection of definable ones.
\end{corollary}
\begin{proof}
We only need to show that the definable semigroup containing $E$ can be made commutative idempotent. For that to we need to demand that all the elements of $S_1$ (in the proof of \Cref{P:product_of_n+1_is_n_is_inside_def}) be idempotents and that they commute, but that can be satisfied by compactness.
\end{proof}
\subsection{Clifford Monoids}\label{Subsec:Cliff}
We assume that $S$ is an $\infty$-definable Clifford semigroup (see \Cref{subsec:Clifford}) inside a stable structure.
The simplest case of Clifford semigroups, commutative idempotent semigroups (semilattices) were considered in \Cref{Subsec:inf-def-with-neg-part}.
Understanding the maximal subgroups of a semigroup is one of the first steps when one wishes to understand the semigroup itself. Lemma \ref{L:max_subgrps_are_inf_definable} is useful and will be used implicitly.
Recall that every Clifford semigroup is a strong semilattice of groups. Between each two maximal subgroups, $G_e$ and $G_{ef}$ there exists a homomorphism $\phi_{e,ef}$ given by multiplication by $f$.
\begin{definition}
By a \emph{surjective Clifford monoid} we mean a Clifford monoid $M$ such that for every $a\in M$ there exist $g\in G(M)$ and $e\in E(M)$ such that $a=ge$.
Surjectivity refers to the fact that these types of Clifford monoids are exactly the ones with $\phi_{e,ef}$ surjective.
\end{definition}
We restrict ourselves to $\infty$-definable surjective Clifford monoids.
\begin{theorem}\label{T:Cliff_with_surj}
Let $M$ be an $\infty$-definable surjective Clifford monoid in a stable structure. Then $M$ is contained in a definable monoid, extending the multiplication on $M$. This monoid is also a surjective Clifford monoid. Furthermore, every such monoid is an intersection of definable surjective Clifford monoids.
\end{theorem}
\begin{proof}
Let $M\subseteq M_0$ be a definable set where the multiplication is defined.
By compactness, there exists a definable subset $M\subseteq M_1\subseteq M_0$
such that
\begin{itemize}
\item Associativity holds for $\leq 6$ elements of $M_1$;
\item Any product of $\leq 6$ elements of $M_1$ is in $M_0$;
\item $1$ is a neutral element of $M_1$;
\item If $x$ and $y$ are elements of $M_1$ with $y$ an idempotent then $xy=yx$.
\end{itemize}
By the standard argument for stable groups, there exists a definable group \[G_1\subseteq G\subseteq M_1,\] where $G_1\subseteq M$ is the maximal subgroup of $M$ associated with the idempotent $1$.
By \Cref{P:idemp_is_inside_definable}, there exists a definable commutative idempotent semigroup $E(M)\subseteq E\subseteq M_1$.
Notice that for every $g\in G$ and $e\in E$, \[ge=eg.\]
Define \[M_2=\{m\in M_0:\exists g\in G,e\in E\quad m=ge\}.\]
$M_2$ is the desired monoid.
The furthermore is a standard corollary of the above proof.
\end{proof}
\begin{remark}\label{C:Cliff_is_inter}
As before, it follows from the proof that any such monoid is an intersection of definable surjective Clifford monoids.
\end{remark}
We don't have an argument for Clifford monoids which are not necessarily surjective. But we do have a proof for a certain kind of inverse monoids. We'll need this result in \Cref{sec:S_G(M)}.
\begin{theorem}\label{T:Needed_for_1_based}
Let $M$ be an $\infty$-definable monoid in a stable structure, such that
\begin{enumerate}
\item Its unit group $G$ is definable,
\item $E(M)$ is commutative, and
\item for every $a\in M$ there exist $g\in G$ and $e\in E(M)$ such that \[a=ge.\]
\end{enumerate}
Then $M$ is contained in a definable monoid, extending the multiplication on $M$. This monoid also has these properties.
\end{theorem}
\begin{remark}
Incidentally $M$ is an inverse monoid (recall the definition from Section \ref{subsec:Clifford}). It is obviously regular and the pseudo-inverse is unique since the idempotents commute (see the preliminaries). Also, as before, every such monoid is an intersection of definable ones.
\end{remark}
\begin{proof}
Let $M\subseteq M_0$ be a definable set where the multiplication if defined and associative.
By compactness, there exists a definable subset $M\subseteq M_1\subseteq M_0$
such that
\begin{itemize}
\item Associativity holds for $\leq 6$ elements of $M_1$;
\item Any product of $\leq 6$ elements of $M_1$ is in $M_0$;
\item $1$ is a neutral element of $M_1$;
\item If $x$ and $y$ are idempotents of $M_1$ then $xy=yx$.
\end{itemize}
By Proposition \ref{P:idemp_is_inside_definable}, there exists a definable commutative idempotent semigroup $E(M)\subseteq E\subseteq M_1$.
Let \[E_1=\{e\in E: \forall g\in G\: g^{-1}eg\in E\}.\]
$E_1$ is still a definable commutatve idempotent semigroup that contains $E(M)$. Moreover for every $e\in E_1$ and $g\in G$, \[g^{-1}eg\in E_1.\]
Define \[M_2=\{m\in M_0:\exists g\in G,e\in E_1\quad m=ge\}.\]
$M_2$ is the desired monoid. Indeed if $g,h\in G$ and $e,f\in E_1$ then there exist $h^\prime\in G$ and $e^\prime\in E_1$ such that
\[eh=h^\prime e^\prime\]
thus \[ge\cdot hf=gh^\prime\cdot e^\prime f.\]
\end{proof}
\section{The space of types $S_G(M)$ on a definable group}\label{sec:S_G(M)}
Let $G$ be a definable group inside a stable structure $M$. Assume that $G$ is definable by a formula $G(x)$. Define
$S_G(M)$ to be all the types of $S(M)$ which are on $G$.
\begin{definition}\label{D:of product}
Let $p,q\in S_G(M)$, define
\[p\cdot q=tp(a\cdot b/M),\]
where $a\models p, b\models q$ and $a\forkindep[M]b$.
\end{definition}
Notice that the above definition may also be stated in the following form:
\[U\in p\cdot q\Leftrightarrow d_q(U)\in p.\]
where $U$ is a formula and $d_q(U):=\{g\in G(M) :g^{-1}U\in q\}$ \cite{Newelski-stable-groups}. Thus, if $\Delta$ is a finite family of formulae, in order to restrict the multiplication to $S_{G,\Delta}(M)$, the set of $\Delta$-types on $G$, we'll need to consider invariant families of formulae:
\begin{definition}
Let $\Delta\subseteq L$ be a finite set of formulae. We'll say that $\Delta$ is ($G$-)invariant if the family of subsets of $G$ definable by instances of formulae from $\Delta$ is invariant under left and right translation in $G$.
\end{definition}
From now on, unless stated otherwise, we'll assume that $\Delta$ is a finite set of invariant formulae. For $\Delta_1\subseteq \Delta_2$ let \[r^{\Delta_2}_{\Delta_1}:S_{G,\Delta_2}(M)\to S_{G,\Delta_1}(M)\] be the restriction map. These are semigroup homomorphisms. Thus \[S_G(M)=\varprojlim_{\Delta}S_{G,\Delta}(M).\]
In \cite{Newelski-stable-groups} Newelski shows that $S_{G,\Delta}(M)$ may be interpreted in $M^{eq}$ as an $\infty$-definable semigroup. Our aim is to show that these $\infty$-definable semigroups are in fact an intersection of definable ones and as a consequence that $S_G(M)$ is an inverse limit of definable semigroups of $M^{eq}$.
\subsection{$S_{G,\Delta}$ is an intersection of definable semigroups}
Let $\varphi(x,y)$ be a $G$-invariant formula.
The proof that $S_{G,\varphi}(M)$ is interpretable as an $\infty$-definable semigroup in $M^{eq}$ is given by Newelski in \cite{Newelski-stable-groups}. We'll show that it may be given as an intersection of definable semigroups.
\begin{proposition}\cite{Pillay}\label{P:def-in-stab}
There exists $n\in\mathbb{N}$ and a formula $d_\varphi(y,u)$ such that for every $p\in S_{G,\varphi}(M)$ there exists a tuple $c_p\subseteq G$ such that
\[d_\varphi(y,c_p)=(d_px)\varphi (x,y).\]
Moreover, $d_\varphi$ may be chosen to be a positive boolean combination of $\varphi$-formulae.
\end{proposition}
Let $E_{d_\varphi}$ be the equivalence relation defined by
\[c_1E_{d_\varphi} c_2 \Longleftrightarrow \forall y(d_\varphi (y,c_1)\leftrightarrow d_\varphi (y,c_2)).\]
Set $Z_{d_\varphi}:=\nicefrac{M}{E_{d_\varphi}}$, it is the sort of canonical parameters for a potential $\varphi$-definition.
\begin{remark}
We may assume that $c_p$ is the canonical parameter for $d_\varphi (M,c_p)$, namely, that it lies in $Z_{d_\varphi}$. Just replace the formula $d_\varphi(y,u)$ with the formula
\[\psi(y,v)=\forall u \left((\pi(u)=v)\to d_\varphi(y,u)\right),\]
where $v$ lies in the sort $\nicefrac{M}{E_{d_\varphi}}$ and $\pi:M\to\nicefrac{M}{E_{d_\varphi}}$.
\end{remark}
Each element $c\in Z_{d_\varphi}$ corresponds to a complete (but not necessarily consistent) set of $\varphi$-formulae:
\[p^0_c:=\{\varphi(x,a): a\in M \text{ and } \models d_\varphi(a,c)\}\cup\] \[\{\neg\varphi(x,a): a\in M \text{ and } \not\models d_\varphi(a,c)\}.\]
\begin{remark}
Notice that $p^0_c$ may not be closed under equivalence of formulae, but the set of canonical parameters $c\in Z_{d_\varphi}$ such that $p^0_c$ is closed under equivalence of formulae is the definable set:
\[\{c\in Z_{d_\varphi}: \forall t_1 \forall t_2 \left(\varphi(x,t_1)\equiv \varphi(x,t_2)\to (d_\varphi(t_1,c)\leftrightarrow d_\varphi(t_2,c)\right)\}.\]
Thus we may assume that we only deal with sets $p^0_c$ which are closed under equivalence of formulae.
\end{remark}
The set of $c\in Z_{d_\varphi}$ such that $p^0_c$ is $k$-consistent is definable: \[Z_{d_\varphi}^k=\{c\in Z_{d_\varphi} : p^0_c \text{ is k-consistent}\}.\]
Define \[Z=\bigcap_{k< \omega} Z_{d_\varphi}^k.\]
There is a bijection ($p\mapsto c_p$) between $S_{G,\varphi}(M)$ and $Z$.
The following is a trivial consequence of \Cref{P:def-in-stab}:
\begin{lemma}
There exists a formula $\Phi(u,v,y)$ with $u,v$ in the sort $Z_{d_\varphi}$, such that \[\Phi(c_p,c_q,a) \Leftrightarrow \varphi(x,a)\in p\cdot q.\] Moreover, $\Phi$ is a positive boolean combination of $d_\varphi$-formulae (and so of $\varphi$-formulae as well).
\end{lemma}
\begin{proof}
Since $\varphi$ is $G$-invariant, for simplicity we'll assume that $\varphi(x,y)$ is in fact of the form $\varphi(l\cdot x\cdot r,y)$.
Let $c_p,c_q\subseteq G$ be tuples whose images in $Z_{d_\varphi}$ correspond to the $\varphi$-types $p,q\in S_{G,\varphi}(M)$, respectively.
Remembering that $u=(u_{ij})_{1\leq i,j\leq n}$ is a tuple of variables, we may write \[d_\varphi(l,r,y,u)=\bigvee_{i<n}\bigwedge_{j<n}\varphi(l\cdot u_{ij}\cdot r,y).\]
Since \[d_q(\varphi(b\cdot x\cdot c,a))=\{g\in G(M):\varphi((b\cdot g)\cdot x\cdot c,a)\in q)\}\] \[=\{g\in G(M) :\models d_\varphi(b\cdot g,c,a,c_q)\}\]
and
\[d_\varphi(b\cdot g,c,a,c_q)=\bigvee_{i<n}\bigwedge_{j<n}\varphi(b\cdot g\cdot ((c_q)_{ij}\cdot c),a),\]
we get that \[\varphi(b\cdot x\cdot c,a)\in p\cdot q \Longleftrightarrow \models \bigvee_{i<n}\bigwedge_{j<n}d_\varphi(b,((c_q)_{ij}\cdot c),a,c_p).\]
\end{proof}
Using this we define a partial binary operation on $Z_{d_\varphi}$:
\begin{definition}
For $c_1,c_2,d\in Z_{d_\varphi}$, we'll say that $c_1\cdot c_2=d$ if $d$ is the unique element of $Z_{d_\varphi}$ that satisfies \[ \models d_\varphi(a,d)\Longleftrightarrow \models \Phi(c_1,c_2,a).\]
for all $a\in M$.
\end{definition}
By compactness, there exists $k\in\mathbb{N}$ such that for all $c_1,c_2\in Z^k_{d_\varphi}$ there exists a unique $d\in Z_{d_\varphi}$ such that $c_1\cdot c_2=d$. For simplicity, we'll assume that this happens for $Z^1_{d_\varphi}$.
\begin{theorem}
$Z$ is contained in a definable semigroup extending the multiplication on $Z$.
\end{theorem}
\begin{proof}
By compactness there exists $k\in\mathbb{N}$ such that the multiplication is associative on $Z^k_{d_\varphi}$ and the product of two elements of $Z^k_{d_\varphi}$ is in $Z^1_{d_\varphi}$. For simplicity, let's assume that this happens for $Z^2_{d_\varphi}$.
\begin{claim}
If $c_p\in Z$ and $c\in Z^2_{d_\varphi}$ then $c_p\cdot c\in Z^2_{d_\varphi}.$
\end{claim}
Let $U_1,U_2\in p_{c_p\cdot c}$, hence \[\{g\in G(M):g^{-1}U_1\in p^0_c\},\{g\in G(M):g^{-1}U_2\in p^0_c\}\in p.\] Since $p$ is consistent, there exists $g\in G(M)$ such that $g^{-1}U_1,g^{-1}U_2\in p^0_c$. Since $c\in Z^2_{d_\varphi}$, $p^0_c$ is $2$-consistent. Thus, the claim follows.
Define \[\widehat{Z^2_{d_\varphi}}=\{c\in Z^2_{d_\varphi}: c\cdot Z^2_{d_\varphi}\subseteq Z^2_{d_\varphi}\}.\]
$\widehat{Z^2_{d_\varphi}}$ is the desired definable semigroup.
\end{proof}
\begin{corollary}
$Z=S_{G,\varphi}(M)$ is an intersection of definable semigroups.
\end{corollary}
Looking even closer at the above proof we may show that $S_G(M)$ is an inverse limit of definable semigroups:
Assume that $\Delta_2=\{\varphi_1,\varphi_2\}$ and $\Delta_1=\{\varphi_1\}$. In the above notations:
\[Z_{\Delta_2}=Z_{d_{\varphi_1}}\times Z_{d_{\varphi_2}}\]
For $c=\langle c_1,c_2\rangle \in Z_{\Delta_2}$ define
\[p^0_c=p^0_{c_1} \cup p^0_{c_2}\] and then \[Z(\Delta_2)=\bigcap Z^k_{\Delta_2}\] similarly.
For $c,c',d\in Z(\Delta_2)$ we'll say that $c\cdot c'=d$ if $d$ is the unique element $d\in Z_(\Delta_2)$ that satisfies:
\[c_1\cdot c'_1=d_1 \text{ and } c_2\cdot c'_2=d_2\]
As before, we assume that such a unique element exists already for any pair of elements in $Z^1_{\Delta_2}=Z^1_{\varphi_1}\times Z^1_{\varphi_2}$. The restriction maps $r^{\Delta_2}_{\Delta_1}:Z^1_{\Delta_2}\to Z^1_{\Delta_1}$ are definable homomorphisms.
Generally, for every $\Delta=\{\varphi_1,\cdots, \varphi_n\}$ and $i<\omega$ \[Z_\Delta^i=Z_{\varphi_1}^i\times\cdots \times Z_{\varphi_n}^i\] the multiplication is coordinate-wise. So the restriction commutes with the inclusion.
As a result,
\begin{theorem}\label{T:invlim}
$S_G(M)$ is an inverse limit of definable semigroups:
\[\varprojlim_{\Delta,i} Z^\Delta_i=\varprojlim_\Delta S_{G,\Delta}(M).\]
\end{theorem}
\subsection{The case where $S_G(M)$ is an inverse monoid}\label{ss:S_G inverse}
We would like to use the Theorems we proved in \Cref{S:big-wedge-defin} to improve the result in the situation where $S_G(M)$ is an inverse monoid. We'll first see that this situation might occur. Notice that the inverse operation $^{-1}$ on $S_G(M)$ is an involution.
\begin{proposition}\cite{Lawson}\label{P:semi-topo semigroup}
Let $S$ be a compact semitopological $*$-semigroup (a semigroup with involution) with a dense unit group $G$. Then the following are equivalent for any element $p\in S$:
\begin{enumerate}
\item $p=pp^*p$;
\item $p$ has a unique quasi-inverse;
\item $p$ has an quasi-inverse.
\end{enumerate}
\end{proposition}
\begin{remark}
In the situation of $G(M)\hookrightarrow S_G(M)$, the above Proposition can be proved directly using model theory and stabilizers.
\end{remark}
Translating the above result to our situation and using results in $1$-based groups (see \cite{Pillay}):
\begin{corollary}
The following are equivalent:
\begin{enumerate}
\item for every $p\in S_G(M)$, $p$ is the generic of a right coset of a connected $M$-$\infty$-definable subgroup of $G$;
\item for every $p\in S_G(M)$, $p\cdot p^{-1}\cdot p=p$;
\item $S_G(M)$ is an inverse monoid;
\item $S_G(M)$ is a regular monoid.
\end{enumerate}
Thus if $G$ is $1$-based then $S_G(M)$ is an inverse monoid.
\end{corollary}
\begin{proof}
$(2),(3)$ and $(4)$ are equivalent by Proposition \ref{P:semi-topo semigroup} and $(1)$ is equivalent to $(2)$ by \cite[Lemma 1.2]{Kowalski}
\end{proof}
With a little more work one may characterise when $S_G(M)$ is a Clifford Monoid.
\begin{definition}
A right-and-left coset of a subgroup $H$ is a right coset $Ha$ such that $aH=Ha$.
\end{definition}
\begin{proposition}\cite{Kowalski}
$p\in S_G(M)$ is a generic of a right-and-left coset of an $M$-$\infty$-definable connected subgroup of $G$ if and only if $p\cdot p\cdot p^{-1}=p$.
\end{proposition}
By using the following easy lemma
\begin{lemma}
Assuming $pp^{-1}p=p$, $$pp^{-1}=p^{-1}p \Leftrightarrow ppp^{-1}=p.$$
\end{lemma}
we get
\begin{proposition}
The following are equivalent:
\begin{enumerate}
\item every $p\in S_G(M)$ is the generic of a right-and-left coset of a connected $M$-$\infty$-definable subgroup of $G$;
\item $S_G(M)$ is a Clifford monoid.
\end{enumerate}
\end{proposition}
As a result, it may happen that $S_G(M)$ is an inverse (or Cliford) monoid. One may wonder if in these cases we may strengthen the result.
\begin{lemma}
If $S_G(M)$ is a Clifford Monoid then so is $S_{G,\Delta}(M)$. They same goes for inverse monoids.
\end{lemma}
\begin{proof}
In order to show that $S_{G,\Delta}(M)$ is a Clifford Monoid we must show that it is regular and that the idempotents are central.
Indeed this follows from the fact that the restriction maps are surjective homomorphisms and that if $q|_\Delta$ is an idempotent there exists an idempotent $p\in S_G(M)$ such that $p|_\Delta=q|_\Delta$ \cite{Newelski-stable-groups}.
\end{proof}
Assume that $\Delta$ is a finite invariant set of formulae. We'll show that if $S_G(M)$ is an inverse monoid then $S_{G,\Delta}(M)$ is an intersection of definable inverse monoids.
We recall some definition from \cite{Newelski-stable-groups}. Since $\Delta$ is invariant for $p\in S_{G,\Delta}(M)$ we have a map
\[d_p:Def_{G,\Delta}(M)\to Def_{G,\Delta}(M)\]
defined by
\[U\mapsto \{g\in G(M): g^{-1}U\in p\}.\]
Here $Def_{G,\Delta}(M)$ are the $\Delta$-$M$-definable subsets of $G(M)$.
Furthermore, for $p\in S_{G,\Delta}(M)$ define
\[Ker(d_p)=\{U\in Def_{G,\Delta}(M) : d_p(U)=\emptyset\}.\]
\begin{lemma}\cite{Newelski-stable-groups}
Let $\Delta$ be a finite invariant set of formulae and $p\in S_{G,\Delta}(M)$ be an idempotent. Then
\[\{q\in S_{G,\Delta}(M):Ker(d_q)=Ker(d_p)\}=\{g\cdot p: g\in G(M)\}=G(M)p.\]
In particular it is definable (in $M^{eq}$).
\end{lemma}
\begin{corollary}
If $S_{G,\Delta}(M)$ is a regular semigroup then
\[S_{G,\Delta}(M)=\bigcup_{p \text{ idempotent}} G(M)p.\]
\end{corollary}
\begin{proof}
Let $q\in S_{G,\Delta}(M)$. By regularity there exists $\tilde{q}$ such that \[q=q\tilde{q}q \text { and } \tilde{q}=\tilde{q}q\tilde{q}.\]
$\tilde{q}q$ is the desired idempotent and $Ker(q)=Ker(\tilde{q}q).$
\end{proof}
Recall \Cref{T:Needed_for_1_based}. Notice that a semigroup $S$ fulfilling the requirements of the Theorem is an inverse monoid. It is obviously regular and the pseudo-inverse is unique since the idempotents commute. We get the following:
\begin{corollary}
If $S_G(M)$ is an inverse semigroup then $S_{G,\Delta}(M)$ is an intersection of definable inverse semigroups.
\end{corollary}
\begin{proof}
Since $S_G(M)$ is inverse so are the $S_{G,\Delta}(M)$. By \cite{Newelski-stable-groups} the unit group of $S_{G,\Delta}(M)$ is definable and by the previous corollary for every $p\in S_{G,\Delta}(M)$ there exists an idempotent $e$ and $g\in G(M)$ such that \[p=ge.\]
\end{proof}
\paragraph*{Acknowledgement}
I would like to thank my PhD advisor, Ehud Hrushovski for our discussions, his ideas and support, and his careful reading of previous drafts.
\end{document} |
\begin{document}
\begin{abstract}
Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces are constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper\,\cite{LeePark:SimplyConnected}. Also, some exceptional bundles on Dolgachev surfaces associated with $\Q$-Gorenstein smoothing have been constructed based on the idea of Hacking\,\cite{Hacking:ExceptionalVectorBundle}. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exists a nontrivial phantom category in the derived category.\par
\end{abstract}
\maketitle
\setcounter{tocdepth}{1}\tableofcontents
\section {Introduction}
In the last few decades, the derived category $\D^{\rm b}(S)$ of a nonsingular projective variety $S$ has been extensively studied by algebraic geometers. One of the attempts is to find an exceptional collection that is a sequence of objects $E_1,\ldots,E_n$ such that
\[
\Ext^k(E_i,E_j) = \left\{
\begin{array}{cl}
0 & \text{if}\ i > j\\
0 & \text{if}\ i=j\ \text{and}\ k\neq 0 \\
\C & \text{if}\ i=j\ \text{and}\ k=0.
\end{array}
\right.
\]
There were many approaches to find exceptional collections of maximal length if $S$ is a nonsingular projective surface with $p_g = q=0$. Gorodentsev and Rudakov\,\cite{GorodenstevRudakov:ExceptionalBundleOnPlane} have classified all possible exceptional collections in the case $S = \P^2$, and exceptional collections on del Pezzo surfaces has been studied by Kuleshov and Orlov\,\cite{KuleshovOrlov:ExceptionalSheavesonDelPezzo}. For Enriques surfaces, Zube \cite{Zube:ExceptionalOnEnriques} gives an exceptional collection of length $10$, and the orthogonal part is studied by Ingalls and Kuznetsov\,\cite{IngallsKuznetsov:EnriquesQuarticDblSolid} for nodal Enriques surfaces. After initiated by the work of B\"ohning, Graf von Bothmer, and Sosna\,\cite{BGvBS:ExeceptCollec_Godeaux}, there also have come numerous results on the surfaces of general type\,({\it e.g.} \cite{GalkinShinder:Beauville,BGvBKS:DeterminantalBarlowAndPhantom,AlexeevOrlov:DerivedOfBurniat,Coughlan:ExceptionalCollectionOfGeneralType,KSLee:Isogenus_1,GalkinKatzarkovMellitShinder:KeumFakeProjective,Keum:FakeProjectivePlanes}).
For surfaces with Kodaira dimension one, such exceptional collections have not been shown to exist, thus it is a natural attempt to find an exceptional collection in $\D^{\rm b}(S)$. In this paper, we use the technique of $\Q$-Gorenstein smoothing to study the case $\kappa(S) = 1$. As far as the authors know, this is the first time to establish an exceptional collection of maximal length on a surface with Kodaira dimension one.
The key ingredient is the method of Hacking\,\cite{Hacking:ExceptionalVectorBundle}, which associates a $T_1$-singularity $(P \in X)$ with an exceptional vector bundle on the general fiber of a $\Q$-Gorenstein smoothing of $X$. A $T_1$-singularity is the cyclic quotient singularity
\[
(0 \in \A^2 \big/ \langle \xi \rangle),\quad \xi \cdot(x,y) = (\xi x, \xi^{na-1}y),
\]
where $n > a > 0$ are coprime integers and $\xi$ is the primitive $n^2$-th root of unity\,(see the works of Koll\'ar and Shepherd-Barron\,\cite{KSB:CompactModuliOfSurfaces}, Manetti\,\cite{Manetti:NormalDegenerationOfPlane}, and Wahl\,\cite{Wahl:EllipticDeform,Wahl:SmoothingsOfNormalSurfaceSings} for the classification of $T_1$-singularities and their smoothings). In the paper\,\cite{LeePark:SimplyConnected}, Lee and Park constructed new surfaces of general type via $\Q$-Gorenstein smoothings of projective normal surfaces with $T_1$-singularities. Motivated by \cite{LeePark:SimplyConnected}, substantial amount of works was carried out, especially on (1) construction of new surfaces of general type\,({\it e.g.}\,\cite{KeumLeePark:GeneralTypeFromElliptic,LeeNakayama:SimplyGenType_PositiveChar,ParkParkShin:SimplyConnectedGenType_K3,ParkParkShin:SimplyConnectedGenType_K4}); (2) investigation of the KSBA boundaries of the moduli of space of surfaces of general type\,({\it e.g.} \cite{HackingTevelevUrzua:FlipSurfaces,Urzua:IdentifyingNeighbors}). Our approach is based on rather different perspective:
\begin{center}
Construct $S$ via a smoothing of a singular surface as in \cite{LeePark:SimplyConnected}, and apply \cite{Hacking:ExceptionalVectorBundle} to investigate $\Pic S$.
\end{center}
We study the case $S={}$a Dolgachev surface with two multiple fibers of multiplicities $2$ and $3$, and give an explicit $\Z$-basis for the N\'eron-Severi lattice of $S$\,(Theorem~\ref{thm:Synop_NSLattice}). Afterwards, we find an exceptional collection of line bundles of maximal length in $\D^{\rm b}(S)$\,(Theorem~\ref{thm:Synop_ExceptCollection_MaxLength}).
\subsection*{Notations and Conventions}
Throughout this paper, everything will be defined over the field of complex numbers. A surface is an irreducible projective variety of dimension two.
If $T$ is a scheme of finite type over $\C$ and $t \in T$ a closed point, then we use $(t \in T)$ to indicate the analytic germ.
Let $n > a > 0$ be coprime integers, and let $\xi$ be the $n^2$-th root of unity. The $T_1$-singularity
\[
( 0 \in \A^2 \big/ \langle \xi \rangle ),\quad \xi\cdot(x,y) = (\xi x , \xi^{na-1}y)
\]
will be denoted by $\bigl( 0 \in \A^2 \big/ \frac{1}{n^2}(1,na-1) \bigr)$.
If two divisors $D_1$ and $D_2$ are linearly equivalent, we write $D_1 = D_2$ if there is no ambiguity. Two $\Q$-Cartier Weil divisors $D_1,D_2$ are $\Q$-linearly equivalent, denoted by $D_1 \equiv D_2$, if there exists $r \in \Z_{>0}$ such that $rD_1 = rD_2$.
Let $S$ be a nonsingular projective variety. The following invariants are associated with $S$.
\begin{itemize}[fullwidth,itemindent=10pt]
\item The geometric genus $p_g(S) = h^2(\mathcal O_S)$.
\item The irregularity $q(S) = h^1(\mathcal O_S)$.
\item The holomorphic Euler characteristic $\chi(S)$.
\item The N\'eron-Severi group $\op{NS}(S) = \Pic S / \Pic^0 S$, where $\Pic^0 S$ is the group of divisors algebraically equivalent to zero.
\end{itemize}
Since the definitions of Dolgachev surfaces vary in literature, we fix our definition.
\begin{definition}
Let $q > p > 0$ be coprime integers. A \emph{Dolgachev surface $S$ of type $(p,q)$} is a minimal, simply connected, nonsingular, projective surface with $p_g(S) = q(S) = 0$ and of Kodaira dimension one such that there are exactly two multiple fibers of multiplicities $p$ and $q$.
\end{definition}
In the sequel, we will be given a degeneration $S \rightsquigarrow X$ from a nonsingular projective surface $S$ to a projective normal surface $X$, and compare information between them. We use the superscript ``$\gen$'' to emphasize this correlation. For example, we use $X^\gen$ instead of $S$.
\subsection*{Synopsis of the paper} In Section~\ref{sec:Construction}, we construct a Dolgachev surface $X^\gen$ of type $(2,n)$ following the technique of Lee and Park\,\cite{LeePark:SimplyConnected}. We begin with a pencil of plane cubics generated by two general nodal cubics, which meet at nine different points. The pencil defines a rational map $\P^2 \dashrightarrow \P^1$, undefined at the nine points of intersection. Blowing up the nine intersection points resolves the indeterminacy of $\P^2 \dashrightarrow \P^1$, hence yields a rational elliptic surface. After additional blow ups, we get two special fibers
\[
F_1 := C_1 \cup E_1,\quad\text{and}\quad F_2:= C_2\cup E_2\cup \ldots \cup E_{r+1}.
\]
Let $Y$ denote the resulting rational elliptic surface with the general fiber $C_0$, and let $p \colon Y \to \P^2$ denote the blow down morphism. Contracting the curves in the $F_1$ fiber\,(resp. $F_2$ fiber) except $E_1$\,(resp. $E_{r+1}$), we get the morphism $\pi \colon Y \to X$ to a projective normal surface $X$ with two $T_1$-singularities of types
\[
(P_1 \in X) \simeq \Bigl( 0 \in \A^2 \Big/ \frac{1}{4}(1,1) \Bigr) \quad \text{and}\quad (P_2 \in X) \simeq \Bigl( 0 \in \A^2 \Big/ \frac{1}{n^2}(1,na-1) \Bigr)
\]
for coprime integers $n > a > 0$. Note that the numbers $n,a$ are determined by the formula
\[
\frac{n^2}{na-1} = k_1 - \frac{1}{ k_2 - \frac{1}{\ldots -\frac{1}{k_r}} },
\]
where $-k_1,\ldots,-k_r$ are the self-intersection numbers of the curves in the chain $\{C_2,\ldots,E_r\}$\,(with the suitable order).
We prove the formula\,(Proposition~\ref{prop:SingularSurfaceX})
\begin{equation}
\pi^* K_X \equiv - C_0 + \frac{1}{2}C_0 + \frac{n-1}{n}C_0, \label{eq:Synop_QuasiCanoncialBdlFormula}
\end{equation}
which resembles the canonical bundle formula for minimal elliptic surfaces\,\cite[p.~213]{BHPVdV:Surfaces}. We then obtain $X^\gen$ by taking a general fiber of a $\Q$-Gorenstein smoothing of $X$. Then, since the divisor $\pi_* C_0$ is away from singularities of $X$, it moves to a nonsingular elliptic curve $C_0^\gen$ along the deformation $X \rightsquigarrow X^\gen$. We prove that the linear system $\lvert C_0^\gen \rvert$ defines an elliptic fibration $f^\gen \colon X^\gen \to \P^1$. Comparing (\ref{eq:Synop_QuasiCanoncialBdlFormula}) with the canonical bundle formula on $X^\gen$, we achieve the following theorem.
\begin{theorem}[see Theorem~\ref{thm:SmoothingX} for details]\label{thm:Synop_NSLattice}
Let $\varphi \colon \mathcal X \to (0 \in T)$ be a one parameter $\Q$-Gorenstein smoothing of $X$ over a smooth curve germ. Then for a general point $0 \neq t_0 \in T$, the fiber $X^\gen := \mathcal X_{t_0}$ is a Dolgachev surface of type $(2,n)$.
\end{theorem}
We jump into the case $a=1$ in Section~\ref{sec:ExcepBundleOnX^g}, and explain the construction of exceptional vector bundles\,(mostly line bundles) on $X^\gen$ associated with the degeneration $X^\gen \rightsquigarrow X$ using the method developed in \cite{Hacking:ExceptionalVectorBundle}. Let $\iota \colon Y \to \tilde X_0$ be the contraction of $E_2,\ldots,E_r$. Then, $Z_1 := \iota(C_1)$ and $Z_2 := \iota(C_2)$ are smooth rational curves. There exists a proper birational morphism $\Phi \colon \tilde{\mathcal X} \to \mathcal X$\,(a weighted blow up at the singularities of $X = \mathcal X_0$) such that the central fiber $\tilde{\mathcal X}_0 := \Phi^{-1}(\varphi^{-1}(0))$ is described as follows: it is the union of $\tilde X_0$, the projective plane $W_1 = \P^2_{x_1,y_1,z_1}$, and the weighted projective plane $W_2 = \P_{x_2,y_2,z_2}(1, n-1, 1)$ attached along
\[
Z_1 \simeq (x_1y_1=z_1^2) \subset W_1,\quad\text{and}\quad Z_2 \simeq (x_2y_2=z_2^n) \subset W_2.
\]
Intersection theory on $W_1$ and $W_2$ tells $\mathcal O_{W_1}(1)\big\vert_{Z_1} = \mathcal O_{Z_1}(2)$ and $\mathcal O_{W_2}(n-1)\big\vert_{Z_2} = \mathcal O_{Z_2}(n)$. The central fiber $\tilde{\mathcal X}_0$ has three irreducible components\,(disadvantage), but each component is more manageable than $X$\,(advantage). We work with the smoothing $\tilde{\mathcal X}/(0 \in T)$ instead of $\mathcal X / (0\in T)$. The general fiber of $\tilde{\mathcal X}/(0\in T)$ does not differ from $\mathcal X/(0\in T)$, hence it is the Dolgachev surface $X^\gen$. If $D$ is a divisor on $Y$ satisfying
\begin{equation}
(D.C_1)=2d_1 \in 2\Z,\ (D.C_2)=nd_2 \in n\Z, \text{ and } (D.E_2) = \ldots = (D.E_r) = 0, \label{eq:Synop_GoodDivisorOnY}
\end{equation}
then there exists a line bundle $\tilde{\mathcal D}$ on $\tilde{\mathcal X}_0$ such that
\[
\tilde{\mathcal D}\big\vert_{\tilde X_0} \simeq \mathcal O_{\tilde X_0}(\iota_*D),\quad \tilde{\mathcal D}\big\vert_{W_1} \simeq \mathcal O_{W_1}(d_1),\quad \text{and}\quad \tilde{\mathcal D}\big\vert_{W_2} \simeq \mathcal O_{W_2}((n-1)d_2).
\]
It can be shown that the line bundle $\tilde{\mathcal D}$ is exceptional, hence it deforms uniquely to give a bundle $\mathscr D$ on the family $\tilde{\mathcal X}$. In this method, we construct $D^\gen \in\Pic X^\gen$ as the divisor associated with the line bundle $\mathscr D\big\vert_{X^\gen}$.
There is a natural topological description of $D^\gen$. Let $B_i \subset X$ be a contractible ball around the singularity $P_i$ and let $M_i$ be the Milnor fiber associated to the smoothing $(P_i \in \mathcal X) / (0 \in T)$. Then $X^\gen$ is diffeomorphic to $(X \setminus (B_1 \cup B_2) ) \cup (M_1 \cup M_2)$, where the union is made by pasting along the natural diffeomorphism $\partial B_i \simeq \partial M_i$\,(see \cite[p.~39]{Manetti:ModuliOfDiffeo}). By Proposition~\ref{prop:Hacking_Specialization}, the relative homology sequence for the pair $(X,\, M_1 \cup M_2)$ reads
\[
0 \to H_2(X^\gen,\Z) \to H_2(X,\Z) \to H_1(M_1,\Z) \oplus H_1(M_2,\Z).
\]
Since $H_1(M_1,\Z) \simeq \Z/2\Z$ and $H_2(M_2,\Z) \simeq \Z/n\Z$, if $D \in \Pic Y$ is a divisor which fits into the condition (\ref{eq:Synop_GoodDivisorOnY}), then $[\pi_*D] \in H_2(X,\Z)$ maps to the zero element in $H_1(M_1, \Z) \oplus H_1(M_2,\Z)$. Thus, there exists a preimage of $[\pi_*D] \in H_2(X,\Z)$ along $H_2(X^\gen, \Z) \to H_2(X,\Z)$, which is nothing but the Poincar\'e dual of the first Chern class of $\mathcal O_{X^\gen}(D^\gen)$.
Section~\ref{sec:NeronSeveri} concerns the case $n=3$ and $a=1$. Let $D$, $\tilde{\mathcal D}$ and $D^\gen$ be chosen as above. There exists a short exact sequence
\begin{equation}
0 \to \tilde{\mathcal D} \to \mathcal O_{\tilde X_0}(\iota_* D) \oplus \mathcal O_{W_1}(d_1) \oplus \mathcal O_{W_2}(2d_2) \to \mathcal O_{Z_1}(2d_1) \oplus \mathcal O_{Z_2}(3d_2) \to 0. \label{eq:Synop_CohomologySequence}
\end{equation}
This expresses $\chi(\tilde{\mathcal D})$ in terms of $\chi(\iota_*D)$, $d_1$, and $d_2$. Since the Euler characteristic is deformation invariant, we get $\chi(D^\gen) = \chi(\tilde{\mathcal D})$. Furthermore, it can be proved that $(C_0. D) = (C_0^\gen . D^\gen)$. This implies that $(C_0 . D) = (6 K_{X^\gen} . D^\gen)$. The Riemann-Roch formula reads
\[
(D^\gen)^2 = \frac{1}{6}(C_0. D) + 2 \chi(\tilde{\mathcal D}) - 2,
\]
which is a clue for discovering the N\'eron-Severi lattice $\op{NS}(X^\gen)$. This leads to the first main theorem of this paper:
\begin{theorem}[$={}$Theorem~\ref{thm:Picard_ofGeneralFiber}]
Let $H \in \Pic \P^2$ be the hyperplane divisor, and let $L_0 = p^*(2H)$. Consider the following correspondences of divisors\,(see Figure~\ref{fig:Configuration_Basic}).
\[
\begin{array}{c|c|c|c}
\Pic Y & F_i - F_j & p^*H - 3F_9 & L_0 \\
\hline
\Pic X^\gen & F_{ij}^\gen & (p^*H - 3F_9)^\gen & L_0^\gen \\[1pt]
\end{array}\raisebox{-0.9\baselineskip}[0pt][0pt]{\,.}
\]\vskip+5pt\noindent
Define the divisors $\{G_i^\gen\}_{i=1}^{10} \subset \Pic X^\gen$ as follows:
\begin{align*}
G_i^\gen &= -L_0^\gen + 10K_{X^\gen} + F_{i9}^\gen,\quad i=1,\ldots,8;\\
G_9^\gen &= -L_0^\gen + 11K_{X^\gen};\\
G_{10}^\gen &= -3L_0^\gen + (p^*H - 3F_9)^\gen + 28K_{X^\gen}.
\end{align*}
Then the intersection matrix $\bigl( ( G_i^\gen . G_j^\gen) \bigr)$ is
\[
\left[
\begin{array}{cccc}
-1 & \cdots & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots \\
0 & \cdots & -1 & 0 \\
0 & \cdots & 0 & 1
\end{array}
\right]\raisebox{-2\baselineskip}[0pt][0pt]{.}
\]
In particular, $\{G_i^\gen\}_{i=1}^{10}$ is a $\Z$-basis for the N\'eron-Severi lattice $\op{NS}(X^\gen)$.
\end{theorem}
\noindent We point out that the assumption $n=3$ is crucial for the definition of $G_{10}^\gen$. Indeed, its definition is motivated by the proof of \cite[Theorem~3.1]{Vial:Exceptional_NeronSeveriLattice}. The divisor $G_{10}^\gen$ has been chosen to satisfy
\[
K_{X^\gen} = G_1^\gen + \ldots + G_9^\gen - 3G_{10}^\gen,
\]
which does not make sense for $n>3$ as $K_{X^\gen}$ is not primitive.
In Section~\ref{sec:ExcepCollectMaxLength} we continue to assume $n=3$, $a=1$. We give the proof of the second main theorem of the paper:
\begin{theorem}[$={}$Theorem~\ref{thm:ExceptCollection_MaxLength} and Corollary~\ref{cor:Phantom}]\label{thm:Synop_ExceptCollection_MaxLength}
Assume that $X^\gen$ is originated from a cubic pencil $\lvert \lambda p_*C_1 + \mu p_*C_2\rvert$ generated by two general nodal cubics. Then, there exists a semiorthogonal decomposition
\[
\bigr\langle \mathcal A,\ \mathcal O_{X^\gen},\ \mathcal O_{X^\gen}(G_1^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_{10}^\gen),\ \mathcal O_{X^\gen}(2G_{10}^\gen) \bigr\rangle
\]
of $\D^{\rm b}(X^\gen)$, where $\mathcal A$ is nontrivial phantom category\,({\it i.e.} $K_0(\mathcal A) = 0$, $\op{HH}_\bullet(\mathcal A) = 0$, but $\mathcal A\not\simeq 0$).
\end{theorem}
The proof contains numerous cohomology computations. As usual, the main ingredients which relate the cohomologies between $X$ and $X^\gen$ are the upper-semicontinuity and the invariance of Euler characteristics. The cohomology long exact sequence of (\ref{eq:Synop_CohomologySequence}) begins with
\[
0 \to H^0(\tilde{\mathcal D}) \to H^0(\iota_*D) \oplus H^0(\mathcal O_{W_1}(d_1)) \oplus H^0(\mathcal O_{W_2}(2d_2)) \to H^0(\mathcal O_{Z_1}(2d_1)) \oplus H^0(\mathcal O_{Z_2}(3d_2)).
\]
We prove that if $(D.C_1) = 2d_1 \leq 2$, $(D.C_2) = 3d_2 \leq 3$, and $(D.E_2)=0$, then $h^0(\tilde{\mathcal D}) \leq h^0(D)$. This gives an upper bound of $h^0(D^\gen)$. By Serre duality, $h^2(D^\gen) = h^0(K_{X^\gen} - D^\gen)$, hence we are able to use the same method to estimate the upper bound. After the computations of upper bounds of $h^0(D^\gen)$ and $h^2(D^\gen)$, the upper bound of $h^1(D^\gen)$ can be examined by looking at $\chi(D^\gen)$. For any divisor $D^\gen$ which appears in the proof of Theorem~\ref{thm:Synop_ExceptCollection_MaxLength}, at least one of $\{h^0(D^\gen), h^2(D^\gen)\}$ is zero, and the other one is bounded by $\chi(D^\gen)$. Then, $h^1(D^\gen)=0$ and all the three numbers $(h^p(D^\gen) : p=0,1,2)$ are exactly evaluated. One obstruction to this argument is the condition $d_1, d_2 \leq 1$, but it can be dealt with the following observation:
\begin{center}
if a line bundle on $X^\gen$ is obtained from either $C_1$ or $2C_2+E_2$, then it is trivial.
\end{center}
Perturbing $D$ by $C_1$ and $2C_2+E_2$, we can adjust the numbers $d_1$, $d_2$.
The proof reduces to find a suitable upper bound of $h^0(D)$. One of the very first trials is to find a smooth rational curve $C \subset Y$ such that $(D.C)$ is small. Then, by the short exact sequence $0 \to \mathcal O_Y(D-C) \to \mathcal O_Y(D) \to \mathcal O_C(D) \to 0$, we get $h^0(D) \leq h^0(D-C) + \min\{ 0,\,(D.C)+1\}$. Replace $D$ by $D-C$ and repeat this procedure. It eventually stops when the value of $h^0(D-C)$ is understood immediately\,({\it e.g.} when $D-C$ is linearly equivalent to a negative sum of effective curves). This will give an upper bound of $h^0(D)$. This method sometimes gives a sharp bound of $h^0(D)$, but sometimes not. Indeed, some cohomologies depend on the configuration of generating cubics $p_*C_1$, $p_*C_2$ of the cubic pencil, while the previous numerical argument cannot capture the configuration of $p_*C_1$ and $p_*C_2$. For those cases, we find an upper bound of $h^0(D)$ as follows. Assume that $D$ is an effective divisor. Then, $p_*D \subset \P^2$ is a plane curve. The divisor form of $D$ determines the degree of $p_*D$ and some conditions that $p_*D$ must admit. For example, consider $D = p^*H - E_1$. The exceptional curve $E_1$ is obtained by blowing up the node of $p_*C_1$. Hence, $p_*D$ must be a line passing through the node of $p_*C_1$. In this way, conditions can be represented by an ideal $\mathcal I \subset \mathcal O_{\P^2}$. Hence, proving $h^0(D) \leq r$ reduces to proving $h^0(\mathcal O_{\P^2}\bigl(\deg p_*D) \otimes \mathcal I\bigr) \leq r$. The latter one can be computed via a computer-based approach\,(Macaulay2). Finally, $\mathcal A\not\simeq 0$ is guaranteed by the argument involving anticanonical pseudoheight due to Kuznetsov\,\cite{Kuznetsov:Height}.
We remark that a (simply connected) Dolgachev surface of type $(2,n)$ cannot have an exceptional collection of maximal length for any $n > 3$ as explained in \cite[Theorem~3.10]{Vial:Exceptional_NeronSeveriLattice}. Also, Theorem~\ref{thm:Synop_ExceptCollection_MaxLength} gives an answer to the question posed in \cite[Remark~3.12]{Vial:Exceptional_NeronSeveriLattice}.
\section{Construction of Dolgachev Surfaces} \label{sec:Construction}
Let $n$ be an odd integer. This section presents the construction of Dolgachev surfaces of type $(2,n)$. The construction follows the technique introduced in \cite{LeePark:SimplyConnected}. Let $C_1,C_2 \subseteq \P^2$ be general nodal cubic curves meeting at $9$ different points, and let $Y' = \op{Bl}_9\P^2 \to \P^2$ be the blow up at the intersection points. Then the cubic pencil $\lvert \lambda C_1 + \mu C_2\rvert$ defines an elliptic fibration $Y' \to \P^1$, with two special fibers $C_1'$ and $C_2'$ (which correspond to the proper transforms of $C_1$ and $C_2$, respectively). Blowing up the nodes of $C_1'$ and $C_2'$, we obtain the $(-1)$-curves, say $E_1$ and $E_2$ respectively. Also, blowing up one of the intersection points of $C_2''$\,(the proper transform of $C_2'$) and $E_2$, we obtain the configuration described in Figure~\ref{fig:Configuration_Basic}.
\begin{figure}
\caption{Configuration of the divisors in the surface obtained by blowing up two points of $Y'$.}
\label{fig:Configuration_Basic}
\end{figure}
The divisors $F_1,\ldots,F_9$ are the proper transforms of the exceptional fibers of $Y' = \op{Bl}_9\P^2 \to \P^2$. The numbers in the parentheses are self-intersection numbers of the corresponding divisors.
On the fiber $C_2'' \cup E_2' \cup E_3$, we can think of two different blow ups as the following dual intersection graphs illustrate.
\[
\begin{tikzpicture}[scale=1]
\draw(0,0) node[anchor=center] (C2) {};
\draw(40pt,0pt) node[anchor=center] (E2) {};
\draw(20pt,15pt) node[anchor=center] (E3) {};
\node[below,shift=(90:1pt)] at (C2.south) {$\scriptstyle -5$};
\node[below,shift=(90:1pt)] at (E2.south) {$\scriptstyle -2$};
\node[above] at (E3.north) {$\scriptstyle -1$};
\fill[red] (C2) circle (1.5pt);
\fill[blue] (E2) circle (1.5pt);
\fill[black] (E3) circle (1.5pt);
\draw[red,-] (C2.north east) -- (E3.south west) node[above left, align=center, midway]{\tiny L};
\draw[blue,-] (E2.north west) -- (E3.south east) node[above right, align=center, midway]{\tiny R};
\draw[-] (C2.east) -- (E2.west);
\begin{scope}[shift={(-170pt,0pt)}]
\draw(0,0) node[anchor=center] (L C2) {};
\draw(40pt,0pt) node[anchor=center] (L E2) {};
\draw(40pt,15pt) node[anchor=center] (L E3) {};
\draw(80pt,0pt) node[anchor=center] (L E4) {};
\node[below] at (L C2.south) {$\scriptstyle -6$};
\node[below] at (L E2.south) {$\scriptstyle -2$};
\node[below] at (L E4.south) {$\scriptstyle -2$};
\node[above] at (L E3.north) {$\scriptstyle -1$};
\fill[red] (L C2) circle (1.5pt);
\fill[blue] (L E2) circle (1.5pt);
\fill[red] (L E3) circle (1.5pt);
\fill[black] (L E4) circle (1.5pt);
\draw[-] (L C2.north east) -- (L E3.south west) node[above, align=center, midway]{\tiny L'};
\draw[-] (L E2.east) -- (L E4.west);
\draw[-] (L C2.east) -- (L E2.west);
\draw[-] (L E4.north west) -- (L E3.south east) node[above, align=center, midway]{\tiny R'};
\end{scope}
\begin{scope}[shift={(130pt,0pt)}]
\draw(0,0) node[anchor=center] (R C2) {};
\draw(40pt,0pt) node[anchor=center] (R E2) {};
\draw(40pt,15pt) node[anchor=center] (R E3) {};
\draw(80pt,0pt) node[anchor=center] (R E4) {};
\node[below] at (R C2.south) {$\scriptstyle -2$};
\node[below] at (R E2.south) {$\scriptstyle -5$};
\node[below] at (R E4.south) {$\scriptstyle -3$};
\node[above] at (R E3.north) {$\scriptstyle -1$};
\fill[black] (R C2) circle (1.5pt);
\fill[red] (R E2) circle (1.5pt);
\fill[blue] (R E3) circle (1.5pt);
\fill[blue] (R E4) circle (1.5pt);
\draw[-] (R C2.north east) -- (R E3.south west) node[above, align=center, midway]{\tiny L'};
\draw[-] (R E2.east) -- (R E4.west);
\draw[-] (R C2.east) -- (R E2.west);
\draw[-] (R E4.north west) -- (R E3.south east) node[above, align=center, midway]{\tiny R'};
\end{scope}
\draw [->,decorate,decoration={snake,amplitude=1pt,segment length=5pt, post length=2pt}] (-15pt,5pt) -- (-75pt, 5pt) node[below, align=center, midway]{$\scriptstyle \op{Bl}_{\rm L}$};
\draw [->,decorate,decoration={snake,amplitude=1pt,segment length=5pt, post length=2pt}] (55pt,5pt) -- (115pt, 5pt) node[below, align=center, midway]{$\scriptstyle \op{Bl}_{\rm R}$};
\end{tikzpicture}
\]
In general, if one has a fiber with configuration\!\!
\raisebox{-11pt}[0pt][13pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(30pt,0pt) node[anchor=center] (E2) {};
\draw(60pt,0pt) node[anchor=center, inner sep=10pt] (E3) {};
\draw(90pt,0pt) node[anchor=center] (E4) {};
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\draw (E3) node[anchor=center]{$\cdots$};
\fill[black] (E4) circle (1.5pt);
\node[black,below,shift=(90:2pt)] at (E1.south) {$\scriptscriptstyle -k_1$};
\node[black,below,shift=(90:2pt)] at (E2.south) {$\scriptscriptstyle -k_2$};
\node[black,below,shift=(90:2pt)] at (E4.south) {$\scriptscriptstyle -k_r$};
\draw[-] (E1.east) -- (E2.west);
\draw[-] (E2.east) -- (E3.west);
\draw[-] (E3.east) -- (E4.west);
\end{tikzpicture} }\!\!\!\!,
then blowing up at L yields\!\!\!\!
\raisebox{-11pt}[0pt][13pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(25pt,0pt) node[anchor=center] (E2) {};
\draw(50pt,0pt) node[anchor=center, inner sep=10pt] (E3) {};
\draw(75pt,0pt) node[anchor=center] (E4) {};
\draw(100pt,0pt) node[anchor=center] (E5) {};
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\draw (E3) node[anchor=center]{$\cdots$};
\fill[black] (E4) circle (1.5pt);
\fill[black] (E5) circle (1.5pt);
\node[black,below,shift=(90:2pt)] at (E1.south) {$\scriptscriptstyle -(k_1+1)$};
\node[black,below,shift=(90:2pt)] at (E2.south) {$\scriptscriptstyle -k_2$};
\node[black,below,shift=(90:2pt)] at (E4.south) {$\scriptscriptstyle -k_r$};
\node[black,below,shift=(90:2pt)] at (E5.south) {$\scriptscriptstyle -2$};
\draw[-] (E1.east) -- (E2.west);
\draw[-] (E2.east) -- (E3.west);
\draw[-] (E3.east) -- (E4.west);
\draw[-] (E4.east) -- (E5.west);
\end{tikzpicture} }\!\!.
Similarly, the blowing up at R yields\!\!
\raisebox{-12pt}[0pt][13pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(25pt,0pt) node[anchor=center] (E2) {};
\draw(50pt,0pt) node[anchor=center, inner sep=10pt] (E3) {};
\draw(75pt,0pt) node[anchor=center] (E4) {};
\draw(108pt,0pt) node[anchor=center] (E5) {};
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\draw (E3) node[anchor=center]{$\cdots$};
\fill[black] (E4) circle (1.5pt);
\fill[black] (E5) circle (1.5pt);
\node[black,below,shift=(90:2pt)] at (E1.south) {$\scriptscriptstyle -2$};
\node[black,below,shift=(90:2pt)] at (E2.south) {$\scriptscriptstyle -k_1$};
\node[black,below,shift=(90:2pt)] at (E4.south) {$\scriptscriptstyle -k_{r-1}$};
\node[black,below,shift=(90:2pt)] at (E5.south) {$\scriptscriptstyle -(k_r+1)$};
\draw[-] (E1.east) -- (E2.west);
\draw[-] (E2.east) -- (E3.west);
\draw[-] (E3.east) -- (E4.west);
\draw[-] (E4.east) -- (E5.west);
\end{tikzpicture} }\!\!\!\!\!\!\!.\ \
These present all possible resolution graphs of $T_1$-singularities\,\cite[Theorem~17]{Manetti:NormalDegenerationOfPlane}. Let $Y$ be the surface obtained after successive blow ups on the second special fiber $C_2'' \cup E_2' \cup E_3$, so that the resulting fiber contains the resolution graph of a $T_1$-singularity of type $\bigl(0 \in \A^2 / \frac{1}{n^2}(1, na-1)\bigr)$ for some odd integer $n$ and an integer $a$ with $\op{gcd}(n,a)=1$.
To simplify notations, we will not distinguish the divisors and their proper transforms unless there arise ambiguities. For instance, the proper transform of $C_1 \in \Pic \P^2$ in $Y$ will be denoted by $C_1$, and so on. We fix this configuration of $Y$ throughout this paper, so it is appropriate to give a summary here:
\begin{enumerate}[label=\normalfont(\arabic{enumi})]
\item the $(-1)$-curves $F_1,\ldots,F_9$ that are proper transforms of the exceptional fibers of $\op{Bl}_9 \P^2 \to \P^2$;
\item the $(-4)$-curve $C_1$ and the $(-1)$-curve $E_1$ arising from the blowing up of the first nodal curve;
\item the negative curves $C_2,\,E_2,\,\ldots,\,E_r,\,E_{r+1}$, where $E_{r+1}^2 = -1$ and $C_2,\,E_2,\,\ldots,\,E_r$ form a resolution graph of a $T_1$-singularity of type $\bigl(0 \in \A^2 \big/\frac{1}{n^2}(1,na-1)\bigr)$.
\end{enumerate}\vskip+5pt
\begin{figure}
\caption{Configuration of the surface $Y$. The sequence $E_{i_k}
\label{fig:Configuration_General}
\end{figure}
Let $C_0$ be the general fiber of the elliptic fibration $Y \to \P^1$. The fibers are linearly equivalent, thus
\begin{align}
C_0 &= C_1 + 2E_1 \nonumber \\
&= C_2 + a_2 E_2 + a_3 E_3 + \ldots + a_{r+1} E_{r+1}, \label{eq:SpecialFiber}
\end{align}
where $a_2,\ldots,a_{r+1}$ are the integers determined by the system of linear equations
\begin{equation}\label{eq:EquationOnFiber}
(C_2.E_i) + \sum_{j=2}^{r+1} a_j (E_j.E_i) = 0,\quad i=2,\ldots, r+1.
\end{equation}
Note that the values $(C_2.E_i)$, $(E_j.E_i)$ are explicitly given in the configuration\,(Figure~\ref{fig:Configuration_General}). The matrix $\bigl( (E_j.E_i) \bigr)_{2\leq i,j \leq r}$ is negative definite\,\cite{Mumford:TopologyOfNormalSurfaceSingularity}, and the number $a_{r+1}$ is determined by Proposition \ref{prop:SingIndexAndFiberCoefficients}, hence the system (\ref{eq:EquationOnFiber}) has a unique solution.
\begin{lemma}\label{lem:CanonicalofY}
In the above situation, the following formula holds:
\[
K_Y = E_1 - C_2 - E_2 - \ldots - E_{r+1}.
\]
\end{lemma}
\begin{proof}
The proof proceeds by an induction on $r$. The minimum value of $r$ is two, the case in which $C_2\cup E_2$ from the chain
\raisebox{0pt}[15pt][0pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(20pt,0pt) node[anchor=center] (E2) {};;
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\node[black,shift=(90:2pt)] at (E1.north) {$\scriptscriptstyle -5$};
\node[black,shift=(90:2pt)] at (E2.north) {$\scriptscriptstyle -2$};
\draw[-] (E1.east) -- (E2.west);
\end{tikzpicture} }\!\!.
Let $H \in \Pic \P^2$ be a hyperplane divisor, and let $p \colon Y \to \P^2$ be the blowing down morphism. Then
\[
K_Y = p^* K_{\P^2} + F_1 + \ldots + F_9 + E_1 + d_2 E_2 + d_3E_3
\]
for some $d_2,d_3 \in \Z$. Since any cubic curve in $\P^2$ is linearly equivalent to $3 H$,
\begin{align*}
p^* ( 3H ) &= C_0 + F_1 + \ldots + F_9 \\
&= (C_2 + a_2 E_2 +a_3 E_3) + F_1 + \ldots + F_9
\end{align*}
where $a_2,a_3$ are the integers introduced in (\ref{eq:SpecialFiber}). Hence,
\begin{align*}
K_Y &= p^* (-3H) + F_1 + \ldots + F_9 + E_1 + d_2 E_2 + d_3 E_3 \\
&= E_1 - C_2 + (d_2-a_2)E_2 + (d_3-a_3)E_3.
\end{align*}
Here, the genus formula shows that $K_Y = E_1 - C_2 - E_2 - E_3$.
Assume the induction hypothesis that $K_Y = E_1 - C_2 - E_2 - \ldots - E_{r+1}$. Let $D \in \{C_2,E_2,\ldots,E_r\}$ be a divisor intersects $E_{r+1}$, and let $\varphi \colon \widetilde Y \to Y$ be the blowing up at the point $D \cap E_{r+1}$. Then,
\[
K_{\widetilde Y} = \varphi^* K_Y + \widetilde E_{r+2},
\]
where $\widetilde E_{r+2}$ is the exceptional divisor of $\varphi$. Let $\widetilde C_2, \widetilde E_1, \ldots, \widetilde E_{r+1}$ denote the proper transforms of the corresponding divisors. Then, $\varphi^*$ maps $D$ to $(\widetilde D + \widetilde E_{r+2})$, maps $E_{r+1}$ to $(\widetilde E_{r+1} + \widetilde E_{r+2})$, and maps the other divisors to their proper transforms. It follows that
\begin{align*}
\varphi^*K_Y &= \varphi^*(E_1 - C_2 - \ldots - E_{r+1}) \\
&= \widetilde E_1 - \widetilde C_2 - \ldots - \widetilde E_{r+1} - 2 \widetilde E_{r+2}.
\end{align*}
Hence, $K_{\widetilde Y} = \varphi^* K_Y + \widetilde E_{r+2} = \widetilde E_1 - \widetilde C_2 - \widetilde E_2 - \ldots - \widetilde E_{r+2}$.
\end{proof}
\begin{proposition}\label{prop:SingularSurfaceX}
Let $\pi \colon Y \to X$ be the contraction of the curves $C_1,\,C_2,\, E_2,\,\ldots,\, E_r$. Let $P_1 = \pi(C_1)$ and $P_2 = \pi(C_2 \cup E_2 \cup \ldots \cup E_r)$ be the singularities of types $\bigl( 0 \in \A^2 \big/ \frac{1}{4}(1,1)\bigr)$ and $\bigl( 0 \in \A^2 \big/ \frac{1}{n^2}(1,na-1)\bigr)$, respectively. Then the following properties of $X$ hold:
\begin{enumerate}[ref=(\alph{enumi})]
\item \label{item:SingularSurfaceX_Cohomologies}$X$ is a projective normal surface with $H^1(\mathcal O_X) = H^2(\mathcal O_X)=0$;
\item $\pi^*K_X \equiv (\frac 12 - \frac 1n)C_0 \equiv C_0 - \frac{1}{2} C_0 - \frac{1}{n} C_0$ as $\Q$-divisors.
\end{enumerate}
In particular, $K_X^2 = 0$, $K_X$ is nef, but $K_X$ is not numerically trivial.
\end{proposition}
\begin{proof}\
\begin{enumerate}
\item Since the singularities of $X$ are rational, $R^q \pi_* \mathcal O_Y = 0$ for $q > 0$. The Leray spectral sequence
\[
E_2^{p,q} = H^p( X, R^q\pi_* \mathcal O_Y ) \Rightarrow H^{p+q}(Y,\mathcal O_Y)
\]
says that $H^p(Y,\mathcal O_Y) \simeq H^p (X, \pi_* \mathcal O_Y) = H^p(X,\mathcal O_X)$ for $p > 0$. The surface $Y$ is obtained from $\P^2$ by a finite sequence of blow ups, hence $H^1(Y,\mathcal O_Y) = H^2(Y,\mathcal O_Y) =0$. One can immediately verify the hypotheses of Artin's criterion for contractibility\,\cite[Theorem~2.3]{Artin:Contractibility} hold, thus $X$ is projective.
\item Since the morphism $\pi$ contracts $C_1,\,C_2,\,E_2,\,\ldots,\,E_r$, we may write
\[
\pi^* K_X \equiv K_Y + c_1 C_1 + c_2 C_2 + b_2 E_2 + \ldots + b_r E_r,
\]
for $c_1,c_2,b_2,\ldots,b_r \in \Q$(the coefficients may not be integral since $X$ is singular). It is easy to see that $c_1 = \frac 12$. By Lemma~\ref{lem:CanonicalofY},
\[
\pi^* K_X \equiv \frac{1}{2}C_0 + (c_2- 1)C_2 + (b_2 -1)E_2+ \ldots + (b_r-1) E_r - E_{r+1}.
\]
Both $\pi^*K_X$ and $C_0$ do not intersect with $C_2,E_2,\ldots,E_r$. Thus, we get
\begin{equation}\label{eq:Aux1}
\left\{
\begin{array}{l@{}l}
0 &{}= (1-c_2)(C_2^2) + \sum_{j =2}^r (1-b_j)(E_j.C_2) + (E_{r+1}.C_2) \\
0 &{}= (1-c_2)(C_2.E_i) + \sum_{j=2}^r (1-b_j)(E_j.E_i) + (E_{r+1}.E_i),\quad \text{for\ }i=2,\ldots,r.
\end{array}
\right.
\end{equation}
After divided by $a_{r+1}$, (\ref{eq:EquationOnFiber}) becomes
\[
0=\frac{1}{a_{r+1}} (C_2. E_i) + \sum_{j=2}^r \frac{a_j}{a_{r+1}} (E_j.E_i) + (E_{r+1}.E_i),\quad \text{for\ }i=2,\ldots,r.
\]
In addition, the equation $( C_2 + a_2 E_2 + \ldots + a_{r+1} E_r \mathbin. C_2 ) = (C_0 . C_2) = 0 $ gives rise to
\[
0=\frac{1}{a_{r+1}} (C_2^2) + \sum_{j=2}^r \frac{a_j}{a_{r+1}} (E_j.C_2) + (E_{r+1}.C_2).
\]
Comparing these equations with (\ref{eq:Aux1}), it is easy to see that the ordered tuples
\[
(1-c_2,\ 1-b_2,\ \ldots,\ 1-b_r)\quad\text{and}\quad (1/a_{r+1},\ a_2/a_{r+1},\ \ldots,\ a_r / a_{r+1})
\]
fit into the same system of linear equations. Since the intersection matrix of the divisors $(C_2,E_2,\ldots,E_r)$ is negative definite,
\[
(1-c_2,\, 1-b_2,\, \ldots,\, 1-b_r) = (1/a_{r+1},\ a_2/a_{r+1},\ \ldots,\ a_r / a_{r+1}).
\]
It follows that
\begin{align*}
\pi^* K_X
&\equiv \frac{1}{2}C_0 + (c_2 -1 )C_2 + (b_2 - 1)E_2 + \ldots + (b_r -1) E_r - E_{r+1} \\
&\equiv \frac{1}{2}C_0 - \frac{1}{a_{r+1}} \bigl( C_2 + a_2 E_2 + \ldots + a_{r+1} E_{r+1} \bigr) \\
&\equiv \Bigl( \frac{1}{2} - \frac{1}{a_{r+1}} \Bigr) C_0.
\end{align*}
It remains to prove $a_{n+1} = n$. This directly follows from Proposition~\ref{prop:SingIndexAndFiberCoefficients}. It is immediate to see that $C_0^2 = 0$, $C_0$ is nef, and $C_0$ is not numerically trivial. The same properties are true for $\pi^*K_X$. \qedhere
\end{enumerate}
\end{proof}
\begin{proposition}\label{prop:SingIndexAndFiberCoefficients}
Suppose that $C_2 \cup E_2 \cup \ldots \cup E_r$ has the configuration\!\!
\raisebox{-11pt}[0pt][13pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(30pt,0pt) node[anchor=center] (E2) {};
\draw(60pt,0pt) node[anchor=center, inner sep=10pt] (E3) {};
\draw(90pt,0pt) node[anchor=center] (E4) {};
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\draw (E3) node[anchor=center]{$\cdots$};
\fill[black] (E4) circle (1.5pt);
\node[black,below,shift=(90:2pt)] at (E1.south) {$\scriptscriptstyle -k_1$};
\node[black,below,shift=(90:2pt)] at (E2.south) {$\scriptscriptstyle -k_2$};
\node[black,below,shift=(90:2pt)] at (E4.south) {$\scriptscriptstyle -k_r$};
\draw[-] (E1.east) -- (E2.west);
\draw[-] (E2.east) -- (E3.west);
\draw[-] (E3.east) -- (E4.west);
\end{tikzpicture} }\!\!,
so that it contracts to give a $T_1$-singularity of type $\bigl( 0 \in \A^2 \big/ \frac{1}{n^2}(1,na-1)\bigr)$. Then, in the expression
\[
C_2 + a_2 E_2 + \ldots + a_{r+1} E_{r+1}
\]
of the fiber (\ref{eq:SpecialFiber}), the coefficient of the $(-k_1)$-curve is $a$, and the coefficient of the $(-k_r)$-curve is $(n-a)$. Furthermore, $a_{r+1}$ equals to the sum of these two coefficients, hence $a_{r+1} = n$.
\end{proposition}
\begin{proof}
The proof proceeds by an induction on $r$. The case $r = 2$ is trivial. Indeed, a simple computations shows that $n = 3$, $a = 1$, and $a_2 = 2$, $a_3= 3$. To make notations simpler, we reindex $\{C_2,\, E_2,\,\ldots,\, E_{r+1}\}$ as follows:
\[
(G_1,\,G_2,\,\ldots,\,G_{r+1}) = (E_{i_k},\,E_{i_{k-1}},\,\ldots,\,E_{i_1},\,C_2,\,E_{j_1},\,\ldots,\,E_{j_\ell},\,E_{r+1}).\hskip-35pt\tag{Figure~\ref{fig:Configuration_General}}
\]
By the induction hypothesis, we may assume
\[
C_2 + a_2E_2 + \ldots + a_{r+1} E_{r+1} = a G_1 + \ldots + (n-a) G_r + n G_{r+1}.
\]
Let $\varphi_1 \colon \widetilde Y \to Y$ be the blow up at the point $G_{r+1} \cap G_1$, let $\widetilde G_i$\,($i=1,\ldots,r+1$) be the proper transform of $G_i$, and let $\widetilde G_{r+2}$ be the exceptional divisor. The $(-1)$-curve $\widetilde G_{r+2}$ meets $\widetilde G_1$ and $\widetilde G_{r+1}$ transversally, so
\begin{align*}
\varphi^*( aG_1 + \ldots + nG_{r+1})
&= a ( \widetilde G_1 + \widetilde G_{r+2}) + g_2 \widetilde G_2 + \ldots + (n-a) \widetilde G_r + n( \widetilde G_{r+1} + \widetilde G_{r+2}) \\
&= a \widetilde G_1 + g_2\widetilde G_2 + \ldots + (n-a) \widetilde G_r + n\widetilde G_{r+1} + (n+a) \widetilde G_{r+2}.
\end{align*}
It is well-known that the contraction of $\widetilde G_1, \ldots, \widetilde G_{r+1} \subset \widetilde Y$ produces a cyclic quotient singularity of type
\[
\Bigl( 0 \in \A^2 \Big/ \frac{1}{(n+a)^2}(1,(n+a)n-1) \Bigr).
\]
This proves the statement for the chain $\widetilde G_1 \cup \ldots \cup \widetilde G_{r+2}$, so we are done by the induction. The same argument also works if one performs the blow up $\varphi_2 \colon \widetilde Y' \to Y$ at the point $G_{r+1} \cap G_r$.
\end{proof}
Now we want to obtain a smooth surface via a $\Q$-Gorenstein smoothing of $X$. It is well-known that $T_1$-singularities admit local $\Q$-Gorenstein smoothings, thus we have to verify that:
\begin{enumerate}
\item every formal deformation of $X$ is algebraizable;
\item every local deformation of $X$ can be globalized.
\end{enumerate}
The answer for (a) is an immediate consequence of Grothendieck's existence theorem\,\cite[Example~21.2.5]{Hartshorne:DeformationTheory} since $H^2(\mathcal O_X)=0$. The next lemma verifies (b).
\begin{lemma}\label{lem:NoObstruction}
Let $Y$ be the nonsingular rational elliptic surface introduced above, and let $\mathcal T_Y$ be the tangent sheaf of $Y$. Then,
\[
H^2(Y, \mathcal T_Y( - C_1 - C_2 - E_2 - \ldots - E_r ) ) = 0.
\]
In particular, $H^2(X,\mathcal T_X) = 0$\,(see \cite[Theorem~2]{LeePark:SimplyConnected}).
\end{lemma}
\begin{proof}
The proof is not very different from \cite[\textsection4, Example~2]{LeePark:SimplyConnected}. The main claim is
\[
H^0(Y, \Omega_Y^1(K_Y + C_1 + C_2 + E_2 + \ldots + E_r)) =0.
\]
By Lemma~\ref{lem:CanonicalofY} and equation~(\ref{eq:SpecialFiber}),
\[
K_Y + C_1 + C_2 + E_2 + \ldots + E_r = C_0 - E_1 - E_{r+1}.
\]
Then, $h^0(Y,\Omega_Y^1(C_0 - E_1 - E_{r+1})) \leq h^0(Y,\Omega_Y^1(C_0)) = h^0(Y',\Omega_{Y'}^1(C_0'))$ where $Y'= \op{Bl}_9\P^2$, and $h^0(Y',\Omega_{Y'}^1(C_0')) =0$ by \cite[\textsection4, Lemma~2]{LeePark:SimplyConnected}. The result directly follows from the Serre duality.
\end{proof}
We have shown that the surface $X$ admits a $\Q$-Gorenstein smoothing $\mathcal X \to T$. The next aim is to show that the general fiber $X^\gen := \mathcal X_t$ is a Dolgachev surface of type $(2,n)$.
\begin{proposition}\label{prop:CohomologyComparison_YtoX}
Let $X$ be a projective normal surface with only rational singularities, let $\pi \colon Y \to X$ be a resolution of singularities, and let $E_1,\ldots,E_r$ be the exceptional divisors. If $D$ is a divisor on $Y$ such that $(D.E_i)=0$ for all $i=1,\ldots,r$, then
\[
H^p(Y,D) \simeq H^p(X,\pi_*D)
\]
for all $p \geq 0$.
\end{proposition}
\begin{proof}
Since the singularities of $X$ are rational, each $E_i$ is a smooth rational curve. The assumption on $D$ in the statement implies that $\pi_*D$ is Cartier\,\cite[Theorem~12.1]{Lipman:RationalSings}, and $\pi^*\mathcal O_X(\pi_*D) = \mathcal O_Y(D)$. By the projection formula, $R^p\pi_*\mathcal O_Y(D) \simeq R^p \pi_*( \mathcal O_Y \otimes \pi^* \mathcal O_X(\pi_*D) ) \simeq (R^p \pi_* \mathcal O_Y) \otimes \mathcal O_X(\pi_*D)$. Since $X$ is normal and has only rational singularities,
\[
R^p\pi_* \mathcal O_Y = \left\{
\begin{array}{ll}
\mathcal O_X & \text{if } p=0\\
0 & \text{if } p > 0.
\end{array}
\right.
\]
Now, the claim is an immediate consequence of the Leray spectral sequence
\[
E_2^{p,q} = H^p(X,\, R^q\pi_*\mathcal O_Y \otimes \mathcal O_X(\pi_*D) ) \Rightarrow H^{p+q}(Y,\, \mathcal O_Y(D)). \qedhere
\]
\end{proof}
\begin{lemma}\label{lem:Cohomologies_ofGeneralFiber_inY}
Let $\pi \colon Y \to X$ be the contraction defined in Proposition~\ref{prop:SingularSurfaceX}. Then,
\[
h^0(X,\pi_*C_0) = 2,\quad h^1(X,\pi_*C_0)=1,\ \text{and}\quad h^2(X,\pi_*C_0)=0.
\]
\end{lemma}
\begin{proof}
It is easy to see that $(C_0.C_1) = (C_0.C_2) = (C_0.E_2) =\ldots = (C_0.E_r) = 0$. Hence by Proposition~\ref{prop:CohomologyComparison_YtoX}, it suffices to compute $h^p(Y,C_0)$. Since $C_0^2 = (K_Y . C_0)=0$, the Riemann-Roch formula shows $\chi(C_0)=1$. By Serre duality, $h^2(C_0) = h^0(K_Y - C_0)$. In the short exact sequence
\[
0 \to \mathcal O_Y(K_Y - C_0 - E_1) \to \mathcal O_Y(K_Y -C_0) \to \mathcal O_{E_1} \otimes \mathcal O_Y(K_Y - C_0)\to 0,
\]
we find that $H^0(\mathcal O_{E_1} \otimes \mathcal O_Y(K_Y - C)) = 0$ since $(K_Y - C_0 \mathbin . E_1) = -1$. It follows that
\[
h^0(K_Y-C_0) = h^0(K_Y-C_0-E_1),
\]
but $K_Y - C_0 - E_1 = - C_0 - C_2 - E_2 - \ldots - E_{r+1}$ by Lemma~\ref{lem:CanonicalofY}. Hence $h^2(C_0)=0$. Since the complete linear system $\lvert C_0 \rvert$ defines the elliptic fibration $Y \to \P^1$, $h^0(C_0) = 2$. Furthermore, $h^1(C_0)=1$ follows from $h^0(C_0)=2$, $h^2(C_0)=0$, and $\chi(C_0)=1$.
\end{proof}
The following proposition, due to Manetti\,\cite{Manetti:NormalDegenerationOfPlane}, is a key ingredient of the proof of Theorem~\ref{thm:SmoothingX}
\begin{proposition}[{\cite[Lemma~2]{Manetti:NormalDegenerationOfPlane}}]\label{prop:Manetti_PicLemma}
Let $\mathcal X \to ( 0 \in T)$ be a smoothing of a normal surface $X$ with $H^1(\mathcal O_X) = H^2(\mathcal O_X)=0$. Then for every $t \in T$, the natural restriction map of the second cohomology groups $H^2(\mathcal X,\Z) \to H^2(\mathcal X_t,\Z)$ induces an injection $\Pic \mathcal X \to \Pic \mathcal X_t$. Furthermore, the restriction to the central fiber $\Pic \mathcal X \to \Pic X$ is an isomorphism.
\end{proposition}
\begin{theorem}\label{thm:SmoothingX}
Let $X$ be the projective normal surface defined in Proposition~\ref{prop:SingularSurfaceX}, and let $\varphi \colon \mathcal X \to (0 \in T)$ be a one parameter $\Q$-Gorenstein smoothing of $X$ over a smooth curve germ $(0 \in T)$. For a general point $0 \neq t_0 \in T$, the fiber $X^\gen := \mathcal X_{t_0}$ satisfies the following:
\begin{enumerate}
\item $p_g(X^\gen) = q(X^\gen) = 0$;
\item $X^\gen$ is a simply connected, minimal, nonsingular surface with Kodaira dimension $1$;
\item there exists an elliptic fibration $f^\gen \colon X^\gen \to \P^1$ such that $K_{X^\gen} \equiv C_0^\gen - \frac{1}{2} C_0^\gen - \frac{1}{n} C_0^\gen$, where $C_0^\gen$ is the general fiber of $f^\gen$. In particular, $X^\gen$ is isomorphic to the Dolgachev surface of type $(2,n)$.
\end{enumerate}
\end{theorem}
\begin{proof}\
\begin{enumerate}[ref=(\normalfont\alph{enumi})]
\item This follows from Proposition~\ref{prop:SingularSurfaceX}\ref{item:SingularSurfaceX_Cohomologies} and the upper-semicontinuity of $h^p$.
\item Shrinking $(0 \in T)$ if necessary, we may assume that $X^\gen$ is simply connected\,\cite[p.~499]{LeePark:SimplyConnected}, and that $K_{X^\gen}$ is nef\,\cite[\textsection5.d]{Nakayama:ZariskiDecomposition}. If $K_{X^\gen}$ is numerically trivial, then $X^\gen$ must be an Enriques surface by the classification theory of surfaces. This violates the simple connectivity of $X^\gen$. It follows that $K_{X^\gen}$ is not numerically trivial, and the Kodaira dimension of $X^\gen$ is $1$.
\item
Since the divisor $\pi_*C_0$ is not supported on the singular points of $X$, $\pi_* C_0 \in \Pic X$. By Proposition~\ref{prop:Manetti_PicLemma}, $\Pic X \simeq \Pic \mathcal X \hookrightarrow \Pic X^\gen$. Let $C_0^\gen\in \Pic X^\gen$ be the image of $\pi_*C_0$ under this correspondence. By \cite[Theorem~4.2]{Kawamata:Moderate}, there exists a smooth complex surface $B$ such that the morphism $\varphi$ factors through $g \colon \mathcal X \to B$ and the general fiber of $g$ is an elliptic curves. In particular, the complete linear system $\lvert C_0^\gen\rvert$ defines the elliptic fibration $f^\gen \colon X^\gen \to \P^1$.
Since $\mathcal X / (0 \in T)$ is a $\Q$-Gorenstein deformation, the map $\Pic X \hookrightarrow \Pic X^\gen$ in Proposition~\ref{prop:Manetti_PicLemma} maps $2nK_X - (n-2) \pi_* C_0$ to $2nK_{X^\gen} - (n-2) C_0^\gen$. Furthermore, $2nK_X - (n-2)\pi_*C_0 \in \Pic X$ is zero, so
\[
K_{X^\gen} \equiv C_0^\gen - \frac{1}{2}C_0^\gen - \frac{1}{n}C_0^\gen.
\]
By \cite[Chapter~2]{Dolgachev:AlgebraicSurfaces}, every minimal simply connected nonsingular surface with $p_g=q=0$ and of Kodaira dimension $1$ has exactly two multiple fibers with coprime multiplicities. Thus, there exist coprime integers $q > p > 0$ such that $X^\gen \simeq X_{p,q}$ where $X_{p,q}$ is a Dolgachev surface of type $(p,q)$. The canonical bundle formula says that $K_{X_{p,q}} \equiv C_0^\gen - \frac 1p C_0^\gen - \frac 1q C_0^\gen$. Since $X^\gen \simeq X_{p,q}$, this leads to the equality
\[
\frac 12 + \frac 1n = \frac 1p + \frac 1q.
\]
Assume $2 < p < q$. Then, $\frac 12 < \frac 12 + \frac 1n = \frac 1p + \frac 1q \leq \frac 13 + \frac 1q$. Hence, $q < 6$. Only the possible candidates are $(p,q,n) = (3,4,12)$, $(3,5,30)$, but all of these cases violate $\op{gcd}(2,n) = 1$. It follows that $p=2$ and $q = n$.\qedhere
\end{enumerate}
\end{proof}
\begin{remark}
Theorem~\ref{thm:SmoothingX} generalizes to the construction of Dolgachev surfaces of type $(m,n)$ for any coprime integers $n>m>0$. Indeed, we shall describe the multiple fiber of multiplicity $n$ associated to the Weil divisor $\pi_* E_{r+1}$. The precise meaning of this sentence will be explained in the next section\,(see Example~\ref{eg:MultipleFibers}). If we perform more blow ups to the $C_1\cup E_1$-fiber so that $X$ contains a $T_1$-singularity of type $\bigl( 0 \in \A^2 \big / \frac{1}{m^2}(1,mb-1)\bigr)$, then the surface $X^\gen$ has two multiple fibers of multiplicities $m$ and $n$. Thus, $X^\gen$ is a Dolgachev surface of type $(m,n)$.
\end{remark}
\section{Exceptional vector bundles on Dolgachev surfaces}\label{sec:ExcepBundleOnX^g}
In general, it is hard to understand how information of the central fiber is carried to the general fiber along the $\Q$-Gorenstein smoothing. Looking at the topology near the singularities of $X$, one can get a clue to relate information between $X$ and $X^\gen$. This section essentially follows the idea of Hacking. Some ingredients of Hacking's method, which are necessary for our application, are included in the appendix(Section~\ref{sec:Appendix}). Readers who want to look up the details are recommended to consult Hacking's original paper\,\cite{Hacking:ExceptionalVectorBundle}.
\subsection{Topology of the singularities of $X$}\label{subsec:TopologyofX}
Let $L_i \subseteq X$\,($i=1,2$) be the link of the singularity $P_i$. Then, $H_1(L_1,\Z) \simeq \Z/4\Z$ and $H_1(L_2,\Z) \simeq \Z/n^2\Z$\,({\it cf.} \cite[Proposition~13]{Manetti:NormalDegenerationOfPlane}). Since $\op{gcd}(2,n)=1$, $H_1(L_1,\Z) \oplus H_1(L_2,\Z) \simeq \Z/ 4n^2 \Z$ is a finite cyclic group. By \cite[p.~1191]{Hacking:ExceptionalVectorBundle}, $H_2(X,\Z) \to H_1(L_i,\Z)$ is surjective for each $i=1,2$, thus the natural map
\[
H_2(X,\Z) \to H_1(L_1,\Z) \oplus H_1(L_2,\Z),\quad \alpha \mapsto ( \alpha \cap L_1 ,\, \alpha \cap L_2)
\]
is surjective. We have further information on groups the $H_1(L_i,\Z)$.
\begin{theorem}[{\cite{Mumford:TopologyOfNormalSurfaceSingularity}}]\label{thm:MumfordTopologyOfLink}
Let $X$ be a projective normal surface containing a $T_1$-singularity $P \in X$. Let $f \colon \widetilde X \to X$ be a good resolution ({\it i.e.} the exceptional divisor is simple normal crossing) of the singularity $P$, and let $E_1,\ldots,E_r$ be the integral exceptional divisors ordered in such a way that $(E_i . E_{i+1}) = 1$ for each $i=1,\ldots,r-1$. Let $\widetilde L\subseteq \widetilde X$ be the plumbing fixture (see Figure~\ref{fig:PlumbingFixture}) around $\bigcup E_i$, and let $\alpha_i \subset \widetilde L$ be the loop around $E_i$ oriented suitably. Then the following statements are true.
\begin{enumerate}
\item The group $H_1(\widetilde L,\Z)$ is generated by the loops $\alpha_i$. The relations are
\[
\sum_j (E_i . E_j) \alpha_j = 0,\quad i=1,\,\ldots,\,r.
\]
\item Let $L \subset X$ be the link of the singularity $P \in X$. Then, $\widetilde L$ is homeomorphic to $L$.
\end{enumerate}
\begin{figure}
\caption{Plumbing fixture around $\bigcup E_i$.}
\label{fig:PlumbingFixture}
\end{figure}
\end{theorem}
Proposition~\ref{prop:Manetti_PicLemma} provides a way to associate a Cartier divisor on $X$ with a Cartier divisor on $X^\gen$. This association can be extended as the following proposition illustrates.
\begin{proposition}[{{\it cf.} \cite[Lemma~5.5]{Hacking:ExceptionalVectorBundle}}]\label{prop:Hacking_Specialization}
Let $X$ be a projective normal surface, and let $(P \in X)$ be a $T_1$-singularity of type $\bigl( 0 \in \A^2 \big/ \frac{1}{n^2}(1,na-1)\bigr)$. Suppose $X$ admits a $\Q$-Gorenstein deformation $\mathcal X/(0 \in T)$ over a smooth curve germ $(0 \in T)$ such that $\mathcal X / (0 \in T)$ is a smoothing of $(P \in X)$, and is locally trivial outside $(P \in X)$. Let $X^\gen$ be the general fiber of $\mathcal X \to (0 \in T)$, and let $\mathcal B \subset \mathcal X$ be a sufficiently small open ball around $P \in \mathcal X$. Then the link $L$ and the Milnor fiber $M$ of $(P \in X)$ given as follows:
\[
L = \partial \mathcal B \cap X^\gen,\qquad M = \mathcal B \cap X^\gen.
\]
In addition, let $B := \mathcal B \cap X$ be the contractible space. Then, the relative homology sequence for the pair $(X^\gen, M)$ yields the exact sequence
\[
0 \to H_2(X^\gen,\Z) \to H_2(X,\Z) \to H_1(M,\Z).
\]
Furthermore, a class in $H_2(X,\Z)$ lifts to a class in $H_2(X^\gen,\Z)$ if and only if its image under the map $H_2(X,\Z) \to H_1(L,\Z)$ is divisible by $n$.
\end{proposition}
\begin{proof}
We have a sequence of isomorphisms
\[
H_2(X^\gen,M) \simeq H_2(X^\gen \setminus M , \partial M) \simeq H_2(X\setminus B , \partial B) \simeq H_2(X,B) \simeq H_2(X).
\]
where the first and the third ones are the excisions, the second one is due to the topological description $X^\gen = (X \setminus B) \cup M$\,(\cite[p.~39]{Manetti:ModuliOfDiffeo}), and the last one is due to the contractibility of $B$. The relative homology sequence for the pair $(X^\gen, M)$ gives
\[
0 \to H_2(X^\gen) \to H_2(X^\gen,M) \simeq H_2(X) \to H_1(M).
\]
The map in the right is the composition $H_2(X) \to H_1(L) \to H_1(M)$, where $H_1(L) \to H_1(M)$ is the natural surjection $\Z/n^2 \Z \to \Z/n\Z$\,(\cite[Lemma~2.1]{Hacking:ExceptionalVectorBundle}). The last assertion follows immediately.
\end{proof}
Recall that $Y$ is the rational elliptic surface constructed in Section~\ref{sec:Construction}, and $\pi \colon Y \to X$ is the contraction of $C_1,\,C_2,\,E_2,\,\ldots,\,E_r$. Proposition~\ref{prop:Hacking_Specialization} gives the short exact sequence
\begin{equation}
0 \to H_2(X^\gen,\Z) \to H_2(X,\Z) \to H_1(M_1,\Z) \oplus H_1(M_2,\Z), \label{eq:SpecializationSequence}
\end{equation}
where $M_1$\,(resp. $M_2$) is the Milnor fiber of the smoothing of $(P_1 \in X)$\,(resp. $(P_2 \in X)$). In this case $H_2(X,\Z) \to H_1(L_1,\Z)\oplus H_1(L_2,\Z)$ is described as follows. If $D \in \Pic Y$, then $[\pi_*D] \in H_2(X,\Z)$ maps to
\[
\bigl( (D.C_1) \alpha_{C_1},\ (D.C_2) \alpha_{C_2} + (D.E_2) \alpha_{E_2} + \ldots + (D.E_r) \alpha_{E_r}\bigr).
\]
Suppose $D \in \Pic Y$ is a divisor such that $(D. C_1) \in 2\Z$, $(D. C_2) \in n\Z$, and $(D.E_2) = \ldots = (D.E_r) = 0$. Then, Theorem~\ref{thm:MumfordTopologyOfLink} and (\ref{eq:SpecializationSequence}) imply that the cycle $[\pi_* D] \in H_2(X,\Z)$ maps to the zero element of $H_1(M_1,\Z) \oplus H_1(M_2,\Z)$. In particular, there is a cycle in $H_2(X^\gen)$, which maps to $[\pi_* D]$. Since $X^\gen$ is a nonsingular surface with $p_g = q = 0$, the first Chern class map and Poincar\'e duality induce the isomorphisms $\Pic X^\gen \simeq H^2(X^\gen,\Z) \simeq H_2(X^\gen,\Z)$\,(\cite[Proposition~4.11]{Kollar:Seifert}). We take the divisor $D^\gen \in \Pic X^\gen$ corresponding to $[\pi_* D] \in H_2(X,\Z)$. More detailed description of $D^\gen$ will be presented in \textsection\ref{subsec:HackingConstr}. We remark that even if $\pi_*D$ is an effective divisor, it does not necessarily mean that the resulting divisor $D^\gen$ is effective.
\begin{example}\label{eg:MultipleFibers}
If $D = E_{r+1}$, then $[\pi_*E_{r+1}] \in H_2(X,\Z)$ maps to $(0, \alpha_{E_r} + \alpha_{E_s} ) \in H_1(L_1,\Z) \oplus H_1(L_2,\Z)$, where $E_s$ is the other end component of the chain $\{C_2,E_2,\ldots,E_r\}$. It can be easily shown that either $\alpha_{E_s} = (na-1) \alpha_{E_r}$ or $\alpha_{E_r} = (na-1) \alpha_{E_s}$. In both cases, $\alpha_{E_r} + \alpha_{E_s}$ maps to the zero cycle along $H_1(L_2,\Z) \to H_1(M_2,\Z)$. It follows that $[\pi_*E_{r+1}] \in H_2(X,\Z)$ admits a preimage $E_{r+1}^\gen$ in $H_2(X^\gen,\Z) \simeq \Pic X^\gen$. By (\ref{eq:EquationOnFiber}) and Proposition~\ref{prop:SingIndexAndFiberCoefficients}, there are integers $a_2,\ldots,a_r \in \Z$ such that
\[
C_0 = C_2 + a_2 E_2 + \ldots + a_r E_r + n E_{r+1}.
\]
This leads to $\pi_* C_0 = \pi_* ( C_2 + a_2 E_2 + \ldots + a_r E_r + n E_{r+1}) = \pi_* ( n E_{r+1})$. Since $\mathcal X/(0 \in T)$ is a $\Q$-Gorenstein deformation, $\pi_*( (n-2) E_{r+1} ) \equiv \frac{n-2}{n}\pi_* C_0 \equiv 2K_X$\,(Proposition~\ref{prop:SingularSurfaceX}) induces $(n-2)E_{r+1}^\gen = 2K_{X^\gen}$. The same argument says that there exists $E_1^\gen \in \Pic X^\gen$ with $(n-2)E_1^\gen = nK_{X^\gen}$. In particular, we find that both $2E_1^\gen$ and $nE_{r+1}^\gen$ are $\Q$-linearly equivalent to $\frac{2n}{n-2}K_{X^\gen}$, which is again $\Q$-linearly equivalent to the general fiber $C_0^\gen$.
\end{example}
The next proposition explains the way to find a preimage along the surjective map $H_2(X,\Z) \to H_1(L_1,\Z)\oplus H_1(L_2,\Z)$.
\begin{proposition}\label{prop:DesiredDivisorOnY}
Let $X$ be a projective normal surface with a cyclic quotient singularity $(P \in X)$, let $\pi \colon Y \to X$ be a resolution of $P \in X$, and let $E_1,\ldots,E_r \subset Y$ be the exceptional divisors over $P$. The first homology group of the link $L$ has the following presentation
\[
\bigl\langle \alpha_1,\ldots,\alpha_r : \sum_{j=1}^r (E_i . E_j) \alpha_j = 0,\ i=1,\ldots,r \bigr\rangle.
\]
Let $D$ be a divisor on $Y$, and let $\ell_1,\ldots,\ell_r$ be integers satisfying
\[
[\pi_* D] \cap L = \ell_1 \alpha_1 + \ldots + \ell_r \alpha_r
\]
in $H_1(L)$. Then there are integers $e_1,\ldots,e_r$ such that $D' := D + \sum_{j=1}^r e_jE_j $ satisfies $(D'.E_i) = \ell_i$ for each $i=1,\ldots,r$.
\end{proposition}
\begin{proof}
Consider the free abelian group $\bigoplus_{j=1}^r \Z \cdot \tilde{\alpha}_j$ and the homomorphism
\[
\bigoplus_{j=1}^r \Z \cdot \tilde\alpha_j \to H_1(L),\quad \tilde\alpha_j \mapsto \alpha_j.
\]
This map is clearly surjective. By Theorem~\ref{thm:MumfordTopologyOfLink}, the kernel is the abelian group generated by
\[
\Bigl\{ R_i := \sum_{j=1}^r(E_i . E_j) \alpha_j : i=1,\ldots,r\Bigr\}.
\]
Since $[\pi_*D] \cap L = \sum_{i=1}^r ( D. E_j) \alpha_j$, the equality $[\pi_*D] \cap L = \sum_{j=1}^r \ell_j \alpha_j$ implies that there are integers $e_1,\ldots,e_r$ such that $\sum_{j=1}^r \ell_j \tilde\alpha_j - \sum_{j=1}^r ( D.E_j) \tilde \alpha_j = e_1 R_1 + \ldots + e_r R_r$. This leads to
\begin{align*}
\sum_{j=1}^r \ell_j \tilde \alpha_j &= \sum_{j=1}^r(D.E_j) \tilde \alpha_j + \sum_{i=1}^r \sum_{j=1}^r (e_i E_i . E_j) \tilde \alpha_j \\
&= \sum_{j=1}^r ( D + e_1 E_1 + \ldots + e_r E_r \mathbin. G_j) \tilde\alpha_j.
\end{align*}
Taking $D' = D + e_1 E_1 + \ldots + E_r$, we get $(D'.E_i)=\ell_i$ for each $i=1,\ldots,r$.
\end{proof}
\subsection{Exceptional vector bundles on $X^\gen$}\label{subsec:HackingConstr}
We keep the notations in Section~\ref{sec:Construction}, namely, $Y$ is the rational elliptic surface\,(Figure~\ref{fig:Configuration_General}), $\pi \colon Y \to X$ is the contraction in Proposition~\ref{prop:SingularSurfaceX}.
Let $(0 \in T)$ be the base space of the versal deformation ${\mathcal X^{\rm ver} / (0 \in T)}$ of $X$, and let $(0 \in T_i)$ be the base space of the versal deformation $(P_i \in \mathcal X^{\rm ver}) / (0 \in T_i)$ of the singularity $(P_i \in X)$. By Lemma~\ref{lem:NoObstruction} and \cite[Lemma~7.2]{Hacking:ExceptionalVectorBundle}, there exists a smooth morphism
\[
\mathfrak T \colon (0 \in T) \to \textstyle\prod_i (0 \in T_i).
\]
For each $i=1,2$, take the base extensions $( 0 \in T_i') \to (0 \in T_i)$ to which Proposition~\ref{prop:HackingWtdBlup} can be applied. Then, there exists a Cartesian diagram
\[
\begin{xy}
(0,0)*+{(0 \in T)}="00";
(30,0)*+{\textstyle \prod_i ( 0 \in T_i)}="10";
(0,15)*+{(0 \in T')}="01";
"10"+"01"-"00"*+{\textstyle \prod_i ( 0 \in T_i')}="11";
{\ar^(0.44){\mathfrak T} "00";"10"};
{\ar^(0.44){\mathfrak T'} "01";"11"};
{\ar "01";"00"};
{\ar "11";"10"};
\end{xy}.
\]
Let $\mathcal X' / (0 \in T')$ be the deformation obtained by pulling back $\mathcal X^{\rm ver} / ( 0 \in T)$ along $(0 \in T') \to (0 \in T)$. By Proposition~\ref{prop:HackingWtdBlup}, there exists a proper birational map $\Phi \colon \tilde{\mathcal X} \to \mathcal X'$ such that the central fiber $\tilde {\mathcal X}_0 = \Phi^{-1}(\mathcal X_0')$ is the union of three irreducible components $\tilde X_0$, $W_1$, $W_2$, where $\tilde X_0$ is the proper transform of $X = \mathcal X_0'$, and $W_1$\,(resp. $W_2$) is the exceptional locus over $P_1$\,(resp. $P_2$). The intersection $Z_i := \tilde X_0 \cap W_i$\,($i=1,2$) is a smooth rational curve.
From now on, assume $a=1$. This is the case in which the resolution graph of the singular point $P_2 \in X$ forms the chain $C_2,\,E_2,\,\ldots,\,E_r$ in this order. Indeed, the resolution graph of a cyclic quotient singularity $\bigl(0 \in \A^2 / \frac{1}{n^2}(1,n-1)\bigr)$ is \!\!\!\!\!
\raisebox{-11pt}[0pt][13pt]{
\begin{tikzpicture}
\draw(0,0) node[anchor=center] (E1) {};
\draw(30pt,0pt) node[anchor=center] (E2) {};
\draw(60pt,0pt) node[anchor=center, inner sep=10pt] (E3) {};
\draw(90pt,0pt) node[anchor=center] (E4) {};
\fill[black] (E1) circle (1.5pt);
\fill[black] (E2) circle (1.5pt);
\draw (E3) node[anchor=center]{$\cdots$};
\fill[black] (E4) circle (1.5pt);
\node[black,below,shift=(90:2pt)] at (E1.south) {$\scriptscriptstyle -(n+2)\ $};
\node[black,below,shift=(90:2pt)] at (E2.south) {$\scriptscriptstyle -2$};
\node[black,below,shift=(90:2pt)] at (E4.south) {$\scriptscriptstyle -2$};
\draw[-] (E1.east) -- (E2.west);
\draw[-] (E2.east) -- (E3.west);
\draw[-] (E3.east) -- (E4.west);
\end{tikzpicture} }\!\!($(n-1)$ vertices).
Let $\iota \colon Y \to \tilde X_0$ be the contraction of $E_2,\ldots,E_r$\,(see Proposition~\ref{prop:HackingWtdBlup}\ref{item:prop:HackingWtdBlup}). As noted in Remark~\ref{rmk:SimplestSingularCase}, $W_1$ is isomorphic to $\P^2$, $Z_1$ is a smooth conic in $W_1$, hence $\mathcal O_{W_1}(1)\big\vert_{Z_1} = \mathcal O_{Z_1}(2)$. Also,
\begin{equation}
W_2 \simeq \P_{x,y,z}(1,n-1,1),\quad Z_2 = (xy=z^n) \subset W_2,\quad \text{and}\quad \mathcal O_{W_2}(n-1)\big\vert_{Z_2} = \mathcal O_{Z_2}(n). \label{eq:SecondWtdBlowupExceptional}
\end{equation}
The last equality can be verified as follows: let $h_{W_2} = c_1(\mathcal O_{W_2}(1))$, then $(n-1)h_{W_2}^2 = 1$, so \linebreak$\bigl( c_1(\mathcal O_{W_2}(n-1)) \mathbin . Z_2\bigr) = \bigl( (n-1)h_{W_2} \mathbin. nh_{W_2} \bigr) = n$.
In what follows, we construct exceptional vector bundles on the reducible surface $\tilde{\mathcal X_0} = \tilde X_0 \cup W_1 \cup W_2$ by gluing suitable vector bundles on each irreducible component which have isomorphic restrictions to the intersection curves $Z_i$.
\begin{proposition}\label{prop:VectorBundleOnReducibleSurface}
Let $D \in \Pic Y$ be a divisor such that $(D.E_i) = 0$ for $i=2,\ldots,r$.
\begin{enumerate}
\item Assume that $(D.C_1) =2d_1 \in 2\Z$, $(D.C_2) = nd_2\in n\Z$. Then, there exists a line bundle $\tilde {\mathcal D}$ on the reducible surface $\tilde{\mathcal X}_0 = \tilde X_0 \cup W_1 \cup W_2$ satisfying
\[
\tilde{\mathcal D}\big\vert_{\tilde X_0} \simeq \mathcal O_{\tilde X_0}(\iota_* D),\quad
\tilde{\mathcal D}\big\vert_{W_1} \simeq \mathcal O_{W_1}(d_1),\quad\text{and}\quad
\tilde{\mathcal D}\big\vert_{W_2} \simeq \mathcal O_{W_2}((n-1)d_2).
\]
\item Assume that $(D.C_1) = 1$, $(D.C_2) = 0$, and that there exists an exceptional vector bundle $G_1$ of rank $2$ on $W_1$ such that $G_1\big\vert_{Z_1} \simeq \mathcal O_{Z_1}(1)^{\oplus 2}$. Then, there exists a vector bundle $\tilde{\mathcal V}_1$ on $\tilde{\mathcal X}_0$ satisfying
\[
\tilde{\mathcal V}_1 \big\vert_{\tilde X_0} \simeq \mathcal O_{\tilde X_0}(\iota_* D)^{\oplus 2},\quad
\tilde{\mathcal V}_1 \big\vert_{W_1} \simeq G_1,\quad\text{and}\quad
\tilde{\mathcal V}_1 \big\vert_{W_2} \simeq \mathcal O_{W_2}^{\oplus 2}.
\]
\item Assume that $(D.C_1)=0$, $(D.C_2) = 1$, and that there exists an exceptional vector bundle $G_2$ of rank $n$ on $W_2$ such that $G_2 \big\vert_{Z_2} \simeq \mathcal O_{Z_2}(1)^{\oplus n}$. Then, there exists a vector bundle $\tilde{\mathcal V}_2$ on $\tilde{\mathcal X}_0$ satisfying
\[
\tilde{\mathcal V}_2 \big\vert_{\tilde X_0} \simeq \mathcal O_{\tilde X_0}(\iota_* D)^{\oplus n},\quad
\tilde{\mathcal V}_2 \big\vert_{W_1} \simeq \mathcal O_{W_1}^{\oplus n},\quad\text{and}\quad
\tilde{\mathcal V}_2 \big\vert_{W_2} \simeq G_2.
\]
\end{enumerate}
Furthermore, all the bundles introduced above are exceptional.
\end{proposition}
\begin{proof}
For all of those three cases, the ``ingredient bundles'' on irreducible components have isomorphic restrictions on $Z_i$, hence $\tilde{\mathcal E}(= \tilde{\mathcal D},\,\tilde{\mathcal V}_1,\,\tilde{\mathcal V}_2)$ exists as a vector bundle in the exact sequence\,(\textit{cf.}~\cite[Lemma~7.3]{Hacking:ExceptionalVectorBundle})
\begin{equation}
0 \to \tilde{\mathcal E} \to \tilde{\mathcal E}\big\vert_{\tilde X_0} \oplus \tilde{\mathcal E}\big\vert_{W_1} \oplus \tilde{\mathcal E}\big\vert_{W_2} \to \tilde{\mathcal E}\big\vert_{Z_1} \oplus \tilde{\mathcal E} \big\vert_{Z_2} \to 0.\label{eq:ExactSeq_onReducibleSurface}
\end{equation}
Conversely, given any vector bundle on $\tilde{\mathcal X}_0$, one can consider the exact sequence of the form (\ref{eq:ExactSeq_onReducibleSurface}). We plug the corresponding endomorphism sheaf into the sequence (\ref{eq:ExactSeq_onReducibleSurface}) to verify that $\tilde{\mathcal E}$ is exceptional.
Replacing $\tilde{\mathcal E}$ by $\varEnd(\tilde{\mathcal D}) \simeq \tilde{\mathcal D}^\vee \otimes \tilde{\mathcal D} \simeq \mathcal O_{\tilde{\mathcal X}_0}$, we rewrite (\ref{eq:ExactSeq_onReducibleSurface}) as
\[
0 \to \mathcal O_{\tilde{\mathcal X}_0} \to \mathcal O_{\tilde X_0} \oplus \mathcal O_{W_1} \oplus \mathcal O_{W_2} \to \mathcal O_{Z_1} \oplus \mathcal O_{Z_2} \to 0.
\]
Looking at cohomologies, we can easily verify that $h^p(\mathcal O_{\tilde{\mathcal X}_0}) = h^p( \mathcal O_{\tilde X_0})$. Using the same argument as in Proposition~\ref{prop:SingularSurfaceX}(a), we find
\[
H^p(\mathcal O_{\tilde X_0}) = \left\{
\begin{array}{ll}
\C & p=0 \\
0 & p\neq 0.
\end{array}
\right.
\]
Since $\tilde{\mathcal D}$ is locally free, $\varExt^q(\tilde{\mathcal D},\tilde{\mathcal D})=0$ for $q \neq 0$. By the local-to-global spectral sequence
\[
E_2^{p,q} = H^p( \varExt^q(\tilde{\mathcal D},\tilde{\mathcal D})) \Rightarrow H^{p+q}( \End(\tilde{\mathcal D})),
\]
$h^p(\End(\tilde{\mathcal D})) \simeq \dim_\C E_2^{p,0} = h^p(\mathcal O_{\tilde{\mathcal X}_0})$, showing that $\tilde{\mathcal D}$ is an exceptional line bundle. Now, we consider (\ref{eq:ExactSeq_onReducibleSurface}) for $\tilde{\mathcal E} = \varEnd(\tilde{\mathcal V}_1)$ which reads
\[
0 \to \varEnd(\tilde{\mathcal V}_1) \to \mathcal O_{\tilde X_0}^{\oplus 4} \oplus \varEnd(G_1) \oplus \mathcal O_{W_2}^{\oplus 4} \to \mathcal O_{Z_1}^{\oplus 4} \oplus \mathcal O_{Z_2}^{\oplus 4} \to 0.
\]
Since the restrictions $H^0(\mathcal O_{\tilde X_0}) \to H^0(\mathcal O_{Z_1})$, $H^0(\mathcal O_{W_2}) \to H^0(\mathcal O_{Z_2})$ are surjective, we have $h^p(\varEnd(\tilde{\mathcal V}_1)) = h^p(\varEnd(G_1))$. Using the local-to-global spectral sequences for the sheaves $\varEnd(\tilde{\mathcal V}_1)$, $\varEnd(G_1)$ and proceed as done in (a), we can conclude that $h^p(\End(\tilde{\mathcal V}_1)) = h^p(\varEnd(\tilde{\mathcal V}_1)) = h^p(\varEnd(G_1)) = h^p(\End(G_1))$, thus $\tilde{\mathcal V}_1$ is an exceptional vector bundle. Similarly, one can prove that $\tilde{\mathcal V}_2$ is an exceptional vector bundle. \qedhere
\end{proof}
We use Proposition~\ref{prop:DesiredDivisorOnY} to find the divisors satisfying the conditions described in Proposition~\ref{prop:VectorBundleOnReducibleSurface}(b) and (c).
\begin{lemma}\label{lem: Divisor for Higher ranks}
Let $N_1,N_2$ be solutions of the systems of congruence equations
\[
N_1 \equiv \left\{
\begin{array}{ll}
1 \mod 4 \\
0 \mod n^2
\end{array}
\right. \qquad
N_2 \equiv \left\{
\begin{array}{ll}
0 \mod 4 \\
1 \mod n^2\,.
\end{array}
\right.
\]
Then,
\begin{enumerate}
\item there are integers $e,e_1,\ldots,e_r \in \Z$ such that $V_1 := N_1 F_1 + e C_1 + e_1 C_2 + e_2 E_2 + \ldots + e_r E_r$ satisfies $(V_1.C_1)=1$ and $(V_1.C_2)=(V_1.E_2)=\ldots=(V_1.E_r)=0$;
\item there are integers $f,f_1,\ldots,f_r \in \Z$ such that $V_2 := N_2 F_1 + f C_1 + f_1 C_2 + f_2 E_2 + \ldots + f_r E_r$ satisfies $(V_2.C_2)=1$ and $(V_2.C_1)=(V_2.E_2)=\ldots=(V_2.E_r)=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
By the choices of $N_1,N_2$, we have
\[
\bigl( [\pi_* (N_i F_1)] \cap L_1 ,\, [\pi_*(N_i F_1)] \cap L_2 \bigr) = \left\{
\begin{array}{ll}
( \alpha_{C_1},\, 0 ) & i=1 \\
( 0,\, \alpha_{C_2}) & i=2
\end{array}
\right.
\]
in $H_1(L_1) \oplus H_1(L_2)$. Applying Proposition~\ref{prop:DesiredDivisorOnY}, we get the desired result.
\end{proof}
Referring to Proposition~\ref{prop:VectorBundleOnReducibleSurface} and Lemma~\ref{lem: Divisor for Higher ranks}, we can assemble several exceptional vector bundles on the reducible surface $\tilde{\mathcal X_0} = \tilde X_0 \cup W_1 \cup W_2$\,(Table~\ref{table:ExcBundles_OnSingular}).
\begin{center}
\begin{tabular}{c|c|c|c}
\raisebox{-5pt}[11pt][6pt]{}$\tilde{\mathcal X}_0$ & $\tilde X_0$ & $W_1$ & $W_2$ \\
\hline \raisebox{-5pt}[11pt][7pt]{}$\mathcal O_{\tilde{\mathcal X}_0}$ & $\mathcal O_{\tilde X_0}$ & $\mathcal O_{W_1}$ & $\mathcal O_{W_2}$ \\
\raisebox{-5pt}[11pt][7pt]{}$\tilde{\mathcal F}_{ij}\,{\scriptstyle (1 \leq i\neq j \leq 9)}$ & $\mathcal O_{\tilde X_0}(\iota_*(F_i - F_j))$ & $\mathcal O_{W_1}$ & $\mathcal O_{W_2}$ \\
\raisebox{-5pt}[11pt][7pt]{}$\tilde{\mathcal C}_0$ & $\mathcal O_{\tilde X_0}(\iota_*C_0)$ & $\mathcal O_{W_1}$ & $\mathcal O_{W_2}$ \\
\raisebox{-1pt}[11pt][7pt]{$\tilde{\mathcal K}$} & $\mathcal O_{\tilde X_0}(\iota_* K_Y)$ & $\mathcal O_{W_1}(1)$ & $\mathcal O_{W_2}(n-1)$ \\
\raisebox{-1pt}[11pt][7pt]{$\tilde{\mathcal V}_1$} & $\mathcal O_{\tilde X_0}(\iota_*V_1)^{\oplus 2}$ & $\mathcal T_{W_1}(-1)$ & $\mathcal O_{W_2}^{\oplus 2}$ \\
\raisebox{-1pt}[11pt][7pt]{$\tilde{\mathcal V}_2$} & $\mathcal O_{\tilde X_0}(\iota_*V_2)^{\oplus n}$ & $\mathcal O_{W_1}^{\oplus n}$ & $\mathcal \mathcal G_2$
\end{tabular}\vskip-0.33\baselineskip
\nopagebreak\captionof{table}{Examples of exceptional vector bundles constructed using Proposition~\ref{prop:VectorBundleOnReducibleSurface}}\label{table:ExcBundles_OnSingular}
\end{center}
Standard arguments, such as \cite[p.~1181]{Hacking:ExceptionalVectorBundle}, in the deformation theory say that if an exceptional vector bundle $\tilde{\mathcal D}$ is given in the central fiber of the family $\tilde{\mathcal X}/(0 \in T)$, then it deforms uniquely in a small neighborhood of the family, \textit{i.e.} there exists a vector bundle $\mathscr D$ on $\tilde{\mathcal X}$\,(shrinking $T$ if necessary) such that $\mathscr D \big\vert_{\tilde{ \mathcal X}_0} = \tilde{\mathcal D}$.
\begin{proposition}\label{prop:Hacking vs Topological}
Let $\tilde {\mathcal D}$ be the exceptional line bundle on the reducible surface $\tilde{\mathcal X}_0$ obtained in Proposition~\ref{prop:VectorBundleOnReducibleSurface}. Let $\mathscr D$ be a line bundle on $\tilde{\mathcal X}$ such that $\mathscr D \big\vert_{\tilde{\mathcal X}_0} = \tilde{\mathcal D}$. Then, $\mathscr D\big\vert_{X^\gen} = \mathcal O_{X^\gen}(D^\gen)$ where $D^\gen$ is the divisor introduced in \textsection\ref{subsec:TopologyofX}.
\end{proposition}
\begin{proof}
Let $\mathcal B \subset \mathcal X$ be the disjoint union of two small balls around $P_i \in \mathcal X$, and let $\tilde{\mathcal B} = \Phi^{-1}\mathcal B$.
Using the argument in \cite[p.~1192]{Hacking:ExceptionalVectorBundle}, we observe that the class $c_1\bigl(\mathscr D\big\vert_{\tilde{\mathcal X}_t \setminus \tilde{\mathcal B}_t}\bigr) \in H^2(\tilde{\mathcal X}_t \setminus \tilde{\mathcal B}_t)$ is independent of $t$ when we identify groups $\{H^2(\tilde{\mathcal X}_t \setminus \tilde{\mathcal B}_t)\}_t$ in the natural way. For $t=0$, Poincar\'e duality on manifolds with boundaries gives a sequence of isomorphisms
\begin{equation}
H^2(\tilde{\mathcal X}_0 \setminus \tilde{\mathcal B}_0) \simeq H_2(X \setminus B, \partial B) \simeq H_2(X,B) \simeq H_2(X), \label{eq: Specialization as Poincare dual}
\end{equation}
which convey $c_1(\mathscr D\big\vert_{\tilde{\mathcal X}_0 \setminus \tilde{\mathcal B}_0})$ to $[\pi_*D] \in H_2(X)$. As topological cycles, both $c_1(D^\gen\big\vert_{\tilde{\mathcal X}_t \setminus \tilde{\mathcal B}_t})$ and $c_1\bigl(\mathscr D\big\vert_{\tilde{\mathcal X}_t \setminus \tilde{\mathcal B}_t}\bigr)$ are obtained from $[\pi_*D\big\vert_{X \setminus B}]$ by the trivial extension, hence they coincide. The injective map $H_2(X^\gen) \to H_2(X)$ defined in Proposition~\ref{prop:Hacking_Specialization} is nothing but the natural restriction $H^2(X^\gen) \to H^2(X^\gen \setminus M)$, where the source and the target are changed by Poincar\'e duality on manifolds with boundaries. Thus, $H^2(X^\gen) \to H^2(X^\gen \setminus M)$ is injective, so $c_1(D^\gen) = c_1(\mathscr D\big\vert_{X^\gen})$. The first Chern class map $c_1 \colon \Pic X^\gen \to H^2(X^\gen,\Z)$ is an isomorphism, hence $\mathcal O_{X^\gen}(D^\gen) = \mathscr D\big\vert_{X^\gen}$.
\end{proof}
We finish this section by presenting an exceptional collection of length $9$ on the Dolgachev surface $X^\gen$. Note that this collection cannot generate the whole category $\D^{\rm b}(X^\gen)$.
\begin{proposition}\label{prop:ExceptCollection_ofLengthNine}
Let $F_{1j}^\gen\, (j>1)$ be the divisor on $X^\gen$, which arises from the deformation of $\tilde {\mathcal F}_{1j}$ along $\tilde{\mathcal X} / (0 \in T')$. Then the ordered tuple
\[
\bigl\langle \mathcal O_{X^\gen},\, \mathcal O_{X^\gen}(F_{12}^\gen),\,\ldots,\, \mathcal O_{X^\gen}(F_{19}^\gen) \bigr\rangle
\]
forms an exceptional collection in the derived category $\D^{\rm b}(X^\gen)$.
\end{proposition}
\begin{proof}
By virtue of upper-semicontinuity, it suffices to prove that $H^p(\tilde{\mathcal X}_0, \tilde{\mathcal F}_{1i} \otimes \tilde{\mathcal F}_{1j}^\vee) = 0$ for $1 \leq i < j \leq 9$ and $p \geq 0$. The sequence (\ref{eq:ExactSeq_onReducibleSurface}) for $\tilde{\mathcal E} = \tilde{\mathcal F}_{1i} \otimes \tilde{\mathcal F}_{1j}^\vee$ reads
\[
0 \to \tilde{\mathcal F}_{1i} \otimes \tilde{\mathcal F}_{1j}^\vee \to \mathcal O_{\tilde X_0}(\iota_* ( F_j - F_i)) \oplus \mathcal O_{W_1} \oplus \mathcal O_{W_2} \to \mathcal O_{Z_1} \oplus \mathcal O_{Z_2} \to 0.
\]
Since $H^0(\mathcal O_{W_k}) \simeq H^0(\mathcal O_{Z_k})$ and $H^p(\mathcal O_{W_k}) = H^p (\mathcal O_{Z_k}) = 0$ for $k=1,2$ and $p > 0$, it suffices to prove that $H^p(\mathcal O_{\tilde X_0} ( \iota_*(F_j - F_i) ) ) = 0$ for all $p\geq 0$ and $i< j$. The surface $\tilde X_0$ is normal({\it cf.} \cite[p.~1178]{Hacking:ExceptionalVectorBundle}) and the divisor $F_j - F_i$ does not intersect with the exceptional locus of $\iota \colon Y \to \tilde X_0$. By Proposition~\ref{prop:CohomologyComparison_YtoX}, $H^p( \tilde X_0, \iota_*(F_j-F_i)) \simeq H^p(Y, F_j-F_i)$ for all $p \geq 0$. It remains to prove that $H^p(Y, F_j-F_i) = 0$ for $p \geq 0$. By Riemann-Roch,
\[
\chi(F_j-F_i) = \frac{1}{2} ( F_j-F_i \mathbin. F_j - F_i - K_Y) + 1,
\]
and this is zero by Lemma~\ref{lem:CanonicalofY}. Since $(F_j\mathbin.F_j - F_i) = -1$ and $F_i \simeq \P^1$, in the short exact sequence
\[
0 \to \mathcal O_Y(-F_i) \to \mathcal O_Y(F_j-F_i) \to \mathcal O_{F_i}(F_j) \to 0,
\]
we obtain $H^0(-F_i) \simeq H^0(F_j-F_i)$. In particular, $H^0(F_j-F_i) = 0$. By Serre duality and Lemma~\ref{lem:CanonicalofY}, $H^2(F_j-F_i) = H^0(E_1 + F_i - F_j - C_2 - \ldots - E_{r+1})^*$. Similarly, since $(E_1\mathbin.E_1 + F_i - F_j - C_2 - \ldots - E_{r+1}) < 0$, $(F_i\mathbin.F_i - F_i - C_2 - \ldots - E_{r+1}) < 0$, and $E_1$, $F_j$ are rational curves, $H^0( E_1 + F_i - F_j - C_2 - \ldots - E_{r+1}) \simeq H^0(-F_j - C_2 - \ldots - E_{r+1}) = 0$. This proves that $H^2(F_j-F_i) = 0$. Finally, $\chi (F_j - F_i) = 0$ implies $H^1(F_j - F_i) =0$.
\end{proof}
\begin{remark}\label{rmk:ExceptCollection_SerreDuality}
In Proposition~\ref{prop:ExceptCollection_ofLengthNine}, the trivial bundle $\mathcal O_{X^\gen}$ can be replaced by the deformation of the line bundle $\tilde{\mathcal K}^\vee$\,(Table~\ref{table:ExcBundles_OnSingular}). The strategy of the proof differs nothing. Since $\tilde{\mathcal K}^\vee$ deforms to $\mathcal O_{X^\gen}(-K_{X^\gen})$, taking dual shows that
\[
\bigl\langle \mathcal O_{X^\gen}(F_{21}^\gen),\,\ldots,\, \mathcal O_{X^\gen}(F_{91}^\gen) ,\, \mathcal O_{X^\gen}(K_{X^\gen}) \bigr\rangle
\]
is also an exceptional collection in $\D^{\rm b}(X^\gen)$. This will be used later\,(see Step~\ref{item:ProofFreePart_thm:ExceptCollection_MaxLength} in the proof of Theorem~\ref{thm:ExceptCollection_MaxLength}).
\end{remark}
\section{The N\'eron-Severi lattices of Dolgachev surfaces of type $(2,3)$}\label{sec:NeronSeveri}
This section is devoted to study the simplest case, namely, the case $n=3$ and $a=1$. The surface $Y$ has the configuration as in Figure~\ref{fig:Configuration_Basic}. We cook up several divisors on $X^\gen$ according to the recipe designed below.
\begin{recipe}\label{recipe:PicardLatticeOfDolgachev}
Recall that $\pi \colon Y \to X$ is the contraction of $C_1, C_2, E_2$ and $\iota \colon Y \to \tilde X_0$ is the contraction of $E_2$.
\begin{enumerate}[label=\normalfont(\arabic{enumi})]
\item Pick a divisor $D \in \Pic Y$ satisfying $(D.C_1) \in 2 \Z$, $(D. C_2) \in 3\Z$, and $(D. E_2) = 0$.
\item As in Proposition~\ref{prop:VectorBundleOnReducibleSurface}, attach suitable line bundles on $W_i$\,($i=1,2$) to $\mathcal O_{\tilde X_0}(\iota_* D)$ to produce a line bundle, say $\tilde{\mathcal D}$, on $\tilde{\mathcal X}_0 = \tilde X_0 \cup W_1 \cup W_2$. It deforms to a line bundle $\mathcal O_{X^\gen}(D^\gen)$ on the Dolgachev surface $X^\gen$.
\item Use the short exact sequence (\ref{eq:ExactSeq_onReducibleSurface}) to compute $\chi(\tilde {\mathcal D})$. By the deformation invariance of Euler characteristics, $\chi(D^\gen) = \chi(\tilde {\mathcal D})$.
\item\label{item:recipe:CanonicalIntersection} Since the divisor $\pi_* C_0$ is away from the singularities of $X$, it is Cartier. By Lemma~\ref{lem:Intersection_withFibers}, $(C_0^\gen.D^\gen) = (C_0.D)$. Furthermore, $C_0^\gen = 6K_{X^\gen}$, thus the Riemann-Roch formula on the surface $X^\gen$ reads
\[
(D^\gen)^2 = \frac{1}{6}( D . C_0) + 2 \chi(\tilde {\mathcal D}) - 2.
\]
This computes the intersections of divisors in $X^\gen$.
\end{enumerate}
\end{recipe}
By Proposition~\ref{prop:Hacking vs Topological}, $D^\gen$ is essentially determined by looking at the preimage of the cycle class $[\pi_*D] \in H_2(X)$ along the map in the sequence (\ref{eq:SpecializationSequence}). This suggests the following use of the terminology.
\begin{definition}
Let $D \in \Pic Y$ and $D^\gen \in \Pic X^\gen$ be as in Recipe~\ref{recipe:PicardLatticeOfDolgachev}. We call $D^\gen$ the \emph{lifting} of $D$.
\end{definition}
We note that this is a slight abuse of terminologies. What lifts to $D^\gen \in \Pic X^\gen$ is $\pi_* D \in \Cl X$, not $D \in \Pic Y$.
\begin{lemma}\label{lem:EulerChar_WtdProj}
Let $h \in H_2(W_2,\Z)$ be the hyperplane class of $W_2 = \P(1,2,1)$. For any even integer $n \in \Z$,
\[
\chi( \mathcal O_{W_2}(n) ) = \frac{1}{4}n(n+4) + 1.
\]
\end{lemma}
\begin{proof}
By well-known properties of weighted projective spaces, $(1 \cdot 2 \cdot 1)h^2 = 1$, $c_1(K_{W_2}) = -(1+2+1)h = -4h$, and $\mathcal O_{W_2}(2)$ is invertible. The Riemann-Roch formula for invertible sheaves\,({\it cf.} \cite[Lemma~7.1]{Hacking:ExceptionalVectorBundle}) says that $\chi(\mathcal O_{W_2}(n)) = \frac{1}{2}( nh \mathbin. (n+4)h) + 1 = \frac{1}{4}n(n+4) + 1$.
\end{proof}
\begin{lemma}\label{lem:EulerCharacteristics}
Let $S$ be a projective normal surface with $\chi(\mathcal O_S) = 1$. Assume that all the divisors below are supported on the smooth locus of $S$. Then,
\begin{enumerate}[ref=(\alph{enumi})]
\item $\chi(D_1 + D_2) = \chi(D_1) + \chi(D_2) + (D_1 . D_2) - 1$;
\item $\chi(-D) = -\chi(D) + D^2 + 2$;
\item $\chi(-D) = p_a(D)$ where $p_a(D)$ is the arithmetic genus of $D$;
\item $\chi(nD) = n\chi(D) + \frac{1}{2} n(n-1)D^2 - n + 1$ for all $n \in \Z$.
\item[\rm(d$'$)] $\chi(nD) = n^2\chi(D) + \frac{1}{2}n(n-1) (K_S .D) - n^2 + 1$ for all $n \in \Z$.
\end{enumerate}
Assume in addition that $D$ is an integral curve with $p_a(D) = 0$. Then
\begin{enumerate}[resume]
\item $\chi(D) = D^2 + 2$, $\chi(-D) = 0$;
\item $\chi(nD) = \frac{1}{2}n(n+1)D^2 + (n+1)$ for all $n \in \Z$.
\end{enumerate}
\end{lemma}
\begin{proof}
All the formula in the statement are simple variants of the Riemann-Roch formula.
\end{proof}
\begin{lemma}\label{lem:Intersection_withFibers}
Let $D$, $\tilde{\mathcal D}$, $D^\gen$ be as in Recipe~\ref{recipe:PicardLatticeOfDolgachev}. Then, $(C_0.D) = (C_0^\gen . D^\gen)$.
\end{lemma}
\begin{proof}
Since $C_0$ does not intersect with $C_1,C_2,E_2$, the corresponding line bundle $\tilde{\mathcal C}_0$ on $\tilde{\mathcal X}_0$ is the gluing of $\mathcal O_{\tilde X_0}(\iota_*C_0)$, $\mathcal O_{W_1}$, and $\mathcal O_{W_2}$. Thus, $(\tilde{\mathcal D} \otimes \tilde{\mathcal C}_0) \big\vert_{W_i} = \tilde{\mathcal D}\big\vert_{W_i}$ for $i=1,2$. From this and (\ref{eq:ExactSeq_onReducibleSurface}), it can be immediately shown that $\chi(\tilde{\mathcal D} \otimes \tilde{\mathcal C}_0 ) - \chi(\tilde{\mathcal D}) = \chi(D + C_0 ) - \chi (D)$. If $D$ is a principal divisor on $Y$, then the previous equation says $\chi(C_0^\gen) = \chi(\tilde{\mathcal C}_0) = \chi(C_0) = 1$. Now, using Lemma~\ref{lem:EulerCharacteristics}(a), we deduce $(C_0^\gen. D^\gen) = \chi(D^\gen + C_0^\gen) - \chi(D^\gen) = \chi(\tilde{\mathcal D} \otimes \tilde{\mathcal C}_0 ) - \chi(\tilde{\mathcal D}) = \chi(D + C_0 ) - \chi (D) = (C_0.D)$. \qedhere
\end{proof}
\begin{definition}
Let $H \in \Pic \P^2$ be a line, let $p \colon Y \to \P^2$ be the blow down morphism, and let $L = p^*(2H)$ be the proper transform of a general plane conic. Then, $( L . C_1) = 6$, $(L . C_2) = 6$ and $(L. E_2) = 0$. Let $L^\gen$ be the lifting of $L$. This means that there exists a line bundle $\tilde{\mathcal L}$ on the reducible surface $\tilde{\mathcal X}_0 = \tilde X_0 \cup W_1 \cup W_2$ such that
\[
\tilde{\mathcal L}\big\vert_{\tilde X_0} = \mathcal O_{\tilde X_0}(\iota_* L),\quad \tilde{\mathcal L}\big\vert_{W_1} = \mathcal O_{W_1}(3),\ \text{and}\quad \tilde{\mathcal L}\big\vert_{W_2} = \mathcal O_{W_2}(4),
\]
which deforms to the line bundle $\mathcal O_{X^\gen}(L^\gen)$ on $X^\gen$. Let $F_{ij}^\gen \in \Pic X^\gen$ be the lifting of $F_i - F_j$, or equivalently, the divisor associated with the deformation of $\tilde{\mathcal F}_{ij}$\,(Table~\ref{table:ExcBundles_OnSingular}). We define
\begin{align*}
G_i^\gen &:= - L^\gen + 10K_{X^\gen} + F_{i9}^\gen \quad \text{for }i=1,\ldots,8; \\
G_9^\gen &:= -L^\gen + 11K_{X^\gen}.
\end{align*}
\end{definition}
\begin{proposition}\label{prop:G_1to9}
The divisors $G_1^\gen,\ldots,G_9^\gen$ satisfy the following numerical properties:
\begin{enumerate}
\item $\chi(G_i^\gen) = 1$ and $(G_i^\gen . K_{X^\gen}) = -1$;
\item for $i < j$, $\chi(G_i^\gen - G_j^\gen)=0$.
\end{enumerate}
In particular, $(G_i^\gen)^2 = -1$ and $(G_i^\gen. G_j^\gen) = 0$ for $1 \leq i < j \leq 9$.
\end{proposition}
\begin{proof}
First, consider the case $i \leq 8$. By Recipe~\ref{recipe:PicardLatticeOfDolgachev}\ref{item:recipe:CanonicalIntersection} and $K_{X^\gen}^2 = 0$, $(K_{X^\gen} . G_i^\gen) = \frac{1}{6}(C_0 \mathbin. -L + F_i - F_9) = -1$.
Since the alternating sum of Euler characteristics in the sequence (\ref{eq:ExactSeq_onReducibleSurface}) is zero, we get the formula
\begin{align*}
\chi( \tilde{\mathcal L}^\vee \otimes \tilde{\mathcal F}_{i9}) ={}& \chi(-L + F_i - F_9) + \chi(\mathcal O_{W_1}(-3)) + \chi(\mathcal O_{W_2}(-4)) \\
&{}- \chi(\mathcal O_{Z_1}(-6)) - \chi(\mathcal O_{Z_2}(-6)).
\end{align*}
From this we compute $\chi( \tilde{\mathcal L}^\vee \otimes \tilde{\mathcal F}_{i9}) = 11$.
The Riemann-Roch formula for $-L^\gen + F_{i9}^\gen = G_i^\gen - 10K_{X^\gen}$ says $(G_i^\gen - 10K_{X^\gen})^2 - (K_{X^\gen} \mathbin. G_i^\gen - K_{X^\gen} ) = 20$, hence $(G_i^\gen)^2 = -1$ and $\chi(G_i^\gen) = 1$. For $1 \leq i < j \leq 8$, $G_i - G_j = F_i - F_j$. Since $(F_i - F_j \mathbin. C_1 ) = (F_i - F_j \mathbin. C_2 ) = (F_i - F_j \mathbin. E_2 ) = 0$, the divisor $F_i - F_j$ lifts to the Cartier divisor $F_{ij}^\gen$. Hence, we can compute $\chi(G_i^\gen - G_j^\gen) = \chi(F_i - F_j) = 0$. This proves the statement for $i,j \leq 8$. The proof of the statement involving $G_9^\gen$ follows the same lines. Since $\chi(\tilde{\mathcal L}^\vee ) = 12$, $( G_9^\gen - 11K_{X^\gen} ) ^2 - (K_{X^\gen}\mathbin . G_9^\gen - 11K_{X^\gen}) = 22$. This leads to $(G_9^\gen)^2 = -1$. For $i \leq 8$,
\begin{align*}
\chi(G_i^\gen - G_9^\gen) ={}& \chi(F_i - F_9 - K_Y) + \chi(\mathcal O_{W_1}(-1)) + \chi(\mathcal O_{W_2}(-2)) \\
&{} - \chi(\mathcal O_{Z_1}(-2)) - \chi(\mathcal O_{Z_2}(-3)),
\end{align*}
and it is immediate to see that the right hand side is zero.
\end{proof}
We complete our list of generators of $\Pic X^\gen$ by introducing $G_{10}^\gen$. The choice of $G_{10}^\gen$ is motivated by the proof of the step (iii)${}\Rightarrow{}$(i) in \cite[Theorem~3.1]{Vial:Exceptional_NeronSeveriLattice}.
\begin{proposition}\label{prop:G_10}
Let $G_{10}^\gen$ be the $\Q$-divisor $\frac{1}{3}( G_1^\gen + G_2^\gen + \ldots + G_9^\gen - K_{X^\gen})$. Then, $G_{10}^\gen$ is Cartier.
\end{proposition}
\begin{proof}
Since
\[
\sum_{i=1}^9 G_i^\gen - K_{X^\gen} = - 9L^\gen + 90 K_{X^\gen} + \sum_{i=1}^8 F_{i9}^\gen,
\]
it suffices to prove that $\sum\limits_{i=1}^8 F_{i9}^\gen = 3D^\gen$ for some $D^\gen \in \Pic X^\gen$. Let $p \colon Y \to \P^2$ be the blowing up morphism and let $H$ be a line in $\P^2$. Since $K_Y = p^* (-3H) + F_1 + F_2 + \ldots + F_9 + E_1 + E_2 + 2E_3$, $K_Y - E_1 - E_2 - 2E_3 = p^*(-3H) + F_1 + \ldots + F_9 = -C_0$, so $F_1 + \ldots + F_9 = 3p^*H - C_0$. Consider the divisor $p^*H - 3F_9$ in $Y$. Clearly, the intersections of $(p^*H - 3F_9)$ with $C_1$, $C_2$, $E_2$ are all zero, hence $p^*H - 3F_9$ lifts to a Cartier divisor $(p^*H-3F_9)^\gen$ in $X^\gen$. Since
\begin{align*}
\sum_{i=1}^8 (F_i - F_9) &= \sum_{i=1}^9 F_i - 9F_9 \\
&= 3(p^*H - 3F_9) - C_0
\end{align*}
and $C_0$ lifts to $6K_{X^\gen}$, $D^\gen := (p^*H - 3F_9)^\gen - 2K_{X^\gen}$ satisfies $\sum_{i=1}^8F_{i9}^\gen = 3D^\gen$.
\end{proof}
Combining the propositions \ref{prop:G_1to9} and \ref{prop:G_10}, we obtain:\nopagebreak
\begin{theorem}\label{thm:Picard_ofGeneralFiber}
The intersection matrix of divisors $\{G_i^\gen\}_{i=1}^{10}$ is
\begin{equation}
\Bigl( (G_i^\gen . G_j^\gen ) \Bigr)_{1 \leq i,j \leq 10} = \left[
\begin{array}{cccc}
-1 & \cdots & 0 & 0 \\
\vdots & \ddots & \vdots & \vdots \\
0 & \cdots & -1 & 0 \\
0 & \cdots & 0 & 1
\end{array}
\right]\raisebox{-2\baselineskip}[0pt][0pt]{.} \label{eq:IntersectionMatrix}
\end{equation}
In particular, the set $G:=\{G_i^\gen\}_{i=1}^{10}$ forms a $\Z$-basis of the N\'eron-Severi lattice $\op{NS}(X^\gen)$. By \cite[p.~137]{Dolgachev:AlgebraicSurfaces}, $\Pic X^\gen$ is torsion-free, thus $G$ forms a $\Z$-basis for $\Pic X^\gen$.
\end{theorem}
\begin{proof}
We claim that the set of divisors $\{G_i^\gen\}_{i=1}^{10}$ generates the N\'eron-Severi lattice. By Hodge index theorem, there is a $\Z$-basis for $\op{NS}(X^\gen)$, say $\alpha = \{\alpha_i\}_{i=1}^{10}$, such that the intersection matrix with respect to $\{\alpha_i\}_{i=1}^{10}$ is the same as (\ref{eq:IntersectionMatrix}). Let $A = ( a_{ij} )_{1 \leq i,j \leq 10}$ be the integral matrix determined by
\[
G_i^\gen = \sum_{j=1}^{10} a_{ij} \alpha_j.
\]
Given $v \in \op{NS}(X^\gen)$, let $[v]_\alpha$ be the column matrix of coordinates with respect to the basis $\alpha$. Then, $[G_i^\gen]_\alpha = Ae_i$ where $e_i$ is the $i$th column vector. For $1 \leq i,j \leq 10$,
\[
(G_i^\gen . G_j^\gen) = (A e_i)^t E (A e_j)
\]
where $E$ is the intersection matrix with respect to the basis $\alpha$. The above equation implies that the intersection matrix with respect to the set $G$ is $A^{\rm t} E A$. Since the intersection matrices with respect to both $G$ and $\alpha$ are the same, $E = A^{\rm t} E A$. This implies that $1 = \det (A^{\rm t}A) = (\det A)^2$, hence $A$ is invertible over $\Z$. This proves that $G$ is a $\Z$-basis of $\op{NS}(X^\gen)$. The last statement on the Picard group follows immediately.
\end{proof}
We close the section with the summary of divisors on $X^\gen$.
\begin{summary}\label{summary:Divisors_onX^g}
Recall that $Y$ is the rational elliptic surface in Section~\ref{sec:Construction}, $p \colon Y \to \P^2$ is the blow down morphism, $H \in \Pic \P^2$ is a hyperplane divisor, and $\pi \colon Y \to X$ is the contraction of $C_1,\,C_2,\,E_2$. Then,
\begin{enumerate}[label=\normalfont(\arabic{enumi})]
\item $F_{ij}^\gen$\,($1\leq i,j\leq 9$) is the lifting of $F_i - F_j$;
\item $(p^*H-3F_9)^\gen$ is the lifting of $p^*H - 3F_9$;
\item $L^\gen$ is the lifting of $p^*(2H)$;
\item $G_i^\gen = - L^\gen + 10 K_{X^\gen} + F_{i9}^\gen$ for $i=1,\ldots,8$;
\item $G_9^\gen = -L^\gen + 11K_{X^\gen}$;
\item $G_{10}^\gen = -3L^\gen + (p^*H - 3F_9)^\gen + 28K_{X^\gen}$.
\end{enumerate}
\end{summary}
\section{Exceptional collections of maximal length on Dolgachev surfaces of type $(2,3)$}\label{sec:ExcepCollectMaxLength}
\subsection{Exceptional collections of maximal length}
We continue to study the case $(n,a)=(3,1)$. Throughout this section, we will prove that there exists an exceptional collection of maximal length in $\D^{\rm b}(X^\gen)$. Proving exceptionality of a given collection usually consists of numerous cohomology computations, so we begin by introducing some computational machineries.
\begin{lemma}\label{lem:DummyBundle}
The liftings $C_1^\gen$, $(2C_2+E_2)^\gen$ exist and they are the zero divisors in $X^\gen$.
\end{lemma}
\begin{proof}
Let $\tilde{\mathcal C}_1$ be the gluing of line bundles $\mathcal O_{\tilde X_0}(\iota_* C_1)$, $\mathcal O_{W_1}(-2)$, and $\mathcal O_{W_2}$, and let $\mathcal O_{X^\gen}(C_1^\gen)$ be its deformation. It is immediate to see that $\chi(C_1^\gen) = 1$ and $\chi(-C_1^\gen)=1$. By Riemann-Roch formula, $(C_1^\gen)^2 = (C_1^\gen . K_{X^\gen}) = 0$. For $i \leq 8$,
\begin{align*}
\chi(G_i^\gen - 10K_{X^\gen} - C_1^\gen ) ={}& \chi( \tilde{\mathcal L}^\vee \otimes \tilde{\mathcal F}_{i9} \otimes \tilde{\mathcal C}_1^\vee ) \\
={}& \chi(-L + F_i - F_9 - C_1) + \chi(\mathcal O_{W_1}(-1)) + \chi(\mathcal O_{W_2}(-4)) \\
&{} - \chi(\mathcal O_{Z_1}(-2)) - \chi(\mathcal O_{Z_2}(-6)),
\end{align*}
which yields $\chi(G_i^\gen - 10K_{X^\gen} - C_1^\gen )=11$. By the Riemann-Roch, $(G_i^\gen - 10K_{X^\gen}- C_1^\gen)^2 - (K_{X^\gen} \mathbin . G_i^\gen - 10K_{X^\gen} - C_1^\gen) = 2\chi(G_i^\gen -10K_{X^\gen} - C_1^\gen)-2 = 20$. The left hand side is $-2(G_i^\gen . C_1^\gen) + 20$, thus $(G_i^\gen . C_1^\gen) =0$. Since $(C_1^\gen . K_{X^\gen}) = 0$ and $3G_{10}^\gen = G_1^\gen + \ldots + G_9^\gen - K_{X^\gen}$, $(G_{10}^\gen . C_1^\gen) = 0$. Hence, $C_1^\gen$ is numerically trivial by Theorem~\ref{thm:Picard_ofGeneralFiber}. This shows that $C_1^\gen$ is trivial since there is no torsion in $\Pic X^\gen$. Exactly the same argument holds for the lifting of $2C_2 + E_2$.
\end{proof}
\begin{example}\label{eg:DivisorVaries_onSingular}
Since $C_0$ lifts to $6K_{X^\gen}$, $2E_1 = C_0 - C_1$ lifts to $6K_{X^\gen}$. Thus $E_1$ lifts to $3K_{X^\gen}$. Similarly, $C_2 + E_2 + E_3$ lifts to $2K_{X^\gen}$. Hence, $K_Y = E_1 - C_2 - E_2 - E_3$ lifts to $3K_{X^\gen} - 2K_{X^\gen} = K_{X^\gen}$. Also, $(E_2 + 2E_3) - E_1$ lifts to $K_{X^\gen}$, whereas $K_Y$ and $(E_2+2E_3)-E_1$ are different in $\Pic Y$. These are essentially due to Lemma~\ref{lem:DummyBundle}. For instance, we have
\begin{align*}
(E_2 + 2E_3) - E_1 - K_Y &= - 2 E_1 + C_2 + 2E_2 + 3E_3 \\
&= -C_1,
\end{align*}
thus $(E_2 + 2E_3)^\gen - E_1^\gen - K_{X^\gen} = -C_1^\gen = 0$.
\end{example}
As Example~\ref{eg:DivisorVaries_onSingular} presents, there are free spaces to choose $D \in \Pic Y$ given a fixed divisor $D^\gen \in \Pic X^\gen$. The following lemma gives a direction to choose $D$. Note that the lemma requires assumptions on $(D.C_1)$ and $(D.C_2)$, but Lemma~\ref{lem:DummyBundle} provides the way to adjust those numbers.
\begin{lemma}\label{lem:H0Computation}
Let $D$ be a divisor in $Y$ such that $(D.C_1) = 2d_1 \in 2\Z$, $(D.C_2) = 3d_2 \in 3\Z$, and $(D . E_2) = 0$.
Let $D^\gen$ be the lifting of $D$. Then,
\[
h^0(X^\gen, D^\gen) \leq h^0(Y,D) + h^0(\mathcal O_{W_1}(d_1)) + h^0(\mathcal O_{W_2}(2d_2)) - h^0(\mathcal O_{Z_1}(2d_1)) - h^0(\mathcal O_{Z_2}(3d_3)).
\]
In particular, if $d_1,d_2 \leq 1$, then $h^0(X^\gen, D^\gen) \leq h^0(Y,D)$.
\end{lemma}
\begin{proof}
Since $(D. E_2) = 0$, we have $H^p(\tilde X_0,\iota_*D) \simeq H^p(Y,D)$ for all $p \geq 0$\,(Proposition~\ref{prop:CohomologyComparison_YtoX}). Recall that there exists a short exact sequence (introduced in (\ref{eq:ExactSeq_onReducibleSurface}))
\begin{equation}
0 \to \tilde{\mathcal D} \to \mathcal O_{\tilde X_0}(\iota_*D) \oplus \mathcal O_{W_1}(d_1) \oplus \mathcal O_{W_2}(2d_2) \to \mathcal O_{Z_1}(2d_1) \oplus \mathcal O_{Z_2}(3d_2) \to 0, \label{eq:ExactSeq_onReducibleSurface_SimplerVer}
\end{equation}
where $\tilde{\mathcal D}$ is the line bundle constructed in Proposition~\ref{prop:VectorBundleOnReducibleSurface}, and the notations $W_i$, $Z_i$ are explained in (\ref{eq:SecondWtdBlowupExceptional}). We first claim the following: if $d_1,d_2 \leq 1$, then the maps $H^0(\mathcal O_{W_1}(d_1)) \to H^0(\mathcal O_{Z_1}(2d_1))$ and $H^0(\mathcal O_{W_2}(2d_2)) \to H^0(\mathcal O_{Z_2}(3d_2))$ are isomorphisms. Only the nontrivial cases are $d_1 = 1$ and $d_2 = 1$. Since $Z_1$ is a smooth conic in $W_1 = \P^2$, there is a short exact sequence
\[
0 \to \mathcal O_{W_1}(-1) \to \mathcal O_{W_1}(1) \to \mathcal O_{Z_1}(2) \to 0.
\]
All the cohomology groups of $\mathcal O_{W_1}(-1)$ vanish, so $H^p( \mathcal O_{W_1}(1)) \simeq H^p(\mathcal O_{Z_1}(2))$ for all $p \geq 0$. In the case $d_2 = 1$, we consider
\[
0 \to \mathcal I_{Z_2}(2) \to \mathcal O_{W_2}(2) \to \mathcal O_{Z_2}(3) \to 0,
\]
where $\mathcal I_{Z_2} \subset \mathcal O_{W_2}$ is the ideal sheaf of the closed subscheme $Z_2 = (xy = z ^3 ) \subset \P_{x,y,z}(1,2,1)$. The ideal $(xy - z^3)$ does not contain any nonzero homogeneous element of degree $2$, so $H^0(\mathcal I_{Z_2}(2)) = 0$. This shows that $H^0( \mathcal O_{W_2}(2)) \to H^0(\mathcal O_{Z_2}(3))$ is injective. Furthermore, $H^0(\mathcal O_{W_2}(2))$ is generated by $x^2, xz, z^2, y$, hence $h^0(\mathcal O_{W_2}(2)) = h^0(\mathcal O_{Z_3}(3)) = 4$. This proves that $H^0(\mathcal O_{W_2}(2)) \simeq H^0(\mathcal O_{Z_3}(3))$, as desired. If $d_1,d_2>1$, it is clear that $H^0(\mathcal O_{W_1}(d_1)) \to H^0(\mathcal O_{Z_1}(2d_1))$ and $H^0(\mathcal O_{W_2}(2d_2)) \to H^0(\mathcal O_{Z_2}(3d_2))$ are surjective.
The cohomology long exact sequence of (\ref{eq:ExactSeq_onReducibleSurface_SimplerVer}) begins with
\begin{align*}
0 \to H^0(\tilde{\mathcal D}) \to H^0(\iota_*D) \oplus H^0( \mathcal O_{W_1}(d_1) ) \oplus H^0(\mathcal O_{W_2}(2d_2)) \\
\to H^0(\mathcal O_{Z_1}(2d_1)) \oplus H^0( \mathcal O_{Z_2}(3d_2)).\qquad\qquad
\end{align*}
By the previous arguments, the last map is surjective. Indeed, the image of $(0, s_1, s_2) \in H^0(\iota_*D) \oplus H^0( \mathcal O_{W_1}(d_1) ) \oplus H^0(\mathcal O_{W_2}(2d_2))$ is $(-s_1\big\vert_{Z_1}, -s_2\big\vert_{Z_2})$. The upper-semicontinuity of cohomologies establishes the inequality in the statement. \qedhere
\end{proof}
By \cite[Theorem~3.1]{Vial:Exceptional_NeronSeveriLattice}, it can be shown that the collection (\ref{eq:ExcColl_MaxLength}) in the theorem below is a numerically exceptional collection. Our aim is to prove that (\ref{eq:ExcColl_MaxLength}) is indeed an exceptional collection in $\D^{\rm b}(X^\gen)$. Before proceed to the theorem, we introduce one terminology.
\begin{definition}
During the construction of $Y$, the node of $p_*C_2$ is blown up twice, which corresponds to one of the two tangent directions\footnote{For example, the nodal curve $y^2 = x^3+x^2z$ has two tangent directions $y=\pm x$ at $(0,0,1)$.} at the node of $p_*C_2$. We refer to the tangent direction corresponding to the second blow up as the \emph{distinguished tangent direction} at the node of $p_*C_2$.
\end{definition}
\begin{theorem}\label{thm:ExceptCollection_MaxLength}
Suppose $X^\gen$ is originated from a cubic pencil $\lvert \lambda p_* C_1 + \mu p_*C_2 \rvert$ which is generated by two general plane nodal cubics. Let $G_1^\gen,\ldots,G_{10}^\gen$ be as in Summary~\ref{summary:Divisors_onX^g}, let $G_0^\gen$ be the zero divisor, and let $G_{11}^\gen = 2G_{10}^\gen$. For notational simplicity, we denote the rank of $\Ext^p(G_i^\gen, G_j^\gen)(=H^p(-G_i^\gen+G_j^\gen))$ by $h^p_{ij}$. The values of $h_{ij}^p$ are described below. For example, the triple of ($G_9^\gen$-row, $G_{10}^\gen$-column), which is $(0\ 0\ 2)$, means that $(h^0_{9,10},\, h^1_{9,10},\, h^2_{9,10}) = (0,0,2)$.
\begin{equation}
\scalebox{0.9}{$
\begin{array}{c|ccccc}
& G_0^\gen & G_{1 \leq i \leq 8}^\gen & G_9^\gen & G_{10}^\gen & G_{11}^\gen \\[2pt] \hline
G_0^\gen & 1\,0\,0 & 0\,0\,1 & 0\,0\,1 & 0\,0\,3 & 0\,0\,6 \\
G_{1 \leq i \leq 8}^\gen & & 1\,0\,0 & & 0\,0\,2 & 0\,0\,5\\
G_9^\gen & & & 1\,0\,0 & 0\,0\,2 & 0\,0\,5 \\
G_{10}^\gen & & & & 1\,0\,0 & 0\,0\,3 \\
G_{11}^\gen & & & & & 1\,0\,0
\end{array}
$}\label{eq:thm:ExceptCollection_MaxLength}
\end{equation}
The symbol $G_{1 \leq i \leq 8}^\gen$ means $G_i^\gen$ for each $i=1,2,\ldots,8$. The blanks stand for $0\,0\,0$, and $h^p_{ij} = 0$ for all $p$ and $1 \leq i\neq j \leq 8$. In particular, the collection
\begin{equation}
\big\langle \mathcal O_{X^\gen}(G_0^\gen),\ \mathcal O_{X^\gen}(G_1^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_{10}^\gen),\ \mathcal O_{X^\gen}(G_{11}^\gen) \big\rangle \label{eq:ExcColl_MaxLength}
\end{equation}
is an exceptional collection of length $12$ in $\D^{\rm b}(X^\gen)$.
\end{theorem}
\begin{proof}
Recall that (Summary~\ref{summary:Divisors_onX^g})
\begin{align*}
G_i^\gen &= -L^\gen + F_{i9}^\gen + 10K_{X^\gen},\ i=1,\ldots,8; \\
G_9^\gen &= -L^\gen + 11K_{X^\gen}; \\
G_{10}^\gen &= -3L^\gen + (p^*H - 3F_9)^\gen + 28K_{X^\gen}; \\
G_{11}^\gen &= -6L^\gen + 2(p^*H - 3F_9)^\gen + 56K_{X^\gen}.
\end{align*}
The proof consists of numerous cohomology vanishings for which we divide into several steps. Note that we can always evaluate $\chi(-G_i^\gen + G_j^\gen) = \sum_p (-1)^p h^p_{ij}$, thus it suffices to compute only two (mostly $h^0$ and $h^2$) of $\{h^p_{ij} : p=0,1,2\}$.
In the first part of the proof, we deduce the following using numerical methods.
\begin{equation}
\scalebox{0.9}{$
\begin{array}{c|ccccc}
& G_0^\gen & G_{1 \leq i \leq 8}^\gen & G_9^\gen & G_{10}^\gen & G_{11}^\gen \\[2pt] \hline
G_0^\gen & 1\,0\,0 & 0\,0\,1 & 0\,0\,1 & \scriptstyle\chi=3 & \scriptstyle\chi=6 \\
G_{1 \leq i \leq 8}^\gen & 0\,0\,0 & 1\,0\,0 & 0\,0\,0 & 0\,0\,2 & \scriptstyle\chi=5 \\
G_9^\gen &\scriptstyle\chi=0 & 0\,0\,0 & 1\,0\,0 & 0\,0\,2 & \scriptstyle\chi=5 \\
G_{10}^\gen &\scriptstyle\chi=0 &\scriptstyle\chi=0 &\scriptstyle\chi=0 & 1\,0\,0 & \scriptstyle\chi=3 \\
G_{11}^\gen &\scriptstyle\chi=0 &\scriptstyle\chi=0 &\scriptstyle\chi=0 &\scriptstyle\chi=0 & 1\,0\,0
\end{array}
$}\label{eq:HumanPart_thm:ExceptCollection_MaxLength}
\end{equation}
The slots with $\chi=d$ means $\chi(-G_i^\gen+G_j^\gen)=\sum_{p}(-1)^ph^p_{ij} =d$. For those slots, we do not compute each $h^p_{ij}$ for the moment. In the end, they will be completed through another approach.
\begin{enumerate}[fullwidth, itemsep=5pt minus 3pt, label=\bf{}Step~\arabic{enumi}., ref=\arabic{enumi}]
\item\label{item:NumericalStep_thm:ExceptCollection_MaxLength} As explained above, the collection (\ref{eq:ExcColl_MaxLength}) is numerically exceptional, hence $\chi(-G_i^\gen + G_j^\gen) = \sum_p (-1)^p h^p_{ij}=0$ for all $0 \leq j < i \leq 11$. Furthermore, the surface $X^\gen$ is minimal, thus $K_{X^\gen}$ is nef. It follows that $h^0(D^\gen) = 0$ if $D^\gen$ is $K_{X^\gen}$-negative, and $h^2(D^\gen)=0$ if $D^\gen$ is $K_{X^\gen}$-positive. Since
\[
(K_{X^\gen} . G_i^\gen) = \left\{
\begin{array}{ll}
-1 & i \leq 9 \\
-3 & i = 10 \\
-6 & i = 11,
\end{array}
\right.
\]
these already enforce a number of cohomologies to be zero. Indeed, all the numbers in the following list are zero:
\[
\{ h^0_{0i} \}_{i \leq 11},\ \{ h^0_{i,10}, h^0_{i,11} \}_{i \leq 9},\ \{ h^2_{i0} \}_{i \leq 11},\ \{ h^2_{10,i}, h^2_{11,i} \}_{i \leq 9}.
\]
Also, since $G_{11}^\gen = 2G_{10}^\gen$, $h^0_{10,11} = h^0_{0,10} = 0$ and $h^2_{11,10} = h^2_{10,0} = 0$.
\item\label{item:ProofFreePart_thm:ExceptCollection_MaxLength} If $1 \leq j \neq i \leq 8$, then $-G_i^\gen + G_j^\gen$ can be realized as the lifting of $-F_i + F_j$. Hence,
\[
\bigl\langle \mathcal O_{X^\gen}(G_1^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_8^\gen) \bigr\rangle
\]
is an exceptional collection by Proposition~\ref{prop:ExceptCollection_ofLengthNine}. This proves that $h^p_{ij}=0$ for all $p \geq 0$ and $1 \leq i \neq j \leq 8$. Also, $-G_9^\gen + G_i^\gen = -K_{X^\gen} + F_{i9}^\gen$ for $1 \leq i \leq 8$. Remark~\ref{rmk:ExceptCollection_SerreDuality} shows that $h^p_{9i} = h^p(-K_{X^\gen} + F_{i9}^\gen)=0$ for $p \geq 0$ and $1 \leq i \leq 8$. Furthermore, by Serre duality, $h^p_{i9} = h^{2-p}(F_{i9}^\gen)=0$ for all $p \geq 0$ and $1 \leq i \leq 8$.
\item\label{item:Strategy_HumanPart_thm:ExceptCollection_MaxLength} We verify (\ref{eq:HumanPart_thm:ExceptCollection_MaxLength}) using the following strategy:
\begin{enumerate}
\item If we want to compute $h^0_{ij}$, then pick $D_{ij}^\gen := -G_i^\gen + G_j^\gen$. If the aim is to evaluate $h^2_{ij}$, then take $D_{ij}^\gen := K_{X^\gen} + G_i^\gen - G_j^\gen$, so that $h^2_{ij} = h^0(D_{ij}^\gen)$ by Serre duality.
\item Express $D^\gen_{ij}$ in terms of $L^\gen$, $(p^*H - 3F_9)^\gen$, $F_{i9}^\gen$, and $K_{X^\gen}$. Via Summary~\ref{summary:Divisors_onX^g}, we can translate $L^\gen$, $(p^*H - 3F_9)^\gen$, $F_{i9}^\gen$ into the divisors on $Y$. Further, we have $6K_{X^\gen} = C_0^\gen$, $3K_{X^\gen} = E_1^\gen$, and $2K_{X^\gen} = (C_2+E_2+E_3)^\gen$, thus an arbitrary integer multiple of $K_{X^\gen}$ also can be translated into divisors on $Y$. Together with these translations, use Lemma~\ref{lem:DummyBundle} to find a Cartier divisor $D_{ij}$ on $Y$ which lifts to $D_{ij}^\gen$, and which satisfies $(D_{ij}.C_1) \leq 2$, $(D_{ij}.C_2) \leq 3$, $(D_{ij}.E_2)=0$.
\item Compute an upper bound of $h^0(D_{ij})$. Then by Lemma~\ref{lem:H0Computation}, $h^0(D_{ij}^\gen) \leq h^0(D_{ij})$.
\item In any occasions, we will find that the upper bound obtained in (3) coincides with $\chi(-G_i^\gen + G_j^\gen)$. Also, at least one of $\{ h^0_{ij}, h^2_{ij}\}$ is zero by Step~\ref{item:NumericalStep_thm:ExceptCollection_MaxLength}. From this we deduce $h^0(D_{ij}^\gen)\geq{}$(the upper bound obtained in (3)), hence the equality holds. Consequently, the numbers $\{h^p_{ij} : p=0,1,2\}$ are evaluated.
\end{enumerate}
\item We follow the strategy in Step~\ref{item:Strategy_HumanPart_thm:ExceptCollection_MaxLength} to verify (\ref{eq:HumanPart_thm:ExceptCollection_MaxLength}). Let $i \in \{1,\ldots,8\}$. To verify $h^0_{i0}=0$, we take $D_{i0}^\gen = -G_i^\gen = L^\gen - F_{i9}^\gen - 10K_{X^\gen}$. Translation into the divisors on $Y$ gives:
\[
D_{i0}' = p^*(2H) + F_9-F_i - 2C_0 + (C_2+E_2+E_3)
\]
Since $(D_{i0}'.C_1) = 6$ and $(D_{i0}' . C_2) = 3$, we replace the divisor $D_{i0}'$ by $D_{i0} := D_{i0}' + C_1$ so that the condition $(D_{i0}.C_1) \leq 2$ is fulfilled. Now, $h^0(D_{i0})=0$ by Dictionary~\ref{dictionary:H0Computations}\ref{dictionary:-G_i}, thus $h^0_{i0} \leq h^0(D_{i0}) = 0$ by Lemma~\ref{lem:H0Computation}. Finally, $\chi(-G_i^\gen)=0$ and $h^2_{i0}=0$\,(Step~\ref{item:NumericalStep_thm:ExceptCollection_MaxLength}), hence $h^1_{i0}=0$.
We repeat this routine to the following divisors:
\begin{align*}
D_{0i} &= p^*(2H) + F_9 - F_i - C_0 + C_1 - E_1 + (2C_2+E_2); \\
D_{09} &= p^*(2H) - 2C_0 + C_1 + (C_2 + E_2 + E_3); \\
D_{i,10} &= p^*(3H) + 2F_9 + F_i - 2C_0 + 2C_1 - E_1 - (C_2 + E_2 + E_3) + 2(2C_2+E_2); \\
D_{9,10} &= p^*(3H) + 3F_9 - 3C_0 + 3C_1 + (C_2+E_2+E_3) + (2C_2+E_2).
\end{align*}
Together with Dictionary~\ref{dictionary:H0Computations}\hyperref[dictionary:Nonvanish_K-G_i]{(2--5)}, all the slots of (\ref{eq:HumanPart_thm:ExceptCollection_MaxLength}) are verified.
\item\label{item:M2Part_thm:ExceptCollection_MaxLength} It is difficult to complete (\ref{eq:thm:ExceptCollection_MaxLength}) using the numerical argument\,(see for example, Remark~\ref{rmk:Configuration_andCohomology}). We introduce another plan to overcome these difficulties.
\begin{enumerate}
\item Take $D_{ij}^\gen \in \Pic X^\gen$ and $D_{ij} \in \Pic Y$ as in Step~\hyperref[item:Strategy_HumanPart_thm:ExceptCollection_MaxLength]{3(1--2)}. We may assume $(D_{ij}.C_1) \in \{0,2\}$ and $(D_{ij}.C_2) \in \{-3,0,3\}$. If $(D_{ij}.C_2) = -3$, then $(D_{ij} - C_2 \mathbin. E_2) = -1$, thus $h^0(D_{ij}) = h^0(D_{ij} - C_2 - E_2)$. Hence, we replace $D_{ij}$ by $D_{ij} - C_2 -E_2$ if $(D.C_2) = -3$. In some occasions, we have $(D_{ij} . F_9) = -1$. We make further replacement $D_{ij} \mapsto D_{ij}-F_9$ for those cases.
\item Rewrite $D_{ij}$ in terms of the $\Z$-basis $\{p^*H, F_1,\ldots, F_9, E_1,E_2,E_3\}$ so that $D_{ij}$ is expressed in the following form:
\[
D_{ij} = p^*(dH) - \bigl( \text{sum of exceptional curves of }p \colon Y \to \P^2\bigr).
\]
\item\label{item:PlaneCurveExistence_thm:ExceptCollection_MaxLength} If $h^0(D_{ij}) > 0$, then we consider an effective divisor $D$ which is linearly equivalent to $D_{ij}$. Then, $p_*D$ is the plane curve of degree $d$ which satisfied the conditions imposed by the exceptional part of $D_{ij}$. Let $\mathcal I_{\rm C} \subset \mathcal O_{\P^2}$ be the ideal sheaf associated with the imposed conditions on $p_*D$. Then the curve $p_*D$ contributes to the number $h^0(\mathcal O_{\P^2}(d) \otimes \mathcal I_{\rm C})$. Indeed, this number gives an upper bound of $h^0(D_{ij})$\,(it is clear that if $D'$ is an effective divisor linearly equivalent to $D$ such that $p_*D$ and $p_*D'$ coincide as plane curves, then $D$ and $D'$ must be the same curve in $Y$).
\item As in Step~\hyperref[item:Strategy_HumanPart_thm:ExceptCollection_MaxLength]{3(4)}, we will see that all the upper bounds $h^0(D_{ij})$ coincide with the numerical invariants $\chi(-G_i^\gen + G_j^\gen)$. Thus, the upper bounds $h^0(D_{ij})$ obtained in (3) determine $\{h^p_{ij} : p=0,1,2\}$ precisely.
\end{enumerate}
\item As explained in Remark~\ref{rmk:Configuration_andCohomology}, the value $h^0(D_{ij})$ might depend on the configuration of $p_*C_1$ and $p_*C_2$. However, for general $p_*C_1 = (h_1=0)$, $p_*C_2 = (h_2=0)$, the minimum value of $h^0(D_{ij})$ is attained. This can be observed in the following way. Let $h = \sum_{\alpha} a_\alpha \mathbf{x}^\alpha$ be a homogeneous equation of degree $d$, where the sum is taken over the $3$-tuples $\alpha = (\alpha_x, \alpha_y, \alpha_z)$ with $\alpha_x + \alpha_y + \alpha_z = d$ and $\mathbf{x}^\alpha = x^{\alpha_x} y^{\alpha_y} z^{\alpha_z}$. Then the ideal $\mathcal I_{\rm C}$ imposes linear relations on $\{a_\alpha\}_\alpha$, thus we get a linear system, or equivalently, a matrix $M$, with the variables $\{a_\alpha\}_\alpha$. After perturbing $h_1$ and $h_2$, the rank of $M$ would not decrease since \{$\op{rank} M \geq r_0$\} is an open condition for any fixed $r_0$. From this we conclude: if $h^0(D_{ij}) \leq r$ for at least one pair of $p_*C_1$ and $p_*C_2$, then $h^0(D_{ij}) \leq r$ for general $p_*C_1$ and $p_*C_2$.
\item\label{item:M2Configuration_thm:ExceptCollection_MaxLength} Let $h_1 = (y-z)^2z - x^3 - x^2z$ and $h_2 = x^3 - 2xy^2 + 2xyz + y^2z$. These equations define plane nodal cubics such that
\begin{enumerate}
\item $p_*C_1$ has the node at $[0,1,1]$, and $p_*C_2$ has the node at $[0,0,1]$;
\item $p_*C_2$ has two tangent directions ($y=0$ and $y=-2x$) at nodes;
\item $p_*C_1 \cap p_*C_2$ contains two $\Q$-rational points, namely $[0,1,0]$ and $[-1,1,1]$.
\end{enumerate}
We take $y=0$ as the distinguished tangent direction at the node of $p_*C_2$, and take $p_*F_9 = [0,1,0]$, $p_*F_8 = [-1,1,1]$. The ideals in Table~\ref{table:Ideal_ofConditions_thm:ExceptCollection_MaxLength} are the building blocks of the ideal $\mathcal I_{\rm C}$ introduced in Step~\hyperref[item:PlaneCurveExistence_thm:ExceptCollection_MaxLength]{5(3)}.
\[
\begin{array}{c|c|c|c}
\text{symbol} & \text{ideal form} & \text{ideal sheaf at the\,...} & \text{divisor on }Y \\ \hline
\mathcal I_{E_1} & (x,y-z) & \text{node of }p_*C_1 & -E_1 \\
\mathcal I_{E_2+E_3} & (x,y) & \text{node of }p_*C_2 & -(E_2+E_3) \\
\mathcal I_{E_2+2E_3} & (x^2,y) &
\begin{tabular}{c}
\footnotesize distinguished tangent \\[-5pt]
\footnotesize at the node of $p_*C_2$
\end{tabular}
& -(E_2+2E_3) \\
\mathcal J_9 & (h_1,h_2) & \text{nine base points} & - \sum_{i\leq9} F_i \\
\mathcal J_7 & \scriptstyle \mathcal J_9 {\textstyle/} (x+z,y-z)(x,z) & \text{seven base points} & -\sum_{i\leq7} F_i \\
\mathcal J_8 & (x+z,y-z)\mathcal J_7 & \text{eight base points} & -\sum_{i\leq8} F_i
\end{array}
\]\nopagebreak\vskip-\baselineskip\captionof{table}{The ideals associated with the exceptional divisors}\label{table:Ideal_ofConditions_thm:ExceptCollection_MaxLength}\vskip+0.33\baselineskip
Note that the nine base points contain $[0,1,0]$ and $[-1,1,1]$, thus there exists an ideal $\mathcal J_7$ such that $\mathcal J_9 = (x+z,y-z)(x,z) \mathcal J_7$.
\item We sketch the proof of $h^p_{10,9}=h^p(-G_{10}^\gen + G_9^\gen)=0$, which illustrates several subtleties. Since $h^2_{10,9}=0$ by Step~\ref{item:NumericalStep_thm:ExceptCollection_MaxLength}, we only have to prove $h^0_{10,9}=0$. Thus, we take $D_{10,9}^\gen := -G_{10}^\gen + G_9^\gen$. As in Step~\hyperref[item:Strategy_HumanPart_thm:ExceptCollection_MaxLength]{3(2)}, take $D_{10,9}' = p^*(3H) + 3F_9 - 2C_0 + 2C_1 - E_1 - (C_2+E_2+E_3) + 2(2C_2+E_2)$. We have $(D_{10,9}'.C_2)=-3$, and $(D_{10,9}'-C_2-E_2 \mathbin. F_9) = -1$. Let $D_{10,9}:= D_{10,9}' - C_2-E_2-F_9$. Then, $h^0(D_{10,9}) = h^0(D_{10,9}') \geq h^0(D_{10,9}^\gen)$. As in Step~\hyperref[item:M2Part_thm:ExceptCollection_MaxLength]{5(2)}, the divisor $D_{10,9}$ can be rewritten as
\[
D_{10,9} = p^*(9H) - 2 \sum_{i=1}^8 F_i - 5E_1 - 4E_2 - 7E_3.
\]
Since $\mathcal I_{E_2+E_3}^2$ imposes more conditions than $\mathcal I_{E_2+2E_3}$, the ideal of (minimal) conditions corresponding to $-4E_2 - 7E_3$ is $\mathcal I_{E_2+E_3} \cdot \mathcal I_{E_2+2E_3}^3$. Thus, the plane curve $p_*D_{10,0}$ corresponds to a nonzero section of
\[
H^0(\mathcal O_{\P^2}(9) \otimes \mathcal J_8^2 \cdot \mathcal I_{E_1}^5 \cdot \mathcal I_{E_2+E_3} \cdot \mathcal I_{E_2+2E_3}^3 ).
\]
Using Macaulay2, we find that the rank of this group is zero. This can be found in \texttt{ExcColl\_Dolgachev.m2}\,\cite{ChoLee:Macaulay2}. In a similar way, we obtain the following table (be aware of the difference with (\ref{eq:thm:ExceptCollection_MaxLength})).
\begin{equation}
\scalebox{0.9}{$
\begin{array}{c|ccccc}
& G_0^\gen & G_8^\gen & G_9^\gen & G_{10}^\gen & G_{11}^\gen \\[2pt] \hline
G_0^\gen & 1\,0\,0 & 0\,0\,1 & 0\,0\,1 & 0\,0\,3 & 0\,0\,6 \\
G_8^\gen & & 1\,0\,0 & & 0\,0\,2 & 0\,0\,5\\
G_9^\gen & & & 1\,0\,0 & 0\,0\,2 & 0\,0\,5 \\
G_{10}^\gen & & & & 1\,0\,0 & 0\,0\,3 \\
G_{11}^\gen & & & & & 1\,0\,0
\end{array}
$}
\label{eq:M2Computation_thm:ExceptCollection_MaxLength}
\end{equation}
Table~\ref{table: Macaualy2 Computations} gives a short summary on the computations done in \texttt{ExcColl\_Dolgachev.m2}\,\cite{ChoLee:Macaulay2}.
\[
\scalebox{0.9}{$
\begin{array}{c|c|l}
(i,j) & \text{result} & \multicolumn{1}{c}{\text{choice of }D_{ij}} \\ \hline
(9,0) & h^0_{9,0}=0 & p^*(5H) - \sum_{i\leq9} F_i - 3E_1 - 2E_2 - 4E_3 \\
(10,0) & h^0_{10,0}=0 & p^*(14H) - 3\sum_{i\leq8} F_i - 8E_1 - 6E_2-11E_3 \\
(10,8) & h^0_{10,8}=0 & p^*(9H) - 2\sum_{i\leq7} F_i - F_8 - 6E_1 - 3E_2 - 6E_3 \\
(10,9) & h^0_{10,9}=0 & p^*(9H) - 2\sum_{i\leq8} F_i - 5E_1 - 4E_2 - 7E_3 \\
(11,0) & h^0_{11,0}=0 & p^*(31H) - 7\sum_{i\leq8} F_i - F_9 - 18E_1 - 11E_2 - 22E_3 \\
(11,8) & h^0_{11,8}=0 & p^*(26H) - 6\sum_{i\leq7} F_i - 5F_8 - F_9 - 14E_1 - 10E_2 - 20E_3 \\
(11,9) & h^0_{11,9}=0 & p^*(26H) - 6\sum_{i\leq8} F_i - 15E_1 - 9E_2 - 18E_3 \\[3pt] \hline
(0,10) & h^2_{0,10}=3 & p^*(17H) - 4\sum_{i\leq8} F_i - F_9 - 9E_1 - 6E_2 - 12E_3 \\
(0,11) & h^2_{0,11}=6 & p^*(31H) - 7\sum_{i\leq8} F_i - F_9 - 17E_1 - 12E_2 - 23E_3 \\
(8,11) & h^2_{8,11}=5 & p^*(26H) - 6\sum_{i\leq7} F_i - 5F_8 - F_9 - 15E_1 - 9E_2 - 18E_3 \\
(9,11) & h^2_{9,11}=5 & p^*(26H) - 6\sum_{i\leq8} F_i - 14E_1 - 10E_2 - 19E_3
\end{array}
$}
\]\nopagebreak\vskip-\baselineskip\captionof{table}{Summary of the Macaulay2 computations}\label{table: Macaualy2 Computations}\vskip+0.33\baselineskip
Note that the numbers $h_{11,10}^p$ and $h_{10,11}^p$ are computed freely; indeed, $-G_{11}^\gen + G_{10}^\gen = -G_{10}^\gen$, thus $h^p_{11,10} = h^p_{10,0}$ and $h^p_{10,11}=h^p_{0,10}$. Finally, perturb the cubics $p_*C_1$ and $p_*C_2$ so that (\ref{eq:M2Computation_thm:ExceptCollection_MaxLength}) remains valid and Lemma~\ref{lem:BasePtPermutation} is applicable. Then, (\ref{eq:thm:ExceptCollection_MaxLength}) is verified immediately. \qedhere
\end{enumerate}
\end{proof}
\begin{remark}\label{rmk:Configuration_andCohomology}
Assume that the nodal curves $p_*C_1$, $p_*C_2$ are in a special position so that the node of $p_*C_1$ is located on the distinguished tangent line at the node of $p_*C_2$. Then, the proper transform $\ell$ of the unique line through the nodes of $p_*C_1$ and $p_*C_2$ has the following divisor expression:
\[
\ell = p^*H - E_1 - (E_2 + 2E_3).
\]
In particular, the divisor $D_{90} = p^*(5H) - \sum_{i\leq9} F_i - 3E_1 - 2(E_2+2E_3)$ is linearly equivalent to $2 \ell + C_1 + E_1$, thus $h^0(D_{90}) > 0$. Consequently, for this particular configuration of $p_*C_1$ and $p_*C_2$, we cannot prove $h^0_{90} = 0$ using upper-semicontinuity. However, the numerical method (Step~\ref{item:Strategy_HumanPart_thm:ExceptCollection_MaxLength} in the proof of the previous theorem) cannot detect such variances originated from the position of nodal cubics, hence it cannot be applied to the proof of $h^0_{90}=0$.
\end{remark}
The following lemma, used in the end of the proof of Theorem~\ref{thm:ExceptCollection_MaxLength}, illustrates the symmetric nature of $F_1,\ldots,F_8$.
\begin{lemma}\label{lem:BasePtPermutation}
Assume that $X^\gen$ is originated from a cubic pencil generated by two general plane nodal cubics $p_*C_1$ and $p_*C_2$. Let $D \in \Pic Y$ be a divisor on the rational elliptic surface $Y$. Assume that in the expression of $D$ in terms of $\Z$-basis $\{p^*H, F_1,\ldots, F_9, E_1, E_2, E_3\}$, the coefficients of $F_1,\ldots, F_8$ are the same. Then, $h^p(D+F_i) = h^p(D+F_j)$ for any $p \geq 0$ and $1 \leq i,j \leq 8$.
\end{lemma}
\begin{proof}
Since $\op{Aut}\P^2 = \op{PGL}(3,\C)$ sends arbitrary 4 points\,(of which any three are not colinear) to arbitrary 4 points\,(of which any three are not colinear), we may assume the following.
\begin{enumerate}
\item\label{item:lem:BasePtPermutation_DistinguishedBasePt} The base point $p_*F_9$ is $\Q$-rational.
\item The nodes of $p_*C_1$ and $p_*C_2$ are $\Q$-rational.
\item The distinguished tangent direction at the node of $p_*C_2$ is defined over $\Q$.
\end{enumerate}
Now, let $K$ be the extension field over $\Q$ which is generated by the coefficients of the cubic forms defining $p_*C_1$, $p_*C_2$. Since $p_*C_1$, $p_*C_2$ are general, we may assume the following:
\begin{enumerate}[resume]
\item\label{item:lem:BasePtPermutation_affineBasePts} The base points $p_*F_1,\ldots,p_*F_9$ are contained in the affine space $(z \neq 0) \subset \P^2_{x,y,z}$.
\item\label{item:lem:BasePtPermutation_Resultant} Let $h_i \in K[x,y,z]$ be the defining equation of $p_*C_i$, and let $\op{res}(h_1,h_2;x)$ be the resultant of $h_1(x,y,1)$, $h_2(x,y,1)$ regarded as elements in $(K[x])[y]$. The irreducible factorization of $\op{res}(h_1,h_2;x)$ over $K$ consists of a linear form and an irreducible polynomial, say $H_x$, of degree $8$. The same holds for $\op{res}(h_1,h_2;y)$, {\it i.e.} $\op{res}(h_1,h_2;y) = (y-c) H_y$ for an irreducible polynomial $H_y \in K[y]$ of degree $8$. We assume further that $H_x \neq H_y$ up to multiplication by $K^\times$.
\end{enumerate}
The last condition has the following interpretation. Let $p_* F_i = [\alpha_i,\beta_i,1] \in \P^2$ for $\alpha_i,\beta_i \in \C$ and $i=1,\ldots,9$. The resultant $\op{res}(h_1,h_2;x) \in K[x]$ is the polynomial having $\{\alpha_i\}_{i=1}^9$ as the solutions. By the conditions \ref{item:lem:BasePtPermutation_DistinguishedBasePt}, $\alpha_9 \in \Q$, so a linear factor must appear in $\op{res}(h_1,h_2;x)$. Hence, \ref{item:lem:BasePtPermutation_Resultant} implies that $\alpha_1,\ldots,\alpha_8$ are Galois conjugate over $K$, which should be true for general $p_*C_1$, $p_*C_2$. The same is assumed to be true for $\beta_1,\ldots,\beta_8$, and the final sentence says that $\{\alpha_1,\ldots,\alpha_8\} \neq \{\beta_1,\ldots,\beta_8\}$.
Let $\tau \in \op{Aut}(\C/K)$ be a field automorphism fixing $K$, and mapping $\alpha_i$ to $\alpha_j$\,($1 \leq i,j \leq 8$). Then $\tau$ induces an automorphism of $\P^2$ which fixes $p_*C_1$ and $p_*C_2$. It follows that $[\alpha_j, \tau(\beta_i), 1]$ is one of the eight base points $\{p_*F_i\}_{i=1}^8$. Since $H_x$ and $H_y$ are different up to multiplication by $K^\times$, there is no point of the form $[\alpha_j, \beta_k, 1]$ in the set $\{p_*F_i\}_{i=1}^8$ except when $k=j$. It follows that $\tau(\beta_i) = \beta_j$. Let $\tau_Y \colon Y \to Y$ be the automorphism induced by $\tau$. According to the assumptions \ref{item:lem:BasePtPermutation_DistinguishedBasePt}--\ref{item:lem:BasePtPermutation_Resultant}, it satisfies the following properties:
\begin{enumerate}[label=(\arabic{enumi})]
\item $\tau_Y$ fixes $F_9, E_1, E_2, E_3$;
\item $\tau_Y$ permutes $F_1,\ldots,F_8$;
\item $\tau_Y$ maps $F_i$ to $F_j$.
\end{enumerate}
Furthermore, since the coefficients of $F_1,\ldots,F_8$ are the same in the expression of $D$, $\tau_Y$ fixes $D$. It follows that $\tau_Y^* \colon \Pic Y \to \Pic Y$ maps $D+F_j$ to $D+F_i$. In particular, $H^p(D+F_j) = H^p(\tau_Y^*(D+F_i)) \simeq H^p( D+F_i)$ for any $1 \leq i,j \leq 8$. \qedhere
\end{proof}
\subsection{Incompleteness of the collection}\label{subsec:Incompleteness} Let $\mathcal A \subset \D^{\rm b}(X^\gen)$ be the orthogonal subcategory
\[
\bigl\langle \mathcal O_{X^\gen}(G_0^\gen),\ \mathcal O_{X^\gen}(G_1^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_{11}^\gen) \bigr\rangle^\perp,
\]
so that there exists a semiorthogonal decomposition
\[
\D^{\rm b}(X^\gen) = \bigl\langle \mathcal A,\ \mathcal O_{X^\gen}(G_0^\gen),\ \mathcal O_{X^\gen}(G_1^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_{11}^\gen) \bigr\rangle.
\]
We will prove that $K_0(\mathcal A) = 0$, $\op{HH}_\bullet(\mathcal A) = 0$, but $\mathcal A\not\simeq 0$. Such a category is called a \emph{phantom} category. To give a proof, we claim that the \emph{pseudoheight} of the collection~(\ref{eq:ExcColl_MaxLength}) is at least $2$. Once we achieve the claim, \cite[Corollary~4.6]{Kuznetsov:Height} implies that $\op{HH}^0(\mathcal A) \simeq \op{HH}^0(X^\gen) = \C$, thus $\mathcal A\not\simeq 0$.
\begin{definition}\
\begin{enumerate}
\item Let $E_1,E_2$ be objects in $\D^{\rm b}(X^\gen)$. The \emph{relative height} $e(E_1,E_2)$ is the minimum of the set
\[
\{ p : \Hom(E_1,E_2[p]) \neq 0 \} \cup \{ \infty \}.
\]
\item Let $\langle F_0,\ldots,F_m\rangle$ be an exceptional collection in $\D^{\rm b}(X^\gen)$. The \emph{anticanonical pseudoheight} is defined by
\[
\op{ph}_{\rm ac}(F_0,\ldots,F_m) = \min \Bigl ( \sum_{i=1}^p e(F_{a_{i-1}}, F_{a_i}) + e(F_{a_p} , F_{a_0} \otimes \mathcal O_{X^\gen}(-K_{X^\gen})) - p \Bigr),
\]
where the minimum is taken over all possible tuples $0 \leq a_0 < \ldots < a_p \leq m$.
\end{enumerate}
\end{definition}
The pseudoheight is given by the formula $\op{ph}(F_0,\ldots,F_m) = \op{ph}_{\rm ac}(F_0,\ldots,F_m) + \dim X^\gen$, thus it suffices to prove that $\op{ph}_{\rm ac}(G_0^\gen,\ldots,G_{11}^\gen) \geq 0$.
\begin{corollary}\label{cor:Phantom}
In the semiorthogonal decomposition
\[
\D^{\rm b}(X^\gen) = \bigl \langle \mathcal A,\ \mathcal O_{X^\gen}(G_0^\gen),\ \ldots,\ \mathcal O_{X^\gen}(G_{11}^\gen)\bigr\rangle,
\]
we have $K_0(\mathcal A) = 0$ and $\op{HH}_\bullet(\mathcal A)=0$. Also, $\op{ph}_{\rm ac}(G_0^\gen,\ldots,G_{11}^\gen) = 2$, thus the restriction map $\op{HH}^p(X^\gen) \to \op{HH}^p(\mathcal A)$ is an isomorphism for $p \leq 2$ and is a monomorphism for $p=3$. In particular, $\op{HH}^0(\mathcal A) \simeq \C$.
\end{corollary}
\begin{proof}
Since $\kappa(X^\gen) = 1$, the Bloch conjecture holds for $X^\gen$\,\cite[\textsection11.1.3]{Voisin:HodgeTheory2}. Thus the Grothendieck group $K_0(X^\gen)$ is a free abelian group of rank $12$\,(see for {\it e.g.} \cite[Lemma~2.7]{GalkinShinder:Beauville}). Furthermore, Hochschild-Kostant-Rosenberg isomorphism for Hochschild homology says
\[
\op{HH}_k(X^\gen) \simeq \bigoplus_{q-p=k} H^{p,q}(X^\gen),
\]
hence, $\op{HH}_\bullet(X^\gen) \simeq \C^{\oplus 12}$. It is well-known that $K_0$ and $\op{HH}_\bullet$ are the additive invariants with respect to semiorthogonal decompositions, thus $K_0(X^\gen) \simeq K_0(\mathcal A) \oplus K_0({}^\perp\mathcal A)$, and $\op{HH}_\bullet(X^\gen) = \op{HH}_\bullet(\mathcal A) \oplus \op{HH}_\bullet({}^\perp \mathcal A)$.\footnote{By definition of $\mathcal A$, ${}^\perp \mathcal A$ is the smallest full triangulated subcategory containing the collection (\ref{eq:ExcColl_MaxLength}) in Theorem~\ref{thm:ExceptCollection_MaxLength}.} If $E$ is an exceptional vector bundle, then $\D^{\rm b}(\langle E\rangle ) \simeq \D^{\rm b}(\Spec \C)$ as $\C$-linear triangulated categories, thus $K_0({}^\perp\mathcal A) \simeq \Z^{\oplus 12}$ and $\op{HH}_\bullet({}^\perp\mathcal A)\simeq \C^{\oplus12}$. It follows that $K_0(\mathcal A) = 0$ and $\op{HH}_\bullet(\mathcal A)=0$.
Assume the chain $0\leq a_0 < \ldots < a_p \leq 11$ has length $p=0$. Then, $e(G_{a_0}^\gen,G_{a_0}^\gen-K_{X^\gen}) = 2$ since $\dim \Ext_{X^\gen}^p(G_i^\gen, G_i^\gen - K_{X^\gen}) = h^p(-K_{X^\gen}) = 1$ for $p=2$ and $0$ otherwise. For any $0 \leq j < i \leq 11$,
\[
e(G_j^\gen, G_i^\gen) =\left\{
\begin{array}{ll}
\infty & \text{if } 1 \leq j < i \leq 9 \\
2 & \text{otherwise}
\end{array}
\right.
\]
by Theorem~\ref{thm:ExceptCollection_MaxLength}. Also, it is easy to see that $H^0(G_i^\gen, G_j^\gen - K_{X^\gen}) = 0$ for $i > j$, thus for any chain $0 \leq a_0 < \ldots< a_p \leq 11$,
\[
e(G_{a_0}^\gen, G_{a_1}^\gen) + \ldots + e(G_{a_{p-1}}^\gen, G_{a_p}^\gen) + e(G_{a_p}^\gen, G_{a_0}^\gen - K_{X^\gen}) - p \geq 2p + 1 - p,
\]
which shows that the value of the left hand side is at least $2$ for any chain of length${}>0$. It follows that $\op{ph}_{\rm ac}(G_0^\gen,\ldots,G_{11}^\gen) =2$. The statements about $\op{HH}^\bullet$ immediately follows by \cite[Corollary~4.6]{Kuznetsov:Height}. \qedhere
\end{proof}
\subsection{Cohomology computations}
We present Dictionary~\ref{dictionary:H0Computations} of cohomology computations that appeared in the proof of Theorem~\ref{thm:ExceptCollection_MaxLength}. It needs the divisors illustrated in Figure~\ref{fig:Configuration_Basic}, together with one more curve, which did not appear in Figure~\ref{fig:Configuration_Basic}. Let $\ell$ be the proper transform of the unique line in $\P^2$ passing through the nodes of $p_* C_1$ and $p_* C_2$. In the divisor form,
\[
\ell = p^*H - E_1 - (E_2 + E_3).
\]
Due to the divisor forms
\[
\begin{array}{r@{}l}
C_1 &{}=p^*(3H) - 2E_1 - \sum_{i=1}^9F_i, \\
C_2 &{}=p^*(3H) - (2E_2 + 3E_3) - \sum_{i=1}^9 F_i,\ \text{and} \\
C_0 &{}=p^*(3H) - \sum_{i=1}^9 F_i,
\end{array}
\]
it is straightforward to write down the intersections involving $\ell$:
\[
\begin{array}{ c | c | c | c | c | c | c | c | c | c }
& p^*H & F_i & C_0 & C_1 & E_1 & C_2 & E_2 & E_3 & \ell \\
\hline
\ell & 1 & 0 & 3 & 1 & 1 & 1 & 1 & 0 & -1
\end{array}
\]
\begin{dictionary}\label{dictionary:H0Computations}
For each of the following Cartier divisors on $Y$, we give upper bounds of $h^0$. The main strategy is the following. We take smooth rational curves $A_1,\ldots, A_r$, and consider the exact sequence
\[
0 \to H^0(D - S_i) \to H^0(D - S_{i-1}) \to H^0(\mathcal O_{A_i}( D - S_{i-1}) ),
\]
where $S_i = \sum_{j\leq i} A_j$. This gives the inequality $h^0(D - S_{i-1}) \leq h^0(D - S_i) + h^0( (D - S_{i-1})\big\vert_{A_i})$. Inductively, we get
\begin{equation}
h^0(D) \leq h^0(D - S_r) + \sum_{i=1}^{r-1} h^0( ( D - S_i)\big\vert_{A_{i+1}}). \label{eq:dictionary:UpperBound}
\end{equation}
In what follows, we take $A_1,\ldots,A_r$ carefully so that $h^0(D- S_r)=0$, and that the values $h^0(D - S_{i-1}\big\vert_{A_i})$ are as small as possible. In each item in the dictionary, we first present the target divisor $D$ and the bound of $h^0(D)$. After then, we give a list of smooth rational curves in the following format:
\[
A_1,\ A_2,\ \ldots,\ A_i\textsuperscript{(\checkmark)},\ \ldots\ , A_r.
\]
The symbol $(\checkmark)$ indicates the situation when $(D - S_{i-1} \mathbin. A_i) = 0$, the case in which the right hand side of (\ref{eq:dictionary:UpperBound}) increases by $1$. The curves without symbols indicate the situations in which $(D - S_{i-1} \mathbin . A_i) < 0$, so that $A_i$ does not contribute to the bound of $h^0(D)$. We conclude by showing that $D - S_r$ is not an effective divisor. The upper bound of $h^0(D)$ will be given by the number of $(\checkmark)$'s in the list. Since all of these calculations are routine, we omit the details. From now on, $i$ is any number between $1,2,\ldots,8$.
\begin{enumerate}[label=\normalfont(\arabic{enumi}), itemsep=7pt plus 5pt minus 0pt]
\item\label{dictionary:-G_i} $D=p^*(2H) + F_9 - F_i - 2C_0 + C_1 + C_2 + E_2 + E_3$
$h^0(D)=0$ \\
The following is the list of curves $A_1,\ldots,A_r$\,(the order is important): $F_9,\, \ell,\, E_2,\, \ell$. The resulting divisor is
\[
D - A_1 - \ldots - A_r = p^*(2H) - F_i - 2C_0 + C_1 + C_2 + E_3 - 2\ell.
\]
Since $\ell = p^*H - E_1 - (E_2 + E_3)$ and $C_0 = C_1 + 2E_1 = C_2 + 2E_2 + 3E_3$, $D - A_1 - \ldots - A_r = -F_i$. It follows that $H^0(D) \simeq H^0( - F_i) = 0$.
\item\label{dictionary:Nonvanish_K-G_i} $D = p^*(2H) + F_9 - F_i - C_0 + C_1 -E_1 + 2C_2 + E_2.$
$h^0(D) \leq 1$ \\
Rule out $C_2,\,E_2,\,\ell\textsuperscript{(\checkmark)},\,C_1,\,F_9,\,C_2,\,\ell,\,E_1$. The resulting divisor is $p^*(2H) - F_i - C_0 - 2E_1 - 2\ell = -F_i - C_2 - E_3$. Since there is only one checkmark, $h^0(D) \leq h^0(-F_i - C_2 - E_3) + 1 = 1$.
\item\label{dictionary:Nonvanish_K-G_9} $D = p^*(2H) - 2C_0 + C_1 + C_2 + E_2 + E_3$
$h^0(D) \leq 1$ \\
Rule out $\ell,\, E_2,\, \ell,\, C_2\textsuperscript{(\checkmark)}$. The remaining part is $ p^*(2H) - 2C_0 + C_1 + E_3 - 2\ell = - C_2$, thus $h^0(D) \leq 1$.
\item\label{dictionary:Nonvanish_K+G_i-G_10} $D = p^*(3H) + 2F_9 + F_i - 2C_0 + 2C_1 - E_1 + 3C_2 + E_2 - E_3$
$h^0(D) \leq 2$ \\
The following is the list of divisors that we have to remove:
\[
C_2,\ E_2,\ \ell\textsuperscript{(\checkmark)},\ E_2,\ F_9\textsuperscript{(\checkmark)},\ C_2,\ E_2,\ \ell,\ C_1,\ F_9,\ F_i,\ \ell.
\]
The remaining part is $p^*(3H) - 2C_0 + C_1 - E_1 + C_2 - E_2 - E_3 - 3\ell = -E_3$, thus $h^0(D) \leq 2$.
\item\label{dictionary:Nonvanish_K+G_9-G_10} $D = p^*(3H) + 3F_9 - 3C_0 + 3C_1 + 3C_2 + 2E_2 + E_3$
$h^0(D) \leq 2$ \\
Rule out the following curves:
\[
F_9\textsuperscript{(\checkmark)},\ C_1,\ C_2,\ E_2,\ F_9,\ \ell,\ E_2\textsuperscript{(\checkmark)},\ \ell,\ C_2,\ \ell,\ E_2,\ E_3,\ F_9,\ C_1,\ E_1.
\]
The remaining part is $p^*(3H) - 3C_0 + C_1 - E_1 + C_2 - E_2 -3\ell = -C_0$, thus $h^0(D) \leq 2$.
\end{enumerate}
\end{dictionary}
\section{Appendix}\label{sec:Appendix}
\subsection{A brief review on Hacking's construction.}\label{subsec:HackingConstruction}
Let $n>a>0$ be coprime integers, let $X$ be a projective normal surface with quotient singularities, and let $(P \in X)$ be a $T_1$-singularity of type $(0 \in \A^2 / \frac{1}{n^2}(1,na-1))$. Suppose there exists a one parameter deformation $\mathcal X / ( 0 \in T)$ of $X$ over a smooth curve germ $(0 \in T)$ such that $(P \in \mathcal X) / (0 \in T)$ is a $\Q$-Gorenstein smoothing of $(P \in X)$.
\begin{proposition}[{\cite[\textsection3]{Hacking:ExceptionalVectorBundle}}]\label{prop:HackingWtdBlup}
Take the base extension $(0 \in T') \to (0 \in T)$ of ramification index $a$, and let $\mathcal X'$ be the pull back along the extension. Then, there exists a proper birational morphism $\Phi \colon \tilde{\mathcal X} \to \mathcal X'$ satisfying the following properties.
\begin{enumerate}
\item The exceptional fiber $W = \Phi^{-1}(P)$ is isomorphic to the projective normal surface
\[
(xy = z^n + t^a) \subset \P_{x,y,z,t}(1,na-1,a,n).
\]
\item The morphism $\Phi$ is an isomorphism outside $W$.
\item\label{item:prop:HackingWtdBlup} The central fiber $\tilde{\mathcal X}_0 = \Phi^{-1}(\mathcal X'_0)$ is reduced and has two irreducible components: $\tilde X_0$ the proper transform of $X$, and $W$. The intersection $Z:=\tilde X_0 \cap W$ is a smooth rational curve given by $(t=0)$ in $W$. Furthermore, the surface $\tilde X_0$ can be obtained in the following way: take a minimal resolution $Y \to X$ of $(P \in X)$, and let $E_1,\ldots,E_r$ be the chain of exceptional curves arranged in such a way that $(E_i . E_{i+1})=1$ and $(E_r^2) = -2$. Then the contraction of $E_2,\ldots,E_r$ defines $\tilde X_0$. Clearly, $E_1$ maps isomorphically onto $Z$ along the contraction $Y \to \tilde X_0$.
\end{enumerate}
\end{proposition}
\begin{proposition}[{\cite[Proposition~5.1]{Hacking:ExceptionalVectorBundle}}]\label{prop:Hacking_BundleG}
There exists an exceptional vector bundle $G$ of rank $n$ on $W$ such that $G \big\vert_{Z} \simeq \mathcal O_Z(1)^{\oplus n}$.
\end{proposition}
\begin{remark}\label{rmk:SimplestSingularCase}
Note that in the decomposition $\tilde{\mathcal X}_0 = \tilde X_0 \cup W$, the surface $W$ is completely determined by the type of singularity $(P \in X)$, whereas $\tilde X_0$ reflects the global geometry of $X$. In some circumstances, $W$ and $G$ have explicit descriptions.
\begin{enumerate}
\item Suppose $a=1$. In $\P_{x,y,z,t}(1,n-1,1,n)$, we have $W_2 =( xy = z^n + t)$ and $Z_2 = (xy=z^n, t=0)$ by Proposition~\ref{prop:HackingWtdBlup}. The projection map $\P_{x,y,z,t}(1,n-1,1,n) \dashrightarrow \P_{x,y,z}(1,n-1,1)$ sends $W_2$ isomorphically onto $\P_{x,y,z}$, thus we get
\[
W_2 \simeq \P_{x,y,z}(1,n-1,1),\quad\text{and}\quad Z_2 \simeq (xy=z^n) \subset \P_{x,y,z}(1,n-1,1).
\]
\item Suppose $(n,a) = (2,1)$, then it can be shown (by following the proof of Proposition~\ref{prop:Hacking_BundleG}) that $W = \P_{x,y,z}^2$, $G = \mathcal T_{\P^2}(-1)$ where $\mathcal T_{\P^2} = (\Omega_{\P^2}^1)^\vee$ is the tangent sheaf of the plane. Moreover, the smooth rational curve $Z = \tilde X_0 \cap W$ is embedded as a smooth conic $(xy = z^2)$ in $W$.
\end{enumerate}
\end{remark}
The final proposition presents how to obtain an exceptional vector bundle on the general fiber of the smoothing.
\begin{proposition}[{\cite[\textsection4]{Hacking:ExceptionalVectorBundle}}]\label{prop:HackingDeformingBundles}
Let $X^\gen$ be the general fiber of the deformation $\mathcal X / (0 \in T)$, and assume $H^2(\mathcal O_{X^\gen}) = H^1(X^\gen,\Z) = 0$.\footnote{Since quotient singularities are Du Bois, we have $H^1(\mathcal O_X) = H^2(\mathcal O_X) = 0$. ({\it cf.} \cite[Lem.~4.1]{Hacking:ExceptionalVectorBundle})} Let $G$ be the exceptional vector bundle on $W$ in Proposition~\ref{prop:Hacking_BundleG}. Suppose there exists a Weil divisor $D \in \Cl X$ such that $D$ does not pass through the singular points of $X$ except $P$, the proper transform $D' \subset \tilde X_0$ of $X$ satisfies $(D'. Z) = 1$, and $\op{Supp} D' \subset \tilde X_0 \setminus \op{Sing} \tilde X_0$. Then the vector bundles $\mathcal O_{\tilde X_0}(D')^{\oplus n}$ and $G$ glue along $\mathcal O_Z(1)^{\oplus n}$ to produce an exceptional vector bundle $\tilde E$ on $\tilde{\mathcal X}_0$. Furthermore, the vector bundle $\tilde E$ deforms uniquely to an exceptional vector bundle $\tilde{\mathcal E}$ on $\tilde{\mathcal X}$. Restriction $\tilde{\mathcal E}\big\vert_{X^\gen}$ to the general fiber is an exceptional vector bundle on $X^\gen$ of rank $n$.
\end{proposition}
\footnotesize
\noindent{\bf Acknowledgments.}
The first author thanks to Kyoung-Seog Lee for helpful comments on derived categories. He also thanks to Alexander Kuznetsov and Pawel Sosna for giving explanation on the technique of height used in Section \ref{subsec:Incompleteness}. The second author thanks to Fabrizio Catanese and Ilya Karzhemanov for useful remarks. The authors would like to appreciate many valuable comments from the anonymous referee.
This work is supported by Global Ph.D. Fellowship Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(No.2013H1A2A1033339)\,(to Y.C.), and is partially supported by the NRF of Korea funded by the Korean government(MSIP)(No.2013006431)\,(to Y.L.).
\end{document} |
\begin{document}
\centerline{\bf\large Riemann Spaces and Pfaff Differential Forms}
\vskip .4in
\centerline{\textbf{Nikos D. Bagis}}
\centerline{\textbf{Aristotele University of Thessaloniki-AUTH-Greece}}
\centerline{\textbf{nikosbagis@hotmail.gr}}
\[
\]
\centerline{\bf Abstract}
In this work we study differential geometry in $N$ dimensional Riemann curved spaces using Pfaff derivatives. Avoiding the classical partial derivative the Pfaff derivatives are constructed in a more sophisticated way and make evaluations become easier. In this way Christofell symbols $\Gamma_{ikj}$ of classical Riemann geometry as also the elements of the metric tensor $g_{ij}$ are replaced with one symbol (the $q_{ikj}$). Actually to describe the space we need no usage of the metric tensor $g_{ij}$ at all. We also don't use Einstein's notation and this quite simplifies things. For example we don't have to use upper and lower indexes, which in eyes of a beginner, is quite messy. Also we don't use the concept of tensor. All quantities of the surface or curve or space which form a tensor field are called invariants or curvatures of the space. Several new ideas are developed in this basis.\\
\\
\textbf{Keywords:} Riemann geometry; Curved spaces; Pfaff derivatives; Differential Operators; Invariant theory
\[
\]
\section{Introduction and Development of the Theory}
Here we assume a $N$ dimensional space $\bf{S}$. The space $\bf{S}$ will be described by the vector
\begin{equation}
\overline{x}=\sum^{N}_{i=1}x_i(u_1,u_2,\ldots,u_N)\overline{\epsilon}_i ,
\end{equation}
where $\overline{\epsilon}_i$ is usual orthonormal base of $\textbf{E}=\textbf{R}^{N}$. We assume that in every point of the space $\bf{S}$ correspond $N$ orthonormal vectors $\{\overline{e}_1,\overline{e}_2,\ldots,\overline{e}_{N}\}$.
These $N$ vectors $\{\overline{e}_1,\overline{e}_2,\ldots,\overline{e}_{N}\}$ span the space $\bf{S}$. We will use Pfaff derivatives to write our equations. We also study some properties of $\bf{S}$ which will need us for the construction of these equations. The Pfaff derivatives related with the structure of the space $\bf{S}$ which produces differential forms $\omega_k$, $k=1,2,\ldots,N$. These are defined as below.\\
It holds that
\begin{equation}
\partial_j\overline{x}=\sum^{N}_{i=1}\partial_jx_i\overline{\epsilon}_i.
\end{equation}
The linear element of $\bf{S}$ is
\begin{equation}
(ds)^2=(d\overline{x})^2=\sum^{N}_{i,j=1}\left\langle \partial_i\overline{x},\partial_j\overline{x}\right\rangle du_idu_j=\sum^{N}_{i=1}g_{ii}du_i^2+2\sum_{i<j}g_{ij}du_idu_j.
\end{equation}
Hence
\begin{equation}
g_{ij}=\left\langle \frac{\partial \overline{x}}{\partial u_i},\frac{\partial \overline{x}}{\partial u_j} \right\rangle,
\end{equation}
are the structure functions of the first linear form.\\
The Pfaff differential forms $\omega_k$ are defined with the help of $\{\overline{e}_k\}$, $k=1,2,\ldots,N$ as
\begin{equation}
d\overline{x}=\sum^{N}_{k=1}\omega_k\overline{e}_k.
\end{equation}
Then the Pfaff derivatives of the function $f$ are $\nabla_kf$, $k=1,2,\ldots,N$ and holds
\begin{equation}
df=\sum^{N}_{k=1}(\partial_kf)du_k=\sum^{N}_{k=1}(\nabla_kf)\omega_k ,
\end{equation}
From (6) we get
\begin{equation}
d\overline{x}=\sum^{N}_{k=1}(\partial_k\overline{x})du_k.
\end{equation}
Also
\begin{equation}
d\overline{x}=\sum^{N}_{k=1}\omega_k\overline{e}_k\Rightarrow \nabla_m(\overline{x})=\overline{e}_m.
\end{equation}
Derivating the vectors $\overline{e}_l$ we can write them as linear composition of their self, since they form a basis of $\bf{E}$:
\begin{equation}
d\overline{e}_i=\sum^{N}_{k=1}\omega_{ik}\overline{e}_k.
\end{equation}
Then we define the connections $q_{ijm}$ and $b_{kl}$ as
\begin{equation}
\omega_{ij}=\sum^{N}_{m=1}q_{ijm}\omega_m\textrm{, }\omega_k=\sum^{N}_{l=1}b_{kl}du_{l}.
\end{equation}
Hence from (10),(5),(7) and $\left\langle \overline{e}_k ,d\overline{x}\right\rangle=\omega_k$, we get that
\begin{equation}
\left\langle \overline{e}_k, \partial_l\overline{x}\right\rangle=b_{kl}\textrm{ and }\partial_{l}\overline{x}=\sum^{N}_{k=1}b_{kl}\overline{e}_k
\end{equation}
and from (4)
\begin{equation}
g_{ij}=\sum^{N}_{k=1}b_{ki}b_{kj}.
\end{equation}
Also
\begin{equation}
b_{sl}=\sum^{N}_{m=1}\frac{\partial x_m}{\partial u_l}\cos\left(\phi_{ms}\right)\textrm{, where }\cos\left(\phi_{ms}\right)=\left\langle \overline{\epsilon}_{m},\overline{e}_{s}\right\rangle
\end{equation}
and $\phi_{ms}$ is the angle formed by $\overline{\epsilon}_m$ and $\overline{e}_s$.
Also it holds
\begin{equation}
\omega_{ij}=\sum^{N}_{l=1}\left(\sum^{N}_{m=1}q_{ijm}b_{ml}\right)du_l
\end{equation}
By this way we get the Christofell symbols
\begin{equation}
\Gamma_{ijl}=\sum^{N}_{m=1}q_{ijm}b_{ml}.
\end{equation}
Thus in view of (19) below
\begin{equation}
\omega_{ij}=\sum^{N}_{l=1}\Gamma_{ijl}du_l\textrm{, }\Gamma_{ijl}+\Gamma_{jil}=0
\end{equation}
and easy (see and Proposition 1 below)
\begin{equation}
\partial_k\overline{e}_m=\sum^{N}_{j=1}\Gamma_{mjk}\overline{e}_j\textrm{ and }\nabla_k\overline{e}_m=\sum^{N}_{j=1}q_{mjk}\overline{e}_j.
\end{equation}
From the orthonormality of $\overline{e}_k$ we have
\begin{equation}
\delta_{ij}=\left \langle \overline{e}_i,\overline{e}_j\right \rangle.
\end{equation}
Derivating the above relation, we get
\begin{equation}
\omega_{ij}+\omega_{ji}=0
\end{equation}
and hence
\begin{equation}
q_{ijm}+q_{jim}=0.
\end{equation}
\\
\textbf{Theorem 1.}\\
The structure equations of $\bf{S}$ are (19) and
\begin{equation}
d\omega_{j}=\sum^{N}_{m=1}\omega_{m}\wedge\omega_{mj}\textrm{ , }d\omega_{ij}=\sum^{N}_{m=1}\omega_{im}\wedge\omega_{mj}.
\end{equation}
\\
\textbf{Proof.}\\
We have
$$
d\left(d\overline{x}\right)=\overline{0}\Rightarrow d\left(\sum^{N}_{i=1}\overline{e}_i\omega_i\right)=\overline{0}\Rightarrow \sum^{N}_{i=1}\left(d\overline{e}_i\wedge\omega_i+\overline{e}_i d\omega_i\right)=\overline{0}.
$$
Hence
$$
\sum^{N}_{i=1}d\omega_i\overline{e}_i+\sum^{N}_{i,k=1}\omega_{ik}\wedge\omega_i\overline{e}_k=\overline{0}\Rightarrow d\omega_i=\sum^{N}_{l=1}\omega_{l}\wedge\omega_{li}.
$$
The same arguments hold and for the second relation of (21). $qed$\\
\\
\textbf{Definition 1.}\\
We write
\begin{equation}
rot_{ij}\left(A_{k\ldots i\ldots j\ldots l}\right)=A_{k\ldots i\ldots j\ldots l}-A_{k\ldots j\ldots i\ldots l}
\end{equation}
\\
\textbf{Definition 2.}\\
We define the Kronecker$-\delta$ symbol as follows:\\
If $\{i_1,i_2,\ldots,i_M\}$, $\{j_1,j_2,\ldots,j_M\}$ are two set of indexes, then
$$
\delta_{{i_1i_2\ldots i_M},{j_1j_2\ldots j_M}}=1,
$$
if $\{i_1,i_2,\ldots,i_M\}$ is even permutation of $\{j_1,j_2,\ldots,j_M\}$.
$$
\delta_{{i_1i_2\ldots i_M},{j_1j_2\ldots j_M}}=-1,
$$
if $\{i_1,i_2,\ldots,i_M\}$ is odd permutation of $\{j_1,j_2,\ldots j_{M}\}$
and
$$
\delta_{{i_1i_2\ldots i_M},{j_1j_2\ldots j_M}}=0
$$
in any other case.\\
\\
Let $a=\sum^{N}_{i=1}a_i\omega_i$ and $b=\sum^{N}_{j,k=1}b_{jk}\omega_j\wedge\omega_k$, then
$$
(a\wedge b)_{123}=\frac{1}{1!}\frac{1}{2!}\sum^{N}_{i,j,k=1}\delta_{{ijk},{123}}a_jb_{jk}=
$$
$$
=\frac{1}{2}[a_1b_{23}\delta_{{123},{123}}+a_1b_{32}\delta_{{132},{123}}+a_2b_{13}\delta_{{213},{123}}+a_2b_{31}\delta_{{231},{123}}+
$$
$$
+a_3b_{12}\delta_{{312},{123}}+a_3b_{21}\delta_{{321},{123}}]=
$$
$$
=\frac{1}{2}\left(a_1b_{23}-a_1b_{32}-a_2b_{13}+a_2b_{31}+a_3b_{12}-a_3b_{21}\right).
$$
We remark here that we don't use Einstein's index notation.\\
\\
By this way equations (21) give
$$
d\omega_j=\sum^{N}_{m=1}\omega_m\wedge\omega_{mj}=\sum^{N}_{m,s=1}q_{mjs}\omega_{m}\wedge\omega_{s}.
$$
Hence
\begin{equation}
d\omega_j=\sum_{m<s}Q_{mjs}\omega_m\wedge\omega_s,
\end{equation}
with
\begin{equation}
Q_{mjs}:=rot_{ms}(q_{mjs})=q_{mjs}-q_{sjm}
\end{equation}
Hence with the above notation holds $Q_{ikl}+Q_{lki}=0$. Also if we set
\begin{equation}
R_{ijkl}:=-\sum^{N}_{m=1}rot_{kl}\left(q_{imk}q_{jml}\right),
\end{equation}
then we have
\begin{equation}
d\omega_{ij}=\sum_{k<l}R_{ijkl}\omega_k\wedge\omega_l.
\end{equation}
\\
Using the structure equations of the space $\bf{S}$ we have the next\\
\\
\textbf{Proposition 1.}\\
\begin{equation}
\nabla_k\nabla_m(\overline{x})=\nabla_k(\overline{e}_m)=\sum^{N}_{j=1}q_{mjk}\overline{e}_j.
\end{equation}
\textbf{Proof.}\\
$$
d\overline{e}_i=\sum^{N}_{k=1}\nabla_k(\overline{e}_i)\omega_k\Leftrightarrow \sum^{N}_{k=1}\omega_{ik}\overline{e}_k=\sum^{N}_{k=1}\nabla_k(\overline{e}_i)\omega_k\Rightarrow
$$
$$ \sum^{N}_{m=1}\left(\sum^{N}_{k=1}q_{ikm}\overline{e}_k\right)\omega_m=\sum^{N}_{m=1}\nabla_{m}(\overline{e}_i)\omega_m.
$$
\\
\textbf{Theorem 2.}\\
For every function $f$ hold the following relations
\begin{equation}
\nabla_l\nabla_mf-\nabla_m\nabla_lf+\sum^{N}_{k=1}(\nabla_kf)Q_{lkm}=0\textrm{, }\forall\mbox{ } l,m\in\{1,2,\ldots,N\}
\end{equation}
or equivalently
\begin{equation}
rot_{lm}\left(\nabla_l\nabla_mf+\sum^{N}_{k=1}(\nabla_kf)q_{lkm}\right)=0.
\end{equation}
\\
\textbf{Proof.}
$$
df=\sum^{N}_{k=1}(\nabla_kf)\omega_k\Rightarrow d(df)=\sum^{N}_{k=1}d(\nabla_kf)\wedge\omega_k+\sum^{N}_{k=1}(\nabla_kf)d\omega_k=0
$$
or
$$
\sum^{N}_{k=1}\left(\sum^{N}_{s=1}(\nabla_s\nabla_kf)\omega_s\right)\wedge\omega_k+\sum^{N}_{k=1}(\nabla_kf)\left(\sum_{l<m}rot_{lm}(q_{l km})\omega_{l}\wedge\omega_{m}\right)=0
$$
or
$$
\sum_{l<m}\left(\nabla_l\nabla_mf-\nabla_m\nabla_lf+\sum^{N}_{k=1}(\nabla_kf)Q_{lkm}\right)\omega_l\wedge\omega_m=0.
$$
\\
\textbf{Corollary 1.}\\
If $\lambda_{ij}=-\lambda_{ji}$, $i,j\in\{1,2,\ldots,N\}$ is any antisymetric field, then
\begin{equation}
\sum^{N}_{i,j=1}\lambda_{ij}\nabla_i\nabla_jf+\sum^{N}_{i,j,k=1}\lambda_{ij}q_{ikj}\nabla_kf=0.
\end{equation}
\\
\textbf{Theorem 3.}
\begin{equation}
R_{iklm}=\nabla_l(q_{ikm})-\nabla_{m}(q_{ikl})+\sum^{N}_{s=1}q_{iks}Q_{lsm}=-\sum^{N}_{s=1}rot_{lm}(q_{isl}q_{ksm})
\end{equation}
and it holds also
\begin{equation}
R_{iklm}=-R_{kilm}\textrm{, }R_{iklm}=-R_{ikml}\textrm{, }
\end{equation}
\\
\textbf{Proof.}\\
We describe the proof.
$$
d\omega_{ij}=\sum^{N}_{k=1}d(q_{ijk})\wedge\omega_k+\sum^{N}_{k=1}q_{ijk}d\omega_k=
$$
$$
=\sum^{N}_{m<l}\left(\nabla_{m}(q_{ijl})-\nabla_{l}(q_{ijm})+\sum^{N}_{s=1}q_{ijs}Q_{msl}\right)\omega_m\wedge\omega_{l}
$$
and
$$
d\omega_{ij}=\sum^{N}_{k=1}\omega_{ik}\wedge\omega_{kj}
=\sum^{N}_{m<l}rot_{ml}\left(\sum^{N}_{s=1}q_{ism}q_{sjl}\right)\omega_{m}\wedge\omega_{l}
$$
Having in mind the above two relations we get the result.\\
\\
\textbf{Note 1.}\\
When exist a field $\Phi_{ij}$ such that $\nabla_{k}\Phi_{ij}=q_{ijk}$ : (a), then from Theorem 2 we have $d(\omega_{ij})=0$ and using Theorem 3:
\begin{equation}
R_{iklm}=-\sum_{s=1}^{N}rot_{lm}\left(q_{isl}q_{ksm}\right)=0
\end{equation}
and in view of (26),(16):
\begin{equation}
\partial_l\Gamma_{ijk}-\partial_k\Gamma_{ijl}=0.
\end{equation}
\\
\textbf{Proposition 2.}
\begin{equation}
\nabla_l\nabla_m(\overline{e}_i)=\sum^{N}_{k=1}\left(\nabla_l\left(q_{ikm}\right)-\sum^{N}_{s=1}q_{ism}q_{ksl}\right)\overline{e}_k
\end{equation}
Hence
\begin{equation}
\left\langle \nabla_k^2(\overline{e}_m),\overline{e}_m \right\rangle=-\sum^{N}_{s=1}q_{msk}^2=\textrm{invariant}
\end{equation}
\begin{equation}
T_{ijlm}:=\left\langle \nabla_{l}\nabla_m(\overline{e}_i)-\nabla_m\nabla_{l}(\overline{e}_i),\overline{e}_{j}\right\rangle=\textrm{invariant},
\end{equation}
and
\begin{equation}
T_{ijlm}=rot_{lm}\left(\nabla_l\left(q_{ijm}\right)+\sum^{N}_{s=1}q_{isl}q_{jsm}\right)
\end{equation}
for all $i,j,l,m\in\{1,2,\ldots,N\}$.\\
\\
\textbf{Proof.}\\
See Lemma 1 below.\\
\\
\textbf{Theorem 4.}
\begin{equation}
T_{ijlm}=-\sum^{N}_{s=1}q_{ijs}Q_{lsm}
\end{equation}
and
\begin{equation}
\nabla_l\left(q_{ijm}\right)-\nabla_m\left(q_{ijl}\right)
\end{equation}
are invariants.\\
\\
\textbf{Proof.}\\
Use Theorem 3 along with Proposition 2.\\
\\
\textbf{Definition 3.}\\
We construct the differential operator
$\Theta^{(1)}_{lm}$, such that for a vector field $\overline{Y}=Y_1\overline{e}_{1}+Y_2\overline{e}_2+\ldots+Y_N\overline{e}_N$ it is
\begin{equation}
\Theta^{(1)}_{lm}(\overline{Y}):=\nabla_l(Y_m)-\nabla_m(Y_l)+\sum^{N}_{k=1}Y_kQ_{lkm}.
\end{equation}
\\
\textbf{Theorem 5.}\\
The derivative of a vector $\overline{Y}=Y_1\overline{e}_{1}+Y_2\overline{e}_2+\ldots+Y_N\overline{e}_N$ is
\begin{equation}
d\overline{Y}=\sum^{N}_{j=1}\left(\sum^{N}_{l=1}Y_{j;l}\omega_l\right)\overline{e}_j
\end{equation}
where
\begin{equation}
Y_{j;l}=\nabla_lY_j-\sum^{N}_{k=1}q_{jkl}Y_k=\textrm{invariant}.
\end{equation}
\\
\textbf{Remark 1.}
\begin{equation}
\Theta_{lm}^{(1)}\left(\overline{Y}\right)=Y_{m;l}-Y_{l;m}
\end{equation}
\\
\textbf{Lemma 1.}\\
For every vector field $\overline{Y}$ we have
\begin{equation}
\left\langle \nabla_k \overline{Y},\overline{e}_l\right\rangle=Y_{l;k}=\textrm{invariant}
\end{equation}
\\
\textbf{Proof.}
$$
\left\langle \nabla_k \overline{Y},\overline{e}_l\right\rangle=\left\langle \nabla_k\left(\sum^{N}_{m=1}Y_m\overline{e}_m\right),\overline{e}_l\right\rangle=
$$
$$
=\sum^{N}_{m=1}\left\langle\nabla_k\left(Y_m\right)\overline{e}_m+Y_m\nabla_k\left(\overline{e}_m\right),\overline{e}_l\right\rangle=
$$
$$
=\nabla_kY_l+\sum^{N}_{m=1}Y_m\sum^{N}_{s=1}q_{msk}\left\langle\overline{e}_s,\overline{e}_l\right\rangle=\nabla_kY_l+\sum^{N}_{m=1}Y_mq_{mlk}=
$$
$$
=\nabla_kY_l-\sum^{N}_{m=1}Y_mq_{lmk}=Y_{l;k}=\textrm{invariant}.
$$
\\
\textbf{Proposition 3.}\\
The connections $q_{ijk}$ are invariants.\\
\\
\textbf{Proof.}
$$
\left\langle \nabla_k\overline{e}_m,\overline{e}_l\right\rangle=\left\langle\sum^{N}_{j=1}q_{mjk}\overline{e}_j,\overline{e}_l\right\rangle=q_{mlk}.
$$
\\
\textbf{Definition 4.}\\
If $\omega=\sum^{N}_{k=1}a_k\omega_k$ is a Pfaff form, then we define
\begin{equation}
\Theta^{(2)}_{lm}(\omega):=\nabla_{l}a_m-\nabla_{m}a_l+\sum^{N}_{k=1}a_kQ_{lkm}
\end{equation}
Hence
\begin{equation}
d\omega=\sum^{N}_{l<m}\Theta^{(2)}_{lm}\left(\omega\right)\omega_{l}\wedge\omega_{m},
\end{equation}
The above derivative operators $X_{m;l}$, $\Theta^{(1)}_{lm}\left(\overline{Y}\right)$ and $\Theta^{(2)}_{lm}(\omega)$ are invariant under all acceptable change of variables. As application of such differentiation are the forms $\omega_{ij}=\sum^{N}_{k=1}q_{ijk}\omega_k$, which we can write
\begin{equation}
\Theta^{(2)}_{lm}\left(\omega_{ij}\right)=R_{ijlm}
\end{equation}
and lead us concluding that $R_{ijlm}$ are invariants of $\textbf{S}$. Actually\\
\\
\textbf{Theorem 6.}\\
$R_{ijlm}$ is the curvature tensor of $\textbf{S}$.\\
\\
\textbf{Theorem 7.}\\
If $\omega=\sum^{N}_{k=1}a_{k}\omega_k$, then
\begin{equation}
d(f\omega)=\sum_{l<m}\left(\left|
\begin{array}{cc}
\nabla_lf\textrm{ }\nabla_{m}f\\
a_l\textrm{ }\textrm{ }a_m
\end{array}
\right|+\Theta^{(2)}_{lm}(\omega)f\right)\omega_{l}\wedge\omega_{m}.
\end{equation}
In particular
\begin{equation}
d(f\omega_{ij})=\sum_{l<m}\left(\left|
\begin{array}{cc}
\nabla_lf\textrm{ }\nabla_{m}f\\
q_{ijl}\textrm{ }\textrm{ }q_{ijm}
\end{array}
\right|+R_{ijlm}f\right)\omega_{l}\wedge\omega_{m}.
\end{equation}
Also
\begin{equation}
\Theta^{(2)}_{lm}(f\omega)=\left|
\begin{array}{cc}
\nabla_lf\textrm{ }\nabla_mf\\
a_l\textrm{ }\textrm{ }a_m
\end{array}
\right|+f\Theta^{(2)}_{lm}(\omega).
\end{equation}
If we set
\begin{equation}
R:=R_1\omega_1+R_2\omega_2+\ldots+R_N\omega_N,
\end{equation}
then
\begin{equation}
\Theta^{(2)}_{lm}\left(R\right)=\nabla_lR_m-\nabla_mR_l+\sum^{N}_{k=1}R_kQ_{lkm}
\end{equation}
and
\begin{equation}
\sum_{l<m}\Theta^{(2)}_{lm}(\omega)=\sum_{l<m}\left(\nabla_la_m-\nabla_ma_l\right)+\sum^{N}_{k=1}a_kA_k,
\end{equation}
where the $A_k$ are defined in Deffinition 6 bellow.\\
\\
\textbf{Theorem 8.}\\
We set
\begin{equation}
\overline{R}:=R_1\overline{e}_1+R_2\overline{e}_2+\ldots+R_N\overline{e}_N,
\end{equation}
where the $R_k$ are as in Definition 6 below, then
\begin{equation}
\sum_{l<m}\left(\nabla_lR_m-\nabla_mR_l\right)=\textrm{invariant}.
\end{equation}
\\
\textbf{Proof.}\\
From Definition 6 below and $\sum^{N}_{k=1}R_kA_k=0$ (relation (76) below), we get
\begin{equation}
\sum_{l<m}\Theta^{(1)}_{lm}\left(\overline{R}\right)=\sum_{l<m}\left(\nabla_lR_m-\nabla_mR_l+\sum^{N}_{k=1}R_kQ_{lkm}\right)
\end{equation}
\\
\textbf{Note 2.}\\
\textbf{i)}
\begin{equation}
\Theta^{(2)}_{lm}(\omega_j)=Q_{ljm}=\textrm{invariant}
\end{equation}
Hence
\begin{equation}
\sum_{l<m}\Theta^{(2)}_{lm}(\omega_j)=A_j=\textrm{invariant}
\end{equation}
\textbf{ii)}
If we assume that $\omega=dg=\sum^{N}_{k=1}(\nabla_{k}g)\omega_k$ and use (27) we get
\begin{equation}
\Theta^{(2)}_{lm}(dg)=0
\end{equation}
and hence for all multivariable functions $f,g$ we have the next\\
\\
\textbf{Theorem 9.}
\begin{equation}
\int_{\partial A}fdg=\sum_{l<m}\int\int_A\left(\nabla_lf\nabla_mg-\nabla_mf\nabla_lg\right)\omega_l\wedge\omega_m
\end{equation}
\\
\textbf{Proposition 4.}\\
If exists multivariable function $f=f(u_1,u_2,\ldots,u_N)\in\textbf{R}$ such that
\begin{equation}
\left|
\begin{array}{cc}
\nabla_1f\textrm{ }\nabla_2f\\
q_{ij1}\textrm{ }q_{ij2}
\end{array}
\right|=\left|
\begin{array}{cc}
\nabla_2f\textrm{ }\nabla_3f\\
q_{ij2}\textrm{ }q_{ij3}
\end{array}
\right|=\ldots=\left|
\begin{array}{cc}
\nabla_{N-1}f\textrm{ }\nabla_Nf\\
q_{ijN-1}\textrm{ }q_{ijN}
\end{array}
\right|=0,
\end{equation}
then exists function $\mu_{ij}$ such that
\begin{equation}
\int_{\partial A} \mu_{ij}df=\sum_{l<m} \int\int_AR_{ijlm}\omega_{l}\wedge\omega_{m}.
\end{equation}
\\
\textbf{Proof.}\\
Obviously we can write
$$
\frac{\nabla_1f}{q_{ij1}}=\frac{\nabla_2f}{q_{ij2}}=\ldots=\frac{\nabla_Nf}{q_{ijN}}=\frac{1}{\mu_{ij}}.
$$
From Theorem 3 we have
$$
\nabla_l(q_{ijm})-\nabla_m(q_{ijl})+\sum^{N}_{k=1}q_{ijk}Q_{lkm}=R_{ijlm}.
$$
Hence
$$
\nabla_{l}(\mu_{ij}\nabla_mf)-\nabla_{m}(\mu_{ij}\nabla_lf)+\sum^{N}_{k=1}(\nabla_kf)\mu_{ij}Q_{lkm}=R_{ijlm},
$$
or equivalently using Theorem 2
$$
\nabla_mf\cdot\nabla_l \mu_{ij}-\nabla_lf\cdot\nabla_{m}\mu_{ij}=R_{ijlm}.
$$
Hence from relation (61) (Stokes formula) we get the result.\\
\\
\textbf{Note 3.}\\
Condition (62) is equivalent to say that exist functions $\mu_{ij}$ and $f$ such that
\begin{equation}
\omega_{ij}=\mu_{ij}df.
\end{equation}
\\
\textbf{Theorem 10.}\\
If exists function field $F_{ij}$ such that
\begin{equation}
\sum_{l<m}\sum_{i,j\in I}F_{ij}\Theta^{(1)}_{lm}\left(\overline{x}\omega_{ij}\right)=\overline{0},
\end{equation}
then
\begin{equation}
\overline{x}=\sum^{N}_{k=1}\left(\frac{\sum_{i,j\in I}F_{ij}r_{ijk}}{\sum_{l<m}\sum_{i,j\in I}F_{ij}R_{ijlm}}\right)\overline{e}_k.
\end{equation}
\\
\textbf{Proof.}\\
From (50) and (47) we have
$$
\sum_{l<m}\Theta_{lm}\left(\overline{x}\omega_{ij}\right)=\sum_{l<m}\left|
\begin{array}{cc}
\nabla_l(\overline{x})\textrm{ }\nabla_m(\overline{x})\\
q_{ijl}\textrm{ }\textrm{ }q_{ijm}
\end{array}
\right|+\overline{x}\sum_{l<m}R_{ijlm}=
$$
$$
=\sum_{l<m}\left|
\begin{array}{cc}
\overline{e}_l\textrm{ }\textrm{ }\overline{e}_m\\
q_{ijl}\textrm{ }\textrm{ }q_{ijm}
\end{array}
\right|+\overline{x}\sum_{l<m}R_{ijlm}=-\sum^{N}_{k,s=1}\epsilon_{ks}q_{ijs}\overline{e}_k+\overline{x}\sum_{l<m}R_{ijlm}=
$$
\begin{equation}
=-\sum^{N}_{k=1}r_{ijk}\overline{e}_k+\overline{x}\sum_{l<m}R_{ijlm},
\end{equation}
where the $r_{ijk}$ are defined in (68) and the $\epsilon_{ks}$ are that of Definition 6 below.
Hence, if exists function field $F_{ij}$ such that
(65) holds, then we get the validity of (66).\\
\\
Above we have set
\begin{equation}
r_{ijk}=\sum^{N}_{s=1}\epsilon_{ks}q_{ijs}.
\end{equation}
Also
\begin{equation}
\left\langle\sum_{l<m}\Theta_{lm}\left(\overline{x}\omega_{ij}\right),\overline{e}_k\right\rangle=-r_{ijk}+w_k\sum_{l<m}R_{ijlm},
\end{equation}
where $w_k=\left\langle \overline{x},\overline{e}_k\right\rangle$ is called support function of the hypersurface.\\
\\
Setting the symbols
$$
f_k=f_{k}(x_1,x_2,\ldots,x_N)\textrm{, }k=1,2,\ldots,N
$$
be such that
\begin{equation}
\overline{x}=\sum^{N}_{k=1}f_k\overline{e}_{k},
\end{equation}
then
$$
d(\overline{x})=\sum^{N}_{k=1}df_k\overline{e}_k+\sum^{N}_{k=1}f_kd(\overline{e}_k)=\sum^{N}_{k=1}(df_k)\overline{e}_k+\sum^{N}_{k=1}f_k\sum^{N}_{j=1}\omega_{kj}\overline{e}_j=
$$
$$
=\sum^{N}_{k=1}(df_k)\overline{e}_k+\sum^{N}_{j=1}f_j\sum^{N}_{k=1}\omega_{jk}\overline{e}_k=\sum^{N}_{k=1}\left(df_k+\sum^{N}_{j=1}f_j\omega_{jk}\right)\overline{e}_k.
$$
Hence from (5) and the above relation we get
\begin{equation}
\omega_k=df_k+\sum^{N}_{j=1}f_j\omega_{jk}.
\end{equation}
If we use the Pfaff expansion of the differential we get
$$
\omega_k=\sum^{N}_{l=1}(\nabla_lf_k)\omega_l+\sum^{N}_{j=1}f_j\sum^{N}_{m=1}q_{jkm}\omega_m
$$
Or equivalent
$$
\omega_k=\sum^{N}_{l=1}(\nabla_lf_k)\omega_l+\sum^{N}_{l=1}\sum^{N}_{j=1}f_jq_{jkl}\omega_l
$$
Hence
$$
\nabla_lf_k-\sum^{N}_{j=1}f_jq_{kjl}=\delta_{lk}
$$
Or equivalently we conclude that:
The necessary conditions such that $f_k$ be the coordinates of the vector $\overline{x}$ (who generates the space $\textbf{S}$), in the moving frame $\overline{e}_{k}$, are
$$
f_{k;l}=\delta_{kl}.
$$
Hence we get the next\\
\\
\textbf{Theorem 11.}\\
If
\begin{equation}
f_{k;l}=\delta_{kl}\textrm{, where }k,l\in\{1,2,\ldots,N\},
\end{equation}
then we have
$$
\overline{x}=\sum^{N}_{k=1}f_k\overline{e}_k+\overline{h}\textrm{, where }d\overline{h}=0
$$
and the opposite.\\
\\
Finally having in mind of (66) we get
\begin{equation}
f_k=\frac{\sum_{i,j\in I}F_{ij}r_{ijk}}{\sum_{l<m}\sum_{i,j\in I}F_{ij}R_{ijlm}}
\end{equation}
For two generalized hyper-vectors $F=F_{ij\ldots k}$ and $G=G_{ij\ldots k}$ we define the generalized inner product as
\begin{equation}
(F,G):=\sum^{N}_{i,j,\ldots,k=1}F_{ij\ldots k}G_{ij\ldots k}.
\end{equation}
Hence relation (73) can be written as
$$
f_k=\sum^{N}_{i,j=1}\frac{F_{ij}}{(F,U)}r_{ijk},
$$
where
$$
U:=U_{ij}:=\sum_{l<m}R_{ijlm}
$$
Hence
$$
f_{k;l}=\sum^{N}_{i,j=1}\nabla_l\left(\frac{F_{ij}}{(F,U)}r_{ijk}\right)-\sum^{N}_{i,j,m=1}\frac{F_{ij}}{(F,U)}r_{ijm}q_{kml}=
$$
$$
=\sum^{N}_{i,j=1}\nabla_l\left(\frac{F_{ij}}{(F,U)}\right)r_{ijk}+\sum^{N}_{i,j=1}\frac{F_{ij}}{(F,U)}\nabla_l\left(r_{ijk}\right)-\sum^{N}_{i,j,m=1}\frac{F_{ij}}{(F,U)}r_{ijm}q_{kml}=
$$
$$
=\sum^{N}_{i,j=1}\nabla_l\left(\frac{F_{ij}}{(F,U)}\right)r_{ijk}+\sum^{N}_{i,j=1}\frac{F_{ij}}{(F,U)}r_{ij\{k\};l}
$$
\\
\textbf{Theorem 12.}\\
If $F_{ij}$ is such that
\begin{equation}
\sum^{N}_{i,j=1}\nabla_l\left(\frac{F_{ij}}{(F,U)}\right)r_{ijk}+\sum^{N}_{i,j=1}\frac{F_{ij}}{(F,U)}r_{ij\{k\};l}=\delta_{kl},
\end{equation}
then exists constant vector $\overline{h}$ such that
\begin{equation}
\overline{x}=\sum^{N}_{k,i,j=1}\frac{F_{ij}}{(F,U)}r_{ijk}\overline{e}_k+\overline{h},
\end{equation}
where $d\overline{h}=0$.\\
\\
\textbf{Theorem 13.}\\
There holds (see note 4 below):
\begin{equation}
q_{ik\{m\};l}-q_{ik\{l\};m}=R_{iklm}.
\end{equation}
and
$$
b_{\{k\}m,l}=b_{\{k\}l,m},\eqno{(77.1)}
$$
where
$$
Y_{\{n\}m,l}=\partial_lY_{nm}-\sum^{N}_{j=1}\Gamma_{njl}Y_{jm}\textrm{, }Y_{\{n\}m,l}=\partial_lY_{nm}-\sum^{N}_{j=1}\Gamma_{mjl}Y_{nj}\eqno{(77.2)}
$$
and
$$
Y_{nm,l}=\partial_lY_{nm}-\sum^{N}_{j=1}\Gamma_{njl}Y_{jm}-\sum^{N}_{j=1}\Gamma_{mjl}Y_{nj}\textrm{, }\ldots\textrm{etc}.\eqno{(77.3)}
$$
\\
\textbf{Proof.}\\
Easy\\
\\
\textbf{Note 4.}\\
In general in $t_{ij\ldots \{k\}\ldots m;l}$ the brackets indicate where the differential acts. Hence
\begin{equation}
t_{ij\ldots \{k\}\ldots m;l}=\nabla_lt_{ij\ldots k\ldots m}-\sum^{N}_{\nu=1}q_{k\nu l}t_{ij\ldots \nu\ldots m}
\end{equation}
Also we can use more brackets $\{\}$ in the vector.
$$
t_{ij\ldots \{k_1\}\ldots\{k_2\}\ldots m;l}=
$$
\begin{equation}
=\nabla_lt_{ij\ldots k_1\ldots k_2\ldots m}-\sum^{N}_{\nu_1=1}q_{k_1\nu_1 l}t_{ij\ldots \nu_1\ldots k_2\ldots m}-\sum^{N}_{\nu_2=1}q_{k_2\nu_2 l}t_{ij\ldots k_1\ldots\nu_2\ldots m}.
\end{equation}
In case we change the ''$;$'' with ''$,$'' then we lead to the classical invariant derivative
$$
t_{ij\ldots \{k\}\ldots m,l}=\partial_lt_{ij\ldots k\ldots m}-\sum^{N}_{\nu=1}\Gamma_{k\nu l}t_{ij\ldots \nu\ldots m},
$$
... etc. The two kinds of derivative lead us to the same evaluations. More precicely it holds
$$
\sum^{N}_{l=1}t_{ij\ldots \{k\}\ldots m;l}\omega_l=\sum^{N}_{l=1}t_{ij\ldots \{k\}\ldots m,l}du_l
$$
\\
\textbf{Theorem 14.}\\
The following forms are invariants of the space $\textbf{S}$:\\
i) The linear element is
\begin{equation}
I=(ds)^2=\sum^{N}_{k=1}\omega_k^2.
\end{equation}
ii) The volume element of $\textbf{S}$ is
\begin{equation}
V=\omega_1\wedge\omega_2\wedge\ldots\wedge\omega_{N-1}\wedge \omega_{N}.
\end{equation}
The area element of the subspace normal to $\overline{e}_{M}$ is
$$
E_{M}=\omega_1\wedge\omega_2\wedge\ldots\wedge\omega_{M-1}\wedge\omega_{M+1}\wedge\ldots\wedge\omega_{N}
$$
iii) The second invariant forms are
\begin{equation}
II_{M}=\left\langle d\overline{x},d\overline{e}_{M}\right\rangle=\sum^{N}_{k=1}\omega_k\omega_{Mk}\textrm{, }M=1,\ldots,N
\end{equation}
iv) The linear element of $\overline{e}_{M}$ is
\begin{equation}
III_{M}=(d\overline{e}_M)^2=\left\langle d\overline{e}_{M},d\overline{e}_{M}\right\rangle=\sum^{N}_{k=1}\omega_{Mk}^2
\end{equation}
v) The Gauss curvature $K_{M}$ which corresponds to the subspace normal to $\overline{e}_M$ vector, is given from
\begin{equation}
K_{M}=det\left(\kappa^{\{M\}}_{ij}\right)=\frac{\omega_{M1}\wedge\omega_{M2}\wedge\ldots\wedge\omega_{M(M-1)}\wedge\omega_{M(M+1)}\wedge\ldots\wedge \omega_{MN}}{E_{M}},
\end{equation}
where
\begin{equation}
\kappa^{\{M\}}_{km}:=q_{kMm}.
\end{equation}
The above forms remain unchanged, in every change of position of $\overline{x}$ and rotation of $\{\overline{e}_j\}_{j=1,2,\ldots,N}$, except for possible change of sign.\\
\\
\textbf{Definition 5.}\\
We define the Beltrami differential operator
\begin{equation}
\Delta_2f:=\sum^{N}_{l<m}\left(
\nabla_{l}^2f+\nabla^2_{m}f+\frac{1}{N-1}\sum^{N}_{i<j}
\left|\begin{array}{cc}
\nabla_if\textrm{ }\nabla_jf\\
Q_{lim}\textrm{ }Q_{ljm}
\end{array}\right|\right).
\end{equation}
We also call $f$ harmonic if $\Delta_2f=0$.\\
\\
\textbf{Remark 2.}\\
In the particular case of $N=2$ where the space $\textbf{S}$ is a two dimensional surface embeded to $\textbf{E}_3$, we have
\begin{equation}
\Delta_2A=\nabla_1\nabla_1A+\nabla_2\nabla_2A+q_2\nabla_1A-q_1\nabla_2A,
\end{equation}
where
\begin{equation}
d\overline{x}=\omega_1\overline{e}_1+\omega_2\overline{e}_2
\end{equation}
and $d\omega_1=q_1\omega_1\wedge\omega_2$ and $d\omega_2=q_2\omega_1\wedge\omega_2$, there exists functions $f,f^{*}$ such that $\nabla_1f=-\nabla_2f^{*}$ and $\nabla_2f=\nabla_1f^{*}$
and $\Delta_2f=\Delta_2f^{*}=0$, ($f,f^{*}$ harmonics). In higher dimensions is not such easy to construct harmonic functions. However we will give here one way of such construction. Before going to study this operator, we simplify the expansion (86) (of the definition of Beltrami differential operator $\Delta_2$). We also generalize it (in a way) as we show below in Definition 8. First we give a definition.\\
\\
\textbf{Definition 6.}\\
Set
\begin{equation}
A_s:=\sum_{l<m}Q_{lsm}\textrm{, }R_k:=\sum^{N}_{s=1}\epsilon_{ks}A_s
\end{equation}
and more generally if
\begin{equation}
a_s=\sum^{N}_{l<m}t_{lsm}\textrm{ and }
r_{s}:=\sum^{N}_{k=1}\epsilon_{sk}a_k,
\end{equation}
where $\epsilon_{ks}:=-1$ if $k<s$, $1$ if $k>s$ and $0$ if $k=s$. Then from identity $\sum^{N_1}_{k,s=1}\epsilon_{ks}f_kf_s=0$, for every choice of $f_k$, we get
\begin{equation}
\sum^{N}_{k=1}R_kA_k=0\textrm{, }\sum^{N}_{s=1}r_s\alpha_s=0
\end{equation}
From the above definition we have the next\\
\\
\textbf{Theorem 15.}
\begin{equation}
\Delta_2f=\sum^{N}_{k=1}\left((N-1)\nabla_k^2f-\frac{1}{N-1}\left
(\nabla_kf\right)R_k\right).
\end{equation}
Also
\begin{equation}
\Delta_2f=(N-1)\sum^{N}_{k=1}\nabla_k^2f-\frac{1}{N-1}\left\langle\overline{\textrm{grad}(f)},\overline{R}\right\rangle.
\end{equation}
\\
\textbf{Proof.}
$$
\Delta_2f=\sum_{l<m}\left(\nabla_l^2f+\nabla_m^2f-\frac{1}{N-1}\sum^{N}_{k,s=1}\left(\nabla_kf\right)\epsilon_{ks}Q_{lsm}\right)=
$$
$$
=(N-1)\sum^{N}_{k=1}\nabla_k^2f-\frac{1}{N-1}\sum^{N}_{k=1}\left(\nabla_kf\right)\sum^{N}_{s=1}\epsilon_{ks}A_s=
$$
$$
=(N-1)\sum^{N}_{k=1}\nabla_k^2f-\frac{1}{N-1}\sum^{N}_{k=1}\left(\nabla_kf\right)R_k.
$$
\\
\textbf{Note 5.}\\
1) If exists $f$ such $\sum^{N}_{k=1}A_k\omega_k=df$, then $\nabla_kf=A_k$ and
$$
\Delta_2f=(N-1)\sum_{k=1}^N\nabla_kA_k.
$$
2) If $h_k$ is any vector such that $\sum^{N}_{k=1}h_k=0$
and $f_k=\nabla_kF$ is such that
\begin{equation}
(N-1)\nabla_kf_k-\frac{1}{N-1}f_kR_k=h_k\textrm{, }\forall k=1,2,\ldots,N,
\end{equation}
then
$$
\Delta_2F=0.
$$
Hence solving first the PDE's
$$
(N-1)\nabla_k\Psi-\frac{1}{N-1}\Psi R_k=h_k\textrm{, }\forall k=1,2,\ldots,N,\eqno{(eq1)}
$$
$$
\sum^{N}_{k=1}h_k=0,\eqno{(eq2)}
$$
we find $N$ solutions $\Psi=f_k$. Then we solve $f_k=\nabla_kF$ and thus we get a solution $F$ of $\Delta_2F=0$.\\
3) This lead us to define the derivative $D_k$, $k=1,2,\ldots,N$, which acts in every function $f$ as
\begin{equation}
D_k\left(f\right):=D_kf:=(N-1)\nabla_kf-\frac{1}{N-1}R_k f.
\end{equation}
Obviously $D_k$ is linear. But
$$
d(fg)=gdf+fdg=g\sum^{N}_{k=1}\nabla_kf\omega_k+f\sum^{N}_{k=1}\nabla_kg\omega_k=\sum^{N}_{k=1}\left(g\nabla_kf+f\nabla_kg\right)\omega_k,
$$
or equivalently
\begin{equation}
\nabla_k(fg)=g\nabla_kf+f\nabla_kg.
\end{equation}
Hence the differential operator $D_k$ acts in the product of $f$ and $g$ as
$$
D_k(fg)=(N-1)\nabla_k(fg)-\frac{1}{N-1}fgR_k=
$$
$$
=(N-1)f\nabla_kg+(N-1)g\nabla_kf-\frac{1}{N-1}fgR_k=fD_kg+(N-1)g\nabla_kf=
$$
$$
=gD_kf+(N-1)f\nabla_kg
$$
and finally we have
\begin{equation}
D_k(fg)=fD_kg+gD_kf+\frac{1}{N-1}f gR_k,
\end{equation}
\begin{equation}
fD_kg-gD_kf=(N-1)(f\nabla_kg-g\nabla_kf).
\end{equation}
The commutator of $D_k$ and $\nabla_k$ acting to a scalar field is
$$
\left[D_k,\nabla_k\right]f
=D_k\nabla_kf-\nabla_kD_kf=
$$
$$
=(N-1)\nabla_k^2f-\frac{1}{N-1}R_k\nabla_kf-\left((N-1)\nabla_k^2f-\frac{1}{N-1}\nabla_k(R_kf)\right).
$$
Hence finally after simplifications
\begin{equation}
\left[D_k,\nabla_k\right]f
=\frac{\nabla_kR_k}{N-1}f.
\end{equation}
i) If $f=const$, then
$$
D_kf=-\frac{1}{N-1}R_kf.
$$
ii) The derivative $D_k$ is such that if exists function $f$ with
$$
D_kf=0\textrm{, }\forall k=1,2,\ldots,N,
$$
then equivalently
$$
\nabla_k(\log f)=\frac{1}{(N-1)^2}R_k\textrm{, }\forall k=1,2,\ldots,N.
$$
But then
\begin{equation}
D_kf=0\Leftrightarrow \exists g\left(=(N-1)^2\log f\right):R_k=\nabla_kg\Leftrightarrow R=dg.
\end{equation}
By these arguments we conclude to the result, that only in specific spaces $\textbf{S}$ exists $f$ such $D_kf=0$, for all $k=1,2,\ldots,N$. Hence we have a definition-theorem:\\
\\
\textbf{Definition 7.}\\
A space is called $\textbf{S}$-$R$ iff exists function $g$ such that $R=dg$.\\
\\
\textbf{Theorem 16.}\\
A space $\textbf{S}$ is $\textbf{S}$-$R$ iff exists $g$ such that $D_kg=0$, for all $k=1,2,\ldots,N$.\\
\\
\textbf{Corollary 2.}\\
In a $\textbf{S}$-$R$ space
\begin{equation}
\Theta_{lm}(R)=0\textrm{, }\forall l,m\in\{1,2,\ldots,N\}.
\end{equation}
\textbf{Proof.}\\
It is $R=\sum^{N}_{k=1}R_k\omega_k$ and $\textbf{S}$ is $\textbf{S}$-$R$. Hence
$$
\Theta_{lm}(R)=\nabla_lR_m-\nabla_mR_l+\sum^{N}_{k=1}R_kQ_{lkm}.
$$
But exists $g$ such that $R_k=\nabla_kg$, for all $k=1,2,\ldots,N$. Hence
$$
\Theta_{lm}(R)=\nabla_l\nabla_mg-\nabla_m\nabla_lg+\sum^{N}_{k=1}(\nabla_kg)Q_{lkm}=0.
$$
The last equality is due to Theorem 2. Hence the result follows.\\
\\
\textbf{Corollary 3.}\\
In a $\textbf{S}$-$R$ space holds
$$
d(fR)=\sum_{l<m}\left(R_m\nabla_lf-R_l\nabla_mf \right)\omega_l\wedge\omega_m.\eqno{(101.1)}
$$
\textbf{Proof.}\\
The result is application of Theorem 7.\\
\\
\textbf{Corollary 4.}\\
In every space $\textbf{S}$ holds
$$
\sum^{N}_{k=1}A_k(D_kf)=(N-1)\sum^{N}_{k=1}A_k(\nabla_kf).\eqno{(101.2)}
$$
\\
\textbf{Theorem 17.}\\
Set
$$
Df:=(D_1f)\omega_1+(D_2f)\omega_2+\ldots+(D_Nf)\omega_N,\eqno{(101.3)}
$$
then
$$
Df=(N-1)df-\frac{1}{N-1}fR.\eqno{(101.4)}
$$
Also
$$
D\left\langle\overline{V}_1,\overline{V}_2\right\rangle=\left\langle D\overline{V}_1,\overline{V}_2\right\rangle+\left\langle \overline{V}_1,D\overline{V}_2\right\rangle+\frac{R}{N-1}\left\langle \overline{V}_1,\overline{V}_2\right\rangle.\eqno{(101.5)}
$$
\\
\textbf{Theorem 18.}\\
If a space $\textbf{S}$ is $\textbf{S}$-$R$, then the PDE $\Delta_2\Psi=\mu \Psi$ have non trivial solution.\\
\\
\textbf{Proof.}\\
If $\textbf{S}$ is $\textbf{S}$-$R$, then from (100) we get that exists $f,g$ such $g=(N-1)^2\log f$ and $R=dg$, $D_jf=0$, $\forall j$. Hence
$$
(N-1)\nabla_jf-\frac{1}{N-1}R_jf=0\Rightarrow
$$
$$
(N-1)\nabla_j\nabla_jf-\frac{1}{N-1}R_j\nabla_jf-\frac{\nabla_jR_j}{N-1}f=0\Rightarrow
$$
$$
\Delta_2f=\frac{f}{N-1}\sum^{N}_{j=1}\nabla_jR_j.
$$
\\
\textbf{Theorem 19.}\\
In every space $\textbf{S}$ we have
\begin{equation}
\overline{Df}:=\sum^{N}_{i=1}(D_kf)\overline{e}_k=(N-1)\overline{\textrm{grad}f}-\frac{1}{N-1}\overline{R}f.
\end{equation}
\\
\textbf{Note 6.}\\
1) If $C$ is a curve, then
$$
\left(\frac{Df}{ds}\right)_C=(N-1)\left(\frac{df}{ds}\right)_C-\frac{1}{N-1}\left(\frac{dR}{ds}\right)_C f
$$
and if $(dR)_C\neq0$, then
$$
\left(\frac{Df}{dR}\right)_C=(N-1)\left(\frac{df}{dR}\right)_C-\frac{1}{N-1}f
$$
and $\left(\frac{Df}{dR}\right)_C=0$ iff
$$
\left(\frac{df}{dR}\right)_C=\frac{1}{(N-1)^2}f.
$$
Hence
$$
f_C=const\cdot\exp\left(\frac{R_C}{(N-1)^2}\right).
$$
Hence if $f=f(x_1,x_2,\ldots,x_N)$ and $C:x_i=x_i(s)$, $s\in\left[a,b\right]$, such $\left(\frac{Df}{ds}\right)_C=0$, then
$$
f\left(x_1(s),x_2(s),\ldots,x_N(s)\right)=const\cdot\exp\left(\frac{R\left(x_1(s),x_2(s),\ldots,x_N(s)\right)}{(N-1)^2}\right).
$$
2) Also assuming $C:x_i=x_i(s)$ is a curve in $\textbf{S}$ and $\overline{\nu}$ is the tangent vector of $C$, which is such that $\left\langle \overline{\nu},\overline{\nu}\right \rangle=1\Rightarrow\left\langle \overline{\nu},\frac{d\overline{\nu}}{ds}\right \rangle=0$, then
$$
\frac{D\overline{\nu}}{ds}=(N-1)\frac{d\overline{\nu}}{ds}-\frac{1}{N-1}\frac{dR}{ds}\overline{\nu}.
$$
Consequently we get
$$
\left\langle\frac{D\overline{\nu}}{ds},\frac{d\overline{\nu}}{ds}\right\rangle=(N-1)k(s)
$$
and
$$
\left\langle\frac{D\overline{\nu}}{ds},\overline{\nu}\right\rangle=-\frac{1}{N-1}\frac{dR}{ds}.
$$
But
$$
D\left\langle \overline{\nu},\overline{\nu}\right \rangle=D1\Leftrightarrow 2\left\langle D\overline{\nu},\overline{\nu}\right \rangle+\frac{dR}{N-1}=-\frac{dR}{N-1}
.
$$
Hence
\begin{equation}
\left\langle D\overline{\nu},\overline{\nu}\right \rangle=-\frac{dR}{N-1}.
\end{equation}
Hence differentiating we get
$$
d\left\langle D\overline{\nu},\overline{\nu}\right \rangle=0.
$$
\\
\textbf{Theorem 20.}\\
For every curve $C:x_i=x_i(s)$ of a general space $\textbf{S}$, with $s$ being its normal parameter, the unitary tangent vector $\overline{\nu}$ of $C$ has the property
$$
\left\langle \left(\frac{D\overline{\nu}}{ds}\right)_C,\overline{\nu}\right\rangle=-\frac{1}{N-1}\left(\frac{dR}{ds}\right)_C.
$$
Moreover we define
$$
k^{*}=\left(k^{*}\right)_C:=\left(\frac{1}{\rho_{R}}\right)_C:=\left|\left(\frac{D \overline{\nu}}{ds}\right)_C\right|,
$$
then
$$
k^{*}(s)=\sqrt{(N-1)^2k^2(s)+\frac{1}{(N-1)^2}\left(\left(\frac{dR}{ds}\right)_C\right)^2}.
$$
If the space is $\textbf{S}$-$R$, then
$$
\left(k^{*}\right)_C=(N-1)k(s).
$$
\\
\textbf{Corollary 5.}\\
Let $C:x_i=x_i(s)$ be a curve of a $\textbf{S}$-$R$ space. If $s$ is the normal parameter of $C$ and $\overline{\nu}$ is the unitary tangent vector of $C$, then
$$
\frac{D\overline{\nu}}{ds}=(N-1)\frac{d\overline{\nu}}{ds}\textrm{
and
}\left|\frac{D\overline{\nu}}{ds}\right|=(N-1)k(s),
$$
where $k(s)$ is the curvature of $C$.\\
\\
Next we generalize the deffinition of $\Delta_2$ operator:\\
\\
\textbf{Definition 8.}\\
\begin{equation}
\Delta^{(\lambda)}(f)=\sum^{N}_{i,j=1}\lambda_{ij}D_i\nabla_jf.
\end{equation}
Hence if $\lambda_{ij}=\delta_{ij}$, then
\begin{equation}
\Delta_2f=\sum^{N}_{i,j=1}\delta_{ij}D_i\nabla_jf=\sum^{N}_{k=1}D_k\nabla_kf.
\end{equation}
\\
\textbf{Theorem 21.}\\
If $\lambda_{ij}=\epsilon_{ij}$, then
\begin{equation}
\Delta^{(\epsilon)}f=\sum^{N}_{k=1}\left((N-1)A_k+\frac{1}{N-1}\sum^{N}_{s=1}\epsilon_{ks}R_s\right)\nabla_kf.
\end{equation}
\\
\textbf{Proof.}\\
For $\lambda_{ij}=\epsilon^{*}_{ij}=-\epsilon_{ij}$ we have
$$
\Delta^{(\epsilon^{*})}f=\sum^{N}_{i,j=1}\epsilon^{*}_{ij}D_i\nabla_jf=\sum_{i<j}\epsilon^{*}_{ij}D_i\nabla_{j}f+\sum_{i>j}\epsilon^{*}_{ij}D_i\nabla_jf=
$$
$$
=\sum_{i<j}\left(D_i\nabla_jf-D_j\nabla_if\right)=
$$
$$
=(N-1)\sum_{i<j}\left(\nabla_i\nabla_jf-\nabla_j\nabla_if\right)-\frac{1}{N-1}\sum_{i<j}\left(R_i\nabla_jf-R_j\nabla_if\right)=
$$
$$
=-(N-1)\sum_{i<j}\sum^{N}_{k=1}(\nabla_kf)Q_{ikj}-\frac{1}{N-1}\sum^{N}_{k,s=1}\epsilon_{ks}R_s\nabla_kf=
$$
$$
=-(N-1)\sum^{N}_{k=1}A_k\nabla_kf-\frac{1}{N-1}\sum^{N}_{k,s=1}\epsilon_{ks}R_s\nabla_kf=
$$
$$
=-\sum^{N}_{k=1}\left((N-1)A_k+\frac{1}{N-1}\sum^{N}_{s=1}\epsilon_{ks}R_s\right)\nabla_kf.
$$
$qed$\\
\\
\textbf{Example 1.}\\
If $\lambda_{ij}=\epsilon_{ij}$ and $\textbf{S}$ is $\textbf{S}$-$R$, then
\begin{equation}
\Delta^{(\epsilon)}g=0,
\end{equation}
where $dg=R$. That is because
$$
\Delta^{(\epsilon)}g=(N-1)\sum^{N}_{k=1}A_kR_k+\frac{1}{N-1}\sum^{N}_{k,s=1}\epsilon_{ks}R_kR_s=0,
$$
since $\sum^{N}_{k=1}A_kR_k=0$ and $\sum^{N}_{k,s=1}\epsilon_{ks}R_kR_s=0$. Also easily we get
\begin{equation}
\left\langle\Delta^{(\epsilon)}\overline{x},\overline{R}\right\rangle=0.
\end{equation}
\\
\textbf{Example 2.}\\
In a $\textbf{S}$-$R$ space we have $R=dg$, for a certain $g$ and thus $\nabla_kg=R_k$. Hence we can write
\begin{equation}
\Delta_2g=(N-1)\sum^{N}_{k=1}\nabla_kR_k-\frac{1}{N-1}\sum^{N}_{k=1}R_k^2.
\end{equation}
\\
\textbf{Theorem 22.}\\
In the general case when $\lambda_{ij}$ is any field we
can write
\begin{equation}
\Delta^{(\lambda)}f=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}\nabla_{i}\nabla_{j}f-\frac{1}{N-1}\sum^{N}_{i,j=1}\lambda_{ij}R_{i}\nabla_{j}f.
\end{equation}
and
\begin{equation}
\left\langle\Delta^{(\lambda)}\overline{x},\overline{e}_k\right\rangle=(N-1)\sum_{i,j=1}^N\lambda_{ij}q_{jki}-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i.
\end{equation}
\\
\textbf{Proof.}\\
The first identity is easy. For the second we have:
Setting $f\rightarrow \overline{x}$, we get
$$
\Delta^{(\lambda)}\overline{x}=\sum^{N}_{i,j=1}\lambda_{ij}D_i\left(\nabla_j\overline{x}\right)=\sum^{N}_{i,j=1}\lambda_{ij}D_i\left(\overline{e}_j\right)=
$$
$$
=\sum^{N}_{i,j=1}(N-1)\lambda_{ij}\nabla_i\overline{e}_j-\sum^{N}_{i,j=1}\frac{\lambda_{ij}}{N-1}R_i\overline{e}_j=
$$
$$
=\sum^{N}_{i,j=1}\lambda_{ij}\left((N-1)\nabla_i\overline{e}_j-\frac{1}{N-1}R_i\overline{e}_j\right)=
$$
$$
=\sum^{N}_{i,j,k=1}(N-1)\lambda_{ij}q_{jki}\overline{e}_k-\sum^{N}_{i,k=1}\frac{\lambda_{ik}}{N-1}R_i\overline{e}_k
$$
$$
=\sum^{N}_{k=1}\left((N-1)\sum^{N}_{i,j=1}\lambda_{ij}q_{jki}-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)\overline{e}_k.
$$
\\
We need some notation to proceed further\\
\\
\textbf{Definition 9.}\\
For any field $\lambda_{ij}$ we define
$$
R^{\{\lambda\}}_k:=\sum^{N}_{i=1}\lambda_{ik}R_{i}\eqno{(111.1)}
$$
$$
A^{(\lambda)}_{k}:=\sum_{i<j}\lambda_{ij}Q_{ikj}\textrm{, }A^{{(\lambda)}{*}}_k:=\sum_{i<j}\lambda_{ij}Q^{*}_{ikj},\eqno{(111.2)}
$$
where
$$
Q_{ikj}:=q_{ikj}-q_{jki}\textrm{, }Q^{*}_{ikj}:=q_{ikj}+q_{jki}\eqno{(111.3)}
$$
and
$$
R^{(\lambda)}_{k}:=\sum^{N}_{s=1}\epsilon_{ks}A^{(\lambda)}_s\textrm{, }R^{{(\lambda)}{*}}_{k}:=\sum^{N}_{s=1}\epsilon_{ks}A^{{(\lambda)}{*}}_s.\eqno{(111.4)}
$$
\\
\textbf{Proposition 5.}
$$
\sum^{N}_{s=1}A^{(\lambda)}_sR^{(\lambda)}_s=0\textrm{, }\sum^{N}_{s=1}A^{{(\lambda)}{*}}_sR^{{(\lambda)}{*}}_s=0.\eqno{(111.5)}
$$
\\
\textbf{Theorem 23.}\\
Let $\lambda_{ij}$ be antisymmetric i.e. $\lambda_{ij}=-\lambda_{ji}$, then we have in general:\\
1)
$$
\Delta^{(\lambda)}f=-\sum^{N}_{k=1}\left((N-1)A^{(\lambda)}_k+\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)\nabla_kf.
$$
2)
$$
\left\langle\Delta^{(\lambda)}\overline{x},\overline{e}_k\right\rangle=-(N-1)A^{(\lambda)}_k-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i.
$$
3)
$$
\sum^{N}_{k=1}R_kR^{\{\lambda\}}_k=0\Leftrightarrow \left\langle \overline{R},\overline{R}^{\{\lambda\}}\right\rangle=0.
$$
Hence
$$
\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle=-(N-1)\sum^{N}_{k=1}A^{(\lambda)}_kR_k
$$
and
$$
\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}^{(\lambda)}\right\rangle=-\frac{1}{N-1}\sum^{N}_{i,k=1}\lambda_{ik}R_iR^{(\lambda)}_k.
$$
4) In case the space is $\textbf{S}$-$R$, with $\nabla_kg=R_k$, then
$$
\Delta^{(\lambda)}g=-(N-1)\sum^{N}_{k=1}A^{(\lambda)}_kR_k=\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle.
$$
\\
\textbf{Proof.}\\
The case of (1) follows with straight forward evaluation. Since $\lambda_{ij}$ is antisymmetric, we have
$$
\Delta^{(\lambda)}f=\sum_{i<j}\lambda_{ij}D_i\nabla_jf-\sum_{i<j}\lambda_{ij}D_j\nabla_if=
$$
$$
=\sum_{i<j}\lambda_{ij}\left[(N-1)\nabla_i\nabla_jf-\frac{1}{N-1}R_i\nabla_jf-\left((N-1)\nabla_j\nabla_if-\frac{1}{N-1}R_j\nabla_if\right)\right]
$$
$$
=(N-1)\sum_{i<j}\lambda_{ij}\left(\nabla_i\nabla_jf-\nabla_j\nabla_if\right)-\frac{1}{N-1}\sum_{i<j}\lambda_{ij}\left(R_i\nabla_jf-R_j\nabla_if\right)=
$$
$$
=(N-1)\sum_{i<j}\lambda_{ij}\left(-\sum^{N}_{k=1}\nabla_kfQ_{ikj}\right)-\frac{1}{N-1}\sum_{i<j}\lambda_{ij}rot_{ij}(R_i\nabla_jf)=
$$
$$
=-(N-1)\sum^{N}_{k=1}A^{(\lambda)}_k\nabla_kf-\frac{1}{N-1}\sum_{i<j}\lambda_{ij}rot_{ij}\left(R_i\nabla_jf\right)=
$$
$$
=-(N-1)\sum^{N}_{k=1}A^{(\lambda)}_k\nabla_kf-\frac{1}{N-1}\sum^{N}_{k=1}\left(\sum^{N}_{i=1}\lambda_{ik}R_i\right)\nabla_kf=
$$
$$
=-\sum^{N}_{k=1}\left((N-1)A^{(\lambda)}_k+\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)\nabla_kf.
$$
In the second we use $\nabla_k\overline{x}=\overline{e}_k$.\\
The third relation can be proved if we consider the formula
$$
\sum^{N}_{i,k=1}\lambda_{ik}R_iR_k=\sum_{i<k}\lambda_{ik}R_{i}R_k-\sum_{i<k}\lambda_{ik}R_kR_i=0.
$$
The fourth case follows easily from the first with $\nabla_kg=R_k$. Note that in the most general case where $\lambda_{ij}$ is arbitrary we still have
$$
\sum^{N}_{k=1}A^{(\lambda)}_kR^{(\lambda)}_k=0.
$$
\\
\textbf{Theorem 24.}\\
In case $\lambda_{ij}$ is any antisymmetric field, we have
\begin{equation}
\Delta^{(\lambda)}f=\sum^{N}_{k=1}\left\langle\Delta^{(\lambda)}\overline{x},\overline{e}_{k}\right\rangle\nabla_kf=\left\langle\Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}(f)}\right\rangle,
\end{equation}
where
\begin{equation}
\overline{\textrm{grad}(f)}:=\sum^{N}_{k=1}(\nabla_kf)\overline{e}_k.
\end{equation}
\\
\textbf{Proof.}\\
If $\lambda_{ij}$ is any antisymmetric field, then using Theorems 22, 23 (see also and Theorem 2 and Corollary 1) we get the result.\\
\\
\textbf{Corollary 6.}\\
If $\lambda_{ij}$ is any antisymmetric field such that
\begin{equation}
\Delta^{(\lambda)}\left(\overline{x}\right)=\overline{0},
\end{equation}
then for every $f$ we have
\begin{equation}
\Delta^{(\lambda)}f=0.
\end{equation}
\\
\textbf{Remark 3.}\\The above corollary is very strange. The Beltrami operator over an antisymmetric field is zero for every function $f$ when (114) holds. This force us to conclude that in any space the vector
\begin{equation}
\overline{\sigma}^{(\lambda)}:=\sum^{N}_{k=1}\sigma^{(\lambda)}_k\overline{e}_k=\Delta^{(\lambda)}\left(\overline{x}\right)
\end{equation}
must play a very prominent role in the geometry of $\textbf{S}$. Then (in any space):
\begin{equation}
\sigma^{(\lambda)}_k=(N-1)\sum_{i,j=1}^N\lambda_{ij}q_{jki}-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i.
\end{equation}
In case $\lambda_{ij}$ is antisymmetric, then we get
$$
\sigma^{(\lambda)}_k=-(N-1)A_k^{(\lambda)}-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i=
$$
$$
=-(N-1)A_k^{(\lambda)}-\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(\lambda)}_{kl}A_l,
$$
where
\begin{equation}
\epsilon^{(\lambda)}_{kl}:=\sum^{N}_{i=1}\lambda_{ik}\epsilon_{il}.
\end{equation}
Hence when $\lambda_{ij}$ is antisymmetric, then
$$
\sigma^{(\lambda)}_{k}=-(N-1)A_k^{(\lambda)}-\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(\lambda)}_{kl}A_l.
$$
Hence
\begin{equation}
\sigma^{(\lambda)}_{k}=-\sum^{N}_{i<j}\left[(N-1)\lambda_{ij}Q_{ikj}+\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(\lambda)}_{kl}Q_{ilj}\right].
\end{equation}
\\
\textbf{Definition 10.}\\
We define
\begin{equation}
\Lambda^{(\lambda)}_{kij}:=(N-1)\lambda_{ij}Q_{ikj}+\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(\lambda)}_{kl}Q_{ilj}
\end{equation}
and
\begin{equation}
M^{(\lambda)}_{kij}:=(N-1)\lambda_{ij}Q^{*}_{ikj}-\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(\lambda)}_{kl}Q_{ilj}.
\end{equation}
Also we define the mean curvature with respect to the $\lambda_{ij}$ as
\begin{equation}
H^{(\lambda)}_k:=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}q_{jki}.
\end{equation}
Then we have
\begin{equation}
H^{(\lambda^{+})}_k=\frac{N-1}{2}\sum^{N}_{i,j=1}\lambda_{ij}Q^{*}_{ikj}
\end{equation}
and
\begin{equation}
H^{(\lambda^{-})}_k=-\frac{N-1}{2}\sum^{N}_{i,j=1}\lambda_{ij}Q_{ikj},
\end{equation}
where $\lambda^{+}$ is the symmetric part of $\lambda_{ij}$ and $\lambda^{-}$ is the antisymmetric part of $\lambda_{ij}$.\\
\\
\textbf{Remark.}\\
It holds
\begin{equation}
\Lambda_{kij}^{(\lambda)}+M_{kij}^{(\lambda)}=2(N-1)\lambda_{ij}q_{ikj}
\end{equation}
\\
\textbf{Theorem 24.1}\\
In any space and any $\lambda_{ij}$, we have
$$
\Delta^{(\lambda)}\left(\overline{x}\right)=\sum^{N}_{k=1}\left(H^{(\lambda)}_k-\frac{1}{N-1}R^{\{\lambda\}}_k\right)\overline{e}_k.\eqno{(125.1)}
$$
Hence also
$$
\Delta^{(\lambda)}\left(\overline{x}\right)=\overline{H}^{(\lambda)}-\frac{1}{N-1}\overline{R}^{\{\lambda\}},\eqno{(125.2)}
$$
where (from Deffinition 9):
$$
\overline{R}^{\{\lambda\}}=\sum^{N}_{k=1}R^{\{\lambda\}}_k\overline{e}_k.
$$
\\
\textbf{Proof.}\\
Easy from the above.\\
\\
\textbf{Theorem 25.}\\
If $\lambda_{ij}$ is antisymmetric we have
\begin{equation}
\sigma^{(\lambda)}_{k}=-\sum^{N}_{i<j}\Lambda^{(\lambda)}_{kij}.
\end{equation}
If $\lambda_{ij}=\lambda^{+}_{ij}+\lambda^{-}_{ij}$, then
$$
\sigma^{(\lambda)}_{k}=\sum^{*}_{i\leq j}M^{(\lambda^+)}_{kij}-\sum_{i<j}\Lambda^{(\lambda^-)}_{kij}=
$$
\begin{equation}
=\sum_{i<j}M^{(\lambda^+)}_{kij}-\sum_{i<j}\Lambda^{(\lambda^-)}_{kij}+(N-1)\sum^{N}_{i=1}\lambda_{ii}q_{iki},
\end{equation}
where the asterisk on the summation means that when $i=j$ we must multiply the summands with $\frac{1}{2}$.\\
\\
\textbf{Corollary 7.}\\
1) We have
\begin{equation}
\Delta^{(\lambda)}\overline{x}=\overline{0},
\end{equation}
iff for all $k=1,2,\ldots,N$ we have
\begin{equation}
H^{(\lambda)}_k-\frac{1}{N-1}R^{\{\lambda\}}_k=0.
\end{equation}
2) For every $\lambda_{ij}$ we have
$$
R^{\{\lambda\}}_k=\sum^{N}_{l,s=1}\epsilon_{ls}\lambda_{lk}\sum_{i<j}Q_{isj}.\eqno{(129.1)}
$$
\\
\textbf{Proof.}\\Actually then we have
\begin{equation}
\sigma_k^{(\lambda)}=H^{(\lambda)}_k-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i,
\end{equation}
where
\begin{equation}
H^{(\lambda)}_k=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}q_{jki}=H^{(\lambda^{+})}_k+H^{(\lambda^{-})}_k.
\end{equation}
In case $\lambda_{ij}=g_{ij}$ is the metric tensor, then we write
\begin{equation}
H_k:=H^{(g)}_k=(N-1)\sum^{N}_{i,j=1}g_{ij}q_{ikj}
\end{equation}
and call $H_k$ mean curvature of the surface $\textbf{S}$.\\
\\
\textbf{Theorem 26.}\\
If $\lambda_{ij}=g_{ij}$ is the metric tensor, then $\Delta^{(g)}\overline{x}=\overline{0}$ iff
\begin{equation}
H_{k}-\frac{1}{N-1}\sum^{N}_{i=1}g_{ik}R_i=0.
\end{equation}
\\
\textbf{Corollary 8.}\\
If $\lambda_{ij}$ is antisymmetric, then
$$
\Delta^{(\lambda)}(fg)=f\Delta^{(\lambda)}g+g\Delta^{(\lambda)}f
$$
\\
\textbf{Proof.}\\
Use Theorem 24 with
$$
\overline{\textrm{grad}(fg)}=f\overline{\textrm{grad}(g)}+g\overline{\textrm{grad}(f)}.
$$
\\
\textbf{Theorem 27.}\\
Assume that $\lambda_{ij}=\epsilon_{ij}-$antisymmetric. Then
\begin{equation}
\Delta^{(\epsilon)}\overline{x}=\overline{0}\Leftrightarrow (N-1)A_k+\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(2)}_{kl}A_l=0,
\end{equation}
where
\begin{equation}
\epsilon^{(2)}_{kl}=\sum^{N}_{i=1}\epsilon_{ik}\epsilon_{il}.
\end{equation}
For every space $\textbf{S}$ with $A_{i}$ as above and for every function $f$, we have
\begin{equation}
\Delta^{(\epsilon)}f=0.
\end{equation}
\\
\textbf{Proof.}\\
Use Theorem 23 with $\lambda_{ij}=\epsilon_{ij}$ and then Definitions 9,6.\\
\\
\textbf{Remark 5.} From the above propositions we conclude that in every space $\textbf{S}$ we have at least one antisymmetric field (the $\lambda_{ij}=\epsilon_{ij}$) that under condition
\begin{equation} (N-1)A_k+\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(2)}_{kl}A_l=0\Leftrightarrow \sum^{N}_{k=1}\Lambda^{(\epsilon)}_{kij}=0,
\end{equation}
we have
\begin{equation}
\Delta^{(\epsilon)}\left(f\right)=0\textrm{, }\forall f.
\end{equation}
Hence in every space $\textbf{S}$ the quantity
\begin{equation}
\sigma^{(\epsilon)}_k:=(N-1)A_k+\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(2)}_{kl}A_l\textrm{, }k=1,2,\ldots,N
\end{equation}
is important. More general the quantities $\sigma^{(\lambda)}_k$ of (119) with $\lambda_{ij}$ antisymmetric are of extreme interest.\\
\\
\textbf{Corollary 9.}\\
If $\lambda_{ij}$ is antisymmetric, then
\begin{equation}
\Delta^{(\lambda)}f=\frac{1}{(N-1)^2}\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle f+\frac{1}{N-1}\left\langle \overline{Df},\Delta^{(\lambda)}\overline{x}\right\rangle.
\end{equation}
\\
\textbf{Proof.}\\
It holds
$$
\overline{Df}=(N-1)\overline{\textrm{grad} f}-\frac{1}{N-1}\overline{R}f.
$$
Hence
$$
\left\langle\overline{Df},\Delta^{(\lambda)}\overline{x}\right\rangle=(N-1)\left\langle\overline{\textrm{grad}f},\Delta^{(\lambda)}\overline{x}\right\rangle-\frac{1}{N-1}\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle f.
$$
Now since $\lambda_{ij}$ is antisymmetric we have from Theorem (24) the result.\\
\\
\textbf{Corollary 10.}\\
If $\lambda_{ij}$ is antisymmetric, then
\begin{equation}
\left\langle\overline{Df},\Delta^{(\lambda)}\overline{x}\right\rangle=0\Leftrightarrow\Delta^{(\lambda)}f=\frac{1}{(N-1)^2}\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle f.
\end{equation}
In particular
\begin{equation}
\overline{Df}=\overline{0}\Rightarrow\Delta^{(\lambda)}f=\frac{1}{(N-1)^2}\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle f.
\end{equation}
\\
\textbf{Corollary 11.}\\If in a space $\textbf{S}$, the $\lambda_{ij}$ is antisymmetric with
\begin{equation}
\left\langle \Delta^{(\lambda)}\overline{x},\overline{R}\right\rangle=0,
\end{equation}
then
$$
\left\langle \overline{H}^{(\lambda)}, \overline{R}\right\rangle=0
$$
and for every $f$ holds
\begin{equation}
\overline{Df}=\overline{0}\Rightarrow \Delta^{(\lambda)}f=0.
\end{equation}
\\
\textbf{Remark.}\\
If $\textbf{S}$ is $S$-$R$, then from Example 1, pg.23, we have $\left\langle \Delta^{(\epsilon)}\overline{x},\overline{R}\right\rangle=0$. Hence in a $S$-$R$ space we have
$$
\Delta^{(\epsilon)}f=\frac{1}{N-1}\left\langle \overline{Df},\Delta^{(\epsilon)}\overline{x}\right\rangle\eqno{(144.1)}
$$
and the equation
\begin{equation}
\left\langle\overline{Df},\Delta^{(\epsilon)}\overline{x}\right\rangle=0\textrm{ is equivalent to }\Delta^{(\epsilon)}f=0.
\end{equation}
In particular if $\overline{Df}=\overline{0}$, then $\Delta^{(\epsilon)}f=0$.\\
\\
\textbf{Theorem 27.1}\\
If $\lambda_{ij}$ is antisymmetric and
\begin{equation}
\Delta^{(\lambda)}\overline{x}=\sum^{N}_{k=1}\sigma^{(\lambda)}_k\overline{e}_{k},
\end{equation}
then
\begin{equation}
\Delta^{(\lambda)}f=\sum^{N}_{k=1}\sigma^{(\lambda)}_{k}\nabla_{k}f
\end{equation}
\\
\textbf{Lemma 2.}\\
If $\lambda_{ij}$ is symmetric, then
\begin{equation}
\Delta^{(\lambda)}f=(N-1)\sum_{i\leq j}^{*}\lambda_{ij}\nabla_i\nabla_jf+\sum^{N}_{k=1}\left((N-1)A^{(\lambda)}_k-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)(\nabla_kf),
\end{equation}
where the asterisk on the summation means that when $i<j$ the summands are multiplied with 2 and when $i=j$ with 1.\\
\\
\textbf{Proof.}\\
Assume that $\lambda_{ij}$ is symmetric, then we can write
$$
\Delta^{(\lambda)}f=\sum^{N}_{i,j=1}\lambda_{ij}D_i\nabla_jf=
$$
$$
=\sum^{N}_{k=1}\lambda_{kk}D_k\nabla_kf+\sum_{i<j}\lambda_{ij}\left(D_{i}\nabla_jf+D_j\nabla_if\right).
$$
But
$$
D_i\nabla_jf+D_j\nabla_if=(N-1)\nabla_i\nabla_jf-\frac{R_i\nabla_jf}{N-1}+(N-1)\nabla_j\nabla_if-\frac{R_j\nabla_if}{N-1}=
$$
$$
=(N-1)\left(\nabla_i\nabla_jf+\nabla_j\nabla_if\right)-\frac{1}{N-1}\left(R_i\nabla_jf+R_j\nabla_if\right)=
$$
$$
(N-1)\left(2\nabla_i\nabla_jf+\sum^{N}_{k=1}\nabla_kfQ_{ikj}\right)-\frac{1}{N-1}\left(R_i\nabla_jf+R_j\nabla_if\right).
$$
Hence
$$
\sum_{i<j}\lambda_{ij}\left(D_{i}\nabla_jf+D_j\nabla_if\right)=
$$
$$
=2(N-1)\sum_{i<j}\lambda_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{k=1}(\nabla_kf)\sum_{i<j}\lambda_{ij}Q_{ikj}-
$$
$$
-\frac{1}{N-1}\sum^{N}_{i,k=1}\lambda_{ik}R_i\nabla_kf+\frac{1}{N-1}\sum^{N}_{k=1}\lambda_{kk}R_k\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{k=1}A^{(\lambda)}_{k}(\nabla_kf)-\frac{1}{N-1}\sum^{N}_{i,k=1}\lambda_{ik}R_i\nabla_kf+
$$
$$
+\frac{1}{N-1}\sum^{N}_{k=1}\lambda_{kk}R_k\nabla_kf.
$$
Also
$$
\sum^{N}_{k=1}\lambda_{kk}D_k\nabla_kf=(N-1)\sum^{N}_{k=1}\lambda_{kk}\nabla^2_kf-\frac{1}{N-1}\sum^{N}_{k=1}\lambda_{kk}R_k(\nabla_kf).
$$
Hence combining the above we get the first result.\\
\\
\textbf{Theorem 28.}\\
If $\lambda_{ij}$ is symmetric, then
\begin{equation}
\Delta^{(\lambda)}f=(N-1)\sum^{*}_{i\leq j}\lambda_{ij}\left(\nabla_jf\right)_{;i}+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle,
\end{equation}
where the asterisk in the sum means that if $i<j$, then the summands are multiplied with 2. In case $i=j$ with 1.\\
\\
\textbf{Proof.}\\
From Lemma we have that if $\lambda_{ij}$ is symmetric, then
$$
\Delta^{(\lambda)}f=(N-1)\sum_{i\leq j}^{*}\lambda_{ij}\nabla_i\nabla_jf+\sum^{N}_{k=1}\left((N-1)A^{(\lambda)}_k-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)(\nabla_kf).
$$
But also from Theorem 22 we have
$$
\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}q_{ikj}-\left\langle \Delta^{(\lambda)}\overline{x},\overline{e}_k\right\rangle
$$
Hence
$$
\sum^{N}_{k=1}\left((N-1)A^{(\lambda)}_k-\frac{1}{N-1}\sum^{N}_{i=1}\lambda_{ik}R_i\right)(\nabla_kf)=
$$
$$
=(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda_{ij}Q_{ikj}\nabla_kf-(N-1)\sum^{N}_{i,j,k=1}\lambda_{ij}q_{ikj}\nabla_kf+\sum^{N}_{k=1}\left\langle \Delta^{(\lambda)}\overline{x},\overline{e}_k\right\rangle\nabla_kf=
$$
$$
=(N-1)\sum^{N}_{k=1}\nabla_kf\left(\sum_{i<j}\lambda_{ij}q_{ikj}-\sum_{i<j}\lambda_{ij}q_{jki}-\sum_{i<j}\lambda_{ij}q_{ikj}-\sum_{i>j}\lambda_{ij}q_{ikj}\right)-
$$
$$
-(N-1)\sum^{N}_{k,i=1}\lambda_{ii}q_{iki}\nabla_kf+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=-2(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda_{ij}q_{jki}\nabla_kf-(N-1)\sum^{N}_{k,i=1}\lambda_{ii}q_{iki}\nabla_kf+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle.
$$
Hence
$$
\Delta^{(\lambda)}f=2(N-1)\sum_{i< j}\lambda_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda_{ii}\nabla_i^2f
-2(N-1)\sum_{i<j}\sum^{N}_{k=1}\lambda_{ij}q_{jki}\nabla_kf-
$$
$$
-(N-1)\sum^{N}_{k,i=1}\lambda_{ii}q_{iki}\nabla_kf+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{*}_{i\leq j}\lambda_{ij}\left(\nabla_jf\right)_{;i}+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle.
$$
\\
\textbf{Definition 11.}\\
Assume that $g_{ij}$ is the metric tensor of the surface $\textbf{S}$. Then
\begin{equation}
\Delta^{(g)}\overline{x}=\overline{0}
\end{equation}
Iff
\begin{equation} (N-1)g_{ij}Q^{*}_{ikj}-\frac{1}{N-1}\sum^{N}_{l=1}\epsilon^{(g)}_{kl}Q_{ilj}=0,
\end{equation}
where
\begin{equation}
\epsilon^{(g)}_{kl}:=\sum^{N}_{i=1}g_{ik}\epsilon_{il}.
\end{equation}
We call a space $\textbf{S}$, $G-$space iff $\Delta^{(g)}(\overline{x})=\overline{0}$.\\
\\
\textbf{Theorem 29.}\\
If $\textbf{S}$ is a $G-$space, then for all functions $f$, we have
\begin{equation}
\Delta^{(g)}f=(N-1)\sum^{*}_{i\leq j}g_{ij}\left(\nabla_jf\right)_{;i}
\end{equation}
\\
\textbf{Theorem 30.}\\
If $\lambda_{ij}$ is any antisymmetric field, then
\begin{equation}
\sum^{N}_{i,j=1}\lambda_{ij}[D_i,\nabla_j]f+\frac{1}{f}\sum^{N}_{k=1}A^{(\lambda)}_kD_k\left(f^2\right)
=\frac{f}{2(N-1)}\sum^{N}_{i,j=1}\lambda_{ij}\Theta^{(2)}_{ij}(R)
.
\end{equation}
In case that $\textbf{S}$ is $\textbf{S}$-$R$, then
\begin{equation}
\sum^{N}_{i,j=1}\lambda_{ij}[D_i,\nabla_j]f=-\frac{1}{f}\sum^{N}_{k=1}A_k^{(\lambda)}D_k\left(f^2\right).
\end{equation}
\\
\textbf{Proof.}\\
We first evaluate the bracket.
$$
[D_i,\nabla_j]f=D_i\nabla_jf-\nabla_jD_if=(N-1)\nabla_i\nabla_jf-\frac{1}{N-1}R_i\nabla_jf-
$$
$$
-\left((N-1)\nabla_j\nabla_if-\frac{1}{N-1}\nabla_j(R_if)\right)=
$$
$$
(N-1)\left(\nabla_i\nabla_jf-\nabla_j\nabla_if\right)-\frac{1}{N-1}R_i\nabla_jf+\frac{1}{N-1}(f\nabla_jR_i+R_i\nabla_jf)=
$$
$$
=-(N-1)\sum^{N}_{k=1}(\nabla_kf)Q_{ikj}+\frac{f}{N-1}\nabla_jR_i.
$$
Hence if we multiply with $\lambda_{ij}$ and sum with respect to $i,j$, we get
$$
P=\sum^{N}_{i,j=1}\lambda_{ij}[D_i,\nabla_j]f=-(N-1)\sum^{N}_{k,i,j=1}\lambda_{ij}\nabla_kfQ_{ikj}+\frac{f}{N-1}\sum^{N}_{i,j=1}\lambda_{ij}\nabla_jR_i.
$$
Also we have from Theorem 7 and the antisymmetric property of $\lambda_{ij}$:
$$
\sum^{N}_{i,j=1}\lambda_{ij}\nabla_jR_i=\frac{1}{2}\sum^{N}_{i,j=1}\lambda_{ij}(\nabla_jR_i-\nabla_iR_j)=-\frac{1}{2}\sum^{N}_{i,j,k=1}\lambda_{ij}R_kQ_{jki}+
$$
$$
+\frac{1}{2}\sum^{N}_{i,j=1}\lambda_{ij}\Theta^{(2)}_{ij}(R)
=\frac{1}{2}\sum^{N}_{i,j,k=1}\lambda_{ij}R_kQ_{ikj}+\frac{1}{2}\sum^{N}_{i,j=1}\lambda_{ij}\Theta^{(2)}_{ij}(R).
$$
Hence combining the above two results, we get
$$
P=-(N-1)\sum^{N}_{k,i,j=1}\lambda_{ij}\nabla_kfQ_{ikj}+\frac{f}{2(N-1)}\sum^{N}_{i,j,k=1}\lambda_{ij}R_kQ_{ikj}+
$$
$$
+\frac{f}{2(N-1)}\sum^{N}_{i,j=1}\lambda_{ij}\Theta^{(2)}_{ij}(R)\Rightarrow
$$
$$
fP=-\frac{1}{2}\sum^{N}_{k,i,j=1}\lambda_{ij}Q_{ikj}\left((N-1)\nabla_k\left(f^2\right)-\frac{R_kf^2}{N-1}\right)+
$$
$$
+\frac{f^2}{2(N-1)}\sum^{N}_{i,j=1}\lambda_{ij}\Theta_{ij}^{(2)}(R)
$$
and the result follows.\\
\\
\textbf{Corollary 11.}\\
If $\lambda_{ij}$ is antisymmetric, then
\begin{equation}
\sum_{i<j}\lambda_{ij}[D_i,\nabla_j]f=-(N-1)\sum^{N}_{k=1}A_k^{(\lambda)}\nabla_kf+\frac{f}{N-1}\sum_{i<j}\lambda_{ij}\nabla_jR_i.
\end{equation}
\\
\textbf{Theorem 31.}\\
If $\lambda_{ij}=\lambda^{+}_{ij}+\lambda^{-}_{ij}$, is any field, then
$$
\Pi^{(\lambda)} f:=\sum^{N}_{i,j=1}\lambda_{ij}\left[D_i,\nabla_j\right]f=\frac{1}{f(N-1)}\sum^{N}_{k=1}H^{(\lambda^{-})}_kD_k\left(f^2\right)+
$$
\begin{equation}
+\frac{f}{2(N-1)}\sum^{N}_{i,j=1}\lambda^{-}_{ij}\Theta^{(2)}_{ij}(R)
+\frac{f}{N-1}\sum^{N}_{i,j=1}\lambda^{+}_{ij}\nabla_jR_i.
\end{equation}
\\
\textbf{Corollary 12.}\\
If $\lambda_{ij}$ is any field and $\overline{D\left(f^2\right)}=\overline{0}$, then exists $\mu$ independed of $f$ such
\begin{equation}
\Pi^{(\lambda)}f=\mu f.
\end{equation}
In particular
\begin{equation}
\mu=\frac{1}{2(N-1)}\sum^{N}_{i,j=1}\lambda^{-}_{ij}\Theta^{(2)}_{ij}(R)
+\frac{1}{N-1}\sum^{N}_{i,j=1}\lambda^{+}_{ij}\nabla_jR_i.
\end{equation}
\\
\textbf{Theorem 32.}\\
If $\lambda_{ij}$ is symmetric, then $\Pi^{(\lambda)}$ is simplified considerably
\begin{equation}
\Pi^{(\lambda)}f=\frac{f}{N-1}\sum^{N}_{i,j=1}\lambda_{ij}\nabla_jR_i.
\end{equation}
\\
\textbf{Theorem 33.}\\
If $\lambda_{ij}$ is any symmetric field, then for every function $f$ we have
\begin{equation}
\Pi^{(\lambda)}f=0\textrm{, }\forall f
\end{equation}
iff
\begin{equation}
\sum^{N}_{i,j=1}\lambda_{ij}\nabla_jR_i=0.
\end{equation}
\\
\textbf{Remark 6.}\\
1) In any space $\textbf{S}$ we define the quantity
\begin{equation}
\eta^{(\lambda)}:=\frac{1}{N-1}\sum^{N}_{i,j=1}\lambda_{ij}\nabla_jR_i.
\end{equation}
2) If $\lambda_{ij}$ is symmetric, then
\begin{equation}
\Pi^{(\lambda)}f=\eta^{(\lambda)}f.
\end{equation}
3) If $\lambda_{ij}=g_{ij}$ (the metric tensor), then $\eta^{(g)}:=\eta$, $\Pi^{(g)}:=\Pi$ and
\begin{equation}
\Pi f:=\sum^{N}_{i,j=1}g_{ij}[D_i,\nabla_j]f=\eta f,
\end{equation}
where
\begin{equation}
\eta=\frac{1}{N-1}\sum^{N}_{i,j=1}g_{ij}\nabla_jR_i.
\end{equation}
\\
\textbf{Theorem 34.}\\
If $\lambda_{ij}$ is any field and $\lambda^{(S)}_{ij}=\frac{1}{2}\left(\lambda_{ij}+\lambda_{ji}\right)$, then
\begin{equation}
\Delta^{(\lambda)}f=(N-1)\sum^{*}_{i\leq j}\lambda^{(S)}_{ij}\left(\nabla_jf\right)_{;i}+\left\langle\Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle.
\end{equation}
\\
\textbf{Proof.}\\
Write $\lambda_{ij}=\lambda^{(S)}_{ij}+\lambda^{(A)}_{ij}$, where $\lambda^{(S)}_{ij}=\frac{1}{2}(\lambda_{ij}+\lambda_{ji})$ is the symmetric part of $\lambda_{ij}$ and $\lambda^{(A)}_{ij}=\frac{1}{2}(\lambda_{ij}-\lambda_{ji})$ is the antisymmetric part of $\lambda_{ij}$. Then we have
$$
\Delta^{(\lambda)}f=\sum^{N}_{i,j=1}\lambda^{(S)}_{ij}D_i\nabla_{j}f+\sum^{N}_{i,j=1}\lambda^{(A)}_{ij}D_i\nabla_{j}f=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-2(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda^{(S)}_{ij}q_{jki}\nabla_kf+
$$
$$
+(N-1)\sum^{N}_{k,i,j=1}\lambda^{(S)}_{ij}q_{ikj}\nabla_kf-\frac{1}{N-1}\sum^{N}_{i,k=1}\lambda^{(S)}_{ik}R_i\nabla_kf-
$$
$$
-(N-1)\sum^{N}_{k=1}A^{(A)}_k\nabla_kf
-\frac{1}{N-1}\sum^{N}_{i,k=1}\lambda^{(A)}_{ik}R_i\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i>j}\lambda_{ij}\left(q_{ikj}-q_{jki}\right)\nabla_{k}f+(N-1)\sum^{N}_{k,i=1}\lambda_{ii}q_{iki}\nabla_kf-
$$
$$
-\frac{1}{N-1}\sum^{N}_{k,i=1}\lambda_{ik}R_i\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i>j}\lambda_{ij}\left(q_{ikj}-q_{jki}\right)\nabla_{k}f+(N-1)\sum^{N}_{k,i=1}\lambda_{ii}q_{iki}\nabla_kf+
$$
$$
+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle-(N-1)\sum^{N}_{k,i,j=1}\lambda_{ij}q_{jki}\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i>j}\lambda_{ij}\left(q_{ikj}-q_{jki}\right)\nabla_{k}f+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda_{ij}q_{jki}\nabla_kf-(N-1)\sum^{N}_{k=1}\sum_{i>j}\lambda_{ij}q_{jki}\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda_{ji}q_{jki}\nabla_{k}f+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle-
$$
$$
-(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda_{ij}q_{jki}\nabla_kf=
$$
$$
=2(N-1)\sum_{i<j}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+(N-1)\sum^{N}_{i=1}\lambda^{(S)}_{ii}\left(\nabla_if\right)_{;i}-
$$
$$
-2(N-1)\sum^{N}_{k=1}\sum_{i<j}\lambda^{(S)}_{ij}q_{jki}\nabla_{k}f+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{*}_{i\leq j}\lambda^{(S)}_{ij}\left(\nabla_jf\right)_{;i}+\left\langle \Delta^{(\lambda)}\overline{x},\overline{\textrm{grad}f}\right\rangle.
$$
\\
\textbf{Definition 12.}\\
We call mean curvature vector of the surface $\textbf{S}$ the vector
\begin{equation}
\overline{H}=\sum^{N}_{k=1}H_k\overline{e}_k,
\end{equation}
where
\begin{equation}
H_k=(N-1)\sum^{N}_{i,j=1}g_{ij}q_{ikj}.
\end{equation}
\\
\textbf{Theorem 35.}\\
In case $\lambda_{ij}=g_{ij}$, then we have
\begin{equation}
\Delta^{(g)} f=(N-1)\sum^{N}_{i,j=1}g_{ij}\nabla_i\nabla_jf+\left\langle \Delta^{(g)}\overline{x}-\overline{H},\overline{\textrm{grad}f}\right\rangle.
\end{equation}
\\
\textbf{Proof.}\\
We know that $(\nabla_jf)_{;i}=(\nabla_if)_{;j}$ and $\lambda_{ij}=g_{ij}$ is symmetric. Hence from Theorem 32 we have
$$
\Delta^{(g)}f=2(N-1)\sum_{i<j}g_{ij}(\nabla_jf)_{;i}+(N-1)\sum^{N}_{i=1}g_{ii}(\nabla_if)_{;i}+\left\langle \Delta^{(g)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{N}_{i<j}g_{ij}(\nabla_jf)_{;i}+(N-1)\sum^{N}_{i>j}g_{ij}(\nabla_jf)_{;i}+(N-1)\sum^{N}_{i=1}g_{ii}(\nabla_if)_{;i}+
$$
$$
+\left\langle \Delta^{(g)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{N}_{i,j=1}g_{ij}(\nabla_jf)_{;i}+\left\langle \Delta^{(g)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{N}_{i,j=1}g_{ij}\nabla_i\nabla_jf-(N-1)\sum^{N}_{i,j=1}g_{ij}\sum^{N}_{k=1}q_{jki}\nabla_kf+\left\langle \Delta^{(g)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{N}_{i,j=1}g_{ij}\nabla_i\nabla_jf-\sum^{N}_{k=1}H_k\nabla_kf+\left\langle \Delta^{(g)}\overline{x},\overline{\textrm{grad}f}\right\rangle=
$$
$$
=(N-1)\sum^{N}_{i,j=1}g_{ij}\nabla_i\nabla_jf+\left\langle \Delta^{(g)}\overline{x}-\overline{H},\overline{\textrm{grad}f}\right\rangle
$$
and the theorem is proved.\\
\\
\textbf{Note 7.}\\
\textbf{i)} Actually the above theorem can be generalized for any $\lambda_{ij}$ as
\begin{equation}
\Delta^{(\lambda)}f=(N-1)\sum^{N}_{i,j=1}\lambda^{(S)}_{ij}\nabla_i\nabla_jf+\left\langle \Delta^{(\lambda)}\overline{x}-\overline{H}^{(S)},\overline{\textrm{grad}f}\right\rangle,
\end{equation}
where $\lambda^{(S)}_{ij}=\frac{\lambda_{ij}+\lambda_{ji}}{2}$, \begin{equation}
H^{(S)}_{k}=(N-1)\sum^{N}_{i,j=1}\lambda^{(S)}_{ij}q_{ikj}
\end{equation}
and
\begin{equation}
\overline{H}^{(S)}=\sum^{N}_{k=1}H^{(S)}_k\overline{e}_k.
\end{equation}
\textbf{ii)} If $\lambda_{ij}$ is symmetric, then
$$
\Delta^{(\lambda)}f=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}\nabla_i\nabla_jf-\frac{1}{N-1}\sum^{N}_{k=1}R^{\{\lambda\}}_k\nabla_kf.\eqno{(173.1)}
$$
\textbf{iii)} If in a space $\textbf{S}$ we have for $\lambda_{ij}$ symmetric $\Delta^{(\lambda)}\left(\overline{x}\right)=\overline{H}^{(\lambda)}\Leftrightarrow\overline{R}^{\{\lambda\}}=\overline{0}$, then
$$
\Delta^{(\lambda)}f=(N-1)\sum^{N}_{i,j=1}\lambda_{ij}\nabla_i\nabla_jf.
$$
\\
\textbf{Definition 13.}\\ We can also expand the definition of Beltrami operator acting to vectors. This can be done as follows:\\
If $\overline{A}=A_1\overline{\epsilon}_1+A_2\overline{\epsilon}_2+\ldots+A_{N}\overline{\epsilon}_{N}$, where $\{\overline{\epsilon}_i\}_{i=1,2,\ldots,N}$ is ''constant'' orthonormal base of $\textbf{E}=\textbf{R}^{N}$, then
\begin{equation}
\Delta_2(\overline{A})=\Delta_2(A_1)\overline{\epsilon}_1+\Delta_2(A_2)\overline{\epsilon}_2+\ldots+\Delta_2(A_{N})\overline{\epsilon}_{N}.
\end{equation}
\\
\textbf{Theorem 36.}\\
If $\overline{V}_1$ and $\overline{V}_2$ are vector fields, then
\begin{equation}
\Delta_2\left\langle \overline{V}_1,\overline{V}_2\right\rangle=\left\langle \Delta_2\overline{V}_1,\overline{V}_2\right\rangle+\left\langle \overline{V}_1,\Delta_2\overline{V}_2\right\rangle+2(N-1)\sum^{N}_{k=1}\left\langle \nabla_k\overline{V}_1,\nabla_k\overline{V}_2\right\rangle.
\end{equation}
\\
\textbf{Proof.}\\
The proof follows by direct use of Theorem 7 and the identity
\begin{equation}
\nabla_k\left\langle \overline{V}_1, \overline{V}_2\right\rangle=\left\langle\nabla_k\overline{V}_1,\overline{V}_2\right\rangle+
\left\langle \overline{V}_1,\nabla_k\overline{V}_2\right\rangle.
\end{equation}
\\
\textbf{Corollary 13.}
\begin{equation}
\left\langle \Delta_2\left(\overline{e}_k\right),\overline{e}_k \right\rangle=-(N-1)\sum^{N}_{j,l=1}q_{kjl}^2=\textrm{invariant.}
\end{equation}
\\
\textbf{Proof.}\\
From Theorem 36 and
$$
\left\langle \overline{e}_k,\overline{e}_k \right\rangle=1,
$$
we have
\begin{equation}
2 \left\langle \Delta_2\left(\overline{e}_k\right),\overline{e}_k \right\rangle+2(N-1)\sum^{N}_{l=1}\left|\nabla_l\left(\overline{e}_k\right)\right|^2=0.
\end{equation}
Hence we get the result.\\
From Theorem 15 and Proposition 1 one can easily see that
\begin{equation}
\Delta_2(\overline{x})=\sum^{N}_{k=1}\left((N-1)\sum^{N}_{l=1}q_{lkl}-\frac{1}{N-1}R_k\right)\overline{e}_k.
\end{equation}
Hence if we define as ''2-mean curvature vector'' the (not to confused with $h_k$ of ($eq1$),($eq2$) in pg.17):
\begin{equation}
\overline{h}=\sum^{N}_{k=1}h_k\overline{e}_k,
\end{equation}
where
\begin{equation}
h_k:=(N-1)\sum^{N}_{l=1}q_{lkl},
\end{equation}
then
\begin{equation}
h_k^{*}=\left\langle \Delta_2(\overline{x}),\overline{e}_k\right \rangle=h_k-\frac{1}{N-1}R_k
\end{equation}
and
\begin{equation}
\Delta_2\left(\overline{x}\right)=\sum^{N}_{k=1}h^{*}_k\overline{e}_k.
\end{equation}
Also we can write
$$
\Delta_2\overline{x}=\overline{h}-\frac{1}{N-1}\overline{R}
$$
and
$$
\left\langle\Delta_2\overline{x},\overline{\textrm{grad}(f)}\right\rangle=\left\langle\overline{H},\overline{\textrm{grad}(f)}\right\rangle+\Delta_2f-(N-1)\sum^{N}_{k=1}\nabla_k^2f.
$$
\\
\textbf{Proposition 6.}\\
The quantities $h_k=(N-1)\sum^{N}_{i=1}q_{iki}$ and $h_{ij}=\sum^{N}_{k=1}q_{ikj}^2$ are invariants.\\
\\
\textbf{Proof.}\\ It follows from invariant property of $\left\langle \nabla_l \overline{Y},\overline{e}_k\right \rangle$ for all $\overline{Y}$.\\
\\
\textbf{Proposition 7.}\\
For the mean curvature holds the following relation
\begin{equation}
h_k=\frac{1}{N-1}R_{k}+\left\langle\Delta_2(\overline{x}),\overline{e}_{k}\right\rangle=\textrm{invariant.}
\end{equation}
\\
\textbf{Definition 14.}\\
We call $\textbf{S}$ 2-minimal if $h_k=0$, $\forall k=1,2,\ldots,N$.\\
\\
\textbf{Theorem 37.}\\
The invariants $R^{\{M\}}_{j}=(N-1)\sum^{N}_{i=1}\left(\kappa^{\{M\}}_{ij}\right)^2$ have interesting properties. The sum $R^{\{M\}}_{o}=\sum^{N}_{j=1}R^{\{M\}}_j$ is also invariant and
\begin{equation}
\left|\nabla_k(\overline{e}_M)\right|^2=\frac{R^{\{M\}}_k}{N-1}
\end{equation}
and
\begin{equation}
\left \langle \Delta_2(\overline{e}_{M}),\overline{e}_{M}\right \rangle=-R^{\{M\}}_{o}.
\end{equation}
\\
\textbf{Proof.}\\
We have
$$
\left\langle\nabla_l\overline{e}_k,\nabla_l\overline{e}_k\right\rangle=\sum^{N}_{s=1}q_{ksl}^2,
$$
which gives (185).\\
From relations (177),(178) and (185) we get (186).\\
\\
\textbf{Definition 15.}\\
We call the $R^{\{M\}}_{o}$ as $R^{\{M\}}_{o}-$curvature.\\
\\
The extremely case $R^{\{M\}}_{o}=0$ for a certain $M$ happens if and only if $\kappa^{\{M\}}_{ij}=0$, for all $i,j=1,2,\ldots,N$. This leads from relation (10) to $\omega_{iM}=0$, for all $i=1,2,\ldots,N$ which means that $III_{M}=0$ and hence $\overline{e}_M$ is constant vector. We call such space flat in the $M$ direction.\\
\\
\textbf{Application 1.}\\
In the case $\overline{x}=\overline{e}_{N}$, then $\textbf{S}$ is a hypersphere and we have:
\begin{equation}
\nabla_{k}\left(\overline{e}_N\right)=\overline{e}_k=\sum^{N}_{j=1}q_{Njk}\overline{e}_j.
\end{equation}
Hence
\begin{equation}
q_{mNk}=-\delta_{mk}.
\end{equation}
From Theorem 36 we have
$$
0=\Delta_2\left \langle \overline{e}_N,\overline{e}_N\right\rangle=2\left\langle \Delta_2\overline{e}_N,\overline{e}_N\right\rangle+2(N-1)N.
$$
Hence
\begin{equation}
R^{\{N\}}_{o}=-\left\langle \Delta_2\overline{e}_N,\overline{e}_N\right\rangle=N(N-1).
\end{equation}
Also
$$
h_{N}=-(N-1)\sum^{N}_{l=1}\delta_{ll}=-N(N-1)
$$
\begin{equation}
h^{*}_{N}=-N(N-1)=h_N-\frac{1}{N-1}R_{N}\Rightarrow R_{N}=0
\end{equation}
and
\begin{equation}
K^{\{N\}}=(-1)^N.
\end{equation}
From
$$
R_{iNml}=-\sum^{N}_{s=1}rot_{lm}\left(q_{isl}q_{Nsm}\right)=\sum^{N}_{s=1}rot_{lm}\left(q_{isl}q_{sNm}\right)=
$$
$$
=-\sum^{N}_{s=1}rot_{lm}\left(q_{isl}\delta_{sm}-q_{ism}\delta_{sl}\right)=-\sum^{N}_{s=1}q_{isl}\delta_{sm}+\sum^{N}_{s=1}q_{ism}\delta_{sl}=
$$
\begin{equation}
=-q_{iml}+q_{ilm}=q_{mil}-q_{lim}=Q_{mil}=-Q_{lim}.
\end{equation}
Hence we get
$$
R_{NNml}=0.
$$
\\
\textbf{Theorem 38.}\\
In general
\begin{equation}
\Delta_2 f=(N-1)\sum^{N}_{k=1}\nabla_k^2f-\frac{1}{N-1}\left\langle \overline{R},\overline{\textrm{grad}f}\right\rangle.
\end{equation}
1) If $\textbf{S}$ is 2-minimal, then
\begin{equation}
\Delta_2\overline{x}=-\frac{1}{N-1}\overline{R}.
\end{equation}
2) If $t_i$ is any vector field, then easily in general
\begin{equation}
\sum^{N}_{i=1}(t_{i})_{;i}=\sum^{N}_{k=1}\nabla_kt_k-\sum^{N}_{k=1}t_kh_k
\end{equation}
and if $\textbf{S}$ is 2-minimal, then
$$
\sum^{N}_{k=1}\left(t_k\right)_{;k}=\sum^{N}_{k=1}\nabla_kt_k.
$$
\\
\textbf{Note 8.}\\
Assume that (with Einstein's notation)
$$
\Delta_2\Phi:=g^{lm}\left(\frac{\partial^2\Phi}{\partial x^l\partial x^m}-\Gamma^{a}_{lm}\frac{\partial \Phi}{\partial x^a}\right).\eqno{(a)}
$$
Then
$$
h_k=\left \langle \Delta_2(\overline{x}),\overline{e}_k\right\rangle=g^{lm}b_{kl,m}.\eqno{(b)}
$$
Hence $h_k$ is invariant. We call $h_k$ mean curvature tensor.\\
\\
\textbf{Proof.}\\
We know that
$$
b_{kl}=\left\langle \frac{\partial \overline{x}}{\partial x^l},\overline{e}_k\right\rangle\textrm{ and }
\partial_l\overline{e}_k=\Gamma^{a}_{lk}\overline{e}_{a}
$$
We differentiate the above first of two identities with respect to $x^m$ and we have
$$
\frac{\partial b_{kl}}{\partial x^m}=\left\langle \partial^2_{lm}\overline{x},\overline{e}_k\right\rangle+\left\langle\partial_{l}\overline{x},\partial_m\overline{e}_k\right\rangle=\left\langle \partial^2_{lm}\overline{x},\overline{e}_k\right\rangle+\Gamma^{a}_{mk}\left\langle\partial_l\overline{x},\overline{e}_a\right\rangle.
$$
Hence
$$
\left\langle \partial^2_{lm}\overline{x},\overline{e}_k\right\rangle=\partial_{m}b_{kl}-\Gamma^a_{mk}b_{al}
$$
But
$$
h_k=\left\langle \Delta_2(\overline{x}),\overline{e}_k\right\rangle=g^{lm}\left\langle\partial^2_{lm}\overline{x},\overline{e}_k\right\rangle-g^{lm}\Gamma^a_{lm}\left\langle\partial_a\overline{x},\overline{e}_k\right\rangle=
$$
$$
g^{lm}\left(\partial_mb_{kl}-\Gamma^a_{mk}b_{al}\right)-g^{lm}\Gamma^a_{lm}b_{ka}=
$$
$$
=g^{lm}\left(\partial_mb_{kl}-\Gamma^a_{km}b_{al}-\Gamma^a_{lm}b_{ka}\right)=g^{lm}b_{kl,m}.
$$
\section{The Spherical Forms}
Consider also the Pfaff derivatives of a function $f$ with respect to the form $\omega_{Mm}$. It holds
\begin{equation}
df=\sum^{N}_{k=1}(\widetilde{\nabla}_{k}f)\omega_{Mk}.
\end{equation}
Assume the connections $q^{(1)}_{mMj}$ such that
\begin{equation}
\omega_{m}=\sum^{N}_{j=1}q^{(1)}_{mMj}\omega_{Mj}.
\end{equation}
Then from (10) we have
\begin{equation}
\omega_m=\sum^{N}_{s,j=1}q^{(1)}_{mMj}q_{Mjs}\omega_{s}.
\end{equation}
Hence
\begin{equation}
\sum^{N}_{j=1}q^{(1)}_{mMj}q_{Mjs}=\delta_{ms},
\end{equation}
where $\delta_{ij}$ is the usual Kronecker symbol. Using this (196) becomes
\begin{equation}
df=\sum^{N}_{k=1}\widetilde{\nabla}_kf\sum^{N}_{s=1}q_{Mks}\omega_s=\sum^{N}_{s=1}\left(\sum^{N}_{k=1}(\widetilde{\nabla}_kf)q_{Mks}\right)\omega_s.
\end{equation}
Hence
\begin{equation}
\nabla_sf=\sum^{N}_{k=1}(\widetilde{\nabla}_kf)q_{Mks}
\end{equation}
and using (199)
\begin{equation}
\widetilde{\nabla}_lf=\sum^{N}_{s=1}{\nabla}_sfq^{(1)}_{sMl}.
\end{equation}
We set $\widetilde{q}_{mjl}$ to be the connection
\begin{equation}
\widetilde{\nabla}_l(\overline{e}_m)=\sum^{N}_{j=1}\widetilde{q}_{mjl}\overline{e}_j.
\end{equation}
One can easily see that
\begin{equation}
\widetilde{q}_{mjl}=\sum^{N}_{s=1}q_{mjs}q^{(1)}_{sMl}.
\end{equation}
From $d\overline{e}_M=\sum^{N}_{k=1}\omega_{Mk} \overline{e}_k$ we get $\widetilde{\nabla}_k(\overline{e}_M)=\overline{e}_k$ and $\left\langle\widetilde{\nabla}^2_k(\overline{e}_M),\overline{e}_M\right\rangle=\widetilde{q}_{kMk}$\\
Also
if $w=\left\langle \overline{x},\overline{e}_M\right\rangle$, then
$$
dw=\left\langle d\overline{x},\overline{e}_M\right\rangle+\left\langle \overline{x},d\overline{e}_M\right\rangle=\omega_{M}+\left\langle \overline{x},\sum^{N}_{k=1}\widetilde{\nabla}_k(\overline{e}_M)\omega_{Mk}\right\rangle=
$$
$$
=\sum^{N}_{j=1}q^{(1)}_{MMj}\omega_{Mj}+\sum^{N}_{k=1}\left\langle \overline{x},\widetilde{\nabla}_k(\overline{e}_M)\right\rangle\omega_{Mk}=
$$
$$
=\sum^{N}_{j=1}q^{(1)}_{MMj}\omega_{Mj}+\sum^{N}_{k=1}\left\langle \overline{x},\overline{e}_k\right\rangle \omega_{Mk}.
$$
Hence
\begin{equation}
\widetilde{\nabla}_kw=q^{(1)}_{MMk}+\left\langle \overline{x},\overline{e}_k\right\rangle .
\end{equation}
Also
$$
d\overline{x}=\sum^{N}_{k=1}(\nabla_k\overline{x})\omega_k=\sum^{N}_{k,j=1}\overline{e}_kq^{(1)}_{kMj}\omega_{Mj}=\sum^{N}_{j=1}\left(\sum^{N}_{k=1}q^{(1)}_{kMj}\overline{e}_k\right)\omega_{Mj}.
$$
From this we get
\begin{equation}
\widetilde{\nabla}_j\overline{x}=\sum^{N}_{k=1}q^{(1)}_{kMj}\overline{e}_k\textrm{, }q^{(1)}_{kMj}\textrm{ is invariant }
\end{equation}
and
$$
\widetilde{\nabla}_l\widetilde{\nabla}_l(w)=\widetilde{\nabla}_l\left(q^{(1)}_{MMl}\right)+\left\langle \widetilde{\nabla}_l\overline{x},\overline{e}_l\right\rangle+\left\langle \overline{x},\widetilde{\nabla}_l\overline{e}_l\right\rangle=
$$
$$
=\widetilde{\nabla}_l\left(q^{(1)}_{MMl}\right)+\left\langle \sum^{N}_{k=1}q^{(1)}_{kMl}\overline{e}_k,\overline{e}_l\right\rangle+\left\langle \overline{x},\sum^{N}_{j=1}\widetilde{q}_{ljl}\overline{e}_j\right\rangle=
$$
$$
=\widetilde{\nabla}_l\left(q^{(1)}_{MMl}\right)+q^{(1)}_{lMl}+\sum^{N}_{j=1}\left\langle\overline{x},\overline{e}_j\right\rangle\widetilde{q}_{ljl}
$$
But
$$
\Delta_2^{III_{M}}w=(N-1)\sum^{N}_{j=1}\widetilde{\nabla}^2_jw-\frac{1}{N-1}\sum^{N}_{j=1}(\widetilde{\nabla}_jw)\widetilde{R}_j=
$$
\begin{equation}
=(N-1)\sum^{N}_{j=1}\widetilde{\nabla}_j\left(q^{(1)}_{MMj}\right)+(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}+(N-1)\sum^{N}_{l,k=1}\left\langle \overline{x},\overline{e}_k\right\rangle \widetilde{q}_{ljl}-
$$
$$
-\frac{1}{N-1}\sum^{N}_{j=1}\left\langle \overline{x},\overline{e}_j\right\rangle \widetilde{R}_j-\frac{1}{N-1}\sum^{N}_{j=1}q^{(1)}_{MMj}\widetilde{R}_j.
\end{equation}
Also
$$
\Delta_2^{III_{M}}\overline{e}_M=(N-1)\sum^{N}_{j=1}\widetilde{\nabla}^2_j\overline{e}_M-\frac{1}{N-1}\sum^{N}_{j=1}(\widetilde{\nabla}_j\overline{e}_M)\widetilde{R}_j=
$$
\begin{equation}
=(N-1)\sum^{N}_{l,j=1}\widetilde{q}_{ljl}\overline{e}_j-\frac{1}{N-1}\sum^{N}_{j=1}\widetilde{R}_j\overline{e}_j
\end{equation}
Hence
\begin{equation}
\left\langle\Delta_2^{III_{M}}\overline{e}_M,\overline{x}\right\rangle=(N-1)\sum^{N}_{l,j=1}\left\langle \overline{x},\overline{e}_j \right\rangle \widetilde{q}_{ljl}-\frac{1}{N-1}\sum^{N}_{j=1}\left\langle \overline{x},\overline{e}_j \right\rangle\widetilde{R}_{j}
\end{equation}
From (207) and (209) we get
$$
\Delta_2^{III_{M}}\left\langle \overline{x},\overline{e}_M\right\rangle-\left\langle \overline{x},\Delta_2^{III_{M}}\overline{e}_M\right\rangle=(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}+
$$
\begin{equation}
+(N-1)\sum^{N}_{j=1}\widetilde{\nabla}_j\left(q^{(1)}_{MMj}\right)-\frac{1}{N-1}\sum^{N}_{j=1}q^{(1)}_{MMj}\widetilde{R}_j.
\end{equation}
From Theorem 36 and formula (197) we get
$$
\Delta_2^{III_M}\left\langle \overline{x},\overline{e}_{M}\right\rangle=\left\langle \Delta_2^{III_M}\overline{x},\overline{e}_M\right\rangle+\left\langle \overline{x},\Delta^{III_M}_2\overline{e}_M\right\rangle+
$$
$$
+2(N-1)\sum^{N}_{k=1}\left\langle \widetilde{\nabla}_k\overline{x},\widetilde{\nabla}_k\overline{e}_M\right\rangle
$$
Or using (210)
$$
(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}+\sum^{N}_{j=1}\widetilde{D}_jq^{(1)}_{MMj}=\left\langle \Delta_2^{III_M}\overline{x},\overline{e}_M\right\rangle+
$$
$$
+2(N-1)\sum^{N}_{k=1}\left\langle \sum^{N}_{j=1}q^{(1)}_{jMk}\overline{e}_j,\overline{e}_k\right\rangle
$$
Or
$$
\sum^{N}_{j=1}\widetilde{D}_jq^{(1)}_{MMj}+(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}=\left\langle \Delta_2^{III_M}\overline{x},\overline{e}_M\right\rangle+
$$
$$
+2(N-1)\sum^{N}_{k=1} q^{(1)}_{kMk}.
$$
Hence we get the next\\
\\
\textbf{Theorem 39.}\\
In $\textbf{S}$ holds
\begin{equation}
\left\langle\Delta_2^{III_M}\overline{x},\overline{e}_M\right\rangle=-(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}+\sum^{N}_{l=1}\widetilde{D}_lq^{(1)}_{MMl}.
\end{equation}
In case $\omega_{M}=0$, then the above formula becomes
$$
\left\langle\Delta_2^{III_M}\overline{x},\overline{e}_M\right\rangle=-(N-1)\sum^{N}_{l=1}q^{(1)}_{lMl}.\eqno{(211.1)}
$$
\section{General Forms and Invariants}
Let $C$ be a curve (one dimensional object) of $\textbf{S}$. Let also $s$ be the physical parameter of $C$. Then
\begin{equation}
\overline{t}=\left(\frac{d\overline{x}}{ds}\right)_C,
\end{equation}
is the tangent vector of $C$ in $P\in \textbf{S}$. If we assume that $C$ lays in a hypersurface $S_{M-1}$ and we choose $\overline{n}=\overline{e}_{M}$ to be the normal vector of the tangent space of $S_{M-1}$, then
\begin{equation}
\left\langle \overline{t},\overline{n}\right\rangle=0.
\end{equation}
\\
\textbf{Definition 16.}\\
We call vertical curvature of $C\in S_{M-1}$ the quantity
\begin{equation}
\left(\frac{1}{\rho^{*}}\right)_C=\left\langle \frac{d\overline{t}}{ds},\overline{n}\right\rangle.
\end{equation}
\\
\textbf{Theorem 40.}\\
The vertical curvature $\left(\frac{1}{\rho^{*}}\right)_C$ is an invariant (by the word invariant we mean that remains unchanged in every acceptable transformation of the coordinates $u_i$).\\
\\
\textbf{Proof.}\\
This can be shown as follows. Derivate (213) to get
$$
\left\langle\frac{d\overline{t}}{ds},\overline{n}
\right\rangle+\left\langle \overline{t},\frac{d\overline{n}}{ds}\right\rangle=0.
$$
Hence
$$
\left(\frac{1}{\rho^{*}}\right)_{C}=-\left\langle \frac{d\overline{x}}{ds},\frac{d\overline{n}}{ds}\right\rangle=-\frac{II_{M}}{I}=-\frac{\sum^{N}_{k=1}\omega_k\omega_{Mk}}{\sum^{N}_{k=1}\omega_k^2}=\textrm{invariant}.
$$
\\
Using relations (8),(10), we get
$$
\left(\frac{1}{\rho^{*}}\right)_{C_i}=-\left\langle \left(\frac{d\overline{x}}{ds}\right)_{C_i},\left(\frac{d\overline{e}_{M}}{ds}\right)_{C_i}\right\rangle=
$$
$$
=-\left\langle \sum^{N}_{k=1}\left(\frac{\omega_k}{ds}\right)_{C_i}\overline{e}_k,\sum^{N}_{l=1}\left(\frac{\omega_{Ml}}{ds}\right)_{C_i}\overline{e}_l\right\rangle
=-\sum^{N}_{k,l=1}\left(\frac{\omega_k}{ds}\right)_{C_i}\left(\frac{\omega_{Ml}}{ds}\right)_{C_l}\delta_{kl}=
$$
$$
=-\sum^{N}_{k=1}\left(\frac{\omega_k}{ds}\right)_{C_i}\left(\frac{\omega_{Mk}}{ds}\right)_{C_i}
=-\sum^{N}_{k,m=1}\left(\frac{\omega_k}{ds}\right)_{C_i}q_{Mkm}\left(\frac{\omega_m}{ds}\right)_{C_i}.
$$
Hence
\begin{equation}
\left(\frac{1}{\rho^{*}}\right)_{C_i}=\sum^{N}_{k,m=1}\kappa^{\{M\}}_{km}\left(\frac{\omega_k}{ds}\right)_{C_i}\left(\frac{\omega_m}{ds}\right)_{C_i}.
\end{equation}
Assume now, in general that exist $N-1$ curves $C_i$, $i=1,2\ldots,N-1$ passing through every point $P$ of $S_{M-1}$ and are vertical to each other. Then in all $P\in S_{M-1}$ we have
\begin{equation}
\left\langle \left(\frac{d\overline{x}}{ds}\right)_{C_i},\left(\frac{d\overline{x}}{ds}\right)_{C_j}\right\rangle=\delta_{ij}\textrm{, }i,j=1,2,\ldots,N-1
\end{equation}
and
$$
\left\langle \left(\frac{d\overline{x}}{ds}\right)_{C_i},\overline{n} \right\rangle=0.
$$
Also for these curves hold
\begin{equation}
\left(\frac{d\overline{x}}{ds}\right)_{C_i}=\sum^{N}_{k=1}\left(\frac{\omega_k}{ds}\right)_{C_i}\overline{e}_k=\sum^{N}_{k=1}\lambda_{ik}\overline{e}_{k},
\end{equation}
where we have set
\begin{equation}
\left(\frac{\omega_k}{ds}\right)_{C_i}=\lambda_{ik}.
\end{equation}
Clearly from the orthogonality of $C_i$ we have that (where we have assumed with no loss of generality that $\omega_{M}=0$ hence $\lambda_{iM}=0$):
$$
\sum_{1\leq k\leq N}^{*}\lambda_{ik}\lambda_{jk}=\delta_{ij}\textrm{, }\forall i,j\in\{1,2,\ldots,N-1\}.
$$
Where the asterisc in the sumation means that the value $k=M$ is omited.\\
Assuming these facts in (215), we get (using the fact that any real unitary matrix is symmetric):
$$
\sum_{1\leq i\leq N}^{*}\left(\frac{1}{\rho^{*}}\right)_{C_i}=\sum^{*}_{1\leq i\leq N}\sum^{N}_{k,m=1}\kappa^{\{M\}}_{km}\lambda_{ik}\lambda_{im}=\sum^{N}_{k,m=1}\kappa^{\{M\}}_{km}\delta_{km}=
$$
$$
=\sum^{N}_{k=1}\kappa^{\{M\}}_{kk}=\textrm{invariant}.
$$
From the above we get the next\\
\\
\textbf{Theorem 41.}\\
From any point $P$ in a hypersurface $S_{M-1}$ of the space $\textbf{S}$, there pass $N-1$ vertical curves $C_i,i=1,2,\ldots,N-1$ and their vertical curvatures satisfy
\begin{equation}
\sum_{1\leq i\leq N}^{*}\left(\frac{1}{\rho^{*}}\right)_{C_i}=\frac{h_M}{N-1}.
\end{equation}
\textbf{Remark.} We mention here that with the word invariant we mean any quantity that remains unchanged in every acceptable choose of parameters $\{u_i\}_{i=1,2,\ldots,N}$ as also change of position and rotation of $\textbf{S}$.\\
\\
Set
\begin{equation}
T=\sum^{N}_{i=1}A_i\omega_{i}^2+\sum^{N}_{i,j=1}B_{ij}\omega_{ij}^2+\sum^{N}_{i,j,k=1}C_{ijk}\omega_{i}\omega_{jk}+\sum^{N}_{i,j,f,m=1}E_{ijfm}\omega_{ij}\omega_{fm}
\end{equation}
and assume the $N-1$ orthogonal curves $C_i$. If the direction of $C_i$ is
\begin{equation}
d_i=\left(\frac{T}{(ds)^2}\right)_{C_i}\textrm{, }i=1,2,\ldots,N-1\textrm{ and }d_{N}=0,
\end{equation}
then summing in all $N$ directions we get, after using the orthogonality and making simplifications
$$
\sum^{N}_{i=1}d_i=\sum^{N}_{i=1}A_i+\sum^{N}_{i,j,m=1}B_{ij}q_{ijm}^2+\sum^{N}_{i,j,k=1}C_{ijk}q_{jki}+
$$
\begin{equation}
+\sum^{N}_{i,j,f,m,n=1}E_{ijfmn}q_{ijn}q_{fmn}
\end{equation}
As application we get that
\begin{equation}
T_0=AI+BII+CIII=A\sum^{N}_{i=1}\omega_{i}^2+B\sum^{N}_{i=1}\omega_{Ni}^2+C\sum^{N}_{i=1}\omega_{i}\omega_{Ni}=\textrm{invariant}.
\end{equation}
Then summing the vertical directions we get
\begin{equation}
\sum^{N-1}_{i=1}d_i=A+B\sum^{N}_{i,j=1}q_{iNj}^2+C\sum^{N}_{i=1}q_{iNi}=\textrm{invariant}
\end{equation}
For every choice of costants $A,B,C$.\\
\\
\textbf{Definition 17.}\\
Let now $C$ be a space curve and $\overline{t}_i$, $i=1,2,\ldots,N$ is a family of orthonormal vectors of $C$. We define
\begin{equation}
\left(\frac{1}{\rho_{ik}}\right)_{C}:=\left\langle\left( \frac{d\overline{t}_i}{ds}\right)_C,\overline{t}_k\right\rangle\textrm{, }\forall i,k\in\{1,2,\ldots,N\}
\end{equation}
and call
$\left(\frac{1}{\rho_{ik}}\right)_C$ as ''$ik-$curvature'' of $C$ and
\begin{equation}
\frac{d\overline{t}_i}{ds}=\sum^{N}_{k=1}\left(\frac{1}{\rho_{ik}}\right)_C\overline{t}_k\textrm{, }\forall i=1,2,\ldots,N.
\end{equation}
\\
\textbf{Theorem 42.}\\
The $\left
(\frac{1}{\rho_{ij}}\right)_C-$curvatures are
semi-invariants of the space. Moreover
\begin{equation}
\left(\frac{1}{\rho_{ij}}\right)_C=-\left(\frac{1}{\rho_{ji}}\right)_C
\end{equation}
and if
$$
\overline{t}_i=\sum^{N}_{k=1}A_{ki}\overline{e}_k,
$$
then
\begin{equation}
\left(\frac{1}{\rho_{ij}}\right)_C=\sum^{N}_{l=1}\sum^{N}_{m=1}A_{\{m\}i;l}A_{mj}\left(\frac{\omega_l}{ds}\right)_{C},
\end{equation}
where $A_{\{m\}i;l}=\nabla_lA_{mi}-\sum^{N}_{k=1}q_{mkl}A_{ki}$.\\
\\
\textbf{Proof.}\\
We express $\overline{t}_i$ in the base $\overline{e}_k$ and differentiate with respect to the canonical parameter $s$ of $C$, hence
$$
\overline{t}_i=\sum^{N}_{k=1}A_{ki}\overline{e}_k,
$$
where $A_{ki}$ is unitary matrix i.e.
$$
\sum^{N}_{i=1}A_{ki}A_{li}=\delta_{kl}.
$$
Hence
\begin{equation}
\overline{e}_j=\sum^{N}_{k=1}A^{T}_{kj}\overline{t}_k,
\end{equation}
where $A^{T}$ is the symmetric of $A$.\\
Differentiating with respect to $s$ of $\overline{t}_i$, we get
$$
\frac{d\overline{t}_i}{ds}=\sum^{N}_{k=1}\frac{dA_{ki}}{ds}\overline{e}_k+\sum^{N}_{k=1}A_{ki}\frac{d\overline{e}_k}{ds}=
$$
$$
=\sum^{N}_{k=1}\frac{dA_{ki}}{ds}\overline{e}_k+\sum^{N}_{k=1}A_{ki}\sum^{N}_{l=1}\nabla_l\overline{e}_k\frac{\omega_l}{ds}=
$$
$$
=\sum^{N}_{k=1}\frac{dA_{ki}}{ds}\overline{e}_k+\sum^{N}_{k,l,m=1}A_{ki}q_{kml}\frac{\omega_l}{ds}\overline{e}_m
=
$$
$$
=\sum^{N}_{k,l=1}\frac{(\nabla_lA_{ki})\omega_l}{ds}\overline{e}_k+\sum^{N}_{k,l,m=1}A_{ki}q_{kml}\frac{\omega_l}{ds}\overline{e}_m=
$$
$$
=\sum^{N}_{m,l=1}\frac{(\nabla_lA_{mi})\omega_l}{ds}\overline{e}_m+\sum^{N}_{k,l,m=1}A_{ki}q_{kml}\frac{\omega_l}{ds}\overline{e}_m.
$$
Hence we can write
\begin{equation}
c_{im}:=\left\langle\frac{d\overline{t}_i}{ds},\overline{e}_m\right\rangle=\sum^{N}_{l=1}\left(\nabla_lA_{mi}-\sum^{N}_{k=1}q_{mkl}A_{ki}\right)\frac{\omega_l}{ds}=\sum^{N}_{l=1}A_{\{m\}i;l}\frac{\omega_l}{ds}.
\end{equation}
Hence
\begin{equation}
c_{im}=\sum^{N}_{l=1}\nabla_lA_{mi}\left(\frac{\omega_l}{ds}\right)_C-\sum^{N}_{l,k=1}A_{ki}q_{lkm}\left(\frac{\omega_l}{ds}\right)_C.
\end{equation}
Hence
\begin{equation}
\frac{dA_{mi}}{ds}=\left\langle\frac{d\overline{t}_i}{ds},\overline{e}_m\right\rangle+\sum^{N}_{l,k=1}A_{ki}q_{lkm}\left(\frac{\omega_l}{ds}\right)_C
\end{equation}
Also
$$
\left\langle \overline{t}_i,\overline{t}_j\right\rangle=\delta_{ij}\Rightarrow \left\langle \frac{d\overline{t}_i}{ds},\overline{t}_j\right\rangle+\left\langle \overline{t}_i,\frac{d\overline{t}_j}{ds}\right\rangle=0.
$$
Hence
\begin{equation}
\left(\frac{1}{\rho_{ij}}\right)_{C}=-\left(\frac{1}{\rho_{ji}}\right)_{C}.
\end{equation}
From the orthonormality of $A_{ij}$ and $A^{T}_{ij}=A_{ji}$ we have
\begin{equation}
\sum^{N}_{i=1}A_{mi}A_{ji}=\delta_{mj}\Leftrightarrow \sum^{N}_{m=1}A_{im}A_{jm}=\delta_{ij}.
\end{equation}
Differentiating we get
\begin{equation}
\sum^{N}_{i=1}A_{\{m\}i;l}A_{ji}+\sum^{N}_{i=1}A_{mi}A_{\{j\}i;l}=0
\end{equation}
Moreover
\begin{equation}
\frac{d\overline{t}_i}{ds}=\sum^{N}_{m=1}c_{im}\overline{e}_m=\sum^{N}_{j=1}\sum^{N}_{l,m=1}A_{\{m\}i;l}A_{mj}\left(\frac{\omega_l}{ds}\right)_C\overline{t}_j
\end{equation}
and the curvatures will be
\begin{equation}
\left(\frac{1}{\rho_{ij}}\right)_C=\sum^{N}_{l=1}\sum^{N}_{m=1}A_{\{m\}i;l}A_{mj}\left(\frac{\omega_{l}}{ds}\right)_{C}.
\end{equation}
Hence
$$
\sum^{N}_{j=1}A_{pj}\left(\frac{1}{\rho_{ij}}\right)_C=\sum^{N}_{l=1}\sum^{N}_{m,j=1}A_{\{m\}i;l}A_{mj}A_{pj}\left(\frac{\omega_{l}}{ds}\right)_{C}=
$$
$$
=\sum^{N}_{l=1}\sum^{N}_{m=1}A_{\{m\}i;l}\delta_{mp}\left(\frac{\omega_{l}}{ds}\right)_{C}=\sum^{N}_{l=1}A_{\{p\}i;l}\left(\frac{\omega_{l}}{ds}\right)_{C}=c_{ip}.
$$
This lead us to write
$$
\sum^{N}_{i,j=1}A_{ni}A_{pj}\left(\frac{1}{\rho_{ij}}\right)_C=\sum^{N}_{i=1}A_{ni}c_{ip}=\sum^{N}_{l,i=1}A_{\{p\}i;l}A_{ni}\left(\frac{\omega_l}{ds}\right)_{C}=
$$
$$
=\sum^{N}_{l,i=1}A^{T}_{\{i\}p;l}A^{T}_{in}\left(\frac{\omega_l}{ds}\right)_C=\sum^{N}_{l,m=1}A^{T}_{\{m\}p;l}A^{T}_{mn}\left(\frac{\omega_l}{ds}\right)_C=
$$
$$
=\left(\sum^{N}_{l,m=1}A_{\{m\}n;l}A_{mp}\left(\frac{\omega_l}{ds}\right)_C\right)^T=\left(\frac{1}{\rho_{np}}\right)^{T}_C=
$$
$$
=\left(\sum^{N}_{l,m=1}\left(\nabla_lA_{\{m\}n}A_{mp}-\sum^{N}_{k=1}q_{mkl}A_{kn}A_{mp}\right)\right)^T\left(\frac{\omega_l}{ds}\right)_C=
$$
$$
=\sum^{N}_{l,m=1}\left(\nabla_lA_{\{m\}p}A_{mn}-\sum^{N}_{k=1}q_{mkl}A_{kp}A_{mn}\right)\left(\frac{\omega_l}{ds}\right)_C=
$$
$$
=\sum^{N}_{l,m=1}A_{\{m\}p;l}A_{mn}\left(\frac{\omega_l}{ds}\right)_C=
$$
$$
=\left(\frac{1}{\rho_{pn}}\right)_C=-\left(\frac{1}{\rho_{np}}\right)_C.
$$
Since we have
$$
\left(A^{T}_{\{i\}j}\right)_{;l}=\nabla_l A_{ji}-\sum^{N}_{k=1}q_{jkl}A_{ki}
$$
and
$$
\left(A_{\{i\}j;l}\right)^{T}=\left(\nabla_lA_{ij}-\sum^{N}_{k=1}q_{ikl}A_{kj}\right)^{T}=\nabla_lA_{ji}-\sum^{N}_{k=1}q_{jkl}A_{ki}
$$
and
\begin{equation}
\left(A^{T}_{\{i\}j}\right)_{;l}=\left(A_{\{i\}j;l}\right)^{T}.
\end{equation}
\[
\]
\centerline{\bf References}
\[
\]
[1]: Nirmala Prakash. ''Differential Geometry An Integrated Approach''. Tata McGraw-Hill Publishing Company Limited. New Delhi. 1981.\\
[2]: Bo-Yu Hou, Bo-Yuan Hou. ''Differential Geometry for Physicists''. World Scientific. Singapore, New Jersey, London, Hong Kong. 1997.\\
[3]: N.K. Stephanidis. ''Differential Geometry''. Vol. I. Zitis Pub. Thessaloniki, Greece. 1995.\\
[4]: N.K. Stephanidis. ''Differential Geometry''. Vol. II. Zitis Pub. Thessaloniki, Greece. 1987.\\
[5]: D. Dimitropoulou-Psomopoulou. ''Calculus of Differential Forms''. 2nd edition. Zitis Pub. Thessaloniki, Greece. 1993.\\
[6]: E.A. Iliopoulou, F. Gouli-Andreou. ''Introduction to Riemann Geometry''. Zitis Pub. Thessaloniki, Greece. 1985.\\
[7]: N.K. Spyrou. ''Introduction to the General Theory of Ralativity''. Gartaganis Pub. Thessaloniki, Greece. 1989.\\
\end{document} |
\begin{document}
\title{Generating Lode Runner Levels by Learning Player Paths with LSTMs}
\author{Kynan Sorochan}
\affiliation{
\institution{University of Alberta}
\city{Edmonton}
\country{Canada}}
\email{ksorocha@ualberta.ca}
\author{Jerry Chen}
\affiliation{
\institution{University of Alberta}
\city{Edmonton}
\country{Canada}}
\email{jerry3@ualberta.ca}
\author{Yakun Yu}
\affiliation{
\institution{University of Alberta}
\city{Edmonton}
\country{Canada}}
\email{yakun2@ualberta.ca}
\author{Matthew Guzdial}
\affiliation{
\institution{University of Alberta}
\city{Edmonton}
\country{Canada}}
\email{guzdial@ualberta.ca}
\renewcommand{Sorochan et al.}{Sorochan et al.}
\begin{abstract}
Machine learning has been a popular tool in many different fields, including procedural content generation.
However, procedural content generation via machine learning (PCGML) approaches can struggle with controllability and coherence.
In this paper, we attempt to address these problems by learning to generate human-like paths, and then generating levels based on these paths.
We extract player path data from gameplay video, train an LSTM to generate new paths based on this data, and then generate game levels based on this path data.
We demonstrate that our approach leads to more coherent levels for the game Lode Runner in comparison to an existing PCGML approach.
\end{abstract}
\begin{CCSXML}
<ccs2012>
<concept>
<concept_id>10010147.10010257.10010293</concept_id>
<concept_desc>Computing methodologies~Machine learning approaches</concept_desc>
<concept_significance>500</concept_significance>
</concept>
<concept>
<concept_id>10010147.10010257.10010293.10010294</concept_id>
<concept_desc>Computing methodologies~Neural networks</concept_desc>
<concept_significance>500</concept_significance>
</concept>
</ccs2012>
\end{CCSXML}
\ccsdesc[500]{Computing methodologies~Machine learning approaches}
\ccsdesc[500]{Computing methodologies~Neural networks}
\keywords{datasets, neural networks, path learning, path detection}
\maketitle
\section{Introduction}
Procedural content generation via machine learning (PCGML) is the study and application of machine learning to procedurally generating content, particularly for games \cite{summerville2018procedural}.
While PCGML has had enjoyed considerable popularity recently, a number of open problems exist.
Particularly in comparison to traditional, non-ML PCG, PCGML approaches struggle with controllability and coherence.
We define controllability as the ability for a user to impact particular attributes of the generated content.
In traditional PCG, since the system is authored by a human, there are a number of strategies to allow a user to impact the output or enforce particular constraints \cite{smith2012case,liapis2016mixed,horswill2019imaginarium}.
There are efforts to make PCGML approaches controllable, but this is still an under-explored problem \cite{mott2019controllable,sarkar2020controllable,cheng2020automatic,chen2020image}.
In particular, we identify a lack of focus on approaches that allow users to specify high-level, intuitive constraints as the input to an ML generator that then outputs game content that matches those constraints.
By coherence we indicate the problem of game content demonstrating global coherence, global structure that fits human understanding of that game content.
Global structure includes a wide range of constraints, and is dependent on the particular game content in question.
For example, playability in game levels, the ability to complete said level, is an example of global structure that we expect from human-authored game levels.
However, a level being playable is not the only element of global structure that we expect.
A completely flat platformer game level would be playable, but would violate other elements of global structure.
Modeling global structure is a common problem in machine learning generally \cite{moon2019unified}.
In PCGML, there have been attempts to model global structure, with the most common approach being to model the player's path through a game level \cite{summerville2016super,sarkar2020exploring}.
However, this area is also under-explored.
In this paper, we investigate a novel PCGML approach that attempts to address these two open problems: controllability and coherence.
Specifically, we introduce an approach to generate Lode Runner levels based on specified paths. Our focus on coherence will be in reference to Lode Runner Levels.
We model these paths to ensure that they are human-like with a Long Short-Term Memory Recurrent Neural Network (LSTM RNN or LSTM).
For training data we extract real human paths on existing Lode Runner levels from gameplay video.
We then use the LSTM to generate novel human-like paths and employ a Markov Chain to generate novel levels based on these generated paths.
We employ a Markov Chain for this initial investigation as it represents a simple ML model that typically struggles to capture global structure.
Therefore, it's an ideal choice to investigate whether this approach improves level coherence.
Our approach is inherently controllable as we can input arbitrary paths, though we focus on our generated, human-like paths in this paper. We acknowledge that we won't be directly evaluating the controllability of this approach, but still contend that it is controllable.
In this paper, we first introduce related prior work.
We then overview our generator, from data extraction to the novel generation of Lode Runner levels.
We compare the performance of our generator to an existing Markov Chain generator without path data to evaluate whether we demonstrate improved coherence \cite{snodgrass2015hierarchical}.
We then present a secondary evaluation of our approach in comparison to the original Lode Runner levels.
We end with a discussion of our limitations and future work.
\section{Related Works}
In this section we overview work in terms of prior PCGML approaches to generate Loderunner levels, controllability in PCGML, and coherence via paths in PCGML.
Many PCGML approaches have been applied to level generation for Lode Runner in recent years.
Thakkar et al. proposed the use of a variational autoencoder to generate new levels for Lode Runner based on a binary encoding of each character in the original levels.
They attempted to improve the playability, one aspect of global structure, by searching the learned, latent space with an evolutionary algorithm \cite{thakkar2019autoencoder}.
We also attempt to increase playability, but based on altering the input to the generation pipeline instead of including search-based PCG within the pipeline.
Snodgrass and Ontan{\'o}n made use of Markov models to generate content for many games including Lode Runner \cite{snodgrass2016learning}, we include this approach as a baseline as we also make use of a Markov chain for our generator.
Markov chains have been a common method for PCGML since its inception \cite{snodgrass2013generating}.
Much of this prior work has focused on \emph{Super Mario Bros.} level generation, a common area of PCGML research \cite{snodgrass2013generating,snodgrass2014hierarchical,Summerville2015MCMCTSP4,snodgrass2015hierarchical}.
We also employ Markov chains, as they typically struggle with global coherence in comparison to other methods \cite{guzdial2018co}.
Many approaches have been made to attempt to improve the coherence of PCGML output \cite{thakkar2019autoencoder}.
Of particular interest to us are approaches that attempt to do this by invoking some representation of the player path \cite{sarkar2020exploring}.
Summerville et al. trained an LSTM on Super Mario Bros. levels that included representations of potential player paths \cite{summerville2016super}.
Follow up work by Summerville et al. extracted player paths from gameplay video and found that these led to significantly different output levels when they were used to train LSTMs \cite{summerville2016learning}.
We use a similar method to extract player paths.
The major difference between Summerville et al.'s approach and ours is that they alter the level representation to include the path information.
Instead, we generate novel paths and use these as input for a PCGML level generator.
The Summerville et al. generator could not take a specified path as input without modification.
This and much of the prior work mentioned above is based on the representations from the Video Game Level Corpus (VGLC) \cite{VGLC}, which we draw on for our level representation.
\section{System Overview}
We aim to show the benefits of human-like paths to improve global coherence in PCGML generators.
Our approach can be divided into several steps: (1) extracting the paths from gameplay videos, (2) training an LSTM on the path data, (3) training our Markov chain on the original Lode Runner levels, and (4) generating a new level from a generated path.
\subsection{Data extraction}
Our goal for this first step is to extract human paths for solving the original Lode Runner levels, which we extract from gameplay video.
We focus on human paths instead of paths generated by an automated level playing agent as in prior work \cite{summerville2016super}.
We made this choice as prior work found that human paths led to different output levels than automated paths \cite{summerville2016super}.
Based on this prior work, we make the assumption that automated paths would lead to levels that were more sparse, but more human-like paths will lead to paths closer to the original game.
We leave a verification of this assumption for future work.
We download a series of videos from YouTube, based on similar approaches in prior work \cite{guzdial2016toward,summerville2016super},
Once we have the videos, we extract the frames and track the location of the player in each frame for each level, based on the same approach used in the above prior work.
We tag each location with the type of movement using OpenCV and pattern matching \cite{bradski2000opencv}.
We do this by hand tagging a series of images or sprites representing the different actions of the player in Lode Runner.
In each frame, we identify the player's action or type of movement based on the image with the highest probability.
We map the player's location in the frame to a 32 by 22 grid, which is the size of the VGLC dataset for each level.
This gives us a sequence of 32 by 22 grids equal in length to the amount of time the player played a given level.
After obtaining this sequence we can identify one out of five possible actions for each grid (moving left as 'l', moving right as 'r', climbing up 'u', climbing down 'c', and falling down as 'f') based on the location change of the player between pairs of grids.
This allows us to store a player's path for each level as a one-dimensional sequence.
\subsection{Path generation}
We could have simply reused the extracted player paths as the input for a Lode Runner level generation process.
However, this would have limited the number of levels our system could generate.
As such, we need some way to generate new human-like paths.
Our extracted human paths vary significantly both in terms of length and patterns.\cite{github}
To address this, we first split the sequences of actions into small chunks of a fixed size (50).
We represent each action as a one-hot encoding of length five for the five actions.
Every action becomes a vector of length five, with every action type represented as an index in that vector.
An action is represented by a 1 at its index if it occurred at this position in the sequence and a 0 otherwise.
Thus our data becomes a series of 50x5 matrices.
We employ a Long Short-Term Memory Recurrent Neural Network (LSTM) for our path modeling, as they have been demonstrated to work well on PCGML tasks with sequence-like data \cite{summerville2016learning}.
An LSTM is designed to better learn long-term dependencies than standard recurrent neural networks.
This is important for our use case as the human paths tended to have many repeated actions in a row, and we didn't want our model to learn to just repeat the same action endlessly.
We employ an LSTM-based Seq2Seq model that is built to generate new paths.
Our input is one 50-length path sequence and the expected output is the next 50-length path sequence.
The model is composed of 2 layers with 512 LSTM cells each.
We employ dropout and gradient clipping to prevent overfitting and gradient explosion, respectively.
The final layer is a fully connected layer of length 50 with softmax activation, to better represent the probability distribution of a single character at each time step.
We trained our model for 60 epochs with a 0.001 learning rate and the adam optimizer\cite{kingma2017adam}.
We probabilistically sample each output action based on treating the softmax activation as a probability distribution.
We note that despite using 50 inputs and 50 outputs this model can be used to create paths of arbitrary lengths by inputting empty (all 0s) actions initially, and then continually generating based on previously generated outputs.
\subsection{Level Structure Learning}
The path information we generate only partially defines a level.
The same level path could be associated with a large, but not infinite number of levels.
This is particularly true in Lode Runner, where the player can directly ``make'' paths through their own actions (e.g. digging holes).
This means that we still need more information about what kind of tiles can be associated with what action, and how to fill out the rest of the level ``away'' from the player path.
Given our interest in demonstrating the impact of player paths in improving global coherency, we employ a multi-dimensional Markov chain \cite{snodgrass2014experiments}.
In our multi-dimensional Markov chain each tile value depends on the tiles to its left and directly below, along with the player action at the current position, to the right, and above.
We note that prior examples of platformer Markov chains have made use of nodes with 3 dependencies: the left, below, and to the left and below.
We experimented with this model, but found that our 2-tile dependency sufficiently modeled level structure and led to increased diversity in the output.
We train our multi-dimensional Markov chain on the Lode Runner levels from the VGLC \cite{VGLC}, with the added information from our extracted paths in the grid-based representation discussed above.
We also record a series of statistics in terms of the number of enemies and gold pieces in the training levels.
For each original level, we found the ratio of the numbers of enemies and gold pieces to the associated player path lengths.
We represent these two distributions as two Gaussian distributions.
This allows us to sample from these two distributions and derive an overall number of desired gold pieces and enemies for a new level.
\subsection{Complete Level Generation}
Our level generation process begins by generating a new path.
From this path we can determine the minimum size of a level necessary to contain this path.
We instantiate this level as an empty grid of the minimum size.
We label every visited tile in this level with the action that the player path indicates, and the remaining tiles with a special token that indicates no actions.
We automatically constrain some tile values based on the action type, based on the existing keys in the Markov Chain.
For example, if the action taken at a tile is to move up, then we know that this tile must contain a ladder.
Or if the action at a tile is to move to the left, then this tile can be empty, have a brick (as the player may break it from above and fall to it, then move left), or a rope.
This gives us some high-level constraints on the tiles and a basic initial structure to work with.
We apply our Markov chain to fill the level out with the final tile types from the partially specified state.
We start from the bottom left corner and move along each row from left to right until we've reached the top of the screen.
If a tile has been specified we do not need to generate a new tile at this location, and we just move on.
If the tile has been partially specified then we remove the tile possibilities that have already been ruled out, and then probabilistically sample from the remaining options based on the learned probability distribution in the Markov chain.
If the tile has not been specified at all, then we simply probabilistically sample from the Markov chain as normal \cite{snodgrass2014experiments}.
In the event that a key does not exist, we remove all dependencies except for the left and below dependency and then sample from this simplified distribution.
We made this choice as this removes all path requirements and only considers the structure of the level.
The final step of our generation process is to place all enemy and gold tiles.
We randomly sample from both of the Gaussian distributions we described above.
We multiply the sampled values by the generated path length to get the final numbers of enemies and gold pieces.
We then randomly place these elements along the player's path, as they are elements that the player would have to avoid or seek out, respectively.
We made this choice as the Lode Runner levels in the VGLC only place enemies and gold pieces in place of empty tiles, but both entities can move.
Further, the Markov chain struggled to learn to place these elements effectively given how rare they are in comparison to the other tile types.
\section{Quantitative Evaluation}
We extracted 66 player paths for 66 game levels from a 4.5-hour long gameplay video.
After training on these sequences, we were able to generate an arbitrary number of levels.
Figure \ref{fig:good1} gives an example of a generated level based on our approach.
This research seeks to determine whether adding a player path as input to a PCGML generator improves global coherence.
We employ Markov chains, a PCGML approach that tends to struggle with global coherence in order to better understand the impact of player paths \cite{guzdial2018co}.
As such, the natural choice for an initial evaluation is investigating how the inclusion of a generated path as input changes the generated levels in comparison to a Markov chain approach without player path information.
We therefore compare against the work of Snodgrass and Ontan{\'o}n, who employed a Markov chain without path information to generate Lode Runner levels \cite{snodgrass2016learning}.
We employed an A* pathfinding agent to test how a player might play the two different sets of Lode Runner levels. \cite{github}
Ideally, we might have used a human subject study to compare the two types of generated levels.
However, we lacked the time and resources for a human subject study, and so use this method for an initial comparison.
The pathfinder starts from the player tile present in both sets of generated levels and attempts to pathfind to each gold piece in turn.
Lode Runner is a complex game, which allows players to momentarily trap enemies, as such, we ignore enemies while pathfinding.
This decision was also motivated by the fact that we lacked a simulator to fully simulate enemy movement in the game, meaning that the enemies would just be treated as impassable obstacles otherwise, which would not be appropriate.
We note that prior work did not make this assumption, and so they reported much lower percentages of playable outputs \cite{snodgrass2016learning}.
However, since this assumption is made for both types of levels it's still helpful for comparison purposes.
The pathfinder tracks the number of nodes explored on the way to each gold as well as each gold that it was able to reach.
It reports the total number of each of those two metrics which we use to compare between the two types of levels.
Ideally, an A* agent should be able to reach each gold piece from the starting location in a playable Lode Runner level.
We had the A* agent report the number of nodes explored on the way to each gold piece as a measure of coherence.
All of the existing Lode Runner levels have clear paths to access each gold piece, essentially acting as puzzles for the player to solve.
Therefore, we take a low number of nodes explore as an indirect measure of global coherence.
We employ the following metrics in this comparative evaluation, reporting the average and standard deviation of each metric across the generated levels:
\begin{itemize}
\item \textit{Gold Total Per Level} - This is the average total number of gold pieces the A* agent needed to find. This is used in determining the percentage of gold collected in a level, as well as the potential difficulty and length of a level. The more gold pieces the more potential for difficulty depending on how they are distributed throughout the level. It also coincides with how long a level may take to play. The more gold, the longer it could take for a player to collect each piece.
\item \textit{Percentage Collected Per Level} - This metric gives how many of the gold pieces could be collected in a level.
Values closer to 100\% indicate more playable levels overall.
A higher average indicates that more of the levels are playable.
\item \textit{Total Nodes Explored} - This metric is the number of nodes the A* agent needed in total to reach all of the reachable gold pieces.
We do not include nodes explored when the pathfinder attempted to reach unreachable gold pieces.
This number can be taken as an approximation for the minimum amount of time a player would need to complete a level.
The higher the number the longer the time needed to complete the level.
\item \textit{Nodes Per Gold} - Since levels do not all have the same number of gold pieces, the total nodes explored metric on it's own could potential leave an inaccurate reflection of each level.
As such, this metric gives the average number of nodes needed to reach each reachable piece of gold per level.
The same rule applies as with the above metric, where the larger the number the longer it potentially could take to reach each gold piece.
\end{itemize}
\noindent
These metrics allow us to compare between our two sets of generated levels.
We are particularly interested in the second metric as a measure of playability and the fourth metric as a measure of global coherence.
We report the other two metrics for context to these two metrics.
If the Snodgrass and Ontan{\'o}n levels outperform our levels in terms of playability, that could indicate that our method for placing gold pieces along the player path is flawed.
If the Snodgrass and Ontan{\'o}n levels outperform our levels or perform similarly in terms of nodes per gold metric, this would indicate that our inclusion of the player path did not improve global coherence, and led to similarly coherent/incoherent levels as a simpler Markov chain approach.
\section{Qualitative Results}
Table 1 includes all of our results in terms of our four metrics.
The first column gives the average and standard deviation of the total number of gold pieces included in each generated level.
This immediately demonstrates the impact of employing a Gaussian distribution to model the total number of gold pieces in a level, as opposed to leaving it up to a Markov chain alone to place gold pieces.
The Snodgrass and Ontan{\'o}n levels have a much higher average and a much larger standard deviation.
This indicates that their levels tended to have many more gold pieces on average compared to existing Lode Runner levels, and that this number varied to a great extent with the largest number of gold pieces being nearly three times as many for the Snodgrass and Ontan{\'o}n levels.
This is an initial indication that our approach led to more coherent levels.
\begin{table*}[tbh]
\begin{tabular}{|l|c|c|c|c|}
\hline
& Gold Total & Percentage Collected & Total Nodes Explored & Nodes per Gold \\
\hline
Snodgrass and Ontan{\'o}n & 18.76\rpm 15.07 & 98.94\rpm 6.98 & 4638.93\rpm 10837.32 & 220.62\rpm 477.89 \\
\hline
Ours & 7.65\rpm 2.98 & 94.68\rpm 19.07 & 910.33\rpm 2047.28 & 116.44\rpm 264.82 \\
\hline
\end{tabular}
\caption{Quantitative Evaluation Results}
\end{table*}
The second column of Table 1 shows the average amount of gold that can be collected per level, indicating how many of the generated levels were playable.
These values indicate that the Snodgrass and Ontan{\'o}n levels are on average more likely to be playable.
However, there is more complexity to this value than might first appear.
Since on average our levels have less gold than the Snodgrass and Ontan{\'o}n levels, the impact of being unable to reach a single gold piece is much higher.
Collecting 6 of 7 available gold pieces will result in a lower percentage than 17 of 18 possible gold pieces.
We also performed a Mann-Whitney U test to determine if these two distributions differed significantly.
The test was unable to reject the null hypothesis ($p=0.05629$) that these two distributions arose from the same underlying distribution.
Thus, we take this to mean that the difference in terms of playability was insignificant.
As mentioned above, this runs counter to prior reported playability values \cite{snodgrass2016learning}, which is due to our choice not to model enemy locations.
Thus, this playability metric should be considered an upper bound.
\begin{figure}
\caption{Boxplot Results of Nodes Explored Metric}
\label{fig:boxplot}
\end{figure}
The third column of Table 1 gives the Total Nodes Explored metric, which we also visualize in Figure \ref{fig:boxplot} for clarity.
It is immediately clear that the Snodgrass and Ontan{\'o}n generator leads to massively more explored nodes than our approach.
However, this is not a fair comparison due to the higher average number of gold pieces among these levels and the larger variance of the number of gold pieces.
Thus, we turn to the fourth column and the average number of nodes explored for each gold piece.
This comparison is closer, only indicating that the Snodgrass and Ontan{\'o}n levels require twice as many nodes to be explored to reach each gold piece.
However, this difference is still substantial.
This indicates that, for each gold piece, it would take a player roughly twice the time to collect it for an average Snodgrass and Ontan{\'o}n level.
This indicates that there is no clear path between the gold pieces in the Snodgrass and Ontan{\'o}n levels.
The standard deviation also signifies that our levels are more consistent in terms of this metric.
We take this as an indication of the greater global coherency of our generated levels, that these levels include clear paths for the player to take to reach most gold pieces, even if the placement of some of these gold pieces makes them unreachable.
Our findings suggest that both our method and the method used by Snodgrass and Ontan{\'o}n produce levels that are roughly equally playable using our A* agent that ignored enemy positions.
Looking at the metric values it would have been very difficult to improve on this metric without achieving 100 percent playability.
Our analysis did show that we improved the consistency and reliability of the solutions to levels.
\section{Qualitative Evaluation}
In this section, we asses the quality of our output levels in comparison to the original levels.
We did this in order to get a more nuanced look as to whether we have actually achieved greater global coherence.
Given the results of our first set of evaluations, it's clear that our approach was able to produce levels with clearer and more consistent solutions.
However, it's possible that our output levels no longer resemble the original Lode Runner levels.
For example, they may have become too simple or have lost other aspects of global coherence.
We use the following metrics to investigate this possibility:
\begin{itemize}
\item \textit{s} - The minimum size of the level to fit the path.
Ideally, we'd like the levels to match the size of the original Lode Runner levels: a 32x22 grid.
Since we do not explicitly enforce this, our hope is that our approach will have led to an implicit bias towards levels of this size.
\item \textit{e} - The proportion of the level taken up by empty space. The original Lode Runner levels have a fair amount of variance when it comes to empty space, and empty space is often used strategically to create shapes or to indicate potential solutions. Thus, if our distribution of empty tiles matches those from the original levels, this would indicate a positive signal in terms of similar global structure.
\item \textit{i} - The proportion of the room taken up by ``interesting'' tiles that are not simply solid or empty.
It goes without saying that the way that the Lode Runner levels employ ladders, ropes, enemies, and gold pieces is very important to the overall design of the level.
\end{itemize}
\section{Results}
\begin{figure}
\caption{A good generated level.}
\label{fig:good1}
\end{figure}
We generated 34 generated levels and compared these to the 34 original levels that we did not use to train our model.
We found that 20 of our generated levels matched the expected size of 32x22, with 14 of our levels having a smaller size.
This means that these levels could be expanded to fit the expected size while retaining their same generated structure.
This is a positive sign, as our approach was able to implicitly lead to levels with the same or similar sizes to the original levels without explicitly modeling this constraint.
Notably, none of the generated levels were larger than the original levels, though this may be due to the fact that we employed a constant generated path size of 103 (which was the average of the paths we extracted from the gameplay video).
However, a generated path of 103 steps could still have led to a level larger than 32x22.
Figure \ref{fig:figure_roi} and Figure \ref{fig:figure_space} show the distribution of ``interesting'' and empty space tiles in the generated and original levels respectively.
The distribution of ``interesting'' tiles does suggest that our levels tended to lead to fewer interesting tiles compared to the original levels.
However, the overall distributions are fairly similar.
Further, it's possible that due to the 14 smaller levels the generated levels look more conservative than they truly are since this distribution does not take into account level size.
The empty space distribution seems to differ more with the original levels employing much more empty space.
This is not an unusual problem for Markov Chain models: filling in too much content.
However, we again note that the 14 smaller levels may be part of the problem here.
By filling the remaining space of these 14 smaller levels with empty space, the two distributions would look much more similar.
\begin{figure}
\caption{The distribution of "interesting" tiles.}
\label{fig:figure_roi}
\end{figure}
\begin{figure}
\caption{The distribution of empty space.}
\label{fig:figure_space}
\end{figure}
\section{Discussion}
There are 150 levels completed in the video we used for this paper.
However, trimming and cropping each level from the video was time-consuming and repetitive.
Therefore we just dealt with 66 levels to extract player paths.
If all levels were processed, the results might be improved.
Extracting path information from videos has a couple of challenges.
Our method works well for most levels, but the levels with lots of stairs and levels with fake bricks (where there appears to be a brick, but it is actually an empty space as the player steps on it) will not work very well.
Due to the low image quality, when the player walks passed the stairs, the combined figure becomes very difficult to recognize.
This may sometimes lead to losing track of the player.
The problem with the fake brick has a similar effect.
When the player falls through a fake brick, the program fails to detect the player, hence losing track of the player.
It is a limitation of our approach that we always assume a fixed path length.
While we made this choice for simplicity, the original levels did not all have the same path length.
As such, it would be better to model this path length value separately.
Alternatively, we could generate the path until we hit the desired level size and then stop.
\begin{figure}
\caption{A bad generated level: badly placed enemies.}
\label{fig:bad1}
\end{figure}
\begin{figure}
\caption{A bad generated level: badly designed structure.}
\label{fig:bad2}
\end{figure}
Figure \ref{fig:bad1} and Figure \ref{fig:bad2} show two typical issues that prevent the player from completing the generated levels.
In Figure \ref{fig:bad1}, an enemy is directly placed beside the player, and there is no other way to go around or trap the enemy.
This shows that randomly placing enemies along the player's path is not ideal, we will also need to take consideration that the player needs to have a way to deal with enemies on the path.
We ignored this problem for our pathfinding-based evaluation, but we will need to confront it for future work.
Figure\ref{fig:bad2} had a different problem.
The generated structure led to the player getting stuck.
This happens when the row where the player is at and the row above it both have path information. Then when filling the tiles, the system would make a mistake that the bottom row can be bricks, and the player can dig along the upper row to create a path to the lower row.
But this will not work if the player started from the lower row.
This situation would trip up our A* pathfinder, and was one of the factors that led to our lower average playability score.
In this initial investigation we assumed controllability due to our input of player paths.
While we can alter these paths and produce new levels based on them, we did not include any evaluation of this aspect of our research.
This would be a difficult thing to evaluate without a human subject study, and so we leave it for future work.
\section{Conclusions}
In this research project, we developed a player path-based method for the generation of Lode Runner levels.
We extracted player paths from gameplay video to serve as training data, then used an LSTM Seq2Seq model to generate new player paths, and applied Markov Chains to produce new levels based on these paths.
Our experimental results show that this approach can lead to improved global coherence of the generated levels while still leading to levels that share a resemblance to the original levels.
For future work, we hope to improve the proposed method to ensure playability, to use a more sophisticated pathfinding agent that takes into account things like enemy placement, and to test this approach on other games.
\begin{acks}
This work was funded by the Canada CIFAR AI Chairs Program. We acknowledge the support of the Alberta Machine Intelligence Institute (Amii).
\end{acks}
\appendix
\end{document} |
\begin{document}
\begin{abstract}
\jz{We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schr\"odinger equation with unknown source and potential terms.
The well-posedness of the direct scattering problem is first established.
Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements.
First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns.
Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source.
Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.}
\noindent{\bf Keywords:}~~random Schr\"odinger equation, inverse scattering, passive/active measurements, \jz{asymptotic expansion, ergodicity}
{\noindent{\bf 2010 Mathematics Subject Classification:}~~35Q60, 35J05, 31B10, 35R30, 78A40}
\end{abstract}
\maketitle
\section{Introduction} \label{sec:Intro-SchroEqu2018}
In this paper, we are mainly concerned with the following random Schr\"odinger system
\begin{subequations} \label{eq:1}
\begin{numcases}{}
\displaystyle{ (-\Delta-E+V(x)) u(x, E, d, \omega)= f(x)+\sigma(x)\dot{B}_x(\omega), \quad x\in\mathbb{R}^3, } \label{eq:1a}
\\
\displaystyle{ u(x, E, d, \omega)=\alpha e^{\mathrm{i}\sqrt E x\cdot d}+u^{sc}(x, E, d,\omega), } \label{eq:1b}
\\
\displaystyle{ \lim_{r\rightarrow\infty} r\left(\frac{\partial u^{sc}}{\partial r}-\mathrm{i}\sqrt{E} u^{sc} \right)=0,\quad r:=|x|, } \label{eq:1c}
\end{numcases}
\end{subequations}
\hy{where $f(x)$ and $\sigma(x)$ in \eqref{eq:1a} are the expectation and standard variance of the source term, $d \in \mathbb{S}^2:=\{ x \in \R^3 \,;\, |x| = 1 \}$ signifies the impinging direction of the incident plane wave, and $E\in\mathbb{R}_+$ is the energy level.
In \eqref{eq:1b}, $\alpha$ takes the value of either 0 or 1 to incur or suppress the presence of the incident wave, respectively.
In the sequel, we follow the convention to replace $E$ with $k^2$, namely $k := \sqrt{E} \in \R_+$, which can be understood as the wave number.
The limit in \eqref{eq:1c} is the Sommerfeld Radiation Condition (SRC) \cite{colton2012inverse} that characterizes the outgoing nature of the scattered wave field $u^{sc}$. The random system \eqref{eq:1} describes the quantum scattering associated with a potential $V$ and a random active source $(f, \sigma)$ at the energy level $k^2$.}
\jz{In the system \eqref{eq:1}, the random parameter $\omega$ belongs to $\Omega$ with $(\Omega, \mathcal{F}, \mathbb{P})$ signifying a complete probability space.
The term $\dot B_x(\omega)$ denotes the three-dimensional spatial Gaussian white noise \cite{dudley2002real}.
The random part $\sigma(x) \dot B_x(\omega)$ within the source term in \eqref{eq:1a} is an ideal mathematical model for noises arising from real world applications \cite{dudley2002real}.
We note that the $\sigma^2(x)$ gives the intensity of the randomness of the source at the point $x$, and can be understood as the variance of $\sigma(x) \dot B(x,\omega)$. In what follows, we call $\sigma^2(x)$ the variance function.
The statistical information of a single zero-mean Gaussian white noise is encoded in its variance function \cite{ross2014introduction}.
In this paper, we are mainly concerned with the recovery of the variance and expectation of the random source as well as the potential function in \eqref{eq:1} by the associated scattering measurements as described in what follows.}
\hy{In order to study the corresponding inverse problems, one needs to have a thorough understanding of the direct scattering problem. In the deterministic case with $\sigma\equiv 0$, the scattering system \eqref{eq:1} is well understood; see, e.g., \cite{colton2012inverse,griffiths2016introduction}. There exists a unique solution $u^{sc}\in H^1_{loc}(\mathbb{R}^3)$, and moreover there holds the following asymptotic expansion as $|x|\rightarrow\infty$,
\begin{equation}\label{eq:farfield}
u^{sc}(x) = \frac{e^{\mathrm{i}k r}}{r} u^\infty(\hat x, k, d) + \mathcal{O} \left( \frac{1}{r^2} \right),
\end{equation}
where $\hat x := x/{|x|} \in \mathbb{S}^2$. The term $u^\infty$ is referred to as the far-field pattern, which encodes the information of the potential $V$ and the source $f$.
In principle, we shall show that the random scattering system \eqref{eq:1} is also well-posed in a proper sense and possesses a far-field pattern.
To that end, throughout the rest of the paper, we assume that $\sigma^2$, $V$, $f$ belong to $L^\infty(\mathbb{R}^3;\R)$, respectively, and that they are compactly supported in a fixed bounded domain $D\subset \mathbb{R}^3$ containing the origin.
Under the aforementioned regularity assumption, we establish that the following mapping of the direct problem (\textbf{DP}) is well-posed in a proper sense,
\begin{equation} \label{eq:dp-SchroEqu2018}
\textbf{DP \ :} \quad (\sigma, V, f) \rightarrow \{u^{sc}(\hat x, k, d, \omega), u^\infty(\hat x, k, d, \omega) \,;\, \omega \in \Omega,\, \hat x \in \mathbb{S}^2, k > 0,\, d \in \mathbb{S}^2 \}.
\end{equation}
The well-posedness of the direct scattering problem paves the way for our further study of the inverse problem ({\bf IP}).}
\sq{In {\bf IP}, we are concerned with the recoveries of the three unknowns $\sigma^2$, $V$, $f$ in a \emph{sequential} way, by knowledge of the associated far-field pattern measurements $u^\infty(\hat x, k, d, \omega)$.
By sequential, we mean the $\sigma^2$, $V$, $f$ are recovered by the corresponding data sets one-by-one.
In addition to this, in the recovery procedure, both the \emph{passive} and \emph{active} measurements are utilized.
When $\alpha = 0$, the incident wave is suppressed and the scattering is solely generated by the unknown source. The corresponding far-field pattern is thus referred to as the passive measurement.
In this case, the far-field pattern is independent of the incident direction $d$, and we denote it as $u^\infty(\hat x, k, \omega)$.
When $\alpha = 1$, the scattering is generated by both the active source and the incident wave, and the far-field pattern is referred to as the active measurement, denoted as $u^\infty(\hat x, k, d, \omega)$.
Under these settings, we formulate our {\bf IP} as}
\sq{\begin{equation}\label{eq:ip-SchroEqu2018}
\textbf{IP \ :}\quad \left\{
\begin{aligned}
\mathcal M_1(\omega) := & \ \{u^\infty(\hat x, k, \omega) \,;\, \forall \hat x \in \mathbb{S}^2,\, \forall k \in \R_+ \} && \rightarrow \quad \sigma^2, \\
\mathcal M_2(\omega) := & \ \{u^\infty(\hat x, k, d, \omega) \,;\, \forall \hat x \in \mathbb{S}^2,\, \forall k \in \R_+,\, \forall d \in \mathbb{S}^2 \} && \rightarrow \quad V, \\
\mathcal M_3 := & \ \{u^\infty(\hat x, k, d, \omega) \,;\, \forall \hat x \in \mathbb{S}^2,\, \forall k \in \R_+,\, d\ \text{fixed},\, \forall \omega \in \Omega\, \} && \rightarrow \quad f.
\end{aligned}
\right.
\end{equation}
The data set $\mathcal M_1(\omega)$ (abbr.~$\mathcal M_1$) corresponds to the passive measurement ($\alpha = 0$), while the data sets $\mathcal M_2(\omega)$ (abbr.~$\mathcal M_2$) and $\mathcal M_3$ correspond to the active measurements ($\alpha = 1$).
Different random sample $\omega$ gives different data sets $\mathcal M_1$ and $\mathcal M_2$.
All of the $\sigma^2$, $V$, $f$ in the {\bf IP} are assumed to be unknown, and our study shows that the data sets $\mathcal M_1$, $\mathcal M_2$, $\mathcal M_3$ can recover $\sigma^2$, $V$, $f$, respectively.
The mathematical arguments of our study are constructive and we derive explicitly recovery formulas, which can be employed for numerical reconstruction in future work.}
In the aforementioned {\bf IP}, we are particularly interested in the case with a single realization, namely the sample $\omega$ is fixed in the recovery of $\sigma^2$ and $V$ in \eqref{eq:ip-SchroEqu2018}.
\sq{Intuitively, a particular realization of $\dot B_x$ provides little information about the statistical properties of the random source.
However, our study indicates that a \emph{single realization} of the far-field measurement can be used to uniquely recover the variance function and the potential in certain scenarios. A crucial assumption to make the single-realization recovery possible is that the randomness is independent of the wave number $k$. Indeed,
there are assorted applications in which the randomness changes slowly or is independent of time \cite{caro2016inverse, Lassas2008}, and by Fourier transforming into the frequency domain, they actually correspond to the aforementioned situation.
The single-realization recovery has been studied in the literature;
see, e.g., \cite{caro2016inverse,Lassas2008,LassasA}.
The idea of this work is mainly motivated by \cite{caro2016inverse}.}
There are abundant literatures for the inverse scattering problem associated with either the passive or active measurements. Given an known potential, the recovery of an unknown source term by the corresponding passive measurement is referred to as the inverse source problem. We refer to \cite{bao2010multi,Bsource,BL2018,ClaKli,GS1,Isakov1990,IsaLu,Klibanov2013,KS1,WangGuo17,Zhang2015} and the references therein for both theoretical uniqueness/stability results and computational methods for the inverse source problem in the deterministic setting, namely $\sigma\equiv 0$.
\sq{The authors are also aware of some study on the inverse source problem concerning the recovery of a random source \cite{LiLiinverse2018,LiHelinLiinverse2018}.
In \cite{LiHelinLiinverse2018}, the homogeneous Helmholtz system with a random source is studied.
Compared with \cite{LiHelinLiinverse2018}, our system \eqref{eq:1} comprises of both unknown source and unknown potential, which make the corresponding study radically more challenging.}
The determination of a random source by the corresponding passive measurement was also recently studied in \cite{bao2016inverse,Lu1,Yuan1}, and the determination of a random potential by the corresponding active measurement was established in \cite{caro2016inverse}. We also refer to \cite{LassasA} and the references therein for more relevant studies on the determination of a random potential. The simultaneous recovery of an unknown source and its surrounding potential was also investigated in the literature. In \cite{KM1,liu2015determining}, motivated by applications in thermo- and photo-acoustic tomography, the simultaneous recovery of an unknown source and its surrounding medium parameter was considered. The simultaneous recovery study in \cite{KM1,liu2015determining} was confined to the deterministic setting and associated mainly with the passive measurement.
In this paper, we consider the
recovery of an unknown random source and an unknown potential term associated with the Schr\"odinger system \eqref{eq:1}.
The major novelty of our unique recovery results compared to those existing ones in the literature is that on the one hand, both the random source and the potential are unknown, and on the other hand, we use both passive and active measurements for the unique recovery. We established three unique recovery results.
\begin{thm} \label{thm:Unisigma-SchroEqu2018}
\sq{Without knowing $V$ and $f$ in system \eqref{eq:1}, the data set $\mathcal M_1$ can recover $\sigma^2$ almost surely.}
\end{thm}
\begin{rem}
Theorem~\ref{thm:Unisigma-SchroEqu2018} implies that the variance function can be uniquely recovered without \emph{a priori} knowledge of $f$ or $V$.
\sq{Moreover, since the passive measurement $\mathcal M_1$ is used, Theorem \ref{thm:Unisigma-SchroEqu2018} indicates that the variance function can be uniquely recovered by a single realization of the passive scattering measurement.
Moreover, for the sake of simplicity, we set the wave number $k$ in the definition of $\mathcal M_1$ to be running over all positive real numbers. But in practice, it is enough to let $k$ be greater than any fixed positive number. This remark equally applies to Theorem \ref{thm:UniPot1-SchroEqu2018}.}
\end{rem}
\begin{thm} \label{thm:UniPot1-SchroEqu2018}
\sq{Without knowing $\sigma$ and $f$ in system \eqref{eq:1}, the data set $\mathcal M_2$ uniquely recovers the potential $V$.}
\end{thm}
\begin{rem}
Theorem \ref{thm:UniPot1-SchroEqu2018} shows that the potential $V$ can be uniquely recovered without knowing the random source, namely $\sigma$ and $f$. Moreover, we only make use of
a single realization of the active scattering measurement.
\end{rem}
\begin{thm} \label{thm:UniSou1-SchroEqu2018}
\sq{In system \eqref{eq:1}, suppose that $\sigma$ is unknown and the potential $V$ is known in advance. Then there exists a positive constant $C$ that depends only on $D$ such that if $\nrm[L^\infty(\R^3)]{V} < C$,
the data set $\mathcal M_3$ can uniquely recover the expectation $f$.}
\end{thm}
\jz{The rest of the paper is outlined as follows. In Section \ref{sec:MADP-SchroEqu2018}, we present the mathematical analysis of the forward scattering problem given in \eqref{eq:1}.
Section \ref{sec:AsyEst-SchroEqu2018} establishes some asymptotic estimates, which are of key importance in the recovery of the variance function.
In Section \ref{sec:RecVar-SchroEqu2018}, we prove the first recovery result of the variance function with a single realization of the passive scattering measurement.
Section \ref{sec:RecPS-SchroEqu2018} is devoted to the second and third recovery results of the potential and the random source. We conclude the work with some discussions in \mbox{Section \ref{sec:Conclusions-SchroEqu2018}.}}
\sq{
\section{Mathematical analysis of the direct problem} \label{sec:MADP-SchroEqu2018}}
\sq{In this section, the uniqueness and existence of a {\it mild solution} is established for the system \eqref{eq:1}. Before analyzing the direct problem, some preparations are made in the beginning.
In Section \ref{subsec:NotandAss-SchroEqu2018}, we introduce some preliminaries which are used throughout the rest of the paper.
Some technical lemmas that are necessary for the analysis of both the direct and inverse problems are presented in Section \ref{subsec:STLemmas-SchroEqu2018}.
In Section \ref{subset:WellDefined-SchroEqu2018}, we give the well-posedness of the direct problem.}
\subsection{Preliminaries} \label{subsec:NotandAss-SchroEqu2018}
\sq{Let us first introduce the generalized Gaussian white noise $\dot B_x(\omega)$ \cite{kusuoka1982support}.}
To give a brief introduction, we write $\dot B_x(\omega)$ temporarily as $\dot B(x,\omega)$.
It is known that $\dot B(\cdot,\omega) \in H_{loc}^{-3/2-\epsilon}(\R^3)$ almost surely for any $\epsilon\in\mathbb{R}_+$ \cite{kusuoka1982support}.
Then $\dot B \colon \omega \in \Omega \mapsto \dot B(\cdot,\omega) \in \mathscr{D}'(D)$ defines a map from the probability space to the space of the generalized functions.
Here, $\mathscr{D}(D)$ signifies the space consisting of smooth functions that are compactly supported in $D$, and $\mathscr{D}'(D)$ signifies its dual space.
For any $\varphi \in \mathscr{D}(D)$, $\dot B \colon \omega \in \Omega \mapsto \agl[\dot B(x,\omega), \varphi(x)] \in \R$ is assumed to be a Gaussian random variable with zero-mean and $\int_{D} |\varphi(x)|^2 \dif{x}$ as its variance.
We also recall that a function $\psi$ in $L_{loc}^1(\R^n)$ defines a distribution through ${\agl[\psi,\varphi] = \int_{\R^n} \psi(x) \varphi(x) \dif{x}}$ \cite{caro2016inverse}. Then $\dot B(x,\omega)$ satisfies:
\[
\agl[\dot B(\cdot,\omega), \varphi(\cdot)] \sim \mathcal{N}(0,\nrm[L^2(D)]{\varphi}^2), \quad \forall \varphi \in \mathscr{D}(D).
\]
Moreover, the covariance of the $\dot B(x,\omega)$ is assumed to satisfy the following property. For every $\varphi$, $\psi$ in $\mathscr{D}(D)$, the covariance between $\agl[\dot B(\cdot,\omega), \varphi]$ and $\agl[\dot B(\cdot,\omega), \psi]$ is defined as $\int_{D} \varphi(x) \psi(x) \dif{x}$:
\begin{equation} \label{eq:ItoIso-SchroEqu2018}
\mathbb{E} \big( \agl[\dot B(\cdot,\omega), \varphi] \agl[\dot B(\cdot,\omega), \psi] \big) := \int_{D} \varphi(x) \psi(x) \dif{x}.
\end{equation}
These aforementioned definitions can be generalized to the case where $\varphi, \psi \in L^2(D)$ by the density arguments.
The $\delta(x) \dot B(x,\omega)$ is defined as
\begin{equation} \label{eq:sigmaB-SchroEqu2018}
\delta(x) \dot B(x,\omega) \colon \varphi \in L^2(D) \mapsto \agl[\dot B(\cdot,\omega), \delta(\cdot)\varphi(\cdot)] \in \R.
\end{equation}
Secondly, let's set
$$\Phi(x,y) = \Phi_k(x,y) := \frac {e^{ik|x-y|}}{4\pi|x-y|}, \quad x\in\mathbb{R}^3\backslash\{y\}.$$
$\Phi_k$ is the outgoing fundamental solution, centered at $y$, to the differential operator $-\Delta-k^2$. Define the resolvent operator ${\mathcal{R}_{k}}$,
\begin{equation} \label{eq:DefnRk-SchroEqu2018}
{\mathcal{R}_{k}}(\varphi)(x) = ({\mathcal{R}_{k}} \varphi)(x) := \int_{\mathop{\rm supp} \varphi} \Phi_k(x,y) \varphi(y) \dif{y}, \quad x \in \R^3,
\end{equation}
where $\varphi$ can be any measurable function on $\mathbb{R}^3$ as long as the \eqref{eq:DefnRk-SchroEqu2018} is well-defined for almost all $x$ in $\R^3$. Similar to the \eqref{eq:DefnRk-SchroEqu2018}, we define ${\mathcal{R}_{k}}(\delta \dot{B}_x)(\omega)$ as
\begin{equation} \label{eq:RkSigmaBDefn-SchroEqu2018}
{\mathcal{R}_{k}}(\delta \dot{B}_x)(\omega) := \agl[\dot B(\cdot,\omega), \delta(\cdot) \Phi(x,\cdot)],
\end{equation}
for any $\delta \in L^{\infty}(\R^3)$ with $\mathop{\rm supp} \delta \subseteq D$. We write ${\mathcal{R}_{k}}(\delta \dot{B}_x)(\omega)$ as ${\mathcal{R}_{k}}(\delta \dot{B}_x)$ for short.
We may also write ${\mathcal{R}_{k}}(\delta \dot{B}_x)$ as $\int_{\R^3} \Phi_k(x,y) \delta(y) \dot B_y \dif{y}$ or $\int_{\R^3} \Phi_k(x,y) \delta(y) \dif{B_y}$.
We may omit the subscript $x$ in ${\mathcal{R}_{k}}(\delta \dot B_x)$ if it is clear in the context.
Write $\agl[x] := (1+|x|^2)^{1/2}$ for $x \in \R^3$. We introduce the following weighted $L^2$-norm and the corresponding function space over $\R^3$ for any $s \in \R$,
\begin{equation} \label{eq:WetdSpace-SchroEqu2018}
\left\{
\begin{aligned}
\nrm[L_{s}^2(\R^3)]{f} & := \nrm[L^2(\R^3)]{\agl[\cdot]^{s} f(\cdot)} = \Big( \int_{\R^3} \agl[x]^{2s} |f|^2 \dif{x} \Big)^{\frac 1 2}, \\
L_{s}^2(\R^3) & := \left\{ f\in L_{loc}^1(\mathbb{R}^3); \nrm[L_{s}^2(\R^3)]{f} < +\infty \right\}.
\end{aligned}\right.
\end{equation}
We also define $L_{s}^2(S)$ for any measurable subset $S$ in $\R^3$ by replacing $\R^3$ in \eqref{eq:WetdSpace-SchroEqu2018} with $S$. In what follows, we may denote $L_s^2(\R^3)$ as $L_s^2$ for short if without ambiguities.
\jz{In the sequel, we write $\mathcal{L}(\mathcal A, \mathcal B)$ to denote the set of all the linear bounded mappings from a norm vector space $\mathcal A$ to a norm vector space $\mathcal B$.
For any mapping $\mathcal K \in \mathcal{L}(\mathcal A, \mathcal B)$, we denote its operator norm as $\nrm[\mathcal{L}(\mathcal A, \mathcal B)]{\mathcal K}$.
We write the identity operator as $I$.
We also use notations $C$ and its variants, such as $C_D$ and $C_{D,f}$ to represent some generic constant(s) whose particular definition may change line by line.
We use $\mathcal{A}\lesssim \mathcal{B}$ to signify $\mathcal{A}\leq C \mathcal{B}$ and $\mathcal{A} \simeq \mathcal{B}$ to signify $\mathcal{A} = C \mathcal{B}$, for some generic positive constant $C$. We denote ``almost everywhere'' as~``a.e.''~and ``almost surely'' as~``a.s.''~for short.
We use $|\mathcal S|$ to denote the Lebesgue measure of any Lebesgue-measurable set $\mathcal S$. Define $M(x) = \sup_{y \in D}|x-y|$, and $\text{diam}\,D := \sup_{x,y \in D} |x-y|$, where $D$ is the bounded domain containing $\mathop{\rm supp} \sigma$, $\mathop{\rm supp} V$, $\mathop{\rm supp} f$ and the origin.
Thus we have
$M(0) \leq \text{diam}\,D < \infty$. It can be verified that
\begin{equation} \label{eq:Contain-SchroEqu2018}
\{ y-x \in \R^3 ; |x| \leq 2 M(0), y \in D \} \subseteq \{ z \in \R^3 ; |z| \leq 3\,\text{diam}\,D \}.
\end{equation}}
This is because $|y-x| \leq |y| + |x| \leq \text{diam}\,D + 2M(0) \leq 3\text{diam}\,D$.
\subsection{Several technical lemmas} \label{subsec:STLemmas-SchroEqu2018}
Several important technical lemmas are presented here.
\begin{lem} \label{lemma:RkBoundedR3-SchroEqu2018}
For any $\varphi \in L^\infty(\R^3)$ with $\mathop{\rm supp} \varphi \subseteq D$ and any $\epsilon \in \mathbb{R}_+$, we have
\[
{\mathcal{R}_{k}} \varphi \in L_{-1/2-\epsilon}^2.
\]
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:RkBoundedR3-SchroEqu2018}]
Assume that $\varphi$ belongs to $L^\infty(\R^3)$ with its support contained in $D$. \jz{Obviously we have that $\nrm[L^2(D)]{\varphi} < +\infty$. Using the Cauchy-Schwarz inequality we have
\begin{align} \label{eq:Rk1-SchroEqu2018}
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} \varphi}^2
& \lesssim \int_{\R^3} \agl[x]^{-1-2\epsilon} \big( \int_D \frac{1}{|x-y|^2} \dif{y} \big) \cdot \big( \int_D |\varphi(y)|^2 \dif{y} \big) \dif{x} \nonumber\\
& \lesssim \nrm[L^2(D)]{\varphi}^2 \Big[ \int_{|x| \leq 2M(0)} \big( \int_D \frac{1}{|x-y|^2} \dif{y} \big) \dif{x} \nonumber\\
& \quad\quad + \int_{|x| > 2M(0)} \agl[x]^{-1-2\epsilon} \agl[x]^{-2} \dif{x} \Big].
\end{align}
By the change of variable, the first term in the square brackets in \eqref{eq:Rk1-SchroEqu2018} satisfies
\begin{equation} \label{eq:Rk2-SchroEqu2018}
\int_{|x| \leq 2M(0)} \big( \int_D \frac{1}{|x-y|^2} \dif{y} \big) \dif{x} = \int_{|x| \leq 2M(0)} \big( \int_{z \in \{ y-x \,;\, y \in D \}} \frac{1}{|z|^2} \dif{z} \big) \dif{x}.
\end{equation}
From \eqref{eq:Contain-SchroEqu2018}, we can continue \eqref{eq:Rk2-SchroEqu2018} as
\begin{align}
\int_{|x| \leq 2M(0)} \big( \int_D \frac{1}{|x-y|^2} \dif{y} \big) \dif{x}
& \leq \int_{|x| \leq 2M(0)} \big( \int_{\{z \,;\, |z| \leq 3\,\text{diam}\,D \}} \frac{1}{|z|^2} \dif{z} \big) \dif{x} \nonumber\\
& = \int_{|x| \leq 2M(0)} \big( 12\pi\,\text{diam}\,D \big) \dif{x} < +\infty. \label{eq:Rk3-SchroEqu2018}
\end{align}
Meanwhile, the second term in the square brackets in \eqref{eq:Rk1-SchroEqu2018} satisfies
\begin{equation} \label{eq:Rk4-SchroEqu2018}
\int_{|x| > 2M(0)} \agl[x]^{-1-2\epsilon} \agl[x]^{-2} \dif{x} \leq \int_{\R^3} \agl[x]^{-3-2\epsilon} \dif{x} < +\infty.
\end{equation}
Note that \eqref{eq:Rk4-SchroEqu2018} holds for every $\epsilon \in \mathbb{R}_+$. Combining \eqref{eq:Rk1-SchroEqu2018}, \eqref{eq:Rk3-SchroEqu2018} and \eqref{eq:Rk4-SchroEqu2018}, we conclude
\[
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} \varphi}^2 < +\infty.
\]
}
The proof is complete.
\end{proof}
\jz{Now we present a special version of Agmon's estimates for the convenience of our reader (cf. \cite{eskin2011lectures}). This special version will be used when proving Lemma \ref{lemma:RkVBoundedR3-SchroEqu2018}.}
\begin{lem}[Agmon's estimates \cite{eskin2011lectures}] \label{lemma:AgmonEst-SchroEqu2018}
For any $\epsilon > 0$, \jz{there exists some $k_0 \geq 2$ such that for any $k > k_0$} we have
\begin{equation} \label{eq:AgmonEst-SchroEqu2018}
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} \varphi} \leq C_\epsilon k^{-1} \nrm[L_{1/2+\epsilon}^2]{\varphi}, \quad \forall \varphi \in L_{1/2+\epsilon}^2
\end{equation}
where $C_\epsilon$ is independent of $k$ {and $\varphi$}.
\end{lem}
\sq{The proof of Lemma \ref{lemma:AgmonEst-SchroEqu2018} can be found in \cite{eskin2011lectures}.
The symbol $k_0$ appearing in Lemma \ref{lemma:AgmonEst-SchroEqu2018} is preserved for future use.}
\begin{lem} \label{lemma:RkVBoundedR3-SchroEqu2018}
For any fixed $\epsilon \geq 0$, when $k > k_0$, we have
$$\nrm[\mathcal{L}(L_{-1/2-\epsilon}^2, L_{-1/2-\epsilon}^2)]{{\mathcal{R}_{k}} \circ V} \leq C_{\epsilon,D,V} k^{-1},$$
where the constant $C_{\epsilon,D,V}$ depends on $\epsilon, D$ and $V$ but is independent of $k$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:RkVBoundedR3-SchroEqu2018}]
By Lemma \ref{lemma:AgmonEst-SchroEqu2018}, when $k > k_0$, we have the following estimate,
$$\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} V u} = \nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} (Vu)} \leq C_\epsilon k^{-1} \nrm[L_{1/2+\epsilon}^2]{Vu}.$$
Due to the boundedness of $\mathop{\rm supp} V$, there holds $\nrm[L_{1/2+\epsilon}^2]{Vu} \leq C_{D,V} \nrm[L_{-1/2-\epsilon}^2]{u}$ for some constant $C_{D,V}$ depending on $D$ and $V$ but independent of $u$ and $\epsilon$. Thus, we have
$$\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} Vu} \leq C_{\epsilon,D,V} k^{-1} \nrm[L_{-1/2-\epsilon}^2]{u}.$$
The proof is complete.
\end{proof}
\sq{In the rest of the paper, we use $k^*$ to represent the maximum between the quantity $k_0$
originated from Lemma \ref{lemma:AgmonEst-SchroEqu2018}
and the quantity
\[
\sup_{k \in \R_+} \{ k \,;\, \nrm[\mathcal{L}{(L_{-1/2-\epsilon}^2, L_{-1/2-\epsilon}^2)}]{\mathcal{R}_k V} \geq 1 \} \ + \ 1.
\]
This choice of $k^*$ guarantees that if $k \geq k^*$, both the inequality \eqref{eq:AgmonEst-SchroEqu2018} and the Neumann expansion
\(
{(I - \mathcal{R}_k V)^{-1} \ = \ \sum_{j \geq 0} (\mathcal{R}_k V)^j}
\)
in $L_{-1/2-\epsilon}^2$ hold.
For the subsequent analysis we also need a local version of Lemma \ref{lemma:RkVBoundedR3-SchroEqu2018}.}
\begin{lem} \label{lemma:RkVBounded-SchroEqu2018}
When $k > k_0$, we have
\begin{equation} \label{eq:RkVbdd-SchroEqu2018}
\nrm[\mathcal{L}(L^2(D), L^2(D))]{{\mathcal{R}_{k}} V} \leq C_{D,V} k^{-1},
\end{equation}
for some constant $C_{D,V}$ depending on $D$ and $V$ but independent of $k$. Moreover, for every $\varphi \in L^2(\R^3)$ with $\mathop{\rm supp} \varphi \subseteq D$, then
\begin{equation} \label{eq:VRkbdd-SchroEqu2018}
\nrm[L^2(D)]{V{\mathcal{R}_{k}} \varphi} \leq C_{D,V} k^{-1} \nrm[L^2(D)]{\varphi},
\end{equation}
for some constant $C_{D,V}$ depending on $D$ and $V$ but independent of $\varphi$ and $k$.
\end{lem}
\begin{proof}
\jz{For any $\varphi \in L^2(D)$, thanks to the boundedness of $D$ we have
\begin{equation} \label{eq:RkVbddInter1-SchroEqu2018}
\nrm[L^2(D)]{{\mathcal{R}_{k}} V\varphi} \leq C_D \nrm[L_{-1}^2]{{\mathcal{R}_{k}} (V\varphi)}.
\end{equation}
By Lemma \ref{lemma:AgmonEst-SchroEqu2018} (letting the $\epsilon$ in Lemma \ref{lemma:AgmonEst-SchroEqu2018} be $\frac 1 2$), we conclude that
\begin{equation} \label{eq:RkVbddInter2-SchroEqu2018}
\nrm[L_{-1}^2]{{\mathcal{R}_{k}} (V\varphi)} \leq C k^{-1} \nrm[L_1^2]{V\varphi}.
\end{equation}
By virtue of the boundedness of $V$, we have
\begin{equation} \label{eq:RkVbddInter3-SchroEqu2018}
\nrm[L_1^2]{V\varphi} \leq C_{D,V}\nrm[L^2(D)]{\varphi}.
\end{equation}
Combining \eqref{eq:RkVbddInter1-SchroEqu2018}-\eqref{eq:RkVbddInter3-SchroEqu2018}, we arrive at \eqref{eq:RkVbdd-SchroEqu2018}.
To prove \eqref{eq:VRkbdd-SchroEqu2018}, by Lemma \ref{lemma:AgmonEst-SchroEqu2018}, we have
$$\nrm[L^2(D)]{{\mathcal{R}_{k}} \varphi} \leq C_D \nrm[L_{-1}^2]{{\mathcal{R}_{k}} \varphi} \leq C_D k^{-1} \nrm[L_1^2]{\varphi} \leq C_D k^{-1} \nrm[L^2(D)]{\varphi}.$$
Therefore,
$$\nrm[L^2(D)]{V{\mathcal{R}_{k}} \varphi} \leq \nrm[L^\infty(D)]{V} \cdot \nrm[L^2(D)]{{\mathcal{R}_{k}} \varphi} \leq C_{D,V} k^{-1} \nrm[L^2(D)]{\varphi}.$$
The proof is complete.}
\end{proof}
Lemma \ref{lemma:RkSigmaB-SchroEqu2018} shows some basic properties of ${\mathcal{R}_{k}}(\sigma \dot{B}_x)$ defined in \eqref{eq:RkSigmaBDefn-SchroEqu2018}.
\begin{lem} \label{lemma:RkSigmaB-SchroEqu2018}
We have
$${\mathcal{R}_{k}}(\sigma \dot B_x) \in L_{-1/2-\epsilon}^2 \quad \textrm{~a.s.~}.$$
Moreover, we have
\begin{equation*}
\mathbb{E} \nrm[L^2(D)]{{\mathcal{R}_{k}}(\sigma \dot B_x)} < C < +\infty
\end{equation*}
for some constant $C$ independent of $k$.
\end{lem}
\begin{proof}
From \eqref{eq:RkSigmaBDefn-SchroEqu2018}, \eqref{eq:sigmaB-SchroEqu2018} and \eqref{eq:ItoIso-SchroEqu2018}, one can compute,
\begin{align*}
\mathbb{E} ( \nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot{B}_x)}^2 )
& = \int_{\R^3} \agl[x]^{-1-2\epsilon} \mathbb{E} \big( \agl[\dot B(\cdot,\omega), \sigma(\cdot) \Phi(x,\cdot)] \agl[\dot B(\cdot,\omega), \sigma(\cdot) \overline{\Phi}(x,\cdot)] \big) \dif{x} \\
& = \int_{\R^3} \agl[x]^{-1-2\epsilon} \int_D \sigma^2(y) \frac 1 {16\pi^2 |x-y|^2} \dif{y} \dif{x} \\
& \leq C \nrm[L^\infty(D)]{\sigma}^2 \int_{\R^3} \agl[x]^{-1-2\epsilon} \int_D |x-y|^{-2} \dif{y} \dif{x}.
\end{align*}
By arguments similar to the ones used in the proof of Lemma \ref{lemma:RkBoundedR3-SchroEqu2018} we arrive at
\begin{equation} \label{eq:Rksigma2Bounded-SchroEqu2018}
\mathbb{E} (\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot{B}_x)}^2) \leq C_D < +\infty,
\end{equation}
for some constant $C_D$ depending on $D$ but not on $k$. By the H\"older inequality applied to the probability measure, (\ref{eq:Rksigma2Bounded-SchroEqu2018}) gives
\begin{equation} \label{eq:RksigmaBoundedCD-SchroEqu2018}
\mathbb{E} (\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot{B}_x)}) \leq [ \mathbb{E} ( \nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot{B}_x)}^2 ) ]^{1/2} \leq \sq{C_D^{1/2}} < +\infty,
\end{equation}
for some constant $C_D$ independent of $k$. The inequality (\ref{eq:RksigmaBoundedCD-SchroEqu2018}) gives
$${\mathcal{R}_{k}}(\sigma \dot B_x) \in L_{-1/2-\epsilon}^2 \quad \textrm{~a.s.~}.$$
By replacing $\R^3$ with $D$ and deleting all the terms $\agl[x]^{-1-2\epsilon}$ in the derivations above, one arrives at $\mathbb{E} \nrm[L^2(D)]{{\mathcal{R}_{k}}(\sigma \dot{B}_x)} < +\infty$.
The proof is done.
\end{proof}
\subsection{The well-posedness of the \textbf{DP}} \label{subset:WellDefined-SchroEqu2018}
For a particular realization of the random sample $\omega \in \Omega$, the term $\dot B_x(\omega)$,
\jz{treated as a function of the spatial argument $x$, could be very rough.
The roughness of this term could make these classical second-order elliptic PDEs theories invalid to \eqref{eq:1}.}
Due to this reason, the notion of the {\em mild solution} is introduced for random PDEs (cf. \cite{bao2016inverse}).
In what follows, we adopt the mild solution in our problem setting, and we show that this mild solution and the corresponding far-field pattern are well-posed in a proper sense.
Reformulating \eqref{eq:1} into the Lippmann-Schwinger equation formally (cf. \cite{colton2012inverse}), we have
\begin{equation} \label{eq:LippSch-SchroEqu2018}
(I - {\mathcal{R}_{k}} V) u = \alpha \cdot u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x),
\end{equation}
where the term ${\mathcal{R}_{k}}(\sigma \dot B_x)$ is defined by (\ref{eq:RkSigmaBDefn-SchroEqu2018}). Recall that $u^{sc} = u - \alpha \cdot u^i$. From (\ref{eq:LippSch-SchroEqu2018}) we have
\begin{equation} \label{eq:uscDefn-SchroEqu2018}
(I - {\mathcal{R}_{k}} V) u^{sc} = \alpha {\mathcal{R}_{k}} V u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x).
\end{equation}
\begin{thm} \label{thm:MildSolUnique-SchroEqu2018}
\sq{When $k > k^*$, there exists a unique stochastic process $u^{sc}(\cdot,\omega) \colon \R^3 \to \mathbb C$ such that $u^{sc}(x)$ satisfies \eqref{eq:uscDefn-SchroEqu2018} a.s.\,, and ${u^{sc}(\cdot,\omega) \in L_{-1/2-\epsilon}^2 \textrm{~a.s.~}}$for any $\epsilon\in\mathbb{R}_+$.
Moreover, we have
\begin{equation} \label{thm:SolPosed-SchroEqu2018}
\nrm[L_{-1/2-\epsilon}^2]{u^{sc}(\cdot,\omega)}
\lesssim
\nrm[L_{1/2+\epsilon}^2]{\alpha V u^i}
+
\nrm[L_{1/2+\epsilon}^2]{f}
+
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot B_x)}.
\end{equation}
Then we call $u(x) := u^{sc} + \alpha \cdot u^i(x)$ the {\em mild solution} to the random scattering system \eqref{eq:1}.}
\end{thm}
\begin{proof}
\sq{By Lemmas \ref{lemma:RkBoundedR3-SchroEqu2018}, \ref{lemma:RkVBoundedR3-SchroEqu2018} and \ref{lemma:RkSigmaB-SchroEqu2018}, we see
$$F := \alpha {\mathcal{R}_{k}} V u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x) \in L_{-1/2-\epsilon}^2.$$
Note that $k > k^*$, so the term $\sum_{j=0}^\infty ({\mathcal{R}_{k}} V)^j$ is well-defined, thus the term $\sum_{j=0}^\infty ({\mathcal{R}_{k}} V)^j F$ belongs to $L_{-1/2-\epsilon}^2$. Because $\sum_{j=0}^\infty ({\mathcal{R}_{k}} V)^j = (I - {\mathcal{R}_{k}} V)^{-1}$, we see $(I - {\mathcal{R}_{k}} V)^{-1} F \in L_{-1/2-\epsilon}^2$. Let $u^{sc} := (I - {\mathcal{R}_{k}} V)^{-1} F \in L_{-1/2-\epsilon}^2$, then $u^{sc}$ is the unique solution of \eqref{eq:uscDefn-SchroEqu2018}. That is, the existence of the mild solution is proved The uniqueness of the mild solution follows from the invertibility of the operator $(I - {\mathcal{R}_{k}} V)^{-1}$.
From \eqref{eq:uscDefn-SchroEqu2018} and Lemmas \ref{lemma:AgmonEst-SchroEqu2018}-\ref{lemma:RkVBoundedR3-SchroEqu2018}, we have
\begin{align*}
\nrm[L_{-1/2-\epsilon}^2]{u^{sc}(\cdot,\omega)}
& = \nrm[L_{-1/2-\epsilon}^2]{(I - {\mathcal{R}_{k}} V)^{-1} (\alpha {\mathcal{R}_{k}} V u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x))} \\
& \leq \sum_{j \geq 0} \nrm[\mathcal L(L_{-1/2-\epsilon}^2,L_{-1/2-\epsilon}^2)]{{\mathcal{R}_{k}} V}^j \cdot \nrm[L_{-1/2-\epsilon}^2]{\alpha {\mathcal{R}_{k}} V u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x)} \\
& \leq C (
\nrm[L_{-1/2-\epsilon}^2]{\alpha {\mathcal{R}_{k}} V u^i}
+
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}} f}
+
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot B_x)}) \\
& \leq C (
\nrm[L_{1/2+\epsilon}^2]{\alpha V u^i}
+
\nrm[L_{1/2+\epsilon}^2]{f}
+
\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot B_x)}
).
\end{align*}
Therefore \eqref{thm:SolPosed-SchroEqu2018} is proved.
The proof is complete.}
\end{proof}
Next we show that the far-field pattern is well-defined in the $L^2$ sense. From \eqref{eq:uscDefn-SchroEqu2018} we derive that
\begin{align*}
u^{sc}
& = (I - {\mathcal{R}_{k}} V)^{-1} \big( \alpha {\mathcal{R}_{k}} V u^i - {\mathcal{R}_{k}} f - {\mathcal{R}_{k}}(\sigma \dot B_x) \big) \\
& = {\mathcal{R}_{k}} (I - V {\mathcal{R}_{k}})^{-1} (\alpha V u^i - f - \sigma \dot B_x).
\end{align*}
Therefore, we define the far-field pattern of the scattered wave $u^{sc}(x,k,d,\omega)$ formally in the following manner,
\begin{equation} \label{eq:uInftyDefn-SchroEqu2018}
u^\infty(\hat x,k,d,\omega) := \frac 1 {4\pi} \int_D e^{-ik\hat x \cdot y} (I - V {\mathcal{R}_{k}})^{-1} (\alpha V u^i - f - \sigma \dot B_y) \dif{y}, \quad \hat x \in \mathbb{S}^2.
\end{equation}
The another result concerning the \textbf{DP} is Theorem \ref{thm:FarFieldWellDefined-SchroEqu2018}, showing that $u^\infty(\hat x,k,d,\omega)$ is well-defined.
\begin{thm} \label{thm:FarFieldWellDefined-SchroEqu2018}
Define the far-field pattern of the mild solution as in \eqref{eq:uInftyDefn-SchroEqu2018}. When \jz{$k > k^*$}, there is a subset \jz{$\Omega_0 \subset \Omega$} with zero measure $\mathbb P (\Omega_0) = 0$, such that there holds
\[
u^\infty(\hat x,k,d,\omega) \in L^2(\mathbb{S}^2),\ \ {\,\forall\,} \omega \in \Omega \backslash \Omega_0.
\]
\end{thm}
\begin{proof}[Proof of Theorem \ref{thm:FarFieldWellDefined-SchroEqu2018}]
\jz{By Lemma \ref{lemma:RkVBounded-SchroEqu2018},
$$\nrm[\mathcal{L}(L^2(D), L^2(D))]{V {\mathcal{R}_{k}}} \leq C k^{-1} < 1$$
when $k$ is sufficiently large. Therefore we have,
\begin{align}
|u^\infty(\hat x)|^2
& \lesssim |D|^2 \cdot \int_D |\sum_{j \geq 0} (V {\mathcal{R}_{k}})^j (\alpha V u^i - f)|^2 \dif{y} \nonumber\\
& \ \ \ \ + \big| \int_D e^{-ik\hat x \cdot y} \sum_{j \geq 1} (V {\mathcal{R}_{k}})^j (\sigma \dot B_y) \dif{y} \big|^2 \nonumber\\
& \ \ \ \ + \big| \int_D e^{-ik\hat x \cdot y} \sigma \dot B_y \dif{y} \big|^2 \nonumber\\
& =: f_1(\hat x, k) + f_2(\hat x, k,\omega) + f_3(\hat x, k,\omega). \label{eq:a1}
\end{align}
We next derive estimates on each term $f_j ~(j=1,2,3)$ defined in \eqref{eq:a1}. For $f_1$, we have
\begin{align}
f_1(\hat x, k)
& \leq C |D|^2 \cdot ( \sum_{j \geq 0} k^{-j} \nrm[L^2(D)]{\alpha V u^i - f} )^2 \leq C |D|^2 (\nrm[L^2(D)]{V} + \nrm[L^2(D)]{f})^2. \label{eq:f1-SchroEqu2018}
\end{align}
For $f_2$, by utilizing \eqref{eq:VRkbdd-SchroEqu2018}, one can compute
\begin{equation} \label{eq:f2Inter-SchroEqu2018}
f_2(\hat x, k, \omega)
\leq C \int_D |\sum_{j \geq 0} (V {\mathcal{R}_{k}})^j V {\mathcal{R}_{k}}(\sigma \dot B_y)|^2 \dif{y} \leq C \big( \sum_{j \geq 0} k^{-j} \nrm[L^2(D)]{V {\mathcal{R}_{k}}(\sigma \dot B_y)} \big)^2.
\end{equation}
By virtue of the boundedness of the support of $V$, we can continue \eqref{eq:f2Inter-SchroEqu2018} as
\begin{equation} \label{eq:f2-SchroEqu2018}
f_2(\hat x, k, \omega) \leq C \big( \sum_{j \geq 0} k^{-j} \nrm[L_{-1/2-\epsilon}^2]{V {\mathcal{R}_{k}}(\sigma \dot B_y)} \big)^2 \leq C_V \nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot B_y)}^2.
\end{equation}}
By (\ref{eq:ItoIso-SchroEqu2018}), the expectation of $f_3(\hat x, k, \omega)$ is
\begin{equation} \label{eq:f3-SchroEqu2018}
\mathbb{E} f_3(\hat x, k, \omega) = \mathbb{E} |\agl[\dot B_y, e^{-ik\hat x \cdot y} \sigma(y)]|^2 = \int_D |\sigma(y)|^2 \dif{y}.
\end{equation}
Combining \eqref{eq:Rksigma2Bounded-SchroEqu2018}, \eqref{eq:a1}-\eqref{eq:f1-SchroEqu2018} and \eqref{eq:f2-SchroEqu2018}-\eqref{eq:f3-SchroEqu2018}, we arrive at
\begin{align}
\mathbb E |u^\infty(\hat x)|^2
& \leq C |D|^2 (\nrm[L^2(D)]{V} + \nrm[L^2(D)]{f})^2 + C_V \mathbb E (\nrm[L_{-1/2-\epsilon}^2]{{\mathcal{R}_{k}}(\sigma \dot B_y)}^2) + \int_D |\sigma(y)|^2 \dif{y} \nonumber\\
& \leq C < +\infty \label{eq:a2-SchroEqu2018}
\end{align}
for some positive constant $C$. From \eqref{eq:a2-SchroEqu2018} we arrive at
\begin{equation} \label{eq:a3-SchroEqu2018}
\mathbb E \int_{\mathbb{S}^2} |u^\infty(\hat x)|^2 \dif{S} \leq C < +\infty.
\end{equation}
Our conclusion follows from \eqref{eq:a3-SchroEqu2018} immediately.
\end{proof}
\sq{\section{Some asymptotic estimates} \label{sec:AsyEst-SchroEqu2018}}
\sq{This section is devoted to some preparations of the recovery of the variance function. To recovery $\sigma^2(x)$, only the passive far-field patterns are utilized.
Therefore, throughout this section, the $\alpha$ in \eqref{eq:1} is set to be 0. Motivated by \cite{caro2016inverse}, our recovery formula of the variance function is of the form
\begin{equation} \label{eq:example-SchroEqu2018}
\frac 1 {K} \int_{K}^{2K} \overline{u^\infty(\hat x,k,\omega)} \cdot u^\infty(\hat x,k+\tau,\omega) \dif{k}.
\end{equation}}
\sq{After expanding $u^\infty(\hat x,k,\omega)$ in the form of Neumann series, there will be several crossover terms in \eqref{eq:example-SchroEqu2018} which decay in different rates in terms of $K$. In this section, we focus on the asymptotic estimates of these terms, which pave the way to the recovery of $\sigma^2(x)$. The recovery of $\sigma^2(x)$ is presented in the next section.}
\sq{To start out, we write
\begin{equation} \label{eq:u1-SchroEqu2018}
u_1^\infty(\hat x,k,\omega) := u^\infty(\hat x,k,\omega) - \mathbb{E} u^\infty(\hat x,k).
\end{equation}
Note that $u_1^\infty$ is independent of the incident direction $d$. Assume that $k > k^*$, then the operator $(I - {\mathcal{R}_{k}} V)^{-1}$ has the Neumann expansion $\sum_{j=0}^{+\infty} ({\mathcal{R}_{k}} V)^j$. By \eqref{eq:uInftyDefn-SchroEqu2018} and \eqref{eq:u1-SchroEqu2018} we have
\begin{align}
u_1^\infty(\hat x,k,\omega)
\ = & \ \frac {-1} {4\pi} \sum_{j=0}^{+\infty} \int_D e^{-ik\hat x \cdot y} ({\mathcal{R}_{k}} V)^j (\sigma \dot B_y) \dif{y}, \quad \hat x \in \mathbb{S}^2 \nonumber\\
:= & \ \frac {-1} {4\pi} \big[ F_0(k,\hat x) + F_1(k,\hat x) \big], \label{eq:u1InftyDefn-SchroEqu2018}
\end{align}
where
\begin{equation} \label{eq:Fjkx-SchroEqu2018}
\left\{\begin{aligned}
F_0(k,\hat x,\omega) & := \int_D e^{-ik \hat x \cdot y} (\sigma \dot{B}_y) \dif{y}, \\
F_1(k,\hat x,\omega) & := \sum_{j \geq 1} \int_D e^{-ik \hat x \cdot y} (V {\mathcal{R}_{k}})^j (\sigma \dot{B}_y) \dif{y}.
\end{aligned}\right.
\end{equation}
Meanwhile, the expectation of the far-field pattern $\mathbb E u^\infty$ is
\begin{equation} \label{eq:u2InftyDefn-SchroEqu2018}
\mathbb E u^\infty(\hat x,k) = \frac {-1} {4\pi} \int_D e^{-ik\hat x \cdot y} (I - V {\mathcal{R}_{k}})^{-1} (f) \dif{y}, \quad \hat x \in \mathbb{S}^2.
\end{equation}}
\sq{
\begin{lem} \label{lemma:FarFieldGoToZero-SchroEqu2018}
We have
\[
\lim_{k \to +\infty} |\mathbb E u^\infty(\hat x,k)| = 0 \quad \text{uniformly in } \hat x \in \mathbb{S}^2.
\]
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:FarFieldGoToZero-SchroEqu2018}]
Due to the fact that $f \in L^\infty(D) \subset L^2(D)$, we know
\begin{equation} \label{eq:fApprox-SchroEqu2018}
{\,\forall\,} \epsilon > 0, {\,\exists\,} \varphi_\epsilon \in \mathscr{D}(D), \textrm{~s.t.~} \nrm[L^2(D)]{f-\varphi_\epsilon} < \epsilon / (2 |D|^{\frac 1 2}).
\end{equation}
Recall that $k > k^*$, so $(I - V {\mathcal{R}_{k}})^{-1}$ equals to $I + \sum_{j=1}^{+\infty} (V{\mathcal{R}_{k}})^j$.
By \eqref{eq:fApprox-SchroEqu2018} and Lemma \ref{lemma:RkVBounded-SchroEqu2018} and utilizing the stationary phase lemma, one can deduce as follows,
\begin{align}
|\mathbb E u^\infty(\hat x,k)|
& \lesssim \big| \int_D e^{-ik \hat x \cdot y} \varphi_\epsilon(y) \dif{y} \big| + \big| \int_D e^{-ik \hat x \cdot y} \big[ f(y) - \varphi_\epsilon(y) + \big( \sum_{j \geq 1} (V{\mathcal{R}_{k}})^j f \big) (y) \big] \dif{y} \big| \nonumber\\
& \lesssim \big| k^{-2} \int_D e^{-ik \hat x \cdot y} \cdot \Delta \varphi_\epsilon(y) \dif{y} \big| + |D|^{\frac 1 2} \cdot \nrm[L^2(D)]{f - \varphi_\epsilon + \sum_{j \geq 1} (V{\mathcal{R}_{k}})^j f } \nonumber\\
& \leq k^{-2} \cdot |D|^{\frac 1 2} \cdot \nrm[L^2(D)]{\Delta \varphi_\epsilon} + |D|^{\frac 1 2} \cdot \big( \epsilon/(2 |D|^{\frac 1 2}) + C \sum_{j \geq 1} k^{-j} \nrm[L^2(D)]{f} \big) \nonumber\\
& = k^{-2} \cdot |D|^{\frac 1 2} \nrm[L^2(D)]{\Delta \varphi_\epsilon} + \epsilon/2 + C (k-1)^{-1} \cdot \nrm[L^2(D)]{f}. \label{eq:u2InftyEst-SchroEqu2018}
\end{align}
Write $\mathcal K := \max\{ K_0, \frac 2 {\sqrt{\epsilon}} |D|^{\frac 1 4} \nrm[L^2(D)]{\Delta \varphi_\epsilon}^{\frac 1 2}, 1 + \frac {4C} \epsilon \nrm[L^2(D)]{f} \}$. From (\ref{eq:u2InftyEst-SchroEqu2018}) we have
$${\,\forall\,} k > \mathcal K, \quad |\mathbb E u^\infty(\hat x,k)| < \frac \epsilon 2 + \frac \epsilon 4 + \frac \epsilon 4 = \epsilon, \quad \text{uniformly for } \forall \hat x \in \mathbb{S}^2.$$
Since the $\epsilon$ is taken arbitrarily, the conclusion follows.
\end{proof}}
\sq{By substituting \eqref{eq:u1-SchroEqu2018}-\eqref{eq:u2InftyDefn-SchroEqu2018} into \eqref{eq:example-SchroEqu2018}, we obtain several crossover terms among $F_0$, $F_1$ and $\mathbb E u^\infty$.
The asymptotic estimates of these crossover terms are the main purpose of Sections \ref{subsec:AELeading-SchroEqu2018} and \ref{subsec:AEHigher-SchroEqu2018}.
Section \ref{subsec:AELeading-SchroEqu2018} focuses on the estimate of the leading order term while the estimates of the higher order terms are presented in Section \ref{subsec:AEHigher-SchroEqu2018}.}
\subsection{Asymptotic estimates of the leading order term} \label{subsec:AELeading-SchroEqu2018}
\jz{Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018} below is the asymptotic estimate of the crossover leading order term. By utilizing the ergodicity, the result of Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018} is also statistically stable. To prove Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018}, we need Lemmas \ref{lem:asconverg-SchroEqu2018}, \ref{lem:IsserlisThm-SchroEqu2018} and \ref{lemma:LeadingTermTechnical-SchroEqu2018}. Lemma \ref{lem:asconverg-SchroEqu2018} is the probabilistic foundation of our single-realization recovery result, and Lemma \ref{lem:IsserlisThm-SchroEqu2018} is called Isserlis' Theorem. In order to keep our arguments flowing, we postpone Lemma \ref{lemma:LeadingTermTechnical-SchroEqu2018} until we finish Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018}.}
\begin{lem} \label{lem:asconverg-SchroEqu2018}
Assume $X$ and $X_n ~(n=1,2,\cdots)$ be complex-valued random variables, then
$$X_n \to X \textrm{~a.s.~}
\quad\text{if and only if}\quad
\lim_{K_0 \to +\infty} P \big( \bigcup_{j \geq K_0} \{ |X_j - X| \geq \epsilon \} \big) = 0 ~{\,\forall\,} \epsilon > 0.$$
\end{lem}
\sq{The proof of Lemma \ref{lem:asconverg-SchroEqu2018} can be found in [\citen{dudley2002real}, Lemma 9.2.4].}
\begin{lem}[Isserlis' Theorem \cite{Michalowicz2009}] \label{lem:IsserlisThm-SchroEqu2018}
Suppose ${(X_{1},\dots, X_{2n})}$ is a zero-mean multi-variate normal random vector, then
\[
\mathbb{E} (X_1 X_2 \cdots X_{2n}) = \sum\prod \mathbb{E} (X_i X_j),
\quad
\mathbb{E} (X_1 X_2 \cdots X_{2n-1}) = 0.
\]
Specially,
\[
\mathbb{E} (\,X_{1}X_{2}X_{3}X_{4}\,)
= \mathbb{E} (X_{1}X_{2})\, \mathbb{E} (X_{3}X_{4}) + \mathbb{E} (X_{1}X_{3})\, \mathbb{E} (X_{2}X_{4}) + \mathbb{E} (X_{1}X_{4})\, \mathbb{E} (X_{2}X_{3}).
\]
\end{lem}
\jz{The proof of Lemma \ref{lem:IsserlisThm-SchroEqu2018} can be found in \cite{Michalowicz2009}.} In what follows, $\widehat{\varphi}$ denotes the Fourier transform of the function $\varphi$ defined as
\begin{equation*}
\widehat{\varphi}(\xi) := (2\pi)^{-n/2} \int_{\R^3} e^{-i x \cdot \xi} \varphi(x) \dif{x}, \quad \xi \in \R^n.
\end{equation*}
For the notational convenience, we use ``$\{K_j\} \in P(t)$'' to mean that the sequence $\{K_j\}_{j \in \mathbb{N}^+}$ satisfies $K_j \geq C j^t ~(j \in \mathbb{N}^+)$ for some fixed constant $C > 0$. Throughout the following context, $\gamma$ stands for any fixed positive real number. Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018} gives the asymptotic estimates of the crossover leading order term.
\begin{lem} \label{lemma:LeadingTermErgo-SchroEqu2018}
Write
\begin{equation*}
X_{0,0}(K,\tau,\hat x,\omega) = \frac 1 K \int_K^{2K} \overline{F_0(k,\hat x,\omega)} \cdot F_0(k+\tau,\hat x,\omega) \dif{k}.
\end{equation*}
Assume $\{K_j\} \in P(2+\gamma)$, then for any $\tau > 0$, we have
\begin{equation*}
\lim_{j \to +\infty} X_{0,0}(K_j,\tau,\hat x,\omega) = (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x) \quad \textrm{~a.s.~}.
\end{equation*}
\end{lem}
We may denote $X_{0,0}(K,\tau,\hat x,\omega)$ as $X_{0,0}$ for short if it is clear in the context.
\begin{proof}[Proof of Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018}]
We have
\begin{align}
& \ \mathbb{E} \big( \overline{F_0(k,\hat x,\omega)} F_0(k+\tau,\hat x,\omega) \big) \nonumber\\
= & \ \mathbb{E} \big( \int_{D_y} e^{ik \hat x \cdot y} \sigma(y) \dif{B_y} \cdot \int_{D_z} e^{-i(k+\tau) \hat x \cdot z} \sigma(z) \dif{B_z} \big) \nonumber\\
= & \ \int_{D} e^{ik \hat x \cdot y} e^{-i(k+\tau) \hat x \cdot y} \sigma(y) \sigma(y) \dif{y} = (2\pi)^{3/2}\, \widehat{\sigma^2} (\tau \hat x). \label{eq:I0-SchroEqu2018}
\end{align}
From \eqref{eq:I0-SchroEqu2018} we conclude that
\begin{align*}
\mathbb{E} ( X_{0,0} )
= \frac 1 K \int_K^{2K} \mathbb{E} \big( \overline{F_0(k,\hat x,\omega)} F_0(k+\tau,\hat x,\omega) \big) \dif{k}
= (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x).
\end{align*}
By Isserlis' Theorem and \eqref{eq:I0-SchroEqu2018}, and note that $\overline{F_j(k,\hat x,\omega)} = F_j(-k,\hat x,\omega)$, one can compute
\begin{align}
& \mathbb{E} \big( | X_{0,0} - (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x) |^2 \big) \nonumber\\
= & \frac 1 {K^2} \int_K^{2K} \int_K^{2K} \mathbb{E} \Big( \overline{F_0(k_1,\hat x,\omega)} F_0(k_1+\tau,\hat x,\omega) F_0(k_2,\hat x,\omega) \overline{F_0(k_2+\tau,\hat x,\omega)} \Big) \dif{k_1} \dif{k_2} \nonumber\\
& - (2\pi)^3 |\widehat{\sigma^2} \big( \tau \hat x \big)|^2 - (2\pi)^3 |\widehat{\sigma^2} \big( \tau \hat x \big)|^2 + (2\pi)^3 |\widehat{\sigma^2} \big( \tau \hat x \big)|^2 \hspace{1.5cm} (\text{by } \eqref{eq:I0-SchroEqu2018}) \nonumber\\
= & \frac 1 {K^2} \int_K^{2K} \int_K^{2K} \mathbb{E} \big( \overline{F_0(k_1,\hat x,\omega)} F_0(k_1+\tau,\hat x,\omega) \big) \cdot \mathbb{E} \big( F_0(k_2,\hat x,\omega) \overline{F_0(k_2+\tau,\hat x,\omega)} \big) \nonumber\\
& + \mathbb{E} \big( \overline{F_0(k_1,\hat x,\omega)} F_0(k_2,\hat x,\omega) \big) \cdot \mathbb{E} \big( F_0(k_1+\tau,\hat x,\omega) \overline{F_0(k_2+\tau,\hat x,\omega)} \big) \nonumber\\
& + \mathbb{E} \big( \overline{F_0(k_1,\hat x,\omega)} F_0(-k_2-\tau,\hat x,\omega) \big) \cdot \mathbb{E} \big( \overline{F_0(-k_1-\tau,\hat x,\omega)} F_0(k_2,\hat x,\omega) \big) \dif{k_1} \dif{k_2} \nonumber\\
& - (2\pi)^3 |\widehat{\sigma^2} \big( \tau \hat x \big)|^2 \nonumber\\
= & \frac {(2\pi)^3} {K^2} \int\limits_K^{2K} \int\limits_K^{2K} |\widehat{\sigma^2}((k_2 - k_1) \hat x)|^2 \dif{k_1} \dif{k_2} + \frac {(2\pi)^3} {K^2} \int\limits_K^{2K} \int\limits_K^{2K} |\widehat{\sigma^2}((k_1 + k_2 + \tau) \hat x)|^2 \dif{k_1} \dif{k_2}. \label{eq:X00Square-SchroEqu2018}
\end{align}
\jz{ Note that $|\widehat{\sigma^2} \big( (k_1 - k_2) \hat x \big)| = |\widehat{\sigma^2} \big( -(k_1 - k_2) \hat x \big)|$.} Combining (\ref{eq:X00Square-SchroEqu2018}) and Lemma \ref{lemma:LeadingTermTechnical-SchroEqu2018}, we have
\begin{equation} \label{eq:X00Bdd-SchroEqu2018}
\mathbb{E} \big( | X_{0,0} - (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x) |^2 \big) = \mathcal{O}(K^{-1/2}), \quad K \to +\infty.
\end{equation}
For any integer $K_0 > 0$, by Chebyshev's inequality and (\ref{eq:X00Bdd-SchroEqu2018}) we have
\begin{align}
& P \big( \bigcup_{j \geq K_0} \{ | X_{0,0}(K_j) - (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x) | \geq \epsilon \} \big) \leq \frac 1 {\epsilon^2} \sum_{j \geq K_0} \mathbb{E} \big( | X_{0,0}(K_j) - (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x) |^2 \big) \nonumber\\
\lesssim & \frac 1 {\epsilon^2} \sum_{j \geq K_0} K_j^{-1/2} = \frac 1 {\epsilon^2} \sum_{j \geq K_0} j^{-1-\gamma/2} \leq \frac 1 {\epsilon^2} \int_{K_0}^{+\infty} (t-1)^{-1-\gamma/2} \dif{t} = \frac 2 {\epsilon^2 \gamma} (K_0-1)^{-\gamma/2}. \label{eq:PX00Epsilon-SchroEqu2018}
\end{align}
Here $X_{0,0}(K_j)$ stands for $X_{0,0}(K_j, \tau, \hat x,\omega)$. By Lemma \ref{lem:asconverg-SchroEqu2018}, formula (\ref{eq:PX00Epsilon-SchroEqu2018}) implies that for any fixed $\tau \geq 0$ and fixed $\hat x \in \mathbb{S}^2$, we have
$$X_{0,0}(K_j,\tau,\hat x,\omega) \to (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x)\quad \textrm{~a.s.~}.$$
The proof is done.
\end{proof}
Lemma \ref{lemma:LeadingTermTechnical-SchroEqu2018} plays a critical role in the estimates of the leading order term.
\begin{lem} \label{lemma:LeadingTermTechnical-SchroEqu2018}
\jz{Assume that} $\tau \geq 0$ is fixed, then $\exists K_0 > \tau$, and $K_0$ is independent of $\hat x$, such that for all $K > K_0$, we have the following estimates:
\begin{align}
\frac {(2\pi)^3} {K^2} \int_K^{2K} \int_K^{2K} \big| \widehat{\sigma^2}((k_1 - k_2) \hat x) \big|^2 \dif{k_1} \dif{k_2} & \leq CK^{-1/2}, \label{eq:F0F0TermOne-SchroEqu2018} \\
\frac {(2\pi)^3} {K^2} \int_K^{2K} \int_K^{2K} \big| \widehat{\sigma^2}((k_1 + k_2 + \tau) \hat x) \big|^2 \dif{k_1} \dif{k_2} & \leq CK^{-1/2}, \label{eq:F0F0TermTwo-SchroEqu2018}
\end{align}
for some constant $C$ independent of $\tau$ and $\hat x$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:LeadingTermTechnical-SchroEqu2018}]
Note that for every $x \in \R^3$, we have
\begin{equation*}
|\widehat{\sigma^2}(x)|^2 \simeq \big| \int_{\R^3} e^{-i x \cdot \xi} \sigma^2(\xi) \dif{\xi} \big|^2
\leq \big( \int_{\R^3} |\sigma^2(\xi)| \dif{\xi} \big)^2
\leq \nrm[L^\infty(D)]{\sigma}^4 \cdot |D|^2.
\end{equation*}
To conclude (\ref{eq:F0F0TermOne-SchroEqu2018}), we make a change of variable,
\begin{equation*}
\left\{\begin{aligned}
s & = k_1 - k_2, \\
t & = k_2.
\end{aligned}\right.
\end{equation*}
Write \jz{$Q = \{(s,t) \in \R^2 \,\big|\, K \leq s+t \leq 2K,\, K \leq t \leq 2K \}$}. $Q$ is illustrated as in Figure \ref{fig:D-SchroEqu2018}.
\begin{figure}
\caption{Illustration of $Q$}
\label{fig:D-SchroEqu2018}
\end{figure}
Recall that $\mathop{\rm supp} \sigma \subseteq D$, so we have
\begin{align}
& \frac 1 {K^2} \int_K^{2K} \int_K^{2K} |\widehat{\sigma^2}((k_1 - k_2) \hat x)|^2 \dif{k_1} \dif{k_2}
= \frac 1 {K^2} \iint_Q \big| \widehat{\sigma^2}(s \hat x) \big|^2 \dif{s} \dif{t}
\nonumber\\
= \ & \frac 1 {K^2} \int_{-K}^0 (K+s) |\widehat{\sigma^2}(s \hat x)|^2 \dif{s} + \frac 1 {K^2} \int_0^{K} (K-s) |\widehat{\sigma^2}(s \hat x)|^2 \dif{s} \nonumber\\
\simeq \ & \int_0^1 \Big( \int_D e^{-iKs \hat x \cdot y} \sigma^2(y) \dif{y} \cdot \int_D e^{iKs \hat x \cdot z} \sigma^2(z) \dif{z} \Big) \dif{s} \nonumber\\
= \ & \int_{(D \times D) \backslash E_\epsilon} \Big( \int_0^1 e^{iK(\hat x \cdot z - \hat x \cdot y)s} \dif{s} \Big) \sigma^2(y) \sigma^2(z) \dif{y} \dif{z} \nonumber\\
& \quad + \int_{E_\epsilon} \Big( \int_0^1 e^{iK(\hat x \cdot z - \hat x \cdot y)s} \dif{s} \Big) \sigma^2(y) \sigma^2(z) \dif{y} \dif{z} \nonumber\\
=: & A_1 + A_2, \label{eq:sigma2Inter-SchroEqu2018}
\end{align}
where $E_\epsilon := \{ (y,z) \in D \times D ; |\hat x \cdot z - \hat x \cdot y| < \epsilon \}$. We first estimate $A_1$,
\begin{align}
|A_1|
& = \Big| \int_{(D \times D) \backslash E_\epsilon} \Big( \int_0^1 e^{iK(\hat x \cdot z - \hat x \cdot y)s} \dif{s} \Big) \sigma^2(y) \sigma^2(z) \dif{y} \dif{z} \Big| \nonumber\\
& \leq \int_{(D \times D) \backslash E_\epsilon} \Big| \frac {e^{iK(\hat x \cdot z - \hat x \cdot y)} - 1} {iK(\hat x \cdot z - \hat x \cdot y)} \sigma^2(y) \sigma^2(z) \Big| \dif{y} \dif{z} \nonumber\\
& \leq \frac 2 {K\epsilon} \nrm[L^\infty(D)]{\sigma}^4 \int_{D \times D} 1 \dif{y} \dif{z} = \frac {2|D|^2} {K\epsilon} \nrm[L^\infty(D)]{\sigma}^4. \label{eq:sigma2InterA1-SchroEqu2018}
\end{align}
\jz{Recall that $\text{diam}\,D < +\infty$ and that the problem setting is in $\R^3$. We can estimate $A_2$ as}
\begin{align}
|A_2|
& \leq \nrm[L^\infty(D)]{\sigma}^4 \int_{E_\epsilon} 1 \dif{y} \dif{z} \nonumber\\
& = \nrm[L^\infty(D)]{\sigma}^4 \int_D \big( \int_{y \in D \,,\, |\hat x \cdot z - \hat x \cdot y| < \epsilon} 1 \dif{y} \big) \dif{z}
\nonumber\\
& \leq \nrm[L^\infty(D)]{\sigma}^4 \int_D 2\epsilon (\text{Diam}\,D)^2 \dif{z} \nonumber\\
& \leq 2\nrm[L^\infty(D)]{\sigma}^4 (\text{Diam}\,D)^2 |D| \cdot \epsilon. \label{eq:sigma2InterA2-SchroEqu2018}
\end{align}
Set $\epsilon = K^{-1/2}$. By \eqref{eq:sigma2Inter-SchroEqu2018}-\eqref{eq:sigma2InterA2-SchroEqu2018}, we arrive at
$$\frac 1 {K^2} \int\limits_K^{2K} \int\limits_K^{2K} |\widehat{\sigma^2}((k_1 - k_2) \hat x)|^2 \dif{k_1} \dif{k_2} \leq C K^{-1/2},$$
for some constant $C$ independent of $\hat x$.
Now we prove \eqref{eq:F0F0TermTwo-SchroEqu2018}. Similarly, we make a change of variable:
\begin{equation*}
\left\{\begin{aligned}
s & = k_1 + k_2 + \tau, \\
t & = k_2.
\end{aligned}\right.
\end{equation*}
Write \jz{$Q' = \{(s,t) \in \R^2 \,\big|\, K \leq s-t-\tau \leq 2K,\, K \leq t \leq 2K \}$}. One can compute
\begin{align*}
& \frac 1 {K^2} \int_K^{2K} \int_K^{2K} | \widehat{\sigma^2}((k_1 + k_2 + \tau) \hat x) |^2 \dif{k_1} \dif{k_2} = \frac 1 {K^2} \iint_{Q'} | \widehat{\sigma^2}(s \hat x) |^2 \dif{s} \dif{s} \\
= & \frac 1 {K^2} \int_{2K+\tau}^{3K+\tau} (s-2K-\tau) | \widehat{\sigma^2}(s \hat x) |^2 \dif{s} + \frac 1 {K^2} \int_{3K+\tau}^{4K+\tau} (4K+\tau-s) | \widehat{\sigma^2}(s \hat x) |^2 \dif{s} \\
\leq & \frac 2 {K} \int_{2K-\tau}^{2K+\tau} | \widehat{\sigma^2}(s \hat x) |^2 \dif{s} = 2 \int_{2+\tau/K}^{4+\tau/K} | \widehat{\sigma^2}(Ks \hat x) |^2 \dif{s}.
\end{align*}
Thus when $K > \tau$,
\begin{equation} \label{eq:sigma2InterTau-SchroEqu2018}
\frac 1 {K^2} \int_K^{2K} \int_K^{2K} | \widehat{\sigma^2}((k_1 + k_2 + \tau) \hat x) |^2 \dif{k_1} \dif{k_2} \leq 2 \int_2^5 | \widehat{\sigma^2}(Ks \hat x) |^2 \dif{s}.
\end{equation}
Following the same manner as in \eqref{eq:sigma2Inter-SchroEqu2018}-\eqref{eq:sigma2InterA2-SchroEqu2018}, from \eqref{eq:sigma2InterTau-SchroEqu2018} we arrive at \eqref{eq:F0F0TermTwo-SchroEqu2018}. The proof is done.
\end{proof}
\subsection{Asymptotic estimates of higher order terms} \label{subsec:AEHigher-SchroEqu2018}
The asymptotic estimates of the higher order terms are presented in Lemma \ref{lemma:HOT-SchroEqu2018}.
\begin{lem} \label{lemma:HOT-SchroEqu2018}
For every $\hat x_1$, $\hat x_2 \in \mathbb{S}^2$ and every $k_1$, $k_2 \geq k$, we have the following estimates ($j = 0,1$) as $k \to +\infty$,
\begin{align}
\big| \mathbb{E} \big( \overline{F_j(k_1,\hat x_1,\omega)} \cdot F_1(k_2,\hat x_2,\omega) \big) \big| & = \mathcal{O}(k^{-1}), \label{eq:hotFjF1-SchroEqu2018}\\
\big| \mathbb{E} \big( F_j(k_1,\hat x_1,\omega) \cdot F_1(k_2,\hat x_2,\omega) \big) \big| & = \mathcal{O}(k^{-1}). \label{eq:hotFjF1Conju-SchroEqu2018}
\end{align}
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:HOT-SchroEqu2018}]
The proof of formulas \eqref{eq:hotFjF1Conju-SchroEqu2018} is similar to that of \eqref{eq:hotFjF1-SchroEqu2018}, so we only present the proof of \eqref{eq:hotFjF1-SchroEqu2018}. In this proof, we may drop the arguments $k$, $\hat x$ or $\omega$ from $F_j$ if it is clear in the context. For the notational convenience, we write
\begin{align*}
G_j(k,\hat x,\omega) & := \int_D e^{-ik \hat x \cdot y} (V {\mathcal{R}_{k}})^j (\sigma \dot{B}_y) \dif{y},
\\
r_j(k,\hat x,\omega) & := \sum_{s \geq j} G_s(k,\hat x,\omega),
\end{align*}
for $j = 0,1,\cdots$. To prove \eqref{eq:hotFjF1-SchroEqu2018} for the case where $j = 0$, we first show that
\begin{equation} \label{eq:hotF0Fj-SchroEqu2018}
\mathbb{E} \big( \overline{G_0(k_1,\hat x_1,\omega)} \cdot G_j(k_2,\hat x_2,\omega) \big) = \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^j \big( e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2 \big) \dif{z}, \quad j \geq 1.
\end{equation}
This can be seen from the following computation
\begin{align}
& \ \mathbb{E} \big( \overline{G_0(k_1,\hat x_1,\omega)} \cdot G_j(k_2,\hat x_2,\omega) \big) \nonumber\\
= & \ \mathbb{E} \big( \int_D e^{ik_1 \hat x_1 \cdot y} \sigma(y) \dif{B_y} \cdot \int_D \big[ e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^{j-1} ( V(\cdot) \int_{D_s} \Phi(\cdot,s) \sigma(s) \dif{B_s} ) \big] \dif{z} \big) \nonumber\\
= & \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^{j-1} \Big\{ V(\cdot) \,\mathbb{E} \big[ \int_{D_y} e^{ik_1 \hat x_1 \cdot y} \sigma(y) \dif{B_y} \cdot \int_{D_s} \Phi(\cdot,s) \sigma(s) \dif{B_s} \big] \Big\} \dif{z} \nonumber\\
= & \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^{j-1} \big( V(\cdot) \mathcal{R}_{k_2}(e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2) \big) \dif{z} \nonumber\\
= & \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^j ( e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2 ) \dif{z}. \label{eq:hotF0FjInter-SchroEqu2018}
\end{align}
From \eqref{eq:hotF0FjInter-SchroEqu2018}, equality \eqref{eq:hotF0Fj-SchroEqu2018} is proved. Using \eqref{eq:hotF0Fj-SchroEqu2018} and Lemma \ref{lemma:RkVBounded-SchroEqu2018}, we have
\begin{align*}
& \ \big| \mathbb{E} \big( \overline{F_0(k_1,\hat x_1,\omega)} \cdot F_1(k_2,\hat x_2,\omega) \big) \big| \nonumber\\
\leq & \ \sum_{j \geq 1} \big| \mathbb{E} \big( G_0(k_1,\hat x_1,\omega) \cdot \overline{G_j(k_2,\hat x_2,\omega)} \big) \big| \nonumber\\
= & \ \sum_{j \geq 1} \Big| \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^j \big( e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2 \big) \dif{z} \Big| \nonumber\\
\leq & \ |D|^{1/2} \cdot \sum_{j \geq 1} \nrm[L^2(D)]{ (V \mathcal{R}_{k_2})^j \big( e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2 \big) } \nonumber\\
\leq & \ C |D|^{1/2} \cdot \sum_{j \geq 1} k_2^{-j} \nrm[L^2(D)]{e^{ik_1 \hat x_1 \cdot (\cdot)} \sigma^2} = \mathcal{O}(k_2^{-1}), \quad k \to +\infty.
\end{align*}
\sq{To prove \eqref{eq:hotFjF1-SchroEqu2018} for the case where $j = 1$, we split $\mathbb{E} (\overline{F_1} F_1)$ into four terms,
\begin{equation} \label{eq:FGr-SchroEqu2018}
\mathbb{E} (\overline{F_1} F_1) = \mathbb{E} (\overline{G_1} G_1) + \mathbb{E} (\overline{r_1} r_2) - \mathbb{E} (\overline{r_2} r_2) + \mathbb{E} (\overline{r_2} r_1).
\end{equation}
We estimate these four terms on the right-hand-side of \eqref{eq:FGr-SchroEqu2018} one by one.}
First, we estimate
\begin{align}
& \ \big| \mathbb{E} \big( \overline{G_1(k_1,\hat x_1,\omega)} \cdot G_1(k_2,\hat x_2,\omega) \big) \big| \nonumber\\
= & \ \Big| \iint_{D_y \times D_z} e^{-ik_1 \hat x_1 \cdot y} e^{ik_2 \hat x_2 \cdot z} V(y) \overline V(z) \cdot \mathbb{E} \big[ \int_{D_s} \Phi(y,s) \sigma(s) \dif{B_s} \cdot \int_{D_t} \overline \Phi(z,t) \sigma(t) \dif{B_t} \big] \dif{y} \dif{z} \Big| \nonumber\\
= & \ \Big| \iint_{D_y \times D_z} e^{-ik_1 \hat x_1 \cdot y} e^{ik_2 \hat x_2 \cdot z} V(y) \overline V(z) \cdot \big[ \int_{D_s} \Phi(y,s) \sigma(s) \overline \Phi(z,s) \sigma(s) \dif{s} \big] \dif{y} \dif{z} \Big| \nonumber\\
= & \ \Big| \int_{D} \sigma^2(s) \cdot \mathcal{R}_{k_1} V( e^{-ik_1 \hat x_1 \cdot (\cdot)} )(s) \cdot \overline{ \mathcal{R}_{k_2} V ( e^{-ik_2 \hat x_2 \cdot (\cdot)} )(s) } \dif{s} \Big| \nonumber\\
\leq & \ C k_1^{-1} k_2^{-1} \nrm[L^\infty(D)]{\sigma}^2 \quad\big( \text{Lemma \ref{lemma:RkVBounded-SchroEqu2018}} \big) \nonumber\\
= & \ \mathcal{O}(k_1^{-1} k_2^{-1}), \quad k \to +\infty. \label{eq:hotG1G1-SchroEqu2018}
\end{align}
Then we estimate
\begin{align}
& \ \big| \mathbb{E} \big( \overline{r_1(k_1,\hat x_1,\omega)} \cdot r_2(k_2,\hat x_2,\omega) \big) \big|
\leq \mathbb{E} \Big( \sum_{j \geq 1} \big| G_j(k_1,\hat x_1,\omega) \big| \times \sum_{\ell \geq 2} \big| G_\ell(k_2,\hat x_2,\omega) \big| \Big) \nonumber\\
= & \ \mathbb{E} \Big( \sum_{j \geq 1} \big| \int_D e^{-ik_1 \hat x_1 \cdot y} (V \mathcal{R}_{k_1})^j (\sigma \dot{B}_y) \dif{y} \big| \times \sum_{\ell \geq 2} \big| \int_D e^{-ik_2 \hat x_2 \cdot z} (V \mathcal{R}_{k_2})^\ell (\sigma \dot{B}_z) \dif{z} \big| \Big) \nonumber\\
= & \ \nrm[L^\infty(D)]{V}^2 |D| \cdot \mathbb{E} \Big( \sum_{j \geq 0} \nrm[L^2(D)]{(\mathcal{R}_{k_1} V)^j [\mathcal{R}_{k_1}(\sigma \dot{B})]} \times \sum_{\ell \geq 1} \nrm[L^2(D)]{(\mathcal{R}_{k_2} V)^\ell [\mathcal{R}_{k_2}(\sigma \dot{B})]} \Big) \nonumber\\
\leq & \ C \nrm[L^\infty(D)]{V}^2 |D| \cdot \mathbb{E} \Big( \sum_{j \geq 0} \big( k_1^{-j} \nrm[L^2(D)]{\mathcal{R}_{k_1}(\sigma \dot{B})} \big) \times \sum_{\ell \geq 1} \big( k_2^{-\ell} \nrm[L^2(D)]{\mathcal{R}_{k_2}(\sigma \dot{B})} \big) \Big) \nonumber\\
\leq & \ \nrm[L^\infty(D)]{V}^2 |D| \cdot \frac {k_1} {k_1-1} \cdot \frac 1 {k_2-1} \cdot \frac 1 2 \mathbb{E} \big( \nrm[L^2(D)]{\mathcal{R}_{k_1}(\sigma \dot{B})}^2 + \nrm[L^2(D)]{\mathcal{R}_{k_2}(\sigma \dot{B})}^2 \big). \label{eq:hotr1r2Inter1-SchroEqu2018}
\end{align}
Utilizing \eqref{eq:Rksigma2Bounded-SchroEqu2018}, we obtain
\begin{equation}
\mathbb{E} \big( \nrm[L^2(D)] {{\mathcal{R}_{k}}(\sigma \dot{B})}^2 \big)
\leq C \mathbb{E} \big( \nrm[L_{-1/2-\epsilon}^2] {{\mathcal{R}_{k}}(\sigma \dot{B})}^2 \big) \leq C_D < +\infty. \label{eq:hotr1r2Inter2-SchroEqu2018}
\end{equation}
From (\ref{eq:hotr1r2Inter1-SchroEqu2018})-(\ref{eq:hotr1r2Inter2-SchroEqu2018}) we arrive at
\begin{equation} \label{eq:hotr1r2-SchroEqu2018}
\big| \mathbb{E} \big( \overline{r_1(k_1,\hat x_1,\omega)} \cdot r_2(k_2,\hat x_2,\omega)\big) \big| \leq \mathcal{O}(k_2^{-1}), \quad k \to +\infty.
\end{equation}
Mimicking (\ref{eq:hotr1r2Inter1-SchroEqu2018})-(\ref{eq:hotr1r2Inter2-SchroEqu2018}), one can obtain
\begin{equation} \label{eq:hotr2r1-SchroEqu2018}
\big| \mathbb{E} \big( \overline{r_2(k_1,\hat x_1,\omega)} \cdot r_1(k_2,\hat x_2,\omega) \big) \big| \leq \mathcal{O}(k_1^{-1}), \quad k \to +\infty.
\end{equation}
By modify $\sum_{j \geq 0} k_1^{-j}$ to $\sum_{j \geq 1} k_1^{-j}$ in (\ref{eq:hotr1r2Inter1-SchroEqu2018}), one can conclude
\begin{equation} \label{eq:hotr2r2-SchroEqu2018}
\big| \mathbb{E} \big( \overline{r_2(k_1,\hat x_1,\omega)} \cdot r_2(k_2,\hat x_2,\omega) \big) \big| \leq \mathcal{O}(k_1^{-1}k_2^{-1}), \quad k \to +\infty.
\end{equation}
Combining \eqref{eq:FGr-SchroEqu2018}-\eqref{eq:hotG1G1-SchroEqu2018} and \eqref{eq:hotr1r2-SchroEqu2018}-\eqref{eq:hotr2r2-SchroEqu2018}, we arrive at \eqref{eq:hotFjF1-SchroEqu2018} for the case where $j = 1$. The proof is complete.
\end{proof}
Lemma \ref{lemma:HOTErgo-SchroEqu2018} is the ergodic version of Lemma \ref{lemma:HOT-SchroEqu2018}.
\begin{lem} \label{lemma:HOTErgo-SchroEqu2018}
Write
\begin{align*}
X_{p,q}(K,\tau,\hat x,\omega) & = \frac 1 K \int_K^{2K} \overline{F_p(k,\hat x,\omega)} \cdot F_q(k+\tau,\hat x,\omega) \dif{k}, \ \ \text{for} \ \ (p,q) \in \{ (0,1), (1,0), (1,1) \}.
\end{align*}
Then for any $\hat x \in \mathbb{S}^2$ and any $\tau \geq 0$, we have the following estimates as $K \to +\infty$,
\begin{align}
\big| \mathbb{E} (X_{p,q}(K,\tau,\hat x,\omega)) \big| & = \mathcal{O}(K^{-1}), \ \mathbb{E} (|X_{p,q}(K,\tau,\hat x,\omega)|^2) = \mathcal{O}(K^{-5/4}),
\label{eq:hotF0F1Ergo-SchroEqu2018} \\
\big| \mathbb{E} (X_{1,1}(K,\tau,\hat x,\omega)) \big| & = \mathcal{O}(K^{-1}), \ \mathbb{E} (|X_{1,1}(K,\tau,\hat x,\omega)|^2) = \mathcal{O}(K^{-2}), \label{eq:hotF1F1Ergo-SchroEqu2018}
\end{align}
for $(p,q) \in \{ (0,1), (1,0) \}$. Let $\{K_j\} \in P(4/5+\gamma)$. Then for any $\tau \geq 0$, we have
\begin{equation} \label{eq:HOTErgoToZero-SchroEqu2018}
\lim_{j \to +\infty} X_{p,q}(K_j,\tau,\hat x,\omega) = 0 \quad \textrm{~a.s.~},
\end{equation}
for every $(p,q) \in \{ (0,1), (1,0), (1,1) \}$.
\end{lem}
We may denote $X_{p,q}(K,\tau,\hat x,\omega)$ as $X_{p,q}$ for short if it is clear in the context.
\begin{proof}[Proof of Lemma \ref{lemma:HOTErgo-SchroEqu2018}] According to Lemma \ref{lemma:HOT-SchroEqu2018}, we have
\begin{align}
\mathbb{E} \big( X_{0,1} \big)
& = \frac 1 K \int_K^{2K} \mathbb{E} \big( \overline{F_0(k,\hat x,\omega)} \cdot F_1(k+\tau,\hat x,\omega) \big) \dif{k} \nonumber\\
& = \mathcal{O}(K^{-1}), \quad K \to +\infty. \label{eq:hotF0F1Ergo1-SchroEqu2018}
\end{align}
By (\ref{eq:I0-SchroEqu2018}), Isserlis' Theorem and Lemma \ref{lemma:LeadingTermTechnical-SchroEqu2018}, we compute the secondary moment of $X_{0,1}$ as
\begin{align}
& \ \mathbb{E} \big( | X_{0,1} |^2 \big) \nonumber\\
= & \ \frac 1 {K^2} \int_K^{2K} \int_K^{2K} \mathbb{E} \big( F_0(k_1,\hat x,\omega) \overline{F_1(k_1+\tau,\hat x,\omega)} \big) \cdot \mathbb{E} \big( \overline{F_0(k_2,\hat x,\omega)} F_1(k_2+\tau,\hat x,\omega) \big) \nonumber\\
& \ + \mathbb{E} \big( F_0(k_1,\hat x,\omega) \overline{F_0(k_2,\hat x,\omega)} \big) \cdot \mathbb{E} \big( \overline{F_1(k_1+\tau,\hat x,\omega)} F_1(k_2+\tau,\hat x,\omega) \big) \nonumber\\
& \ + \mathbb{E} \big( F_0(k_1,\hat x,\omega) F_1(k_2+\tau,\hat x,\omega) \big) \cdot \mathbb{E} \big( \overline{F_1(k_1+\tau,\hat x,\omega)} \, \overline{F_0(k_2,\hat x,\omega)} \big) \dif{k_1} \dif{k_2} \nonumber\\
= & \ \frac 1 {K^2} \int_K^{2K} \int_K^{2K} \mathcal{O}(K^{-2}) + (2\pi)^{3/2} \widehat{\sigma^2} ((k_1-k_2) \hat x) \cdot \mathcal{O}(K^{-1}) + \mathcal{O}(K^{-2}) \dif{k_1} \dif{k_2} \nonumber\\
= & \ \mathcal{O}(K^{-1/4}) \cdot \mathcal{O}(K^{-1}) + \mathcal{O}(K^{-2}) \quad(\text{H\"older ineq. and Lemma } \ref{lemma:LeadingTermTechnical-SchroEqu2018}) \nonumber\\
= & \ \mathcal{O}(K^{-5/4}), \quad K \to +\infty. \label{eq:hotF0F1Ergo2-SchroEqu2018}
\end{align}
From \eqref{eq:hotF0F1Ergo1-SchroEqu2018}-\eqref{eq:hotF0F1Ergo2-SchroEqu2018} we obtain \eqref{eq:hotF0F1Ergo-SchroEqu2018} for the case where $(p,q) = (0,1)$.
Using similar arguments, formula \eqref{eq:hotF0F1Ergo-SchroEqu2018} for $(p,q) = (1,0)$ can be proved and we skip the details.
By Chebyshev's inequality and (\ref{eq:hotF0F1Ergo2-SchroEqu2018}), for any $\epsilon > 0$, we have
\begin{align}
\qquad & \ P \big( \bigcup_{j \geq K_0} \{ |X_{0,1}(K_j, \tau, \hat x,\omega) - 0| \geq \epsilon \} \big) \leq \frac C {\epsilon^2} \sum_{j \geq K_0} K_j^{-5/4} \leq \frac C {\epsilon^2} \sum_{j \geq K_0} j^{-1-5\gamma/4} \nonumber\\
\leq & \ \frac C {\epsilon^2} \int_{K_0}^{+\infty} (t-1)^{-1-5\gamma/4} \dif{t} = \frac C {\epsilon^2 \gamma} (K_0-1)^{-5\gamma/4} \to 0, \quad K_0 \to +\infty. \label{eq:X01Ergo-SchroEqu2018}
\end{align}
According to Lemma \ref{lem:asconverg-SchroEqu2018}, inequality \eqref{eq:X01Ergo-SchroEqu2018} implies \eqref{eq:HOTErgoToZero-SchroEqu2018} for the case where $(p,q) = (0,1)$. Similarly, formula \eqref{eq:HOTErgoToZero-SchroEqu2018} can be proved for the case where $(p,q) = (1,0)$.
We now prove \eqref{eq:hotF1F1Ergo-SchroEqu2018}. We have
\begin{align}
\mathbb{E} \big( X_{1,1} \big)
& = \frac 1 K \int_K^{2K} \mathbb{E} \big( \overline{F_1(k,\hat x,\omega)} \cdot F_1(k+\tau,\hat x,\omega) \big) \dif{k}
= \mathcal{O}(K^{-1}). \label{eq:hotF1F1Ergo1-SchroEqu2018}
\end{align}
Similar to \eqref{eq:hotF0F1Ergo2-SchroEqu2018}, we compute the secondary moment of $X_{1,1}$ as
\begin{align}
& \ \mathbb{E} \big( | X_{1,1} |^2 \big) \nonumber\\
= & \ \mathbb{E} \big( \frac 1 K \int_K^{2K} F_1(k_1,\hat x,\omega) \cdot \overline{F_1(k_1+\tau,\hat x,\omega)} \dif{k_1} \cdot \frac 1 K \int_K^{2K} \overline{ F_1(k_2,\hat x,\omega) } \cdot F_1(k_2+\tau,\hat x,\omega) \dif{k_2} \big) \nonumber\\
= & \ \frac 1 {K^2} \int_K^{2K} \int_K^{2K} \mathcal{O}(K^{-1}) \cdot \mathcal{O}(K^{-1}) \dif{k_1} \dif{k_2} \quad (\text{Lemma } \ref{lemma:HOT-SchroEqu2018}) \nonumber\\
= & \ \mathcal{O}(K^{-2}), \quad K \to +\infty. \label{eq:hotF1F1Ergo2-SchroEqu2018}
\end{align}
Formulae \eqref{eq:hotF1F1Ergo1-SchroEqu2018} and \eqref{eq:hotF1F1Ergo2-SchroEqu2018} give \eqref{eq:hotF1F1Ergo-SchroEqu2018}.
By Chebyshev's inequality and \eqref{eq:hotF1F1Ergo2-SchroEqu2018}, for any $\epsilon > 0$, we have
\begin{align}
& P \big( \bigcup_{j \geq K_0} \{ |X_{1,1} - 0| \geq \epsilon \} \big) \leq \frac C {\epsilon^2} \sum_{j \geq K_0} K_j^{-2} \leq \frac C {\epsilon^2} \sum_{j \geq K_0} j^{-8/5-2\gamma} \nonumber\\
\leq & \frac C {\epsilon^2} \int_{K_0}^{+\infty} (t-1)^{-8/5-2\gamma} \dif{t} = \frac {C (K_0-1)^{-3/5-2\gamma}} {\epsilon^2 (3+10\gamma)} \to 0, \quad K_0 \to +\infty. \label{eq:X11Ergo-SchroEqu2018}
\end{align}
Lemma \ref{lem:asconverg-SchroEqu2018} together with \eqref{eq:X11Ergo-SchroEqu2018} implies \eqref{eq:HOTErgoToZero-SchroEqu2018} for the case that $(p,q) = (1,1)$. The proof is thus completed.
\end{proof}
\section{The recovery of the variance function} \label{sec:RecVar-SchroEqu2018}
In this section we focus on the recovery of the variance function. We employ only a single passive scattering measurement.
Namely, there is no incident plane wave sent and the random sample $\omega$ is fixed. Throughout this section, $\alpha$ is set to be 0.
The data set $\mathcal M_1$ is utilized to achieve the unique recovery result.
We present the main results of recovering the variance function in Section \ref{subsec:MainSteps-SchroEqu2018}, and put the corresponding proofs in Section \ref{subsec:ProofsToMainSteps-SchroEqu2018}.
\subsection{Main unique recovery results} \label{subsec:MainSteps-SchroEqu2018}
To make it clearer, we use three lemmas, i.e., Lemmas \ref{lem:sigmaHatRec-SchroEqu2018}, \ref{lem:sigmaHatRecErgo-SchroEqu2018} and \ref{lem:sigmaHatRecSingle-SchroEqu2018}, to illustrate our recovering scheme of the variance function. The first main result is as follows.
\begin{lem} \label{lem:sigmaHatRec-SchroEqu2018}
We have the following asymptotic identity,
\begin{equation} \label{eq:sigmaHatRec-SchroEqu2018}
4\sqrt{2\pi} \lim_{k \to +\infty} \mathbb{E} \Big( \big[ \overline{u^\infty(\hat x, k, \omega)} - \overline{\mathbb{E} u^\infty(\hat x,k)}\, \big] \cdot \big[ u^\infty(\hat x, k+\tau, \omega) - \mathbb{E} u^\infty(\hat x,k+\tau) \big] \Big) = \widehat{\sigma^2}(\tau \hat x),
\end{equation}
where $\tau \geq 0,~ \hat x \in \mathbb{S}^2$.
\end{lem}
Lemma \ref{lem:sigmaHatRec-SchroEqu2018} clearly yields a recovery formula for the variance function. However, it requires many realizations. The result in Lemma \ref{lem:sigmaHatRec-SchroEqu2018} can be improved by using the ergodicity. See, e.g., \cite{caro2016inverse, Lassas2008, Helin2018}.
\begin{lem} \label{lem:sigmaHatRecErgo-SchroEqu2018}
Assume $\{K_j\} \in P(2+\gamma)$. Then $\exists\, \Omega_0 \subset \Omega \colon \mathbb{P}(\Omega_0) = 0$, $\Omega_0$ depends only on $\{K_j\}_{j \in \mathbb{N}^+}$, such that for any $\omega \in \Omega \backslash \Omega_0$, there exists $S_\omega \subset \R^3 \colon m(S_\omega) = 0$, such that for $\forall x \in \R^3 \backslash S_\omega$,
\begin{align}
& 4\sqrt{2\pi} \lim_{j \to +\infty} \frac 1 {K_j} \int_{K_j}^{2K_j} \big[ \overline{u^\infty(\hat x,k,\omega)} - \overline{\mathbb{E} u^\infty(\hat x,k)}\, \big] \cdot \big[ u^\infty(\hat x,k+\tau,\omega) - \mathbb{E} u^\infty(\hat x,k+\tau) \big] \dif{k} \nonumber\\
& = \widehat{\sigma^2} (x), \label{eq:SecondOrderErgo-SchroEqu2018}
\end{align}
where $\tau = |x|$ and $\hat x := x / |x|$.
\end{lem}
\sq{The recovering formula \eqref{eq:SecondOrderErgo-SchroEqu2018} holds for any $\hat x \in \mathbb S^2$ when $x = 0$.}
The recovery formulae presented in Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018} still involves every realization of the random sample $\omega$.
To recover the variance function by only one realization, the term $\mathbb{E} u^\infty(\hat x,k)$ should be further relaxed in Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018}, and this is achieved by Lemma \ref{lem:sigmaHatRecSingle-SchroEqu2018}.
\begin{lem} \label{lem:sigmaHatRecSingle-SchroEqu2018}
Under the same condition as in Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018}, we have
\begin{equation} \label{eq:sigmaHatRecSingle-SchroEqu2018}
4\sqrt{2\pi} \lim_{j \to +\infty} \frac 1 {K_j} \int_{K_j}^{2K_j} \overline{u^\infty(\hat x,k,\omega)} \cdot u^\infty(\hat x,k+\tau,\omega) \dif{k} = \widehat{\sigma^2} (x), \quad \textrm{~a.s.~}.
\end{equation}
\end{lem}
\sq{
\begin{rem}
In Lemma \ref{lem:sigmaHatRecSingle-SchroEqu2018}, it should be noted that the left-hand-side of \eqref{eq:sigmaHatRecSingle-SchroEqu2018} contains the random sample $\omega$, while the right-hand-side does not. This means that the limit in \eqref{eq:sigmaHatRecSingle-SchroEqu2018} is statistically stable.
\end{rem}}
Now Theorem \ref{thm:Unisigma-SchroEqu2018} becomes a direct consequence of Lemma \ref{lem:sigmaHatRecSingle-SchroEqu2018}.
\begin{proof}[Proof of Theorem \ref{thm:Unisigma-SchroEqu2018}]
Lemma \ref{lem:sigmaHatRecSingle-SchroEqu2018} provides a recovery formula for the variance function $\sigma^2$ by the data set $\mathcal M_1$.
\end{proof}
\subsection{Proofs of the main results} \label{subsec:ProofsToMainSteps-SchroEqu2018}
In this subsection, we present proofs of Lemmas \ref{lem:sigmaHatRec-SchroEqu2018}, \ref{lem:sigmaHatRecErgo-SchroEqu2018} and \ref{lem:sigmaHatRecSingle-SchroEqu2018}.
\begin{proof}[Proof of Lemma \ref{lem:sigmaHatRec-SchroEqu2018}]
\sq{Write $u_1^\infty(\hat x,k,\omega) = u^\infty(\hat x,k,\omega) - \mathbb{E} u^\infty(\hat x,k)$ as in \eqref{eq:u1-SchroEqu2018}. Therefore $4\pi u_1^\infty(\hat x,k,\omega)$ equals to $(-1) \sum_{j=0}^{+\infty} \int_D e^{-ik \hat x \cdot y} (V {\mathcal{R}_{k}})^j (\sigma \dot{B}_y)\dif{y}$. Recall the definition of $F_j(k,\hat x,\omega)$ $(j = 0,1)$ in \eqref{eq:Fjkx-SchroEqu2018}.}
Let $k_1, k_2 > k > k^*$. One can compute
\begin{align}
16\pi^2 \mathbb{E} \big( \overline{u_1^\infty(\hat x,k_1,\omega)} u_1^\infty(\hat x,k_2,\omega) \big)
& = \sq{\sum_{j,\ell = 0,1}} \mathbb{E} \big( \overline{F_j(k_1,\hat x,\omega)} F_\ell(k_2,\hat x,\omega) \big) \nonumber\\
& =: I_0 + I_1 + I_2 + I_3. \label{eq:Thm1uI-SchroEqu2018}
\end{align}
From Lemma \ref{lemma:HOT-SchroEqu2018}, we have
$I_1,\, I_2,\, I_3$ are all of order $k^{-1}$, hence
\begin{equation} \label{eq:u1Infty-SchroEqu2018}
16\pi^2 \mathbb{E} \big( \overline{u_1^\infty(\hat x,k_1,\omega)} u_1^\infty(\hat x,k_2,\omega) \big) = I_0 + \mathcal{O}(k^{-1}), \quad k \to +\infty.
\end{equation}
By \eqref{eq:I0-SchroEqu2018}, \eqref{eq:Thm1uI-SchroEqu2018} and \eqref{eq:u1Infty-SchroEqu2018}, we have
$$16\pi^2 \lim_{k \to +\infty} \mathbb{E} \big( \overline{u_1^\infty (\hat x,k_1,\omega)} u_1^\infty (\hat x,k_2,\omega) \big) = (2\pi)^{3/2} \,\widehat{\sigma^2}((k_2 - k_1) \hat x),$$
which implies (\ref{eq:sigmaHatRec-SchroEqu2018}).
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018}]
Our proof is divided into two steps. In the first step we give a basic result, i.e., the conclusion \eqref{eq:SecondOrderErgo2-SchroEqu2018}, and in the second step the logical order between $y$ and $\omega$ in \eqref{eq:SecondOrderErgo2-SchroEqu2018} is exchanged.
\noindent \textbf{Step 1}: give a basic result.
We denote by $\mathcal{E}_k$ the averaging operation w.r.t. $k$: ${{\mathcal{E}_k f} = \frac 1 K \int_K^{2K} f(k) \dif{k}}$. Following the notation conventions in the proof of Lemma \ref{lem:sigmaHatRec-SchroEqu2018}, we have
\begin{align}
16\pi^2 \mathcal{E}_k \big( \overline{u_1^\infty(\hat x,k,\omega)} u_1^\infty(\hat x,k+\tau,\omega) \big)
& = \jz{\sum_{j,\ell = 0,1}} \mathcal{E}_k \big( \overline{F_j(k,\hat x,\omega)} F_\ell(k+\tau,\hat x,\omega) \big) \nonumber\\
& =: X_{0,0} + X_{0,1}+ X_{1,0} + X_{1,1}. \label{eq:Thm2uX-SchroEqu2018}
\end{align}
Recall that $\{K_j\} \in P(2+\gamma)$.~Then, for $\forall \tau \geq 0$ and $\forall \hat x \in \mathbb{S}^2$, Lemma \ref{lemma:LeadingTermErgo-SchroEqu2018} implies that $\exists\, \Omega_{\tau,\hat x}^{0,0} \subset \Omega \colon \mathbb{P}(\Omega_{\tau,\hat x}^{0,0}) = 0$, $\Omega_{\tau,\hat x}^{0,0}$ depending on $\tau$ and $\hat x$, such that
\begin{equation} \label{eq:Thm2X00-SchroEqu2018}
\lim_{j \to +\infty} X_{0,0}(K_j,\tau,\hat x,\omega) = (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x), \quad \forall \omega \in \Omega \backslash \Omega_{\tau,\hat x}^{0,0}.
\end{equation}
$\{K_j\} \in P(2+\gamma)$ implies $\{K_j\} \in P(5/4+\gamma)$, so Lemma \ref{lemma:HOTErgo-SchroEqu2018} implies the existence of the sets $\Omega_{\tau,\hat x}^{p,q} ~\big( (p,q) \in \{ (0,1),\, (1,0),\, (1,1) \} \big)$ with zero probability measures such that $\forall \tau \geq 0$ and $\forall \hat x \in \mathbb{S}^2$,
\begin{equation} \label{eq:Thm2Xpq-SchroEqu2018}
\lim_{j \to +\infty} X_{p,q}(K_j,\tau,\hat x,\omega) = 0, \quad \forall \omega \in \Omega \backslash \Omega_{\tau,\hat x}^{p,q}.
\end{equation}
for all $(p,q) \in \{ (0,1),\, (1,0),\, (1,1) \}$. Write $\Omega_{\tau,\hat x} = \bigcup_{p,q = 0,1} \Omega_{\tau,\hat x}^{p,q}$\,, then $\mathbb{P} (\Omega_{\tau,\hat x}) = 0$. From Lemmas \ref{lemma:LeadingTermErgo-SchroEqu2018} and \ref{lemma:HOTErgo-SchroEqu2018} we note that $\Omega_{\tau,\hat x}^{p,q}$ also depends on $K_j$, so does $\Omega_{\tau,\hat x}$, but we omit this dependence in the notation. Write
\[
Z(\tau\hat x,\omega) := \lim_{j \to +\infty} \frac {16\pi^2} {K_j} \int_{K_j}^{2K_j} \overline{u_1^\infty(\hat x,k,\omega)} u_1^\infty(\hat x,k+\tau,\omega) \dif{k} - (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x)
\]
for short. By (\ref{eq:Thm2uX-SchroEqu2018})-(\ref{eq:Thm2Xpq-SchroEqu2018}), we conclude that,
\begin{equation} \label{eq:SecondOrderErgo2-SchroEqu2018}
{\,\forall\,} y \in \R^3, {\,\exists\,} \Omega_y \subset \Omega \colon \mathbb P (\Omega_y) = 0, \textrm{~s.t.~} \forall\, \omega \in \Omega \backslash \Omega_y,\, Z(y,\omega) = 0.
\end{equation}
\noindent \textbf{Step 2}: exchange the logical order.
To conclude \eqref{eq:SecondOrderErgo-SchroEqu2018} from \eqref{eq:SecondOrderErgo2-SchroEqu2018}, we should exchange the logical order between $y$ and $\omega$. To achieve this, we utilize Fubini's Theorem. Denote the usual Lebesgue measure on $\R^3$ as $\mathbb L$ and the product measure $\mathbb L \times \mathbb P$ as $\mu$, and construct the product measure space $\mathbb M := (\R^3 \times \Omega, \mathcal G, \mu)$ in the canonical way, where $\mathcal G$ is the corresponding complete $\sigma$-algebra. Write
\[
\mathcal{A} := \{ (y,\omega) \in \R^3 \times \Omega \,;\, Z(y, \omega) \neq 0 \},
\]
then $\mathcal{A}$ is a subset of $\mathbb M$. Set $\chi_\mathcal{A}$ as the characteristic function of $\mathcal{A}$ in $\mathbb M$. By \eqref{eq:SecondOrderErgo2-SchroEqu2018} we obtain
\begin{equation} \label{eq:FubiniEq0-SchroEqu2018}
\int_{R^3} \big( \int_\Omega \chi_{\mathcal A}(y,\omega) \dif{\mathbb P(\omega)} \big) \dif{\mathbb L(y)} = 0.
\end{equation}
By \eqref{eq:FubiniEq0-SchroEqu2018} and [Corollary 7 in Section 20.1, \citen{royden2000real}] we obtain
\begin{equation} \label{eq:FubiniEq1-SchroEqu2018}
\int_{\mathbb M} \chi_{\mathcal A}(y,\omega) \dif{\mathbb \mu} = \int_\Omega \big( \int_{R^3} \chi_{\mathcal A}(y,\omega) \dif{\mathbb L(y)} \big) \dif{\mathbb P(\omega)} = 0.
\end{equation}
Because $\chi_{\mathcal A}(y,\omega)$ is non-negative, (\ref{eq:FubiniEq1-SchroEqu2018}) implies
\begin{equation} \label{eq:FubiniEq2-SchroEqu2018}
{\,\exists\,} \Omega_0 \colon \mathbb P (\Omega_0) = 0, \textrm{~s.t.~} \forall\, \omega \in \Omega \backslash \Omega_0,\, \int_{R^3} \chi_{\mathcal A}(y,\omega) \dif{\mathbb L(y)} = 0.
\end{equation}
Formula (\ref{eq:FubiniEq2-SchroEqu2018}) further implies for every $\omega \in \Omega \backslash \Omega_0$,
\begin{equation} \label{eq:FubiniEq3-SchroEqu2018}
{\,\exists\,} S_\omega \subset \R^3 \colon \mathbb L (S_\omega) = 0, \textrm{~s.t.~} \forall\, y \in \R^3 \backslash S_\omega,\, Z(y,\omega) = 0.
\end{equation}
From (\ref{eq:FubiniEq3-SchroEqu2018}) we arrive at (\ref{eq:SecondOrderErgo-SchroEqu2018}).
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:sigmaHatRecSingle-SchroEqu2018}]
The symbol $\mathcal{E}_k$ is defined the same as in the proof of Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018}. We have
\begin{align}
& 16\pi^2 \mathcal{E}_k \big( \overline{u^\infty(\hat x,k,\omega)} u^\infty(\hat x,k+\tau,\omega) \big) \nonumber\\
= \ & 16\pi^2 \mathcal{E}_k \big( \overline{u_1^\infty(\hat x,k,\omega)} \cdot u_1^\infty(\hat x,k,\omega) \big) + 16\pi^2 \mathcal{E}_k \big( \overline{u_1^\infty(\hat x,k,\omega)} \cdot \mathbb E u^\infty(\hat x,k+\tau) \big) \nonumber\\
& + 16\pi^2 \mathcal{E}_k \big( \overline{\mathbb E u^\infty(\hat x,k)} \cdot u_1^\infty(\hat x,k+\tau,\omega) \big) + 16\pi^2 \mathcal{E}_k \big( \overline{\mathbb E u^\infty(\hat x,k)} \cdot \mathbb E u^\infty(\hat x,k+\tau) \big) \nonumber\\
=: & J_0 + J_1 + J_2 + J_3. \label{eq:J-SchroEqu2018}
\end{align}
From Lemma \ref{lem:sigmaHatRecErgo-SchroEqu2018} we obtain
\begin{equation} \label{eq:J0-SchroEqu2018}
\begin{split}
& \lim_{j \to +\infty} J_0 = \lim_{j \to +\infty} \jz{\frac {16\pi^2} {K_j}} \int_{K_j}^{2K_j} \overline{u_1^\infty(\hat x,k,\omega)} \cdot u_1^\infty(\hat x,k+\tau,\omega) \dif{k} = (2\pi)^{3/2} \widehat{\sigma^2} (\tau \hat x), \\
& \quad \tau \hat x \textrm{~a.e.~} \! \in \R^3, \quad \omega \textrm{~a.s.~} \! \in \Omega.
\end{split}
\end{equation}
We now estimate $J_1$,
\begin{align}
|J_1|^2
& \simeq \big| \mathcal{E}_k \big( \overline{u_1^\infty(\hat x,k,\omega)} \cdot \mathbb E u^\infty(\hat x,k+\tau) \big) \big|^2
= \big| \frac 1 {K_j} \int_{K_j}^{2K_j} \overline{u_1^\infty(\hat x,k,\omega)} \cdot \mathbb E u^\infty(\hat x,k+\tau) \dif{k} \big|^2 \nonumber\\
& \leq \frac 1 {K_j} \int_{K_j}^{2K_j} |u^\infty(\hat x,k,\omega) - \mathbb{E} u^\infty(\hat x,k)|^2 \dif{k} \cdot \frac 1 {K_j} \int_{K_j}^{2K_j} |\mathbb E u^\infty(\hat x,k+\tau)|^2 \dif{k}. \label{eq:J1One-SchroEqu2018}
\end{align}
Combining \eqref{eq:J1One-SchroEqu2018} with Lemmas \ref{lemma:FarFieldGoToZero-SchroEqu2018} and \ref{lem:sigmaHatRecErgo-SchroEqu2018}, we have
\begin{equation} \label{eq:J1-SchroEqu2018}
|J_1|^2 \lesssim (\widehat{\sigma^2}(0) + o(1)) \cdot o(1) = o(1) \to 0, \quad j \to +\infty.
\end{equation}
The analysis of $J_2$ is similar to that of $J_1$ so we skip the details.
Finally, by Lemma \ref{lemma:FarFieldGoToZero-SchroEqu2018}, the $J_3$ can be estimated as
\begin{align}
|J_3|^2
& \simeq \big| \mathcal{E}_k \big( \overline{\mathbb E u^\infty(\hat x,k)} \cdot \mathbb E u^\infty(\hat x,k+\tau) \big) \big|^2 \nonumber\\
& \leq \frac 1 {K_j} \int_{K_j}^{2K_j} \sup_{\kappa \geq K_j} \big| \mathbb E u^\infty(\hat x,\kappa) \big|^2 \dif{k} \cdot \frac 1 {K_j} \int_{K_j}^{2K_j} \sup_{\kappa \geq K_j+\tau} \big| \mathbb E u^\infty(\hat x,\kappa) \big|^2 \dif{k} \nonumber\\
& = \sup_{\kappa \geq K_j} |\mathbb E u^\infty(\hat x,\kappa)|^2 \cdot \sup_{\kappa \geq K_j+\tau} |\mathbb E u^\infty(\hat x,\kappa)|^2 \to 0, \quad j \to +\infty. \label{eq:J3-SchroEqu2018}
\end{align}
Combining \eqref{eq:J-SchroEqu2018}, \eqref{eq:J0-SchroEqu2018}, \eqref{eq:J1-SchroEqu2018} and \eqref{eq:J3-SchroEqu2018}, we arrive at \eqref{eq:sigmaHatRecSingle-SchroEqu2018}. Our proof is done.
\end{proof}
\section{Uniqueness of the potential and the random source} \label{sec:RecPS-SchroEqu2018}
In this section, we focus on the recovery of the potential term and the expectation of the random source. Due to the highly nonlinear relation between the total wave and the potential, the active scattering measurements are thus utilized to recover the potential.
In the recovery of the potential, the random sample $\omega$ is set to be fixed so that a single realization of the random term $\dot B_x$ is enough to obtain the unique recovery.
Different from the recovery of the potential, the uniqueness of the expectation requires all realizations of the random sample $\omega$.
\jz{Because the deterministic and random parts of the source are entangled together so that only one realization of the random source cannot reveal exact values of the expectation at each spatial point $x$.}
\subsection{Recovery of the potential}
Now we are in the position to prove Theorem \ref{thm:UniPot1-SchroEqu2018}. We are to use the incident plane wave, so $\alpha$ is set to be 1 throughout this section.
\begin{proof}[Proof of Theorem \ref{thm:UniPot1-SchroEqu2018}]
The random sample $\omega$ is assumed to be fixed. Given two direction $d_1$ and $d_2$ of the incident plane waves, we denote the corresponding total wave as $u_{d_1}$ and $u_{d_2}$, respectively. Then, from \eqref{eq:1}, we have
\begin{equation} \label{eq:uSubtract-SchroEqu2018}
\begin{cases}
(-\Delta - k^2)(u_{d_1} - u_{d_2}) = V (u_{d_1} - u_{d_2}) & \\
u_{d_1} - u_{d_2} = e^{ikd_1 \cdot x} - e^{ikd_2 \cdot x} + u_{d_1}^{sc}(x) - u_{d_2}^{sc}(x) & \\
u_{d_1}^{sc}(x) - u_{d_2}^{sc}(x): \text{ SRC} &
\end{cases}
\end{equation}
From \eqref{eq:uSubtract-SchroEqu2018} we have the Lippmann-Schwinger equation,
\begin{equation} \label{eq:uLippSchw-SchroEqu2018}
\big( I - {\mathcal{R}_{k}} V \big) (u_{d_1} - u_{d_2}) = e^{ikd_1 \cdot x} - e^{ikd_2 \cdot x}.
\end{equation}
When $k > k^*$, equality \eqref{eq:uLippSchw-SchroEqu2018} gives
\[
u_{d_1}^{sc} - u_{d_2}^{sc}
= {\mathcal{R}_{k}} V (e^{ikd_1 \cdot x} - e^{ikd_2 \cdot x}) + \sum_{j=2}^\infty ({\mathcal{R}_{k}} V)^j (e^{ikd_1 \cdot x} - e^{ikd_2 \cdot x}).
\]
Therefore the difference between the far-field patterns is
\begin{align}
& u^{\infty}(\hat{x},k,d_1) - u^{\infty}(\hat{x},k,d_2) \nonumber\\
= \ & \int_{D} \frac{e^{-ik\hat{x} \cdot y}}{4\pi} V(y) (e^{ikd_1 \cdot y} - e^{ikd_2 \cdot y}) \dif{y} + \sum_{j=1}^\infty \int_{D} \frac{e^{-ik\hat{x} \cdot y}}{4\pi} V(y) ({\mathcal{R}_{k}} V)^j (e^{ikd_1 \cdot (\cdot)} - e^{ikd_2 \cdot (\cdot)}) \dif{y} \nonumber\\
=: & \sqrt{\frac{\pi}{2}} \widehat{V} \big( k(\hat{x} - d_1) \big) - \sqrt{\frac{\pi}{2}} \widehat{V} \big( k(\hat{x} - d_2) \big) + \sum_{j=1}^\infty H_j(k), \label{eq:uFarfied-SchroEqu2018}
\end{align}
where
\begin{equation} \label{eq:Fjk-SchroEqu2018}
H_j(k) := \int_{D} \frac{e^{-ik\hat{x} \cdot y}}{4\pi} V(y) ({\mathcal{R}_{k}} V)^j (e^{ikd_1 \cdot (\cdot)} - e^{ikd_2 \cdot (\cdot)}) \dif{y}, \quad j = 1,2, \cdots.
\end{equation}
For any $p \in \R^3$, when $p = 0$, we let $\hat x = (1,0,0),$ $d_1 = (1,0,0),$ $d_2 = (0,1,0)$; when $p \neq 0$, we can always find a $p^\perp \in \R^3$ which is perpendicular to $p$. Let
\begin{equation*}
e = p^\perp / \nrm{p^\perp}
\quad\text{ and }\quad
\left\{\begin{aligned}
\hat x & = \sqrt{1 - \nrm{p}^2 / (4k^2)} \cdot e + p / (2k), \\
d_1 & = \sqrt{1 - \nrm{p}^2 / (4k^2)} \cdot e - p / (2k), \\
d_2 & = p/\nrm{p},
\end{aligned}\right.
\end{equation*}
when $k > \nrm{p}/2$, we have
\begin{equation} \label{eq:xhatd1Property-SchroEqu2018}
\left\{\begin{aligned}
& \hat x, d_1, d_2 \in \mathbb{S}^2, \\
& k(\hat{x} - d_1) = p, \\
& |k(\hat{x} - d_2)| \to \infty ~(k \to \infty).
\end{aligned}\right.
\end{equation}
Note that the choices of these two unit vectors $\hat x$, $d_1$ depend on $k$. For different values of $k$, we pick up different directions $\hat x$, $d_1$ to guarantee \eqref{eq:xhatd1Property-SchroEqu2018}. Then,
\begin{equation} \label{eq:Vxd1d2-SchroEqu2018}
\sqrt{\frac \pi 2}\widehat{V}(p) = \lim_{k \to +\infty} \big( \sqrt{\frac \pi 2} \widehat{V} (k(\hat{x} - d_1)) - \sqrt{\frac{\pi}{2}} \widehat{V} (k(\hat{x} - d_2)) \big).
\end{equation}
Combining \eqref{eq:uFarfied-SchroEqu2018}, \eqref{eq:Vxd1d2-SchroEqu2018} and Lemma \ref{lemma:FjkEstimated-SchroEqu2018}, we conclude
\begin{equation} \label{eq:PotnFourier-SchroEqu2018}
\widehat{V}(p) = \sqrt{\frac{2}{\pi}} \lim_{k \to +\infty} \big( u^{\infty}(\hat{x},k,d_1) - u^{\infty}(\hat{x},k,d_2) \big).
\end{equation}
Formula \eqref{eq:PotnFourier-SchroEqu2018} completes the proof.
\end{proof}
It remains to give the estimates of these high-order terms $H_j(k)$, and this is done by Lemma \ref{lemma:FjkEstimated-SchroEqu2018}.
\begin{lem} \label{lemma:FjkEstimated-SchroEqu2018}
The sum of high-order terms $H_j(k)$ defined in \eqref{eq:Fjk-SchroEqu2018} satisfies the following estimate,
$$\big| \sum_{j \geq 1} H_j(k) \big| \leq C k^{-1},$$
for some constant $C$ independent of $k$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{lemma:FjkEstimated-SchroEqu2018}]
According to Lemma \ref{lemma:RkVBounded-SchroEqu2018}, we have
\begin{align*}
|H_j(k)|
& \lesssim \int_D |V(y)| \cdot \big| [({\mathcal{R}_{k}} V)^{j} e^{ik d_1 \cdot (\cdot)}] (y) \big| \dif{y} + \int_D |V(y)| \cdot \big| [({\mathcal{R}_{k}} V)^{j} e^{ik d_2 \cdot (\cdot)}] (y) \big| \dif{y} \\
& \lesssim \nrm[L^\infty]{V} \cdot |D|^{1/2} \cdot \big( k^{-j} \nrm[L^2(D)]{ e^{ik d_1 \cdot (\cdot)} } + k^{-j} \nrm[L^2(D)]{ e^{ik d_2 \cdot (\cdot)} } \big) \\
& = 2\nrm[L^\infty]{V} \cdot |D| \cdot k^{-j}.
\end{align*}
Therefore,
$$|\sum_{j=1}^{\infty} H_j(k)|
\leq \sum_{j=1}^{\infty} |H_j(k)|
\leq 2C \nrm[L^\infty]{V} \cdot |D| \cdot \sum_{j=1}^{\infty} k^{-j} \leq C k^{-1}, \quad k \to +\infty.$$
The proof is done.
\end{proof}
\subsection{Recovery of the random source}
The variance function of the random source is recovered in Section \ref{sec:RecVar-SchroEqu2018}, and now we recover its expectation.
\begin{proof}[Proof to Theorem \ref{thm:UniSou1-SchroEqu2018}]
According to Theorem \ref{thm:UniPot1-SchroEqu2018}, we have the uniqueness of the potential. Assume that two source $f$, $f'$ generate same far-field patterns for all $k > 0$. We denote the restriction on $D$ of the corresponding total waves as $u$ and $u'$. Then,
\begin{equation} \label{eq:uuprime-SchroEqu2018}
\left\{\begin{aligned}
(\Delta + k^2 + V) (\mathbb{E} u - \mathbb{E} u') & = f - f' && \text{ in } D \\
\mathbb{E} u - \mathbb{E} u' = \partial_\nu (\mathbb{E} u) - \partial_\nu (\mathbb{E} u') & = 0 && \text{ on } \partial D
\end{aligned}\right.
\end{equation}
where $\nu$ is the outer normal to $\partial D$. Let test functions $v_k \in H_0^1(D)$ be the weak solutions of the boundary value problem
\begin{equation} \label{eq:DiriLap-SchroEqu2018}
\left\{\begin{aligned}
(-\Delta - V)v_k & = k^2 v_k && \text{ in } D \\
v_k & = 0 && \text{ on } \partial D
\end{aligned}\right.
\end{equation}
for delicately picked $k$. The solutions $v_k$ are eigenvectors of the system \eqref{eq:DiriLap-SchroEqu2018}. From \eqref{eq:uuprime-SchroEqu2018} we have
\begin{equation} \label{eq:uuprimev-SchroEqu2018}
\int_D (\Delta + V + k^2) (\mathbb{E}u - \mathbb{E}u') \cdot v_k \dif{x} = \int_D (f - f')v_k \dif{x}.
\end{equation}
Using integral by parts and noting that the $v_k$'s in \eqref{eq:uuprimev-SchroEqu2018} satisfy \eqref{eq:DiriLap-SchroEqu2018}, we have
\begin{equation} \label{eq:ffprime-SchroEqu2018}
\int_D (f - f')v_k \dif{x} = 0.
\end{equation}
When $\nrm[L^\infty(D)]{V}$ is less than some constant depending on $D$, the set of eigenvectors $\{v_k\}$ corresponding to different eigenvalues $k^2$ forms an orthonormal basis of $L^2(D)$ [Theorem 2.37, \citen{mclean2000strongly}]. Therefore, from \eqref{eq:ffprime-SchroEqu2018} we conclude that
$$f = f' \text{ in } L^2(D).$$
The proof is done.
\end{proof}
\section{Conclusions} \label{sec:Conclusions-SchroEqu2018}
In this paper, we are concerned with a random Schr\"odinger equation.
First, the well-posedness of the direct problem is studied.
Then, the variance function of the random source is recovered by using a single passive scattering measurement.
By further utilizing active scattering measurements under a single realization of the random sample, the potential is recovered.
Finally, with the help of multiple realizations of the random sample, the expectation of the random source are recovered.
The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active measurements are used to recover all of the unknowns.
\sq{While the direct problem in this paper is well-formulated in the space $L_{-1/2-\epsilon}^2$, the regularity of the solution of the random Schr\"odinger system is not taken into consideration.
A different formulation of the direct problem, which takes the regularity of the solution into consideration, is possible.
And this new formulation gives possibility to handle the case where both the source and potential are random.
We shall report our finding in this aspect in a forthcoming article.
}
\sq{
\end{document} |
\begin{document}
\title{On Coercivity and the Frequency Domain Condition in Indefinite LQ-Control}
\pagestyle{myheadings}
\markboth{T. Damm, B. Jacob
}{Coercivity and Frequency Domain Condition}
\begin{abstract}
We introduce a coercivity condition as a time domain analogue of the
frequency criterion provided by the famous Kalman-Yakubovich-Popov
lemma. For a simple stochastic linear quadratic control problem we show how the
coercivity condition characterizes the solvability of Riccati equations.
\textit{Keywords:} linear quadratic control, Riccati equation,
frequency domain condition, stochastic system
\textit{MSC2020:} 93C80,
49N10,
15A24,
93E03
\end{abstract}
\section{Introduction}
\label{sec:introduction}
Since the formulation of the Kalman-Yakubovich-Popov-lemma in the 1960s
the interplay of time domain and frequency domain methods has always
been fruitful and appealing in linear control theory. For the
linear-quadratic control problem and the algebraic Riccati equation
this has been worked out to a large extent already in \cite{Will71}.
However, the applicability of frequency domain methods is mostly
limited to linear time-invariant deterministic models. In the
consideration of time-varying or stochastic systems it is often
necessary to find suitable substitutes.
In this note we want to draw the attention to an equivalent
formulation of the frequency domain condition, which to our knowledge
is not very present in the literature. We call it the {\em coercivity
condition}. As our two main contributions, we first establish the equivalence of
the coercivity condition and the frequency domain condition and show
second that the coercivity condition plays the same role for the
solvability of the Riccati equation of a stochastic linear quadratic
control problem as the frequency condition does for the corresponding
deterministic problem. To simplify the presentation we choose the most
simple setup for the stochastic problem. A detailed discussion of the
analogous result for time-varying linear systems is to be found
in the forthcoming book \cite{HinrPrit}.
It is a great honour for us to dedicate this note to Vasile
Dr\u{a}gan at the occasion of his 70-th birthday. We had the pleasure to collaborate with Vasile e.g.~in \cite{DragDamm05} and \cite{JacoDrag98}.
Vasile Dr\u{a}gan has made
numerous and substantial contributions in the context of our topic.
Together with Aristide Halanay, he was among the first to study stochastic disturbance attenuation
problems \cite{DragHala96, DragHala97, DragHala99}, and in still
ongoing work (e.g.\ \cite{DragIvan20}) he extended the theory in many different directions. The textbook \cite{DragMoro13} is closely related to this
note.
\section{Preliminaries}
\label{sec:preliminaries}
Consider the time-invariant finite-dimensional linear control system
\begin{eqnarray*}
\dot x(t)&=&Ax(t)+Bu(t),\quad t\ge 0,\\
x(0)&=&x_0,
\end{eqnarray*}
together with the {\em quadratic cost functional}
\begin{displaymath}
J(x_0,u)=\int_0^\infty \left[
\begin{array}{c}
x(t)\\u(t)
\end{array}
\right]^*M \left[
\begin{array}{c}
x(t)\\u(t)
\end{array}
\right]\,dt\;,
\end{displaymath}
where $A$, $B$ and $M$ are complex matrices of suitable size.
Assume that $M=M^*=\left[
\begin{array}{cc}
W&V^*\\V&R
\end{array}
\right]$ where $R>0$, but not necessarily $M\ge 0$ or $W\ge 0$.
With these data we associate the {\em algebraic Riccati equation}
\begin{equation}
\label{eq:are}
A^*P+PA+W-(B^*P+V)^*R^{-1}(B^*P+V)=0\;.
\end{equation}
Moreover, for $\omega\in{\bf R}$ with
$\imath\omega\not\in\sigma(A)$ we define the {\em frequency function}
(or Popov function, \cite{Popo73})
\begin{equation}
\label{eq:freqfunc}
\Phi(\omega)=\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right]^* M\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right]\;.
\end{equation}
Here $\sigma(A)$ denotes the spectrum of the matrix $A$.
Then the {\em strict frequency domain condition} requires
\begin{equation}
\label{eq:fdc_strict}
\exists \varepsilon>0: \forall\omega\in{\bf R},
\imath\omega\not\in\sigma(A):\quad \Phi(\omega)\ge \varepsilon^2
B ^*(\imath\omega I-A)^* (\imath \omega I-A)^{-1}B\;.
\end{equation}
The {\em nonstrict frequency domain condition} is just
\begin{equation}
\label{eq:fdc_nonstrict}
\forall\omega\in{\bf R},
\imath\omega\not\in\sigma(A):\quad \Phi(\omega)\ge 0\;.
\end{equation}
Note that (\ref{eq:fdc_strict}) holds with a given $M$ and fixed
$ \varepsilon>0$, if and only if (\ref{eq:fdc_nonstrict}) holds with $M$
replaced by
\begin{displaymath}
M_ \varepsilon=M-\left[
\begin{array}{cc}
\varepsilon^2I&0\\0&0
\end{array}
\right]\;.
\end{displaymath}
\begin{remark}\label{rem:fdc_are}
If $(A,B)$ is stabilizable, then it is well-known (e.g.\
\cite{Will71}) that (\ref{eq:are}) possesses a {\em stabilizing solution}
(i.e.\ a solution $P$ with the additional property that
$\sigma(A-BR^{-1}(B^*P+V))\subset{\bf C}_-$, where ${\bf C}_-$
denotes the open left half plane), if and only if the frequency
condition (\ref{eq:fdc_strict}) holds. There exists an {\em almost
stabilizing solution} (satisfying
$\sigma(A-BR^{-1}(B^*P+V))\subset{\bf C}_-\cup \imath{\bf R}$), if
and only if (\ref{eq:fdc_nonstrict}) holds.
\end{remark}
However, there are other classes of linear systems for which quadratic
cost functionals can be formulated, which do not allow for an
analogous frequency domain interpretation. These are, for instance,
time-varying or stochastic systems e.g.\ \cite{DragMoro13}.
It is therefore useful to have a time domain condition which is
equivalent to (\ref{eq:fdc_strict}). Such a condition can be obtained by
applying the inverse Laplace
transformation, but we choose a more elementary
approach here.
For an initial value $x_0$ and a square-integrable input function $u\in L^2({\bf R}_+)$ we denote by $x(t,x_0,u)$ the unique solution
of our time-invariant finite-dimensional linear control system at time $t$.
Let
\begin{displaymath}
U=\{u\in L^2({\bf R}_+)\;\big|\; x(\cdot,0,u)\in L^2({\bf R}_+)\}
\end{displaymath}
denote the set of {\em admissible inputs}. For $u\in U$ we consider
the cost associated to zero initial state
\begin{equation}
\label{eq:J0u}
J(0,u)= \int_0^\infty \left[
\begin{array}{c}
x(t,0,u)\\u(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,0,u)\\u(t)
\end{array}
\right] \,dt \;.
\end{equation}
Then we say that $J$ satisfies the {\em strict coercivity condition},
if
\begin{equation}
\label{eq:ccstrict}
\exists \varepsilon >0 : \forall u\in U: \quad J(0,u)\ge \varepsilon^2\|x(\cdot,0,u)\|_{L^2}^2\;.
\end{equation}
We say that $J$ satisfies the {\em nonstrict coercivity condition}, if
\begin{equation}
\label{eq:cc}
\forall u\in U:\quad J(0,u)\ge 0\;.
\end{equation}
As for the frequency domain conditions, note that (\ref{eq:ccstrict}) holds with a given $M$ and fixed
$ \varepsilon>0$, if and only if (\ref{eq:cc}) holds with $M$
replaced by $M_ \varepsilon$.\\
In the next section, we prove the equivalence of (\ref{eq:fdc_nonstrict}) and
(\ref{eq:cc}). Since in the strict cases with given $ \varepsilon>0$ we
can replace $M$ by $M_ \varepsilon$ as indicated above, this also
establishes the equivalence of (\ref{eq:fdc_strict}) and
(\ref{eq:ccstrict}).
Then, in Section \ref{sec:stoch-lq-contr}, we show for a stochastic LQ-problem that
(\ref{eq:ccstrict}) is a natural time domain replacement of (\ref{eq:fdc_strict}).
\section{Equivalence of frequency domain and coercivity condition}
In this section we consider the system $\dot x=Ax+Bu$ and we assume that the pair
$(A,B)\in{\bf C}^{n\times n}\times {\bf C}^{n\times m}$ is stabilizable.
Solutions with initial value $x(0)=x_0$ and input $u\in
L^2({\bf R}_+)$ are denoted by $x(\cdot,x_0,u)$. As above, let
$$U=\{u\in L^2({\bf R}_+)\;\big|\; x(\cdot,0,u)\in L^2({\bf R}_+)
\}$$ be the set of admissible inputs and let $M\in{\bf C}^{(n+m)\times(n+m)}$ be a weight matrix of the form $M=M^*=\left[
\begin{array}{cc}
W&V^*\\V&R
\end{array}
\right]$ where $R>0$.
\begin{theorem} The following statements are equivalent.
\begin{itemize}
\item[(a)] For all $u\in U$ it holds that
\begin{displaymath}
J(0,u)= \int_0^\infty \left[ \begin{array}{c}
x(t,0,u)\\u(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,0,u)\\u(t)
\end{array}
\right] \,dt \ge 0\;.
\end{displaymath}
\item[(b)] For all $\omega\in{\bf R}$ with
$\imath\omega\not\in\sigma(A)$ it holds that
\begin{displaymath}
\Phi(\omega)=\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right]^* M\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right] \ge 0\;.
\end{displaymath}
\end{itemize}
\end{theorem}
\bf Proof: \nopagebreak \rm
(a)$\Rightarrow$(b) Let $\eta\in{\bf C}^m$ be arbitrary and $\omega>0$,
$\imath\omega\not\in\sigma(A)$. We have to
show that $\eta^*\Phi(\omega)\eta\ge 0$.
Let $\xi=(\imath\omega I-A)^{-1}B\eta$. Then $\xi$ is reachable from
$0$ and there exists a control input $u_0\in L^2([0,1])$ such that $x(1,0,u_0)=\xi$. Since $(A,B)$ is stabilizable, there also exists $u_\infty\in
L^2({\bf R}_+)$ with $x(\cdot,\xi,u_\infty)\in L^2({\bf R}_+)$.
For $k\in {\bf N}$, $k>0$, and $T_k=\frac{2k\pi}\omega+1 $, we define
\begin{eqnarray*}
u_k(t)&=&\left\{
\begin{array}{ll}
u_0(t)&t\in[0,1[\\
\eta e^{\imath\omega (t-1)}&t\in[1,T_k]\\
u_\infty(t-T_k)& t\in\;]T_k,\infty[
\end{array}
\right.\,.
\end{eqnarray*}
Then $x(1,0,u_k)=\xi$. An easy calculation shows that on $[1,T_k]$ we have the
resonance solution $x(t,0,u_k)=\xi e^{\imath\omega (t-1)}$ with
$x(T_k,0,u_k)=\xi$, such that it is stabilized by $u_\infty$ on
$]T_k,\infty[$. The integrals
\begin{eqnarray*}
&&\int_0^1 \left[
\begin{array}{c}
x(t,0,u_k)\\u_k(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,0,u_k) \\u_k(t)
\end{array}
\right] \,dt\\&&+\int_{T_k}^\infty \left[
\begin{array}{c}
x(t,0,u_k) \\u_k(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,0,u_k) \\u_k(t)
\end{array}
\right] \,dt=c <\infty
\end{eqnarray*}
are independent of $k$.
By (a) we have
\begin{eqnarray*}
0&\le& J(0,u_k)=c+\int_1^{T_k} \left[
\begin{array}{c}
x(t,0,u_k)\\u_k(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,0,u_k)\\u_k(t)
\end{array}
\right] \,dt\\
&=&c+\int_1^{T_k}\left[
\begin{array}{c}
\xi e^{\imath\omega t}\\\eta e^{\imath\omega t}
\end{array}
\right]^* M \left[
\begin{array}{c}
\xi e^{\imath\omega t}\\\eta e^{\imath\omega t}
\end{array}
\right] \,dt\\
&=&c+\int_1^{T_k}\left[
\begin{array}{c}
\xi \\\eta
\end{array}
\right]^* M \left[
\begin{array}{c}
\xi \\\eta
\end{array}
\right] \,dt\\ &=&c+\int_1^{T_k}\eta^*\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right]^* M\left[
\begin{array}{c}
(\imath \omega I-A)^{-1}B\\I
\end{array}
\right]\eta \,dt \;.
\end{eqnarray*}
Since $T_k$ can be arbitrarily large, the integrand must be
nonnegative. This proves (b).\\
(b)$\Rightarrow$(a)
Note first that
\begin{equation}
\left[
\begin{array}{c}
\xi\\\eta
\end{array}
\right]^*M \left[\begin{array}{c}
\xi\\\eta
\end{array}
\right]\ge 0, \mbox{ if }
(\imath\omega I-A) \xi=B\eta \mbox{ for some } \omega\in{\bf R}.\label{eq:initial_observation}
\end{equation}
Let now $u\in U$ be given and assume by way of contradiction that
$J(0,u)<0$.
For $T>0$, $x_0\in{\bf C}^n$, we set $$J_T(x_0,u)= \int_0 ^T \left[
\begin{array}{c}
x(t,x_0,u)\\u(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,x_0,u)\\u(t)
\end{array}
\right] \,dt \;.$$
Then there exists $\delta>0,T_0>1$ such that $J_{T-1}(0,u)<-2\delta$ for
all $T\ge T_0$. For $x_T=x(T-1,0,u)$ there exists a control
input $u_T\in L^2([0,1])$ such that $x(1,x_T,u_T)=0$.
In fact, one can choose $u_T(t)=-e^{A ^*(1-t)}P_1^\dagger
e^Ax_T$, where $P_1$
denotes the finite-time controllability Gramian over the interval $[0,1]$ and $P_1^\dagger$ its
Moore-Penrose pseudoinverse. Then, e.g.\ \cite{BennDamm11},
$$\|u_T\|_{L^2([0,1])}^2=x_T ^*e^{A ^*}P_1^\dagger e^Ax_T={\cal
O}(\|x_T\|^2)\mbox{ for } x_T\to 0\;. $$
This implies that also $J_1(x_{T},u_{T})={\cal O}(\|x_T\|^2)$. Since
$x(\cdot,0,u)\in L^2({\bf R}_+)$,
we can fix $T>T_0$ such that $\|x_{T}\|$ is small enough to ensure
$J_1(x_{T},u_{T})<\delta$.
We concatenate $u\big|_{[0,T-1]}$ and $u_T$ to a new input
$\tilde u\in L^2([0,T])$ with
\begin{eqnarray*}
\tilde u(t)&=&\left\{
\begin{array}{ll}
u(t) & t\in[0,T-1[\\
u_T(t-T+1)& t\in[T-1,T]
\end{array}
\right.\;.
\end{eqnarray*}
By construction, we have
\begin{equation}
J_T(0,\tilde u)<-\delta<0\quad\mbox{ and }\quad 0=x(0,0,\tilde u) =x(T,0,\tilde
u).\label{eq:JT}
\end{equation}
By definition $\tilde u, x\in L^2([0,T])$, and the equation $\dot x=
Ax +B\tilde u$ implies that $x$ is absolutely continuous and $\dot
x\in L^2([0,T])$. Thus, on $[0,T]$, the Fourier series of $\tilde
u$, $x$ and $\dot x$ converge in $ L^2([0,T])$ to $\tilde u$, $x$ and
$\dot x$, respectively. On $[0,T]$, let
\begin{displaymath}
x(t,0,u)=\sum_{k=-\infty}^\infty
\xi_ke^{\imath\frac{2\pi k t}T}\quad\mbox{ and }\quad\tilde u(t)=\sum_{k=-\infty}^\infty
\eta_ke^{\imath\frac{2\pi k t}T}.
\end{displaymath}
Then we get
\begin{eqnarray}
\nonumber
\sum_{k=-\infty}^\infty
B\eta_ke^{\imath\frac{2\pi k t}T} =B\tilde u(t) &=&\dot x(t,0,\tilde u)-Ax(t,0,\tilde u)\\&=&\sum_{k=-\infty}^\infty \left(\imath\frac{2\pi}T k
I-A\right)\xi_ke^{\imath\frac{2\pi k t}T}\;.
\label{eq:formalderivative}
\end{eqnarray}
Note that the periodicity condition $x(0)=x(T)$ in (\ref{eq:JT})
justifies the formal differentiation of the Fourier series in (\ref{eq:formalderivative}),
e.g.\ \cite[Theorem 1]{Tayl44}.\\
Comparing the coefficients in (\ref{eq:formalderivative}), we have
\begin{equation}
\label{eq:xikBetak}
\left(\imath\frac{2\pi}T k I-A\right)\xi_k=B\eta_k\;.
\end{equation}
In the expression of $J_T(0,\tilde u)$ we replace $x(t,0,\tilde u)$ and $\tilde
u(t)$ by
their Fourier-series representations. Exploiting orthogonality we have
\begin{eqnarray*}
J_T(0,\tilde u)&=&
T\sum_{k=-\infty}^\infty \left[
\begin{array}{c}
\xi_k\\\eta_k
\end{array}
\right]^*M \left[\begin{array}{c}
\xi_k\\\eta_k
\end{array}
\right].
\end{eqnarray*}
Together with (\ref{eq:xikBetak}) and (\ref{eq:initial_observation}) this
implies $J_T(0,\tilde u)\ge 0$ contradicting the first condition in
(\ref{eq:JT}). Thus our initial assumption was wrong, and we have shown that
(b) implies (a).\eprf
\section{An indefinite stochastic LQ-control problem}
\label{sec:stoch-lq-contr}
Consider the It\^o-type linear stochastic system
\begin{equation}
\label{eq:Ito}
dx=(Ax+Bu)\,dt + Nx\,dw\;.
\end{equation}
Here $w$ is a Wiener process and by $L^2_w$ we denote the space of square
integrable stochastic processes adapted to $L^2_w$. For the
appropriate definitions see textbooks such as \cite{Arno73, DragMoro13}.
Let further the cost functional
\begin{equation}
\label{eq:JWR}
J(x_0,u)={\bf E}\int_0^\infty \left[
\begin{array}{c}
x(t,x_0,u)\\u(t)
\end{array}
\right]^* M \left[
\begin{array}{c}
x(t,x_0,u)\\u(t)
\end{array}
\right] \,dt
\end{equation}
be given, where ${\bf E}$ denotes expectation. \\
For simplicity of presentation let $M=\left[
\begin{array}{cc}
W&0\\0&I
\end{array}
\right]$ which can always be achieved by a suitable transformation, if
the lower right block of $M$ is positive definite, e.g.\ \cite[Section
5.1.7]{Damm04}. We do not impose
any definiteness conditions on $W$. Note that we might include further noise processes or control
dependent noise in (\ref{eq:Ito}) at the price of increasing the
technical burden.
\begin{definition}
Equation (\ref{eq:Ito}) is {\em internally mean square asymptotically
stable}, if for all initial conditions $x_0$ the uncontrolled
solution converges to zero in mean square, that is ${\bf
E}\|x(t,x_0,0)\|^2\to 0$ for $t\to\infty$.
In this case, for brevity, we also
call the pair $(A,N)$ {\em asymptotically stable}. We call an input
signal $u\in L^2_w$ admissible, if also $x(\cdot,0,u)\in L^2_w$.
\end{definition}
It is well known, that the pair $(A,N)$ is asymptotically stable, if
and only if
\begin{displaymath}
\sigma(I\otimes A+A\otimes I+N\otimes N)\subset{\bf C}_-\;,
\end{displaymath}
where $\otimes$ denotes the Kronecker product, \cite{Klei69}.
With (\ref{eq:Ito}) and (\ref{eq:JWR}) we associate the algebraic Riccati equation
\begin{equation}
\label{eq:Riccati}
A ^*P+PA+N ^*PN+W-PBB ^*P=0\;.
\end{equation}
\begin{definition}
A solution $P$ of (\ref{eq:Riccati}) is {\em stabilizing}, if the
pair $(A-BB ^*P,N)$ is asymptotically stable. We call the triple
$(A,N,B)$ {\em stabilizable}, if there exists a matrix $F$, such
that $(A+BF,N)$ is asymptotically stable.
\end{definition}
We now relate the existence of stabilizing solutions of
(\ref{eq:Riccati}) to a coercivity condition. Recall from Remark
\ref{rem:fdc_are} that the frequency condition is used for this
purpose in the deterministic case. For stochastic systems, however,
there is no obvious way to define a transfer function.
\begin{theorem}
Let $(A,N,B)$ be {\em stabilizable}.\\
The Riccati equation (\ref{eq:Riccati}) possesses a stabilizing
solution, if and only if for some $ \varepsilon>0$ and all admissible
$u$ the coercivity condition $J(0,u)\ge
\varepsilon\|x\|_{L_w^2}^2$ holds.
\end{theorem}
\bf Proof: \nopagebreak \rm
We develop the proof along results available in the literature.\\
Let $W=W_1-W_2$, where both $W_1,W_2>0$, and consider first the
definite LQ-problem with the cost functional
\begin{displaymath}
J_{W_1}(x_0,u)={\bf E}\int_0^\infty (x ^*W_1x+\|u\|^2)\,dt\;.
\end{displaymath}
Then it is known from \cite{Wonh68}, that a minimizing control $u_1$
for $J_{W_1}$ is given in the form $u_1=Fx=-B ^*P_1x$, where $P_1$ is the
unique stabilizing solution of the Riccati equation
\begin{equation}
\label{eq:Riccati1}
A ^*P+PA+N ^*PN+W_1-PBB ^*P=0\;.
\end{equation}
For a control of the form $u=-B ^*P_1x+u_2$ it follows that
\begin{equation}\label{eq:J1x0}
J(x_0,u)=x_0 ^*Px_0+{\bf E}\int_0^\infty \left(\|u_2(t)\|^2-x(t) ^*W_2x(t)\right)\,dt\;,
\end{equation}
where now $x(t)$ is the solution of the closed loop equation
\begin{equation}\label{eq:P1closed}
dx=(A-BB ^*P_1)x\,dt+Nx\,dw+Bu_2\,dt\;
\end{equation}
with initial value $x_0$. Our next goal is to minimize
\begin{displaymath}
J_{W_2}(x_0,u_2)={\bf E}\int_0^\infty \left(\|u_2(t)\|^2-x(t) ^*W_2x(t)\right)\,dt
\end{displaymath}
subject to (\ref{eq:P1closed}).
If we factorize $W_2=C_2 ^*C_2$ and set $y(t)=C_2x(t)$, then we recognize $J_{W_2}$ as the cost
functional related to the stochastic bounded real lemma, \cite[Theorem
2.8]{HinrPrit98}, see also e.g.\ \cite{DragMoro13}.
The associated Riccati {\em inequality}
\begin{equation}
\label{eq:Riccati2}
(A-BB ^*P_1) ^*P+P(A-BB ^*P_1)+N ^*PN+W_2-PBB ^*P>0\;,
\end{equation}
possesses a solution $\hat P<0$, if and only if there exists a $\delta>0$,
such that
\begin{equation}\label{eq:BRL_condition}
J_{W_2}(0,u_2)\ge\delta\|u_2\|_{L^2_w}^2\mbox{ for all } u_2\in {L^2_w}
\end{equation}
see \cite[Corollary 2.14]{HinrPrit98}. By \cite[Theorem 5.3.1]{Damm04}
this is equivalent to the
corresponding Riccati {\em equation} having a stabilizing solution
$P_2<0$. \\
Note now that for $P=P_1+P_2$, the Riccati equation
(\ref{eq:Riccati}) holds because
\begin{eqnarray*}
0&=&(A-BB ^*P_1) ^*P_2+P_2(A-BB ^*P_1)+N ^*P_2N+W_2-P_2BB ^*P_2\\
&=&A ^*P_2+P_2A+N ^*P_2N-W_2-PBB ^*P+P_1BB ^*P_1\\
&=&A ^*P+PA+N ^*PN+W-PBB ^*P\;.
\end{eqnarray*}
Moreover, the pair $(A-BB ^*P,N)=(A-BB ^*P_1-BB ^*P_2,N)$ is
stabilizing.
It remains to show that (\ref{eq:BRL_condition}) is equivalent to the
coercivity condition.
As above, let $u\in {L^2_w}$ be of the
form $u=-B ^*P_1x+u_2$.
Assume first that the coercivity condition holds. By (\ref{eq:J1x0}) we have
\begin{displaymath}
J(0,u)=J_{W_2}(0,u_2)=\|u_2\|_{L^2_w}^2-\|y\|_{L^2_w}^2\ge \varepsilon^2 \|x\|_{L^2_w}^2\;,
\end{displaymath}
where $x$ solves (\ref{eq:P1closed}) and $y=C_2x$. It follows that
$\|x\|_{L^2_w}\le \frac1{\|C_2\|}\|y\|_{L^2_w}$, whence
\begin{displaymath}
\|u_2\|_{L^2_w}^2\ge \left(1+\frac{ \varepsilon^2}{\|C_2\|^2}\right)
\|y\|_{L^2_w}^2=\alpha \|y\|_{L^2_w}^2
\end{displaymath}
with $\alpha>1$. Hence, with $\delta^2=1-\frac1\alpha>0$, we have
\begin{displaymath}
J_{W_2}(0,u_2)= \|u_2\|_{L^2_w}^2-\|y\|_{L^2_w}^2=\frac1\alpha
\|u_2\|_{L^2_w}^2-\|y\|_{L^2_w}^2+\delta^2 \|u_2\|_{L^2_w}^2\ge \delta^2 \|u_2\|_{L^2_w}^2
\end{displaymath}
for all $u_2\in L^2_w$, which is (\ref{eq:BRL_condition}).
Vice versa, assume (\ref{eq:BRL_condition}).
Since
(\ref{eq:P1closed}) is asymptotically stable, the system has finite
input to state gain $\gamma$, such that $\|x\|_{L^2_w}\le \gamma
\|u_2\|_{L^2_w}$. Therefore
\begin{displaymath}
J(0,u)=J_{W_2}(0,u_2)\ge \frac{\delta^2}{\gamma^2} \|x\|_{L^2_w}^2=
\varepsilon^2\|x\|_{L^2_w}^2
\end{displaymath}
with $ \varepsilon=\frac{\delta}{\gamma}$. This is the coercivity condition.
\eprf
\section{Conclusion}
\label{sec:conclusion}
We have provided a time domain substitute for the frequency domain
condition of the Kalman-Yakubovich-Popov lemma. The equivalence of the
two criteria has been proven and the applicability has been
demonstrated for a stochastic linear quadratic control problem.
\end{document} |
\begin{document}
\title{The Use of Deep Learning for Symbolic
Integration \
A Review of (Lample and Charton, 2019)}
\begin{abstract}
Lample and Charton (2019) describe a system that uses deep learning
technology to compute symbolic, indefinite integrals, and to find symbolic
solutions to first- and second-order
ordinary differential equations, when the solutions are elementary
functions.
They found that, over a particular test set,
the system could find solutions more successfully than sophisticated packages
for symbolic mathematics such as Mathematica run with a long time-out.
This is an impressive accomplishment, as far as it goes. However,
the system can handle only a quite limited subset of the problems that
Mathematica deals with, and the test set has significant built-in biases.
Therefore the claim that this outperforms
Mathematica on symbolic integration needs to be very much qualified.
\end{abstract}
Lample and Charton (2019) describe a system (henceforth LC)
that uses deep learning
technology to compute symbolic, indefinite integrals, and to find symbolic
solutions to first- and second-order
ordinary differential equations, when the solutions are elementary
functions (i.e. compositions of the arithmetic operators with the exponential
and trigonometric functions and their inverses). They found that, over a
particular test set,
LC could find solutions more successfully than sophisticated packages
for symbolic mathematics such as Mathematica given a long time-out.
This is an impressive accomplishment; however, it is important to understand
its scope and limits.
We will begin by discussing the case of symbolic integration, which is simpler.
Our discussion of ODE's is much the same; however, that introduces
technical complications that are largely extraneous to the points we want to
make.
\section{Symbolic integration}
There are three categories of computational symbolic mathematics that are
important here:
\begin{itemize}
\item {\bf Symbolic differentiation.} Using the standard rules for
differential calculus, this is easy to program and efficient to execute.
\item {\bf Symbolic integration}
This is difficult. In most cases, the integral of an elementary function
that is not extremely simple is not, itself, an elementary function. In
principle, the decision problem whether the integral of an elementary function
is itself elementary is undecidable (Richardson, 1969).
Even
in the cases where the integral is elementary, finding it can be very
difficult. Nonetheless powerful modules for symbolic integration have been
incorporated in systems for symbolic math like Mathematica, Maple, and
Matlab.
\item {\bf Simplification of symbolic expressions.}
The decision problem of determining whether an elementary expression is
identically equal to zero is undecidable (Richardson, 1969).
Symbolic math platforms incorporate powerful modules, but
but building a high-quality system is a substantial undertaking.
\end{itemize}
If one can, in one way or another, conjecture that the integral of elementary
function $f$ is elementary function $g$ (both functions being specified
symbolically) then verifying that conjecture involves,
first computing the derivative $h = f^{\prime}$ and, second, determining that
the expression $h-g$ simplifies to 0. As we have stated, the first step is
easy; the second step is hard in principle but often reasonably
straightforward in practice.
Given an elementary expression $f$, finding an elementary symbolic integral
is, in general, a search in an enormous and strange state space for something
that most of the time does not even exist. Even if you happen to know that it
exists, as is the case with the test examples used by Lample and Charton,
it remains a very hard problem.
\section{What LC does and how it works}
At a high level, LC works as follows:
\begin{itemize}
\item A large corpus of examples (80 million)
was created synthetically by generating random,
complex pairs of symbolic expressions and their derivatives. We will discuss
below how that was done.
\item A seq2seq transformer model is trained on the corpus.
\item At testing time, given a function $g$ to integrate, the model was
executed, using a beam search of width either 1, 10, or 50.
An answer $f$ produced by the model
was checked using the procedure described above:
the symbolic differentiator was applied to $f$, and then the symbolic
simplifier tested whether $f'=g$.
\end{itemize}
In effect, the process of integration is being treated as something like
machine translation: the source is the integrand, the target is the
integral.
Three techniques were used to generate integral/derivative pairs:
\begin{itemize}
\item {\bf Forward generation} (FWD). Randomly generate a symbolic function;
give it to a preexisting symbolic integrator; if it finds an answer, then
record the pair. 20 million such pairs were created. This tends to generate
pairs with comparatively small derivative and large integrals, in terms of
the size of the symbolic expression.
\item {\bf Backward generation} (BWD). Randomly generate a symbolic function;
compute its derivative; and record the pair. 40 million such pairs were
created. The form of the derivative is simplified by symbolic techniques
before the pair is recorded.
This approach
tends to generate pairs with comparatively small integrals and
derivatives that are almost always much larger.
\item {\bf Integration by parts} (IBP). If, for some functions $f$ and $G$,
LC has computed that the
integral of $f$ is $F$ and that the integral of the product $fG=H$, then,
by the rule of integration by parts.
\[ \int Fg = FG - H \]
where $g$ is the derivative of $G$.
It can now record $Fg$ and its integral as a
new pair. 20 million such pairs were created.
\end{itemize}
The comparisons to Mathematica, Maple, and Matlab were carried out using
entirely items generated by BWD,
They found that LC was able to solve
a much higher percentage of this test set than Mathematica, Maple, or Matlab
giving an extended time out to all of these.
Mathematica, the highest scoring of these, was able to solve 84\% of the
problems in the test set, whereas, running with a beam size of 1,
LC produced the correct solution
98.4\% of the time.
\section{No integration without simplification!}
There are many problems where it is critical to simplify an integrand
before carrying out the process of integration.
Consider the following integral:
\begin{equation}
\int \sin^{2}(e^{e^x}) + \cos^{2}(e^{e^x}) \: dx
\end{equation}
At first glance, that looks scary, but in fact it is just a ``trick question''
that some malevolent calculus teacher might pose. The integrand is identically
equal to 1, so the integral is $x+c$.
Obviously, one can spin out examples of this kind to arbitrary complexity.
The reader might enjoy evaluating this (2)
\[
\int \sin(e^{x}+\frac{e^{2x}-1}{2\cos^{2}(\sin(x))-1)}) -
\cos(\frac{(e^{x}+1)(e^{x}-1)}{\cos(2\sin(x))})\sin(e^{x}) -
\cos(e^{(x^{3}+3x^{2}+3x+1)^{1/3}-1})
\sin(\frac{e^{2x}-1}{1-2\sin^{2}(\sin(x))})
\]
or not. Anyway, rule-based symbolic systems can and do quite easily carry out a
sequence of transformations to do the simplification here
This raises two issues:
\begin{itemize}
\item It is safe to assume that few examples of this form, of any significant
complexity, were included in Lample and Charton's corpus. BWD and IBP cannot
possibly generate these. FWD could, in principle, but the integrand would have
to be generated at random, which is extremely improbable. Therefore, LC
was not tested on them.
\item Could LC
have found a solution to such a problem if it were tested on one?
The answer to that question is certainly yes, if the test procedure
begins by applying a high quality simplifier and reduces the integrand to
a simple form.
Alternatively, the answer is also yes if, with integral (1),
LC proposes ``$f(x)=x$'' as a candidate solution
then the simplifier verifies that, indeed, $f'$ is equal to the complex
integrand. It does seem rather unlikely that LC, operating on
the integrand in equation (1), would propose $f(x)=x$ as a candidate. If
one were to construct a problem
where a mess like integral (2) has some moderately
complicated solution --- say, $\log(x^{2}/\sin(x))$ --- which, of course,
is easily done, the likelihood that LC will find it seems still
smaller; though certainly there is no way to know until you try.
\end{itemize}
Fran\c{c}ois Charton informs me (personal communication) that in fact
LC did not do simplifications at this stage and therefore would not
have been able to solve these problem.
This is not a serious strike against their general methodology.
The problem is easily fixed; they can add easily add calls
to the simplifier at the appropriate steps, and they could automatically
generate examples of this kind for their corpus
by using an ``uglifier'' that turns simple
expressions into equivalent complicated ones.
But the point is that there a class of problems whose solution inherently
requires a high quality simplifier, and which currently is not being tested.
One might object that problems of this form are very artificial. But the entire
problem that LC addresses is very artificial. If there is any
natural application that tends to generate lots of problems of integrating
novel complicated
symbolic expressions with the property that a significant fraction of those
have integrals that are elementary functions, I should like to
hear about it. As far as I know, the problem is purely of mathematical
interest.
Another situation where simplification is critical:
Suppose that you take an integrand
produced by BWD and make some small change. Almost certainly, the new function
$f$ has no elementary integral. Now give it to LC. LC will produce an
answer $g$,
because it always produces an answer, and that answer will be wrong,
because there is no right answer. In a situation where you actually cared about
the integral of $f$, it would be undesirable to accept $g$ as an answer.
So you can add a check; you differentiate $g$ and check whether $g'=f$. But
now you are checking for the equivalence of two very complicated expressions,
and again you would need a very high-powered simplifier.
\section{Differential equations}
Lample and Charton have developed a very ingenious technique in which you
can input any elementary function $f(x,c)$ with a single occurrence
of parameter $c$, and find a
an ODE whose solution is $f(x,c)$ where $c$ is the free parameter of the
integral. They also can do the corresponding thing for second-order
equations.
Their overall procedure was then essentially the same as for integrals:
They generated a large corpus of pairs of equations and solutions,
and trained a seq2seq neural network. At testing time, LC used the neural
network to carry out a beam search which generated candidates; each
candidate was tested to see whether it was a solution to the problem.
Here the results were more mixed. With first-order equations, LC with a
beam size of 1 comes out slightly ahead of Mathematica (81.2\% to 77.2\%)
with a beam size of 50, it comes out well ahead (97\%). With second
order equations, LC with a beam size of 1 does not do as well as Mathematica
(40.8\% to 61.6\%) but with a beam size of 50 it attains 81.0\%.
The concerns that we have raised in the context of integration apply
here as well, suitably adapted.
\section{Special functions}
Systems such as Mathematica, Maple, and Matlab are able to solve symbolically
many symbolic integration problems and many differential equations in which
the solution is a special function (i.e. a non-elementary function with
a standard name). For instance Mathematica can
integrate the function $\log(1-x/x)$ to get the answer $-$PolyLog(2,x),
It can solve the equation
\[ x^{2}+y^{\prime \prime}(s) + xy^{\prime}(x) +(x^{2}-16)y(x) = 0 \]
to find the solution
$y(x) = c_{1}\mbox{BesselJ}(4,x) + c_{2}\mbox{BesselY}(4,x)$
In principle, LC could be extended to handle these.
In FWD, it would be a matter of including the pairs where the automated
integrator being called generates an expression with a special function.
In BWD, it would be a matter of generating expressions with special
functions and computing their derivatives.
With integration, the impact on performance might be small;
special functions that are integrals of elementary functions
are mostly unary
(though PolyLog is binary), and therefore have only a moderate
impact on the size of the state space. But the ODE solver is a different matter;
many of the functions that arise in solving
ODEs, such as the many variants of Bessel functions, are binary, and
adding expressions that include these expands the search space exponentially.
To put the point another way: When the ODE solver was tested,
Mathematica
was searching through a space of solutions that are includes the
special functions, whereas LC was limited to the much smaller
space of the elementary functions. The tests were designed so that the
solution was always in the smaller space.
LC thus had an entirely unfair advantage.
\section{The Test Set}
There are also issues with the test set. The comparison with
Mathematica, Matlab, and Maple used a test set consisting entirely of problems
generated by BWD (problems generated by FWD by definition can be solved by
symbolic integrators). These inevitably tend to have comparatively small
integrals (in expression size) and long integrals. Unless you are very lucky,
or unless an expression is full of addition and subtraction, the derivative
of an expression of size $n$ has length
$\Omega(n^2)$. For example the derivative
of the function $\sin(\sin(\sin(\sin(x))))$ is
\[
\cos(\sin(\sin(\sin(x)))) \cdot \cos(\sin(\sin(x))) \cdot \cos(\sin(x))
\cdot \cos(x)
\]
And in fact the average length of an integrand in the test set was 70 symbols
with a standard deviation of 47 symbols; thus, a large fraction of the test
examples had 120 symbols or so. (Table 1 of Lample and Charton).
So what the comparison with Mathematica establishes is that,
given a really long expression, which happens
to have a much shorter, exact symbolic integral, LC is awfully good at
finding it. But that is a really special class. One can certainly understand
why the teams building Mathematica and so on have not considered
this niche category of problem much of a priority.
Another point that does not seem to have been tested is whether LC may have
been picking up on arbitrary artifacts of the differentiation process, such
as the order in which parts of a derivative are presented. For instance, the
derivative of a three level composed function $f(g(h(x)))$ is a product of
three terms $h'(x) \cdot g'(h(x)) \cdot f'(g(h(x))$. Any particular
symbolic differentiator will probably generate these in a fixed order, such
as the one above. This particular choice of orderings will then be
consistent thoughout the corpus, so LC will be trained and then tested
only with this ordering. A system like LC may have much more difficulty
finding the integral if the multiplicands are presented in any of the five
other possible orders.
The techniques that LC learns from BWD and FWD are very different. If LC is
trained only on BWD and tested on problems in FWD,
then running with a beam size of 1, it finds the correct solution
to a problem in FWD only 18.9\% of the time; with a beam size of 50, the
correct solution is among its top 50 candidates only 27.5\% of the time.
Training it only on problems in FWD and testing it on problems in BWD it does
even worse, with corresponding success rates of 10.9\% and 17.2\%.
The procedure for generating corpus example will not succeed in creating
examples that combine features from FWD and BWD. For instance, if $f', f$
is a pair that would be naturally generated by FWD, and $g', g$ is a pair
that would be naturally generated by BWD, then the sum $f'+g', f+g$ will
not be included in the corpus, and therefore will not be tested.
In fact: If one were to put together a test set of random, enormously complex
integrands, LC would certainly give a wrong answer on nearly all of them,
because only a small fraction would have an elementary integral. Mathematica,
certainly, would also fail to find an integral, but presumably it would not
give a wrong answer; it would either give up or time out. If you consider
that a wrong answer is worse than no answer, then on this test set, Mathematica
would beat LC by a enormous margin.
\section{Summary}
The fact that LC ``beat'' Mathematica on the test set of integration problems
produced by BWD is certainly impressive. But Lample and Charton's claim
\begin{quote}
[This] transformer model \ldots can perform extremely well both at computing
function integrals and and solving differential equations, outperforming \ldots
Matlab or Mathematica \ldots .
\end{quote}
is very much overstated, and requires significant qualification. The correct
statement, as regards integration, is as follows:
\begin{quote}
The transformer model outperforms Mathematica and Matlab in computing
symbolic
indefinite integrals of enormously complex functions of a single variable `$x$'
whose integral is a much smaller
elementary function containing no constant symbols
other than the integers $-5$ to 5.
\end{quote}
Since both BWD and FWD were limited to functions of a single
variable $x$, it is
unknown whether LC can handle $\int t \: dt$ or $\int a \: dx$ (it's not clear
whether LC's input includes any way to specify the variable of integration) and
essentially certain that it cannnot handle $\int 1/(x^{2} + a^{2}) \: dx$. On
problems like these, far from outperforming Mathematica and Matlab, it
falls far short of a high-school calculus student.
It is important to emphasize that
{\em the construction of LC is entirely dependent on
the pre-existing
symbolic processors developed over the last 50 years by experts in symbolic
mathematics.} Moreover, as things now stand, extending LC to fill in some of
its gaps (e.g. the simplification problems described in section 3) would
make it even less of a stand-alone system and more dependent on
conventional symbolic processors. There is no reason whatever to suppose
that NN-based systems will supercede symbolic mathematics systems any time
in the foreseeable future.
It goes without saying that LC has no understanding of the significance of
an integral or a derivative or even a function or a number.
In fact, occasionally, it
outputs a solution that is not even a well-formed expression. LC is like
the worst possible student in a calculus class: it doesn't understand the
concepts, it doesn't learned the rules, it has no idea what is the
significance of what it is doing, but it has looked at 80 million examples
and gotten a feeling of what integrands and their integrals look like.
Finally LC, like the recent successes in
game-playing AI, depends on the ability to generate enormous quantities
of high-quality (in the case of LC, flawless)
synthetic labelled data. In open-world domains, this is effectively
impossible. Therefore, the success of LC is in no way evidence that
deep learning or other such methods will suffice for high-level reasoning
in real-world situations.
\subsection*{References}
Lample, Guillaume and Fran\c{c}ois Charton, 2019.
``Deep Learning for Symbolic Mathematics''
{\em NeurIPS-2019.}
https://arxiv.org/abs/1912.01412
Richardson, Daniel, 1969. ``Some undecidable problems involving elementary
functions of a real variable."
{\em The Journal of Symbolic Logic,} {\bf 33}:4, 514-520.
WolframAlpha, ``Integrals that Can and Cannot be Done'' \\
https://reference.wolfram.com/language/tutorial/IntegralsThatCanAndCannotBeDone.html
WolframAlpha, ``Symbolic Differential Equation Solving'' \\
https://reference.wolfram.com/language/tutorial/DSolveOverview.html
\end{document} |
{\texttt{b}}egin{document}
\title{Finite and symmetrized colored multiple zeta values}
{\texttt{b}}egin{abstract}
Colored multiple zeta values are special values of multiple polylogarithms evaluated at $N$th roots of unity.
In this paper, we define both the finite and the symmetrized versions of these values and
show that they both satisfy the double shuffle relations.
Further, we provide strong evidence for an isomorphism connecting the two spaces
generated by these two kinds of values.
This is a generalization of a recent work of Kaneko and Zagier on finite and symmetrized multiple zeta
values and of the second author on finite and symmetrized Euler sums.
\end{abstract}
\tableofcontents
\section{Introduction}
\label{sec:Intro}
Let $\gG_N$ be the group of $N$th roots of unity. For ${\texttt{b}}fs:=(s_1,\dots,s_d) \in \mathbb{N}^d$ and ${\texttt{b}}feta:=(\eta_1,\dots,\eta_d)\in (\gG_N)^d$, we define
\emph{colored multiple zeta values} (CMZVs) by
{\texttt{b}}egin{align}\label{def:CMZV}
\zeta\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}:=\sum_{k_1>k_2>{\texttt{c}}dots>k_d>0}
\frac{\eta_1^{k_1}{\texttt{c}}dots\eta_d^{k_d}}{k_1^{s_1}{\texttt{c}}dots k_d^{s_d}}.
\end{align}
We call $d$ the \emph{depth} and $|{\texttt{b}}fs|:=s_1+\dots+s_d$ the \emph{weight}.
These objects were first systematically studied by Deligne, Goncharov, Racinet,
Arakawa and Kaneko {\texttt{c}}ite{Arakawa99,Deligne05,Goncharov01,Racinet02}.
It is not hard to see that the series in \eqref{def:CMZV} diverge if and only if $(s_1,\eta_1)=(1,1)$. By multiplying these series we get the so called stuffle (or quasi-shuffle) relations. Additionally, it turns out that these values can also be expressed by iterated integrals
which lead to the shuffle relations. Note that for $N=1$ we rediscover \emph{multiple zeta values} (MZVs)
{\texttt{b}}egin{align}\label{def:MZV}
\zeta({\texttt{b}}fs):= \sum_{k_1>k_2>{\texttt{c}}dots>k_d>0}
\frac{1}{k_1^{s_1}{\texttt{c}}dots k_d^{s_d}} = \zeta\lrp{{\texttt{b}}fs}{\{1\}^d},
\end{align}
where $\{s\}^d:=(s,\ldots,s)\in \mathbb{N}^d$. When $N=2$ the colored multiple zeta values are usually called \emph{Euler sums}.
In {\texttt{c}}ite{Ihara06} Ihara, Kaneko and Zagier defined the regularized MZVs in two different ways and then obtained the regularized double shuffle relations. Racinet {\texttt{c}}ite{Racinet02} and Arakawa and Kaneko {\texttt{c}}ite{Arakawa04}
further generalized these regularized values to arbitrary levels, which we denote by $\displaystyle \zeta_{\texttt{a}}st\lrpT{{\texttt{b}}fs}{{\texttt{b}}feta}$ and $\displaystyle\zeta_\sha\lrpT{{\texttt{b}}fs}{{\texttt{b}}feta}$,
which are in general polynomials of $T$. Using these values,
we define the \emph{symmetrized colored multiple zeta values} (SCVs) of level $N$ by
{\texttt{b}}egin{align}
\zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}:=&\, \sum_{j=0}^d
(-1)^{s_1+{\texttt{c}}dots+s_j} \ol{\eta_1}{\texttt{c}}dots\ol{\eta_j}\,\zeta_{\texttt{a}}st\lrpT{s_j,\ldots,s_1}{\ol{\eta_j},\ldots,\ol{\eta_1}} \zeta_{\texttt{a}}st\lrpT{s_{j+1},\dots,s_d}{\eta_{j+1},\dots,\eta_d}, \label{equ:astSCV}\\
\zeta_\sha^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}:=&\, \sum_{j=0}^d
(-1)^{s_1+\dots+s_j} \ol{\eta_1}{\texttt{c}}dots\ol{\eta_j}\,\zeta_\sha\lrpT{s_j,\dots,s_1}{\ol{\eta_j},\dots,\ol{\eta_1}} \zeta_\sha\lrpT{s_{j+1},\dots,s_d}{\eta_{j+1},\dots,\eta_d} \label{equ:shaSCV}
\end{align}
for all ${\texttt{b}}fs\in \mathbb{N}^d$ and ${\texttt{b}}feta \in (\gG_N)^d$. This definition includes as special cases the \emph{symmetrized MZVs} (when $N=1$) introduced by Kaneko and Zagier (see also Jarossay {\texttt{c}}ite{Jarossay14}) and the \emph{symmetrized Euler sums} (when $N=2$) established
in {\texttt{c}}ite{Zhao15} by the second author.
We will show that SCVs are actually independent of $T$ (Proposition \ref{prop:SCVconst}) and the two versions are essentially the same modulo $\zeta(2)$ (Theorem \ref{theo:moduloz2}). Furthermore, we prove that they satisfy both the stuffle relations (Theorem \ref{thm:astSCVmorphism}) and the shuffle relations (Theorem \ref{thm:shuffleSCV})
by using two different Hopf algebra structures, respectively.
Let ${\texttt{c}}alP$ be the set of rational primes and $\F_p$ the finite field of $p$ elements. Set
{\texttt{b}}egin{align*}
{\texttt{c}}alP(N):=\{p\in {\texttt{c}}alP {\texttt{c}}olon p\equiv -1 \pmod{N} \},
\end{align*}
which is of infinite cardinality with density $1/\varphi(N)$
by Chebotarev's Density Theorem {\texttt{c}}ite{Tschebotareff26}, where $\varphi(N)$ is
Euler's totient function. Further, we define
{\texttt{b}}egin{equation*}
\F_p[\xi_N]:=\frac{\F_p[X]}{(X^N-1)}=
\left\{ \sum_{j=0}^{N-1} c_j \xi_N^j{\texttt{c}}olon c_0,\dots,c_{N-1}\in \F_p\right\},
\end{equation*}
where $\xi_N:=\xi_{N,p}$ is a fixed primitive root of $X^N-1\in\F_p[X]$. Moreover, we denote
{\texttt{b}}egin{equation*}
{\texttt{c}}alA(N):=\prod_{p\in{\texttt{c}}alP(N)} \F_p[\xi_N] {\texttt{b}}igg/{\texttt{b}}igoplus_{p\in{\texttt{c}}alP(N)} \F_p[\xi_N].
\end{equation*}
We usually remove the dependence of $\xi_N$ on $p$ by abuse of notation. For convenience,
we also identify
$\prod_{p\in{\texttt{c}}alP(N),p>k} \F_p[\xi_N] {\texttt{b}}igg/{\texttt{b}}igoplus_{p\in{\texttt{c}}alP(N),p>k} \F_p[\xi_N]$
with ${\texttt{c}}alA(N)$ by setting the components $a_p=0$ for all $p\le k$.
Now we define the \emph{finite colored multiple zeta values} (FCVs) of level $N$ by
{\texttt{b}}egin{equation}\label{equ:defnFCV}
\zeta_{{\texttt{c}}alA(N)} \lrp{{\texttt{b}}fs}{{\texttt{b}}feta}:=\left( \sum_{p>k_1>k_2>{\texttt{c}}dots>k_d>0}
\frac{\eta_1^{k_1}{\texttt{c}}dots\eta_d^{k_d}}{k_1^{s_1}{\texttt{c}}dots k_d^{s_d}}\right)_{p\in{\texttt{c}}alP(N)} \in {\texttt{c}}alA(N)
\end{equation}
for all ${\texttt{b}}fs\in\N^d$ and ${\texttt{b}}feta\in(\gG_N)^d$. Again, this definition includes as special cases the \emph{finite MZVs} (when $N=1$) and the \emph{finite Euler sums} (when $N=2$).
{\texttt{b}}egin{rem}\label{rem:subtleDefn}
In fact, we have abused the notation in \eqref{equ:defnFCV}. All $\eta_j=\eta_{j,p}$ depend on $p$ so that by a fixed choice ${\texttt{b}}feta\in (\gG_N)^d$ we really mean a fixed choice
of $(e_1,\dots,e_d)\in(\Z/N\Z)^d$ independent of $p$ such that $\eta_{j,p}:=\xi_{N,p}^{e_j}$
for all $p$.
\end{rem}
Similar to SCVs, we shall show that FCVs satisfy both the stuffle and the shuffle relations in Theorem \ref{thm:stuffleFCV} and Theorem \ref{thm:shuffleFCV}, respectively.
The primary motivation for this paper is the following conjectural relation between
SCVs and FCVs. Let $\CMZV_{w,N}$ (resp.\ $\FCV_{w,N}$, resp.\ $\SCV_{w,N}$)
be the $\Q(\gG_N)$-space generated by CMZVs (resp.\ FCVs, resp.\ SCVs) of weight $w$
and level $N$. Set $\nSCV_{0,N}:=\Q(\gG_N)$ and define
$\nSCV_{w,N}:=\SCV_{w,N}+2\pi i\, \SCV_{w-1,N}$ for all $w\ge 1$.
{\texttt{b}}egin{conj}\label{conj:Main}
Let $N\ge 3$ and weight $w\ge 1$. Then:
{\texttt{b}}egin{itemize}
\item [\upshape{(i)}] $\nSCV_{w,N}=\CMZV_{w,N}$ as vector spaces over $\Q(\xi_N)$.
\item [\upshape{(ii)}] We have a $\Q(\gG_N)$-algebra isomorphism
{\texttt{b}}egin{align*}
f{\texttt{c}}olon \FCV_{w,N} &\, \overset{\sim}{\lra} \frac{\CMZV_{w,N}}{2\pi i\, \CMZV_{w-1,N}} \\
\zeta_{{\texttt{c}}alA(N)}\lrp{{\texttt{b}}fs}{{\texttt{b}}feta} &\,\lmaps \ \zeta_{\sha}^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}.
\end{align*}
\end{itemize}
\end{conj}
It follows from this conjecture that
{\texttt{b}}egin{align*}
\frac{\CMZV_{w,N}}{2\pi i\, \CMZV_{w-1,N}}
{\texttt{c}}ong \frac{\nSCV_{w,N}}{2\pi i\, \nSCV_{w-1,N}}
{\texttt{c}}ong \frac{\SCV_{w,N}}{{\texttt{b}}ig(2\pi i\, \SCV_{w-1,N}+\zeta(2)\SCV_{w-2,N}{\texttt{b}}ig){\texttt{c}}ap \SCV_{w,N}}.
\end{align*}
So the map $f$ in Conjecture~\ref{conj:Main} (ii) is surjective. The analogous result
for MZVs at level $1$ corresponding to Conjecture~\ref{conj:Main} (i) has
been proved by Yasuda {\texttt{c}}ite{Yasuda2014}.
For numerical examples supporting Conjecture~\ref{conj:Main} at level 3 and 4, see Section \ref{sec:numex}.
This conjecture should be regarded as a generalization of the corresponding conjectures
of Kaneko and Zagier for the MZVs and of the second author for the Euler sums {\texttt{c}}ite{Zhao15},
where $2\pi i$ is replaced by $\zeta(2)$ since the MZVs and Euler sums are all real numbers.
We are sure the final proof of Conjecture~\ref{conj:Main} will need the $p$-adic version
of the generalized Drinfeld associators whose coefficients should satisfy the same algebraic
relations as those of CMZVs plus the equation ``$2\pi i=0$'' when level $N\ge 3$.
{\texttt{b}}igskip
{{\texttt{b}}f{Acknowledgements.}}
The authors would like to thank the ICMAT at Madrid, Spain, for its warm hospitality and
gratefully acknowledge the support by the Severo Ochoa Excellence Program.
{\texttt{b}}igskip
\section{Algebraic framework}
\label{sec:algframe}
The study of MZVs with word algebras was initiated by Hoffman {\texttt{c}}ite{Hoffman97}
and generalized by Racinet {\texttt{c}}ite{Racinet02} to deal with colored MZVs, which
we now review briefly.
Fix a positive integer $N$ as the level. Define the alphabet $X_N:=\{x_\eta{\texttt{c}}olon \eta \in \gG_N{\texttt{c}}up \{0\}\}$ and
let $X_N^{\texttt{a}}st$ be the set of words over $X_N$ including the empty word ${\texttt{b}}e$.
Denoted by $\fA_N$ the free noncommutative polynomial algebra in $X_N$, i.e., the algebra of words on $X_N^{\texttt{a}}st$.
The \emph{weight} of a word ${\texttt{b}}fw \in \fA_N$, denoted by $|{\texttt{b}}fw|$,
is the number of letters contained in ${\texttt{b}}fw$, and its \emph{depth}, denoted by $\dep({\texttt{b}}fw)$,
is the number of letters $x_\eta$ ($\eta\in\gG_N$) contained in ${\texttt{b}}fw$.
Further, let $\fA_N^1$ denote the subalgebra of $\fA_N$ consisting of words not ending with $x_0$. Hence, $\fA_N^1$ is generated by words of the form $y_{m,\mu}:=x_0^{m-1}x_\mu$ for $m\in \mathbb N$ and $\mu \in \gG_N$.
Define the alphabet $Y_N=\{y_{k,\mu}: k\in \mathbb N,\mu \in \gG_N\}$ and $Y_N^{\texttt{a}}st$ is
the set of words (including the empty word) over $Y_N$. Additionally,
let $\fA_N^0$ denote the subalgebra of $\fA_N^1$ with words not beginning with $x_1$ and not ending with $x_0$.
The words in $\fA_N^0$ are called \emph{admissible words.}
We now equip $\fA_N^1$ with a Hopf algebra structure $(\fA_N^1,{\texttt{a}}st,\widetilde{\Delta}_{\texttt{a}}st)$.
The \emph{stuffle product} ${\texttt{a}}st {\texttt{c}}olon \fA_N^1 \otimes \fA_N^1 \to \fA_N^1$ is defined as follows:
{\texttt{b}}egin{enumerate}[(ST1)]
\item ${\texttt{b}}e {\texttt{a}}st w := w {\texttt{a}}st {\texttt{b}}e := w$,
\item $y_{m,\mu}u {\texttt{a}}st y_{n,\nu}v :=y_{m,\mu}(u {\texttt{a}}st y_{n,\nu}v)+y_{n,\nu}(y_{m,\mu}u {\texttt{a}}st v) + y_{m+n,\mu\nu}(u{\texttt{a}}st v)$,
\end{enumerate}
for any word $u,v,w\in \fA_N^1$, $m,n\in \mathbb N$ and $\mu,\nu\in \gG_N$. Then linearly extend it to $\fA_N^1$.
The coproduct $\widetilde{\Delta}_{\texttt{a}}st {\texttt{c}}olon \fA_N^1 \to \fA_N^1\otimes \fA_N^1$ is defined by deconcatenation:
{\texttt{b}}egin{align*}
\widetilde{\Delta}_{\texttt{a}}st(y_{s_1,\eta_1}{\texttt{c}}dots y_{s_d,\eta_d}):= \sum_{j=0}^d y_{s_1,\eta_1}{\texttt{c}}dots y_{s_j,\eta_j}\otimes y_{s_{j+1},\eta_{j+1}} {\texttt{c}}dots y_{s_d,\eta_d}.
\end{align*}
Note that $(\fA_N^0,{\texttt{a}}st)$ is a sub-algebra (but not a sub-Hopf algebra).
We also need another Hopf algebra structure $(\fA_N,\sha, \widetilde{\Delta}_\sha)$ which
will provide the shuffle relations. Here, the \emph{shuffle product} $\sha{\texttt{c}}olon \fA_N\otimes \fA_N\to \fA_N$
is defined as follows:
{\texttt{b}}egin{enumerate}[(SH1)]
\item ${\texttt{b}}e \sha w := w \sha {\texttt{b}}e := w$,
\item $au \sha b v := a(u\sha bv) + b(au\sha v)$,
\end{enumerate}
for any word $u,v,w \in \fA_N$ and $a,b\in X_N$. Then linearly extend it to $\fA_N$.
The coproduct $\widetilde{\Delta}_\sha {\texttt{c}}olon \fA_N \to \fA_N\otimes \fA_N$ is
defined (again) by deconcatenation:
{\texttt{b}}egin{align*}
\widetilde{\Delta}_\sha (x_{\eta_1}{\texttt{c}}dots x_{\eta_d}):= \sum_{j=0}^d x_{\eta_1}{\texttt{c}}dots x_{\eta_j}\otimes x_{\eta_{j+1}}{\texttt{c}}dots x_{\eta_{d}},
\end{align*}
where $\eta_1,\ldots,\eta_d\in \gG_N{\texttt{c}}up \{0\}$.
Note that both $(\fA_N^1,\sha)$ and $(\fA_N^0,\sha)$ are sub-algebras
(but not as sub-Hopf algebras).
Finally, we remark that both $(\fA_N^1,{\texttt{a}}st)$ and $(\fA_N,\sha)$ are commutative
and associative algebras.
\section{Finite colored multiple zeta values}
We recall that finite MZVs and finite Euler sums are elements of the $\Q$-ring
{\texttt{b}}egin{align*}
{\texttt{c}}alA:=\prod_{p\in{\texttt{c}}alP} \F_p {\texttt{b}}igg/{\texttt{b}}igoplus_{p\in{\texttt{c}}alP} \F_p.
\end{align*}
Note that for $N=1,2$, ${\texttt{c}}alA$ can be identified with ${\texttt{c}}alA(N)$
since all primes greater than 2 are odd and we can safely disregard the prime $p=2$.
By our choice of the primes in ${\texttt{c}}alP(N)$ it follows immediately from Fermat's Little Theorem
that the Frobenius endomorphism ($p$-power map at prime $p$-component)
is given by the \emph{conjugation} $(\ol{a_p})_p\in {\texttt{c}}alA(N)$, where
{\texttt{b}}egin{align}\label{equ:pPower=inv}
\ol{\sum_{j=0}^{N-1} c_j \xi_N^j}:=\sum_{j=0}^{N-1} c_j \xi_N^{N-j}, \quad \quad (c_j\in \F_p).
\end{align}
The following lemma implies that ${\texttt{c}}alA(N)$ is in fact a $\Q(\gG_N)$-vector space.
{\texttt{b}}egin{lem}
The field $\Q(\gG_N)$ can be embedded into ${\texttt{c}}alA(N)$ diagonally.
\end{lem}
{\texttt{b}}egin{proof}
The map $\phi{\texttt{c}}olon \mathbb Q \to {\texttt{c}}alA(N)$, $r\mapsto (\phi_p(r))_{p\in {\texttt{c}}alP(N)}$ given by $\phi(0)=(0)_p$ and
{\texttt{b}}egin{align*}
\phi_p(r):={\texttt{b}}egin{cases}
r \pmod p & \text{~if~} \ord_p(r)\geq 0, \\
0 & \text{~otherwise},
\end{cases}
\end{align*}
embeds $\mathbb Q$ diagonally in ${\texttt{c}}alA(N)$ due to the fundamental theorem of arithmetic. The cyclotomic field $\mathbb Q(\gG_N)$ is given by
{\texttt{b}}egin{align*}
\mathbb Q(\gG_N):=\left\{\sum_{j=0}^{N-1}a_j\xi_N^j: a_j\in \mathbb Q \right\},
\end{align*}
where $\xi_N\in\gG_N $ is a primitive element. Therefore $\varphi{\texttt{c}}olon \Q(\gG_N) \to {\texttt{c}}alA(N)$ defined by
{\texttt{b}}egin{align*}
\sum_{j=0}^{N-1}a_j\xi_N^j \longmapsto \left(\sum_{j=0}^{N-1} \phi_p(a_j)\xi_N^j \right)_{p\in {\texttt{c}}alP(N)}
\end{align*}
is an embedding.
\end{proof}
In this section, we study $\mathbb Q(\gG_N)$-linear relations among FCVs by developing a double shuffle picture.
First, similar to MZVs, the stuffle product is simply induced by the defining series of
FCVs \eqref{equ:defnFCV}. For example, we have
{\texttt{b}}egin{align*}
\sum_{p>k>0}\frac{{\texttt{a}}lpha^k}{k^a} \sum_{p>l>01}\frac{{\texttt{b}}eta^l}{l^b} = \sum_{p>k>l>0}\frac{{\texttt{a}}lpha^k{\texttt{b}}eta ^l}{k^al^b} + \sum_{p>l>k>0}\frac{{\texttt{a}}lpha^k{\texttt{b}}eta ^l}{k^al^b} + \sum_{p>k>0}\frac{({\texttt{a}}lpha{\texttt{b}}eta)^k}{k^{a+b}}
\end{align*}
for $a,b\in \mathbb N$ and ${\texttt{a}}lpha,{\texttt{b}}eta \in \gG_N$.
On the other hand, the shuffle product is more involved than that for the MZVs
since apparently there is no integral representation for FCVs available. However, we
will deduce the shuffle relations by using the integral representation of the single variable
multiple polylogarithms.
Define the $\Q(\gG_N)$-linear map $\zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}{\texttt{c}}olon \fA_N^1\to{\texttt{c}}alA(N)$ by setting
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}(w):=\zeta_{{\texttt{c}}alA(N)}\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}
\end{align*}
for any word $\displaystyle w=W\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}:=y_{s_1,\eta_1} {\texttt{c}}dots y_{s_d,\eta_d}
\in \fA_N^1$, where ${\texttt{b}}fs=(s_1,\ldots,s_d)\in \N^d$ and ${\texttt{b}}feta=(\eta_1,\dots,\eta_d)\in(\gG_N)^d$.
{\texttt{b}}egin{thm}\label{thm:stuffleFCV}
The map $\zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}{\texttt{c}}olon (\fA_N^1,{\texttt{a}}st)\to{\texttt{c}}alA(N)$ is an algebra homomorphism.
\end{thm}
{\texttt{b}}egin{proof}
Let $u,v\in \fA_N^1$. It is easily seen by induction on $|u|+|v|$ that
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}(u{\texttt{a}}st v)= \zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}(u)\zeta_{{\texttt{c}}alA(N),{\texttt{a}}st}(v),
\end{align*}
which concludes the proof.
\end{proof}
We now define a map ${\texttt{b}}fp{\texttt{c}}olon \fA_N^1\to \fA_N^1$ by setting
{\texttt{b}}egin{align*}
{\texttt{b}}fp(y_{s_1,\eta_1}y_{s_2,\eta_2}{\texttt{c}}dots y_{s_d,\eta_d})= y_{s_1,\eta_1}y_{s_2,\eta_1\eta_2}{\texttt{c}}dots y_{s_d,\eta_1\eta_2{\texttt{c}}dots \eta_d}
\end{align*}
and its inverse ${\texttt{b}}fq{\texttt{c}}olon \fA_N^1\to \fA_N^1$ by setting
{\texttt{b}}egin{align*}
{\texttt{b}}fq(y_{s_1,\eta_1}y_{s_2,\eta_2}{\texttt{c}}dots y_{s_d,\eta_d})= y_{s_1,\eta_1}y_{s_2,\eta_2 \eta_1^{-1}}{\texttt{c}}dots y_{s_d,\eta_d\eta_{d-1}^{-1}}.
\end{align*}
Further, set the map $\tau{\texttt{c}}olon \fA_1^1 \to \fA_1^1$ by defining $\tau({\texttt{b}}e):={\texttt{b}}e$ and
{\texttt{b}}egin{align*}
\tau(x_0^{s_1-1}x_1{\texttt{c}}dots x_0^{s_d-1}x_1) := (-1)^{s_1+{\texttt{c}}dots+s_d} x_0^{s_d-1}x_1{\texttt{c}}dots x_0^{s_1-1}x_1
\end{align*}
and extended to $\fA_1^1$ by linearity.
The next theorem can be proved in the same manner as that for {\texttt{c}}ite[Thm.~3.11]{Zhao15}.
In order to be self-contained, we present its complete proof. For any
$s,s_1,\dots,s_d\in\N$ and $\xi,\xi_1,\dots,\xi_d\in\gG_N$, we define
the function with complex variable $z$ with $|z|<1$ by
{\texttt{b}}egin{align*}
\zeta_\sha(y_{s,\xi} y_{s_1,\xi_1} \dots y_{s_d,\xi_d};z)
:=& \Li_{s,s_1,\dots,s_d}(z\xi,\xi_1/\xi,\xi_2/\xi_1,\dots,\xi_d/\xi_{d-1}) \\
=&\, \int_{[0,z]} \left(\frac{dt}{t}\right)^{s-1} \frac{dt}{\xi^{-1}-t}
\left(\frac{dt}{t}\right)^{s_1-1} \frac{dt}{\xi_1^{-1}-t} {\texttt{c}}dots
\left(\frac{dt}{t}\right)^{s_d-1} \frac{dt}{\xi_d^{-1}-t},
\end{align*}
where $[0,z]$ is the straight line from 0 to $z$. It is clear that
$\zeta_\sha(-;z)$ is well-defined.
By the shuffle relation of iterated integrals, we get
{\texttt{b}}egin{align}\label{equ:zetaShaz}
\zeta_\sha(u;z)\zeta_\sha(v;z)=\zeta_\sha(u\sha v;z)
\end{align}
for all $u,v\in Y_N^{\texttt{a}}st$.
{\texttt{b}}egin{thm}\label{thm:shuffleFCV}
Let $N\ge 1$. Define the map $\zeta_{{\texttt{c}}alA(N),\sha}:= \zeta_{{\texttt{c}}alA(N),{\texttt{a}}st} {\texttt{c}}irc {\texttt{b}}fq{\texttt{c}}olon \fA_N^1\to{\texttt{c}}alA(N)$.
Then we have
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(N),\sha}(u \sha v)=\zeta_{{\texttt{c}}alA(N),\sha}(\tau(u)v)
\end{align*}
for any $u\in \fA_1^1$ and $v\in \fA_N^1$. These relations are called linear shuffle relations
for the FCVs.
\end{thm}
{\texttt{b}}egin{proof}
Obviously, it suffices to prove
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(N),\sha}((x_0^{s-1}x_1 u) \sha v) =(-1)^s \zeta_{{\texttt{c}}alA(N),\sha}(u \sha (x_0^{s-1}x_1v))
\end{align*}
for $s\in \mathbb N$, $u\in \fA_1^1$ and $v\in \fA_N^1$. For simplicity we use the notation $a:=x_0$ and $b:=x_1$ in the rest of this proof.
Let $w$ be a word in $\fA_N^1$, i.e., there exist ${\texttt{b}}fs \in \mathbb{N}^d$ and ${\texttt{b}}fxi\in (\gG_N)^d$ such that $w=y_{s_1,\xi_1} \dots y_{s_d,\xi_d}$.
Then there exists ${\texttt{b}}feta\in (\gG_N)^d$, such that $\displaystyle {\texttt{b}}fq(w)=W\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}$.
For any prime $p>2$, the coefficient of
$z^p$ in $\zeta_\sha(bw;z)$ is given by
{\texttt{b}}egin{equation*}
{\rm Coeff}_{z^p}\Big[\zeta_\sha\lrp{1,s_1,\dots,s_d}{z,\xi_1,\dots,\xi_d} \Big]=
\frac{1}{p}\sum_{p>k_1>{\texttt{c}}dots>k_d>0}
\frac{\eta_1^{k_1}{\texttt{c}}dots \eta_d^{k_d}}{k_1^{s_1}{\texttt{c}}dots k_d^{s_d}}
=\frac{1}{p} H_{p-1}({\texttt{b}}fq(w)),
\end{equation*}
where
{\texttt{b}}egin{align*}
H_k(y_{s_1,\eta_1} \dots y_{s_d,\eta_d}):=\sum_{k\geq k_1>{\texttt{c}}dots>k_d>0}
\frac{\eta_1^{k_1}{\texttt{c}}dots \eta_d^{k_d}}{k_1^{s_1}{\texttt{c}}dots k_d^{s_d}}.
\end{align*}
Observe that
{\texttt{b}}egin{align}\label{eq:shdecomp}
b \Big( (a^{s-1}b u) \sha v -(-1)^s u \sha (a^{s-1}b v)\Big)
=\sum_{j=0}^{s-1} (-1)^{j} (a^{s-1-j}b u) \sha (a^j b v).
\end{align}
By applying $\zeta_\sha(-;z)$ to \eqref{eq:shdecomp} and then extracting
the coefficients of $z^p$ from both sides we obtain
{\texttt{b}}egin{align*}
&\, \frac{1}{p}\Big( H_{p-1}{\texttt{c}}irc{\texttt{b}}fq{\texttt{b}}ig((a^{s-1}b u) \sha v{\texttt{b}}ig)
-(-1)^s H_{p-1}{\texttt{c}}irc{\texttt{b}}fq{\texttt{b}}ig(u \sha (a^{s-1}b v){\texttt{b}}ig)\Big)\\
=&\,\sum_{j=0}^{s-1} (-1)^{j} {\rm Coeff}_{z^p}
{\texttt{b}}ig[\zeta_\sha(a^{s-1-j}b u;z)\zeta_\sha(a^{j}b v;z){\texttt{b}}ig] \\
=&\,\sum_{j=0}^{s-1} (-1)^{j}\sum_{j=1}^{p-1}
{\rm Coeff}_{z^j}{\texttt{b}}ig[\zeta_\sha(a^{s-1-j}b u;z) {\texttt{b}}ig]
{\rm Coeff}_{z^{p-j}}{\texttt{b}}ig[\zeta_\sha(a^{j}b v;z){\texttt{b}}ig]
\end{align*}
by \eqref{equ:zetaShaz}. Since $p-j<p$ and $j<p$ the last sum
is $p$-integral. Therefore we get
{\texttt{b}}egin{equation*}
H_{p-1}{\texttt{c}}irc{\texttt{b}}fq((a^{s-1}b u) \sha v)\equiv
(-1)^s H_{p-1}{\texttt{c}}irc{\texttt{b}}fq(u \sha (a^{s-1}b v)) \pmod{p}
\end{equation*}
which completes the proof of the theorem.
\end{proof}
\section{Symmetrized colored multiple zeta values}
\subsection{Regularizations of colored MZVs}
Since the regularized colored MZVs (recalled in Theorem \ref{theo:Racinet} below)
are polynomials of $T$, both of the two kinds of SCVs defined by
\eqref{equ:astSCV} and \eqref{equ:shaSCV} are \emph{a priori} also polynomials of $T$.
We show first that they are in fact constant complex numbers. To this end,
we define two maps $\zeta_{\texttt{a}}st,\zeta_\sha {\texttt{c}}olon \fA_N^0 \to \CC$ such that for any word
$\displaystyle w=W\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}\in \fA_N^0$
{\texttt{b}}egin{align*}
\zeta_{\texttt{a}}st(w):=\zeta\lrp{{\texttt{b}}fs}{{\texttt{b}}feta} \quad \quad\text{~and~}\quad \quad \zeta_\sha(w):=\zeta_\sha \Lrp{s_1,s_2,\ldots,\ \, s_d\ \, }{\eta_1,\frac{\eta_{2}}{\eta_1},\ldots, \frac{\eta_{d}}{\eta_{d-1}}}.
\end{align*}
{\texttt{b}}egin{lem}[{\texttt{c}}ite{Arakawa04,Racinet02}]
The maps $\zeta_{\texttt{a}}st {\texttt{c}}olon (\fA_N^0,{\texttt{a}}st) \to \CC$ and
$\zeta_\sha {\texttt{c}}olon (\fA_N^0,\sha) \to \CC$ are algebra homomorphisms.
\end{lem}
This first preliminary result is well-known. The map $\zeta_{\texttt{a}}st$ originates from
the series definition \eqref{def:CMZV} while the map $\zeta_\sha$ comes from
the integral representation by setting $z=1$ in \eqref{equ:zetaShaz}.
The following results of Racinet {\texttt{c}}ite{Racinet02} addressing the regularization of colored MZVs
generalize those for the MZVs first discovered by Ihara, Kaneko and Zagier {\texttt{c}}ite{Ihara06}.
{\texttt{b}}egin{thm}\label{theo:Racinet}
Let $N$ be a positive integer.
{\texttt{b}}egin{enumerate}[(i)]
\item The algebra homomorphism $\zeta_{\texttt{a}}st {\texttt{c}}olon (\fA_N^0,{\texttt{a}}st) \to \CC$ can be extended to a homomorphism $\zeta_{{\texttt{a}}st}(-;T){\texttt{c}}olon(\fA_N^1,{\texttt{a}}st)\to\CC[T]$, where $\zeta_{{\texttt{a}}st}(y_{1,1};T)=T$.
\item The algebra homomorphism $\zeta_\sha {\texttt{c}}olon (\fA_N^0,\sha) \to \CC$ can be extended to a homomorphism $\zeta_{\sha}(-;T){\texttt{c}}olon(\fA_N^1,\sha)\to\CC[T]$, where $\zeta_{\sha}(x_1;T)=T$.
\item For all $w\in\fA_N^1$, we have
{\texttt{b}}egin{align*}
\zeta_{\sha}{\texttt{b}}ig({\texttt{b}}fp(w);T{\texttt{b}}ig)=\rho {\texttt{b}}ig(\zeta_{{\texttt{a}}st}(w;T){\texttt{b}}ig),
\end{align*}
where $\rho{\texttt{c}}olon \CC[T]\to \CC[T]$ is a $\CC$-linear map such that
{\texttt{b}}egin{align*}
\rho(e^{Tu})=\exp\left(\sum_{n=2}^\infty \frac{(-1)^n}{n}\zeta(n)u^n\right)e^{Tu}
\end{align*}
for all $|u|<1$.
\end{enumerate}
\end{thm}
In the above theorem, to signify the fact that the images of $\zeta_{\sha}$ and $\zeta_{{\texttt{a}}st}$ are polynomials of $T$, we have used the notation $\zeta_{\sha}(w;T)$ and $\zeta_{{\texttt{a}}st}(w;T)$.
{\texttt{b}}egin{prop}\label{prop:SCVconst}
For all ${\texttt{b}}fs\in \N^d$ and ${\texttt{b}}feta\in (\gG_N)^d$, both
$\displaystyle \zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}$
and $\displaystyle \zeta_\sha^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}$ are constant complex values, i.e.,
they are independent of $T$.
\end{prop}
{\texttt{b}}egin{proof}
We start with the shuffle version
$\displaystyle \zeta_\sha^\Sy\lrp{s_1,\ldots, s_d}{\eta_1,\ldots,\eta_d}$.
If $(s_j,\eta_j)\ne (1,1)$ for all $j=1,\ldots,d$ then clearly it is finite.
In the case that $(s_j,\eta_j)=(1,1)$ for all $j=1,\ldots,d$ then the binomial theorem implies
{\texttt{b}}egin{equation*}
\zeta_\sha^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}=\sum_{j=0}^d (-1)^{d-j} \frac{T^j}{j!}\frac{T^{d-j}}{(d-j)!} =0.
\end{equation*}
Otherwise, we may assume for some $k$ and $l\ge 1$ we have $(s_k,\eta_k)\ne(1,1)$,
$(s_{k+1},\eta_{k+1})={\texttt{c}}dots=(s_{k+l},\eta_{k+l})=(1,1)$,
and $(s_{k+l+1},\eta_{k+l+1})\ne(1,1)$. We only need to show that
{\texttt{b}}egin{equation*}
\sum_{j=0}^l
(-1)^{j} \zeta_\sha\lrpT{s_{k+j},\dots,s_1}{\eta_{k+j},\ldots,\eta_1} \zeta_\sha\lrpT{s_{k+j+1},\dots,s_d}{\eta_{k+j+1},\ldots,\eta_d}
\end{equation*}
is finite. Consider the words in $\fA_\sha^1$ corresponding to the
above values. We may assume
$x_\gl u=x_0^{s_k-1}x_{\eta_k} {\texttt{c}}dots x_0^{s_1-1}x_{\eta_1}$ (which is
the empty word if $k=0$) and
$x_\mu v=x_0^{s_{k+l+1}-1}x_{\eta_{k+l+1}} {\texttt{c}}dots x_0^{s_d-1}x_{\eta_d}$
(which is the empty word if $k+l=d$),
where $\gl,\mu\ne 1$. So we only need to show that
{\texttt{b}}egin{equation}\label{equ:symmtrizeWordsAdm}
\sum_{j=0}^l (-1)^j (x_1^jx_\gl u)\sha(x_1^{l-j} x_\mu v) \in \fA_N^0.
\end{equation}
Let $q_0=0$ and $q_m=1$ for all $m\ne 0$. Then we have
{\texttt{b}}egin{align*}
&\,\sum_{j=0}^l (-1)^j (x_1^jx_\gl u) \sha (x_1^{l-j} x_\mu v)\\
=&\, q_k x_\gl(u\sha x_1^l x_\mu v)
+\sum_{j=1}^l (-1)^{j}x_1 (x_1^{j-1} x_\gl u\sha x_1^{l-j} x_\mu v)\\
+&\, q_{k+l-d}(-1)^l x_\mu(x_1^l x_\gl u \sha v) +
\sum_{j=0}^{l-1} (-1)^j x_1 (x_1^j x_\gl u\sha x_1^{l-j-1}x_\mu v) \\
=&\,q_k x_\gl (u\sha x_1^l x_\mu v)+q_{k+l-d}(-1)^l x_\mu( x_1^l x_\gl u \sha v)\in \fA_N^0,
\end{align*}
where we have used the substitution $j\to j+1$ in the first sigma
summation which is canceled by the second sigma summation.
For the stuffle version $ \zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}$ we can use
the same idea because the stuffing parts, i.e., the contraction of two beginning letters,
always produce admissible words. We leave the details to the interested reader.
\end{proof}
\subsection{Generating series}
Using the algebraic framework of Section \ref{sec:algframe}, we define for
${\texttt{b}}fs:=(s_1,\ldots,s_d)\in \N^d$ and ${\texttt{b}}feta:=(\eta_1,\ldots,\eta_d)\in (\gG_N)^d$
the following maps:
{\texttt{b}}egin{align*}
\zeta_{\texttt{a}}st^\Sy &{\texttt{c}}olon \fA_N^1 \to \CC, \quad \quad \zeta_{\texttt{a}}st^\Sy(y_{s_1,\eta_1}{\texttt{c}}dots y_{s_d,\eta_d}):=\zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}, \\
\zeta_{\sha}^\Sy &{\texttt{c}}olon \fA_N^1 \to \CC, \quad \quad \zeta_\sha^\Sy(y_{s_1,\eta_1}{\texttt{c}}dots y_{s_d,\eta_d}):=\zeta_\sha^\Sy\Lrp{s_1,s_2,\ldots,\ \, s_d\ \, }{\eta_1,\frac{\eta_{2}}{\eta_1},\ldots, \frac{\eta_{d}}{\eta_{d-1}}}.
\end{align*}
In order to study the SCVs effectively, we need to utilize their generating series, which
should be associated with some dual objects of the Hopf algebras
$(\fA_N^1,{\texttt{a}}st,\widetilde{\Delta}_{\texttt{a}}st)$ and $(\fA_N,\sha, \widetilde{\Delta}_\sha)$.
So we denote by $\hfA_N$ the completion of $\fA_N$ with respect to the weight
(and define $\hfA_N^1$ and $\hfA_N^0$ similarly).
Let $R$ be any $\Q$-algebra. Define the coproduct $\gD_{\texttt{a}}st$ on
$R\langle\!\langle Y_N^{\texttt{a}}st \rangle\!\rangle:=\hfA_N^1\otimes_\Q R$ by
$\gD_{\texttt{a}}st({\texttt{b}}e):={\texttt{b}}e \otimes{\texttt{b}}e$ and
{\texttt{b}}egin{align*}
\gD_{\texttt{a}}st(y_{k,\xi}):={\texttt{b}}e \otimes y_{k,\xi}+y_{k,\xi}\otimes {\texttt{b}}e
+\sum_{a+b=k,\,a,b\in\N {\texttt{a}}top
\gl\eta=\xi,\,\gl,\eta\in\gG_N} y_{a,\gl}\otimes y_{b,\eta}
\end{align*}
for all $k\in\N$ and $\xi\in\gG_N$. Then extend it $R$-linearly.
One can check easily that
$(R\langle\!\langle Y_N^{\texttt{a}}st \rangle\!\rangle, {\texttt{a}}st,\gD_{\texttt{a}}st)$
is the dual to the Hopf algebra $(R\langle Y_N^{\texttt{a}}st \rangle, {\texttt{a}}st,\tilde{\gD}_{\texttt{a}}st)$.
Take $R:=\CC[T]$. Let ${\mathbb P}si_{\texttt{a}}st^T$ be the generating series of
${\texttt{a}}st$-regularized colored MZVs, i.e.,
{\texttt{b}}egin{align*}
{\mathbb P}si_{\texttt{a}}st^T:=\sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\texttt{a}}st(w;T) w \in \CC[T]\langle\!\langle Y_N^{\texttt{a}}st \rangle\!\rangle,
\end{align*}
which can be regarded as an element in the regular ring of functions
of $\CC[T]\langle Y_N^{\texttt{a}}st\rangle$ by the above consideration.
Namely, we can define ${\mathbb P}si_{\texttt{a}}st^T[w]$ to be the coefficient of $w$ in ${\mathbb P}si_{\texttt{a}}st^T$.
Further we set ${\mathbb P}si_{\texttt{a}}st:={\mathbb P}si_{\texttt{a}}st^0$.
The shuffle version of ${\mathbb P}si_{\texttt{a}}st$ is more involved even though the basic idea is the same.
First, for any $\Q$-algebra $R$ we can define the coproduct $\gD_\sha$ on
$R\langle\!\langle X_N^{\texttt{a}}st \rangle\!\rangle:=\hfA_N\otimes_\Q R$ by
{\texttt{b}}egin{align*}
\gD_\sha({\texttt{b}}e):={\texttt{b}}e \otimes {\texttt{b}}e \quad \text{and} \quad \gD_\sha(x_\gl):=x_\gl\otimes{\texttt{b}}e+{\texttt{b}}e \otimes x_\gl
\end{align*}
for all $\gl\in\gG_N{\texttt{c}}up\{0\}$. Let $\eps_\sha$ be the counit such that
$\eps_\sha({\texttt{b}}e)=1$ and $\eps_\sha(w)=0$ for all $w\ne{\texttt{b}}e$.
Then the Hopf algebra $(R\langle\!\langle X_N^{\texttt{a}}st \rangle\!\rangle,\sha, \gD_\sha)$
is dual to $(R\langle X_N^{\texttt{a}}st \rangle,\sha,\tilde{\gD}_\sha)$.
Now we can define
{\texttt{b}}egin{align*}
{\mathbb P}si_\sha^T:=\sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\sha}({\texttt{b}}fp(w);T) w \in \CC[T]\langle\!\langle Y_N^{\texttt{a}}st \rangle\!\rangle,
\end{align*}
{\texttt{b}}egin{lem}\label{lem:group-likePsi}
We have
{\texttt{b}}egin{align*}
{\mathbb P}si_{\texttt{a}}st^T=\exp(T y_{1,1}) {\mathbb P}si_{\texttt{a}}st \quad \text{and} \quad
{\mathbb P}si_\sha^T=\exp(T y_{1,1}) {\mathbb P}si_\sha.
\end{align*}
\end{lem}
{\texttt{b}}egin{proof}
Set $\widehat{{\mathbb P}si}_\sha(T):=\sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\sha}(w;T) w$.
Then $\widehat{{\mathbb P}si}_\sha(T)$ is group-like for $\gD_\sha$. Further, ${\mathbb P}si_{\texttt{a}}st^T$ is group-like
for $\gD_{\texttt{a}}st$. It follows from Cor.~2.4.4 and Cor.~2.4.5 of {\texttt{c}}ite{Racinet02} that
{\texttt{b}}egin{align*}
{\mathbb P}si_{\texttt{a}}st^T=\exp(T y_{1,1}) {\mathbb P}si_{\texttt{a}}st \quad \text{and} \quad
\widehat{{\mathbb P}si}_\sha(T)=\exp(T y_{1,1}) \widehat{{\mathbb P}si}_\sha(0),
\end{align*}
since $x_1=y_{1,1}$.
Since ${\texttt{b}}fq(y_{1,1}^n w)=y_{1,1}^n {\texttt{b}}fq(w)$ for all $n\in\Z_{\ge 0}$ and $w\in Y_N^{\texttt{a}}st$, applying ${\texttt{b}}fq$ to the second equality leads to
{\texttt{b}}egin{align*}
\sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\sha}(w;T) {\texttt{b}}fq(w)
=\exp(T y_{1,1}){\texttt{b}}igg( \sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\sha}(w;0) {\texttt{b}}fq(w) {\texttt{b}}igg),
\end{align*}
which is equivalent to ${\mathbb P}si_\sha^T=\exp(T y_{1,1}) {\mathbb P}si_\sha$, as desired.
\end{proof}
For all $s_1,\dots,s_d\in\N$ and $\eta_1,\dots,\eta_d\in\gG_N$, we define the
anti-automorphism $\inv{\texttt{c}}olon Y_N^{\texttt{a}}st \to Y_N^{\texttt{a}}st$ by
{\texttt{b}}egin{equation*}
\inv(y_{s_1,\eta_1} {\texttt{c}}dots y_{s_d,\eta_d}):=
(-1)^{s_1+{\texttt{c}}dots+s_d}\, \ol{\eta_1}{\texttt{c}}dots \ol{\eta_d}\, y_{s_d,\eta_d}{\texttt{c}}dots y_{s_1,\eta_1} .
\end{equation*}
and then extend it to $\CC\langle\!\langle Y_N^{\texttt{a}}st \rangle\!\rangle$ by linearity.
{\texttt{b}}egin{lem}\label{lem:invPsi}
We have
{\texttt{b}}egin{align*}
\inv({\mathbb P}si_{\texttt{a}}st){\mathbb P}si_{\texttt{a}}st=\inv({\mathbb P}si_{\texttt{a}}st^T){\mathbb P}si_{\texttt{a}}st^T=&\,\sum_{w\in Y_N^{\texttt{a}}st} \zeta_{\texttt{a}}st^\Sy(w) w, \\
\inv({\mathbb P}si_\sha){\mathbb P}si_\sha=\inv({\mathbb P}si_\sha^T){\mathbb P}si_\sha^T=&\,\sum_{w\in Y_N^{\texttt{a}}st} \zeta_\sha^\Sy({\texttt{b}}fp(w)) w.
\end{align*}
\end{lem}
{\texttt{b}}egin{proof}
In each of two lines above, the second equality follows directly from the definition
while the first is an immediate consequence of Proposition~\ref{prop:SCVconst}.
\end{proof}
{\texttt{b}}egin{thm}\label{theo:moduloz2}
For any ${\texttt{b}}fs\in\N^d,{\texttt{b}}feta\in(\gG_N)^d$, we have
{\texttt{b}}egin{align*}
\zeta_\sha^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}\equiv \zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{{\texttt{b}}feta} \pmod{\zeta(2)}.
\end{align*}
\end{thm}
{\texttt{b}}egin{proof}
By Lemma~\ref{lem:group-likePsi} and Theorem~\ref{theo:Racinet}, we get
{\texttt{b}}egin{align*}
\exp(T y_{1,1}) {\mathbb P}si_\sha= {\mathbb P}si_\sha^T=\rho({\mathbb P}si_{\texttt{a}}st^T)=\exp(T y_{1,1}) \gL(y_{1,1}){\mathbb P}si_{\texttt{a}}st,
\end{align*}
where $\gL(y_{1,1}):=\exp\left(\sum_{n=2}^\infty\frac{(-1)^n}{n}\zeta(n) y_{1,1}^n\right)$.
Therefore ${\mathbb P}si_\sha= \gL(y_{1,1}){\mathbb P}si_{\texttt{a}}st$.
Using the fact $\zeta(2n)\in \zeta(2)^n\mathbb{Q}$ for $n\in \mathbb{N}$ implies
{\texttt{b}}egin{align*}
\inv({\mathbb P}si_\sha){\mathbb P}si_\sha=\inv({\mathbb P}si_{\texttt{a}}st)\gL(-y_{1,1})\gL(y_{1,1}){\mathbb P}si_{\texttt{a}}st
\equiv \inv({\mathbb P}si_{\texttt{a}}st){\mathbb P}si_{\texttt{a}}st \pmod{\zeta(2)}.
\end{align*}
Hence, the theorem follows from Lemma~\ref{lem:invPsi}.
\end{proof}
\subsection{Shuffle and stuffle relations}
We first prove the stuffle relations of the SCVs.
{\texttt{b}}egin{thm}\label{thm:astSCVmorphism}
The map $\zeta_{\texttt{a}}st^\Sy{\texttt{c}}olon (\fA_N^1,{\texttt{a}}st) \to \CC$ is a homomorphism of algebras, i.e.
{\texttt{b}}egin{align*}
\zeta_{\texttt{a}}st^\Sy(w{\texttt{a}}st w') = \zeta_{\texttt{a}}st^\Sy(w)\zeta_{\texttt{a}}st^\Sy(w')
\end{align*}
for all $w,w'\in \fA_N^1$.
\end{thm}
{\texttt{b}}egin{proof}
Since $\zeta_{\texttt{a}}st{\texttt{c}}olon (\fA_N^1,{\texttt{a}}st)\to \CC[T]$ is an algebra homomorphism, its generating series ${\mathbb P}si_{\texttt{a}}st^T$ must be a group-like element of $\gD_*$, i.e., $\gD_*({\mathbb P}si_{\texttt{a}}st^T) = {\mathbb P}si_{\texttt{a}}st^T \otimes {\mathbb P}si_{\texttt{a}}st^T$. Further, it can be checked in a straight-forward manner that $\Delta_{\texttt{a}}st {\texttt{c}}irc\inv = (\inv \otimes \inv){\texttt{c}}irc \Delta_{\texttt{a}}st$. Thus we get
{\texttt{b}}egin{align*}
\Delta_{\texttt{a}}st{\texttt{b}}ig(\inv({\mathbb P}si_{\texttt{a}}st^T){\mathbb P}si_{\texttt{a}}st^T{\texttt{b}}ig) = {\texttt{b}}ig(\inv({\mathbb P}si_{\texttt{a}}st^T){\mathbb P}si_{\texttt{a}}st^T{\texttt{b}}ig) \otimes {\texttt{b}}ig(\inv({\mathbb P}si_{\texttt{a}}st^T){\mathbb P}si_{\texttt{a}}st^T{\texttt{b}}ig)
\end{align*}
and Lemma \ref{lem:invPsi} implies the claim.
\end{proof}
For the shuffle relations we need the \emph{generalized Drinfeld associator
${\mathbb P}hi={\mathbb P}hi_N$ at level $N$}.
Enriquez {\texttt{c}}ite{Enriquez2007} defined it as the renormalized holonomy from 0 to 1 of
{\texttt{b}}egin{equation} \label{equ:generalizedDrinfeld}
H'(z) = \left( \sum_{\eta\in \gG_N {\texttt{c}}up \{0\}}
\frac{x_\eta}{z-\eta} \right) H(z),
\end{equation}
i.e., ${\mathbb P}hi:=H_1^{-1}H_0$, where $H_0,H_1$ are the solutions
of \eqref{equ:generalizedDrinfeld} on the open interval $(0,1)$
such that $H_0(z) \sim z^{x_0}=\exp(x_0 \log z)$ when $z\to 0^+$,
$H_1(z) \sim (1-z)^{x_1}=\exp(x_1 \log(1-z))$ when $z\to 1^-$.
{\texttt{b}}egin{thm}
The generalized Drinfeld associator ${\mathbb P}hi$ is the unique element in the Hopf algebra $(\CC\langle\!\langle X_N^{\texttt{a}}st \rangle\!\rangle,\sha,\gD_\sha,\eps_\sha)$ such that
{\texttt{b}}egin{itemize}
\item[\upshape{(i)}] ${\mathbb P}hi$ is group-like, i.e., $\eps_\sha({\mathbb P}hi)=1$ and $\gD_\sha({\mathbb P}hi)={\mathbb P}hi\otimes{\mathbb P}hi$,
\item[\upshape{(ii)}] ${\mathbb P}hi[x_0]={\mathbb P}hi[x_1]=0$,
\item[\upshape{(iii)}] $\displaystyle {\mathbb P}hi[{\texttt{b}}fp(x_0^{s_1-1} x_{\eta_1}\dots x_0^{s_d-1} x_{\eta_d})]
=(-1)^d \zeta \lrpTZ{{\texttt{b}}fs}{{\texttt{b}}feta}$
for any ${\texttt{b}}fs:=(s_1,\dots,s_d)\in\N^d$ and ${\texttt{b}}feta:=(\eta_1,\dots,\eta_d)\in(\gG_N)^d$.
\end{itemize}
\end{thm}
{\texttt{b}}egin{proof}
The uniqueness, the statements in (i), (ii) and the case $(s_1,\eta_1)\ne(1,1)$ of (iii)
of the theorem follow directly from {\texttt{c}}ite[App.]{Enriquez2007} and {\texttt{c}}ite[Prop. 5.17]{Deligne05}.
By Theorem~\ref{theo:Racinet} (ii), if $(s_1,\eta_1)=(1,1)$ then $\zeta \lrpTZ{{\texttt{b}}fs}{{\texttt{b}}feta}$
is determined uniquely by the admissible values from the shuffle structure by using (ii). But ${\mathbb P}hi[{\texttt{b}}fp(x_0^{s_1-1} x_{\eta_1}\dots x_0^{s_d-1} x_{\eta_d})]$ is also determined uniquely
by the coefficients of admissible words from the same
shuffle structure so that (iii) still holds even if $(s_1,\eta_1)=(1,1)$. This completes
the proof of the theorem.
\end{proof}
For any $\eta\in\gG_N$, we define the map $r_\eta{\texttt{c}}olon X_N^*\to X_N^*$ by
setting
{\texttt{b}}egin{align*}
r_\eta(x_0^{a_1} x_{\eta_1}^{b_1}\dots x_0^{a_d} x_{\eta_d}^{b_d})
:=x_0^{a_1} x_{\eta_1/\eta}^{b_1}\dots x_0^{a_d} x_{\eta_d/\eta}^{b_d}
\end{align*}
for all $a_1,b_1,\dots,a_d,b_d\in\Z_{\ge 0}$ and $\eta_1,\dots,\eta_d\in\gG_N$.
{\texttt{b}}egin{lem}\label{lem:eta-twistPhi}
For any $\eta\in\gG_N$, the \emph{$\eta$-twist ${\mathbb P}hi_{\eta}$ of ${\mathbb P}hi$} defined by
{\texttt{b}}egin{align*}
{\mathbb P}hi_{\eta}:=\sum_{w\in X_N^{\texttt{a}}st} {\mathbb P}hi[r_\eta(w)] w
\end{align*}
is group-like for $\gD_\sha$ and ${\mathbb P}hi_{\eta}^{-1}$ is well-defined.
\end{lem}
{\texttt{b}}egin{proof}
For any words $u,v\in X_N^*$, we have
{\texttt{b}}egin{multline*}
\gD_\sha({\mathbb P}hi_\eta) [u\otimes v]={\mathbb P}hi_\eta[ u\sha v]={\mathbb P}hi[r_\eta ( u\sha v)]\\
={\mathbb P}hi[r_\eta (u) \sha r_\eta (v)]
={\mathbb P}hi[r_\eta (u)]{\mathbb P}hi[r_\eta(v)]
=({\mathbb P}hi_\eta\otimes{\mathbb P}hi_\eta)[u\otimes v],
\end{multline*}
since ${\mathbb P}hi$ is group-like. Thus $\gD_\sha({\mathbb P}hi_\eta)={\mathbb P}hi_\eta\otimes{\mathbb P}hi_\eta$.
Further, ${\mathbb P}hi_{\eta}^{-1}$ is well-defined since
{\texttt{b}}egin{align*}
({\mathbb P}hi_{\eta})^{-1}=\sum_{w\in X_N^{\texttt{a}}st} (-1)^{|w|}{\mathbb P}hi[r_\eta(w)] \revs{w}
=\sum_{w\in X_N^{\texttt{a}}st} (-1)^{|w|}{\mathbb P}hi[r_\eta(\revs{w})]w=({\mathbb P}hi^{-1})_{\eta},
\end{align*}
where $\revs{w}=\ga_d\ga_{d-1}{\texttt{c}}dots\ga_1$ is the reversal of the word
$w=\ga_1{\texttt{c}}dots \ga_{d-1}\ga_d\in X_N^{\texttt{a}}st$ with the letters $\ga_j\in X_N$ for $j=1,\ldots,d$.
\end{proof}
{\texttt{b}}egin{thm}\label{thm:SCVDrinfeldAss}
For any word $w\in \fA_N^1$, we have
{\texttt{b}}egin{align*}
\zeta_\sha^\Sy(w)=(-1)^d\sum_{\eta\in\gG_N} \ol{\eta}\, {\mathbb P}hi_{\eta}^{-1} x_{\eta} {\mathbb P}hi_{\eta} [x_1 w].
\end{align*}
\end{thm}
{\texttt{b}}egin{proof}
First we observe that
{\texttt{b}}egin{align}\label{eq:invphi}
{\mathbb P}hi_{\eta}^{-1}[x_1 x_0^{s_1-1} x_{\eta_1}{\texttt{c}}dots x_{\eta_{j-1}} x_0^{s_j-1}]
= (-1)^{s_1+{\texttt{c}}dots+s_j}{\mathbb P}hi_{\eta}[ x_0^{s_j-1} x_{\eta_{j-1}}{\texttt{c}}dots x_{\eta_1} x_0^{s_1-1}x_1].
\end{align}
for $j=1,\ldots,d$. Then we obtain (by setting $\eta_0:=1$)
{\texttt{b}}egin{align*}
& (-1)^d\sum_{\eta\in\gG_N} \ol{\eta}\, {\mathbb P}hi_{\eta}^{-1} x_{\eta} {\mathbb P}hi_{\eta} {\texttt{b}}ig[x_1 x_0^{s_1-1} x_{\eta_1}\dots x_0^{s_d-1} x_{\eta_d}{\texttt{b}}ig] \\
= &~ (-1)^d \sum_{j=0}^d \ol{\eta_j}\, {\mathbb P}hi_{\eta_j}^{-1}{\texttt{b}}ig[x_1 x_0^{s_1-1} x_{\eta_1}{\texttt{c}}dots x_{\eta_{j-1}}x_0^{s_{j}}{\texttt{b}}ig] {\mathbb P}hi_{\eta_j}{\texttt{b}}ig[x_0^{s_{j+1}-1}x_{\eta_{j+1}}{\texttt{c}}dots x_0^{s_d-1}x_{\eta_d}{\texttt{b}}ig]\\
= &~ (-1)^d \sum_{j=0}^d (-1)^{s_1+{\texttt{c}}dots+s_j}\ol{\eta_j}\, {\mathbb P}hi_{\eta_j}{\texttt{b}}ig[x_0^{s_{j}}x_{\eta_{j-1}}{\texttt{c}}dots x_0^{s_1-1} x_1{\texttt{b}}ig] {\mathbb P}hi_{\eta_j}{\texttt{b}}ig[x_0^{s_{j+1}-1}x_{\eta_{j+1}}{\texttt{c}}dots x_0^{s_d-1}x_{\eta_d}{\texttt{b}}ig]\\
= &~ (-1)^d \sum_{j=0}^d (-1)^{s_1+{\texttt{c}}dots+s_j}\ol{\eta_j}\, {\mathbb P}hi{\texttt{b}}ig[x_0^{s_{j}}x_{\eta_{j-1}/\eta_j}{\texttt{c}}dots x_0^{s_1-1} x_{\eta_0/\eta_j}{\texttt{b}}ig] {\mathbb P}hi{\texttt{b}}ig[x_0^{s_{j+1}-1}x_{\eta_{j+1}/\eta_j}{\texttt{c}}dots x_0^{s_d-1}x_{\eta_d/\eta_j}{\texttt{b}}ig]\\
= &~ (-1)^d \sum_{j=0}^d (-1)^{s_1+{\texttt{c}}dots+s_j}\ol{\eta_j}\,
{\texttt{c}}dot (-1)^j \zeta_\sha\LrpTZ{\ s_j\ , \ldots,s_1}
{\frac{\eta_{j-1}}{\eta_{j}},\ldots,\frac{\eta_0}{\eta_1} }
{\texttt{c}}dot (-1)^{d-j} \zeta_\sha\LrpTZ{s_{j+1},\ldots,\ s_d\ }
{\frac{\eta_{j+1}}{\eta_j},\ldots,\frac{\eta_d}{\eta_{d-1}} }\\
= & ~ \zeta_\sha^\Sy \Lrp{s_1,s_2,\ldots,\ \, s_d\ \, }{\eta_1,\frac{\eta_{2}}{\eta_1},\ldots, \frac{\eta_{d}}{\eta_{d-1}}},
\end{align*}
by Proposition~\ref{prop:SCVconst}. We have completed our proof.
\end{proof}
{\texttt{b}}egin{thm}\label{thm:shuffleSCV}
Let $N\ge 1$. For any $w,u\in\fA_1^1$ and $v\in\fA_N^1$, we have
{\texttt{b}}egin{align*}
\zeta_\sha^\Sy(u\sha v)= \zeta_\sha^\Sy(\tau(u)v).
\end{align*}
These relations are called linear shuffle relations for the SCVs.
\end{thm}
{\texttt{b}}egin{proof}
It suffices to prove
{\texttt{b}}egin{align*}
\zeta_\sha^\Sy( (x_0^{s-1}x_1 u) \sha v) =(-1)^s \zeta_\sha^\Sy(u \sha (x_0^{s-1}x_1 v)).
\end{align*}
for all $s\in\N$. We observe that
{\texttt{b}}egin{equation}\label{equ:sumOfShuffles}
x_1 \Big( (x_0^{s-1}x_1 u)\sha v-(-1)^s u\sha(x_0^{s-1} x_1 v)\Big)
=\sum_{i=0}^{s-1} (-1)^i (x_0^{s-1-i}x_1 u)\sha(x_0^{i}x_1v).
\end{equation}
By Theorem.~\ref{thm:SCVDrinfeldAss}, it suffices to show that the image of \eqref{equ:sumOfShuffles} under $E:={\mathbb P}hi_{\eta}^{-1}x_1{\mathbb P}hi_{\eta}$ vanishes
for all $\eta\in\gG_N$. By Lemma~\ref{lem:eta-twistPhi}
{\texttt{b}}egin{equation*}
\gD_\sha(E)=({\mathbb P}hi_{\eta}^{-1}\ot{\mathbb P}hi_{\eta}^{-1})
(x_1\ot {\texttt{b}}e+{\texttt{b}}e\ot x_1)({\mathbb P}hi_{\eta}\ot{\mathbb P}hi_{\eta})
=E\ot {\texttt{b}}e+{\texttt{b}}e\ot E.
\end{equation*}
Therefore $E$ is a primitive element for $\gD_\sha$ so that
we can regard it as a Lie element, namely, it acts on shuffle products like a derivation.
Hence, for any nonempty words $u,v\in X_N^*$,
{\texttt{b}}egin{equation*}
E[u \sha v]=E[u]\epsilon_\sha[v]+ \epsilon_\sha[u]E[v]=0.
\end{equation*}
This completes the proof by Theorem~\ref{thm:SCVDrinfeldAss}
since none of the factors in the shuffle products on the
right-hand side of Eq.~\eqref{equ:sumOfShuffles} is the
empty word as the letter $x_1$ appears in every factor.
\end{proof}
\section{Reversal relations of FCVs and SCVs}
One of the simplest but very important relations among FCVs and SCVs
are the following reversal relations.
{\texttt{b}}egin{prop}\label{prop:reversalFCVSCV}
Let ${\texttt{b}}fs \in \N^d$, ${\texttt{b}}feta \in (\gG_N)^d$,
and define $\pr({\texttt{b}}feta):=\prod_{j=1}^d \eta_j$. Then we have
{\texttt{b}}egin{align}\label{equ:reversalFCV}
\zeta_{{\texttt{c}}alA(N)} \lrp{\revs{{\texttt{b}}fs}}{\revs{{\texttt{b}}feta}}
=&\,(-1)^{|{\texttt{b}}fs|} \pr(\ol{{\texttt{b}}feta}) \zeta_{{\texttt{c}}alA(N)} \lrp{{\texttt{b}}fs}{\ol{{\texttt{b}}feta}}, \\
\zeta_\sha^\Sy\lrp{\revs{{\texttt{b}}fs}}{\revs{{\texttt{b}}feta}}
=&\,(-1)^{|{\texttt{b}}fs|} \pr(\ol{{\texttt{b}}feta}) \zeta_\sha^\Sy\lrp{{\texttt{b}}fs}{\ol{{\texttt{b}}feta}},\label{equ:reversalSCVsha}\\
\zeta_{\texttt{a}}st^\Sy\lrp{\revs{{\texttt{b}}fs}}{\revs{{\texttt{b}}feta}}
=&\,(-1)^{|{\texttt{b}}fs|} \pr(\ol{{\texttt{b}}feta}) \zeta_{\texttt{a}}st^\Sy\lrp{{\texttt{b}}fs}{\ol{{\texttt{b}}feta}},\label{equ:reversalSCVast}
\end{align}
where $\revs{{\texttt{b}}fa}:=(a_d,\dots,a_1)$ is the reversal of ${\texttt{b}}fa:=(a_1,\ldots,a_d)$ and $\ol{{\texttt{b}}fa}:=(\ol{a_1},\dots,\ol{a_d})$ is the componentwise conjugation of ${\texttt{b}}fa$.
\end{prop}
{\texttt{b}}egin{proof}
Equation \eqref{equ:reversalFCV} follows easily from the substitution $k_j\to p-k_j$
for the indices in \eqref{equ:defnFCV} by the condition $p\equiv -1\pmod{N}$.
Equations~\eqref{equ:reversalSCVsha}
and \eqref{equ:reversalSCVast} follow easily from the definitions.
\end{proof}
\section{Numerical examples}\label{sec:numex}
In this last section, we provide some numerical examples in support of
Conjecture~\ref{conj:Main}.
We will need some results from the level 1 case.
{\texttt{b}}egin{prop}\label{prop:homogeneousWols1} \emph{({\texttt{c}}ite[Theo.~2.13]{Zhao08b})}
Let $s$, $d$ and $N$ be positive integers. Then
{\texttt{b}}egin{align}\label{equ:homogeneousWols}
\zeta_{{\texttt{c}}alA(N)}\Lrp{\{s\}^d}{\{1\}^d}=0.
\end{align}
\end{prop}
{\texttt{b}}egin{eg}\label{eg:level3}
At level 3, by Proposition~\ref{prop:reversalFCVSCV} and
Proposition~\ref{prop:homogeneousWols1}, for all $w\in \N$, we have
{\texttt{b}}egin{alignat*}{3}
\zeta_{{\texttt{c}}alA(3)}\lrp{w}{1}=&\, 0, \quad &\zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{1,1}=&\, 0, \quad & \zeta_{{\texttt{c}}alA(3)}\lrp{w}{\xi_3}=&\, \xi_3^2 (-1)^w \zeta_{{\texttt{c}}alA(3)}\lrp{w}{\xi_3^2},\\
\zeta_{\texttt{a}}st^\Sy\lrp{w}{1}=&\, \gd_w\zeta(w), \quad &\zeta_{\texttt{a}}st^\Sy\lrp{w,w}{1,1}=&\,
\gd_w\zeta(w)^2-\zeta(2w), \quad & \zeta_{\texttt{a}}st^\Sy\lrp{w}{\xi_3}=&\, \xi_3^2(-1)^w \zeta_{\texttt{a}}st^\Sy\lrp{w}{\xi_3^2},
\end{alignat*}
where $\gd_w= (1+(-1)^w)$, and
{\texttt{b}}egin{alignat*}{3}
\zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{1,\xi_3}=&\, \xi_3^2 \zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{\xi_3^2,1}, \quad &
\zeta_{\texttt{a}}st^\Sy\lrp{w,w}{1,\xi_3}=&\, \xi_3^2 \zeta_{\texttt{a}}st^\Sy\lrp{w,w}{\xi_3^2,1}, \\ \zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{1,\xi_3^2}=&\,\xi_3\zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{\xi_3,1},\quad & \zeta_{\texttt{a}}st^\Sy\lrp{w,w}{1,\xi_3^2}=&\,\xi_3 \zeta_{\texttt{a}}st^\Sy\lrp{w,w}{\xi_3,1}, \\ \zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{\xi_3,\xi_3}=&\,\xi_3\zeta_{{\texttt{c}}alA(3)}\lrp{w,w}{\xi_3^2,\xi_3^2},\quad &
\zeta_{\texttt{a}}st^\Sy\lrp{w,w}{\xi_3,\xi_3}=&\,\xi_3 \zeta_{\texttt{a}}st^\Sy\lrp{w,w}{\xi_3^2,\xi_3^2}.
\end{alignat*}
\end{eg}
{\texttt{b}}egin{eg}\label{eg:level4}
At level 4, by {\texttt{c}}ite[Cor.\ 2.3]{Tauraso10}, we have
{\texttt{b}}egin{align}
\zeta_{{\texttt{c}}alA(4)}\lrp{w}{1}=0, \quad
\zeta_{{\texttt{c}}alA(4)}\lrp{w}{i^2}=
\left\{
{\texttt{b}}egin{array}{ll}
0, & \hbox{if $w$ is even;} \\
-2q_2 , & \hbox{if $w=1$;} \\
-2(1-2^{1-w}) \gb_w, & \hbox{otherwise,}
\end{array}
\right. \notag\\
\zeta_{{\texttt{c}}alA(4)}\lrp{w}{i}=\sum_{p>k>0} \frac{i^k}{k^w} =E^{(w)}_p+iO^{(w)}_p,\quad
\zeta_{{\texttt{c}}alA(4)}\lrp{w}{i^3}=\sum_{p>k>0} \frac{i^{3k}}{k^w}=E^{(w)}_p-iO^{(w)}_p,\label{equ:depth1N=4}
\end{align}
where $q_2:=((2^{p-1}-1)/p )_{p\in{\texttt{c}}alA(4)}$ is the ${\texttt{c}}alA(4)$-Fermat quotient, $\gb_w:=(B_{p-w}/w)_{p\in{\texttt{c}}alA(4),p>w}$ is the ${\texttt{c}}alA(4)$-Bernoulli number and
{\texttt{b}}egin{align*}
E^{(w)}_p:=\sum_{p>2k>0}\frac{(-1)^k}{(2k)^w},\quad \quad O^{(w)}_p:=\sum_{p>2k+1>0}\frac{(-1)^k}{(2k+1)^w}.
\end{align*}
To find more relations, first we observe that
{\texttt{b}}egin{align*}
\sum_{k=(p-1)/2}^{p-1} \frac{(-1)^k} k
\equiv \sum_{p-k=(p-1)/2}^{p-1} \frac{(-1)^{p-k}}{p-k}
\equiv \sum_{p>2k>0}\frac{(-1)^k}k\equiv \frac12E^{(1)}_p \pmod{p}.
\end{align*}
By Eq.~\eqref{equ:depth1N=4}, we have
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(4)}\lrp{1}{i^2}=2 \zeta_{{\texttt{c}}alA(4)}\lrp{1}{i}+2 \zeta_{{\texttt{c}}alA(4)}\lrp{1}{i^3}.
\end{align*}
This is consistent with Conjecture~\ref{conj:Main} since
{\texttt{b}}egin{align}\label{equ:piInSCV14}
\zeta_{\texttt{a}}st^\Sy\lrp{1}{i^2}=
2\zeta_{\texttt{a}}st^\Sy\lrp{1}{i}+2\zeta_{\texttt{a}}st^\Sy\lrp{1}{i^3}+2\pi
\equiv
2\zeta_{\texttt{a}}st^\Sy\lrp{1}{i}+2\zeta_{\texttt{a}}st^\Sy\lrp{1}{i^3} \pmod{2\pi i \Q(i)}.
\end{align}
Indeed, we have
{\texttt{b}}egin{align*}
\zeta_{\texttt{a}}st^\Sy\lrp{1}{i^2}=&\, \Li_1(i^2)-\ol{i^2} \Li_1(\ol{i^2})=2\Li_1(-1)=-2\log(2),\\
\zeta_{\texttt{a}}st^\Sy\lrp{1}{i}=&\, \Li_1(i)+i \Li_1(i^3)=-\log(1-i)-i\log(1+i)
=-\frac12\log 2+\frac{\pi i}{2}-\frac{i}{2}\log 2+\frac{\pi}{2},\\
\zeta_{\texttt{a}}st^\Sy\lrp{1}{i^3}=&\, \Li_1(i^3)-i \Li_1(i)=-\log(1+i)+i\log(1-i)
=-\frac12\log 2-\frac{\pi i}{2}+\frac{i}{2}\log 2+\frac{\pi}{2}.
\end{align*}
\end{eg}
{}From numerical evidence, we form the following conjecture:
{\texttt{b}}egin{conj}\label{conj:level4FCVbasis}
Let $w\ge 1$. When the level $N=3,4$, the $\Q(\xi_N)$-vector space $\FCV_{w,N}$ has
the following basis:
{\texttt{b}}egin{align}\label{equ:BasisN=3,4}
\left\{ \zeta_{{\texttt{c}}alA(N)}\Lrp{\{1\}^w}{\xi_N,\xi_N^{\gd_2},\dots,\xi_N^{\gd_w}}{\texttt{c}}olon \gd_2,\dots,\gd_{w}\in \{0,1\} \right\}.
\end{align}
\end{conj}
To find as many $\Q(\gG_N)$-linear relations as possible in weight $w$
we may choose all the known relations in weight $k<w$, multiply
them by $\displaystyle \zeta_{{\texttt{c}}alA(N)}\lrp{{\texttt{b}}fs}{{\texttt{b}}feta}$ for all ${\texttt{b}}fs$ of weight $w-k$
and all ${\texttt{b}}feta$, and then expand all the products
using the stuffle relation proved in Theorem \ref{thm:stuffleFCV}. All the $\Q(\gG_N)$-linear
relations among FCVs of the same weight produced in this way are
called \emph{linear stuffle relations of FCVs}.
We can similarly define linear stuffle relations of SCVs
By using linear shuffle and stuffle relations and the reversal relations we can show that
Eq.~\eqref{equ:BasisN=3,4} in Conjecture~\ref{conj:level4FCVbasis} are generating sets
in the cases $1\le w\le 4$ and $N=3,4$ but cannot show their linear independence at the moment.
Concerning Conjecture~\ref{conj:Main} (i), the inclusion $\nSCV_{w,N}\subseteq \CMZV_{w,N}$ is
trivial but the opposite inclusion seems difficult. Note that since $2\pi i\in \SCV_{1,4}$ by
Eq.~\eqref{equ:piInSCV14} we have $\nSCV_{w,4}=\SCV_{w,4}$ for all $w$
by the stuffle relations. But from Example~\ref{eg:level3} we see that $\SCV_{1,3}$
is generated by
{\texttt{b}}egin{align*}
\zeta_{\texttt{a}}st^\Sy\lrp{1}{\xi_3}=\zeta\lrp{1}{\xi_3}-\xi_3^2\zeta\lrp{1}{\xi_3^2}
\equiv (1-\xi_3^2)\zeta\lrp{1}{\xi_3} \pmod{2\pi i \Q(\xi_3)},
\end{align*}
which should imply that
$\SCV_{1,3}=\Big\langle \zeta\lrp{1}{\xi_3} \Big\rangle \ne
\nSCV_{1,3}$, since conjecturally $\zeta\lrp{1}{\xi_3}=(\pi i-3\log 3)/6$
and $\pi$ are algebraically independent.
Moreover, when the weight $w\le 3$ and the level $N=3,4$ we have numerically
verified in both spaces of Conjecture~\ref{conj:Main} (ii), exactly the same
linear relations leading to the dimension upper bound $2^{w-1}$ hold
(with error bounded by $10^{-99}$ for SCVs
and with congruence checked for all primes $p<312$ and $p=1019$ in ${\texttt{c}}alP(N)$ for FCVs).
{\texttt{b}}egin{eg}\label{eg:higherWtlevel3,4}
In weight 2 level 3 and 4, we can prove rigorously that
{\texttt{b}}egin{align*}
3\zeta_{{\texttt{c}}alA(3)}\lrp{2}{\xi_3}=&\, 2\zeta_{{\texttt{c}}alA(3)}\lrp{1,1}{\xi_3,\xi_3}(1-\xi_3)
-6\zeta_{{\texttt{c}}alA(3)}\lrp{1,1}{\xi_3,1},\\
3\zeta_{\texttt{a}}st^\Sy\lrp{2}{\xi_3}=&\, 2\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{\xi_3,\xi_3}(1-\xi_3)
-6\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{\xi_3,1}+\frac{(2\pi i)^2}{12}(2+\xi_3),\\
\zeta_{{\texttt{c}}alA(4)}\lrp{2}{i}=&\, (i-1)\zeta_{{\texttt{c}}alA(4)}\lrp{1,1}{i,1}-i\zeta_{{\texttt{c}}alA(4)}\lrp{1,1}{i,i}, \\
\zeta_{\texttt{a}}st^\Sy\lrp{2}{i}=&\, (i-1)\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{i,1}-i\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{i,i}
+ \frac{2\pi i}{12} \Big( 2(1+i)\zeta_{\texttt{a}}st^\Sy\lrp{1}{i}-i\zeta_{\texttt{a}}st^\Sy\lrp{1}{-1} \Big).
\end{align*}
In weight 3 level $N=3$ and 4, we have verified numerically
{\texttt{b}}egin{align*}
\zeta_{{\texttt{c}}alA(3)}\lrp{1,1,1}{1,\xi_3^2,\xi_3}=&\,3\xi_3\zeta_{{\texttt{c}}alA(3)}\lrp{1,1,1}{\xi_3,1,1},\\
\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{1,\xi_3^2,\xi_3}=&\,3\xi_3\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{\xi_3,1,1}+
\frac{(2\pi i)^2}{24}(1-\xi_3)\zeta_{\texttt{a}}st^\Sy\lrp{1}{\xi_3}-\frac{(2\pi i)^3}{144}(1-\xi_3),
\end{align*}
and
{\texttt{b}}egin{align*}
(15-66i)\zeta_{{\texttt{c}}alA(4)}\lrp{1,2}{1,1}=&\, 48\Big(
(1+i)\zeta_{{\texttt{c}}alA(4)}\lrp{1,1,1}{i,i,1}
-(1+2i)\zeta_{{\texttt{c}}alA(4)}\lrp{1,1,1}{i,i,i} \\
+&\,2(1-i)\zeta_{{\texttt{c}}alA(4)}\lrp{1,1,1}{i,1,i}
-3(1-i)\zeta_{{\texttt{c}}alA(4)}\lrp{1,1,1}{i,1,1} \Big),
\end{align*}
while
{\texttt{b}}egin{multline*}
(15-66i)\zeta_{\texttt{a}}st^\Sy\lrp{1,2}{1,1}=48\Big(
(1+i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{i,i,1}
-(1+2i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{i,i,i}\\
+2(1-i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{i,1,i}
-3(1-i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1,1}{i,1,1}\Big)\\
+2\pi i \Big[
(106+2i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{i,i}
-(2-88i) \zeta_{\texttt{a}}st^\Sy\lrp{1,1}{i,1}
-(\frac{3}2-11i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{-1,-1}
-(64+26i)\zeta_{\texttt{a}}st^\Sy\lrp{1,1}{i,-1} \Big].
\end{multline*}
\end{eg}
We end this paper by an intriguing mystery. During our Maple computation, we found
that FCVs should have an interesting structure over $\Q$. For example, numerical
evidence suggests that all the weight $w$ and level 4 FCVs should generate a
dimension $2^w$ vector space over $\Q$. We wonder how to relate this $\Q$ structure
to that of the CMZVs at level 4.
{\texttt{b}}ibliographystyle{alpha}
{\texttt{b}}ibliography{library}
\end{document} |
\begin{document}
\title[On orthogonal polynomials described by Chebyshev polynomials]{A note on orthogonal polynomials described by Chebyshev polynomials}
\author{K. Castillo}
\address{CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal}
\email{kenier@mat.uc.pt}
\author{M. N. de Jesus}
\address{CI$\&$DETS/IPV, Polytechnic Institute of Viseu, ESTGV, Campus Polit\'ecnico de Repeses, 3504-510 Viseu, Portugal}
\email{mnasce@estv.ipv.pt}
\author{J. Petronilho}
\address{CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal}
\email{josep@mat.uc.pt}
\subjclass[2010]{42C05, 33C45}
\date{\today}
\keywords{Orthogonal polynomials, Chebyshev polynomials, polynomial mappings, positive measures, semiclassical orthogonal polynomials}
\begin{abstract}
The purpose of this note is to extend in a simple and unified way
some results on orthogonal polynomials with respect to the weight function
$$\frac{|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;,$$
where $T_m$ is the Chebyshev polynomial of the first kind of degree $m$ and $p>-1$.
\end{abstract}
\maketitle
\section{Main result}
Let $T_n$ and $U_n$ denote the Chebyshev polynomials of first and second kind,
that is, $T_n(\cos\theta)=\cos(n\theta)$ and $U_n(\cos\theta)=\sin\big((n+1)\theta\big)/\sin\theta$
for each nonnegative integer $n$ and $0<\theta<\pi$.
Let $\widehat{T}_n$ and $\widehat{U}_n$ denote the corresponding monic polynomials,
so that $\widehat{T}_0:=\widehat{U}_0:=1$, $\widehat{T}_n(x):=2^{1-n}T_n(x)$
and $\widehat{U}_n(x):=2^{-n}U_n(x)$ for each positive integer $n$.
Set $T_{-n}:=U_{-n}:=\widehat{T}_{-n}:=\widehat{U}_{-n}:=0$ for each positive integer $n$.
The reader is assumed familiar with basic properties of Chebyshev polynomials.
We prove the following
\begin{proposition}\label{main}
Fix an integer $m\geq2$. Define $t_0:=0$ and let $(t_n)_{n\geq1}$ be a sequence of nonzero complex numbers such that
\begin{align*}
& t_{2mn+j}=\mbox{$\frac14$}\;,\quad j\in\{0,1,\dots,2m-1\}\setminus\{0,1,m,m+1\}\;, \\
& t_{2mn}+t_{2mn+1}=t_{2mn+m}+t_{2mn+m+1}=\mbox{$\frac12$}\;,
\end{align*}
for each nonnegative integer $n$.
Let $(P_n)_{n\geq0}$ be the sequence of monic orthogonal polynomials given by
\begin{align}\label{pn}
P_{n+1}(x)=xP_n(x)-t_nP_{n-1}(x)\;.
\end{align}
Then $P_j(x)=\widehat{T}_{j}(x)$ for each $j\in\{0,1,\dots,m\}$ and
\begin{equation}\label{p2mn+m+j+1}
P_{2mn+m+j+1}(x) =\frac{A_j(n;x)Q_{n+1}\big(\widehat{T}_{2m}(x)\big)
+4^{-j}t_{2mn+m+1}B_j(n;x)Q_{n}\big(\widehat{T}_{2m}(x)\big)}{\widehat{U}_{m-1}(x)}
\end{equation}
for each $j\in\{0,1,\dots,2m-1\}$, where
\begin{align*}
A_j(n;x)&:=\widehat{U}_j(x)+\Big(\mbox{$\frac14$}-t_{2m(n+1)}\Big)
\Big(\widehat{U}_{j-m}(x)\widehat{U}_{m-2}(x)-\widehat{U}_{j-m-1}(x)\widehat{U}_{m-1}(x)\Big)\,, \\
B_j(n;x)&:=\widehat{U}_{2m-2-j}(x) \\
&\quad+\Big(\mbox{$\frac14$}-t_{2m(n+1)}\Big)
\Big(\widehat{U}_{m-j-3}(x)\widehat{U}_{m-1}(x)-\widehat{U}_{m-j-2}(x)\widehat{U}_{m-2}(x)\Big)\,,
\end{align*}
and $(Q_n)_{n\geq0}$ is the sequence of monic orthogonal polynomials given by
\begin{align*}
Q_{n+1}(x)=(x-r_n)Q_n(x)-s_nQ_{n-1}(x)\;,
\end{align*}
with
\begin{align*}
r_n&:=\frac{1}{\displaystyle 2^{2m-4}}\Big(t_{2mn+m}t_{2mn+1}+t_{2m(n+1)}t_{2mn+m+1}-\mbox{$\frac18$}\Big)\;,\\
s_n&:=\frac{1}{\displaystyle 4^{2m-4}}\,t_{2mn}t_{2mn+1}t_{2mn+m}t_{2m(n-1)+m+1}\;.
\end{align*}
Assume furthermore that $t_n>0$ for each positive integer $n$.
Then $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$ are orthogonal polynomial sequences with respect to
certain positive measures, say $\mu_P$ and $\mu_Q$ respectively.
Suppose that $\mu_Q$ is absolutely continuous
with weight function $w_Q$ on $[\xi,\eta]$, with $-2^{1-2m}\leq\xi<\eta\leq2^{1-2m}$, i.e.,
$$
{\rm d}\mu_Q(x)=w_Q(x)\chi_{(\xi,\eta)}(x){\rm d}x\;.
$$
Suppose in addition that $w_Q$ satisfies the condition
$$C:=\int_\xi^\eta\frac{w_Q(x)}{x+2^{1-2m}}\,{\rm d}x<\infty\;.$$
Then $($up to a positive constant factor$)$
\begin{align}\label{muP}
{\rm d}\mu_P(x)=w_P(x)\chi_{E}(x){\rm d}x +
M\sum_{j=1}^m\delta\left(x-\cos\frac{(2j-1)\pi}{2m}\right)\;,
\end{align}
where
$$
w_P(x):=\left|\frac{U_{m-1}(x)}{T_m(x)}\right|w_Q\big(\widehat{T}_{2m}(x)\big)\;,\quad
x\in E:=\widehat{T}_{2m}^{-1}\big((\xi,\eta)\big)\,,
$$
and $M$ is a nonnegative number given by
$$
M:=\frac{1}{m}\left(\frac{2^{2m-3}\mu_Q(\mathbb{R})}{t_m}-C\right)\;.
$$
\end{proposition}
\begin{remark}
For $j=2m-1$, \eqref{p2mn+m+j+1} reduces to
$$
P_{2mn+m}(x)=\widehat{T}_m(x)\,Q_n\big(\widehat{T}_{2m}(x)\big)\;.
$$
\end{remark}
\begin{remark}\label{Zmzi}
In general, the set $E$ is the union of $2m$ disjoint open intervals
separated by the zeros of $U_{2m-1}$.
$($If all these intervals have the zeros of $U_{2m-1}$ as boundary points,
then $\overline{E}$ reduces to a single interval.$)$
Moreover,
$\mbox{\rm supp}\big(\mu_P\big)=\overline{E}$ if $M=0$, and
$\mbox{\rm supp}\big(\mu_P\big)=\overline{E}\cup Z_m$ if $M>0$,
where $Z_m$ is the set of zeros of $T_m$, i.e.,
$Z_m:=\big\{z_i:=\cos\big((2i-1)\pi/(2m)\big)\,|\, 1\leq i\leq m\big\}$.
\end{remark}
\begin{remark}
Proposition \ref{main} cannot be considered for $m=1$.
Indeed, if $m=1$ the conditions imposed on the sequence $(t_n)_{n\geq1}$ imply $t_2=0$.
In any case, after reading Section 2, interested readers are able to derive the corresponding
result for $m=1$.
\end{remark}
In order to illustrate the practical effectiveness of Proposition \ref{main},
we consider a simple example.
Fix $p\in\mathbb{C}\setminus\{-1,-2,\ldots\}$ and an integer $m\geq2$. Define
$t_{2mn+j}:=1/4$ for each $j\in\{0,1,\ldots,2m-1\}\setminus\{0,1,m,m+1\}$ and
\begin{align*}
\qquad t_{2mn}&:=\frac{n}{4n+p}\;, &
t_{2mn+1}&:=\frac{2n+p}{2(4n+p)}\;, & \quad \\
\qquad t_{2mn+m}&:=\frac{2n+p+1}{2(4n+p+2)}\;, &
t_{2mn+m+1}&:=\frac{2n+1}{2(4n+p+2)} &
\end{align*}
for each nonnegative integer $n$. (If $p=0$ it is understood that $t_1:=1/2$.)
These parameters satisfy the hypothesis of Proposition \ref{main}.
In this case
\begin{align*}
r_n&=\frac{1}{2^{2m-1}}
\frac{p(p+2)}{(4n+p)(4n+p+4)}\;, \\
s_n&=\frac{1}{4^{2m-1}}\frac{2n(2n-1)(2n+p)(2n+p+1)}
{(4n+p-2)(4n+p)^2(4n+p+2)}\;,
\end{align*}
and so
$$Q_n(x)=2^{(1-2m)n}\widehat{P}_n^{\big(-\frac12,\frac{p+1}{2}\big)}\big(2^{2m-1}x\big)\;,$$
where $\widehat{P}_n^{(\alpha,\beta)}$ denotes the monic Jacobi polynomial of degree $n$
(see \cite[Chapter 4]{I2005} and \cite[(6.11)]{M1991}).
Hence, by Proposition \ref{main},
\begin{align}\label{ExamplePn}
&P_{2mn+m+j+1}(x) \\
&\quad=\frac{(2n+2+p)U_j(x)-pT_m(x)U_{j-m}(x)}{2^{(2m-1)n+m+j-1}(4n+p+4)U_{m-1}(x)}
\widehat{P}_{n+1}^{\big(-\frac12,\frac{p+1}{2}\big)}\big(T_{2m}(x)\big) \nonumber \\
&\quad\quad
+\frac{(2n+1)\big((2n+2)U_{2m-2-j}(x)-pU_{m-1}(x)T_{m-j-1}(x)\big)}
{2^{(2m-1)n+m+j}(4n+p+2)(4n+p+4)U_{m-1}(x)}
\widehat{P}_{n}^{\big(-\frac12,\frac{p+1}{2}\big)}\big(T_{2m}(x)\big) \nonumber
\end{align}
for each nonnegative integer $n$ and each $j\in\{0,1,\dots,2m-1\}$.
Furthermore, if $p>-1$ then $(Q_n)_{n\geq0}$ is a sequence of orthogonal polynomials associated with the weight function
$w_Q$ on $\big[-2^{1-2m},2^{1-2m}\big]$ given by
$$
w_Q(x)=2^{-p/2}\big(1-2^{2m-1}x\big)^{-\frac12}\big(1+2^{2m-1}x\big)^{\frac{p+1}{2}}\;,\quad -2^{1-2m}<x<2^{1-2m}\;.
$$
Since
\begin{align*}
\mu_Q(\mathbb{R})&=\int_{-2^{1-2m}}^{2^{1-2m}}w_Q(x)\,{\rm d}x
=2^{2-2m}\,\frac{p+1}{p+2}B\Big(\mbox{$\frac{p+1}{2}$},\mbox{$\frac{1}{2}$}\Big)\;,\\
C&=\int_{-2^{1-2m}}^{2^{1-2m}}\frac{w_Q(x)}{x+2^{1-2m}}\,{\rm d}x
=B\Big(\mbox{$\frac{p+1}{2}$},\mbox{$\frac{1}{2}$}\Big)\;,
\end{align*}
$B$ being the Beta function (see \cite[p. 8]{I2005}), we get $M=0$.
Taking into account that
$\overline{E}=\widehat{T}_{2m}^{-1}\big(\big[-2^{1-2m},2^{1-2m}\big]\big)={T}_{2m}^{-1}\big([-1,1]\big)=[-1,1]$,
we conclude that $(P_n)_{n\geq0}$ is a sequence of orthogonal polynomials associated with the weight function
$w_P$ on $[-1,1]$ given by
\begin{align}
w_P(x)&=2^{-p/2}\,\left|\frac{U_{m-1}(x)}{T_m(x)}\right|
\big(1-T_{2m}(x)\big)^{-\frac12}\big(1+T_{2m}(x)\big)^{\frac{p+1}{2}} \nonumber \\
&=\frac{\;|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;. \label{measureTmp}
\end{align}
\begin{remark}
The search of the recurrence coefficients from the weight function \eqref{measureTmp}
has arouse interest in recent and not recent years.
For $p=2$ see \cite{MVA1989};
for $p=1$ and $m=2$ see \cite{GLi1988};
for $p=2s$ with $s\in\mathbb{N}$ and $m=2$ see \cite{CMM2016};
and for $p=2s$ with $s>-1/2$ and $m=2$ see \cite{CMV2018}.
It is worth mentioning that in these works
the explicit representation \eqref{ExamplePn} has not been established.
While it is true that a representation given by a different polynomial mapping appears in \cite{MVA1989} for $p=2$
using results from \cite{GVA1988}.
\end{remark}
The sequence $(P_n)_{n\geq0}$ in Proposition \ref{main} may be regarded as an example of sieved orthogonal polynomials
$($see e.g. \cite{AAA1984,CI1986,CI1993,CIM1994}$)$. It is important to highlight that we can go even further and show more distinctly
how some recent developments of the theory of polynomial mappings (see \cite{MP2010,KMP2017,KMP2019}) apply to this
kind of problems, but that would demand a more extended discussion in which we
need not now get involved. Indeed, using the general results stated in \cite{KMP2017},
it can be shown that if $(P_n)_{n\geq0}$ or $(Q_n)_{n\geq0}$ in Proposition \ref{main} is semiclassical (see \cite{M1991}),
then so is the other one. In this case, we may obtain also the functional (distributional) equation
fulfilled by the (moment) regular functional associated with $(P_n)_{n\geq0}$ or $(Q_n)_{n\geq0}$.
In particular, as in \cite{KMP2019} or directly from \eqref{ExamplePn}, we may easily derive the linear homogeneous second order ordinary differential equation that the orthogonal polynomials with respect to \eqref{measureTmp} fulfil, which in turn leads to interesting electrostatic models (see \cite[Section 6]{KMP2019} and \cite[Section 3.5]{I2005}).
\section{Proof of Proposition \ref{main}}
Set $k:=2m$ and rewrite \eqref{pn} as a system of blocks of recurrence relations
\begin{align*}
x P_{nk+j}(x)=P_{nk+j+1}(x)+a_n^{(j)}P_{nk+j-1}(x) \;,\quad 0\leq j\leq k-1\;,
\end{align*}
where $P_{-1}:=0$ and $a_n^{(j)}:=t_{kn+j}$ whenever $(n,j)\neq(0,0)$.
Following \cite{CI1993,CIM1994}, we introduce the notation
\begin{equation}\nonumber
\Delta_n(i,j;x):=\left\{
\begin{array}{rl}
0\,,&j<i-2 \\
1\,,& j=i-2 \\
x\,,& j=i-1
\end{array}
\right.
\end{equation}
and
\begin{equation}\nonumber
\Delta_n(i,j;x):=\det\left(
\begin{array}{cccccc}
x & 1 & 0 & \dots & 0 & 0 \\
a_n^{(i)} & x & 1 & \dots & 0 & 0 \\
0 & a_n^{(i+1)} & x & \dots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ldots & x & 1 \\
0 & 0 & 0 & \ldots & a_n^{(j)} & x
\end{array}
\right)\;,\quad j\geq i\geq 1\;.
\end{equation}
The reader should satisfy himself that
$$
\Delta_n(m+2,m+j;x)=A_j(n;x)\,, \quad
\Delta_{n}(m+j+3,m+k-1;x)=B_j(n;x)
$$
for each $j\in\{0,1,\ldots,2m-1\}$. In particular,
\begin{align*}
\Delta_n(m+2,m+k-1;x)&=\widehat{U}_{2m-1}(x)=\widehat{T}_m(x)\widehat{U}_{m-1}(x) \\
&=\Delta_0(1,m-1;x)\eta_{k-1-m}(x)\,,\quad \eta_{k-1-m}(x):=\widehat{U}_{m-1}(x)\,.
\end{align*}
Moreover,
\begin{align*}
r_n(x)&:=a_{n}^{(m+1)}\Delta_{n}(m+3,m+k-1;x)-a_0^{(m+1)}\Delta_{0}(m+3,m+k-1;x) \\
&\quad+a_{n}^{(m)}\Delta_{n-1}(m+2,m+k-2;x)-a_0^{(m)}\Delta_{0}(1,m-2;x)\,\eta_{k-1-m}(x) \\
&=\frac{1}{\displaystyle 2^{2m-4}}\Big(t_{2mn+m}t_{2mn+1}+t_{2m(n+1)}t_{2mn+m+1}
-\mbox{$\frac12$}\,t_m-t_{m+1}t_{2m}\Big)
\end{align*}
for each positive integer $n$, and so
$$
\quad
r+r_n(0)=r_n\;,\quad r_0=r:=\frac{1}{\displaystyle 2^{2m-4}}\Big(\mbox{$\frac18$}-t_{m+1}t_{2m+1}\Big)\;.
$$
Note also that
$a_{n}^{(m)}a_{n-1}^{(m+1)}\cdots a_{n-1}^{(m+k-1)}=s_n$.
In addition,
\begin{align*}
\pi_k(x)&:=\Delta_0(1,m;x)\,\eta_{k-1-m}(x)-a_0^{(m+1)}\,\Delta_0(m+3,m+k-1;x)+r\\
&=\widehat{T}_{2m}(x)\;.
\end{align*}
Consequently, the hypotheses of \cite[Theorem 2.1]{MP2010} are satisfied
with $\theta_m(x)=\widehat{T}_{m}(x)$, and so (\ref{p2mn+m+j+1}) follows.
Finally, the explicit representation of $\mu_P$ appearing in (\ref{muP}) follows by
\cite[Theorem 3.4\footnote{In \cite[Theorem 3.4]{MP2010}, $r=0$. Nevertheless, inspection
of its proof shows that the theorem remains true if instead $r=0$ we assume that the polynomials $\pi_k$ and $\theta_m\eta_{k-m-1}$ have real and distinct zeros.} and Remark 3.5]{MP2010}, after noting that
\begin{align*}
\pi_k'& =2m\widehat{U}_{2m-1}=2m\widehat{T}_{m}\widehat{U}_{m-1}=2m\theta_m\eta_{k-m-1}\;, \\
\pi_k(z_i)& =2^{1-2m}\,T_2\big(T_m(z_i)\big)=2^{1-2m}\,T_2(0)=-2^{1-2m}\;, \\
M_i & :=\frac{ \mu_Q(\mathbb{R})\,\Delta_0(2,m-1;z_i)/
\prod_{j=1}^m a_0^{(j)}-\eta_{k-1-m}(z_i)\,C}{ \theta_m^\prime(z_i)}\\
&=\frac{ \mu_Q(\mathbb{R})\,\widehat{U}_{m-1}(z_i)\cdot 2^{2m-3}t_m^{-1}
-\widehat{U}_{m-1}(z_i)\,C}{ m\widehat{U}_{m-1}(z_i)}=M
\end{align*}
for each $i\in\{1,\ldots,m\}$, where $z_i$ is a zero of $T_m$ (see Remark \ref{Zmzi}).
\end{document} |
\begin{document}
\newcommand{1608.04435}{1608.04435}
\renewcommand{042}{042}
{\mathbb F}irstPageHeading
\ShortArticleName{On the Equivalence of Module Categories over a Group-Theoretical Fusion Category}
{\mathcal A}rticleName{On the Equivalence of Module Categories\\ over a Group-Theoretical Fusion Category}
{\mathcal A}uthor{Sonia NATALE}
{\mathcal A}uthorNameForHeading{S.~Natale}
{\mathcal A}ddress{Facultad de Matem\'atica, Astronom\'{\i}a, F\'{\i}sica y Computaci\'on, Universidad Nacional de C\'ordoba,\\ CIEM-CONICET, C\'ordoba, Argentina}
{\mathcal E}mail{\href{mailto:natale@famaf.unc.edu.ar}{natale@famaf.unc.edu.ar}}
{\mathcal U}RLaddress{\url{http://www.famaf.unc.edu.ar/~natale/}}
{\mathcal A}rticleDates{Received April 28, 2017, in f\/inal form June 14, 2017; Published online June 17, 2017}
{\mathcal A}bstract{We give a necessary and suf\/f\/icient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classif\/ication of such module categories.}
{\mathcal K}eywords{fusion category; module category; group-theoretical fusion category}
{\mathcal C}lassification{18D10; 16T05}
\section{Introduction}
Throughout this paper we shall work over an algebraically closed f\/ield $k$ of characteristic zero. Let $\mathcal{C}$ be a fusion category over~$k$. The notion of a ${\mathcal C}$-module category provides a natural categorif\/ication of the notion of representation of a group. The problem of classifying module categories plays a fundamental role in the theory of tensor categories.
Two fusion categories ${\mathcal C}$ and ${\mathcal D}$ are called \emph{categorically Morita equivalent} if there exists an indecomposable $\mathcal{C}$-module category $\mathcal{M}$ such that $\mathcal{D}^{\rm op}$ is equivalent as a fusion category to the category $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})$ of ${\mathcal C}$-module endofunctors of~${\mathcal M}$. This def\/ines an equivalence relation in the class of all fusion categories.
Recall that a fusion category ${\mathcal C}$ is called \emph{pointed} if every simple object of ${\mathcal C}$ is invertible. A~basic class of fusion categories consists of those which are categorically Morita equivalent to a pointed fusion category; a fusion category in this class is called \emph{group-theoretical}. Group-theoretical fusion categories can be described in terms of f\/inite groups and their cohomology.
The purpose of this note is to give a necessary and suf\/f\/icient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category to be equivalent. For this, it is enough to solve the same problem for indecomposable module categories over pointed fusion categories.
Let ${\mathcal C}$ be a pointed fusion category. Then there exist a f\/inite group $G$ and a~3-cocycle~$\omega$ on~$G$ such that ${\mathcal C} \cong {\mathcal C}(G, \omega)$, where ${\mathcal C}(G, \omega)$ is the category of f\/inite-dimensional $G$-graded vector spaces with associativity constraint def\/ined by $\omega$ (see Section~\ref{cgomega} for a precise def\/inition). Let~${\mathcal M}$ be an indecomposable right ${\mathcal C}$-module category. Then there exists a~subgroup~$H$ of~$G$ and a~2-cochain $\psi \in C^2(H, k^{\times})$ satisfying
\begin{gather}
\label{cond-alfa}d\psi = \omega\vert_{H \times H \times H},
\end{gather}
such that ${\mathcal M}$ is equivalent as a ${\mathcal C}$-module category to the category ${\mathcal M}_0(H, \psi)$ of left $A(H, \psi)$-modules in ${\mathcal C}$, where $A(H, \psi) = k_\psi H$ is the group algebra of $H$ with multiplication twisted by~$\psi$~\cite{ostrik}, \cite[Example~9.7.2]{egno}.
The main result of this paper is the following theorem.
\begin{Theorem}\label{main} Let $H, L$ be subgroups of $G$ and let $\psi \in C^2(H, k^{\times})$ and $\xi \in C^2(L, k^{\times})$ be $2$-cochains satisfying condition~\eqref{cond-alfa}. Then ${\mathcal M}_0(H, \psi)$ and ${\mathcal M}_0(L, \xi)$ are equivalent as ${\mathcal C}$-module categories if and only if there exists an element $g \in G$ such that $H = {}^gL$ and the class of the $2$-cocycle
\begin{gather}\label{cond-equiv} {\xi}^{-1}{\psi}^g \Omega_g\vert_{L\times L}
\end{gather}
is trivial in $H^2(L, k^{\times})$.
\end{Theorem}
Here we use the notation ${}^g x = gxg^{-1}$ and $^gL = \{{}^g x\colon x\in L\}$. The 2-cochain ${\psi}^g \in C^2(L, k^{\times})$ is def\/ined by ${\psi}^g(g_1, g_2) = {\psi}({}^gg_1, {}^gg_2)$, for all $g_1, g_2 \in L$, and $\Omega_g\colon G \times G \to k^{\times}$ is given by
\begin{gather*}\Omega_g(g_1, g_2) = \frac{\omega({}^gg_1, {}^gg_2, g) \omega(g, g_1, g_2)}{\omega({}^gg_1, g, g_2)}.
\end{gather*}
Observe that \cite[Theorem 3.1]{ostrik} states that the indecomposable module categories considered in Theorem~\ref{main} are parameterized by conjugacy classes of pairs $(H, \psi)$. However, this conjugation relation is not described loc.\ cit.\ (compare also with \cite{nikshych} and \cite[Section~9.7]{egno}).
\looseness=-1 Consider for instance the case where ${\mathcal C}$ is the category of f\/inite-dimensional representations of the 8-dimensional Kac Paljutkin Hopf algebra. Then ${\mathcal C}$ is group-theoretical. In fact, ${\mathcal C} \cong {\mathcal C}(G, \omega, C, 1)$, where $G \cong D_8$ is a semidirect product of the group $L = \mathbb Z_2 \times \mathbb Z_2$ by $C = \mathbb Z_2$ and~$\omega$ is a certain (nontrivial) 3-cocycle on $G$~\cite{schauenburg}. Let $\xi$ represent a nontrivial cohomology class in $H^2(L, k^\times)$. According to the usual conjugation relation among pairs $(L, \psi)$, the result in \cite[Theorem~3.1]{ostrik} would imply that the pairs $(L, 1)$ and $(L, \xi)$, not being conjugated under the adjoint action of $G$, give rise to two inequivalent ${\mathcal C}$-module categories. These module categories both have rank one, whence they give rise to non-isomorphic f\/iber functors on ${\mathcal C}$. However, it follows from \cite[Theorem~4.8(1)]{ma-contemp} that the category ${\mathcal C}$ has a unique f\/iber functor up to isomorphism. In fact, in this example there exists $g \in G$ such that $\Omega_g\vert_{L\times L}$ is a 2-cocycle cohomologous to~$\xi$. See Example~\ref{kp}.
Certainly, the condition given in Theorem \ref{main} and the usual conjugacy relation agree in the case where the 3-cocycle $\omega$ is trivial, and it reduces to the conjugation relation among subgroups when they happen to be cyclic.
As explained in Section~\ref{adj-action}, condition~\eqref{cond-equiv} is equivalent to the condition that $A(L, \xi)$ and ${}^gA(H, \psi)$ be isomorphic as algebras in~${\mathcal C}$, where $\underline{G} \to \underline{\text{Aut}}_\otimes{\mathcal C}$, $g \mapsto {}^g( \ )$, is the adjoint action of~$G$ on~${\mathcal C}$ (see Lemma~\ref{gdea}).
Theorem \ref{main} can be reformulated as follows.
\begin{Theorem} Two ${\mathcal C}$-module categories ${\mathcal M}_0(H, \psi)$ and ${\mathcal M}_0(L, \xi)$ are equivalent if and only if the algebras $A(H, \psi)$ and $A(L, \xi)$ are conjugated under the adjoint action of $G$ on ${\mathcal C}$.
\end{Theorem}
Theorem \ref{main} is proved in Section~\ref{demo}. Our proof relies on the fact that, as happens with group actions on vector spaces, the adjoint action of the group $G$ in the set of equivalence classes of ${\mathcal C}$-module categories is trivial (Lemma~\ref{adj-triv}). In the course of the proof we establish a relation between the 2-cocycle in~\eqref{cond-equiv} and a 2-cocycle attached to $g$, $\psi$ and $\xi$ in~\cite{ostrik} (Remark~\ref{rmk-alfag} and Lemma~\ref{rel-cociclos}).
We refer the reader to~\cite{egno} for the main notions on fusion categories and their module categories used throughout.
\section{Preliminaries and notation}\label{prels}
\subsection{}
Let $\mathcal{C}$ be a fusion category over $k$. A (\emph{right}) ${\mathcal C}$-\emph{module category} is a f\/inite semisimple $k$-linear abelian category $\mathcal{M}$ equipped with a bifunctor $\bar\otimes\colon \mathcal{M}\times \mathcal{C} \rightarrow\mathcal{M}$ and natural isomorphisms
\begin{gather*}\mu_{M, X,Y}\colon \ M \bar\otimes (X\otimes Y)\rightarrow (M\bar\otimes X)\bar\otimes Y,\qquad r_M\colon \ M \bar\otimes \textbf{1}\rightarrow M, \end{gather*}
$X, Y \in {\mathcal C}$, $M \in \mathcal M$, satisfying the following conditions:
\begin{gather} \label{modcat1} \mu_{M\bar\otimes X, Y, Z} \mu_{M, X, Y \otimes Z} ( \id_M \bar\otimes a_{X, Y, Z} ) =
(\mu_{M, X, Y} \otimes \id_Z ) \mu_{M, X\otimes Y, Z},\\
\label{modcat2} (r_M \otimes \id_Y ) \mu_{M, \textbf{1}, Y} = \id_M \bar\otimes l_Y,
\end{gather}
for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$, where $a\colon \otimes \circ (\otimes \times \id_{\mathcal C}) \to \otimes \circ (\id_{\mathcal C} \times \otimes)$ and $l\colon \textbf{1} \otimes ? \to \id_{\mathcal C}$, denote the associativity and left unit constraints in ${\mathcal C}$, respectively.
Let $A$ be an algebra in ${\mathcal C}$. Then the category $_A{\mathcal C}$ of left $A$-modules in ${\mathcal C}$ is a right ${\mathcal C}$-module category with action $\bar\otimes\colon _A{\mathcal C} \times {\mathcal C} \to {\mathcal C}_A$, given by $M\bar\otimes X = M\otimes X$ endowed with the left $A$-module structure $(m_M \otimes \id_X) a_{A, M, X}^{-1}\colon A\otimes (M\otimes X) \to M\otimes X$, where $m_M\colon A\otimes M \to M$ is the $A$-module structure in $M$. The associativity constraint of $_A{\mathcal C}$ is given by $a^{-1}_{M, X, Y}\colon M\bar\otimes (X \otimes Y) \to (M\bar\otimes X) \bar\otimes Y$, for all $M \in {} _A{\mathcal C}$, $X, Y \in {\mathcal C}$.
A ${\mathcal C}$-module functor ${\mathcal M} \to {\mathcal M}'$ between right ${\mathcal C}$-module categories $({\mathcal M}, \bar\otimes)$ and $({\mathcal M}', \bar\otimes')$ is a pair $(F, \zeta)$, where $F\colon {\mathcal M}\to {\mathcal M}'$ is a functor and $\zeta_{M, X}\colon F(M \bar\otimes X) \to F(M) \bar\otimes' X$ is a natural isomorphism satisfying
\begin{gather}\label{uno-z} (\zeta_{M, X} \otimes \id_Y ) \zeta_{M \bar\otimes X, Y} F(\mu_{M, X, Y}) = {\mu'}_{F(M), X, Y} \zeta_{M, X \otimes Y},\\
\label{dos-z} {r'}_{F(M)} \zeta_{M, \textbf{1}} = F(r_M),
\end{gather}
for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$.
Let ${\mathcal M}$ and ${\mathcal M}'$ be ${\mathcal C}$-module categories. An \emph{equivalence} of ${\mathcal C}$-module categories ${\mathcal M} \to {\mathcal M}'$ is a ${\mathcal C}$-module functor $(F, \zeta)\colon {\mathcal M}\to {\mathcal M}'$ such that $F$ is an equivalence of categories. If such an equivalence exists, ${\mathcal M}$ and ${\mathcal M}'$ are called \emph{equivalent ${\mathcal C}$-module categories}. A~${\mathcal C}$-module category is called \emph{indecomposable} if it is not equivalent to a direct sum of two nontrivial $\mathcal{C}$-submodule categories.
Let ${\mathcal M}, {\mathcal M}'$ be indecomposable ${\mathcal C}$-module categories. Then $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})$ is a fusion category with tensor product given by composition of functors and the category $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M}')$ is an indecomposable module category over $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})$ in a natural way. If $A$ and $B$ are indecomposable algebras in ${\mathcal C}$ such that ${\mathcal M} \cong _A\!{\mathcal C}$ and ${\mathcal M}' \cong {}_B{\mathcal C}$, then $\operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M})^{\rm op}$ is equivalent to the fusion category $_A{\mathcal C}_A$ of $(A, A)$-bimodules in ${\mathcal C}$ and there is an equivalence of $_A{\mathcal C}_A$-module categories $_B{\mathcal C}_A \cong \operatorname{Fun}_{\mathcal C}({\mathcal M}, {\mathcal M}')$, where $_B{\mathcal C}_A$ is the category of $(B, A)$-bimodules in ${\mathcal C}$.
\subsection{}\label{action-mc} Let ${\mathcal M}$ be a ${\mathcal C}$-module category. Every tensor autoequivalence $\rho\colon {\mathcal C} \to {\mathcal C}$ induces a ${\mathcal C}$-module category structure ${\mathcal M}^\rho$ on ${\mathcal M}$ in the form $M \bar\otimes^\rho X = M\bar\otimes \rho(X)$, with associativity constraint
\begin{gather*}\mu^\rho_{M, X, Y} = \mu_{M, \rho(X) \otimes \rho(Y)} \big(\id_M \bar\otimes {\rho^2_{X, Y}}^{-1}\big)\colon \ M \bar\otimes \rho(X\otimes Y) \to (M\bar\otimes \rho(X)) \bar\otimes \rho(Y),\end{gather*} for all $M \in {\mathcal M}$, $X, Y \in {\mathcal C}$, where $\rho^2_{X, Y}\colon \rho(X) \otimes \rho(Y) \to \rho(X \otimes Y)$ is the monoidal structure of~$\rho$. See \cite[Section~3.2]{nikshych}.
Suppose that $A$ is an algebra in ${\mathcal C}$. Then $\rho(A)$ is an algebra in ${\mathcal C}$ with multiplication
\begin{gather*}m_{\rho(A)} = \rho(m_A) \rho^2_{A, A}\colon \ \rho(A) \otimes \rho(A) \to \rho(A).\end{gather*}
The functor $\rho$ induces an equivalence of ${\mathcal C}$-module categories $_{\rho(A)}{\mathcal C} \to (_A{\mathcal C})^\rho$ with intertwining isomorphisms
\begin{gather*} {\rho^2_{M, X}}^{-1}\colon \ \rho(M \bar\otimes X) \to \rho(M) \bar\otimes^\rho X.\end{gather*}
\subsection{}\label{cgomega} Let $G$ be a f\/inite group.
Let $X$ be a $G$-module. Given an $n$-cochain $f \in C^n(G, X)$ (where $C^0(G, M) = M$), the coboundary of $f$ is the $(n+1)$-cochain $df = d^nf \in C^{n+1}(G, X)$ def\/ined by
\begin{gather*}d^nf(g_1, \dots, g_{n+1}) = g_1.f(g_2, \dots, g_{n+1}) + \sum_{i = 1}^{n}f(g_1, \dots, g_ig_{i+1}, \dots, g_n)\\
\hphantom{d^nf(g_1, \dots, g_{n+1}) =}{} + (-1)^{n+1} f(g_1, \dots, g_n),\end{gather*}
for all $g_1, \dots, g_{n+1} \in G$. The kernel of $d^n$ is denoted $Z^n(G, M)$; an element of $Z^n(G, M)$ is an $n$-cocycle. We have $d^nd^{n-1} = 0$, for all $n \geq 1$. The $n$th cohomology group of $G$ with coef\/f\/icients in $M$ is $H^n(G, M) = Z^n(G, M)/d^{n-1}(C^{n-1}(G, M))$. We shall write $f \equiv f'$ when the cochains $f, f' \in C^n(G, k^\times)$ dif\/fer by a~coboundary.
We shall assume that every cochain $f$ is \emph{normalized}, that is, $f(g_1, \dots, g_n) = 1$, whenever one of the arguments $g_1, \dots, g_n$ is the identity. If $H$ is a subgroup of $G$ and $f \in C^n(H, k^\times)$, we shall indicate by~$f^g$ the $n$-cochain in $^{g^{-1}\!}H$ given by $f^g(h_1, \dots, h_n) = f({}^g h_1, \dots, {}^g h_n)$, $h_1, \dots, h_n \in H$.
Let $\omega \colon G \times G \times G \to k^\times$ be a 3-cocycle on $G$. Let ${\mathcal C}(G, \omega)$ denote the fusion category of f\/inite-dimensional $G$-graded vector spaces with associativity constraint def\/ined, for all $U, V, W\in {\mathcal C}(G, \omega)$, as
\begin{gather*}a_{X, Y, Z} ((u \otimes v)\otimes w) = \omega^{-1}(g_1, g_2, g_3) u \otimes (v\otimes w),\end{gather*}
for all homogeneous vectors $u \in U_{g_1}$, $v \in V_{g_2}$, $w \in W_{g_3}$, $g_1, g_2, g_3 \in G$. Any pointed fusion category is equivalent to a~category of the form ${\mathcal C}(G, \omega)$.
A fusion category ${\mathcal C}$ is called \emph{group-theoretical} if it is categorically Morita equivalent to a~pointed fusion category. Equivalently, ${\mathcal C}$ is group-theoretical if and only if there exist a f\/inite group $G$ and a $3$-cocycle $\omega\colon G \times G \times G \to k^{\times}$ such that~${\mathcal C}$ is equivalent to the fusion category ${\mathcal C}(G, \omega, H, \psi) = _{A(H, \psi)\!}{\mathcal C}(G, \omega)_{A(H, \psi)}$, where $H$ is a subgroup of $G$ such that the class of $\omega\vert_{H\times H \times H}$ is trivial and $\psi\colon H \times H \to k^{\times}$ is a 2-cochain on~$H$ satisfying condition~\eqref{cond-alfa}.
Let ${\mathcal C}(G, \omega, H, \psi) \cong {\mathcal C}(G, \omega)^*_{{\mathcal M}_0(H, \psi)}$ be a group-theoretical fusion category. Then there is a bijective correspondence between equivalence classes of indecomposable ${\mathcal C}(G, \omega, H, \psi)$-module categories and equivalence classes of indecomposable ${\mathcal C}(G, \omega)$-module categories. This correspondence attaches to every indecomposable ${\mathcal C}(G, \omega)$-module category ${\mathcal M}$ the ${\mathcal C}(G, \omega, H, \psi)$-module category
\begin{gather*}{\mathcal M}(H, \psi) = {\mathbb F}un_{{\mathcal C}(G, \omega)}({\mathcal M}_0(H, \psi), {\mathcal M}). \end{gather*}
\section[Indecomposable module categories over ${\mathcal C}(G, \omega)$]{Indecomposable module categories over $\boldsymbol{{\mathcal C}(G, \omega)}$}\label{ptd-mc}
Throughout this section $G$ is a f\/inite group and $\omega \colon G \times G \times G \to k^\times$ is a 3-cocycle on $G$.
\subsection{}\label{adj-action} Let $g \in G$. Consider the 2-cochain $\Omega_g\colon G \times G \to k^{\times}$ given by
\begin{gather*}\Omega_g(g_1, g_2) = \frac{\omega({}^gg_1, {}^gg_2, g) \omega(g, g_1, g_2)}{\omega({}^gg_1, g, g_2)}.
\end{gather*}
For all $g \in G$ we have the relation
\begin{gather}\label{o-oy}d\Omega_g = \frac{\omega}{\omega^g}.
\end{gather}
Let ${\mathcal C} = {\mathcal C}(G, \omega)$ and let $g \in G$. For every object $V$ of ${\mathcal C}$ let ${}^g V$ be the object of ${\mathcal C}$ such that $^gV = V$ as a vector space with $G$-grading def\/ined as $({}^gV)_x = V_{^g x}$, $x \in G$. For every $g \in G$, we have a functor $\ad_g \colon {\mathcal C} \to {\mathcal C}$, given by $\ad_g(V) = {}^gV$ and $\ad_g(f) = f$, for every object $V$ and morphism $f$ of ${\mathcal C}$.
Relation~\eqref{o-oy} implies that $\ad_g$ is a tensor functor with monoidal structure def\/ined by
\begin{gather*}\big(\ad_g^2\big)_{U, V}\colon \ {}^gU \otimes {}^gV \to {}^g (U \otimes V), \qquad \big(\ad_g^2\big)_{U, V} (u \otimes v) = \Omega_g(h, h')^{-1} u\otimes v,\end{gather*}
for all $h, h' \in G$, and for all homogeneous vectors $u \in U_h$, $v \in V_{h'}$.
For every $g, g_1, g_2 \in G$, let $\gamma(g_1, g_2)\colon G \to k^\times$ be the map def\/ined in the form
\begin{gather*}
\gamma(g_1, g_2)(g) = \frac{\omega(g_1, g_2, g) \omega({}^{g_1g_2}g, g_1, g_2)}{\omega(g_1,{}^{g_2}g, g_2)}.
\end{gather*}
The following relation holds, for all $g_1, g_2 \in G$:
\begin{gather}\label{rel-omega}
\Omega_{g_1g_2} = \Omega_{g_1}^{g_2} \Omega_{g_2} d \gamma(g_1, g_2) .
\end{gather}
In this way, $\ad\colon \underline G \to \underline{\text{Aut}}_\otimes{\mathcal C}$, $\ad(g) = \big(\ad_g, \ad_g^2\big)$, gives rise to an action by tensor autoequivalences of $G$ on ${\mathcal C}$ where, for every $g, x \in G$, $V \in {\mathcal C}(G, \omega)$, the monoidal isomorphisms ${\ad^2}_V\colon {}^g({}^{g'}V) \to {}^{gg'}V$ are given by
\begin{gather*}\ad^2_V(v) = \gamma(g, g') (x) v,\end{gather*} for all homogeneous vectors $v \in V_x$, $h \in G$. The equivariantization ${\mathcal C}^G$ with respect to this action is equivalent to the category of f\/inite-dimensional representations of the twisted quantum double $D^\omega G$ (see \cite[Lemma~6.3]{naidu}).
For each $g \in G$, and for each ${\mathcal C}$-module category ${\mathcal M}$, let ${\mathcal M}^g$ denote the module category induced by the functor $\ad_g$ as in Section~\ref{action-mc}. Recall that the action of ${\mathcal C}$ on ${\mathcal M}^g$ is def\/ined by $M\bar\otimes^g V = M\bar\otimes ({}^gV)$, for all objects~$V$ of~${\mathcal C}$.
\begin{Lemma}\label{adj-triv} Let $g \in G$ and let ${\mathcal M}$ be a ${\mathcal C}$-module category. Then ${\mathcal M}^g \cong {\mathcal M}$ as ${\mathcal C}$-module categories.
\end{Lemma}
\begin{proof} For each $g \in G$, let $\{g\}$ denote the object of ${\mathcal C}$ such that $\{g\} = k$ with degree $g$. In what follows, by abuse of notation, we identify $\{g\} \otimes \{h\}$ and $\{gh\}$, $g, h \in G$, by means of the canonical isomorphisms of vector spaces.
Let $R_g\colon {\mathcal M}^g \to {\mathcal M}$ be the functor def\/ined by the right action of $\{g\}$: $R_g(M) = M \bar\otimes \{g\}$. Consider the natural isomorphism $\zeta\colon R_g \circ \bar\otimes^g \to \bar\otimes \circ (R_g \times \id_{\mathcal C})$, def\/ined as
\begin{gather*}\zeta_{M, V} = \mu_{M, \{g\}, V} \mu^{-1}_{M, {}^g V, \{g\}}\colon \ R_g(M\bar\otimes^g V) \to R_g(M) \bar\otimes V,
\end{gather*} for all objects $M$ of ${\mathcal M}$ and $V$ of ${\mathcal C}$, where $\mu$ is the associativity constraint of~${\mathcal M}$.
The functor $R_g$ is an equivalence of categories with quasi-inverse given by the functor $R_{g^{-1}}\colon$ ${\mathcal M} \to {\mathcal M}^g$.
A direct calculation, using the coherence conditions~\eqref{modcat1} and~\eqref{modcat2} for the module category~${\mathcal M}$, shows that $\zeta$ satisf\/ies conditions~\eqref{uno-z} and~\eqref{dos-z}. Hence $(R_g, \zeta)$ is a ${\mathcal C}$-module functor. Therefore ${\mathcal M}^g \cong {\mathcal M}$ as ${\mathcal C}$-module categories, as claimed. \end{proof}
\begin{Lemma}\label{gdea} Let $H$ be a subgroup of $G$ and let $\psi$ be a $2$-cochain on $H$ satisfying~\eqref{cond-alfa}. Let $A(H, \psi)$ denote the corresponding indecomposable algebra in~${\mathcal C}$. Then, for all $g \in G$, ${}^gA(H, \psi) \cong A({}^gH, \psi^{g^{-1}} \Omega_{g^{-1}})$ as algebras in~${\mathcal C}$.
\end{Lemma}
\begin{proof} By def\/inition, ${}^gA(H, \psi) = A\big({}^gH, \psi^{g^{-1}} \big(\Omega_{g}^{g^{-1}}\big)^{-1}\big)$. It follows from formula~\eqref{rel-omega} that $\big(\Omega_{g}^{g^{-1}}\big)^{-1}$ and $\Omega_{g^{-1}}$ dif\/fer by a coboundary. This implies the lemma. \end{proof}
\subsection{} Let $H$, $L$ be subgroups of $G$ and let $\psi \in C^2(H, k^\times)$, $\xi \in C^2(L, k^\times)$, be 2-cochains such that $\omega\vert_{H \times H \times H} = d\psi$ and $\omega\vert_{L \times L \times L} = d\xi$.
Let $B$ be an object of the category $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ of $(A(H, \psi), A(L, \xi))$-bimodules in ${\mathcal C}$. For each $z\in G$, let $\pi_l(h)\colon B_z \to B_{hz}$ and $\pi_r(s)\colon B_z \to B_{zs}$, denote the linear maps induced by the actions of $h \in H$ and $s \in L$, respectively. Then the following relations hold, for all $h, h' \in H$, $s, s' \in L$:
\begin{gather}\label{uno}\pi_l(h)\pi_l(h') = \omega(h, h', z) \psi(h, h') \pi_l(hh'), \\
\label{dos}\pi_r(s')\pi_r(s) = \omega(z, s, s')^{-1} \xi(s, s') \pi_r(ss'),\\
\label{tres}\pi_l(h)\pi_r(s) = \omega(h, z, s) \pi_r(s)\pi_l(h).\end{gather}
\begin{Lemma}\label{alfa-g} Let $g \in G$ and let $B_g$ denote the homogeneous component of degree $g$ of $B$. Then the map $\pi\colon H \cap {}^gL \to \text{\rm GL}(B_g)$, defined as $\pi(x) = \pi_r\big({}^{g^{-1}}x\big)^{-1}\pi_l(x)$ is a projective representation of $H \cap {}^gL$ with cocycle~$\alpha_g$ given, for all $x, y \in H \cap {}^gL$, as follows:
\begin{gather*}
\alpha_g(x, y) = \psi(x, y) \xi^{-1}\big({}^{g^{-1}}x, {}^{g^{-1}}y\big) \frac{\omega(x, y, g) \omega\big(x, yg, {}^{g^{-1}}\big(y^{-1}\big)\big)}
{\omega\big(xyg, {}^{g^{-1}}\big(y^{-1}\big),{} ^{g^{-1}}\big(x^{-1}\big)\big)} du_g(x, y) \\
\hphantom{\alpha_g(x, y) =}{} \times \frac{\omega\big({}^{g^{-1} } y, {}^{g^{-1}}\big(y^{-1}\big), {}^{g^{-1}}\big(x^{-1}\big)\big)} {\omega\big({}^{g^{-1}}x, {} ^{g^{-1}}y, {} ^{g^{-1}}\big(y^{-1}x^{-1}\big)\big)},
\end{gather*}
where the $1$-cochain $u_g$ is defined as $u_g(x) = \omega\big(xg, {} ^{g^{-1}} x, {}^{g^{-1}} \big(x^{-1}\big)\big)$. \end{Lemma}
\begin{proof} It follows from~\eqref{dos} that $\pi_r(s)^{-1} = \omega\big(z, s, s^{-1}\big) \xi\big(s, s^{-1}\big)^{-1} \pi_r\big(s^{-1}\big)$, for all $z\in G$, $s \in L$. In addition, for all $h, h' \in L$, we have the following relation:
\begin{gather*}\xi \big({h'}^{-1}, h^{-1}\big) \xi (h, h') = df(h, h')\frac{\omega\big(h', {h'}^{-1}, h^{-1}\big)}
{\omega\big(h, h', {h'}^{-1}h^{-1}\big)},\end{gather*}
where $f$ is the 1-cochain given by $f(h) = \xi\big(h, h^{-1}\big)$. A straightforward computation, using this relation and conditions~\eqref{uno},~\eqref{dos} and~\eqref{tres}, shows that $\pi(x) \pi(y) = \alpha_g(x, y) \pi(xy)$, for all $x, y \in H \cap {}^gL$. This proves the lemma. \end{proof}
\begin{Remark}\label{rmk-alfag} Lemma \ref{alfa-g} is a~version of \cite[Proposition~3.2]{ostrik}, where it is shown that $B$ is a~simple object of~$_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ if and only if~$B$ is supported on a single double coset $HgL$ and the projective representation~$\pi$ in the component~$B_g$ is irreducible.
\end{Remark}
For all $g \in G$, $\psi^g \Omega_g$ is a 2-cochain in $^{g^{-1}\!}\!H$ such that $\omega\vert_{{}^{g^{-1}}H \times {}^{g^{-1}}H \times {}^{g^{-1}} H} = d(\psi^g \Omega_g)$. Then the product $\xi^{-1} \psi^g \Omega_g$ def\/ines a 2-cocycle of ${}^{g^{-1}} H \cap L$.
\begin{Lemma}\label{rel-cociclos} The class of the $2$-cocycle $\big(\xi^{-1} \psi^g \Omega_g\big)^{g^{-1}}$ in $H^2(H\cap {}^gL, k^\times)$ coincides with the class of the $2$-cocycle $\alpha_g$ in Lemma~{\rm \ref{alfa-g}}.
\end{Lemma}
\begin{proof}
A direct calculation shows that for all $x, y \in G$,
\begin{gather*}\frac{\omega\big(y, y^{-1}, x^{-1}\big)}
{\omega\big(x, y, y^{-1}x^{-1}\big)}
\frac{\omega\big({}^gx, {}^gy, g\big) \omega\big({}^gx, {}^gyg, {} y^{-1}\big)}
{\omega\big({}^gx ^gyg, {}y^{-1}, x^{-1}\big)}
= \Omega_g(x, y) d\theta_g (x, y),\end{gather*}
where the 1-cochain $\theta_g$ is def\/ined as $\theta_g(x) = \omega\big(g, x, x^{-1}\big)^{-1}$. This implies that $\alpha_g^g \equiv \xi^{-1} \psi^g \Omega_g$, as was to be proved.
\end{proof}
\subsection{}\label{demo} In this subsection we give a proof of the main result of this paper.
\begin{proof}[Proof of Theorem \ref{main}] Let $H, L$ be subgroups of $G$ and let $\psi \in C^2(H, k^{\times})$ and $\xi \in C^2(L, k^{\times})$ be 2-cochains satisfying condition~\eqref{cond-alfa}. Let $A(H, \psi)$, $A(L, \xi)$ be the associated algebras in ${\mathcal C}$ and let ${\mathcal M}_0(H, \psi)$, ${\mathcal M}_0(L, \xi)$ be the corresponding ${\mathcal C}$-module categories.
Let ${\mathcal M} = {\mathcal M}_0(L, \xi)$. For every $g \in G$, let ${\mathcal M}^g$ denote the module category induced by the autoequivalence $\ad_g\colon {\mathcal C} \to {\mathcal C}$.
The ${\mathcal C}$-module category ${\mathcal M}^g$ is equivalent to $_{^gA(L, \xi)}{\mathcal C}$. Hence, by Lemma~\ref{gdea}, ${\mathcal M}^g \cong {\mathcal M}_0\big({}^gL, \xi^{g^{-1}} \Omega_{g^{-1}}\big)$.
Suppose that there exists an element $g \in G$ such that $H = {}^gL$ and the class of the cocycle $\xi^{-1}\psi^g\Omega_g$ is trivial on $L$. Relation~\eqref{rel-omega} implies that $\Omega_g^{g^{-1}} = \Omega_{g^{-1}}^{-1}$, and thus the class of $\psi^{-1}\xi^{g^{-1}}\Omega_{g^{-1}}$ is trivial on~$H$. Then $\psi = \xi^{g^{-1}}\Omega_{g^{-1}} df$, for some 1-cochain $f \in C^1(H, k^{\times})$. Therefore $^gA(L, \xi) = A\big(H, \xi^{g^{-1}} \Omega_{g^{-1}}\big) \cong A(H, \psi)$ as algebras in ${\mathcal C}$. Thus we obtain equivalences of ${\mathcal C}$-module categories
\begin{gather*}{\mathcal M}_0(L, \xi) \cong {\mathcal M}_0(L, \xi)^g \cong {} _{^gA(L, \xi)}{\mathcal C} \cong {\mathcal M}_0(H, \psi),\end{gather*}
where the f\/irst equivalence is deduced from Lemma~\ref{adj-triv}.
Conversely, suppose that $F\colon {\mathcal M}_0(L, \xi) \to {\mathcal M}_0(H, \psi)$ is an equivalence of ${\mathcal C}$-module categories. Recall that there is an equivalence
\begin{gather*}\label{equiv-fun-1} {\mathbb F}un_{\mathcal C}\left({\mathcal M}_0(L, \xi), {\mathcal M}_0(H, \psi)\right) \cong {} _{A(H, \psi)}{\mathcal C}_{A(L, \xi)}.\end{gather*}
Under this equivalence, the functor $F$ corresponds to an object $B$ of $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ such that there exists an object $B'$ of $_{A(L, \xi)}{\mathcal C}_{A(H, \psi)}$ satisfying
\begin{gather}\label{dim-b}B \otimes_{A(L, \xi)}B' \cong A(H, \psi),\end{gather} as $A(H, \psi)$-bimodules in ${\mathcal C}$, and
\begin{gather}\label{bbprime}B' \otimes_{A(H, \psi)}B \cong A(L, \xi), \end{gather} as $A(L, \xi)$-bimodules in ${\mathcal C}$.
Let ${\mathbb F}Pdim_{A(H, \psi)}M$ denote the Frobenius--Perron dimension of an object $M$ of $_{A(H, \psi)}{\mathcal C}_{A(H, \psi)}$. Then we have
\begin{gather*}\dim M = \dim A(H, \psi) {\mathbb F}Pdim_{A(H, \psi)}M = |H| {\mathbb F}Pdim_{A(H, \psi)}M.\end{gather*}
Taking Frobenius--Perron dimensions in both sides of~\eqref{dim-b} and using this relation we obtain that $\dim \left(B \otimes_{A(L, \xi)}B'\right) = |H|$.
On the other hand, $\dim (B \otimes_{A(H, \psi)}B' ) = \frac{\dim B \dim B'}{\dim{A(L, \xi)}} = \frac{\dim B \dim B'}{|L|}$. Thus
\begin{gather}\label{dimbb'} \dim B \dim B' = |H| |L|. \end{gather}
Since $A(H, \psi)$ is an indecomposable algebra in ${\mathcal C}$, then it is a simple object of $_{A(H, \psi)}{\mathcal C}_{A(H, \psi)}$. Then~\eqref{bbprime} implies that $B$ is a simple object of $_{A(H, \psi)}{\mathcal C}_{A(L, \xi)}$ and $B'$ is a simple object of $_{A(L, \xi)}{\mathcal C}_{A(H, \psi)}$.
In view of \cite[Proposition~3.2]{ostrik}, the support of $B$ is a two sided $(H, L)$-double coset, that is, $B = \bigoplus_{(h,h') \in H \times L} B_{hgh'}$, where $g \in G$ is a representative of the double coset that supports~$B$. Moreover, the homogeneous component $B_g$ is an irreducible $\alpha_g$-projective representation of the group $^gL \cap H$, where the 2-cocycle $\alpha_g$ satisf\/ies~$\alpha_g \equiv \big(\xi^{-1}\psi^g \Omega_g\big)^{g^{-1}}$; see Remark~\ref{rmk-alfag} and Lemmas~\ref{alfa-g} and~\ref{rel-cociclos}.
Notice that the actions of $h \in H$ and $h'\in L$ induce isomorphisms of vector spaces $B_{g} \cong B_{hg}$ and $B_{g} \cong B_{gh'}$. Hence
\begin{gather}\label{dimb}\dim B = |HgL| \dim B_g = \frac{|H| |L|}{|H \cap {} ^gL|} \dim B_g = [H: H \cap {}^gL] |L| \dim B_g.\end{gather}
In particular, $\dim B \geq |L|$. Reversing the roles of $H$ and $L$, the same argument implies that $\dim B' \geq |H|$. Combined with relations~\eqref{dimbb'} and~\eqref{dimb} this implies
\begin{gather*}|H| |L| = \dim B \dim B' \geq |H| [H : H \cap {}^gL] |L| \dim B_g.\end{gather*}
Hence $[H\colon H \cap {}^gL] \dim B_g = 1$, and therefore $[H\colon H \cap {}^gL] = 1$ and $\dim B_g = 1$. The f\/irst condition means that $H \subseteq {}^gL$, while the second condition implies that the class of $\alpha_g$ is trivial in $H^2(H \cap {}^gL, k^\times)$. Since the rank of ${\mathcal M}_0(H, \psi)$ equals the index $[G:H]$ and the rank of ${\mathcal M}_0(H, \xi)$ equals the index $[G:L]$, then $|H| = |L|$. Thus we get that $H = {}^gL$ and that the class of the 2-cocycle~\eqref{cond-equiv} is trivial in $H^2(L, k^\times)$. This f\/inishes the proof of the theorem.
\end{proof}
\begin{Example}\label{kp}
Let $B_8$ be the 8-dimensional Kac Paljutkin Hopf algebra. The Hopf algebra $B_8$ f\/its into an exact sequence
\begin{gather*}
k \longrightarrow k^C \longrightarrow B_8 \longrightarrow kL \longrightarrow k,
\end{gather*}
where $C = \mathbb Z_2$ and $L = \mathbb Z_2 \times \mathbb Z_2$. See~\cite{masuoka-6-8}. This exact sequence gives rise to mutual actions by permutations
\begin{gather*}C \overset{\vartriangleleft}\longleftarrow C \times L
\overset{\vartriangleright}\longrightarrow L,\end{gather*}
and compatible cocycles $\tau\colon L \times L \to \big(k^C\big)^\times$, $\sigma\colon C \times C \to (k^L)^\times$, such that $B_8$ is isomorphic to the bicrossed product $kC {}^{\tau}\#_\sigma kL$. The data $\lhd$, $\rhd$, $\sigma$ and $\tau$ are explicitly determined in \cite[Proposition~3.11]{ma-contemp} as follows. Let $C = \langle x\colon x^2 = 1 \rangle$, $L = \langle z, t\colon z^2 = t^2 = ztz^{-1}t^{-1} = 1\rangle$. Then $\lhd\colon C\times L \to C$ is the trivial action of $L$ on $C$, $\rhd\colon C \times L \to L$ is the action def\/ined by $x \rhd z = z$ and $x\rhd t = zt$,
\begin{gather*}\tau_{x^n}\big(z^it^j, z^{i'}t^{j'}\big) = (-1)^{nji'},\end{gather*} for all $0\leq n, i, i', j, j' \leq 1$,
and
\begin{gather*}\sigma_{z^it^j}\big(x^n, x^{n'}\big) = (\sqrt{-1})^{j\big(\frac{n+n'-\langle n+n'\rangle}{2}\big)},\end{gather*}
for all $0 \leq i, j, n, n' \leq 1$, where $\langle n+n'\rangle$ denotes the remainder of $n+n'$ in the division by~$2$. Here we use the notation $\tau(a, a')(y) = : \tau_y(a, a')$ and, similarly, $\sigma(y, y')(a) = : \sigma_a(y, y')$, $a, a' \in L$, $y, y' \in C$.
In view of \cite[Theorem~3.3.5]{schauenburg} (see \cite[Proposition~4.3]{gttic}), the fusion category of f\/inite-dimen\-sio\-nal representations of $B_8^{\rm op} \cong B_8$ is equivalent to the category ${\mathcal C}(G, \omega, L, 1)$, where $G = L \rtimes C$ is the semidirect product with respect to the action~$\rhd$, and $\omega$ is the 3-cocycle arising from the pair $(\tau, \sigma)$ under one of the maps of the so-called \emph{Kac exact sequence} associated to the matched pair.
In this example $G$ is isomorphic to the dihedral group $D_8$ of order 8. The 3-cocycle $\omega$ is determined by the formula
\begin{gather}\label{kac}\omega\big(x^{n}z^{i}t^{j}, x^{n'}z^{i'}t^{j'}, x^{n''}z^{i''}t^{j''}\big) =
\tau_{x^n}\big(z^{i'}t^{j'}, x^{n'} \rhd z^{i''}t^{j''}\big) \sigma_{z^{i''}t^{j''}}\big(x^n, x^{n'}\big),\end{gather}
for all $0 \leq i, j, i', j', i'', j'', n, n', n'' \leq 1$.
Notice that $\omega\vert_{L \times L\times L} = 1$. Hence, for every 2-cocycle $\xi$ on $L$, the pair $(L, \xi)$ gives rise to an indecomposable ${\mathcal C}$-module category ${\mathcal M}(L, \xi)$.
Formula~\eqref{kac} implies that $\Omega_x\vert_{L \times L}$ is given by
\begin{gather*}\Omega_x\big(z^{i}t^{j}, z^{i'}t^{j'}\big) = (-1)^{ji'}, \qquad 0\leq i, i', j, j' \leq 1.\end{gather*}
Then $\Omega_x$ is a 2-cocycle representing the unique nontrivial cohomology class in $H^2(L, k^\times)$. By Theorem~\ref{main}, for any 2-cocycle~$\xi$ on $L$, ${\mathcal M}_0(L, 1)$ and ${\mathcal M}_0(L, \xi)$ are equivalent as ${\mathcal C}(G, \omega)$-module categories, and therefore so are the corresponding ${\mathcal C}$-module categories ${\mathcal M}(L, 1)$ and ${\mathcal M}(L, \xi)$. This implies that indecomposable ${\mathcal C}$-module categories are in this example parameterized by conjugacy classes of subgroups of $D_8$ on which $\omega$ has trivial restriction, as claimed in \cite[Section~6.4]{MN}.
\end{Example}
\LastPageEnding
\end{document} |
\begin{document}
\title{SHIP: A Scalable High-performance IPv6 Lookup Algorithm that Exploits Prefix Characteristics}
\author{Thibaut Stimpfling, Normand Belanger,
J.M. Pierre~Langlois~\IEEEmembership{~Member~IEEE}
and~Yvon~Savaria~\IEEEmembership{~Fellow~IEEE}\thanks{Thibaut Stimpfling, Normand Bélanger, J.M. Pierre Langlois, and Yvon Savaria are with École Polytechnique de Montréal (e-mail: \{thibaut.stimpfling, normand.belanger, pierre.langlois, yvon.savaria\}@polymtl.ca).}}
\maketitle
\begin{abstract}
Due to the emergence of new network applications, current IP lookup engines must support high-bandwidth, low lookup latency and the ongoing growth of IPv6 networks.
However, existing solutions are not designed to address jointly those three requirements.
This paper introduces SHIP, an IPv6 lookup algorithm that exploits prefix characteristics to build a two-level data structure designed to meet future application requirements. Using both prefix length distribution and prefix density, SHIP first clusters prefixes into groups sharing similar characteristics, then it builds a hybrid trie-tree for each prefix group. The compact and scalable data structure built can be stored in on-chip low-latency memories, and allows the traversal process to be parallelized and pipelined at each level in order to support high packet bandwidth.
Evaluated on real and synthetic prefix tables holding up to 580 k IPv6 prefixes, SHIP has a logarithmic scaling factor in terms of the number of memory accesses, and a linear memory consumption scaling. Using the largest synthetic prefix table, simulations show that compared to other well-known approaches, SHIP uses at least 44\% less memory per prefix, while reducing the memory latency by 61\%.
\end{abstract}
\begin{IEEEkeywords}
Algorithm, Routing, IPv6 Lookup, Networking.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}
\IEEEPARstart{G}{lobal} IP traffic carried by networks is continuously growing,
as around a zettabyte total traffic is expected for the whole of 2016, and it is envisioned to increase threefold between 2015 and 2019~\cite{cisco_forecast}. To handle this increasing Internet traffic, network link working groups have ratified the 100-gigabit Ethernet standard (IEEE P802.3ba), and are studying the 400-gigabit Ethernet standard (IEEE P802.3bs). As a result, network nodes have to process packets at those line rates which puts pressure on IP address lookup engines used in the routing process. Indeed, less than $6$ ns is available to determine the IP address lookup result for an IPv6 packet~\cite{scalable_ipv6_vk}.
The IP lookup task consists of identifying the next hop information (NHI) to which a packet should be forwarded. The lookup process starts by extracting the destination IP field from the packet header, and then matching it against a list of entries stored in a lookup table, called the forwarding information base (FIB). Each entry in the lookup table represents a network defined by its prefix address. While a lookup key may match multiple entries in the FIB, only the longest prefix and its NHI are returned for result as IP lookup is based on the Longest Prefix Match (LPM)~\cite{high_performance_routers_switches}.
IP lookup algorithms and architectures that have been tailored for IPv4 technology are not performing well with IPv6~\cite{scalable_ipv6_vk, mem_eff}, due to the fourfold increase in the number of bits in IPv6 addresses over IPv4. Thus, dedicated IPv6 lookup methods are needed to support upcoming IPv6 traffic.
IP lookup engines must be optimized for high bandwidth, low latency, and scalability for two reasons. First, due to the convergence of wired and mobile networks, many future applications that require a high bandwidth and a low latency, such as virtual reality, remote object manipulation, eHealth, autonomous driving, and the Internet of Things, will be carried on both wired and mobile networks~\cite{5g_whitepaper}. Second, the number of IPv6 networks is expected to grow, and so is the size of the IPv6 routing tables, as IPv6 technology is still being deployed in production networks~\cite{potaroo,ris_raw_data}. However, current solutions presented in the literature are not jointly addressing these three performance requirements.
In this paper, we introduce SHIP: a Scalable and High Performance IPv6 lookup algorithm designed to meet current and future performance requirements. SHIP is built around the analysis of prefix characteristics. Two main contributions are presented: 1) two-level prefix grouping, that clusters prefixes in groups sharing common properties, based on the prefix length distribution and the prefix density, 2) a hybrid trie-tree tailored to handle prefix distribution variations.
SHIP builds a compact and scalable data structure that is suitable for on-chip low-latency memories, and allows the traversal process to be parallelized and pipelined at each level in order to support high packet bandwidth. SHIP stores 580 k prefixes and the associated NHI using less than $5.9$ MB of memory, with a linear memory consumption scaling. SHIP achieves logarithmic latency scaling and requires in the worst case 10 memory accesses per lookup. For both metrics, SHIP outperforms known methods by over 44\% for the memory footprint, and by over 61\% for the memory latency.
The remainder of this paper is organized as follows. Section~\ref{sec:related_work} introduces common approaches used for IP lookup and Section~\ref{sec:overview} gives an overview of SHIP. Then, two-level prefix grouping is presented in Section~\ref{sec:two_level_prefix_grouping}, while the proposed hybrid trie-tree is covered in Section~\ref{sec:trie_tree_hybrid_data_structure}. Section~\ref{sec:method} introduces the method and metrics used for performance evaluation and Section~\ref{sec:results} presents the simulation results. Section~\ref{sec:disscusion} shows that SHIP fulfills the properties for hardware implementability, and compares SHIP performance with other methods. Lastly, we conclude the work by summarizing our main results in Section~\ref{sec:conclusion}.
\section{Related Work}\label{sec:related_work}
Many data structures have been proposed for the LPM operation applied to IP addresses. We can classify them in four main types: hash tables, Bloom filters, tries and trees. Those data structures are encoding prefixes that are loosely structured. First, not only prefix length distribution is highly nonuniform, but it also varies with the prefix table used. Second, for any given prefix length, prefix density ranges from sparse to very dense. Thus, each of the four main data structures type comes with a different tradeoff between time and storage complexity.
Interest for IP lookup with hash table is twofold. First, a hash function aims at distributing uniformly a large number of entries over a number of bins, independently of the structure of the data stored. Second, a hash table provides $O(1)$ lookup time and $O(N)$ space complexity. However, a pure hash based LPM solution can require up to one hash table per IP prefix length. An alternative to reduce the number of hash tables is to use prefix expansion~\cite{flashlook}, but it increases memory consumption. Two main types of hash functions can be selected to build a hash table: perfect or non-perfect hash functions. A Hash table built with a perfect hash functions offers a fixed time complexity that is independent from the prefixes used as no collision is generated. Nevertheless, a perfect hash function cannot handle dynamic prefix tables, making it unattractive for a pure hash based LPM solution. On the other hand, a non-perfect hash function leads to collisions and cannot provide a fixed time complexity. Extra-matching sequences are required with collisions that drastically decrease performance~\cite{flashtrie_conf,distributed_bloom_filters}. In addition, not only the number of collisions is determined after the creation of the hash table but it also depends on the prefix distribution characteristics. In order to reduce the number of collisions independently of the characteristics of the prefix table used, a method has been proposed that exploits multiple hash tables~\cite{flashlook,flashtrie_conf}. This method divides the prefix table into groups of prefixes, and selects a hash function such that it minimizes the number of collisions within each prefix group~\cite{flashlook,flashtrie_conf}. Still, the hash function selection for each prefix group requires to probe all the hash functions, making it unattractive for dynamic prefix tables. Finally, no scaling evaluation has been completed in recent publications~\cite{flashlook, flashtrie_journal} making it unclear whether the proposed hash-based data structures can address forthcoming challenges.
Low-memory footprint hashing schemes known as Bloom filters have also been covered in the literature~\cite{distributed_bloom_filters,bloom_filters}. Bloom filters are used to select a subgroup of prefixes that may match the input IP address. However, Bloom filters suffer from two drawbacks. First, by design, this data structure generates false positives independent of the configuration parameters used. Thus, a Bloom filter can improve the average lookup time, but it can also lead to poor performance in the worst case, as many sub-groups need to be matched. Second, the selection of a hash function that minimizes the number of false positives is highly dependent of the prefix distribution characteristics used. Hence, its complexity is similar to that of of a hash function that minimizes the number of collisions in a regular hash table.
Tree solutions based on binary search trees (BST) or generalized B-trees have also been explored in~\cite{mem_eff,scalable_ipv6_vk}. Such data structures are tailored to store loosely structured data such as prefixes, as their time complexity is independent from the prefix distribution characteristics. Indeed, BST and 2-3 Trees have a time complexity of respectively $log_2(N)$ and $log_3(N)$, with $N$ being the number of entries~\cite{scalable_ipv6_vk}. Nevertheless, such data structure provides a solution at the cost of a large memory consumption. Indeed, each node stores a full-size prefix, leading to memory waste. Hence, their memory footprint makes them unsuitable for the very large prefix tables that are anticipated in future networks.
At the other end of the tree spectrum, decision-trees (D-Trees) have been proposed in~\cite{hicuts,scalable_packet_classification} for the field of packet classification. D-Trees were found to offer a good tradeoff between memory footprint and the number of memory accesses. However, no work has been conducted yet on using this data structure for IPv6 lookup.
The trie data structure, also known as radix tree, has regained interest with tree bitmap~\cite{bitmap_tree}. Indeed, a $k$-bit trie requires $k/W$ memory accesses, but has very poor memory efficiency when built with unevenly distributed prefixes. A tree bitmap improves the memory efficiency over a multi-bit trie, independently of the prefix distribution characteristics, by using a bitmap to encode each level of a multi-bit trie. However, tree bitmaps cannot be used with large strides, as the node size grows exponentially with the stride size, leading to multiple wide memory accesses to read a single node. An improved tree bitmap, the PC-trie, is proposed for the FlashTrie architecture~\cite{flashtrie_journal}. A PC-Trie reduces the size of bitmap nodes using a multi-level leaf pushing method. This data structure is used jointly with a pre-processing hashing stage to reduce the total number of memory accesses. Nevertheless, the main shortcoming of the Flashtrie architecture lies in its pre-processing hashing module. First, similar to other hashing solutions, its performance highly depends on the distribution characteristics of the prefixes used. Second, the hashing module does not scale well with the number of prefixes used.
At the other end of the spectrum of algorithmic solutions, TCAMs have been proposed as a pure hardware solution, achieving $O(1)$ lookup time by matching the input key simultaneously against all prefixes, independently of their distribution characteristics. However, these solutions use a very large amount of hardware resources, leading to large power consumption and high cost, and making them unattractive for routers holding a large number of prefixes~\cite{a_tcam_based_distributed_parallel,flashtrie_journal}.
Recently, information-theoretic and compressed data structures have been applied to IP lookup, yielding very compact data structures, that can handle a very large number of prefixes~\cite{compressing_ip_forwarding_tables}. Even though this work is limited to IPv4 addresses, it is an important shift in terms of concepts. However, the hardware implementation of the architecture achieves 7 million lookups per second. In order to support a 100-Gbps bandwidth, this would require many lookup engines, leading to a memory consumption that is similar or higher than previous trie or tree algorithms~\cite{flashlook,flashtrie_conf,flashtrie_journal,scalable_ipv6_vk}.
In summary, the existing data structures are not exploiting the full potential of the prefix distribution characteristics.
In addition, none of the existing data structures were shown to optimize jointly the time complexity, the storage complexity, and the scalability.
\section{SHIP Overview}\label{sec:overview}
SHIP consists of two procedures: the first one is used to build a two-level data structure, and the second one is used to traverse the two-level data structure. The procedure to build the data structure is called two-level prefix grouping, while the traversal procedure is called the lookup algorithm.
Two-level prefix grouping clusters prefixes upon their characteristics to build an efficient two-level data structure, presented in Fig~\ref{fig:global_lookup_seq}. At the first level, SHIP leverages the low density of the IPv6 prefix MSBs to divide prefixes into M address block bins (ABBs). A pointer to each ABB is stored in an N-entry hash table. At the second level, SHIP uses the uneven prefix length distribution to sort prefixes held in each ABB into K Prefix Length Sorted (PLS) groups. For each non-empty K$\cdot$M PLS groups, SHIP further exploits the prefix length distribution and the prefix density variation to encode prefixes into a hybrid trie-tree (HTT).
The lookup algorithm, which identifies the NHI associated to the longest prefix matched, is presented in Fig~\ref{fig:global_lookup_seq}. First, the MSBs of the destination IP address are hashed to select an ABB pointer stored in an $N$ entry hash table. The selected ABB pointer in this figure is held in the $n$-th entry of the hash table, represented with a dashed rectangle. This pointer identifies bin $m$, represented with a dashed rectangle. Second, the HTTs associated to each PLS group of the $m$-th bin, are traversed in parallel, using portions of the destination IP address. Each HTT can output a NHI, if a match occurs with its associated portion of the destination IP address. Thus, up to $K$ HHT results can occur, and a priority resolution module is used to select the NHI associated to the longest prefix.
\begin{figure}
\caption{SHIP two-level data structure organization and its lookup process with $M$ address block bins and $K$ prefix length sorting groups.}
\label{fig:global_lookup_seq}
\end{figure}
In the following section, we present the two-level prefix grouping procedure.
\section{Two-level Prefix Grouping}\label{sec:two_level_prefix_grouping}
This section introduces two-level prefix grouping, that clusters and sort prefixes into groups, and then builds the two-level data structure. First, prefixes are binned and the first level of the two-level data structure is built with the address block binning method. Second, inside each bin, prefixes are sorted into groups, and then the HTTs are built with the prefix length sorting method.
\subsection{Address block binning}
The proposed address block binning method exploits both the structure of IP addresses and the low density of the IPv6 prefix MSBs to cluster the prefixes into bins, and then build the hash table used at the first level of SHIP data structure.
IPv6 addresses are structured into IP address blocks, managed by the Internet Assigned Numbers Authority (IANA) that assigns blocks of IPv6 addresses ranging in size from $/16$ to $/23$, that are then further divided into smaller address blocks. However, the prefix density on the first 23 bits is low, and the prefix distribution is sparse~\cite{global_unicast_address_assig}. Therefore, the address block binning method bins prefixes based on their first 23 bits. Before prefixes are clustered, all prefixes with a prefix length that is less than $/23$ are converted into $/23$. The pseudo-code used for this binning method is presented in Algorithm~\ref{alg:build_address_block_binning}. For each prefix held in the prefix table, this method checks whether a bin already exists for the first $23$ bits. If none exists, a new bin is created, and the prefix is added to the new bin. Otherwise, the prefix is simply added to the existing bin.
This method only keeps track of the $M$ created bins. The prefixes held in each bin are further grouped by the prefix length sorting method that is presented next.
Address block binning utilizes a perfect hash function~\cite{perfect_hashing_function} to store a pointer to each valid bin. Let $N$ be the size of the hash table used. The first 23 MSBs of the IP address represents the key of the perfect hash table. A perfect hashing function is chosen for three reasons. First, the number of valid keys on the first 23 bits is relatively small compared to the number of bits used, making hashing attractive. Second, a perfect hashing function is favoured when the data hashed is static, which is the case here, for the first 23 bits only, because it represents blocks of addresses allocated to regional internet registries that are unlikely to be updated on a short time scale. Finally, no resolution module is required because no collisions are generated.
\begin{algorithm}
\caption{Building the Address Block binning data structure}
\label{alg:build_address_block_binning}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Prefix table
\Ensure Address Block binning data structure
\For{each prefix held in the prefix table}
\State {Extract its 23 MSBs}
\If {no bin already exists for the extracted bits}
\parState{Create a bin that is associated to the value of the extracted bits}
\State{Add the current prefix to the bin}
\Else
\parState {Select the bin that is associated to the value of the extracted bits}
\State{Add the current prefix to the selected bin}
\EndIf
\EndFor
\State {Build a hash table that stores a pointer to each address block bin, with the 23 MSBs of each created bin as a key. Empty entries are holding invalid pointers.}
\State \textbf{return} {the hash table}
\end{algorithmic}
\end{algorithm}
The lookup procedure is presented in Algorithm~\ref{alg:lookup_address_block_binning}. It uses the 23 MSBs of the destination IP address as a key for the hash table and returns a pointer to an address block bin. If no valid pointer exists in the hash table for the key, then a null pointer is returned.
\begin{algorithm}
\caption{Lookup in the Address Block binning data structure}
\label{alg:lookup_address_block_binning}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require{Address Block binning hash table, destination IP Address}
\Ensure{Pointer to an address block bin}
\State{Extract the 23 MSBs of the destination IP address}
\State {Hash the extracted 23 MSBs}
\If{the hash table entry pointed by the hashed key holds a valid pointer}
\State{ \textbf{return} {pointer to the address block bin}}
\Else
\State{ \textbf{return} {null pointer}}
\EndIf
\end{algorithmic}
\end{algorithm}
The perfect hash table created with the address block binning method is an efficient data structure to perform a lookup on the first 23 MSBs of the IP address.
However, within the ABBs the prefix length distribution can be highly uneven, which degrades the performance of the hybrid trie-trees at the second level. Therefore, the prefix length sorting method, described next, is proposed to address that problem.
\subsection{Prefix length sorting}
Prefix length sorting (PLS) aims at reducing the impact of the uneven prefix length distribution on the number of overlaps between prefixes held in each address block bin. By reducing the number of prefix overlaps, the performance of the HTTs is improved, as it will be shown later. The PLS method sorts the prefixes held in each address block bin by their length, into $K$ PLS groups that cover disjoints prefix length ranges. Each range consists of contiguous prefix lengths that are associated to a large number of prefixes with respect to the prefix table size. For each PLS group, a hybrid trie-tree is built.
The number of PLS groups, $K$, is chosen to maximize the HTT's performance. As will be shown experimentally in section~\ref{sec:results}, beyond a threshold value, increasing the value of $K$ does not further improve performance. The prefix length range selection is based on the prefix length distribution and it is guided by two principles. First, to minimize prefix overlap, when a prefix length covers a large percentage of the total number of prefixes, this prefix length must be used as an upper bound of the considered group. Second, prefix lengths included in a group are selected such that group sizes are as balanced as possible.
To illustrate those two principles, an analysis of prefix length distribution using a real prefix table is presented in Fig.~\ref{fig:prefix_distribution}. The prefix table extracted from~\cite{ris_raw_data} holds approximately 25 k prefixes. The first $23$ prefix lengths are omitted in Fig.~\ref{fig:prefix_distribution}, as the address block binning method already bins prefixes on their $23$ MSBs. It can be observed in Fig.~\ref{fig:prefix_distribution} that the prefix lengths with the largest cardinality are $/32$ and $/48$ for this example. Applying the two principles of prefix length sorting to this example, the first group covers prefix lengths from $/24$ to $/32$, and the second group covers the second peak, from $/33$ to $/48$. Finally, all remaining prefix lengths, from $/49$ to $/64$ are left in the third prefix length sorting group.
\begin{figure}
\caption{The uneven prefix length distribution of real a prefix table used by the PLS method to create 3 PLS groups.}
\label{fig:prefix_distribution}
\end{figure}
For each of the $K$ PLS group created, an HTT is built. Thus, the lookup step associated to the prefix length sorting method consists of traversing the $K$ HTTs held in the selected address block bin.
To summarize, the created PLS groups cover disjoint prefix length ranges by construction. Therefore, the PLS method directly reduces prefix overlaps in each address block bin that increases the performance of HTT. However, within each PLS group, the prefix density variation remains uneven. Hence, a hybrid-trie tree is proposed that exploits the local prefix characteristics to build an efficient data structure.
\section{Hybrid Trie-Tree data structure}\label{sec:trie_tree_hybrid_data_structure}
The hybrid trie-tree proposed in this work is designed to leverage the prefix density variation. This hybrid data structure uses a density-adaptive trie, and a reduced D-Tree leaf when the number of prefixes covered by a density-adaptive trie node is below a fixed threshold value. A description of the two data structures is first presented, then the procedure to build the hybrid trie-tree is formulated, and finally the lookup procedure is introduced.
\subsection{Density-Adaptive Trie}
The proposed density-adaptive trie is a data structure that is built upon the prefix density distribution. A density-adaptive trie combines a trie data structure with the Selective Node Merge (SNM) method.
While a trie or multi-bit trie creates equi-sized regions whose size is independent of the prefix density distribution, the proposed SNM method adapts the size of the equi-sized regions to the prefix density. Low-density equi-sized regions created with a trie data structure are merged into variable region sizes by the SNM method. Two equi-sized regions are merged if the total number of prefixes after merging is equal to the largest number of prefixes held by the two equi-sized regions, or if it is less than a fixed threshold value. The SNM method merges equi-sized regions both from the low indices to the highest ones and from the high indices to the lowest ones. For both directions, the SNM method first selects the lowest and highest index equi-sized regions, respectively. Second, it evaluates if each selected region can be merged with its next contiguous equi-sized region. The two steps are repeated until the selected region can no longer be merged with its next contiguous equi-sized region. Else, the two previous steps are repeated from the last equi-sized region that was left un-merged. The SNM method has two constraints with respect to the number of merged regions. First, each merged region covers a number of equi-sized regions that is restricted to powers of two, as the space covered by the merged region is described with the prefix notation. Second, the total number of merged regions is bounded by the size of a node header field of the adaptive trie.
By merging equi-sized regions together, the SNM method reduces the number of regions that are stored in the data structure. As a result, the SNM method improves the memory efficiency of the data structure.
The benefit of the SNM method on the memory efficiency is presented in Fig.~\ref{fig:with_selective_node_merge} for the first level of a multi-bit trie. As a reference, the first level of a multi-bit trie without the SNM method is also presented in Fig.~\ref{fig:wo_selective_node_merge}. In both figures, IP addresses are defined on $3$ bits for the following prefix set $P_{1} = 110/3$, $P_{2} = 111/3 $ and $P_{3} = 0/0$. In both figures, the region is initially partitioned into four equi-sized regions, each corresponding to a different bit combination, called $0$ to $3$. In Fig.~\ref{fig:with_selective_node_merge}, the SNM method merges the two leftmost equi-sized regions $0$ and $1$, separated by a dashed line, as they fulfill the constraints of SNM. In Fig.~\ref{fig:with_selective_node_merge}, not only the SNM method reduces the number of nodes held in memory by 25\% compared to the multi-bit trie presented in Fig.~\ref{fig:wo_selective_node_merge} but also prefix $P_{3}$ is replicated twice, that is a 33\% reduction of the prefix replication factor. As a result, the SNM method increases the memory efficiency of the multi-bit trie data structures.
\begin{figure}
\caption{Impact of Selective Node Merge on the replication factor for the first level of a trie data structure.}
\label{fig:with_selective_node_merge}
\label{fig:wo_selective_node_merge}
\label{fig:selective_node_merge_example}
\end{figure}
The regions that are traversed by the SNM method (merged or not) are stored in a SNM field of the adaptive-trie node header. The SNM field is divided into a $LtoH$ and $HtoL$ array. The $LtoH$ and $HtoL$ arrays hold the indices of the regions traversed respectively from low to high index values, and high to low index values. For each region traversed by the SNM method, merged or equi-sized, one index is stored either in the $LtoH$ or the $HtoL$ array. Indeed, as a merged region holds two or more multiple contiguous equi-sized regions, a merged region can be described with the indices of the first and the last equi-sized region it holds. In addition, the SNM method traverses the equi-sized regions contiguously. Therefore, the index of the last equi-sized region held in a merged region can be determined implicitly using the index of the next region traversed by the SNM method. The index value of a non-merged region is sufficient to fully describe it.
\subsection{Reduced D-Tree leaf}
A reduced D-Tree leaf is built when the number of prefixes held in a region of the density-adaptive trie is below a fixed threshold value \textit{b}. The proposed leaf is based on a D-tree leaf~\cite{hicuts,scalable_packet_classification} that is extended with the Leaf Size Reduction technique (LSR).
A D-Tree leaf is a bucket that stores the prefixes and their associated NHI held in a given region. A D-Tree leaf has a memory complexity and time complexity of $O(n)$ for $n$ prefixes stored. A D-Tree leaf is used for the regions at the bottom of the density-adaptive trie because of its higher memory efficiency in those regions. Indeed, we observed that most of the bottom level regions of an density-adaptive trie hold highly unevenly distributed prefixes. Moreover, a D-Tree leaf has better memory efficiency with highly unevenly distributed prefixes over a density-adaptive trie. Whereas a density-adaptive trie can create prefix replication, which reduces the memory efficiency, no prefix replication is created with a D-Tree leaf. However, the D-Tree leaf comes at the cost of higher time complexity compared to a density-adaptive trie.
As a consequence, the LSR technique is introduced to reduce the time complexity of a D-Tree leaf by reducing the amount of information stored in a D-Tree leaf. In fact, a D-Tree leaf stores entirely each prefix even the bits that have already been matched by the density-adaptive trie before reaching the leaf. On the other hand, the LSR technique stores in the reduced D-Tree leaf only the prefix bits that are left unmatched. To specify the number of bits that are left unmatched, a new LSR leaf header field is added, coded on $6$ bits. The LSR technique reduces the amount of information that is stored in each reduced D-Tree leaf. As a result, not only does the reduced D-Tree leaf requires fewer memory accesses but it also has a better memory efficiency over a D-Tree leaf.
\subsection{HTT build procedure}
The hybrid trie-tree build procedure is presented in Algorithm~\ref{alg:hybrid_data_structure_build_proc}, starting with the root region that holds all the prefixes (line $1$). If the number of prefixes stored in the root region is below a fixed threshold \textit{b}, then a reduced D-Tree leaf is built (line $2 - 3$). Else, the algorithm iteratively partitions this region into equi-sized regions (lines $4 - 10$). The SNM method is then applied on the equi-sized regions (line $11$). Next, for each region, if the number of prefixes is below the threshold value (line $12$), a reduced D-Tree leaf is built (line $13$), else a density-adaptive trie node is built (line $14 - 16$) and the region is again partitioned.
\begin{algorithm}
\caption{Hybrid Trie-Tree build procedure}
\label{alg:hybrid_data_structure_build_proc}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Prefix Table, stack Q
\Ensure Hybrid Trie-Tree
\State{Create a root region covering all prefixes \;}
\If{the number of prefixes held in that region is below the threshold value \textit{b}}
\State Create a reduced D-Tree leaf for those prefixes \;
\Else
\State Push the root region onto Q \;
\EndIf
\While{Q is not empty}
\parState{Remove the top node in Q and use it as the reference region}
\parState{Compute the number of partitions in the reference region}
\parState{Partition the reference region according to the previous step}
\parState{Apply the SNM method on the partitioned reference regions}
\For{each partitioned reference region}
\If{it holds a number of prefixes that is below the threshold value}
\State{Create a reduced D-Tree leaf for those prefixes}
\Else
\parState{Build an adaptive-density trie node for those prefixes}
\State Push this region onto Q
\EndIf
\EndFor
\EndWhile
\State \Return the Hybrid Trie-Tree \;
\end{algorithmic}
\end{algorithm}
The number of partitions in a region (line $9$ of Algorithm~\ref{alg:hybrid_data_structure_build_proc}) is computed by a greedy heuristic proposed in~\cite{hicuts}. The heuristic uses the prefix distribution to adapt the number of partitions, as expressed in Algorithm~\ref{alg:bi_heuristic}. An objective function, the \textit{Space Measurement (Sm)} is evaluated at each iteration (lines $4$ and $5$) and compared to a threshold value, the \textit{Space Measurement Factor (Smpf)} evaluated in the first step (line $1$). The number of partitions increases by a factor of two at each iteration (line $3$), until the value of the objective function \textit{Sm} (line $4$) becomes greater than the threshold value (line $5$). The objective function estimates the memory usage efficiency with the prefix replication factor by summing the number of prefixes held in each $j$ equi-sized region created $\sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j} )$ (line $4$). The prefix replication factor is impacted by the prefix distribution. If prefixes are evenly distributed, the replication factor remains very low until the equi-sized regions become smaller than the average prefix size. Then, the prefix replication factor increases exponentially. Thus, to avoid over-partitioning a region if the replication factor remains low for many iterations, the number of partitions $N_{p}$ and the result of the previous iterations $Sm(N_{p-1})$ are used as a penalty term that is added to the objective function (line $4$). On the other hand, if prefixes are unevenly distributed, the prefix replication factor increases linearly until the largest prefixes in the region partitioned become slightly smaller compared to an equi-sized region. Passed this point, an exponential growth of the replication factor is observed. The heuristic creates fine-grained partition size in a dense region, and coarse-grained partition size in a sparse region.
\begin{algorithm}
\caption{Heuristic used to compute the number of partitions in a region}
\label{alg:bi_heuristic}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Region to be cut
\Ensure Number of partitions ($N_{p}$)
\label{alg:bi_obj_greedy}
\State $N_{p} = 1; Smpf = Num_{Prefixes} \cdot 8; Sm(N_{p}) = 0 ;$
\Do
\State $N_{p} = N_{p} \cdot 2; $
\parState{$ Sm(N_{p})= \sum_{j=0}^{N_{p}} Num_{Prefixes}(equi-sized region_{j}) + N_{p} + Sm(N_{p-1}) ; $}
\doWhile $Sm(N_{p}) \leq Smpf $ \\
\Return $N_{p}$
\end{algorithmic}
\end{algorithm}
The number of partitions in a region is a power of two. Thus, the base-2 logarithm of the number of partitions represents the number of bits from the IP address used to select the equi-sized region covering this IP address.
\subsection{HTT lookup procedure}\label{sec:lookup_fixed}
The hybrid trie-tree lookup algorithm starts with a traversal of the density-adaptive trie until a reduced D-Tree leaf is reached. Next, the reduced D-Tree leaf is traversed to identify the matching prefix and its NHI.
The traversal of the density-adaptive trie consists in computing the memory address of the child node that matches the destination IP address, calculated with Algorithm~\ref{alg:children_node_address}. This algorithm uses as input parameters the memory base address, the destination-IP bit-sequence, the $LtoH$ and the $HtoL$ arrays that are extracted from the node header. The SNM method can merge multiple equi-sized nodes into a single node in memory, and thus the destination-IP bit-sequences cannot be used directly as an index to the child node. Therefore, Algorithm~\ref{alg:children_node_address} computes for each destination-IP bit-sequence the number of equi-sized nodes that are skipped in memory based on the characteristics of the merged regions described in the $LtoH$ and the $HtoL$ arrays. The value of the destination-IP bit-sequence can point to a region that is either included 1) in a merged region described in the $LtoH$ array (line $1$), or 2) in a merged region described in the $HtoL$ array (line $4$), or 3) in a equi-sized region that has not been traversed by the SNM method (line $7$).
The following notation is introduced: $L$ represents the size of the $HtoL$ and $LtoH$ arrays, $LtoH [i]$ and $HtoL [i]$ are respectively the $i-th$ entry of the $LtoH$ and the $HtoL$ arrays. In the first case, each entry of the $LtoH$ array is traversed to find the closest $LtoH [i]$ that is less than or equal to the destination-IP bit-sequence (line $1$). The index of the matched child node is equal to $index_{LtoH}$ (line $2$), where $index_{LtoH}$ is the index of the $LtoH$ array that fulfills this condition. In the second case, each entry of the $HtoL$ array is similarly traversed to find the closest $HtoL [i]$ that is greater than or equal to the destination-IP bit-sequence (line $4$). The $index_{HtoL}$ in the $LtoH$ array that fulfills this condition is combined with the characteristics of the $LtoH$ and $HtoL$ arrays to compute the index of the selected child node (line $5$). In the third case, the algorithm evaluates only the number of equi-sized nodes that are skipped in memory based on the characteristics of the $LtoH$ array and the destination IP address bit sequence of the matched child node (line $7$). Finally, the index that is added to the base address points in memory to the matched child node.
\begin{algorithm}
\caption{Memory address of the matched child node using SNM method}
\label{alg:children_node_address}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Children base address, destination IP address bit sequence, $LtoH$ and $HtoL$ arrays
\Ensure Child node address
\Comment{Index included in a region using SNM method}
\If{destination IP address bit sequence $\leq LtoH [L-1]$}
\Comment{In LtoH array}
\State{Index = $Index_{LtoH}$}
\Else{
\If{destination IP address bit sequence $\geq HtoL[0] $}
\Comment{In HtoL array}
\parState{Index = $index_{HtoL} + HtoL [0] - LtoH [L-1]+ L-1$}
\Else{
\Comment{destination IP address bit sequence included in an equi-sized region that has not been traversed by the SNM method}
\parState{Index = destination IP address bit sequence $- LtoH [L-1]+ L-1$}
\EndIf}
}
\EndIf
\\
\Return{Child~node~address = base~address + Index}
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{alg:children_node_address} is illustrated with Figures~\ref{fig:example_SNM} in which $L = 3$ and the destination IP address bit sequence is arbitrarily set to $10$. Based on Fig.~\ref{fig:example_SNM}, the destination IP address bit sequence $10$ matches the equi-sized region with the index $10$ before the SNM method is applied. However, after the SNM method is applied, the destination IP address bit sequence matches a merged node with the index $9$. Based on the SNM header, the destination IP address bit sequence $10$ is greater than both $LtoH [L-1]= 3$ and $HtoL [0]= 9$. Thus, we must identify the number of equi-sized nodes that are skipped in memory with the $LtoH$ and $HtoL$ arrays. Because $LtoH [L-1] = 3$, two equi-sized nodes have been merged. As one node is skipped in memory, any child index greater than $3$ is stored at offset $index - 1$ in memory. Moreover, the destination IP address bit sequence is greater than $HtoL [0]= 9$. However, $HtoL [1] = 10$, meaning that indices $9$ and $10$ are not merged, and no entry is skipped in memory for the first two regions held in the $HtoL$ array. As a consequence, only one node is skipped in memory, and thus the child node index is $10 - 1 = 9$.
\begin{figure}
\caption{SNM method applied to a region that holds 11 nodes after merging, and its associated SNM field}
\label{fig:example_SNM}
\end{figure}
\begin{algorithm}
\caption{Lookup in the Reduced D-tree leaf}
\label{alg:lookup_reduced_leaf}
\begin{algorithmic}[1]
\renewcommand{\textbf{Input: }}{\textbf{Input: }}
\renewcommand{\textbf{Output: }}{\textbf{Output: }}
\Require Reduced D-Tree leaf, destination IP Address
\Ensure LPM and its NHI
\State {Parse the leaf header}
\State {Read the prefixes held in the leaf}
\For {each prefix held in the leaf}
\parState{Match the destination IP address against the selected prefix}
\If{Positive Match}
\State{Record the prefix length of the matched prefix}
\EndIf
\EndFor
\State{Identify the longest prefix match amongst all positive matches}\\
\Return {the longest prefix match and its NHI }
\end{algorithmic}
\end{algorithm}
The density-adaptive trie is traversed until a reduced D-Tree leaf is reached. The lookup procedure of a reduced D-Tree leaf is presented in Algorithm~\ref{alg:lookup_reduced_leaf}. The leaf header is first parsed, and then prefixes are read (lines $1$ to $2$). Next, all prefixes are matched against the destination IP address, and their prefix length is recorded if matches are positive (lines $3$ to $6$). When all the prefixes are matched, only the longest prefix match is returned with its NHI (lines $7-8$).
\section{Performance Measurement Methodology}\label{sec:method}
This section describes the methodology used to evaluate SHIP performance using both real and synthetic prefix tables. Eleven real prefix tables were extracted using the RIS remote route collectors~\cite{ris_raw_data}, and each one holds approximately $25$ k prefixes. Each scenario, noted $rrc$ followed by a two-digit number, characterizes the location in the network of the remote route collector used. For prefix tables holding up to $580$ k entries, synthetic prefixes were generated with a method that uses IPv4 prefixes to generate IPv6 prefixes, in a one-to-one-mapping~\cite{non_random_generation}. The IPv4 prefixes used were also extracted from~\cite{ris_raw_data}. Using the IPv6 prefix table holding $580$ k prefixes, four smaller prefix tables were created, with a similar prefix length distribution, holding respectively $290$ k, $116$ k, $58$ k and $29$ k prefixes.
The performance of SHIP was evaluated using two metrics: the number of memory accesses to traverse its data structure and its memory consumption. For the two metrics, the performance is reported separately for the hash table used by the address block binning method, and the HTTs built by the prefix length sorting method. SHIP performance is characterized using $1$ to $6$ groups for two-level prefix grouping, and as a reference the performance of a single HTT without grouping is also presented.The number of groups is limited to six, as we have observed with simulations that increasing it further does not improve the performance.
For the evaluation of the number of memory accesses, it is assumed that the selected hybrid trie-trees within an address block bin are traversed in parallel, using dedicated traversal engines. Therefore, the reported number of memory accesses is the largest number of memory accesses of all the hybrid trie-trees amongst all address block bins. It is also assumed that the memory bus width is equal to a node size, in order to transfer one node per memory clock cycle.
The memory consumption is evaluated as the sum of all nodes held in the hybrid trie-tree for the prefix length sorting method, and of the size of the perfect hash table used for the address block binning method. In order to evaluate the data structure overhead, this metric is given in bytes per byte of prefix. This metric is evaluated as the size of the data structure divided by the size of the prefixes held in the prefix table.
The format and size of a non-terminal node and a leaf header used in a hybrid trie-tree are detailed respectively in Table~\ref{tab:node_format} and in Table~\ref{tab:SHIP_leaf_header}. The node type field, coded with $1$ bit, specifies whether the node is a leaf or a non-terminal node. The following fields are used only for non-terminal nodes. Up to $10$ bits can be matched at each node, corresponding to a node header field coded with $4$ bits. The fourth field is used for SNM, to store the index value of the traversed regions. Each index is restricted to $10$ bits, while the $HtoL$ and $LtoH$ arrays each store up to $5$ indices. The third field, coded in $16$ bits, stores the base address of the first child node associated with its parent's node.
\begin{table}[htb]
\renewcommand{1.3}{1.3}
\caption{ Non-terminal node header field sizes in bits}
\label{tab:node_format}
\centering
\begin{tabular}{|c|c|c|}
\hline
\bf Header Field & \bf Size \\
\hline
Node type & $1$\\
\hline
Number of cuts & $4$ \\
\hline
Pointer to child node & $16$ \\
\hline
Size of selective node merge array & $5 \cdot 10 + 5 \cdot 10$ \\
\hline
\end{tabular}
\end{table}
The leaf node format is presented in Table~\ref{tab:SHIP_leaf_header}. A leaf can be split over multiple nodes to store all its prefixes. Therefore, two bits are used in the leaf header to specify whether the current leaf node is a terminal leaf or not. The next field gives the number of prefixes stored in the leaf. It is coded with $4$ bits because in this work, the largest number of prefixes held in a leaf is set to $12$ for each hybrid trie-tree. The LSR field stores the number of bits that need to be matched, using 6 bits. If a leaf is split over multiple nodes, a pointer coded with $16$ bits points at the remaining nodes that are part of the leaf. Inside a leaf, prefixes are stored alongside their prefix length and with their NHI. The prefix length is coded with the number of bits specified by the LSR field while the NHI is coded with $8$ bits.
\begin{table}[htb]
\renewcommand{1.3}{1.3}
\caption{Leaf header field sizes in bits}
\label{tab:SHIP_leaf_header}
\centering
\begin{tabular}{|c|c|c|}
\hline
\bf Header Field & Size \\
\hline
Node type & $2$ \\
\hline
Number of prefixes stored & $4$ \\
\hline
LSR field & $6$ \\
\hline
Pointer to remaining leaf entries & $16$ \\
\hline
Prefix and NHI & Value specified in the LSR field + $8$ \\
\hline
\end{tabular}
\end{table}
\section{Results}\label{sec:results}
SHIP performance is first evaluated using real prefixes, and then with synthetic prefixes, for both the number of memory accesses and the memory consumption.
\subsection{Real Prefixes}
The performance analysis is first made for the perfect hash table used by the address block binning method. In Table~\ref{tab:result_bin23_real},
the memory consumption and the number of memory accesses for the hash table are shown. The ABB method uses between $19$ kB and $24$ kB, that is between $0.7$ and $0.9$ bytes per prefix byte for the real prefix tables evaluated. The memory consumption is similar across all the scenarios tested as prefixes share most of the $23$ MSBs. On the other hand, the number of memory accesses is by construction independent of the number of prefixes used, and constant to $2$.
\begin{table}[htbp]
\renewcommand{1.3}{1.3}
\caption{Memory consumption of the address block binning method for real prefix tables}
\label{tab:result_bin23_real}
\centering
\begin{tabular}{|l|c|c|}
\hline
\bf Scenario & \bf Hashing Table size (kB) & \bf Memory Accesses \\
\hline
$rrc00$ & 20 & 2 \\
\hline
$rrc01$ & 19 & 2 \\
\hline
$rrc04$ & 24 & 2 \\
\hline
$rrc05$ & 19 & 2 \\
\hline
$rrc06$ & 19 & 2 \\
\hline
$rrc07$ & 20 & 2 \\
\hline
$rrc10$ & 21 & 2 \\
\hline
$rrc11$ & 20 & 2 \\
\hline
$rrc12$ & 20 & 2 \\
\hline
$rrc13$ & 22 & 2 \\
\hline
$rrc14$ & 21 & 2 \\
\hline
\end{tabular}
\end{table}
In Figures~\ref{mem_cons_real_prefix_table} and~\ref{mem_acc_real_prefix_table}, the performance of the HTTs is evaluated respectively on the memory consumption and the number of memory accesses. In both figures, $1$ to $6$ groups are used for two-level prefix grouping. As a reference, the performance of the HTT without grouping is also presented.
In Fig.~\ref{mem_cons_real_prefix_table}, the memory consumption of the HTTs ranges from $1.36$ to $1.60$ bytes per prefix byte for all scenarios, while it ranges between $1.22$ up to $3.15$ bytes per byte of prefix for a single HTT. Thus, using two-level prefix grouping, the overhead of the HTTs ranges from $0.36$ to $0.6$ byte per byte of prefix. However, a single HTT leads to an overhead of $0.85$ on average, and up to $3.15$ bytes per byte of prefix for scenario $rrc13$. Thus, two-level grouping reduces the memory consumption and smooths its variability, but it also reduces the hybrid trie-tree overhead.
Fig.~\ref{mem_cons_real_prefix_table} shows that increasing the number $K$ of groups up to three reduces the memory consumption. However, using more groups does not improve the memory consumption, and even worsens it. Indeed, it was observed experimentally that when increasing the value of $K$ most groups hold very few prefixes, leading to a hybrid trie-tree holding a single leaf with part of the allocated node memory left unused. Thus, using too many groups increases memory consumption.
\begin{figure}
\caption{Real prefix tables: impact of the number of groups on the memory consumption (a) and the number of memory accesses (b) of the HTTs.}
\label{mem_cons_real_prefix_table}
\label{mem_acc_real_prefix_table}
\label{fig:result_memory_consumption}
\end{figure}
It can be observed in Fig.~\ref{mem_acc_real_prefix_table} that the number of memory accesses to traverse the HTTs ranges from $6$ to $9$ with two-level prefix grouping, whereas it varies between $9$ and $18$ with a single HTT. So, two-level prefix grouping smooths the number of memory accesses variability, but it also reduces on average the number of memory accesses approximatively by a factor 2.
However, increasing the number $K$ of groups used by two-level prefix grouping, from $1$ to $6$, yields little gain on the number of memory accesses, as seen in Fig.~\ref{mem_acc_real_prefix_table}. Indeed, for most scenarios, one memory access is saved, and up to two memory accesses are saved in two scenarios, by increasing the number $K$ of groups from $1$ to $6$. Indeed, for each scenario, the performance is limited by a prefix length that cannot be divided in smaller sets by increasing the number of groups. Still, using two or more groups, in the worst case, $8$ memory accesses are required for all scenarios. The performance is similar across all scenarios evaluated, as few variations exist between the prefix groups created using two-level grouping for those scenarios.
\subsection{Synthetic Prefixes}
The complexity of the perfect hash table used for the address block binning method is presented in Table~\ref{tab:result_bin23} with synthetic prefix tables. It requires on average $2.7$ bytes per prefix byte for the $5$ scenarios tested, holding from $29$ k up to $580$ k prefixes. The perfect hash table used shows linear memory consumption scaling. For the number of memory accesses, its value is independent of the prefix table, and is equal to $2$.
\begin{table}[htbp]
\renewcommand{1.3}{1.3}
\caption{Cost of binning on the first 23 bits for synthetic prefix tables}
\label{tab:result_bin23}
\centering
\begin{tabular}{|l|c|c|}
\hline
\bf Prefix Table Size & \bf Hashing Table size (kB) & \bf Memory Accesses \\
\hline
580 k & 1282 & 2 \\
\hline
290 k & 642 & 2 \\
\hline
110 k & 322 & 2 \\
\hline
50 k & 162 & 2 \\
\hline
29 k & 82 & 2 \\
\hline
\end{tabular}
\end{table}
The performance of the HTTs with synthetic prefixes is evaluated for the number of memory accesses, the memory consumption, and the memory consumption scaling, respectively in Fig.~\ref{mem_acc_synth_prefix_table},~\ref{mem_cons_synth_prefix_table}, and~\ref{fig:mem_cons_synth_prefix_table_scaling}. For each of the three figures, $1$ to $6$ groups are used for two-level prefix grouping. The performance of the HTT without grouping is also presented in the three figures, and is used as a reference.
Two behaviors can be observed for the memory consumption in Fig.~\ref{mem_cons_synth_prefix_table}.
First, for prefix tables with $290$ k prefixes and more, it can be seen that two-level prefix grouping used with $2$ groups slightly decreases the memory consumption over a single HTT. Using this method with two groups, the HTTs consumes between $1.18$ and $1.09$ byte per byte of prefix, whereas the memory consumption for a single HTT lies between $1.18$ and $1.20$ byte per byte of prefix. However, increasing the number of groups to more than two does not improve memory efficiency, as it was observed that most prefix length sorting groups hold very few prefixes, leading to hybrid trie-tree holding a single leaf, with part of the allocated node memory that is left unused. Even though the memory consumption reduction brought by two-level prefix grouping over a single HTT is small for large synthetic prefix tables, it will be shown in this paper that the memory consumption remains lower when compared to other solutions. Moreover, it will be demonstrated that two-level prefix grouping reduces the number of memory accesses to traverse the HTT with the worst case performance over a single HTT, for all synthetic prefix table sizes. Second, for smaller prefix tables with up to $116$ k prefixes, a lower memory consumption is achieved using only a single HTT for two reasons. First, the synthetic prefixes used have fewer overlaps and are more distributed than real prefixes for small to medium size prefix tables, making two-level prefix grouping less advantageous in terms of memory consumption. Indeed, a larger number $M$ of address block bins has been observed compared to real prefix tables with respect to the number of prefixes held the prefix tables, for small and medium prefix tables. Thus, on average, each bin holds fewer prefixes compared to real prefix tables. As a consequence, we observe that the average and maximum number of prefixes held in each PLS group is smaller for prefix tables holding up to $116$ k prefixes. It then leads to hybrid trie-trees where the allocated leaf memory is less utilized, achieving lower memory efficiency and lower memory consumption.
\begin{figure}
\caption{Synthetic prefix tables: impact of the number of groups on the number of memory accesses (a), the memory consumption (b) and scaling (c) of the HTTs.}
\label{mem_acc_synth_prefix_table}
\label{mem_cons_synth_prefix_table}
\label{fig:mem_cons_synth_prefix_table_scaling}
\label{fig:result_memory_accesses}
\end{figure}
In order to observe the memory consumption scaling of the HTTs, Fig.~\ref{fig:mem_cons_synth_prefix_table_scaling} shows the total size of the HTTs using synthetic prefix tables, two-level prefix grouping, and a number $K$ of groups that ranges from $1$ to $6$. The memory consumption of the HTTs with and without two-level prefix grouping grows exponentially for prefix tables larger than $116$ k. However, because the abscissa uses a logarithmic scale, the memory consumption scaling of the proposed HTT is linear with and without two-level prefix grouping. In addition, the memory consumption of the HTTs is $4,753$ kB for the largest scenario with $580$ k prefixes, two-level prefix grouping, and $K = 2$.
Next, we analyze in Fig.~\ref{mem_acc_synth_prefix_table} the number of memory accesses required to traverse the HTT leading to the worst-case performance, for synthetic prefix tables, with two-level prefix grouping using $1$ to $6$ groups. It can be observed that two-level prefix grouping reduces the number of memory accesses over a single HTT, for all the number of groups and all prefix table sizes. The impact of two-level prefix grouping is more pronounced when using two groups or more, as the number of memory accesses is reduced by 40\% over a single HTT. Using more than $3$ groups does not further reduce the number of memory accesses as the group leading to the worst-case scenario cannot be reduced in size by increasing the number of groups. Finally, it can be observed in Fig.~\ref{mem_acc_synth_prefix_table} that the increase in the number of memory accesses for a search is at most logarithmic with the number of prefixes, since each curve is approximately linear and the x-axis is logarithmic.
The performance analysis presented for synthetic prefixes has shown that two-level prefix grouping improves the performance over a single HTT for the two metrics evaluated. Although the performance improvement of the memory consumption is limited to large prefix tables, using few groups, the number of memory accesses is reduced for all prefix table sizes and for all numbers of groups. In addition, it has been observed experimentally that the HTTs used with two-level prefix grouping have a linear memory consumption scaling, and a logarithmic scaling for the number of memory accesses. The hash table used in the address block binning method has shown to offer a linear memory consumption scaling and a fixed number of memory accesses. Thus, SHIP has a linear memory consumption scaling, and a logarithm scaling for the number of memory accesses.
\section{Discussion}\label{sec:disscusion}
This section first demonstrates that SHIP is optimized for a fast hardware implementation. Then, the performance of SHIP is compared with previously reported results.
\subsection{SHIP hardware implementability}
We demonstrate that SHIP is optimized for a fast hardware implementation, as it complies with the following two properties; 1) pipeline-able and parallelizable processing to maximize the number of packets forwarded per second, 2) use of a data structure that can fit within on-chip memory to minimize the total memory latency.
A data structure traversal can be pipelined if it can be decomposed into a fixed number of stages, and for each stage both the information read from memory and the processing are fixed. First, the HTT traversal can be decomposed into a pipeline, where each pipeline stage is associated to a HTT level. Indeed, the next node to be traversed in the HTT depends only on the current node selected and the value of the packet header. Second, for each pipeline stage of the HTT both the information read from memory and the processing are fixed. Indeed, the information is stored in memory using a fixed node size for both the adaptive-density trie and the reduced D-Tree leaf. In addition, the processing of a node is constant for each data structure and depends only on its type, as presented in Section~\ref{sec:lookup_fixed}. As a result, the HTT traversal is pipeline-able. Moreover, the HTTs within the $K$ PLS groups are independent, thus their traversal is by nature parallelizable. As a consequence, by combining a parallel traversal of the HTTs with a pipelined traversal of each HTT, property $1$ is fulfilled. The hash table data structure used for the address block binning technique has been implemented in hardware in previous work~\cite{mem_eff}, and thus it already complies with property $1$.
For the second property, SHIP uses $5.9$ MB of memory for $580$ k prefixes, with two-level prefix grouping, and $K = 2$ . Therefore, SHIP data structure can fit within on-chip memory of the current generation of FPGAs and ASICs~\cite{altera,xilinx,pisa}. Hence, SHIP fulfills property $2$. As both the hash table used by two-level prefix grouping, and the hybrid trie-tree comply with properties $1$ and $2$ required for a fast hardware implementation, SHIP is optimized for a fast hardware implementation.
\subsection{Comparison with previously reported results}
\begin{table*}[htbp]
\renewcommand{1.3}{1.3}
\caption{Comparison Results}
\label{tab:result_comparison}
\centering
\begin{tabular}{|l|c|c|c|c|}
\hline
\bf Method & \bf Memory Consumption & \bf Latency (ns) & \multicolumn{2}{|c|}{\bf Complexity} \\
& \bf (in bytes per prefix) & & Memory Consumption & Memory latency \\
\hline
Tree-based \cite{scalable_ipv6_vk} & $19.0$ & 90 & $O(N)$ & $O(log_2 (N) ) \leq Latency \leq 2 \cdot O(log_3 (N) )$\\
\hline
CLIPS \cite{mem_eff} & $27.6$ & N/A & N/A & N/A \\
\hline
FlashTrie \cite{flashtrie_journal} & $124.2$ & 80 & N/A & N/A \\
\hline
FlashLook \cite{flashlook} & $1010.0$ & 90 & N/A & N/A \\
\hline
\bf SHIP & $10.64$ & 31 & $O(N)$ & $O(log(N))$ \\
\hline
\end{tabular}
\end{table*}
Table~\ref{tab:result_comparison} compares the performance of SHIP and previous work in terms of memory consumption and worst case memory latency. If available, the time and space complexity are also shown. In order to use a common metric between all reported results, the memory consumption is expressed in bytes per prefix, obtained by dividing the size of the data structure by the number of prefixes used. The memory latency is based on the worst-case number of memory accesses to traverse a data structure.
For the following comparison, it is assumed that on-chip SRAM memory running at 322 MHz~\cite{scalable_ipv6_vk} is used, and off-chip DDR3-1600 memory running at 200 MHz is used.
Using both synthetic and real benchmarks, SHIP requires in the worst case $10$ memory accesses, and consumes $5.9$ MB of memory for the largest prefix table, with $2$ groups for two-level prefix grouping. Hence, the memory latency to complete a lookup with on-chip memory is equal to $10 \cdot 3.1$ = 31 ns.
FlashTrie has a high memory consumption, as reported in Table~\ref{tab:result_comparison}. The results presented were reevaluated using the node size equation presented in~\cite{flashtrie_conf} due to incoherence with equations shown in~\cite{flashtrie_journal}. This algorithm leads to a memory consumption per prefix that is around $11 \times$ higher than the SHIP method, as multiple copies of the data structure have to be spread over DDR3 DRAM banks. In terms of latency, in the worst case, two on-chip memory accesses are required, followed by three DDR3 memory bursts. However, DRAM memory access comes at a high cost in terms of latency for the FlashTrie method. First, independently of the algorithm, a delay is incurred to send the address off-chip to be read by the DDR3 memory controller. Second, the latency to complete a burst access for a given bank, added to the maximum number of bank-activate commands that can be issued in a given period of time, limits the memory latency to $80$ ns and reduces the maximum lookup frequency to $84$ MHz. Thus, FlashTrie memory latency is $2.5 \times$ higher than SHIP.
The FlashLook~\cite{flashlook} architecture uses multiple copies of data structures in order to sustain a bandwidth of 100 Gbps, leading to a very large memory consumption compared to SHIP. Moreover, the memory consumption of this architecture is highly sensitive to the prefix distribution used. For the memory latency, in the worst case, when a collision is detected, two on-chip memory accesses are required, followed by three memory bursts pipelined in a single off-chip DRAM, leading to a total latency of~80~ns. The observed latency of SHIP is 61\% smaller. Finally, no scaling study is presented, making it difficult to appreciate the performance of FlashLook for future applications.
The method proposed in~\cite{scalable_ipv6_vk} uses a tree-based solution that requires $19$ bytes per prefix, which is $78\%$ larger than the proposed SHIP algorithm. Regarding the memory accesses, in the worst case, using a prefix table holding $580$ k prefixes, $22$ memory accesses are required, which is more than twice the number of memory accesses required by SHIP. In terms of latency, their implementation leads to a total latency of $90$ ns for a prefix table holding $580$ k prefixes, that is $2.9 \times$ higher than the proposed SHIP solution. Nevertheless, similar to SHIP, this solution has a logarithmic scaling factor in terms of memory accesses, and scales linearly in terms of memory consumption.
Finally, Tong et al.~\cite{mem_eff} present the CLIPS architecture~\cite{CLIPS} extended to IPv6. Their method uses $27.6$ bytes per prefix, which is about $2.5 \times$ larger than SHIP. The data structure is stored in both on-chip and off-chip memory, but the number of memory accesses per module is not presented by the authors, making it impossible to give an estimate of the memory latency. Finally, the scalability of this architecture has not been discussed by the authors.
These results show that SHIP reduces the memory consumption over other solutions and decreases the total memory latency to perform a lookup. It also offers a logarithmic scaling factor for the number of memory accesses, and it has a linear memory consumption scaling.
\section{Conclusion}\label{sec:conclusion}
In this paper, SHIP, a scalable and high performance IPv6 lookup algorithm, has been proposed to address current and future application performance requirements. SHIP exploits prefix characteristics to create a shallow and compact data structure. First, two-level prefix grouping leverages the prefix length distribution and prefix density to cluster prefixes into groups that share common characteristics. Then, for each prefix group, a hybrid trie-tree is built. The proposed hybrid trie-tree is tailored to handle local prefix density variations using a density-adaptive trie and a reduced D-Tree leaf structure.
Evaluated with real and synthetic prefix tables holding up to 580 k IPv6 prefixes, SHIP builds a compact data structure that can fit within current on-chip memory, with very low memory lookup latency. Even for the largest prefix table, the memory consumption per prefix is $10.64$ bytes, with a maximum number of $10$ on-chip memory accesses. Moreover, SHIP provides a logarithmic scaling factor in terms of the number of memory accesses and a linear memory consumption scaling. Compared to other approaches, SHIP uses at least 44\% less memory per prefix, while reducing the memory latency by 61\%.
\section*{Acknowledgments}
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC), Prompt,
and Ericsson Canada for financial support to this research.
\end{document} |
\begin{document}
\title[Induced Representation]{Induced representations of Hilbert $C^*$-modules }
\author{Gh. Abbaspour tabadkan and S. Farhangi}
\address{Department of pure mathematics, school of mathematics and computer science, Damghan university, Damghan, Iran.}
\email{abbaspour@du.ac.ir, sfarmdst@gmail.com.}
\subjclass[2010]{46L08, 46L05}
\keywords{Hilbert modules, Morita equivalent and Induced representations. }
\date{\today}
\begin{abstract}
In this paper, we define the notion of induced representations of a Hilbert $C^{*}$-module and we show that Morita equivalence of two Hilbert modules (in the sense of Moslehian and Joita \cite{JOI}), implies the equivalence of categories of non-degenerate representations of two Hilbert modules.
\end{abstract}
\maketitle
\section{Introduction}
The concept of Morita equivalence was first made by Morita \cite{MOR} in a purely algebraic content. Two unital rings are called Morita equivalent if their categories of left modules are equivalent.\\
This concept has been applied to many different categories in mathematics. And investigate the relationship between an "object", and its "representation theory".\\
In the category of $C^{*}$-algebras, Rieffel \cite{RIE1,RIE2} defined the notions of induced representations and (strong) Morita equivalence. The notion of induced representations of $C^*$-algebras, now called Rieffel induction, is to constructing functors between the categories of non-degenerate representations of two $C^*$-algebras. Bursztyn and Waldmann \cite{BUR}, generalized this notion to $*$-algebras and in 2005 Joita \cite{JOT1} introduced this notion for locally $C^*$-algebras.\\ Two $C^*$-algebras $A$ and $B$ are called Morita equivalent if there exist an $A-B$-imprimitivity. This notion is weaker than isomorphism. There are many valuable papers which study properties of $C^*$-algebras are invariant under the Morita equivalence. (see for examples \cite{ BEE, RIE1, Z})\\
The notion of Morita equivalence in the category of Hilbert $C^*$-modules is defined by Skeide \cite{MSK1} and Joita, Moslehian \cite{JOI}, in two different form. In \cite{JOI}, two Hilbert $A$-module $V$ and Hilbert $B$-module $W$ are called Morita equivalent if the $C^*$-algebras $K(V)$ and $K(W)$ are Morita equivalent as $C^*$-algebras. This notion is weaker than the notion of Morita equivalence defined by Skeide, where he called $V$ and $W$ Morita equivalent when the $C^*$-algebras $K(V)$ and $K(W)$ are isomorphic as $C^*$-algebras.\\
In this paper we show that the weaker notion of Morita equivalence is enough to Hilbert modules have same categories of non-degenerate representations.\\
In section 2, we fix our terminologies and discuss preliminaries about representations of Hilbert modules and Morita equivalence of $C^*$-algebras.\\ In section 3, we introduce the notion of induced representations for Hilbert modules
and we show that Morita equivalence Hilbert module in the sense of Joita and Moslehian \cite{JOI}, have same categories of non-degenerate representations. \\
----------------------------------------------------------------
\section{Preliminary}
A (right) Hilbert $C^*$-module $V$ over a $C^*$-algebra $A$ (or a Hilbert $A$-module ) is by definition
a linear space that is a right $A$-module, together with an $A$-valued inner
product $\langle . , . \rangle$ on $V \times V$ that is $A$-linear in the second and conjugate linear in the first variable, such that $V$ is a Banach space with the norm define by $\Vert x\Vert_{A}:= \Vert \langle x , x \rangle_{A} \Vert^{\frac{1}{2}}.$ A Hilbert $A$-module $V$ is a full Hilbert $A$-module if the ideal
\begin{center}
I = $span\lbrace\langle x , y \rangle_{A} ; x,y \in{X} \rbrace $
\end{center}
is dense in $A$. The notion of left Hilbert $A$-module is defined in similar way.\\
We denote the $C^*$-algebras of adjointable and compact operators on Hilbert $C^*$-module $V$ by $L(V)$ and $K(V)$, respectively. See \cite{LAN} for more details on Hilbert modules.\\
Now we have a quick review on the notion of Rieffel induction. Let $X$ be a right Hilbert $B$-module and let $\pi:B\rightarrow B(H)$ be a representation. Then $X\otimes_{alg}H$ is a Hilbert space with inner product
\begin{center}
$\langle x \otimes h,y \otimes k \rangle$:=$\langle \pi(\langle y,x\rangle_{B})h,k \rangle$
\end{center}
for $x,y \in X$ and $h,k \in H$ [ \cite{RAE}, Proposition 2.64 ].\\
If $A$ acts as adjointable operators on a Hilbert $B$-module $X$, and $\pi$ is a non-degenerate representation of $B$ on $H$. Then $Ind\pi$ defined by $$Ind\pi(a)(x\otimes_{B}h):=(ax)\otimes_{B}h$$ is a representation of $A$ on $X\otimes_{B}H$. If $X$ is non-degenerate as an $A$-module, then $Ind\pi$ is a non-degenerate representation of $A$ [ \cite{RAE}, Proposition 2.66 ].\\ This is a functor from the non-degenerate representations of $B$ to the non-degenerate representations of $A$. Now if we want to get back from representations of $A$ to representations of $B$, we need also an $A$-valued inner product on $X$. This lead us to the following definition.
\begin{defn}
An $A-B$\emph{-imprimitivity bimodule} is an $A-B$-bimodule such that:\\
$(a)$ $X$ is a full left Hilbert $A$-module, and is a full right Hilbert $B$-module;\\
$(b)$ for all $ x,y \in X$, $a \in A$, $b \in B$
\begin{center}
$\langle ax , y \rangle_{B}$=$ \langle x , a^{*}y \rangle_{B} $ and $_{A}\langle xb , y \rangle$=$ _{A}\langle x , yb^{*} \rangle $
\end{center}
$(c)$ for all $ x,y,z \in X$
\begin{center}
$_{A}\langle x , y \rangle z $=$ x \langle y,z \rangle_{B} $.
\end{center}
\end{defn}
If $X$ is an $A-B$-imprimitivity bimodule, let $\widetilde{X}$ be the conjugate vector space, so that there is by definition an additive bijection $b:X\rightarrow \widetilde{X}$ such that $b(\lambda x):= \overline{\lambda}b(x)$. Then $\widetilde{X}$ is a $B-A$-imprimitivity bimodule with\\
$\begin{array}{cc}
bb(x):=b(xb^{*}) & b(x)a:=b(a^{*}x) \\
_{B}\langle b(x),b(y)\rangle:=\langle x,y\rangle_{B} & \langle b(x),b(y)\rangle _{A}:=_{A} \langle x,y\rangle
\end{array}$\\
for $x,y \in X$, $a\in A$ and $b \in B$. $\widetilde{X}$ called the \emph{dual module} of $X$.
\begin{exa}
A Hilbert space $H$ is a $K(H)-\textbf{C}$-impirimitivity bimodule with $_{K(H)}\langle h,k \rangle $:=$h \otimes \overline{k}$, where $h \otimes \overline{k}$ denote the rank one operator $g \mapsto \langle g,k\rangle h$.
\end{exa}
\begin{exa}
Every $C^*$-algebra $A$ is an $A-A$-imprimitivity bimodule for the bimodule structure given by the multiplication in $A$, with the inner product $ _{A}\langle a , b \rangle$=$ab^{*}$ and $ \langle a , b \rangle_{A}$=$a^{*}b$ .
\end{exa}
Two $C^*$-algebras $A$ and $B$ are Morita equivalent if there is an $A-B$-imprimitivity bimodule $X$; we shall say that $X$ implements the Morita equivalence of $A$ and $B$ .\\
Morita equivalence is weaker than isomorphism. If $\varphi$ is an isomorphism of $A$ onto $B$, we can construct an imprimitivity bimodule $_{A}X_{B}$ with underlying space $B$ by
\begin{center}
$xb:=xb$, $ax:= \varphi(a)x$, $\langle x,y \rangle_{B}:=x^{*}y$ and $_{A}\langle x,y \rangle:=\varphi^{-1}(xy^{*})$.
\end{center}
Morita equivalence is an equivalence relation on $C^*$-algebras. If $A$ and $B$ are Morita equivalent then the functor mentioned above which comes from tensoring by $X$ has an inverse functor. In fact its inverse is functor comes from tensoring by $\widetilde{X}$, the dual module of $X$ [ \cite{RAE}, Proposition 3.29 ]. So $A$ and $B$ have the same categories of non-degenerate representations.\\
In this paper we will prove that two full Hilbert modules on Morita equivalent $C^*$-algebras have the same categories of non-degenerate representations. But let us first say some facts about representations of Hilbert modules.\\
Let $V$ and $W$ be Hilbert $C^{*}$-modules over $C^{*}$-algebras $A$ and $B$, respectively, and $ \varphi:A\longrightarrow B$ a morphism of $C^*$-algebras.\\
A map $\Phi :V \longrightarrow W$ is said to be a $\varphi$-morphism of Hilbert $C^*$-modules if\\ $ \langle \Phi(x), \Phi (y) \rangle=\varphi(\langle x,y \rangle)$ is satisfied for all $x,y \in V$.\\
A $\varphi$-morphism $ \Phi :V \longrightarrow B(H,K)$, where $ \varphi :A \longrightarrow B(H) $ is a representation of $A$ is called a representation of $V$. We will say that a representation $ \Phi :V \longrightarrow B(H,K)$ is a faithful representation of $V$ if $\Phi$ is injective.\\
Throughout the paper, when we say that $\Phi$ is a representation of $V$, we will assume that an associated representation of $A$ is denoted by the same small case letter $ \varphi$.\\
Let $ \Phi :V \longrightarrow B(H,K)$ be a representation of a Hilbert $A$-module $V$. $\Phi$ is said to be\emph{ non-degenerate} if $\overline{\Phi (V)H}$=$K$ and $\overline{\Phi (V)^{*}K}$=$H$. (Or equivalently, if $\xi_{1} \in H$,$\xi_{2} \in K$ are such that $\Phi (V)\xi_{1}=0$ and $\Phi(V)^{*}\xi_{2}=0$, then $\xi_{1}=0$ and $\xi_{2}=0$ ). If $\Phi$ is non-degenerate, then $\varphi$ is non-degenerate [ \cite{ARA}, Lemma 3.4 ].\\
Let $ \Phi :V \longrightarrow B(H,K)$ be a representation of a Hilbert $A$-module $V$ and $ K_{1} \prec H $ , $ K_{2} \prec K $ be closed subspaces.
A pair of subspaces $(K_{1},K_{2})$ is said to be $\Phi$\emph{-invariant} if
\begin{center}
$\Phi (V)K_{1}\subseteq K_{2}$ and $\Phi (V)^{*}K_{2}\subseteq K_{1}$.
\end{center}
$ \Phi $ is said to be \emph{irreducible} of $(0,0)$ and $(H,K)$ are the only $\Phi$-invariant pairs.\\
Two representations $\Phi_{i}: V \rightarrow B(H_{i};K_{i})$ of V, $i =1,2$ are said to be (unitarily) equivalent, if there are unitary operators $U_{1}:H_{1}\rightarrow H_{2}$ and $U_{2}:K_{1}\rightarrow K_{2}$; such that $U_{2}\Phi_{1}(v)=\Phi_{2}(v)U_{1}$
for all $v \in V$. For more details on representations of Hilbert modules see \cite{ARA}.\\
Finally we need the interior tensor product of Hilbert modules, we mention it here briefly. For more details one can refer to the Lance book \cite{LAN}.
Suppose that $V$ and $W$ are Hilbert $A$-module and Hilbert $B$-module, respectively, and $\rho :A \longrightarrow L(W)$ is a *-homomorphism, we can regard $W$ as a left $A$-module, the action being given by $(a,y) \longmapsto \rho(a)y$ for all $a \in A$, $y \in W$.\\
We can form the algebraic tensor product of $V$ and $W$ over $A$, $V \otimes_{alg} W$, which is a right $B$-module. The action of $B$ being given by $(x \otimes y)b$ := $x \otimes yb$ for $b \in B$.\\
In fact it is the quotient space of the vector space tensor product $V \otimes_{alg} W$ by the subspace generated by elements of the form
\begin{center}
$xa \otimes y-x \otimes \rho(a)y$, $(x \in V , y \in W , a \in A)$.
\end{center}
$V\otimes_{alg}W$ is an inner product $B$-module under the inner product
\begin{center}
$\langle x_{1} \otimes y_{1},x_{2} \otimes y_{2} \rangle$=$\langle y_{1},\rho( \langle x_{1},x_{2} \rangle)y_{2}\rangle$
\end{center}
for $x_{1},x_{2} \in V$, $y_{1},y_{2} \in W$.\\
And $V\otimes_{A}W$, which is called the interior tensor product of $V$ and $W$, obtained by completing $V\otimes_{alg}W$ with respect to this inner product.\\
\section{Induced Representation}
In this section we discussed about Morita equivalence of Hilbert $C^{*}$-modules and speak about the notion of induced representation of a Hilbert $C^{*}$-module and then we prove the imprimitivity theorem for induced representations of Hilbert $C^{*}$-modules.
\begin{prop}\label{prop:ind}
Let $V$ and $W$ be two full Hilbert $C^{*}$-modules over $C^{*}$-algebras $A$ and $B$, respectively. Let $X$ be a $B$-module and $A$ acts as adjointable operators on Hilbert $C^{*}$-module $X$, and $\Phi : W \rightarrow B(H,K)$ is a non-degenerate representation. Then the formula,
\begin{center}
$ Ind_{X}\Phi(v)(x\otimes g) := v\otimes x \otimes h $
\end{center}
extends to give a representation of $V$ as bounded operator of Hilbert space $X\otimes_{B} H$ to Hilbert space $V\otimes_{A} X\otimes_{B} H$. If $X$ is non-degenerate as an $A$-module, then $Ind_{X}\Phi$ is a non-degenerate representation.
\end{prop}
\begin{proof}
Since $A$ acts as adjointable operators on the Hilbert $B$-module $X$, so we may construct interior tensor product $V\otimes_{A} X$, which is a $B$-module. Then $V\otimes_{A} X\otimes_{B} H$ and $X\otimes_{B} H $ are Hilbert spaces.\\
Let $\Phi :W\rightarrow B(H,K)$ be a non-degenerate representation, so there is a representation $\varphi :B\rightarrow B(H)$ such that $\langle \Phi(x),\Phi(y) \rangle$=$\varphi (\langle x,y \rangle_{B}) $ for all $x,y \in W$. $\varphi$ is non-degenerate so by Rieffel induction we get a non-degenerate representation,
\begin{center}
$Ind_{X}\varphi:A\rightarrow B(X\otimes_{B} H)$.
\end{center}
Now we want to construct a representation, $Ind_{X}\Phi$ of $V$.\\
The mapping $(x,h)\mapsto v\otimes x \otimes h$ is bilinear, thus there is a linear transformation $\eta_{v}: X\otimes_{alg}H \rightarrow V\otimes_{A} X\otimes_{B} H$ such that $\eta _{v}(x\otimes h)$=$v\otimes x \otimes h$.\\
To see that $\eta_{v}$ is bounded, as in the $C^*$-algebraic case, we may suppose that $\varphi$ is cyclic, with cyclic vector $h$. Then for any $x_{i} \in X , b_{i} \in B$ we have
\begin{align*}
\|\eta_{v}(\sum_{i=1}^{n}x_{i} \otimes \varphi(b_{i})h) \|^{2} &= \sum_{i} \sum_{j}\langle v\otimes x_{i} \otimes \varphi(b_{i})h,v\otimes x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle x_{i} \otimes \varphi(b_{i})h,Ind_{X}\varphi (_{A}\langle v,v \rangle) x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle x_{i} \otimes \varphi(b_{i})h,_{A}\langle v,v \rangle x_{j} \otimes \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle \varphi(b_{i})h,\varphi (\langle _{A}\langle v,v \rangle x_{i}, x_{j}\rangle_{B}) \varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle h,\varphi(b_{i}^{*})\varphi (\langle _{A}\langle v,v \rangle x_{i}, x_{j}\rangle_{B})\varphi(b_{j})h \rangle \\
& = \sum_{i} \sum_{j}\langle h,\varphi (\langle _{A}\langle v,v \rangle x_{i}b_{i}, x_{j}b_{j}\rangle_{B})h \rangle \\ & = \sum_{i} \sum_{j}\langle h,\varphi(\langle _{A}\langle v,v \rangle^{\frac{1}{2}} x_{i}b_{i},_{A}\langle v,v \rangle^{\frac{1}{2}} x_{j}b_{j}\rangle_{B})h \rangle \\
& = \langle h,\varphi(\langle _{A}\langle v,v \rangle^\frac{1}{2}\sum_{i} x_{i}b_{i},_{A}\langle v,v \rangle^\frac{1}{2} \sum_{j} x_{j}b_{j}\rangle_{B})h \rangle \\
& \leq \|_{A}\langle v,v \rangle^\frac{1}{2}\|^{2}\langle h,\varphi(\langle \sum_{i} x_{i}b_{i}, \sum_{j}x_{j}b_{j}\rangle_{B})h \rangle \\
& = \|v\|_{A}^{2}\langle \sum_{i} x_{i}b_{i}\otimes h,\sum_{j} x_{j}b_{j}\otimes h \rangle \\
& = \|v\|_{A}^{2}\| \sum_{i} x_{i} \otimes \varphi(b_{i})h\|^{2} .
\end{align*}
So $\eta_{v}$ is bounded and $\|\eta_{v}\|^{2}\leq \|v\|_{A}^{2}$.\\
Hence $\eta_{v}$ extends to an operator $Ind_{X}\Phi(v)$ on $X\otimes_{B} H$ and we have\\
\begin{align*}
\langle x \otimes h,Ind_{X}\Phi^{*}(v) Ind_{X}\Phi(v^{'}) x^{'}\otimes h^{'} \rangle & = \langle Ind_{X}\Phi(v)(x\otimes h),Ind_{X}\Phi(v^{'})(x^{'}\otimes h^{'}) \rangle \\
& = \langle v\otimes x\otimes h,v^{'}\otimes x^{'}\otimes h^{'} \rangle \\
& = \langle x\otimes h, Ind_{X}\varphi(_{A}\langle v,v^{'}\rangle)x^{'}\otimes h^{'} \rangle .
\end{align*}
Thus $$\langle Ind_{X}\Phi(v), Ind_{X}\Phi(v^{'}) \rangle=Ind_{X}\Phi^{*}(v) Ind_{X}\Phi(v^{'})=Ind_{X}\varphi(_{A}\langle v,v^{'}\rangle).$$So $Ind_{X}\Phi:V \rightarrow B(X\otimes_{B} H,V\otimes_{A} X\otimes_{B} H)$ is an $Ind_{X}\varphi$-morphism and hence a representation of $V$.\\
Now we show that $Ind_{X}\Phi$ is non-degenerate. For this we must to show that\\ $\overline{Ind_{X}\Phi(V)X\otimes_{B} H}$=$V\otimes_{A} X\otimes_{B} H$ and $\overline{Ind_{X}\Phi(V)^{*}(V\otimes X\otimes h)}$=$X\otimes H$.\\
By definition of $Ind_{X}\Phi$ it is easy to see that $\overline{Ind_{X}\Phi(V)X\otimes_{B} H}$=$V\otimes_{A} X\otimes_{B} H$.\\
By hypotheses, $A$ acts as adjointable operators on Hilbert $C^{*}$-module $X$ and this action is non-degenerate, that is, $\overline{AX}$=$X$ and $V$ is full, $\overline{\langle V, V\rangle}$=$A$, so $\overline{\langle V,V\rangle X}$=$X$.\\
For all $x\otimes h\in X\otimes_{alg}H$ we have:
$$\|x\otimes h\|^{2}=|\langle h,(\varphi(\langle x,x\rangle_{B}))h\rangle |\leq \|\langle x,x\rangle_{B}\|\|h\|^{2}=\|x\|_{B}^{2}\|h\|^{2},$$
so if $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}$ approximates $x$, then $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}\otimes h$ approximates $x\otimes h$.\\
But, $\sum _{A}\langle v_{i},v^{'}_{i}\rangle x_{i}\otimes h$=$\sum Ind_{X}\varphi(_{A}\langle v_{i},v^{'}_{i}\rangle)x_{i}\otimes h$=$\sum Ind_{X}\Phi(v_{i})^{*}Ind_{X}\Phi(v^{'}_{i})(x_{i}\otimes h)$.\\
So every elementary tensor $x\otimes h$ in $X\otimes_{alg}H$ can be approximated by a sum of the form $\sum Ind_{X}\Phi(v_{i})^{*}Ind_{X}\Phi(v^{'}_{i})(x_{i})\otimes h$=$\sum Ind_{X}\Phi(v_{i})^{*}(v^{'}_{i}\otimes x_{i}\otimes h)$.\\
Thus $Ind_{X}\Phi$ is a non-degenerate representation.
\end{proof}
\begin{defn}
We call the representation $ Ind_{X}\Phi$ constructed above, the Rieffel-induced representation from $W$ to $V$ via $X$.
\end{defn}
\begin{prop}
Let $\Phi_{1} :W\rightarrow B(H_{1},K_{1})$ and $\Phi_{2} :W\rightarrow B(H_{2},K_{2})$ be two non-degenerate
representations. If $\Phi_{1}$ and $\Phi_{2}$ are unitarily equivalent, then $Ind_{X}\Phi_{1}$ and $Ind_{X}\Phi_{2}$ are unitarily equivalent.
\end{prop}
\begin{proof} Suppose $U_{1}:H_{1}\rightarrow H_{2}$ and $U_{2}:K_{1}\rightarrow K_{2}$ be unitary operators
such that $U_{2}\Phi_{1}(w)=\Phi_{2}(w)U_{1}$. Then $id_{X}\otimes U_{1}:X\otimes_{alg} H_{1}\rightarrow X\otimes_{alg} H_{2}$ given by $x\otimes h\mapsto x\otimes U_{1}(h)$ and $id_{V}\otimes id_{X}\otimes U_{2}:V\otimes_{A} X\otimes_{alg} H_{1}\rightarrow V\otimes_{A} X\otimes_{alg} H_{2}$ given by $v\otimes x\otimes h\mapsto v\otimes x\otimes U_{1}(h)$ may be extended to unitary operators $V_{1}$ from $X\otimes_{B} H_{1} $ onto $X\otimes_{B} H_{2} $ and $V_{2}$ from $V\otimes_{A}X\otimes_{B} H_{1}$ onto $V\otimes_{A}X\otimes_{B} H_{1}$ and moreover, $V_{2}Ind_{X}\Phi_{1}(v)=Ind_{X}\Phi_{2}(v)V_{1}$. So $Ind_{X}\Phi_{1}$ and $Ind_{X}\Phi_{2}$ are unitarily equivalent.
\end{proof}
\begin{cor}
Suppose $\Phi:W \rightarrow B(H,K)$ and $\oplus_{s}\Phi_{s}:W \rightarrow B(\oplus_{s}H_{s},K)$ are unitary equivalent, then $Ind_{X}\Phi:V\rightarrow B(X\otimes_{B}H,V\otimes_{A}X\otimes_{B}H)$ is unitary equivalent to $\oplus_{s}Ind_{X}\Phi_{s}:V\rightarrow B(X\otimes_{B}\oplus_{s}H_{s},V\otimes_{A}X\otimes_{B}\oplus_{s}H_{s})$.\\
\end{cor}
\begin{defn}[ \cite{JOI}, Definition 2.1 ]
Two Hilbert $C^*$-modules $V$ and $W$ , respectively, over $C^*$-algebras
$A$ and $B$ are called Morita equivalent, if the $C^*$-algebras $K(V)$ and $K(W)$ are Morita equivalent as $C^*$-algebras.
\end{defn}
It is well known that for Hilbert $C^*$-module $V$, $K(V)$ is Morita equivalent to $\overline{\langle V, V\rangle}$, so if $V$ and $W$ are full, then they are Morita equivalent if and only if their underlying $C^*$-algebras are Morita equivalent [ \cite{JOI}, Proposition 2.8 ].\\
The following theorem show that there is a bijection between non-degenerate representations of two Morita equivalent full Hilbert $C^*$-modules. The fullness property is not crucial, if necessary we can replace underlying $C^*$-algebra by a suitable ones, thus two Morita equivalent Hilbert modules in the above sense have same categories of non-degenerate representations.
\begin{thm} \label{thm:cor}
Suppose that $X$ is an $A-B$-imprimitivity bimodule, and $\Phi$ and $\Psi$ are non-degenerate representation of $W$ and $V$, respectively. Then $Ind_{\widetilde{X}}(Ind_{X}\Phi)$ is naturally unitary equivalent to $\Phi$, and $Ind_{X}(Ind_{\widetilde{X}}\Psi)$ is naturally unitary equivalent to $\Psi$.
\end{thm}
\begin{proof} If $\Phi:W\rightarrow B(H,K)$ is a non-degenerate representation by proposition \ref{prop:ind}, $Ind_{X}\Phi:V\rightarrow B(X\otimes_{B}H,V\otimes_{A}X\otimes_{B}H)$ is a non-degenerate representation of $V$.\\ Again usage of proposition \ref{prop:ind} to $Ind_{X}\Phi$ instead of $\Phi$, give us the following non-degenerate representation of $W$, $$Ind_{\widetilde{X}}(Ind_{X}\Phi):W\rightarrow B(\widetilde{X}\otimes_{A}X\otimes_{B}H,W\otimes_{B}\widetilde{X}\otimes_{A}X\otimes_{B}H).$$ Now we want to show that $\Phi$ is unitary equivalent to $Ind_{\widetilde{X}}(Ind_{X}\Phi)$. By the proof of Theorem 3.29 \cite{RAE}; $U_{1}:\widetilde{X}\otimes_{A}X\otimes_{B}H\rightarrow H$ defined by $b(x)\otimes y\otimes h\mapsto \varphi(\langle x,y \rangle_{B})h$ is a unitary operator.\\ We define $$U_{2}:W\otimes_{B}\widetilde{X}\otimes_{A}X\otimes_{B}H\rightarrow K$$ given by $$w\otimes b(x)\otimes y\otimes h\mapsto\Phi(w)\varphi(\langle x,y \rangle_{B})h.$$ $U_{2}$ is a unitary operator, and we have,
\begin{align*}
U_{2}Ind_{\widetilde{X}}(Ind_{X}\Phi(w))(b(x)\otimes y\otimes h)&=U_{2}(w\otimes \varphi(\langle x,y\rangle_{B})h)\\&=\Phi(w)\varphi(\langle x,y \rangle_{B})h \\&= \Phi(w)U_{1}(b(x)\otimes y\otimes h).
\end{align*}
So $$U_{2}Ind_{\widetilde{X}}(Ind_{X}\Phi(w))=\Phi(w)U_{1}.$$ Hence $\Phi$ and $Ind_{\widetilde{X}}(Ind_{X}\Phi)$ are unitary equivalent.\\
For the equivalence of $\Psi$ and $Ind_{X}(Ind_{\widetilde{X}}\Psi)$, apply the first part to $_{B}\widetilde{X}_{A}$ instead of $_{A}X_{B}$.
\end{proof}
The Financial Support of the Research Council of Damghan University
with the Grant Number 92093 is Acknowledged.
\end{document} |
\begin{document}
\begin{abstract}
In 2008, Haglund, Morse and Zabrocki \cite{classComp} formulated a Compositional form of the Shuffle Conjecture of Haglund {\em et al.} \cite{Shuffle}. In very recent work, Gorsky and Negut by combining their discoveries \cite{GorskyNegut}, \cite{NegutFlags} and \cite{NegutShuffle}, with the work of Schiffmann-Vasserot \cite{SchiffVassMac} and \cite{SchiffVassK} on the symmetric function side and the work of Hikita \cite{Hikita} and Gorsky-Mazin \cite{GorskyMazin} on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of \cite{Shuffle}. In fact, they formulated one conjecture for each pair $(m,n)$ of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair $(km,kn)$, with $(m,n)$ co-prime and $k > 1$.
\end{abstract}
\title{Compositional \lowercase{$(km,kn)$}
\operatorname{comp}arskip=0pt
{ \setcounter{tocdepth}{1}\operatorname{comp}arskip=0pt\footnotesize Teofcontents}
\operatorname{comp}arskip=8pt
\operatorname{comp}arindent=20pt
\section*{Introduction}
The subject of the present investigation has its origin, circa 1990, in a effort to obtain a representation theoretical setting for the Macdonald $q,t$-Kotska coefficients. This effort culminated in Haimain's proof, circa 2000, of the $n!$ conjecture (see \cite{natBigraded})
by means of the Algebraic Geometry of the Hilbert Scheme.
In the 1990's, a concerted effort by many researchers led to a variety of conjectures tying the theory of Macdonald Polynomials to
the representation theory of Diagonal Harmonics and the combinatorics
of parking functions. More recently, this subject has been literally flooded
with connections with other areas of mathematics such as: the Elliptic Hall Algebra of Shiffmann-Vasserot, the Algebraic Geometry of Springer Fibers of Hikita, the Double Affine Hecke Algebras of Cherednik, the HOMFLY polynomials, and the truly fascinating Shuffle Algebra of symmetric functions. This has brought to the fore a variety of symmetric function operators with close connection to the extended notion of \define{rational parking functions}. The present work results from an ongoing effort to express and deal with these new developments in a
language that is more accessible to the algebraic
combinatorial audience.
This area of investigation involves many aspects of symmetric function theory, including a central role played by Macdonald polynomials, as well as some of their closely related symmetric function operators.
One of the alluring characteristic features of these operators is that they appear to control in a rather surprising manner combinatorial properties of
rational parking functions. A close investigation of these connections led us to a variety of new discoveries and conjectures in this area which in turn
should open up a variety of open problems in
Algebraic Combinatorics as well as in the above
mentioned areas.
\section{The previous shuffle conjectures}
We begin by reviewing the statement of the Shuffle Conjecture of Haglund {\em et al.} (see \cite{Shuffle}). In Figure \ref{fig:Park2} we have an example of two convenient ways to represent a parking function: a two-line array and a tableau.
\begin{figure}
\caption{Two representations of a parking function}
\label{fig:Park2}
\end{figure}
The tableau on the right is constructed by first choosing a \hbox{Dyck path.} Recall that this is a path in the $n\times n$ lattice square that goes from $(0,0)$ to $(n,n)$ by \define{north} and \define{east} steps, always remaining weakly above the main diagonal (the shaded cells). The lattice cells adjacent and to the east of north steps are filled with {\define{cars} $ 1,2,\dots,n $} in a column-increasing manner. The numbers on the top of the two-line array are the cars as we read them by rows, from bottom to top. The numbers on the bottom of the two line array are the \define{area} numbers, which are obtained by successively counting the number of lattice cells between a \define{north} step and the main diagonal.
All the necessary statistics of a parking function $\operatorname{comp}ark$ can be immediately obtained from the corresponding two line array
$$
\operatorname{comp}ark:= \Big[\begin{matrix}
v_1 & v_2 & \cdots & v_n\\[-4pt]
u_1 & u_2 & \cdots & u_n\\
\end{matrix}\Big].$$
To begin we let
\begin{eqnarray*}
\operatorname{area}(\operatorname{comp}ark)&:=&\sum_{i=1}^n u_i,
\qquad {\rm and}\\
\operatorname{dinv}(\operatorname{comp}ark)&:=&
\sum_{1 \leq i<j \leq n } \raise 2pt\hbox{\large$\chi$}\big(\,( u_i=u_j\ \&\ v_i<v_j)\quad {\rm or}\quad
(u_i=u_j+1\ \&\ v_i>v_j)\,\big),
\end{eqnarray*}
where $\raise 2pt\hbox{\large$\chi$}(-)$ denotes the function that takes value $1$ if its argument is true, and $0$ otherwise.
Next we define $\sigma(\operatorname{comp}ark)$ to be the permutation obtained by successive right to left readings of the components of the vector $(v_1 , v_2 , \dots , v_n)$ according to decreasing values of $u_1 , u_2 , \dots, u_n$. Alternatively, $\sigma(\operatorname{comp}ark)$ is also obtained by reading the cars, in the tableau, from right to left by diagonals and from the highest diagonal to the lowest. Finally, we denote by $\operatorname{ides}(\operatorname{comp}ark)$ the descent set of the inverse of $\sigma(\operatorname{comp}ark)$.
This given, in \cite{Shuffle} Haglund {\em et al.} stated the following.
\begin{conj}[HHLRU-2005] \label{conjHHLRU}
For all $n \geq 1$,
\begin{equation} \label{HHLRU}
\nabla e_n(\mathbf{x}) \,=\, \sum_{\operatorname{comp}ark \in\mathrm{Park}_n } t^{\operatorname{area}(\operatorname{comp}ark) }q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}]
\end{equation}
\end{conj}
Here, $\mathrm{Park}_n$ stands for the set of all parking functions in the $n\times n$ lattice square. Moreover, for a subset $S \subseteq \{1,2,\cdots, ,n-1\}$, we denote by $F_S[\mathbf{x}]$ the corresponding Gessel fundamental quasi-symmetric function homogeneous of degree $n$ (see \cite{Gessel}). Finally, $\nabla$ is the symmetric function\footnote{See Section~\ref{secsymm} for a quick review of the usual tools for calculations with symmetric functions, and operators on them.} operator introduced in \cite{SciFi}, with eigenfunctions the modified Macdonald polynomial basis $\{\widetilde{H}_\mu[\mathbf{x};q,t] \}_{\mu}$, indexed by partitions $\mu$.
Let us now recall that a result of Gessel implies
that
a homogeneous symmetric function $f[\mathbf{x}]$ of degree $n$ has an expansion of the form
\begin{displaymath}
f[\mathbf{x}] \,=\, \sum_{\sigma \in S_n} c_\sigma F_{\operatorname{ides}(\sigma)}[\mathbf{x}],
\qquad\qquad ( \, \operatorname{ides}(\sigma) = \operatorname{des}(\sigma^{-1}) \, )
\end{displaymath}
if and only, if for all partitions $\mu =(\mu_1,\mu_2,\dots \mu_k)$ of $n$, we have
\begin{displaymath}
\left\langle f \,,\, h_\mu \right\rangle
\,=\, \sum_{\sigma \in S_n} c_\sigma \, \raise 2pt\hbox{\large$\chi$} \big(\sigma \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big),
\end{displaymath}
where $\langle- \,,\,- \rangle$ denotes the Hall scalar product of symmetric functions, $E_1, E_2, \dots, E_k$ are successive segments of the word $123\cdots n$ of respective lengths $\mu_1,\mu_2,\dots ,\mu_k$,
and the symbol $E_1\shuffle E_2 \shuffle \cdots \shuffle E_k$ denotes the collection of permutations obtained by shuffling in all possible ways the words $E_1,E_2,\dots ,E_k$.
Thus \operatorname{comp}ref{HHLRU} may be restated as
\begin{equation} \label{I.2}
\left\langle \nabla e_n \,,\, h_\mu \right\rangle
\, = \sum_{\operatorname{comp}ark \in\mathrm{Park}_n} t^{\operatorname{area} (\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \big( \sigma(\operatorname{comp}ark) \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big)
\end{equation}
for all $ \mu \vdash n$, which is the original form of the Shuffle Conjecture. Recall that it is customary to write $\mu\vdash n$, when $\mu$ is a partition of $n$.
For about 5 years from its formulation, the Shuffle Conjecture appeared untouchable for lack of any recursion satisfied by both sides of the equality. However, in the fall of 2008, Haglund, Morse, and Zabrocki \cite{classComp} made a discovery that is nothing short of spectacular. They discovered that two slight deformations $\BC_a$ and $\BB_a$ of the well-known Hall-Littlewood operators, combined with $\nabla$, yield considerably finer versions of the Shuffle conjecture. For any $a\in\mathbb{N}$, they define the operators $\BC_a$ and $\BB_a$, acting on symmetric polynomials $f[\mathbf{x}]$, as follows.
\begin{eqnarray}
\BC_a f[\mathbf{x}] &=& (-q)^{1-a} f\!\left[\mathbf{x}-{(q-1)}/{(qz)}\right]\, \sum_{m \geq 0} z^m h_m[\mathbf{x}]\, \Big|_{z^a}, \qquad {\rm and} \label{defC}\\
\BB_b f[\mathbf{x}] &=& f\!\left[ \mathbf{x} + \epsilon\,(1-q)/z\right]\, \sum_{m \geq 0} z^m e_m[\mathbf{x}]\, \Big|_{z^b},
\end{eqnarray}
where $(-)\big|_{z^a}$ means that we take the coefficient of $z^a$ in the series considered. We use here ``plethystic'' notation which is described in more details in Section~\ref{secsymm}.
Haglund, Morse, and Zabrocki also introduce a new statistic on paths (or parking functions), the \define{return composition}
$$\operatorname{comp}(\operatorname{comp}ark)=(a_1,a_2,\dots, a_\ell),$$
whose parts are the sizes of the intervals between successive diagonal hits of the Dyck path of $\operatorname{comp}ark$, reading from left to right. As usual we write $\alpha\models n$, when $\alpha$ is a composition of $n$, {\em i.e.} $n=a_1+\ldots+a_\ell$ with the $a_i$ positive integers, and set $\BC_\alpha$ for the product $\BC_{a_1} \BC_{a_2} \cdots \BC_{a_\ell}$, with a similar convention for $\BB_\alpha$.
This given, their discoveries led them to state the following two conjectures\footnote{To make it clear that we are applying an operator to a constant function such as ``$\mathbf{1}$'', we add a dot between this operator and its argument.}.
\begin{conj} [HMZ-2008] \label{conjHMZC}
For any composition $\alpha$ of $n$,
$$
\nabla \BC_\alpha \cdot\mathbf{1} \,=\sum_{\operatorname{comp}(\operatorname{comp}ark)=\alpha} t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}(\operatorname{comp}ark)}F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
$$
where the sum is over parking functions in the $n\times n$ lattice square with composition \hbox{equal to $\alpha$.}
\end{conj}
\begin{conj} [HMZ-2008]\label{conjHMZB}
For any composition $\alpha$ of $n$,
$$
\nabla \BB_\alpha \cdot\mathbf{1} \,=\sum_{\operatorname{comp}(\operatorname{comp}ark)\operatorname{comp}receq \alpha}t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}_\alpha(\operatorname{comp}ark)}F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
$$
where ``$\operatorname{dinv}_\alpha$'' is a suitably $\alpha$-modified $\operatorname{dinv}$ statistic, and the sum is over parking functions in the $n\times n$ lattice square with composition finer than $\alpha$.
\end{conj}
We first discuss Conjecture \ref{conjHMZC}, referred to as the Compositional Shuffle Conjecture, and we will later come back to Conjecture 1.3. Yet another use of Gessel's Theorem shows that Conjecture \ref{conjHMZC} is equivalent to the family of identities
\begin{equation} \label{I.6}
\left\langle \nabla \BC_\alpha \cdot\mathbf{1}, h_\mu\right\rangle =
\sum_{\operatorname{comp}(\operatorname{comp}ark)=\alpha}
t^{\operatorname{area}(\operatorname{comp}ark)}q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \big( \sigma(\operatorname{comp}ark) \in E_1\shuffle E_2 \shuffle \cdots \shuffle E_k \big),
\end{equation}
where, as before, the parts of $\mu$ correspond to the cardinalities of the $E_i$.
The fact that Conjecture \ref{conjHMZC} refines the Shuffle Conjecture is due to the identity
\begin{equation} \label{I.7}
\sum_{\alpha \models n} \BC_\alpha \cdot\mathbf{1} \,=\, e_n,
\end{equation}
hence summing \operatorname{comp}ref{I.6} over all compositions $\alpha\models n$ we obtain \operatorname{comp}ref{I.2}.
Our main contribution here is to show that a suitable extension of the Gorsky-Negut Conjectures (NG Conjectures) to the non-coprime case leads to the formulation of an infinite variety of new Compositional Shuffle conjectures, widely extending both the NG and the HMZ Conjectures. To state them we need to briefly review the Gorsky-Negut Conjectures in a manner that most closely resembles the classical Shuffle conjecture.
\section{The coprime case} \label{sec:coprime}
Our main actors on the symmetric function side are the operators $\D_k$ and $\D_k^*$, introduced in \cite{qtCatPos}, whose action on a symmetric function $f[\mathbf{x}]$ are defined by setting respectively
\begin{eqnarray}
\D_k f[\mathbf{x}] &:=& f\!\left[\mathbf{x}+{M}/{z}\right] \sum_{i \geq 0}(-z)^i e_i[\mathbf{x}] \Big|_{z^k},\qquad {\rm and} \label{defD}\\
\D_k^*f[\mathbf{x}]&:=& f\!\left[\mathbf{x}-{\widetilde{M}}/{z}\right] \sum_{i \geq 0}z^i h_i[\mathbf{x}] \Big|_{z^k}.\label{defDetoile}
\end{eqnarray}
with $M:=(1-t)(1-q)$ and $\widetilde{M}:=(1-1/t)(1-1/q)$.
The focus of the present work is the algebra of symmetric function operators generated by the family $\{ \D_k \}_{k \geq 0}$. Its connection to the algebraic geometrical developments is that this algebra is a concrete realization of a portion of the Elliptic Hall Algebra studied Schiffmann and Vasserot in \cite{elliptic2}, \cite{SchiffVassMac}, and \cite{SchiffVassK}. Our conjectures are expressed in terms of a family of operators $\Qop_{a,b}$ indexed by pairs of positive integers $a,b$. Here, and in the following, we use the notation $\Qop_{km,kn}$, with $(m,n)$ a coprime pair of non-negative integers and $k$ an arbitrary positive integer. In other words, $k$ is the greatest common divisor of $a$ and $b$, and $(a,b)=(km,kn)$.
\begin{wrapfigure}[8]{R}{2.8cm} \centering
\vskip-5pt
\operatorname{des}sin{height=1.5 in}{dia35.pdf}
\end{wrapfigure}
Restricted to the coprime case, the definition of the operators $\Qop_{m,n}$ is first illustrated in a special case. For instance, to obtain $\Qop_{3,5}$ we start by drawing the $3\times 5$ lattice with its diagonal (the line $(0,0) \to (3,5)$) as depictedf in the adjacent figure. Then we look for the lattice point $(a,b)$ that is closest to and below the diagonal. In this case $(a,b)=(2,3)$. This yields the decomposition $(3,5)=(2,3)+(1,2)$, and unfolding the recursivity we get
\begin{eqnarray}
\Qop_{3,5} &=& \frac{1}{M} \left[ \Qop_{1,2} , \Qop_{2,3} \right]\nonumber\\
&=& \frac{1}{M} \left(
\Qop_{1,2} \Qop_{2,3} - \Qop_{2,3} \Qop_{1,2} \right).\displaywidth=\parshapelength\numexpr\prevgraf+2\relax \label{1.2}
\end{eqnarray}
\begin{wrapfigure}[5]{R}{2.8cm} \centering
\vskip-15pt
\operatorname{des}sin{height=1 in}{dia23.pdf}
\end{wrapfigure}
We must next work precisely in the same way with the $2\times 3$ rectangle, as indicated in the adjacent figure. We obtain the decomposition $(2,3)=(1,1)+(1,2)$ and recursively set
\begin{equation} \displaywidth=\parshapelength\numexpr\prevgraf+2\relax \label{1.3}
\Qop_{2,3} = \frac{1}{M} \left[ \Qop_{1,2} , \Qop_{1,1} \right]
= \frac{1}{M} \left(
\Qop_{1,2} \Qop_{1,1} - \Qop_{1,1} \Qop_{1,2} \right).
\end{equation}
Now, in this case, we are done, since it turns out that we can set
\begin{equation} \label{1.4}
\Qop_{1,k} = \D_k.
\end{equation}
In particular by combining \operatorname{comp}ref{1.2}, \operatorname{comp}ref{1.3} and \operatorname{comp}ref{1.4} we obtain
\begin{equation} \label{1.5}
\Qop_{3,5}
= \frac{1}{M^2}
\left( \D_2 \D_2 \D_1 - 2 \D_2 \D_1 \D_2 + \D_1 \D_2 \D_2 \right).
\end{equation}
To give a precise general definition of the $\Qop$ operators we use the following elementary number theoretical characterization of the closest lattice point $(a,b)$ below the line $(0,0) \to (m,n)$. We observe that by construction $(a,b)$ is coprime. See \cite{newPleth} for a proof.
\begin{prop} \label{prop3.3}
For any pair of coprime integers $m,n> 1$ there is a unique pair $a,b$ satisfying the following three conditions
\begin{equation} \label{3.18}
(1) \quad 1\leq a\leq m-1, \qquad\qquad
(2) \quad 1\leq b\leq n-1, \qquad\qquad
(3) \quad mb+1=na
\end{equation}
In particular, setting $(c,d):=(m,n)-(a,b)$ we will write, for $m,n>1$,
\begin{equation} \label{3.19}
\spl(m,n):= (a,b)+(c,d).
\end{equation}
Otherwise, we set
\begin{equation} \label{3.20}
a) \quad \spl(1,n):=(1,n-1)+(0,1),
\qquad
b) \quad \spl(m,1):=(1,0)+(m-1,1).
\end{equation}
\end{prop}
All pairs considered being coprime, we are now in a position to give the definition of the operators $\Qop_{m,n}$ (restricted for the moment to the coprime case) that is most suitable in the present writing.
\begin{defn} \label{defnQ}
For any coprime pair $(m,n)$, we set
\begin{equation} \label{eqdefnQ}
\Qop_{m,n} :=
\begin{cases}
\frac{1}{M}[\Qop_{c,d},\Qop_{a,b}] & \hbox{if }m>1 \hbox{ and } \spl(m,n)=(a,b)+(c,d), \\[6pt]
\D_n & \hbox{if }m=1.
\end{cases}
\end{equation}
\end{defn}
The combinatorial side of the upcoming conjecture is constructed in \cite{Hikita} by Hikita as the Frobenius characteristic of a bi-graded $S_n$ module whose precise definition is not needed in this development. For our purposes it is sufficient to directly define the \define{Hikita polynomial}, which we denote by $H_{m,n}[\mathbf{x};q,t]$, using a process that closely follows our present rendition of the right hand side of \operatorname{comp}ref{HHLRU}. That is, we set
\begin{equation} \label{defHikita}
H_{m,n}[\mathbf{x};q,t] :=
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{m,n} } t^{\operatorname{area}(\operatorname{comp}ark) } q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}[\mathbf{x}],
\end{equation}
with suitable definitions for all the ingredients occurring in this formula. We will start
with the collection of $(m,n)$-\define{parking functions} which we have denoted $\mathrm{Park}_{m,n}$. Again, a simple example will suffice.
\begin{figure}
\caption{First combinatorial ingredients for the Hikita polynomial.}
\label{fig:1to3}
\end{figure}
Figures \ref{fig:1to3} and \ref{fig:4to5} contain all the information needed to construct the polynomial $H_{7,9}[\mathbf{x};q,t]$. The first object in Figure~\ref{fig:1to3} is a $5 \times 7$ lattice rectangle with its main diagonal $(0,0) \to (5,7)$. In a darker color we have the lattice cells cut by the main diagonal, which we will call the \define{lattice diagonal}. Because of the coprimality of $(m,n)$, the main diagonal, and any line parallel to it, can touch at most a single lattice point inside the $m \times n$ lattice. Thus the main diagonal (except for its end points) remains interior to the lattice cells that it touches. Since the path joining the centers of the touched cells has $n-1$ north steps and $m-1$ east steps, it follows that the lattice diagonal has $m+n-1$ cells. This gives that the number of cells above (or below) the lattice diagonal is $(m-1)(n-1)/2$.
A path in the $m \times n$ lattice that proceeds by north and east steps from $(0,0)$ to $(m,n)$, always remaining weakly above the lattice diagonal, is said to be an $(m,n)$-Dyck path. For example, the second object in Figure \ref{fig:1to3} is a $(5,7)$-Dyck path. The number of cells between a path $\operatorname{comp}ath$ and the lattice diagonal is denoted $\operatorname{area}(\operatorname{comp}ath)$. In the third object of Figure \ref{fig:1to3}, we have an $(11,10)$-Dyck path. Notice that the collection of cells above the path may be viewed as an english Ferrers diagram. We also show there the \define{leg} and the \define{arm} of one of its cells (see Section~\ref{secsymm} for more details). Denoting by $\lambda(\operatorname{comp}ath)$ the Ferrers diagram above the path $\operatorname{comp}ath$, we define
\begin{equation} \label{1.8}
\operatorname{dinv}(\operatorname{comp}ath) := \sum_{c \in \lambda(\operatorname{comp}ath)} \raise 2pt\hbox{\large$\chi$}\!\left( \frac{\operatorname{arm}(c)}{\operatorname{leg}(c)+1} < \frac{m}{n} < \frac{\operatorname{arm}(c)+1}{\operatorname{leg}(c)}\right).
\end{equation}
As in the classical case an $(m,n)$-parking function is the tableau obtained by labeling the cells east of and adjacent to the north steps of an $(m,n)$-Dyck path with cars $1,2,\dots,n$ in a column-increasing manner. We denote by $\mathrm{Park}_{m,n}$ the set of $(m,n)$-parking functions. When $(m,n)$ is a pair of coprime integers, it is easy to show that there are $m^{n-1}$ such parking functions. For more on the coprime case, see \cite{armstrong}. We will discuss further aspects of the more general case in \cite{newPleth}, a paper in preparation.
\begin{figure}
\caption{Last combinatorial ingredients for the Hikita polynomial.}
\label{fig:4to5}
\end{figure}
The first object in Figure \ref{fig:4to5} gives a $(7,9)$-parking function and the second object gives a $7 \times 9$ table of \define{ranks}. In the general case, this table is obtained by placing in the \define{north-west} corner of the $m\times n$ lattice a number of one's choice. Here we have used $47=(m-1)(n-1)-1$, but the choice is immaterial. We then fill the cells by subtracting $n$ for each east step and adding $m$ for each north step. Denoting by $\operatorname{rank}(i)$ the \define{rank} of the cell that contains car $i$, we define the \define{temporary dinv} of an $(m,n)$-Parking function $\operatorname{comp}ark$ to be the statistic
\begin{equation} \label{1.9}
\operatorname{tdinv}(\operatorname{comp}ark) := \sum_{1\leq i<j \leq n} \raise 2pt\hbox{\large$\chi$}\!\left( \operatorname{rank}(i)< \operatorname{rank}(j) < \operatorname{rank}(i)+m \right).
\end{equation}
Next let us set for any $(m,n)$-path $\operatorname{comp}ath$
\begin{equation} \label{1.10}
\operatorname{maxtdinv}(\operatorname{comp}ath) := \max \{ \operatorname{tdinv}(\operatorname{comp}ark) : \operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath \},
\end{equation}
where the symbol $\operatorname{comp}ath(\operatorname{comp}ark)=\operatorname{comp}ath$ simply means that $\operatorname{comp}ark$ is obtained by labeling the path $\operatorname{comp}ath$. It will also be convenient to refer to $\operatorname{comp}ath(\operatorname{comp}ark)$ as the \define{support} of $\operatorname{comp}ark$. This given, we can now set
\begin{equation} \label{1.11}
\operatorname{dinv}(\operatorname{comp}ark) := \operatorname{dinv}(\operatorname{comp}ath(\operatorname{comp}ark)) + \operatorname{tdinv}(\operatorname{comp}ark) - \operatorname{maxtdinv}(\operatorname{comp}ath(\operatorname{comp}ark)).
\end{equation}
This is a reformulation of Hikita's definition of the \define{dinv} of an $(m,n)$-parking function first introduced by Gorsky and Mazin in \cite{GorskyMazin}.
To complete the construction of the Hikita polynomials we need the notion of the \define{word} of a parking function, which we denote by $\sigma(\operatorname{comp}ark)$. This is the permutation obtained by reading the cars of $\operatorname{comp}ark$ by decreasing ranks. Geometrically, $\sigma(\operatorname{comp}ark)$ can be obtained simply by having a line parallel to the main diagonal sweep the cars from left to right, reading a car the moment the moving line passes through the south end of its adjacent north step. For instance, for the parking function in Figure \ref{fig:4to5} we have $\sigma(\operatorname{comp}ark)= 784615923.$
Letting $\operatorname{ides}(\operatorname{comp}ark)$ denote the \define{descent set of the inverse} of the permutation $\sigma(\operatorname{comp}ark)$ and setting, as in the classical case, $\operatorname{area}(\operatorname{comp}ark) = \operatorname{area}(\operatorname{comp}ath(\operatorname{comp}ark))$, we finally have all of the ingredients necessary for \operatorname{comp}ref{defHikita} to be a complete definition of the Hikita polynomial. The Gorsky-Negut $(m,n)$-Shuffle Conjecture may now be stated as follows.
\begin{conj}[GN-2013] \label{conjNG}
For all coprime pairs of positive integers $(m,n)$, we have
\begin{equation} \label{1.12}
\Qop_{m,n} \cdot\mathbf{1}n = H_{m,n}[\mathbf{x};q,t].
\end{equation}
\end{conj}
Of course we can use the word ``Shuffle'' again since another use of Gessel's theorem allows us to rewrite \operatorname{comp}ref{1.12} in the equivalent form
$$
\left\langle \Qop_{m,n}\cdot\mathbf{1}n , h_\alpha \right\rangle = \hskip -10pt
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{m,n}} \hskip -10pt
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} \raise 2pt\hbox{\large$\chi$} \left( \sigma(\operatorname{comp}ark) \in E_1\shuffle\cdots \shuffle E_k \right),\quad\hbox{for all}\quad \alpha\models n,
$$
where $a_i=|E_i|$ when $\alpha=(a_1,\ldots,a_k)$, and writing $h_\alpha$ for the product $h_{a_1}\cdots h_{a_k}$.
We must point out that it can be shown that \operatorname{comp}ref{1.12} reduces to \operatorname{comp}ref{HHLRU} when $m=n+1$. In fact, it easily follows from the definition in \operatorname{comp}ref{defnQ} that $\Qop_{n+1,n} = \nabla \D_n \nabla^{-1}.$ This, together with the fact that $\nabla^{-1}\cdot\mathbf{1} = 1$ and the definition in \operatorname{comp}ref{defD}, yields $\Qop_{n+1,n} \cdot\mathbf{1}n = \nabla e_n.$ The equality of the right hand sides of \operatorname{comp}ref{1.12} and \operatorname{comp}ref{HHLRU} for $m=n+1$ is obtained by a combinatorial argument which is not too difficult.
\operatorname{comp}agebreak
\section{Our Compositional \texorpdfstring{$(km,kn)$}--Shuffle Conjectures}
The present developments result from theoretical and computer explorations of what takes place in the non-coprime case. Notice first that there is no difficulty in extending the definition of parking functions to the
$km \times kn$ lattice square, including the $\operatorname{area}$ statistic. Problems arise in extending the definition of the $\operatorname{dinv}$ and $\sigma$ statistics. Previous experience strongly suggested to use the symmetric function side as a guide to the construction of these two statistics. We will soon see that we may remove the coprimality condition in the definition of the $\Qop$ operators, thus allowing us to consider operators $\Qop_{km,kn}$ which, for $k>1$, may be simply obtained by bracketing two $\Qop$ operators indexed by coprime pairs.
However one quickly discovers, by a simple parking function count, that these $\Qop_{km,kn}$ operators do not provide the desired symmetric function side.
Our search for the natural extension
of the symmetric function side led us to focus on the following general construction of symmetric function operators indexed by non-coprime pairs $(km,kn)$.
This construction is based
on a simple commutator identity satisfied by the operators
$D_k$ and $D_k^*$ which shows that the $Q_{0,k}$ operator
is none other than multiplication by a rescaled plethystic version of the ordinary symmetric function $h_k$.
This implies that
the family
$\{\operatorname{comp}rod_i \Qop_{0,\lambda_i} \}_\lambda$
is a basis for the space of symmetric functions (viewed as multiplication operators).
Our definition also uses a commutativity property
(proved in [4]) between $\Qop$ operators indexed by collinear vectors, {\em i.e.} $\Qop_{km,kn}$ and $\Qop_{jm,jn}$ commute for all $k$, $j$, $m$ and $n$ (see Theorem~\ref{thmQind}).
For our purpose, it is convenient to denote by $\cdot\mathbf{1}derline{f}$ the operator of \define{multiplication by} $f$ for any symmetric function $f$.
We can now give our general construction.
\begin{alg} \label{algF}
Given any symmetric function $f$ that is homogeneous of degree $k$, and any coprime pair $(m,n)$, we proceed as follows
\begin{enumerate}\itemsep=-4pt
\item[]{\bf Step 1:} calculate the expansion
\begin{equation}
f = \sum_{\lambda \vdash k} c_\lambda(q,t)\, \operatorname{comp}rod_{i=1}^{\ell(\lambda)} \Qop_{0,\lambda_i},
\end{equation}
\item[]{\bf Step 2:} using the coefficients $c_\lambda(q,t)$, set
\begin{equation} \label{defnF}
\Bf_{km,kn} := \sum_{\lambda \vdash k} c_\lambda(q,t) \operatorname{comp}rod_{i=1}^{\ell(\lambda)}\Qop_{m \lambda_i, n\lambda_i}.
\end{equation}
\end{enumerate}
\end{alg}
Theoretical considerations reveal, and extensive computer experimentations confirm, that the operators that we should use to extend the \define{rational parking function} theory to all pairs $(km,kn)$, are none other than the operators $\Be_{km,kn}$ obtained by taking $f=e_k$ in Algorithm~\ref{algF}. This led us to look for the construction of natural extensions of the definitions of $\operatorname{dinv}(\operatorname{comp}ark)$ and $\sigma(\operatorname{comp}ark)$, that would ensure the validity of the following sequence of increasingly refined conjectures. The coarsest one of which is as follows.
\begin{conj}\label{conjE} For all coprime pair of positive integers $(m,n)$, and any $k\in\mathbb{N}$, we have
\begin{equation} \label{2.3}
\Be_{km,kn}\cdot {(-\mathbf{1})^{k(n+1)}}= \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
\end{conj}
\begin{wrapfigure}[12]{r}{4cm} \centering
\operatorname{des}sin{height=2.3 in}{multimn.pdf}
\end{wrapfigure}
To understand our first refinement, we focus on a special case. In the figure displayed on the right, we have depicted a $12 \times 20$ lattice. The pair in this case has a $\gcd$ of $4$. Thus here $(m,n)=(3,5)$ and $k=4$. Note that in the general case, $(km,kn)$-Dyck paths can hit the diagonal in $k-1$ places within the $km\times kn$ lattice square. In this case, in $3$ places. We have depicted here a Dyck path which hits the diagonal in the first and third places.
At this point the classical decomposition (discovered in \cite{qtCatPos})
\begin{equation}\label{edecompE}\displaywidth=\parshapelength\numexpr\prevgraf+2\relax
e_k= E_{1,k}+E_{2,k}+\cdots +E_{k,k},
\end{equation}
combined with extensive computer experimentations, suggested that we have the following refinement of Conjecture~\ref{conjE}.
\begin{conj}\label{conjEr} For all coprime pair of positive integers $(m,n)$, all $k\in\mathbb{N}$, and if $1\leq r\leq k$, we have
\begin{equation} \label{2.4}
\BE_{km,kn}^{r}\cdot (-\mathbf{1})^{k(n+1)} = \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{r}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
where $\BE_{km,kn}^{r}$ is the operator obtained by setting $f=E_{k,r}$ in Algorithm~\ref{algF}.
Here $\mathrm{Park}_{km,kn}^{r}$ denotes the set of parking functions, in the $km \times kn$ lattice, whose Dyck path hit the diagonal in $r$ places {\rm (}including $(0,0)${\rm )}.
\end{conj}
Clearly, \operatorname{comp}ref{edecompE} implies that Conjecture~\ref{conjE} follows from Conjecture~\ref{conjEr}. For example, the parking functions supported by the path in the above figure would be picked up by the operator $\BE_{4\times 3,4\times 5 }^{3}$.
Our ultimate refinement is suggested by the decomposition (proved in \cite{classComp})
\begin{equation} \label{2.5}
E_{k,r} = \sum_{\alpha \models k}
C_{\alpha_1}C_{\alpha_2}\cdots C_{\alpha_r}
\cdot\mathbf{1}.
\end{equation}
What emerges is the following most general conjecture that clearly subsumes our two previous conjectures, as well as Conjectures~\ref{conjHHLRU}, \ref{conjHMZC}, and \ref{conjNG}.
\begin{conj}[Compositional $(km,kn)$-Shuffle Conjecture] \label{conjBGLX}
For all compositions $\alpha=(a_1,a_2, \dots ,a_r)\models k$ we have
\begin{equation} \label{2.6}
\BC_{km,kn}^{(\alpha)}\cdot(-\mathbf{1})^{k(n+1)}
= \sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{(\alpha)}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)},
\end{equation}
where $\BC_{km,kn}^{(\alpha)}$ is the operator obtained by setting $f=\BC_\alpha\cdot\mathbf{1}$ in Algorithm~\ref{algF} and $\mathrm{Park}_{km,kn}^{(\alpha)}$ denotes the collection of parking functions in the $km\times kn$ lattice whose Dyck path hits the diagonal according to the composition $\alpha$.
\end{conj}
For example, the parking functions supported by the path in the above figure would be picked up by the operator $\BC_{4\times 3 ,4\times 5}^{(1,2,1)}$.
We will later see that an analogous conjecture may be stated for the operator $\BB_{km,kn}^{(\alpha)}$ obtained by taking $f=\BB_\alpha\cdot\mathbf{1}$ in our general Algorithm~\ref{algF}, with $\alpha$ any composition of $k$.
We will make extensive use in the sequel of a collection of results stated and perhaps even proved in the works of Schiffmann and Vasserot.
Unfortunately most of this material is written in a language that is nearly inaccessible to most practitioners of Algebraic Combinatorics. We were fortunate that the two young researchers E. Gorsky and A. Negut, in a period of several months, made us aware of some of the contents of the latter publications as well as the results in their papers (\cite{GorskyNegut}, \cite{NegutFlags} and \cite{NegutShuffle}) in a language we could understand. The present developments are based on these results. Nevertheless, for sake of completeness we have put together in \cite{newPleth} a purely Algebraic Combinatorial treatment of all the background needed here with proofs that use only the Macdonald polynomial ``tool kit'' derived in the 90's in \cite{SciFi}, \cite{IdPosCon}, \cite{plethMac} and\cite{explicit}, with some additional identities discovered in \cite{HLOpsPF}.
The remainder of this paper is divided into three further sections. In the next section we review some notation and recall some identities from Symmetric Function Theory, and our Macdonald polynomial tool kit.
This done, we state some basic identities that will be instrumental in extending the definition of the $\Qop$ operators to the non-coprime case.
In the following section we describe how the modular group $\mathrm{SL}_2(\mathbb{Z})$ acts on the operators $\Qop_{m,n}$ and use this action to justify our definition of the operators $\Qop_{km,kn}$. Elementary proofs that justify the uses we make of this action are given in \cite{newPleth}. Here we also show how these operators can be efficiently programmed on the computer. This done, we give a precise construction of the operators $\BC_{km,kn}^{(\alpha)}$ and $\BB_{km,kn}^{(\alpha)}$, and workout some examples. We also give a compelling argument which shows the inevitability of Conjecture \ref{conjBGLX}.
In the last section we complete our definitions for all the combinatorial ingredients occurring in the right hand sides of \operatorname{comp}ref{2.3}, \operatorname{comp}ref{2.4} and \operatorname{comp}ref{2.6}. Finally, we derive some consequences of our conjectures and discuss some possible further extensions.
\section{Symmetric function basics, and necessary operators}\label{secsymm}
In dealing with symmetric function identities, especially those arising in the theory of Macdonald Polynomials, it is convenient and often indispensable to use plethystic notation. This device has a straightforward definition which can be implemented almost verbatim in any computer algebra software. We simply set for any expression $E = E(t_1,t_2 ,\dots )$ and any symmetric function $f$
\begin{equation} \label{3.1}
f[E] := \Qop_f(p_1,p_2, \dots )
\Big|_{p_k \to E( t_1^k,t_2^k,\dots )},
\end{equation}
where $(-)\big|_{p_k \to E( t_1^k,t_2^k,\dots )}$ means that we replace each $p_k$ by $E( t_1^k,t_2^k,\dots )$, for $k\geq 1$.
Here $\Qop_f$ stands for the polynomial yielding the expansion of $f$ in terms of the power basis. We say that we have a \define{plethystic substitution} of $E$ in $f$.
The above definition of plethystic substitutions implicitly requires that
$p_k[-E]= -p_k[E]$, and we say that this is the \define{plethystic} minus sign rule. This notwithstanding, we also need to carry out ``ordinary'' changes of signs. To distinguish the later from the plethystic minus sign, we obtain the \define{ordinary} sign change by multiplying our expressions by a new variable ``$\epsilon$'' which, outside of the plethystic bracket, is replaced by $-1$. Thus we have
\begin{displaymath}
p_k[\epsilon E]= \epsilon^k p_k[ E]= (-1)^k p_k[E].
\end{displaymath}
In particular we see that, with this notation, for any expression $E$ and any symmetric function $f$ we have
\begin{equation} \label{3.2}
(\omega f)[E]= f[-\epsilon E],
\end{equation}
where, as customary, ``$\omega$'' denotes the involution that interchanges the elementary and homogeneous symmetric function bases.
Many symmetric function identities can be considerably simplified by means of the $\Omega$-notation, allied with plethystic calculations. For any expression $E = E(t_1,t_2,\cdots )$ set
\begin{displaymath}
\Omega[E] := \exp\! \left(
\sum_{k \geq 1}{p_k[E]\over k}
\right) = \exp \! \left(
\sum_{k \geq 1} \frac{E(t_1^k,t_2^k,\cdots )}{k}
\right).
\end{displaymath}
In particular, for $\mathbf{x}=x_1+x_2+\cdots$, we see that
\begin{equation} \label{3.3}
\Omega[z\mathbf{x}]= \sum_{m \geq 0} z^m h_m[\mathbf{x}]
\end{equation}
and for $M=(1-t)(1-q)$ we have
\begin{equation} \label{3.4}
\Omega[-uM] = \frac{(1-u)(1-qtu)}{(1-tu)(1-qu)}.
\end{equation}
\begin{wrapfigure}[7]{r}{4.5cm} \centering
\vskip-10pt
\operatorname{des}sin{height=1.2 in}{ninesix.pdf}
\begin{picture}(0,0)(-3,0)\setlength{\cdot\mathbf{1}itlength}{3mm}
\operatorname{comp}ut(-2,7.8){$\scriptstyle\operatorname{arm}$}
\operatorname{comp}ut(-7,9){$\scriptstyle\operatorname{coarm}$}
\operatorname{comp}ut(-4.8,6.3){$\scriptstyle\operatorname{leg}$}
\operatorname{comp}ut(-3,10){$\scriptstyle\operatorname{coleg}$}
\end{picture}
\end{wrapfigure}
Drawing the cells of the Ferrers diagram of a partition $\mu$ as in \cite{Macdonald}, For a cell $c$ in $\mu$, (in symbols $c\in\mu$), we have parameters $\operatorname{leg}(c)$, and $\operatorname{arm}(c)$,
which respectively give the number of cells of $\mu$ strictly south of $c$, and strictly east of $c$.
Likewise we have parameters $\operatorname{coleg}(c)$, and $\operatorname{coarm}(c)$, which respectively give the number of cells of $\mu$ strictly north of $c$, and strictly west of $c$.
This is illustrated in the adjacent figure for the partition that sits above a path.
Denoting by $\mu'$ the conjugate of $\mu$, the basic ingredients we need to keep in mind here are
$$\begin{array}{lll}\displaystyle
\displaystyle n(\mu):= \sum_{k=1}^{\ell(\mu)} (k-1) \mu_k, \qquad
& \displaystyle T_\mu:= t^{n(\mu)}q^{n(\mu')},
\qquad M:=(1-t)(1-q),\\[8pt]
\displaystyle B_\mu(q,t):= \sum_{c \in \mu} t^{\operatorname{coleg}(c)} q^{\operatorname{coarm}(c)} ,
&\displaystyle \displaystyle\Pi_\mu(q,t):=\operatorname{comp}rod_{{c\in\mu\atop c\not=(0,0)}} (1-t^{\operatorname{coleg}(c)} q^{\operatorname{coarm}}),
\end{array}$$
and
$$w_\mu(q,t) := \operatorname{comp}rod_{c \in \mu} (q^{\operatorname{arm}(c)} - t^{\operatorname{leg}(c)+1})(t^{\operatorname{leg}(c)} - q^{\operatorname{arm}(c)+1})$$
Let us recall that the Hall scalar product is defined by setting
\begin{displaymath}
\left\langle p_\lambda, p_\mu \right\rangle\ := \
z_\mu \, \chi(\lambda=\mu),
\end{displaymath}
where $z_\mu$ gives the order of the stabilizer of a permutation with cycle structure $\mu$.
The Macdonald polynomials we work with here are the unique (\cite{natBigraded}) symmetric function basis $\{\widetilde{H}_\mu[\mathbf{x};q,t]\}_\mu$ which is upper-triangularly related (in dominance order) to the modified Schur basis $\{s_\lambda[\frac{\mathbf{x}}{t-1}] \}_\lambda$ and satisfies the orthogonality condition
\begin{equation} \label{3.5}
\left\langle \widetilde{H}_\lambda, \widetilde{H}_\mu \right\rangle_* =\ \raise 2pt\hbox{\large$\chi$}(\lambda=\mu)\, w_\mu(q,t),
\end{equation}
where $\left\langle-,- \right\rangle_*$ denotes a deformation of the Hall scalar product defined by setting
\begin{equation} \label{3.6}
\left\langle p_\lambda, p_\mu \right\rangle_*
:= (-1)^{|\mu|-\ell(\mu)} \operatorname{comp}rod_i (1-t^{\mu_i})(1-q^{\mu_i})
\, z_\mu \, \raise 2pt\hbox{\large$\chi$}(\lambda =\mu).
\end{equation}
We will use here the operator $\nabla$, introduced in \cite{SciFi}, obtained by setting
\begin{equation} \label{3.7}
\nabla \widetilde{H}_\mu[\mathbf{x};q,t]= T_\mu\, \widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
We also set, for any symmetric function $f[\mathbf{x}]$,
\begin{equation} \label{3.8}
\Delta_f \widetilde{H}_\mu[\mathbf{x};q,t]= f[B_\mu]\, \widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
These families of operators were intensively studied in the $90's$ (see \cite{IdPosCon} and \cite{explicit}) where they gave rise to a variety of conjectures, some of which are still open to this date. In particular it is shown in \cite{explicit} that the operators $\D_k$, $\D_k^*$, $\nabla$ and the modified Macdonald polynomials $\widetilde{H}_\mu[\mathbf{x};q,t]$ are related by the following identities.
\begin{equation} \label{formulaoper}
\begin{array}{clcl}
{\rm (i)} & \D_0 \widetilde{H}_\mu = -D_\mu(q,t)\, \widetilde{H}_\mu,
\qquad\qquad &
{\rm (i)}^* & \D_0^* \widetilde{H}_\mu = -D_\mu(1/q,1/t) \widetilde{H}_\mu,
\\[5pt]
{\rm (ii)} & \D_k \underline{e}_1 - \underline{e}_1 \D_k = M \D_{k+1} ,&
{\rm (ii)}^* & \D_k^* \underline{e}_1 - \underline{e}_1 \D_k^* = -\widetilde{M} \D_{k+1}^*,
\\[5pt]
{\rm (iii)} & \nabla \underline{e}_1 \nabla^{-1} = -\D_1, &
{\rm (iii )}^* & \nabla \D_1^* \nabla^{-1} = \underline{e}_1,
\\[5pt]
{\rm (iv)} & \nabla^{-1} \underline{e}_1^\operatorname{comp}erp \nabla = {\textstyle \frac{1}{M}} \D_{-1}, &
{\rm (iv )}^* & \nabla^{-1} \D_{-1}^* \nabla = -{\widetilde M}\, \underline{e}_1^\operatorname{comp}erp,
\end{array}
\end{equation}
with $\underline{e}_1^\operatorname{comp}erp$ denoting the Hall scalar product adjoint of multiplication by $e_1$, and
\begin{equation} \label{formDmu}
D_\mu(q,t)= MB_\mu(q,t)-1.
\end{equation}
We should mention that recursive applications of \operatorname{comp}ref{formulaoper} ${\rm (ii)}$ and ${\rm (ii)}^*$ give
\begin{eqnarray}
\D_{k} &=&
\frac{1}{M^k} \sum_{i=0}^k {k \choose r}(-1)^r \underline{e}_1^r \D_0 \underline{e}_1^{k-r},\qquad{\rm and}\\
\D_{k}^* &=&
\frac{1}{\widetilde{M}^k} \sum_{i=0}^k {k \choose r}(-1)^{k-r} \underline{e}_1^r \D_0^* \underline{e}_1^{k-r}.
\end{eqnarray}
For future use, it is convenient to set
\begin{eqnarray}
\Phi_k &:=& \nabla \D_k \nabla^{-1}
\qquad {\rm and}\label{3.12a} \\
\Psi_k &:=& -(qt)^{1-k} \nabla \D_k^* \nabla^{-1}.\label{3.12b}
\end{eqnarray}
The following identities are then immediate consequences of identities \operatorname{comp}ref{formulaoper}. See \cite{newPleth} for details.
\begin{prop} \label{propphipsi}
The operators $\Phi_k$ and $\Psi_k$ are uniquely determined by the recursions
\begin{equation} \label{3.13}
{\rm a)} \quad \Phi_{k+1} = \frac{1}{M}[ \D_1,\Phi_{k} ]
\qquad {\rm and} \qquad
{\rm b)} \quad \Psi_{k+1} = \frac{1}{M}[\Psi_{k},\D_1]
\end{equation}
with initial conditions
\begin{equation}
{\rm a)}\quad \Phi_1 = \frac{1}{M} [\D_1,\D_0]
\qquad {\rm and} \qquad
{\rm b)}\quad \Psi_1=-e_1.
\end{equation}
\end{prop}
Next, we must include the following fundamental identity, proved in \cite{newPleth}.
\begin{prop} \label{propDD}
For $a,b \in \mathbb{Z}$ with $n=a+b>0$ and any symmetric function $f[\mathbf{x}]$, we have
\begin{equation} \label{3.15}
\frac{1}{M} ( \D_a \D_b^* - \D_b^* \D_a) f[\mathbf{x}] = \frac{(qt)^b}{qt-1} h_{n} \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] f[\mathbf{x}].
\end{equation}
\end{prop}
As a corollary we obtain the following.
\begin{prop} \label{propcrochetphipsi}
The operators $\Phi_k$ and $\Psi_k$, defined in \operatorname{comp}ref{3.12a} and \operatorname{comp}ref{3.12b}, satisfy the following identity when $a,b$ are any positive integers with sum equal to $n$.
\begin{equation} \label{3.16}
\frac{1}{M} [\Psi_b, \Phi_a] = \frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The identity in \operatorname{comp}ref{3.15} essentially says that under the given hypotheses the operator $\frac{1}{M}(\D_b^* \D_a - \D_a \D_b^*)$ acts as multiplication by the symmetric function $\frac{(qt)^b}{qt-1} h_{n} \!\left[(1-qt)\mathbf{x}/(qt)\right]$. Thus, with our notational conventions, \operatorname{comp}ref{3.15} may be rewritten as
\begin{displaymath}
- \frac{(qt)^{1-b}}{M} \left( \D_b^* \D_a - \D_a \D_b^* \right) = \frac{qt}{qt-1}\ \underline{h}_n\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{displaymath}
Conjugating both sides by $\nabla$, and using
\operatorname{comp}ref{3.12a} and \operatorname{comp}ref{3.12b}, gives \operatorname{comp}ref{3.16}.
\end{proof}
In the sequel, we will need to keep in mind the following identity which expresses the action of a sequence of $\D_k$ operators on a symmetric function $f[\mathbf{x}]$.
\begin{prop} \label{prop3.2} For all composition $\alpha=(a_1,a_2,\ldots,a_m)$ we have
\begin{equation}
\D_{a_m} \cdots \D_{a_2} \D_{a_1} f[\mathbf{x}]
= f\!\left[\mathbf{x}+{\textstyle{\sum_{i=1}^m {M}/{z_i}}}\right] \,
\frac{\Omega[-\mathbf{z}X] }{\mathbf{z}^\alpha} \,
\operatorname{comp}rod_{1\leq i<j \leq m} \Omega \left[-M \textstyle{z_i/ z_j} \right] \Big|_{\mathbf{z}^0},
\end{equation}
where, for $\mathbf{z}=z_1+\ldots +z_m$, we write $\mathbf{z}^\alpha=z_1^{a_1}\cdots z_m^{a_m}$, and in particular $\mathbf{z}^0=z_1^0z_2^0\cdots z_m^0$.
\end{prop}
\begin{proof}[\bf Proof]
It suffices to see what happens when we use \operatorname{comp}ref{defD} twice.
\begin{eqnarray*}
\D_{a_2} \D_{a_1}f[\mathbf{x}]
&=& \D_{a_2}f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}\right] \Omega[-z_1\mathbf{x}] \Big|_{z_1^{a_1}}\\
&=& f\!\left[\mathbf{x}+{\textstyle \frac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\Omega[-z_1(\mathbf{x}+{\tfrac{M}{z_2}})]\Omega[-z_2\mathbf{x}]\Big|_{z_1^{a_1}z_2^{a_2}} \\
&=& f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\Omega[-z_1\mathbf{x}]\,\Omega[-z_2\mathbf{x}]\, \Omega[-Mz_1/ z_2]
\Big|_{z_1^{a_1}z_2^{a_2}}\\
&=& f\!\left[\mathbf{x}+{\tfrac{M}{z_1}}+{\tfrac{M}{z_2}}\right]\,\frac{\Omega[-(z_1+z_2)\mathbf{x}]}{z_1^{a_1}z_2^{a_2}}\, \Omega[-Mz_1/ z_2]
\Big|_{z_1^0z_2^0},
\end{eqnarray*}
and the pattern of the general result clearly emerges.
\end{proof}
\section{The \texorpdfstring{$\mathrm{SL}_2[\mathbb{Z}]$}--action and the \texorpdfstring{$ \Qop$}- operators indexed by pairs \texorpdfstring{$(km,kn)$}.}
To extend the definition of the $\Qop$ operators to any non-coprime pairs of indices we need to make use of the action of $\mathrm{SL}_2[ \mathbb{Z} ]$ on the operators $\Qop_{m,n}$. In \cite{newPleth}, $\mathrm{SL}_2[\mathbb{Z}]$ is shown to act on the algebra generated by the $\D_k$ operators by setting, for its generators
\begin{equation} \label{4.1}
N:=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}
\qquad {\rm and} \qquad
S:=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix},
\end{equation}
\begin{equation} \label{4.2}
N( \D_{k_1} \D_{k_2} \cdots \D_{k_r} )
= \nabla ( \D_{k_1} \D_{k_2} \cdots \D_{k_r} ) \nabla ^{-1},
\end{equation}
and
\begin{equation} \label{4.3}
S( \D_{k_1} \D_{k_2} \cdots \D_{k_r})
= \D_{k_1+1} \D_{k_2+1} \cdots \D_{k_r+1}.
\end{equation}
It is easily seen that \operatorname{comp}ref{4.2} is a well-defined action since any polynomial in the $\D_k$ that acts by zero on symmetric functions has an image under $N$ which also acts by zero. In \cite{newPleth}, the same property is shown to hold true for the action of $S$ as defined by \operatorname{comp}ref{4.3}.
Since $\D_k = \Qop_{1,k}$, and thus $S \Qop_{1,k}= \Qop_{1,k+1}$, it recursively follows that
\begin{equation} \label{4.4}
S\Qop_{m,n}= \Qop_{m,n+m}.
\end{equation}
On the other hand, it turns out that the property
\begin{equation} \label{4.5}
N\Qop_{m,n}=\Qop_{m+n,n},
\end{equation}
is a consequence of the following general result proved in \cite{newPleth}
\begin{prop} \label{prop4.1}
For any coprime pair $m,n$ we have
\begin{equation} \label{4.6}
\Qop_{m+n,n} = \nabla \Qop_{m,n} \nabla^{-1}.
\end{equation}
It then follows from \operatorname{comp}ref{4.4} and \operatorname{comp}ref{4.5} that for any $\Big[\begin{matrix} a & c \\[-2pt] b & d \end{matrix}\Big] \in \mathrm{SL}_2[\mathbb{Z}]$, we have
\begin{equation} \label{4.7}
\begin{bmatrix} a & c \\ b & d \end{bmatrix} \Qop_{m,n} = \Qop_{am+cn,bn+dn}.
\end{equation}
\end{prop}
The following identity has a variety of consequences
in the present development.
\begin{prop} \label{prop4.2}
For any $k \geq 1$ we have $\Qop_{k+1,k}= \Phi_k$ and $\Qop_{k-1,k} = \Psi_k$. In particular, for all pairs $a,b$, of positive integers with sum equal to $n$, it follows that
\begin{equation} \label{4.8}
\frac{1}{M}[\Qop_{b+1,b}, \Qop_{a-1,a}] =
\frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
In view of \operatorname{comp}ref{3.12a} and the second case of \operatorname{comp}ref{eqdefnQ}, the first equality is a special instance of \operatorname{comp}ref{4.6}. To prove the second equality, by Proposition \ref{propphipsi}), we only need to show that the operators $\Qop_{k-1,k}$ satisfy the same recursions and base cases as the $\Psi_k$ operators. To begin, note that since $\spl(k,k+1)=(1,1)+(k-1,k)$ it follows that
\begin{equation}\label{4.10}
\Qop_{k,k+1} = \frac{1}{M}\left[\Qop_{k-1,k},\Qop_{1,1} \right]
= \frac{1}{M}\left[\Qop_{k-1,k}, \D_1 \right],
\end{equation}
which is (\ref{3.13}b) for $\Qop_{k,k+1}$. However the base case is trivial since by definition $\Qop_{0,1}= -\underline{e}_1$.
The identity in \operatorname{comp}ref{4.10} is another way of stating \operatorname{comp}ref{3.16}.
\end{proof}
\begin{figure}
\caption{The four splits of (12,8).}
\label{fig:quadrop}
\end{figure}
Our first application is best illustrated by an example. In Figure \ref{fig:quadrop} we have depicted $k$ versions of the $km\times kn$ rectangle for the case $k=4$ and $(m,n)=(3,2)$. These illustrate that there are $4$ ways to split the vector $(0,0)\to (4\times 3,4\times 2)$ by choosing a closest lattice point below the diagonal. Namely,
\begin{displaymath}
(12,8) = (2,1)+(10,7) = (5,3)+(7,5) = (8,5)+(4,3) = (11,7)+(1,1).
\end{displaymath}
Now, it turns out that the corresponding four bracketings
\begin{displaymath}
[\Qop_{10,7},\Qop_{2,1}], \qquad
[\Qop_{7,5},\Qop_{5,3}], \qquad
[\Qop_{4,3},\Qop_{8,5}], \qquad {\rm and} \qquad
[\Qop_{1,1},\Qop_{11,7}],
\end{displaymath}
give the same symmetric function operator. This is one of the consequences of the identity in \operatorname{comp}ref{4.8}. In fact, the reader should not have any difficulty checking that these four bracketings are the images of the bracketings in \operatorname{comp}ref{4.8} for $n=4$ by $\big[\begin{matrix}\scriptstyle 2 & \scriptstyle1\\[-5pt] \scriptstyle1 &\scriptstyle 1 \end{matrix}\big]$. Therefore they must also give the same symmetric function operator since our action of $\mathrm{SL}_2[\mathbb{Z}]$ preserves all the identities satisfied by the $\D_k$ operators.
In the general case if $\spl(m,n)=(a,b)+(c,d)$, the $k$ ways are given by
\begin{displaymath}
\left( (u-1)m+a,(u-1)n+b \right) + \left( (k-u)m+c,(k-u)n+d \right),
\end{displaymath}
with $u$ going from $1$ to $k$.
The definition of the $\Qop$ operators in the non-coprime case, as well as some of their remarkable properties, appear in the following result proved in \cite{newPleth}.
\begin{thm} \label{thmQind}
If $\spl(m,n)=(a,b)+(c,d)$ then we may set, for $k> 1$ and any $1\leq u\leq k$,
\begin{displaymath}
\Qop_{km,kn}= \frac{1}{M}
\left[\Qop_{ (k-u)m+c,(k-u)n+d},
\Qop_{(u-1)m+a,(u-1)n+b} \right].
\end{displaymath}
Moreover, letting
$\Gamma := \Big[\begin{matrix} a & c \\[-2pt] b & d\end{matrix}\Big]$
we also have\footnote{Notice $\Gamma \in \mathrm{SL}_2[\mathbb{Z}]$ since (3) of \operatorname{comp}ref{3.13} gives $ad-bc=1$.}
\begin{displaymath}
{\rm a)} \quad \Qop_{k,k}= \frac{qt}{qt-1}
\nabla \underline{h}_k \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1},
\qquad {\rm and} \qquad
{\rm b)} \quad \Qop_{km,kn}= \Gamma \Qop_{k,k}.
\end{displaymath}
In particular, it follows that for any fixed $(m,n)$ the operators $\left\{ \Qop_{km,kn} \right\}_{k\geq1}$ form a commuting family.
\end{thm}
An immediate consequence of Theorem \ref{thmQind} is a very efficient recursive algorithm for computing the action of the operators $\Qop_{km,kn}$ on a symmetric function $f$.
Let us recall that the \define{Lie derivative} of an operator $X$ by an operator $Y$, which we will denote $\delta_Y X$, is simply defined by setting $\delta_Y X = [X,Y]:=XY-YX$. It follows, for instance, that
\begin{displaymath}
\delta_Y^2X= [[X,Y],Y], \qquad \delta_Y^3X= [[[X,Y],Y],Y], \qquad \ldots
\end{displaymath}
Now, our definition gives
\begin{eqnarray*}
\Qop_{2,1} &=& \frac{1}{M} [\Qop_{1,1},\Qop_{1,0}] =\frac{1}{M}[\D_1,\D_0], \qquad{\rm and}\\
\Qop_{3,1} &=& \frac{1}{M} [\Qop_{2,1},\Qop_{1,0}] = \frac{1}{M^2}[ [\D_1,\D_0], \D_0],
\end{eqnarray*}
and by induction we obtain
$$\Qop_{k,1} = \frac{1}{M^{k-1}}\, \delta_{\D_0}^{k-1} \D_1.$$
Thus the action of the matrix $S$ gives
$\Qop_{k,k+1} = \tfrac{1}{M^{k-1}} \delta_{\D_1}^{k-1} \D_2$.
In conclusion we may write
\begin{displaymath}
\Qop_{k,k} = \frac{1}{M}[\Qop_{k-1,k},\Qop_{1,0}]
= \frac{1}{M}[\Qop_{k-1,k},\D_0]
= \frac{1}{M^{k-1}}\left[ \delta_{\D_1}^{k-2} \D_2, \D_0 \right].
\end{displaymath}
This leads to the following recursive general construction of the operator $\Qop_{u,v}$.
\begin{alg} \label{alg1}
Given a pair $(u,v)$ of positive integers:
\begin{enumerate}\itemsep3pt
\item[] {\bf If} $u=1$ {\bf then} $\Qop_{u,v}:= \D_v$
\item[] \qquad {\bf else if} $u<v$ {\bf then} $\Qop_{u,v}:= S \Qop_{u,v-u}$
\item[] \qquad {\bf else if} $u>v$, {\bf then} $ \Qop_{u,v}:= N\Qop_{u-v,v}$
\item[] \qquad {\bf else} $\Qop_{u,v}:= \frac{1}{M^{u-1}} \left[ \delta_{\D_1}^{u-2} \D_2, \D_0 \right]$.
\end{enumerate}
\end{alg}
\noindent
This assumes that $S$ acts on a polynomial in the $\D$ operators by the replacement $\D_k \mapsto \D_{k+1}$, and $N$ acts by the replacement
$$\D_k\ \mapsto\ (-1)^{k}\frac{1}{M^{k}}\delta_{\D_1}^{k} \D_0.$$
We are now finally in a position to validate our construction (see Algorithm~\ref{algF}) of the operators $\BF_{km,kn}$. To this end, for any partition $\lambda =(\lambda_1,\lambda_2, \dots, \lambda_\ell)$, of length $\ell(\lambda)=\ell$, it is convenient to set
\begin{equation} \label{4.14}
h_\lambda[\mathbf{x};q,t] = \left(\frac{qt}{qt-1}\right)^\ell
\operatorname{comp}rod_{i=1}^\ell h_{\lambda_i}\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{equation}
Notice that the collection $\left\{ h_\lambda[\mathbf{x};q,t] \right\}_\lambda$ is a symmetric function basis. Thus we may carry out step one of Algorithm~\ref{algF}.
It may be good to illustrate this in a special case. For instance, when $f=e_3$ we proceed as follows. Note first that for any two expressions $A,B$ we have
\begin{displaymath}
h_3[AB]= \sum_{\lambda \vdash 3} h_\lambda[A]\, m_\lambda[B].
\end{displaymath}
Letting $A=\mathbf{x}(1-qt)/(qt)$ and $B=qt/(qt-1)$ gives
\begin{displaymath}
(-1)^3 e_3[\mathbf{x}] = h_3[-\mathbf{x}] = \sum_{\lambda \vdash 3} h_\lambda[\mathbf{x};q,t]\, m_\lambda\! \left[ \frac{qt}{qt-1} \right] \left( \textstyle\frac{qt-1}{qt} \right)^{\ell(\lambda)}.
\end{displaymath}
Thus
\begin{equation} \label{4.16}
\Be_{3m,3n} = -\sum_{\lambda \vdash 3}
m_\lambda\! \left[ \frac{qt}{qt-1} \right]
\left(\frac{qt-1}{qt} \right)^{\ell(\lambda)}
\operatorname{comp}rod_{i=1}^{\ell(\lambda)} \Qop_{m \lambda_i, n \lambda_i}.
\end{equation}
Carrying this out gives
\begin{displaymath}
\Be_{3m,3n}=
-\frac{1}{[3]_{qt}[2]_{qt}} \Qop_{m,n}^3
-\frac{qt(2+qt)}{[3]_{qt}[2]_{qt}} \Qop_{m,n}\Qop_{2m,2n}
-\frac{(qt)^2}{[3]_{qt}} \Qop_{3m,3n},
\end{displaymath}
where for convenience we have set $[a]_{qt} = ({1-(qt)^a})/({1-qt})$.
To illustrate, by direct computer assisted calculation, we get
\begin{eqnarray*}
\Be_{3,6}\cdot(-\mathbf{1}) &=& s_{{33}}[\mathbf{x}]+ ( {q}^{2}+qt+{t}^{2}+q+t ) s_{{321}}[\mathbf{x}]\\
&&\quad + ( {q}^{3}+{q}^{2}t+q{t}^{2}+{t}^{3}+qt ) s_{{3111}}[\mathbf{x}]\\
&&\quad + ( {q}^
{3}+{q}^{2}t+q{t}^{2}+{t}^{3}+qt ) s_{{222}}[\mathbf{x}]\\
&&\quad + ( {q}^{4}+{
q}^{3}t+{q}^{2}{t}^{2}+q{t}^{3}+{t}^{4}\\
&&\qquad\qquad+{q}^{3}+2\,{q}^{2}t+2\,q{t}^{2
}+{t}^{3}+{q}^{2}+qt+{t}^{2} ) s_{{2211}}[\mathbf{x}]\\
&&\quad + ( {q}^{5}+{q}
^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q{t}^{4}+{t}^{5}\\
&&\qquad\qquad+{q}^{4}+2\,{q}^{3
}t+2\,{q}^{2}{t}^{2}+2\,q{t}^{3}+{t}^{4}+{q}^{2}t+q{t}^{2} ) s_{
{21111}}[\mathbf{x}]\\
&&\quad + ( {q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q
}^{2}{t}^{4}+q{t}^{5}+{t}^{6}\\
&&\qquad\qquad+{q}^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q
{t}^{4}+{q}^{2}{t}^{2} ) s_{{111111}}[\mathbf{x}]
\end{eqnarray*}
Conjecturally, the symmetric polynomial $\Be_{km,kn}\cdot(-\mathbf{1})^{k(n+1)}$ should be the Frobenius characteristic of a bi-graded $S_{kn}$ module.
In particular the two expressions
\begin{displaymath}
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), e_1^6 \right\rangle
\qquad {\rm and} \qquad
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), s_{1^6} \right\rangle
\end{displaymath}
should respectively give the Hilbert series of the corresponding $S_6$ module and the Hilbert series of its alternants.
\begin{figure}
\caption{Dyck paths in the $3 \times 6$ lattice.}
\label{fig:ntwelves}
\end{figure}
Our conjecture states that we should have (as in the case of the module of Diagonal Harmonics)
\begin{displaymath}
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), e_1^6 \right\rangle
= \sum_{\operatorname{comp}ark\in\mathrm{Park}_{3,6}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)},
\qquad {\rm and} \qquad
\left\langle \Be_{3,6}\cdot(-\mathbf{1}), s_{1^6} \right\rangle
= \sum_{\operatorname{comp}ath \in \mathcal{D}_{3,6}} t^{\operatorname{area}(\operatorname{comp}ath)}q^{\operatorname{dinv}(\operatorname{comp}ath)},
\end{displaymath}
where the first sum is over all parking functions and the second is over all Dyck paths in the $3 \times 6$ lattice rectangle.
The first turns out to be the polynomial
\begin{eqnarray*}
&&{q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q}^{2}{t}^{4}+q{t}^{5}
+{t}^{6}\\
&&\qquad +5\,{q}^{5}+6\,{q}^{4}t+6\,{q}^{3}{t}^{2}+6\,{q}^{2}{t}^{3}+6
\,q{t}^{4}+5\,{t}^{5}\\
&&\qquad +14\,{q}^{4}+19\,{q}^{3}t+20\,{q}^{2}{t}^{2}+19\,
q{t}^{3}+14\,{t}^{4}\\
&&\qquad +24\,{q}^{3}+38\,{q}^{2}t+38\,q{t}^{2}+24\,{t}^{3}
+\\
&&\qquad 25\,{q}^{2}+40\,qt+25\,{t}^{2}+16\,q+16\,t+5.
\end{eqnarray*}
Setting first $q=1$, we get
$${t}^{6}+6\,{t}^{5}+21\,{t}^{4}+50\,{t}^{3}+90\,{t}^{2}+120\,t+90,$$
which evaluates to $378$ at $t=1$; the second polynomial
\begin{equation} \label{4.17}
{q}^{6}+{q}^{5}t+{q}^{4}{t}^{2}+{q}^{3}{t}^{3}+{q
}^{2}{t}^{4}+q{t}^{5}+{t}^{6} +{q}^{4}t+{q}^{3}{t}^{2}+{q}^{2}{t}^{3}+q
{t}^{4}+{q}^{2}{t}^{2} ,
\end{equation}
evaluates to $12$, at $q=t=1$.
All this is beautifully confirmed on the combinatorial side. Indeed, there are 12 Dyck paths in the $3\times 6$ lattice, as presented in Figure \ref{fig:ntwelves}. A simple calculation verifies that the total number of column-increasing labelings of the north steps of these Dyck paths (as recorded in Figure~\ref{fig:ntwelves} below each path) by a permutation of $\{1,2,\dots,6\}$, is indeed $378$. One may carefully check that this coincidences still holds true when one takes into account the statistics area and dinv.
In the next section we give the construction of the parking function statistics that must be used to obtain the polynomial $\Be_{3,6}\cdot(-\mathbf{1})$ by purely combinatorial methods. Figure~\ref{fig:ntwelves} shows the result of a procedure that places a square in each lattice cell, above the path, that contributes a unit to the dinv of that path. Taking into account that the area is the number of lattice squares between the path and the lattice diagonal, the reader should have no difficulty seeing that the polynomial in \operatorname{comp}ref{4.17} is indeed the $q,t$-enumerator of the above Dyck paths by $\operatorname{dinv}$ and $\operatorname{area}$.
\begin{rmk} In retrospect, our construction of the operators $\BC_{km,kn}^{(\alpha)}$ has a certain degree of inevitability. In fact, since the multiplication operator
$$\frac{qt}{qt-1}\, \underline{h}_k\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]$$
corresponds to the operator $\Qop_{0,k}$, and since we must have
\begin{displaymath}
\Qop_{k,k} = \nabla \Qop_{0,k} \nabla ^{-1} = \frac{qt}{qt-1} \nabla \underline{h}_k\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla ^{-1}
\end{displaymath}
by Proposition \ref{prop4.1},
then it becomes natural to set
\begin{displaymath}
\Bf_{k,k} := \nabla \cdot\mathbf{1}derline{f} \nabla ^{-1}.
\end{displaymath}
for any symmetric function $f$ homogeneous of degree $k$.
Therefore using the matrix $\Gamma$ of Theorem \ref{thmQind}, we obtain
\begin{equation} \label{4.18}
\Bf_{km,kn}= \Gamma \Bf_{k,k}
\end{equation}
In particular, it follows (choosing $f=e_k$) that
\begin{displaymath}
\Be_{k,k}\cdot\mathbf{1} = \nabla e_k\nabla^{-1} \cdot\mathbf{1}= \nabla e_k.
\end{displaymath}
The expansion
$e_k = \sum_{\alpha\models k} C_\alpha \cdot\mathbf{1}$
yields the decomposition $\Be_{k,k} = \sum_{\alpha\models k} \BC_{k ,k }^{(\alpha)}$,
and \operatorname{comp}ref{4.18} then yields
$$
\Be_{km,kn}= \sum_{\alpha\models k}\BC_{km ,kn }^{(\alpha)}.
$$
\qed
\end{rmk}
\section{The combinatorial side, further extensions and conjectures.}
Our construction of the parking function statistics in the non-coprime case closely follows what we did in section 1 with appropriate modifications necessary to
resolve conflicts that did not arise in the coprime case. For clarity we will present our definitions as a collection of algorithms which can be directly implemented on a computer.
The symmetric polynomials arising from the right hand sides of our conjectures may also be viewed as Frobenius characteristics of certain bi-graded $S_n$ modules. Indeed, they are shown to be by \cite{Hikita} in the coprime case. Later in this section we will present some conjectures to this effect.
As for Diagonal Harmonics (see \cite{Shuffle}), all these Frobenius Characteristics are sums of LLT polynomials. More precisely, a $(km,kn)$-Dyck path $\operatorname{comp}ath$ may be represented by a vector
\begin{equation} \label{5.1}
\mathbf{u}=(u_1,u_2,\dots,u_{kn}) \qquad
\quad {\rm with}\quad u_0=0,
\qquad{\rm and}\qquad u_{i-1} \leq u_i \leq (i-1)\frac{m}{n},
\end{equation}
for all $2\leq i\leq kn$.
This given, we set
\begin{equation} \label{5.2}
\mathrm{LLT}(m,n,k;\mathbf{u}) := \sum_{\operatorname{comp}ark \in\mathrm{Park}(\mathbf{u})}
t^{\operatorname{area}(\mathbf{u})} q^{\operatorname{dinv}(\mathbf{u}) + \operatorname{tdinv}(\operatorname{comp}ark) - \operatorname{maxtdinv}(\mathbf{u})} s_{\operatorname{pides}(\operatorname{comp}ark)}[\mathbf{x}],
\end{equation}
where $\mathrm{Park}(\mathbf{u})$ denotes the collection of parking functions supported by the path corresponding to $\mathbf{u}$. We also use here the Egge-Loehr-Warrington (see \cite{expansions}) result and substitute the Gessel fundamental by the Schur function indexed by $\operatorname{pides}(\operatorname{comp}ark)$ (the descent composition of the inverse of the permutation $\sigma(\operatorname{comp}ark)$).
Here $\sigma(\operatorname{comp}ark)$ and the other statistics used in \operatorname{comp}ref{5.2} are constructed according to the following algorithm.
\begin{alg} \label{algL}
\begin{enumerate}
\item Construct the collection $\mathrm{Park}(\mathbf{u})$ of vectors $\mathbf{v}=( v_1, v_2, \dots, v_{kn} )$ which are the permutations of $1, 2, \dots, kn$ that satisfy the conditions
\begin{displaymath}
u_{i-1}=u_i \quad\implies\quad v_{i-1}<v_{i}.
\end{displaymath}
\item Compute the $\operatorname{area}$ of the path, that is the number of full cells between the path and the
main diagonal of the $km\times kn$ rectangle, by the formula
\begin{displaymath}
\operatorname{area}(\mathbf{u})= (kmkn - km - kn + k)/2 - \sum_{i=1}^{kn} u_i .
\end{displaymath}
\item Denoting by $\lambda(\mathbf{u})$ the partition above the path, set
\begin{displaymath}
\operatorname{dinv}(\mathbf{u}) = \sum_{c \in \lambda(\mathbf{u})} \raise 2pt\hbox{\large$\chi$}\!\left( \frac{\operatorname{arm}(c)}{\operatorname{leg}(c)+1} \le \frac{m}{n} < \frac{\operatorname{arm}(c)+1}{\operatorname{leg}(c)}\right).
\end{displaymath}
\item Define the $\operatorname{rank}$ of the $i^{th}$ north step by $km(i-1) - kn u_i + u_i/(km+1)$ and accordingly use this number as the $\operatorname{rank}$ of car $v_i$, which we will denote as $\operatorname{rank}(v_i)$. This given, we set, for $\operatorname{comp}ark = \Big[\begin{matrix} \mathbf{u} \\[-3pt] \mathbf{v} \end{matrix}\Big]$
\begin{displaymath}
\operatorname{tdinv}(\operatorname{comp}ark) = \sum_{1\leq r<s \leq kn}
\raise 2pt\hbox{\large$\chi$}\left( \operatorname{rank}(r) < \operatorname{rank}(s) < \operatorname{rank}(r)+km \right).
\end{displaymath}
\item Define $\sigma(\operatorname{comp}ark)$ as the permutation of $1, 2, \dots, kn$ obtained by reading the cars by decreasing ranks. Let $\operatorname{pides}(\operatorname{comp}ark)$ be the composition whose first $kn-1$ partial sums give the descent set of the inverse of $\sigma(\operatorname{comp}ark)$.
\item Finally $\operatorname{maxtdinv}(\mathbf{u})$ may be computed as $\max \{\operatorname{tdinv}(\operatorname{comp}ark) : \operatorname{comp}ark \in\mathrm{Park}(\mathbf{u}) \}$, or more efficiently as the $\operatorname{tdinv}$ of the $\operatorname{comp}ark$ whose word $\sigma(\operatorname{comp}ark)$ is the reverse permutation $(kn)\cdots 3 2 1$.
\end{enumerate}
\end{alg}
This completes our definition of the polynomial $\mathrm{LLT}(m,n,k;\mathbf{u})$, which may be shown to expand as a linear combination of Schur functions with coefficients in $\mathbb{N}[q,t]$. It may be also be shown that, at $q=1$, the polynomial $\mathrm{LLT}(m,n,k;\mathbf{u})$ specializes to $t^{\operatorname{area}(\mathbf{u})}e_{\lambda(\mathbf{u})}[\mathbf{x}]$, with $\lambda(\mathbf{u})$ the partition giving the multiplicities of the components of $\mathbf{u}_{km,kn}-\mathbf{u}$, with $\mathbf{u}_{km,kn}$ the vector encoding the unique $0$-area $(km,kn)$-Dyck path.
With the above definition at hand, Conjecture \ref{conjBGLX} can be restated as
\begin{equation} \label{5.3}
\BC^{(\alpha)}_{km,kn}\cdot (-\mathbf{1})^{k(n+1)}
=\sum_{\mathbf{u} \in \mathcal{U}(\alpha)}\mathrm{LLT}(m,n,k;\mathbf{u})
\end{equation}
where $\mathcal{U}(\alpha)$ denotes the collection of all $\mathbf{u}$ vectors satisfying \operatorname{comp}ref{5.1} whose corresponding Dyck path hits the diagonal of the $km\times kn$ rectangle according to the composition $\alpha\models k$. Alternatively we can simply require that $\alpha$ be the composition of $k$ corresponding to the subset
\begin{displaymath}
\left\{ 1\leq i\leq k-1 : u_{ni+1}=i m \right\}.
\end{displaymath}
Note that, although the operators $\BB_b$ and $ \sum_{\beta \models b}
\BC_{\beta}$ are different, in view of definition \operatorname{comp}ref{defC} they take the same value on constant symmetric functions
\begin{equation} \label{5.4}
\BB_b \cdot\mathbf{1} = e_b[\mathbf{x}] = \sum_{\beta \models b}
\BC_{\beta} \cdot\mathbf{1}.
\end{equation}
This circumstance, combined with the commutativity relation (proved in \cite{classComp})
\begin{equation} \label{5.5}
\BB_b \BC_\gamma= q^{\ell(\gamma)} \BC_\gamma \BB_b,
\end{equation}
for all compositions $\gamma$,
enables us to derive a $(km, kn)$ version of the
Haglund-Morse-Zabrocki conjecture \ref{conjHMZB}. To see how this comes about we briefly reproduce an argument first given in \cite{classComp}.
Exploiting \operatorname{comp}ref{5.4} and \operatorname{comp}ref{5.5}, we calculate that
\begin{align*}
\BB_a \BB_b \BB_c \cdot\mathbf{1} &= \BB_a \BB_b \sum_{\gamma \models c} \BC_{\gamma} \cdot\mathbf{1} \\
&= \BB_a \sum_{\gamma \models c}q^{\ell(\gamma)} \BC_{\gamma} \BB_b \cdot\mathbf{1} \\
&= \BB_a \sum_{\gamma\models c} q^{\ell(\gamma)} \BC_{\gamma}\sum_{\beta \models b} \BC_{\beta} \cdot\mathbf{1} \\
&= \sum_{\gamma\models c} q^{2 \ell(\gamma)} \BC_{\gamma}\sum_{\beta\models b} q^{\ell(\beta)} \BC_{\beta}\sum_{\alpha \models a} \BC_{\alpha} \cdot\mathbf{1}.
\end{align*}
From this example one can easily derive that the following identity holds true in full generality.
\begin{prop} \label{5.1}
For any $\beta=(\beta_1,\beta_2,\dots ,\beta_k)$, we have
\begin{equation} \label{5.6}
\BB_{\beta} \cdot\mathbf{1}
= \sum_{\alpha \operatorname{comp}receq \beta} q^{c(\alpha,\beta)}
\BC_{\alpha} \cdot\mathbf{1}
\end{equation}
where $\alpha \operatorname{comp}receq \beta$ here means that $\alpha$ is a refinement of the reverse of $\beta$. That is $\alpha = \alpha^{(k)} \cdots \alpha^{(2)}\alpha^{(1)}$ with $\alpha^{(i)} \models \beta_{i}$, and in that case
\begin{displaymath}
c(\alpha,\beta) = \sum_{i=1}^k (i-1)\, \ell(\alpha^{(i)}).
\end{displaymath}
\end{prop}
Using \operatorname{comp}ref{5.6} we can easily derive the following.
\begin{prop} Assuming that Conjecture~\ref{conjBGLX} holds, then
for all compositions
$$\beta=(\beta_1,\beta_2,\dots ,\beta_{\ell)}\models k$$ we have
\begin{displaymath}
\BB_{km,kn}^{(\beta)}\cdot (-\mathbf{1})^{k(n+1)}
= \sum_{\alpha \operatorname{comp}receq \beta} q^{c(\alpha,\beta)}
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{km,kn}^{(\alpha)}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\operatorname{comp}ark)}.
\end{displaymath}
Here $\BB_{km,kn}^{(\beta)}$ is the operator obtained by setting $f=\BB_{\beta} \cdot\mathbf{1}$ in Algorithm~\ref{algF} and, as before, $\mathrm{Park}_{km,kn}^{(\alpha)}$ denotes the collection of parking functions in the $km\times kn$ lattice whose Dyck path hits the diagonal according to the composition $\alpha$.
\end{prop}
A natural question that arises next is what parking function interpretation may be given to the
polynomials $\Qop_{km,kn} \cdot(-\mathbf{1})^{k(n+1)}$. Our attempts to answer this question lead to a variety of interesting identities. We start with a known identity and two new ones.
\begin{thm} \label{thmQsquare} For all $n$, we have
\begin{eqnarray}
\Qop_{n,n+1} \cdot\mathbf{1}n &=& \nabla e_n,\qquad {\rm and}\label{5.7a}\\
\Qop_{n,n} \cdot\mathbf{1}n &=& (-qt)^{1-n}\nabla \Delta_{e_1} h_n=\Delta_{e_{n-1}} e_n.\label{5.7b}
\end{eqnarray}
\end{thm}
\begin{proof}[\bf Proof]
Since $\Qop_{n+1,n} = \nabla \Qop_{1,n}\nabla ^{-1}$ and $\Qop_{1,n} = \D_n$, then
\begin{eqnarray*}
\Qop_{n,n+1} \cdot\mathbf{1}n &=& \nabla \D_n \nabla ^{-1} \cdot\mathbf{1}n\\
&=& \nabla \D_n \cdot\mathbf{1}n.
\end{eqnarray*}
Recalling that $\nabla^{-1} \, \cdot\mathbf{1}n =(-1)^{n}$ and $\D_n \cdot\mathbf{1} = (-1)^n e_n$, this gives \operatorname{comp}ref{5.7a}. The proof of \operatorname{comp}ref{5.7b} is a bit more laborious. We will obtain it below, by combining a few auxiliary identities.
\end{proof}
\begin{prop} \label{prop5.2}
For any monomial $m$ and $\lambda \vdash n$
\begin{equation} \label{5.8}
s_\lambda[1-m] =
\begin{cases}
(-m)^k (1-m) & \hbox{\rm if}\ \lambda = (n-k,1^k)\ \hbox{\rm for some }\ 0\leq k\leq n-1,\\[4pt]
0 & \hbox{\rm otherwise.}
\end{cases}
\end{equation}
\end{prop}
\noindent
This is an easy consequence of the addition formula for Schur functions.
\begin{prop}\label{prop5.3} For all $n$,
\begin{equation} \label{5.9}
\frac{qt}{qt-1} h_n\! \left[ \frac{1-qt}{qt}\,\mathbf{x}\right]
= -(qt)^{1-n} \sum_{k=0}^{n-1} (-qt)^k s_{n-k,1^k}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The Cauchy formula gives
\begin{displaymath}
\frac{qt}{qt-1} h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]
= \frac{(qt)^{1-n}}{qt-1} \sum_{\lambda \vdash n} s_\lambda[\mathbf{x}] s_\lambda[1-qt]
\end{displaymath}
and \operatorname{comp}ref{5.8} with $m=qt$ proves \operatorname{comp}ref{5.9}.
\end{proof}
\noindent
Observe that when we set $qt=1$, the right hand side of \operatorname{comp}ref{5.9} specializes to $-p_n$.
\begin{prop} \label{prop5.4} For all $n$,
\begin{equation} \label{5.10}
\Delta_{e_1} h_n[\mathbf{x}] = \sum_{k=0}^{n-1} (-qt)^k\,s_{n-k,1^k}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
By \operatorname{comp}ref{3.8} and \operatorname{comp}ref{formulaoper} $(i)$ we have $\Delta_{e_1} = (I-\D_0)/M$. Thus, by \operatorname{comp}ref{defD}
\begin{eqnarray*}
M \Delta_{e_1} h_n[\mathbf{x}]
&=& h_n[\mathbf{x}] - h_n[\mathbf{x}+{M}/{z}]\, \Omega[-z\mathbf{x}]\Big|_{z^0}\\
&=& - \sum_{k=1}^{n} (-1)^k h_{n-k}[\mathbf{x}] \, e_k[\mathbf{x}]\, h_k[M] \\
&=& - \sum_{k=1}^{n-1} (-1)^k s_{n-k,1^k}[\mathbf{x}]\, h_k[M] + \sum_{k=0}^{n-1} (-1)^{k} s_{n-k,1^k}[\mathbf{x}]\, h_{k+1}[M] \\
&=& \sum_{k=1}^{n-1} (-1)^k s_{n-k,1^k}[\mathbf{x}]\, (h_{k+1}[M]-h_{k}[M]) + s_n[\mathbf{x}] M.
\end{eqnarray*}
This proves \operatorname{comp}ref{5.10} since the Cauchy formula and \operatorname{comp}ref{5.8} give
\begin{eqnarray*}
h_n[M] &=& \sum_{k=0}^{n-1}(-t)^k(1-t)(-q)^k(1-q)\\
&=& M\sum_{k=0}^{n-1}(qt)^k.
\end{eqnarray*}
\end{proof}
Now, it is shown in \cite{explicit} that $e_n[\mathbf{x}]$ and $h_n[\mathbf{x}]$ have the expansions
\begin{align}
e_n[\mathbf{x}] &= \sum_{\mu \vdash n}\frac{M\, B_\mu(q,t) \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t], \qquad {\rm and} \label{5.11a}\\
h_n[\mathbf{x}] &= (-qt)^{n-1} \sum_{\mu\vdash n}\frac{M\,B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t]. \label{5.11b}
\end{align}
We are then ready to proceed with the rest of our proof.
\begin{proof}[\bf Proof][Proof of Theorem \ref{thmQsquare} continued]
The combination of \operatorname{comp}ref{5.9}, \operatorname{comp}ref{5.10} and
\operatorname{comp}ref{5.11a} gives
\begin{eqnarray}
\frac{qt}{qt-1} h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right]
&=& -(qt)^{1-n} \Delta_{e_1} h_n\label{5.12}\\
&=& (-1)^{n} \sum_{\mu\vdash n} \frac{ M B_\mu(q,t) B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t] \label{5.13}
\end{eqnarray}
Now from Theorem \ref{thmQind} we derive that
\begin{eqnarray*}
\Qop_{n,n}\cdot\mathbf{1}n &=& \frac{qt}{qt-1} \nabla \underline{h}_n \!\left[\frac{1-qt}{qt}\,\mathbf{x}\right] \nabla^{-1} \cdot\mathbf{1}n\\
&=& (-1)^n \frac{qt}{qt-1} \nabla h_n\!\left[\frac{1-qt}{qt}\,\mathbf{x}\right].
\end{eqnarray*}
Thus \operatorname{comp}ref{5.12} gives
\begin{displaymath}
\Qop_{n,n} \cdot\mathbf{1}n = (-qt)^{1-n} \nabla \Delta_{e_1}h_n.
\end{displaymath}
This proves the first equality in \operatorname{comp}ref{5.7b}. The second equality in \operatorname{comp}ref{5.13} gives
\begin{equation} \label{5.13}
\Qop_{n,n} \cdot\mathbf{1}n = \sum_{\mu\vdash n}\frac{T_\mu\, M B_\mu(q,t) B_\mu(1/q,1/t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t].
\end{equation}
But it is not difficult to see that we may write (for any $\mu\vdash n$)
\begin{displaymath}
T_\mu \,B_\mu(1/q,1/t)= e_{n-1}\left[B_\mu(q,t)\right]
\end{displaymath}
and \operatorname{comp}ref{5.13} becomes (using \operatorname{comp}ref{5.11a})
\begin{eqnarray*}
\Qop_{n,n} \cdot\mathbf{1}n &=& \Delta_{e_{n-1}} \sum_{\mu\vdash n} \frac{ M B_\mu(q,t)\, \Pi_\mu(q,t)}{w_\mu}\,\widetilde{H}_\mu[\mathbf{x};q,t]\\
&=& \Delta_{e_{n-1}} e_n.
\end{eqnarray*}
\end{proof}
To obtain a combinatorial version of \operatorname{comp}ref{5.7b} we need some auxiliary facts.
\begin{prop} For all positive integers $a$ and $b$, we have
\begin{equation} \label{5.15}
\BC_a \BB_b \cdot\mathbf{1} = \BC_a e_b[\mathbf{x}] = (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}],
\end{equation}
and
\begin{equation} \label{5.16}
\sum_{\beta\models n-a} \BC_a \BC_\beta \cdot\mathbf{1}
= (-1/q)^{a-1} s_{a,1^{n-a}}[\mathbf{x}] - (-1/q)^a s_{1+a,1^{n-a-1}}[\mathbf{x}].
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
The equality in \operatorname{comp}ref{5.4} gives the first equality in \operatorname{comp}ref{5.15} and the equivalence of \operatorname{comp}ref{5.15} to \operatorname{comp}ref{5.16}, for $b=n-a$. Using \operatorname{comp}ref{defC} and \operatorname{comp}ref{5.8} we derive that
\begin{eqnarray*}
\BC_a e_b[\mathbf{x}]
&=& (-1/q)^{a-1} \sum_{r=0}^b e_{b-r}[\mathbf{x}] (-1)^r\, h_r \left[ (1-1/q)/z \right]\, \Omega[z\mathbf{x}] \Big|_{z^a} \\
&=& (-1/{q})^{a-1} \left(e_b[\mathbf{x}]\, h_a[\mathbf{x}] + ( 1- 1/q) \sum_{r=1}^b (-1)^r e_{b-r}[\mathbf{x}]\, h_{r+a}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} \Big(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] + \\
&& \qquad\qquad + (1-1/q) \left( \textstyle \sum_{r=1}^b (-1)^r s_{r+a,1^{b-r}}[\mathbf{x}] -
\sum_{r=2}^{b} (-1)^r s_{r+a ,1^{b-r}}[\mathbf{x}] \right) \Big) \\
&=& (-1/q)^{a-1} \left(s_{a,1^{b}}[\mathbf{x}]+s_{a+1,1^{b-1}}[\mathbf{x}] - (1-1/q)\, s_{1+a,1^{b-1}}[\mathbf{x}] \right) \\
&=& (-1/q)^{a-1} s_{a,1^{b}}[\mathbf{x}]-(-1/q)^a s_{1+a,1^{b-1}}[\mathbf{x}].
\end{eqnarray*}
\end{proof}
The following conjectured identity is well known and is also stated in \cite{Shuffle}. We will derive it from Conjecture \ref{conjHMZC} for sake of completeness.
\begin{thm} \label{thm5.3}
Upon the validity of the Compositional Shuffle conjecture we have
\begin{equation} \label{5.17}
\nabla (-q)^{1-a} s_{a,1^{n-a}} =
\sum_{\operatorname{comp}ark \in\mathrm{Park}_{n, \geq a}}
t^{\operatorname{area}(\operatorname{comp}ark)} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n, \geq a}$ signifies that the sum is to be extended over the parking functions in the $n\times n$ lattice whose Dyck path's first return to the main diagonal occurs in a row $y \geq a$.
\end{thm}
\begin{proof}[\bf Proof]
An application of $\nabla$ to both sides of \operatorname{comp}ref{5.16} yields
\begin{equation} \label{5.18}
\sum_{\beta\models n-a} \nabla \BC_a \BC _\beta\cdot\mathbf{1}
= ( \tfrac{1}{q})^{a-1} \nabla (- 1)^{a-1} s_{a,1^{n-a}} - ( \tfrac{1}{q})^{a} \nabla (-1)^{a} s_{1+a,1^{n-a-1}}.
\end{equation}
Furthermore, \operatorname{comp}ref{5.17} is an immediate consequence of the fact that the Compositional Shuffle Conjecture implies
\begin{equation} \label{5.19}
\sum_{\beta\models n-a; } \nabla \, \BC_a \BC _\beta\cdot\mathbf{1}
= \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n,= a}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))},
\end{equation}
where the symbol $\mathrm{Park}_{n,= a}$ signifies that the sum is to be extended over the parking functions whose Dyck path's first return to the diagonal occurs
exactly at row $a$.
\end{proof}
All these findings lead us to the following surprising identity.
\begin{thm} \label{thm5.4}
Let $\operatorname{ret}(\operatorname{comp}ark)$ denote the first row where the supporting Dyck path of $\operatorname{comp}ark$ hits the diagonal. We have, for all $n\geq 1$, that
\begin{equation} \label{5.20}
\Qop_{n,n} \cdot\mathbf{1}n = \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}} [\operatorname{ret}(\operatorname{comp}ark)]_t\, t^{\operatorname{area}(\operatorname{comp}ark))-\operatorname{ret}(\operatorname{comp}ark)+1} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}.
\end{equation}
\end{thm}
\begin{proof}[\bf Proof]
Combining \operatorname{comp}ref{5.7a} with \operatorname{comp}ref{5.10} gives
\begin{equation} \label{5.21}
\Qop_{n,n} \cdot\mathbf{1}n = (-qt)^{1-n} \sum_{k=0}^{n-1} \nabla (-qt)^k s_{n-k,1^k}.
\end{equation}
Now this may be rewritten as
\begin{equation} \label{5.22}
\Qop_{n,n} \cdot\mathbf{1}n
= (-qt)^{1-n} \sum_{a=1}^{n} \nabla (-qt)^{n-a} s_{a,1^{n-a}}
= \sum_{a=1}^{n} \nabla (-qt)^{1-a} s_{a,1^{n-a}},
\end{equation}
and \operatorname{comp}ref{5.17} gives
\begin{eqnarray}
\Qop_{n,n} \cdot\mathbf{1}n
&=& \sum_{a=1}^{n} t^{1-a} \sum_{\operatorname{comp}ark \in\mathrm{Park}_{n}} t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))}\, \raise 2pt\hbox{\large$\chi$}\left(\operatorname{ret}(\operatorname{comp}ark)\geq a \right) \label{5.23}\\
&=& \sum_{\operatorname{comp}ark\in\mathrm{Park}_{n}}
t^{\operatorname{area}(\operatorname{comp}ark))} q^{\operatorname{dinv}(\operatorname{comp}ark)} F_{\operatorname{ides}(\sigma(\operatorname{comp}ark))} \sum_{a=1}^n t^{1-a}\, \raise 2pt\hbox{\large$\chi$} \left( a\leq \operatorname{ret}(\operatorname{comp}ark) \right).
\end{eqnarray}
This proves \operatorname{comp}ref{5.20}.
\end{proof}
\begin{rmk} \label{rmk5.3}
Extensive computer experiments have revealed that the following difference is Schur positive
\begin{equation} \label{5.24}
\Be_{km+1,kn} \cdot (-\mathbf{1}) - t^{d(km,kn)} \Be_{km,kn} \cdot\mathbf{1},
\end{equation}
where $d(km,kn)$ is the number of integral points between the diagonals for $(km+1,kn)$ and $(km,kn)$. Assuming that $m\leq n$ for simplicity sake, this implies that the following difference is also Schur positive
\begin{equation} \label{5.25}
\Be_{kn,kn} \cdot\mathbf{1} - t^{a(km,kn)} \Be_{km,kn} \cdot\mathbf{1},
\end{equation}
with $a(km,kn)$ equal to the area between the diagonal $(km,kn)$ and the diagonal $(kn,kn)$. This suggests that there is a nice interpretation of $t^{a(km,kn)} \Be_{km,kn} \cdot\mathbf{1}$ as a new sub-module of the space of diagonal harmonic polynomials. We believe that we have a good candidate for this submodule, at least in the coprime case.
\end{rmk}
We terminate with some surprising observations concerning the so-called \define{Rational $(q,t)$-Catalan}. In the present notation, this remarkable generalization of the $q,t$-Catalan polynomial (see \cite{qtCatPos}) may be defined by setting, for a coprime pair $(m,n)$
\begin{equation} \label{5.26}
C_{m,n}(q,t):= \left\langle \Qop_{m,n}\cdot\mathbf{1}n, e_n\right\rangle_*.
\end{equation}
It is shown in \cite{NegutShuffle}, by methods which are still beyond our reach, that this polynomial may also be obtained by the following identity.
\begin{thm} [A. Negut] \label{thm5.8}
\begin{equation} \label{5.27}
C_{m,n}(q,t)= \operatorname{comp}rod_{i=1}^n \frac{1}{(1-z_i)z_i^{a_i(m,n)}}
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{(1-qtz_i/z_{i+1})}
\operatorname{comp}rod_{1\leq i<j\leq n}\Omega[-M z_i/z_j]
\Big|_{z_1^0z_2^0\cdots z_n^0},
\end{equation}
with
\begin{equation} \label{5.28}
a_i(m,n):=\left\lfloor i\frac{m}{n}\right\rfloor- \left\lfloor (i-1)\frac{m}{n}\right\rfloor.
\end{equation}
\end{thm}
By a parallel but distinct path Negut has obtained the same polynomial as a weighted sum of standard tableaux. However his version of this result turns out to be difficult to program on the computer. Fortunately, in \cite{constantTesler}, in a different but in a closely related context a similar sum over standard tableaux has been obtained. It turns out that basically the same method used in \cite{constantTesler} can be used in the present context to derive a standard tableaux expansion for $C_{m,n}(q,t)$ directly from \operatorname{comp}ref{5.27}.
Let us write
\begin{eqnarray*}
\mathbf{z}_{m,n}&:=&\operatorname{comp}rod_{i=1}^n z_i^{a_{n+1-i}(m,n)}, \qquad {\rm and\ then}\\
\mathcal{N}_{m,n} [\mathbf{z};q,t]&:=& \frac{\Omega[\mathbf{z}]}{\mathbf{z}_{m,n}}
\, \operatorname{comp}rod_{i=2}^{n} \frac{1}{(1-qt z_i/z_{i-1})}
\operatorname{comp}rod_{1\leq i <j \leq n} \Omega[-uMz_j/z_i],
\end{eqnarray*}
where $\mathbf{z}$ stands for the set of variables $z_1,z_2,\ldots,z_n$. Equivalently, $\mathbf{z}=z_1+z_2+\ldots+z_n$ in the plethystic setup.
The resulting rendition of the Negut's result can then be stated as follows.
\begin{prop} \label{thm5.9}
Let $T_n$ be the set of all standard tableaux with labels $1,2,\dots,n$.
For a given $T \in T_n$, we set $w_T(k) = q^{j-1} t^{i-1}$ if the label $k$ of $T$ is in the $i$-th row $j$-th column.
We also denote by $S_T$ the substitution set
\begin{equation} \label{5.29}
\{z_k = w_T^{-1}(k): 1\leq k\leq n \}.
\end{equation}
We have
\begin{equation} \label{5.30}
C_{m,n}(q,t) = \sum_{T\in T_n} \mathcal{N}_{m,n} [\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^n (1-z_i w_T(i)) \Big|_{S_T},
\end{equation}
where the sum ranges over all standard tableaux of size $n$, and the $S_T$ substitution should be carried out in the iterative manner. That is, we successively do the substitution for $z_1$ followed by cancellation, and then do the substitution
for $z_2$ followed by cancellation, and so on.
\end{prop}
\begin{proof}[\bf Proof]
For convenience, let us write $f(u)$ for $\Omega[-Mu]$ and $b_i$ for $a_{n+1-i}(m,n)$ this gives
$$
\mathcal{N}_{m,n} [\mathbf{z};q,t] =
\operatorname{comp}rod_{i=1}^n{1\over (1- z_i)z_i^{b_i}}\operatorname{comp}rod_{i=2}^n{1\over (1-qtz_i/z_{i-1})}\operatorname{comp}rod_{1\le i<j\le n}f(z_i/z_j).
$$
We will show by induction that
\begin{displaymath}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \Big|_{z_1^0\cdots z_{d}^0} = \sum_{T} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_T(i)) \Big|_{S_T},
\end{displaymath}
where $T$ ranges over all standard tableaux of size $d$. Then the proposition is just the $d=n$ case. The $d=1$ case is straightforward. Thus, assume the equality holds for $d-1$.
Now for any term corresponding to a tableau $T$ of size $d-1$, the factors containing $z_d$ are
\begin{eqnarray*}
&& \frac{1}{(1-z_d) z_d^{b_d}}\frac{1}{(1-qt\, {z_d/z_{d-1}})(1- qt \, {z_{d+1}/z_d})^{\chi(d<n)}}
\operatorname{comp}rod_{1\leq i <d } f({z_d/z_i})\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d}) \Big|_{S_T} \\
&&=\ \frac{1}{(1-z_d) z_d^{b_d}}
\frac{1}{\left(1-qt\, z_d\,w_T(d-1)\right)\left(1- qt \,{z_{d+1}}/{z_d} \right)^{\chi(d<n)} }
\operatorname{comp}rod_{1\leq i <d } f(z_d\,w_T(i))\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d})\\
&&=\ \frac{1}
{z_d^{b_d} }\, \Omega\!\Big[z_d(1+qt\,w_T(d-1) -M\sum_{i=1}^{d-1} w_T(i)) \Big]
\operatorname{comp}rod_{d <j \leq n} f({z_j}/{z_d})\times\frac{1}{\big(1- qt \, {z_{d+1} \over{z_d}} \big)^{\chi(d<n)} }.
\end{eqnarray*}
By a simple calculation, first carried out in \cite{plethMac} we may write
\begin{displaymath}
- M B_{\operatorname{sh}(T)} = \bigg(\sum_{(i,j) \in \OC{\operatorname{sh}(T)}} t^{i-1} q^{j-1} - \sum_{(i,j) \in \IC{\operatorname{sh}(T)}} t^i q^j \bigg) - 1
\end{displaymath}
where for a partition $\lambda$ we respectively denote by ``$\OC{\lambda}$'' and ``$\IC{\lambda}$'' the \define{outer} and \define{inner} corners of the Ferrers diagram of $\lambda$, as depicted in Figure~\ref{fig:corners} using the french convention.
\begin{figure}
\caption{Inner and outer corners of a partition.}
\label{fig:corners}
\end{figure}
This given, the rational function object of the constant term becomes the following proper rational function in $z_d$ (provided $b_d \geq -1$):
$$
\frac{1}{z_d^{b_d}}
\frac{ \operatorname{comp}rod_{(i,j) \in \IC{\operatorname{sh}(T)}} (1- t^i q^j z_{d}) }
{ (1-{ z_d}\, qt\, w_T(d-1))
\operatorname{comp}rod_{(i,j)\in \OC{\operatorname{sh}(T)}} (1- t^{i-1} q^{j-1} z_{d})}
\operatorname{comp}rod_{d <j \leq n}f({z_j}/{z_d})
\times
\frac{1}{\left(1- qt\, {z_{d+1} \over{z_d}} \right)^{\chi(d<n)}}.
$$
Since $d-1$ must appear in $T$ in an inner corner of $\operatorname{sh}(T)$, the factor $1-{ z_d}\, qt\, w_T(d-1)$ in the denominator cancels with a factor in the numerator.
Therefore, the only factors, in the denominator that contribute to the constant term\footnote{By the partial fraction algorithm in \cite{fastAlg}.} are those of the form $(1-q^{j-1}t^{i-1} z_{d})$, for $(i,j)$ an outer corner of $T$. For each such $(i,j)$, construct $T'$ by adding $d$ to $T$ at the cell $(i,j)$.
Thus, by the partial fraction algorithm, we obtain
\begin{eqnarray*}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^{d-1} (1-z_i w_T(i)) \Big|_{S_T} \Big|_{z_d^0}
&=&\sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_T} \Big|_{z_d={1\over w_{T'}(d)} }\\
&=&\sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}},
\end{eqnarray*}
where the sum ranges over all $T'$ obtained from $T$ by adding $d$ at one of its outer corners.
Applying the above formula to all $T$ of size $d-1$, and using the induction hypothesis, we obtain:
\begin{eqnarray*}
\mathcal{N}_{m,n}[\mathbf{z};q,t] \Big|_{z_1^0\cdots z_{d}^0}
&=& \sum_{T} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^{d-1} (1-z_i w_T(i)) \Big|_{S_T} \Big|_{z_d^0} \\
&=& \sum_T \sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}} \\
&=& \sum_{T'} \mathcal{N}_{m,n}[\mathbf{z};q,t] \operatorname{comp}rod_{i=1}^d (1-z_i w_{T'}(i)) \Big|_{S_{T'}},
\end{eqnarray*}
where the final sum ranges over all $T'$ of size $d$.
\end{proof}
\begin{rmk} \label{rmk5.4}
We can see that the above argument only needs $b_1\geq 0$, and $b_i\geq -1$ for $i=2,3,\dots,n$. Thus the equality of the right hand sides of \operatorname{comp}ref{5.27} and \operatorname{comp}ref{5.30} holds true also if the sequence $\{a_i(m,n)\}_{i=1}^n$ is replaced by any of these sequences. In fact computer data reveals that the constant term in \operatorname{comp}ref{5.27} yields a polynomial with positive integral coefficients for a variety of choices of $(b_1,b_2,\dots,b_n)$ replacing the sequence $\{a_{i}(m,n)\}_{i=1}^n$. Trying to investigate the nature of these sequences and the possible combinatorial interpretations of the resulting polynomial led to the following construction.
\end{rmk}
Given a path $\operatorname{comp}ath$ in the $m\times n$ lattice rectangle, we define the monomial of $\operatorname{comp}ath$ by setting
\begin{equation} \label{5.31}
\mathbf{z}_\operatorname{comp}ath:=\operatorname{comp}rod_{j=1}^n z_j^{e_j},
\end{equation}
where $e_j=e_j(\operatorname{comp}ath)$ gives the number of east steps taken by $\operatorname{comp}ath$ at height $j$. Note that, by the nature of \operatorname{comp}ref{5.31}, we are tacitly assuming that the path takes no east steps at height $0$. Note also that if $\operatorname{comp}ath$ remains above the diagonal $(0,0)\to (m,n)$ then for each east step $(i-1,j)\to(i,j)$ we must have $i/j \leq m/n$. In particular for the path $\operatorname{comp}ath_0$ that remains closest to the diagonal $(0,0)\to (m,n)$, the last east step at height $j$ must be given by $i=\lfloor j{m}/{n}\rfloor$. Thus
\begin{displaymath}
\mathbf{z}_{\operatorname{comp}ath_0}=\mathbf{z}_{m,n}=\operatorname{comp}rod_{j=1}^n
z_j^{ \lfloor j{m}/{n}\rfloor - \lfloor (j-1){m}/{n}\rfloor }.
\end{displaymath}
We can easily see that the series
\begin{displaymath}
\Omega[\mathbf{z}]= \operatorname{comp}rod_{j=1}^n\frac{1}{1- z_j}
\end{displaymath}
may be viewed as the generating function of all monomials of paths with north and east steps that end at height $n$ and start with a north step. We will refer to the later as the \define{NE-paths}. Our aim is to obtain a formula for the $q$-enumeration of the NE-paths in the $m\times n$ lattice rectangle that remain weakly above a given NE-path $\operatorname{comp}ath$.
Notice that if
\begin{equation} \label{5.32}
\mathbf{z}_\operatorname{comp}ath = z_{r_1}z_{r_2}\cdots z_{r_m},
\qquad {\rm and} \qquad
\mathbf{z}_\delta=z_{s_1}z_{s_2}\cdots z_{s_m}.
\end{equation}
Then $\delta$ remains weakly above $\operatorname{comp}ath$ if and only if
\begin{displaymath}
s_i \geq r_i \qquad\qquad \hbox{for }1\leq i\leq m.
\end{displaymath}
When this happens let us write $\delta \geq \operatorname{comp}ath$. This given, let us set
\begin{displaymath}
C_\operatorname{comp}ath(t):=\sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}(\delta / \operatorname{comp}ath)},
\end{displaymath}
where for $\operatorname{comp}ath$ and $\delta$ as in \operatorname{comp}ref{5.32}, we let $\operatorname{area}(\delta/\operatorname{comp}ath)$ denote the number of lattice cells between $\delta$ and $\operatorname{comp}ath$. In particular, for $\delta$ as in \operatorname{comp}ref{5.32}, we have
\begin{displaymath}
\operatorname{area}(\delta/\operatorname{comp}ath)= \sum_{i=1}^m (s_i-r_i).
\end{displaymath}
Now we have the following fact
\begin{prop} \label{prop5.5} For all path $\gamma$, we have
\begin{equation} \label{5.33}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}\,
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{1-tz_i/z_{i+1}}
\Big|_{z_1^0z_2^0\cdots z_n^0}
=\sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}(\delta/\operatorname{comp}ath)}.
\end{equation}
\end{prop}
\begin{proof}[\bf Proof]
Notice that each Laurent monomial produced by expansion of the product
\begin{equation} \label{5.34}
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{1-tz_i/z_{i+1}}=
\operatorname{comp}rod_{i=1}^{n-1} \left(1+t \frac{z_i}{z_{i+1}}+\left(t\frac{z_i}{z_{i+1}} \right)^2+\cdots
\right)
\end{equation}
may be written in the form
\begin{eqnarray} \label{5.35}
\operatorname{comp}rod_{i=1}^{n-1}\left(t \frac{z_i}{z_{i+1}}\right)^{c_i}
&=& \frac{z_1^{c_1}
\operatorname{comp}rod_{i=2}^{n-1} z_i^{(c_i-c_{i-1})^+} }
{z_{n}^{c_{n-1}} \operatorname{comp}rod_{i=2}^{n-1} z_i^{(c_i-c_{i-1})^-} }\ \operatorname{comp}rod_{i=1}^{n-1} t^{c_i} \nonumber\\[4pt]
&=& \frac{z_{a_1}z_{a_2}\cdots z_{a_\ell}}
{z_{b_1}z_{b_2}\cdots z_{b_\ell}}\ t^{\sum_{i=1}^{n-1}c_i}, \label{5.35}
\end{eqnarray}
with
\begin{eqnarray*}
\ell &=&c_1+\sum_{i=2}^{n-1}(c_i-c_{i-1})^+ = \sum_{i=2}^{n-1}(c_i-c_{i-1})^-+c_{n-1},\\
a_r &=& \min \left\{ j : c_1+\sum_{i=2}^j(c_i-c_{i-1})^+ \geq r \right\},
\qquad {\rm and} \\
b_r &=& \min \left\{ j : \sum_{i=2}^j(c_i-c_{i-1})^- \geq r\right\}.
\end{eqnarray*}
Since $j=a_r$ forces $c_j>0$, we see that the equality
\begin{displaymath}
c_1+\sum_{i=2}^j(c_i-c_{i-1})^+=c_j+\sum_{i=2}^j(c_i-c_{i-1})^-
\end{displaymath}
yields that in \operatorname{comp}ref{5.35} we must have
\begin{equation} \label{5.36}
a_r< b_r , \qquad \hbox{for}\qquad 1\leq r\leq \ell.
\end{equation}
Now for the ratio in \operatorname{comp}ref{5.35} to contribute to the constant term in \operatorname{comp}ref{5.33}, it is necessary and sufficient that the reciprocal of this ratio should come out of the expansion
\begin{equation} \label{5.37}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}
= \frac{1}{z_{r_1}z_{r_2}\cdots z_{r_m}}
\sum_{d_i\geq 0} z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}.
\end{equation}
That is for some $d_1,d_2,\dots,d_n$ we must have
\begin{equation} \label{5.38}
{ z_{r_1}z_{r_2}\cdots z_{r_m}\over z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}}
= {z_{a_1}z_{a_2}\cdots z_{a_\ell}\over z_{b_1}z_{b_2}\cdots z_{b_\ell}}.
\end{equation}
Notice that $z_{a_1}z_{a_2}\cdots z_{a_\ell}$ and $z_{b_1}z_{b_2}\cdots z_{b_\ell}$ have no factor in common, since from the second expression in \operatorname{comp}ref{5.35} we derive that each variable $z_i$ can appear only in one of these two monomials. Thus $z_{a_1}z_{a_2}\cdots z_{a_\ell}$ divides $z_{r_1}z_{r_2}\cdots z_{r_m}$ and $z_{b_1}z_{b_2}\cdots z_{b_\ell}$ divides $z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}$ and in particular $\ell \leq m$. But this, together with the inequalities in \operatorname{comp}ref{5.36} shows that we must have
\begin{displaymath}
z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}
= z_{s_1}z_{s_2} \cdots z_{s_m},
\qquad \hbox{with}\quad s_i\geq r_i \quad\hbox{for}\quad 1\leq i \leq m.
\end{displaymath}
In other words $z_1^{d_1}z_2^{d_2}\cdots z_n^{d_n}$ must be the monomial of a NE-path $\delta\geq \operatorname{comp}ath$. Moreover from the identity in \operatorname{comp}ref{5.38} we derive that
\begin{displaymath}
\operatorname{area}(\delta/\operatorname{comp}ath) = (b_1-a_1)+(b_2-a_2)+\cdots +(b_\ell - a_\ell)
= - \sum_{i=1}^\ell a_i + \sum_{i=1}^\ell b_i.
\end{displaymath}
Thus from the middle expression in 5.35 it follows that
\begin{eqnarray*}
\operatorname{area}(\delta/\operatorname{comp}ath)
&=& -c_1-\sum_{i=2}^{n-1}i(c_i-c_{i-1})^+
+ \sum_{i=2}^{n-1}i(c_i-c_{i-1})^-+nc_{n-1} \\
&=& -c_1-\Big(\sum_{i=2}^{n-1}i(c_i-c_{i-1}) \Big)+c_{n-1}\\
&=& -c_1- \sum_{i=2}^{n-1}i c_i + \sum_{i=1}^{n-2}(i+1) c_i+nc_{n-1}\\
&=&\sum_{i=1}^{n-1}c_i,
\end{eqnarray*}
which is precisely the power of $t$ contributed by the Laurent monomial in \operatorname{comp}ref{5.35}.
Finally, suppose that $\delta$ is a NE-path weakly above $\operatorname{comp}ath$ as in \operatorname{comp}ref{5.32}. This given, let us weight each lattice cell with southeast corner $(a,b)$ with the Laurent monomial $t z_b/z_{b+1}$. Then it is easily seen that for each fixed column $1\leq i\leq m$, the product of the weights of the lattice cells lying between $\delta$ and $\operatorname{comp}ath$ is precisely $t^{s_i-r_i}x_{r_i}/x_{s_i}$. Thus
\begin{displaymath}
\operatorname{comp}rod_{i=1}^m t^{s_i-r_i} \frac{x_{r_i}}{x_{s_i}}
= t^{\operatorname{area}(\delta/\operatorname{comp}ath)} \frac{\mathbf{z}_\operatorname{comp}ath}{\mathbf{z}_\delta}.
\end{displaymath}
Since the left hand side of this identity is in the form given in \operatorname{comp}ref{5.35}, we clearly see that every summand of $C_\operatorname{comp}ath(t)$ will come out of the constant term.
\end{proof}
\begin{rmk} \label{rmk5.5}
It is easy to see that, for $q=1$, the constant term in \operatorname{comp}ref{5.27}, reduces to the one in \operatorname{comp}ref{5.33} with $\operatorname{comp}ath=\operatorname{comp}ath_0$ (the closest path to the diagonal $(0,0)\to (m,n)$). This is simply due to the identity
\begin{displaymath}
\Omega[-uM]\Big|_{q=1}= {(1-u)(1-qtu)\over (1-tu)(1-qu)}\Big|_{q=1}= 1.
\end{displaymath}
Moreover, since the coprimality of the pair $(m,n)$ had no role in the proof of Proposition \ref{prop5.5}, we were led to the formulation of the following conjecture, widely supported by our computer data.
\end{rmk}
\begin{conj} \label{conjV}
For any pair of positive integers $(u,v)$ and any NE-path $\operatorname{comp}ath$ in the $u\times v$ lattice that remains weakly above the lattice diagonal $(0,0)\to (u,v)$ we have
\begin{equation} \label{5.39}
\frac{\Omega[\mathbf{z}]}{\mathbf{z}_\operatorname{comp}ath}\,
\operatorname{comp}rod_{i=1}^{n-1} \frac{1}{(1-qtz_i/z_{i+1})}
\operatorname{comp}rod_{1\leq i<j\leq n} \Omega[-M z_i/z_j]
\Big|_{z_1^0z_2^0\cdots z_n^0}
= \sum_{\delta \geq \operatorname{comp}ath} t^{\operatorname{area}((\delta/\operatorname{comp}ath)} q^{\operatorname{dinv}(\delta)},
\end{equation}
where $\operatorname{dinv}(\delta)$ is computed as in step (3) of Algorithm \ref{algL} for $(km,kn)=(u,v)$.
\end{conj}
Remarkably, the equality in \operatorname{comp}ref{5.39} is still unproven even for general coprime pairs $(m,n)$, except of course for the cases $m=n+1$ proved in \cite{qtCatPos}. What is really intriguing is to explain how the inclusion of the expression
\begin{displaymath}
\operatorname{comp}rod_{1\leq i<j\leq n}\Omega[-M z_i/z_j]
\end{displaymath}
accounts for the insertion of the factor
\begin{displaymath}
q^{\operatorname{dinv}(\operatorname{comp}ath)-\operatorname{area}(\delta/\operatorname{comp}ath)}
\end{displaymath}
in the right hand side of \operatorname{comp}ref{5.33}). A combinatorial explanation of this phenomenon would lead to an avalanche of consequences in this area, in addition to proving Conjecture \ref{conjV}.
\end{document} |
\begin{document}
\title{Three Approaches to the
Quantitative Definition of Information in an Individual Pure
Quantum State}
\author{Paul Vitanyi\thanks{Partially supported by the EU fifth framework
project QAIP, IST--1999--11234, the NoE QUIPROCONE IST--1999--29064,
the ESF QiT Programmme, and ESPRIT BRA IV NeuroCOLT II Working Group
EP 27150.
Part of this work was done
during the author's 1998 stay at Tokyo Institute of Technology,
Tokyo, Japan, as Gaikoku-Jin Kenkyuin at INCOCSAT. A preliminary version
was archived as quant-ph/9907035. Address:
CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands. Email:
{\tt paulv@cwi.nl}}\\
CWI and University of Amsterdam}
\date{}
\maketitle
\thispagestyle{empty}
\begin{abstract}
In analogy of classical Kolmogorov complexity
we develop a theory of the algorithmic information in bits contained
in any one of continuously many pure quantum states:
quantum Kolmogorov complexity.
Classical Kolmogorov complexity
coincides with the new quantum Kolmogorov complexity restricted to
the classical domain. Quantum Kolmogorov complexity is upper
bounded and can be effectively approximated from above.
With high probability a quantum object is incompressible.
There are two alternative approaches possible:
to define the complexity as the length of the shortest
qubit program that effectively describes the object, and to
use classical descriptions with
computable real parameters.
\end{abstract}
\section{Introduction}
While Kolmogorov complexity is the accepted absolute measure
of information content in a {\em classical} individual finite object,
a similar absolute notion is needed for the information
content of a pure quantum state.
\footnote{
For definitions and theory of Kolmogorov complexity consult \cite{LiVi97},
and for quantum theory consult \cite{Pe95}.
}
Quantum theory assumes that every complex vector, except the null vector,
represents a realizable pure quantum state.\footnote{That is,
every complex vector that can be normalized to unit length.}
This leaves open the question
of how to design the equipment that prepares such a pure
state. While there are continuously many pure states in
a finite-dimensional complex vector space---corresponding to all vectors of
unit length---we can finitely
describe only a countable subset. Imposing effectiveness on
such descriptions leads to constructive procedures.
The most general such procedures satisfying universally agreed-upon
logical principles of effectiveness are quantum Turing machines, \cite{BV97}.
To define quantum Kolmogorov complexity
by way of quantum Turing machines leaves
essentially two options:
\begin{enumerate}
\item
We want to describe every quantum superposition exactly; or
\item
we want to take
into account the number of bits/qubits in the specification
as well the accuracy of the quantum state produced.
\end{enumerate}
We have to deal with three problems:
\begin{itemize}
\item
There are continuously many quantum Turing machines;
\item
There are continously many pure quantum states;
\item
There are continuously many qubit descriptions.
\end{itemize}
There are uncountably many quantum Turing machines
only if we allow arbitrary real rotations in the definition of
machines. Then, a quantum Turing machine can only be universal
in the sense that it can approximate the computation of an
arbitrary machine, \cite{BV97}. In descriptions using universal
quantum Turing machines
we would have to account for the closeness of approximation,
the number of steps required to get this precision, and the like.
In contrast, if we fix the rotation
of all contemplated machines to a single primitive rotation $\theta$
with $\cos \theta = 3/5$ and $\sin \theta = 4/5$ then there
are only countably many Turing machines and the universal machine
simulates the others exactly \cite{ADH97}.
Every quantum
Turing machine computation using arbitrary real rotations
can be approximated to any precision by machines with fixed
rotation $\theta$ but in general cannot be simulated
exactly---just like in the case of the simulation of
arbitrary quantum Turing machines by a universal
quantum Turing machine. Since exact simulation is impossible
by a fixed universal quantum Turing machine anyhow, but arbitrarily
close approximations are possible by Turing machines using
a fixed rotation like $\theta$, we are motivated to fix
$Q_1 , Q_2 , \ldots$ as a standard enumeration of
quantum Turing machines using only rotation $\theta$.
Our next question is whether we want programs
(descriptions) to be in classical bits
or in qubits?
The intuitive notion of computability requires
the programs to be classical. Namely, to prepare a quantum state
requires a physical apparatus that ``computes'' this quantum state
from classical specifications.
Since such specifications have effective descriptions,
every quantum state that can be prepared can
be described effectively in descriptions consisting of classical bits.
Descriptions consisting of arbitrary pure quantum states
allows noncomputable (or hard to compute)
information to be hidden in the bits of
the amplitudes.
In Definition~\ref{def.pqscomp} we call a pure quantum state {\em directly
computable} if there is a (classical) program such that
the universal quantum Turing machine computes that state from
the program and then halts in an appropriate fashion.
In a computational setting we naturally
require that directly computable pure quantum states can be
prepared.
By repeating the preparation we can obtain
arbitrarily many copies of the pure quantum state.
\footnote{See the discussion in \cite{Pe95}, pp. 49--51.
If descriptions are not effective then we are not going to use them in our
algorithms except possibly on inputs from an ``unprepared''
origin. Every quantum state used in a quantum computation
arises from some classically preparation or is possibly
captured from some unknown origin. If the latter, then we can consume
it as conditional side-information or an oracle.}
Restricting ourselves to an effective enumeration of
quantum Turing machines and classical descriptions
to describe by approximation continuously many pure quantum states is
reminiscent of the construction of continuously many real numbers
from Cauchy sequences of rational numbers, the rationals being
effectively enumerable.
The second approach
considers the shortest effective qubit description
of a pure quantum state. This can also be properly
formulated in terms of the conditional version
of the first approach. An advantage of this version is that the
upper bound on the complexity of a pure
quantum state is immediately given by the number of qubits involved in the
literal description of that pure quantum state.
The status
of incompressibility and degree of uncomputability is as yet
unknown and potentially a source of problems with this approach.
The third approach is to give programs for the $2^{n+1}$ real numbers
involved in the precise description of the $n$-qubit state. Then
the question reduces to the problem of describing lists of
real numbers.
In the classical situation there are also several variants
of Kolmogorov complexity that are very meaningful in their
respective settings: plain Kolmogorov complexity, prefix complexity,
monotone complexity, uniform complexity,
negative logarithm of universal measure, and so on \cite{LiVi97}.
It is therefore not surprising that in the more complicated situation
of quantum information several different choices of complexity
can be meaningful and unavoidable in different settings.
\section{Classical Descriptions}
The complex
quantity $\bracket{x}{z}$ is the inner product of vectors $\ket{x}$
and $\ket{z}$.
Since pure quantum states $\ket{x}, \ket{z}$ have unit length,
$|\bracket{x}{z}| = | \cos \theta |$ where $\theta$ is the
angle between vectors $\ket{x}$ and $\ket{z}$
and $|\bracket{x}{z}|^2$ is the probability of outcome
$\ket{x}$ being
measured from state $\ket{z}$, \cite{Pe95}. The idea is as follows.
A {\em von Neumann measurement} is a decomposition of the Hilbert space
into subspaces that are mutually orthogonal, for example an
orthonormal basis is an observable. Physicists like to
specify observables as Hermitian matrices, where the
understanding is that the eigenspaces of the matrices
(which will always be orthogonal) are the actual subspaces.
When a measurement is performed, the state is projected
into one of the subspaces (with probability equal to the square of
the projection). So the subspaces correspond to the possible {\em outcomes}
of a measurement. In the above case we project $\ket{z}$ on outcome
$\ket{x}$ using projection $\shar{x}{x}$ resulting in
$\bracket{x}{z} \ket{x}$.
Our model of computation is a quantum
Turing machine with classical binary program $p$ on the input tape
and a quantum auxiliary input
on a special conditional input facility. We think of this auxiliary input
as being given as a pure quantum state $\ket{y}$
(in which case it can be used only once),
as a mixture density matrix $\rho$, or
(perhaps partially) as a classical program from which
it can be computed. In the last case, the classical program can of course
be used indefinitely often.\footnote{We can even allow that the conditional
information $y$ is infinite or noncomputable, or an oracle.
But we will not need this in the present paper.
}
It is therefore not only important {\em what} information is
given conditionally, but also {\em how} it is described---like
this is the sometimes the case in the classical version
of Kolmogorov complexity for other reasons that would additionally
hold in the quantum case.
We impose the condition that the set of {\em halting
programs} ${\cal P}_y = \{p: T(p | y) < \infty \}$ is {\em prefix-free}:
no program in ${\cal P}_y$ is a proper prefix of another program in
${\cal P}_y$. Put differently, the Turing machine scans all of a
halting program $p$ but never scans the bit following the last
bit of $p$: it is {\em self-delimiting}.
\footnote{One can also use a model were the input $p$ is delimited
by distinguished markers. Then the Turing machine always knows where
the input ends. In the self-delimiting case the endmarker must be
implicit in the halting program $p$ itself. This encoding of the
endmarker carries an inherent penalty in the form of increased length:
typically a prefix code of an $n$-length binary string has length
about $n+ \log n + 2 \log \log n$ bits, \cite{LiVi97}.}
\footnote{
There are two possible interpretations for the computation relation
$Q(p, y) = \ket{x}$. In the narrow interpretation
we require that $Q$ with $p$ on the input tape
and $y$ on the conditional tape halts with $\ket{x}$
on the output tape. In the wide interpretation we can
define pure quantum states by requiring
that for every precision $\delta > 0$ the computation
of $Q$ with $p$ on the input tape
and $y$ on the conditional tape
and $\delta$ on a tape where the precision is to be supplied
halts with $\ket{x'}$
on the output tape and $|\bracket{x}{x'}|^2 \geq 1-\delta$.
Such a notion of ``computable''
or ``recursive'' pure quantum states is similar to Turing's
notion of ``computable numbers.''
In the remainder of this section we use the narrow interpretation.
}
\begin{definition}
\rm
The {\em (self-delimiting) complexity} of $\ket{x}$
with respect to quantum Turing machine $Q$
with $y$ as conditional input given for free is
\[
K_Q (\ket{x} | y ) :=
\min_{p} \{ l(p) + \lceil - \log(|\bracket{z}{x}|^2) \rceil :
Q(p, y) = \ket{z} \}
\]
where $l(p)$ is the number of bits in the specification $p$,
$y$ is an input quantum state and
$\ket{z}$ is
the quantum state produced by the computation $Q(p, y)$,
and $\ket{x}$ is the target state that one is
trying to describe.
\end{definition}
\begin{theorem}\label{theo.inv}
There is a universal machine
\footnote{
We use ``$U$'' to denote a universal (quantum) Turing machine
rather than a unitary matrix.}
$U$ such that for all machines $Q$
there is a constant $c_Q$ (the length of the
description of the index of $Q$ in the enumeration)
such that for all quantum states $\ket{x}$ we have
$K_U (\ket{x} |y) \leq K_Q (\ket{x}|y) + c_Q$.
\end{theorem}
\begin{proof}
There is a universal quantum Turing machine $U$ in the standard enumeration
$Q_1 , Q_2, \ldots$ such that for every quantum Turing machine
$Q$ in the enumeration there is a self-delimiting program $i_Q$
(the index of $Q$) and $U(i_Q p , y) = Q(p,y)$ for all $p,y$.
Setting $c_Q = l(i_Q)$ proves the theorem.
\end{proof}
We fix once and for all a reference universal quantum Turing machine $U$
and define the quantum Kolmogorov complexity as
\begin{eqnarray*}
&& K (\ket{x} | y) := K_U (\ket{x}|y), \\
&& K (\ket{x}) := K_U (\ket{x} | \epsilon ),
\end{eqnarray*}
where $\epsilon$ denotes the absence of any conditional
information.
The definition is continuous:
If two quantum states are very close then their quantum Kolmogorov
complexities are very close. Furthermore, since we can approximate
every (pure quantum) state $\ket{x}$ to arbitrary closeness, \cite{BV97},
in particular, for every constant $\epsilon > 0$
we can compute a (pure quantum) state $\ket{z}$
such that
$|\bracket{z}{x}|^2 > 1-\epsilon$.
\footnote{We can view this as the probability of the possibly
noncomputable outcome $\ket{x}$
when executing projection $\shar{x}{x}$ on $\ket{z}$
and measuring outcome $\ket{x}$.}
For this definition to be
useful it should satisfy:
\begin{itemize}
\item
The complexity of a pure state that can be directly computed should be the
length of the shortest program that computes that state. (If the
complexity is less then this may lead to discontinuities when we restrict
quantum Kolmogorov complexity to the domain of classical objects.)
\item
The quantum Kolmogorov complexity of a classical object should
equal the classical Kolmogorov complexity of that object (up to
a constant additive term).
\item
The quantum Kolmogorov complexity of a quantum object should
have an upper bound. (This is necessary for the complexity
to be approximable from above, even if the quantum object is
available in as many copies as we require.)
\item
Most objects should be ``incompressible'' in terms of quantum
Kolmogorov complexity.
\item
In a probabilistic ensemble the expected quantum Kolmogorov
complexity should be about equal (or have another meaningful
relation) to the von Neumann entropy.
\footnote{In the classical case the average self-delimiting
Kolmogorov complexity
equals the Shannon entropy up to an additive constant depending
on the complexity of the distribution concerned.}
\end{itemize}
For a quantum system
$\ket{z}$
the quantity $P(x):= |\bracket{z}{x}|^2$ is the probability that
the system passes a test for $\ket{x}$, and vice versa.
The term $\lceil - \log(|\bracket{z}{x}|^2) \rceil$ can be viewed
as the
code word length to redescribe $\ket{x}$ given $\ket{z}$
and an orthonormal basis with $\ket{x}$ as one of the basis vectors
using
the well-known Shannon-Fano prefix code.
This works as follows: For every state $\ket{z}$ in
$N :=2^n$-dimensional Hilbert space
with basis vectors ${\cal B} = \{ \ket{e_0}, \ldots , \ket{e_{N-1}}\}$ we have
$\sum_{i=0}^{N-1} |\bracket{e_i }{z}|^2 =1$. If the basis has
$\ket{x}$ as one of the basis vectors, then we can
consider $\ket{z}$ as a random variable that assumes value $\ket{x}$
with probability $|\bracket{x}{z}|^2$. The Shannon-Fano code word
for $\ket{x}$ in the probabilistic ensemble
${\cal B}, (|\bracket{e_i}{z}|^2)_i$ is
based on the probability $|\bracket{x}{z}|^2$ of $\ket{x}$
given $\ket{z}$ and has length
$\lceil - \log(|\bracket{x}{z}|^2) \rceil$. Considering a canonical
method of constructing an orthonormal basis
${\cal B} = \ket{e_0}, \ldots, \ket{e_{N-1}}$
from a given basis
vector, we can choose ${\cal B}$ such that
$K({\cal B}) = \min_i \{ K(\ket{e_i}) \} +O(1)$.
The Shannon-Fano code is appropriate for our purpose since it is optimal
in that it achieves the least expected code word
length---the expectation taken over the probability of the
source words---up to 1 bit by Shannon's Noiseless Coding Theorem.
\subsection{Consistency with Classical Complexity}
Our proposal would not be useful if it were the case that for
a directly computable object the complexity is less than the
shortest program to compute that object. This would imply
that the code corresponding to the
probabilistic component in the description is possibly shorter than
the difference in program lengths for programs for an approximation
of the object and the
object itself. This would penalize definite description compared
to probabilistic description and in case of classical objects
would make quantum Kolmogorov complexity less than classical
Kolmogorov complexity.
\begin{theorem}\label{theo.equiv}
Let $U$ be the reference universal quantum Turing machine
and let $\ket{x}$ be a basis vector in a directly computable orthonormal
basis ${\cal B}$ given $y$: there is
a program $p$ such that $U(p, y)= \ket{x}$.
Then $K(\ket{x} | y)= \min_p \{l(p): U(p, y)= \ket{x} \}$
up to $K({\cal B}|y) +O(1)$.
\end{theorem}
\begin{proof}
Let $\ket{z}$ be such that
\[
K (\ket{x} | y ) =
\min_{q} \{ l(q) + \lceil - \log(|\bracket{z}{x}|^2) \rceil :
U(q, y) = \ket{z} \} .
\]
Denote the program $q$ that minimizes the righthand side
by $q_{\min}$
and the program $p$ that minimizes the expression in the statement
of the theorem by $p_{\min}$.
By running $U$ on all binary strings (candidate programs)
simultaneously dovetailed-fashion
\footnote{A {\em dovetailed} computation is a method related
to Cantor's diagonalization to run all programs alternatingly
in such a way that every program eventually makes progress. On
an list of programs $p_1, p_2, \ldots$ one divides
the overall computation into stages $k:=1,2, \ldots$.
In stage $k$ of the overall computation one
executes the $i$th computation step of every program $p_{k-i+1}$
for $i:=1, \ldots , k$.}
one can enumerate all objects that are directly computable given $y$
in order of their halting programs. Assume that $U$ is also
given a $K({\cal B}|y)$ length program $b$ to compute
${\cal B}$---that is, enumerate the basis
vectors in ${\cal B}$.
This way $q_{\min}$ computes
$\ket{z}$, the program $b$ computes ${\cal B}$.
Now since the vectors of ${\cal B}$ are mutually orthogonal
\[ \sum_{\ket{e} \in {\cal B}} | \bracket{z}{e}|^2 = 1 .
\]
Since $\ket{x}$ is one of the basis vectors
we have $- \log |\bracket{z}{x}|^2$ is the length of
a prefix code (the Shannon-Fano code) to compute $\ket{x}$ from $\ket{z}$
and ${\cal B}$.
Denoting this code by $r$ we have that the concatenation $q_{\min} b r$
is a program to compute $\ket{x}$: parse it into
$q_{\min}, b,$ and $r$ using the self-delimiting
property of $q_{\min}$ and $b$. Use
$q_{\min}$ to compute $\ket{z}$ and use $b$ to compute ${\cal B}$,
determine the
probabilities $|\bracket{z}{e}|^2$ for all basis vectors
$\ket{e}$ in ${\cal B}$. Determine the Shannon-Fano code words
for all the basis vectors from these probabilities.
Since $r$ is the code word for $\ket{x}$ we can now
decode $\ket{x}$. Therefore,
\[ l(q_{\min} ) + \lceil - \log(|\bracket{z}{x}|^2) \rceil
\geq l( p_{\min}) - K({\cal B}|y) - O(1) \]
which was what we had to prove.
\end{proof}
\begin{corollary}\label{cor.clasquant}
\rm
On classical objects (that is, the natural numbers
or finite binary strings that are all directly computable) the
quantum Kolmogorov complexity coincides up
to a fixed additional constant with the self-delimiting
Kolmogorov complexity since $K({\cal B}|n) = O(1)$ for the standard
classical basis ${\cal B}= \{0,1\}^n$.
\footnote{
This proof does not show that it coincide up to an additive constant term
with the original plain complexity defined by Kolmogorov, \cite{LiVi97},
based on Turing machines where the input is delited by distinguished markers.
The same proof for the plain Kolmogorov complexity shows
that it coincides up to a logarithmic additive term.
}
(We assume that the information about the dimensionality
of the Hilbert space is given conditionally.)
\end{corollary}
\begin{remark}
\rm
Fixed additional constants are no problem since
the complexity also varies by fixed additional constants due to the choice of
reference universal Turing machine.
\end{remark}
\subsection{Upper Bound on Complexity}
A priori, in the worst case $K(\ket{x} |n )$
is possibly $\infty$. We show that the worst-case has a $2n$ upper bound.
\begin{lemma}
For all $n$-qubit quantum states $\ket{x}$ we
have $K(\ket{x} |n)\leq 2n+O(1)$.
\end{lemma}
\begin{proof}
For every state $\ket{x}$ in $N :=2^n$-dimensional Hilbert space
with basis vectors $\ket{e_0}, \ldots , \ket{e_{N-1}}$ we have
$\sum_{i=0}^{N-1} |\bracket{e_i }{x}|^2 =1$. Hence there is an $i$
such that $|\bracket{e_i }{x}|^2 \geq 1/N$.
Let $p$ be a $K(i|n)+O(1)$-bit program to construct a
basis state $\ket{e_i}$ given $n$.
Then $l(p) \leq n + O(1)$.
Then $K ( \ket{x} |n ) \leq l(p) - \log (1/N) \leq 2n + O(1)$.
\end{proof}
\begin{comment}
\begin{remark}
\rm
One may think that $K(\ket{x} ) \leq 3n/2 + O(1)$.
Namely, for every diagonal $d \in \{ d_0, \ldots , d_{2^n - 1} \}$
for some basis vector $e$ we have $|\bracket{e}{d}|^2 \geq 1/2^{n/2}$.
Moreover, for every vector $x$ there is a $d$ such that
$| \bracket{e}{x}|^2 \geq | \bracket{e}{d}|^2$.
But the argument seems wrong because there is equal probability
for all basis vectors that a diagonal is observed?
\end{remark}
\end{comment}
\subsection{Computability}
In the classical case Kolmogorov complexity is not computable
but can be approximated from above by a computable process.
The non-cloning property prevents us from copying an unknown pure
quantum state given to us. Therefore, an approximation from
above that requires checking every output state against the
target state destroys the latter. To overcome the fragility of
the pure quantum target state one has to postulate that it
is available as an outcome in a measurement.
\begin{theorem}
Let $\ket{x}$ be the pure quantum state we want to describe.
{\rm (i)} The quantum Kolmogorov complexity $K(\ket{x})$ is not computable.
{\rm (ii)}
If we can repeatedly execute the projection $\shar{x}{x}$
and perform a measurement with outcome $\ket{x}$, then
the quantum Kolmogorov complexity $K(\ket{x})$
can be approximated
from above by a computable process with arbitrarily
small probability of error $\alpha$ of giving a too small value.
\end{theorem}
\begin{proof}
The uncomputability follows a fortiori from the classical case.
The semicomputability follows because we have established an upper
bound on the quantum Kolmogorov complexity, and we can simply
enumerate all halting classical programs up to that length by running their
computations dovetailed fashion. The idea is as follows:
Let the target state be $\ket{x}$
of $n$ qubits. Then, $K(\ket{x}|n) \leq 2n + O(1)$. (The
unconditional case $K(\ket{x})$ is similar
with $2n$ replaced by $2(n + \log n)$.)
We want to identify a program $x^*$ such that $p:=x^*$ minimizes
$l(p) - \log |\bracket{x}{U(p,n)}|^2$ among all candidate programs.
To identify it in the limit,
for some fixed $k$ satisfying (\ref{eq.alpha}) below
for given $n, \alpha , \epsilon$,
repeat the computation of every halting program
$p$ with $l(p) \leq 2n+O(1)$ at least $k$ times and perform the assumed
projection and measurement. For every halting program $p$ in the dovetailing
process we estimate the probability
$q :=|\bracket{x}{U(p,n)}|^2$ from the fraction $m/k$:
the fraction of $m$ positive outcomes out of $k$ measurements.
The probability that the estimate $m/k$ is off from the real
value $q$ by more than
an $\epsilon q$ is given by Chernoff's bound:
for
$0 \leq \epsilon \leq 1$,
\begin{equation}
\label{chernoff}
P ( |m- qk | > \epsilon qk )
\leq 2e^{ - \epsilon^2 qk /3}.
\end{equation}
This means that the probability that the deviation $|m/k - q|$
exceeds $\epsilon q$ vanishes exponentially with growing $k$.
Every candidate program $p$ satisfies
(\ref{chernoff}) with its own $q$ or $1-q$. There are $O(2^{2n})$
candidate programs $p$ and hence also $O(2^{2n})$ outcomes $U(p,n)$
with halting computations.
We use this estimate to upper bound the probability of error $\alpha$.
For given $k$, the probability
that {\em some} halting candidate program $p$ satisfies
$ |m- qk | > \epsilon qk$
is at most $\alpha$ with
\[ \alpha \leq \sum_{U(p,n) < \infty } 2e^{ - \epsilon^2 q k /3} .\]
The probability that {\em no} halting program does so is
at least $1- \alpha$. That is, with probability
at least $1-\alpha$ we have
\[ (1- \epsilon)q \leq \frac{m}{k} \leq (1+ \epsilon)q \]
for every halting program $p$.
It is convenient to restrict attention to the case that all $q$'s are large.
Without loss of generality,
if $q < \frac{1}{2}$ then consider $1-q$ instead of $q$.
Then,
\begin{equation}\label{eq.alpha}
\log \alpha \leq 2n- (\epsilon^2 k \log e )/ 6 +O(1).
\end{equation}
The approximation algorithm is as follows:
{\bf Step 0:} Set the required degree of approximation $\epsilon < 1/2$
and the number of trials $k$ to achieve the required probability of error $\alpha$.
{\bf Step 1:} Dovetail the running of all candidate programs until the
next halting program is enumerated.
Repeat the computation of the new halting program $k$ times
{\bf Step 2:} If there is more than one program $p$ that achieves the
current minimum then choose the program with the smaller length
(and hence least number of successfull observations).
If $p$ is the selected program with $m$ successes out of $k$ trials
then set the current approximation of $K(\ket{x})$ to
\[l(p) - \log \frac{m}{(1+\epsilon)k} .\]
This exceeds the proper value
of the approximation based on the real $q$ instead
of $m/k$ by at most 1 bit for all $\epsilon < 1$.
{\bf Step 3:} Goto {\bf Step 1}.
\end{proof}
\subsection{Incompressibility}
\begin{definition}\label{def.pqscomp}
\rm
A pure quantum state $\ket{x}$ is {\em computable} if
$K(\ket{x}) < \infty$. Hence all finite-dimensional pure
quantum states are computable.
We call a pure quantum state {\em directly
computable} if there is a program $p$ such that
$U(p)= \ket{x}$.
\end{definition}
The standard orthonormal basis---consisting of all $n$-bit strings---of
the $2^n$-dimensional
Hilbert space ${\cal H}_N$ has
at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
that satisfy $K(\ket{e_i} |n) \geq n-c$. This is the standard counting argument
in \cite{LiVi97}. But what about nonclassical orthonormal bases?
\begin{lemma}\label{lem.lowb}
There is a (possibly nonclassical) orthonormal basis of the $2^n$-dimensional
Hilbert space ${\cal H}_N$ such
that at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
satisfy $K(\ket{e_i} |n) \geq n-c$.
\end{lemma}
\begin{proof}
Every orthonormal basis of ${\cal H}_N$
has $2^n$ basis vectors and there are at most
$m \leq \sum_{i=0}^{n-c-1} 2^i = 2^{n-c}-1$ programs of length less than
$n-c$. Hence there are at most $m$ programs
available to approximate the basis vectors.
We construct an orthonormal basis satisfying the lemma:
The set of directly computed pure quantum states
$\ket{x_0}, \ldots , \ket{x_{m-1}}$
span an $m'$-dimensional subspace ${\cal A}$ with $m' \leq m$
in the $2^n$-dimensional Hilbert space ${\cal H}_N$ such
that ${\cal H}_N = {\cal A} \oplus {\cal A}^{\perp}$.
Here ${\cal A}^{\perp}$ is a $(2^n - m')$-dimensional
subspace of ${\cal H}_N$ such that every vector in it is
perpendicular to every vector in ${\cal A}$.
We can write every element $\ket{x} \in {\cal H}_N$ as
\[
\sum_{i=0}^{m'-1} \alpha_i \ket{a_i}+ \sum_{i=0}^{2^n-m'-1} \beta_i \ket{b_i}
\]
where the $\ket{a_i}$'s form an orthonormal basis
of ${\cal A}$ and the $\ket{b_i}$'s form an
orthonormal basis of $ {\cal A}^{\perp}$ so that
the $\ket{a_i}$'s and $\ket{b_i}$'s form an orthonormal basis $K$
for ${\cal H}_N$. For every directly computable state
$\ket{x_j} \in {\cal A}$ and basis vector $\ket{b_i} \in A^{\perp}$ we have
$|\bracket{x_j}{b_i} |^2 = 0$
implying that
$K(\ket{x_j}|n) - \log | \bracket{x_j}{b_i} |^2 = \infty$
and therefore $K(\ket{b_i}|n) > n-c$
($0 \leq j < m, 0 \leq i < 2^n - m'$).
This proves the lemma.
\end{proof}
We generalize this lemma to arbitrary bases:
\begin{theorem}\label{theo.lowb}
Every orthonormal basis $\ket{e_0}, \dots ,
\ket{e_{2^n-1}}$ of the $2^n$-dimensional
Hilbert space ${\cal H}_N$ has
at least $2^n (1-2^{-c})$ basis vectors $\ket{e_i}$
that satisfy $K(\ket{e_i}|n) \geq n-c$.
\end{theorem}
\begin{proof}
Use the notation of the proof of Lemma~\ref{lem.lowb}.
Assume to the contrary that there are $>2^{n-c}$
basis vectors $\ket{e_i}$ with $K(\ket{e_i}|n) < n-c$.
Then at least two of them, say $\ket{e_0}$
and $\ket{e_1}$ and some
pure quantum state $\ket{x}$ directly computed from a $<(n-c)$-length program
satisfy
\begin{equation}\label{eq.ex}
K(\ket{e_i}|n) = K(\ket{x}|n) + \lceil - \log |\bracket{e_i}{x}|^2 \rceil .
\end{equation}
($i=0,1$). This means that $K(\ket{x}|n)<n-c-1$ since not both
$\ket{e_0}$ and $\ket{e_1}$ can be equal to $\ket{x}$.
Hence for every directly computed pure quantum state of complexity
$n-c-1$ there is at most one basis state of the same complexity
(in fact only if that basis state is identical with the directly
computed state.)
Now eliminate all directly computed pure quantum states $\ket{x}$ of
complexity $n-c-1$ together with the basis states $\ket{e}$
that stand in relation Equation~\ref{eq.ex}. We are now
left with $> 2^{n-c-1}$ basis states that stand in relation
of Equation~\ref{eq.ex} with the remaining at most
$2^{n-c-1}-1$ remaining directly computable pure quantum states
of complexity $\leq n-c-2$.
Repeating the same argument we end up with $>1$ basis vector
that stand in relation of Equation~\ref{eq.ex} with 0
directly computable pure quantum states of complexity $\leq 0$
which is impossible.
\end{proof}
\begin{corollary}
The uniform probability
$\Pr\{\ket{x}: K(\ket{x}|n) \geq n-c \} \geq 1-1/2^c$.
\end{corollary}
\begin{example}
\rm
We elucidate the role of the $- \log | \bracket{x}{z} |^2$
term.
Let $x$ be a random classical string with $K(x) \geq l(x)$
and let $y$ be a string obtained from $x$
by complementing one bit. It is known (Exercise 2.2.8 in \cite{LiVi97})
that for every such $x$ of length $n$ there is such a $y$ with complexity
$K(y|n) = n - \log n +O(1)$. Now let $\ket{z}$ be a pure quantum state which has
classical bits except the difference bit between $x$ and $y$ that has
equal probabilities of being observed as ``1'' and as ``0.''
We can prepare $\ket{z}$ by giving $y$ and the position of the
difference bit (in $\log n$ bits)
and therefore $K(\ket{z}|n) \leq n + O(1)$. Since from
$\ket{z}$ we have probability $\frac{1}{2}$ of obtaining $x$
by observing the particular bit in superposition
and $K(x|n) \geq n$ it follows $K(\ket{z} |n) \geq n + O(1)$ and
therefore $K(\ket{z} |n) = n + O(1)$.
\begin{comment}
Since $\ket{z}$ is
a directly computable state this is consistent with Corollary~\ref{cor.clasquant}.
\end{comment}
From $\ket{z}$ we have probability $\frac{1}{2}$ of obtaining $y$
by observing the particular bit in superposition which (correctly) yields that
$K(y|n) \leq n +O(1)$.
\end{example}
\subsection{Conditional Complexity}
We have used the conditional complexity $K(\ket{x}|y)$
to mean the minimum sum of the length of a classical program to compute
$\ket{z}$ plus the negative logarithm of the probability of outcome
$\ket{x}$ when executing projection $\shar{x}{x}$ on $\ket{z}$
and measuring, given the pure quantum
state $y$ as input on a separate input tape.
In the quantum situation the notion of inputs consisting
of pure quantum states is subject to very special rules.
Firstly, if we are given an unknown pure quantum state $\ket{y}$ as
input it can be used only once, that is, it is irrevocably consumed
and lost in the computation. It cannot be copied or cloned without
destroying the original \cite{Pe95}. This phenomenon is
subject to the so-called {\em no-cloning theorem} and means that there is
a profound difference between giving a directly computable pure quantum
state as a classical program or giving it literally. Given as a
classical program we can prepare and use arbitrarily many copies of it.
Given as an (unknown) pure quantum state in superposition it can be
used as start of a computation only once---unless of course we
deal with an identity computation in which the input state is simply
transported to the output state. This latter computation nonetheless
destroys the input state.
If an unknown state $\ket{y}$ is given as input (in the conditional for example)
then the no-cloning theorem of quantum computing says it can be used
only {\em once}. Thus, for a non-classical pure quantum state $\ket{x}$
we have
\[ K(\ket{x},\ket{x} | \ket{x}) \leq K(\ket{x})+O(1) \]
rather than
$K(x,x|x)=O(1)$ as in the case for classical objects $x$. This
holds even if $\ket{x}$ is directly computable but is
given in the conditional in the form of an unknown pure quantum state. However,
if $\ket{x}$ is directly computable and
the conditional is a classical program to compute
this directly computable state, then
that program can be used over and over again.
In the previous example,
if the conditional $\ket{x}$ is directly computable, for example
by a classical program $p$, then we have
both $K(\ket{x}|p) = O(1)$ and
$ K(\ket{x},\ket{x} | p) = O(1)$.
In particular, for a classical program $p$ that
computes a directly computable state $\ket{x}$ we have
\[ K(\ket{x},\ket{x} | p) = O(1) .\]
It is important here to notice that a classical program for
computing a directly computable quantum state carries {\em more information}
than the directly computable quantum state itself---much like a
shortest program for a classical object carries more information than the
object itself. In the latter case it consists in partial information
about the halting problem. In the quantum case of a directly
computable pure state we have the additional
information that the state is directly computable {\em and}
in case of a shortest classical program additional information
about the halting problem.
\subsection{Sub-Additivity}
Quantum Kolmogorov complexity of directly computable pure
quantum states in simple orthonormal bases is {\em sub-additive}:
\begin{lemma}\label{lem.additive}
For directly computable $\ket{x}, \ket{y}$ both of which
belong to (possibly different) orthonormal bases of
Kolmogorov complexity $O(1)$ we have
\[ K(\ket{x}, \ket{y} ) \leq K(\ket{x}|\ket{y}) + K(\ket{y}) \]
up to an additive constant term.
\end{lemma}
\begin{proof}
By Theorem~\ref{theo.equiv} we there is a program $p_y$ to compute $\ket{y}$
with $l(p)= K(\ket{y})$ and a program
$p_{y \rightarrow x}$ to compute $\ket{x}$ from $\ket{y}$
with $l(p_{y \rightarrow x}) = K(\ket{x}|\ket{y})$ up
to additional constants. Use $p_y$ to
construct two copies of $\ket{y}$ and $p_{y \rightarrow x}$ to construct
$\ket{x}$ from one of the copies of $\ket{y}$.
The separation between
these concatenated binary programs is taken care of
by the self-delimiting property
of the subprograms. The additional constant term
takes care of the couple of $O(1)$-bit
programs that are required.
\end{proof}
\begin{comment}
\begin{definition}
\rm
Define the information $I(\ket{x}:\ket{y})$ in pure quantum state $\ket{x}$
about pure quantum state $\ket{y}$ by
$I(\ket{x}: \ket{y}):= K(\ket{x})-K(\ket{x}| \ket{y})$.
\end{definition}
The proof is identical to that of the same relations in the classical
case since we are dealing with directly computable states. The last displayed
equation is known as ``symmetry of information'' since
it states that the information in a classical program for $\ket{x}$
about a classical program for $\ket{y}$ is
the same as the information in a classical program
for $\ket{y}$ about a classical program
for $\ket{x}$ up to the
additional logarithmic term.
\[ I(\ket{x}:\ket{y}) = I(\ket{y}:\ket{x}) .\]
\end{comment}
\begin{remark}
\rm
In the classical case we have equality in the theorem (up
to an additive logarithmic term).
The proof of the remaining inequality, as given in the classical case,
doesn't hold directly for the quantum case. It would require
a decision procedure that establishes equality between
two pure quantum states without error. While the sub-additivity
property holds in case of directly computable states,
is easy to see that for the general case of pure states
the subadditivity property fails
due to the ``non-cloning'' property.
For example for pure states $\ket{x}$ that are not ``clonable'' we
have:
\[ K(\ket{x}, \ket{x} ) > K(\ket{x}| \ket{x}) + K(\ket{x}) =
K(\ket{x}) + O(1) .\]
\end{remark}
We additionally note:
\begin{lemma}
For all directly computable pure states $\ket{x}$ and $\ket{y}$ we have
$K(\ket{x}, \ket{y}) \leq K(\ket{y}) - \log | \bracket{x}{y}|^2$
up to an additive logarithmic term.
\end{lemma}
\begin{proof}
$ K(\ket{x}|\ket{y}) \leq - \log | \bracket{x}{y}|^2$ by the proof
of Theorem~\ref{theo.equiv}.
Then, the lemma follows by Lemma~\ref{lem.additive}.
\end{proof}
\begin{comment}
\section{Application to QFA}
A one-way {\em quantum finite automaton} (QFA)
is the quantum version of the classical finite automaton.
We follow te definition in \cite{ANTSV99}.
A QFA has a finite set of {\em basis states} $Q$ consisting
of three disjoint subsets $Q_1$ (accepting states), $Q_0$ (rejecting
states) and $Q_n$ (non-halting states). The set $Q_h= Q_0 \bigcup Q_1$
is the set of halting states. The QFA has a finite
nonempty {\em input alphabet} $\Sigma$. The input is placed
between special input markers
$\#, \$ \notin \Sigma$ indicating the left end and the right end of the
input and the working alphabet is $\Gamma = \Sigma \bigcup \{\#,\$$.
For every $a \in \Gamma$ there is a unitary transformation $U_a$
on the space ${\cal C}^{|Q|}$ where ${\cal C}$ denotes the complex reals.
The QFA starts in a distinguished
basis state $q_0 \in Q$. A {\em state} of the QFA is a superposition
of basis states, in particular the QFA starts in state $\ket{q_0}$.
The computation of the QFA on input $\#a_1 \ldots a_n \$$
applies $U_{\#} U_{a_1} \ldots U_{a_n} U_{\$}$ in sequence to
$\ket{q_0}$ until a halting state occurs. In particular,
a transformation according to $a \in \Gamma$ does the following:
\begin{enumerate}
\item
If $\ket{\phi}$ is the current state then $\phi' := U_a (\phi)$.
\item
We measure $\phi'$ according to the observables
$E_i = \mbox{ span}\{\ket{q}: q \in Q_i \}$,
$i \in \{0,1,n\}$. The probability of observing $E_i$ equals
the squared norm of the projection of $\ket{\phi '}$ on $E_i$.
On measurement the superposition of the automaton collapses to the
projection of one of the spaces observed and is renormalized.
If we observe $E_1$ or $E_0$ then the input is accepted or rejected,
otherwise the computation continues.
\end{enumerate}
A QFA {\em accepts} or {\em rejects} a language $L \subseteq \Sigma^*$
with probability $p > \frac{1}{2}$ if it accepts every word in $L$
with probability at least $p$ and rejects every word not in $L$
with probability at least $p$.
A priori it is by no means obvious that QFA cannot accept nonregular
languages like $L_1 = \{0^k1^k: k > 0 \}$.
Using incompressibility this is very simple to show and in fact
we can give a characterization of QFA languages.
\begin{lemma}
The language $L_1$ is not accepted by a QFA.
\end{lemma}
\begin{proof}
Suppose a QFA accepts $L_1$. Then for every input word $0^k1^k$
we can do the following. First run the QFA until we have processed
all of $0^k$. Then the QFA is in a superposition say $\ket{\psi}$.
If we now continue running the QFA feeding it only $1$'s
then the first accepting state it meets must be after precisely $k$
such $1$'s with probability more than $\frac{1}{2}$.
The description of the QFA together with the description of $\ket{\psi}$
is a description of $KQ$-complexity $O(1)$. But we can choose $k$
such that $KQ(k) \geq \log k$: contradiction for $k$ large enough.
\end{proof}
\end{comment}
\begin{comment}
\appendix
\section{Potentially Useful Ideas}
\begin{remark}
\rm
The following arguments may be useful later:
Use the notation of the proof of Lemma~\ref{lem.lowb}.
Let $P$ be the linear operator that projects a state $\ket{x}$
on the subspace $A$: If $\ket{x} = \ket{x_A} + \ket{x_{A^{\perp}}}$
then $\ket{Px} = \ket{x_A}$. The {\em trace} of $P$ is defined by
$\trace{P} = \sum_{\ket{e} \in K} \bracket{e}{Pe}$ where
$K$ is an orthonormal basis of the Hilbert space ${\cal H}_N$.
It is known that the trace is invariant under change of $K$.
The orthonormal basis $K$ constructed in the proof of Lemma~\ref{lem.lowb}
demonstrates
\[ \trace{P} = \sum_{i=0}^{m'-1} \bracket{a_i}{a_i}=m' \leq 2^{n-c}-1 \] since
$\ket{Pa_i}=\ket{a_i}$ if $\ket{a_i}$ is an element of the orthonormal basis
for $A$ and $ \sum_{i=0}^{2^n -m'-1} \bracket{b_i}{Pb_i}=0$ since all the
$\ket{Pb_i}$'s are 0. Then, for every basis $K'= (\ket{e_0}, \ldots , \ket{e_{2^n-1}})$
we have
\[ \sum_{\ket{e} \in K'} \bracket{e}{Pe} \leq 2^{n-c}-1 , \]
where the $\bracket{e}{Pe}= | \bracket{e}{Pe}|^2 = \cos \theta$ where $\theta$ is the
angle between $\ket{e}$ and $\ket{Pe}$.
By the concavity of the logarithm function
\[ - \sum_{\ket{e} \in K'} \log | \bracket{e}{Pe}|^2 > 2^n c . \]
So if the directly computable vector $\ket{x^e}$
is involved in the shortest program for $\ket{e}$
\[ \sum_{\ket{e} \in K'} ( C(\ket{e}) \geq
\sum_{\ket{e} \in K'} ( C( \ket{x^e}) + c) ,\]
and
\begin{equation}\label{eq.ub1}
\sum_{\ket{e} \in K'} C(\ket{x^e}) \leq 2^n (n-c).
\end{equation}
Let $\alpha_i := |A_i|$ where $A_i := \{\ket{e}: \ket{x^e} = \ket{x_i} \}$.
That is, $\alpha_i$ is the number of basis vectors
that use $\ket{x_i}$ as directly computable vector involved in the
shortest program. Without loss of generality we can assume there is only
one such directly computable vector for every basis vector.
From Equation~\ref{eq.ub1} we have
\[ \sum_{i=0}^{2^{n-c}-1} \frac{ \alpha_i}{2^n} C(\ket{x_i} ) \leq n-c .\]
Note that $p_i = \alpha_i /2^n$ is the probability that a uniformly chosen
basis vector $e$ belongs to $A_i$. It is known that the $p_i$-expectation
of the complexity equals (up to an additive constant) the $p_i$-expectation
of $\log p_i$ (the entropy) \cite{LiVi97}:
\[ \sum_{i=0}^{2^{n-c}-1} p_i C(\ket{x_i} ) =
- ( \sum_{i=0}^{2^{n-c}-1} p_i \log p_i ) +O(1) \leq n-c+O(1) .\]
Consequently, the $p_i$-expectation of $\alpha_i$ is at least $2^{c-O(1)}$.
This means that if we uniformly at random pick a basis vector then
the expectation is that it belongs to an $A_i$ of size at least $2^{c-O(1)}$.
\end{remark}
\begin{theorem}
With uniform probability of at least $1-1/2^c$ an $n$-qubit quantum
state $\ket{x}$ satisfies $C(\ket{x}) \geq n-c$.
\end{theorem}
\begin{proof}
There are $2^n$ basis vectors and there are at most
$m \leq \sum_{i=0}^{n-c-1} 2^i = 2^{n-c}-1$ programs of length less than
$n-c$. Hence there are at most $m$ programs
available to approximate the basis vectors.
The set of directly computed pure quantum states is
$\ket{x_0}, \ldots , \ket{x_{m-1}}$
spanning a $m'$-dimensional subspace ${\cal A}$ with $m' \leq m$
in the $2^n$-dimensional Hilbert space ${\cal H}_N$ such
that ${\cal H}_N = {\cal A} \oplus {\cal A}^{\perp}$.
Here ${\cal A}^{\perp}$ is a $\geq (2^n - m')$-dimensional
subspace of ${\cal H}_N$ such that every vector in it is
perpendicular to every vector in ${\cal A}$. Consider the
directly computable states as observables.
We can write every element $\ket{x} \in {\cal H}_N$ as
\[
\sum_{i=0}^{m'-1} \alpha_i \ket{a_i}+ \sum_{i=0}^{2^n-m'-1} \beta_i \ket{b_i}
\]
where the $\ket{a_i}$'s form an orthonormal basis
of ${\cal A}$ and the $\ket{b_i}$'s form an
orthonormal basis of $ {\cal A}^{\perp}$.
Note that the $\ket{a_i}$'s and $\ket{b_i}$'s form an orthonormal basis
for ${\cal H}_N$. The uniform probability
that the state $\ket{x}$ will not be observed in ${\cal A}$ is
(by symmetry):
\[
\sum_{i=0}^{2^n-m'-1} | \beta_i \overline{\beta_i}|^2
= \frac{2^n-m'}{2^n} > 1 - 2^{-c}.
\]
Hence the uniform expectation of the probability that state $\ket{x}$
is observed in ${\cal A}$ at all, and hence the uniform expectation
of the maximal probability that it is observed as any vector in
${\cal A}$ is $< 2^{-c}$.
***********
Let $p_{i} = | \bracket{x}{x_i}|^2$ be the probability
of observing basis vector $\ket{x_i}$ when we are in state $\ket{x}$.
The set of $\ket{x}$ with complexity exceeding $n-c$ is
\[
\{\ket{x}: C (\ket{x_i}) - \log | \bracket{x}{x_i}|^2 \geq n-c
\mbox{ for all $i$ } (0 \leq i \leq m-1) \}.
\]
Since $C ( \ket{x_i} ) = C(i)+O(1)$,
if $\theta_i$ is the angle between the vectors
$\ket{x}$ and $\ket{x_i}$ then $\cos^2 \theta_i \leq 1/2^{n-c-C(i)-O(1)}$
such that
\[
-2^{-\frac{n-c-C(i)}{2}}
\leq \cos \theta_i
\leq 2^{-\frac{n-c-C(i)}{2}}
\]
for all $0 \leq i < 2^{n-c}-1$.
\end{proof}
\begin{remark}
\rm
What about $C(x,y)=C(x)+C(y|x)$ up to log term?
If the answer to previous remark is positive than
this equality holds certainly for directly computed
sattes and also wrt symmetry of information.
\end{remark}
\begin{remark}
\rm
What about the expected quantum KC? Is it equal
to the entropy of the probability distribution?
\end{remark}
\end{comment}
\section{Qubit Descriptions}
One way to avoid two-part descriptions as we used above
is to allow qubit programs as input. This leads to the
following definitions, results, and problems.
\begin{definition}
\rm
The {\em qubit complexity}
of $\ket{x}$ with respect to quantum Turing machine $Q$
with $y$ as conditional input given for free is
\[
KQ_Q (\ket{x} | y ) :=
\min_{p} \{ l(\ket{p}) :
Q(\ket{p}, y) = \ket{x} \}
\]
where $l(\ket{p})$ is the number of qubits in
the qubit specification $\ket{p}$,
$\ket{p}$ is an input quantum state,
$y$ is given conditionally, and
$\ket{x}$ is
the quantum state produced by the computation $Q(\ket{p}, y)$:
the target state that one
describes.
\end{definition}
Note that here too
there are two possible interpretations for the computation relation
$Q(\ket{p}, y) = \ket{x}$. In the narrow interpretation
we require that $Q$ with $\ket{p}$ on the input tape
and $y$ on the conditional tape halts with $\ket{x}$
on the output tape. In the wide interpretation we require
that for every precision $\delta > 0$ the computation
of $Q$ with $\ket{p}$ on the input tape
and $y$ on the conditional tape
and $\delta$ on a tape where the precision is to be supplied
halts with $\ket{x'}$
on the output tape and $|\bracket{x}{x'}|^2 \geq 1-\delta$.
Additionally one can require that the approximation finishes
in a certain time, say, polynomial in
$l(\ket{x})$ and $1/\delta$.
In the remainder of this section we can allow either interpretation
(note that the ``narrow'' complexity will always be at least as
large as the ``wide''
complexity).
Fix an enumeration of quantum Turing machines like in Theorem~\ref{theo.inv},
this time with Turing machines that use qubit programs.
Just like before it is now straightforward to derive an Invariance Theorem:
\begin{theorem}
There is a universal machine $U$ such that for all machines $Q$
there is a constant $c$ (the length of a self-delimiting
encoding of the index of $Q$ in the enumeration)
such that for all quantum states $\ket{x}$ we have
$KQ_U (\ket{x} |y) \leq KQ_Q (\ket{x}|y) + c$.
\end{theorem}
We fix once and for all a reference universal quantum Turing machine $U$
and express the {\em qubit quantum Kolmogorov complexity} as
\begin{eqnarray*}
&& KQ (\ket{x} | y) := KQ_U (\ket{x}|y), \\
&& KQ (\ket{x}) := KQ_U (\ket{x} | \epsilon ),
\end{eqnarray*}
where $\epsilon$ indicates the absence of conditional
information (the conditional tape contains the ``quantum state''
with 0 qubits). We now have immediately:
\begin{lemma}
$KQ ( \ket{x} ) \leq l(\ket{x})+O(1)$.
\end{lemma}
\begin{proof}
Give the reference universal machine $\ket{1^n 0} \otimes \ket{x}$
as input where $n$ is the index of the identity quantum Turing machine
that transports the attached pure quantum state $\ket{x}$ to
the output.
\end{proof}
It is possible to define unconditional $KQ$-complexity
in terms of conditional $K$-complexity as follows:
Even for pure quantum states that are not directly computable from
effective descriptions we have
$K( \ket{x} | \ket{x}) = O(1)$. This naturaly gives:
\begin{lemma}
The qubit quantum Kolmogorov complexity of
$\ket{x}$ satisfies
\[ KQ ( \ket{x} ) = \min_{p}
\{ l( \ket{p}): K(\ket{x} | \ket{p} ) \} + O(1),\]
where $l(\ket{p})$ denotes the number of qubits in $\ket{p}$.
\end{lemma}
\begin{proof}
Transfer the conditional $\ket{p}$ to the input using an $O(1)$-bit
program.
\end{proof}
We can generalize this definition
to obtain conditional $KQ$-complexity.
\subsection{Potential Problems of Qubit Complexity}
While it is clear that (just as with the previous aproach)
the qubit complexity is not computable, it is unknown to the author
whether one can approximate the qubit complexity from above by
a computable process in any meaningful sense.
In particular, the dovetailing approach
we used in the first approach now doesn't seem applicable due
to the non-countability of the potentential qubit program candidates.
While it is clear that the qubit complexity of a pure quantum
state is at least 1, why would it need to be more than
one qubit since the probability amplitude can be any complex number?
In case the target pure quantum state is a classical binary string,
as observed by Harry Buhrman,
Holevo's theorem \cite{Pe95} tells us that on average one cannot transmit
more than $n$ bits of classical information by $n$-qubit messages
(without using entangled qubits on the side).
This suggests that for every $n$ there exist classical binary
strings of length $n$ that have qubit complexity at least $n$.
This of course leaves open the case of the non-classical pure quantum
states---a set of measure one---and
of how to prove incompressibility
of the overwhelming majority of states. These matters have since been
investigated by A. Berthiaume, S. Laplante, and W. van Dam
(paper in preparation).
\section{Real Descriptions}
A final version of quantum Kolmogorov complexity uses
computable real parameters to describe the pure quantum state
with complex probability amplitudes.
This requires two reals per complex probability amplitude, that is,
for $n$ qubits one requires $2^{n+1}$ real numbers in the worst case.
Since every computable real number may require a separate program,
a computable $n$ qubit state may require $2^{n+1}$ finite programs.
While this approach does not allow the development of a clean
theory in the sense of the previous approaches, it can be directly
developed in terms of algorithmic thermodynamics---an extension
of Kolmogorov complexity to randomness of infinite sequences
(such as binary expansions of real numbers)
in terms of coarse-graining and sequential Martin-L\"off tests, completely
analogous to Peter G\'acs theory \cite{Ga94,LiVi97}.
\end{document} |
\begin{document}
\vspace*{0.5in}
{\Large
\centerline {\bf Quantum Wavelet Transforms: Fast Algorithms}
\centerline {\bf and Complete Circuits\footnote{Presented at 1st NASA Int. Conf. on Quantum Computing and Communication, Palm Spring, CA, Feb. 17-21, 1998.}}
}
\centerline {\bf Amir Fijany and Colin P. Williams}
\centerline {\it Jet Propulsion Laboratory, California Institute of Technology}
\centerline {4800 Oak Grove Drive, Pasadena, CA 91109}
\centerline {Email: Amir.Fijany@jpl.nasa.gov \, and \, Colin.P.Williams@jpl.nasa.gov}
\centerline {\bf Abstract}
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the
multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies $D^{(4)}$ transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
\noindent {\it Key Words: Quantum Computing, Quantum Algorithms, Quantum
circuits, Wavelet Transforms}
\section{Introduction}
The field of quantum computing has undergone an explosion of activity over the past few years. Several important quantum algorithms are now known. Moreover, prototypical quantum computers have already been built using nuclear magnetic resonance [1, 2] and nonlinear optics technologies [3]. Such devices are far from being general-purpose computers.
Nevertheless, they constitute significant milestones along the road to practical quantum computing.
A quantum computer is a physical device whose natural evolution over time can be interpreted as the execution of a useful computation. The basic element of a quantum computer is the quantum bit or "qubit", implemented physically as the state of some convenient 2-state quantum system such as the spin of an electron. Whereas a classical bit must be either a 0 or a 1
at any instant, a qubit is allowed to be an arbitrary superposition of a 0 and a 1 simultaneously. To make a quantum memory register we simply consider the simultaneous state of (possibly entangled) tuples of qubits.
The state of a quantum memory register, or any other isolated quantum system, evolves in time according to some unitary operator. Hence, if the evolved state of a quantum memory register is interpreted as having implemented some computation, that computation must be describable as a unitary operator. If the quantum memory register consists of $n$ qubits, this operator will be represented, mathematically, as some $2^n \times 2^n$ dimensional unitary matrix.
Several quantum algorithms are now known, the most famous examples being Deutsch and Jozsa's algorithm for deciding whether a function is even or balanced [4], Shor's algorithm for factoring a composite integer [5] and Grover's algorithm for finding an item in an unstructured database [6]. However, the field is growing rapidly and new quantum algorithms are
being discovered every year. Some recent examples include Brassard, Hoyer, and Tapp's quantum algorithm for counting the number of solutions to a problem [7], Cerf, Grover and Williams quantum algorithm for solving NP-complete problems by nesting one quantum search within another [8] and van Dam, Hoyer, and Tapp's algorithm for distributed quantum computing [9].
The fact that quantum algorithms are describable in terms of unitary transformations is both good news and bad for quantum computing. The good news is that knowing that a quantum computer must perform a unitary transformation allows theorems to be proved about the tasks that quantum computers can and cannot do. For example, Zalka has proved that Grover's algorithm is optimal [10]. Aharonov, Kitaev, and Nisan have proved that a quantum algorithm that involves intermediate measurements is no more powerful than one that postpones all measurements until the end of the unitary evolution stage [11]. Both these proofs rely upon quantum algorithms being unitary transformations. On the other hand, the bad news is that many computations that we would like to perform are not originally described in terms of unitary operators. For example, a desired computation might be nonlinear, irreversible or both nonlinear and irreversible. As a unitary transformation must be linear and reversible we might need to be quite creative in encoding a desired computation on a quantum computer. Irreversibility can be handled by incorporating extra "ancilla" qubits that permit us to remember the input corresponding to each output. But nonlinear transformations are still problematic.
Fortunately, there is an important class of computations, the unitary transforms, such as the Fourier transform, Walsh-Hadamard transform and assorted wavelet transforms, that are describable, naturally, in terms of unitary operators. Of these, the Fourier and Walsh-Hadamard transforms have been the ones studied most extensively by the quantum computing community. In fact, the quantum Fourier transform (QFT) is now recognized as being pivotal in many known quantum algorithms [12]. The quantum Walsh-Hadamard transform is a critical component of both Shor's algorithm [5] and Grover's algorithm [6]. However, the wavelet transforms are every bit as useful as the Fourier transform, at least in the context of classical computing. For example, wavelet transforms are particularly suited to exposing the multi-scale structure of a signal. They are likely to be useful for quantum image processing and quantum data compression. It is natural therefore to consider how to achieve a quantum wavelet transform.
Starting with the unitary operator for the wavelet transform, the next step in the process of finding a quantum circuit that implements it, is to factor the wavelet operator into the direct sum, direct product and dot product of smaller unitary operators. These operators correspond to 1-qubit and 2-qubit quantum gates. For such a circuit to be physically realizable, the number of gates within it must be bounded above by a polynomial in the number of qubits, $n$. Finding such a factorization can be extremely challenging. For example, although there are known algebraic techniques for factoring an arbitrary $2^n \times 2^n$ operator, e.g. [13], they are guaranteed to produce $O(2^n)$, i.e., exponentially many, terms in the factorization. Hence, although such a factorization is mathematically valid, it is physically unrealizable because, when treated as a quantum circuit design, would require too many quantum gates. Indeed, Knill has {\it proved} that an arbitrary unitary matrix will require exponentially many quantum gates if we restrict ourselves to using only gates that correspond to all 1-qubit rotations and XOR [14]. It is therefore clear that the key enabling factor for achieving an efficient quantum implementation, i.e., with a polynomial time and space complexity, is to exploit the specific structure of the given unitary operator.
Perhaps the most striking example of the potential for achieving compact and efficient quantum circuits is the case of the Walsh-Hadamard transform. In quantum computing, this transform arises whenever a quantum register is loaded with all integers in the range 0 to $2^n-1$. Classically, application of the Walsh-Hadamard transform on a vector of length $2^n$
involves a complexity of $O(2^n)$. Yet, by exploiting the factorization of the Walsh-Hadamard operator in terms of the Kroenecker product, it can implemented with a complexity of $O(1)$ by $n$ identical 1-qubit quantum gates. Likewise, the classical FFT algorithm has been found to be implementable in a polynomial space and time complexity, quantum circuit
[15] (see also Sec. 2.3). However, exploitation of the operator structure arising in the wavelet transforms (and perhaps other unitary transforms) is more challenging.
A key technique, in classical computing, for exposing and exploiting specific structure of a given unitary transform is the use of permutation matrices. In fact, there is an extensive literature in classical computing on the use of permutation matrices for factorizing unitary transforms into simpler forms that enable efficient implementations to be
devised (see, for example, [16] and [17]). However, the underlying assumption in using the permutation matrices in classical computation is that they can be implemented easily and inexpensively. Indeed, they are considered so trivial that the cost of their implementation is often not included in the complexity analysis. This is because any permutation matrix
can be described by its effect on the ordering of the elements of a vector. Hence, it can simply be implemented by re-ordering the elements of the vector involving only data movement and without performing any arithmetic operations. As is shown in this paper, the permutation matrices also play a pivotal role in the factorization of the unitary operators that arise in the wavelet transforms. However, unlike the classical computing, the cost of implementation of the permutation matrices cannot be neglected in quantum computing. Indeed, the main issue in deriving feasible and efficient quantum circuits for the quantum wavelet transforms considered in this paper, is the design of efficient quantum circuits for certain permutation matrices. Note that, any permutation matrix acting on $n$ qubits can mathematically be represented by a $2^n \times 2^n$ unitary operator. Hence, it is possible to factor any permutation matrix by using general techniques such as [13] but this would lead to an exponential time and space complexity. However, the permutation matrices, due to their specific structure (i.e., sparsity pattern), represents a very special subclass of unitary matrices. Therefore, the key to achieve an efficient quantum implementation of permutation matrices is the exploitation of this specific structure.
In this paper, we first develop efficient quantum circuits for a set of permutation matrices arising in the development of the quantum wavelet transforms (and the quantum Fourier transform). We propose three techniques for an efficient quantum implementation of permutation matrices, depending on the permutation matrix considered. In the first technique, we show that a certain class of permutation matrices, designated as {\it qubit permutation matrices}, can directly be described by their effect on the ordering of qubits. This quantum description is very similar to classical description of the permutation matrices. We show that the {\it Perfect Shuffle} permutation matrix, designated as $\Pi_{2^n}$, and the {\it Bit Reversal} permutation matrix, designated as $P_{2^n}$, which arise in the quantum wavelet and Fourier transforms (as well as in many other classical computations) belong to this class. We present a new gate, designated as the {\it qubit swap gate} or $\Pi_4$, which can be used to directly derive efficient quantum circuits for implementation of the qubit permutation matrices. Interestingly, such circuits for quantum implementation of $\Pi_{2^n}$ and $P_{2^n}$ lead to new factorizations of these two permutation matrices which were not previously know in classical computation. A second technique is based on a {\it quantum arithmetic description} of permutation matrices. In particular, we consider the {\it downshift} permutation matrix, designated as $Q_{2^n}$, which plays a major role in derivation of quantum wavelet transforms and also frequently arises in many classical computations [16]. We show that a quantum description of $Q_{2^n}$ can be given as a {\it quantum arithmetic operator}. This description then allows the quantum implementation of $Q_{2^n}$ by using the quantum arithmetic circuits proposed in [18].
A third technique is based on developing totally new factorizations of the permutation matrices. This technique is the most case dependent, challenging, and even counterintuitive (from a classical computing point of view). For this technique, we again consider the permutation matrix $Q_{2^n}$ and we show that it can be factored in terms of FFT which then allows its implementation by using the circuits for QFT. More interestingly, however, we derive a recursive factorization of $Q_{2^n}$ which was not previously known in classical computation. This new factorization enables a direct and efficient implementation of $Q_{2^n}$. Our analysis of though a limited set of permutation matrices reveals some of the surprises of quantum computing in contrast to classical computing. That is, certain operations that are hard to implement in classical computing are much easier to implement on quantum computing and vice versa. As a specific example, while the classical implementation of $\Pi_{2^n}$ and $P_{2^n}$ are much harder (in terms of the data movement pattern) than $Q_{2^n}$, their quantum implementation is much easier and more straightforward than $Q_{2^n}$.
Given a wavelet kernel, its application is usually performed according to the packet or pyramid algorithms. Efficient quantum implementation of theses two algorithms requires efficient circuits for operators of the form $I_{2^{n-i}} \otimes \Pi_{2^i}$ and $\Pi_{2^i} \oplus I_{2^n - 2^i}$, for some $i$, where $\otimes$ and $\oplus$ designate, respectively, the kronecker product and the direct sum operator. We show that these operators can be efficiently implemented by using our proposed circuits for implementation of $\Pi_{2^i}$. We then consider two representative wavelet kernels, the Haar [17] and Daubechies $D^{(4)}$ [19] wavelets which have previously been considered by Hoyer [20]. For the Haar wavelet, we show that Hoyer's proposed solution is incomplete since it does not lead to a gate-level circuit and, consequently, it does not allow the analysis of time and space complexity. We propose a scheme for design of a complete gate-level circuit for the Haar wavelet and analyze its time and space complexity. For the Daubechies $D^{(4)}$ wavelet, we develop three new factorizations which lead to three gate-level circuits for its implementation. Interestingly, one of this factorization allows efficient implementation of Daubechies $D^{(4)}$ wavelets by using the circuit for QFT.
\section{Efficient Quantum Circuits for two Fundamental Qubits Permutation Matrices: Perfect Shuffle and Bit-Reversal}
In this section, we develop quantum circuits for two fundamental permutation matrices, the perfect shuffle, $\Pi_{2^n}$, and the bit reversal, $P_{2^n}$, permutation matrices, which arise in quantum wavelet and Fourier transforms as well as many classical computations involving unitary transforms for signal and image processing [16]. For quantum computing, these two permutation matrices can directly be described in terms of their effect on ordering of qubits. This enables the design of efficient circuits for their implementation. Interestingly, these circuits lead to the discovery of new factorizations for these two permutation matrices.
\subsection{Perfect Shuffle Permutation Matrices}
A classical description of $\Pi_{2^n}$ can be given by describing its effect on a given vector. If $Z$ is a $2^n$-dimensional vector, then the vector $Y = \Pi_{2^n}Z$ is obtained by splitting $Z$ in half and then shuffling the top and bottom halves of the deck. Alternatively, a description of the matrix $\Pi_{2^n}$, in terms of its elements $\Pi_{ij}$, for $i$
and $j = 0, 1, \cdots, 2^n-1$, can be given as
\begin{equation}
\Pi_{ij} = \left\{ \begin{array}{ll}
1 & \mbox{ if $j = i/2$ and $i$ is even, or if $j = (i - 1)/2 +2^{n-1}$ and
$i$ is odd} \\
0 & \mbox{ otherwise}
\end{array}
\right.
\end{equation}
As first noted by Hoyer [20], a quantum description of $\Pi_{2^n}$ can be given by
\begin{equation}
\Pi_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots \, a_1 \, a_0 \rangle \,
\longmapsto \vert a_0 \, a_{n-1} \, a_{n-2} \, \cdots a_1 \rangle
\end{equation}
That is, for quantum computation, $\Pi_{2^n}$ is the operator which performs the left qubit-shift operation on $n$ qubits. Note that, $\Pi_{2^n}^t$ ($t$ indicates the transpose) performs the right qubit-shift operation, i.e.,
\begin{equation}
\Pi_{2^n}^t: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert a_{n-2} \, \cdots a_1 \, a_0 \, a_{n-1} \rangle
\end{equation}
\subsection{Bit-Reversal Permutation Matrices}
A classical description of $P_{2^n}$ can be given by describing its effect on a given vector. If $Z$ is a $2^n$-dimensional vector and $Y = P_{2^n}Z$, then $Y_i = Z_j$, for $i = 0, 1, \cdots, 2^n-1$, wherein $j$ is obtained by reversing the bits in the binary representation of index $i$. Therefore, a description of the matrix $P_{2^n}$, in terms of its elements $P_{ij}$, for $i$ and $j = 0, 1, \cdots, 2^n-1$, is given as
\begin{equation}
P_{ij} = \left\{ \begin{array}{cc}
1 & \mbox{if $j$ is bit reversal of $i$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
A factorization of $P_{2^n}$ in terms of $\Pi_{2^i}$ is given as [16]
\begin{equation}
P_{2^n} = \Pi_{2^n}(I_2 \otimes \Pi_{2^{n-1}}) \cdots (I_{2^i} \otimes \Pi_{2^{n-i}}) \, \cdots (I_{2^{n-3}} \otimes \Pi_8)(I_{2^{n-2}} \otimes \Pi_4)
\end{equation}
A quantum description of $P_{2^n}$ is given as
\begin{equation}
P_{2^n}: \, \vert a_{n-1} \, a_{n-2}, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert a_0 \, a_1 \, \cdots a_{n-2} \, a_{n-1} \rangle
\end{equation}
That is, $P_{2^n}$ is the operator which reverses the order of $n$ qubits. This quantum description can be seen from the factorization of $P_{2^n}$, given by (5), and quantum description of permutation matrices $\Pi_{2^i}$. It is interesting to note that for classical computation the term "bit-reversal" refers to reversing the bits in the binary representation of index of the elements of a vector while, for quantum computation, the matrix $P_{2^n}$ literally performs a reversal of the order of qubits.
Note that, $P_{2^n}$ is symmetric, i.e., $P_{2^n} = P_{2^n}^t$ [16]. This can be also easily proved based on the quantum description of $P_{2^n}$ since if the qubits are reversed twice then the original ordering of the qubits is restored. This implies that, $P_{2^n}P_{2^n} = I_{2^n}$ and since $P_{2^n}$ is orthogonal, i.e., $P_{2^n}P_{2^n}^t = I_{2^n}$, it then follows that $P_{2^n} = P_{2^n}^t$.
\subsection{Quantum FFT and Bit-Reversal Permutation Matrix}
Here, we review the quantum FFT algorithm since it not only arises in derivation of the quantum wavelet transforms (see Sec. 4.3) but also it represents a case in which the roles of permutation matrices $\Pi_{2^n}$ and $P_{2^n}$ seems to have been overlooked in quantum computing literature.
The classical Cooley-Tukey FFT factorization for a $2^n$-dimensional vector is given by [16]
\begin{equation}
F_{2^n} = A_n A_{n-1} \cdots A_1 P_{2^n} =
{\underline F}_{2^n} P_{2^n}
\end{equation}
where $A_i = I_{2^{n-i}} \otimes B_{2^i}$, $B_{2^i} = \frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & \Omega_{2^{i-1}} \\
I_{2^{i-1}} & - \Omega_{2^{i-1}}
\end{array} \right)
$ and $\Omega_{2^{i-1}} = \mbox {Diag} \{1, \, \omega_{2^i}, \, \omega_{2^i}^2, \, \ldots , \omega_{2^i}^{2^{i-1} -1} \}$ with
$\omega_{2^i} = e^{-{2 \iota \pi} \over {2^i}}$ and $\iota = \sqrt {-1}$. We have that $F_2 = W =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
1 & 1 \\
1 & - 1
\end{array} \right)
$. The operator
\begin{equation}
{\underline F}_{2^n} = A_n A_{n-1} \cdots A_1
\end{equation}
represents the computational kernel of Cooley-Tukey FFT while $P_{2^n}$ represents the permutation which needs to be performed on the elements of the input vector before feeding that vector into the computational kernel. Note that, the presence of $P_{2^n}$ in (7) is due to the accumulation of its factors, i.e., the terms $(I_{2^i} \otimes \Pi_{2^{n-i}})$, as given by (5).
The Gentleman-Sande FFT factorization is obtained by exploiting the symmetry of $F_{2^n}$ and transposing the Cooley-Tukey factorization [16] leading to
\begin{equation}
F_{2^n} = P_{2^n} A_1^t \cdots A_{n-1}^t A_n^t =
P_{2^n} {\underline F}_{2^n}^t
\end{equation}
where
\begin{equation}
{\underline F}_{2^n}^t = A_1^t \cdots A_{n-1}^t A_n^t
\end{equation}
represents the computational kernel of the Gentleman-Sande FFT while $P_{2^n}$ represents the permutation which needs to be performed to obtain the elements of the output vector in the correct order.
In [15] a quantum circuit for the implementation of ${\underline F}_{2^n}$, given by (8), is presented by developing a factorization of the operators $B_{2^i}$ as
\begin{equation}
B_{2^i} =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & \Omega_{2^{i-1}} \\
I_{2^{i-1}} & -\Omega_{2^{i-1}}
\end{array} \right) =
\frac {1} {\sqrt {2}}
\left( \begin{array}{cc}
I_{2^{i-1}} & I_{2^{i-1}} \\
I_{2^{i-1}} & - I_{2^{i-1}}
\end{array} \right)
\left( \begin{array}{cc}
I_{2^{i-1}} & 0 \\
0 & \Omega_{2^{i-1}}
\end{array} \right)
\end{equation}
Let $C_{2^i} =
\left( \begin{array}{cc}
I_{2^{i-1}} & 0 \\
0 & \Omega_{2^{i-1}}
\end{array} \right)
$. It then follows that
\begin{equation}
B_{2^i} = (W \otimes I_{2^{i-1}}) C_{2^i}
\end{equation}
\begin{equation}
A_i = I_{2^{n-i}} \otimes B_{2^i} =
(I_{2^{n-i}} \otimes W \otimes I_{2^{i-1}})(I_{2^{n-i}} \otimes C_{2^i})
\end{equation}
In [15] a factorization of the operators $C_{2^i}$ is developed as
\begin{equation}
C_{2^i} = \theta_{n-1, n-i}\theta_{n-2, n-i} \cdots \theta_{n-i+1, n-i}
\end{equation}
where $\theta_{jk}$ is a two-bit gate acting on $j$th and $k$th qubits.
Using (13)-(14) a circuit for implementation of (8) is developed in [15] and presented in Fig. 1. However, there is an error in the corresponding figure in [15] since it implies that, with a correct ordering of the input qubits, the output qubits are obtained in a reverse order. Note that, as can be seen from (7), the operator ${\underline F}_{2^n}$ performs the FFT operation and provides the output qubits in a correct order if the input qubits are presented in a reverse order.
The quantum circuit for Gentleman-Sande FFT can be obtained from the circuit of Fig. 1 by first reversing the order of gates that build the operator block $A_i$ (and thus building operators $A_i^t$) and then reversing the order of the blocks representing operators $A_i$. By using the Gentleman-Sande circuit, with the input qubits in the correct order the output qubits are obtained in reverse order.
For an efficient and correct implementation of the quantum FFT, one needs to take into account the ordering of the input and output qubits, particularly if the FFT is used as a block box in a quantum computation. If the FFT is used as a stand-alone block or as the last stage in the computation (and hence its output is sampled directly), then it is more efficient to use the Gentleman-Sande FFT since the ordering of the output qubits does not cause any problem. If the FFT is used as the first stage of the computation, then it is more efficient to use the Cooley-Tukey factorization by preparing the input qubits in a reverse order. Note that, as in classical computation, each or a combination of the Cooley-Tukey or Gentleman-Sande FFT factorization can be chosen in a given quantum computation to avoid explicit implementation of $P_{2^n}$ (or, any other mechanism) for reversing the order of qubits and hence achieve a greater efficiency. As an example, in Sec. 4.3 we will show that the use of the Cooley-Tukey rather than the Gentleman-Sande factorization leads to a greater efficiency in quantum implementation by eliminating the need for an explicit implementation of $P_{2^n}$ (or, any other mechanism) for reversing the order of qubits.
\subsection{A Basic Quantum Gate for Efficient Implementation of Qubits Permutation Matrices}
If a permutation matrix can be described by its effect on the ordering of the qubits then it might be possible to devise circuits for its implementation directly. We call the class of such permutation matrices as "Qubit Permutation Matrices". A set of efficient and practically realizable circuits for implementation of Qubit Permutation Matrices can be built by using a new quantum gate, called
{\it the qubit swap gate}, $\Pi_4$, where \begin{equation}
\Pi_4 =
\left( \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{array} \right)
\end{equation}
For quantum computation, $\Pi_4$ is the "qubit swap operator", i.e.,
\begin{equation}
\Pi_4: \, \vert a_1 \, a_0 \rangle \, \longmapsto \, \vert a_0 \, a_1 \rangle
\end{equation}
The $\Pi_4$ gate, shown in Fig. 2.a, can be implemented with three XOR (or Controlled-NOT) gates as shown in Fig. 2.b. The $\Pi_4$ gate offers two major advantages for practical implementation:
\begin{itemize}
\item It performs a local operation, i.e., swapping the two neighboring qubits. This locality can be advantageous in
practical realizations of quantum circuits, and
\item Given the fact that $\Pi_4$ can be implemented using three XOR (or, Controlled-NOT) gates, it is possible to implement conditional operators involving $\Pi_4$, for example, operators of the form $\Pi_4 \oplus I_{2^n - 4}$, by using Controlled$^k$-NOT gates [21].
\end{itemize}
A circuit for implementation of $\Pi_{2^n}$ by using $\Pi_4$ gates is shown in Fig. 3. This circuit is based on an intuitively simple idea of successive swapping of the neighboring qubits, and implements $\Pi_{2^n}$ with a complexity of $O(n)$ by using an $O(n)$ number of $\Pi_4$ gates. It is interesting to note that, this circuit leads to a new (to our knowledge) factorization of $\Pi_{2^n}$ in terms of $\Pi_4$ as
\begin{equation}
\Pi_{2^n} = (I_{2^{n-2}} \otimes \Pi_4)(I_{2^{n-3}} \otimes \Pi_4 \otimes I_2)
\cdots (I_{2^{n-i}} \otimes \Pi_4 \otimes I_{2^{i-2}}) \, \cdots \,
(I_2 \otimes \Pi_4 \otimes I_{2^{n-3}})(\Pi_4 \otimes I_{2^{n-2}})
\end{equation}
This new factorization of $\Pi_{2^n}$ is less efficient than other schemes (see, for example, [16]) for a {\it classical implementation} of $\Pi_{2^n}$. Interestingly, it is derived here as a result of our search for an efficient {\it quantum implementation} of $\Pi_{2^n}$, and in this sense it is only efficient for a quantum implementation. Note also, that a new (to our knowledge) recursive factorization of $\Pi_{2^i}$ directly results from Fig. (3) as
\begin{equation}
\Pi_{2^i} = (I_{2^{i-2}} \otimes \Pi_4)(\Pi_{2^{i-1}} \otimes I_2)
\end{equation}
A circuit for implementation of $P_{2^n}$ by using $\Pi_4$ gates is shown in Fig. 4. Again, this circuit is based on an intuitively simple idea, that is, successive and parallel swapping of the neighboring qubits, and implements $P_{2^n}$ with a complexity of $O(n)$ by using $O(n^2)$ $\Pi_4$ gates. This circuit leads to a new (to our knowledge) factorization of $P_{2^n}$ in terms of $\Pi_4$ as
\begin{equation}
P_{2^n} = ((\underbrace{\Pi_4 \otimes \Pi_4 \cdots \otimes \Pi_4}_{\frac{n}{2}})
(I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n}{2} -1} \otimes I_2))^{\frac{n}{2}}
\end{equation}
for $n$ even, and
\begin{equation}
P_{2^n} = ((I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n-1}{2}})
(\underbrace{\Pi_4 \otimes \cdots \, \Pi_4}_{\frac{n-1}{2}} \otimes I_2))^{\frac{n-1}{2}}
(I_2 \otimes \underbrace{\Pi_4 \otimes \cdots \otimes \Pi_4}_{\frac{n-1}{2}})
\end{equation}
for $n$ odd.
It should be emphasized that this new factorization of $P_{2^n}$ is less efficient than other schemes, e.g., the use of (5) for a {\it classical implementation} (see also [16] for further discussion). However, this factorization is more efficient for a {\it quantum implementation} of $P_{2^n}$. In fact, a quantum implementation of $P_{2^n}$ by using (5) and (17) will result in a complexity of $O(n^2)$ by using $O(n^2)$ $\Pi_4$ gates.
As will be shown, the development of {\it complete} and efficient circuits for implementation of wavelet transforms requires a mechanism for implementation of conditional operators of the forms $\Pi_{2^i} \oplus I_{2^n - 2^i}$ and $P_{2^i} \oplus I_{2^n - 2^i}$, for some $i$. The key enabling factor for a successful implementation of such conditional operators is the use of factorizations similar to (17) and (19)-(20) or, alternatively, circuits similar to those in Figures 3 and 4, along with the conditional operators involving $\Pi_4$ gates.
\section{Quantum Wavelet Algorithms}
\subsection{Wavelet Pyramidal and Packet Algorithms}
Given a wavelet kernel, its corresponding wavelet transform is usually performed according to a packet algorithm (PAA) or a pyramid algorithm (PYA). The first step in devising quantum counterparts of these algorithms is the development of suitable factorizations. Consider the Daubechies fourth-order wavelet kernel of dimension $2^i$, denoted as $D^{(4)}_{2^i}$. First level factorizations of PAA and PYA for a $2^n$-dimensional vector are given as
\begin{equation}
PAA = (I_{2^{n-2}} \otimes D^{(4)}_4)(I_{2^{n-3}} \otimes \Pi_8) \cdots
(I_{2^{n-i}} \otimes D^{(4)}_{2^i})(I_{2^{n-i-1}} \otimes \Pi_{2^{i+1}}) \cdots (I_2 \otimes D^{(4)}_{2^{n-1}}) \Pi_{2^n}D^{(4)}_{2^n}
\end{equation}
\begin{equation}
PYA = (D^{(4)}_4 \oplus I_{2^n-4})(\Pi_8 \oplus I_{2^n-8}) \cdots
(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})(\Pi_{2^{i+1}} \oplus I_{2^n - 2^{i+1}}) \cdots
\Pi_{2^n}D^{(4)}_{2^n}
\end{equation}
These factorizations allow a first level analysis of the feasibility and efficiency of quantum implementations of the packet and pyramid algorithms. To see this, suppose we have a practically realizable and efficient, i.e., $O(i)$, quantum algorithm for implementation of $D^{(4)}_{2^i}$. For the packet algorithm, the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ can be directly and efficiently implemented by using the algorithm for $D^{(4)}_{2^i}$. Also, using the factorization of $\Pi_{2^i}$, given by (17), the operators $(I_{2^{n-i}} \otimes \Pi_{2^i})$ can be implemented efficiently in $O(i)$.
For the pyramid algorithm, the existence of an algorithm for $D^{(4)}_{2^i}$ does not automatically imply an efficient algorithm for implementation of the conditional operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$. An example of such a case is discussed in Sec. 4.4. Thus, careful analysis is needed to establish both the feasibility and efficiency of implementation of the conditional operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ by using the algorithm for $D^{(4)}_{2^i}$. Note, however, that the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be efficiently implemented in $O(i)$ by using the factorization in (17) and the conditional $\Pi_4$ gates.
The above analysis can be extended to any wavelet kernel (WK) and summarized as follows:
\begin{itemize}
\item Packet algorithm: A physically realizable and efficient algorithm for the WK along with the use of (17) leads to a physically realizable and efficient implementation of the packet algorithm.
\item Pyramid algorithm: A physically realizable and efficient algorithm for the WK does not automatically lead to an
implementation of the conditional operators involving WK (and hence the pyramid algorithm) but the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be efficiently implemented by using the factorization in (17) and the conditional $\Pi_4$ gates.
\end{itemize}
\subsection{Haar Wavelet Factorization and Implementation}
The Haar transform can be defined from the Haar functions [17]. Hoyer [20] used a recursive definition of Haar matrices based on the {\it generalized Kronecker product} (see also [17] for similar definitions) and developed a factorization of $H_{2^n}$ as
\begin{eqnarray}
H_{2^n} = & (I_{2^{n-1}} \otimes W) \cdots (I_{2^{n-i}} \otimes W \oplus
I_{2^n - 2^{n-i+1}}) \, \cdots \, (W \oplus I_{2^n - 2}) \times \nonumber \\
& (\Pi_4 \oplus I_{2^n - 4}) \, \cdots \, (\Pi_{2^i} \oplus I_{2^n - 2^i}) \, \cdots \, (\Pi_{2^{n-1}} \oplus I_{2^{n-1}}) \Pi_{2^n}
\end{eqnarray}
Hoyer's circuit for implementation of (23) is shown in Fig 5. However, this represents an {\it incomplete} solution for
quantum implementation and subsequent complexity analysis of the Haar transform. To see this, let
\begin{equation}
H^{(1)}_{2^n} = (I_{2^{n-1}} \otimes W) \cdots (I_{2^{n-i}} \otimes W \oplus
I_{2^n - 2^{n-i+1}}) \, \cdots \, (W \oplus I_{2^n - 2})
\end{equation}
\begin{equation}
H^{(2)}_{2^n} = (\Pi_4 \oplus I_{2^n - 4}) \, \cdots \, (\Pi_{2^i} \oplus I_{2^n - 2^i}) \, \cdots \, (\Pi_{2^{n-1}} \oplus I_{2^{n-1}}) \Pi_{2^n}
\end{equation}
Clearly, the operator $H^{(1)}_{2^n}$ can be implemented in $O(n)$ by using $O(n)$ conditional $W$ gates. But the feasibility of practical implementation of the operator $H^{(2)}_{2^n}$ and its complexity (and consequently those of the factorization in (23)) cannot be assessed unless a mechanism for implementation of the terms $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ is devised.
However, by using the factorizations and circuits similar to (17) and Figure 3, it can be easily shown that the operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$ can be implemented in $O(i)$ by using $O(i)$ conditional $\Pi_4$ gates (or, Controlled$^k$-NOT gates). This leads to the implementation of $H^{(2)}_{2^n}$ and consequently $H_{2^n}$ in $O(n^2)$ by using $O(n^2)$ gates. This represents not only the first practically feasible quantum circuit for implementation of $H_{2^n}$ but also the first complete analysis of complexity of its time and space (gates) quantum implementation. Note that, both operators $(I_{2^{n-i}} \otimes H_{2^i})$ and $(H_{2^i} \oplus I_{2^n - 2^i})$ can be directly and efficiently implemented by using the above algorithm and circuit for implementation of $H_{2^i}$. This implies both the feasibility and efficiency of the quantum implementation of the packet and pyramid algorithms by using our factorization for Haar wavelet kernel.
\subsection{Daubechies $D^{(4)}$ Wavelet and Hoyer's Factorization}
The Daubechies fourth-order wavelet kernel of dimension $2^n$ is given in a matrix form as [22]
\begin{equation}
D^{(4)}_{2^n} =
\left( \begin{array}{ccccccccccc}
c_0 & c_1 & c_2 & c_3 \\
c_3 & -c_2 & c_1 & -c_0 \\
& & c_0 & c_1 & c_2 & c_3 \\
& & c_3 & -c_2 & c_1 & -c_0 \\
\vdots & \vdots & & & & & \ddots \\
& & & & & & & c_0 & c_1 & c_2 & c_3 \\
& & & & & & & c_3 & -c_2 & c_1 & -c_0 \\
c_2 & c_3 & & & & & & & & c_0 & c_1 \\
c_1 & -c_0 & & & & & & & & c_3 & -c_2
\end{array} \right)
\end{equation}
where $c_0 = \frac {(1 + \sqrt {3})} {4 \sqrt {2}}$, $c_1 = \frac {(3 + \sqrt {3})} {4 \sqrt {2}}$, $c_2 = \frac {(3 - \sqrt {3})} {4 \sqrt {2}}$, and $c_3 = \frac {(1 - \sqrt {3})} {4 \sqrt {2}}$. For classical computation and given its sparse structure, the application of $D^{(4)}_{2^n}$ can be performed with an optimal cost of $O(2^n)$. However, the matrix $D^{(4)}_{2^n}$, as given by (26), is not suitable for a quantum implementation. To achieve a feasible and efficient quantum implementation, a suitable factorization of $D^{(4)}_{2^n}$ needs to be developed. Hoyer [20] proposed a factorization of $D^{(4)}_{2^n}$ as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) S_{2^n}(I_{2^{n-1}} \otimes C_0)
\end{equation}
where
\begin{equation}
C_0 =
2 \left( \begin{array}{cc}
c_4 & -c_2 \\
-c_2 & c_4
\end{array} \right)
\mbox{ and }
C_1 = \frac{1}{2}
\left( \begin{array}{cc}
\frac{c_0}{c_4} & 1 \\
1 & \frac{c_1}{c_2}
\end{array} \right)
\end{equation}
and $S_{2^n}$ is a permutation matrix with a classical description given by
\begin{equation}
S_{ij} = \left\{ \begin{array}{cc}
1 & \mbox{ if $i = j$ and $i$ is even, or if $i+2 = j$ \, (mod $2^n$)} \\
0 & \mbox{ otherwise}
\end{array} \right.
\end{equation}
Hoyer's block-level circuit for implementation of (27) is shown in Figure 6. Clearly, the main issue for a practical quantum implementation and subsequent complexity analysis of (27) is the quantum implementation of matrix $S_{2^n}$. To this end,
Hoyer discovered a quantum arithmetic description of $S_{2^n}$ as
\begin{equation}
S_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert b_{n-1} \, b_{n-2} \, \cdots b_1 \, b_0 \rangle
\end{equation}
where
\begin{equation}
b_i = \left\{ \begin{array}{cc}
a_i - 2 \mbox{ \, (mod $n$)}, & \mbox{if $i$ is odd} \\
a_i & \mbox{otherwise}
\end{array} \right.
\end{equation}
As suggested by Hoyer, this description of $S_{2^n}$ then allows its quantum implementation by using quantum arithmetic circuits of [18] with a complexity of $O(n)$. This algorithm can be directly extended for implementation of the operators
$(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and hence the packet algorithm. However, the feasibility and efficiency of an implementation of the operators $(I_{2^{n-i}} \oplus D^{(4)}_{2^i})$ and thus the pyramid algorithm needs further analysis.
\section{Fast Quantum Algorithms and Circuits for Implementation of Daubechies $D^{(4)}$ Wavelet}
In this section, we develop a new factorization of the Daubechies $D^{(4)}$ wavelet. This factorization leads to three new and efficient circuits, including one using the circuit for QFT, for implementation of Daubechies $D^{(4)}$ wavelet.
\subsection{A New Factorization of Daubechies $D^{(4)}$ Wavelet}
We develop a new factorization of the Daubechies $D^{(4)}$ wavelet transform by showing that the permutation matrix $S_{2^n}$ can be written as a product of two permutation matrices as
\begin{equation}
S_{2^n} = Q_{2^n}R_{2^n}
\end{equation}
where $Q_{2^n}$ is the {\it downshift permutation matrix} [16] given by
\begin{equation}
Q_{2^n} = \left( \begin{array}{ccccccc}
0 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
\vdots & \vdots & \vdots & & \ddots \\
0 & 0 & \cdots & 0 & 0 & 1 \cr
1 & 0 & \cdots & 0 & 0 & 0
\end{array} \right)
\end{equation}
and $R_{2^n}$ is a permutation matrix given by
\begin{equation}
R_{2^n} = \left( \begin{array}{cccccccc}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 \\
& \ddots & \ddots & \ddots & \ddots \\
& & & & & & 0 & 1 \cr
& & & & & & 1 & 0
\end{array} \right)
\end{equation}
The matrix $R_{2^n}$ can be written as
\begin{equation}
R_{2^n} = I_{2^{n-1}} \otimes N
\end{equation}
where $
N =
\left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right)
$.
Substituting (35) and (32) into (27), a new factorization of $D^{(4)}_{2^n}$ is derived as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) Q_{2^n}(I_{2^{n-1}} \otimes N)
(I_{2^{n-1}} \otimes C_0) = (I_{2^{n-1}} \otimes C_1) Q_{2^n}
(I_{2^{n-1}} \otimes C_0^\prime)
\end{equation}
where
\begin{equation}
C_0^\prime = N.C_0 = 2
\left( \begin{array}{cc}
-c_2 & c_4 \\
c_4 & -c_2
\end{array} \right)
\end{equation}
Fig. 7 shows a block-level implementation of (36). Clearly, the main issue for a practical quantum gate-level implementation and subsequent complexity analysis of (36) is the quantum implementation of matrix $Q_{2^n}$. In the following, we present three circuits for quantum implementation of matrix $Q_{2^n}$.
\subsection{Quantum Arithmetic Implementation of Permutation Matrix $Q_{2^n}$}
A first circuit for implementation of matrix $Q_{2^n}$ is developed based on its description as a {\it quantum arithmetic operator}. We have discovered such a quantum arithmetic description of $Q_{2^n}$ as
\begin{equation}
Q_{2^n}: \, \vert a_{n-1} \, a_{n-2} \, \cdots a_1 \, a_0 \rangle \,
\longmapsto \vert b_{n-1} \, b_{n-2} \, \cdots b_1 \, b_0 \rangle
\end{equation}
where
\begin{equation}
b_i = a_i - 1 \mbox { \, (mod $n$)}
\end{equation}
This description of $Q_{2^n}$ allows its quantum implementation by using quantum arithmetic circuit of [18] with a complexity of $O(n)$. Note, however, that the arithmetic description of $Q_{2^n}$ is simpler than that of $S_{2^n}$ since it does not involve conditional quantum arithmetic operations (i.e., the same operation is applied to all qubits). This algorithm for quantum implementation of $Q_{2^n}$ and hence $D^{(4)}_{2^n}$ can be directly extended for implementation of the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and hence the packet algorithm. However, the feasibility and efficiency of an implementation of the operators $(I_{2^{n-i}} \oplus D^{(4)}_{2^i})$ and thus the pyramid algorithm needs further analysis.
\subsection{Quantum FFT Factorization of Permutation Matrix $Q_{2^n}$}
A direct and efficient factorization and subsequent circuit for implementation of $Q_{2^n}$ (and hence Daubechies $D^{(4)}$ wavelet) can be derived by using the FFT algorithm. This factorization is based on the observation that $Q_{2^n}$ can be described in terms of FFT as [16]
\begin{equation}
Q_{2^n} = F_{2^n} T_{2^n} F^*_{2^n}
\end{equation}
where $T_{2^n}$ is a diagonal matrix given as
$T_{2^n} = \mbox {Diag} \{1, \, \omega_{2^n}, \, \omega_{2^n}^2, \, \ldots ,
\omega_{2^n}^{2^n -1} \}$ with
$\omega_{2^n} = e^{{-2 \iota \pi} \over {2^n}}$ (* indicates conjugate transpose). As will be seen, it is more efficient to use the Cooley-Tukey factorization, given by (7), and write (40) as
\begin{equation}
Q_{2^n} = {\underline F}_{2^n} P_{2^n} T_{2^n}P_{2^n}{\underline F}^*_{2^n}
\end{equation}
It can be shown that the matrix $T_{2^n}$ has a factorization as
\begin{equation}
T_{2^n} = (G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}}) \cdots
(I_{2^{i-1}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{n-i}}) \cdots
(I_{2^{n-1}} \otimes G(\omega_{2^n}))
\end{equation}
where $G(\omega_{2^n}^k) = \mbox {Diag} \{1, \, \omega_{2^n}^k \} =
\left( \begin{array}{cc}
1 & 0 \\
0 & \omega_{2^n}^k
\end{array} \right)
$. This factorization leads to an efficient implementation of $T_{2^n}$ by using $n$ single qubit $G(\omega_{2^n}^k)$ gates as shown in Fig. 8. Together with the circuit for implementation of $P_{2^n}$ (Fig. 4) and the circuit for implementation of FFT (Fig. 1), they represent a complete gate-level implementation of $D^{(4)}_{2^n}$.
However, a more efficient circuit can be derived by avoiding the explicit implementation of $P_{2^n}$ by showing that the operator
\begin{equation}
P_{2^n}T_{2^n}P_{2^n} = P_{2^n}(G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}})\cdots
(I_{2^{i-1}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{n-i}}) \cdots
(I_{2^{n-1}} \otimes G(\omega_{2^n}))P_{2^n}
\end{equation}
can be efficiently implemented by simply reversing the order of gates in Fig. 8. This is established by the following lemma:
\noindent {\bf Lemma 1.}
\begin{equation}
P_{2^n}(G(\omega_{2^n}^{2^{n-1}}) \otimes I_{2^{n-1}}) =
(I_{2^{n-1}} \otimes G(\omega_{2^n}^{2^{n-1}}))P_{2^n}
\end{equation}
\begin{equation}
P_{2^n} (I_{2^{n-j}} \otimes G(\omega_{2^n}^{2^{j-1}}) \otimes I_{2^{j-1}}) =
(I_{2^{j-1}} \otimes G(\omega_{2^n}^{2^{j-1}}) \otimes I_{2^{n-j}})P_{2^n}
\end{equation}
\begin{equation}
P_{2^n}(I_{2^{n-1}} \otimes G(\omega_{2^n})) = (G(\omega_{2^n}) \otimes
I_{2^{n-1}})P_{2^n}
\end{equation}
\noindent {\bf\it Proof.} This lemma can be easily proved based on the physical interpretation of operations in (44)-(46). The left-hand side of (44) implies first an operation, i.e., application of $G(\omega_{2^n}^{2^{n-1}})$, on the last qubit and then application of $P_{2^n}$ on all the qubits, i.e., reversing the order of qubits. However, this is equivalent to first reversing the order of qubits, i.e., applying $P_{2^n}$, and then applying $G(\omega_{2^n}^{2^{n-1}})$, on the first qubit which is the operation described by the right-hand side of (44). Similarly, the left-hand side of (45) implies first application of $G(\omega_{2^n}^{2^{i-1}})$ on the $(n-i)$th qubit and then reversing the order of qubits. This is equivalent to first reversing the order of qubits and then applying $G(\omega_{2^n}^{2^{i-1}})$ on the $i$th qubit which is the operations described by the right hand side of (45). In a same fashion, the left hand side of (46) implies first application of $G(\omega_{2^n})$ on the first qubit and then reversing the order of qubits which is equivalent to first reversing the order of qubits and then applying $G(\omega_{2^n}^{2^{n-1}})$ on the last qubit, that is, the operations in right-hand side of (46).
Applying (44)-(46) to (43) from left to right and noting that, due to the symmetry of $P_{2^n}$, we have
$P_{2^n}P_{2^n} = I_{2^n}$, it then follows that
\begin{equation}
P_{2^n}T_{2^n}P_{2^n} = (I_{2^{n-1}} \otimes G(\omega_{2^n}^{2^{n-1}})) \cdots
(I_{2^{n-i}} \otimes G(\omega_{2^n}^{2^{n-i}}) \otimes I_{2^{i-1}}) \cdots
(G(\omega_{2^n}) \otimes I_{2^{n-1}})
\end{equation}
The circuit for implementation of (47) is shown in Fig.9 which, as can be seen, has been obtained by reversing the order of gates in Fig. 8. Note that, the use of (47), which is a direct consequence of using the Cooley-Tukey factorization, enables the implementation of (40) without explicit implementation of $ P_{2^n}$.
Using (40) and (47), the complexity of the implementation of $Q_{2^n}$ and thus $D^{(4)}_{2^n}$ is the same as of the quantum FFT, that is, $O(n^2)$ for an exact implementation and $O(nm)$ for an approximation of order $m$ [15]. Note
that, by using (47), (40), and (36) both operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ and
$(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ can be directly implemented. This implies both the feasibility and efficiency
of the quantum implementation of the packet and pyramid algorithms by using this algorithm for quantum implementation of $D^{(4)}_{2^n}$.
\subsection{A Direct Recursive Factorization of Permutation Matrix $Q_{2^n}$}
A new direct and recursive factorization of $Q_{2^n}$ can be derived based on a similarity transformation of $Q_{2^n}$ by using $\Pi_{2^n}$ as
\begin{equation}
\Pi^t_{2^n}Q_{2^n}\Pi_{2^n} =
\left( \begin{array}{cc}
0 & I_{2^{n-1}} \\
Q_{2^{n-1}} & 0
\end{array} \right)
\end{equation}
which can be written as
\begin{equation}
\Pi^t_{2^n}Q_{2^n}\Pi_{2^n} =
\left( \begin{array}{cc}
0 & I_{2^{n-1}} \\
I_{2^{n-1}} & 0
\end{array} \right)
\left( \begin{array}{cc}
Q_{2^{n-1}} & 0 \\
0 & I_{2^{n-1}}
\end{array} \right) =
(N \otimes I_{2^{n-1}})(Q_{2^{n-1}} \oplus I_{2^{n-1}})
\end{equation}
from which $Q_{2^n}$ can be calculated as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(Q_{2^{n-1}} \oplus
I_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Replacing a similar factorization of $Q_{2^{n-1}}$ into (50), we get
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})
(\Pi_{2^{n-1}}(N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}}) \Pi^t_{2^{n-1}} \oplus I_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
By using the identity
\begin{equation}
\Pi_{2^{n-1}} A \Pi^t_{2^{n-1}} \oplus I_{2^{n-1}} =
(I_2 \otimes \Pi_{2^{n-1}})(A \oplus I_{2^{n-1}})(I_2 \otimes \Pi^t_{2^{n-1}})
\end{equation}
for any matrix $A \varepsilon \Re^{2^{n-1} \times 2^{n-1}}$, (51) can be then written as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
((N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}})\oplus I_{2^{n-1}})
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Using the identity
\begin{eqnarray}
(N \otimes I_{2^{n-2}})(Q_{2^{n-2}} \oplus I_{2^{n-2}})\oplus I_{2^{n-1}} & = &
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus
I_{2^{n-2}} \oplus I_{2^{n-1}}) \nonumber \\
& = & (N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus I_{3.2^{n-2}})
\end{eqnarray}
(53) is now written as
\begin{equation}
Q_{2^n} = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(Q_{2^{n-2}} \oplus I_{2^n - 2^{n-2}})
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{equation}
Repeating the same procedures for all $Q_{2^i}$, for $i = n-3$ to 1, and noting that $Q_2 = N$, it then follows
\begin{eqnarray}
Q_{2^n} & = \Pi_{2^n}(N \otimes I_{2^{n-1}})(I_2 \otimes \Pi_{2^{n-1}})
(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}})(I_4 \otimes \Pi_{2^{n-2}})
(N \otimes I_{2^{n-3}} \oplus I_{2^n - 2^{n-2}}) \cdots \nonumber \\
& (I_{2^{n-2}} \otimes \Pi_4)(N \otimes I_2 \oplus I_{2^n -4})
(N \oplus I_{2^n -2})(I_{2^{n-2}} \otimes \Pi^t_4) \cdots
(I_2 \otimes \Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{eqnarray}
The above expression of $Q_{2^n}$ can be further simplified by exploiting the fact that (see Appendix for the proof) every operator of the form $(I_{2^i} \otimes \Pi_{2^{n-i}})$, for $i = n-2$ to $1$, commutes with all operators of the form $(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})$, for $j = i$ to $1$. Using this commutative property, (56) can be now written as
\begin{eqnarray}
Q_{2^n} & = \Pi_{2^n}(I_2 \otimes \Pi_{2^{n-1}})(I_4 \otimes
\Pi_{2^{n-2}}) \cdots (I_{2^{n-2}} \otimes \Pi_{4}) (N \otimes
I_{2^{n-1}}) (N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots \nonumber \\
& \qquad (N \otimes I_2 \oplus I_{2^n -4}) (N \oplus I_{2^n
-2})(I_{2^{n-2}} \otimes \Pi^t_4) \cdots (I_2 \otimes
\Pi^t_{2^{n-1}})\Pi^t_{2^n}
\end{eqnarray}
Using the factorization of $P_{2^n}$ given in (5), we then have
\begin{equation}
Q_{2^n} = P_{2^n}(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})P_{2^n}
\end{equation}
Substituting (58) into (36), a factorization of $D^{(4)}_{2^n}$ is then obtained as
\begin{equation}
D^{(4)}_{2^n} = (I_{2^{n-1}} \otimes C_1) P_{2^n}(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})P_{2^n} (I_{2^{n-1}} \otimes C_0^\prime)
\end{equation}
Using Lemma 1, it then follows that
\begin{equation}
D^{(4)}_{2^n} = P_{2^n}(C_1 \otimes I_{2^{n-1}})(N \otimes I_{2^{n-1}})(N \otimes I_{2^{n-2}} \oplus I_{2^{n-1}}) \cdots (N \otimes I_2 \oplus I_{2^n -4})(N \oplus I_{2^n -2})(C_0^\prime \otimes I_{2^{n-1}})P_{2^n}
\end{equation}
A circuit for implementation of $D^{(4)}_{2^n}$, based on (60), is shown in Fig. 10. Together with the circuit for implementation of $P_{2^n}$, shown in Fig. 4, they represent a complete gate-level circuit for implementation of $D^{(4)}_{2^n}$ with an optimal complexity of $O(n)$.
Using (60) and (19)-(20), the operators $(I_{2^{n-i}} \otimes D^{(4)}_{2^i})$ can be directly and efficiently implemented with a complexity of $O(i)$. This implies both the feasibility and efficiency of the implementation of the packet algorithm by using this algorithm for $D^{(4)}_{2^n}$ wavelet kernel. However, this algorithm is less efficient for implementation of the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$ and hence the pyramid algorithm. To see this, note that, the implementation of the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$, by using (60), requires the implementation of the conditional operators $(P_{2^i} \oplus I_{2^n - 2^i})$. However, these conditional operators cannot be directly implemented by using (19) and (20). An alternative solution is to use the factorization of $P_{2^i}$ in (5) and the conditional operators $(\Pi_{2^i} \oplus I_{2^n - 2^i})$. However, this leads to a complexity of $O(i^2)$ for implementation of operators $(P_{2^i} \oplus I_{2^n - 2^i})$ and hence the operators $(D^{(4)}_{2^i} \oplus I_{2^n - 2^i})$. Therefore, while (60) is optimal for implementation of $D^{(4)}_{2^i}$ and the packet algorithm, it is not efficient for implementation of the pyramid algorithm.
It should be emphasized that this recursive factorization of $Q_{2^n}$, originated by the similarity transformation in (48) and given by (56) and (58), was not previously known in classical computing. Note that, the permutation matrices $\Pi_{2^n}$ and, particularly, $P_{2^n}$ are much harder (in terms of data movement pattern) for a classical implementation than $Q_{2^n}$. In this sense, such a factorization of $Q_{2^n}$ is rather counterintuitive from a classical computing point of view since it involves the use of permutation matrices $\Pi_{2^n}$ and $P_{2^n}$ and thus it is highly inefficient for a classical implementation.
\section{Discussion and Conclusion}
In this paper, we developed fast algorithms and efficient circuits for quantum wavelet transforms. Assuming an efficient quantum circuit for a given wavelet kernel and starting with a high level description of the packet and pyramid algorithms, we analyzed the feasibility and efficiency of the implementation of the packet and pyramid algorithms by using the given wavelet kernel. We also developed efficient and complete gate-level circuits for two representative wavelet kernels, the Haar and Daubechies $D^{(4)}$ kernels. We gave the first complete time and space complexity analysis of the quantum Haar wavelet transform. We also described three complete circuits for Daubechies $D^{(4)}$ wavelet kernel. In particular, we showed that Daubechies $D^{(4)}$ kernel can be implemented by using the circuit for QFT. Given the problem of decoherence, exploitation of parallelism in quantum computation is a key issue in practical implementation of a given computation. To this end, we are currently analyzing the algorithms of this paper in terms of their parallel efficiency and developing more efficient parallel quantum wavelet algorithms.
As shown in this paper, permutation matrices play a pivotal role in the development of quantum wavelet transforms. In fact, not only they arise explicitly in the packet and pyramid algorithms but also they play a key role in factorization of wavelet kernels. For classical computing, the implementation of permutation matrices is trivial. However, for quantum computing, it represents a challenging task and demands new, unconventional, and even counterintuitive (from a classical computing view point) techniques. For example, note that most of the factorizations developed in paper for permutation matrices $\Pi_{2^n}$, $P_{2^n}$, and $Q_{2^n}$ were not previously known in classical computing and, in fact, they are not at all efficient for a classical implementation. Also, implementation of the permutation matrices reveals some of the surprises of quantum computing in contrast to classical computing. In the sense that, certain operations that are hard to implement in classical computing are easier to implement in quantum computing and vice versa. As a concrete example, note that while the classical implementation of permutation matrices $\Pi_{2^n}$ and (particularly) $P_{2^n}$ is much harder (in terms of data movement pattern) than the permutation matrix $Q_{2^n}$, their quantum implementation is much easier and more straightforward than $Q_{2^n}$.
In this paper, we focussed on the set of permutation matrices arising in the development of quantum wavelet transforms and analyzed three techniques for their quantum implementation. However, it is clear that the permutation matrices will also play a major role in deriving compact and efficient factorizations, i.e., with polynomial time and space complexity, for other unitary operators by exposing and exploiting their specific structure. Therefore, we believe strongly that a more systematic study of permutation matrices is needed in order to develop further insight into efficient techniques for their implementation in quantum circuits. Such a study might eventually lead to the discovery of new and more efficient approaches for the implementation of unitary transformations and therefore quantum computation.
\noindent {\bf Acknowledgement}
The research described in this paper was performed at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under contract with National Aeronautics and Space Administration (NASA). This work was supported by the NASA/JPL Center for Integrated Space Microsystems (CISM), NASA/JPL Advanced Concepts Office, and NASA/JPL Autonomy and Information Technology Management Program.
\noindent {\bf Appendix: Commutation of the Operators $I_{2^i} \otimes \Pi_{2^{n-i}}$ with $N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}$}
We first prove that every operator of the form $I_{2^i} \otimes \Pi_{2^{n-i}}$, for $i = n-2$ to $1$, commutes with all the operators of the form $N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}$, for $j = i$ to $2$, by simply showing that
\begin{equation}
(I_{2^i} \otimes \Pi_{2^{n-i}})(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^(n-j+1}) = (N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})(I_{2^i} \otimes \Pi_{2^{n-i}})
\end{equation}
The matrix $I_{2^i} \otimes \Pi_{2^{n-i}}$ is a block diagonal matrix and therefore can be written as
\begin{equation}
I_{2^i} \otimes \Pi_{2^{n-i}} = I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
It can be then shown that
\begin{equation}
(I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}})(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}}) = N \otimes \Pi_{2^{n-j}} \oplus
I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
and
\begin{equation}
(N \otimes I_{2^{n-j}} \oplus I_{2^n - 2^{n-j+1}})(I_2 \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}) = N \otimes \Pi_{2^{n-j}} \oplus I_{2^j - 2} \otimes \Pi_{2^{n-j}}
\end{equation}
It now remains to show that every operator of the form $I_{2^i} \otimes \Pi_{2^{n-i}}$ commutes with the operator $N \otimes I_{2^{n-1}}$. This is simply proved by first using the fact that
\begin{equation}
I_{2^i} \otimes \Pi_{2^{n-i}} = I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}})
\end{equation}
and then showing that
\begin{equation}
(I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}}))( N \otimes I_{2^{n-1}}) =
(N \otimes I_{2^{n-1}})(I_2 \otimes (I_{2^{i-1}} \otimes \Pi_{2^{n-i}})) =
N \otimes I_{2^{i-1}} \otimes \Pi_{2^{n-i}}
\end{equation}
\input{epsf}
\begin{figure}
\caption{A circuit for implementation of quantum Fourier transform, QFT (from [15]).}
\label{fig:one}
\end{figure}
\begin{figure}
\caption{The $\Pi_4$ gate (a) and its implementation by using three XOR (Controlled-NOT) gates (b).}
\label{fig:two}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of Perfect Shuffle permutation matrix, $\Pi_{2^n}
\label{fig:three}
\end{figure}
\begin{figure}
\caption{Circuits for implementation of Bit Reversal permutation matrix, $P_{2^n}
\label{fig:four}
\end{figure}
\begin{figure}
\caption{A block-level circuit for Haar wavelet (from [20]).}
\label{fig:five}
\end{figure}
\begin{figure}
\caption{A block-level circuit for implementation of Hoyer's factorization of $D^{(4)}
\label{fig:six}
\end{figure}
\begin{figure}
\caption{A block-level circuit for implementation of new factorization of $D^{(4)}
\label{fig:seven}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of operator $T_{2^n}
\label{fig:eight}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of operator $P_{2^n}
\label{fig:nine}
\end{figure}
\begin{figure}
\caption{A circuit for implementation of $D^{(4)}
\label{fig:ten}
\end{figure}
\end{document} |
\begin{document}
\title{A host-parasite model for a two-type cell population}
\begin{abstract}
We consider a host-parasite model for a population of cells that can be of two types, $\sfA$ or $\sfB$, and exhibits unilateral reproduction: while a $\sfB$-cell always splits into two cells of the same type, the two daughter cells of an $\sfA$-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of $\sfA$-cells and its parasites, our model differs from the single-type model recently studied by \textsc{Bansaye} \cite{Bansaye:08} in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as $n\to\infty$, of the number $\sfA$-parasites in generation $n$ and the relative proportion of $\sfA$- and $\sfB$-cells in this generation which host a given number of parasites. As in \cite{Bansaye:08}, proofs will
make use of a so-called random cell line which, when conditioned to be of type $\sfA$, behaves like a branching process in random environment.
\end{abstract}
\section{Introduction}\label{Section.Introduction}
The reciprocal adaptive genetic change of two antagonists (e.g.\ different species or genes)
through reciprocal selective pressures is known as host-parasite coevolution. It may be observed even in real-time under both, field and laboratory conditions, if reciprocal adaptations take place rapidly and generation times are short. For more information see e.g.\ \cite{Laine:09, Woolhouse:02}.
The present work studies a host-parasite branching model with two types of cells (the hosts), here called $\sfA$ and $\sfB$, and proliferating parasites colonizing the cells. Adopting a genealogical perspective, we are interested in the evolution of certain characteristics over generations and under the following assumptions on the reproductive behavior of cells and parasites. All cells behave independently and split into two daughter cells after one unit of time. The types of the daughter cells of a type-$\sfA$ cell are chosen in accordance with a random mechanism which is the same for all mother cells of this type whereas both daughter cells of a type-$\sfB$ cell are again of type $\sfB$. Parasites within a cell multiply in an iid manner to produce a random number of offspring the distribution of which may depend on the type of this cell as well as on those of its daughter cells. The same holds true for the random mechanism by which the offspring is shared into these daughter cells.
The described model grew out of a discussion with biologists in an attempt to provide a first very simple setup that allows to study coevolutionary adaptations, here due to the presence of two different cell types. It may also be viewed as a simple multi-type extension of a model studied by \textsc{Bansaye} \cite{Bansaye:08} which in turn forms a discrete-time version of a model introduced by \textsc{Kimmel} \cite{Kimmel:97}. Bansaye himself extended his results in \cite{Bansaye:09} by allowing immigration and random environments, the latter meaning that each cell chooses the reproduction law for the parasites it hosts in an iid manner.
Let us further mention related recent work by \textsc{Guyon} \cite{Guyon:07} who studied another discrete-time model with asymmetric sharing and obtained limit theorems under ergodic hypotheses which, however, exclude an extinction-explosion principle for the parasites which is valid in our model.
We continue with the introduction of some necessary notation which is similar to the one in \cite{Bansaye:08}. Making the usual assumption of starting from one ancestor cell, denoted as $\varnothing$, we put $\G_{0}:=\{\varnothing\}$, $\G_{n}:=\{0,1\}^{n}$ for $n\ge 1$, and let
\begin{equation*}
\T := \bigcup_{n\in\mathbb N_0} \G_n\quad\text{with}\quad\G_n := \{0,1\}^n
\end{equation*}
be the binary Ulam-Harris tree rooted at $\varnothing$ which provides the label set of all cells in the considered population. Plainly, $\G_{n}$ contains the labels of all cells of generation $n$. For any cell $v\in\T$, let $T_v\in\{\sfA,\sfB\}$ denote its type and $Z_v$ the number of parasites it contains. \emph{Unless stated otherwise, the ancestor cell is assumed to be of type $\sfA$ and to contain one parasite, i.e.}
\begin{equation}\label{SA1}\tag{SA1}
T_{\varnothing}=\sfA\quad\text{and}\quad Z_{\varnothing}=1.
\end{equation}
Then, for $\sft\in\{\sfA,\sfB\}$ and $n\ge 0$, define
\begin{equation*}
\G_n(\sft) := \{v\in\G_n : T_v=\sft\}\quad\text{and}\quad\G^*_n(\sft) := \{v\in\G_n (\sft) : Z_v>0\}
\end{equation*}
as the sets of type-$\sft$ cells and type-$\sft$ contaminated cells in generation $n$, respectively. The set of all contaminated cells in generation $n$ are denoted $\G^*_n$,
thus $\G^*_n = \G^*_n(\sfA)\cup\G^*_n(\sfB)$.
As common, we write $v_{1}...v_{n}$ for $v=(v_1,...,v_n)\in\G_n$, $uv$ for the concatenation of $u,v\in\T$, i.e.
\begin{equation*}
uv=u_1...,u_m v_1...v_n\ \ \text{if}\ u=u_1...u_m\ \text{and}\ v=v_1...v_n,
\end{equation*}
and $v|k$ for the ancestor of $v=v_{1}...v_{n}$ in generation $k\le n$, thus $v|k=v_1,...,v_k$. Finally, if $v|k=u$ for some $k$ and $u\ne v$, we write $u<v$.
The process $(T_v)_{v\in\T}$ is a Markov process indexed by the tree $\T$ as defined in \cite{BenPeres:94}. It has transition probabilities
\begin{align*}
&\mathbb{P}(T_{v0}=\sfx,T_{v1}=\sfy|T_v = \sfA) = p_{\sfx\sfy},
\quad (\sfx,\sfy)\in\{(\sfA,\sfA),(\sfA,\sfB),(\sfB,\sfB)\},\\[1ex]
&\mathbb{P}(T_{v0}=\sfB,T_{v1}=\sfB|T_v = \sfB) = 1,
\end{align*}
and we denote by
\begin{equation*}
p_0 := p_{\sfAA} + p_{\sfAB}=1-p_{\sfBB}\quad\text{and}\quad p_1 := p_{\sfAA}
\end{equation*}
the probabilities that the first and the second daughter cell are of type $\sfA$, respectively.
In order to rule out total segregation of type-$\sfA$ and type-$\sfB$ cells, which would just lead back to the model studied in \cite{Bansaye:08}, it will be assumed throughout that
\begin{equation}\label{SA2}\tag{SA2}
p_{\sfAA}<1.
\end{equation}
The sequence $(\#\G_n(\sfA))_{n\ge 0}$ obviously forming a Galton-Watson branching process with one ancestor (as $T_{\varnothing}=\sfA$) and mean
\begin{equation*}
\nu := p_{0}+p_{1}=2 p_{\sfAA}+p_{\sfAB}=1+(p_{\sfAA}-p_{\sfBB})<2,
\end{equation*}
it is a standard fact that (see e.g.\ \cite{Athreya+Ney:72})
\begin{equation*}
\#\G_n(\sfA)\rightarrow 0\text{ a.s.}\quad\text{iff}\quad p_{\sfAA}\leq p_{\sfBB}\quad\text{and}\quad p_{\sfAB}<1.
\end{equation*}
To describe the multiplication of parasites, let $Z_{v}$ denote the number of parasites in cell $v$ and, for $\sft\in\{\sfA,\sfB\}$, $\sfs\in\{\sfAA,\sfAB,\sfBB\}$, let
$$ \left(X^{(0)}_{k,v}(\sft, \sfs),X^{(1)}_{k,v}(\sft, \sfs)\right)_{k\in\mathbb N, v\in\T},\quad\sft\in\{\sfA,\sfB\},\ \sfs\in\{\sfAA,\sfAB,\sfBB\}$$
be independent families of iid $\mathbb N^2_0$-valued random vectors with respective generic copies $(X^{(0)}(\sft, \sfs),X^{(1)}(\sft, \sfs))$. If $v$ is of type $\sft$ and their daughter cells are of type $\sfx$ and $\sfy$, then $X^{(i)}_{k,v}(\sft, \sfx\!\sfy)$ gives the offspring number of the $k^{\rm th}$ parasite in cell $v$ that is shared into the daughter cell $vi$ of $v$. Since type-$\sfB$ cells can only produce daughter cells of the same type, we will write $(X^{(0)}_{k,v}(\sfB),X^{(1)}_{k,v}(\sfB))$ as shorthand for $(X^{(0)}_{k,v}(\sfB, \sfBB),X^{(1)}_{k,v}(\sfB, \sfBB))$. To avoid trivialities, it is always assumed hereafter that
\begin{equation}\label{SA3}\tag{SA3}
\mathbb{P}\left(X^{(0)}(\sfA, \sfAA) \leq 1,\ X^{(1)}(\sfA, \sfAA)\leq1\right)<1
\end{equation}
and
\begin{equation}\label{SA4}\tag{SA4}
\mathbb{P}\left(X^{(0)}(\sfB)\leq1,\ X^{(1)}(\sfB)\leq1\right)<1.
\end{equation}
Next, observe that
\begin{equation*}
(Z_{v0}, Z_{v1}) = \sum_{\sft\in\{\sfA,\sfB\}}\1_{\{T_{v}=\sft\}}\sum_{\sfs\in\{\sfAA,\sfAB,\sfBB\}}\1_{\{(T_{v0},T_{v1})=\sfs\}}\sum_{k=1}^{Z_v}(X^{(0)}_{k,v}(\sft, \sfs),X^{(1)}_{k,v}(\sft, \sfs)).
\end{equation*}
We put $\mu_{i,\sft}(\sfs) := \mathbb{E} X^{(i)}(\sft, \sfs)$ for $i\in\{0,1\}$ and $\sft,\sfs$ as before, write $\mu_{i,\sfB}$ as shorthand for $\mu_{i,\sfB}(\sfBB)$ and assume throughout that $\mu_{i,\sft}(\sfs)$ are finite and
\begin{equation}\label{SA5}\tag{SA5}
\mu_{0,\sfA}(\sfAA),\ \mu_{1,\sfA}(\sfAA),\ \mathbb{E}\left(\#\G^*_1(\sfB)\right)>0,\ \mu_{0,\sfB},\ \mu_{1,\sfB}\ >\ 0.
\end{equation}
The total number of parasites in cells of type $\sft\in\{\sfA, \sfB\}$ at generation $n$ is denoted by
\begin{equation*}
\mathcal{Z}_n(\sft) := \sum_{v\in\G_n(\sft)} Z_v,
\end{equation*}
and we put $\mathcal{Z}_n := \mathcal{Z}_n(\sfA)+\mathcal{Z}_n(\sfB)$, plainly the total number of all parasites at generation $n$. Both, $(\mathcal{Z}_n)_{n\ge 0}$ and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$, are transient Markov chains with absorbing state $0$ and satisfy the extinction-explosion principle (see Section I.5 in \cite{Athreya+Ney:72} for a standard argument), i.e.
\begin{equation*}
\mathbb{P}(\mathcal{Z}_n\rightarrow 0)+\mathbb{P}(\mathcal{Z}_n\rightarrow\infty)=1\quad\text{and}\quad \mathbb{P}(\mathcal{Z}_n(\sfA)\rightarrow 0)+\mathbb{P}(\mathcal{Z}_n(\sfA)\rightarrow\infty)=1.
\end{equation*}
The extinction events are defined as
\begin{equation*}
\mathbb{E}xt := \{\mathcal{Z}_n\rightarrow 0\}\quad\text{and}\quad\mathbb{E}xt(\sft):=\{\mathcal{Z}_n(\sft)\rightarrow 0\},\quad\sft\in\{\sfA,\sfB\},
\end{equation*}
their complements by $\Surv$ and $\Surv(\sft)$, respectively.
As in \cite{Bansaye:08}, we are interested in the statistical properties of an infinite \emph{random cell line}, picked however from those lines consisting of $\sfA$-cells only.
This leads to a so-called \emph{random $\sfA$-cell line}. Since $\sfB$-cells produce only daughter cells of the same type, the properties of a random $\sfB$-cell line may be deduced from the afore-mentioned work and are therefore not studied hereafter.
For the definition of a random $\sfA$-cell line, a little more care than in \cite{Bansaye:08} is needed because cells occur in two types and parasitic reproduction may depend on the types of the host and both its daughter cells. On the other hand, we will show in Section \ref{Section.Preliminaries} that a random $\sfA$-cell line still behaves like a branching process in iid random environment (BPRE) which has been a fundamental observation in \cite{Bansaye:08} for a random cell line in the single-type situation.
Let $U=(U_n)_{n\in\mathbb N}$ be an iid sequence of symmetric Bernoulli variables independent of the parasitic evolution and put $V_{n}:=U_{1}...U_{n}$. Then
$$ \varnothing=:V_{0}\to V_{1}\to V_{2}\to...\to V_{n}\to... $$
provides us with a random cell line in the binary Ulam-Haris tree, and we denote by
\begin{equation*}
T_{[n]} = T_{V_n}\quad\text{and}\quad Z_{[n]} = Z_{V_n}\quad n\ge 0,
\end{equation*}
the cell types and the number of parasites along that random cell line. A random $\sfA$-cell line up to generation $n$ is obtained when $T_{[n]}=\sfA$, for then $T_{[k]}=\sfA$ for any
$k=0,...,n-1$ as well. As will be shown in Prop.\ \ref{prop:random cell line=BPRE}, the conditional law of $(Z_{[0]},...,Z_{[n]})$ given $T_{[n]}=\sfA$, i.e., given an $\sfA$-cell line up to generation $n$ is picked at random, equals the law of a certain BPRE $(Z_{k}(\sfA))_{k\ge 0}$ up to generation $n$, for each $n\in\N$. It should be clear that this cannot be generally true for the unconditional law of $(Z_{[0]},...,Z_{[n]})$, due to the multi-type structure of the cell population.
Aiming at a study of host-parasite coevolution in the framework of a multitype host population, our model may be viewed as the simplest possible alternative. There are only two types of host cells and reproduction is unilateral in the sense that cells of type $\sfA$ may give birth to both, $\sfA$- and $\sfB$-cells, but those of type $\sfB$ will never produce cells of the opposite type.
The basic idea behind this restriction is that of irreversible mutations that generate new types of cells but never lead back to already existing ones. Observe that our setup could readily be generalized without changing much the mathematical structure by allowing the occurrence of further irreversible mutations from cells of type $\sfB$ to cells of type $\mathsf{C}$, and so on.
The rest of this paper is organized as follows. We focus on the case of non-extinction of
contaminated $\sfA$-cells, that is $\mathbb{P}(\mathbb{E}xt(\sfA))<1$. Basic results on $\mathcal{Z}_n(\sfA)$, $Z_{[n]}$, $\#\G^*_n(\sfA)$ and $\#\G^*_n$ including the afore-mentioned one will be shown in Section \ref{Section.Preliminaries} and be partly instrumental for the proofs of our results on the asymptotic behavior of the relative proportion of contaminated cells with $k$ parasites within the population of all contaminated cells. These results are stated in Section \ref{Section.Results} and proved in Section \ref{Section.Proofs}. A glossary of the most important notation used throughout may be found at the end of this article.
\section{Basic Results}\label{Section.Preliminaries}
We begin with a number of basic properties of and results about the quantities $\G^*_n(\sfA)$, $\G^*_n$, $\mathcal{Z}_n(\sfA)$ and $Z_{[n]}$.
\subsection{The random $\sfA$-cell line and its associated sequence $(Z_{[n]})_{n\ge 0}$}
In \cite{Bansaye:08}, a random cell line was obtained by simply picking a random path in the infinite binary Ulam-Harris tree representing the cell population. Due to the multi-type structure here, we must proceed in a different manner when restricting to a specific cell type, here type $\sfA$. In order to study the properties of a ''typical'' $\sfA$-cell in generation $n$ for large $n$, i.e., an $\sfA$-cell picked at random from this generation, a convenient (but not the only) way is to first pick at random a cell line up to generation $n$ from the full height $n$ binary tree as in \cite{Bansaye:08} and then to condition upon the event that the cell picked at generation $n$ is of type $\sfA$. This naturally leads to a random $\sfA$-cell line up to generation $n$, for $\sfA$-cells can only stem from cells of the same type. Then looking at the conditional distribution of the associated parasitic random vector $(Z_{[0]},...,Z_{[n]})$ leads to a BPRE not depending on $n$ and thus to an analogous situation as in
\cite{Bansaye:08}. The precise result is stated next.
\begin{Proposition}\label{prop:random cell line=BPRE}
Let $(Z_{n}(\sfA))_{n\ge 0}$ be a BPRE with one ancestor and iid environmental sequence
$(\Lambda_{n})_{n\ge 1}$ taking values in $\{\mathcal L(X^{(0)}(\sfA, \sfAA)), \mathcal L(X^{(1)}(\sfA, \sfAA)), \mathcal L(X^{(0)}(\sfA, \sfAB))\}$ such that
\begin{equation*}
\mathbb{P}\left(\Lambda_1=\mathcal L(X^{(0)}(\sfA, \sfAB))\right)=\frac{p_{\sfAB}}{\nu}\quad\text{and}\quad\mathbb{P}\left(\Lambda_1=\mathcal L(X^{(i)}(\sfA, \sfAA))\right)=\frac{p_{\sfAA}}{\nu},
\end{equation*}
for $i\in\{0,1\}$.
Then the conditional law of $(Z_{[0]},...,Z_{[n]})$ given $T_{[n]}=\sfA$ equals the law of $(Z_{0}(\sfA),...,Z_{n}(\sfA))$, for each $n\ge 0$.
\end{Proposition}
\begin{proof}
We use induction over $n$ and begin by noting that nothing has to be shown if $n=0$.
For $n\ge 1$ and $(z_0,...,z_n)\in\N^{n+1}_0$, we introduce the notation
\begin{equation*}
C_{z_0,...,z_n}:=\{(Z_{[0]},...,Z_{[n]})=(z_0,...,z_n)\}\quad\text{and}\quad C_{z_0,...,z_n}^{\sfA}:=C_{z_0,...,z_n}\cap\{T_{[n]}=\sfA\}
\end{equation*}
and note that
\begin{equation*}
\mathbb{P}\left(T_{[n]}=\sfA\right)=2^{-n}\, \mathbb{E}\left(\sum_{v\in\G_n}\1_{\{T_v=\sfA\}}\right)=\left(\frac{\nu}{2}\right)^n,
\end{equation*}
for each $n\in\N$, in particluar
$$ \mathbb{P}(T_{[n]}=\sfA|T_{[n-1]}=\sfA)=\frac{\mathbb{P}(T_{[n]}=\sfA)}{\mathbb{P}(T_{[n-1]}=\sfA)}=\frac{\nu}{2}. $$
Assuming the assertion holds for $n-1$ (inductive hypothesis), thus
$$ \mathbb{P}(C_{z_{0},...,z_{n-1}}|T_{[n-1]}=\sfA)=\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right) $$
for any $(z_{0},...,z_{n-1})\in\N_{0}^{n}$, we infer with the help of the Markov property that
\begin{align*}
\mathbb{P}&\left((Z_{[0]},...,Z_{[n]})=(z_0,...,z_n)| T_{[n]}=\sfA\right)\\[1ex]
&=~\frac{\mathbb{P}(C^{\sfA}_{z_0,...,z_n})}{\mathbb{P}(T_{[n]}=\sfA)}\\[1ex]
&=~\mathbb{P}\left(C_{z_0,...,z_{n-1}}|T_{[n-1]}=\sfA\right)\,
\mathbb{P}(Z_{[n]}=z_n,T_{[n]}=\sfA|C^{\sfA}_{z_0,...,z_{n-1}})\,\frac{\mathbb{P}(T_{[n-1]}=
\sfA)}{\mathbb{P}(T_{[n]}=\sfA)}\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\,
\frac{\mathbb{P}(Z_{[1]}=z_n, T_{[1]}=\sfA | Z_{[0]}=z_{n-1},
T_{[0]}=\sfA)}{\mathbb{P}(T_{[n]}=\sfA|T_{[n-1]}=\sfA)}\\
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\\
&\hspace{4ex}\times\frac{2}{\nu}\left(\frac{p_{\sfA\sfB}}{2}\left(\mathbb{P}^{X^{(0)}
(\sfA, \sfAB)}\right)^{*z_{n-1}}(\{z_n\})+\sum_{i\in\{0,1\}}\frac{p_{\sfA\sfA}}
{2}\left(\mathbb{P}^{X^{(i)}(\sfA, \sfAA)}\right)^{*z_{n-1}}(\{z_n\})\right)\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n-1}(\sfA)=z_{n-1}\right)\,\mathbb{P}\left(Z_
{[n]}(\sfA)=z_{n}|Z_{[n-1]}(\sfA)=z_{n-1}\right)\\[1ex]
&=~\mathbb{P}\left(Z_{0}(\sfA)=z_0,...,Z_{n}(\sfA)=z_{n}\right).
\end{align*}
This proves the assertion.
\end{proof}
The connection between the distribution of $Z_{n}(\sfA)$ and the expected number of $\sfA$-cells in generation $n$ with a specified number of parasites is stated in the next result.
\begin{Proposition}\label{Prop:dist of Zn(A)}
For all $n\in\N$ and $k\in\N_{0}$,
\begin{equation}\label{Eq.BPRE.EG}
\mathbb{P}\left(Z_n(\sfA)=k\right)=\nu^{-n}\,\mathbb{E}\left(\#\{v\in\G_n(\sfA):Z_v=k\}\right),
\end{equation}
in particular
\begin{equation}\label{Eq.BPRE.EG,k>0}
\mathbb{P}\left(Z_n(\sfA)>0\right)=\nu^{-n}\,\mathbb{E}\#\G_{n}^{*}(\sfA).
\end{equation}
\end{Proposition}
\begin{proof}
For all $n,k\in\N$, we find that
\begin{align*}
\mathbb{E}\left(\#\{v\in\G_n(\sfA):Z_v=k\}\right)
~&=~\sum_{v\in\G_n}\mathbb{P}(Z_v=k, T_v=\sfA)\\[1ex]
&=~2^n\mathbb{P}\left(Z_{[n]}=k, T_{[n]}=\sfA\right)\\[1ex]
&=~2^n\mathbb{P}(T_{[n]}=\sfA)\mathbb{P}\left(Z_{[n]}=k| T_{[n]}=\sfA\right)\\[1ex]
&=~\nu^n\mathbb{P}\left(Z_{[n]}=k| T_{[n]}=\sfA\right)\\[1ex]
&=~\nu^n\mathbb{P}\left(Z_n(\sfA)=k\right),
\end{align*}
and this proves the result.
\end{proof}
For $n\in\N$ and $s\in [0,1]$, let
\begin{equation*}
f_n(s|\Lambda) := \mathbb{E}(s^{Z_n(\sfA)}|\Lambda)\quad\text{and}\quad f_n(s) := \mathbb{E} s^{Z_n(\sfA)}=\mathbb{E} f_{n}(s|\Lambda)
\end{equation*}
denote the quenched and annealed generating function of $Z_n(\sfA)$, respectively, where $\Lambda:=(\Lambda_{n})_{n\ge 1}$.
Then the theory of BPRE (see \cite{Athreya:71.1, Athreya:71.2, GeKeVa:03, Smith+Wilkinson:69} for more details) provides us with the following facts: For each $n\in\N$,
\begin{align*}
f_n(s|\Lambda)=g_{\Lambda_{1}}\circ...\circ g_{\Lambda_{n}}(s),
\quad g_{\lambda}(s):=\mathbb{E}(s^{Z_{1}(\sfA)}|\Lambda_{1}=\lambda)=\sum_{n\ge 0}\lambda_{n}s^{n}
\end{align*}
for any distribution $\lambda=(\lambda_{n})_{n\ge 0}$ on $\N_{0}$. Moreover, the $g_{\Lambda_{n}}$ are iid with
\begin{align*}
\mathbb{E} g_{\Lambda_{1}}'(1)&=\mathbb{E} Z_{1}(\sfA)\\
&=\frac{p_{\sfAA}}{\nu}\Big(\mu_{0,\sfA}(\sfAA)+\mu_{1,\sfA}(\sfAA)\Big)+\frac{p_{\sfAB}}{\nu}\mu_{0,\sfA}(\sfAB)=\frac{\gamma}{\nu},
\end{align*}
where
\begin{equation*}
\gamma := \mathbb{E}\mathcal{Z}_1(\sfA) = p_{\sfAA}\left(\mu_{0,\sfA}(\sfAA)+\mu_{1,\sfA}(\sfAA)\right)+p_{\sfAB}\mu_{0,\sfA}(\sfAB)
\end{equation*}
denotes the expected total number of parasites in cells of type $\sfA$ in the first generation (recall from \eqref{SA1} that $Z_{\varnothing}=Z_{\varnothing}(\sfA)=1$). As a consequence,
\begin{align*}
\mathbb{E}(Z_{[n]}|T_{[n]}=\sfA)&=\mathbb{E} Z_n(\sfA)
=f_n'(1)=\prod_{k=1}^{n}\mathbb{E} g_{\Lambda_{k}}'(1)=\left(\frac{\gamma}{\nu}\right)^n
\end{align*}
for each $n\in\N$. It is also well-known that $(Z_{n}(A))_{n\ge 0}$ dies out a.s., which in terms of $(Z_{[n]})_{n\ge 0}$ means that $\lim_{n\to\infty}\mathbb{P}(Z_{[n]}=0|T_{[n]}=\sfA)=1$, iff
\begin{align}\label{Eq.BPRE.Aussterben}
\mathbb{E}\log g_{\Lambda_{1}}'(1)=\frac{p_{\sfAA}}{\nu}\Big(\log\mu_{0,\sfA}(\sfAA)+\log
\mu_{1,\sfA}(\sfAA)\Big)+\frac{p_{\sfAB}}{\nu}\log\mu_{0,\sfA}(\sfAB)\le 0.
\end{align}
\subsection{Properties of $\#\G^*_n(\sfA)$ and $\#\G^*_n$:}
We proceed to the statement of a number of results on the asymptotic behavior of $\G^*_n(\sfA)$ and $\G^*_n$ conditioned upon $\Surv(\sfA)$ and $\Surv$, respectively. It turns out that, if the number of parasites tends to infinity, then so does the number of contaminated cells.
\begin{Theorem}\label{Satz.ExplosionInfZellen} ${}$
\begin{enumerate}[(a)]
\item\label{Item.ExplosionInfZellenB} If $\mathbb{P}(\Surv(\sfA))>0$ and $p_{\sfAA}>0$, then $\mathbb{P}(\#\G^*_n(\sfA)\to\infty|\Surv(\sfA))=1$.
\item\label{Item.ExplosionInfZellenAlle} If $\mathbb{P}(\Surv)>0$, then $\mathbb{P}(\#\G^*_n\to\infty|\Surv)=1.$
\end{enumerate}
\end{Theorem}
\begin{proof}
The proof of assertion \eqref{Item.ExplosionInfZellenB} is the same as for Theorem 4.1 in \cite{Bansaye:08} and thus omitted.
\eqref{Item.ExplosionInfZellenAlle} We first note that, given $\Surv$, a contaminated $\sfB$-cell is eventually created with probability one and then spawns a single-type cell process (as $\mathbb{E}\mathcal{Z}_1(\sfB)>0$ by \eqref{SA5}). Hence the assertion follows again from Theorem 4.1 in \cite{Bansaye:08} if $\mu_{\sfB}:=\mu_{0,\sfB}+\mu_{1,\sfB}>1$.
Left with the case $\mu_{\sfB}\le 1$, it follows that
$$ \mathbb{P}(\Surv(\sfA)|\Surv)=1,$$
for otherwise, given $\Surv$, only $\sfB$-parasites would eventually be left with positive probability which however would die out almost surely. Next, $p_{\sfAA}>0$ leads back to
\eqref{Item.ExplosionInfZellenB} so that it remains to consider the situation when $p_{\sfAA}=0$. In this case there is a single line of $\sfA$-cells, namely $\varnothing\to 0\to 00\to ...$, and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ is an ordinary Galton-Watson branching process
tending $\mathbb{P}(\cdot|\Surv(\sfA))$-a.s.\ to infinity. For $n,k\in\N$, let
\begin{equation*}
\mathcal{Z}_k(n,\sfB) := \sum_{v\in\G_{n+k+1}(\sfB):v|n+1=0^{n}1}Z_v
\end{equation*}
denote the number of $\sfB$-parasites at generation $k$ sitting in cells of the subpopulation stemming from the cell $0^{n}1$, where $0^n:=0...0$ ($n$-times). Using $p_{\sfAB}=1$ and \eqref{SA5}, notably $\mu_{1,\sfA}(\sfAB)>0,\mu_{0,\sfB}>0$ and $\mu_{1,\sfB}>0$, it
is readily seen that
\begin{equation*}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_0(n-k,\sfB)=\infty|\Surv(\sfA)\right)=1
\end{equation*}
and thus
\begin{equation*}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_K(n-k,\sfB)=0|\Surv(\sfA)\right)=0
\end{equation*}
for all $K\in\N$ and $k\leq K$. Consequently,
\begin{align*}
&\mathbb{P}\left( \liminf_{n\to\infty}\#\G^*_n\leq K|\Surv(\sfA)\right)\\
&\hspace{1cm}\leq\ \mathbb{P}\left(\lim_{n\to\infty}\max_{0\le k\le K}
\mathcal{Z}_k(n-k,\sfB)=0|\Surv(\sfA)\right)\\
&\hspace{1cm}\leq\ \sum_{k=0}^{K}
\mathbb{P}\left(\lim_{n\to\infty}\mathcal{Z}_K(n-k,\sfB)=0|\Surv(\sfA)\right)\\
&\hspace{1cm}=~0
\end{align*}
for all $K\in\N$
\end{proof}
The next result provides us with the geometric rate at which the number of contaminated cells tends to infinity.
\begin{Theorem}\label{Satz.ErholungB} $(\nu^{-n}\#\G^*_n(\sfA))_{n\ge 0}$ is a non-negative supermartingale and therefore a.s.\ convergent to a random variable $L(\sfA)$ as $n\to\infty$. Furthermore,
\begin{enumerate}[(a)]
\item\label{Item.ErholungB.FSExtinction} $L(\sfA)=0$ a.s.\ iff $\mathbb{E}\log g_{\Lambda_
{1}}'(1)\leq 0$ or $\nu\leq1$
\item\label{Item.ErholungB.Extinction} $\mathbb{P}(L(\sfA)=0)<1$ implies $\{L(\sfA)=0\}=
\mathbb{E}xt(\sfA)$ a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
That $(\nu^{-n}\#\G^*_n(\sfA))_{n\ge 0}$ forms a supermartingale follows by an easy calculation and therefore a.s. convergence to an integrable random variable $L(\sfA)$ is ensured. This supermartingale is even uniformaly integrable in the case $\nu>1$, which follows because the obvious majorant $(\nu^{-n}\#\G_n(\sfA))_{n\ge 0}$ is a normalized Galton-Watson branching process having a reproduction law with finite variance and is thus $L^2$-bounded (see Section I.6 in \cite{Athreya+Ney:72}). Consequently, $(\nu^{-n}\#\G^*_n(\sfA))_{n\geq0}$ is uniformaly integrable and
\begin{align}\label{eq:EL(A)}
\mathbb{E} L(\sfA) = \lim_{n\to\infty}\mathbb{E}\frac{\#\G_{n}^{*}(\sfA)}{\nu^n}
= \lim_{n\to\infty}\mathbb{P}(Z_n(\sfA)>0),
\end{align}
the last equality following from \eqref{Eq.BPRE.EG,k>0} in Proposition \ref{Prop:dist of Zn(A)}.
As for \eqref{Item.ErholungB.FSExtinction}, $L(\sfA)=0$ a.s.\ occurs iff either $\nu\le 1$, in which case $\#\G_n^*(\sfA)\le\#\G_{n}(\sfA)=0$ eventually, or $\nu>1$ and $\mathbb{E}\log g_{\Lambda_{1}}'(1)\leq 0$, in which case almost certain extinction of $(Z_{n}(\sfA))_{n\ge 0}$ in combination with \eqref{eq:EL(A)} yields the conclusion.
\eqref{Item.ErholungB.Extinction} Defining $\tau_n = \inf\{m\in\N:\#\G^*_m(\sfA)\geq n\}$,
we find that
\begin{align*}
\mathbb{P}(L(\sfA)=0) &\leq~ \mathbb{P}(L(\sfA)=0|\tau_n<\infty)+\mathbb{P}(\tau_n=\infty)\\
&\leq\ \mathbb{P}\left(\bigcap_{k=1}^{\#\G^*_{\tau_n}(\sfA)}\{\#\G^*_{m,k}(\sfA)/\nu^m
\to0\}\bigg|\tau_n<\infty\right)+\mathbb{P}(\tau_n=\infty)\\
&\leq\ \mathbb{P}(L(\sfA)=0)^n+\mathbb{P}(\tau_n=\infty)
\end{align*}
for all $n\ge 1$, where the $\#\G^*_{m,k}(\sfA)$, $k\ge 1$, are independent copies of $\#\G^*_m(\sfA)$. Since $\mathbb{P}(L(\sfA)=0)<1$, Theorem \ref{Satz.ExplosionInfZellen} implies
\begin{equation*}
\mathbb{P}(L(\sfA)=0)\leq \lim_{n\to\infty}\mathbb{P}(\tau_n=\infty)=\mathbb{P}\left(\sup_{n\ge 1}
\#\G^*_{n}(\sfA)<\infty\right)=\mathbb{P}(\mathbb{E}xt(\sfA))
\end{equation*}
which in combination with $\mathbb{E}xt(\sfA)\subset\{L(\sfA)=0\}$ a.s. proves the assertion.
\end{proof}
Since $\nu<2$ and $(\nu^{-n}\#\G_{n}(\sfA))_{n\ge 0}$ is a nonnegative, a.s.\ convergent martingale, we see that $2^{-n}\#\G^{*}_n(\sfA)\le 2^{-n}\#\G_n(\sfA)\to 0$ a.s. and therefore
\begin{equation*}
\frac{\#\G^*_n}{2^n}\ \simeq\ \frac{\#\G^*_n(\sfB)}{2^n},\quad\text{as }n\to\infty.
\end{equation*}
that is, the asymptotic proportion of all contaminated cells is the same as the asymptotic proportion of contaminated $\sfB$-cells. Note also that
\begin{equation}\label{eq:T[n]to zero}
\mathbb{P}(T_{[n]}=\sfA)=\mathbb{E}\left(\frac{\#\G_n(\sfA)}{2^n}\right)\to 0,\quad\text{as }n\to\infty.
\end{equation}
Further information is provided by the next result.
\begin{Theorem}\label{Satz.Erholung}
There exists a r.v. $L\in[0,1]$ such that $\#\G^*_n/2^n\to L$ a.s. Furthermore,
\begin{enumerate}[(a)]
\item\label{Item.Erholung.AlleFSExtinction} $L=0$ a.s. iff
$\mu_{0,\sfB}\mu_{1,\sfB}\leq1$.
\item\label{Item.Erholung.AlleExtinction} If $\mathbb{P}(L=0)<1$, then $\{L=0\}=\mathbb{E}xt$
a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
The existence of $L$ follows because $2^{-n}\#\G^*_n$ is obviously decreasing.
As for \eqref{Item.Erholung.AlleFSExtinction}, suppose first that $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ and note that this is equivalent to almost sure extinction of a random $\sfB$-cell line, i.e.
$$ \lim_{n\to\infty}\mathbb{P}(Z_{[n]}>0|Z_{\varnothing}=k,T_{[0]}=\sfB)=0 $$
for any $k\in\N$. This follows because, starting from a $\sfB$-cell, we are in the one-type model studied in \cite{Bansaye:08}. There it is stated that $(Z_{[n]})_{n\geq0}$ forms a BPRE which dies out a.s.\ iff $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ (see \cite[Prop.\ 2.1]{Bansaye:08}). Fix any $\varepsilon>0$ and choose $m\in\N$ so large that $\mathbb{P}(T_{[m]}=\sfA)\leq\varepsilon$, which is possible by \eqref{eq:T[n]to zero}. Then, by the monotone convergence theorem, we find that for sufficiently large $K\in\N$
\begin{align*}
\mathbb{E} L ~&=~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+m]}>0)\\
&\leq~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+m]}>0, T_{[m]}=\sfB) + \varepsilon\\
&\leq~ \lim_{n\to\infty}\sum_{k\ge 0}\mathbb{P}(Z_{[n+m]}>0, Z_{[m]}=k,
T_{[m]}=\sfB) + \varepsilon\\
&\leq~ \lim_{n\to\infty}\sum_{k=0}^{K}\mathbb{P}(Z_{[n]}>0|Z_{[0]}=k,
T_{[0]}=\sfB) + 2\varepsilon\\
&\leq~ 2\varepsilon
\end{align*}
and thus $\mathbb{E} L=0$. For the converse, note that
\begin{align*}
0 ~&=~ \mathbb{E} L\\
&=~ \lim_{n\to\infty}\mathbb{P}(Z_{[n+1]}>0)\\
&\geq~ \lim_{n\to\infty}\mathbb{P}(Z_{[1]}>0, T_{[1]}=\sfB)\mathbb{P}(Z_{[n]}>0|T_{[0]}=\sfB)
\end{align*}
implies $0=\lim_{n\to\infty}\mathbb{P}(Z_{[n]}>0|T_{[0]}=\sfB)$ and thus $\mu_{0,\sfB}\mu_{1,\sfB}\leq1$ as well.
The proof of \eqref{Item.Erholung.AlleExtinction} follows along similar lines as Theorem \ref{Satz.ErholungB}\eqref{Item.ErholungB.Extinction} and is therefore omitted.
\end{proof}
\subsection{Properties of $\mathcal{Z}_n(\sfA)$}
We continue with some results on $\mathcal{Z}_n(\sfA)$, the number of $\sfA$-parasites at generation $n$, and point out first that $(\gamma^{-n}\mathcal{Z}_n(\sfA))_{n\ge 0}$ constitutes a nonnegative, mean one martingale which is a.s.\ convergent to a finite random variable $W$. In particular, $\mathbb{E}\mathcal{Z}_n(\sfA)=\gamma^n$ for all $n\in\N_{0}$. If $\mathbb{E}\mathcal{Z}_1(\sfA)^2<\infty$, $\gamma>1$ and
$$ \hat\gamma\ :=\ \nu\,\mathbb{E} g^{\prime}_{\Lambda_1}(1)^2=p_{\sfAA}\left(\mu^2_{0,\sfA}(\sfAA)+\mu^2_{1,\sfA}(\sfAA)\right)+p_{\sfAB}\mu^2_{0,\sfA}(\sfAB)\ \leq\ \gamma, $$
then the martingale is further $L^{2}$-bounded as may be assessed by a straightforward but tedious computation. The main difference between a standard Galton-Watson process and the $\sfA$-parasite process $(\mathcal{Z}_n(\sfA))_{n\geq0}$ is the dependence of the offspring numbers of parasites living in the same cell, which (by some elementary calculations) leads to an additional term in the recursive formula for the variance, viz.
\begin{equation*}
\mathbb{V}ar\left(\mathcal{Z}_{n+1}(\sfA)\right)~=~\gamma^2\,\mathbb{V}ar\left(\mathcal{Z}_{n}(\sfA)\right)+\gamma^n\,\mathbb{V}ar(\mathcal{Z}_1(\sfA))+c_{1} \nu^n f_{n}''(1)
\end{equation*}
for all $n\geq0$ and some finite positive constant $c_{1}$. Here it should be recalled that $f_n(s)=\mathbb{E} s^{Z_n(\sfA)}$. Consequently, by calculating the second derivative of $f_{n}$ and using $\hat\gamma\le\gamma$, we obtain
\begin{equation*}
f^{\prime\prime}_n(1)\ =\ \mathbb{E} g^{\prime\prime}_{\Lambda_1}(1) \sum_{i=1}^n\left(\frac{\hat\gamma}{\nu}\right)^{n-i}\left(\frac{\gamma}{\nu}\right)^{i-1}\ \leq\ c_{2} n \left(\frac{\gamma}{\nu}\right)^n
\end{equation*}
for some finite positive constant $c_2$. A combination of this inequality with the above recursion for the variance of $\mathcal{Z}_{n}(\sfA)$ finally provides us with
\begin{equation*}
\mathbb{V}ar\left(\gamma^{-n}\mathcal{Z}_{n}(\sfA)\right)~\leq~1+\gamma^{-2}\sum_{k=0}^{\infty}\gamma^{-k}\big(\mathbb{V}ar(\mathcal{Z}_1(\sfA))+c_{1} c_{2} k\big)\ <\ \infty
\end{equation*}
for all $n\ge 0$ and thus the $L^2$-boundness of $(\gamma^{-n}\mathcal{Z}_{n}(\sfA))_{n\geq0}$.
Recalling that $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ and $(\mathcal{Z}_n)_{n\ge 0}$ satisfy the extinction-explosion principle, the next theorem gives conditions for almost sure extinction, that is, for
$\mathbb{P}(\mathbb{E}xt(\sfA))=1$ and $\mathbb{P}(\mathbb{E}xt)=1$.
\begin{Theorem}\label{Satz.Aussterben}
\begin{enumerate}[(a)]
\item\label{Extinction_p=0} If $p_{\sfAA}=0$, then
\begin{equation*}
\mathbb{P}(\mathbb{E}xt(\sfA))=1\quad\text{iff}\quad \mu_{0,\sfA}(\sfAB)\le 1\quad\text{or}\quad\nu<1.
\end{equation*}
\item\label{Extinction_p>0} If $p_{\sfAA}>0$, then the following statements are
equivalent:
\begin{enumerate}[(1)]\setlength{\itemsep}{1ex}
\item\label{Item.Extinction.1} $\mathbb{P}(\mathbb{E}xt(\sfA))=1$
\item\label{Item.Extinction.EG} $\mathbb{E}\#\G^*_n(\sfA)\leq1$ for
all $n\in\N\quad$
\item\label{Item.Extinction.gamma} $\nu\leq1$, or
\begin{equation*}
\nu>1,\quad\mathbb{E}\log g_{\Lambda_{1}}'(1)<0\quad\text{and}\quad\inf_{0\leq\theta
\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\leq \frac{1}{\nu}.
\end{equation*}
\end{enumerate}
\item\label{Extinction_all} $\mathbb{P}(\mathbb{E}xt)=1\quad\text{iff}\quad \mathbb{P}(\mathbb{E}xt
(\sfA))=1\ \text{and}\ \mu_{0,\sfB}+\mu_{1,\sfB}\leq1$
\end{enumerate}
\end{Theorem}
\begin{Remark}
Let us point out the following useful facts before proceeding to the proof of the theorem. We first note that, if $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$ and $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)\leq 0$, then the convexity of
$\theta\mapsto\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}$ implies that
\begin{equation*}
\mathbb{E} g_{\Lambda_{1}}'(1)=\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}.
\end{equation*}
If $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$, Geiger et al. \cite[Theorems 1.1--1.3]{GeKeVa:03} showed that
\begin{equation}\label{eq:Geiger survival estimate}
\mathbb{P}(Z_{n}(\sfA)>0)\simeq cn^{-\kappa}\left(\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^{n}\quad\text{as }n\to\infty
\end{equation}
for some $c\in (0,\infty)$, where
\begin{itemize}
\item[] $\kappa=0$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)<0$,
(strongly subcritical case)
\item[] $\kappa=1/2$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)=0$,
(intermediately subcritical case)
\item[] $\kappa=3/2$ if $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$,
(weakly subcritical case)
\end{itemize}
A combination of \eqref{Eq.BPRE.EG,k>0} and \eqref{eq:Geiger survival estimate} provides us with the asymptotic relation
\begin{equation}\label{eq:Gn and Geiger}
\mathbb{E}\#\G^*_{n}(\sfA)\simeq cn^{-\kappa}\nu^{n}\left(\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^{n}\quad\text{as }n\to\infty,
\end{equation}
in particular (with $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$ still being in force)
\begin{equation}\label{eq:Gn and Geiger2}
\inf_{0\leq\theta\leq 1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\le\frac{1}{\nu}\quad\text{if}\quad\sup_{n\ge 1}\mathbb{E}\#\G^*_{n}(\sfA)<\infty.
\end{equation}
\end{Remark}
\begin{proof}
\eqref{Extinction_p=0} If $p_{\sfAA}=0$ and $\nu=p_{\sfAB}=1$, each generation possesses exactly one $\sfA$-cell and $(\mathcal{Z}_n(\sfA))_{n\ge 0}$ thus forms a Galton-Watson branching process with offspring mean $\mu_{0,\sfA}(\sfAB)$ and positive offspring variance (by \eqref{SA3}). Hence a.s.\ extinction occurs iff $\mu_{0,\sfA}(\sfAB)\le 1$ as claimed. If $\nu<1$, type $\sfA$ cells die out a.s. and so do type $\sfA$ parasites.
``\eqref{Item.Extinction.1}$\Rightarrow$\eqref{Item.Extinction.EG}''
(by contraposition) We fix $m\in\N$ such that $\mathbb{E}\left(\#\G^*_m(\sfA)\right)>1$ and consider a supercritical Galton-Watson branching process $(S_n)_{n\ge 0}$ with $S_0=1$ and offspring distribution
\begin{equation*}
\mathbb{P}(S_1=k) = \mathbb{P}(\#\G^*_m(\sfA)=k),\quad k\in\N_{0}.
\end{equation*}
Obviously,
\begin{equation*}
\mathbb{P}(S_n>k)\leq \mathbb{P}(\#\G^*_{nm}(\sfA)>k)
\end{equation*}
for all $k,n\in\N_0$, hence
\begin{equation*}
\lim_{n\to\infty}\mathbb{P}(\#\G^*_{nm}(\sfA)>0)\geq\lim_{n\to\infty}\mathbb{P}(S_n>0)>0,
\end{equation*}
i.e.\ $\sfA$-parasites survive with positive probability.
``\eqref{Item.Extinction.EG}$\Rightarrow$\eqref{Item.Extinction.1}''
If $\mathbb{E}\#\G^*_n(\sfA)\leq1$ for all $n\in\N$, then Fatou's lemma implies
\begin{equation*}
1\geq\liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n
(\sfA)\right)
\end{equation*}
giving $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by an appeal to Theorem \ref{Satz.ExplosionInfZellen}.
``\eqref{Item.Extinction.gamma}$\Rightarrow$\eqref{Item.Extinction.1},\eqref{Item.Extinction.EG}''
If $\nu\leq1$ then $\mathbb{E}\#\G_{n}^{*}(\sfA)\le\mathbb{E}\#\G_n(\sfA)=\nu^{n}\le 1$ for all $n\in\N$. So let us consider the situation when
$$\nu>1,\quad\mathbb{E}\log g_{\Lambda_{1}}'(1)<0\quad\text{and}\quad\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\leq\frac{1}{\nu} $$
is valid. By \eqref{Eq.BPRE.EG,k>0},
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA) = \nu^n \mathbb{P}(Z_n(\sfA)>0)
\end{equation*}
for all $n\in\N$. We must distinguish three cases:
\textsc{Case A}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)\leq 0$.
By what has been pointed out in the above remark, we then infer
\begin{equation*}
\frac{\gamma}{\nu}=\mathbb{E} g_{\Lambda_{1}}'(1)=\inf_{0\leq\theta\leq1}\mathbb{E}
g_{\Lambda_{1}}'(1)^{\theta}\le\frac{1}{\nu}
\end{equation*}
and thus $\gamma\le 1$, which in turn entails
\begin{equation*}
\mathbb{E}\#\G^*_n(\sfA)\leq \mathbb{E}\mathcal{Z}_n(\sfA) = \gamma^n \leq 1
\end{equation*}
for all $n\in\N$ as required.
\textsc{Case B}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$ and $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$. Then, by \eqref{eq:Geiger survival estimate},
\begin{equation*}
\mathbb{P}(Z_n(\sfA)>0)\ \simeq\ c n^{-3/2} \left(\inf_{0\leq\theta\leq 1}
\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\right)^n\quad\text{as } n\to\infty
\end{equation*}
holds true for a suitable constant $c\in (0,\infty)$ and therefore
\begin{equation*}
0 = \lim_{n\to\infty} \nu^n \mathbb{P}(Z_n(\sfA)>0) = \liminf_{n\to\infty} \mathbb{E}\#\G^*_n(\sfA)
\geq \mathbb{E}\left(\liminf_{n\to\infty}\#\G^*_n(\sfA)\right),
\end{equation*}
implying $\mathbb{P}(\mathbb{E}xt(\sfA))=1$ by Theorem \ref{Satz.ExplosionInfZellen}.
\textsc{Case C}. $\mathbb{E} g_{\Lambda_{1}}'(1)\log g_{\Lambda_{1}}'(1)>0$ and $\mathbb{E} Z_{1}(\sfA)^{2}=\infty$. Using contraposition, suppose that $\sup_{n\in\N}\mathbb{E}\#\G^*_n(\sfA)>1$.
Fix any vector $\alpha=(\alpha^{(u)}_{\sfs})_{u\in\{0,1\},\sfs\in\{\sfAA,\sfAB,\sfBB\}}$
of distributions on $\N_{0}$ satisfying
$$ \alpha_{\sfs,x}^{(u)}\ \le\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)=x\right)\quad\text{for }x\geq 1 $$
and $u,\sfs$ as stated, hence
$$ \alpha_{\sfs,0}^{(u)}\ \ge\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)=0\right)\quad\text{and}\quad\sum_{x\ge n}\alpha_{\sfs,x}^{(u)}\ \le\ \mathbb{P}\left(X^{(u)}_{1,v}(\sfA,\sfs)\ge n\right) $$
for each $n\ge 0$. Possibly after enlarging the underlying probability space, we can then construct a cell division process $(Z_{\alpha,v}, T_v)_{v\in\T}$ coupled with and of the same kind as $(Z_{v}, T_v)_{v\in\T}$ such that
\begin{align*}
&X^{(u)}_{\alpha, k,v}(\sfA,\sfs)\ \leq\ X^{(u)}_{k,v}(\sfA,\sfs)\quad\text{a.s.}\\
\text{and}\quad
&\mathbb{P}\left(X^{(u)}_{\alpha,k,v}(\sfA,\sfs) = x \right)\ =\ \alpha_{\sfs,x}^{(u)}
\end{align*}
for each $u\in\{0,1\}$, $\sfs\in\{\sfAA,\sfAB,\sfBB\}$, $v\in\T$, $k\geq1$ and $x\ge 1$.
To have $(Z_{\alpha,v}, T_v)_{v\in\T}$ completely defined, put also
$$ (X^{(0)}_{\alpha,k,v}(\sfB),X^{(1)}_{\alpha,k,v}(\sfB)):=(X^{(0)}_{k,v}(\sfB),X^{(1)}_{k,v}(\sfB)) $$
for all $v\in\T$ and $k\ge 1$. Then $Z_{\alpha,v}\leq Z_v$ a.s.\ and thus
\begin{equation}\label{Eq.GestutzterProzess.G}
\mathbb{E} g_{\alpha,\Lambda_{1}}'(1)^{\theta}\ \leq\ \mathbb{E} g_{\Lambda_{1}}'(1)^{\theta},\quad \theta
\in[0,1],
\end{equation}
where $Z_{\alpha,k}(\sfA)$ and $g_{\alpha,\Lambda_{1}}$ have the obvious meaning. Since the choice of $\alpha$ has no affect on the cell splitting process, we have $\nu_{\alpha}=\nu>1$, while \eqref{Eq.GestutzterProzess.G} ensures
\begin{equation}\label{eq:truncation gf}
\mathbb{E}\log g_{\alpha,\Lambda_{1}}'(1)\le\mathbb{E}\log g_{\Lambda_{1}}'(1)<0.
\end{equation}
For $N\in\N$ let $\alpha(N)=(\alpha_{\sfs}^{(u)}(N))_{u\in\{0,1\},\sfs\in\{\sfAA,\sfAB,\sfBB\}}$ be the vector specified by
\begin{equation*}
\alpha_{\sfs,x}^{(u)}(N)\ :=\
\begin{cases}
\mathbb{P}\left(X^{(u)}_{k,v}(\sfA,\sfs) = x \right), &\text{if $1\leq x\leq N$}\\
0, &\text{if $x>N$}.
\end{cases}
\end{equation*}
Then $\mathbb{E} Z_{\alpha(N),1}(\sfA)^2<\infty$ and we can fix $N\in\N$ such that $\sup_{n\in\N}\mathbb{E}\#\G^*_{\alpha(N),n}(\sfA)>1$, because $\#\G^*_{\alpha(N),n}(\sfA)\uparrow\#\G^*_{n}(\sfA)$ as $N\to\infty$. Then, by what has already been proved under Case B in combination with \eqref{Eq.GestutzterProzess.G},\eqref{eq:truncation gf} and $\nu_{\alpha(N)}>1$, we infer
\begin{equation*}
\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq\inf_{0\leq\theta\leq1}\mathbb{E} g_{\alpha(N),\Lambda_{1}}'(1)^{\theta}>\frac{1}{\nu}.
\end{equation*}
and thus violation of \eqref{Item.Extinction.EG}.
``\eqref{Item.Extinction.EG}$\Rightarrow$\eqref{Item.Extinction.gamma}''
Suppose $\mathbb{E}\#\G_{n}^{*}(\sfA)\le 1$ for all $n\in\N$ and further $\nu>1$ which, by
\eqref{Eq.BPRE.EG,k>0}, entails $\lim_{n\to\infty}\mathbb{P}(Z_{n}(\sfA)>0)=0$ and thus $\mathbb{E}\log g_{\Lambda_{1}}'(1)\le 0$. We must show that $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$ and $\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\le\nu^{-1}$. But given $\mathbb{E}\log g_{\Lambda_{1}}'(1)<0$, the second condition follows from \eqref{eq:Gn and Geiger2} if $\mathbb{E} Z_{1}(\sfA)^{2}<\infty$, and by a suitable ``$\alpha$-coupling'' as described under Case C above if $\mathbb{E} Z_{1}(\sfA)^{2}=\infty$. Hence it remains to rule out that $\mathbb{E}\log g_{\Lambda_{1}}'(1)=0$. Assuming the latter, we find with the help of Jensen's inequality that
\begin{equation*}
\inf_{0\leq\theta\leq 1}\log \mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq \inf_{0\leq\theta\leq1}\theta \,\mathbb{E}\log g_{\Lambda_{1}}'(1) = 0
\end{equation*}
or, equivalently,
\begin{equation*}
\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}\geq 1>\frac{1}{\nu}
\end{equation*}
(which implies $\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta}=1$).
Use once more a suitable ``$\alpha$-coupling'' and fix $\alpha$ in such a way that
\begin{equation*}
1=\inf_{0\leq\theta\leq1}\mathbb{E} g_{\Lambda_{1}}'(1)^{\theta} >\inf_{0\leq\theta\leq1}\mathbb{E} g_{\alpha,\Lambda_{1}}'(1)^{\theta}>\frac{1}{\nu}
\end{equation*}
which implies subcriticality of the associated BPRE $(Z_{\alpha,n}(\sfA))_{n\geq0}$.
By another appeal to \eqref{eq:Gn and Geiger2}, we thus arrive at the contradiction
\begin{equation*}
\sup_{n\in\N}\mathbb{E}\#\G^*_{n}(\sfA)\ge\sup_{n\in\N}\mathbb{E}\#\G^*_{\alpha,n}(\sfA)=\infty.
\end{equation*}
This completes the proof of \eqref{Extinction_p>0}.
\eqref{Extinction_all}
Since $\mathbb{E}xt\subseteq \mathbb{E}xt(\sfA)$, we see that $\mathbb{P}(\mathbb{E}xt)=1$ holds iff $\mathbb{P}(\mathbb{E}xt(\sfA))=1$
and the population of $\sfB$-parasites dies out a.s.\ as well. But the latter form a Galton-Watson branching process with offspring mean $\mu_{0,\sfB}+\mu_{1,\sfB}$ once all $\sfA$-parasites have disappeared and hence die out as well iff $\mu_{0,\sfB}+\mu_{1,\sfB}\leq1$.
\end{proof}
\begin{Theorem}\label{Satz.W}
Assuming $\mathbb{P}(\Surv(\sfA))>0$ and thus particularly $\gamma>1$, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.W>0} If $\mathbb{E}\mathcal{Z}_1(\sfA)^2<\infty$ and $\hat{\gamma}\leq\gamma$, then $\mathbb{P}(W>0)>0$ and $\mathbb{E} W=1$.
\item\label{Item.W=0} If $\mathbb{P}(W=0)<1$, then $\mathbb{E}xt(\sfA)=\{W=0\}$ a.s.
\end{enumerate}
\end{Theorem}
\begin{proof}
\eqref{Item.W>0}
As pointed out at the beginning of this subsection, $(\mathcal{Z}_n(\sfA)/\gamma^n)_n$ is a $L^2$-bounded martingale and thus uniformly integrable. It therefore converges in $L^1$ to its limit $W$ satisfying $\mathbb{E} W=1$ as well as $\mathbb{P}(W>0)>0$.
\eqref{Item.W=0} follows in the same manner as Theorem \ref{Satz.ErholungB}\eqref{Item.ErholungB.Extinction}.
\end{proof}
\section{Relative proportions of contaminated cells}\label{Section.Results}
We now turn to a statement of our main results that are concerned with the long-run behavior of relative proportions of contaminated cells containing a given number of parasites, viz.
\begin{equation*}
F_k(n):=\frac{\#\{v\in\G^*_n|Z_v=k\}}{\#\G^*_n}
\end{equation*}
for $k\in\N$ and $n\to\infty$, and of the corresponding quantities when restricting to contaminated cells of a given type $\sft$, viz.
\begin{equation*}
F_k(n,\sft):=\frac{\#\{v\in\G^*_n(\sft)|Z_v=k\}}{\#\G^*_n(\sft)}
\end{equation*}
for $\sft\in\{\sfA,\sfB\}$. Note that
\begin{equation*}
F_k(n) = F_k(n,\sfA)\,\frac{\#\G^*_n(\sfA)}{\#\G^*_n}+F_k(n,\sfB)\,\frac{\#\G^*_n (\sfB)}{\#\G^*_n}.
\end{equation*}
Given survival of type-$\sfA$ parasites, i.e.\ conditioned upon the event $\Surv(\sfA)$, our results, devoted to regimes where at least one of $\sfA$- or $\sfB$-parasites multiply at a high rate, describe the limit behavior of $F_k(n,\sfA)$, $\#\G^*_n(\sfA)/\#\G^*_n$ and $F_k(n,\sfB)$, which depends on that of $\mathcal{Z}_n(\sfA)$ and the BPRE $Z_n(\sfA)$ in a crucial way.
For convenience, we define
\begin{equation*}
\mathbb{P}_{z,\sft} := \mathbb{P}(\cdot|Z_{\varnothing}=z, T_{\varnothing}=\sft),\quad z\in\N,\ \sft\in\{\sfA,\sfB\},
\end{equation*}
and use $\mathbb{E}_{z,\sft}$ for expectation under $\mathbb{P}_{z,\sft}$. Recalling that $\mathbb{P}$ stands for $\mathbb{P}_{1,\sfA}$,
we put $\mathbb{P}^{*}:=\mathbb{P}(\cdot|\Surv(\sfA))$ and, furthermore,
\begin{equation*}
\mathbb{P}^*_{z,\sft}:=\mathbb{P}_{z,\sft}(\cdot|\Surv(\sfA))\quad \text{and}\quad \mathbb{P}^n_{z,\sft}=\mathbb{P}_{z,\sft}(\cdot|\mathcal{Z}_n(\sfA)>0)
\end{equation*}
for $z\in\N$ and $\sft\in\{\sfA,\sfB\}$. Convergence in probability with respect to $\mathbb{P}^{*}$ is shortly expressed as $\mathop{\Prob^{*}\text{\rm -lim}\,}$.
Theorem \ref{Satz.Proportion.A1} deals with the situation when $\sfB$-parasites multiply at a high rate, viz.\
$$ \mu_{0,\sfB}\mu_{1,\sfB}>1, $$
In essence, it asserts that among all contaminated cells in generation $n$ those of type $\sfB$ prevail as $n\to\infty$. This may be surprising at first glance because multiplication of $\sfA$-parasites may also be high (or even higher), namely if
\begin{equation}\tag{SupC}\label{Eq.Supercritical}
\mu_{0,\sfA}(\sfAA)^{p_{\sfAA}}\mu_{1,\sfA}(\sfAA)^{p_{\sfAA}}\mu_{0,\sfA}(\sfAB)^{p_{\sfAB}}>1,
\end{equation}
i.e., if the BPRE $(Z_n(\sfA))_{n\ge 0}$ is supercritical. On the other hand, it should be recalled that the subpopulation of $\sfA$-cells grows at rate $\nu<2$ only, whereas the growth rate of $\sfB$-cells is 2. Hence, prevalence of $\sfB$-cells in the subpopulation of all contaminated cells is observed whenever $\#\G^*_n(\sfB)/\#\G_{n}(\sfB)$, the relative proportion of contaminated cells within the $n^{th}$ generation of all $\sfB$-cells, is asymptotically positive as $n\to\infty$.
\begin{Theorem}\label{Satz.Proportion.A1}
Assuming $\mu_{0,\sfB}\mu_{1,\sfB}>1$, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.Satz.A1.G}
\begin{equation*}
\frac{\#\G^*_n(\sfA)}{\#\G^*_n}\rightarrow0\quad \mathbb{P}^*\text{-a.s.}
\end{equation*}
\item\label{Item.Satz.A1.FA} Conditioned upon survival of $\sfA$-cells, $F_k(n,
\sfB)$ converges to $0$ in probability for any $k\in\N$, i.e.
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty} F_k(n,\sfB) = 0.
\end{equation*}
\end{enumerate}
\end{Theorem}
Properties attributed to a high multiplication rate of $\sfA$-parasites are given in Theorem \ref{Satz.Proportion.B1}. First of all, contaminated $\sfB$-cells still prevail in the long-run because, roughly speaking, highly infected $\sfA$-cells eventually produce highly infected $\sfB$-cells whose offspring $m$ generations later for any fixed $m$ are all contaminated (thus $2^{m}$ in number). Furthermore, the $F_{k}(n,\sfA)$ behave as described in \cite{Bansaye:08} for the single-type case: as $n\to\infty$, the number of parasites in any contaminated $\sfA$-cell in generation $n$ tends to infinity and $F_k(n,\sfA)$ to $0$ in probability. Finally, if we additionally assume that type-$\sfB$ parasites multiply faster than type-$\sfA$ parasites, i.e.
$$\mu_{\sfB}:=\mu_{0,\sfB}+\mu_{1,\sfB}>\gamma,$$
then type-$\sfB$ parasites become predominant and $F_k(n,\sfB)$ behaves again in Bansaye's single-type model \cite{Bansaye:08}.
\begin{Theorem}\label{Satz.Proportion.B1}
Assuming \eqref{Eq.Supercritical}, the following assertions hold true:
\begin{enumerate}[(a)]
\item\label{Item.Satz.B1.FB} Conditioned upon survival of $\sfA$-cells, $F_k(n,
\sfA)$ converges to $0$ in probability for any $k\in\N$, i.e.
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty}F_k(n,\sfA)=0.
\end{equation*}
\item\label{Item.Satz.B1.G} \begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\rightarrow\infty}\frac{\#\G^*_n(\sfA)}{\#\G^*_n}=0.
\end{equation*}
\item If $\mathbb{E}_{1,\sfB}\mathcal{Z}_1^2<\infty$, $\mu_{\sfB}>\gamma$ and $ \mu_{0,\sfB}\log\mu_{0,\sfB}+\mu_{1,\sfB}\log\mu_{1,\sfB}<0$,
then
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\rightarrow\infty}F_k(n,\sfB)=\mathbb{P}(\mathcal{Y}(\sfB)=k)
\end{equation*}
for all $k\in\N$, where $\mathbb{P}(\mathcal{Y}(\sfB)=k)=\lim_{n\rightarrow\infty}
\mathbb{P}_{1,\sfB}(Z_{[n]}=k|Z_{[n]}>0)$.
\end{enumerate}
\end{Theorem}
\section{Proofs}\label{Section.Proofs}
\begin{Beweis}[Theorem \ref{Satz.Proportion.A1}]
\eqref{Item.Satz.A1.G}
By Theorem \ref{Satz.Erholung}, $2^{-n}\#\G^*_n\to L$ $\mathbb{P}^{*}$-a.s. and $\mathbb{P}^{*}(L>0)=1$, while Theorem \ref{Satz.ErholungB} shows that $\nu^{-n}\#\G^*_n(\sfA)\to L(\sfA)$ $\mathbb{P}$-a.s.\ for an a.s. finite random variable $L(\sfA)$. Consequently,
\begin{equation*}
\frac{\#\G^*_n(\sfA)}{\#\G^*_n} ~=~ \left(\frac{\nu}{2}\right)^{n}\left(\frac{2^n}
{\#\G^*_n}\right)\left(\frac
{\#\G^*_n(\sfA)}{\nu^n}\right) ~\simeq~ \frac{1}{L}\left(\frac{\nu}{2}\right)^{n}
\frac{\#\G^*_n(\sfA)}{\nu^n}\to 0\quad \mathbb{P}^{*}\text{-a.s.}
\end{equation*}
as $n\to\infty$, for $\nu<2$.
\eqref{Item.Satz.A1.FA}
Fix arbitrary $\varepsilon, \delta>0$ and $K\in\N$ and define
\begin{equation*}
D_n := \left \{ \sum_{k=1}^{K}F_k(n,\sfB)>\delta \right \}\cap \Surv(\sfA).
\end{equation*}
By another appeal to Theorem \ref{Satz.Erholung}, $\#\G^*_n(\sfB)\geq 2^nL$ $\mathbb{P}^*$-a.s.\ for all $n\in\N$ and $L$ as above. It follows that
\begin{align*}
\#\{v\in\G_n(\sfB) : 0< Z_v\leq K\} ~\geq~ \delta\,\#\G^*_n(\sfB)\1_{D_n}
~\geq~ \delta\,2^n\,L\1_{D_n},
\end{align*}
and by taking the expectation, we obtain for $m\leq n$
\begin{align*}
\delta\, &\mathbb{E}\left( L\1_{D_n}\right)
~\leq~ \frac{1}{2^{n}}\,\mathbb{E}\left(\sum_{v\in\G_n}\1_{\{0<Z_v\leq K, T_v=\sfB
\}}\right)\\
&~\leq~ \frac{1}{2^{n}}\,\mathbb{E}\left(\sum_{v\in\G_n}\1_{\{0<Z_v\leq K, T_{v|m}
=\sfB\}}+\,\#\big\{v\in\G_n:T_{v|m}=\sfA, T_v=\sfB\big\}\right)\\
&~\leq~ \frac{1}{2^{n}}\,\sum_{v\in\G_n}\mathbb{P}\left(0<Z_v\leq K, T_{v|m}=
\sfB\right)+\frac{1}{2^{m}}\,\mathbb{E}\#\G_{m}(\sfA)\\
&~\leq~ \frac{1}{2^{n}}\,\sum_{z\ge 1}\sum_{v\in\G_n}\mathbb{P}\left(0<Z_v\leq
K, Z_{v|m}=z, T_{v|m}=\sfB\right)+\left(\frac{\nu}{2}\right)^m\\
&~\leq~ \sum_{z=1}^{\infty}\left(\sum_{u\in\G_m}\frac{\mathbb{P}(Z_u=z,
T_u=\sfB)}{2^{m}}\right)\Bigg(\sum_{u\in\G_{n-m}}\frac{\mathbb{P}_{z,\sfB}
(0<Z_v\leq K)}{2^{n-m}}\Bigg)+\left(\frac{\nu}{2}\right)^m\\
&~\leq~ \sum_{z=1}^{\infty}\mathbb{P}(Z_{[m]}=z, T_{[m]}=\sfB)\,\mathbb{P}_{z,\sfB}
\left(0<Z_{[n-m]}\leq K\right)+\left(\frac{\nu}{2}\right)^m.
\end{align*}
Since $\nu<2$ we can fix $m\in\N$ such that $(\nu/2)^{m}\le\varepsilon$.
Also fix $z_0\in\N$ such that
\begin{equation*}
\mathbb{P}(Z_{[m]}>z_0)\leq\varepsilon.
\end{equation*}
Then
\begin{align*}
\delta\,\mathbb{E}\left( L\1_{D_n}\right) ~&\leq~ \sum_{z\ge 1}\mathbb{P}(Z_{[m]}=z, T_
{[m]}=\sfB)\,\mathbb{P}_{z,\sfB}\left(0<Z_{[n-m]}\leq K\right)+\left(\frac{\nu}
{2}\right)^m\\
&\leq~\sum_{z=1}^{z_0}\mathbb{P}_{z,\sfB}\left(0<Z_{[n-m]}\leq K\right)
+2\varepsilon.
\end{align*}
But the last sum converges to zero as $n\to\infty$ because, under $\mathbb{P}_{z,\sfB}$,
$(Z_{[n]})_{n\ge 0}$ is a single-type BPRE (see \cite{Bansaye:08}) and thus satisfies the extinction-explosion principle. So we have shown that $\mathbb{E} L\1_{D_n}\to 0$
implying $\mathbb{P}(D_{n})\to 0$ because $L>0$ on $\Surv$. This completes the proof of the theorem.
\end{Beweis}
Turning to the proof of Theorem \ref{Satz.Proportion.B1}, we first note that part (a) can be directly inferred from Theorem 5.1 in \cite{Bansaye:08} after some minor modifications owing to the fact that $\sfA$-cells do not form a binary tree here but rather a Galton-Watson subtree of it. Thus left with the proof of parts (b) and (c), we first give an auxiliary lemma after the following notation:
For $v\in\G_n$ and $k\in\N$, let
\begin{equation*}
\G^*_k(\sft,v):=\{u\in\mathbb G^*_{n+k}(\sft):v<u\}
\end{equation*}
denote the set of all infected $\sft$-cells in generation $n+k$ stemming from $v$.
Let further be
\begin{equation*}
\mathbb G^*_n(\sfA,\sfB) := \{u\in\G^*_{n+1}(\sfB):T_{u|n}=\sfA\}
\end{equation*}
which is the set of all infected $\sfB$-cells in generation $n+1$ with mother cells of type $\sfA$.
\begin{Lemma}\label{Lemma.Proportion.B1}
If \eqref{Eq.Supercritical} holds true, then
\begin{equation*}
\mathop{\Prob^{*}\text{\rm -lim}\,}_{n\to\infty}\frac{\#\mathbb G^*_n(\sfA,\sfB)}{\#\mathbb G^*_n(\sfA)} ~=~ \beta ~>~0,
\end{equation*}
where $\beta := \lim_{z\to\infty}\mathbb{E}_{z,\sfA}\#\G^{*}_{1}(\sfB)$.
\end{Lemma}
\begin{proof}
Since $z\mapsto\mathbb{E}_{z,\sfA}\#\G^{*}_{1}(\sfB)$ is increasing and $\mathbb{E}_{1,\sfA}\#\G^{*}_{1}(\sfB)>0$ by our standing assumption \eqref{SA5}, we see that $\beta$ must be positive. Next observe that, for each $n\in\N$,
\begin{equation*}
\#\G^*_n(\sfA,\sfB) = \sum_{v\in\G^*_{n-1}(\sfA)}\#\G^*_1(\sfB,v),
\end{equation*}
where the $\#\G^*_1(\sfB,v)$ are conditionally independent given $\mathcal{Z}_n(\sfA)>0$. Since $\#\G^*_n(\sfA)\to\infty$ $\mathbb{P}^*$-a.s. (Theorem \ref{Satz.ExplosionInfZellen}) and $\mathbb{P}^n=\mathbb{P}(\cdot|\mathcal{Z}_n(\sfA)>0)\xrightarrow{w} \mathbb{P}^*$, it is not difficult to infer with the help of the SLLN that
\begin{equation*}
\frac{\#\mathbb G^*_n(\sfA,\sfB)}{\#\mathbb G^*_n(\sfA)} ~\overset{\mathbb{P}^*}\simeq~
\frac{1}{\#\G^*_n(\sfA)}\sum_{v\in\G^*_n(\sfA)}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB),\quad n
\to\infty.
\end{equation*}
where $a_{n}\stackrel{\mathbb{P}^{*}}{\simeq}b_{n}$ means that $\mathbb{P}^{*}(a_{n}/b_{n}\to 1)=1$. Now use $\mathbb{E}_{z,\sfA}\#\G^*_1(\sfB)\uparrow\beta$ to infer the existence of a $z_0\in\N$ such that for all $z\geq z_0$
\begin{equation*}
\mathbb{E}_{z,\sfA}\#\G^*_1(\sfB) \geq \beta(1-\varepsilon).
\end{equation*}
After these observations we finally obtain by an appeal to Theorem \ref{Satz.Proportion.B1}(a) that
\begin{align*}
\beta~&\geq~\frac{1}{\#\G^*_n(\sfA)}\sum_{v\in\G^*_n(\sfA)}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB)\\
&\geq~\sum_{z\geq z_0}\frac{F_z(n,\sfA)}{\#\{v\in\G^*_n(\sfA)|Z_v\geq z_0\}}\sum_{v\in\{u\in\G^*_n(\sfA)|Z_u\geq z_0\}}\mathbb{E}_{Z_v,\sfA}\#\G^*_1(\sfB)\\
&\geq~\beta(1-\varepsilon)\sum_{z\geq z_0}F_z(n,\sfA)\\
&\to~ \beta(1-\varepsilon),\quad n\to\infty.
\end{align*}
This completes the proof of the lemma.
\end{proof}
\begin{Beweis}[Theorem \ref{Satz.Proportion.B1}(b) and (c)]
Let $\varepsilon>0$ and $N\in\N$. Then
\begin{align*}
\#\G^*_n(\sfB) ~&=~ \sum_{k=0}^{n-1}\sum_{v\in\G^*_k(\sfA,\sfB)}\#\mathbb
G^*_{n-k-1}(\sfB,v)\\
&\geq~\sum_{k=0}^{n-1}\sum_{v\in\{u\in\mathbb G^*_k(\sfA,\sfB) | Z_u\geq z
\}}\#\mathbb G^*_{n-k-1}(\sfB,v)\\
&\geq~\sum_{v\in\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB) | Z_u\geq z\}}\#\mathbb
G^*_{m}(\sfB,v)
\end{align*}
a.s. for all $n>m\ge 1$ and $z\in\N$, and thus
\begin{equation}\label{Eq.GA.nur ein m}
\begin{split}
\mathbb{P}^*&\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)~=~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA) >
\#\mathbb G^*_n(\sfB)\right)\\
&\leq~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA)>\sum_{v\in\{u\in\mathbb G^*
_{n-1-m}(\sfA,\sfB) | Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)\right).
\end{split}
\end{equation}
Fix $m$ so large that
\begin{equation*}
2^{m}(1-\varepsilon)>\frac{4N}{\beta}.
\end{equation*}
Then, since
\begin{equation*}
\lim_{z\rightarrow\infty}\mathbb{P}_{z,\sfB}(\#\mathbb G^*_{m}=2^{m})=1,
\end{equation*}
there exists $z_0\in\N$ such that
\begin{equation*}
\mathbb{P}_{z,\sfB}(\#\mathbb G^*_{m}=2^{m})~\geq~1-\varepsilon
\end{equation*}
and therefore
\begin{equation}\label{Eq.Th.B1.1}
\mathbb{E}_{z,\sfB}\#\mathbb G^*_{m}~\geq~(1-\varepsilon)2^{m} ~>~\frac{4N}{\beta}
\end{equation}
for all $z\geq z_0$. Moreover, $\sum_{k\geq z_0}F_k(n,\sfA)\xrightarrow{\mathbb{P}^*} 1$
by part (a), whence
\begin{equation*}
\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA,\sfB)}\xrightarrow
{\mathbb{P}^*}1.
\end{equation*}
This together with Lemma \ref{Lemma.Proportion.B1} yields
\begin{equation*}
\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA)}\xrightarrow{\mathbb{P}^*}\beta
\end{equation*}
and thereupon
\begin{equation}\label{Eq.Th.B1.2}
\mathbb{P}^*\left(\frac{\#\{v\in\G^*_n(\sfA,\sfB):Z_v\geq z_0\}}{\#\G^*_n(\sfA)}\geq
\frac{\beta}{2}\right)\geq1-\varepsilon
\end{equation}
for all $n\geq n_0$ and some $n_{0}\in\N$. By combining \eqref{Eq.GA.nur ein m} and \eqref{Eq.Th.B1.2}, we now infer for all $n\geq n_0+m$
\begin{align*}
&\mathbb{P}^*\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)\\
&\leq~\mathbb{P}^*\left(N\,\#\mathbb G^*_n(\sfA) > \sum_{v\in\{u\in\mathbb G^*_{n-1-
m}(\sfA,\sfB) : Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)\right)\\
&\leq~\mathbb{P}^*\left(\frac{2N}{\beta} > \frac{\sum_{v\in\{u\in\mathbb
G^*_{n-1-m}(\sfA,\sfB) : Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)+\varepsilon\\
&\leq~\mathbb{P}^{\,n-m}\left(\frac{2N}{\beta} > \frac{\sum_{v\in\{u\in
\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\#\mathbb G^*_{m}(\sfB,v)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)\frac
{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
&\leq~\mathbb{P}^{\,n-m}\left(\frac{2N}{\beta} > \frac{\sum_{i=1}^{\#\{u
\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\mathcal G_{i,m}(z_0)}
{\displaystyle\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):Z_u\geq z\}}\right)\frac
{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
\end{align*}
where the $\mathcal G_{i,m}(z_0)$ are iid with the same law as $\#\{v\in\G^*_{m}(\sfB):Z_{\varnothing}=z_0, T_{\varnothing}=\sfB\}$. The LLN provides us with $n_1\geq n_0+m$ such that for all $n\geq n_1$
\begin{equation*}
\mathbb{P}^{n-m}\left(\frac{\sum_{i=1}^{\#\{u\in\mathbb G^*_{n-1-m}(\sfA,\sfB):
Z_u\geq z\}}\mathcal G_{i,m}(z_0)}{\#\{u\in\mathbb G^*_{n-1-m}(\sfA,
\sfB) :Z_u\geq z\}}\geq \mathbb{E}\mathcal G_{i,m}(z_0)/2\right)\geq 1-\varepsilon.
\end{equation*}
By combining this with \eqref{Eq.Th.B1.1}, we can further estimate in the above inequality
\begin{align*}
\mathbb{P}^*&\left(\frac{\#\mathbb G^*_n(\sfA)}{\#\mathbb G^*_n}>\frac{1}{N
+1}\right)\\
&\leq~\left(\mathbb{P}^{n-m}\left(\frac{2N}{\beta} > \mathbb{E}\mathcal G_{i,m}(z_0)/2 > \frac{2N}{\beta}\right) +\varepsilon\right)
\frac{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+\varepsilon\\
&=~\left(\frac{\mathbb{P}(\mathcal{Z}_{n-m}(\sfA)>0)}{\mathbb{P}(\Surv(\sfA))}+1\right)\varepsilon~\xrightarrow{n\to\infty}~2\varepsilon.
\end{align*}
This completes the proof of part (b).
As for (c), we will show that all conditions needed by Bansaye \cite{Bansaye:08} to prove his Theorem 5.2 are fulfilled. Our assertions then follow along the same arguments as provided there.
\textsc{Step 1:} $(\mu_{\sfB}^{-n}\mathcal{Z}_n(\sfB))_{n\ge 0}$ is a submartingale and converges a.s.\ to a finite limit $W(\sfB)$. The submartingale property follows from
\begin{align*}
&\mathbb{E}(\mathcal{Z}_{n+1}(\sfB)|\mathcal{Z}_n(\sfB))
~=~\mathbb{E}\left(\sum_{v\in\G^*_n}\big(Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\big)\Bigg|\mathcal{Z}_n(\sfB)\right)\\
&=~\mathcal{Z}_n(\sfB)\mathbb{E}\left(X^{(0)}(\sfB)+X^{(1)}(\sfB)\right)+\mathbb{E}\left(\sum_{v\in\G^*_n(\sfA)}\hspace{-.2cm}\big(Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\big)\Bigg|\mathcal{Z}_n(\sfB)\right)\\
&\geq~\mathcal{Z}_n(\sfB)\mu_{\sfB}
\end{align*}
for any $n\in\N$, while the a.s.\ convergence is a consequence of
\begin{equation*}
\sup_{n\in\N}\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)~<~\infty
\end{equation*}
which, using our assumption $\gamma<\mu_{\sfB}$, follows from
\begin{align*}
\mathbb{E}\left(\frac{\mathcal{Z}_{n+1}(\sfB)}{\mu_{\sfB}^{n+1}}\right) ~&=~ \mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\mathbb{E}\left(\frac{1}{\mu_{\sfB}^{n+1}}\sum_{v\in\G^*_n(\sfA)}Z_{v0}\1_{\{T_{v0}=\sfB\}}+Z_{v1}\1_{\{T_{v1}=\sfB\}}\right)\\
&=~\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\frac{1}{\mu_{\sfB}^{n+1}}\mathbb{E}\left(\mathcal{Z}_n(\sfA)\underbrace{\mathbb{E}(Z_{0}\1_{\{T_{0}=\sfB\}}+Z_{1}\1_{\{T_{1}=\sfB\}})}_{=:\mu_{\sfAB}}\right)\\
&=~\mathbb{E}\left(\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}\right)+\frac{\mu_{\sfAB}}{\mu_{\sfB}}\left(\frac{\gamma}{\mu_{\sfB}}\right)^n\\
=...&=~\frac{\mu_{\sfAB}}{\mu_{\sfB}}\sum_{k=0}^n\left(\frac{\gamma}{\mu_{\sfB}}\right)^k\
\ \leq~\frac{\mu_{\sfAB}}{\mu_{\sfB}}\sum_{k=0}^{\infty}\left(\frac{\gamma}{\mu_{\sfB}}\right)^k~<~\infty
\end{align*}
for any $n\in\N$.
\textsc{Step 2:} $\{W(\sfB)=0\}=\mathbb{E}xt$ a.s.\\
The inclusion $\supseteq$ being trivial, we must only show that $\mathbb{P}(W(\sfB)>0)\geq\mathbb{P}(\Surv)$. For $i\ge 1$, let $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ be iid copies of $(\mathcal{Z}_{n}(\sfB))_{n\ge 0}$ under $\mathbb{P}_{1,\sfB}$. Each $(\mathcal{Z}_{i,n}(\sfB))_{n\ge 0}$ forms a Galton-Watson process which dies out iff $\mu_{\sfB}^{-n}\mathcal{Z}_{i,n}(\sfB)\to 0$ (see \cite{Bansaye:08}). Then for all $m,N\in\N$, we obtain
\begin{align*}
\mathbb{P}(W(\sfB)>0) ~&=~\mathbb{P}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{m+n}(\sfB)}{\mu_{\sfB}^{m+n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^m}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\frac{1}{\mu_{\sfB}^{m}}\sum_{i=1}^{\mathcal{Z}_{m}(\sfB)}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_m(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}>0, \mathcal{Z}_{m}(\sfB)\geq N\right)\\
&\geq~\mathbb{P}\left(\mathcal{Z}_m(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\sum_{i=1}^{N}\frac{\mathcal{Z}_{i,n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N\right)-\mathbb{P}_{1,\sfB}\left(\lim_{n\to\infty}\frac{\mathcal{Z}_{n}(\sfB)}{\mu_{\sfB}^{n}}=0\right)^N\\
&=~\mathbb{P}\left(\mathcal{Z}_{m}(\sfB)\geq N|\Surv\right)\,\mathbb{P}(\Surv)-\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)^N.
\end{align*}
and then, upon letting $m$ and $N$ tend to infinity,
\begin{equation*}
\mathbb{P}(W(\sfB)>0)~\geq~\mathbb{P}(\Surv)
\end{equation*}
because $\mathbb{P}_{1,\sfB}\left(\mathbb{E}xt\right)<1$ and by Theorem \ref{Satz.ExplosionInfZellen}.
\textsc{Step 3:} $\sup_{n\ge 0}\mathbb{E}\xi_{n}<\infty$, where $\xi_{n}:=\left(\mu_{\sfB}/2\right)^{-n} Z_{[n]}$.
\noindent
First, we note that $(Z_{[n]})_{n\ge 0}$, when starting with a $\sfB$-cell hosting one parasite (under $\mathbb{P}_{1,\sfB}$), is a BPRE with mean $\mu_{\sfB}/2$ (see \cite{Bansaye:08}). Second, we have
\begin{equation*}
\mathbb{E} Z_{[n]}\1_{\{T_{[n]}=\sfA\}} ~=~ \mathbb{P}(T_{[n]}=\sfA)\,\mathbb{E} Z_{n}(\sfA) ~=~ \left(\frac{\gamma}{2}\right)^n
\end{equation*}
and thus
\begin{align*}
\mathbb{E} Z_{[n]} ~&=~ \mathbb{E} Z_{[n]}\1_{\{T_{[n]}=\sfA\}}+\sum_{m=0}^{n-1}\mathbb{E} Z_{[n]}\1_{\{T_{[m]}=\sfA,T_{[m+1]}=\sfB\}}\\
&=~\left(\frac{\gamma}{2}\right)^n+\sum_{m=0}^{n-1}\mathbb{E} Z_{[m]}\1_{\{T_{[m]}=\sfA\}}\,\mathbb{E}_{1,\sfA}Z_{[1]}\1_{\{T_{[1]}=\sfB\}}\,\mathbb{E}_{1,\sfB}Z_{[n-m-1]}\\
&=~\left(\frac{\gamma}{2}\right)^n+\eta\sum_{m=0}^{n-1}\left(\frac{\gamma}{2}\right)^m\left(\frac{\mu_{\sfB}}{2}\right)^{n-m-1}
\end{align*}
for all $n\in\N$, where $\eta:=\mathbb{E}_{1,\sfA}Z_{[1]}\1_{\{T_{[1]}=\sfB\}}$. This implies
\begin{equation}\label{Eq.BPRE.l1beschr}
\sup_{n\in\N}\mathbb{E}\xi_n ~=~ \left(\frac{\gamma}{\mu_{\sfB}}\right)^n+\frac{2\eta}{\mu_{\sfB}}\sum_{m=0}^{n-1}\left(\frac{\gamma}{\mu_{\sfB}}\right)^m~\leq~c\sum_{m=0}^{\infty}\left(\frac{\gamma}{\mu_{\sfB}}\right)^m~<\infty
\end{equation}
for some $c<\infty$.
\textsc{Step 4:} $\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\xi_n\1_{\{Z_{[n]}\geq K\}}=0$.
\noindent
By our assumptions, $(Z_{[n]})_{n\ge 0}$, when starting in a $\sfB$-cell with one parasite, is a strongly subcritical BPRE with mean $\mu_{\sfB}/2$ (see \cite{Bansaye:08}). Hence, by \cite[Corollary 2.3]{Afanasyev+etal:05},
\begin{equation}\label{Eq.B1c.gi}
\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}_{1,\sfB}\xi_n\1_{\{Z_{[n]}> K\}} ~=~0,
\end{equation}
which together with \eqref{Eq.BPRE.l1beschr} implies for $n,m\in\N$
\begin{align*}
\lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\, &\xi_{n+m}\1_{\{Z_{[n+m]}>K\}}\\
~&\leq~ \lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}\,\xi_{n+m}\1_{\{Z_{[n+m]}>K\}}\1_{\{T_{[m]}=\sfB\}}+\sup_{n\ge 0}\mathbb{E}\,\xi_{n+m}\1_{\{T_{[m]}=\sfA\}}\\
&\leq~ \lim_{K\to\infty}\sup_{n\ge 0}\mathbb{E}_{1,\sfB}\xi_n\1_{\{Z_{[n]}>K\}}\,\mathbb{E} \xi_m+\mathbb{E} \xi_m\1_{\{T_{[m]}=\sfA\}}\,\sup_{n\in\N}\mathbb{E}\,\xi_n\\
&\leq~ \left(\frac{\gamma}{\mu_{\sfB}}\right)^m\sup_{n\in\N}\mathbb{E}\,\xi_n
\end{align*}
and the last expression can be made arbitrarily small by choosing $m$ sufficiently large, for $\gamma<\mu_{\sfB}$. This proves Step 4.
\textsc{Final step:}
Having verified all conditions needed for the proof of Theorem 5.2 in \cite{Bansaye:08}, one can essentially follow his arguments to prove Theorem \ref{Satz.Proportion.B1}(c). We refrain from supplying all details here and restrict ourselves to an outline of the main ideas. First use
what has been shown as \textsc{Step 1 - 4} to prove an analogue of \cite[Lemma 6.5]{Bansaye:08}, i.e. \textit{(control of filled-in cells)}
\begin{equation}\label{control.filled.in.cells}
\lim_{K\to\infty}\sup_{n,q\geq0}\mathbb{P}^*\left(\frac{\#\{v\in\G^*_{n+q}(\sfB):Z_{v|n}>K\}}{\#\G^*_{n+q}(\sfB)}\geq\eta\right)=0
\end{equation}
for all $\eta>0$, and of \cite[Prop.\ 6.4]{Bansaye:08}, i.e. \textit{(separation of descendants of parasites)}
\begin{equation}\label{sep.of.descendants}
\lim_{q\to\infty}\sup_{n\geq0}\mathbb{P}^*\left(\frac{\#\{v\in\G^*_{n+q}(\sfB):Z_{v|n}\leq K, N_n(v)\geq2\}}{\#\G^*_{n+q}(\sfB)}\geq\eta\right)=0
\end{equation}
for all $\eta>0, K\in\N$, where $N_n(v)$ denotes the number of parasites in cell $v|n$ with at least one descendant in cell $v$. In particular, \eqref{control.filled.in.cells} (with $q=0$) combined with $\#\G^*_n(\sfB)\to\infty$ $\mathbb{P}^*$-a.s. implies the existence of a $K_0\geq0$ such that for all $N\in\N$
\begin{equation}\label{concentration.of.parasites}
\lim_{n\to\infty}\inf_{K\geq K_0}\mathbb{P}^*\left(\sum_{v\in\G^*_n(\sfB)}Z_v\1_{\{Z_v\leq K\}}\geq N\right) = 1.
\end{equation}
Using \eqref{control.filled.in.cells} and \eqref{sep.of.descendants}, we infer that, for all $\eta, \varepsilon>0$, there exist $K_1\geq K_0$ and $q_0\in\N$ such that for all $n\in\N$
\begin{equation*}
\mathbb{P}^*\Bigg(\Bigg|F_k(n+q_0,\sfB)-\underbrace{\frac{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_v=k, Z_{v|n}\leq K_1, N_n(v)=1\}}{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_{v|n}\leq K_1, N_n(v)=1\}}}_{=: J_{n}}\Bigg|\geq\eta\Bigg)~\leq~\varepsilon.
\end{equation*}
Since $\#\G^*_n(\sfA)/\#\G^*_n(\sfB)\xrightarrow{\mathbb{P}^*}0$, we further get
\begin{align*}
J_{n} ~&\overset{\mathbb{P}^*}{{\underset{n\to\infty}{\simeq}}}~ \frac{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_v=k, Z_{v|n}\leq K_1, T_{v|n}=\sfB, N_n(v)=1\}}{\#\{v\in\G^*_{n+q_0}(\sfB) | Z_{v|n}\leq K_1, T_{v|n}=\sfB, N_n(v)=1\}}
\end{align*}
as $n\to\infty$, which puts us in the same situation as in the proof of \cite[Thm.\ 5.2]{Bansaye:08}. Now, by using \eqref{concentration.of.parasites} and the LLN, we can identify the limit of $J_n$ which is in fact the same as in Step 1 of the proof of \cite[Thm.\ 5.2]{Bansaye:08}. A reproduction of the subsequent arguments from there finally establishes the result.
\end{Beweis}
\section*{Acknowledgment}
We are indebted to Joachim Kurtz (Institute of Evolutionary Biology, University of M\"unster)
for sharing with us his biological expertise of host-parasite coevolution and many fruitful discussions that helped us to develop the model studied in this paper.
\section{Glossary}
\begin{tabular}[c]{p{0.25\textwidth} p{0.75\textwidth}}
$\T$ & cell tree\\
$\G_n$ & set of cells in generation $n$\\
$\G_n(\sft)$ & set of cells of type $\sft$ in gegeration $n$\\
$\G^*_n$ & set of contaminated cells in generation $n$\\
$\G^*_n(\sft)$ & set of contaminated cells of type $\sft$ in generation $n$\\
$T_v$ & type of cell $v$\\
$p_{\sfs}$ & probability that the daughter cell of an $\sfA$-cell is of type $\sfs$\\
$p_0$ & probability that the $1^{st}$ daughter cell of an $\sfA$-cell
is of type $\sfA$\\
$p_1$ & probability that the $2^{nd}$ daughter cell of an $\sfA$-cell
is of type $\sfA$\\
$\nu$ & mean number of type-$\sfA$ daughter cells of an $\sfA$-cell\\
$(X^{(0)}(\sfA,\sfs),X^{(1)}(\sfA,\sfs))$ & offspring numbers of an $\sfA$-parasite with
daughter cells of type $\sfs\in\{\sfAA,\sfAB,\sfBB\}$\\
$(X^{(0)}(\sfB),X^{(1)}(\sfB))$ & offspring numbers of a $\sfB$-parasite\\
$Z_v$ & number of parasites in cell $v$\\
$\mu_{i,\sft}(\sfs)$ & mean number of offspring of a $\sft$-parasite which goes in
daughter cell $i\in\{0,1\}$ if daughter cells are of type
$\sfs\in\{\sfAA,\sfAB,\sfBB\}$\\
$\mu_{i,\sfB}$ & mean offspring number of $\sfB$-parasites which go in
daughter cell $i\in\{0,1\}$\\
$\mu_{\sfB}$ & reproduction mean of a parasite in a $\sfB$-cell\\
$\mathcal{Z}_n$ & number of parasites in generation $n$\\
$\Z_n(\sft)$ & number of parasites in $\sft$-cells in generation $n$\\
$\mathbb{E}xt/\Surv$ & event of extinction/survival of parasites\\
$\mathbb{E}xt(\sft)/\Surv(\sft)$ & event of extinction/survival of $\sft$-parasites\\
$Z_{[n]}$ & number of parasites in a random cell in generation $n$\\
$Z_n(\sfA)$ & number of parasites of a random $\sfA$-cell in generation $n$\\
$f_{n}(s|\Lambda),\ f_{n}(s)$ &quenched and annealed generating function of $Z_{n}(\sfA)$, respectively\\
$g_{\Lambda_n}(s)$ & generating function giving the $n$-th reproduction law of the
process of a random $\sfA$ cell line\\
$\gamma$ & mean number of offspring of an $\sfA$-parasite which go in an $\sfA$-cell\\
$\hat\gamma$ & $:=\nu\,\mathbb{E} g^{\prime}_{\Lambda_1}(1)^2
=p_{\sfAA}\left(\mu^2_{0,\sfA}(\sfAA)+\mu^2_{1,\sfA}(\sfAA)\right)
+p_{\sfAB}\,\mu^2_{0,\sfA}(\sfAB)$\\
$\mathbb{P}_{z,\sft}$ & probability measure under which the process starts with one $\sft$-cell
containing $z$ parasites\\
$\mathbb{P}^*_{z,\sft}$ & the same as before but conditioned upon $\Surv(\sfA)$\\
$\mathbb{P}^n_{z,\sft}$ & the same as before but conditioned upon survival of
$\sfA$-parasites in generation $n$
\end{tabular}
\input{host-parasite.bbl}
\end{document} |
\begin{document}
\mkcoverpage
\setcounter{page}{1}
\title{Heuristics for Packing Semifluids}
\author{Jo{\~a}o Pedro Pedroso}
\date{June 2015}
\maketitle
\begin{abstract}
Physical properties of materials are seldom studied in the context of packing problems. In this work we study the behavior of semifluids: materials with particular characteristics, that share properties both with solids and with fluids. We describe the importance of some specific semifluids in an industrial context, and propose methods for tackling the problem of packing them, taking into account several practical requirements and physical constraints. Although the focus of this paper is on the computation of practical solutions, it also uncovers interesting mathematical properties of this problem, which differentiate it from other packing problems.
\ \\
\noindent \textbf{Keywords:}
Packing; Semifluid; Heuristics; Tree search.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Semifluids are materials having characteristics of both fluids and solids. In the context of this paper, we will consider materials that cannot flow in one direction, though they are fluid in the other directions. As an example, consider tubes, which correspond to the industrial origin of this problem. Placed in a container, they can flow in the directions perpendicular to their length, but \emph{not} in the direction of their length (see Figure~\ref{fig:pipes1}). Assuming the tubes will be positioned perpendicularly to the Cartesian axes, depending on the direction of their placement they will flow either in the $x$ or in the $y$ dimension. Pipes, having positive radii, are imperfect semifluids, as they will not fully occupy the space available in the $z$ dimension; however, they approximate a perfect fluid as the radii becomes smaller. We will consider that the material is a perfect semifluid, and hence the volume occupied is constant and divisible.
This paper describes several possibilities for packing semifluids in a container, and presents heuristics for the variant which closer corresponds to an industrial application.
\section{Problem description}
\label{sec:problem}
Even though packing problems may be generalized into a single problem, they are usually divided in two categories: minimizing the number of bins, and maximizing the load to pack in a bin (see, \emph{e.g.}, \cite{Baldi20121205}). Given an index set $\mathcal{S}$ of semifluid items, with each item $i$ characterized by a fixed length $\ell_i$ and a volume $v_i$, and dimensions $D, W, H$ of containers, these two variants for the problem of packing a semifluid are:
\begin{enumerate}
\item bin packing variant: find the minimum number of containers to accommodate all the items;
\item knapsack variant: given, additionally, a value $w_i$ for the available volume $v_i$ of each item $i$, find the packing of maximum value that can be inserted in a container.
\end{enumerate}
In this paper we will focus on the knapsack variant.
\begin{figure}
\caption{A container accommodating a semifluid: tubes (left); coordinate system used (right).}
\label{fig:pipes1}
\end{figure}
\subsection{Semifluid packing problems}
There are several possibilities for packing a semifluid orthogonally in a container, as shown in Figure~\ref{fig:pipes2}. Both the length $\ell$ (corresponding to the length of the tubes) and the volume $v$ occupied by the semifluid are constant; in this figure, this means that $a \times b \times \ell = c \times d \times \ell = v$. Assuming that, except the container itself, there are no walls, a semifluid will take on all the available horizontal space in the direction where it freely flows. In the case presented, $a$ would take the depth $D$ of the container, and $c$ would take its width $W$, and hence the corresponding heights are $b = \frac{v}{D \ell}$ and $d = \frac{v}{W \ell}$.
\begin{figure}
\caption{Two possibilities for accommodating a semifluid in a container.}
\label{fig:pipes2}
\end{figure}
After a semifluid is placed, others may be put on top of it, but they must not protrude (as detailed next). Hence, one may think of the space above a semifluid as a ``container'', which can be filled up with the same rules as the original container; in this sense, this is a recursive problem.
Depending on the application, it may be allowed or not that, when packing a semifluid, it overflows others previously packed, as illustrated in Figure~\ref{fig:pipes3}. In general, allowing overflow makes packing solutions more difficult to implement in practice, and brings the problem more difficult to tackle; overflow will not be considered here.
\begin{figure}
\caption{Packing a semifluid without overflowing another previously packed (left), and overflowing it (right).}
\label{fig:pipes3}
\end{figure}
We will focus on packing semifluids by positioning the fixed dimension parallel to the $x$ axis, as shown in Figure~\ref{fig:pipes1}. This is the relevant variant when the container must be loaded from a lateral door at $x=D$: if the semifluids were rotated and be placed along the $y$ axis, they would flow out of the door.
An important, practical packing rule restricts what can be placed on top of what. Indeed, for cargo stability and for facilitating loading, it is usually acceptable that shorter tubes are placed on top of longer tubes, but not the inverse; more precisely, there must be no holders protruding with respect to holders below them.
In semifluid packing, any fraction of an item's available volume may be packed; this is major difference with respect to other packing problems.
We call the problem of maximizing the value of semifluids packed in the container in these conditions the \emph{basic semifluid packing problem}.
\subsection{Background}
\label{sec:background}
Three-dimensional packing has recently been studied under several different perspectives; a recent survey can be found in~\cite{Crainic2012}. The problem of allocating a given set of three-dimensional rectangular items to the minimum number of identical finite bins without overlapping has been addressed with tabu search in~\cite{Lodi2002410}: items are packed in several layers, the floor of the container being the first. A heuristic method for the situation where there is no requirement for packed boxes to form flat layers, keeping track of empty space seen from different perspectives and using a look-ahead scheme for positioning, is presented in~\cite{Lim2003471}.
However, the nature of the basic semifluid packing problem is rather different of these three-dimensional packing problems.
As will be seen later, there is more similarity between our problem and two-dimensional cutting. The most closely related problem is the orthogonal two-dimensional knapsack problem with guillotine patterns. Methods for tackling this problem are often based on a discretization of possible positions for the rectangles in the Cartesian plane (see, \emph{e.g.}, \cite{Puchinger2007,Elsa2010,dolatabadi2012}). A different approach is proposed in~\cite{fekete2007}, providing an exact algorithm for higher-dimensional orthogonal packing; the algorithm is based on bounding procedures which make use of dual feasible functions, within a tree search procedure. With respect to these problems, semifluid packing has the property that it is not required to pack all the available volume of each item; in rectangle packing, this would correspond to being able to cut some of the rectangles at the time of packing. Another difference between semifluid packing and previously studied problems concerns the requirement of no protuberance of items above others; this requirement is naturally respected in two-staged guillotine cuts, but usually is not enforced in general guillotine patterns.
To the best of our knowledge, basic semifluid packing or equivalent problems have not been studied before.
\subsection{Mathematical model}
We are not aware of previous attempts to formulate the semifluid packing problem as a mathematical optimization model, but there are some related problems. Integer programming models for two-dimensional two-stage bin packing problem have been proposed
in~\cite{Lodi2004} and extended by~\cite{Puchinger2007} to the three-stage problem. In both cases, decision variables are related to the assignment of the items to bins, stripes or stacks. Models for the related cutting stock problem, providing better linear relaxation bounds, are presented in~\cite{Elsa2010}, where a set of small rectangular items of given sizes is to be cut from a set of larger rectangular plates, in such a way that the total number of used plates is minimized. Despite some similarities, none of these models is adequate for our problem, mainly for two reasons: in semifluid packing the number of stages is in general much larger, and items may be partially assigned to a position (\emph{i.e.}, a position may hold a fraction of the available volume of an item).
The formulation proposed next is not compact, as it requires an exponential number of variables; however, it hopefully conveys the characteristics of the problem. For the sake of clarity, we start with a simplified model, and later describe how it could be extended to the general case; the simplification consists of assuming that only one stack of each item is allowed on each layer. A layer, in this context, is either the floor of the container or the space above a previously packed item. Figures \ref{fig:example} and~\ref{fig:instances} may be of help for visualizing the model.
The first set of binary variables indicates which items are packed in the first layer: $y_i=1$ if item $i$ is packed directly on the container, $y_i=0$ otherwise. To each variable $y_i$ there is a corresponding continuous variable $0 \leq x_i \leq 1$ which represents the fraction of item $i$ being packed at this place. Before introducing more variables, let us specify a constraint related to the length $D$ of the container, which limits the length of items packed in this layer:
\begin{alignat*}{27}
& \sum_i \ell_i y_i \leq D.
\end{alignat*}
Variable $x_i$ may be positive only if $y_i=1$:
\begin{alignat*}{27}
& x_i \leq y_i, && \quad \forall i.
\end{alignat*}
The height of the first layer is limited by the height of the container:
\begin{alignat*}{27}
& h_i x_i \leq H, && \quad \forall i,
\end{alignat*}
where $h_i = v_i/W$ is the total height that item $i$ would take on a container of width~$W$ (this will be later be replaced by a stronger constraint).
We now introduce variables concerning the placement of items $j$ on the second layer, \emph{i.e.}, directly above some previously packed item~$i$. Variables $y_{ij}$ are the indicators for this, and the corresponding $x_{ij}$ represent the fraction of $j$ packed at this place. The solution must, therefore, observe:
\begin{alignat*}{27}
& y_{ij} \leq y_{i}, && \quad \forall i,j,\\
& x_{ij} \leq y_{ij}, && \quad \forall i,j,\\
& \sum_{j} \ell_{ij} y_{ij} \leq \ell_{i} y_{i}, && \quad \forall i,
\end{alignat*}
where the last constraint limits the length of items placed directly above item~$i$. For each pair $i,j$ the height of the corresponding stack is limited to the height of the container:
\begin{alignat*}{27}
& h_i x_i + h_j x_{ij} \leq H, && \quad \forall i,j.
\end{alignat*}
The fraction of $i$ used in the two first layers is limited by one (this and the previous constraints will later be extended):
\begin{alignat*}{27}
& 0 \leq x_{i} + \sum_{j} x_{ji} \leq 1.
\end{alignat*}
We now have all the components to complete the model, by extending the number of layers. Notice that, as layers cannot protrude and items with identical length should be placed by decreasing value, there may be at most $N$ layers, where $N$ is the number of items. We may assume that the items are reversely ordered by length, \emph{i.e.}, $\ell_1 \geq \ell_2 \geq \ldots \geq \ell_N$; this allows us to define variables with indices $i,i'$ only for $i'>i$. Notice also that the number of indices indicates the level at which the item corresponding to a variable is being packed:
variables for layer $1\leq K\leq N$ will have $K$ indices $i,j,\ldots,m,n$, with $i < j < \ldots < m < n$.
The entire model is presented in Figure~\ref{fig:model}.
\begin{figure}
\caption{Mathematical optimization model.}
\label{eq:obj}
\label{eq:totalx}
\label{eq:D}
\label{eq:Dn}
\label{eq:x1}
\label{eq:xn}
\label{eq:y1}
\label{eq:yn}
\label{eq:H}
\label{eq:beginvar}
\label{eq:endvar}
\label{fig:model}
\end{figure}
Equations~(\ref{eq:totalx}) determines the total quantity of item $i$ packed, which allows determining the total value packed in~(\ref{eq:obj}).
Constraints (\ref{eq:D}) to~(\ref{eq:Dn}) guarantee that the total length of what is packed on top of the container, or of a packed item, does exceed the respective lengths.
Constraints (\ref{eq:x1}) to~(\ref{eq:xn}) allow a positive quantity of an item to packed only if the corresponding indicator variable is equal to~1.
Constraints (\ref{eq:y1}) to~(\ref{eq:yn}) allow packing only on top of previously packed items.
Inequalities (\ref{eq:H}) determines the height of all stacks, and limits it to the height of the container.
Finally, (\ref{eq:beginvar}) to~(\ref{eq:endvar}) define the domain for each of the variables.
This model is rather clumsy, but it is not yet complete: it does not take into account the possibility of packing several stacks of each item on a given layer. For make the model complete one would have to create, for each layer and each compatible item, a number of variables equal to the number of times that item would fit in the layer, if it was packed alone. It is obvious that direct usage of this model is implausible, except for a rather small number of items; realistic usage would require a column generation approach.
\section{Heuristic and complete search}
\label{sec:heur}
For solving the basic semifluid packing problem, we firstly propose a heuristic method --- which will later be improved --- for dividing a container into smaller parallelepipeds, which we call \emph{holders}. Each holder has a fixed depth, determined by the length of the semifluid it will accommodate. Due to the possibility for the semifluid to flow downwards, along the $z$ dimension, and also along the $y$ dimension, a semifluid will fully use the width of the physical container. The height of a filled holder is determined either by the volume of its semifluid or by the height of the physical container; in the latter case, the semifluid left over will possibly be packed in a different holder.
In this situation, one may think of the packing process as a division of, say, the container's wall at $y=0$, into rectangles. Each rectangle corresponds to the volume of a particular item when projected into the $y=0$ plane. For example, consider the placement of a semifluid as in Figure~\ref{fig:pipes1}; a projection of the volume occupied is represented as a rectangle, alike~1 in the left diagram of Figure~\ref{fig:example}. Upon placing this item, the container is divided into three partitions: one where item~1 is held (which is \emph{closed}, in the sense that it may not be used for other items), and the \emph{open} holders above (A) and besides the item~(B). Upon placing three more items in this example, the open holders are A, B, C, D in the right diagram of Figure~\ref{fig:example}.
\begin{figure}
\caption{Section of a container through the $y=0$ plane: open holders (shaded) after placing one item (left), and after placing four items (right).}
\label{fig:example}
\end{figure}
\subsection{Simple packing}
\label{packing}
A heuristic method for packing semifluids in these conditions can hence be though of as the process of choosing an item to pack, and an open holder for putting it (if some is available). For a semifluid of length $\ell$, candidate holders $j$ must have depth $D_j \geq \ell$. If the volume of a semifluid does not completely fit in the selected holder, the full height of the holder will be used (as for item 4 in the right diagram of Figure~\ref{fig:example}), and the remaining fluid is left to (possibly) pack later.
Given the characteristics of this problem, one might think of adapting known heuristics for bin packing and knapsack problems, as has been done for the two-dimensional knapsack problem (see, \emph{e.g.},~\cite{coffman1980,dolatabadi2012}); however, the geometric constraint forbidding longer lengths on top of shorter leads to possibly unexpected performance, as we will see shortly. Several alternative heuristic rules have been tried:
\begin{enumerate}
\item Best fit (BF): select the item/holder pair $(i,j)$ which leads to the minimum difference $D_j - \ell_i$, \emph{i.e.}, which leads to minimum currently unused space along~$x$;
\item Longest item first, first fit (LFF): select the longest item that can be packed in some open holder (\emph{i.e.}, item $i$ with largest $\ell_i$ for which there exists a holder $j$ such that $D_j - \ell_i \geq 0$), and insert it in the last open holder where it fits;
\item Longest item first, best fit (LBF): as LFF, but select the \emph{smallest} open holder in which the item fits;
\item Worthiest item first, first fit (WFF): as LFF, but select most valuable items (per unit volume) first;
\item Worthiest item first, best fit (WBF): as LBF, but select most valuable items first.
\end{enumerate}
\begin{algorithm}[htbp]
\begin{footnotesize}
\DontPrintSemicolon
\SetKwFunction{algo}{algo}\SetKwFunction{pack}{pack}
\SetKwFunction{algo}{algo}\SetKwFunction{h}{h}
\KwData{instance:
\begin{itemize}
\item set $\mathcal{S}$ of items to pack\;
\item item's length $\ell_i$, volume $v_i$, and value $w_i$, $\forall i \in \mathcal{S}$\;
\item physical container's width $W$, height $H$, and depth $D$;
\end{itemize}
}
\KwResult{
\begin{itemize}
\item set of holders $\mathcal{H}$ and their dimensions and position inside the container;
\item for each item $i$, the set $x_i$ of holders where it is packed.
\end{itemize}
}
\SetKwProg{myproc}{procedure}{}{}
\SetKw{Break}{break}
\myproc{\pack{$D, W, H, \mathcal{S}, \ell, v, w$}}{
$x_i \leftarrow \{\}, \quad \forall i \in \mathcal{S}$ \tcp*{initialize holders packing item $i$ as empty sets}
$\mathcal{H} \leftarrow \{$holder with dimensions $D \times W \times H\}$ \tcp*{open main holder}
\While{some item in $\mathcal{S}$ fits in an holder in $\mathcal{H}$}{
$(i,j) \leftarrow \h(\mathcal{S},\mathcal{H},\ell,v,w)$ \tcp*{heuristic choice of item $i$ and holder $j$} \label{alg:rule}
let $D_j, W_j, H_j$ be the current dimensions of holder $j$ \;
$z \leftarrow v_i / (\ell_i W_j)$ \;
\If(\tcp*[f]{all volume of $i$ fits}){$z \leq H_j$}{
$v_i \leftarrow 0$\;
$\mathcal{S} \leftarrow \mathcal{S} \setminus \{i\}$ \;
$(D_j,W_j,H_j) \leftarrow (\ell_i,W_j,z)$ \tcp*{adjust $j$'s dimensions}
}
\Else{
$v_i \leftarrow (v_i - \ell_i W_j H_j)$ \tcp*{update volume of $i$ remaining unpacked}
$(D_j,W_j,H_j) \leftarrow (\ell_i,W_j,H_j)$ \tcp*{adjust $j$'s dimensions}
}
$x_i \leftarrow x_i \cup \{j\}$ \tcp*{add $j$ to set of holders packing $i$}
$\mathcal{H} \leftarrow \mathcal{H} \setminus \{j\}$ \tcp*{remove $j$ from open holders}
\If{$D_j > \ell_i$}{
$\mathcal{H} \leftarrow \mathcal{H} \cup \{$holder with dimensions $(D_j-\ell_i) \times W_j \times H_j\}$ \tcp*{open holder besides $j$}
}
\If{$H_j > z$}{
$\mathcal{H} \leftarrow \mathcal{H} \cup \{$holder with dimensions $\ell_i \times W_j \times (H_j-z)\}$ \tcp*{open holder on top of $j$}
}
}
\Return{$x$}\;
}
\end{footnotesize}
\caption{Simple heuristic method for packing semifluids.}
\label{alg:pack}
\end{algorithm}
These rules are used in the heuristic method detailed in Algorithm~\ref{alg:pack}; we are abusing of notation, by allowing items and holders to be represented also by indices in their respective sets. The algorithm returns a map associating each item to the set of holders that contain it (which is empty for items that are not packed). The heuristic rule to be used is specified in line~\ref{alg:rule}, and holders are created accordingly in the subsequent lines. The algorithm iterates as long as there is an open holder where some unpacked item fits.
The full description of the computational setup is deferred to Section~\ref{sec:results}; for the time being, we just present in Table~\ref{tab:pack} a comparison of the solutions obtained with these simple rules on a set of 3000 test instances. We have counted the number of times that heuristic construction with a rule is \emph{strictly} better than with another, for all the combinations.
\begin{table}[h!tbp]
\centering
\caption{Comparison of simple rules for a data set of 3000 instances. Left table: $n_{ij}$, the number of times rule $i$ was strictly better (\emph{i.e.}, found a better solution) than rule $j$. Right table: $n_{ij} - n_{ji}$; positive values mean that rule on line $i$ is better for more instances than the rule in column $j$.}
\label{tab:pack}
\begin{tabular}{l|*{5}{r@{~~~~}}}
& BF & LFF & LBF & WFF & WBF \\\hline
BF & 0 & 225 & 51 & 2526 & 2397 \\
LFF & 338 & 0 & 101 & 2525 & 2396 \\
LBF & 338 & 237 & 0 & 2529 & 2404 \\
WFF & 339 & 342 & 336 & 0 & 91 \\
WBF & 398 & 401 & 391 & 1737 & 0 \\
\end{tabular}
~~~~~
\begin{tabular}{l|*{5}{r@{~~~~}}}
& BF & LFF & LBF & WFF & WBF \\\hline
BF & 0 & -113 & -287 & 2187 & 1999 \\
LFF & 113 & 0 & -136 & 2183 & 1995 \\
LBF & 287 & 136 & 0 & 2193 & 2013 \\
WFF & -2187 & -2183 & -2193 & 0 & -1646 \\
WBF & -1999 & -1995 & -2013 & 1646 & 0 \\
\end{tabular}
\end{table}
The results obtained are rather surprising: rules based on the value of the items, very effective for the knapsack problem, are clearly outclassed by rules based on the length of the semifluid. The simple rule of selecting the longest semifluid, independently of its value, and placing it in the open holder that leads to less used space along the $x$ axis (LBF) has generated the best results. This is the heuristic rule selected for comparison with more elaborate methods.
\subsection{Local ascent}
\label{sec:local}
The previous packing algorithm can be easily extended to encompass local ascent, as proposed in Algorithm~\ref{alg:ascent}. The idea is very simple: after finding a packing with the previous heuristics, attempt another construction forbidding items packed in the current solution, one at a time. As soon as an improving solution is found, it is adopted as incumbent (\emph{first-improve}). This process stops when all the neighbors of the current solution have been attempted, and they all lead to inferior solutions.
\begin{algorithm}[htbp]
\begin{footnotesize}
\DontPrintSemicolon
\SetKwFunction{algo}{algo}\SetKwFunction{ascent}{ascent}
\SetKwFunction{algo}{algo}\SetKwFunction{pack}{pack}
\SetKwProg{myproc}{procedure}{}{}
\SetKw{True}{true}
\SetKw{False}{false}
\SetKw{Break}{break}
\myproc{\ascent{$D, W, H, \mathcal{S}, \ell, v, w$}}{
$x \leftarrow \pack(D, W, H, \mathcal{S}, \ell, v, w)$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
$\mathcal{T} \leftarrow \{\}$\;
\Repeat{not improved}{
improved = \False\;
\For{$i \in \mathcal{I} \setminus \mathcal{T}$}{
$\mathcal{T} \leftarrow \mathcal{T} \cup \{ i\}$ \;
$x' \leftarrow \pack(D, W, H, \mathcal{S}\setminus\{i\}, \ell, v, w)$ \;
\If{value of $x'$ is greater than value of $x$}{
$x \leftarrow x'$ \;
let $\mathcal{I}$ be the set of items packed in $x$ \;
improved = \True \;
\Break \;
}
}
}
\Return{$x$}\;
}
\caption{Local ascent for packing semifluids.}
\label{alg:ascent}
\end{footnotesize}
\end{algorithm}
This method is simple, and obviously finds a solution which is at least as good as that of Algorithm~\ref{alg:pack}. As local ascent is still very fast, it is suitable for demanding situations (\emph{e.g.}, interactive processes).
\subsection{Complete search}
\label{sec:tree}
There are two reasons why the previous methods may be unsatisfactory. The first reason concerns some rare, small instances for which a better solution can easily be found by inspection; the second reason concerns proving that the solution found is optimal. We next propose some variants for doing complete search, based on tree search.
Let us start with a caveat. In the packing process we are considering, division of the semifluid occurs only when it does not fit vertically, and the amount left is possibly packed in another holder. However, it may be optimal to fill only a part of the available amount of a semifluid. This case is illustrated in Figure~\ref{fig:subopt}; if item 2 is more valuable than 1, it would be optimal to fill all the volume of item 2 over a part of 1, and leave the remaining 1 unpacked, as shown in the rightmost diagram. However, visited solutions in a complete tree search are only the leftmost and the one in the center; hence, an \emph{``optimum''} for tree search many not be truly optimal for the original problem.
\begin{figure}
\caption{An instance for which complete search does not find the optimum (shown in the rightmost diagram). A vertical section of the container is represented with a bold line, and the item left over is shown beside it.}
\label{fig:subopt}
\end{figure}
Complete search is an extension of Algorithm~\ref{alg:pack} where, instead of considering only packing the item chosen by the heuristic rule in line~\ref{alg:rule}, we consider all the possibilities of placing available items in open containers; each of these possibilities leads to a new node in the search tree. Notice that the branching factor is very large, and hence straightforward complete search is prohibitive even for small instances. Next, we present three relevant tree search alternatives for dealing with this difficulty; a visual insight of the differences between them is provided in Figure~\ref{fig:queue}.
\begin{figure}
\caption{Queueing methods: branch-and-bound (left), where nodes in the queue are sorted by their upper bound; breadth-first search with diving (center), where no information about about the nodes entering the queue is used (at each expansion, one node generated is the diving node); and limited discrepancy search (right), where nodes are sorted by discrepancy (at each expansion, nodes are generated in this order).}
\label{fig:queue}
\end{figure}
\subsubsection{Branch-and-bound}
\label{sec:bb}
Branch-and-bound (BB) is the standard method for searching a tree in optimization (see, \emph{e.g.},~\cite{lawler66} for an early survey). For a maximization problem, the comparison of an upper bound of the objective that can be reached from a given node, to a known lower bound of the objective, is used to eliminate from consideration parts of the search tree. The best solution visited so far is commonly used as the lower bound. In the case of the basic semifluid packing problem, an upper bound can be obtained by sorting the items by decreasing unit value, and filling the space still available in the container by this order, assuming no shape constraints (this is similar to the linear relaxation bound for the knapsack problem; see~\cite{martello1990}). For a given partial solution, holders that cannot be filled due to having no unpacked items that fit inside them are withdrawn from the list of open holders; their volume is subtracted from the space available when computing the corresponding upper bound.
Another important factor for having a reasonably effective branch-and-bound concerns avoiding symmetric, or otherwise equivalent solutions. This is done with the following rules:
\begin{itemize}
\item items placed at the same horizontal level must have increasing indices in the set $\mathcal{S}$ of semifluids to pack;
\item items placed on top of given item $i$ having the same length as $i$ cannot have a larger unit value than~$i$.
\end{itemize}
The main steps of the branch-and-bound algorithm are outlined in Algorithm~\ref{alg:bb} (see also Appendix~\ref{sec:data}). The algorithm is based on the iteration over elements in a queue ($Q$) until it becomes empty. Nodes whose upper bound is inferior to the objective value of the best known solution are discarded (line~\ref{alg:pruning}). Branching is carried out in lines \ref{alg:branchS}--\ref{alg:branchE}. As all the possible assignments of yet unpacked items to open holders must be considered, the main limitation of the algorithm concerns the large number of nodes added in these lines.
The algorithm has two parameters, limiting CPU time and the size of the queue. The latter is used when restricting the number of open nodes is required for keeping memory usage acceptable; in such cases, we provide the possibility of removing a part of the queue (\emph{chopping}, lines \ref{alg:chopS}--\ref{alg:chopE}). When this occurs, as well as when the time limit is reached, the solution returned may be not optimal. In the experiment reported in the Section~\ref{sec:results}, the maximum number of nodes is set to infinity, making CPU time the only factor limiting the search.
\newcommand{P}{P}
\newcommand{S}{S}
\newcommand{\textit{UB}}{\textit{UB}}
\newcommand{\textit{LB}}{\textit{LB}}
\newcommand{\textit{OPT}}{\textit{OPT}}
\begin{algorithm}[htbp!]
\SetKw{True}{true}
\SetKw{False}{false}
\SetKw{Continue}{continue}
\DontPrintSemicolon
create a queue $Q$ with one node (the root relaxation) \tcp*{Initialization}
set upper bound $\textit{UB} \leftarrow \infty$, lower bound $\textit{LB} \leftarrow -\infty$, optimality flag $\textit{OPT} \leftarrow \True$ \;
\Repeat{$Q = \{\}$ or time limit has been reached}{
select and remove from $Q$ node $k$ with largest $\textit{UB}$ \label{alg:pruning} \tcp*{Subproblem selection}
\If(\tcp*[f]{Pruning and fathoming}){$\textit{UB}^k \leq \textit{LB}$ or no items fit in open holders}{
\Continue
}
\ForEach(\tcp*[f]{Partitioning}){feasible assignment of unpacked items to open holders \label{alg:branchS}}{
add new node $n$ to $Q$\;
\If{$\textit{LB}^n > \textit{LB}$}{
update $\textit{LB} \leftarrow \textit{LB}^n$\; \label{alg:branchE}
}
}
\While(\tcp*[f]{Chopping}){size of $Q$ is larger than the allowed limit \label{alg:chopS}}{
remove from $Q$ node with smallest $\textit{UB}$\;
$\textit{OPT} \leftarrow \False$ \; \label{alg:chopE}
}
\If(\tcp*[f]{Termination}){time limit has been reached} {
$\textit{OPT} \leftarrow \False$ \;
}
}
\Return{solution that yielded $\textit{LB}$, with optimality flag $\textit{OPT}$}
\caption{Main steps of the branch-and-bound algorithm.}\label{alg:bb}
\end{algorithm}
\subsubsection{Breadth-first search with diving}
\label{sec:bfs}
As the branching factor is very large, standard branch-and-bound may not be allowed the time and space to produce a good solution, even for relatively small instances. Indeed, as will be seen in the next session, in a limited time the solution of branch-and-bound is often worse than that of the simple heuristics. For overcoming this issue, several alternatives have been proposed in the literature; these are usually based on \emph{diving} (see, \emph{e.g.}, \cite{achterberg2008,pochet2006}).
We firstly propose what we call \emph{breadth-first with diving (BFD)}, which consists of the following:
\begin{enumerate}
\item Keep two search queues: the main queue $Q$ and the \emph{diving queue} $R$;
\item If $R$ is not empty, at the current iteration explore the last element added to this queue (\emph{i.e.}, explore $R$ in a last in, first out manner);
\item If $R$ is empty, at the current iteration explore the first element added to $Q$ (\emph{i.e.}, explore $Q$ in a first in, first out manner);
\item When creating children of the current node, append the one that corresponds to the heuristic rule (LBF) to the queue $R$, and the remaining children (generated by decreasing item length) to $Q$.
\end{enumerate}
Hence, $R$ is searched in a last in, first out fashion, corresponding to the order of the LBF heuristic rule (longest item first, best fit container); therefore, the first leaf visited is the LBF solution. The exploration of $Q$ in a breadth-first (first in, first out) fashion introduces diversity in the search, which balances well with the intensive search of the dive; this is important for time-limited executions, where parts of the tree are left unexplored. Furthermore, quickly finding solutions of good quality allows pruning more nodes in the search tree. Notice that as long as the item list is initially sorted by length, we can generate new nodes to add to $Q$ without further sorting (however, sorting available items by value is required for computing the upper bound of a new node).
Diving does not interact well with the symmetry breaking rules: if the diving item was forbidden for avoiding symmetry, the first dive would be interrupted, and the corresponding heuristic solution would not be reached. In order to assure that we reach that solution, rules for avoiding symmetry are not enforced during diving.
In our implementation of BFS we are using the bounds described in Section~\ref{sec:bb}, which in most cases allow pruning significant parts of the search tree.
\subsubsection{Limited discrepancy search}
\label{sec:lds}
Another alternative to standard branch-and-bound is \emph{limited discrepancy search (LDS)}, where the tree is searched by increasing order of the \emph{number of violations} of the heuristic rule, as proposed in \cite{harvey1995}. This method has been attempted with the LBF heuristic rule, but the computation of discrepancy in this case requires sorting the moves available, using considerable computational time. A better alternative is to base the search in the longest item first, first fit rule (LFF); this allows a very quick expansion of nodes at each iteration, and exploring much larger parts of the tree in a limited time.
As in the standard version of LDS, this method uses a parameter specifying the discrepancy level above which search is abandoned. This usually allows adjusting the part of the tree that is explored to the resources available, as an alternative to simply interrupting the execution after a certain time has elapsed. We acknowledge that better solutions are often found with such an adjustment, and that memory usage will make the search impractical for long-running executions without limiting discrepancy; however, for an easier comparison with the other methods, we have set the discrepancy limit to infinity. Due to this choice, whenever LDS ends before reaching the limit CPU time, its solution is optimal.
In our implementation of LDS we are using the bounds described in Section~\ref{sec:bb}, which in most cases allow pruning significant parts of the search tree.
\section{Computational results}
\label{sec:results}
In order to assess the performance of the methods proposed, we have created a set of instances based on the characteristics of the real-world application. Practical instances we are aware of are small, as is the number of different semifluid lengths (tubes are usually cut in standard lengths). Instances with more than 20 items go beyond the application's requirement, but are useful for testing the behavior of the different algorithms. Instances are classified into two main families:
\begin{itemize}
\item Easy instances: generated in such a way that in the optimum there are no items left unpacked; for these instances, an optimal solution completely occupying the container is known.
\item Hard instances: no optimum is known in advance; the volume of available items corresponds either to 100\% of the container (as for easy instances, though now it is unlikely that all items can be packed), or to 150\% of it.
\end{itemize}
The number of semifluids considered are 5, 10, 20, 50, and~100. Some instances have just a few distinct item lengths, other have more diverse lengths.
For each combination of these characteristics, 100 different instances have been generated, totaling 3000 instances. A visualization of instances from the easy and hard subsets, with corresponding optimal and heuristic solutions, is provided in Figure~\ref{fig:instances} (details on the instance generator are available in Appendix~\ref{sec:data}).
\begin{figure}
\caption{An optimal solution (left), and a heuristic solution (right) for instances with ten items: an easy instance (top) and a hard instance (bottom).}
\label{fig:instances}
\end{figure}
Our programs use exact arithmetic for all operations (hence, values in the instance files are written as fractions). All the executions were limited to 60 seconds of CPU time, and both the maximum number of nodes and the discrepancy limit were set to infinity.
We start recalling the comparison among simple heuristics (Table~\ref{tab:pack}). Having selected LBF, we now compare it to more elaborate methods in Table~\ref{tab:all}. As expected, local ascent is always at least as good as LBF, being strictly superior for a massive share of instances. As the CPU time limitation is rather severe, local ascent is also often better than tree search methods. The best results overall have been obtained by limited discrepancy search.
\begin{table}[h!tbp]
\centering
\caption{Comparison of simple rule (LBF), local ascent (LA), and tree search --- standard branch-and-bound version (BB), breadth-first search with diving (BFD), and limited discrepancy search (LDS) --- for a data set of 3000 instances. Left table: $n_{ij}$, the number of times method $i$ was strictly better (\emph{i.e.}, found a better solution) than method $j$. Right table: $n_{ij} - n_{ji}$; positive values mean the method on line $i$ is better for more instances than the method in column $j$.}
\label{tab:all}
\begin{tabular}{l|*{7}{r@{~~~}}}
& LBF & LA & BB & BFS & LDS \\\hline
LBF & 0 & 0 & 1627 & 306 & 42 \\
LA & 2041 & 0 & 1744 & 849 & 187 \\
BB & 909 & 633 & 0 & 108 & 101 \\
BFS & 1520 & 1092 & 1895 & 0 & 329 \\
LDS & 2007 & 1350 & 1915 & 1192 & 0 \\
\end{tabular}
~~~~~
\begin{tabular}{l|*{7}{r@{~~~}}}
& LBF & LA & BB & BFS & LDS \\\hline
LBF & 0 & -2041 & 718 & -1214 & -1965 \\
LA & 2041 & 0 & 1111 & -243 & -1163 \\
BB & -718 & -1111 & 0 & -1787 & -1814 \\
BFS & 1214 & 243 & 1787 & 0 & -863 \\
LDS & 1965 & 1163 & 1814 & 863 & 0 \\
\end{tabular}
\end{table}
Figure~\ref{fig:apercu} graphically summarizes the results obtained. On each sub-figure, results for instances of type easy and hard are separated into two rows. Each bar (or curve, in the bottom sub-figure) represents percentages or averages considering all the instances of each size. Methods considered are, as before, the simple heuristic rule (LBF), local ascent based on this rule, branch-and-bound, breadth-first search with diving, and limited discrepancy search; to each method corresponds a column in the top three sub-figures, and a line in the bottom sub-figure. The abscissa for the three top sub-figures is the instance size, and for the bottom sub-figure is the CPU time used. For each instance we have identified the best solution found by all the methods (which in some cases is optimal); on the top sub-figure, the ordinate is the percentage of instances for which each method finds such solution. The next set of plots shows the percentage of instances for which the search tree was completely explored (for the relevant methods). Follows a plot of the average CPU time used in the solution process, for all the instances of each size/type; the time for each run was limited to 60 seconds, but in many cases was smaller. Finally, the sub-figure in the bottom shows the evolution of the average value of the objective for the best solution found by each method, in terms of the CPU time used; here we can observe how the gap between the different methods progresses.
As can be seen in Figure~\ref{fig:apercu}, ``standard'' branch-and-bound (taking the node with the highest upper bound at each iteration) quickly becomes very limited, when the instance size increases; this is due to the very high branching factor. Crossing information on that figure with that of Table~\ref{tab:nnodes}, we see that when instance size increases, a very large number of nodes are open, but, due to the time limitation, only a small part of them can be explored. This can be observed for all except smallest, easy instances. For finding good solutions in a limited time, methods fully exploiting the heuristics (LA, BFS and LDS) have a much better performance. Note that, for larger values of the CPU limit, the number of open nodes may have to be limited for avoiding memory overflow.
\begin{figure}
\caption{Overall aper{\c c}
\label{fig:apercu}
\end{figure}
We have seen in Table~\ref{tab:all} that the method that is able to find strictly better solutions than the others for more instances is limited discrepancy search. This is corroborated by the evolution over time of the average solution, for all instances of a given size, presented at the bottom of Figure~\ref{fig:apercu}. The general tendency is to have LDS finding good solutions more quickly than the other tree search methods; however, near the CPU limit imposed, LDS is closely followed by BFD (\emph{e.g.}, for hard instances of size 50). In terms of the ability to complete the search, and hence to prove optimality, BFD and LDS are roughly equivalent; these methods appear to be considerably better than BB for easy instances, though slightly inferior for hard instances.
The main factor for LDS to be able to explore much more nodes than BFD is the ability to easily keep nodes organized by increasing discrepancy; for technical details, please consult the implementation code (see Appendix~\ref{sec:data}).
Another interesting observation concerns the performance of local ascent. For small instances, LA quickly finds the best solution (often proven optimal by tree search); however, LA is outclassed by tree search methods for mid-sized instances, to regain a relative good performance for large instances, as can be seen in the top graphic of Figure~\ref{fig:apercu}. This is because local ascent is very fast, and hence the time constraint is not limiting it in our experiment, even for large instances.
\begin{table}
\centering
\caption{Average number of nodes explored, remaining in the queue at the end of the search, and created, for each of the tree search methods.}
\label{tab:nnodes}
\sisetup{
round-integer-to-decimal,
table-format = 5.0,
round-mode = places,
round-precision = 0,
}
\begin{small}
\begin{tabular}{lr|*{3}S|*{3}S|*{3}S}
\multicolumn{2}{c|}{Instance} & \multicolumn{3}{c|}{Nodes explored} & \multicolumn{3}{c|}{Nodes in queue} & \multicolumn{3}{c}{Nodes created} \\
Type & Size & {BB} & {BFS} & {LDS} & {BB} & {BFS} & {LDS} & {BB} & {BFS} & {LDS}\\ \hline
easy & 5 & 10.38 & 5 & 25.96 & 0 & 0 & 0 & 27.84 & 16.545 & 16.695 \\
easy & 10 & 3193.46 & 11.97 & 103.195 & 4379.28 & 0 & 0 & 13790 & 82.42 & 78.09 \\
easy & 20 & 8113.42 & 165.94 & 1895.53 & 17430.8 & 998.705 & 3699.11 & 37510.8 & 3348.65 & 5502.44 \\
easy & 50 & 1973.79 & 139.17 & 6052.78 & 17811.1 & 6167.86 & 109864 & 23525.5 & 8231.42 & 115697 \\
easy & 100 & 574.07 & 54.485 & 4487.73 & 13399.5 & 7131.38 & 208455 & 14544.6 & 7811.54 & 212709 \\
hard & 5 & 1895.21 & 4691.01 & 9091.66 & 1279.15 & 2830.75 & 4108.38 & 4563.49 & 9964.85 & 12813.2 \\
hard & 10 & 5586.23 & 7454.29 & 28896.2 & 7979.26 & 12363.2 & 23270.6 & 20654& 32700.7 & 51551.2 \\
hard & 20 & 4951 & 3429.5 & 34053.7 & 25272.2 & 22851 & 73126.3 & 40219.9 & 40261.3 & 106639 \\
hard & 50 & 890.347 & 446.882 & 10487.9 & 26709.5 & 16993.2 & 268792 & 27603.2 & 18228.7 & 278513 \\
hard & 100 & 290.43 & 83.8375 & 4649.9 & 18534.9 & 7564.85 & 256590 & 18824.5 & 7675.5 & 260851 \\
\end{tabular}
\end{small}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
Semifluids are materials having both fluid and solid characteristics. In this paper, we studied the problem of packing a particular type of semifluid which cannot flow in one direction, though it is fluid in the other directions; this is the case when tubes of a small diameter are packed in parallel. In this context, a packing item --- a semifluid --- is a set of identical tubes. Different items have different length and/or value, and any fraction of an item may be used, with the objective of obtaining the maximum value packed.
Given the assumption of continuity, \emph{i.e.}, that one may arbitrarily divide a given volume, the problem of packing a set of tubes of different lengths in a container is surprisingly difficult. This paper presents heuristics and complete search for the variant which closer corresponds to the industrial application: all the semifluids must be packed in the same direction, and a semifluid placed on top of another must not protrude. In this paper, we have considered divisions of the volume of an item only when it reaches the ceiling of the container.
Several methods, from simple heuristics to complete search, are proposed. The choice among them depends on the application. Simple heuristics are very quick, but often fail to find good solutions. Local ascent based on simple heuristics often finds very good solutions, and is likely to be the best method for large instances and limited CPU time. Among tree search methods, limited discrepancy search is often superior to the others, finding solutions of very good quality and frequently proving them optimal.
Semifluid packing under assumptions not considered in this paper is an interesting subject for future research; in particular, exploring different packing directions and the possibility of overflow. The complexity class of this problem is unknown; determining it is an interesting research topic. Another interesting research direction concerns developing compact mathematical models for optimization, taking full consideration of the possibility of packing fractions of each item.
The proposed heuristics could be extended and refined, in particular for taking into account the possibility of diversifying the point of division of an item into several packing places, further than the top of the container. Yet another unexplored possibility for improvement concerns using the objective value of a heuristic solution as a lower bound, at the root node.
\section{Supplementary programs and data}
\label{sec:data}
Supplementary programs and data associated with this article can be found online at \url{http://www.dcc.fc.up.pt/~jpp/code/semifluid}. That page contains an implementation of all the algorithms described in this paper and the program used for generating the instances, as well as the generated data.
In the real-world application of this problem the number of different tube lengths in catalog is small. To a smaller extent, this is also true for other numeric values in the required data. We simulate this by limiting the number of digits in the random numbers generated: 3 digits in general, 2 or 3 digits for tube lengths. All the values are normalized, so that the container dimensions are $1 \times 1 \times 1$. As we use exact arithmetic for all operations, values generated and stored in the files are fractions. The combinations of parameters used for instance generation are summarized in Table~\ref{tab:data}.
\begin{table}[h!tbp]
\centering
\caption{Characteristics of benchmark instances used: for each set of parameters 100 independent random instances have been generated, totaling 3000 instances.}
\begin{tabular}{lccccc}
Type & Number of items & Digits in $\ell_i$ & Volume of items (\% of $D \times W \times H $) & Total \\ \hline
easy & 5, 10, 20, 50, 100 & 2, 3 & 100\% & 1000 instances\\
hard & 5, 10, 20, 50, 100 & 2, 3 & 100\%, 150\% & 2000 instances\\
\end{tabular}
\label{tab:data}
\end{table}
For hard instances no optimum is known in advance. These instances have been generated by simply drawing random numbers for lengths and volumes with the required number of digits, and afterwards updating the volumes so that the total volume will be the desired factor of the container's volume (in our data set, 1 or 3/2).
Easy instances have the space of the container completely filled. This is done by successive divisions of the container, as shown in Algorithm~\ref{alg:geneasy}. To each holder generated this way there will correspond a different item. Using this procedure, the total volume of items will always equal the volume of the container.
Instances closer to the real-world application that motivated this paper are hard instances with 10 to 20 items, two digits in $\ell_i$ and items occupying 100\% to 150\% of the container's volume.
\begin{algorithm}[htbp!]
\DontPrintSemicolon
create a holder $h$ with the size of the container\;
$\mathcal{H} \leftarrow [h]$ \;
\Repeat{number of holders is equal to the number of desired items}{
randomly select a holder $h$ from $\mathcal{H}$\;
randomly select $r$ with uniform distribution in $[0,1]$\;
\If(\tcp*[f]{With 50\% probability}){$r < 1/2$}{
\If{$h$ has no other holders on top}{
divide $h$ vertically\;
replace $h$ by the two newly created holders
}
}
\Else{
divide $h$ horizontally\;
replace $h$ by the two newly created holders
}
}
\caption{Main steps for generating an easy instance.}\label{alg:geneasy}
\end{algorithm}
\end{document} |
\begin{document}
\title[Infinite dimensional oscillatory integrals]{Infinite dimensional oscillatory integrals with polynomial phase and applications to high order heat-type equations}
\author{S. Mazzucchi}
\address{ Dipartimento di Matematica, Universit\`a di Trento, 38123 Povo, Italia}
\maketitle
\begin{abstract}
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution to a general class of high order heat-type equations.\\
\noindent {\it Key words:} Infinite dimensional integration, partial differential equations, representations of solutions.
\noindent {\it AMS classification }: 35C15, 35G05, 28C20, 47D06.
\end{abstract}
\vskip 1\baselineskip
\section{Introduction}
Functional integration is a powerful tool for the study of dynamical systems \cite{Sim}. The main example is the celebrated Feynman-Kac formula \eqref{Fey-Kac}, which provides a probabilistic representation of the solution to the heat equation
\begin{equation} \label{heat}
\left\{\begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=\frac{1}{2}\Delta u(t,x) -V(x)u(t,x),\qquad t\in {\mathbb{R}}^+, x\in{\mathbb{R}}^d, \\
u (0,x)&=u _0(x),
\end{aligned}\right. \end{equation}
in terms of the expectation with respect to the distribution of the Wiener process $W$ starting at $x$ (see, e.g. \cite{KarSh}),
\begin{equation}\label{Fey-Kac}
u (t,x)={\mathbb{E}}^x[ e^{-\int_0^tV(W(s))ds}u_0(W(t))].\end{equation}
Formula \eqref{Fey-Kac} can be established under rather mild requirements on the potential $V$ and the initial datum $u_0$ (see, e.g. , \cite{Sim}) and provides an important instrument in the study of heat equation and its solutions.
More generally, an extensively developed theory relates stochastic processes with the solution to parabolic equations associated to second-order elliptic operators \cite{Dyn}.
However, that theory cannot be applied to more general PDEs such as, for instance, the Schr\"odinger equation
\begin{equation} \label{schroedinger}\left\{ \begin{aligned}
i\frac{\partial}{\partial t}u(t,x)& =-\frac{1}{2}\Delta u(t,x) +V(x)u(t,x),\qquad t\in {\mathbb{R}}^+, x\in{\mathbb{R}}^d\\
u (0,x)&=u _0(x)
\end{aligned}\right. \end{equation}
describing the time evolution of the state of a nonrelativistic quantum particle,
or also heat-type equations associated to high-order differential operators, such as for instance
\begin{equation}\label{eqLapl2}
\frac{\partial}{\partial t} u(t)=- \Delta^2u(t)-V(x)u(t,x).
\end{equation}
Indeed, a Markov process $\{X(s)\, :\, 0\leq s\leq t\}$ playing the same role for Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} as the Brownian motion for the heat equation doesn't exist. Hence there is no ``generalized Feynman Kac formula"
\begin{eqnarray}\label{Fey-KacN}
u (t,x)&=&{\mathbb{E}}^x[ e^{-\int_0^tV(X(s))ds}u_0( X(t))]\\
&=&\int _{{\mathbb{R}}^[0,t]}e^{-\int_0^tV(\omega(s))ds}u_0( \omega (t))dP(\omega),\nonumber
\end{eqnarray} representing the solution of Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} in terms of a (Lebesgue type) integral with respect to a probability measure $P$ on ${\mathbb{R}}^{[0,t]}$ associated to the process $X(s)$.\\
Contrarily to the heat equation case, for both Eq. \eqref{schroedinger} and Eq. \eqref{eqLapl2} the fundamental solution $G_t(x,y)$ is not real and positive, even in the simplest case $V\equiv 0$. In particular the Green function $G_t(x,y)$ of the Schr\"odinger equation is complex, while for the high-order heat-type equation \eqref{eqLapl2} $G_t(x,y)$ is real and attains both positive and negative values \cite{Hochberg1978}. Therefore it cannot be interpreted as the density of a transition probability measure.
As a troublesome consequence,
the complex (resp. signed) finitely-additive measure $\mu$ on $\Omega={\mathbb{R}}^{[0,t]}$ defined on the algebra of ``cylinder sets"
$I_k\subset\Omega$ (where $\Omega\equiv {\mathbb{R}}^{[0,+\infty)}$) of the form $$I_k:=\{\omega\in \Omega: \omega (t_j)\in [a_j,b_j], j=1,\dots k\},\quad 0<t_1<t_2<\dots t_k,$$ by
\begin{equation}\label{CylMeas}
\mu(I_k)=\int_{a_1}^{b_1}...\int_{a_k}^{b_k}\prod_{j=0}^{k-1}G_{t_{j+1}-t_j}(x_{j+1},x_j)dx_1...dx_{k},
\end{equation} doesn't extend to a corresponding $\sigma$-additive measure on the generated $\sigma$-algebra. As a matter of fact, if this measure existed, it would have infinite total variation.\\
This problem was addressed in 1960 by Cameron \cite{Cam} for the Schr\"odinger equation and by Krylov \cite{Kry} for Eq. \eqref{eqLapl2}. These results may be viewed as particular cases of a general theorem later established by E. Thomas \cite{Thomas}, extending Kolmogorov existence theorem to limits of projective systems of signed or complex measures, instead of probability ones.\\
In fact, these no-go results forbid a functional integral representation of the solution of Eq. \eqref{schroedinger} or Eq. \eqref{eqLapl2} in terms of a Lebesgue-type integral with respect to a $\sigma$-additive complex or signed measure with finite total variation. Consequently, the integral appearing in the generalized Feynman-Kac formula \eqref{Fey-KacN} has to be thought in a weaker sense. One possibility is the definition of the ``integral" in terms of a linear continuous functional on a suitable Banach algebra of ``integrable functions", in the spirit of Riez-Markov theorem, that provides a one-to-one correspondence between complex bounded measures (on suitable topological spaces $X$) and linear continuous functionals on $C_\infty(X)$ (the continuous functions on $X$ vanishing at $\infty$).\\
Referring to Schr\"odinger equation, this issue has been extensively studied, producing a number of different mathematical definitions of Feynman path integrals (see \cite{Ma} for an account). We mention in particular for future reference the {\em Parseval approach}, introduced by It\^o \cite{Ito1,Ito2} in the 60s and developed in the 70s by S. Albeverio and R. Hoegh-Krohn \cite{AlHK, AlHKMa}, and by D. Elworthy and A. Truman \cite{ELT}.
Dealing with the parabolic equation \eqref{eqLapl2} associated to the bilaplacian, various formulations have been proposed. One of the first was introduced by Krylov \cite{Kry} and extended by Hochberg \cite{Hochberg1978}. Defining a suitable stochastic pseudo-process whose transition probability function is not positive definite, the authors realized formula \eqref{Fey-KacN} in terms of the expectation with respect to a signed measure on ${\mathbb{R}}^{[0,t]}$ with infinite total variation. That is the reason way the integral in \eqref{Fey-KacN} is not defined in Lebesgue sense, but is meant as the limit of finite dimensional cylindrical approximations \cite{BeHocOr}.
It is worthwhile mentioning the work by D. Levin and T. Lyons relying on the ``rough paths" theory. Indeed, in \cite{LevLyo} the authors conjecture that the signed measure (with infinite total variation) associated to the Krylov-Hochberg pseudo-process could became finite if defined on a certain quotient space on the path space (two path paths are equivalent if they differ for reparametrization). \\
A different approach was proposed by Funaki \cite{Funaki1979} and continued by Burdzy \cite{Burdzy1993}. It is based
on the construction of a complex-valued stochastic process with dependent increments, obtained by composing two independent Brownian motions. In \cite{Funaki1979}, formula \eqref{Fey-KacN} with $V=0$ is realized as an integral with respect to a well defined positive probability measure on a complex space for a suitable class of analytic initial data $u_0$ at least. These results have been further developed in \cite{Funaki1979,HocOr96,OrZha} and are related to Bochner's subordination theory \cite{Boch}.
Complex-valued processes, related to PDEs of the form \eqref{eqLapl2}, were also proposed by other authors exploiting various techniques \cite{Burdzy1995,ManRy93,BurMan96,Sainty1992}. A new construction for the solution of a general class of high order heat-type equations has been recently proposed, where formula \eqref{Fey-KacN} has been realized as limit of expectations with respect to a sequence of suitable random walks in the complex plane \cite{BoMa14}.\\
We also mention a completely different approach proposed by R. L\'eandre \cite{Lea}, which shares some analogies with the mathematical construction of Feynman path integrals with the white-noise-calculus approach \cite{HKPS}.\\
It is worthwhile remarking that most of the results appearing in the literature are restricted to the cases where either $V=0$ or $V$ is linear. The construction of a generalized Feynman-Kac type formula is still lacking for the solution of high-order heat-type equations similar to \eqref{eqLapl2} with a more general $V$. \par
This work aims to construct a Feynman-Kac formula for the solution of a general class of high-order heat-type equations of the form
\begin{equation}\label{PDE-N}
\frac{\partial}{\partial t}u(t,x)=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)+V(x)u(t,x), \quad t\in [0, +\infty), \; x\in {\mathbb{R}},
\end{equation}
where $p\in{\mathbb{N}}$, $p>2$, $\alpha\in{\mathbb{C}}$ is a complex constant and $V:{\mathbb{R}}\to{\mathbb{C}}$ a continuous bounded function Fourier transform of a complex Borel measure on ${\mathbb{R}}$. \\
Adopting the {\em Fresnel integral } formulation of the mathematical definition of Feynman path integrals \cite{AlHKMa, AlHK}, we introduce infinite dimensional Fresnel integrals with polynomial phase, generalizing the existing results valid for quadratic phase functions.
If the phase function is an homogeneous polynomial of order $p$, we show in particular how this new kind of functional integral
is related to the fundamental solution of Eq. \eqref{PDE-N} with $V\equiv 0$. This relation will be eventually exploited in the proof of a functional integral representation of the solution of Eq. \eqref{PDE-N}, for a suitable class of potentials $V$ and initial data $u_0$, giving rise to a new type of generalized Feynman-Kac formula.
In section 2, a detailed study of the fundamental solution of Eq. \eqref{PDE-N} takes place in the case $V=0$. In section 3, we introduce the definition of infinite dimensional Fresnel integral with polynomial phase function showing that a particular example is related to the PDE \eqref{PDE-N} with $V\equiv 0$. In section 4, we build up a representation of the solution of \eqref{PDE-N} with $V\neq 0$ in terms of an infinite dimensional Fresnel integral.
\section{The fundamental solution of high-order heat-type equations}
Let us consider the $p$-order heat-type equation:
\begin{equation} \label{PDE-p}\left\{ \begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)\\
u(0,x) &= u_0(x),\qquad x\in{\mathbb{R}}, t\in [0,+\infty)
\end{aligned}\right. \end{equation}
where $p\in{\mathbb{N}}$, $p\geq 2$, and $\alpha\in{\mathbb{C}}$ is a complex constant. In the following we shall assume that $|e^{\alpha tx^p}|\leq 1$ for all $x\in{\mathbb{R}}$ and $ t\in [0,+\infty)$. In particular, if $p$ is even this condition is fulfilled if ${\mathbb{R}}ea(\alpha)\leq 0$, while if $p$ is odd then $\alpha$ will be taken purely imaginary.\\
In the case where $p=2$ and $\alpha \in {\mathbb{R}}$, $\alpha<0$, we obtain the heat equation, while for $p=2$ and $\alpha =i$ Eq. \eqref{PDE-p} is the Schr\"oedinger equation. Since both cases are extensively studied, in the following we shall mainly focus ourselves on the case where $p\geq 3$.
Let $G^p_t(x,y)$ be the fundamental solution of Eq.\eqref{PDE-p}. Given an initial datum $u_0$ belonging to the space $S({\mathbb{R}})$ of Schwartz test functions, the solution of the Cauchy problem \eqref{PDE-p} is given by:
\begin{equation}\label{Green}
u(t,x)=\int_{\mathbb{R}} G^p_t(x,y)u_0(y)dy.
\end{equation}
In particular the following equality holds: $$G^p_t(x-y)=g^p_t(x-y),$$ where $g^p_t\in S'({\mathbb{R}})$ is the Schwartz distribution defined by the Fourier transform
\begin{equation}\label{Green2}
g^p_t(x):=\frac{1}{2\pi}\int e^{ikx}e^{\alpha tk^p}dk,\qquad x\in{\mathbb{R}}.
\end{equation}
The following lemmas state some regularity properties of the distribution $g^p_t$ that will be used in the next section.
\begin{lemma}\label{lemma1}
The tempered distribution \eqref{Green2} is a $C^\infty$ function.
\end{lemma}
\begin{proof}
A priori $g^p_t$ is an element of $S'({\mathbb{R}})$, the Schwartz space of distribution, but we shall prove that $g^p_t$ is a $C^\infty$ function defined by an absolutely convergent Lebesgue integral. This can be easily proved in the case where $p$ is even and ${\mathbb{R}}ea(\alpha) <0$, since the function $k\mapsto e^{\alpha tk^p}$ is an element of $L^1({\mathbb{R}})$.\\
In the case where ${\mathbb{R}}ea(\alpha)= 0$, i.e. $\alpha=ic$ with $c\in{\mathbb{R}}$, the function $k\mapsto e^{\alpha tk^p}$ is not summable.
Let us denote by $\psi\in S'({\mathbb{R}})$ the tempered distribution defined by this map and by $\chi_{[-R,R]}$ the characteristic function of the interval $[-R,R]\subset {\mathbb{R}}$. By the convergence of $\chi_{[-R,R]}\psi$ to $ \psi $ in $S'({\mathbb{R}})$ as $R\to +\infty$ and the continuity of the Fourier transform as a map from $S'({\mathbb{R}})$ to $S'({\mathbb{R}})$ we have that $$g^p_t=\hat \psi=\lim _{R\to+\infty }\widehat{\chi_{[-R,R]}\psi}.$$
On the other hand, by a change in the integration path in the complex $k$-plane, in the case where $p$ is even and $c>0$ we have:
\begin{eqnarray}
g^p_t(x)&=&\lim_{R\to\infty}\frac{1}{2\pi}\int_{-R}^R e^{ikx}e^{ict k^p}dk=\lim_{R\to\infty}\frac{1}{2\pi}\int_{0}^R (e^{ikx}+e^{-ikx})e^{ict k^p}dk\nonumber\\
&=&\lim_{R\to\infty}\frac{e^{i\pi/2p}}{2\pi}\int_{0}^R (e^{ie^{i\pi/2p}kx}+e^{-ie^{i\pi/2p}kx})e^{-c tk^p}dk\nonumber\\
&=&\frac{e^{i\pi/2p}}{2\pi} \int_{\mathbb{R}} e^{ie^{i\pi/2p}kx}e^{-ct k^p}dk,\label{G1}
\end{eqnarray}
while in the case where $p$ is even and $c<0$:
\begin{equation}
g^p_t(x)=\frac{e^{-i\pi/2p}}{2\pi} \int_{\mathbb{R}} e^{ie^{-i\pi/2p}kx}e^{-c tk^p}dk.\label{G2}
\end{equation}
In the case $p$ is odd, a different integration contour in the complex $k-$plane yields the following representation:
\begin{eqnarray}
g^p_t(x)&=&\lim_{R\to\infty}\frac{1}{2\pi}\int_{-R}^R e^{ikx}e^{ict k^p}dk\nonumber\\
&=&\frac{1}{2\pi}\int_{{\mathbb{R}}+i\eta} e^{ixz}e^{ic tz^p}dz\label{G3}
\end{eqnarray}
where $\eta>0$ if $c>0$ while $\eta<0$ if $c<0$. The integrand in the second line of \eqref{G3} is absolutely convergent since $ |e^{ict ({\mathbb{R}}ea(z)+i\eta)^p}|\sim e^{-ct\eta ({\mathbb{R}}ea(z))^{p-1}}$ as $|{\mathbb{R}}ea(z)|\to\infty$. \\
Eventually representations \eqref{G1}, \eqref{G2} and \eqref{G3} show that $g^p_t$ is a $C^\infty $ function of the variable $x$.
\end{proof}
\begin{Remark}
The proof of lemma \eqref{lemma1} shows that $g^p_t:{\mathbb{R}}\to {\mathbb{C}}$ can be extended to an entire analytic function of $z\in{\mathbb{C}}$.
The analyticity of $g^p_t$ follows by the application of Fubini's and Morera's theorems.
\end{Remark}
\begin{Remark}
A formula similar to \eqref{G1} has also been proved in \cite{AlMa2005} and applied to the study of some asymptotic properties of finite dimensional Fresnel integral with polynomial phase function.
\end{Remark}
The following lemma relies on the study of the detailed asymptotic behaviour of $g^p_t(x)$ for $x\to \infty$.
\begin{lemma}\label{lemma1-asy}
The function $g^p_t$ is bounded. In particular if $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then $g^p_t\in L^1({\mathbb{R}})$.
\end{lemma}
\begin{proof}
By lemma \ref{lemma1} the function $g^p_t$ is continuous, hence the proof of its boundedness can be based only on the study of its asymptotic behavior for $x\to \infty$. This task is accomplished by means of the stationary phase method \cite{Mur,Hor}.\\
For $x\to +\infty$, a change of variables in \eqref{Green2} gives:
\begin{equation}\label{int1}
g^p_t(x)=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{x^{p/p-1}(i\xi+\alpha t\xi^p)}d\xi=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{x^{p/p-1}\phi(\xi)}d\xi,
\end{equation}
$\phi:{\mathbb{R}}\to {\mathbb{C}}$ being the complex phase function
$$\phi(\xi)=i\xi+\alpha t\xi^p, \qquad \xi\in{\mathbb{R}}.$$
If either ${\mathbb{R}}ea (\alpha) \neq 0$ or
$p$ is odd and $\alpha =ic$, with $c\in {\mathbb{R}}^+$, then the phase function $\phi$ has no stationary points on the real line, i.e. there are no real solutions of the equation $\phi'(\xi)=0$. In this cases an integration by parts argument yelds:
$$\int e^{x^{\frac{p}{p-1}}\phi(\xi)}d\xi=\int \frac{1}{x^{\frac{p}{p-1}}\phi'(\xi)} \frac{d}{d\xi} e^{x^{\frac{p}{p-1}}\phi(\xi)}d\xi=\frac{1}{x^{\frac{p}{p-1}}}\int e^{x^{\frac{p}{p-1}}\phi(\xi)}\frac{\phi''(\xi)}{(\phi'(\xi))^2}d\xi.$$
By iterating this procedure we obtain that for all $N\in {\mathbb{N}}$:
$$g^p_t(x)\stackrel{x\to +\infty}{<<}(x^{\frac{p}{p-1}})^{-N}.$$
In the case where $\alpha =ic$ with $c\in {\mathbb{R}}$, Eq. \eqref{int1} can be written as
\begin{equation}\nonumber
g^p_t(x)=\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+c t\xi^p)}d\xi, \qquad x>0.
\end{equation}
If $p$ is even, an application of the stationary phase method \cite{Mur,Hor} gives:
\begin{eqnarray}
g^p_t(x)&=&\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+ct\xi^p)}d\xi \nonumber\\
&\stackrel{x\to +\infty}{\sim}& e^{sign (c)i\frac{\pi}{4}} \frac{x^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}} e^{-ix^{p/p-1}\frac{p-1}{p}\left(\frac{1}{pct}\right)^{1/p-1}} \sqrt{ \frac{ (pct)^{ \frac{p-2}{p-1}} }{|c|tp(p-1) } } .\nonumber
\end{eqnarray}
In the case where $p$ is odd and $c<0$, the same technique yields:
\begin{eqnarray}
g^p_t(x)&=&\frac{x^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{ix^{p/p-1}(\xi+ct\xi^p)}d\xi \nonumber\\
&\stackrel{x\to +\infty}{\sim} & e^{-i\frac{\pi}{4}}(p-1)^{-1/2}(p|c|t)^{-\frac{1}{2(p-1)}} \frac{x^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}} e^{ix^{p/p-1}\frac{p-1}{p}\left(-\frac{1}{pct}\right)^{\frac{1}{p-1}}}.\nonumber
\end{eqnarray}
The case where $x\to -\infty$ can be studied in the same way. In particular, if $p$ is an even integer the behaviour of $g^p_t$ for $x\to -\infty$ coincides with the one for $x\to +\infty$.\\
For $p$ odd and
$x<0$, a change of variable argument gives:
$$g^p_t(x)=\frac{(-x)^{\frac{1}{p-1}}}{2\pi}\int_{\mathbb{R}} e^{i(-x)^{p/p-1}(-\xi+ct\xi^p)}d\xi. $$
If $c<0$ then the phase function $\phi(\xi)=-\xi+ct\xi^p$ has no real stationary points, hence
$$g^p_t(x)\stackrel{x\to -\infty}{<<}x^{-N}, \qquad \forall N\in{\mathbb{N}}.$$ In the case where $c>0$ and $x\to -\infty$ the stationary phase method yields
$$g^p_t(x)\stackrel{x\to -\infty}{\sim} e^{i\frac{\pi}{4}}(p-1)^{-1/2}(pct)^{-\frac{1}{2(p-1)}} \frac{(-x)^{\frac{2-p}{2(p-1)}}}{\sqrt{2\pi}}e^{i(-x)^{p/p-1}\frac{1-p}{p}(pct)^{-\frac{1}{p-1}}}.$$
Eventually these results give the boundedness of the function $g^p_t$. Furthermore, if $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then $g^p_t$ is even summable.
\end{proof}
\section{Infinite dimensional Fresnel integrals with polynomial phase}\label{sez2}
Classical oscillatory integrals on ${\mathbb{R}}^n$ are objects of this form
\begin{equation}\label{int-osc}\int _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx,
\end{equation}
where $\Phi$ and $ f$ are complex Borel functions. The interesting case where the {\em phase function} $\Phi$ is real valued has been extensively studied in connection with the theory of Fourier integral operator \cite{Hor}. If the function $f$ is not summable the integral \eqref{int-osc} is not defined in Lebesgue sense. In \cite{Hor}, H\"ormander proposes and exploits an alternative definition which can handle the case where $f\notin L^1({\mathbb{R}}^n)$. We present here a formulation of H\"ormander's definition of oscillatory integral, which was applied to the mathematical construction of Feynman path integrals in \cite{ELT,AlBr}.
\begin{definition}\label{def-int-osc}Let $f:{\mathbb{R}}^n\to {\mathbb{C}}$ and $\Phi:{\mathbb{R}}^n\to {\mathbb{R}}$ be Borel functions. Assuming that:
\begin{enumerate}
\item for any Schwartz test function $\phi\in S({\mathbb{R}}^n)$ such that $\phi(0)=1$ the function $g_\epsilon (x):=\phi(\epsilon x)f(x)e^{i\Phi(x)}$ is summable,
\item the limit $\lim_{\epsilon \to 0}\int g_\epsilon (x)dx $ exists and is independent of $\phi$.
\end{enumerate}
Then the {\em oscillatory integral } $ \int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx$ is defined as:
$$\int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx:=\lim_{\epsilon \to 0}\int_{{\mathbb{R}}^n} \phi(\epsilon x)f(x)e^{i\Phi(x)}dx$$
\end{definition}
In the case where $f\in L^1({\mathbb{R}}^n)$ the oscillatory integral reduces to a Lebesgue integral, i.e. $\int^o _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx =\int _{{\mathbb{R}}^n}f(x)e^{i\Phi(x)}dx$. \\
Definition \ref{def-int-osc} gives sense to classical Fresnel integrals such as
$\int_{{\mathbb{R}}^n}f(x)e^{\frac{i}{2}\|x\|^2} dx$ which are extensively applied in the theory of wave diffraction. In particular, for $f=1$ definition \eqref{int-osc} yields the equality $\int_{{\mathbb{R}}^n}e^{\frac{i}{2}\|x\|^2} dx=(2\pi i)^{n/2}$.\\
In \cite{AlHKMa} oscillatory integration is generalized to the case where ${\mathbb{R}}^n$ is replaced by a real separable Hilbert space $({\mathcal{H}}, \langle \;,\;\rangle)$ and the definition of {\em infinite dimensional Fresnel integral} is introduced. The construction relies upon a generalization of the Parseval equality
\begin{equation}\label{Parseval1}
\int_{{\mathbb{R}}^n}\frac{e^{\frac{i}{2}\|x\|^2}}{(2\pi i)^{n/2}}f(x)dx=\int_{{\mathbb{R}}^n}e^{-\frac{i}{2}\|x\|^2}\hat f(x)dx,
\end{equation}
(valid for Schwartz test functions functions $f\in S({\mathbb{R}}^n)$, where $\hat f(x)=\int_{{\mathbb{R}}^n}e^{ixy}f(y)dy$). In fact (see \cite{ELT}) equality \eqref{Parseval1} can be generalized to the case the function $f:{\mathbb{R}}^n\to {\mathbb{C}}$ is the Fourier transform of a complex bounded Borel measure $\mu_f$ on ${\mathbb{R}}^n$, giving the following Parseval equality for the oscillatory integral
\begin{equation}\label{Parseval2}
\int^o_{{\mathbb{R}}^n}\frac{e^{\frac{i}{2}\|x\|^2}}{(2\pi i)^{n/2}}f(x)dx=\int_{{\mathbb{R}}^n}e^{-\frac{i}{2}\|x\|^2}d\mu_f(x),
\end{equation}
with $f(x)=\int_{{\mathbb{R}}^n}e^{ixy}d\mu_f(y)$.
Formula \eqref{Parseval2} is crucial for the extension of oscillatory integration theory to an infinite dimensional setting.\\
Let us introduce the Banach space ${\mathcal{M}}({\mathcal{H}})$ of complex Borel measures on ${\mathcal{H}}$ with finite total variation, endowed with the total variation norm $\|\mu\|_{{\mathcal{M}}({\mathcal{H}})}$. ${\mathcal{M}}({\mathcal{H}})$ is a commutative Banach algebra under convolution, the unit being the Dirac point measure at $0$.\\ Let ${\mathcal{F}}({\mathcal{H}})$ be the space of complex functions $f:{\mathcal{H}}\to {\mathbb{C}}$ the form:
\begin{equation}\label{Fo}f(x)=\int_{\mathcal{H}} e^{i\langle x,y\rangle}d\mu(y)\equiv \hat \mu (x),\qquad x\in {\mathcal{H}}\end{equation}
for some $\mu \in {\mathcal{M}}({\mathcal{H}})$. The map ${\mathcal{F}}:{\mathcal{M}}({\mathcal{H}})\to {\mathcal{F}}({\mathcal{H}})$ sending a complex measure $\mu \in{\mathcal{M}}({\mathcal{H}})$ to its Fourier transform $\hat \mu$ defined by Eq. \eqref{Fo} is linear and one to one.
By endowing the space ${\mathcal{F}}({\mathcal{H}})$ with the norm $\|f\|_{\mathcal{F}}:=\|{\mathcal{F}}^{-1}(f)\|_{{\mathcal{M}}({\mathcal{H}})}$, ${\mathcal{F}}({\mathcal{H}})$ becomes a commutative Banach algebra of continuous functions and the map ${\mathcal{F}}:{\mathcal{M}}({\mathcal{H}})\to {\mathcal{F}}({\mathcal{H}})$ is an isometry.\\
In \cite{AlHKMa,AlHK,ELT} the Parseval equality \eqref{Parseval2} is generalized to the case where $f\in {\mathcal{F}}({\mathcal{H}})$. The {\em infinite dimensional Fresnel integral} of a function $f\in {\mathcal{F}}({\mathcal{H}})$ is denoted by $\widetilde{ \int} e^{\frac{i}{2}\|x\|^2}f(x)dx$ and defined as
\begin{equation}\label{Parseval2}
\widetilde{\int} e^{\frac{i}{2}\|x\|^2}f(x)dx:=\int_{{\mathcal{H}}}e^{-\frac{i}{2}\|x\|^2}d\mu(x),
\end{equation}
where $f(x)=\int_{\mathcal{H}} e^{i\langle x,y\rangle}d\mu(y)$ and
the right hand side of \eqref{Parseval2} is a well defined (absolutely convergent) Lebesgue integral.\\
Infinite dimensional Fresnel integrals have been successfully applied to the representation of the solution of Schr\"odinger equation \eqref{schroedinger} (see i.e. \cite{AlHKMa,Ma} and references therein). Let us denote with ${\mathcal{H}}_t$ the real Hilbert space of absolutely continuous paths $\gamma:[0,t]\to{\mathbb{R}}^d$, such that $\int_0^t\dot\gamma(s)^2ds<\infty$ and $\gamma(t)=0$. The inner product in ${\mathcal{H}}_t$ is defined as $\langle\gamma, \eta\rangle=\int_0^t\dot \gamma(s)\dot \eta(s)ds$. By assuming that the initial datum $u_0$ and the potential $V$ in Eq. \eqref{schroedinger} belong to ${\mathcal{F}}({\mathbb{R}}^d)$, it is possible to prove that the function on ${\mathcal{H}}_t$:
$$\gamma\mapsto u_0(\gamma(0)+x)e^{-i\int_0^tV(\gamma(s)+x)ds},\qquad \gamma\in H_t,\; x\in {\mathbb{R}}^d,$$
belongs to ${\mathcal{F}}({\mathcal{H}}_t)$. Further the infinite dimensional Fresnel integral
$$\widetilde{\int} e^{\frac{i}{2}\|\gamma\|^2}e^{-i\int_0^tV(\gamma(s)+x)ds}u_0(\gamma(0)+x)d\gamma$$
provides a functional integral representation of the solution to the Schr\"odinger equation \eqref{schroedinger}.\\
A partial generalization of the definition of infinite dimensional Fresnel integrals and of formula \eqref{Parseval2} was developed in \cite{AlMa2}, where the quadratic phase function $\Phi(x)=\frac{i}{2}\|x\|^2$ was replaced with a fourth order polynomial. This new functional integral allows the mathematical definition of the Feynman path integrals for the Schr\"odinger equation with a quartic-oscillator potential \cite{AlMa2,AlMa3,Ma2008}.\par
In the following we are going to generalize the definition in \eqref{Parseval2} to polynomial phase functions of any order and apply these {\em generalized Fresnel integrals} to the construction of a Feynman-Kac formula for the solution of high-order heat-type equations \eqref{PDE-N}.
Let us consider a real separable Banach space $({\mathcal{B}}, \|\,\|)$. Let ${\mathcal{M}}({\mathcal{B}})$ be the space of complex bounded variation measures on ${\mathcal{B}}$, endowed with the total variation norm. As remarked above, ${\mathcal{M}}({\mathcal{B}})$ is a Banach algebra under convolution. Let ${\mathcal{B}}^*$ be the topological dual of ${\mathcal{B}}$ and ${\mathcal{F}}({\mathcal{B}})$ the Banach algebra of complex-valued functions $f:{\mathcal{B}}^*\to {\mathbb{C}}$ of the form
\begin{equation}\label{Fo-Ba}f(x)=\int_{{\mathcal{B}}}e^{i\langle x,y\rangle}d\mu(y)\equiv \hat\mu(x), \qquad x\in {\mathcal{B}}^*,\, \mu\in {\mathcal{M}}({\mathcal{B}}), \end{equation}
where $\langle \, ,\,\rangle$ denotes the dual pairing between ${\mathcal{B}}$ and ${\mathcal{B}}^*$. The space ${\mathcal{F}}({\mathcal{B}})$ endowed with the norm $\|\hat \mu\|_{{\mathcal{F}}}:=\|\mu\|_{{\mathcal{M}}({\mathcal{B}})}$ and the pointwise multiplication is a Banach algebra of functions.\\
In the following we are going to define a class of linear continuous functionals on ${\mathcal{F}}({\mathcal{B}})$, by generalizing the construction of infinite dimensional Fresnel integrals defined by Eq. \eqref{Parseval2}.
\begin{definition}
Let $\Phi:{\mathcal{B}}\to {\mathbb{C}}$ be a continuous map such that ${\mathbb{R}}ea (\Phi_p(x))\leq 0$ for all $x\in{\mathcal{B}}$.
The infinite dimensional Fresnel integral on ${\mathcal{B}}^*$ with phase function $\Phi$ is the functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$, given by
\begin{equation}\label{Parseval-p}I_{\Phi}(f):=\int_{\mathcal{B}} e^{\Phi(x)}d\mu(x), \qquad f\in{\mathcal{F}}({\mathcal{B}}), f=\hat\mu.\end{equation}
\end{definition}
By construction, the functional $I_{\Phi}$ is linear and continuous, indeed:
$$|I_{\Phi_p}(f)|\leq \int_{\mathcal{B}} |e^{\Phi_p}|d|\mu|(x)\leq \|\mu\|=\|f\|_{{\mathcal{F}} }$$
Further $I_{\Phi}$ is normalized, i.e., $I_{\Phi_p}(1)=1$.
We summarize these properties in the following proposition.
\begin{proposition}
The space ${\mathcal{F}}({\mathcal{B}})$ of Fresnel integrable functions is a Banach function algebra in the norm $\|\,\|_{\mathcal{F}}$. The infinite dimensional Fresnel integral with phase function $\Phi$ is a continuous bounded linear functional $I_{\Phi}:{\mathcal{F}}({\mathcal{B}})\to {\mathbb{C}}$ such that $|I_{\Phi_p}(f)|\leq \|f\|_{{\mathcal{F}} }$ and $I_{\Phi_p}(1)=1$.
\end{proposition}
We can now present an interesting example of infinite dimensional Fresnel integral with polynomial phase function.\par
Fixed a $p\in {\mathbb{N}}$, with $p\geq 2$, let us consider the Banach space
${\mathcal{B}}_p$ of absolutely continuous maps $\gamma:[0,t]\to{\mathbb{R}}$, with $\gamma(t)=0$ and a weak derivative $\dot \gamma $ belonging to $L^p([0,t])$, endowed with the norm:
$$\|\gamma\|_{{\mathcal{B}}_p}=\left(\int_0^t|\dot \gamma(s)|^pds\right)^{1/p}. $$
The application $T:{\mathcal{B}}_p\to L^p([0,t])$ mapping an element $\gamma\in {\mathcal{B}}_p$ to its weak derivative $\dot \gamma \in L^p([0,t])$ is an isomorphism and its inverse $T^{-1}:L^p([0,t])\to{\mathcal{B}}_p$ is given by:
\begin{equation}\label{T-1}
T^{-1}(v)(s)=-\int_s^t v(u)du\qquad v\in L^p([0,t]) .
\end{equation}
Analogously the dual space ${\mathcal{B}}_p^*$ is isomorphic to $ L^q([0,t])=(L^{p}([0,t]))^*$, with $\frac{1}{p}+\frac{1}{q}=1$, and the pairing $\langle \eta, \gamma\rangle$ between $\eta\in{\mathcal{B}}_p^*$ and $\gamma\in {\mathcal{B}}_p$ can be written in the following form:
$$\langle \eta, \gamma\rangle=\int_0^t\dot \eta(s)\dot\gamma(s)ds\qquad \dot\eta\in L_{q}([0,t]), \gamma\in{\mathcal{B}}_p.$$
Further ${\mathcal{B}}_p^*$ is isomorphic to ${\mathcal{B}}_q$.\\
Let us consider the space ${\mathcal{F}}({\mathcal{B}}_q) $ of functions $f:{\mathcal{B}}_q\to{\mathbb{C}}$ of the form
$$f(\eta)=\int_{{\mathcal{B}}_p}e^{i\int_0^t\dot \eta(s)\dot \gamma(s)ds}d\mu_f(\gamma),\, \quad \eta \in {\mathcal{B}}_q, \mu_f\in {\mathcal{M}}({\mathcal{B}}_p).$$
Let $\Phi_p:{\mathcal{B}}_p\to{\mathbb{C}}$ be the phase function defined as
$$\Phi_p(\gamma):=(-1)^p\alpha\int_0^t\dot\gamma(s)^pds,$$
where $\alpha\in{\mathbb{C}}$ is a complex constant such that
\begin{itemize}
\item ${\mathbb{R}}ea(\alpha)\leq 0$ if $p$ is even,
\item ${\mathbb{R}}ea(\alpha)= 0$ if $p$ is odd.
\end{itemize}
The infinite dimensional Fresnel integral on $B_q$ with phase function $\Phi_p$ is the functional $I _{\Phi_p} : {\mathcal{F}}({\mathcal{B}}_q)\to{\mathbb{C}}$ given by
\begin{equation}\label{funzI-p}
I _{\Phi_p}(f)=\int_{{\mathcal{B}}_p}e^{(-1)^p\alpha \int_0^t\dot \gamma(s)^pds}d\mu_f(\gamma), \quad f\in {\mathcal{F}}({\mathcal{B}}_q),\,f=\hat\mu_f.
\end{equation}
The following lemma states an interesting connection between the functional \eqref{funzI-p} and the high-order PDE \eqref{PDE-p}.
\begin{lemma}\label{lemmacyl}
Let $f: {\mathcal{B}}_q\to{\mathbb{C}}$ be a cylinder function of the
following form:
$$f(\eta)=F(\eta(t_1), \eta(t_2), ...,\eta(t_n)),\qquad \eta \in {\mathcal{B}}_q,$$ with $0\leq t_1<t_2<...<t_n< t$ and $F:{\mathbb{R}}^n\to {\mathbb{C}}$, $F\in {\mathcal{F}}({\mathbb{R}}^n)$:
$$F(x_1,x_2, ..., x_n)=\int_{{\mathbb{R}}^n}e^{i\sum_{k=1}^ny_kx_k}d\nu_F(y_1,...,y_n), \qquad \nu_F\in{\mathcal{M}}({\mathbb{R}}^n).$$
Then $f\in{\mathcal{F}}({\mathcal{B}}_p)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ is given by
\begin{equation}\label{IntIphi-p}
I _{\Phi_p}(f)=\int^o_{{\mathbb{R}}^n}F(x_1,x_2, ...,x_n)\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})dx_1...dx_n,
\end{equation}
where $x_{n+1}\equiv 0$, $t_{n+1}\equiv t$ , $G^p_s$ is the fundamental solution \eqref{Green} of the high order heat-type equation \eqref{PDE-p} and the integral on the right hand side of \eqref{IntIphi-p} is an oscillatory integral in the sense of definition \ref{def-int-osc}.
\end{lemma}
\begin{Remark}
In the case $p$ is even and ${\mathbb{R}}ea(\alpha) <0$ the integral \eqref{IntIphi-p} is an absolutely convergent Lebesgue integral because of the boundedness of the function $F\in {\mathcal{F}}({\mathbb{R}}^n)$ and the summability of the function $g^p_t$ stated in lemma \ref{lemma1-asy}.
\end{Remark}
\begin{proof}[Proof of lemma \ref{lemmacyl}]
The proof that $f\in{\mathcal{F}}({\mathcal{B}}_p)$ follows froms the explicit form of the function $f$
$$f(\eta)=F(\eta(t_1), \eta(t_2), ...,\eta(t_n))=\int_{{\mathbb{R}}^n}e^{i\sum_{k=1}^ny_k\eta(t_k)}d\nu_F(y_1,...,y_n)),\quad \eta\in {\mathcal{B}}_q.$$
and the identity
$$e^{iy\eta(s)}=\int _{{\mathcal{B}}_p}e^{i\langle \eta,\gamma \rangle}\delta_{yv_{s}}(\gamma),$$ where $v_s\in{\mathcal{B}}_p$ is the vector of ${\mathcal{B}}_p$ defined by $$\langle \eta, v_s\rangle=\eta(s), \qquad \forall \eta \in {\mathcal{B}}_q,$$ which can be explicitly written as $$v_s(\tau)=\chi_{[0,s]}(t-s)+\chi_{(s,t]}(t-\tau)s.$$
By the definition of the functional $I_{\Phi_p}$ we have
\begin{eqnarray}
I _{\Phi_p}(f)
&=&\int_{{\mathbb{R}}^n} e^{(-1)^p\alpha \int_0^t \left(\sum_{k=1}^ny_k\dot v_{t_k}(\tau)\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha \int_0^t \left(\sum_{k=1}^ny_k\chi_{(t_k,t]}(\tau)\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha \int_0^t \left(\sum_{k=1}^n\chi_{(t_k,t_{k+1}]}(\tau)\sum_{j=1}^ky_j\right)^p d\tau}d\nu_F(y_1,...,y_n)\nonumber\\
&=&\int_{{\mathbb{R}}^n} e^{\alpha\sum_{k=1}^n ( \sum_{j=1}^k y_j)^p(t_{k+1}-t_{k})}d\nu_F(y_1,...,y_n)\label{eq16}
\end{eqnarray}
On the other hand the last line of Eq. \eqref{eq16} coincides with the oscillatory integral
\begin{equation}\int^o_{{\mathbb{R}}^n}F(x_1,x_2, ...,x_n)\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})dx_1...dx_n.
\end{equation}
Indeed, taken an arbitrary test function $\phi\in S({\mathbb{R}}^n)$ such that $\phi(0)=1$, the the function $F_\epsilon:{\mathbb{R}}^n\to{\mathbb{C}}$
$$F_\epsilon (x_1,x_2, ...,x_n)\equiv F(x_1,x_2, ...,x_n)\phi(\epsilon x_1,\epsilon x_2, ...,\epsilon x_n))\Pi_{k=1}^nG^p_{t_{k+1}-t_{k}}(x_{k+1},x_{k})$$
is summable because of the boundedness of $F\in{\mathcal{F}}({\mathbb{R}}^n)$ and the decaying properties at infinity stated in lemma \ref{lemma1-asy}. Further a change of variable argument and Fubini theorem yield:
$$\int_{{\mathbb{R}}^n}F_\epsilon(x)dx
=\int_{{\mathbb{R}}^n} \left( \int_{{\mathbb{R}}^n}e^{\alpha\sum_{k=1}^n(t_{k+1}-t_k)(\sum_{j=1}^ky_j+\epsilon \xi_j)^p}\hat \phi (\xi
)d\xi\right) d\nu_F(y),$$
where $\phi(x)=\int_{{\mathbb{R}}^n}e^{ix \xi }\hat \phi (\xi)d\xi$. By dominated convergence theorem and the condition $\phi(0)=\int_{{\mathbb{R}}^n}\hat \phi (\xi)d\xi=1$, we eventually obtain
$$\lim_{\epsilon\to 0}\int_{{\mathbb{R}}^n}F_\epsilon(x)dx =\int_{{\mathbb{R}}^n} e^{\alpha\sum_{k=1}^n(t_{k+1}-t_k)(\sum_{j=1}^ky_j)^p}d\nu_F(y),$$
\end{proof}
\begin{corollary}\label{cor1}
Let $u_0\in {\mathcal{F}}({\mathbb{R}})$. Then the cylinder function $f_0:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by
$$f_0(\eta):=u_0(x+\eta(0)), \qquad x\in {\mathbb{R}}, \eta\in {\mathcal{B}}_q,$$
belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ provides a representation for the solution of the Cauchy problem \eqref{PDE-p}, in the sense that the function
$u(t,x):=I_{\Phi_p}(f_0)$ has the form \begin{equation}\label{sol-free}u(t,x)=\int^o _{\mathbb{R}} G_t(x,y)u_0(y)dy.\end{equation}
In the case $p$ is even and ${\mathbb{R}}ea(\alpha)<0$ then the integral \eqref{sol-free} is absolutely convergent, while in the general case it is meant in the oscillatory sense of definition \ref{def-int-osc}.
\end{corollary}
\section{A generalized Feynman-Kac formula}
In the present section, we consider a Cauchy problem of the form
\begin{equation} \label{PDE-p-V}\left\{ \begin{aligned}
\frac{\partial}{\partial t}u(t,x)&=(-i)^p \alpha \frac{\partial^p}{\partial x^p}u(t,x)+V(x)u(t,x)\\
u(0,x) &= u_0(x),\qquad x\in{\mathbb{R}}, t\in [0,+\infty)
\end{aligned}\right. \end{equation}
where $p\in{\mathbb{N}}$, $p\geq 2$, and $\alpha\in{\mathbb{C}}$ is a complex constant such that $|e^{\alpha tx^p}|\leq 1$ forall $x\in{\mathbb{R}}, t\in [0,+\infty)$, while $V:{\mathbb{R}}\to{\mathbb{C}}$ is a bounded continuous function. Under these assumption the Cauchy problem \eqref{PDE-p-V} is well posed in $L^2({\mathbb{R}})$. Indeed the operator ${\mathcal{D}}_p:D({\mathcal{D}}_p)\subset L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ defined by
\begin{eqnarray*}
D({\mathcal{D}}_p)&:=& H^p=\{u\in L^2({\mathbb{R}}), k\mapsto k^p\hat u(k)\in L^2 ({\mathbb{R}}) \},\\
\widehat{{\mathcal{D}}_pu}(k)&:=&k^p\hat u(k), \, u\in D({\mathcal{D}}_p),
\end{eqnarray*}
($\hat u$ denoting the Fourier transform of $u$) is self-adjoint. For $\alpha\in{\mathbb{C}}$, with $|e^{\alpha tx^p}|\leq 1$ forall $x\in{\mathbb{R}}, t\in [0,+\infty)$, one has that the operator $A:=\alpha D_p$ generates a strongly continuous semigroup $(e^{tA})_{t\geq 0}$ on $L^2(R)$. By denoting with $B:L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ the bounded multiplication operator defined by
$$Bu(x)=V(x)u(x), \qquad u\in L^2({\mathbb{R}}),$$
one has that the operator sum $A+B:D(A)\subset L^2({\mathbb{R}})\to L^2({\mathbb{R}})$ generates a strongly continuous semigroup $(T(t))_{t\geq 0}$ on $L^2({\mathbb{R}})$. Moreover, given a $u\in L^2({\mathbb{R}})$, the vector $T(t)u$
can be computed by means of the convergent (in the $L^2({\mathbb{R}})$-norm) Dyson series (see \cite{HiPhi}, Th. 13.4.1):
\begin{equation}\label{Dyson}
T(t)u=\sum_{n=0}^\infty S_n(t)u,
\end{equation}
where $S_0(t)u=e^{tA}u$ and $S_n(t)u=\int_0^t e^{(t-s)A}VS_{n-1}(s)uds$.
By passing to a subsequence, the series above converges also a.e. in $x\in{\mathbb{R}}$ giving
\begin{multline}\label{Dyson2}
T(t)u (x)=\\
=\sum_{n=0}^\infty \;\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t }\int_{{\mathbb{R}}^{n+1}}V(x_1)\dots V(x_n) G_{t-s_n}(x,x_n)
G_{s_n-s_{n-1}}(x_n,x_{n-1})\\ \dots G_{s_1}(x_1,x_0)
u_0(x_0)dx_0\dots dx_n\,ds_1\dots ds_n ,\qquad a.e.\; x\in {\mathbb{R}}.
\end{multline}
Under suitable assumptions on the initial datum $u_0$ and the potential $V$, we are going to construct a representation of the solution of equation \eqref{PDE-p-V} in $L^2({\mathbb{R}})$ in terms of an infinite dimensional oscillatory integral with polynomial phase.
\begin{teorema}
Let $u_0\in {\mathcal{F}}({\mathbb{R}})\cap L^2({\mathbb{R}})$ and $V\in {\mathcal{F}}({\mathbb{R}})$, with $u_0(x)=\int_{\mathbb{R}} e^{ixy}d\mu_0(y)$ and $V(x)=\int_{\mathbb{R}} e^{ixy}d\nu(y)$, $\mu_0,\nu\in{\mathcal{M}}({\mathbb{R}})$. Then the functional $f_{t,x}:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by
\begin{equation}\label{f-tx}
f_{t,x}(\eta):=u_0(x+\eta(0))e^{\int_0^tV(x+\eta(s))ds}, \qquad x\in {\mathbb{R}}, \eta\in {\mathcal{B}}_q,
\end{equation}
belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral with phase function $\Phi_p$ provides a representation for the solution of the Cauchy problem \eqref{PDE-p-V}.
\end{teorema}
\begin{Remark}
By Plancherel's theorem the assumption that $u_0\in {\mathcal{F}}({\mathbb{R}})\cap L^2({\mathbb{R}})$ is equivalent to the fact that $u_0$ is the Fourier transform of a function $\hat u_0\in L^1({\mathbb{R}})\cap L^2({\mathbb{R}})$.
\end{Remark}
\begin{proof}
Let $\mu_V\in {\mathcal{M}}({\mathcal{B}}_p)$ be the measure defined by
$$\int_{{\mathcal{B}}_p}f(\gamma)d\mu_V(\gamma)=\int_0^t\int_{\mathbb{R}} e^{ixy}f(y\,v_s)d\nu(y)ds, \qquad f\in C_b({\mathcal{B}}_p),$$
where $v_s\in {\mathcal{B}}_p$ is the function $v_s(\tau)=\chi_{[0,s]}(\tau)(t-s)+\chi_{(s,t]}(t-\tau)s$. One can easily verify that $\|\mu_V\|_{{\mathcal{M}}({\mathcal{B}}_p)}\leq t\|\nu\|_{{\mathcal{M}}({\mathbb{R}})}$ and the map $\eta \in {\mathcal{B}}_q\mapsto \int_0^tV(x+\eta(s))ds$ is the Fourier transform of $\mu_V$. Analogously the map $\eta \in {\mathcal{B}}_q\mapsto \exp(\int_0^tV(x+\eta(s))ds) $ is the Fourier transform of the measure $\nu_V\in {\mathcal{M}}({\mathcal{B}}_p)$ given by $\nu_V=\sum _{n=0}^\infty\frac{1}{n!}\mu_V^{*n}$, where $\mu_V^{*n}$ denotes the $n$-fold convolution of $\mu_V$ with itself.
The series is convergent in the ${\mathcal{M}}({\mathcal{B}}_p)$-norm and one has $\|\nu_V\|_{{\mathcal{M}}({\mathcal{B}}_p)}\leq e^{t\|\nu\|_{{\mathcal{M}}({\mathbb{R}})}}$. Further, by lemma \ref{lemmacyl} the cylinder function $\eta\mapsto u_0(x+\eta(0))$, $\eta \in {\mathcal{B}}_q$, is an element of ${\mathcal{F}}({\mathcal{B}}_q)$. More precisely, it is the Fourier transform of the measure $\nu_{u_0}$ defined by
$$\int_{{\mathcal{B}}_p}f(\gamma)d\nu_{u_0}(\gamma)=\int _{\mathbb{R}} e^{ixy}f(y\,v_0)d\mu_0(y), \qquad f\in C_b({\mathcal{B}}_p).$$
We can then conclude that the map
$f_{t,x}:{\mathcal{B}}_q\to{\mathbb{C}}$ defined by \eqref{f-tx} belongs to ${\mathcal{F}}({\mathcal{B}}_q)$ and its infinite dimensional Fresnel integral $ I_{\Phi_p}(f_{t,x})$ with phase function $\Phi_p$ is given by
\begin{multline}\nonumber
\sum_{n=0}^\infty \frac{1}{n!}\int_{{\mathcal{B}}_p}e^{(-1)^p\alpha \int_0^y \dot \gamma(s)^pds}d\nu_{u_0}*\mu_V*\cdots *\mu_V=\\
\sum_{n=0}^\infty \frac{1}{n!}\int_0^t...\int_0^t I_{\Phi_p}\big(u_0(x+\eta(0))V(x+\eta(s_1))\dots V(x+\eta(s_n))\big)ds_1 \cdots ds_n
\end{multline}
By the symmetry of the integrand the latter is equal to
$$\sum_{n=0}^\infty\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t } I_{\Phi_p}\big(u_0(x+\eta(0))V(x+\eta(s_1))\dots V(x+\eta(s_n))\big)ds_1 \cdots ds_n$$
By lemma \ref{lemmacyl} we eventually obtain
\begin{multline}\nonumber \sum_{n=0}^\infty\; \idotsint\limits _{0\leq s_1 \leq \dots \leq s_n \leq t }\int_{{\mathbb{R}}^{n+1}} u_0(x+x_0)V(x+x_1)\dots V(x+x_n)G_{s_1}(x_1,x_0)\\G_{s_2-s_1}(x_2,x_1)\dots G_{t-s_n}(0,x_{n})dx_0 dx_1\cdots dx_n ds_1 \cdots ds_n,
\end{multline}
that coincides with the Dyson series \eqref{Dyson2} for the solution of the high-order PDE \eqref{PDE-p-V}, as one can easily verify by means of a change of variables argument.
\end{proof}
\section*{Acknowledgments}
Many interesting discussions with Prof. S. Albeverio, S. Bonaccorsi, G. Da Prato and L.Tubaro are gratefully acknowledged, as well as the financial support of CIRM-Fondazione Bruno Kessler to the project { \em Functional integration and applications to quantum dynamical systems}.
\end{document} |
\begin{document}
\begin{flushleft}
JOURNAL OF COMBINATORIAL THEORY, Series B \textbf{42}, 146-155 (1987)
\end{flushleft}
$$$$$$$$
\begin{center}
\textbf{\large Circuit Preserving Edge Maps II}
\end{center}
\begin{center}
\textbf{Jon Henry Sanders }
\end{center}
$$$$
\begin{center}
JHS Consulting jon\_sanders@partech.com\\
\end{center}
\noindent\par In Chapter 1 of this article we prove the following. Let $f:G\rightarrow G^\prime$ be a \textit{circuit surjection,} i.e., a mapping of the edge set of $G$ onto the edge set of $G^\prime$ which maps circuits of $G$ onto circuits of $G^\prime,$ where $G,G^\prime$ are graphs without loops or multiple edges and $G^\prime$ has no isolated vertices. We show that if $G$ is assumed finite and 3-connected, then $f$ is induced by a vertex isomorphism. If $G$ is assumed 3-connected but not necessarily finite and $G^\prime$ is assumed to not be a circuit, then $f$ is induced by a vertex isomorphism. Examples of circuit surjections $f:G\rightarrow G^\prime$ where $G^\prime$ is a circuit and $G$ is an infinite graph of arbitrarily large connectivity are given. In general if we assume $G$ two-connected and $G^\prime$ not a circuit then any circuit surjection $f:G \rightarrow G^\prime$ may be written as the composite of three maps, $f(G)=q(h(k(G))),$ where $k$ is a $1-1$ onto edge map which preserves circuits in both directions (the``2-isomorphism'' of Whitney(\textit{Amer. J. Math.} 55(1993), 245-254 ) when $G$ is finite), $h$ is an onto edge \textit{circuit injection} (a 1-1 circuit surjection). Let $f: G\rightarrow M$ be a 1-1 onto mapping of the edges of $G$ onto the cells of $M$ which takes circuits of $G$ onto circuits of $M$ where $G$ is a graph with no isolated vertices, $M$ a matroid. If there exists a circuit $C$ of $M$ which is not the image of a circuit in $G$, we call $f$ \textit{nontrivial}, otherwise \textit{trivial}. In Chapter 2 we show the following. Let $G$ be a graph of even order. Then the statement `` no nontrivial map $f: g\rightarrow M$ exists, where $M$ is a binary matroid,'' is equivalent to ``$G$ is Hamiltonian.'' If $G$ is a graph of odd order, then the statement ``no nontrivial map $f:G\rightarrow M$ exists, where $M$ is a binary matroid'' is equivalent to ``$G$ is almost Hamiltonian'', where we define a graph $G$ of order $n$ to be \textit{almost Hamiltonian} if every subset of vertices of order $n-1$ is contained in some circuit of $G$.
\begin{center}
\textbf{INTRODUCTION AND PRELIMINARY DEFINITIONS}
\end{center}
\noindent\par The results obtained in this paper grew from an attempt to generalize the main theorem of [1]. There it was shown that any \textit{circuit injection} (a 1-1 onto edge map $f$ such that if $C$ is a circuit then $f(C)$ is a circuit from a 3-connected (not necessarily finite)) graph $G$ onto a graph $G^\prime$ is induced by a vertex isomorphism, where $G^\prime$ is assumed to not have any isolated vertices. In the present article we examine the situation when the 1-1 condition is dropped (Chapter 1). An interesting result then is that the theorem remains true for finite (3-connected ) graphs $G$ but not for infinite $G$.
\par In Chapter 2 we retain the 1-1 condition but allow the image of $f$ to be first an arbitrary matroid and second a binary matroid.
\par Throughout this paper we will assume that graphs are undirected without loops or multiple edges and not necessarily finite unless otherwise stated. We will denote the set of edges of a graph $G$ by $E(G)$ and the set of vertices of $G$ by $V(G)$. We will also use the notation $G=(V,E)$ to indicate $V=V(G), E=E(G)$ when $G$ is a graph. The graph $G: A$ will be the graph with edge set $A$ and vertex set $V(G)$. The abuse of language of referring to a set of edges $S$ as a graph (usually a subgraph of a given graph) will be tolerated where it is understood that the set of vertices of such a graph is simply the set of all vertices adjacent to any edge of $S$.
\par A subgraph $P$ of a graph $G$ is a \textit{suspended chain} of $G$ if $|V|\geq 3, |V|$ finite and there exists two distinct vertices $v_1,v_2\in V$, the endpoints of $P$ such that $\deg_Pv_1=1,~~ \deg_P v_2=1$, and $\deg_Pv=\deg_Gv=2$ for $v\in V, v\neq v_1,v_2$, where $V=V(P).$ We shall also refer to the set of edges of $P$ as a suspended chain. The notation $\mathscr{C}(v)$ will be used to indicated the set of edges adjacent to the vertex $v$ in a given graph.
\par A \textit{circuit surjection f} of $G$ onto $G^\prime,$ denoted by $f:G\rightarrow G^\prime$, is an onto map of the edge set of $G$ onto the edge set of $G^\prime$ such that if $C$ is a circuit of $G$ then $f(C)$ is a circuit of $G^\prime$. We also understand the terminology $f:G\rightarrow G^\prime$ is a circuit surjrction to preclude the possibility of $G^\prime$ having isolated vertices.
\chapter{\small 1. CIRCUIT SURJECTIONS ONTO GRAPHS}
\numberwithin{theorem}{chapter}
\begin{lemma}
Let $f:G\rightarrow G^\prime$ be circuit surjection where $G$ is 2-connected and $G^\prime$ is not a circuit. Let $e$ be an edge of $G^\prime$. Then if $C$ is circuit of $G$ such that $C$ contains at least one element of $f^{-1}(e)$ then $C$ contains every element of $f^{-1}(e).$
\end{lemma}
\noindent\par\textit{Proof.}~~~~ First, we note that $G^\prime$ is 2-connected since if $e_1,e_2$ are two distinct edges of $G^\prime$ then $f(C)$ is a circuit which contains $e_1$ and $e_2$ where $C$ is any circuit of $G$ which contains $h_1,h_2$ such that $h_1\in f^{-1}(e_1),h_2\in f^{-1}(e_2).$ Let $v_1,v_2$ be the vertices adjacent to $e$. Let $P(v_1,v^\prime)$ be a path in $G^\prime$ of minimal length such that $v^\prime$ is a vertex of degree greater than 2. Define $S=\mathscr{C}(v^\prime)-\{h\}$ if $v^\prime\neq v_1,~~ S=\mathscr{C}(v^\prime)-\{e\}$ if $v^\prime=v_1,$ where $h$ is the edge in $P(v_1,v^\prime)$ adjacent to $v^\prime.$\\
FACT 1.~~ Any circuit of $G^\prime$ which contains $e$ must contain one and only one element of $S$.
\par Let $a_\alpha,\alpha\in I$ be the elements of $S$ and let $A=f^{-1}(e),A_\alpha=f^{-1}(a_\alpha),\alpha\in I.$ Then Fact 1 implies\\
FACT 2.~~ If $C\cap A\neq\O$ for $C$ a circuit of $G$ then $C\cap A_\alpha\neq\O$ is true for one and only $\alpha\in I.$
\par Let $C_0$ be a circuit which contains an edge of $A$. We will show that the assumption $C_0\not\supset A$ leads to a contradiction of Fact2. Denote by $B$ the unique set $A_{\alpha 0}, \alpha_0\in I$ such that $C_0\cap A_{\alpha 0}\neq\O.$ Let $D=A_{\alpha 1}, \alpha_1\neq \alpha_0$ (since $|I|=|S|\geq 2$, this is possible) and let $d\in D$. Since $G$ is 2-connected and $d\notin C_0$ there is a path $P_3(q_0,q_1),d\in P_3(q_0,q_1)$ where $q_0,q_1$ are distinct vertices of $C_0$ and $P_3(q_0,q_1)$ is edge disjoint from $C_0.$ Denote by $P_1(q_0,q_1)$ and $P_2(q_0,q_1)$ the two paths such that $C_0=P_1(q_0,q_1)\cup P_2(q_0,q_1).$ Now $P_i\cap A\neq\O$ and $P_i\cap B\neq\O$ is not possible, $i=1$ or $2$, since then $P_3\cup P_i$ would be a circuit which violates Fact 2. Thus $P_i\cap A\neq\O (P_i\cap B=\O),$ and $P_j\cap B\neq\O (P_j\cap A=\O)$ where either $i=1, j=2,$ or $j=1, i=2,$ say, the former (Fig. 1).
\par Suppose now there exists an edge $k\in A,k\notin C_0$. Now $k\in P_3$ is impossible since if that were the case then $P_3\cup P_2$ would be a circuit which violates Fact 2. Thus $k$ is edge disjoint from $G^{\prime\prime}$, where $G^{\prime\prime}$ is the subgraph of $G$ consisting of $P_3\cup P_1\cup P_2.$ Since $G$ is 2-connected there exists a path $P_4(t_0,t_1)$ in $G$ such that $k\in P_4(t_0,t_1),t_0,t_1$ are distinct vertices of $G^{\prime\prime}$ and $P_4(t_0,t_1)$ is edge disjoint from $G^{\prime\prime}$. We now show that no matter where $t_0,t_1$ fall on $G^{\prime\prime}$ a contradiction to Fact 2 arises. For if $G^{\prime\prime}$ has a $t_0-t_1$ path $P_5$ disjoint from $B\cup D,$ then $P_4\cup P_5$ is a circuit intersecting $A$ and hence $P_4$ intersects some $A_\alpha$. Since $P_4$ can be extended to a circuit intersecting $B$ (resp. $D$) this contradicts Fact 2. If $G^{\prime\prime}$ has no such path $P_5$, then it has a $t_0-t_1$ path intersecting both $B$ and $D$ and that path union $P_4$ contradicts Fact 2.\\
\begin{theorem}
Let $f:G\rightarrow H$ be a circuit surjection, where $G$ is 2-connected and $H$ is not a circuit. Then $f$ is the composite of three maps $f(G)=g(h(k(G))),$ where $k$ is a 1-1 onto edge map which preserves circuits in both directions (a ``2-isomorphism'' of [8] when $G$ is finite), $h$ is an onto edge map obtained by replacing suspended chains by single edges (which preserves circuits in both directions) and $q$ is a circuit injection.
\end{theorem}
$$$$$$$$$$$$$$$$$$$$
\includegraphics[scale=1.0]{Images/fig1.jpg}
\noindent\par We note that the theorem implies that $f^{-1}(e)$ is a finite set for each edge of $H$ and thus $H$ must be infinite if $G$ is infinite.
\par Theorem 1.1 follows from the fact that (by Lemma 1.1) for any $e\in H$, any two edges of $f^{-1}(e)$ form a minimal cut set (cocycle) It is apparent that $f^{-1}(e)$ can thus be transformed into a suspended chain by a sequence of 2-switchings. This establishes Theorem 1.1 for finite $G$. Theorem 1.1 also holds for infinite $G$ by the same method used in Theorem 4.1 of [3] (where Whitney's 2-isomorphism theorem [8] is extended to the infinite case).
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is finite and 3-connected. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ We will show that $G^\prime$ cannot be a circuit. For assume $G^\prime$ is a $k$-circuit, $k\geq3.$ Write $G=(V,E)$ and $|V|=n$. Now $f^{-1}(G^\prime-\{e\})$ contains no circuit and thus $|f^{-1}(G^\prime-\{e_i\})|<n,i=1,\ldots,k,$ ~~ where $e_1,\ldots,e_k$ are the edges of $G^\prime.$ But each of $G,$ i.e., each element of $E$ occurs in exactly $k-1$ of the $k$ sets $f^{-1}(G^\prime-\{e_i\},i=1),\ldots,k,$ and $E=\displaystyle \bigcup_{i=1,\ldots,k} f^{-1}(G^\prime-\{e_i\}).$ Thus $(k-1)|E|<kn,$ or $|E|<(k/(k-1))n,$ and thus $|E|<\frac{3}{2}n.$ But $|E|\geq\frac{3}{2}n$ for any (finite) graph each vertex of which is of degree three or greater and thus for any 3-connected finite graph,$\Rightarrow\Leftarrow$. Thus $G^\prime$ cannot be a circuit. Theorem 1.1 thus implies that $f$ is 1-1 so the result follows from [1].
\begin{theorem}
Let $f:G\rightarrow G^\prime$ be a circuit surjection, where $G$ is 3-connected, not necessarily finite and $G^\prime$ is not a circuit. Then $f$ is induced by a vertex isomorphism.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ Theorem 1.1 implies that $f$ must be a 1-1 map so the result follows from [1].\\
\textit{Construction}\\
\noindent\par An $n$-connected graph which has a circuit surjection onto a 3-circuit may be obtained from a sequence of disjoint 2-way infinite paths $P_1,P_2,\ldots,$ such that each vertex of $P_i$ is ``connected'' to $P_{i+1}$ by a tree as indicated in Fig. 2 for $n=4$. (The mapping which takes each edge labeled $i$ onto $e_i,i=1,2,3,$ defines the circuit surjection onto the 3-circuit with edges $e_1,e_2,$ and $e_3$)
\chapter{\small 2. CIRCUIT INJECTIONS ONTO MATROIDS}
\textit{Terminology and Notation}\\
\noindent\par A \textit{matroid} $M$ is an ordered pair of sets $\{S,\mathscr{C}\},$ where $S\neq\O,\mathscr{C}\subseteq 2^S$, which satisfies the following two axioms. Axiom I. $A,B\in\mathscr{C}, A\subseteq B$ implies $A=B$. Axiom II. $A,B\in\mathscr{C}, a\in A\cap B,~~ b\in(A\cup B)-(A\cap B)$ implies there
\includegraphics{Images/fig22.jpg}
exists $D\in\mathscr{C}$ such that $D\subseteq A\cup B,a\notin D, b\in D.$ The elements of $S$ are called the cells of $M$, the elements of $\mathscr{C}$ are called the circuits of $M.$
\par The matroid associated with a graph $G_M$, is the matroid whose cells are the edges of $G$ and whose circuits are the circuits of $G$.
\par Let $M=\{S,\mathscr{C}\}, M^\prime=\{S^\prime,\mathscr{C}^\prime\}$ be matroids, and let $f:S\rightarrow S^\prime$ be a 1-1 onto map such that $f(A)\in\mathscr{C}^\prime$ whenever $A\in\mathscr{C}$. Such an $f$ is called a circuit injection of $M$ onto $M^\prime$ denoted by $f:M\rightarrow M^\prime.$ The circuit injection injection $f$ is called nontrivial if there exists $B\in\mathscr{C}^\prime$ such that $B\neq f(A)$ for all $A\in\mathscr{C}.$
\par We can assume without loss of generality that $S=S^\prime, f$ is the identity map and $\mathscr{C}\subseteq\mathscr{C}^\prime$ for a circuit injection $f$. Then $f$ is nontrivial if $\mathscr{C}$ is properly contained in $\mathscr{C}^\prime$.
\par We denote by $A\oplus B$ the $mod$ 2 $addition$ of set $A$ and $B$ which is defined to be the set $(A\cup B)-(a\cap B).$
\par A matroid $(S,\mathscr{C})$ is a binary matroid if for all $A,B\in\mathscr{C}, A\oplus B =\displaystyle \bigcup_{i=1}^k C_i$ for $C_i\in \mathscr{C}, i=1,\ldots,k,~~ C_i\cap C_j=\O, i\neq j,1\leq i, j\leq k$. Given a set $S$ and an arbitrary set $\mathscr{C}\subseteq 2^S$ we denote by $<\mathscr{C}>$ the collection of all sets $A$ such that there exists $k\geq1, C_1,\ldots, C_k\in\mathscr{C}$ and $A=C_1\oplus\cdots\oplus C_k.$
\par We denote by $<\mathscr{C}>_{\min}$ the minimal elements of $<\mathscr{C}>,$~~i.e., the elements $A\in<\mathscr{C}>$ such that $B\in<\mathscr{C}>, B\subseteq A\Rightarrow B=A$. A useful theorem of matroid theory [5, Sects. 1 and 5.3] is that $\{S,<\mathscr{C}>_{\min}\}$ is a binary matroid for arbitrary $\mathscr{C}\subseteq 2^S.$
\par We denote the rank of a matroid by $r(M)$. If $A\in\mathscr{C}$ exists such that $|A|=r(M)+1$ we call $A$ a $Hamiltonian$ $circuit$ of $M$, and we call $M$ $Hamiltonian$.\\
\textit{Condition for Trivial/Nontrivial Circuit Injections}\\
\par We would like to establish conditions on a graph $G$ such that all circuit injections $f:G_M\rightarrow N$ are trivial, where $N$ is first assumed to be an arbitrary matroid and second assumed to be a binary matroid. (We note that if $N$ is assumed to be a graphic matroid, i.e., $N=G^\prime_M$ for some graph $G^\prime$ then the theorem of [1] implies that $G$ 3-connected is a condition when ensures no nontrivial circuit injection exists).
\par Since the addition of an isolated vertex to a graph $G$ has no effect on $G_M$ we assume (without loss of generality) that $G$ has no isolated vertices throughout this section to simplify the statements of the theorems.\\
\textit{Remark.}~~~~ The fact that if $M$ is a Hamiltonian matroid (or in particular $G_M$, where $G$ is a Hamiltonian graph) then the only circuit injections $f: M\rightarrow M^\prime$ are trivial, where $M^\prime$ is an arbitrary matroid follows from the fact that $r(M^\prime)=r(M)$ in this case. The converse is also easily established as follows.
\begin{theorem}
If $G$ is a non-Hamiltonian matroid (or in particular the matroid associated with a graph without Hamiltonian circuits) there exists a nontrivial circuit injection $f:G\rightarrow M$, where $M$ is a (not in general binary) matroid.
\end{theorem}
\noindent\par\textit{Proof.}~~~~ Let the cells of $M$ be the edges of $G$; let the circuits of $M$ be $\mathscr{C}\cup\mathscr{L},$ where $\mathscr{C}$ is the set of circuits of $G$ and $\mathscr{L}$ is the set of all bases of $G$, and let $f$ be the identity map. Then $f$ is a nontrivial circuit injection (the matroid $M$ is the so-called truncation of $G$ see [7]).
\par \textit{Remark}.~~~~ Since matroids of arbitrarily large connectivity exist without Hamiltonian circuits (the duals of complete graphs are one example \footnote{We take the definition of connectivity for matroids from [4, 6]. A property of this definition is that the connectivity of a matroid equals the connectivity of its dual and also the connectivity of the matroid $G^n_M$ associated with the complete graph on n vertices $G^n$ approaches $\infty $ as $n \rightarrow \infty $. Thus the duals of the complete graphs have arbitrarily large connectivity.}) there is no general matroid analogue to the result of [1]. We note that $M$ is never a binary matroid in the construction of Theorem 2.1.
\par A more interesting result is obtained when we restrict $M$ to be an arbitrary matroid, $G$ a graphic matroid.\\
\par DEFINITION.~~~~Let the order of a graph $G$ be $n$. We say $G$ is almost Hamiltonian if every subset of $n-1$ vertices is contained in a circuit.
\begin{theorem}
Let the order of $G$ be even. Then ``no nontrivial circuit injection $f$ exists, $f:G_M\rightarrow B,$ where $B$ is binary'' is true iff $G$ is Hamiltonian. Let the order of $G$ be odd. Then ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists where $B$ is binary'' is true iff $G$ is almost Hamiltonian.
\end{theorem}
We abbreviate ``no nontrivial circuit injection $f:G_M\rightarrow B$ exists, where $B$ is binary'' by saying ``$G$ has no nontrivial map.'' To prove the theorem we need the following
\begin{lemma}
$G$ has no nontrivial map implies ``if $v_1,\ldots,v_n$ are vertices of odd degree in $S$, for any subgraph $S$ of $G$, then there exists a circuit $C$ of $G$ such that $v_1,\ldots,v_n$ are vertices of C.''
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $\mathscr{C}$ be the set of circuits of $G$, $S$ a subset of edges of $G$. Let $\mathscr{C}^\prime=<\mathscr{C}\cup\{S\}>_{\min}$. Then $f:\{E,\mathscr{C}\}\rightarrow\{E,\mathscr{C}^\prime\},$ where $f$ is the identity map, will be a circuit injection unless $\mathscr{C}\not\subseteq\mathscr{C}^\prime$, i.e., unless there exists $A\in<\mathscr{C}\cup\{S\}>_{\min},C\in\mathscr{C}$ and $A$ is properly contained in $C$, i.e., unless
\begin{equation}
S\oplus C_1\oplus\cdots\oplus C_k\subset C \qquad\qquad \mbox{ for }\qquad C_i\in\mathscr{C}, i=1,\ldots,k.
\end{equation}
Now if $S$ has a vertex $v$ of odd degree in $S$ then $\mathscr{C}\neq<\mathscr{C}\cup\{S\}>_{\min}$ so $f$ will be a nontrivial circuit injection unless $(2.1)$ holds. But $v$ of odd degree in $S$ implies $v$ will be of odd degree in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $v$ must be contained in $C$. (If vertex $q$ is of even degree in $S$ then all edges adjacent to it could cancel in $S\oplus C_1\oplus\cdots\oplus C_k$ and thus $q\notin C$ is possible).
\begin{corollary}
$G$ has no nontrivial map implies $G$ is 2-connected.
\end{corollary}
\noindent\par\textit{Proof.}~~~~ We show given $q_1\neq q_2,$ vertices of $G$, there exist $C\in\mathscr{C}$ with $q_1,q_2$ vertices of $C$. First assume there exists the edge $e=(q_1,q_2)$ in $G$. Then taking $S=\{e\}$ in the hypothesis of Lemma 2.3 yields $C$. Otherwise choose an edge a adjacent to $q_1$ and an edge $b$ adjacent to $q_2$ (since $G$ has no isolated vertices this is possible) and put $S=\{a,b\}$ to get $C$.
\par We prove the implications of Theorem 2.2 separately in the following two lemmas.
\begin{lemma}
$|G|=2N$ and $G$ has no nontrivial map $\Rightarrow G$ is Hamiltonian; $|G|=2N+1$ and $G$ has no nontrivial map $\Rightarrow G$ is almost Hamiltonian.
\end{lemma}
\noindent\par\textit{Proof.}~~~~ Let $C$ be a circuit of $G$ and let $G$ have no nontrivial map, $|G|$ odd or even.
\par FACT 1.~~ If $C$ is even and there exist two distinct vertices $v_1,v_2$ of $G$ not on $C$ then $C$ is not of maximal order.
\par \textit{Proof of Fact 1.}~~ Let $q_1,q_2$ be two distinct vertices of $C$. Then by Menger's Theorem (since $G$ is 2-connected ) there exists a pair of vertex disjoint paths $P(v_1,q_1), P(v_2,q_2)$ or $P(v_1,q_2), P(v_2,q_1)$. In either case there exists a pair of distinct vertices $v_1^\prime,v_2^\prime$ not on $C$ such that $(v_1^\prime, q_1),(v_2^\prime,q_2)$ are edges of $G$. If $q_1,q_2$ are separated by an odd (even) number of edges in $C$ there exists a subgraph of $G$ having $|C|+2$ odd vertices as in Fig. 3(A) (3(B)) and thus $C$ is not maximal by Lemma 2.1.
\begin{center}
\includegraphics[width=1\textwidth]{Images/fig3.jpg}
\end{center}
$$$$$$$$
\begin{flushright}
\includegraphics{Images/fig4.jpg}
\end{flushright}
\par FACT 2.~~ If $|C|$ is odd and there exists a vertex $v_1\in G$ not on $C$ then $C$ is not maximal.
\par \textit{Proof of Fact 2.}~~ By the connectivity of $G$ we have $(v_1,q)$ is an edge for some vertex $q$ on $C$. We construct a subgraph having $|C|+1$ odd vertices as in Fig. 4 and apply Lemma 2.1.
\par If $|G|=2N$, Facts 1 and 2 imply that a circuit of maximal length is a Hamiltonian circuit. If $|G|=2N+1$, Facts 1 and 2 imply either $G$ is Hamiltonian (in which case it is also almost Hamiltonian) or a maximal circuit is of length $2N$. Let $C$ be a circuit of length $2N,v$ the vertex of $G$ not on $C, q$ a vertex on $C$ such that $(v,q)$ is an edge. We can find a subgraph of $G$ all vertices of which are of odd degree containing $v$ and all other vertices of $C$ other than an arbitrary vertex $v^\prime$ of $C$ as in Fig. 5. Thus $G$ is almost Hamiltonian by Lemma 2.1.
\begin{lemma}
Let $G$ be an almost Hamiltonian graph, $|G|=2N+1$. Then $G$ has no nontrivial map. Let $G$ be a Hamiltonian graph, $|G|=2N$. Then $G$ has no nontrivial map.
\end{lemma}
\noindent \par \textit{Proof.}~~~~ Case 1. $|G|=2N+1.$ Suppose otherwise, i.e., let $f:(E,\mathscr{C})\rightarrow(E,\mathscr{C}^\prime)$ be a nontrivial circuit injection, where $E$ are the edges of $G,\mathscr{C}$ are the circuits of G, and $\mathscr{C}^\prime$ properly contains $\mathscr{C}.$ Let $C$ be a circuit of $G$, \\
\includegraphics[width=1\textwidth]{Images/fig5.jpg}
$$$$
$|C|=2N, q$ a vertex of $G$ not on $C,e^\prime$ an edge of $G$ adjacent to $q$ and some vertex $v$ of $C$, and $e$ an edge of $C$ adjacent to $v$.
\par Then $P=(C-\{e\})\cup\{e^\prime\}$ is a Hamiltonian path of $G$ (i.e., a path which contains every vertex) and $P$ is a dependent set of $\{E,\mathscr{C}^\prime\}$(since otherwise $r(E,\mathscr{C})=r(E,\mathscr{C}^\prime)=2N$ and $f$ must be trivial). Let $T\in\mathscr{C}^\prime, T\notin\mathscr{C}, T\subseteq P.$ Now $T$ has at most $2N$ odd vertices, $v_1,\ldots,v_s$, since the sum of the degrees of all the vertices of $T$ is even and $T$ has at most $2N+1$ vertices. Let $C^\prime$ be a circuit of $G$ which contains $v_1,\ldots,v_s$. Let $T\subseteq T$ be the set of edges of $T$ not contained in $C^\prime.$ Then $T^\prime\subseteq P$ is the union of vertex disjoint paths $P_1,\ldots,P_k$ and the endpoints $b_i,e_i$ of $P_i$ are on $C^\prime$.Let $C_i^\prime$ be one of the two paths in $C^\prime$ with endpoints $b_i,e_i$ of $P_i$ are on $C^\prime$. Let $C_i^\prime$ be one of the two paths in $C^\prime$ with endpoints $b_i,e_i$ and define $k$ circuits of $G$ by $C_i=C_i^\prime\cup P_i, i=1,\ldots,k.$ Then $T\oplus C_i\oplus\cdots\oplus C_k\subseteq C^\prime$ contradicting the definition of $T$.\\
\par Case 2.~~ $|G|=2N$. If $G$ is Hamiltonian of arbitrary order then $G$ has no nontrivial map as noted in an earlier remark.
\par Lemmas 2.2 and 2.3 establish Theorem 2.2. The existence of almost Hamiltonian graphs of odd order which are not Hamiltonian is shown in [2]. Thus there are graphs which are not Hamiltonian for which no nontrivial map exists.
\par\textit{Remark}.~~ The duals of the matroids of complete graphs of order 5 or more provide a counter example to the assertion that an $n$ exists such that if a binary matroid $M$ has a connectivity $n$ no nontrivial map $f:M\rightarrow M^\prime$ exists, where $M^\prime$ is a binary matroid. For if $G_n$ is the complete graph of $n$ vertices let $M_n^\prime=<B_n\cup\{E_n\}>_{\min},$ where $E_n=E(G_n)$ and $B_n$ is the set of bonds of $G_n.$ Then $f:M_n\rightarrow M_n^\prime,$ where $M_n$ is the dual of $G_n$, and $f$ is the identity map, is a nontrivial map, since $a\oplus E_n\not\subset b$ for $a,b\in B_n$ when $n\geq 5$ and $a_1\oplus\cdots\oplus a_k$ where $a_i\in B_n,$ $1\leq i \leq k.$
\section*{\centering\small ACKNOWLEDGMENT}
The author would like to thank the referee for many helpful suggestions.
$$$$
\section*{\centering\small REFERENCES}
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\item 1. J.H. SANDERS AND D. SANDERS, Circuit preserving edge maps,\textit{J. Combin. Theory Ser}. B \textbf{22} (1977),91-96.
\item 2. C. THOMASSEN, Planner and infinite hypohamiltonian and hypotraceable graphs, \textit{Discrete math.} \textbf{14 } (1976),377-389.
\item 3. C. THOMASSEN, Duality of infinite graphs, \textit{J. Combin. Theory Ser.} B \textbf{33} (1982), 137-160.
\item 4. W.T. TUTTE, Menger's theorem for matroids, \textit{J. Res. Nat. Bur. Standards} B \textbf{69}(1964, 49-53).
\item 5. W.T. TUTTE, Lectures on matroids,\textit{J. Res. Nat. Bur. Standards} B \textbf{68}(1965),1-47.
\item 6. W.T. TUTTE, Connectivity in matroids, \textit{Canad. J. Math.} \textbf{18} (1966), 1301-1324
\item 7. D.J. A. WELSH, ``Matroid Theory,'' Academic Press, London/ New York, 1976.
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\end{description}
\end{document} |
\begin{document}
\title{The rationality of dynamical zeta functions and Woods Hole fixed point formula}
\begin{abstract}
For one variable rational function $\phi\in K(z)$ over a field $K$, we can define a discrete dynamical system by regarding $\phi$ as a self morphism of $\mathbb{P}^{1}_K$. Hatjispyros and Vivaldi defined a dynamical zeta function for this dynamical system using multipliers of periodic points, that is, an invariant which indicates the local behavior of dynamical systems. In this paper, we prove the rationality of dynamical zeta functions of this type for a large class of rational functions $\phi\in K(z)$. The proof here relies on Woods Hole fixed point formula and some basic facts on the trace of a linear map acting on cohomology of a coherent sheaf on $\mathbb{P}^{1}_{K}$.
\end{abstract}
\tableofcontents
\section{Introduction}
In this paper, we study the rationality of the dynamical zeta function introduced by Hatjispyros and Vivaldi in \cite{Hatjispyros-Vivaldi}. In this section, we introduce the dynamical zeta function $\dzeta{m}{\phi}$ without explaining details and state our main result.
Let $K$ be an algebraically closed field with characteristic 0. For a rational function $\phi\in K(z)$ and a non-negative integer $m\in\mathbb{Z}_{\ge 0}$, we define the \textit{dynamical zeta function} as
\[\dzeta{m}{\phi}=\exp\left(\sum_{n=1}^{\infty}\frac{t^n}{n}\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m\right)\in K\llbracket t\rrbracket.\]
$\per{n}{\phi}=\{x\in \mathbb{P}^{1}_{K} : \phi^{n}(x)=x\}=\fixpt{\phi^n}$ is the set of periodic points of period $n$ and
\begin{align*}
\mult{\phi}{x}=
\begin{cases}
\phi'(x) & \text{if } x\neq \infty \\
\psi'(0) & \text{if } x=\infty
\end{cases},
\end{align*}
where $\psi(z)=1/\phi(1/z)$, is the \textit{multiplier} of the fixed point $x\in \fixpt{\phi}$.
For $m=0$, $\dzeta{0}{\phi}$ is Artin-Mazur zeta function defined by Artin and Mazur in \cite{Artin-Mazur}.
\[\dzeta{0}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n}\# \per{n}{\phi}\right).\]
Hinkkanen showed that $\dzeta{0}{\phi}$ is a rational function for $K=\mathbb{C}$ in \cite[Theorem 1]{Hinkkanen}. More precisely, he showed that there exists $N\in\mathbb{Z}_{\ge 0}$, $p_1,\dots p_N,q_1,\dots, q_N, l_1,\dots, l_N \in \mathbb{Z}_{\ge 0}$ such that
\[\dzeta{0}{\phi}=\frac{1}{(1-t)(1-dt)}\prod_{i=1}^{N}(1-t^{p_i q_i})^{l_i}.\]
He used techniques from complex dynamics. Lee pointed out the Hinkkanen's result is still valid for any algebraically closed field $K$ with characteristic $0$ in \cite[Theorem 1.1.]{Lee}
For $m>0$, Hatjispyros and Vivaldi showed the rationality of $\dzeta{1}{\phi}$ for a polynomial of special type $\phi(z)=z^d+c$ in \cite[Lemma 3.1.]{Hatjispyros-Vivaldi} and conjectured that $\dzeta{m}{\phi}$ is a rational function for quadratic polynomials in the same paper. Predrag Cvitanovic, Kim Hansen, Juri Rolf, and G{\'{a}}bor Vattay showed the rationality of $\dzeta{m}{\phi}$ for quadratic polynomials in \cite[Section 4.1.]{Cvitanovic}. Eremenko and Levin showed that $\dzeta{1}{\phi}$ is a rational function for each polynomial in \cite[Lemma 1]{Eremenko-Levin}.
For $m>0$, the rationality of $\dzeta{m}{\phi}$ in previous results is restricted to the case that $\phi$ is a polynomial. Our main result is the rationality of the zeta function $\dzeta{m}{\phi}$ for a rational function $\phi\in K(z)$ which is not supposed to be a polynomial.
\begin{thm}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a rational function of degree $\ge 2$. Assume that $\phi$ has no periodic point with multiplier 1. Then the dynamical zeta function $\dzeta{m}{\phi}$ is a rational function over $K$ for all $m\in \mathbb{Z}_{\ge 0}$ .
\end{thm}
This paper will be organized as follows. In Section \ref{preliminaries}, we recall some basic notions of the discrete dynamics on $\mathbb{P}^{1}$ and some results on dynamical zeta functions. We devote Section \ref{woodshole} to recall the Woods Hole fixed point formula, which is the key tool of our proof for the main theorem. In Section \ref{main}, we prove the main theorem after some observation. In Section \ref{examples}, We construct some examples of rational function $\phi\in \mathbb{C}(z)$ that we can apply our main theorem and we give the explicit formula of $\dzeta{1}{\phi}$ for such $\phi$.
\section{Prelimiaries}\label{preliminaries}
In this section, we recall some basic notions on dynamical systems on $\mathbb{P}^{1}$ associated with a rational function and we see some examples and results of dynamical zeta functions. Notation in this section follows Silverman's textbook \cite{Silverman}.
\subsection{Periodic points and its multipliers}
Let $K$ be a field. For a rational function $\phi\in K(z)$, we can regard $\phi$ as an endomorphism $\phi:\mathbb{P}^{1}_K\to \mathbb{P}^{1}_K$. For any integer $n\ge 0$, $\phi^n:\mathbb{P}^{1}_K\to\mathbb{P}^{1}_K$ denotes the \textit{n-th iterate} of $\phi$,
\begin{align*}
\phi^n = \underbrace{\phi\circ\phi\circ \cdots \circ \phi }_{n}.
\end{align*}
\begin{defin}
Let $\phi \in K(z)$ be a rational function.
\begin{itemize}
\item[$(1)$]A point $x\in\mathbb{P}^{1}_K$ is said to be a \textit{fixed point} of $\phi$ if $\phi(x)=x$. We denote by $\fixpt{\phi}$ the set of fixed points of $\phi$.
\item[$(2)$]We say a point $x\in\mathbb{P}^{1}_K$ is a \textit{periodic point} of period $n$ if $\phi^{n}(x)=x$. We denote by $\per{n}{\phi}=\fixpt{\phi^n}$ the set of periodic points of period $n$.
\item[$(3)$]A periodic point $x$ is said to be \textit{of minimal period} $n$ if $x\in \per{n}{\phi}$ and $x\not\in \per{m}{\phi}$ for any $0<m<n$. We denote by $\mathrm{Per}_{d}^{**}(\phi)$ the set of periodic points of minimal period $n$.
\item[$(4)$]The \textit{forward orbit} of $x\in\mathbb{P}^{1}_K$ is defined by $\mathcal{O}_{\phi}(x)=\{\phi^{n}(x)\colon n\in\mathbb{Z}_{\ge 0}\}$.
\end{itemize}
\end{defin}
It is easy to show that we can decompose $\per{n}{\phi}$ into the disjoint union $\per{n}{\phi}=\bigcup_{d|n}\mathrm{Per}_{d}^{**}(\phi)$. Next, we define the multiplier of fixed points. Recall that the linear fractional transformation $\theta(z)=\dfrac{az+b}{cz+d}$ for $ \begin{pmatrix}a & b\\ c & d\end{pmatrix}\in\mathrm{GL}_{2}(K)$ defines an automorphism of $\mathbb{P}^{1}_K$.
\begin{defin}
For $\phi\in K(z)$ and $\theta\in \mathrm{PGL}_2(K)$ the \textit{linear conjugate} of $\phi$ by $\theta$ is the map $\theta\circ\phi\circ \theta^{-1}$.
\end{defin}
The linear conjugation of $\phi$ at $\theta$ yields the following commutative diagram.
\begin{center}
\begin{tikzcd}
\mathbb{P}^{1}_K \arrow[d,"\theta"']\arrow[r,"\phi"] & \mathbb{P}^{1}_K \arrow[d,"\theta"]& \\
\mathbb{P}^{1}_K \arrow[r,"\theta\circ\phi\circ \theta^{-1}"'] & \mathbb{P}^{1}_K.
\end{tikzcd}
\end{center}
Given $x\in \fixpt{\phi}\setminus\{\infty\}$, we define the \textit{multiplier} $\mult{\phi}{x}$ of $\phi$ at $x$ by $\mult{\phi}{x}=\phi'(x)$. The multiplier is invariant under the linear conjugation.
\begin{prop}
Let $\phi\in K(z)$ be a rational function and $\theta\in\mathrm{PGL}_2(K)$ be a fractional linear transformation. Then the followings hold.
\begin{itemize}
\item[$(1)$]$\fixpt{\theta\circ\phi\circ \theta^{-1}}=\theta(\fixpt{\phi})$.
\item[$(2)$]For any $x\in\fixpt{\phi}$ , $\mult{\theta\circ\phi\circ \theta^{-1}}{\theta(x)}=\mult{\phi}{x}$ if $\theta(x) \neq \infty$.
\end{itemize}
\end{prop}
\begin{proof}
The first statement is trivial. The second statement is an easy consequence of the chain rule.
\end{proof}
Using this property, we can extend the definition of the multiplier as below.
\begin{defin}
Let $\phi\in K(z)$ be a rational function. We define the \textit{multiplier} of $\phi$ at each $x\in\fixpt{\phi}$ as
\begin{align*}
\mult{\phi}{x}=
\begin{cases}
\phi'(x) & \text{if } x\neq \infty \\
\psi'(0) & \text{if } x=\infty
\end{cases},
\end{align*}
where $\psi(z)=1/\phi(1/z)$, which is the linear conjugation of $\phi$ at $\theta(z)=1/z$.
\end{defin}
\subsection{Dynamical zeta function}
\begin{defin}
Let $K$ be an algebraically closed field with characteristic 0. For a rational function $\phi\in K(z)$ and a non-negative integer $m\in\mathbb{Z}_{\ge 0}$, we define the \textit{dynamical zeta function} as
\[\dzeta{m}{\phi}=\exp\left(\sum_{n=1}^{\infty}\frac{t^n}{n}\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m\right)\in K\llbracket t\rrbracket.\]
\end{defin}
\begin{rem}
Since the multiplier is invariant under the linear conjugation, $\dzeta{m}{\phi}$ is also invariant under the linear conjugation.
\end{rem}
In \cite[Section 2]{Hatjispyros-Vivaldi}, Hatjispyros and Vivaldi pointed out that the dynamical zeta function $\dzeta{m}{\phi}$ has the Eulerian product as follows. We denote by $\mathrm{Per}_{n}^{**}(\phi)/\sim$ the quotient set of $\mathrm{Per}_{n}^{**}(\phi)$ by the equivalent relation $x\sim y \Leftrightarrow \mathcal{O}_{\phi}(x)=\mathcal{O}_{\phi}(y)$.
\begin{thm}
\[\dzeta{m}{\phi}=\prod_{n=1}^{\infty}\prod_{x\in \mathrm{Per}_{n}^{**}(\phi)/\sim} (1-\mult{\phi^n}{x}^{m}t^{n})^{-1}\]
where the second product runs over a complete system of representatives of $\mathrm{Per}_{n}^{**}(\phi)/\sim$.
\end{thm}
\begin{proof}
Using the chain rule, we have
\begin{itemize}
\item $\mult{\phi^{md}}{x}=\mult{\phi^{d}}{x}^m$ for any $x\in \per{d}{\phi}$,
\item $\mult{\phi^n}{x}=\mult{\phi^n}{y}$ if $x\sim y$.
\end{itemize}
Since we can decompose $\per{n}{\phi}$ into the disjoint union $\per{n}{\phi}=\bigcup_{d|n}\mathrm{Per}_{d}^{**}(\phi)$, we obtain
\begin{align*}
\sum_{x\in \per{n}{\phi}}\mult{\phi^n}{x}^m &= \sum_{d|n}\sum_{x\in \mathrm{Per}_{d}^{**}(\phi)}\mult{\phi^n}{x}^m\\
&=\sum_{d|n}\sum_{x\in \mathrm{Per}_{d}^{**}(\phi)/\sim} d(\mult{\phi^{d}}{x}^m)^{n/d}.
\end{align*}
Therefore,
\begin{align*}
\dzeta{m}{\phi}&=\exp\left(\sum_{n=1}^{\infty}\sum_{d|n}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim} \frac{(\mult{\phi^{d}}{x}^m)^{n/d}}{n/d}t^n\right)\\
&=\exp\left(\sum_{l=1}^{\infty}\sum_{d=1}^{\infty}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim} \frac{(\mult{\phi^{d}}{x}^mt^d)^{l}}{l}\right)\\
&=\exp\left(-\sum_{d=1}^{\infty}\sum_{\mathrm{Per}_{d}^{**}(\phi)/\sim}\log (1-\mult{\phi^d}{x}^mt^d)\right)\\
&=\prod_{d=1}^{\infty}\prod_{\mathrm{Per}_{d}^{**}(\phi)/\sim}(1-\mult{\phi^d}{x}^mt^d)^{-1}.\\
\end{align*}
\end{proof}
\begin{ex} Let $\phi(z)=z^d$ for an integer $d\ge 2$. Since $\phi^n(z)=z^{d^{n}}$, we have $\per{n}{\phi}=\{0\}\cup \mu_{d^{n}-1}=\{0\}\cup \{\zeta\in K \colon \zeta^{d^{n}-1}=1\}$. Therefore,
\begin{align*}
\mult{\phi^{n}}{x}=
\begin{cases}
0 & (x=0) \\
d^n & (x\in \mu_{d^{n}-1})
\end{cases}
\end{align*}
and
\[\sum_{x\in\per{n}{\phi}} \mult{\phi^{n}}{x}^m=(d^n-1)d^{nm}=d^{n(m+1)}-d^{nm}.\]
So we obtain
\[\dzeta{m}{\phi}=\frac{1-d^{m} t}{1-d^{m+1} t}.\]
\end{ex}
\begin{ex}
Let $T_{d}$ be the \textit{d-th Chebyshev polynomial} satisfying $2\cos (d\theta)=T_d(2\cos \theta)$. It is well known that $T_{d}^{n}=T_{d^n}$, $\per{n}{T_d}=\{\zeta+\zeta^{-1}\colon \zeta \in \mu_{d-1}\}\cup\{\zeta+\zeta^{-1}\colon \zeta \in \mu_{d+1}\}\cup\{\infty\}$ and
\begin{align*}
\mult{T_d}{x}=
\begin{dcases}
d & (x=\zeta+\zeta^{-1} , \zeta\in \mu_{d-1}\setminus\{\pm 1\})\\
-d & (x=\zeta+\zeta^{-1} , \zeta\in \mu_{d+1}\setminus\{\pm 1\})\\
d^2 & (x= \pm 2)\\
0 & (x=\infty)
\end{dcases}.
\end{align*}
For details, see \cite[Section~6.2]{Silverman}. Therefore, we obtain
\begin{align*}
\dzeta{m}{T_d}=
\begin{dcases}
\frac{1-d^{m}t}{1-d^{2m}t} \frac{1}{1-d^{m+1}t} & (d:\text{even},m:\text{even})\\
\frac{1-d^{m}t}{1-d^{2m}t} & (d:\text{even},m:\text{odd})\\
\left(\frac{1-d^{m}t}{1-d^{2m}t}\right)^2 \frac{1}{1-d^{m+1}t} & (d:\text{odd},m:\text{even})\\
\frac{1-d^{m}t}{(1-d^{2m}t)^2} & (d:\text{odd},m:\text{odd})
\end{dcases}.
\end{align*}
This result is firstly obtained by Hatjispyros in \cite[Theorem 1]{Hatjispyros}.
\end{ex}
\section{Woods Hole fixed point formula}\label{woodshole}
In this section, we review the Woods Hole fixed point formula which is the key tool of our proof. This formula is also called \textit{Atiyah-Bott fixed point formula} since Atiyah and Bott proved this formula in differential geometry in \cite[Theorem A]{AtBo} and \cite[Theorem A]{AB}. A purely algebraic proof can be found in SGA5 \cite[Expos{\'{e}} III, Corollaire 6.12]{SGA5} and in \cite[Theorem A.4.]{Taelman}. For details and other applications of this formula, see \cite{SGA5}, \cite{Taelman}, \cite{Beauville}, \cite{Kond}, and \cite{Ramirez}.
Let $X$ be a Noetherian scheme over an algebraically closed field $K$, $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $X$, and $\varphi: \mathcal{G}\to\mathcal{F}$ is a sheaf homomorphism. For $x\in X$, we define $\mathcal{F}(x)=\mathcal{F}_x\otimes_{\mathcal{O}_{X,x}}\mathcal{O}_{X,x}/\mathfrak{m}_x$. Then we obtain a natural $\mathcal{O}_{X,x}/\mathfrak{m}_x$-linear map $\varphi(x): \mathcal{G}(x)\to \mathcal{F}(x)$. If $f:X\to X$ is an endomorphism and $\mathcal{G}=f^{*}\mathcal{F}$, we have a map $\widetilde{\varphi}^{(p)}:H^{p}(X,\mathcal{F})\to H^{p}(X,\mathcal{F})$ satisfying
\begin{center}
\begin{tikzcd}
H^{p}(X,\mathcal{F}) \arrow[rd,"\widetilde{\varphi}^{(p)}"]\arrow[d]& \\
H^{p}(X,f^{*}\mathcal{F}) \arrow[r]& H^{p}(X,\mathcal{F})
\end{tikzcd}.
\end{center}
$H^{p}(X,\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is the pull-back on cohomology and $H^{p}(X,f^{*}\mathcal{F})\to H^{p}(X,\mathcal{F})$ is the homomorphism induced by $\varphi:f^{*}\mathcal{F}\to\mathcal{F}$. Note that the pull-back $H^{p}(X,\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is decomposed as $H^{p}(X,\mathcal{F})\to H^{p}(X,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$, where $H^{p}(X,\mathcal{F})\to H^{p}(X,f_{*}f^{*}\mathcal{F})$ is induced by the sheaf homomorphism $\mathcal{F}\to f_{*}f^{*}\mathcal{F}$. Moreover, note that if $f$ is affine, $H^{p}(X,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is an isomorphism \cite[III, Excercise 8.2.]{Hartshorne}.
\begin{thm}(Woods Hole Fixed Point Formula, \cite[Theorem A.4.]{Taelman})
Let $X$ be a smooth proper scheme over an algebraically closed field $K$ and let $f: X \to X$ be an endomorphism. Let $\mathcal{F}$ be a locally free $\mathcal{O}_{X}$ module of finite rank and let $\varphi: f^{*}\mathcal{F}\to \mathcal{F}$ be a homomorphism of $\mathcal{O}_{X}$ module. Assume that the graph $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$. Then the following identity holds.
\[\sum_{p}(-1)^{p}\tr{\widetilde{\varphi}^{(p)}}{H^{p}(X,\mathcal{F})}=\sum_{x\in \fixpt{f}}\frac{\tr{\varphi(x)}{\mathcal{F}(x)}}{\det (1-df(x)\colon \Omega_{X/K}(x))} \]
where $df:f^{*}\Omega_X\to\Omega_X$ is the differential of $f$.
\end{thm}
For a morphism $f:X\to Y$, the differential $df:f^{*}\Omega_Y\to\Omega_X$ locally comes from
\[ \Omega_B\otimes_{B}A\to \Omega_A; db\otimes a\mapsto ad(\varphi(b)),\]
where $\spec{B}\subset X$ and $\spec{A} \subset Y$ are affine open subsets and $f$ corresponds to a ring homomorphism $\varphi: B\to A$.
The statement that $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$ means that $\Gamma_{f}$ and $\Delta$ meet with intersection multiplicity $1$ at each point $x\in \Gamma_{f}\cap\Delta$.
\begin{rem}
It is known that the followings are equivalent.
\begin{itemize}
\item[$(1)$]The graph $\Gamma_{f}\subset X\times X$ and the diagonal $\Delta\subset X\times X$ intersect transversally in $X\times X$.
\item[$(2)$]$\det (1-df(x))\neq 0$ for all $x$ satisfying $f(x)=x$.
\end{itemize}
For details, see \cite[Proposition I.1]{Beauville}. Therefore, in the case that $X=\mathbb{P}^{1}_K$ and $\phi\in K(z)$, the transversality condition equals to that $\phi$ has no fixed point with multiplier $1$.
\end{rem}
\section{Main theorem}\label{main}
In this section, we prove, by applying Woods Hole fixed point formula, the rationality of dynamical zeta functions $\dzeta{m}{\phi}$ for each \textit{completely transversal} rational function $\phi \in K(z)$.
\subsection{Notation}\label{preparation}
\begin{defin}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a rational function. We say that $\phi$ is \textit{completely transversal} if $\mult{\phi^n}{x}\neq 1$ for any $n\in\mathbb{Z}_{\ge 0}$ and $x\in\per{n}{\phi}$.
\end{defin}
\begin{defin}
For $\phi\in K(z)$ with no fixed point with multiplier $1$ and $m\in\mathbb{Z}_{\ge 0}$, we define
\[T_{m}(\phi)=\sum_{x\in \fixpt{\phi}}\frac{\mult{\phi}{x}^{m}}{1-\mult{\phi}{x}}.\]
\end{defin}
\begin{rem}\label{decomposition}
Note that
\[ \sum_{x\in\fixpt{\phi}}\mult{\phi}{x}^{m}=T_{m}(\phi)-T_{m+1}(\phi)\]
for all rational functions $\phi \in K(z)$ with no fixed point with multiplier $1$. So we have
\[ \sum_{x\in\per{n}{\phi}}\mult{\phi^n}{x}^{m}=T_{m}(\phi^n)-T_{m+1}(\phi^n)\]
for all $n\in\mathbb{Z}_{>0}$ if $\phi$ is completely transversal.
\end{rem}
\begin{defin}
Let $K$ be an algebraically closed field with characteristic $0$ and let $\phi \in K(z)$ be a completely transversal rational function of degree $\ge 2$.
We define $\lzeta{m}{\phi}\in K\llbracket t\rrbracket$ by
\[\lzeta{m}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} T_{m}(\phi^n)\right).\]
\end{defin}
\begin{rem}\label{quotient}
By Remark \ref{decomposition}, we have
\[\dzeta{m}{\phi}=\frac{\lzeta{m}{\phi}}{\lzeta{m+1}{\phi}}\]
if $\phi$ is completely transversal.
\end{rem}
We state the main theorem as below. A proof will given in Section \ref{conclution} after some observation.
\begin{thm}\label{maintheorem}
Let $K$ be an algebraiclly closed field with characteristic $0$ and let $\phi \in K(z)$ be a completely transversal rational function of degree $\ge 2$. Then the following hold.
\[\lzeta{m}{\phi}=\dfrac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))}\in K(t),\]
where $D_m^p(\phi): H^{p}(\mathbb{P}^{1}_K,\Omega_{\mathbb{P}^{1}_K}^{\otimes m})\to H^{p}(\mathbb{P}^{1}_K,\Omega_{\mathbb{P}^{1}_K}^{\otimes m})$ is the map associated to the sheaf homomorphism $(d\phi)^{\otimes m}:{\phi}^{*}\Omega_{\mathbb{P}^{1}_K}^{\otimes m}\to \Omega_{\mathbb{P}^{1}_K}^{\otimes m}$.
Especially, $\dzeta{m}{\phi}\in K(t)$ for any $m\in\mathbb{Z}_{\ge0}$.
\end{thm}
\begin{rem}
For $m>0$, we have
\[\dzeta{m}{\phi}=\frac{\dyndet{m}}{\dyndet{m+1}}\]
since $\Omega_{\mathbb{P}^{1}_K} \cong \mathcal{O}_{\mathbb{P}^{1}_K}(-2)$, $\Omega_{\mathbb{P}^{1}_K}^{\otimes m} \cong \mathcal{O}_{\mathbb{P}^{1}_K}(-2m)$ and
\begin{align*}
\dim_K H^{i}(\mathbb{P}^{1}_K,\mathcal{O}_{\mathbb{P}^{1}_K}(-2m))=
\begin{cases}
2m-1 & (i=1)\\
0 & (\text{otherwise}).
\end{cases}
\end{align*}
\end{rem}
\subsection{Connecton between multipliers and sheaf cohomologies}
\begin{lem}\label{lem_localvsgrobal}
Let $\phi \in K(z)$ be a rational function with no fixed point with multiplier $1$. Then for any positive integer $m\in\mathbb{Z}_{>0}$, the following holds.
\[T_m(\phi)=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}},\]
where $D_m^i(\phi):\cohomo{i}{m}\to\cohomo{i}{m}$ is the $K$-linear map assosiated to the sheaf homomorphism $(d\phi)^{\otimes m}:\phi^{*}\Omega_{\mathbb{P}^{1}_K}^{\otimes m}\to \Omega_{\mathbb{P}^{1}_K}^{\otimes m}$.
\end{lem}
\begin{proof}
After changing a coordinate, we may assume that $\infty \not\in \fixpt{\phi}$. We apply Woods Hole fixed point formula for $X=\mathbb{P}^{1}_K$, $\mathcal{F}=\Omega_{X}^{\otimes m}$, $f=\phi$ and $\varphi=(d\phi)^{\otimes m}:\phi^{*}\Omega_{X}^{\otimes m}\to\Omega_{X}^{\otimes m}$ and obtain
\[\sum_{x\in\fixpt{\phi}}\frac{\tr{d\phi^{\otimes m}(x)}{\Omega_{\mathbb{P}^{1}_K}^{\otimes m}(x)}}{\det (1-d\phi(x)\colon \Omega_{\mathbb{P}^{1}_K}(x))}=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}}.\]
For $x\in\fixpt{\phi}$, we take a local parameter $t=z-x$ of $\mathbb{P}^{1}_K$ at $x$. Then we have \begin{itemize}
\item $\mathcal{O}_{\mathbb{P}^{1}_K,x}\cong K[z]_{(t)}$,
\item $\mathcal{O}_{\mathbb{P}^{1}_K,x}/\mathfrak{m}_{x}\cong K$ via the evalutation at $t=0$, and
\item $\Omega_{\mathbb{P}^{1}_K,x}\cong K[z]_{(t)}dt$.
\end{itemize}
Therefore, we have
\[(d\phi)_{x}(dt)=d(\phi^{*}t)=d\phi(t+x)=\phi'(t+x)dt\]
and $\Omega_{\mathbb{P}^{1}_K}(x)=K[z]_{(t)}dt\otimes_{K[z]_{(t)}}(K[z]_{(t)}/(t)K[z]_{(t)})\cong Kdt$ via $a(t)dt\otimes 1\mapsto a(0)dt$. So
\[d\phi(x)(dt)=\phi'(x)dt=\mult{\phi}{x}dt.\]
This yields that $d\phi(x)=\mult{\phi}{x} \mathrm{id}$. So we have
\begin{align*}
T_m(\phi)&=\sum_{x\in \fixpt{\phi}}\frac{\mult{\phi}{x}^{m}}{1-\mult{\phi}{x}}\\
&=\sum_{x\in\fixpt{\phi}}\frac{\tr{d\phi^{\otimes m}(x)}{\Omega_{\mathbb{P}^{1}_K}^{\otimes m}(x)}}{\det (1-d\phi(x)\colon \Omega_{\mathbb{P}^{1}_K}(x))}.
\end{align*}
Summing up, we conclude
\[T_m(\phi)=\sum_{i=0}^{1}(-1)^{i}\tr{D_m^i(\phi)}{\cohomo{i}{m}}.\]
\end{proof}
\subsection{Lemmas on sheaf cohomologies}
We prepare some lemmas for cohomology of coherent sheaves on schemes. First we recall that there are an adjunction $f^{*}\dashv f_{*}$ for a morphism $f:X\to Y$ of schemes and there are two natural transformations $\varepsilon_{f}: f^{*}f_{*} \to 1$ called the \textit{counit} and $\eta_{f}: 1\to f_{*}f^{*}$ called the \textit{unit}. Although it is abusing notation, we use the same notation $\eta_{f}$ for corresponding sheaf homomorphism $\eta_{f}:\mathcal{F}\to f_{*}f^{*}\mathcal{F}$ for a coherent sheaf $\mathcal{F}$ on $Y$.
The next lemma shows that $\eta_{f}$ has the functoriality as following.
\begin{lem}\label{lem_adjunction}
Let $X$,$Y$ and $Z$ be Noetherian schemes and $f:X \to Y$ and $g:Y \to Z$ be morphisms . Let $\mathcal{F}$ be a quasi-coherent sheaf on $Z$. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
\mathcal{F} \arrow[rd,"\eta_{g\circ f}"]\arrow[d,"\eta_{g}"']& \\
g_{*}g^{*}\mathcal{F} \arrow[r,"g_{*}\eta_{f}"']& g_{*}f_{*}f^{*}g^{*}\mathcal{F}.
\end{tikzcd}
\end{center}
\end{lem}
\begin{proof}
Since the problem is local, we may assume that $X$,$Y$ and $Z$ are affine. We assume that $X=\spec{A}$, $Y=\spec{B}$, $Z=\spec{C}$ and $f:X \to Y$ and $g:Y \to Z$ come from $\varphi: B\to A$ and $\psi: C\to B$, respectively. We use the notation ${}_{C}N$ (\textit{resp}. ${}_{C}L$) for $B$-module $N$(\textit{resp}. $A$-module $L$) if we regard $N$ (\textit{resp}. $L$) as $C$-module via $\psi: C\to B$ (\textit{resp}. $\phi\circ\psi:C\to A$).
Then there exsists a $C$ module $M$ such that $\mathcal{F}=\tilde{M}$ and we have $g_{*}g^{*}\mathcal{F}= {}_{C}(M\otimes_{C}B)^{\sim}$, $g_{*}f_{*}f^{*}g^{*}\mathcal{F}={}_{C}(M\otimes_{C}A)^{\sim}={}_{C}((M\otimes_{C}B)\otimes_B A)^{\sim}$. So the desired commutative diagram comes from the following diagrams.
\begin{center}
\begin{tikzcd}
M \arrow[d] \arrow[r] & {}_{C}(M\otimes_{C}A) \arrow[d, no head, equal] & m \arrow[d, maps to] \arrow[r, maps to] & m\otimes 1 \arrow[d, maps to] \\
{}_{C}(M\otimes_{C}B) \arrow[r] & {}_{C}((M\otimes_CB)\otimes_BA), & m\otimes 1 \arrow[r, maps to] & (m\otimes 1)\otimes 1 .
\end{tikzcd}
\end{center}
\end{proof}
The next lemma shows that the differential $df$ has the functoriality as following.
\begin{lem}\label{lem_differential}
Let $X$,$Y$ and $Z$ be Noetherian schemes over $K$ and $f:X \to Y$ and $g:Y \to Z$ be morphisms over $K$. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
f^{*}g^{*}\Omega_{Z} \arrow[rd,"d(g\circ f)"]\arrow[d,"f^{*}dg"']& \\
f^{*}\Omega_{Y} \arrow[r,"df"']& \Omega_X.
\end{tikzcd}
\end{center}
\end{lem}
\begin{proof}
Since the problem is local, we may assume that $X$,$Y$ and $Z$ are affine. We assume that $X=\spec{A}$, $Y=\spec{B}$, $Z=\spec{C}$ and $f:X \to Y$ and $g:Y \to Z$ come from $\varphi: B\to A$ and $\psi: C\to B$, respectively. Then $df:f^{*}\Omega_{Y}\to\Omega_{X}$, $dg:g^{*}\Omega_{Z}\to\Omega_{Y}$ and $d(g\circ f):f^{*}g^{*}\Omega_{Z}\to\Omega_{X}$ correspond to $\Omega_B\otimes_{B}A\to \Omega_A; db\otimes a\mapsto ad(\varphi(b))$, $\Omega_C\otimes_{C}B\to \Omega_B; dc\otimes b\mapsto bd(\psi(c))$ and $\Omega_C\otimes_{C}A\to \Omega_A; dc\otimes a\mapsto ad(\varphi\psi(c))$, respectively. So the desired commutative diagram comes from the following diagrams.
\begin{center}
\begin{tikzcd}
\Omega_{C}\otimes_CA \arrow[d] \arrow[rd] & & dc\otimes 1 \arrow[d, maps to] \arrow[rd, maps to] & \\
\Omega_{B}\otimes_BA \arrow[r] & \Omega_A, & d(\psi(c))\otimes 1 \arrow[r, maps to] & d(\varphi\psi(c))\otimes 1.
\end{tikzcd}
\end{center}
\end{proof}
\begin{lem}\label{comparison}
Let $\mathcal{A}$ be an abelian category and $f:A\to B$ be a morphism in $\mathcal{A}$. Let $0\to A\to I^{\bullet}$ and $0\to B\to J^{\bullet}$ be complexes in $\mathcal{A}$. If each $J^{n}$ is injective, and if $0\to A\to I^{\bullet}$ is exact, then there exists a chain map $I^{\bullet}\to J^{\bullet}$ making the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & A \arrow[r] \arrow[d, "f"] & I^{0} \arrow[r] \arrow[d] & I^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & B \arrow[r] & J^{0} \arrow[r] & J^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
Moreover, any two such chain maps are homotopic.
\end{lem}
\begin{proof}
See \cite[Theorem 6.16]{Rotman}.
\end{proof}
We review how the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ is constructed in general. Let $0\to f^{*}\mathcal{F}\to J^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{F}\to I^{\bullet}$ be injective resolutions of $f^{*}\mathcal{F}$ and $f_{*}f^{*}\mathcal{F}$, respectively. Applying the functor $f_{*}$, we obtain a complex $0\to f_{*}f^{*}\mathcal{F}\to f_{*}J^{\bullet}$. Note that the complex $f_{*}J^{\bullet}$ consists of injective sheaves. By Lemma \ref{comparison}, we obtain a chain map $I^{\bullet}\to f_{*}J^{\bullet}$ and this chain map gives the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$.
The next lemma shows that the map $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{F})$ has the functoriality as following.
\begin{lem}\label{functoriality_of_pull-back}
\begin{itemize}
\item[$(1)$]Let $X$ and $Y$ be Noetherian schemes and $f:X\to Y$ be a morpshim. Let $\mathcal{F}$ and $\mathcal{G}$ be sheaves of $\mathcal{O}_{Y}$ module and $\varphi : \mathcal{F}\to \mathcal{G}$ be a morphism of $\mathcal{O}_{Y}$ module. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Y,f_{*}f^{*}\mathcal{F})} \arrow[d] \arrow[r] & {H^{p}(X,f^{*}\mathcal{F})} \arrow[d] \\
{H^{p}(Y,f_{*}f^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}\mathcal{G})},
\end{tikzcd}
\end{center}
where $H^{p}(Y,f_{*}f^{*}\mathcal{F})\to H^{p}(Y,f_{*}f^{*}\mathcal{G})$ and $H^{p}(X,f^{*}\mathcal{F})\to H^{p}(X,f^{*}\mathcal{G})$ are induced by $f_{*}f^{*}\varphi : f_{*}f^{*}\mathcal{F}\to f_{*}f^{*}\mathcal{G}$ and $f^{*}\varphi: f^{*}\mathcal{F}\to f^{*}\mathcal{G}$, respectively.
\item[$(2)$]Let $X$,$Y$ and $Z$ be Noetherian schemes and let $f:X \to Y$ and $g:Y \to Z$ be morphisms. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{Z}$ module. Then the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Z,g_{*}f_{*}f^{*}g^{*}\mathcal{F})} \arrow[d] \arrow[rd] & \\
{H^{p}(Y,f_{*}f^{*}g^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}g^{*}\mathcal{F})}.
\end{tikzcd}
\end{center}
\end{itemize}
\end{lem}
\begin{proof}
For the first claim, let $0\to f^{*}\mathcal{F}\to I^{\bullet}$ and $0\to f^{*}\mathcal{G}\to J^{\bullet}$ be injective resolutions of $f^{*}\mathcal{F}$ and $f^{*}\mathcal{G}$, respectively. Then we obtain complexes $0\to f_{*}f^{*}\mathcal{F}\to f_{*}I^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to f_{*}I^{\bullet}$. By Lemma \ref{comparison}, we have a chain map $I^{\bullet}\to J^{\bullet}$ which makes the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & f^{*}\mathcal{F} \arrow[r] \arrow[d, "f^{*}\varphi"] & I^{0} \arrow[r] \arrow[d] & I^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & f^{*}\mathcal{G} \arrow[r] & J^{0} \arrow[r] & J^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
Applying the functor $f_{*}$, we obtain two complexes $0\to f_{*}f^{*}\mathcal{F}\to f_{*}I^{\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to f_{*}J^{\bullet}$ and a chain map $f_{*}I^{\bullet}\to f_{*}J^{\bullet}$. Note that $f_{*}I^{\bullet}$ and $f_{*}J^{\bullet}$ consist of injective sheaves. Let $0\to f_{*}f^{*}\mathcal{F}\to I^{'\bullet}$ and $0\to f_{*}f^{*}\mathcal{G}\to J^{'\bullet}$ be injective resolutions of $f_{*}f^{*}\mathcal{F}$ and $f_{*}f^{*}\mathcal{G}$, respectively. By Lemma \ref{comparison}, we have chain maps $I^{'\bullet}\to J^{'\bullet}$, $I^{'\bullet}\to f_{*}I^{\bullet}$, and $J^{'\bullet}\to f_{*}J^{\bullet}$. Moreover, since the chain map $I^{'\bullet}\to f_{*}J^{\bullet}$ is unique up to homotopy, the following diagram commutes up to homotopy.
\begin{center}
\begin{tikzcd}
I^{'\bullet} \arrow[d] \arrow[r] & f_{*}I^{\bullet} \arrow[d] \\
J^{'\bullet} \arrow[r] & f_{*}J^{\bullet}.
\end{tikzcd}
\end{center}
Therefore, the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Y,f_{*}f^{*}\mathcal{F})} \arrow[d] \arrow[r] & {H^{p}(X,f^{*}\mathcal{F})} \arrow[d] \\
{H^{p}(Y,f_{*}f^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}\mathcal{G})}.
\end{tikzcd}
\end{center}
Next we prove the second claim. Let $0\to f^{*}g^{*}\mathcal{F}\to K^{\bullet}$ be an injective resolution of $f^{*}g^{*}\mathcal{F}$. Applying the functor $f_{*}$, we have a complex $0\to f_{*}f^{*}g^{*}\mathcal{F}\to f_{*}K^{\bullet}$. Let $0\to f_{*}f^{*}g^{*}\mathcal{F}\to J^{\bullet}$ be an injective resolution of $f_{*}f^{*}g^{*}\mathcal{F}$. By Lemma \ref{comparison}, we have a chain map $J^{\bullet}\to f_{*}K^{\bullet}$ which makes the following diagram commute.
\begin{center}
\begin{tikzcd}
0 \arrow[r] & f_{*}f^{*}g^{*}\mathcal{F} \arrow[r] \arrow[d, equal] & J^{0} \arrow[r] \arrow[d] & J^{1} \arrow[r] \arrow[d] & {\cdots} \\
0 \arrow[r] & f_{*}f^{*}g^{*}\mathcal{F} \arrow[r] & f_{*}K^{0} \arrow[r] & f_{*}K^{1} \arrow[r] & {\cdots}.
\end{tikzcd}
\end{center}
For an injective resolution $0\to g_{*}f^{*}f^{*}g^{*}\mathcal{F}\to I^{\bullet}$ of $g_{*}f^{*}f^{*}g^{*}\mathcal{F}$, we obtain chain maps $I^{\bullet}\to g_{*}J^{\bullet}$ and $I^{\bullet}\to g_{*}f_{*}K^{\bullet}$. Since the chain map $I^{\bullet}\to g_{*}f_{*}K^{\bullet}$ is unique up to homotopy, the following diagram commutes up to homotopy.
\begin{center}
\begin{tikzcd}
I^{\bullet} \arrow[d] \arrow[rd] & \\
g_{*}J^{\bullet} \arrow[r] & g_{*}f_{*}K^{\bullet}.
\end{tikzcd}
\end{center}
Therefore, the following diagram commutes.
\begin{center}
\begin{tikzcd}
{H^{p}(Z,g_{*}f_{*}f^{*}g^{*}\mathcal{F})} \arrow[d] \arrow[rd] & \\
{H^{p}(Y,f_{*}f^{*}g^{*}\mathcal{G})} \arrow[r] & {H^{p}(X,f^{*}g^{*}\mathcal{F})}.
\end{tikzcd}
\end{center}
\end{proof}
\begin{prop}\label{lem_cohomology}
Let $X$ be a Noetherian scheme over $K$ and let $f:X\to X$ and $g:X\to X$ be endomorphisms over $K$. Then for all $p,m \in \mathbb{Z}_{\ge 0}$ the following diagram commutes.
\begin{center}
\begin{tikzcd}
H^{p}(X,\Omega_{X}^{\otimes m}) \arrow[d, "D_m^p(g)"'] \arrow[rd, "D_m^p(g\circ f)"]& \\
H^{p}(X,\Omega_{X}^{\otimes m}) \arrow[r, "D_m^p(f)"'] & H^{p}(X,\Omega_{X}^{\otimes m}),
\end{tikzcd}
\end{center}
where $D_m^p(f): H^{p}(X,\Omega_{X}^{\otimes m})\to H^{p}(X,\Omega_{X}^{\otimes m})$ is the map associated to the sheaf homomorphism $(df)^{\otimes m}:f^{*}\Omega_{X}^{\otimes m}\to \Omega_{X}^{\otimes m}$.
\end{prop}
\begin{proof}
By combining Lemma \ref{lem_adjunction}, Lemma \ref{lem_differential}, and the naturality of unit $\eta_{f}: 1\to f_{*}f^{*}$, we have following three commutative diagrams.
\begin{center}
\begin{tikzcd}
\Omega_{X}^{\otimes m} \arrow[d, "\eta_{g}"'] \arrow[rd, "\eta_{g\circ f}"] & & & \\
g_{*}g^{*}\Omega_{X}^{\otimes m} \arrow[r, "g_{*}\eta_{f}"'] & g_{*}f_{*}f^{*}g^{*}\Omega_{X}^{\otimes m}, & & \\
g^{*}\Omega_{X}^{\otimes m} \arrow[d, "dg^{\otimes m}"'] \arrow[r, "\eta_{f}"] & f_{*}f^{*}g^{*}\Omega_{X}^{\otimes m} \arrow[d, "f_{*}f^{*}dg^{\otimes m}"] & f^{*}g^{*}\Omega_{X}^{\otimes m} \arrow[d, "f^{*}dg^{\otimes m}"'] \arrow[rd, "d{(g \circ f)^{\otimes m}}"] & \\
\Omega_{X}^{\otimes m} \arrow[r, "\eta_{f}"'] & f_{*}f^{*}\Omega_{X}^{\otimes m}, & f^{*}\Omega_{X}^{\otimes m} \arrow[r, "df^{\otimes m}"'] & \Omega_{X}^{\otimes m}.
\end{tikzcd}
\end{center}
Taking cohomology $H^{p}(X,-)$ and using Lemma \ref{functoriality_of_pull-back}, we obtain
\begin{center}
\begin{tikzcd}
{H^{p}(X,\Omega_{X}^{\otimes m})} \arrow[d] \arrow[rd] \arrow[dd, "D_{m}^{p}(g)"', bend right=67] \arrow[rrdd, "D_{m}^{p}(g\circ f)", bend left] & & \\
{H^{p}(X,g^{*}\Omega_{X}^{\otimes m})} \arrow[d] \arrow[r] & {H^{p}(X,g^{*}f^{*}\Omega_{X}^{\otimes m})} \arrow[d] \arrow[rd] & \\
{H^{p}(X,\Omega_{X}^{\otimes m})} \arrow[r] \arrow[rr, "D_{m}^{p}(f)"', bend right] & {H^{p}(X,f^{*}\Omega_{X}^{\otimes m})} \arrow[r] & {H^{p}(X,\Omega_{X}^{\otimes m})}.
\end{tikzcd}
\end{center}.
\end{proof}
\subsection{Conclusion of the proof}\label{conclution}
Now we are ready to prove Theorem \ref{maintheorem}.
\begin{proof}
Using Lemma \ref{lem_localvsgrobal}, we obtain
\begin{align*}
T_m(\phi^n)&=\sum_{i=0}^{1}(-1)^i\tr{D_m^i(\phi^n)}{\cohomo{i}{m}}.
\end{align*}
Therefore,
\begin{align*}
\lzeta{m}{\phi}=\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi^n)\right).
\end{align*}
Using Proposition \ref{lem_cohomology}, we have $D_m^i(\phi^n)=D_m^i(\phi)^n$. Note that $D_m^i(\phi)$ is a $K$-linear operator acting on the finite dimensional $K$-vector space $\cohomo{i}{m}$. Therefore, we obtain
\begin{align*}
&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi^n)\right)\\
=&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \sum_{i=0}^{1}(-1)^{i}\mathrm{tr}D_m^i(\phi)^n\right)\\
=&\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \mathrm{tr}D_m^0(\phi)^n\right)\exp\left( \sum_{n=1}^{\infty}\frac{t^n}{n} \mathrm{tr}D_m^1(\phi)^n\right)^{-1}\\
=&\frac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))}\in K(t).
\end{align*}
Summing up, we conclude
\[\lzeta{m}{\phi}=\dfrac{\det (1-tD_m^1(\phi))}{\det (1-tD_{m}^0(\phi))} .\]
\end{proof}
\section{Examples}\label{examples}
In this section, we construct some examples of completely transversal rational functions and calculate an explicit form of $\dzeta{m}{\phi}$ using Theorem \ref{maintheorem}. We consider the case $K=\mathbb{C}$ since we use the results on complex dynamics.
\begin{defin}
Let $x$ be a periodic point with minimal period $n$ for a rational function $\phi\in \mathbb{C}(z)$, and $\lambda=\mult{\phi^n}{x}$ be the multiplier of $\phi$ at $x$. Then $x$ is:
\begin{itemize}
\item[$(1)$]\textit{attracting} if $|\lambda|<1$;
\item[$(2)$]\textit{repelling} if $|\lambda|>1$;
\item[$(3)$]\textit{rationally indifferent} if $\lambda$ is a root of unity;
\item[$(4)$]\textit{irrationally indifferent} if $|\lambda|=1$, but $\lambda$ is not a root of unity.
\end{itemize}
\end{defin}
\begin{defin}
Let $\phi\in\mathbb{C}(z)$ be a rational function and $x\in\per{n}{\phi}$. For a point $c\in\mathbb{P}^{1}_{\mathbb{C}}$, we say that $c$ \textit{is attracted to the orbit of} $x$ if there exists $i\in\{0,1,\dots,n-1\}$ such that $\displaystyle \lim_{m\to\infty}\phi^{mn}(c)=\phi^{i}(x)$ with respect to the classical topology of $\mathbb{P}^{1}_{\mathbb{C}}$.
\end{defin}
\begin{thm}\label{att_rat_ind}
Let $\phi\in\mathbb{C}(z)$ be a rational function. If $x$ is a periodic point of $\phi$ and $x$ is attracting or rationally indifferent, then there exists a critical point $c$ of $\phi$ which is attracted to the orbit of $x$.
\end{thm}
\begin{proof}
See \cite[Theorem 9.3.1. and Theorem 9.3.2.]{Beardon}.
\end{proof}
\begin{rem}\label{crit}
If $\phi(z)=F(z)/G(z)$ in lowest terms, the \textit{degree} of $\phi$ is $\deg \phi=\mathrm{max}\{\deg F,\deg G\}$. By Riemann-Hurwitz formula, $\phi$ has at most $2d-2$ critical points. See \cite[Section 1.2]{Silverman}.
\end{rem}
Using Theorem \ref{att_rat_ind}, we can find a sufficient condition for the complete transversality.
\begin{cor}\label{sufficient}
Let $\phi\in\mathbb{C}(z)$ be a rational function of degree $d$. If $\phi$ has $2d-2$ attracting periodic points whose orbits are pairwise distinct, then $\phi$ is completely transversal.
\end{cor}
\begin{proof}
By the definition of the complete transversality, $\phi$ is completely transversal if $\phi$ has no rationally indifferent periodic points. By Theorem \ref{att_rat_ind} and \ref{crit}, $\phi$ has no rationally indifferent periodic points if $\phi$ has $2d-2$ attracting periodic points whose orbits are pairwise distinct.
\end{proof}
We can construct completely transversal rational functions using Corollary \ref{sufficient}.
\begin{ex}
Let $\lambda_0,\lambda_{\infty}\in \mathbb{C}$ be complex numbers such that $|\lambda_0|<1$ and $|\lambda_{\infty}|<1$. We define $\phi\in\mathbb{C}(z)$ by
\[\phi(z)=\frac{z^2+\lambda_0 z}{\lambda_{\infty} z+1}\in\mathbb{C}(z).\]
Then we have
\[\fixpt{\phi}=\left\{0,\infty,\alpha=\frac{1-\lambda_{0}}{1-\lambda_{\infty}}\right\}.\]
The multipliers of $\phi$ at $0$ and $\infty$ are $\mult{\phi}{0}=\lambda_0$ and $\mult{\phi}{\infty}=\lambda_{\infty}$, respectively. Since $|\lambda_0|<1$ and $|\lambda_{\infty}|<1$, $\phi$ has two attracting fixed points. On the other hand, $\phi$ has at most $2=2d-2$ critical points since $d=\deg \phi =2$. Therefore, $\phi$ is completely transversal.
\end{ex}
\begin{rem}
All nonconstant polynomials have a fixed point at $\infty$ and the multiplier at $\infty$ is $0$. Therefore, if $\psi\in\mathbb{C}(z)$ is conjugate to a polynomial, then $\psi$ has a fixed point with multiplier $0$. Since $\mult{\phi}{0}=\lambda_0$, $\mult{\phi}{\infty}=\lambda_{\infty}$, and $\mult{\phi}{\alpha}=\dfrac{2-\lambda_{0}-\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}$, $\phi$ is not conjugate to any polynomial if $\lambda_{0}\lambda_{\infty}\neq 0$.
\end{rem}
Next, we calculate the dynamical zeta function $\dzeta{1}{\phi}$ using the formula
\[\dzeta{1}{\phi}=\frac{\dyndet{1}}{\dyndet{2}}.\]
\begin{ex}
We use {\v{C}}ech cohomology to compute the linear maps $D_{1}^{1}(\phi)$ and $D_{2}^{1}(\phi)$ explicitly.
We define $F_{0}(z),F_{\infty}(z)\in \mathbb{C}[z]$, and $G_0(w)\in\mathbb{C}[w]$ by $F_{0}(z)=z+\lambda_{0}$, $F_{\infty}(z)=\lambda_{\infty} z +1$, and $G_0(w)=\lambda_{0} w +1$, respectively. Note that $\phi(z)=zF_{0}/F_{\infty}$ and $G_0(1/z)=F_{0}(z)/z$. We take open coverings $\mathcal{U}=\{U_0, U_1\}$ and $\mathcal{V}=\{V_0, V_1\}$ of $\mathbb{P}^{1}$, where $U_0=\mathbb{P}^{1}\setminus\{\infty\}=\spe{\mathbb{C}[z]}$, $U_1=\mathbb{P}^{1}\setminus\{0\}=\spe{\mathbb{C}[w]}$, $V_0=\phi^{-1}(U_0)=\mathbb{P}^{1}\setminus\{\infty, -\lambda_{\infty}^{-1}\}=\spe{\mathbb{C}[z,F_{\infty}^{-1}]}$, and $V_1=\phi^{-1}(U_1)=\mathbb{P}^{1}\setminus\{0, -\lambda_{0}\}=\spe{\mathbb{C}[w,G_0^{-1}]}$. Then the differential induces the map $\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})\to \check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and this map is identified with $D_{m}^{1}(\phi)$ via the isomrphisms $H^{1}(\mathbb{P}^{1},\Omega_{\mathbb{P}^{1}}^{\otimes m})\cong \check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and $H^{1}(\mathbb{P}^{1},\Omega_{\mathbb{P}^{1}}^{\otimes m})\cong \check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$.
$\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ is the first cohomology of the complex
\[\mathbb{C}[z](dz)^{\otimes m}\times \mathbb{C}[w](dw)^{\otimes m}\to \mathbb{C}[z^{\pm 1}]\dform{m}.\]
$\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ is the first cohomology of the complex
\[\mathbb{C}[z,F_{\infty}^{-1}](dz)^{\otimes m}\times \mathbb{C}[w,G_{0}^{-1}](dw)^{\otimes m}\to \mathbb{C}[z^{\pm 1},F_{0}^
{-1},F_{\infty}^{-1}]\dform{m}.\]
Note that $\dfrac{z^m a(z)}{F_{\infty}^i}\left(\dfrac{dz}{z}\right)^{\otimes m}=0$ and $\dfrac{b(1/z)}{z^{m-j}F_{0}^j}\left(\dfrac{dz}{z}\right)^{\otimes m}=0$ in $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ for all $a(z)\in\mathbb{C}[z]$ and $b(w)\in\mathbb{C}[w]$ since they are the image of $\dfrac{a(z)}{F_{\infty}^i}(dz)^{\otimes m}$ and $\dfrac{b(w)}{G_0^j}(dw)^{\otimes m}$, respectively.
Both $\check{H}^{1}(\mathcal{U},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ and $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}}^{\otimes m})$ have a $\mathbb{C}$-basis
\[\left\{ z^i \dform{m} \colon |i|<m\right \}.\]
The image of $z^i\left(\dfrac{dz}{z}\right)^{\otimes m}$ by $(d\phi)^{\otimes m}$ is
\begin{align*}
(d\phi)^{\otimes m}\left(z^{i}\dform{m}\right)&=\phi^{i}\frac{d\phi}{\phi}\\
&=z^i\frac{F_{0}^{i}}{F_{\infty}^{i}}\left(1+\frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}\right)^{m}\dform{m}\\
&=\sum_{j=0}^{m}\binom{m}{j}(1-\lambda_{0}\lambda_{\infty})^j\frac{z^{i+j}}{F_{0}^{j-i} F_{\infty}^{i+j}}\dform{m}.
\end{align*}
We can compute $\dfrac{z^{i+j}}{F_{0}^{j-i} F_{\infty}^{i+j}}\left(\dfrac{dz}{z}\right)^{\otimes m}$ by
\[ \frac{\lambda_{0}}{F_{0}}=-\sum_{n=1}^{m-1}\left(\frac{-\lambda_{0}}{z}\right)^n-\frac{(-\lambda_{0})^m}{z^{m-1}F_{0}},\]
\[ \frac{1}{F_{\infty}}=\sum_{n=0}^{m-1}(-\lambda_{\infty} z)^n+\frac{(-\lambda_{\infty} z)^{m}}{F_{\infty}}, \,\text{and}\]
\[ \frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}=\frac{1}{F_{\infty}}-\frac{\lambda_{0}}{F_{0}}.\]
For example, if $m=1$ then $\dfrac{1}{F_{\infty}}\dfrac{dz}{z}=\dfrac{dz}{z}$ and $\dfrac{\lambda_{0}}{F_{0}}\dfrac{dz}{z}=0$ in $\check{H}^{1}(\mathcal{V},\Omega_{\mathbb{P}^{1}})$. Therefore,
\begin{align*}
d\phi\left(\frac{dz}{z}\right)&=\sum_{j=0}^{1}\binom{1}{j}(1-\lambda_{0}\lambda_{\infty})^j\frac{z^{j}}{F_{0}^{j} F_{\infty}^{j}}\frac{dz}{z}\\
&=\frac{dz}{z}+\frac{(1-\lambda_{0}\lambda_{\infty})z}{F_{0}F_{\infty}}\frac{dz}{z}\\
&=\frac{dz}{z}+\left(\frac{1}{F_{\infty}}-\frac{\lambda_{0}}{F_{0}}\right)\frac{dz}{z}\\
&=2\frac{dz}{z}.
\end{align*}
The characteristic polynomial of $D_{1}^{1}(\phi)$ is $\det(1-tD_{1}^{1}(\phi))=1-2t$.
A similar but complicated computation shows that the representation matrix of $D_{2}^{1}(\phi)$ with respect to the basis $\{z^i(dz/z)^{\otimes 2} \colon |i|<2\}$ is
\begin{align*}
\begin{pmatrix}
-\dfrac{\lambda_{0}\lambda_{\infty}^{2}}{1-\lambda_{0}\lambda_{\infty}} & -2\lambda_{0}\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & -\dfrac{\lambda_{0}^{3}}{1-\lambda_{0}\lambda_{\infty}} \\[8pt]
\dfrac{\lambda_{\infty}^{2}}{1-\lambda_{0}\lambda_{\infty}} & 2\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & \dfrac{\lambda_{0}^{2}}{1-\lambda_{0}\lambda_{\infty}} \\
-\dfrac{\lambda_{0}^{2}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} & -2\lambda_{\infty}\dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} &
-\dfrac{\lambda_{0}^{2}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}} \\
\end{pmatrix}
=
\begin{pmatrix}
\lambda_{0} A & \lambda_{0} B & \lambda_{0} C\\
-A & -B & -C\\
\lambda_{\infty} A & \lambda_{\infty} B & \lambda_{\infty} C\\
\end{pmatrix},
\end{align*}
where
\[A=-\frac{\lambda_{\infty}^2}{1-\lambda_{0}\lambda_{\infty}},\, B=-2 \dfrac{2-\lambda_{0}\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}, \text{and}\, C=-\dfrac{\lambda_{0}^{2}}{1-\lambda_{0}\lambda_{\infty}}.\]
Therefore, the characteristic polynomial of $D_{2}^{1}(\phi)$ is
\begin{align*}
\det(1-tD_{2}^{1}(\phi))&=1+(B-\lambda_{0} A- \lambda_{\infty} C)t\\
&=1-\left(2+\lambda_{0}+\lambda_{\infty}+\frac{2-\lambda_{0}-\lambda_{\infty}}{1-\lambda_{0}\lambda_{\infty}}\right)t\\
&=1-(2+\sigma_{1}(\phi))t,
\end{align*}
where $\sigma_{1}(\phi)=\mult{\phi}{0}+\mult{\phi}{\infty}+\mult{\phi}{\alpha}$ is the sum of the multipliers at fixed points of $\phi$.
Summing up, we obtain an explicit formula for $\dzeta{1}{\phi}$.
\begin{align*}
\dzeta{1}{\phi}&=\frac{\dyndet{1}}{\dyndet{2}}\\
&=\frac{1-2t}{1-(2+\sigma_1(\phi))t}.
\end{align*}
\end{ex}
\end{document} |
\begin{document}
\maketitle
\centerline{\scshape{Silvia Frassu$^{\sharp,*}$, Tongxing Li$^{\natural}$ \and Giuseppe Viglialoro$^{\sharp}$}}
{
\centerline{$^\sharp$Dipartimento di Matematica e Informatica}
\centerline{Universit\`{a} di Cagliari}
\centerline{Via Ospedale 72, 09124. Cagliari (Italy)}
}
{
\centerline{$^{\natural}$School of Control Science and Engineering}
\centerline{Shandong University}
\centerline{Jinan, Shandong, 250061 (P. R. China)}
}
\begin{abstract}
We enter the details of two recent articles concerning as many chemotaxis models, one nonlinear and the other linear, and both with produced chemoattractant and saturated chemorepellent. More precisely, we are referring respectively to the papers ``Boundedness in a nonlinear attraction-repulsion Keller--Segel system with production and consumption'', by S. Frassu, C. van der Mee and G. Viglialoro [\textit{J. Math.\ Anal.\ Appl.} {\bf 504}(2):125428, 2021] and ``Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent'', by S. Frassu and G. Viglialoro [\textit{Nonlinear Anal. }{\bf 213}:112505, 2021]. These works, when properly analyzed, leave open room for some improvement of their results. We generalize the outcomes of the mentioned articles, establish other statements and put all the claims together; in particular, we select the sharpest ones and schematize them. Moreover, we complement our research also when logistic sources are considered in the overall study.
\end{abstract}
\section{Preamble}
For details and discussions on the meaning of the forthcoming model, especially in the frame of chemotaxis phenomena and related variants, as well as for mathematical motivations and connected state of the art, we refer to \cite{FrassuCorViglialoro,frassuviglialoro}. These articles will be often cited throughout this work.
\section{Presentation of the Theorems}\label{IntroSection}
Let $\Omega \subset \mathbb R} \def\N{\mathbb N^n$, $n \geq 2$, be a bounded and smooth domain, $\chi,\xi,\delta>0$, $m_1,m_2,m_3\in\mathbb R} \def\N{\mathbb N$, $f(u), g(u)$ and $h(u)$ be reasonably regular functions generalizing the prototypes $f(u)=K u^\alpha$, $g(u)=\gamma u^l$, and $h(u)=k u - \mu u^{\beta}$ with $K,\gamma, \mu>0$, $k \in \mathbb R} \def\N{\mathbb N$ and suitable $\alpha, l, \beta>0$. Once nonnegative initial configurations $u_0$ and $v_0$ are fixed, we aim at deriving sufficient conditions involving the above data so to ensure that the following attraction-repulsion chemotaxis model
\begin{equation}\label{problem}
\begin{cases}
u_t= \nabla \cdot ((u+1)^{m_1-1}\nabla u - \chi u(u+1)^{m_2-1}\nabla v+\xi u(u+1)^{m_3-1}\nabla w) + h(u) & \text{ in } \Omega \times (0,T_{\rm{max}}),\\
v_t=\Delta v-f(u)v & \text{ in } \Omega \times (0,T_{\rm{max}}),\\
0= \Delta w - \delta w + g(u)& \text{ in } \Omega \times (0,T_{\rm{max}}),\\
u_{\nu}=v_{\nu}=w_{\nu}=0 & \text{ on } \partial \Omega \times (0,T_{\rm{max}}),\\
u(x,0)=u_0(x), \; v(x,0)=v_0(x) & x \in \bar\Omega,
\end{cases}
\end{equation}
admits classical solutions which are global and uniformly bounded in time. Specifically, we look for nonnegative functions $u=u(x,t), v=v(x,t), w=w(x,t)$ defined for $(x,t) \in \bar{\Omega}\times [0,T_{\rm{max}})$, and $T_{\rm{max}}=\infty$, with the properties that
\begin{equation}\label{ClassicalAndGlobability}
\begin{cases}
u,v\in C^0(\bar{\Omega}\times [0,\infty))\cap C^{2,1}(\bar{\Omega}\times (0,\infty)), w\in C^0(\bar{\Omega}\times [0,\infty))\cap C^{2,0}(\bar{\Omega}\times (0,\infty)),\\ (u,v,w) \in (L^\infty((0, \infty);L^{\infty}(\Omega)))^3,
\end{cases}
\end{equation}
and pointwisely satisfying all the relations in problem \eqref{problem}.
To this scope, let $f$, $g$ and $h$ be such that
\begin{equation}\label{f}
f,g \in C^1(\mathbb R} \def\N{\mathbb N) \quad \textrm{with} \quad 0\leq f(s)\leq Ks^{\alpha} \textrm{ and } \gamma s^l\leq g(s)\leq \gamma s(s+1)^{l-1},\; \textrm{for some}\; K,\,\gamma,\,\alpha>0,\, l\geq 1 \quad \textrm{and all } s \geq 0,
\end{equation}
and
\begin{equation}\label{h}
h \in C^1(\mathbb R} \def\N{\mathbb N) \quad \textrm{with} \quad h(0)\geq 0 \textrm{ and } h(s)\leq k s-\mu s^{\beta}, \quad \textrm{for some}\quad k \in \mathbb R} \def\N{\mathbb N,\,\mu>0,\, \beta>1\, \quad \textrm{and all } s \geq 0.
\end{equation}
Then we prove these two theorems.
\begin{theorem}\label{MainTheorem}
Let $\Omega$ be a smooth and bounded domain of $\mathbb{R}^n$, with $n\geq 2$, $\chi, \xi, \delta$ positive reals, $l \geq 1$ and $h \equiv 0$. Moreover, for $\alpha >0$ and $m_1, m_2, m_3 \in \mathbb R} \def\N{\mathbb N$, let $f$ and $g$ fulfill \eqref{f} for each of the following cases:
\begin{enumerate}[label=$A_{\roman*}$)]
\item \label{A1} $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>\min\left\{2m_2-(m_3+l),\max\left\{2m_2-1,\frac{n-2}{n}\right\}, m_2 - \frac{1}{n}\right\}=:\mathcal{A}$,
\item \label{A2} $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right)$ and $m_1>m_2 + \alpha - \frac{2}{n}=:\mathcal{B}$,
\item \label{A3} $\alpha \in \left[\frac{2}{n},1\right]$ and $m_1>m_2 + \frac{n\alpha-2}{n\alpha-1}=:\mathcal{C}$.
\end{enumerate}
Then for any initial data $(u_0,v_0)\in (W^{1,\infty}(\Omega))^2$, with $u_0, v_0\geq 0$ on $\bar{\Omega}$, there exists a unique triplet $(u,v,w)$ of nonnegative functions, uniformly bounded in time and classically solving problem \eqref{problem}.
\end{theorem}
\begin{theorem}\label{MainTheorem1}
Under the same hypotheses of Theorem \ref{MainTheorem} and $\beta>1$, let $h$ comply with \eqref{h}.
Moreover, for $\alpha >0$ and $m_1, m_2, m_3 \in \mathbb R} \def\N{\mathbb N$, let $f$ and $g$ fulfill \eqref{f} for each of the following cases:
\begin{enumerate}[label=$A_{\roman*}$)]
\setcounter{enumi}{3}
\item \label{A4} $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>\min\left\{2m_2-(m_3+l), 2m_2-\beta\right\}=:\mathcal{D}$,
\item \label{A5} $\alpha \in \left(\frac{1}{n},1\right)$ and $m_1>\min\left\{2m_2+1-(m_3+l), 2m_2+1-\beta\right\}=:\mathcal{E}$.
\end{enumerate}
Then the same claim holds true.
\end{theorem}
When the logistic term $h$ does not take part in the model, problem \eqref{problem} has been already analyzed in \cite{FrassuCorViglialoro} for the nonlinear diffusion and sensitivities case, and in \cite{frassuviglialoro} for the linear scenario; nevertheless, in these papers only small values of $\alpha$ are considered. Precisely, for $\alpha$ belonging to $(0,\frac{1}{2}+\frac{1}{n}),$ boundedness is ensured:
\begin{itemize}
\item in \cite[Theorem 2.1]{FrassuCorViglialoro} for $m_1,m_2,m_3\in \mathbb R} \def\N{\mathbb N$ and $l\geq 1$, under the assumption
$$m_1>\min\left\{2m_2+1-(m_3+l),\max\left\{2m_2,\frac{n-2}{n}\right\}\right\}=:\mathcal{F};$$
\item in \cite[Theorem 2.1]{frassuviglialoro} for either $m_1=m_2=m_3=l=1$, under the assumption
$$\xi>\left(\frac{8}{n}
\frac{2^\frac{2}{n}\frac{2}{n}^{n+1}(\frac{2}{n}-1)(n^2+n)}{(\frac{2}{n}+1)^{\frac{2}{n}+1}}\right)^\frac{2}{n}\lVert v_0\rVert_{L^\infty(\Omega)}^\frac{4}{n}=:\mathcal{G},$$
or in \cite[Theorem 2.2]{frassuviglialoro} for $m_1=m_2=m_3=1$ and any $l>1$.
\end{itemize}
In light of Theorems \ref{MainTheorem} and \ref{MainTheorem1}, herein we develop an analysis dealing also with values of $\alpha$ larger than $\frac{1}{2}+\frac{1}{n}$. Additionally, for $\alpha$ belonging to some sub-intervals of $(0,\frac{1}{2}+\frac{1}{n})$ we improve \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro}. On the other hand, the introduction of $h$ allows us to obtain further generalizations and/or claims.
All this aspects are put together into Table \ref{Table_ResultUnified}. It, when possible, also indicates which of the assumptions taken from \cite[Theorem 2.1]{FrassuCorViglialoro}, \cite[Theorems 2.1 and 2.2]{frassuviglialoro}, and Theorems \ref{MainTheorem} and \ref{MainTheorem1} are the mildest leading to boundedness.
\setlength\extrarowheight{4.8pt}
\begin{table}[h!]
\makegapedcells
\centering
\begin{tabular}{ccccccccccc}
\hline
$m_2$ &$m_3$&$l$&\multicolumn{1}{c|}{$\alpha$}&\multicolumn{1}{c|}{$m_1$}&$\chi$&$\xi$& Reference & \quad \quad Implication\\
\hline
$1$ &$1$&$1$&\multicolumn{1}{c|}{$[\frac{2}{n},1)$}&\multicolumn{1}{c|}{$1$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$ >\mathcal{G}$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.1]{frassuviglialoro}\\
$1$ &$1$&$>1$&\multicolumn{1}{c|}{$ [\frac{2}{n},1)$}&\multicolumn{1}{c|}{$1$}&$ \mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.2]{frassuviglialoro}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$ (\frac{1}{n},1)$}&\multicolumn{1}{c|}{$>\mathcal{F}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}& Remark \ref{Alpha}, generalizing \cite[Th. 2.1]{FrassuCorViglialoro}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$ (0,\frac{1}{n}]$}&\multicolumn{1}{c|}{$>\mathcal{A}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}& \quad \quad $\ast$ and $\ast\ast$\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$(\frac{1}{n},\frac{2}{n})$}&\multicolumn{1}{c|}{$>\mathcal{B}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}& \quad \quad $\ast$\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&\multicolumn{1}{c|}{$[\frac{2}{n},1]$}&\multicolumn{1}{c|}{$>\mathcal{C}$}&$\mathbb R} \def\N{\mathbb N^+$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem}\\
\hline
$m_2$ &$m_3$&$l$&$\beta$&\multicolumn{1}{c|}{$\alpha$}&\multicolumn{1}{c|}{$m_1$}&$\chi$&$\xi$&$k$&$\mu$& Reference \\
\hline
$1$ &$1$&$1$&$>2$&\multicolumn{1}{c|}{$(\frac{1}{n},1)$}&\multicolumn{1}{c|}{$1$}&$\mathbb R} \def\N{\mathbb N^+$&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
$ \mathbb R} \def\N{\mathbb N$ &$\mathbb R} \def\N{\mathbb N$&$\geq 1$&$>1$&\multicolumn{1}{c|}{$ (0,\frac{1}{n}]$}&\multicolumn{1}{c|}{$>\mathcal{D}$}&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
$\mathbb R} \def\N{\mathbb N$& $\mathbb R} \def\N{\mathbb N$&$\geq 1$&$>1$&\multicolumn{1}{c|}{$(\frac{1}{n},1)$}&\multicolumn{1}{c|}{$>\mathcal{E}$}&$\mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N^+$&$ \mathbb R} \def\N{\mathbb N$&\multicolumn{1}{c|}{$\mathbb R} \def\N{\mathbb N^+$}&Th. \ref{MainTheorem1}\\
\hline
\end{tabular}
\caption{Schematization collecting the ranges of the parameters involved in model \eqref{problem} for which boundedness of its solutions is established for any fixed initial distribution $u_0$ and $v_0$. The symbol $\ast$ stands for ``improves \cite[Th. 2.1]{frassuviglialoro} and recovers \cite[Th. 2.2]{frassuviglialoro}'' and $\ast\ast$ for ``improves \cite[Th. 2.1]{FrassuCorViglialoro}''. ($\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ are defined above.)
}\label{Table_ResultUnified}
\end{table}
\section{Local well posedness, boundedness criterion, main estimates and analysis of parameters}\label{SectionLocalInTime}
For $\Omega$, $\chi,\xi,\delta$, $m_1,m_2,m_3$ and $f, g, h$ as above, from now on with $u, v, w \geq 0$ we refer to functions of $(x,t) \in \bar{\Omega}\times [0,T_{\rm{max}})$, for some finite $T_{\rm{max}}$, classically solving problem \eqref{problem} when nonnegative initial data $(u_0,v_0)\in (W^{1,\infty}(\Omega))^2$ are provided. In particular, $u$ satisfies
\begin{equation}\label{massConservation}
\int_\Omega u(x, t)dx \leq m_0 \quad \textrm{for all }\, t \in (0,T_{\rm{max}}),
\end{equation}
whilst $v$ is such that
\begin{equation*}\label{MaxPrincV}
0 \leq v\leq \lVert v_0\rVert_{L^\infty(\Omega)}\quad \textrm{in}\quad \Omega \times (0,T_{\rm{max}}).
\end{equation*}
Further, globality and boundedness of $(u,v,w)$ (in the sense of \eqref{ClassicalAndGlobability}) are ensured whenever (boundedness criterion) the $u$-component belongs to $L^\infty((0,T_{\rm{max}});L^p(\Omega))$, with $p>1$ arbitrarily large, and uniformly with respect $t\in (0,T_{\rm{max}})$.
These basic statements can be proved by standard reasoning; in particular, when $h\equiv 0$ they verbatim follow from \cite[Lemmas 4.1 and 4.2]{FrassuCorViglialoro} and relation \eqref{massConservation} is the well-known mass conservation property. Conversely, in the presence of the logistic terms $h$ as in \eqref{h}, some straightforward adjustments have to be considered and the $L^1$-bound of $u$ is consequence of an integration of the first equation in \eqref{problem} and an application of the H\"{o}lder inequality: precisely for $k_+=\max\{k,0\}$
\[
\frac{d}{dt} \int_\Omega u = \int_\Omega h(u) =k \int_\Omega u - \mu \int_\Omega u^{\beta} \leq k_+ \int_\Omega u - \frac{\mu}{|\Omega|^{\beta-1}} \left(\int_\Omega u\right)^{\beta}
\quad \textrm{for all }\, t \in (0,T_{\rm{max}}),
\]
and we can conclude by invoking an ODI-comparison argument.
In our computations, beyond the above positions, some uniform bounds of $\|v(\cdot,t)\|_{W^{1,s}(\Omega)}$ are required. In this sense, the following lemma gets
the most out from $L^p$-$L^q$ (parabolic) maximal regularity; this is a cornerstone and for some small values of $\alpha$ the succeeding $W^{1,s}$-estimates are sharper than the $W^{1,2}$-estimates derived in \cite{FrassuCorViglialoro,frassuviglialoro}, and therein employed.
\begin{lemma}\label{LocalV}
There exists $c_0>0$ such that $v$ fulfills
\begin{equation}\label{Cg}
\int_\Omega |\nabla v(\cdot, t)|^s\leq c_0 \quad \textrm{on } \, (0,T_{\rm{max}})
\begin{cases}
\; \textrm{for all } s \in [1,\infty) & \textrm{if } \alpha \in \left(0, \frac{1}{n}\right],\\
\; \textrm{for all } s \in \left[1, \frac{n}{(n\alpha-1)}\right) & \textrm{if } \alpha \in \left(\frac{1}{n},1\right].
\end{cases}
\end{equation}
\begin{proof}
For each $\alpha \in (0,1]$, there is $\rho >\frac{1}{2}$ such that for all $s \in \left[\frac{1}{\alpha},\frac{n}{(n\alpha-1)_+}\right)$ we have
$\frac{1}{2}<\rho <1-\frac{n}{2}\big(\alpha-\frac{1}{s}\big)$. From $1-\rho-\frac{n}{2}\big(\alpha-\frac{1}{s}\big)>0$, the claim follows invoking properties related to the Neumann heat semigroup, exactly as done in the second part of \cite[Lemma 5.1]{FrassuCorViglialoro}.
\end{proof}
\end{lemma}
We will make use of this technical result.
\begin{lemma}\label{LemmaCoefficientAiAndExponents}
Let $n\in \N$, with $n\geq 2$, $m_1>\frac{n-2}{n}$, $m_2,m_3\in \mathbb R} \def\N{\mathbb N$ and $\alpha \in (0,1]$. Then there is $s \in [1,\infty)$, such that for proper
$p,q\in[1,\infty)$, $\theta$ and $\theta'$, $\mu$ and $\mu'$ conjugate exponents, we have that
\begin{align}
a_1&= \frac{\frac{m_1+p-1}{2}\left(1-\frac{1}{(p+2m_2-m_1-1)\theta}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}},
& a_2&=\frac{q\left(\frac{1}{s}-\frac{1}{2\theta'}\right)}{\frac{q}{s}+\frac{1}{n}-\frac{1}{2}}, \nonumber \\
a_3 &= \frac{\frac{m_1+p-1}{2}\left(1-\frac{1}{2\alpha\mu}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}},
& a_4 &= \frac{q\left(\frac{1}{s}-\frac{1}{2(q-1)\mu'}\right)}{\frac{q}{s}+\frac{1}{n}-\frac{1}{2}},\nonumber\\
\kappa_1 & =\frac{\frac{p}{2}\left(1- \frac{1}{p}\right)}{\frac{m_1+p-1}{2}+\frac{1}{n}-\frac{1}{2}} \nonumber, & \kappa_2 & = \frac{q - \frac{1}{2}}{q+\frac{1}{n}-\frac{1}{2}},
\end{align}
belong to the interval $(0,1)$. If, additionally,
\begin{equation}\label{Restrizionem1-m2-alphaPiccolo}
\alpha \in \left(0,\frac{1}{n}\right] \; \text{and}\; m_2-m_1<\frac{1}{n},
\end{equation}
\begin{equation}\label{Restrizionem1-m2-alphaGrande}
\alpha \in \left(\frac{1}{n},\frac{2}{n}\right) \; \text{and}\; m_2-m_1<\frac{2}{n}-\alpha,
\end{equation}
or
\begin{equation}\label{Restrizionem1-m2-alphaGrandeBis}
\alpha \in \left[\frac{2}{n},1\right] \; \text{and}\; m_2-m_1<\frac{2-n\alpha}{n\alpha-1},
\end{equation}
these futher relations hold true:
\begin{equation*}\label{MainInequalityExponents}
\beta_1 + \gamma_1 =\frac{p+2m_2-m_1-1}{m_1+p-1}a_1+\frac{1}{q}a_2\in (0,1) \;\textrm{ and }\; \beta_2 + \gamma_2= \frac{2 \alpha }{m_1+p-1}a_3+\frac{q-1}{q}a_4
\in (0,1).
\end{equation*}
\begin{proof}
For any $s\geq 1$, let $\theta'>\max\left\{\frac{n}{2},\frac{s}{2}\right\}$ and $\mu>\max\left\{\frac{1}{2\alpha},\frac{n}{2}\right\}$. Thereafter, for
\begin{equation}\label{Prt_q}
\begin{cases}
q > \max \left\{\frac{(n-2)}{n}\theta', \frac{s}{2\mu'}+1\right\} \\
p>\max \left\{2-\frac{2}{n}-m_1,\frac{1}{\theta}-2m_2+m_1+1, \frac{(2m_2-m_1-1)(n-2)\theta-nm_1+n}{n-(n-2)\theta},\frac{2\alpha \mu(n-2)}{n} -m_1+1\right\},
\end{cases}
\end{equation}
it can be seen that $a_i,k_2\in (0,1)$, for any $i=1,2,3,4.$ On the other hand, $k_1\in (0,1)$ also thanks to the assumption $m_1>\frac{n-2}{n}.$
As to the second part, we distinguish three cases: $\alpha \in \left(0,\frac{1}{n}\right]$, $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right)$ and
$\alpha \in \left[\frac{2}{n},1\right].$ (We insert Figure \ref{FigureSpiegazioneLemma} to clarify the proof, by focusing on the relation involving the values of $\alpha$, $s$ and
$\theta'$ in terms of assumptions \eqref{Restrizionem1-m2-alphaPiccolo}, \eqref{Restrizionem1-m2-alphaGrande}, \eqref{Restrizionem1-m2-alphaGrandeBis}.)
\begin{itemize}
\item [$\circ$] $\alpha \in \left(0,\frac{1}{n}\right]$.
For $s>\frac{2\mu'}{2\mu'-1}$ arbitrarily large, consistently with \eqref{Prt_q}, we take $p=q=s$ and $\theta'=s\omega$, for some $\omega>\frac{1}{2}$. Computations provide
\begin{equation*}
0< \beta_1+\gamma_1=\frac{s+2m_2-m_1-1-\frac{1}{\theta}}{m_1+s-2+\frac{2}{n}}+\frac{2-\frac{1}{\omega}}{s+\frac{2s}{n}},
\end{equation*}
and
\begin{equation*}
0< \beta_2+\gamma_2=\frac{2\alpha-\frac{1}{\mu}}{m_1+s-2+\frac{2}{n}}+\frac{2s-2-\frac{s}{\mu'}}{s+\frac{2s}{n}}.
\end{equation*}
In light of the above positions, the largeness of $s$ infers $\theta$ arbitrarily close to $1$. Further, by choosing $\omega$ approaching $\frac{1}{2}$, continuity arguments imply that $\beta_1+\gamma_1<1$ whenever restriction \eqref{Restrizionem1-m2-alphaPiccolo} is satisfied, whereas $\beta_2+\gamma_2<1$ comes from $\mu>\frac{n}{2}.$
\item [$\circ$] $\alpha \in \left(\frac{1}{n},\frac{2}{n}\right).$ First let $s$ be arbitrarily close to $\frac{n}{n\alpha-1}$ and let $q=\frac{p}{2}$ fulfill \eqref{Prt_q}. Then, in these circumstances it holds that $\max\left\{\frac{s}{2},\frac{n}{2}\right\}=\frac{s}{2}$, so that restriction on $\theta'$ reads $\theta'>\frac{s}{2}$. Subsequently,
\begin{equation*}
0< \beta_1+\gamma_1=\frac{p+2m_2-m_1-1-\frac{1}{\theta}}{m_1+p-2+\frac{2}{n}}+\frac{2-\frac{s}{\theta'}}{p+\frac{2s}{n}-s},
\end{equation*}
and
\begin{equation*}
0< \beta_2+\gamma_2=\frac{2\alpha-\frac{1}{\mu}}{m_1+p-2+\frac{2}{n}}+\frac{p-2-\frac{s}{\mu'}}{p+\frac{2s}{n}-s}.
\end{equation*}
Since from $\theta'>\frac{s}{2}$ we have that $\theta'$ approaches $\frac{n}{2(n\alpha-1)}$, similar arguments used above imply that upon enlarging $p$ one can see that condition \eqref{Restrizionem1-m2-alphaGrande} leads to $\beta_1+\gamma_1<1$. On the other hand, in order to have $\beta_2+\gamma_2<1$ we have to invoke the above constrain on $\mu$, i.e., $\mu>\frac{1}{2\alpha}.$
\item [$\circ$] $\alpha \in \left[\frac{2}{n},1\right].$ We only have to consider in the previous case that now $\theta'>\frac{n}{2}$, so concluding thanks to
\eqref{Restrizionem1-m2-alphaGrandeBis}.
\end{itemize}
\end{proof}
\end{lemma}
\begin{figure}
\caption{The colored lines, functions of $\alpha$, represent the supremum of the difference $m_2-m_1$ for some space dimension $n$. Moreover, for the sub-intervals $(0,1/n]$, $(1/n,2/n)$ and $[2/n,1]$ of $\alpha$, the corresponding range of $s$ and choice of $\theta'$ are also indicated.}
\label{FigureSpiegazioneLemma}
\end{figure}
\begin{remark}\label{RemarkOnS}
In view of its importance in the computations, we have to point out that from the above lemma $s$ can be chosen arbitrarily large only when $\alpha \in \left(0,\frac{1}{n}\right]$.
In particular, as we will see, in such an interval the terms $\int_{\Omega} (u+1)^{p+2m_2-m_1-1} \vert \nabla v\lvert^2$ and $\int_{\Omega} (u+1)^{2\alpha} \vert \nabla v\lvert^{2(q-1)}$, appearing in our reasoning, can be treated in two alternative ways: either invoking the Young inequality or the Gagliardo--Nirenberg one.
\end{remark}
\section{A priori estimates and proof of the Theorems}\label{EstimatesAndProofSection}
\subsection{The non-logistic case}\label{NonLog}
Recalling the globality criterion mentioned in $\S$\ref{SectionLocalInTime}, let us define the functional $y(t):=\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}$,
with $p,q>1$ properly large (and, when required, with $p=q$), and let us dedicate to derive the desired uniform bound of $\int_\Omega u^p$.
In the spirit of Remark \ref{RemarkOnS}, let us start by analyzing the evolution in time of the functional $y(t)$ by relying on the Young inequality.
\begin{lemma}\label{Estim_general_For_u^p_nablav^2qLemma}
Let $\alpha \in \left(0,\frac{1}{n}\right]$. If $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with either $m_1>2m_2-(m_3+l)$ or $m_1>\max\left\{2m_2-1,\frac{n-2}{n}\right\}$,
then there exist $p, q>1$ such that $(u,v,w)$ satisfies for some $c_{16}, c_{17}, c_{18}>0$
\begin{equation}\label{MainInequality}
\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + c_{16} \int_\Omega |\nabla |\nabla v|^q|^2 + c_{17} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 \leq c_{18} \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
\begin{proof}
Let $p=q>1$ sufficiently large; moreover, in view of Remark \ref{RemarkOnS}, from now on, when necessary we will tacitly enlarge these parameters.
In the first part of the proof we focus on the estimate of the term $\frac{d}{dt} \int_\Omega (u+1)^p$.
Standard testing procedures provide
\begin{equation*}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p =\int_\Omega p(u+1)^{p-1}u_t &= -p(p-1) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+p(p-1)\chi \int_\Omega u(u+1)^{m_2+p-3} \nabla u \cdot \nabla v \\
&-p(p-1)\xi \int_\Omega u(u+1)^{m_3+p-3} \nabla u \cdot \nabla w \quad \text{on } (0,T_{\rm{max}}).
\end{split}
\end{equation*}
By reasoning as in \cite[Lemma 5.2]{FrassuCorViglialoro}, we obtain for $\epsilon_1, \epsilon_2, \tilde{\sigma}$ positive, and for all $t \in (0,T_{\rm{max}})$, some $c_1>0$ such that
\begin{equation}\label{Estim_1_For_u^p}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p & \leq -p(p-1) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 +p(p-1)\chi \int_\Omega u(u+1)^{m_2+p-3} \nabla u \cdot \nabla v \\
&+ \left(\epsilon_1 + \tilde{\sigma} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}
+ \epsilon_2 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2 + c_1.
\end{split}
\end{equation}
Let us now discuss the cases $m_1>2m_2-(m_3+l)$ and $m_1>\max\left\{2m_2-1,\frac{n-2}{n}\right\}$, respectively.
A double application of the Young inequality in \eqref{Estim_1_For_u^p} and bound \eqref{Cg} give
\begin{equation}\label{Young}
\begin{split}
&p(p-1)\chi \int_\Omega u (u+1)^{m_2+p-3} \nabla u \cdot \nabla v \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + \epsilon_4 \int_\Omega |\nabla v|^{s} + c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}}\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} + c_4 \quad \text{on }\, (0,T_{\rm{max}}),
\end{split}
\end{equation}
with $\epsilon_3, \epsilon_4 >0$ and some positive $c_2, c_3, c_4$.
From $m_1> 2m_2-(m_3+l)$, we have $\frac{(p+2m_2-m_1-1)s}{s-2} < (m_3+p+l-1)$, and for every $\epsilon_5>0$, Young's inequality yields some $c_5>0$ entailing
\begin{equation}\label{Young1}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \epsilon_5 \int_\Omega (u+1)^{m_3+p+l-1} + c_5 \quad \text{for all }\, (0,T_{\rm{max}}).
\end{equation}
Now, we note that $m_1>2 m_2-1$ implies $\frac{(p+2m_2-m_1-1)s}{s-2} < p$, and the Young inequality allows us to rephrase \eqref{Young1} in an alternative way:
\begin{equation}\label{Young4}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \epsilon_6 \int_\Omega (u+1)^p + c_6 \quad \text{on }\, (0,T_{\rm{max}}),
\end{equation}
with $\epsilon_6>0$ and positive $c_6$.
Further, an application of the Gagliardo--Nirenberg inequality and property \eqref{massConservation} yield
\begin{equation*}\label{Theta_2}
\theta=\frac{\frac{n(m_1+p-1)}{2}\left(1-\frac{1}{p}\right)}{1-\frac{n}{2}+\frac{n(m_1+p-1)}{2}}\in (0,1),
\end{equation*}
so giving for $c_7, c_8>0$
\begin{equation*}
\begin{split}
\int_{\Omega} (u+1)^p&= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2p}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}}\\
&\leq c_7 \|\nabla (u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{2p}{m_1+p-1}\theta} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}(1-\theta)} + c_7 \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2p}{m_1+p-1}}\\
& \leq c_8 \Big(\int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\Big)^{\kappa_1}+ c_8 \quad \text{ for all } t \in(0,T_{\rm{max}}).
\end{split}
\end{equation*}
Since $\kappa_1 \in (0,1)$ (see Lemma \ref{LemmaCoefficientAiAndExponents}), for any positive $\epsilon_7$ thanks to the Young inequality we arrive for some positive $c_9>0$ at
\begin{equation}\label{GN2}
\begin{split}
\epsilon_6 \int_\Omega (u+1)^p &\leq \epsilon_7 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2 + c_9 \quad \text{on } (0,T_{\rm{max}}).
\end{split}
\end{equation}
By plugging estimate \eqref{Young} into relation \eqref{Estim_1_For_u^p}, and by relying on bound \eqref{Young1} (or, alternatively to \eqref{Young1}, relations \eqref{Young4} and \eqref{GN2}), infer for appropriate $\tilde{\epsilon}_1, \tilde{\epsilon}_2>0$ and proper $c_{10}>0$
\begin{equation}\label{ClaimU}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p &\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \tilde{\epsilon}_1 \right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\\
&+ \left(\tilde{\epsilon}_2 - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)}\right) \int_\Omega (u+1)^{m_3+p+l-1} + c_{10} \quad \text{for all } t \in (0,T_{\rm{max}}),
\end{split}
\end{equation}
where we also exploited that
\begin{equation}\label{GradU}
\int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 = \frac{4}{(m_1+p-1)^2} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
Now, as to the term $\frac{d}{dt} \int_\Omega |\nabla v|^{2q}$ of the functional $y(t)$, reasoning similarly as in \cite[Lemma 5.3]{FrassuCorViglialoro}, we obtain for some
$c_{11}, c_{12}>0$
\begin{equation}\label{Estim_gradV}
\frac{d}{dt}\int_\Omega |\nabla v|^{2q}+ q \int_\Omega |\nabla v|^{2q-2} |D^2v|^2 \leq c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} +c_{12} \quad \textrm{on } \; (0,T_{\rm{max}}).
\end{equation}
Moreover, Young's inequalities and bound \eqref{Cg} give for every arbitrary $\epsilon_8, \epsilon_9>0$ and some $c_{13}, c_{14}, c_{15}>0$
\begin{equation}\label{Estimat_nablav^2q+2}
\begin{split}
&c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} \leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + c_{13} \int_\Omega |\nabla v|^{\frac{2(q-1)(m_3+p+l-1)}{m_3+p+l-1-2\alpha}}\\
& \leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + \epsilon_9 \int_\Omega |\nabla v|^s + c_{14}
\leq \epsilon_8 \int_\Omega (u+1)^{m_3+p+l-1} + c_{15} \quad \textrm{for all} \quad t \in (0,T_{\rm{max}}).
\end{split}
\end{equation}
Therefore, by inserting relation \eqref{Estimat_nablav^2q+2} into \eqref{Estim_gradV} and adding \eqref{ClaimU}, we have the claim for a proper choice of
$\tilde{\epsilon}_1, \tilde{\epsilon}_2, \epsilon_8$ and some positive $c_{16}, c_{17}, c_{18}$, also by taking into account the relation (see \cite[page 17]{FrassuCorViglialoro})
\begin{equation}\label{GradV}
\vert \nabla \lvert \nabla v\rvert^q\rvert^2=\frac{q^2}{4}\lvert \nabla v \rvert^{2q-4}\vert \nabla \lvert \nabla v\rvert^2\rvert^2=q^2\lvert \nabla v \rvert^{2q-4}\lvert D^2v \nabla v \rvert^2\leq q^2|\nabla v|^{2q-2} |D^2v|^2.
\end{equation}
\end{proof}
\end{lemma}
Let us now turn our attention when, as mentioned before, the Gagliardo--Nirenberg inequality is employed. In this case, we can derive information not only for
$\alpha \in \left(0,\frac{1}{n}\right]$ but also for $\alpha \in \left(\frac{1}{n},1\right]$.
\begin{lemma}\label{Met_GN}
If $m_1,m_2\in \mathbb R} \def\N{\mathbb N$ and $\alpha>0$ are taken accordingly to \eqref{Restrizionem1-m2-alphaPiccolo}, \eqref{Restrizionem1-m2-alphaGrande},
\eqref{Restrizionem1-m2-alphaGrandeBis}, then there exist $p, q>1$ such that $(u,v,w)$ satisfies a similar inequality as in \eqref{MainInequality}.
\begin{proof}
For $s$, $p$ and $q$ taken accordingly to Lemma \ref{LemmaCoefficientAiAndExponents} (in particular, $p=q$ for $\alpha \in \left(0,\frac{1}{n}\right]$, and $q=\frac{p}{2}$
for $\alpha \in \left(\frac{1}{n},1\right]$), let $\theta, \theta', \mu, \mu'$, $a_1,a_2, a_3, a_4$ and $\beta_1, \beta_2, \gamma_1, \gamma_2$ be therein defined.
With a view to Lemma \ref{Estim_general_For_u^p_nablav^2qLemma}, by manipulating relation \eqref{Estim_1_For_u^p} and focusing on the first inequality in \eqref{Young} and on \eqref{Estim_gradV}, proper $\epsilon_1, \tilde{\sigma}$ lead to
\begin{equation}\label{Somma}
\begin{split}
&\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + q \int_\Omega |\nabla v|^{2q-2} |D^2v|^2
\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \tilde{\epsilon}_1 \right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\\
&+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2 + c_{11} \int_\Omega u^{2\alpha} |\nabla v|^{2q-2} +c_{19} \quad \textrm{on } \; (0,T_{\rm{max}}),
\end{split}
\end{equation}
for some $c_{19}>0$ (we also used relation \eqref{GradU}).
In this way, we can estimate the second and third integral on the right-hand side of \eqref{Somma} by applying the H\"{o}lder inequality so to have
\begin{equation} \label{H1}
\int_{\Omega} (u+1)^{p+2m_2-m_1-1} |\nabla v|^2 \leq \left(\int_{\Omega} (u+1)^{(p+2m_2-m_1-1)\theta}\right)^{\frac{1}{\theta}}
\left(\int_{\Omega} |\nabla v|^{2 \theta'}\right)^{\frac{1}{\theta'}} \quad \textrm{ on } (0, T_{\rm{max}}),
\end{equation}
and
\begin{equation} \label{H2}
\int_{\Omega} (u+1)^{2\alpha} |\nabla v|^{2q-2} \leq
\left(\int_{\Omega} (u+1)^{2\alpha\mu}\right)^{\frac{1}{\mu}} \left(\int_{\Omega} |\nabla v|^{2(q-1)\mu'}\right)^{\frac{1}{\mu'}}\quad \textrm{ on } (0,T_{\rm{max}}).
\end{equation}
By invoking the Gagliardo--Nirenberg inequality and bound \eqref{massConservation}, we obtain for some $c_{20}, c_{21}>0$
\begin{align}\label{a1}
&\left(\int_{\Omega} (u+1)^{(p+2m_2-m_1-1)\theta}\right)^{\frac{1}{\theta}}= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}\theta}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}}\\ \nonumber
& \leq c_{20} \|\nabla(u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1} a_1} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1} (1-a_1)} + c_{20} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{2(p+2m_2-m_1-1)}{m_1+p-1}} \\ \nonumber
&\leq c_{21} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}+ c_{21} \quad \textrm{ for all } \,t\in(0,T_{\rm{max}}),
\end{align}
and for some $c_{22}, c_{23}>0$
\begin{align}\label{a3}
&\left(\int_{\Omega} (u+1)^{2\alpha\mu}\right)^{\frac{1}{\mu}}= \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{4\alpha\mu}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1}}\\ \nonumber
& \leq c_{22} \|\nabla(u+1)^{\frac{m_1+p-1}{2}}\|_{L^2(\Omega)}^{\frac{4\alpha}{m_1+p-1} a_3} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1} (1-a_3)} + c_{22} \|(u+1)^{\frac{m_1+p-1}{2}}\|_{L^{\frac{2}{m_1+p-1}}(\Omega)}^{\frac{4\alpha}{m_1+p-1}}\\ \nonumber
&\leq c_{23} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2}+ c_{23}
\quad \textrm{for all }\, t \in (0,T_{\rm{max}}).
\end{align}
In a similar way, we can again apply the Gagliardo--Nirenberg inequality and bound \eqref{Cg} and get for some $c_{24}, c_{25}>0$
\begin{align}\label{a2}
&\left(\int_{\Omega} |\nabla v|^{2 \theta'}\right)^{\frac{1}{\theta'}} =\| |\nabla v|^q \|_{L^{\frac{2 \theta'}{q}}(\Omega)}^{\frac{2}{q}}\leq c_{24} \|\nabla |\nabla v|^q\|_{L^2(\Omega)}^{\frac{2}{q}a_2} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2}{q}(1-a_2)} + c_{24} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2}{q}} \\ \nonumber
& \leq c_{25} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1}+ c_{25}
\quad \textrm{ for all } t\in (0, T_{\rm{max}}),
\end{align}
and for some $c_{26}, c_{27}>0$
\begin{align}\label{a4}
&\left(\int_{\Omega} |\nabla v|^{2(q-1) \mu'}\right)^{\frac{1}{\mu'}} =\| |\nabla v|^q \|_{L^{\frac{2(q-1)}{q}\mu'}(\Omega)}^{\frac{2(q-1)}{q}}
\leq c_{26} \|\nabla |\nabla v|^q\|_{L^2(\Omega)}^{\frac{2(q-1)}{q} a_4} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2(q-1)}{q} (1-a_4)}
+ c_{26} \| |\nabla v|^q\|_{L^{\frac{s}{q}}(\Omega)}^{\frac{2(q-1)}{q}} \\ \nonumber
&\leq c_{27} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2}+ c_{27} \quad \textrm{ for every } t\in (0,T_{\rm{max}}).
\end{align}
By plugging \eqref{H1} and \eqref{H2} into \eqref{Somma} and taking into account \eqref{a1}, \eqref{a3}, \eqref{a2}, \eqref{a4}, we deduce for a proper
$\tilde{\epsilon}_1$, once inequality \eqref{GradV} is considered, the following estimate for some $c_{28}, c_{29}, c_{30}, c_{31}, c_{32}>0$:
\begin{equation}\label{Somma1}
\begin{split}
&\frac{d}{dt} \left(\int_\Omega (u+1)^p + \int_\Omega |\nabla v|^{2q}\right) + c_{28} \int_\Omega |\nabla |\nabla v|^q|^2
+ c_{29} \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 - c_{30}\\
&\leq c_{31} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}
\left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1} + c_{31} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_1}\\
& + c_{31} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_1}
+ c_{32} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2}\\
&+ c_{32} \left(\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2\right)^{\beta_2} + c_{32} \left(\int_{\Omega} |\nabla |\nabla v|^q|^2 \right)^{\gamma_2} \quad \text{on }
(0,T_{\rm{max}}).
\end{split}
\end{equation}
Since by Lemma \ref{LemmaCoefficientAiAndExponents} we have that $\beta_1 + \gamma_1 <1$ and $\beta_2 + \gamma_2 <1$, and in particular $\beta_1, \gamma_1, \beta_2, \gamma_2 \in (0,1)$, we can treat the two integral products and the remaining four addenda of the right-hand side in a such way that eventually they are absorbed by the two integral terms involving the gradients in the left one. More exactly, to the products we apply
\[
a^{d_1}b^{d_2} \leq \epsilon(a+b)+c \quad \textrm{with } a,b\geq0, d_1,d_2 >0 \; \textrm{such that } d_1+d_2<1, \; \textrm{for all } \epsilon>0 \; \textrm{and some } c>0
\]
(achievable by means of applications of Young's inequality), and to the other terms the Young inequality. In this way, the resulting linear combination of $\int_{\Omega} |\nabla |\nabla v|^q|^2$ and $\int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2$ can be turned into $\frac{c_{28}}{2} \int_{\Omega} |\nabla |\nabla v|^q|^2 + \frac{c_{29}}{2} \int_{\Omega} |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2$, which coming back to \eqref{Somma1} infers what claimed.
\end{proof}
\end{lemma}
\subsection{The logistic case}\label{Log}
For the logistic case we retrace part of the computations above connected to the usage of the Young inequality only.
\begin{lemma}\label{Estim_general_For_u^p_nablav^2qLemmaLog}
If $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with $m_1>2m_2-(m_3+l)$ or $m_1>2m_2-\beta$ whenever $\alpha \in \left(0, \frac{1}{n}\right]$, or
$m_1>2m_2 +1-(m_3+l)$ or $m_1>2m_2+1-\beta$ whenever $\alpha \in \left(\frac{1}{n},1\right)$, then there exist $p,q>1$ such that $(u,v,w)$ satisfies
a similar inequality as in \eqref{MainInequality}.
\begin{proof}
As in Lemma \ref{Estim_general_For_u^p_nablav^2qLemma}, in view of inequality \eqref{Young} and the properties of the logistic $h$ in \eqref{h}, relation \eqref{Estim_1_For_u^p} now becomes for some positive $c_{33}$
\begin{equation}\label{Estim_1_For_u^p1}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p & \leq (-p(p-1)+\epsilon_3) \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 +c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}}\\
&+ \left(\epsilon_1 + \tilde{\sigma} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}
+ \epsilon_2 \int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\\
&+ pk_+ \int_\Omega (u+1)^p - p \mu \int_\Omega (u+1)^{p-1}u^{\beta} + c_{33} \quad \text{ for all } t \in (0,T_{\rm{max}}).
\end{split}
\end{equation}
Applying the inequality $(A+B)^p \leq 2^{p-1} (A^p+B^p)$ with $A,B \geq 0$ and $p>1$ to the last integral in \eqref{Estim_1_For_u^p1}, implies that
$-u^{\beta} \leq -\frac{1}{2^{\beta-1}} (u+1)^{\beta}+1$; therefore
\begin{equation}\label{beta}
- p \mu \int_\Omega (u+1)^{p-1} u^{\beta} \leq -\frac{p \mu}{2^{\beta-1}} \int_\Omega (u+1)^{p-1+\beta} + p \mu \int_\Omega (u+1)^{p-1} \quad \text{on } (0,T_{\rm{max}}).
\end{equation}
Henceforth, by taking into account the Young inequality, we have that for $t \in (0,T_{\rm{max}})$
\begin{equation} \label{k}
pk_+ \int_\Omega (u+1)^p \leq \delta_1 \int_\Omega (u+1)^{p-1+\beta} + c_{34} \quad \text{and} \quad
p \mu \int_\Omega (u+1)^{p-1} \leq \delta_2 \int_\Omega (u+1)^{p-1+\beta} + c_{35},
\end{equation}
with $\delta_1, \delta_2 >0$ and some $c_{34}, c_{35} >0$.
\quad \textbf{Case $1$}: $\alpha \in \left(0, \frac{1}{n}\right]$ and $m_1>2m_2-(m_3+l)$ or $m_1>2m_2-\beta$.
For $m_1> 2m_2-(m_3+l)$ we refer to Lemma \ref{Estim_general_For_u^p_nablav^2qLemma} and we take in mind inequality \eqref{Young1}.
Conversely, when $m_1>2 m_2-\beta$, we have that (recall $s$ may be arbitrary large) $\frac{(p+2m_2-m_1-1)s}{s-2} < p-1+\beta$, and by means of the Young inequality estimate \eqref{Young1} can alternatively read
\begin{equation}\label{young4}
c_3 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)s}{s-2}} \leq \delta_3 \int_\Omega (u+1)^{p-1+\beta} + c_{36} \quad \text{on }\, (0,T_{\rm{max}}),
\end{equation}
with $\delta_3>0$ and positive $c_{36}$.
By inserting estimates \eqref{beta} and \eqref{k} into relation \eqref{Estim_1_For_u^p1}, as well as taking into account \eqref{Young1}
(or, alternatively to \eqref{Young1}, bound \eqref{young4}), for suitable $\hat{\epsilon}, \tilde{\epsilon}_2, \tilde{\delta} >0$ and some $c_{37}>0$ we arrive at
\begin{equation*}\label{ClaimUlog}
\begin{split}
\frac{d}{dt} \int_\Omega (u+1)^p &\leq \left(-\frac{4p(p-1)}{(m_1+p-1)^2} + \hat{\epsilon}\right) \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2
+ \left(\tilde{\epsilon_2} - \frac{p(p-1)\xi \gamma}{2^{p-1}(m_3+p-1)} \right) \int_\Omega (u+1)^{m_3+p+l-1}\\
& + \left(\tilde{\delta} - \frac{p \mu}{2^{\beta-1}} \right) \int_\Omega (u+1)^{p-1+\beta} + c_{37} \quad \text{ for all } t \in (0,T_{\rm{max}}),
\end{split}
\end{equation*}
where we used again relation \eqref{GradU}. We conclude reasoning exactly as in the second part of the proof of Lemma \ref{Estim_general_For_u^p_nablav^2qLemma} and by choosing suitable
$\hat{\epsilon}, \tilde{\epsilon}_2, \tilde{\delta}, \epsilon_8$.
\textbf{Case $2$}: $\alpha \in \left(\frac{1}{n},1\right)$ and $m_1>2m_2 +1-(m_3+l)$ or $m_1>2m_2+1-\beta$.
Accordingly to Remark \ref{RemarkOnS}, since now $s$ cannot increase arbitrarily, relations \eqref{Young1} and \eqref{young4} have to be differently discussed.
In particular, for some $\bar{c}_1>0$ we can estimate relation \eqref{Young} as follows:
\begin{equation*}
\begin{split}
&p(p-1)\chi \int_\Omega u (u+1)^{m_2+p-3} \nabla u \cdot \nabla v \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2
+ c_2 \int_\Omega (u+1)^{p+2m_2-m_1-1} |\nabla v|^2\\
& \leq \epsilon_3 \int_\Omega (u+1)^{p+m_1-3} |\nabla u|^2 + \epsilon_4 \int_\Omega |\nabla v|^{2(p+1)} + \bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}}
\quad \text{on }\, (0,T_{\rm{max}}).
\end{split}
\end{equation*}
Now, if $m_1>2m_2+1-(m_3+l)$, then some $p$ sufficiently large infers to $\frac{(p+2m_2-m_1-1)(p+1)}{p}< p+m_3+l-1$, so that for any positive $\bar{\epsilon}_1$
and some $\bar{c}_2>0$ we have
\[
\bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}} \leq \bar{\epsilon}_1 \int_\Omega (u+1)^{p+m_3+l-1} + \bar{c}_2 \quad \text{on } (0,T_{\rm{max}}).
\]
Conversely, and in a similar way, for $m_1>2m_2+1-\beta$ we have for any positive $\bar{\epsilon}_2$ and some $\bar{c}_3>0$
\[
\bar{c}_1 \int_\Omega (u+1)^{\frac{(p+2m_2-m_1-1)(p+1)}{p}} \leq \bar{\epsilon}_2 \int_\Omega (u+1)^{p-1+\beta} + \bar{c}_3 \quad \text{on } (0,T_{\rm{max}}).
\]
The remaining part of the proof follows as the previous case, by taking into account \cite[Lemma 5.3-Lemma 5.4]{FrassuCorViglialoro} for the term dealing with
$ \int_\Omega |\nabla v|^{2(p+1)}$.
\end{proof}
\end{lemma}
As a by-product of what now obtained we are in a position to conclude.
\subsection{Proof of Theorems \ref{MainTheorem} and \ref{MainTheorem1}}
\begin{proof}
Let $(u_0,v_0) \in (W^{1,\infty}(\Omega))^2$ with $u_0, v_0 \geq 0$ on $\bar{\Omega}$. For $f$ and $g$ as in \eqref{f} and, respectively, for $f$, $g$ as in \eqref{f} and
$h$ as in \eqref{h}, let $\alpha >0$ and let $m_1,m_2,m_3 \in \mathbb R} \def\N{\mathbb N$ comply with \ref{A1}, \ref{A2} and \ref{A3}, respectively, \ref{A4} and \ref{A5}. Then, we refer to Lemmas \ref{Estim_general_For_u^p_nablav^2qLemma} and \ref{Met_GN}, respectively, Lemma \ref{Estim_general_For_u^p_nablav^2qLemmaLog} and obtain for some
$C_1,C_2,C_3>0$
\begin{equation}\label{Estim_general_For_y_2}
y'(t) + C_1 \int_\Omega |\nabla (u+1)^{\frac{m_1+p-1}{2}}|^2 + C_2 \int_\Omega |\nabla |\nabla v|^q|^2 \leq C_3
\quad \text{ on } (0, T_{\rm{max}}).
\end{equation}
Successively, the Gagliardo--Nirenberg inequality again makes that some positive constants $c_{38}, c_{39}$ imply from the one hand
\begin{equation*}\label{G_N2}
\int_\Omega (u+1)^p \leq c_{38} \Big(\int_\Omega |\nabla (u+1)^\frac{m_1+p-1}{2}|^2\Big)^{\kappa_1}+ c_{38} \quad \text{for all } t \in (0,T_{\rm{max}}),
\end{equation*}
(as already done in \eqref{GN2}), and from the other
\begin{equation}\label{Estim_Nabla nabla v^q}
\int_\Omega \lvert \nabla v\rvert^{2q}=\lvert \lvert \lvert \nabla v\rvert^q\lvert \lvert_{L^2(\Omega)}^2
\leq c_{39} \lvert \lvert\nabla \lvert \nabla v \rvert^q\rvert \lvert_{L^2(\Omega)}^{2\kappa_2} \lvert \lvert\lvert \nabla v \rvert^q\lvert \lvert_{L^\frac{1}{q}(\Omega)}^{2(1-\kappa_2)} +c_{39} \lvert \lvert \lvert \nabla v \rvert^q\lvert \lvert^2_{L^\frac{1}{q}(\Omega)}\quad \textrm{on } (0,T_{\rm{max}}),
\end{equation}
with $\kappa_2$ already defined in Lemma \ref{LemmaCoefficientAiAndExponents}.
Subsequently, the $L^s$-bound of $\nabla v$ in \eqref{Cg} infers some $c_{40}>0$ such that
\begin{equation*}\label{Estim_Nabla nabla v^p^2}
\int_\Omega \lvert \nabla v\rvert^{2q}\leq c_{40} \Big(\int_\Omega \lvert \nabla \lvert \nabla v \rvert^q\rvert^2\Big)^{\kappa_2}+c_{40} \quad \text{for all } t \in (0,T_{\rm{max}}).
\end{equation*}
In the same flavour of \cite[Lemma 5.4]{FrassuCorViglialoro}, by using the estimates involving $\int_\Omega (u+1)^p$ and $\int_\Omega \rvert\nabla v\lvert^{2q}$,
relation \eqref{Estim_general_For_y_2} provides positive constants $c_{41}$ and $c_{42}$, and $\kappa=\min\{\frac{1}{\kappa_1},\frac{1}{\kappa_2}\}$ such that
\begin{equation*}\label{MainInitialProblemWithM}
\begin{cases}
y'(t)\leq c_{41}-c_{42} y^{\kappa}(t)\quad \textrm{for all } t \in (0,T_{\rm{max}}),\\
y(0)=\int_\Omega (u_0+1)^p+ \int_\Omega |\nabla v_0|^{2q}.
\end{cases}
\end{equation*}
Finally, ODE comparison principles imply $u \in L^\infty((0,T_{\rm{max}});L^p(\Omega))$, and the conclusion is a consequence of the boundedness criterion in
$\S$\ref{SectionLocalInTime}.
\end{proof}
\begin{remark}[On the validity of the theorems in \cite{FrassuCorViglialoro} and \cite{frassuviglialoro} for $\alpha \geq \frac{1}{2}+\frac{1}{n}$]\label{Alpha}
In the proofs of \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro}, it is seen that the $L^2$ uniform estimate of $\nabla v$ is used to control some integral on $\partial \Omega$ (and this allows us to avoid to restrict our study to convex domains), as well as to deal with the term
$\int_\Omega \rvert \nabla v \lvert^{2p}$ with the Gagliardo--Nirenberg inequality; for instance we are referring to \cite[(28) and (39)]{frassuviglialoro}, respectively.
Such finiteness of $\int_\Omega \rvert \nabla v \lvert^2$ is related to the values of $\alpha$ in these articles:
$\alpha \in \left(0, \frac{1}{2}+\frac{1}{n}\right)$ (see \cite[Lemma 4.1]{frassuviglialoro}). Apparently only $\nabla v \in L^\infty((0,T_{\rm{max}});L^1(\Omega))$ suffices to address these issues. Indeed, as far as the topological property of $\Omega$ is concerned, we can invoke \cite[(3.10) of Proposition 8]{YokotaEtAlNonCONVEX} with $s=1$; on the other hand, for the question tied to the employment of the Gagliardo--Nirenberg inequality, we may operate as done in \eqref{Estim_Nabla nabla v^q}. As a consequence, in view of Lemma \ref{LocalV}, we have that $\nabla v \in L^\infty((0,T_{\rm{max}});L^1(\Omega))$, so that \cite[Theorem 2.1]{FrassuCorViglialoro} and \cite[Theorems 2.1 and 2.2]{frassuviglialoro} hold true for any $\alpha \in (0,1)$.
\end{remark}
\subsubsection*{Acknowledgments}
SF and GV are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partially supported by the research project \emph{Evolutive and Stationary Partial Differential Equations with a Focus on Biomathematics}, funded by Fondazione di Sardegna (2019). GV is partially supported by MIUR (Italian Ministry of Education, University and Research) Prin 2017 \emph{Nonlinear Differential Problems via Variational, Topological and Set-valued Methods} (Grant Number: 2017AYM8XW). TL is partially supported by NNSF of P. R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), and NSF of Shandong Province (Grant No. ZR2016JL021).
\end{document} |
\begin{document}
\mainmatter
\title{Efficiency in Multi-objective Games}
\titlerunning{Efficiency in Multi-objective Games}
\author{Anisse Ismaili$^{1,2}$}
\authorrunning{Anisse Ismaili}
\institute{$^1$ Universit\'{e} Pierre et Marie Curie,
Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France\\
$^2$ Paris Dauphine University, Place du Mal de Lattre de Tassigny,
75775 Paris Cedex 16, France\\
\email{anisse.ismaili@lip6.fr}}
\maketitle
\vspace*{-0.5cm}
\begin{abstract}
In a multi-objective game, each agent individually evaluates each overall action-profile on multiple objectives. I generalize the price of anarchy to multi-objective games and provide a polynomial-time algorithm to assess it$^1$.\\
This work asserts that policies on tobacco promote a higher economic efficiency.
\end{abstract}
\vspace*{-0.8cm}
\section{Introduction}
Economic agents, for each individual decision, make a trade off between multiple objectives, like for instance: time, resources, goods, financial income, sustainability, happiness and life. This motivated the introduction of a super-class of games: multi-objective (MO) games \cite{blackwell1956analog,shapley1959equilibrium}. Each agent evaluates each overall action profile by a \emph{vector}. His individual preference is a \emph{partial} rationality modelled by the Pareto-dominance. It induces Pareto-Nash-equilibria (PN) as the overall selfish outcomes.
Furthermore, concerning economic models, such vectorial evaluations are a humble backtrack from the intrinsic and subjective theories of value, towards a non-theory of value where the evaluations are maintained vectorial, in order to enable partial rationalities and to avoid losses of information in the model.
In this more realistic (behaviourally less assumptive) framework, in order to avoid critical losses of information on the several objectives in the model, thoroughly computing efficiency is a tremendous necessity \cite{madeley1999big,sloan2004price}.
The literature on MO games is disparate and will be presented where relevant.
After the preliminaries below, Section \ref{sec:mopoa} generalizes the \textit{coordination ratio} (CR, better known as ``price of anarchy'') to MO games. Section \ref{sec:application} applies it to the efficiency of tobacco economy. Section \ref{sec:computation} provides algorithms\footnote{Appendix \ref{app:smooth} shows that ``smoothness'' analysis \cite{roughgarden2009intrinsic} cannot be applied to MO games.} to assess the MO-CR.
Let $N=\{1,\ldots,n\}$ denote the \textit{set of agents}.
Let $A^i$ denote each agent $i$'s \textit{action-set} (discrete, finite).
Each agent $i$ decides an \textit{action} $a^i\in A^i$.
Given a subset of agents $M\subseteq N$, let $A^M$ denote $\times_{i\in M} A^i$ and let $A=A^N$ denote the \textit{set of overall action-profiles}.
Let ${\mathcal{O}}=\{1,,\ldots,d\}$ denote the \textit{set of all the objectives}, with $d$ fixed.
Let $v^i: A\rightarrow \mathbb{R}p$ denote an agent $i$'s \textit{individual MO evaluation function}, which maps each overall action-profile $a=(a^1,\ldots,a^n)\in A$ to an MO evaluation $v^i(a)\in\mathbb{R}p$. Hence, agent $i$'s evaluation for objective $k$ is $v^i_k(a)\in\mathbb{R}_+$. Given an overall action-profile $a\in A$, $a^M$ is the restriction of $a$ to $A^M$, and $a^{-i}$ to $A^{N\setminus\{i\}}$.
\begin{definition}\quad A Multi-objective Game (MOG) is a tuple $\left(N, \{A^i\}_{i\in N}, {\mathcal{O}}, \{v^i\}_{i\in N}\right)$.
\end{definition}
For instance, MO games encompass single-objective (discrete) optimization problems, MO optimization problems and non-cooperative games.
Assuming $\alpha=|A^i|\in\mathbb{N}$ for each agent, the representation of an MOG requires $n\alpha^n$ $d$-dimensional vectors.
Let us now supply the vectors with a preference relation. Assuming a \textit{maximization} setting, given $x,y\in\mathbb{R}p$, the following relations state respectively that $y$ (\ref{eq:wpp}) weakly-Pareto-dominates and (\ref{eq:pp}) Pareto-dominates $x$:
\begin{eqnarray}
y\succsim x &\hspace{0.5cm}\Leftrightarrow\hspace{0.5cm}& \forall k\in{\mathcal{O}},~~ y_k\geq x_k
\label{eq:wpp}\\
y\succ x &\Leftrightarrow & \forall k\in{\mathcal{O}},~~ y_k\geq x_k\text{~~and~~}\exists k\in{\mathcal{O}},~~ y_k> x_k
\label{eq:pp}
\end{eqnarray}
The Pareto-dominance is a \emph{partial} order, inducing a multiplicity of Pareto-efficient outcomes. Formally, the set of efficient vectors is defined as follows:
\begin{definition}[Pareto-efficiency]
For $Y\subseteq\mathbb{R}p$, the efficient vectors $\text{EFF}[Y]\subseteq Y$ are:
$$\text{EFF}[Y]=\{y^\ast\in Y~~|~~ \forall y\in Y, \mbox{~not~} (y\succ y^\ast)\}$$
\end{definition}
(Similarly, let $\text{WST}[Y]=\{y^{-}\in Y\mbox{~s.t.~} \forall y\in Y, \mbox{~not~} (y^{-}\succ y)\}$ denote the subset of worst vectors.)
Pareto-efficiency enables to define as efficient all the trade-offs that cannot be improved on one objective without being downgraded on another one, that is: the best compromises between objectives (see e.g. Figure \ref{fig:eff}).
At the individual scale, Pareto-efficiency defines a \textit{partial rationality}, enabling to model behaviours that single-objective (SO) games would not model consistently.
\begin{definition}[Pareto-Nash equilibrium \cite{shapley1959equilibrium}]\label{def:PE}
In an MOG, an action-profile $a\in A$ is a Pareto-Nash equilibrium (denoted by $a\in\text{PN}$), if and only if, for each agent $i\in N$:
$$v^i(a^i,a^{-i})\quad\in\quad\text{EFF}\left[\quad v^i(A^i,a^{-i})\quad\right]$$
where $v^i(A^i,a^{-i})$ denotes $\{v^i(b^i,a^{-i})\mid b^i\in A^i\}$.
\end{definition}
Pareto-Nash equilibria encompass most behaviourally possible action-profiles. For instance, whatever an agent's subjective linear positive weighted combination of the objectives, his decision is Pareto-efficient. One can distinguish behavioural objectives inducing $\text{PN}$ and also objectives on which to focus an efficiency study.
{\it Equilibrium existence.}
In many sound probabilistic settings \cite{daskalakis2011connectivity,dresher1970probability,rinott2000number}, Pareto efficiency is not demanding on the conditions of individual rationality, hence there are multiple Pareto-efficient responses. Consequently, pure PN are numerous in average: $|\text{PN}|\in\Theta(\alpha^{\frac{d-1}{d}n})$, justifying their existence in a probabilistic manner. Furthermore, in MO games with MO potentials \cite{monderer1996potential,patrone2007multicriteria,rosenthal1973class}, the existence is guaranteed.
\begin{example}[A didactic toy-example in Ocean Shores]
Five shops in Ocean Shores (the nodes) can decide upon two activities: renting bikes or buggies, selling clams or fruits, etc. Each agent evaluates his local action-profile depending on the actions of his inner-neighbours and according to two objectives: financial revenue and sustainability.\\
\hspace*{-31cm}\includegraphics[scale=0.8]{Ocean4.pdf}\\
For instance, we have $(b^1,b^2,a^3,b^4,b^5)\in\text{PN}$, since each of these individual actions, given the adversary local action profile (column), is Pareto-efficient among the two actions of the agent (row). Even if the relative values of the objectives cannot be certainly ascertained, all the subjectively efficient vectors are encompassed by the individual Pareto-efficiency. In this MO game, there are $13$ Pareto-Nash-equilibria, which utilitarian evaluations are depicted in Figure \ref{fig:eff} (Section \ref{sec:mopoa}).
\label{ex:Ocean}
\end{example}
\section{The Multi-objective Coordination Ratio}
\label{sec:mopoa}
It is well known in game theory that an equilibrium can be overall inefficient with regard to the sum of the individual evaluations. This loss of efficiency is measured by the {\it coordination ratio}\footnote{As Smoothness \cite{roughgarden2009intrinsic} cannot be applied to MO games, I cannot use the term \textit{Price of Anarchy}.} (CR) \cite{aland2006exact,awerbuch2005price,christodoulou2005price,guo2005price,koutsoupias1999worst,roughgarden2009intrinsic,roughgarden2007introduction} $\min[u(PE)]/\max[u(A)]$. Regrettably, when focusing on one sole objective (e.g. making money or a higher GDP), there are losses of efficiency that are not measured (e.g. non-sustainability of productions or production of addictive carcinogens). This appeals for a more thorough analysis of the loss of efficiency at equilibrium and the definition of a {\it multi-objective} coordination ratio.
The utilitarian social welfare $u:A\rightarrow\mathbb{R}p$ is a vector-valued function measuring social welfare with respect to the $d$ objectives: $u(a)=\sum_{i\in N} v^i(a)$, excluding the purely behavioural objectives that cause irrationality \cite{sloan2004price}.
Given a function $f:A\rightarrow Z$, the \textit{image set} $f(E)$ of a subset $E\subseteq A$ is defined by $f(E)=\{f(a)|a\in E\}\subseteq Z$. Given $\rho,y,z\in\mathbb{R}p$, the vector $\rho\star y\in\mathbb{R}p$ is defined by $\forall k\in{\mathcal{O}}, (\rho\star y)_k=\rho_k y_k$ and the vector $y / z\in\mathbb{R}p$ is defined by $\forall k\in{\mathcal{O}}, (y / z)_k=y_k / z_k$.
For $x\in\mathbb{R}p$, $x\star Y$ denotes $\{x\star y\in\mathbb{R}p ~~|~~ y\in Y\}$.
Given $x\in\mathbb{R}p$, $\mathcal{C}(x)$ denotes $\{y\in\mathbb{R}p~|~x\succsim y\}$.
I also introduce\footnote{To enable ratios, one can do the minor assumption $\mathcal{F}\subseteq\mathbb{R}pp$.} the notations $\mathcal{E}$ and $\mathcal{F}$, illustrated in Figures \ref{fig:eff} and \ref{fig:mopoa}:
\noindent
\begin{minipage}{0.58\columnwidth}
\noindent
\begin{itemize}
\item \textcolor{darkblue}{$\mathcal{A}=u(A)$} the set of \textit{outcomes}. $\textcolor{darkblue}{(\bullet)}$
\item \textcolor{darkgreen}{$\mathcal{E}=u(\text{PN})$} the \textit{equilibria outcomes}. $\textcolor{darkgreen}{(\blacklozenge)}$
\item \textcolor{darkred}{$\mathcal{F}=\text{EFF}[u(A)]$} the \textit{efficient outcomes}. $\textcolor{darkred}{(\times)}$
\end{itemize}
For SO games, the worst-case efficiency of equilibria is measured by the CR $\min[u(PE)]/\max[u(A)]$. However, for MO games, there are many equilibria and optima, and a ratio of the (green) \textit{set} \textcolor{darkgreen}{$\mathcal{E}$} over the (red) \textit{set} \textcolor{darkred}{$\mathcal{F}$} is not defined yet and ought to maintain the information on each objective without introducing dictatorial choices.
\end{minipage}
~~~
\begin{minipage}{0.38\columnwidth}
\centering
\includegraphics[scale=0.36]{utilitarian_ocean4.png}
\captionof{figure}{The bi-objective utilitarian vectors of Ocean Shores}
\label{fig:eff}
\end{minipage}\\[0.5ex]
I introduce a multi-objective CR.
Firstly, the efficiency of one equilibrium $y\in\mathcal{E}$ is quantified without taking side with any efficient outcome, by defining with flexibility and no dictatorship, a \textit{disjunctive set} of guaranteed ratios of efficiency $R[y,\mathcal{F}]=\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$.
Secondly, in MOGs, in average, there are many Pareto-Nash-equilibria. An efficiency \textit{guarantee} $\rho\in\mathbb{R}p$, must hold \textit{for each} equilibrium-outcome, inducing the conjunctive definition of the set of guaranteed ratios $R[\mathcal{E},\mathcal{F}]=\bigcap_{y\in\mathcal{E}} R[y,\mathcal{F}]$.
Technically, $R[\mathcal{E},\mathcal{F}]$ only depends on $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$.
Finally, if two bounds on the efficiency $\rho$ and $\rho'$ are such that $\rho\succ\rho'$, then $\rho'$ brings no more information, hence, MO-CR is defined by using $\text{EFF}$ on the guaranteed efficiency ratios $R[\text{WST}[\mathcal{E}],\mathcal{F}]$.
This MO-CR satisfies a set of key properties detailed in Appendix \ref{app:axioms}.
\begin{definition}[MO Coordination Ratio]\label{def:mopoa}
Given an MOG, a vector $\rho\in\mathbb{R}p$ bounds the MOG's inefficiency if and only if it holds that:\quad
$\forall y\in\mathcal{E},\quad \exists z\in\mathcal{F},\quad y/z\succsim \rho$.
Consequently, the set of guaranteed ratios is defined by:
$$R[\mathcal{E},\mathcal{F}]\quad=\quad\bigcap_{y\in\mathcal{E}}\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$$
and the MO-CR is defined by:\quad
$
\text{MO-CR}[\mathcal{E},\mathcal{F}]=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]
$
\end{definition}
\begin{example}[The Efficiency ratios of Example \ref{ex:Ocean}]\label{ex:mopoashores}
I depict the efficiency ratios of Ocean Shores (intersected with $[0,1]^d$) which depend of $\text{WST}[\mathcal{E}]=\{(30,53), (40,38)\}$ and $\mathcal{F}=\{(46,61),\ldots, (69,31)\}$.
The part below the red line corresponds to $R[(30,53),\mathcal{F}]$, the part below the blue line to $R[(40,38),\mathcal{F}]$ and the yellow part below both lines is the conjunction on both equilibria $R[\text{WST}[\mathcal{E}],\mathcal{F}]$.
The freedom degree of deciding what
\vspace*{0.1cm}
\noindent
\begin{minipage}{0.55\columnwidth}
the overall efficiency should be is left free (no dictatorship) which results in several ratios in the MO-CR.
Firstly, for each $\rho\in R[\mathcal{E},\mathcal{F}]$, we have $\rho_1\leq 65\%$. Hence, whatever the choices of overall efficiency, one cannot guarantee more than \textit{65\% of efficiency on objective 1}.
Secondly, there are some subjectivities for which the efficiency on objective 2 is already total (100\%, if not more) while situation on objective 1 is worse and only 50\% can be obtained.
Thirdly, from 50\% to 65\% of subjective efficiency on objective 1, the various subjectivities range the efficiency on objective 2 from 100\% to 75\%.
\end{minipage}
~~
\begin{minipage}{0.43\columnwidth}
\includegraphics[scale=0.44]{yellow.pdf}
\captionof{figure}{The MO-CR of Ocean Shores}
\end{minipage}
\end{example}
\vspace*{1cm}
\noindent
\begin{minipage}{0.46\columnwidth}
\centering
\includegraphics[scale=0.8]{MOCR.pdf}
\end{minipage}
~~~
\begin{minipage}{0.5\columnwidth}
Having $\rho$ in $\text{MO-CR}$ means that for each $y\in\mathcal{E}$, there is an efficient outcome $z^{(y)}\in\mathcal{F}$ such that $y$ dominates $\rho\star z^{(y)}$. In other words, if $\rho\in R[\mathcal{E},\mathcal{F}]$, then each equilibrium satisfies the ratio of efficiency $\rho$. This means that equilibria-outcomes are at least as good as $\rho\star\mathcal{F}$. That is: $\mathcal{E}\subseteq(\rho\star\mathcal{F})+\mathbb{R}p$. Moreover, since $\rho$ is tight, $\mathcal{E}$ sticks to $\rho\star\mathcal{F}$.
\end{minipage}
\captionof{figure}{$\rho\in\text{MO-CR}$ bounds below $\mathcal{E}$'s inefficiency:\quad $\mathcal{E}~~\subseteq~~(\rho\star\mathcal{F})+\mathbb{R}p$}
\label{fig:mopoa}~\\
\section{Application to Tobacco Economy}\label{sec:application}
Tobacco consumption is a striking example of economic inefficiency induced by bounded rationalities.
According to the World Health Organisation \cite{world2011report}, 17.000 humans die each day of smoking related diseases (one person per 5 seconds). Meanwhile, addictive satisfaction and the financial revenue of the tobacco industry fosters consumption and production.
According to the subjective theory of value \cite{walras1896elements}, some economists would say: ``Since consumers value the product, then the industry creates value.'' According to other health economists \cite{sloan2004price}, most consumers become addict before age 18, and as adults, would prefer a healthier life, but fail to opt-out.
\hspace*{0.5cm}The theory of MO games, based on a non-theory of value, just maintains vectorial evaluations and properly considers dollars, addiction and life expectancy as distinct objectives, with PN equilibria encompassing the relevant behaviours, even irrational.
We \textbf{modelled} the tobacco industry and its consumers \cite{globalissuestobacco,madeley1999big} by a succinct MOG, with the help of (..) the association\footnote{I am grateful to Cl\'{e}mence Cagnat-Lardeau for her help on modelling tobacco economy.} ``\textit{Alliance contre le tabac}''.
The set of agents is $N=\{\text{industry}, \nu \text{ consumers}\}$, where there are about $\nu=6.10^9$ prospective consumers.
Each consumer decides in $A^{\text{consumer}}=\{\text{not-smoking},\text{smoking}\}$ and cares about money, his addictive pleasure, and living. The industry only cares about money and decides in $A^{\text{industry}}=\{\text{not-active},\text{active},\text{advertise\&active}\}$. We have ${\mathcal{O}}=\{\text{money},\text{reward},\text{life-expectancy}\}$.
The tables below depict the evaluation vectors (over a life-time and ordered as in ${\mathcal{O}}$) of one prospective consumer and the evaluations of the industry with respect to the number $\theta\in\{0,\ldots,\nu\}$ of consumers who decide to smoke. The money budget (already an aggregation) is expressed in kilo-dollars\footnote{Note that most states set the prices of tobacco, hence prices do not follow supply/demand.}\footnote{These numbers differ from \cite{sloan2004price} which aggregates everything (e.g. life expectancy) into money.}; the addictive reward is on an ordinal scale $\{1,2,3,4\}$; life-expectancy is in years.
\begin{center}
\begin{tabular}{r|ccc}
$v^{\text{consumer}}$ & not-active & active & advertise\&active\\
\hline
not-smoking &
$(48,1,75)$ & $(48,1,75)$ & $(48,1,75)$\\
smoking &
$(48,1,75)$ & $(12,3,65)$ & $(0,4,55)$\\
\multicolumn{4}{c}{}\\
$v^{\text{industry}}(\theta)$ & not-active & active & advertise\&active \\
\hline
$(\nu-\theta)\times$ &
$(0,-,-)$ & $(0,-,-)$ & $(0,-,-)$ \\
$+$ \hspace*{0.8cm}$\theta~~\times$ &
$(0,-,-)$ & $(26,-,-)$ & $(36,-,-)$ \\
\end{tabular}
\end{center}
\textbf{Pareto-Nash equilibria.} If the industry is active, then for the consumer, deciding to smoke or not depends on how the consumer subjectively values/weighs money, addiction and life expectancy: both decisions are encompassed by Pareto-efficiency. For the industry, advertise\&active is a dominant strategy. Consequently, Pareto-Nash-equilibria are all the action-profiles in which the industry decides advertise\&active.
\textbf{Efficiency.}
Since addiction is irrational (detailed in Appendix \ref{app:tabac}), I focus on money and life-expectancy.
We have
$
\mathcal{E}
=
\{\theta(36,55)+(\nu-\theta)(48,75) \mid 0\leq\theta\leq \nu\}
$
and
$\mathcal{F}=\{\nu(48,75)\}$,
where $\nu$ is the world's population, and $\theta$ the number of smokers. Since $\text{WST}[\mathcal{E}]=\{(36,55)\}$, the MO-CR is the singleton $\{(75\%,73\%)\}$: in the worst case, we lose 12k\$ and 20 years of life-expectancy per-consumer. These Pareto-Nash-equilibria are the worst action-profiles for money and life-expectancy, a critical information that was not lost by this MOG and its MO-CR.
\textbf{Practical lessons.} Advertising tobacco fosters consumption. The association ``\textit{Alliance contre le tabac}'' passed a law for standardized neutral packets (April 3rd 2015), in order to annihilate all the benefits of branding, but only in France. The model indicates that:
\begin{center}
\emph{This law will promote a higher economic efficiency}.
\end{center}
\section{Computation of the MO-CR}
\label{sec:computation}
In this section, I provide a polynomial-time algorithm for the computation of MO-CR which relies on a very general procedure based on two phases:
\begin{enumerate}
\item Given a MOG, compute the worst equilibria $\text{WST}[\mathcal{E}]$ and the efficient outcomes $\mathcal{F}$.
\item Given $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$, compute $\text{MO-CR}=\text{EFF}[~R[~\text{WST}[\mathcal{E}]~,~\mathcal{F}~]~]$.
\end{enumerate}
Depending on the input (normal form or compact representation), it adapts as follows.
\subsection{Computation of the MO-CR for Multi-objective Normal Forms}
For a MOG given in MO normal form (which representation length is $L=n\alpha^n d$), Phase 1 (computing $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$) is easy and takes time $O(L^2)$ (see Appendix \ref{app:proof}). For $d=2$, this lowers to $O(L\log_2(L))$. Let us denote the sizes of the outputs $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$. For normal forms, it holds that $q,m=O(|\mathcal{A}|)=O(L)$.
\hspace*{0.5cm}For Phase 2, at first glance, the development of the intersection of unions $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ causes an exponential $m^q$. But fortunately, one can compute the $\text{MO-CR}$ in polynomial time. Below, $D^t$ is a set of vectors. Given two vectors $x,y\in\mathbb{R}p$,
let $x\wedge y$ denote the vector defined by $\forall k\in{\mathcal{O}},~(x\wedge y)_k=\min\{x_k,y_k\}$ and recall that $\forall k\in{\mathcal{O}},~(x/y)_k=x_k/y_k$. Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} is the development of $\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$, on a set-algebra of cone-unions.
Appendix \ref{app:proof} shows that Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes time $O((qm)^{2d-1}d)$, or $O((qm)^{2}\log_2(qm))$ for $d=2$.
\vspace*{-0.4cm}
\restylealgo{ruled}
\begin{algorithm}[!h]
\KwIn{$\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$}
\KwOut{$\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$}
\ \\ [-1.5ex]
{\bf create } $D^1\leftarrow \{y^1/z\in\mathbb{R}p~|~z\in\mathcal{F}\}$\\
\For{$t=2,\ldots,q$}{
$D^t\leftarrow\text{EFF}[\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}]$
}
{\bf return } $D^q$
\caption{Computing MO-CR in polynomial-time $O((qm)^{2d-1}d)$}
\label{alg:givenEFoutputMOPoA:polynomial}
\end{algorithm}
\vspace*{-0.4cm}
Having specified Phase 1 and 2 for normal forms, Theorem 1 follows:
\begin{theorem}[Computation of MO-CR]\label{th:master}
Given a MO normal form, one can compute the MO-CR in polynomial time $O(L^{4d-2})$.\quad If $d=2$, it lowers to $O(L^4\log_2(L))$.
\end{theorem}
\subsection{Computation of the MO-CR for Multi-objective Compact Representations}
Compact representations of massively multi-agent games (e.g. MO graphical games, MO action-graph games) have a representation length $L$ that is polynomial with respect to the number of agents $n$ and the sizes of the action-sets $\alpha$. As $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$ can be exponentials $\alpha^n$ of this representation length, compact representations are algorithmically more challenging, leaving open the computation of $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$ in Phase 1, and complicating the use of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} in Phase 2.
To overcome this, one can do MO approximations \cite{papadimitriou2000approximability}, by implementing an approximate Phase 1 which precision transfers to Phase 2 in polynomial time, as follows.
\begin{lemma}\label{lem:approx} Given $\varepsilon_1,\varepsilon_2>0$ and approximations $E$ of $\mathcal{E}$ and $F$ of $\mathcal{F}$ in the sense that:
\begin{eqnarray}
\forall y\in\mathcal{E}, \exists y'\in E,\quad y\succsim y' &
\quad \text{and}\quad &
\forall y'\in E, \exists y\in\mathcal{E},\quad (1+\varepsilon_1)y'\succsim y\label{eq:approx:E}\\
\forall z'\in F, \exists z\in\mathcal{F},\quad z'\succsim z &
\quad \text{and}\quad &
\forall z\in\mathcal{F}, \exists z'\in F,\quad (1+\varepsilon_2)z\succsim z'\label{eq:approx:F}
\end{eqnarray}
it holds that $R[E,F]\subseteq R[\mathcal{E},\mathcal{F}]$ and:
\begin{eqnarray}
\forall \rho\in R[\mathcal{E},\mathcal{F}], \exists \rho'\in R[E,F],\quad (1+\varepsilon_1)(1+\varepsilon_2)\rho'\succsim\rho\label{eq:approx:R}
\end{eqnarray}
\end{lemma}
Equations (\ref{eq:approx:E}) and (\ref{eq:approx:F}) state approximation bounds. Equations (\ref{eq:approx:E}) state that $(1+\varepsilon_1)^{-1}\mathcal{E}$ bounds below $E$ which bounds below $\mathcal{E}$. Equations (\ref{eq:approx:F}) state that $\mathcal{F}$ bounds below $F$ which bounds below $(1+\varepsilon_2)\mathcal{F}$. Crucially, whatever the sizes of $\mathcal{E}$ and $\mathcal{F}$, there exist such approximations $E$ and $F$ that are $O((1/\varepsilon_1)^{d-1})$ and $O((1/\varepsilon_2)^{d-1})$ sized \cite{papadimitriou2000approximability}, yielding the approximation scheme below.
\begin{theorem}[Approximation Scheme for MO-CR]\label{th:approx}
Given a compact MOG of representation length $L$, precisions $\varepsilon_1,\varepsilon_2>0$ and two algorithms to compute approximations $E$ of $\mathcal{E}$ and $F$ of $\mathcal{F}$ in the sense of Equations (\ref{eq:approx:E}) and (\ref{eq:approx:F}) that take time $\theta_{\mathcal{E}}(\varepsilon_1,L)$ and $\theta_{\mathcal{F}}(\varepsilon_2,L)$, one can approximate $R[\mathcal{E},\mathcal{F}]$ in the sense of Equation (\ref{eq:approx:R}) in time:
$$O\left(\theta_{\mathcal{E}}(\varepsilon_1,L) \quad+\quad \theta_{\mathcal{F}}(\varepsilon_2,L) \quad+\quad {(\varepsilon_1 \varepsilon_2)^{-(d-1)(2d-1)}}\right)$$
\end{theorem}
For MO graphical games, Phase 1 could be instantiated with approximate junction-tree algorithms on MO graphical models \cite{dubus2009multiobjective}. For MO symmetric action-graph games, in the same fashion, one could generalize existing algorithms \cite{jiang2007computing}. More generally, for $\text{WST}[\mathcal{E}]$ and $\mathcal{F}$, one can also use meta-heuristics with experimental guarantees.
\section{Prospects}
Multi-objective games can be used as a behaviourally more realistic framework to model a wide set of games occurring in business situations ranging from carpooling websites to combinatorial auctions.
Also, studying the efficiency of MO generalizations of routing or Cournot-competitions \cite{guo2005price} could provide realistic economic insights.
\appendix
\section{Why Smoothness will not work on multi-objective games}\label{app:smooth}
Most single-objective price of anarchy analytic results rely on a smoothness-analysis \cite{roughgarden2009intrinsic}. A crucial step for ``smoothness'' is to sum the best response inequalities: For a single-objective game and an equilibrium $a\in\text{PN}$, from the best-response conditions $\forall i\in N, \forall b^i\in A^i, v^i(a)\geq v^i(b^i,a^{-i})$, one has:
$\forall b\in A, \sum_{i=1}^{n} v^i(a)\geq\sum_{i=1}^{n} v^i(b^i,a^{-i})$.
However, the Pareto-Nash-equilibrium conditions are rather:
$\forall i\in N, \forall b^i\in A^i, v^i(b^i,a^{-i}) \not\succ v^i(a)$.
As shown in the following counter-example, such $\not\succ$ relations cannot be summed:
$$
\left(\begin{array}{c} 2\\4 \end{array}\right)
\not\succ
\left(\begin{array}{c} 3\\1 \end{array}\right)
\text{ and }
\left(\begin{array}{c} 3\\1 \end{array}\right)
\not\succ
\left(\begin{array}{c} 1\\2 \end{array}\right)
~~~~~~~~\text{ but }~~~~~~~~
\left(\begin{array}{c} 2\\4 \end{array}\right)
+
\left(\begin{array}{c} 3\\1 \end{array}\right)
\succ
\left(\begin{array}{c} 3\\1 \end{array}\right)
+
\left(\begin{array}{c} 1\\2 \end{array}\right)
$$
Consequently, Smoothness-analysis does not encompass Pareto-Nash equilibria, regardless of the efficiency measurement chosen.
\section{Properties of the Multi-objective Coordination Ratio}\label{app:axioms}
The Multi-objective Coordination Ratio fulfils a list of key good properties for the thorough measurement of the multi-objective efficiency of MO games.
\subsection{Worst case guarantee on equilibria outcomes\\ and No dictatorship on efficient outcomes}
Each vectorial efficiency ratio that the MO-CR states, bounds below the efficiency for each equilibrium outcome, compared to an existing efficient outcome:
$$
\forall \rho\in\text{MO-CR}[\mathcal{E},\mathcal{F}],\quad
\forall y\in\mathcal{E},\quad
\exists z\in\mathcal{F},\quad
y/z\succsim\rho
$$
The process of measuring efficiency by MO-CR does not imply any choice in $\mathcal{F}$ that would impose a point-of-view telling what efficiency should be (e.g. no five-year plans).
\subsection{Multi-objective ratio-scale}
Given $\mathcal{E}$, $\mathcal{F}$ and $r\in\mathbb{R}pp$, it holds that:
\begin{eqnarray}
\text{MO-CR}[\mathcal{E},\mathcal{F}] &\quad\subseteq\quad & \mathbb{R}p\label{eq:ratio:6}\\
\text{MO-CR}[\{(0,\ldots,0)\},\mathcal{F}] & \quad=\quad & \{(0,\ldots,0)\}\label{eq:ratio:7}\\
\text{MO-CR}[r\star\mathcal{E},\mathcal{F}] & \quad=\quad & r\star\text{MO-CR}[\mathcal{E},\mathcal{F}]\label{eq:ratio:8}\\
\text{MO-CR}[\mathcal{E},r\star\mathcal{F}] & \quad=\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}]/r\label{eq:ratio:9}\\
\mathcal{E}\subseteq\mathcal{F} & \quad\Leftrightarrow\quad & (1,\ldots,1)\in \text{MO-CR}[\mathcal{E},\mathcal{F}]\label{eq:ratio:10}
\end{eqnarray}
Equation (\ref{eq:ratio:6}) states that the MO-CR is expressed in the multi-objective space. It is worth noting that while $\text{MO-CR}[\mathcal{E},\mathcal{F}]\subseteq[0,1]^d$ is a more classical choice, MO-CR also allows for measurements of over-efficiencies. (E.g. if $\mathcal{F}$ is a family-car and $\mathcal{E}$ is a Lamborghini, then there is over-efficiency on the speed objective.)
\hspace*{0.5cm} Equations (\ref{eq:ratio:7}), (\ref{eq:ratio:8}) and (\ref{eq:ratio:9}) state that MO-CR is sensitive on each objective to multiplications of the outcomes. For instance, if $\mathcal{E}$ is three times better on objective $k$, then so is MO-CR. If there are twice better opportunities of efficiency in $\mathcal{F}$ on objective $k'$, then MO-CR is one half on objective $k'$. In other words, the efficiency of each objective independently reflects into the MO-CR in a ratio-scale.
\hspace*{0.5cm} If all equilibria outcomes are efficient (i.e. $\mathcal{E}\subseteq\mathcal{F}$), then this must imply that according to the MO-CR, the MO game is fully efficient, that is: $(1,\ldots,1)\in \text{MO-CR}[\mathcal{E},\mathcal{F}]$. The MO-CR seems to be the only multi-objective ratio-scale measurement that fulfils Equation (\ref{eq:ratio:10}) while being a worst case guarantee on equilibria outcomes with no dictatorship on what efficiency should be.
\hspace*{0.5cm} It is also worth noting that MO-CR is MO-monotonic with respect to $\mathcal{E}$ and $\mathcal{F}$. For $X,Y\subseteq\mathbb{R}p$, let $X\unrhd Y$ denote $Y\subseteq\mathcal{C}(X)$ where $\mathcal{C}(X)=\cup_{x\in X}\mathcal{C}(x)$ (i.e. $X$ dominates $Y$). Then it holds that:
\begin{eqnarray}
\mathcal{E}\unrhd\mathcal{E}' &\quad\Rightarrow\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}]\unrhd\text{MO-CR}[\mathcal{E}',\mathcal{F}]\\
\mathcal{F}\unrhd\mathcal{F}' &\quad\Rightarrow\quad & \text{MO-CR}[\mathcal{E},\mathcal{F}']\unrhd\text{MO-CR}[\mathcal{E},\mathcal{F}]
\end{eqnarray}
\section{Bounded Rationality in Tobacco Consumption}\label{app:tabac}
According to the \textit{intrinsic theory of value} \cite{adam1776inquiry}, the value of a cigarette objectively amounts to the quantities of raw materials used for its production, or is the combination of the labour times put into it \cite{marx1867kapital}. However, each economic agent needs to keep the freedom to evaluate and act how he pleases, in order to keep his good will and some economic efficiency, as observed in the end of the Soviet Union.
According to the \textit{subjective theory of value} \cite{adam1776inquiry}, the value of a cigarette amounts to the price an agent is willing to pay for it. Since the consumers value the product, then the industry creates value \cite{walras1896elements}.
However, this disregards what the disastrous consequence is on life expectancy, belittles 7.500.000 deaths-per-year and emphasizes the bounded rationality of behaviours.
While for some health economists, consuming a cigarette is a rational choice, as one values pleasure more than life expectancy, for others, consumers are stuck into addiction before becoming adults. The truth is likely between these two extreme points of view \cite{sloan2004price}: Economic agents discount the future at a rate of 6\% per-year, hence a day of life in 40 years is valued 10 times less than now, leading to overweighting the actual smoking pleasure and to irrational behaviours with respect to preferences over a full lifetime. Agents behave according to objectives (e.g. addictive satisfaction) that they would avoid if they had the full experience of their lifetime (e.g. a lung cancer with probability $1/2$) and a sufficient will (e.g. quit smoking). Time discounting also explains other non-sustainable behaviours like over-fishing catastrophes.
\section{Proofs}\label{app:proof}
\subsection{Phase 1 for Normal Forms, Correctness of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} and Theorem \ref{th:master}}
\label{sec:mopoa:poly}
Phase 1 is easy, if the MOG is given in normal form. The MOG is made of the MO evaluations of each agent on each action-profile, that is: $O(n\alpha^n)$ vectors. Hence, the computation of $u(A)$ requires
for each $a\in A$, the addition of $n$ vectors. Therefore, the computation of $u(A)$ takes time $O(\alpha^n n d)$ (linear in the size of the input) and yields $O(\alpha^n)$ vectors. The computation of $\mathcal{F}=\text{EFF}[u(A)]$ given $u(A)$ takes time $O(|u(A)|^2 d)=O(\alpha^{2n} d)$. To conclude, the computation of $\mathcal{F}$ takes time $O(n \alpha^n d + \alpha^{2n} d)$, which is polynomial (quadratic) in the size of the input. If $d=2$, this can be significantly lowered to $O(n \alpha^n d + \alpha^{n}\log_2(\alpha^n)d)=O(n\alpha^n \log_2(\alpha) )$.
\hspace*{0.7cm}The computation of $\text{WST}[\mathcal{E}]$ can be achieved by first computing $\text{PN}$. For this purpose, for each agent $i\in N$ and each adversary action profile $a^{-i}\in A^{-i}$, one has to compute which individual actions give a Pareto-efficient evaluation in $v^i(A^i,a^{-i})$, in order to mark which action-profiles can be a $\text{PN}$ from $i$'s point of view. Overall, computing $\text{PN}$ takes time $O(n \alpha^{n-1} \alpha^2 d)$. Using back $u(A)$, computing $\mathcal{E}=u(\text{PN})$ is straightforward. Again, the computation of $\text{WST}[\mathcal{E}]$ given $\mathcal{E}$ takes time $O(|\mathcal{E}|^2 d)=O(\alpha^{2n} d)$. To sum up, the computation of $\text{WST}[\mathcal{E}]$ takes time $O(n \alpha^{n+1} d + \alpha^{2n} d)$. If $d=2$, this lowers to $O(n\alpha^n \log_2(\alpha))$.\\
\hspace*{0.7cm}In order to compute $\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$,
let us study the structure of $\bigcap_{y\in\text{WST}[\mathcal{E}]}\bigcup_{z\in\mathcal{F}}\mathcal{C}(y/z)$, by restricting a set-algebra to the following objects:
\begin{definition}[Cone-Union]
\label{def:coneunion}
For a set of vectors $X\subseteq\mathbb{R}p$, the Cone-Union $\mathcal{C}(X)$ is:
$$
\mathcal{C}(X)
~~=~~\bigcup_{x\in X}\mathcal{C}(x)
~~~~=\{y\in\mathbb{R}p ~~|~~ \exists x\in X, x\succsim y\}
$$
Let $\mathcal{C}$ denote the set of all cone-unions of $\mathbb{R}p$.
\end{definition}
To define an algebra on $\mathcal{C}$, one can supply $\mathcal{C}$ with $\cup$ and $\cap$.
\begin{lemma}[On the Set-Algebra $(\mathcal{C},\cup,\cap)$]\label{prop:algebra}~\\
Given two descriptions of cone-unions $X^1,X^2\subseteq\mathbb{R}p$, we have:
$$\mathcal{C}(X^1)\cup\mathcal{C}(X^2)=\mathcal{C}( X^1\cup X^2 )$$
Given two descriptions of cones $x^1,x^2\in\mathbb{R}p$, we have:
$$\mathcal{C}(x^1)\cap\mathcal{C}(x^2)=\mathcal{C}(x^1\wedge x^2)$$
where $x^1\wedge x^2\in\mathbb{R}p$ is: $\forall k\in{\mathcal{O}}, (x^1\wedge x^2)_k=\min\{x^1_k,x^2_k\}$.\\
Given two descriptions of cone-unions $X^1,X^2\subseteq\mathbb{R}p$, we have:
\begin{eqnarray*}
\mathcal{C}(X^1)\cap\mathcal{C}(X^2)
&=&\left(\cup_{x^1\in X^1}\mathcal{C}(x^1)\right)\cap\left(\cup_{x^2\in X^2}\mathcal{C}(x^2)\right)\\
&=&\bigcup_{(x^1,x^2)\in X^1\times X^2}\mathcal{C}(x^1)\cap\mathcal{C}(x^2)\\
&=&\bigcup_{(x^1,x^2)\in X^1\times X^2}\mathcal{C}(x^1\wedge x^2)\\
&=&\mathcal{C}( X^1\wedge X^2 )
\end{eqnarray*}
where $X^1\wedge X^2=\{x^1\wedge x^2~|~x^1\in X^1,~x^2\in X^2\}\subseteq\mathbb{R}p$.\\
Therefore, $(\mathcal{C},\cup,\cap)$ is stable, and then is a set-algebra.
\end{lemma}
\begin{proof}
The three properties derive from set calculus.
\end{proof}
The main consequence of Lemma \ref{prop:algebra} is that $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ is a cone-union. Moreover, one can do the development for $\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ within the cone-unions, using distributions and developments.
\begin{remark}\label{rk:app:cone}
For a finite set $X\subseteq\mathbb{R}p$, we have: $\mathcal{C}(X)=\mathcal{C}(\text{EFF}[X])$.
\end{remark}
\begin{proof}
Firstly, we prove $\mathcal{C}(X)\subseteq\mathcal{C}(\text{EFF}[X])$.
If $y\in\mathcal{C}(X)$, then there exists $x\in X$ such that $x\succsim y$. There are two cases, $x\in\text{EFF}[X]$ and $x\not\in\text{EFF}[X]$. If $x\in\text{EFF}[X]$, then $y\in\mathcal{C}(\text{EFF}[X])$, by definition of a cone-union.
Otherwise, if $x\not\in\text{EFF}[X]$, then there exists $z\in X$ such that $z\succ x$. And since $X$ is finite, we can find such a $z$ in $\text{EFF}[X]$, by iteratively taking $z'\succ z$ until $z\in\text{EFF}[X]$, which will happen because $X$ is finite and $\succ$ is transitive and irreflexive. Hence, there exists $z\in\text{EFF}[X]$ such that $z\succ x\succsim y$ and then $z\succsim y$. Consequently, $y\in\mathcal{C}(\text{EFF}[X])$, by definition of a cone-union.
Conversely, $Y\subseteq X\Rightarrow \mathcal{C}(Y)\subseteq\mathcal{C}(X)$ proves $\mathcal{C}(\text{EFF}[X])\subseteq\mathcal{C}(X)$.
\end{proof}
As a consequence of Remark \ref{rk:app:cone}, for $x\in\mathbb{R}p$, a simple cone $\mathcal{C}(x)$ is fully described by its summit $x$. The main consequence of this remark is that $\mathcal{C}(X)$ can be fully described and represented by $\text{EFF}[X]$.
For instance, since $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is a cone-union (thanks to Lemma \ref{prop:algebra}), and since $\text{MO-CR}=\text{EFF}[R[\text{WST}[\mathcal{E}],\mathcal{F}]]$ (by definition of the MO-CR), then $R[\text{WST}[\mathcal{E}],\mathcal{F}]$ is fully represented (as a cone-union) by the MO-CR, which means that $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$.\\
Recall that $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$. In this subsection, we also denote $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$.
Let $\mathcal{A}_q^m$ denote the set of functions $\pi$ from $\{1,\ldots,q\}$ to $\{1,\ldots,m\}$. (We have: $|\mathcal{A}_q^m|=m^q$.)
\begin{corollary}[The cone-union of MO-CR]~\\
Given $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$, we have:
$$R[\text{WST}[\mathcal{E}],\mathcal{F}]=\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$$
and therefore:
$$\text{MO-CR}=\text{EFF}\left[\left\{\bigwedge\nolimits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right]$$
\end{corollary}
\begin{proof}
For the first statement, just think to a development. We write down $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\cap_{y\in\text{WST}[\mathcal{E}]}\cup_{z\in\mathcal{F}}\mathcal{C}(y/z)$ into the layers just below. There is one layer per $y^t$ in $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^t,\ldots,y^q\}$:
$$
\begin{array}{ccccccccccl}
& ( & \mathcal{C}(\frac{y^1}{z^1}) & \cup & \mathcal{C}(\frac{y^1}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^1}{z^m})& )&\text{layer 1}\\
\bigcap & ( & \mathcal{C}(\frac{y^2}{z^1}) & \cup & \mathcal{C}(\frac{y^2}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^2}{z^m})& )&\text{layer 2}\\
&&&&&\vdots\\
\bigcap & ( & \mathcal{C}(\frac{y^q}{z^1}) & \cup & \mathcal{C}(\frac{y^q}{z^2}) & \cup & \ldots & \cup & \mathcal{C}(\frac{y^q}{z^m})& )&\text{layer q}
\end{array}
$$
Imagine the simple cones as vertices and imagine edges going from each vertex of layer $t$ to each vertex of the next layer $(t+1)$. The development into a union outputs as many intersection-terms as paths from the first layer to the last one.
Let the function $\pi:\{1,\ldots,q\}\rightarrow\{1,\ldots,m\}$ denote a path from layer $1$ to layer $q$, where $\pi(t)$ is the vertex chosen in layer $t$.
Consequently, in the result of the development into an union, each term is an intersection $\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})$.
The second statement results from the first statement, Lemma \ref{prop:algebra} and Remark \ref{rk:app:cone}.
\begin{eqnarray*}
R[\text{WST}[\mathcal{E}],\mathcal{F}]
&=&\bigcup_{\pi\in\mathcal{A}_q^m}\bigcap_{t=1}^{q} \mathcal{C}(y^t / z^{\pi(t)})\\
&=&\bigcup_{\pi\in\mathcal{A}_q^m} \mathcal{C}\left(\bigwedge_{t=1}^{q} y^t / z^{\pi(t)}\right)\\
&=&\mathcal{C}\left(\left\{\bigwedge\limits_{t=1}^{q}y^t / z^{\pi(t)}~~|~~\pi\in\mathcal{A}_q^m\right\}\right)
\end{eqnarray*}
That $R[\text{WST}[\mathcal{E}],\mathcal{F}]=\mathcal{C}(\text{MO-CR})$ concludes the proof.
\end{proof}
Ultimately, this proves the \textbf{correctness} of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} for the computation of MO-CR, given $\text{WST}[\mathcal{E}]=\{y^1,\ldots,y^q\}$ and $\mathcal{F}=\{z^1,\ldots,z^m\}$. It consists in the iterative development of the intersection $R(\mathcal{E},\mathcal{F})$, which can be seen as dynamic programming on the paths of the layer graph.
For $k\in\{1,\ldots,q\}$, we denote $D^t$ the description of the cone-union corresponding to the intersection:
$$\mathcal{C}(D^t)=\cap_{l=1}^{t} \cup_{z\in\mathcal{F}} \mathcal{C}(y^l/z)$$
Recursively, for $t>1$, $\mathcal{C}(D^t)=\mathcal{C}(D^{t-1})~\cap~(\cup_{z\in\mathcal{F}}~\mathcal{C}(y^{t}/z))$.
From Lemma \ref{prop:algebra} and Remark \ref{rk:app:cone}, in order to develop, we
then have to iterate the following:
$$
D^t=\text{EFF}[\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}]
$$
We now proceed with the \textbf{time complexity} of Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}. At first glance, since there are $m^q$ paths in the layer graph, then there are $O(m^q)$ elements in MO-CR. Fortunately, they are much less, because we have:
\begin{theorem}[MO-CR is polynomially-sized]~\\
\label{th:mopoa:poly}
Given a MOG and denoting $d=|{\mathcal{O}}|$, $q=|\text{WST}[\mathcal{E}]|$ and $m=|\mathcal{F}|$, we have:
$$|\text{MO-CR}|\leq (qm)^{d-1}$$
\end{theorem}
\begin{proof}
Given $\rho\in\text{MO-CR}$, for some $\pi\in\mathcal{A}_q^m$, we have $\rho=\bigwedge\nolimits_{t=1}^{q}y^t / z^{\pi(t)}$, and then $\forall k\in{\mathcal{O}}, \rho_k=\min_{t=1\ldots q}\{y^t_k / z^{\pi(t)}_k\}$. Therefore, $\rho_k$ is exactly realized by the $k$th component of at least one cone summit $y^t / z^{\pi(t)}$ in the layer graph (that is a vertex in the layer-graph above). Consequently, there are at most as many possible values for the $k$th component of $\rho$, as the number of vertices in the layer graph, that is $qm$. This holds for the $d$ components of $\rho$; hence there are at most $(qm)^d$ vectors in MO-CR. More precisely, by Lemma \ref{lem:eff} (below), since MO-CR is an efficient set, then there are at most $(qm)^{d-1}$ vectors in MO-CR.
\end{proof}
\begin{lemma}\label{lem:eff}
Let $Y\subseteq\mathbb{R}p$ be a set of vectors, with at most $M$ values on each component:
$$|~\text{EFF}[Y]~|\leq M^{d-1}$$
\end{lemma}
\begin{proof} At most $M^{d-1}$ valuations are realized on the $d-1$ first components. If you fix the $d-1$ first components, there is at most one Pareto-efficient vector which maximizes the last component.
\end{proof}
In Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}, there are $\Theta(q)$ steps. At each step $t$, from Theorem \ref{th:mopoa:poly}, we know that $|D^{t-1}|\leq (qm)^{d-1}$. Hence, $|\{\rho ~\wedge~ (y^t/z)~~|~~\rho\in D^{t-1},~~z\in\mathcal{F}\}|\leq q^{d-1} m^d$, and the computation of the efficient set $D^t$ requires time $O((q^{d-1} m^d)^2 d)$.
However, by using an insertion process, since there are at most $B=|D^{t}|\leq (qm)^{d-1}$ Pareto-efficient vectors at each insertion, then we only need $O(q^{d-1} m^d \times (qm)^{d-1})$ Pareto-comparisons. If $d=2$, time lowers to $O(q^{d-1} m^d\log_2(q m) d^2)=O(q m^2\log_2(q m))$.
Ultimately, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes $q$ steps and then time $O(q (q^{d-1} m^d)(qm)^{d-1} d)=O((qm)^{2d-1}d)$. If $d=2$, this lowers to $O((q m)^2\log_2(q m))$.
\subsection{Approximations: Proof of Lemma \ref{lem:approx} and Theorem \ref{th:approx}}
\begin{proof}
(1) First, let us show $R[E,F]\subseteq R[\text{WST}[\mathcal{E}],\mathcal{F}]$. Let $\rho'$ be a ratio of $R[E,F]$ and let us show that:
$$
\forall y\in \text{WST}[\mathcal{E}],~~
\exists z\in \mathcal{F},~~
\text{ s.t.: } y\succsim\rho'\star z
$$
Take $y\in\text{WST}[\mathcal{E}]$. From Equation (\ref{eq:approx:E}) (first condition), there is a $y'\in E$ such that $y\succsim y'$.
From Definition \ref{def:mopoa}, there is a $z'$ such that $y'\succsim \rho'\star z'$. From Equation (\ref{eq:approx:F}) on $z'$ (first condition), there exists $z\in\mathcal{F}$ such that $z'\succsim z$. Recap: $y\succsim y'\succsim \rho'\star z'\succsim \rho'\star z$.
(2) Then, let $\rho$ be a ratio of $R[\text{WST}[\mathcal{E}],\mathcal{F}]$,
and let us show that $\rho'=(1+\varepsilon_1)^{-1}(1+\varepsilon_2)^{-1}\rho$~~ is in $R[E,F]$, that is:
$$
\forall y'\in E,~~
\exists z'\in F,~~
(1+\varepsilon_1) y'\succsim (1+\varepsilon_2)^{-1} \rho\star z'
$$
Take an element $y'$ of $E$.
From Equation (\ref{eq:approx:E}) (second condition), there is $y\in\text{WST}[\mathcal{E}]$ such that $(1+\varepsilon_1)y'\succsim y$.
From Definition \ref{def:mopoa}, there is $z\in\mathcal{F}$ such that $y\succsim\rho\star z$.
From Equation (\ref{eq:approx:E}) on $z$ (second condition), there exists $z'\in F$ s.t. $z\succsim (1+\varepsilon_2)^{-1} z'$.
Recap: $(1+\varepsilon_1)y'\succsim y\succsim \rho\star z\succsim (1+\varepsilon_2)^{-1} \rho\star z'$.
\end{proof}
\begin{proof}[Theorem \ref{th:approx}]
For the first claim, since Algorithm \ref{alg:givenEFoutputMOPoA:polynomial}, given $E$ and $F$, outputs the MO-PoA corresponding to $R[E,F]$, by Theorem \ref{th:approx}, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} outputs an $((1+\varepsilon_1)(1+\varepsilon_2))$-covering of $R(\text{WST}[\mathcal{E}],\mathcal{F})$.
For the second claim, from Lemma \ref{lem:approx}, applying Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} on $E$ and $F$ outputs an $((1+\varepsilon_1)(1+\varepsilon_2))$-covering of $R(\text{WST}[\mathcal{E}],\mathcal{F})$.
Moreover, since we have $|E|=O((1/\varepsilon_1)^{d-1})$ and $|F|=O((1/\varepsilon_2)^{d-1})$, Algorithm \ref{alg:givenEFoutputMOPoA:polynomial} takes time $O\left(d/(\varepsilon_1\varepsilon_2)^{(d-1)(2d-1)}\right)$.
\end{proof}
\end{document} |
\begin{document}
\maketitle
\begin{abstract} Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds \cite{Wag1}. Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections and algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved \cite{V10} (see also \cite{V11}) that such a compatible linear connection must be uniquely determined.
The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.
\end{abstract}
\section{Compatibility equations in Finsler geometry}
Let $M$ be a smooth connected manifold with a local coordinate system $u^1, \ldots, u^n$. The induced local coordinate system on the tangent manifold $TM$ consists of the functions $x^1, \ldots, x^n$ and $y^1, \ldots, y^n$ given by $x^i(v):=u^i\circ \pi (v)=u^i(p)=:p^i$, where $\pi \colon TM \to M$ is the canonical projection and $y^i(v)=v(u^i)$, $i=1, \ldots, n$. Throughout the paper, we will use the shorthand notations
\begin{equation}
\partial_i := \dfrac{\partial}{\partial x^i} \hspace{1cm} \text{and} \hspace{1cm} \dot{\partial}_i:= \dfrac{\partial}{\partial y^i}.
\end{equation}
A Finsler metric \cite{BSC} on a manifold is a smoothly varying family of Minkowski norms in the tangent spaces. It is a direct generalization of Riemannian metrics, with inner products (quadratic indicatrices) in the tangent spaces replaced by Minkowski norms (smooth strictly convex bodies).
\begin{defi} A \textbf{Finsler metric} is a non-negative continuous function $F\colon TM\to \mathbb{R}$ satisfying the following conditions: $\displaystyle{F}$ is smooth on the complement of the zero section (\emph{regularity}), $\displaystyle{F(tv)=tF(v)}$ for all $\displaystyle{t> 0}$ (\emph{positive homogeneity}), $F(v)= 0$ if and only if $v={\bf 0}$ (\emph{definiteness}) and the Hessian $\displaystyle{g_{ij}=\dot{\partial}_{i}\dot{\partial}_{j}E}$ of the energy function $E=F^2/2$ is positive definite at all non-zero elements $\displaystyle{v\in T_pM}$ (\emph{strong convexity}). The pair $(M,F)$ is called a \textbf{Finsler manifold}.
\end{defi}
On a Riemannian manifold we obviously have compatible linear connections in the sense that the induced parallel transports preserve the Riemannian length of tangent vectors (metric linear connections). Following the classical Christoffel process it is clear that such a linear connection is uniquely determined by the torsion tensor. In contrast to the Riemannian case, non-Riemannian Finsler manifolds admitting compatible linear connections form a special class of spaces in Finsler geometry. They are called generalized Berwald manifolds \cite{Wag1}. It is known that some Finsler manifolds do not admit any compatible linear connections because of topological constraints, some have infinitely many compatible linear connections and it can also happen that the compatible linear connection is uniquely determined \cite{VOM}, see also \cite{RandersGBM}.
\begin{defi} A linear connection is \textbf{compatible} with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called \textbf{generalized Berwald manifolds}.
\end{defi}
In terms of local coordinates, equations
\begin{equation}
\label{ceq_christoffel}
X_i^h F := \partial_i F-y^j \left( {\Gamma}^k_{ij}\circ \pi \right) \dot{\partial}_k F=0 \hspace{1cm} (i=1,\dots,n)
\end{equation}
form necessary and sufficient conditions for a linear connection $\nabla$ to be compatible with the Finsler metric $F$. Equations (\ref{ceq_christoffel}) are called \textbf{compatibility equations} or CEQ for short. The fundamental result of generalized Berwald manifold theory states that a compatible linear connection $\nabla$ is always Riemann metrizable \cite{V5}, i.e. $\nabla$ must be a metric linear connection with respect to a Riemannian metric $\gamma$. Such a Riemannian metric can be given by integration of $g_{ij}$ on the indicatrix hypersurfaces \cite{V5}, see also \cite{Cram} and \cite{Mat1}. It is the so-called \textbf{averaged Riemannian metric}. Therefore CEQ can be reformulated by replacing the Christoffel symbols $\Gamma_{ij}^k$ by the torsion tensor components \cite{V14}, see also \cite{RandersGBM}. Using the horizontal vector fields
\[ X_i^{h*}:=\partial_i -y^j \left( {\Gamma}^{k*}_{ij}\circ \pi \right) \dot{\partial}_k \hspace{1cm} (i=1,\dots,n), \]
where ${\Gamma}^{k*}_{ij}$ are the Christoffel symbols of the L\'{e}vi-Civita connection of the averaged Riemannian metric $\gamma$, CEQ takes the form
\begin{equation}
y^j \left(T^{l}_{jk}\gamma^{kr}\gamma_{il}+T^{l}_{ik}\gamma^{kr}\gamma_{jl}-T_{ij}^r\right) \dot{\partial}_r F=-2X_i^{h^*}F\hspace{1cm} (i=1,\dots,n).
\end{equation}
The unknown quantities $T_{ab}^c$ are the torsion tensor components of the compatible linear connection.
In what follows we are going to use normal coordinates with respect to the Riemannian metric $\gamma$ around a given point $p\in M$. The coordinate vector fields $\partial/\partial u^1, \dots, \partial/\partial u^n$ form an orthonormal basis in $T_p M$, i.e. $\gamma_{ij}(p)=\delta_{ij}$ and ${\Gamma}^{k*}_{ij}(p)=0$. Therefore $X_i^{h*}(v)=\partial_i(v)$ for any $v\in T_pM$ and CEQ takes the form
\begin{equation}
\label{CEQ-tors2}
\sum_{a<b,c} \sigma_{ab;i}^c T_{ab}^{c} = -2\partial_i F \hspace{1cm} (i=1, \dots, n),
\end{equation}
where the coefficients are
\begin{equation}
\label{CEQ-coeff2}
\sigma_{ab;i}^c := \delta_i^a f_{cb} + \delta_i^b f_{ac} + \delta_i^c f_{ab}, \hspace{1cm} f_{ij} := y^i \dot{\partial}_j F - y^j \dot{\partial}_i F.
\end{equation}
If none of the indices $a,b,c$ are equal to $i$ then $\sigma_{ab;i}^c=0$. Otherwise the table shows the possible cases, where indices are separated according to their values (equal indices are put into the same cell and different cells contain different values).
\begin{center}
{\tabulinesep=1pt \begin{tabu} {|c||c|c|c||l|}
\hline
& \multicolumn{3}{c||}{\textrm{indices}} & \textrm{the coefficients} \\
\hline
\hline
1. & $i=a$ & $b$ & $c$ & $\sigma_{ib;i}^{c}=f_{cb}$ \\
\hline
2. & $i=a$ & $b=c$ & & $\sigma_{ib;i}^{b}=0$ \\
\hline
3. & $i=b$ & $a$ & $c$ & $\sigma_{ai;i}^{c}=f_{ac}$ \\
\hline
4. & $i=b$ & $a=c$ & & $\sigma_{ai;i}^{a}=0$ \\
\hline
5. & $i=c$ & $a$ & $b$ & $\sigma_{ab;i}^{i}=f_{ab}$ \\
\hline
6. & $i=a=c$ & $b$ & & $\sigma_{ib;i}^{i}=2f_{ib}$ \\
\hline
7. & $i=b=c$ & $a$ & & $\sigma_{ai;i}^{i}=2f_{ai}$ \\
\hline
\end{tabu} }
\end{center}
Therefore the $i$-th compatibility equation at the point $p$ is
\begin{equation} \label{CEQ-general}
\sideset{}{'}\sum_{a} 2f_{ia} T_{ia}^i + \sideset{}{'}\sum_{a<b} f_{ab} \left( T_{ib}^a + T_{ab}^i + T_{ai}^b \right) = -2\, \partial_i F,
\end{equation}
where the primed summation means summing for $a \neq i$ in the first one, and in the second one, $a$ and $b$ where $i\neq a, b$.
\section{The geometry of the tangent spaces}
The following table shows a panoramic view about the geometric structures of the tangent space $T_pM$ due to the simultaneously existing Finsler and Riemannian metrics.
\begin{center}
\begin{tabularx}{\textwidth}{|Y|Y|Y|}
\hline
\textbf{Finsler structure} & & \textbf{Riemannian structure} \\
\hline
Minkowski norm & metric on $T_p M$ & Euclidean norm and inner product \\
\hline
Finslerian spheres $F_p(\lambda)$ & level sets of $\lambda\in \mathbb{R}_+$ & Euclidean spheres $R_p(\lambda)$ \\
\hline
$\mathcal{F}_v$ & tangent hyperplanes of level sets at $v\in T_pM$ & $\mathcal{R}_v$ \\
\hline
$\mathcal{LF}_v:= \mathcal{F}_v - v$ & linear tangent hyperplanes at $v\in T_pM$ & $\mathcal{LR}_v := \mathcal{R}_v - v$ \\
\hline
$G := \mathrm{grad} \, F = [ \dot{\partial}_1 F, \dots, \dot{\partial}_n F]$ & normal vector fields (w.r.t. $\gamma$) & $C := [y^1, \dots, y^n]$ \\
\hline
\end{tabularx}
\end{center}
Both gradient vector fields $G$ and $C$ are nonzero everywhere on $T_p^{\circ} M := T_p M \backslash \{ \textbf{0} \}$. For every element $v \in T_p^{\circ} M$ the tangent hyperplanes of the Finslerian and the Riemannian (Euclidean) spheres passing through $v$ can be related as follows.
\begin{itemize}
\item If $\mathcal{F}_v=\mathcal{R}_v$, i.e. $G_v \parallel C_v$, we call the point $v$ a \textbf{vertical contact point} of the metrics.
\item If $\mathcal{F}_v \neq \mathcal{R}_v$, i.e. $G_v$ and $C_v$ are linearly independent, then the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ is the orthogonal complement of $\mathrm{span}(C_v, G_v)$, and thus $\mathcal{F}_v \cap \mathcal{R}_v$ is an affine subspace of dimension $n-2$.
\end{itemize}
Let us define the vector field
\begin{equation} \label{H-def}
H_v := \left[ \partial_1 F(v), \dots, \partial_n F(v) \right].
\end{equation}
An element $v \in T_p^{\circ} M$ is a \textbf{horizontal contact point} of the metrics if $H_v$ is the zero vector. It can be easily seen that if $T_p M$ is a vertical contact tangent space, i. e. all of its non-zero elements are vertical contact, then the Finsler metric is a scalar multiple of $\gamma$ at the point $p\in M$. The quadratic indicatrix of a generalized Berwald metric at a single point means quadratic indicatrices at all points of the (connected) base manifold because the tangent spaces are related by linear isometries due to the parallel transports with respect to the compatible linear connection with the Finsler metric. Therefore such a generalized Berwald manifold reduces to a Riemannian manifold. At a horizontal contact point $v\in T_pM$, equations of CEQ are homogeneous. If $T_p M$ is a horizontal contact tangent space, i. e. all of its non-zero elements are horizontal contact, then $T \equiv 0$ is a solution of CEQ at $p\in M$.
\subsection{A useful family of vector fields} Let us define the vector fields
\begin{equation} \label{rowvector}
f_i(v) := \left[ f_{i1}(v), f_{i2}(v), \dots, f_{in}(v) \right]^T \hspace{1cm} (i=1,\dots, n)
\end{equation}
on $T_p^{\circ} M$ to help in proving some elementary properties and solving CEQ.
\begin{lemm} \label{rowvectorspan} For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq \mathrm{span}(G_v, C_v)$.
\end{lemm}
\begin{proof} Observe that $f_i$ can be written as
\begin{equation} \label{rowvector-lincomb}
f_i = \begin{bmatrix} f_{i1} \\ f_{i2} \\ \vdots \\ f_{in}\end{bmatrix} =
\begin{bmatrix} y^i \dot{\partial}_1 F - y^1 \dot{\partial}_i F \\ y^i \dot{\partial}_2 F - y^2 \dot{\partial}_i F \\ \vdots \\ y^i \dot{\partial}_n F - y^n \dot{\partial}_i F \end{bmatrix} =
y^i \begin{bmatrix} \dot{\partial}_1 F \\ \dot{\partial}_2 F \\ \vdots \\ \dot{\partial}_n F \end{bmatrix} - \dot{\partial}_i F \begin{bmatrix}
y^1 \\ y^2 \\ \vdots \\ y^n \end{bmatrix},
\end{equation}
i.e. $f_i = y^i \cdot G - \dot{\partial}_i F \cdot C$.
\end{proof}
\begin{lemm} \label{rowvector-vertcont} At a vertical contact point $v \in T_p^{\circ} M$, $f_i(v)=0$ and CEQ takes the form $0=\partial_i F(v)$ $(i=1, \ldots, n)$.
\end{lemm}
\begin{proof} If $v$ is vertically contact, then $G_v = \lambda C_v$ for some nonzero $\lambda \in \mathbb{R}$. Substituting into \eqref{rowvector-lincomb}, we get
\[ f_i(v) = v^i \cdot \lambda C_v - \lambda v^i \cdot C_v = 0 \hspace{1cm} (i=1, \dots, n), \]
i.e. the coordinates of $f_i$ are all zero at $v$ and the reformulation \eqref{CEQ-general} of CEQ implies the statement.
\end{proof}
\begin{cor} In order for CEQ to have a solution, all vertically contact points must be horizontal contact.
\end{cor}
\begin{lemm} \label{rowvector-notvertcont} At a not vertically contact $v \in T_p^{\circ} M$ there are indices such that $f_{ij}(v)\neq 0$ and the vectors $f_i$ and $f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$.
\end{lemm}
\begin{proof} Suppose that all the $f_{ij}(v)$, and consequently, all the vectors $f_i(v)$ are zero. Since $v$ is not vertically contact, $G_v$ and $C_v$ are linearly independent. In particular, neither of them is the zero vector, so one of their coordinates is nonzero, meaning that for some index $i$, \eqref{rowvector-lincomb} gives the zero vector as a linear combination of the independent vectors $C_v$ and $G_v$ with nonzero coefficients. This is a contradiction, so there must be an $f_{ij}(v)$, and thus two vectors $f_i(v)$ and $f_j(v)$ different from zero. Since the matrix
\[ \begin{bmatrix} f_i \\ f_j \end{bmatrix} =
\begin{bmatrix} f_{i1} & \dots & 0 & \dots & f_{ij} & \dots & f_{in} \\ f_{j1} & \dots & f_{ji} & \dots & 0 & \dots & f_{jn}\\ \end{bmatrix} \]
has rank 2 at $v$ (choose the $i$-th and $j$-th columns), they are linearly independent at $v$ and the same is true at the points of some adequately small neighborhood of $v$ in $T_pM$ by a continuity argument.
\end{proof}
\begin{cor} \label{rowvector-ultimate} At a point $v \in T_p^{\circ} M$ the vectors defined by \eqref{rowvector} span the subspace
\[ \mathrm{span}(f_1(v), \dots, f_n(v)) = \left\{\begin{array}{cl}
\{ \bm{0} \} & \text{if} \ v \ \text{is vertically contact}, \\
\mathrm{span}(G_v, C_v) & \text{if} \ v \ \text{is not vertically contact}.
\end{array}\right. \]
\end{cor}
\begin{proof} For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq \mathrm{span}(G_v, C_v)$ by Lemma \ref{rowvectorspan}. If $v$ is vertically contact, all the vectors $f_i(v)$ are zero according to Lemma \ref{rowvector-vertcont}. If not, there are 2 independent vectors among them according to Lemma \ref{rowvector-notvertcont}, thus generating the whole $\mathrm{span}(G_v, C_v)$.
\end{proof}
\begin{lemm} \label{rowvector-basis} At a not vertically contact $v \in T_p^{\circ} M$, let us choose indices $i \neq j$ such that $f_{ij}(v)\neq 0$. Then $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$ over some neighborhood $U$ of $v$ in $T_pM$ and
\begin{equation}\label{rowvector-lincomb2}
f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot f_j \hspace{1cm} (k=1, \dots, n)
\end{equation}
at any point of $U$.
\end{lemm}
\begin{proof} By Lemma \ref{rowvector-notvertcont}, we know that $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$ at the points of $U$. Let us choose an index $k \in \{ 1, \dots, n\}$ and write $f_k = \lambda_1 f_i + \lambda_2 f_j$. By \eqref{rowvector-lincomb}, we can write that
\[f_k= y^k \cdot G - \dot{\partial}_k F \cdot C = \lambda_1 \left( y^i \cdot G - \dot{\partial}_i F \cdot C \right) + \lambda_2 \left( y^j \cdot G - \dot{\partial}_j F \cdot C \right). \]
By comparing the coefficients in the basis $(G, C)$,
\[ \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \begin{bmatrix} y^i & y^j \\ \dot{\partial}_i F & \dot{\partial}_j F \end{bmatrix} \cdot \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} \]
and, consequently,
\[ \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} \dot{\partial}_j F & -y^j \\ -\dot{\partial}_i F & y^i \end{bmatrix} \cdot \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} y^k \dot{\partial}_j F - y^j \dot{\partial}_k F \\ y^i \dot{\partial}_k F - y^k \dot{\partial}_i F \end{bmatrix}. \qedhere \]
\end{proof}
\section{The semi-symmetric compatible linear connection and its unicity}
Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections and algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor.
\begin{defi} A linear connection is called \textbf{semi-symmetric} if its torsion tensor can be written as
\begin{equation}\label{semi-symm-tors}
T(X,Y) = \rho(Y) X - \rho(X)Y
\end{equation}
for some differential 1-form $\rho$ on the base manifold.
\end{defi}
It is proved \cite{V10} that a semi-symmetric compatible linear connection must be uniquely determined.
\begin{thm}\label{original} \emph{\cite{V10}} A non-Riemannian Finsler manifold admits at most one semi-symmetric compatible linear connection.
\end{thm}
The original proof is based on averaging in the sense that the 1-form $\rho$ can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.
\subsection{The proof of Theorem \ref{original}} Since
$$T(\partial/\partial u^i, \partial/\partial u^j) = \rho(\partial/\partial u^j) \partial/\partial u^i - \rho(\partial/\partial u^i) \partial/\partial u^j=$$
$$= \rho_j \partial/\partial u^i - \rho_i \partial/\partial u^j = \left( \delta_i^k \rho_j - \delta_j^k \rho_i \right) \partial/\partial u^k,$$
the torsion components are
\begin{equation} \label{semi-symm-torscomp}
T_{ij}^k = \delta_i^k \rho_j - \delta_j^k \rho_i.
\end{equation}
In particular, all torsion components with 3 different indices are zero, and
\[ T_{ij}^i = \delta_i^i \rho_j - \delta_j^i \rho_i =\rho_j \hspace{1cm }(j \neq i). \]
Substituting the torsion components into the general form \eqref{CEQ-general} of CEQ at a point $p$, it takes the (matrix) form
\begin{equation} \label{CEQ}
{\tabulinesep=2pt
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12} & f_{13} & \cdots & f_{1n} & -\partial_1 F \\
f_{21} & 0 & f_{23} & \cdots & f_{2n} & -\partial_2 F \\
f_{31} & f_{32} & 0 & \cdots & f_{3n} & -\partial_3 F \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} & f_{n2} & f_{n3} & \cdots & 0 & -\partial_n F \\
\hline
\end{tabu}}
\end{equation}
The problem is to solve \eqref{CEQ} for $\rho_1, \dots, \rho_n$, considered as the coordinates of a vector $\rho \in T_p M$, as $v$ ranges over $T_p^{\circ} M$. To prove Theorem \ref{original} it is enough to consider the homogeneous version H-CEQ with vanishing right hand side of \eqref{CEQ}. We are going to verify that the only solution of H-CEQ is $\rho_1=\dots=\rho_n=0$. Since the rows of the matrix on the left hand side are exactly the vectors $f_1, \dots, f_n$ defined in \eqref{rowvector}, solving H-CEQ at a fixed element $v$ means finding the orthogonal complement of $\mathrm{span}(f_1(v), \dots, f_n(v))$. By Corollary \ref{rowvector-ultimate},
\begin{itemize}
\item it is $T_p M$ for any vertically contact element $v$,
\item it is the orthogonal complement of $\mathrm{span}(G_v, C_v)$ for any not vertically contact element $v$, i.e. the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ of the linear tangent hyperplanes of the Finslerian and Riemannian spheres.
\end{itemize}
The solution of H-CEQ at the point $p$ is the intersection of all the solution spaces as the element $v$ ranges over $T_p^{\circ} M$. Note that the homogeneity of the coefficients imply that it is enough to consider the intersection of all the solution spaces as the element $v$ ranges over the Finslerian (or the Riemannian) unit sphere.
\begin{itemize}
\item If all the elements of $T_p M$ are vertically contact and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold.
\item If there is a not vertically contact element $v$, then, by a continuity argument, we can consider a neighborhood $U\subseteq T_pM$ containing not vertically contact elements. For the solution vector $\rho$ we have
\[ \rho \in \bigcap_{v \in U} \left( \mathcal{LF}_v \cap \mathcal{LR}_v \right) \subseteq \big(\bigcap_{v \in U} \mathcal{LF}_v \big) \cap \big(\bigcap_{v \in U} \mathcal{LR}_v \big). \]
It is clear that the right hand side contains only the zero vector because the normal vectors at the points of any open set on the boundary of a Euclidean sphere (or any smooth strictly convex body) span the entire space. Therefore $\rho=0$ is the only solution of H-CEQ at $p$ and, consequently, CEQ admits at most one solution for the components of the torsion tensor of a semi-symmetric linear connection point by point. \qedhere
\end{itemize}
\subsection{Intrinsic equations and $v$-solvability of CEQ} In this section we investigate \eqref{CEQ} evaluated at non-zero tangent vectors in $T_pM$:
\begin{equation} \label{v-CEQ}
{\tabulinesep=2pt
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12}(v) & f_{13}(v) & \cdots & f_{1n}(v) & -\partial_1 F(v) \\
f_{21}(v) & 0 & f_{23}(v) & \cdots & f_{2n}(v) & -\partial_2 F(v) \\
f_{31} (v)& f_{32}(v) & 0 & \cdots & f_{3n}(v) & -\partial_3 F(v) \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} (v)& f_{n2}(v) & f_{n3} (v)& \cdots & 0 & -\partial_n F(v) \\
\hline
\end{tabu}}
\end{equation}
\begin{defi} The system of the compatibility equations is called \textbf{$v$-solvable} for $\rho$ at the point $p\in M$ if \eqref{v-CEQ} is solvable for any non-zero element $v\in T_pM$.
\end{defi}
\begin{rem}
The system of the compatibility equations is $v$-solvable if and only if all vertical contact vectors are horizontal contact. It is an obvious necessary condition because the coefficient matrix of CEQ is zero at a vertically contact point and the system must be homogeneous with vanishing horizontal derivatives of the Finsler metric with respect to the compatible linear connection. The sufficiency is based on the idea of the extremal compatible linear connection \cite{V14}. The extremal solution typically depends on the reference element $v\in T_pM$ but does not take a decomposable form in general. Therefore $v$-solvability for $\rho$ needs additional conditions. Using Corollary \ref{rowvector-ultimate} and basic linear algebra we can formulate the following characterizations of $v$-solvability in case of semi-symmetric compatible linear connections.
\end{rem}
\begin{lemm} \label{v-solvability1} The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:
\begin{itemize}
\item all vertically contact elements are also horizontal contact in $T_pM$ and
\item the rank of the augmented matrix of system \eqref{v-CEQ} is 2 at all not vertically contact elements $v\in T_pM$, i.e. $H_v \in \mathrm{span}(f_1(v), \dots, f_n(v))$, where the vector $H_v$ is defined by formula \eqref{H-def}.
\end{itemize}
\end{lemm}
\begin{prop} \label{v-solvability2} The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:
\begin{itemize}
\item all vertically contact elements are also horizontal contact in $T_pM$ and
\item for any triplets of distinct indices $i,j,k$ we have
\begin{equation} \label{solv-symm}
f_{ij}(v) \, \partial_k F (v)+ f_{jk}(v) \, \partial_i F (v)+ f_{ki}(v) \, \partial_j F(v) = 0
\end{equation}
provided that $f_{ij}(v)\neq 0$.
\end{itemize}
\end{prop}
\begin{proof} If $v$ is vertically contact, \eqref{solv-symm} stands trivially. Otherwise we are going to show that it is equivalent to the augmented matrix having rank 2. Suppose that $v$ is not vertically contact and choose indices $i\neq j$ such that $f_{ij}(v)\neq 0$ and $(f_i(v), f_j(v))$ is a basis of $\mathrm{span}(G_v, C_v)$. By Lemma \ref{rowvector-basis}, if $k\neq i, j$ then
\[ f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot f_j \hspace{1cm} (k=1, \dots, n). \]
In other words we can eliminate the $k$-th row for any $k\neq i, j$. The elimination must also yield zeroes on the right-hand side of \eqref{v-CEQ} to have a solution, i.e. for $k \neq i, j$ we must have
\[ \begin{array}{c}
-\partial_k F - \dfrac{f_{kj}}{f_{ij}} \cdot (-\partial_i F) - \dfrac{f_{ik}}{f_{ij}} \cdot (-\partial_j F) = 0 \\[12pt]
f_{ij} \, \partial_k F - f_{kj} \, \partial_i F - f_{ik} \, \partial_j F = 0.
\end{array} \]
Equation \eqref{solv-symm} follows by interchanging the indices in $f_{kj}$ and $f_{ik}$. Since the coefficient matrix is of maximal rank $2$, the augmented matrix is of maximal rank $2$ as the extension of the coefficient matrix with $-\partial_i F$ and $-\partial_j F$ in the corresponding rows.
\end{proof}
\begin{rem} Equations \eqref{solv-symm} do not contain unknown quantities. They are intrinsic conditions of the solvability. Taking $f_{ij}(v)\neq 0$ for some fixed indices $i$ and $j$ at a not vertically contact element, they provide $n-2$ equations to be automatically satisfied because $k=1, \ldots, n$, but $k\neq i, j$. The missing equations are
\begin{equation} \label{CEQ-elim}
\left.\begin{array}{rcl}
\dotprod{f_i,\rho} & = & -\partial_i F \\[4pt]
\dotprod{f_j,\rho} & = & -\partial_j F \\
\end{array}\right\},
\end{equation}
where $\dotprod{f_i,\rho}$ and $ \dotprod{f_j,\rho}$ stand for the inner product at $p\in M$ coming from the Riemannian metric $\gamma$. They provide the only possible solution $\rho$ in an explicit form.
\end{rem}
\subsection{The only possible solution of CEQ at the point $p\in M$} \label{lastsect} Recall that if the tangent space at $p\in M$ contains only vertically contact non-zero elements (vertically contact tangent space) and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold. Therefore we present the solution of CEQ in the generic case of non-Riemannian Finsler manifolds. Let us choose a not vertically contact element $v \in T_p^{\circ} M$ and indices $i\neq j$ with $f_{ij}(v)\neq 0$, i.e. $f_i, f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$. Using \eqref{rowvector-lincomb}, the eliminated form \eqref{CEQ-elim} of CEQ gives that
\[ {\arraycolsep=1pt \left.\begin{array}{ccccl}
y^i \dotprod{G,\rho} & - & \dot{\partial}_i F \dotprod{C,\rho} & = & -\partial_i F \\[5pt]
y^j \dotprod{G,\rho} & - & \dot{\partial}_j F \dotprod{C,\rho} & = & -\partial_j F \\
\end{array}\right\} } \Longleftrightarrow
\begin{bmatrix} y^i & -\dot{\partial}_i F \\[5pt] y^j & -\dot{\partial}_j F \end{bmatrix} \cdot \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}, \]
and, consequently,
\[ \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \dfrac{1}{f_{ji}}\begin{bmatrix}
-\dot{\partial}_j F &\dot{\partial}_i F \\[5pt] -y^j & y^i
\end{bmatrix} \cdot \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}. \]
We are going to concentrate on the second row
\begin{equation} \label{eq-c}
\dotprod{C,\rho} = \dfrac{1}{f_{ji}} \left( y^j \partial_i F - y^i \partial_j F \right) =: \dfrac{f_{ji}^h}{f_{ji}}
\end{equation}
at the points of the open neighborhood $U$ around $v$. Let us choose a value $\varepsilon>0$ such that all the elements
\[ \begin{array}{rcccl}
w_1 & := & v-\varepsilon \cdot \partial/\partial u^1(p) & = & [v^1-\varepsilon, v^2, v^3, \dots, v^n] \\
w_2 & := & v-\varepsilon \cdot \partial/\partial u^2(p) &= & [v^1, v^2-\varepsilon, v^3, \dots, v^n] \\
& \vdots &&& \\
w_n & := & v-\varepsilon \cdot \partial/\partial u^n(p) & = & [v^1, v^2, v^3, \dots, v^n-\varepsilon]
\end{array} \]
are contained in $U$. Then \eqref{eq-c} implies the system
\begin{equation} \label{ceq-indep} \begin{bmatrix}
v^1-\varepsilon & v^2 & v^3 & \cdots & v^n \\[4pt]
v^1 & v^2-\varepsilon & v^3 & \dots & v^n \\[4pt]
\vdots & \vdots & \vdots & \ddots & \vdots \\[4pt]
v^1 & v^2 & v^3 & \dots & v^n-\varepsilon
\end{bmatrix} \cdot \begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} = \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}
\end{equation}
of linear equations. Using the notation
\[ V:= \begin{bmatrix}
v^1 & v^2 & \cdots & v^n \\
\vdots & \vdots & \ddots & \vdots \\
v^1 & v^2 & \cdots & v^n
\end{bmatrix}, \]
we have to investigate the regularity of the matrix $V-\varepsilon I$, where $I$ denotes the identity matrix of the same type as $V$.
\begin{lemm} The matrix $V-\varepsilon I$ is regular if and only if $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n \}$.
\end{lemm}
\begin{proof} The determinant $\mathrm{det}(V-\varepsilon I)$ is the characteristic polynomial of $V$ with $\varepsilon$ as the variable. It is zero if and only if $\varepsilon$ is an eigenvalue of $V$. Consider the transpose of $V$ as the matrix of a linear transformation $\varphi$ (the eigenvalues are the same as those of $V$). Since the image of $\varphi$ is the line generated by $v$, it follows that $\varphi$ has a kernel of dimension $n-1$ and $v$ is an eigenvector corresponding to the eigenvalue $\widetilde{v}:=v^1+\dots+v^n$ because of
\[ \varphi(v)= \begin{bmatrix}
v^1 & \cdots & v^1 \\ \vdots & \ddots & \vdots \\ v^n & \cdots & v^n \end{bmatrix} \cdot \begin{bmatrix} v^1 \\ \vdots \\ v^n\end{bmatrix} = \begin{bmatrix}
v^1 (v^1+\dots+v^n) \\ \vdots \\v^n(v^1+\dots+v^n)
\end{bmatrix} = \widetilde{v} \cdot v. \qedhere\]
\end{proof}
\begin{lemm} Choosing $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$, we have
\[ \left[ V-\varepsilon I \right]^{-1} = \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \left[ V-(\widetilde{v}-\varepsilon) I \right]. \]
\end{lemm}
\begin{proof} We shall prove the formula
\[ \left[V-\varepsilon I\right] \cdot \left[ V-(\widetilde{v}-\varepsilon)I\right] = (\widetilde{v}-\varepsilon)\varepsilon I.\]
Rearranging the left-hand side,
\[ V \cdot V - (\widetilde{v}-\varepsilon)V-\varepsilon V + \varepsilon (\widetilde{v}-\varepsilon)I=(\widetilde{v}-\varepsilon)\varepsilon I \]
because of $V\cdot V=\widetilde{v} \cdot V$.
\end{proof}
Returning to \eqref{ceq-indep},
\[\begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} = \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \begin{bmatrix}
v^1+\varepsilon-\widetilde{v} & v^2 & \cdots & v^n \\[4pt]
v^1 & v^2+\varepsilon-\widetilde{v} & \dots & v^n \\[4pt]
\vdots & \vdots & \ddots & \vdots \\[4pt]
v^1 & v^2 & \dots & v^n+\varepsilon-\widetilde{v}
\end{bmatrix} \cdot \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}. \]
\begin{thm} If a non-Riemannian Finsler manifold admits a semi-symmetric compatible linear connection, then the only possible values of the components $\rho_k$ in formula \eqref{semi-symm-torscomp} for its torsion at the point $p\in M$ are
\begin{equation}
\rho_k = \dfrac{1}{\varepsilon} \left( \dfrac{1}{\widetilde{v}-\varepsilon} \sum_{l=1}^n v^l \dfrac{f^h_{ji}}{f_{ji}}(w_l) - \dfrac{f^h_{ji}}{f_{ji}}(w_k) \right),
\end{equation}
where,
\begin{itemize}
\item $v=[v^1, \dots, v^n]$ is a not vertically contact vector in $T^{\circ}_p M$,
\item $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$ is a radius of a closed ball around $v$ in $T_pM$ all of whose elements are also not vertically contact,
\end{itemize}
\begin{itemize}[itemsep=6pt]
\item $w_i=[v^1, \dots, v^{i-1}, v^i-\varepsilon, v^{i+1}, \dots, v^n]$, $i=1, \ldots, n$,
\item $f^h_{ji}= y^j \partial_i F - y^i \partial_j F = y^j \dfrac{\partial F}{\partial x^i} - y^i \dfrac{\partial F}{\partial x^j}$,
\item $f_{ji}= y^j \dot{\partial}_i F - y^i \dot{\partial}_j F = y^j \dfrac{\partial F}{\partial y^i} - y^i \dfrac{\partial F}{\partial y^j}$ and
\item the coordinates on the base manifold form a normal coordinate system with respect to the averaged Riemannian metric $\gamma$ around the point $p\in M$.
\end{itemize}
\end{thm}
\section{Acknowledgments}
Márk Oláh is supported by the UNKP-21-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund of Hungary.
\end{document} |
\begin{document}
\maketitle
\newif\ifarx
\arxfalse
\begin{abstract}
In the field of reinforcement learning (RL), agents are often tasked with solving a variety of problems differing only in their reward functions.
In order to quickly obtain solutions to unseen problems with new reward functions, a popular approach involves functional composition of previously solved tasks.
However, previous work using such functional composition has primarily focused on specific instances of composition functions whose limiting assumptions allow for exact zero-shot composition.
Our work unifies these examples and provides a more general framework for compositionality in both standard and entropy-regularized RL.
We find that, for a broad class of functions, the optimal solution for the composite task of interest can be related to the known primitive task solutions. Specifically, we present double-sided inequalities relating the optimal composite value function to the value functions for the primitive tasks. We also show that the regret of using a zero-shot policy can be bounded for this class of functions.
The derived bounds can be used to develop clipping approaches for reducing uncertainty during training, allowing agents to quickly adapt to new tasks.
\end{abstract}
\mathcal{S}ection{Introduction}\label{sec:intro}
Reinforcement learning has seen great success recently, but still suffers from poor sample complexity and task generalization.
Generalizing and transferring domain knowledge to similar tasks remains a major challenge in the field.
To combat this, different methods of transfer learning have been proposed; such as the option framework \citep{sutton_4room, barreto2019option}, successor features \citep{dayan1993improving, barreto_sf, hunt_diverg, nemecek}, and functional composition \citep{Todorov, Haarnoja2018, peng_MCP, boolean_stoch, vanNiekerk}. In this work, we focus on the latter method of ``compositionality'' for transfer learning.
Research in compositionality has focused on the development of approaches to combine previously learned optimal behaviors to obtain solutions to new tasks. In the process, many instances of functional composition in the literature have required limiting assumptions on the dynamics and allowable class of reward functions (goal-based rewards in \citep{boolean}) in order to derive exact results. Furthermore, previous work has focused on isolated examples of particular functions in either standard or entropy-regularized RL and a framework for studying a general class of composition functions without limiting assumptions is currently lacking. One of the main contributions of our work is to provide a unifying general framework to study compositionality in reinforcement learning.
In our approach, we focus on ``primitive'' tasks which differ only in their associated reward functions.
More specifically, we consider those downstream tasks whose reward functions can be written as a global function of the known source tasks' reward functions.
To maintain generality, we do not assume that transition dynamics are deterministic. We also do not assume that reward functions are limited to the goal-based setting, in which there are a limited number of absorbing ``goal'' states \citep{Todorov, vanNiekerk} defining the primitive task. Given the generality of this setting, we cannot expect to obtain exact solutions for compositions as in prior work. Instead, we provide a class of functions which can be used to obtain approximate solutions and bounds on the corresponding downstream tasks.
Given the solutions to a set of primitive tasks, we show that it is possible to leverage such information to obtain approximate solutions for a large class of compositely-defined tasks.
To do so, we relate the solution of the downstream composite task to the solved primitive (source) tasks. Specifically, we derive relations on the optimal value function of interest. From such a relation, a ``zero-shot'' (i.e. not requiring further training) policy can be extracted for use in the composite domain of interest.
We then show that the suboptimality (regret) of this zero-shot policy is upper bounded.
Our results support the idea that RL agents can focus on obtaining domain knowledge for simpler tasks, and later use this knowledge to effectively solve more difficult tasks. The primary contributions of the present work are as follows:
\textbf{Main contributions}
\begin{itemize}
\item Establishing a general framework for analyzing reward transformations and compositions for the case of stochastic dynamics, globally varying reward structures, and continuing tasks.
\item Derivation of bounds on the respective optimal value functions for transformed and composite tasks.
\item Demonstration of zero-shot approximate solutions and value-based clipping of new tasks based on the known optimal solutions for primitive tasks.
\end{itemize}
\mathcal{S}ection{Background}
In this work, we analyze the case of finite, discrete state and action spaces, with the Markov Decision Process (MDP) model \citep{suttonBook}. Let $\Delta(X)$ represent the set of probability distributions over $X$. Then the MDP is represented as a tuple $\mathcal{T}=\langle \mathcal{S},\mathcal{A},p, \mu, r,\gamma \rangle$ where $\mathcal{S}$ is the set of available states; $\mathcal{A}$ is the set of possible actions; $p: \mathcal{S} \times \mathcal{A} \to \Delta(\mathcal{S})$ is a transition function describing the system dynamics; $\mu \in \Delta(\mathcal{S})$ is the initial state distribution; $r: \mathcal{S} \times \mathcal{A} \to \mathbb{R}$ is a (bounded) reward function which associates a reward (or cost) with each state-action pair; and $\gamma \in (0,1)$ is a discount factor which discounts future rewards and guarantees the convergence of total reward for infinitely long trajectories ($T \to \infty$).
In ``standard'' (un-regularized) RL, the agent maximizes an objective function which is the expected future reward:
\begin{equation}\label{eq:std_rl_obj}
J(\pi) = \underset{\tau \mathcal{S}im{} p, \pi}{\mathbb{E}}
\left[ \mathcal{S}um_{t=0}^{\infty} \gamma^{t} r(s_t,a_t) \right].
\end{equation}
This objective has since been generalized for the setting of entropy-regularized RL \citep{ZiebartThesis, LevineTutorial}, which augments the standard RL objective in Eq. \eqref{eq:std_rl_obj} by appending an entropic regularization term for the policy:
\begin{equation}
J(\pi)=
\underset{\tau \mathcal{S}im{} p, \pi}{\mathbb{E}}
\left[ \mathcal{S}um_{t=0}^{\infty} \gamma^{t} \left( r(s_t,a_t) - f\left(r(s,a)\right)ac{1}{\beta}
\log\left(f\left(r(s,a)\right)ac{\pi(a_t|s_t)}{\pi_0(a_t|s_t)} \right) \right) \right]
\label{eq:ent_rl_obj}
\end{equation}
where $\pi_0: \mathcal{S} \to \Delta(\mathcal{A})$ is the fixed prior policy. The inverse temperature parameter, $\beta \in (0, \infty)$, regulates the contribution of entropic costs relative to the accumulated rewards.
The additional entropic control cost discourages the agent from choosing policies that deviate too much from the prior policy. Importantly, entropy-regularized MDPs lead to stochastic optimal policies that are provably robust to perturbations of rewards and dynamics \citep{eysenbach}; making them a more suitable choice for real-world problems.
By ``solution to the RL problem'', we hereon refer to the corresponding \textit{optimal} action-value function $Q(s,a)$ from which an \textit{optimal} control policy can be derived: $\pi(s) \in \text{argmax}_a Q(s,a)$ for standard RL; and $\pi(a|s) \propto \exp(\beta Q(s,a))$ for entropy-regularized RL. Note that these definitions are consistent with the limit $\beta \to \infty$ in which the standard RL objective is recovered from Eq.~\eqref{eq:ent_rl_obj}. For both standard and entropy-regularized RL, the optimal $Q$-function can be obtained by iterating a recursive Bellman equation. For standard RL, the Bellman optimality equation is given by \citep{suttonBook}:
\begin{equation}
Q(s,a) = r(s,a) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(\cdot|s,a)} \max_{a'} \left( Q(s',a') \right)
\end{equation}
The entropy term in the objective function for entropy-regularized RL modifies the previous optimality equation to \citep{ZiebartThesis, Haarnoja_SAC}:
\begin{equation}
Q(s,a) = r(s,a) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(\cdot|s')} e^{ \beta Q(s',a') }
\end{equation}
One of the primary goals of research in compositionality and transfer learning is deriving results for the optimal $Q$ function for new tasks based on the known optimal $Q$ function(s) for primitive tasks. There exist many forms of composition and transfer learning in RL, as discussed by \cite{taylor_survey}. In this paper, we focus on the case of concurrent skill composition by a single agent as opposed to an options-based approach \citep{sutton_4room}, or other hierarchical compositions \citep{hierarchy, saxe_hier}. We elaborate on this point with the definitions below.
To formalize our problem setup, we adopt the relevant definitions provided by \citep{ourAAAI}:
\begin{definition}
A \textbf{primitive RL task} is specified by an MDP $\mathcal{T} = \langle \mathcal{S},\mathcal{A},p,r, \gamma \rangle$ for which the optimal $Q$ function is known.
\end{definition}
In this work, we focus on primitive tasks with general reward functions, i.e. including both goal-based (sparse rewards on absorbing sets such as in the linearly solvable MDP framework of \citep{Todorov, boolean, vanNiekerk}) \textit{and} arbitrary reward landscapes \citep{Haarnoja2018}.
\begin{definition}
The \textbf{transformation of an RL task} is defined by its (bounded and continuous) transformation function: $f: \mathbb{R} \to \mathbb{R}$ and a primitive task $\mathcal{T}$. The transformed task shares the same states, actions, dynamics, and discount factor as $\mathcal{T}$ but has a transformed reward function $\widetilde{r}(s,a) = f(r(s,a))$.
\end{definition}
\begin{definition}
The \textbf{composition of $M$ RL tasks} is defined by a (bounded and continuous) function $F: \mathbb{R}^M \to \mathbb{R}$ and a set of primitive tasks $\{\mathcal{T}^{(k)}\}$. The \textbf{composite} RL task is defined by a new reward function $\widetilde{r}(s,a)~=~F(\{r^{(k)}(s,a)\})$; and shares the same states, actions, dynamics, and discount factor as all the primitive RL tasks.
\end{definition}
Finally, we define the Transfer Library, the set of functions which obey the hypotheses of our subsequent results (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas}). This definition serves to facilitate the general discussion of results obtained.
\begin{definition}
Given a set of primitive tasks $\{\mathcal{T}^{(k)}\}$, the \textbf{Transfer Library}, denoted by $\mathcal{T}L$, is the set of all transformation (or composition, when $M > 1$) functions $f$ which admit double-sided bounds (see Sections \ref{sec:std_lemmas} and \ref{sec:entropy-regularized_lemmas})
on the composite task's optimal $Q$ function ($\widetilde{Q}$).
\end{definition}
Specifically, $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 4.1 or 4.3}\}$ for standard RL and $\mathcal{T}L = \{f\ | f\ \textrm{satisfies Lemma 5.1 or 5.3}\}$ for entropy-regularized RL.
We have empirically found (cf. Fig.~\ref{fig:tasks} and Supplementary Material) that by using the derived bounds for the optimal value function, the agent can learn the optimal policy more efficiently for tasks in the Transfer Library.
\mathcal{S}ection{Previous Work}\label{sec:prev_work}
There is much previous work concerning compositionality and transfer learning in reinforcement learning. In this section we will give a brief overview by highlighting the work most relevant for the current discussion.
In this work, we focus on value-based composition; rather than policy-based composition \cite{peng_MCP}, features-based composition \citep{barreto_sf}, or hierarchical (e.g. options-based) composition \citep{alver, sutton_4room, barreto2019option}.
Value based methods of composition use the optimal value functions of lower-level or simpler ``primitive'' tasks to derive an approximation (or in some cases exact solution) for the composite task of interest.
In the optimal control framework, \citep{Todorov}
has shown that optimal value functions can be composed exactly for linearly-solvable MDPs with a $\textrm{LogSumExp}$ or ``soft OR'' composition over primitive tasks; assuming that tasks share the same absorbing set (boundary states). With a similar assumption of the shared absorbing set, \citep{boolean} show that exact optimal value functions for Boolean compositions may be recovered from primitive task solutions; thereby allowing an exponential improvement in knowledge acquisition.
In more recent work, in the context of MaxEnt RL, \cite{Haarnoja2018} have shown that linear convex-weighted compositions in stochastic environments result in a bound on optimal value functions, and the policy extracted from this zero-shot bound is indeed useful for solving the composite task. The same premise of convex-weighted reward structures was studied by \cite{hunt_diverg} where the difference between the bound of \citep{Haarnoja2018} and the optimal value function can itself be learned, effectively tightening the bound until convergence. This notion of a corrective function was subsequently generalized by \cite{ourAAAI} to allow for arbitrary functions of composition in entropy-regularized RL.
Other authors have considered the question of linear task decomposition, for instance \citep{barreto_sf} where a convex weight vector over learned \textit{features} can be calculated to solve the transfer problem over linearly-decomposable reward functions in standard RL. More recent developments on this line of research include \citep{hong2022bilinear} where a more general ``bilinear value decomposition'', conditioned on various goals, is learned. In \citep{kimconstrained}, the authors consider the successor features (SFs) framework of \citep{barreto_sf}, and propose lower and upper bounds on the optimal value function of interest. They show that by replacing standard generalized policy improvement (GPI) with a constrained version which respects their bounds, they are able to transfer knowledge more successfully to future tasks in the successor features framework.
With our reduced assumptions (any constant dynamics, constant discount factor, any rewards) it is not generally possible to solve the transformed or composed tasks based only on primitive knowledge. Nevertheless, we are able to derive bounds on the optimal $Q$-functions in both standard and entropy-regularized RL, from which we can immediately derive policies which fare well in the transformed and composed problem settings. Additionally, we are able to prove that the derived policies have a bounded regret, in a similar form as \cite{Haarnoja2018}'s Theorem 1; but in a more general setting.
\mathcal{S}ection{Standard RL}\label{sec:std_lemmas}
\mathcal{S}ubsection{Transformation of Primitive Task}
In this section, we consider \textbf{transformations of a primitive task} in the ``standard'' (un-regularized) RL setting. We assume a solved primitive task is given with reward function $r(s,a)$. Transforming this underlying reward function gives rise to a new reward function, $f(r(s,a))$ which specifies a new RL task to solve. All other variables defining the MDP ($ \mathcal{S}, \mathcal{A}, p, \mu, \gamma $) are assumed to be fixed. In this new setting, we consider how to use the solution to the primitive task (that is, with rewards $r$) to inform the solution of the new, transfer task (that is, with rewards $f(r)$). The set of all applicable functions $f$ for which we can derive bounds, forms the aforementioned \textit{Transfer Library} with respect to the primitive task, for standard RL.
For a general class of transformations of reward functions (as defined below), we show that the optimal value function for the transformed task is bounded by an analogous functional transformation of the optimal value function for the primitive task. (The proofs for all theoretical results are provided in the Supplementary Material.)
We use the following definitions in the subsequent (standard RL) results: Let $X$ be the codomain for the $Q$ function of the primitive task ($Q: \mathcal{S} \times \mathcal{A} \to X \mathcal{S}ubseteq \mathbb{R}$). Let $V_f$ denote the state-value function derived from the transformation function $f(Q)$: $V_f(s) = \max_a f(Q(s,a))$.
\begin{lemma}[Convex Conditions]\label{thm:convex_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$\footnote{This condition is not required for deterministic dynamics.\label{dynamics condition}};
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \leq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$\footnote{Although this condition is automatically satisfied, it allows for a smoother connection to the analogous hypotheses in Lemmas \ref{thm:concave_cond_std}, \ref{thm:forward_cond_entropy-regularized}, \ref{thm:reverse_cond_entropy-regularized} and compositional results in the Supplementary Material.}
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eqn:convex_std}
f(Q(s,a)) \leq \widetilde{Q}(s,a) \leq f(Q(s,a)) + C(s,a)
\end{equation}
where $C(s,a)$ is the optimal value function for a task with reward
\begin{equation}\label{eq:std_convex_C_def}
r_C(s,a) = f(r(s,a)) + \gamma \E_{s' \mathcal{S}im{} p } V_f(s') - f(Q(s,a)).
\end{equation}
that is, $C$ satisfies the following recursive equation:
\begin{equation}
C(s,a) = r_C(s,a) + \gamma \E_{s' \mathcal{S}im{} p} \max_{a'} C(s',a').
\end{equation}
\end{lemma}
With this result, we have a double-sided bound on the values of the optimal $Q$-function for the composite task.
In particular, the lower bound ($f(Q)$) provides a zero-shot approximation for the optimal $Q$-function. It is thus of interest to analyze how well a policy $\pi_f$ extracted from such an estimate ($f(Q)$) might perform.
To this end, we provide the following result which bounds the suboptimality of $\pi_f$ as compared to the optimal policy.
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq D(s,a)
\end{equation}
where $D$ is the value of the policy $\pi_f$ in a task with reward
\begin{align*}
r_D(s,a) = \gamma \E_{s'\mathcal{S}im{} p}\E_{a' \mathcal{S}im{} \pi_f} &\biggr[ \max_{b} \big\{ f(Q(s',b)) + C(s',b) \big\} \\ &- f(Q(s',a')) \biggr]
\end{align*}
that is, $D$ satisfies the following recursive equation:
\begin{equation}
D(s,a) = r_D(s,a) + \gamma \E_{s'\mathcal{S}im{}p} \E_{a' \mathcal{S}im{} \pi_f} D(s',a').
\end{equation}
\end{lemma}
Interestingly, the previous result shows that for functions $f$ admitting a tight double-sided bound (that is, a relatively small value of $C$), the associated zero-shot policy $\pi_f$ can be expected to perform near-optimally in the composite domain.
Another class of functions for which general bounds can be derived arises when $f$ satisfies the following ``reverse'' conditions.
\begin{lemma}[Concave Conditions]\label{thm:concave_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$\textsuperscript{\ref{dynamics condition}};
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \geq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$
\end{enumerate}
then the optimal action-value functions are now related in the following way:
\begin{equation}\label{eqn:concave_std}
f(Q(s,a)) - \hat{C}(s,a) \leq \widetilde{Q}(s,a) \leq f(Q(s,a))
\end{equation}
where $\hat{C}$ is the optimal value function for a task with reward
\begin{equation}
\hat{r}_C(s,a) = f(Q(s,a)) - f(r(s,a)) - \gamma \E_{s' \mathcal{S}im{} p} V_f(s')
\end{equation}
\end{lemma}
One obvious way to satisfy the final condition in the preceding lemma is to consider functions $f(x)$ which are monotonically increasing.
Note that the definitions of $C$ and $\hat{C}$ guarantee them to be positive, as is required for the bounds to be meaningful (this statement is shown explicitly in the Supplementary Material).
Furthermore, by again considering the derived policy $\pi_f(a|s)$, we next provide a similar result for concave conditions, noting the difference in definitions between $D$ and $\hat{D}$.
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq \hat{D}(s,a)
\end{equation}
where $\hat{D}$ is the value of the policy $\pi_f$ in a task with reward
\begin{equation*}
\hat{r}_D = \gamma \underset{s'\mathcal{S}im{} p\ }{\mathbb{E}} \underset{a' \mathcal{S}im{} \pi_f}{\mathbb{E}} \biggr[ V_f(s') - f(Q(s',a')) + \hat{C}(s',a') \biggr]
\end{equation*}
\end{lemma}
\renewcommand{1.5}{1.5}
\begin{table}[ht]
\centering
\begin{tabular}{ll}
\toprule
\multicolumn{2}{c}{Standard RL Results} \\
\cmidrule(r){1-2}
Transformation & Result \\
\midrule
Linear Map: & $\widetilde{Q}(s,a) = k Q(s,a)$ \\
Convex conditions: & $\widetilde{Q}(s,a) \geq f(Q(s,a))$ \\
Concave conditions: & $\widetilde{Q}(s,a) \leq f(Q(s,a))$ \\
OR Composition: & $\widetilde{Q}(s,a) \geq \max_k \{Q^{(k)}(s,a)\}$ \\
AND Composition: & $\widetilde{Q}(s,a) \leq \min_{k} \{ Q^{(k)}(s,a)\}$ \\
NOT Gate:
& $ \widetilde{Q}(s,a) \geq
- Q(s,a)$ \\
Conical combination: & $\widetilde{Q}(s,a) \leq \mathcal{S}um_k \alpha_k Q^{(k)}(s,a)$ \\
\bottomrule
\end{tabular}
\caption{\textbf{Standard Transfer Library.} Lemmas \ref{thm:convex_cond_std}, \ref{thm:concave_cond_std} stated in Section \ref{sec:std_lemmas} lead to a broad class of applicable transfer functions in standard RL. In this table we list several common examples which are demonstrated throughout the paper and in the Supplementary Materials. We show only one side of the bounds from Eq. \eqref{eqn:convex_std}, \eqref{eqn:concave_std} which requires no additional training.}\label{tab:std_rl}
\end{table}
We remark that the conditions imposed on the function $f$ are not very restrictive. For example, the Boolean functions and linear combinations considered in previous work are all included in our framework, while we also include novel transformations not considered in previous work (see Table~\ref{tab:std_rl}). Furthermore, the conditions for $f$ can be further relaxed if specific conditions are met. For the case of deterministic dynamics, the first condition is not required ($f$ need not be convex nor concave).
We have shown that in the standard RL case, quite general conditions (convexity and sublinearity) lead to a wide class of applicable functions defining the Transfer Library. The conditions given in Lemmas \ref{thm:convex_cond_std} and \ref{thm:concave_cond_std} are straightforward to check for general functions. When given a primitive task defined by a reward function $r$, one can therefore bound the optimal $Q$ function for a general transformation of the rewards, $f(r)$, when $f$ obeys the conditions above. This new set of transformed tasks defines the Transfer Library from a given set of primitive tasks.
The previous (and following) results are presented for the case in which the primitive task $Q$-values are known \textit{exactly}. In practice, however, this is not typically the case, even in tabular settings. In continuous environments where the use of function approximators is necessary, the error that is present in learned $Q$-values is further increased. To address this issue, we provide an extension of all double-sided bounds for the case where an $\varepsilon$-optimal estimate of the primitive task's $Q$-values is known, such that $|Q(s,a)~-~\bar{Q}(s,a)|~\leq~\varepsilon$ for all $s,a$. To derive such an extension, we further require that the composition function $f$ is $L$-Lipschitz continuous (essentially a bounded first derivative), i.e. $|f(x_1)-f(x_2)|\leq L|x_1-x_2|$ for all $x_1,x_2 \in X$, the domain of $f$ (in the present case, the $x_i$ are the primitive task's $Q$-values). To maintain the focus of the main text, we provide these results and the corresponding proofs in the Supplementary Material. We note that all functions listed in Table \ref{tab:std_rl} and \ref{tab:entropy-regularized} are indeed $1$-Lipschitz continuous.
\mathcal{S}ubsection{Generalization to Composition of Primitive Tasks}\label{sec:comp_std}
The previous lemmas can be extended to the case of multivariable transformations (see Supplementary Material for details), where $X \to \bigotimes X^{(k)}$ (the Cartesian product of primitive codomains). That is, with a function $F: \bigotimes X^{(k)} \to \mathbb{R}$ and a collection of $M$ subtasks, $\{r^{(k)}(s,a)\}_{k=1}^{M}$, one can synthesize a new, \textbf{composition of subtasks}, with reward defined by $r^{(c)}(s,a) = F(r^{(1)}(s,a), \dotsc, r^{(M)}(s,a))$.
In this vectorized format, $F$ must obey the above conditions in each argument:
\begin{itemize}
\item $F$ is convex (concave) in each argument,
\item $F$ is sublinear (superlinear) in each argument.
\end{itemize}
For the final conditions, we also require a similar vectorized inequality, which we spell out in detail in the Supplementary Material.
As an example of composition in standard RL, we consider the possible sums of reward functions, with each task having a positive weight associated to it.
In such a setup, the agent has learned to solve a set of primitive tasks, then it must solve a task with a new compositely-defined reward function, say $f\left(r^{(1)}, \dotsc, r^{(M)}\right)~\doteq~\mathcal{S}um_{k=1}^{M} \alpha_k r^{(k)}$ for (possibly many) target tasks defined by the weights $\{\alpha_k\}$.
To determine which bound is satisfied for such a composition function, we look to the vectorized conditions above. This function is linear in all arguments, so we must only check the final condition. Since the inequality
\begin{equation}
\mathcal{S}um_k \alpha_k \max_a Q^{(k)}(s,a) \geq \max_a \mathcal{S}um_k \alpha_k Q^{(k)}(s,a)
\end{equation}
holds for any set of $\alpha_k > 0$, this function conforms to the concave vectorized conditions, implying that $\widetilde{Q}(s,a)~\leq~f(Q^{(k)}(s,a))=\mathcal{S}um_k~\alpha_k Q^{(k)}(s,a)$. We can then use the right-hand side of this bound to calculate the associated state-value function ($V_f(s) = \max_a f(Q^{(k)}(s,a))$) and the associated greedy policy ($\pi_f(s) = \text{argmax}_a f(Q^{(k)}(s,a))$). This result agrees with an independent result by \cite{nemecek} (the upper bound in Theorem 1 therein) without accounting for approximation errors.
\mathcal{S}ection{entropy-regularized RL}\label{sec:entropy-regularized_lemmas}
\mathcal{S}ubsection{Transformation of Primitive Task}
We will now extend the results obtained in the previous section to the case of entropy-regularized RL.
Again we first consider the single-reward transformation $f(r)$ for some function $f$. Here we state the conditions that must be met by functions $f$, which define the Transfer Library for entropy-regularized RL.
We now use the following definitions in the subsequent (entropy-regularized RL) results.
In the following results, we set $\beta=1$ for brevity, and the expectation in the final condition is understood to be over actions, sampled from the prior policy. Full details can be found in the proofs provided in the Supplementary Material.
\begin{lemma}[Convex Conditions]
\label{thm:forward_cond_entropy-regularized}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$\textsuperscript{\ref{dynamics condition}};
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \log \E \exp \mathcal{Q}(s,a) \right) \leq \log \E \exp f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eq:convex_entropy-regularized}
f \left( Q(s,a) \right) \leq \widetilde{Q}(s,a) \leq f \left( Q(s,a) \right) + C(s,a)
\end{equation}
\end{lemma}
\begin{table}[ht]
\centering
\begin{tabular}{ll}
\toprule
\multicolumn{2}{c}{Entropy-Regularized RL Results} \\
\cmidrule(r){1-2}
Transformation& Result\\
\midrule
Linear Map, $k \in (0,1)$\tablefootnote{Note that linear reward scaling can also be viewed as a linear scaling in the temperature parameter.}:& $\widetilde{Q}(s,a) \geq k Q(s,a) $ \\
Linear Map, $k > 1$:& $\widetilde{Q}(s,a) \leq k Q(s,a)$ \\
Convex conditions: & $\widetilde{Q} \geq f(Q(s,a))$ \\
Concave conditions: & $\widetilde{Q} \leq f(Q(s,a))$ \\
OR Composition: & $\widetilde{Q}(s,a) \geq \max_k \{Q^{(k)}(s,a)\}$ \\
AND Composition: & $ \widetilde{Q}(s,a) \leq \min_k \{ Q^{(k)}(s,a)\}$ \\
NOT Gate: & $\widetilde{Q}(s,a) \geq
-Q(s,a)$ \\
Convex combination\tablefootnote{This extends to the case $\mathcal{S}um_k \alpha_k \geq 1$ by composing with a linear scaling, which respects the same inequality.}: & $\widetilde{Q}(s,a) \leq \mathcal{S}um_k \alpha_k Q^{(k)}(s,a)$ \\
\bottomrule
\end{tabular}
\caption{\textbf{Entropy-Regularized Transfer Library.} Lemmas \ref{thm:forward_cond_entropy-regularized}, \ref{thm:reverse_cond_entropy-regularized} lead to a broad class of applicable transfer functions in entropy-regularized RL. In this table we list several common examples which are demonstrated throughout the paper and in the Supplementary Materials. We show only one side of the bounds from Eq. \eqref{eq:convex_entropy-regularized}, \eqref{eq:concave_entropy-regularized} which requires no additional training.}\label{tab:entropy-regularized}
\end{table}
We note that $C$ has the same definition as before, but with $V_f$ replaced by its entropy-regularized analog: $V_f(s)~\doteq~\log \E_{a \mathcal{S}im{} \pi_0} \exp f\left(Q(s,a)\right)$.
\begin{lemma}
Consider the soft value of the policy $\pi_f(a|s)~=~\pi_0(a|s)\exp\left( f(Q(s,a)) - V_f(s) \right)$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq D(s,a)
\end{equation}
where $D$ is the soft value of the policy $\pi_f$
with reward
\begin{equation*}
r_D(s,a) = \gamma \E_{s'}
\left[ \max_{b} \left\{ f\left(Q(s',b)\right) + C(s',b) \right\} -V_f(s') \right].
\end{equation*}
\end{lemma}
Conversely, for concave conditions we have
\begin{lemma}[Concave Conditions]
\label{thm:reverse_cond_entropy-regularized}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$\textsuperscript{\ref{dynamics condition}};
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \log \E \exp \mathcal{Q}(s,a) \right) \geq \log \E \exp f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X$.
\end{enumerate}
then the optimal action-value function for the transformed rewards obeys the following inequality:
\begin{equation}\label{eq:concave_entropy-regularized}
f\left( Q(s,a) \right) - \hat{C}(s,a) \leq \widetilde{Q}(s,a) \leq f \left( Q(s,a) \right)
\end{equation}
\end{lemma}
As in the preceding section, we provide a similar result for the derived policy $\pi_{f}$, given the concave conditions provided.
\begin{lemma}
Consider the soft value of the policy $\pi_f(a|s)$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq \hat{D}(s,a)
\end{equation}
where $\hat{D}$ satisfies the following recursive equation
\begin{equation}
\hat{D}(s,a) = \gamma \E_{s' \mathcal{S}im{} p}\E_{a' \mathcal{S}im{} \pi_f} \left( \hat{C}(s',a') + \hat{D}(s',a') \right).
\end{equation}
\label{lem:concave_regret_maxent}
\end{lemma}
Now, by taking $V_f(s)$ as the previously defined \textit{soft} value function, the fixed points $C$ and $\hat{C}$ have the same definitions as presented in Lemma \ref{thm:convex_cond_std} and \ref{thm:concave_cond_std}, respectively with this new definition of $V_f$.
This final constraint (in Lemma 5.1 and 5.3) on $f$ arises out of the requirements for extending the previous results to entropy-regularized RL.
Although the final condition (similar to a log-convexity) appears somewhat cumbersome, we show that it is nevertheless possible to satisfy it for several non-trivial functions (Table~\ref{tab:entropy-regularized}). For instance, functions defining Boolean composition over subtasks ($\max(\cdot), \ \min(\cdot)$), which have not been considered in previous entropy-regularized results \citep{Haarnoja2018, vanNiekerk} as well as new functional transformations such as the NOT gate (Table~\ref{tab:entropy-regularized}).
\begin{figure*}
\caption{In the first panel, we show learning curves for each of the clipping methods proposed, averaged over $50$ trials, with a 95\% confidence interval shown in the shaded region. In the next two panels, we depict the primitive tasks with rewarding states (orange diamonds) on the left side and bottom side of the maze, respectively. In the rightmost panel, we show the composite task of interest, with the multivariable ``OR'' composition function $\max_k\{...\}
\label{fig:tasks}
\end{figure*}
\mathcal{S}ubsection{Generalization to Composition of Primitive Tasks}
As we have done in the standard RL setting (Section \ref{sec:comp_std}), we can also extend the previous results to include compositionality: functions operating over multiple primitive tasks.
In this case, \cite{Haarnoja2018} have demonstrated a special case of Lemma \ref{thm:reverse_cond_entropy-regularized} for the composition function $f(\{r^{(k)}\}) = \mathcal{S}um_k \alpha_k r^{(k)}$ for convex weights $\alpha_k$. This can also be shown in our framework by proving the final condition of Lemma \ref{thm:reverse_cond_entropy-regularized} (since the others are automatic given that $f$ is linear). This vectorized condition can be proven via H\"older's inequality.
Besides this previously studied composition function, we can now readily derive value function bounds for other transformations and compositions, for example Boolean compositions as defined previously. The corresponding results for entropy-regularized RL are summarized in Table \ref{tab:entropy-regularized}.
\mathcal{S}ection{Experiments}
To test our theoretical results using function approximators (FAs), we consider a deterministic ``gridworld'' MDP amenable to task composition\footnote{Source code available at \url{https://github.com/JacobHA/Q-Bounding-in-Compositional-RL}}. Figure ~\ref{fig:tasks} shows the environments of the trained primitive tasks ``$6\times6\ \text{L}$'' and ``$6\times6\ \text{D}$'', whose reward functions are then combined to produce a composite task, ``$6\times6\ \text{L OR D}$''. The agent has $4$ possible actions (in each of the cardinal directions) and begins at the green circle in all cases. The agent's goal is to navigate to the orange states which provide a reward. We note that these states are \textit{not absorbing} unlike the cases considered in prior work. The red ``X'' indicates a penalizing state where the agent's episode is immediately terminated. Finally, a wall (black square) is added for the agent to navigate around. The primitive tasks are assumed to be solved with high accuracy (i.e. we assume the $Q$-values for primitive tasks to be exactly known). Although the domain is rather simple, we use such an experiment as a means of validating our theoretical results while gaining insight on the experimental effects of \textit{clipping} (discussed below) during training.
\begin{figure}
\caption{Mean bound violation, shown with shaded 95\% confidence intervals. The bound violation measures the difference between the Q-network's estimate and the allowed bound $\widetilde{Q}
\label{fig:bound_viol}
\end{figure}
With the primitive tasks solved, we now consider training on a target composite (``OR'') task. We learn from scratch (with no prior information or bounds being applied) as our baseline (blue line, denoted ``none'', in Fig.~\ref{fig:tasks} and Fig.~\ref{fig:bound_viol}).
To implement the derived bounds, we consider the one-sided bound, thereby not requiring further training. In this case (standard RL, ``OR'' composition) we have the following lower bound (see Table~\ref{tab:std_rl}): $Q^{(\text{OR})} \ge \max\{Q^{(\text{L})}, Q^{(\text{D})}\}$. There are many ways to implement such a bound in practice. One na\"ive method is to simply clip the target network's new (proposed target) value to be within the allowed region for each of the $\widetilde{Q}(s,a)$ that are currently being updated. We term this method ``hard clipping''. Inspired by Section 3.2 of \citep{kimconstrained}, we can also use an additional penalty by adding to the loss function the absolute value of bound violations that occur (the quantity ``$\textrm{BV}$'' defined in Eq.~\eqref{eq:bound_violations}). We term this method as ``soft clipping''. As mentioned by \citep{kimconstrained}, this method of clipping could produce a new hyperparameter (the relative weight for this term relative to the Bellman residual). We keep this coefficient fixed (to unity) for simplicity, and we intend on exploring the possibility of a variable weight in future work.
\begin{equation}
\textrm{BV} \doteq || \widetilde{Q}(s,a) - f(\{Q^{(k)}(s,a)\}_{k=1}^M) ||
\label{eq:bound_violations}
\end{equation}
Similar to Eq. (21) of \citep{kimconstrained} we also considered a clipping at test-time only, with some differences in how the bounds are applied. This discrepancy is due to the difference in frameworks: \citep{kimconstrained} leverages the GPI framework, and in our setting we are learning a new policy from scratch while imposing said bounds. Our method is as follows:
Whenever the agent acts greedily and samples from the policy network, it first applies (hard) clipping to the network's value; then the agent extracts an action via greedy argmax. We term this method of clipping as ``test clipping''. The results for each method (as well as a combination of both hard and soft clipping) are shown in Fig.~\ref{fig:tasks} and \ref{fig:bound_viol}.
Interestingly, we find that by directly incorporating the bound violations into the loss function (via the ``soft'' clipping mechanism); the bound violations most quickly become (and remain) zero (Fig.~\ref{fig:bound_viol}) as opposed to the other methods considered. We find that reduction in bound violation also generally correlates with a high evaluation reward during training. One exception to this observation (for the particular environment shown) is the case of ``test'' clipping.
For this particular composition, either primitive task will solve the composite task, thus yielding high evaluation rewards (Fig.~\ref{fig:tasks}). However, the $Q$-values are not accurate, which leads to a high frequency of clipping, comparable to the baseline without clipping (Fig.~\ref{fig:bound_viol}).
In order to ensure the agent has learned accurate $Q$-values, it is therefore important to monitor the bound violations rather than only the evaluation performance which may not be representative of convergence of $Q$-values.
\mathcal{S}ection{Discussion}
In summary, we have established a general theoretical treatment for functional transformation and composition of primitive tasks. This extends the scope of previous work, which has primarily focused on isolated instances of reward transformations and compositions without general structure. Additionally, we have theoretically addressed the broader setting of stochastic dynamics, with rewards varying on both terminal and non-terminal (i.e. boundary and interior) states. In this work, we have shown that it is possible to derive a general class of functions which obey transfer bounds in standard and entropy-regularized RL beyond those cases discussed in previous work. In particular, we show that by using the same functional form on the optimal $Q$ functions as used on the reward, we can bound the transformed optimal $Q$ function. The derived bound can then be used to calculate a zero-shot solution. We have used these functions to define a Transfer Library: a set of tasks which can immediately be addressed by our bounds. Since our approach via the optimal backup equation is general, we apply it to both standard RL and entropy-regularized RL.
The newly-defined fixed point $C$ ($\hat{C}$) has an interesting interpretation. Rather than simply being an arbitrary function, for both the standard RL and entropy-regularized RL bounds, $C$ represents an optimal value function for a standard RL task with reward function given by $r_C$ ($\hat{r}_C$ for $\hat{C}$).
The function $C$ bounds the total gap between $f(Q(s,a))$ and $\widetilde{Q}(s,a)$ at the level of state-actions. We also note the simple relationship between reward functions $r_C = -\hat{r}_C$.
The fixed point $D$ ($\hat{D}$) is not an optimal value function, but the value of the zero-shot policy $\pi_f$ in some other auxiliary task. The auxiliary task takes various ``rewards'', e.g. the function $\gamma \hat{C}$ in Lemma \ref{lem:concave_regret_maxent}. Although for general functions $f$, the rewards do not have a simple interpretation (i.e. R\'enyi divergence between two policies as in \citep{Haarnoja2018}), we see that $r_C$ essentially measures the non-linearities of the composition function $f$ with respect to the given dynamics, and hence accounts for the errors made in using the bounding conditions of $f$.
Furthermore, we can bound $C$ (and thus the difference between the optimal value and the suggested zero-shot approximation $f(Q)$) in a simple way: by bounding the rewards corresponding to $C$. By simply calculating the maximum of $r_C$ for example, one easily finds $C(s,a) \le f\left(r(s,a)\right)ac{1}{1-\gamma} \max_{s,a} r_C(s,a)$ (and similarly for $\hat{C}$).
Interestingly, \cite{TL_bound} have shown the provable usefulness of using an upper bound when used for ``warmstarting'' the training in new domains. In particular, it appears that $f(Q)$ (for the concave conditions) is related to their proposed ``$\alpha$-weak admissible heuristic'' for $\widetilde{\mathcal{T}}$. In future work, we hope to precisely connect to such theoretical results in order to obtain provable benefits to our derived bounds.
Experimentally we have observed that this warmstarting procedure does indeed improve convergence times, however a detailed study of this effect is beyond the scope of the present work and will be explored in future work. The derived results have also been used to devise protocols for clipping which improve performance and reduce variance during training based on the experiments presented.
In the future, we hope that the class of functions discussed in this work will be broadened further, allowing for a larger class of non-trivial zero-shot bounds for the Transfer Library. By adding these known transformations and compositions to the Transfer Library, the RL agent will be able to approach significantly more novel tasks without the need for further training.
The current research has also emphasized questions for transfer learning in this context, such as:
\textit{Which primitives should be prioritized for learning?} (Discussed in \cite{boolean_stoch, nemecek, alver}.) \textit{What other functions can be used for transfer?} \textit{How tight are these bounds?} \textit{How does the Transfer Library depend on the parameters $\gamma$ and $\beta$?}
In this work, we provide general bounds for the discrete MDP setting and an extension of the theory to continuous state-action spaces is deferred to future work. It would be of interest to explore if it is possible to prove general bounds for this extension, given sufficient smoothness conditions on the dynamics and the function of transformation. Other extensions can be considered as well, for instance: the applicability to other value-based or actor-critic methods, the warmstarting of function approximators, learning the $C$ and $D$ functions, and adjusting the ``soft'' clipping weight parameter.
In future work, we also aim to discover other functions satisfying the derived conditions; and will attempt to find necessary (rather than sufficient) conditions that classify the functions $f \in \mathcal{T}L$.
It would be of interest to explore if extensions of the current approach can further enable agents to expand and generalize their knowledge base to solve complex dynamic tasks in Deep RL.
\nocite{haarnoja2019_learn}
\nocite{hardy1952inequalities}
\nocite{lee2021sunrise}
\nocite{openAI}
\title{Bounding the Optimal Value Function \\in Compositional Reinforcement Learning\\(Supplementary Material)}
\onecolumn
\maketitle
\mathcal{S}ection{Introduction}
In the following, we discuss the results of additional experiments in the four room domain. In these experiments, we want to answer the following questions:
\begin{itemize}
\item How do the optimal policies and value functions compare to those calculated from the zero-shot approximations using the derived bounds?
\item What are other examples of compositions and functional transformations that can be analyzed using our approach?
\item Does warmstarting (using the derived bounds for initialization) in the tabular case improve the convergence?
\end{itemize}
To address these issues, we modify OpenAI's frozen lake environment \cite{openAI} to allow for stochastic dynamics.
In the tabular experiments, numerical solutions for the optimal $Q$ functions were obtained by solving the Bellman backup equations iteratively. Iterations are considered converged once the maximum difference between successive iterates is less than $10^{-10}$.
Beyond the motivating example shown in the main text, we have included video files demonstrating a full range of zero-shot compositions with convex weights between the Bottom Left (BL) room and Bottom Right (BR) room subtasks, in both entropy-regularized ($\beta=5$) and standard RL with deterministic dynamics. These videos, along with all code for the above experiments are made publicly available at a repository on \url{https://github.com/JacobHA/Q-Bounding-in-Compositional-RL}.
\mathcal{S}ection{Experiments}
\mathcal{S}ubsection{Function Approximators}
For function approximator experiments (as shown in the main text), we use the DQN implementation from Stable-Baselines3 \cite{stable-baselines3}. We first fully train the subtasks (seen in Fig. 1 of the main text). Then, we perform hyperparameter sweeps for each possible clipping option. Several hyperparameters are kept fixed (Table~\ref{tab:shared}), and we sweep with the range and distribution shown below in Table~\ref{tab:sweep}. Finally, we use the optimal hyperparameters (as measured by those which maximize the accumulated reward throughout training). These values are shown in Table~\ref{tab:optimal}.
\begin{center}
\captionof{table}{Hyperparameters shared by all Deep Q Networks}
\begin{tabular}{||c c||}
\hline
Hyperparameter & Value \\ [0.5ex]
\hline\hline
Buffer Size & 1,000,000 \\
\hline
Discount factor, $\gamma$ & 0.99 \\
\hline
$\epsilon_{\text{initial}}$ & 1.0 \\
\hline
$\epsilon_{\text{final}}$ & 0.05 \\
\hline
``learning starts'' & 5,000 \\
\hline
Target Update Interval & 10000 \\ [1ex]
\hline
\end{tabular}
\label{tab:shared}
\end{center}
\begin{center}
\captionof{table}{Hyperparameter Ranges Used for Finetuning}
\begin{tabular}{||c c c c||}
\hline
Hyperparameter & Sampling Distribution & Min. Value & Max. Value\\ [0.5ex]
\hline
Learning Rate & Log Uniform & $10^{-4}$ & $10^{-1}$ \\
\hline
Batch Size & Uniform & $32$ & $256$ \\
\hline
Exploration Fraction & Uniform & $0.1$ & $0.3$ \\
\hline
Polyak Update, $\tau$ & Uniform & $0.5$ & $1.0$ \\ [1ex]
\hline
\end{tabular}
\label{tab:sweep}
\end{center}
\begin{center}
\captionof{table}{Hyperparameters used for different clipping methods}
\begin{tabular}{||c c c c c||}
\hline
Hyperparameter & None & Soft & Hard & Soft-Hard \\ [0.5ex]
\hline
Learning Rate & $7.825\times10^{-4}$ & $3.732\times10^{-3}$ & $1.457\times10^{-3}$ & $3.184\times10^{-3}$ \\
\hline
Batch Size & 245 & 247 & 146 & 138\\
\hline
Exploration Fraction & 0.137 & 0.1075 & 0.1243 & 0.1207\\
\hline
Polyak Update, $\tau$ & 0.9107 & 0.9898 & 0.5545 & 0.7682 \\ [1ex]
\hline
\end{tabular}
\label{tab:optimal}
\end{center}
\mathcal{S}ubsection{Tabular experiments}
In these experiments we will demonstrate on simple discrete environments the effect of increasingly stochastic dynamics and increasingly dense rewards. As a proxy for measuring the usefulness or accuracy of the bound $f(Q)$, we calculate the mean difference between $f\left(Q(s,a)\right)-\widetilde{Q}(s,a)$, as well as the mean Kullback-Liebler (KL) divergence between $\pi$ (the true optimal policy) and $\pi_f$, the policy derived from the bound $f(Q)$. The proceeding experiments are situated in the entropy-regularized formalism (unless $\beta=\inf$ as shown in Fig.~\ref{fig:beta_inf_stoch_expt}) with the uniform prior policy $\pi_0(a|s) = 1/ |\mathcal{A}|$.
\mathcal{S}ubsubsection{Stochasticity of Dynamics}
In this experiment, we investigate the effect of stochastic dynamics on the bounds. Specifically, we vary the probability that taking an action will result in the intended action. This is equivalent to a slip probability.
\begin{figure}
\caption{Reward functions for a simple maze domain; used for stochasticity experiments. We place reward (whose cost is half the default step penalty of $-1$) at the edges of the room, denoted by an orange diamond. }
\label{fig:stochastic_desc}
\end{figure}
We notice in the following plots that at near-deterministic dynamics the bound becomes tighter. We also remark that the Kullback-Liebler divergence is lowest in very highly-stochastic environments. This is because for any $\beta>0$, the cost of changing the policy $\pi$ away from the prior policy is not worth it: the dynamics are so stochastic that there will be no considerable difference in trajectories even if significant controls (nearly deterministic choices) are applied via $\pi$.
\begin{figure}
\caption{$\beta=1$ KL divergence between $\pi$ and $\pi_f$ and average difference between optimal $Q$ function and presented bound.}
\label{fig:b1}
\end{figure}
\begin{figure}
\caption{$\beta=3$ KL divergence between $\pi$ and $\pi_f$ and average difference between optimal $Q$ function and presented bound.}
\label{fig:b3}
\end{figure}
\begin{figure}
\caption{$\beta=5$ KL divergence between $\pi$ and $\pi_f$ and average difference between optimal $Q$ function and presented bound.}
\label{fig:b5}
\end{figure}
\begin{figure}
\caption{$\beta=\inf$, standard RL.
Average difference between optimal $Q$ function and presented bound. Note that we do not plot a KL divergence for this case as $\pi$ is greedy and hence the divergence is always infinite.}
\label{fig:beta_inf_stoch_expt}
\end{figure}
\mathcal{S}ubsection{Sparsity of Rewards}
In this experiment, we consider an empty environment ($|S|\times |S|$ empty square) with reward $r=0$ everywhere and deterministic dynamics. No other rewards or obstacles are present. Then fix an integer $0<n<|S|$. Drawing randomly (without repetition), we choose one of the states of the environment to grant a reward, drawn uniformly between $(0, 1)$. We do this again for another copy of the empty environment.
We then compose these two (randomly generated as described) subtasks by using a simple average $F(r^{(1)}, r^{(2)}) = 0.5r^{(1)} + 0.5 r^{(2)}$. We have used $\beta=5$ for all experiments in this subsection.
\begin{figure}
\caption{$6\times 6$ environment. KL divergence between $\pi$ and $\pi_f$ and average difference between optimal $Q$ function and presented bound, with the shaded region representing one standard deviation over 1000 runs.}
\label{fig:6x6}
\end{figure}
\begin{figure}
\caption{$10\times 10$ environment. KL divergence between $\pi$ and $\pi_f$ and average difference between optimal $Q$ function and presented bound, with the shaded region representing one standard deviation over 1000 runs.}
\label{fig:10x10}
\end{figure}
Interestingly, we find a somewhat universal behavior, in that there is a certain level of density which makes the bound a poor approximation to the true $Q$ function. We also note that the bound is a better approximation at low densities.
\mathcal{S}ection{Boolean Composition Definitions}
In this section, we explicitly define the action of Boolean operators on subtask reward functions. These definitions are similar to those used by \cite{boolean}.
\begin{definition}[OR Composition]
Given subtask rewards $\{r^{(1)}, r^{(2)}, \dotsc , r^{(M)} \}$, the OR composition among them is given by the \textit{maximum} over all subtasks, at each state-action pair:
\begin{equation}
r^{(\text{OR})}(s,a) = \max_k r^{(k)}(s,a).
\end{equation}
\end{definition}
\begin{definition}[AND Composition]
Given subtask rewards $\{r^{(1)}, r^{(2)}, \dotsc , r^{(M)} \}$, the AND composition among them is given by the \textit{minimum} over all subtasks, at each state-action:
\begin{equation}
r^{(\text{AND})}(s,a) = \min_k r^{(k)}(s,a).
\end{equation}
\end{definition}
\begin{definition}[NOT Gate]
Given a subtask reward function $r$, applying the NOT gate transforms the reward function by negating all rewards (i.e. rewards $\to$ costs):
\begin{equation}
r^{(\text{NOT})}(s,a) = - r(s,a),
\end{equation}
\end{definition}
The proofs in all subsequent sections follow an inductive form based on the Bellman backup equation, whose solution converges to the optimal $Q$ function. This is a similar approach as employed by \cite{Haarnoja2018} and \cite{hunt_diverg}, but with the extension to all applicable functions; rather than (linear) convex combinations.
\mathcal{S}ection{Proofs for Standard RL}
Let $X$ be the codomain for the $Q$ function of the primitive task ($Q: \mathcal{S} \times \mathcal{A} \to X \mathcal{S}ubseteq \mathbb{R}$).
\begin{lemma}[Convex Conditions]\label{thm:convex_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$ (for stochastic dynamics);
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \leq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all $\mathcal{Q}: \mathcal{S} \times \mathcal{A} \to \mathbb{R}.$
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eqn:convex_std}
f(Q(s,a)) \leq \widetilde{Q}(s,a) \leq f(Q(s,a)) + C(s,a)
\end{equation}
where $C$ is the optimal value function for a task with reward
\begin{equation}\label{eq:std_convex_C_def}
r_C(s,a) = f(r(s,a)) + \gamma \mathbb{E}_{s'} V_f(s') - f(Q(s,a)).
\end{equation}
\end{lemma}
\begin{proof}
We will prove all inequalities by induction on the number of backup steps, $N$. We start with the lower bound $\widetilde{Q} \ge f(Q)$. The base case, $N=1$ is trivial since $f(r(s,a))=f(r(s,a))$. The inductive step is the assumption $\widetilde{Q}^{(N)}(s,a) \geq f(Q^{(N)}(s,a))$ for some $N>1$.
In the case of standard RL, the Bellman backup equation for transformed rewards is given by:
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) = f\left(r(s,a)\right) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} \max_{a'} \widetilde{Q}^{(N)}(s',a')
\end{equation}
Using the inductive assumption,
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f\left(r(s,a)\right) +
\gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} \max_{a'} f \left({Q}^{(N)}(s',a') \right)
\end{equation}
The condition $v_f(s) \ge f(v(s)) $ is used on the right hand side to give:
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f\left(r(s,a)\right) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} f \left( \max_{a'} {Q}^{(N)}(s',a') \right)
\end{equation}
Since $f$ is convex, we use Jensen's inequality to factor it out of the expectation. Note that this condition on $f$ is only required for stochastic dynamics. The error introduced by swapping these operators is characterized by the ``Jensen's gap'' for the transformation function $f$.
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f\left(r(s,a)\right) + \gamma f \left( \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} \max_{a'} {Q}^{(N)}(s',a') \right)
\end{equation}
Finally, using both sublinearity conditions
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f\left(r(s,a) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} \max_{a'} {Q}^{(N)}(s',a') \right)
\label{eq:pf:last_line_induction}
\end{equation}
where the right-hand side is simply $f(Q^{(N+1)}(s,a))$. Since this inequality holds for all $N$, we take the limit $N \to \infty$ wherein $Q^{(N)}$ converges to the optimal $Q$-function. For the right-hand side of Eq. \eqref{eq:pf:last_line_induction}, we thus have (by continuity of $f$):
\begin{equation}
\lim_{N \to \infty} f\left(Q^{(N)}(s,a)\right) = f\left(\lim_{N \to \infty}Q^{(N)}(s,a)\right) = f(Q(s,a))
\end{equation}
where $Q(s,a)$ is the optimal action value function for the primitive task.
Combined with the limit of the left-hand side, we arrive at the desired inequality:
\begin{equation}
\widetilde{Q}(s,a) \geq f\left(Q(s,a)\right).
\end{equation}
This completes the proof of the lower bound. To prove the upper bound we again use induction on the backup equation of $\widetilde{Q}^{(N)}$. We wish to show $\widetilde{Q}^{(N)} \le f\left(Q(s,a)\right) + C^{(N)}(s,a)$ holds for all $N$, with the definition of $C$ provided in Lemma~4.1.
Let $f$ satisfy the convex conditions.
Consider the backup equation for $\widetilde{Q}$.
Again, the base case ($N=1$) is trivially satisfied with equality. Using the inductive assumption, we find
\begin{align*}
\widetilde{Q}^{(N+1)}(s,a) &= f(r(s,a)) + \gamma \E_{s'} \max_{a'} \widetilde{Q}^{(N)}(s',a')
\\
&\le f(r(s,a)) + \gamma \E_{s'} \max_{a'} \left( f(Q(s',a')) + C^{(N)}(s',a')\right)
\\
&\le f(r(s,a)) + \gamma \E_{s'} \max_{a'} f(Q(s',a')) + \gamma \E \max_{a'}C^{(N)}(s',a')
\\
&= f(Q(s,a)) + \left[ f(r_i) + \gamma \E_{s'} V_f(s') - f(Q(s,a)) \right] + \gamma \E_{s'} \max_{a'}C^{(N)}(s',a')
\\
&= f(Q(s,a)) + C^{(N+1)}(s,a)
\end{align*}
\end{proof}
At this point, we verify that $C(s,a)>0$ which ensures the double-sided bounds above are valid.
To do so, we can simply bound the reward function $r_C(s,a)$. By determining $r_C(s,a)>0$, this will ensure $C(s,a) > \min r_C / (1-\gamma) > 0$.
\begin{align*}
r_C(s,a) &= f(r(s,a)) + \gamma \mathbb{E}_{s'} V_f(s') - f(Q(s,a)) \\
&\geq f(r(s,a)) + \gamma \mathbb{E}_{s'} f(V(s')) - f(Q(s,a)) \\
&\geq f(r(s,a)) + f(\gamma \mathbb{E}_{s'} V(s')) - f(Q(s,a)) \\
&\geq f(r(s,a) + \gamma \mathbb{E}_{s'} V(s')) - f(Q(s,a))\\
&\geq 0
\end{align*}
where each line follows from the required conditions in Lemma \ref{thm:convex_cond_std}. A similar proof holds for showing the quantities $\hat{C}$, $D$, $\hat{D}$ are all non-negative.
We now prove the policy evaluation bound for standard RL.
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq D(s,a)
\end{equation}
where $D$ is the value of the policy $\pi_f$ in a task with reward
\begin{align*}
r_D = \gamma \mathbb{E}_{s',a' \mathcal{S}im{} \pi_f} \biggr[ \max_{a} \big( f(Q(s',a')) + C(s',a') \big) - f(Q(s,a)) \biggr]
\end{align*}
\end{lemma}
\def\widetilde{Q}^{\pi_f}(s,a){\widetilde{Q}^{\pi_f}(s,a)}
\def\widetilde{Q}^{\pi_f}(s,a)p{\widetilde{Q}^{\pi_f}(s',a')}
\deff\left(r(s,a)\right){f\left(r(s,a)\right)}
\begin{proof}
We will again prove this bound by induction on steps in the Bellman backup equation for the value of $\pi_f$, as given by the following fixed point equation:
\begin{equation}
\widetilde{Q}^{\pi_f}(s,a) = f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\widetilde{Q}^{\pi_f}(s,a)p
\end{equation}
We consider the following initial conditions: $ \widetilde{Q}^{\pi_f(0)}(s,a) = \widetilde{Q}(s,a), D(s,a)=0$.
We note that there is freedom in the choice of initial conditions, as the final statement (regarding the optimal value functions) holds regardless of initialization. As usual, the base case is trivially satisfied. We will now show that the equivalent inequality
\begin{equation}
\widetilde{Q}^{\pi_f (N)}(s,a) \ge \widetilde{Q}(s,a) - D^{(N)}(s,a)
\end{equation}
holds for all $N$. Similar to the previous proofs, we will subsequently take the limit $N \to \infty$ to recover the desired result.
To do so, we consider the next step of the Bellman backup, and apply the inductive hypothesis:
\begin{align}
\widetilde{Q}^{\pi_f (N+1)}(s,a) &= f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( \widetilde{Q}^{\pi_f (N)}(s',a') \right) \\
&\geq f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( \widetilde{Q}(s',a') - D^{(N)}(s',a') \right) \\
&\geq f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - D^{(N)}(s',a') \right) \\
&= f\left(r(s,a)\right) + \gamma \E_{s'} \widetilde{V}(s') + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - D^{(N)}(s',a') - \widetilde{V}(s') \right) \\
&\geq \widetilde{Q}(s,a) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - D^{(N)}(s',a') - \max_{a'} \left\{ f\left(Q(s',a')\right) + C(s',a') \right\} \right) \\
&= \widetilde{Q}(s,a) - \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( \max_{a'} \left\{ f\left(Q(s',a')\right) + C(s',a') \right\} - f\left(Q(s',a')\right) + D^{(N)}(s',a')\right) \\
&= \widetilde{Q}(s,a) - \left( r_D(s,a) + \gamma \E_{s',a' \mathcal{S}im{} \pi_f} D^{(N)}(s',a') \right) \\
&= \widetilde{Q}(s,a) - D^{(N+1)}(s,a)
\end{align}
The third and fifth line follow from the previous bounds (Lemma 4.1). In the limit $N \to \infty$, we can thus see that the fixed point $D$ corresponds to the policy evaluation for $\pi_f$ in an environment with reward function $r_D$.
\end{proof}
Now we prove similar results, but for the ``concave conditions'' presented in the main text.
\begin{lemma}[Concave Conditions]\label{thm:concave_cond_std}
Given a primitive task with discount factor $\gamma$ and a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$ (for stochastic dynamics);
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \geq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$
\end{enumerate}
then the optimal action-value functions are now related in the following way:
\begin{equation}\label{eqn:concave_std}
f(Q(s,a)) - \hat{C}(s,a) \leq \widetilde{Q}(s,a) \leq f(Q(s,a))
\end{equation}
where $\hat{C}$ is the optimal value function for a task with reward
\begin{equation}
\hat{r}_C(s,a) = f(Q(s,a)) - f(r(s,a)) - \gamma \E_{s'} V_f(s')
\end{equation}
\end{lemma}
\begin{proof}
The proof of $\widetilde{Q} \le f(Q)$ is the same as the preceding theorem's lower bound but with all inequalities reversed. To prove the upper bound involving $\hat{C}$, we use a similar approach
\begin{align*}
\widetilde{Q}^{(N+1)}(s,a) &= f(r(s,a)) + \gamma \E_{s'} \max_{a'} \widetilde{Q}^{(N)}(s',a')
\\
&\ge f(r(s,a)) + \gamma \E_{s'} \max_{a'} \left( f(Q(s',a')) - \hat{C}^{(N)}(s',a')\right)
\\
&\ge f(r(s,a)) + \gamma \E_{s'} \left( \max_{a'} f(Q(s',a')) - \max_{a'} \hat{C}^{(N)}(s',a') \right)
\\
&= f(Q(s,a)) - \left[f(Q(s,a)) - f(r(s,a)) - \gamma \E_{s'} V_f(s') + \gamma \E_{s'} \max_{a'} \hat{C}^{(N)}(s',a')\right]
\\
&= f(Q(s,a)) - \hat{C}^{(N+1)}(s,a)
\end{align*}
The second line follows from the inductive hypothesis. The third line follows from the $\max$ of a difference. In the penultimate line, we add and subtract $f(Q)$, and identify the definitions for $V_f$ and the backup equation for $\hat{C}$. In the limit $N \to \infty$, we have the desired result.
\end{proof}
\begin{lemma}
Consider the value of the policy $\pi_f(s) = \max_{a} f(Q(s,a))$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}(s,a)$.
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq \hat{D}(s,a)
\end{equation}
where $\hat{D}$ is the value of the policy $\pi_f$ in a task with reward
\begin{equation}
\hat{r}_D = \gamma \mathbb{E}_{s',a' \mathcal{S}im{} \pi_f} \biggr[ V_f(s') - f(Q(s',a')) + \hat{C}(s',a') \biggr]
\end{equation}
\end{lemma}
\begin{proof}
The proof of this result is similar to that of Lemma 4.2, except now we must employ the corresponding results of Lemma 4.3. Beginning with a substitution of the inductive hypothesis:
\begin{align}
\widetilde{Q}^{\pi_f (N+1)}(s,a) &= f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( \widetilde{Q}^{\pi_f (N)}(s',a') \right) \\
&\geq f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( \widetilde{Q}(s',a') - \hat{D}^{(N)}(s',a') \right) \\
&\geq f\left(r(s,a)\right) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - \hat{C}(s',a') - \hat{D}^{(N)}(s',a') \right) \\
&= f\left(r(s,a)\right) + \gamma \E_{s'} \widetilde{V}(s') + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - \hat{C}(s',a') - \hat{D}^{(N)}(s',a') - \widetilde{V}(s') \right) \\
&\geq \widetilde{Q}(s,a) + \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( f\left(Q(s',a')\right) - \hat{C}(s',a') - \hat{D}^{(N)}(s',a') - V_f(s') \right) \\
&= \widetilde{Q}(s,a) - \gamma \E_{s', a' \mathcal{S}im{} \pi_f}\left( V_f(s') - f\left(Q(s',a')\right)+ \hat{C}(s',a') + \hat{D}^{(N)}(s',a')\right) \\
&= \widetilde{Q}(s,a) - \left( \hat{r}_D(s,a) + \gamma \E_{s',a' \mathcal{S}im{} \pi_f} \hat{D}^{(N)}(s',a') \right) \\
&= \widetilde{Q}(s,a) - \hat{D}^{(N+1)}(s,a)
\end{align}
\end{proof}
Now we provide further details on the technical conditions for compositions (rather than transformations) of primitive tasks to satisfy the derived bounds.
\begin{lemma}[Convex Composition of Primitive Tasks]
Suppose $F:\bigotimes_k X^{(k)} \to \mathbb{R}$ is convex on its domain and is sublinear (separately in each argument), that is:
\begin{align}
F(x^{(1)}+y^{(1)},x^{(2)}, \dotsc, x^{(M)}) &\leq F(x^{(1)},x^{(2)},\dotsc, x^{(M)}) + F(y^{(1)},x^{(2)},\dotsc, x^{(M)}) \\
F(x^{(1)},x^{(2)}+y^{(2)}, \dotsc, x^{(M)}) &\leq F(x^{(1)},y^{(2)},\dotsc, x^{(M)}) + F(x^{(1)},y^{(2)},\dotsc, x^{(M)})
\end{align}
and similarly for the remaining arguments.
\begin{equation}
F(\gamma x^{(1)}, \dotsc, \gamma x^{(M)}) \leq \gamma F(x^{(1)}, \dotsc x^{(M)})
\end{equation}and also satisfies
\begin{equation}
F\left( \max_a \mathcal{Q}^{(1)}(s,a), \dotsc , \max_a \mathcal{Q}^{(M)}(s,a) \right) \leq \max_a F\left(\mathcal{Q}^{(1)}(s,a), \dotsc , \mathcal{Q}^{(M)}(s,a)\right)
\end{equation}
for all functions $\mathcal{Q}^{(k)}:\mathcal{S} \times \mathcal{A} \to \mathbb{R}$.
Then,
\begin{equation}
F(\vec{Q}(s,a)) \le \widetilde{Q}(s,a) \le F(\vec{Q}(s,a)) + C(s,a)
\end{equation}
where we use a vector notation to emphasize that the function acts over the set of optimal value functions $\{Q^{(k)}\}$ corresponding to each primitive task, defined by $r^{(k)}$.
\end{lemma}
\begin{proof}
The proof of this statement is identical to the proof of Lemma \ref{thm:convex_cond_std}, now using the fact that $F$ is a multivariable function $F: X^N \to Y$, with each argument obeying the required conditions. $C$ takes the analogous definition as provided for the original result.
\end{proof}
\begin{lemma}[Concave Composition of Primitive Tasks]\label{thm:compos_concave_std}
If on the other hand $F$ is concave and superlinear in each argument, and also satisfies
\begin{equation}
F\left( \max_a \mathcal{Q}^{(1)}(s,a), \dotsc , \max_a \mathcal{Q}^{(M)}(s,a) \right) \leq \max_a F\left(\mathcal{Q}^{(1)}(s,a), \dotsc , \mathcal{Q}^{(M)}(s,a)\right)
\end{equation}
for all functions $\mathcal{Q}^{(k)}:\mathcal{S} \times \mathcal{A} \to \mathbb{R}$, then
\begin{equation}
F(\vec{Q}(s,a)) - \hat{C}(s,a) \le \widetilde{Q}(s,a) \le F(\vec{Q}(s,a)).
\end{equation}
\end{lemma}
\begin{proof}
Again, the proof of this statement is identical to the proof of Lemma \ref{thm:concave_cond_std}, now using the fact that $F$ is a multivariable function $F: X^N \to Y$, with each argument obeying the required conditions.
\end{proof}
\mathcal{S}ubsection{Examples of transformations and compositions}
In this section, we consider the examples of transformations and compositions mentioned in the main text, and discuss the corresponding results in standard RL.
\begin{remark}
Given the convex composition of subtasks $r^{(c)} \equiv f(\{r^{(k)}\}) = \mathcal{S}um_k \alpha_k r^{(k)}$ considered by \cite{Haarnoja2018} and \cite{hunt_diverg}, we can use the results of Lemma \ref{thm:compos_concave_std} to bound the optimal $Q$ function by using the optimal $Q$ functions for the primitive tasks:
\begin{equation}
Q^{(c)}(s,a) \leq \mathcal{S}um_k \alpha_k Q^{(k)}(s,a)
\end{equation}
\end{remark}
\begin{proof}
In standard RL, we need only show that $f( \max_i x_{1i}, \dotsc , \max_i x_{ni} ) \geq \max_i f(x_i, \dotsc , x_n)$:
\begin{equation}
\mathcal{S}um_k \alpha_k \max_i x^{(k)}_{i} \geq \max_i \mathcal{S}um_k \alpha_k x^{(k)}_i
\end{equation} which holds given $\alpha_k \geq 0$ for all $k$.
We also note that in this case the result clearly holds for general $\alpha_k \geq 0$ not necessarily with $\mathcal{S}um_k \alpha_k = 1$ (as assumed in \cite{Haarnoja2018} and \cite{hunt_diverg}).
\end{proof}
\begin{remark}
Given the AND composition defined above and considered in \cite{boolean}, we have the following result in standard RL:
\begin{equation}
Q^{\text{AND}}(s,a) \leq \min_k \left\{Q^{(k)}(s,a)\right\}
\end{equation}
\end{remark}
\begin{proof}
We could proceed via induction as in the previous proofs, or simply use the above remark, and prove the necessary conditions on the function $f(\cdot) = \min(\cdot)$.
The function $\min(\cdot)$ is concave in each argument. It is also straightforward to show that $\min(\cdot)$ is subadditive over all arguments.
\end{proof}
\begin{remark}
Result of (hard) OR composition result in standard RL:
\begin{equation}
Q^{\text{OR}}(s,a) \geq \max_k \left\{Q^{(k)}(s,a)\right\}
\end{equation}
\end{remark}
\begin{proof}
The proof is analogous to the (hard) AND result: $\max$ is a convex, superadditive function.
\end{proof}
\begin{remark}
Result for NOT operation in standard RL:
\begin{equation}
Q^{\text{NOT}}(s,a) \geq - Q(s,a)
\end{equation}
\end{remark}
\begin{proof}
Since the ``NOT'' gate is a unary function, and we are in the standard RL setting, we must check the conditions of Lemma 4.1 or 4.3. Moreoever, since the transformation function applied to the rewards, $f(r)=-r$ is linear, we must check the final condition: $\max_i\{-x_i\} = -\min_i\{x_i\} \geq -\max_i\{x_i\}$. This is the condition required by the concave conditions.
\end{proof}
\mathcal{S}ection{Proofs for Entropy-Regularized RL}
Let $X$ be the codomain for the $Q$ function of the primitive task ($Q: \mathcal{S} \times \mathcal{A} \to X \mathcal{S}ubseteq \mathbb{R}$).
\begin{lemma}[Convex Conditions]
\label{thm:forward_cond_entropy-regularized}
Given a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$ (for stochastic dynamics);
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \log \E \exp \mathcal{Q}(s,a) \right) \leq \log \E \exp f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to \mathbb{R}.$
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eq:convex_entropy-regularized}
f \left( Q(s,a) \right) \leq \widetilde{Q}(s,a) \leq f \left( Q(s,a) \right) + C(s,a)
\end{equation}
\end{lemma}
\begin{proof}
We will again prove the result with induction, beginning by writing the backup equation for the optimal soft $Q$ function in the transformed reward environment to prove the upper bound on $\widetilde{Q}$:
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) = f(r(s,a)) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} f\left(r(s,a)\right)ac{1}{\beta} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(a'|s')} \exp \left(\beta Q^{(N)}(s',a')\right)
\end{equation}
where $p$ is the dynamics and $\pi_0$ is the prior policy. Applying the inductive assumption,
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f(r(s,a)) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} f\left(r(s,a)\right)ac{1}{\beta} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(a'|s')}\exp \left( f\left(\beta Q^{(N)}(s',a')\right) \right)
\end{equation}
Next, using the third condition on $f$ as well as its convexity, we may factor $f$ out of the expectations by Jensen's inequality:
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f(r(s,a)) + \gamma f\left( \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} f\left(r(s,a)\right)ac{1}{\beta} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(a'|s')}\exp \left(\beta Q^{(N)}(s',a')\right) \right)
\end{equation}
Finally, using the sublinearity conditions of $f$, we arrive at
\begin{equation}
\widetilde{Q}^{(N+1)}(s,a) \geq f \left(r(s,a) + \gamma \mathbb{E}_{s' \mathcal{S}im{} p(s'|s,a)} f\left(r(s,a)\right)ac{1}{\beta} \log \mathbb{E}_{a' \mathcal{S}im{} \pi_0(a'|s')}\exp \left(\beta Q^{(N)}(s',a')\right) \right)
\end{equation}
The right hand side is $f \left(Q^{(N+1)}(s,a) \right)$. In the limit $N\to \infty,\ Q^{(N)}(s,a) \to Q(s,a)$ so the inductive proof for the upper bound is complete.
Let $f$ satisfy the ``convex conditions''.
Consider the backup equation for $\widetilde{Q}$. For the initialization (base case) we let $\widetilde{Q}^{(0)}(s,a)=f\left(Q(s,a)\right)$ and $C^{(0)}(s,a)=0$.
Using the inductive assumption,
\begin{align*}
\widetilde{Q}^{(N+1)}(s,a) &= f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s' \mathcal{S}im{} p} \log \E_{a' \mathcal{S}im{} \pi_0} \exp \beta \widetilde{Q}^{(N)}(s',a')
\\
&\leq f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \log \E_{a'} \exp \beta \left( f(Q(s',a')) + C^{(N)}(s',a')\right)
\\
&\leq f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \left(\log \E_{a'} \exp \beta f(Q(s',a')) + \max_{a'} C^{(N)}(s',a')\right)
\\
&= f(Q(s,a)) + f(r(s,a)) + \gamma \E_{s'} V_f(s') - f(Q(s,a)) + \gamma \E_{s'} \max_{a'} C^{(N)}(s',a')
\\
&= f(Q(s,a)) + C^{(N+1)}(s,a)
\end{align*}
Therefore in the limit $N \to \infty$, we have:
$\widetilde{Q}(s,a) \leq f(Q(s,a)) + C(s,a)$ as desired. We note that since $f(r(s,a))~+~\gamma~\E_{s'}~V_f(s')~\geq~f(Q(s,a))$, we immediately have $C(s,a) \ge 0$, as is required for the bound to be non-vacuous.
\end{proof}
\begin{lemma}
Consider the soft value of the policy $\pi_f \propto \exp \beta f(Q)$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}$(s,a).
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq D(s,a)
\end{equation}
where $D$ is the value of the policy $\pi_f$
with reward
\begin{equation}
r_D(s,a) = \gamma \mathbb{E}_{s' \mathcal{S}im{} p}
\left[ \max_{b} \left\{ f\left(Q(s',b)\right) + C(s',b) \right\} -V_f(s') \right]
\end{equation}
\end{lemma}
\begin{proof}
To prove the (soft) policy evaluation bound, we use iterations of soft-policy evaluation \cite{Haarnoja_SAC} and denote iteration $N$ of the evaluation of $\pi_f$ in the composite environment as $\widetilde{Q}^{\pi_f(N)}$. Beginning with the definitions $\widetilde{Q}^{\pi_f(0)}(s,a) = Q(s,a)$ (since the evaluation is independent of the initialization), and $D^{(0)}=0$, the $N=0$ step is trivially satisfied. Assuming the inductive hypothesis, we consider the next step of soft policy evaluation:
As in the previous policy evaluation results, we prove an equivalent result with induction.
\begin{align*}
\widetilde{Q}^{\pi_f(N+1)}(s,a) &= f(r(s,a)) + \gamma \E_{s' \mathcal{S}im{} p}\E_{ a'\mathcal{S}im{} \pi_f} \left[\widetilde{Q}^{\pi_f(N)}(s',a') - f\left(r(s,a)\right)ac{1}{\beta} \log f\left(r(s,a)\right)ac{\pi_f(a'|s')}{\pi_0(a'|s')} \right]
\\
&\geq f(r(s,a)) + \gamma \E_{s',a'} \left[\widetilde{Q}(s',a') - D^{(N)}(s',a') - f(Q(s',a')) + V_f(s') \right]
\\
&= f(r(s,a)) + \gamma \E_{s'}\widetilde{V}(s') + \gamma \E_{s',a'} \left[\widetilde{Q}(s',a') - D^{(N)}(s',a') - f(Q(s',a')) + V_f(s') - \widetilde{V}(s')\right] \\
&\geq \widetilde{Q}(s,a) + \gamma \E_{s',a'} \left[f(Q(s',a')) - D^{(N)}(s',a') - f(Q(s',a')) + V_f(s') - \widetilde{V}(s')\right] \\
&\geq \widetilde{Q}(s,a) + \gamma \E_{s',a'} \left[ - D^{(N)}(s',a') + V_f(s') - \max_{b} \left\{ f\left(Q(s',b)\right) + C(s',b) \right\} \right] \\
&\geq \widetilde{Q}(s,a) -D^{(N+1)}(s,a)
\\
\end{align*}
where we have used $\widetilde{Q}(s,a) \geq f(Q(s,a))$ in the fourth line.
where we have used the fact that $ \widetilde{V}(s) \leq \max_b \left\{ f\left(Q(s,b)\right) + \max_a C(s,a)\right\}$ and $\widetilde{Q}(s,a) - f(Q(s,a)) \geq 0$ which both follow from the previously stated bounds.
\end{proof}
\begin{lemma}[Concave Conditions]
\label{thm:reverse_cond_entropy-regularized}
Given a bounded, continuous transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$ (for stochastic dynamics);
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \log \E \exp \mathcal{Q}(s,a) \right) \geq \log \E \exp f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to \mathbb{R}.$
\end{enumerate}
then the optimal action-value function for the transformed rewards obeys the following inequality:
\begin{equation}\label{eq:concave_entropy-regularized}
f\left( Q(s,a) \right) - \hat{C}(s,a) \leq \widetilde{Q}(s,a) \leq f \left( Q(s,a) \right)
\end{equation}
\end{lemma}
\begin{proof}
The proof of the upper bound is the same as the preceding theorem's lower bound with all inequalities reversed.
For the lower bound involving $C$,
Again consider the backup equation for $\widetilde{Q}$.
Using the definitions and inductive assumption as before, we have
\begin{align*}
\widetilde{Q}^{(N+1)}(s,a) &= f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s' \mathcal{S}im{} p} \log \E_{a' \mathcal{S}im{} \pi_0} \exp \beta \widetilde{Q}^{(N)}(s',a')
\\
&\geq f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \log \E_{a'} \exp \beta \left( f(Q(s',a')) -\hat{C}^{(N)}(s',a')\right)
\\
&\geq f(r(s,a)) + f\left(r(s,a)\right)ac{\gamma}{\beta} \E_{s'} \left(\log \E_{a'} \exp \beta f(Q(s',a')) - \max_{a'} \hat{C}^{(N)}(s',a')\right)
\\
&= f(Q(s,a)) - \left[f(Q(s,a)) - f(r(s,a)) - \gamma \E_{s'} V_f(s') + \gamma \E_{s'} \max_{a'} \hat{C}^{(N)}(s',a')\right]
\\
&= f(Q(s,a)) - \hat{C}^{(N+1)}(s,a)
\end{align*}
Therefore in the limit $N \to \infty$, we have:
$\widetilde{Q}(s,a) \geq f(Q(s,a)) - \hat{C}(s,a)$ as desired.
\end{proof}
\begin{lemma}
Consider the soft value of the policy $\pi_f \propto \exp \beta f(Q)$ on the transformed task of interest, denoted by $\widetilde{Q}^{\pi_f}$(s,a).
The sub-optimality of $\pi_f$ is then upper bounded by:
\begin{equation}
\widetilde{Q}(s,a) - \widetilde{Q}^{\pi_f}(s,a) \leq \hat{D}(s,a)
\end{equation}
where $\hat{D}$ is the fixed point of
\begin{equation}
\hat{D}(s,a) \xleftarrow{} \gamma \mathbb{E}_{s' \mathcal{S}im{} p}\mathbb{E}_{a' \mathcal{S}im{} \pi_f} \left[ \hat{C}(s',a') + \hat{D}(s',a') \right]
\end{equation}
\end{lemma}
\begin{proof}
We will show the policy evaluation result by induction, by evaluating $\pi_f \propto \exp(\beta f(Q))$ in the environment with rewards $f(r)$. We shall denote iterations of policy evaluation for $\pi_f$ in the environment with rewards $f(r)$ by $\widetilde{Q}^{\pi_f(N)}(s,a)$.
\begin{align*}
\widetilde{Q}^{\pi_f(N+1)}(s,a) &= f(r(s,a)) + \gamma \E_{s'\mathcal{S}im{}p} \E_{a'\mathcal{S}im{} \pi_f} \left[\widetilde{Q}^{\pi_f(N)}(s',a') - f\left(r(s,a)\right)ac{1}{\beta} \log f\left(r(s,a)\right)ac{\pi_f(a'|s')}{\pi_0(a'|s')} \right]
\\
&\geq f(r(s,a)) + \gamma \E_{s',a'} \left[\widetilde{Q}(s',a')-\hat{D}^{(N)}(s',a') - (f(Q(s',a')) - V_f(s')) \right]
\\
&\geq f(r(s,a)) + \gamma \E_{s',a'} \left[\widetilde{Q}(s',a')-\hat{D}^{(N)}(s',a') - \widetilde{Q}(s',a') -\hat{C}(s',a') + V_f(s') \right]
\\
&\ge f(r(s,a)) + \gamma \E_{s'} \widetilde{V}(s') - \gamma \E_{s',a'} \left[\hat{D}^{(N)}(s',a') + \hat{C}(s',a') \right]
\\
&= \widetilde{Q}(s,a)) - \hat{D}^{(N+1)}(s,a)
\\
\end{align*}
where we have used the inductive assumption and $V_f(s) \ge \widetilde{V}(s)$ and which follows from the previously stated bounds.
Therefore in the limit $N \to \infty$, we have:
$ \widetilde{Q}^{\pi_f}(s,a) \geq \widetilde{Q}(s,a) - \hat{D}(s,a)
$ as desired.
\end{proof}
\begin{lemma}[Convex Composition of Primitive Tasks]\label{thm:compos_convex_maxent}
Suppose $F:X^N \to Y$ is convex on its domain $X^N$ and satisfies all conditions of Lemma 5.1 (Main Text) component-wise. Then,
\begin{equation}
F(\vec{Q}(s,a)) \le \widetilde{Q}(s,a) \le F(\vec{Q}(s,a)) + C(s,a)
\end{equation}
and
\begin{equation}
\widetilde{Q}^{\pi_f}(s,a) \geq \widetilde{Q}(s,a) - D(s,a)
\end{equation}
where we use a vector notation to emphasize that the function acts over the set of optimal $\{Q_k\}$ functions corresponding to each subtask, defined by $r_k$.
\end{lemma}
\begin{proof}
The proof of this statement is identical to the previous proofs, now using the fact that $F$ is a multivariable function $F: X^N \to Y$, with each argument obeying the required conditions.
\end{proof}
\begin{lemma}[Concave Composition of Primitive Tasks]\label{thm:compos_concave_maxent}
If on the other hand $F$ is concave and and satisfies all conditions of Lemma 5.2 (Main Text) component-wise, then
\begin{equation}
F(\vec{Q}(s,a)) - \hat{C}(s,a) \le \widetilde{Q}(s,a) \le F(\vec{Q}(s,a)).
\end{equation}
and
\begin{equation}
\widetilde{Q}^{\pi_f}(s,a) \geq \widetilde{Q}(s,a) - \hat{D}(s,a)
\end{equation}
\end{lemma}
\begin{proof}
Again, the proof of this statement is identical to the previous proofs, now using the fact that $F$ is a multivariable function $F: X^N \to Y$, with each argument obeying the required conditions.
\end{proof}
\mathcal{S}ubsection{Examples of Transformations and Compositions}
In this section we consider several examples mentioned in the main text, and show how they are proved with our results in entropy-regularized RL.
\begin{remark}
Given the convex composition of subtasks $r^{(c)} \equiv F(\{r^{(k)}\}) = \mathcal{S}um_k \alpha_k r^{(k)}$ considered by \cite{Haarnoja2018} and \cite{hunt_diverg}, we can use the results of Lemma \ref{thm:compos_concave_maxent} to bound the optimal $Q$ function by using the optimal $Q$ functions for the primitive tasks:
\begin{equation}
Q^{(c)}(s,a) \leq \mathcal{S}um_k \alpha_k Q^{(k)}(s,a)
\end{equation}
\end{remark}
\begin{proof}
In entropy-regularized RL we need to show that the final condition holds (in vectorized form). This is simply H\"older's inequality \cite{hardy1952inequalities} for vector-valued functions in a probability space (with measure defined by $\pi_0$).
\end{proof}
\begin{remark}
Given the AND composition defined above and considered in \cite{boolean}, we have the following result in standard RL:
\begin{equation}
Q^{\text{AND}}(s,a) \leq \min_k \left\{Q^{(k)}(s,a)\right\}
\end{equation}
\end{remark}
\begin{proof}
The function $\min(\cdot)$ is concave in each argument. It is also straightforward to show that $\min(\cdot)$ is subadditive over all arguments. For the final condition, the $\min$ acts globally over all subtasks:
\begin{equation}
\min_k \left\{ f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp\left(\beta \mathcal{Q}^{(k)}(s,a)\right)\right\} \leq f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp\left( \beta \min_k \left\{\mathcal{Q}^{(k)}(s,a)\right\}\right).
\end{equation}
\end{proof}
\begin{remark}
Result of (hard) OR composition result in standard RL:
\begin{equation}
Q^{\text{OR}}(s,a) \geq \max_k \left\{Q^{(k)}(s,a)\right\}
\end{equation}
\end{remark}
\begin{proof}
The proof is analogous to the (hard) AND result: $\max$ is a convex, superadditive function.
For the final condition, the $\max$ again acts globally over all subtasks:
\begin{equation}
\max_k \left\{ f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp\left(\beta
\mathcal{Q}^{(k)}(s,a)\right)\right\} \ge f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp\left( \beta \max_k \left\{\mathcal{Q}^{(k)}(s,a)\right\}\right).
\end{equation}
\end{proof}
\begin{remark}
Again we consider the NOT operation defined above, now in entropy-regularized RL, which yields the bound:
\begin{equation}
Q^{\text{NOT}}(s,a) \geq - Q(s,a)
\end{equation}
\end{remark}
\begin{proof}
As in the standard RL case, we need only consider the third condition of either Lemma 5.1 or 5.3.
In particular, we show
\begin{equation}
f\left( \log \E \exp \mathcal{Q}(s,a) \right) \leq \log \E \exp f\left( \mathcal{Q}(s,a) \right)
\end{equation}
for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to \mathbb{R}$. This follows from
\begin{align}
f\left(r(s,a)\right)ac{1}{\E \exp \mathcal{Q}(s,a)} \leq \E f\left(r(s,a)\right)ac{1}{\exp \mathcal{Q}(s,a) }
\end{align}
which is given by Jensen's inequality, since the function $f(x)=1/x$ is convex.
\end{proof}
\begin{remark}[Linear Scaling]
\label{thm:scaling}
Given some $k \in (0,1)$ the function $f(x) = k x$ satisfies the results of the first theorem. Conversely, if $k \geq 1$, $f(x) = k x$ satisfies the results of the second theorem.
\end{remark}
\begin{proof}
This result (specifically the third condition of Lemma 5.1, 5.3) follows from the monotonicity of $\ell_p$ norms.
\end{proof}
Since we have already shown the case of $k=-1$ (NOT gate), with the result of Theorem \ref{thm:compos}, the case for all $k \in \mathbb{R}$ has been characterized.
\mathcal{S}ection{Extension for Error-Prone $Q$-Values}
In this section, we provide some discussion on the case of inexact $Q$-values, as often occurs in practice (discussed at the end of Section 4.1 in the main text). We focus on the case of task transformation in standard RL. The corresponding statements in the settings of composition and entropy-regularized RL follow similarly.
As our starting point, we assume that an ``$\varepsilon$-optimal estimate'' $\overbar{Q}(s,a)$ for a primitive task's exact value function $Q(s,a)$ is known.
\begin{definition}
An $\varepsilon$-optimal $Q$-function, $\overbar{Q}$, satisfies
\begin{equation}
|Q(s,a)-\overbar{Q}(s,a)|\leq \varepsilon
\end{equation}
for all $s \in \mathcal{S}, a \in \mathcal{A}$.
\end{definition}
To allow the derived double-sided bounds on the transformed tasks' $Q$-values to carry over to this more general setting, we assume that the transformation function is $L$-Lipschitz continuous. With these assumptions, we prove the following extensions of Lemma 4.1 and 4.3:
\begin{customlemma}{4.1A}[Convex Conditions, Error-Prone]\label{thm:convex_cond_std_err}
Given a primitive task with discount factor $\gamma$, corresponding $\varepsilon$-optimal value function $\overbar{Q}$, and a bounded, continuous, $L$-Lipschitz transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is convex on its domain $X$ (for stochastic dynamics);
\item $f$ is sublinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \leq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \leq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \leq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all $\mathcal{Q}: \mathcal{S} \times \mathcal{A} \to \mathbb{R}.$
\end{enumerate}
then the optimal action-value function for the transformed rewards, $\widetilde{Q}$, is now related to the optimal action-value function with respect to the original rewards by:
\begin{equation}\label{eqn:convex_std_err}
f(\overbar{Q}(s,a)) - L \varepsilon \leq \widetilde{Q}(s,a) \leq f(\overbar{Q}(s,a)) + \overbar{C}(s,a) + f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon
\end{equation}
where $\overbar{C}$ is the optimal value function for a task with reward
\begin{equation}\label{eq:std_convex_C_def_err}
\overbar{r_C}(s,a) = f(r(s,a)) + \gamma \mathbb{E}_{s'} \overbar{V_f}(s') - f(\overbar{Q}(s,a)).
\end{equation}
with $\overbar{V_f}(s)=\max_a f(\overbar{Q}(s,a))$.
\end{customlemma}
Note that as $\varepsilon \to 0$, the exact result (Lemma 4.1) is recovered. If the function $\overbar{C}$ is not known exactly, one can similarly exchange $\overbar{C}$ for $\overbar{\overbar{C}}$, an $\varepsilon$-optimal estimate for $\overbar{C}$. This consideration loosens the upper-bound by an addition of $\varepsilon$, shown at the end of the proof.
We will make use a well-known result (cf. proof of Lemma 1 in \cite{barreto_sf}) that bounds the difference in optimal $Q$-values for two tasks with different reward functions.
\begin{lemma}
Let two tasks, only differing in their reward functions, be given with reward $r_1(s,a)$ and $r_2(s,a)$, respectively. Suppose $|r_1(s,a)-r_2(s,a)|\leq \delta$ Then, the optimal value functions for the tasks satisfies:
\begin{equation}
|Q_1(s,a)-Q_2(s,a)|\leq f\left(r(s,a)\right)ac{\delta}{1-\gamma}
\end{equation}
\label{lem:bounded_q_diff}
\end{lemma}
Now we are in a position to prove Lemma \ref{thm:convex_cond_std_err}:
\begin{proof}
To prove the lower bound, we begin with the original lower bound in Lemma 4.1, for the optimal primitive task $Q$-values:
\begin{equation}
\widetilde{Q}(s,a) \geq f(Q(s,a)),
\end{equation}
or equivalently
\begin{align}
-\widetilde{Q}(s,a) &\leq -f(Q(s,a)) \\
-\widetilde{Q}(s,a) &\leq -f(Q(s,a)) + f(\overbar{Q}(s,a)) -f(\overbar{Q}(s,a)) \\
-\widetilde{Q}(s,a) &\leq |f(Q(s,a)) - f(\overbar{Q}(s,a))| - f(\overbar{Q}(s,a)) \\
\widetilde{Q}(s,a) &\geq -|f(Q(s,a)) - f(\overbar{Q}(s,a))| + f(\overbar{Q}(s,a)) \\
\widetilde{Q}(s,a) &\geq -L|Q(s,a) - \overbar{Q}(s,a)| + f(\overbar{Q}(s,a)) \\
\widetilde{Q}(s,a) &\geq f(\overbar{Q}(s,a)) - L \varepsilon \\
\end{align}
Where the final steps follow from the function $f$ being $L$-Lipschitz and the definition of $\varepsilon$-optimality of $\overbar{Q}(s,a)$.
To prove the upper bound, we take a similar approach, noting that the reward function $r_C$ in Lemma 4.1 must be updated to account for the inexact $Q$-values. Therefore, we must account for the following error propagations:
\begin{align*}
Q(s,a) &\to \overbar{Q}(s,a) \\
V_f(s) &\to \overbar{V_f}(s)\\
r_C(s,a) &\to \overbar{r_C}(s,a).
\end{align*}
We first find the difference between $r_C$ and $\overbar{r_C}$ to be bounded by $(1+\gamma)L\varepsilon$:
\begin{align}
|r_C(s,a)-\overbar{r_C}(s,a)| &= |\gamma \E_{s' \mathcal{S}im{} p } V_f^*(s') - f(Q^*(s,a)) - \gamma \E_{s' \mathcal{S}im{} p } V_f(s') + f(Q(s,a))| \\
&\leq \gamma \E_{s'} |V_f^*(s') - V_f(s')| + |f(Q^*(s,a)) - f(Q(s,a))| \\
&\leq \gamma \E_{s'} \max_{a'} |f(Q^*(s',a')) - f(Q(s',a'))| + |f(Q^*(s,a)) - f(Q(s,a))| \\
&\leq (1+\gamma)L \varepsilon
\end{align}
where in the third line we have used the bound $|\max_{x} f(x) - \max_{x} g(x)| \leq \max_{x} |f(x)-g(x)|$.
Now, applying Lemma \ref{lem:bounded_q_diff} to the reward functions $r_C$ and $\overbar{r_C}$:
\begin{equation}
|C(s,a) - \overbar{C}(s,a)| \leq f\left(r(s,a)\right)ac{ (1+\gamma)}{1-\gamma}L \varepsilon
\end{equation}
With the same technique as was used above for the lower bound, we find:
\begin{align}
\widetilde{Q}(s,a) &\leq f(Q(s,a)) + C(s,a) \\
&\leq f(\overbar{Q}(s,a)) + L \varepsilon + C(s,a) \\
&= f(\overbar{Q}(s,a)) + L \varepsilon + \overbar{C}(s,a) - \overbar{C}(s,a) + C(s,a) \\
&\leq f(\overbar{Q}(s,a)) + L \varepsilon + |C(s,a) - \overbar{C}(s,a)| + \overbar{C}(s,a) \\
&\leq f(\overbar{Q}(s,a)) + L \varepsilon + \overbar{C}(s,a) + f\left(r(s,a)\right)ac{ (1+\gamma)}{1-\gamma}L \varepsilon \\
&= f(\overbar{Q}(s,a)) + \overbar{C}(s,a) + f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon
\end{align}
Further extending the result to the case where only an $\varepsilon$-optimal estimate of $\overbar{C}$ is known, denoted by $\overbar{\overbar{C}}$, we find:
\begin{align}
\widetilde{Q}(s,a) &\leq f(\overbar{Q}(s,a)) + \overbar{C}(s,a) + f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon \\
&\leq f(\overbar{Q}(s,a)) + \overbar{\overbar{C}}(s,a) + |\overbar{\overbar{C}}(s,a)- \overbar{C}(s,a)| + f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon \\
&\leq f(\overbar{Q}(s,a)) + \overbar{\overbar{C}}(s,a) + \varepsilon + f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon \\
&= f(\overbar{Q}(s,a)) + \overbar{\overbar{C}}(s,a) + \left(1+ f\left(r(s,a)\right)ac{2}{1-\gamma}L \right)\varepsilon
\end{align}
\end{proof}
Similarly, Lemma 4.3 from the main text can be extended under the same conditions:
\begin{customlemma}{4.3A}[Concave Conditions, Error-Prone]\label{thm:concave_cond_std_err}
Given a primitive task with discount factor $\gamma$, corresponding $\varepsilon$-optimal value function $\overbar{Q}$, and a bounded, continuous, $L$-Lipschitz transformation function $f~:~X~\to~\mathbb{R}$ which satisfies:
\begin{enumerate}
\item $f$ is concave on its domain $X$ (for stochastic dynamics);
\item $f$ is superlinear:
\begin{enumerate}[label=(\roman*)]
\item $f(x+y) \geq f(x) + f(y)$ for all $x,y \in X$
\item $f(\gamma x) \geq \gamma f(x)$ for all $x \in X$
\end{enumerate}
\item $f\left( \max_{a} \mathcal{Q}(s,a) \right) \geq \max_{a}~f\left( \mathcal{Q}(s,a) \right)$ for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X.$
\end{enumerate}
then the optimal action-value functions are now related in the following way:
\begin{equation}\label{eqn:concave_std_err}
f(\overbar{Q}(s,a)) - \overbar{\hat{C}}(s,a)-f\left(r(s,a)\right)ac{2}{1-\gamma}L \varepsilon \leq \widetilde{Q}(s,a) \leq f(\overbar{Q}(s,a)) + L \varepsilon
\end{equation}
where $\overbar{\hat{C}}$ is the optimal value function for a task with reward
\begin{equation}
\overbar{\hat{r}_C}(s,a) = f(\overbar{Q}(s,a)) - f(r(s,a)) - \gamma \E_{s'\mathcal{S}im{}p} \overbar{V_f}(s')
\end{equation}
with $\overbar{V_f}(s)=\max_a f(\overbar{Q}(s,a))$.
\end{customlemma}
The proof of Lemma \ref{thm:concave_cond_std_err} is the same as that given above for Lemma \ref{thm:convex_cond_std_err}, with all signs flipped.
Finally, we note that both extensions of Lemma \ref{thm:convex_cond_std_err} and \ref{thm:concave_cond_std_err} hold for the entropy-regularized case. The only differences required to prove the results are showing that Lemma \ref{lem:bounded_q_diff} and $|V_f(s)-\overbar{V_f}(s)|\leq L\varepsilon$ hold in entropy-regularized RL. Both statements are trivial given that the necessary soft-max operation is $1$-Lipschitz. Similar results can be derived for the case of compositions, when each subtasks' $Q$-function is replaced by an $\varepsilon$-optimal estimate thereof.
\mathcal{S}ection{Results Applying to Both Entropy-Regularized and Standard RL}
As we have discussed in the main text; an agent with a large library of accessible functions will be able to transform and compose their primitive knowledge in a wider variety of ways. Therefore, we would like to extend $\mathcal{F}$ to encompass as many functions as possible. Below, we will show that the functions $f\in \mathcal{F}$ characterizing the Transfer MDP Library have two closure properties (additivity and function composition) which enables more accessible transfer functions.
First, let $\mathcal{F}^+$ denote the set of functions $f \in \mathcal{F}$ obeying the convex conditions, and similarly let $\mathcal{F}^-$ denote the set of functions obeying the concave conditions.
In standard RL, we have the following closure property for addition of functions.
\begin{theorem}
Let $f,g \in \mathcal{F}^+$. Then $f+g \in \mathcal{F}^+$. Similarly, if $f,g \in \mathcal{F}^-$, then $f+g \in \mathcal{F}^-$.
\end{theorem}
\begin{proof}
Let $f,g \in \mathcal{F}^+$.
Convexity:
The sum of two convex functions is convex.
Subadditive:
$(f+g)(x+y) = f(x+y)+g(x+y)\leq f(x)+g(x)+f(y)+g(y)=(f+g)(x)+(f+g)(y)$.
Submultiplicative:
$(f+g)(\gamma x) = f(\gamma x) + g(\gamma x) \le \gamma f(x) + \gamma g(x) = \gamma(f+g)(x)$.
The proof for $f,g \in \mathcal{F}^-$ is the same with all signs flipped, except for the additional final condition:
$(f+g)(\max_i x_i) = f(\max_i x_i) + g(\max_i x_i) = \max f(x) + \max g(x) \ge \max f(x) + g(x).$ Although this is not equality as shown in the main text, the condition still suffices. For the case of a single function (no addition, as seen in main text), it can never be the cases that $\max_i f(x_i)~>~\max f(x)$ and therefore was excluded. (Just as $\max_i f(x_i) \le \max f(x)$ is automatically satisfied for the convex conditions.)
\end{proof}
\begin{theorem}[Function Composition]
\label{thm:compos}
For any reward-mapping functions $f$, $g \in \mathcal{F}^+$ ($\mathcal{F}^-$) with $f$ non-decreasing, the composition of functions $f$ and $g$, $h(x) = f(g(x)) \in \mathcal{F}^+ (\mathcal{F}^-)$.
\end{theorem}
\begin{proof}
Let $f,g \in \mathcal{F}^+$ assume $f: B \to C$ and $g:A \to B$, and let $f$ be non-decreasing. This guarantees that $f(g(x))$ is convex.
Additionally, $f(g(x+y)) \leq f(g(x)+g(y)) \leq f(g(x)) + f(g(y))$ by the sublinearity of $g,f$ respectively. Similarly $f(g(\gamma x)) \leq f(\gamma g(x)) \leq \gamma f(g(x))$.
For the standard RL (concave) condition, note that for all functions $\mathcal{Q}:~\mathcal{S}~\times~\mathcal{A} \to X$:
\begin{equation}
f\left( g\left( \max_{a} \mathcal{Q}(s,a)\right) \right) \geq f\left(\max_{a}~g\left( \mathcal{Q}(s,a)\right) \right) \geq \max_{a}~f\left(g\left( \mathcal{Q}(s,a)\right) \right)
\end{equation}
For the entropy-regularized condition, we first apply the condition to $g$:
\begin{equation}
f\left( g\left( f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp(\beta \mathcal{Q}(s,a))\right) \right) \le f\left( f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp(\beta g(\mathcal{Q}(s,a)) )\right)
\end{equation}
Then to $f$:
\begin{equation}
f\left( g\left( f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp(\beta \mathcal{Q}(s,a))\right) \right) \le f\left(r(s,a)\right)ac{1}{\beta}\log\mathbb{E}_{a \mathcal{S}im{} \pi_0(a|s)} \exp\left(\beta f\left( g\left(\mathcal{Q}(s,a)\right) \right) \right)
\end{equation}
The reversed statement, when $f,g \in \mathcal{F}^-$ with $f$ non-decreasing has a similar proof and is omitted.
\end{proof}
With this result established, we are now able to concatenate multiple transformations. This allows for multiple gates in Boolean logic statements, for example. As stated in the main text, this ability to compose multiple functions will greatly expand the number of tasks in the Transfer MDP Library which the agent may (approximately) solve.
\nocite{openAI}
\begin{@fileswfalse}
\end{@fileswfalse}
\end{document} |
\begin{document}
\title[A splitter theorem for $3$-connected $2$-polymatroids]{A splitter theorem for $3$-connected $2$-polymatroids}
\thanks{The second and third
authors were supported
by the New Zealand Marsden Fund.}
\author{James Oxley}
\address{Department of
Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA}
\email{oxley@math.lsu.edu}
\author{Charles Semple}
\address{Department of Mathematics and Statistics,
University of Canterbury,
Christchurch,
New Zealand}
\email{c.semple@math.canterbury.ac.nz}
\author{Geoff Whittle}
\address{School of Mathematics, Statistics and Operations Research,
Victoria University,
Wellington,
New Zealand}
\email{Geoff.Whittle@mcs.vuw.ac.nz}
\subjclass{05B35}
\downarrowte{\today}
\begin{abstract} Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A $2$-polymatroid $N$ is an s-minor of a $2$-polymatroid $M$ if $N$ can be obtained from $M$ by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions.
The main result proves that if $M$ and $N$ are $3$-connected $2$-polymatroids such that $N$ is an s-minor of $M$, then $M$ has a $3$-connected s-minor $M'$ that has an s-minor isomorphic to $N$ and has $|E(M)| - 1$ elements unless $M$ is a whirl or the cycle matroid of a wheel. In the exceptional case, such an $M'$ can be found with $|E(M)| - 2$ elements.
\end{abstract}
\maketitle
\vspace*{-30pt}
\section{Introduction}
\label{intro}
Let $M$ be a $3$-connected matroid other than a wheel or a whirl. Tutte~\cite{wtt} proved that $M$ has an element whose deletion or contraction is $3$-connected. Seymour~\cite{pds} extended this theorem by showing that, for a proper $3$-connected minor $N$ of $M$, the matroid $M$ has an element whose deletion or contraction is $3$-connected and has an $N$-minor.
These theorems have been powerful inductive tools for working with $3$-connected matroids. In \cite{oswww}, with a view to attacking representability problems for 2-polymatroids, we generalized the Wheels-and-Whirls Theorem to $2$-polymatroids. In this paper, we prove a generalization of the Splitter Theorem for $2$-polymatroids.
A basic example of a matroid is a set of points in a projective space. If, instead, we take a finite set of points and lines in a projective space, we get an example of a $2$-polymatroid. Whereas each element of a matroid has rank zero or one, an individual element in a $2$-polymatroid can also have rank two. Formally, for a positive integer $k$,
a {\em $k$-polymatroid} $M$ is a pair
$(E,r)$ consisting of a finite set $E$, called the {\it ground set}, and a function $r$,
called the {\it rank function}, from the power set of $E$ into the integers satisfying the
following conditions:
\begin{itemize}
\item[(i)] $r(\emptyset) = 0$;
\item[(ii)] if $X \subseteq Y \subseteq E$, then $r(X) \le r(Y)$;
\item[(iii)] if $X$ and $Y$ are subsets of $E$, then $r(X) + r(Y) \ge r(X \cup Y) + r(X \cap Y)$;
and
\item[(iv)] $r(\{e\})\leq k$ for all $e\in E$.
\end{itemize}
A matroid is just a 1-polymatroid. Equivalently, it is a $2$-polymatroid in which every element has rank at most one.
Our focus in this paper will be on $2$-polymatroids.
From a graph $G$, in addition to its cycle matroid, we can derive a second $2$-polymatroid on $E(G)$, which we denote by $M_2(G)$. The latter is defined
by letting the rank of a set $A$ of edges be the number of vertices incident with edges in $A$.
Observe that non-loop edges of $G$ have rank two in $M_2(G)$.
Matroid connectivity generalizes naturally to $2$-polymatroids. In particular,
3-connectivity for matroids extends routinely to a notion of 3-connectivity for
$2$-polymatroids. A simple $3$-connected
graph $G$ has a $3$-connected cycle matroid. On the other hand, $M_2(G)$ is $3$-connected whenever
$G$ is a $2$-connected loopless graph.
Deletion and contraction for matroids
extend easily to $2$-polymatroids. This gives a notion of minor for $2$-polymatroids that extends
that of minor for matroids, and, via cycle matroids, that of minor for graphs. But what
happens when we consider the $2$-polymatroid $M_2(G)$? If $e$ is an edge of $G$, then deletion
in $M_2(G)$ corresponds to deletion in $G$, but it is not the same with contraction.
However, there is an operation on $M_2(G)$ that corresponds to contraction in $G$.
Specifically, if $e$ is an element of the $2$-polymatroid $M$ and $r(\{e\}) > 0$, then
the {\em compression} of $e$ from $M$, denoted $M\downarrow e$, is obtained by
placing a rank-$1$ element $x$ freely on $e$, contracting $x$, and then deleting $e$ from the resulting $2$-polymatroid.
In particular, $M_2(G)\downarrow e=M_2(G/e)$
for a non-loop edge $e$ of the graph $G$.
Representability of matroids extends easily to representability of polymatroids over fields. Indeed, much of the motivation for this paper is derived from our desire to develop tools for attacking representability problems for $2$-polymatroids.
The class of $2$-polymatroids representable over a field $\mathbb F$ is closed under both
deletion and contraction. When $\mathbb F$ is finite, this is not the case for compression in general although it is the case for a restricted type of compression.
In \cite{oswww}, we defined a certain type of 3-separator, which we called a `prickly' 3-separator. A series pair in a graph $G$ is a 2-element prickly 3-separator of $M_2(G)$. Larger prickly 3-separators do not arise from graphs, but do arise in more general settings. Compressing elements from prickly 3-separators preserves representability. We gave examples in \cite{oswww} to show that, if we wish to generalize Tutte's Wheels-and-Whirls Theorem to $2$-polymatroids, it is necessary to allow compression of elements from prickly 3-separators. The main result of \cite{oswww} proves such a generalization by showing that a $3$-connected non-empty $2$-polymatroid that not a whirl or the cycle matroid of a
wheel has an element $e$ such that either $M\backslash e$ or $M/e$ is $3$-connected, or
$e$ belongs to a prickly $3$-separator, and $M\downarrow e$ is $3$-connected.
Geelen, Gerards, and Whittle~\cite{ggwrota} have announced that Rota's Conjecture~\cite{rot} is true, that is, for every finite field, there is a finite set of minor-minimal matroids that are not representable over that field. In \cite{oswww}, we showed that, for every field ${\mathbb F}$, the set of minor-minimal 2-polymatroids that are not representable over ${\mathbb F}$ is infinite, so one generalization of Rota's Conjecture for 2-polymatroids fails. We believe, however, that an alternative generalization of the conjecture does hold. Specifically, we conjectured in \cite{oswww} that, when ${\mathbb F}$ is finite, there are only finitely many 2-polymatroids that are minimal with the property of being non-representable over ${\mathbb F}$ where we allow, as reduction operations, not only deletion and contaction but also compression of elements from prickly 3-separators.
Our main result appears at the end of this section. We now give the rest of the background needed to understand that result.
The matroid terminology used here will follow Oxley~\cite{oxbook}. Lov\'{a}sz and Plummer~\cite[Chapter 11]{lovaplum} have given an interesting discussion of $2$-polymatroids and some of their properties. We call $(E,r)$ a {\it polymatroid} if it is a $k$-polymatroid for some positive integer $k$.
In a $2$-polymatroid $(E,r)$, an element $x$ will be called a {\it line}, a {\it point}, or a {\it loop} when its rank is $2$, $1$, or $0$, respectively. For readers accustomed to using the terms `point' and `line' for flats in a matroid of rank one and two, respectively, this may create some potential confusion. However, in this paper, we shall never use the terms `point' and `line' in this alternative way. Indeed, we will not even define a flat of a $2$-polymatroid.
Let $M$ be a polymatroid $(E,r)$. For a subset $X$ of $E$, the {\it deletion} $M\backslash X$ and the {\it contraction} $M/X$ of $X$ from $M$ are the pairs $(E-X,r_1)$ and \linebreak$(E-X,r_2)$ where, for all subsets $Y$ of $E-X$, we have $r_1(Y) = r(Y)$ and $r_2(Y) = r(Y \cup X) - r(X)$.
We shall also write $M|(E- X)$ for $M\backslash X$.
A {\it minor} of the polymatroid $M$ is any polymatroid that can be obtained from $M$ by a sequence of operations each of which is a deletion or a contraction.
It is straightforward to check that every minor of a $k$-polymatroid is also a $k$-polymatroid. The {\it closure} ${\rm cl}(X)$ of a set $X$ in $M$ is, as for matroids, the set $\{x \in E: r(X \cup x) = r(X)\}$. Two polymatroids $(E_1,r_1)$ and $(E_2,r_2)$ are {\it isomorphic} if there is a bijection $\phi$ from $E_1$ onto $E_2$ such that $r_1(X) = r_2(\phi(X))$ for all subsets $X$ of $E_1$.
One natural way to obtain a polymatroid is from a collection of flats of a matroid $M$. Indeed, every polymatroid arises in this way \cite{helg, lova, mcd}. More precisely, we have the following.
\begin{theorem}
\label{herepoly}
Let $t$ be a function defined on the power set of a finite set $E$. Then $(E,t)$ is a polymatroid if and only if, for some matroid $M$, there is a function $\psi$ from $E$ into the set of flats of $M$ such that $t(X) = r_M(\cup_{x \in X} \psi(x))$ for all subsets $X$ of $E$.
\end{theorem}
The key idea in proving this theorem is that of freely adding a point to an element of a polymatroid. Let $(E,r)$ be a polymatroid, let $x$ be an element of $E$, and let $x'$ be an element that is not in $E$. We can extend the domain of $r$ to include all subsets of $E \cup x'$ by letting
\begin{equation*}
r(X \cup x') =
\begin{cases}
r(X), & \text{if $r(X \cup x) = r(X)$};\\
r(X) + 1, & \text{if $r(X \cup x) > r(X)$}.
\end{cases}
\end{equation*}
Then it is not difficult to check that $(E \cup x', r)$ is a polymatroid. We say that it has been obtained from $(E,r)$ by {\it freely adding} $x'$ to $x$. If we repeat this construction by freely adding a new element $y'$ to some element $y$ of $E$, we can show that the order in which these two operations is performed is irrelevant.
Using this idea, we can associate a matroid with every $2$-polymatroid $M$ as follows.
Let $L$ be the set of lines of $M$. For each $\ell$ in $L$, freely add two points $s_{\ell}$ and $t_{\ell}$ to $\ell$. Let $M^+$ be the $2$-polymatroid obtained after performing all of
these $2|L|$ operations. Let $M'$ be $M^+ \backslash L$. We call $M'$ the {\it natural matroid derived from $M$}.
Given a graph $G$ with edge set $E$, as noted earlier, one can define a $2$-polymatroid $M_2(G)$ on $E$ by, for each subset $X$ of $E$, letting $r(X)$ be $|V(X)|$ where $V(X)$ is the set of vertices of $G$ that have at least one endpoint in $X$. A polymatroid $(E',r')$ is {\it Boolean} if is isomorphic to the $2$-polymatroid that is obtained in this way from some graph.
One attractive feature of $M_2(G)$ is that, except for the possible presence of isolated vertices, it uniquely determines $G$. More precisely, if $G_1$ and $G_2$ are graphs neither of which has any isolated vertices and if $M_2(G_1) = M_2(G_2)$, then there is a labelling of the vertices of $G_2$ such that $G_1 = G_2$. This contrasts with the situation for matroids where quite different graphs can have the same cycle matroids.
Let $M$ be a polymatroid $(E,r)$. The {\it connectivity function}, $\lambda_M$ or $\lambda$, of $M$ is defined, for all subsets $X$ of $E$, by $\lambda_M(X) = r(X) + r(E-X) - r(M)$. Observe that $\lambda_M(E-X) = \lambda_M(X)$.
It is routine to check, using the submodularity of the rank function, that the connectivity function is submodular, that is, for all subsets $Y$ and $Z$ of $E$,
$$\lambda_M(Y) + \lambda_M(Z) \ge \lambda_M(Y \cup Z) + \lambda_M(Y \cap Z).$$
Let $M$ be a polymatroid. For a positive integer $n$, a subset $X$ of $E(M)$ is {\it $n$-separating} if $\lambda_M(X) \le n-1$ and is {\it exactly $n$-separating} if $\lambda_M(X) = n-1$ We say that $M$ is
{\it $2$-connected} if it has no proper non-empty $1$-separating subset. We will also say that $M$ is {\it disconnected} if it is not 2-connected.
We call $M$ {\it $3$-connected} if $M$ is $2$-connected and $M$ has no {\it $2$-separation}, that is, $M$ has no partition $(X,Y)$ with $\max\{|X|, r(X)\} > 1$ and $\max\{|Y|, r(Y)\} > 1$ but $\lambda(X) \le 1$. When $M$ is a $3$-connected $2$-polymatroid $(E,r)$, a {\it $3$-separation} of $M$ is a partition
$(X,Y)$ of $E$ such that
$\lambda(X) = 2$ and both $r(X)$ and $r(Y)$ exceed $2$.
Duality plays a fundamental role in matroid theory and will also be important in our work with $2$-polymatroids. Whereas there is a standard notion of what constitutes the dual of a matroid, for $2$-polymatroids, there is more than one choice. Let $M$ be a $k$-polymatroid $(E,r)$. The {\it $k$-dual} of $M$ is the pair $(E,r^*_k)$ defined by
$r^*_k(Y) = k|Y| + r(E-Y) - r(M)$. This notion of duality was used, for example, in Oxley and Whittle's treatment \cite{ow2p} of Tutte invariants for $2$-polymatroids. An {\it involution} on the class ${\mathcal M}_k$ of $k$-polymatroids is a function $\zeta$ from ${\mathcal M}_k$ into ${\mathcal M}_k$ such that $\zeta(\zeta(M)) = M$ for all $M$ in ${\mathcal M}_k$. Whittle~\cite{gpw} showed that the $k$-dual is the only involution on ${\mathcal M}_k$ under which deletion and contraction are interchanged in the familiar way. However, a disadvantage of this duality operation is that, for a matroid $M$, we can view $M$ as a $k$-polymatroid for all $k\ge 1$. Hence $M$ has a $1$-dual, which is its usual matroid dual. But it also has a $2$-dual, a $3$-dual, and so on.
In \cite{oswww}, we used a duality operation on the class of all polymatroids that, when applied to a $k$-polymatroid, produces another $k$-polymatroid and that, when applied to a matroid produces its usual matroid dual. In this paper, we will use a variant on that operation that agrees with it when applied to $3$-connected $2$-polymatroids with at least two elements.
Both of these versions of duality are members of a family of potential duals for a polymatroid $(E,r)$ that were defined by McDiarmid~\cite{mcd} and were based on assigning a weight $w(e)$ to each element $e$ of $E$ where $w(e) \ge r(\{e\})$ for all $e$ in $E$. For a set $X$, we shall write $||X||$ for the sum $\sum_{e \in X} w(e)$. In \cite{oswww}, we took $w(e)$ to be $\max\{r(\{e\}),1\}$. Here, instead, we will take $w(e) = r(\{e\})$ and
define
the {\it dual} of a polymatroid $(E,r)$ to be the pair $(E,r^*)$ where, for all subsets $Y$ of $E$,
$$r^*(Y) = ||Y|| + r(E- Y) - r(E) = \sum_{e \in Y} r(\{e\}) + r(E- Y) - r(E).$$
It is straightforward to check that, when $(E,r)$ is a $k$-polymatroid, so too is $(E,r^*)$. When $M = (E,r)$, we shall write $M^*$ for $(E,r^*)$. When the polymatroid $M$ is a matroid, its dual as just defined coincides with its usual matroid dual provided $M$ has no loops. However, if $e$ is a loop of $M$, then $e$ is a loop of $M^*$. The definition of dual used in \cite{oswww} (where we took $||Y|| = \sum_{e \in Y} \max\{1,r(\{e\})\}$) was chosen to ensure that, when $M$ is a matroid, its polymatroid dual coincides with its matroid dual. Here, however, we are giving up on that, albeit in a rather specialized case. Note, however, that the two definitions of dual coincide unless $M$ has a loop so, in particular, they coincide when $M$ is $3$-connected having at least two elements.
Moreover, as noted in \cite{oswww}, these two versions of duality share a number of important properties, the proofs of which are very similar. For example, $\lambda_M(X) = \lambda_{M^*}(X)$. Next we discuss the reason for the use of the above definition of duality, which follows \cite{susan, jmw}.
Consider the following example, which will guide how we proceed. Begin with the matroid that is the direct sum of $PG(r-1,q)$ and $PG(k-2,q)$ viewing this as a restriction of $PG(r+k-2,q)$. Let $N$ be the restriction of $PG(r-1,q)$ to the complement of a hyperplane $H$ of it, so $N {\rm co}ng AG(r-1,q)$. Take $k$ distinct points, $x_1,x_2,\ldots,x_k$, of $PG(r-1,q)$ that are in $H$, and let $\{y_1,y_2,\ldots,y_k\}$ be a spanning circuit in $PG(k-2,q)$. For each $i$ in $\{1,2,\ldots,k\}$, let $\ell_i$ be the line of $PG(r+k-2,q)$ that is spanned by $\{x_i,y_i\}$. Let $M$ be the $2$-polymatroid whose elements are the points of $N$ along with the set $L$ consisting of the lines $\ell_1,\ell_2,\ldots,\ell_k$. It is straightforward to check that $M$ and $N$ are $3$-connected. The only way to obtain an $N$-minor of $M$ is to delete all the elements of $L$ since contracting any member of $L$ has the effect of reducing the rank of $E(N)$. But, in each of the $2$-polymatroids $M\backslash L'$, where $L'$ is a proper non-empty subset of $L - \ell_k$, the set $\ell_k$ is $2$-separating. Since our goal is a splitter theorem, where we can remove some bounded number of elements from $M$ maintaining both $3$-connectivity and an $N$-minor, we will need a strategy for dealing with this example. One significant feature of this example is the very constrained nature of the 2-separations in each $M\backslash L'$ with one side of each such 2-separation consisting of a single line. This is reminiscent of what happens in Bixby's Lemma~\cite{bixby} for $3$-connected matroids where, for every element $e$ of such a matroid $N$, either $N\backslash e$ is $3$-connected except for some possible series pairs, or $N/e$ is $3$-connected except for some possible parallel pairs. Indeed, in the matroid derived from $M\backslash L'$, each 2-separating line yields a series pair in the derived matroid.
The strategy that we will adopt is intimately linked to our choice of definition for the dual of a polymatroid. It is well known that, under the familiar definition of duality for matroids, taking the dual of the dual returns us to the original matroid. We now consider the relationship between a polymatroid $M$ and the polymatroid $(M^*)^*$. If $M$ is a $3$-connected $2$-polymatroid with at least two elements, then $(M^*)^* = M$. To see what happens in general, we follow \cite{jmw}. Let $M$ be the polymatroid $(E,r)$. An element $e$ of $M$ is {\it compact} if $r(\{e\}) = \lambda_M(\{e\})$ or, equivalently, if $r(E - \{e\}) = r(E)$. We call $M$ {\it compact} if every element is compact. Thus, for example, a matroid is compact if it has no coloops. In the example in the last paragraph, although $M$ is compact, $M\backslash \{\ell_1\}$ is not since, for each $i \ge 2$, we have $r(\{\ell_i\}) = 2$ whereas $\lambda_{M\backslash \{\ell_1\}}(\{\ell_i\}) = 1$.
The {\it compactification} $M^{\flat}$ of the polymatroid $M$ is the pair $(E,r^{\flat})$ where
$$r^{\flat}(X) = r(X) + \sum_{x \in X} [\lambda (\{x\}) - r(\{x\})]$$ for all subsets $X$ of $E$. It is shown in \cite{jmw} that $M^{\flat}$ is a compact polymatroid and it is clear that if $M$ is a $2$-polymatroid, then so is $M^{\flat}$. The next result \cite{jmw} encapsulates some key properties of this compactification operation and justifies the approach we take here.
\begin{lemma}
\label{compact0}
Let $(E,r)$ be a polymatroid $M$. Then
\begin{enumerate}
\item[(i)] $M^*$ is compact;
\item[(ii)] $(M^*)^* = M^{\flat}$;
\item[(iii)] $\lambda_M = \lambda_{M^*} = \lambda_{M^{\flat}}$; and
\item[(iv)] $M/X$ is compact for all non-empty subsets $X$ of $E$ and
$$(M/X)^* = (M^*\backslash X)^{\flat}.$$
\end{enumerate}
\end{lemma}
Returning to our guiding example above, although $M\backslash \{\ell_1\}$ is neither compact nor $3$-connected, its compactification is both. Observe that this compactification can be obtained from the restriction of the matroid $PG(r-1,q)$ to $E(N) \cup \{x_2,x_3,\ldots,x_k\}$ by relabelling each $x_i$ by $\ell_i$ noting that these $\ell_i$ are now points rather than lines. Thus compactification here has an analogous effect to cosimplification in matroids.
By incorporating compactification as part of the deletion operation, which is justified by (iv) of the last lemma, we see that, after deleting a single element, we have both maintained $3$-connectivity and kept an $N$-minor. This is precisely what we want in a splitter theorem.
In $2$-polymatroids, the behaviour of contraction differs significantly from that for matroids. In particular, consider the $2$-polymatroid $M_2(G)$ obtained from a graph $G$, where $G$ has vertex set $V$ and edge set $E$. Let $e$ be an edge of $G$. Deleting $e$ from $G$ has an unsurprising effect; specifically, $M_2(G) \backslash e = M_2(G \backslash e)$. But, to find $M_2(G)/e$, we cannot simply look at $M_2(G/e)$. In particular, what do we do with elements whose rank is reduced to zero in the contraction? To deal with this situation, it is standard to extend the definition of a graph to allow the presence of {\it free loops}, that is, edges with no endpoints. This terminology is due to Zaslavsky \cite{zas}. For a graph $G$ with free loops, the associated $2$-polymatroid $M_2(G)$ is defined, as before, to have rank function $r(X) = |V(X)|$. The deletion of a free loop $f$ from a graph just removes $f$ from the graph. We define the contraction of $f$ to be
the same as its deletion. For an edge $e$ that is not a free loop,
to obtain a graph $H$ so that $M_2(G) /e = M_2(H)$, we let $H$ have edge set $E - e$ and vertex set $V - V(\{e\})$.
An edge $x$ of $H$ is incident with the vertices in $V(\{x\}) - V(\{e\})$.
The difference between $M_2(G)/e$ and $M_2(G/e)$ motivated us to introduce an operation for $2$-polymatroids in \cite{oswww} that mimics the effect of the usual operation of contraction of an edge from the graph.
Let $(E,r)$ be a $2$-polymatroid $M$, and let $x$ be an element of $E$. We have described already what it means to add an element $x'$ freely to $x$. Our new operation $M\downarrow x$ is obtained from $M$ by freely adding $x'$ to $x$ in $M$, then contracting $x'$ from the resulting extension, and finally deleting $x$. Because each of the steps in this process results in a $2$-polymatroid, we have a well-defined operation on $2$-polymatroids. When $x$ has rank at most one in $M$, one easily checks that $M\downarrow x = M/x$. When $x$ is a line in $M$,
we see that $M\downarrow x$ and $M/x$ are different as their ranks are $r(M) - 1$ and $r(M) - 2$, respectively. Combining the different parts of the definition, we see that
$M\downarrow x$ is the $2$-polymatroid with ground set $E - \{x\}$ and rank function given, for all subsets $X$ of $E - \{x\}$, by
\begin{equation}
\label{getdown}
r_{M\downarrow x}(X) =
\begin{cases}
r(X), & \text{if $r(x) = 0$, or $r(X \cup x) > r(X)$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation}
We shall say that $M\downarrow x$ has been obtained from $M$ by {\it compressing} $x$, and $M\downarrow x$ will be called the {\it compression} of $x$. We showed in \cite{oswww} that $M_2(G)\downarrow e = M_2(G/e).$ Songbao Mo~\cite{smo} established a number of properties of a generalization of this operation that he defines for connectivity functions and calls {\it elision}.
Instead of treating arbitrary minors, much of graph theory restricts attention to topological minors in which the only allowed contractions involve edges that meet vertices of degree two. When $e$ and $f$ are the only edges in a $2$-connected graph $G$ meeting a vertex $v$, and $G$ has at least four vertices, $\{e,f\}$ is a $3$-separating set in $M_2(G)$. This $3$-separating set is an example of a special type of $3$-separating set that we introduced in \cite{oswww}. In a $2$-polymatroid $M$, a 3-separating set $Z$ is {\it prickly} if it obeys the following conditions:
\begin{itemize}
\item[(i)] Each element of $Z$ is a line;
\item[(ii)] $|Z| \ge 2$ and $\lambda(Z) = 2$;
\item[(iii)] $r((E -Z) \cup Z') = r(E - Z) + |Z'|$ for all proper subsets $Z'$ of $Z$; and
\item[(iv)] if $Z'$ is a non-empty subset of $Z$, then
\begin{equation*}
r(Z') =
\begin{cases}
2 & \text{if $|Z'| = 1$};\\
|Z'| + 2 & \text{if $1 < |Z'| < |Z|$; and}\\
|Z| + 1 & \text{if $|Z'| = |Z|.$}
\end{cases}
\end{equation*}
\end{itemize}
A prickly $3$-separating set of $M$ will also be called a {\it prickly $3$-separator} of $M$.
Observe that, when $Z$ is a prickly $3$-separating set, for all distinct $z$ and $z'$ in $Z$, the $2$-polymatroid $M\backslash z$ has $(\{z'\}, E - \{z,z'\})$ as a $2$-separation.
We are now able to formally state the main result of \cite{oswww}.
Recall that a $2$-polymatroid is {\it pure} if every individual element has rank $2$. It is {\it non-empty} if its ground set is non-empty.
\begin{theorem}
\label{lastmainone}
Let $M$ be a $3$-connected non-empty $2$-polymatroid. Then one of the following holds.
\begin{itemize}
\item[(i)] $M$ has an element $e$ such that $M\backslash e$ or $M/e$ is $3$-connected;
\item[(ii)] $M$ has rank at least three and is a whirl or the cycle matroid of a wheel; or
\item[(iii)] $M$ is a pure $2$-polymatroid having a prickly $3$-separating set. Indeed, every minimal $3$-separating set $Z$ with at least two elements is prickly, and $M\downarrow z$ is $3$-connected and pure for all $z$ in $Z$.
\end{itemize}
\end{theorem}
In \cite{oswww}, we gave a number of examples to show the need for the third part of the above theorem. It is worth noting here, since it contrasts with what we have already mentioned and what will feature in the main result of this paper, the operation of deletion used in the last theorem does not incorporate compactification. In the main result of this paper, we will incorporate compactification as part of deletion but we will no longer need to allow arbitrary prickly compressions, only those that arise from a $2$-element prickly $3$-separator.
Let $Z$ be such a set in a $2$-polymatroid $M$. For $z$ in $Z$, we will call $M\downarrow z$ a {\it series compression} of $M$.
For a compact $2$-polymatroid $M_1$, we call $M_2$ an {\it s-minor} of $M_1$ if $M_2$ can be obtained from $M_1$ by a sequence of contractions, deletions followed by compactifications, and series compressions. The next result is the main theorem of the paper. It concerns s-minors of 3-connected 2-polymatroids. Such a 2-polymatroid is compact provided it has at least three elements.
\begin{theorem}
\label{mainone}
Let $M$ be a $3$-connected $2$-polymatroid and $N$ be a $3$-connected proper s-minor of $M$ having at least four elements. Then one of the following holds.
\begin{itemize}
\item[(i)] $M$ has an element $e$ such that $M/ e$ is $3$-connected having an s-minor isomorphic to $N$; or
\item[(ii)] $M$ has an element $e$ such that $(M\backslash e)^{\flat} $ is $3$-connected having an s-minor isomorphic to $N$; or
\item[(iii)] $M$ has a two-element prickly $3$-separating set $Z$ such that, for each $z$ in $Z$, the series compression $M\downarrow z$ is $3$-connected having an s-minor isomorphic to $N$; or
\item[(iv)] $r(M) \ge 3$ and $M$ is a whirl or the cycle matroid of a wheel.
\end{itemize}
\end{theorem}
For compact $2$-polymatroids $M_1$ and $M_2$, we call $M_2$ a {\it c-minor} of $M_1$ if $M_2$ can be obtained from $M_1$ by a sequence of operations each consisting of a contraction or of a deletion followed by a compactification. As we shall show in Section~\ref{redc}, the last theorem can be proved by establishing the following result.
\begin{theorem}
\label{modc0}
Let $M$ be a $3$-connected $2$-polymatroid and $N$ be a $3$-connected proper c-minor of $M$ having at least four elements. Then one of the following holds.
\begin{itemize}
\item[(i)] $M$ has an element $e$ such that $M/ e$ is $3$-connected having a c-minor isomorphic to $N$; or
\item[(ii)] $M$ has an element $e$ such that $(M\backslash e)^{\flat} $ is $3$-connected having a c-minor isomorphic to $N$; or
\item[(iii)] $M$ has a prickly $3$-separator $\{y,z\}$ such that $M\downarrow y$ is $3$-connected having a c-minor isomorphic to $N$; or
\item[(iv)] $r(M) \ge 3$ and $M$ is a whirl or the cycle matroid of a wheel.
\end{itemize}
\end{theorem}
The paper is structured as follows. The next section includes some basic preliminaries. In Sections~\ref{clc} and \ref{pc2s}, we develop a number of results relating to connectivity and local connectivity, and to parallel connection and 2-sums. In Section~\ref{strat}, we describe the strategy for proving Theorem~\ref{mainone}. That section serves as a guide to the remaining sections of the paper, with the purpose of each of these sections being to complete an identified step in the proof. Section~\ref{redc} plays an important role in this proof by showing that the main theorem can be proved by adding the assumption that all series compressions are performed last in the production of an s-minor of $M$ isomorphic to $N$. That result is helpful but it cannot obscure the fact that the proof of Theorem~\ref{mainone} is complex with some subtleties in the logic that need to be carefully negotiated.
\section{Preliminaries}
\label{prelim}
Much of the terminology for matroids carries over to $2$-polymatroids. For example, suppose $x$ and $y$ are distinct points of a $2$-polymatroid $M$, that is, $r(\{x\}) = 1 = r(\{y\})$. If $r(\{x,y\}) = 1$, then $x$ and $y$ are {\it parallel points} of $M$. On the other hand, if
$r(E - \{x,y\}) = r(E) -1 < r(E - x) = r(E-y)$, then $\{x,y\}$ is a {\it series pair of points} of $M$. Evidently, if $\{x,y\}$ is a parallel or series pair of points, then
$\lambda_M(\{x,y\}) \le 1$. If $x$ and $y$ are distinct lines of $M$ and $r(\{x,y\}) = 2$, then $x$ and $y$ are {\it parallel lines} of $M$.
One tool that is used repeatedly in our earlier work is the submodularity of the connectivity function. Once again, this will play a vital role here. Partitions $(X_1,X_2)$ and $(Y_1,Y_2)$ of a set $E$ are said to {\it cross} if all four of the sets \linebreak $X_1\cap Y_1,$ $X_1 \cap Y_2, X_2 \cap Y_1$, and $X_2 \cap Y_2$ are non-empty. We shall frequently encounter crossing partitions of the ground set of a $2$-polymatroid. We shall use the term {\it by uncrossing} to refer to an application of the submodularity of the connectivity function.
In this paper, we shall frequently switch between considering the deletion $M\backslash X$ of a set $X$ of elements of a $2$-polymatroid $M$ and the compactification $(M\backslash X)^{\flat}$ of this deletion, which we shall sometimes call the {\it compactified deletion of $X$}. We shall often use the following abbreviated notation for the latter:
$$(M\backslash X)^{\flat} = M\backslashba X.$$
We shall often encounter the situation when we have a $2$-polymatroid $M$ such that $M^{\flat}$ is $3$-connected although $M$ itself is not. This occurs when $M$ has a line $\ell$ such that $(\{\ell\},E - \ell)$ is a $2$-separation. We call such a $2$-separation of $M$ {\it trivial}. Thus, in general, a partition $(X,Y)$ of $E$ is a {\it non-trivial $2$-separation} of $M$ if $\lambda_M(X) \le 1$ and $\min\{|X|,|Y|\} \ge 2$.
For a $2$-polymatroid $M$, we recall that a minor of $M$ is any $2$-polymatroid that can be obtained from $M$ by a sequence of contractions and deletions where, here, deletions are not automatically accompanied by compactifications. When $M$ and $N$ are compact, we defined $N$ to be a c-minor of $M$ if it can be obtained from $M$ by a sequence of contractions and deletions followed by compactifications. In the proof of Theorem~\ref{modc0}, it is convenient to be able to separate the compactifications from the deletions. Thus we define a {\it c-minor} of an arbitrary 2-polymatroid $M$ to be any $2$-polymatroid that can be obtained from $M$ by a sequence of contractions, deletions, and compactifications. As we shall show in Corollary~\ref{complast2}, this extension of the definition is consistent with our original definition.
For a $2$-polymatroid $N$, a {\it special $N$-minor} of $M$ is any c-minor of $M$ that is either equal to $N$ or differs from $N$ by having a single point relabelled.
\begin{lemma}
\label{complast}
Let $P$ and $Q$ be $2$-polymatroids such that $Q$ can be obtained from $P$ by a sequence of deletions, contractions, and compactifications with the last move being a compactification. Then $Q$ can be obtained from $P$ by the same sequence of deletions and contractions with none of the compactifications being done except for the last move.
\end{lemma}
To prove this lemma, we shall require a preliminary result.
\begin{lemma}
\label{complast1}
Let $P$ be the $2$-polymatroid $(E,r)$. For $A \subseteq E$,
\begin{itemize}
\item[(i)] $(P^{\flat}\backslash A)^{\flat} = (P\backslash A)^{\flat}$; and
\item[(ii)] $(P^{\flat}/ A)^{\flat} = (P/ A)^{\flat}$.
\end{itemize}
\end{lemma}
\begin{proof} Let $P_1$ be a $2$-polymatroid with ground set $E$ and rank function $r_1$. Then, for $X \subseteq E - A$, we have
\begin{eqnarray}
\label{xae}
r_{(P_1 \backslash A)^{\flat}}(X) & = &r_{P_1 \backslash A}(X) + \sum_{x \in X} [\lambda_{P_1 \backslash A}(\{x\}) - r_{P_1\backslash A}(\{x\})] \nonumber \\
&=& r_1(X) + \sum_{x \in X} [r_1(E - A - x) - r_1(E - A)].
\end{eqnarray}
Thus
\begin{equation}
\label{xae0}
r_{(P \backslash A)^{\flat}}(X) = r(X) + \sum_{x \in X} [r(E - A - x) - r(E - A)].
\end{equation}
Next we observe that, for $x$ in $X$,
\begin{align}
\label{xae1}
r_{P^{\flat}}(E-A - x) - r_{P^{\flat}}(E-A) & = r(E - A - x) + \sum_{y \in E-A-x} [\lambda(\{y\}) - r(\{y\})] \nonumber \\
& \hspace*{0.75in} - r(E - A) - \sum_{y \in E-A} [\lambda(\{y\}) - r(\{y\})] \nonumber \\
& = r(E - A - x) - r(E - A) - \lambda(\{x\}) + r(\{x\}).
\end{align}
Thus, by (\ref{xae}), (\ref{xae1}), and (\ref{xae0}),
\begin{align*}
r_{(P^{\flat} \backslash A)^{\flat}}(X) &= r_{P^{\flat}}(X) + \sum_{x \in X} [r_{P^{\flat}}(E - A - x) - r_{P^{\flat}}(E - A)]\\
& = r(X) + \sum_{x \in X} [\lambda(\{x\}) - r(\{x\}) +r(E - A - x) - r(E - A) \\
& \hspace*{2.5in} - \lambda(\{x\}) + r(\{x\})]\\
& = r(X) + \sum_{x \in X} [r(E - A - x) - r(E - A)]\\
& = r_{(P \backslash A)^{\flat}}(X).
\end{align*}
We conclude that (i) holds.
Again, for $X \subseteq E - A$, we have
\begin{eqnarray}
\label{xae2}
r_{(P_1 / A)^{\flat}}(X) & = &r_{P_1 / A}(X) + \sum_{x \in X} [\lambda_{P_1 / A}(\{x\}) - r_{P_1/ A}(\{x\})] \nonumber \\
&=& r_1(X \cup A) -r_1(A) + \sum_{x \in X} [r_{P_1 / A}(E - A - x) - r_{P_1 / A}(E - A)] \nonumber \\
& = & r_1(X \cup A) -r_1(A) + \sum_{x \in X} [r_{1}(E - x) - r_{1}(E)].
\end{eqnarray}
Thus
\begin{equation}
\label{xae3}
r_{(P / A)^{\flat}}(X) = r(X \cup A) -r(A) + \sum_{x \in X} [r(E - x) - r(E)].
\end{equation}
Therefore, by (\ref{xae2}), (\ref{xae1}), and (\ref{xae3})
\begin{align*}
r_{(P^{\flat} /A)^{\flat}}(X) &= r_{P^{\flat}}(X \cup A) -r_{P^{\flat}}(A) + \sum_{x \in X} [r_{P^{\flat}}(E - x) - r_{P^{\flat}}(E)]\\
& = r(X \cup A) -r (A) + \sum_{x \in X} [ \lambda(\{x\}) - r(\{x\}) +r(E - x) - r(E)\\
& \hspace*{2.6in} - \lambda(\{x\}) + r(\{x\})]\\
& = r(X \cup A) -r (A) + \sum_{x \in X} [r(E - x) - r(E)]\\
& = r_{(P / A)^{\flat}}(X).
\end{align*}
Hence (ii) holds.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{complast}.]
We may assume that there are disjoint subsets $A_1,A_2,\ldots,A_n$ of $E$ such that, in forming $Q$ from $P$, these sets are removed in order via deletion or contraction with the possibility that, after each such move, a compactification is performed. To prove the lemma, we argue by induction on $n$.
It follows immediately from Lemma~\ref{complast1} that the lemma holds if $n= 1$. Assume the result holds for $n < m$ and let $n = m\ge 2$. Then there is a $2$-polymatroid $R$ such that $Q$ is $(R \backslash A_n)^{\flat}$ or $(R / A_n)^{\flat}$, so, by Lemma~\ref{complast1}, $Q$ is $(R^{\flat} \backslash A_n)^{\flat}$ or $(R^{\flat} / A_n)^{\flat}$, respectively. In forming $R$, a certain sequence of deletions, contractions, and compactifications is performed. Let $R_0$ be the $2$-polymatroid that is obtained from $P$ by performing the same sequence of operations except for the compactifications. Then, by the induction assumption, $R^{\flat} = R_0^{\flat}$. Since
$(R^{\flat} \backslash A_n)^{\flat} = (R_0^{\flat} \backslash A_n)^{\flat} = (R_0 \backslash A_n)^{\flat}$ and $(R^{\flat} / A_n)^{\flat} =(R_0^{\flat} / A_n)^{\flat} = (R_0 / A_n)^{\flat}$, the lemma follows by induction.
\end{proof}
The following are straightforward consequences of Lemma~\ref{complast}. We prove only the second of these.
\begin{corollary}
\label{complast3}
Let $P$ and $Q$ be $2$-polymatroids such that $Q$ is compact. Then $Q$ is a c-minor of $P$ if and only if $Q$ can be obtained from $P$ by a sequence of deletions and contractions followed by a single compactification.
\end{corollary}
\begin{corollary}
\label{complast2}
Let $P$ and $Q$ be compact $2$-polymatroids. Then $Q$ is a c-minor of $P$ if and only if $Q$ can be obtained from $P$ by a sequence of operations each of which consists of either a contraction or a deletion followed by a compactification.
\end{corollary}
\begin{proof} We need to show that if $Q$ is a c-minor of $P$, then $Q$ can be obtained as described. Now $Q^{\flat} = Q$. Thus, by Lemma~\ref{complast}, $Q$ can be obtained from $P$ by a sequence of deletions and contractions with one compactification being done as the final move. By Lemma~\ref{complast1}, we can perform a compactification after each deletion and still obtain $Q$ at the end of the process. Since $P$ is compact and each contraction of a compact $2$-polymatroid is compact, we retain compactness throughout this sequence of moves, so the result holds.
\end{proof}
\begin{lemma}
\label{compact2}
Let $M$ be a polymatroid. Then
$$(M^{\flat})^* = M^* = (M^*)^{\flat}.$$
\end{lemma}
\begin{proof}
By Lemma~\ref{compact0}(i), $M^*$ is compact, so $M^* = (M^*)^{\flat}.$ Also, by Lemma~\ref{compact0}(ii),
$(M^{\flat})^* = ((M^*)^*)^* = (M^*)^{\flat}$.
\end{proof}
\begin{lemma}
\label{csm}
Let $P$ and $Q$ be $2$-polymatroids, where $Q$ is compact. Then
$P$ has a c-minor isomorphic to $Q$ if and only if $P^*$ has a c-minor isomorphic to $Q^*$.
\end{lemma}
\begin{proof} Suppose $P$ has a c-minor isomorphic to $Q$. By Corollary~\ref{complast3}, $Q$ can be obtained from $P$ by a sequence of deletions and contractions with one compactification being done as the final move. By Lemma~\ref{complast1}, we can perform a compactification after each deletion and after each contraction and still obtain $Q$ at the end of the process. Indeed, since $(P^{\flat}\backslash A)^{\flat} = (P\backslash A)^{\flat}$ and
$(P^{\flat}/ A)^{\flat} = (P/ A)^{\flat}$, we see that $P^{\flat}$ has a c-minor isomorphic to $Q$. Thus we may assume that, in forming $Q$ from $P^{\flat}$, we remove, in order, disjoint sets $A_1, A_2,\ldots, A_n$ where each such removal is followed by a compactification. To prove that
$P^*$ has a c-minor isomorphic to $Q^*$, we shall argue by induction on $n$.
Suppose $n = 1$. Then $Q$ is $(P^{\flat}\backslash A_1)^{\flat}$ or $(P^{\flat}/ A_1)^{\flat}$.
Then, by Lemmas~\ref{compact0} and \ref{compact2},
$$((P^{\flat}\backslash A_1)^{\flat})^* = (((P^*)^*\backslash A_1)^{\flat})^*= ((P^*/A_1)^*)^* = (P^*/A_1)^{\flat} = P^*/A_1$$
and
$$((P^{\flat}/ A_1)^{\flat})^* = (P^{\flat}/ A_1)^* = ((P^{\flat})^*\backslash A_1)^{\flat} = (P^*\backslash A_1)^{\flat}.$$
Since $Q$ is compact, we deduce that the result holds for $n = 1$. Assume it holds for $n< k$ and let $n = k \ge 2$. Then there is a compact $2$-polymatroid $R$ that is a c-minor of $P$ such that
$Q$ is $(R\backslash A_n)^{\flat}$ or $(R/A_n)^{\flat}$. By the induction assumption, $R^*$ is a c-minor of $P^*$, and $Q^*$ is a c-minor of $R^*$. Hence
$Q^*$ is a c-minor of $P^*$.
For the converse, we note that, by what we have just proved, if $Q^*$ is a c-minor of $P^*$, then $(Q^*)^*$ is a c-minor of $(P^*)^*$, that is,
$Q^{\flat}$ is a c-minor of $P^{\flat}$. But $Q$ is compact so $Q$ is a c-minor of $P^{\flat}$. Hence $Q$ is a c-minor of $P$.
\end{proof}
When we compactify a $2$-polymatroid, loosely speaking what we are doing is dealing simultaneously with a number of $2$-separations. It will be helpful to be able to treat these $2$-separations one at a time. In the introduction, we defined the compression $M\downarrow x$ for an element $x$ of a $2$-polymatroid $M$. Ultimately, that operation removes $x$. Let $M\underline{\downarrow} x$ be the $2$-polymatroid that is obtained from $M$ by freely adding an element $x'$ on $x$ and then contracting $x'$. Thus $M\underline{\downarrow} x$ has ground set $E$ and rank function given, for all subsets $X$ of $E$, by
\begin{equation}
\label{getdown4 }
r_{M\underline{\downarrow} x}(X) =
\begin{cases}
r(X), & \text{if $r(x) = 0$, or $r(X \cup x) > r(X)$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation}
We shall say that $M\underline{\downarrow} x$ has been obtained by {\it compactifying} $x$. Evidently
$$M\downarrow x = (M{\underline{\downarrow}\,} x) \backslash x.$$
\begin{lemma}
\label{compel}
Let $M$ be a $2$-connected $2$-polymatroid that is not compact. Let $Z$ be the set of lines $z$ of $M$ such that $\lambda(\{z\}) = 1$. Then
$$M^{\flat} = ((\ldots((M\underline{\downarrow} z_1)\underline{\downarrow} z_2)\ldots)\underline{{\downarrow}} z_n)$$
where $Z = \{z_1,z_2,\ldots,z_n\}$.
\end{lemma}
\begin{proof} We argue by induction on $n$. Suppose $n = 1$. Let $X \subseteq E(M)$. Then
\begin{equation*}
\label{getdown2}
r_{M^{\flat}}(X) =
\begin{cases}
r(X), & \text{if $z_1 \not \in X$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
On the other hand,
\begin{equation*}
\label{getdown3}
r_{M\underline{\downarrow} z_1}(X) =
\begin{cases}
r(X), & \text{if $r(X \cup z_1) > r(X)$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
The result is easily checked in this case.
Now assume that $n \ge 2$ and that the lemma holds if $|Z| \le n-1$. Let $M_1 = M\underline{\downarrow} z_1$. Then $M_1$ is easily shown to be $2$-connected having $\{z_2,z_3,\ldots,z_n\}$ as its set of lines $z$ for which $\lambda_{M_1}(\{z\}) = 1$. Thus, by the induction assumption,
$$M_1^{\flat} = ((\ldots((M_1\underline{\downarrow} z_2)\underline{\downarrow} z_3)\ldots)\underline{{\downarrow}} z_n).$$
Since $M_1 = M\underline{\downarrow} z_1$, it suffices to show that $M_1^{\flat} = M^{\flat}.$
Suppose $X \subseteq E$. Then
$$r^{\flat}(X) = r(X) + \sum_{x \in X}(\lambda(\{x\}) - r(\{x\})).$$
Now
\begin{equation*}
\label{getdown5.1}
r_{M_1}(X) =
\begin{cases}
r(X), & \text{if $r(X \cup z_1) > r(X)$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
Thus
\begin{equation*}
\label{getdown6}
r_{M_1}(X) =
\begin{cases}
r(X), & \text{if $z_1 \not \in X$; and}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
Hence
\begin{eqnarray*}
r_{M_1^{\flat}}(X) & = & r_{M_1}(X) + \sum_{x \in X}(\lambda_{M_1}(\{x\}) - r_{M_1}(\{x\}))\\
& = & r_{M_1}(X) + \sum_{x \in X \cap (Z - z_1)}(\lambda_{M}(\{x\}) - r_{M}(\{x\}))\\
& = & r_{M}(X) + \sum_{x \in X \cap Z}(\lambda_{M}(\{x\}) - r_{M}(\{x\}))\\
& = & r^{\flat}(X).
\end{eqnarray*}
We conclude, by induction, that the lemma holds.
\end{proof}
We will need some elementary properties of deletion, contraction, and series compression.
\begin{lemma}
\label{elemprop}
Let $A$ and $B$ be disjoint subsets of the ground set $E$ of a $2$-polymatroid $P$. Then
\begin{itemize}
\item[(i)] $P/A/B = P/ (A \cup B) = P/B/A$;
\item[(ii)] $P\backslashba A\backslashba B = P\backslashba (A \cup B) = P\backslashba B \backslashba A$; and
\item[(iii)] $P/ A\backslashba B = P\backslashba B / A$.
\end{itemize}
\end{lemma}
\begin{proof} Because the proofs of all three parts are routine, we only include a proof of (iii).
Suppose $X \subseteq E- (A \cup B)$. Then
\begin{align*}
r_{P/A\backslashba B}(X) &= r_{((P/A)\backslash B)^{\flat}}(X)\\
&= r_{P/A}(X) + \sum_{x \in X}[\lambda_{P/A\backslash B}(\{x\}) - r_{P/A\backslash B}(\{x\})]\\
& = r_{P/A}(X) + \sum_{x \in X}[r_{P/A}(E-A-B-x) - r_{P/A}(E-A-B)]\\
& = r(X\cup A) - r(A) + \sum_{x \in X}[r(E-B-x) - r(E-B)]\\
& = r(X\cup A) - r(A) + \sum_{x \in X}[\lambda_{P\backslash B}(\{x\}) - r_{P\backslash B}(\{x\})]\\
& = r_{P\backslashba B}(X\cup A) - r_{P\backslashba B}(A)\\
& = r_{P\backslashba B/A}(X).
\end{align*}
We conclude that (iii) holds.
\end{proof}
The remainder of this section presents a number of basic properties of 2-element prickly 3-separators and of the compression operation.
\begin{lemma}
\label{atlast}
Let $P$ be a $2$-polymatroid having $j$ and $k$ as lines and with $r(\{j,k\}) = 3$. Suppose $X \subseteq E(P) - k$ and $j \in X$. Then $r_{P\downarrow k}(X) = r(X \cup k) - 1$.
\end{lemma}
\begin {proof}
By definition,
\begin{equation*}
r_{P\downarrow k}(X) =
\begin{cases}
r(X), & \text{if $r(X \cup k) > r(X)$;}\\
r(X) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
As $j \in X$ and $\sqcap(j,k) = 1$, it follows that $r(X \cup k)$ is $r(X)$ or $r(X) + 1$. It follows that
$r_{P\downarrow k}(X) = r(X \cup k) - 1$.
\end{proof}
\begin{lemma}
\label{atlast2}
Let $P$ be a $2$-polymatroid having $j$ and $k$ as lines and with $r(\{j,k\}) = 3$. Suppose $\ell$ is a line of $P$ that is not in $\{j,k\}$ and is not parallel to $k$. Then $\{\ell\}$ is $2$-separating in $P$ if and only if it is $2$-separating in $P\downarrow k$.
\end{lemma}
\begin {proof} Clearly $\{\ell\}$ is $2$-separating in $P$ if and only if $r(E- \ell) \le r(E) - 1$. Since $\ell$ is not parallel to $k$, we see that $r_{P\downarrow k}(\ell) = r(\ell) = 2$. Now $\{\ell\}$ is $2$-separating in $P\downarrow k$ if and only if $r_{P\downarrow k}(E- \{k,\ell\}) \le r_{P\downarrow k}(E- k) - 1$. By Lemma~\ref{atlast}, the last inequality holds if and only if $r(E - \ell) - 1 \le r(E) - 1 - 1.$ We conclude that the lemma holds.
\end{proof}
\begin{lemma}
\label{symjk}
Let $\{j,k\}$ be a prickly $3$-separator in a $2$-polymatroid $P$.
Then $P\downarrow j$ can be obtained from $P\downarrow k$ by relabelling $j$ as $k$.
\end{lemma}
\begin{proof}
Suppose $X \subseteq E - \{j,k\}$. Then, since both $r(X \cup j)$ and $r(X \cup k)$ exceed $r(X)$,
$$r_{P\downarrow j}(X) = r_P(X) = r_{P\downarrow k}(X).$$
Now, as $\sqcap(j,k) = 1$, it follows that either
$r(X \cup j \cup k) = r(X \cup j) + 1$, or $r(X \cup j \cup k) = r(X \cup j)$. Thus
$r_{P\downarrow k}(X \cup j) = r(X \cup j \cup k) - 1$. By symmetry,
$r_{P\downarrow j}(X \cup k) = r(X \cup j \cup k) - 1$, and the lemma follows.
\end{proof}
The first part of the next lemma was proved by Jowett, Mo, and Whittle~\cite[Lemma 3.6]{jmw}.
\begin{lemma}
\label{elemprop23}
Let $P$ be a compact polymatroid $(E,r)$. For $A\subseteq E$,
\begin{itemize}
\item[(i)] $P/A$ is compact; and
\item[(ii)] if $P$ is a $2$-polymatroid and $\{j,k\}$ is a prickly $3$-separator of $P$, then $P\downarrow k$ is compact.
\end{itemize}
\end{lemma}
\begin{proof} We prove (ii).
It suffices to show that $r_{P\downarrow k}(E - k - y) = r_{P\downarrow k}(E-k)$ for all $y$ in $E - k$. Since $P$ is compact, $r(E- k) = r(E)$, so
$r_{P\downarrow k}(E-k) = r(E) - 1$. Now
\begin{equation*}
\label{getdown5}
r_{P\downarrow k}(E - k - y) =
\begin{cases}
r(E-k-y), & \text{if $r(E-y) -1 \ge r(E-y - k)$; and}\\
r(E-k-y) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
It follows that
$r_{P\downarrow k}(E - k - y) = r(E) - 1 = r_{P\downarrow k}(E-k)$ unless $r(E- k - y) = r(E-y) - 2$. Consider the exceptional case. Evidently $y \neq j$ as $r(E- k - j) = r(E-j) - 1$. Thus $j \in E- k -y$. Since $\sqcap(j,k) = 1$, it follows that $r(E- y) \le r(E-k-y) + 1$. This contradiction completes the proof of (ii).
\end{proof}
\begin{lemma}
\label{pricklytime0}
In a $2$-polymatroid $P$, let $k$ and $y$ be distinct elements. Then
\begin{itemize}
\item[(i)] $P\downarrow k \backslash y = P\backslash y \downarrow k$; and
\item[(ii)] $P\downarrow k /y = P/y \downarrow k.$
\end{itemize}
\end{lemma}
\begin{proof} Part (i) is essentially immediate. We now prove (ii).
If $r(\{k\}) \le 1$, then $P\downarrow k = P/k$, so
$$P\downarrow k /y = P/k/y = P/y/k = P/y \downarrow k.$$
Thus we may assume that $r(\{k\}) = 2$.
Suppose $y$ is a line such that $r(\{y,k\}) = 2$. Then
$$P/y\downarrow k = P/y/k = P/k/y = P\downarrow k/y$$
where the last equality follows by considering how $P\downarrow k$ is constructed.
Thus we may assume that $y$ is not a line that is parallel to $k$. Hence
$$r(\{y,k\}) > r(\{y\}).$$
Let $X$ be a subset of $E - k - y$. Then
$$r_{P\downarrow k /y}(X) = r_{P\downarrow k}(X \cup y) - r_{P\downarrow k}(\{y\}) = r_{P\downarrow k}(X \cup y) - r(\{y\})$$
where the second equality follows because $r(\{y,k\}) > r(\{y\}).$ We deduce that
\begin{equation*}
r_{P\downarrow k /y}(X) =
\begin{cases}
r(X\cup y) - r(\{y\}), & \text{if $r(X \cup y \cup k) > r(X \cup y)$};\\
r(X\cup y) - r(\{y\}) - 1, & \text{otherwise}.
\end{cases}
\end{equation*}
On the other hand,
since $r_{P/y}(X \cup k) = r(X \cup k \cup y) - r(\{y\})$ and $r_{P/y}(X) = r(X \cup y) - r(\{y\})$, we see that
\begin{equation*}
r_{P/y\downarrow k }(X) =
\begin{cases}
r(X\cup y) - r(\{y\}), & \text{if $r(X \cup y \cup k) > r(X \cup y)$};\\
r(X\cup y) - r(\{y\}) - 1, & \text{otherwise}.
\end{cases}
\end{equation*}
Thus
$$r_{P\downarrow k /y}(X) = r_{P/y\downarrow k }(X)$$
so the lemma holds.
\end{proof}
\begin{lemma}
\label{elemprop24}
Let $P$ be a compact $2$-polymatroid $(E,r)$ having $\{j,k\}$ as a prickly $3$-separator. Suppose $y \in E - \{j,k\}$. If $\{j,k\}$ is not a prickly $3$-separator of $P/y$, then
\begin{itemize}
\item[(i)] $r(\{j,k,y\}) = 3$ and $P/y$ has $\{j,k\}$ as a $1$-separating set; or
\item[(ii)] $P\downarrow k/y = P/y\backslashba k$; or
\item[(iii)] $P\downarrow k/y = P/y/k$; or
\item[(iv)] $P\downarrow k/y$ can be obtained from $P/y\backslashba j$ by relabelling $k$ as $j$.
\end{itemize}
\end{lemma}
\begin{proof} Suppose first that $r_{P/y}(\{j,k\}) = 1$. Then $y$ is a line of $P$ that is in the closure of $\{j,k\}$. Thus $\lambda_{P/y}(\{j,k\}) = 0$ and (i) holds.
Next assume that $r_{P/y}(\{j,k\}) = 2$. Then $\lambda_{P/y}(\{j,k\}) = 1$. Thus $P/y$ can be written as the 2-sum, with basepoint $p$ of two polymatroids, one of which, $P_1$, has ground set $\{j,k,p\}$ and has rank $2$. As $P$ is compact, so is $P/y$. There are four choices for $P_1$:
\begin{itemize}
\item[(a)] $j$ and $k$ are parallel lines and $p$ is a point lying on them both;
\item[(b)] $P_1$ is isomorphic to the matroid $U_{2,3}$;
\item[(c)] $P_1$ has $k$ as a line and has $j$ and $p$ as distinct points on this line; or
\item[(d)] $P_1$ has $j$ as a line and has $k$ and $p$ as distinct points on this line;
\end{itemize}
By Lemma~\ref{pricklytime0}, $P\downarrow k/y = P/y \downarrow k$. If $P_1$ is one of the $2$-polymatroids in (b) or (d), then, as $k$ is a point of $P/y$, it follows that $P/y \downarrow k = P/y / k$, so (iii) holds. Next suppose that $P_1$ is the $2$-polymatroid in (a). Then, as $P/y$ is compact, it follows that $P/y \downarrow k = P/y \backslashba k$, so (ii) holds. Finally, suppose that $P_1$ is the $2$-polymatroid in (c). Then $P\downarrow j/y = P/y \downarrow j = P/y \backslashba j$. By Lemma~\ref{symjk},
$P\downarrow j$ can be obtained from $P\downarrow k$ by relabelling $j$ as $k$. Thus $P\downarrow j/y$ can be obtained from $P/y \backslashba j$ by relabelling $k$ as $j$, that is, (iv) holds.
We may now assume that $r_{P/y}(\{j,k\}) = 3$. Then $\sqcap(y, \{j,k\}) = 0$ and one easily checks that $\{j,k\}$ is a prickly $3$-separator of $P/y$; a contradiction.
\end{proof}
\begin{lemma}
\label{elemprop25}
Let $\{j,k\}$ be a prickly $3$-separator in a $2$-polymatroid $P$. If $P\downarrow k$ is $3$-connected, then so is $P$.
\end{lemma}
\begin{proof}
Let $(X,Y)$ be an exact $m$-separation of $P$ for some $m$ in $\{1,2\}$ where $k \in X$.
Then $r(X) + r(Y) - r(P) = m-1$. Now $r(P\downarrow k) = r(P) -1$.
Consider $r_{P\downarrow k}(X-k) + r_{P\downarrow k}(Y)$. Suppose first that $j \in X - k$. Then, by Lemma~\ref{atlast}, $r_{P\downarrow k}(X - k) = r(X) - 1$ and $r_{P\downarrow k}(Y) = r(Y)$. Hence
$$r_{P\downarrow k}(X - k) + r_{P\downarrow k}(Y) - r(P\downarrow k) = m-1.$$
As $P\downarrow k$ is $2$-connected, we cannot have $m = 1$ since both $X - k$ and $Y$ are non-empty. Thus $m = 2$. Now $\max\{|X|,r(X)\} \ge 2$ and
$\max\{|Y|,r(Y)\} \ge 2$. Thus $\max\{|Y|,r_{P\downarrow k}(Y)\} \ge 2$. If $X = \{j,k\}$, then $r(X) + r(Y) - r(P) = 1$; a contradiction to the fact that $\{j,k\}$ is a $3$-separator of $P$. We deduce that $|X - k| \ge 2$, so $(X-k,Y)$ is a $2$-separation of $P\downarrow k$; a contradiction.
We may now assume that $j \in Y$. Then
$$r_{P\downarrow k}(X - k) + r_{P\downarrow k}(Y) - r(P\downarrow k) \le r(X) - 1 + r(Y) - 1 -r(P) + 1 = m-2.$$
As $P\downarrow k$ is $2$-connected, it follows that $X - k$ is empty. Then $r(\{k\}) + r(E-k) - r(E) = 1$; a contradiction. Thus the lemma holds.
\end{proof}
\section{Some results for connectivity and local connectivity}
\label{clc}
This section notes a number of properties of the connectivity and local-connectivity functions that will be used in the proof of the main theorem.
First we show that compression is, in most situations, a self-dual operation. We proved this result in \cite[Proposition 3.1]{oswww} for the variant of duality used there. By making the obvious replacements in that proof, it is straightforward to check that the result holds with the modified definition of duality used here. We omit the details.
\begin{proposition}
\label{compdual}
Let $e$ be a line of a $2$-polymatroid $M$ and suppose that $M$ contains no line parallel to $e$. Then
$$M^*\downarrow e = (M\downarrow e)^*.$$
\end{proposition}
The next result implies that the main theorem is a self-dual result.
\begin{proposition}
\label{sminordual}
Let $P$ and $Q$ be compact $2$-polymatroids. Then $Q$ is an s-minor of $P$ if and only if $Q^*$ is an s-minor of $P^*$.
\end{proposition}
\begin{proof} By Lemma~\ref{compact0}, both $P^*$ and $Q^*$ are compact. Moreover, $(P^*)^* = P$ and $(Q^*)^* = Q$.
Assume $Q$ is an s-minor of $P$. To prove the lemma, it suffices to show that $Q^*$ is an s-minor of $P^*$. By Lemma~\ref{compact0} again, for an element $\ell$ of $P$, we have that
$(P\backslashba \ell)^* = P^*/ \ell$ and $(P/ \ell)^* = P^*\backslashba \ell$. Moreover, if $\{j,k\}$ is a prickly 3-separator of $P$, then one easily checks that it is a prickly 3-separator of $P^*$. By Lemma~\ref{elemprop23}, $P\downarrow k$ is compact and, by Proposition~\ref{compdual}, $(P\downarrow k)^* = P^*\downarrow k$. Because the dual of each allowable move on $P$ produces a 2-polymatroid that is obtained from $P^*$ by an allowable move, the lemma follows by a straightforward induction argument.
\end{proof}
There is an attractive link between the connectivity of a $2$-polymatroid $M$ and the connectivity of the natural matroid associated with $M$.
\begin{lemma}
\label{missinglink}
Let $M$ be a $2$-polymatroid with at least two elements and let $M'$ be the natural matroid derived from $M$. Then
\begin{itemize}
\item[(i)] $M$ is $2$-connected if and only if $M'$ is $2$-connected; and
\item[(ii)] $M$ is $3$-connected if and only if $M'$ is $3$-connected.
\end{itemize}
\end{lemma}
\begin{proof}
The result is immediate if $M$ is a matroid or has a loop, so we may assume that $M$ is loopless and has at least one line.
Let $L$ be the set of lines of $M$ and let $M^+$ be the matroid that is obtained from $M$ by freely adding two points on each line in $L$. Then $M' = M^+\backslash L$.
Suppose that $M$ has a $k$-separation $(X,Y)$ for some $k$ in $\{1,2\}$. Replacing each line in each of $X$ and $Y$ by two points freely placed on the line gives sets $X'$ and $Y'$ that partition $E(M')$ such that $r(X') = r(X)$ and $r(Y') = r(Y)$. Hence $(X',Y')$ is a $k$-separation of $M'$.
Now suppose that $M'$ has a $k$-separation for some $k$ in $\{1,2\}$. Choose such a $k$-separation $(X',Y')$ to minimize the number $m$ of lines of $M$ that have exactly one of the corresponding points of $M'$ in $X'$. If $m = 0$, then there is a $k$-separation of $M$ that corresponds naturally to $(X',Y')$. Thus we may assume that $M$ has a line $\ell$ whose corresponding points, $s_{\ell}$ and $t_{\ell}$, are in $X'$ and $Y'$, respectively. Now
\begin{equation}
\label{kay}
r(X') + r(Y') - r(M') = k-1.
\end{equation}
Suppose $|E(M')| = 3$. Then $M$ consists of a point and a line. For each $n$ in $\{2,3\}$, both $M$ and $M'$ are $n$-connected if and only if the point lies on the line. Thus the result holds if $|E(M')| = 3$. Now assume that $|E(M')| = 4$. Then $M$ consists of either two lines, or a line and two points. Again the result is easily checked. Thus we may assume that $|E(M')| \ge 5$.
We may also assume that $|X'| \ge |Y'|$. Then $|X'|\ge 3$. Now $r(X'- s_{\ell}) + r(Y' \cup s_{\ell}) - r(M') \ge k$, otherwise the choice of $(X',Y')$ is contradicted. Thus $r(X' - s_{\ell}) = r(X')$ and $r(Y' \cup s_{\ell}) = r(Y') + 1$. Hence, in $M^+$, as $s_{\ell}$ and $t_{\ell}$ are freely placed on $\ell$, we see that
$r((X' - s_{\ell}) \cup \ell) = r(X' - s_{\ell})$, so
$$r(X' - s_{\ell}) = r((X' - s_{\ell}) \cup t_{\ell}) = r(X' \cup t_{\ell}).$$ Hence $(X' \cup t_{\ell},Y'-t_{\ell})$ violates the choice of $(X',Y')$ unless either $k=1$ and $Y' = \{t_{\ell}\}$, or $k =2$ and $Y'$ consists of two points. In the first case, $r(X') = r(M')$, so $r(X') + r(Y') - r(M') = 1,$ a contradiction to (\ref{kay}). In the second case, since one of the points in $Y'$ is $t_{\ell}$, the points are not parallel so $r(Y') = 2$ and $r(X') = r(M') - 1$. Thus $r(X' \cup t_{\ell}) = r(M') - 1$ and $r(Y'-t_{\ell}) = 1$; a contradiction to (\ref{kay}).
\end{proof}
Let $M$ be a polymatroid $(E,r)$. If $X$ and $Y$ are subsets of $E$, the {\it local connectivity}
$\sqcap(X,Y)$ between $X$ and $Y$ is defined by $\sqcap(X,Y) = r(X) + r(Y) - r(X \cup Y)$.
Sometimes we will write $\sqcap_M$ for $\sqcap$, and $\sqcap^*$ for $\sqcap_{M^*}$. It is straightforward to prove the following. Again this holds for both the version of duality used here and the variant used in \cite{oswww}.
\begin{lemma}
\label{sqcapdual}
Let $M$ be a polymatroid $(E,r)$. For disjoint subsets $X$ and $Y$ of $E$,
$$\sqcap_{M^*}(X,Y) = \sqcap_{M/(E - (X \cup Y))}(X,Y).$$
\end{lemma}
The next lemma will be used repeatedly, often without explicit reference. Two sets $X$ and $Y$ in a polymatroid $M$ are {\it skew} if $\sqcap(X,Y) = 0$.
\begin{lemma}
\label{skewer}
Let $M$ be a $2$-polymatroid and $z$ be an element of $M$ such that $(A,B)$ is a $2$-separation of $M/z$. Suppose $z$ is skew to $A$. Then
$(A,B\cup z)$ is a $2$-separation of $M$. Moreover, if $M$ is $3$-connected, then $A$ is not a single line in $M/z$.
\end{lemma}
\begin{proof} Clearly $r(A \cup z) - r(z) = r(A)$, so $(A,B \cup z)$ is a $2$-separation\ of $M$. If $M$ is $3$-connected\ and $A$ consists of a single line $a$ of $M/z$, then $a$ is a line of $M$, so $a$ and $z$ are skew, and we obtain the contradiction\ that $M$ has a $2$-separation.
\end{proof}
Numerous properties of the connectivity function of a matroid are proved simply by applying properties of the rank function; they do not rely on the requirement that $r(\{e\}) \le 1$ for all elements $e$. Evidently, such properties also hold for the connectivity function of a polymatroid. The next few lemmas note some of these properties.
The first two are proved in \cite[Lemmas 8.2.3 and 8.2.4]{oxbook}.
\begin{lemma}
\label{8.2.3} Let $(E,r)$ be a polymatroid and
let $X_1, X_2, Y_1$, and $Y_2$ be subsets of $E$ with $Y_1 \subseteq X_1$ and $Y_2 \subseteq X_2$.
Then
$$\sqcap(Y_1,Y_2) \le \sqcap(X_1,X_2).$$
\end{lemma}
\begin{lemma}
\label{8.2.4} Let $(E,r)$ be a polymatroid $M$ and
let $X,C$, and $D$ be disjoint subsets of $E$.
Then
$$\lambda_{M\backslash D/C}(X) \le \lambda_M(X).$$
Moreover, equality holds if and only if
$$r(X \cup C) = r(X) + r(C)$$ and
$$r(E - X) + r(E-D) = r(E) + r(E- (X \cup D)).$$
\end{lemma}
The following \cite[Corollary 8.7.6]{oxbook} is a straightforward consequence of the last
lemma.
\begin{corollary}
\label{8.7.6}
Let $X$ and $D$ be disjoint subsets of the ground set $E$ of a polymatroid $M$. Suppose that $r(M\backslash D) = r(M)$. Then
\begin{itemize}
\item[(i)] $\lambda_{M\backslash D}(X) = \lambda_M(X)$ if and only if $D \subseteq {\rm cl}_M(E - (X \cup D))$; and
\item[(ii)] $\lambda_{M\backslash D}(X) = \lambda_M(X\cup D)$ if and only if
$D \subseteq {\rm cl}_M(X)$.
\end{itemize}
\end{corollary}
It is well known that, when $M$ is a matroid, for all subsets $X$ of $E(M)$,
$$\lambda_M(X) = r_M(X) + r_{M^*}(X) - |X|.$$
It is easy to check that the following variant on this holds for polymatroids.
Recall that $||X|| = \sum_{x \in X} r(\{x\}).$
\begin{lemma}
\label{rr*}
In a polymatroid $M$, for all subsets $X$ of $E(M)$,
$$\lambda_M(X) = r_M(X) + r_{M^*}(X) - ||X||.$$
In particular, if every element of $X$ has rank one, then
$$\lambda_M(X) = r_M(X) + r_{M^*}(X) - |X|.$$
\end{lemma}
The next lemma contains another useful equation whose proof is straightforward.
\begin{lemma}
\label{obs1}
Let $(X,Y)$ be a partition of the ground set of a polymatroid $M$. Suppose $z \in Y$. Then
$$\sqcap(X,\{z\}) + \sqcap_{M/z}(X,Y-z) = \sqcap(X,\{z\}) + \lambda_{M/z}(X) = \lambda_M(X).$$
\end{lemma}
The next two lemmas are natural generalizations of matroid results that appear in \cite{osw}.
\begin{lemma}
\label{univ}
Let $(E,r)$ be a polymatroid and
let $X$ and $Y$ be disjoint subsets of $E$.
Then
$$\lambda(X \cup Y) = \lambda(X) + \lambda(Y) - \sqcap(X,Y) - \sqcap^*(X,Y).$$
\end{lemma}
\begin{lemma}
\label{oswrules} Let $A, B, C$, and $D$ be subsets of the ground set of a polymatroid. Then
\begin{itemize}
\item[(i)] $\sqcap(A \cup B, C \cup D) + \sqcap(A,B) + \sqcap(C,D) =
\sqcap(A \cup C, B \cup D) + \sqcap(A,C) + \sqcap(B,D)$; and
\item[(ii)] $\sqcap(A \cup B, C) + \sqcap(A,B) =
\sqcap(A \cup C, B) + \sqcap(A,C).$
\end{itemize}
\end{lemma}
\begin{lemma}
\label{general}
Let $M$ be a polymatroid and $(A,B,Z)$ be a partition of its ground set into possibly empty subsets.
Then
$$\lambda_{M/Z}(A) = \lambda_{M\backslash Z}(A) - \sqcap_M(A,Z) - \sqcap_M(B,Z) + \lambda_M(Z).$$
\end{lemma}
\begin{proof}
We have $B = A \cup Z$ and $A = B \cup Z$. Then
\begin{align*}
\lambda_{M/Z}(A) &= r_{M/Z}(A) + r_{M/Z}(B) - r(M/Z)\\
& = r(A \cup Z) - r(Z) + r(B\cup Z) - r(Z) -r(M) + r(Z)\\
& = r(B) + r(A) - r(M) - r(Z)\\
&= r(A) + r(B) - r(M\backslash Z) + r(M\backslash Z) - r(M) - r(Z)\\
& = \lambda_{M\backslash Z}(A) + r(M\backslash Z) - r(M) - r(Z).
\end{align*}
The required result holds if and only if
$$\sqcap_M(A,Z) + \sqcap_M(B,Z) - \lambda_M(Z) = r(M) + r(Z) - r(M\backslash Z).$$
Now
\begin{align*}
\sqcap(A,Z) + \sqcap(B,Z) - \lambda_M(Z) &= r(A) + r(Z) - r(A \cup Z) + r(B) + r(Z) - r(B\cup Z)\\
& \hspace*{1.7in} - r(Z) - r(M\backslash Z) + r(M)\\
& = r(A) + r(Z) - r(B) + r(B) + r(Z) - r(A) - r(Z)\\
& \hspace*{1.7in} - r(M\backslash Z) + r(M)\\
& = r(M) + r(Z) - r(M\backslash Z),
\end{align*}
as required.
\end{proof}
\begin{corollary}
\label{general2}
Let $M$ be a polymatroid and $(A,B,Z)$ be a partition of its ground set into possibly empty subsets.
Suppose $r(M\backslash Z) = r(M)$. Then
$$\lambda_{M/Z}(A) = \lambda_{M\backslash Z}(A) - \sqcap_M(A,Z) - \sqcap_M(B,Z) + r(Z).$$
\end{corollary}
\begin{proof}
As $\lambda_M(Z) = r(Z) + r(M\backslash Z) - r(M)$ and $r(M\backslash Z) = r(M)$, the result is an immediate consequence of the last lemma.
\end{proof}
\begin{lemma}
\label{general3}
Let $M$ be a polymatroid and $(A,B,C)$ be a partition of its ground set into possibly empty subsets.
Suppose $\lambda(A) = 1 = \lambda(C)$ and $\lambda(B) = 2$. Then $\sqcap(A,B) = 1$.
\end{lemma}
\begin{proof}
We have
$$2= r(B) + r(A\cup C) -r(M)$$ and
\begin{align*}
r(M) &= r(A\cup B) + r(C) - 1\\
& = r(A) + r(B) - \sqcap(A,B) + r(C) - 1\\
& = r(A) + r(B) + r(C) - 1 - \sqcap (A,B).
\end{align*}
Thus
$$2 = r(B) + r(A \cup C) - r(A) - r(B) - r(C) + 1 + \sqcap(A,B)$$
so $\sqcap(A,B) = 1 + \sqcap(A,C) \ge 1$. By Lemma~\ref{8.2.3}, $\sqcap(A,B) \le \sqcap(A,B\cup C) = 1$, so $\sqcap(A,B) = 1$.
\end{proof}
\begin{lemma}
\label{general4}
Let $M$ be a polymatroid and $(A,B,C)$ be a partition of its ground set into possibly empty subsets.
Then $\sqcap^*(A,B) = \lambda_{M/C}(A)$.
\end{lemma}
\begin{proof}
By making repeated use of Lemma~\ref{compact0}, we have
\begin{align*}
\sqcap^*(A,B) & = \sqcap_{M^*}(A,B)\\
& = \lambda_{M^*\backslash C}(A)\\
& = \lambda_{(M^*\backslash C)^{\flat}}(A)\\
& = \lambda_{(M/C)^*}(A)\\
& = \lambda_{M/C}(A).
\end{align*}
\end{proof}
The following is a consequence of a result of Oxley and Whittle~\cite[Lemma 3.1]{owconn}.
\begin{lemma}
\label{Tutte2}
Let $M$ be a $2$-connected $2$-polymatroid with $|E(M)| \ge 2$. If $e$ is a point of $M$, then $M\backslash e$ or $M/e$ is $2$-connected.
\end{lemma}
The next result is another straightforward extension of a matroid result.
\begin{lemma}
\label{matroidef}
Let $M$ be a $2$-connected $2$-polymatroid having $e$ and $f$ as points. Then
\begin{itemize}
\item[(i)] $\lambda_{M/ f}(\{e\}) = 0$ if and only if $e$ and $f$ are parallel points; and
\item[(ii)] $\lambda_{M\backslash f}(\{e\}) = 0$ if and only if $e$ and $f$ form a series pair.
\end{itemize}
\end{lemma}
\begin{proof} We prove (i) omitting the similar proof of (ii). If $e$ and $f$ are parallel points of $M$, then $\lambda_{M/ f}(\{e\}) = 0$. Now assume that $\lambda_{M/ f}(\{e\}) = 0$. Let $M'$ be the natural matroid derived from $M$. Then $M'/f$ has $\{e\}$ as a component. Hence $\{e,f\}$ is a series or parallel pair in $M'$. But if $\{e,f\}$ is a series pair, then $M'/f$ is $2$-connected; a contradiction. We conclude that $\{e,f\}$ is a parallel pair of points in $M$, so (i) holds.
\end{proof}
The next result is a generalization of a lemma of Bixby \cite{bixby} (see also \cite[Lemma 8.7.3]{oxbook}) that is widely used when dealing with $3$-connected matroids.
\begin{lemma}
\label{newbix}
Let $M$ be a $3$-connected $2$-polymatroid and $z$ be a point of $M$. Then either
\begin{itemize}
\item[(i)] $M/z$ is $2$-connected having one side of every $2$-separation being a pair of points of $M$ that are parallel in $M/z$; or
\item[(ii)] $M\backslash z$ is $2$-connected having one side of every $2$-separation being either a single line of $M$, or a pair of points of $M$ that form a series pair in $M\backslash z$.
\end{itemize}
\end{lemma}
\begin{proof}
If $z$ lies on a line in $M$, then $M\backslash z$ is $3$-connected. Thus we may assume that $z$ does not lie on a line in $M$. Take the matroid $M'$ that is naturally derived from $M$. Then, by Bixby's Lemma, either $M'/z$ is $2$-connected having one side of every $2$-separation being a pair of parallel points of $M'$, or $M'\backslash z$ is $2$-connected having one side of every $2$-separation being a series pair of points of $M'$. In the first case, if $\{a,b\}$ is a parallel pair of points of $M'/z$, then $\{a,b,z\}$ is a circuit of $M'$. Because the points added to $M$ to form $M'$ are freely placed on lines, we cannot have a circuit containing just one of them. Since $z$ is not on a line of $M$, we deduce that $a$ and $b$ are points of $M$. We conclude that, in the first case, (i) holds.
Now suppose that $M'\backslash z$ is not $3$-connected\ and has $\{u,v\}$ as a series pair. Then either $u$ and $v$ are both matroid points of $M$, or $M$ has a line on which the points $u$ and $v$ are freely placed in the formation of $M'$. We deduce that (ii) holds.
\end{proof}
We recall from \cite{oswww} that, when $\{a,b,c\}$ is a set of three points in a $2$-polymatroid $Q$, we call $\{a,b,c\}$ a {\it triangle} if every subset of $\{a,b,c\}$ of size at least two has rank two. If, instead, $r(E - \{a,b,c\}) = r(Q) - 1$ but $r(X) = r(Q)$ for all proper supersets $X$ of $E - \{a,b,c\}$, then we call $\{a,b,c\}$ a {\it triad} of $Q$. When $Q$ is $3$-connected, $\{a,b,c\}$ is a triad of $Q$ if and only if $\{a,b,c\}$ is a triangle of $Q^*$. It is straightforward to check that a triangle and a triad of $Q$ cannot have exactly one common element. Just as for matroids, we call a sequence $x_1,x_2,\dots,x_k$ of distinct points of a $2$-polymatroid $Q$ a {\it fan} of {\it length} $k$ if $k \ge 3$ and the sets $\{x_1,x_2,x_3\}, \{x_2,x_3,x_4\},\dots,\{x_{k-2},x_{k-1},x_k\}$ are alternately triangles and triads beginning with either a triangle or a triad.
The following lemma will be helpful in proving our main result when fans arise in the argument.
\begin{lemma}
\label{fantan}
Let $M$ and $N$ be $3$-connected $2$-polymatroids where $|E(N)| \ge 4$ and $M$ is not a whirl or the cycle matroid of a wheel. Suppose $M$ has a fan $ x_1,x_2,x_3,x_4$ where $\{x_1,x_2,x_3\}$ is a triangle and $M/x_2$ has a c-minor isomorphic to $N$. Then $M$ has a point $z$ such that either $M\backslash z$ or $M/z$ is $3$-connected having a c-minor isomorphic to $N$, or both $M\backslash z$ and $M/z$ have c-minors isomorphic to $N$.
\end{lemma}
\begin{proof}
Assume that the lemma fails. Extend $x_1,x_2,x_3,x_4$ to a maximal fan $x_1,x_2,\dots,x_n$.
Since $M/x_2$ has a c-minor isomorphic to $N$ and has $x_1$ and $x_3$ as a parallel pair of points, it follows that each of $M/x_2\backslash x_1$ and $M/x_2\backslash x_3$ has a c-minor isomorphic to $N$. Thus each of $M\backslash x_1$ and $M\backslash x_3$ has a c-minor isomorphic to $N$. Hence $M/x_4$ has a c-minor isomorphic to $N$. A straightforward induction argument establishes that $M/x_i$ has a c-minor isomorphic to $N$ for all even $i$, while $M\backslash x_i$ has a c-minor isomorphic to $N$ for all odd $i$. Then $M/x_i$ is not $3$-connected\ when $i$ is even, while $M\backslash x_i$ is not $3$-connected\ when $i$ is odd.
Next we show that
\begin{sublemma}
\label{whereare}
$M$ has no
triangle that contains more than one element $x_i$ with $i$ even; and $M$ has no
triad that contains more than one element $x_i$ with $i$ odd.
\end{sublemma}
Suppose $M$ has a triangle that contains $x_i$ and $x_j$ where $i$ and $j$ are distinct even integers. Since $M/x_i$ has $x_j$ in a parallel pair of points, $M/x_i\backslash x_j$, and hence $M\backslash x_j$, has a c-minor isomorphic to $N$. As $M/ x_j$ also has a c-minor isomorphic to $N$, we have a contradiction. Thus the first part of \ref{whereare} holds. A similar argument proves the second part.
Suppose $n$ is odd. Then, since neither $M\backslash x_n$ nor $M\backslash x_{n-2}$ is $3$-connected, by \cite[Lemma 4.2]{oswww}, $M$ has a triad $T^*$ containing $x_n$ and exactly one of $x_{n-1}$ and $x_{n-2}$. By \ref{whereare}, $T^*$ contains $x_{n-1}$. Then, by the maximality of the fan, the third element of $T^*$ lies in $\{x_1,x_2,\dots,x_{n-2}\}$. But, as each of the points in the last set is in a triangle that is contained in that set, we obtain the contradiction\ that
$M$ has a triangle having a single element in common with the triad $T^*$.
We may now assume that $n$ is even. As neither $M/x_n$ nor $M/x_{n-2}$ is $3$-connected, by \cite[Lemma 4.2]{oswww}, $M$ has a triangle $T$ that contains $x_n$ and exactly one of $x_{n-1}$ and $x_{n-2}$. By \ref{whereare}, $x_{n-1} \in T$. The maximality of the fan again implies that the third element of $T$ is in $\{x_1,x_2,x_3,\dots,x_{n-2}\}$. As every element of the last set, except $x_1$, is in a triad that is contained in the set, to avoid having $T$ meet such a triad in a single element, we must have that $T = \{x_n,x_{n-1},x_1\}$.
If $n= 4$, then $M|\{x_1,x_2,x_3,x_4\} {\rm co}ng U_{2,4}$ so $\{x_2,x_3,x_4\}$ is a triangle; a contradiction\ to \ref{whereare}. We deduce that $n> 4$. Now neither $M\backslash x_1$ nor $M\backslash x_{n-1}$ is $3$-connected. Thus, by \cite[Lemma 4.2]{oswww}, $M$ has a triad $T^*_2$ containing $x_1$ and exactly one of $x_n$ and $x_{n-1}$. By \ref{whereare}, $x_n \in T^*_2$. The triangles $\{x_1,x_2,x_3\}$ and $\{x_3,x_4,x_5\}$ imply that $x_2 \in T^*_2$. Let $X = \{x_1,x_2,\dots, x_n \}$. Then, using the triangles we know, including $\{x_n,x_{n-1},x_1\}$, we deduce that $r(X) \le \tfrac{n}{2}$. Similarly, the triads in $M$, which are triangles in $M^*$, imply that $r^*(X) \le \tfrac{n}{2}$. Thus, by Lemma~\ref{rr*}, $\lambda(X) = 0$. Hence $X = E(M)$. As every element of $M$ is a point, $M$ is a matroid. Since every point of $M$ is in both a triangle and a triad, by Tutte's Wheels-and-Whirls-Theorem \cite{wtt}, we obtain the contradiction\ that $M$ is a whirl or the cycle matroid of a wheel.
\end{proof}
\begin{lemma}
\label{hath}
Let $M$ be a $3$-connected $2$-polymatroid having $a$ and $\ell$ as distinct elements. Then $(E(M) - \{a,\ell\},\{\ell\})$ is not a $2$-separation of $M/a$.
\end{lemma}
\begin{proof}
Assume the contrary. Then $\ell$ is a line in $M/a$, so $\sqcap(a,\ell) = 0$. We have
$$r_{M/a}(E(M) - \{a,\ell\}) + r_{M/ a}(\ell) = r(M/a) + 1.$$
As $\sqcap(a,\ell) = 0$, it follows that $(E(M) - \ell,\{\ell\})$ is a $2$-separation of $M$; a contradiction.
\end{proof}
\section{Parallel connection and $2$-sum}
\label{pc2s}
In this section, we follow Mat\'{u}\v{s}~\cite{fm} and Hall~\cite{hall} in defining the parallel connection and $2$-sum of polymatroids.
For a positive integer $k$, let $M_1$ and $M_2$ be $k$-polymatroids $(E_1,r_1)$ and $(E_2,r_2)$. Suppose first that $E_1 \cap E_2 = \emptyset$.
The {\it direct sum} $M_1 \oplus M_2$ of $M_1$ and $M_2$ is the $k$-polymatroid $(E_1 \cup E_2,r)$ where, for all subsets $A$ of $E_1 \cup E_2$, we have $r(A) = r(A\cap E_1) + r(A \cap E_2)$. The following result is easily checked.
\begin{lemma}
\label{dualsum}
For $k$-polymatroids $M_1$ and $M_2$ on disjoint sets,
$$(M_1 \oplus M_2)^* = M_1^* \oplus M_2^*.$$
\end{lemma}
Clearly a $2$-polymatroid is $2$-connected if and only if it cannot be written as the direct sum of two non-empty $2$-polymatroids.
Now suppose that $E_1 \cap E_2 = \{p\}$ and $r_1(\{p\}) = r_2(\{p\})$. Let $P(M_1,M_2)$ be $(E_1 \cup E_2, r)$ where $r$ is defined for all subsets $A$ of $E_1 \cup E_2$ by
$$r(A) = \min\{r_1(A \cap E_1) + r_2(A\cap E_2), r_1((A \cap E_1)\cup p) + r_2((A \cap E_2)\cup p) - r_1(\{p\})\}.$$
As Hall notes, it is routine to check that $P(M_1,M_2)$ is a $k$-polymatroid. We call it the {\it parallel connection} of $M_1$ and $M_2$ with respect to the {\it basepoint} $p$. When $M_1$ and $M_2$ are both matroids, this definition coincides with the usual definition of the parallel connection of matroids. Hall extends the definition of parallel connection to deal with the case when $r_1(\{p\}) \neq r_2(\{p\})$ but we shall not do that here.
Now suppose that $M_1$ and $M_2$ are $2$-polymatroids having at least two elements, that $E(M_1) \cap E(M_2) = \{p\}$,
that neither $\lambda_{M_1}(\{p\})$ nor $\lambda_{M_2}(\{p\})$ is $0$, and that
$r_1(\{p\}) = r_2(\{p\}) = 1$. We define the {\it $2$-sum}, $M_1 \oplus_2 M_2$, of $M_1$ and $M_2$ to be $P(M_1,M_2)\backslash p$. We remark that this extends Hall's definition since, to ensure that $M_1 \oplus_2 M_2$ has more elements than each of $M_1$ and $M_2$, he requires that they each have at least three elements. He imposes the same requirement in his Proposition 3.6. The next result is that result with this restriction omitted. Hall's proof \cite{hall} remains valid.
\begin{proposition}
\label{dennis3.6}
Let $M$ be a $2$-polymatroid $(E,r)$ having a partition $(X_1,X_2)$ such that $r(X_1) + r(X_2) = r(E) + 1$. Then there are $2$-polymatroids $M_1$ and $M_2$ with ground sets $X_1 \cup p$ and $X_2 \cup p$, where $p$ is a new element not in $E$, such that
$M = P(M_1,M_2)\backslash p$. In particular, for all $A \subseteq X_1 \cup p$,
\begin{equation*}
r_1(A) =
\begin{cases}
r(A), & \text{if $p \not\in A$;}\\
r((A-p) \cup X_2) - r(X_2) + 1, & \text{if $p \in A$.}
\end{cases}
\end{equation*}
\end{proposition}
\begin{lemma}
\label{dennisplus}
Let $(X,Y)$ be a partition of the ground set of a $2$-polymatroid $M$ such that $\lambda(X) = 1$. Then, for some element $p$ not in $E(M)$, there are $2$-polymatroids $M_X$ and $M_Y$ on $X \cup p$ and $Y \cup p$, respectively, such that $M = M_X \oplus_2 M_Y$. Moreover, for $y \in Y$,
\begin{itemize}
\item[(i)] $\lambda_{M_Y \backslash y}(\{p\}) = \sqcap(X,Y-y)$;
\item[(ii)] $\lambda_{M_Y/y}(\{p\}) + \sqcap(X,\{y\}) = \lambda(X) = 1$;
\item[(iii)] if $\sqcap(X,Y-y) = 1$, then $M\backslash y = M_X \oplus_2 (M_Y\backslash y)$;
\item[(iv)] if $\sqcap(X,\{y\}) = 0$, then $M/ y = M_X \oplus_2 (M_Y/ y)$; and
\item[(v)] if $r(\{y\}) \le 1$, then
\begin{equation*}
M\downarrow y =
\begin{cases}
(M_X/p) \oplus (M_Y\backslash y /p), & \text{if $\sqcap(X,\{y\}) = 1$;}\\
M_X \oplus_2 (M_Y\downarrow y), & \text{if $\sqcap(X,\{y\}) = 0$.}
\end{cases}
\end{equation*}
\item[(vi)] if $y$ is a line, then
\begin{equation*}
M\downarrow y =
\begin{cases}
(M_X\backslash p) \oplus (M_Y\downarrow y \backslash p), & \text{if $r(Y) = r(Y - y) +2$;}\\
M_X \oplus_2 (M_Y \downarrow y) & \text{if $r(Y) \le r(Y - y) +1$.}
\end{cases}
\end{equation*}
In particular, $M\downarrow y = M_X \oplus_2 (M_Y \downarrow y)$ when $\sqcap_{M\downarrow y}(X,Y-y) = 1.$
\end{itemize}
\end{lemma}
\begin{proof} The existence of $M_X$ and $M_Y$ such that $M = P(M_X,M_Y)\backslash p$ is an immediate consequence of Proposition~\ref{dennis3.6}.
To see that $P(M_X,M_Y)\backslash p = M_X \oplus_2 M_Y$, one needs only to check that
$r_{M_X}(\{p\}) = 1 = r_{M_Y}(\{p\})$ and $\lambda_{M_X}(\{p\}) = 1 = \lambda_{M_Y}(\{p\})$.
The proof of (i) follows by a straightforward application of the rank formula in Proposition~\ref{dennis3.6}. We omit the details. To see that (ii) holds, note that
\begin{align*}
\lambda_{M_Y/y}(\{p\}) & = r_{M_Y}(\{p,y\}) - r(\{y\}) + r_{M_Y}(Y) - r_{M_Y}(Y\cup p)\\
& = r(y \cup X) - r(X) + 1 - r(\{y\}) + r(Y) - r(X \cup Y) + r(X) - 1\\
& = r(y \cup X) - r(\{y\}) + r(Y) - r(X \cup Y) \\
& = r(X) - \sqcap(X,\{y\}) + r(Y) - r(X \cup Y)\\
& = \lambda_M(X) - \sqcap(X,\{y\}).
\end{align*}
By Hall~\cite[Proposition 3.1]{hall}, $M\backslash y = P(M_X,M_Y\backslash y)\backslash p$. If $\sqcap(X,Y-y) = 1$, then, by (i), $\lambda_{M_Y \backslash y}(\{p\}) = 1$.
Hence, by Hall~\cite[Proposition 3.1]{hall}, $M\backslash y = M_X \oplus_2(M_Y \backslash y)$; that is, (iii) holds.
To prove (iv), assume that $\sqcap(X,\{y\}) = 0$. We could again follow Hall~\cite[Proposition 3.1]{hall} to get that $M/y = P(M_X, M_Y/y)\backslash p$. But since he omits a full proof of this fact, we include it for completeness.
By Proposition~\ref{dennis3.6}, $M/y = P(M_1,M_2)\backslash p$ for some $M_1$ and $M_2$.
For $A \subseteq X \cup p$,
\begin{align*}
r_{M_1}(A) &=
\begin{cases}
r_{M/y}(A), & \text{if $p \not\in A$;}\\
r_{M/y}((A-p) \cup (Y-y)) - r_{M/y}(Y-y) + 1, & \text{if $p \in A$;}
\end{cases}\\
&=
\begin{cases}
r(A\cup y) - r(\{y\}), & \text{if $p \not\in A$;}\\
r((A-p) \cup Y) - r(Y) + 1, & \text{if $p \in A$;}
\end{cases}\\
& = r_{M_X}(A).
\end{align*}
Thus $M_1 = M_X$.
Now, for $A \subseteq (Y-y) \cup p$,
\begin{align*}
r_{M_2}(A) &=
\begin{cases}
r_{M/y}(A), & \text{if $p \not\in A$;}\\
r_{M/y}((A-p) \cup X) - r_{M/y}(X) + 1, & \text{if $p \in A$;}
\end{cases}\\
&=
\begin{cases}
r(A\cup y) - r(\{y\}), & \text{if $p \not\in A$;}\\
r((A-p) \cup X \cup y) - r(\{y\}) - r(X \cup y) + r(\{y\}) + 1, & \text{if $p \in A$;}
\end{cases}\\
& =
\begin{cases}
r(A\cup y) - r(\{y\}), & \text{if $p \not\in A$;}\\
r((A-p) \cup X \cup y) - r(\{y\}) - r(X) + 1, & \text{if $p \in A$.}
\end{cases}
\end{align*}
But
\begin{align*}
r_{M_Y/y}(A) &= r_{M_Y}(A \cup y) - r_{M_Y}(\{y\})\\
&=
\begin{cases}
r(A\cup y) - r(\{y\}), & \text{if $p \not\in A$;}\\
r((A-p) \cup X \cup y) - r(\{y\}) - r(X) + 1, & \text{if $p \in A$;}
\end{cases}\\
&= r_{M_2}(A).
\end{align*}
Thus $M_2 = M_Y/y$, so $M/y = P(M_X,M_Y/y)\backslash p$. As $\sqcap(X,\{y\}) = 0$, we see, by (ii), that $\lambda_{M_Y/y}(\{p\}) = 1$. Hence $M/y = M_X \oplus_2 (M_Y/y)$; that is, (iv) holds.
For (v), since $r(\{y\}) \le 1$, we have $M\downarrow y = M/y$. If $\sqcap(X,\{y\}) = 1$, then $y$ is parallel to $p$ in $M_Y$, so, by \cite[Proposition 3.1]{hall},
$M\downarrow y = (M_X/p) \oplus (M_Y/p)$. If $\sqcap(X,\{y\}) = 0$, then, as $M_Y \downarrow y = M_Y/y$, it follows by (iv) that
$$M\downarrow y = M/y = M_X \oplus_2 (M_Y/y) = M_X \oplus_2 (M_Y \downarrow y).$$
To prove (vi), suppose first that $r(Y) = r(Y- y) +2$. We have
$$r_{M_Y}(\{y,p\}) = r(y \cup X) - r(X) + 1 = 3 - \sqcap(X, \{y\}).$$
Assume $\sqcap(X, \{y\}) = 0$. Then $M_Y$ is the 2-sum, with basepoint $q$, say, of two 2-polymatroids, one of which has ground set $\{q,y,p\}$ and consists of two points and the line $y$ freely placed in the plane. Clearly, $M\downarrow y = (M_X \backslash p) \oplus (M_Y \backslash y \backslash p)$. Now assume that $\sqcap(X, \{y\}) = 1$. Then $M_Y$ is the direct sum of two 2-polymatroids, one of which has rank $2$ and consists of the line $y$ with the point $p$ on it. Once again, we see that $M\downarrow y = (M_X \backslash p) \oplus (M_Y \backslash y \backslash p)$.
We may now assume that $r(Y) \le r(Y-y) + 1$. Hence $r_{M\downarrow y}(Y-y) = r(Y) - 1$. Clearly $r(X \cup y) > r(X)$. Thus
\begin{equation}
\label{numbth}
\sqcap_{M\downarrow y}(X,Y-y) = 1.
\end{equation}
By Proposition~\ref{dennis3.6}, $M\downarrow y = P(M_1,M_2)\backslash p$ for some $2$-polymatroids $M_1$ and $M_2$ with ground sets $X \cup p$ and $(Y-y) \cup p$, respectively. We shall show that $M_1 = M_X$ and $M_2 = M_Y\downarrow y$.
First observe that, for $A \subseteq X$, we have
\begin{equation*}
r_{M_1}(A) =
\begin{cases}
r_{M\downarrow y}(A), & \text{if $p \not\in A$;}\\
r_{M\downarrow y}((A-p) \cup (Y- y)) - r_{M\downarrow y}(Y-y) + 1, & \text{if $p \in A$.}
\end{cases}
\end{equation*}
Since $r(X \cup y) > r(X)$, we see that if $p \not\in A$, then $r_{M_1}(A) = r_{M\downarrow y}(A) = r_M(A) = r_{M_X}(A)$.
Now suppose $p \in A$. Assume $r((A-p) \cup (Y-y)) = r((A-p) \cup Y)$. Then
$$r_{M\downarrow y}((A-p) \cup (Y-y)) = r((A-p) \cup (Y-y)) - 1 = r((A-p) \cup Y) - 1.$$
Moreover, $r_{M\downarrow y}(Y-y) = r(Y) -1.$ Hence
$$r_{M_1}(A) = r_M((A-p) \cup Y) - r_M(Y) + 1 = r_{M_X}(A).$$
To show that $M_1 = M_X$, it remains to consider when $p \in A$ and $r((A-p) \cup (Y-y)) < r((A-p) \cup Y)$. Then, as $r(Y-y) \ge r(Y) - 1$, we deduce that
$r((A-p) \cup (Y-y)) = r((A-p) \cup Y) - 1$, so $r(Y -y) = r(Y) - 1$. Thus we have
\begin{eqnarray*}
r_{M_1}(A) & = & r_{M\downarrow y}((A-p) \cup (Y-y)) - r_{M\downarrow y}(Y-y) + 1\\
& = & r((A-p) \cup (Y-y)) - r(Y-y) + 1\\
& = & r((A-p) \cup Y) - 1 - r(Y) + 1 + 1\\
& = & r_{M_X}(A).
\end{eqnarray*}
We conclude that $M_1 = M_X$.
To show that $M_2 = M_Y\downarrow y$, suppose that $A \subseteq (Y-y) \cup p$. Now
\begin{equation*}
r_{M_2}(A) =
\begin{cases}
r_{M\downarrow y}(A), & \text{if $p \not\in A$;}\\
r_{M\downarrow y}((A-p) \cup X) - r_{M\downarrow y}(X) + 1, & \text{if $p \in A$.}
\end{cases}
\end{equation*}
Suppose $p \not\in A$. Then
\begin{align*}
r_{M_2}(A) &=
\begin{cases}
r(A), & \text{if $r(A \cup y) > r(A)$;}\\
r(A) - 1, & \text{otherwise;}
\end{cases}\\
&=r_{M_Y\downarrow y}(A).
\end{align*}
Now assume that $p \in A$. Then $r_{M\downarrow y}(X) = r(X)$. Thus
\begin{equation*}
r_{M_2}(A) =
\begin{cases}
r((A- p) \cup X) - r(X) + 1, & \text{if $r((A - p) \cup X \cup y) > r((A-p) \cup X)$;}\\
r((A- p) \cup X) - 1 - r(X) + 1, & \text{otherwise.}
\end{cases}
\end{equation*}
Moreover,
\begin{equation*}
r_{M_Y\downarrow y}(A) =
\begin{cases}
r_{M_Y}(A), & \text{if $r_{M_Y}(A\cup y) > r_{M_Y}(A)$;}\\
r_{M_Y}(A) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
Now $r_{M_Y}(A) = r((A-p) \cup X) - r(X) + 1$. Thus
\begin{align*}
r_{M_Y}(A\cup y) - r_{M_Y}(A) &= r((A-p) \cup y \cup X) - r(X) + 1 - r((A-p) \cup X) \\
& \hspace*{2.5in}+r(X) -1\\
& = r((A-p) \cup y \cup X) - r((A-p) \cup X).
\end{align*}
We conclude that, when $p \in A$, we have $r_{M_Y\downarrow y}(A) = r_{M_2}(A)$. Thus $M_Y\downarrow y = M_2.$ Hence
$M\downarrow y = P(M_X, M_Y\downarrow y) \backslash p$. Using (\ref{numbth}), it is straightforward to show that $\lambda_{M_Y\downarrow y}(\{p\}) = 1$. It follows that
$M\downarrow y = M_X \oplus_2 (M_Y \downarrow y)$.
\end{proof}
The following was shown by Hall~\cite[Corollary 3.5]{hall}.
\begin{proposition}
\label{connconn}
Let $M_1$ and $M_2$ be $2$-polymatroids $(E_1,r_1)$ and $(E_2,r_2)$ where $E_1 \cap E_2 = \{p\}$ and $r_1(\{p\}) = r_2(\{p\}) = 1$ and each of $M_1$ and $M_2$ has at least two elements. Then the following are equivalent.
\begin{itemize}
\item[(i)] $M_1$ and $M_2$ are both $2$-connected;
\item[(ii)] $M_1 \oplus_2 M_2$ is $2$-connected;
\item[(iii)] $P(M_1,M_2)$ is $2$-connected.
\end{itemize}
\end{proposition}
One situation that will often occur will be when we have a certain $3$-connected $2$-polymatroid $N$ arising as a c-minor of a $2$-polymatroid $M$ that has a $2$-separation. Recall that a special $N$-minor of $M$ is a c-minor of $M$ that either equals $N$ or differs from $N$ by having a single point relabelled.
\begin{lemma}
\label{p49}
Let $M$ be a $2$-polymatroid that can be written as the $2$-sum $M_X \oplus_2 M_Y$ of $2$-polymatroids $M_X$ and $M_Y$ with ground sets $X \cup p$ and $Y \cup p$, respectively. Let $N$ be a $3$-connected $2$-polymatroid with $|E(N)| \ge 4$ and $E(N) \subseteq E(M)$. If $M_X$ has a special $N$-minor, then $M$ has a special $N$-minor.
\end{lemma}
\begin{proof} Since $M_X\backslash p = M\backslash Y$ and $M_X/p = M/Y$, we may assume that the special $N$-minor of $M_X$ uses $p$. Hence every other element of the special $N$-minor of $M_X$ is in $E(N)$. For $y$ in $Y$, we will denote by $M_X(y)$ the $2$-polymatroid that is obtained from $M_X$ by relabelling $p$ by $y$. We argue by induction on $|Y|$.
Suppose $|Y| = 1$ and let $y$ be the element of $Y$. If $y$ is a point, then the result is immediate since $M = M_X(y)$. If $y$ is a line, then compactifying this line gives $M_X(y)$ and again the result holds.
Now suppose that $|Y| > 1$ and choose $y$ in $Y$. Suppose $\sqcap(\{y\},X) = 1$, which is certainly true if $|Y| = 1$. Then $M|(X \cup y) = M_X(y)$ if $y$ is a point. If $y$ is a line, then compactifying $y$ in $M|(X \cup y)$ gives $M_X(y)$. In each case, the result holds. We may now assume that $|Y| > 1$ and $\sqcap(\{y\},X) = 0$. Then, by
Lemma~\ref{dennisplus}(iv), $M/y = M_X \oplus_2 (M_Y/y)$ so the result follows by induction.
\end{proof}
\begin{lemma}
\label{p69}
Let $M$ be a $2$-polymatroid that can be written as the $2$-sum $M_X \oplus_2 M_Y$ of $2$-polymatroids $M_X$ and $M_Y$ with ground sets $X \cup p$ and $Y \cup p$, respectively. Let $N$ be a $3$-connected $2$-polymatroid with $|E(N)| \ge 4$ such that $N$ is a c-minor of $M$. If $|E(N) \cap X| \ge |E(N)| - 1$, then $M_X$ has a special $N$-minor that uses $E(N) \cap X$.
\end{lemma}
\begin{proof}
As $N$ is a c-minor of $M$, it follows by Corollary~\ref{complast3} that $N$ can be obtained from $M$ by a sequence of deletions and contractions followed by one compactification at the end. Let $N_1$ be the $2$-polymatroid that is obtained prior to the last compactification.
We know that we can shuffle these deletions and contractions at will. In producing $N_1$ from $M$, let $C_Y$ and $D_Y$ be the sets of elements of $Y$ that are contracted and deleted, respectively.
Suppose $\sqcap(X,C_Y) = 1$. Now $M = P(M_X,M_Y) \backslash p$. Consider $P(M_X,M_Y) /C_Y\backslash D_Y$. This has $p$ as a loop, so
$P(M_X,M_Y) \backslash p /C_Y\backslash D_Y = P(M_X,M_Y)/ p /C_Y\backslash D_Y$. Since $P(M_X,M_Y)/ p= (M_X/p) \oplus (M_Y/p)$, we deduce that $Y = D_Y \cup C_Y$, so $N_1$ is a c-minor of $M_X/p$. Thus $N$ is a c-minor of $(M_X)^{\flat}$ and hence of $M_X$.
We may now assume that $\sqcap(X,C_Y) = 0$. Suppose $Y \cap E(N) = \emptyset$. Then $M\backslash D_Y/C_Y = M\backslash Y = M_X\backslash p$. Hence $N_1$ is a c-minor of $M_X$. As we can perform a compactification whenever we want, we deduce that $N$ is a c-minor of $(M_X)^{\flat}$. It remains to consider the case when $Y \cap E(N)$ consists of a single element, $y$. In $M/C_Y\backslash D_Y$, we must have $\sqcap(X,\{y\}) = 1$, otherwise $\sqcap(X,\{y\}) = 0$ and $\{y\}$ is 1-separating in $N_1$ and hence in $N$; a contradiction. We deduce that, in $M_Y/C_Y\backslash D_Y$, the element $y$ is either a point parallel to the basepoint $p$ or a line through $p$. In the latter case, $(M/C_Y\backslash D_Y)^{\flat}$ is $(M_X(y))^{\flat}$ where $M_X(y)$ is obtained from $M_X$ by relabelling $p$ by $y$. In both cases, $(M_X(y))^{\flat}$ has $N$ as a c-minor so $(M_X)^{\flat}$ and hence $M_X$ has a special $N$-minor.
\end{proof}
\begin{lemma}
\label{useful}
Let $p$ be a point in a $2$-polymatroid $P$ having ground set $E$. If $\sqcap(p,E-p) = 1$, then $P$ has as a minor a $2$-element $2$-connected $2$-polymatroid using $p$.
\end{lemma}
\begin{proof}
We argue by induction on $|E-p|$. the result is certainly true if $|E-p| = 1$. Assume it true for $|E-p| < n$ and let $|E-p| = n$. If $E-p$ contains an element $z$ such that $\sqcap(p,z) = 1$, then the result is immediate. Thus $E- p$ contains an element $z$ such that $\sqcap (p,z) = 0$. Then
$\sqcap_{P/z}(p,E - \{p,z\}) = r(p) + r(E-p) - r(P) = 1.$ Thus, by the induction assumption, $P/z$ and hence $P$ has, as a minor, a $2$-element $2$-connected $2$-polymatroid using $p$.
\end{proof}
\begin{lemma}
\label{claim1}
Let $(X,Y)$ be an exact $2$-separation of a $2$-polymatroid $M$ and let $N$ be a $3$-connected $2$-polymatroid that is a c-minor of $M$. Suppose that $|X - E(N)| \le 1$ and
$y \in Y$.
\begin{itemize}
\item[(i)] If $\sqcap_{M\backslash y}(X,Y-y) = 1$, then $M\backslash y$ has a special $N$-minor.
\item[(ii)] If $\sqcap_{M/ y}(X,Y-y) = 1$, then $M/y$ has a special $N$-minor.
\item[(iii)] If $\sqcap_{M\downarrow y}(X,Y-y) = 1$, then $M\downarrow y$ has a special $N$-minor.
\end{itemize}
\end{lemma}
\begin{proof}
By Lemma~\ref{dennisplus}, $M = M_X \oplus_2 M_Y$ where $M_X$ and $M_Y$have ground sets $X \cup p$ and $Y\cup p$, respectively.
By Lemma~\ref{p69}, $M_X$ has a special $N$-minor using $E(N) \cap X$. Suppose $\sqcap_{M\backslash y}(X,Y-y) = 1$. Then $\sqcap_{M_Y}(\{p\},Y-y) = 1$. Thus, by Lemma~\ref{useful}, $M_Y\backslash y$ has as a minor a $2$-polymatroid with ground set $\{p,z\}$ for some $z$ in $Y-y$ where either $p$ and $z$ are parallel points, or $z$ is a line and $p$ is a point on this line. It follows that $(M \backslash y)^{\flat}$ has as a c-minor the $2$-polymatroid that is obtained from $(M_X)^{\flat}$ by relabelling $p$ by $z$. Hence $M\backslash y$ has a special $N$-minor and (i) holds.
Now suppose that $\sqcap_{M/ y}(X,Y-y) = 1$. Then, by Lemma~\ref{obs1}, $\sqcap(X,\{y\}) = 0$. Thus, by Lemma~\ref{dennisplus}(iv),
$M/y = M_X \oplus_2 (M_Y/y)$. Then, by replacing $M_Y\backslash y$ by $M_Y/ y$ in the argument in the previous paragraph, we deduce that (ii) holds.
Finally, suppose that $\sqcap_{M\downarrow y}(X,Y-y) = 1$. Assume first that $r(\{y\}) \le 1$. Then $M\downarrow y = M/y$, so $\sqcap_{M/ y}(X,Y-y) = 1$, and the result follows by (ii). Now let $y$ be a line of $M$. Then, by Lemma~\ref{dennisplus}(vi), $M\downarrow y = M_X \oplus_2 (M_Y \downarrow y)$. Again, by replacing $M_Y \backslash y$ by $M_Y \downarrow y$ in the argument in the first paragraph, we get that (iii) holds.
\end{proof}
\begin{lemma}
\label{switch}
Let $Q$ be a $2$-polymatroid having $k$ and $\ell$ as distinct elements and suppose that $\ell$ is a $2$-separating line.
Then
$$Q {\underline{\downarrow}\,} \ell \downarrow k = Q \downarrow k {\underline{\downarrow}\,} \ell.$$
\end{lemma}
\begin{proof}
The result is easily checked if $\lambda(\ell) = 0$, so assume that $\lambda(\ell) = 1$. Then, by Lemma~\ref{dennisplus}, $Q = P(Q_1,Q_2)\backslash p$ for some $2$-polymatroids $Q_1$ and $Q_2$ with ground sets $(E(Q) - \ell) \cup p$ and $\{\ell, p\}$ where $Q_2$ consists of the line $\ell$ with the point freely placed on it. Moreover, either
\begin{itemize}
\item[(i)] $k$ is a point that is parallel to $p$ in $Q_1$; or
\item[(ii)] $Q \downarrow k = P(Q_1 \downarrow k,Q_2)\backslash p$.
\end{itemize}
Consider the first case. Then $Q\downarrow k = Q/k$ and $Q\downarrow k {\underline{\downarrow}\,} \ell$ can be obtained from $Q_1/p$ by adjoining $\ell$ as a loop. On the other hand, $Q {\underline{\downarrow}\,} \ell$ can be obtained from $Q_1$ by relabelling $p$ as $\ell$. Thus $Q {\underline{\downarrow}\,} \ell \downarrow k$, which equals $Q {\underline{\downarrow}\,} \ell/k$, can be obtained from $Q_1/p$ by adjoining $\ell$ as a loop. Hence the result holds in case (i).
Now suppose (ii) holds. Then $Q \downarrow k {\underline{\downarrow}\,} \ell$ can be obtained from $Q_1 \downarrow k$ by relabelling $p$ as $\ell$.
On the other hand, $Q{\underline{\downarrow}\,} \ell$ can be obtained from $Q_1$ by relabelling $p$ as $\ell$. Hence $Q{\underline{\downarrow}\,} \ell \downarrow k$ can be obtained from $Q_1\downarrow k$ by relabelling $p$ as $\ell$. Thus the lemma holds.
\end{proof}
We end this section with three lemmas concerning $2$-element prickly $3$-separators.
\begin{lemma}
\label{portia}
Let $\{j,k\}$ be a prickly $3$-separator in a $3$-connected $2$-polymatroid $M$. Then $M\downarrow j$ and $M\downarrow k$ are $3$-connected.
\end{lemma}
\begin{proof} It suffices to show that $M\downarrow j$ is $3$-connected. We form $M\downarrow j$ by freely adding a point $j'$ to $j$, deleting $j$, and contracting $j'$. As $M$ is $3$-connected, so is the $2$-polymatroid $M'$ we get by adding $j'$. Now $M\downarrow j = M'\backslash j/j'$. Assume this $2$-polymatroid is not $3$-connected, letting $(U,V)$ be an $m$-separation of it for some $m$ in $\{1,2\}$. Then
$$r_{M'/j'}(U) + r_{M'/j'}(V) = r(M'/j') + m-1.$$
Thus
$$r_{M'}(U\cup j') + r_{M'}(V\cup j') = r(M') + m.$$
Without loss of generality, $k \in V$. Then $r_{M'}(V\cup j') = r_M(V \cup j)$ and $r_{M'}(U\cup j') = r_M(U)+ 1$. Therefore
$$r_{M}(U) + r_{M}(V\cup j) = r(M) + m- 1.$$
As $M$ is $3$-connected, we deduce that $m = 2$. Then $\max\{|U|, r_{M'/j'}(U)\} \ge 2$. Hence $(U, V \cup j)$ is a $2$-separation of $M$; a contradiction.
\end{proof}
\begin{lemma}
\label{pricklytime}
The set $\{j,k\}$ is a prickly $3$-separator of the $2$-polymatroid $M$ if and only if it is a prickly $3$-separator in $M^*$.
\end{lemma}
\begin{proof} Suppose $\{j,k\}$ is a prickly $3$-separator of $M$. By Lemma~\ref{compact0}, $\lambda_{M^*}(\{j,k\}) = \lambda_{M}(\{j,k\}) = 2$. Moreover, it is straightforward to check that
$r_{M^*}(\{j\}) = 2 = r_{M^*}(\{k\})$, that $r_{M^*}(\{j,k\}) = 3$, and that $\sqcap_{M^*}(\{j\},E-\{j,k\}) = 1 = \sqcap_{M^*}(k,E-\{j,k\})$. Hence $\{j,k\}$ is a prickly $3$-separator of $M^*$. Conversely, suppose that $\{j,k\}$ is a prickly $3$-separator of $M^*$. Then, by what we have just shown, $\{j,k\}$ is a prickly $3$-separator of $(M^*)^*$, that is, of $M^{\flat}$. Now $2 = \lambda_{M^{\flat}}(\{j,k\}) = \lambda_{M}(\{j,k\})$. Moreover, since $r_{M^{\flat}}(\{j\}) = 2$, it follows that $\lambda(\{j\}) = 2$, so $r(\{j\}) = 2$ and $r(E-j) = r(E)$. Similarly, $\lambda(\{k\}) = 2 = r(\{k\})$ and $r(E-k) = r(E)$. It follows, since $r_{M^{\flat}}(\{j,k\}) = 3$, that $r(\{j,k\}) = 3$. By using the fact that $\sqcap_{M^{\flat}}(\{j\},E-\{j,k\}) = 1 = \sqcap_{M^{\flat}}(\{k\},E-\{j,k\})$, it is not difficult to check that
$\sqcap(\{j\},E-\{j,k\}) = 1 = \sqcap(\{k\},E-\{j,k\})$. We conclude that $\{j,k\}$ is a prickly $3$-separator of $M$, so the lemma holds.
\end{proof}
\begin{lemma}
\label{ess3}
Let $\{j,k\}$ be a prickly $3$-separator in a $2$-polymatroid $P$. Then
\begin{itemize}
\item[(i)] $P\downarrow k \backslash j = P\backslash k,j$; and
\item[(ii)] $P\downarrow k / j = P/ k,j$.
\end{itemize}
\end{lemma}
\begin{proof} Suppose $X \subseteq E(P) - \{j,k\}$. Then
$r_{P\downarrow k}(X) = r_P(X)$ as $r(X \cup k) > r(X)$. Thus (i) holds.
To see (ii), observe that $r_{P\downarrow k /j}(X) = r_{P\downarrow k}(X \cup j) - r(\{j\})$ since $r(\{j,k\}) > r(\{j\})$. Now, as $\sqcap(\{j\},\{k\}) = 1$, we deduce that
$r(X \cup j) \le r(X \cup j \cup k) \le r(X \cup j) +1$. Thus
\begin{equation*}
r_{P\downarrow k}(X \cup j) =
\begin{cases}
r(X \cup j), & \text{if $r(X \cup j \cup k) = r(X \cup j) + 1$;}\\
r(X \cup j) - 1, & \text{if $r(X \cup j \cup k) = r(X \cup j)$.}
\end{cases}
\end{equation*}
Hence
$r_{P\downarrow k}(X \cup j) = r(X \cup j \cup k) - 1.$
Thus
$r_{P\downarrow k /j}(X) = r(X \cup j \cup k) - 3 = r_{P/k,j}(X)$, so (ii) holds.
\end{proof}
\section{The strategy of the proof}
\label{strat}
The proof of Theorem~\ref{mainone} is long and will occupy the rest of the paper. In this section, we outline the steps in the proof.
We shall assume that the theorem fails for $M$. Hence $|E(M)| \ge |E(N)| + 2$. As $|E(N)| \ge 4$, we deduce that $|E(M)| \ge 6$.
We know that $M$ has $N$ as an s-minor. This means, of course, that $N$ can be obtained from $M$ by a sequence of contractions, deletions accompanied by compactifications, and series compressions. Our first goal will be to prove the following.
\begin{lemma}
\label{endtime}
The $2$-polymatroid $M$ has an s-minor that is isomorphic to $N$ such that, in the production of this s-minor, all of the series compressions are done last in the process.
\end{lemma}
Next we focus on the c-minor $N_0$ of $M$ that is obtained in the above process after all of the contractions and compactified deletions are done but before doing any of the series compressions. By Lemma~\ref{elemprop25}, $N_0$ is $3$-connected. In view of this, we see that, to prove Theorem~\ref{mainone}, it suffices to prove Theorem~\ref{modc0}, which we restate here for the reader's convenience.
\begin{theorem}
\label{modc}
Let $M$ and $N$ be distinct $3$-connected $2$-polymatroids such that $N$ is a c-minor of $M$ and $|E(N)| \ge 4$. Then
\begin{itemize}
\item[(i)] $r(M) \ge 3$ and $M$ is a whirl or the cycle matroid of a wheel; or
\item[(ii)] $M$ has an element $\ell$ such that $M\backslashba \ell$ or $M/\ell$ is $3$-connected having a c-minor isomorphic to $N$; or
\item[(iii)] $M$ has a prickly $3$-separator $\{y,z\}$ such that $M\downarrow y$ is $3$-connected having a c-minor isomorphic to $N$.
\end{itemize}
\end{theorem}
Our focus then becomes proving Theorem~\ref{modc}. For the rest of this section, we assume that the pair $(M,N)$ a counterexample to that theorem. The first two steps in the argument, whose proofs appear in Section~\ref{edlp}, are as follows.
\begin{lemma}
\label{Step0}
$M$ has no point $z$ such that both $M\backslashba z$ and $M/z$ have c-minors isomorphic to $N$.
\end{lemma}
\begin{lemma}
\label{Step1}
$M$ has no element $\ell$ such that $M\backslash \ell$ or $M/ \ell$ is disconnected having a c-minor isomorphic to $N$.
\end{lemma}
Note that the use of $\ell$ above, and in what follows, does not imply that $\ell$ is a line, although most of our attention will be focused on that case.
Now $N$ occurs as a c-minor of $M$. Although we will often work with c-minors of $M$ that are isomorphic to $N$, at a certain point in the argument, we will settle on a particular labelled c-minor of $M$ that is isomorphic to $N$.
When $M$ has $N$ as a c-minor and has a 2-separation $(X,Y)$, either $X$ or $Y$, say $X$, contains at least $|E(N)| - 1$ elements of $N$. We call $X$ the {\it $N$-side} of the 2-separation and $Y$ the {\it non-$N$-side}.
Suppose $M\backslashba \ell$ has $N$ as a c-minor. Because the theorem fails, $M\backslashba \ell$ is not $3$-connected. Now, by Lemma~\ref{compact0}(iii),
$\lambda_{M\backslashba \ell} = \lambda_{M\backslash \ell}$. Thus a partition $(X,Y)$ of $E - \ell$ with $\min\{|X|,|Y|\} \ge 2$ is a 2-separation of $M\backslashba \ell$ if and only if it is a 2-separation of $M\backslash \ell$. It follows that we can label the $N$- and non-$N$-sides of a non-trivial $2$-separation of $M\backslash \ell$ based on their labels in the corresponding $2$-separation of $M\backslashba \ell$. Among all $2$-separations of $M\backslashba \ell$, let the maximum cardinality of the non-$N$-side be $\mu(\ell)$. Similarly, if $M/\ell$ has $N$ as a c-minor, let $\mu^*(\ell)$ be the maximum cardinality of the non-$N$-side of a $2$-separation of $M/\ell$. We observe that $\mu(\ell)$ and $\mu^*(\ell)$ are not defined unless $M\backslashba \ell$ and $M/ \ell$, respectively, have $N$ as a c-minor.
The next step in the argument establishes the following.
\begin{noname}
\label{Step2.2}
$M$ has no element $\ell$ for which $\mu(\ell) = 2$ or $\mu^*(\ell)= 2$.
\end{noname}
The argument for (\ref{Step2.2}) is quite long since it involves a detailed analysis of the various structures that can arise on the non-$N$-side when $\mu(\ell) = 2$. We then use duality to eliminate the cases when $\mu^*(\ell)= 2$. These arguments appear in Section~\ref{alltwos}.
Recall that a special $N$-minor of $M$ is any c-minor of $M$ that is either equal to $N$ or differs from $N$ by having a single point relabelled. The next major step in the argument, which is dealt with in Lemma~\ref{bubbly}, proves the following.
\begin{noname}
\label{Step3}
If $(X,Y)$ is a $2$-separation of $M\backslash \ell$ where $X$ is the $N$-side and $|Y| = \mu(\ell)$, then $Y$ contains an element $y$ such that both $M\backslashba y$ and $M/ y$ have special $N$-minors.
\end{noname}
In Lemma~\ref{nonN}, we use the element found in the last step to prove the following.
\begin{noname}
\label{Step4}
There is a c-minor $N'$ of $M$ that is isomorphic to $N$ such that $M$ has a $3$-separator $(X,Y)$ with $|E(N') \cap Y| \le 1$ such that if $|Y| = 2$, then both elements of $Y$ are lines.
\end{noname}
The particular c-minor $N'$ whose existence is proved in (\ref{Step4}) is the one used throughout the rest of the argument. From that point on in the argument, we use $N$ to denote $N'$. An exactly 3-separating set $Y$ is called a {\it non-$N$-$3$-separator} if $|E(N) \cap Y| \le 1$ and, when $|Y| = 2$, both elements of $Y$ are lines. By (\ref{Step4}), a non-$N$-$3$-separator exists. Hence there is a minimal such set.
At the beginning of Section~\ref{bigtime}, we prove that
\begin{noname}
\label{Step5}
$M$ has a minimal non-$N$-$3$-separator with at least three elements.
\end{noname}
The rest of Section~\ref{bigtime} is devoted to showing the following.
\begin{noname}
\label{Step5.5}
A minimal non-$N$-$3$-separator of $M$ with exactly three elements consists of three lines.
\end{noname}
The purpose of Section~\ref{threeel} is to prove that
\begin{noname}
\label{Step6}
$M$ has a minimal non-$N$-$3$-separator with at least four elements.
\end{noname}
The argument to show (\ref{Step6}) is quite long since it involves treating all non-$N$-$3$-separators that consist of exactly three lines.
We say that an element $\ell$ of $M$ is {\it doubly labelled} if both $M\backslash \ell$ and $M/\ell$ have special $N$-minors. The next step, which is shown in Section~\ref{fourel}, establishes the following.
\begin{noname}
\label{Step7}
If $Y_1$ is a minimal non-$N$-$3$-separator of $M$ with at least four elements, then $Y_1$ contains a doubly labelled element.
\end{noname}
Next we take the doubly labelled element $\ell$ identified in the last step. We then take non-trivial $2$-separations $(D_1,D_2)$ and $(C_1,C_2)$ of $M\backslash \ell$ and $M/ \ell$, respectively, having $D_1$ and $C_1$ as their $N$-sides. We show that these 2-separations can be chosen so that each of $D_2$ and $C_2$ is contained in $Y_1 - \ell$, and neither contains any points of $M$.
We then show that each of $D_1\cap C_2, D_2 \cap C_1$, and $D_2 \cap C_2$ consists of a single line of $M$, that the union of these lines spans $\ell$, and these four lines together make up $Y_1$.
The final contradiction is obtained by showing that $M/\ell_{22}$ is $3$-connected\ having a c-minor isomorphic to $N$, where $\ell_{22}$ is the unique element in $D_2 \cap C_2$.
\section{The reduction to c-minors}
\label{redc}
\setcounter{theorem}{1}
The goal of this section is to prove Lemma~\ref{endtime} and thereby show that Theorem~\ref{mainone} can be proved by verifying Theorem~\ref{modc}.
\begin{proof}[Proof of Lemma~\ref{endtime}.]
Consider the s-minors of $M$ that are isomorphic to $N$ and are obtained using the minimum number of series compressions. Suppose $N_1$ is such an s-minor and let the number of series compressions used in its production be $m$. If $m = 0$, then $N_1$ is an s-minor of $M$ satisfying the requirements of the lemma. Hence we may assume that $m > 0$. Let $n_1$ be the number of elements that are removed after the last series compression has been completed. For $2 \le i \le m$, let $n_i$ be the number of elements that are removed via deletion or contraction between the $(m-i+1)$st and the $(m-i+2)$nd series compressions. Consider the sequence $(n_1,n_2,\dots,n_m)$ and let $N_0$ be a choice for $N_1$ for which the corresponding sequence is lexicographically minimal. If each $n_i$ is zero, then we have found, as desired, an s-minor of $M$ in which all of the series compressions are performed after all of the contractions and compactified deletions. Assume then that $n_i$ is the first non-zero $n_j$. Let $P$ be the $2$-polymatroid that we have immediately prior to the $(m-i+1)$st series compression, with this series compression involving compressing the line $k$ from the prickly $3$-separator $\{j,k\}$ of $P$. Let $Q$ be the 2-polymatroid we have immediately prior to the $(m - i +2)$nd series compression.
By Lemma~\ref{ess3}, we may assume that $j$ is neither deleted or contracted in producing $N_0$
otherwise we can replace the compression of $k$ by a deletion followed by a compactification or by a contraction.
By Lemma~\ref{elemprop}, we may assume that either
\begin{itemize}
\item[(a)] all of the elements removed in producing $Q$ from $P\downarrow k$ are done so by deletion followed by compactification; or
\item[(b)] the next move in the production of $Q$ is the contraction of an element, say $y$.
\end{itemize}
Assume that (b) holds.
By Lemma~\ref{pricklytime0}, $P\downarrow k /y = P/y \downarrow k.$ Assume that $\{j,k\}$ is not a prickly $3$-separator of $P/y$. We now apply Lemma~\ref{elemprop24}. If $r(\{y,j,k\}) = 3$, then $j$ is a loop of $P\downarrow k/y$ so $j$ must be deleted or contracted to produce $N_0$; a contradiction.
If $P\downarrow k/y$ is $P/y\backslashba k$ or $P/y/k$, then we do not need to compress $k$ in the production of $N_0$, so the choice of $N_0$ is contradicted. We
are left with the possibility that
$P\downarrow k/y$ can be obtained from $P/y\backslashba j$ by relabelling $k$ as $j$. Again we obtain the contradiction\ that we can reduce the number of series compressions where, if $j \in E(N_0)$, we replace $N_0$ by the 2-polymatroid in which $j$ is relabelled by $k$.
We conclude that $\{j,k\}$ is a prickly $3$-separator of $P/y$. In that case, interchanging the compression of $k$ and the contraction of $y$ in $P$ produces a $2$-polymatroid in which $n_i$ is reduced and so the choice of $N_0$ is contradicted. We deduce that (b) does not hold, so (a) holds.
In the construction of $N_0$, let $y$ be the first element that is deleted following the compression of $k$. Now, by Lemma~\ref{pricklytime0},
$P\downarrow k \backslashba y = (P\downarrow k \backslash y)^{\flat} = (P\backslash y \downarrow k)^{\flat}.$ As $P$ is compact, $r(E - y) = r(E)$. If $r(E - \{y,j,k\}) = r(E) - 3$, then $P\backslash y$ has $\{j,k\}$ as a $1$-separating set. This is a contradiction as $j$ cannot be deleted or contracted in the production of $N_0$ from $P$. Hence
\begin{equation}
\label{2down}
r(E - \{y,j,k\}) \ge r(E) - 2.
\end{equation}
Let $S$ be the set of $2$-separating lines in $P\backslash y$. Clearly no member of $S - k$ is parallel to $k$.
We show next that
\begin{sublemma}
\label{newjk}
$S \cap \{j,k\} \neq \emptyset.$
\end{sublemma}
Suppose, instead, that neither $j$ nor $k$ is in $S$. Then, by Lemma~\ref{atlast2}, $S$ is the set of $2$-separating lines of $P\backslash y \downarrow k$. Let $S = \{\ell_1,\ell_2,\dots,\ell_t\}$. Then
$$P\backslashba y \downarrow k = P\backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \downarrow k.$$
Thus, by repeated application of Lemma~\ref{switch} and using Lemma~\ref{pricklytime0}, we see that
\begin{align*}
P\backslashba y \downarrow k & = P\backslash y \downarrow k {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \\
& = P\downarrow k \backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \\
& = P\downarrow k \backslashba y.
\end{align*}
To prevent us from being able to reduce $n_i$, we must have that
\begin{sublemma}
\label{newjk2}
$\{j,k\}$ is not a prickly $3$-separator of $P\backslashba y$.
\end{sublemma}
Continuing with the proof of \ref{newjk}, suppose $\lambda_{P\backslash y}(\{j,k\}) = 1$. Then $P\backslash y$ is the $2$-sum with basepoint $p$ of two $2$-polymatroids $P_1$ and $P_2$ having ground sets $(E - \{y,j,k\}) \cup p$ and $\{j,k,p\}$, respectively. Since neither $j$ nor $k$ is 2-separating in $P\backslash y$, it follows that, in the rank-$3$ $2$-polymatroid $P_2$, the point $p$ does not lie on either of the lines $j$ or $k$. By Lemma~\ref{dennisplus}(iv), $P\backslash y \downarrow k = P_1 \oplus (P_2 \downarrow k)$. Now $P_2 \downarrow k$ consists of the line $j$ with the point $p$ lying on it. Hence $j$ is a $2$-separating line of $P\backslash y \downarrow k$, so $S \cup j$ is the set of $2$-separating lines of $P\backslash y \downarrow k$. Since $P_2 \downarrow k{\underline{\downarrow}\,} j = P_2 /k$, we deduce that $P\backslash y \downarrow k{\underline{\downarrow}\,} j = P\backslash y/k$. It follows that $S$ is the set of 2-separating lines of $P\backslash y/k$.
Thus
\begin{align*}
P\downarrow k \backslashba y & = (P\backslash y \downarrow k)^{\flat}\\
& = P\backslash y \downarrow k {\underline{\downarrow}\,} j {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \\
& = (P\backslash y \downarrow k {\underline{\downarrow}\,} j) {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \\
& = P \backslash y /k {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t\\
& = P/k \backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t\\
& = P/k\backslashba y.
\end{align*}
We conclude that, instead of compressing $k$, we can contract it, which contradicts that choice of $N_0$. We conclude that $\lambda_{P\backslash y}(\{j,k\}) = 2$. Moreover, $\sqcap(j, E-\{y,j,k\}) = 1 = \sqcap(k, E-\{y,j,k\})$ since neither $j$ nor $k$ is in $S$. Thus $\{j,k\}$ is a prickly $3$-separator of $P\backslash y$. It follows without difficulty that $\{j,k\}$ is a prickly $3$-separator of $P\backslashba y$; a contradiction to \ref{newjk2}. We conclude that \ref{newjk} holds.
We now know that $j$ or $k$ is in $S$. Suppose next that both $j$ and $k$ are in $S$. Thus $r(E - \{y,j\}) = r(E) - 1 = r(E - \{y,k\})$. By submodularity and (\ref{2down}), we deduce that $r(E - \{y,j,k\}) = r(E) - 2$. Hence $P\backslash y$ is the $2$-sum with basepoint $p$ of two $2$-polymatroids $P_1$ and $P_2$ having ground sets $(E - \{y,j,k\}) \cup p$ and $\{j,k,p\}$, respectively. Moreover, in $P_2$, the point $p$ lies on both $j$ and $k$.
Now $P\backslashba y = P\backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t {\underline{\downarrow}\,} j {\underline{\downarrow}\,} k$. Hence $P\backslashba y$ has $j$ and $k$ as parallel points.
Thus
\begin{align*}
P\downarrow k \backslashba y & = (P\backslash y \downarrow k)^{\flat}\\
& = P\backslash y \downarrow k {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t {\underline{\downarrow}\,} j \\
& = P \backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t {\underline{\downarrow}\,} j \downarrow k ~\text{~~~~by Lemma~\ref{switch};} \\
& = P \backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t {\underline{\downarrow}\,} j {\underline{\downarrow}\,} k \backslash k ~\text{~~~~as $Q\downarrow x = Q {\underline{\downarrow}\,} x \backslash x$ when $r(\{x\}) = 2$;} \\
& = P\backslashba y\backslash k\\
& = P\backslashba y\backslashba k,
\end{align*}
where the last step follows because $P\downarrow k \backslashba y$ is compact and so $P\backslashba y\backslash k$ is compact. Again we have a contradiction since we have managed to remove $k$ via deletion rather than by series compression.
Now assume that $k$ is in $S$ but $j$ is not. Then
\begin{align*}
P\downarrow k \backslashba y & = (P\backslash y \downarrow k)^{\flat}\\
& = P\backslash y \ {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t \downarrow k \\
& = P \backslash y {\underline{\downarrow}\,} \ell_1 {\underline{\downarrow}\,} \ell_2{\underline{\downarrow}\,} \dots{\underline{\downarrow}\,} \ell_t {\underline{\downarrow}\,} k \backslash k\\
& = P\backslashba y \backslash k\\
& = P\backslashba y \backslashba k.
\end{align*}
Once again we have managed to avoid the need to perform a series compression on $k$; a contradiction.
Finally, suppose $j$ is in $S$ but $k$ is not. Then we use the fact that $P\downarrow j$ is $P\downarrow k$ with $j$ relabelled as $k$.
The argument in the last paragraph yields a contradiction\ where, when $j \in E(N_0)$, we replace $N_0$ by the 2-polymatroid in which $j$ is labelled as $k$.
\end{proof}
\section{Eliminating doubly labelled points}
\label{edlp}
In this section, we prove that, when $(M,N)$ is a counterexample to Theorem~\ref{modc}, $M$ has no doubly labelled point and has no element whose deletion or contraction is disconnected having a c-minor isomorphic to $N$.
The following elementary lemmas will be helpful.
\begin{lemma}
\label{helpful}
Let $T$ be a set of three points in a $2$-polymatroid $Q$ and suppose $x \in T$.
\begin{itemize}
\item[(i)] If $T$ is a triangle of $Q$, then $\lambda_{Q/ x}(T-x) \le 1$.
\item[(ii)] If $T$ is a triad of $Q$, then $\lambda_{Q\backslash x}(T-x) \le 1$.
\end{itemize}
\end{lemma}
\begin{lemma}
\label{tryto}
Let $T_1$ and $T_2$ be distinct triads in a $2$-polymatroid $Q$. Then $r(E(Q) - (T_1 \cup T_2)) \le r(Q) - 2.$
\end{lemma}
\begin{proof}
We know that $r(E(Q) - T_i) = r(Q) - 1$ for each $i$. The lemma follows easily by applying the submodularity of the rank function.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{Step0}.]
Suppose $M$ has a point $z$ such that both $M\backslashba z$ and $M/z$ have c-minors isomorphic to $N$. Then neither $M\backslash z$ nor $M/z$ is $3$-connected. We may also assume that $M$ is neither a whirl nor the cycle matroid of a wheel. By \cite[Lemma 4.1]{oswww}, $M$ has points $s$ and $t$ such that $\{z,s,t\}$ is a triangle or a triad of $M$. By replacing $M$ by $M^*$ if necessary, we may assume that $\{z,s,t\}$ is a triangle of $M$. Then $M/z$ has $s$ and $t$ as a pair of parallel points. Thus both $M/z\backslash s$ and $M/z\backslash t$ have c-minors isomorphic to $N$. As the theorem fails, neither $M\backslash s$ nor $M\backslash t$ is $3$-connected. Thus, by \cite[Lemma 4.2(i)]{oswww}, $M$ has a triad that contains $z$ and exactly one of $s$ and $t$. We may assume that
the triad is $\{z,s,u\}$. Then $t,z,s,u$ is a fan in $M$.
Now take a fan $x_1,x_2,\dots,x_k$ in $M$ of maximal length such that both $M\backslash x_2$ and $M/x_2$ have c-minors isomorphic to $N$. Then $k \ge 4$. A straightforward induction argument, whose details we omit, gives the following.
\begin{sublemma}
\label{fant}
For all $i$ in $\{2,3,\dots,k-1\}$, both $M\backslash x_i$ and $M/x_i$ have c-minors isomorphic to $N$.
\end{sublemma}
Now consider $\{x_{k-2},x_{k-1},x_k\}$. Suppose first that it is a triangle. As $M/x_{k-1}$ has a c-minor isomorphic to $N$, so do $M/x_{k-1}\backslash x_k$ and hence $M\backslash x_k$. As $M\backslash x_{k-1}$ also has a c-minor isomorphic to $N$, neither $M\backslash x_k$ nor $M\backslash x_{k-1}$ is $3$-connected. Thus, by \cite[Lemma 4.2]{oswww}, it follows that $M$ has a triad $T^*$ containing $x_k$ and exactly one of $x_{k-2}$ and $x_{k-1}$. Let its third element be $x_{k+1}$. By the choice of $k$, it follows that $x_{k+1} \in \{x_1,x_2,\dots,x_{k-3}\}$. Suppose $k = 4$. Then $x_1 \in T^*$. Then $\{x_1,x_2,x_3,x_4\}$ contains two distinct triads so, by Lemma~\ref{tryto},
$r(E - \{x_1,x_2,x_3,x_4\}) \le r(M) - 2$. Thus $\lambda(\{x_1,x_2,x_3,x_4\}) \le 1$; a contradiction since $|E| \ge 6$. We deduce that $k \ge 5$.
As $M$ cannot have a triangle and a triad that meet in a single element, either
\begin{itemize}
\item[(i)] $x_{k+1} = x_1$ and $\{x_1,x_2,x_3\}$ is a triad; or
\item[(ii)] $T^*$ contains $\{x_k,x_{k-2}\}$, and $x_{k+1} \in \{x_{k-3},x_{k-4}\}$.
\end{itemize}
In the latter case, let $X = \{x_{k-4},x_{k-3},x_{k-2},x_{k-1},x_k\}$. Then, by Lemma~\ref{tryto},
$$r(X) + r(E-X) - r(M) \le 3 + r(M) - 2 - r(M) = 1.$$
Since $M$ is $3$-connected, we obtain a contradiction\ unless $E-X$ is empty or contains a single element, which must be a point. In the exceptional case, $M$ is a $3$-connected matroid having 5 or 6 elements and containing a 5-element subset that contains two triangles and two triads. But there is no $3$-connected\ matroid with these properties. We deduce that (ii) does not hold.
We now know that (i) holds and that $T^*$ contains $\{x_k,x_{k-1}\}$. Then $k$ is even. Let $X = \{x_1,x_2,\dots,x_k\}$. As $M \backslash x_2$ has a c-minor isomorphic to $N$ and has $\{x_1,x_3\}$ as a series pair of points, it follows that $M \backslash x_2/x_1$, and hence, $M/x_1$ has a c-minor isomorphic to $N$. Thus, by \cite[Lemma 4.2]{oswww}, $M$ has a triangle containing $x_1$ and exactly one of $x_2$ and $x_3$. This triangle must also contain $x_k$ or $x_{k-1}$. Hence
$r(X) \le r(\{x_2,x_4,x_6,\dots,x_k\}) \le \tfrac{k}{2}.$ Also $r^*(X) \le r(\{x_1,x_3,x_5,\dots,x_{k-1}\})\le \tfrac{k}{2}.$ Thus, by Lemma~\ref{rr*}, $\lambda(X) = 0$, so $X = E(M)$.
Hence $M$ is a $3$-connected\ matroid in which every element is in both a triangle and a triad, so $M$ is a whirl or the cycle matroid of a wheel; a contradiction.
We still need to consider the case when $\{x_{k-2},x_{k-1},x_k\}$ is a triad of $M$. Then it is a triangle of $M^*$ and the result follows by replacing $M$ by $M^*$ in the argument above.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{Step1}.]
Suppose $M\backslash \ell$ is disconnected having a c-minor isomorphic to $N$. Then $E(M\backslash \ell)$ has a non-empty proper subset $X$
such that $\lambda_{M\backslash \ell}(X) = 0$ and $M\backslash \ell \backslash X$ has $N$ as a c-minor. Then, by Lemma~\ref{Step0}, every element of $X$ must be a line. Let $Y = E(M\backslash \ell) - X$. Since $r(M\backslash \ell) = r(M),$ we deduce that
\begin{equation}
\label{xym}
r(X) + r(Y) = r(M).
\end{equation}
As $r(X) \ge 2$ and $(X, Y \cup \ell)$ is not a $2$-separation\ of $M$, we deduce that $r(Y \cup \ell) = r(Y) + 2$. It follows, since $Y$ and $\ell$ are skew and $M\backslash X$ has $N$ as a c-minor, that $M/ \ell$ has $N$ as a c-minor. Since $(M,N)$ is a counterexample to the theorem, $M/ \ell$ is not $3$-connected. Thus there is a partition $(C_1,C_2)$ of $E(M) - \ell$ such that, for some $k$ in $\{1,2\}$,
\begin{equation}
\label{eqk}
r_{M/ \ell}(C_1) + r_{M/ \ell}(C_2) \le r(M/ \ell) + k-1
\end{equation}
where, if $k = 2$, we may assume that $\min\{|C_1|, r_{M/ \ell}(C_1),|C_2|, r_{M/ \ell}(C_2)\} \ge 2$.
Hence
\begin{equation}
\label{c12}
r(C_1 \cup \ell) + r(C_2\cup \ell) \le r(M) + 3.
\end{equation}
By (\ref{xym}), (\ref{c12}), and submodularity,
$r(X \cup C_1 \cup \ell) + r(X \cap C_1) + r(Y \cup C_2 \cup \ell) + r(Y \cap C_2) \le 2r(M) + 3$.
Then
$$r(X \cup C_1 \cup \ell) + r(Y \cap C_2) \le r(M) + 1 \text{~~ or}$$
$$r(Y \cup C_2 \cup \ell) + r(X \cap C_1) \le r(M) + 1,$$
so
$$r(Y \cap C_2) \le 1 \text{~~or~~} r(X \cap C_1) \le 1.$$
By symmetry,
$$r(Y \cap C_1) \le 1 \text{~~or~~} r(X \cap C_2) \le 1.$$
Since $X$ does not contain any points, either $r(Y \cap C_2) \le 1$ and $r(Y \cap C_1) \le 1$; or, for some $i$ in $\{1,2\}$,
$$X \cap C_i = \emptyset \text{~~ and ~~} r(Y \cap C_i) \le 1.$$
In the former case, $ |Y| \le 2$; a contradiction\ since $Y$ contains $E(N)$. In the latter case, we may assume that $C_1$ consists of a single point $p$. Then we deduce that $k = 1$ in (\ref{eqk}). Thus $p$ is a point of $M$ and $\{p\}$ is a component of $M/ \ell$. Hence both $M\backslash p$ and $M/p$ have $N$ as c-minors; a contradiction\ to Lemma~\ref{Step0}. We conclude that if $M\backslash \ell$ has a c-minor isomorphic to $N$, then $M\backslash \ell$ is $2$-connected.
Now suppose that $M/ \ell$ is disconnected having a c-minor isomorphic to $N$. By Lemma~\ref{compact0},
$$\lambda_{M/ \ell} = \lambda_{(M/ \ell)^*} = \lambda_{(M^*\backslash \ell)^{\flat}} = \lambda_{M^*\backslash \ell}.$$
Thus, by replacing $M$ by $M^*$ in the argument above, we deduce that if $M/ \ell$ has a c-minor isomorphic to $N$, then $M/ \ell$ is $2$-connected.
\end{proof}
\section{If all 2-separations have a side with at most two elements}
\label{alltwos}
The purpose of this section is to treat (\ref{Step2.2}). The argument here is long as it involves analyzing numerous cases. The setup is that $M$ and $N$ are $3$-connected\ $2$-polymatroids such that $|E(N)| \ge 4$. The pair $(M,N)$ is a counterexample to Theorem~\ref{modc} and $M$ has an element $\ell$ such that $M\backslash \ell$ has $N$ as a c-minor. Thus $M\backslashba \ell$ is not $3$-connected. We assume that the non-$N$-side of every non-trivial $2$-separation of $M\backslash \ell$ has exactly two elements. Thus $\mu(\ell) = 2$. Let $(X,Y)$ be a non-trivial 2-separation of $M\backslash \ell$ in which $Y$ is the non-$N$-side. Now $M\backslash \ell$ can be written as the 2-sum, with basepoint $p$, of $2$-polymatroids $M_X$ and $M_Y$ having ground sets $X \cup p$ and $Y \cup p$.
The first lemma identifies the various possibilities for $M_Y$.
\begin{lemma}
\label{old2}
Let $P$ be a $2$-connected $2$-polymatroid with three elements and rank at least two. Suppose $P$ has a distinguished point $p$. Then $P$ is one of the nine $2$-polymatroids, $P_1, P_2,\dots,P_9$, depicted in Figure~\ref{9lives}.
\end{lemma}
\begin{figure}
\caption{The nine possible $3$-element $2$-polymatroids in Lemma~\ref{old2}
\label{9lives}
\end{figure}
\begin{proof} As $P$ is $2$-connected having rank at least $2$, we see that $2 \le r(P) \le 4$. If $r(P) = 2$, then $P$ is one of $P_1,P_2,P_3,$ or $P_4$; if $r(P) = 3$, then $P$ is one of $P_5,P_6,P_7,$ or $P_8$; if $r(P) = 4$, then $P$ is $P_9$.
\end{proof}
We shall systematically eliminate the various possibilities for $M_Y$. In each case, we will label the two elements of $M_Y$ other than $p$ by $a$ and $b$.
\begin{lemma}
\label{not23}
$M_Y$ is not isomorphic to $P_2$ or $P_3$.
\end{lemma}
\begin{proof}
Assume the contrary. Then $M_Y$ and hence $M$ has a point $q$ on a line $y$ where $q\neq p$. Thus $M\backslash q$ is $3$-connected. As $M\backslashba \ell$ has $N$ as a c-minor, it follows that $M\backslashba \ell\backslash q$, and hence $M\backslash q$, has a c-minor isomorphic to $N$; a contradiction.
\end{proof}
\begin{lemma}
\label{fourmost}
$M_Y$ is not isomorphic to $P_4$.
\end{lemma}
\begin{proof}
Assume the contrary. Let the two parallel lines in $M_Y$ be $y$ and $y'$ where we may assume that $y \not\in E(N)$. Now $M\backslash y$ is $3$-connected, so $M\backslash y$ does not have $N$ as a c-minor. Thus $M/y$ has $N$ as a c-minor. But $y'$ is a loop of $M/y$, so $y' \not\in E(N)$ and $M\backslash y'$ has $N$ as a c-minor. Since $M\backslash y'$ is $3$-connected, we have a contradiction.
\end{proof}
The next lemma is designed to facilitate the elimination of the cases when $M_Y$ is one of $P_1$, $P_7$, or $P_9$.
\begin{lemma}
\label{179}
Suppose both $a$ and $b$ are skew to $p$ in $M_Y$, and both $M_Y/a$ and $M_Y/b$ are $2$-connected. Then $M/a$ and $M/b$ have $2$-separations $(X_a,Y_a)$ and $(X_b,Y_b)$ such that $\ell \in Y_a \cap Y_b$. Moreover, both $M/a$ and $M/b$ have special $N$-minors, and
\begin{itemize}
\item[(i)] $b \in X_a$ and $a \in X_b$;
\item[(ii)] both $Y_a$ and $Y_b$ properly contain $\{\ell\}$;
\item[(iii)] $(X_a,Y_a - \ell)$ and $(X_b,Y_b-\ell)$ are $2$-separating partitions of $M/a\backslash \ell$ and $M/b\backslash \ell$, respectively, and $\ell \in {\rm cl}_{M/a}(Y_a - \ell)$ and $\ell \in {\rm cl}_{M/b}(Y_b - \ell)$;
\item[(iv)] $(X_a \cup a,Y_a - \ell)$ and $(X_b \cup b,Y_b-\ell)$ are $2$-separating partitions of $M\backslash \ell$;
\item[(v)] for $c$ in $\{a,b\}$, provided $a$ or $b$ is a point, $(X_c,Y_c - \ell)$ is a $2$-separation of $M/c\backslash \ell$ and $(X_c \cup c,Y_c - \ell)$ is a $2$-separation of $M\backslash \ell$;
\item[(vi)] either $(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$; or each of $X_b \cap (Y_a - \ell)$ and $X_a \cap (Y_b - \ell)$ consists of a single point, both $a$ and $b$ are lines of $M$, and, when $r(\{a,b\}) = 4$, the element $\ell$ is a point of $M$.
\end{itemize}
\end{lemma}
\begin{proof} Since both $M_Y/a$ and $M_Y/b$ are $2$-connected, it follows by Lemma~\ref{claim1} that both $M\backslash \ell/a$ and $M\backslash \ell/b$ have special $N$-minors. Hence so do both $M/a$ and $M/b$. Since the theorem fails, $M/a$ and $M/b$ have $2$-separations $(X_a,Y_a)$ and $(X_b,Y_b)$ such that $\ell \in Y_a \cap Y_b$.
To see that (i) holds, it suffices to show that $b \in X_a$. Assume $b\in Y_a$. Then
$$r_{M/a}(X_a) + r_{M/a}(Y_a) = r(M/a) + 1,$$ so
$r_{M}(X_a \cup a) - r_M(\{a\})+ r_{M}(Y_a \cup a) = r(M) + 1$. As $a$ is skew to $p$ in $M_Y$, it follows that $a$ is skew to $X$ in $M$. Since $X_a \subseteq X$, it follows that $(X_a,Y_a \cup a)$ is a $2$-separation\ of $M$; a contradiction. Hence (i) holds.
Part (ii) is an immediate consequence of Lemma~\ref{hath}. To prove (iii), first observe that, by Proposition~\ref{connconn}, $M/a\backslash \ell$ is $2$-connected. We show next that
\begin{equation}
\label{aell}
r(M/a \backslash \ell) = r(M/a).
\end{equation}
Suppose not. Then $r(M/a \backslash \ell) \le r(M/a) - 1.$ Since $M/a$ is $2$-connected, it follows that equality must hold here and $\ell$ is a line of $M/a$. This gives a contradiction\ to Lemma~\ref{hath}. Hence (\ref{aell}) holds.
Now
\begin{align*}
r(M/a\backslash \ell) + 1 & \le r_{M/a\backslash \ell}(X_a) + r_{M/a\backslash \ell}(Y_a - \ell)\\
& \le r_{M/a}(X_a) + r_{M/a}(Y_a)\\
& = r(M/a) +1\\
& = r(M/a \backslash \ell) +1,
\end{align*}
where the last equality follows from (\ref{aell}). We see that equality must hold throughout the last chain of inequalities. Hence
$(X_a, Y_a - \ell)$ is a $2$-separating partition of $M/a\backslash \ell$, and $\ell \in {\rm cl}_{M/a}(Y_a - \ell)$. Using symmetry, we deduce that (iii) holds.
Since $b\in X_a$, we see that $Y_a - \ell \subseteq X$, so $a$ is skew to $Y_a - \ell$. It follows by (iii) that $(X_a \cup a,Y_a - \ell)$ is a 2-separating partition of $M\backslash \ell$, and (iv) follows by symmetry.
To show (v), observe that, since $Y_c - \ell$ avoids $\{a,b\}$, it follows that $c$ is skew to $Y_c - \ell$. Thus it suffices to show that $(X_c,Y_c - \ell)$ is a $2$-separation of $M/c\backslash \ell$. Assume it is not. Then $Y_c - \ell$ consists of a single point $e$ of $M/c\backslash \ell$. Then $e$ is a point of $M$ and, by (iii), $r_{M/c}(\{e,\ell\}) = r_{M/c}(\{e\}) = 1$, so
\begin{equation}
\label{cel}
r_M(\{c,e,\ell\})= r_M(\{c,e\}) = 1 + r(\{c\}).
\end{equation}
Suppose $c$ is a point. If $\ell$ is a line, then $c$ and $e$ are on $\ell$, so $M\backslash e$ is $3$-connected. Since $M/c$ has $e$ and $\ell$ as parallel points, $M\backslash e$ is $3$-connected\ having a c-minor isomorphic to $N$; a contradiction. Thus we may assume that $\ell$ is a point. Then $\{e,\ell,c\}$ is a triangle in $M$. Thus, for $\{c,d\} = \{a,b\}$, we see that $(X \cup \ell \cup c,\{d\})$ is a $2$-separation\ of $M$ unless $d$ is a point of $M$. In the exceptional case, $M$ has $e,c,\ell,d$ as a fan with $M/c$ having a c-minor isomorphic to $N$. Thus, by Lemmas~\ref{fantan} and \ref{Step0}, we have a contradiction.
We may now assume that $c$ is a line. Then $r(\{c,e\}) = 3$, so, by (\ref{cel}), $r(\{c,\ell\}) = 3$. Thus $(X,\{c,\ell\})$ is a $2$-separation\ of $M\backslash d$ where $\{c,d\} = \{a,b\}$. Moreover, by hypothesis, $d$ is a point. Thus, by Lemma~\ref{newbix}, we obtain the contradiction\ that $M/d$ is $3$-connected\ unless $M/d$ has a parallel pair $\{z_1,z_2\}$ of points. In the exceptional case, we deduce that $z_1$, say, is $\ell$. Hence $(X \cup \ell \cup d,\{c\})$ is a $2$-separation\ of $M$; a contradiction. We conclude that (v) holds.
To prove (vi), assume that $(Y_a - \ell) \cap (Y_b - \ell) = \emptyset$. Then $Y_b - \ell \subseteq X_a \cup a$. But $\ell \in {\rm cl}((Y_b - \ell) \cup b)$ and $b \in X_a$, so $\ell \in {\rm cl}(X_a \cup a)$. Because $M$ is $3$-connected, it follows that $Y_a - \ell$ consists of a single point $a'$. By symmetry, $Y_b - \ell$ consists of a single point $b'$. Then $(X_a \cup a, Y_a - \ell)$ is not a $2$-separation\ of $M\backslash \ell$, so, by (v), each of $a$ and $b$ is a line of $M$.
To finish the proof of (vi), it remains to show that, when $r(\{a,b\}) = 4$, the element $\ell$ is a point of $M$. Assume $\ell$ is a line. Then, in $M/a$, we have $a'$ and $\ell$ as parallel points, so $M/a \backslash a'$, and hence $M\backslash a'$, has a c-minor isomorphic to $N$. As $\ell \in {\rm cl}_{M/a}(\{a'\})$, it follows that $\ell \in {\rm cl}_M(X \cup a)$. By symmetry, $\ell \in {\rm cl}_M(X \cup b)$. Thus
\begin{align*}
r(X) + 2 + r(X) + 2 & = r(X \cup a) + r(X \cup b)\\
& = r(X \cup a \cup \ell) + r(X \cup b \cup \ell)\\
& \ge r(X \cup \ell) + r(M)\\
& = r(X \cup \ell) + r(X) + 3.
\end{align*}
Thus
$$r(X \cup \ell) \le r(X) + 1.$$
Then
\begin{align*}
3 + r(X) + 1 & \ge r(\{a',a,\ell\}) + r(X \cup \ell)\\
& \ge r(\{a',\ell\}) + r(X \cup \{a',a,\ell\})\\
& = r(\{a',\ell\}) + r(X \cup a)\\
& = r(\{a',\ell\}) + r(X) + 2.
\end{align*}
We deduce that $r(\{a',\ell\}) = 2$, so $a'$ is a point on the line $\ell$. Thus $M\backslash a'$ is $3$-connected\ having a c-minor isomorphic to $N$; a contradiction.
\end{proof}
Next we eliminate the possibility that $M_Y$ is $P_1$.
\begin{lemma}
\label{noone}
$M_Y$ is not isomorphic to $P_1$.
\end{lemma}
\begin{proof}
Assume $M_Y$ is isomorphic to $P_1$. Since $\{a,b\}$ is a series pair in $M\backslash \ell$, it follows that both $M/a$ and $M/b$ have c-minors isomorphic to $N$. Hence neither $M/a$ nor $M/b$ is $3$-connected.
We show next that
\begin{sublemma}
\label{nonesub0}
$\ell$ is a line of $M$.
\end{sublemma}
Assume $\ell$ is a point. Then $\{\ell,a,b\}$ is a triad of $M$. Since neither $M/a$ nor $M/b$ is $3$-connected, it follows by \cite[Lemma 4.2]{oswww} that $M$ has a triangle containing $a$ and exactly one of $b$ and $\ell$. If $M$ has $\{a,b,c\}$ as a triangle, then $M/a$ has $\{b,c\}$ as a parallel pair of points. Thus $M/a\backslash b$, and hence $M\backslash b$, has a c-minor isomorphic to $N$. Thus $b$ is a doubly labelled point; a contradiction\ to Lemma~\ref{Step0}. We deduce that $M$ has $\{a,\ell\}$ in a triangle with a point $d$, say. Then $M$ has $d,a,\ell,b$ as a fan with $M/a$ having a c-minor isomorphic to $N$. Thus, by Lemmas~\ref{fantan} and \ref{Step0}, we have a contradiction. We conclude that \ref{nonesub0} holds.
By Lemma~\ref{179}, $M/a$ and $M/b$ have $2$-separations $(X_a,Y_a)$ and $(X_b,Y_b)$ such that $\ell \in Y_a \cap Y_b$. Moreover, both $M/a$ and $M/b$ have special $N$-minors, and
\begin{itemize}
\item[(i)] $b \in X_a$ and $a \in X_b$;
\item[(ii)] both $Y_a$ and $Y_b$ properly contain $\{\ell\}$;
\item[(iii)] $(X_a,Y_a - \ell)$ and $(X_b,Y_b-\ell)$ are $2$-separating partitions of $M/a\backslash \ell$ and $M/b\backslash \ell$, respectively, and $\ell \in {\rm cl}_{M/a}(Y_a - \ell)$ and $\ell \in {\rm cl}_{M/b}(Y_b - \ell)$;
\item[(iv)] $(X_a \cup a,Y_a - \ell)$ and $(X_b \cup b,Y_b-\ell)$ are $2$-separating partitions of $M\backslash \ell$; and
\item[(v)] $(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$.
\end{itemize}
\begin{sublemma}
\label{noonesub3}
$(Y_a - \ell)\cup (Y_b - \ell) = E - \{a,b,\ell\}$.
\end{sublemma}
We know that $\lambda_{M\backslash \ell}(Y_a - \ell) = 1 = \lambda_{M\backslash \ell}(Y_b - \ell)$ and $(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$, so
$\lambda_{M\backslash \ell}((Y_a - \ell)\cap (Y_b - \ell)) \geq 1$. Thus, by applying the submodularity of the connectivity function, we see that
\begin{align*}
1 + 1 & = \lambda_{M\backslash \ell}(Y_a - \ell) + \lambda_{M\backslash \ell}(Y_b - \ell)\\
& \ge \lambda_{M\backslash \ell}((Y_a - \ell)\cap (Y_b - \ell)) + \lambda_{M\backslash \ell}((Y_a - \ell)\cup (Y_b - \ell))\\
& \ge 1 + \lambda_{M\backslash \ell}((Y_a - \ell)\cup (Y_b - \ell)).
\end{align*}
Since $M\backslash \ell$ is $2$-connected, we deduce that $\lambda_{M\backslash \ell}((Y_a - \ell)\cup (Y_b - \ell)) = 1$.
This application of the submodularity of the connectivity function is an example of an `uncrossing' argument. For the rest of the paper, we will omit the details of such arguments and will follow our stated practice of using the abbreviation `by uncrossing' to mean `by applying the submodularity of the connectivity function.'
Now $(X_a \cup a, Y_a)$ is not a 2-separation of $M$, so, as $\ell \in {\rm cl}_{M/a}(Y_a- \ell)$, we see that
$$r(Y_a - \ell) < r(Y_a) \le r(Y_a \cup a) = r((Y_a - \ell) \cup a) \le r(Y_a - \ell) + 1.$$
Hence
$$r(Y_a) = r(Y_a \cup a) = r((Y_a - \ell) \cup a) = r(Y_a - \ell) + 1.$$
Thus $r((Y_a - \ell) \cup (Y_b - \ell) \cup \{a,b\}) = r(Y_a \cup Y_b \cup \{a,b\}) = r(Y_a \cup Y_b) \le r((Y_a - \ell)\cup (Y_b - \ell)) + 1.$
Also, as $\{a,b\}$ is a series pair of points in $M\backslash \ell$, we see that
$r(X_a \cap X_b) \le r((X_a \cap X_b) \cup \{a,b\}) - 1$. Therefore,
$\lambda_{M}(Y_a \cup Y_b \cup \{a,b\}) \leq 1$. Thus, we may assume that
$X_a \cap X_b$ consists of a single matroid point, $z$, otherwise $(Y_a - \ell)\cup (Y_b - \ell) = E- \{a,b,\ell\}$ as desired.
Now $\lambda_{M\backslash \ell}(X_a \cap X_b) = \lambda_{M\backslash \ell}(\{a,b,z\}) = 1$. If $a \notin {\rm cl}(\{b,z\})$, then
$\lambda_{M\backslash \ell}((Y_a - \ell) \cup (Y_b - \ell) \cup a) \le 1$, so
$\lambda_{M}((Y_a-\ell) \cup (Y_b - \ell) \cup a \cup \ell) = \lambda_{M}(Y_a \cup Y_b \cup a) \le 1$; a contradiction. Thus $a \in {\rm cl}(\{b,z\})$. Hence $\{a,b,z\}$ is a triangle of $M$. It follows that the point $b$ is doubly labelled; a contradiction\ to Lemma~\ref{Step0}. We conclude that \ref{noonesub3} holds.
\begin{sublemma}
\label{noonesub4}
$Y_a - Y_b \neq \emptyset$ and $Y_b - Y_a \neq \emptyset$.
\end{sublemma}
By symmetry, it suffices to prove the first of these. Assume $Y_a - Y_b = \emptyset$. Then, as $(Y_a - \ell) \cup (Y_b - \ell) = E - \{a,b,\ell\}$, we deduce that $X_b = \{a\}$, so $(X_b,Y_b - \ell)$ is not a $2$-separation\ of $M\backslash \ell/b$; a contradiction. Thus \ref{noonesub4} holds.
By \ref{noonesub4} and the fact that $(X_a \cup a) \cap (X_b\cup b)$ contains $\{a,b\}$, we see that each of $X_a\cup a$ and $X_b\cup b$ has at least three elements. It follows by the definition of $\mu(\ell)$ that each of $Y_a- \ell$ and $Y_b- \ell$ has exactly two elements. Since each of $Y_a - Y_b, (Y_a \cap Y_b) - \ell,$ and $Y_b - Y_a$ is non-empty, each of these sets has exactly one element. As the union of these sets is $E - \{\ell,a,b\}$, we deduce that $|E(M)| = 6$ and $|X_a \cup a| = 3$. Since at least one of $a$ and $b$ is not in $E(N)$, we deduce that each of $X_a\cup a$ and
$Y_a - \ell$ contains at most two elements of $N$; a contradiction\ as one of these sets must contain at least three elements of $E(N)$. We conclude that Lemma~\ref{noone} holds.
\end{proof}
\begin{lemma}
\label{no7}
$M_Y$ is not isomorphic to $P_7$.
\end{lemma}
\begin{proof}
Assume that $M_Y$ is isomorphic to $P_7$, letting $a$ be the line. Then, by Lemma~\ref{179}, $M/a$ and $M/b$ have $2$-separations $(X_a,Y_a)$ and $(X_b,Y_b)$ such that $\ell \in Y_a \cap Y_b$. Moreover, both $M/a$ and $M/b$ have special $N$-minors, and
\begin{itemize}
\item[(i)] $b \in X_a$ and $a \in X_b$;
\item[(ii)] both $Y_a$ and $Y_b$ properly contain $\{\ell\}$;
\item[(iii)] $(X_a,Y_a - \ell)$ and $(X_b,Y_b-\ell)$ are $2$-separating partitions of $M/a\backslash \ell$ and $M/b\backslash \ell$, respectively, and $\ell \in {\rm cl}_{M/a}(Y_a - \ell)$ and $\ell \in {\rm cl}_{M/b}(Y_b - \ell)$;
\item[(iv)] $(X_a \cup a,Y_a - \ell)$ and $(X_b \cup b,Y_b-\ell)$ are $2$-separating partitions of $M\backslash \ell$; and
\item[(v)] $(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$.
\end{itemize}
We show next that
\begin{sublemma}
\label{*5}
$X_a \cap X_b$ is empty or consists of a single point.
\end{sublemma}
Suppose $b \not\in {\rm cl}(Y_b)$. Then $r(Y_b \cup b) = r(Y_b) + 1$. Thus $(X_b \cup b, Y_b)$ is a $2$-separation\ of $M$; a contradiction.
Hence $r(Y_b \cup b) = r(Y_b)$. Now
\begin{align*}
r((Y_a - \ell) \cup (Y_b - \ell)) + 2 & \ge r((Y_a - \ell) \cup (Y_b - \ell) \cup a)\\
& = r(Y_a \cup (Y_b - \ell) \cup a)\\
& = r(Y_a \cup Y_b \cup a)\\
& = r(Y_a \cup Y_b \cup a \cup b).
\end{align*}
Also $r(X_a \cap X_b) \le r((X_a \cup a) \cap (X_b \cup b)) - 2$ since $X_a \cap X_b \subseteq X$ and $\sqcap_M(X,Y) = 1$ while $r_M(\{a,b\}) = 3$.
Thus
\begin{align*}
\lambda_M(X_a \cap X_b) & = r(Y_a \cup Y_b \cup a \cup b) + r(X_a \cap X_b) - r(M)\\
& \le r((Y_a - \ell) \cup (Y_b - \ell)) + 2 + r((X_a \cup a) \cap (X_b \cup b)) - 2 - r(M\backslash \ell)\\
& = \lambda_{M\backslash \ell}((X_a\cup a) \cap (X_b\cup b))\\
& = 1,
\end{align*}
where the second-last step follows by uncrossing $(X_a \cup a, Y_a - \ell)$ and $(X_b \cup b, Y_b - \ell)$.
We deduce that \ref{*5} holds.
\begin{sublemma}
\label{*6}
$E(M) - \{\ell,a,b\}$ contains no point $\gamma$ such that $\{a,b,\gamma\}$ is $2$-separating in $M\backslash \ell$.
\end{sublemma}
To see this, suppose that such a point $\gamma$ exists. Recall that $M\backslash \ell$ has $N$ as a c-minor so at most one element of $\{a,b\}$ is in $E(N)$.
Thus at most two elements of $\{a,b,\gamma\}$ are in $E(N)$. But $|E(N)| \ge 4$. Hence $\{a,b,\gamma\}$ is the non-$N$-side of a $2$-separation of $M\backslash \ell$ contradicting the fact that $\mu(\ell) = 2$. We conclude that \ref{*6} holds.
An immediate consequence of \ref{*6} is that $X_a \cap X_b$ does not consist of a single point. Hence, by \ref{*5}, $X_a\cap X_b = \emptyset$.
As $(X_a,Y_a)$ is a $2$-separation\ of $M/a$, it follows that $X_a$ cannot contain just the element $b$. Thus
$(X_a \cup a) \cap (Y_b - \ell) \neq \emptyset$. We show next that
\begin{sublemma}
\label{*7}
$(X_b \cup b) \cap (Y_a - \ell) \neq \emptyset$.
\end{sublemma}
Suppose $(X_b \cup b) \cap (Y_a - \ell) = \emptyset$. Then $Y_b - \ell = E(M) - \{a,b,\ell\} = X$ so $r(Y_b - \ell) = r(M) - 2$. Hence
$r((Y_b - \ell) \cup b) \le r(M) - 1$. But $\ell \in {\rm cl}_{M/b}(Y_b - \ell)$. Thus $r(Y_b \cup b) \le r(M) - 1$, so $\{a\}$ is 2-separating in $M$; a contradiction. We deduce that \ref{*7} holds.
By uncrossing, $\lambda_{M\backslash \ell}((X_b\cup b) \cap (Y_a - \ell)) = 1 = \lambda_{M\backslash \ell}((X_a\cup a) \cap (Y_b - \ell))$. As $\ell$ is in both
${\rm cl}((Y_a - \ell) \cup a)$ and ${\rm cl}((Y_b - \ell) \cup b)$, we deduce that each of $(X_a \cup a) \cap (Y_b - \ell)$ and $(X_b \cup b) \cap (Y_a - \ell)$ consists of a single point. Thus we get a contradiction\ to \ref{*6} that completes the proof of Lemma~\ref{no7}.
\end{proof}
On combining Lemmas~\ref{not23}, \ref{noone}, and \ref{no7}, we immediately obtain the following.
\begin{corollary}
\label{pointless}
The non-$N$-side of every $2$-separation of $M\backslash \ell$ does not contain any points.
\end{corollary}
\begin{lemma}
\label{no9}
$M_Y$ is not isomorphic to $P_9$.
\end{lemma}
\begin{proof}
Assume $M_Y$ is isomorphic to $P_9$. Since each of $M_Y\backslash \ell/a$ and $M_Y\backslash \ell/b$ consists of a line through $p$, it follows that both $M/a$ and $M/b$ have c-minors isomorphic to $N$. Hence neither $M/a$ nor $M/b$ is $3$-connected.
Then $M/a$ and $M/b$ have $2$-separations $(X_a,Y_a)$ and $(X_b,Y_b)$ such that $\ell \in Y_a \cap Y_b$. Moreover, by Lemma~\ref{179}, \begin{itemize}
\item[(i)] $b \in X_a$ and $a \in X_b$;
\item[(ii)] both $Y_a$ and $Y_b$ properly contain $\{\ell\}$;
\item[(iii)] $(X_a,Y_a - \ell)$ and $(X_b,Y_b-\ell)$ are $2$-separating partitions of $M/a\backslash \ell$ and $M/b\backslash \ell$, respectively, and $\ell \in {\rm cl}_{M/a}(Y_a - \ell)$ and $\ell \in {\rm cl}_{M/b}(Y_b - \ell)$;
\item[(iv)] $(X_a \cup a,Y_a - \ell)$ and $(X_b \cup b,Y_b-\ell)$ are $2$-separating partitions of $M\backslash \ell$; and
\item[(v)] either $(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$; or each of $X_b \cap (Y_a - \ell)$ and $X_a \cap (Y_b - \ell)$ consists of a single point, both $a$ and $b$ are lines of $M$, and $\ell$ is a point of $M$.
\end{itemize}
\begin{sublemma}
\label{no9.3}
$(Y_a - \ell) \cap (Y_b - \ell) \neq \emptyset$.
\end{sublemma}
Assume the contrary. Then, by (v), $X_b \cap (Y_a - \ell)$ consists of a point, $a'$, say. By (iii), $\ell \in {\rm cl}_{M/a}(\{a'\})$, so $\ell \in {\rm cl}(\{a',a\})$. As $r(M) - 3 = r(X)$, it follows that $r(X \cup a \cup \ell) \le r(M) - 1$. Hence the line $\{b\}$ is 2-separating in $M$; a contradiction. Thus \ref{no9.3} holds.
\begin{sublemma}
\label{no9.3.5}
$|(Y_a - \ell) \cup (Y_b - \ell)| \geq 2$.
\end{sublemma}
Assume $(Y_a - \ell) \cup (Y_b - \ell)$ contains a unique element, $z$. Then, by \ref{no9.3}, $z \in (Y_a - \ell) \cap (Y_b - \ell)$. Now $\ell \in {\rm cl}_{M/a}(\{z\})$, so $\ell \in {\rm cl}_M(\{z,a\})$. Thus
$$r(X \cup a \cup \ell) = r(X \cup a) = r(X) + 2 = r(M) -1,$$
so$(X \cup a \cup \ell, \{b\})$ is a $2$-separation\ of $M$; a contradiction. Thus \ref{no9.3.5} holds.
By \ref{no9.3} and uncrossing, we see that
$\lambda_{M\backslash \ell}((X_a \cup a) \cap (X_b \cup b)) = 1$. Next we show the following.
\begin{sublemma}
\label{no9.4}
$(Y_a - \ell) \cup (Y_b - \ell)$ is the non-$N$-side of a $2$-separation of $M\backslash \ell$ and it is a $2$-element set, both members of which are lines.
\end{sublemma}
By \ref{no9.3.5}, $((X_a \cup a) \cap (X_b \cup b),(Y_a - \ell) \cup (Y_b - \ell))$ is a $2$-separation\ of $M\backslash \ell$.
Suppose $(X_a \cup a) \cap (X_b \cup b)$ is the non-$N$-side of this 2-separation. Then, as $\mu(\ell) = 2$, we deduce that $(X_a \cup a) \cap (X_b \cup b) = \{a,b\}$.
Thus, as $\ell \in {\rm cl}((Y_a - \ell) \cup a)$,
\begin{align*}
r(M) + 1 & = r((Y_a - \ell) \cup (Y_b - \ell)) + r(\{a,b\})\\
& = r((Y_a - \ell) \cup (Y_b - \ell) \cup a) + r(\{b\})\\
& = r(Y_a \cup Y_b \cup a) + r(\{b\}).
\end{align*}
Hence $\{b\}$ is 2-separating in $M$; a contradiction.
Thus $(Y_a - \ell) \cup (Y_b - \ell)$ must be the non-$N$-side of a 2-separation of $M\backslash \ell$, so this set has cardinality two. Moreover,
by Corollary~\ref{pointless}, both elements of this set are lines. Thus \ref{no9.4} holds.
We deduce from \ref{no9.4} that $Y_a - \ell$ and $Y_b - \ell$ are the non-$N$-sides of 2-separations of $M\backslash \ell$. Thus, by symmetry, we may assume that
$Y_b - \ell \subseteq Y_a - \ell$. Hence
\begin{equation}
\label{unc}
(Y_a - \ell) \cup (Y_b - \ell) = Y_a - \ell.
\end{equation}
\begin{sublemma}
\label{no9.5}
$(Y_a \cup \{a,b\}, X_a \cap X_b)$ is a $2$-separation of $M$.
\end{sublemma}
Since $Y_a - \ell \supseteq Y_b - \ell$, we have
$$r(Y_a \cup a) = r((Y_a - \ell) \cup a) = r(Y_a - \ell) + 2$$
and
$$r(Y_a \cup b) = r((Y_a - \ell) \cup b) = r(Y_a - \ell) + 2.$$
Moreover,
$$r(Y_a \cup \{a,b\}) = r((Y_a- \ell) \cup \{a,b\}) \ge r(Y_a - \ell) + 3.$$
Thus, by submodularity,
\begin{align*}
r(Y_a - \ell) + 2 + r(Y_a - \ell) + 2 & = r(Y_a \cup a) + r(Y_a \cup b)\\
& \ge r(Y_a \cup a \cup b) + r(Y_a)\\
& \ge r(Y_a - \ell) + 3 + r(Y_a)\\
& \ge r(Y_a - \ell) + 3 + r(Y_a - \ell) + 1,
\end{align*}
where the last step follows because $\ell \notin {\rm cl}(Y_a - \ell)$.
We see that equality must hold throughout the last chain of inequalities.
Hence $r(Y_a) = r(Y_a - \ell) + 1$ and $r(Y_a \cup \{a,b\}) = r(Y_a - \ell) + 3 = r(Y_a) + 2$.
As $\lambda_{M\backslash \ell}(Y_a - \ell) = 1$, it follows that $\lambda_{M}(Y_a) = 2$, that is,
$$r(Y_a) + r((X_a \cup a) \cap (X_b \cup b)) - r(M) = 2.$$
Hence
\begin{align*}
r(Y_a \cup \{a,b\}) + r(X_a \cap X_b) - r(M) & \le r(Y_a) + 2 + r((X_a \cup a) \cap (X_b \cup b))\\
& \hspace*{2in} - 3 - r(M)\\
& = 1.
\end{align*}
Thus $(Y_a \cup \{a,b\}, X_a \cap X_b)$ is a 2-separating partition of $M$. Since $(X_a \cup a) \cap (X_b \cup b)$ is the $N$-side of a 2-separation of $M\backslash \ell$, it follows that $X_a \cap X_b$ contains at least two elements of $E(N)$ as $\{a,b\}$ contains at most one element of $E(N)$. Thus
$(Y_a \cup \{a,b\}, X_a \cap X_b)$ is a 2-separation of $M$, that is, \ref{no9.5} holds.
But \ref{no9.5} gives a contradiction\ and thereby completes the proof of Lemma~\ref{no9}.
\end{proof}
We now know that there are only three possibilities for $M_Y$, namely $P_5$, $P_6$, or $P_8$. The next few lemmas will be useful in treating all three cases.
\begin{lemma}
\label{cactus}
Assume $M\backslash \ell$ has $(X,\{a,b\})$ as a $2$-separation where $r(\{a,b\}) = 3$ and each of $a$ and $b$ is a line. Then $r(X \cup \ell) = r(X) + 1$ if and only if $\{a,b\}$ is a prickly $3$-separating set in $M$.
\end{lemma}
\begin{proof}
If $\{a,b\}$ is a $3$-separating set in $M$, then $r(X \cup \ell) = r(M) - 1$. But $r(X) = r(M) - 2$, so $r(X \cup \ell) = r(X) + 1$. Conversely, if $r(X \cup \ell) = r(X) + 1$, then $r(X \cup \ell) = r(M) - 1$, so $\{a,b\}$ is a 3-separating set in $M$. Now $r(X \cup \ell \cup a) = r(M)$ otherwise $\{b\}$ is 2-separating in $M$. By symmetry, $r(X \cup \ell \cup b) = r(M)$. Hence $\{a,b\}$ is a prickly $3$-separating set in $M$.
\end{proof}
\begin{lemma}
\label{cactus2}
Assume $M$ has $\{a,b\}$ as a prickly $3$-separating set that is $2$-separating in $M\backslash \ell$. Then $M\downarrow a$ and $M\downarrow b$ are $3$-connected having c-minors isomorphic to $N$.
\end{lemma}
\begin{proof} By Lemma~\ref{portia}, $M\downarrow a$ and $M\downarrow b$ are $3$-connected. Since $M_X$ and $M_Y$ have ground sets $X \cup p$ and $\{a,b,p\}$, we see that $r(M_Y) = 3$. By Lemma~\ref{pricklytime0}, $M\downarrow a \backslash \ell = M\backslash \ell \downarrow a$. But $M\backslash \ell \downarrow a$ equals the 2-sum of $M_X$ and the $2$-polymatroid consisting of a line $b$ through the point $p$. Compactifying $b$ in $M\backslash \ell \downarrow a$ gives the $2$-polymatroid that is obtained from $M_X$ by relabelling $p$ by $b$. Hence $M\downarrow a \backslashba \ell$ has a c-minor isomorphic to $N$. Thus, using symmetry, so do $M\downarrow a$ and $M\downarrow b$.
\end{proof}
\begin{lemma}
\label{pixl}
If $M_Y$ is $P_5$, $P_6$, or $P_8$, then $r(X \cup \ell) = r(X) +2$, so $\ell$ is a line.
\end{lemma}
\begin{proof} Assume $r(X \cup \ell) = r(X) +1$. Then, by Lemma~\ref{cactus}, $\{a,b\}$ is a prickly 3-separating set in $M$. Then, by Lemma~\ref{cactus2}, $M\downarrow a$ and $M\downarrow b$ are $3$-connected having c-minors isomorphic to $N$; a contradiction\ to the fact that $(M,N)$ is a counterexample to Theorem~\ref{modc}. Thus $r(X \cup \ell) \neq r(X) +1$. Since $\ell \not\in {\rm cl}(X)$, we deduce that $r(X \cup \ell) = r(X) +2$, so $\ell$ is a line.
\end{proof}
Next we deal with the case when $M\backslash \ell$ has $(X,Y)$ as its only $2$-separation\ with $|Y| = 2$, beginning with the possibility that $M_Y= P_6$.
\begin{lemma}
\label{duh}
Suppose $M_Y = P_6$ and $(X,Y)$ is the only non-trivial $2$-separation of $M\backslash \ell$. Then
\begin{itemize}
\item[(i)] $M\backslashba a$ or $M\backslashba b$ is $3$-connected having a special $N$-minor; or
\item[(ii)] each of $\{a,\ell\}$ and $\{b,\ell\}$ is a prickly $3$-separator of $M$, and each of $M\downarrow a$ and $M\downarrow b$ is $3$-connected having a c-minor isomorphic to $N$.
\end{itemize}
\end{lemma}
\begin{proof} By Lemma~\ref{pixl}, $\ell$ is a line of $M$ and $\sqcap(X,\ell) = 0$. In $M\backslashba \ell$, we see that $a$ and $b$ are parallel points. Hence each of $M\backslash a$ or $M\backslash b$ has a special $N$-minor. But
$r(E - \{a,b,\ell\}) = r(M) - 2$ and $r(E - \{a,\ell\}) = r(M) - 1$, so $\{\ell\}$ is 2-separating in $M\backslash a$. Now both $M\backslashba a$ or $M\backslashba b$ have special $N$-minors. Hence we may assume that neither of these matroids is $3$-connected.
Next we show that
\begin{sublemma}
\label{duh2}
$r(\{a,\ell\}) = 3 = r(\{b,\ell\})$.
\end{sublemma}
We shall show that $r(\{b,\ell\}) = 3$, which, by symmetry, will suffice. As $M\backslashba a$ is not $3$-connected, $M\backslash a$ has a non-trivial 2-separation $(A,B)$ in which $A$ contains $\ell$. Then $(A - \ell,B)$ is a 2-separating partition of $M\backslash a \backslash \ell$. Observe that $r(M\backslash a \backslash \ell) = r(M) - 1$. Suppose $b \in B$. Then $r(B \cup a) = r(B) + 1$. Thus $(A - \ell, B\cup a)$ is a 2-separating partition of $M \backslash \ell$. Since $B\cup a \neq \{a,b\}$, we deduce that $A - \ell$ contains a unique element. Moreover, as $\sqcap(X,\ell) = 0$, it follows that $r(A) = r(A - \ell) + 2$. Thus $(A- \ell, B \cup a)$ is a 1-separating partition of $M\backslash a$; a contradiction\ to Lemma~\ref{Step1}.
We may now assume that $b \in A - \ell$. Then $((A - \ell) \cup a, B)$ is a non-trivial $2$-separation\ of $M\backslash \ell$. Thus $(A - \ell) \cup a = \{a,b\}$, so $A = \{b,\ell\}$. Hence $B = X$ and $r(\{b,\ell\}) = 3$. Thus \ref{duh2} holds.
As $r(X \cup a) = r(M) - 1$, we deduce that $\{b, \ell\}$ is a prickly $3$-separator of $M$. Now $M\backslash \ell \downarrow b$, which, by Lemma~\ref{pricklytime0}, equals $M \downarrow b \backslash \ell$, has a c-minor isomorphic to $N$. Hence so does $M\downarrow b$ and, by symmetry, $M\downarrow a$. Thus, by Lemma~\ref{portia},
part~(ii) of the lemma holds.
\end{proof}
\begin{lemma}
\label{duh85}
Suppose $M_Y$ is $P_5$ or $P_8$. Let $a$ be an element of $Y$ for which $\sqcap(\{a\},\{p\}) = 0$. Then
\begin{itemize}
\item[(i)] $M/a$ has a $2$-separation; and
\item[(ii)] for every $2$-separation $(A,B)$ of $M/a$ with $\ell$ in $A$,
\begin{itemize}
\item[(a)] $b \in B$;
\item[(b)] $(A - \ell,B \cup a)$ is a $2$-separation of $M\backslash \ell$ and $|B-b| \ge 2$;
\item[(c)] $|A - \ell| \le 2$ and if $|A - \ell| = 1$, then $A - \ell$ consists of a line of $M/a$;
\item[(d)] $r_{M/a}(A - \ell) = r_{M/a}(A)$; and
\item[(e)] $\sqcap(\{a,b\}, A - \ell) = 0$.
\end{itemize}
\end{itemize}
Moreover, if $(X,Y)$ is the unique non-trivial $2$-separation of $M\backslash \ell$, then $M/a$ has a unique $2$-separation $(A,B)$ with $\ell$ in $A$. Further, $A - \ell$ consists of a line of $M/a$.
\end{lemma}
\begin{proof}
Certainly $M\backslash \ell/a$ and hence $M/a$ has a c-minor isomorphic to $N$. By Lemma~\ref{pixl}, $\ell$ is a line and $\sqcap(X,\ell) = 0$. As the theorem fails, $M/a$ is not $3$-connected, but, by Lemma~\ref{Step1}, it is $2$-connected. Let $(A,B)$ be a $2$-separation\ of $M/a$ with $\ell$ in $A$.
\begin{sublemma}
\label{bB}
$b \in B$.
\end{sublemma}
Suppose $b \in A$. Then $a$ is skew to $B$ in $M$, so $(A\cup a,B)$ is a $2$-separation\ of $M$; a contradiction. Thus \ref{bB} holds.
\begin{sublemma}
\label{mab}
$M$ does not have a point $c$ such that $B = \{b,c\}$.
\end{sublemma}
Assume the contrary. We have $r_{M/a}(A)+r_{M/a}(B)-r(M/a)=1$, that is, $r(A\cup a)-2+r(\{a, b, c\})-r(M)=1$. But $r(A-\ell)\le r(A\cup a)-2$ and $A - \ell = X - c$. Hence $r(X-c)+r(\{a, b, c\})-r(M)\le 1$. Since $r(M)=r(M\backslash \ell)$, this implies that $(X-c, \{a, b, c\})$ is a $2$-separation of $M\backslash \ell$ that violates the fact that $\mu(\ell) = 2$.
If such a point $c$ exists, then $A \cup a \supseteq X \cup a$, so $r(A \cup a) = r(M)$. Hence $r(\{a,b,c\}) = 3 = r(\{a,b\})$, so $(X - c, \{a,b,c\})$ is a $2$-separation\ of $M\backslash \ell$ that violates the choice of $Y$. Thus \ref{mab} holds.
Next we show that
\begin{sublemma}
\label{aminusl}
$(A - \ell, B)$ is a $2$-separation of $M\backslash \ell/a$.
\end{sublemma}
Certainly $(A - \ell, B)$ is $2$-separating in $M\backslash \ell/a$. We need to show that $\max\{|A - \ell|, r(A - \ell)\} \ge 2$. By Lemma~\ref{skewer}, $A \neq \{\ell\}$. Assume $A = \{\ell,c\}$ where $c$ is a point of $M/a$. Then $c$ is a point in $M$ as $a$ is skew to $X$. Moreover,
\begin{equation}
\label{seec}
c \in {\rm cl}_M(X - c)
\end{equation}
otherwise $(X - c, \{a,b,c\})$ is a $2$-separation\ of $M\backslash \ell$ that violates the choice of $Y$.
By Lemma~\ref{skewer}, $a$ is not skew to $\{c,\ell\}$, so $r_{M/a}(\{c, \ell\}) < r_{M}(\{c, \ell\}) \le 3$. Suppose $r_{M/a}(\{c,\ell\}) = 2$. Then
$r_M(B \cup a) = r(M) - 1$, so $(\{c\}, B \cup a)$ is a 1-separation of $M\backslash \ell$; a contradiction. We conclude that
\begin{sublemma}
\label{aminusl.1}
$r_{M/a}(\{c,\ell\}) = 1$, so $r_M(\{a,c,\ell\}) = 3$ and $r(M\backslash \ell/a) = r(M/a)$.
\end{sublemma}
Since $c$ and $\ell$ are parallel points in $M/a$, we deduce that $M\backslash c$ has a c-minor isomorphic to $N$. Thus $M\backslash c$ has a $2$-separation\ $(U,V)$ where we may assume that $\ell \in U$ and $a \in V$ otherwise $M$ has a $2$-separation.
Continuing with the proof of \ref{aminusl}, next we show that
\begin{sublemma}
\label{bisinP}
$b \in U$.
\end{sublemma}
Suppose $b \in V$. Then, as $a \in V$, we see that $r(V \cup \ell) \le r(V) + 1$ and $r(U - \ell) = r(U) - 2$. Thus $U = \{\ell\}$ otherwise $(U - \ell, V \cup \ell)$ is a 1-separation of $M\backslash c$. But, by (\ref{seec}), $c \in {\rm cl}(E - c - \ell)$. Hence $(U,V \cup c)$ is a $2$-separation\ of $M$; a contradiction. Hence \ref{bisinP} holds.
\begin{sublemma}
\label{pnotlb}
$U \neq \{\ell,b\}$.
\end{sublemma}
Assume $U = \{\ell,b\}$. Then $V = (X - c) \cup a$. Thus $r(V) \ge r(X) + 1 = r(M) - 1$. But $r(U) \ge 3$ so $(U,V)$ is not a $2$-separation\ of $M\backslash c$. Thus contradiction\ completes the proof of \ref{pnotlb}.
\begin{sublemma}
\label{pnotlbd}
$M$ does not have a point $d$ such that $U = \{\ell,b,d\}$.
\end{sublemma}
Assume the contrary. Then
\begin{equation}
\label{veeq}
r(V) = r((X - \{c,d\}) \cup a) \ge r(M) - 2
\end{equation}
so, as $r(U) + r(V) = r(M) +1$, we must have that
\begin{equation}
\label{peep}
r(\{\ell,b,d\}) = r(U) \le 3.
\end{equation}
Thus equality must hold in each of (\ref{veeq}) and (\ref{peep}).
As $r(\{a,c,\ell\}) = 3$, we have
\begin{align*}
r(\{b,d\}) + r((X - d) \cup \{a,\ell\}) & = r(\{b,d\}) + r((X - \{c,d\}) \cup \{a,\ell\})\\
& \le r(\{\ell,b,d\}) + r((X - \{c,d\}) \cup a) + 1\\
& = r(M) + 2.
\end{align*}
Now $r(\{b,d\}) = 3$, otherwise $\{a,b,d\}$ contradicts the choice of $Y$ since at most one of $a$ and $b$ is in $E(N)$.
Hence $(\{b,d\}, (X - d) \cup \{a,\ell\})$ is a 3-separation of $M$.
Thus
$r((X - d) \cup \{a,\ell\}) = r(M) - 1$, so $r((X - d) \cup a) \le r(M) - 1$. Hence $r(X - d) \le r(M) - 3$, while $r(X) = r(M) - 2$. Thus $(X - d, \{a,b,d\})$ is a $2$-separation\ of $M\backslash \ell$ contradicting the choice of $Y$. We conclude that \ref{pnotlbd} holds.
Now recall that $\{\ell,b\} \subseteq U$ and $a \in V$. Moreover, $r(\{a,c,\ell\}) = 3$ and $\sqcap(a,b) = 1$. Thus
$$r(V \cup \{\ell,b\}) \le r(V) + 2.$$
Also $\ell \not\in {\rm cl}(X \cup b)$ otherwise $\{a\}$ is 2-separating in $M$; a contradiction. Thus
$$r(U - \{\ell,b\}) \le r(U) - 2.$$
It follows by \ref{pnotlb} and \ref{pnotlbd} that $(U - \{\ell,b\}, V \cup \{\ell,b\})$ is a $2$-separation\ of $M\backslash c$, so $(U - \{\ell,b\}, V \cup \{\ell,b\}\cup c)$ is a $2$-separation\ of $M$. This contradiction\ completes the proof of \ref{aminusl}.
We deduce from \ref{aminusl} that (ii)(d) of the lemma holds, that is,
\begin{sublemma}
\label{rankla}
$r((A - \ell) \cup a) = r(A \cup a).$
\end{sublemma}
Moreover, since $a$ is skew to $X$, and $A - \ell \subseteq X$, it follows, by Lemma~\ref{skewer}, that
\begin{sublemma}
\label{aminuslba}
$(A - \ell, B \cup a)$ is a $2$-separation of $M\backslash \ell$.
\end{sublemma}
Now $(A,B)$ is a $2$-separation\ of $M/a$ and $b \in B$. Since $b$ is a point of $M/a$, it follows that $|B| \ge 2$, so $|B \cup a| \ge 3$. Hence $B \cup a$ is the $N$-side of the $2$-separation\ $(A - \ell, B \cup a)$ of $M\backslash \ell$. At most one member of $\{a,b\}$ is in $E(N)$. Since $|E(N)| \ge 4$, it follows that at least two elements of $N$ are in $B- b$, so $|B-b| \ge 2$. Thus (ii)(b) of the lemma holds.
Moreover, $|A - \ell| \le 2$. Since $A - \ell$ is one side of a $2$-separation, if it contains a single element, that element is a line of $M/a$. Thus (ii)(c) of the lemma holds.
Next we observe that
\begin{sublemma}
\label{piab}
$\sqcap(\{a,b\}, A - \ell) = 0.$
\end{sublemma}
Since $\sqcap(\{a,b\}, X) = 1$, we see that $\sqcap(\{a,b\}, A - \ell) \le 1$. Assume $\sqcap(\{a,b\}, A - \ell) = 1$. Then
$r((A - \ell) \cup \{a,b\}) = r(A - \ell) + 2$. But $r(A - \ell) + r(B \cup a) = r(M\backslash \ell) + 1$. Thus
\begin{equation}
\label{eqbb}
r((A - \ell) \cup \{a,b\}) + r(B - b) \le r(M\backslash \ell) + 1.
\end{equation}
By \ref{rankla}, $r((A - \ell) \cup a) = r(A \cup a)$. Hence we obtain the contradiction\ that $(A \cup \{a,b\}, B- b)$ is a $2$-separation\ of $M$. Thus \ref{piab} holds.
Now suppose that $(X,Y)$ is the unique non-trivial $2$-separation of $M\backslash \ell$. We complete the proof of the lemma by showing that
\begin{sublemma}
\label{lonely}
$M/a$ has a unique $2$-separation $(A,B)$ with $\ell$ in $A$. Moreover, $A - \ell$ consists of a line of $M/a$.
\end{sublemma}
Let $(A_1, B_1)$ and $(A_2,B_2)$ be distinct 2-separations of $M/a$ with $\ell$ in $A_1 \cap A_2$. Then $b \in B_1 \cap B_2$. By (ii)(c), $|A_i - \ell| \le 2$. Suppose $|A_i - \ell| = 2$. Then, by (ii)(b), $(A_i - \ell, B_i \cup a)$ is a non-trivial $2$-separation\ of $M\backslash \ell$, so $A_i - \ell = Y$; a contradiction\ as $a \not\in A_i - \ell$. We deduce that $|A_i - \ell| = 1$, so $A_i - \ell$ consists of a line $m_i$ of $M/a$.
Now $(\{m_1\}, B_1 \cup a)$ and $(\{m_2\}, B_2 \cup a)$ are 2-separations of $M\backslash \ell$. Thus $r(\{m_1,m_2\}) = 4$ otherwise one easily checks that
$(\{m_1,m_2\}, (B_1\cap B_2) \cup a)$ is a $2$-separation\ of $M\backslash \ell$ that contradicts the uniqueness of $(X,Y)$. Now $\sqcap(a,X) = 0$, so $\sqcap(a,\{m_1,m_2\}) = 0$. Thus $r_{M/a}(\{m_1,m_2\}) = 4$. But, by (ii)(d) of the lemma,
\begin{align*}
2 + 2 & = r_{M/a}(\{m_1,\ell\}) + r_{M/a}(\{m_2,\ell\})\\
& \ge r_{M/a}(\{m_1,m_2,\ell\}) + r_{M/a}(\{\ell\})\\
& \ge 4 + 1.
\end{align*}
This contradiction\ finishes the proof of \ref{lonely} and thereby completes the proof of the lemma.
\end{proof}
\begin{lemma}
\label{duh8}
If $M_Y = P_8$, then $(X,Y)$ is not the only non-trivial $2$-separation of $M\backslash \ell$.
\end{lemma}
\begin{proof}
Assume $(X,Y)$ is the unique such $2$-separation. By Lemma~\ref{duh85}, $M/a$ and $M/b$ have unique $2$-separations $(A_1,B_1)$ and $(A_2,B_2)$ with
$\ell$ in $A_1 \cap A_2$. Moreover, $A_1 - \ell$ and $A_2 - \ell$ consist of lines $\ell_1$ and $\ell_2$ in $M/a$ and $M/b$; and $M\backslash \ell$ has $(A_1 - \ell, B_1 \cup a)$ and $(A_2 - \ell, B_2 \cup b)$ as $2$-separations.
Assume $\ell_1 \neq \ell_2$. Then $\{b,\ell_2\} \subseteq B_1 \cup a$, so $\ell \in {\rm cl}(B_1 \cup a)$. Hence $(A_1 - \ell, B_1 \cup a \cup \ell)$ is a $2$-separation\ of $M$; a contradiction. Thus $\ell_1 = \ell_2$. Hence $r(\{\ell_1,b,\ell\}) = r(\{\ell_1,b\}) = 4$. But we also know that $r(\{\ell_1,a,\ell\}) = r(\{\ell_1,a\}) = 4$. By Lemma~\ref{duh85}(ii)(a) and (b), we see that $b \notin {\rm cl}_{M/a}(A_1)$, so $r(\{\ell_1,\ell,b,a\}) \ge 5$. Thus
\begin{align*}
4+4 & = r(\{\ell_1,\ell, b\}) + r(\{\ell_1,\ell, a\})\\
& \ge r(\{\ell_1,\ell, b,a\}) + r(\{\ell_1,\ell\})\\
& \ge 5 + r(\{\ell_1,\ell\}).
\end{align*}
Therefore $r(\{\ell_1,\ell\}) \le 3$. As $\sqcap(\{\ell_1\},\{\ell\}) = 0$, we deduce that $r(\{\ell\}) = 1$; a contradiction\ to Lemma~\ref{pixl}.
\end{proof}
\begin{lemma}
\label{duh5}
If $M_Y = P_5$, then $(X,Y)$ is not the only non-trivial $2$-separation of $M\backslash \ell$.
\end{lemma}
\begin{proof}
Assume $(X,Y)$ is the unique such $2$-separation. Label $Y$ so that $\sqcap(a,X) = 0$ and $\sqcap(b,X) = 1$.
\begin{sublemma}
\label{duh5a}
$\sqcap(a,\ell) = 0$.
\end{sublemma}
Suppose $\sqcap(a,\ell) = 1$. Then $r(\{a,\ell\}) = 3$. Now $r(E - \ell) = r(E)$ and $r(E - \{\ell,a\}) = r(E) - 1$. Thus, by Lemma~\ref{cactus}, $\{a,\ell\}$ is a prickly 3-separator of $M$. Now $M\backslash \ell \downarrow a$ has a c-minor isomorphic to $N$ since it is the 2-sum of $M_X$ and the $2$-polymatroid consisting of the line $b$ with the point $p$ on it. But, by Lemma~\ref{pricklytime0}, $M\backslash \ell \downarrow a = M\downarrow a \backslash \ell$. Thus, by Lemma~\ref{portia}, $M\downarrow a$ is $3$-connected\ having a c-minor isomorphic to $N$; a contradiction. We conclude that \ref{duh5a} holds.
By Lemma~\ref{duh85}, $M/a$ has a unique $2$-separation\ and it has the form $(\{\ell_1,\ell\}, E - \{\ell_1,\ell,a\})$ where $\ell_1$ is a line of $M/a$. Moreover, $r(\{\ell_1,\ell,a\}) = r(\{\ell_1,\ell\})$.
Now $\sqcap(\ell,a) = 0$ and, by Lemma~\ref{pixl}, $\ell$ is a line of $M$. Thus
\begin{equation}
\label{ral}
r(\{a,\ell\}) = 4.
\end{equation}
Now $M\backslash \ell \backslash a$, and hence $M\backslash a$, has a c-minor isomorphic to $N$. Thus $M\backslash a$ has a non-trivial $2$-separation\ $(U,V)$. Without loss of generality, we may assume that
$\ell_1 \in U$ and $\ell \in V$ since $r(\{\ell_1,\ell\}) = 4 = r(\{\ell_1,\ell,a\})$.
\begin{sublemma}
\label{binV}
$b \in V$.
\end{sublemma}
Suppose $b \in U$. Then, as $\sqcap(X,\ell) = 0$, we see that, unless $V = \{\ell,c\}$ for some point $c$, the partition $(U \cup \ell,V-\ell)$ is a $2$-separation\ of $M\backslash a$, so $(U \cup \ell \cup a,V-\ell)$ is a $2$-separation\ of $M$.
Consider the exceptional case. Then $r(V-\ell) = r(V) - 2 = 1$. Now $r(M\backslash a, \ell) = r(M) - 1$ and $r(U) + r(V) = r(M) + 1$. We see that $r(U) = r(E - \{a,\ell,c\}) = r(M) - 2.$ Hence $\lambda_{M\backslash a,\ell}(\{c\}) = 0$; a contradiction. We conclude that \ref{binV} holds.
We now have that $V \supseteq \{\ell,b\}$. Next observe that
\begin{sublemma}
\label{new2s}
$(U,(V - \ell) \cup a)$ is a $2$-separation of $M\backslash \ell$, and $r((V - \ell) \cup a) = r(V)$.
\end{sublemma}
To see this, first note that, since $b \in V- \ell$, we have
\begin{equation}
\label{vee1}
r((V- \ell) \cup a) \le r(V - \ell) + 1.
\end{equation}
We also have
\begin{equation}
\label{vee2}
r(V - \ell) \le r(V) - 1
\end{equation}
otherwise $r(V - \ell) = r(V)$ so $\ell \in {\rm cl}(E - \{a,\ell\})$. But $r(E - \{a,\ell\}) = r(E) - 1$, so $(E-a,\{a\})$ is a $2$-separation\ of $M$; a contradiction.
Combining (\ref{vee1}) and (\ref{vee2}) gives \ref{new2s}.
Since $(X,Y)$ is the unique non-trivial $2$-separation\ of $M\backslash \ell$, we deduce that $(V - \ell) \cup a = \{a,b\}$. Moreover, by \ref{new2s},
$r(\{a,b\}) = 3 = r(\{b,\ell\})$. It follows using submodularity that $r(\{a,b,\ell\}) = 4$. Thus $b \in {\rm cl}_{M/a}(\{\ell\})$. Hence
$(\{\ell_1,\ell,b\}, E- \{\ell_1,\ell,a,b\})$ is a $2$-separation\ of $M/a$, which contradicts the fact that $(\{\ell_1,\ell\}, E - \{\ell_1,\ell,a\})$ is the unique $2$-separation\ of $M/a$.
This completes the proof of Lemma~\ref{duh5}.
\end{proof}
By Lemma~\ref{cactus2}, $M\backslash \ell$ has no 2-element 2-separating set that is a prickly 3-separating set in $M$.
\begin{lemma}
\label{oldstep5}
Let $\{a,b\}$ and $\{c,d\}$ be disjoint $2$-separating sets of $M\backslash \ell$ where each of $a$, $b$, $c$, and $d$ is a line, $r(\{a,b\}) = 3 = r(\{c,d\})$, and $\sqcap(\{a\},E - \{a,b,\ell\}) = 0$.
Then either
\begin{itemize}
\item[(i)] $M/a$ is $3$-connected having a c-minor isomorphic to $N$; or
\item[(ii)] $M/ \ell$ has a c-minor isomorphic to $N$ and $\ell \in {\rm cl}_{M/a}(\{c,d\})$.
\end{itemize}
\end{lemma}
\begin{proof} Assume that the lemma fails. Let $Z = E - \{\ell,a,b,c,d\}$. Then, as neither $\{a,b\}$ nor $\{c,d\}$ is a prickly 3-separating set of $M$, by Lemma~\ref{cactus}, we see that
$$\sqcap(Z \cup \{c,d\}, \{\ell\}) = 0 = \sqcap(Z \cup \{a,b\}, \{\ell\}),$$
so $\sqcap(Z,\{\ell\}) = 0$ and $\sqcap(\{a,b\},\{\ell\}) = 0$. It follows, as $\sqcap(\{a\}, Z) = 0$, that
\begin{equation}
\label{mazl}
\sqcap_{M/a}(Z,\{\ell\}) = 0.
\end{equation}
Let $X = E - \{a,b,\ell\}$ and $Y = \{a,b\}$. Then $M\backslash \ell = M_X \oplus_2 M_Y$ where $M_Y$ has ground set $\{p,a,b\}$. Then $M_X$ has a c-minor isomorphic to $N$. As $\sqcap(\{a\}, X) = 0$, it follows that $M\backslash \ell/a$, and hence $M/a$, has a c-minor isomorphic to $N$.
\begin{sublemma}
\label{lcd}
$\ell \in {\rm cl}_{M/a}(\{c,d\})$.
\end{sublemma}
Assume $\ell \not\in {\rm cl}_{M/a}(\{c,d\})$. Since $M/a$ is not $3$-connected, it has a $2$-separation\ $(A,B)$ with $\ell \in A$ and $b \in B$. Moreover, by Lemma~\ref{duh85}, we know that $(A-\ell, B\cup a)$ is a 2-separation of $M\backslash \ell$, that $|B - b| \ge 2$, that $|A - \ell| \le 2$, and that $\ell \in {\rm cl}_{M/a}(A - \ell)$.
Suppose $|A - \ell| = 1$. Then, by Lemma~\ref{duh85} again, $A - \ell$ consists of a line $m$ of $M/a$ and $\ell \in {\rm cl}_{M/a}(\{m\})$. Thus $m \not \in \{c,d\}$, so $m \in Z$ and we have a contradiction to (\ref{mazl}). Now suppose that $|A - \ell| = 2$. Then $\ell \in {\rm cl}_{M/a}(A - \ell)$. Thus $\{c,d\} \neq A - \ell$. If $\{c,d\}$ avoids $A - \ell$, then we again get a contradiction\ to (\ref{mazl}). Thus $A- \ell$ meets $\{c,d\}$ in a single element. Then, by uncrossing the 2-separations $(A - \ell, B \cup a)$ and $(\{c,d\}, E - \{\ell,c,d\})$ of $M\backslash \ell$, we see that $(A - \ell) \cup \{c,d\})$ is a 3-element 2-separating set in $M\backslash \ell$. At most one element of $\{c,d\}$ is in $E(N)$. Thus $(A - \ell) \cup \{c,d\}$ is the non-$N$-side of a $2$-separation\ of $M\backslash \ell$. This is a contradiction\ as this set has three elements. We conclude that \ref{lcd} holds.
We shall complete the proof of Lemma~\ref{oldstep5} by showing that $M/ \ell$ has a c-minor isomorphic to $N$. In the argument that follows, it helps to think in terms of the matroids that are naturally derived from the $2$-polymatroids we are considering. We know that $M\backslash \ell = M_X \oplus_2 M_Y$ where $M_Y$ has ground set $\{a,b,p\}$ with $p$ being the basepoint of the 2-sum. As $\{c,d\}$ is 2-separating in $M\backslash \ell$, it is also 2-separating in $M_X$. Thus $M_X = M_Z \oplus_2 M_W$ where $M_W$ has ground set $\{c,d,q\}$ with $q$ being the basepoint of this 2-sum. Now $\{c,d\}$ does not span $p$ otherwise $\{a,b,c,d\}$ is 2-separating in $M\backslash \ell$ and contains at most two elements of $N$, a contradiction\ to the definition of $Y$. By two applications of Lemma~\ref{claim1}, we see that $M_X$, and hence $M_Z$, has a c-minor isomorphic to $N$.
Now $M\backslash \ell/a$ equals $M_X$ after relabelling the element $p$ of the latter by $b$. We will call this relabelled $2$-polymatroid $M_X'$. By \ref{lcd}, $M/a$ is obtained from $M'_X$ by adding $\ell$ to the closure of $\{c,d\}$ as a point or a line. Thus $M/a$ is the 2-sum with basepoint $q$ of $M'_Z$ and $M'_W$ where $M'_Z$ is obtained from $M_Z$ by relabelling $p$ as $b$, while $M'_W$ is obtained from $M_W$ by adding $\ell$. By (\ref{mazl}), $\ell$ is skew to $Z$ in $M/a$, so $\ell$ is skew to $q$ in $M'_W$. Now $\ell$ is a not a line of $M'_W$, otherwise at least one of $c$ and $d$ is parallel to the basepoint $q$ in $M'_W$, so $M/a/ \ell$ and hence $M/\ell$ has a c-minor isomorphic to $N$.
Hence $\ell$ is a point of $M'_W$, so $M'_W / \ell$ has rank $2$. It has no point parallel to $q$ otherwise $M/a /\ell$ has a c-minor isomorphic to $N$. Thus $M'_W/ \ell$ can be obtained from one of $P_1, P_2$, or $P_4$ by relabelling the element $p$ by $q$. In the first two cases, we can contract a point from $M'_W/ \ell$ to obtain a $2$-polymatroid consisting of two parallel points, one of which is $q$, so we get the contradiction\ that $M/a / \ell$ has a c-minor isomorphic to $N$. In the third case, deleting one of the lines, say $c$, of $M'_W/ \ell$ leaves $d$ as a line through $q$. Thus $\{d\}$ is $2$-separating in $M/a \backslash \ell \backslash c$. Compactifying $d$, we obtain a $2$-polymatroid having a c-minor isomorphic to $N$. Again we obtain the contradiction\ that $M/a \backslash \ell$ has a c-minor isomorphic to $N$.
\end{proof}
\begin{lemma}
\label{3connel}
Let $\{a,b\}$ and $\{c,d\}$ be disjoint $2$-separating sets of $M\backslash \ell$ where each of $a$, $b$, $c$, and $d$ is a line, $r(\{a,b\}) = 3 = r(\{c,d\})$.
Assume $M/ \ell$ has a c-minor isomorphic to $N$.
Then at least one of $\sqcap(\{a\}, E - \{\ell,a,b\})$ and $\sqcap(\{b\}, E - \{\ell,a,b\})$ is not equal to one.
\end{lemma}
\begin{proof}
As before, let $Z = E - \{a,b,c,d,\ell\}$. Since the theorem fails, it follows by Lemmas~\ref{Step1} and \ref{cactus2} that $M/ \ell$ is $2$-connected and neither $\{a,b\}$ nor $\{c,d\}$ is a prickly 3-separating set of $M$. Moreover, by Lemma~\ref{pixl}, $\ell$ is a line that is skew to each of $Z \cup \{a,b\}$ and $Z \cup \{c,d\}$. Thus, if $(R,B)$ is a $2$-separation\ of $M/ \ell$, then, by
Lemma~\ref{skewer}, $\sqcap(R,\{\ell\}) \ge 1$ and $\sqcap(B,\{\ell\}) \ge 1$.
By Lemma~\ref{general},
$$\sqcap(R,\{\ell\}) + \sqcap(B,\{\ell\}) + \lambda_{M/ \ell}(R) = \lambda_{M\backslash \ell}(R) + \lambda_M(\{\ell\}),$$
so
\begin{equation}
\label{eq1rb}
\sqcap(R,\{\ell\}) + \sqcap(B,\{\ell\}) = \lambda_{M\backslash \ell}(R) + 1.
\end{equation}
As $\sqcap(\{\ell\}, Z \cup \{a,b\}) = 0 = \sqcap(\{\ell\}, Z \cup \{c,d\})$, it follows by Lemma~\ref{8.2.3} that
both $R$ and $B$ meet both $\{a,b\}$ and $\{c,d\}$. Without loss of generality, we may assume that $\{a,c\} \subseteq R$ and $\{b,d\} \subseteq B$.
Now suppose that $\sqcap(\{a\}, E - \{\ell,a,b\}) = 1 = \sqcap(\{b\}, E - \{\ell,a,b\})$.
By Lemma~\ref{oswrules}(i),
\begin{align*}
\sqcap(\{a,c\}, \{b,d\}) + \sqcap(\{a\},\{c\}) + \sqcap(\{b\},\{d\}) & = \sqcap(\{a,b\}, \{c,d\}) + \sqcap(\{a\},\{b\})\\
&\hspace*{1.5in} + \sqcap(\{c\},\{d\}).
\end{align*}
As $\mu(\ell) = 2$, we see that $\sqcap(\{a,b\}, \{c,d\}) = 0$, so $\sqcap(\{a\},\{c\}) = 0 = \sqcap(\{b\},\{d\})$. Thus
$$\sqcap(\{a,c\}, \{b,d\}) = \sqcap(\{a\},\{b\}) + \sqcap(\{c\},\{d\}) = 2.$$
Hence $\sqcap(R,B) \ge 2$, that is, $\lambda_{M\backslash \ell}(R) \ge 2$. Thus, by (\ref{eq1rb}), $\sqcap(R,\{\ell\}) = 2$ or $\sqcap(B,\{\ell\}) = 2$. By symmetry, we may assume the former. But, as $\sqcap(\{c,d\} \cup Z, \{\ell\}) = 0$ and $\sqcap(\{c,d\} \cup Z, \{a\}) = 1$, by Lemma~\ref{oswrules}(ii),
\begin{align*}
\sqcap(\{c,d\} \cup Z \cup a, \{\ell\}) + 1 & = \sqcap(\{c,d\} \cup Z \cup a, \{\ell\}) + \sqcap(\{c,d\} \cup Z, \{a\})\\
& = \sqcap(\{c,d\} \cup Z \cup \ell, \{a\}) + \sqcap(\{c,d\} \cup Z, \{\ell\})\\
& \le 2 + 0.
\end{align*}
Thus $\sqcap(\{c,d\} \cup Z \cup a, \{\ell\}) \le 1$. But $R \subseteq Z \cup \{a,c\}$ so $\sqcap(R, \{\ell\}) \le 1$; a contradiction.
\end{proof}
\begin{lemma}
\label{22sep}
The $2$-polymatroid $M\backslash \ell$ does not have two disjoint $2$-element $2$-separating sets.
\end{lemma}
\begin{proof} Assume that $M\backslash \ell$ has $\{a,b\}$ and $\{c,d\}$ as disjoint $2$-separating sets. Then each of $a, b, c$, and $d$ is a line and $r(\{a,b\}) = 3 = r(\{c,d\})$. As before, let $Z = E - \{a,b,c,d,\ell\}$. Suppose $Y$ is $\{a,b\}$ or $\{c,d\}$, and $X = E - \ell - Y$. Then $M\backslash \ell = M_X \oplus_2 M_Y$. By Lemmas~\ref{not23}, \ref{fourmost}, \ref{noone}, \ref{no7}, and \ref{no9}, we know that $M_Y$ is isomorphic to $P_5$, $P_6$, or $P_8$. By Lemma~\ref{pixl},
\begin{sublemma}
\label{skewy}
$\ell$ is skew to $X$, so $\ell$ is skew to each of $a$, $b$, $c$, and $d$.
\end{sublemma}
When $M_Y {\rm co}ng P_n$, we shall say that $Y$ is a {\it type-$n$ $2$-separator} of $M\backslash \ell$.
\begin{sublemma}
\label{not66}
Neither $\{a,b\}$ nor $\{c,d\}$ is of type-$6$.
\end{sublemma}
Assume the contrary. Suppose $\{a,b\}$ is of type-6. Then, by Lemma~\ref{3connel}, $M/ \ell$ does not have a c-minor isomorphic to $N$. Thus, by Lemma~\ref{oldstep5}, neither $\sqcap(\{c\},X)$ nor $\sqcap(\{d\},X)$ is $0$. Hence $\{c,d\}$ is also of type-6.
Suppose $\alpha \in \{a,b\}$ and $\gamma \in \{c,d\}$. Then $r(Z \cup \{\alpha, \gamma\}) = r(Z) + 2$. Of course, $r(M) = r(Z) + 4$.
Suppose $r(Z \cup \{\alpha, \gamma\} \cup \ell) = r(M)$. Then $\sqcap(Z \cup \{\alpha, \gamma\}, \ell) = 0$. Let the elements of $\{a,b,c,d\} - \{\alpha, \gamma\}$ be $\beta$ and $\delta$. In $M\backslash \beta \backslash \delta$, the set $\{\ell\}$ is $1$-separating. Thus $M\backslash \beta \backslash \delta\backslash \ell = M\backslash \beta \backslash \delta/ \ell$. As $M\backslash \beta \backslash \delta\backslash \ell$ has a c-minor isomorphic to $N$, so does $M/\ell$. We then get a contradiction\ to Lemma~\ref{3connel} since
$\sqcap(a, E - \{\ell,a,b\}) = 1 = \sqcap(b, E - \{\ell,a,b\})$.
We may now assume that $r(Z \cup \{\alpha, \gamma\} \cup \ell) \le r(M) - 1.$ By \ref{skewy}, $\ell$ is skew to $Z \cup \{a,b\}$, so
$r(Z \cup a \cup \ell) = r(M) - 1.$ Thus, using the submodularity of $r$, we have
\begin{align*}
2r(M) - 1 & = r(Z \cup a \cup \ell) + r(M)\\
& \le r(Z \cup \{a,c\} \cup \ell) + r(Z \cup \{a,d\} \cup \ell)\\
& \le 2r(M) - 2.
\end{align*}
This contradiction\ establishes \ref{not66}.
We now know that each of $\{a,b\}$ and $\{c,d\}$ is of type-5 or of type-8.
In particular, we may assume that
$\sqcap(\{a\}, Z \cup \{c,d\}) = 0 = \sqcap(\{c\}, Z \cup \{a,b\})$.
Since $\mu(\ell) = 2$ and $\{a,b,c,d\}$ contains at most two elements of $N$, we see that
\begin{equation}
\label{abcd6}
r(\{a,b,c,d\}) = 6.
\end{equation}
By Lemma~\ref{oldstep5},
\begin{sublemma}
\label{mell}
$\ell \in {\rm cl}_{M/a}(\{c,d\})$ and $\ell \in {\rm cl}_{M/c}(\{a,b\})$.
\end{sublemma}
We deduce that $r(\{a,c,d,\ell\}) = r(\{a,c,d\}) = 5$ and
$r(\{a,b,c,\ell\}) = r(\{a,b,c\}) = 5$. By submodularity and (\ref{abcd6}),
\begin{align*}
10 & = r(\{a,c,d,\ell\}) + r(\{a,b,c,\ell\})\\
& \ge r(\{a,b,c,d,\ell\}) + r(\{a,c,\ell\})\\
& \ge 6 + 4 = 10.
\end{align*}
We conclude that
\begin{equation}
\label{eqnacl}
r(\{a,c,\ell\}) = 4.
\end{equation}
Next we show the following.
\begin{sublemma}
\label{type5}
Both $\{a,b\}$ and $\{c,d\}$ are of type-$5$.
\end{sublemma}
Suppose $\{a,b\}$ is of type-8. Then $\sqcap(\{b\}, Z \cup \{c,d\}) = 0$. Thus we can replace $a$ by $b$ in the argument used to prove (\ref{eqnacl}) to get that $r(\{b,c,\ell\}) = 4$. Hence
\begin{align*}
4 + 4 & = r(\{a,c,\ell\}) + r(\{b,c,\ell\})\\
& \ge r(\{a,b,c,\ell\}) + r(\{c,\ell\})\\
& \ge 5 + 4.
\end{align*}
This contradiction\ and symmetry implies that
\ref{type5} holds.
Now, by Lemma~\ref{claim1}, $M\backslash \ell \backslash a$, and hence $M\backslash a$, has a c-minor isomorphic to $N$. Thus $M\backslash a$ is not $3$-connected. Let $(U,V)$ be a non-trivial $2$-separation\ of $M\backslash a$. Then we may assume that $\ell \in U$ and $c\in V$ otherwise $M$ has a $2$-separation.
Suppose $d \in U$. Then, by \ref{skewy}, $(U \cup c, V - c)$ is a 1-separation of $M\backslash a$; a contradiction. Thus $d \in V$. By \ref{skewy} again, $r(U- \ell) = r(U) - 2$, so we obtain the contradiction\ that $(U - \ell, V \cup \ell \cup a)$ is a 1- or $2$-separation\ of $M$ unless $U - \ell$
consists of a single point, $u$, and $r(U) = 3$. In the exceptional case, since $M\backslash a\backslash \ell$ is $2$-connected, we see that $u \in {\rm cl}(V)$, so $(U - u, V \cup u)$ is a $1$-separation of $M\backslash a$; a contradiction.
\end{proof}
\begin{lemma}
\label{muend}
Suppose that $M$ has an element $\ell$ such that $M\backslash \ell$ has $N$ as a c-minor.
Then the largest non-$N$-side in a $2$-separation of $M\backslash \ell$ has size exceeding two.
\end{lemma}
\begin{proof}
Assume $\mu(\ell) = 2$. Then $M\backslash \ell = M_X \oplus_2 M_Y$ where $|Y| = 2$. In Lemma~\ref{old2}, we identified the nine possibilities for $M_Y$. We showed in Lemmas~\ref{not23}, \ref{fourmost}, \ref{noone}, \ref{no7}, and \ref{no9} that $M_Y$ must be isomorphic to $P_5$, $P_6$, or $P_8$. In Lemmas~\ref{duh}, \ref{duh8}, and \ref{duh5}, we showed that $(X,Y)$ cannot be the sole non-trivial 2-separation of $M\backslash \ell$. Lemma~\ref{22sep} completes the proof by showing that $M\backslash \ell$ cannot have a second non-trivial 2-separation.
\end{proof}
\begin{lemma}
\label{dualmu}
Suppose that $M$ has an element $\ell$ such that $M/\ell$ has $N$ as a c-minor.
Then the largest non-$N$-side in a $2$-separation of $M/ \ell$ has size exceeding two.
\end{lemma}
\begin{proof}
By Lemma~\ref{csm}, $(M/ \ell)^*$ has a c-minor isomorphic to $N^*$. By Lemma~\ref{compact0}, $(M/ \ell)^* = (M^*\backslash \ell)^{\flat}$. Thus $M^*\backslash \ell$ has a c-minor isomorphic to $N^*$. Let $Y$ be a largest non-$N$-side in a $2$-separation\ of $M/ \ell$. By Lemma~\ref{compact0} again, $Y$ is a largest non-$N^*$-side in a $2$-separation\ of $M^*\backslash \ell$. Replacing $(M,N)$ by $(M^*,N^*)$ in Lemma~\ref{muend}, we deduce that $|Y| > 2$.
\end{proof}
\section{Finding a doubly labelled line}
\label{fdll}
Recall that we are assuming that $(M,N)$ is a counterexample to Theorem~\ref{modc} where $N$ is a $3$-connected $2$-polymatroid that is a c-minor of $M$. In this section, we prove some lemmas that will eventually enable us to deduce that $M$ has a doubly labelled line. The first step in this process is to prove the following elementary but useful lemma.
\begin{lemma}
\label{predichotomy}
Suppose $y \in E(M) - E(N)$. If $y$ is not a doubly labelled element of $M$, and $M'$ has a special $N$-minor for some $M'$ in $\{M\backslash y,M/ y\}$, then $M'$ has $N$ as a c-minor.
\end{lemma}
\begin{proof} Since $y \in E(M) - E(N)$, some $M''$ in $\{M\backslash y,M/ y\}$ has $N$ as a c-minor. Since $y$ is not doubly labelled, we see that $M'' = M'$.
\end{proof}
The next lemma identifies an important dichotomy.
\begin{lemma}
\label{dichotomy}
Let $M'$ be a c-minor of $M$ having $N$ as a c-minor and let $(X',Y')$ be a $2$-separation of $M'$ having $X'$ as the $N$-side.
Assume that, for all elements $y$ of $Y'$, at least one of $M'\backslash y$ and $M'/y$ does not have a special $N$-minor. Then either
\begin{itemize}
\item[(i)] $\sqcap_{M'}(\{y\},X') = 1$ for all $y$ in $Y'$; or
\item[(ii)] $\sqcap_{M'}(Y'-y,X') = 0$ for all $y$ in $Y'$.
\end{itemize}
\end{lemma}
\begin{proof} Suppose $y \in Y'$. If $\sqcap_{M'}(\{y\},X) = 0$, then, by Lemma~\ref{obs1}, $\sqcap_{M'/y}(X',Y'-y) = 1$, so, by Lemma~\ref{claim1}(ii), $M'/y$ has a special $N$-minor. If $\sqcap_{M'}(Y'-y,X') = 1$, then, by Lemma~\ref{claim1}(i), $M'\backslash y$ has a special $N$-minor. By hypothesis, $M'\backslash y$ or $M'/y$ has no special $N$-minor.
We deduce the following.
\begin{sublemma}
\label{dich1}
Either $\sqcap_{M'}(\{y\},X') = 1$ or $\sqcap_{M'}(Y'-y,X') = 0$.
\end{sublemma}
Next we show that all the elements of $Y'$ behave similarly.
\begin{sublemma}
\label{dich2}
If $\sqcap_{M'}(\{y\},X') = 1$, then $\sqcap_{M'}(\{z\},X) = 1$ for all $z$ in $Y'$.
\end{sublemma}
To see this, note first that $M' = M'_{X'} \oplus_2 M'_{Y'}$. Since $\sqcap_{M'}(\{y\},X') = 1$, it follows that $p \in {\rm cl}_{M'_{Y'}}(\{y\})$. Suppose $z \in Y'-y$. Then $p \in {\rm cl}_{M'_{Y'}}(Y' - z)$. Hence
$\sqcap_{M'}(X',Y'-z) = 1$ so $M'\backslash z$ has a special $N$-minor. Thus $M'/z$ does not have a special $N$-minor. Hence, by Lemma~\ref{claim1}(ii),
$\sqcap_{M'/z}(X',Y'-z) = 0$, so, by Lemma~\ref{obs1}, $\sqcap_{M'}(X',\{z\}) = 1$, and \ref{dich2} holds.
Now suppose that $\sqcap_{M'}(\{y\},X') = 0$. Then, by \ref{dich2}, $\sqcap_{M'}(\{z\},X') = 0$ for all $z$ in $Y'$. Thus $M'/z$ has a special $N$-minor for all $z$ in $Y'$. The hypothesis implies that $M'\backslash z$ has no special $N$-minor for all $z$ in $Y'$. Then, by Lemma~\ref{claim1}(i), $\sqcap_{M'}(Y'-z,X') = 0$ and the lemma follows.
\end{proof}
The next lemma describes what happens when (i) of Lemma~\ref{dichotomy} holds.
\begin{lemma}
\label{p63rev}
Suppose $M\backslashba \ell$ has $N$ as a c-minor. Let $(X,Y)$ be a $2$-separation of $M\backslash \ell$ in which $X$ is the $N$-side and $|Y| \ge 3$. Then
\begin{itemize}
\item[(i)] $Y$ contains a doubly labelled element; or
\item[(ii)] $\sqcap(\{y\},X) \neq 1$ for some $y$ in $Y$; or
\item[(iii)] $Y$ contains an element $y$ such that $M\backslashba y$ has $N$ as a c-minor and every non-trivial $2$-separation of $M\backslash y$ has the form $(Z_1,Z_2)$ where $Z_1$ is the $N$-side and $Z_2 \subseteq Y - y$.
\end{itemize}
\end{lemma}
\begin{proof} Suppose that $\sqcap(\{y\},X) = 1$ for all $y$ in $Y$ and that $Y$ does not contain any doubly labelled elements. As usual, we write $M\backslash \ell$ as the $2$-sum with basepoint $p$ of the $2$-polymatroids $M_X$ and $M_Y$ having ground sets $X \cup p$ and $Y \cup p$, respectively. First we show that
\begin{sublemma}
\label{nopoint}
$Y$ does not contain a point.
\end{sublemma}
Assume that $Y$ does contain a point, $z$. Then, since $\sqcap(\{z\},X) = 1$, we see that $z$ is parallel to $p$ in $M_Y$.
By Proposition~\ref{connconn}, $M\backslash \ell\backslash z$ is $2$-connected. Hence $M\backslash z$ is $2$-connected. Also, in $M_X$ and $M_Y$, the sets $X$ and $Y-z$ span $p$, and hence span $z$. We show next that
\begin{sublemma}
\label{nopointsub}
$M\backslash z$ is $3$-connected.
\end{sublemma}
Suppose that $M\backslash z$ has a $2$-separation $(R,B)$ where $\ell \in R$. Then $(R-\ell,B)$ is 2-separating in $M\backslash z \backslash \ell$. Note that $r(M\backslash \ell) = r(M)$, so $r(M\backslash \ell\backslash z) = r(M)$. We have
$$r(R) + r(B) = r(M\backslash z) + 1.$$
Thus
$$r(R - \ell) + r(B) \le r(M\backslash z,\ell) +1.$$
Now $R \neq \{\ell\}$ otherwise $Y - z \subseteq B$ and we obtain the contradiction that $(R,B \cup z)$ is a $2$-separation of $M$. Observe that, since $M\backslash \ell\backslash z$ is $2$-connected, $r(R - \ell) = r(R)$. As $M$ is $3$-connected, neither $B$ nor $R - \ell$ spans $z$. Thus neither $X$ nor $Y - z$ is contained in $B$ or $R - \ell$. Hence $(X,Y-z)$ and $(R - \ell,B)$ cross.
Now $\lambda_{M\backslash \ell \backslash z}(Y-z) = \lambda_{M\backslash \ell}(Y) = 1$ and $\lambda_{M\backslash \ell \backslash z}(B) = 1$. Thus, by uncrossing,
$\lambda_{M\backslash \ell \backslash z}(B\cap (Y-z)) = 1$. Since $\ell \in {\rm cl}(R - \ell)$ and $z \in {\rm cl}(X)$, we deduce that $\lambda_M(B \cap (Y-z)) = 1$. As $M$ is $3$-connected, it follows that $B \cap (Y - z)$ consists of a single point $y$. Then, by assumption, $\sqcap(X,\{y\}) = 1$. But $\sqcap(X,\{z\}) = 1$. Thus $y$ is parallel to $p$ in $M_Y$. Hence $y$ and $z$ are parallel points in $M$; a contradiction. We conclude that \ref{nopointsub} holds.
To complete the proof of \ref{nopoint}, we shall show that $M\backslashba z$ has a special $N$-minor. We know that $M\backslash \ell = M_X \oplus_2 M_Y$ where $z$ is parallel in $M_Y$ to the basepoint $p$ of the $2$-sum. Moreover, by Lemma~\ref{p69}, $M_X$ has a special $N$-minor.
Now $M\backslash \ell\backslash z$ is $2$-connected and, by \cite[Proposition 3.1]{hall}, $M\backslash \ell\backslash z = M_X \oplus_2 (M_Y \backslash z)$.
Hence $M\backslash z$ has a special $N$-minor. Thus $M\backslashba z$ is $3$-connected having a c-minor isomorphic to $N$; a contradiction. We deduce that \ref{nopoint} holds.
We now know that every element of $Y$ is a line $y$ with $\sqcap(X,\{y\}) = 1$. Hence, in $M_Y$, the basepoint $p$ lies on $y$. Thus, for all $y$ in $Y$, we see that $M\backslash \ell\backslash y$ is $2$-connected. Then,
by Lemma~\ref{p49} again, we deduce that
\begin{sublemma}
\label{usey}
for all $y$ in $Y$, both $M\backslash \ell\backslash y$ and $M\backslash y$ have special $N$-minors.
\end{sublemma}
Since every line in $Y$ contains $p$, it follows that $M_Y/p$ is a matroid. Next we show that
\begin{sublemma}
\label{caseD1.1}
$M_Y/p$ has a circuit.
\end{sublemma}
Assume that $M_Y/p$ has no circuits. Let $y$ and $y'$ be two distinct elements of $Y$. Then $r(X \cup (Y - \{y,y'\})) = r(X) + |Y - \{y,y'\}|$ and
$r(X \cup Y) = r(X) + |Y|$.
As a step towards \ref{caseD1.1}, we show that
\begin{sublemma}
\label{caseD1.1sub}
$\sqcap((X\cup Y) - \{y,y'\},\{\ell\}) = 0$.
\end{sublemma}
Suppose that $\sqcap((X\cup Y) - \{y,y'\},\{\ell\}) \ge 1$.
Then, as $r(Y) = |Y| + 1$,
\begin{align*}
\lambda_M(\{y,y'\}) & = r(X \cup (Y - \{y,y'\})\cup \ell) + r(\{y,y'\}) - r(M)\\
& \le r(X) + |Y - \{y,y'\}| +1 + 3 - r(M)\\
& = r(X) + r(Y) - r(M) + 1 = 2.
\end{align*}
As $M$ is $3$-connected, we see that $\lambda_M(\{y,y'\}) = 2$, so equality holds thoughout the last chain of inequalities. Thus $\{y,y'\}$ is a prickly 3-separator of $M$ and $\lambda_{M\backslash \ell}(\{y,y'\}) = 1$. By Lemma~\ref{portia}, $M\downarrow y$ is $3$-connected.
By Lemma~\ref{dennisplus}(vi), $(M\backslash \ell)\downarrow y = M_X \oplus_2 (M_Y \downarrow y)$. Thus $\sqcap_{M\backslash \ell \downarrow y}(X,Y-y) = 1$ so, by Lemma~\ref{claim1}(iii), $(M\downarrow y)\backslash \ell$, and hence $M\downarrow y$, has a special $N$-minor. This contradiction
implies that \ref{caseD1.1sub} holds for all distinct $y$ and $y'$ in $Y$.
As the next step towards proving \ref{caseD1.1}, we now show that
\begin{sublemma}
\label{lcn}
$M/ \ell$ has a c-minor isomorphic to $N$.
\end{sublemma}
In $M\backslash \ell$, deleting all but one element, $y$, of $Y$ leaves the $2$-polymatroid that, when $y$ is compactified, equals $M_X$ with $p$ relabelled as $y$. Hence $M\backslash \ell \backslash (Y - y)$ has a c-minor isomorphic to $N$. By \ref{caseD1.1sub}, since $|Y| \ge 3$, we deduce that $\{\ell\}$ is 1-separating in $M\backslash (Y - y)$. Hence $M\backslash (Y - y) \backslash \ell = M\backslash (Y - y) / \ell$, so, by \ref{usey}, we deduce that \ref{lcn} holds.
Still continuing towards the proof of \ref{caseD1.1}, next we observe that
\begin{sublemma}
\label{linear}
$\ell$ is a line of $M$.
\end{sublemma}
Suppose $\ell$ is a point. By Lemma~\ref{newbix}, $M/ \ell$ is $2$-connected having one side of every $2$-separation being a pair of points of $M$ that are parallel in $M/ \ell$. By \ref{lcn}, $M$ must have such a pair $\{u,v\}$ of points. Then both $M\backslash u$ and $M\backslash v$ have c-minors isomorphic to $N$. By \cite[Lemma 4.2]{oswww}, $M$ has a triad of points containing $\ell$ and one of $u$ and $v$, say $u$. Let $w$ be the third point in this triad. Then $M\backslash \ell$ has $\{u,w\}$ as a series pair of points, so $M\backslash \ell/u$, and hence $M/u$, has a c-minor isomorphic to $N$. Thus the point $u$ contradicts Lemma~\ref{Step0}.
By \ref{lcn}, $M/ \ell$ has a 2-separation $(U,V)$. Thus $r(U \cup \ell) + r(V \cup \ell) - r(M) = 3$. By symmetry, we may assume that
$U \subseteq (X \cup Y) - \{y,y'\}$ for some $y'$ in $Y - y$. Then, by \ref{caseD1.1sub} and \ref{linear}, $r(U \cup \ell) = r(U) + 2$. Hence $(U, V \cup \ell)$ is a $2$-separation\ of $M$. This contradiction\ completes the proof of \ref{caseD1.1}.
Choose $y$ in $Y$ such that $y$ is in a circuit of $M_Y/p$ and $y \in E(M) - E(N)$. By \ref{usey}, $M\backslash y$ has a special $N$-minor. Thus, by Lemma~\ref{predichotomy}, $M\backslash y$ has $N$ as a c-minor. Now $r(M\backslash \ell\backslash y) = r(M\backslash \ell) = r(M) = r(M\backslash y)$. Hence $\ell \in {\rm cl}_{M\backslash y}(X \cup (Y - y))$ and $M\backslash \ell \backslash y$ is $2$-connected. Next we show the following.
\begin{sublemma}
\label{caseD1.2}
Every non-trivial $2$-separation of $M\backslash y$ has the form $(X \cup Y' \cup \ell, Y'')$ where $Y'$ and $Y''$ are disjoint and $Y' \cup Y'' = Y-y$.
\end{sublemma}
Let $(A,B)$ be a non-trivial $2$-separation of $M\backslash y$ that is not in the stated form. Without loss of generality, $\ell \in A$. Then $X \not \subseteq A$. Since $M\backslash \ell\backslash y$ is $2$-connected having the same rank as $M\backslash y$, it follows that $r(A - \ell) = r(A)$ and $(A- \ell,B)$ is a 2-separation of $M\backslash \ell\backslash y$. We also know that $(X,Y-y)$ is a 2-separation of $M\backslash \ell \backslash y$. Now $\ell \not\in {\rm cl}(X)$ and $\ell \not\in {\rm cl}(Y-y)$. But $\ell \in {\rm cl}(A - \ell)$, so $(A - \ell) \cap (Y -y) \neq \emptyset \neq (A - \ell) \cap X$. By uncrossing, $\lambda_{M\backslash \ell\backslash y}(B \cap X) = 1$. As $\ell \in {\rm cl}(A - \ell)$ and $y \in {\rm cl}(Y-y)$, we deduce that $\lambda_M(B\cap X) = 1$. Thus $B \cap X$ consists of a single point $x$ of $M$. Then $B \cap (Y-y) \neq \emptyset$. Therefore, by uncrossing again, $\lambda_{M\backslash \ell\backslash y}(X \cap (A - \ell)) = 1$, so $\lambda_{M\backslash \ell}(X \cap (A - \ell)) = 1$. Thus $(X-x,Y\cup x)$ is a 2-separation of $M\backslash \ell$. If $r(Y \cup x) = r(Y)$, then $x$ is parallel to $p$ in $M_X$. Hence, we see that $x$ lies on $y$. Then $M\backslash x$ is $3$-connected having a special $N$-minor; a contradiction. Thus we may assume that $r(Y \cup x) = r(Y) + 1$. Then $r(X- x) = r(X) - 1$. Hence, in $M_X$, the points $p$ and $x$ are a series pair. Thus $M_X$ is the 2-sum with basepoint $q$ of a $2$-polymatroid $M'_X$, say, and a copy of $U_{2,3}$ with ground set $\{q,p,x\}$. Moreover, every element of $Y$ is a line through $p$ in $M_Y$. Thus we see that both $M\backslash y$ and $M/y$ have special $N$-minors; a contradiction. We conclude that \ref{caseD1.2} holds, so (iii) of the lemma holds, and the proof of the lemma is complete.
\end{proof}
\begin{lemma}
\label{prep65rev}
Suppose $M\backslashba \ell$ has $N$ as a c-minor. Let $(X,Y)$ be a $2$-separation of $M\backslash \ell$ in which $X$ is the $N$-side and $|Y| \ge 3$.
Let $M_X \oplus_2 M_Y$ be the associated $2$-sum decomposition of $M\backslash \ell$ with respect to the basepoint $p$.
Then
\begin{itemize}
\item[(i)] $Y$ contains a doubly labelled element; or
\item[(ii)] $\sqcap(Y-y,X) > 0$ for some $y$ in $Y$; or
\item[(iii)] $r(X\cup \ell \cup y_0) > r(X \cup y_0)$ for some $y_0$ in $Y$, and $M/y_0$ has a special $N$-minor. Moreover, either
\begin{itemize}
\item[(a)] every non-trivial $2$-separation of $M/y_0$ has the form $(Z_1,Z_2)$ where $Z_1$ is the $N$-side and $Z_2 \subseteq Y - y_0$; or
\item[(b)] $M_X$ is the $2$-sum with basepoint $q$ of two $2$-polymatroids, one of which is a copy of $U_{2,3}$ with ground set $\{p,z,q\}$.
\end{itemize}
\end{itemize}
\end{lemma}
\begin{proof} Assume that neither (i) nor (ii) holds. Suppose $y \in Y$. As $\sqcap(Y,X) = 1$, it follows that $r(Y) > r(Y-y)$ so
\begin{sublemma}
\label{*1}
$r(Y-y) \le r(Y) - 1.$
\end{sublemma}
Next we show that
\begin{sublemma}
\label{MYy}
$\lambda_{M_Y}(\{y\}) = \lambda_{M\backslash (X \cup \ell)}(\{y\}) + 1.$
\end{sublemma}
We see that $\lambda_{M_Y}(\{y\}) = r_M(\{y\}) + r_{M_Y}((Y-y) \cup p) - r(M_Y).$ Since $\sqcap(Y-y, X) = 0$, we deduce that
$r_{M_Y}((Y-y) \cup p) = r_M(Y-y) + 1.$ As $M_Y$ is $2$-connected, $r(Y) = r(M_Y)$ and \ref{MYy} follows.
We now extend \ref{*1} as follows.
\begin{sublemma}
\label{*2}
Let $\{y_1,y_2,\dots,y_k\}$ be a subset of $Y$. Then
$$r(Y - \{y_1,y_2,\dots,y_k\}) \le r(Y) - k.$$
\end{sublemma}
By \ref{*1}, $r(Y-y_1) \le r(Y) - 1$ and $r(Y-y_2) \le r(Y) - 1$. Thus, by submodularity, $r(Y-\{y_1,y_2\}) \le r(Y) - 2$. Repeating this argument gives \ref{*2}.
Next we show the following.
\begin{sublemma}
\label{mcony}
For all $y$ in $Y$, the $2$-polymatroid $M\backslash \ell/y$ has a special $N$-minor and $\lambda_{M\backslash \ell/y}(X) = 1$.
\end{sublemma}
Let $M' = M\backslash \ell$. By Corollary~\ref{general2},
\begin{align}
\label{Yyy}
\lambda_{M'/y}(X) & = \lambda_{M'\backslash y}(X) - \sqcap_{M'}(X,y) - \sqcap_{M'}(Y-y,y) + r(\{y\}) \nonumber \\
& = \lambda_{M'\backslash y}(X) - r_{M'}(Y-y) + r_{M'}(Y) \text{~as $\sqcap(X,Y-y') = 0$ for all $y'$ in $Y$;} \nonumber\\
& = r_{M'}(Y) - r_{M'}(Y-y).
\end{align}
But
\begin{align*}
1 & = \lambda_{M'}(X)\\
& = r(X) + r(Y) - r(M')\\
& \ge r(X \cup y) - r(\{y\}) + r(Y) - r(\{y\}) - r(M') + r(\{y\})\\
& = \lambda_{M'/y}(X).
\end{align*}
We conclude, using (\ref{Yyy}) that, since $r(Y) \neq r(Y-y)$, we have $\lambda_{M'/y}(X) = 1$ for all $y$ in $Y$. Then, by Lemma~\ref{claim1}(ii), $M'/y$ has a special $N$-minor. Hence $M\backslash \ell /y$ has a special $N$-minor, that is, \ref{mcony} holds.
\begin{sublemma}
\label{neworg}
If $y \in Y$ and $\ell$ is in a parallel pair of points in $M/y$, then $r(X \cup \ell \cup y) = r(X \cup y)$.
\end{sublemma}
To see this, observe that, as $M$ is $3$-connected, $\ell \not\in {\rm cl}_M(Y)$. Thus $\ell$ is parallel to a point of $X$ in $M/y$, and \ref{neworg} follows.
\begin{sublemma}
\label{subsume}
Let $Y = \{y_1,y_2,\dots,y_n\}$. If $r(X \cup \ell \cup y_i) = r(X \cup y_i)$ for all $i$ in $\{1,2,\dots,n\}$, then $\{y_{n-1},y_n\}$ is a prickly $3$-separator of $M$, and $M\downarrow y_n$ is $3$-connected having a special $N$-minor.
\end{sublemma}
First observe that each $y_i$ in $Y$ is a line for if $y_i$ is a point, then
$$r(X \cup \ell \cup y_i) = r(X \cup y_i) = r(X) + r(\{y_i\}) = r(X) + 1.$$
As $r(Y-y_i) \le r(Y) - 1$, we deduce that $(X \cup \ell \cup y_i,Y-y_i)$ is a 2-separation of $M$; a contradiction.
Continuing with the proof of \ref{subsume}, next we show the following.
\begin{sublemma}
\label{subsume2}
For $1 \le k \le n-1$,
\begin{align*}
r(X \cup \ell \cup \{y_1,y_2,\dots,y_k\}) & = r(X) + 1 + k \text{~~and}\\
r(Y - \{y_1,y_2,\dots,y_k\}) & = r(Y) - k.
\end{align*}
\end{sublemma}
We argue by induction on $k$. By assumption, $r(X \cup \ell \cup y_1) = r(X) + r(\{y_1\}) = r(X) + 2$. Moreover, $r(Y-y_1) \le r(Y) - 1$. Equality must hold otherwise we get the contradiction that $(X \cup \ell \cup y_1, Y- y_1)$ is a 2-separation of $M$. We deduce that the result holds for $k = 1$. Assume it holds for $k < m$ and let $k = m\ge 2$. Then
\begin{multline*}
r(X \cup \{y_1,y_2,\dots,y_{m-1}\} \cup \ell) + r(X \cup \{y_2,y_3,\dots,y_{m}\} \cup \ell)\\
\shoveleft{\hspace*{1in}\ge r(X \cup \{y_2,y_3,\dots,y_{m-1}\} \cup \ell) + r(X \cup \{y_1,y_2,\dots,y_{m}\} \cup \ell).}
\end{multline*}
If $m = 2$, then $r(X \cup \{y_2,y_3,\dots,y_{m-1}\} \cup \ell) = r(X \cup \ell) \ge r(X) + 1$. If $m>2$, then
$r(X \cup \{y_2,y_3,\dots,y_{m-1}\} \cup \ell) = r(X) + m - 1$ by the induction assumption. Thus
\begin{align}
\label{eq1}
r(X \cup \{y_1,y_2,\dots,y_m\}\cup \ell) & \le r(X) + m + r(X) + m - (r(X) + m-1) \nonumber \\
& = r(X) + m + 1.
\end{align}
But
\begin{equation}
\label{eq2}
r(Y - \{y_1,y_2,\dots,y_m\}) \le r(Y) - m.
\end{equation}
It follows that equality must hold in (\ref{eq1}) and (\ref{eq2}). Thus, by induction, \ref{subsume2} holds.
By \ref{subsume2}, $r(Y - \{y_1,y_2,\dots,y_{n-1}\}) = r(Y) - (n-1).$ But $r(Y - \{y_1,y_2,\dots,y_{n-1}\}) = r(\{y_n\}) = 2$. Thus $r(Y) = n+1$,
and it follows by \ref{subsume2} that $r(\{y_{n-1},y_n\}) = 3$ and $\{y_{n-1},y_n\}$ is a prickly 3-separating set in $M$. Hence, by Lemma~\ref{portia},
$M\downarrow y_n$ is $3$-connected. Recall that
\begin{equation*}
r_{M\downarrow y_n}(Z) =
\begin{cases}
r(Z), & \text{if $r(Z \cup y_n) > r(Z)$; and}\\
r(Z) - 1, & \text{otherwise.}
\end{cases}
\end{equation*}
Thus
\begin{align*}
\sqcap_{M\downarrow y_n}(X,Y-y_n) & = r_{M\downarrow y_n}(X) + r_{M\downarrow y_n}(Y -y_n) - r_{M\downarrow y_n}(X \cup (Y-y_n))\\
& = r(X) + r(Y-y_n) - r(M) + 1\\
& = r(X) + r(Y) - r(M) \text{~~by \ref{subsume2};}\\
& = 1.
\end{align*}
It follows by Lemma~\ref{dennisplus}(vi) that
$(M\backslash \ell)\downarrow y_n = M_X \oplus_2 M_Y \downarrow y_n$. Then, by Lemma~\ref{claim1}(iii), $(M\backslash \ell)\downarrow y_n$ has a special $N$-minor. We deduce that $M\downarrow y_n$ is $3$-connected having a special $N$-minor. Thus \ref{subsume} holds.
Since we have assumed that the theorem fails, it follows by \ref{subsume} that, for some element $y_0$ of $Y$,
$$r(X \cup \ell \cup y_0) > r(X \cup y_0).$$
By \ref{mcony}, $M/y_0$ has a special $N$-minor.
Thus $M/y_0$ is not $3$-connected.
Moreover, by \ref{neworg}, the element $\ell$ is not in a pair of parallel points of $M/y_0$.
Let $(A \cup \ell, B)$ be a $2$-separation\ of $M/y_0$ with $\ell \not\in A$.
Next we show that
\begin{sublemma}
\label{2sepab}
$(A,B)$ is an exact $2$-separation of $M/y_0\backslash \ell$, and $\ell \in {\rm cl}_{M/y_0}(A).$
\end{sublemma}
If $(A,B)$ is not exactly 2-separating in $M/y_0\backslash \ell$, then, by Proposition~\ref{connconn}, $M_Y/y_0$ is not $2$-connected, so we obtain the contradiction\ that $Y$ contains a doubly labelled element. Thus $r_{M/y_0}(A \cup \ell) = r_{M/y_0}(A)$ and
\ref{2sepab} holds.
We shall show that
\begin{sublemma}
\label{crosspath}
either (iii)(b) holds, or $(A,B)$ does not cross $(X,Y-y_0)$.
\end{sublemma}
Assume each of $A$ and $B$ meets each of $X$ and $Y - y_0$. Then, by uncrossing, $\lambda_{M\backslash \ell/y_0}(X \cap B) = 1$. But $\sqcap(X,\{y_0\}) = 0$, so $r_M(X \cap B) = r_{M/y_0}(X \cap B).$ Also $r_M((Y-y_0) \cup A \cup \ell \cup y_0) = r_{M/y_0}((Y-y_0) \cup A \cup \ell) + r(\{y_0\}).$ Then
\begin{multline*}
r(X\cap B)) + r((Y-y_0) \cup A \cup \ell \cup y_0) - r(M)\\
\shoveleft{= r_{M/y_0}(X \cap B) + r_{M/y_0}((Y-y_0) \cup A \cup \ell) + r(\{y_0\}) - r(M/y_0) - r(\{y_0\})}\\
\shoveleft{= \lambda_{M/y_0}(X \cap B)}\\
\shoveleft{= \lambda_{M/y_0\backslash \ell}(X \cap B) \text{~~ as $\ell \in {\rm cl}_{M/y_0}(A)$;}}\\
\shoveleft{= 1.}\\
\end{multline*}
Since $M$ is $3$-connected, it follows that $X \cap B$ consists of a point $z$ of $M$.
Now $\lambda_{M\backslash \ell /y_0}((Y- y_0) \cup z) = 1$, so
\begin{align*}
1 & = r_{M/y_0}((Y-y_0) \cup z) + r_{M/y_0}(A \cap X) - r(M/y_0)\\
& = r(Y \cup z) - r(\{y_0\}) + r((A \cap X)\cup y_0) - r(\{y_0\}) - r(M) + r(\{y_0\})\\
& = r(Y \cup z)) + r(A \cap X) - r(M\backslash \ell) \text{~~since $\sqcap(X,\{y_0\}) = 0$.}\\
\end{align*}
Thus $Y \cup z$ is $2$-separating in $M\backslash \ell$. If $r(Y \cup z) = r(Y)$, then $z$ is parallel to the basepoint $p$ of the 2-sum. Hence each element of $Y$ is doubly labelled; a contradiction. Thus we may assume that $r(Y \cup z) = r(Y) + 1$. Then $r(X- z) = r(X) - 1$. Now $M_X$ is $2$-connected, so $r(M_X) = r(X)$ and $M_X$ has $\{p,z\}$ as a series pair of points. It follows that $M_X$ is the 2-sum with basepoint $q$ of a $2$-polymatroid $M'_X$ and a copy of $U_{2,3}$ with ground set $\{q,z,p\}$. Thus (iii)(b) of the lemma holds. Hence so does \ref{crosspath}.
We shall now assume that (iii)(b) does not hold.
\begin{sublemma}
\label{notsubset}
$A\not \subseteq Y - y_0$ and $B\not \subseteq X$ and $A\not \subseteq X$.
\end{sublemma}
To see this, first suppose that $A \subseteq Y - y_0$. Then, as $\ell \in {\rm cl}_{M/y_0}(A)$, we deduce that $\ell \in {\rm cl}_M(Y)$; a contradiction. Thus $A\not \subseteq Y - y_0$.
Now suppose that $B \subseteq X$. We have
\begin{align*}
1 & = \lambda_{M/y_0}(B)\\
& = r_{M/y_0}(B) + r_{M/y_0}(A \cup \ell) - r(M/y_0)\\
& = r(B \cup y_0) - r(\{y_0\}) + r(A \cup \ell \cup y_0) - r(\{y_0\}) - r(M) + r(\{y_0\})\\
& = r(B) + r(A \cup \ell \cup y_0) - r(M) \text{~~as $B \subseteq X$.}
\end{align*}
Thus $(A \cup \ell \cup y_0,B)$ is a $2$-separation\ of $M$; a contradiction. Thus $B\not\subseteq X$.
Next suppose that $A \subseteq X$. As $(A \cup \ell,B)$ is a $2$-separation\ of $M/y_0$, we have
\begin{align*}
1 & = r_{M/y_0}(A \cup \ell) + r_{M/y_0}(B) - r(M/y_0)\\
& = r(A \cup \ell \cup y_0) - r(\{y_0\}) + r(B \cup y_0) - r(\{y_0\}) - r(M) + r(\{y_0\})\\
& \ge r(A \cup y_0) - r(\{y_0\}) + r(B \cup y_0) - r(M)\\
& \ge r(A) + r(B \cup y_0) - r(M\backslash \ell) \text{~~as $A \subseteq X$;}\\
& \ge 1 \text{~~as $M\backslash \ell$ is $2$-connected.}
\end{align*}
We deduce that equality holds throughout, so $r(A \cup \ell \cup y_0) = r(A \cup y_0)$. But $A \subseteq X$, so $r(X \cup \ell \cup y_0) = r(X \cup y_0)$, which contradicts the choice of $y_0$. Hence $A\not\subseteq X$, so \ref{notsubset} holds.
By \ref{crosspath}, we deduce that $B \subseteq Y - y_0$. Since, by \ref{mcony}, $M/y_0$ has a special $N$-minor, we see that (iii)(a) of the lemma holds, so the lemma is proved.
\end{proof}
We now combine the above lemmas to prove one of the two main results of this section.
\begin{lemma}
\label{bubbly}
Suppose $M\backslash \ell$ has $N$ as a c-minor. Let $(X,Y)$ be a $2$-separation of $M\backslash \ell$ having $X$ as the $N$-side and $|Y| = \mu(\ell)$. Then $Y$ contains a doubly labelled element.
\end{lemma}
\begin{proof}
By Lemma~\ref{muend}, $|Y| \ge 3$. Assume that $Y$ does not contain a doubly labelled element. Then, by Lemma~\ref{p63rev},
\begin{itemize}
\item[(i)(a)] $\sqcap(\{y\},X) \neq 1$ for some $y$ in $Y$; or
\item[(i)(b)] $Y$ contains an element $y$ such that $M\backslashba y$ has $N$ as a c-minor and every non-trivial $2$-separation of $M\backslash y$ has the form $(Z_1,Z_2)$ where $Z_1$ is the $N$-side and $Z_2 \subseteq Y - y$.
\end{itemize}
Now, since $|Y| = \mu(\ell)$, outcome (iii)(b) of Lemma~\ref{prep65rev} does not arise. Thus, by that lemma and Lemma~\ref{predichotomy},
\begin{itemize}
\item[(ii)(a)] $\sqcap(Y-y,X) > 0$ for some $y$ in $Y$; or
\item[(ii)(b)] $Y$ contains an element $y$ such that $M/ y$ has $N$ as a c-minor and every non-trivial $2$-separation of $M/y$ has the form $(Z_1,Z_2)$ where $Z_1$ is the $N$-side and $Z_2 \subseteq Y - y$.
\end{itemize}
By Lemma~\ref{dichotomy}, (i)(a) and (ii)(a) cannot both hold. Thus (i)(b) or (ii)(b) holds. Therefore, for some $y$ in $Y$, either $M\backslashba y$ has $N$ as a c-minor and has a $2$-separation\ $(Z_1,Z_2)$ where $Z_1$ is the $N$-side, $Z_2 \subseteq Y - y$, and $|Z_2| = \mu(y) < \mu(\ell)$, or $M/ y$ has $N$ as a c-minor and has a $2$-separation\ $(Z_1,Z_2)$ where $Z_1$ is the $N$-side, $Z_2 \subseteq Y - y$, and $|Z_2| = \mu^*(y) < \mu(\ell)$. We can now repeat the argument above using $(y,Z_2)$ in place of $(\ell,Y)$ and, in the latter case, $M^*$ in place of $M$. Since we have eliminated the possibility that $\mu(\ell) = 2$ or $\mu^*(\ell) = 2$, after finitely many repetitions of this argument, we obtain a contradiction\ that completes the proof.
\end{proof}
\begin{corollary}
\label{doubly}
The $2$-polymatroid $M$ contains a doubly labelled element.
\end{corollary}
\begin{proof} Take $\ell$ in $E(M) - E(N)$. Then $M\backslash \ell$ or $M/ \ell$ has $N$ as a c-minor, so applying the last lemma to $M$ or its dual gives the result.
\end{proof}
\section{Non-$N$-$3$-separators exist}
\label{keylargo}
The purpose of this section is prove the existence of a non-$N$-3-separating set in $M$ where we recall that such a set $Y$ is exactly $3$-separating, meets $E(N)$ in at most one element, and, when it has exactly two elements, both of these elements are lines. The following lemma will be key in what follows.
\begin{lemma}
\label{key}
Let $(X,Y)$ be a $2$-separation of $M\backslash \ell$ where $X$ is the $N$-side, $|Y| \ge 2$, and $Y$ is not a series pair of points in $M\backslash \ell$. Then $Y$ contains no points.
\end{lemma}
\begin{proof}
Assume that $Y$ contains a point $y$. Then, by Lemma~\ref{Step0}, $y$ is not doubly labelled.
\begin{sublemma}
\label{key1}
$M\backslash y$ or $M/y$ has a special $N$-minor.
\end{sublemma}
To see this, consider the $2$-connected $2$-polymatroid $M_Y$. By Lemma~\ref{Tutte2}, $M_Y\backslash y$ or $M_Y/y$ is $2$-connected, so $\sqcap_{M\backslash y}(X,Y-y) = 1$ or $\sqcap_{M/ y}(X,Y-y) = 1$. As $M_X$ has a special $N$-minor, so does $M\backslash y$ or $M/y$.
\begin{sublemma}
\label{key2}
$M\backslash y$ does not have a special $N$-minor.
\end{sublemma}
Assume $M\backslash y$ does have a special $N$-minor. Then, as $y$ is not doubly labelled, $M/y$ does not have a special $N$-minor. Then, by Lemma~\ref{claim1}(ii), $\sqcap_{M/y}(X,Y-y) = 0$, that is, $r_{M/y}(X) + r_{M/y}(Y-y) - r(M/y) = 0$, so
$r(X \cup y) + r(Y) = r(M) + r(\{y\}) = r(M) + 1$. But $r(X) + r(Y) = r(M) +1$, so $r(X \cup y) = r(X)$ and $r(Y - y) = r(Y)$ otherwise $(X\cup y, Y-y)$ is a 1-separation of $M\backslash \ell$; a contradiction\ to Lemma~\ref{Step1}. Since $y \in Y$ and $r(X \cup y) = r(X)$, we see that $\sqcap(X,\{y\}) = 1$. But $\sqcap(X,Y) = 1$. Thus, in $M_Y$, the point $y$ is parallel to the basepoint $p$ of the 2-sum. Hence $M\backslash \ell \backslash y$ is $2$-connected and $r(M\backslash \ell \backslash y) = r(M)$.
Let $(A \cup \ell, B)$ be a non-trivial $2$-separation\ of $M\backslash y$ where $\ell \not \in A$. Now
\begin{align*}
1 & \le r(A) + r(B) - r(M\backslash \ell,y)\\
& \le r(A \cup \ell) + r(B) - r(M\backslash y)\\
& = 1.
\end{align*}
Thus $r(A) = r(A \cup \ell)$. Hence $\ell \in {\rm cl}(A)$ so $r(A) \ge 2$. Continuing with the proof of \ref{key2}, we now show the following.
\begin{sublemma}
\label{key3}
$(A,B)$ crosses $(X,Y-y)$.
\end{sublemma}
Because $y \in {\rm cl}(X) \cap {\rm cl}(Y-y)$ but $y \notin {\rm cl}(A) \cup {\rm cl}(B)$, we deduce that neither $A$ nor $B$ contains $X$ or $Y-y$, so \ref{key3} holds.
By uncrossing,
$\lambda_{M\backslash \ell,y}(B \cap (Y-y)) = 1$. But $\ell \in {\rm cl}(A)$ and $y \in {\rm cl}(X)$ so $\lambda_M(B \cap (Y - y)) = 1$. Hence $B \cap (Y - y)$ consists of a single point, say $z$. As $z$ is not parallel to $y$, we deduce that $\sqcap(X,\{z\}) = 0$. Thus, by Lemma~\ref{obs1},
$\sqcap_{M/z}(X,Y-z) = \lambda_{M\backslash \ell/z}(X) = 1$. Hence, by Lemma~\ref{claim1}(ii), $M\backslash \ell/z$, and hence $M/z$, has a special $N$-minor. On the other hand,
$$1 = \sqcap(X,\{y\}) \le \sqcap(X,Y-z) \le \sqcap(X,Y) = 1.$$
Thus $\sqcap_{M\backslash z}(X,Y-z) = 1$ so $M\backslash z$ has a special $N$-minor. Since $z$ is a point, we have a contradiction\ to Lemma~\ref{Step0} that proves \ref{key2}.
By combining \ref{key1} and \ref{key2}, we deduce that $M/y$ has a special $N$-minor but $M\backslash y$ does not. Since $(M,N)$ is a counterexample, $M/y$ is not $3$-connected. By Lemma~\ref{Step1}, $M/y$ is $2$-connected.
As $M\backslash y$ does not have a special $N$-minor, by Lemma~\ref{claim1}(i), $\sqcap(X,Y-y) = 0$. But $\sqcap(X,Y) = 1$. As $y$ is a point, it follows that $$r(Y-y) = r(Y) - 1$$
and $r(X \cup (Y-y)) = r(X \cup Y)$.
Moreover, as $(X \cup y,Y-y)$ is not a 1-separation of $M\backslash \ell$, we deduce that
\begin{sublemma}
\label{xy1}
$r(X \cup y) = r(X) + 1.$
\end{sublemma}
Now $r(M_Y \backslash p,y) = r(Y - y) = r(Y) - 1$. But $r(M_Y \backslash p) = r(Y)$. If $r(M_Y\backslash y) = r(Y) - 1$, then $\{y\}$ is a 1-separating set in $M_Y$. We deduce that $\{p,y\}$ is a series pair of points in $M_Y$. Thus $M_Y\backslash y$ is not 2-connected but $M_Y$ is, so, by Lemma~\ref{Tutte2}, $M_Y/y$ is 2-connected. Hence, by Proposition~\ref{connconn}, $M\backslash \ell/y$ is 2-connected.
\begin{sublemma}
\label{notel}
$(\{\ell\}, X \cup (Y-y))$ is not a $2$-separation of $M/y$.
\end{sublemma}
Assume the contrary. Then
$r(\{\ell,y\}) + r(X \cup Y) = r(M) + 2.$
But $r_{M/y}(\{\ell\}) = 2$ otherwise we do not have a $2$-separation. Thus $r(\{\ell,y\}) = 3$, so $(\{\ell\}, X \cup Y)$ is a $2$-separation\ of $M$; a contradiction. Therefore \ref{notel} holds.
Let $(A\cup \ell,B)$ be a $2$-separation\ of $M/y$ with $\ell$ not in $A$. By \ref{notel}, $A \neq \emptyset.$ Since $M/y \backslash \ell$ is 2-connected, $\lambda_{M/y\backslash \ell}(A) > 0$. Hence $\lambda_{M/y\backslash \ell}(A) = 1$, so $\ell \in {\rm cl}_{M/y}(A).$ Hence one easily checks that
\begin{sublemma}
\label{crank1}
\begin{itemize}
\item[(i)] $r(A \cup y \cup \ell) = r(A \cup y)$; and
\item[(ii)] $r(A \cup y) + r(B \cup y) = r(M\backslash \ell) + 2$.
\end{itemize}
\end{sublemma}
Next we show that
\begin{sublemma}
\label{crossagain}
$(A,B)$ crosses $(X,Y-y)$.
\end{sublemma}
Assume $B \cap (Y - y) = \emptyset$ or $B \cap X = \emptyset$. As $r(X \cup y) = r(X) + 1$ and $r(Y) = r(Y-y) + 1$, we have $r(B \cup y) = r(B) + 1$. Then, as $r(A\cup y \cup \ell) = r(A \cup y)$, we have, by \ref{crank1},
$$r(A \cup y \cup \ell) + r(B) = r(M) + 1,$$
that is, $(A \cup y \cup \ell, B)$ is a $2$-separation\ of $M$; a contradiction. We deduce that $B \cap (Y-y) \neq \emptyset \neq B \cap X$.
Now assume that $A \cap (Y-y) = \emptyset.$ Then $A \subseteq X$ and $Y-y \subseteq B$, so $r(X \cup y \cup \ell) = r(X \cup y)$. As
$r(X \cup y) = r(X) + 1$ and
$r(Y-y) = r(Y) -1$, it follows that $(X \cup y \cup \ell, Y-y)$ is 2-separating in $M$. Hence $Y-y$ consists of a single point $z$. Now $r(X) + r(Y) = r(M\backslash \ell) + 1$, so $r(X) = r(M\backslash \ell) - 1$. As $M\backslash \ell$ is connected, neither $y$ nor $z$ is in ${\rm cl}(X)$ so $\{y,z\}$ is a series pair of points in $M\backslash \ell$; a contradiction. Hence $A \cap (Y - y) \neq \emptyset$.
Finally, assume that $X \cap A = \emptyset$. Then $A \subseteq Y -y$, so, as $r(A \cup y \cup \ell) = r(A \cup y)$, it follows that $r(Y \cup \ell) = r(Y)$, so $(X,Y \cup \ell)$ is a $2$-separation\ of $M$; a contradiction. We conclude that \ref{crossagain} holds.
Next we determine the structure of the set $B$.
\begin{sublemma}
\label{whatsb}
In $M$, the set $B$ consists of two points, $x'$ and $y'$, that lie in $B\cap X$ and $B\cap (Y-y)$, respectively.
\end{sublemma}
By uncrossing, $\lambda_{M\backslash \ell/y}(X \cap B) = 1$, so
$$r((X \cap B) \cup y) + r(A \cup Y) - r(M\backslash \ell) = 2.$$ As $X \cap B \subseteq X$, we deduce that $r((X \cap B) \cup y) = r(X \cap B) + 1$. Also $y \in Y$, so $r(A \cup Y) = r(A \cup Y \cup \ell)$. Thus $(X\cap B, A \cup Y \cup \ell)$ is $2$-separating in $M$. Hence $X \cap B$ consists of a point, say $x'$.
By uncrossing again, we see that $\lambda_{M\backslash \ell/y}((Y-y) \cap B) = 1$, so
$$r(((Y-y) \cap B) \cup y) + r(A \cup X \cup y) - r(M\backslash \ell) = 2.$$
Thus
$$r((Y-y) \cap B) + r(A \cup X \cup y\cup \ell) = r(M) + 1$$
since $r(((Y-y) \cap B) \cup y) = r((Y-y) \cap B) + 1$ and $r(A \cup X \cup y) = r(A \cup X \cup y \cup \ell).$ Hence
$((Y-y) \cap B, A \cup X \cup y \cup \ell)$ is 2-separating in $M$, so $(Y-y) \cap B$ consists of a single matroid point, $y'$. We deduce that \ref{whatsb} holds.
\begin{sublemma}
\label{doubleup}
The element $y'$ is doubly labelled.
\end{sublemma}
To see this, first observe that, in $M/y$, the set $B$ is a 2-separating set consisting of two matroid points, $x'$ and $y'$. Suppose $r_{M/y}(B) = 2$. Then $r_{M/y}(A \cup \ell) = r(M/y) - 1$, so $r(A \cup \ell \cup y) = r(M) -1$. Hence $(A \cup \ell \cup y, \{x',y'\})$ is a $2$-separation\ of $M$; a contradiction. We deduce that $r_{M/y}(B) = 1$ so $\{x',y'\}$ is a pair of parallel points in $M/y$. Then $M/y\backslash y'$, and so $M\backslash y'$, has a special $N$-minor.
Now $r_{M/y}(\{x',y'\}) = 1$, so $r(\{x',y',y\}) = 2$. Thus $y \in {\rm cl}_{M/y'}(X)$, so $r(X \cup y' \cup y) = r(X \cup y')$. But, by \ref{xy1}, $r(X \cup y) > r(X)$, so $r(X \cup y') > r(X)$.
Hence
$\sqcap(X,\{y'\}) = 0$. Thus, by Lemma~\ref{obs1}, $\sqcap_{M/y'}(X,Y- y') = \sqcap(X,Y) = 1$. We conclude by Lemma~\ref{claim1} that $M/y'$ has a special $N$-minor. Therefore \ref{doubleup} holds.
As \ref{doubleup} contradicts Lemma~\ref{Step0}, we deduce that Lemma~\ref{key} holds.
\end{proof}
\begin{lemma}
\label{nonN}
There is a c-minor $N_0$ of $M$ that is isomorphic to $N$ such that $M$ has a non-$N_0$-$3$-separating set.
\end{lemma}
\begin{proof} By Corollary~\ref{doubly}, $M$ has a doubly labelled element $\ell$. By Lemma~\ref{Step0}, $\ell$ is a line. Moreover, by Lemma~\ref{Step1}, each of $M\backslash \ell$ and $M/ \ell$ is 2-connected.
Assume the lemma fails. Let $N_D$ and $N_C$ be special $N$-minors of $M\backslash \ell$ and $M/ \ell$, respectively. We now apply what we have learned earlier using $N_D$ in place of $N$. Let $(X,Y)$ be a $2$-separation\ of $M\backslash \ell$ in which $X$ is the $N_D$-side and $|Y| = \mu(\ell)$. Then $|Y| \ge 3$. Now $\sqcap(X,\{\ell\}) \in \{0,1\}$.
We show next that
\begin{sublemma}
\label{pixel}
$\sqcap(X,\{\ell\}) = 0$ and $\sqcap(Y,\{\ell\}) = 0$.
\end{sublemma}
Assume that $\sqcap(X,\{\ell\}) = 1$. Then $r(X \cup \ell) = r(X) + 1$, so $\lambda_M(Y) = 2$. Thus $Y$ is a non-$N_D$-3-separating set; a contradiction. Thus $\sqcap(X,\{\ell\}) = 0$. Similarly, if $\sqcap(Y,\{\ell\}) = 1$, then $\lambda_M(X) = 2$, so $Y \cup \ell$ is a non-$N_D$-3-separating set. This contradiction\ completes the proof of \ref{pixel}.
We deduce that $M\backslash \ell$ has a $2$-separation\ $(D_1,D_2)$ where $D_1$ is the $N_D$-side, $|D_2| = \mu(\ell) \ge 3$, and $\sqcap(D_1,\ell) = 0 = \sqcap(D_2,\ell)$. A similar argument to that used to show \ref{pixel} shows that $M/ \ell$ has a $2$-separation\ $(C_1,C_2)$ where $C_1$ is the $N_C$-side, $|C_2| = \mu^*(\ell) \ge 3$, and $\sqcap(C_1,\ell) = 2 = \sqcap(C_2,\ell)$. We observe here that the definition of $\mu^*(\ell)$ depends on $N_C$ here rather than on $N_D$.
By the local connectivity conditions between $\ell$ and each of $D_1,D_2, C_1$, and $C_2$,
\begin{sublemma}
\label{crosscd}
$(C_1,C_2)$ and $(D_1,D_2)$ cross.
\end{sublemma}
We have
$r(D_1) + r(D_2) = r(E - \ell ) + 1$ and $r(C_1) + r(C_2) = r(E - \ell) + 3$. By uncrossing,
$$\lambda_{M\backslash \ell}(D_2 \cap C_2) + \lambda_{M\backslash \ell}(D_1 \cap C_1) \le 4.$$
Suppose $\lambda_{M\backslash \ell}(D_2 \cap C_2) \le 1$. Since $\ell \in {\rm cl}(D_1 \cup C_1)$, it follows that
$\lambda_{M}(D_2 \cap C_2) \le 1$. Thus $D_2 \cap C_2$ consists of a single point, $z$. Then
$$2 = \sqcap(C_2,\{\ell\}) \le \sqcap(D_1\cup z, \{\ell\}) \le \sqcap(D_1,\{\ell\}) + 1 = 1;$$
a contradiction. We deduce that $\lambda_{M\backslash \ell}(D_2 \cap C_2) = 2 = \lambda_{M\backslash \ell}(D_1 \cap C_1)$, so
$\lambda_{M}(D_2 \cap C_2) = 2 = \lambda_{M}(D_1 \cap C_1)$. By symmetry,
$\lambda_{M}(D_1 \cap C_2) = 2 = \lambda_{M}(D_2 \cap C_1)$.
Clearly each of $D_2 \cap C_1$ and $D_2 \cap C_2$ contains at most one element of $N_D$. As $|D_2| \ge 3$, we deduce from Lemma~\ref{key} that $D_2$ contains no points.
Hence, some $Z$ in
$\{D_2 \cap C_1, D_2 \cap C_2\}$ contains at least two elements. Then $Z$ is a non-$N_D$-3-separator of $M$.
\end{proof}
For the rest of the proof of Theorem~\ref{modc}, we will use the c-minor $N_0$ of $M$ found in the last lemma. To avoid cluttering the notation, we will relabel $N_0$ as $N$.
\begin{lemma}
\label{p124}
Let $Y_1$ be a minimal non-$N$-$3$-separating set in $M$ with $|Y_1| \ge 3$, and let $X_1 = E(M) - Y_1$. Let $\ell$ be an element of $Y_1$ such that $M\backslash \ell$ has $N$ as a c-minor. Let $(A,B)$ be a $2$-separation of $M\backslash \ell$ where $A$ is the $N$-side and $|B| = \mu(\ell)$. Then one of the following holds.
\begin{itemize}
\item[(i)] $\lambda_{M\backslash \ell}(Y_1 - \ell) = 1$; or
\item[(ii)] $B \subseteq Y_1 - \ell$; or
\item[(iii)] $(A,B)$ crosses $(X_1, Y_1 - \ell)$ and $\lambda_{M\backslash \ell}(A \cap (Y_1 - \ell)) = 1 = \lambda_{M\backslash \ell}(B \cap (Y_1 - \ell))$, while
$\lambda_{M\backslash \ell}(A \cap X_1) = 2 = \lambda_{M\backslash \ell}(B \cap X_1) = \lambda_{M\backslash \ell}(Y_1 - \ell)$.
\end{itemize}
\end{lemma}
\begin{proof} Assume neither (i) nor (ii) holds. Then $\ell \in {\rm cl}(Y_1 - \ell)$ and $B \not \subseteq Y_1 - \ell$. If $B \subseteq X_1$, then $\lambda_M(B) = 1$; a contradiction. If $B \supseteq Y_1 - \ell$, then $\lambda_M(A) = 1$; a contradiction. Finally, observe that $|X_1 \cap A| \ge 2$ since $|E(N)| \ge 4$ and $X_1$ and $A$ are the $N$-sides of their separations. We conclude that $(A,B)$ crosses $(X_1, Y_1 - \ell)$.
By Lemma~\ref{muend}, $|B| \ge 3$. By Lemma~\ref{key}, $B$ contains no points. Now $\lambda_{M\backslash \ell}(B \cap X_1) \ge 2$ otherwise, as $\ell \in {\rm cl}(Y_1 - \ell)$, we get the contradiction\ that
$\lambda_{M}(B \cap X_1)= 1$. By uncrossing, we deduce that $\lambda_{M\backslash \ell}(A \cap (Y_1 - \ell)) \le 1$. Since $|X_1 \cap A| \ge 2$, we get, similarly, that
$\lambda_{M\backslash \ell}(A \cap X_1) \ge 2$, so $\lambda_{M\backslash \ell}(B \cap (Y_1- \ell)) \le 1$. As $M\backslash \ell$ is $2$-connected, we deduce that
$\lambda_{M\backslash \ell}(A \cap (Y_1 - \ell)) = 1 = \lambda_{M\backslash \ell}(B \cap (Y_1 - \ell))$. Hence $\lambda_{M\backslash \ell}(A \cap X_1) = 2 = \lambda_{M\backslash \ell}(B \cap X_1)$. We conclude that (iii) holds. Hence so does the lemma.
\end{proof}
\section{Finding big enough $3$-separators}
\label{bigtime}
In this section, we first establish (\ref{Step5}) and then we start the proof of (\ref{Step6}). Specifically, we begin by showing the following.
\begin{lemma}
\label{Step5+}
$M$ has a minimal non-$N$-$3$-separator with at least three elements.
\end{lemma}
\begin{proof}
Assume every minimal non-$N$-3-separating set has exactly two elements. Let $\{a,b\}$ be such a set, $Z$. Then both of its members are lines.
We may assume that $b \not\in E(N)$. Suppose first that $r(Z) = 2$. Then $a$ and $b$ are parallel lines. Suppose that $N$ is a c-minor of $M/b$. Since $a$ is a loop of $M/b$, we deduce that $a \notin E(N)$ so $M\backslash a$ has $N$ as a c-minor. Since $M\backslash a$ is $3$-connected, this is a contradiction. We may now assume that $M\backslash b$ has $N$ as a c-minor. Since it is $3$-connected, we have a contradiction\ that implies that $r(Z) > 2$.
Suppose next that $r(Z) = 4$. Then $r^*(Z) = ||Z|| + r(E - Z) - r(M)= 4 -2 = 2$. Hence $Z$ consists of a pair of parallel lines in $M^*$, so we obtain a contradiction\ as above. We may now assume that $r(Z) = 3$. Then $Z$ is a prickly $3$-separating set and, by Lemma~\ref{portia}, $M\downarrow b$ is $3$-connected. Hence $M\downarrow b$ has no c-minor isomorphic to $N$.
Now $M\backslash b$ or $M/b$ has a c-minor isomorphic to $N$. We begin by assuming the former. Let $(S \cup a, T)$ be a non-trivial $2$-separation\ of $M\backslash b$ with $a \not\in S$. Suppose the non-$N$-side of $(S \cup a, T)$ has $\mu(b)$ elements. By Lemma~\ref{muend}, $\mu(b) \ge 3$. We have
$r(S \cup a) + r(T) - r(M) = 1$. As $\sqcap(\{a\},\{b\}) = 1$ and $M$ is $3$-connected, $r(S \cup a \cup b) = r(S \cup a) + 1$, so
\begin{equation}
\label{eqmt}
\lambda_M(T) = 2.
\end{equation}
Moreover,
$$r(S \cup a) \ge r(S) + 1$$
otherwise $r(S \cup a) = r(S)$ so $r(E - b) = r(E - \{a,b\})$; a contradiction.
Next we show the following.
\begin{sublemma}
\label{addon}
Suppose $M\backslash b$ has a $2$-separation $(S_1,S_2)$ where $S_1$ is the $N$-side and $S_2$ contains a prickly $3$-separator $\{u,v\}$ where $u \notin E(N)$. Then $M\backslash b \downarrow u$ is not $2$-connected.
\end{sublemma}
Suppose $M\backslash b \downarrow u$ is $2$-connected. Now $M\backslash b = M_1 \oplus_2 M_2$ where $M_i$ has ground set $S_i \cup p$. Since $M\backslash b \downarrow u$ is $2$-connected, $\sqcap_{M\backslash b\downarrow u}(S_1, S_2 - u) = 1$. Then, by Lemma`\ref{claim1}(iii), $M\backslash b \downarrow u$ has a special $N$-minor. By Lemma~\ref{portia}, $M\downarrow u$ is $3$-connected. Since it has a c-minor isomorphic to $N$, we have a contradiction. Thus \ref{addon} holds.
Now suppose that $T$ is the $N$-side of $(S \cup a, T)$. Then, by Lemma~\ref{key}, $S \cup a$ contains no points. Assume that $r(S \cup a) = r(S) + 1$.
As $r(S \cup a) + r(T) - r(M\backslash b) = 1$, we see that
$$[r(S) + 1] + r(T) - [r(M\backslash b,a) + 1] = 1.$$ Hence $\sqcap(S,T) = 1$, so, by Lemma~\ref{claim1}(i), $M\backslash b\backslash a$ has a special $N$-minor.
As $\{a,b\}$ is a prickly 3-separating set, we see that $M\backslash b \backslash a = M\downarrow b \backslash a$ so $M\downarrow b$ has a c-minor isomorphic to $N$; a contradiction.
Next we consider the case when $T$ is the $N$-side of $(S \cup a, T)$, and $r(S \cup a) = r(S) + 2$. Then
$r(S) + r(T \cup a \cup b) = r(M) + 2$. Thus $S$ is a non-$N$-$3$-separator and so contains a minimal such set, $\{u,v\}$ where $u \notin E(N)$. From above, we know that $\{u,v\}$ is a prickly $3$-separator of $M$. By \ref{addon},
$M\backslash b \downarrow u$ is not $2$-connected.
Now $M\backslash b \downarrow u = M\downarrow u\backslash b$. Let $(J,K)$ be a $1$-separation of $M\downarrow u \backslash b$ with $a \in J$. Then $r_{M\downarrow u}(J \cup b) \le r_{M\downarrow u}(J) + 1$. Thus
\begin{align*}
r_{M\downarrow u}(J \cup b) + r_{M\downarrow u}(K) - r(M\downarrow u)
& \le [r_{M\downarrow u}(J) + r_{M\downarrow u}(K) - r(M\downarrow u\backslash b)] \\
& \hspace*{0.9in} + [1 + r(M\downarrow u\backslash b) - r(M\downarrow u)]\\
& = 1 + r(M\downarrow u\backslash b) - r(M\downarrow u).
\end{align*}
By Lemma~\ref{portia}, $M\downarrow u$ is $3$-connected, so $r(M\downarrow u\backslash b) = r(M\downarrow u)$, and $K$ consists of a single point, $k$, of $M\downarrow u$. Then
\begin{align*}
1 & = r_{M\downarrow u}(J) + r_{M\downarrow u}(\{k\}) - r(M\downarrow u\backslash b)\\
& = r_{M\downarrow u}(E - \{b,u,k\}) + r_{M\downarrow u}(\{k\}) - r(M\downarrow u)\\
& = r(E - \{b,k\}) - 1 + r(\{k\}) - r(M) + 1\\
& = r(E - \{b,k\}) + r(\{k\}) - r(M\backslash b).
\end{align*}
Hence $\{k\}$ is $1$-separating in $M\backslash b$. Thus $k$ contradicts Lemma~\ref{Step0}.
When $M\backslash b$ has a c-minor isomorphic to $N$, it remains to consider the case when $S \cup a$ is the $N$-side of $(S \cup a, T)$. As $\mu(b) \ge 3$, it follows that $|T| \ge 3$. By (\ref{eqmt}), $\lambda_M(T) = 2$. By assumption, $T$ contains a minimal non-$N$-$3$-separating set $T'$. The latter consists of a pair, $\{u,v\}$, of lines that form a prickly 3-separating set. We may assume that $u \not\in E(N)$. Now $M\backslash b$ is certainly $2$-connected. By Lemma~\ref{portia}, $M\downarrow u$ is $3$-connected. Since $\sqcap(\{a\},\{b\}) = 1$, it follows that $M\downarrow u \backslash b$ is $2$-connected; a contradiction. We conclude that
$M\backslash b$ does not have a c-minor isomorphic to $N$.
We now know that $M/b$ has a c-minor isomorphic to $N$. Moreover, $M^*$ has a c-minor isomorphic to $N^*$ and has $\{a,b\}$ as a prickly 3-separating set; and $(M/b)^* = (M^* \backslash b)^{\flat}$. We use $M^* \backslash b$ in place of $M\backslash b$ in the argument above to complete the proof of the lemma.
\end{proof}
The argument to establish that $M$ has a minimal non-$N$-$3$-separator with at least four elements is much longer than that just given since it involves analyzing a number of cases. We shall use three preliminary results. In each, we denote $E(M) - Y_1$ by $X_1$.
\begin{lemma}
\label{pre3lines}
Let $Y_1$ be a minimal-non-$N$-$3$-separator with exactly three elements. Suppose $\ell \in Y_1$ and $M\backslash \ell$ has $N$ as a c-minor. Let $(A,B)$ be a $2$-separation\ of $M\backslash \ell$ where $A$ is the $N$-side and $|B| \ge 3$. Suppose $\ell \in {\rm cl}(Y_1 - \ell)$. Then $(A,B)$ crosses $(X_1, Y_1 - \ell)$ and $\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$. Moreover, $Y_1 \cap B$ consists of a single line.
\end{lemma}
\begin{proof} As $\ell \in {\rm cl}(Y_1 - \ell)$, we see that $\lambda_{M_1 \backslash \ell}(Y_1 - \ell) = 2$. To see that $(A,B)$ crosses
$(X_1, Y_1 - \ell)$,
note first that, as $|Y_1 - \ell| = 2$ and $|A|, |B| \ge 3$, neither $A$ nor $B$ is contained in $Y_1 - \ell$. Moreover, $Y_1 - \ell$ is not contained in $A$ or $B$ otherwise $(A \cup \ell,B)$ or $(A,B \cup \ell)$ is a $2$-separation\ of $M$; a contradiction. Hence $(A,B)$ crosses
$(X_1, Y_1 - \ell)$.
As $|E(N)| \ge 4$, we see that $|X_1 \cap A| \ge 2$. Then
$$\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$$
otherwise, as $\ell \in {\rm cl}(Y_1 - \ell)$, we get the contradiction\ that $\lambda_{M}(X_1 \cap A) \le 1$. By uncrossing, $\lambda_{M\backslash \ell}(Y_1 \cap B) \le 1$. By Lemma~\ref{key}, $B$ contains no points, so $Y_1 \cap B$ contains no points. As $|Y_1 - \ell| = 2$, we see that $Y_1 \cap B$ consists of a single line.
\end{proof}
\begin{lemma}
\label{3lines}
Let $Y_1$ be a minimal-non-$N$-$3$-separator with exactly three elements. If $Y_1$ contains a line $\ell$ such that $M\backslash \ell$ has $N$ as a c-minor, then $Y_1$ consists of three lines.
\end{lemma}
\begin{proof} Assume that the lemma fails. Let $(A,B)$ be a $2$-separation\ of $M\backslash \ell$ where $A$ is the $N$-side and $|B| \ge 3$. First we show that
\begin{sublemma}
\label{3lines1}
$\ell \in {\rm cl}(Y_1 - \ell)$.
\end{sublemma}
Assume that $\ell \not\in {\rm cl}(Y_1 - \ell)$. Then $(X_1, Y_1 - \ell)$ is a $2$-separation\ of $M\backslash \ell$ with $|Y_1 - \ell| = 2$. By Lemma~\ref{key}, we may assume that $Y_1 - \ell$ consists of a series pair $\{y_1,y_2\}$ of points. Now $r(M\backslash \ell) = r(M) = r(X_1) + 1$, so $r(\{\ell,y_1,y_2\}) = 3$. Moreover, for each $i$ in $\{1,2\}$, we see that $M\backslash \ell/y_i$, and hence $M/y_i$, has a special $N$-minor.
As the theorem fails for $M$, we know that $M/y_i$ is not $3$-connected. Now $M/y_i$ is certainly $2$-connected. Let $(J,K)$ be a $2$-separation\ of it where we may assume that $\ell \in J$. Now $r_{M/y_i}(\{\ell,y_j\}) = 2$ where $\{i,j\} = \{1,2\}$.
Suppose $r_{M/y_i}(\{\ell\}) = 2$. Assume $y_j \in K$. Then $(J \cup y_j, K- y_j)$ is a $2$-separation\ of $M/y_i$ unless $K-y_j$ consists of a single point. In the exceptional case, $y_j$ is in a parallel pair of points in $M/y_i$. Hence $M\backslash y_j$ has a special $N$-minor. As $M/y_j$ also has such a minor, we contradict Lemma~\ref{Step0}. We deduce that we may assume that $J$ contains $\{\ell,y_j\}$. Then $r(J \cup y_i) + r(K \cup y_i) = r(M) + 2$, so $r(J \cup y_i,K)$ is a $2$-separation\ of $M$; a contradiction.
We may now assume that $r_{M/y_i}(\{\ell\}) = 1$. Then $y_i$ lies on the line $\ell$. Since this must be true for each $i$ in $\{1,2\}$, we see that $r(\{\ell,y_1,y_2\}) = 2$; a contradiction. We deduce that \ref{3lines1} holds.
By Lemma~\ref{pre3lines}, we know that $(A,B)$ crosses $(X_1,Y_1)$, that $\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$, and that $Y_1 \cap B$ consists of a single line. As the lemma fails, $A \cap (Y_1 - \ell)$ consists of a single point, $a$. As $\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$ and $\lambda_{M\backslash \ell}(A) = 1$, we deduce that $r(A-a) = r(A)$ and $r(B \cup a) = r(B) + 1$. Hence $a \in {\rm cl}(X_1)$. Thus $Y_1 - a$ is a minimal non-$N$-$3$-separator; a contradiction.
\end{proof}
The next lemma verifies (\ref{Step5.5}).
\begin{lemma}
\label{y13}
Let $Y_1$ be a minimal-non-$N$-$3$-separator having exactly three elements. Then $Y_1$ consists of three lines.
\end{lemma}
\begin{proof}
As $|Y_1 \cap E(N)| \le 1$, at least two of the elements of $Y_1$ are not in $E(N)$. Let $\ell$ be one of these elements. Suppose $\ell$ is a line. If $M\backslash \ell$ has $N$ as a c-minor, then the result follows by Lemma~\ref{3lines}. If $M/\ell$ has $N$ as a c-minor, then $(M^*\backslash \ell)^{\flat}$, and hence
$M^*\backslash \ell$ has $N^*$ as a c-minor and again the result follows by Lemma~\ref{3lines}.
We may now assume that $\ell$ is a point. By switching to the dual if necessary, we may assume that $M\backslash \ell$ has $N$ as a c-minor. Let $(A,B)$ be a $2$-separation\ of $M\backslash \ell$ where $A$ is the $N$-side and $|B| \ge 3$. Next we show that
\begin{sublemma}
\label{ellisnot}
$\ell \notin {\rm cl}(Y_1 - \ell)$.
\end{sublemma}
Assume $\ell \in {\rm cl}(Y_1 - \ell)$. Then, by Lemma~\ref{pre3lines}, we know that $(A,B)$ crosses $(X_1,Y_1- \ell)$, that $\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$, and that $Y_1 \cap B$ consists of a single line, say $m$. Now $|B \cap X_1| \ge 2$ since $|B| \ge 3$. Then
$$\lambda_{M\backslash \ell}(B \cap X_1) \ge 2$$
otherwise, since $\ell \in {\rm cl}(Y_1 - \ell)$, we deduce that $\lambda_{M}(B \cap X_1) \ge 1$; a contradiction.
By uncrossing, $\lambda_{M\backslash \ell}(Y_1 \cap A) \le 1$.
Since $|Y_1| = 3$ and $Y_1 \cap B$ consists of the line $m$, we deduce that $A \cap (Y_1 - \ell)$ consists of a single point, say $a$, otherwise one of the elements of $Y_1 - \ell$ is a line that is not in $E(N)$ and we have already dealt with that case. As $\lambda_{M\backslash \ell}(X_1 \cap A) \ge 2$ and $\lambda_{M\backslash \ell}(A) = 1$, we deduce that
\begin{equation}
\label{aab}
r(A-a) = r(A) \text{~~and~~} r(B \cup a) = r(B) + 1.
\end{equation}
Hence
\begin{equation}
\label{ax1}
a \in {\rm cl}(X_1).
\end{equation}
We may assume that $m \in E(N)$ otherwise $m$ is removed in forming $N$ and that case was dealt with in the first paragraph.
Now $Y_1 = \{a,\ell,m\}$. As $m \in B$, it follows by (\ref{aab}) that $r(\{m,a\}) = 3$. Moreover, as $\{m,a\} = Y_1 - \ell$ and $\ell \in {\rm cl}(Y_1 - \ell)$, we deduce that $r(Y_1) = 3$. By (\ref{ax1}), $r(X_1 \cup a) = r(X_1)$. We deduce that
\begin{equation}
\label{lmn}
r(\{a,\ell,m\}) = r(\{\ell,m\}) = 3 \text{~~and~~} r(X_1 \cup a) = r(M) - 1.
\end{equation}
Since $m \in E(N)$, it follows that $a \not\in E(N)$. Suppose that $M\backslash \ell /a$ has $N$ as a c-minor. Still as part of the proof of \ref{ellisnot}, we show next that
\begin{sublemma}
\label{am2s}
$M/a$ is the $2$-sum with basepoint $q$ of two $2$-polymatroids, one of which consists of the line $m$ having non-parallel points $q$ and $\ell$ on it.
\end{sublemma}
By (\ref{lmn}), $(\{\ell,m\},X_1)$ is a 2-separation of $M/a$. Thus $M/a$ is the 2-sum with basepoint $q$ of two $2$-polymatroids, one of which, $Q$ say, consists of the line $m$ having points $q$ and $\ell$ on it. Suppose $q$ and $\ell$ are parallel points in $Q$. Then $(\{m\},X_1 \cup \ell)$ is a 2-separation of $M/a$. It follows that
$(\{m\},X_1 \cup \ell \cup a)$ is a 2-separation of $M$; a contradiction. Thus \ref{am2s} holds.
By \ref{am2s}, both $M/\ell$ and $M\backslash \ell$ have $N$ as a c-minor; a contradiction\ to Lemma~\ref{Step0}.
We now know that $N$ is a c-minor of $M\backslash \ell\backslash a$. In that $2$-polymatroid, $\{m\}$ is $2$-separating so, in the formation of $N$, the element $m$ is compactified. As the next step towards showing \ref{ellisnot}, we now show that
\begin{sublemma}
\label{m1m'}
$M{\underline{\downarrow}\,} m$ is $3$-connected.
\end{sublemma}
To see this, it will be helpful to consider the $2$-polymatroid $M_1$ that is obtained from $M$ by freely adding the point $m'$ on $m$. By definition, $M{\underline{\downarrow}\,} m = M_1/m_1$. Certainly $M_1$ is $3$-connected, so $M_1/m'$ is $2$-connected. Assume it has a $2$-separation\ $(U,V)$ where $m \in U$. Then
$$r(U \cup m') + r(V \cup m') - r(M_1) = 2.$$
But $r(U \cup m') = r(U).$ Hence $r(V \cup m') = r(V)$ otherwise $M_1$ has a $2$-separation; a contradiction. But, as $m'$ was freely placed on $m$, we deduce that
$r(V\cup m' \cup m) = r(V \cup m') = r(V)$. Now, in $M_1\backslash m$, we see that $\{\ell,m'\}$ is a series pair of points. As $m' \in {\rm cl}(V)$, it follows that $\ell \in V$. Then $r(U - m) < r(U)$ since $\{m\}$ is 2-separating in $M\backslash \ell$. Now $r(U - m) = r(U) - 1$ otherwise $r(U - m) = r(U) - 2$ and
$(U-m,V\cup \{m',m\})$ is a $1$-separation of $M_1$. As $(U-m,V\cup \{m',m\})$ is not a $2$-separation of $M_1$, it follows that $U - m$ consists of a single point $u$ and $r(\{u,m\}) = 2$. Thus, in $M\backslash \ell$, when we compactify $m$, we find that $u$ and $m$ are parallel. Since $m\in E(N)$, we see that $u \not\in E(N)$. Moreover, $M\backslash u$ has $N$ as a c-minor. Since $u$ lies on $m$ in $M$, we deduce that $M\backslash u$ is $3$-connected\ having $N$ as a c-minor. This contradiction\ completes the proof of \ref{m1m'}.
Now, in $M_1/m'$, the elements $a, \ell$, and $m$ form a triangle of points. We know that $M_1/m'\backslash \ell$ is not $3$-connected\ otherwise $(M\backslash \ell)^{\flat}$ is $3$-connected\ having $N$ as a c-minor. Because $M\backslash a$ has $N$ as a c-minor, $M\backslash a$ is not $3$-connected, so $M_1 \backslash a$ is not $3$-connected. Still continuing with the proof of
\ref{ellisnot}, we show next that
\begin{sublemma}
\label{m1m'a}
$M_1\backslash a/m'$ is not $3$-connected.
\end{sublemma}
Let $(G,H)$ be a $2$-separation of $M_1 \backslash a$ with $m$ in $G$. Then $(G \cup m',H-m')$ is a $2$-separation\ of $M_1\backslash a$ unless $H$ consists of two points. In the exceptional case, $r(H) = 2$ so $r(G) = r(M) - 1$. But then $a \cup (H - m')$ is a series pair in $M$; a contradiction. We conclude that we may assume that $m' \in G$. Then $\ell \in H$, otherwise, by (\ref{lmn}), $(G \cup a, H)$ is a $2$-separation\ of $M_1$; a contradiction.
Observe that $G \neq \{m,m'\}$ otherwise $\{m\}$ is 2-separating in $M\backslash a$ and so, as $a \in {\rm cl}(X_1)$ , we obtain the contradiction\ that $\{m\}$ is 2-separating in $M$.
Now
\begin{equation}
\label{m1m1m1}
r_{M_1 \backslash a/m'}(G - m') + r_{M_1 \backslash a/m'}(H) - r(M_1 \backslash a/m')= r(G) + r(H \cup m') - 1 - r(M_1\backslash a).
\end{equation}
Suppose that $r(H \cup m') = r(H)$. Then $r(H \cup m'\cup m) = r(H)$ as $m'$ is freely placed on $m$. Thus, as $G \supsetneqq \{m,m'\}$ and $\{m\}$ is 2-separating in $M\backslash \ell \backslash a$, we see that $(G- m- m', H \cup \{m,m'\})$ is a 1-separation of $M_1\backslash a$. Therefore $(G- m- m', H \cup \{m,a\})$ is a 1-separation of $M$; a contradiction.
We now know that $r(H \cup m') = r(H) + 1$. Then, as $(G,H)$ is a $2$-separation\ of $M_1 \backslash a$, it follows by (\ref{m1m1m1}) that $(G-m',H)$ is a $2$-separation\ of $M_1 \backslash a /m'$ unless either $|H| = 1$ and $r_{M_1/m'}(H) = 1$, or $|G-m'| = 1$ and $r_{M_1/m'}(G-m') = 1$. Consider the exceptional cases. The first of these cannot occur since $m'$ is freely placed on $m$; the second cannot occur since it implies that $G = \{m,m'\}$, which we eliminated above. As neither of the exceptional cases occurs, $M_1 \backslash a /m'$ has a 2-separation and so
\ref{m1m'a} holds.
Recall that $M_1/m' = M{\underline{\downarrow}\,} m$. In this
$2$-polymatroid, we have $\{a, \ell,m\}$ as a triangle such that the deletion of either $a$ or $\ell$ destroys $3$-connectedness. Hence, by \cite[Lemma 4.2]{oswww}, there is a triad of $M_1/m'$ that contains $a$ and exactly one of $\ell$ and $m$. Assume this triad contains $\ell$.
Thus, in $M\backslash \ell{\underline{\downarrow}\,} m$, we have that $a$ is in a series pair with some element $b$. Then $M\backslash \ell/a$ has $N$ as a c-minor, so $a$ is a doubly labelled point of $M$; a contradiction\ to Lemma~\ref{Step0}. We deduce that $M_1/m'$ has a triad containing $\{a,m\}$ but not $\ell$.
Then $M_1/m'\backslash \ell$, which equals $M\backslash \ell{\underline{\downarrow}\,} m$, either has a triad containing $\{a,m\}$ or has $a$ in a series pair. This is straightforward to see by considering the matroid that is naturally derived from $M\backslash \ell{\underline{\downarrow}\,} m$ and using properties of the cocircuits in this matroid. Now $a$ is not in a series pair in $M\backslash \ell{\underline{\downarrow}\,} m$ otherwise we again obtain the contradiction\ that $a$ is a doubly labelled point. We deduce that $M\backslash \ell{\underline{\downarrow}\,} m$ has a triad containing $\{a,m\}$. Since $m \in B$ and, by (\ref{aab}), $r(A - a) = r(A)$, we must have that the third point, $b$, of this triad is in $A-a$.
Now $M\backslash \ell {\underline{\downarrow}\,} m$ has $(A,B)$ as a $2$-separation and has $\{a,b,m\}$ as a triad with $\{a,b\} \subseteq A$. Thus
$(A \cup m,B - m)$ is a $2$-separation\ of $M\backslash \ell {\underline{\downarrow}\,} m$. Since $\ell$ is in the triangle $\{a,m,\ell\}$ in $M {\underline{\downarrow}\,} m$, it follows that $(A \cup m\cup \ell,B - m)$ is a $2$-separation\ of $M {\underline{\downarrow}\,} m$. This contradiction\ to \ref{m1m'} completes the proof of \ref{ellisnot}.
Since $\ell \not\in {\rm cl}(Y_1 - \ell)$, we deduce that $(X_1\cup \ell,Y_1 - \ell)$ is 3-separating in $M$. Because $Y_1$ is a minimal non-$N$-3-separating set, $Y_1 - \ell$ does not consist of two lines. Moreover, $(X_1,Y_1 - \ell)$ is a 2-separation in $M\backslash \ell$.
\begin{sublemma}
\label{pointline}
$Y_1 - \ell$ does not consist of a point and a line.
\end{sublemma}
Assume that $Y_1- \ell$ consists of a line $k$ and a point $y$. If $k \not\in E(N)$, then the argument in the first paragraph of the proof of the lemma gives a contradiction. Thus $k \in E(N)$, so $y \not\in E(N)$. If $r(Y_1 - \ell) = 2$, then $M\backslash \ell \backslash y$, and hence $M\backslash y$, has $N$ as a c-minor. Since $y$ is on the line $k$, we see that $M\backslash y$ is $3$-connected; a contradiction. We deduce that $r(Y_1 - \ell) = 3$. Hence
\begin{equation}
\label{x1rk}
r(X_1) = r(M) - 2 \text{~~and~~} r(X_1 \cup \ell) = r(M) - 1.
\end{equation}
Now $M\backslash \ell$ is the 2-sum with basepoint $p$, say, of two $2$-polymatroids, $M_X$ and $M_Y$, with ground sets $X_1 \cup p$ and $(Y_1 - \ell) \cup p$, respectively. Then $r(M_Y) = 3$. Moreover, $y$ does not lie on $k$ in $M_Y$, otherwise $M_Y$ is not $2$-connected, a contradiction\ to Proposition~\ref{connconn}.
Thus $M^*\backslash y$ has $N^*$ as a c-minor. Then, by applying \ref{ellisnot} to $M^*\backslash y$, we deduce that $y \not\in {\rm cl}_{M^*}(Y_1 - y)$. Thus
$r^*(Y_1 - y) = r^*(Y_1) - 1$. It follows that $r(X_1 \cup \ell \cup y) = r(X_1 \cup \ell)$. But $r(X_1 \cup \ell \cup y) = r(M\backslash k) = r(M)$ yet $r(X_1 \cup \ell) = r(M) - 1$. This contradiction\ completes the proof of \ref{pointline}
We now know that $Y_1 - \ell$ consists of a series pair of points, say $y_1$ and $y_2$. Now $r(M\backslash \ell) = r(M) = r(X_1) + 1$. Also $r(\{\ell,y_1,y_2\}) = 3$. Thus $\{\ell,y_1,y_2\}$ is a triad of $M$. Moreover, both $M/y_1$ and $M/y_2$ have special $N$-minors. Thus neither is $3$-connected. By \cite[Lemma 4.2]{oswww}, $M$ has a triangle that contains $y_1$ and exactly one of $y_2$ and $\ell$. Likewise, $M$ has a triangle that contains $y_2$ and exactly one of $y_1$ and $\ell$. Thus either
\begin{itemize}
\item[(i)] $M$ has a triangle $\{y_1,y_2,z\}$; or
\item[(ii)] $M$ has triangles $\{y_1,\ell,z_1\}$ and $\{y_2,\ell,z_2\}$ but no triangle containing $\{y_1,y_2\}$.
\end{itemize}
In the first case, $M/y_1$ has $\{y_2,z\}$ as a pair of parallel points. Hence $M\backslash y_2$ has a special $N$-minor. Thus $y_2$ is doubly labelled; a contradiction. We deduce that (ii) holds. Thus $M$ contains a fan $x_1,x_2,\dots,x_n$ where $(x_1,x_2,x_3,x_4,x_5) = (z_2,y_2,\ell,y_1,z_1)$. Hence $M/x_2$ has a c-minor isomorphic to $N$. Then, by Lemmas~\ref{fantan} and \ref{Step0}, we obtain a contradiction.
\end{proof}
We complete the proof of Lemma~\ref{Step6} by analyzing the various possibilities for a minimal non-$N$-3-separator consisting of exactly three lines.
\section{A minimal non-$N$-3-separator consisting of exactly three lines}
\label{threeel}
In this section, we finish the proof of (\ref{Step6}). We begin by restating that assertion.
\begin{lemma}
\label{Step6+}
$M$ has a minimal non-$N$-$3$-separator with at least four elements.
\end{lemma}
We have $(X_1,Y_1)$ as a $3$-separation of the $3$-connected\ $2$-polymatroid $M$. We shall consider the extension $M+z$ of $M$ that is obtained by adjoining the line $z$ to $M$ so that $z$ is in the closure of each of $X_1$ and $Y_1$ in $M+z$. To see that this extension exists, we note that, by building on a result of Geelen, Gerards, and Whittle~\cite{ggwtconn}, Beavers~\cite[Proposition~2.2.2]{beavs}
showed that, when $(A,B)$ is a $3$-separation\ in a $3$-connected\ matroid $Q$, we can extend $Q$ by an independent set $\{z_1,z_2\}$ of size two so that these two points are clones, and each lies in the closure of both $A$ and $B$ in the extension $Q'$.
By working in the matroid naturally derived from $M$, we can add $z_1$ and $z_2$. This corresponds to adding the line $z$ to $M$ to form $M+z$ where $z = \{z_1,z_2\}$.
More formally, recall that the natural matroid $M'$ derived from $M$ is obtained from $M$ by freely adding two points, $s_{\ell}$ and $t_{\ell}$, on each line $\ell$ of $M$ and then deleting all such lines $\ell$. After we have extended $M'$ by $z_1$ and $z_2$, we have a matroid with points $\{z_1,z_2\} \cup \{p:\text{~$p$ is a point of $M$}\} \cup \{s_{\ell},t_{\ell}:\text{~$\ell$ is a line of $M$}\}$. Taking $z= \{z_1,z_2\}$, we see that $M+ z$ is the $2$-polymatroid with elements $\{z\} \cup \{p:\text{~$p$ is a point of $M$}\} \cup \{\ell:\text{~$\ell$ is a line of $M$}\} = \{z\} \cup E(M)$. We call $M+z$ the $2$-polymatroid that is obtained from $M$ by {\it adding the guts line $z$ of $(X_1,Y_1)$.}
When we have $Y_1$ as a minimal non-$N$-$3$-separator of $M$ consisting of three lines, we look at $(M+z)|(Y_1 \cup z)$. This $2$-polymatroid consists of exactly four lines.
\begin{lemma}
\label{claim1y1}
$(M+z)|(Y_1 \cup z)$ has no parallel lines, so $r_{M+z}(Y_1 \cup z) \ge 3$.
\end{lemma}
\begin{proof}
Suppose $a$ and $b$ are parallel lines in $Y_1$. Then we may assume that $b\not\in E(N)$. Now $M\backslash b$ or $M/b$ has $N$ as a c-minor. In the latter case, as $a$ is a loop of $M/b$, it follows that $a \not \in E(N)$ and $M\backslash a$ has $N$ as a c-minor. We conclude that $M\backslash b$ or $M\backslash a$ has $N$ as a c-minor. Since each of $M\backslash b$ and $M\backslash a$ is $3$-connected, we obtain the contradiction\ that the theorem holds. Thus $Y_1$ contains no pair of parallel lines.
Suppose $z$ is parallel to some element $y$ of $Y_1$. Then $(X_1 \cup y, Y_1 - y)$ is a non-$N$-$3$-separator of $M$ contradicting the minimality of $Y_1$. Thus $(M+z)|(Y_1 \cup z)$ has no parallel lines and the lemma holds.
\end{proof}
\begin{lemma}
\label{claim3}
$r(Y_1) > 3$.
\end{lemma}
\begin{proof}
Assume that
$r(Y_1) = 3$. Then $r_{M+z}(Y_1 \cup z) = 3$, so $\sqcap(z,y) = 1$ for all $y$ in $Y_1$. Moreover, $r(Y_1 - y) = 3 = r(Y_1)$ for all $y$ in $Y_1$.
Suppose that $y \in Y_1 - E(N)$ and $N$ is a c-minor of $M/y$. Then the remaining two elements, $y_1$ and $y_2$, of $Y$ are parallel points in $M/y$. We may assume that $y_1 \not \in E(N)$. Thus $M\backslash y_1$ has $N$ as a c-minor. We conclude that $N$ is a c-minor of $M\backslash y$ for some element $y$ of $Y_1$. We now focus on this element $y$.
Let $(R,G)$ be a non-trivial $2$-separation\ of $M\backslash y$, that is, $\lambda_{M\backslash y}(R) = 1$ and $\min\{|R|,|G|\} \ge 2$. We show next that
\begin{sublemma}
\label{claim4}
$(R,G)$ crosses $(X_1,Y_1 - y)$.
\end{sublemma}
If $R \subseteq X_1$, then $G \supseteq Y_1 - y$ so $y \in {\rm cl}_M(G)$ and $(R,G \cup y)$ is a $2$-separation\ of $M$. This contradiction\ implies, using symmetry, that both $R$ and $G$ meet $Y_1 - y$.
Suppose $R \cap X_1 = \emptyset$. Then $R$ consists of single line, so $(R,G)$ is a trivial $2$-separation. This contradiction, combined with symmetry, completes the proof of \ref{claim4}.
Let $Y_1 - y = \{a,b\}$. We may assume that $a \in R$ and $b \in G$. Now, as $y \in {\rm cl}(Y_1 - y)$, we see that $\lambda_{M\backslash y}(Y_1 - y) = 2$. Thus
\begin{align*}
1+2 & = \lambda_{M\backslash y}(R) + \lambda_{M\backslash y}(Y_1 - y)\\
& \ge \lambda_{M\backslash y}(\{a\}) + \lambda_{M\backslash y}(R\cup (Y_1 - y))\\
& = \lambda_{M\backslash y}(\{a\}) + \lambda_{M\backslash y}(G \cap X_1).
\end{align*}
We know $r(E- Y_1) = r(X_1) = r(M) - 1$ since $r(Y_1) = 3$. Thus $r(E - \{y,a\}) = r(X_1 \cup b) = r(M)$. Hence
$$ \lambda_{M\backslash y}(\{a\}) = r(\{a\}) + r(E - \{y,a\}) - r(E - y) = r(\{a\}) = 2,$$
so $\lambda_{M\backslash y}(G\cap X_1) \le 1$. But $y \in {\rm cl}(\{a,b\})$ so $\lambda_M(G \cap X_1) \le 1$. By symmetry, $\lambda_M(R \cap X_1) \le 1$. We conclude that $|G \cap X_1| \le 1$ and $|R \cap X_1| \le 1$, so $|X_1| \le 2$. This is a contradiction\ since $|E(N)| \ge 4$. We conclude that the lemma holds.
\end{proof}
\begin{lemma}
\label{claim5}
$r_{M+z}(Y_1 \cup z) =r_M(Y_1) = 4$.
\end{lemma}
\begin{proof} We know that $r_{M+z}(Y_1 \cup z) = r_M(Y_1) \ge 4$. Suppose $r_M(Y_1) \ge 5$. Then $r_M(X_1) \le r(M) - 3$, so
$$r_{M^*}(Y_1) = \sum_{y \in Y_1} r_M(\{y\}) + r_M(X_1) - r(M) \le 6 + r(M) - 3 - r(M) = 3.$$ By using $M^*$ in place of $M$, we get a contradiction\ to Lemma~\ref{claim3}. We conclude that the lemma holds.
\end{proof}
We will now work with the $2$-polymatroid $(M+z)|(Y_1 \cup z)$, which we rename $P$. This has rank 4 and consists of four lines, $z, a,b,$ and $c$.
\begin{lemma}
\label{claim5.5}
If $B \subseteq Y_1$ and $A = Y_1 - B$, then
$$\sqcap_P(A \cup z, B) = \sqcap_{M+z}(A \cup X_1 \cup z, B).$$
\end{lemma}
\begin{proof} Since $P = (M+z)|(Y_1 \cup z)$, we can do all of these local connectivity calculations in $M+z$. Now
$ \sqcap(A \cup z, X_1) = \sqcap(A \cup z, X_1\cup z)$, so
$$2 = \sqcap(Y_1 \cup z, X_1) \ge \sqcap(A \cup z, X_1) = \sqcap(A \cup z, X_1\cup z) \ge 2.$$
Thus
$$r(A \cup z) - 2 = r(A \cup z \cup X_1) - r(X_1).$$
Hence
\begin{align*}
\sqcap(A \cup z, B) & = r(A \cup z) + r(B) - r(A\cup z \cup B)\\
& = r(A \cup z \cup X_1) - r(X_1) + 2 + r(B) - r(Y_1)\\
& = r(A \cup z \cup X_1) + r(B) - [r(X_1) + r(Y_1) - 2]\\
& = r(A \cup z \cup X_1) + r(B) - r(M)\\
& = \sqcap(A \cup X_1 \cup z, B).
\end{align*}
\end{proof}
\begin{lemma}
\label{claim6}
$P$ is $3$-connected.
\end{lemma}
\begin{proof} From the last lemma, if $(A,B)$ is a $k$-separation of $P$ for some $k$ in $\{1,2\}$ and $z \in A$, then $ (A\cup X_1\cup z,B)$ is a $k$-separation of $M+z$; a contradiction.
\end{proof}
\begin{lemma}
\label{3one}
If $y \in Y_1$ and $\sqcap(X_1,\{y\}) = 1$, then $r(Y_1 - y) = 4$.
\end{lemma}
\begin{proof}
By Lemma~\ref{claim1y1}, $r(Y_1 - y) > 2$. If $r(Y_1 - y) = 3$, then $(X_1 \cup y, Y_1 - y)$ is a $3$-separation\ violating the choice of $(X_1,Y_1)$.
\end{proof}
\begin{lemma}
\label{3m}
Suppose $y \in Y_1$ and $r(Y_1 - y) = 4$. If $m$ is a line such that $\{m\}$ is $2$-separating in $M\backslash y$, then $m \in Y_1 - y$.
\end{lemma}
\begin{proof}
We have $1 = r(\{m\}) + r(E - \{y,m\}) - r(M\backslash y)$. Thus $r(E - \{y,m\}) = r(M) - 1$. Suppose $m \not\in Y_1 - y$. Then $E- \{y,m\}$ contains $Y_1 - y$ and so spans $y$. Thus $r(E - \{y,m\}) = r(M\backslash m) = r(M)$; a contradiction.
\end{proof}
The next four lemmas will help eliminate many of the possibilities for $P$.
\begin{lemma}
\label{parallel}
If $c$ is skew to $X_1$ in $M$, and $M/c$ has $a$ and $b$ as parallel lines, then $M/c$ is $3$-connected.
\end{lemma}
\begin{proof}
Assume $(A,B)$ is a $k$-separation of $M/c$ for some $k$ in $\{1,2\}$ where $|A| \le |B|$. If $\{a,b\} \subseteq Z$ for some $Z$ in $\{A,B\}$, and $\{Z,W\} = \{A,B\}$, then $r(Z \cup c) + r(W \cup c) - r(M) = k+1$. But $c$ is skew to $W$ since $W \subseteq X_1$, so $(Z \cup c,W)$ is a $k$-separation of $M$; a contradiction. We may now assume that $a \in A$ and $b \in B$. Then $(A \cup b, B-b)$ is a $k$-separation of $M/c$ with $\{a,b\} \subseteq A \cup b$ and this possibility has already been eliminated.
\end{proof}
\begin{lemma}
\label{earlier}
If $c$ is skew to each of $a$, $b$, and $X_1$ in $M$, then $M/c$ has no c-minor isomorphic to $N$.
\end{lemma}
\begin{proof}
We see that $M/c$ has $a$ and $b$ as parallel lines. Since $(M,N)$ is a counterexample to the theorem, we obtain this lemma as a direct consequence of the last one.
\end{proof}
\begin{lemma}
\label{earlybird}
Assume that $M\backslash b$ has a c-minor isomorphic to $N$ and that $P\backslash b$ has rank $4$, has $c$ skew to each of $a$ and $z$, and has $\sqcap(\{a\},\{z\}) = 1$. Then $M/c$ has a c-minor isomorphic to $N$.
\end{lemma}
\begin{proof}
Let $(A,C)$ be a non-trivial $2$-separation\ of $M\backslash b$. If $\{a,c\}$ is contained in $A$ or $C$, then $M$ has a $2$-separation; a contradiction. Thus we may assume that $a \in A$ and $c \in C$. Now $c$ is skew to $C - c$ so $(A \cup c, C-c)$ is 2-separating in $M\backslash b$. Hence $(A \cup c \cup b, C - c)$ is 2-separating in $M$. Thus $C - c$ consists of a point $d$ of $M$. Now, by Lemma~\ref{3m}, the only $2$-separating lines in $M\backslash b$ can be $a$ and $c$. But $a$ is not 2-separating. Thus $(M\backslash b)^{\flat} = M\backslash b{\underline{\downarrow}\,} c$, so $c$ is a point of $M\backslash b{\underline{\downarrow}\,} c$. The rank of this $2$-polymatroid is $r(M) - 1$, and it has $\{c,d\}$ as a series pair since $A$ has rank $r(M) - 2$ in it. Thus $M\backslash b{\underline{\downarrow}\,} c/c$, and hence $M/c$, has a c-minor isomorphic to $N$.
\end{proof}
\begin{lemma}
\label{earlybird2}
If $\sqcap(\{a\},\{z\}) = 1$ and both $b$ and $c$ are skew to each other and to $z$, then $M\backslash a$ has no c-minor isomorphic to $N$.
\end{lemma}
\begin{proof} Assume that $M\backslash a$ has a c-minor isomorphic to $N$. Let $(B,C)$ be a $k$-separation of $M\backslash a$ for some $k$ in $\{1,2\}$. If $B$ or $C$ contains $\{b,c\}$, then $M$ has a $k$-separation. Thus we may assume that $b \in B$, that $c \in C$, and that $|B| \ge |C|$. Then $b$ is skew to $B- b$, so the partition $(B-b,C \cup b \cup a)$ of $E(M)$ shows that $M$ is not $3$-connected; a contradiction.
\end{proof}
By Lemma~\ref{claim1y1}, for all $y$ in $Y_1$, we have $\sqcap(\{y\},\{z\}) \in \{0,1\}$. We shall treat the possibilities for $P$ based on the number $\theta$ of
members $y$ of $Y_1$ for which $\sqcap(\{y\},\{z\}) = 1$. The most difficult case is when $\theta = 3$ and we will treat that after we deal with the cases when $\theta=2$ and when $\theta = 1$.
\begin{lemma}
\label{3two}
$\theta \neq 2$.
\end{lemma}
\begin{proof}
Suppose that $\sqcap(\{a\},\{z\}) = 1 = \sqcap(\{b\},\{z\})$ and $\sqcap(\{c\},\{z\}) = 0$. Then, by Lemma~\ref{3one}, $r(\{b,c\}) = 4 = r(\{a,c\})$. Thus, by Lemma~\ref{earlier}, $M/c$ has no c-minor isomorphic to $N$. By Lemma~\ref{earlybird}, neither $M\backslash a$ nor $M\backslash b$ has a c-minor isomorphic to $N$. Thus, without loss of generality, we may assume that $M/a$ has a c-minor isomorphic to $N$. Now, in $M/a$, we have $\{b,c\}$ as a $2$-separating set where $c$ is a line and $b$ is either a point on that line or is a parallel line. Thus, by Lemma~\ref{claim1}, $M/a\backslash b$, and hence $M\backslash b$, has a c-minor isomorphic to $N$; a contradiction.
\end{proof}
We can exploit duality to eliminate the case when $\theta = 1$.
\begin{lemma}
\label{3three}
$\theta \neq 1$.
\end{lemma}
\begin{proof}
Suppose that $\sqcap(\{a\},\{z\}) = 1$ and $\sqcap(\{b\},\{z\}) = 0 = \sqcap(\{c\},\{z\})$. Then, by Lemma~\ref{3one}, $r(\{b,c\}) = 4$.
By Lemma~\ref{general4}, for $y$ in $Y_1$, we have $\sqcap^*(\{y\}, X_1) = \lambda_{M/(Y-y_1)}(\{y\})$. Since $\{b,c\}$ spans $a$ in $M$, we deduce that $\sqcap^*(\{a\},X_1) = 0$. If $\sqcap^*(\{b\},X_1) = 1 = \sqcap^*(\{c\},X_1)$, then $\theta = 2$ in $M^*$ so the result follows by Lemma~\ref{3two}. Thus, we may assume, by symmetry, that $\sqcap^*(\{b\},X_1) = 0$. Hence $\{a,c\}$ spans $b$ in $M$, so $r(\{a,c\}) = 4$. Thus, by Lemma~\ref{earlier}, $M/c$ does not have a c-minor isomorphic to $N$. By Lemma~\ref{earlybird}, $M\backslash b$ has no c-minor isomorphic to $N$. If $\sqcap(\{a\},\{b\}) = 0$, then, by symmetry, the argument of the last two sentences shows that neither $M/b$ nor $M\backslash c$ has a c-minor isomorphic to $N$. Thus both $b$ and $c$ must be in every c-minor of $M$ isomorphic to $N$; a contradiction. We deduce that $\sqcap(\{a\},\{b\}) = 1$.
By Lemma~\ref{earlybird2}, $M\backslash a$ has no c-minor isomorphic to $N$. Suppose $M/a$ has a c-minor isomorphic to $N$. In $M/a$, we see that $\{c,b\}$ is a 2-separating set with $c$ as a line and $b$ as a point on it. Hence, by Lemma~\ref{claim1}, $M/a\backslash b$, and so $M\backslash b$, has a c-minor isomorphic to $N$; a contradiction. We conclude that $M/a$ has no c-minor isomorphic to $N$. It follows that $a$ is in every c-minor of $M$ isomorphic to $N$. Thus $M/b$ has $N$ as a c-minor. In $M/b$, we see that $a$ is a point on the line $c$. Suppose that $a$ is parallel to some point $e$, say. Then $e \in X_1$. Moreover, $M/b\backslash e$, and hence $M\backslash e$, has a c-minor isomorphic to $N$. Now $r(X_1 \cup \{a,b\}) = r(X_1) + 2$. Thus
\begin{align*}
r(X_1) + 1 + 3 & = r(X_1 \cup a) + r(\{a,b,e\})\\
& \ge r(\{a,e\}) + r(X_1 \cup \{a,b\})\\
& = r(\{a,e\}) + r(X_1) + 2.
\end{align*}
Hence $r(\{a,e\}) = 2$, so
$e$ lies on $a$ in $M$. Thus $M\backslash e$ is $3$-connected\ having a c-minor isomorphic to $N$; a contradiction. We deduce that, in $M/b$, the point $a$ is not parallel to another point, so $M/b$ is simple.
We complete the proof by showing that $M/b$ is $3$-connected. Suppose it has $(A,C)$ as a $2$-separation.
If $A$ or $C$, say $A$, contains $\{a,c\}$, then $b$ is skew to $C$, so $(A\cup b,C)$ is a $2$-separation of $M$; a contradiction. Thus, we may assume that $a \in A$ and $c \in C$. Then, as $a$ is a point on the line $c$ in $M/b$, we see that $(A - a,C\cup a)$ is 2-separating in $M/b$. It is not a 2-separation otherwise we obtain a contradiction\ as before. It follows that $A$ is a parallel pair of points in $M/b$, contradicting the fact that $M/b$ is simple.
\end{proof}
Next we eliminate the case when $\theta = 3$. The core of the argument in this case mimics the argument used to prove Tutte's Triangle Lemma for matroids (see, for example, \cite[Lemma 8.7.7]{oxbook}).
\begin{lemma}
\label{3five}
$\theta \neq 3$.
\end{lemma}
\begin{proof}
Assume that $\sqcap(\{a\},\{z\}) = \sqcap(\{b\},\{z\}) = \sqcap(\{c\},\{z\}) = 1$. Then, by Lemma~\ref{3one}, $r(\{a,b\}) = r(\{b,c\}) = r(\{a,c\}) = 4$. First we show the following.
\begin{sublemma}
\label{two2}
There are at least two members $y$ of $Y_1$ such that $M\backslash y$ has a c-minor isomorphic to $N$.
\end{sublemma}
Assume that this fails. Since $|Y_1 - E(N)| \ge 2$, there is an element, say $a$, of $Y_1 - E(N)$ such that $M/a$ has $N$ as a c-minor. In $M/a$, we see that $b$ and $c$ are parallel lines and $\{b,c\}$ is 2-separating. Thus, by Lemma~\ref{claim1}, each of $M/a\backslash b$ and $M/a\backslash c$ have special $N$-minors. This contradiction\ implies that \ref{two2} holds.
We now assume that both $M\backslash a$ and $M\backslash b$ have special $N$-minors. Clearly, $M\backslash a$ has $b$ and $c$ as 2-separating lines, and, by Lemma~\ref{3m}, these are the only 2-separating lines in $M\backslash a$. Thus $(M\backslash a)^{\flat} = M\backslash a {\underline{\downarrow}\,} b {\underline{\downarrow}\,} c$. Symmetrically,
$(M\backslash b)^{\flat} = M\backslash b {\underline{\downarrow}\,} a {\underline{\downarrow}\,} c$. As the theorem fails, neither $(M\backslash a)^{\flat}$ nor $(M\backslash b)^{\flat}$ is $3$-connected. Thus each of $M{\underline{\downarrow}\,} c \backslash a$ and
$M{\underline{\downarrow}\,} c \backslash b$ have non-trivial 2-separations. It will be convenient to work in the $2$-polymatroid $M{\underline{\downarrow}\,} c$, which we shall rename $M_c$. Let $(X_a,Y_a)$ and $(X_b,Y_b)$ be non-trivial 2-separations of $M_c\backslash a$ and $M_c\backslash b$, respectively, with $b$ in $Y_a$ and $a$ in $Y_b$.
Now it is straightforward to check the following.
\begin{sublemma}
\label{zex}
If $Z \subseteq X_1$ and $e \in \{a,b\}$, then
$\sqcap_{M}(Z, \{e\}) = \sqcap_{M_c}(Z,\{e\})$.
\end{sublemma}
We deduce that
\begin{sublemma}
\label{two2.5}
$\sqcap_{M_c}(X_1,\{a\}) = 1 = \sqcap_{M_c}(X_1,\{b\})$.
\end{sublemma}
Next we show that
\begin{sublemma}
\label{two3}
$c \in X_a \cap X_b$.
\end{sublemma}
Suppose $c$ in $Y_a$. Since $\{c,b\}$ spans $a$ in $M_c$, it follows that $(X_a,Y_a \cup a)$ is a 2-separation of $M_c$ and hence of $M$; a contradiction. We deduce that $c \in X_a$ and, by symmetry, \ref{two3} holds.
\begin{sublemma}
\label{zztop}
For $Z \subseteq X_1$, if $\sqcap_M(Z,\{a\}) = 1 = \sqcap_M(Z,\{b\})$, then $\sqcap_M(Z,\{a,b\}) = 2.$
\end{sublemma}
Assume $\sqcap_M(Z,\{a,b\}) < 2.$ Then $\sqcap_M(Z,\{a,b\}) = \sqcap_M(Z,\{a\}) = 1$. Thus
$$r(Z) + r(\{a,b\}) - r(Z \cup \{a,b\}) = r(Z) + r(\{a\}) - r(Z \cup a),$$
so $r(\{a,b\}) - r(\{a\}) = r(Z \cup \{a,b\}) - r(Z \cup a)$. Hence $b$ is skew to $Z \cup a$, so $b$ is skew to $Z$; a contradiction. We deduce that
\ref{zztop} holds.
\begin{sublemma}
\label{zztop2}
For $Z \subseteq X_1$, if $\sqcap_{M_c}(Z,\{a\}) = 1 = \sqcap_{M_c}(Z,\{b\})$, then $\sqcap_{M_c}(Z,\{a,b\}) = 2.$
\end{sublemma}
By \ref{zex}, $\sqcap_{M_c}(Z,\{a\}) = \sqcap_{M}(Z,\{a\})$. Moreover,
\begin{align*}
\sqcap_{M_c}(Z,\{a,b\}) & = r_{M_c}(Z) + r_{M_c}(\{a,b\}) - r_{M_c}( Z \cup \{a,b\})\\
& = r_M(Z) + [r_M(\{a,b\}) - 1] - [ r_{M}( Z \cup \{a,b\}) -1]\\
& = \sqcap_{M}(Z,\{a,b\}).
\end{align*}
Thus \ref{zztop2} follows immediately from \ref{zztop}.
\begin{sublemma}
\label{zztop3}
Assume $Z \subseteq X_1$ and $\sqcap_{M_c}(Z,\{a,b\}) = 2.$ Then $c \in {\rm cl}_{M_c}(Z).$
\end{sublemma}
To see this, note that
$$r_{M_c}(Z \cup \{a,b,c\}) = r_{M_c}(Z \cup \{a,b\}) = r_{M_c}(Z) + r_{M_c}(\{a,b\}) - 2 = r_{M_c}(Z) + 1.$$
By submodularity,
$$r_{M_c}(E - \{a,b\}) + r_{M_c}(Z \cup \{a,b,c\}) \ge r(M_c) + r_{M_c}(Z \cup c).$$
Thus
$$r(M_c) - 1 + r_{M_c}(Z) + 1 \ge r(M_c) + r_{M_c}(Z \cup c).$$
Hence $r_{M_c}(Z) \ge r_{M_c}(Z \cup c)$ and \ref{zztop3} holds.
\begin{sublemma}
\label{two4}
Neither $a$ nor $b$ has a point on it in either $M$ or $M_c$.
\end{sublemma}
Assume there is a point $e$ on $a$ in $M$. Then $M\backslash e$ is $3$-connected. Moreover, in $(M\backslash b)^{\flat}$, we see that $e$ is parallel to $a$ so $(M\backslash b)^{\flat}\backslash e$, and hence $M\backslash e$, has a c-minor isomorphic to $N$; a contradiction. We conclude that \ref{two4} holds.
The next step in the proof of Lemma~\ref{3five} is to show that
\begin{sublemma}
\label{two5}
$M_c\backslash a,b$ is $2$-connected.
\end{sublemma}
Suppose $(A,B)$ be a 1-separation of $M_c\backslash a,b$ having $c$ in $A$. Then
\begin{equation}
\label{abc}
r_{M_c}(A) + r_{M_c}(B) = r(M_c \backslash a,b) = r(M_c) - 1 = r(M) - 2.
\end{equation}
Thus
\begin{multline*}
r_{M_c}(A \cup a) + r_{M_c}(B) - r(M_c)\\
\shoveleft{\hspace*{0.8in}= r_{M_c}(A) + r_{M_c}(\{a\}) - \sqcap_{M_c}(A,\{a\}) + r_{M_c}(B) - r(M_c)}\\
\shoveleft{\hspace*{0.8in}= [r_{M_c}(A) + r_{M_c}(B) - r(M_c) + 1] - 1 + r_{M_c}(\{a\}) - \sqcap_{M_c}(A,\{a\})}\\
\shoveleft{\hspace*{0.635in}= 0 - 1 + 2 - \sqcap_{M_c}(A,\{a\}) = 1 - \sqcap_{M_c}(A,\{a\}).}
\end{multline*}
If $\sqcap_{M_c}(A,\{a\}) = 1$, then $(A \cup a \cup b, B)$ is a 1-separation of $M_c$ and hence of $M$; a contradiction. We deduce that $\sqcap_{M_c}(A,\{a\}) = 0$ and $(A\cup a \cup b, B)$ is 2-separating in $M_c$ and hence in $M$. Thus $B$ consists of a point, say $d$, of $M$. Moreover,
$r_{M_c}(A \cup a) = r(M_c)$. Thus, as $\sqcap_{M_c}(A,\{a\}) = 0$, we see that
\begin{equation}
\label{abcd}
r_{M_c}(A) = r(M_c) - 2.
\end{equation}
Still working towards proving \ref{two5}, we show next that
\begin{sublemma}
\label{two6}
$\{b,d\}$ is a series pair of points in $(M\backslash a)^{\flat}$.
\end{sublemma}
Recall that $(M\backslash a)^{\flat} = M_c\backslash a {\underline{\downarrow}\,} b$. Now
$$r_{M_c}(\{d,b\}) + r_{M_c}(A) - r(M_c\backslash a) \le 3 + r(M_c) - 2 - r(M_c) = 1.$$
Thus $\{d,b\}$ is 2-separating in $M_c\backslash a$. It follows that it is also 2-separating in $M_c\backslash a {\underline{\downarrow}\,} b$, that is, in $(M\backslash a)^{\flat}$.
But $d$ and $b$ are points in $(M\backslash a)^{\flat}$, which is $2$-connected. We deduce by \ref{two4} that \ref{two6} holds.
By \ref{two6}, $(M\backslash a)^{\flat}/d$, and hence $M/d$, has a c-minor isomorphic to $N$. Next we show that
\begin{sublemma}
\label{two7}
$(A- c,\{a,b,c\})$ is a $2$-separation of $M\backslash d$.
\end{sublemma}
By (\ref{abcd}), $r_{M_c}(A-c) \le r(M_c) - 2 = r(M) - 3$ and \ref{two7} follows.
It follows from \ref{two7} and Lemma~\ref{newbix} that $M/d$ is $3$-connected\ unless
$M$ has a pair $\{e,f\}$ of points such that $e$ and $f$ are parallel in $M/d$.
Consider the exceptional case. Then $M$ has $\{d,e,f\}$ as a triangle. Then $\{e,f\} \subseteq A - c$. Thus, by \ref{two7}, $((A- c) \cup d,\{a,b,c\})$ is a $2$-separation of $M$; a contradiction. We conclude that \ref{two5} holds.
By \ref{two5}, we deduce that
\begin{sublemma}
\label{har0}
$\lambda_{M_c\backslash a,b}(X_a) = 1 = \lambda_{M_c\backslash a}(X_a)$ and $\lambda_{M_c\backslash a,b}(X_b) = 1 = \lambda_{M_c\backslash b}(X_b)$.
\end{sublemma}
Since $r(M_c \backslash a,b) = r(M_c\backslash a) - 1$, it follows from \ref{har0} and symmetry that
\begin{sublemma}
\label{har1}
$r_{M_c}(Y_a - b) = r_{M_c}(Y_a) - 1$ and $r_{M_c}(Y_b - a) = r_{M_c}(Y_b) - 1$.
\end{sublemma}
It follows from this, symmetry, and the fact that $r_M(Y_a \cup c) > r_M(Y_a)$ that
\begin{sublemma}
\label{har2}
$r_{M}(Y_a - b) = r_{M}(Y_a) - 1$ and $r_{M}(Y_b - a) = r_{M}(Y_b) - 1$.
\end{sublemma}
By uncrossing,
\begin{align}
\label{subm}
2 & = \lambda_{M_c\backslash a,b}(X_a) + \lambda_{M_c\backslash a,b}(Y_b - a) \nonumber\\
& \ge \lambda_{M_c\backslash a,b}(X_a\cap (Y_b - a)) + \lambda_{M_c\backslash a,b}(X_a \cup (Y_b - a)).
\end{align}
\begin{sublemma}
\label{xeyfnot}
$X_a\cap Y_b \neq \emptyset \neq X_b\cap Y_a.$
\end{sublemma}
Suppose $X_a\cap Y_b = \emptyset$. Then $Y_b - a \subseteq Y_a - b$. Thus, by \ref{har1},
$\sqcap_{M_c}(Y_a - b,\{b\}) = 1 = \sqcap_{M_c}(Y_a - b,\{a\})$. Hence, by \ref{zztop2},
$\sqcap_{M_c}(Y_a - b,\{a,b\}) = 2$. Thus, by \ref{zztop3}, $c \in {\rm cl}_{M_c}(Y_a - b).$
It follows that $(Y_a \cup c, X_a - c)$ is 2-separating in $M_c \backslash a$. Thus
$(Y_a \cup c \cup a, X_a - c)$ is 2-separating in $M$. As $M$ is $3$-connected, we deduce that
$X_a$ consists of exactly two points, $c$ and $x$, say. If $r_{M_c}(\{x,c\}) = 1$, then, in $M$, we see that $x$ is a point that lies on the line $c$. Thus $M\backslash x$ is $3$-connected. As $(M\backslash a)^{\flat}$ has a c-minor isomorphic to $N$ and has $x$ and $c$ as a parallel pair of points, we deduce that $M\backslash x$ has a c-minor isomorphic to $N$; a contradiction. We conclude that $r_{M_c}(\{x,c\}) = 2$. Thus $\{x\}$ is 1-separating in $M_c$; a contradiction. We deduce that $X_a\cap Y_b \neq \emptyset$ and \ref{xeyfnot} follows by symmetry.
We now choose the non-trivial 2-separation $(X_a, Y_a)$ of $M_c \backslash a$ such that $|X_a|$ is a minimum subject to the condition that $b \in Y_a$. Since $X_a\cap Y_b$ and $X_b\cap Y_a$ are both non-empty, we deduce from (\ref{subm}) and symmetry that
$$\lambda_{M_c\backslash a,b}(X_a\cap Y_b) = 1 = \lambda_{M_c\backslash a,b}(X_b \cap Y_a).$$
We show next that
\begin{sublemma}
\label{haz5}
$\lambda_{M_c\backslash a}(X_a\cap Y_b) = 1 = \lambda_{M_c\backslash b}(X_b \cap Y_a).$
\end{sublemma}
We have $1 = r_{M_c}(X_a \cap Y_b) + r_{M_c}((Y_a - b) \cup X_b) - r(M_c \backslash a,b).$ But
$r(M_c \backslash a,b) = r(M_c \backslash a) - 1$ and, by \ref{har1}, $r_{M_c}(Y_a - b) = r_{M_c}(Y_a) - 1$. Hence $r_{M_c}((Y_a - b) \cup X_b) = r_{M_c}(Y_a \cup X_b) - 1$. Thus
\ref{haz5} follows by symmetry.
By the choice of $X_a$ and the fact that $b$ and $c$ are the only 2-separating lines of $M\backslash a$, we deduce that $X_a \cap Y_b$ consists of a single point, say $w$.
\begin{sublemma}
\label{haz6}
$X_a$ consists of a series pair $\{w,c\}$ in $M_c\backslash a$.
\end{sublemma}
Suppose $w \notin {\rm cl}_{M_c}(X_a - w)$. Then $( X_a - w, Y_a \cup w)$ violates the choice of $(X_a,Y_a)$ unless $|X_a - w| = 1$. In the exceptional case, $\{w,c\}$ is a series pair in $M_c\backslash a$.
Now suppose that $w \in {\rm cl}_{M_c}(X_a - w)$. Then $w \in {\rm cl}_{M_c}(X_b)$. Thus $(X_b \cup w, Y_b - w)$ is a 2-separation of $M_c \backslash b$. But $Y_b - w$ avoids $X_a$ so we have a contradiction\ to \ref{xeyfnot} when we replace $(X_b,Y_b)$ by $(X_b \cup w, Y_b - w)$ unless $Y_b = \{a,w\}$. In the exceptional case, by \ref{har1}, $r(Y_b) = 2$ and we have a contradiction\ to \ref{two4}. We conclude that \ref{haz6} holds.
Since $M_c \backslash a$ has $\{w,c\}$ as a series pair. It follows that $M_c \backslash a/w$ has a c-minor isomorphic to $N$. Thus so do $(M\backslash a)^{\flat}/w$ and $M/w$. In $M\backslash a$, we have $\{c,w\}$ and $\{b\}$ as 2-separating sets. Now $w \notin {\rm cl}_{M_c \backslash a}(X_1 - w)$.
Hence $r_M(X_1 - w) = r_M(X_1) - 1 = r(M) - 3$. As $r(Y_1) = 4$, we deduce that $(X_1 - w, Y_1)$ is a $2$-separation\ in $M\backslash w$. Thus, by Lemma~\ref{newbix}, $M/w$ is $3$-connected\ unless $M$ has a triangle $T$ of points including $w$. In the exceptional case, $T- w \subseteq X_1 - w$, so
$(X_1, Y_1)$ is a $2$-separation\ of $M$. This contradiction\ completes the proof of Lemma~\ref{3five}.
\end{proof}
\begin{lemma}
\label{3four}
$\theta \neq 0$.
\end{lemma}
\begin{proof}
Assume that $\theta= 0$. Thus $\sqcap(X_1,\{y\}) = 0$ for all $y$ in $Y_1$. We may assume that $\sqcap^*(X_1,\{y\}) = 0$ for all $y$ in $Y_1$ otherwise, in $M^*$, we have $\theta \in \{1,2,3\}$. Thus, for all $y$ in $Y_1$, we have $r(Y_1 - y) = r(Y_1) = 4$.
Then, by Lemma~\ref{earlier}, none of $M/a$, $M/b$, nor $M/c$ has a c-minor isomorphic to $N$. Hence we may assume that $a$ and $b$ are deleted to get $N$. But, in $M\backslash a,b$, we see that $\{c\}$ is a component, so $c$ can be contracted to get $N$; a contradiction.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{Step6+}.]
By Lemma~\ref{y13}, a minimal non-$N$-$3$-separator $Y_1$ of $M$ having exactly three elements consists of three lines. Above, we looked at the number $\theta$ of members $y$ of $Y_1$ for which $\sqcap(X_1,\{y\}) = 1$. In Lemmas~\ref{3two} and \ref{3three}, we showed that $\theta \neq 2$ and $\theta \neq 1$, while Lemmas~\ref{3five} and \ref{3four} showed that $\theta \neq 3$ and $\theta \neq 0$. There
are no remaining possibilities for $\theta$, so Lemma~\ref{Step6+} holds.
\end{proof}
\section{A minimal non-$N$-$3$-separator with at least four elements}
\label{fourel}
By \ref{Step6}, we may now assume that $M$ has a minimal non-$N$-$3$-separator $Y_1$ having at least four elements. As before, we write $X_1$ for $E(M) - Y_1$. Our next goal is to prove \ref{Step7}, which we restate here for convenience.
\begin{lemma}
\label{dubya}
Let $Y_1$ be a minimal non-$N$-$3$-separating set having at least four elements. Then $Y_1$ contains a doubly labelled element.
\end{lemma}
\begin{proof} Assume that the lemma fails. For each $e$ in $Y_1 - E(N)$, let $\nu(e)$ be equal to the unique member of $\{\mu(e), \mu^*(e)\}$ that is defined. Choose $\ell$ in $Y_1 - E(N)$ to minimize $\nu(\ell)$. By switching to the dual if necessary, we may suppose that $\nu(\ell) = \mu(\ell)$.
Let $(A,B)$ be a $2$-separation\ of $M\backslash \ell$ where $A$ is the $N$-side and $|B| = \mu(\ell)$. We now apply Lemma~\ref{p124}. Part (ii) of that lemma does not hold otherwise, by Lemma~\ref{bubbly}, $Y_1 - \ell$ contains a doubly labelled element.
Assume next that (iii) of Lemma~\ref{p124} holds. Then $\lambda_{M\backslash \ell}(Y_1 - \ell) = 2$ and
$\lambda_{M\backslash \ell}(A \cap(Y_1 - \ell)) = 1= \lambda_{M\backslash \ell}(B \cap(Y_1 - \ell))$, while
$\lambda_{M\backslash \ell}(A \cap X_1) = 2= \lambda_{M\backslash \ell}(B \cap X_1)$. Then using the partitions $(A\cap (Y_1 - \ell), A \cap X_1,B)$
and $(B\cap (Y_1 - \ell), B \cap X_1,A)$ as $(A,B,C)$ in Lemma~\ref{general3}, we deduce that
$\sqcap(A \cap (Y_1 - \ell), A\cap X_1) = 1$ and
$\sqcap(B \cap (Y_1 - \ell), B\cap X_1) = 1$.
Now $M\backslash \ell$ is the 2-sum of $2$-polymatroids $M_A$ and $M_B$ having ground sets $A \cup q$ and $B \cup q$, respectively. Since $M\backslash \ell$ is $2$-connected, it follows by Proposition~\ref{connconn}, that each of $M_A$ and $M_B$ is $2$-connected.
Now $\lambda_{M\backslash \ell}(B \cap (Y_1 - \ell))= \sqcap_{M\backslash \ell}(B \cap (Y_1 - \ell), (B \cap X_1) \cup A) = 1$ and $\sqcap_{M_B}(B \cap (Y_1 - \ell), B \cap X_1) = 1$. Noting that $M\backslash \ell = P(M_A,M_B)\backslash q$, we see that, in $P(M_A,M_B)$, we have $\sqcap(B \cap (Y_1 - \ell), (B \cap X_1) \cup A\cup q) = 1$. Hence
$\sqcap_{M_B}(B \cap (Y_1 - \ell), (B \cap X_1) \cup q) = 1$. Thus $M_B$ is the 2-sum of two $2$-connected $2$-polymatroids $M_{B,Y}$ and $M_{B,X}$ having ground sets $(B \cap (Y_1 - \ell)) \cup s$ and $(B \cap X_1) \cup q \cup s$. Note that $M_B = P(M_{B,X},M_{B,Y})\backslash s$.
Let $M'_B = P(M_{B,X},M_{B,Y})$ and consider $P(M_A,M'_B)$ noting that deleting $q$ and $s$ from this $2$-polymatroid gives $M\backslash \ell$.
By Lemma~\ref{oswrules}(ii),
$$\sqcap(A,B) + \sqcap(B \cap X_1, B \cap (Y_1 - \ell)) = \sqcap(A \cup (B \cap X_1),B\cap (Y_1 - \ell)) + \sqcap(A, B \cap X_1).$$
Since the first three terms in this equation equal one,
\begin{equation}
\label{labx}
\sqcap(A, B \cap X_1) = 1.
\end{equation}
We deduce, by Lemma~\ref{claim1}(i) that if $y \in B \cap (Y_1 - \ell)$, then $M\backslash \ell\backslash y$ has a special $N$-minor.
Now $M_{B,X}$ has $q$ and $s$ as points. We show next that
\begin{sublemma}
\label{notzero} $\lambda_{M_{B,X}/ s}(\{q\}) = 0$.
\end{sublemma}
Assume that $\lambda_{M_{B,X}/ s}(\{q\}) \neq 0$. When we contract $s$ in $M'_B$, the set $B \cap (Y_1 - \ell)$ becomes 1-separating. Moreover, in $M'_B/(B \cap (Y_1 - \ell))$, the element $s$ is a loop, so $M'_B\backslash s/(B \cap (Y_1 - \ell)) = M'_B/s/(B \cap (Y_1 - \ell))$. It follows that
$\sqcap_{M\backslash \ell/(B \cap (Y_1 - \ell)}(A, B\cap X_1) = 1$. Hence, by Lemma~\ref{claim1}(ii), if $y \in B \cap (Y_1 - \ell)$, then $M\backslash \ell/ y$ has a special $N$-minor.
Thus each $y$ in $B \cap (Y_1 - \ell)$ is doubly labelled. This contradiction\ completes the proof of \ref{notzero}.
By \ref{notzero}, $\{q,s\}$ is a parallel pair of points in $M_{B,X}$. From considering $P(M_A,M'_B)$, we deduce that
$\lambda_{M\backslash \ell}(B \cap X_1) = 1$.
This contradiction\ implies that (iii) of Lemma~\ref{p124} does not hold.
It remains to consider when (i) of Lemma~\ref{p124} holds. We now apply Lemma~\ref{p63rev} to get, because of the choice of $\ell$, that $\sqcap(\{y\},X_1) \neq 1$ for some $y$ in $Y_1 - \ell$. Then, by Lemma~\ref{dichotomy}, $\sqcap(Y_1 - y, X_1) = 0$ for all $y$ in $Y_1 - \ell$.
Thus, by Lemma~\ref{prep65rev}
and the choice of $\ell$, (iii)(b) rather than (iii)(a) of that lemma holds. Then
$(X_1 - z, (Y_1 - \ell) \cup z$ is a 2-separation of $M\backslash \ell$ having $X_1 - z$ as the $N$-side. Since $z$ is a point, we have a contradiction\ to Lemma~\ref{key}.
\end{proof}
The doubly labelled element found in the last lemma will be crucial in completing the proof of Theorem~\ref{modc}. We shall need another preliminary lemma.
\begin{lemma}
\label{series}
Let $\ell$ be a doubly labelled element of $M$. Then $M\backslash \ell$ does not have a series pair of points $\{a,b\}$ such that $r(\{a,b,\ell\}) = 3$.
\end{lemma}
\begin{proof} Assume that $M\backslash \ell$ does have such a series pair $\{a,b\}$. By Lemma~\ref{Step0}, $\ell$ is a line. Thus $M/\ell$ has $\{a,b\}$ as a parallel pair of points or has $a$ or $b$ as a loop. In each case, $M$ has $a$ or $b$ as a doubly labelled point; a contradiction.
\end{proof}
We will now take $\ell$ to be a doubly labelled element of $Y_1$, a minimal non-$N$-$3$-separating set having at least four elements.
\begin{lemma}
\label{p125} There is a $2$-separating set $Q$ in $M\backslash \ell$ such that $Q \subseteq Y_1 - \ell$ and $|Q| \ge 2$ and contains no points.
\end{lemma}
\begin{proof} Suppose $\ell \not\in {\rm cl}(Y_1 - \ell)$. Then $(X_1, Y_1 - \ell)$ is a $2$-separation\ of $M\backslash \ell$ and $r(Y_1) = r(Y_1 - \ell) + 1$. Then, by Lemma~\ref{series}, $Y_1 - \ell$ does not consist of a series pair of points. Hence, by Lemma~\ref{key}, $Y_1 - \ell$ contains no points so the result holds by taking $Q =Y_1 - \ell$.
We may now assume that $\ell \in {\rm cl}(Y_1 - \ell)$. Let $(A,B)$ be a $2$-separation\ of $M\backslash \ell$ where $A$ is the $N$-side and $|B| = \mu(\ell)$. Since $|B| \ge 3$, it follows by Lemma~\ref{key} that $B$ contains no points. If $B \subseteq Y_1 - \ell$, then the lemma holds by taking $Q = B$. Thus we may assume that $B \cap X_1 \neq \emptyset$.
Since $X_1$ and $A$ are the $N$-sides of their respective separations and $|E(N)| \ge 4$, we see that $|A \cap X_1| \ge 2$. If $A \subseteq X_1$, then $B \supseteq Y_1 - \ell$, so $(A,B \cup \ell)$ is a $2$-separation\ of $M$; a contradiction. Likewise, if $B \subseteq X_1$, then $(A \cup \ell,B)$ is a $2$-separation\ of $M$; a contradiction. We conclude that $(A,B)$ crosses $(X_1,Y_1 - \ell)$.
Since $|A \cap X_1| \ge 2$ and $\ell \in {\rm cl}(Y_1 - \ell)$, it follows that $\lambda_{M\backslash \ell}(A \cap X_1) \ge 2$ otherwise
$(A \cap X_1, B \cup Y_1)$ is a $2$-separation\ of $M$; a contradiction. Then, by uncrossing, we deduce that $\lambda_{M\backslash \ell}(B \cap Y_1) \le 1$. Since $B$ contains no points, the lemma holds with $Q = B \cap Y_1$ unless this set contains a single line.
Consider the exceptional case. As $|B| \ge 3$, we deduce that $|B \cap X_1| \ge 2$. Now $\lambda_{M\backslash \ell}(B \cap X_1) \ge 2$ otherwise, as $\ell \in {\rm cl}(Y_1 - \ell)$, we obtain the contradiction\ that $(B\cap X_1,A \cup Y_1)$ is a $2$-separation\ of $M$. Hence, by uncrossing, $\lambda_{M\backslash \ell}(A \cap Y_1) = 1$ and, as $|Y_1| \ge 4$, it follows using Lemma~\ref{key} that the lemma holds by taking $Q = A \cap Y_1$.
\end{proof}
\begin{lemma}
\label{ab} The $2$-polymatroid $M\backslash \ell$ has a $2$-separation $(D_1,D_2)$ where $D_2$ has at least two elements, is contained in $Y_1 - \ell$, and contains no points. Moreover, either
\begin{itemize}
\item[(i)] $D_2 \cup \ell = Y_1$; and $\sqcap(D_1,\{\ell\}) = 0$ and $\sqcap(D_2,\{\ell\}) = 1$; or
\item[(ii)] $Y_1 - \ell - D_2 \neq \emptyset$ and $\sqcap(D_1,\{\ell\}) = 0 = \sqcap(D_2,\{\ell\})$.
\end{itemize}
\end{lemma}
\begin{proof} Let $D_2$ be the set $Q$ found in Lemma~\ref{p125} and let $D_1 = E(M\backslash \ell) - D_2$.
Now $(D_1,D_2)$ is a $2$-separation\ of $M\backslash \ell$. Thus, there are the following four possibilities.
\begin{itemize}
\item[(I)] $\sqcap(D_1,\{\ell\}) = 1 = \sqcap(D_2,\{\ell\})$;
\item[(II)] $\sqcap(D_1,\{\ell\}) = 1$ and $\sqcap(D_2,\{\ell\}) = 0$;
\item[(III)] $\sqcap(D_1,\{\ell\}) = 0$ and $\sqcap(D_2,\{\ell\}) = 1$; and
\item[(IV)] $\sqcap(D_1,\{\ell\}) = 0 = \sqcap(D_2,\{\ell\})$.
\end{itemize}
\begin{sublemma}
\label{elim12}
Neither (I) nor (II) holds.
\end{sublemma}
Suppose (I) or (II) holds. Then $\lambda_M(D_1 \cup \ell) = 2$, so $\lambda_M(D_2) =2$ and $|D_2| \ge 2$. Since $D_2$ contains no points and $D_2 \subseteq Y_1 - \ell$, we get a contradiction to the minimality of $Y_1$. Thus \ref{elim12} holds.
\begin{sublemma}
\label{case3}
If (III) holds, then $D_1 \cap Y_1 = \emptyset$.
\end{sublemma}
As $\lambda_M(D_2 \cup \ell) = 2$, we must have that $D_1 \cap Y_1 = \emptyset$ otherwise $D_2 \cup \ell$ violates the minimality of $Y_1$. Thus \ref{case3} holds.
\begin{sublemma}
\label{case4}
If (IV) holds, then $D_1 \cap Y_1 \neq \emptyset$.
\end{sublemma}
Suppose $D_1 \cap Y_1 = \emptyset$. Then $D_1 = X_1$ and $D_2 = Y_1 - \ell$. Thus $\lambda_M(X_1) > 2$ as $\sqcap(D_2,\{\ell\}) = 0$. This contradiction\ establishes that \ref{case4} holds and thereby completes the proof of the lemma.
\end{proof}
\begin{lemma}
\label{abdual} The $2$-polymatroid $M/ \ell$ has a $2$-separation $(C_1,C_2)$ where $C_2$ contains at least two elements and is contained in $Y_1 - \ell$, and contains no points of $M$. Moreover, either
\begin{itemize}
\item[(i)] $C_2 \cup \ell = Y_1$; and $\sqcap(C_1,\{\ell\}) = 1$ and $\sqcap(C_2,\{\ell\}) = 2$; or
\item[(ii)] $Y_1 - \ell - C_2 \neq \emptyset$ and $\sqcap(C_1,\{\ell\}) = 2 = \sqcap(C_2,\{\ell\})$.
\end{itemize}
\end{lemma}
\begin{proof} We apply the preceding lemma to $M^*\backslash \ell$ recalling that $(M^*\backslash \ell)^{\flat} = (M/\ell)^*$ and that the connectivity functions of
$M^*\backslash \ell$ and $M/\ell$ are equal. Thus $M/ \ell$ does indeed have a $2$-separation $(C_1,C_2)$ where $C_2$ contains at least two elements,
is contained in $Y_1 - \ell$, and contains no points of $M^*$. Thus $C_2$ contains no points of $M$. Since $r(E - \ell) = r(M)$, one easily checks that
$\sqcap^*(C_i, \{\ell\}) + \sqcap(C_j,\{\ell\}) = 2$ where $\{i,j\} = \{1,2\}$. The lemma now follows from the preceding one.
\end{proof}
\begin{lemma}
\label{bb} The $2$-polymatroids $M\backslash \ell$ and $M/ \ell$ have $2$-separations $(D_1,D_2)$ and $(C_1,C_2)$, respectively, such that each of $D_2$ and $C_2$ contains at least two elements, both $D_2$ and $C_2$ are contained in $Y_1 - \ell$, and neither $D_2$ nor $C_2$ contains any points of $M$. Moreover, $Y_1 - \ell - D_2 \neq \emptyset \neq Y_1 - \ell - C_2$ and $\sqcap(D_1,\{\ell\}) = 0 = \sqcap(D_2,\{\ell\})$ while
$\sqcap(C_1,\{\ell\}) = 2 = \sqcap(C_2,\{\ell\})$.
\end{lemma}
\begin{proof}
Assume that (i) of Lemma~\ref{ab} holds. Then, as $D_2 = Y_1 - \ell$, we see that $\sqcap(Y_1 - \ell,\{\ell\}) = 1$. Thus (i) of Lemma~\ref{abdual} cannot hold. Moreover, if (ii) of Lemma~\ref{abdual} holds, then $\sqcap(C_2, \{\ell\}) = 2$. This is a contradiction\ as $\sqcap(D_2,\{\ell\}) = 1$ and $C_2 \subseteq D_2$. We conclude that (ii) of Lemma~\ref{ab} holds. If (i) of Lemma~\ref{abdual} holds, then
$\sqcap(X_1, \{\ell\}) = 1$. But $X_1 \subseteq D_1$ and $\sqcap(D_1, \{\ell\}) = 0$; a contradiction.
\end{proof}
We now use the $2$-separations $(D_1,D_2)$ and $(C_1,C_2)$ of $M\backslash \ell$ and $M/ \ell$, respectively, found in the last lemma.
\begin{lemma}
\label{linemup} The partitions $(D_1,D_2)$ and $(C_1,C_2)$ have the following properties.
\begin{itemize}
\item[(i)] $\lambda_{M\backslash \ell}(C_1) = 3 = \lambda_{M\backslash \ell}(C_2)$;
\item[(ii)] $(D_1,D_2)$ and $(C_1,C_2)$ cross;
\item[(iii)] each of $D_1 \cap C_2, D_2 \cap C_2$, and $D_2 \cap C_1$ consists of a single line, and $C_2 \cup D_2 = Y_1 - \ell$; and
\item[(iv)] $\lambda_{M\backslash \ell}(D_1 \cap C_1) = \lambda_{M}(D_1 \cap C_1) = 2.$
\end{itemize}
\end{lemma}
\begin{proof} We have $\sqcap(D_1,\{\ell\}) = 0 = \sqcap(D_2,\{\ell\})$ and
$\sqcap(C_1,\{\ell\}) = 2 = \sqcap(C_2,\{\ell\})$. Thus neither $C_1$ nor $C_2$ is contained in $D_1$ or $D_2$, so (ii) holds. Moreover, as $r(C_1 \cup \ell) + r(C_2 \cup \ell) - r(M) = 3$, we see that (i) holds. To prove (iii) and (iv), we use an uncrossing argument. We have, for each $i$ in $\{1,2\}$,
\begin{align*}
1 + 3 & = \lambda_{M\backslash \ell}(D_2) + \lambda_{M\backslash \ell}(C_i)\\
& \ge\lambda_{M\backslash \ell}(D_2\cap C_i) + \lambda_{M\backslash \ell}(D_2 \cup C_i).
\end{align*}
Since $D_2 \cap C_i \neq \emptyset$ and contains no points and $\ell \in {\rm cl}(C_j)$ where $j \neq i$, we deduce that
$\lambda_{M\backslash \ell}(D_2 \cap C_i) = \lambda_{M}(D_2 \cap C_i) \ge 2$. Thus
$\lambda_{M\backslash \ell}(D_2 \cup C_i) \le 2.$
Hence, as $\ell \in {\rm cl}(C_i)$, we see that
\begin{equation}
\label{addno}
2 \ge \lambda_{M\backslash \ell}(D_2 \cup C_i) = \lambda_M(D_2 \cup C_i \cup \ell).
\end{equation}
But $D_2 \cup C_2 \subseteq Y_1 - \ell$, so, by the definition of $Y_1$, we deduce that
\begin{equation}
\label{addno2}
D_2 \cup C_2 = Y_1 - \ell \text{~and~} D_1 \cap C_1 = X_1.
\end{equation}
Moreover, as $\lambda_{M\backslash \ell}(D_2 \cup C_i) = 2$, we see that $\lambda_{M\backslash \ell}(D_2 \cap C_i) = 2$. Hence
\begin{equation}
\label{addno4}
\lambda_M(D_2 \cap C_i) = 2.
\end{equation}
Since $D_1 \cap C_1 = X_1$ and $D_1 \cap C_2$ is non-empty containing no points, it follows from (\ref{addno}) that
\begin{equation}
\label{addno3}
2 = \lambda_{M\backslash \ell}(D_2 \cup C_i) = \lambda_M(D_2 \cup C_i \cup \ell) = \lambda_{M\backslash \ell}(D_1 \cap C_j) = \lambda_M(D_1 \cap C_j).
\end{equation}
Thus (iv) holds. By that and (\ref{addno4}), it follows, using the minimality of $Y_1$, that each of $D_1 \cap C_2, D_2 \cap C_2$, and $D_2 \cap C_1$ consist of a single line in $M$. Hence (iii) holds.
\end{proof}
For each $(i,j)$ in $\{(1,2),(2,2),(2,1)\}$, let $C_i \cap D_j = \{\ell_{ij}\}$.
\begin{lemma}
\label{ranks}
The following hold.
\begin{itemize}
\item[(i)] $r(D_2) = 3$ so $r(D_1) = r(M) - 2$;
\item[(ii)] $r(D_1) = r(X_1) + 1$;
\item[(iii)] $r(Y_1) = 5$; and
\item[(iv)] $r(C_2) = 4$.
\end{itemize}
\end{lemma}
\begin{proof}
Now $D_2$ consists of two lines, $\ell_{12}$ and $\ell_{22}$. Suppose first that $r(D_2) = 2$. Then both $M\backslash \ell_{12}$ and $M\backslash \ell_{22}$ are $3$-connected. Without loss of generality, $M/ \ell_{12}$ has $N$ as a c-minor. But $M/ \ell_{12}$ has $\ell_{22}$ as a loop, so $M\backslash \ell_{22}$ is $3$-connected\ having a c-minor isomorphic to $N$. Thus $r(D_2) \ge 3$.
Now suppose that $r(D_2) = 4$. Then $r(D_1) = r(M) - 3$. Clearly $r(D_1 \cup \ell_{12}) \le r(M) - 1$. Now $D_1 \cup \ell_{12} \supseteq C_2$ so
$r(D_1 \cup \ell_{12} \cup \ell) \le r(M) - 1$. Hence $\{\ell_{22}\}$ is 2-separating in $M$; a contradiction. Hence (i) holds.
Since $C_2 \cup D_2 = Y_1 - \ell$, we see that $D_1 = X_1 \cup \ell_{21}$. Suppose $r(D_1) = r(X_1)$. As $X_1 \subseteq C_1$, we deduce that
$\ell_{21} \in {\rm cl}(C_1)$. But $\ell \in {\rm cl}(C_1)$. Hence
\begin{align*}
r(M) + 3 & = r(C_1) + r(C_2)\\
& =r(C_1 \cup \ell) + r(C_2 \cup \ell)\\
& = r(C_1 \cup \ell\cup \ell_{21}) + r(C_2 \cup \ell)\\
& \ge r(C_1 \cup C_2 \cup \ell) + r(\{\ell,\ell_{21}\}).
\end{align*}
Thus $r(\{\ell,\ell_{21}\}) \le 3$, so $\sqcap(D_1,\{\ell\}) \ge 1$; a contradiction. Hence $r(D_1) \ge r(X_1) + 1$.
Suppose $r(D_1) = r(X_1) + 2.$ Then
$$r(M) + 1 = r(D_1) + r(D_2) = r(X_1) + 2 + 3,$$
so $r(X_1) = r(M) - 4$. Thus $r(Y_1) = 6$. Now $r(C_2) = r(C_2 \cup \ell)$. Thus $6 = r(Y_1) = r(Y_1 - \ell)$. Since $Y_1 - \ell$ consists of three lines, two of which are in $D_2$, we deduce that $r(D_2) = 4$; a contradiction\ to (i). We conclude that $r(D_1) = r(X_1) + 1$, that is, (ii) holds.
Finally, as $r(D_2) = 3$, we see that $r(M) = r(D_1) + r(D_2) - 1 = [r(X_1) +1] + 3 - 1$. But $r(M) = r(X_1) + r(Y_1) - 2$. Thus $r(Y_1) = 5$, so (iii) holds. Moreover, $r(Y_1 - \ell) = 5$, that is, $r(C_2 \cup D_2) = 5.$ Now $C_2$ consists of two lines so $r(C_2) \le 4$. Thus
\begin{align*}
4 + 3 & \ge r(C_2) + 3\\
& = r(C_2) + r(D_2)\\
& \ge r(C_2 \cup D_2) + r(C_2 \cap D_2)\\
& = 5 + r(\{\ell_{22}\})\\
& = 5 + 2.
\end{align*}
We deduce that $r(C_2) = 4$ so (iv) holds.
\end{proof}
By proving the following lemma, we will establish the final contradiction\ that completes the proof of Theorem~\ref{modc}.
\begin{lemma}
\label{final} The $2$-polymatroid $M/ \ell_{22}$ is $3$-connected having a c-minor isomorphic to $N$.
\end{lemma}
\begin{proof}
First we show the following.
\begin{sublemma}
\label{finalfirst}
$\sqcap(D_1,\{\ell_{i2}\}) = 0$ for each $i$ in $\{1,2\}$.
\end{sublemma}
Suppose $\sqcap(D_1,\{\ell_{i2}\}) \ge 1$. Then $r(D_1 \cup \ell_{i2}) \le r(D_1) + 1$. But $\ell \in {\rm cl}(C_i) \subseteq {\rm cl}(D_1 \cup \ell_{i2})$, so
$r(D_1 \cup \ell \cup \ell_{i2}) \le r(D_1) + 1$. Also $r(\{\ell_{j2}\}) = 2$ where $\{i,j\} = \{1,2\}$. Thus
\begin{align*}
r(D_1 \cup \ell \cup \ell_{i2})+ r(\{\ell_{j2}\}) & \le r(D_1) + 1 + 2\\
& = r(M) - 2 + 1 +2\\
& = r(M) + 1.
\end{align*}
Hence $\{\ell_{j2}\}$ is 2-separating in $M$; a contradiction. Hence \ref{finalfirst} holds.
Now $M\backslash \ell$ has a c-minor isomorphic to $N$ and $\sqcap(D_1,D_2) = 1$.
As $\sqcap(D_1,\{\ell_{i2}\}) = 0$, Lemma~\ref{obs1} implies that $\sqcap_{M/\ell_{i2}}(D_1,D_2 - \ell_{i2}) = 1$ for each $i$ in $\{1,2\}$. Thus, by Lemma~\ref{claim1}(ii),
\begin{sublemma}
\label{finalsecond}
$M\backslash \ell/\ell_{i2}$ has a c-minor isomorphic to $N$ for each $i$ in $\{1,2\}$.
\end{sublemma}
It remains to show that $M/\ell_{22}$ is $3$-connected. This matroid is certainly $2$-connected. Next we show that
\begin{sublemma}
\label{end1}
$\ell$ and $\ell_{21}$ are parallel lines in $M/\ell_{22}$.
\end{sublemma}
To see this, note that, by Lemma~\ref{ranks}(iv),
$$r(C_2 \cup \ell) = r(C_2) = r(\{\ell_{21},\ell_{22}\}.$$
Also, for each $i$ in $\{1,2\}$, we have $\sqcap(\{\ell\}, \{\ell_{2i}\}) \le \sqcap(\{\ell\},D_i) = 0$, so \ref{end1} holds.
Now take a fixed c-minor of $M\backslash \ell/\ell_{22}$ isomorphic to $N$; call it $N_1$. Let $(A' \cup \ell, B')$ be a $2$-separation\ of $M/\ell_{22}$ in which the non-$N_1$-side has maximum size and $\ell \not\in A'$. By Lemma~\ref{dualmu}, both $A' \cup \ell$ and $B'$ have at least three elements.
\begin{sublemma}
\label{ell12a'}
$\ell_{21} \in A'$.
\end{sublemma}
To see this, note that, since $\ell$ and $\ell_{21}$ are parallel lines in $M/\ell_{22}$, if $\ell_{21} \in B'$, then $\ell \in {\rm cl}_{M/\ell_{22}}(B')$, so
$\sqcap_{M/\ell_{22}}(A' \cup \ell, B') \ge 2$; a contradiction.
\begin{sublemma}
\label{ranksagain}
$r_M(D_1 \cap C_1) = r(M) -3= r_{M/\ell_{22}}(D_1 \cap C_1)$.
\end{sublemma}
By Lemma~\ref{linemup}(iii), $D_1 \cap C_1 = X_1$. By Lemma~\ref{ranks}(i) and (ii), $r(D_1) = r(M) - 2$ and $r(D_1) = r(D_1 \cap C_1) + 1$, so
$r_M(D_1 \cap C_1) = r(M) -3$. By \ref{finalfirst}, $\sqcap(D_1 \cap C_1, \{\ell_{22}\}) = 0$, so $r_M(D_1 \cap C_1) = r_{M/\ell_{22}}(D_1 \cap C_1)$.
\begin{sublemma}
\label{ranksagain2}
$r_{M/\ell_{22}}((D_1 \cap C_1)\cup \ell_{12})= r(M) -2$.
\end{sublemma}
To see this, observe that
\begin{align*}
r_{M/\ell_{22}}((D_1 \cap C_1)\cup \ell_{12}) & = r((D_1 \cap C_1)\cup \ell_{12}\cup \ell_{22}) - 2\\
& = r((D_1 \cap C_1)\cup \ell_{12}\cup \ell \cup \ell_{22}) - 2 \text{~~as $\ell \in {\rm cl}(C_1)$;}\\
& = r(M\backslash \ell_{21}) - 2\\
&= r(M) - 2.
\end{align*}
By combining \ref{ranksagain} and \ref{ranksagain2}, we deduce that
\begin{sublemma}
\label{ell12not}
$r_{M/\ell_{22}}(D_1 \cap C_1) = r_{M/\ell_{22}}((D_1 \cap C_1)\cup \ell_{12}) - 1$.
\end{sublemma}
Next we show that
\begin{sublemma}
\label{onetime}
$\lambda_{M/\ell_{22}}(\{\ell_{12}, \ell_{21},\ell\}) = 1$ or $\lambda_{M/\ell_{22}}(\{\ell_{21},\ell\}) = 1$.
\end{sublemma}
Recall that $X_1 = D_1 \cap C_1$ and $Y_1 = \{\ell,\ell_{12},\ell_{21},\ell_{22}\}$. By
uncrossing, we have
\begin{align*}
1 + 2 & = \lambda_{M/\ell_{22}}(B') + \lambda_{M/\ell_{22}}(X_1)\\
& \ge \lambda_{M/\ell_{22}}(B' \cup X_1) + \lambda_{M/\ell_{22}}(B' \cap X_1).
\end{align*}
As $|B'| \ge 3$, it follows by \ref{ell12a'} that $|B'\cap X_1| \ge 2$. Suppose $\lambda_{M/\ell_{22}}(B' \cap X_1) = 1$. Now $B'\cap X_1 \subseteq D_1$, so $\sqcap(B' \cap X_1,\{\ell_{22}\}) \le \sqcap(D_1,\{\ell_{22}\}) \le 0$. Thus, by Lemma~\ref{obs1}, $ \lambda_{M}(B' \cap X_1) = 1$. This contradiction\ implies that $\lambda_{M/\ell_{22}}(B' \cap X_1) = 2$, so $1 = \lambda_{M/\ell_{22}}(B' \cup X_1)$. Now, by \ref{ell12a'}, $\ell_{21} \in A'$, so
$E- \ell - (B' \cup X_1)$ is $\{\ell_{12}, \ell_{21},\ell\}$ or $\{\ell_{21},\ell\}$. Thus \ref{onetime} holds.
Suppose $\lambda_{M/\ell_{22}}(\{\ell_{12}, \ell_{21},\ell\}) = 1$. Then, as $\ell_{22}$ is skew to $X_1$ in $M$, we deduce that $\lambda_{M}(\{\ell_{12}, \ell_{21},\ell, \ell_{22}\}) = 1$, that is, $\lambda_{M}(Y_1) = 1$; a contradiction. We conclude that $\lambda_{M/\ell_{22}}(\{\ell_{21},\ell\}) = 1$.
By Lemma~\ref{obs1}, as $\lambda_{M}(D_1 \cap C_1) = 2$ and $\sqcap(D_1 \cap C_1, \{\ell_{22}\}) = 0$, we see that
$\lambda_{M/\ell_{22}}(D_1 \cap C_1) = 2$. Thus, by \ref{ell12not} and Lemma~\ref{ranks},
\begin{align*}
2 & = r_{M/\ell_{22}}(D_1 \cap C_1) + r_{M/\ell_{22}}(\{\ell_{12},\ell_{21},\ell\}) - r(M/\ell_{22})\\
& = [r_{M/\ell_{22}}(D_1 \cap C_1) + 1] + [r_{M/\ell_{22}}(\{\ell_{12},\ell_{21},\ell\}) - 1] - r(M/\ell_{22})\\
& = r_{M/\ell_{22}}((D_1 \cap C_1) \cup \ell_{12}) + [r(Y_1) - 2 - 1] - r(M/\ell_{22})\\
& = r_{M/\ell_{22}}((D_1 \cap C_1) \cup \ell_{12}) + [r(C_2) - 2] - r(M/\ell_{22})\\
& = r_{M/\ell_{22}}((D_1 \cap C_1) \cup \ell_{12}) + r_{M/\ell_{22}}(\{\ell_{21},\ell\}) - r(M/\ell_{22})\\
& = \lambda_{M/\ell_{22}}(\{\ell_{21},\ell\}).
\end{align*}
This contradiction\ to \ref{onetime} completes the proof of the lemma and thereby finishes the proof of Theorem~\ref{modc}.
\end{proof}
\end{document} |
\begin{document}
\title{On a series of finite automata\defining free transformation groups}
\begin{abstract}
We introduce two series of finite automata starting from the so-called
Aleshin and Bellaterra automata. We prove that each automaton in the first
series defines a free non-Abelian group while each automaton in the second
series defines the free product of groups of order $2$. Furthermore, these
properties are shared by disjoint unions of any number of distinct automata
from either series.
\end{abstract}
\section{Introduction}\lambdabel{main}
A (Mealy) automaton over a finite alphabet $X$ is determined by the set of
internal states, the state transition function and the output function. A
finite (or finite-state) automaton has finitely many internal states. An
initial automaton has a distinguished initial state. Any initial automaton
over $X$ defines a transformation $T$ of the set $X^*$ of finite words in
the alphabet $X$. That is, the automaton transduces any input word $w\in
X^*$ into the output word $T(w)$. The transformation $T$ preserves the
lengths of words and common beginnings. The set $X^*$ is endowed with the
structure of a regular rooted tree so that $T$ is an endomorphism of the
tree. A detailed account of the theory of Mealy automata is given in
\cite{GNS}.
The set of all endomorphisms of the regular rooted tree $X^*$ is of
continuum cardinality. Any endomorphism can be defined by an automaton.
However the most interesting are finite automaton transformations that
constitute a countable subset. If $T_1$ and $T_2$ are mappings defined by
finite initial automata over the same alphabet $X$, then their composition
is also defined by a finite automaton over $X$. If a finite automaton
transformation $T$ is invertible, then the inverse transformation is also
defined by a finite automaton. Furthermore, there are simple algorithms to
construct the corresponding composition automaton and inverse automaton.
In particular, all invertible transformations defined by finite automata
over $X$ constitute a transformation group $\mathcal{G}(X)$. This fact was
probably first observed by Ho\v{r}ej\v{s} \cite{H}.
A finite non-initial automaton $A$ over an alphabet $X$ defines a finite
collection of transformations of $X^*$ corresponding to various choices of
the initial state. Assuming all of them are invertible, these
transformations generate a group $G(A)$, which is a finitely generated
subgroup of $\mathcal{G}(X)$. We say that the group $G(A)$ is defined by the
automaton $A$. The groups defined by finite automata were introduced by
Grigorchuk \cite{G} in connection with the Grigorchuk group of intermediate
growth. The finite automaton nature of this group has great impact on
its properties. The formalization of these properties has resulted in the
notions of a branch group (see \cite{BGS}), a fractal group (see
\cite{BGN}), and, finally, the most general notion of a self-similar group
\cite{N}, which covers all automaton groups.
The main issue of this paper are free non-Abelian groups of finite
automaton transformations. Also, we are interested in the free products of
groups of order $2$ (such a product contains a free subgroup of index $2$).
Brunner and Sidki \cite{BS} proved that the free group embeds into the
group of finite automaton transformations over a $4$-letter alphabet.
Olijnyk \cite{O1}, \cite{O2} showed that the group of finite automaton
transformations over a $2$-letter alphabet contains a free group as well as
free products of groups of order $2$. In the above examples, all automata
are of linear algebraic origin.
A harder problem is to present the free group as the group defined by a
single finite non-initial automaton. This problem was solved by Glasner
and Mozes \cite{GM}. They constructed infinitely many finite automata of
algebraic origin that define transformation groups with various
properties, in particular, free groups. A finite automaton that defines
the free product of $3$ groups of order $2$ was found by Muntyan and
Savchuk (see \cite{N} and Theorem \ref{main4} below).
Actually, the first attempt to embed the free non-Abelian group into a
group of finite automaton transformations was made by Aleshin \cite{A} a
long ago. He introduced two finite initial automata over alphabet
$\{0,1\}$ and claimed that two automorphisms of the rooted binary tree
$\{0,1\}^*$ defined by these automata generate a free group. However the
argument in \cite{A} seems to be incomplete. Aleshin's automata are
depicted in Figure \ref{fig1} by means of Moore diagrams. The Moore
diagram of an automaton is a directed graph with labeled edges. The
vertices are the states of the automaton and edges are state transition
routes. Each label consists of two letters from the alphabet. The left
one is the input field, it is used to choose a transition route. The right
one is the output generated by the automaton. Aleshin considered these
automata as initial, with initial state $b$.
\begin{figure}
\caption{
\lambdabel{fig1}
\end{figure}
The Aleshin automata are examples of bi-reversible automata. This notion,
which generalizes the notion of invertibility, was introduced in
\cite{MNS} (see also \cite{GM}). The class of bi-reversible automata is in
a sense opposite to the class of automata defining branch groups. All
automata considered in this paper are bi-reversible.
In this paper, we are looking for finite automata that define free
non-Abelian groups of maximal rank, i.e., the free rank of the group is
equal to the number of states of the automaton. Note that the automata
constructed by Glasner and Mozes do not enjoy this property. For any of
those automata, the transformations assigned to various internal states
form a symmetric generating set so that the free rank of the group is half
of the number of the states. Brunner and Sidki conjectured (see \cite{S})
that the first of two Aleshin's automata shown in Figure \ref{fig1} is the
required one. The conjecture was proved in \cite{VV}.
\begin{theorem}[\cite{VV}]\lambdabel{main1}
The first Aleshin automaton defines a free group on $3$ generators.
\end{theorem}
In this paper we generalize and extend Theorem \ref{main1} in several
directions.
The two automata of Aleshin are related as follows. When the first
automaton is in the state $c$, it is going to make transition to the state
$a$ independently of the next input letter, which is sent directly to the
output. The second automaton is obtained from the first one by inserting
two additional states on the route from $c$ to $a$ (see Figure \ref{fig1}).
For any integer $n\ge1$ we define a $(2n+1)$-state automaton $A^{(n)}$ of
Aleshin type. Up to renaming of internal states, $A^{(n)}$ is obtained
from the first Aleshin automaton by inserting $2n-2$ additional states on
the route from $c$ to $a$ (for a precise definition, see Section
\ref{series}); in particular, $A^{(1)}$ and $A^{(2)}$ are the Aleshin
automata. The Moore diagram of the automaton $A^{(3)}$ is depicted in
Figure \ref{fig6} below. Note that the number of internal states of an
Aleshin type automaton is always odd. This is crucial for the proof of the
following theorem.
\begin{theorem}\lambdabel{main2}
For any $n\ge1$ the automaton $A^{(n)}$ defines a free group on $2n+1$
generators.
\end{theorem}
Given a finite number of automata $Y^{(1)},\dots,Y^{(k)}$ over the same
alphabet with disjoint sets of internal states $S_1,\dots,S_k$, we can
regard them as a single automaton $Y$ with the set of internal states
$S_1\cup\dots\cup S_k$. The automaton $Y$ is called the disjoint union of
the automata $Y^{(1)},\dots,Y^{(k)}$ as its Moore diagram is the disjoint
union of the Moore diagrams of $Y^{(1)},\dots,Y^{(k)}$. The group defined
by $Y$ is generated by the groups $G(Y^{(1)}),\dots,G(Y^{(k)})$.
We define the Aleshin type automata so that their sets of internal states
are disjoint. Hence the disjoint union of any finite number of distinct
automata of Aleshin type is well defined.
\begin{theorem}\lambdabel{main3}
Let $N$ be a nonempty set of positive integers and denote by $A^{(N)}$
the disjoint union of automata $A^{(n)}$, $n\in N$. Then the automaton
$A^{(N)}$ defines a free group on $\sum_{n\in N}(2n+1)$ generators.
\end{theorem}
One consequence of Theorem \ref{main3} is that the $8$ transformations
defined by the two Aleshin automata generate a free group on $8$
generators. In particular, any two of them generate a free non-Abelian
group. Thus Aleshin's claim is finally justified.
\begin{figure}
\caption{
\lambdabel{fig2}
\end{figure}
The Bellaterra automaton $B$ is a $3$-state automaton over a $2$-letter
alphabet. Its Moore diagram is depicted in Figure \ref{fig2}. The
automaton $B$ coincides with its inverse automaton and hence all $3$
transformations defined by $B$ are involutions. Otherwise there are no
more relations in the group $G(B)$.
\begin{theorem}[\cite{N}]\lambdabel{main4}
The Bellaterra automaton defines the free product of $3$ groups of order
$2$.
\end{theorem}
Theorem \ref{main4} is due to Muntyan and Savchuk. It was proved during
the 2004 summer school on automata groups at the Autonomous University of
Barcelona and so the automaton $B$ was named after the location of the
university.
The Bellaterra automaton $B$ is closely related to the Aleshin automaton
$A$. Namely, the two automata share the alphabet, internal states, and
the state transition function while their output functions never coincide.
We use this relation to define a series $B^{(1)},B^{(2)},\dots$ of automata
of Bellaterra type. By definition, $B^{(n)}$ is a $(2n+1)$-state automaton
obtained from $A^{(n)}$ by changing values of the output function at all
elements of its domain. Also, we define a one-state automaton $B^{(0)}$
that interchanges letters $0$ and $1$ of the alphabet. All transformations
defined by a Bellaterra type automaton are involutions.
\begin{theorem}\lambdabel{main5}
For any $n\ge0$ the automaton $B^{(n)}$ defines the free product of $2n+1$
groups of order $2$.
\end{theorem}
\begin{theorem}\lambdabel{main6}
Let $N$ be a nonempty set of nonnegative integers and denote by $B^{(N)}$
the disjoint union of automata $B^{(n)}$, $n\in N$. Then the automaton
$B^{(N)}$ defines the free product of $\sum_{n\in N}(2n+1)$ groups of
order $2$.
\end{theorem}
Theorems \ref{main3} and \ref{main6} have the following obvious corollary.
\begin{corollary}\lambdabel{main7}
(i) Let $n$ be an integer such that $n=3$ or $n=5$ or $n\ge7$. Then there
exists an $n$-state automaton over alphabet $\{0,1\}$ that define a free
transformation group on $n$ generators.
(ii) For any integer $n\ge3$ there exists an $n$-state automaton over
alphabet $\{0,1\}$ that define a transformation group freely generated by
$n$ involutions.
\end{corollary}
We prove Theorems \ref{main1}, \ref{main2}, and \ref{main3} using the dual
automaton approach. Namely, each finite automaton $Y$ is assigned a dual
automaton $Y'$ obtained from $Y$ by interchanging the alphabet with the set
of internal states and the state transition function with the output
function. It turns out that there is a connection between transformation
groups defined by $Y$ and $Y'$. As intermediate results, we obtain some
information on the dual automata of the Aleshin type automata.
\begin{proposition}\lambdabel{main8}
(i) The dual automaton of the Aleshin automaton defines a group that acts
transitively on each level of the rooted ternary tree $\{a,b,c\}^*$.
(ii) For any $n\ge1$ the dual automaton of $A^{(n)}$ defines a group that
acts transitively on each level of the rooted $(2n+1)$-regular tree
$Q_n^*$.
\end{proposition}
The proof of Theorem \ref{main4} given in \cite{N} also relies on the dual
automaton approach. In particular, it involves a statement on the dual
automaton $\widehat D$ of $B$. Since the group $G(B)$ is generated by
involutions, it follows that the set of double letter words over the
alphabet $\{a,b,c\}$ is invariant under the action of the group $G(\widehat D)$.
Hence $G(\widehat D)$ does not act transitively on levels of the rooted tree
$\{a,b,c\}$.
\begin{proposition}[\cite{N}]\lambdabel{main9}
The dual automaton of the Bellaterra automaton defines a transformation
group that acts transitively on each level of the rooted subtree of
$\{a,b,c\}^*$ formed by no-double-letter words.
\end{proposition}
We derive Theorems \ref{main4}, \ref{main5}, and \ref{main6} from Theorem
\ref{main3}. This does not involve dual automata. Nonetheless we obtain a
new proof of Proposition \ref{main9} that also works for all Bellaterra
type automata.
\begin{proposition}\lambdabel{main10}
For any $n\ge1$ the dual automaton of $B^{(n)}$ defines a group that
acts transitively on each level of the rooted subtree of $Q_n^*$ formed
by no-double-letter words.
\end{proposition}
Finally, we establish relations between groups defined by automata of
Aleshin type and of Bellaterra type.
\begin{proposition}\lambdabel{main11}
(i) The group $G(A)$ is an index $2$ subgroup of $G(B^{(\{0,1\})})$;
(ii) for any $n\ge1$ the group $G(A^{(n)})$ is an index $2$ subgroup of
$G(B^{(\{0,n\})})$;
(iii) for any nonempty set $N$ of positive integers the group $G(A^{(N)})$
is an index $2$ subgroup of $G(B^{(N\cup\{0\})})$.
\end{proposition}
\begin{proposition}\lambdabel{main12}
(i) $G(A)\cap G(B)$ is a free group on $2$ generators and an index $2$
subgroup of $G(B)$.
(ii) For any $n\ge1$, $G(A^{(n)})\cap G(B^{(n)})$ is a free group on $2n$
generators and an index $2$ subgroup of $G(B^{(n)})$.
(ii) For any nonempty set $N$ of positive integers, $G(A^{(N)})\cap
G(B^{(N)})$ is an index $2$ subgroup of $G(B^{(N)})$. Also,
$G(A^{(N)})\cap G(B^{(N)})$ is a free group of rank less by $1$ than the
free rank of $G(A^{(N)})$.
\end{proposition}
The paper is organized as follows. Section \ref{auto} addresses some
general constructions concerning automata and their properties. In Section
\ref{a} we recall constructions and arguments of the paper \cite{VV} where
Theorem \ref{main1} was proved. In Section \ref{series} they are applied
to the Aleshin type automata, which results in the proof of Theorem
\ref{main2} (Theorem \ref{series7}). Besides, Proposition \ref{main8} is
established in Sections \ref{a} and \ref{series} (see Corollaries
\ref{a5plus} and \ref{series6plus}). In Section \ref{union} we consider
disjoint unions of Aleshin type automata and obtain Theorem \ref{main3}
(Theorem \ref{union6}). Section \ref{b} is devoted to the study of the
Bellaterra automaton, automata of Bellaterra type, and their relation to
automata of Aleshin type. Here we prove Theorems \ref{main4}, \ref{main5},
and \ref{main6} (Theorems \ref{b3} and \ref{b4}), Propositions \ref{main9}
and \ref{main10} (Propositions \ref{b9} and \ref{b10}), Proposition
\ref{main11} (Proposition \ref{b2}), and Proposition \ref{main12}
(Propositions \ref{b6}, \ref{b7}, and \ref{b8}).
\section{Automata}\lambdabel{auto}
An {\em automaton\/} $A$ is a quadruple $(Q,X,\phi,\psi)$ formed by two
nonempty sets $Q$ and $X$ along with two maps $\phi:Q\times X\to Q$ and
$\psi:Q\times X\to X$. The set $X$ is to be finite, it is called the {\em
(input/output) alphabet\/} of the automaton. We say that $A$ is an
automaton over the alphabet $X$. $Q$ is called the set of {\em internal
states\/} of $A$. The automaton $A$ is called {\em finite\/} (or {\em
finite-state\/}) if the set $Q$ is finite. $\phi$ and $\psi$ are called
the {\em state transition function\/} and the {\em output function},
respectively. One may regard these functions as a single map $(\phi,\psi):
Q\times X\to Q\times X$.
The automaton $A$ canonically defines a collection of transformations.
First we introduce the set on which these transformations act. This is the
set of words over the alphabet $X$, which is denoted by $X^*$. A {\em
word\/} $w\in X^*$ is merely a finite sequence whose elements belong to
$X$. The elements of $w$ are called {\em letters\/} and $w$ is usually
written so that its elements are not separated by delimiters. The number
of letters of $w$ is called its {\em length\/}. It is assumed that $X^*$
contains the empty word $\varnothing$. The set $X$ is embedded in $X^*$ as
the subset of one-letter words. If $w_1=x_1\dots x_n$ and $w_2=y_1\dots
y_m$ are words over the alphabet $X$ then $w_1w_2$ denotes their
concatenation $x_1\dots x_ny_1\dots y_m$. The operation $(w_1,w_2)\mapsto
w_1w_2$ makes $X^*$ into the free monoid generated by all elements of $X$.
The unit element of the monoid $X^*$ is the empty word. Another structure
on $X^*$ is that of a rooted $k$-regular tree, where $k$ is the cardinality
of $X$. Namely, we consider a graph with the set of vertices $X^*$ where
two vertices $w_1,w_2\in X^*$ are joined by an edge if $w_1=w_2x$ or
$w_2=w_1x$ for some $x\in X$. The root of the tree is the empty word. For
any integer $n\ge0$ the $n$-th {\em level\/} of a rooted tree is the set of
vertices that are at distance $n$ from the root. Clearly, the $n$-th level
of the rooted tree $X^*$ is formed by all words of length $n$ in the
alphabet $X$.
Now let us explain how the automaton $A$ functions. First we choose an
{\em initial state\/} $q\in Q$ and prepare an {\em input word\/}
$w=x_1x_2\dots x_n\in X^*$. Then we set the automaton to the state $q$ and
start inputting the word $w$ into it, letter by letter. After reading a
letter $x'$ in a state $q'$, the automaton produces the output letter
$\psi(q',x')$ and makes transition to the state $\phi(q',x')$. Hence the
automaton's job results in two sequences: a sequence of states
$q_0=q,q_1,\dots,q_n$, which describes the internal work of the automaton,
and the {\em output word\/} $v=y_1y_2\dots y_n\in X^*$. Here
$q_i=\phi(q_{i-1},x_i)$ and $y_i=\psi(q_{i-1},x_i)$ for $1\le i\le n$.
For every choice of the initial state $q\in Q$ of $A$ we get a mapping
$A_q:X^*\to X^*$ that sends any input word to the corresponding output
word. We say that $A_q$ is the transformation defined by the automaton $A$
with the initial state $q$. Clearly, $A_q$ preserves the length of words.
Besides, $A_q$ transforms words from the left to the right, that is, the
first $n$ letters of $A_q(w)$ depend only on the first $n$ letters of $w$.
This implies that $A_q$ is an endomorphism of $X^*$ as a rooted tree. If
$A_q$ is invertible then it belongs to the group $\mathop{\mathrm{Aut}}(X^*)$ of
automorphisms of the rooted tree $X^*$. The set of transformations $A_q$,
$q\in Q$ is self-similar in the following sense. For any $q\in Q$, $x\in
X$, and $w\in X^*$ we have that $A_q(xw)=yA_p(w)$, where $p=\phi(q,x)$,
$y=\psi(q,x)$.
The semigroup of transformations of $X^*$ generated by $A_q$, $q\in Q$ is
denoted by $S(A)$. The automaton $A$ is called {\em invertible\/} if $A_q$
is invertible for all $q\in Q$. If $A$ is invertible then $A_q$, $q\in Q$
generate a transformation group $G(A)$, which is a subgroup of $\mathop{\mathrm{Aut}}(X^*)$.
We say that $S(A)$ (resp. $G(A)$) is the semigroup (resp. group) defined by
the automaton $A$.
\begin{lemma}[\cite{VV}]\lambdabel{auto1}
Suppose the automaton $A$ is invertible. Then the actions of the semigroup
$S(A)$ and the group $G(A)$ on $X^*$ have the same orbits.
\end{lemma}
One way to picture an automaton, which we use in this paper, is the {\em
Moore diagram}. The Moore diagram of an automaton $A=(Q,X,\phi,\psi)$ is
a directed graph with labeled edges defined as follows. The vertices of
the graph are states of the automaton $A$. Every edge carries a
label of the form $x|y$, where $x,y\in X$. The left field $x$ of the label
is referred to as the {\em input field\/} while the right field $y$ is
referred to as the {\em output field}. The set of edges of the graph is in
a one-to-one correspondence with the set $Q\times X$. Namely, for any
$q\in Q$ and $x\in X$ there is an edge that goes from the vertex $q$ to
$\phi(q,x)$ and carries the label $x|\psi(q,x)$. The Moore diagram of an
automaton can have loops (edges joining a vertex to itself) and multiple
edges. To simplify pictures, we do not draw multiple edges in this paper.
Instead, we use multiple labels.
The transformations $A_q$, $q\in Q$ can be defined in terms of the Moore
diagram of the automaton $A$. For any $q\in Q$ and $w\in X^*$ we find a
path $\gamma$ in the Moore diagram such that $\gamma$ starts at the vertex
$q$ and the word $w$ can be obtained by reading the input fields of labels
along $\gamma$. Such a path exists and is unique. Then the word $A_q(w)$
is obtained by reading the output fields of labels along the path $\gamma$.
Let $\Gamma$ denote the Moore diagram of the automaton $A$. We associate
to $\Gamma$ two directed graphs $\Gamma_1$ and $\Gamma_2$ with labeled
edges. $\Gamma_1$ is obtained from $\Gamma$ by interchanging the input and
output fields of all labels. That is, a label $x|y$ is replaced by $y|x$.
$\Gamma_2$ is obtained from $\Gamma$ by reversing all edges. The {\em
inverse automaton\/} of $A$ is the automaton whose Moore diagram is
$\Gamma_1$. The {\em reverse automaton\/} of $A$ is the automaton whose
Moore diagram is $\Gamma_2$. The inverse and reverse automata of $A$ share
the alphabet and internal states with $A$. Notice that any automaton is
completely determined by its Moore diagram. However neither $\Gamma_1$ nor
$\Gamma_2$ must be the Moore diagram of an automaton. So it is possible
that the inverse automaton or the reverse automaton (or both) of $A$ is not
well defined.
\begin{lemma}[\cite{GNS}]\lambdabel{auto2}
An automaton $A=(Q,X,\phi,\psi)$ is invertible if and only if for any $q\in
Q$ the map $\psi(q,\cdot):X\to X$ is bijective. The inverse automaton $I$
of $A$ is well defined if and only if $A$ is invertible. If this is the
case, then $I_q=A_q^{-1}$ for all $q\in Q$.
\end{lemma}
An automaton $A$ is called {\em reversible\/} if the reverse automaton of
$A$ is well defined.
\begin{lemma}[\cite{VV}]\lambdabel{auto3}
An automaton $A=(Q,X,\phi,\psi)$ is reversible if and only if for any $x\in
X$ the map $\phi(\cdot,x):Q\to Q$ is bijective.
\end{lemma}
Let $A=(Q,X,\phi,\psi)$ be an automaton. For any nonempty word
$\xi=q_1q_2\dots q_n\in Q^*$ we let $A_\xi=A_{q_n}\dots A_{q_2}A_{q_1}$.
Also, we let $A_\varnothing=1$ (here $1$ stands for the unit element of the
group $\mathop{\mathrm{Aut}}(X^*)$, i.e., the identity mapping on $X^*$). Clearly, any
element of the semigroup $S(A)$ is represented as $A_\xi$ for a nonempty
word $\xi\in Q^*$. The map $X^*\times Q^*\to X^*$ given by $(w,\xi)\mapsto
A_\xi(w)$ defines a right action of the monoid $Q^*$ on the rooted regular
tree $X^*$. That is, $A_{\xi_1\xi_2}(w)=A_{\xi_2}(A_{\xi_1}(w))$ for all
$\xi_1,\xi_2\in Q^*$ and $w\in X^*$.
To each finite automaton $A=(Q,X,\phi,\psi)$ we associate a {\em dual
automaton\/} $D$, which is obtained from $A$ by interchanging the alphabet
with the set of internal states and the state transition function with the
output function. To be precise, $D=(X,Q,\tilde\phi,\tilde\psi)$, where
$\tilde\phi(x,q)=\psi(q,x)$ and $\tilde\psi(x,q)=\phi(q,x)$ for all $x\in
X$ and $q\in Q$. Unlike the inverse and reverse automata, the dual
automaton is always well defined. It is easy to see that $A$ is the dual
automaton of $D$.
The dual automaton $D$ defines a right action of the monoid $X^*$ on $Q^*$
given by $(\xi,w)\mapsto D_w(\xi)$. This action and the action of $Q^*$ on
$X^*$ defined by the automaton $A$ are related in the following way.
\begin{proposition}[\cite{VV}]\lambdabel{auto4}
For any $w,u\in X^*$ and $\xi\in Q^*$,
$$
A_\xi(wu)=A_\xi(w)A_{D_w(\xi)}(u).
$$
\end{proposition}
\begin{corollary}[\cite{VV}]\lambdabel{auto5}
Suppose $A_\xi=1$ for some $\xi\in Q^*$. Then $A_{g(\xi)}=1$ for every
$g\in S(D)$.
\end{corollary}
A finite automaton $A=(Q,X,\phi,\psi)$ is called {\em bi-reversible\/} if
the map $\phi(\cdot,x):Q\to Q$ is bijective for any $x\in X$, the map
$\psi(q,\cdot):X\to X$ is bijective for any $q\in Q$, and the map
$(\phi,\psi):Q\times X\to Q\times X$ is bijective as well. All automata
that we consider in this paper are bi-reversible. Below we formulate some
basic properties of bi-reversible automata (see also \cite{N}).
\begin{lemma}\lambdabel{auto6}
Given a finite automaton $A$, the following are equivalent:
(i) $A$ is bi-reversible;
(ii) $A$ is invertible, reversible, and its reverse automaton is
invertible;
(iii) $A$ is invertible, reversible, and its inverse automaton is
reversible;
(iv) $A$ is invertible, its dual automaton is invertible, and the dual
automaton of its inverse is invertible.
\end{lemma}
\begin{proof}
Suppose $A=(Q,X,\phi,\psi)$ is a finite automaton. By Lemma \ref{auto2},
$A$ is invertible if and only if maps $\psi(q,\cdot):X\to X$ are bijective
for all $q\in Q$. By Lemma \ref{auto3}, $A$ is reversible if and only if
maps $\phi(\cdot,x):Q\to Q$ are bijective for all $x\in X$. Let $\Gamma$
be the Moore diagram of $A$ and $\Gamma'$ be the graph obtained from
$\Gamma$ by reversing all edges and interchanging fields of all labels.
The graph $\Gamma'$ is the Moore diagram of an automaton if for any $q\in
Q$ and $x\in X$ there is exactly one edge of $\Gamma'$ that starts at the
vertex $q$ and has $x$ as the input field of its label. By definition of
$\Gamma'$ the number of edges with the latter property is equal to the
number of pairs $(p,y)\in Q\times X$ such that $q=\phi(p,y)$ and
$x=\psi(p,y)$. Therefore $\Gamma'$ is the Moore diagram of an automaton if
and only if the map $(\phi,\psi):Q\times X\to Q\times X$ is bijective.
Thus $A$ is bi-reversible if and only if it is invertible, reversible, and
$\Gamma'$ is the Moore diagram of an automaton.
Assume that the automaton $A$ is invertible and reversible. Let $I$ and
$R$ be the inverse and reverse automata of $A$, respectively. If the graph
$\Gamma'$ is the Moore diagram of an automaton then the automaton is both
the inverse automaton of $R$ and the reverse automaton of $I$. On the
other hand, if $\Gamma'$ is not the Moore diagram of an automaton then $R$
is not invertible and $I$ is not reversible. It follows that conditions
(i), (ii), and (iii) are equivalent.
It follows from Lemmas \ref{auto2} and \ref{auto3} that a finite automaton
is reversible if and only if its dual automaton is invertible. This
implies that conditions (iii) and (iv) are equivalent.
\end{proof}
\begin{lemma}\lambdabel{auto7}
If an automaton is bi-reversible then its inverse, reverse, and dual
automata are also bi-reversible.
\end{lemma}
\begin{proof}
It follows directly from definitions that an automaton is bi-reversible if
and only if its dual automaton is bi-reversible.
Suppose $A$ is a bi-reversible automaton. By Lemma \ref{auto6}, $A$ is
invertible and reversible. Let $I$ and $R$ denote the inverse and reverse
automata of $A$, respectively. By Lemma \ref{auto6}, $I$ is reversible and
$R$ is invertible. It is easy to see that $A$ is both the inverse
automaton of $I$ and the reverse automaton of $R$. Therefore the automata
$I$ and $R$ are invertible and reversible. Moreover, the inverse automaton
of $I$ is reversible and the reverse automaton of $R$ is invertible. By
Lemma \ref{auto6}, the automata $I$ and $R$ are bi-reversible.
\end{proof}
Suppose $A^{(1)}=(Q_1,X,\phi_1,\psi_1),\dots,A^{(k)}=(Q_k,X,\phi_k,\psi_k)$
are automata over the same alphabet $X$ such that their sets of internal
states $Q_1,Q_2,\dots,Q_k$ are disjoint. The {\em disjoint union\/} of
automata $A^{(1)},A^{(2)},\dots,A^{(k)}$ is an automaton
$U=(Q_1\cup\dots\cup Q_k,X,\phi,\psi)$, where the functions $\phi$, $\psi$
are defined so that $\phi=\phi_i$ and $\psi=\psi_i$ on $Q_i\times X$ for
$1\le i\le k$. Obviously, $U_q=A^{(i)}_q$ for all $q\in Q_i$, $1\le i\le
k$. The Moore diagram of the automaton $U$ is the disjoint union of the
Moore diagrams of $A^{(1)},A^{(2)},\dots,A^{(k)}$.
\begin{lemma}\lambdabel{auto8}
The disjoint union of automata $A^{(1)},A^{(2)},\dots,A^{(k)}$ is
invertible (resp. reversible, bi-reversible) if and only if each $A^{(i)}$
is invertible (resp. reversible, bi-reversible).
\end{lemma}
\begin{proof}
Suppose that an automaton $U$ is the disjoint union of automata $A^{(1)},
\dots,A^{(k)}$. Note that the disjoint union of graphs $\Gamma_1,\dots,
\Gamma_k$ is the Moore diagram of an automaton over an alphabet $X$ if and
only if each $\Gamma_i$ is the Moore diagram of an automaton defined over
$X$. Since the Moore diagram of $U$ is the disjoint union of the Moore
diagrams of $A^{(1)},\dots,A^{(k)}$, it follows that $U$ is invertible
(resp. reversible) if and only if each $A^{(i)}$ is invertible (resp.
reversible). Moreover, if $U$ is invertible then its inverse automaton is
the disjoint union of the inverse automata of $A^{(1)},\dots,A^{(k)}$.
Hence the inverse automaton of $U$ is reversible if and only if the inverse
automaton of each $A^{(i)}$ is reversible. Now Lemma \ref{auto6} implies
that $U$ is bi-reversible if and only if each $A^{(i)}$ is bi-reversible.
\end{proof}
\section{The Aleshin automaton}\lambdabel{a}
In this section we recall constructions and results of the paper \cite{VV}
where the Aleshin automaton was studied. Some constructions are slightly
modified.
The Aleshin automaton is an automaton $A$ over the alphabet $X=\{0,1\}$
with the set of internal states $Q=\{a,b,c\}$. The state transition
function $\phi$ and the output function $\psi$ of $A$ are defined as
follows: $\phi(a,0)=\phi(b,1)=c$, $\phi(a,1)=\phi(b,0)=b$, $\phi(c,0)=
\phi(c,1)=a$; $\psi(a,0)=\psi(b,0)=\psi(c,1)=1$, $\psi(a,1)=\psi(b,1)=
\psi(c,0)=0$. The Moore diagram of $A$ is depicted in Figure \ref{fig1}.
It is easy to verify that the automaton $A$ is invertible and reversible.
Moreover, the inverse automaton of $A$ can be obtained from $A$ by
renaming letters $0$ and $1$ of the alphabet to $1$ and $0$, respectively.
The reverse automaton of $A$ can be obtained from $A$ by renaming its
states $a$ and $c$ to $c$ and $a$, respectively. Lemma \ref{auto6}
implies that $A$ is bi-reversible.
\begin{figure}
\caption{
\lambdabel{fig3}
\end{figure}
Let $I$ denote the automaton obtained from the inverse of $A$ by renaming
its states $a$, $b$, $c$ to $a^{-1}$, $b^{-1}$, $c^{-1}$, respectively.
Here, $a^{-1}$, $b^{-1}$, and $c^{-1}$ are assumed to be elements of the
free group on generators $a$, $b$, $c$. Further, let $U$ denote the
disjoint union of automata $A$ and $I$. The automaton $U$ is defined over
the alphabet $X=\{0,1\}$, with the set of internal states $Q^\pm=
\{a,b,c,a^{-1},b^{-1},c^{-1}\}$. By definition, $U_a=A_a$, $U_b=A_b$,
$U_c=A_c$, $U_{a^{-1}}=A_a^{-1}$, $U_{b^{-1}}=A_b^{-1}$, $U_{c^{-1}}=
A_c^{-1}$.
\begin{figure}
\caption{
\lambdabel{fig4}
\end{figure}
Let $D$ denote the dual automaton of the automaton $U$. The automaton $D$
is defined over the alphabet $Q^\pm$, with two internal states $0$ and $1$.
By $\phi_D$ denote its transition function. Then $\phi_D(0,q)=1$ and
$\phi_D(1,q)=0$ for $q\in\{a,b,a^{-1},b^{-1}\}$, while $\phi_D(0,q)=0$ and
$\phi_D(1,q)=1$ for $q\in\{c,c^{-1}\}$. Also, we consider an auxiliary
automaton $E$ that is closely related to $D$. By definition, the automaton
$E$ shares with $D$ the alphabet, the set of internal states, and the state
transition function. The output function $\psi_E$ of $E$ is defined so
that $\psi_E(0,q)=\sigma_0(q)$ and $\psi_E(1,q)=\sigma_1(q)$ for all $q\in
Q^\pm$, where $\sigma_0=(a^{-1}b^{-1})$ and $\sigma_1=(ab)$ are permutations on
the set $Q^\pm$.
\begin{figure}
\caption{
\lambdabel{fig5}
\end{figure}
Lemmas \ref{auto7} and \ref{auto8} imply that $I$, $U$, and $D$ are
bi-reversible automata. As for the automaton $E$, it is easy to verify
that $E$ coincides with its inverse automaton while the reverse automaton
of $E$ can be obtained from $E$ by renaming its states $0$ and $1$ to $1$
and $0$, respectively. Hence $E$ is bi-reversible due to Lemma
\ref{auto6}.
To each permutation $\tau$ on the set $Q=\{a,b,c\}$ we assign an
automorphism $\pi_\tau$ of the free monoid $(Q^\pm)^*$. The automorphism
$\pi_\tau$ is uniquely defined by $\pi_\tau(q)=\tau(q)$, $\pi_\tau(q^{-1})=
(\tau(q))^{-1}$ for all $q\in Q$. Let $\lambdangle a,b,c\rangle$ denote the
free group on generators $a$, $b$, and $c$, let $\delta:(Q^\pm)^*\to\lambdangle
a,b,c\rangle$ be the homomorphism that sends each element of $Q^\pm\subset
(Q^\pm)^*$ to itself, and let $p_\tau$ be the automorphism of $\lambdangle
a,b,c\rangle$ defined by $p_\tau(q)=\tau(q)$, $q\in Q$. Then
$\delta(\pi_\tau(\xi))=p_\tau(\delta(\xi))$ for all $\xi\in(Q^\pm)^*$.
\begin{lemma}[\cite{VV}]\lambdabel{a1}
(i) $E_0^2=E_1^2=1$, $E_0E_1=E_1E_0=\pi_{(ab)}$;
(ii) $D_0=\pi_{(ac)}E_0=\pi_{(abc)}E_1$, $D_1=\pi_{(abc)}E_0=
\pi_{(ac)}E_1$.
\end{lemma}
\begin{proposition}[\cite{VV}]\lambdabel{a2}
The group $G(D)$ contains $E_0$, $E_1$, and all transformations of the form
$\pi_\tau$. Moreover, $G(D)$ is generated by $E_0$, $\pi_{(ab)}$, and
$\pi_{(bc)}$.
\end{proposition}
As shown in Section \ref{auto}, the automaton $U$ defines a right action
$X^*\times(Q^\pm)^*\to X^*$ of the monoid $(Q^\pm)^*$ on the rooted binary
tree $X^*$ given by $(w,\xi)\mapsto U_\xi(w)$. Let $\chi:(Q^\pm)^*\to
\{-1,1\}$ be the unique homomorphism such that $\chi(a)=\chi(b)=
\chi(a^{-1})=\chi(b^{-1})=-1$, $\chi(c)=\chi(c^{-1})=1$.
\begin{lemma}[\cite{VV}]\lambdabel{a3}
Given $\xi\in(Q^\pm)^*$, the automorphism $U_\xi$ of the rooted binary tree
$\{0,1\}^*$ acts trivially on the first level of the tree (i.e., on
one-letter words) if and only if $\chi(\xi)=1$.
\end{lemma}
Now we introduce an alphabet consisting of two symbols $*$ and $*^{-1}$. A
word over the alphabet $\{*,*^{-1}\}$ is called a {\em pattern}. Every
word $\xi$ over the alphabet $Q^\pm$ is assigned a pattern $v$ that is
obtained from $\xi$ by substituting $*$ for each occurrence of letters
$a,b,c$ and substituting $*^{-1}$ for each occurrence of letters
$a^{-1},b^{-1},c^{-1}$. We say that $v$ is the pattern of $\xi$ or that
$\xi$ follows the pattern $v$.
A word $\xi=q_1q_2\dots q_n\in(Q^\pm)^*$ is called {\em freely
irreducible\/} if none of its two-letter subwords $q_1q_2,q_2q_3,\dots,
q_{n-1}q_n$ coincides with one of the following words: $aa^{-1},bb^{-1},
cc^{-1},a^{-1}a,b^{-1}b,c^{-1}c$. Otherwise $\xi$ is called {\em freely
reducible}.
\begin{lemma}[\cite{VV}]\lambdabel{a4}
For any nonempty pattern $v$ there exist words $\xi_1,\xi_2\in(Q^\pm)^*$
such that $\xi_1$ and $\xi_2$ are freely irreducible, follow the pattern
$v$, and $\chi(\xi_2)=-\chi(\xi_1)$.
\end{lemma}
\begin{proposition}[\cite{VV}]\lambdabel{a5}
Suppose $\xi\in(Q^\pm)^*$ is a freely irreducible word. Then the orbit of
$\xi$ under the action of the group $G(D)$ on $(Q^\pm)^*$ consists of all
freely irreducible words following the same pattern as $\xi$.
\end{proposition}
\begin{corollary}\lambdabel{a5plus}
The group defined by the dual automaton of $A$ acts transitively on each
level of the rooted ternary tree $Q^*$.
\end{corollary}
\begin{proof}
Let $D^+$ denote the dual automaton of $A$. The rooted tree $Q^*$ is a
subtree of $(Q^\pm)^*$. It is easy to see that $Q^*$ is invariant under
transformations $D_0$, $D_1$ and the restrictions of these transformations
to $Q^*$ are $D^+_0$, $D^+_1$. In particular, the orbits of the $G(D^+)$
action on $Q^*$ are those orbits of the $G(D)$ action on $(Q^\pm)^*$ that
are contained in $Q^*$. Any level of the tree $Q^*$ consists of words of a
fixed length over the alphabet $Q$. As elements of $(Q^\pm)^*$, all these
words are freely irreducible and follow the same pattern. Proposition
\ref{a5} implies that they are in the same orbit of the $G(D^+)$ action.
\end{proof}
Lemmas \ref{a3}, \ref{a4} and Proposition \ref{a5} lead to the following
statement.
\begin{theorem}[\cite{VV}]\lambdabel{a6}
The group $G(A)$ is the free non-Abelian group on generators $A_a$, $A_b$,
$A_c$.
\end{theorem}
\section{Series of finite automata of Aleshin type}\lambdabel{series}
In this section we consider a series of finite automata starting from the
Aleshin automaton. We use the notation of the previous section.
For any integer $n\ge1$ we define an Aleshin type automaton $A^{(n)}$.
This is an automaton over the alphabet $X=\{0,1\}$ with a set of states
$Q_n$ of cardinality $2n+1$. The states of $A^{(n)}$ are denoted so that
$Q_1=\{a_1,b_1,c_1\}$ and $Q_n=\{a_n,b_n,c_n,q_{n1},\dots,q_{n,2n-2}\}$ for
$n\ge2$. The state transition function $\phi_n$ of $A^{(n)}$ is defined as
follows: $\phi_n(a_n,0)=\phi_n(b_n,1)=c_n$, $\phi_n(a_n,1)=\phi_n(b_n,0)=
b_n$, and $\phi_n(q_{ni},0)=\phi_n(q_{ni},1)=q_{n,i+1}$ for $0\le i\le
2n-2$, where by definition $q_{n0}=c_n$ and $q_{n,2n-1}=a_n$. The output
function $\psi_n$ of $A^{(n)}$ is defined so that for any $x\in X$ we have
$\psi_n(q,x)=1-x$ if $q\in\{a_n,b_n\}$ and $\psi_n(q,x)=x$ if $q\in
Q_n\setminus\{a_n,b_n\}$.
\begin{figure}
\caption{
\lambdabel{fig6}
\end{figure}
Up to renaming of the internal states, $A^{(1)}$ and $A^{(2)}$ are the two
automata introduced by Aleshin \cite{A} (see Figure \ref{fig1}).
We shall deal with automata $A^{(n)}$ by following the framework developed
in the paper \cite{VV} and described in Section \ref{a}.
Let us fix a positive integer $n$. It is easy to see that the inverse
automaton of the automaton $A^{(n)}$ can be obtained from $A^{(n)}$ by
renaming letters $0$ and $1$ of the alphabet to $1$ and $0$, respectively.
Besides, the reverse automaton of $A^{(n)}$ can be obtained from $A^{(n)}$
by renaming its states $c_n,q_{n1},\dots,q_{n,2n-2},a_n$ to
$a_n,q_{n,2n-2},\dots,q_{n1},c_n$, respectively. Lemma \ref{auto6} implies
that $A^{(n)}$ is bi-reversible.
Let $I^{(n)}$ denote the automaton obtained from the inverse of $A^{(n)}$
by renaming each state $q\in Q_n$ to $q^{-1}$, where $q^{-1}$ is regarded
as an element of the free group on generators $a_n,b_n,c_n,q_{n1},\dots,
q_{n,2n-2}$. Further, let $U^{(n)}$ denote the disjoint union of automata
$A^{(n)}$ and $I^{(n)}$. The automaton $U^{(n)}$ is defined over the
alphabet $X=\{0,1\}$, with the set of internal states $Q_n^\pm=
\bigcup_{q\in Q_n}\{q,q^{-1}\}$. By definition, $U^{(n)}_q=A^{(n)}_q$ and
$U^{(n)}_{q^{-1}}=(A^{(n)}_q)^{-1}$ for all $q\in Q_n$.
Let $D^{(n)}$ denote the dual automaton of the automaton $U^{(n)}$. The
automaton $D^{(n)}$ is defined over the alphabet $Q_n^\pm$, with two
internal states $0$ and $1$. By $\lambda_n$ denote its transition function.
Then $\lambda_n(0,q)=1$ and $\lambda_n(1,q)=0$ if $q\in\{a_n,b_n,a_n^{-1},
b_n^{-1}\}$ while $\lambda_n(0,q)=0$ and $\lambda_n(1,q)=1$ otherwise. Also, we
consider an auxiliary automaton $E^{(n)}$. By definition, the automaton
$E^{(n)}$ shares with $D^{(n)}$ the alphabet, the set of internal states,
and the state transition function. The output function $\mu_n$ of
$E^{(n)}$ is defined so that $\mu_n(0,q)=\sigma_0(q)$ and $\mu_n(1,q)=
\sigma_1(q)$ for all $q\in Q_n^\pm$, where $\sigma_0=(a_n^{-1}b_n^{-1})$ and
$\sigma_1=(a_nb_n)$ are permutations on the set $Q_n^\pm$.
Lemmas \ref{auto7} and \ref{auto8} imply that $I^{(n)}$, $U^{(n)}$, and
$D^{(n)}$ are bi-reversible automata. Further, it is easy to see that the
automaton $E^{(n)}$ coincides with its inverse automaton while the reverse
automaton of $E^{(n)}$ can be obtained from $E^{(n)}$ by renaming its
states $0$ and $1$ to $1$ and $0$, respectively. By Lemma \ref{auto6},
$E^{(n)}$ is bi-reversible.
To each permutation $\tau$ on the set $Q_n$ we assign an automorphism
$\pi^{(n)}_\tau$ of the free monoid $(Q_n^\pm)^*$ such that
$\pi^{(n)}_\tau(q)=\tau(q)$, $\pi^{(n)}_\tau(q^{-1})=(\tau(q))^{-1}$ for
all $q\in Q_n$. The automorphism $\pi^{(n)}_\tau$ is uniquely determined
by $\tau$.
\begin{lemma}\lambdabel{series1}
(i) $(E^{(n)}_0)^2=(E^{(n)}_1)^2=1$, $E^{(n)}_0E^{(n)}_1=
E^{(n)}_1E^{(n)}_0=\pi^{(n)}_{(a_nb_n)}$;
(ii) $D^{(n)}_0=\pi^{(n)}_{\tau_0}E^{(n)}_0=\pi^{(n)}_{\tau_1}E^{(n)}_1$,
$D^{(n)}_1=\pi^{(n)}_{\tau_1}E^{(n)}_0=\pi^{(n)}_{\tau_0}E^{(n)}_1$, where
$\tau_0=(a_nc_nq_{n1}\dots q_{n,2n-2})$, $\tau_1=(a_nb_nc_nq_{n1}\dots
q_{n,2n-2})$.
\end{lemma}
\begin{proof}
Since the inverse automaton of $E^{(n)}$ coincides with $E^{(n)}$, Lemma
\ref{auto2} implies that $(E^{(n)}_0)^2=(E^{(n)}_1)^2=1$.
We have that $E^{(n)}=(X,Q_n^\pm,\lambda_n,\mu_n)$, where the functions $\lambda_n$
and $\mu_n$ are defined above. Note that the function $\lambda_n$ does not
change when elements $0$ and $1$ of the set $X$ are renamed to $1$ and $0$,
respectively. For any permutation $\sigma$ on the set $Q_n^\pm$ we define an
automaton $Y^\sigma=(X,Q_n^\pm,\lambda_n,\sigma\mu_n)$. The Moore diagram of $Y^\sigma$
is obtained from the Moore diagram of $E^{(n)}$ by applying $\sigma$ to the
output fields of all labels. It is easy to observe that $Y^\tau_0=
\alpha_\sigma E^{(n)}_0$ and $Y^\tau_1=\alpha_\sigma E^{(n)}_1$, where
$\alpha_\sigma$ is the unique automorphism of the monoid $(Q_n^\pm)^*$ such
that $\alpha_\sigma(q)=\sigma(q)$ for all $q\in Q_n^\pm$.
Let us consider the following permutations on $Q_n^\pm$:
\begin{eqnarray*}
& \sigma_0=(a_n^{-1}b_n^{-1}), \qquad \sigma_1=(a_nb_n), \qquad
\sigma_2=(a_nb_n)(a_n^{-1}b_n^{-1}),\\
& \sigma_3=(a_nc_nq_{n1}\dots q_{n,2n-2})(a_n^{-1}b_n^{-1}c_n^{-1}q_{n1}^{-1}
\dots q_{n,2n-2}^{-1}),\\
& \sigma_4=(a_nb_nc_nq_{n1}\dots q_{n,2n-2})(a_n^{-1}c_n^{-1}q_{n1}^{-1}\dots
q_{n,2n-2}^{-1}),\\
& \sigma_5=(a_nc_nq_{n1}\dots q_{n,2n-2})(a_n^{-1}c_n^{-1}q_{n1}^{-1}\dots
q_{n,2n-2}^{-1}),\\
& \sigma_6=(a_nb_nc_nq_{n1}\dots q_{n,2n-2})
(a_n^{-1}b_n^{-1}c_n^{-1}q_{n1}^{-1}\dots q_{n,2n-2}^{-1}).
\end{eqnarray*}
Since $\sigma_2\sigma_0=\sigma_1$ and $\sigma_2\sigma_1=\sigma_0$, it follows that the
automaton $Y^{\sigma_2}$ can be obtained from $E^{(n)}$ by renaming its states
$0$ and $1$ to $1$ and $0$, respectively. Therefore $E^{(n)}_0=
Y^{\sigma_2}_1=\alpha_{\sigma_2}E^{(n)}_1$ and $E^{(n)}_1=Y^{\sigma_2}_0=
\alpha_{\sigma_2}E^{(n)}_0$. Consequently, $E^{(n)}_0E^{(n)}_1=\alpha_{\sigma_2}
(E^{(n)}_1)^2=\alpha_{\sigma_2}$ and $E^{(n)}_1E^{(n)}_0=\alpha_{\sigma_2}
(E^{(n)}_0)^2=\alpha_{\sigma_2}$. Clearly, $\alpha_{\sigma_2}=
\pi^{(n)}_{(a_nb_n)}$.
Since $\sigma_5\sigma_0=\sigma_3$ and $\sigma_5\sigma_1=\sigma_4$, it follows that
$Y^{\sigma_5}=D^{(n)}$. Hence $D^{(n)}_0=\alpha_{\sigma_5}E^{(n)}_0$ and
$D^{(n)}_1=\alpha_{\sigma_5}E^{(n)}_1$. Furthermore, the equalities
$\sigma_6\sigma_0=\sigma_4$ and $\sigma_6\sigma_1=\sigma_3$ imply that the automaton
$Y^{\sigma_6}$ can be obtained from $D^{(n)}$ by renaming its states $0$ and
$1$ to $1$ and $0$, respectively. Therefore $D^{(n)}_0=Y^{\sigma_6}_1=
\alpha_{\sigma_6}E^{(n)}_1$ and $D^{(n)}_1=Y^{\sigma_6}_0=\alpha_{\sigma_6}
E^{(n)}_0$. It remains to notice that $\alpha_{\sigma_5}=\pi^{(n)}_{\tau_0}$
and $\alpha_{\sigma_6}=\pi^{(n)}_{\tau_1}$.
\end{proof}
\begin{lemma}\lambdabel{series2}
For any integer $M\ge3$ the group of permutations on the set $\{1,2,\dots,
M\}$ is generated by permutations $(12)$ and $(123\dots M)$.
\end{lemma}
\begin{proof}
Let $\tau_0=(12)$, $\tau_1=(123\dots M)$, and $\tau_2=(23\dots M)$. Then
$\tau_2=\tau_0\tau_1$. For any $k$, $2\le k\le M$ we have $(1k)=
\tau_2^{k-2}\tau_0\tau_2^{-(k-2)}$. Further, for any $l$ and $m$, $1\le l<
m\le M$ we have $(lm)=\tau_1^{l-1}(1k)\tau_1^{-(l-1)}$, where $k=m-l+1$.
Therefore the group generated by $\tau_0$ and $\tau_1$ contains all
transpositions $(lm)$, $1\le l<m\le M$. It remains to notice that any
permutation on $\{1,2,\dots,M\}$ is a product of transpositions.
\end{proof}
\begin{proposition}\lambdabel{series3}
The group $G(D^{(n)})$ contains $E^{(n)}_0$, $E^{(n)}_1$, and all
transformations of the form $\pi^{(n)}_\tau$. Moreover, $G(D^{(n)})$ is
generated by $E^{(n)}_0$, $\pi^{(n)}_{\tau_0}$, and $\pi^{(n)}_{\tau_1}$,
where $\tau_0=(a_nc_nq_{n1}\dots q_{n,2n-2})$, $\tau_1=
(a_nb_nc_nq_{n1}\dots q_{n,2n-2})$.
\end{proposition}
\begin{proof}
It is easy to see that $\pi^{(n)}_{\tau\sigma}=\pi^{(n)}_\tau\pi^{(n)}_\sigma$
for any permutations $\tau$ and $\sigma$ on the set $Q_n$. It follows that
$\pi^{(n)}_{\tau^{-1}}=(\pi^{(n)}_\tau)^{-1}$ for any permutation $\tau$ on
$Q_n$.
By Lemma \ref{series1}, the group generated by $E^{(n)}_0$,
$\pi^{(n)}_{\tau_0}$, and $\pi^{(n)}_{\tau_1}$ contains $G(D^{(n)})$.
Besides, $D^{(n)}_0(D^{(n)}_1)^{-1}=\pi^{(n)}_{\tau_0}E^{(n)}_0
(\pi^{(n)}_{\tau_1}E^{(n)}_0)^{-1}=\pi^{(n)}_{\tau_0}
(\pi^{(n)}_{\tau_1})^{-1}$. By the above $\pi^{(n)}_{\tau_0}
(\pi^{(n)}_{\tau_1})^{-1}=\pi^{(n)}_{\tau_2}$, where $\tau_2=
\tau_0\tau_1^{-1}=(b_nc_n)$. Similarly,
$$
(D^{(n)}_0)^{-1}D^{(n)}_1=(\pi^{(n)}_{\tau_0}E^{(n)}_0)^{-1}
\pi^{(n)}_{\tau_1}E^{(n)}_0=(E^{(n)}_0)^{-1}\pi^{(n)}_{\tau_3}E^{(n)}_0,
$$
where $\tau_3=\tau_0^{-1}\tau_1=(a_nb_n)$. Lemma \ref{series1} implies
that $E^{(n)}_0$ and $\pi^{(n)}_{\tau_3}$ commute, hence
$(D^{(n)}_0)^{-1}D^{(n)}_1=\pi^{(n)}_{\tau_3}$. Consider two more
permutations on $Q_n$: $\tau_4=(a_nc_n)$ and $\tau_5=(c_nq_{n1}\dots
q_{n,2n-2})$. Note that $\tau_4=\tau_2\tau_3\tau_2$ and $\tau_5=
\tau_4\tau_0$. By the above $\pi^{(n)}_{\tau_2},\pi^{(n)}_{\tau_3}\in
G(D^{(n)})$, hence $\pi^{(n)}_{\tau_4}\in G(D^{(n)})$. Then
$\pi^{(n)}_{\tau_5}E^{(n)}_0=\pi^{(n)}_{\tau_4}\pi^{(n)}_{\tau_0}
E^{(n)}_0=\pi^{(n)}_{\tau_4}D^{(n)}_0\in G(D^{(n)})$. Since $\tau_5(a_n)=
a_n$ and $\tau_5(b_n)=b_n$, it easily follows that transformations
$\pi^{(n)}_{\tau_5}$ and $E^{(n)}_0$ commute. As $\tau_5$ is a permutation
of odd order $2n-1$ while $E^{(n)}_0$ is an involution, we have that
$(\pi^{(n)}_{\tau_5}E^{(n)}_0)^{2n-1}=E^{(n)}_0$. In particular,
$E^{(n)}_0\in G(D^{(n)})$. Now Lemma \ref{series1} implies that
$\pi^{(n)}_{\tau_0},\pi^{(n)}_{\tau_1},E^{(n)}_1\in G(D^{(n)})$.
By Lemma \ref{series2}, the group of all permutations on the set $Q_n$ is
generated by permutations $\tau_1$ and $\tau_3$. Since
$\pi^{(n)}_{\tau_1},\pi^{(n)}_{\tau_3}\in G(D^{(n)})$, it follows that
$G(D^{(n)})$ contains all transformations of the form $\pi^{(n)}_\tau$.
\end{proof}
Recall that words over the alphabet $\{*,*^{-1}\}$ are called patterns.
Every word $\xi\in(Q_n^\pm)^*$ is assigned a pattern $v$ that is obtained
from $\xi$ by substituting $*$ for each occurrence of letters $a_n,b_n,c_n,
q_{n1},\dots,q_{n,2n-2}$ and substituting $*^{-1}$ for each occurrence of
letters $a_n^{-1},b_n^{-1},c_n^{-1},q_{n1}^{-1},\dots,q_{n,2n-2}^{-1}$. We
say that $\xi$ follows the pattern $v$.
A word $\xi=q_1q_2\dots q_k\in(Q_n^\pm)^*$ is called freely irreducible if
none of its two-letter subwords $q_1q_2,q_2q_3,\dots,q_{k-1}q_k$ is of the
form $qq^{-1}$ or $q^{-1}q$, where $q\in Q_n$. Otherwise $\xi$ is called
freely reducible.
\begin{lemma}\lambdabel{series4}
For any nonempty pattern $v$ there exists a freely irreducible word
$\xi\in(Q_n^\pm)^*$ such that $v$ is the pattern of $\xi$ and the
transformation $U^{(n)}_\xi$ acts nontrivially on the first level of the
rooted binary tree $X^*$.
\end{lemma}
\begin{proof}
Given a nonempty pattern $v$, let us substitute $a_n$ for each occurrence
of $*$ in $v$ and $b_n^{-1}$ for each occurrence of $*^{-1}$. We get a
word $\xi\in(Q_n^\pm)^*$ that follows the pattern $v$. Now let us modify
$\xi$ by changing its last letter. If this letter is $a_n$, we change it
to $c_n$. If the last letter of $\xi$ is $b_n^{-1}$, we change it to
$c_n^{-1}$. This yields another word $\eta\in(Q_n^\pm)^*$ that follows the
pattern $v$. By construction, $\xi$ and $\eta$ are freely irreducible.
Furthermore, $U^{(n)}_\eta=A^{(n)}_{c_n}(A^{(n)}_{a_n})^{-1}U^{(n)}_\xi$ if
the last letter of $v$ is $*$ while $U^{(n)}_\eta=(A^{(n)}_{c_n})^{-1}
A^{(n)}_{b_n}U^{(n)}_\xi$ if the last letter of $v$ is $*^{-1}$. Both
$A^{(n)}_{c_n}(A^{(n)}_{a_n})^{-1}$ and $(A^{(n)}_{c_n})^{-1}A^{(n)}_{b_n}$
interchange one-letter words $0$ and $1$. It follows that one of the
transformations $U^{(n)}_\xi$ and $U^{(n)}_\eta$ also acts nontrivially on
the first level of the rooted tree $\{0,1\}^*$.
\end{proof}
Given a nonempty, freely irreducible word $\xi\in(Q_n^\pm)^*$, let
$Z_n(\xi)$ denote the set of all freely irreducible words in $(Q_n^\pm)^*$
that follow the same pattern as $\xi$ and match $\xi$ completely or except
for the last letter. Obviously, $\xi\in Z_n(\xi)$, and $\eta\in Z_n(\xi)$
if and only if $\xi\in Z_n(\eta)$. The set $Z_n(\xi)$ consists of $2n$ or
$2n+1$ words. Namely, there are exactly $2n+1$ words in $(Q_n^\pm)^*$ that
follow the same pattern as $\xi$ and match $\xi$ completely or except for
the last letter. However if the last two letters in the pattern of $\xi$
are distinct then one of these $2n+1$ words is freely reducible.
\begin{lemma}\lambdabel{series5}
For any nonempty pattern $v$ there exists a freely irreducible word $\xi\in
(Q_n^\pm)^*$ such that $v$ is the pattern of $\xi$ and the set $Z_n(\xi)$
is contained in one orbit of the $G(D^{(n)})$ action on $(Q_n^\pm)^*$.
\end{lemma}
\begin{proof}
Let $h_n:(Q^\pm)^*\to(Q_n^\pm)^*$ be the homomorphism of monoids such that
$h_n(a)=a_n$, $h_n(b)=b_n$, $h_n(c)=c_n$, $h_n(a^{-1})=a_n^{-1}$,
$h_n(b^{-1})=b_n^{-1}$, $h_n(c^{-1})=c_n^{-1}$. The range of $h_n$
consists of words over alphabet $\{a_n,b_n,c_n,a_n^{-1},b_n^{-1},
c_n^{-1}\}$. For any $\zeta\in(Q^\pm)^*$ the word $h_n(\zeta)$ follows the
same pattern as $\zeta$. Besides, $h_n(\zeta)$ is freely irreducible if
and only if $\zeta$ is. It is easy to see that $h_n(\pi_{(ab)}(\zeta))=
\pi^{(n)}_{(a_nb_n)}(h_n(\zeta))$, $h_n(\pi_{(bc)}(\zeta))=
\pi^{(n)}_{(b_nc_n)}(h_n(\zeta))$, and $h_n(E_0(\zeta))=
E^{(n)}_0(h_n(\zeta))$. By Proposition \ref{a2}, the group $G(D)$ is
generated by $\pi_{(ab)}$, $\pi_{(bc)}$, and $E_0$. On the other hand,
$\pi^{(n)}_{(a_nb_n)},\pi^{(n)}_{(b_nc_n)},E^{(n)}_0\in G(D^{(n)})$ due to
Proposition \ref{series3}. It follows that for any $g_0\in G(D)$ there
exists $g\in G(D^{(n)})$ such that $h_n(g_0(\zeta))=g(h_n(\zeta))$ for all
$\zeta\in(Q^\pm)^*$. Now Proposition \ref{a5} implies that two words over
alphabet $\{a_n,b_n,c_n,a_n^{-1},b_n^{-1},c_n^{-1}\}$ are in the same orbit
of the $G(D^{(n)})$ action on $(Q_n^\pm)^*$ whenever they are freely
irreducible and follow the same pattern.
Let $v_0$ be the pattern obtained by deleting the last letter of $v$. We
substitute $a_n$ for each occurrence of $*$ in $v_0$ and $b_n^{-1}$ for
each occurrence of $*^{-1}$. This yields a word $\eta\in(Q_n^\pm)^*$ that
follows the pattern $v_0$. Now let $\xi=\eta c_n$ if the last letter of
$v$ is $*$ and let $\xi=\eta c_n^{-1}$ otherwise. Clearly, $\xi$ is a
freely irreducible word following the pattern $v$. Take any $\zeta\in
Z_n(\xi)$. If both $\zeta$ and $\xi$ are words over alphabet $\{a_n,b_n,
c_n,a_n^{-1},b_n^{-1},c_n^{-1}\}$, then it follows from the above that
$\zeta=g(\xi)$ for some $g\in G(D^{(n)})$. Otherwise the last letter of
$\zeta$ is $q_{ni}$ or $q_{ni}^{-1}$, where $1\le i\le 2n-2$. In this case
we have $\zeta=(\pi^{(n)}_\tau)^i(\xi)$, where $\tau=(c_nq_{n1}\dots
q_{n,2n-2})$. By Proposition \ref{series3}, $\pi^{(n)}_\tau\in
G(D^{(n)})$.
\end{proof}
\begin{proposition}\lambdabel{series6}
Suppose $\xi\in(Q_n^\pm)^*$ is a freely irreducible word. Then the orbit
of $\xi$ under the action of the group $G(D^{(n)})$ on $(Q_n^\pm)^*$
consists of all freely irreducible words following the same pattern as
$\xi$.
\end{proposition}
\begin{proof}
First we shall show that the $G(D^{(n)})$ action on $(Q_n^\pm)^*$
preserves patterns and free irreducibility of words. Let $\phi_n^\pm$ and
$\psi_n^\pm$ denote the state transition and output functions of the
automaton $U^{(n)}$. By $\tilde\phi_n$ and $\tilde\psi_n$ denote the state
transition and output functions of its dual $D^{(n)}$. Take any $q\in
Q_n^\pm$ and $x\in X$. By definition of $U^{(n)}$ we have that
$\phi_n^\pm(q,x)\in Q_n$ if and only if $q\in Q_n$. Since
$\phi_n^\pm(q,x)=\tilde\psi_n(x,q)$, it follows that transformations
$D^{(n)}_0$ and $D^{(n)}_1$ preserve patterns of words. So does any $g\in
G(D^{(n)})$. Further, let $p=\phi_n^\pm(q,x)$ and $y=\psi_n^\pm(q,x)$.
Then $\phi_n^\pm(q^{-1},y)=p^{-1}$ and $\psi_n^\pm(q^{-1},y)=x$.
Consequently, $D^{(n)}_x(qq^{-1})=\tilde\psi_n(x,q)
\tilde\psi_n(\tilde\phi_n(x,q),q^{-1})=\phi_n^\pm(q,x)
\phi_n^\pm(q^{-1},\psi_n^\pm(q,x))=pp^{-1}$. It follows that the set
$P=\{qq^{-1}\mid q\in Q_n^\pm\}\subset(Q_n^\pm)^*$ is invariant under
$D^{(n)}_0$ and $D^{(n)}_1$. Any freely reducible word $\xi\in(Q_n^\pm)^*$
is represented as $\xi_1\xi_0\xi_2$, where $\xi_0\in P$ and $\xi_1,\xi_2\in
(Q_n^\pm)^*$. For any $x\in X$ we have $D^{(n)}_x(\xi)=D^{(n)}_x(\xi_1)
D^{(n)}_{x_0}(\xi_0)D^{(n)}_{x_1}(\xi_2)$, where $x_0,x_1\in X$. By the
above $D^{(n)}_x(\xi)$ is freely reducible. Thus $D^{(n)}_0$ and
$D^{(n)}_1$ preserve free reducibility of words. Since these
transformations are invertible, they also preserve free irreducibility, and
so does any $g\in G(D^{(n)})$.
Now we are going to prove that for any freely irreducible words
$\xi_1,\xi_2\in(Q_n^\pm)^*$ following the same pattern $v$ there exists
$g\in G(D^{(n)})$ such that $\xi_2=g(\xi_1)$. The claim is proved by
induction on the length of the pattern $v$. The empty pattern is followed
only by the empty word. Now let $k\ge1$ and assume that the claim holds
for all patterns of length less than $k$. Take any pattern $v$ of length
$k$. By Lemma \ref{series5}, the pattern $v$ is followed by a freely
irreducible word $\xi\in(Q_n^\pm)^*$ such that the set $Z_n(\xi)$ is
contained in an orbit of the $G(D^{(n)})$ action. Suppose $\xi_1,\xi_2\in
(Q_n^\pm)^*$ are freely irreducible words following the pattern $v$. Let
$\eta,\eta_1,\eta_2$ be the words obtained by deleting the last letter of
$\xi,\xi_1,\xi_2$, respectively. Then $\eta,\eta_1,\eta_2$ are freely
irreducible and follow the same pattern of length $k-1$. By the inductive
assumption there are $g_1,g_2\in G(D^{(n)})$ such that $\eta=g_1(\eta_1)=
g_2(\eta_2)$. Since the $G(D^{(n)})$ action preserves patterns and free
irreducibility, it follows that $g_1(\xi_1),g_2(\xi_2)\in Z_n(\xi)$. As
$Z_n(\xi)$ is contained in an orbit, there exists $g_0\in G(D^{(n)})$ such
that $g_0(g_1(\xi_1))=g_2(\xi_2)$. Then $\xi_2=g(\xi_1)$, where
$g=g_2^{-1}g_0g_1\in G(D^{(n)})$.
\end{proof}
\begin{corollary}\lambdabel{series6plus}
The group defined by the dual automaton of $A^{(n)}$ acts transitively on
each level of the rooted tree $Q_n^*$.
\end{corollary}
Corollary \ref{series6plus} follows from Proposition \ref{series6} in the
same way as Corollary \ref{a5plus} follows from Proposition \ref{a5}. We
omit the proof.
\begin{theorem}\lambdabel{series7}
The group $G(A^{(n)})$ is the free non-Abelian group on $2n+1$ generators
$A^{(n)}_q$, $q\in Q_n$.
\end{theorem}
\begin{proof}
The group $G(A^{(n)})$ is the free non-Abelian group on generators $A_q$,
$q\in Q_n$ if and only if $(A^{(n)}_{q_1})^{m_1}(A^{(n)}_{q_2})^{m_2}\dots
(A^{(n)}_{q_k})^{m_k}\ne1$ for any pair of sequences $q_1,\dots,q_k$ and
$m_1,\dots,m_k$ such that $k>0$, $q_i\in Q_n$ and $m_i\in\mathbb{Z}\setminus\{0\}$
for $1\le i\le k$, and $q_i\ne q_{i+1}$ for $1\le i\le k-1$. Since
$U^{(n)}_q=A^{(n)}_q$ and $U^{(n)}_{q^{-1}}=(A^{(n)}_q)^{-1}$ for all $q\in
Q_n$, an equivalent condition is that $U^{(n)}_\xi\ne1$ for any nonempty
freely irreducible word $\xi\in(Q_n^\pm)^*$.
Suppose $U^{(n)}_\xi=1$ for some freely irreducible word $\xi\in
(Q_n^\pm)^*$. By Corollary \ref{auto5}, $U^{(n)}_{g(\xi)}=1$ for all $g\in
S(D^{(n)})$. Then Proposition \ref{auto1} imply that $U^{(n)}_{g(\xi)}=1$
for all $g\in G(D^{(n)})$. Now it follows from Proposition \ref{series6}
that $U^{(n)}_\eta=1$ for any freely irreducible word $\eta\in(Q_n^\pm)^*$
following the same pattern as $\xi$. In particular, $U^{(n)}_\eta$ acts
trivially on the first level of the rooted binary tree $\{0,1\}^*$.
Finally, Lemma \ref{series4} implies that $\xi$ follows the empty pattern.
Then $\xi$ itself is the empty word.
\end{proof}
\section{Disjoint unions}\lambdabel{union}
In this section we consider disjoint unions of Aleshin type automata. We
use the notation of Sections \ref{a} and \ref{series}.
Let $N$ be a nonempty set of positive integers. We denote by $A^{(N)}$ the
disjoint union of automata $A^{(n)}$, $n\in N$. Then $A^{(N)}$ is an
automaton over the alphabet $X=\{0,1\}$ with the set of internal states
$Q_N=\bigcup_{n\in N}Q_n$. It is bi-reversible since each $A^{(n)}$ is
bi-reversible.
Let $I^{(N)}$ denote the disjoint union of automata $I^{(n)}$, $n\in N$.
The automaton $I^{(N)}$ can be obtained from the inverse of $A^{(N)}$ by
renaming each state $q\in Q_N$ to $q^{-1}$. Further, let $U^{(N)}$ denote
the disjoint union of automata $A^{(N)}$ and $I^{(N)}$. Obviously, the
automaton $U^{(N)}$ is the disjoint union of automata $U^{(n)}$, $n\in N$.
It is defined over the alphabet $X=\{0,1\}$, with the set of internal
states $Q_N^\pm=\bigcup_{n\in N}Q_n^\pm$. Clearly, $U^{(N)}_q=A^{(N)}_q$
and $U^{(N)}_{q^{-1}}=(A^{(N)}_q)^{-1}$ for all $q\in Q_N$.
Let $D^{(N)}$ denote the dual automaton of the automaton $U^{(N)}$. The
automaton $D^{(N)}$ is defined over the alphabet $Q_N^\pm$, with two
internal states $0$ and $1$. Also, we consider an auxiliary automaton
$E^{(N)}$. By definition, the automaton $E^{(N)}$ shares with $D^{(N)}$
the alphabet, the set of internal states, and the state transition
function. The output function $\mu_N$ of $E^{(N)}$ is defined so that
$\mu_N(0,q)=\sigma_0(q)$ and $\mu_N(1,q)=\sigma_1(q)$ for all $q\in Q_N^\pm$,
where $\sigma_0=\prod_{n\in N}(a_n^{-1}b_n^{-1})$ and $\sigma_1=\prod_{n\in N}
(a_nb_n)$ are permutations on the set $Q_N^\pm$.
Lemmas \ref{auto7} and \ref{auto8} imply that $I^{(N)}$, $U^{(N)}$, and
$D^{(N)}$ are bi-reversible automata. Further, it is easy to see that the
automaton $E^{(N)}$ coincides with its inverse automaton while the reverse
automaton of $E^{(N)}$ can be obtained from $E^{(N)}$ by renaming its
states $0$ and $1$ to $1$ and $0$, respectively. By Lemma \ref{auto6},
$E^{(N)}$ is bi-reversible.
To each permutation $\tau$ on the set $Q_N$ we assign an automorphism
$\pi^{(N)}_\tau$ of the free monoid $(Q_N^\pm)^*$ such that
$\pi^{(N)}_\tau(q)=\tau(q)$, $\pi^{(N)}_\tau(q^{-1})=(\tau(q))^{-1}$ for
all $q\in Q_N$. The automorphism $\pi^{(N)}_\tau$ is uniquely determined
by $\tau$.
\begin{lemma}\lambdabel{union1}
(i) $(E^{(N)}_0)^2=(E^{(N)}_1)^2=1$, $E^{(N)}_0E^{(N)}_1=E^{(N)}_1
E^{(N)}_0=\pi^{(N)}_\tau$, where $\tau=\prod_{n\in N}(a_nb_n)$;
(ii) $D^{(N)}_0=\pi^{(N)}_{\tau_0}E^{(N)}_0=\pi^{(N)}_{\tau_1}E^{(N)}_1$,
$D^{(N)}_1=\pi^{(N)}_{\tau_1}E^{(N)}_0=\pi^{(N)}_{\tau_0}E^{(N)}_1$, where
$\tau_0=\prod_{n\in N}(a_nc_nq_{n1}\dots q_{n,2n-2})$, $\tau_1=\prod_{n\in
N}(a_nb_nc_nq_{n1}\dots q_{n,2n-2})$.
\end{lemma}
The proof of Lemma \ref{union1} is completely analogous to the proof of
Lemma \ref{series1} and we omit it.
\begin{proposition}\lambdabel{union2}
The group $G(D^{(N)})$ contains transformations $E^{(N)}_0$, $E^{(N)}_1$,
$\pi^{(N)}_{\tau_1}$, $\pi^{(N)}_{\tau_2}$, $\pi^{(N)}_{\tau_3}$, and
$\pi^{(N)}_{\tau_4}$, where $\tau_1=\prod_{n\in N}(a_nb_nc_nq_{n1}\dots
q_{n,2n-2})$, $\tau_2=\prod_{n\in N}(c_nq_{n1}\dots q_{n,2n-2})$, $\tau_3=
\prod_{n\in N}(a_nb_n)$, and $\tau_4=\prod_{n\in N}(b_nc_n)$.
\end{proposition}
\begin{proof}
It is easy to see that $\pi^{(N)}_{\tau\sigma}=\pi^{(N)}_\tau\pi^{(N)}_\sigma$
for any permutations $\tau$ and $\sigma$ on the set $Q_N$. It follows that
$\pi^{(N)}_{\tau^{-1}}=(\pi^{(N)}_\tau)^{-1}$ for any permutation $\tau$ on
$Q_N$.
By Lemma \ref{union1}, $D^{(N)}_0(D^{(N)}_1)^{-1}=\pi^{(N)}_{\tau_0}
E^{(N)}_0(\pi^{(N)}_{\tau_1}E^{(N)}_0)^{-1}=\pi^{(N)}_{\tau_0}
(\pi^{(N)}_{\tau_1})^{-1}$, where $\tau_0=\prod_{n\in N}(a_nc_nq_{n1}\dots
q_{n,2n-2})$. Since $\tau_0\tau_1^{-1}=\tau_4$, it follows that $D^{(N)}_0
(D^{(N)}_1)^{-1}=\pi^{(N)}_{\tau_4}$. Similarly,
$$
(D^{(N)}_0)^{-1}D^{(N)}_1=(\pi^{(N)}_{\tau_0}E^{(N)}_0)^{-1}
\pi^{(N)}_{\tau_1}E^{(N)}_0=(E^{(N)}_0)^{-1}\pi^{(N)}_{\tau_3}E^{(N)}_0
$$
since $\tau_3=\tau_0^{-1}\tau_1$. Lemma \ref{union1} implies that
$E^{(N)}_0$ and $\pi^{(N)}_{\tau_3}$ commute, hence $(D^{(N)}_0)^{-1}
D^{(N)}_1=\pi^{(N)}_{\tau_3}$. Consider the permutation $\tau_5=
\prod_{n\in N}(a_nc_n)$ on $Q_N$. Notice that $\tau_5=\tau_4\tau_3\tau_4$
and $\tau_2=\tau_5\tau_0$. By the above $\pi^{(N)}_{\tau_3},
\pi^{(N)}_{\tau_4}\in G(D^{(N)})$, hence $\pi^{(N)}_{\tau_5}\in
G(D^{(N)})$. Then $\pi^{(N)}_{\tau_2}E^{(N)}_0=\pi^{(N)}_{\tau_5}
\pi^{(N)}_{\tau_0}E^{(N)}_0=\pi^{(N)}_{\tau_5}D^{(N)}_0\in G(D^{(N)})$.
Since $\tau_2(a_n)=a_n$ and $\tau_2(b_n)=b_n$ for all $n\in N$, it easily
follows that transformations $\pi^{(N)}_{\tau_2}$ and $E^{(N)}_0$ commute.
As $\tau_2$ is the product of commuting permutations of odd orders $2n-1$,
$n\in N$, while $E^{(N)}_0$ is an involution, we have that
$(\pi^{(N)}_{\tau_2}E^{(N)}_0)^m=E^{(N)}_0$, where $m=\prod_{n\in N}
(2n-1)$. In particular, $E^{(N)}_0$ and $\pi^{(N)}_{\tau_2}$ are contained
in $G(D^{(N)})$. Now Lemma \ref{union1} implies that $\pi^{(N)}_{\tau_1},
E^{(N)}_1\in G(D^{(N)})$.
\end{proof}
Every word $\xi\in(Q_N^\pm)^*$ is assigned a pattern $v$ (i.e., a word in
the alphabet $\{*,*^{-1}\}$) that is obtained from $\xi$ by substituting
$*$ for each occurrence of letters $q\in Q_N$ and substituting $*^{-1}$ for
each occurrence of letters $q^{-1}$, $q\in Q_N$. We say that $\xi$ follows
the pattern $v$.
Now we introduce an alphabet $P_N^\pm$ that consists of symbols $*_n$ and
$*_n^{-1}$ for all $n\in N$. A word over the alphabet $P_N^\pm$ is called
a {\em marked pattern}. Every word $\xi\in(Q_N^\pm)^*$ is assigned a
marked pattern $v\in(P_N^\pm)^*$ that is obtained from $\xi$ as follows.
For any $n\in N$ we substitute $*_n$ for each occurrence of letters $q\in
Q_n$ in $\xi$ and substitute $*_n^{-1}$ for each occurrence of letters
$q^{-1}$, $q\in Q_n$. We say that $\xi$ follows the marked pattern $v$.
Clearly, the pattern of $\xi$ is uniquely determined by its marked pattern.
Notice that each letter of the alphabet $P_N^\pm$ corresponds to a
connected component of the Moore diagram of the automaton $U^{(N)}$. Since
$D^{(N)}$ is the dual automaton of $U^{(N)}$, it easily follows that the
$G(D^{(N)})$ action on $(Q_N^\pm)^*$ preserves marked patterns of words.
A word $\xi=q_1q_2\dots q_k\in(Q_N^\pm)^*$ is called freely irreducible if
none of its two-letter subwords $q_1q_2,q_2q_3,\dots,q_{k-1}q_k$ is of the
form $qq^{-1}$ or $q^{-1}q$, where $q\in Q_N$. Otherwise $\xi$ is called
freely reducible.
\begin{lemma}\lambdabel{union3}
For any nonempty word $v\in(P_N^\pm)^*$ there exists a freely irreducible
word $\xi\in(Q_N^\pm)^*$ such that $v$ is the marked pattern of $\xi$ and
the transformation $U^{(N)}_\xi$ acts nontrivially on the first level of
the rooted binary tree $X^*$.
\end{lemma}
\begin{proof}
For any $n\in N$ let us substitute $a_n$ for each occurrence of $*_n$ in
$v$ and $b_n^{-1}$ for each occurrence of $*_n^{-1}$. We get a nonempty
word $\xi\in(Q_N^\pm)^*$ that follows the marked pattern $v$. Now let us
modify $\xi$ by changing its last letter. If this letter is $a_n$ ($n\in
N$), we change it to $c_n$. If the last letter of $\xi$ is $b_n^{-1}$, we
change it to $c_n^{-1}$. This yields another word $\eta\in(Q_N^\pm)^*$
that follows the marked pattern $v$. By construction, $\xi$ and $\eta$ are
freely irreducible. Furthermore, $U^{(N)}_\eta=A^{(n)}_{c_n}
(A^{(n)}_{a_n})^{-1}U^{(N)}_\xi$ if the last letter of $v$ is $*_n$, $n\in
N$ while $U^{(N)}_\eta=(A^{(n)}_{c_n})^{-1}A^{(n)}_{b_n}U^{(N)}_\xi$ if the
last letter of $v$ is $*_n^{-1}$. For any $n\in N$ both $A^{(n)}_{c_n}
(A^{(n)}_{a_n})^{-1}$ and $(A^{(n)}_{c_n})^{-1}A^{(n)}_{b_n}$ interchange
one-letter words $0$ and $1$. It follows that one of the transformations
$U^{(N)}_\xi$ and $U^{(N)}_\eta$ also acts nontrivially on the first level
of the rooted tree $\{0,1\}^*$.
\end{proof}
Given a nonempty, freely irreducible word $\xi\in(Q_N^\pm)^*$, let
$Z_N(\xi)$ denote the set of all freely irreducible words in $(Q_N^\pm)^*$
that follow the same marked pattern as $\xi$ and match $\xi$ completely or
except for the last letter. Obviously, $\xi\in Z_N(\xi)$, and $\eta\in
Z_N(\xi)$ if and only if $\xi\in Z_N(\eta)$.
\begin{lemma}\lambdabel{union4}
For any nonempty word $v\in(P_N^\pm)^*$ there exists a freely irreducible
word $\xi\in(Q_N^\pm)^*$ such that $v$ is the marked pattern of $\xi$ and
the set $Z_N(\xi)$ is contained in one orbit of the $G(D^{(N)})$ action on
$(Q_N^\pm)^*$.
\end{lemma}
\begin{proof}
Let $\widetilde Q_N^\pm=\bigcup_{n\in N}\{a_n,b_n,c_n,a_n^{-1},b_n^{-1},c_n^{-1}\}$.
The set $(\widetilde Q_N^\pm)^*$ of words in the alphabet $\widetilde Q_N^\pm$ is a submonoid
of $(Q_N^\pm)^*$. Let $h_N:(\widetilde Q_N^\pm)^*\to(Q^\pm)^*$ be the homomorphism
of monoids such that $h_N(a_n)=a$, $h_N(b_n)=b$, $h_N(c_n)=c$,
$h_N(a_n^{-1})=a^{-1}$, $h_N(b_n^{-1})=b^{-1}$, $h_N(c_n^{-1})=c^{-1}$ for
all $n\in N$. For any $\zeta\in(\widetilde Q_N^\pm)^*$ the word $h_N(\zeta)$
follows the same pattern as $\zeta$. The word $\zeta$ is uniquely
determined by $h_N(\zeta)$ and the marked pattern of $\zeta$. If
$h_N(\zeta)$ is freely irreducible then so is $\zeta$ (however $h_N(\zeta)$
can be freely reducible even if $\zeta$ is freely irreducible). It is easy
to see that $E_0(h_N(\zeta))=h_N(E^{(N)}_0(\zeta))$,
$\pi_{(ab)}(h_N(\zeta))=h_N(\pi^{(N)}_{\sigma_1}(\zeta))$, and
$\pi_{(bc)}(h_N(\zeta))=h_N(\pi^{(N)}_{\sigma_2}(\zeta))$, where $\sigma_1=
\prod_{n\in N}(a_nb_n)$ and $\sigma_2=\prod_{n\in N}(b_nc_n)$ are permutations
on $Q_N$. By Proposition \ref{a2}, the group $G(D)$ is generated by $E_0$,
$\pi_{(ab)}$, and $\pi_{(bc)}$. On the other hand, $E^{(N)}_0,
\pi^{(N)}_{\sigma_1},\pi^{(N)}_{\sigma_2}\in G(D^{(N)})$ due to Proposition
\ref{union2}. Let $\widetilde G$ denote the subgroup of $G(D^{(N)})$ generated by
$E^{(N)}_0$, $\pi^{(N)}_{\sigma_1}$, and $\pi^{(N)}_{\sigma_2}$. It follows that
for any $g_0\in G(D)$ there exists $g\in\widetilde G$ such that $g_0(h_N(\zeta))=
h_N(g(\zeta))$ for all $\zeta\in(\widetilde Q_N^\pm)^*$. Now Proposition \ref{a5}
implies that words $\zeta_1,\zeta_2\in(\widetilde Q_N^\pm)^*$ are in the same orbit
of the $G(D^{(N)})$ action on $(Q_N^\pm)^*$ whenever they follow the same
marked pattern and the words $h_N(\zeta_1)$, $h_N(\zeta_2)$ are freely
irreducible.
Given a nonempty marked pattern $v\in(P_N^\pm)^*$, let $v_0$ be the word
obtained by deleting the last letter of $v$. For any $n\in N$ we
substitute $a_n$ for each occurrence of $*_n$ in $v_0$ and $b_n^{-1}$ for
each occurrence of $*_n^{-1}$. This yields a word $\eta\in(Q_N^\pm)^*$
that follows the marked pattern $v_0$. Now let $\xi=\eta c_n$ if the last
letter of $v$ is $*_n$, $n\in N$ and let $\xi=\eta c_n^{-1}$ if the last
letter of $v$ is $*_n^{-1}$. Clearly, $\xi$ is a freely irreducible word
following the marked pattern $v$. Moreover, $\xi\in(\widetilde Q_N^\pm)^*$ and the
word $h_N(\xi)$ is also freely irreducible.
We shall show that the set $Z_N(\xi)$ is contained in the orbit of $\xi$
under the $G(D^{(N)})$ action on $(Q_N^\pm)^*$. Take any $\zeta\in
Z_N(\xi)$. If $\zeta$ is a word over the alphabet $\widetilde Q_N^\pm$ and
$h_N(\zeta)$ is freely irreducible, then it follows from the above that
$\zeta=g(\xi)$ for some $g\in\widetilde G\subset G(D^{(N)})$. On the other hand,
suppose that the last letter of $\zeta$ is $q_{ni}$ or $q_{ni}^{-1}$, where
$n\in N$, $1\le i\le 2n-2$. In this case we have $\zeta=
(\pi^{(N)}_\tau)^i(\xi)$, where $\tau=\prod_{n\in N}(c_nq_{n1}\dots
q_{n,2n-2})$. By Proposition \ref{union2}, $\pi^{(N)}_\tau\in G(D^{(n)})$.
It remains to consider the case when the last letter of $\zeta$ belongs to
$\widetilde Q_N^\pm$ but the word $h_N(\zeta)$ is freely reducible. There is at
most one $\zeta\in Z_N(\xi)$ with such properties. It exists if the last
two letters of $v$ are of the form $*_l*_m^{-1}$ or $*_l^{-1}*_m$, where
$l,m\in N$, $l\ne m$. Assume this is the case. Then the last letter of
the word $\eta$ is either $a_l$ or $b_l^{-1}$. Let us change this letter
to $c_l$ or $c_l^{-1}$, respectively. The resulting word $\eta_1$ follows
the marked pattern $v_0$. Also, the words $h_N(\eta)$ and $h_N(\eta_1)$
are freely irreducible. By Proposition \ref{a5}, $h_N(\eta_1)=
g_1(h_N(\eta))$ for some $g_1\in G(D)$. There exists a unique $\zeta_1\in
(\widetilde Q_N^\pm)^*$ such that $h_N(\zeta_1)=g_1(h_N(\zeta))$ and $v$ is the
marked pattern of $\zeta_1$. By the above there exists $\tilde g_1\in\widetilde G$
such that $\tilde g_1(\eta)=\eta_1$ and $\tilde g_1(\zeta)=\zeta_1$. Since
the word $h_N(\zeta)$ is freely reducible, so is $h_N(\zeta_1)$. On the
other hand, the word $h_N(\eta_1)$, which can be obtained by deleting the
last letter of $h_N(\zeta_1)$, is freely irreducible. It follows that the
last two letters of $h_N(\zeta_1)$ are $cc^{-1}$ or $c^{-1}c$. Then the
last two letters of $\zeta_1$ are $c_lc_m^{-1}$ or $c_l^{-1}c_m$. If
$2m-1$ does not divide $2l-1$ then the word
$(\pi^{(N)}_\tau)^{2l-1}(\zeta_1)$ matches $\zeta_1$ except for the last
letter. Consequently, the word $\zeta'=\tilde g_1^{-1}
(\pi^{(N)}_\tau)^{2l-1}\tilde g_1(\zeta)$ matches $\zeta$ except for the
last letter. Since the $G(D^{(N)})$ action preserves marked patterns, the
word $\zeta'$ follows the marked pattern $v$. Hence $\zeta'\in Z_N(\xi)$.
As $\zeta'\ne\zeta$, it follows from the above that $\zeta'=g(\xi)$ for
some $g\in G(D^{(N)})$. Then $\zeta=g_0(\xi)$, where $g_0=\tilde g_1^{-1}
(\pi^{(N)}_\tau)^{1-2l}\tilde g_1g\in G(D^{(N)})$.
Now suppose that $2m-1$ divides $2l-1$. Then $(\pi^{(N)}_\tau)^{2l-1}
(\zeta_1)=\zeta_1$ and the above argument does not apply. Recall that the
last two letters of $h_N(\zeta_1)$ are $cc^{-1}$ or $c^{-1}c$. If these
letters are preceded by $b^{-1}$, we let $\zeta_2=\pi^{(N)}_{\sigma_1}
(\zeta_1)$. Otherwise they are preceded by $a$ or $h_N(\zeta_1)$ has
length $2$. In this case, we let $\zeta_2=\zeta_1$. Further, consider the
permutation $\tau_1=\tau^{2m-1}\sigma_2\tau^{-(2m-1)}\sigma_2\tau^{2m-1}$ on
$Q_N$. Since $\pi^{(N)}_\tau,\pi^{(N)}_{\sigma_2}\in G(D^{(n)})$, we have
that $\pi^{(N)}_{\tau_1}=(\pi^{(N)}_\tau)^{2m-1}\pi^{(N)}_{\sigma_2}
(\pi^{(N)}_\tau)^{1-2m}\pi^{(N)}_{\sigma_2}(\pi^{(N)}_\tau)^{2m-1}\in
G(D^{(N)})$. It is easy to see that $\tau_1(c_m)=c_m$ and $\tau_1(a_n)=
a_n$ for all $n\in N$. Since $2m-1<2l-1$, we have $\tau_1(c_l)=b_l$.
Also, for any $n\in N$ we have $\tau_1(b_n)=b_n$ if $2n-1$ divides $2m-1$
and $\tau_1(b_n)=c_n$ otherwise. It follows that $\zeta_3=
\pi^{(N)}_{\tau_1}(\zeta_2)$ is a word in the alphabet $\widetilde Q_N^\pm$ such
that $h_N(\zeta_3)$ is freely irreducible. Since $\zeta_3$ follows the
marked pattern $v$, we obtain that $\zeta_3$ belongs to the orbit of $\xi$
under the $G(D^{(N)})$ action. So does the word $\zeta$.
\end{proof}
\begin{proposition}\lambdabel{union5}
Suppose $\xi\in(Q_N^\pm)^*$ is a freely irreducible word. Then the orbit
of $\xi$ under the action of the group $G(D^{(N)})$ on $(Q_N^\pm)^*$
consists of all freely irreducible words following the same marked pattern
as $\xi$.
\end{proposition}
\begin{theorem}\lambdabel{union6}
The group $G(A^{(N)})$ is the free non-Abelian group on generators
$A^{(N)}_q$, $q\in Q_N$.
\end{theorem}
Proposition \ref{union5} is derived from Lemma \ref{union4} in the same way
as Proposition \ref{series6} was derived from Lemma \ref{series5}. Then
Theorem \ref{union6} is derived from Proposition \ref{union5} and Lemma
\ref{union3} in the same way as Theorem \ref{series7} was derived from
Proposition \ref{series6} and Lemma \ref{series4}. We omit both proofs.
\section{The Bellaterra automaton and its series}\lambdabel{b}
In this section we consider the Bellaterra automaton, a series of automata
of Bellaterra type, and their disjoint unions. We use the notation of
Sections \ref{a}, \ref{series}, and \ref{union}.
The Bellaterra automaton $B$ is an automaton over the alphabet $X=\{0,1\}$
with the set of internal states $Q=\{a,b,c\}$. The state transition
function $\widehat\phi$ and the output function $\widehat\psi$ of $B$ are defined as
follows: $\widehat\phi(a,0)=\widehat\phi(b,1)=c$, $\widehat\phi(a,1)=\widehat\phi(b,0)=b$, $\widehat\phi(c,0)=
\widehat\phi(c,1)=a$; $\widehat\psi(a,0)=\widehat\psi(b,0)=\widehat\psi(c,1)=0$, $\widehat\psi(a,1)=
\widehat\psi(b,1)=\widehat\psi(c,0)=1$. The Moore diagram of $B$ is depicted in Figure
\ref{fig2}. It is easy to verify that the inverse automaton of $B$
coincides with $B$. Besides, the reverse automaton of $B$ can be obtained
from $B$ by renaming its states $a$ and $c$ to $c$ and $a$, respectively.
Lemma \ref{auto6} implies that $B$ is bi-reversible.
The Bellaterra automaton $B$ is closely related to the Aleshin automaton
$A$. Namely, the two automata share the alphabet, the set of internal
states, and the state transition function. On the other hand, the output
function $\widehat\psi$ of $B$ never coincides with the output function $\psi$ of
$A$, that is, $\widehat\psi(q,x)\ne\psi(q,x)$ for all $q\in Q$ and $x\in X$.
For any integer $n\ge1$ we define a Bellaterra type automaton $B^{(n)}$ as
the automaton that is related to the Aleshin type automaton $A^{(n)}$ in
the same way as the automaton $B$ is related to $A$. To be precise,
$B^{(n)}$ is an automaton over the alphabet $X=\{0,1\}$ with the set of
states $Q_n$. The state transition function of $B^{(n)}$ coincides with
that of $A^{(n)}$. The output function $\widehat\psi_n$ of $B^{(n)}$ is defined
so that for any $x\in X$ we have $\widehat\psi_n(q,x)=x$ if $q\in\{a_n,b_n\}$ and
$\widehat\psi_n(q,x)=1-x$ if $q\in Q_n\setminus\{a_n,b_n\}$. Then $\widehat\psi_n(q,x)=
1-\psi_n(q,x)$ for all $q\in Q_n$ and $x\in X$, where $\psi_n$ is the
output function of $A^{(n)}$. Note that the automaton $B^{(1)}$ coincides
with $B$ up to renaming of the internal states.
In addition, we define a Bellaterra type automaton $B^{(0)}$. This is an
automaton over the alphabet $X$ with the set of internal states $Q_0$
consisting of a single element $c_0$. The state transition function
$\widehat\phi_0$ and the output function $\widehat\psi_0$ of $B^{(0)}$ are defined as
follows: $\widehat\phi_0(c_0,0)=\widehat\phi_0(c_0,1)=c_0$; $\widehat\psi_0(c_0,0)=1$,
$\widehat\psi_0(c_0,1)=0$.
It is easy to see that each Bellaterra type automaton $B^{(n)}$ coincides
with its inverse automaton. The reverse automaton of $B^{(0)}$ coincides
with $B^{(0)}$ as well. In the case $n\ge1$, the reverse automaton of
$B^{(n)}$ can be obtained from $B^{(n)}$ by renaming its states
$c_n,q_{n1},\dots,q_{n,2n-2},a_n$ to $a_n,q_{n,2n-2},\dots,q_{n1},c_n$,
respectively. Lemma \ref{auto6} implies that each $B^{(n)}$ is
bi-reversible.
\begin{figure}
\caption{
\lambdabel{fig7}
\end{figure}
Let $N$ be a nonempty set of nonnegative integers. We denote by $B^{(N)}$
the disjoint union of automata $B^{(n)}$, $n\in N$. Then $B^{(N)}$ is an
automaton over the alphabet $X=\{0,1\}$ with the set of internal states
$Q_N=\bigcup_{n\in N}Q_n$. It is bi-reversible since each $B^{(n)}$ is
bi-reversible. If $0\notin N$, then the automaton $B^{(N)}$ shares its
alphabet, its internal states, and its state transition function with the
automaton $A^{(N)}$ while the output functions of these automata never
coincide.
The relation between automata of Aleshin type and of Bellaterra type
induces a relation between transformations defined by automata of these two
types.
\begin{lemma}\lambdabel{b1}
Let $h=B^{(0)}_{c_0}$. Then
(i) $A_q=hB_q$ and $B_q=hA_q$ for any $q\in\{a,b,c\}$;
(ii) $A^{(n)}_q=hB^{(n)}_q$ and $B^{(n)}_q=hA^{(n)}_q$ for any $n\ge1$ and
$q\in Q_n$;
(iii) $A^{(N)}_q=hB^{(N)}_q$ and $B^{(N)}_q=hA^{(N)}_q$ for any nonempty
set $N$ of positive integers and any $q\in Q_N$.
\end{lemma}
\begin{proof}
The transformation $h$ is the automorphism of the free monoid $\{0,1\}^*$
that interchanges the free generators $0$ and $1$. For any $w\in X^*$ the
word $h(w)$ can be obtained from $w$ by changing all letters $0$ to $1$ and
all letters $1$ to $0$.
Suppose $\widetilde A$ and $\widetilde B$ are two automata over the
alphabet $X$ such that their sets of internal states and state transition
functions are the same but their output functions never coincide. It is
easy to see that $\widetilde A_q=h\widetilde B_q$ and $\widetilde B_q=
h\widetilde A_q$ for any internal state $q$ of the automata $\widetilde A$
and $\widetilde B$. The lemma follows.
\end{proof}
\begin{proposition}\lambdabel{b2}
(i) The group $G(A)$ is an index $2$ subgroup of $G(B^{(\{0,1\})})$;
(ii) for any $n\ge1$ the group $G(A^{(n)})$ is an index $2$ subgroup of
$G(B^{(\{0,n\})})$;
(iii) for any nonempty set $N$ of positive integers the group $G(A^{(N)})$
is an index $2$ subgroup of $G(B^{(N\cup\{0\})})$.
\end{proposition}
\begin{proof}
Note that the statement (i) is a particular case of the statement (ii) as
$G(A)=G(A^{(1)})$. Furthermore, the statement (ii) is a particular case of
the statement (iii) since $A^{(n)}=A^{(\{n\})}$ for any integer $n\ge1$.
Suppose $N$ is a nonempty set of positive integers. The group $G(A^{(N)})$
is generated by transformations $A^{(N)}_q$, $q\in Q_N$. The group
$G(B^{(N\cup\{0\})})$ is generated by transformations $h=B^{(0)}_{c_0}$ and
$B^{(N)}_q$, $q\in Q_N$. By Lemma \ref{b1}, $A^{(N)}_q=hB^{(N)}_q$ and
$B^{(N)}_q=hA^{(N)}_q$ for any $q\in Q_N$. It follows that the group
$G(B^{(N\cup\{0\})})$ is generated by transformations $h$ and $A^{(N)}_q$,
$q\in Q_N$. In particular, $G(A^{(N)})\subset G(B^{(N\cup\{0\})})$.
For any $n\ge0$ the automaton $B^{(n)}$ coincides with its inverse. Lemma
\ref{auto2} implies that $h^2=1$ and $(B^{(N)}_q)^2=1$, $q\in Q_N$. Then
$hA^{(N)}_qh^{-1}=B^{(N)}_qh=(A^{(N)}_q)^{-1}$ for any $q\in Q_N$. It
follows that $G(A^{(N)})$ is a normal subgroup of $G(B^{(N\cup\{0\})})$.
Since $h^2=1$, the index of the group $G(A^{(N)})$ in $G(B^{(N\cup\{0\})})$
is at most $2$. On the other hand, $G(A^{(N)})\ne G(B^{(N\cup\{0\})})$ as
$G(B^{(N\cup\{0\})})$ contains a nontrivial involution $h$ while
$G(A^{(N)})$ is a free group due to Theorem \ref{union6}. Thus
$G(A^{(N)})$ is an index $2$ subgroup of $G(B^{(N\cup\{0\})})$.
\end{proof}
The relation between groups defined by automata of Aleshin type and of
Bellaterra type allows us to establish the structure of the groups defined
by automata of the latter type. As the following two theorems show, these
groups are free products of groups of order $2$.
\begin{theorem}[\cite{N}]\lambdabel{b3}
The group $G(B)$ is freely generated by involutions $B_a$, $B_b$, $B_c$.
\end{theorem}
\begin{theorem}\lambdabel{b4}
(i) For any $n\ge1$ the group $G(B^{(n)})$ is freely generated by $2n+1$
involutions $B^{(n)}_q$, $q\in Q_n$;
(ii) for any nonempty set $N$ of nonnegative integers the group
$G(B^{(N)})$ is freely generated by involutions $B^{(N)}_q$, $q\in Q_N$.
\end{theorem}
To prove Theorems \ref{b3} and \ref{b4}, we need the following lemma.
\begin{lemma}\lambdabel{b5}
Suppose that a group $G$ is generated by elements $g_0,g_1,\dots,g_k$
($k\ge1$) of order at most $2$. Let $H$ be the subgroup of $G$ generated
by elements $h_i=g_0g_i$, $1\le i\le k$. Then $G$ is freely generated by
$k+1$ involutions $g_0,g_1,\dots,g_k$ if and only if $H$ is the free group
on $k$ generators $h_1,\dots,h_k$.
\end{lemma}
\begin{proof}
Consider an element $h=h_{i_1}^{\varepsilon_1}h_{i_2}^{\varepsilon_2}\dots
h_{i_l}^{\varepsilon_l}$, where $l\ge1$, $1\le i_j\le k$, $\varepsilon_j\in\{-1,1\}$, and
$\varepsilon_j=\varepsilon_{j+1}$ whenever $i_j=i_{j+1}$. Since $h_i=g_0g_i$ and
$h_i^{-1}=g_ig_0$ for $1\le i\le k$, and $g_0^2=1$, we obtain that
$h=g'_0g_{i_1}g'_1\dots g_{i_l}g'_l$, where each $g'_j$ is equal to $g_0$
or $1$. Moreover, $g'_j=g_0$ whenever $\varepsilon_j=\varepsilon_{j+1}$. In particular,
$h\ne1$ if $G$ is freely generated by involutions $g_0,g_1,\dots,g_k$. It
follows that $H$ is the free group on generators $h_1,\dots,h_k$ if $G$ is
freely generated by involutions $g_0,g_1,\dots,g_k$.
Now assume that $H$ is the free group on generators $h_1,\dots,h_k$. Then
each $h_i$ has infinite order. Since $h_i=g_0g_i$ and $g_0^2=g_i^2=1$, it
follows that $g_0\ne1$ and $g_i\ne1$. Hence each of the elements
$g_0,g_1,\dots,g_k$ has order $2$. In particular, none of these elements
belongs to the free group $H$.
The group $G$ is freely generated by involutions $g_0,g_1,\dots,g_k$ if
$g\ne1$ for any $g=g_{i_1}\dots g_{i_l}$ such that $l\ge1$, $0\le i_j\le
k$, and $i_j\ne i_{j+1}$. First consider the case when $l$ is even. Note
that $g_ig_j=h_i^{-1}h_j$ for $0\le i,j\le n$, where by definition $h_0=1$.
Therefore $g=h_{i_1}^{-1}h_{i_2}\dots h_{i_{l-1}}^{-1}h_{i_l}\in H$. Since
$h_0=1$, the sequence $h_{i_1}^{-1},h_{i_2},\dots,h_{i_{l-1}}^{-1},h_{i_l}$
can contain the unit elements. After removing all of them, we obtain a
nonempty sequence in which neighboring elements are not inverses of each
other. Since $h_1,\dots,h_k$ are free generators, we conclude that
$g\ne1$. In the case when $l$ is odd, it follows from the above that
$g=g_{i_1}h$, where $h\in H$. Since $g_{i_1}\notin H$, we have that
$g\notin H$, in particular, $g\ne1$.
\end{proof}
\begin{proofof}{Theorems \ref{b3} and \ref{b4}}
First we observe that Theorem \ref{b3} is a particular case of Theorem
\ref{b4} since the automata $B$ and $B^{(1)}$ coincide up to renaming of
their internal states. Further, the statement (i) of Theorem \ref{b4} is a
particular case of the statement (ii) since $B^{(n)}=B^{(\{n\})}$ for any
$n\ge1$.
Suppose $N$ is a nonempty set of nonnegative integers such that $0\in N$.
For any $n\in N$ the automaton $B^{(n)}$ coincides with its inverse. Lemma
\ref{auto2} implies that $(B^{(N)}_q)^2=1$ for all $q\in Q_N$. If
$N=\{0\}$ then $Q_N=\{c_0\}$ and $G(B^{(N)})$ is a group of order $2$
generated by the involution $h=B^{(0)}_{c_0}$. Now assume that $N\ne
\{0\}$. Then $K=N\setminus\{0\}$ is a nonempty set of positive integers.
The group $G(B^{(N)})$ is generated by transformations $h$ and $B^{(K)}_q$,
$q\in Q_K$. All generators are of order at most $2$. The group
$G(A^{(K)})$ is the free group on generators $A^{(K)}_q$, $q\in Q_K$ due to
Theorem \ref{union6}. By Lemma \ref{b1}, $A^{(K)}_q=hB^{(K)}_q$ for any
$q\in Q_K$. Then Lemma \ref{b5} implies that $G(B^{(N)})$ is freely
generated by involutions $h$ and $B^{(K)}_q$, $q\in Q_K$.
Now consider the case when $N$ is a nonempty set of positive integers. By
the above the group $G(B^{(N\cup\{0\})})$ is freely generated by
involutions $h$ and $B^{(N)}_q$, $q\in Q_N$. Clearly, this implies that
the group $G(B^{(N)})$ is freely generated by involutions $B^{(N)}_q$,
$q\in Q_N$.
\end{proofof}
Now we shall establish a relation between transformation groups defined by
the Aleshin type and the Bellaterra type automata with the same set of
internal states.
Since $G(A)$ is the free group on generators $A_a$, $A_b$, $A_c$, there is
a unique homomorphism $\Delta:G(A)\to G(B)$ such that $\Delta(A_a)=B_a$,
$\Delta(A_b)=B_b$, $\Delta(A_c)=B_c$. Likewise, for any $n\ge1$ there is a
unique homomorphism $\Delta_n:G(A^{(n)})\to G(B^{(n)})$ such that
$\Delta_n(A^{(n)}_q)=B^{(n)}_q$ for all $q\in Q_n$. Also, for any nonempty
set $N$ of positive integers there is a unique homomorphism $\Delta_N:
G(A^{(N)})\to G(B^{(N)})$ such that $\Delta_N(A^{(N)}_q)=B^{(N)}_q$ for all
$q\in Q_N$.
\begin{proposition}\lambdabel{b6}
(i) $G(A)\cap G(B)=\{g\in G(A)\mid \Delta(g)=g\}$;
(ii) $G(A)\cap G(B)$ is the free group on generators $B_aB_b$ and
$B_aB_c$;
(iii) $G(A)\cap G(B)$ is an index $2$ subgroup of $G(B)$;
(iv) $A_p^{-1}A_q=B_pB_q$ for all $p,q\in\{a,b,c\}$.
\end{proposition}
\begin{proof}
Let $h=B^{(0)}_{c_0}$. By Lemma \ref{b1}, $A_q=hB_q$ for all $q\in
\{a,b,c\}$. Since the inverse automaton of $B$ coincides with $B$, Lemma
\ref{auto2} implies that $B_a^2=B_b^2=B_c^2=1$. Then for any $p,q\in
\{a,b,c\}$ we have $A_p^{-1}A_q=(hB_p)^{-1}hB_q=B_p^{-1}B_q=B_pB_q$.
It is easy to see that $\{g\in G(A)\mid \Delta(g)=g\}$ is a subgroup of
$G(A)\cap G(B)$. Let $\widetilde G$ be the group generated by transformations
$B_aB_b$ and $B_aB_c$. By the above $\Delta(A_a^{-1}A_b)=B_a^{-1}B_b=
B_aB_b=A_a^{-1}A_b$ and $\Delta(A_a^{-1}A_c)=B_a^{-1}B_c=B_aB_c=
A_a^{-1}A_c$. It follows that $\widetilde G$ is a subgroup of $\{g\in G(A)\mid
\Delta(g)=g\}$.
By Theorem \ref{b3}, the group $G(B)$ is freely generated by involutions
$B_a$, $B_b$, $B_c$. Then Lemma \ref{b5} implies that $\widetilde G$ is the free
group on generators $B_aB_b$ and $B_aB_c$. Note that $B_aB_q\in\widetilde G$ for
all $q\in Q$. Then for any $p,q\in Q$ we have $B_pB_q=(B_aB_p)^{-1}
B_aB_q\in\widetilde G$. It follows that for any $g\in G(B)$ at least one of the
transformations $g$ and $B_ag$ belongs to $\widetilde G$. Therefore the index of
$\widetilde G$ in $G(B)$ is at most $2$.
Note that $B_a\notin G(A)$ as $B_a$ is a nontrivial involution while $G(A)$
is a free group. Hence $G(A)\cap G(B)\ne G(B)$. Now it follows from the
above that $\widetilde G=\{g\in G(A)\mid \Delta(g)=g\}=G(A)\cap G(B)$ and this is an
index $2$ subgroup of $G(B)$.
\end{proof}
\begin{proposition}\lambdabel{b7}
Let $n$ be a positive integer. Then
(i) $G(A^{(n)})\cap G(B^{(n)})=\{g\in G(A^{(n)})\mid \Delta_n(g)=g\}$;
(ii) $G(A^{(n)})\cap G(B^{(n)})$ is the free group on $2n$ generators
$B^{(n)}_{a_n}B^{(n)}_q$, $q\in Q_n\setminus\{a_n\}$;
(iii) $G(A^{(n)})\cap G(B^{(n)})$ is an index $2$ subgroup of $G(B^{(n)})$;
(iv) $(A^{(n)}_p)^{-1}A^{(n)}_q=B^{(n)}_pB^{(n)}_q$ for all $p,q\in Q_n$.
\end{proposition}
\begin{proposition}\lambdabel{b8}
Let $N$ be a nonempty set of positive integers. Then
(i) $G(A^{(N)})\cap G(B^{(N)})=\{g\in G(A^{(N)})\mid \Delta_N(g)=g\}$;
(ii) for any $n\in N$ the group $G(A^{(N)})\cap G(B^{(N)})$ is the free
group on generators $B^{(N)}_{a_n}B^{(N)}_q$, $q\in Q_N\setminus\{a_n\}$;
(iii) $G(A^{(N)})\cap G(B^{(N)})$ is an index $2$ subgroup of $G(B^{(N)})$;
(iv) $(A^{(N)}_p)^{-1}A^{(N)}_q=B^{(N)}_pB^{(N)}_q$ for all $p,q\in Q_N$.
\end{proposition}
The proofs of Propositions \ref{b7} and \ref{b8} are completely analogous
to the proof of Proposition \ref{b6} and we omit them.
Now let us consider the dual automata of the Bellaterra automaton and
automata of Bellaterra type.
Let $\widehat D$ denote the dual automaton of the Bellaterra automaton $B$. The
automaton $\widehat D$ is defined over the alphabet $Q=\{a,b,c\}$, with two
internal states $0$ and $1$. The Moore diagram of $\widehat D$ is depicted in
Figure \ref{fig8}. The automaton $\widehat D$ is bi-reversible since $B$ is
bi-reversible.
\begin{figure}
\caption{
\lambdabel{fig8}
\end{figure}
A word $\xi$ over an arbitrary alphabet is called a {\em double letter
word\/} if there are two adjacent letters in $\xi$ that coincide.
Otherwise we call $\xi$ a {\em no-double-letter word}.
The set of no-double-letter words over the alphabet $Q$ forms a subtree of
the rooted ternary tree $Q^*$. As an unrooted tree, this subtree is
$3$-regular. However it is not regular as a rooted tree. The following
proposition shows that the group $G(\widehat D)$ acts transitively on each level
of the subtree.
\begin{proposition}[\cite{N}]\lambdabel{b9}
Suppose $\xi\in Q^*$ is a no-double-letter word. Then the orbit of $\xi$
under the action of the group $G(\widehat D)$ on $Q^*$ consists of all
no-double-letter words of the same length as $\xi$.
\end{proposition}
\begin{proof}
Let $\lambda$ and $\mu$ denote the state transition and output functions of the
automaton $B$. By $\tilde\lambda$ and $\tilde\mu$ denote the state transition
and output functions of its dual $\widehat D$. Take any $q\in Q$ and $x\in X$.
Let $p=\lambda(q,x)$ and $y=\mu(q,x)$. Since $B$ coincides with its inverse
automaton, it follows that $p=\lambda(q,y)$. Consequently, $\widehat D_x(qq)=
\tilde\mu(x,q)\tilde\mu(\tilde\lambda(x,q),q)=\lambda(q,x)\lambda(q,\mu(q,x))=pp$. It
follows that the set $P=\{qq\mid q\in Q\}\subset Q^*$ is invariant under
$\widehat D_0$ and $\widehat D_1$. Any double letter word $\xi\in Q^*$ is represented as
$\xi_1\xi_0\xi_2$, where $\xi_0\in P$ and $\xi_1,\xi_2\in Q^*$. For any
$x\in X$ we have $\widehat D_x(\xi)=\widehat D_x(\xi_1)\widehat D_{x_0}(\xi_0)\widehat D_{x_1}(\xi_2)$,
where $x_0,x_1\in X$. By the above $\widehat D_x(\xi)$ is a double letter word.
Thus $\widehat D_0$ and $\widehat D_1$ map double letter words to double letter words.
Since these transformations are invertible, they also map no-double-letter
words to no-double-letter words, and so does any $g\in G(\widehat D)$.
Now we are going to prove that for any no-double-letter words
$\xi_1,\xi_2\in Q^*$ of the same length $l$ there exists $g\in G(\widehat D)$ such
that $\xi_2=g(\xi_1)$. The empty word is the only word of length $0$ so it
is no loss to assume that $l>0$. First consider the case when $l$ is even.
We have $\xi_1=q_1q_2\dots q_{l-1}q_l$ and $\xi_2=p_1p_2\dots p_{l-1}p_l$
for some $q_i,p_i\in Q$, $1\le i\le l$. Consider two words $\eta_1=
q_1q_2^{-1}\dots q_{l-1}q_l^{-1}$ and $\eta_2=p_1p_2^{-1}\dots
p_{l-1}p_l^{-1}$ over the alphabet $Q^\pm$. Clearly, $\eta_1$ and $\eta_2$
follow the same pattern. Furthermore, they are freely irreducible since
$\xi_1$ and $\xi_2$ are no-double-letter words. By Proposition \ref{a5},
$\eta_2=g_0(\eta_1)$ for some $g_0\in G(D)$. By Lemma \ref{auto1}, we can
assume that $g_0\in S(D)$. Then $g_0=D_w$ for some word $w\in X^*$.
Proposition \ref{auto4} implies that $U_{\eta_1}(wu)=U_{\eta_1}(w)
U_{\eta_2}(u)$ for any $u\in X^*$. By Proposition \ref{b6}, $A_p^{-1}A_q=
B_pB_q$ for all $p,q\in Q$. It follows that $U_{\eta_1}=B_{\xi_1}$ and
$U_{\eta_2}=B_{\xi_2}$. In particular, $B_{\xi_1}(wu)=B_{\xi_1}(w)
B_{\xi_2}(u)$ for any $u\in X^*$. Now Proposition \ref{auto4} implies that
$B_{\xi_2}=B_{g(\xi_1)}$, where $g=\widehat D_w\in G(\widehat D)$. By the above
$g(\xi_1)$ is a no-double-letter word. By Theorem \ref{b3}, the group
$G(B)$ is freely generated by involutions $B_q$, $q\in Q$. Since $\xi_2$
and $g(\xi_1)$ are no-double-letter words in the alphabet $Q$, the equality
$B_{\xi_2}=B_{g(\xi_1)}$ implies that $\xi_2=g(\xi_1)$.
Now consider the case when $\xi_1$ and $\xi_2$ have odd length. Obviously,
there exist letters $q_0,p_0\in Q$ such that $\xi_1q_0$ and $\xi_2p_0$ are
no-double-letter words. Since $\xi_1q_0$ and $\xi_2p_0$ are of the same
even length, it follows from the above that $\xi_2p_0=g(\xi_1q_0)$ for some
$g\in G(\widehat D)$. Then $\xi_2=g(\xi_1)$.
\end{proof}
For any integer $n\ge0$ let $\widehat D^{(n)}$ denote the dual automaton of the
automaton $B^{(n)}$. The automaton $\widehat D^{(n)}$ is defined over the
alphabet $Q_n$, with two internal states $0$ and $1$. It is bi-reversible
since $B^{(n)}$ is bi-reversible.
\begin{proposition}\lambdabel{b10}
Let $n\ge1$ and suppose $\xi\in Q_n^*$ is a no-double-letter word. Then
the orbit of $\xi$ under the action of the group $G(\widehat D^{(n)})$ on $Q_n^*$
consists of all no-double-letter words of the same length as $\xi$.
\end{proposition}
The proof of Proposition \ref{b10} is completely analogous to the above
proof of Proposition \ref{b9} and we omit it.
{\sc
\begin{raggedright}
Department of Mathematics\\
Texas A\&M University\\
College Station, TX 77843--3368
\end{raggedright}
}
\end{document} |
\begin{document}
\title{On the Partition Set Cover Problem}
\begin{abstract}
Several algorithms with an approximation guarantee of $O(\log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ \emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $\mathcal{C}_t$, using the minimum number of sets. We give a randomized \textsf{LP}\xspace-rounding algorithm that is an $O(\beta + \log r)$ approximation for the Partition Set Cover problem. Here $\beta$ denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard \textsf{LP}\xspace. As a corollary, we obtain improved approximation guarantees for various set systems for which $\beta$ is known to be sublogarithmic in $n$. We also extend the \textsf{LP}\xspace rounding algorithm to obtain $O(\log r)$ approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is \mathcal{N}P-hard to obtain an $o(\log r)$-approximation for any of these problems.
\end{abstract}
\section{Introduction}
We first consider the Set Cover problem. The input of the Set Cover problem consists of a set system $(X, \mathcal{R})$, where $X$ is a set of $n$ elements and $\mathcal{R}$ is a collection of subsets of $X$. Each set $S_i \in \mathcal{R}$ has a non-negative weight $w_i$. The goal of the set cover problem is to find a minimum-weight sub-collection of sets from $\mathcal{R}$, that \emph{covers} $X$. In the unweighted version, all weights are assumed to be $1$. We state the standard \textsf{LP}\xspace relaxation for the Set Cover problem.
\begin{mdframed}[backgroundcolor=gray!9]
(Set Cover \textsf{LP}\xspace)
\begin{alignat}{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in \mathcal{R}} w_{i}x_{i} & \nonumber \\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq 1, \quad & \forall e_j \in X \label[constr]{constr:sc-cover-ej}\\
\displaystyle &x_i \in [0, 1], & \forall S_i \in \mathcal{R} \label[constr]{constr:sc-fractional-x}
\end{alignat}
\end{mdframed}
It is well-known that a simple greedy algorithm, or an \textsf{LP}\xspace rounding algorithm gives an $O(\log n)$ approximation. This can be improved to $O(\log \mathsf{D}elta)$, where $\mathsf{D}elta$ is the maximum size of a set -- see \cite{VaziraniBook} for further details and references. It is also known that it is not possible to obtain an approximation guarantee that is asymptotically smaller than $O(\log n)$ in general, under certain standard complexity theoretic assumptions (\cite{Feige1998,DS2014}). A simple \textsf{LP}\xspace rounding algorithm is also known to give an $f$ approximation (see \cite{VaziraniBook}), where $f$ is the maximum \emph{frequency} of an element, i.e., the maximum number of sets any element is contained in. For several set systems such as geometric set systems, however, sublogarithmic\,---\,or even constant\,---\,approximation guarantees are known. For a detailed discussion of such results, see \cite{Inamdar2018partial}.
Partial Set Cover problem (\textsf{PSC}\xspace) is a generalization of the Set Cover problem. Here, along with the set system $(X, \mathcal{R})$, we are also given the coverage requirement $1 \le k \le n$. The objective of \textsf{PSC}\xspace is to find a minimum-weight cover for at least $k$ of the given $n$ elements. It is easy to see that when the coverage requirement $k$ equals $n$, \textsf{PSC}\xspace reduces to the Set Cover problem. However, for a general $k$, \textsf{PSC}\xspace introduces the additional difficulty of discovering a subset with $k$ of the $n$ elements that one must aim to cover in order to obtain a least expensive solution. Despite this, approximation guarantees matching that for the standard Set Cover are known in many cases. For example, a slight modification of the greedy algorithm can be shown to be an $O(\log \mathsf{D}elta)$ approximation (\cite{Kearns1990,Slavik1997}). Algorithms achieving the approximation guarantee of $f$ are known via various techniques -- see \cite{Fujito04} and the references therein. For several instances of \textsf{PSC}\xspace, an $O(\beta)$ approximation algorithm was described in \cite{Inamdar2018partial}, where $\beta$ is the integrality gap of the standard set cover \textsf{LP}\xspace for a related set system. As a corollary, they give improved approximation guarantees for the geometric instances of \textsf{PSC}\xspace, for which $\beta$ is known to be sublogarithmic (or even a constant).
\paragraph{Partition Set Cover Problem.}
Now we consider a further generalization of the Partial Set Cover problem, called the Partition Set Cover problem. Again, the input contains a set system $(X, \mathcal{R})$, with weights on the sets, where $X = \{e_1, \ldots, e_n\}$ and $\mathcal{R} = \{S_1, \ldots, S_m\}$. We are also given $r$ non-empty subsets of $X$: $\mathcal{C}_1, \ldots, \mathcal{C}_r$, where each $\mathcal{C}_t$ is referred to as a \emph{color class}. These $r$ color classes form a partition of $X$. Each color class $\mathcal{C}_t$ has a coverage requirement $1 \le k_t \le |\mathcal{C}_t|$ that is also a part of the input. The objective of the Partition Set Cover problem is to find a minimum-weight sub-collection $\mathcal{R}' \subseteq \mathcal{R}$, such that it meets the coverage requirement of each color class, i.e., for each color class $\mathcal{C}_t$, we have that $|(\bigcup \mathcal{R}') \cap \mathcal{C}_t| \ge k_t$. Here, for any $\mathcal{R}' \subseteq \mathcal{R}$, we use the shorthand $\bigcup \mathcal{R}'$ for referring to the union of all sets in $\mathcal{R}'$, i.e., $\bigcup \mathcal{R}' \coloneqq \bigcup_{S_i \in \mathcal{R}'} S_i$.
\citet{bera2014approximation} give an $O(\log r)$-approximation for an analogous version of the Vertex Cover problem, called the Partition Vertex Cover problem. This is a special case of the Partition Set Cover problem where each element is contained in exactly two sets. The Vertex Cover and Partial Vertex Cover problems are, respectively, special cases of the Set Cover and Partial Set Cover problems.
Various $2$ approximations are known for Vertex Cover (see references in \cite{VaziraniBook}) as well as Partial Vertex Cover (\cite{BshoutyL1998,Hochbaum98,Gandhi2004}).
\citet{bera2014approximation} note that for the Partition Set Cover problem, an extension of the greedy algorithm of \citet{Slavik1997} gives an $O(\log (\sum_{t = 1}^r k_t))$ approximation. On the negative side, they (\cite{bera2014approximation}) show that it is \mathcal{N}P-hard to obtain an approximation guarantee asymptotically better than $O(\log r)$ for the Partition Vertex Cover problem. Since Partition Vertex Cover is a special case of Partition Set Cover, the same hardness result holds for the Partition Set Cover problem as well.
\citet{Har2018few} consider a problem concerned with breaking up a collection of point sets using a minimum number of hyperplanes. They reduce this problem to an instance of the Partition Set Cover problem, where the color classes are no longer required to form a partition of $X$. For this problem, which they call `Partial Cover for Multiple Sets', they describe an $O(\log (nr))$ approximation. We note that the algorithm of \citet{bera2014approximation} as well as our algorithm easily extends to this more general setting.
\subsection{Natural \textsf{LP}\xspace Relaxation and Its Integrality Gap} \label{subsec:nat-lp}
Given the success of algorithms based on the natural \textsf{LP}\xspace relaxation for \textsf{PSC}\xspace, let us first consider the natural \textsf{LP}\xspace for the Partition Set Cover problem. We first state this natural \textsf{LP}\xspace relaxation.
\begin{mdframed}[backgroundcolor=gray!9]
(Natural \textsf{LP}\xspace)
\begin{alignat}{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in \mathcal{R}} w_{i}x_{i} & \nonumber \\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq z_j, \quad & \forall e_j \in X \label[constr]{constr:cover-ej}\\
\displaystyle&\sum_{e_j \in \mathcal{C}_r}z_j \ge k_t, & \forall \mathcal{C}_t \in \{\mathcal{C}_1, \ldots, \mathcal{C}_r\} \label[constr]{constr:cover-ci}\\
\displaystyle &z_j \in [0, 1], & \forall e_j \in X \label[constr]{constr:fractional-z}\\
\displaystyle &x_i \in [0, 1], & \forall S_i \in \mathcal{R} \label[constr]{constr:fractional-x}
\end{alignat}
\end{mdframed}
In the corresponding integer program, the variable $x_i$ denotes whether or not the set $S_i \in \mathcal{R}$ is included in the solution. Similarly, the variable $z_j$ denotes whether or not an element $e_j \in X$ is covered in the solution. Both types of variables are restricted to $\{0, 1\}$ in the integer program. The \textsf{LP}\xspace relaxation stated above relaxes this condition by allowing those variables to take any value from $[0, 1]$.
One important way the Standard \textsf{LP}\xspace differs from the Partial Cover \textsf{LP}\xspace is that we have a coverage constraint (\mathcal{C}ref{constr:cover-ci}) for each color class $\mathcal{C}_1, \ldots, \mathcal{C}_r$. \iffalse If an element $e_j \in X$ belongs to multiple color classes, then its $z_j$ appears in multiple such constraints.\fi Unfortunately, this \textsf{LP}\xspace has a large integrality gap as demonstrated by the following simple construction.
\paragraph{Integrality Gap.}
Let $(X, \mathcal{R})$ be the given set system, where $X = \{e_1, e_2, \ldots, e_n\}$ and $\mathcal{R} = \{S_1, S_2, \ldots, S_{\sqrt{n}}\}$ -- assuming $n$ is a perfect square. The sets $S_i$ form a partition of $X$, such that each $S_i$ contains exactly $\sqrt{n}$ elements. For any $1 \le i \le \sqrt{n}$, the color class $\mathcal{C}_i$ equals $S_i$, and its coverage requirement, $k_i = 1$. Also, for each set $S_i$, $w_i = 1$. Clearly any integral optimal solution must choose all sets $S_1, \ldots, S_{\sqrt{n}}$, with cost $\sqrt{n}$.
On the other hand, consider a fractional solution $(x, z)$, where for any $S_i \in \mathcal{R}$, $x_i = \frac{1}{\sqrt{n}}$; and for any $e_j \in X$, $z_j = \frac{1}{\sqrt{n}}$. It is easy to see that this solution satisfies all constraints, and has cost $1$. We emphasize here that even the natural \textsf{PSC}\xspace \textsf{LP}\xspace has a large integrality gap, but it can be easily circumvented via parametric search; see \cite{Gandhi2004,Inamdar2018partial} for examples. For the Partition Set Cover problem, however, similar techniques do not seem to work.
\subsection{Our Results and Techniques}
For any subset $Y \subseteq X$, let $(Y, \mathcal{R}_{|Y})$ denote the projection of $(X, \mathcal{R})$ on $Y$, where $\mathcal{R}_{|Y} = \{S_i \cap Y \mid S_i \in \mathcal{R} \}$. Suppose there exists an algorithm that can round a feasible \emph{Set Cover} \textsf{LP}\xspace solution for any projection $(Y, \mathcal{R}_{|Y})$, within a factor of $\beta$. Then, we show that there exists an $O(\beta + \log r)$ approximation for the Partition Set Cover problem on the original set system $(X, \mathcal{R})$.
Given the integrality gap of the natural \textsf{LP}\xspace for the Partition Set Cover problem, we strengthen it by adding the knapsack cover inequalities (first introduced by \citet{carrStrengthening}) to the \textsf{LP}\xspace relaxation -- the details are given in the following section. This approach is similar to that used for the Partition Vertex Cover problem considered in \citet{bera2014approximation}, and for the Partial Set Cover problem in \citet{Fujito04}. A similar technique was also used for a scheduling with outliers problem by \citet{gupta2009scheduling}.
Once we have a solution to this strengthened \textsf{LP}\xspace, we partition the elements of $X$ as follows. The elements that are covered to an extent of at least a positive constant in this \textsf{LP}\xspace solution are said to be \emph{heavy} (the precise definition is given in the following section), and the rest of the elements are \emph{light}. For the heavy elements, we obtain a standard \emph{Set Cover} \textsf{LP}\xspace solution, and round it using a black-box rounding algorithm with a guarantee $\beta$.
To meet the residual coverage requirements of the color classes, we use a randomized algorithm that covers some of the light elements. This randomized rounding consists of $O(\log r)$ independent iterations of a simple randomized rounding process. Let $\mathcal{R}igma_\ell$ be a random collection of sets, which is the outcome of an iteration $\ell$. The cover for the heavy elements from previous step, plus $\bigcup_\ell \mathcal{R}igma_\ell$, which is the result of randomized rounding, make up the final output of our algorithm. The analysis of this solution hinges on showing the following two properties of $\mathcal{R}igma_\ell$.
\begin{enumerate}
\item Expected cost of $\mathcal{R}igma_\ell$ is no more than some constant times the optimal cost, and
\item For any color class $\mathcal{C}_t$, the sets in $\mathcal{R}igma_\ell$ satisfy the residual coverage of $\mathcal{C}_t$ with at least a constant probability.
\end{enumerate}
The first property of $\mathcal{R}igma_\ell$ follows easily, given the description of the randomized rounding process. Much of the analysis is devoted to showing the second property. The randomized rounding algorithm ensures that in each iteration, the residual requirement of each color class is satisfied in expectation. However, showing the second property, i.e., the residual coverage is satisfied with at least a constant probability, is rather tricky for the Partition Set Cover problem. The analogous claim about the Partition Vertex Cover problem in \cite{bera2014approximation} is easier to prove, because each edge is incident on exactly two vertices. Despite the fact that here, an element can belong to any number of sets, we are able to show that the second property holds, using a careful analysis of the randomized rounding process.
From these two properties, it is straightforward to show that the algorithm is an $O(\beta + \log r)$ approximation for the Partition Set Cover problem (for details, see \mathcal{C}ref{thm:main-theorem}). As shown in \cite{bera2014approximation}, $\Omega(\log r)$ is necessary, even for the Partition Vertex Cover problem, and we extend this result to the Partition Set Cover problem induced by various geometric set systems (see \mathcal{C}ref{sec:hardness}). Also $\beta$ is the integrality gap of the standard Set Cover \textsf{LP}\xspace for this set system. Our approximation guarantee of $O(\beta + \log r)$ should be viewed in the context of these facts. We also note here that an extension of the \textsf{LP}\xspace rounding algorithm of \cite{Inamdar2018partial} for the \textsf{PSC}\xspace can be shown to be an $O(\beta + r)$ approximation. In particular, this extension involves doing a separate rounding for the light elements in each color class, which does not take advantage of the fact that a set in $\mathcal{R}$ may cover elements from multiple color classes. When $\beta$ is a constant, our guarantee is an exponential improvement over $O(\beta + r)$.
Our analysis of the randomized rounding process may be of independent interest, and at its core establishes the following type of claim. Suppose we have a set system $(X, \mathcal{R})$, where each set $S_i \in \mathcal{R}$ has at most $k$ elements; here $k$ is a parameter that can be much smaller than $|X|$. Suppose that we construct a collection of subsets $\mathcal{R}igma \subseteq \mathcal{R}$ by independently picking each set $S_i \in \mathcal{R}$ with a certain probability $0 < \mu_i < 1$. Assume that the set system and the $\mu_i$ ensure the following properties:
\begin{enumerate}
\item The expected number of elements that $\mathcal{R}igma$ covers is large in terms of $k$, say at least $6k$.
\item For any element in $X$, the probability that it is covered by $\mathcal{R}igma$ is at most a constant strictly smaller than $1$.
\end{enumerate}
Given these conditions, our analysis shows that $\mathcal{R}igma$ covers at least $k$ elements with probability at least a positive constant.
\paragraph{Applications.}
As described earlier, for several geometric set systems, the approximation guarantee $\beta$ for Set Cover based on the natural LP relaxation is sublogarithmic. For example, when $X$ is a set of points in $\mathbb{R}^2$ (resp. $\mathbb{R}^3$), and each set in $\mathcal{R}$ consists of the points in $X$ contained in some input disk (resp. halfspace), an \textsf{LP}\xspace rounding algorithm with a guarantee $\beta = O(1)$ is known. Therefore, for the corresponding Partition Cover instances, we get an $O(\log r)$ approximation. If $X$ is a set of $n$ rectangles in $\mathbb{R}^3$, and each set in $\mathcal{R}$ is the subset of rectangles that are stabbed by an input point, then an \textsf{LP}\xspace rounding algorithm with a guarantee $\beta = O(\log \log n)$ is known. In the corresponding Partition Set Cover instance, we get an $O(\log \log n + \log r)$ approximation. We summarize some of these results in the following table. We remind the reader that it is \mathcal{N}P-hard to obtain an $o(\log r)$ approximation for all of these set systems, as shown in \mathcal{C}ref{sec:hardness}. Therefore, when $\beta = O(\log r)$, improving on the $O(\beta + \log r)$ approximation is \mathcal{N}P-hard. Otherwise, when $\beta = \omega(\log r)$, improving $O(\beta + \log r)$ also involves improving the approximation guarantee for the corresponding Set Cover problem, for example, hitting rectangles in $\mathbb{R}^3$, or covering by fat triangles in $\mathbb{R}^2$.
\begin{table}[H]
\mathsf{cen}tering
\caption{Some of the approximation guarantees for Partition Set Cover (last column). In the third column, we have $\beta$, the \textsf{LP}\xspace-based approximation guarantee for the corresponding Set Cover instance. See \cite{ClarksonV2007,AronovES2010,VaradarajanWGSC2010, EASFat, ChanGKS12,ElbTerrain} for the references establishing these bounds on $\beta$.}
\begin{tabular}{|c|c|c|c|}
\hline
$X$ & Geometric objects inducing $\mathcal{R}$ & $\beta$ & Our guarantee\\
\hline
\multirow{2}{*}{Points in $\mathbb{R}^2$} & Disks (via containment) & $O(1)$ & $O(\log r)$ \\
\hhline{|~|---}
& Fat triangles (containment) & $O(\log \log^* n)$ & $O(\log \log^* n + \log r)$ \\
\hline
\multirow{2}{*}{Points in $\mathbb{R}^3$} & Unit cubes (containment) & $O(1)$ & $O(\log r)$ \\
\hhline{|~|---}
& Halfspaces (containment) & $O(1)$ & $O(\log r)$ \\
\hline
Rectangles in $\mathbb{R}^3$ & Points (via stabbing) & $O(\log \log n)$ & $O(\log \log n + \log r)$ \\
\hline
Points on 1.5D terrain & Points on terrain (via visibility) & $O(1)$ & $O(\log r)$ \\
\hline
\end{tabular}
\end{table}
As a combinatorial application, consider the combinatorial version of the Partition Set Cover problem, where each element is contained in at most $f$ sets of $\mathcal{R}$. We note that the algorithm of \citet{bera2014approximation} can be extended in a straightforward way to obtain an $O(f \log r)$ approximation in this case. However, recall that the Set Cover \textsf{LP}\xspace can be rounded to give an $f$ approximation. Therefore, using our result, we can get an $O(f + \log r)$ approximation for the Partition Set Cover problem, which is an improvement over the earlier result.
Finally, in \mathcal{C}ref{sec:fl-mcc}, we consider analogous generalizations of the (Metric Uncapacitated) Facility Location Problem, and the so-called Minimum Cost Covering Problem (\cite{CharikarP04}). Various \textsf{LP}\xspace-based $O(1)$ approximations are known for these problems (\cite{JainVazirani2001,ByrkaFLLP,Li2013}, and \cite{CharikarP04} respectively), i.e., for these problems, $\beta = O(1)$. We show how to adapt the algorithm from \mathcal{C}ref{sec:LPRounding} to obtain $O(\log r)$ approximations for the generalizations of these problems with $r$ color classes.
\paragraph{Organization.}
In the following section, we first describe the strengthened \textsf{LP}\xspace and then discuss the randomized rounding algorithm. In \mathcal{C}ref{subsec:coverage}, we prove the second property of $\mathcal{R}igma_\ell$ as stated above. In \mathcal{C}ref{subsec:expectation} we prove a technical lemma that is required in the \mathcal{C}ref{subsec:coverage}. In \mathcal{C}ref{subsec:ellipsoid}, we show how to obtain a feasible solution to the strengthened \textsf{LP}\xspace, despite exponentially many constraints in the \textsf{LP}\xspace. In \mathcal{C}ref{sec:fl-mcc}, we give the $O(\log r)$ approximations for the generalizations of the Facility Location and the Minimum Cost Covering problems. Finally, in \mathcal{C}ref{sec:hardness}, we show how to extend the $\Omega(\log r)$ hardness result from \cite{bera2014approximation}, for all these problems.
\iffalse
\section{Problem Definition and the Standard \textsf{LP}\xspace}
We first consider the following \textsf{LP}\xspace relaxation for Partition Set Cover problem, which is a simple extension of the standard \textsf{LP}\xspace relaxation for the Partial Set Cover problem.
\begin{mdframed}[backgroundcolor=gray!9]
(Standard \textsf{LP}\xspace)
\begin{alignat}{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in \mathcal{R}} w_{i}x_{i} & \nonumber \\
\text{subject to} \displaystyle&\sum\limits_{i:e_{j} \in S_{i}} x_{i} \geq z_j, \quad & e_j \in X \label[constr]{constr:cover-ej}\\
\displaystyle&\sum_{e_j \in \mathcal{C}_r}z_j \ge k_t, & \mathcal{C}_t \in \{\mathcal{C}_1, \ldots, \mathcal{C}_r\} \label[constr]{constr:cover-ci}\\
\displaystyle &z_j \in [0, 1], & e_j \in X \label[constr]{constr:fractional-z}\\
\displaystyle &x_i \in [0, 1], & S_i \in \mathcal{R} \label[constr]{constr:fractional-x}
\end{alignat}
\end{mdframed}
One important way the Standard \textsf{LP}\xspace differs from the Partial Cover \textsf{LP}\xspace is that we have a coverage constraint (\mathcal{C}ref{constr:cover-ci}) for each color class $\mathcal{C}_1, \ldots, \mathcal{C}_r$. If an element $e_j \in X$ belongs to multiple color classes, then its $z_j$ appears in multiple such constraints. Unfortunately, this \textsf{LP}\xspace has a large integrality gap as demonstrated by the following simple construction.
\subsection{Integrality Gap}
Let $(X, \mathcal{R})$ be the given set system, where $X = \{e_1, e_2, \ldots, e_n\}$ and $\mathcal{R} = \{S_1, S_2, \ldots, S_{\sqrt{n}}\}$ -- assuming $n$ is a perfect square. The sets $S_i$ form a partition of $X$, such that each $S_i$ contains exactly $\sqrt{n}$ elements. For any $1 \le i \le \sqrt{n}$, the color class $\mathcal{C}_i$ equals $S_i$, and its coverage requirement, $k_i = 1$. Also, for each set $S_i$, $w_i = 1$. Clearly any integral optimal solution must choose all sets $S_1, \ldots, S_{\sqrt{n}}$, with cost $\sqrt{n}$.
However, consider a fractional solution $(x, z)$, where for any $S_i \in \mathcal{R}$, $x_i = \frac{1}{\sqrt{n}}$; and for any $e_j \in X$, $z_j = \frac{1}{\sqrt{n}}$. It is easy to see that this solution satisfies all constraints, and has cost $1$. Note that the sub-instance induced by each color class is equivalent to a Partial Set Cover instance, and hence such an integrality gap can be overcome if we knew the heaviest set from each color class. However, the standard technique of guessing the heaviest set from each color class will take time that is exponential in the number of color classes. Therefore, we strengthen this \textsf{LP}\xspace by adding extra constraints.
\fi
\section{Strenghtened \textsf{LP}\xspace and Randomized Rounding} \label{sec:LPRounding}
Recall that the input contains a set system $(X, \mathcal{R})$, with weights on the sets, where $X = \{e_1, \ldots, e_n\}$ and $\mathcal{R} = \{S_1, \ldots, S_m\}$. We are also given $r$ non-empty subsets of $X$: $\mathcal{C}_1, \ldots, \mathcal{C}_r$, where each $\mathcal{C}_t$ is referred to as a \emph{color class}. Here, we consider a generalization of the Partition Set Cover problem, where the color classes cover $X$ but are no longer required to form a partition of $X$, i.e., an element $e_j$ can belong to multiple color classes. Each color class $\mathcal{C}_t$ has a coverage requirement $1 \le k_t \le |\mathcal{C}_t|$. The objective of the Partition Set Cover problem is to find a minimum-weight sub-collection $\mathcal{R}' \subseteq \mathcal{R}$, such that it meets the coverage requirement of each color class, i.e., for each color class $\mathcal{C}_t$, we have that $|(\bigcup \mathcal{R}') \cap \mathcal{C}_t| \ge k_t$. Let $OPT$ denote the cost of an optimal solution for this problem.
Now, we describe the strengthened \textsf{LP}\xspace. Imagine $\mathcal{A} \subseteq \mathcal{R}$ is a collection of sets that we have decided to add to our solution. The sets in $\mathcal{A}$ may cover some elements from each color class $\mathcal{C}_t$, potentially reducing the remaining coverage requirement. For a color class $\mathcal{C}_t$, define $\mathcal{C}_t(\mathcal{A}) \coloneqq (\bigcup \mathcal{A}) \cap \mathcal{C}_t$ to be the set of elements of color $\mathcal{C}_t$, covered by the sets in $\mathcal{A}$. Then, let $k_t(\mathcal{A}) \coloneqq \max\{0, k_t - |\mathcal{C}_t(\mathcal{A})| \}$ be the residual coverage requirement of $\mathcal{C}_t$ with respect to the collection $\mathcal{A}$. Finally, for a set $S_i \not\in \mathcal{A}$, and a color class $\mathcal{C}_t$, define $\textsf{deg}_t(S_i, \mathcal{A}) \coloneqq |S_i \cap (\mathcal{C}_t \setminus \mathcal{C}_t(\mathcal{A}))|$ to be the additional number of elements from $\mathcal{C}_t$ covered by $S_i$, provided that $\mathcal{A}$ is already a part of the solution. For any $\mathcal{C}_t$ and for any collection $\mathcal{A} \subseteq \mathcal{R}$, the following constraint is satisfied by any feasible integral solution:
$$\sum_{S_i \not\in \mathcal{A}} x_i \cdot \min\{\textsf{deg}_t(S_i, \mathcal{A}), k_t(\mathcal{A})\} \ge k_t(\mathcal{A})$$
For a detailed explanation of why this constraint holds, we refer the reader to the discussion in \citet{bera2014approximation}. Adding such constraints for each $\mathcal{A} \subseteq \mathcal{R}$ and for each $\mathcal{C}_t$, gives the following strengthened \textsf{LP}\xspace.
\begin{mdframed}[backgroundcolor=gray!9]
(Strengthened \textsf{LP}\xspace)
\begin{alignat}{3}
\text{minimize} \displaystyle&\sum\limits_{S_i \in \mathcal{R}} w_{i}x_{i} & \nonumber \\
\text{subject to\quad }
&\text{\mathcal{C}ref{constr:cover-ej,constr:cover-ci,constr:fractional-z,constr:fractional-x}, and } \nonumber
\\\displaystyle&\sum_{S_i \not\in \mathcal{A}} x_i \cdot \min\{\textsf{deg}_t(S_i, \mathcal{A}), k_t(\mathcal{A})\} \ge k_t(\mathcal{A}), &\quad \forall \mathcal{C}_t \in \{\mathcal{C}_1, \ldots, \mathcal{C}_r\} \text{ and } \forall \mathcal{A} \subseteq \mathcal{R} \label[constr]{constr:s-cover-ci-a}
\end{alignat}
\end{mdframed}
Let $(x, z)$ be an \textsf{LP}\xspace solution to the natural \textsf{LP}\xspace\,---\, i.e., $(x, z)$ satisfies \mathcal{C}ref{constr:cover-ej,constr:cover-ci,constr:fractional-z,constr:fractional-x}. Let $H = \{e_j \in X \mid \sum_{ S_i \ni e_j} x_i \ge \frac{1}{6 \alpha} \}$ be the set of \emph{heavy} elements, where $\alpha > 1$ is some constant to be fixed later. Let $(\widetilde{x})$ be a solution defined as $\widetilde{x}_i = \min\{6\alpha \cdot x_i, 1\}$, for all $S_i \in \mathcal{R}$. By definition, for any heavy element $e_j \in H$, we have that $\sum_{S_i \ni e_j} \widetilde{x}_i \ge 1$, so $(\widetilde{x})$ is a feasible Set Cover \textsf{LP}\xspace solution for the projected set system $(H, \mathcal{R}_{|H})$, with cost at most $6\alpha \cdot \sum_{S_i \in \mathcal{R}} x_i$. Let $\mathcal{A}'$ be the collection of sets returned by a $\beta$-approximate Set Cover \textsf{LP}\xspace rounding algorithm. Starting from an empty solution, we add the sets from $\mathcal{A}'$ to the solution. Let $\mathcal{A} = \mathcal{A}' \cup \{S_i \in \mathcal{R} \mid x_i \ge \frac{1}{6\alpha} \}$. However, notice that if $x_i \ge \frac{1}{6\alpha}$ for some set $S_i$, then all elements in $S_i$ are heavy by definition, and hence are covered by $\mathcal{A}'$. Therefore, $\mathcal{A}$ and $\mathcal{A}'$ cover the same set of elements from $X$, and we may pretend that we have added the sets in $\mathcal{A}$ to our solution.
In \mathcal{C}ref{subsec:ellipsoid}, we discuss how to obtain a fractional \textsf{LP}\xspace solution $(x, z)$ satisfying the following properties, in polynomial time.
\begin{enumerate}
\item $(x, z)$ satisfies \mathcal{C}ref{constr:cover-ej,constr:cover-ci,constr:fractional-z,constr:fractional-x}.
\item $\sum_{S_i \in \mathcal{R}} w_i x_i \le 2 \cdot OPT$
\item $\sum_{S_i \not\in \mathcal{A}} x_i \cdot \min\{ \textsf{deg}_{t}(S_i, \mathcal{A}), k_t(\mathcal{A}) \} \ge k_t(\mathcal{A}) \qquad \forall \mathcal{C}_t \in \{ \mathcal{C}_1, \ldots, \mathcal{C}_t \}$, where $\mathcal{A}$ is obtained from $(x, z)$ as described above.
\end{enumerate}
We assume that we have such a fractional \textsf{LP}\xspace solution $(x, z)$. Now, for any uncovered element $e_j \in X \setminus \bigcup \mathcal{A}$, we have that $\sum_{ S_i \ni e_j} x_i < \frac{1}{6\alpha}$. Furthermore, for any set $S_i \in \mathcal{R} \setminus \mathcal{A}$, we have $x_i < \frac{1}{6\alpha}$. However, even after adding $\mathcal{A}$ to the solution, each color class $\mathcal{C}_t$ is left with some residual coverage $k_t(\mathcal{A})$. In order to satisfy this residual coverage requirement, we use the following randomized rounding algorithm.
Consider an iteration $\ell$ of the algorithm. In this iteration, for each set $S_i \in \mathcal{R} \setminus \mathcal{A}$, we independently add $S_i$ to the solution with probability $6x_i$. Let $\mathcal{R}igma_\ell$ be the collection of sets added during this iteration in this manner. We perform $c \log r$ independent iterations of this algorithm for some constant $c$, and let $\mathcal{R}igma = \bigcup_{\ell = 1}^{c \log r} \mathcal{R}igma_\ell$. In \mathcal{C}ref{subsec:coverage}, we prove the following property about $\mathcal{R}igma_\ell$.
\begin{restatable}{lemma}{coverage}
\label{lem:constant-prob}
For any color class $\mathcal{C}_t$, the solution $\mathcal{R}igma_\ell$ covers at least $k_t(\mathcal{A})$ elements from $\mathcal{C}_t \setminus \mathcal{C}_t(\mathcal{A})$ with probability at least a positive constant.
\end{restatable}
This lemma implies the following result.
\begin{theorem} \label{thm:main-theorem}
Suppose there exists a polynomial time \textsf{LP}\xspace rounding algorithm that rounds a given Set Cover \textsf{LP}\xspace solution on any projection of $(X, \mathcal{R})$, within a $\beta$ factor. Then, there exists a polynomial time randomized algorithm that returns a solution with approximation guarantee of $O(\beta + \log r)$ for the Partition Set Cover problem, with at least a constant probability.
\end{theorem}
\begin{proof}
First, we argue about the running time. As described earlier, the discussion of how to obtain the required \textsf{LP}\xspace solution $(x, z)$ in polynomial time is deferred to \mathcal{C}ref{subsec:ellipsoid}. It is easy to see that the rest of the steps take polynomial time, and hence the overall time taken by this algorithm is polynomial.
Now, we argue about the feasibility of this solution. The sets in $\mathcal{A}'$ form a cover for the heavy elements by assumption. Furthermore, from \mathcal{C}ref{lem:constant-prob}, in any iteration $\ell$, the sets in $\mathcal{R}igma_\ell$ satisfy the remaining coverage requirement of any color class $\mathcal{C}_t$, with at least a constant probability. It follows from union bound that the probability that there exists a color class with unmet coverage requirement after $c \log r$ independent iterations is at most $1/(2r)$, for appropriately chosen constant $c$. Therefore, the solution $\mathcal{A}' \cup \mathcal{R}igma$ is feasible with probability at least $1-1/(2r).$
As argued earlier, $w(\mathcal{A}') \le 6\alpha \beta \cdot \sum_{S_i \in \mathcal{R}} w_i x_i = O(\beta) \cdot OPT$, since $\alpha$ is a constant. It is also easy to see that for any iteration $\ell$ of the randomized rounding, $\operatorname{E}[w(\mathcal{R}igma_\ell)] \le 6 \cdot \sum_{S_i \in \mathcal{R}} w_i x_i = O(1) \cdot OPT$. Therefore, $\operatorname{E}[w(\mathcal{R}igma)] \le c' \log r \cdot OPT$, for some constant $c'$. Therefore, $\Pr[w(\mathcal{R}igma) \le 3c' \log r \cdot OPT] \ge \frac{2}{3}$, using Markov's inequality. The theorem follows from an application of union bound over the events concerning the feasibility and the approximation guarantee of the solution.
\end{proof}
\iffalse
Let $\mathcal{R}igma_\ell$ be the collection of sets added during this iteration. We show that with at least a constant probability, $\mathcal{R}igma_\ell$ covers at least $k_t(\mathcal{A})$ elements from $\mathcal{C}_t$, for any color class $\mathcal{C}_t$. Let $\mathcal{R}igma$ denote the collection of sets added throughout $O(\log r)$ independent iterations. It then follows from the standard arguments that, the probability that there exists a color class with unmet coverage requirement is at most $1/r$. Furthermore, the expected weight of the sets added to the solution during any iteration, $\operatorname{E}[w(\mathcal{R}igma_\ell)] \le 6 \cdot w(x, z)$. Again, using standard arguments involving Chernoff Bounds, it follows that $w(\mathcal{R}igma) \le O(\log r) \cdot w(x, z)$ with high probability. Therefore, $\mathcal{R}igma \cup \mathcal{A}'$ is an $O(\beta + \log r)$-approximation. In the rest of the section, we prove the preceding claim about $\mathcal{R}igma_\ell$.
\fi
\iffalse
\section{Randomized Rounding Algorithm}
\begin{algorithm}
\caption{RandomizedRounding($x$)}
\begin{algorithmic}[1]
\mathcal{R}tate Let $H = \{e_j \in X \mid \sum_{ S_i \ni e_j} x_i \ge \frac{1}{6 \alpha} \}$ for some constant $\alpha$.
\mathcal{R}tate Let $\widetilde{x}$ be a Set Cover \textsf{LP}\xspace solution, where $\widetilde{x}_i \gets \min\{1, 6\alpha \cdot x_i\}$ for each $S_i \in \mathcal{R}$.
\mathcal{R}tatex Let $\mathcal{A}'$ be the solution returned by the Set Cover \textsf{LP}\xspace rounding algorithm.
\mathcal{R}tate Let $\mathcal{A} \gets \mathcal{A}' \cup \{S_i \in \mathcal{R} \mid x_i \ge \frac{1}{6\alpha} \}$
\mathcal{R}tate $\mathcal{R}igma \gets \emptyset$
\For{$\ell = 1$ \textbf{to} $O(\log r)$}
\mathcal{R}tate $\mathcal{R}igma_\ell \gets \emptyset$
\For{Each $S_i \in \mathcal{R} \setminus \mathcal{A}$}
\mathcal{R}tate Add $S_i$ to $\mathcal{R}igma_\ell$ with probability $6\cdot x_i$.
\operatorname{E}ndFor
\mathcal{R}tate $\mathcal{R}igma \gets \mathcal{R}igma \cup \mathcal{R}igma_\ell$.
\operatorname{E}ndFor
\mathcal{R}tate \mathcal{R}eturn $\mathcal{R}igma \cup \mathcal{A}'$
\end{algorithmic}
\end{algorithm}
Here, $H$ is a set of ``heavy'' elements that are covered to an extent of at least $\frac{1}{\alpha}$. We create a standard Set Cover \textsf{LP}\xspace solution on the set system $(H, S_{|H})$, and use a known Set Cover \textsf{LP}\xspace rounding algorithm. Let $\mathcal{A}'$ be a $\beta$-approximate solution returned by this algorithm. Note that if $x_i \ge \frac{1}{6\alpha}$, then $S_i \subseteq H$, and so the elements in $S_i$ are covered by $\mathcal{A}'$. Therefore, we can safely remove any such set from consideration without any effect. Let $X' \coloneqq X \setminus \bigcup \mathcal{A}$. Notice that for any $e_j \in X'$, $\sum_{S_i \ni e_j} x_i < \frac{1}{6\alpha}$. Furthermore, for any set $S_i \in \mathcal{R} \setminus \mathcal{A}$, we have that $x_i < \frac{1}{6\alpha}$.
We perform $O(\log r)$ iterations of randomized rounding for the sets in $\mathcal{R} \setminus \mathcal{A}$. In each such iteration, we add a set $S_i \in \mathcal{R} \setminus \mathcal{A}$ to the solution with probability $6 \cdot x_i$ -- it is easy to see that $6 x_i < 1$. Let $\mathcal{R}igma_\ell$ be the collection of sets added in an arbitrary iteration $\ell$. We show in the following that $\mathcal{R}igma_\ell$ covers at least $k_t(\mathcal{A})$ elements from the color class $\mathcal{C}_t$, with at least a constant probability. Using standard arguments, this implies that, after $O(\log r)$ iterations, $\mathcal{R}igma = \bigcup \mathcal{R}igma_\ell$ will cover at least $k_t(\mathcal{A})$ elements from each color class $\mathcal{C}_1, \ldots, \mathcal{C}_r$, with high probability. Combining with the fact that $\operatorname{E}[w(\mathcal{R}igma_\ell)] \le 6 \cdot OPT$, this readily implies that $\mathcal{R}igma \cup \mathcal{A}'$ is an $O(\beta + \log r)$-approximation. In the rest of the section, we prove the preceding claim about each $\mathcal{R}igma_\ell$.
\fi
\subsection{Analyzing Coverage of \texorpdfstring{$\mathcal{R}igma_\ell$}{Σ\_l}} \label{subsec:coverage}
As stated earlier, we show that for any color class $\mathcal{C}_t$, the probability that $\mathcal{R}igma_\ell$ covers at least $k_t(\mathcal{A})$ elements is at least a positive constant. Let $\mathcal{A}$ be the collection of sets as defined earlier. Recall that for any color class $\mathcal{C}_t$, the solution $(x, z)$ satisfies:
$$\sum_{S_i \not\in \mathcal{A}} x_i \cdot \min\{\textsf{deg}_t(S_i, \mathcal{A}), k_t(\mathcal{A})\} \ge k_t(\mathcal{A}).$$
Henceforth, fix a color class $\mathcal{C}_t$. To simplify notation, let us use the following shorthands: $\mathcal{C} \coloneqq \mathcal{C}_t \setminus \mathcal{C}_t(\mathcal{A})$ is the set of uncovered elements from $\mathcal{C}_t$. $k \coloneqq k_t(\mathcal{A})$ is the residual coverage requirement. For any set $S_i \in \mathcal{R}$, we restrict it to its projection on $\mathcal{C}_t \setminus \mathcal{C}_t(\mathcal{A})$. Similarly, for a collection of sets $\mathcal{R}' \subseteq \mathcal{R} \setminus \mathcal{A}$, we restrict $\bigcup \mathcal{R}'$ to mean the set of ``uncovered elements'' from this color class $\mathcal{C}_t$, i.e., $\bigcup \mathcal{R}' = \bigcup_{S_i \in \mathcal{R}'} S_i$ (where each $S_i \in \mathcal{R}'$ is the projection of the original set, as in the previous sentence). Finally, let $\delta_i \coloneqq \frac{\min\{\textsf{deg}_t(S_i, \mathcal{A}), k\}}{k}$. Notice that using this notation, the preceding constraint is equivalent to $$\sum_{S_i \not\in \mathcal{A}} \delta_i x_i \ge 1.$$
For a set $S_i$, let $\hat{x}_i$ be an indicator random variable that denotes whether or not $S_i$ was added to $\mathcal{R}igma_\ell$. It is easy to see that $\operatorname{E}[\hat{x}_i] = \Pr[S_i \text{ is added}] = 6x_i$. For an element $e_j \in \mathcal{C}$, let $Z_j \coloneqq \sum_{S_i \ni e_j} \hat{x}_i$ be a random variable that denotes the number of sets containing $e_j$ that are added to $\mathcal{R}igma_\ell$. Notice that $e_j$ is covered by $\mathcal{R}igma_\ell$ iff $Z_j \ge 1$. Note that, $\operatorname{E}[Z_j] = \sum_{ S_i \ni e_j} \operatorname{E}[\hat{x}_i] = \sum_{S_i \ni e_j} 6 x_i < 6 \cdot \frac{1}{6\alpha} = \frac{1}{\alpha}$.
Let $Z \coloneqq \sum_{S_i \not\in \mathcal{A}} \delta_i \cdot \hat{x}_i $ be a random variable. Notice that
\begin{equation}
Z = \sum_{S_i \not\in \mathcal{A}} \delta_i \cdot \hat{x}_i \le \frac{1}{k} \sum_{S_i \not\in \mathcal{A}} \hat{x}_i \cdot \textsf{deg}_{t}(S_i, \mathcal{A}) = \frac{1}{k} \sum_{S_i \not\in \mathcal{A}} \sum_{e_j \in S_i} \hat{x}_i = \frac{1}{k} \sum_{e_j \in \mathcal{C}} Z_j\ . \label[ineq]{ineq:Z}
\end{equation}
Here, the second equality follows from the fact that $\textsf{deg}_t(S_i, \mathcal{A})$ is exactly the number of elements in $S_i$ (that are not covered by $\mathcal{A}$).
Now following \citet{bera2014approximation}, we show the following fact about $Z$.
\begin{claim} \label{lem:Z-var}
$\Pr[Z < 2] \le \frac{3}{8}$.
\end{claim}
\begin{proof}
First, notice that $$\operatorname{E}[Z] = \sum_{S_i \not\in \mathcal{A}} \delta_i \cdot \operatorname{E}[\hat{x_i}] = 6 \cdot \sum_{S_i \not\in \mathcal{A}} \delta_i x_i \ge 6.$$
Now, consider
\begin{align*}
\mathrm{Var}[Z] = \sum_{S_i \not\in \mathcal{A}} \delta_i^2 \cdot \mathrm{Var}[\hat{x}_i] = \sum_{S_i \not\in \mathcal{A}} \delta_i^2 \cdot 6x_i (1-6x_i) \le 6 \sum_{S_i \not\in \mathcal{A}} \delta_i x_i\ . \tag{$\because$ $0 \le \delta_i \le 1$.}
\end{align*}
Using Chebyshev's inequality,
\begin{align*}
\Pr[Z < 2] \le \Pr\left[\big|Z - \operatorname{E}[Z]\big| \ge \frac{2\operatorname{E}[Z]}{3} \right] \le \frac{9}{4} \cdot \frac{\mathrm{Var}[Z]}{\operatorname{E}[Z]^2} \le \frac{9}{4} \cdot \frac{6 \sum_{S_i \not\in \mathcal{A}} \delta_i x_i}{(6 \sum_{S_i \not\in \mathcal{A}} \delta_i x_i)^2} \le \frac{3}{8}\ . \tag{$\because \sum_{S_i \not\in \mathcal{A}} \delta_i x_i \ge 1$.}
\end{align*}
\end{proof}
For convenience, let us use the following notation for some events of interest:
\begin{align*}
\mathcal{K} &\equiv \mathcal{R}igma_\ell \text{ covers at most $k-1$ elements from $\mathcal{C}$}
\\\mathcal{M} &\equiv Z < 2
\end{align*}
Recall that the objective is to show that $\Pr[\mathcal{K}]$ is upper bounded by a constant less than $1$. To this end, we first analyze $\Pr[\mathcal{M}|\mathcal{K}]$.
\begin{claim} \label{lem:p-given-q}
$\Pr[\mathcal{M}|\mathcal{K}] \ge \frac{2}{5}$
\end{claim}
\begin{proof}
For an element $e_j \in \mathcal{C}$, define an event $$\mathcal{L}_j \equiv Z_j \ge 1 \text{ (i.e., $e_j$ is covered)}.$$
First, consider the following conditional expectation:
\begin{align}
\operatorname{E}[Z|\mathcal{K}] &\le \frac{1}{k}\sum_{e_j \in \mathcal{C}} \operatorname{E}[Z_j | \mathcal{K}] \tag{From \ref{ineq:Z}}
\\&= \frac{1}{k} \sum_{e_j \in \mathcal{C}} \Pr[\bar{\mathcal{L}_j} | \mathcal{K}] \cdot \operatorname{E}[Z_j | \mathcal{K} \cap \bar{\mathcal{L}_j}] + \Pr[\mathcal{L}_j | \mathcal{K}] \cdot \operatorname{E}[Z_j | \mathcal{K} \cap \mathcal{L}_j] \nonumber
\\&= \frac{1}{k} \sum_{e_j \in \mathcal{C}} \Pr[\mathcal{L}_j | \mathcal{K}] \cdot \operatorname{E}[Z_j | \mathcal{K} \cap \mathcal{L}_j ]\ . \tag{$\because \operatorname{E}[Z_j | \mathcal{K} \cap \bar{\mathcal{L}_j}] = 0.$}
\end{align}
In \mathcal{C}ref{subsec:expectation}, we show that the conditional expectation $E[Z_j | \mathcal{K} \cap \mathcal{L}_j]$ is upper bounded by $\frac{6}{5}$. Then, it follows that,
\begin{align*}
\operatorname{E}[Z|\mathcal{K}] &\le \frac{6}{5k} \sum_{e_j \in \mathcal{C}} \Pr[Z_j \ge 1 | \mathcal{K}]
\\&= \frac{6}{5k} \sum_{\mathcal{R}' \subseteq \mathcal{R} \setminus \mathcal{A}} \Pr[\mathcal{R}igma_\ell = \mathcal{R}' | \mathcal{K}] \cdot \left|\bigcup \mathcal{R}'\right| \tag{Where we sum over collections $\mathcal{R}' \subseteq \mathcal{R} \setminus \mathcal{A}$ s.t. $|\bigcup \mathcal{R}'| \le k-1$}
\\&\le \frac{6(k-1)}{5k} \sum_{\mathcal{R}' \subseteq \mathcal{R} \setminus \mathcal{A}} \Pr[\mathcal{R}igma_\ell = \mathcal{R}' | \mathcal{K}]
\\&< \frac{6}{5}\ .
\end{align*}
Now, using Markov's inequality, we have that
$$\Pr[\mathcal{M}|\mathcal{K}] = \Pr[Z < 2 | \mathcal{K}] \ge 1 - \frac{\operatorname{E}[Z|\mathcal{K}]}{2} \ge 1- \frac{3}{5} = \frac{2}{5}\ .$$
\end{proof}
We conclude with proving the main result of the section, which follows from \mathcal{C}ref{lem:Z-var} and \mathcal{C}ref{lem:p-given-q}.
\coverage*
\begin{proof}
Consider
$$\Pr[\mathcal{K}] = \frac{\Pr[\mathcal{K}|\mathcal{M}] \cdot \Pr[\mathcal{M}]}{\Pr[\mathcal{M}|\mathcal{K}]} \le \frac{\Pr[\mathcal{M}]}{\Pr[\mathcal{M}|\mathcal{K}]} \le \frac{3/8}{2/5} = \frac{15}{16}\ .$$
Therefore, $\Pr[\mathcal{R}igma_\ell \text{ covers at least $k_t(\mathcal{A})$ elements from }\mathcal{C}_t \setminus \mathcal{C}_t(\mathcal{A}) ] = 1 - \Pr[\mathcal{K}] \ge \frac{1}{16}\ .$
\end{proof}
\iffalse
\begin{theorem} \label{thm:main}
$\mathcal{R}igma \cup \mathcal{A}'$ is an $O(\beta + \log r)$-approximation with probability at least $1-1/r$.
\end{theorem}
\begin{proof}
By assumption, $\mathcal{A}'$ is a cover for $H$ with cost at most $6\alpha\beta \cdot OPT$, where $\alpha$ is a constant. Therefore, we argue about $\mathcal{R}igma = \bigcup_{\ell = 1}^{c \log r} \mathcal{R}igma_\ell$. From \mathcal{C}ref{lem:constant-prob}, we have that, for each color class $\mathcal{C}_t$, $\mathcal{R}igma_\ell$ covers at least $k_t$ elements, with constant probability. Therefore, for appropriately chosen constant $c$, we have
$$\Pr[\mathcal{R}igma \text{ covers $k_t$ elements from $\mathcal{C}_t$}] \ge 1 - \lr{\frac{15}{16}}^{c \log r} = 1 - \frac{1}{r^{3}}$$
By union bound, $$\Pr[\mathcal{R}igma \cup \mathcal{A} \text{ is a feasible solution }] \ge 1 - \frac{1}{r^2}.$$
This shows that $\mathcal{R}igma \cup \mathcal{A}$ is a feasible solution with probability at least $1 - \frac{1}{r}$. As argued earlier, $E[w(\mathcal{R}igma_\ell)] \le 6 \cdot OPT$. By using standard techniques and Chernoff Bounds, we can show that the approximation guarantee of the algorithm is $O(\beta + \log r)$. Using union bound on both the events, the statement of the theorem follows.
\end{proof}
\fi
\subsection{Analyzing \texorpdfstring{$\operatorname{E}[Z_j | \mathcal{K} \cap \mathcal{L}_j]$}{E[Z\_j | K ∩ L\_j]}} \label{subsec:expectation}
In this section, we show that for any $e_j \in \mathcal{C}$, the conditional expectation $\operatorname{E}[Z_j | \mathcal{K} \cap \mathcal{L}_j]$ is bounded by $\frac{6}{5}$. Recall that $\mathcal{L}_j$ denotes the event $Z_j \ge 1$ (equivalently, $e_j$ is covered by $\mathcal{R}igma_\ell$), and that $\mathcal{K}$ denotes the event that $\mathcal{R}igma_\ell$ covers at most $k-1$ elements. For notational convenience, we shorten $\mathcal{L}_j$ to $\mathcal{L}$.
Partition the sets $\mathcal{R} \setminus \mathcal{A}$ into disjoint collections $\mathcal{R}_1, \mathcal{R}_2$, where $\mathcal{R}_1$ consists of sets that do not contain $e_j$, and $\mathcal{R}_2$ consists of sets that contain $e_j$. Fix an arbitrary ordering $\sigma$ of sets in $\mathcal{R} \setminus \mathcal{A}$, where the sets in $\mathcal{R}_1$ appear before the sets in $\mathcal{R}_2$. We view the algorithm for choosing $\mathcal{R}igma_\ell$, as considering the sets in $\mathcal{R} \setminus \mathcal{A}$ according to this ordering $\sigma$, and making a random decision of whether to add each set in $\mathcal{R}igma_\ell$. Let $\mathcal{R}igma'_\ell$ be the random collection of sets added according to the ordering $\sigma$, until the first set containing $e_j$, say $S_i$, is added to $\mathcal{R}igma_\ell$. Note that if we condition on the event $\mathcal{L}$, such an $S_i$ must exist.
Let \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace denote the event that (i) $S_i$ is the first set containing $e_j$ that is added by the algorithm, and (ii) $\mathcal{R}igma'_\ell$ is the collection added by the algorithm, just after $S_i$ was added. Note that \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace contains the history of the choices made by the algorithm, until the point just after $S_i$ is considered. For a history \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace, $k-1-|\bigcup \mathcal{R}igma'_\ell|$ is the maximum number of additional elements that can still be covered, without violating the condition $\mathcal{K}$ (which says that $\mathcal{R}igma_\ell$ covers at most $k-1$ elements). We say that a history \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace is relevant, if $k-1-|\bigcup \mathcal{R}igma'_\ell| \ge 0$. Thus,
\begin{equation} \label{eqn:exp-1}
\operatorname{E}[Z_j \mid \mathcal{K} \cap \mathcal{L}] = \sum_{\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace} \Pr[\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace \mid \mathcal{K} \cap \mathcal{L}] \cdot \operatorname{E}[Z_j \mid \mathcal{K} \cap \mathcal{L} \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace],
\end{equation}
where we only sum over the relevant histories.
Now, once $S_i$ has been added to the solution, $e_j$ is covered, thereby satisfying the condition $\mathcal{L}$. That is, the event \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace implies the event $\mathcal{L}$. It follows that,
\begin{equation} \label{eqn:exp-2}
\operatorname{E}[Z_j \mid \mathcal{K} \cap \mathcal{L} \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace] = \operatorname{E}[Z_j \mid \mathcal{K} \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace].
\end{equation}
Let $\mathcal{K}'$ denote the event that $\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell$ covers at most $p \coloneqq k-1-|\bigcup \mathcal{R}igma'_\ell|$ elements. Then,
\begin{equation} \label{eqn:exp-3}
\operatorname{E}[Z_j \mid \mathcal{K} \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace] = \operatorname{E}[Z_j \mid \mathcal{K}' \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace].
\end{equation}
Now, let $\bar{Z_j}$ be the sum of the indicator random variables $\hat{x}_{i'}$, over the the sets $S_{i'} \in \mathcal{R}_2$, that occur after $S_i$ in the ordering $\sigma$. Clearly, $\operatorname{E}[\bar{Z_j}] \le \operatorname{E}[Z_j]$. We also have,
\begin{equation} \label{eqn:exp-4}
\operatorname{E}[Z_j \mid \mathcal{K}' \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace] = 1 + \operatorname{E}[\bar{Z_j} \mid \mathcal{K}' \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace].
\end{equation}
This is because, $Z_j$ denotes the number of sets containing $e_j$ that are added to $\mathcal{R}igma_\ell$, including $S_i$; whereas $\bar{Z_j}$ does not count $S_i$. Now, $\bar{Z_j}$ and $\mathcal{K}'$ are concerned with the sets after $S_i$ according to $\sigma$, whereas \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace concerns the history upto $S_i$. Therefore,
\begin{align}
\operatorname{E}[\bar{Z_j} \mid \mathcal{K}' \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace] &= \operatorname{E}[\bar{Z_j} \mid \mathcal{K}'] \nonumber
\\&= \operatorname{E}[\bar{Z_j} |\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell \text{ covers at most } p \text{ additional elements}] \nonumber
\\&\le \frac{\operatorname{E}[\bar{Z_j}]}{\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell \text{ covers at most }p \text{ additional elements}]} \nonumber
\\&\le \frac{\operatorname{E}[\bar{Z_j]}}{\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset]} \tag{$\because$ ``$\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell$ covers at most $p$ additional elements'' $\supseteq$ ``$\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset$'' } \nonumber
\\&\le \frac{1}{\alpha - 1}\ . \label{eqn:exp-5}
\end{align}
This follows from (i) $\operatorname{E}[\bar{Z_j}] \le \operatorname{E}[Z_j] \le \frac{1}{\alpha}$ as argued earlier, and (ii) $\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset] \ge \mathsf{pr}od_{S_{i'} \ni e_j }(1 - 6x_{i'}) \ge 1 - \sum_{S_{i'} \ni e_j} 6x_{i'} \ge \frac{\alpha - 1}{\alpha}$, where we use Weierstrass product inequality in the second step.
Now, combining \mathcal{C}ref{eqn:exp-2,eqn:exp-3,eqn:exp-4,eqn:exp-5}, we conclude that
$$\operatorname{E}[Z_j \mid \mathcal{K} \cap \mathcal{L} \cap \ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace] \le 1 + \frac{1}{\alpha-1} = \frac{\alpha}{\alpha -1}.$$
Plugging this into \mathcal{C}ref{eqn:exp-1}, we get that,
$$\operatorname{E}[Z_j \mid \mathcal{L}\cap \mathcal{K}] \le \sum_{\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace} \frac{\alpha}{\alpha - 1} \cdot \Pr[\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace \mid \mathcal{K}\cap \mathcal{L}] = \frac{\alpha}{\alpha - 1} \sum_{\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace} \Pr[\ensuremath{\langle S_i, \mathcal{R}igma'_\ell \rangle}\xspace \mid \mathcal{K}\cap \mathcal{L}] = \frac{\alpha}{\alpha-1}.$$
\iffalse
Now we look at an iteration of the algorithm as first making the random decisions for the sets $\mathcal{R}_1$ (considered in an arbitrary order), and then for the sets in $\mathcal{R}_2$. Let $\mathcal{R}igma'_\ell$ be the random collection of sets added according to this ordering until the first set containing $e_j$, say $S_i$, is added to $\mathcal{R}igma_\ell$. Note that because of the condition $\mathcal{L}$, such an $S_i$ must exist. Now, let $p \coloneqq k-1 - \left| \bigcup \mathcal{R}igma'_\ell \right|$ denote the maximum number of elements that can still be covered, without violating the condition $\mathcal{K}$. Without loss of generality, we assume that we only consider the collections $\mathcal{R}igma'_\ell$, where $p \ge 0$ (since otherwise the condition $\mathcal{K}$ is violated). Note that $S_i \ni e_j$ is a random set, and hence we can write the original conditional expectation as follows:
\begin{align}
\operatorname{E}[Z_j | \mathcal{K} \cap \mathcal{L} ] &= \sum_{S_i \ni e_j} \Pr[S_i \text{ is the first set} | \mathcal{K} \cap \mathcal{L}] \cdot \operatorname{E}[Z_j | \mathcal{K}' \cap \mathcal{L} \cap S_i \text{ is the first set} ] \label{eqn:lcapn}
\end{align}
Where $\mathcal{K}'$ denotes the event that $\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell$ covers at most $p$ new elements.
Note that since $S_i$ has already been added to the solution, we can see that $E[Z_j | \mathcal{K}' \cap \mathcal{L} \cap S_i \text{ is the first set}] = 1 + \operatorname{E}[\bar{Z_j} | \mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell \text{ covers at most } p \text{ new elements}]$, where $\bar{Z_j} \coloneqq \sum_{S_{i'} \in \mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell} \hat{x}_{i'}$. This is because, once $S_i$ has been added to the solution, $e_j$ is covered, satisfying condition $\mathcal{L}$. It is also easy to see that $\operatorname{E}[\bar{Z_j}] \le \operatorname{E}[Z_j]$. Therefore, we have,
\begin{align}
\operatorname{E}[Z_j | \mathcal{K}' \cap \mathcal{L} \cap S_i \text{ is the first set}] &= 1 + \operatorname{E}[\bar{Z_j} |\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell \text{ covers at most } p \text{ new elements}] \nonumber
\\&\le 1 +\frac{\operatorname{E}[\bar{Z_j}]}{\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell \text{ covers at most }p \text{ new elements}]} \nonumber
\\&\le 1 + \frac{\operatorname{E}[\bar{Z_j]}}{\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset]} \tag{$\because$ ``$\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell$ covers at most $p$ new elements'' $\subseteq$ ``$\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset$'' } \nonumber
\\&\le 1 + \frac{1}{\alpha - 1} \label{ineq:lcapncapsi}
\end{align}
This follows from (i) $\operatorname{E}[\bar{Z_j}] \le \operatorname{E}[Z_j] \le \frac{1}{\alpha}$ as argued earlier, and (ii) $\Pr[\mathcal{R}igma_\ell \setminus \mathcal{R}igma'_\ell = \emptyset] \ge \mathsf{pr}od_{S_{i'} \ni e_j }(1 - 6x_{i'}) \ge 1 - \sum_{S_{i'} \ni e_j} 6x_{i'} \ge \frac{\alpha - 1}{\alpha}$, where we use Weierstrass product inequality in the second step.
Combining \ref{eqn:lcapn} and \ref{ineq:lcapncapsi}, we get that,
\begin{align*}
E[Z_j | \mathcal{L}\cap \mathcal{K}] &\le \sum_{S_i \ni e_j} \frac{\alpha}{\alpha-1} \cdot \Pr[S_i \text{ is the first set} | \mathcal{K} \cap \mathcal{L}] = \frac{\alpha}{\alpha - 1}
\end{align*}
\fi
Choosing $\alpha = 6$, it follows that $E[Z_j | \mathcal{L}\cap \mathcal{K}] \le \frac{6}{5}$, as claimed.
\subsection{Solving the \textsf{LP}\xspace} \label{subsec:ellipsoid}
Recall that the strengthened \textsf{LP}\xspace has exponentially many constraints, and hence we cannot use a standard \textsf{LP}\xspace algorithm directly. We guess the cost of the integral optimal solution up to a factor of $2$ (this can be done by binary search), say $\mathsf{D}elta$. Then, we convert the \textsf{LP}\xspace into a feasibility \textsf{LP}\xspace by removing the objective function, and adding a constraint $\sum_{S_i \in \mathcal{R}} w_i x_i \le \mathsf{D}elta$. We then use Ellipsoid algorithm to find a feasible solution to this \textsf{LP}\xspace and let $(x, z)$ be a candidate solution returned by the Ellipsoid algorithm. If it does not satisfy the preceding constraint, we report it as a violated constraint. We also check \mathcal{C}ref{constr:cover-ej,constr:cover-ci,constr:fractional-z,constr:fractional-x} (the number of these constraints is polynomial in the input size), and report if any of these constraints is violated.
Otherwise, let $H$ be the set of heavy elements with respect to $(x, z)$ (as defined earlier), and let $\mathcal{A} = \mathcal{A}' \cup \{S_i \in \mathcal{R} \mid x_i \ge \frac{1}{6\alpha}\}$ be the collection of sets as defined in \mathcal{C}ref{sec:LPRounding} (where $\mathcal{A}'$ is the Set Cover solution for the heavy elements $H$ returned by the rounding algorithm). Then, we check if the following constraint is satisfied with respect to this $\mathcal{A}$, for all color classes $\mathcal{C}_t$:
$$\sum_{S_i \not\in \mathcal{A}} x_i \min\{ \textsf{deg}_{t}(S_i, \mathcal{A}), k_t(\mathcal{A}) \} \ge k_t(\mathcal{A})\ .$$
If this constraint is not satisfied for some color class $\mathcal{C}_t$, we report it as a violated constraint. Otherwise we stop the Ellipsoid algorithm and proceed with the randomized rounding algorithm with the current \textsf{LP}\xspace solution $(x, z)$, as described in \mathcal{C}ref{sec:LPRounding}. Note that $(x, z)$ may not satisfy \mathcal{C}ref{constr:s-cover-ci-a} with respect to all collections $\mathcal{A}$. However, our randomized rounding algorithm requires that it is satisfied with respect to the specific collection $\mathcal{A}$ as defined earlier. Therefore, we do not need to check the feasibility of $(x, z)$ with respect to exponentially many constraints.
\section{Facility Location and Minimum Cost Covering with Multiple Outliers}
\label{sec:fl-mcc}
We consider generalizations of the Facility Location and Minimum Cost Covering problems. These generalizations are analogous to the Partition Set Cover problem considered in the previous section. That is, the set of ``clients'' (which are the objects to be covered) is partitioned into $r$ color classes, and each color class has a coverage requirement. We note that for the (standard) Facility Location and Minimum Cost Covering problems, \textsf{LP}\xspace-based $O(1)$ approximation algorithms are known. In the following, we first state the generalizations formally, and then show how the Randomized Rounding framework from the previous section can be adapted to obtain $O(\log r)$ approximations for these problems. These guarantees are asymptotically tight, in light of the hardness results given in \mathcal{C}ref{sec:hardness}.
\subsection{Facility Location with Multiple Outliers}
In the Facility Location with Multiple Outliers problem, we are given a set of Facilities $F$, a set of Clients $C$, belonging to a metric space $(F \cup C, d)$. Each facility $i \in F$ has a non-negative opening cost $f_i$. We are given $r$ non-empty subsets of clients (or ``color classes'') $\mathcal{C}_1, \ldots, \mathcal{C}_r$, that partition the set of clients. Each color class $\mathcal{C}_t$ has a connection requirement $1 \le k_t \le |\mathcal{C}_t|$. The objective of the Facility Location with Multiple Outliers problem is to find a solution $(F^*, C^*)$, such that $\sum_{i \in F'} f_i + \sum_{j \in C'} d(j, F')$ is minimized over all feasible solutions $(F', C')$. A solution $(F', C')$ is feasible if (i) $|F'| \ge 1$ and (ii) For all color classes $\mathcal{C}_t$, $|\mathcal{C}_t \cap C'| \ge k_t$. Note that this is a generalization of the Robust Facility Location problem, first considered by \citet{Charikar2001FLwO}.
A natural \textsf{LP}\xspace formulation of this problem is as follows.
\begin{mdframed}[backgroundcolor=gray!9]
(Natural \textsf{LP}\xspace for Facility Location with Multiple Outliers)
\begin{alignat}{3}
\text{minimize} &\ \sum\limits_{i \in F} f_{i}x_{i} + \sum_{i \in F, j \in C} y_{ij} \cdot d(i, j) & \nonumber \\
\text{subject to} \displaystyle&\sum\limits_{i \in F} y_{ij} \geq z_j, \quad & \forall j \in C \label[constr]{constr:fl-cover-j}\\
\displaystyle&\sum_{j \in \mathcal{C}_r}z_j \ge k_t, & \forall \mathcal{C}_t \in \{\mathcal{C}_1, \ldots, \mathcal{C}_r\} \label[constr]{constr:fl-cover-ci}\\
\displaystyle&0 \le y_{ij} \le x_i \le 1, & \forall i \in F, \forall j \in C \label[constr]{constr:fl-atmost-ci}\\
\displaystyle &z_j \in [0, 1], & \forall j \in C \label[constr]{constr:fl-fractional-z}
\end{alignat}
\end{mdframed}
We note that the integrality gap example from \mathcal{C}ref{subsec:nat-lp} can be easily converted to show a similar gap for the Facility Location with Multiple Outliers problem. Therefore, we strengthen the \textsf{LP}\xspace in a manner similar to the previous section.
First, we convert the \textsf{LP}\xspace to a feasibility \textsf{LP}\xspace by guessing the optimal cost up to a factor of $2$, say $\mathsf{D}elta$, and by adding a constraint $\sum\limits_{i \in F} f_{i}x_{i} + \sum_{i \in F, j \in C} y_{ij} \cdot d(i, j) \le \mathsf{D}elta$. Similar to \mathcal{C}ref{subsec:ellipsoid}, we use the Ellipsoid algorithm to find a feasible \textsf{LP}\xspace solution that satisfies this constraint, as well \mathcal{C}refrange{constr:fl-cover-j}{constr:fl-fractional-z}. Let $H = \{j \in C \mid \sum_{i \in F} y_{ij} \ge \frac{1}{6\alpha} \}$ be the set of \emph{heavy} clients. For any $i \in F$, let $\widetilde{x}_i \coloneqq \min \{1, 6\alpha \cdot x_i \}$, and for any $i \in F, j \in H$, let $y_{ij} \coloneqq \min\{1, 6\alpha \cdot y_{ij}\}$. It is easy to see that $(\widetilde x, \widetilde y)$ is a feasible Facility Location (without outliers) solution for the instance induced by the heavy clients, and its cost is at most $6\alpha \mathsf{D}elta$. We use an \textsf{LP}\xspace-based algorithm (such as \cite{ByrkaFLLP}) with a constant approximation guarantee to round this solution to an integral solution $(F_H, H)$, where $F_H \subseteq F$.
Let $L = C \setminus H$ be the set of \emph{light} clients. Note that for any light client $j \in L$, $z_j \le \sum_{i \in F} y_{ij} < \frac{1}{6\alpha}$. Also, for a color class $\mathcal{C}_t$, let $\mathcal{C}_t(H) \coloneqq \mathcal{C}_t \setminus H$ denote the uncovered (light) elements from $\mathcal{C}_t$, and let $k_t(H) \coloneqq k_t - |\mathcal{C}_t \cap H|$ denote its residual coverage requirement. Wlog, we assume that $k_t(H)$ is positive, otherwise we can ignore the color class $\mathcal{C}_t$ from consideration in the remaining part. Now, we check whether the following constraint holds for all color classes $\mathcal{C}_t$:
\begin{equation}
\sum_{i \in F} \min\bigg\{x_i \cdot k_t(H), \sum_{j \in \mathcal{C}_t(H)} y_{ij} \bigg\} \ge k_t(H) \label[constr]{eqn:fl-constraint}
\end{equation}
First, note that this can be easily formulated as an \textsf{LP}\xspace constraint by introducing auxiliary variables. If this constraint is not satisfied for some color class $\mathcal{C}_t$, we report it as a violated constraint. Consider the integral \textsf{LP}\xspace solution $(x', y', z')$ corresponding to a feasible integral solution $(F', C')$. We argue that $(x', y', z')$ satisfies this constraint. Note that at most $|\mathcal{C}_t \cap H|$ clients are connected from the set $\mathcal{C}_t \cap H$. Therefore, by feasibility of the solution, at least $k_t(H)$ clients must be connected from $\mathcal{C}_t(H)$. For a facility $i \in F'$, the quantity $\sum_{j \in \mathcal{C}_t(H)} y'_{ij}$ denotes the number of clients connected to $i$. However, even if the number of clients connected to $i$ is more than $k_t(H)$, only $k_t(H)$ of them count towards satisfying the residual connection requirement. Therefore, $(x', y', z')$ satisfies this constraint for all color classes, and hence it is a valid constraint.
Now, suppose we have an \textsf{LP}\xspace solution $(x, y, z)$ that satisfies \mathcal{C}refrange{constr:fl-cover-j}{eqn:fl-constraint}, and has cost at most $\mathsf{D}elta$. By ``splitting'' the facilities into multiple co-located copies if necessary, we ensure the following two conditions hold:
\begin{enumerate}
\item For any facility $i \in F$, $x_i < \frac{1}{6\alpha}$.
\item For any client $j \in L$ and any facility $i \in F$, $y_{ij} > 0\ \implies\ y_{ij} = x_i$.
\end{enumerate}
This has to be done in a careful manner, since we also want to maintain \mathcal{C}ref{eqn:fl-constraint} after the facilities have been split. This procedure results in a feasible \textsf{LP}\xspace solution of the same cost. Henceforth, we treat all co-located copies of a facility as distinct facilities for the sake of the analysis. We now show that the rounding for the light clients can be reduced to the Randomized Rounding algorithm from the previous section.
For any facility $i \in F$, let $S_i \coloneqq \{j \in L \mid x_{i} = y_{ij} \}$ denote the set of light clients that are fractionally connected to $i$. The cost of opening facility $i$ and connecting all $j \in S_i$ to $i$ is equal to $w_i \coloneqq f_i + \sum_{j \in S_i} d(i, j)$. Consider an instance $(L, \mathcal{R})$ of the Partition Set Cover problem, where $\mathcal{R} = \{S_i \mid i \in F\}$ with weights $w_i$, and residual coverage requirement $k_t(H)$ for each color class $\mathcal{C}_t(H)$, and consider the corresponding \textsf{LP}\xspace solution $(x, z)$. The following properties are satisfied by the \textsf{LP}\xspace solution.
\begin{enumerate}
\item All the elements are light, and all the sets $S_i \in \mathcal{R}$ have $x_i < \frac{1}{6\alpha}$.
\item The costs of the two \textsf{LP}\xspace solutions are equal: $$\sum_{S_i \in \mathcal{R}} w_i x_i = \sum_{i \in F} x_i \cdot \bigg(f_i + \sum_{j \in S_i} d(i, j)\bigg) = \sum_{i \in F} f_i x_i + \sum_{i \in F,\ j \in L} y_{ij} \cdot d(i, j).$$
\item \mathcal{C}ref{eqn:fl-constraint} is equivalent to:
$$\sum_{S_i \in \mathcal{R}} x_i \cdot \min\left\{ k_t(H), |S_i \cap \mathcal{C}_t| \right\} \ge k_t(H) \quad \forall \mathcal{C}_t.$$
\end{enumerate}
Therefore, we can use the Randomized Rounding algorithm from the previous section to obtain a solution $\mathcal{R}igma = \bigcup_{\ell = 1}^{O(\log r)} \mathcal{R}igma_\ell$. It has cost at most $O(\log r) \cdot \mathsf{D}elta$, and for each color class $\mathcal{C}_t(H)$, it covers at least $k_t(H)$ clients, with at least a constant probability. To obtain a solution for the Facility Location with Multiple Outliers problem, we open any facility $i \in F$, if its corresponding set $S_i$ is selected in $\mathcal{R}igma$. Furthermore, we connect $k_t(H)$ clients from $\mathcal{C}_t(H)$ to the set of opened facilities. Note that the cost of this solution is upper bounded by $w(\mathcal{R}igma) \le O(\log r) \cdot \mathsf{D}elta$. Combining this with the solution $(F_H, H)$ for the heavy clients with cost at most $O(1) \cdot \mathsf{D}elta$, we obtain our overall solution for the given instance. It is easy to see that this is an $O(\log r)$ approximation.
\subsection{Minimum Cost Covering with Multiple Outliers}
Here, we are given a set of Facilities $F$, a set of Clients $C$, belonging to a metric space $(F \cup C, d)$. Each facility $i \in F$ has a non-negative opening cost $f_i$. We are given $r$ subsets of clients (or ``color classes'') $\mathcal{C}_1, \ldots, \mathcal{C}_r$, where any client $j \in C$ belongs to at least one color class. Each color class $\mathcal{C}_t$ has a coverage requirement $1 \le k_t \le |\mathcal{C}_t|$. A ball centered at a facility $i \in F$ of radius $r \ge 0$ is the set $B(i, r) \coloneqq \{j \in C \mid d(i, j) \le r \}$. The goal is to select a set of balls $\mathcal{B} = \{B_i = B(i, r_i) \mid i \in F' \subseteq F \}$ centered at some subset of facilities $F' \subseteq F$, such that (i) The set of balls $\mathcal{B}$ satisfies the coverage requirement of each color class and (ii) the sum $\sum_{i \in F'} (f_i + r_i^\gamma)$ is minimized. Here, $\gamma \ge 1$ is a constant, and is a parameter of the problem.
Note that even though the radius of a ball centered at $i \in F$ is allowed to be any non-negative real number, it can be restricted to the following set of ``relevant'' radii: $R_i \coloneqq \{d(i, j)\mid j \in C\}$. Now, define a set system $(C, \mathcal{R})$. Here, $C$ is the set of clients, and $\mathcal{R} = \{ B(i, r) \mid i \in F, r \in R_i \}$, with weight of the set corresponding to a ball $B(i, r)$ being defined as $f_i + r^\gamma$. Now, we use the algorithm from the previous section for this set system. Let $H$ be the set of heavy clients (or elements) as defined in \mathcal{C}ref{sec:LPRounding}. We use the Primal-Dual algorithm of \citet{CharikarP04} \footnote{\citet{CharikarP04} consider the special case of $\gamma = 1$, however their algorithm easily generalizes to arbitrary $\gamma$.} with an approximation guarantee of $\beta = 3^\gamma$ (which is a constant) to obtain a cover for the heavy clients. For the remaining light clients, we use the Randomized Rounding algorithm as is. Note that this reduction from the Minimum Cost Covering with Multiple Outliers Problem to the Partition Set Cover Problem is not exact, since the solution thus obtained may select sets corresponding to concentric balls in the original instance. However, from each set of concentric balls, we can choose the largest radius ball. This pruning process does not affect the coverage, and can only decrease the cost of the solution. Therefore, it is easy to see that the resulting solution is an $O(\log r)$ approximation.
\section{\texorpdfstring{$\Omega(\log r)$}{Ω(log r)} Hardness Results} \label{sec:hardness}
In this section, we show that it is \mathcal{N}P-hard to obtain approximation guarantees better than $O(\log r)$ for the Partition Set Cover for several geometric set systems, as well as the problems considered in \mathcal{C}ref{sec:fl-mcc}. The reductions are from (unweighted) Set Cover, and are straightforward extensions of a similar hardness result shown in \cite{bera2014approximation}.
\subsection*{Geometric Set Systems}
Suppose we are given an instance of Set Cover $(X, \mathcal{R})$, where $X = \{e_1, \ldots, e_n\}$. For each set $S_i \in \mathcal{R}$, add a unit interval $I_i$ in $\mathbb{R}$, such that all intervals are disjoint. Add a point $p_{ij}$ (of color class $\mathcal{C}_j$) inside an interval $I_i$, corresponding to an element $e_j \in X$, and a set $S_i \ni e_j$. Thus, there are $n$ disjoint color classes, partitioning the set of points. The coverage requirement of each color class is $1$. It is easy to see that a feasible solution to the Partition Set Cover instance corresponds to a feasible solution to the original Set Cover instance, of the same cost. The $\Omega(\log r)$ hardness follows from the $\Omega(\log n)$ hardness for Set Cover (\cite{DS2014}), and $r = n$ is the number of color classes. Therefore, $\Omega(\log r)$ hardness follows, for the Partition Set Cover problem where the sets are unit intervals in $\mathbb{R}$. This is easily generalized to any other type of geometric objects, as all that is needed is the disjointness of the geometric objects.
\subsection*{Facility Location and Minimum Cost Covering}
First, consider the Facility Location with Multiple Outliers problem. Given an instance $(X, \mathcal{R})$ of the unweighted Set Cover problem, we add facilities $i \in F$, corresponding to sets $S_i \in R$, uniformly separated on the real line, such that the distance between the facilities is at least $|X| \cdot |\mathcal{R}|$. The opening cost of each facility is $1$. Similar to the reduction above, we add a client $c_{ij}$ co-located with facility $i \in F$, corresponding to an element $e_j \in X$, and a set $S_i \ni e_j$. The coverage requirement of each color class is set to $1$. It is easy to see the one-to-one correspondence between optimal solutions to both of these problems.
Now, we tweak the above instance for obtaining the same result for the Minimum Cost Covering with Multiple Outliers problem on a line, even when all the opening costs are $0$ (otherwise, we can use the reduction from the paragraph above as is). For each facility $i \in F$, add the clients $c_{ij}$ (corresponding to the elements in $S_i$) at a distance of $1$ from $i$ (instead of being co-located, as in the previous reduction). The coverage requirement of each color class is $1$, as before. Note that the facility $i$ and the clients $\{c_{ij} \mid e_j \in S_i \}$ form a ``cluster'', and the inter-cluster distance is large enough to ensure that an optimal solution to the resulting instance consists of disjoint clusters, which then exactly corresponds to an optimal solution to the Set Cover problem.
\subsection*{Acknowledgment}
We thank Sariel Har-Peled and Timothy M. Chan for preliminary discussions on this problem.
\end{document} |
\begin{document}
\title{Identification and well-posedness in nonparametric models with
independence conditions}
\author{Victoria Zinde-Walsh\thanks{
The support of the Social Sciences and Humanities Research Council of Canada
(SSHRC) and the \textit{Fonds\ qu\'{e}becois de la recherche sur la soci\'{e}
t\'{e} et la culture} (FRQSC) is gratefully acknowledged. } \\
\\
McGill University and CIREQ\\
victoria.zinde-walsh@mcgill.ca}
\maketitle
\date{}
\begin{center}
\pagebreak
{\LARGE Abstract}
\end{center}
This paper provides a nonparametric analysis for several classes of models,
with cases such as classical measurement error, regression with errors in
variables, and other models that may be represented in a form involving
convolution equations. The focus here is on conditions for existence of
solutions, nonparametric identification and well-posedness in the space $
S^{\ast }$ of generalized functions (tempered distributions). This space
provides advantages over working in function spaces by relaxing assumptions
and extending the results to include a wider variety of models, for example
by not requiring existence of density. Classes of (generalized) functions
for which solutions exist are defined; identification conditions, partial
identification and its implications are discussed. Conditions for
well-posedness are given and the related issues of plug-in estimation and
regularization are examined.
\section{Introduction}
Many statistical and econometric models involve independence (or conditional
independence) conditions that can be expressed via convolution. Examples are
independent errors, classical measurement error and Berkson error,
regressions involving data measured with these types of errors, common
factor models and models that conditionally on some variables can be
represented in similar forms, such as a nonparametric panel data model with
errors conditionally on observables independent of the idiosyncratic
component.
Although the convolution operator is well known, this paper provides
explicitly convolution equations for a wide list of models for the first
time. In many cases the analysis in the literature takes Fourier transforms
as the starting point, e.g. characteristic functions for distributions of
random vectors (as in the famous Kotlyarski lemma, 1967). The emphasis here
on convolution equations for the models provides the opportunity to
explicitly state nonparametric classes of functions defined by the model for
which such equations hold, in particular, for densities, conditional
densities and regression functions. The statistical model may give rise to
different systems of convolution equations and may be over-identified in
terms of convolution equations; some choices may be better suited to
different situations, for example, here in Section 2 two sets of convolution
equations (4 and 4a in Table 1) are provided for the same classical
measurement error model with two measurements; it turns out that one of
those allows to relax some independence conditions, while the other makes it
possible to relax a support assumption in identification. Many of the
convolution equations derived here are based on density-weighted conditional
averages of the observables.
The main distinguishing feature is that here all the functions defined by
the model are considered within the space of generalized functions $S^{\ast
},$ the space of so-called tempered distributions (they will be referred to
as generalized functions). This is the dual space, the space of linear
continuous functionals, on the space $S$ of well-behaved functions: the
functions in $S$ are infinitely differentiable and all the derivatives go to
zero at infinity faster than any power. An important advantage of assuming
the functions are in the space of generalized functions is that in that
space any distribution function has a density (generalized function) that
continuously depends on the distribution function, so that distributions
with mass points and fractal measures have well-defined generalized
densities.
Any regular function majorized by a polynomial belongs to $S^{\ast }$; this
includes polynomially growing regression functions and binary choice
regression as well as many conditional density functions. Another advantage
is that Fourier transform is an isomorphism of this space, and thus the
usual approaches in the literature that employ characteristic functions are
also included. Details about the space $S^{\ast }$ are in Schwartz (1966)
and are summarized in Zinde-Walsh (2012).
The model classes examined here lead to convolution equations that are
similar to each other in form; the main focus of this paper is on existence,
identification, partial identification and well-posedness conditions.
Existence and uniqueness of solutions to some systems of convolution
equations in the space $S^{\ast }$ were established in Zinde-Walsh (2012).
Those results are used here to state identification in each of the models.
Identification requires examining support of the functions and generalized
functions that enter into the models; if support excludes an open set then
identification at least for some unknown functions in the model fails,
however, some isolated points or lower-dimensional manifolds where the e.g.
the characteristic function takes zero values (an example is the uniform
distribution) does not preclude identification in some of the models. This
point was made in e.g. Carrasco and Florens (2010), Evdokimov and White
(2011) and is expressed here in the context of operating in $S^{\ast }.$
Support restriction for the solution may imply that only partial
identification will be provided. However, even in partially identified
models some features of interest (see, e.g. Matzkin, 2007) could be
identified thus some questions could be addressed even in the absence of
full identification. A common example of incomplete identification which
nevertheless provides important information is Gaussian deconvolution of a
blurred image of a car obtained from a traffic camera; the filtered image is
still not very good, but the licence plate number is visible for forensics.
Well-posedness conditions are emphasized here. The well-known definition by
Hadamard (1923) defines well-posedness via three conditions: existence of a
solution, uniqueness of the solution and continuity in some suitable
topology. The first two are essentially identification. Since here we shall
be defining the functions in subclasses of $S^{\ast }$ we shall consider
continuity in the topology of this generalized functions space. This
topology is weaker than the topologies in functions spaces, such as the
uniform or $L_{p}$ topologies; thus differentiating the distribution
function to obtain a density is a well-posed problem in $S^{\ast },$ by
contrast, even in the class of absolutely continuous distributions with
uniform metric where identification for density in the space $L_{1}$ holds,
well-posedness however does not obtain (see discussion in Zinde-Walsh,
2011). But even though in the weaker topology of $S^{\ast }$ well-posedness
obtains more widely, for the problems considered here some additional
restrictions may be required for well-posedness.
Well-posedness is important for plug-in estimation since if the estimators
are in a class where the problem is well-posed they are consistent, and
conversely, if well-posedness does not hold consistency will fail for some
cases. Lack of well-posedness can be remedied by regularization, but the
price is often more extensive requirements on the model and slower
convergence. For example, in deconvolution (see e.g. Fan, 1991, and most
other papers cited here) spectral cut-off regularization is utilized; it
crucially depends on knowing the rate of the decay at infinity of the
density.
Often non-parametric identification is used to justify parametric or
semi-parametric estimation; the claim here is that well-posedness should be
an important part of this justification. The reason for that is that in
estimating a possibly misspecified parametric model, the misspecified
functions of the observables belong in a nonparametric neighborhood of the
true functions; if the model is non-parametrically identified, the unique
solution to the true model exists, but without well-posedness the solution
to the parametric model and to the true one may be far apart.
For deconvolution An and Hu (2012) demonstrate well-posedness in spaces of
integrable density functions when the measurement error has a mass point;
this may happen in surveys when probability of truthful reporting is
non-zero. The conditions for well-posedness here are provided in $S^{\ast }$
; this then additionally does not exclude mass points in the distribution of
the mismeasured variable itself; there is some empirical evidence of mass
points in earnings and income. The results here show that in $S^{\ast }$
well-posedness holds more generally: as long as the error distribution is
not super-smooth.
The solutions for the systems of convolution equations can be used in
plug-in estimation. Properties of nonparametric plug-in estimators are based
on results on stochastic convergence in $S^{\ast }$ for the solutions that
are stochastic functions expressed via the estimators of the known functions
of the observables.
Section 2 of the paper enumerates the classes of models considered here.
They are divided into three groups: 1. measurement error models with
classical and Berkson errors and possibly an additional measurement, and
common factor models that transform into those models; 2. nonparametric
regression models with classical measurement and Berkson errors in
variables; 3. measurement error and regression models with conditional
independence. The corresponding convolution equations and systems of
equations are provided and discussed. Section 3 is devoted to describing the
solutions to the convolution equations of the models. The main mathematical
aspect of the different models is that they require solving equations of a
similar form. Section 4 provides a table of identified solutions and
discusses partial identification and well-posedness. Section 5 examines
plug-in estimation. A brief conclusion follows.
\section{Convolution equations in classes of models with independence or
conditional independence}
This section derives systems of convolution equations for some important
classes of models. The first class of model is measurement error models with
some independence (classical or Berkson error) and possibly a second
measurement; the second class is regression models with classical or Berkson
type error; the third is models with conditional independence. For the first
two classes the distributional assumptions for each model and the
corresponding convolution equations are summarized in tables; it is
indicated which of the functions are known and which unknown; a brief
discussion of each model and derivation of the convolution equations
follows. The last part of this section discusses convolution equations for
two specific models with conditional independence; one is a panel data model
studied by Evdokimov (2011), the other a regression model where independence
of measurement error of some regressors obtains conditionally on a covariate.
The general assumption made here is that all the functions in the
convolution equations belong to the space of generalized functions $S^{\ast
}.$
\textbf{Assumption 1. }\textit{All the functions defined by the statistical
model are in the space of generalized functions }$S^{\ast }.$
This space of generalized function includes functions from most of the
function classes that are usually considered, but allows for some useful
generalizations. The next subsection provides the necessary definitions and
some of the implications of working in the space $S^{\ast }.$
\subsection{The space of generalized functions $S^{\ast }.$}
The space $S^{\ast }$ is the dual space, i.e. the space of continuous linear
functionals on the space $S$ of functions. The theory of generalized
functions is in Schwartz (1966); relevant details are summarized in
Zinde-Walsh (2012). In this subsection the main definitions and properties
are reproduced.
Recall the definition of $S.$
For any vector of non-negative integers $m=(m_{1},...m_{d})$ and vector $
t\in R^{d}$ denote by $t^{m}$ the product $t_{1}^{m_{1}}...t_{d}^{m_{d}}$
and by $\partial ^{m}$ the differentiation operator $\frac{\partial ^{m_{1}}
}{\partial x_{1}^{m_{1}}}...\frac{\partial ^{m_{d}}}{\partial x_{d}^{m_{d}}}
; $ $C_{\infty }$ is the space of infinitely differentiable (real or
complex-valued) functions on $R^{d}.$ The space $S\subset C_{\infty }$ of
test functions is defined as:
\begin{equation*}
S=\left\{ \psi \in C_{\infty }(R^{d}):|t^{l}\partial ^{k}\psi (t)|=o(1)\text{
as }t\rightarrow \infty \right\} ,
\end{equation*}
for any $k=(k_{1},...k_{d}),l=(l_{1},...l_{d}),$ where $k=(0,...0)$
corresponds to the function itself, $t\rightarrow \infty $ coordinate-wise;
thus \ the functions in $S$ go to zero at infinity faster than any power as
do their derivatives; they are rapidly decreasing functions. A sequence in $
S $ converges if in every bounded region each $\left\vert t^{l}\partial
^{k}\psi (t)\right\vert $ converges uniformly.
Then in the dual space $S^{\ast }$ any $b\in S^{\ast }$ represents a linear
functional on $S;$ the value of this functional for $\psi \in S$ is denoted
by $\left( b,\psi \right) .$ When $b$ is an ordinary (point-wise defined)
real-valued function, such as a density of an absolutely continuous
distribution or a regression function, the value of the functional on
real-valued $\psi $ defines it and is given by
\begin{equation*}
\left( b,\psi \right) =\int b(x)\psi (x)dx.
\end{equation*}
If $b$ is a characteristic function it may be complex-valued, then the value
of the functional $b$ applied to $\psi \in S$ where $S$ is the space of
complex-valued functions, is
\begin{equation*}
\left( b,\psi \right) =\int b(x)\overline{\psi (x)}dx,
\end{equation*}
where overbar denotes complex conjugate. The integrals are taken over the
whole space $R^{d}.$
The generalized functions in the space $S^{\ast }$ are continuously
differentiable and the differentiation operator is continuous; Fourier
transforms and their inverses are defined for all $b\in S^{\ast },$ the
operator is a (continuos) isomorphism of the space $S^{\ast }.$ However,
convolutions and products are not defined for all pairs of elements of $
S^{\ast },$ unlike, say, the space $L_{1};$ on the other hand, in $L_{1}$
differentiation is not defined and not every distribution has a density that
is an element of $L_{1}.$
Assumption 1 places no restrictions on the distributions, since in $S^{\ast
} $ any distribution function is differentiable and the differentiation
operator is continuous. The advantage of not restricting distributions to be
absolutely continuous is that mass points need not be excluded;
distributions representing fractal measures such as the Cantor distribution
are also allowed. This means that mixtures of discrete and continuous
distributions e.g. such as those examined by An and Hu (2012) for
measurement error in survey responses, some of which may be
error-contaminated, but some may be truthful leading to a mixture with a
mass point distribution are included. Moreover, in $S^{\ast }$ the case of
mass points in the distribution of the mismeasured variable is also easily
handled; in the literature such mass points are documented for income or
work hours distributions in the presence of rigidities such as unemployment
compensation rules (e.g. Green and Riddell, 1997). Fractal distributions may
arise in some situations, e.g. Karlin's (1958) example of the equilibrium
price distribution in an oligopolistic game.
For regression functions the assumption $g\in S^{\ast }$ implies that growth
at infinity is allowed but is somewhat restricted. In particular for any
ordinary point-wise defined function $b\in S^{\ast }$ the condition
\begin{equation}
\int ...\int \Pi _{i=1}^{d}\left( (1+t_{i}^{2}\right)
^{-1})^{m_{i}}\left\vert b(t)\right\vert dt_{1}...dt_{d}<\infty ,
\label{condition}
\end{equation}
needs to be satisfied for some non-negative valued $m_{1},...,m_{d}.$ If a
locally integrable function $g$ is such that its growth at infinity is
majorized by a polynomial, then $b\equiv g$ satisfies this condition. While
restrictive this still widens the applicability of many currently available
approaches. For example in Berkson regression the common assumption is that
the regression function be absolutely integrable (Meister, 2009); this
excludes binary choice, linear and polynomial regression functions that
belong to $S^{\ast }$ and satisfy Assumption 1. Also, it is advantageous to
allow for functions that may not belong to any ordinary function classes,
such as sums of $\delta -$functions ("sum of peaks") or (mixture) cases with
sparse parts of support, such as isolated points; such functions are in $
S^{\ast }.$ Distributions with mass points can arise when the response to a
survey questions may be only partially contaminated; regression "sum of
peaks" functions arise e.g. in spectroscopy and astrophysics where isolated
point supports are common.
\subsection{Measurement error and related models}
Current reviews for measurement error models are in Carrol et al, (2006),
Chen et al (2011), Meister (2009).
Here and everywhere below the variables $x,z,x^{\ast },u,u_{x}$ are assumed
to be in $R^{d};y,v$ are in $R^{1};$ all the integrals are over the
corresponding space; density of $\nu $ for any $\nu $ is denoted by $f_{v};$
independence is denoted by $\bot $; expectation of $x$ conditional on $z$ is
denoted by $E(x|z).$
\subsubsection{List of models and corresponding equations}
The table below lists various models and corresponding convolution
equations. Many of the equations are derived from density weighted
conditional expectations of the observables.
Recall that for two functions, $f$ and $g$ convolution $f\ast g$ is defined
by
\begin{equation*}
(f\ast g)\left( x\right) =\int f(w)g(x-w)dw;
\end{equation*}
this expression is not always defined. A similar expression (with some abuse
of notation since generalized functions are not defined pointwise) may hold
for generalized functions in $S^{\ast };$ similarly, it is not always
defined. With Assumption 1 for the models considered here we show that
convolution equations given in the Tables below hold in $S^{\ast }.$
\begin{center}
\textbf{Table 1.} Measurement error models: 1. Classical measurement error;
2. Berkson measurement error; 3. Classical measurement error with additional
observation (with zero conditional mean error); 4., 4a. Classical error with
additional observation (full independence).
\begin{tabular}{|c|c|c|c|c|}
\hline
Model & $
\begin{array}{c}
\text{Distributional} \\
\text{assumptions}
\end{array}
$ & $
\begin{array}{c}
\text{Convolution } \\
\text{equations}
\end{array}
$ & $
\begin{array}{c}
\text{Known} \\
\text{ functions}
\end{array}
$ & $
\begin{array}{c}
\text{Unknown} \\
\text{ functions}
\end{array}
$ \\ \hline
\multicolumn{1}{|l|}{$\ \ \ $1.} & \multicolumn{1}{|l|}{$\ \ \ \ \ \ \
\begin{array}{c}
z=x^{\ast }+u \\
x^{\ast }\bot u
\end{array}
$} & \multicolumn{1}{|l|}{$\ \ \ \ \ \ \ f_{x^{\ast }}\ast f_{u}=f_{z}$} &
\multicolumn{1}{|l|}{$\ \ \ \ \ \ \ f_{z},f_{u}$} & \multicolumn{1}{|l|}{$\
\ \ \ \ \ \ f_{x^{\ast }}$} \\ \hline
2. & $\
\begin{array}{c}
z=x^{\ast }+u \\
z\bot u
\end{array}
$ & $f_{z}\ast f_{-u}=f_{x^{\ast }}$ & $f_{z},f_{u}$ & $f_{x^{\ast }}$ \\
\hline
\multicolumn{1}{|l|}{$\ $\ 3.} & \multicolumn{1}{|l|}{$\ \
\begin{array}{c}
z=x^{\ast }+u; \\
x=x^{\ast }+u_{x} \\
x^{\ast }\bot u; \\
E(u_{x}|x^{\ast },u)=0; \\
E\left\Vert z\right\Vert <\infty ;E\left\Vert u\right\Vert <\infty .
\end{array}
$} & \multicolumn{1}{|l|}{$
\begin{array}{c}
f_{x^{\ast }}\ast f_{u}=f_{z}; \\
h_{k}\ast f_{u}=w_{k}, \\
\text{with }h_{k}(x)\equiv x_{k}f_{x^{\ast }}(x); \\
k=1,2...d
\end{array}
$} & \multicolumn{1}{|l|}{$
\begin{array}{c}
f_{z},w_{k}, \\
k=1,2...d
\end{array}
$} & \multicolumn{1}{|l|}{$f_{x^{\ast }}$; $f_{u}$} \\ \hline
4. & $
\begin{array}{c}
z=x^{\ast }+u; \\
x=x^{\ast }+u_{x};x^{\ast }\bot u; \\
x^{\ast }\bot u_{x};E(u_{x})=0; \\
u\bot u_{x}; \\
E\left\Vert z\right\Vert <\infty ;E\left\Vert u\right\Vert <\infty .
\end{array}
$ & $
\begin{array}{c}
f_{x^{\ast }}\ast f_{u}=f_{z}; \\
h_{k}\ast f_{u}=w_{k}; \\
f_{x^{\ast }}\ast f_{u_{x}}=f_{x}; \\
\text{with }h_{k}(x)\equiv x_{k}f_{x^{\ast }}(x); \\
k=1,2...d
\end{array}
$ & $
\begin{array}{c}
f_{z}\text{, }f_{x};w;w_{k} \\
k=1,2...d
\end{array}
$ & $f_{x^{\ast }};f_{u},$ $f_{u_{x}}$ \\ \hline
4a. & $
\begin{array}{c}
\text{Same model as 4.,} \\
\text{alternative} \\
\text{equations:}
\end{array}
$ & $
\begin{array}{c}
f_{x^{\ast }}\ast f_{u}=f_{z}; \\
f_{u_{x}}\ast f_{-u}=w; \\
h_{k}\ast f_{-u}=w_{k}, \\
\text{with }h_{k}(x)\equiv x_{k}f_{u_{x}}(x); \\
k=1,2...d
\end{array}
$ & --"-- & --"-- \\ \hline
\end{tabular}
\end{center}
Notation: $k=1,2,...,d;$ in 3. and 4, $w_{k}=E(x_{k}f_{z}(z)|z);$ in 4a $
w=f_{z-x};w_{k}=E(x_{k}w(z-x)|\left( z-x\right) ).$
\textbf{Theorem 1.} \textit{Under Assumption 1 for each of the models 1-4
the corresponding convolution equations of Table 1 hold in the generalized
functions space }$S^{\ast }$\textit{.}
The proof is in the derivations of the following subsection.
Assumption 1 requires considering all the functions defined by the model as
elements of the space $S^{\ast },$ but if the functions (e.g. densities, the
conditional moments) exist as regular functions, the convolutions are just
the usual convolutions of functions, on the other hand, the assumption
allows to consider convolutions for cases where distributions are not
absolutely continuous.
\subsubsection{\protect
Measurement error models and derivation of
the corresponding equations.}
1. The classical measurement error model.
The case of the classical measurement error is well known in the literature.
The concept of error independent of the variable of interest is applicable
to many problems in seismology, image processing, where it may be assumed
that the source of the error is unrelated to the signal. In e.g. Cunha et
al. (2010) it is assumed that some constructed measurement of ability of a
child derived from test scores fits into this framework. As is well-known in
regression a measurement error in the regressor can result in a biased
estimator (attenuation bias).
Typically the convolution equation
\begin{equation*}
f_{x^{\ast }}\ast f_{u}=f_{z}
\end{equation*}
is written for density functions when the distribution function is
absolutely continuous. The usual approach to possible non-existence of
density avoids considering the convolution and focuses on the characteristic
functions. Since density always exists as a generalized function and
convolution for such generalized functions is always defined it is possible
to write convolution equations in $S^{\ast }$ for any distributions in model
1. The error distribution (and thus generalized density $f_{u})$ is assumed
known thus the solution can be obtained by "deconvolution" (Carrol et al
(2006), Meister (2009), the review of Chen et al (2011) and papers by Fan
(1991), Carrasco and Florens(2010) among others).
2. The Berkson error model.{}
For Berkson error the convolution equation is also well-known. Berkson error
of measurement arises when the measurement is somehow controlled and the
error is caused by independent factors, e.g. amount of fertilizer applied is
given but the absorption into soil is partially determined by factors
independent of that, or students' grade distribution in a course is given in
advance, or distribution of categories for evaluation of grant proposals is
determined by the granting agency. The properties of Berkson error are very
different from that of classical error of measurement, e.g. it does not lead
to attenuation bias in regression; also in the convolution equation the
unknown function is directly expressed via the known ones when the
distribution of Berkson error is known. For discussion see Carrol et al
(2006), Meister (2009), and Wang (2004).
Models 3. and 4. The classical measurement error with another observation.
In 3., 4. in the classical measurement error model the error distribution is
not known but another observation for the mis-measured variable is
available; this case has been treated in the literature and is reviewed in
Carrol et al (2006), Chen et al \ (2011). In econometrics such models were
examined by Li and Vuong (1998), Li (2002), Schennach (2004) and
subsequently others (see e.g. the review by Chen et al, 2011). In case 3 the
additional observation contains an error that is not necessarily
independent, just has conditional mean zero.
Note that here the multivariate case is treated where arbitrary dependence
for the components of vectors is allowed. For example, it may be of interest
to consider the vector of not necessarily independent latent abilities or
skills as measured by different sections of an IQ test, or the GRE scores.
Extra measurements provide additional equations. Consider for any $k=1,...d$
the function of observables $w_{k}$ defined by density weighted expectation $
E(x_{k}f_{z}(z)|z)$ as a generalized function; it is then determined by the
values of the functional $\left( w_{k},\psi \right) $ for every $\psi \in S.$
Note that by assumption $E(x_{k}f_{z}(z)|z)=E(x_{k}^{\ast }f_{z}(z)|z);$
then for any $\psi \in S$ the value of the functional:
\begin{eqnarray*}
(E(x_{k}^{\ast }f_{z}(z)|z),\psi ) &=&\int [\int x_{k}^{\ast }f_{x^{\ast
},z}(x^{\ast },z)dx^{\ast }]\psi (z)dz= \\
\int \int x_{k}^{\ast }f_{x^{\ast },z}(x^{\ast },z)\psi (z)dx^{\ast }dz
&=&\int \int x_{k}^{\ast }\psi (x^{\ast }+u)f_{x^{\ast },u}(x^{\ast
},u)dx^{\ast }du= \\
\int \int x_{k}^{\ast }f_{x^{\ast }}(x^{\ast })f_{u}(u)\psi (x^{\ast
}+u)dx^{\ast }du &=&(h_{k}\ast f_{u},\psi ).
\end{eqnarray*}
The third expression is a double integral which always exists if $
E\left\Vert x^{\ast }\right\Vert <\infty $; this is a consequence of
boundedness of the expectations of $z$ and $u.$ The fourth is a result of
change of variables $\left( x^{\ast },z\right) $ into $\left( x^{\ast
},u\right) ,$ the fifth uses independence of $x^{\ast }$and $u,$ and the
sixth expression follows from the corresponding expression for the
convolution of generalized functions (Schwartz, 1967, p.246). The conditions
of model 3 are not sufficient to identify the distribution of $u_{x};$ this
is treated as a nuisance part in model 3.
The model in 4 with all the errors and mis-measured variable independent of
each other was investigated by Kotlyarski (1967) who worked with the joint
characteristic function. In 4 consider in addition to the equations written
for model 3 another that uses the independence between $x^{\ast }$ and $
u_{x} $ and involves $f_{u_{x}}.$
In representation 4a the convolution equations involving the density $
f_{u_{x}}$ are obtained by applying the derivations that were used here for
the model in 3.:
\begin{equation*}
\begin{array}{c}
z=x^{\ast }+u; \\
x=x^{\ast }+u_{x},
\end{array}
\end{equation*}
to the model in 4 with $x-z$ playing the role of $z,$ $u_{x}$ playing the
role of $x^{\ast },$ $-u$ playing the role of $u,$ and $x^{\ast }$ playing
the role of $u_{x}.$ The additional convolution equations arising from the
extra independence conditions provide extra equations and involve the
unknown density $f_{u_{x}}.$ This representation leads to a generalization
of Kotlyarski's identification result similar to that obtained by Evdokimov
(2011) who used the joint characteristic function. The equations in 4a make
it possible to identify $f_{u},f_{u_{x}}$ ahead of $f_{x^{\ast }};$ for
identification this will require less restrictive conditions on the support
of the characteristic function for $x^{\ast }.$
\subsubsection{Some extensions}
\textbf{A. Common factor models.}
Consider a model $\tilde{z}=AU,$ with $A$ a matrix of known constants and $
\tilde{z}$ a $m\times 1$ vector of observables, $\ U$ a vector of
unobservable variables. Usually, $A$ is a block matrix and $AU$ can be
represented via a combination of mutually independent vectors. Then without
loss of generality consider the model
\begin{equation}
\tilde{z}=\tilde{A}x^{\ast }+\tilde{u}, \label{factormod}
\end{equation}
where $\tilde{A}$ is a $m\times d$ known matrix of constants, $\tilde{z}$ is
a $m\times 1$ vector of observables, unobserved $x^{\ast }$ is $d\times 1$
and unobserved $\tilde{u}$ is $m\times 1.$ If the model $\left( \ref
{factormod}\right) $ can be transformed to model 3 considered above, then $
x^{\ast }$ will be identified whenever identification holds for model 3.
Once some components are identified identification of other factors could be
considered sequentially.
\textbf{Lemma 1. }\textit{If in }$\left( \ref{factormod}\right) $ \textit{
the vectors }$x^{\ast }$\textit{\ and }$\tilde{u}$\textit{\ are independent
and all the components of the vector }$\tilde{u}$\textit{\ are mean
independent of each other and are mean zero and the matrix }$A$ \textit{can
be partitioned after possibly some permutation of rows as }$\left(
\begin{array}{c}
A_{1} \\
A_{2}
\end{array}
\right) $\textit{\ with }$rankA_{1}=rankA_{2}=d,$\textit{\ then the model }$
\left( \ref{factormod}\right) $\textit{\ implies model 3.}
Proof. Define $z=T_{1}\tilde{z},$ where conformably to the partition of $A$
the partitioned $T_{1}=\left(
\begin{array}{c}
\tilde{T}_{1} \\
0
\end{array}
\right) ,$ with $\tilde{T}_{1}A_{1}x^{\ast }=x^{\ast }$ (such a $\tilde{T}
_{1}$ always exists by the rank condition); then $z=x^{\ast }+u,$ where $
u=T_{1}\tilde{u}$ is independent of $x^{\ast }.$ Next define $T_{2}=\left(
\begin{array}{c}
0 \\
\tilde{T}_{2}
\end{array}
\right) $ similarly with $\tilde{T}_{2}A_{2}x^{\ast }=x^{\ast }$.
Then $x=T_{2}\tilde{z}$ is such that $x=x^{\ast }+u_{x},$ where $u_{x}=T_{2}
\tilde{u}$ and does not include any components from $u.$ This implies $
Eu_{x}|(x^{\ast },u)=0.$ Model 3 holds. $\blacksquare $
Here dependence in components of $x^{\ast }$ is arbitrary. A general
structure with subvectors of $U$ independent of each other but with
components which may be only mean independent (as $\tilde{u}$ here) or
arbitrarily dependent (as in $x^{\ast })$ is examined by Ben-Moshe (2012).
Models of linear systems with full independence were examined by e.g. Li and
Vuong (1998). These models lead to systems of first-order differential
equations for the characteristic functions.
It may be that there are no independent components $x^{\ast }$ and $\tilde{u}
$ for which the conditions of Lemma 1 are satisfied. Bonhomme and Robin
(2010) proposed to consider products of the observables to increase the
number of equations in the system and analyzed conditions for
identification; Ben-Moshe (2012) provided necessary and sufficient
conditions under which this strategy leads to identification when there may
be some dependence.
\textbf{B. Error correlations with more observables.}
The extension to non-zero $E(u_{x}|z)$ in model 3 is trivial if this
expectation is a known function. A more interesting case results if the
errors $u_{x}$ and $u$ are related, e.g.
\begin{equation*}
u_{x}=\rho u+\eta ;\eta \bot z.
\end{equation*}
With an unknown parameter (or function of observables) $\rho $ if more
observations are available more convolution equations can be written to
identify all the unknown functions. Suppose that additionally a observation $
y$ is available with
\begin{eqnarray*}
y &=&x^{\ast }+u_{y}; \\
u_{y} &=&\rho u_{x}+\eta _{1};\eta _{1}\bot ,\eta ,z.
\end{eqnarray*}
Without loss of generality consider the univariate case and define $
w_{x}=E(xf(z)|z);w_{y}=E(yf(z)|z).\,\ $Then the system of convolution
equations expands to
\begin{equation}
\left\{
\begin{array}{ccc}
f_{x^{\ast }}\ast f_{u} & & =w; \\
(1-\rho )h_{x^{\ast }}\ast f_{u} & +\rho zf(z) & =w_{x}; \\
(1-\rho ^{2})h_{x^{\ast }}\ast f_{u} & +\rho ^{2}zf(z) & =w_{y}.
\end{array}
\right. \label{ar(1)}
\end{equation}
The three equations have three unknown functions, $f_{x^{\ast }},f_{u}$ and $
\rho .$ Assuming that support of $\rho $ does not include the point 1, $\rho
$ can be expressed as a solution to a linear algebraic equation derived from
the two equations in $\left( \ref{ar(1)}\right) $ that include $\rho :$
\begin{equation*}
\rho =(w_{x}-zf(z))^{-1}\left( w_{y}-w_{x}\right) .
\end{equation*}
\subsection{Regression models with classical and Berkson errors and the
convolution equations}
\subsubsection{The list of models}
The table below provides several regression models and the corresponding
convolution equations involving density weighted conditional expectations.
\begin{center}
Table 2. Regression models: 5. Regression with classical measurement error
and an additional observation; 6. Regression with Berkson error ($x,y,z$ are
observable); 7. Regression with zero mean measurement error and Berkson
instruments.
\end{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Model & $
\begin{array}{c}
\text{Distributional} \\
\text{assumptions}
\end{array}
$ & $
\begin{array}{c}
\text{Convolution } \\
\text{equations}
\end{array}
$ & $
\begin{array}{c}
\text{Known} \\
\text{ functions}
\end{array}
$ & $
\begin{array}{c}
\text{Unknown} \\
\text{ functions}
\end{array}
$ \\ \hline
\multicolumn{1}{|l|}{$\ \ $5.} & \multicolumn{1}{|l|}{$\ \
\begin{array}{c}
y=g(x^{\ast })+v \\
z=x^{\ast }+u; \\
x=x^{\ast }+u_{x} \\
x^{\ast }\bot u;E(u)=0; \\
E(u_{x}|x^{\ast },u)=0; \\
E(v|x^{\ast },u,u_{x})=0.
\end{array}
$} & \multicolumn{1}{|l|}{$
\begin{array}{c}
f_{x^{\ast }}\ast f_{u}=f_{z}; \\
\left( gf_{x^{\ast }}\right) \ast f_{u}=w, \\
h_{k}\ast f_{u}=w_{k}; \\
\text{with }h_{k}(x)\equiv x_{k}g(x)f_{x^{\ast }}(x); \\
k=1,2...d
\end{array}
$} & \multicolumn{1}{|l|}{$f_{z};$ $w;w_{k}$} & \multicolumn{1}{|l|}{$
f_{x^{\ast }}$; $f_{u}$; $g.$} \\ \hline
\multicolumn{1}{|l|}{$\ \ $6.} & \multicolumn{1}{|l|}{$
\begin{array}{c}
y=g(x)+v \\
z=x+u;E(v|z)=0; \\
z\bot u;E(u)=0.
\end{array}
$} & \multicolumn{1}{|l|}{$\ \ \ \ \
\begin{array}{c}
f_{x}=f_{-u}\ast f_{z}; \\
g\ast f_{-u}=w
\end{array}
$} & \multicolumn{1}{|l|}{$\ f_{z};f_{x},w$} & \multicolumn{1}{|l|}{$f_{u}$;
$g.$} \\ \hline
\multicolumn{1}{|l|}{$\ \ $7.} & \multicolumn{1}{|l|}{$\ \
\begin{array}{c}
y=g(x^{\ast })+v; \\
x=x^{\ast }+u_{x}; \\
z=x^{\ast }+u;z\bot u; \\
E(v|z,u,u_{x})=0; \\
E(u_{x}|z,v)=0.
\end{array}
$} & \multicolumn{1}{|l|}{$
\begin{array}{c}
g\ast f_{u}=w; \\
h_{k}\ast f_{u}=w_{k}, \\
\text{with }h_{k}(x)\equiv x_{k}g(x); \\
k=1,2...d
\end{array}
$} & \multicolumn{1}{|l|}{$w,w_{k}$} & \multicolumn{1}{|l|}{$f_{u}$; $g.$}
\\ \hline
\end{tabular}
Notes. Notation: $k=1,2...d;$ in model 5.$
w=E(yf_{z}(z)|z);w_{k}=E(x_{k}f_{z}(z)|z);$ in model 6. $w=E(y|z);$ in model
7. $w=E(y|z);w_{k}=E(x_{k}y|z).$
\textbf{Theorem 2.} \textit{Under Assumption 1 for each of the models 5-7
the corresponding convolution equations hold.}
The proof is in the derivations of the next subsection.
\subsubsection{\protect
Discussion of the regression models and
derivation of the convolution equations.}
5. The nonparametric regression model with classical measurement error and
an additional observation.
This type of model was examined by Li (2002) and Li and Hsiao (2004); the
convolution equations derived here provide a convenient representation.
Often models of this type were considered in semiparametric settings.
Butucea and Taupin (2008) (extending the earlier approach by Taupin, 2001)
consider a regression function known up to a finite dimensional parameter
with the mismeasured variable observed with independent error where the
error distribution is known. Under the latter condition the model 5 here
would reduce to the two first equations
\begin{equation*}
f_{x^{\ast }}\ast f_{u}=f_{z};\text{ }\left( gf_{x^{\ast }}\right) \ast
f_{u}=w,
\end{equation*}
where $f_{u}$ is known and two unknown functions are $g$ (here
nonparametric) and $f_{x^{\ast }}.$
The model 5 incorporates model 3 for the regressor and thus the convolution
equations from that model apply. An additional convolution equation is
derived here; it is obtained from considering the value of the density
weighted conditional expectation in the dual space of generalized functions,
$S^{\ast },$ applied to arbitrary $\psi \in S,$
\begin{equation*}
(w,\psi )=(E(f(z)y|z),\psi )=(E(f(z)g(x^{\ast })|z),\psi );
\end{equation*}
this equals
\begin{eqnarray*}
&&\int \int g(x^{\ast })f_{x^{\ast },z}(x^{\ast },z)\psi (z)dx^{\ast }dz \\
&=&\int \int g(x^{\ast })f_{x^{\ast },u}(x^{\ast },u)\psi (x^{\ast
}+u)dx^{\ast }du \\
&=&\int g(x^{\ast })f_{x^{\ast }}(x^{\ast })f_{u}(u)dx^{\ast }\psi (x^{\ast
}+u)dx^{\ast }du=((gf_{x^{\ast }})\ast f_{u},\psi ).
\end{eqnarray*}
Conditional moments for the regression function need not be integrable or
bounded functions of $z$; we require them to be in the space of generalized
functions $S^{\ast }.$
6. Regression with Berkson error.
This model may represent the situation when the regressor (observed) $x$ is
correlated with the error $v,$ but $z$ is a (vector) possibly representing
an instrument uncorrelated with the regression error.
Then as is known in addition to the Berkson error convolution equation the
equation
\begin{equation*}
w=E(y|z)=E(g(x)|z)=\int g(x)\frac{f_{x,z}(x,z)}{f_{z}(z)}dx=\int
g(z-u)f_{u}(u)dx=g\ast f_{u}
\end{equation*}
holds. This is stated in Meister (2008); however, the approach there is to
consider $g$ to be absolutely integrable so that convolution can be defined
in the $L_{1}$ space. Here by working in the space of generalized functions $
S^{\ast }$ a much wider nonparametric class of functions that includes
regression functions with polynomial growth is allowed.
7. Nonparametric regression with error in the regressor, where Berkson type
instruments are assumed available.
This model was proposed by Newey (2001), examined in the univarite case by
Schennach (2007) and Zinde-Walsh (2009), in the multivariate case in
Zinde-Walsh (2012), where the convolution equations given here in Table 2
were derived.
\subsection{\textbf{Convolution equations in models with conditional
independence conditions.}}
All the models 1-7 can be extended to include some additional variables
where conditionally on those variables, the functions in the model (e.g.
conditional distributions) are defined and all the model assumptions hold
conditionally.
Evdokimov (2011) derived the conditional version of the model 4 from a very
general nonparametric panel data model. Model 8 below describes the panel
data set-up and how it transforms to conditional model 4 and 4a and possibly
model 3 with relaxed independence condition (if the focus is on identifying
the regression function).
Model 8. Panel data model with conditional independence.
Consider a two-period panel data model with an unknown regression function $
m $ and an idiosyncratic (unobserved) $\alpha :$
\begin{eqnarray*}
Y_{i1} &=&m(X_{i1},\alpha _{i})+U_{i1}; \\
Y_{i2} &=&m(X_{i2},\alpha _{i})+U_{i2}.
\end{eqnarray*}
To be able to work with various conditional characteristic functions
corresponding assumptions ensuring existence of the conditional
distributions need to be made and in what follows we assume that all the
conditional density functions and moments exist as generalized functions in $
S^{\ast }$.
In Evdokimov (2011) independence (conditional on the corresponding period $
X^{\prime }s)$ of the regression error from $\alpha ,$ and from the $
X^{\prime }s$ and error of the other period is assumed:
\begin{equation*}
f_{t}=f_{Uit}|_{X_{it},\alpha
_{i},X_{i(-t)},U_{i(-t)}}(u_{t}|x,...)=f_{Uit}|_{X_{it}}(u_{t}|x),t=1,2
\end{equation*}
with $f_{\cdot |\cdot }$ denoting corresponding conditional densities.
Conditionally on $X_{i2}=X_{i1}=x$ the model takes the form 4
\begin{equation*}
\begin{array}{c}
z=x^{\ast }+u; \\
x=x^{\ast }+u_{x}
\end{array}
\end{equation*}
with $z$ representing $Y_{1},x$ representing $Y_{2},$ $x^{\ast }$ standing
in for $m(x,\alpha ),$ $u$ for $U_{1}$ and $u_{x}$ for $U_{2}.$ The
convolution equations derived here for 4 or 4a now apply to conditional
densities.
The convolution equations in 4a are similar to Evdokimov; they allow for
equations for $f_{u},$ $f_{u_{x}}$ that do not rely on $f_{x^{\ast }}.$ The
advantage of those lies in the possibility of identifying the conditional
error distributions without placing the usual non-zero restrictions on the
characteristic function of $x^{\ast }$ (that represents the function $m$ for
the panel model).
The panel model can be considered with relaxed independence assumptions.
Here in the two-period model we look at forms of dependence that assume zero
conditional mean of the second period error, rather than full independence
of the first period error:
\begin{eqnarray*}
f_{Ui1}|_{X_{i1},\alpha _{i},X_{i2},Ui2}(u_{t}|x,...)
&=&f_{Ui1}|_{Xi1}(u_{t}|x); \\
E(U_{i2}|X_{i1},\alpha _{i},X_{i2},U_{i1}) &=&0; \\
f_{Ui2}|_{\alpha _{i},X_{i2}=X_{i1}=x}(u_{t}|x,...)
&=&f_{Ui2}|_{Xi2}(u_{t}|x).
\end{eqnarray*}
Then the model maps into the model 3 with the functions in the convolution
equations representing conditional densities and allows to identify
distribution of $x^{\ast }$ (function $\ m$ in the model). But the
conditional distribution of the second-period error in this set-up is not
identified.
Evdokimov introduced parametric AR(1) or MA(1) dependence in the errors $U$
and to accommodate that extended the model to three periods. Here this would
lead in the AR case to the equations in $\left( \ref{ar(1)}\right) .$
Model 9. Errors in variables regression with classical measurement error
conditionally on covariates.
Consider the regression model
\begin{equation*}
y=g(x^{\ast },t)+v,
\end{equation*}
with a measurement of unobserved $x^{\ast }$ given by $\ \tilde{z}=x^{\ast }+
\tilde{u},$ with $x^{\ast }\bot \tilde{u}$ conditionally on $t$. Assume that
$E(\tilde{u}|t)=0$ and that $E(v|x^{\ast },t)=0.$ Then redefining all the
densities and conditional expectations to be conditional on $t$ we get the
same system of convolution equations as in Table 2 for model 5 with the
unknown functions now being conditional densities and the regression
function, $g.$
Conditioning requires assumptions that provide for existence of conditional
distribution functions in $S^{\ast }$.
\section{\textbf{Solutions for the models.}}
\subsection{Existence of solutions}
To state results for nonparametric models it is important first to clearly
indicate the classes of functions where the solution is sought. Assumption 1
requires that all the (generalized) functions considered are elements in the
space of generalized functions $S^{\ast }.$ This implies that in the
equations the operation of convolution applied to the two functions from $
S^{\ast }$ provides an element in the space $S^{\ast }.$ This subsection
gives high level assumptions on the nonparametric classes of the unknown
functions where the solutions can be sought: any functions from these
classes that enter into the convolution provide a result in $S^{\ast }.$
No assumptions are needed for existence of convolution and full generality
of identification conditions in models 1,2 where the model assumptions imply
that the functions represent generalized densities. For the other models
including regression models convolution is not always defined in $S^{\ast }.$
Zinde-Walsh (2012) defines the concept of convolution pairs of classes of
functions in $S^{\ast }$ where convolution can be applied.
To solve the convolution equations a Fourier transform is usually employed,
so that e.g. one transforms generalized density functions into
characteristic functions. Fourier transform is an isomorphism of the space $
S^{\ast }.$ The Fourier transform of a generalized function $a\in S^{\ast }$
, $Ft(a),$ is defined as follows. For any $\psi \in S,$ as usual $Ft(\psi
)(s)=\int \psi (x)e^{isx}dx;$ then the functional $Ft(a)$ is defined by
\begin{equation*}
(Ft(a),\psi )\equiv (a,Ft(\psi )).
\end{equation*}
The advantage of applying Fourier transform is that integral convolution
equations transform into algebraic equations when the "exchange formula"
applies:
\begin{equation}
a\ast b=c\Longleftrightarrow Ft(a)\cdot Ft(b)=Ft(c). \label{exchange}
\end{equation}
In the space of generalized functions $S^{\ast },$ the Fourier transform and
inverse Fourier transform always exist. As shown in Zinde-Walsh (2012) there
is a dichotomy between convolution pairs of subspaces in $S^{\ast }$ and the
corresponding product pairs of subspaces of their Fourier transforms.
The classical pairs of spaces (Schwartz, 1966) are the convolution pair $
\left( S^{\ast },O_{C}^{\ast }\right) $ and the corresponding product pair $
\left( S^{\ast },O_{M}\right) ,$ where $O_{C}^{\ast }$ is the subspace of $
S^{\ast }$ that contains rapidly decreasing (faster than any polynomial)
generalized functions and $\mathit{O}_{M}$ is the space of infinitely
differentiable functions with every derivative growing no faster than a
polynomial at infinity. These pairs are important in that no restriction is
placed on one of the generalized functions that could be any element of
space $S^{\ast }$; the other belongs to a space that needs to be
correspondingly restricted. A disadvantage of the classical pairs is that
the restriction is fairly severe, for example, the requirement that a
characteristic function be in $O_{M}\,\ $implies existence of all moments
for the random variable. Relaxing this restriction would require placing
constraints on the other space in the pair; Zinde-Walsh (2012) introduces
some pairs that incorporate such trade-offs.
In some models the product of a function with a component of the vector of
arguments is involved,such as $d(x)=x_{k}a(x),$ then for Fourier transforms $
Ft(d)\left( s\right) =-i\frac{\partial }{\partial s_{k}}Ft(a)(s);$ the
multiplication by a variable is transformed into ($-i)$ times the
corresponding partial derivative. Since the differentiation operators are
continuous in $S^{\ast }$ this transformation does not present a problem.
\textbf{Assumption 2.} \textit{The functions }$a\in A,b\in B,$\textit{\ are
such that }$\left( A,B\right) $\textit{\ form a convolution pair in }$
S^{\ast }$\textit{.}
Equivalently, $Ft(a),$ $Ft(b)$ are in the corresponding product pair of
spaces.
Assumption 2 is applied to model 1 for $a=f_{x^{\ast }},b=f_{u};$ to model 2
with $a=f_{z},b=f_{u};$ to model 3 with $a=f_{x^{\ast }},b=f_{u}$ and with $
a=h_{k},b=f_{u},$ for all $k=1,...,d;$ to model 4a for $a=f_{x^{\ast }},$ or
$f_{u_{x}},$ or $h_{k}$ for all $k$ and $b=f_{u};$ to model 5 with $
a=f_{x^{\ast }},$ or $gf_{x^{\ast }},$ or $h_{k}f_{x^{\ast }}$ and $b=f_{u};$
to model 6 with $a=f_{z},$ or $g$ and $b=f_{u};$ to model 7 with $a=g$ or $
h_{k}$ and $b=f_{u}.$
Assumption 2 is a high-level assumption that is a sufficient condition for a
solution to the models 1-4 and 6-7 to exist. Some additional conditions are
needed for model 5 and are provided below.
Assumption 2 is automatically satisfied for generalized density functions,
so is not needed for models 1 and 2. Denote by $\bar{D}\subset S^{\ast }$
the subset of generalized derivatives of distribution functions
(corresponding to Borel probability measures in $R^{d}$) then in models 1
and 2 $A=B=\bar{D};$ and for the characteristic functions there are
correspondingly no restrictions; denote the set of all characteristic
functions, $Ft\left( \bar{D}\right) \subset S^{\ast },$ by $\bar{C}.$
Below a (non-exhaustive) list of nonparametric classes of generalized
functions that provide sufficient conditions for existence of solutions to
the models here is given. The classes are such that they provide minimal or
often no restrictions on one of the functions and restrict the class of the
other in order that the assumptions be satisfied.
In models 3 and 4 the functions $h_{k}$ are transformed into derivatives of
continuous characteristic functions. An assumption that either the
characteristic function of $x^{\ast }$ or the characteristic function of $u$
be continuously differentiable is sufficient, without any restrictions on
the other to ensure that Assumption 2 holds. Define the subset of all
continuously differentiable characteristic functions by $\bar{C}^{(1)}.$
In model 5 equations involve a product of the regression function $g$ with $
f_{x^{\ast }}.$ Products of generalized functions in $S^{\ast }$ do not
always exist and so additional restrictions are needed in that model. If $g$
is an arbitrary element of $S^{\ast },$ then for the product to exist, $
f_{x^{\ast }}$ should be in $\mathit{O}_{M}$. On the other hand, if $
f_{x^{\ast }}$ is an arbitrary generalized density it is sufficient that $g$
and $h_{k}$ belong to the space of $d$ times continuously differentiable
functions with derivatives that are majorized by polynomial functions for $
gf_{x^{\ast }},h_{k}f_{x^{\ast }}$ to be elements of $S^{\ast }.$ Indeed,
the value of the functional $h_{k}f_{x^{\ast }}$ for an arbitrary $\psi \in
S $ is defined by
\begin{equation*}
(h_{k}f_{x^{\ast }},\psi )=\left( -1\right) ^{d}\int F_{x^{\ast
}}(x)\partial ^{(1,...,1)}(h_{k}(x)\psi (x))dx;
\end{equation*}
here $F$ is the distribution (ordinary bounded) function and this integral
exists because $\psi $ and all its derivatives go to zero at infinity faster
than any polynomial function. Denote by $\bar{S}^{B,1}$ the space of
continuously differentiable functions $g\in S^{\ast }$ such that the
functions $h_{k}(x)=x_{k}g(x)$ are also continuously differentiable with all
derivatives majorized by polynomial functions$.$ Since the products are in $
S^{\ast }$ then the Fourier transforms of the products are defined in $
S^{\ast }.$ Further restrictions requiring the Fourier transforms of the
products $gf_{x^{\ast }}$\ and $h_{k}f_{x^{\ast }}$ to be continuously
differentiable functions in $S^{\ast }$ would remove any restrictions on $
f_{u}$ for the convolution to exist. Denote the space of all continuously
differentiable functions in $S^{\ast }$ by $\bar{S}^{(1)}.$
If $g$ is an ordinary function that represents a regular element in $S^{\ast
}$ the infinite differentiability condition on $f_{x^{\ast }}$ can be
relaxed to simply requiring continuous first derivatives.
In models 6 and 7 if the generalized density function for the error, $f_{u},$
decreases faster than any polynomial (all moments need to exist for that),
so that $f_{u}\in \mathit{O}_{C}^{\ast },$ \ then $g$ could be any
generalized function in $S^{\ast };$ this will of course hold if $f_{u}$ has
bounded support. Generally, the more moments the error is assumed to have,
the fewer restrictions on the regression function $g$ are needed to satisfy
the convolution equations of the model and the exchange formula. The models
6, 7 satisfy the assumptions for any error $u$ when support of generalized
function $g$ is compact (as for the "sum of peaks"), then $g\in E^{\ast
}\subset S^{\ast },$ where $E^{\ast }$ is the space of generalized functions
with compact support. More generally the functions $g$ and all the $h_{k}$
could belong to the space $\mathit{O}_{C}^{\ast }$ of generalized functions
that decrease at infinity faster than any polynomial, and still no
restrictions need to be placed on $u.$
Denote for any generalized density function $f_{\cdot }$ the corresponding
characteristic function, $Ft(f_{\cdot }),$ by $\phi _{\cdot }.$ Denote
Fourier transform of the (generalized) regression function $g,$ $Ft(g),$ by $
\gamma .$
The following table summarizes some fairly general sufficient conditions on
the models that place restrictions on the functions themselves or on the
characteristic functions of distributions in the models that will ensure
that Assumption 2 is satisfied and a solution exists. The nature of these
assumptions is to provide restrictions on some of the functions that allow
the others to be completely unrestricted for the corresponding model.
\textbf{Table 3.} Some nonparametric classes of generalized functions for
which the convolution equations of the models are defined in $S^{\ast }$.
\begin{tabular}{|c|c|c|}
\hline
Model & Sufficient & assumptions \\ \hline
1 & no restrictions: & $\phi _{x^{\ast }}\in \bar{C};\phi _{u}\in \bar{C}$
\\ \hline
2 & no restrictions: & $\phi _{x^{\ast }}\in \bar{C};\phi _{u}\in \bar{C}$
\\ \hline
& Assumptions A & Assumptions B \\ \hline
\multicolumn{1}{|l|}{$\ \ \ \ $3} & \multicolumn{1}{|l|}{any$\ \ \phi
_{x^{\ast }}\in \bar{C};\phi _{u}\in \bar{C}^{(1)}$} & \multicolumn{1}{|l|}{
any $\phi _{u}\in \bar{C};\phi _{x^{\ast }}\in \bar{C}^{(1)}$} \\ \hline
4 & any $\phi _{u_{x}},\phi _{x^{\ast }}\in \bar{C};\phi _{u}\in \bar{C}
^{(1)}$ & any $\phi _{u},\phi _{x^{\ast }}\in \bar{C};\phi _{u_{x}}\in \bar{C
}^{(1)}$ \\ \hline
4a & any $\phi _{u_{x}},\phi _{x^{\ast }}\in \bar{C};\phi _{u}\in \bar{C}
^{(1)}$ & any $\phi _{u},\phi _{u_{x}}\in \bar{C};\phi _{x^{\ast }}\in \bar{C
}^{(1)}$ \\ \hline
\multicolumn{1}{|l|}{$\ \ \ \ $5} & \multicolumn{1}{|l|}{any $g\in S^{\ast
};f_{x^{\ast }}\in O_{M};f_{u}\in O_{C}^{\ast }$} & \multicolumn{1}{|l|}{$\ $
any $\ f_{x^{\ast }}\in \bar{D};\ g,h_{k}\in \bar{S}^{B,1};f_{u}\in
O_{C}^{\ast }$} \\ \hline
\multicolumn{1}{|l|}{$\ \ \ \ $6} & \multicolumn{1}{|l|}{any$\ g\in S^{\ast
};f_{u}\in O_{C}^{\ast }$} & \multicolumn{1}{|l|}{$\ g\in O_{C}^{\ast };$
any $f_{u}:\phi _{u}\in \bar{C}$} \\ \hline
7 & any $g\in S^{\ast };f_{u}\in O_{C}^{\ast }$ & $g\in O_{C}^{\ast };$ any $
f_{u}:\phi _{u}\in \bar{C}$ \\ \hline
\end{tabular}
The next table states the equations and systems of equations for Fourier
transforms that follow from the convolution equations.
\textbf{Table 4.} The form of the equations for the Fourier transforms:
\begin{tabular}{|c|c|c|}
\hline
Model & Eq's for Fourier transforms & Unknown functions \\ \hline
1 & $\phi _{x^{\ast }}\phi _{u}=\phi _{z};$ & $\phi _{x^{\ast }}$ \\ \hline
2 & $\phi _{x^{\ast }}=\phi _{z}\phi _{-u};$ & $\phi _{x^{\ast }}$ \\ \hline
3 & $\left\{
\begin{array}{c}
\phi _{x^{\ast }}\phi _{u}=\phi _{z}; \\
\left( \phi _{x^{\ast }}\right) _{k}^{\prime }\phi _{u}=\varepsilon
_{k},k=1,...,d.
\end{array}
\right. $ & $\phi _{x^{\ast }},\phi _{u}$ \\ \hline
4 & $\left\{
\begin{array}{c}
\phi _{x^{\ast }}\phi _{u}=\phi _{z}; \\
\left( \phi _{x^{\ast }}\right) _{k}^{\prime }\phi _{u}=\varepsilon
_{k},k=1,...,d; \\
\phi _{x^{\ast }}\phi _{u_{x}}=\phi _{x}.
\end{array}
\right. $ & $\phi _{x^{\ast }},\phi _{u},\phi _{u_{x}}$ \\ \hline
4a & $\left\{
\begin{array}{c}
\phi _{u_{x}}\phi _{u}=\phi _{z-x}; \\
\left( \phi _{u_{x}}\right) _{k}^{\prime }\phi _{u}=\varepsilon
_{k},k=1,...,d. \\
\phi _{x^{\ast }}\phi _{u_{x}}=\phi _{x}.
\end{array}
\right. $ & --"-- \\ \hline
5 & $\left\{
\begin{array}{c}
\phi _{x^{\ast }}\phi _{u}=\phi _{z}; \\
Ft\left( gf_{x^{\ast }}\right) \phi _{u}=\varepsilon \\
\left( Ft\left( gf_{x^{\ast }}\right) \right) _{k}^{\prime }\phi
_{u}=\varepsilon _{k},k=1,...,d.
\end{array}
\right. $ & $\phi _{x^{\ast }},\phi _{u},g$ \\ \hline
6 & $\left\{
\begin{array}{c}
\phi _{x}=\phi _{-u}\phi _{z}; \\
Ft(g)\phi _{-u}=\varepsilon .
\end{array}
\right. $ & $\phi _{u},g$ \\ \hline
7 & $\left\{
\begin{array}{c}
Ft(g)\phi _{u}=\varepsilon ; \\
\left( Ft\left( g\right) \right) _{k}^{\prime }\phi _{u}=\varepsilon
_{k},k=1,...,d.
\end{array}
\right. $ & $\phi _{u},g$ \\ \hline
\end{tabular}
Notes. Notation $\left( \cdot \right) _{k}^{\prime }$ denotes the k-th
partial derivative of the function. The functions $\varepsilon $ are Fourier
transforms of the corresponding $w,$ and $\varepsilon _{k}=-iFt(w_{k})$
defined for the models in Tables 1 and 2.
Assumption 2 (that is fulfilled e.g. by generalized functions classes of
Table 3) ensures existence of solutions to the convolution equations for
models 1-7; this does not exclude multiple solutions and the next section
provides a discussion of solutions for equations in Table 4.
\subsection{Classes of solutions; support and multiplicity of solutions}
Typically, support assumptions are required to restrict multiplicity of
solutions; here we examine the dependence of solutions on the support of the
functions. The results here also give conditions under which some zeros,
e.g. in the characteristic functions, are allowed. Thus in common with e.g.
Carrasco and Florens (2010), Evdokimov and White (2011), distributions such
as the uniform or triangular for which the characteristic function has
isolated zeros are not excluded. The difference here is the extension of the
consideration of the solutions to $S^{\ast }$ and to models such as the
regression model where this approach to relaxing support assumptions was not
previously considered.
\ Recall that for a continuous function $\psi (x)$ on $R^{d}$ support is
defined as the set $W=$supp($\psi ),$ such that
\begin{equation*}
\psi (x)=\left\{
\begin{array}{cc}
a\neq 0 & \text{for }x\in W \\
0 & \text{for }x\in R^{d}\backslash W.
\end{array}
\right.
\end{equation*}
Support of a continuous function is an open set.
Generalized functions are functionals on the space $S$ and support of a
generalized function $b\in S^{\ast }$ is defined as follows (Schwartz, 1967,
p. 28). Denote by $\left( b,\psi \right) $ the value of the functional $b$
for $\psi \in S.$ Define a null set for $b\in S^{\ast }$ as the union of
supports of all functions in $S$ for which the value of the functional is
zero$:$ $\Omega =\{\cup $supp$\left( \psi \right) ,$ $\psi \in S,$ such that
$\left( b,\psi \right) =0\}.$ Then supp$\left( b\right) =R^{d}\backslash
\Omega .$ Note that a generalized function has support in a closed set, for
example, support of the $\delta -function$ is just one point 0.
Note that for model 2 Table 4 gives the solution for $\phi _{x^{\ast }}$
directly and the inverse Fourier transform can provide the (generalized)
density function, $f_{x^{\ast }}.$
In Zinde-Walsh (2012) identification conditions in $S^{\ast }$ were given
for models 1 and 7 under assumptions that include the ones in Table 3 but
could also be more flexible.
The equations in Table 3 for models 1,3, 4, 4a, 5, 6 and 7 are of two types,
similar to those solved in Zinde-Walsh (2012). One is a convolution with one
unknown function; the other is a system of equations with two unknown
functions, each leading to the corresponding equations for their Fourier
transforms.
\subsubsection{Solutions to the equation $\protect\alpha \protect\beta =
\protect\gamma .$}
Consider the equation
\begin{equation}
\alpha \beta =\gamma , \label{product}
\end{equation}
with one unknown function $\alpha ;$ $\beta $ is a given continuous
function. By assumption 2 the non-parametric class for $\alpha $ is such
that the equation holds in $S^{\ast }$ on $R^{d}$; it is also possible to
consider a nonparametric class for $\alpha $ with restricted support, $\bar{W
}.$ Of course without any restrictions $\bar{W}=R^{d}.$ Recall the
differentiation operator, $\partial ^{m},$ for $m=(m_{1},...m_{d}\dot{)}$
and denote by $supp(\beta ,\partial )$ the set $\cup _{\Sigma
m_{i}=0}^{\infty }supp(\partial ^{m}\beta );$ where $supp(\partial ^{m}\beta
)$ is an open set where a continuous derivative $\partial ^{m}\beta $
exists. Any point where $\beta $ is zero belongs to this set if some
finite-order partial continuous derivative of $\beta $ is not zero at that
point (and in some open neighborhood); for $\beta $ itself $supp(\beta
)\equiv supp(\beta ,0).$
Define the functions
\begin{equation}
\alpha _{1}=\beta ^{-1}\gamma I\left( supp(\beta ,\partial )\right) ;\alpha
_{2}(x)=\left\{
\begin{array}{cc}
1 & \text{for }x\in supp(\beta ,\partial ); \\
\tilde{\alpha} & \text{for }x\in \bar{W}\backslash (supp(\beta ,\partial ))
\end{array}
\right. \label{division}
\end{equation}
with any $\tilde{\alpha}$ such that $\alpha _{1}\alpha _{2}\in Ft\left(
A\right) .$
Consider the case when $\alpha ,\beta $ and thus $\gamma $ are continuous.
For any point $x_{0}$ if $\beta (x_{0})\neq 0,$ there is a neighborhood $
N(x_{0})$ where $\beta \neq 0,$ and division by $\beta $ is possible. If $
\beta (x_{0})$ has a zero, it could only be of finite order and in some
neighborhood, $N(x_{0})\in supp(\partial ^{m}\beta )$ a representation
\begin{equation}
\beta =\eta (x)\Pi _{i=1}^{d}\left( x_{i}-x_{0i}\right) ^{m_{i}}
\label{finitezero}
\end{equation}
holds for some continuous function $\eta $ in $S^{\ast },$ such that $\eta
>c_{\eta }>0$ on $supp(\eta ).$Then $\eta ^{-1}\gamma $ in $N(x_{0})$ is a
non-zero continuous function; division of such a function by $\Pi
_{i=1}^{d}\left( x_{i}-x_{0i}\right) ^{m_{i}}$ in $S^{\ast }$ is defined
(Schwartz, 1967, pp. 125-126), thus division by $\beta $ is defined in this
neighborhood $N(x_{0})$. For the set $supp(\beta ,\partial )$ consider a
covering of every point by such neighborhoods, the possibility of division
in each neighborhood leads to possibility of division globally on the whole $
supp(\beta ,\partial ).$ Then $a_{1}$ as defined in $\left( \ref{division}
\right) $ exists in $S^{\ast }.$
In the case where $\gamma $ is an arbitrary generalized function, if $\beta $
is infinitely differentiable then then by (Schwartz, 1967, pp.126-127)
division by $\beta $ is defined on $supp(\beta ,\partial )$ and the solution
is given by $\left( \ref{division}\right) .$
For the cases where $\gamma $ is not continuous and $\beta $ is not
infinitely differentiable the solution is provided by
\begin{equation*}
\alpha _{1}=\beta ^{-1}\gamma I\left( supp(\beta ,0)\right) ;\alpha
_{2}(x)=\left\{
\begin{array}{cc}
1 & \text{for }x\in supp(\beta ,0); \\
\tilde{\alpha} & \text{for }x\in \bar{W}\backslash (supp(\beta ,0))
\end{array}
\right.
\end{equation*}
with any $\tilde{\alpha}$ such that $\alpha _{1}\alpha _{2}\in Ft\left(
A\right) .$
Theorem 2 in Zinde-Walsh (2012) implies that the solution to $\left( \ref
{product}\right) $ is $a=Ft^{-1}(\alpha _{1}\alpha _{2});$ the sufficient
condition for the solution to be unique is $supp(\beta ,0)\supset \bar{W};$
if additionally either $\gamma $ is a continuous function or $\beta $ is an
infinitely continuously differentiable function it is sufficient for
uniqueness that $supp(\beta ,\partial )\supset \bar{W}.$
This provides solutions for models 1 and 6 where only equations of this type
appear.
\subsubsection{Solutions to the system of equations}
For models 3,4,5 and 7 a system of equations of the form
\begin{eqnarray}
&&
\begin{array}{cc}
\alpha \beta & =\gamma ; \\
\alpha \beta _{k}^{\prime } & =\gamma _{k},
\end{array}
\label{twoeq} \\
k &=&1,...,d. \notag
\end{eqnarray}
(with $\beta $ continuously differentiable) arises. Theorem 3 in Zinde-Walsh
(2012) provides the solution and uniqueness conditions for this system of
equations. It is first established that a set of continuous functions $
\varkappa _{k},k=1,...,d,$ that solves the equation
\begin{equation}
\varkappa _{k}\gamma -\gamma _{k}=0 \label{difeq}
\end{equation}
in the space $S^{\ast }$ exists and is unique on $W=supp(\gamma )$ as long
as $supp(\beta )\supset W.$ Then $\beta _{k}^{\prime }\beta ^{-1}=\varkappa
_{k}$ and substitution into $\left( \ref{difeq}\right) $ leads to a system
of first-order differential equations in $\beta .$
Case 1. Continuous functions; $W$ is an open set.
For the models 3 and 4 the system $\left( \ref{twoeq}\right) $ involves
continuous characteristic functions thus there $W$ is an open set. In some
cases $W$ can be an open set under conditions of models 5 and 7, e.g. if the
regression function is integrable in model 7.
For this case represent the open set $W$ as a union of (maximal) connected
components $\cup _{v}W_{v}.$
Then by the same arguments as in the proof of Theorem 3 in Zinde-Walsh
(2012)\ the solution can be given uniquely on $W$ as long as at some point $
\zeta _{0v}\in (W_{v}\cap W)$ the value $\beta \left( \zeta _{0\nu }\right) $
is known for each of the connected components . Consider then $\beta
_{1}(\zeta )=\Sigma _{\nu }[\beta \left( \zeta _{0\nu }\right) \exp
\int_{\zeta _{0}}^{\zeta }\tsum\limits_{k=1}^{d}\varkappa _{k}(\xi )d\xi
]I(W_{\nu }),$ where integration is along any arc within the component that
connects $\zeta $ to $\zeta _{0\nu }.$ Then $\alpha _{1}=\beta
_{1}^{-1}\gamma ,$ and $\alpha _{2},\beta _{2}$ are defined as above by
being $1$ on $\cup _{v}W_{v}$ and arbitrary outside of this set.
When $\beta (0)=1$ as is the case for the characteristic function, the
function is uniquely determined on the connected component that includes 0.
Evdokimov and White (2012) provide a construction that permits in the
univariate case to extend the solution $\beta \left( \zeta _{0\nu }\right)
[\exp \int_{\zeta _{0}}^{\zeta }\tsum\limits_{k=1}^{d}\varkappa _{k}(\xi
)d\xi ]I(W_{\nu })$ from a connected component of support where $\beta
\left( \zeta _{0\nu }\right) $ is known (e.g. at 0 for a characteristic
function) to a contiguous connected component when on the border between the
two where $\beta =0,$ at least some finite order derivative of $\beta $ is
not zero. In the multivariate case this approach can be extended to the same
construction along a one-dimensional arc from one connected component to the
other. Thus identification is possible on a connected component of $
supp(\beta ,\partial ).$
Case 2. $W$ is a closed set.
Generally for models 5 and 7, $W$ is the support of a generalized function
and is a closed set. It may intersect with several connected components of
support of $\beta .$ Denote by $W_{v\text{ }}$ here the intersection of a
connected component of support of $\beta $ and $W.$ Then similarly $\beta
_{1}(\zeta )=\tsum\limits_{\nu }[\beta \left( \zeta _{0\nu }\right) \exp
\int_{\zeta _{0}}^{\zeta }\tsum\limits_{k=1}^{d}\varkappa _{k}(\xi )d\xi
]I(W_{\nu }),$ where integration is along any arc within the component that
connects $\zeta $ to $\zeta _{0\nu }.$ Then $\alpha _{1}=\beta
_{1}^{-1}\varepsilon ,$ and $\alpha _{2},\beta _{2}$ are defined as above by
being $1$ on $\cup _{v}W_{v}$ and arbitrary outside of this set. The issue
of the value of $\beta $ at some point within each connected component
arises. In the case of $\beta $ being a characteristic function if there is
only one connected component, $W$ and $0\in W$ the solution is unique, since
then $\beta (0)=1.$
Note that for model 5 the solution to equations of the type $\left( \ref
{twoeq}\right) $ would only provide $Ft(gf_{x^{\ast }})$ and $\phi _{u};$
then from the first equation for this model in Table 4 $\phi _{x^{\ast }}$
can be obtained; it is unique if supp$\phi _{x^{\ast }}=$supp$\phi _{z}$. To
solve for $g$ find $g=Ft^{-1}\left( Ft\left( gf_{x^{\ast }}\right) \right)
\cdot \left( f_{x^{\ast }}\right) ^{-1}.$
\section{Identification, partial identification and well-posedness}
\subsection{Identified solutions for the models 1-7}
As follows from the discussion of the solutions uniqueness in models
1,2,3,4,4a,5,6 holds (in a few cases up to a value of a function at a point)
if all the Fourier transforms are supported over the whole $R^{d};$ in many
cases it is sufficient that $supp(\beta ,\partial )=R^{d}.$
The classes of functions could be defined with Fourier transforms supported
on some known subset $\bar{W}$ of $R^{d},$ rather than on the whole space;
if all the functions considered have $\bar{W}$ as their support, and the
support consists of one connected component that includes 0 as an interior
point then identification for the solutions holds. For the next table assume
that $\bar{W}$ is a single connected component with $0$ as an interior
point; again $\bar{W}$ could coincide with $supp(\beta ,\partial )$. For
model 5 under Assumption B assume additionally that the value at zero: $
Ft(gf_{x^{\ast }})(0)$ is known; similarly for model 7 under assumption B
additionally assume that $Ft(g)(0)$ is known.
Table 5. The solutions for identified models on $\bar{W}.$
\begin{tabular}{|c|c|}
\hline
Model & $
\begin{array}{c}
\text{Solution to } \\
\text{equations}
\end{array}
$ \\ \hline
\multicolumn{1}{|l|}{$\ \ \ $1.} & \multicolumn{1}{|l|}{$\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f_{x^{\ast }}=Ft^{-1}\left( \phi
_{u}^{-1}\phi _{z}\right) .$} \\ \hline
2. & $f_{x^{\ast }}=Ft^{-1}\left( \phi _{-u}\phi _{z}\right) .$ \\ \hline
\multicolumn{1}{|l|}{$\ $\ 3.} & \multicolumn{1}{|l|}{$
\begin{array}{c}
\text{Under Assumption A} \\
f_{x^{\ast }}=Ft^{-1}(\exp \int_{\zeta _{0}}^{\zeta
}\tsum\limits_{k=1}^{d}\varkappa _{k}(\xi )d\xi ), \\
\text{where }\varkappa _{k}\text{ solves }\varkappa _{k}\phi _{z}-[\left(
\phi _{z}\right) _{k}^{\prime }-\varepsilon _{k}]=0; \\
f_{u}=Ft^{-1}(\phi _{x^{\ast }}^{-1}\varepsilon ). \\
\text{Under Assumption B} \\
f_{u}=Ft^{-1}(\exp \int_{\zeta _{0}}^{\zeta }\tsum\limits_{k=1}^{d}\varkappa
_{k}(\xi )d\xi ); \\
\varkappa _{k}\text{ solves }\varkappa _{k}\phi _{z}-\varepsilon _{k}=0; \\
f_{x^{\ast }}=Ft^{-1}(\phi _{u}^{-1}\varepsilon ).
\end{array}
$} \\ \hline
4 & $
\begin{array}{c}
f_{x^{\ast }},f_{u}\text{ obtained similarly to those in 3.;} \\
\phi _{u_{x}}=\phi _{x^{\ast }}^{-1}\phi _{x}.
\end{array}
$ \\ \hline
4a. & $
\begin{array}{c}
f_{u_{x}},f_{u}\text{ obtained similarly to }\phi _{x^{\ast }},\phi _{u}
\text{ in 3.;} \\
\phi _{x^{\ast }}=\phi _{u_{x}}^{-1}\phi _{x}.
\end{array}
$ \\ \hline
5. & $
\begin{array}{c}
\text{Three steps:} \\
\text{1. (a) Get }Ft(gf_{x^{\ast }}),\phi _{u}\text{ similarly to }\phi
_{x^{\ast }},\phi _{u}\text{ in model 3} \\
\text{(under Assumption A use }Ft(gf_{x^{\ast }})(0))\text{;} \\
\text{2. Obtain }\phi _{x^{\ast }}=\phi _{u}^{-1}\phi _{z}; \\
\text{3. Get }g=\left[ Ft^{-1}\left( \phi _{x^{\ast }}\right) \right]
^{-1}Ft^{-1}(Ft(gf_{x^{\ast }})).
\end{array}
$ \\ \hline
6. & $\phi _{-u}=\phi _{z}^{-1}\phi _{x}$ and $g=Ft^{-1}(\phi _{x}^{-1}\phi
_{z}\varepsilon ).$ \\ \hline
7. & $
\begin{array}{c}
\phi _{x^{\ast }},Ft(g)\text{obtained similarly to }\phi _{x^{\ast }},\phi
_{u}\text{in }3 \\
\text{(under Assumption A use }Ft(g)(0)).
\end{array}
$ \\ \hline
\end{tabular}
\subsection{Implications of partial identification.}
Consider the case of Model 1. Essentially lack of identification, say in the
case when the error distribution has characteristic function supported on a
convex domain $W_{u}$ around zero results in the solution for $\phi
_{x^{\ast }}=\phi _{1}\phi _{2},$ with $\phi _{1}$ non-zero and unique on $
W_{u},$ and thus captures the lower-frequency components of $x^{\ast },$ and
with $\phi _{2}$ is a characteristic function of a distribution with
arbitrary high frequency components. Transforming back to densities provides
a corresponding model with independent components
\begin{equation*}
z=x_{1}^{\ast }+x_{2}^{\ast }+u,
\end{equation*}
where $x_{1}^{\ast }$ uniquely extracts the lower frequency part of observed
$z.$ The more important the contribution of $x_{1}^{\ast }$ to $x^{\ast }$
the less important is lack of identification.
If the feature of interest as discussed e.g. by Matzkin (2007) involves only
low frequency components of $x^{\ast },$ it may still be fully identified
even when the distribution for $x^{\ast }$ itself is not. An example of that
is a deconvolution applied to an image of a car captured by a traffic
camera; although even after deconvolution the image may still appear blurry
the licence plate number may be clearly visible. In nonparametric regression
the polynomial growth of the regression or the expectation of the response
function may be identifiable even if the regression function is not fully
identified.
Features that are identified include any functional, $\Phi ,$ linear or
non-linear on a class of functions of interest, such that in the frequency
domain $\Phi $ is supported on $W_{u}.$
\subsection{Well-posedness in $S^{\ast }$}
Conditions for well-posedness in $S^{\ast }$ for solutions of the equations
entering in models 1-7 were established in Zinde-Walsh (2012).
Well-posedness is needed to ensure that if a sequence of functions converges
(in the topology of $S^{\ast })$ to the known functions of the equations
characterizing the models 1-7 in tables 1 and 2, then the corresponding
sequence of solutions will converge to the solution for the limit functions.
A feature of well-posedness in $S^{\ast }$ is that the solutions are
considered in a class of functions that is a bounded set in $S^{\ast }.$
The properties that differentiation is a continuous operation, and that the
Fourier transform is an isomorphism of the topological space $S^{\ast },$
make conditions for convergence in this space much weaker than those in
functions spaces, say, $L_{1},$ $L_{2}.$ Thus for density that is given by
the generalized derivative of the distribution function well-posedness holds
in spaces of generalized functions by the continuity of the differentiation
operator$.$
For the problems here however, well-posedness does not always obtain. The
main sufficient condition is that the inverse of the characteristic function
of the measurement error satisfy the condition $\left( \ref{condition}
\right) $ with $b=\phi _{u}^{-1}$ on the corresponding support. This holds
if either the support is bounded or if the distribution is not super-smooth.
If $\phi _{u}$ has some zeros but satisfies the identification conditions so
that it has local representation $\left( \ref{finitezero}\right) $ where $
\left( \ref{condition}\right) $ is satisfied for $b=\eta ^{-1}$
well-posedness will hold.
Example in Zinde-Walsh (2012) demonstrates that well-posedness of
deconvolution will not hold even in the weak topology of $S^{\ast }$ for
super-smooth (e.g. Gaussian) distributions on unbounded support. On the
other hand, well-posedness of deconvolution in $S^{\ast }$ obtains for
ordinary smooth distributions and thus under less restrictive conditions
than in function spaces, such as $L_{1}$ or $L_{2}$ usually considered.
In the models 3-7 with several unknown functions, more conditions are
required to ensure that all the operations by which the solutions are
obtained are continuous in the topology of $S^{\ast }.$ It may not be
sufficient to assume $\left( \ref{condition}\right) $ for the inverses of
unknown functions where the solution requires division; for continuity of
the solution the condition may need to apply uniformly.
Define a class of ordinary functions on $R^{d},$ $\Phi (m,V)$ (with $m$ a
vector of integers, $V$ a positive constant) where $b\in \Phi (m,V)$ if
\begin{equation}
\int \Pi \left( (1+t_{i}^{2})^{-1}\right) ^{m_{i}}\left\vert b(t)\right\vert
dt<V<\infty .\text{ } \label{condb}
\end{equation}
Then in Zinde-Walsh (2012) well-posedness is proved for model 7 as long as
in addition to Assumption A or B, for some $\Phi (m,V)$ both $\phi _{u}$ and
$\phi _{u}^{-1}$ belong to the class $\Phi (m,V)$. This condition is
fulfilled by non-supersmooth $\phi _{u};$ this could be an ordinary smooth
distribution or a mixture with some mass point.
A convenient way of imposing well-posedness is to restrict the support of
functions considered to a bounded $\bar{W}.$ If the features of interest are
associated with low-frequency components only, then if the functions are
restricted to a bounded space the low-frequency part can be identified and
is well-posed.
\section{Implications for estimation}
\subsection{Plug-in non-parametric estimation}
Solutions in Table 5 for the equations that express the unknown functions
via known functions of observables give scope for plug-in estimation. As
seen e.g. in the example of Model 4, 4 and 4a are different expressions that
will provide different plug-in estimators for the same functions.
The functions of the observables here are characteristic functions and
Fourier transforms of density-weighted conditional expectations and in some
cases their derivatives, that can be estimated by non-parametric methods.
There are some direct estimators, e.g. for characteristic functions. In the
space $S^{\ast }$ the Fourier transform and inverse Fourier transform are
continuous operations thus using standard estimators of density weighted
expectations and applying the Fourier transform would provide consistency in
$S^{\ast }$; the details are provided in Zinde-Walsh (2012). Then the
solutions can be expressed via those estimators by the operations from Table
5 and, as long as the problem is well-posed, the estimators will be
consistent and the convergence will obtain at the appropriate rate. As in An
and Hu (2012), the convergence rate may be even faster for well-posed
problems in $S^{\ast }$ than the usual nonparametric rate in (ordinary)
function spaces. For example, as demonstrated in Zinde-Walsh (2008) kernel
estimators of density that may diverge if the distribution function is not
absolutely continuous, are always (under the usual assumptions on
kernel/bandwidth) consistent in the weak topology of the space of
generalized functions, where the density problem is well-posed. Here,
well-posedness holds for deconvolution as long as the error density is not
super-smooth.
\subsection{Regularization in plug-in estimation}
When well-posedness cannot be ensured, plug-in estimation will not provide
consistent results and some regularization is required; usually spectral
cut-off is employed for the problems considered here. In the context of
these non-parametric models regularization requires extra information: the
knowledge of the rate of decay of the Fourier transform of some of the
functions.
For model 1 this is not a problem since $\phi _{u}$ is assumed known; the
regularization uses the information about the decay of this characteristic
function to construct a sequence of compactly supported solutions with
support increasing at a corresponding rate. In $S^{\ast }$ no regularization
is required for plug-in estimation unless the error distribution is
super-smooth. Exponential growth in $\phi _{u}^{-1}$ provides a logarithmic
rate of convergence in function classes for the estimator (Fan, 1991). Below
we examine spectral cut-off regularization for the deconvolution in $S^{\ast
}$ when the error density is super-smooth.
With super-smooth error in $S^{\ast }$ define a class of generalized
functions $\Phi (\Lambda ,m,V)$ for some non-negative-valued function $
\Lambda $; a generalized function $b\in \Phi (\Lambda ,m,V)$ if there exists
a function $\bar{b}(\zeta )\in \Phi (m,V)$ such that also $\bar{b}(\zeta
)^{-1}\in \Phi (m,V)$ and $b=\bar{b}(\zeta )\exp \left( -\Lambda (\zeta
)\right) .$ Note that a linear combination of functions in $\Phi (\Lambda
,m,V)$ belongs to the same class. Define convergence: a sequence of $
b_{n}\in \Phi (\Lambda ,m,V)$ converges to zero if the corresponding
sequence $\bar{b}_{n}$ converges to zero in $S^{\ast }.$
Convergence in probability for a sequence of random functions, $\varepsilon
_{n},$ in $S^{\ast }$ is defined as follows: $(\varepsilon _{n}-\varepsilon
)\rightarrow _{p}0$ in $S^{\ast }$ if for any set $\psi _{1},...,\psi
_{v}\in S$ the random vector of the values of the functionals converges: $
\left( (\varepsilon _{n}-\varepsilon ,\psi _{1}),...,(\varepsilon
_{n}-\varepsilon ,\psi _{v})\right) \rightarrow _{p}0.$
\textbf{Lemma 2.} \textit{If in model 1 }$\phi _{u}=b\in \Phi (\Lambda ,m,V),
$\textit{\ where }$\Lambda $\textit{\ is a polynomial function of order no
more than }$k,$\textit{\ and }$\varepsilon _{n}$\textit{\ is a sequence of
estimators of }$\varepsilon $\textit{\ that are consistent in }$S^{\ast
}:r_{n}(\varepsilon _{n}-\varepsilon )\rightarrow _{p}0$\textit{\ in }$
S^{\ast }$\textit{\ at some rate }$r_{n}\rightarrow \infty ,$\textit{\ then
for any sequence of constants }$\bar{B}_{n}:$\textit{\ }$0<\bar{B}
_{n}<\left( \ln r_{n}\right) ^{\frac{1}{k}}$\textit{\ and the corresponding
set }$B_{n}=\left\{ \zeta :\left\Vert \zeta \right\Vert <\bar{B}_{n}\right\}
$\textit{\ the sequence of regularized estimators }$\phi
_{u}^{-1}(\varepsilon _{n}-\varepsilon )I(B_{n})$\textit{\ converges to zero
in probability in }$S^{\ast }.$\textit{\ }
Proof. For $n$ the value of the random functional
\begin{equation*}
(\phi _{u}^{-1}(\varepsilon _{n}-\varepsilon )I(B_{n}),\psi )=\int \bar{b}
^{-1}(\zeta )r_{n}(\varepsilon _{n}-\varepsilon )r_{n}^{-1}I(B_{n})\exp
\left( \Lambda (\zeta )\right) \psi (\zeta )d\zeta .
\end{equation*}
Multiplication by $\bar{b}^{-1}\in \Phi (m,V),$ that corresponds to $\phi
_{u}=b$ does not affect convergence thus $\bar{b}^{-1}(\zeta
)r_{n}(\varepsilon _{n}-\varepsilon )$ converges to zero in probability in $
S^{\ast }.$ To show that $(\phi _{u}^{-1}(\varepsilon _{n}-\varepsilon
)I(B_{n}),\psi )$ converges to zero it is sufficient to show that the
function $r_{n}^{-1}I(B_{n})\exp \left( \Lambda (\zeta )\right) \psi (\zeta
) $ is bounded$.$ It is then sufficient to find $B_{n}$ such that $
r_{n}^{-1}I(B_{n})\exp \left( \Lambda (\zeta )\right) $ is bounded (by
possibly a polynomial), thus it is sufficient that $\underset{B_{n}}{\sup }
\left\vert \exp \left( \Lambda (\zeta )\right) r_{n}^{-1}\right\vert $ be
bounded. This will hold if $\exp \left( \bar{B}_{n}^{k}\right) <r_{n},$ $
\bar{B}_{n}^{k}<\ln r_{n}.\blacksquare $
Of course an even slower growth for spectral cut-off would result from $
\Lambda $ that grows faster than a polynomial. The consequence of the slow
growth of the support is usually a correspondingly slow rate of convergence
for $\phi _{u}^{-1}\varepsilon _{n}I(B_{n}).$ Additional conditions (as in
function spaces) are needed for the regularized estimators to converge to
the true $\gamma $.
It may be advantageous to focus on lower frequency components and ignore the
contribution from high frequencies when the features of interest depend on
the contribution at low frequency.
\section{Concluding remarks}
Working in spaces of generalized functions extends the results on
nonparametric identification and well-posedness for a wide class of models.
Here identification in deconvolution is extended to generalized densities in
the class of all distributions from the usually considered classes of
integrable density functions. In regression with Berkson error nonparametric
identification in $S^{\ast }$\ holds for functions of polynomial growth,
extending the usual results obtained in $L_{1};$ a similar extension applies
to regression with measurement error and Berkson type measurement; this
allows to consider binary choice and polynomial regression models. Also,
identification in models with sum-of-peaks regression function that cannot
be represented in function spaces is included. Well-posedness results in $
S^{\ast }$ also extend the results in the literature provided in function
spaces; well-posedness of deconvolution holds as long as the characteristic
function of the error distribution does not go to zero at infinity too fast
(as e.g. super-smooth) and a similar condition provides well-posedness in
the other models considered here.
Further investigation of the properties of estimators in spaces of
generalized functions requires deriving the generalized limit process for
the function being estimated and investigating when it can be described as a
generalized Gaussian process. A generalized Gaussian limit process holds for
kernel estimator of the generalized density function (Zinde-Walsh, 2008).
Determining the properties of inference based on the limit process for
generalized random functions requires both further theoretical development
and simulations evidence.
\end{document} |
\begin{document}
\def{\mathbb{R}}{{\mathbb{R}}}
\newcommand{ \Om }{ \Omega}
\newcommand{ \pOm}{\partial \Omega}
\newcommand{ \RO}{\mathbb R^n\setminus \Omega}
\def{\mathbb{R}}{{\mathbb{R}}}
\title{Regularity of extremal solutions of nonlocal elliptic systems}
\author{Mostafa Fazly}
\address{Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA}
\email{mostafa.fazly@utsa.edu}
\maketitle
\begin{abstract} We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem
\begin{eqnarray*}
\left\{ \begin{array}{lcl}
\mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\
u,v &=&0 \qquad \qquad \text{on} \ \ \RO ,
\end{array}\right.
\end{eqnarray*}
with an integro-differential operator, including the fractional Laplacian, of the form
\begin{equation*}\label{}
\mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz ,
\end{equation*}
when $J$ is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of $J(y)=\frac{a(y/|y|)}{|y|^{n+2s}}$ where $s\in (0,1)$ and $a$ is any nonnegative even measurable function in $L^1(\mathbb {S}^{n-1})$ that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions $n < 10s$ and
$ n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$ for the Gelfand and Lane-Emden systems when $p>1$ (with positive and negative exponents), respectively. When $s\to 1$, these dimensions are optimal. However, for the case of $s\in(0,1)$ getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions $n<4s$. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.
\end{abstract}
\noindent
{\it \footnotesize 2010 Mathematics Subject Classification}. {\scriptsize 35R09, 35R11, 35B45, 35B65, 35J50}\\
{\it \footnotesize Key words: Nonlocal elliptic systems, regularity of extremal solutions, stable solutions, nonlinear eigenvalue problems}. {\scriptsize }
\section{Introduction and main results}\label{secin}
Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain. Consider the nonlinear nonlocal eigenvalue problem
\begin{eqnarray*}
(P)_{\lambda,\gamma} \qquad
\left\{ \begin{array}{lcl}
\mathcal L u &=& \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\
u,v &=&0 \qquad \qquad \text{on} \ \ \RO ,
\end{array}\right.
\end{eqnarray*}
where $ \lambda, \gamma$ are positive parameters, $F,G$ are smooth functions and
the operator $\mathcal L$ is an integral operator of convolution type
\begin{equation}\label{Lui}
\mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz .
\end{equation}
Here, $J$ is a nonnegative measurable even jump kernel such that
\begin{equation}
\int_{\mathbb R^n} \min\{|y|^2,1\} J(y) dy<\infty.
\end{equation}
The above nonlocal operator with a measurable kernel
\begin{equation}\label{Jump}
J(x,z)= \frac{c(x,z)}{|x-z|^{n+2s}},
\end{equation}
when the function $c(x,z)$ is bounded between two positive constants, $0<c_1 \le c_2$, is studied extensively in the literature from both theory of partial differential equations and theory of probability points of view, see the book of Bass \cite{bass} and references therein. Integro-differential equations and systems, of the above form, arise naturally
in the study of stochastic processes with jumps, and more precisely in L\'{e}vy processes. A L\'{e}vy process is a stochastic process with independent and stationary increments. A special class of such processes is the so called stable
processes. These are the processes that satisfy self-similarity properties, and they
are also the ones appearing in the Generalized Central Limit Theorem. We refer interested readers to the book of Bertoin \cite{ber} for more information. The infinitesimal generator of any isotropically symmetric stable L\'{e}vy process in $\mathbb R^n$ is
\begin{equation}\label{}
\mathcal Lu(x)=\int_{\mathbb S^{n-1}} \int_{-\infty}^\infty [u(x+r\theta)+u(x-r\theta) -2 u(x)]\frac{dr}{r^{1+2s}} d\mu(\theta),
\end{equation}
where $\mu$ is any nonnegative and finite measure on the unit sphere $\mathbb S^{n-1}$ called the spectral measure and $s\in (0, 1)$. When the spectral measure is absolutely continuous, $d\mu(\theta) = a(\theta)d\theta$, the above operators can be rewritten in the form of
\begin{equation}\label{La}
\mathcal L(u (x))= \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [ u(x+y)+u(x-y)-2u(x)]
\frac{a(y/|y|)}{|y|^{n+2s}} dy ,
\end{equation}
where $s\in (0,1)$ and $a$ is any nonnegative even function in $L^1(\mathbb S^{n-1})$. Note that the fractional Laplacian operator $\mathcal L={(-\Delta)}^{s}$ with $0<s<1$ that is
\begin{equation}\label{Luj}
\mathcal L u(x) = \lim_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] \frac{c_{n,s}}{|x-z|^{n+2s}} dz,
\end{equation}
for a positive constant $c_{n,s}$ is the simplest stable L\'{e}vy process for $d\mu(\theta) = c_{n,s} d\theta$. Note that the above operator can be written in the form of (\ref{Lui}) due to the fact that $a$ is even. The regularity of solutions for equation $\mathcal Lu=f$ has been studied thoroughly in the literature by many experts and in this regard we refer interested to \cite{bass, cs, fro1, rs1, si} and references therein. The most common assumption on the jump kernel in this context is $0<c_1\le a(\theta) \le c_2$ in $\mathbb S^{n-1}$ and occasionally $a(\theta)\ge c_1>0$ in a subset of $\mathbb S^{n-1}$ with positive measure. In this article, we consider the ellipticity assumption on the operator $\mathcal L$ of the form
\begin{equation}\label{c1c2}
0<c_1 \le \inf_{\nu\in \mathbb S^{n-1}} \int_{\mathbb S^{n-1}} |\nu\cdot\theta|^{2s} a(\theta)d\theta \ \ \text{and} \ \ 0 \le a(\theta) < c_2 \ \ \text{for all}\ \ \theta\in \mathbb S^{n-1},
\end{equation}
where $c_1$ and $c_2$ are constants. Note that regularity results (interior and boundary) under such an assumption on general operator $\mathcal L$ is studied in the literature, and in this regard we refer interested readers to \cite{rs2, rs1} and references therein. For particular nonlinearities of $F$ and $G$, we consider the following Gelfand system
\begin{eqnarray*}
(G)_{\lambda,\gamma}\qquad \left\{ \begin{array}{lcl}
\mathcal L u &=& \lambda e^v \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma e^u \qquad \text{in} \ \ \Omega, \\
u, v &=&0 \qquad \text{on} \ \ \RO ,
\end{array}\right.
\end{eqnarray*}
and the Lane-Emden system, when $p>1$
\begin{eqnarray*}
(E)_{\lambda,\gamma}\qquad \left\{ \begin{array}{lcl}
\mathcal L u &=& \lambda (1+v)^p \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma (1+u)^p \qquad \text{in} \ \ \Omega, \\
u, v &=&0 \qquad \qquad \text{on} \ \ \RO,
\end{array}\right.
\end{eqnarray*}
and the Lane-Emden system with singular nonlinearity, for $p>1$ and when $0<u,v<1$
\begin{eqnarray*}
(M)_{\lambda,\gamma}\qquad \left\{ \begin{array}{lcl}
\mathcal L u &=& \frac{\lambda}{(1-v)^p} \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \frac{\gamma}{(1-u)^p} \qquad \text{in} \ \ \Omega, \\
u, v &=&0 \qquad \qquad \text{on} \ \ \RO.
\end{array}\right.
\end{eqnarray*}
Note that for the case of $p=2$ the above singular nonlinearity and system is known as the MicroElectroMechanical Systems (MEMS), see \cite{egg2,lw} and references therein for the mathematical analysis of such equations. In addition, we study the following gradient system with more general nonlinearities
\begin{eqnarray*}
(H)_{\lambda,\gamma}\qquad \left\{ \begin{array}{lcl}
\mathcal L u &=& \lambda f'(u) g(v) \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma f(u) g'(v) \qquad \text{in} \ \ \Omega, \\
u, v &=&0 \qquad \qquad \text{on} \ \ \RO.
\end{array}\right.
\end{eqnarray*}
The nonlinearities $f$ and $g$ will satisfy various properties but will always at least satisfy
\begin{equation} \label{R}
f \text{ is smooth, increasing and convex with } f(0)=1 \text{ and } f \text{ superlinear at infinity}.
\end{equation}
A bounded weak solution pair $(u,v)$ is called a classical solution when both components $u,v$ are regular in the interior of $\Omega$ and $(P)_{\lambda,\gamma}$ holds. Given a nonlinearity $ f$ which satisfies (\ref{R}), the following nonlinear eigenvalue problem
\begin{eqnarray*}
\hbox{$(Q)_{\lambda}$}\hskip 50pt \left\{ \begin{array}{lcl}
-\Delta u &=& \lambda f(u)\qquad \text{in} \ \ \Omega, \\
u &=&0 \qquad \qquad \text{on} \ \ \partial \Omega,
\end{array}\right.
\end{eqnarray*}
is now well-understood. Brezis and V\'{a}zquez in \cite{BV} raised the question of determining the boundedness of $u^*$ for general nonlinearities $f$ satisfying (\ref{R}). See, for instance, \cite{BV,Cabre,CC, CR, cro, rs, cdds, Nedev,bcmr,v2,ns} for both local and nonlocal cases. It is known that there exists a critical parameter $ \lambda^* \in (0,\infty)$, called the extremal parameter, such that for all $ 0<\lambda < \lambda^*$ there exists a smooth, minimal solution $u_\lambda$ of $(Q)_\lambda$. Here the minimal solution means in the pointwise sense. In addition for each $ x \in \Omega$ the map $ \lambda \mapsto u_\lambda(x)$ is increasing in $ (0,\lambda^*)$. This allows one to define the pointwise limit $ u^*(x):= \lim_{\lambda \nearrow \lambda^*} u_\lambda(x)$ which can be shown to be a weak solution, in a suitably defined sense, of $(Q)_{\lambda^*}$. For this reason $ u^*$ is called the extremal solution. It is also known that for $ \lambda >\lambda^*$, there are no weak solutions of $(Q)_\lambda$. The regularity of the extremal solution has been of great interests in the literature. There have several attempts to tackle the problem and here we list a few.
For a general nonlinearity $f$ satisfying (\ref{R}), Nedev in \cite{Nedev} proved that $u^*$ is bounded when $ n \le 3$. This was extended to fourth dimensions when $ \Omega$ is a convex domain by Cabr\'{e} in \cite{Cabre}. The convexity of the domain was relaxed by Villegas in \cite{v2}. Most recently, Cabr\'{e} et al. in \cite{cfsr} claimed the regularity result when $n\le 9$. For the particular nonlinearity $f(u)=e^u$, known as the Gelfand equation, the regularity is shown $u^*\in L^{\infty}(\Omega)$ for dimensions $n<10$ by Crandall and Rabinowitz in \cite{CR}, see also \cite{far2}. If $ \Omega$ is a radial domain in $ {\mathbb{R}}^n$ with $ n <10$ the regularity is shown in \cite{CC} when $f$ is a general nonlinearity satisfying conditions (\ref{R}) but without the convexity assumption. In view of the above result for the exponential nonlinearity,
this is optimal. Note that for the case of $\Omega=B_1$, the classification of all radial solutions to this problem was originally done by Liouville in \cite{liou} for $n=2$ and then in higher dimensions in \cite{ns, jl,MP1} and references therein. For power nonlinearity $f(u)=(1+u)^p$ and for singular nonlinearity $f(u)=(1-u)^{-p}$ when $0<u<1$ for $p>1$, known as the Lane-Emden equation and MEMS equation respectively, the regularity of extremal solutions is established for the Joseph-Lundgren exponent, see \cite{jl}, in the literature. We refer interested readers to \cite{gg,egg1,egg2,far1,CR} and references therein for regularity results and Liouville theorems.
The regularity of extremal solutions for
nonlocal eigenvalue problem,
\begin{eqnarray*}
\hbox{$(S)_{\lambda}$}\hskip 50pt \left\{ \begin{array}{lcl}
{(-\Delta)}^s u &=& \lambda f(u)\qquad \text{in} \ \ \Omega, \\
u &=&0 \qquad \text{on} \ \ \RO,
\end{array}\right.
\end{eqnarray*}
is studied in the literature, see \cite{rs,r1, sp, cdds}, when $0<s<1$. However, there are various questions remaining as open problems. Ros-Oton and Serra in \cite{rs} showed that for a general nonlinearity $f$, $u^*$ is bounded when $n<4s$. In addition, if the following limit exists
\begin{equation}\label{conf}
\lim_{t\to\infty} \frac{f(t)f''(t)}{\left|f'(t)\right|^2} <\infty ,
\end{equation}
then $u^*$ is bounded when $n<10s$. Note that specific nonlinearities $f(u)=e^u$, $f(u)=(1+u)^p$ and $f(u)=(1-u)^{-p}$ for $p>1$ satisfy the above condition (\ref{conf}). When $s\to 1$, the dimension $n<10$ is optimal. However, for the fractional Laplacian $n<10s$ is not optimal, see Remark \ref{rem1}. In the current article, we prove counterparts of these regularity results for system of nonlocal equations. Later in \cite{r1}, Ros-Oton considered the fractional Gelfand problem, $f(u)=e^u$, on a domain $\Omega$ that is convex in the $x_i$-direction and symmetric with respect to $\{x_i=0\}$, for $1\le i\le n$. As an example, the unit ball satisfies these conditions. And he proved that $u^*$ is bounded for either $n\le 2s$ or $n>2s$ and
\begin{equation}
\frac{ \Gamma(\frac{n}{2}) \Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} >
\frac{ \Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})} .
\end{equation}
This, in particular, implies that $u^*$ is bounded in dimensions $ n\le 7$ for all $s\in (0,1)$. The above inequality is expected to provide the optimal dimension, see Remark \ref{rem1}. Relaxing the convexity and symmetry conditions on the domain remains an open problem. Capella et al. in \cite{cdds} studied the extremal solution of a problem related to $(S)_{\lambda}$ in the unit ball $B_1$ with a spectral fractional Laplacian operator that is defined using Dirichlet eigenvalues and eigenfunctions of the Laplacian operator in $B_1$. They showed that $u^*\in L^\infty(B_1)$ when $2\le n< 2\left[s+2+\sqrt{2(s+1)}\right]$.
More recently, Sanz-Perela in \cite{sp} proved regularity of the extremal solution of $(S)_{\lambda}$ with the fractional Laplacian operator in the unit ball with the same condition on $n$ and $s$. This implies that $u^*$ is bounded in dimensions $2\le n\le 6$ for all $s\in (0,1)$. Note also that it is well-known that there is a correspondence between the regularity of stable solutions on bounded domains and the Liouville theorems for stable solutions on the entire space, via rescaling and a blow-up procedure. For the
classification of solutions of above nonlocal equations on the entire space we refer interested readers to \cite{clo,ddw,fw,li}, and for the local equations to \cite{egg2,far1,far2} and references therein.
For the case of systems, as discussed in \cite{cf,Mont,faz}, set $ \mathcal{Q}:=\{ (\lambda,\gamma): \lambda, \gamma >0 \}$ and define
\begin{equation}
\mathcal{U}:= \left\{ (\lambda,\gamma) \in \mathcal{Q}: \mbox{ there exists a smooth solution $(u,v)$ of $(P)_{\lambda,\gamma}$} \right\}.
\end{equation}
We assume that $F(0,0),G(0,0)>0$.
A simple argument shows that if $F$ is superlinear at infinity for $ u$, uniformly in $v$, then the set of $ \lambda$ in $\mathcal{U}$ is bounded. Similarly we assume that $ G$ is superlinear at infinity for $ v$, uniformly in $u$, and hence we get $ \mathcal{U}$ is bounded. We also assume that $F,G$ are increasing in each variable. This allows the use of a sub/supersolution approach and one easily sees that if $ (\lambda,\gamma) \in \mathcal{U}$ then so is $ (0,\lambda] \times (0,\gamma]$. One also sees that $ \mathcal{U}$ is nonempty. We now define
$ \Upsilon:= \partial \mathcal{U} \cap \mathcal{Q}$, which plays the role of the extremal parameter $ \lambda^*$. Various properties of $ \Upsilon$ are known, see \cite{Mont}. Given $ (\lambda^*,\gamma^*) \in \Upsilon$ set $ \sigma:= \frac{\gamma^*}{\lambda^*} \in (0,\infty)$ and define
\begin{equation}
\Gamma_\sigma:=\{ (\lambda, \lambda \sigma): \frac{\lambda^*}{2} < \lambda < \lambda^*\}.
\end{equation}
We let $ (u_\lambda,v_\lambda)$ denote the minimal solution $(P)_{\lambda, \sigma \lambda}$ for $ \frac{\lambda^*}{2} < \lambda < \lambda^*$. One easily sees that for each $ x \in \Omega$ that $u_\lambda(x), v_\lambda(x)$ are increasing in $ \lambda$ and hence we define
\begin{equation}
u^*(x):= \lim_{\lambda \nearrow \lambda^*} u_\lambda(x) \ \ \text{and} \ \ v^*(x):= \lim_{\lambda \nearrow \lambda^*} v_\lambda(x),
\end{equation}
and we call $(u^*,v^*)$ the extremal solution associated with $ (\lambda^*,\gamma^*) \in \Upsilon$.
Under some very minor growth assumptions on $F$ and $G$ one can show that $(u^*,v^*)$ is a weak solution of $(P)_{\lambda^*,\gamma^*}$. For the rest of this article we refer to $(u^*,v^*)$ as $(u,v)$. For the case of local Laplacian operator, Cowan and the author in \cite{cf} proved that the extremal solution of $(H)_{\lambda,\gamma}$ when $\Omega$ is a convex domain is regular provided $1\le n \le 3$ for general nonlinearities $f,g\in C^1(\mathbb R)$ that satisfying (\ref{R}). This can be seen as a counterpart of the Nedev's result for elliptic gradient systems. For radial solutions, it is also shown in \cite{cf} that stable solutions are regular in dimensions $1\le n <10$ for general nonlinearities. This is a counterpart of the regularity result of Cabr\'{e}-Capella \cite{CC} and Villegas \cite{v2} for elliptic gradient systems. For the local Gelfand system, regularity of the extremal solutions is given by Cowan in \cite{cow} and by Dupaigne et al. in \cite{dfs} when $n<10$.
Here are our main results. The following theorem deals with regularity of the extremal solution of nonlocal Gelfand, Lane-Emden and MEMS systems.
\begin{thm}\label{thmg}
Suppose that $ \Omega$ is a bounded smooth domain in $ {\mathbb{R}}^n$. Let $(\lambda^*,\gamma^*) \in \Upsilon$ and $\mathcal L$ is given by (\ref{La}) where the ellipticity condition (\ref{c1c2}) holds and $0<s<1$. Then, the associated extremal solution of $ (G)_{\lambda^*,\gamma^*}$, $ (E)_{\lambda^*,\gamma^*}$ and $ (M)_{\lambda^*,\gamma^*}$ is bounded when
\begin{eqnarray}\label{dimg}
&&n < 10s,
\\&&\label{dime} n<2s+\frac{4s}{p-1}[p+\sqrt{p(p-1)}],
\\&&\label{dimm} n<2s+\frac{4s}{p+1}[p+\sqrt{p(p+1)}],
\end{eqnarray}
respectively.
\end{thm}
The following theorem is a counterpart of the Nedev regularity result for nonlocal system $(H)_{\lambda,\gamma}$.
\begin{thm} \label{nedev} Suppose that $ \Omega$ is a bounded smooth convex domain in $ {\mathbb{R}}^n$. Assume that $f$ and $g$ satisfy condition (\ref{R}) and $f'(0), g'(0)>0$ when $ f'(\cdot),g'(\cdot)$ are convex and
\begin{equation} \label{deltaeps}
\liminf_{s \rightarrow \infty} \frac{\left[f''(s)\right]^2}{f'''(s)f'(s)} >0 \ \ \text{and} \ \ \liminf_{s \rightarrow \infty} \frac{\left[g''(s)\right]^2}{g'''(s)g'(s)} >0,
\end{equation}
holds. Let $(\lambda^*,\gamma^*) \in \Upsilon$ and $\mathcal L={(-\Delta)}^s$ for $0<s<1$. Then, the associated extremal solution $(u,v)$ of $ (H)_{\lambda^*,\gamma^*}$ is bounded where $n < 4s$.
\end{thm}
In order to prove the above results, we first establish integral estimates for minimal solutions of systems with a general nonlocal operator $\mathcal L$ given by (\ref{Lui}) when $J$ is a nonnegative measurable even jump kernel. Then, we apply nonlocal Sobolev embedding arguments to conclude boundedness of the extremal solutions. Note that when $\lambda=\gamma$ the systems $(G)_{\lambda,\gamma}$, $(E)_{\lambda,\gamma}$ and $(M)_{\lambda,\gamma}$ turn into scalar equations.
Here is how this article is structured. In Section \ref{secpre}, we provide regularity theory for nonlocal operators. In Section \ref{secstab}, we establish various stability inequalities for minimal solutions of systems with a general nonlocal operator of the form (\ref{Lui}) with a nonnegative measurable even jump kernel. In Section \ref{secint}, we provide some technical integral estimates for stable solutions of systems introduced in the above. In Section \ref{secreg}, we apply the integral estimates to establish regularity of extremal solutions for nonlocal Gelfand, Lane-Emden and MEMS systems with exponential and power nonlinearities. In addition, we provide regularity of the extremal solution for the gradient system $(H)_{\lambda,\gamma}$ with general nonlinearities and also for particular power nonlinearities.
\section{Preliminaries}\label{secpre}
In this section, we provide regularity results
not only to the fractional Laplacian, but also to more general integro-differential
equations. We omit the proofs in this section and refer interested readers to corresponding references. Let us start with the following classical regularity result concerning embeddings for the Riesz potential, see the book of Stein \cite{st}.
\begin{thm}
Suppose that $0<s<1$, $n>2s$ and $f$ and $u$ satisfy
\begin{equation*}
u = {(-\Delta)}^{-s} f \qquad \text{in} \ \ \mathbb R^n,
\end{equation*}
in the sense that $u$ is the Riesz potential of order $2s$ of $f$. Let $u,f\in L^p(\mathbb R^n)$ when $1\le p<\infty$.
\begin{enumerate}
\item[(i)] For $p=1$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^q(\mathbb R^n)} \le C ||f||_{L^1(\mathbb R^n)} \ \ \text{for} \ \ q=\frac{n}{n-2s}.
\end{equation*}
\item[(ii)] For $1 < p<\frac{n}{2s}$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^q(\mathbb R^n)} \le C ||f||_{L^p(\mathbb R^n)} \ \ \text{for} \ \ q=\frac{np}{n-2ps}.
\end{equation*}
\item[(iii)] For $\frac{n}{2s}<p<\infty$, there exists a positive constant $C$ such that
\begin{equation*}
[u]_{C^\beta(\mathbb R^n)} \le C ||f||_{L^p(\mathbb R^n)} \ \ \text{for} \ \ \beta=2s-\frac{n}{p} ,
\end{equation*}
where $[\cdot]_{C^\beta(\mathbb R^n)}$ denotes the $C^\beta$ seminorm.
\end{enumerate}
Here the constant $C$ depending only on $n$, $s$ and $p$.
\end{thm}
The above theorem is applied by Ros-Oton and Serra in \cite{rs} to establish the following regularity theory for the fractional Laplacian. See also \cite{rs1} for the boundary regularity results.
\begin{prop}
Suppose that $0<s<1$, $n>2s$ and $f\in C(\bar \Omega)$ where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,1}$ domain. Let $u$ be the solution of
\begin{eqnarray*}
\left\{ \begin{array}{lcl}
{(-\Delta)}^s u &=& f \qquad \text{in} \ \ \Omega, \\
u&=&0 \qquad \text{in} \ \ \RO.
\end{array}\right.
\end{eqnarray*}
\begin{enumerate}
\item[(i)] For $1\le r<\frac{n}{n-2s}$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^r(\Omega)} \le C ||f||_{L^1(\Omega)} \ \ \text{for} \ \ r<\frac{n}{n-2s}.
\end{equation*}
\item[(ii)] For $1 < p<\frac{n}{2s}$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^q(\Omega)} \le C ||f||_{L^p(\Omega)} \ \ \text{for} \ \ q=\frac{np}{n-2ps}.
\end{equation*}
\item[(iii)] For $\frac{n}{2s}<p<\infty$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{C^\beta(\Omega)} \le C ||f||_{L^p(\Omega)} \ \ \text{for} \ \ \beta=\min\left\{s,2s-\frac{n}{p}\right\}.
\end{equation*}
\end{enumerate}
Here the constant $C$ depending only on $n$, $s$, $p$, $r$ and $\Omega$.
\end{prop}
For the case of $n\le 2s$, the fact that $0<s<1$ implies that $n=1$ and $s\ge \frac{1}{2}$. Note that in this case the Green function $G(x,y)$ is explicitly known. Therefore, $G(\cdot,y)\in L^\infty(\Omega)$ for $s>\frac{1}{2}$ and in $ L^p(\Omega)$ for all $p<\infty$ when $s=\frac{1}{2}$. We summarize this as $||u||_{L^\infty(\Omega)}\le C ||f||_{L^1(\Omega)}$ when $n<2s$. In addition, for the case of $n=2s$, we conclude that
$||u||_{L^p(\Omega)}\le C ||f||_{L^1(\Omega)}$ for all $p<\infty$ and $||u||_{L^\infty(\Omega)}\le C ||f||_{L^p(\Omega)}$ for $p>1$.
In what follows we provide a counterpart of the above regularity result for general integro-differential operators given by (\ref{La}). These operators are infinitessimal generators of stable and symmetric L\'{e}vy processes and they are uniquely determined by a finite measure on the unit
sphere $\mathbb S^{n-1}$, often referred as the spectral measure of the process. When this measure
is absolutely continuous, symmetric stable processes have generators of the
form (\ref{La}) where $0<s<1$ and $a$ is any nonnegative function $L^1(\mathbb S^{n-1})$ satisfying $a(\theta)=a(-\theta)$ for $\theta\in \mathbb S^{n-1}$. The regularity theory for general operators of the form (\ref{La}) has been recently
developed by Fern\'{a}ndez-Real and Ros-Oton in \cite{fro1}. In order to prove this result, authors apply results of \cite{gh} to study the fundamental solution
associated to the operator $\mathcal L$ in view of the one of the fractional Laplacian.
\begin{prop}\label{propregL}
Let $\Omega\subset \mathbb R^{n}$ be any bounded domain,
$0<s<1$ and $f\in L^2(\Omega)$. Let $u$ be any weak solution of
\begin{eqnarray*}
\left\{ \begin{array}{lcl}
\mathcal L u &=& f \qquad \text{in} \ \ \Omega, \\
u&=&0 \qquad \text{in} \ \ \RO ,
\end{array}\right.
\end{eqnarray*}
where the operator $\mathcal L$ is given by (\ref{La}) and the ellipticity condition (\ref{c1c2}) holds. Assume that $f\in L^r(\Omega)$ for some $r$.
\begin{enumerate}
\item[(i)] For $1< r<\frac{n}{2s}$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^q(\Omega)} \le C ||f||_{L^r(\Omega)} \ \ \text{for} \ \ q=\frac{nr}{n-2rs}.
\end{equation*}
\item[(ii)] For $r=\frac{n}{2s}$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^q(\Omega)} \le C ||f||_{L^r(\Omega)} \ \ \text{for} \ \ q<\infty.
\end{equation*}
\item[(iii)] For $\frac{n}{2s}<r<\infty$, there exists a positive constant $C$ such that
\begin{equation*}
||u||_{L^\infty(\Omega)} \le C ||f||_{L^r(\Omega)}.
\end{equation*}
\end{enumerate}
Here the constant $C$ depending only on $n$, $s$, $r$, $\Omega$ and ellipticity constants.
\end{prop}
We end this section with this point that $(u,v)$ is a weak solution of $(P)_{\lambda,\gamma}$ for $u,v\in L^1(\Omega)$ if $ F(u,v)\delta^s \in L^1(\Omega)$ and $G(u,v) \delta^s \in L^1(\Omega)$ where $\delta(x)=\text{dist}(x,\Omega)$ and
$$ \int_{\Omega} u \mathcal L \zeta = \int_{\Omega} \lambda F(u,v) \zeta \ \ \text{and} \ \ \int_{\Omega} v \mathcal L \eta = \int_{\Omega} \gamma G(u,v) \eta , $$
when $\zeta,\eta$ and $\mathcal \zeta,\mathcal \eta$ are bounded in $\Omega$ and $\zeta,\eta\equiv 0$ on $\partial\Omega$. Any bounded weak solution is a classical solution, in the sense that it is regular in the
interior of $\Omega$, continuous up to the boundary, and $(P)_{\lambda,\gamma}$ holds pointwise. Note that for the case of local operators, that is $s = 1$, the above notion of weak solution is consistent with the one introduced by Brezis et al. in \cite{bcmr,BV}.
\section{Stability inequalities}\label{secstab}
In this section, we provide stability inequalities for minimal solutions of system $(P)_{\lambda,\gamma}$ for various nonlinearities $F$ and $G$. We start with the following technical lemma in regards to nonlocal operator $\mathcal L$ with even symmetric kernel $J$.
\begin{lemma}\label{fgprop}
Assume that an operator $\mathcal L$ is given by (\ref{Luj}) with a measurable symmetric kernel $J(x,z)=J(x-z)$ that is even. Then,
\begin{eqnarray*}
&&\mathcal L(f(x)g(x)) = f(x)\mathcal L(g(x))+g(x)\mathcal L(f(x)) - \int_{\mathbb R^n} \left[f(x)-f(z) \right] \left[g(x)-g(z) \right] J(x-z) dz,\\
&&\int_{\mathbb R^n} g(x)\mathcal L(f(x)) dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[f(x)-f(z) \right] \left[g(x)-g(z) \right] J(x-z) dx dz,
\end{eqnarray*}
where $f,g\in C^1(\mathbb R^n)$ and the integrals are finite.
\end{lemma}
\begin{proof} The proof is elementary and we omit it here.
\end{proof}
We now establish a stability inequality for minimal solutions of system $(P)_{\lambda,\gamma}$. Note that for the case of local operators this inequality is established by the author and Cowan in \cite{cf} and in \cite{faz}.
\begin{prop}\label{stablein}
Let $(u_\lambda,v_\lambda)$ be a minimal solution of system $(P)_{\lambda,\gamma}$ such that $u_\lambda,v_\lambda$ are increasing in $\lambda$. Assume that $J$ is a measurable even kernel and $F_vG_u\ge 0$. Then,
\begin{eqnarray} \label{stability}
&& \int_{\Omega} F_u \zeta^2 +G_v \eta^2 + 2\sqrt{F_vG_u} \zeta\eta dx
\\& \le& \nonumber \frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n}\left( \frac{1}{\lambda} [\zeta(x)- \zeta(z)]^2 +\frac{1}{\gamma} [\eta(x)- \eta(z)]^2\right) J(x-z) dz dx ,
\end{eqnarray}
for test functions $\zeta,\eta$ so that $\zeta,\eta=0$ in $\RO$.
\end{prop}
\begin{proof}
Since $u_\lambda,v_\lambda$ are increasing in $\lambda$, differentiating $(P)_{\lambda,\gamma}$ with respect to $\lambda$ we get
\begin{eqnarray*}
\mathcal L (\partial_\lambda u_\lambda) &=& F + \lambda F_u \partial_\lambda u_\lambda + \lambda F_v \partial_\lambda v_\lambda , \\
\mathcal L (\partial_\lambda v_\lambda) &=&\sigma G + \gamma G_u \partial_\lambda u_\lambda + \gamma G_v \partial_\lambda v_\lambda,
\end{eqnarray*}
where $u_\lambda,v_\lambda>0$. Multiply both sides with $\frac{1}{\lambda}\frac{\zeta^2}{\partial_\lambda u_\lambda}$ and $\frac{1}{\gamma}\frac{\eta^2}{\partial_\lambda v_\lambda}$ to get
\begin{eqnarray*} \label{L1}
\frac{1}{\lambda} \mathcal L (\partial_\lambda u_\lambda) \frac{\zeta^2}{\partial_\lambda u_\lambda} + \frac{1}{\gamma}\mathcal L (\partial_\lambda v_\lambda) \frac{\eta^2}{\partial_\lambda v_\lambda}
&=&
\frac{1}{\lambda} F\frac{\zeta^2}{\partial_\lambda u_\lambda} + F_u \zeta^2 + F_v \partial_\lambda v_\lambda \frac{\zeta^2}{\partial_\lambda u_\lambda}
\\&& +
\frac{\sigma}{\gamma} G\frac{\zeta^2}{\partial_\lambda v_\lambda} + G_v \eta^2 + G_u \partial_\lambda u_\lambda \frac{\eta^2}{\partial_\lambda v_\lambda} .
\end{eqnarray*}
Note that the following lower-bound holds for the left-hand side of the above equality
\begin{eqnarray*}
RHS& \ge & F_u \zeta^2 + G_v \eta^2
+ F_v \partial_\lambda v_\lambda \frac{\zeta^2}{\partial_\lambda u_\lambda}
+ G_u \partial_\lambda u_\lambda \frac{\eta^2}{\partial_\lambda v_\lambda}
\ge F_u \zeta^2 + G_v \eta^2
+ 2 \sqrt{ F_vG_u} \zeta\eta.
\end{eqnarray*}
Integrating the above we end up with
\begin{equation} \label{LL1}
\int_{\Omega} F_u \zeta^2 + G_v \eta^2 + 2 \sqrt{ F_vG_u} \zeta\eta dx
\le \int_{\mathbb R^n}
\frac{1}{\lambda} L (\partial_\lambda u_\lambda) \frac{\zeta^2}{\partial_\lambda u_\lambda} + \frac{1}{\gamma} L (\partial_\lambda v_\lambda) \frac{\eta^2}{\partial_\lambda v_\lambda} dx .
\end{equation}
Applying Lemma \ref{fgprop}, we have
\begin{eqnarray*}\label{Lphi}
&&\int_{{\mathbb{R}}^n} \mathcal L (\partial_\lambda u_\lambda(x)) \frac{\zeta^2(x)}{\partial_\lambda u_\lambda(x)} dx
\\&=&
\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} [\partial_\lambda u_\lambda(x) - \partial_\lambda u_\lambda(z)] \left[ \frac{\zeta^2(x)}{\partial_\lambda u_\lambda(x)}- \frac{\zeta^2(z)}{\partial_\lambda u_\lambda(z)} \right] J(x-z) dx dz.
\end{eqnarray*}
Note that for $a,b,c,d\in\mathbb R$ when $ab<0$ we have
\begin{equation*}
(a+b)\left[ \frac{c^2}{a} + \frac{d^2}{b} \right] \le (c-d)^2 .
\end{equation*}
Since each $\partial_\lambda u_\lambda$ does not change sign, we have $\partial_\lambda u_\lambda(x)\partial_\lambda u_\lambda(z)>0$. Setting $a=\partial_\lambda u_\lambda(x)$, $b=-\partial_\lambda u_\lambda(z)$, $c=\zeta(x)$ and $d=\zeta(z)$ in the above inequality and from the fact that $ab=- \partial_\lambda u_\lambda(x) \partial_\lambda u_\lambda(z)<0$, we conclude
\begin{equation*}
[\partial_\lambda u_\lambda(x) - \partial_\lambda u_\lambda(z)] \left[ \frac{\zeta^2(x)}{\partial_\lambda u_\lambda(x)}- \frac{\zeta^2(z)}{\partial_\lambda u_\lambda(z)} \right]
\le [\zeta(x)- \zeta(z)]^2 .
\end{equation*}
Therefore,
\begin{equation*}
\int_{{\mathbb{R}}^n}\mathcal L (\partial_\lambda u_\lambda(x)) \frac{\zeta^2(x)}{\partial_\lambda u_\lambda(x)} dx\le \frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} [\zeta(x)- \zeta(z)]^2 J(z-x) dz dx.
\end{equation*}
This together with (\ref{LL1}) complete the proof.
\end{proof}
Following ideas provided in the above, we provide stability inequalities for minimal solutions of Gelfand, Lane-Emden and MEMS systems with exponential and power-type nonlinearities.
\begin{cor}\label{stablein}
Let $(u,v)$ be the extremal solution of system $(G)_{\lambda,\gamma}$, $(E)_{\lambda,\gamma}$ and $(M)_{\lambda,\gamma}$. Then,
\begin{eqnarray}\label{stabilityG}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}}\zeta^2 dx &\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} {|\zeta(x)- \zeta(z)|^2} J(x-z) dz dx ,
\\
\label{stabilityE}
p\sqrt{\lambda\gamma} \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} \zeta^2 dx
&\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} {|\zeta(x)- \zeta(z)|^2} J(x-z) dz dx ,
\\ \label{stabilityM}
p\sqrt{\lambda\gamma} \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} \zeta^2 dx &\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} {|\zeta(x)- \zeta(z)|^2} J(x-z) dz dx ,
\end{eqnarray}
for test functions $\zeta$ so that $\zeta=0$ in $\RO$.
\end{cor}
\begin{cor}\label{stablein1}
Let $(u,v)$ be the extremal solution of system $(H)_{\lambda,\gamma}$ when $f'g'\ge 0$. Then,
\begin{eqnarray}\label{stabilityH}
&& \int_{\Omega} f''g \zeta^2 +fg'' \eta^2 + 2f'g' \zeta\eta dx
\\& \le&\nonumber \frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n}\left( \frac{1}{\lambda} |\zeta(x)- \zeta(z)|^2 + \frac{1}{\gamma} |\eta(x)- \eta(z)|^2 \right) J(x-z) dz dx ,
\end{eqnarray}
for test functions $\zeta,\eta$ so that $\zeta,\eta=0$ in $\RO$.
\end{cor}
\section{Integral estimates for stable solutions}\label{secint}
In this section, we establish some technical integral estimates for stable solutions of systems. Most of the ideas and methods applied in this section are inspired by the ones developed in the literature, see for example \cite{dfs, far1, far2, fg, fw}. We start with the Gelfand system.
\begin{lemma}\label{lemuvg} Suppose that $(u,v)$ is a solution of $ (G)_{\lambda,\gamma}$ when
the associated stability inequality (\ref{stabilityG}) holds. Then, there exists a positive constant $C_{\lambda,\gamma,|\Omega|}=C({\lambda,\gamma,|\Omega|})$ such that
\begin{equation*}
\int_{\Omega} e^{u+v} dx \le C_{\lambda,\gamma,|\Omega|} .
\end{equation*}
\end{lemma}
\begin{proof} Multiply the second equation of $ (G)_{\lambda,\gamma}$ with $e^{u}-1$ and integrate to get
\begin{equation*}
\lambda \int_{\Omega} (e^{u}-1) e^v dx= \int_{\Omega} \mathcal L u (e^{u}-1) dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u (e^{u}-1) dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[e^{u(x)}-e^{u(z)} \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
\left| e^{\frac{\beta}{2}} - e^{\frac{\alpha}{2}} \right|^2 \le \frac{1}{4} (e^\beta-e^\alpha)(\beta-\alpha).
\end{equation*}
Applying the above inequality for $\alpha=u(z)$ and $\beta=u(x)$, we obtain
\begin{equation*}
\left| e^{\frac{u(x)}{2}} - e^{\frac{u(z)}{2}} \right|^2 \le \frac{1}{4} (e^{u(x)}-e^{u(z)})(u(x)-u(z)).
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} e^{u+v} dx \ge \lambda \int_{\Omega} (e^{u}-1) e^v dx \ge 2 \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| e^{\frac{u(x)}{2}} - e^{\frac{u(z)}{2}} \right|^2J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=e^{\frac{u}{2}}-1$ to get
\begin{equation*}\label{}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} (e^{\frac{u}{2}}-1)^2 dx \le \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left|e^{\frac{u(x)}{2}}- e^{\frac{u(z)}{2}} \right|^2 J(x-z) dz dx .
\end{equation*}
Combining above inequalities, we conclude
\begin{equation}\label{rruv}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} (e^{\frac{u}{2}}-1)^2 dx \le \frac{1}{4} \lambda \int_{\Omega} e^{u+v} dx .
\end{equation}
Applying the Young's inequality $e^{u/2}\le \frac{e^u}{4}+1$, we conclude
\begin{equation*}\label{}
\int_{\Omega} e^{\frac{u+v}{2}} e^{\frac{u}{2}} dx \le \frac{1}{4} \int_{\Omega} e^{\frac{u+v}{2}} e^u dx + \int_{\Omega} e^{\frac{u+v}{2}} dx .
\end{equation*}
From this and expanding the left-hand side of (\ref{rruv}), we obtain
\begin{eqnarray*}
\lambda \int_{\Omega} e^{u+v} dx + 8 \sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} dx &\ge& 2 \sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} e^{u} dx ,
\\
\gamma \int_{\Omega} e^{u+v} dx + 8 \sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} dx & \ge& 2 \sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} e^{v} dx .
\end{eqnarray*}
Multiplying these inequalities and applying the Cauchy-Schwarz inequality, i.e.
\begin{equation*}\label{}
\int_{\Omega} e^{\frac{u+v}{2}} e^{{u}} dx \int_{\Omega} e^{\frac{u+v}{2}} e^{{v}} dx \ge
\left(\int_{\Omega} e^{u+v} dx\right)^2,
\end{equation*}
we complete the proof.
\end{proof}
We now provide a counterpart of the above estimate for stable solutions of $ (E)_{\lambda,\gamma}$.
\begin{lemma}\label{lemuve} Suppose that $(u,v)$ is a solution of $ (E)_{\lambda,\gamma}$ when
the associated stability inequality (\ref{stabilityE}) holds. Then, there exists a positive constant $C_{\lambda,\gamma,|\Omega|}=C({\lambda,\gamma,|\Omega|})$ such that
\begin{equation}\label{inuvp}
\int_{\Omega} (1+u)^p(1+v)^p dx \le C_{\lambda,\gamma,|\Omega|} .
\end{equation}
\end{lemma}
\begin{proof}
Multiply the second equation of $ (E)_{\lambda,\gamma}$ with $(1+u)^p-1$ and integrate to get
\begin{equation*}
\lambda \int_{\Omega} [(1+u)^p-1] (1+v)^p dx= \int_{\Omega} \mathcal L u [(1+u)^p-1] dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u [(1+u)^p-1] dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[(1+u(x))^p-(1+u(z))^p \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
[(1+\alpha)^p-(1+\beta)^p](\alpha-\beta) \ge \frac{4p}{(p+1)^2} \left| (1+\alpha)^{\frac{p+1}{2}} - (1+\beta)^{\frac{p+1}{2}} \right|^2 .
\end{equation*}
Applying the above inequality for $\alpha=u(x)$ and $\beta=u(z)$, we obtain
\begin{equation*}
[(1+u(x))^p-(1+u(z))^p][u(x)-u(z)] \ge \frac{4p}{(p+1)^2} \left| (1+u(x))^{\frac{p+1}{2}} - (1+u(z))^{\frac{p+1}{2}} \right|^2 .
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} (1+u)^p (1+v)^p dx \ge \frac{4p}{(p+1)^2} \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| (1+u(x))^{\frac{p+1}{2}} - (1+u(z))^{\frac{p+1}{2}} \right|^2 J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=(1+u)^{\frac{p+1}{2}}-1$ to get
\begin{eqnarray*}\label{}
&&\sqrt{\lambda\gamma} p \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} [(1+u)^{\frac{p+1}{2}}-1]^2 dx
\\&\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left| (1+u(x))^{\frac{p+1}{2}}- (1+u(z))^{\frac{p+1}{2}} \right|^2 J(x-z) dz dx .
\end{eqnarray*}
Combining above inequalities, we conclude
\begin{equation}\label{rruv}
\sqrt{\lambda\gamma} p \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} [(1+u)^{\frac{p+1}{2}}-1]^2 dx
\le \frac{(p+1)^2}{4p} \lambda \int_{\Omega} (1+u)^p (1+v)^p dx .
\end{equation}
Expanding the left-hand side of the inequality and rearranging we get
\begin{eqnarray}\label{l1}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} (1+u)^{p+1} dx
\\&&\nonumber \le \frac{(p+1)^2}{4p} \lambda \int_{\Omega} (1+u)^p (1+v)^p dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} dx,
\end{eqnarray}
where we have used the inequality $a\le \frac{\epsilon}{2}a^2+\frac{1}{2\epsilon}$ for any $\epsilon>0$. Similarly,
\begin{eqnarray}\label{l2}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1+v)^{\frac{p-1}{2}} (1+u)^{\frac{p-1}{2}} (1+v)^{p+1} dx
\\&&\nonumber \le \frac{(p+1)^2}{4p} \gamma \int_{\Omega} (1+u)^p (1+v)^p dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} dx .
\end{eqnarray}
Note that from the Cauchy-Schwarz inequality we get
\begin{equation*}\label{}
\int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} (1+u)^{p+1} dx \int_{\Omega} (1+v)^{\frac{p-1}{2}} (1+u)^{\frac{p-1}{2}} (1+v)^{p+1} dx \ge \left( \int_{\Omega} (1+u)^p (1+v)^p dx \right)^2 .
\end{equation*}
From this and
multiplying both sides of the above (\ref{l1}) and (\ref{l2}) we conclude
\begin{equation*}\label{}
\lambda\gamma \left[ p^2(1-\epsilon)^2- \left(\frac{(p+1)^2}{4p}\right)^2\right]\left( \int_{\Omega} (1+u)^p (1+v)^p dx \right)^2 \le C_{\epsilon,\lambda,\gamma} \left[ \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} dx\right]^2 ,
\end{equation*}
for small $\epsilon>0$. Note that $p^2- \left(\frac{(p+1)^2}{4p}\right)^2>0$ when $p>1$. Therefore, taking small enough $\epsilon>0$ and applying the H\"{o}lder's inequality, we complete the proof.
\end{proof}
Here is a counterpart of the above estimate for stable solutions of $ (M)_{\lambda,\gamma}$.
\begin{lemma}\label{lemuvm} Suppose that $(u,v)$ is a solution of $ (M)_{\lambda,\gamma}$ when
the associated stability inequality (\ref{stabilityM}) holds. Then, there exists a positive constant $C_{\lambda,\gamma,|\Omega|}=C({\lambda,\gamma,|\Omega|})$ such that
\begin{equation}
\int_{\Omega} (1-u)^{-p}(1-v)^{-p} dx \le C_{\lambda,\gamma,|\Omega|}.
\end{equation}
\end{lemma}
\begin{proof} The proof is similar to the one provide in Lemma \ref{lemuve}. Multiply the second equation of $ (M)_{\lambda,\gamma}$ with $(1-u)^{-p}-1$ and integrate to get
\begin{equation*}
\lambda \int_{\Omega} [(1-u)^{-p}-1] (1-v)^{-p} dx= \int_{\Omega} \mathcal L u [(1-u)^{-p}-1] dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u [(1-u)^{-p}-1] dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[(1-u(x))^{-p}-(1-u(z))^{-p} \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
[(1-\alpha)^{-p}-(1-\beta)^{-p}](\alpha-\beta) \ge \frac{4p}{(p-1)^2} \left| (1-\alpha)^{\frac{1-p}{2}} - (1-\beta)^{\frac{1-p}{2}} \right|^2 .
\end{equation*}
Applying the above inequality for $\alpha=u(x)$ and $\beta=u(z)$, we obtain
\begin{equation*}
[(1-u(x))^{-p}-(1-u(z))^{-p}][u(x)-u(z)] \ge \frac{4p}{(p-1)^2} \left| (1-u(x))^{\frac{-p+1}{2}} - (1-u(z))^{\frac{-p+1}{2}} \right|^2 .
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} (1-u)^{-p} (1-v)^{-p} dx \ge \frac{4p}{(p-1)^2} \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| (1-u(x))^{\frac{-p+1}{2}} - (1-u(z))^{\frac{-p+1}{2}} \right|^2 J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=(1-u)^{\frac{-p+1}{2}}-1$ to get
\begin{eqnarray*}\label{}
&&\sqrt{\lambda\gamma} p \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} [(1-u)^{\frac{1-p}{2}}-1]^2 dx
\\&\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left| (1-u(x))^{\frac{-p+1}{2}}- (1-u(z))^{\frac{-p+1}{2}} \right|^2 J(x-z) dz dx .
\end{eqnarray*}
Combining above inequalities, we conclude
\begin{equation}\label{rruv}
\sqrt{\lambda\gamma} p \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} [(1-u)^{\frac{1-p}{2}}-1]^2 dx
\le \frac{(p-1)^2}{4p} \lambda \int_{\Omega} (1-u)^{-p }(1-v)^{-p} dx .
\end{equation}
Expanding the left-hand side of the inequality and rearranging we get
\begin{eqnarray}\label{l1}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} (1-u)^{-p+1} dx
\\&&\nonumber \le \frac{(p-1)^2}{4p} \lambda \int_{\Omega} (1-u)^{-p} (1-v)^{-p} dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{\frac{p+1}{2}} dx,
\end{eqnarray}
where we have used the inequality $a\le \frac{\epsilon}{2}a^2+\frac{1}{2\epsilon}$ for any $\epsilon>0$. Similarly,
\begin{eqnarray}\label{l2}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1-v)^{-\frac{p+1}{2}} (1-u)^{-\frac{p+1}{2}} (1-v)^{-p+1} dx
\\&&\nonumber \le \frac{(p-1)^2}{4p} \gamma \int_{\Omega} (1-u)^{-p} (1-v)^{-p} dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} dx .
\end{eqnarray}
Note that from the Cauchy-Schwarz inequality we get
\begin{eqnarray*}\label{}
&&\int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} (1-u)^{-p+1} dx \int_{\Omega} (1-v)^{-\frac{p+1}{2}} (1-u)^{-\frac{p+1}{2}} (1-v)^{-p+1} dx \\&& \ge \left( \int_{\Omega} (1-u)^{-p} (1-v)^{-p} dx \right)^2 .
\end{eqnarray*}
From this and
multiplying both sides of the above (\ref{l1}) and (\ref{l2}) we conclude
\begin{equation*}\label{}
\lambda\gamma \left[ p^2(1-\epsilon)^2- \left(\frac{(p-1)^2}{4p}\right)^2\right]\left( \int_{\Omega} (1-u)^{-p} (1-v)^{-p} dx \right)^2 \le C_{\epsilon,\lambda,\gamma} \left[ \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} dx\right]^2 ,
\end{equation*}
for small $\epsilon>0$. Note that $p^2- (\frac{(p-1)^2}{4p})^2>0$ when $p>0$. Therefore, taking small enough $\epsilon>0$ and applying the H\"{o}lder's inequality when $p>1$, we complete the proof.
\end{proof}
In the next lemmata, we provide integral $L^q(\Omega)$ estimates for Gelfand, Lane-Emden and MEMS systems. We start with the Gelfand system and establish a relation between $\int_{\Omega} e^{\frac{2t+1}{2}u} e^{\frac{v}{2}} dx$ and $\int_{\Omega} e^{\frac{2t+1}{2}v} e^{\frac{u}{2}} dx$ for some constant $t>\frac{1}{2}$.
\begin{lemma}\label{lemuvgt} Under the same assumptions as Lemma \ref{lemuvg}, set
\begin{equation*}
X:=\int_{\Omega} e^{\frac{2t+1}{2}u} e^{\frac{v}{2}} dx, Y:=\int_{\Omega} e^{\frac{2t+1}{2}v} e^{\frac{u}{2}} dx, Z:=\int_{\Omega} e^{u} dx , W:= \int_{\Omega} e^{v} dx,
\end{equation*}
where $t>\frac{1}{2}$. Then,
\begin{eqnarray}\label{lgXg}
\begin{array}{lcl}
\sqrt{\lambda\gamma} X \le (\frac{t }{4}+\epsilon) \lambda X^{\frac{2t-1}{2t}} Y^{\frac{1}{2t}} + C_{\epsilon,\lambda,\gamma,|\Omega|} Z, \\
\sqrt{\lambda\gamma} Y \le (\frac{t }{4}+\epsilon) \gamma Y^{\frac{2t-1}{2t}} X^{\frac{1}{2t}} + C_{\epsilon,\lambda,\gamma,|\Omega|} W,
\end{array}
\end{eqnarray}
where $C_{\epsilon,\lambda,\gamma,|\Omega|} $ is a positive constant.
\end{lemma}
\begin{proof} Multiply the second equation of $ (G)_{\lambda,\gamma}$ with $e^{tu}-1$ when $t>\frac{1}{2}$ is a constant. Integrating implies that
\begin{equation*}
\lambda \int_{\Omega} (e^{tu}-1) e^v dx= \int_{\Omega} \mathcal L u (e^{tu}-1) dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u (e^{tu}-1) dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[e^{tu(x)}-e^{tu(z)} \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
\left| e^{\frac{\beta}{2}} - e^{\frac{\alpha}{2}} \right|^2 \le \frac{1}{4} (e^\beta-e^\alpha)(\beta-\alpha).
\end{equation*}
Applying the above inequality for $\alpha=tu(z)$ and $\beta=tu(x)$, we obtain
\begin{equation*}
\left| e^{\frac{tu(x)}{2}} - e^{\frac{tu(z)}{2}} \right|^2 \le \frac{t}{4} (e^{tu(x)}-e^{tu(z)})(u(x)-u(z)).
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} (e^{tu}-1) e^v dx \ge \frac{2}{t} \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| e^{\frac{tu(x)}{2}} - e^{\frac{tu(z)}{2}} \right|^2J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=e^{\frac{tu}{2}}-1$ to get
\begin{equation*}\label{}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} (e^{\frac{tu}{2}}-1)^2 dx \le \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left|e^{\frac{tu(x)}{2}}- e^{\frac{tu(z)}{2}} \right|^2 J(x-z) dz dx .
\end{equation*}
Combining above inequalities, we conclude
\begin{equation}\label{uvt2}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{u+v}{2}} (e^{\frac{tu}{2}}-1)^2 dx \le \frac{t}{4} \lambda \int_{\Omega} (e^{tu}-1) e^v dx .
\end{equation}
On the other hand, from the Young inequality we have
\begin{equation}\label{uvte1}
\int_{\Omega} e^{\frac{t+1}{2}u} e^{\frac{v}{2}} dx \le \frac{\epsilon}{2} \sqrt{\frac{\lambda }{\gamma}} \int_{\Omega} e^{tu} e^{v} dx + \frac{1}{2\epsilon} \sqrt{\frac{\gamma }{\lambda}} \int_{\Omega} e^{u} dx ,
\end{equation}
where $\epsilon$ is a positive constant. In addition, from the H\"{o}lder inequality we get
\begin{equation}\label{uvte2}
\int_{\Omega} e^{t u} e^v dx \le \left( \int_{\Omega} e^{\frac{2t+1}{2}u} e^{\frac{v}{2}} dx \right)^{\frac{2t-1}{2t}} \left( \int_{\Omega} e^{\frac{2t+1}{2}v} e^{\frac{u}{2}} dx \right)^{\frac{1}{2t}} .
\end{equation}
Now, expanding both sides of (\ref{uvt2}) we have
\begin{equation}\label{uvte3}
\sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{2t+1}{2}u} e^{\frac{v}{2}} dx \le \frac{t}{4} \lambda \int_{\Omega} e^{tu} e^v dx + 2 \sqrt{\lambda\gamma} \int_{\Omega} e^{\frac{t+1}{2}u} e^{\frac{v}{2}} dx .
\end{equation}
Combining (\ref{uvte1}), (\ref{uvte2}) and (\ref{uvte3}) proves the first inequality in (\ref{lgXg}). With similar arguments one can show the second inequality.
\end{proof}
We now consider the Lane-Emden system and establish a relation between $\int_{\Omega} (1+u)^{\frac{p+1}{2}+t} (1+v)^{\frac{p-1}{2}} dx$ and $\int_{\Omega} (1+v)^{\frac{p+1}{2}+t} (1+u)^{\frac{p-1}{2}} dx$ for some constant $t>{1}$.
\begin{lemma}\label{lemuvet} Under the same assumptions as Lemma \ref{lemuve}, set
\begin{eqnarray*}
&&X:=\int_{\Omega} (1+u)^{\frac{p-1}{2}+t+1} (1+v)^{\frac{p-1}{2}} dx, \ \ Y:=\int_{\Omega} (1+v)^{\frac{p-1}{2}+t+1} (1+u)^{\frac{p-1}{2}} dx,
\\&&Z:=\int_{\Omega} (1+u)^{\frac{p-1}{2}}(1+v)^{\frac{p-1}{2}} dx ,
\end{eqnarray*}
for $t>1$. Then, for some constant $0<\epsilon<1$ we get
\begin{eqnarray}\label{lgXet}
\begin{array}{lcl}
\sqrt{\lambda\gamma} p(1-\epsilon) X \le \frac{(t+1)^2 }{4t} \lambda X^{\frac{2t-p+1}{2(t+1)}} Y^{\frac{p+1}{2(t+1)}} + C_{\epsilon,\lambda,\gamma,|\Omega|} Z, \\
\sqrt{\lambda\gamma} p(1-\epsilon) Y \le \frac{(t+1)^2 }{4t} \gamma Y^{\frac{2t-p+1}{2(t+1)}} X^{\frac{p+1}{2(t+1)}} + C_{\epsilon,\lambda,\gamma,|\Omega|} Z,
\end{array}
\end{eqnarray}
where $C_{\epsilon,\lambda,\gamma,|\Omega|} $ is a positive constant.
\end{lemma}
\begin{proof} Let $t>1$ be a constant. Multiply the second equation of $ (E)_{\lambda,\gamma}$ with $(1+u)^t-1$ and integrate to get
\begin{equation*}
\lambda \int_{\Omega} [(1+u)^t-1] (1+v)^p dx= \int_{\Omega} \mathcal L u [(1+u)^t-1] dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u [(1+u)^t-1] dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[(1+u(x))^t-(1+u(z))^t \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
[(1+\alpha)^t-(1+\beta)^t](\alpha-\beta) \ge \frac{4t}{(t+1)^2} \left| (1+\alpha)^{\frac{t+1}{2}} - (1+\beta)^{\frac{t+1}{2}} \right|^2 .
\end{equation*}
Applying the above inequality for $\alpha=u(x)$ and $\beta=u(z)$, we obtain
\begin{equation*}
[(1+u(x))^t-(1+u(z))^t][u(x)-u(z)] \ge \frac{4t}{(t+1)^2} \left| (1+u(x))^{\frac{t+1}{2}} - (1+u(z))^{\frac{t+1}{2}} \right|^2 .
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} (1+u)^t (1+v)^p dx \ge \frac{4t}{(t+1)^2} \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| (1+u(x))^{\frac{t+1}{2}} - (1+u(z))^{\frac{t+1}{2}} \right|^2 J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=(1+u)^{\frac{t+1}{2}}-1$ to get
\begin{eqnarray*}\label{}
&&\sqrt{\lambda\gamma} p \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} [(1+u)^{\frac{t+1}{2}}-1]^2 dx
\\&\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left| (1+u(x))^{\frac{t+1}{2}}- (1+u(z))^{\frac{t+1}{2}} \right|^2 J(x-z) dz dx ,
\end{eqnarray*}
Combining above inequalities, we conclude
\begin{equation}\label{rruv}
\sqrt{\lambda\gamma} p \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} [(1+u)^{\frac{t+1}{2}}-1]^2 dx
\le \frac{(t+1)^2}{4t} \lambda \int_{\Omega} (1+u)^t (1+v)^p dx .
\end{equation}
Expanding the left-hand side of the inequality and rearranging we get
\begin{eqnarray}\label{l1e}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} (1+u)^{t+1} dx
\\&&\nonumber \le \frac{(t+1)^2}{4t} \lambda \int_{\Omega} (1+u)^t (1+v)^p dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} dx,
\end{eqnarray}
where we have used the inequality $a\le \frac{\epsilon}{2}a^2+\frac{1}{2\epsilon}$ for any $\epsilon>0$. From the H\"{o}lder's inequality we get
\begin{eqnarray*}\label{}
&& \int_{\Omega} (1+u)^t (1+v)^p dx
\\& \le &\left[ \int_{\Omega} (1+u)^{\frac{p-1}{2}} (1+v)^{\frac{p-1}{2}} (1+u)^{t+1} dx \right]^{\frac{1}{\beta}}
\left[ \int_{\Omega} (1+v)^{\frac{p-1}{2}} (1+u)^{\frac{p-1}{2}} (1+v)^{t+1} dx \right]^{1-\frac{1}{\beta}} ,
\end{eqnarray*}
where $\beta=\frac{2(t+1)}{2t-p+1}$. This and (\ref{l1e}) completes the proof of the first estimate in (\ref{lgXet}). Similarly, one can show the second estimate.
\end{proof}
We now consider the MEMS system with singular power nonlinearities and establish a relation between $\int_{\Omega} (1-u)^{\frac{1-p}{2}-t} (1-v)^{-\frac{p+1}{2}} dx$ and $\int_{\Omega} (1-v)^{\frac{1-p}{2}-t} (1-u)^{-\frac{p+1}{2}} dx$ for some constant $t>{1}$.
\begin{lemma}\label{lemuvmt} Under the same assumptions as Lemma \ref{lemuvm}, set
\begin{eqnarray*}
&& X:=\int_{\Omega} (1-u)^{-\frac{p+1}{2}-t+1} (1-v)^{-\frac{p+1}{2}} dx, \ \ Y:=\int_{\Omega} (1-v)^{-\frac{p+1}{2}-t+1} (1-u)^{-\frac{p+1}{2}} dx,
\\&& Z:=\int_{\Omega} (1-u)^{-\frac{p+1}{2}}(1-v)^{-\frac{p+1}{2}} ,
\end{eqnarray*}
for $t>1$. Then, for some constant $0<\epsilon<1$ we get
\begin{eqnarray}\label{lgXmt}
\begin{array}{lcl}
\sqrt{\lambda\gamma} p(1-\epsilon) X \le \frac{(t-1)^2 }{4t} \lambda X^{\frac{2t-p-1}{2(t-1)}} Y^{\frac{p-1}{2(t-1)}} + C_{\epsilon,\lambda,\gamma,|\Omega|} Z, \\
\sqrt{\lambda\gamma} p(1-\epsilon) Y \le \frac{(t-1)^2 }{4t} \gamma Y^{\frac{2t-p-1}{2(t-1)}} X^{\frac{p-1}{2(t-1)}} + C_{\epsilon,\lambda,\gamma,|\Omega|} Z,
\end{array}
\end{eqnarray}
where $C_{\epsilon,\lambda,\gamma,|\Omega|} $ is a positive constant.
\end{lemma}
\begin{proof} Let $t>1$ be a constant. Multiply the second equation of $ (M)_{\lambda,\gamma}$ with $(1-u)^{-t}-1$ and integrate to get
\begin{equation*}
\lambda \int_{\Omega} [(1-u)^{-t}-1] (1-v)^p dx= \int_{\Omega} \mathcal L u [(1-u)^{-t}-1] dx.
\end{equation*}
From Lemma \ref{fgprop}, we get
\begin{equation*}
\int_{\Omega} \mathcal L u [(1-u)^{-t}-1] dx = \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n} \left[u(x)-u(z) \right] \left[(1-u(x))^{-t}-(1-u(z))^{-t} \right] J(x-z) dx dz .
\end{equation*}
Note that for $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}
[(1-\alpha)^{-t}-(1-\beta)^{-t}](\alpha-\beta) \ge \frac{4t}{(t-1)^2} \left| (1-\alpha)^{\frac{-t+1}{2}} - (1-\beta)^{\frac{-t+1}{2}} \right|^2 .
\end{equation*}
Applying the above inequality for $\alpha=u(x)$ and $\beta=u(z)$, we obtain
\begin{equation*}
[(1-u(x))^{-t}-(1-u(z))^{-t}][u(x)-u(z)] \ge \frac{4t}{(t-1)^2} \left| (1-u(x))^{\frac{-t+1}{2}} - (1-u(z))^{\frac{-t+1}{2}} \right|^2 .
\end{equation*}
From the above, we conclude
\begin{equation*}
\lambda \int_{\Omega} (1-u)^{-t} (1-v)^{-p} dx \ge \frac{4t}{(t-1)^2} \frac{1}{2} \int_{\mathbb R^n} \int_{\mathbb R^n}
\left| (1-u(x))^{\frac{-t+1}{2}} - (1-u(z))^{\frac{-t+1}{2}} \right|^2 J(x-z) dx dz .
\end{equation*}
Test the stability inequality, Corollary \ref{stablein}, on $\zeta=(1-u)^{\frac{-t+1}{2}}-1$ to get
\begin{eqnarray*}\label{}
&&\sqrt{\lambda\gamma} p \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} [(1-u)^{\frac{-t+1}{2}}-1]^2 dx
\\&\le& \frac{1}{2}\int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \left| (1-u(x))^{\frac{-t+1}{2}}- (1-u(z))^{\frac{-t+1}{2}} \right|^2 J(x-z) dz dx .
\end{eqnarray*}
Combining above inequalities, we conclude
\begin{equation}\label{rruv}
\sqrt{\lambda\gamma} p \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} [(1-u)^{\frac{-t+1}{2}}-1]^2 dx
\le \frac{(t-1)^2}{4t} \lambda \int_{\Omega} (1-u)^{-t} (1-v)^{-p} dx .
\end{equation}
Expanding the left-hand side of the inequality and rearranging we get
\begin{eqnarray}\label{l1e}
&&\sqrt{\lambda\gamma} p(1-\epsilon) \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} (1-u)^{-t+1} dx
\\&&\nonumber \le \frac{(t-1)^2}{4t} \lambda \int_{\Omega} (1-u)^{-t} (1-v)^{-p} dx
+ \frac{ \sqrt{\lambda\gamma} p}{\epsilon} \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} dx,
\end{eqnarray}
where we have used the inequality $a\le \frac{\epsilon}{2}a^2+\frac{1}{2\epsilon}$ for any $\epsilon>0$ and $a\in\mathbb R$. From the H\"{o}lder's inequality we get
\begin{eqnarray*}\label{}
&& \int_{\Omega} (1-u)^{-t} (1-v)^{-p} dx
\\& \le &\left[ \int_{\Omega} (1-u)^{-\frac{p+1}{2}} (1-v)^{-\frac{p+1}{2}} (1-u)^{-t+1} dx \right]^{\frac{1}{\beta}}
\left[ \int_{\Omega} (1-v)^{-\frac{p+1}{2}} (1-u)^{-\frac{p+1}{2}} (1-v)^{-t+1} dx \right]^{1-\frac{1}{\beta}} ,
\end{eqnarray*}
where $\beta=\frac{2(t-1)}{2t-p-1}$. This and (\ref{l1e}) completes the proof of the first estimate in (\ref{lgXet}). Similarly, one can show the second estimate.
\end{proof}
In regards to the gradient system with superlinear nonlinearities satisfying (\ref{R}) we establish an integral estimate that yields $L^2(\Omega)$ of the function $f'(u)g'(v)$. We then use this to conclude estimates on the nonlinearities of the gradient system. Our methods and ideas in the proof are inspired by the ones developed in \cite{cf} and originally by Nedev in \cite{Nedev}.
\begin{lemma}\label{lemab}
Suppose that $f$ and
$g$ both satisfy condition (\ref{R}) and $a:=f'(0)>0 $ and $ b:=g'(0)>0$. Assume that $ f',g'$ are convex and (\ref{deltaeps}) holds. Let $ (\lambda^*,\gamma^*) \in \Upsilon$ and $ (u,v)$ denote the extremal solution associated with $ (H)_{\lambda^*,\gamma^*}$.
Then, there exists a positive constant $ C < \infty$ such that
\begin{equation*}
\int_{\Omega} f'(u) g'(v) (f'(u)-a) (g'(v)-b) \le C.
\end{equation*}
\end{lemma}
\begin{proof} We obtain uniform estimates for any minimal solution $(u,v)$ of $(H)_{\lambda,\gamma}$ on the ray $ \Gamma_\sigma$ and then one sends $ \lambda \nearrow \lambda^*$ to obtain the same estimate for $ (u^*,v^*)$. Let $(u,v)$ denote a smooth minimal solution of $(H)_{\lambda,\gamma}$ on the ray $\Gamma_\sigma$ and put $ \zeta:= f'(u)-a$ and $ \eta:=g'(v)-b$ into (\ref{stabilityH}) to obtain
\begin{eqnarray*}\label{}
&&\int_{\Omega} \left[f''(u) g(v) (f'(u)-a)^2 + f(u) g''(v) (g'(v)-b)^2 + 2 f'(u) g'(v) (f'(u)-a)(g'(v)-b)\right]dx
\\ &\le&\nonumber
\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n}\left( \frac{1}{\lambda} |f'(u(x))- f'(u(z))|^2 + \frac{1}{\gamma} |g'(v(x))- g'(v(z))|^2 \right) J(x-z) dz dx .
\end{eqnarray*}
Note that for all $\alpha,\beta\in\mathbb R$, one can see that
\begin{equation*}\label{}
|f'(\beta)-f'(\alpha)|^2=\left|\int_\alpha^\beta f''(s) ds\right|^2 \le \int_\alpha^\beta |f''(s)|^2 ds (\beta-\alpha)= (h_1(\beta)-h_1(\alpha))(\beta-\alpha),
\end{equation*}
when $h_1(s):=\int_0^s |f''(w)|^2 dw$. Similar inequality holds for the function $g$
that is
\begin{equation*}\label{}
|g'(\beta)-g'(\alpha)|^2 \le (h_2(\beta)-h_2(\alpha))(\beta-\alpha),
\end{equation*}
when $h_2(s):=\int_0^s |g''(w)|^2 dw$. Set $\beta=u(x)$ and $\alpha=u(z)$ and $\beta=v(x)$ and $\alpha=v(z)$ in the above inequalities to conclude
\begin{eqnarray*}\label{}
&&\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n}\left( \frac{1}{\lambda} |f'(u(x))- f'(u(z))|^2 + \frac{1}{\gamma} |g'(v(x))- g'(v(z))|^2 \right) J(x-z) dz dx
\\&\le&
\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \frac{1}{\lambda} [h_1(u(x))- h_1(u(z))][u(x)- u(z)] J(x-z) dz dx
\\&&+
\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n} \frac{1}{\gamma} [h_2(v(x))- h_2(v(z))][v(x)- v(z)] J(x-z) dz dx.
\end{eqnarray*}
From the equation of system and Lemma \ref{fgprop}, we get
\begin{eqnarray*}\label{}
&&\frac{1}{2} \int_{{\mathbb{R}}^n} \int_{{\mathbb{R}}^n}\left( \frac{1}{\lambda} |f'(u(x)) - f'(u(z))|^2 + \frac{1}{\gamma} |g'(v(x))- g'(v(z))|^2 \right) J(x-z) dz dx
\\&\le&
\int_{\Omega} \left[ h_1(u) \frac{1}{\lambda} \mathcal L(u) + h_2(v) \frac{1}{\gamma} \mathcal L(v) \right]dx
\\&=&
\int_{\Omega} \left[ h_1(u)f'(u)g(v) dx + h_2(v)f(u)g'(v) \right]dx .
\end{eqnarray*}
From this and we conclude that
\begin{eqnarray}\label{stafg}
&&\int_{\Omega} \left[f''(u) g(v) (f'(u)-a)^2 + f(u) g''(v) (g'(v)-b)^2 + 2 f'(u) g'(v) (f'(u)-a)(g'(v)-b)\right]dx
\\ &\le&\nonumber
\int_{\Omega} \left[ h_1(u)f'(u)g(v) dx + h_2(v)f(u)g'(v) \right] dx .
\end{eqnarray}
Given the assumptions, there is some $ M>1$ large enough and $0<\delta<1$ that for all $ u \ge M$ we have
$ h_1(u) \le \delta f''(u)(f'(u)-a)$ for all $ u \ge M$. Then, we have
\begin{equation*}
\int_{\Omega} h_1(u) g(v) f'(u) = \int_{u \ge M} + \int_{u <M} \le \delta \int f''(u) g(v) (f'(u)-a)^2 + \int_{u<M} \int h_1(u) g(v) f'(u) .
\end{equation*}
We now estimate the last integral in the above. Let $ k \ge 1$ denote a natural number. Then,
\begin{equation*}
\int_{u<M} h_1(u) g(v) f'(u) = \int_{u<M, v<kM} + \int_{u<M, v \ge kM} = C(k,M) + \int_{u<M, v \ge kM}h_1(u) g(v) f'(u) .
\end{equation*}
Note that this integral is bounded above by
\begin{equation*}
\sup_{u<M} \frac{h_1(u)}{(f'(u)-a)} \sup_{v >kM} \frac{g(v)}{(g'(v)-b) g'(v)} \int (f'(u)-a) (g'(v)-b) f'(u) g'(v).
\end{equation*}
From the above estimates, we conclude that for sufficiently large $ M$ and for all $1 \le k$ there is some positive constant $ C(k,M)$ and $0<\delta <1$ that
\begin{eqnarray}\label{h1gf}
\ \ \ \ \ \int_{\Omega} h_1(u) g(v) f'(u) &\le & \delta \int f''(u) g(v) (f'(u)-a)^2
+ C(k,M)
\\&& \nonumber+ \sup_{u<M} \frac{h_1(u)}{(f'(u)-a)} \sup_{v >kM} \frac{g(v)}{(g'(v)-b) g'(v)} \int (f'(u)-a) (g'(v)-b) f'(u) g'(v).
\end{eqnarray}
Applying the same argument, one can show that for sufficiently large $M$ and for all $ 1 \le k$ there is some positive constant $ D(k,M)$ and $0<\epsilon <1$ that
\begin{eqnarray}\label{h2gf}
\ \ \ \ \ \int_{\Omega} h_2(v) g'(v) f(u) &\le & \epsilon \int f(u) g''(v) (g'(v)-b)^2
+ D(k,M)
\\&& \nonumber + \sup_{v<M} \frac{h_2(v)}{(g'(v)-b)} \sup_{u >kM} \frac{f(u)}{(f'(u)-a) f'(u)} \int (f'(u)-a) (g'(v)-b) f'(u) g'(v).
\end{eqnarray}
Note that $ f''(u),g''(v) \rightarrow \infty$ when $u,v\to\infty$. This implies that
\begin{equation*}
\lim_{k \rightarrow \infty} \sup_{u >kM} \frac{f(u)}{(f'(u)-a) f'(u)} =0 \ \ \text{and} \ \ \lim_{k \rightarrow \infty} \sup_{v >kM} \frac{g(v)}{(g'(v)-b) g'(v)} =0 .
\end{equation*}
Now set $ k$ to be sufficiently large and substitute (\ref{h1gf}) and (\ref{h2gf}) in (\ref{stafg}) to
complete the proof. Note that see that all the integrals in (\ref{stafg}) are bounded independent of $ \lambda$ and $\gamma$.
\end{proof}
\section{Regularity of the extremal solution; Proof of Theorem \ref{thmg}-\ref{nedev}}\label{secreg}
In this section, we apply the integral estimates established in the latter section to prove regularity results for extremal solutions of systems mentioned in the introduction earlier.
\\
\\
\noindent {\it Proof of Theorem \ref{thmg}}.
We shall provide the proof for the case of $n>2s$, since otherwise is straightforward. Let $(u,v)$ be the smooth minimal solution of $ (G)_{\lambda,\gamma}$ for $\frac{\lambda^*}{2}<\lambda<\lambda^*$ and $\frac{\gamma^*}{2}<\gamma<\gamma^*$. From Lemma \ref{lemuvg} we conclude that
\begin{equation}
\int_{\Omega} e^{u+v} dx \le C_{\lambda,\gamma,|\Omega|} .
\end{equation}
From this and Lemma \ref{lemuvgt}, we conclude that for $t>\frac{1}{2}$
\begin{equation}
\lambda\gamma \left[1-\left(\frac{t }{4}+\epsilon\right)^2\right] XY \le C_{\epsilon,\lambda,\gamma,|\Omega|}
\left(1+X^{\frac{2t-1}{2t}} Y^{\frac{1}{2t}} + Y^{\frac{2t-1}{2t}} X^{\frac{1}{2t}}\right) .
\end{equation}
Therefore, for every $t<4$ either $X$ or $Y$ must be
bounded where $X$ and $Y$ are given by
\begin{equation}
X:=\int_{\Omega} e^{\frac{2t+1}{2}u} e^{\frac{v}{2}} dx \ \ \text{and} \ \ Y:=\int_{\Omega} e^{\frac{2t+1}{2}v} e^{\frac{u}{2}} dx.
\end{equation}
Without loss of generality, assume that $\lambda\le \gamma$ implies that $u\le v$ and therefore $e^u$ is bounded in $L^{q}(\Omega)$ for $q=t+1<5$. Therefore, in light of Proposition \ref{propregL} we have $u\in L^\infty(\Omega)$ for $\frac
{n}{2s}<5$ that is $n<10s$.
Now, let $(u,v)$ be the smooth minimal solution of $ (E)_{\lambda,\gamma}$ for $\frac{\lambda^*}{2}<\lambda<\lambda^*$ and $\frac{\gamma^*}{2}<\gamma<\gamma^*$. From Lemma \ref{lemuve} we conclude that
\begin{equation}\label{}
\int_{\Omega} (1+u)^p(1+v)^p dx \le C_{\lambda,\gamma,|\Omega|} .
\end{equation}
From this and Lemma \ref{lemuvet}, we conclude that
\begin{equation}
\lambda\gamma \left[p^2(1-\epsilon)^2 -\left( \frac{(t+1)^2 }{4t} \right)^2\right] XY \le C_{\epsilon,\lambda,\gamma,|\Omega|}
\left(1+X^{\frac{2t-p+1}{2(t+1)}} Y^{\frac{p+1}{2(t+1)}} + Y^{\frac{2t-p+1}{2(t+1)}} X^{\frac{p+1}{2(t+1)}} \right) .
\end{equation}
Therefore, for every $1\le t< 2p+2\sqrt{p(p-1)}-1$ either $X$ or $Y$ must be
bounded where $X$ and $Y$ are given by
\begin{equation}
X:=\int_{\Omega} (1+u)^{\frac{p-1}{2}+t+1} (1+v)^{\frac{p-1}{2}} dx \ \ \text{and} \ \ Y:=\int_{\Omega} (1+v)^{\frac{p-1}{2}+t+1} (1+u)^{\frac{p-1}{2}} dx.
\end{equation}
Without loss of generality, assume that $\lambda\le \gamma$ implies that $u\le v$ and therefore $(1+u)$ is bounded in $L^{q}(\Omega)$ for $q=p+t$. We now rewrite the system $ (E)_{\lambda,\gamma}$ for the extremal solution $(u,v)$ that is
\begin{eqnarray*}
\left\{ \begin{array}{lcl}
\mathcal L u &=&\lambda^* c_{v}(x) v + \lambda^* \qquad \text{in} \ \ \Omega, \\
\mathcal L v &=& \gamma^* c_{u}(x) u +\gamma^* \qquad \text{in} \ \ \Omega, \\
\end{array}\right.
\end{eqnarray*}
when $0\le c_{v}(x)= \frac{(1+v)^{p}-1}{v} \le p v^{p-1}$ and $0\le c_{u}(x)= \frac{(1+u)^{p}-1}{u} \le p u^{p-1}$ where convexity argument is applied. From the regularity theory, Proposition \ref{propregL}, we conclude that $v\in L^\infty(\Omega)$ provided $ c_{u}(x)\in L^{r}(\Omega)$ when $r>\frac{n}{2s}$. This implies that $\frac{n}{2s} <\frac{p+t}{p-1}$ for $1\le t<2p+2\sqrt{p(p-1)}-1$. This completes the proof for the case of Lane-Emden system $ (E)_{\lambda,\gamma}$. The proof for the case of $ (M)_{\lambda,\gamma}$ is very similar and replies on applying Lemma \ref{lemuvm} and Lemma \ref{lemuvmt}.
$\Box$
\begin{remark}\label{rem1}
Even though the above theorem is optimal as $s\to 1$, it is not optimal for smaller values of $0<s<1$.
\end{remark}
In this regard, consider the case of $\lambda=\gamma$ and the Gelfand system turns into ${(-\Delta)}^s u= \lambda e^u$ in the entire space $\mathbb R^n$. It is known that the explicit singular solution $u^*(x)=\log \frac{1}{|x|^{2s}}$ is stable solution of the scalar Gelfand equation if and only if
\begin{equation*}
\frac{ \Gamma(\frac{n}{2}) \Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} \le
\frac{ \Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})} ,
\end{equation*}
for the constant
$$\lambda=2^{2s}\frac{ \Gamma(\frac{n}{2}) \Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} .
$$
This implies that the extremal solution of the fractional Gelfand equation should be regular for
\begin{equation*}
\frac{ \Gamma(\frac{n}{2}) \Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} >
\frac{ \Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})} .
\end{equation*}
In particular, the extremal solution should be bounded when $n\le 7$ for $0<s<1$. We refer interested to \cite{rs,r1} for more details. Now, consider the case of Lane-Emden equation $(-\Delta)^s u= \lambda u^p$ in the entire space $\mathbb R^n$. It is also known that the explicit singular solution $u_s(x) = A |x|^{-\frac{2s}{p-1}}$ where the constant $A$ is given by
\begin{equation*}
A^{p-1} =\frac{\Gamma(\frac{n}{2}-\frac{s}{p-1}) \Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1}) \Gamma(\frac{n-2s}{2}-\frac{s}{p-1})} ,
\end{equation*}
is a stable solution of the scalar Lane-Emden equation if and only if
\begin{equation*}\label{}
p \frac{\Gamma(\frac{n}{2}-\frac{s}{p-1}) \Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1}) \Gamma(\frac{n-2s}{2}-\frac{s}{p-1})}
\le
\frac{ \Gamma^2(\frac{n+2s}{4}) }{\Gamma^2(\frac{n-2s}{4})}.
\end{equation*}
This yields that the extremal solution of the above equation should be regular for
\begin{equation*}\label{}
p \frac{\Gamma(\frac{n}{2}-\frac{s}{p-1}) \Gamma(s+\frac{s}{p-1})}{\Gamma(\frac{s}{p-1}) \Gamma(\frac{n-2s}{2}-\frac{s}{p-1})}
>
\frac{ \Gamma^2(\frac{n+2s}{4}) }{\Gamma^2(\frac{n-2s}{4})}.
\end{equation*}
As $s\to 1$, the above inequality is consistent with the dimensions given in (\ref{dime}). For more information, we refer interested readers to Wei and the author in \cite{fw} and Davila et al. in \cite{ddw}. Given above, proof of the optimal dimension for regularity of extremal solutions remains an open problem.
We now provide a proof for Theorem \ref{nedev} that deals with a regularity result for the gradient system $ (H)_{\lambda,\gamma}$ with general nonlinearities $f$ and $g$. Note that for the case of local scalar equations such results are provided by Nedev in \cite{Nedev} for $n=3$ and Cabr\'{e} in \cite{Cabre} for $n=4$. For the case of scalar equation with the fractional Laplacian operator, Ros-Oton and Serra in \cite{rs} established regularity results for dimensions $n<4s$ when $0<s<1$. For the case of local gradient systems, that is when $s=1$, such a regularity result is established by the author and Cowan in \cite{cf} in dimensions $n\le 3$.
\\
\\
\noindent {\it Proof of Theorem \ref{nedev}}.
We suppose that $ (\lambda^*,\gamma^*) \in \Upsilon$ and $(u,v)$ is the associated extremal solution of $(G)_{\lambda^*,\gamma^*}$. Set $ \sigma=\frac{\gamma^*}{\lambda^*}$.
From Lemma \ref{lemab}, we conclude that $ f'(u) g'(v) \in L^2(\Omega)$. Note that this and the convexity of $ g$ show that
\begin{equation}
\int_\Omega \frac{f'(u)^2 g(v)^2}{(v+1)^2} \le C.
\end{equation}
Note that $ (-\Delta)^s u\in L^1$ and $ (-\Delta)^s v\in L^1$ and hence we have $ u,v \in L^{p}$ for any $ p <\frac{n}{n-2s}$. We now use the domain decomposition method as in \cite{Nedev,cf}. Set
\begin{eqnarray*}
\Omega_1 &:=& \left\{ x: \frac{f'(u)^2 g(v)^2}{(v+1)^2} \ge f'(u)^{2-\alpha} g(v)^{2-\alpha} \right\},\\
\Omega_2 &:=& \Omega \backslash \Omega_1 = \left\{ x : f'(u) g(v) \le (v+1)^\frac{2}{\alpha} \right\},
\end{eqnarray*}
where $ \alpha $ is a positive constant and will be fixed later. First note that
\begin{equation}
\int_{\Omega_1} (f'(u) g(v))^{2-\alpha} \le \int_\Omega \frac{f'(u)^2 g(v)^2}{(v+1)^2} \le C.
\end{equation}
Similarly we have
\begin{equation}
\int_{\Omega_2} (f'(u) g(v))^p \le \int_\Omega (v+1)^\frac{2p}{\alpha}.
\end{equation}
We shall consider the case of $n>2s$, and $n\le 2s$ is straightforward as discussed in Section \ref{secpre}. Taking $ \alpha= \frac{4(n-2s)}{3n-4s}$ and using the $ L^{p}$-bound on $ v$ for $p<\frac{n}{n-2s}$ shows that $ f'(u) g(v) \in L^{p}(\Omega)$ for $p<\frac{2n}{3n-4s}$. By a symmetry argument we also have $ f(u) g'(v) \in L^{p}(\Omega)$ for $p<\frac{2n}{3n-4s}$. Therefore, $(-\Delta )^s u,(-\Delta )^s v\in L^p(\Omega)$ when $p<\frac{2n}{3n-4s}$. From elliptic estimates we conclude that $u,v\in L^p(\Omega)$ when $p<\frac{2n}{3n-8s}$ for $n>\frac{8}{3}s$ and when $p<\infty$ for $n=\frac{8}{3}s$ and when $p\le \infty$ for $n<\frac{8}{3}s$. This completes the proof when $2s\le n<\frac{8}{3}s$.
\\
Now, set $ \alpha= \frac{3n-8s}{2(n-2s)}$. From the above estimate $u,v\in L^p(\Omega)$ when $p<\frac{2n}{3n-8s}$ for $n>\frac{8}{3}s$ on $v$ and domain decomposition arguments, we get $ f(u) g'(v), f'(u) g(v)\in L^{p}(\Omega)$ for $p<\frac{n}{2(n-2s)}$. Therefore, $(-\Delta )^s u,(-\Delta )^s v\in L^p(\Omega)$ with the latter bounds for $p$. From elliptic estimates we get
$u,v\in L^p(\Omega)$ when $p<\frac{n}{2(n-3s)}$ for $n>3s$ and when $p<\infty$ for $n=3s$ and when $p\le \infty$ for $n<3s$. We perform the above arguments once more to arrive at
$u,v\in L^p(\Omega)$ when $p<\frac{2n}{5n-16s}$ for $n>\frac{16}{5}s$ and when $p<\infty$ for $n=\frac{16}{5}s$ and when $p\le \infty$ for $n<\frac{16}{5}s$. This completes the proof for
$\frac{8}{3}s\le n < \frac{16}{5}s$ containing $n=3s$.
\\
Now, suppose that $u,v\in L^p(\Omega)$ for $p<p_*$. Then, notice that
\begin{equation*}
\int_{\Omega_2} (f'(u) g(v))^p \le \int_\Omega (v+1)^\frac{2p}{\alpha} \le C \ \ \text{when} \ \ p<\frac{\alpha p_*}{2}.
\end{equation*}
Set $ \alpha= \frac{4}{p_*+2}$. Then, from the above we conclude that $ f'(u) g(v) \in L^{p}(\Omega)$ for $p<\frac{2p_*}{p_*+2}$ and similarly $ f(u) g'(v) \in L^{p}(\Omega)$ for $p<\frac{2p_*}{p_*+2}$. Applying the fact that $(-\Delta)^s u,(-\Delta)^s v\in L^p(\Omega)$ for the same range of $p$. From elliptic estimates we conclude that
\begin{equation*}
u,v\in L^{p}(\Omega) \ \ \text{when} \ \ p<\frac{2p_*n}{p_*( n-4s) +2n} .
\end{equation*}
Applying the above elliptic estimates arguments, we conclude the boundedness of solutions when $n<4s$.
$\Box$
We end this section with power polynomial nonlinearities for the gradient system $ (H)_{\lambda^*,\gamma^*}$ and we provide regularity of the extremal solution. For the case of local systems, that is when $s=1$, a similar result is given in \cite{cf}. Due to the technicality of the proof we omit it here.
\begin{thm} \label{grade} Let $ f(u)=(1+u)^p $ and $ g(v)=(1+v)^q$ when $ p,q>2$. Assume that $(\lambda^*,\gamma^*) \in \Upsilon$. Then, the associated extremal solution of $ (H)_{\lambda^*,\gamma^*}$ is bounded provided
\begin{equation} \label{gradp}
n < 2s + \frac{4s}{p+q-2} \max\{ T(p-1), T(q-1)\} ,
\end{equation}
when $ T(t):= t+ \sqrt{t(t-1)}$.
\end{thm}
\vspace*{.4 cm }
\noindent {\it Acknowledgment}. The author would like to thank Professor Xavier Ros-Oton for online communications and comments in regards to Section \ref{secpre}. The author is grateful to Professor Xavier Cabr\'{e} for bringing reference \cite{sp} to his attention and for the comments in regards to Section \ref{secin}.
\end{document} |
\begin{document}
\title{On the group structure of $[\Omega \mathbb S^2, \Omega Y]$}
\author{Marek Golasi\'nski}
\address{Institute of Mathematics, Casimir the Great University,
pl.\ Weyssenhoffa 11, 85-072 Bydgoszcz, Poland}
\email{marek@ukw.edu.pl}
\author{Daciberg Gon\c calves}
\address{Dept. de Matem\'atica - IME - USP, Caixa Postal 66.281 - CEP 05314-970,
S\~ao Paulo - SP, Brasil}
\email{dlgoncal@ime.usp.br}
\author{Peter Wong}
\address{Department of Mathematics, Bates College, Lewiston, ME 04240, U.S.A.}
\email{pwong@bates.edu}
\thanks{}
\begin{abstract} Let $J(X)$ denote the James
construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),\Omega Y]\cong \lim\limits_{\leftarrow}
[J_n(X),\Omega Y]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=\Omega {\mathbb S}igma \mathbb S^1=\Omega \mathbb S^2$. Furthermore,
there is a bijection between $[J(\mathbb S^1),\Omega Y]$ and the product $\prod_{i=2}^\infty \pi_i(Y)$, as sets.
In this paper, we describe the group structure of $[J_n(\mathbb S^1),\Omega Y]$ by
determining the co-multiplication structure on the suspension ${\mathbb S}igma J_n(\mathbb S^1)$.
\end{abstract}
\date {\today}
\keywords{Cohen groups, Fox torus homotopy groups, James construction, Whitehead products}
\subjclass[2010]{Primary: 55Q05, 55Q15, 55Q20; secondary: 55P35}
\maketitle
\section*{Introduction}\setcounter{section}{0}
Groups of homotopy classes of maps $[\Omega {\mathbb S}igma X, \Omega Y]$ from the loop space of the suspension ${\mathbb S}igma X$ to the loop space of $Y$ play an important role in classical homotopy theory. These groups have been used to give functorial homotopy decompositions of loop suspensions via modular representation theory and to investigate the intricate relationship between Hopf invariants and looped Whitehead products. In the special case when $X={\mathbb S}^1$ and $Y={\mathbb S}^2$, the authors in \cite{cohen,cohen-sato,cohen-wu} explored the relationship between $[\Omega {\mathbb S}^2, \Omega {\mathbb S}^2]$ and the Artin's pure braid groups via Milnor's
free group construction $F[K]$ (see \cite{M2}) for a simplicial set $K$. For $K={\mathbb S}^1$, the geometric realization of $F[{\mathbb S}^1]$ has the homotopy type of $\Omega {\mathbb S}igma {\mathbb S}^1=J({\mathbb S}^1)$, the James construction on ${\mathbb S}^1$. In general, the James construction $J(X)$ on $X$ admits a filtration $J_1(X) \subseteq J_2(X) \subseteq\cdots$ so that
$\displaystyle{[J(X),\Omega Y]\cong\lim_{\leftarrow} [J_n(X),\Omega Y]}$. The {\it Cohen groups} $[J_n(X),\Omega Y]$ and the {\it total Cohen group} $[J(X),\Omega Y]$ have been studied in \cite{wu2}
via the simplicial group $\{[X^n, \Omega Y]\}_{n\ge 1}$. In his original paper \cite{J}, James introduced $J(X)$ as a model for $\Omega{\mathbb S}igma X$, the loop
space of the suspension ${\mathbb S}igma X$ of the space $X$ and showed that ${\mathbb S}igma J(X)$ has the homotopy type of the suspension of the wedge of self smash products of $X$. This shows that $[\Omega {\mathbb S}^2, \Omega Y]=[J({\mathbb S}^1),\Omega Y]$, {\it as a set}, is in one-to-one correspondence with the direct product $\prod_{i= 2}^\infty \pi_i(Y)$ of the higher homotopy groups of $Y$. However, the group structure of $[J({\mathbb S}^1),\Omega Y]$ is far from being abelian.
In \cite{cohen-sato}, the group $[J_n({\mathbb S}^1),\Omega Y]$ is shown to be a central extension with kernel $\pi_{n+1}(Y)$ and quotient
$[J_{n-1}({\mathbb S}^1),\Omega Y]$. In \cite{ggw5}, it is shown that $\pi_{n+1}(Y)$ is in fact central in a larger group $\tauu_{n+1}(Y)$,
the Fox torus homotopy group. The proof in \cite{ggw5} relies on embedding $[J_n({\mathbb S}^1),\Omega Y]$ into $\tauu_{n+1}(Y)$. Fox \cite{fox}
introduced the torus homotopy groups such that the Whitehead products, when embedded as elements of a torus homotopy group, become commutators.
Indeed, the Fox torus homotopy group $\tauu_n(Y)$ is completely determined by the homotopy groups $\pi_i(Y)$ for $1\le i\le n$ and the Whitehead products.
Furthermore, Fox determined whether $\alpha \in \pi_{k+1}(Y), \beta\in \pi_{l+1}(Y)$, when embedded in
$\tauu_n(Y)$ commute. Following \cite{fox}, we consider a $k$-subset ${\bf a}$ and an $l$-subset ${\bf b}$ of the set of indices $\{1,2,\ldots, n\}$
for some $n\ge k+l$. The sets ${\bf a}$ and ${\bf b}$ determine two embeddings $\pi_{i+1}(Y) \to \tauu_{n+1}(Y)$ for $i=k,l$. Denote by $\alpha^{{\bf a}}$ and $\beta^{{\bf b}}$ the corresponding images of $\alpha$ and $\beta$ in $\tauu_{n+1}(Y)$.
\begin{proM}
\begin{enumerate}
\item[(1)] If ${\bf a} \cap {\bf b}=\emptyset$ then $(\alpha^{{\bf a}},\beta^{{\bf b}})=(-1)^{w+(|{\bf a}|-1)}[\alpha, \beta]^{{\bf a}\cup {\bf b}}$.\\
\item[(2)] If ${\bf a} \cap {\bf b}\ne \emptyset$ then $(\alpha^{{\bf a}},\beta^{{\bf b}})=1$.
\end{enumerate}
Here $(x,y)$ denotes the commutator $xyx^{-1}y^{-1}$ of $x$ and $y$ and $w={\mathbb S}igma_{i\in {\bf a}, j\in {\bf b}} w_{i,j}$, where $w_{i,j}=1$ if $j<i$ and $w_{i,j}=0$ otherwise.
\end{proM}
Since the groups $[J_n({\mathbb S}^1),\Omega Y]$ can be embedded as subgroups of the Fox torus homotopy groups $\tauu_{n+1}(Y)$, the group structure of
$[J_n({\mathbb S}^1),\Omega Y]$ is induced by that of $\tauu_{n+1}(Y)$. In this paper, we obtain the group structure of $[J_n(\mathbb S^1),\Omega Y]$ via the torus homotopy group studied in \cite{ggw4} together with a recent result by Arkowitz and Lee \cite{AL2} on the co-$H$-structures of a wedge of spheres. The following is our main theorem.
\begin{thmM} The suspension co-$H$-structure $$\overline{\mu}_n : {\mathbb S}igma J_n(\mathbb S^1)
\to {\mathbb S}igma J_n(\mathbb S^1) \vee {\mathbb S}igma J_n(\mathbb S^1)$$ for $n\ge 1$ is given by $\overline{\mu}_n k_i\simeq\iota_1k_i+\iota_2k_i+P_i$,
where the perturbation $P_i\simeq\sum_{l=0}^i\varphii(i-l,i-1)P_{l,i}$ with the Fox function $\varphii$ and
$$P_{l,i}: \mathbb S^{i+1}\to \mathbb S^{i-l+1}\vee \mathbb S^{l+1}\stackrel{k_{i-l}\vee k_l}{\hookrightarrow}{\mathbb S}igma J_n(\mathbb S^1)\vee {\mathbb S}igma J_n(\mathbb S^1)$$
determined by the Whitehead product map $\mathbb S^{i+1}\to \mathbb S^{i-l+1}\vee \mathbb S^{l+1}$ for $i=0,\ldots,n$ and $l=0,\ldots,i$.
\end{thmM}
Here the perturbations $P_i$ are used in \cite{AL2} to measure the deviation of the co-$H$-structure of a space of homotopy type of a wedge of spheres from the usual coproduct of the suspension co-$H$-structures of spheres. The function $\varphii$ derived from Proposition \ref{gen-Fox-W-product} is used to determine the coefficients of Whitehead products.
This paper is organized as follows. In Section 1, we recall the Fox torus homotopy groups $\tauu_{n+1}(Y)$ and give examples of $[J_n({\mathbb S}^1),\Omega Y]$,
where the group structure can be obtained via $\tauu_{n+1}(Y)$. In Section $2$, we obtain a closed form solution to a recurrence relation on the function
$\varphii$ which gives the coefficient of the Whitehead product in Proposition \ref{gen-Fox-W-product}. Section $3$ combines the result of \cite{AL2}
and the result of Section $2$ to obtain the co-$H$-structure of ${\mathbb S}igma J_n({\mathbb S}^1)$. In Section 4, we revisit and generalize the examples of Section $1$.
In particular, we give necessary and sufficient conditions (see Proposition \ref{3_stem}) for $[J_{4n+1}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ to be abelian when $n$
is not a power of $2$. We also investigate when two torsion elements in $[J({\mathbb S}^1),\Omega Y]$ commute. To end
this introduction, we want to point out that the Fox torus homotopy groups $\tauu_{n+1}({\rm Conf}(n))$ of the
configuration space of $n$ distinct points in $\mathbb R^3$ were used by F.\ Cohen et al.\ \cite{cohen-et-al} to
give an alternative proof of a result of U.\ Koschorke, which is related to Milnor's link homotopy \cite{M} and
homotopy string links examined by Habegger-Lin \cite{H-L}, that the $\kappa$-invariant for the Brunnian links
in $\mathbb R^3$ is injective.
We thank the anonymous referee for his/her careful reading of the earlier version of the manuscript as well as helpful comments that lead to a better exposition of the paper.
\section{Fox torus homotopy groups}
In this section, we make some calculation on $[J_n({\mathbb S}^1),\Omega Y]$ using the Fox torus homotopy groups. First, we recall from \cite{fox} the definition of the $n$-th Fox torus homotopy group of
a connected pointed space $Y$, for $n\ge 1$. Let $y_0$ be a basepoint of $Y$, then
$$
\tauu_n(Y)\cong \tauu_n(Y,y_0)=\pi_1(Y^{{\mathbb T}^{n-1}},\overline{y_0}),
$$
where $Y^{{\mathbb T}^{n-1}}$ denotes the space of unbased maps from the $(n-1)$-torus ${\mathbb T}^{n-1}$ to $Y$ and
$\overline{y_0}$ is the constant map at $y_0$. When $n=1$, $\tauu_1(Y)=\pi_1(Y)$.
To re-interpret Fox's result, we showed in \cite{ggw1} that
$$
\tauu_n(Y)\cong [F_n({\mathbb S}^1),Y]
$$
the group of homotopy classes of basepoint preserving maps from the
reduced suspension $F_n({\mathbb S}^1):={\mathbb S}igma ({\mathbb T}^{n-1}\sqcup *)$ of ${\mathbb T}^{n-1}$ adjoined with a distinguished point
to $Y$.
\addtocounter{theorem}{1}
One of the main results of \cite{fox} is the following split exact sequence:
\begin{equation}\label{fox-split}
0\to \prod_{i=2}^n \pi_i(Y)^{\sigma_i} \to \tauu_n(Y) \stackrel{\dashleftarrow}{\to} \tauu_{n-1}(Y) \to 0,
\end{equation}
where $\sigma_i=\binom{n-2}{i-2}$, the binomial coefficient.
With the isomorphism $\tauu_{n-1}(\Omega Y)\cong \prod_{i=2}^n \pi_i(Y)^{\sigma_i}$ shown in \cite[Theorem 1.1]{ggw1},
the sequence \eqref{fox-split} becomes
\begin{equation}\label{general-fox-split}
0\to \tauu_{n-1}(\Omega Y) \to \tauu_n(Y) \stackrel{\dashleftarrow}{\to} \tauu_{n-1}(Y) \to 0.
\end{equation}
Here, the projection $\tauu_n(Y) \to \tauu_{n-1}(Y)$ is induced by the suspension of the inclusion
${\mathbb T}^{n-2}\sqcup * \hookrightarrow {\mathbb T}^{n-1}\sqcup *$ given
by $(t_1,\ldots,t_{n-2})\mapsto (1,t_1,\ldots,t_{n-2})$ and the
section $\tauu_{n-1}(Y) \to \tauu_n(Y)$ is the homomorphism induced by the suspension of the
projection ${\mathbb T}^{n-1} \sqcup * \to {\mathbb T}^{n-2}\sqcup *$ given by $(t_1,\ldots,t_{n-1}) \mapsto (t_2,\ldots,t_{n-1})$. This
splitting (section) gives the semi-direct product structure so that the action
$$\bullet : \tauu_{n-1}(Y)\times \tauu_{n-1}(\Omega Y)\longrightarrow\tauu_{n-1}(\Omega Y)$$
of the quotient $\tauu_{n-1}(Y)$ on the kernel $\tauu_{n-1}(\Omega Y)$ is simply conjugation
in $\tauu_n(Y)$ by the image of $\tauu_{n-1}(Y)$ under the section. It follows from the work
of Fox (in particular Proposition \ref{gen-Fox-W-product}) that the action is determined by the
Whitehead products. More precisely, given $\alpha \in \pi_i(Y), \beta \in \pi_j(Y)$, let
$\hat \alpha$ and $\hat \beta$ be the respective images of $\alpha$ in $\tauu_{n-1}(Y)$ and
of $\beta$ in $\tauu_{n-1}(\Omega Y)$. Then
$$\hat \alpha \bullet \hat \beta =\widehat {[\alpha,\beta]} \hat \beta,$$
where $\widehat{[\alpha,\beta]}$ denotes the image in $\tauu_n(Y)$ of the Whitehead product $[\alpha,\beta]$.
Note that $F_n({\mathbb S}^1)\simeq ({\mathbb S}igma {\mathbb T}^{n-1}) \vee {\mathbb S}^1$ but
$[F_n({\mathbb S}^1),Y]\cong [{\mathbb S}igma {\mathbb T}^{n-1},Y] \rtimes \pi_1(Y)$.
\begin{example}\label{ex1}
The group $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ is abelian (in fact isomorphic to $\mathbb Z \oplus \mathbb Z$, see Example \ref{ex1-revisit}). We now examine the multiplication in $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ by embedding this inside the group $\tauu_3(\mathbb S^2)$. First, we note that $\tauu_3(\mathbb S^2) \cong (\pi_3(\mathbb S^2) \oplus \pi_2(\mathbb S^2)_{\{1\}}) \rtimes \pi_2(\mathbb S^2)_{\{2\}}$. Here we follow the work of \cite{fox} by using the indexing set $\{1,2\}$ for the two copies of $\pi_2(\mathbb S^2)$ in $\tauu_3(\mathbb S^2)$. Let $\alpha, \beta \in \pi_2(\mathbb S^2)$ and we write $\alpha^{\{i\}}, \beta^{\{i\}} \in \pi_2(\mathbb S^2)_{\{i\}}$ for $i=1,2$. Using the semi-direct product structure and the representation of elements of $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ in $\tauu_3(\mathbb S^2)$, the product of two elements of $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ is given by
\begin{equation*}
\begin{aligned}
&(\alpha^{\{1\}},\alpha^{\{2\}})\cdot (\beta^{\{1\}}, \beta^{\{2\}})\\
=&(\alpha^{\{1\}}+(\alpha^{\{2\}}\bullet \beta^{\{1\}}), \alpha^{\{2\}}+\beta^{\{2\}})\\
=&(\alpha^{\{1\}}+\beta^{\{1\}}+(-1)^{w+(m-1)}[\alpha, \beta]^{\{1,2\}}, \alpha^{\{2\}}+\beta^{\{2\}}) \quad \text{(here $m=1,w=1$)} \\
=&(\alpha^{\{1\}}+\beta^{\{1\}}+(-1)[\alpha, \beta]^{\{1,2\}}, \alpha^{\{2\}}+\beta^{\{2\}}).
\end{aligned}
\end{equation*}
Conversely,
\begin{equation*}
\begin{aligned}
&(\beta^{\{1\}}, \beta^{\{2\}})\cdot (\alpha^{\{1\}},\alpha^{\{2\}})\\
=&(\beta^{\{1\}}+\alpha^{\{1\}}+(-1)^{w+(m-1)}[\beta,\alpha]^{\{1,2\}}, \beta^{\{2\}}+\alpha^{\{2\}}) \quad \text{here $m=1$} \\
=&(\alpha^{\{1\}}+\beta^{\{1\}}+(-1)[\beta,\alpha]^{\{1,2\}}, \alpha^{\{2\}}+\beta^{\{2\}}).
\end{aligned}
\end{equation*}
Since both $\alpha$ and $\beta$ have dimension $2$, the Whitehead product $[\alpha,\beta]$ coincides with $[\beta, \alpha]$.
Thus the above calculation shows that $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ is abelian.
\end{example}
However, the groups $[J_n(X),\Omega Y]$ are non-abelian in general.
\begin{example}\label{ex2}
The group $[J_3(\mathbb S^1), \Omega Y]$ is non-abelian for $Y=\mathbb S^2 \vee \mathbb S^3$. First, the Fox homotopy group $\tauu_4(Y)\cong \tauu_3(\Omega Y) \rtimes \tauu_3(Y)$. Since $Y$ is 1-connected, using the Fox short split exact sequence \eqref{general-fox-split},
we obtain
\begin{equation}
\begin{aligned}
\tauu_4(Y) &\cong \left( \pi_4(Y) \oplus 2\pi_3(Y) \oplus \pi_2(Y)\right)\rtimes \left((\pi_3(Y)\oplus \pi_2(Y))\rtimes \pi_2(Y)\right).
\end{aligned}
\end{equation}
Let $\iota_1:\mathbb S^2 \hookrightarrow \mathbb S^2 \vee \mathbb S^3$ and $\iota_2:\mathbb S^3 \hookrightarrow \mathbb S^2 \vee \mathbb S^3$ denote the canonical inclusions. The (basic) Whitehead product $[\iota_1,\iota_2]$ is a non-trivial element in $\pi_4(\mathbb S^2 \vee \mathbb S^3)$. When regarded as an element of the group $\tauu_4(Y)$, the image of $[\iota_1,\iota_2]$ in $\tauu_4(Y)$ is the commutator $(\hat a, \hat b)$, where
\begin{equation}
\begin{aligned}
\hat a&=\left( (1\oplus (1\oplus 1) \oplus a), ((1\oplus a),a)\right) \\
\hat b&=\left( (1\oplus (b\oplus b) \oplus 1), ((b\oplus 1),1)\right)
\end{aligned}
\end{equation}
with $a=[\iota_1]\in \pi_2(Y)$ and $b=[\iota_2]\in \pi_3(Y)$. By \cite[Theorem 2.2]{ggw5}, $\hat a, \hat b$ are
in the image of $[J_3(\mathbb S^1),\Omega Y]\hookrightarrow\tauu_4(Y)$. Since the commutator $(\hat a, \hat b)$
is non-trivial, it follows that $[J_3(\mathbb S^1),\Omega Y]$ is non-abelian.
\end{example}
Now, we analyze the central extension
\begin{equation}\label{cohen-sequence}
0 \to \pi_{n+1}(Y) \to [J_n(\mathbb S^1),\Omega Y] \to [J_{n-1}(\mathbb S^1),\Omega Y] \to 0
\end{equation}
and give an example in which this extension does not split.
The calculation below makes use of torus homotopy groups.
\begin{example}\label{ex3}
Take $Y=\mathbb S^4$. Since ${\mathbb S}^4$ is $3$-connected, we have
$[J_1(\mathbb S^1),\Omega Y]=[J_2(\mathbb S^1),\Omega Y]=0$ and
$[J_3(\mathbb S^1),\Omega \mathbb S^4]\cong \pi_4(\mathbb S^4)\cong \mathbb Z$.
For $n=4$, the sequence
\eqref{cohen-sequence} becomes
$$0\to \pi_5(\mathbb S^4) \to [J_4(\mathbb S^1),\Omega \mathbb S^4] \to [J_3(\mathbb S^1),\Omega \mathbb S^4] \to 0.
$$
Note that the corresponding Fox split exact sequence is
$$
0\to \pi_5(\mathbb S^4)\oplus 3 \pi_4(\mathbb S^4) \to \tauu_5(\mathbb S^4) \stackrel{\dashleftarrow}{\to} \tauu_4(\mathbb S^4) \to 0
$$
and $\tauu_4(\mathbb S^4)\cong \pi_4(\mathbb S^4)$. For dimensional reasons, $\tauu_5(\mathbb S^4)$ does not contain any non-trivial
Whitehead products so that $\pi_5(\mathbb S^4)\oplus 3 \pi_4(\mathbb S^4)$ is central in $\tauu_5(\mathbb S^4)$.
Thus, $\tauu_5(\mathbb S^4)\cong \pi_5(\mathbb S^4)\oplus 4 \pi_4(\mathbb S^4)$ is abelian. It follows
that $$[J_4(\mathbb S^1),\Omega \mathbb S^4]\cong \pi_5({\mathbb S}^ 4)\oplus\pi_4({\mathbb S}^4)$$ is abelian.
For $n=5$, the corresponding Fox split exact sequence is
$$
0\to \pi_6(\mathbb S^4)\oplus 4 \pi_5(\mathbb S^4) \oplus 6 \pi_4(\mathbb S^4) \to \tauu_6(\mathbb S^4) \stackrel{\dashleftarrow}{\to} \tauu_5(\mathbb S^4) \to 0.$$
Again, for dimensional reasons, $\tauu_6(\mathbb S^4)$ contains no non-trivial Whitehead products so that
$$[J_5({\mathbb S}^1),\Omega {\mathbb S}^4]\cong \pi_6({\mathbb S}^4) \oplus \pi_5({\mathbb S}^4) \oplus \pi_4({\mathbb S}^4).$$ It follows that $[J_5(\mathbb S^1),\Omega\mathbb S^4]$ is
abelian and the sequence \eqref{cohen-sequence} splits for $n=5$. To see this, we note that $[J_5(\mathbb S^1),\Omega\mathbb S^4]$ is of rank $1$ and
so it is either $\mathbb Z \oplus \mathbb Z_2 \oplus \mathbb Z_2$ or $\mathbb Z \oplus \mathbb Z_4$, where $\mathbb{Z}_n$ is the
cyclic group of order $n$. Since $\tauu_6(\mathbb S^4)$ has no elements of order $4$ so we have $[J_5(\mathbb S^1),\Omega\mathbb S^4] \cong \mathbb Z \oplus \mathbb Z_2 \oplus \mathbb Z_2$.
When $n=6$, the sequence \eqref{cohen-sequence} becomes
$$
0\to \pi_7(\mathbb S^4) \to [J_6(\mathbb S^1),\Omega \mathbb S^4] \to [J_5(\mathbb S^1),\Omega \mathbb S^4] \to 0.
$$
By projecting $[J_6(\mathbb S^1),\Omega \mathbb S^4]$ onto $[J_3(\mathbb S^1),\Omega \mathbb S^4]$, the above sequence
gives rise to the following exact sequence
\begin{equation}\label{alt-exact}
0\to \mathcal W \to [J_6(\mathbb S^1),\Omega \mathbb S^4] \to [J_3(\mathbb S^1),\Omega \mathbb S^4]\cong \pi_4(\mathbb S^4)\cong \mathbb Z \to 0
\end{equation}
so that $[J_6(\mathbb S^1),\Omega \mathbb S^4] \cong \mathcal W \rtimes \pi_4(\mathbb S^4)$. Similarly, the corresponding Fox
split exact sequence can be written as
$$
0\to \widehat {\mathcal W} \to \tauu_7({\mathbb S}^4) \to \tauu_4(\mathbb S^4)=\pi_4(\mathbb S^4) \to 0.
$$
Here, $\mathcal W$ is generated by elements of $\pi_i(\mathbb S^4)$ for $i=5,6,7$ while $\widehat {\mathcal W}$ is generated by elements of $\pi_i(\mathbb S^4)$ for $i=4,5,6,7$. It follows
that the action of $\pi_4(\mathbb S^4)$ on $\mathcal W$ is the same as the action of $\pi_4(\mathbb S^4)=\tauu_4(\mathbb S^4)$ on
$\widehat {\mathcal W}$ and is determined by the Whitehead products. For dimensional reasons, if $x\in \mathcal W$, the Whitehead
product of $x$ with any element in $\pi_4(\mathbb S^4)$ will be zero in $\tauu_7(\mathbb S^4)$ and hence $\pi_4(\mathbb S^4)$
acts trivially on $\mathcal W$. Thus,
$$
[J_6(\mathbb S^1),\Omega \mathbb S^4] \cong \mathcal W \times \pi_4(\mathbb S^4)\cong \bigoplus_{i=4}^7 \pi_i(\mathbb S^4)
$$
while $\tauu_7(\mathbb S^4)$ is non-abelian. Moreover, the central extension
$$
0\to \pi_7(\mathbb S^4) \to [J_6(\mathbb S^1),\Omega \mathbb S^4] \to [J_5(\mathbb S^1),\Omega \mathbb S^4] \to 0
$$
splits because $\mathcal W\subset \pi_5(\mathbb S^4) \oplus \pi_6(\mathbb S^4) \subset \pi_4(\mathbb S^4) \oplus \pi_5(\mathbb S^4) \oplus \pi_6(\mathbb S^4) \cong [J_5(\mathbb S^1),\Omega \mathbb S^4]$.
Next, consider the case when $n=7$. Then, the sequence \eqref{cohen-sequence} becomes
$$
0\to \pi_8(\mathbb S^4) \to [J_7(\mathbb S^1),\Omega \mathbb S^4] \to [J_6(\mathbb S^1),\Omega \mathbb S^4] \to 0.
$$
The corresponding Fox split exact sequence is
$$
0\to \pi_8({\mathbb S}^4) \oplus 6 \pi_7({\mathbb S}^4) \oplus 15 \pi_6({\mathbb S}^4)\oplus
20 \pi_5({\mathbb S}^4) \oplus 15 \pi_4({\mathbb S}^4) \to \tauu_8(\mathbb S^4) \stackrel{\dashleftarrow}{\to} \tauu_7(\mathbb S^4) \to 0.
$$
Again, for dimensional reasons, the only non-trivial Whitehead products lie in $\pi_8({\mathbb S}^4)$ between the elements in $\pi_4({\mathbb S}^4)$ and
$\pi_5({\mathbb S}^4)$. Let $\iota_1, \iota_2$ be the generators of the cyclic groups $\pi_4(\mathbb S^4)\cong \mathbb Z$ and
$\pi_5(\mathbb S^4)\cong \mathbb Z_2$, respectively. Since there are $15$ copies of $\pi_5(\mathbb S^4)$ and $20$ copies of $\pi_4({\mathbb S}^4)$ in
$\tauu_7({\mathbb S}^4)$, there are a total of $35$ copies of $\pi_4({\mathbb S}^4)$ and $35$ copies of $\pi_5({\mathbb S}^4)$ in $\tauu_8({\mathbb S}^4)$. According to \cite{fox},
the Whitehead products are determined by embedding $\pi_n({\mathbb S}^4)$ in $\tauu_r({\mathbb S}^4)$ using $\binom{r-1}{n-1}$ embeddings. With $r=8$ and
$n=4,5$, there are $(35)^2$ possible pairings $(\iota_1',\iota_2')$, where $\iota_i'$ corresponds to the image of $\iota_i$
under one of $35$ embeddings.
\par Now, by the result of \cite{fox}, once an embedding for $\pi_4({\mathbb S}^4)$ is chosen, there is a unique
embedding of $\pi_5({\mathbb S}^4)$ so that non-trivial Whitehead products can be formed. Thus, there are exactly $35$ such products (commutators)
each of which is the generator of $\pi_8({\mathbb S}^4)\cong \mathbb Z_2$. The product of these $35$ commutators
is non-trivial since $35\not\equiv\, 0\,(\bmod\, 2)$. If $\tilde \iota_i$ is a preimage of $\iota_i$ in
$[J_7(\mathbb S^1), \Omega \mathbb S^4]$ for $i=1,2$, the commutator $[\tilde \iota_1, \tilde \iota_2]$
is independent of choice of the preimages since $\pi_8(\mathbb S^4)$ is central in $[J_7(\mathbb S^1), \Omega \mathbb S^4]$.
This commutator is non-trivial in $[J_7(\mathbb S^1), \Omega \mathbb S^4]$ and this shows that the projection
$[J_7(\mathbb S^1), \Omega \mathbb S^4] \to [J_6(\mathbb S^1), \Omega \mathbb S^4]$ cannot have a section.
\par We point out that such non-trivial Whitehead products will persist and induce non-trivial commutators in
$[J_k(\mathbb S^1), \Omega \mathbb S^4]$ for any $k>7$. Thus, we conclude that $[J_k(\mathbb S^1), \Omega \mathbb S^4]$ is non-abelian for any $k\ge7$.
\end{example}
\section{The Fox number and the function $\varphii$}
The Fox torus homotopy group $\tauu_n(Y)$ is completely determined by the homotopy groups $\pi_i(Y)$ for $1\le i\le n$ and the Whitehead products.
Furthermore, Fox determined whether $\alpha \in \pi_{k+1}(Y), \beta\in \pi_{l+1}(Y)$, when embedded in
$\tauu_n(Y)$ commute. Following \cite{fox}, we consider a $k$-subset ${\bf a}$ and an $l$-subset ${\bf b}$ of the set of indices $\{1,2,\ldots, n\}$
for some $n\ge k+l$. The sets ${\bf a}$ and ${\bf b}$ determine two embeddings $\pi_{i+1}(Y) \to \tauu_{n+1}(Y)$ for $i=k,l$. Denote by $\alpha^{{\bf a}}$ and $\beta^{{\bf b}}$ the corresponding images of $\alpha$ and $\beta$ in $\tauu_{n+1}(Y)$.
\begin{proposition}\label{gen-Fox-W-product}{\em
\begin{enumerate}
\item[(1)] If ${\bf a} \cap {\bf b}=\emptyset$ then $(\alpha^{{\bf a}},\beta^{{\bf b}})=(-1)^{w+(|{\bf a}|-1)}[\alpha, \beta]^{{\bf a}\cup {\bf b}}$.\\
\item[(2)] If ${\bf a} \cap {\bf b}\ne \emptyset$ then $(\alpha^{{\bf a}},\beta^{{\bf b}})=1$.
\end{enumerate}
Here $(x,y)$ denotes the commutator $xyx^{-1}y^{-1}$ of $x$ and $y$ and $w={\mathbb S}igma_{i\in {\bf a}, j\in {\bf b}} w_{i,j}$, where $w_{i,j}=1$ if $j<i$ and $w_{i,j}=0$ otherwise.}
\end{proposition}
The number $(-1)^{w+(|{\bf a}|-1)}$ as in Proposition \ref{gen-Fox-W-product}, which is crucial in determining the structure of the torus homotopy groups, depends on the parameters $l$ and $k$ and we shall call this the {\it Fox number} of the partition $\{{\bf a},{\bf b}\}$.
For any integers $l,k$ with $k>l>0$, let
$$
\varphii(l,k)=\sum_{{\bf a}, |{\bf a}|=l} (-1)^{w+(|{\bf a}|-1)}.
$$
Moreover, for all $k\ge 0$, we let $\varphii(k,k)=(-1)^k$ and $\varphii(0,k)=1$. We shall call $\varphii$ the {\it Fox function}.
In order to compute $\varphii(l,k)$ in terms of a simple algebraic expression, we state our main lemma.
\begin{lemma}\label{main-lemma} The function $\varphii$ satisfies the following recurrence relations.
For $k>l>0$,
$$\varphii(l,k)=(-1)^{k-l+1}\varphii(l-1,k-1)+\varphii(l,k-1)$$
and
$$\varphii(k,k)=-\varphii(k-1,k-1).$$
\end{lemma}
\begin{proof} The formula for $l=k$ follows from the definition of $\varphii(k , k)$.
Now assume that $k>l>0$. Let $L$ denote the family of all subsets of $\{ 1,2,\ldots,k\}$
with cardinality $l$. Divide this family into two subfamilies, $L_1$ and $L_2$. A subset
belongs to $L_1$ if it contains $k$, otherwise it belongs to $L_2$. Recall that the Fox number for one partition is given by $(-1)^{l-1+w}$. Summing of the Fox numbers over all partitions $\{{\bf a},{\bf b}\}$ with ${\bf a}$ belonging to $L_2$ amounts to
computing the Fox function for the pair $(l,k-1)$ since the element
$k$ does not play a role in the calculation because $k$ is the last element of the set of $k$ elements and it does not belong to any subset of $l$ elements. Thus we conclude that the Fox function restricted to those partitions, where ${\bf a}$ belongs to $L_2$ coincides with $\varphii(l,k-1)$.
For the sum of the Fox numbers over all partitions with ${\bf a}$ belonging to $L_1$, since $k$ belongs to the subset ${\bf a}$ the number $w$ contains a summand $k-l$ independent of the subset in $L_1$ since $\sum_j w_{k,j}=\sum_{j\in {\bf b}}1=k-l$. Then the remaining part of $w$ is obtained by considering subsets of $l-1$ elements in a set of cardinality $k-1$. So this coincides with
the calculation of $\varphii(l-1,k-1)$ except that the computation for $\varphii(l-1,k-1)$ uses subsets of length
$l-1$ and the one for elements of $L_1$ uses subsets of length $l$. Thus we conclude that the Fox number restricted
to those partitions with ${\bf a}$ belonging to $L_1$ coincides with $ (-1)^{k-l+1}\varphii(l-1,k-1)$ and the result follows.
\end{proof}
Our function $\varphii$ by definition satisfies $\varphii(0,k)=1$ and certainly satisfies the equality $\varphii(0,2k+1)=\varphii(0,2k)$. It is not difficult to see from
the definition of $\varphii$ that $\varphii(1,2k)=0$ and
$\varphii(1,2k+1)=-\varphii(0,2k)=-1$.
The following three propositions give the basic properties in order to compute $\varphii$.
\begin{proposition}\label{odd-even} {\em \mbox{\em (1)} For $l$ odd and $k>l/2$ we have $$\varphii(l,2k)=0.$$
\mbox{\em (2)} For $l$ even and $k\geq l/2$ we have $$\varphii(l,2k)=\varphii(l,2k+1).$$
\mbox{\em (3)} For $l$ even and $k\geq l/2$ we have $$ \varphii(l+1,2k+1)=-\varphii(l,2k+1).$$}
\end{proposition}
\begin{proof} The proof is by induction. We say that
the inductive hypothesis holds for an integer $m$ if (1) holds for all $l\leq m$ with $l$ odd, and (2) and (3) hold for all $l\leq m$ with $l$ even.
First, we show that the inductive hypothesis holds for $m=1$. Part (2) follows from the definition of $\varphii$ where both sides of the equation are $1$.
For the parts ($1$) and ($3$) we use the following equations
\begin{equation}\label{formulaI}
\varphii(1,2k+1)=(-1)^{2k+1}\varphii(0,2k)+\varphii(1,2k)
\end{equation}
and
\begin{equation}\label{formulaII}
\varphii(1,2k)=(-1)^{2k}\varphii(0,2k-1)+\varphii(1,2k-1)
\end{equation}
obtained from Lemma \ref{main-lemma}.
\par By induction on $k$, we prove simultaneously that $\varphii(1,2k)=0$,
and $\varphii(1,2k-1)=0-1=-1$, where the latter equality is equivalent to part (3) for $l=0$.
By definition $\varphii(1,1)=-1$ and by formula \eqref{formulaII} above
$\varphii(1,2)=(-1)^{2}\varphii(0,1)+\varphii(1,1)=1-1=0$. So the result holds for $k=1$.
Suppose that $\varphii(1,2r)=0$, $\varphii(1,2r-1)=-1$, for $r\leq k$.
Next we prove that $\varphii(1,2k+2)=0$ and $\varphii(1,2k+1)=-1$. By the inductive hypothesis and equation \eqref{formulaI} it follows that
$\varphii(1,2k+1)=(-1)^{2k+1}\varphii(0,2k)+\varphii(1,2k)=-1+0=-1$. Now the inductive hypothesis and formula \eqref{formulaII} yield
$\varphii(1,2k+2)=(-1)^{2k+2}\varphii(0,2k+1)+\varphii(1,2k+1)=1-1=0$ and the result follows.
Now assume the assertions for parts (1) - (3) hold for $m=2s+1$. Then, we show that the result holds for $m=2s+3$.
The proof is similar to the arguments above. First, we have
$$\varphii(2s+2,2k+1)=\varphii(2s+1,2k)+\varphii(2s+2,2k)=\varphii(2s+2,2k)$$
where the first equality follows from Lemma \ref{main-lemma} and the second equality holds by inductive hypothesis about part (1). So part (2) follows.
For parts ($1$) and ($3$), we use the following equations
\begin{equation}\label{formulaIII}
\varphii(2s+3,2k+1)=-\varphii(2s+2,2k)+\varphii(2s+3,2k),
\end{equation}
\begin{equation}\label{formulaIV}
\varphii(2s+3,2k)=\varphii(2s+2,2k-1)+\varphii(2s+3,2k-1)
\end{equation} and
\begin{equation}\label{formulaV}
\varphii(2s+2,2k+1)=\varphii(2s+1,2k)+\varphii(2s+2,2k)
\end{equation}
obtained from Lemma \ref{main-lemma}.
\par By induction on $k$, we prove simultaneously that $\varphii(2s+3,2k)=0$,
and $\varphii(2s+3,2k-1)=-\varphii(2s+2,2k-1)$. We have $k\geq s+2$ and take
$k=s+2$.
By definition of $\varphii$, we have
$\varphii(2s+3,2s+3)=-\varphii(2s+2,2s+2)$ and by equation \eqref{formulaV}, $\varphii(2s+2,2s+3)=\varphii(2s+1,2s+2)+\varphii(2s+2,2s+2)=\varphii(2s+2,2s+2)$ where the last equality
follows from the inductive hypothesis. Therefore $\varphii(2s+3,2s+3)=-\varphii(2s+2,2s+3)$ and (3) follows. The following equation holds
$$\varphii(2s+3,2s+4)=\varphii(2s+2,2s+3)+\varphii(2s+3,2s+3)=-\varphii(2s+3,2s+3)+\varphii(2s+3,2s+3)=0,$$
where the first equality follows from equation \eqref{formulaIV}, and the second equality
follows from part (3). So the result holds for part (1).
Now, suppose that the statement holds for $k$ and
let us prove for $k+1$.
From equation \eqref{formulaIII} we have
$\varphii(2s+3,2k+1)=-\varphii(2s+2,2k)+\varphii(2s+3,2k)$. Since $\varphii(2s+3,2k)=0$ by inductive hypothesis,
and $\varphii(2s+2,2k)=\varphii(2s+2, 2k+1)$ it follows that (3) holds. It remains to show that $\varphii(2s+3,2k+2)=0$ in order for (1) to hold.
From \eqref{formulaIV} we have $\varphii(2s+3,2k+2)=\varphii(2s+2,2k+1)+\varphii(2s+3,2k+1)$, and from (3), $\varphii(2s+3,2k+1)-\varphii(2s+2,2k+1)$, so it follows that
$\varphii(2s+3,2k+2)=0$.
Therefore (1) and (3) hold for $k+1$ and this concludes the proof.
\end{proof}
\begin{proposition}\label{rec} {\em The function $\varphii$ satisfies the recursive formula
\begin{equation}\label{ggw-formula}
\varphii(2l,2k)=\varphii(2l ,2k-2)+\varphii(2l -2, 2k-2).
\end{equation}}
\end{proposition}
\begin{proof} From Lemma \ref{main-lemma}, we have
$$\varphii(2l,2k)=(-1)^{2k-2l+1}\varphii(2l-1,2k-1)+\varphii(2l,2k-1)=(-1)\varphii(2l-1,2k-1)+\varphii(2l,2k-1).$$
From Proposition \ref{odd-even}, we have
$$(-1)\varphii(2l-1,2k-1)=\varphii(2l-2,2k-2)$$
and
$$\varphii(2l,2k-1)=\varphii(2l,2k-2).$$
Hence, the result follows.
\end{proof}
Next, we give a simple expression for $\varphii(l,k)$ when $l$ and $k$ are even.
\begin{proposition}\label{Pascal}{\em
For any $l,k\ge 1$,
$$
\varphii(2l,2k)=-\binom{k}{l}.
$$}
\end{proposition}
\begin{proof}
Let $c(n,k)$ be a function of two integer variables such that
\begin{equation}\label{2-var}
c(n+1,k)=c(n,k)+c(n,k-1)
\end{equation}
for $n,k\ge 1$. Suppose that $c(n,0)=a_n$ and $c(1,k)=b_k$, where $\{a_n\}_{n\ge 1}$ and $\{b_k\}_{k\ge 1}$ are two arbitrary sequences. Then H. Gupta \cite{gupta} showed that
\begin{equation}\label{gupta}
c(n,k)=\sum_{r=k}^{n-1} \binom{r-1}{k-1} a_{n-r} + \sum_{r=0}^{k-1} \binom{n-1}{r} b_{k-r}.
\end{equation}
Now, if we let $c(n,k)=\varphii(2k,2n)$ then \eqref{ggw-formula} shows that \eqref{2-var} holds. Moreover, $a_n=c(n,0)=\varphii(0,2n)=-1$ for all $n\ge 0$
and $b_k=c(1,k)=\varphii(2k,2)$. It follows that $b_1=\varphii(2,2)=-1$ and $b_k=0$ for all $k\ge 2$. Now, the formula \eqref{gupta} becomes
\begin{equation*}
\varphii(2l , 2k)=\sum_{r=l}^{k-1} \binom{r-1}{l-1} a_{k-r} + \sum_{r=0}^{l-1} \binom{k-1}{r} b_{l-r}.
\end{equation*}
Since $b_k=0$ for $k\ge 2$, it follows that
\begin{equation}\label{ggw-recurrence}
\begin{aligned}
\varphii(2l , 2k)&=\sum_{r=l}^{k-1} \binom{r-1}{l-1} a_{k-r} + \binom{k-1}{l -1} b_{1} \\
&=\sum_{r=l}^{k-1} \binom{r-1}{l-1} (-1) + \binom{k-1}{l -1} (-1) \\
&=-\sum_{r=l}^{k} \binom{r-1}{l-1}.
\end{aligned}
\end{equation}
The following equality
\begin{equation}\label{pascal}
\binom{0}{k} + \binom{1}{k} + \cdots + \binom{n}{k}=\binom{n+1}{k+1}
\end{equation}
can be derived using the Pascal triangle and the general form of the binomial coefficient $\binom{n}{k}$.
Now,
\begin{equation*}
\begin{aligned}
\sum_{r=l}^{k} \binom{r-1}{l-1}&=\binom{l -1}{l -1} + \cdots +\binom{k-1}{l -1}\\
&=\left[\binom{0}{l -1} + \cdots +\binom{l -2}{l -1} + \binom{l -1}{l -1} + \cdots +\binom{k-1}{l -1}\right] - \left[\binom{0}{l -1} + \cdots +\binom{l -2}{l -1}\right] \\
&=\binom{k}{l} -\binom{l -1}{l} \quad \text{by \eqref{pascal}} \\
&=\binom{k}{l} -0 = \binom{k}{l}.
\end{aligned}
\end{equation*}
Hence, we have
\begin{equation}\label{simple-phi}
\varphii(2l, 2k)=-\binom{k}{l}
\end{equation}
and the proof is complete.
\end{proof}
Lemma \ref{main-lemma} and all propositions in this section yield its main result.
\begin{theorem}\label{main-phi}
For any integers $l, k$ with $1\le l \le k$, we have
$$
\varphii(l,k) \quad = \quad
\left\{
\aligned
& -\binom{\frac{k}{2}}{\frac{l}{2}}, \qquad & \text{if $l$ is even and $k$ is even;} \\
& 0, \qquad & \text{if $l$ is odd and $k$ is even;} \\
& \binom{\frac{k-1}{2}}{\frac{l-1}{2}}, \qquad & \text{if $l$ is odd and $k$ is odd;} \\
& -\binom{\frac{k-1}{2}}{\frac{l}{2}}, \qquad & \text{if $l$ is even and $k$ is odd.}
\endaligned
\right.
$$
\end{theorem}
\section{Group structure of $[J_n(\mathbb S^1),\Omega Y]$}
The group structure of $[J_n({\mathbb S}^1),\Omega Y]$ is determined by the suspension co-$H$-structure on ${\mathbb S}igma J_n({\mathbb S}^1)$.
But, in view of \cite{ggw5}, the suspension $p_n : F_{n+1}({\mathbb S}^1)\to {\mathbb S}igma J_n({\mathbb S}^1)$ of the projection map ${\mathbb T}^n\sqcup *\to J_n({\mathbb S}^1)$
leads to a monomorphism of groups $$[{\mathbb S}igma J_n({\mathbb S}^1),Y] \hookrightarrow [F_{n+1}({\mathbb S}^1),Y]=\tauu_{n+1}(Y)$$ for any pointed space $Y$.
Thus, the group structure of $[{\mathbb S}igma J_n({\mathbb S}^1),Y]$ is detrmined by the multiplication of $[F_{n+1}({\mathbb S}^1),Y]$.
\par To relate those structures, first notice that the cofibration
$${\mathbb S}^1\stackrel{j}{\to} \mathcal P_n({\mathbb S}^1):={\mathbb T}^n/{\mathbb T}^{n-1}\stackrel{q}{\to} {\mathbb S}^1\wedge {\mathbb T}^{n-1}$$
has a retraction $p : \mathcal P_n({\mathbb S}^1)\to {\mathbb S}^1$.
Hence, the map $${\mathbb S}igma p+{\mathbb S}igma q :{\mathbb S}igma \mathcal P_n({\mathbb S}^1)\stackrel{\simeq}{\longrightarrow} {\mathbb S}igma {\mathbb S}^1\vee {\mathbb S}^1\wedge {\mathbb S}igma{\mathbb T}^{n-1}$$
is a homotopy equivalence for $n\ge 1$. Because ${\mathbb S}igma(X_1\times X_2)\simeq {\mathbb S}igma X_1\vee {\mathbb S}igma X_2\vee {\mathbb S}igma(X_1\wedge X_2)$
for any pointed spaces $X_1$ and $X_2$, by an inductive argument, we derive:
\begin{equation}\label{eq1}
\begin{array}{l}
{\mathbb S}igma \mathcal P_n({\mathbb S}^1)\simeq \bigvee_{k=1}^n\binom{n-1}{k-1}{\mathbb S}^{k+1},\\
F_n({\mathbb S}^1)\simeq {\mathbb S}igma{\mathbb T}^{n-1}\vee {\mathbb S}^1 \simeq\bigvee_{k=0}^{n-1}\binom{n-1}{k}{\mathbb S}^{k+1}
\end{array}
\end{equation}
for $n\ge 1$.
Further, recall from \cite{J} that
\begin{equation}\label{eq2}
{\mathbb S}igma J_n({\mathbb S}^1)\simeq\bigvee_{k=1}^n{\mathbb S}^{k+1}
\end{equation}
for $n\ge 1$ and notice that (up to homotopy equivalences above) the suspension map $$p_n : F_{n+1}({\mathbb S}^1)\longrightarrow {\mathbb S}igma J_n({\mathbb S}^1)$$
restricts to ${p_n}|_{{\mathbb S}^1}=\ast$ and $p_n|_{{\mathbb S}^{k+1}} : {\mathbb S}^{k+1} \to {\mathbb S}igma J_n({\mathbb S}^1)$ to the inclusion map for $k=1,\ldots,n$.
\par In view of \cite[Theorem 3.1]{ggw1}, the suspension co-$H$-structure $$\hat{\mu}_n : F_n({\mathbb S}^1)\to F_n({\mathbb S}^1)\vee F_n({\mathbb S}^1)$$ on $F_n({\mathbb S}^1)$ leads to
the following split exact sequence
$$
1\to [{\mathbb S}igma {\mathcal P}_{n-1}({\mathbb S}^1),Y] \to [F_n({\mathbb S}^1),Y]\stackrel{\dashleftarrow}{\to} [F_{n-1}({\mathbb S}^1),Y] \to 1
$$
for any pointed space $Y$.
Hence, $[F_n({\mathbb S}^1),Y]\cong [{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y]\rtimes[F_{n-1}({\mathbb S}^1),Y]$ is
the semi-direct product with respect to the natural action
$$[F_{n-1}({\mathbb S}^1),Y]\times[{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y]\to[{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y].$$
In particular, for $Y=F_{n-1}({\mathbb S}^1)\vee{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1)$, by the natural bijection
$[F_{n-1}({\mathbb S}^1),Y]\times[{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y]\cong[F_{n-1}({\mathbb S}^1)\vee{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y]$,
the identity map $\mbox{id}_Y$ is sent to the corresponding co-action
$$\alpha_{n-1} : {\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1)\to F_{n-1}({\mathbb S}^1)\vee {\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1).$$
Furthermore, the natural bijection $[{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1),Y]\times[F_{n-1}({\mathbb S}^1),Y]\cong
[F_n({\mathbb S}^1),Y]$ for any pointed space $Y$ yields a homotopy equivalence
$$F_n({\mathbb S}^1)\simeq F_{n-1}({\mathbb S}^1)\vee{\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1).$$
\par By means of \cite[Theorem 2.2]{ggw4}, the suspension co-$H$-structure
$$\hat{\mu}_n : F_n({\mathbb S}^1)\longrightarrow F_n({\mathbb S}^1)\vee F_n({\mathbb S}^1)$$
is described inductively and determined by
$$\hat{\mu}^1_n : F_{n-1}({\mathbb S}^1)\stackrel{\hat{\mu}_{n-1}}{\to}F_{n-1}({\mathbb S}^1)\vee F_{n-1}({\mathbb S}^1)\hookrightarrow F_n({\mathbb S}^1)\vee F_n({\mathbb S}^1)$$
and a map
$$\hat{\mu}_n^2: {\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1)\to F_n({\mathbb S}^1)\vee F_n({\mathbb S}^1)$$
defined via the co-action $\alpha_{n-1} : {\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1)\to F_{n-1}({\mathbb S}^1)\vee {\mathbb S}igma \mathcal P_{n-1}({\mathbb S}^1)$.
\par Similarly to $F_{n+1}({\mathbb S}^1)$, the suspension co-$H$-structure
$$\overline{\mu}_n: {\mathbb S}igma J_n({\mathbb S}^1)\to {\mathbb S}igma J_n({\mathbb S}^1) \vee {\mathbb S}igma J_n({\mathbb S}^1)$$
is also described inductively.
Because ${\mathbb S}igma J_1({\mathbb S}^1)=F_2({\mathbb S}^1)$, the co-$H$-structure
$\overline{\mu}_1=\hat{\mu}_2 : {\mathbb S}igma J_1({\mathbb S}^1)\to {\mathbb S}igma J_1({\mathbb S}^1)\vee {\mathbb S}igma J_1({\mathbb S}^1)$.
Given the co-$H$-structure $\overline \mu_n : {\mathbb S}igma J_n({\mathbb S}^1)
\to {\mathbb S}igma J_n({\mathbb S}^1) \vee {\mathbb S}igma J_n({\mathbb S}^1)$
write ${\mathbb S}igma J_{n+1}({\mathbb S}^1) \simeq {\mathbb S}igma J_n({\mathbb S}^1) \vee {\mathbb S}^{n+2}$ and
$F_{n+2}({\mathbb S}^1) \simeq F_{n+1}({\mathbb S}^1)
\vee {\mathbb S}igma \mathcal P_{n+1}({\mathbb S}^1)$. Then, one can easily verify that the composite maps
$$\overline{\mu}_{n+1}^1 :
{\mathbb S}igma J_n({\mathbb S}^1) \stackrel{\overline
{\mu}_n}{\longrightarrow} {\mathbb S}igma J_n({\mathbb S}^1) \vee
{\mathbb S}igma J_n({\mathbb S}^1) \hookrightarrow {\mathbb S}igma J_{n+1}({\mathbb S}^1) \vee
{\mathbb S}igma J_{n+1}({\mathbb S}^1)$$
and $$\overline{\mu}_{n+1}^2 :
{\mathbb S}^{n+2} \hookrightarrow {\mathbb S}igma \mathcal P_{n+1}({\mathbb S}^1)
\stackrel{\hat{\mu}_{n+2}^2}{\longrightarrow}F_{n+2}({\mathbb S}^1)\vee F_{n+2}({\mathbb S}^1)\stackrel{p_{n+1}\vee
p_{n+1}}{\longrightarrow} {\mathbb S}igma J_{n+1}({\mathbb S}^1) \vee {\mathbb S}igma J_{n+1}({\mathbb S}^1)$$
lead to the suspension co-$H$-structure $$\overline \mu_{n+1}=\overline{\mu}_{n+1}^1\vee\overline{\mu}_{n+1}^2 : {\mathbb S}igma J_{n+1}({\mathbb S}^1)
\longrightarrow {\mathbb S}igma J_{n+1}({\mathbb S}^1) \vee {\mathbb S}igma J_{n+1}({\mathbb S}^1)$$
on the space ${\mathbb S}igma J_{n+1}({\mathbb S}^1)\simeq{\mathbb S}igma J_n({\mathbb S}^1)\vee {\mathbb S}^{n+2}$.
\par To analyze the suspension co-$H$-structure on ${\mathbb S}igma J_n(\mathbb S^1)$, we recall the recent work \cite{AL2} on co-$H$-structures
on a wedge of spheres ${\mathbb S}=\bigvee_{i=1}^t\mathbb{S}^{n_i}$. Write
$k_i :\mathbb{S}^{n_i}\hookrightarrow {\mathbb S}$ for the inclusion maps with $i=1,\ldots,t$.
Further, set $\iota_j :{\mathbb S}\hookrightarrow {\mathbb S}\vee {\mathbb S}$ for the inclusion, and
$p_j : {\mathbb S}\vee {\mathbb S}\to {\mathbb S}$ for the projection maps with $j=1,2$. In \cite{AL2}, Arkowitz and Lee have proved the following result.
\begin{proposition}\mbox{$($\cite[Lemma 3.3]{AL2}$)$}\label{n-spheres}{\em
Let $\varphi:{\mathbb S}\to {\mathbb S}\vee {\mathbb S}$ be a co-action. Then, $\varphi$ is a co-$H$-structure on ${\mathbb S}$
if and only if $\varphi k_i=\iota_1k_i+\iota_2k_i+P_i$, where
$P_i : \mathbb{S}^{n_i}\to {\mathbb S}\vee {\mathbb S}$ has the property $p_1P_i=0=p_2P_i$ for $i=1,\ldots,t$.}
\end{proposition}
Adapting Proposition \ref{n-spheres} to our setting, we prove the first of our main results.
\begin{theorem} \label{DMP} The suspension co-$H$-structure $$\overline{\mu}_n : {\mathbb S}igma J_n(\mathbb S^1)
\to {\mathbb S}igma J_n(\mathbb S^1) \vee {\mathbb S}igma J_n(\mathbb S^1)$$ for $n\ge 1$ is given by $\overline{\mu}_n k_i\simeq\iota_1k_i+\iota_2k_i+P_i$, where the perturbation $P_i\simeq\sum_{l=0}^i\varphii(i-l,i-1)P_{l,i}$ with the Fox function $\varphii$ and
$$P_{l,i}: \mathbb S^{i+1}\to \mathbb S^{i-l+1}\vee \mathbb S^{l+1}\stackrel{k_{i-l}\vee k_l}{\hookrightarrow}{\mathbb S}igma J_n(\mathbb S^1)\vee {\mathbb S}igma J_n(\mathbb S^1)$$
determined by the Whitehead product map $\mathbb S^{i+1}\to \mathbb S^{i-l+1}\vee \mathbb S^{l+1}$ for $i=0,\ldots,n$ and $l=0,\ldots,i$.
\end{theorem}
\begin{proof} We proceed inductively on $n\ge 1$. Since the space
$J_n(\mathbb S^1)$ is a $CW$-complex, by the Cellular Approximation Theorem,
$\overline{\mu}_n\simeq \overline{\mu}'_n : {\mathbb S}igma J_n(\mathbb S^1)
\to {\mathbb S}igma J_n(\mathbb S^1) \vee {\mathbb S}igma J_n(\mathbb S^1)$, where $\overline{\mu}'_n$ is cellular.
Hence, from now on, we may assume that $\overline{\mu}_n$ is a cellular map.
\par If $n=1$ then $P_0=0$ and, by definition, $\bar{\mu}_1k_0=\iota_1k_0+\iota_2k_0$.
\par Because $J_n(\mathbb S^1)\subseteq J_{n+1}(\mathbb S^1)$, for dimensional reasons,
we derive that the restriction $\overline{\mu}_{n+1}|_{J_n(\mathbb S^1)}=\overline{\mu}_n$.
Consequently, we must analyze the map $\overline{\mu}_{n+1} k_{n+1}: \mathbb S^{n+2}\to
{\mathbb S}igma J_{n+1}(\mathbb S^1)\vee {\mathbb S}igma J_{n+1}(\mathbb S^1)$ determined by the composition
$$\overline{\mu}_{n+1}^2 :
\mathbb S^{n+2} \hookrightarrow {\mathbb S}igma \mathcal P_{n+1}(\mathbb S^1)
\stackrel{\hat{\mu}_{n+2}^2}{\longrightarrow} F_{n+2}({\mathbb S}^1)\vee F_{n+1}({\mathbb S}^1)\stackrel{p_{n+1}\vee
p_{n+1}}{\longrightarrow} {\mathbb S}igma J_{n+1}(\mathbb S^1)\vee {\mathbb S}igma J_{n+1}(\mathbb S^1).
$$
But, in view of the decompositions (\ref{eq1}) and (\ref{eq2}), we derive that
$$p_{n+1}|_{{\mathbb S}igma \mathcal P_{n+1}(\mathbb S^1)} : {\mathbb S}igma \mathcal P_{n+1}(\mathbb S^1)\simeq \bigvee_{k=1}^{n+1}\binom{n}{k-1}\mathbb S^{k+1}
\to \bigvee _{k=1}^{n+1}\mathbb S^{k+1}$$
restricts to the inclusion map ${\mathbb S}^{k+1}\hookrightarrow \bigvee _{k=1}^{n+1}\mathbb S^{k+1}$ for $k=1,\ldots,n+1$.
\par Because $\hat{\mu}_{n+2}^2$ is defined via the co-action $\alpha_{n+1}: {\mathbb S}igma \mathcal P_{n+1}(\mathbb S^1) \to F_{n+1}(\mathbb S^1) \vee {\mathbb S}igma \mathcal P_{n+1}(\mathbb S^1)$,
granting the monomorphism $[{\mathbb S}igma J_n({\mathbb S}^1),Y] \to [F_{n+1}({\mathbb S}^1),Y]=\tauu_{n+1}(Y)$ analysed in \cite[Theorem 2.2]{ggw5}, we have
to include all possible ways of contributing the same Whitehead product and thus the sum of all those $(-1)^{w+(|{\bf a}|-1)}$
given by Proposition \ref{gen-Fox-W-product}.
This leads to $\overline{\mu}_{n+1}k_{n+1}-\iota_1k_{n+1}-\iota_2k_{n+1}\simeq P_{n+1}\simeq\sum_{i=0}^{n+1}\varphii(n+1-i,n) P_{n+1,i}$
with $P_{n+1,i}: \mathbb S^{n+2}\to \mathbb S^{n+2-i}\vee \mathbb S^{i+1}\hookrightarrow J_{n+1}(\mathbb S^1)\vee J_{n+1}(\mathbb S^1)$
determined by the Whitehead product map $\mathbb S^{n+2}\to \mathbb S^{n+2-i}\vee \mathbb S^{i+1}$
and the proof is complete.
\end{proof}
When $Y$ is a $W$-space, that is, all Whitehead products vanish, it has
been shown in \cite[Corollary 3.6]{ggw5} that $[J({\mathbb S}^1),\Omega Y]$ is isomorphic to the direct product of $\prod_{i= 2}^\infty\pi_i(Y)$
so that $[J({\mathbb S}^1),\Omega Y]$ is abelian, in particular. As an easy consequence of Theorem \ref{DMP}, we have
the following corollary.
\begin{corollary}\label{abelian-cohen-groups}{\em
Let $Y$ be a path connected pointed space such that the Whitehead products
$[f,g]: \mathbb S^{k+l+1}\to Y$ for $f: \mathbb S^{k+1}\to Y$
and $g : \mathbb S^{l+1}\to Y$ vanish with $k,l$ even. Then,
the groups $[J_n(\mathbb S^1),\Omega Y]$ for $n\ge 1$ and $[J(\mathbb S^1),\Omega Y]$
are abelian provided $Y$ is a path connected pointed
space with $\pi_i(Y)=0$ for $i$ odd.}
\end{corollary}
Next, we make use of the function $\varphii$ of Section 2 to determine whether $\alpha \in \pi_{n+1}(Y)$ and $\beta \in \pi_{m+1}(Y)$ commute in the group $[J(\mathbb S^1),\Omega Y]$.
\begin{theorem}\label{abelian}
Let $\alpha\in \pi_{n+1}(Y), \beta\in \pi_{m+1}(Y)$. Suppose the Whitehead product $[\alpha,\beta]\ne 0$ and has order $k$. Then:
\begin{itemize}
\item if $k=\infty$ then the product $\alpha \#\beta \in [J(\mathbb S^1),\Omega Y]$ coincides with $\beta \#\alpha$ iff both $n$ and $m$ are odd; \\
\item if $k<\infty$ then the product $\alpha \#\beta \in [J(\mathbb S^1),\Omega Y]$ coincides with $\beta \#\alpha$ iff both $n$ and $m$ are odd or
\[
k~\text{divides} \begin{cases}
\binom{\frac{n+m-1}{2}}{\frac{m}{2}} & \text{if $n$ is odd and $m$ is even;} \\
\binom{\frac{n+m-1}{2}}{\frac{n}{2}} & \text{if $n$ is even and $m$ is odd;} \\
\binom{\frac{n+m}{2}}{\frac{n}{2}} & \text{if $n$ and $m$ is both even.}
\end{cases}
\]
\end{itemize}
\end{theorem}
\begin{proof}
First of all, if there are non-trivial perturbations $P_{l,i}$ determined by the Whitehead product $[\alpha,\beta]$ then $l=m$ and $i-l=n$. In other words, $P_{l,i}=P_{m,n+m}$. It follows that $P_{n+m}=\varphii(n,n+m-1)P_{m,n+m}$. Now, we have
$$
\alpha \# \beta =\alpha + \beta + \varphii(n,n+m-1)[\alpha,\beta] \qquad \text{and} \qquad \beta \# \alpha =\alpha + \beta + \varphii(m,n+m-1)[\beta,\alpha].
$$
Since $[\beta,\alpha]=(-1)^{(n+1)(m+1)}[\alpha,\beta]$, it suffices to compare $\varphii(n,n+m-1)$ with $\varphii(m,n+m-1)(-1)^{(n+1)(m+1)}$.
Let $\Delta=|\varphii(n,n+m-1)-\varphii(m,n+m-1)(-1)^{(n+1)(m+1)}|$. Depending on the parity of $n$ and $m$, we have the following table:
\begin{center}
\begin{tabular}{|c|c|c|c|c|l|}
\hline
$n$ & $m$ & $\varphii(n,n+m-1)$ & $\varphii(m,n+m-1)(-1)^{(n+1)(m+1)}$ & $\Delta$ \\ \hline
{odd} & {odd} & $\binom{\frac{n+m-2}{2}}{\frac{n-1}{2}}$ & $\binom{\frac{n+m-2}{2}}{\frac{m-1}{2}}$ & $0$\\ \hline
{odd} & {even} & $0$ & $-\binom{\frac{n+m-1}{2}}{\frac{m}{2}}$ & $\binom{\frac{n+m-1}{2}}{\frac{m}{2}}$\\ \hline
{even} & {odd} & $-\binom{\frac{n+m-1}{2}}{\frac{n}{2}}$ & $0$ & $\binom{\frac{n+m-1}{2}}{\frac{n}{2}}$\\ \hline
{even} & {even} & $-\binom{\frac{n+m-2}{2}}{\frac{n}{2}}$ & $\binom{\frac{n+m-2}{2}}{\frac{m}{2}}$ & $\binom{\frac{n+m}{2}}{\frac{n}{2}}$\\ \hline
\end{tabular}
\end{center}
\begin{equation}\label{table}
\text{Table for }\Delta
\end{equation}
Because of $\binom{a}{b}=\binom{a}{a-b}$, the equality $\Delta=0$ holds exactly when $n$ and $m$ are both odd.
For the case when both $n$ and $m$ are even, we note
that $\binom{\frac{n+m-2}{2}}{\frac{m}{2}}=\binom{\frac{n+m-2}{2}}{\frac{n}{2}-1}$ and the Pascal triangle
asserts that
$$
\Delta = \binom{\frac{n+m-2}{2}}{\frac{n}{2}-1} + \binom{\frac{n+m-2}{2}}{\frac{n}{2}} = \binom{\frac{n+m}{2}}{\frac{n}{2}}.
$$
This completes the proof.
\end{proof}
Now, using the fact that the group structure of $[J_n({\mathbb S}^1),\Omega Y]$ is induced by the group structure of
the torus homotopy group $\tauu_{n+1}(Y)$ which in turn is determined completely by the homotopy
groups $\{\pi_i(Y)\}_{1\le i\le n+1}$ and their Whitehead products, we give explicitly the group
structure of $[J_n({\mathbb S}^1),\Omega Y]$ following Theorem \ref{DMP} and using the function $\varphii$.
Since $[J_n({\mathbb S}^1),\Omega Y]$ is in one-to-one correspondence with $\displaystyle{\prod_{i=2}^{n+1}\pi_i(Y)}$ as sets, we denote the group multiplication determined
by the co-$H$-structure on ${\mathbb S}igma J_n(\mathbb{S}^1)$ by $\#_n$, i.e.,
$$
[J_n({\mathbb S}^1),\Omega Y]\cong \left(\prod_{i=2}^{n+1} \pi_i(Y), \#_n\right).
$$
Now, we describe $\#_n$ inductively as follows. First write
$$\prod_{i=2}^{n+1} \pi_i(Y)=\left(\prod_{i=2}^{n} \pi_i(Y)\right)\times \pi_{n+1}(Y).$$ For any $\displaystyle{\alpha \in \prod_{i=2}^{n+1} \pi_i(Y)}$, write $\alpha=(\alpha_1,\alpha_2)$, where $\displaystyle{\alpha_1\in \prod_{i=2}^{n} \pi_i(Y)}$ and $\alpha_2\in \pi_{n+1}(Y)$. Moreover, $(\alpha)_k$ denotes the coordinate of $\alpha$ in $\pi_k(Y)$. For $\displaystyle{(\alpha_1,\alpha_2), (\beta_1,\beta_2)\in \left(\prod_{i=2}^{n} \pi_i(Y)\right)\times \pi_{n+1}(Y)}$,
we have
\begin{equation}\label{J_n-product}
(\alpha_1,\alpha_2)\#_n(\beta_1,\beta_2):=\left(\alpha_1\#_{n-1}\beta_1, \alpha_2+\beta_2+\sum_{k+j=n+2}\varphii(k-1,k+j-3)[(\alpha_1)_k,(\beta_1)_j]\right).
\end{equation}
Recall that the natural inclusion $j_n :J_{n-1}({\mathbb S}^1) \hookrightarrow J_n({\mathbb S}^1)$ induces a surjective homomorphism $j_n^\ast: [J_n({\mathbb S}^1),\Omega Y] \to [J_{n-1}({\mathbb S}^1),\Omega Y]$, where $\mbox{Ker}\, j_n^\ast\cong \pi_{n+1}(Y)$ is central. In other words,
$$
j_n^\ast((\alpha)_1, (\alpha)_2,\ldots,(\alpha)_n, (\alpha)_{n+1})=((\alpha)_1, (\alpha)_2,\ldots,(\alpha)_n)
$$
for $((\alpha)_1, (\alpha)_2,\ldots,(\alpha)_n, (\alpha)_{n+1})\in[{\mathbb S}igma J_n(\mathbb{S}^1),Y]$.
\section{Computations}
In this section, we revisit, simplify, and generalize the examples from Section 1 using the group structure of $[J_n({\mathbb S}^1),\Omega Y]$ in the previous
section together with the function $\varphii$. Furthermore, we determine whether $[J_k(\mathbb S^1),\Omega \mathbb S^{2n}]$ is abelian for certain values of $k$.
\begin{example}\label{ex1-revisit}
The multiplication in $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ as in Example \ref{ex1} is given by the following rule:
$$(\alpha_1,\alpha_2) \#_2 (\beta_1,\beta_2)=(\alpha_1+\beta_1,\alpha_2+\beta_2+2\alpha_1\beta_1),
$$
where $\alpha_1,\beta_1\in \pi_2({\mathbb S}^2)\cong \mathbb Z$ and $\alpha_2,\beta_2 \in \pi_3({\mathbb S}^2)
\cong \mathbb Z$.
The perturbation $P: {\mathbb S}^3\to {\mathbb S}igma J_2({\mathbb S}^1)\vee {\mathbb S}igma J_2({\mathbb S}^1)$ in this case is
determined by the basic Whitehead product $[i_1,i_2]$, where
$i_j :{\mathbb S}^2\to{\mathbb S}^2\vee{\mathbb S}^2$ is the corresponding inclusion for $j=1,2$
which yields in the group $[J_2(\mathbb S^1),\Omega \mathbb S^2]$ the relation $[\iota_2,\iota_2]=2\etaa_2$
for the generators $\iota_2\in\pi_2({\mathbb S}^2)$ and $\etaa_2\in\pi_3({\mathbb S}^2)$
given by the Hopf map. Further, the isomorphism
$\mathbb Z\oplus \mathbb Z\cong [J_2({\mathbb S}^1),\Omega {\mathbb S}^2]$ is given
by $(m,n) \mapsto(m,m(m-1)+n)$ for $(m,n)\in\mathbb Z\oplus \mathbb Z$.
\end{example}
More generally, we can use Theorem \ref{DMP} to analyze the co-$H$-structure of ${\mathbb S}igma J_2(\mathbb S^1)$.
The perturbation $P: \mathbb S^{3}\to {\mathbb S}igma J_2(\mathbb S^1)\vee {\mathbb S}igma J_2(\mathbb S^1)$ is given by the
Whitehead product $\mathbb S^{3}\to \mathbb S^2\vee \mathbb S^2$.
Therefore, the multiplication on the set $[J_2(\mathbb S^1), \Omega Y]=\pi_2(Y) \times \pi_3(Y)$
is given by
$$(\alpha_1,\alpha_2)\#_2(\beta_1,\beta_2)=(\alpha_1+\beta_1,\alpha_2+\beta_2+[\alpha_1,\beta_1]),$$
where $\alpha_1,\beta_1\in \pi_2(Y)$, $\alpha_2,\beta_2\in \pi_3(Y)$ and
$[\alpha_1,\beta_1]$ denotes the Whitehead product.
Note that $[\beta_1,\alpha_1]=(-1)^4[\alpha_1,\beta_1]=[\alpha_1,\beta_1]$. Consequently, we generalize Example \ref{ex1} by the following proposition.
\begin{proposition}\label{J_2-abelian}{\em
The group $[J_2(\mathbb S^1),\Omega Y]$ is an abelian group.}
\end{proposition}
\begin{remark}
Since $[J_2(\mathbb S^1),\Omega Y]$ is abelian, using induction on the central extension $0\to \pi_{n+1}(Y) \to [J_n({\mathbb S}^1),\Omega Y]\to [J_{n-1}({\mathbb S}^1),\Omega Y]\to 1$ shows that the group $[J_n({\mathbb S}^1),\Omega Y]$ is nilpotent with nilpotency class $\le n-1$.
\end{remark}
\begin{proposition} {\em If $Y$ is a $(2n-1)$-connected space then there is
an isomorphism of groups $[J_{4n-3}({\mathbb S}^1),\Omega Y]\cong\pi_{2n}(Y)\oplus\pi_{2n+1}(Y)\oplus\cdots\oplus\pi_{4n-2}(Y)$,
the group $[J_{4n-2}({\mathbb S}^1),\Omega Y]$ is abelian and the short exact sequence
$$0\to\pi_{4n-1}(Y)\longrightarrow[J_{4n-2}({\mathbb S}^1),\Omega Y]\longrightarrow[J_{4n-3}({\mathbb S}^1),\Omega Y]\to 0$$
splits provided $\pi_{2n}(Y)$ is free abelian.
In particular, this holds for $Y={\mathbb S}^{2n}$.}
\end{proposition}
\begin{proof} For dimensional reasons and by the connectivity of $Y$, there are no non-trivial Whitehead products obtained from the elements in $[J_{4n-3}({\mathbb S}^1),\Omega Y]$, which in turn is isomorphic to $\pi_{2n}(Y)\oplus\pi_{2n+1}(Y)\oplus\cdots\oplus\pi_{4n-2}(Y)$. The only possibly non-trivial Whitehead products in $[J_{4n-2}({\mathbb S}^1),\Omega Y]$ come from elements $\alpha, \beta \in \pi_{2n}(Y)$. It follows from Table \eqref{table} that $\Delta =0$, that is, $[\alpha, \beta]=[\beta,\alpha]$. This implies that $[J_{4n-2}({\mathbb S}^1),\Omega Y]$ is abelian. Finally, when $\pi_{2n}(Y)$ is free abelian, one can find a section $\pi_{2n}(Y) \to [J_{4n-2}({\mathbb S}^1),\Omega Y]$ so that $\alpha \#_{4n-3} \beta \mapsto (\alpha \#_{4n-3} \beta, [\alpha,\beta])$.
Hence, this gives rise to a section $\pi_{2n}(Y)\oplus\pi_{2n+1}(Y)\oplus\cdots\oplus\pi_{4n-2}(Y) \to[J_{4n-2}({\mathbb S}^1),\Omega Y]$.
\end{proof}
\begin{remark}
The assumption that $\pi_{2n}(Y)$ being free abelian above is only sufficient for the splitting of the short exact sequence as we illustrate in the following examples.
\end{remark}
\par It is well-known that the $(n-2)$-suspension $M^n={\mathbb S}igma^{n-2}\mathbb R P^2$ of the projective plane $\mathbb R P^2$
is the Moore space of type $({\mathbb Z}_2,n-1)$ for $n\ge 3$ so that $M^n$ is $(n-2)$-connected.
Given the inclusion $i_2 : {\mathbb S}^1\hookrightarrow \mathbb R P^2$ and the collapsing map $p_2 : \mathbb R P^2\to \mathbb R P^2/{\mathbb S}^1={\mathbb S}^2$,
we write $i_n ={\mathbb S}igma^{n-2}i_2 : {\mathbb S}^{n-1}\to M^n$ and $p_n = {\mathbb S}igma^{n-2}p_2 : M^n\to {\mathbb S}^n$ for the $(n-2)$-suspension maps with $n\ge 2$, respectively.
\begin{example}\label{JM-ex1}
Take $Y=M^3$. In view of \cite[Proposition 3.6]{G-M}, $\pi_2(M^3)= \mathbb Z_2\langle i_3\rangle$ and $\pi_3(M^3)=\mathbb Z_4\langle i_3\circ \etaa_2\rangle$.
Then, the following short exact sequence
$$0\to\pi_{3}(M^3)\longrightarrow[J_{2}({\mathbb S}^1),\Omega M^3]\longrightarrow[J_{1}({\mathbb S}^1),\Omega M^3]\to 0$$
becomes
\begin{equation}\label{Mukai_ex1}
0\to \mathbb Z_4 \to [J_{2}({\mathbb S}^1),\Omega M^3] \to \mathbb Z_2 \to 0.
\end{equation}
By Proposition \ref{J_2-abelian}, $[J_{2}({\mathbb S}^1),\Omega M^3]$ is abelian and thus is isomorphic to either $\mathbb Z_8$ or $\mathbb Z_4 \oplus \mathbb Z_2$.
It was pointed out to us by J.\ Mukai that the Whitehead product $[i_3,i_3]\ne 0$ and its order is $2$. It is straighforward to see, based upon the group structure of $[J_{2}({\mathbb S}^1),\Omega M^3]$, that there are no elements of order $8$ in $[J_{2}({\mathbb S}^1),\Omega M^3]$.
Hence, the short exact sequence \eqref{Mukai_ex1} splits.
\end{example}
Next, we provide a similar example \footnote{The authors are grateful to Juno Mukai for providing the example below.} as above, where the sequence does not split.
\begin{example}\label{JM-ex2}
The short exact sequence
$$0\to \pi_{11}(M^7)\longrightarrow[J_{10}({\mathbb S}^1),\Omega M^7]\longrightarrow[J_9({\mathbb S}^1),\Omega M^7]\cong\pi_6(M^7)\oplus\cdots\oplus\pi_{10}(M^7)\to 0$$
is central, does not split, and the group $[J_{10}({\mathbb S}^1),\Omega M^7]$ is abelian.
To see this, we recall that $\pi_{6}(M^7)={\mathbb Z}_2\langle i_7\rangle$, and in view of \cite{MS,wu}, it follows
that $\pi_{11}(M^7)\cong {\mathbb Z}_2\langle [i_7,i_7]\rangle$. Consider the element $\alpha =(0,0,0,0,i_7,0,0,0,0)$. Then
by \eqref{J_n-product}, we get
$$(\alpha,0) \#_{10} (\alpha,0) = (2\alpha, [i_7,i_7]) =(0,[i_7,i_7]) \quad \text{since $\alpha$ has order $2$.}$$
It follows that
$$(\alpha,0) \#_{10} (\alpha,0) \#_{10} (\alpha,0) \#_{10} (\alpha,0) =(0,0) \quad \text{since the Whitehead product $[i_7,i_7]$ has order 2.}$$
This shows that the group $[J_{10}({\mathbb S}^1),\Omega M^7]$ contains a copy of the group ${\mathbb Z}_4$ determined
by the element $\alpha$.
Hence, $[J_{10}({\mathbb S}^1),\Omega M^7]$ is abelian but it is not isomorphic to the direct sum $\pi_6(M^7)\oplus\cdots\oplus\pi_{11}(M^7)$. Consequently,
the short exact sequence
$$0\longrightarrow \pi_{11}(M^7)\longrightarrow[J_{10}({\mathbb S}^1),\Omega M^7]\longrightarrow[J_9({\mathbb S}^1),\Omega M^7]\cong\pi_6(M^7)\oplus\cdots\oplus\pi_{10}(M^7)\longrightarrow 0$$
does not split.
\end{example}
In Example \ref{ex3}, the group $[J_{k}({\mathbb S}^1),\Omega {\mathbb S}^{4}]$ is non-abelian for $k\ge 7$. We now generalize this example in the following proposition.
\begin{proposition} \label{1-2_stems}{\em
\mbox{\em (1)} The group $[J_{4n-1}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is non-abelian if and only if
$n$ is a power of $2$. In particular, when $n$ is a power of $2$, $[J_k({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is non-abelian for $k\ge 4n-1$.
When $n$ is not a power of $2$, $[J_k({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is abelian for $k\leq 4n$.
\par \mbox{\em (2)} The group $[J_k({\mathbb S}^1),\Omega M^{2n}]$ is non-abelian for $k\ge 4n-1$ provided $n$ is a power of $2$.}
\end{proposition}
\begin{proof}
(1): Given a prime $p$, consider the base $p$ expansions of the integers $m$ and $n$, where we
assume $0\le m\le n$:
\noindent
$n=a_0+pa_1+\cdots+p^ka_k,$ $0\le a_i<p$ and $m=b_0+pb_1+\cdots+p^kb_k$, $0\le b_i< p$.
Then, by Lukas' Theorem (see e.g., \cite{fine}),
$${n\choose m}\equiv \prod^k_{i=0}{a_i\choose b_i}(\bmod\; p).$$
Hence, ${n\choose m}\not\equiv 0\;(\bmod\; p)$ if and only if $b_i\le a_i$ for all $0\le i\le k$.
\par In particular, for $0\le m\le n$ and $p=2$ with expansions
$n=a_0+2a_1+\cdots+2^ka_k,$ $0\le a_i<2$ and $m=b_0+2b_1+\cdots+2^kb_k$, $0\le b_i< 2$,
the number ${n\choose m}$ is odd if and only if $b_i\le a_i$ for all $0\le i\le k$.
Thus, following Table (\ref{table}), $\Delta={2n-1\choose n}$ is odd if and only if $n=2^l$ for some $l\ge 0$.
Since $[\etaa_{2n},\iota_{2n}]\ne 0$ (see \cite[p.\ 404]{GM}) and $\Delta\ne 0$, we have $(0,\ldots,0,\iota_{2n},0\ldots,0)\#_k(0,\ldots,0,\etaa_{2n},0\ldots,0)\ne(0,\etaa_{2n},0\ldots,0)\#_k(\iota_{2n},0\ldots,0)$
and consequently $[J_{4n-1}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is non-abelian if, and only if,
$n$ is a power of $2$. Hence, $[J_k({\mathbb S}^1),\Omega{\mathbb S}^{2n}]$ is non-abelian for $k\ge 4n-1$ when $n=2^l$ for some $l>0$.
When $n$ is not a power of $2$, we only need to consider the Whitehead product $[\iota_{2n},\etaa_{2n}^2]$ because other Whitehead products from lower
dimensions lie in the abelian group $[J_{4n-1}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$. Since both $2n-1$ and $(2n+2)-1$ are both odd, it follows from Table (\ref{table})
that $\Delta=0$, i.e., $\iota_{2n}\#_{4n}\etaa_{2n}^2=\etaa_{2n}^2\#_{4n}\iota_{2n}$. Thus, we conclude that $[J_{4n}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is abelian and hence so is $[J_{k}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ for any $k\le 4n$.
(2): Let $\tilde{\etaa}_2\in \pi_4(M^3)$ be a lift of $\etaa_3\in\pi_4({\mathbb S}^3)$ satisfying $2\tilde{\etaa}_2=i_n \etaa^2_2$,
$p_3\tilde{\etaa}_2=\etaa_3$ and set $\tilde{\etaa}_n={\mathbb S}igma^{n-2}\tilde{\etaa}_2$ for $n\ge 2$.
Then $\pi_{2n+1}(M^{2n})= {\mathbb Z}_4\langle \tilde{\etaa}_{2n-1}\rangle$ and, by \cite[Lemma 3.8]{G-M}, it holds
$[i_{2n}\etaa_{2n-1},\tilde{\etaa}_{2n-1}]\ne 0$ for $n\ge 2$. Then, we can deduce as in (1)
that $[J_k({\mathbb S}^1),\Omega M^{2n}]$ is non-abelian for $k\ge 4n-1$ and $n=2^l$ for some $l>0$.
\end{proof}
Based upon the results in Proposition \ref{1-2_stems}, the next question is whether $[J_{4n+1}({\mathbb S}^1),\Omega{\mathbb S}^{2n}]$ is abelian when $n$ is not a power of $2$. Proposition \ref{1-2_stems} (1) depends on certain divisibility properties of certain types of binomial coefficients. In the next result, we answer
this question by exploring further such divisibility results concerning the Catalan numbers and thereby strengthen Proposition \ref{1-2_stems}.
Let $T^*(01)$ denote the set of natural numbers $n$ with $(n)_3=(n_i)$ and $n_i\in \{0,1\}$ for $i\ge 1$, where $(n)_3$ is the base $3$ expansion of $n$. Further, denote by $T^*(01)-1$ the set $\{n-1|n\in T^*(01)\}$.
\par Following Table (\ref{table}), we get $\Delta=\binom{2n}{n-1}$. Since $\binom{2n}{n-1}+\binom{2n}{n}=\binom{2n+1}{n}=\frac{2n+1}{n+1}\binom{2n}{n}$,
it follows that
\begin{equation}\label{catalan}
\Delta=\binom{2n}{n-1}=\binom{2n}{n}\left(\frac{2n+1}{n+1}-1\right)=\frac{n}{n+1}\binom{2n}{n}=nC_n,
\end{equation}
where $C_n$ is the $n$-th Catalan number.
\par Write $\nu_{2n}$ for a generator of $\pi_{2n+3}({\mathbb S}^{2n})$ with $n\ge 2$ and consider the Whitehead product $[\iota_{2n},\nu_{2n}]$.
Suppose $n$ is not a power of $2$. According to \cite[p.\ 405]{GM}, the order of $[\iota_{2n},\nu_{2n}]$ is $12$ if $n$ is odd or of order $24$ if $n$ is even.
\begin{proposition}\label{3_stem}{\em
Suppose $n\ne 2^{\ell}$ for any $\ell>0$. Then $[J_{4n+1}({\mathbb S}^1),\Omega {\mathbb S}^{2n}]$ is abelian if and only if,
\begin{itemize}
\item[(1)] \mbox{\em (i)} $n\ne 2^a-1$ and $n\ne 2^a+2^b-1$ for some $b>a\ge 0$;
\noindent
and
\noindent
\mbox{\em (ii)} $n\equiv 0\, (\bmod\, 3)$ or $n\notin T^*(01)-1$ when $n$ is odd;
\item[(2)] $n\equiv 0\, (\bmod \,3)$ or $n\notin T^*(01)-1$ when $n$ is even.
\end{itemize}}
\end{proposition}
\begin{proof}
When $n$ is odd, $\Delta=nC_n$ is divisible by $12$ if and only if, $C_n$ is divisible by $4$ and either $n$ or $C_n$ is divisible by $3$. Similarly, when $n$ is even, $\Delta=nC_n$ is divisible by $24$ if and only if, $C_n$ is divisible by $4$ (since $n$ is even) and either $n$ or $C_n$ is divisible by $3$.
The result follows from \cite[Theorem 5.2]{ds} for the divisibility of $C_n$ by $3$ and from \cite[Theorem 2.3]{ely} for the divisibility of $C_n$ by $4$.
\end{proof}
\begin{remark}
Let $n=29$. Then, $C_{29}$ is divisible by $4$ but $29$ is not divisible by $3$ and $C_{29}$ is not divisible by $3$ since $29\in T^*(01)-1$. Thus, in case
(1) in Proposition \ref{3_stem}, (i) is satisfied but (ii) is not.
Let $n=34$. Then, $C_{34}$ is divisible by $4$ and by $3$ but $34$ is not divisible by $3$.
\end{remark}
The following result generalizes Example \ref{ex2}.
\begin{proposition}{\em
The group $[J_{4n-1}({\mathbb S}^1),\Omega({\mathbb S}^{2n}\vee{\mathbb S}^{2n+1})]$ is non-abelian.}
\end{proposition}
\begin{proof}
Consider the inclusion maps $i_1 : {\mathbb S}^{2n}\hookrightarrow{\mathbb S}^{2n}\vee{\mathbb S}^{2n+1}$ and $i_2 :{\mathbb S}^{2n+1}\hookrightarrow{\mathbb S}^{2n}\vee{\mathbb S}^{2n+1}$.
Then, Hilton's result \cite{H} asserts that for every positive integer $k$, there is an isomorphism
$${\mathbb T}heta :\bigoplus_{l=1}^\infty\pi_k({\mathbb S}^{n_l})\stackrel{\cong}{\longrightarrow} \pi_k({\mathbb S}^{2n}\vee{\mathbb S}^{2n+1}),$$
where the restriction ${\mathbb T}heta|_{\pi_k({\mathbb S}^{n_l})}=\omegaega_{l\ast} :\pi_k({\mathbb S}^{n_l})\to \pi_k({\mathbb S}^{2n}\vee{\mathbb S}^{2n+1})$
is determined by the iterated Whitehead product of the maps $i_1$ and $i_2$.
In particular, the Whitehead product $[i_1,i_2] : {\mathbb S}^{4n}\to {\mathbb S}^{2n}\vee{\mathbb S}^{2n+1}$ is non-trivial.
Furthermore, $$(0,\ldots,0,i_1,0,\ldots)\#_{4n-1}(0,\ldots,0,i_2,0,\ldots)=(0,\ldots,0,i_1,0,\ldots,0,i_2,0,\ldots)$$ and
$$(0,\ldots,0,i_2,0,\ldots)\#_{4n-1}(0,\ldots,0,i_1,0,\ldots)=(0,\ldots,0,i_1,0,\ldots,0,i_2,0,\ldots,0,[i_1,i_2],0\ldots).$$
The above implies that the group $[J_{4n-1}({\mathbb S}^1),\Omega({\mathbb S}^{2n}\vee{\mathbb S}^{2n+1})]$ is non-abelian.
\end{proof}
To close the paper, we derive few simple properties about the torsion elements in $[J({\mathbb S}^1), \Omega Y]$.
\begin{proposition} {\em Let $\alpha, \beta\in [J({\mathbb S}^1), \Omega Y]$ be two elements which correspond to
homogeneous sequences with $\alpha\in \pi_m(Y)$ and $\beta\in \pi_n(Y)$. Then:
\mbox{\em (1)} if $\alpha$ has order $k$ in $\pi_m(Y)$, then $\alpha$, regarded as an element
of $[J({\mathbb S}^1), \Omega Y]$, has order $k$ or $k^2$;
\mbox{\em (2)} if $\alpha,\beta$ are torsion elements of order $|\alpha|$ and $|\beta|$, respectively such that ${\rm gcd}(|\alpha|,|\beta|)=1$, then $\alpha$ and $\beta$ commute in $[J({\mathbb S}^1), \Omega Y]$.}
\end{proposition}
\begin{proof} (1): By \cite[Chapter XI, Section 8, Theorem 8.8]{Whi}, all Whitehead products of weight $\geq 3$ of an element of odd dimension and all Whitehead products of weight $\geq 4$ of an element of even dimension, vanish. Therefore, using our formula and the result above,
we obtain that $\alpha^k$ (as an element of $[J({\mathbb S}^1), \Omega Y]$) is a sequence of the form $(0,\ldots,0, k\alpha, \lambda_1[\alpha, \alpha], \lambda_2[\alpha, [\alpha, \alpha]],0,\ldots)=
(0\ldots,0, \lambda_1[\alpha, \alpha], \lambda_2[\alpha, [\alpha, \alpha]],0,\ldots)$. Again, by the result cited above, we obtain that
\begin{equation*}
\begin{aligned}
(0,\ldots,0,\lambda_1[\alpha, \alpha], \lambda_2[\alpha, [\alpha, \alpha]],0,\ldots)^k&=(0,\ldots,0, k\lambda_1[\alpha, \alpha], k\lambda_2[\alpha, [\alpha, \alpha]],0,\ldots) \\
&=(0,\ldots,0, \lambda_1[k\alpha, \alpha], \lambda_2[k\alpha, [\alpha, \alpha]],0,\ldots) \\
&=0
\end{aligned}
\end{equation*} and (1) follows.
(2): It suffices to observe that the Whitehead product $[\alpha, \beta]$ vanishes. Since $|\alpha|[\alpha, \beta]=[|\alpha|\alpha, \beta]=0$,
$|\beta|[\alpha, \beta]=[\alpha, |\beta|\beta]=0$ and ${\rm gcd}(|\alpha|,|\beta|)=1$, it follows that $[\alpha, \beta]=0$ and
the proof is complete.
\end{proof}
\end{document} |
\begin{document}
\title{Generalized Laplacian decomposition of vector fields on fractal surfaces}
\author{Daniel Gonz\'alez-Campos$^{(1)}$, Marco Antonio P\'erez-de la Rosa$^{(2)}$\\and\\ Juan Bory-Reyes$^{(3)}$}
\date{ \small $^{(1)}$ Escuela Superior de F\'isica y Matem\'aticas. Instituto Polit\'ecnico Nacional. CDMX. 07738. M\'exico. \\ E-mail: daniel\_uz13@hotmail.com \\
$^{(2)}$ Department of Actuarial Sciences, Physics and Mathematics, Universidad de las Am\'ericas Puebla.
San Andr\'es Cholula, Puebla. 72810. M\'exico. \\ Email: marco.perez@udlap.mx \\
$^{(3)}$ ESIME-Zacatenco. Instituto Polit\'ecnico Nacional. CDMX. 07738. M\'exico. \\ E-mail: juanboryreyes@yahoo.com }
\maketitle
\begin{abstract}
We consider the behavior of generalized Laplacian vector fields on a Jordan domain of $\mathbb{R}^{3}$ with fractal boundary. Our approach is based on properties of the Teodorescu transform and suitable extension of the vector fields. Specifically, the present article addresses the decomposition problem of a H\"older continuous vector field on the boundary (also called reconstruction problem) into the sum of two generalized Laplacian vector fields in the domain and in the complement of its closure, respectively. In addition, conditions on a H\"older continuous vector field on the boundary to be the trace of a generalized Laplacian vector field in the domain are also established.
\end{abstract}
\small{
\noindent
\textbf{Keywords.} Quaternionic analysis; vector field theory; fractals.\\
\noindent
\textbf{Mathematics Subject Classification (2020).} 30G35, 32A30, 28A80.}
\section{Introduction}
Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field $\mathbb{H}$, which would somehow resemble the classical complex one-dimensional function theory.
An ordered set of quaternions $\psi:=(\psi_1, \psi_2, \psi_3)\in \mathbb{H}^{3}$, which form an orthonormal (in the usual Euclidean sense) basis in $\mathbb{R}^{3}$ is called a structural $\mathbb{H}$-vector.
The foundation of the so-called $\psi$-hyperholomorphic quaternion valued function theory, see \cite{NM, VSMV, MS} and elsewhere, is that the structural $\mathbb{H}$-vector $\psi$ must be chosen in a way that the factorization of the quaternionic Laplacian holds for $\psi$-Cauchy-Riemann operators. This question goes back at least as far as N\^{o}no's work \cite{Nono1, Nono2}.
The use of a general orthonormal basis introducing a generalized Moisil-Teodorescu system is the cornerstone of a generalized quaternionic analysis, where the generalized Cauchy-Riemann operator with respect to the standard basis in $\mathbb{R}^3$ are submitted to an orthogonal transformation. Despite the fact that some of the results in the present work can be obtained after the action of an orthogonal transformation on the standard basis; we keep their proofs in the work for the sake of completeness.
The $\psi$-hyperholomorphic functions theory by itself is not much of a novelty since it can be reduced by an orthogonal transformation to the standard case. In the face of this, the picture changes entirely by studying some important operators involving a pair of different orthonormal basis.
Moreover, the possibility to study simultaneously several conventional known theories, which can be embedded into a corresponding version of $\psi$-hyperholomorphic functions theory, again cannot be reduced to the standard context and reveal indeed the relevance of the $\psi$-hyperholomorphic functions theory.
The advantageous idea behind the unified study of particular cases of a generalized Moisil-Teodorescu system in $\psi$-hyperholomorphic functions theory simultaneously is considered in the present work.
The special case of structural $\mathbb{H}$-vector $\psi^\theta:=\{\textbf{i},\, \textbf{i}e^{\textbf{i}\theta}\textbf{j},\, e^{\textbf{i}\theta}\textbf{j}\}$ for $\theta\in[0,2\pi)$ fixed and its associated $\psi^\theta$-Cauchy-Riemann operator
\begin{equation*}
{^{\psi^{\theta}}}D:=\displaystyle\frac{\partial}{\partial x_{1}}\textbf{i}+\frac{\partial}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial}{\partial x_{3}} e^{\textbf{i}\theta}\textbf{j},
\end{equation*}
are used in \cite{BAPS} to give a quaternionic treatment of inhomogeneous case of the system
\begin{equation}\label{sedi}
\left\{
\begin{array}{rcl}
-\displaystyle \frac{\partial f_{1}}{\partial x_{1}}+\left(\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}\right)\sin\theta-\left(\frac{\partial f_{3}}{\partial x_{2}}+\frac{\partial f_{2}}{\partial x_{3}}\right)\cos\theta & = & 0,
\\ {}\\ \displaystyle {\left(\frac{\partial f_{3}}{\partial x_{3}}-\frac{\partial f_{2}}{\partial x_{2}}\right)}\cos\theta-\left(\frac{\partial f_{3}}{\partial x_{2}}+\frac{\partial f_{2}}{\partial x_{3}}\right)\sin\theta & = & 0,
\\ {}\\ \displaystyle {-\frac{\partial f_{3}}{\partial x_{1}}+\frac{\partial f_{1}}{\partial x_{3}}\sin\theta+\frac{\partial f_{1}}{\partial x_{2}}\cos\theta} & = & 0, \\ {}\\
\displaystyle {\frac{\partial f_{2}}{\partial x_{1}}-\frac{\partial f_{1}}{\partial x_{3}}\cos\theta+\frac{\partial f_{1}}{\partial x_{2}}\sin\theta} & = & 0,
\end{array}
\right.
\end{equation}
wherein the unknown well-behaved functions $f_m: \Omega \rightarrow \mathbb{C}, m=1,2,3$ are prescribed in an smooth domain $\Omega\subset\mathbb{R}^{3}$.
From now on, an smooth vector field $\vec{f}=(f_{1}, f_{2}, f_{3})$ that satisfies \eqref{sedi}, will said to be a generalized Laplacian vector field.
We will consider complex quaternionic valued functions (a detailed exposition of notations and definitions will be given in Section 2) to be expressed by
\begin{equation}
\notag
f=f_{0}+f_{1}\textbf{i}+f_{2}\textbf{j}+f_{3}\textbf{k},
\end{equation}
where $\textbf{i}$, $\textbf{j}$ and $\textbf{k}$ denote the quaternionic imaginary units.
On the other hand, the one-to-one correspondence
\begin{equation}\label{corre}
\mathbf{f}=f_1\mathbf{i}+f_2\mathbf{j}+f_3\mathbf{k}\, \longleftrightarrow \vec{f}=(f_{1}, f_{2}, f_{3})
\end{equation}
makes it obvious that $\eqref{sedi}$ can be obtained from the classical Moisil-Theodorescu system after the action of some element in $O(3)$ as:
$${^{\psi^{\theta}}}D[\mathbf{f}]= 0.$$
System \eqref{sedi} contains as a particular case the well-known solenoidal and irrotational, or harmonic system of vector fields (see \cite{ABS, ABMP} and the references given there). Indeed, under the correspondence $\mathbf{f}=f_1\mathbf{i}+f_3\mathbf{j}+f_2\mathbf{k}\, \longleftrightarrow \vec{f}=(f_{1}, f_{2}, f_{3})\,$ we have for $\theta=0$:
\begin{equation}\label{equi}
{}{^{\psi^{0}}}D[\mathbf{f}]=0\,\Longleftrightarrow \,
\begin{cases}
\text{div} \vec{f}=0,\cr
\text{rot} \vec{f}=0.
\end{cases}
\end{equation}
Besides, the system \eqref{sedi} includes other partial differential equations systems (see \cite{BAPS} for more details): A particular case of the inhomogeneous Cimmino system (\cite{C}) when one looks for a solution $(f_1,f_2,f_3)$, where each $f_m,\,m=1,2,3$ does not depend on $x_0$. This system is obtained from \eqref{sedi} for $\theta=\frac{\pi}{2}$. Also, an equivalent system to the so-called the Riesz system \cite{Riesz} studied in \cite{Gur, Gur2}, which can be obtained from \eqref{sedi} for $\theta=\pi$ and the convenient embedding in $\mathbb{R}^3$.
In order to get more generalized results than those of \cite{ABMP}, it is assumed in this paper that $\Omega\subset \mathbb{R}^{3}$ is a Jordan domain (\cite{HN}) with fractal boundary $\Gamma$ in the Mandelbrot sense, see \cite{FKJ, FJ}.
Let us introduce the temporary notations $\Omega_{+}:=\Omega$ and $\Omega_{-}:=\mathbb{R}^{3}\setminus \{\Omega_{+}\cup\Gamma\}$. We are interested in the following problems: Given a continuous three-dimensional vector field $\vec{f}: \Gamma \rightarrow \mathbb{C}^{3}$:
\begin{itemize}
\item [$(I)$]
(Problem of reconstruction) Under which conditions can $\vec{f}$ be decomposed on $\Gamma$ into the sum:
\begin{equation} \label{des}
\vec{f}(t)=\vec{f}^{+}(t)+\vec{f}^{-}(t), \quad \forall \, t\in\Gamma,
\end{equation}
where $\vec{f}^{\pm}$ are extendable to generalized Laplacian vector fields $\vec{F}^{\pm}$ in $\Omega_{\pm}$, with $\vec{F}^{-}(\infty)=0$?
\item [$(II)$] When $\vec{f}$ is the trace on $\Gamma$ of a generalized Laplacian vector field $\vec{F}^{\pm}$ in $\Omega_{\pm}\cup\Gamma$?
\end{itemize}
In what follows, we deal with problems $(I)$ and $(II)$ using the quaternionic analysis tools and working with $\mathbf{f}$ instead of $\vec{f}$ under the one-to-one correspondence (\ref{corre}). It will cause no confusion if we call $\mathbf{f}$ also vector field.
In the case of a rectifiable surface $\Gamma$ (the Lipschitz image of some bounded subset of $\mathbb{R}^{2}$) these problems have been investigated in \cite{GPB}.
\section{Preliminaries.}
\subsection{Basics of $\psi^{\theta}$-hyperholomorphic function theory.}
Let $\mathbb{H}:=\mathbb{H(\mathbb{R})}$ and $\mathbb{H(\mathbb{C})}$ denote the sets of real and complex quaternions respectively. If $a\in\mathbb{H}$ or $a\in\mathbb{H(\mathbb{C})}$, then $a=a_{0}+a_{1}\textbf{i}+a_{2}\textbf{j}+a_{3}\textbf{k}$, where the coefficients $a_{k}\in\mathbb{R}$ if $a\in\mathbb{H}$ and $a_{k}\in\mathbb{C}$ if $a\in\mathbb{H(\mathbb{C})}$. The symbols
$\textbf{i}$, $\textbf{j}$ and $\textbf{k}$ denote different imaginary units, i. e. $\textbf{i}^{2}=\textbf{j}^{2}=\textbf{k}^{2}=-1$ and they satisfy the following multiplication rules $\textbf{i}\textbf{j}=-\textbf{j}\textbf{i}=\textbf{k}$; $\textbf{j}\textbf{k}=-\textbf{k}\textbf{j}=\textbf{i}$; $\textbf{k}\textbf{i}=-\textbf{i}\textbf{k}=\textbf{j}$. The unit imaginary $i\in\mathbb{C}$ commutes with every quaternionic unit imaginary.
It is known that $\mathbb{H}$ is a skew-field and $\mathbb{H(\mathbb{C})}$ is an associative, non-commutative complex algebra with zero divisors.
If $a\in\mathbb{H}$ or $a\in\mathbb{H(\mathbb{C})}$, $a$ can be represented as $a=a_{0}+\vec{a}$, with $\vec{a}=a_{1}\textbf{i}+a_{2}\textbf{j}+a_{3}\textbf{k}$,
$\text{Sc}(a):=a_{0}$ is called the scalar part and $\text{Vec}(a):=\vec{a}$ is called the vector part of the quaternion $a$.
Also, if $a\in\mathbb{H(\mathbb{C})}$, $a$ can be represented as $a=\alpha_{1}+i\alpha_{2}$ with $\alpha_{1},\,\alpha_{2}\in\mathbb{H}$.
Let $a,\,b\in\mathbb{H(\mathbb{C})}$, the product between these quaternions can be calculated by the formula:
\begin{equation} \label{pc2}
ab=a_{0}b_{0}-\langle\vec{a},\vec{b}\rangle+a_{0}\vec{b}+b_{0}\vec{a}+[\vec{a},\vec{b}],
\end{equation}
where
\begin{equation} \label{proint}
\langle\vec{a},\vec{b}\rangle:=\sum_{k=1}^{3} a_{k}b_{k}, \quad
[\vec{a},\vec{b}]:= \left|\begin{matrix}
\textbf{i} & \textbf{j} & \textbf{k}\\
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}
\end{matrix}\right|.
\end{equation}
We define the conjugate of $a=a_{0}+\vec{a}\in\mathbb{H(\mathbb{C})}$ by $\overline{a}:=a_{0}-\vec{a}$.
The Euclidean norm of a quaternion $a\in\mathbb{H}$ is the number $\abs{a}$ given by:
\begin{equation}\label{normar}
\abs{a}=\sqrt{a\overline{a}}=\sqrt{\overline{a}a}.
\end{equation}
We define the quaternionic norm of $a\in\mathbb{H(\mathbb{C})} $ by:
\begin{equation}
\abs{a}_{c}:=\sqrt{{{\abs {a_{0}}}_{\mathbb{C}}}^{2}+{{\abs {a_{1}}}_{\mathbb{C}}}^{2}+{{\abs {a_{2}}}_{\mathbb{C}}}^{2}+{{\abs {a_{3}}}_{\mathbb{C}}}^{2}},
\end{equation}
where ${\abs {a_{k}}}_{\mathbb{C}}$ denotes the complex norm of each component of the quaternion $a$.The norm of a complex quaternion $a=a_{1}+ia_{2}$ with $a_{1}, a_{2} \in \mathbb{H}$ can be rewritten in the form
\begin{equation} \label{nc2}
{\abs{a}_{c}}=\sqrt{\abs{\alpha_{1}}^2+\abs{\alpha_{2}}^2}.
\end{equation}
If $a \in \mathbb{H}$, $b \in \mathbb{H(\mathbb{C})}$, then
\begin{equation}
{\abs{ab}}_{c}=\abs{a}{\abs{b}}_{c}.
\end{equation}
If $a\in\mathbb{H(\mathbb{C})}$ is not a zero divisor then $\displaystyle a^{-1}:=\frac{\overline{a}}{a\overline{a}}$ is the inverse of the complex quaternion $a$.
\begin{subsection}{Notations}
\begin{itemize}
\item We say that $f:\Omega \rightarrow \mathbb{H(\mathbb{C}})$ has properties in $\Omega$ such as continuity and real differentiability of order $p$ whenever all $f_{j}$ have these properties. These spaces are usually denoted by $C^{p}(\Omega,\, \mathbb{H(\mathbb{C})})$ with $p\in \mathbb{N}\cup\{0\}$.
\item Throughout this work, $\text{Lip}_{\mu}(\Omega,\, \mathbb{H(\mathbb{C})})$, $0<\mu\leq 1$, denotes the set of H\"older continuous functions $f:\Omega \rightarrow \mathbb{H(\mathbb{C}})$ with H\"older exponent $\mu$. By abuse of notation, when $f_{0}=0$ we write $\mathbf{Lip}_{\mu}(\Omega,\, \mathbb{C}^{3})$ instead of $\text{Lip}_{\mu}(\Omega,\, \mathbb{H(\mathbb{C})})$.
\end{itemize}
\end{subsection}
In this paper, we consider the structural set $\psi^\theta:=\{\textbf{i},\, \textbf{i}e^{\textbf{i}\theta}\textbf{j},\, e^{\textbf{i}\theta}\textbf{j}\}$ for $\theta\in[0,2\pi)$ fixed, and the associated operators ${^{\psi^\theta}}D$ and $D{^{\psi^\theta}}$ on $C^{1}(\Omega,\, \mathbb{H(\mathbb{C})})$ defined by
\begin{equation}
{^{\psi^{\theta}}}D[f]:=\textbf{i}\frac{\partial f}{\partial x_{1}}+\textbf{i}e^{\textbf{i}\theta}\textbf{j}\frac{\partial f}{\partial x_{2}}+e^{\textbf{i}\theta}\textbf{j}\frac{\partial f}{\partial x_{3}},
\end{equation}
\begin{equation}
D{^{\psi^\theta}}[f]:=\frac{\partial f}{\partial x_{1}}\textbf{i}+\frac{\partial f}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial f}{\partial x_{3}} e^{\textbf{i}\theta}\textbf{j},
\end{equation}
which linearize the Laplace operator $\Delta_{\mathbb{R}^{3}}$ in the sense that
\begin{equation}
{^{\psi^{\theta}}}D^{2}= \left[D{^{\psi^\theta}}\right]^{2}=-\Delta_{\mathbb{R}^{3}}.
\end{equation}
All functions belong to $\ker \left({^{\psi^{\theta}}}D\right) := \left\{f : {^{\psi^{\theta}}}D[f]=0\right\}$ are called left-$\psi^{\theta}$-hyperholomorphic in $\Omega$. Similarly, those functions which belong to $\ker \left(D{^{\psi^{\theta}}}\right):= \left\{f : D{^{\psi^{\theta}}}[f]=0\right\}$ will be called right-$\psi^{\theta}$-hyperholomorphic in $\Omega$. For a deeper discussion of the hyperholomorphic function theory we refer the reader to \cite{KVS}.
The function
\begin{equation} \label{kernel}
\mathscr{K}_{\psi^{\theta}}(x):=-\frac{1}{4\pi}\frac{(x)_{\psi^{\theta}}}{\abs{x}^3}, \quad x\in\mathbb{R}^{3}\setminus\{0\},
\end{equation}
where
\begin{equation}
(x)_{\psi^{\theta}}:=x_{1}\textbf{i}+x_{2} \textbf{i}e^{\textbf{i}\theta}\textbf{j}+x_{3}e^{\textbf{i}\theta}\textbf{j},
\end{equation}
is a both-side-$\psi^{\theta}$-hyperholomorphic fundamental solution of $^{\psi^{\theta}}D$. Observe that $\abs{(x)_{\psi^{\theta}}}=\abs{x}$ for all $ x \in \mathbb{R}^{3}$.
For $f=f_{0}+\mathbf{f}\in C^1(\Omega,\mathbb{H(\mathbb{C})})$ let us define
\begin{equation}
{^{\psi^{\theta}}}\text{div}[\mathbf{f}]:=\frac{\partial f_{1}}{\partial x_{1}}+\left({\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}}\right)\textbf{i}e^{\textbf{i}\theta},
\end{equation}
\begin{equation}
{^{\psi^{\theta}}}\text{grad}[f_{0}]:=\frac{\partial f_{0}}{\partial x_{1}}\textbf{i}+\frac{\partial f_{0}}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}\textbf{j}+\frac{\partial f_{0}}{\partial x_{3}}e^{\textbf{i}\theta}\textbf{j},
\end{equation}
\begin{equation}
\begin{split}
{^{\psi^{\theta}}}\text{rot}[\mathbf{f}]:=\left({-\frac{\partial f_{3}}{\partial x_{2}}-\frac{\partial f_{2}}{\partial x_{3}}}\right)e^{\textbf{i}\theta}+\left({-\frac{\partial f_{1}}{\partial x_{3}}\textbf{i}e^{\textbf{i}\theta}-\frac{\partial f_{3}}{\partial x_{1}}}\right)\textbf{j} +\left({\frac{\partial f_{2}}{\partial x_{1}}-\frac{\partial f_{1}}{\partial x_{2}}\textbf{i}e^{\textbf{i}\theta}}\right)\textbf{k}.
\end{split}
\end{equation}
The action of ${^{\psi^{\theta}}}D$ on $f\in C^1(\Omega, \, \mathbb{H(\mathbb{C})})$ yields
\begin{equation}
{^{\psi^{\theta}}}D[f]=-{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+ {^{\psi^{\theta}}}\text{rot}[\mathbf{f}],
\end{equation}
which implies that $f \in \ker ({^{\psi^{\theta}}}D) $ is equivalent to
\begin{equation} \label{eq1}
-{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+ {^{\psi^{\theta}}}\text{rot}[\mathbf{f}]=0.
\end{equation}
If $f_{0}=0$, \eqref{eq1} reduces to
\begin{equation} \label{eq2}
-{^{\psi^{\theta}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}]=0.
\end{equation}
We check at once that \eqref{sedi} is equivalent to \eqref{eq2}.
Similar considerations apply to $D^{\psi^{\theta}}$, for this case one obtains
\begin{equation} \label{eq3}
D^{\psi^{\theta}}[f]=-{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\psi^{\theta}}}\text{grad}[f_{0}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}],
\end{equation}
where
\begin{equation}
{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]:=\frac{\partial f_{1}}{\partial x_{1}}+\left({\frac{\partial f_{2}}{\partial x_{2}}-\frac{\partial f_{3}}{\partial x_{3}}}\right)\overline{\textbf{i}e^{\textbf{i}\theta}},
\end{equation}
\begin{equation}
\begin{split}
{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}]:=\left({-\frac{\partial f_{3} }{\partial x_{2}}-\frac{\partial f_{2}}{\partial x_{3}}}\right) \overline{e^{\textbf{i}\theta}}-{\frac{\partial f_{1}}{\partial x_{3}}\overline{\textbf{i}e^{\textbf{i}\theta}\textbf{j}}+\frac{\partial f_{3}}{\partial x_{1}}}\textbf{j} -\frac{\partial f_{2}}{\partial x_{1}}\textbf{k}-\frac{\partial f_{1}}{\partial x_{2}}\overline{\textbf{i}e^{\textbf{i}\theta}\textbf{k}}.
\end{split}
\end{equation}
If $f_{0}=0$, \eqref{eq3} reduces to
\begin{equation} \label{eq4}
D^{\psi^{\theta}}[f]=-{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}].
\end{equation}
It follows easily that
\begin{equation} \label{eq5}
-{^{\overline{\psi^{\theta}}}}\text{div}[\mathbf{f}]+{^{\overline{\psi^{\theta}}}}\text{rot}[\mathbf{f}]=0,
\end{equation}
is also equivalent to \eqref{sedi}.
\begin{lemma} \label{two-sided} Let $f=f_{0}+\mathbf{f}\in C^{1}(\Omega, \, \mathbb{H(\mathbb{C})})$. Then $f$ is both-side-$\psi^\theta$-hyperholomorphic in $\Omega$ if and only if ${^{\psi^{\theta}}}\text{grad}[f_{0}](x)\equiv 0$ in $\Omega$ and $\mathbf{f}$ is a generalized Laplacian vector field in $\Omega$.
\begin{proof}
The proof is based on the fact that \eqref{eq2} and \eqref{eq5} are equivalent to \eqref{sedi}.
\end{proof}
\end{lemma}
\subsection{Fractal dimension and the Whitney operator}
Let $E$ a subset in $\mathbb{R}^{3}$, we denote by $\mathcal{H}_{\lambda}(E)$ the $\lambda$-Hausdorff measure of $E$ (\cite{GJ}).
Assume that $E$ is a bounded set, the Hausdorff dimension of $E$ (denoted by $\lambda(E)$) is the infimum $\lambda$ such that $\mathcal{H}_{\lambda}(E)<\infty$.
Frequently, the Minkowski dimension of $E$ (also called box dimension and denoted by $\alpha(E)$) is more appropriate than the Hausdorff dimension to measure the roughness of E (\cite{ABMP,ABS}).
It is known that Minkowski and Hausdorff dimensions can be equal, for example, for rectifiable surfaces (the Lipschitz image of some bounded subset of $\mathbb{R}^{2}$). But in general, if $E$ is a two-dimensional set in $\mathbb{R}^{3}$
\begin{equation}
2\leq \lambda(E)\leq \alpha(E)\leq3.
\end{equation}
If $2<\lambda(E)$, $E$ is called a fractal set in the Mandelbrot sense. For more information about the Hausdorff and Minkowski dimension, see \cite{FKJ,FJ}.
Let $f\in \text{Lip}_{\mu}(\Gamma, \mathbb{H(\mathbb{C})})$, then $f=f_{1}+if_{2}$ with $f_{k}\in \text{Lip}_{\mu}(\Gamma, \mathbb{H(\mathbb{R})})$ and $\mathcal{E}_{0}(f):=\mathcal{E}_{0}(f_{1})+i\mathcal{E}_{0}(f_{2})$. Write
\begin{equation}
f^{w}:=\mathcal{X}\mathcal{E}_{0}(f),
\end{equation}
where $\mathcal{E}_{0}$ is the Whitney operator and $\mathcal{X}$ denotes the characteristic function in $\Omega_{+}\cup\Gamma$.
For completeness, we recall the main lines in the construction of the Whitney decomposition $\mathcal W$ of the Jordan domain $\Omega$ with boundary $\Gamma$ by squares $Q$ of diameter $||Q||_{\mathbb{R}^{3}}$ and the notion of Whitney operator. This can be found in \cite[Ch VI]{SEM}.
Consider the lattice $\mathbb Z^{3}$ in $\mathbb R^{3}$ and the collection of closed unit cubes defined by it; let $\mathcal{M}_1$ be the mesh consisting of those unit cubes having a non-empty intersection with $\Omega$. Then, we recursively define the meshes $\mathcal{M}_k$, $k=2,3,\ldots$, each time bisecting the sides of the cubes of the previous one. The cubes in $\mathcal{M}_k$ thus have side length $2^{-k+1}$ and diameter $||Q||_{\mathbb{R}^{3}} = (\sqrt{3})\, 2^{-k+1}$. Define, for $k=2,3,\ldots$,
\begin{eqnarray*}
\mathcal{W}^1 & := & \left \{ Q\in \mathcal{M}_1 \, | \, \mbox{$Q$ and every cube of $\mathcal{M}_1$ touching $Q$ are contained in $\Omega$} \right \}, \\
\mathcal{W}^k & := & \left \{ Q\in \mathcal{M}_k \, | \, \mbox{$Q$ and every cube of $\mathcal{M}_k$ touching $Q$ are contained in $\Omega$} \right .\\
& & \hspace*{50mm} \left . \mbox{and}\,\not \exists \, Q^\ast \in \mathcal{W}^{k-1}: Q \subset Q^\ast \right \},
\end{eqnarray*}
for which it can be proven that
$$
\Omega = \bigcup_{k=1}^{+\infty} \mathcal{W}^k = \bigcup_{k=1}^{+\infty} \bigcup_{Q \in \mathcal{W}^k} Q \equiv \bigcup_{Q \in \mathcal{W}} Q,
$$
all cubes $Q$ in the Whitney decomposition $\mathcal{W}$ of $\Omega$ having disjoint interiors.
We denote by $Q_{0}$ the unit cube with center at the origin and fix a $C^{\infty}$ function with the properties: $0\leq \varphi \leq 1$; $\varphi(x)=1$ if $x\in Q_{0}$; and $\varphi(x)=0 $ if $x\notin Q^*_{0}$.
Let $\varphi_{k}$ the function $\varphi(x)$ adjusted to the cube $Q_{k}\in\mathcal{W}$, that is
\begin{equation}
\varphi_{k}(x):=\varphi\bigg(\frac{x-x^{k}}{l_{k}}\bigg),
\end{equation}
where $x^{k}$ is the center of $Q_{k}$ and $l_{k}$ the common length of its sides.
Function $\varphi_{k}$ satisfies that $0\leq \varphi_{k} \leq 1$, $\varphi_{k}(x)=1$ if $x\in Q_{k}$ and $\varphi_{k}(x)=0 $ if $x\notin Q^*_{k}$. Let ${\varphi_{k}^*}(x)$ be defined for $x\in \Omega$ by
\begin{equation} \label{pdu}
{\varphi_{k}^*}(x):=\frac{\varphi_{k}(x)}{\Phi (x)},
\end{equation}
with
\begin{equation}
\Phi(x):=\sum_{k}^{}\varphi_{k}(x)
\end{equation}
and
$\sum_{k}^{}\varphi_{k}^{*}(x)=1$ for $x\in \Omega$.
For each cube $Q_{k}$ let $p_{k}$ be a point fixed in $\Gamma$ such that $dist(Q_{k}, \Gamma)=dist(Q_{k}, p_{k})$. Then the Whitney operator is defined as follows
\begin{equation}
\mathcal{E}_{0}(f)(x):=f(x), \quad \text{if} \quad x\in \Gamma,
\end{equation}
\begin{equation} \label{suma}
\mathcal{E}_{0}(f)(x):=\sum_{k}f(p_{k})\varphi_{k}^{*}(x), \quad \text{if} \quad x\in \Omega.
\end{equation}
Similar construction may be made for the domain $\mathbb{R}^{3}\setminus \{\Omega\cup\Gamma\}$.
The operator $\mathcal{E}_{0}$ extends functions $f$ defined in $\Gamma$ to functions defined in $\mathbb{R}^{3}$. Its main properties are given as follows:
\begin{itemize}
\item Assume $f\in\text{Lip}_{\mu}(\Omega\cup\Gamma, \mathbb{H(\mathbb{C})})$. Then $\mathcal{E}_{0}(f) \in \text{Lip}_{\mu}(\mathbb{R}^{3}, \mathbb{H(\mathbb{C})})$ and in fact is $C^{\infty}$ in $\mathbb{R}^{3}\setminus\Gamma$, see \cite[Proposition, pag. 172]{SEM}.
\item The following quantitative estimate holds (see \cite[(14), pag. 174]{SEM})
\begin{equation}
\absol{\frac{\partial{\mathcal{E}_{0}(f)}}{ { \partial x_{i} } } (x)}\leq c (dist(x, \Gamma))^{\mu-1}, \, \text{for}\, x \in \mathbb{R}^{3}\setminus\Gamma.
\end{equation}
\end{itemize}
It is necessary to go further and to express the essential fact that under some specific relation between $\mu$ and $\alpha(\Gamma)$ we have that
\begin{equation}\label{integrability}
{^{\psi^{\theta}}D}[f^{w}]\in L_{p}(\mathbb{R}^{3}, \mathbb{H(\mathbb{R})})\ \mbox{for}\; \displaystyle p<\frac{3-\alpha(\Gamma)}{1-\mu}.
\end{equation}
This follows in much by the same methods as \cite[Proposition 4.1]{AB}.
\section{Auxiliary results on $\psi^{\theta}$-hyperholomorphic function theory.}
It is a well-known fact that in proving the existence of the boundary value of the Cauchy transform via the Plemelj-Sokhotski formulas, the solvability of the jump problem is an easy task whenever the data is a H\"older continuous function and the boundary of the considered domain is assumed sufficiently smooth. But by far much more subtle is the case where it can be thought of as a fractal surface. Then the standard method is no longer applicable, and it is necessary to introduce an alternative way of defining Cauchy transform, where a central role is played by the Teodorescu operator involving fractal dimensions. This is the idea behind the proofs of the following auxiliary results.
\begin{theorem} \label{thm 6} Let $f\in \text{Lip}_{\mu}(\Gamma,\, \mathbb{H(\mathbb{C})})$, $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$. Then the function $f$ can be represented as $f=\left.F^{+}\right|_{\Gamma}-\left.F^{-}\right|_{\Gamma}$, where $F^{\pm}\in \text{Lip}_{\nu}(\Omega_{\pm}\cup\Gamma)\cap \ker\left( {^{\psi^{\theta}}}D\right)$ for some $\nu<\mu$, $F^{\pm}$ are given by
\begin{equation}\label{tc}
F^{\pm}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right](x)+ f^{w}(x), \quad x\in\big({\Omega}_{\pm}\cup\Gamma\big),
\end{equation}
where
\begin{equation}
{^{\psi^{\theta}}}T[v](x):=\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(x-\xi) \,v(\xi) }\,dm(\xi), \quad x\in \mathbb{R}^{3}.
\end{equation}
is the well-defined Teodorescu transform for the $\mathbb{H(\mathbb{C})}$-valued function $v$, see \cite{KVS}.
\end{theorem}
\begin{proof}
Since ${f}^{w}={f}_{1}^{w}+i {f}_{2}^{w}$ with ${f}_{k}^{w}:\Omega\cup\Gamma\to\mathbb{H}$, $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$, and by (\ref{integrability}) ${^{\psi^{\theta}}}D[{f}_{k}^{w}]\in L_{p}(\Omega,\, \mathbb{H})$ for some $p\in\left( 3, \,\displaystyle\frac{3-\alpha(\Gamma)}{1-\mu}\right)$. Then the integral on the right side of \eqref{tc} exists and represents a continuous function in the whole $\mathbb{R}^{3}$ (see \cite[Theorem 2.8]{GPB}). Hence, the functions $F^{\pm}$ possess continuous extensions to the closures of the domains $\Omega_{\pm}$ and they satisfy that $\left.F^{+}\right|_{\Gamma}-\left.F^{-}\right|_{\Gamma}=f$. By the property of the Teodorescu operator to still being a right inverse to the Cauchy-Riemann operator (see \cite{KVS}, p. 73), ${^{\psi^{\theta}}}D[F^{+}]=0$ and ${^{\psi^{\theta}}}D[F^{-}]=0$ in the domains $\Omega_{\pm}$, respectively.
\end{proof}
\begin{remark}
Uniqueness in the statement of Theorem 3.1 could be ensured introducing an additional requirement analogous to that in \cite[Theorem 6.6]{ABJ}
\end{remark}
In the remainder of this section we assume that $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$.
The following results are related to the problem of extending $\psi^{\theta}$-hyperholomorphically a $\mathbb{H(\mathbb{C})}$-valued H\"older continuous function.
\begin{theorem} \label{t1}
Let $f\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$ the trace of $F\in \text{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right).$ Then
\begin{equation}\label{c1}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}=0.
\end{equation}
Conversely, if \eqref{c1} is satisfied, then $f$ is the trace of $F\in \text{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$ for some $\nu<\mu$.
\begin{proof}
Sufficiency. As we can write $f=f_{1}+if_{2}$ and $F=F_{1}+iF_{2}$ with $f_{r}\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{R})}), r=1,2$ and $F_{r}\in \text{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\mathbb{H(\mathbb{R})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$. Then $f^{w}=f^{w}_{1}+if^{w}_{2}$ and
\begin{equation}
{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right]={^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f_{1}^{w}]\right]+i\;{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f_{2}^{w}]\right].
\end{equation}
Following \cite[Theorem 3.1]{ABMT}, let $F_{r}^*=f_{r}^{w}-F_{r}$, $\tilde{Q}_{k}$ the union of cubes of the mesh $\mathcal{M}_{k}$ intersecting $\Gamma$, $\Omega_{k}=\Omega_{+}\setminus \tilde{Q}_{k}$, $\Delta_{k}=\Omega_{+}\setminus\Omega_{k}$ and denote by $\Gamma_{k}$ the boundary of $\Omega_{k}$. Applying the definition of $\alpha(\Gamma)$, given $\varepsilon>0$ there is a constant $C(\varepsilon)$ such that $\mathcal{H}^{2}(\Gamma_{k})$ (the Hausdorff measure of $\Gamma_{k}$) is less or equal than $6C(\varepsilon)2^{k(\alpha(\Gamma)-2+\varepsilon)}$.
Since $F_{r}^*\in\text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$, $F_{r}^*|_{\Gamma}=0$ and any point of $\Gamma_{k}$ is distant by no more than $C_{1}2^{-k}$, then
\begin{equation*}
\text{max}_{\xi\in\Gamma_{k}}\abs{F_{r}^*(\xi)}\leq C_{2}2^{-\mu k}
\end{equation*}
where $C_{1}$, $C_{2}$ denoted absolute constants.
Therefore, for $x\in\Omega_{-}$, let $s=dist(x,\Gamma)$
\begin{equation*}
\abso{\int_{\Gamma_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dS(\xi)}\leq C_{2}C(\varepsilon)\frac{6}{s^{2}}2^{(\alpha(\Gamma)-2-\mu+\varepsilon)}.
\end{equation*}
As $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$ the right-hand side of the previous inequality tends to zero as $k\to \infty$. By the Stokes formula, we have that
\begin{equation*}
\begin{split}
&\int_{\Omega_{+}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)=\lim_{k\to\infty}\bigg( \int_{\Delta_{k}}+\int_{\Omega_{k}}\bigg)\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)\\ &=\lim_{k\to\infty}\bigg( \int_{\Delta_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dm(\xi)-\int_{\Gamma_{k}}\mathscr{K}_{\psi^{\theta}}(\xi-x){^{\psi^{\theta}}}D[F_{r}^{*}](\xi)dS(\xi)\bigg)=0.
\end{split}
\end{equation*}
Then
\begin{equation}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{r}^{w}]\right]\right|_{\Gamma}={^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[F_{r}]\right]\right|_{\Gamma}=0.
\end{equation}
Necessity. If \eqref{c1} is satisfied we have
\begin{equation}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}={^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{1}^{w}]\right]\right|_{\Gamma}+i\;{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f_{2}^{w}]\right]\right|_{\Gamma}=0,
\end{equation}
and we take
\begin{equation}
\begin{split}
F(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[f^{w}]\right](x)+ f^{w}(x), \quad x\in \Omega_{+}\cup\Gamma.
\end{split}
\end{equation}
\end{proof}
\end{theorem}
In the same manner next theorem can be verified
\begin{theorem}
Let $f\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})$. If $f$ is the trace of a function $F\in \text{Lip}_{\mu}(\Omega_{-}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{-}}\right)$
\begin{equation}\label{c2}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}=-f.
\end{equation}
Conversely, if \eqref{c2} is satisfied, then $f$ is the trace of a function $F\in \text{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{-}}\right)$ for some $\nu<\mu$.
\end{theorem}
These two results generalize those of\cite[Theorem 3.1, Theorem 3.2]{ABMT}.
\begin{remark}
Similar results can be drawn for the case of right $\psi^{\theta}$-hyperholomorphic extensions. The only necessity being to replace in both theorems $\ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{\pm}}\right)$ by $\ker\left(\left.D^{\psi^{\theta}}\right|_{\Omega_{\pm}}\right)$ and ${^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}$ by $\left[D^{\psi^{\theta}}[f^{w}]\right]\, \left.{{^{\psi^{\theta}}}T}\right|_{\Gamma}$, where for every $\mathbb{H(\mathbb{C})}$-valued function $v$ we have set
\begin{equation}
[v]\, {{^{\psi^{\theta}}}T}=\int_{\Omega_{+}}{ v(\xi)\, \mathscr{K}_{\psi^{\theta}}(x-\xi) }\,dm(\xi), \quad x\in \mathbb{R}^{3}.
\end{equation}
The following theorem presents a result connecting two-sided $\psi^{\theta}$-hyperholomorphicity in the domain $\Omega_{+}$ and it is obtained by application of the previous results
\begin{theorem}
If $F\in \text{Lip}_{\mu}(\Gamma,\mathbb{H(\mathbb{C})})\cap \ker\left(\left.^{\psi^{\theta}}D\right|_{\Omega_{+}}\right)$ has trace $\left.F\right|_{\Gamma}=f$, then the following assertions are equivalent:
\begin{itemize}
\item [1.] F is left and right $\psi^{\theta}$-hyperholomorphic in $\Omega_{+}$,
\item [2.] ${^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[f^{w}]\right]\right|_{\Gamma}= \left[D^{\psi^{\theta}}[f^{w}]\right]\, \left.{{^{\psi^{\theta}}}T}\right|_{\Gamma}$.
\end{itemize}
\begin{proof}
The proof is obtained reasoning as in \cite[Theorem 3.3]{ABMP}.
\end{proof}
\end{theorem}
\end{remark}
\section{Main results}
In this section our main results are stated and proved. They give sufficient conditions for solving the Problems $(I)$ and $(II)$.
Let $\mathscr{M}_{\psi^{\theta}}^{*}$ be the subclass of vector fields $\mathbf{f}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ defined by
\begin{equation} \label{set}
\mathscr{M}_{\psi^{\theta}}^{*}:=\left\{\mathbf{f}: \int_{\Omega_{+}}{\left\langle\mathscr{K}_{\psi^{\theta}}(x-\xi)\,,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0, \; x\in\Gamma \right\},
\end{equation}
where $m$ denotes the Lebesgue measure in $\mathbb{R}^{3}$. The set $\mathscr{M}_{\psi^{\theta}}^{*}$ can be seen as a fractal version of the corresponding class in \cite{ZMS}, which can be described in purely physical terms.
\begin{theorem} \label{TH1}
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$. Then the problem (I) is solvable if
\begin{equation}
\begin{split}
\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)&:=\left({\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}\right)}\cos\theta-\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}+\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\sin\theta\right)\textbf{i}\\ & +\left(\displaystyle {-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}\sin\theta+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}\cos\theta}\right)\textbf{j}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}\cos\theta+\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}\sin\theta}\right)\textbf{k}\in\mathscr{M}_{\psi^{\theta}}^{*}.
\end{split}
\end{equation}
\begin{proof}
It is enough to prove that
\begin{equation}
\mathbf{F^{\pm}}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in\big({\Omega}_{\pm}\cup\Gamma\big),
\end{equation}
are vector fields.
Observe that
\begin{equation}
\notag
\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)(x)=-\int_{\Omega_{+}}{\left\langle \mathscr{K}_{\psi^{\theta}}(x-\xi),\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) \right\rangle }\,dm(\xi), \quad x\in \Omega_{\pm},
\end{equation}
\begin{equation}
\notag
\Delta\left(\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\right)(x)=0, \quad x\in \Omega_{\pm}
\end{equation}
and
\begin{equation}
\notag
\text{Sc}\left.\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\right|_{\Gamma}=0,
\end{equation}
because $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. Therefore $\text{Sc}\left({^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right)\equiv 0$ in $\Omega_{\pm}$. Then $\mathbf{F^{\pm}}(x)$
are vector fields.
\end{proof}
\end{theorem}
\begin{theorem} \label{TH2}
Let $\mathbf{f}$ $\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that\\ $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. If $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$, then
\begin{equation}\label{c3}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }dm(\xi)\\ &=\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }dm(\xi), \quad t\in \Gamma,
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)&=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}+\left(\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}\right)\sin\theta-\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}+\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\cos\theta.
\end{split}
\end{equation}
Conversely, if \eqref{c3} is satisfied, then $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$.
\begin{proof}
Suppose that $\mathbf{f}$ $\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$. Therefore
\begin{equation*}
{^{\psi^{\theta}}}T\left.\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right]\right|_{\Gamma}=0,
\end{equation*}
by Theorem \ref{t1}.
Of course
\begin{equation*}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }\,dm(\xi)\\ &=\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }\,dm(\xi), \quad t\in \Gamma,
\end{split}
\end{equation*}
as is easy to check.
Now, if \eqref{c3} is satisfied. Set
\begin{equation}
\mathbf{F^{+}}(x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in\big(\Omega_{+}\cup\Gamma\big).
\end{equation}
As $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$, $\mathbf{F^{+}}$ is a generalized Laplacian vector field in $\Omega_{+}$. By Theorem \ref{thm 6}, $\left.\mathbf{F^{+}}\right|_{\Gamma}=\mathbf{f}$, which completes the proof.
\end{proof}
\end{theorem}
The method of proof carries to domain $\Omega_{-}$. Indeed, we have
\begin{theorem} \label{TH3}
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that\\ $\text{Vec}\left( -{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\in\mathscr{M}_{\psi^{\theta}}^{*}$. If $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ which vanishes at infinity, then
\begin{equation}\label{c4}
\begin{split}
&\int_{\Omega_{+}}{\mathscr{K}_{\psi^{\theta}}(t-\xi)\; \text{Sc}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right) }\,dm(\xi)\\ &-\int_{\Omega_{+}}{\left[\mathscr{K}_{\psi^{\theta}}(t-\xi)\, ,\,\text{Vec}\left(-{^{\psi^{\theta}}}\text{div}[\mathbf{f}^{w}]+{^{\psi^{\theta}}}\text{rot}[\mathbf{f}^{w}]\right)\right] }\,dm(\xi)=-\mathbf{f}(t), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c4} is satisfied, then $\bf{f}$ is the trace of a generalized Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$, which vanishes at infinity.
\end{theorem}
\begin{remark} The mains results of this paper are generalizations of those in \cite{ABMP}, where is considered the operator Moisil-Teodorescu
\begin{equation}
D_{MT}:=\textbf{i}\frac{\partial }{\partial x_{1}}+\textbf{j}\frac{\partial }{\partial x_{2}}+\textbf{k}\frac{\partial }{\partial x_{3}}.
\end{equation}
Applying the operator $D_{MT}$ to $\mathbf{h}^{w}:=\mathbf{f}^{w}_{1}\textbf{i}+\mathbf{f}^{w}_{2}\textbf{j}+\mathbf{f}^{w}_{3}\textbf{k}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ we get
\begin{equation}
\begin{split}
D_{MT}[\mathbf{h}^{w}]&=-\text{div}[\mathbf{h}^{w}]+\text{rot}[\mathbf{h}^{w}]\\ &=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}+\left(\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}\right)\textbf{i}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}}\right)\textbf{j}+\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}}\right)\textbf{k}.
\end{split}
\end{equation}
For abbreviation, we let $D_{MT}[\mathbf{h}^{w}]$ stand for
\begin{equation} \label{2}
\begin{split}
D_{MT}[\mathbf{h}^{w}]=\left[D_{MT}[\mathbf{h}^{w}]\right]_{0}+\left[D_{MT}[\mathbf{h}^{w}]\right]_{1}\textbf{i} +\left[D_{MT}[\mathbf{h}^{w}]\right]_{2}\textbf{j}+\left[D_{MT}[\mathbf{h}^{w}]\right]_{3}\textbf{k}.
\end{split}
\end{equation}
On the other hand, setting $\mathbf{f}^{w}:=\mathbf{f}^{w}_{1}\textbf{i}+\mathbf{f}^{w}_{3}\textbf{j}+\mathbf{f}^{w}_{2}\textbf{k}\in C^{1}(\Omega, \mathbb{C}^{3})\cap\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ it follows that
\begin{equation}
\begin{split}
{^{\psi^{0}}D}[\mathbf{f}^{w}]&=-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{3}}+\left({\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{3}}-\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{2}}}\right)\textbf{i}\\ & +\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{2}}-\frac{\partial \mathbf{f}^{w}_{2}}{\partial x_{1}}}\right)\textbf{j}+\left(\displaystyle {\frac{\partial \mathbf{f}^{w}_{3}}{\partial x_{1}}-\frac{\partial \mathbf{f}^{w}_{1}}{\partial x_{3}}}\right)\textbf{k}.
\end{split}
\end{equation}
The above expression may be written as
\begin{equation} \label{1}
\begin{split}
{^{\psi^{0}}D}[\mathbf{f}^{w}]=\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{0}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{1}\textbf{i}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{2}\textbf{j}+\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{3}\textbf{k}.
\end{split}
\end{equation}
It is worth noting that under the correspondence $\left(\mathbf{f}^{w}_{1},\,\mathbf{f}^{w}_{2},\,\mathbf{f}^{w}_{3}\right)\, \leftrightarrow \, \left(\mathbf{f}^{w}_{1},\,\mathbf{f}^{w}_{3},\,\mathbf{f}^{w}_{2}\right)$ we can assert that
\begin{equation}\label{equiv}
D_{MT}[\mathbf{h}^{w}]=0\,\Longleftrightarrow \, {}{^{\psi^{0}}D}[\mathbf{f}^{w}]=0,
\end{equation}
which follow from
\begin{align*}
\left[D_{MT}[\mathbf{h}^{w}]\right]_{0} &=\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{0} ,\\
\left[D_{MT}[\mathbf{h}^{w}]\right]_{1} & =- \left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{1},\\
\left[D_{MT}[\mathbf{h}^{w}]\right]_{2} & =-\left[{^{\psi^{0}}D}[\mathbf{f}^{w}\right]_{3}, \\
\left[D_{MT}[\mathbf{h}^{w}]\right]_{3} & =-\left[{^{\psi^{0}}D}[\mathbf{f}^{w}]\right]_{2}.
\end{align*}
\end{remark}
\begin{remark}
In \cite{ABMP} is defined
\begin{equation}
\mathscr{M}^{*}:=\left\{\mathbf{f}: \frac{1}{4\pi}\int_{\Omega_{+}}{\left\langle \text{grad}\;\frac{1}{\abs{t-\xi}}\, ,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0, \, t\in\Gamma \right\}.
\end{equation}
For $\mathbf{h}:=\mathbf{f_{1}}\textbf{i}+\mathbf{f_{2}}\textbf{j}+\mathbf{f_{3}}\textbf{k} \in \mathscr{M}^{*}$ it is clear that
\begin{equation}
\begin{split}
\frac{1}{4\pi}\int_{\Omega_{+}}{\left\langle \text{grad}\;\frac{1}{\abs{t-\xi}}\, ,\,\mathbf{h}(\xi)\right\rangle}\,dm(\xi)=\int_{\Omega_{+}}{\left\langle\mathscr{K}_{\psi^{0}}(t-\xi)\, ,\,\mathbf{f}(\xi)\right\rangle}\,dm(\xi)=0,
\end{split}
\end{equation}
where $\mathbf{f}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{3}\textbf{j}+\mathbf{f}_{2}\textbf{k} \in \mathscr{M}^{*}_{\psi^{0}}$. Hence
$$\mathbf{h}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{2}\textbf{j}+\mathbf{f}_{3}\textbf{k}\in \mathscr{M}^{*} \iff \mathbf{f}:=\mathbf{f}_{1}\textbf{i}+\mathbf{f}_{3}\textbf{j}+\mathbf{f}_{2}\textbf{k} \in \mathscr{M}^{*}_{\psi^{0}}.$$
\end{remark}
From Theorems \ref{TH1}, \ref{TH2}, \ref{TH3} and the previous remarks the followings corollaries are obtained.
\begin{corollary} \cite[Theorem 2.2]{ABMP}.
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$. Then the reconstruction problem for the div-rot system is solvable if $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$.
\end{corollary}
\begin{corollary} \cite[Theorem 2.3]{ABMP}.
Let $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$. If $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$, then
\begin{equation}\label{c31}
\begin{split}
&\frac{1}{4\pi}\int_{\Omega_{+}}{ \text{grad}\;\frac{1}{\abs{t-\xi}}\; \text{div}[\mathbf{f}^{w}]}\,dm(\xi)\\ &=\frac{1}{4\pi}\int_{\Omega_{+}}{\left[ \text{grad}\;\frac{1}{\abs{t-\xi}}\,,\, \text{rot}[\mathbf{f}^{w}]\right] }\,dm(\xi), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c31} is satisfied, then $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{+}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$.
\end{corollary}
\begin{corollary} \cite[Theorem 2.4]{ABMP}.
Let $\mathbf{f}\in\mathbf{Lip}_{\mu}(\Gamma,\, \mathbb{C}^{3})$ such that $\displaystyle \mu>\frac{\alpha(\Gamma)}{3}$ and suppose that $\text{rot}[\mathbf{f}^{w}]\in\mathscr{M}^{*}$. If $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\mu}(\Omega_{-}\cup\Gamma, \, \mathbb{C}^{3})$ which vanishes at infinity, then
\begin{equation}\label{c42}
\begin{split}
&\frac{1}{4\pi}\int_{\Omega_{+}}{ \text{grad}\;\frac{1}{\abs{t-\xi}}\;\text{div}[\mathbf{f}^{w}]}\,dm(\xi)\\ &-\frac{1}{4\pi}\int_{\Omega_{+}}{\left[ \text{grad}\;\frac{1}{\abs{t-\xi}}\,,\,\text{rot}[\mathbf{f}^{w}]\right] }\,dm(\xi)=-\mathbf{f}(t), \quad t\in \Gamma.
\end{split}
\end{equation}
Conversely, if \eqref{c42} is satisfied, then $\bf{f}$ is the trace of a Laplacian vector field in $\mathbf{Lip}_{\nu}(\Omega_{-}\cup\Gamma,\, \mathbb{C}^{3})$ for some $\nu<\mu$, which vanishes at infinity.
\end{corollary}
\section*{Appendix. Criteria for the generalized Laplacianness of a vector field}
We continue to assume that $\Omega\subset \mathbb{R}^{3}$ is a Jordan domain with a fractal boundary $\Gamma$. Our interest here is to find necessary and sufficient conditions for the generalized Laplacianness of an vector field $\mathbf{F}\in\mathbf{Lip}_{\nu}(\Omega\cup\Gamma,\, \mathbb{C}^{3})$ in terms of its boundary value $\mathbf {f}:=\left.\mathbf {F}\right|_\Gamma$.
The inspiration for the following definition is that in \cite[Definition 2.1]{ARBR}.
\begin{defi} \label{dtc1}
Let $\Omega$ a Jordan domain with fractal boundary $\Gamma$. Then we define the Cauchy transform of $\mathbf{f}\in \mathbf{Lip}_{\mu}(\Gamma,\,\mathbb{C}^{3})$ by
\begin{equation}\label{tc2}
K_{\Gamma}^{*}[\mathbf{f}](x):=-{^{\psi^{\theta}}}T\left[{^{\psi^{\theta}}}D[\mathbf{f}^{w}]\right](x)+ \mathbf{f}^{w}(x), \quad x\in \mathbb{R}^{3}\setminus\Gamma.
\end{equation}
\end{defi}
Under condition $\displaystyle \frac{\alpha(\Gamma)}{3}<\mu \leq 1$ the Cauchy transform $K_{\Gamma}^{*}[\mathbf{f}]$ has continuous extension to $\Omega\cup\Gamma$ for every vector field $\mathbf {f}\in \mathbf{Lip}_{\mu}(\Gamma,\,\mathbb{C}^{3})$ (take a fresh look at Theorem \ref{thm 6}). On the other hand, using the properties of the Theodorescu operator (see \cite{KVS}, p. 73) we obtain that $K_{\Gamma}^{*}[\bf{f}]$ is left-$\psi^{\theta}$-hyperholomorphic in $\mathbb{R}^{3}\setminus\Gamma$. Note that $K_{\Gamma}^{*}[\mathbf{f}](x)$ vanishes at infinity.
Let us introduce the following fractal version of the Cauchy singular integral operator
\begin{equation*}
\mathcal{S}_{\Gamma}^{*}[\mathbf{f}](x):=2K^{*}_{\Gamma}[\mathbf{f}]^{+}(x)-f(x), \quad x\in\Gamma.
\end{equation*}
Here and subsequently, $K^{*}_{\Gamma}[\mathbf{f}]^{+}$ denotes the trace on $\Gamma$ of the continuous extension of $K^{*}_{\Gamma}[\mathbf{f}]$ to $\Omega\cup\Gamma$.
Let us now establish and prove the main result of this appendix, which gives necessary and sufficient conditions for the generalized Laplacianness of a vector field in terms of its boundary value.
\begin{theorem}
Let $\mathbf{F}\in\mathbf{Lip}_{\mu}(\Omega\cup\Gamma,\mathbb{C}^{3})$ with trace $\mathbf{f}=\left.\mathbf{F}\right|_{\Gamma}$. Then the following sentences are equivalent:
\begin{itemize}
\item [(i)] $\mathbf{F}$ is a generalized Laplacian vector field.
\item [(ii)] $\mathbf{F}$ is harmonic in $\Omega$ and $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$.
\end{itemize}
\begin{proof} Let $\mathbf{F}^{w}$ be the Whitney extension of $\mathbf{F}$ in $\mathbf{Lip}_{\mu}(\Omega\cup\Gamma,\mathbb{C}^{3})$. Suppose that $\mathbf{F}$ is a generalized Laplacian vector field in $\Omega$. Since ${^{\psi^{\theta}}}D[\mathbf{F}]=0$ in $\Omega$, it follows that $\mathbf{F}$ is harmonic. Also $\mathbf{F}^{w}$ is a Whitney extension of $\mathbf{f},$ i.e. $\mathbf{f}=\left.\mathbf{F}^{w}\right|_{\Gamma}$.
According to Definition \ref{dtc1}, with $\mathbf{f}^{w}$ replaced by $\mathbf{F}^{w}$, we get
\begin{equation*}
K_{\Gamma}^{*}[\mathbf{f}](x)=-\int_{\Omega}{\mathscr{K}_{\psi^{\theta}}(x-\xi)\, {{^{\psi^{\theta}}}D}[\mathbf{F}^{w}](\xi) }\,dm(\xi)+ \mathbf{F}^{w}(x)=\mathbf{F}(x), \quad x\in\Omega,
\end{equation*}
which imply that $K_{\Gamma}^{*}[\mathbf{f}]^{+}=\mathbf{f}$ and $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$.
Conversely, assume that $(ii)$ holds and define
\begin{equation}
\Psi(x):= \left\{
\begin{array}{ll}
K_{\Gamma}^{*}[\mathbf{f}](x), & x \in\Omega, \\
\mathbf{f}(x), & x \in \Gamma.
\end{array}
\right.
\end{equation}
Note that $\Psi(x)$ is left-$\psi^{\theta}$-hyperholomorphic function, hence harmonic in $\Omega$. Since $\mathcal{S}_{\Gamma}^{*}[\mathbf{f}]=\mathbf{f}$ in $\Gamma$, it follows that $K_{\Gamma}^{*}[\mathbf{f}]^{+}=\mathbf{f}$. Therefore $K_{\Gamma}^{*}[\mathbf{f}]$ is also continuous on $\Omega\cup\Gamma$.
As $\mathbf{F}-\Psi$ is harmonic in $\Omega$ and $\left.(\mathbf{F}-\Psi)\right|_{\Gamma}=0$ we have that $\mathbf{F}(x)=K_{\Gamma}^{*}[\mathbf{f}](x)$ for all $x \in\Omega,$ which follows from the harmonic maximum principle. Lemma \ref{two-sided} now forces $\mathbf{F}$ to be a generalized Laplacian vector field in $\Omega,$ and the proof is complete.
\end{proof}
\end{theorem}
\end{document} |
\begin{document}
\title{The role of coherence on two-particle quantum walks}
\author{Li-Hua Lu$^1$, Shan Zhu and You-Quan Li$^{1,2}$}
\affiliation{1. Zhejiang Institute of Modern Physics and Department of Physics,\\ Zhejiang
University, Hangzhou 310027, P. R. China\\
2. Collaborative Innovation Center of Advanced Microstructures, Nanjing, P. R. China
}
\begin{abstract}
We investigate the dynamical properties of the two-bosons quantum walk in system with different degrees of coherence, where the effect of the coherence on the two-bosons quantum walk can be naturally introduced. A general analytical expression of the two-bosons correlation function for both pure states and mixed states is given. We propose a possible two-photon quantum-walk scheme with a mixed initial state and find that the two-photon correlation function and the average distance between two photons can be influenced by either the initial photon distribution, or the relative phase, or the degree of coherence. The propagation features of our numerical results can be explained by our analytical two-photon correlation function.
Keywords: two-particle quantum walk, degree of coherence, two-photon correlation function, pure state, mixed state
\end{abstract}
\pacs{03.67.Ac, 03.67.Lx, 05.40.Fb, 05.90.+m}
\received{\today}
\maketitle
\section{Introduction}
As the quantum mechanical counterparts of the classical random walk~\cite{Ahar},
the quantum random walk has been increasingly receiving attentions
because of their potential applications range from quantum information to simulation of physical phenomena. For example, the quantum walk offers an advanced tool for building quantum algorithm~\cite{Moh,Shen,Sal}
that is shown to be of the primitive for universal quantum computations~\cite{und,Lov,chi,and}.
We know that the quantum walk include two main classes that are discrete-time quantum walk and continuous-time quantum walk~\cite{Jwa,Far}.
The continuous-time quantum walk can evolve continuously with time through tunneling between neighbors sites and does not require quantum coin to generate superposition of states.
This means the continuous-time quantum walk can be implemented
via a constant tunneling of quantum particles in several possible lattice sites.
So far, the quantum walks of single particles have been studied in experiments by using either classical waves~\cite{peret}, single photons~\cite{Sch,broo}, or single atoms~\cite{Kars,Wei}.
Additionally, quantum walks of two correlated photons trapped in waveguide lattices
were also studied in experiments~\cite{Yar,Alberto}.
Note that many-particle quantum walks can exhibit more fascinating quantum features in contrast to single-particle quantum walks. The reason is that single-particle quantum walks can be exactly mapped to classical wave phenomena~\cite{Knight} but for quantum walks of more than one indistinguishable particle, the classical theory can not provide sufficient descriptions. In Ref.~\cite{omar} , the authors theoretically studied the discrete-time quantum walk of two particles and demonstrated the distinctly nonclassical correlations. Meanwhile, the effect of interactions between particles on quantum walk of two indistinguishable particles was theoretically studied in Ref.~\cite{lahini,Qin}, where the system was assumed to be completely coherent and the influence of the degree of coherence was not studied. We know that except for the interaction between particles, the other factors, {\it e.g.}, the initial states, the quantum-walk parameters and the degree of coherence of the system, can also affect the features of two-particle quantum walks. Especially, we know that the major challenge to experimentally realize quantum walks of correlated particles is to find a low-decoherence system that can preserves the nonclassical features of quantum walks~\cite{Alberto}, which implies that the influence of the degree of coherence of the system on two-particle quantum walks is important. Then it is worthwhile to investigate the properties of two-particle quantum walks with attention to different degrees of coherence since the decoherence effects in quantum walks have potential algorithmic applications~\cite{kendon}.
In this paper, we propose a density matrix formulism to study the two-particle quantum walk
where the degrees of coherence can be naturally introduced.
With the help of Heisenberg equation of motion, we derive a general analytical expression
of the two-particle correlation function.
As a concrete example, we propose a possible two-photon quantum-walk scheme with a mixed initial state to exhibit the quantum features of the two-particle quantum walk
via the two-particle correlation and the average distance between the two particles.
Our result exhibits that the propagation of the two particles depends
not only on the initial distribution of the two particles
but also on the relative phase and the degree of coherence of the system.
Such a propagation feature can be
explained by our analytical two-particle correlation function. In the next section, we present the model and derive the analytical expression of the two-particle correlation. In Sec.~\ref{sec:two-photon}, we propose a concrete scheme to show some dynamical features of two-particle quantum walks. Our main conclusions are summarized in Sec.~\ref{sec:conc}.
\section{A general formulation }
We consider a two-particle quantum walk in a one-dimensional lattice space.
The propagation of the two particles is described by the evolution of the state
of a tight-binding model,
\begin{equation}
H=-\sum_{q}T_{q,q+1}(\hat{a}_q^\dagger\hat{a}_{q+1} + \mathrm{h.c.})
+\sum_q \beta_q\hat{a}_q^\dagger\hat{a}_q,
\end{equation}
where the operators $a_{q}^\dag$ and $a_{q}$ create and annihilate a bosonic particle at site $q$, respectively.
Here the parameter $T_{q,q+1}$ refers to the tunneling strength of particles between the nearest neighbor sites, and
\begin{equation}
\beta_q=T_{q+1,q}+T_{q-1,q}.
\end{equation}
Note that the above form of $\beta_q$ was picked to keep the probability conservation in the proposal of continuous-time quantum walk via decision tree~\cite{Far} . Now more generally, the value of $\beta_q$ can be arbitrary due to the probability conservation is naturally satisfied in quantum mechanics.
If the tunneling strength $T_{q,q+1}$ is a constant, $\beta_q$ will become a constant for the periodical boundary condition. In this case, the value of $\beta_q$ does not affect the dynamical properties of the quantum-walk system. Whereas, for the open boundary condition, the values of $\beta_q$ for the two boundary sites are different from that for the other sites. This can naturally introduce two defects to the quantum-walk system. Note that the effect of defects on single-particle quantum walks was studied in Ref.~\cite{Li}.
Since we consider a two-bosons system,
the Fock bases describing the system are
\begin{equation}
\label{eq:fockstate}
\ket{1}_q\ket{1}_r=\frac{1}{\sqrt{1+\delta_{q,r}}}\hat{a}_q^\dagger\hat{a}_r^\dagger\ket{\mathrm{vac}},
\end{equation}
where $\delta_{q,r}$ denotes the Kronecker delta.
Equation (\ref{eq:fockstate}) represents a two-particle state with one
on the $q$th site and the other one on the $r$th site.
Note that $\ket{1}_q\ket{1}_r$ is regarded as identical to $\ket{1}_r\ket{1}_q$
for indistinguishable particles that we considered.
Meanwhile, the two particles can be in the same site ({\it i.e.}, $q=r$)
for bosonic particle that we considered.
Thus the Hilbert space expanded by the aforementioned Fock bases is of $D=L(L+1)/2$ dimension.
Here $L$ denotes the number of the sites.
We know that the propagation of the two particles is determined not only by the property of the waveguide lattice but also by the two-particle input state.
If the two particles are in a pure state at the initial time,
the two-particle input state can be expressed as a wavefunction, namely, a coherent superposition of the Fock bases,
\begin{equation}\label{eq:purstates}
\ket{\psi}=\sum_{q,r}c_{q,r}\ket{1}_q\ket{1}_r,
\end{equation}
where $\sum_{q,r}|c_{q,r}|^2=1$.
However, if the two particles are in a mixed state at the initial time,
the two-particle input state needs to be described by a density matrix
rather than wavefunction.
Such a density matrix is given by
\begin{equation}
\label{eq:densitymatix}
\rho=\sum_{qr,q'r'}\rho_{qr,q'r'}\bigl(\ket{1}_q\ket{1}_r\bigr)\bigl(\bra{1}_{q'}\bra{1}_{r'}\bigr),
\end{equation}
which is a $D\times D$ matrix.
We know that $\textrm{Tr}\rho^2\leq(\textrm{Tr}\rho)^2$
where the equal sign holds only for pure states.
In the following,
we will focus on the two-particle quantum walk for the mixed input states.
Now we are in the position to study the propagation of the two particles
with the help of Heisenberg equation of motion for the creation operators,
namely,
\begin{equation}
\label{eq:dye}
i\frac{\partial \hat{a}_q^\dagger}{\partial t}=\beta_q\hat{a}_q^\dagger+T_{q,q+1}\hat{a}_{q+1}^\dagger
+T_{q,q-1}\hat{a}_{q-1}^\dagger,
\end{equation}
where we set $\hbar=1$ for simplicity in calculation.
The creation operator $\hat{a}_q^\dagger$ at any time can be obtained with the help of Eq.~(\ref{eq:dye}),
\begin{equation}
\label{eq:creation}
\hat{a}_q^\dagger(t)=\sum_r U_{q,r}(t)\hat{a}_r^\dagger(0),\quad
U(t)=e^{-iHt},
\end{equation}
where $U_{q,r}(t)$ is the probability amplitude of a single particle transiting
from the $q$th waveguide to the $r$th one.
To exhibit the quantum behaviors of the two-particle quantum walk,
let us firstly evaluate the two-particle correlation function $\Gamma_{k,l}(t)=\ave{\hat{a}_k^\dagger(t)\hat{a}_l^\dagger(t)\hat{a}_l(t)\hat{a}_k(t)}$
which manifests the probability that the two particles are coincident in the $k$th and the $l$th waveguide~\cite{Yar,Mattle}.
Since the two-particle input state can be described by the density matrix given
in Eq.~(\ref{eq:densitymatix}), the expectation value of any observable of the system
can be calculated via $\ave{\hat{O}(t)}= \textrm{Tr}(\hat{O}(t)\rho)$.
Then we obtain an expression of
two-particle correlation function
\begin{widetext}
\begin{eqnarray}\label{eq:twocorre}
&&\Gamma_{k,l}(t)=\sum_{q\neq r,q'\neq r'}\rho_{qr,q'r'}\Bigl(U_{kq'}U_{lr'}U_{lq}^*U_{kr}^*+U_{kq'}U_{lr'}U_{lr}^*U_{kq}^*
+U_{kr'}U_{lq'}U_{lq}^*U_{kr}^*+U_{kr'}U_{lq'}U_{lr}^*U_{kq}\Bigr)
\nonumber\\
&&+\sum_{q,q'\neq r'}\sqrt{2}\rho_{qq,q'r'}\Bigl(U_{kq'}U_{lr'}U_{lq}^*U_{kq}^*+U_{kr'}U_{lq'}U_{lq}^*U_{kq}^*\Bigr)
+\sum_{q\neq r,q'}\sqrt{2}\rho_{qr,q'q'}\Bigl(U_{kq'}U_{lq'}U_{lq}^*U_{kr}^*+U_{kq'}U_{lq'}U_{lr}^*U_{kq}^*\Bigr)
\nonumber\\
&&+\sum_{q,q'}2\rho_{qq,q'q'}U_{kq'}U_{lq'}U_{lq}^*U_{kq}^*,
\end{eqnarray}
\end{widetext}
which presents a general form for either pure initial input states or mixed ones.
One can obtain the two-particle correlation at any time
as long as the density matrix corresponding to the input state is given. with the help of such an expression of two-particle correlation function, many dynamical features of two-particle quantum walk can be explained.
\section{Two-photon quantum walk for a concrete input state}\label{sec:two-photon}
In order to expose the quantum properties of two-particle quantum walks more clearly, we turn to a concrete example where the particles are assumed to be photons. We know that each beam can become two coherent beams after propagating through a grating~\cite{sza} and the pure two-photon input states can be experimentally realized via injecting two coherent beams into waveguide lattice~\cite{Yar}. Then we suppose that there are two incoherent light beams and the relation of their intensity is $\cos^2{\delta}$:$\sin^2{\delta}$. The two incoherent beams propagate through two gratings, respectively, and then simultaneously inject into the waveguide arrays. The two incoherent beams will create two pure two-photon states $\psi_1$ and $\psi_2$, respectively~\cite{Yar,sza}. Because the initial two beams are not coherent, the initial state of the system needs to be described by the density matrix
\begin{equation}
\rho=\cos^2\delta \ket{\psi_1}\bra{\psi_1}
+\sin^2\delta \ket{\psi_2}\bra{\psi_2}.
\end{equation}
As an example, we take $\psi_1=\cos{\frac{\theta}{2}}\ket{2}_1+\sin{\frac{\theta}{2}}e^{i\phi}\ket{2}_0$
and $\psi_2=\ket{2}_1$, then the initial density matrix is
\begin{align}
\label{eq:exrho}
\rho = \rho_{00,11}\ket{2}_0\bra{2}_1 + \rho_{11,00}\ket{2}_1\bra{2}_0
+ \rho_{11,11}\ket{2}_1\bra{2}_1 + \rho_{00,00}\ket{2}_0\bra{2}_0
\end{align}
with $\rho_{00,11}=\cos^2\delta\cos{\frac{\theta}{2}}\sin{\frac{\theta}{2}}e^{i\phi}$,
$\rho_{11,00}=\rho_{00,11}^*$,
$\rho_{11,11}=\cos^2\delta\cos^2{\frac{\theta}{2}}+\sin^2\delta$,
and
$\rho_{00,00}=\cos^2\delta\sin^2{\frac{\theta}{2}}$.
Here $\ket{2}_q$ stands for
$\ket{1}_q\ket{1}_q$ whose definition has been given
in Eq.~(\ref{eq:fockstate}). We can find that the other matrix elements of $\rho$
are zeros except for the above four elements. Since the above four elements can be changed via $\delta$, $\theta$ and $\phi$,
without losing the generality, we redefine them as
$\rho_{00,00}=\alpha$, $\rho_{11,11}=1-\alpha$ and $\rho_{00,11}=\rho_{11,00}^*=e^{i\phi}\sqrt{\eta-1-4\alpha^2+4\alpha}/2$ with $0\leq\alpha\leq 1$. Here the parameter $\eta=2\textrm{Tr}(\rho^2)-1$ $(0\leq\eta\leq1)$ is introduced to characterize the degree of coherence~\cite{lhlu}.
We have $\eta=1$ when the system is in a pure state, otherwise $\eta<1$. With the help of Eq.~(\ref{eq:twocorre}),
the two-photon correlation of the system for the mixed state shown in Eq.~(\ref{eq:exrho}) yields
\begin{align}
\label{eq:extwocre}
\Gamma_{q,r} = 2\gamma\textrm{Re}(e^{i\phi}U_{q0}U_{r0}U^*_{r1}U^*_{q1})
+2\alpha |U_{r0}U_{q0}|^2+2(1-\alpha)|U_{r1}U_{q1}|^2,
\end{align}
where $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$.
This implies that the two-photon correlation depends not only on the initial probability distribution of the two photons but also on the degree of coherence and the relative phase of the system at the initial time. The first term in Eq.~(\ref{eq:extwocre}) is a coherent one that reveals well the quantum nature of two-photon quantum walk. Taking a pure initial state ({\it i.e.}, $\eta=1$) as an example, the two-photon correlation function~(\ref{eq:extwocre}) becomes $\Gamma=|\sqrt{2\alpha}e^{i\phi}U_{r0}U_{q0}+\sqrt{2(1-\alpha)}U_{r1}U_{q1}|^2$, which implies that the two-photon correlation can take place when the two photons from the 0th site propagate to the $q$th and the $r$th sites, respectively, or when the two photons from the 1th site to the $q$th and the $r$th sites, respectively. Due to the two photons are indistinguishable, the two paths can interfere, which is essentially the Hanbury Brown Twiss(HBT) interference~\cite{Han}.
Now we investigate the quantum features of a two-photon quantum walk
by considering a waveguide arrays consisting of (2$l$+1) identical waveguides.
In this case, the tunneling strengths between nearest-neighbor arrays are all the same, {\it i.e.}, $T_{q,r}=C$ with $C$ being a constant, and $\beta_q$ becomes a constant $2C$ for the periodical boundary condition we considered.
Then $U_{q,r}(t)$ becomes $e^{i2Ct}i^{q-r}J_{q-r}(2Ct)$
where $J_q$ is the $q$th order Bessel function~\cite{Led,Yariv}.
With the help of Eq.~(\ref{eq:extwocre}),
we can write out the two-photon correlation function in terms of Bessel functions.
\begin{widetext}
\begin{align}
\label{eq:twocrebf}
\Gamma_{q,r}(\tau) =-2\gamma\cos{\phi} J_q(\tau)J_r(\tau)J_{r-1}(\tau)J_{q-1}(\tau)
+2\alpha [ J_q(\tau) J_r(\tau) ]^2
+2(1-\alpha)[ J_{r-1}(\tau) J_{q-1}(\tau) ]^2.
\end{align}
\end{widetext}
where $\tau=2Ct$.
\begin{figure}
\caption{(Color online) Two-photon correlation at time $t=4(1/C)$ for different initial conditions. The initial condition is (a) $\alpha=1$, $\eta=1$, $\phi=0$ (b) $\alpha=0.5$, $\eta=0$, $\phi=0$
(c) $\alpha=0.5$, $\eta=0.5$, $\phi=0$ (d) $\alpha=0.5$, $\eta=1$, $\phi=0$ (e) $\alpha=0.5$, $\eta=0.5$, $\phi=\pi$, and (f) $\alpha=0.5$, $\eta=1$, $\phi=\pi$. }
\label{fig:tcou}
\end{figure}
In Fig.~\ref{fig:tcou},
we plotted the two-photon correlation matrix at time $t=4({1}/{C})$ for different initial conditions, where $1/C$ is the unit of time ({\it i.e.}, the time $t$ is in the unit of the inverse of tunneling strength). We can see that each particle can be found on either side of origin after propagation, which is reflected in the four symmetric peaks in Fig.~\ref{fig:tcou} (a). For this case ({\it i.e.}, $\alpha=1$ and $\eta=1$), the system is in a pure state and Eq.~(\ref{eq:extwocre}) becomes $\Gamma_{q,r}=2|U_{r0}U_{q0}|^2$ which is the same as the result in Ref.~\cite{Yar}. Such a correlation function is just a product of the two classical probability distribution, so there is no interference and the photons propagate in the ballistic direction. From Fig.~\ref{fig:tcou} (b), we can find that just like Fig.~\ref{fig:tcou} (a), the two photons also favor to localize at the four corners of the correlation map. The reason is that there is no interference because the system is completely incoherent, which is confirmed by that the coherent term in Eq.~(\ref{eq:extwocre}) vanishes in the case of $\eta=0$. Whereas, with the increase of the degree of coherence, the coherent term emerges in the two-photon correlation function $\Gamma_{q,r}$, so the $\Gamma_{q,r}$ exhibits the properties of interference. Due to the existence of the Hanbury Brown-Twiss (HBT) interference, two local maximums emerge in the off-diagonal regions of the correlation matrix (see Fig.~\ref{fig:tcou} (c) and (d)). That implies that the two photons favor to far from each other which is in contrast to the case of Fig.~\ref{fig:tcou} (e) and (f) where except the initial relative phase $\phi$, the other parameters are the same as those in Fig.~\ref{fig:tcou} (c) and (d), respectively. This is reasonable because the coherent term in Eq.~(\ref{eq:twocrebf}) is in proportion to $\cos{\phi}$ for the periodical waveguide lattice we considered.
Additionally, comparing the values of the maximums in Fig.~\ref{fig:tcou} (c) and (d),
it is easy to find
that the larger the degree of coherence is,
the more distinct the interference effect of the system will be.
\begin{figure}
\caption{(Color online) The time evolution of the distance between two photons for different initial conditions. The parameters are $\alpha=0.5$, and $\eta=1$ (left panel), $\eta=0.5$ (right panel).}
\label{fig:distanceevo}
\end{figure}
\begin{figure}
\caption{(Color online) The dependence of the distance between two photons at time $t=4(1/C)$ on the degree of coherence (left panel) and the initial relative phase (right panel). The parameter is $\alpha=0.5$.}
\label{fig:distance}
\end{figure}
To exhibit the propagation properties of the two photons,
we also calculate the average distance between the two photons,
\begin{eqnarray}\label{eq:distance}
\displaystyle d&=&-2\gamma \cos{\phi}\sum_{q>r}(q-r)J_q(\tau)J_r(\tau)J_{r-1}(\tau)J_{q-1}(\tau)\nonumber\\
&&+\sum_{q>r}(q-r)\Bigl(2\alpha [ J_q(\tau) J_r(\tau) ]^2
+2(1-\alpha)[ J_{r-1}(\tau) J_{q-1}(\tau) ]^2\Bigr).
\end{eqnarray}
Here the first term is a coherence one that is affected by the degree of coherence of the system due to $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$.
We plot the time evolution of the distance between the two photons in Fig.~\ref{fig:distanceevo},
we can see that the distance between two photons is affected not only by the relative phase but also by the degree of coherence.
In Fig.~\ref{fig:distance}, we plot the dependence of the
distance $d$ at time $t=4(1/C)$ on the degree of coherence and the initial relative phase.
From the left panel of this figure, we can see that the distance between two photons becomes larger with the increase of the degree of coherence when $\phi=0$, which is contrast to the case of $\phi=\pi$.
The reason for this phenomenon is that the HBT interference makes the two photons far from each when $0\leq\phi<\pi/2$,
which can be confirmed by Fig.~\ref{fig:tcou} (c) and (d) where there are two maximums in the off-diagonal regions.
Therefore, the distance between the two photons becomes larger with the increase of the degree of coherence due to the fact that the increase of the degree of coherence makes the interference effect more significant.
Whereas, the case of $\pi/2\leq\phi<\pi$ is in contrast to that of $0\leq\phi<\pi/2$ because the two photons favor to stay together when $\pi/2\leq\phi<\pi$, which can be confirmed by Fig.~\ref{fig:tcou} (e) and (f).
Note that when $\phi=\pi/2$, the interference term in the two-photon correlation function becomes zeros, so the degree of coherence does not affect the distance between two particles (see the dot-symbol line in the left panel of Fig.~\ref{fig:distance}).
The right panel of Fig.~\ref{fig:distance} exhibits that
the relative phase of the system at the initial time can affect the distance between two photons in the case of $\eta>0$ but such an effect vanishes in the case of $\eta=0$.
That is reasonable because the distance between two photons is in proportion to $\cos{\phi}\sqrt{\eta-1-4\alpha^2+4\alpha}$ which can be found in Eq.~(\ref{eq:distance}). Additionally, we calculate the von Neumann entropy to show the evolution of the entanglement of the system. We split the system into two halves, $L$ and $R$, in the center of the system, and build the reduced density matrix $\rho_L$ of the subsystem $L$ at any time~\cite{Schach}. Then we can calculate the von Neumann entropy of $\rho_L$ as
\begin{equation}
S=-\sum_i\lambda_i\log_2\lambda_i,
\end{equation}
where $\lambda_i$ are the non-zero eigenvalues of the matrix $\rho_L$. In Fig.~\ref{fig:von}, we plot the time evolution of the von Neumann entropy of the left half of the system for different initial conditions.
\begin{figure}
\caption{(Color online) The time evolution of the von Neumann entropy of the set of sites on the left part of the system. The parameters are $\eta=1$, $\phi=0$, and $L=15$.}
\label{fig:von}
\end{figure}
\section{conclusion}\label{sec:conc}
We proposed a density matrix formulism to study the properties of two-particle quantum walks where the effect of coherence was introduced naturally. We gave the general analytical expression of the two-particle correlation function which is correct for systems in both mixed states and pure states. We suggested a possible two-photon scheme to exhibit the more fascinating quantum features of two-particle random walks with mixed initial states. For such a concrete scheme, we calculated the two-photon correlation and the average distance between the two photons. The corresponding results manifested that the propagation of the two photons depends not only on the initial distribution of the two photons but also on the relative phase and the degree of coherence of the system. Such propagation features of the two photons were explained with the help of the analytical expression of the two-particle correlation function we obtained.
The work is supported by the NBRP of China (2014CB921201), the NSFC (11104244 and 11274272, 11434008), and by the Fundamental Research Funds for Central Universities.
\end{document} |
\begin{equation}gin{itemize}n{document}
\title{\bf A Posteriori Error Estimates for Self-Similar Solutions to the Euler
Equations
}
\vskip 4emip 2emkip 1em
\author{
Alberto Bressan and Wen Shen \\ \, \\
Department of Mathematics, Penn State University.\\
University Park, PA~16802, USA.\\
\\ e-mails:~axb62@psu.edu, ~wxs27@psu.edu}
\date{Dec 15, 2019}
\maketitle
\begin{equation}gin{itemize}n{abstract}
The main goal of this paper is to analyze a family of ``simplest possible" initial data for which,
as shown by numerical simulations,
the incompressible Euler equations have multiple solutions.
We take here a first step toward a rigorous validation of these numerical results.
Namely, we consider the system of equations corresponding to a self-similar solution,
restricted to a bounded domain with smooth boundary.
Given an approximate solution
obtained via a finite dimensional Galerkin method,
we establish a posteriori error bounds on the distance between the numerical
approximation and the exact solution having the same boundary data.
\epsilonnd{abstract}
\vskip 4emip 2emkip 1em
\section{Introduction}
\setcounter{equation}{0}
The flow of a homogeneous, incompressible, non-viscous fluid
in ${\mathbb R}^2$ is modeled by the
Euler equations
\begin{equation}l{E}
\left\{\begin{equation}gin{itemize}n{array}{rll}
u_t +(u\centerlinedot\noindentabla) u&=~-\noindentabla p &\qquad \textrm{(balance of momentum),}\\
\textrm{div}\, u&= ~0 & \qquad\textrm{(incompressibility
condition).}
\epsilonnd{array}\right.
\epsilonnd{equation}
Here $u=u(t,x)$ denotes the velocity of the fluid, while the scalar function
$p$ is a pressure.
The condition $\hbox{div } u=0$ implies the existence of a stream
function $\psi$ such that
\begin{equation}l{psi}
u~=~\noindentabla^\perp\psi,\qquad\qquad (u_1,u_2)~=~(-\psi_{x_2}, \, \psi_{x_1}).\epsiloneq
Denoting by $\omega = \centerlineurl u = (- u_{1, x_2}+ u_{2, x_1})$
the vorticity of the fluid,
it is well known that
the Euler equations (\ref{E}) can be reformulated
in terms of the system
\begin{equation}l{E2}
\left\{\begin{equation}gin{itemize}n{array}{rl} \omega_t + \noindentabla^\perp \psi\centerlinedot\noindentabla\omega
&=~0,\\[3mm]
{\mathcal D}elta\psi&=~\omega.\epsilonnda\right.\epsiloneq
The velocity $u$ is then recovered from the vorticity
by the Biot-Savart formula
\begin{equation}l{BS}u(x)~=~{1\overlineer 2\pi}\itemnt_{{\mathbb R}^2} {(x-y)^\perp\overlineer|x-y|^2}
\,\omega(y)\, dy.
\epsiloneq
Our eventual goal is to construct ``simplest possible" initial data
for which the equations (\ref{E}) have multiple solutions. Numerical
simulations, shown in Figures~\ref{f:EC1spi} and \ref{f:EC2spi},
indicate that two distinct solutions can be achieved
for initial data where the vorticity
\begin{equation}l{om0}\overline \omega(x)~=~ \omega(0,x)~=~\centerlineurl u(0,x)\epsiloneq has the form
\begin{equation}l{ssv}
\overline \omega(x)~=~r^{-{1/\mu}} \,\overline {\cal O}mega(\theta),
\qquad\qquad x= (x_1, x_2) = (r\centerlineos \theta, \,r\sin\theta).\epsiloneq
Here ${1\overlineer 2}<\mu<+\itemnfty$, while $\overline{\cal O}mega\itemn {\mathcal C}^\itemnfty({\mathbb R})$ is a non-negative, smooth, periodic function
which satisfies
\begin{equation}l{ovo}
\overline{\cal O}mega(\theta) ~=~\overline{\cal O}mega(\pi+\theta),\qquad\qquad
\overline{\cal O}mega(\theta)~=~0\quad\hbox{if}~~\theta\itemn \left[{\pi\overlineer 4}\,,\, \pi\right].\epsiloneq
As shown in Figure~\ref{f:e51}, left, the initial vorticity $\overline\omega$ is supported on two wedges,
and becomes arbitrarily large as $|x|\to 0$.
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.38]{e51}
\centerlineaption{\small The supports of the initial vorticity considered in (\ref{ssv})-(\ref{ID2}).}
\bigl\langlebel{f:e51}
\epsilonnd{figure}
One can approximate $\overline\omega$ by two families of initial data
$\overline \omega_\vskip 4emip 2emkip 1emarepsilon, \overline\omega_\vskip 4emip 2emkip 1emarepsilon^\dagger\itemn {\bf L}^\itemnfty({\mathbb R}^2)$, taking
\begin{equation}l{ID2}\overline \omega_\vskip 4emip 2emkip 1emarepsilon(x)~=~\left\{\begin{equation}gin{itemize}n{array}{cl} \omega(x)\quad&\hbox{if} ~~|x|>\vskip 4emip 2emkip 1emarepsilon,\centerliner
\vskip 4emip 2emkip 1emarepsilon^{-1/\mu}\quad&\hbox{if} ~~|x|\leq\vskip 4emip 2emkip 1emarepsilon,\epsilonnda\right.\qquad\qquad
\overline \omega^\dagger_\vskip 4emip 2emkip 1emarepsilon(x)~=~\left\{\begin{equation}gin{itemize}n{array}{cl} \omega(x)\quad&\hbox{if} ~~|x|>\vskip 4emip 2emkip 1emarepsilon,\centerliner
0\quad&\hbox{if} ~~|x|\leq\vskip 4emip 2emkip 1emarepsilon.\epsilonnda\right.
\epsiloneq
As $\vskip 4emip 2emkip 1emarepsilon\to 0$, we have $\overline \omega_\vskip 4emip 2emkip 1emarepsilon, \overline\omega_\vskip 4emip 2emkip 1emarepsilon^\dagger \to \overline\omega$
in ${\bf L}^p_{loc}$, for a suitable $p$ depending on the parameter $\mu$ in (\ref{ssv}).
By Yudovich's theorem \centerlineite{Y}, for every $\vskip 4emip 2emkip 1emarepsilon>0$ these initial data yield a unique solution. However, the numerical simulations indicate that, as $\vskip 4emip 2emkip 1emarepsilon\to 0$,
two distinct limit solutions are obtained. In the first solution, shown in Figure~\ref{f:EC1spi}, both wedges wind up together in a single spiral. In the second solution, shown in Figure~\ref{f:EC2spi},
each wedge curls up on itself, and two distinct spirals are observed.
\begin{equation}gin{itemize}n{remark} {\rm
The fact that all approximate solutions $\omega_\vskip 4emip 2emkip 1emarepsilon, \omega_\vskip 4emip 2emkip 1emarepsilon^\dagger$ are uniquely determined by their initial data
implies that the ill-posedness exhibited by this example is ``uncurable". Namely, there is no way to
select one solution with initial datum as in (\ref{ssv})-(\ref{ovo}), preserving the continuous dependence on initial data. }
\epsilonnd{remark}
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.8]{EC1spi.pdf}
\centerlineaption{\small
The vorticity distribution at time $t=1$, for a solution to (\ref{E2})
with initial vorticity $\overline \omega_\vskip 4emip 2emkip 1emarepsilon$.}
\bigl\langlebel{f:EC1spi}
\epsilonnd{figure}
\vskip 4emip 2emkip 1em
We observe that both of these limit solutions are self-similar,
i.e.~they have the form
\begin{equation}l{SS3}
\left\{\begin{equation}gin{itemize}n{array}{rl} u(t,x)&=~t^{\mu-1} U\left({x\overlineer t^\mu}\right),\\[4mm]
\omega(t,x)&=~t^{-1} {\cal O}mega\left({x\overlineer t^\mu}\right),\\[4mm]
\psi(t,x)&=~t^{2\mu-1} \Psi \left({x\overlineer t^\mu}\right)
.\epsilonnda\right.\epsiloneq
Notice that, by self-similarity, these solutions are completely determined as soon as
we know their values at time $t=1$. Indeed, these are given by $U, {\cal O}mega,\Psi$.
Inserting (\ref{SS3}) in (\ref{E2}), one obtains the equations
\begin{equation}l{SSE}\left\{
\begin{equation}gin{itemize}n{array}{rl}\Big(\noindentabla^\perp \Psi -\mu y\Big)\centerlinedot \noindentabla{\cal O}mega &=~ {\cal O}mega\,,\\[3mm]
{\mathcal D}elta \Psi&=~{\cal O}mega\,,
\epsilonnda
\right.
\epsiloneq
while the velocity is recovered by
\begin{equation}l{U}U~=~\noindentabla^\perp \Psi.\epsiloneq
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.8]{EC2spi.pdf}
\centerlineaption{\small
The vorticity distribution at time $t=1$, for a solution to (\ref{E2})
with initial vorticity $\overline\omega_\vskip 4emip 2emkip 1emarepsilon^\dagger$.}
\bigl\langlebel{f:EC2spi}
\epsilonnd{figure}
Constructing two distinct self-similar solutions of (\ref{E2}) with the same initial data (\ref{ssv})
amounts to finding two distinct solutions $({\cal O}mega,\Psi)$, $({\cal O}mega^\dagger,
\Psi^\dagger) $ of (\ref{SSE}) with the same
asymptotic behavior as $|x|\to + \itemnfty$. More precisely, writing the vorticity ${\cal O}mega$ in polar coordinates, this
means
\begin{equation}l{asy1}\lim_{r\to +\itemnfty} r^{1\overlineer\mu}\,{\cal O}mega(r,\theta)~=~\lim_{r\to +\itemnfty} r^{1\overlineer\mu}\,{\cal O}mega^\dagger(r,\theta)
~\doteq~\overline{\cal O}mega(\theta),\epsiloneq
for some smooth function $\overline{\cal O}mega$ as in (\ref{ovo}).
Since the two solutions in Figures~\ref{f:EC1spi} and \ref{f:EC2spi}
are produced by numerical computations,
a natural question is whether an exact self-similar solution of the
Euler equations exists, close to each computed one.
This requires suitable a posteriori error bounds.
Toward this goal, two difficulties arise:
\begin{equation}gin{itemize}
\itemtem[(i)] The self similar solution $({\cal O}mega,\Psi)$ is defined on the entire plane
${\mathbb R}^2$, while a numerical solution is computed only on some bounded domain.
\itemtem[(ii)] The solution is smooth, with the exception of one or two
points corresponding to the spirals' centers. In a neighborhood of these points
the standard error estimates break down.
\epsilonndi
To address these issues, we propose a domain decomposition method.
As shown in Fig.~\ref{f:e64},
the plane can be decomposed
into an outer domain $D^\sharp\supseteq\{x\itemn{\mathbb R}^2\,; |x|>R\}$,
an inner domain ${\mathcal D}^\flat$ containing
a neighborhood of the spirals' centers where the solution has singularities, and a bounded intermediate domain ${\mathcal D}^\noindentatural$ where the solution is smooth.
The solution is constructed analytically on ${\mathcal D}^\sharp$ and on ${\mathcal D}^\flat$,
and numerically on ${\mathcal D}^\noindentatural$.
These three components are then patched together
by suitable matching conditions.
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.22]{e64}
\centerlineaption{\small Decomposing the plane ${\mathbb R}^2={\mathcal D}^\sharp\centerlineup{\mathcal D}^\noindentatural\centerlineup{\mathcal D}^\flat$
into an outer, a middle, and an inner domain. Left: the case of a single spiraling vortex, as in Fig.~\ref{f:EC1spi}. Right: the case of two spiraling vortices, as in Fig.~\ref{f:EC2spi}.}
\bigl\langlebel{f:e64}
\epsilonnd{figure}
A detailed analysis of the solution to (\ref{SSE}) in a neighborhood of
infinity and near the spirals' centers will appear in the companion paper
\centerlineite{BM}, relying on the approach developed in \centerlineite{E1, E2, E3}.
In the present paper we focus on the derivation of a posteriori
error estimates for a numerically computed solution to (\ref{SSE}), on
a bounded domain ${\mathcal D}\subset{\mathbb R}^2$ with smooth boundary $\partial{\mathcal D}$.
As shown in Fig.~\ref{f:e88}, we assume that this boundary
can be decomposed as the union of two closed, disjoint components:
\begin{equation}l{dbd}\partial{\mathcal D}~=~\Sigma_1\centerlineup\Sigma_2\,.\epsiloneq
We seek a solution to (\ref{SSE}), satisfying boundary conditions of the form
\begin{equation}l{BC}\left\{
\begin{equation}gin{itemize}n{array}{rl}\Psi(x)&=~g(x),\qquad\qquad x\itemn \partial {\mathcal D},\\[3mm]
{\cal O}mega(x)&=~h(x),\qquad\qquad x\itemn\Sigma_1\,,\epsilonnda\right.\epsiloneq
where $g,h$ are given smooth functions.
Given an approximate solution
$({\cal O}mega_0,\Psi_0)\itemn {\mathcal C}^{0,\alpha}({\mathcal D})\times {\mathcal C}^{2,\alpha}({\mathcal D})$, computed
by a Galerkin finite dimensional approximation,
we want to prove the existence
of an exact solution close to the approximate one.
The remainder of the paper is organized as follows. Section~\ref{s:2} introduces the basic framework,
specifying the main assumptions on the numerical scheme and on the approximate solution.
Section~\ref{s:3} begins by analyzing the first equation in (\ref{lam1}), regarded as a linear
PDE for the vorticity function ${\cal O}mega$. In this direction, Lemma~\ref{l:42}
provides a detailed estimate on how the solution depends on the vector field $\noindentabla^\perp \Phi$.
In addition, Lemma~\ref{l:interp}
yields a sharper regularity estimate on the solutions, deriving an
a priori bound on their ${\mathcal C}^{0,\alpha}$ norm.
Finally, in Section~\ref{s:4} the exact solution ${\cal O}mega$ is constructed as the fixed point of a transformation
which is contractive w.r.t.~a norm equivalent to the ${\bf L}^2$ norm. We remark that,
in order to achieve this contractivity, a bound on the norms $\|{\cal O}mega\|_{{\bf L}^2}$ and $\|\Phi\|_{H^2}$
is not good enough. Indeed, we need an a priori bound on $\|{\cal O}mega\|_{{\mathcal C}^{0,\alpha}}$ and on
$\|\Phi\|_{{\mathcal C}^2}$.
This is achieved by means of Lemma~\ref{l:interp}.
At the end of Section~\ref{s:4} we collect all the various constants appearing in the estimates,
and summarize our analysis by stating a theorem on the existence of an exact solution, close to the
computed one.
For results on the uniqueness of solutions to the incompressible Euler equations we refer to
\centerlineite{BH, MP, Y}. Examples showing the non-uniqueness of solutions to the incompressible Euler equations
were first constructed in \centerlineite{Sch, Shn}. See also
\centerlineite{V} for a different approach, yielding more regular solutions.
Following the major breakthrough in
\centerlineite{DS09} several examples
of multiple solutions for Euler's equations have recently been provided \centerlineite{Dan, DRS, DanSz, DS10}.
These solutions are obtained by means of convex integration and a Baire category argument.
They have turbulent nature and their physical significance is unclear.
Our numerical simulations, on the other hand, suggest that ``uncurable" non-uniqueness
can arise from quite simple initial data. In our case, both solutions can be easily visualized;
they remain everywhere smooth (with the exception of one or two points), and conserve energy at all times.
\section{Setting of the problem}
\setcounter{equation}{0}
\bigl\langlebel{s:2}
Throughout the following, we consider the boundary value problem (\ref{SSE}), (\ref{BC}) on
a bounded, open domain ${\mathcal D}\subset {\mathbb R}^2$, with smooth boundary $\partial{\mathcal D}$ decomposed as in (\ref{dbd}).
The unit outer normal
at the boundary point $y\itemn \partial {\mathcal D}$ will be denoted by ${\bf n}(y)$.
We call $\Psi^g$ the solution to the non-homogeneous boundary value problem
\begin{equation}l{Pg}\left\{\begin{equation}gin{itemize}n{array}{rll}
{\mathcal D}elta\Psi(x)&=~0\qquad & x\itemn{\mathcal D},\\[3mm]
\Psi(x)&=~g(x)\qquad &x\itemn\partial{\mathcal D},\epsilonnda\right.\epsiloneq
and define the vector field
\begin{equation}l{bbv}
{\bf v}(x)~\doteq~\noindentabla^\perp \Psi^g(x) -\mu x\,.\epsiloneq
Assuming that $g$ is smooth, the same is true of ${\bf v}$.
Given ${\cal O}mega\itemn {\bf L}^2({\mathcal D})$, we define
\begin{equation}l{D-1}\Phi~=~{\mathcal D}elta^{-1}{\cal O}mega\epsiloneq
to be the solution of
\begin{equation}l{lam2}
\left\{\begin{equation}gin{itemize}n{array}{rll} {\mathcal D}elta\Phi (x)&=~{\cal O}mega(x),\qquad &x\itemn {\mathcal D},\\[3mm]
\Phi(x)&=~0,\qquad &x\itemn \partial {\mathcal D}.\epsilonnda\right.\epsiloneq
Our problem thus amounts to finding a function ${\cal O}mega$ such that
\begin{equation}l{lam1}
\left\{\begin{equation}gin{itemize}n{array}{rll} \bigl(\noindentabla^\perp \Phi(x)+{\bf v}(x)\bigr)\centerlinedot \noindentabla{\cal O}mega(x)&=~{\cal O}mega(x),\qquad\qquad
&x\itemn {\mathcal D},\\[3mm]
{\cal O}mega(x)&=~h(x), &x\itemn \Sigma_1\,,\epsilonnda\right.\epsiloneq
where $\Phi={\mathcal D}elta^{-1}{\cal O}mega$ and ${\bf v}$ is the vector field at (\ref{bbv}).
In the following, for any given function $\Phi\itemn {\mathcal C}^{2}({\mathcal D})$,
we denote by ${\cal O}mega={\mathcal G}amma(\Phi)$ the solution to (\ref{lam1}).
We seek a fixed point of the composed map
\begin{equation}l{ldef}
{\cal O}mega~\mapsto~{\bf L}ambda({\cal O}mega)~\doteq~{\mathcal G}amma({\mathcal D}elta^{-1}{\cal O}mega).\epsiloneq
\vskip 4emip 2emkip 1em
We shall consider approximate solutions of (\ref{lam1}) which are obtained
by finite dimensional Galerkin
approximations. More precisely, given a finite set of linearly independent functions
$\{\phi_j\,;~~1\leq j\leq N\}\subset{\bf L}^2({\mathcal D})$, consider the orthogonal decomposition
${\bf L}^2({\mathcal D})~=~U\times V$, where
\begin{equation}l{UV} U~=~\hbox{span}\{\phi_1,\ldots, \phi_N\},\qquad \qquad V= U^\perp,\epsiloneq
with orthogonal projections
\begin{equation}l{proj}
P:{\bf L}^2({\mathcal D})~\mapsto~ U,\qquad\quad (I-P): {\bf L}^2({\mathcal D})~\mapsto ~V.\epsiloneq
\vskip 4emip 2emkip 1em
{\bf Example 1.} We can choose the functions
$\phi_j$ to be piecewise affine, obtained from a
triangulation of the domain ${\mathcal D}$. This implies
\begin{equation}l{pjp}
\phi_j\itemn W^{1,\itemnfty}({\mathcal D})\subset {\mathcal C}^{0,\alpha}({\mathcal D})\epsiloneq
for every $0<\alpha\leq 1$. In this case, the gradient $\noindentabla\phi_j\itemn{\bf L}^\itemnfty\centerlineap BV$
is piecewise constant, with jumps
along finitely many segments.
\vskip 4emip 2emkip 1em
{\bf Example 2.} In alternative, one can choose $\phi_1,\ldots,\phi_N\itemn {\bf L}^2({\mathcal D})$
to be the first $N$ normalized eigenfunctions of the Laplacian. More precisely,
for $j=1,\ldots,N$ we require that the functions $\phi_j$ satisfy
\begin{equation}l{EL}\left\{\begin{equation}gin{itemize}n{array}{rl} {\mathcal D}elta\phi_j + \bigl\langlembda_j \phi_j~=~0\qquad & x\itemn {\mathcal D},\\[3mm]
\phi_j~=~0\qquad & x\itemn \Sigma_1\,,\\[3mm]
{\bf n}\centerlinedot \noindentabla\phi_j~=~0\qquad & x\itemn \Sigma_2\,,\epsilonnda\right.\epsiloneq
with eigenvalues $0<\bigl\langlembda_1\leq \bigl\langlembda_2\leq\centerlinedots\leq \bigl\langlembda_N$. Moreover, $\|\phi_j\|_{{\bf L}^2}=1$.
\vskip 4emip 2emkip 1em
\begin{equation}gin{itemize}n{remark}\bigl\langlebel{r:1}{\rm In both of the above cases, there may be no linear combination $\sum_j c_j \phi_j$
of the basis functions that matches the boundary data $h$ along $\Sigma_1$.
This issue can be addressed simply by adding to our basis an additional function $\phi_0\itemn {\mathcal C}^\itemnfty({\mathcal D})$,
chosen so that $\phi_0=h$ on $\Sigma_1$.}
\epsilonnd{remark}
Our basic question can be formulated as follows.
\begin{equation}gin{itemize}
\itemtem[{\bf (Q)}]
{\itemt
Assume we can find a finite dimensional approximation
\begin{equation}l{UU} {\cal O}mega_0 ~=~\sum_{j=1}^N c_j\phi_j\,,\qquad\qquad
\Phi_0~=~{\mathcal D}elta^{-1}{\cal O}mega_0\,,\epsiloneq
with error
\begin{equation}l{AA} \Big\| {\cal O}mega_0 -P\,{\bf L}ambda({\cal O}mega_0)\Big\|_{{\bf L}^2({\mathcal D})}~=~\delta_0\,.\epsiloneq
How small should $\delta_0$ be, to make sure that an exact solution
$({\cal O}mega,\Phi)$ of (\ref{lam1}), (\ref{D-1}) exists,
close to $({\cal O}mega_0,\Phi_0)$ ?}
\epsilonndi
\vskip 4emip 2emkip 1em
Given a function $\Phi\itemn {\mathcal C}^2({\mathcal D})$, the linear, first order PDE (\ref{lam1}) for ${\cal O}mega$ can be solved
by the method of characteristics.
Namely, consider the
vector field
\begin{equation}l{qd}{\bf q}(x)~=~\noindentabla^\perp \Phi(x)+{\bf v}(x)
,\epsiloneq
whose divergence is
\begin{equation}l{divq}
\hbox{div } {\bf q}~=~-2\mu\,.\epsiloneq
We shall denote by $t\mapsto \epsilonxp(t{\bf q})(y)$ the solution to the ODE
$$\dot x(t)~=~{\bf q}(x(t)),\qquad x(0)=y.$$
For convenience, we shall use the notation
\begin{equation}l{xty}t~\mapsto ~x(t,y)~\doteq~\epsilonxp(-t{\bf q})(y)\epsiloneq
for the solution to the ODE
$$\dot x(t)~=~-{\bf q}(x(t)),\qquad x(0)=y.$$
Consider the set
\begin{equation}l{D*} {\mathcal D}^*~\doteq~\Big\{ y\itemn \overline{\mathcal D}\,;~~x(\tau,y)\itemn \Sigma_1\centerlineap\hbox{Supp}(h)\quad\hbox{for some}~\tau\geq 0\Big\}.\epsiloneq
In other words, $y\itemn {\mathcal D}^*$ if the characteristic through $y$ reaches a boundary point in the support of $h$ within
finite time. Calling
\begin{equation}l{tauy}\tau(y)~\doteq~\min\,\bigl\{t\geq 0\,;~~ x(t,y)\itemn \Sigma_1\bigr\}\epsiloneq
the first time when the characteristic starting at $y$ reaches the boundary $\Sigma_1$,
the solution to (\ref{lam1}) is computed by
\begin{equation}l{OR}{\cal O}mega(y)~=~\left\{ \begin{equation}gin{itemize}n{array}{cl} e^{\tau(y)}\, h(x(\tau(y), y))\qquad &\hbox{if} ~~x\itemn {\mathcal D}^*,\\[3mm]
0\qquad&\hbox{if}~~x\noindentotin{\mathcal D}^*.\epsilonnda\right.\epsiloneq
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.45]{e88.pdf}
\centerlineaption{\small According to {\bf (A1)}, every characteristic starting at a point
$y\itemn \Sigma_1\centerlineap {\rm Supp }(h)$ exits from the domain ${\mathcal D}$ at some boundary point
$z=z(y)\itemn \Sigma_2$, at a time $T(y)\leq T^*$. The shaded region represents the subdomain ${\mathcal D}^*$ in
(\ref{D*}).}
\bigl\langlebel{f:e88}
\epsilonnd{figure}
The following transversality assumption will play a key role in the sequel (see Fig.~\ref{f:e88}).
\vskip 4emip 2emkip 1em
\begin{equation}gin{itemize}
\itemtem[{\bf (A1)}] {\itemt There exists constants $T^*, c_1>0$ such that, for
every boundary point $y\itemn \Sigma_1\centerlineap \hbox{Supp}(h)$, the following holds.
\begin{equation}gin{itemize}
\itemtem[(i)] The vector ${\bf q}$
is strictly inward pointing:
\begin{equation}l{OT3}\bigl\bigl\langlengle {\bf n}(y)\,,\, {\bf q}(y)\bigr\bigr\ranglengle~\leq~-c_1\,.
\epsiloneq
\itemtem[(ii)] The characteristic $t\mapsto \epsilonxp(t{\bf q})(y)$ remains inside ${\mathcal D}$
until it reaches a boundary point
\begin{equation}l{exitp}z(y)\,=\, \epsilonxp\bigl(T(y){\bf q}\bigr)(y)~\itemn ~\Sigma_2\epsiloneq
within a finite time $T(y)\leq T^*$, and exits transversally:
\begin{equation}l{OT2}
\Big\bigl\langlengle {\bf n}(z(y))\,,\, {\bf q}(z(y))\Big\bigr\ranglengle~\geq ~c_1\,.\epsiloneq
\epsilonndi
}
\epsilonndi
As in (\ref{UV})-(\ref{proj}), we consider the decomposition $H\doteq {\bf L}^2({\mathcal D})=U\times V$, with perpendicular projections
$P$ and $ I-P$, and write ${\cal O}mega= (u,v)$. The partial derivatives of the map
${\bf L}ambda={\bf L}ambda(u,v)$ introduced at (\ref{ldef}) will be denoted by
$D_u{\bf L}ambda$, $D_v{\bf L}ambda$.
Let a finite dimensional approximate solution ${\cal O}mega_0= (u_0,0)\itemn U$ of (\ref{lam1}) be given,
with $\Phi_0={\mathcal D}elta^{-1}{\cal O}mega_0$. Throughout the following, we denote by
\begin{equation}l{Adef} A~\doteq~P\centerlineirc D_u{\bf L}ambda({\cal O}mega_0).\epsiloneq
the partial differential w.r.t.~$u$ of the map $(u,v)\mapsto P{\bf L}ambda(u,v)$, computed at the point
${\cal O}mega_0$.
Notice that $A$ is a linear map from the finite dimensional space $U$ into itself.
Since we are seeking a fixed point, in the same
spirit of the Newton-Kantorovich theorem~\centerlineite{Berger, Ciarlet, Deimling}
we shall assume the invertibility of the
Jacobian map, restricted to the finite dimensional subspace $U$.
\begin{equation}gin{itemize}
\itemtem[{\bf (A2)}] {\itemt The operator $I - A$
from $U$ into itself has a bounded inverse, with }
\begin{equation}l{IA}\big\|(I-A)^{-1}\bigr\|_{{\centerlineal L}(U)}~\leq~\gamma~<~+\itemnfty\epsiloneq
for some $\gamma\geq 1$.
\epsilonndi
Here the left hand side refers to the operator norm, in the space of linear operators on $U\subset{\bf L}^2({\mathcal D})$.
Notice that (\ref{IA}) implies that the operator $I-AP:H\mapsto H$ is also invertible,
with
\begin{equation}l{IA2}
\big\|(I-A\,P)^{-1}\bigr\|_{{\centerlineal L}(H)}~\leq~\gamma.\epsiloneq
Concerning the $V$-component, a key assumption used in our analysis will be
\begin{equation}gin{itemize}
\itemtem[{\bf (A3)}] {\itemt The orthogonal spaces $U,V$ in (\ref{UV}) are chosen so that
\begin{equation}l{LVsmall}
\| {\mathcal D}elta^{-1}\centerlineirc (I-P)\|~<~\vskip 4emip 2emkip 1emarepsilon_0~<\!\!<~1\,.\epsiloneq
}
\epsilonndi
Intuitively this means that, in the decomposition ${\cal O}mega = (u,v)\itemn U\times V$, the
component $v\itemn V$ captures the high frequency modes,
which are heavily damped by the inverse Laplace operator \centerlineite{BRS, Cle, KS}.
\vskip 4emip 2emkip 1em
{}From an abstract point of view, proving the existence of a fixed point of the map ${\cal O}mega\mapsto
{\bf L}ambda({\cal O}mega)$ in (\ref{ldef}) is a simple matter.
Using the decomposition ${\cal O}mega = (u,v)\itemn U\times V$ with orthogonal projections as in (\ref{proj}),
we start with an initial guess ${\cal O}mega_0 = (u_0,0)$.
Assuming (\ref{IA}), we write (\ref{lam1}) in the form
$${\cal O}mega-AP\,{\cal O}mega~=~{\bf L}ambda({\cal O}mega)- AP\,{\cal O}mega\,.$$
Equivalently,
\begin{equation}l{eqeq}
{\cal O}mega~=~{\mathcal U}psilon({\cal O}mega)~\doteq~(I-AP)^{-1}\bigl({\bf L}ambda({\cal O}mega)- AP\,{\cal O}mega\bigr).\epsiloneq
The heart of the matter is to show that, on a neighborhood ${\centerlineal N}$ of ${\cal O}mega_0$,
the map ${\cal O}mega\mapsto{\mathcal U}psilon({\cal O}mega)$ is a strict contraction.
If the initial error $\|{\mathcal U}psilon({\cal O}mega_0)-{\cal O}mega_0\|$
is sufficiently small, the iterates ${\cal O}mega_n={\mathcal U}psilon^n({\cal O}mega_0)$ will thus remain inside ${\centerlineal N}$
and converge to a fixed point. The contraction property is proved as follows.
By (\ref{IA2}) one has $\|(I-AP)^{-1}\|\leq\gamma$. On the other hand, computing the differential of the map
${\cal O}mega\mapsto {\bf L}ambda({\cal O}mega)- AP\,{\cal O}mega$
w.r.t.~the components $(u,v)\itemn U\times V$, at the point ${\cal O}mega={\cal O}mega_0= (u_0,0)$ one obtains
\begin{equation}l{diff}
D\bigl({\bf L}ambda(u,v)-Au\bigr)
~=~\left(
\begin{equation}gin{itemize}n{array}{ccc} 0 && P D_v{\bf L}ambda\\[4mm]
(I-P) D_u{\bf L}ambda && (I-P) D_v{\bf L}ambda\epsilonnda\right).\epsiloneq
Because of (\ref{LVsmall}), we expect
\begin{equation}l{dism}\|D_v{\bf L}ambda\|~\leq~\|D{\mathcal G}amma\|\centerlinedot \bigl\| {\mathcal D}elta^{-1} \centerlineirc(I-P)\bigr\|~<\!<~1.\epsiloneq
By possibly using an equivalent norm on the product space $U\times V$ (see (\ref{n*}) for details),
we thus achieve the strict contractivity of the map ${\mathcal U}psilon$ in (\ref{eqeq}).
\vskip 4emip 2emkip 1em
In the next section, more careful estimates will be derived on the differentials of ${\mathcal G}amma$ and
${\bf L}ambda$ in (\ref{ldef}). In this direction we remark that,
while the operator ${\mathcal D}elta^{-1}$ is well defined on ${\bf L}^2({\mathcal D})$,
a difficulty arises in connection with the differential of ${\mathcal G}amma$.
Indeed, to compute this differential, we need to perturb the function $\Phi$ in (\ref{lam1})
and estimate the change in the corresponding solution ${\cal O}mega$.
This can be done assuming that $\Phi\itemn{\mathcal C}^2$,
hence $\noindentabla^\perp\Phi\itemn {\mathcal C}^1$. Unfortunately, the assumption ${\cal O}mega\itemn {\bf L}^2$
only implies $\Phi= {\mathcal D}elta^{-1}{\cal O}mega\itemn H^2$, which does not yield any bound on $\|\Phi\|_{{\mathcal C}^2}$.
In order to establish the desired a posteriori error bound,
an additional argument will thus be needed, showing that our approximate solutions actually
enjoy some additional regularity.
\section{Preliminary lemmas}
\setcounter{equation}{0}
\bigl\langlebel{s:3}
\begin{equation}gin{itemize}n{figure}[htbp]
\centerlineentering
\itemncludegraphics[scale=0.5]{e86.pdf}
\centerlineaption{\small Computing the perturbed solution ${\cal O}mega^\vskip 4emip 2emkip 1emarepsilon(y)$ in (\ref{om4}), by estimating the change in
the characteristic through $y$.}
\bigl\langlebel{f:e86}
\epsilonnd{figure}
\subsection{Continuous dependence of solutions to a first order linear PDE.}
As remarked in Section~\ref{s:2}, solutions to the linear PDE (\ref{lam1}) can be found by the method of characteristics (\ref{OR}). Our present goal is to understand how the solution changes,
depending on the vector field ${\bf q}$ in (\ref{qd}). To fix the ideas,
consider the boundary value problem
\begin{equation}l{lam3}
\left\{\begin{equation}gin{itemize}n{array}{rll} {\bf q}(x)\centerlinedot \noindentabla{\cal O}mega&=~{\cal O}mega,\qquad\qquad
&x\itemn {\mathcal D},\\[3mm]
{\cal O}mega&=~h, &x\itemn \Sigma_1\,,\epsilonnda\right.\epsiloneq
assuming that ${\bf q}:{\mathcal D}\mapsto{\mathbb R}^2$ is a ${\mathcal C}^1$ vector field
satisfying {\bf (A1)} together with
\begin{equation}l{qass}\|{\bf q}\|_{{\mathcal C}^1({\mathcal D})}~\leq~M,\qquad\qquad \hbox{div } {\bf q}(x) ~=~ -2\mu\qquad\hbox{for all }~ x\itemn {\mathcal D}\,.\epsiloneq
Given a second vector field $\widetilde{\bf q}\itemn {\mathcal C}^1$, consider the family of perturbations
\begin{equation}l{qep}{\bf q}^\vskip 4emip 2emkip 1emarepsilon(x)~=~{\bf q}(x) + \vskip 4emip 2emkip 1emarepsilon \,\widetilde{\bf q} (x),\epsiloneq
and let
\begin{equation}l{om4}{\cal O}mega^\vskip 4emip 2emkip 1emarepsilon(x)~=~{\cal O}mega(x) +\vskip 4emip 2emkip 1emarepsilon\, \widetilde{\cal O}mega (x) + o(\vskip 4emip 2emkip 1emarepsilon)\epsiloneq
be the corresponding solutions of (\ref{lam3}).
Here and in the sequel, the notation $o(\vskip 4emip 2emkip 1emarepsilon)$ indicates a higher order infinitesimal,
so that $\vskip 4emip 2emkip 1emarepsilon^{-1}o(\vskip 4emip 2emkip 1emarepsilon)\to 0$ as $\vskip 4emip 2emkip 1emarepsilon\to 0$.
The next lemma provides an ${\bf L}^2$~estimate on size of the first order perturbation $\widetilde{\cal O}mega$.
Setting $\Sigma^*_1\doteq\Sigma_1\centerlineap\hbox{Supp}(h)$, we introduce the constant
\begin{equation}l{TC}
\widetilde C~=~ \sup_{x\itemn \Sigma_1^*} {1\overlineer | \bigl\langlengle {\bf n}(x), {\bf q}(x)\bigr\ranglengle|}
\centerlinedot \|h\|_{{\mathcal C}^0}+\left( 1+\,\sup_{x\itemn \Sigma_1^*} {|{\bf q}(x)|\overlineer | \bigl\langlengle {\bf n}(x), {\bf q}(x)\bigr\ranglengle|}\right) \|Dh\|_{{\mathcal C}^0} \,,
\epsiloneq
and define
\begin{equation}l{pera} K(M, t)~=~e^t\left({e^{(2M+1)t}- e^{-2\mu t}\overlineer 4M+4\mu+2}\right)^{1/2}
.\epsiloneq
\begin{equation}gin{itemize}n{lemma}\bigl\langlebel{l:42} Recalling (\ref{D*}), assume that ${\bf q}\itemn {\mathcal C}^1({\mathcal D})$ satisfies (\ref{qass})
together with {\bf (A1)}.
In the setting considered at (\ref{qep})-(\ref{om4}), the first order perturbation $\widetilde {\cal O}mega$ satisfies
\begin{equation}l{oper}
\|\widetilde{\cal O}mega \|_{{\bf L}^2({\mathcal D})}~\leq~
\widetilde C \centerlinedot K\Big(\|D{\bf q}\|_{{\mathcal C}^0({\mathcal D}^*)},\,T^*\Big)\centerlinedot \|\widetilde{\bf q}\|_{{\bf L}^2({\mathcal D}^*)}\,.\epsiloneq
\epsilonnd{lemma}
{\bf Proof.} {\bf 1.} In analogy with (\ref{xty}),
we denote by
$$ t\,\mapsto\, x^\vskip 4emip 2emkip 1emarepsilon(t,y)\,=\,\epsilonxp(-t{\bf q}^\vskip 4emip 2emkip 1emarepsilon)(y)$$
the solution to
\begin{equation}l{od2}\dot x^\vskip 4emip 2emkip 1emarepsilon~=~-{\bf q}(x)-\vskip 4emip 2emkip 1emarepsilon \widetilde{\bf q}(x),\qquad x(0)=y.\epsiloneq
For $0\leq t<\tau(y)$, the tangent vector
\begin{equation}l{tanv}{\bf w}(t,y)~\doteq~\lim_{\vskip 4emip 2emkip 1emarepsilon\to 0} {x^\vskip 4emip 2emkip 1emarepsilon(t,y)-x(t,y)\overlineer\vskip 4emip 2emkip 1emarepsilon}\epsiloneq
provides a solution to the linearized equation
\begin{equation}l{leq}
\dot {\bf w}(t,y)~=~-D{\bf q}(x(t,y))\centerlinedot {\bf w}(t,y)- \widetilde{\bf q}(x(t,y)),\qquad\qquad {\bf w}(0,y)=0.\epsiloneq
For notational convenience,
we extend the definition of the vector ${\bf w}(t,y)$ by setting
\begin{equation}l{tT}{\bf w}(t,y)~=~{\bf w}(\tau(y), y)\qquad\hbox{if}\qquad t\itemn [\tau(y), \,T^*].\epsiloneq
We denote by
\begin{equation}l{xie}\begin{equation}gin{itemize}n{array}{l} \xi(y)\,=\,x(\tau(y),y)\,=\,\epsilonxp\bigl(-\tau(y){\bf q}\bigr)(y),
\\[4mm] \xi^\vskip 4emip 2emkip 1emarepsilon(y)\,=\,x^\vskip 4emip 2emkip 1emarepsilon(\tau^\vskip 4emip 2emkip 1emarepsilon(y),y)\, =\,\epsilonxp\bigl(-\tau^\vskip 4emip 2emkip 1emarepsilon(y){\bf q}^\vskip 4emip 2emkip 1emarepsilon\bigr)(y)\,,\epsilonnda\epsiloneq
the points where the characteristic through $y$ crosses the boundary $\Sigma_1$,
and consider the expansions
\begin{equation}l{txx}
\xi^\vskip 4emip 2emkip 1emarepsilon(y)~=~\xi(y)+\vskip 4emip 2emkip 1emarepsilon\tilde\xi(y)+o(\vskip 4emip 2emkip 1emarepsilon),\qquad\qquad \tau^\vskip 4emip 2emkip 1emarepsilon(y)~=~\tau(y)+\vskip 4emip 2emkip 1emarepsilon \widetilde\tau(y)+o(\vskip 4emip 2emkip 1emarepsilon).\epsiloneq
Observing that
\begin{equation}l{wwy}
\Big\bigl\langlengle {\bf n}\bigl(x(\tau(y),y)\bigr)~,~{\bf w}(\tau(y),y) -\widetilde\tau(y)\centerlinedot {\bf q}\bigl(x(\tau(y),y)\bigr)
\Big\bigr\ranglengle~=~0,
\epsiloneq
we obtain
\begin{equation}l{ttau}\widetilde\tau(y)~=~{\Big\bigl\langlengle {\bf n}\bigl(\xi(y)\bigr)~,~{\bf w}(\tau(y),y)\Big\bigr\ranglengle \overlineer
\Big\bigl\langlengle {\bf n}\bigl(\xi(y)\bigr)~,~{\bf q}\bigl(\xi(y)\bigr)\Big\bigr\ranglengle}\,,\epsiloneq
\begin{equation}l{txi}
\tilde \xi(y)~=~{\bf w}(\tau(y),y) -\widetilde\tau(y)\, {\bf q}\bigl(\xi(y)\bigr)
\,,\epsiloneq
\begin{equation}l{tx2}
|\tilde \xi(y)|~\leq~|{\bf w}(\tau(y),y)|\centerlinedot \left( 1+ \left| { {\bf q}(\xi(y))\overlineer \bigl\langle {\bf n}(\xi(y)),\,{\bf q}(\xi(y))
\bigr\rangle}\right| \,\right). \epsiloneq
Finally, for $y\itemn {\mathcal D}^*$ we have
\begin{equation}l{TO5}
\widetilde{\cal O}mega(y)~=~\widetilde \tau(y)\,e^{\tau(y)} h(\xi(y))
+ e^{\tau(y)}\,\noindentabla h(\xi(y))\centerlinedot \tilde\xi(y) .\epsiloneq
In view of (\ref{ttau})--(\ref{tx2}) and the definition of $\widetilde C$ at (\ref{TC}), this yields
\begin{equation}l{TO6}\begin{equation}gin{itemize}n{array}{rl}\displaystyle
\bigl|\widetilde{\cal O}mega(y)\bigr|&\displaystyle\leq~e^{\tau(y)}\|h\|_{{\bf L}^\itemnfty}
\centerlinedot{\bigl|{\bf w}(\tau(y),y)\bigr|\overlineer\left| \bigl\langle {\bf n}(\xi(y)))\,,~{\bf q}(\xi(y)) \bigr\rangle\right|} \\[4mm]
&\displaystyle\qquad\qquad +
e^{\tau(y)}\|\noindentabla h\|_{{\bf L}^\itemnfty}\centerlinedot \left( 1+ { |{\bf q}(\xi(y))|\overlineer \left| \bigl\langle {\bf n}(\xi(y)),\,{\bf q}(\xi(y))
\bigr\rangle\right| } \right)\,\bigl|{\bf w}(\tau(y),y)\bigr|\\[5mm]
&\leq~e^{\tau(y)}\widetilde C\,\bigl|{\bf w}(\tau(y),y)\bigr|~\leq~e^{T^*}\widetilde C\,\bigl|{\bf w}(\tau(y),y)\bigr|.\epsilonnda\epsiloneq
\vskip 4emip 2emkip 1em
{\bf 2.}
It now remains to derive a bound on the ${\bf L}^2$ norm of
${\bf w}$.
Keeping (\ref{tT}) in mind, define
$$Z(t)~\doteq~\itemnt_{\mathcal D} |{\bf w}(t, y)|^2\, dy.$$
For any $t\itemn [0, T^*]$, by (\ref{qass})
the Jacobian determinant of the map $y\mapsto x(t,y)$ satisfies
\begin{equation}l{det}\det \Big({\partial x(t,y)\overlineer\partial y}\Big) ~=~e^{2\mu t}.\epsiloneq
Using the above identity to change variables of integration,
setting
$${\mathcal D}^*_t~\doteq~\bigl\{ y\itemn {\mathcal D}^*\,;~~\tau(y)<t\bigr\},$$
we obtain
\begin{equation}l{i3}\itemnt_{{\mathcal D}^*_t} |\widetilde{\bf q} (x(t,y))|^2\, dy~\leq~e^{-2\mu t} \itemnt_{{\mathcal D}^*} |\widetilde{\bf q} (x)|^2\, dx~
=~e^{-2\mu t}\,\|\widetilde{\bf q} \|_{{\bf L}^2({\mathcal D}^*)}^2\,.\epsiloneq
In turn, by the elementary inequality $ab\leq {1\overlineer 2} (a^2+b^2)$, this yields
$$\begin{equation}gin{itemize}n{array}{rl}\displaystyle {d\overlineer dt} Z(t)&\displaystyle\leq~2\itemnt_{{\mathcal D}^*_t} \Big|\bigl\langle {\bf w}(t,y), ~\dot{\bf w}(t,y)\bigr\rangle\Big|\, dy
\\[4mm]
&\displaystyle\leq~2\itemnt_{{\mathcal D}^*_t}
\Big\{ \bigl|D{\bf q}(x(t,y))\bigr|\, |{\bf w}(t,y)|^2 + |{\bf w}(t,y)|\, |\widetilde{\bf q} (x(t,y))|\Big\}\, dy
\\[4mm]
&\displaystyle\leq~2 \Big\{ \|D{\bf q}\|_{{\mathcal C}^0} \centerlinedot \|{\bf w}(t,\centerlinedot)\|_{{\bf L}^2({\mathcal D}^*_t)}^2 + \|{\bf w}(t,\centerlinedot)\|_{{\bf L}^2({\mathcal D}^*_t)}
e^{-\mu t}\|\widetilde {\bf q}\|_{{\bf L}^2({\mathcal D}^*)}\Big\}\\[4mm]
&\leq~\displaystyle 2 \Big\{ C\, Z(t) + {Z(t)\overlineer 2} +
{e^{-2\mu t}\overlineer 2} \|\widetilde{\bf q} \|^2_{{\bf L}^2({\mathcal D}^*)}\Big\}
\,.
\epsilonnda$$
Hence
\begin{equation}l{wl2}\|{\bf w}(t,\centerlinedot)\|_{{\bf L}^2({\mathcal D})}^2~=~Z(t)~\leq~\kappa (t)\centerlinedot \|\widetilde {\bf q}\|^2_{{\bf L}^2({\mathcal D})}, \epsiloneq
where we set
\begin{equation}l{kapdef} \kappa (t)~\doteq~{1\overlineer 2}
\itemnt_0^t e^{(2M+1)(t-s)} \,e^{-2\mu s}\, ds~=~{e^{(2M+1)t}- e^{-2\mu t}\overlineer 4M+4\mu+2}\,.\epsiloneq
Taking $t=T^*$ we conclude
\begin{equation}l{wl3}\itemnt_{{\mathcal D}^*} \bigl|{\bf w}(\tau(y),y)\bigr|^2\, dy~\leq~{e^{(2M+1)T^*}- e^{-2\mu T^*}\overlineer 4M+4\mu+2}\centerlinedot \|\widetilde {\bf q}\|^2_{{\bf L}^2({\mathcal D})}
\,.\epsiloneq
Using this bound, from (\ref{TO6}) one obtains
(\ref{oper}).
\epsilonndproof
\subsection{A regularity estimate.}
Given a H\"older continuous function
$f:{\mathcal D}\mapsto{\mathbb R}$, for $0<\alpha,\delta<1$, we introduce the notation
\begin{equation}l{39}
\|f\|_{\alpha,\delta}~\doteq~
\sup_{x,y\itemn{\mathcal D},~0<|x-y|<\delta} ~{|f(x)-f(y)|\overlineer |x-y|^\alpha}\,.\epsiloneq
Notice that, compared with the standard definition of the norm in the H\"older space ${\mathcal C}^{0,\alpha}$
(see for example \centerlineite{BFA, Evans}), in (\ref{39}) the supremum is taken only over couples with $|x-y|<\delta$.
From the above definition it immediately follows
\begin{equation}l{ades}
\|f\|_{\alpha,\delta}~\leq~\|f\|_{{\mathcal C}^{0,\alpha}}\,,
\qquad\quad \|f\|_{\alpha,\delta}~\leq~
\delta^{1-\alpha} \|\noindentabla f\|_{{\bf L}^\itemnfty}\,.
\epsiloneq
The next lemma shows that the ${\mathcal C}^{0,\alpha}$ norm can be controlled in terms
of the seminorm
$\|\centerlinedot\|_{\alpha,\delta}$ together with the ${\bf L}^2$ norm.
As a preliminary we observe that, since ${\mathcal D}$ is an open set with smooth boundary, it has a positive inner radius $\rho>0$.
Namely, every point $y\itemn {\mathcal D}$ lies in an open ball
of radius $\rho$, entirely contained inside ${\mathcal D}$.
\begin{equation}gin{itemize}n{lemma}\bigl\langlebel{l:interp}
Let ${\mathcal D}\subset{\mathbb R}^2$ be a bounded open set with inner radius $\rho>0$, and let $\alpha,\delta\itemn \,]0,1[\,$ and $c>0$ be given. Then there exists $\delta_c>0$ such that the following holds.
If
\begin{equation}l{40} \|f\|_{\alpha,\delta}~\leq ~c,\qquad\qquad \|f\|_{{\bf L}^2({\mathcal D})}~\leq~\delta_c\,,\epsiloneq
then \begin{equation}l{42}
\|f\|_{{\mathcal C}^0({\mathcal D})}~\leq~{c\overlineer 2} \,\delta^\alpha.\epsiloneq
In turn, this implies
\begin{equation}l{41}\|f\|_{{\mathcal C}^{0,\alpha}({\mathcal D})}~\leq~c.
\epsiloneq
\epsilonnd{lemma}
{\bf Proof.}
{\bf 1.} We claim that, by choosing $\delta_c>0$ small enough, the inequalities
(\ref{40}) imply (\ref{42}).
Indeed, consider the function
$$\phi(x)~\doteq~\max\Big\{ 0,~{c\overlineer 2} \delta^\alpha - c|x|^\alpha\Big\}.$$
and the disc
$$B~=~\{(x_1,x_2)\,;~~(x_1-\rho)^2 + x_2^2<\rho^2\}.$$
Setting
\begin{equation}l{doc}\delta_c~\doteq~\|\phi\|_{{\bf L}^2(B)} ~=~\left(\itemnt_B\phi^2(x)\, dx\right)^{1/2},\epsiloneq
we claim that the assumptions (\ref{40}) imply (\ref{42}).
Indeed, assume that $f(y_0)> c\delta^\alpha/2$ at some point $y_0\itemn {\mathcal D}$.
Let $B_0\subset{\mathcal D}$ be a disc of radius $\rho$ which contains $y_0$.
A comparison argument now yields
$$\begin{equation}gin{itemize}n{array}{rl}\displaystyle\|f\|^2_{{\bf L}^2({\mathcal D})}&\displaystyle\ge~\itemnt_{B_0} f^2(x)\, dx~\geq~\itemnt_{B_0}
\left(\max\Big\{ 0,~|f(y_0)|-c|x-y_0|^\alpha\Big\}\right)^2\, dx \\[4mm]
\displaystyle\qquad &\displaystyle >~
\itemnt_B\phi^2(x)\, dx~=~\delta_c^2\,,\epsilonnda$$
reaching a contradiction. Hence (\ref{42}) holds.
\vskip 4emip 2emkip 1em
{\bf 2.} By the assumptions, we already know that
\begin{equation}l{44} {|f(x)-f(y)|\overlineer |x-y|^\alpha}~\leq~c\epsiloneq
when $0<|x-y|<\delta$. It remains to prove that the same holds when
$|x-y|\geq\delta$. But in this case by (\ref{42}) one trivially has
$${|f(x)-f(y)|\overlineer |x-y|^\alpha}~\leq~{|f(x)|+|f(y)|\overlineer \delta^\alpha}
~\leq~c\,.$$
Together with (\ref{42}), this yields
$$\|f\|_{{\mathcal C}^{0,\alpha}({\mathcal D})}~\doteq~\max\left\{ \sup_x~|f(x)|\,,~~\sup_{x\noindentot= y} {|f(x)-f(y)|\overlineer |x-y|^\alpha}\right\}
~\leq~c\,,$$
proving (\ref{41}).
\epsilonndproof
\section{Construction of an exact solution}
\setcounter{equation}{0}
\bigl\langlebel{s:4}
As in (\ref{UV})-(\ref{proj}), we consider the decomposition $H\doteq {\bf L}^2({\mathcal D})=U\times V$, with perpendicular projections
$P$ and $ I-P$, and write ${\cal O}mega= (u,v)$. We recall that $D_u{\bf L}ambda$, $D_v{\bf L}ambda$ denote the partial derivatives of the map
${\bf L}ambda={\bf L}ambda(u,v)$ introduced at (\ref{ldef}).
As anticipated in Section~\ref{s:2}, we write the fixed point problem ${\cal O}mega = {\bf L}ambda({\cal O}mega)$ in the
equivalent form
\begin{equation}l{ups}
{\cal O}mega~=~{\mathcal U}psilon({\cal O}mega)~\doteq~(I-AP)^{-1}\bigl({\bf L}ambda({\cal O}mega)- AP\,{\cal O}mega\bigr).\epsiloneq
In terms of the components $(u,v)$, at
${\cal O}mega={\cal O}mega_0= (u_0,0)$, the differential of the map ${\cal O}mega\mapsto {\bf L}ambda({\cal O}mega)- AP\,{\cal O}mega$
has the form (\ref{diff}).
To achieve the contraction property, on the product space $H=U\times V$ we consider
the equivalent inner product
\begin{equation}l{ip*}\bigl\langle(u,v),(u',v')\bigr\rangle_*~=~\bigl\langlengle u, u'\bigr\ranglengle + \epsilonta_0 \bigl\langlengle v, v'\bigr\ranglengle,\epsiloneq
for a suitable constant $0<\epsilonta_0\leq 1$. The corresponding norm is
\begin{equation}l{n*}\|(u,v)\|_*~=~\bigl( \|u\|^2 + \epsilonta_0 \|v\|^2\bigr)^{1/2}.\epsiloneq
Based on (\ref{dism}), we assume that a constant $\epsilonta_0$ can be chosen so that, at the point
${\cal O}mega_0$, the corresponding norm of the linear operator (\ref{diff}) is $\leq 1/4\gamma$.
By (\ref{IA2}) and the definition of ${\mathcal U}psilon$ at (\ref{ups}), this implies
$$\|D{\mathcal U}psilon({\cal O}mega_0)\|_*~\leq~{1\overlineer 4}\,.$$
Here and in the sequel, we also denote by $\|\centerlinedot\|_*$ the norm of a linear operator, corresponding to the
norm (\ref{n*}) on the product space $H=U\times V$.
By continuity, we can determine a radius $r_0>0$ such that,
denoting by $B_*({\cal O}mega_0, r_0)$ a ball centered at ${\cal O}mega_0$ with radius $r_0$
w.r.t.~the equivalent norm $\|\centerlinedot\|_*$, one has the implication
\begin{equation}l{c23}
{\cal O}mega,{\cal O}mega'\itemn B_*({\cal O}mega_0, r_0)\qquad \itemmplies\qquad \|{\mathcal U}psilon({\cal O}mega)-{\mathcal U}psilon({\cal O}mega')\|_*~\leq~
{1\overlineer 2} \|{\cal O}mega-{\cal O}mega'\|_*\,.\epsiloneq
If the approximate solution ${\cal O}mega_0$ satisfies
\begin{equation}l{ig}
\|{\mathcal U}psilon({\cal O}mega_0)-{\cal O}mega_0\|_*~\leq~{r_0\overlineer 2}\,,\epsiloneq
we can then define the iterates
\begin{equation}l{OPN}
{\cal O}mega_n~\doteq~{\mathcal U}psilon({\cal O}mega_{n-1}),\qquad\qquad \Phi_n~=~{\mathcal D}elta^{-1}{\cal O}mega_n\,.\epsiloneq
Taking the limit
\begin{equation}l{UPN}
\overline{\cal O}mega~=~\lim_{n\to\itemnfty} {\cal O}mega_n\,,
\epsiloneq
we thus obtain a fixed point
$\overline{\cal O}mega = {\mathcal U}psilon(\overline{\cal O}mega)$, with
\begin{equation}l{fixo}
\|\overline{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2({\mathcal D})}~\leq~\|\overline{\cal O}mega-{\cal O}mega_0\|_*~\leq~r_0\,.\epsiloneq
While this approach is entirely straightforward, our main concern here is to
derive more precise estimates on the various constants, which guarantee that an exact solution actually exists.
Notice that the contraction property, in a norm equivalent to ${\bf L}^2$, implies that
$\|{\mathcal U}psilon^n({\cal O}mega_0)-{\cal O}mega_0\|_{{\bf L}^2}$ will be small, for every $n\geq 1$.
In turn, recalling (\ref{OPN}), we conclude that
$\|\Phi_n-\Phi_0\|_{H^2}$ also remains small. However, this estimate
is not enough to provide an a priori bound on $\|\noindentabla^\perp \Phi_n\|_{{\mathcal C}^1}$, which is needed to
estimate the differential $D{\bf L}ambda$. For this reason, an additional regularity estimate for the
iterates ${\cal O}mega_n={\mathcal U}psilon^n({\cal O}mega_0)$ will be derived.
\subsection{Regularity estimates.}
Assuming that the ${\mathcal C}^1$ vector field ${\bf q}_0= \noindentabla^\perp \Phi_0+{\bf v}$ satisfies the transversality
assumptions {\bf (A1)}, we can find $0<\delta_1\leq 1$ with the following property.
\begin{equation}gin{itemize}
\itemtem[{\bf (P1)}]
{\itemt If $\|\Phi-\Phi_0\|_{{\mathcal C}^2}\leq\delta_1$, then the vector field ${\bf q}= \noindentabla^\perp \Phi+{\bf v}$
still satisfies (\ref{OT3})--(\ref{OT2}), possibly with a smaller constant $c_1>0$ and with $T^*$ replaced by $T^*+1$.
}
\epsilonndi
In particular, all characteristics starting from a point $y\itemn\Sigma_1$ in the support of $h$ still exit from the domain ${\mathcal D}$
within time $T^*+1$.
Moreover, the right hand side of (\ref{TC}) remains uniformly bounded by some constant,
which we still denote by $\widetilde C$.
Our next goal is to ensure that all our approximations satisfy
\begin{equation}l{pno}
\|\Phi_n-\Phi_0\|_{{\mathcal C}^2}~\leq~ \delta_1\qquad\qquad \hbox{for all }~ n\geq 1.\epsiloneq
This bound will be achieved by an inductive argument, in several steps.
\vskip 4emip 2emkip 1em
{\bf 1.}
If (\ref{pno}) holds,
then
\begin{equation}l{qn}\|{\bf q}_n\|_{{\mathcal C}^1}~=~\bigl\|\noindentabla^\perp \Phi_n+ {\bf v}\bigr\|_{{\mathcal C}^1}~\leq~
\bigl\|\noindentabla^\perp \Phi_0+ {\bf v}\bigr\|_{{\mathcal C}^1} + \bigl\|\noindentabla^\perp \Phi_n- \noindentabla^\perp \Phi_0\bigr\|_{{\mathcal C}^1}
~\leq~M+1\,.\epsiloneq
Assuming (\ref{pno}), the solution ${\cal O}mega = {\mathcal G}amma(\Phi_n)$ of (\ref{lam1}) satisfies
\begin{equation}l{Gn1}
\|{\cal O}mega\|_{{\mathcal C}^0}~\leq~e^{T^*+1}\, \|h\|_{{\mathcal C}^0}\,.\epsiloneq
To provide a bound on the gradient $\noindentabla{\cal O}mega$, recalling the notation introduced at
(\ref{xty})--(\ref{tauy}),
fix any $x_0\itemn {\mathcal D}$ such that $x_0 = \epsilonxp(t{\bf q}_n)(y)$ for some $y\itemn \Sigma_1\centerlineap \hbox{Supp}(h)$ and $t= \tau(x_0)>0$.
Using the notation $x(t, x_0) \doteq \epsilonxp(-t{\bf q}_n)(x_0)$, consider the vector
$${\bf w}(t)~\doteq~\lim_{\vskip 4emip 2emkip 1emarepsilon\to 0} {x(t, x_0+\vskip 4emip 2emkip 1emarepsilon {\bf e}) - x(t,x_0)\overlineer\vskip 4emip 2emkip 1emarepsilon}\,,$$
where ${\bf e}\itemn {\mathbb R}^2$ is any unit vector.
Then
$$\dot {\bf w}(t)~=~D{\bf q}_n(x(t, x_0))\centerlinedot {\bf w}(t),\qquad\qquad {\bf w}(0)~=~{\bf e}.$$
Hence
\begin{equation}l{wn}
|{\bf w}(t)|~\leq ~\epsilonxp\bigl\{ t\|D{\bf q}_n\|_{{\mathcal C}^0}\bigr\}~\leq~e^{ (T^*+1) (M+1)}.\epsiloneq
By (\ref{OR}), the same computations performed at (\ref{wwy})--(\ref{TO6})
now yield
$$|\noindentabla {\cal O}mega(x_0)\centerlinedot {\bf e}|~\leq~e^{T^*+1} \,\widetilde C\, \bigl|{\bf w}(\tau(x_0))\bigr|~\leq~\widetilde C\, e^{(T^*+1)(M+2)}.$$
Since the unit vector ${\bf e}$ was arbitrary, this implies that the gradient of ${\cal O}mega={\mathcal G}amma(\Phi_n)$ satisfies
\begin{equation}l{nom}
\bigl\|\noindentabla {\cal O}mega\bigr\|_{{\mathcal C}^0}~\leq~\widetilde C\, e^{(T^*+1)(M+2)}.\epsiloneq
\vskip 4emip 2emkip 1em
{\bf 2.}
Recalling (\ref{ups}) and the definition of ${\mathcal G}amma$ at (\ref{lam1})-(\ref{ldef}),
by (\ref{Gn1}) and (\ref{nom}) we now obtain
\begin{equation}l{on1}
\bigl\|{\mathcal G}amma({\mathcal D}elta^{-1}{\cal O}mega_n)\bigr\|_{{\mathcal C}^1}~\leq~e^{T^*+1} \|h\|_{{\mathcal C}^0} +\widetilde C\, e^{(T^*+1)(M+2)},\epsiloneq
as long as (\ref{pno}) holds. In turn, this yields a bound of the form
\begin{equation}l{oc1}
\|{\cal O}mega_{n+1}\|_{{\mathcal C}^1}~=~\Big\| (I-AP)^{-1} \bigl({\mathcal G}amma({\mathcal D}elta^{-1}{\cal O}mega_n)- AP \,{\cal O}mega_n\bigr)\Big\|_{{\mathcal C}^1}\leq~C_1\,,\epsiloneq
where the constant $C_1$ can be estimated in terms of (\ref{on1})
and the properties of the linear operator $A$.
\vskip 4emip 2emkip 1em
{\bf 3.}
Since the domain ${\mathcal D}$ has smooth boundary, Schauder's regularity estimates \centerlineite{Evans, GT} with $\alpha=1/2$
yield a bound of the form
\begin{equation}l{pnes}
\|\Phi_n-\Phi_0\|_{{\mathcal C}^2}~=~\|{\mathcal D}elta^{-1}({\cal O}mega_n-{\cal O}mega_0)\|_{{\mathcal C}^{2}} ~\leq~C_2\, \|{\cal O}mega_n-{\cal O}mega_0\|_{{\mathcal C}^{0,
1/2}}\,,\epsiloneq
for some constant $C_2$ depending only on ${\mathcal D}$.
Assume we have the inductive estimate
\begin{equation}l{ie2}
\|{\cal O}mega_n-{\cal O}mega_0\|_{{\mathcal C}^1} ~\leq~2C_1\,.
\epsiloneq
Choosing $0<\delta<1$ so that $\delta\leq (2C_1 C_2)^{1\overlineer\alpha-1} = (2 C_1C_2)^{-2}$,
we obtain
\begin{equation}l{del}\|{\cal O}mega_n-{\cal O}mega_0\|_{\alpha,\delta} ~\leq~\delta^{1/2}\|\noindentabla{\cal O}mega_n-\noindentabla{\cal O}mega_0\|_{C^0}
~\leq~2C_1\delta^{1/2}~\leq~{1\overlineer C_2} \,.\epsiloneq
\vskip 4emip 2emkip 1em
{\bf 4.} Let $\delta_1>0$ be the constant introduced in {\bf (P1)}.
We now use Lemma~\ref{l:interp}, with $\delta>0$ as in (\ref{del}) and $c=\delta_1/C_2$.
This yields a constant $\delta_c\itemn \, ]0, \delta_1]$ such that
(\ref{40}) implies (\ref{42})-(\ref{41}). In the present setting, this means that the two inequalities
\begin{equation}l{ndc}\|{\cal O}mega_n-{\cal O}mega_0\|_{{\mathcal C}^1} ~\leq~2C_1\,,\qquad\qquad
\|{\cal O}mega_n-{\cal O}mega_0\|_{{\bf L}^2} ~\leq~\delta_c\,,\epsiloneq
together imply
\begin{equation}l{onz}\|{\cal O}mega_n-{\cal O}mega_0\|_{{\mathcal C}^{0,1/2}}~\leq~2C_1\,\delta^{1/2} ~\leq~{\delta_1\overlineer C_2}\,,
\qquad\qquad \|\Phi_n-\Phi_0\|_{{\mathcal C}^2}~\leq
~\delta_1.\epsiloneq
In other words, if the ${\bf L}^2$ distance between ${\cal O}mega_0$ and every ${\cal O}mega_n$
remains small, then all these approximate solutions have uniformly bounded ${\mathcal C}^{0,1/2}$ norm.
In turn, property {\bf (P1)} applies.
\subsection{Convergence of the approximations.}
As long as $\|{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2}\leq \delta_c$ we have $\|\Phi\|_{{\mathcal C}^2}\leq \delta_1$, and hence
\begin{equation}l{kod}\|D{\mathcal G}amma({\cal O}mega)\|_{{\bf L}^2}~\leq~\widetilde C\centerlinedot K(M+\delta_1, \, T^*+1)~\doteq~\kappa_0\,.\epsiloneq
\begin{equation}l{kol}\|D{\bf L}ambda({\cal O}mega)\|_{{\bf L}^2}~\leq~\|D{\mathcal G}amma({\cal O}mega)\|_{{\bf L}^2}
\centerlinedot \|{\mathcal D}elta^{-1}\|_{{\bf L}^2} ~\leq~~\kappa_0/\bigl\langlembda_1\,,
\epsiloneq
where $\bigl\langlembda_1>0$ denotes the first eigenvalue of the Laplace operator on ${\mathcal D}$.
On the other hand, choosing a sufficiently large number $N$ of functions in the orthogonal basis
at (\ref{UV}), we achieve (\ref{LVsmall}). When $(u,v)=(u_0,0)\itemn U\times V$,
the four blocks in the matrix of partial derivatives (\ref{diff}) have norms which can be dominated
respectively by the entries of the matrix
$\left(\begin{equation}gin{itemize}n{array}{cc} 0 & \kappa_0\vskip 4emip 2emkip 1emarepsilon_0\centerliner\kappa_0/\bigl\langlembda_1 & \kappa_0\vskip 4emip 2emkip 1emarepsilon_0\epsilonnda\right)$.
More generally, we can determine $\delta_2\itemn \,]0, \delta_c]$ such that
\begin{equation}l{imp3}
\|{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2}~\leq~\delta_2\qquad\itemmplies\qquad \|P \,D_u {\bf L}ambda({\cal O}mega)- P\, D_u{\bf L}ambda({\cal O}mega_0)\|_{{\bf L}^2}
~\leq~\kappa_0\vskip 4emip 2emkip 1emarepsilon_0\,.\epsiloneq
Computing the matrix of partial derivatives at (\ref{diff}) at such a point ${\cal O}mega=(u,v)$, and recalling
that $A\doteq P\, D_u{\bf L}ambda({\cal O}mega_0)$, we obtain the relation
\begin{equation}l{cbo}
D\bigl({\bf L}ambda(u,v)-Au\bigr)
~=~\left(\begin{equation}gin{itemize}n{array}{ccc} P \,D_u {\bf L}ambda({\cal O}mega)- P\, D_u{\bf L}ambda({\cal O}mega_0) && P D_v{\bf L}ambda\\[4mm]
(I-P) D_u{\bf L}ambda && (I-P) D_v{\bf L}ambda\epsilonnda\right)~~\prec~~
\left(\begin{equation}gin{itemize}n{array}{cc} \kappa_0\vskip 4emip 2emkip 1emarepsilon_0 & \kappa_0\vskip 4emip 2emkip 1emarepsilon_0\centerliner\kappa_0/\bigl\langlembda_1 & \kappa_0\vskip 4emip 2emkip 1emarepsilon_0\epsilonnda\right).
\epsiloneq
Here we have used the notation $A\prec B$, meaning that every entry in the $2\times 2$ matrix $A$ has norm
bounded by the corresponding entry in the matrix $B$. Notice that the constant $\delta_2$ in (\ref{imp3})
involves only
the behavior of ${\mathcal G}amma\centerlineirc {\mathcal D}elta^{-1}$ on a neighborhood of ${\cal O}mega_0$ in the finite dimensional subspace
$U$, and can be directly estimated.
According to {\bf (A3)}, we now make the key assumption that the constant
$\vskip 4emip 2emkip 1emarepsilon_0>0$ in (\ref{LVsmall}) is small enough so that
\begin{equation}l{A3}\sqrt 2\, \kappa_0\left( \vskip 4emip 2emkip 1emarepsilon_0^2+{\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}
\right)^{1/2}
\leq ~{1\overlineer 2\gamma}\,.\epsiloneq
This allows us to introduce on $U\times V$ the equivalent norm (\ref{n*}), choosing
$$\epsilonta_0~\doteq~\bigl\langlembda_1\vskip 4emip 2emkip 1emarepsilon_0\,.$$
Without loss of generality, we can assume that $0<\epsilonta_0\leq 1$, so that
\begin{equation}l{eqn}\|\centerlinedot\|_*~\leq~\|\centerlinedot\|_{{\bf L}^2}~\leq~{1\overlineer\sqrt{\epsilonta_0}} \|\centerlinedot\|_*\,.\epsiloneq
In term of this new norm, a direct computation shows, at any point $(u,v)$ where (\ref{cbo}) holds,
the corresponding operator norm satisfies\footnote{Indeed, assume $u,v\itemn {\mathbb R}$, $u^2+\epsilonta_0v^2\leq 1$.
Set $z=\sqrt{\epsilonta_0} v$, so that $u^2+z^2\leq 1$.
This implies
$$\begin{equation}gin{itemize}n{array}{l}\displaystyle(\kappa_0 \vskip 4emip 2emkip 1emarepsilon_0 u + \kappa_0\vskip 4emip 2emkip 1emarepsilon_0 v)^2 + \epsilonta_0\left({\kappa_0\overlineer \bigl\langlembda_1} u +\kappa_0\vskip 4emip 2emkip 1emarepsilon_0 v
\right)^2~=~\left( \kappa_0 \vskip 4emip 2emkip 1emarepsilon_0 u + {\kappa_0\vskip 4emip 2emkip 1emarepsilon_0 \overlineer\sqrt {\bigl\langlembda_1\vskip 4emip 2emkip 1emarepsilon_0 }}z\right)^2
+ \bigl\langlembda_1\vskip 4emip 2emkip 1emarepsilon_0\left({\kappa_0\overlineer \bigl\langlembda_1} u +{\kappa_0\vskip 4emip 2emkip 1emarepsilon_0 \overlineer\sqrt {\bigl\langlembda_1\vskip 4emip 2emkip 1emarepsilon_0 }}z
\right)^2\\[4mm]
\displaystyle\qquad =~\kappa_0^2\left[ \Big( \vskip 4emip 2emkip 1emarepsilon_0 u + \sqrt{\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1} z\Big)^2 +
\Big(\sqrt{\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}\, u
+ \vskip 4emip 2emkip 1emarepsilon_0 z\Big)^2\right]~=~\kappa_0^2\left[ \vskip 4emip 2emkip 1emarepsilon_0^2( u^2+z^2) + {\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}(u^2+z^2)
+ 4\vskip 4emip 2emkip 1emarepsilon_0\sqrt{\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}\, uz \right]\\[4mm]
\qquad \leq~\displaystyle 2\kappa_0^2 \left( \vskip 4emip 2emkip 1emarepsilon_0^2 + {\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}\right) (u^2+z^2).
\epsilonnda$$
An entirely similar computation, with the numbers $u,v$ replaced by $\|u\|$ and $\|v\|$ respectively, shows that
the corresponding norm of the linear operator (\ref{cbo}) is bounded by $\sqrt 2\, \kappa_0\left( \vskip 4emip 2emkip 1emarepsilon_0^2+\displaystyle{\vskip 4emip 2emkip 1emarepsilon_0\overlineer\bigl\langlembda_1}
\right)^{1/2}$. }
\begin{equation}l{opn3}\Big\|
D\bigl({\bf L}ambda(u,v)-Au\bigr)\Big\|_*~\leq~{1\overlineer 2\gamma}\,.\epsiloneq
Hence
\begin{equation}l{co2}\|D{\mathcal U}psilon\|_*~\leq~\|(I-PA)^{-1}\|\centerlinedot \Big\|D\bigl({\bf L}ambda(u,v)-Au\bigr)\Big\|_*~\leq~{1\overlineer 2}\,.\epsiloneq
By the previous analysis, restricted to the set
$${\centerlineal S}~\doteq~\left\{ {\cal O}mega\,;~~\|{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2}~\leq~ \delta_2\,,\quad \|{\cal O}mega-{\cal O}mega_0
\|_{{\mathcal C}^{0,\alpha}}\,\leq\, {\delta_1\overlineer C_2}
\right\},$$
the map ${\cal O}mega\mapsto {\mathcal U}psilon({\cal O}mega)$ is a strict contraction, w.r.t.~the equivalent norm $\|\centerlinedot \|_*$ introduced at
(\ref{n*}). Indeed, by (\ref{co2}),
\begin{equation}l{co1}
{\cal O}mega,{\cal O}mega'\,\itemn\, {\centerlineal S}\qquad\itemmplies\qquad \|{\mathcal U}psilon({\cal O}mega)-{\mathcal U}psilon({\cal O}mega')\|_*
~\leq~{1\overlineer 2} \|{\cal O}mega-{\cal O}mega'\|_*\,.\epsiloneq
Recalling that $\delta_1$ is the constant in {\bf (P1)}, we now have the chain of implications
$$\begin{equation}gin{itemize}n{array}{l}\displaystyle \|{\cal O}mega-{\cal O}mega_0\|_*~\leq~ {\delta_2\overlineer\sqrt{\epsilonta_0}} \quad\itemmplies\quad
\|{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2}
~\leq~\delta_2\\[4mm]\displaystyle
\quad\itemmplies\quad \|{\cal O}mega-{\cal O}mega_0
\|_{{\mathcal C}^{0,\alpha}}\,\leq\, {\delta_1\overlineer C_2}\quad\itemmplies\quad \|{\mathcal D}elta^{-1}({\cal O}mega-{\cal O}mega_0)\|_{{\mathcal C}^2}
~\leq~\delta_1\,.\epsilonnda$$
If the initial guess ${\cal O}mega_0$ satisfies
\begin{equation}l{igss}\|{\mathcal U}psilon({\cal O}mega_0) - {\cal O}mega_0\|_{{\bf L}^2}~\leq~{\delta_2\sqrt{\epsilonta_0}\overlineer 2}\,,\epsiloneq
then by (\ref{eqn}) and (\ref{co1}) all the iterates ${\cal O}mega_n$ in (\ref{OPN}) will remain inside ${\centerlineal S}$.
Namely,
$$\|{\cal O}mega_n-{\cal O}mega_0\|_{{\bf L}^2}~\leq~{1\overlineer\sqrt{\epsilonta_0}}\,\|{\cal O}mega_n-{\cal O}mega_0\|_*~\leq~
{1\overlineer\sqrt{\epsilonta_0}}\centerlinedot 2\|{\mathcal U}psilon({\cal O}mega_0)-{\cal O}mega_0\|_*~\leq~\delta_2\,.$$
Letting $n\to \itemnfty$ we have the convergence ${\cal O}mega_n\to \overline{\cal O}mega\itemn {\centerlineal S}$.
This limit function $\overline{\cal O}mega$ provides a solution to the boundary value problem
(\ref{lam1}), (\ref{D-1}), with
\begin{equation}l{cl1}\|\overline{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2}~\leq~\delta_2\,.\epsiloneq
\subsection{An existence theorem.} For readers' convenience,
we summarize the previous analysis, recalling the various constants introduced along the way.
All these constants can be estimated in terms of the domain ${\mathcal D}$, the boundary data $g,h$ in (\ref{BC}),
and the finite dimensional
approximation ${\cal O}mega_0$ produced by the numerical algorithm.
\begin{equation}gin{itemize}
\itemtem $\widetilde C$ is the constant at (\ref{TC}), related to the stability of the ODEs for the characteristics of the linear PDE
(\ref{lam3}). This applies to any vector field ${\bf q}= \noindentabla^\perp\Phi+{\bf v}$ with $\|\Phi-\Phi_0\|_{{\mathcal C}^2}\leq \delta_1$.
\itemtem $K(M,t)$ is the function in (\ref{pera}). Together with $\widetilde C$, it provides an estimate (\ref{oper}) on
how the solution to the linear PDE (\ref{lam3}) varies in the ${\bf L}^2$ distance,
depending on the vector field $\noindentabla^\perp \Phi$.
\itemtem $C_2$ is the Schauder regularity constant for the smooth domain ${\mathcal D}$, introduced at (\ref{pnes}).
\itemtem $\bigl\langlembda_1>0$ is the lowest eigenvalue of the Laplace operator on ${\mathcal D}$.
\itemtem $\vskip 4emip 2emkip 1emarepsilon_0$ is the small constant in (\ref{LVsmall}), determining the rate at which the components in the orthogonal
space $V$ are damped by the inverse Laplace operator ${\mathcal D}elta^{-1}$.
\itemtem $\kappa_0$ is the constant in (\ref{kod}), estimating how the solution of the linear PDE (\ref{lam1})
varies, depending on the vector field $\noindentabla^\perp\Phi_1$, w.r.t.~the ${\bf L}^2$ norm.
\itemtem $\delta_1$ is the constant in {\bf (P1)}, measuring by how much we can perturb the vector field
${\bf q}= \noindentabla^\perp\Phi+{\bf v}$, and still retain the transversality conditions at the boundary, and a finite exit time.
\itemtem $\delta_c$ is the constant in the (\ref{ndc}), providing the regularity estimate (\ref{onz}).
\itemtem $\gamma$ is a bound on the norm $\|(I-PD_u{\bf L}ambda({\cal O}mega_0))^{-1}\|$ of the inverse Jacobian matrix in (\ref{IA2}).
It gives a measure of stability for the finite dimensional fixed point problem ${\cal O}mega = P\centerlineirc
{\bf L}ambda({\cal O}mega)$.
\itemtem $\delta_2$ is the constant in (\ref{imp3}), determining a neighborhood of the initial approximation ${\cal O}mega_0$,
where the differential $D{\bf L}ambda({\cal O}mega)$ remains close to $D{\bf L}ambda({\cal O}mega_0)$.
\epsilonndi
Recalling the definition of the finite dimensional linear operator $A$ at (\ref{Adef}) and the equivalent formulation
of the boundary value problem (\ref{lam2})-(\ref{lam1}) as a fixed point problem (\ref{ups}),
we can now summarize all of the previous analysis as follows.
\begin{equation}gin{itemize}n{theorem} Let ${\mathcal D}\subset{\mathbb R}^2$ be a bounded open domain with smooth boundary, decomposed as
$\partial{\mathcal D}=\Sigma_1\centerlineup\Sigma_2$. Given smooth boundary data $g,h$, consider the boundary value problem
(\ref{lam2})-(\ref{lam1}), where ${\bf v}$ is the vector field in (\ref{bbv}).
Assume that the properties {\bf (A1)-(A2)} hold, together with {\bf (P1)}.
Consider the orthogonal decomposition ${\bf L}^2({\mathcal D})= U\times V$ as in (\ref{UV})-(\ref{proj}),
and let
${\cal O}mega_0= (u_0, 0)\itemn U\times V$ be an approximate solution such that
\begin{equation}l{igs2}\|{\mathcal U}psilon({\cal O}mega_0) - {\cal O}mega_0\|_{{\bf L}^2({\mathcal D})}~\leq~{\delta_2\sqrt{\bigl\langlembda_1\vskip 4emip 2emkip 1emarepsilon_0}\overlineer 2}\,.\epsiloneq
Then an exact solution $\overline{\cal O}mega$ exists, with
\begin{equation}l{OO}
\|\overline{\cal O}mega-{\cal O}mega_0\|_{{\bf L}^2({\mathcal D})}~\leq~\delta_2\,.
\epsiloneq
\epsilonnd{theorem}
\vskip 4emip 2emkip 1em
{\bf Acknowledgment.} The authors would like to thank Ludmil Zikatanov for useful discussions.
\vskip 4emip 2emkip 1em
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